LMFDB - Embedded newform 120.3.l.a.41.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,3,Mod(41,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("120.41");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 120 = 2^{3} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	120.l (of order $2$, degree $1$, 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:	$3.26976317232$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.681615360000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 $ x^8 - 4x^7 - 2x^6 + 20x^5 + 49x^4 - 136x^3 + 168x^2 - 96*x + 864
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.4
Root			$-0.542939 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	120.41
Dual form			120.3.l.a.41.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.40140 + 1.79813i) q^{3} -2.23607i q^{5} -10.2132 q^{7} +(2.53346 - 8.63606i) q^{9} +O(q^{10})$	\(q+(-2.40140 + 1.79813i) q^{3} -2.23607i q^{5} -10.2132 q^{7} +(2.53346 - 8.63606i) q^{9} -8.19300i q^{11} -13.5822 q^{13} +(4.02074 + 5.36970i) q^{15} -15.4710i q^{17} -25.4934 q^{19} +(24.5261 - 18.3647i) q^{21} +17.9156i q^{23} -5.00000 q^{25} +(9.44491 + 25.2941i) q^{27} +42.0022i q^{29} +38.4878 q^{31} +(14.7321 + 19.6747i) q^{33} +22.8375i q^{35} +11.8387 q^{37} +(32.6163 - 24.4225i) q^{39} -46.3781i q^{41} -54.0181 q^{43} +(-19.3108 - 5.66499i) q^{45} -43.0955i q^{47} +55.3102 q^{49} +(27.8188 + 37.1521i) q^{51} +82.7421i q^{53} -18.3201 q^{55} +(61.2199 - 45.8404i) q^{57} -45.8928i q^{59} -93.6873 q^{61} +(-25.8748 + 88.2022i) q^{63} +30.3707i q^{65} +34.4995 q^{67} +(-32.2147 - 43.0227i) q^{69} -68.0061i q^{71} -44.7191 q^{73} +(12.0070 - 8.99065i) q^{75} +83.6770i q^{77} -11.7499 q^{79} +(-68.1632 - 43.7582i) q^{81} -144.436i q^{83} -34.5942 q^{85} +(-75.5255 - 100.864i) q^{87} +63.7094i q^{89} +138.718 q^{91} +(-92.4247 + 69.2061i) q^{93} +57.0050i q^{95} +63.9013 q^{97} +(-70.7552 - 20.7566i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 + 16 * q^7 + 20 * q^9	$ 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} - 8 q^{13} - 8 q^{19} + 28 q^{21} - 40 q^{25} + 20 q^{27} + 120 q^{31} - 112 q^{33} + 8 q^{37} - 72 q^{39} - 328 q^{43} - 60 q^{45} + 64 q^{49} + 64 q^{51} - 40 q^{55} + 72 q^{57} + 8 q^{61} + 88 q^{63} + 152 q^{67} + 100 q^{69} + 32 q^{73} + 20 q^{75} + 88 q^{79} + 224 q^{81} - 152 q^{87} + 560 q^{91} - 368 q^{93} + 144 q^{97} + 32 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 + 16 * q^7 + 20 * q^9 - 8 * q^13 - 8 * q^19 + 28 * q^21 - 40 * q^25 + 20 * q^27 + 120 * q^31 - 112 * q^33 + 8 * q^37 - 72 * q^39 - 328 * q^43 - 60 * q^45 + 64 * q^49 + 64 * q^51 - 40 * q^55 + 72 * q^57 + 8 * q^61 + 88 * q^63 + 152 * q^67 + 100 * q^69 + 32 * q^73 + 20 * q^75 + 88 * q^79 + 224 * q^81 - 152 * q^87 + 560 * q^91 - 368 * q^93 + 144 * q^97 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$.

$n$	$31$	$41$	$61$	$97$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	−2.40140	+	1.79813i	−0.800467	+	0.599377i
$3$	−2.40140	+	1.79813i	−0.800467	+	0.599377i
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	−10.2132			−1.45903			−0.729517	−	0.683963i	$-0.760255\pi$
$7$	−10.2132			−1.45903			−0.729517	+	0.683963i	$0.760255\pi$
$8$		0			0
$9$	2.53346	−	8.63606i	0.281496	−	0.959563i
$10$		0			0
$11$		−	8.19300i		−	0.744818i	−0.928069	−	0.372409i	$-0.878532\pi$
$11$		−	8.19300i		−	0.744818i	0.928069	−	0.372409i	$-0.121468\pi$
$12$		0			0
$13$	−13.5822			−1.04478			−0.522392	−	0.852705i	$-0.674960\pi$
$13$	−13.5822			−1.04478			−0.522392	+	0.852705i	$0.674960\pi$
$14$		0			0
$15$	4.02074	+	5.36970i	0.268049	+	0.357980i
$16$		0			0
$17$		−	15.4710i		−	0.910058i	−0.890477	−	0.455029i	$-0.849629\pi$
$17$		−	15.4710i		−	0.910058i	0.890477	−	0.455029i	$-0.150371\pi$
$18$		0			0
$19$	−25.4934			−1.34176			−0.670879	−	0.741567i	$-0.734083\pi$
$19$	−25.4934			−1.34176			−0.670879	+	0.741567i	$0.734083\pi$
$20$		0			0
$21$	24.5261	−	18.3647i	1.16791	−	0.874511i
$22$		0			0
$23$			17.9156i			0.778941i	0.921039	+	0.389471i	$0.127342\pi$
$23$			17.9156i			0.778941i	−0.921039	+	0.389471i	$0.872658\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	9.44491	+	25.2941i	0.349811	+	0.936820i
$28$		0			0
$29$			42.0022i			1.44835i	0.689615	+	0.724177i	$0.257780\pi$
$29$			42.0022i			1.44835i	−0.689615	+	0.724177i	$0.742220\pi$
$30$		0			0
$31$	38.4878			1.24154			0.620771	−	0.783992i	$-0.286820\pi$
$31$	38.4878			1.24154			0.620771	+	0.783992i	$0.286820\pi$
$32$		0			0
$33$	14.7321	+	19.6747i	0.446426	+	0.596202i
$34$		0			0
$35$			22.8375i			0.652500i
$36$		0			0
$37$	11.8387			0.319965			0.159982	−	0.987120i	$-0.448856\pi$
$37$	11.8387			0.319965			0.159982	+	0.987120i	$0.448856\pi$
$38$		0			0
$39$	32.6163	−	24.4225i	0.836315	−	0.626219i
$40$		0			0
$41$		−	46.3781i		−	1.13117i	−0.824689	−	0.565587i	$-0.808650\pi$
$41$		−	46.3781i		−	1.13117i	0.824689	−	0.565587i	$-0.191350\pi$
$42$		0			0
$43$	−54.0181			−1.25623			−0.628117	−	0.778119i	$-0.716174\pi$
$43$	−54.0181			−1.25623			−0.628117	+	0.778119i	$0.716174\pi$
$44$		0			0
$45$	−19.3108	−	5.66499i	−0.429129	−	0.125889i
$46$		0			0
$47$		−	43.0955i		−	0.916925i	−0.888714	−	0.458462i	$-0.848400\pi$
$47$		−	43.0955i		−	0.916925i	0.888714	−	0.458462i	$-0.151600\pi$
$48$		0			0
$49$	55.3102			1.12878
$50$		0			0
$51$	27.8188	+	37.1521i	0.545468	+	0.728472i
$52$		0			0
$53$			82.7421i			1.56117i	0.625049	+	0.780586i	$0.285079\pi$
$53$			82.7421i			1.56117i	−0.625049	+	0.780586i	$0.714921\pi$
$54$		0			0
$55$	−18.3201			−0.333093
$56$		0			0
$57$	61.2199	−	45.8404i	1.07403	−	0.804218i
$58$		0			0
$59$		−	45.8928i		−	0.777844i	−0.921271	−	0.388922i	$-0.872848\pi$
$59$		−	45.8928i		−	0.777844i	0.921271	−	0.388922i	$-0.127152\pi$
$60$		0			0
$61$	−93.6873			−1.53586			−0.767929	−	0.640535i	$-0.778713\pi$
$61$	−93.6873			−1.53586			−0.767929	+	0.640535i	$0.778713\pi$
$62$		0			0
$63$	−25.8748	+	88.2022i	−0.410712	+	1.40003i
$64$		0			0
$65$			30.3707i			0.467242i
$66$		0			0
$67$	34.4995			0.514917			0.257459	−	0.966289i	$-0.417115\pi$
$67$	34.4995			0.514917			0.257459	+	0.966289i	$0.417115\pi$
$68$		0			0
$69$	−32.2147	−	43.0227i	−0.466879	−	0.623517i
$70$		0			0
$71$		−	68.0061i		−	0.957832i	−0.877861	−	0.478916i	$-0.841030\pi$
$71$		−	68.0061i		−	0.957832i	0.877861	−	0.478916i	$-0.158970\pi$
$72$		0			0
$73$	−44.7191			−0.612591			−0.306295	−	0.951937i	$-0.599089\pi$
$73$	−44.7191			−0.612591			−0.306295	+	0.951937i	$0.599089\pi$
$74$		0			0
$75$	12.0070	−	8.99065i	0.160093	−	0.119875i
$76$		0			0
$77$			83.6770i			1.08671i
$78$		0			0
$79$	−11.7499			−0.148733			−0.0743665	−	0.997231i	$-0.523693\pi$
$79$	−11.7499			−0.148733			−0.0743665	+	0.997231i	$0.523693\pi$
$80$		0			0
$81$	−68.1632	−	43.7582i	−0.841521	−	0.540225i
$82$		0			0
$83$		−	144.436i		−	1.74020i	−0.492877	−	0.870099i	$-0.664055\pi$
$83$		−	144.436i		−	1.74020i	0.492877	−	0.870099i	$-0.335945\pi$
$84$		0			0
$85$	−34.5942			−0.406990
$86$		0			0
$87$	−75.5255	−	100.864i	−0.868109	−	1.15936i
$88$		0			0
$89$			63.7094i			0.715836i	0.933753	+	0.357918i	$0.116513\pi$
$89$			63.7094i			0.715836i	−0.933753	+	0.357918i	$0.883487\pi$
$90$		0			0
$91$	138.718			1.52438
$92$		0			0
$93$	−92.4247	+	69.2061i	−0.993814	+	0.744151i
$94$		0			0
$95$			57.0050i			0.600052i
$96$		0			0
$97$	63.9013			0.658776			0.329388	−	0.944195i	$-0.393158\pi$
$97$	63.9013			0.658776			0.329388	+	0.944195i	$0.393158\pi$
$98$		0			0
$99$	−70.7552	−	20.7566i	−0.714699	−	0.209663i
$100$		0			0
$101$		−	50.8769i		−	0.503731i	−0.967762	−	0.251866i	$-0.918956\pi$
$101$		−	50.8769i		−	0.503731i	0.967762	−	0.251866i	$-0.0810441\pi$
$102$		0			0
$103$	45.9491			0.446108			0.223054	−	0.974806i	$-0.428397\pi$
$103$	45.9491			0.446108			0.223054	+	0.974806i	$0.428397\pi$
$104$		0			0
$105$	−41.0648	−	54.8420i	−0.391093	−	0.522305i
$106$		0			0
$107$		−	110.249i		−	1.03036i	−0.857082	−	0.515181i	$-0.827725\pi$
$107$		−	110.249i		−	1.03036i	0.857082	−	0.515181i	$-0.172275\pi$
$108$		0			0
$109$	−49.2797			−0.452107			−0.226054	−	0.974115i	$-0.572582\pi$
$109$	−49.2797			−0.452107			−0.226054	+	0.974115i	$0.572582\pi$
$110$		0			0
$111$	−28.4295	+	21.2875i	−0.256121	+	0.191779i
$112$		0			0
$113$		−	157.450i		−	1.39337i	−0.717379	−	0.696683i	$-0.754658\pi$
$113$		−	157.450i		−	1.39337i	0.717379	−	0.696683i	$-0.245342\pi$
$114$		0			0
$115$	40.0606			0.348353
$116$		0			0
$117$	−34.4099	+	117.297i	−0.294102	+	1.00254i
$118$		0			0
$119$			158.009i			1.32781i
$120$		0			0
$121$	53.8748			0.445246
$122$		0			0
$123$	83.3939	+	111.373i	0.677999	+	0.905468i
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−135.065			−1.06351			−0.531753	−	0.846900i	$-0.678466\pi$
$127$	−135.065			−1.06351			−0.531753	+	0.846900i	$0.678466\pi$
$128$		0			0
$129$	129.719	−	97.1315i	1.00557	−	0.752957i
$130$		0			0
$131$			150.653i			1.15002i	0.818146	+	0.575010i	$0.195002\pi$
$131$			150.653i			1.15002i	−0.818146	+	0.575010i	$0.804998\pi$
$132$		0			0
$133$	260.370			1.95767
$134$		0			0
$135$	56.5594	−	21.1195i	0.418959	−	0.156440i
$136$		0			0
$137$			139.086i			1.01523i	0.861585	+	0.507613i	$0.169472\pi$
$137$			139.086i			1.01523i	−0.861585	+	0.507613i	$0.830528\pi$
$138$		0			0
$139$	110.296			0.793496			0.396748	−	0.917928i	$-0.370139\pi$
$139$	110.296			0.793496			0.396748	+	0.917928i	$0.370139\pi$
$140$		0			0
$141$	77.4912	+	103.490i	0.549583	+	0.733968i
$142$		0			0
$143$			111.279i			0.778174i
$144$		0			0
$145$	93.9199			0.647723
$146$		0			0
$147$	−132.822	+	99.4549i	−0.903551	+	0.676564i
$148$		0			0
$149$			90.8076i			0.609447i	0.952441	+	0.304723i	$0.0985640\pi$
$149$			90.8076i			0.609447i	−0.952441	+	0.304723i	$0.901436\pi$
$150$		0			0
$151$	111.286			0.736992			0.368496	−	0.929629i	$-0.379873\pi$
$151$	111.286			0.736992			0.368496	+	0.929629i	$0.379873\pi$
$152$		0			0
$153$	−133.608	−	39.1951i	−0.873258	−	0.256177i
$154$		0			0
$155$		−	86.0614i		−	0.555235i
$156$		0			0
$157$	142.755			0.909269			0.454635	−	0.890678i	$-0.349770\pi$
$157$	142.755			0.909269			0.454635	+	0.890678i	$0.349770\pi$
$158$		0			0
$159$	−148.781	−	198.697i	−0.935730	−	1.24967i
$160$		0			0
$161$		−	182.977i		−	1.13650i
$162$		0			0
$163$	−170.179			−1.04405			−0.522023	−	0.852932i	$-0.674822\pi$
$163$	−170.179			−1.04405			−0.522023	+	0.852932i	$0.674822\pi$
$164$		0			0
$165$	43.9939	−	32.9419i	0.266630	−	0.199648i
$166$		0			0
$167$		−	6.38780i		−	0.0382503i	−0.999817	−	0.0191251i	$-0.993912\pi$
$167$		−	6.38780i		−	0.0382503i	0.999817	−	0.0191251i	$-0.00608809\pi$
$168$		0			0
$169$	15.4759			0.0915734
$170$		0			0
$171$	−64.5865	+	220.163i	−0.377699	+	1.28750i
$172$		0			0
$173$		−	238.403i		−	1.37805i	−0.724736	−	0.689027i	$-0.758038\pi$
$173$		−	238.403i		−	1.37805i	0.724736	−	0.689027i	$-0.241962\pi$
$174$		0			0
$175$	51.0662			0.291807
$176$		0			0
$177$	82.5212	+	110.207i	0.466221	+	0.622638i
$178$		0			0
$179$			243.541i			1.36057i	0.732950	+	0.680283i	$0.238143\pi$
$179$			243.541i			1.36057i	−0.732950	+	0.680283i	$0.761857\pi$
$180$		0			0
$181$	−325.449			−1.79806			−0.899030	−	0.437887i	$-0.855727\pi$
$181$	−325.449			−1.79806			−0.899030	+	0.437887i	$0.855727\pi$
$182$		0			0
$183$	224.981	−	168.462i	1.22940	−	0.920557i
$184$		0			0
$185$		−	26.4721i		−	0.143093i
$186$		0			0
$187$	−126.754			−0.677828
$188$		0			0
$189$	−96.4631	−	258.335i	−0.510387	−	1.36685i
$190$		0			0
$191$			166.001i			0.869113i	0.900645	+	0.434556i	$0.143095\pi$
$191$			166.001i			0.869113i	−0.900645	+	0.434556i	$0.856905\pi$
$192$		0			0
$193$	239.408			1.24045			0.620227	−	0.784422i	$-0.287041\pi$
$193$	239.408			1.24045			0.620227	+	0.784422i	$0.287041\pi$
$194$		0			0
$195$	−54.6105	−	72.9323i	−0.280054	−	0.374012i
$196$		0			0
$197$		−	257.518i		−	1.30720i	−0.756840	−	0.653600i	$-0.773258\pi$
$197$		−	257.518i		−	1.30720i	0.756840	−	0.653600i	$-0.226742\pi$
$198$		0			0
$199$	−78.4735			−0.394339			−0.197170	−	0.980369i	$-0.563175\pi$
$199$	−78.4735			−0.394339			−0.197170	+	0.980369i	$0.563175\pi$
$200$		0			0
$201$	−82.8471	+	62.0345i	−0.412174	+	0.308629i
$202$		0			0
$203$		−	428.979i		−	2.11320i
$204$		0			0
$205$	−103.705			−0.505876
$206$		0			0
$207$	154.721	+	45.3886i	0.747443	+	0.219268i
$208$		0			0
$209$			208.867i			0.999365i
$210$		0			0
$211$	−112.724			−0.534237			−0.267119	−	0.963664i	$-0.586072\pi$
$211$	−112.724			−0.534237			−0.267119	+	0.963664i	$0.586072\pi$
$212$		0			0
$213$	122.284	+	163.310i	0.574102	+	0.766713i
$214$		0			0
$215$			120.788i			0.561805i
$216$		0			0
$217$	−393.085			−1.81145
$218$		0			0
$219$	107.389	−	80.4108i	0.490359	−	0.367172i
$220$		0			0
$221$			210.130i			0.950814i
$222$		0			0
$223$	−204.686			−0.917872			−0.458936	−	0.888469i	$-0.651769\pi$
$223$	−204.686			−0.917872			−0.458936	+	0.888469i	$0.651769\pi$
$224$		0			0
$225$	−12.6673	+	43.1803i	−0.0562991	+	0.191913i
$226$		0			0
$227$			115.071i			0.506920i	0.967346	+	0.253460i	$0.0815685\pi$
$227$			115.071i			0.506920i	−0.967346	+	0.253460i	$0.918431\pi$
$228$		0			0
$229$	−129.458			−0.565317			−0.282659	−	0.959221i	$-0.591216\pi$
$229$	−129.458			−0.565317			−0.282659	+	0.959221i	$0.591216\pi$
$230$		0			0
$231$	−150.462	−	200.942i	−0.651351	−	0.869879i
$232$		0			0
$233$		−	250.150i		−	1.07361i	−0.843708	−	0.536803i	$-0.819632\pi$
$233$		−	250.150i		−	1.07361i	0.843708	−	0.536803i	$-0.180368\pi$
$234$		0			0
$235$	−96.3644			−0.410061
$236$		0			0
$237$	28.2162	−	21.1278i	0.119056	−	0.0891470i
$238$		0			0
$239$			49.2556i			0.206090i	0.994677	+	0.103045i	$0.0328586\pi$
$239$			49.2556i			0.206090i	−0.994677	+	0.103045i	$0.967141\pi$
$240$		0			0
$241$	−457.672			−1.89905			−0.949526	−	0.313688i	$-0.898435\pi$
$241$	−457.672			−1.89905			−0.949526	+	0.313688i	$0.898435\pi$
$242$		0			0
$243$	242.370	−	17.4851i	0.997408	−	0.0719551i
$244$		0			0
$245$		−	123.677i		−	0.504806i
$246$		0			0
$247$	346.256			1.40185
$248$		0			0
$249$	259.715	+	346.850i	1.04303	+	1.39297i
$250$		0			0
$251$		−	119.717i		−	0.476959i	−0.971148	−	0.238479i	$-0.923351\pi$
$251$		−	119.717i		−	0.476959i	0.971148	−	0.238479i	$-0.0766489\pi$
$252$		0			0
$253$	146.783			0.580169
$254$		0			0
$255$	83.0745	−	62.2048i	0.325782	−	0.243940i
$256$		0			0
$257$		−	111.805i		−	0.435041i	−0.976056	−	0.217520i	$-0.930203\pi$
$257$		−	111.805i		−	0.435041i	0.976056	−	0.217520i	$-0.0697969\pi$
$258$		0			0
$259$	−120.911			−0.466839
$260$		0			0
$261$	362.734	+	106.411i	1.38979	+	0.407705i
$262$		0			0
$263$		−	55.3652i		−	0.210514i	−0.994445	−	0.105257i	$-0.966433\pi$
$263$		−	55.3652i		−	0.210514i	0.994445	−	0.105257i	$-0.0335665\pi$
$264$		0			0
$265$	185.017			0.698177
$266$		0			0
$267$	−114.558	−	152.992i	−0.429055	−	0.573003i
$268$		0			0
$269$		−	309.427i		−	1.15028i	−0.818053	−	0.575142i	$-0.804947\pi$
$269$		−	309.427i		−	1.15028i	0.818053	−	0.575142i	$-0.195053\pi$
$270$		0			0
$271$	−194.062			−0.716096			−0.358048	−	0.933703i	$-0.616558\pi$
$271$	−194.062			−0.716096			−0.358048	+	0.933703i	$0.616558\pi$
$272$		0			0
$273$	−333.118	+	249.433i	−1.22021	+	0.913675i
$274$		0			0
$275$			40.9650i			0.148964i
$276$		0			0
$277$	5.20416			0.0187876			0.00939379	−	0.999956i	$-0.497010\pi$
$277$	5.20416			0.0187876			0.00939379	+	0.999956i	$0.497010\pi$
$278$		0			0
$279$	97.5073	−	332.383i	0.349489	−	1.19134i
$280$		0			0
$281$			240.004i			0.854105i	0.904227	+	0.427053i	$0.140448\pi$
$281$			240.004i			0.854105i	−0.904227	+	0.427053i	$0.859552\pi$
$282$		0			0
$283$	153.109			0.541022			0.270511	−	0.962717i	$-0.412807\pi$
$283$	153.109			0.541022			0.270511	+	0.962717i	$0.412807\pi$
$284$		0			0
$285$	−102.502	−	136.892i	−0.359657	−	0.480322i
$286$		0			0
$287$			473.671i			1.65042i
$288$		0			0
$289$	49.6485			0.171794
$290$		0			0
$291$	−153.453	+	114.903i	−0.527329	+	0.394855i
$292$		0			0
$293$			169.252i			0.577653i	0.957381	+	0.288827i	$0.0932651\pi$
$293$			169.252i			0.577653i	−0.957381	+	0.288827i	$0.906735\pi$
$294$		0			0
$295$	−102.619			−0.347862
$296$		0			0
$297$	207.235	−	77.3821i	0.697760	−	0.260546i
$298$		0			0
$299$		−	243.334i		−	0.813825i
$300$		0			0
$301$	551.699			1.83289
$302$		0			0
$303$	91.4832	+	122.176i	0.301925	+	0.403220i
$304$		0			0
$305$			209.491i			0.686857i
$306$		0			0
$307$	330.401			1.07623			0.538113	−	0.842873i	$-0.319137\pi$
$307$	330.401			1.07623			0.538113	+	0.842873i	$0.319137\pi$
$308$		0			0
$309$	−110.342	+	82.6224i	−0.357095	+	0.267387i
$310$		0			0
$311$		−	133.596i		−	0.429570i	−0.976661	−	0.214785i	$-0.931095\pi$
$311$		−	133.596i		−	0.429570i	0.976661	−	0.214785i	$-0.0689050\pi$
$312$		0			0
$313$	−459.981			−1.46959			−0.734794	−	0.678290i	$-0.762721\pi$
$313$	−459.981			−1.46959			−0.734794	+	0.678290i	$0.762721\pi$
$314$		0			0
$315$	197.226	+	57.8579i	0.626114	+	0.183676i
$316$		0			0
$317$			298.150i			0.940536i	0.882524	+	0.470268i	$0.155843\pi$
$317$			298.150i			0.940536i	−0.882524	+	0.470268i	$0.844157\pi$
$318$		0			0
$319$	344.124			1.07876
$320$		0			0
$321$	198.241	+	264.751i	0.617574	+	0.824771i
$322$		0			0
$323$			394.408i			1.22108i
$324$		0			0
$325$	67.9110			0.208957
$326$		0			0
$327$	118.340	−	88.6112i	0.361897	−	0.270982i
$328$		0			0
$329$			440.144i			1.33782i
$330$		0			0
$331$	397.599			1.20120			0.600602	−	0.799548i	$-0.294927\pi$
$331$	397.599			1.20120			0.600602	+	0.799548i	$0.294927\pi$
$332$		0			0
$333$	29.9929	−	102.240i	0.0900687	−	0.307026i
$334$		0			0
$335$		−	77.1432i		−	0.230278i
$336$		0			0
$337$	−427.942			−1.26986			−0.634929	−	0.772571i	$-0.718970\pi$
$337$	−427.942			−1.26986			−0.634929	+	0.772571i	$0.718970\pi$
$338$		0			0
$339$	283.116	+	378.101i	0.835151	+	1.11534i
$340$		0			0
$341$		−	315.331i		−	0.924723i
$342$		0			0
$343$	−64.4477			−0.187894
$344$		0			0
$345$	−96.2016	+	72.0342i	−0.278845	+	0.208795i
$346$		0			0
$347$		−	138.137i		−	0.398090i	−0.979990	−	0.199045i	$-0.936216\pi$
$347$		−	138.137i		−	0.398090i	0.979990	−	0.199045i	$-0.0637840\pi$
$348$		0			0
$349$	−301.421			−0.863669			−0.431835	−	0.901953i	$-0.642134\pi$
$349$	−301.421			−0.863669			−0.431835	+	0.901953i	$0.642134\pi$
$350$		0			0
$351$	−128.283	−	343.550i	−0.365477	−	0.978775i
$352$		0			0
$353$			142.622i			0.404028i	0.979383	+	0.202014i	$0.0647487\pi$
$353$			142.622i			0.404028i	−0.979383	+	0.202014i	$0.935251\pi$
$354$		0			0
$355$	−152.066			−0.428356
$356$		0			0
$357$	−284.120	−	379.443i	−0.795856	−	1.06286i
$358$		0			0
$359$			3.05663i			0.00851429i	0.999991	+	0.00425715i	$0.00135510\pi$
$359$			3.05663i			0.00851429i	−0.999991	+	0.00425715i	$0.998645\pi$
$360$		0			0
$361$	288.913			0.800313
$362$		0			0
$363$	−129.375	+	96.8739i	−0.356405	+	0.266870i
$364$		0			0
$365$			99.9950i			0.273959i
$366$		0			0
$367$	−370.072			−1.00837			−0.504186	−	0.863595i	$-0.668207\pi$
$367$	−370.072			−1.00837			−0.504186	+	0.863595i	$0.668207\pi$
$368$		0			0
$369$	−400.525	−	117.497i	−1.08543	−	0.318420i
$370$		0			0
$371$		−	845.065i		−	2.27780i
$372$		0			0
$373$	455.556			1.22133			0.610665	−	0.791889i	$-0.290902\pi$
$373$	455.556			1.22133			0.610665	+	0.791889i	$0.290902\pi$
$374$		0			0
$375$	−20.1037	−	26.8485i	−0.0536099	−	0.0715960i
$376$		0			0
$377$		−	570.482i		−	1.51322i
$378$		0			0
$379$	−380.051			−1.00277			−0.501387	−	0.865223i	$-0.667177\pi$
$379$	−380.051			−1.00277			−0.501387	+	0.865223i	$0.667177\pi$
$380$		0			0
$381$	324.346	−	242.865i	0.851301	−	0.637440i
$382$		0			0
$383$			164.078i			0.428402i	0.976790	+	0.214201i	$0.0687147\pi$
$383$			164.078i			0.428402i	−0.976790	+	0.214201i	$0.931285\pi$
$384$		0			0
$385$	187.108			0.485994
$386$		0			0
$387$	−136.853	+	466.503i	−0.353624	+	1.20544i
$388$		0			0
$389$		−	562.622i		−	1.44633i	−0.690676	−	0.723164i	$-0.742687\pi$
$389$		−	562.622i		−	1.44633i	0.690676	−	0.723164i	$-0.257313\pi$
$390$		0			0
$391$	277.173			0.708882
$392$		0			0
$393$	−270.893	−	361.778i	−0.689295	−	0.920553i
$394$		0			0
$395$			26.2736i			0.0665154i
$396$		0			0
$397$	−93.4668			−0.235433			−0.117716	−	0.993047i	$-0.537557\pi$
$397$	−93.4668			−0.235433			−0.117716	+	0.993047i	$0.537557\pi$
$398$		0			0
$399$	−625.253	+	468.179i	−1.56705	+	1.17338i
$400$		0			0
$401$		−	23.4831i		−	0.0585613i	−0.999571	−	0.0292807i	$-0.990678\pi$
$401$		−	23.4831i		−	0.0585613i	0.999571	−	0.0292807i	$-0.00932166\pi$
$402$		0			0
$403$	−522.749			−1.29714
$404$		0			0
$405$	−97.8464	+	152.417i	−0.241596	+	0.376339i
$406$		0			0
$407$		−	96.9944i		−	0.238316i
$408$		0			0
$409$	697.472			1.70531			0.852655	−	0.522474i	$-0.174991\pi$
$409$	697.472			1.70531			0.852655	+	0.522474i	$0.174991\pi$
$410$		0			0
$411$	−250.095	−	334.001i	−0.608503	−	0.812656i
$412$		0			0
$413$			468.714i			1.13490i
$414$		0			0
$415$	−322.970			−0.778240
$416$		0			0
$417$	−264.865	+	198.326i	−0.635167	+	0.475603i
$418$		0			0
$419$		−	115.537i		−	0.275744i	−0.990450	−	0.137872i	$-0.955974\pi$
$419$		−	115.537i		−	0.275744i	0.990450	−	0.137872i	$-0.0440262\pi$
$420$		0			0
$421$	117.396			0.278850			0.139425	−	0.990233i	$-0.455475\pi$
$421$	117.396			0.278850			0.139425	+	0.990233i	$0.455475\pi$
$422$		0			0
$423$	−372.175	−	109.181i	−0.879847	−	0.258110i
$424$		0			0
$425$			77.3549i			0.182012i
$426$		0			0
$427$	956.851			2.24087
$428$		0			0
$429$	−200.094	−	267.225i	−0.466419	−	0.622903i
$430$		0			0
$431$			673.123i			1.56177i	0.624675	+	0.780885i	$0.285232\pi$
$431$			673.123i			1.56177i	−0.624675	+	0.780885i	$0.714768\pi$
$432$		0			0
$433$	−392.655			−0.906824			−0.453412	−	0.891301i	$-0.649793\pi$
$433$	−392.655			−0.906824			−0.453412	+	0.891301i	$0.649793\pi$
$434$		0			0
$435$	−225.539	+	168.880i	−0.518481	+	0.388230i
$436$		0			0
$437$		−	456.731i		−	1.04515i
$438$		0			0
$439$	409.353			0.932468			0.466234	−	0.884661i	$-0.345611\pi$
$439$	409.353			0.932468			0.466234	+	0.884661i	$0.345611\pi$
$440$		0			0
$441$	140.126	−	477.662i	0.317746	−	1.08313i
$442$		0			0
$443$			355.935i			0.803464i	0.915757	+	0.401732i	$0.131592\pi$
$443$			355.935i			0.803464i	−0.915757	+	0.401732i	$0.868408\pi$
$444$		0			0
$445$	142.459			0.320132
$446$		0			0
$447$	−163.284	−	218.065i	−0.365288	−	0.487842i
$448$		0			0
$449$		−	294.636i		−	0.656205i	−0.944642	−	0.328103i	$-0.893591\pi$
$449$		−	294.636i		−	0.656205i	0.944642	−	0.328103i	$-0.106409\pi$
$450$		0			0
$451$	−379.976			−0.842519
$452$		0			0
$453$	−267.242	+	200.106i	−0.589938	+	0.441736i
$454$		0			0
$455$		−	310.183i		−	0.681721i
$456$		0			0
$457$	491.814			1.07618			0.538090	−	0.842887i	$-0.319146\pi$
$457$	491.814			1.07618			0.538090	+	0.842887i	$0.319146\pi$
$458$		0			0
$459$	391.325	−	146.122i	0.852561	−	0.318349i
$460$		0			0
$461$			42.6832i			0.0925883i	0.998928	+	0.0462942i	$0.0147412\pi$
$461$			42.6832i			0.0925883i	−0.998928	+	0.0462942i	$0.985259\pi$
$462$		0			0
$463$	−422.582			−0.912705			−0.456352	−	0.889799i	$-0.650844\pi$
$463$	−422.582			−0.912705			−0.456352	+	0.889799i	$0.650844\pi$
$464$		0			0
$465$	154.749	+	206.668i	0.332795	+	0.444447i
$466$		0			0
$467$		−	203.497i		−	0.435754i	−0.975976	−	0.217877i	$-0.930087\pi$
$467$		−	203.497i		−	0.435754i	0.975976	−	0.217877i	$-0.0699131\pi$
$468$		0			0
$469$	−352.351			−0.751282
$470$		0			0
$471$	−342.813	+	256.693i	−0.727840	+	0.544995i
$472$		0			0
$473$			442.570i			0.935666i
$474$		0			0
$475$	127.467			0.268352
$476$		0			0
$477$	714.566	+	209.624i	1.49804	+	0.439463i
$478$		0			0
$479$			485.253i			1.01305i	0.862224	+	0.506527i	$0.169071\pi$
$479$			485.253i			1.01305i	−0.862224	+	0.506527i	$0.830929\pi$
$480$		0			0
$481$	−160.795			−0.334294
$482$		0			0
$483$	329.016	+	439.401i	0.681192	+	0.909732i
$484$		0			0
$485$		−	142.888i		−	0.294614i
$486$		0			0
$487$	140.013			0.287500			0.143750	−	0.989614i	$-0.454084\pi$
$487$	140.013			0.287500			0.143750	+	0.989614i	$0.454084\pi$
$488$		0			0
$489$	408.669	−	306.005i	0.835724	−	0.625776i
$490$		0			0
$491$		−	20.4450i		−	0.0416396i	−0.999783	−	0.0208198i	$-0.993372\pi$
$491$		−	20.4450i		−	0.0416396i	0.999783	−	0.0208198i	$-0.00662763\pi$
$492$		0			0
$493$	649.816			1.31809
$494$		0			0
$495$	−46.4132	+	158.214i	−0.0937641	+	0.319623i
$496$		0			0
$497$			694.562i			1.39751i
$498$		0			0
$499$	49.2051			0.0986074			0.0493037	−	0.998784i	$-0.484300\pi$
$499$	49.2051			0.0986074			0.0493037	+	0.998784i	$0.484300\pi$
$500$		0			0
$501$	11.4861	+	15.3397i	0.0229263	+	0.0306181i
$502$		0			0
$503$			149.060i			0.296343i	0.988962	+	0.148171i	$0.0473387\pi$
$503$			149.060i			0.296343i	−0.988962	+	0.148171i	$0.952661\pi$
$504$		0			0
$505$	−113.764			−0.225275
$506$		0			0
$507$	−37.1639	+	27.8277i	−0.0733015	+	0.0548870i
$508$		0			0
$509$		−	655.375i		−	1.28757i	−0.765205	−	0.643787i	$-0.777362\pi$
$509$		−	655.375i		−	1.28757i	0.765205	−	0.643787i	$-0.222638\pi$
$510$		0			0
$511$	456.727			0.893790
$512$		0			0
$513$	−240.783	−	644.834i	−0.469362	−	1.25699i
$514$		0			0
$515$		−	102.745i		−	0.199505i
$516$		0			0
$517$	−353.081			−0.682942
$518$		0			0
$519$	428.680	+	572.502i	0.825973	+	1.10309i
$520$		0			0
$521$			419.868i			0.805890i	0.915224	+	0.402945i	$0.132013\pi$
$521$			419.868i			0.805890i	−0.915224	+	0.402945i	$0.867987\pi$
$522$		0			0
$523$	146.678			0.280454			0.140227	−	0.990119i	$-0.455217\pi$
$523$	146.678			0.280454			0.140227	+	0.990119i	$0.455217\pi$
$524$		0			0
$525$	−122.630	+	91.8236i	−0.233582	+	0.174902i
$526$		0			0
$527$		−	595.444i		−	1.12988i
$528$		0			0
$529$	208.030			0.393251
$530$		0			0
$531$	−396.333	−	116.268i	−0.746390	−	0.218960i
$532$		0			0
$533$			629.917i			1.18183i
$534$		0			0
$535$	−246.524			−0.460792
$536$		0			0
$537$	−437.919	−	584.840i	−0.815491	−	1.08909i
$538$		0			0
$539$		−	453.156i		−	0.840735i
$540$		0			0
$541$	7.07300			0.0130739			0.00653697	−	0.999979i	$-0.497919\pi$
$541$	7.07300			0.0130739			0.00653697	+	0.999979i	$0.497919\pi$
$542$		0			0
$543$	781.534	−	585.199i	1.43929	−	1.07772i
$544$		0			0
$545$			110.193i			0.202188i
$546$		0			0
$547$	3.46995			0.00634360			0.00317180	−	0.999995i	$-0.498990\pi$
$547$	3.46995			0.00634360			0.00317180	+	0.999995i	$0.498990\pi$
$548$		0			0
$549$	−237.353	+	809.090i	−0.432337	+	1.47375i
$550$		0			0
$551$		−	1070.78i		−	1.94334i
$552$		0			0
$553$	120.005			0.217006
$554$		0			0
$555$	47.6003	+	63.5702i	0.0857663	+	0.114541i
$556$		0			0
$557$			654.467i			1.17499i	0.809230	+	0.587493i	$0.199885\pi$
$557$			654.467i			1.17499i	−0.809230	+	0.587493i	$0.800115\pi$
$558$		0			0
$559$	733.684			1.31249
$560$		0			0
$561$	304.387	−	227.920i	0.542579	−	0.406274i
$562$		0			0
$563$			546.529i			0.970745i	0.874307	+	0.485373i	$0.161316\pi$
$563$			546.529i			0.970745i	−0.874307	+	0.485373i	$0.838684\pi$
$564$		0			0
$565$	−352.070			−0.623132
$566$		0			0
$567$	696.167	+	446.913i	1.22781	+	0.788207i
$568$		0			0
$569$			746.774i			1.31243i	0.754573	+	0.656216i	$0.227844\pi$
$569$			746.774i			1.31243i	−0.754573	+	0.656216i	$0.772156\pi$
$570$		0			0
$571$	−230.358			−0.403430			−0.201715	−	0.979444i	$-0.564651\pi$
$571$	−230.358			−0.403430			−0.201715	+	0.979444i	$0.564651\pi$
$572$		0			0
$573$	−298.491	−	398.634i	−0.520926	−	0.695696i
$574$		0			0
$575$		−	89.5782i		−	0.155788i
$576$		0			0
$577$	525.585			0.910893			0.455446	−	0.890263i	$-0.349480\pi$
$577$	525.585			0.910893			0.455446	+	0.890263i	$0.349480\pi$
$578$		0			0
$579$	−574.914	+	430.486i	−0.992943	+	0.743499i
$580$		0			0
$581$			1475.16i			2.53901i
$582$		0			0
$583$	677.906			1.16279
$584$		0			0
$585$	262.283	+	76.9430i	0.448348	+	0.131526i
$586$		0			0
$587$			75.3619i			0.128385i	0.997938	+	0.0641924i	$0.0204471\pi$
$587$			75.3619i			0.128385i	−0.997938	+	0.0641924i	$0.979553\pi$
$588$		0			0
$589$	−981.185			−1.66585
$590$		0			0
$591$	463.051	+	618.405i	0.783505	+	1.04637i
$592$		0			0
$593$		−	1142.87i		−	1.92727i	−0.267230	−	0.963633i	$-0.586108\pi$
$593$		−	1142.87i		−	1.92727i	0.267230	−	0.963633i	$-0.413892\pi$
$594$		0			0
$595$	353.319			0.593813
$596$		0			0
$597$	188.446	−	141.106i	0.315656	−	0.236358i
$598$		0			0
$599$		−	1060.02i		−	1.76965i	−0.465927	−	0.884823i	$-0.654279\pi$
$599$		−	1060.02i		−	1.76965i	0.465927	−	0.884823i	$-0.345721\pi$
$600$		0			0
$601$	−530.900			−0.883361			−0.441681	−	0.897172i	$-0.645618\pi$
$601$	−530.900			−0.883361			−0.441681	+	0.897172i	$0.645618\pi$
$602$		0			0
$603$	87.4030	−	297.940i	0.144947	−	0.494095i
$604$		0			0
$605$		−	120.468i		−	0.199120i
$606$		0			0
$607$	−719.995			−1.18615			−0.593077	−	0.805146i	$-0.702087\pi$
$607$	−719.995			−1.18615			−0.593077	+	0.805146i	$0.702087\pi$
$608$		0			0
$609$	771.360	+	1030.15i	1.26660	+	1.69154i
$610$		0			0
$611$			585.331i			0.957988i
$612$		0			0
$613$	−216.348			−0.352932			−0.176466	−	0.984307i	$-0.556467\pi$
$613$	−216.348			−0.352932			−0.176466	+	0.984307i	$0.556467\pi$
$614$		0			0
$615$	249.037	−	186.474i	0.404938	−	0.303210i
$616$		0			0
$617$		−	231.190i		−	0.374701i	−0.982293	−	0.187350i	$-0.940010\pi$
$617$		−	231.190i		−	0.374701i	0.982293	−	0.187350i	$-0.0599900\pi$
$618$		0			0
$619$	−689.633			−1.11411			−0.557054	−	0.830476i	$-0.688068\pi$
$619$	−689.633			−1.11411			−0.557054	+	0.830476i	$0.688068\pi$
$620$		0			0
$621$	−453.161	+	169.212i	−0.729728	+	0.272482i
$622$		0			0
$623$		−	650.679i		−	1.04443i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−375.570	−	501.574i	−0.598996	−	0.799959i
$628$		0			0
$629$		−	183.156i		−	0.291187i
$630$		0			0
$631$	−616.973			−0.977770			−0.488885	−	0.872348i	$-0.662596\pi$
$631$	−616.973			−0.977770			−0.488885	+	0.872348i	$0.662596\pi$
$632$		0			0
$633$	270.696	−	202.692i	0.427639	−	0.320209i
$634$		0			0
$635$			302.015i			0.475614i
$636$		0			0
$637$	−751.234			−1.17933
$638$		0			0
$639$	−587.305	−	172.291i	−0.919100	−	0.269626i
$640$		0			0
$641$		−	1131.85i		−	1.76576i	−0.469598	−	0.882880i	$-0.655601\pi$
$641$		−	1131.85i		−	1.76576i	0.469598	−	0.882880i	$-0.344399\pi$
$642$		0			0
$643$	−637.481			−0.991418			−0.495709	−	0.868489i	$-0.665092\pi$
$643$	−637.481			−0.991418			−0.495709	+	0.868489i	$0.665092\pi$
$644$		0			0
$645$	−217.193	−	290.061i	−0.336733	−	0.449706i
$646$		0			0
$647$			800.782i			1.23769i	0.785515	+	0.618843i	$0.212398\pi$
$647$			800.782i			1.23769i	−0.785515	+	0.618843i	$0.787602\pi$
$648$		0			0
$649$	−375.999			−0.579352
$650$		0			0
$651$	943.955	−	706.818i	1.45001	−	1.08574i
$652$		0			0
$653$			313.185i			0.479610i	0.970821	+	0.239805i	$0.0770835\pi$
$653$			313.185i			0.479610i	−0.970821	+	0.239805i	$0.922916\pi$
$654$		0			0
$655$	336.870			0.514305
$656$		0			0
$657$	−113.294	+	386.197i	−0.172442	+	0.587819i
$658$		0			0
$659$		−	878.311i		−	1.33279i	−0.745597	−	0.666397i	$-0.767836\pi$
$659$		−	878.311i		−	1.33279i	0.745597	−	0.666397i	$-0.232164\pi$
$660$		0			0
$661$	200.421			0.303208			0.151604	−	0.988441i	$-0.451556\pi$
$661$	200.421			0.303208			0.151604	+	0.988441i	$0.451556\pi$
$662$		0			0
$663$	−377.841	−	504.606i	−0.569896	−	0.761096i
$664$		0			0
$665$		−	582.205i		−	0.875497i
$666$		0			0
$667$	−752.497			−1.12818
$668$		0			0
$669$	491.532	−	368.051i	0.734727	−	0.550151i
$670$		0			0
$671$			767.580i			1.14393i
$672$		0			0
$673$	−60.8622			−0.0904341			−0.0452171	−	0.998977i	$-0.514398\pi$
$673$	−60.8622			−0.0904341			−0.0452171	+	0.998977i	$0.514398\pi$
$674$		0			0
$675$	−47.2245	−	126.471i	−0.0699623	−	0.187364i
$676$		0			0
$677$			415.595i			0.613877i	0.951729	+	0.306939i	$0.0993047\pi$
$677$			415.595i			0.613877i	−0.951729	+	0.306939i	$0.900695\pi$
$678$		0			0
$679$	−652.639			−0.961177
$680$		0			0
$681$	−206.912	−	276.331i	−0.303836	−	0.405772i
$682$		0			0
$683$		−	434.504i		−	0.636170i	−0.948062	−	0.318085i	$-0.896960\pi$
$683$		−	434.504i		−	0.636170i	0.948062	−	0.318085i	$-0.103040\pi$
$684$		0			0
$685$	311.006			0.454023
$686$		0			0
$687$	310.880	−	232.782i	0.452518	−	0.338838i
$688$		0			0
$689$		−	1123.82i		−	1.63109i
$690$		0			0
$691$	−166.630			−0.241144			−0.120572	−	0.992705i	$-0.538473\pi$
$691$	−166.630			−0.241144			−0.120572	+	0.992705i	$0.538473\pi$
$692$		0			0
$693$	722.640	+	211.992i	1.04277	+	0.305905i
$694$		0			0
$695$		−	246.629i		−	0.354862i
$696$		0			0
$697$	−717.516			−1.02943
$698$		0			0
$699$	449.802	+	600.711i	0.643494	+	0.859386i
$700$		0			0
$701$			248.345i			0.354272i	0.984186	+	0.177136i	$0.0566833\pi$
$701$			248.345i			0.354272i	−0.984186	+	0.177136i	$0.943317\pi$
$702$		0			0
$703$	−301.809			−0.429315
$704$		0			0
$705$	231.410	−	173.276i	0.328241	−	0.245781i
$706$		0			0
$707$			519.617i			0.734961i
$708$		0			0
$709$	−24.1911			−0.0341200			−0.0170600	−	0.999854i	$-0.505431\pi$
$709$	−24.1911			−0.0341200			−0.0170600	+	0.999854i	$0.505431\pi$
$710$		0			0
$711$	−29.7679	+	101.473i	−0.0418677	+	0.142719i
$712$		0			0
$713$			689.534i			0.967088i
$714$		0			0
$715$	248.827			0.348010
$716$		0			0
$717$	−88.5679	−	118.282i	−0.123526	−	0.164969i
$718$		0			0
$719$		−	323.584i		−	0.450048i	−0.974353	−	0.225024i	$-0.927754\pi$
$719$		−	323.584i		−	0.450048i	0.974353	−	0.225024i	$-0.0722460\pi$
$720$		0			0
$721$	−469.289			−0.650886
$722$		0			0
$723$	1099.05	−	822.953i	1.52013	−	1.13825i
$724$		0			0
$725$		−	210.011i		−	0.289671i
$726$		0			0
$727$	−454.949			−0.625789			−0.312895	−	0.949788i	$-0.601299\pi$
$727$	−454.949			−0.625789			−0.312895	+	0.949788i	$0.601299\pi$
$728$		0			0
$729$	−550.588	+	477.802i	−0.755264	+	0.655421i
$730$		0			0
$731$			835.713i			1.14325i
$732$		0			0
$733$	131.588			0.179520			0.0897600	−	0.995963i	$-0.471390\pi$
$733$	131.588			0.179520			0.0897600	+	0.995963i	$0.471390\pi$
$734$		0			0
$735$	222.388	+	296.999i	0.302569	+	0.404080i
$736$		0			0
$737$		−	282.654i		−	0.383520i
$738$		0			0
$739$	1033.59			1.39864			0.699319	−	0.714810i	$-0.253487\pi$
$739$	1033.59			1.39864			0.699319	+	0.714810i	$0.253487\pi$
$740$		0			0
$741$	−831.500	+	622.613i	−1.12213	+	0.840234i
$742$		0			0
$743$		−	439.535i		−	0.591569i	−0.955255	−	0.295784i	$-0.904419\pi$
$743$		−	439.535i		−	0.591569i	0.955255	−	0.295784i	$-0.0955810\pi$
$744$		0			0
$745$	203.052			0.272553
$746$		0			0
$747$	−1247.36	−	365.924i	−1.66983	−	0.489858i
$748$		0			0
$749$			1126.00i			1.50333i
$750$		0			0
$751$	496.343			0.660909			0.330455	−	0.943822i	$-0.392798\pi$
$751$	496.343			0.660909			0.330455	+	0.943822i	$0.392798\pi$
$752$		0			0
$753$	215.266	+	287.488i	0.285878	+	0.381790i
$754$		0			0
$755$		−	248.843i		−	0.329593i
$756$		0			0
$757$	468.694			0.619147			0.309574	−	0.950875i	$-0.399814\pi$
$757$	468.694			0.619147			0.309574	+	0.950875i	$0.399814\pi$
$758$		0			0
$759$	−352.485	+	263.935i	−0.464406	+	0.347740i
$760$		0			0
$761$		−	156.940i		−	0.206229i	−0.994669	−	0.103115i	$-0.967119\pi$
$761$		−	156.940i		−	0.206229i	0.994669	−	0.103115i	$-0.0328808\pi$
$762$		0			0
$763$	503.305			0.659640
$764$		0			0
$765$	−87.6430	+	298.758i	−0.114566	+	0.390533i
$766$		0			0
$767$			623.325i			0.812679i
$768$		0			0
$769$	1188.63			1.54569			0.772844	−	0.634596i	$-0.218834\pi$
$769$	1188.63			1.54569			0.772844	+	0.634596i	$0.218834\pi$
$770$		0			0
$771$	201.041	+	268.490i	0.260753	+	0.348236i
$772$		0			0
$773$		−	950.291i		−	1.22935i	−0.788779	−	0.614677i	$-0.789286\pi$
$773$		−	950.291i		−	1.22935i	0.788779	−	0.614677i	$-0.210714\pi$
$774$		0			0
$775$	−192.439			−0.248308
$776$		0			0
$777$	290.357	−	217.414i	0.373690	−	0.279813i
$778$		0			0
$779$			1182.34i			1.51776i
$780$		0			0
$781$	−557.174			−0.713411
$782$		0			0
$783$	−1062.41	+	396.707i	−1.35685	+	0.506650i
$784$		0			0
$785$		−	319.211i		−	0.406638i
$786$		0			0
$787$	1056.60			1.34257			0.671284	−	0.741200i	$-0.265743\pi$
$787$	1056.60			1.34257			0.671284	+	0.741200i	$0.265743\pi$
$788$		0			0
$789$	99.5538	+	132.954i	0.126177	+	0.168510i
$790$		0			0
$791$			1608.08i			2.03297i
$792$		0			0
$793$	1272.48			1.60464
$794$		0			0
$795$	−444.300	+	332.684i	−0.558868	+	0.418471i
$796$		0			0
$797$			78.7126i			0.0987612i	0.998780	+	0.0493806i	$0.0157247\pi$
$797$			78.7126i			0.0987612i	−0.998780	+	0.0493806i	$0.984275\pi$
$798$		0			0
$799$	−666.730			−0.834455
$800$		0			0
$801$	550.198	+	161.405i	0.686889	+	0.201505i
$802$		0			0
$803$			366.384i			0.456268i
$804$		0			0
$805$	−409.148			−0.508259
$806$		0			0
$807$	556.389	+	743.058i	0.689454	+	0.920765i
$808$		0			0
$809$		−	176.199i		−	0.217799i	−0.994053	−	0.108899i	$-0.965267\pi$
$809$		−	176.199i		−	0.217799i	0.994053	−	0.108899i	$-0.0347327\pi$
$810$		0			0
$811$	1103.89			1.36115			0.680574	−	0.732679i	$-0.261730\pi$
$811$	1103.89			1.36115			0.680574	+	0.732679i	$0.261730\pi$
$812$		0			0
$813$	466.021	−	348.949i	0.573211	−	0.429211i
$814$		0			0
$815$			380.533i			0.466911i
$816$		0			0
$817$	1377.10			1.68556
$818$		0			0
$819$	351.437	−	1197.98i	0.429105	−	1.46273i
$820$		0			0
$821$		−	1321.32i		−	1.60940i	−0.593679	−	0.804702i	$-0.702325\pi$
$821$		−	1321.32i		−	1.60940i	0.593679	−	0.804702i	$-0.297675\pi$
$822$		0			0
$823$	760.814			0.924439			0.462220	−	0.886765i	$-0.347053\pi$
$823$	760.814			0.924439			0.462220	+	0.886765i	$0.347053\pi$
$824$		0			0
$825$	−73.6604	−	98.3734i	−0.0892853	−	0.119240i
$826$		0			0
$827$		−	610.599i		−	0.738331i	−0.929364	−	0.369165i	$-0.879644\pi$
$827$		−	610.599i		−	0.738331i	0.929364	−	0.369165i	$-0.120356\pi$
$828$		0			0
$829$	265.766			0.320586			0.160293	−	0.987069i	$-0.448756\pi$
$829$	265.766			0.320586			0.160293	+	0.987069i	$0.448756\pi$
$830$		0			0
$831$	−12.4973	+	9.35776i	−0.0150388	+	0.0112608i
$832$		0			0
$833$		−	855.704i		−	1.02726i
$834$		0			0
$835$	−14.2836			−0.0171061
$836$		0			0
$837$	363.514	+	973.516i	0.434306	+	1.16310i
$838$		0			0
$839$		−	48.5299i		−	0.0578425i	−0.999582	−	0.0289213i	$-0.990793\pi$
$839$		−	48.5299i		−	0.0578425i	0.999582	−	0.0289213i	$-0.00920721\pi$
$840$		0			0
$841$	−923.188			−1.09773
$842$		0			0
$843$	−431.558	−	576.345i	−0.511931	−	0.683683i
$844$		0			0
$845$		−	34.6052i		−	0.0409529i
$846$		0			0
$847$	−550.236			−0.649629
$848$		0			0
$849$	−367.677	+	275.310i	−0.433071	+	0.324276i
$850$		0			0
$851$			212.098i			0.249234i
$852$		0			0
$853$	−331.180			−0.388253			−0.194127	−	0.980976i	$-0.562187\pi$
$853$	−331.180			−0.388253			−0.194127	+	0.980976i	$0.562187\pi$
$854$		0			0
$855$	492.298	+	144.420i	0.575788	+	0.168912i
$856$		0			0
$857$		−	159.395i		−	0.185992i	−0.995666	−	0.0929961i	$-0.970356\pi$
$857$		−	159.395i		−	0.185992i	0.995666	−	0.0929961i	$-0.0296444\pi$
$858$		0			0
$859$	−1635.01			−1.90339			−0.951694	−	0.307049i	$-0.900658\pi$
$859$	−1635.01			−1.90339			−0.951694	+	0.307049i	$0.900658\pi$
$860$		0			0
$861$	−851.722	−	1137.47i	−0.989224	−	1.32111i
$862$		0			0
$863$			152.237i			0.176405i	0.996103	+	0.0882025i	$0.0281123\pi$
$863$			152.237i			0.176405i	−0.996103	+	0.0882025i	$0.971888\pi$
$864$		0			0
$865$	−533.086			−0.616284
$866$		0			0
$867$	−119.226	+	89.2744i	−0.137516	+	0.102969i
$868$		0			0
$869$			96.2669i			0.110779i
$870$		0			0
$871$	−468.578			−0.537977
$872$		0			0
$873$	161.891	−	551.856i	0.185443	−	0.632137i
$874$		0			0
$875$		−	114.187i		−	0.130500i
$876$		0			0
$877$	521.588			0.594742			0.297371	−	0.954762i	$-0.403890\pi$
$877$	521.588			0.594742			0.297371	+	0.954762i	$0.403890\pi$
$878$		0			0
$879$	−304.338	−	406.443i	−0.346232	−	0.462392i
$880$		0			0
$881$			821.701i			0.932692i	0.884603	+	0.466346i	$0.154430\pi$
$881$			821.701i			0.932692i	−0.884603	+	0.466346i	$0.845570\pi$
$882$		0			0
$883$	747.791			0.846876			0.423438	−	0.905925i	$-0.360823\pi$
$883$	747.791			0.846876			0.423438	+	0.905925i	$0.360823\pi$
$884$		0			0
$885$	246.430	−	184.523i	0.278452	−	0.208501i
$886$		0			0
$887$		−	1160.21i		−	1.30801i	−0.756490	−	0.654006i	$-0.773087\pi$
$887$		−	1160.21i		−	1.30801i	0.756490	−	0.654006i	$-0.226913\pi$
$888$		0			0
$889$	1379.45			1.55169
$890$		0			0
$891$	−358.511	+	558.461i	−0.402369	+	0.626780i
$892$		0			0
$893$			1098.65i			1.23029i
$894$		0			0
$895$	544.575			0.608463
$896$		0			0
$897$	437.546	+	584.342i	0.487788	+	0.651440i
$898$		0			0
$899$			1616.57i			1.79819i
$900$		0			0
$901$	1280.10			1.42076
$902$		0			0
$903$	−1324.85	+	992.027i	−1.46717	+	1.09859i
$904$		0			0
$905$			727.726i			0.804117i
$906$		0			0
$907$	−354.207			−0.390526			−0.195263	−	0.980751i	$-0.562556\pi$
$907$	−354.207			−0.390526			−0.195263	+	0.980751i	$0.562556\pi$
$908$		0			0
$909$	−439.376	−	128.894i	−0.483362	−	0.141798i
$910$		0			0
$911$		−	1461.82i		−	1.60463i	−0.596900	−	0.802316i	$-0.703601\pi$
$911$		−	1461.82i		−	1.60463i	0.596900	−	0.802316i	$-0.296399\pi$
$912$		0			0
$913$	−1183.37			−1.29613
$914$		0			0
$915$	−376.692	−	503.073i	−0.411686	−	0.549806i
$916$		0			0
$917$		−	1538.65i		−	1.67792i
$918$		0			0
$919$	600.104			0.652997			0.326499	−	0.945198i	$-0.394131\pi$
$919$	600.104			0.652997			0.326499	+	0.945198i	$0.394131\pi$
$920$		0			0
$921$	−793.427	+	594.105i	−0.861484	+	0.645065i
$922$		0			0
$923$			923.672i			1.00073i
$924$		0			0
$925$	−59.1935			−0.0639930
$926$		0			0
$927$	116.410	−	396.819i	0.125577	−	0.428068i
$928$		0			0
$929$		−	243.828i		−	0.262462i	−0.991352	−	0.131231i	$-0.958107\pi$
$929$		−	243.828i		−	0.262462i	0.991352	−	0.131231i	$-0.0418930\pi$
$930$		0			0
$931$	−1410.05			−1.51455
$932$		0			0
$933$	240.223	+	320.818i	0.257474	+	0.343856i
$934$		0			0
$935$			283.430i			0.303134i
$936$		0			0
$937$	894.637			0.954789			0.477395	−	0.878689i	$-0.341581\pi$
$937$	894.637			0.954789			0.477395	+	0.878689i	$0.341581\pi$
$938$		0			0
$939$	1104.60	−	827.105i	1.17636	−	0.880837i
$940$		0			0
$941$		−	592.336i		−	0.629475i	−0.949179	−	0.314737i	$-0.898084\pi$
$941$		−	592.336i		−	0.629475i	0.949179	−	0.314737i	$-0.101916\pi$
$942$		0			0
$943$	830.894			0.881118
$944$		0			0
$945$	−577.655	+	215.698i	−0.611275	+	0.228252i
$946$		0			0
$947$			526.526i			0.555994i	0.960582	+	0.277997i	$0.0896705\pi$
$947$			526.526i			0.555994i	−0.960582	+	0.277997i	$0.910330\pi$
$948$		0			0
$949$	607.383			0.640025
$950$		0			0
$951$	−536.112	−	715.977i	−0.563735	−	0.752868i
$952$		0			0
$953$			117.158i			0.122936i	0.998109	+	0.0614678i	$0.0195781\pi$
$953$			117.158i			0.122936i	−0.998109	+	0.0614678i	$0.980422\pi$
$954$		0			0
$955$	371.189			0.388679
$956$		0			0
$957$	−826.381	+	618.780i	−0.863512	+	0.646583i
$958$		0			0
$959$		−	1420.52i		−	1.48125i
$960$		0			0
$961$	520.311			0.541427
$962$		0			0
$963$	−952.114	−	279.311i	−0.988696	−	0.290042i
$964$		0			0
$965$		−	535.332i		−	0.554748i
$966$		0			0
$967$	−811.294			−0.838981			−0.419490	−	0.907760i	$-0.637791\pi$
$967$	−811.294			−0.838981			−0.419490	+	0.907760i	$0.637791\pi$
$968$		0			0
$969$	−709.197	−	947.132i	−0.731885	−	0.977432i
$970$		0			0
$971$			1052.66i			1.08410i	0.840346	+	0.542050i	$0.182352\pi$
$971$			1052.66i			1.08410i	−0.840346	+	0.542050i	$0.817648\pi$
$972$		0			0
$973$	−1126.48			−1.15774
$974$		0			0
$975$	−163.081	+	122.113i	−0.167263	+	0.125244i
$976$		0			0
$977$			830.832i			0.850391i	0.905102	+	0.425195i	$0.139795\pi$
$977$			830.832i			0.850391i	−0.905102	+	0.425195i	$0.860205\pi$
$978$		0			0
$979$	521.971			0.533167
$980$		0			0
$981$	−124.848	+	425.582i	−0.127266	+	0.433825i
$982$		0			0
$983$			1056.30i			1.07457i	0.843402	+	0.537283i	$0.180549\pi$
$983$			1056.30i			1.07457i	−0.843402	+	0.537283i	$0.819451\pi$
$984$		0			0
$985$	−575.829			−0.584598
$986$		0			0
$987$	−791.436	−	1056.96i	−0.801861	−	1.07088i
$988$		0			0
$989$		−	967.768i		−	0.978532i
$990$		0			0
$991$	1592.65			1.60712			0.803559	−	0.595225i	$-0.202937\pi$
$991$	1592.65			1.60712			0.803559	+	0.595225i	$0.202937\pi$
$992$		0			0
$993$	−954.795	+	714.934i	−0.961525	+	0.719974i
$994$		0			0
$995$			175.472i			0.176354i
$996$		0			0
$997$	978.023			0.980966			0.490483	−	0.871451i	$-0.336820\pi$
$997$	978.023			0.980966			0.490483	+	0.871451i	$0.336820\pi$
$998$		0			0
$999$	111.815	+	299.450i	0.111927	+	0.299749i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.3.l.a.41.4	yes	8
3.2	odd	2	inner	120.3.l.a.41.3	✓	8
4.3	odd	2		240.3.l.d.161.5		8
5.2	odd	4		600.3.c.d.449.14		16
5.3	odd	4		600.3.c.d.449.3		16
5.4	even	2		600.3.l.f.401.5		8
8.3	odd	2		960.3.l.g.641.4		8
8.5	even	2		960.3.l.h.641.5		8
12.11	even	2		240.3.l.d.161.6		8
15.2	even	4		600.3.c.d.449.4		16
15.8	even	4		600.3.c.d.449.13		16
15.14	odd	2		600.3.l.f.401.6		8
20.3	even	4		1200.3.c.m.449.14		16
20.7	even	4		1200.3.c.m.449.3		16
20.19	odd	2		1200.3.l.x.401.4		8
24.5	odd	2		960.3.l.h.641.6		8
24.11	even	2		960.3.l.g.641.3		8
60.23	odd	4		1200.3.c.m.449.4		16
60.47	odd	4		1200.3.c.m.449.13		16
60.59	even	2		1200.3.l.x.401.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.3	✓	8	3.2	odd	2	inner
120.3.l.a.41.4	yes	8	1.1	even	1	trivial
240.3.l.d.161.5		8	4.3	odd	2
240.3.l.d.161.6		8	12.11	even	2
600.3.c.d.449.3		16	5.3	odd	4
600.3.c.d.449.4		16	15.2	even	4
600.3.c.d.449.13		16	15.8	even	4
600.3.c.d.449.14		16	5.2	odd	4
600.3.l.f.401.5		8	5.4	even	2
600.3.l.f.401.6		8	15.14	odd	2
960.3.l.g.641.3		8	24.11	even	2
960.3.l.g.641.4		8	8.3	odd	2
960.3.l.h.641.5		8	8.5	even	2
960.3.l.h.641.6		8	24.5	odd	2
1200.3.c.m.449.3		16	20.7	even	4
1200.3.c.m.449.4		16	60.23	odd	4
1200.3.c.m.449.13		16	60.47	odd	4
1200.3.c.m.449.14		16	20.3	even	4
1200.3.l.x.401.3		8	60.59	even	2
1200.3.l.x.401.4		8	20.19	odd	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:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	120.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.40140 + 1.79813i) q^{3} -2.23607i q^{5} -10.2132 q^{7} +(2.53346 - 8.63606i) q^{9} +O(q^{10})\)	\(q+(-2.40140 + 1.79813i) q^{3} -2.23607i q^{5} -10.2132 q^{7} +(2.53346 - 8.63606i) q^{9} -8.19300i q^{11} -13.5822 q^{13} +(4.02074 + 5.36970i) q^{15} -15.4710i q^{17} -25.4934 q^{19} +(24.5261 - 18.3647i) q^{21} +17.9156i q^{23} -5.00000 q^{25} +(9.44491 + 25.2941i) q^{27} +42.0022i q^{29} +38.4878 q^{31} +(14.7321 + 19.6747i) q^{33} +22.8375i q^{35} +11.8387 q^{37} +(32.6163 - 24.4225i) q^{39} -46.3781i q^{41} -54.0181 q^{43} +(-19.3108 - 5.66499i) q^{45} -43.0955i q^{47} +55.3102 q^{49} +(27.8188 + 37.1521i) q^{51} +82.7421i q^{53} -18.3201 q^{55} +(61.2199 - 45.8404i) q^{57} -45.8928i q^{59} -93.6873 q^{61} +(-25.8748 + 88.2022i) q^{63} +30.3707i q^{65} +34.4995 q^{67} +(-32.2147 - 43.0227i) q^{69} -68.0061i q^{71} -44.7191 q^{73} +(12.0070 - 8.99065i) q^{75} +83.6770i q^{77} -11.7499 q^{79} +(-68.1632 - 43.7582i) q^{81} -144.436i q^{83} -34.5942 q^{85} +(-75.5255 - 100.864i) q^{87} +63.7094i q^{89} +138.718 q^{91} +(-92.4247 + 69.2061i) q^{93} +57.0050i q^{95} +63.9013 q^{97} +(-70.7552 - 20.7566i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 + 16 * q^7 + 20 * q^9	\( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} - 8 q^{13} - 8 q^{19} + 28 q^{21} - 40 q^{25} + 20 q^{27} + 120 q^{31} - 112 q^{33} + 8 q^{37} - 72 q^{39} - 328 q^{43} - 60 q^{45} + 64 q^{49} + 64 q^{51} - 40 q^{55} + 72 q^{57} + 8 q^{61} + 88 q^{63} + 152 q^{67} + 100 q^{69} + 32 q^{73} + 20 q^{75} + 88 q^{79} + 224 q^{81} - 152 q^{87} + 560 q^{91} - 368 q^{93} + 144 q^{97} + 32 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 + 16 * q^7 + 20 * q^9 - 8 * q^13 - 8 * q^19 + 28 * q^21 - 40 * q^25 + 20 * q^27 + 120 * q^31 - 112 * q^33 + 8 * q^37 - 72 * q^39 - 328 * q^43 - 60 * q^45 + 64 * q^49 + 64 * q^51 - 40 * q^55 + 72 * q^57 + 8 * q^61 + 88 * q^63 + 152 * q^67 + 100 * q^69 + 32 * q^73 + 20 * q^75 + 88 * q^79 + 224 * q^81 - 152 * q^87 + 560 * q^91 - 368 * q^93 + 144 * q^97 + 32 * q^99

Introduction

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