LMFDB - Embedded newform 1200.3.l.x.401.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(401,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.l (of order $2$, degree $1$, not 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:	$32.6976317232$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.4
Root			$-0.542939 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	1200.401
Dual form			1200.3.l.x.401.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.40140 + 1.79813i) q^{3} -10.2132 q^{7} +(2.53346 - 8.63606i) q^{9} +O(q^{10})$	$q+(-2.40140 + 1.79813i) q^{3} -10.2132 q^{7} +(2.53346 - 8.63606i) q^{9} +8.19300i q^{11} +13.5822 q^{13} +15.4710i q^{17} +25.4934 q^{19} +(24.5261 - 18.3647i) q^{21} +17.9156i q^{23} +(9.44491 + 25.2941i) q^{27} +42.0022i q^{29} -38.4878 q^{31} +(-14.7321 - 19.6747i) q^{33} -11.8387 q^{37} +(-32.6163 + 24.4225i) q^{39} -46.3781i q^{41} -54.0181 q^{43} -43.0955i q^{47} +55.3102 q^{49} +(-27.8188 - 37.1521i) q^{51} -82.7421i q^{53} +(-61.2199 + 45.8404i) q^{57} +45.8928i q^{59} -93.6873 q^{61} +(-25.8748 + 88.2022i) q^{63} +34.4995 q^{67} +(-32.2147 - 43.0227i) q^{69} +68.0061i q^{71} +44.7191 q^{73} -83.6770i q^{77} +11.7499 q^{79} +(-68.1632 - 43.7582i) q^{81} -144.436i q^{83} +(-75.5255 - 100.864i) q^{87} +63.7094i q^{89} -138.718 q^{91} +(92.4247 - 69.2061i) q^{93} -63.9013 q^{97} +(70.7552 + 20.7566i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 20 q^{27} - 120 q^{31} + 112 q^{33} - 8 q^{37} + 72 q^{39} - 328 q^{43} + 64 q^{49} - 64 q^{51} - 72 q^{57} + 8 q^{61} + 88 q^{63} + 152 q^{67} + 100 q^{69} - 32 q^{73} - 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 + 20 * q^27 - 120 * q^31 + 112 * q^33 - 8 * q^37 + 72 * q^39 - 328 * q^43 + 64 * q^49 - 64 * q^51 - 72 * q^57 + 8 * q^61 + 88 * q^63 + 152 * q^67 + 100 * q^69 - 32 * q^73 - 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 $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$		0			0
$5$		0			0
$6$		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	$0.121468\pi$
$11$			8.19300i			0.744818i	−0.928069	+	0.372409i	$0.878532\pi$
$12$		0			0
$13$	13.5822			1.04478			0.522392	−	0.852705i	$-0.325040\pi$
$13$	13.5822			1.04478			0.522392	+	0.852705i	$0.325040\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			15.4710i			0.910058i	0.890477	+	0.455029i	$0.150371\pi$
$17$			15.4710i			0.910058i	−0.890477	+	0.455029i	$0.849629\pi$
$18$		0			0
$19$	25.4934			1.34176			0.670879	−	0.741567i	$-0.265917\pi$
$19$	25.4934			1.34176			0.670879	+	0.741567i	$0.265917\pi$
$20$		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
$25$		0			0
$26$		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	$-0.713180\pi$
$31$	−38.4878			−1.24154			−0.620771	+	0.783992i	$0.713180\pi$
$32$		0			0
$33$	−14.7321	−	19.6747i	−0.446426	−	0.596202i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−11.8387			−0.319965			−0.159982	−	0.987120i	$-0.551144\pi$
$37$	−11.8387			−0.319965			−0.159982	+	0.987120i	$0.551144\pi$
$38$		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
$45$		0			0
$46$		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	$-0.714921\pi$
$53$		−	82.7421i		−	1.56117i	0.625049	−	0.780586i	$-0.285079\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−61.2199	+	45.8404i	−1.07403	+	0.804218i
$58$		0			0
$59$			45.8928i			0.777844i	0.921271	+	0.388922i	$0.127152\pi$
$59$			45.8928i			0.777844i	−0.921271	+	0.388922i	$0.872848\pi$
$60$		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
$65$		0			0
$66$		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	$0.158970\pi$
$71$			68.0061i			0.957832i	−0.877861	+	0.478916i	$0.841030\pi$
$72$		0			0
$73$	44.7191			0.612591			0.306295	−	0.951937i	$-0.400911\pi$
$73$	44.7191			0.612591			0.306295	+	0.951937i	$0.400911\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	83.6770i		−	1.08671i
$78$		0			0
$79$	11.7499			0.148733			0.0743665	−	0.997231i	$-0.476307\pi$
$79$	11.7499			0.148733			0.0743665	+	0.997231i	$0.476307\pi$
$80$		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
$85$		0			0
$86$		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
$95$		0			0
$96$		0			0
$97$	−63.9013			−0.658776			−0.329388	−	0.944195i	$-0.606842\pi$
$97$	−63.9013			−0.658776			−0.329388	+	0.944195i	$0.606842\pi$
$98$		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
$105$		0			0
$106$		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	$0.245342\pi$
$113$			157.450i			1.39337i	−0.717379	+	0.696683i	$0.754658\pi$
$114$		0			0
$115$		0			0
$116$		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
$125$		0			0
$126$		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	$-0.804998\pi$
$131$		−	150.653i		−	1.15002i	0.818146	−	0.575010i	$-0.195002\pi$
$132$		0			0
$133$	−260.370			−1.95767
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	139.086i		−	1.01523i	−0.861585	−	0.507613i	$-0.830528\pi$
$137$		−	139.086i		−	1.01523i	0.861585	−	0.507613i	$-0.169472\pi$
$138$		0			0
$139$	−110.296			−0.793496			−0.396748	−	0.917928i	$-0.629861\pi$
$139$	−110.296			−0.793496			−0.396748	+	0.917928i	$0.629861\pi$
$140$		0			0
$141$	77.4912	+	103.490i	0.549583	+	0.733968i
$142$		0			0
$143$			111.279i			0.778174i
$144$		0			0
$145$		0			0
$146$		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	$-0.620127\pi$
$151$	−111.286			−0.736992			−0.368496	+	0.929629i	$0.620127\pi$
$152$		0			0
$153$	133.608	+	39.1951i	0.873258	+	0.256177i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−142.755			−0.909269			−0.454635	−	0.890678i	$-0.650230\pi$
$157$	−142.755			−0.909269			−0.454635	+	0.890678i	$0.650230\pi$
$158$		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
$165$		0			0
$166$		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	$0.241962\pi$
$173$			238.403i			1.37805i	−0.724736	+	0.689027i	$0.758038\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−82.5212	−	110.207i	−0.466221	−	0.622638i
$178$		0			0
$179$		−	243.541i		−	1.36057i	−0.732950	−	0.680283i	$-0.761857\pi$
$179$		−	243.541i		−	1.36057i	0.732950	−	0.680283i	$-0.238143\pi$
$180$		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
$185$		0			0
$186$		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	$-0.856905\pi$
$191$		−	166.001i		−	0.869113i	0.900645	−	0.434556i	$-0.143095\pi$
$192$		0			0
$193$	−239.408			−1.24045			−0.620227	−	0.784422i	$-0.712959\pi$
$193$	−239.408			−1.24045			−0.620227	+	0.784422i	$0.712959\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			257.518i			1.30720i	0.756840	+	0.653600i	$0.226742\pi$
$197$			257.518i			1.30720i	−0.756840	+	0.653600i	$0.773258\pi$
$198$		0			0
$199$	78.4735			0.394339			0.197170	−	0.980369i	$-0.436825\pi$
$199$	78.4735			0.394339			0.197170	+	0.980369i	$0.436825\pi$
$200$		0			0
$201$	−82.8471	+	62.0345i	−0.412174	+	0.308629i
$202$		0			0
$203$		−	428.979i		−	2.11320i
$204$		0			0
$205$		0			0
$206$		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	$-0.413928\pi$
$211$	112.724			0.534237			0.267119	+	0.963664i	$0.413928\pi$
$212$		0			0
$213$	−122.284	−	163.310i	−0.574102	−	0.766713i
$214$		0			0
$215$		0			0
$216$		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
$225$		0			0
$226$		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	$0.180368\pi$
$233$			250.150i			1.07361i	−0.843708	+	0.536803i	$0.819632\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−28.2162	+	21.1278i	−0.119056	+	0.0891470i
$238$		0			0
$239$		−	49.2556i		−	0.206090i	−0.994677	−	0.103045i	$-0.967141\pi$
$239$		−	49.2556i		−	0.206090i	0.994677	−	0.103045i	$-0.0328586\pi$
$240$		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
$245$		0			0
$246$		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	$0.0766489\pi$
$251$			119.717i			0.476959i	−0.971148	+	0.238479i	$0.923351\pi$
$252$		0			0
$253$	−146.783			−0.580169
$254$		0			0
$255$		0			0
$256$		0			0
$257$			111.805i			0.435041i	0.976056	+	0.217520i	$0.0697969\pi$
$257$			111.805i			0.435041i	−0.976056	+	0.217520i	$0.930203\pi$
$258$		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
$265$		0			0
$266$		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	$-0.383442\pi$
$271$	194.062			0.716096			0.358048	+	0.933703i	$0.383442\pi$
$272$		0			0
$273$	333.118	−	249.433i	1.22021	−	0.913675i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.20416			−0.0187876			−0.00939379	−	0.999956i	$-0.502990\pi$
$277$	−5.20416			−0.0187876			−0.00939379	+	0.999956i	$0.502990\pi$
$278$		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
$285$		0			0
$286$		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	$-0.906735\pi$
$293$		−	169.252i		−	0.577653i	0.957381	−	0.288827i	$-0.0932651\pi$
$294$		0			0
$295$		0			0
$296$		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
$305$		0			0
$306$		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	$0.0689050\pi$
$311$			133.596i			0.429570i	−0.976661	+	0.214785i	$0.931095\pi$
$312$		0			0
$313$	459.981			1.46959			0.734794	−	0.678290i	$-0.237279\pi$
$313$	459.981			1.46959			0.734794	+	0.678290i	$0.237279\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	298.150i		−	0.940536i	−0.882524	−	0.470268i	$-0.844157\pi$
$317$		−	298.150i		−	0.940536i	0.882524	−	0.470268i	$-0.155843\pi$
$318$		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
$325$		0			0
$326$		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	$-0.705073\pi$
$331$	−397.599			−1.20120			−0.600602	+	0.799548i	$0.705073\pi$
$332$		0			0
$333$	−29.9929	+	102.240i	−0.0900687	+	0.307026i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	427.942			1.26986			0.634929	−	0.772571i	$-0.281030\pi$
$337$	427.942			1.26986			0.634929	+	0.772571i	$0.281030\pi$
$338$		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
$345$		0			0
$346$		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	$-0.935251\pi$
$353$		−	142.622i		−	0.404028i	0.979383	−	0.202014i	$-0.0647487\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	284.120	+	379.443i	0.795856	+	1.06286i
$358$		0			0
$359$		−	3.05663i		−	0.00851429i	−0.999991	−	0.00425715i	$-0.998645\pi$
$359$		−	3.05663i		−	0.00851429i	0.999991	−	0.00425715i	$-0.00135510\pi$
$360$		0			0
$361$	288.913			0.800313
$362$		0			0
$363$	−129.375	+	96.8739i	−0.356405	+	0.266870i
$364$		0			0
$365$		0			0
$366$		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	$-0.709098\pi$
$373$	−455.556			−1.22133			−0.610665	+	0.791889i	$0.709098\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			570.482i			1.51322i
$378$		0			0
$379$	380.051			1.00277			0.501387	−	0.865223i	$-0.332823\pi$
$379$	380.051			1.00277			0.501387	+	0.865223i	$0.332823\pi$
$380$		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
$385$		0			0
$386$		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
$395$		0			0
$396$		0			0
$397$	93.4668			0.235433			0.117716	−	0.993047i	$-0.462443\pi$
$397$	93.4668			0.235433			0.117716	+	0.993047i	$0.462443\pi$
$398$		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
$405$		0			0
$406$		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
$415$		0			0
$416$		0			0
$417$	264.865	−	198.326i	0.635167	−	0.475603i
$418$		0			0
$419$			115.537i			0.275744i	0.990450	+	0.137872i	$0.0440262\pi$
$419$			115.537i			0.275744i	−0.990450	+	0.137872i	$0.955974\pi$
$420$		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
$425$		0			0
$426$		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	$-0.714768\pi$
$431$		−	673.123i		−	1.56177i	0.624675	−	0.780885i	$-0.285232\pi$
$432$		0			0
$433$	392.655			0.906824			0.453412	−	0.891301i	$-0.350207\pi$
$433$	392.655			0.906824			0.453412	+	0.891301i	$0.350207\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			456.731i			1.04515i
$438$		0			0
$439$	−409.353			−0.932468			−0.466234	−	0.884661i	$-0.654389\pi$
$439$	−409.353			−0.932468			−0.466234	+	0.884661i	$0.654389\pi$
$440$		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
$445$		0			0
$446$		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
$455$		0			0
$456$		0			0
$457$	−491.814			−1.07618			−0.538090	−	0.842887i	$-0.680854\pi$
$457$	−491.814			−1.07618			−0.538090	+	0.842887i	$0.680854\pi$
$458$		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
$465$		0			0
$466$		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
$475$		0			0
$476$		0			0
$477$	−714.566	−	209.624i	−1.49804	−	0.439463i
$478$		0			0
$479$		−	485.253i		−	1.01305i	−0.862224	−	0.506527i	$-0.830929\pi$
$479$		−	485.253i		−	1.01305i	0.862224	−	0.506527i	$-0.169071\pi$
$480$		0			0
$481$	−160.795			−0.334294
$482$		0			0
$483$	329.016	+	439.401i	0.681192	+	0.909732i
$484$		0			0
$485$		0			0
$486$		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	$0.00662763\pi$
$491$			20.4450i			0.0416396i	−0.999783	+	0.0208198i	$0.993372\pi$
$492$		0			0
$493$	−649.816			−1.31809
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	694.562i		−	1.39751i
$498$		0			0
$499$	−49.2051			−0.0986074			−0.0493037	−	0.998784i	$-0.515700\pi$
$499$	−49.2051			−0.0986074			−0.0493037	+	0.998784i	$0.515700\pi$
$500$		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
$505$		0			0
$506$		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
$515$		0			0
$516$		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
$525$		0			0
$526$		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
$535$		0			0
$536$		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
$545$		0			0
$546$		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
$555$		0			0
$556$		0			0
$557$		−	654.467i		−	1.17499i	−0.809230	−	0.587493i	$-0.800115\pi$
$557$		−	654.467i		−	1.17499i	0.809230	−	0.587493i	$-0.199885\pi$
$558$		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
$565$		0			0
$566$		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	$-0.435349\pi$
$571$	230.358			0.403430			0.201715	+	0.979444i	$0.435349\pi$
$572$		0			0
$573$	298.491	+	398.634i	0.520926	+	0.695696i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−525.585			−0.910893			−0.455446	−	0.890263i	$-0.650520\pi$
$577$	−525.585			−0.910893			−0.455446	+	0.890263i	$0.650520\pi$
$578$		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
$585$		0			0
$586$		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	$0.413892\pi$
$593$			1142.87i			1.92727i	−0.267230	+	0.963633i	$0.586108\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−188.446	+	141.106i	−0.315656	+	0.236358i
$598$		0			0
$599$			1060.02i			1.76965i	0.465927	+	0.884823i	$0.345721\pi$
$599$			1060.02i			1.76965i	−0.465927	+	0.884823i	$0.654279\pi$
$600$		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
$605$		0			0
$606$		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	$-0.443533\pi$
$613$	216.348			0.352932			0.176466	+	0.984307i	$0.443533\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			231.190i			0.374701i	0.982293	+	0.187350i	$0.0599900\pi$
$617$			231.190i			0.374701i	−0.982293	+	0.187350i	$0.940010\pi$
$618$		0			0
$619$	689.633			1.11411			0.557054	−	0.830476i	$-0.311932\pi$
$619$	689.633			1.11411			0.557054	+	0.830476i	$0.311932\pi$
$620$		0			0
$621$	−453.161	+	169.212i	−0.729728	+	0.272482i
$622$		0			0
$623$		−	650.679i		−	1.04443i
$624$		0			0
$625$		0			0
$626$		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	$-0.337404\pi$
$631$	616.973			0.977770			0.488885	+	0.872348i	$0.337404\pi$
$632$		0			0
$633$	−270.696	+	202.692i	−0.427639	+	0.320209i
$634$		0			0
$635$		0			0
$636$		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
$645$		0			0
$646$		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	$-0.922916\pi$
$653$		−	313.185i		−	0.479610i	0.970821	−	0.239805i	$-0.0770835\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	113.294	−	386.197i	0.172442	−	0.587819i
$658$		0			0
$659$			878.311i			1.33279i	0.745597	+	0.666397i	$0.232164\pi$
$659$			878.311i			1.33279i	−0.745597	+	0.666397i	$0.767836\pi$
$660$		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
$665$		0			0
$666$		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	$-0.485602\pi$
$673$	60.8622			0.0904341			0.0452171	+	0.998977i	$0.485602\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	415.595i		−	0.613877i	−0.951729	−	0.306939i	$-0.900695\pi$
$677$		−	415.595i		−	0.613877i	0.951729	−	0.306939i	$-0.0993047\pi$
$678$		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
$685$		0			0
$686$		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	$-0.461527\pi$
$691$	166.630			0.241144			0.120572	+	0.992705i	$0.461527\pi$
$692$		0			0
$693$	−722.640	−	211.992i	−1.04277	−	0.305905i
$694$		0			0
$695$		0			0
$696$		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
$705$		0			0
$706$		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
$715$		0			0
$716$		0			0
$717$	88.5679	+	118.282i	0.123526	+	0.164969i
$718$		0			0
$719$			323.584i			0.450048i	0.974353	+	0.225024i	$0.0722460\pi$
$719$			323.584i			0.450048i	−0.974353	+	0.225024i	$0.927754\pi$
$720$		0			0
$721$	−469.289			−0.650886
$722$		0			0
$723$	1099.05	−	822.953i	1.52013	−	1.13825i
$724$		0			0
$725$		0			0
$726$		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	$-0.528610\pi$
$733$	−131.588			−0.179520			−0.0897600	+	0.995963i	$0.528610\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			282.654i			0.383520i
$738$		0			0
$739$	−1033.59			−1.39864			−0.699319	−	0.714810i	$-0.746513\pi$
$739$	−1033.59			−1.39864			−0.699319	+	0.714810i	$0.746513\pi$
$740$		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
$745$		0			0
$746$		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	$-0.607202\pi$
$751$	−496.343			−0.660909			−0.330455	+	0.943822i	$0.607202\pi$
$752$		0			0
$753$	−215.266	−	287.488i	−0.285878	−	0.381790i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−468.694			−0.619147			−0.309574	−	0.950875i	$-0.600186\pi$
$757$	−468.694			−0.619147			−0.309574	+	0.950875i	$0.600186\pi$
$758$		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
$765$		0			0
$766$		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	$0.210714\pi$
$773$			950.291i			1.22935i	−0.788779	+	0.614677i	$0.789286\pi$
$774$		0			0
$775$		0			0
$776$		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
$785$		0			0
$786$		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
$795$		0			0
$796$		0			0
$797$		−	78.7126i		−	0.0987612i	−0.998780	−	0.0493806i	$-0.984275\pi$
$797$		−	78.7126i		−	0.0987612i	0.998780	−	0.0493806i	$-0.0157247\pi$
$798$		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
$805$		0			0
$806$		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	$-0.738270\pi$
$811$	−1103.89			−1.36115			−0.680574	+	0.732679i	$0.738270\pi$
$812$		0			0
$813$	−466.021	+	348.949i	−0.573211	+	0.429211i
$814$		0			0
$815$		0			0
$816$		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
$825$		0			0
$826$		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
$835$		0			0
$836$		0			0
$837$	−363.514	−	973.516i	−0.434306	−	1.16310i
$838$		0			0
$839$			48.5299i			0.0578425i	0.999582	+	0.0289213i	$0.00920721\pi$
$839$			48.5299i			0.0578425i	−0.999582	+	0.0289213i	$0.990793\pi$
$840$		0			0
$841$	−923.188			−1.09773
$842$		0			0
$843$	−431.558	−	576.345i	−0.511931	−	0.683683i
$844$		0			0
$845$		0			0
$846$		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	$-0.437813\pi$
$853$	331.180			0.388253			0.194127	+	0.980976i	$0.437813\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			159.395i			0.185992i	0.995666	+	0.0929961i	$0.0296444\pi$
$857$			159.395i			0.185992i	−0.995666	+	0.0929961i	$0.970356\pi$
$858$		0			0
$859$	1635.01			1.90339			0.951694	−	0.307049i	$-0.0993416\pi$
$859$	1635.01			1.90339			0.951694	+	0.307049i	$0.0993416\pi$
$860$		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
$865$		0			0
$866$		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
$875$		0			0
$876$		0			0
$877$	−521.588			−0.594742			−0.297371	−	0.954762i	$-0.596110\pi$
$877$	−521.588			−0.594742			−0.297371	+	0.954762i	$0.596110\pi$
$878$		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
$885$		0			0
$886$		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
$895$		0			0
$896$		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
$905$		0			0
$906$		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	$0.296399\pi$
$911$			1461.82i			1.60463i	−0.596900	+	0.802316i	$0.703601\pi$
$912$		0			0
$913$	1183.37			1.29613
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1538.65i			1.67792i
$918$		0			0
$919$	−600.104			−0.652997			−0.326499	−	0.945198i	$-0.605869\pi$
$919$	−600.104			−0.652997			−0.326499	+	0.945198i	$0.605869\pi$
$920$		0			0
$921$	−793.427	+	594.105i	−0.861484	+	0.645065i
$922$		0			0
$923$			923.672i			1.00073i
$924$		0			0
$925$		0			0
$926$		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
$935$		0			0
$936$		0			0
$937$	−894.637			−0.954789			−0.477395	−	0.878689i	$-0.658419\pi$
$937$	−894.637			−0.954789			−0.477395	+	0.878689i	$0.658419\pi$
$938$		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
$945$		0			0
$946$		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	$-0.980422\pi$
$953$		−	117.158i		−	0.122936i	0.998109	−	0.0614678i	$-0.0195781\pi$
$954$		0			0
$955$		0			0
$956$		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
$965$		0			0
$966$		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	$-0.817648\pi$
$971$		−	1052.66i		−	1.08410i	0.840346	−	0.542050i	$-0.182352\pi$
$972$		0			0
$973$	1126.48			1.15774
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	830.832i		−	0.850391i	−0.905102	−	0.425195i	$-0.860205\pi$
$977$		−	830.832i		−	0.850391i	0.905102	−	0.425195i	$-0.139795\pi$
$978$		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
$985$		0			0
$986$		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	$-0.797063\pi$
$991$	−1592.65			−1.60712			−0.803559	+	0.595225i	$0.797063\pi$
$992$		0			0
$993$	954.795	−	714.934i	0.961525	−	0.719974i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−978.023			−0.980966			−0.490483	−	0.871451i	$-0.663180\pi$
$997$	−978.023			−0.980966			−0.490483	+	0.871451i	$0.663180\pi$
$998$		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	1200.3.l.x.401.4		8
3.2	odd	2	inner	1200.3.l.x.401.3		8
4.3	odd	2		600.3.l.f.401.5		8
5.2	odd	4		1200.3.c.m.449.14		16
5.3	odd	4		1200.3.c.m.449.3		16
5.4	even	2		240.3.l.d.161.5		8
12.11	even	2		600.3.l.f.401.6		8
15.2	even	4		1200.3.c.m.449.4		16
15.8	even	4		1200.3.c.m.449.13		16
15.14	odd	2		240.3.l.d.161.6		8
20.3	even	4		600.3.c.d.449.14		16
20.7	even	4		600.3.c.d.449.3		16
20.19	odd	2		120.3.l.a.41.4	yes	8
40.19	odd	2		960.3.l.h.641.5		8
40.29	even	2		960.3.l.g.641.4		8
60.23	odd	4		600.3.c.d.449.4		16
60.47	odd	4		600.3.c.d.449.13		16
60.59	even	2		120.3.l.a.41.3	✓	8
120.29	odd	2		960.3.l.g.641.3		8
120.59	even	2		960.3.l.h.641.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.3	✓	8	60.59	even	2
120.3.l.a.41.4	yes	8	20.19	odd	2
240.3.l.d.161.5		8	5.4	even	2
240.3.l.d.161.6		8	15.14	odd	2
600.3.c.d.449.3		16	20.7	even	4
600.3.c.d.449.4		16	60.23	odd	4
600.3.c.d.449.13		16	60.47	odd	4
600.3.c.d.449.14		16	20.3	even	4
600.3.l.f.401.5		8	4.3	odd	2
600.3.l.f.401.6		8	12.11	even	2
960.3.l.g.641.3		8	120.29	odd	2
960.3.l.g.641.4		8	40.29	even	2
960.3.l.h.641.5		8	40.19	odd	2
960.3.l.h.641.6		8	120.59	even	2
1200.3.c.m.449.3		16	5.3	odd	4
1200.3.c.m.449.4		16	15.2	even	4
1200.3.c.m.449.13		16	15.8	even	4
1200.3.c.m.449.14		16	5.2	odd	4
1200.3.l.x.401.3		8	3.2	odd	2	inner
1200.3.l.x.401.4		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.l (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

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