LMFDB - Embedded newform 900.3.l.d.757.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(757,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("900.757");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.l (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			757.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	900.757
Dual form			900.3.l.d.793.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.67423 - 3.67423i) q^{7} +O(q^{10})$	$q+(-3.67423 - 3.67423i) q^{7} -6.00000 q^{11} +(-6.12372 + 6.12372i) q^{13} +(22.0454 + 22.0454i) q^{17} -25.0000i q^{19} +(-7.34847 + 7.34847i) q^{23} +42.0000i q^{29} +49.0000 q^{31} +(-4.89898 - 4.89898i) q^{37} +60.0000 q^{41} +(1.22474 - 1.22474i) q^{43} +(51.4393 + 51.4393i) q^{47} -22.0000i q^{49} +(-14.6969 + 14.6969i) q^{53} -78.0000i q^{59} -13.0000 q^{61} +(52.6640 + 52.6640i) q^{67} +60.0000 q^{71} +(-63.6867 + 63.6867i) q^{73} +(22.0454 + 22.0454i) q^{77} +106.000i q^{79} +(-80.8332 + 80.8332i) q^{83} -60.0000i q^{89} +45.0000 q^{91} +(121.250 + 121.250i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{11} + 196 q^{31} + 240 q^{41} - 52 q^{61} + 240 q^{71} + 180 q^{91}+O(q^{100}) $ 4 * q - 24 * q^11 + 196 * q^31 + 240 * q^41 - 52 * q^61 + 240 * q^71 + 180 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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
$7$	−3.67423	−	3.67423i	−0.524891	−	0.524891i	0.394154	−	0.919044i	$-0.371038\pi$
$7$	−3.67423	−	3.67423i	−0.524891	−	0.524891i	−0.919044	+	0.394154i	$0.871038\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.00000			−0.545455			−0.272727	−	0.962091i	$-0.587926\pi$
$11$	−6.00000			−0.545455			−0.272727	+	0.962091i	$0.587926\pi$
$12$		0			0
$13$	−6.12372	+	6.12372i	−0.471056	+	0.471056i	−0.902256	−	0.431200i	$-0.858090\pi$
$13$	−6.12372	+	6.12372i	−0.471056	+	0.471056i	0.431200	+	0.902256i	$0.358090\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	22.0454	+	22.0454i	1.29679	+	1.29679i	0.930502	+	0.366287i	$0.119371\pi$
$17$	22.0454	+	22.0454i	1.29679	+	1.29679i	0.366287	+	0.930502i	$0.380629\pi$
$18$		0			0
$19$		−	25.0000i		−	1.31579i	−0.753110	−	0.657895i	$-0.771447\pi$
$19$		−	25.0000i		−	1.31579i	0.753110	−	0.657895i	$-0.228553\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.34847	+	7.34847i	−0.319499	+	0.319499i	−0.848575	−	0.529076i	$-0.822539\pi$
$23$	−7.34847	+	7.34847i	−0.319499	+	0.319499i	0.529076	+	0.848575i	$0.322539\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			42.0000i			1.44828i	0.689655	+	0.724138i	$0.257762\pi$
$29$			42.0000i			1.44828i	−0.689655	+	0.724138i	$0.742238\pi$
$30$		0			0
$31$	49.0000			1.58065			0.790323	−	0.612691i	$-0.209913\pi$
$31$	49.0000			1.58065			0.790323	+	0.612691i	$0.209913\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.89898	−	4.89898i	−0.132405	−	0.132405i	0.637798	−	0.770203i	$-0.279845\pi$
$37$	−4.89898	−	4.89898i	−0.132405	−	0.132405i	−0.770203	+	0.637798i	$0.779845\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	60.0000			1.46341			0.731707	−	0.681619i	$-0.238724\pi$
$41$	60.0000			1.46341			0.731707	+	0.681619i	$0.238724\pi$
$42$		0			0
$43$	1.22474	−	1.22474i	0.0284824	−	0.0284824i	−0.692722	−	0.721205i	$-0.743589\pi$
$43$	1.22474	−	1.22474i	0.0284824	−	0.0284824i	0.721205	+	0.692722i	$0.243589\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	51.4393	+	51.4393i	1.09445	+	1.09445i	0.995047	+	0.0994059i	$0.0316942\pi$
$47$	51.4393	+	51.4393i	1.09445	+	1.09445i	0.0994059	+	0.995047i	$0.468306\pi$
$48$		0			0
$49$		−	22.0000i		−	0.448980i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−14.6969	+	14.6969i	−0.277301	+	0.277301i	−0.832031	−	0.554730i	$-0.812822\pi$
$53$	−14.6969	+	14.6969i	−0.277301	+	0.277301i	0.554730	+	0.832031i	$0.312822\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	78.0000i		−	1.32203i	−0.750371	−	0.661017i	$-0.770125\pi$
$59$		−	78.0000i		−	1.32203i	0.750371	−	0.661017i	$-0.229875\pi$
$60$		0			0
$61$	−13.0000			−0.213115			−0.106557	−	0.994307i	$-0.533983\pi$
$61$	−13.0000			−0.213115			−0.106557	+	0.994307i	$0.533983\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	52.6640	+	52.6640i	0.786030	+	0.786030i	0.980841	−	0.194811i	$-0.0624092\pi$
$67$	52.6640	+	52.6640i	0.786030	+	0.786030i	−0.194811	+	0.980841i	$0.562409\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	60.0000			0.845070			0.422535	−	0.906347i	$-0.361140\pi$
$71$	60.0000			0.845070			0.422535	+	0.906347i	$0.361140\pi$
$72$		0			0
$73$	−63.6867	+	63.6867i	−0.872421	+	0.872421i	−0.992736	−	0.120315i	$-0.961610\pi$
$73$	−63.6867	+	63.6867i	−0.872421	+	0.872421i	0.120315	+	0.992736i	$0.461610\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	22.0454	+	22.0454i	0.286304	+	0.286304i
$78$		0			0
$79$			106.000i			1.34177i	0.741560	+	0.670886i	$0.234086\pi$
$79$			106.000i			1.34177i	−0.741560	+	0.670886i	$0.765914\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−80.8332	+	80.8332i	−0.973894	+	0.973894i	−0.999668	−	0.0257743i	$-0.991795\pi$
$83$	−80.8332	+	80.8332i	−0.973894	+	0.973894i	0.0257743	+	0.999668i	$0.491795\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	60.0000i		−	0.674157i	−0.941476	−	0.337079i	$-0.890561\pi$
$89$		−	60.0000i		−	0.674157i	0.941476	−	0.337079i	$-0.109439\pi$
$90$		0			0
$91$	45.0000			0.494505
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	121.250	+	121.250i	1.25000	+	1.25000i	0.955720	+	0.294277i	$0.0950789\pi$
$97$	121.250	+	121.250i	1.25000	+	1.25000i	0.294277	+	0.955720i	$0.404921\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	192.000			1.90099			0.950495	−	0.310740i	$-0.100577\pi$
$101$	192.000			1.90099			0.950495	+	0.310740i	$0.100577\pi$
$102$		0			0
$103$	78.3837	−	78.3837i	0.761007	−	0.761007i	−0.215498	−	0.976504i	$-0.569137\pi$
$103$	78.3837	−	78.3837i	0.761007	−	0.761007i	0.976504	+	0.215498i	$0.0691374\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	29.3939	+	29.3939i	0.274709	+	0.274709i	0.830993	−	0.556283i	$-0.187773\pi$
$107$	29.3939	+	29.3939i	0.274709	+	0.274709i	−0.556283	+	0.830993i	$0.687773\pi$
$108$		0			0
$109$			1.00000i			0.00917431i	0.999989	+	0.00458716i	$0.00146014\pi$
$109$			1.00000i			0.00917431i	−0.999989	+	0.00458716i	$0.998540\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−58.7878	+	58.7878i	−0.520246	+	0.520246i	−0.917646	−	0.397400i	$-0.869913\pi$
$113$	−58.7878	+	58.7878i	−0.520246	+	0.520246i	0.397400	+	0.917646i	$0.369913\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	162.000i		−	1.36134i
$120$		0			0
$121$	−85.0000			−0.702479
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	14.6969	+	14.6969i	0.115724	+	0.115724i	0.762597	−	0.646873i	$-0.223924\pi$
$127$	14.6969	+	14.6969i	0.115724	+	0.115724i	−0.646873	+	0.762597i	$0.723924\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.0000			−0.0916031			−0.0458015	−	0.998951i	$-0.514584\pi$
$131$	−12.0000			−0.0916031			−0.0458015	+	0.998951i	$0.514584\pi$
$132$		0			0
$133$	−91.8559	+	91.8559i	−0.690646	+	0.690646i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−80.8332	−	80.8332i	−0.590023	−	0.590023i	0.347614	−	0.937638i	$-0.386992\pi$
$137$	−80.8332	−	80.8332i	−0.590023	−	0.590023i	−0.937638	+	0.347614i	$0.886992\pi$
$138$		0			0
$139$		−	58.0000i		−	0.417266i	−0.977994	−	0.208633i	$-0.933099\pi$
$139$		−	58.0000i		−	0.417266i	0.977994	−	0.208633i	$-0.0669014\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	36.7423	−	36.7423i	0.256939	−	0.256939i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			234.000i			1.57047i	0.619198	+	0.785235i	$0.287458\pi$
$149$			234.000i			1.57047i	−0.619198	+	0.785235i	$0.712542\pi$
$150$		0			0
$151$	85.0000			0.562914			0.281457	−	0.959574i	$-0.409182\pi$
$151$	85.0000			0.562914			0.281457	+	0.959574i	$0.409182\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−74.7094	−	74.7094i	−0.475856	−	0.475856i	0.427947	−	0.903804i	$-0.359237\pi$
$157$	−74.7094	−	74.7094i	−0.475856	−	0.475856i	−0.903804	+	0.427947i	$0.859237\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	54.0000			0.335404
$162$		0			0
$163$	79.6084	−	79.6084i	0.488395	−	0.488395i	−0.419404	−	0.907800i	$-0.637761\pi$
$163$	79.6084	−	79.6084i	0.488395	−	0.488395i	0.907800	+	0.419404i	$0.137761\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	176.363	+	176.363i	1.05607	+	1.05607i	0.998332	+	0.0577355i	$0.0183880\pi$
$167$	176.363	+	176.363i	1.05607	+	1.05607i	0.0577355	+	0.998332i	$0.481612\pi$
$168$		0			0
$169$			94.0000i			0.556213i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	80.8332	−	80.8332i	0.467244	−	0.467244i	−0.433777	−	0.901020i	$-0.642819\pi$
$173$	80.8332	−	80.8332i	0.467244	−	0.467244i	0.901020	+	0.433777i	$0.142819\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			42.0000i			0.234637i	0.993094	+	0.117318i	$0.0374298\pi$
$179$			42.0000i			0.234637i	−0.993094	+	0.117318i	$0.962570\pi$
$180$		0			0
$181$	145.000			0.801105			0.400552	−	0.916274i	$-0.368818\pi$
$181$	145.000			0.801105			0.400552	+	0.916274i	$0.368818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−132.272	−	132.272i	−0.707339	−	0.707339i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−294.000			−1.53927			−0.769634	−	0.638486i	$-0.779561\pi$
$191$	−294.000			−1.53927			−0.769634	+	0.638486i	$0.779561\pi$
$192$		0			0
$193$	−25.7196	+	25.7196i	−0.133262	+	0.133262i	−0.770592	−	0.637329i	$-0.780039\pi$
$193$	−25.7196	+	25.7196i	−0.133262	+	0.133262i	0.637329	+	0.770592i	$0.280039\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.0454	−	22.0454i	−0.111906	−	0.111906i	0.648937	−	0.760842i	$-0.275214\pi$
$197$	−22.0454	−	22.0454i	−0.111906	−	0.111906i	−0.760842	+	0.648937i	$0.775214\pi$
$198$		0			0
$199$		−	179.000i		−	0.899497i	−0.893155	−	0.449749i	$-0.851514\pi$
$199$		−	179.000i		−	0.899497i	0.893155	−	0.449749i	$-0.148486\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	154.318	−	154.318i	0.760186	−	0.760186i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			150.000i			0.717703i
$210$		0			0
$211$	−85.0000			−0.402844			−0.201422	−	0.979505i	$-0.564556\pi$
$211$	−85.0000			−0.402844			−0.201422	+	0.979505i	$0.564556\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−180.037	−	180.037i	−0.829666	−	0.829666i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−270.000			−1.22172
$222$		0			0
$223$	−104.103	+	104.103i	−0.466831	+	0.466831i	−0.900886	−	0.434055i	$-0.857082\pi$
$223$	−104.103	+	104.103i	−0.466831	+	0.466831i	0.434055	+	0.900886i	$0.357082\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	373.000i		−	1.62882i	−0.580289	−	0.814410i	$-0.697060\pi$
$229$		−	373.000i		−	1.62882i	0.580289	−	0.814410i	$-0.302940\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−220.454	+	220.454i	−0.946155	+	0.946155i	−0.998623	−	0.0524678i	$-0.983291\pi$
$233$	−220.454	+	220.454i	−0.946155	+	0.946155i	0.0524678	+	0.998623i	$0.483291\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	36.0000i		−	0.150628i	−0.997160	−	0.0753138i	$-0.976004\pi$
$239$		−	36.0000i		−	0.150628i	0.997160	−	0.0753138i	$-0.0239959\pi$
$240$		0			0
$241$	−97.0000			−0.402490			−0.201245	−	0.979541i	$-0.564499\pi$
$241$	−97.0000			−0.402490			−0.201245	+	0.979541i	$0.564499\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	153.093	+	153.093i	0.619810	+	0.619810i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−288.000			−1.14741			−0.573705	−	0.819062i	$-0.694494\pi$
$251$	−288.000			−1.14741			−0.573705	+	0.819062i	$0.694494\pi$
$252$		0			0
$253$	44.0908	−	44.0908i	0.174272	−	0.174272i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−338.030	−	338.030i	−1.31529	−	1.31529i	−0.917458	−	0.397832i	$-0.869763\pi$
$257$	−338.030	−	338.030i	−1.31529	−	1.31529i	−0.397832	−	0.917458i	$-0.630237\pi$
$258$		0			0
$259$			36.0000i			0.138996i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	249.848	−	249.848i	0.949992	−	0.949992i	−0.0488156	−	0.998808i	$-0.515545\pi$
$263$	249.848	−	249.848i	0.949992	−	0.949992i	0.998808	+	0.0488156i	$0.0155447\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			318.000i			1.18216i	0.806614	+	0.591078i	$0.201297\pi$
$269$			318.000i			1.18216i	−0.806614	+	0.591078i	$0.798703\pi$
$270$		0			0
$271$	−226.000			−0.833948			−0.416974	−	0.908918i	$-0.636909\pi$
$271$	−226.000			−0.833948			−0.416974	+	0.908918i	$0.636909\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−380.896	−	380.896i	−1.37507	−	1.37507i	−0.852742	−	0.522333i	$-0.825062\pi$
$277$	−380.896	−	380.896i	−1.37507	−	1.37507i	−0.522333	−	0.852742i	$-0.674938\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−126.000			−0.448399			−0.224199	−	0.974543i	$-0.571977\pi$
$281$	−126.000			−0.448399			−0.224199	+	0.974543i	$0.571977\pi$
$282$		0			0
$283$	395.593	−	395.593i	1.39785	−	1.39785i	0.591683	−	0.806171i	$-0.298464\pi$
$283$	395.593	−	395.593i	1.39785	−	1.39785i	0.806171	−	0.591683i	$-0.201536\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−220.454	−	220.454i	−0.768133	−	0.768133i
$288$		0			0
$289$			683.000i			2.36332i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−22.0454	+	22.0454i	−0.0752403	+	0.0752403i	−0.743725	−	0.668485i	$-0.766943\pi$
$293$	−22.0454	+	22.0454i	−0.0752403	+	0.0752403i	0.668485	+	0.743725i	$0.266943\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	90.0000i		−	0.301003i
$300$		0			0
$301$	−9.00000			−0.0299003
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	86.9569	+	86.9569i	0.283247	+	0.283247i	0.834403	−	0.551155i	$-0.185813\pi$
$307$	86.9569	+	86.9569i	0.283247	+	0.283247i	−0.551155	+	0.834403i	$0.685813\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	246.000			0.790997			0.395498	−	0.918467i	$-0.370572\pi$
$311$	246.000			0.790997			0.395498	+	0.918467i	$0.370572\pi$
$312$		0			0
$313$	−37.9671	+	37.9671i	−0.121301	+	0.121301i	−0.765151	−	0.643851i	$-0.777336\pi$
$313$	−37.9671	+	37.9671i	−0.121301	+	0.121301i	0.643851	+	0.765151i	$0.277336\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.6969	−	14.6969i	−0.0463626	−	0.0463626i	0.683545	−	0.729908i	$-0.260437\pi$
$317$	−14.6969	−	14.6969i	−0.0463626	−	0.0463626i	−0.729908	+	0.683545i	$0.760437\pi$
$318$		0			0
$319$		−	252.000i		−	0.789969i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	551.135	−	551.135i	1.70630	−	1.70630i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	378.000i		−	1.14894i
$330$		0			0
$331$	−178.000			−0.537764			−0.268882	−	0.963173i	$-0.586654\pi$
$331$	−178.000			−0.537764			−0.268882	+	0.963173i	$0.586654\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	417.638	+	417.638i	1.23928	+	1.23928i	0.960295	+	0.278987i	$0.0899987\pi$
$337$	417.638	+	417.638i	1.23928	+	1.23928i	0.278987	+	0.960295i	$0.410001\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−294.000			−0.862170
$342$		0			0
$343$	−260.871	+	260.871i	−0.760556	+	0.760556i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−448.257	−	448.257i	−1.29181	−	1.29181i	−0.933669	−	0.358137i	$-0.883412\pi$
$347$	−448.257	−	448.257i	−1.29181	−	1.29181i	−0.358137	−	0.933669i	$-0.616588\pi$
$348$		0			0
$349$			542.000i			1.55301i	0.630112	+	0.776504i	$0.283009\pi$
$349$			542.000i			1.55301i	−0.630112	+	0.776504i	$0.716991\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−242.499	+	242.499i	−0.686967	+	0.686967i	−0.961561	−	0.274593i	$-0.911457\pi$
$353$	−242.499	+	242.499i	−0.686967	+	0.686967i	0.274593	+	0.961561i	$0.411457\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	408.000i		−	1.13649i	−0.822859	−	0.568245i	$-0.807623\pi$
$359$		−	408.000i		−	1.13649i	0.822859	−	0.568245i	$-0.192377\pi$
$360$		0			0
$361$	−264.000			−0.731302
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−199.633	−	199.633i	−0.543960	−	0.543960i	0.380727	−	0.924687i	$-0.375674\pi$
$367$	−199.633	−	199.633i	−0.543960	−	0.543960i	−0.924687	+	0.380727i	$0.875674\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	108.000			0.291105
$372$		0			0
$373$	319.658	−	319.658i	0.856993	−	0.856993i	−0.133990	−	0.990983i	$-0.542779\pi$
$373$	319.658	−	319.658i	0.856993	−	0.856993i	0.990983	+	0.133990i	$0.0427789\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−257.196	−	257.196i	−0.682219	−	0.682219i
$378$		0			0
$379$			71.0000i			0.187335i	0.995604	+	0.0936675i	$0.0298591\pi$
$379$			71.0000i			0.187335i	−0.995604	+	0.0936675i	$0.970141\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−58.7878	+	58.7878i	−0.153493	+	0.153493i	−0.779676	−	0.626183i	$-0.784616\pi$
$383$	−58.7878	+	58.7878i	−0.153493	+	0.153493i	0.626183	+	0.779676i	$0.284616\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			288.000i			0.740360i	0.928960	+	0.370180i	$0.120704\pi$
$389$			288.000i			0.740360i	−0.928960	+	0.370180i	$0.879296\pi$
$390$		0			0
$391$	−324.000			−0.828645
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	74.7094	+	74.7094i	0.188185	+	0.188185i	0.794911	−	0.606726i	$-0.207517\pi$
$397$	74.7094	+	74.7094i	0.188185	+	0.188185i	−0.606726	+	0.794911i	$0.707517\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−36.0000			−0.0897756			−0.0448878	−	0.998992i	$-0.514293\pi$
$401$	−36.0000			−0.0897756			−0.0448878	+	0.998992i	$0.514293\pi$
$402$		0			0
$403$	−300.062	+	300.062i	−0.744572	+	0.744572i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	29.3939	+	29.3939i	0.0722208	+	0.0722208i
$408$		0			0
$409$		−	13.0000i		−	0.0317848i	−0.999874	−	0.0158924i	$-0.994941\pi$
$409$		−	13.0000i		−	0.0317848i	0.999874	−	0.0158924i	$-0.00505893\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−286.590	+	286.590i	−0.693923	+	0.693923i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	144.000i		−	0.343675i	−0.985125	−	0.171838i	$-0.945030\pi$
$419$		−	144.000i		−	0.343675i	0.985125	−	0.171838i	$-0.0549705\pi$
$420$		0			0
$421$	610.000			1.44893			0.724466	−	0.689311i	$-0.242087\pi$
$421$	610.000			1.44893			0.724466	+	0.689311i	$0.242087\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	47.7650	+	47.7650i	0.111862	+	0.111862i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	402.000			0.932715			0.466357	−	0.884596i	$-0.345566\pi$
$431$	402.000			0.932715			0.466357	+	0.884596i	$0.345566\pi$
$432$		0			0
$433$	292.714	−	292.714i	0.676014	−	0.676014i	−0.283082	−	0.959096i	$-0.591357\pi$
$433$	292.714	−	292.714i	0.676014	−	0.676014i	0.959096	+	0.283082i	$0.0913568\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	183.712	+	183.712i	0.420393	+	0.420393i
$438$		0			0
$439$		−	119.000i		−	0.271071i	−0.990773	−	0.135535i	$-0.956725\pi$
$439$		−	119.000i		−	0.271071i	0.990773	−	0.135535i	$-0.0432754\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	191.060	−	191.060i	0.431287	−	0.431287i	−0.457779	−	0.889066i	$-0.651355\pi$
$443$	191.060	−	191.060i	0.431287	−	0.431287i	0.889066	+	0.457779i	$0.151355\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			84.0000i			0.187082i	0.995615	+	0.0935412i	$0.0298187\pi$
$449$			84.0000i			0.187082i	−0.995615	+	0.0935412i	$0.970181\pi$
$450$		0			0
$451$	−360.000			−0.798226
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−146.969	−	146.969i	−0.321596	−	0.321596i	0.527783	−	0.849379i	$-0.323023\pi$
$457$	−146.969	−	146.969i	−0.321596	−	0.321596i	−0.849379	+	0.527783i	$0.823023\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−516.000			−1.11931			−0.559653	−	0.828727i	$-0.689066\pi$
$461$	−516.000			−1.11931			−0.559653	+	0.828727i	$0.689066\pi$
$462$		0			0
$463$	524.191	−	524.191i	1.13216	−	1.13216i	0.142344	−	0.989817i	$-0.454536\pi$
$463$	524.191	−	524.191i	1.13216	−	1.13216i	0.989817	−	0.142344i	$-0.0454640\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	507.044	+	507.044i	1.08575	+	1.08575i	0.995961	+	0.0897872i	$0.0286187\pi$
$467$	507.044	+	507.044i	1.08575	+	1.08575i	0.0897872	+	0.995961i	$0.471381\pi$
$468$		0			0
$469$		−	387.000i		−	0.825160i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.34847	+	7.34847i	−0.0155359	+	0.0155359i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			102.000i			0.212944i	0.994316	+	0.106472i	$0.0339554\pi$
$479$			102.000i			0.212944i	−0.994316	+	0.106472i	$0.966045\pi$
$480$		0			0
$481$	60.0000			0.124740
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	243.724	+	243.724i	0.500460	+	0.500460i	0.911581	−	0.411121i	$-0.134862\pi$
$487$	243.724	+	243.724i	0.500460	+	0.500460i	−0.411121	+	0.911581i	$0.634862\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−444.000			−0.904277			−0.452138	−	0.891948i	$-0.649339\pi$
$491$	−444.000			−0.904277			−0.452138	+	0.891948i	$0.649339\pi$
$492$		0			0
$493$	−925.907	+	925.907i	−1.87811	+	1.87811i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−220.454	−	220.454i	−0.443570	−	0.443570i
$498$		0			0
$499$		−	707.000i		−	1.41683i	−0.705794	−	0.708417i	$-0.749410\pi$
$499$		−	707.000i		−	1.41683i	0.705794	−	0.708417i	$-0.250590\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−176.363	+	176.363i	−0.350623	+	0.350623i	−0.860341	−	0.509718i	$-0.829750\pi$
$503$	−176.363	+	176.363i	−0.350623	+	0.350623i	0.509718	+	0.860341i	$0.329750\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$			666.000i			1.30845i	0.756301	+	0.654224i	$0.227005\pi$
$509$			666.000i			1.30845i	−0.756301	+	0.654224i	$0.772995\pi$
$510$		0			0
$511$	468.000			0.915851
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−308.636	−	308.636i	−0.596974	−	0.596974i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	558.000			1.07102			0.535509	−	0.844530i	$-0.320120\pi$
$521$	558.000			1.07102			0.535509	+	0.844530i	$0.320120\pi$
$522$		0			0
$523$	−547.461	+	547.461i	−1.04677	+	1.04677i	−0.0479193	+	0.998851i	$0.515259\pi$
$523$	−547.461	+	547.461i	−1.04677	+	1.04677i	−0.998851	+	0.0479193i	$0.984741\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1080.22	+	1080.22i	2.04976	+	2.04976i
$528$		0			0
$529$			421.000i			0.795841i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−367.423	+	367.423i	−0.689350	+	0.689350i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			132.000i			0.244898i
$540$		0			0
$541$	803.000			1.48429			0.742144	−	0.670240i	$-0.233809\pi$
$541$	803.000			1.48429			0.742144	+	0.670240i	$0.233809\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−215.555	−	215.555i	−0.394068	−	0.394068i	0.482067	−	0.876135i	$-0.339886\pi$
$547$	−215.555	−	215.555i	−0.394068	−	0.394068i	−0.876135	+	0.482067i	$0.839886\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1050.00			1.90563
$552$		0			0
$553$	389.469	−	389.469i	0.704284	−	0.704284i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	448.257	+	448.257i	0.804770	+	0.804770i	0.983837	−	0.179067i	$-0.0573080\pi$
$557$	448.257	+	448.257i	0.804770	+	0.804770i	−0.179067	+	0.983837i	$0.557308\pi$
$558$		0			0
$559$			15.0000i			0.0268336i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	257.196	−	257.196i	0.456832	−	0.456832i	−0.440782	−	0.897614i	$-0.645299\pi$
$563$	257.196	−	257.196i	0.456832	−	0.456832i	0.897614	+	0.440782i	$0.145299\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	582.000i		−	1.02285i	−0.859329	−	0.511424i	$-0.829118\pi$
$569$		−	582.000i		−	1.02285i	0.859329	−	0.511424i	$-0.170882\pi$
$570$		0			0
$571$	215.000			0.376532			0.188266	−	0.982118i	$-0.439713\pi$
$571$	215.000			0.376532			0.188266	+	0.982118i	$0.439713\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−246.174	−	246.174i	−0.426644	−	0.426644i	0.460839	−	0.887484i	$-0.347548\pi$
$577$	−246.174	−	246.174i	−0.426644	−	0.426644i	−0.887484	+	0.460839i	$0.847548\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	594.000			1.02238
$582$		0			0
$583$	88.1816	−	88.1816i	0.151255	−	0.151255i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.6969	−	14.6969i	−0.0250374	−	0.0250374i	0.694477	−	0.719515i	$-0.255636\pi$
$587$	−14.6969	−	14.6969i	−0.0250374	−	0.0250374i	−0.719515	+	0.694477i	$0.755636\pi$
$588$		0			0
$589$		−	1225.00i		−	2.07980i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	323.333	−	323.333i	0.545249	−	0.545249i	−0.379814	−	0.925063i	$-0.624012\pi$
$593$	323.333	−	323.333i	0.545249	−	0.545249i	0.925063	+	0.379814i	$0.124012\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			528.000i			0.881469i	0.897637	+	0.440735i	$0.145282\pi$
$599$			528.000i			0.881469i	−0.897637	+	0.440735i	$0.854718\pi$
$600$		0			0
$601$	493.000			0.820300			0.410150	−	0.912018i	$-0.365476\pi$
$601$	493.000			0.820300			0.410150	+	0.912018i	$0.365476\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	729.948	+	729.948i	1.20255	+	1.20255i	0.973388	+	0.229162i	$0.0735985\pi$
$607$	729.948	+	729.948i	1.20255	+	1.20255i	0.229162	+	0.973388i	$0.426402\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−630.000			−1.03110
$612$		0			0
$613$	−112.677	+	112.677i	−0.183812	+	0.183812i	−0.793014	−	0.609203i	$-0.791490\pi$
$613$	−112.677	+	112.677i	−0.183812	+	0.183812i	0.609203	+	0.793014i	$0.291490\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	382.120	+	382.120i	0.619320	+	0.619320i	0.945357	−	0.326037i	$-0.105713\pi$
$617$	382.120	+	382.120i	0.619320	+	0.619320i	−0.326037	+	0.945357i	$0.605713\pi$
$618$		0			0
$619$			733.000i			1.18417i	0.805876	+	0.592084i	$0.201695\pi$
$619$			733.000i			1.18417i	−0.805876	+	0.592084i	$0.798305\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−220.454	+	220.454i	−0.353859	+	0.353859i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	216.000i		−	0.343402i
$630$		0			0
$631$	647.000			1.02536			0.512678	−	0.858581i	$-0.328653\pi$
$631$	647.000			1.02536			0.512678	+	0.858581i	$0.328653\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	134.722	+	134.722i	0.211494	+	0.211494i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	636.000			0.992200			0.496100	−	0.868265i	$-0.334765\pi$
$641$	636.000			0.992200			0.496100	+	0.868265i	$0.334765\pi$
$642$		0			0
$643$	396.817	−	396.817i	0.617134	−	0.617134i	−0.327661	−	0.944795i	$-0.606260\pi$
$643$	396.817	−	396.817i	0.617134	−	0.617134i	0.944795	+	0.327661i	$0.106260\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	29.3939	+	29.3939i	0.0454310	+	0.0454310i	0.729457	−	0.684026i	$-0.239773\pi$
$647$	29.3939	+	29.3939i	0.0454310	+	0.0454310i	−0.684026	+	0.729457i	$0.739773\pi$
$648$		0			0
$649$			468.000i			0.721109i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−521.741	+	521.741i	−0.798991	+	0.798991i	−0.982936	−	0.183945i	$-0.941113\pi$
$653$	−521.741	+	521.741i	−0.798991	+	0.798991i	0.183945	+	0.982936i	$0.441113\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	36.0000i		−	0.0546282i	−0.999627	−	0.0273141i	$-0.991305\pi$
$659$		−	36.0000i		−	0.0546282i	0.999627	−	0.0273141i	$-0.00869543\pi$
$660$		0			0
$661$	−1022.00			−1.54614			−0.773071	−	0.634319i	$-0.781280\pi$
$661$	−1022.00			−1.54614			−0.773071	+	0.634319i	$0.781280\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−308.636	−	308.636i	−0.462722	−	0.462722i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	78.0000			0.116244
$672$		0			0
$673$	−857.321	+	857.321i	−1.27388	+	1.27388i	−0.329845	+	0.944035i	$0.606997\pi$
$673$	−857.321	+	857.321i	−1.27388	+	1.27388i	−0.944035	+	0.329845i	$0.893003\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	536.438	+	536.438i	0.792376	+	0.792376i	0.981880	−	0.189504i	$-0.0606881\pi$
$677$	536.438	+	536.438i	0.792376	+	0.792376i	−0.189504	+	0.981880i	$0.560688\pi$
$678$		0			0
$679$		−	891.000i		−	1.31222i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−764.241	+	764.241i	−1.11895	+	1.11895i	−0.127051	+	0.991896i	$0.540551\pi$
$683$	−764.241	+	764.241i	−1.11895	+	1.11895i	−0.991896	+	0.127051i	$0.959449\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	180.000i		−	0.261248i
$690$		0			0
$691$	−842.000			−1.21852			−0.609262	−	0.792969i	$-0.708534\pi$
$691$	−842.000			−1.21852			−0.609262	+	0.792969i	$0.708534\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1322.72	+	1322.72i	1.89774	+	1.89774i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	534.000			0.761769			0.380884	−	0.924623i	$-0.375620\pi$
$701$	534.000			0.761769			0.380884	+	0.924623i	$0.375620\pi$
$702$		0			0
$703$	−122.474	+	122.474i	−0.174217	+	0.174217i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−705.453	−	705.453i	−0.997812	−	0.997812i
$708$		0			0
$709$			743.000i			1.04795i	0.851732	+	0.523977i	$0.175552\pi$
$709$			743.000i			1.04795i	−0.851732	+	0.523977i	$0.824448\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−360.075	+	360.075i	−0.505014	+	0.505014i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	906.000i		−	1.26008i	−0.776561	−	0.630042i	$-0.783038\pi$
$719$		−	906.000i		−	1.26008i	0.776561	−	0.630042i	$-0.216962\pi$
$720$		0			0
$721$	−576.000			−0.798890
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−74.7094	−	74.7094i	−0.102764	−	0.102764i	0.653855	−	0.756619i	$-0.273150\pi$
$727$	−74.7094	−	74.7094i	−0.102764	−	0.102764i	−0.756619	+	0.653855i	$0.773150\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	54.0000			0.0738714
$732$		0			0
$733$	631.968	−	631.968i	0.862167	−	0.862167i	−0.129423	−	0.991590i	$-0.541312\pi$
$733$	631.968	−	631.968i	0.862167	−	0.862167i	0.991590	+	0.129423i	$0.0413124\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−315.984	−	315.984i	−0.428744	−	0.428744i
$738$		0			0
$739$			470.000i			0.635995i	0.948092	+	0.317997i	$0.103010\pi$
$739$			470.000i			0.635995i	−0.948092	+	0.317997i	$0.896990\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−249.848	+	249.848i	−0.336269	+	0.336269i	−0.854961	−	0.518692i	$-0.826419\pi$
$743$	−249.848	+	249.848i	−0.336269	+	0.336269i	0.518692	+	0.854961i	$0.326419\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	216.000i		−	0.288385i
$750$		0			0
$751$	1282.00			1.70706			0.853529	−	0.521046i	$-0.174458\pi$
$751$	1282.00			1.70706			0.853529	+	0.521046i	$0.174458\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−106.553	−	106.553i	−0.140757	−	0.140757i	0.633217	−	0.773974i	$-0.281734\pi$
$757$	−106.553	−	106.553i	−0.140757	−	0.140757i	−0.773974	+	0.633217i	$0.781734\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	768.000			1.00920			0.504599	−	0.863354i	$-0.331640\pi$
$761$	768.000			1.00920			0.504599	+	0.863354i	$0.331640\pi$
$762$		0			0
$763$	3.67423	−	3.67423i	0.00481551	−	0.00481551i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	477.650	+	477.650i	0.622752	+	0.622752i
$768$		0			0
$769$		−	491.000i		−	0.638492i	−0.947672	−	0.319246i	$-0.896570\pi$
$769$		−	491.000i		−	0.638492i	0.947672	−	0.319246i	$-0.103430\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	793.635	−	793.635i	1.02669	−	1.02669i	0.0270605	−	0.999634i	$-0.491385\pi$
$773$	793.635	−	793.635i	1.02669	−	1.02669i	0.999634	−	0.0270605i	$-0.00861467\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1500.00i		−	1.92555i
$780$		0			0
$781$	−360.000			−0.460948
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	378.446	+	378.446i	0.480872	+	0.480872i	0.905410	−	0.424538i	$-0.139564\pi$
$787$	378.446	+	378.446i	0.480872	+	0.480872i	−0.424538	+	0.905410i	$0.639564\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	432.000			0.546144
$792$		0			0
$793$	79.6084	−	79.6084i	0.100389	−	0.100389i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−617.271	−	617.271i	−0.774494	−	0.774494i	0.204395	−	0.978889i	$-0.434477\pi$
$797$	−617.271	−	617.271i	−0.774494	−	0.774494i	−0.978889	+	0.204395i	$0.934477\pi$
$798$		0			0
$799$			2268.00i			2.83855i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	382.120	−	382.120i	0.475866	−	0.475866i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			348.000i			0.430161i	0.976596	+	0.215080i	$0.0690013\pi$
$809$			348.000i			0.430161i	−0.976596	+	0.215080i	$0.930999\pi$
$810$		0			0
$811$	1045.00			1.28853			0.644266	−	0.764801i	$-0.277163\pi$
$811$	1045.00			1.28853			0.644266	+	0.764801i	$0.277163\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−30.6186	−	30.6186i	−0.0374769	−	0.0374769i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−606.000			−0.738124			−0.369062	−	0.929405i	$-0.620321\pi$
$821$	−606.000			−0.738124			−0.369062	+	0.929405i	$0.620321\pi$
$822$		0			0
$823$	−302.512	+	302.512i	−0.367572	+	0.367572i	−0.866591	−	0.499019i	$-0.833694\pi$
$823$	−302.512	+	302.512i	−0.367572	+	0.367572i	0.499019	+	0.866591i	$0.333694\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−396.817	−	396.817i	−0.479827	−	0.479827i	0.425249	−	0.905076i	$-0.360187\pi$
$827$	−396.817	−	396.817i	−0.479827	−	0.479827i	−0.905076	+	0.425249i	$0.860187\pi$
$828$		0			0
$829$		−	406.000i		−	0.489747i	−0.969555	−	0.244873i	$-0.921254\pi$
$829$		−	406.000i		−	0.489747i	0.969555	−	0.244873i	$-0.0787464\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	484.999	−	484.999i	0.582232	−	0.582232i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			462.000i			0.550656i	0.961350	+	0.275328i	$0.0887864\pi$
$839$			462.000i			0.550656i	−0.961350	+	0.275328i	$0.911214\pi$
$840$		0			0
$841$	−923.000			−1.09750
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	312.310	+	312.310i	0.368725	+	0.368725i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	72.0000			0.0846063
$852$		0			0
$853$	−285.366	+	285.366i	−0.334543	+	0.334543i	−0.854309	−	0.519766i	$-0.826019\pi$
$853$	−285.366	+	285.366i	−0.334543	+	0.334543i	0.519766	+	0.854309i	$0.326019\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	191.060	+	191.060i	0.222941	+	0.222941i	0.809736	−	0.586795i	$-0.199611\pi$
$857$	191.060	+	191.060i	0.222941	+	0.222941i	−0.586795	+	0.809736i	$0.699611\pi$
$858$		0			0
$859$		−	410.000i		−	0.477299i	−0.971106	−	0.238650i	$-0.923295\pi$
$859$		−	410.000i		−	0.477299i	0.971106	−	0.238650i	$-0.0767047\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1116.97	+	1116.97i	−1.29428	+	1.29428i	−0.362174	+	0.932111i	$0.617965\pi$
$863$	−1116.97	+	1116.97i	−1.29428	+	1.29428i	−0.932111	+	0.362174i	$0.882035\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	636.000i		−	0.731876i
$870$		0			0
$871$	−645.000			−0.740528
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−180.037	−	180.037i	−0.205288	−	0.205288i	0.596973	−	0.802261i	$-0.296370\pi$
$877$	−180.037	−	180.037i	−0.205288	−	0.205288i	−0.802261	+	0.596973i	$0.796370\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	120.000			0.136209			0.0681044	−	0.997678i	$-0.478305\pi$
$881$	120.000			0.136209			0.0681044	+	0.997678i	$0.478305\pi$
$882$		0			0
$883$	584.203	−	584.203i	0.661612	−	0.661612i	−0.294148	−	0.955760i	$-0.595036\pi$
$883$	584.203	−	584.203i	0.661612	−	0.661612i	0.955760	+	0.294148i	$0.0950359\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−198.409	−	198.409i	−0.223685	−	0.223685i	0.586363	−	0.810048i	$-0.300559\pi$
$887$	−198.409	−	198.409i	−0.223685	−	0.223685i	−0.810048	+	0.586363i	$0.800559\pi$
$888$		0			0
$889$		−	108.000i		−	0.121485i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1285.98	−	1285.98i	1.44007	−	1.44007i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			2058.00i			2.28921i
$900$		0			0
$901$	−648.000			−0.719201
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1033.68	−	1033.68i	−1.13967	−	1.13967i	−0.988508	−	0.151166i	$-0.951697\pi$
$907$	−1033.68	−	1033.68i	−1.13967	−	1.13967i	−0.151166	−	0.988508i	$-0.548303\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1218.00			−1.33699			−0.668496	−	0.743716i	$-0.733062\pi$
$911$	−1218.00			−1.33699			−0.668496	+	0.743716i	$0.733062\pi$
$912$		0			0
$913$	484.999	−	484.999i	0.531215	−	0.531215i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	44.0908	+	44.0908i	0.0480816	+	0.0480816i
$918$		0			0
$919$		−	469.000i		−	0.510337i	−0.966897	−	0.255169i	$-0.917869\pi$
$919$		−	469.000i		−	0.510337i	0.966897	−	0.255169i	$-0.0821310\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−367.423	+	367.423i	−0.398075	+	0.398075i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			474.000i			0.510226i	0.966911	+	0.255113i	$0.0821127\pi$
$929$			474.000i			0.510226i	−0.966911	+	0.255113i	$0.917887\pi$
$930$		0			0
$931$	−550.000			−0.590763
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−709.127	−	709.127i	−0.756806	−	0.756806i	0.218934	−	0.975740i	$-0.429742\pi$
$937$	−709.127	−	709.127i	−0.756806	−	0.756806i	−0.975740	+	0.218934i	$0.929742\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−990.000			−1.05207			−0.526036	−	0.850462i	$-0.676322\pi$
$941$	−990.000			−1.05207			−0.526036	+	0.850462i	$0.676322\pi$
$942$		0			0
$943$	−440.908	+	440.908i	−0.467559	+	0.467559i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1036.13	−	1036.13i	−1.09412	−	1.09412i	−0.995084	−	0.0990391i	$-0.968423\pi$
$947$	−1036.13	−	1036.13i	−1.09412	−	1.09412i	−0.0990391	−	0.995084i	$-0.531577\pi$
$948$		0			0
$949$		−	780.000i		−	0.821918i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	631.968	−	631.968i	0.663136	−	0.663136i	−0.292982	−	0.956118i	$-0.594648\pi$
$953$	631.968	−	631.968i	0.663136	−	0.663136i	0.956118	+	0.292982i	$0.0946477\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			594.000i			0.619395i
$960$		0			0
$961$	1440.00			1.49844
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−563.383	−	563.383i	−0.582609	−	0.582609i	0.353011	−	0.935619i	$-0.385158\pi$
$967$	−563.383	−	563.383i	−0.582609	−	0.582609i	−0.935619	+	0.353011i	$0.885158\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1338.00			−1.37796			−0.688980	−	0.724780i	$-0.741941\pi$
$971$	−1338.00			−1.37796			−0.688980	+	0.724780i	$0.741941\pi$
$972$		0			0
$973$	−213.106	+	213.106i	−0.219019	+	0.219019i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	301.287	+	301.287i	0.308380	+	0.308380i	0.844281	−	0.535901i	$-0.180028\pi$
$977$	301.287	+	301.287i	0.308380	+	0.308380i	−0.535901	+	0.844281i	$0.680028\pi$
$978$		0			0
$979$			360.000i			0.367722i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−426.211	+	426.211i	−0.433582	+	0.433582i	−0.889845	−	0.456263i	$-0.849188\pi$
$983$	−426.211	+	426.211i	−0.433582	+	0.433582i	0.456263	+	0.889845i	$0.349188\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			18.0000i			0.0182002i
$990$		0			0
$991$	83.0000			0.0837538			0.0418769	−	0.999123i	$-0.486666\pi$
$991$	83.0000			0.0837538			0.0418769	+	0.999123i	$0.486666\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	573.181	+	573.181i	0.574905	+	0.574905i	0.933495	−	0.358590i	$-0.116742\pi$
$997$	573.181	+	573.181i	0.574905	+	0.574905i	−0.358590	+	0.933495i	$0.616742\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.l.d.757.1		4
3.2	odd	2		300.3.k.c.157.1	✓	4
5.2	odd	4	inner	900.3.l.d.793.2		4
5.3	odd	4	inner	900.3.l.d.793.1		4
5.4	even	2	inner	900.3.l.d.757.2		4
12.11	even	2		1200.3.bg.i.1057.2		4
15.2	even	4		300.3.k.c.193.2	yes	4
15.8	even	4		300.3.k.c.193.1	yes	4
15.14	odd	2		300.3.k.c.157.2	yes	4
60.23	odd	4		1200.3.bg.i.193.2		4
60.47	odd	4		1200.3.bg.i.193.1		4
60.59	even	2		1200.3.bg.i.1057.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.3.k.c.157.1	✓	4	3.2	odd	2
300.3.k.c.157.2	yes	4	15.14	odd	2
300.3.k.c.193.1	yes	4	15.8	even	4
300.3.k.c.193.2	yes	4	15.2	even	4
900.3.l.d.757.1		4	1.1	even	1	trivial
900.3.l.d.757.2		4	5.4	even	2	inner
900.3.l.d.793.1		4	5.3	odd	4	inner
900.3.l.d.793.2		4	5.2	odd	4	inner
1200.3.bg.i.193.1		4	60.47	odd	4
1200.3.bg.i.193.2		4	60.23	odd	4
1200.3.bg.i.1057.1		4	60.59	even	2
1200.3.bg.i.1057.2		4	12.11	even	2

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.67423 - 3.67423i) q^{7} +O(q^{10})\)	\(q+(-3.67423 - 3.67423i) q^{7} -6.00000 q^{11} +(-6.12372 + 6.12372i) q^{13} +(22.0454 + 22.0454i) q^{17} -25.0000i q^{19} +(-7.34847 + 7.34847i) q^{23} +42.0000i q^{29} +49.0000 q^{31} +(-4.89898 - 4.89898i) q^{37} +60.0000 q^{41} +(1.22474 - 1.22474i) q^{43} +(51.4393 + 51.4393i) q^{47} -22.0000i q^{49} +(-14.6969 + 14.6969i) q^{53} -78.0000i q^{59} -13.0000 q^{61} +(52.6640 + 52.6640i) q^{67} +60.0000 q^{71} +(-63.6867 + 63.6867i) q^{73} +(22.0454 + 22.0454i) q^{77} +106.000i q^{79} +(-80.8332 + 80.8332i) q^{83} -60.0000i q^{89} +45.0000 q^{91} +(121.250 + 121.250i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{11} + 196 q^{31} + 240 q^{41} - 52 q^{61} + 240 q^{71} + 180 q^{91}+O(q^{100}) \) 4 * q - 24 * q^11 + 196 * q^31 + 240 * q^41 - 52 * q^61 + 240 * q^71 + 180 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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