LMFDB - Embedded newform 2052.3.m.a.881.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(881,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 2]))

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

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

chi := DirichletCharacter("2052.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.m (of order $6$, degree $2$, 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:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			881.18
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.944063i q^{5} +(-5.01764 - 8.69081i) q^{7} +O(q^{10})$	$q-0.944063i q^{5} +(-5.01764 - 8.69081i) q^{7} +(-2.50935 + 1.44878i) q^{11} +(3.63747 + 6.30028i) q^{13} +(-15.0969 + 8.71618i) q^{17} +(-18.3970 - 4.74856i) q^{19} +(11.5334 - 6.65879i) q^{23} +24.1087 q^{25} -16.6225i q^{29} +(4.71890 - 8.17338i) q^{31} +(-8.20467 + 4.73697i) q^{35} -63.1336 q^{37} +34.1661i q^{41} +(-23.8541 + 41.3166i) q^{43} -42.6230i q^{47} +(-25.8534 + 44.7794i) q^{49} +(69.9633 + 40.3933i) q^{53} +(1.36774 + 2.36899i) q^{55} +92.1179i q^{59} -56.9441 q^{61} +(5.94786 - 3.43400i) q^{65} +(-39.6354 - 68.6505i) q^{67} +(61.9380 - 35.7599i) q^{71} +(63.6907 + 110.316i) q^{73} +(25.1820 + 14.5389i) q^{77} +(-3.36504 + 5.82843i) q^{79} +(101.821 - 58.7865i) q^{83} +(8.22863 + 14.2524i) q^{85} +(87.6080 + 50.5805i) q^{89} +(36.5030 - 63.2251i) q^{91} +(-4.48294 + 17.3680i) q^{95} +(-42.5419 + 73.6847i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	0.944063i		−	0.188813i	−0.995534	−	0.0944063i	$-0.969905\pi$
$5$		−	0.944063i		−	0.188813i	0.995534	−	0.0944063i	$-0.0300953\pi$
$6$		0			0
$7$	−5.01764	−	8.69081i	−0.716806	−	1.24154i	−0.962259	−	0.272135i	$-0.912270\pi$
$7$	−5.01764	−	8.69081i	−0.716806	−	1.24154i	0.245453	−	0.969408i	$-0.421063\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.50935	+	1.44878i	−0.228123	+	0.131707i	−0.609706	−	0.792628i	$-0.708712\pi$
$11$	−2.50935	+	1.44878i	−0.228123	+	0.131707i	0.381583	+	0.924335i	$0.375379\pi$
$12$		0			0
$13$	3.63747	+	6.30028i	0.279805	+	0.484637i	0.971336	−	0.237710i	$-0.0763967\pi$
$13$	3.63747	+	6.30028i	0.279805	+	0.484637i	−0.691531	+	0.722347i	$0.743063\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.0969	+	8.71618i	−0.888051	+	0.512717i	−0.873305	−	0.487175i	$-0.838028\pi$
$17$	−15.0969	+	8.71618i	−0.888051	+	0.512717i	−0.0147468	+	0.999891i	$0.504694\pi$
$18$		0			0
$19$	−18.3970	−	4.74856i	−0.968265	−	0.249924i
$19$	−18.3970	−	4.74856i	−0.968265	−	0.249924i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	11.5334	−	6.65879i	0.501450	−	0.289512i	−0.227862	−	0.973693i	$-0.573174\pi$
$23$	11.5334	−	6.65879i	0.501450	−	0.289512i	0.729312	+	0.684181i	$0.239840\pi$
$24$		0			0
$25$	24.1087			0.964350
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	16.6225i		−	0.573190i	−0.958052	−	0.286595i	$-0.907477\pi$
$29$		−	16.6225i		−	0.573190i	0.958052	−	0.286595i	$-0.0925234\pi$
$30$		0			0
$31$	4.71890	−	8.17338i	0.152223	−	0.263657i	−0.779822	−	0.626002i	$-0.784690\pi$
$31$	4.71890	−	8.17338i	0.152223	−	0.263657i	0.932044	+	0.362344i	$0.118024\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.20467	+	4.73697i	−0.234419	+	0.135342i
$36$		0			0
$37$	−63.1336			−1.70631			−0.853156	−	0.521655i	$-0.825315\pi$
$37$	−63.1336			−1.70631			−0.853156	+	0.521655i	$0.825315\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			34.1661i			0.833319i	0.909063	+	0.416659i	$0.136799\pi$
$41$			34.1661i			0.833319i	−0.909063	+	0.416659i	$0.863201\pi$
$42$		0			0
$43$	−23.8541	+	41.3166i	−0.554747	+	0.960850i	0.443176	+	0.896435i	$0.353852\pi$
$43$	−23.8541	+	41.3166i	−0.554747	+	0.960850i	−0.997923	+	0.0644157i	$0.979482\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	42.6230i		−	0.906873i	−0.891289	−	0.453437i	$-0.850198\pi$
$47$		−	42.6230i		−	0.906873i	0.891289	−	0.453437i	$-0.149802\pi$
$48$		0			0
$49$	−25.8534	+	44.7794i	−0.527620	+	0.913865i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	69.9633	+	40.3933i	1.32006	+	0.762138i	0.983738	−	0.179609i	$-0.0574832\pi$
$53$	69.9633	+	40.3933i	1.32006	+	0.762138i	0.336323	+	0.941747i	$0.390817\pi$
$54$		0			0
$55$	1.36774	+	2.36899i	0.0248679	+	0.0430725i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			92.1179i			1.56132i	0.624955	+	0.780661i	$0.285117\pi$
$59$			92.1179i			1.56132i	−0.624955	+	0.780661i	$0.714883\pi$
$60$		0			0
$61$	−56.9441			−0.933509			−0.466755	−	0.884387i	$-0.654577\pi$
$61$	−56.9441			−0.933509			−0.466755	+	0.884387i	$0.654577\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.94786	−	3.43400i	0.0915056	−	0.0528308i
$66$		0			0
$67$	−39.6354	−	68.6505i	−0.591573	−	1.02463i	−0.994021	−	0.109192i	$-0.965174\pi$
$67$	−39.6354	−	68.6505i	−0.591573	−	1.02463i	0.402448	−	0.915443i	$-0.368160\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	61.9380	−	35.7599i	0.872366	−	0.503661i	0.00423248	−	0.999991i	$-0.498653\pi$
$71$	61.9380	−	35.7599i	0.872366	−	0.503661i	0.868134	+	0.496330i	$0.165319\pi$
$72$		0			0
$73$	63.6907	+	110.316i	0.872475	+	1.51117i	0.859428	+	0.511257i	$0.170820\pi$
$73$	63.6907	+	110.316i	0.872475	+	1.51117i	0.0130472	+	0.999915i	$0.495847\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	25.1820	+	14.5389i	0.327040	+	0.188816i
$78$		0			0
$79$	−3.36504	+	5.82843i	−0.0425955	+	0.0737776i	−0.886537	−	0.462657i	$-0.846896\pi$
$79$	−3.36504	+	5.82843i	−0.0425955	+	0.0737776i	0.843942	+	0.536435i	$0.180229\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	101.821	−	58.7865i	1.22676	−	0.708271i	0.260410	−	0.965498i	$-0.416142\pi$
$83$	101.821	−	58.7865i	1.22676	−	0.708271i	0.966351	+	0.257227i	$0.0828089\pi$
$84$		0			0
$85$	8.22863	+	14.2524i	0.0968074	+	0.167675i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	87.6080	+	50.5805i	0.984360	+	0.568320i	0.903584	−	0.428412i	$-0.140927\pi$
$89$	87.6080	+	50.5805i	0.984360	+	0.568320i	0.0807762	+	0.996732i	$0.474260\pi$
$90$		0			0
$91$	36.5030	−	63.2251i	0.401132	−	0.694781i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.48294	+	17.3680i	−0.0471889	+	0.182821i
$96$		0			0
$97$	−42.5419	+	73.6847i	−0.438576	+	0.759637i	−0.997580	−	0.0695287i	$-0.977850\pi$
$97$	−42.5419	+	73.6847i	−0.438576	+	0.759637i	0.559004	+	0.829165i	$0.311184\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	196.653i		−	1.94706i	−0.228552	−	0.973532i	$-0.573399\pi$
$101$		−	196.653i		−	1.94706i	0.228552	−	0.973532i	$-0.426601\pi$
$102$		0			0
$103$	−32.8937	+	56.9735i	−0.319356	+	0.553141i	−0.980354	−	0.197247i	$-0.936800\pi$
$103$	−32.8937	+	56.9735i	−0.319356	+	0.553141i	0.660998	+	0.750388i	$0.270133\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.00186i		−	0.0187090i	−0.999956	−	0.00935448i	$-0.997022\pi$
$107$		−	2.00186i		−	0.0187090i	0.999956	−	0.00935448i	$-0.00297767\pi$
$108$		0			0
$109$	33.9399	+	58.7857i	0.311375	+	0.539318i	0.978660	−	0.205485i	$-0.0658770\pi$
$109$	33.9399	+	58.7857i	0.311375	+	0.539318i	−0.667285	+	0.744802i	$0.732544\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	92.2373	+	53.2532i	0.816260	+	0.471268i	0.849125	−	0.528192i	$-0.177130\pi$
$113$	92.2373	+	53.2532i	0.816260	+	0.471268i	−0.0328653	+	0.999460i	$0.510463\pi$
$114$		0			0
$115$	−6.28631	−	10.8882i	−0.0546636	−	0.0946801i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	151.501	+	87.4693i	1.27312	+	0.735036i
$120$		0			0
$121$	−56.3021	+	97.5181i	−0.465307	+	0.805935i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	46.3617i		−	0.370894i
$126$		0			0
$127$	−8.57128	+	14.8459i	−0.0674904	+	0.116897i	−0.897796	−	0.440412i	$-0.854833\pi$
$127$	−8.57128	+	14.8459i	−0.0674904	+	0.116897i	0.830306	+	0.557308i	$0.188166\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			175.291i			1.33810i	0.743218	+	0.669049i	$0.233298\pi$
$131$			175.291i			1.33810i	−0.743218	+	0.669049i	$0.766702\pi$
$132$		0			0
$133$	51.0409	+	183.712i	0.383766	+	1.38129i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			171.435i			1.25135i	0.780084	+	0.625675i	$0.215176\pi$
$137$			171.435i			1.25135i	−0.780084	+	0.625675i	$0.784824\pi$
$138$		0			0
$139$	31.9396	+	55.3210i	0.229781	+	0.397993i	0.957743	−	0.287625i	$-0.0928655\pi$
$139$	31.9396	+	55.3210i	0.229781	+	0.397993i	−0.727962	+	0.685618i	$0.759532\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−18.2554	−	10.5398i	−0.127660	−	0.0737046i
$144$		0			0
$145$	−15.6927			−0.108225
$146$		0			0
$147$		0			0
$148$		0			0
$149$			161.148i			1.08153i	0.841173	+	0.540766i	$0.181866\pi$
$149$			161.148i			1.08153i	−0.841173	+	0.540766i	$0.818134\pi$
$150$		0			0
$151$	97.2259	+	168.400i	0.643880	+	1.11523i	0.984559	+	0.175053i	$0.0560098\pi$
$151$	97.2259	+	168.400i	0.643880	+	1.11523i	−0.340679	+	0.940180i	$0.610657\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.71618	−	4.45494i	−0.0497818	−	0.0287415i
$156$		0			0
$157$	−60.0978			−0.382789			−0.191394	−	0.981513i	$-0.561301\pi$
$157$	−60.0978			−0.382789			−0.191394	+	0.981513i	$0.561301\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−115.740	−	66.8228i	−0.718885	−	0.415048i
$162$		0			0
$163$	37.0420			0.227252			0.113626	−	0.993524i	$-0.463753\pi$
$163$	37.0420			0.227252			0.113626	+	0.993524i	$0.463753\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−19.3428	+	11.1676i	−0.115825	+	0.0668718i	−0.556793	−	0.830651i	$-0.687969\pi$
$167$	−19.3428	+	11.1676i	−0.115825	+	0.0668718i	0.440968	+	0.897523i	$0.354635\pi$
$168$		0			0
$169$	58.0376	−	100.524i	0.343418	−	0.594817i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−189.603	−	109.467i	−1.09597	−	0.632759i	−0.160811	−	0.986985i	$-0.551411\pi$
$173$	−189.603	−	109.467i	−1.09597	−	0.632759i	−0.935160	+	0.354227i	$0.884744\pi$
$174$		0			0
$175$	−120.969	−	209.524i	−0.691251	−	1.19728i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			71.3531i			0.398621i	0.979936	+	0.199310i	$0.0638702\pi$
$179$			71.3531i			0.398621i	−0.979936	+	0.199310i	$0.936130\pi$
$180$		0			0
$181$	42.9974	−	74.4737i	0.237555	−	0.411457i	−0.722457	−	0.691416i	$-0.756987\pi$
$181$	42.9974	−	74.4737i	0.237555	−	0.411457i	0.960012	+	0.279959i	$0.0903207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			59.6021i			0.322173i
$186$		0			0
$187$	25.2556	−	43.7440i	0.135057	−	0.233925i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−276.487	+	159.630i	−1.44757	+	0.835757i	−0.998337	−	0.0576547i	$-0.981638\pi$
$191$	−276.487	+	159.630i	−1.44757	+	0.835757i	−0.449238	+	0.893412i	$0.648304\pi$
$192$		0			0
$193$	112.770			0.584300			0.292150	−	0.956372i	$-0.405629\pi$
$193$	112.770			0.584300			0.292150	+	0.956372i	$0.405629\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			94.3781i			0.479077i	0.970887	+	0.239538i	$0.0769961\pi$
$197$			94.3781i			0.479077i	−0.970887	+	0.239538i	$0.923004\pi$
$198$		0			0
$199$	149.404	−	258.776i	0.750775	−	1.30038i	−0.196672	−	0.980469i	$-0.563013\pi$
$199$	149.404	−	258.776i	0.750775	−	1.30038i	0.947447	−	0.319911i	$-0.103653\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−144.463	+	83.4057i	−0.711640	+	0.410866i
$204$		0			0
$205$	32.2549			0.157341
$206$		0			0
$207$		0			0
$208$		0			0
$209$	53.0443	−	14.7374i	0.253800	−	0.0705137i
$210$		0			0
$211$	−32.9241			−0.156039			−0.0780193	−	0.996952i	$-0.524860\pi$
$211$	−32.9241			−0.156039			−0.0780193	+	0.996952i	$0.524860\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	39.0054	+	22.5198i	0.181421	+	0.104743i
$216$		0			0
$217$	−94.7110			−0.436456
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−109.829	−	63.4097i	−0.496963	−	0.286922i
$222$		0			0
$223$	−122.138	+	211.549i	−0.547704	+	0.948652i	0.450727	+	0.892662i	$0.351165\pi$
$223$	−122.138	+	211.549i	−0.547704	+	0.948652i	−0.998431	+	0.0559898i	$0.982169\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	376.013	−	217.091i	1.65644	−	0.956348i	0.682107	−	0.731253i	$-0.261064\pi$
$227$	376.013	−	217.091i	1.65644	−	0.956348i	0.974337	−	0.225095i	$-0.0722694\pi$
$228$		0			0
$229$	−202.602	+	350.917i	−0.884724	+	1.53239i	−0.0386942	+	0.999251i	$0.512320\pi$
$229$	−202.602	+	350.917i	−0.884724	+	1.53239i	−0.846030	+	0.533136i	$0.821014\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−200.063	+	115.506i	−0.858639	+	0.495735i	−0.863556	−	0.504253i	$-0.831768\pi$
$233$	−200.063	+	115.506i	−0.858639	+	0.495735i	0.00491747	+	0.999988i	$0.498435\pi$
$234$		0			0
$235$	−40.2388			−0.171229
$236$		0			0
$237$		0			0
$238$		0			0
$239$	83.8959	+	48.4373i	0.351029	+	0.202667i	0.665138	−	0.746720i	$-0.268373\pi$
$239$	83.8959	+	48.4373i	0.351029	+	0.202667i	−0.314110	+	0.949387i	$0.601706\pi$
$240$		0			0
$241$	−298.299			−1.23775			−0.618877	−	0.785488i	$-0.712412\pi$
$241$	−298.299			−1.23775			−0.618877	+	0.785488i	$0.712412\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	42.2746	+	24.4072i	0.172549	+	0.0996214i
$246$		0			0
$247$	−37.0014	−	133.179i	−0.149803	−	0.539187i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.82420	+	1.63055i	0.0112518	+	0.00649622i	0.505615	−	0.862759i	$-0.331266\pi$
$251$	2.82420	+	1.63055i	0.0112518	+	0.00649622i	−0.494364	+	0.869255i	$0.664599\pi$
$252$		0			0
$253$	−19.2942	+	33.4185i	−0.0762615	+	0.132089i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	75.6315	−	43.6659i	0.294286	−	0.169906i	−0.345587	−	0.938387i	$-0.612320\pi$
$257$	75.6315	−	43.6659i	0.294286	−	0.169906i	0.639873	+	0.768481i	$0.278987\pi$
$258$		0			0
$259$	316.781	+	548.682i	1.22309	+	2.11846i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	146.393	+	84.5202i	0.556629	+	0.321370i	0.751791	−	0.659401i	$-0.229190\pi$
$263$	146.393	+	84.5202i	0.556629	+	0.321370i	−0.195163	+	0.980771i	$0.562523\pi$
$264$		0			0
$265$	38.1338	−	66.0497i	0.143901	−	0.249244i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	28.8312	−	16.6457i	0.107179	−	0.0618798i	−0.445452	−	0.895306i	$-0.646957\pi$
$269$	28.8312	−	16.6457i	0.107179	−	0.0618798i	0.552631	+	0.833426i	$0.313624\pi$
$270$		0			0
$271$	101.879	+	176.460i	0.375938	+	0.651144i	0.990467	−	0.137750i	$-0.0439871\pi$
$271$	101.879	+	176.460i	0.375938	+	0.651144i	−0.614529	+	0.788894i	$0.710654\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−60.4973	+	34.9282i	−0.219990	+	0.127011i
$276$		0			0
$277$	69.4690	+	120.324i	0.250791	+	0.434382i	0.963744	−	0.266830i	$-0.0859761\pi$
$277$	69.4690	+	120.324i	0.250791	+	0.434382i	−0.712953	+	0.701212i	$0.752643\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			145.827i			0.518956i	0.965749	+	0.259478i	$0.0835505\pi$
$281$			145.827i			0.518956i	−0.965749	+	0.259478i	$0.916450\pi$
$282$		0			0
$283$	−333.171			−1.17728			−0.588641	−	0.808395i	$-0.700337\pi$
$283$	−333.171			−1.17728			−0.588641	+	0.808395i	$0.700337\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	296.931	−	171.433i	1.03460	−	0.597328i
$288$		0			0
$289$	7.44374	−	12.8929i	0.0257569	−	0.0446123i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	163.865	+	94.6077i	0.559267	+	0.322893i	0.752851	−	0.658191i	$-0.228678\pi$
$293$	163.865	+	94.6077i	0.559267	+	0.322893i	−0.193584	+	0.981084i	$0.562011\pi$
$294$		0			0
$295$	86.9651			0.294797
$296$		0			0
$297$		0			0
$298$		0			0
$299$	83.9044	+	48.4423i	0.280617	+	0.162014i
$300$		0			0
$301$	478.766			1.59058
$302$		0			0
$303$		0			0
$304$		0			0
$305$			53.7588i			0.176258i
$306$		0			0
$307$	42.5147	+	73.6377i	0.138484	+	0.239862i	0.926923	−	0.375251i	$-0.122444\pi$
$307$	42.5147	+	73.6377i	0.138484	+	0.239862i	−0.788439	+	0.615113i	$0.789110\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−119.243	−	68.8452i	−0.383420	−	0.221367i	0.295885	−	0.955223i	$-0.404385\pi$
$311$	−119.243	−	68.8452i	−0.383420	−	0.221367i	−0.679305	+	0.733856i	$0.737719\pi$
$312$		0			0
$313$	38.4710			0.122910			0.0614552	−	0.998110i	$-0.480426\pi$
$313$	38.4710			0.122910			0.0614552	+	0.998110i	$0.480426\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			343.114i			1.08238i	0.840901	+	0.541189i	$0.182026\pi$
$317$			343.114i			1.08238i	−0.840901	+	0.541189i	$0.817974\pi$
$318$		0			0
$319$	24.0823	+	41.7117i	0.0754930	+	0.130758i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	319.127	−	88.6636i	0.988010	−	0.274500i
$324$		0			0
$325$	87.6948	+	151.892i	0.269830	+	0.467360i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−370.429	+	213.867i	−1.12592	+	0.650052i
$330$		0			0
$331$	−59.9385	−	103.817i	−0.181083	−	0.313645i	0.761167	−	0.648556i	$-0.224627\pi$
$331$	−59.9385	−	103.817i	−0.181083	−	0.313645i	−0.942250	+	0.334911i	$0.891294\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−64.8104	+	37.4183i	−0.193464	+	0.111696i
$336$		0			0
$337$	−228.092			−0.676830			−0.338415	−	0.940997i	$-0.609891\pi$
$337$	−228.092			−0.676830			−0.338415	+	0.940997i	$0.609891\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			27.3465i			0.0801951i
$342$		0			0
$343$	27.1635			0.0791938
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	153.282i		−	0.441734i	−0.975304	−	0.220867i	$-0.929111\pi$
$347$		−	153.282i		−	0.441734i	0.975304	−	0.220867i	$-0.0708887\pi$
$348$		0			0
$349$	−166.507	−	288.398i	−0.477097	−	0.826356i	0.522559	−	0.852603i	$-0.324978\pi$
$349$	−166.507	−	288.398i	−0.477097	−	0.826356i	−0.999655	+	0.0262473i	$0.991644\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	158.739	−	91.6479i	0.449685	−	0.259626i	−0.258012	−	0.966142i	$-0.583067\pi$
$353$	158.739	−	91.6479i	0.449685	−	0.259626i	0.707697	+	0.706516i	$0.249734\pi$
$354$		0			0
$355$	−33.7596	−	58.4734i	−0.0950975	−	0.164714i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−420.734	+	242.911i	−1.17196	+	0.676633i	−0.954141	−	0.299356i	$-0.903228\pi$
$359$	−420.734	+	242.911i	−1.17196	+	0.676633i	−0.217821	+	0.975989i	$0.569895\pi$
$360$		0			0
$361$	315.902	+	174.719i	0.875076	+	0.483986i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	104.145	−	60.1280i	0.285328	−	0.164734i
$366$		0			0
$367$	−173.687			−0.473262			−0.236631	−	0.971600i	$-0.576043\pi$
$367$	−173.687			−0.473262			−0.236631	+	0.971600i	$0.576043\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	810.716i		−	2.18522i
$372$		0			0
$373$	69.7815	−	120.865i	0.187082	−	0.324035i	−0.757194	−	0.653190i	$-0.773430\pi$
$373$	69.7815	−	120.865i	0.187082	−	0.324035i	0.944276	+	0.329155i	$0.106764\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	104.726	−	60.4639i	0.277789	−	0.160382i
$378$		0			0
$379$	307.458			0.811234			0.405617	−	0.914043i	$-0.367057\pi$
$379$	307.458			0.811234			0.405617	+	0.914043i	$0.367057\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			187.428i			0.489369i	0.969603	+	0.244685i	$0.0786844\pi$
$383$			187.428i			0.489369i	−0.969603	+	0.244685i	$0.921316\pi$
$384$		0			0
$385$	13.7256	−	23.7734i	0.0356509	−	0.0617492i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	658.884i		−	1.69379i	−0.531761	−	0.846895i	$-0.678469\pi$
$389$		−	658.884i		−	1.69379i	0.531761	−	0.846895i	$-0.321531\pi$
$390$		0			0
$391$	−116.078	+	201.054i	−0.296876	+	0.514204i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.50240	+	3.17681i	0.0139301	+	0.00804257i
$396$		0			0
$397$	−357.219	−	618.721i	−0.899796	−	1.55849i	−0.827755	−	0.561090i	$-0.810382\pi$
$397$	−357.219	−	618.721i	−0.899796	−	1.55849i	−0.0720408	−	0.997402i	$-0.522951\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			712.138i			1.77590i	0.459936	+	0.887952i	$0.347872\pi$
$401$			712.138i			1.77590i	−0.459936	+	0.887952i	$0.652128\pi$
$402$		0			0
$403$	68.6594			0.170371
$404$		0			0
$405$		0			0
$406$		0			0
$407$	158.424	−	91.4664i	0.389249	−	0.224733i
$408$		0			0
$409$	−35.7805	−	61.9736i	−0.0874828	−	0.151525i	0.818964	−	0.573845i	$-0.194549\pi$
$409$	−35.7805	−	61.9736i	−0.0874828	−	0.151525i	−0.906446	+	0.422321i	$0.861216\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	800.579	−	462.215i	1.93845	−	1.11916i
$414$		0			0
$415$	−55.4981	−	96.1256i	−0.133730	−	0.231628i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	339.757	+	196.159i	0.810875	+	0.468159i	0.847260	−	0.531179i	$-0.178251\pi$
$419$	339.757	+	196.159i	0.810875	+	0.468159i	−0.0363848	+	0.999338i	$0.511584\pi$
$420$		0			0
$421$	100.615	−	174.271i	0.238991	−	0.413945i	−0.721434	−	0.692484i	$-0.756517\pi$
$421$	100.615	−	174.271i	0.238991	−	0.413945i	0.960425	+	0.278538i	$0.0898499\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−363.967	+	210.136i	−0.856392	+	0.494438i
$426$		0			0
$427$	285.725	+	494.890i	0.669144	+	1.15899i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	158.719	+	91.6364i	0.368257	+	0.212613i	0.672697	−	0.739918i	$-0.265136\pi$
$431$	158.719	+	91.6364i	0.368257	+	0.212613i	−0.304440	+	0.952532i	$0.598469\pi$
$432$		0			0
$433$	−341.233	+	591.034i	−0.788068	+	1.36497i	0.139081	+	0.990281i	$0.455585\pi$
$433$	−341.233	+	591.034i	−0.788068	+	1.36497i	−0.927149	+	0.374693i	$0.877748\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−243.799	+	67.7351i	−0.557893	+	0.155000i
$438$		0			0
$439$	135.974	−	235.514i	0.309736	−	0.536478i	−0.668569	−	0.743650i	$-0.733093\pi$
$439$	135.974	−	235.514i	0.309736	−	0.536478i	0.978304	+	0.207172i	$0.0664261\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	139.879i		−	0.315754i	−0.987459	−	0.157877i	$-0.949535\pi$
$443$		−	139.879i		−	0.315754i	0.987459	−	0.157877i	$-0.0504649\pi$
$444$		0			0
$445$	47.7512	−	82.7075i	0.107306	−	0.185860i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	290.887i		−	0.647855i	−0.946082	−	0.323928i	$-0.894997\pi$
$449$		−	290.887i		−	0.647855i	0.946082	−	0.323928i	$-0.105003\pi$
$450$		0			0
$451$	−49.4990	−	85.7347i	−0.109754	−	0.190099i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−59.6884	−	34.4611i	−0.131183	−	0.0757388i
$456$		0			0
$457$	−256.031	−	443.459i	−0.560244	−	0.970371i	−0.997475	−	0.0710214i	$-0.977374\pi$
$457$	−256.031	−	443.459i	−0.560244	−	0.970371i	0.437231	−	0.899349i	$-0.355959\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−655.852	−	378.657i	−1.42267	−	0.821381i	−0.426147	−	0.904654i	$-0.640129\pi$
$461$	−655.852	−	378.657i	−1.42267	−	0.821381i	−0.996527	+	0.0832733i	$0.973463\pi$
$462$		0			0
$463$	2.97566	−	5.15400i	0.00642692	−	0.0111318i	−0.862794	−	0.505556i	$-0.831288\pi$
$463$	2.97566	−	5.15400i	0.00642692	−	0.0111318i	0.869221	+	0.494424i	$0.164621\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			329.416i			0.705387i	0.935739	+	0.352693i	$0.114734\pi$
$467$			329.416i			0.705387i	−0.935739	+	0.352693i	$0.885266\pi$
$468$		0			0
$469$	−397.752	+	688.927i	−0.848086	+	1.46893i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	138.237i		−	0.292256i
$474$		0			0
$475$	−443.530	−	114.482i	−0.933747	−	0.241014i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	63.0086i		−	0.131542i	−0.997835	−	0.0657710i	$-0.979049\pi$
$479$		−	63.0086i		−	0.131542i	0.997835	−	0.0657710i	$-0.0209507\pi$
$480$		0			0
$481$	−229.646	−	397.759i	−0.477435	−	0.826942i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	69.5630	+	40.1622i	0.143429	+	0.0828087i
$486$		0			0
$487$	−83.6068			−0.171677			−0.0858386	−	0.996309i	$-0.527357\pi$
$487$	−83.6068			−0.171677			−0.0858386	+	0.996309i	$0.527357\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	15.2612i		−	0.0310819i	−0.999879	−	0.0155409i	$-0.995053\pi$
$491$		−	15.2612i		−	0.0310819i	0.999879	−	0.0155409i	$-0.00494703\pi$
$492$		0			0
$493$	144.885	+	250.948i	0.293884	+	0.509022i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−621.565	−	358.861i	−1.25063	−	0.722054i
$498$		0			0
$499$	38.5770			0.0773086			0.0386543	−	0.999253i	$-0.487693\pi$
$499$	38.5770			0.0773086			0.0386543	+	0.999253i	$0.487693\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	146.505	+	84.5850i	0.291263	+	0.168161i	0.638511	−	0.769612i	$-0.279551\pi$
$503$	146.505	+	84.5850i	0.291263	+	0.168161i	−0.347248	+	0.937773i	$0.612884\pi$
$504$		0			0
$505$	−185.653			−0.367630
$506$		0			0
$507$		0			0
$508$		0			0
$509$	403.328	−	232.861i	0.792392	−	0.457488i	−0.0484118	−	0.998827i	$-0.515416\pi$
$509$	403.328	−	232.861i	0.792392	−	0.457488i	0.840804	+	0.541340i	$0.182083\pi$
$510$		0			0
$511$	639.154	−	1107.05i	1.25079	−	2.16643i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	53.7866	+	31.0537i	0.104440	+	0.0602984i
$516$		0			0
$517$	61.7512	+	106.956i	0.119441	+	0.206879i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			940.532i			1.80524i	0.430433	+	0.902622i	$0.358361\pi$
$521$			940.532i			1.80524i	−0.430433	+	0.902622i	$0.641639\pi$
$522$		0			0
$523$	180.183	−	312.086i	0.344518	−	0.596723i	−0.640748	−	0.767751i	$-0.721376\pi$
$523$	180.183	−	312.086i	0.344518	−	0.596723i	0.985266	+	0.171028i	$0.0547090\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			164.523i			0.312188i
$528$		0			0
$529$	−175.821	+	304.531i	−0.332365	+	0.575673i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−215.256	+	124.278i	−0.403857	+	0.233167i
$534$		0			0
$535$	−1.88988			−0.00353249
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	149.823i		−	0.277965i
$540$		0			0
$541$	242.147	−	419.411i	0.447591	−	0.775251i	−0.550638	−	0.834744i	$-0.685615\pi$
$541$	242.147	−	419.411i	0.447591	−	0.775251i	0.998229	+	0.0594939i	$0.0189487\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	55.4974	−	32.0414i	0.101830	−	0.0587916i
$546$		0			0
$547$	31.1222			0.0568961			0.0284481	−	0.999595i	$-0.490943\pi$
$547$	31.1222			0.0568961			0.0284481	+	0.999595i	$0.490943\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−78.9330	+	305.805i	−0.143254	+	0.555000i
$552$		0			0
$553$	67.5383			0.122131
$554$		0			0
$555$		0			0
$556$		0			0
$557$	267.742	+	154.581i	0.480685	+	0.277524i	0.720702	−	0.693245i	$-0.243820\pi$
$557$	267.742	+	154.581i	0.480685	+	0.277524i	−0.240017	+	0.970769i	$0.577153\pi$
$558$		0			0
$559$	−347.075			−0.620885
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−694.453	−	400.943i	−1.23349	−	0.712154i	−0.265732	−	0.964047i	$-0.585614\pi$
$563$	−694.453	−	400.943i	−1.23349	−	0.712154i	−0.967755	+	0.251893i	$0.918947\pi$
$564$		0			0
$565$	50.2744	−	87.0778i	0.0889813	−	0.154120i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−94.4488	+	54.5300i	−0.165991	+	0.0958349i	−0.580694	−	0.814122i	$-0.697219\pi$
$569$	−94.4488	+	54.5300i	−0.165991	+	0.0958349i	0.414703	+	0.909957i	$0.363885\pi$
$570$		0			0
$571$	240.420	−	416.419i	0.421050	−	0.729281i	−0.574992	−	0.818159i	$-0.694995\pi$
$571$	240.420	−	416.419i	0.421050	−	0.729281i	0.996043	+	0.0888782i	$0.0283282\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	278.055	−	160.535i	0.483573	−	0.279191i
$576$		0			0
$577$	−1108.78			−1.92164			−0.960818	−	0.277179i	$-0.910601\pi$
$577$	−1108.78			−1.92164			−0.960818	+	0.277179i	$0.910601\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1021.80	−	589.939i	−1.75870	−	1.01538i
$582$		0			0
$583$	−234.083			−0.401515
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−840.894	−	485.490i	−1.43253	−	0.827070i	−0.435215	−	0.900327i	$-0.643328\pi$
$587$	−840.894	−	485.490i	−1.43253	−	0.827070i	−0.997313	+	0.0732563i	$0.976661\pi$
$588$		0			0
$589$	−125.626	+	127.958i	−0.213286	+	0.217246i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	399.441	+	230.618i	0.673594	+	0.388900i	0.797437	−	0.603402i	$-0.206189\pi$
$593$	399.441	+	230.618i	0.673594	+	0.388900i	−0.123843	+	0.992302i	$0.539522\pi$
$594$		0			0
$595$	82.5765	−	143.027i	0.138784	−	0.240381i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	930.550	−	537.253i	1.55351	−	0.896917i	0.555653	−	0.831415i	$-0.312468\pi$
$599$	930.550	−	537.253i	1.55351	−	0.896917i	0.997852	−	0.0655020i	$-0.0208649\pi$
$600$		0			0
$601$	−352.905	−	611.249i	−0.587196	−	1.01705i	−0.994598	−	0.103805i	$-0.966898\pi$
$601$	−352.905	−	611.249i	−0.587196	−	1.01705i	0.407401	−	0.913249i	$-0.366435\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	92.0632	+	53.1527i	0.152171	+	0.0878557i
$606$		0			0
$607$	499.764	−	865.617i	0.823335	−	1.42606i	−0.0798504	−	0.996807i	$-0.525444\pi$
$607$	499.764	−	865.617i	0.823335	−	1.42606i	0.903185	−	0.429251i	$-0.141222\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	268.537	−	155.040i	0.439504	−	0.253748i
$612$		0			0
$613$	−26.0192	−	45.0666i	−0.0424457	−	0.0735181i	0.844022	−	0.536308i	$-0.180182\pi$
$613$	−26.0192	−	45.0666i	−0.0424457	−	0.0735181i	−0.886468	+	0.462790i	$0.846848\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1025.06	+	591.821i	−1.66137	+	0.959191i	−0.689305	+	0.724471i	$0.742084\pi$
$617$	−1025.06	+	591.821i	−1.66137	+	0.959191i	−0.972063	+	0.234721i	$0.924583\pi$
$618$		0			0
$619$	−94.2234	−	163.200i	−0.152219	−	0.263650i	0.779824	−	0.625999i	$-0.215308\pi$
$619$	−94.2234	−	163.200i	−0.152219	−	0.263650i	−0.932043	+	0.362348i	$0.881975\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1015.18i		−	1.62950i
$624$		0			0
$625$	558.950			0.894320
$626$		0			0
$627$		0			0
$628$		0			0
$629$	953.120	−	550.284i	1.51529	−	0.874855i
$630$		0			0
$631$	80.3672	−	139.200i	0.127365	−	0.220602i	−0.795290	−	0.606229i	$-0.792681\pi$
$631$	80.3672	−	139.200i	0.127365	−	0.220602i	0.922655	+	0.385627i	$0.126015\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.0155	+	8.09183i	0.0220716	+	0.0127430i
$636$		0			0
$637$	−376.164			−0.590524
$638$		0			0
$639$		0			0
$640$		0			0
$641$	29.1362	+	16.8218i	0.0454543	+	0.0262430i	0.522555	−	0.852606i	$-0.324979\pi$
$641$	29.1362	+	16.8218i	0.0454543	+	0.0262430i	−0.477101	+	0.878849i	$0.658312\pi$
$642$		0			0
$643$	−681.548			−1.05995			−0.529975	−	0.848013i	$-0.677799\pi$
$643$	−681.548			−1.05995			−0.529975	+	0.848013i	$0.677799\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	823.205i		−	1.27234i	−0.771548	−	0.636171i	$-0.780517\pi$
$647$		−	823.205i		−	1.27234i	0.771548	−	0.636171i	$-0.219483\pi$
$648$		0			0
$649$	−133.458	−	231.156i	−0.205637	−	0.356173i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	252.547	+	145.808i	0.386750	+	0.223290i	0.680751	−	0.732515i	$-0.261654\pi$
$653$	252.547	+	145.808i	0.386750	+	0.223290i	−0.294001	+	0.955805i	$0.594987\pi$
$654$		0			0
$655$	165.486			0.252650
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	578.433i		−	0.877743i	−0.898550	−	0.438871i	$-0.855378\pi$
$659$		−	578.433i		−	0.877743i	0.898550	−	0.438871i	$-0.144622\pi$
$660$		0			0
$661$	552.534	+	957.017i	0.835906	+	1.44783i	0.893291	+	0.449479i	$0.148390\pi$
$661$	552.534	+	957.017i	0.835906	+	1.44783i	−0.0573850	+	0.998352i	$0.518276\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	173.435	−	48.1858i	0.260805	−	0.0724599i
$666$		0			0
$667$	−110.686	−	191.713i	−0.165946	−	0.287426i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	142.893	−	82.4991i	0.212955	−	0.122950i
$672$		0			0
$673$	−373.740	−	647.337i	−0.555334	−	0.961867i	−0.997877	−	0.0651201i	$-0.979257\pi$
$673$	−373.740	−	647.337i	−0.555334	−	0.961867i	0.442543	−	0.896747i	$-0.354076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−630.167	+	363.827i	−0.930823	+	0.537411i	−0.887072	−	0.461632i	$-0.847264\pi$
$677$	−630.167	+	363.827i	−0.930823	+	0.537411i	−0.0437512	+	0.999042i	$0.513931\pi$
$678$		0			0
$679$	853.840			1.25750
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	123.112i		−	0.180252i	−0.995930	−	0.0901260i	$-0.971273\pi$
$683$		−	123.112i		−	0.180252i	0.995930	−	0.0901260i	$-0.0287270\pi$
$684$		0			0
$685$	161.845			0.236271
$686$		0			0
$687$		0			0
$688$		0			0
$689$			587.718i			0.853001i
$690$		0			0
$691$	471.063	+	815.905i	0.681712	+	1.18076i	0.974458	+	0.224569i	$0.0720975\pi$
$691$	471.063	+	815.905i	0.681712	+	1.18076i	−0.292746	+	0.956190i	$0.594569\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	52.2265	−	30.1530i	0.0751461	−	0.0433856i
$696$		0			0
$697$	−297.798	−	515.801i	−0.427257	−	0.740030i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−557.020	+	321.596i	−0.794608	+	0.458767i	−0.841582	−	0.540129i	$-0.818375\pi$
$701$	−557.020	+	321.596i	−0.794608	+	0.458767i	0.0469742	+	0.998896i	$0.485042\pi$
$702$		0			0
$703$	1161.47	+	299.794i	1.65216	+	0.426449i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1709.08	+	986.736i	−2.41736	+	1.39567i
$708$		0			0
$709$	508.567			0.717301			0.358651	−	0.933472i	$-0.383237\pi$
$709$	508.567			0.717301			0.358651	+	0.933472i	$0.383237\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	125.689i		−	0.176281i
$714$		0			0
$715$	−9.95019	+	17.2342i	−0.0139163	+	0.0241038i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	581.261	−	335.591i	0.808430	−	0.466747i	−0.0379806	−	0.999278i	$-0.512093\pi$
$719$	581.261	−	335.591i	0.808430	−	0.466747i	0.846410	+	0.532531i	$0.178759\pi$
$720$		0			0
$721$	660.194			0.915665
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	400.748i		−	0.552756i
$726$		0			0
$727$	−243.419	+	421.613i	−0.334826	+	0.579936i	−0.983451	−	0.181172i	$-0.942011\pi$
$727$	−243.419	+	421.613i	−0.334826	+	0.579936i	0.648625	+	0.761108i	$0.275344\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	831.668i		−	1.13771i
$732$		0			0
$733$	244.666	−	423.774i	0.333787	−	0.578136i	−0.649464	−	0.760392i	$-0.725007\pi$
$733$	244.666	−	423.774i	0.333787	−	0.578136i	0.983251	+	0.182256i	$0.0583400\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	198.918	+	114.846i	0.269903	+	0.155828i
$738$		0			0
$739$	−313.307	−	542.664i	−0.423961	−	0.734322i	0.572362	−	0.820001i	$-0.306027\pi$
$739$	−313.307	−	542.664i	−0.423961	−	0.734322i	−0.996323	+	0.0856793i	$0.972694\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1256.65i			1.69132i	0.533725	+	0.845658i	$0.320792\pi$
$743$			1256.65i			1.69132i	−0.533725	+	0.845658i	$0.679208\pi$
$744$		0			0
$745$	152.134			0.204207
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−17.3978	+	10.0446i	−0.0232280	+	0.0134107i
$750$		0			0
$751$	396.432	+	686.641i	0.527873	+	0.914302i	0.999472	+	0.0324894i	$0.0103435\pi$
$751$	396.432	+	686.641i	0.527873	+	0.914302i	−0.471599	+	0.881813i	$0.656323\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	158.980	−	91.7874i	0.210570	−	0.121573i
$756$		0			0
$757$	341.349	+	591.233i	0.450923	+	0.781021i	0.998444	−	0.0557708i	$-0.0177616\pi$
$757$	341.349	+	591.233i	0.450923	+	0.781021i	−0.547521	+	0.836792i	$0.684428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−931.990	−	538.085i	−1.22469	−	0.707076i	−0.258777	−	0.965937i	$-0.583319\pi$
$761$	−931.990	−	538.085i	−1.22469	−	0.707076i	−0.965915	+	0.258861i	$0.916653\pi$
$762$		0			0
$763$	340.596	−	589.930i	0.446391	−	0.773172i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−580.369	+	335.076i	−0.756674	+	0.436866i
$768$		0			0
$769$	14.6952	+	25.4528i	0.0191095	+	0.0330986i	0.875422	−	0.483359i	$-0.160584\pi$
$769$	14.6952	+	25.4528i	0.0191095	+	0.0330986i	−0.856313	+	0.516458i	$0.827250\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	587.271	+	339.061i	0.759729	+	0.438630i	0.829198	−	0.558954i	$-0.188797\pi$
$773$	587.271	+	339.061i	0.759729	+	0.438630i	−0.0694693	+	0.997584i	$0.522131\pi$
$774$		0			0
$775$	113.767	−	197.050i	0.146796	−	0.254258i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	162.240	−	628.555i	0.208267	−	0.806874i
$780$		0			0
$781$	−103.616	+	179.469i	−0.132671	+	0.229793i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			56.7361i			0.0722753i
$786$		0			0
$787$	472.469	−	818.341i	0.600342	−	1.03982i	−0.392427	−	0.919783i	$-0.628364\pi$
$787$	472.469	−	818.341i	0.600342	−	1.03982i	0.992769	−	0.120040i	$-0.0383022\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1068.82i		−	1.35123i
$792$		0			0
$793$	−207.132	−	358.764i	−0.261201	−	0.452413i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1032.66	−	596.209i	−1.29569	−	0.748066i	−0.316032	−	0.948748i	$-0.602351\pi$
$797$	−1032.66	−	596.209i	−1.29569	−	0.748066i	−0.979656	+	0.200682i	$0.935684\pi$
$798$		0			0
$799$	371.510	+	643.475i	0.464969	+	0.805350i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−319.645	−	184.547i	−0.398063	−	0.229822i
$804$		0			0
$805$	−63.0849	+	109.266i	−0.0783663	+	0.135734i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1351.83i			1.67099i	0.549502	+	0.835493i	$0.314818\pi$
$809$			1351.83i			1.67099i	−0.549502	+	0.835493i	$0.685182\pi$
$810$		0			0
$811$	−79.2739	+	137.307i	−0.0977484	+	0.169305i	−0.910752	−	0.412953i	$-0.864497\pi$
$811$	−79.2739	+	137.307i	−0.0977484	+	0.169305i	0.813004	+	0.582258i	$0.197831\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	34.9700i		−	0.0429080i
$816$		0			0
$817$	635.040	−	646.830i	0.777282	−	0.791713i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			975.246i			1.18788i	0.804511	+	0.593938i	$0.202427\pi$
$821$			975.246i			1.18788i	−0.804511	+	0.593938i	$0.797573\pi$
$822$		0			0
$823$	−59.2050	−	102.546i	−0.0719380	−	0.124600i	0.827813	−	0.561005i	$-0.189585\pi$
$823$	−59.2050	−	102.546i	−0.0719380	−	0.124600i	−0.899751	+	0.436404i	$0.856252\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−893.922	−	516.106i	−1.08092	−	0.624070i	−0.149777	−	0.988720i	$-0.547855\pi$
$827$	−893.922	−	516.106i	−1.08092	−	0.624070i	−0.931145	+	0.364650i	$0.881189\pi$
$828$		0			0
$829$	21.5992			0.0260546			0.0130273	−	0.999915i	$-0.495853\pi$
$829$	21.5992			0.0260546			0.0130273	+	0.999915i	$0.495853\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	901.372i		−	1.08208i
$834$		0			0
$835$	10.5429	+	18.2609i	0.0126262	+	0.0218693i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1030.30	−	594.841i	−1.22800	−	0.708988i	−0.261392	−	0.965233i	$-0.584182\pi$
$839$	−1030.30	−	594.841i	−1.22800	−	0.708988i	−0.966612	+	0.256244i	$0.917515\pi$
$840$		0			0
$841$	564.692			0.671453
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−94.9011	−	54.7912i	−0.112309	−	0.0648416i
$846$		0			0
$847$	1130.01			1.33414
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−728.142	+	420.393i	−0.855631	+	0.493999i
$852$		0			0
$853$	320.869	−	555.761i	0.376165	−	0.651537i	−0.614336	−	0.789045i	$-0.710576\pi$
$853$	320.869	−	555.761i	0.376165	−	0.651537i	0.990501	+	0.137508i	$0.0439093\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	217.083	+	125.333i	0.253305	+	0.146246i	0.621277	−	0.783591i	$-0.286614\pi$
$857$	217.083	+	125.333i	0.253305	+	0.146246i	−0.367972	+	0.929837i	$0.619948\pi$
$858$		0			0
$859$	−19.1802	−	33.2211i	−0.0223285	−	0.0386741i	0.854645	−	0.519212i	$-0.173775\pi$
$859$	−19.1802	−	33.2211i	−0.0223285	−	0.0386741i	−0.876974	+	0.480538i	$0.840441\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1063.03i		−	1.23178i	−0.787832	−	0.615890i	$-0.788796\pi$
$863$		−	1063.03i		−	1.23178i	0.787832	−	0.615890i	$-0.211204\pi$
$864$		0			0
$865$	−103.344	+	178.997i	−0.119473	+	0.206933i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	19.5008i		−	0.0224405i
$870$		0			0
$871$	288.345	−	499.428i	0.331051	−	0.573396i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−402.921	+	232.626i	−0.460481	+	0.265859i
$876$		0			0
$877$	−1249.88			−1.42518			−0.712588	−	0.701583i	$-0.752477\pi$
$877$	−1249.88			−1.42518			−0.712588	+	0.701583i	$0.752477\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1354.58i			1.53755i	0.639522	+	0.768773i	$0.279132\pi$
$881$			1354.58i			1.53755i	−0.639522	+	0.768773i	$0.720868\pi$
$882$		0			0
$883$	124.127	−	214.994i	0.140574	−	0.243481i	−0.787139	−	0.616776i	$-0.788439\pi$
$883$	124.127	−	214.994i	0.140574	−	0.243481i	0.927713	+	0.373294i	$0.121772\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−323.107	+	186.546i	−0.364269	+	0.210311i	−0.670952	−	0.741501i	$-0.734114\pi$
$887$	−323.107	+	186.546i	−0.364269	+	0.210311i	0.306683	+	0.951812i	$0.400781\pi$
$888$		0			0
$889$	172.030			0.193510
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−202.398	+	784.138i	−0.226650	+	0.878094i
$894$		0			0
$895$	67.3618			0.0752646
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−135.862	−	78.4400i	−0.151126	−	0.0872525i
$900$		0			0
$901$	−1408.30			−1.56304
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−70.3079	−	40.5923i	−0.0776883	−	0.0448533i
$906$		0			0
$907$	418.749	−	725.295i	0.461686	−	0.799664i	−0.537359	−	0.843354i	$-0.680578\pi$
$907$	418.749	−	725.295i	0.461686	−	0.799664i	0.999045	+	0.0436898i	$0.0139113\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	806.747	−	465.775i	0.885562	−	0.511279i	0.0130736	−	0.999915i	$-0.495838\pi$
$911$	806.747	−	465.775i	0.885562	−	0.511279i	0.872488	+	0.488635i	$0.162505\pi$
$912$		0			0
$913$	−170.337	+	295.032i	−0.186568	+	0.323146i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1523.42	−	879.546i	1.66131	−	0.959156i
$918$		0			0
$919$	−72.4463			−0.0788316			−0.0394158	−	0.999223i	$-0.512550\pi$
$919$	−72.4463			−0.0788316			−0.0394158	+	0.999223i	$0.512550\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	450.595	+	260.151i	0.488185	+	0.281854i
$924$		0			0
$925$	−1522.07			−1.64548
$926$		0			0
$927$		0			0
$928$		0			0
$929$	788.117	+	455.020i	0.848350	+	0.489795i	0.860094	−	0.510136i	$-0.170405\pi$
$929$	788.117	+	455.020i	0.848350	+	0.489795i	−0.0117435	+	0.999931i	$0.503738\pi$
$930$		0			0
$931$	688.264	−	701.042i	0.739274	−	0.752999i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−41.2970	−	23.8429i	−0.0441680	−	0.0255004i
$936$		0			0
$937$	−248.636	+	430.651i	−0.265354	+	0.459606i	−0.967656	−	0.252273i	$-0.918822\pi$
$937$	−248.636	+	430.651i	−0.265354	+	0.459606i	0.702303	+	0.711879i	$0.252155\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1037.45	−	598.972i	1.10250	−	0.636527i	0.165622	−	0.986189i	$-0.447037\pi$
$941$	1037.45	−	598.972i	1.10250	−	0.636527i	0.936876	+	0.349662i	$0.113704\pi$
$942$		0			0
$943$	227.505	+	394.049i	0.241256	+	0.417868i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	282.836	+	163.295i	0.298665	+	0.172434i	0.641843	−	0.766836i	$-0.278170\pi$
$947$	282.836	+	163.295i	0.298665	+	0.172434i	−0.343178	+	0.939270i	$0.611503\pi$
$948$		0			0
$949$	−463.346	+	802.539i	−0.488246	+	0.845668i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.86935	−	2.23397i	0.00406018	−	0.00234415i	−0.497969	−	0.867195i	$-0.665921\pi$
$953$	3.86935	−	2.23397i	0.00406018	−	0.00234415i	0.502029	+	0.864851i	$0.332587\pi$
$954$		0			0
$955$	150.700	+	261.021i	0.157802	+	0.273320i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1489.91	−	860.199i	1.55361	−	0.896975i
$960$		0			0
$961$	435.964	+	755.112i	0.453657	+	0.785756i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	106.462i		−	0.110323i
$966$		0			0
$967$	−68.7657			−0.0711124			−0.0355562	−	0.999368i	$-0.511320\pi$
$967$	−68.7657			−0.0711124			−0.0355562	+	0.999368i	$0.511320\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1310.58	+	756.661i	−1.34972	+	0.779260i	−0.988209	−	0.153109i	$-0.951071\pi$
$971$	−1310.58	+	756.661i	−1.34972	+	0.779260i	−0.361508	+	0.932369i	$0.617738\pi$
$972$		0			0
$973$	320.523	−	555.162i	0.329417	−	0.570567i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	360.077	+	207.891i	0.368554	+	0.212785i	0.672826	−	0.739800i	$-0.265080\pi$
$977$	360.077	+	207.891i	0.368554	+	0.212785i	−0.304273	+	0.952585i	$0.598413\pi$
$978$		0			0
$979$	−293.119			−0.299407
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1468.69	−	847.950i	−1.49409	−	0.862615i	−0.494115	−	0.869396i	$-0.664508\pi$
$983$	−1468.69	−	847.950i	−1.49409	−	0.862615i	−0.999977	+	0.00678179i	$0.997841\pi$
$984$		0			0
$985$	89.0989			0.0904557
$986$		0			0
$987$		0			0
$988$		0			0
$989$			635.358i			0.642425i
$990$		0			0
$991$	488.526	+	846.152i	0.492963	+	0.853837i	0.999967	−	0.00810664i	$-0.00258045\pi$
$991$	488.526	+	846.152i	0.492963	+	0.853837i	−0.507004	+	0.861944i	$0.669247\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−244.301	−	141.047i	−0.245528	−	0.141756i
$996$		0			0
$997$	−219.619			−0.220280			−0.110140	−	0.993916i	$-0.535130\pi$
$997$	−219.619			−0.220280			−0.110140	+	0.993916i	$0.535130\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.m.a.881.18		80
3.2	odd	2		684.3.m.a.653.2	yes	80
9.2	odd	6		2052.3.be.a.197.18		80
9.7	even	3		684.3.be.a.425.28	yes	80
19.11	even	3		2052.3.be.a.125.18		80
57.11	odd	6		684.3.be.a.581.28	yes	80
171.11	odd	6	inner	2052.3.m.a.1493.23		80
171.106	even	3		684.3.m.a.353.2	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.2	✓	80	171.106	even	3
684.3.m.a.653.2	yes	80	3.2	odd	2
684.3.be.a.425.28	yes	80	9.7	even	3
684.3.be.a.581.28	yes	80	57.11	odd	6
2052.3.m.a.881.18		80	1.1	even	1	trivial
2052.3.m.a.1493.23		80	171.11	odd	6	inner
2052.3.be.a.125.18		80	19.11	even	3
2052.3.be.a.197.18		80	9.2	odd	6

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:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-0.944063i q^{5} +(-5.01764 - 8.69081i) q^{7} +O(q^{10})\)	\(q-0.944063i q^{5} +(-5.01764 - 8.69081i) q^{7} +(-2.50935 + 1.44878i) q^{11} +(3.63747 + 6.30028i) q^{13} +(-15.0969 + 8.71618i) q^{17} +(-18.3970 - 4.74856i) q^{19} +(11.5334 - 6.65879i) q^{23} +24.1087 q^{25} -16.6225i q^{29} +(4.71890 - 8.17338i) q^{31} +(-8.20467 + 4.73697i) q^{35} -63.1336 q^{37} +34.1661i q^{41} +(-23.8541 + 41.3166i) q^{43} -42.6230i q^{47} +(-25.8534 + 44.7794i) q^{49} +(69.9633 + 40.3933i) q^{53} +(1.36774 + 2.36899i) q^{55} +92.1179i q^{59} -56.9441 q^{61} +(5.94786 - 3.43400i) q^{65} +(-39.6354 - 68.6505i) q^{67} +(61.9380 - 35.7599i) q^{71} +(63.6907 + 110.316i) q^{73} +(25.1820 + 14.5389i) q^{77} +(-3.36504 + 5.82843i) q^{79} +(101.821 - 58.7865i) q^{83} +(8.22863 + 14.2524i) q^{85} +(87.6080 + 50.5805i) q^{89} +(36.5030 - 63.2251i) q^{91} +(-4.48294 + 17.3680i) q^{95} +(-42.5419 + 73.6847i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

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