LMFDB - Embedded newform 600.3.c.d.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,3,Mod(449,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("600.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.c (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:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 138x^{12} + 3393x^{8} + 15208x^{4} + 1296 $ x^16 + 138x^12 + 3393x^8 + 15208*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-1.09102 - 1.09102i$ of defining polynomial
Character	$\chi$	$=$	600.449
Dual form			600.3.c.d.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.79813 + 2.40140i) q^{3} -10.2132i q^{7} +(-2.53346 - 8.63606i) q^{9} +O(q^{10})$	$q+(-1.79813 + 2.40140i) q^{3} -10.2132i q^{7} +(-2.53346 - 8.63606i) q^{9} +8.19300i q^{11} +13.5822i q^{13} -15.4710 q^{17} +25.4934 q^{19} +(24.5261 + 18.3647i) q^{21} -17.9156 q^{23} +(25.2941 + 9.44491i) q^{27} +42.0022i q^{29} +38.4878 q^{31} +(-19.6747 - 14.7321i) q^{33} +11.8387i q^{37} +(-32.6163 - 24.4225i) q^{39} +46.3781i q^{41} +54.0181i q^{43} -43.0955 q^{47} -55.3102 q^{49} +(27.8188 - 37.1521i) q^{51} -82.7421 q^{53} +(-45.8404 + 61.2199i) q^{57} -45.8928i q^{59} -93.6873 q^{61} +(-88.2022 + 25.8748i) q^{63} +34.4995i q^{67} +(32.2147 - 43.0227i) q^{69} +68.0061i q^{71} +44.7191i q^{73} +83.6770 q^{77} +11.7499 q^{79} +(-68.1632 + 43.7582i) q^{81} +144.436 q^{83} +(-100.864 - 75.5255i) q^{87} +63.7094i q^{89} +138.718 q^{91} +(-69.2061 + 92.4247i) q^{93} +63.9013i q^{97} +(70.7552 - 20.7566i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 40 q^{9}+O(q^{10}) $ 16 * q - 40 * q^9	$ 16 q - 40 q^{9} + 16 q^{19} + 56 q^{21} + 240 q^{31} + 144 q^{39} - 128 q^{49} + 128 q^{51} + 16 q^{61} - 200 q^{69} - 176 q^{79} + 448 q^{81} + 1120 q^{91} - 64 q^{99}+O(q^{100}) $ 16 * q - 40 * q^9 + 16 * q^19 + 56 * q^21 + 240 * q^31 + 144 * q^39 - 128 * q^49 + 128 * q^51 + 16 * q^61 - 200 * q^69 - 176 * q^79 + 448 * q^81 + 1120 * q^91 - 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−1.79813	+	2.40140i	−0.599377	+	0.800467i
$3$	−1.79813	+	2.40140i	−0.599377	+	0.800467i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	10.2132i		−	1.45903i	−0.683963	−	0.729517i	$-0.739745\pi$
$7$		−	10.2132i		−	1.45903i	0.683963	−	0.729517i	$-0.260255\pi$
$8$		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
$13$			13.5822i			1.04478i	0.852705	+	0.522392i	$0.174960\pi$
$13$			13.5822i			1.04478i	−0.852705	+	0.522392i	$0.825040\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.4710			−0.910058			−0.455029	−	0.890477i	$-0.650371\pi$
$17$	−15.4710			−0.910058			−0.455029	+	0.890477i	$0.650371\pi$
$18$		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
$23$	−17.9156			−0.778941			−0.389471	−	0.921039i	$-0.627342\pi$
$23$	−17.9156			−0.778941			−0.389471	+	0.921039i	$0.627342\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	25.2941	+	9.44491i	0.936820	+	0.349811i
$28$		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
$33$	−19.6747	−	14.7321i	−0.596202	−	0.446426i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			11.8387i			0.319965i	0.987120	+	0.159982i	$0.0511437\pi$
$37$			11.8387i			0.319965i	−0.987120	+	0.159982i	$0.948856\pi$
$38$		0			0
$39$	−32.6163	−	24.4225i	−0.836315	−	0.626219i
$40$		0			0
$41$			46.3781i			1.13117i	0.824689	+	0.565587i	$0.191350\pi$
$41$			46.3781i			1.13117i	−0.824689	+	0.565587i	$0.808650\pi$
$42$		0			0
$43$			54.0181i			1.25623i	0.778119	+	0.628117i	$0.216174\pi$
$43$			54.0181i			1.25623i	−0.778119	+	0.628117i	$0.783826\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−43.0955			−0.916925			−0.458462	−	0.888714i	$-0.651600\pi$
$47$	−43.0955			−0.916925			−0.458462	+	0.888714i	$0.651600\pi$
$48$		0			0
$49$	−55.3102			−1.12878
$50$		0			0
$51$	27.8188	−	37.1521i	0.545468	−	0.728472i
$52$		0			0
$53$	−82.7421			−1.56117			−0.780586	−	0.625049i	$-0.785079\pi$
$53$	−82.7421			−1.56117			−0.780586	+	0.625049i	$0.785079\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−45.8404	+	61.2199i	−0.804218	+	1.07403i
$58$		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
$63$	−88.2022	+	25.8748i	−1.40003	+	0.410712i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			34.4995i			0.514917i	0.966289	+	0.257459i	$0.0828851\pi$
$67$			34.4995i			0.514917i	−0.966289	+	0.257459i	$0.917115\pi$
$68$		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
$73$			44.7191i			0.612591i	0.951937	+	0.306295i	$0.0990895\pi$
$73$			44.7191i			0.612591i	−0.951937	+	0.306295i	$0.900911\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	83.6770			1.08671
$78$		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
$83$	144.436			1.74020			0.870099	−	0.492877i	$-0.164055\pi$
$83$	144.436			1.74020			0.870099	+	0.492877i	$0.164055\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−100.864	−	75.5255i	−1.15936	−	0.868109i
$88$		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
$93$	−69.2061	+	92.4247i	−0.744151	+	0.993814i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			63.9013i			0.658776i	0.944195	+	0.329388i	$0.106842\pi$
$97$			63.9013i			0.658776i	−0.944195	+	0.329388i	$0.893158\pi$
$98$		0			0
$99$	70.7552	−	20.7566i	0.714699	−	0.209663i
$100$		0			0
$101$			50.8769i			0.503731i	0.967762	+	0.251866i	$0.0810441\pi$
$101$			50.8769i			0.503731i	−0.967762	+	0.251866i	$0.918956\pi$
$102$		0			0
$103$		−	45.9491i		−	0.446108i	−0.974806	−	0.223054i	$-0.928397\pi$
$103$		−	45.9491i		−	0.446108i	0.974806	−	0.223054i	$-0.0716026\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−110.249			−1.03036			−0.515181	−	0.857082i	$-0.672275\pi$
$107$	−110.249			−1.03036			−0.515181	+	0.857082i	$0.672275\pi$
$108$		0			0
$109$	49.2797			0.452107			0.226054	−	0.974115i	$-0.427418\pi$
$109$	49.2797			0.452107			0.226054	+	0.974115i	$0.427418\pi$
$110$		0			0
$111$	−28.4295	−	21.2875i	−0.256121	−	0.191779i
$112$		0			0
$113$	157.450			1.39337			0.696683	−	0.717379i	$-0.254658\pi$
$113$	157.450			1.39337			0.696683	+	0.717379i	$0.254658\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	117.297	−	34.4099i	1.00254	−	0.294102i
$118$		0			0
$119$			158.009i			1.32781i
$120$		0			0
$121$	53.8748			0.445246
$122$		0			0
$123$	−111.373	−	83.3939i	−0.905468	−	0.677999i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	135.065i		−	1.06351i	−0.846900	−	0.531753i	$-0.821534\pi$
$127$		−	135.065i		−	1.06351i	0.846900	−	0.531753i	$-0.178466\pi$
$128$		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
$133$		−	260.370i		−	1.95767i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	139.086			1.01523			0.507613	−	0.861585i	$-0.330528\pi$
$137$	139.086			1.01523			0.507613	+	0.861585i	$0.330528\pi$
$138$		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
$143$	−111.279			−0.778174
$144$		0			0
$145$		0			0
$146$		0			0
$147$	99.4549	−	132.822i	0.676564	−	0.903551i
$148$		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
$153$	39.1951	+	133.608i	0.256177	+	0.873258i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			142.755i			0.909269i	0.890678	+	0.454635i	$0.150230\pi$
$157$			142.755i			0.909269i	−0.890678	+	0.454635i	$0.849770\pi$
$158$		0			0
$159$	148.781	−	198.697i	0.935730	−	1.24967i
$160$		0			0
$161$			182.977i			1.13650i
$162$		0			0
$163$			170.179i			1.04405i	0.852932	+	0.522023i	$0.174822\pi$
$163$			170.179i			1.04405i	−0.852932	+	0.522023i	$0.825178\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.38780			−0.0382503			−0.0191251	−	0.999817i	$-0.506088\pi$
$167$	−6.38780			−0.0382503			−0.0191251	+	0.999817i	$0.506088\pi$
$168$		0			0
$169$	−15.4759			−0.0915734
$170$		0			0
$171$	−64.5865	−	220.163i	−0.377699	−	1.28750i
$172$		0			0
$173$	238.403			1.37805			0.689027	−	0.724736i	$-0.258038\pi$
$173$	238.403			1.37805			0.689027	+	0.724736i	$0.258038\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	110.207	+	82.5212i	0.622638	+	0.466221i
$178$		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
$183$	168.462	−	224.981i	0.920557	−	1.22940i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	126.754i		−	0.677828i
$188$		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
$193$		−	239.408i		−	1.24045i	−0.784422	−	0.620227i	$-0.787041\pi$
$193$		−	239.408i		−	1.24045i	0.784422	−	0.620227i	$-0.212959\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−257.518			−1.30720			−0.653600	−	0.756840i	$-0.726742\pi$
$197$	−257.518			−1.30720			−0.653600	+	0.756840i	$0.726742\pi$
$198$		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
$203$	428.979			2.11320
$204$		0			0
$205$		0			0
$206$		0			0
$207$	45.3886	+	154.721i	0.219268	+	0.747443i
$208$		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
$213$	−163.310	−	122.284i	−0.766713	−	0.574102i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	393.085i		−	1.81145i
$218$		0			0
$219$	−107.389	−	80.4108i	−0.490359	−	0.367172i
$220$		0			0
$221$		−	210.130i		−	0.950814i
$222$		0			0
$223$			204.686i			0.917872i	0.888469	+	0.458936i	$0.151769\pi$
$223$			204.686i			0.917872i	−0.888469	+	0.458936i	$0.848231\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	115.071			0.506920			0.253460	−	0.967346i	$-0.418431\pi$
$227$	115.071			0.506920			0.253460	+	0.967346i	$0.418431\pi$
$228$		0			0
$229$	129.458			0.565317			0.282659	−	0.959221i	$-0.408784\pi$
$229$	129.458			0.565317			0.282659	+	0.959221i	$0.408784\pi$
$230$		0			0
$231$	−150.462	+	200.942i	−0.651351	+	0.869879i
$232$		0			0
$233$	250.150			1.07361			0.536803	−	0.843708i	$-0.319632\pi$
$233$	250.150			1.07361			0.536803	+	0.843708i	$0.319632\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−21.1278	+	28.2162i	−0.0891470	+	0.119056i
$238$		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
$243$	17.4851	−	242.370i	0.0719551	−	0.997408i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			346.256i			1.40185i
$248$		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
$253$		−	146.783i		−	0.580169i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−111.805			−0.435041			−0.217520	−	0.976056i	$-0.569797\pi$
$257$	−111.805			−0.435041			−0.217520	+	0.976056i	$0.569797\pi$
$258$		0			0
$259$	120.911			0.466839
$260$		0			0
$261$	362.734	−	106.411i	1.38979	−	0.407705i
$262$		0			0
$263$	55.3652			0.210514			0.105257	−	0.994445i	$-0.466433\pi$
$263$	55.3652			0.210514			0.105257	+	0.994445i	$0.466433\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−152.992	−	114.558i	−0.573003	−	0.429055i
$268$		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
$273$	−249.433	+	333.118i	−0.913675	+	1.22021i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			5.20416i			0.0187876i	0.999956	+	0.00939379i	$0.00299018\pi$
$277$			5.20416i			0.0187876i	−0.999956	+	0.00939379i	$0.997010\pi$
$278$		0			0
$279$	−97.5073	−	332.383i	−0.349489	−	1.19134i
$280$		0			0
$281$		−	240.004i		−	0.854105i	−0.904227	−	0.427053i	$-0.859552\pi$
$281$		−	240.004i		−	0.854105i	0.904227	−	0.427053i	$-0.140448\pi$
$282$		0			0
$283$		−	153.109i		−	0.541022i	−0.962717	−	0.270511i	$-0.912807\pi$
$283$		−	153.109i		−	0.541022i	0.962717	−	0.270511i	$-0.0871927\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	473.671			1.65042
$288$		0			0
$289$	−49.6485			−0.171794
$290$		0			0
$291$	−153.453	−	114.903i	−0.527329	−	0.394855i
$292$		0			0
$293$	−169.252			−0.577653			−0.288827	−	0.957381i	$-0.593265\pi$
$293$	−169.252			−0.577653			−0.288827	+	0.957381i	$0.593265\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−77.3821	+	207.235i	−0.260546	+	0.697760i
$298$		0			0
$299$		−	243.334i		−	0.813825i
$300$		0			0
$301$	551.699			1.83289
$302$		0			0
$303$	−122.176	−	91.4832i	−0.403220	−	0.301925i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			330.401i			1.07623i	0.842873	+	0.538113i	$0.180863\pi$
$307$			330.401i			1.07623i	−0.842873	+	0.538113i	$0.819137\pi$
$308$		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
$313$			459.981i			1.46959i	0.678290	+	0.734794i	$0.262721\pi$
$313$			459.981i			1.46959i	−0.678290	+	0.734794i	$0.737279\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	298.150			0.940536			0.470268	−	0.882524i	$-0.344157\pi$
$317$	298.150			0.940536			0.470268	+	0.882524i	$0.344157\pi$
$318$		0			0
$319$	−344.124			−1.07876
$320$		0			0
$321$	198.241	−	264.751i	0.617574	−	0.824771i
$322$		0			0
$323$	−394.408			−1.22108
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−88.6112	+	118.340i	−0.270982	+	0.361897i
$328$		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
$333$	102.240	−	29.9929i	0.307026	−	0.0900687i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	427.942i		−	1.26986i	−0.772571	−	0.634929i	$-0.781030\pi$
$337$		−	427.942i		−	1.26986i	0.772571	−	0.634929i	$-0.218970\pi$
$338$		0			0
$339$	−283.116	+	378.101i	−0.835151	+	1.11534i
$340$		0			0
$341$			315.331i			0.924723i
$342$		0			0
$343$			64.4477i			0.187894i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−138.137			−0.398090			−0.199045	−	0.979990i	$-0.563784\pi$
$347$	−138.137			−0.398090			−0.199045	+	0.979990i	$0.563784\pi$
$348$		0			0
$349$	301.421			0.863669			0.431835	−	0.901953i	$-0.357866\pi$
$349$	301.421			0.863669			0.431835	+	0.901953i	$0.357866\pi$
$350$		0			0
$351$	−128.283	+	343.550i	−0.365477	+	0.978775i
$352$		0			0
$353$	−142.622			−0.404028			−0.202014	−	0.979383i	$-0.564749\pi$
$353$	−142.622			−0.404028			−0.202014	+	0.979383i	$0.564749\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−379.443	−	284.120i	−1.06286	−	0.795856i
$358$		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
$363$	−96.8739	+	129.375i	−0.266870	+	0.356405i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	370.072i		−	1.00837i	−0.863595	−	0.504186i	$-0.831793\pi$
$367$		−	370.072i		−	1.00837i	0.863595	−	0.504186i	$-0.168207\pi$
$368$		0			0
$369$	400.525	−	117.497i	1.08543	−	0.318420i
$370$		0			0
$371$			845.065i			2.27780i
$372$		0			0
$373$		−	455.556i		−	1.22133i	−0.791889	−	0.610665i	$-0.790902\pi$
$373$		−	455.556i		−	1.22133i	0.791889	−	0.610665i	$-0.209098\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−570.482			−1.51322
$378$		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
$383$	−164.078			−0.428402			−0.214201	−	0.976790i	$-0.568715\pi$
$383$	−164.078			−0.428402			−0.214201	+	0.976790i	$0.568715\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	466.503	−	136.853i	1.20544	−	0.353624i
$388$		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
$393$	361.778	+	270.893i	0.920553	+	0.689295i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	93.4668i		−	0.235433i	−0.993047	−	0.117716i	$-0.962443\pi$
$397$		−	93.4668i		−	0.235433i	0.993047	−	0.117716i	$-0.0375574\pi$
$398$		0			0
$399$	625.253	+	468.179i	1.56705	+	1.17338i
$400$		0			0
$401$			23.4831i			0.0585613i	0.999571	+	0.0292807i	$0.00932166\pi$
$401$			23.4831i			0.0585613i	−0.999571	+	0.0292807i	$0.990678\pi$
$402$		0			0
$403$			522.749i			1.29714i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−96.9944			−0.238316
$408$		0			0
$409$	−697.472			−1.70531			−0.852655	−	0.522474i	$-0.825009\pi$
$409$	−697.472			−1.70531			−0.852655	+	0.522474i	$0.825009\pi$
$410$		0			0
$411$	−250.095	+	334.001i	−0.608503	+	0.812656i
$412$		0			0
$413$	−468.714			−1.13490
$414$		0			0
$415$		0			0
$416$		0			0
$417$	198.326	−	264.865i	0.475603	−	0.635167i
$418$		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
$423$	109.181	+	372.175i	0.258110	+	0.879847i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			956.851i			2.24087i
$428$		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
$433$			392.655i			0.906824i	0.891301	+	0.453412i	$0.149793\pi$
$433$			392.655i			0.906824i	−0.891301	+	0.453412i	$0.850207\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−456.731			−1.04515
$438$		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
$443$	−355.935			−0.803464			−0.401732	−	0.915757i	$-0.631592\pi$
$443$	−355.935			−0.803464			−0.401732	+	0.915757i	$0.631592\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−218.065	−	163.284i	−0.487842	−	0.365288i
$448$		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
$453$	−200.106	+	267.242i	−0.441736	+	0.589938i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			491.814i			1.07618i	0.842887	+	0.538090i	$0.180854\pi$
$457$			491.814i			1.07618i	−0.842887	+	0.538090i	$0.819146\pi$
$458$		0			0
$459$	−391.325	−	146.122i	−0.852561	−	0.318349i
$460$		0			0
$461$		−	42.6832i		−	0.0925883i	−0.998928	−	0.0462942i	$-0.985259\pi$
$461$		−	42.6832i		−	0.0925883i	0.998928	−	0.0462942i	$-0.0147412\pi$
$462$		0			0
$463$			422.582i			0.912705i	0.889799	+	0.456352i	$0.150844\pi$
$463$			422.582i			0.912705i	−0.889799	+	0.456352i	$0.849156\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−203.497			−0.435754			−0.217877	−	0.975976i	$-0.569913\pi$
$467$	−203.497			−0.435754			−0.217877	+	0.975976i	$0.569913\pi$
$468$		0			0
$469$	352.351			0.751282
$470$		0			0
$471$	−342.813	−	256.693i	−0.727840	−	0.544995i
$472$		0			0
$473$	−442.570			−0.935666
$474$		0			0
$475$		0			0
$476$		0			0
$477$	209.624	+	714.566i	0.439463	+	1.49804i
$478$		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
$483$	−439.401	−	329.016i	−0.909732	−	0.681192i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			140.013i			0.287500i	0.989614	+	0.143750i	$0.0459162\pi$
$487$			140.013i			0.287500i	−0.989614	+	0.143750i	$0.954084\pi$
$488$		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
$493$		−	649.816i		−	1.31809i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	694.562			1.39751
$498$		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
$503$	−149.060			−0.296343			−0.148171	−	0.988962i	$-0.547339\pi$
$503$	−149.060			−0.296343			−0.148171	+	0.988962i	$0.547339\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	27.8277	−	37.1639i	0.0548870	−	0.0733015i
$508$		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
$513$	644.834	+	240.783i	1.25699	+	0.469362i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	353.081i		−	0.682942i
$518$		0			0
$519$	−428.680	+	572.502i	−0.825973	+	1.10309i
$520$		0			0
$521$		−	419.868i		−	0.805890i	−0.915224	−	0.402945i	$-0.867987\pi$
$521$		−	419.868i		−	0.805890i	0.915224	−	0.402945i	$-0.132013\pi$
$522$		0			0
$523$		−	146.678i		−	0.280454i	−0.990119	−	0.140227i	$-0.955217\pi$
$523$		−	146.678i		−	0.280454i	0.990119	−	0.140227i	$-0.0447833\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−595.444			−1.12988
$528$		0			0
$529$	−208.030			−0.393251
$530$		0			0
$531$	−396.333	+	116.268i	−0.746390	+	0.218960i
$532$		0			0
$533$	−629.917			−1.18183
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−584.840	−	437.919i	−1.08909	−	0.815491i
$538$		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
$543$	585.199	−	781.534i	1.07772	−	1.43929i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			3.46995i			0.00634360i	0.999995	+	0.00317180i	$0.00100962\pi$
$547$			3.46995i			0.00634360i	−0.999995	+	0.00317180i	$0.998990\pi$
$548$		0			0
$549$	237.353	+	809.090i	0.432337	+	1.47375i
$550$		0			0
$551$			1070.78i			1.94334i
$552$		0			0
$553$		−	120.005i		−	0.217006i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	654.467			1.17499			0.587493	−	0.809230i	$-0.300115\pi$
$557$	654.467			1.17499			0.587493	+	0.809230i	$0.300115\pi$
$558$		0			0
$559$	−733.684			−1.31249
$560$		0			0
$561$	304.387	+	227.920i	0.542579	+	0.406274i
$562$		0			0
$563$	−546.529			−0.970745			−0.485373	−	0.874307i	$-0.661316\pi$
$563$	−546.529			−0.970745			−0.485373	+	0.874307i	$0.661316\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	446.913	+	696.167i	0.788207	+	1.22781i
$568$		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
$573$	398.634	+	298.491i	0.695696	+	0.520926i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			525.585i			0.910893i	0.890263	+	0.455446i	$0.150520\pi$
$577$			525.585i			0.910893i	−0.890263	+	0.455446i	$0.849480\pi$
$578$		0			0
$579$	574.914	+	430.486i	0.992943	+	0.743499i
$580$		0			0
$581$		−	1475.16i		−	2.53901i
$582$		0			0
$583$		−	677.906i		−	1.16279i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	75.3619			0.128385			0.0641924	−	0.997938i	$-0.479553\pi$
$587$	75.3619			0.128385			0.0641924	+	0.997938i	$0.479553\pi$
$588$		0			0
$589$	981.185			1.66585
$590$		0			0
$591$	463.051	−	618.405i	0.783505	−	1.04637i
$592$		0			0
$593$	1142.87			1.92727			0.963633	−	0.267230i	$-0.0861084\pi$
$593$	1142.87			1.92727			0.963633	+	0.267230i	$0.0861084\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−141.106	+	188.446i	−0.236358	+	0.315656i
$598$		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
$603$	297.940	−	87.4030i	0.494095	−	0.144947i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	719.995i		−	1.18615i	−0.805146	−	0.593077i	$-0.797913\pi$
$607$		−	719.995i		−	1.18615i	0.805146	−	0.593077i	$-0.202087\pi$
$608$		0			0
$609$	−771.360	+	1030.15i	−1.26660	+	1.69154i
$610$		0			0
$611$		−	585.331i		−	0.957988i
$612$		0			0
$613$			216.348i			0.352932i	0.984307	+	0.176466i	$0.0564666\pi$
$613$			216.348i			0.352932i	−0.984307	+	0.176466i	$0.943533\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−231.190			−0.374701			−0.187350	−	0.982293i	$-0.559990\pi$
$617$	−231.190			−0.374701			−0.187350	+	0.982293i	$0.559990\pi$
$618$		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
$623$	650.679			1.04443
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−501.574	−	375.570i	−0.799959	−	0.598996i
$628$		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
$633$	202.692	−	270.696i	0.320209	−	0.427639i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	751.234i		−	1.17933i
$638$		0			0
$639$	587.305	−	172.291i	0.919100	−	0.269626i
$640$		0			0
$641$			1131.85i			1.76576i	0.469598	+	0.882880i	$0.344399\pi$
$641$			1131.85i			1.76576i	−0.469598	+	0.882880i	$0.655601\pi$
$642$		0			0
$643$			637.481i			0.991418i	0.868489	+	0.495709i	$0.165092\pi$
$643$			637.481i			0.991418i	−0.868489	+	0.495709i	$0.834908\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	800.782			1.23769			0.618843	−	0.785515i	$-0.287602\pi$
$647$	800.782			1.23769			0.618843	+	0.785515i	$0.287602\pi$
$648$		0			0
$649$	375.999			0.579352
$650$		0			0
$651$	943.955	+	706.818i	1.45001	+	1.08574i
$652$		0			0
$653$	−313.185			−0.479610			−0.239805	−	0.970821i	$-0.577084\pi$
$653$	−313.185			−0.479610			−0.239805	+	0.970821i	$0.577084\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	386.197	−	113.294i	0.587819	−	0.172442i
$658$		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
$663$	504.606	+	377.841i	0.761096	+	0.569896i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	752.497i		−	1.12818i
$668$		0			0
$669$	−491.532	−	368.051i	−0.734727	−	0.550151i
$670$		0			0
$671$		−	767.580i		−	1.14393i
$672$		0			0
$673$			60.8622i			0.0904341i	0.998977	+	0.0452171i	$0.0143980\pi$
$673$			60.8622i			0.0904341i	−0.998977	+	0.0452171i	$0.985602\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	415.595			0.613877			0.306939	−	0.951729i	$-0.400695\pi$
$677$	415.595			0.613877			0.306939	+	0.951729i	$0.400695\pi$
$678$		0			0
$679$	652.639			0.961177
$680$		0			0
$681$	−206.912	+	276.331i	−0.303836	+	0.405772i
$682$		0			0
$683$	434.504			0.636170			0.318085	−	0.948062i	$-0.396960\pi$
$683$	434.504			0.636170			0.318085	+	0.948062i	$0.396960\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−232.782	+	310.880i	−0.338838	+	0.452518i
$688$		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
$693$	−211.992	−	722.640i	−0.305905	−	1.04277i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	717.516i		−	1.02943i
$698$		0			0
$699$	−449.802	+	600.711i	−0.643494	+	0.859386i
$700$		0			0
$701$		−	248.345i		−	0.354272i	−0.984186	−	0.177136i	$-0.943317\pi$
$701$		−	248.345i		−	0.354272i	0.984186	−	0.177136i	$-0.0566833\pi$
$702$		0			0
$703$			301.809i			0.429315i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	519.617			0.734961
$708$		0			0
$709$	24.1911			0.0341200			0.0170600	−	0.999854i	$-0.494569\pi$
$709$	24.1911			0.0341200			0.0170600	+	0.999854i	$0.494569\pi$
$710$		0			0
$711$	−29.7679	−	101.473i	−0.0418677	−	0.142719i
$712$		0			0
$713$	−689.534			−0.967088
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−118.282	−	88.5679i	−0.164969	−	0.123526i
$718$		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
$723$	822.953	−	1099.05i	1.13825	−	1.52013i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	454.949i		−	0.625789i	−0.949788	−	0.312895i	$-0.898701\pi$
$727$		−	454.949i		−	0.625789i	0.949788	−	0.312895i	$-0.101299\pi$
$728$		0			0
$729$	550.588	+	477.802i	0.755264	+	0.655421i
$730$		0			0
$731$		−	835.713i		−	1.14325i
$732$		0			0
$733$		−	131.588i		−	0.179520i	−0.995963	−	0.0897600i	$-0.971390\pi$
$733$		−	131.588i		−	0.179520i	0.995963	−	0.0897600i	$-0.0286100\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−282.654			−0.383520
$738$		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
$743$	439.535			0.591569			0.295784	−	0.955255i	$-0.404419\pi$
$743$	439.535			0.591569			0.295784	+	0.955255i	$0.404419\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−365.924	−	1247.36i	−0.489858	−	1.66983i
$748$		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
$753$	−287.488	−	215.266i	−0.381790	−	0.285878i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			468.694i			0.619147i	0.950875	+	0.309574i	$0.100186\pi$
$757$			468.694i			0.619147i	−0.950875	+	0.309574i	$0.899814\pi$
$758$		0			0
$759$	352.485	+	263.935i	0.464406	+	0.347740i
$760$		0			0
$761$			156.940i			0.206229i	0.994669	+	0.103115i	$0.0328808\pi$
$761$			156.940i			0.206229i	−0.994669	+	0.103115i	$0.967119\pi$
$762$		0			0
$763$		−	503.305i		−	0.659640i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	623.325			0.812679
$768$		0			0
$769$	−1188.63			−1.54569			−0.772844	−	0.634596i	$-0.781166\pi$
$769$	−1188.63			−1.54569			−0.772844	+	0.634596i	$0.781166\pi$
$770$		0			0
$771$	201.041	−	268.490i	0.260753	−	0.348236i
$772$		0			0
$773$	950.291			1.22935			0.614677	−	0.788779i	$-0.289286\pi$
$773$	950.291			1.22935			0.614677	+	0.788779i	$0.289286\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−217.414	+	290.357i	−0.279813	+	0.373690i
$778$		0			0
$779$			1182.34i			1.51776i
$780$		0			0
$781$	−557.174			−0.713411
$782$		0			0
$783$	−396.707	+	1062.41i	−0.506650	+	1.35685i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1056.60i			1.34257i	0.741200	+	0.671284i	$0.234257\pi$
$787$			1056.60i			1.34257i	−0.741200	+	0.671284i	$0.765743\pi$
$788$		0			0
$789$	−99.5538	+	132.954i	−0.126177	+	0.168510i
$790$		0			0
$791$		−	1608.08i		−	2.03297i
$792$		0			0
$793$		−	1272.48i		−	1.60464i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	78.7126			0.0987612			0.0493806	−	0.998780i	$-0.484275\pi$
$797$	78.7126			0.0987612			0.0493806	+	0.998780i	$0.484275\pi$
$798$		0			0
$799$	666.730			0.834455
$800$		0			0
$801$	550.198	−	161.405i	0.686889	−	0.201505i
$802$		0			0
$803$	−366.384			−0.456268
$804$		0			0
$805$		0			0
$806$		0			0
$807$	743.058	+	556.389i	0.920765	+	0.689454i
$808$		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
$813$	348.949	−	466.021i	0.429211	−	0.573211i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			1377.10i			1.68556i
$818$		0			0
$819$	−351.437	−	1197.98i	−0.429105	−	1.46273i
$820$		0			0
$821$			1321.32i			1.60940i	0.593679	+	0.804702i	$0.297675\pi$
$821$			1321.32i			1.60940i	−0.593679	+	0.804702i	$0.702325\pi$
$822$		0			0
$823$		−	760.814i		−	0.924439i	−0.886765	−	0.462220i	$-0.847053\pi$
$823$		−	760.814i		−	0.924439i	0.886765	−	0.462220i	$-0.152947\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−610.599			−0.738331			−0.369165	−	0.929364i	$-0.620356\pi$
$827$	−610.599			−0.738331			−0.369165	+	0.929364i	$0.620356\pi$
$828$		0			0
$829$	−265.766			−0.320586			−0.160293	−	0.987069i	$-0.551244\pi$
$829$	−265.766			−0.320586			−0.160293	+	0.987069i	$0.551244\pi$
$830$		0			0
$831$	−12.4973	−	9.35776i	−0.0150388	−	0.0112608i
$832$		0			0
$833$	855.704			1.02726
$834$		0			0
$835$		0			0
$836$		0			0
$837$	973.516	+	363.514i	1.16310	+	0.434306i
$838$		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
$843$	576.345	+	431.558i	0.683683	+	0.511931i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	550.236i		−	0.649629i
$848$		0			0
$849$	367.677	+	275.310i	0.433071	+	0.324276i
$850$		0			0
$851$		−	212.098i		−	0.249234i
$852$		0			0
$853$			331.180i			0.388253i	0.980976	+	0.194127i	$0.0621873\pi$
$853$			331.180i			0.388253i	−0.980976	+	0.194127i	$0.937813\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−159.395			−0.185992			−0.0929961	−	0.995666i	$-0.529644\pi$
$857$	−159.395			−0.185992			−0.0929961	+	0.995666i	$0.529644\pi$
$858$		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
$863$	−152.237			−0.176405			−0.0882025	−	0.996103i	$-0.528112\pi$
$863$	−152.237			−0.176405			−0.0882025	+	0.996103i	$0.528112\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	89.2744	−	119.226i	0.102969	−	0.137516i
$868$		0			0
$869$			96.2669i			0.110779i
$870$		0			0
$871$	−468.578			−0.537977
$872$		0			0
$873$	551.856	−	161.891i	0.632137	−	0.185443i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			521.588i			0.594742i	0.954762	+	0.297371i	$0.0961097\pi$
$877$			521.588i			0.594742i	−0.954762	+	0.297371i	$0.903890\pi$
$878$		0			0
$879$	304.338	−	406.443i	0.346232	−	0.462392i
$880$		0			0
$881$		−	821.701i		−	0.932692i	−0.884603	−	0.466346i	$-0.845570\pi$
$881$		−	821.701i		−	0.932692i	0.884603	−	0.466346i	$-0.154430\pi$
$882$		0			0
$883$		−	747.791i		−	0.846876i	−0.905925	−	0.423438i	$-0.860823\pi$
$883$		−	747.791i		−	0.846876i	0.905925	−	0.423438i	$-0.139177\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1160.21			−1.30801			−0.654006	−	0.756490i	$-0.726913\pi$
$887$	−1160.21			−1.30801			−0.654006	+	0.756490i	$0.726913\pi$
$888$		0			0
$889$	−1379.45			−1.55169
$890$		0			0
$891$	−358.511	−	558.461i	−0.402369	−	0.626780i
$892$		0			0
$893$	−1098.65			−1.23029
$894$		0			0
$895$		0			0
$896$		0			0
$897$	584.342	+	437.546i	0.651440	+	0.487788i
$898$		0			0
$899$			1616.57i			1.79819i
$900$		0			0
$901$	1280.10			1.42076
$902$		0			0
$903$	−992.027	+	1324.85i	−1.09859	+	1.46717i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	354.207i		−	0.390526i	−0.980751	−	0.195263i	$-0.937444\pi$
$907$		−	354.207i		−	0.390526i	0.980751	−	0.195263i	$-0.0625560\pi$
$908$		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
$913$			1183.37i			1.29613i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1538.65			−1.67792
$918$		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
$923$	−923.672			−1.00073
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−396.819	+	116.410i	−0.428068	+	0.125577i
$928$		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
$933$	−320.818	−	240.223i	−0.343856	−	0.257474i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			894.637i			0.954789i	0.878689	+	0.477395i	$0.158419\pi$
$937$			894.637i			0.954789i	−0.878689	+	0.477395i	$0.841581\pi$
$938$		0			0
$939$	−1104.60	−	827.105i	−1.17636	−	0.880837i
$940$		0			0
$941$			592.336i			0.629475i	0.949179	+	0.314737i	$0.101916\pi$
$941$			592.336i			0.629475i	−0.949179	+	0.314737i	$0.898084\pi$
$942$		0			0
$943$		−	830.894i		−	0.881118i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	526.526			0.555994			0.277997	−	0.960582i	$-0.410330\pi$
$947$	526.526			0.555994			0.277997	+	0.960582i	$0.410330\pi$
$948$		0			0
$949$	−607.383			−0.640025
$950$		0			0
$951$	−536.112	+	715.977i	−0.563735	+	0.752868i
$952$		0			0
$953$	−117.158			−0.122936			−0.0614678	−	0.998109i	$-0.519578\pi$
$953$	−117.158			−0.122936			−0.0614678	+	0.998109i	$0.519578\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	618.780	−	826.381i	0.646583	−	0.863512i
$958$		0			0
$959$		−	1420.52i		−	1.48125i
$960$		0			0
$961$	520.311			0.541427
$962$		0			0
$963$	279.311	+	952.114i	0.290042	+	0.988696i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	811.294i		−	0.838981i	−0.907760	−	0.419490i	$-0.862209\pi$
$967$		−	811.294i		−	0.838981i	0.907760	−	0.419490i	$-0.137791\pi$
$968$		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
$973$			1126.48i			1.15774i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	830.832			0.850391			0.425195	−	0.905102i	$-0.360205\pi$
$977$	830.832			0.850391			0.425195	+	0.905102i	$0.360205\pi$
$978$		0			0
$979$	−521.971			−0.533167
$980$		0			0
$981$	−124.848	−	425.582i	−0.127266	−	0.433825i
$982$		0			0
$983$	−1056.30			−1.07457			−0.537283	−	0.843402i	$-0.680549\pi$
$983$	−1056.30			−1.07457			−0.537283	+	0.843402i	$0.680549\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1056.96	−	791.436i	−1.07088	−	0.801861i
$988$		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
$993$	−714.934	+	954.795i	−0.719974	+	0.961525i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			978.023i			0.980966i	0.871451	+	0.490483i	$0.163180\pi$
$997$			978.023i			0.980966i	−0.871451	+	0.490483i	$0.836820\pi$
$998$		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	600.3.c.d.449.4		16
3.2	odd	2	inner	600.3.c.d.449.14		16
4.3	odd	2		1200.3.c.m.449.13		16
5.2	odd	4		600.3.l.f.401.6		8
5.3	odd	4		120.3.l.a.41.3	✓	8
5.4	even	2	inner	600.3.c.d.449.13		16
12.11	even	2		1200.3.c.m.449.3		16
15.2	even	4		600.3.l.f.401.5		8
15.8	even	4		120.3.l.a.41.4	yes	8
15.14	odd	2	inner	600.3.c.d.449.3		16
20.3	even	4		240.3.l.d.161.6		8
20.7	even	4		1200.3.l.x.401.3		8
20.19	odd	2		1200.3.c.m.449.4		16
40.3	even	4		960.3.l.g.641.3		8
40.13	odd	4		960.3.l.h.641.6		8
60.23	odd	4		240.3.l.d.161.5		8
60.47	odd	4		1200.3.l.x.401.4		8
60.59	even	2		1200.3.c.m.449.14		16
120.53	even	4		960.3.l.h.641.5		8
120.83	odd	4		960.3.l.g.641.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.3	✓	8	5.3	odd	4
120.3.l.a.41.4	yes	8	15.8	even	4
240.3.l.d.161.5		8	60.23	odd	4
240.3.l.d.161.6		8	20.3	even	4
600.3.c.d.449.3		16	15.14	odd	2	inner
600.3.c.d.449.4		16	1.1	even	1	trivial
600.3.c.d.449.13		16	5.4	even	2	inner
600.3.c.d.449.14		16	3.2	odd	2	inner
600.3.l.f.401.5		8	15.2	even	4
600.3.l.f.401.6		8	5.2	odd	4
960.3.l.g.641.3		8	40.3	even	4
960.3.l.g.641.4		8	120.83	odd	4
960.3.l.h.641.5		8	120.53	even	4
960.3.l.h.641.6		8	40.13	odd	4
1200.3.c.m.449.3		16	12.11	even	2
1200.3.c.m.449.4		16	20.19	odd	2
1200.3.c.m.449.13		16	4.3	odd	2
1200.3.c.m.449.14		16	60.59	even	2
1200.3.l.x.401.3		8	20.7	even	4
1200.3.l.x.401.4		8	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.79813 + 2.40140i) q^{3} -10.2132i q^{7} +(-2.53346 - 8.63606i) q^{9} +O(q^{10})\)	\(q+(-1.79813 + 2.40140i) q^{3} -10.2132i q^{7} +(-2.53346 - 8.63606i) q^{9} +8.19300i q^{11} +13.5822i q^{13} -15.4710 q^{17} +25.4934 q^{19} +(24.5261 + 18.3647i) q^{21} -17.9156 q^{23} +(25.2941 + 9.44491i) q^{27} +42.0022i q^{29} +38.4878 q^{31} +(-19.6747 - 14.7321i) q^{33} +11.8387i q^{37} +(-32.6163 - 24.4225i) q^{39} +46.3781i q^{41} +54.0181i q^{43} -43.0955 q^{47} -55.3102 q^{49} +(27.8188 - 37.1521i) q^{51} -82.7421 q^{53} +(-45.8404 + 61.2199i) q^{57} -45.8928i q^{59} -93.6873 q^{61} +(-88.2022 + 25.8748i) q^{63} +34.4995i q^{67} +(32.2147 - 43.0227i) q^{69} +68.0061i q^{71} +44.7191i q^{73} +83.6770 q^{77} +11.7499 q^{79} +(-68.1632 + 43.7582i) q^{81} +144.436 q^{83} +(-100.864 - 75.5255i) q^{87} +63.7094i q^{89} +138.718 q^{91} +(-69.2061 + 92.4247i) q^{93} +63.9013i q^{97} +(70.7552 - 20.7566i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 40 q^{9}+O(q^{10}) \) 16 * q - 40 * q^9	\( 16 q - 40 q^{9} + 16 q^{19} + 56 q^{21} + 240 q^{31} + 144 q^{39} - 128 q^{49} + 128 q^{51} + 16 q^{61} - 200 q^{69} - 176 q^{79} + 448 q^{81} + 1120 q^{91} - 64 q^{99}+O(q^{100}) \) 16 * q - 40 * q^9 + 16 * q^19 + 56 * q^21 + 240 * q^31 + 144 * q^39 - 128 * q^49 + 128 * q^51 + 16 * q^61 - 200 * q^69 - 176 * q^79 + 448 * q^81 + 1120 * q^91 - 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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