LMFDB - Embedded newform 1440.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(1,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.a (trivial)

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:	yes
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +4.47214 q^{7} +4.47214 q^{11} +4.00000 q^{13} -2.00000 q^{17} -8.94427 q^{23} +1.00000 q^{25} +6.00000 q^{29} -8.94427 q^{31} -4.47214 q^{35} +8.00000 q^{37} -8.00000 q^{41} +8.94427 q^{47} +13.0000 q^{49} +6.00000 q^{53} -4.47214 q^{55} +4.47214 q^{59} +10.0000 q^{61} -4.00000 q^{65} -8.94427 q^{67} +8.94427 q^{71} +6.00000 q^{73} +20.0000 q^{77} +8.94427 q^{79} -8.94427 q^{83} +2.00000 q^{85} -4.00000 q^{89} +17.8885 q^{91} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 8 q^{13} - 4 q^{17} + 2 q^{25} + 12 q^{29} + 16 q^{37} - 16 q^{41} + 26 q^{49} + 12 q^{53} + 20 q^{61} - 8 q^{65} + 12 q^{73} + 40 q^{77} + 4 q^{85} - 8 q^{89} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 8 * q^13 - 4 * q^17 + 2 * q^25 + 12 * q^29 + 16 * q^37 - 16 * q^41 + 26 * q^49 + 12 * q^53 + 20 * q^61 - 8 * q^65 + 12 * q^73 + 40 * q^77 + 4 * q^85 - 8 * q^89 + 4 * q^97

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.47214	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.94427	−1.86501	−0.932505	−	0.361158i	$-0.882382\pi$
$23$	−8.94427	−1.86501	−0.932505	+	0.361158i	$0.882382\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.94427	−1.60644	−0.803219	−	0.595683i	$-0.796881\pi$
$31$	−8.94427	−1.60644	−0.803219	+	0.595683i	$0.796881\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.47214	−0.755929
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.94427	1.30466	0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427	1.30466	0.652328	+	0.757937i	$0.273792\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−4.47214	−0.603023
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−8.94427	−1.09272	−0.546358	−	0.837552i	$-0.683986\pi$
$67$	−8.94427	−1.09272	−0.546358	+	0.837552i	$0.683986\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	20.0000	2.27921
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.94427	−0.981761	−0.490881	−	0.871227i	$-0.663325\pi$
$83$	−8.94427	−0.981761	−0.490881	+	0.871227i	$0.663325\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	17.8885	1.87523
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	4.47214	0.440653	0.220326	−	0.975426i	$-0.429288\pi$
$103$	4.47214	0.440653	0.220326	+	0.975426i	$0.429288\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	8.94427	0.834058
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.94427	−0.819920
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.47214	0.396838	0.198419	−	0.980117i	$-0.436419\pi$
$127$	4.47214	0.396838	0.198419	+	0.980117i	$0.436419\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−17.8885	−1.51729	−0.758643	−	0.651506i	$-0.774137\pi$
$139$	−17.8885	−1.51729	−0.758643	+	0.651506i	$0.774137\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	17.8885	1.49592
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.94427	0.718421
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−40.0000	−3.15244
$162$	0	0
$163$	−8.94427	−0.700569	−0.350285	−	0.936643i	$-0.613915\pi$
$163$	−8.94427	−0.700569	−0.350285	+	0.936643i	$0.613915\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.94427	−0.692129	−0.346064	−	0.938211i	$-0.612482\pi$
$167$	−8.94427	−0.692129	−0.346064	+	0.938211i	$0.612482\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	4.47214	0.338062
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.47214	0.334263	0.167132	−	0.985935i	$-0.446550\pi$
$179$	4.47214	0.334263	0.167132	+	0.985935i	$0.446550\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	−8.94427	−0.654070
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.8328	−1.94155	−0.970777	−	0.239983i	$-0.922858\pi$
$191$	−26.8328	−1.94155	−0.970777	+	0.239983i	$0.922858\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−17.8885	−1.26809	−0.634043	−	0.773298i	$-0.718606\pi$
$199$	−17.8885	−1.26809	−0.634043	+	0.773298i	$0.718606\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	26.8328	1.88329
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−40.0000	−2.71538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−13.4164	−0.898429	−0.449215	−	0.893424i	$-0.648296\pi$
$223$	−13.4164	−0.898429	−0.449215	+	0.893424i	$0.648296\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.8885	1.18730	0.593652	−	0.804722i	$-0.297686\pi$
$227$	17.8885	1.18730	0.593652	+	0.804722i	$0.297686\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	−8.94427	−0.583460
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.94427	0.578557	0.289278	−	0.957245i	$-0.406585\pi$
$239$	8.94427	0.578557	0.289278	+	0.957245i	$0.406585\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−13.0000	−0.830540
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−22.3607	−1.41139	−0.705697	−	0.708514i	$-0.749366\pi$
$251$	−22.3607	−1.41139	−0.705697	+	0.708514i	$0.749366\pi$
$252$	0	0
$253$	−40.0000	−2.51478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	35.7771	2.22308
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.94427	0.551527	0.275764	−	0.961225i	$-0.411069\pi$
$263$	8.94427	0.551527	0.275764	+	0.961225i	$0.411069\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	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
$275$	4.47214	0.269680
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	26.8328	1.59505	0.797523	−	0.603289i	$-0.206143\pi$
$283$	26.8328	1.59505	0.797523	+	0.603289i	$0.206143\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−35.7771	−2.11185
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−4.47214	−0.260378
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−35.7771	−2.06904
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10.0000	−0.572598
$306$	0	0
$307$	17.8885	1.02095	0.510477	−	0.859892i	$-0.329469\pi$
$307$	17.8885	1.02095	0.510477	+	0.859892i	$0.329469\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.94427	0.507183	0.253592	−	0.967311i	$-0.418388\pi$
$311$	8.94427	0.507183	0.253592	+	0.967311i	$0.418388\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	26.8328	1.50235
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	0	0
$328$	0	0
$329$	40.0000	2.20527
$330$	0	0
$331$	−35.7771	−1.96649	−0.983243	−	0.182298i	$-0.941646\pi$
$331$	−35.7771	−1.96649	−0.983243	+	0.182298i	$0.941646\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.94427	0.488678
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−40.0000	−2.16612
$342$	0	0
$343$	26.8328	1.44884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−8.94427	−0.474713
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.7771	1.88824	0.944121	−	0.329598i	$-0.106913\pi$
$359$	35.7771	1.88824	0.944121	+	0.329598i	$0.106913\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−13.4164	−0.700331	−0.350165	−	0.936688i	$-0.613875\pi$
$367$	−13.4164	−0.700331	−0.350165	+	0.936688i	$0.613875\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	26.8328	1.39309
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	−17.8885	−0.918873	−0.459436	−	0.888211i	$-0.651949\pi$
$379$	−17.8885	−0.918873	−0.459436	+	0.888211i	$0.651949\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.94427	−0.457031	−0.228515	−	0.973540i	$-0.573387\pi$
$383$	−8.94427	−0.457031	−0.228515	+	0.973540i	$0.573387\pi$
$384$	0	0
$385$	−20.0000	−1.01929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	17.8885	0.904663
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.94427	−0.450035
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	0	0
$403$	−35.7771	−1.78218
$404$	0	0
$405$	0	0
$406$	0	0
$407$	35.7771	1.77340
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	20.0000	0.984136
$414$	0	0
$415$	8.94427	0.439057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.47214	−0.218478	−0.109239	−	0.994016i	$-0.534841\pi$
$419$	−4.47214	−0.218478	−0.109239	+	0.994016i	$0.534841\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	44.7214	2.16422
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.8885	0.861661	0.430830	−	0.902433i	$-0.358221\pi$
$431$	17.8885	0.861661	0.430830	+	0.902433i	$0.358221\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−35.7771	−1.70755	−0.853774	−	0.520644i	$-0.825692\pi$
$439$	−35.7771	−1.70755	−0.853774	+	0.520644i	$0.825692\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−35.7771	−1.69982	−0.849910	−	0.526927i	$-0.823344\pi$
$443$	−35.7771	−1.69982	−0.849910	+	0.526927i	$0.823344\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	−35.7771	−1.68468
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.8885	−0.838628
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−40.2492	−1.87054	−0.935270	−	0.353935i	$-0.884843\pi$
$463$	−40.2492	−1.87054	−0.935270	+	0.353935i	$0.884843\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.94427	−0.413892	−0.206946	−	0.978352i	$-0.566352\pi$
$467$	−8.94427	−0.413892	−0.206946	+	0.978352i	$0.566352\pi$
$468$	0	0
$469$	−40.0000	−1.84703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.8328	−1.22602	−0.613011	−	0.790074i	$-0.710042\pi$
$479$	−26.8328	−1.22602	−0.613011	+	0.790074i	$0.710042\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−22.3607	−1.01326	−0.506630	−	0.862164i	$-0.669109\pi$
$487$	−22.3607	−1.01326	−0.506630	+	0.862164i	$0.669109\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.47214	−0.201825	−0.100912	−	0.994895i	$-0.532176\pi$
$491$	−4.47214	−0.201825	−0.100912	+	0.994895i	$0.532176\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.0000	1.79425
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−26.8328	−1.19642	−0.598208	−	0.801341i	$-0.704120\pi$
$503$	−26.8328	−1.19642	−0.598208	+	0.801341i	$0.704120\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	26.8328	1.18701
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.47214	−0.197066
$516$	0	0
$517$	40.0000	1.75920
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	0	0
$523$	26.8328	1.17332	0.586659	−	0.809834i	$-0.300443\pi$
$523$	26.8328	1.17332	0.586659	+	0.809834i	$0.300443\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17.8885	0.779237
$528$	0	0
$529$	57.0000	2.47826
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−32.0000	−1.38607
$534$	0	0
$535$	8.94427	0.386695
$536$	0	0
$537$	0	0
$538$	0	0
$539$	58.1378	2.50417
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	40.0000	1.70097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.8885	−0.753912	−0.376956	−	0.926231i	$-0.623029\pi$
$563$	−17.8885	−0.753912	−0.376956	+	0.926231i	$0.623029\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	35.7771	1.49722	0.748612	−	0.663008i	$-0.230720\pi$
$571$	35.7771	1.49722	0.748612	+	0.663008i	$0.230720\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.94427	−0.373002
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−40.0000	−1.65948
$582$	0	0
$583$	26.8328	1.11130
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.8328	1.10751	0.553754	−	0.832680i	$-0.313195\pi$
$587$	26.8328	1.10751	0.553754	+	0.832680i	$0.313195\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	8.94427	0.366679
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−35.7771	−1.46181	−0.730906	−	0.682478i	$-0.760902\pi$
$599$	−35.7771	−1.46181	−0.730906	+	0.682478i	$0.760902\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	−22.3607	−0.907592	−0.453796	−	0.891106i	$-0.649931\pi$
$607$	−22.3607	−0.907592	−0.453796	+	0.891106i	$0.649931\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.7771	1.44739
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	0	0
$619$	17.8885	0.719001	0.359501	−	0.933145i	$-0.382947\pi$
$619$	17.8885	0.719001	0.359501	+	0.933145i	$0.382947\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17.8885	−0.716689
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	26.8328	1.06820	0.534099	−	0.845422i	$-0.320651\pi$
$631$	26.8328	1.06820	0.534099	+	0.845422i	$0.320651\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.47214	−0.177471
$636$	0	0
$637$	52.0000	2.06032
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	0	0
$643$	8.94427	0.352728	0.176364	−	0.984325i	$-0.443566\pi$
$643$	8.94427	0.352728	0.176364	+	0.984325i	$0.443566\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.94427	−0.351636	−0.175818	−	0.984423i	$-0.556257\pi$
$647$	−8.94427	−0.351636	−0.175818	+	0.984423i	$0.556257\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	4.47214	0.174741
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13.4164	0.522629	0.261315	−	0.965254i	$-0.415844\pi$
$659$	13.4164	0.522629	0.261315	+	0.965254i	$0.415844\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−53.6656	−2.07794
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.7214	1.72645
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	8.94427	0.343250
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.94427	0.342243	0.171122	−	0.985250i	$-0.445261\pi$
$683$	8.94427	0.342243	0.171122	+	0.985250i	$0.445261\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	0	0
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	17.8885	0.680512	0.340256	−	0.940333i	$-0.389486\pi$
$691$	17.8885	0.680512	0.340256	+	0.940333i	$0.389486\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.8885	0.678551
$696$	0	0
$697$	16.0000	0.606043
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−44.7214	−1.68192
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	80.0000	2.99602
$714$	0	0
$715$	−17.8885	−0.668994
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17.8885	−0.667130	−0.333565	−	0.942727i	$-0.608252\pi$
$719$	−17.8885	−0.667130	−0.333565	+	0.942727i	$0.608252\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	13.4164	0.497587	0.248794	−	0.968557i	$-0.419966\pi$
$727$	13.4164	0.497587	0.248794	+	0.968557i	$0.419966\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	16.0000	0.590973	0.295487	−	0.955347i	$-0.404518\pi$
$733$	16.0000	0.590973	0.295487	+	0.955347i	$0.404518\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−40.0000	−1.47342
$738$	0	0
$739$	17.8885	0.658041	0.329020	−	0.944323i	$-0.393282\pi$
$739$	17.8885	0.658041	0.329020	+	0.944323i	$0.393282\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.94427	0.328134	0.164067	−	0.986449i	$-0.447539\pi$
$743$	8.94427	0.328134	0.164067	+	0.986449i	$0.447539\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−40.0000	−1.46157
$750$	0	0
$751$	−26.8328	−0.979143	−0.489572	−	0.871963i	$-0.662847\pi$
$751$	−26.8328	−0.979143	−0.489572	+	0.871963i	$0.662847\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	−26.8328	−0.971413
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17.8885	0.645918
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	−8.94427	−0.321288
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.0000	−0.428298
$786$	0	0
$787$	−26.8328	−0.956487	−0.478243	−	0.878227i	$-0.658726\pi$
$787$	−26.8328	−0.956487	−0.478243	+	0.878227i	$0.658726\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−62.6099	−2.22615
$792$	0	0
$793$	40.0000	1.42044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−17.8885	−0.632851
$800$	0	0
$801$	0	0
$802$	0	0
$803$	26.8328	0.946910
$804$	0	0
$805$	40.0000	1.40981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	−17.8885	−0.628152	−0.314076	−	0.949398i	$-0.601695\pi$
$811$	−17.8885	−0.628152	−0.314076	+	0.949398i	$0.601695\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.94427	0.313304
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	−40.2492	−1.40300	−0.701500	−	0.712670i	$-0.747486\pi$
$823$	−40.2492	−1.40300	−0.701500	+	0.712670i	$0.747486\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.8885	−0.622046	−0.311023	−	0.950402i	$-0.600672\pi$
$827$	−17.8885	−0.622046	−0.311023	+	0.950402i	$0.600672\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−26.0000	−0.900847
$834$	0	0
$835$	8.94427	0.309529
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.8885	0.617581	0.308791	−	0.951130i	$-0.400076\pi$
$839$	17.8885	0.617581	0.308791	+	0.951130i	$0.400076\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	40.2492	1.38298
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−71.5542	−2.45285
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−53.6656	−1.83105	−0.915524	−	0.402264i	$-0.868224\pi$
$859$	−53.6656	−1.83105	−0.915524	+	0.402264i	$0.868224\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.8328	0.913400	0.456700	−	0.889621i	$-0.349031\pi$
$863$	26.8328	0.913400	0.456700	+	0.889621i	$0.349031\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	0	0
$868$	0	0
$869$	40.0000	1.35691
$870$	0	0
$871$	−35.7771	−1.21226
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.47214	−0.151186
$876$	0	0
$877$	52.0000	1.75592	0.877958	−	0.478738i	$-0.158906\pi$
$877$	52.0000	1.75592	0.877958	+	0.478738i	$0.158906\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	0	0
$883$	8.94427	0.300999	0.150499	−	0.988610i	$-0.451912\pi$
$883$	8.94427	0.300999	0.150499	+	0.988610i	$0.451912\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.94427	0.300319	0.150160	−	0.988662i	$-0.452021\pi$
$887$	8.94427	0.300319	0.150160	+	0.988662i	$0.452021\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−4.47214	−0.149487
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−53.6656	−1.78985
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	53.6656	1.78194	0.890969	−	0.454064i	$-0.150026\pi$
$907$	53.6656	1.78194	0.890969	+	0.454064i	$0.150026\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.8885	−0.592674	−0.296337	−	0.955083i	$-0.595765\pi$
$911$	−17.8885	−0.592674	−0.296337	+	0.955083i	$0.595765\pi$
$912$	0	0
$913$	−40.0000	−1.32381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.0000	−0.660458
$918$	0	0
$919$	26.8328	0.885133	0.442566	−	0.896736i	$-0.354068\pi$
$919$	26.8328	0.885133	0.442566	+	0.896736i	$0.354068\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	35.7771	1.17762
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.94427	0.292509
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	71.5542	2.33012
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	0	0
$955$	26.8328	0.868290
$956$	0	0
$957$	0	0
$958$	0	0
$959$	80.4984	2.59943
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.00000	−0.193147
$966$	0	0
$967$	40.2492	1.29433	0.647164	−	0.762351i	$-0.275955\pi$
$967$	40.2492	1.29433	0.647164	+	0.762351i	$0.275955\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.3050	1.00462	0.502312	−	0.864687i	$-0.332483\pi$
$971$	31.3050	1.00462	0.502312	+	0.864687i	$0.332483\pi$
$972$	0	0
$973$	−80.0000	−2.56468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−17.8885	−0.571720
$980$	0	0
$981$	0	0
$982$	0	0
$983$	44.7214	1.42639	0.713195	−	0.700966i	$-0.247247\pi$
$983$	44.7214	1.42639	0.713195	+	0.700966i	$0.247247\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−53.6656	−1.70474	−0.852372	−	0.522935i	$-0.824837\pi$
$991$	−53.6656	−1.70474	−0.852372	+	0.522935i	$0.824837\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.8885	0.567105
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.a.p.1.2	yes	2
3.2	odd	2		1440.2.a.q.1.2	yes	2
4.3	odd	2	inner	1440.2.a.p.1.1	✓	2
5.2	odd	4		7200.2.f.be.6049.4		4
5.3	odd	4		7200.2.f.be.6049.2		4
5.4	even	2		7200.2.a.ch.1.1		2
8.3	odd	2		2880.2.a.bj.1.1		2
8.5	even	2		2880.2.a.bj.1.2		2
12.11	even	2		1440.2.a.q.1.1	yes	2
15.2	even	4		7200.2.f.bj.6049.3		4
15.8	even	4		7200.2.f.bj.6049.1		4
15.14	odd	2		7200.2.a.cg.1.1		2
20.3	even	4		7200.2.f.be.6049.3		4
20.7	even	4		7200.2.f.be.6049.1		4
20.19	odd	2		7200.2.a.ch.1.2		2
24.5	odd	2		2880.2.a.bi.1.2		2
24.11	even	2		2880.2.a.bi.1.1		2
60.23	odd	4		7200.2.f.bj.6049.4		4
60.47	odd	4		7200.2.f.bj.6049.2		4
60.59	even	2		7200.2.a.cg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.a.p.1.1	✓	2	4.3	odd	2	inner
1440.2.a.p.1.2	yes	2	1.1	even	1	trivial
1440.2.a.q.1.1	yes	2	12.11	even	2
1440.2.a.q.1.2	yes	2	3.2	odd	2
2880.2.a.bi.1.1		2	24.11	even	2
2880.2.a.bi.1.2		2	24.5	odd	2
2880.2.a.bj.1.1		2	8.3	odd	2
2880.2.a.bj.1.2		2	8.5	even	2
7200.2.a.cg.1.1		2	15.14	odd	2
7200.2.a.cg.1.2		2	60.59	even	2
7200.2.a.ch.1.1		2	5.4	even	2
7200.2.a.ch.1.2		2	20.19	odd	2
7200.2.f.be.6049.1		4	20.7	even	4
7200.2.f.be.6049.2		4	5.3	odd	4
7200.2.f.be.6049.3		4	20.3	even	4
7200.2.f.be.6049.4		4	5.2	odd	4
7200.2.f.bj.6049.1		4	15.8	even	4
7200.2.f.bj.6049.2		4	60.47	odd	4
7200.2.f.bj.6049.3		4	15.2	even	4
7200.2.f.bj.6049.4		4	60.23	odd	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +4.47214 q^{7} +4.47214 q^{11} +4.00000 q^{13} -2.00000 q^{17} -8.94427 q^{23} +1.00000 q^{25} +6.00000 q^{29} -8.94427 q^{31} -4.47214 q^{35} +8.00000 q^{37} -8.00000 q^{41} +8.94427 q^{47} +13.0000 q^{49} +6.00000 q^{53} -4.47214 q^{55} +4.47214 q^{59} +10.0000 q^{61} -4.00000 q^{65} -8.94427 q^{67} +8.94427 q^{71} +6.00000 q^{73} +20.0000 q^{77} +8.94427 q^{79} -8.94427 q^{83} +2.00000 q^{85} -4.00000 q^{89} +17.8885 q^{91} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 8 q^{13} - 4 q^{17} + 2 q^{25} + 12 q^{29} + 16 q^{37} - 16 q^{41} + 26 q^{49} + 12 q^{53} + 20 q^{61} - 8 q^{65} + 12 q^{73} + 40 q^{77} + 4 q^{85} - 8 q^{89} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 8 * q^13 - 4 * q^17 + 2 * q^25 + 12 * q^29 + 16 * q^37 - 16 * q^41 + 26 * q^49 + 12 * q^53 + 20 * q^61 - 8 * q^65 + 12 * q^73 + 40 * q^77 + 4 * q^85 - 8 * q^89 + 4 * q^97

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