LMFDB - Embedded newform 1800.2.s.b.593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,2,Mod(593,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 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("1800.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1800.s (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:	$14.3730723638$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1800.593
Dual form			1800.2.s.b.1457.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q-5.65685i q^{11} +(-3.00000 - 3.00000i) q^{13} +4.00000i q^{19} +(-2.82843 + 2.82843i) q^{23} +1.41421 q^{29} -8.00000 q^{31} +(-7.00000 + 7.00000i) q^{37} +1.41421i q^{41} +(-4.00000 - 4.00000i) q^{43} +(-2.82843 - 2.82843i) q^{47} +7.00000i q^{49} +(8.48528 - 8.48528i) q^{53} -11.3137 q^{59} -12.0000 q^{61} +(8.00000 - 8.00000i) q^{67} -5.65685i q^{71} +(3.00000 + 3.00000i) q^{73} -8.00000i q^{79} +(11.3137 - 11.3137i) q^{83} -7.07107 q^{89} +(-5.00000 + 5.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{13} - 32 q^{31} - 28 q^{37} - 16 q^{43} - 48 q^{61} + 32 q^{67} + 12 q^{73} - 20 q^{97}+O(q^{100}) $ 4 * q - 12 * q^13 - 32 * q^31 - 28 * q^37 - 16 * q^43 - 48 * q^61 + 32 * q^67 + 12 * q^73 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.65685i		−	1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$		−	5.65685i		−	1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$		0			0
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	0.155747	−	0.987797i	$-0.450222\pi$
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	−0.987797	+	0.155747i	$0.950222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.82843	+	2.82843i	−0.589768	+	0.589768i	−0.937568	−	0.347801i	$-0.886929\pi$
$23$	−2.82843	+	2.82843i	−0.589768	+	0.589768i	0.347801	+	0.937568i	$0.386929\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.41421			0.262613			0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421			0.262613			0.131306	+	0.991342i	$0.458083\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−7.00000	+	7.00000i	−1.15079	+	1.15079i	−0.164399	+	0.986394i	$0.552568\pi$
$37$	−7.00000	+	7.00000i	−1.15079	+	1.15079i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.41421i			0.220863i	0.993884	+	0.110432i	$0.0352233\pi$
$41$			1.41421i			0.220863i	−0.993884	+	0.110432i	$0.964777\pi$
$42$		0			0
$43$	−4.00000	−	4.00000i	−0.609994	−	0.609994i	0.332950	−	0.942944i	$-0.391956\pi$
$43$	−4.00000	−	4.00000i	−0.609994	−	0.609994i	−0.942944	+	0.332950i	$0.891956\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.82843	−	2.82843i	−0.412568	−	0.412568i	0.470064	−	0.882632i	$-0.344231\pi$
$47$	−2.82843	−	2.82843i	−0.412568	−	0.412568i	−0.882632	+	0.470064i	$0.844231\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.48528	−	8.48528i	1.16554	−	1.16554i	0.182300	−	0.983243i	$-0.441646\pi$
$53$	8.48528	−	8.48528i	1.16554	−	1.16554i	0.983243	−	0.182300i	$-0.0583542\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.3137			−1.47292			−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137			−1.47292			−0.736460	+	0.676481i	$0.763504\pi$
$60$		0			0
$61$	−12.0000			−1.53644			−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000			−1.53644			−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	8.00000	−	8.00000i	0.977356	−	0.977356i	−0.0223937	−	0.999749i	$-0.507129\pi$
$67$	8.00000	−	8.00000i	0.977356	−	0.977356i	0.999749	+	0.0223937i	$0.00712872\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.65685i		−	0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$		−	5.65685i		−	0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$		0			0
$73$	3.00000	+	3.00000i	0.351123	+	0.351123i	0.860527	−	0.509404i	$-0.170134\pi$
$73$	3.00000	+	3.00000i	0.351123	+	0.351123i	−0.509404	+	0.860527i	$0.670134\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	8.00000i		−	0.900070i	−0.893011	−	0.450035i	$-0.851411\pi$
$79$		−	8.00000i		−	0.900070i	0.893011	−	0.450035i	$-0.148589\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.3137	−	11.3137i	1.24184	−	1.24184i	0.282604	−	0.959237i	$-0.408802\pi$
$83$	11.3137	−	11.3137i	1.24184	−	1.24184i	0.959237	−	0.282604i	$-0.0911983\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.07107			−0.749532			−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107			−0.749532			−0.374766	+	0.927119i	$0.622277\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.00000	+	5.00000i	−0.507673	+	0.507673i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−5.00000	+	5.00000i	−0.507673	+	0.507673i	0.406138	+	0.913812i	$0.366875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	1.41421i		−	0.140720i	−0.997522	−	0.0703598i	$-0.977585\pi$
$101$		−	1.41421i		−	0.140720i	0.997522	−	0.0703598i	$-0.0224147\pi$
$102$		0			0
$103$	−4.00000	−	4.00000i	−0.394132	−	0.394132i	0.482025	−	0.876157i	$-0.339901\pi$
$103$	−4.00000	−	4.00000i	−0.394132	−	0.394132i	−0.876157	+	0.482025i	$0.839901\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.3137	−	11.3137i	−1.09374	−	1.09374i	−0.995126	−	0.0986115i	$-0.968560\pi$
$107$	−11.3137	−	11.3137i	−1.09374	−	1.09374i	−0.0986115	−	0.995126i	$-0.531440\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.24264	−	4.24264i	0.399114	−	0.399114i	−0.478806	−	0.877920i	$-0.658930\pi$
$113$	4.24264	−	4.24264i	0.399114	−	0.399114i	0.877920	+	0.478806i	$0.158930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−21.0000			−1.90909
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−12.0000	+	12.0000i	−1.06483	+	1.06483i	−0.0670802	+	0.997748i	$0.521368\pi$
$127$	−12.0000	+	12.0000i	−1.06483	+	1.06483i	−0.997748	+	0.0670802i	$0.978632\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	5.65685i		−	0.494242i	−0.968985	−	0.247121i	$-0.920516\pi$
$131$		−	5.65685i		−	0.494242i	0.968985	−	0.247121i	$-0.0794845\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.7279	+	12.7279i	1.08742	+	1.08742i	0.995793	+	0.0916263i	$0.0292065\pi$
$137$	12.7279	+	12.7279i	1.08742	+	1.08742i	0.0916263	+	0.995793i	$0.470793\pi$
$138$		0			0
$139$			20.0000i			1.69638i	0.529694	+	0.848189i	$0.322307\pi$
$139$			20.0000i			1.69638i	−0.529694	+	0.848189i	$0.677693\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−16.9706	+	16.9706i	−1.41915	+	1.41915i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.24264			0.347571			0.173785	−	0.984784i	$-0.444400\pi$
$149$	4.24264			0.347571			0.173785	+	0.984784i	$0.444400\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.00000	−	7.00000i	0.558661	−	0.558661i	−0.370265	−	0.928926i	$-0.620733\pi$
$157$	7.00000	−	7.00000i	0.558661	−	0.558661i	0.928926	+	0.370265i	$0.120733\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−12.0000	−	12.0000i	−0.939913	−	0.939913i	0.0583818	−	0.998294i	$-0.481406\pi$
$163$	−12.0000	−	12.0000i	−0.939913	−	0.939913i	−0.998294	+	0.0583818i	$0.981406\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.48528	−	8.48528i	−0.656611	−	0.656611i	0.297966	−	0.954577i	$-0.403692\pi$
$167$	−8.48528	−	8.48528i	−0.656611	−	0.656611i	−0.954577	+	0.297966i	$0.903692\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.07107	+	7.07107i	−0.537603	+	0.537603i	−0.922824	−	0.385221i	$-0.874125\pi$
$173$	−7.07107	+	7.07107i	−0.537603	+	0.537603i	0.385221	+	0.922824i	$0.374125\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.65685			0.422813			0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685			0.422813			0.211407	+	0.977398i	$0.432196\pi$
$180$		0			0
$181$	4.00000			0.297318			0.148659	−	0.988889i	$-0.452504\pi$
$181$	4.00000			0.297318			0.148659	+	0.988889i	$0.452504\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	11.3137i		−	0.818631i	−0.912393	−	0.409316i	$-0.865768\pi$
$191$		−	11.3137i		−	0.818631i	0.912393	−	0.409316i	$-0.134232\pi$
$192$		0			0
$193$	−5.00000	−	5.00000i	−0.359908	−	0.359908i	0.503871	−	0.863779i	$-0.331909\pi$
$193$	−5.00000	−	5.00000i	−0.359908	−	0.359908i	−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.7279	+	12.7279i	0.906827	+	0.906827i	0.996015	−	0.0891879i	$-0.0284272\pi$
$197$	12.7279	+	12.7279i	0.906827	+	0.906827i	−0.0891879	+	0.996015i	$0.528427\pi$
$198$		0			0
$199$		−	24.0000i		−	1.70131i	−0.525720	−	0.850657i	$-0.676204\pi$
$199$		−	24.0000i		−	1.70131i	0.525720	−	0.850657i	$-0.323796\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	22.6274			1.56517
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		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
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.65685	−	5.65685i	0.370593	−	0.370593i	−0.497100	−	0.867693i	$-0.665602\pi$
$233$	5.65685	−	5.65685i	0.370593	−	0.370593i	0.867693	+	0.497100i	$0.165602\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.9706			−1.09773			−0.548867	−	0.835910i	$-0.684941\pi$
$239$	−16.9706			−1.09773			−0.548867	+	0.835910i	$0.684941\pi$
$240$		0			0
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	12.0000	−	12.0000i	0.763542	−	0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	16.0000	+	16.0000i	1.00591	+	1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.07107	−	7.07107i	−0.441081	−	0.441081i	0.451294	−	0.892375i	$-0.350963\pi$
$257$	−7.07107	−	7.07107i	−0.441081	−	0.441081i	−0.892375	+	0.451294i	$0.850963\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.82843	+	2.82843i	−0.174408	+	0.174408i	−0.788913	−	0.614505i	$-0.789356\pi$
$263$	−2.82843	+	2.82843i	−0.174408	+	0.174408i	0.614505	+	0.788913i	$0.289356\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.5563			−0.948487			−0.474244	−	0.880394i	$-0.657278\pi$
$269$	−15.5563			−0.948487			−0.474244	+	0.880394i	$0.657278\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	−0.955598	−	0.294672i	$-0.904789\pi$
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	0.294672	+	0.955598i	$0.404789\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			4.24264i			0.253095i	0.991961	+	0.126547i	$0.0403896\pi$
$281$			4.24264i			0.253095i	−0.991961	+	0.126547i	$0.959610\pi$
$282$		0			0
$283$	16.0000	+	16.0000i	0.951101	+	0.951101i	0.998859	−	0.0477577i	$-0.0152075\pi$
$283$	16.0000	+	16.0000i	0.951101	+	0.951101i	−0.0477577	+	0.998859i	$0.515208\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.41421	+	1.41421i	−0.0826192	+	0.0826192i	−0.747209	−	0.664589i	$-0.768606\pi$
$293$	−1.41421	+	1.41421i	−0.0826192	+	0.0826192i	0.664589	+	0.747209i	$0.268606\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.9706			0.981433
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	20.0000	−	20.0000i	1.14146	−	1.14146i	0.153277	−	0.988183i	$-0.451017\pi$
$307$	20.0000	−	20.0000i	1.14146	−	1.14146i	0.988183	−	0.153277i	$-0.0489827\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			28.2843i			1.60385i	0.597422	+	0.801927i	$0.296192\pi$
$311$			28.2843i			1.60385i	−0.597422	+	0.801927i	$0.703808\pi$
$312$		0			0
$313$	9.00000	+	9.00000i	0.508710	+	0.508710i	0.914130	−	0.405420i	$-0.132875\pi$
$313$	9.00000	+	9.00000i	0.508710	+	0.508710i	−0.405420	+	0.914130i	$0.632875\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.82843	+	2.82843i	0.158860	+	0.158860i	0.782062	−	0.623201i	$-0.214168\pi$
$317$	2.82843	+	2.82843i	0.158860	+	0.158860i	−0.623201	+	0.782062i	$0.714168\pi$
$318$		0			0
$319$		−	8.00000i		−	0.447914i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−15.0000	+	15.0000i	−0.817102	+	0.817102i	−0.985687	−	0.168585i	$-0.946080\pi$
$337$	−15.0000	+	15.0000i	−0.817102	+	0.817102i	0.168585	+	0.985687i	$0.446080\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			45.2548i			2.45069i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	4.00000i		−	0.214115i	−0.994253	−	0.107058i	$-0.965857\pi$
$349$		−	4.00000i		−	0.214115i	0.994253	−	0.107058i	$-0.0341429\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.65685	−	5.65685i	0.301084	−	0.301084i	−0.540354	−	0.841438i	$-0.681710\pi$
$353$	5.65685	−	5.65685i	0.301084	−	0.301084i	0.841438	+	0.540354i	$0.181710\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5.65685			−0.298557			−0.149279	−	0.988795i	$-0.547695\pi$
$359$	−5.65685			−0.298557			−0.149279	+	0.988795i	$0.547695\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.00000	−	4.00000i	0.208798	−	0.208798i	−0.594958	−	0.803757i	$-0.702831\pi$
$367$	4.00000	−	4.00000i	0.208798	−	0.208798i	0.803757	+	0.594958i	$0.202831\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	3.00000	+	3.00000i	0.155334	+	0.155334i	0.780496	−	0.625161i	$-0.214967\pi$
$373$	3.00000	+	3.00000i	0.155334	+	0.155334i	−0.625161	+	0.780496i	$0.714967\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.24264	−	4.24264i	−0.218507	−	0.218507i
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.48528	+	8.48528i	−0.433578	+	0.433578i	−0.889843	−	0.456266i	$-0.849187\pi$
$383$	−8.48528	+	8.48528i	−0.433578	+	0.433578i	0.456266	+	0.889843i	$0.349187\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	29.6985			1.50577			0.752886	−	0.658150i	$-0.228661\pi$
$389$	29.6985			1.50577			0.752886	+	0.658150i	$0.228661\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	19.0000	−	19.0000i	0.953583	−	0.953583i	−0.0453868	−	0.998969i	$-0.514452\pi$
$397$	19.0000	−	19.0000i	0.953583	−	0.953583i	0.998969	+	0.0453868i	$0.0144520\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			26.8701i			1.34183i	0.741536	+	0.670913i	$0.234098\pi$
$401$			26.8701i			1.34183i	−0.741536	+	0.670913i	$0.765902\pi$
$402$		0			0
$403$	24.0000	+	24.0000i	1.19553	+	1.19553i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	39.5980	+	39.5980i	1.96280	+	1.96280i
$408$		0			0
$409$		−	24.0000i		−	1.18672i	−0.804936	−	0.593362i	$-0.797800\pi$
$409$		−	24.0000i		−	1.18672i	0.804936	−	0.593362i	$-0.202200\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11.3137			0.552711			0.276355	−	0.961056i	$-0.410873\pi$
$419$	11.3137			0.552711			0.276355	+	0.961056i	$0.410873\pi$
$420$		0			0
$421$	38.0000			1.85201			0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000			1.85201			0.926003	+	0.377515i	$0.123221\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			5.65685i			0.272481i	0.990676	+	0.136241i	$0.0435020\pi$
$431$			5.65685i			0.272481i	−0.990676	+	0.136241i	$0.956498\pi$
$432$		0			0
$433$	−11.0000	−	11.0000i	−0.528626	−	0.528626i	0.391536	−	0.920163i	$-0.371944\pi$
$433$	−11.0000	−	11.0000i	−0.528626	−	0.528626i	−0.920163	+	0.391536i	$0.871944\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.3137	−	11.3137i	−0.541208	−	0.541208i
$438$		0			0
$439$			8.00000i			0.381819i	0.981608	+	0.190910i	$0.0611437\pi$
$439$			8.00000i			0.381819i	−0.981608	+	0.190910i	$0.938856\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.3137	−	11.3137i	0.537531	−	0.537531i	−0.385272	−	0.922803i	$-0.625893\pi$
$443$	11.3137	−	11.3137i	0.537531	−	0.537531i	0.922803	+	0.385272i	$0.125893\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.3848			−0.867631			−0.433816	−	0.901002i	$-0.642833\pi$
$449$	−18.3848			−0.867631			−0.433816	+	0.901002i	$0.642833\pi$
$450$		0			0
$451$	8.00000			0.376705
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	23.0000	−	23.0000i	1.07589	−	1.07589i	0.0790217	−	0.996873i	$-0.474820\pi$
$457$	23.0000	−	23.0000i	1.07589	−	1.07589i	0.996873	−	0.0790217i	$-0.0251796\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	32.5269i		−	1.51493i	−0.652876	−	0.757465i	$-0.726438\pi$
$461$		−	32.5269i		−	1.51493i	0.652876	−	0.757465i	$-0.273562\pi$
$462$		0			0
$463$	−4.00000	−	4.00000i	−0.185896	−	0.185896i	0.608023	−	0.793919i	$-0.291963\pi$
$463$	−4.00000	−	4.00000i	−0.185896	−	0.185896i	−0.793919	+	0.608023i	$0.791963\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.9706	−	16.9706i	−0.785304	−	0.785304i	0.195416	−	0.980720i	$-0.437394\pi$
$467$	−16.9706	−	16.9706i	−0.785304	−	0.785304i	−0.980720	+	0.195416i	$0.937394\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−22.6274	+	22.6274i	−1.04041	+	1.04041i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.65685			−0.258468			−0.129234	−	0.991614i	$-0.541252\pi$
$479$	−5.65685			−0.258468			−0.129234	+	0.991614i	$0.541252\pi$
$480$		0			0
$481$	42.0000			1.91504
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−12.0000	+	12.0000i	−0.543772	+	0.543772i	−0.924632	−	0.380861i	$-0.875628\pi$
$487$	−12.0000	+	12.0000i	−0.543772	+	0.543772i	0.380861	+	0.924632i	$0.375628\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	22.6274i		−	1.02116i	−0.859830	−	0.510581i	$-0.829431\pi$
$491$		−	22.6274i		−	1.02116i	0.859830	−	0.510581i	$-0.170569\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	4.00000i		−	0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$		−	4.00000i		−	0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−25.4558	+	25.4558i	−1.13502	+	1.13502i	−0.145690	+	0.989330i	$0.546540\pi$
$503$	−25.4558	+	25.4558i	−1.13502	+	1.13502i	−0.989330	+	0.145690i	$0.953460\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.7279			−0.564155			−0.282078	−	0.959392i	$-0.591024\pi$
$509$	−12.7279			−0.564155			−0.282078	+	0.959392i	$0.591024\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−16.0000	+	16.0000i	−0.703679	+	0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	21.2132i		−	0.929367i	−0.885477	−	0.464684i	$-0.846168\pi$
$521$		−	21.2132i		−	0.929367i	0.885477	−	0.464684i	$-0.153832\pi$
$522$		0			0
$523$	24.0000	+	24.0000i	1.04945	+	1.04945i	0.998712	+	0.0507346i	$0.0161562\pi$
$523$	24.0000	+	24.0000i	1.04945	+	1.04945i	0.0507346	+	0.998712i	$0.483844\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			7.00000i			0.304348i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.24264	−	4.24264i	0.183769	−	0.183769i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	39.5980			1.70561
$540$		0			0
$541$	−4.00000			−0.171973			−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000			−0.171973			−0.0859867	+	0.996296i	$0.527404\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12.0000	+	12.0000i	−0.513083	+	0.513083i	−0.915470	−	0.402387i	$-0.868181\pi$
$547$	−12.0000	+	12.0000i	−0.513083	+	0.513083i	0.402387	+	0.915470i	$0.368181\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			5.65685i			0.240990i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.48528	+	8.48528i	0.359533	+	0.359533i	0.863641	−	0.504108i	$-0.168179\pi$
$557$	8.48528	+	8.48528i	0.359533	+	0.359533i	−0.504108	+	0.863641i	$0.668179\pi$
$558$		0			0
$559$			24.0000i			1.01509i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.65685	−	5.65685i	0.238408	−	0.238408i	−0.577783	−	0.816191i	$-0.696082\pi$
$563$	5.65685	−	5.65685i	0.238408	−	0.238408i	0.816191	+	0.577783i	$0.196082\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	35.3553			1.48217			0.741086	−	0.671410i	$-0.234311\pi$
$569$	35.3553			1.48217			0.741086	+	0.671410i	$0.234311\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	9.00000	−	9.00000i	0.374675	−	0.374675i	−0.494502	−	0.869177i	$-0.664649\pi$
$577$	9.00000	−	9.00000i	0.374675	−	0.374675i	0.869177	+	0.494502i	$0.164649\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−48.0000	−	48.0000i	−1.98796	−	1.98796i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.65685	−	5.65685i	−0.233483	−	0.233483i	0.580662	−	0.814145i	$-0.302794\pi$
$587$	−5.65685	−	5.65685i	−0.233483	−	0.233483i	−0.814145	+	0.580662i	$0.802794\pi$
$588$		0			0
$589$		−	32.0000i		−	1.31854i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.3848	+	18.3848i	−0.754972	+	0.754972i	−0.975403	−	0.220430i	$-0.929254\pi$
$593$	−18.3848	+	18.3848i	−0.754972	+	0.754972i	0.220430	+	0.975403i	$0.429254\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−28.2843			−1.15566			−0.577832	−	0.816156i	$-0.696101\pi$
$599$	−28.2843			−1.15566			−0.577832	+	0.816156i	$0.696101\pi$
$600$		0			0
$601$	−32.0000			−1.30531			−0.652654	−	0.757656i	$-0.726344\pi$
$601$	−32.0000			−1.30531			−0.652654	+	0.757656i	$0.726344\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−8.00000	+	8.00000i	−0.324710	+	0.324710i	−0.850571	−	0.525861i	$-0.823743\pi$
$607$	−8.00000	+	8.00000i	−0.324710	+	0.324710i	0.525861	+	0.850571i	$0.323743\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			16.9706i			0.686555i
$612$		0			0
$613$	−15.0000	−	15.0000i	−0.605844	−	0.605844i	0.336013	−	0.941857i	$-0.390921\pi$
$613$	−15.0000	−	15.0000i	−0.605844	−	0.605844i	−0.941857	+	0.336013i	$0.890921\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.9706	+	16.9706i	0.683209	+	0.683209i	0.960722	−	0.277513i	$-0.0895101\pi$
$617$	16.9706	+	16.9706i	0.683209	+	0.683209i	−0.277513	+	0.960722i	$0.589510\pi$
$618$		0			0
$619$		−	12.0000i		−	0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$		−	12.0000i		−	0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	21.0000	−	21.0000i	0.832050	−	0.832050i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	26.8701i		−	1.06130i	−0.847590	−	0.530652i	$-0.821947\pi$
$641$		−	26.8701i		−	1.06130i	0.847590	−	0.530652i	$-0.178053\pi$
$642$		0			0
$643$	28.0000	+	28.0000i	1.10421	+	1.10421i	0.993897	+	0.110316i	$0.0351862\pi$
$643$	28.0000	+	28.0000i	1.10421	+	1.10421i	0.110316	+	0.993897i	$0.464814\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.7990	+	19.7990i	0.778379	+	0.778379i	0.979555	−	0.201176i	$-0.0644765\pi$
$647$	19.7990	+	19.7990i	0.778379	+	0.778379i	−0.201176	+	0.979555i	$0.564476\pi$
$648$		0			0
$649$			64.0000i			2.51222i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−25.4558	+	25.4558i	−0.996164	+	0.996164i	−0.999993	−	0.00382851i	$-0.998781\pi$
$653$	−25.4558	+	25.4558i	−0.996164	+	0.996164i	0.00382851	+	0.999993i	$0.498781\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	33.9411			1.32216			0.661079	−	0.750316i	$-0.270099\pi$
$659$	33.9411			1.32216			0.661079	+	0.750316i	$0.270099\pi$
$660$		0			0
$661$	−20.0000			−0.777910			−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000			−0.777910			−0.388955	+	0.921257i	$0.627164\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4.00000	+	4.00000i	−0.154881	+	0.154881i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			67.8823i			2.62057i
$672$		0			0
$673$	19.0000	+	19.0000i	0.732396	+	0.732396i	0.971094	−	0.238698i	$-0.0767205\pi$
$673$	19.0000	+	19.0000i	0.732396	+	0.732396i	−0.238698	+	0.971094i	$0.576721\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19.7990	−	19.7990i	−0.760937	−	0.760937i	0.215555	−	0.976492i	$-0.430844\pi$
$677$	−19.7990	−	19.7990i	−0.760937	−	0.760937i	−0.976492	+	0.215555i	$0.930844\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	33.9411	−	33.9411i	1.29872	−	1.29872i	0.369484	−	0.929237i	$-0.379534\pi$
$683$	33.9411	−	33.9411i	1.29872	−	1.29872i	0.929237	−	0.369484i	$-0.120466\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−50.9117			−1.93958
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	32.5269i		−	1.22852i	−0.789102	−	0.614262i	$-0.789454\pi$
$701$		−	32.5269i		−	1.22852i	0.789102	−	0.614262i	$-0.210546\pi$
$702$		0			0
$703$	−28.0000	−	28.0000i	−1.05604	−	1.05604i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	22.6274	−	22.6274i	0.847403	−	0.847403i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−22.6274			−0.843860			−0.421930	−	0.906628i	$-0.638647\pi$
$719$	−22.6274			−0.843860			−0.421930	+	0.906628i	$0.638647\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−12.0000	+	12.0000i	−0.445055	+	0.445055i	−0.893707	−	0.448651i	$-0.851904\pi$
$727$	−12.0000	+	12.0000i	−0.445055	+	0.445055i	0.448651	+	0.893707i	$0.351904\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−23.0000	−	23.0000i	−0.849524	−	0.849524i	0.140549	−	0.990074i	$-0.455113\pi$
$733$	−23.0000	−	23.0000i	−0.849524	−	0.849524i	−0.990074	+	0.140549i	$0.955113\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−45.2548	−	45.2548i	−1.66698	−	1.66698i
$738$		0			0
$739$		−	4.00000i		−	0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$		−	4.00000i		−	0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	25.4558	−	25.4558i	0.933884	−	0.933884i	−0.0640616	−	0.997946i	$-0.520405\pi$
$743$	25.4558	−	25.4558i	0.933884	−	0.933884i	0.997946	+	0.0640616i	$0.0204054\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	3.00000	−	3.00000i	0.109037	−	0.109037i	−0.650484	−	0.759520i	$-0.725434\pi$
$757$	3.00000	−	3.00000i	0.109037	−	0.109037i	0.759520	+	0.650484i	$0.225434\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1.41421i			0.0512652i	0.999671	+	0.0256326i	$0.00816000\pi$
$761$			1.41421i			0.0512652i	−0.999671	+	0.0256326i	$0.991840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	33.9411	+	33.9411i	1.22554	+	1.22554i
$768$		0			0
$769$		−	40.0000i		−	1.44244i	−0.692708	−	0.721218i	$-0.743582\pi$
$769$		−	40.0000i		−	1.44244i	0.692708	−	0.721218i	$-0.256418\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	14.1421	−	14.1421i	0.508657	−	0.508657i	−0.405457	−	0.914114i	$-0.632888\pi$
$773$	14.1421	−	14.1421i	0.508657	−	0.508657i	0.914114	+	0.405457i	$0.132888\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.65685			−0.202678
$780$		0			0
$781$	−32.0000			−1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−24.0000	+	24.0000i	−0.855508	+	0.855508i	−0.990805	−	0.135297i	$-0.956801\pi$
$787$	−24.0000	+	24.0000i	−0.855508	+	0.855508i	0.135297	+	0.990805i	$0.456801\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	36.0000	+	36.0000i	1.27840	+	1.27840i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−29.6985	−	29.6985i	−1.05197	−	1.05197i	−0.998573	−	0.0534012i	$-0.982994\pi$
$797$	−29.6985	−	29.6985i	−1.05197	−	1.05197i	−0.0534012	−	0.998573i	$-0.517006\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	16.9706	−	16.9706i	0.598878	−	0.598878i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−32.5269			−1.14359			−0.571793	−	0.820398i	$-0.693752\pi$
$809$	−32.5269			−1.14359			−0.571793	+	0.820398i	$0.693752\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	16.0000	−	16.0000i	0.559769	−	0.559769i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	18.3848i		−	0.641633i	−0.947141	−	0.320817i	$-0.896043\pi$
$821$		−	18.3848i		−	0.641633i	0.947141	−	0.320817i	$-0.103957\pi$
$822$		0			0
$823$	32.0000	+	32.0000i	1.11545	+	1.11545i	0.992401	+	0.123049i	$0.0392673\pi$
$823$	32.0000	+	32.0000i	1.11545	+	1.11545i	0.123049	+	0.992401i	$0.460733\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.9706	−	16.9706i	−0.590124	−	0.590124i	0.347541	−	0.937665i	$-0.387017\pi$
$827$	−16.9706	−	16.9706i	−0.590124	−	0.590124i	−0.937665	+	0.347541i	$0.887017\pi$
$828$		0			0
$829$			28.0000i			0.972480i	0.873825	+	0.486240i	$0.161632\pi$
$829$			28.0000i			0.972480i	−0.873825	+	0.486240i	$0.838368\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.65685			0.195296			0.0976481	−	0.995221i	$-0.468868\pi$
$839$	5.65685			0.195296			0.0976481	+	0.995221i	$0.468868\pi$
$840$		0			0
$841$	−27.0000			−0.931034
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	39.5980i		−	1.35740i
$852$		0			0
$853$	−9.00000	−	9.00000i	−0.308154	−	0.308154i	0.536039	−	0.844193i	$-0.319920\pi$
$853$	−9.00000	−	9.00000i	−0.308154	−	0.308154i	−0.844193	+	0.536039i	$0.819920\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	33.9411	+	33.9411i	1.15941	+	1.15941i	0.984603	+	0.174803i	$0.0559290\pi$
$857$	33.9411	+	33.9411i	1.15941	+	1.15941i	0.174803	+	0.984603i	$0.444071\pi$
$858$		0			0
$859$			28.0000i			0.955348i	0.878537	+	0.477674i	$0.158520\pi$
$859$			28.0000i			0.955348i	−0.878537	+	0.477674i	$0.841480\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	25.4558	−	25.4558i	0.866527	−	0.866527i	−0.125559	−	0.992086i	$-0.540072\pi$
$863$	25.4558	−	25.4558i	0.866527	−	0.866527i	0.992086	+	0.125559i	$0.0400725\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−45.2548			−1.53517
$870$		0			0
$871$	−48.0000			−1.62642
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−15.0000	+	15.0000i	−0.506514	+	0.506514i	−0.913455	−	0.406941i	$-0.866596\pi$
$877$	−15.0000	+	15.0000i	−0.506514	+	0.506514i	0.406941	+	0.913455i	$0.366596\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1.41421i		−	0.0476461i	−0.999716	−	0.0238230i	$-0.992416\pi$
$881$		−	1.41421i		−	0.0476461i	0.999716	−	0.0238230i	$-0.00758382\pi$
$882$		0			0
$883$	−12.0000	−	12.0000i	−0.403832	−	0.403832i	0.475749	−	0.879581i	$-0.342177\pi$
$883$	−12.0000	−	12.0000i	−0.403832	−	0.403832i	−0.879581	+	0.475749i	$0.842177\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−31.1127	−	31.1127i	−1.04466	−	1.04466i	−0.998955	−	0.0457073i	$-0.985446\pi$
$887$	−31.1127	−	31.1127i	−1.04466	−	1.04466i	−0.0457073	−	0.998955i	$-0.514554\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11.3137	−	11.3137i	0.378599	−	0.378599i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.3137			−0.377333
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	8.00000	−	8.00000i	0.265636	−	0.265636i	−0.561703	−	0.827339i	$-0.689854\pi$
$907$	8.00000	−	8.00000i	0.265636	−	0.265636i	0.827339	+	0.561703i	$0.189854\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	28.2843i		−	0.937100i	−0.883437	−	0.468550i	$-0.844777\pi$
$911$		−	28.2843i		−	0.937100i	0.883437	−	0.468550i	$-0.155223\pi$
$912$		0			0
$913$	−64.0000	−	64.0000i	−2.11809	−	2.11809i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			32.0000i			1.05558i	0.849374	+	0.527791i	$0.176980\pi$
$919$			32.0000i			1.05558i	−0.849374	+	0.527791i	$0.823020\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16.9706	+	16.9706i	−0.558593	+	0.558593i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.5563			0.510387			0.255194	−	0.966890i	$-0.417861\pi$
$929$	15.5563			0.510387			0.255194	+	0.966890i	$0.417861\pi$
$930$		0			0
$931$	−28.0000			−0.917663
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−5.00000	+	5.00000i	−0.163343	+	0.163343i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−5.00000	+	5.00000i	−0.163343	+	0.163343i	0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	12.7279i		−	0.414918i	−0.978244	−	0.207459i	$-0.933481\pi$
$941$		−	12.7279i		−	0.414918i	0.978244	−	0.207459i	$-0.0665194\pi$
$942$		0			0
$943$	−4.00000	−	4.00000i	−0.130258	−	0.130258i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.3137	+	11.3137i	0.367646	+	0.367646i	0.866618	−	0.498972i	$-0.166289\pi$
$947$	11.3137	+	11.3137i	0.367646	+	0.367646i	−0.498972	+	0.866618i	$0.666289\pi$
$948$		0			0
$949$		−	18.0000i		−	0.584305i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21.2132	+	21.2132i	−0.687163	+	0.687163i	−0.961604	−	0.274441i	$-0.911507\pi$
$953$	−21.2132	+	21.2132i	−0.687163	+	0.687163i	0.274441	+	0.961604i	$0.411507\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−24.0000	+	24.0000i	−0.771788	+	0.771788i	−0.978419	−	0.206631i	$-0.933750\pi$
$967$	−24.0000	+	24.0000i	−0.771788	+	0.771788i	0.206631	+	0.978419i	$0.433750\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			22.6274i			0.726148i	0.931760	+	0.363074i	$0.118273\pi$
$971$			22.6274i			0.726148i	−0.931760	+	0.363074i	$0.881727\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−33.9411	−	33.9411i	−1.08587	−	1.08587i	−0.995949	−	0.0899242i	$-0.971338\pi$
$977$	−33.9411	−	33.9411i	−1.08587	−	1.08587i	−0.0899242	−	0.995949i	$-0.528662\pi$
$978$		0			0
$979$			40.0000i			1.27841i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−25.4558	+	25.4558i	−0.811915	+	0.811915i	−0.984921	−	0.173006i	$-0.944652\pi$
$983$	−25.4558	+	25.4558i	−0.811915	+	0.811915i	0.173006	+	0.984921i	$0.444652\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	22.6274			0.719510
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−3.00000	+	3.00000i	−0.0950110	+	0.0950110i	−0.753015	−	0.658004i	$-0.771401\pi$
$997$	−3.00000	+	3.00000i	−0.0950110	+	0.0950110i	0.658004	+	0.753015i	$0.271401\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.2.s.b.593.1		4
3.2	odd	2	inner	1800.2.s.b.593.2		4
4.3	odd	2		3600.2.w.d.593.2		4
5.2	odd	4	inner	1800.2.s.b.1457.1		4
5.3	odd	4		360.2.s.a.17.1	✓	4
5.4	even	2		360.2.s.a.233.2	yes	4
12.11	even	2		3600.2.w.d.593.1		4
15.2	even	4	inner	1800.2.s.b.1457.2		4
15.8	even	4		360.2.s.a.17.2	yes	4
15.14	odd	2		360.2.s.a.233.1	yes	4
20.3	even	4		720.2.w.c.17.1		4
20.7	even	4		3600.2.w.d.1457.2		4
20.19	odd	2		720.2.w.c.593.2		4
40.3	even	4		2880.2.w.f.2177.2		4
40.13	odd	4		2880.2.w.e.2177.2		4
40.19	odd	2		2880.2.w.f.2753.1		4
40.29	even	2		2880.2.w.e.2753.1		4
60.23	odd	4		720.2.w.c.17.2		4
60.47	odd	4		3600.2.w.d.1457.1		4
60.59	even	2		720.2.w.c.593.1		4
120.29	odd	2		2880.2.w.e.2753.2		4
120.53	even	4		2880.2.w.e.2177.1		4
120.59	even	2		2880.2.w.f.2753.2		4
120.83	odd	4		2880.2.w.f.2177.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.s.a.17.1	✓	4	5.3	odd	4
360.2.s.a.17.2	yes	4	15.8	even	4
360.2.s.a.233.1	yes	4	15.14	odd	2
360.2.s.a.233.2	yes	4	5.4	even	2
720.2.w.c.17.1		4	20.3	even	4
720.2.w.c.17.2		4	60.23	odd	4
720.2.w.c.593.1		4	60.59	even	2
720.2.w.c.593.2		4	20.19	odd	2
1800.2.s.b.593.1		4	1.1	even	1	trivial
1800.2.s.b.593.2		4	3.2	odd	2	inner
1800.2.s.b.1457.1		4	5.2	odd	4	inner
1800.2.s.b.1457.2		4	15.2	even	4	inner
2880.2.w.e.2177.1		4	120.53	even	4
2880.2.w.e.2177.2		4	40.13	odd	4
2880.2.w.e.2753.1		4	40.29	even	2
2880.2.w.e.2753.2		4	120.29	odd	2
2880.2.w.f.2177.1		4	120.83	odd	4
2880.2.w.f.2177.2		4	40.3	even	4
2880.2.w.f.2753.1		4	40.19	odd	2
2880.2.w.f.2753.2		4	120.59	even	2
3600.2.w.d.593.1		4	12.11	even	2
3600.2.w.d.593.2		4	4.3	odd	2
3600.2.w.d.1457.1		4	60.47	odd	4
3600.2.w.d.1457.2		4	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q-5.65685i q^{11} +(-3.00000 - 3.00000i) q^{13} +4.00000i q^{19} +(-2.82843 + 2.82843i) q^{23} +1.41421 q^{29} -8.00000 q^{31} +(-7.00000 + 7.00000i) q^{37} +1.41421i q^{41} +(-4.00000 - 4.00000i) q^{43} +(-2.82843 - 2.82843i) q^{47} +7.00000i q^{49} +(8.48528 - 8.48528i) q^{53} -11.3137 q^{59} -12.0000 q^{61} +(8.00000 - 8.00000i) q^{67} -5.65685i q^{71} +(3.00000 + 3.00000i) q^{73} -8.00000i q^{79} +(11.3137 - 11.3137i) q^{83} -7.07107 q^{89} +(-5.00000 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{13} - 32 q^{31} - 28 q^{37} - 16 q^{43} - 48 q^{61} + 32 q^{67} + 12 q^{73} - 20 q^{97}+O(q^{100}) \) 4 * q - 12 * q^13 - 32 * q^31 - 28 * q^37 - 16 * q^43 - 48 * q^61 + 32 * q^67 + 12 * q^73 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more