LMFDB - Embedded newform 756.3.m.a.557.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,3,Mod(557,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 4]))

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

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

chi := DirichletCharacter("756.557");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.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:	$20.5995079856$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			557.16
Character	$\chi$	$=$	756.557
Dual form			756.3.m.a.737.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.81763i q^{5} +(-1.80016 - 6.76457i) q^{7} +O(q^{10})$	$q+7.81763i q^{5} +(-1.80016 - 6.76457i) q^{7} -9.90888i q^{11} +(-5.60780 + 9.71299i) q^{13} +(-12.1030 - 6.98765i) q^{17} +(7.19662 + 12.4649i) q^{19} +38.2153i q^{23} -36.1153 q^{25} +(-20.5328 + 11.8546i) q^{29} +(-12.5727 - 21.7766i) q^{31} +(52.8829 - 14.0730i) q^{35} +(-19.6806 - 34.0879i) q^{37} +(-45.5225 - 26.2824i) q^{41} +(-35.6209 - 61.6972i) q^{43} +(46.9954 + 27.1328i) q^{47} +(-42.5188 + 24.3547i) q^{49} +(-47.3589 - 27.3427i) q^{53} +77.4639 q^{55} +(-33.8465 + 19.5413i) q^{59} +(32.5574 - 56.3910i) q^{61} +(-75.9325 - 43.8397i) q^{65} +(-53.4942 - 92.6547i) q^{67} -27.3419i q^{71} +(-0.465523 + 0.806310i) q^{73} +(-67.0293 + 17.8376i) q^{77} +(-47.1616 + 81.6863i) q^{79} +(-81.5203 + 47.0658i) q^{83} +(54.6269 - 94.6165i) q^{85} +(17.9546 - 10.3661i) q^{89} +(75.7992 + 20.4494i) q^{91} +(-97.4460 + 56.2605i) q^{95} +(77.0276 + 133.416i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - q^{7}+O(q^{10}) $ 32 * q - q^7	$ 32 q - q^{7} - 5 q^{13} + 27 q^{17} - 14 q^{19} - 160 q^{25} - 36 q^{29} - 8 q^{31} + 45 q^{35} - 11 q^{37} - 72 q^{41} + 16 q^{43} + 108 q^{47} + 35 q^{49} - 180 q^{53} - 24 q^{55} - 45 q^{59} - 41 q^{61} + 81 q^{65} - 35 q^{67} - 98 q^{73} - 225 q^{77} - 71 q^{79} - 30 q^{85} + 189 q^{89} + 109 q^{91} + 288 q^{95} + 19 q^{97}+O(q^{100}) $ 32 * q - q^7 - 5 * q^13 + 27 * q^17 - 14 * q^19 - 160 * q^25 - 36 * q^29 - 8 * q^31 + 45 * q^35 - 11 * q^37 - 72 * q^41 + 16 * q^43 + 108 * q^47 + 35 * q^49 - 180 * q^53 - 24 * q^55 - 45 * q^59 - 41 * q^61 + 81 * q^65 - 35 * q^67 - 98 * q^73 - 225 * q^77 - 71 * q^79 - 30 * q^85 + 189 * q^89 + 109 * q^91 + 288 * q^95 + 19 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$			7.81763i			1.56353i	0.623576	+	0.781763i	$0.285679\pi$
$5$			7.81763i			1.56353i	−0.623576	+	0.781763i	$0.714321\pi$
$6$		0			0
$7$	−1.80016	−	6.76457i	−0.257166	−	0.966367i
$7$	−1.80016	−	6.76457i	−0.257166	−	0.966367i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	9.90888i		−	0.900808i	−0.892825	−	0.450404i	$-0.851280\pi$
$11$		−	9.90888i		−	0.900808i	0.892825	−	0.450404i	$-0.148720\pi$
$12$		0			0
$13$	−5.60780	+	9.71299i	−0.431369	+	0.747153i	−0.996991	−	0.0775113i	$-0.975303\pi$
$13$	−5.60780	+	9.71299i	−0.431369	+	0.747153i	0.565622	+	0.824664i	$0.308636\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.1030	−	6.98765i	−0.711939	−	0.411038i	0.0998393	−	0.995004i	$-0.468167\pi$
$17$	−12.1030	−	6.98765i	−0.711939	−	0.411038i	−0.811779	+	0.583965i	$0.801500\pi$
$18$		0			0
$19$	7.19662	+	12.4649i	0.378769	+	0.656048i	0.990883	−	0.134722i	$-0.0430141\pi$
$19$	7.19662	+	12.4649i	0.378769	+	0.656048i	−0.612114	+	0.790769i	$0.709681\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			38.2153i			1.66154i	0.556620	+	0.830768i	$0.312098\pi$
$23$			38.2153i			1.66154i	−0.556620	+	0.830768i	$0.687902\pi$
$24$		0			0
$25$	−36.1153			−1.44461
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−20.5328	+	11.8546i	−0.708026	+	0.408779i	−0.810330	−	0.585974i	$-0.800712\pi$
$29$	−20.5328	+	11.8546i	−0.708026	+	0.408779i	0.102304	+	0.994753i	$0.467379\pi$
$30$		0			0
$31$	−12.5727	−	21.7766i	−0.405571	−	0.702470i	0.588816	−	0.808267i	$-0.299594\pi$
$31$	−12.5727	−	21.7766i	−0.405571	−	0.702470i	−0.994388	+	0.105797i	$0.966261\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	52.8829	−	14.0730i	1.51094	−	0.402086i
$36$		0			0
$37$	−19.6806	−	34.0879i	−0.531909	−	0.921294i	−0.999306	−	0.0372462i	$-0.988141\pi$
$37$	−19.6806	−	34.0879i	−0.531909	−	0.921294i	0.467397	−	0.884048i	$-0.345192\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−45.5225	−	26.2824i	−1.11030	−	0.641035i	−0.171396	−	0.985202i	$-0.554828\pi$
$41$	−45.5225	−	26.2824i	−1.11030	−	0.641035i	−0.938908	+	0.344168i	$0.888161\pi$
$42$		0			0
$43$	−35.6209	−	61.6972i	−0.828393	−	1.43482i	−0.899298	−	0.437335i	$-0.855922\pi$
$43$	−35.6209	−	61.6972i	−0.828393	−	1.43482i	0.0709056	−	0.997483i	$-0.477411\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	46.9954	+	27.1328i	0.999902	+	0.577294i	0.908219	−	0.418494i	$-0.137442\pi$
$47$	46.9954	+	27.1328i	0.999902	+	0.577294i	0.0916829	+	0.995788i	$0.470775\pi$
$48$		0			0
$49$	−42.5188	+	24.3547i	−0.867731	+	0.497034i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−47.3589	−	27.3427i	−0.893565	−	0.515900i	−0.0184578	−	0.999830i	$-0.505876\pi$
$53$	−47.3589	−	27.3427i	−0.893565	−	0.515900i	−0.875107	+	0.483930i	$0.839209\pi$
$54$		0			0
$55$	77.4639			1.40844
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−33.8465	+	19.5413i	−0.573670	+	0.331209i	−0.758614	−	0.651540i	$-0.774123\pi$
$59$	−33.8465	+	19.5413i	−0.573670	+	0.331209i	0.184944	+	0.982749i	$0.440790\pi$
$60$		0			0
$61$	32.5574	−	56.3910i	0.533727	−	0.924443i	−0.465497	−	0.885050i	$-0.654124\pi$
$61$	32.5574	−	56.3910i	0.533727	−	0.924443i	0.999224	−	0.0393930i	$-0.0125424\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−75.9325	−	43.8397i	−1.16819	−	0.674456i
$66$		0			0
$67$	−53.4942	−	92.6547i	−0.798421	−	1.38291i	−0.920644	−	0.390403i	$-0.872336\pi$
$67$	−53.4942	−	92.6547i	−0.798421	−	1.38291i	0.122223	−	0.992503i	$-0.460998\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	27.3419i		−	0.385097i	−0.981287	−	0.192549i	$-0.938325\pi$
$71$		−	27.3419i		−	0.385097i	0.981287	−	0.192549i	$-0.0616753\pi$
$72$		0			0
$73$	−0.465523	+	0.806310i	−0.00637703	+	0.0110453i	−0.869196	−	0.494467i	$-0.835363\pi$
$73$	−0.465523	+	0.806310i	−0.00637703	+	0.0110453i	0.862819	+	0.505513i	$0.168697\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−67.0293	+	17.8376i	−0.870511	+	0.231657i
$78$		0			0
$79$	−47.1616	+	81.6863i	−0.596983	+	1.03400i	0.396281	+	0.918129i	$0.370301\pi$
$79$	−47.1616	+	81.6863i	−0.596983	+	1.03400i	−0.993264	+	0.115875i	$0.963033\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−81.5203	+	47.0658i	−0.982172	+	0.567057i	−0.902925	−	0.429798i	$-0.858585\pi$
$83$	−81.5203	+	47.0658i	−0.982172	+	0.567057i	−0.0792470	+	0.996855i	$0.525252\pi$
$84$		0			0
$85$	54.6269	−	94.6165i	0.642669	−	1.11314i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	17.9546	−	10.3661i	0.201738	−	0.116473i	−0.395728	−	0.918368i	$-0.629508\pi$
$89$	17.9546	−	10.3661i	0.201738	−	0.116473i	0.597466	+	0.801894i	$0.296174\pi$
$90$		0			0
$91$	75.7992	+	20.4494i	0.832958	+	0.224718i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−97.4460	+	56.2605i	−1.02575	+	0.592215i
$96$		0			0
$97$	77.0276	+	133.416i	0.794099	+	1.37542i	0.923409	+	0.383816i	$0.125390\pi$
$97$	77.0276	+	133.416i	0.794099	+	1.37542i	−0.129310	+	0.991604i	$0.541276\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	36.8047i		−	0.364403i	−0.983261	−	0.182201i	$-0.941678\pi$
$101$		−	36.8047i		−	0.364403i	0.983261	−	0.182201i	$-0.0583223\pi$
$102$		0			0
$103$	75.4332			0.732361			0.366181	−	0.930544i	$-0.380665\pi$
$103$	75.4332			0.732361			0.366181	+	0.930544i	$0.380665\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−72.7667	+	42.0119i	−0.680062	+	0.392634i	−0.799879	−	0.600162i	$-0.795103\pi$
$107$	−72.7667	+	42.0119i	−0.680062	+	0.392634i	0.119816	+	0.992796i	$0.461769\pi$
$108$		0			0
$109$	−81.0516	+	140.385i	−0.743592	+	1.28794i	0.207257	+	0.978287i	$0.433546\pi$
$109$	−81.0516	+	140.385i	−0.743592	+	1.28794i	−0.950850	+	0.309653i	$0.899787\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.5728	+	10.7230i	0.164361	+	0.0948940i	0.579924	−	0.814670i	$-0.303082\pi$
$113$	18.5728	+	10.7230i	0.164361	+	0.0948940i	−0.415563	+	0.909564i	$0.636415\pi$
$114$		0			0
$115$	−298.753			−2.59785
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−25.4811	+	94.4503i	−0.214127	+	0.793700i
$120$		0			0
$121$	22.8140			0.188546
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	86.8950i		−	0.695160i
$126$		0			0
$127$	−53.7261			−0.423040			−0.211520	−	0.977374i	$-0.567841\pi$
$127$	−53.7261			−0.423040			−0.211520	+	0.977374i	$0.567841\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			61.0950i			0.466374i	0.972432	+	0.233187i	$0.0749154\pi$
$131$			61.0950i			0.466374i	−0.972432	+	0.233187i	$0.925085\pi$
$132$		0			0
$133$	71.3646	−	71.1209i	0.536576	−	0.534744i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	122.968i		−	0.897580i	−0.893637	−	0.448790i	$-0.851855\pi$
$137$		−	122.968i		−	0.897580i	0.893637	−	0.448790i	$-0.148145\pi$
$138$		0			0
$139$	3.77631	−	6.54076i	0.0271677	−	0.0470558i	−0.852122	−	0.523343i	$-0.824685\pi$
$139$	3.77631	−	6.54076i	0.0271677	−	0.0470558i	0.879290	+	0.476288i	$0.158018\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	96.2449	+	55.5670i	0.673041	+	0.388580i
$144$		0			0
$145$	−92.6748	−	160.517i	−0.639136	−	1.10702i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			217.813i			1.46183i	0.682467	+	0.730917i	$0.260907\pi$
$149$			217.813i			1.46183i	−0.682467	+	0.730917i	$0.739093\pi$
$150$		0			0
$151$	225.421			1.49285			0.746426	−	0.665468i	$-0.231768\pi$
$151$	225.421			1.49285			0.746426	+	0.665468i	$0.231768\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	170.241	−	98.2888i	1.09833	−	0.634121i
$156$		0			0
$157$	−53.5259	−	92.7095i	−0.340929	−	0.590506i	0.643677	−	0.765298i	$-0.277408\pi$
$157$	−53.5259	−	92.7095i	−0.340929	−	0.590506i	−0.984605	+	0.174791i	$0.944075\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	258.510	−	68.7938i	1.60565	−	0.427291i
$162$		0			0
$163$	−121.650	−	210.704i	−0.746321	−	1.29267i	−0.949575	−	0.313539i	$-0.898485\pi$
$163$	−121.650	−	210.704i	−0.746321	−	1.29267i	0.203254	−	0.979126i	$-0.434848\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.49683	+	4.32830i	0.0448912	+	0.0259179i	0.522278	−	0.852776i	$-0.325082\pi$
$167$	7.49683	+	4.32830i	0.0448912	+	0.0259179i	−0.477386	+	0.878693i	$0.658416\pi$
$168$		0			0
$169$	21.6052	+	37.4214i	0.127842	+	0.221428i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−150.424	−	86.8472i	−0.869501	−	0.502007i	−0.00231889	−	0.999997i	$-0.500738\pi$
$173$	−150.424	−	86.8472i	−0.869501	−	0.502007i	−0.867183	+	0.497990i	$0.834071\pi$
$174$		0			0
$175$	65.0134	+	244.304i	0.371505	+	1.39602i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	143.928	+	83.0966i	0.804065	+	0.464227i	0.844890	−	0.534939i	$-0.179666\pi$
$179$	143.928	+	83.0966i	0.804065	+	0.464227i	−0.0408259	+	0.999166i	$0.512999\pi$
$180$		0			0
$181$	−46.4431			−0.256592			−0.128296	−	0.991736i	$-0.540951\pi$
$181$	−46.4431			−0.256592			−0.128296	+	0.991736i	$0.540951\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	266.486	−	153.856i	1.44047	−	0.831653i
$186$		0			0
$187$	−69.2398	+	119.927i	−0.370267	+	0.641320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−213.409	−	123.212i	−1.11732	−	0.645088i	−0.176608	−	0.984281i	$-0.556512\pi$
$191$	−213.409	−	123.212i	−1.11732	−	0.645088i	−0.940717	+	0.339194i	$0.889846\pi$
$192$		0			0
$193$	30.5002	+	52.8278i	0.158032	+	0.273719i	0.934159	−	0.356857i	$-0.116152\pi$
$193$	30.5002	+	52.8278i	0.158032	+	0.273719i	−0.776127	+	0.630577i	$0.782818\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			63.3390i			0.321518i	0.986994	+	0.160759i	$0.0513942\pi$
$197$			63.3390i			0.321518i	−0.986994	+	0.160759i	$0.948606\pi$
$198$		0			0
$199$	69.4520	−	120.294i	0.349005	−	0.604494i	−0.637068	−	0.770808i	$-0.719853\pi$
$199$	69.4520	−	120.294i	0.349005	−	0.604494i	0.986073	+	0.166313i	$0.0531863\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	117.154	+	117.555i	0.577111	+	0.579089i
$204$		0			0
$205$	205.466	−	355.878i	1.00227	−	1.73599i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	123.513	−	71.3104i	0.590973	−	0.341198i
$210$		0			0
$211$	−113.387	+	196.391i	−0.537378	+	0.930765i	0.461667	+	0.887054i	$0.347252\pi$
$211$	−113.387	+	196.391i	−0.537378	+	0.930765i	−0.999044	+	0.0437118i	$0.986082\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	482.326	−	278.471i	2.24337	−	1.29521i
$216$		0			0
$217$	−124.676	+	124.250i	−0.574545	+	0.572583i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	135.742	−	78.3707i	0.614217	−	0.354618i
$222$		0			0
$223$	151.921	+	263.135i	0.681261	+	1.17998i	0.974596	+	0.223968i	$0.0719013\pi$
$223$	151.921	+	263.135i	0.681261	+	1.17998i	−0.293336	+	0.956009i	$0.594765\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	90.7509i		−	0.399784i	−0.979818	−	0.199892i	$-0.935941\pi$
$227$		−	90.7509i		−	0.399784i	0.979818	−	0.199892i	$-0.0640591\pi$
$228$		0			0
$229$	193.955			0.846966			0.423483	−	0.905904i	$-0.360807\pi$
$229$	193.955			0.846966			0.423483	+	0.905904i	$0.360807\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−342.650	+	197.829i	−1.47060	+	0.849053i	−0.999455	−	0.0330016i	$-0.989493\pi$
$233$	−342.650	+	197.829i	−1.47060	+	0.849053i	−0.471147	+	0.882054i	$0.656160\pi$
$234$		0			0
$235$	−212.114	+	367.392i	−0.902613	+	1.56337i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	121.457	+	70.1232i	0.508188	+	0.293403i	0.732089	−	0.681209i	$-0.238546\pi$
$239$	121.457	+	70.1232i	0.508188	+	0.293403i	−0.223900	+	0.974612i	$0.571879\pi$
$240$		0			0
$241$	155.209			0.644019			0.322010	−	0.946736i	$-0.395642\pi$
$241$	155.209			0.644019			0.322010	+	0.946736i	$0.395642\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−190.396	−	332.396i	−0.777125	−	1.35672i
$246$		0			0
$247$	−161.429			−0.653557
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	316.992i		−	1.26292i	−0.775409	−	0.631459i	$-0.782456\pi$
$251$		−	316.992i		−	1.26292i	0.775409	−	0.631459i	$-0.217544\pi$
$252$		0			0
$253$	378.671			1.49672
$254$		0			0
$255$		0			0
$256$		0			0
$257$			302.702i			1.17783i	0.808195	+	0.588914i	$0.200444\pi$
$257$			302.702i			1.17783i	−0.808195	+	0.588914i	$0.799556\pi$
$258$		0			0
$259$	−195.161	+	194.495i	−0.753519	+	0.750945i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			343.878i			1.30752i	0.756702	+	0.653760i	$0.226809\pi$
$263$			343.878i			1.30752i	−0.756702	+	0.653760i	$0.773191\pi$
$264$		0			0
$265$	213.755	−	370.234i	0.806622	−	1.39711i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	230.930	+	133.327i	0.858474	+	0.495640i	0.863501	−	0.504347i	$-0.168267\pi$
$269$	230.930	+	133.327i	0.858474	+	0.495640i	−0.00502670	+	0.999987i	$0.501600\pi$
$270$		0			0
$271$	201.627	+	349.227i	0.744009	+	1.28866i	0.950656	+	0.310247i	$0.100412\pi$
$271$	201.627	+	349.227i	0.744009	+	1.28866i	−0.206647	+	0.978416i	$0.566255\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			357.862i			1.30132i
$276$		0			0
$277$	358.434			1.29399			0.646993	−	0.762496i	$-0.276026\pi$
$277$	358.434			1.29399			0.646993	+	0.762496i	$0.276026\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−284.206	+	164.087i	−1.01141	+	0.583938i	−0.911604	−	0.411069i	$-0.865156\pi$
$281$	−284.206	+	164.087i	−1.01141	+	0.583938i	−0.0998062	+	0.995007i	$0.531822\pi$
$282$		0			0
$283$	189.174	+	327.660i	0.668461	+	1.15781i	0.978334	+	0.207031i	$0.0663800\pi$
$283$	189.174	+	327.660i	0.668461	+	1.15781i	−0.309873	+	0.950778i	$0.600287\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−95.8413	+	355.253i	−0.333942	+	1.23781i
$288$		0			0
$289$	−46.8454	−	81.1386i	−0.162095	−	0.280757i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−343.533	−	198.339i	−1.17247	−	0.676924i	−0.218207	−	0.975903i	$-0.570021\pi$
$293$	−343.533	−	198.339i	−1.17247	−	0.676924i	−0.954260	+	0.298979i	$0.903354\pi$
$294$		0			0
$295$	−152.767	−	264.600i	−0.517853	−	0.896948i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−371.185	−	214.304i	−1.24142	−	0.716735i
$300$		0			0
$301$	−353.232	+	352.025i	−1.17353	+	1.16952i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	440.844	+	254.521i	1.44539	+	0.834496i
$306$		0			0
$307$	298.494			0.972292			0.486146	−	0.873878i	$-0.338402\pi$
$307$	298.494			0.972292			0.486146	+	0.873878i	$0.338402\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−292.420	+	168.829i	−0.940258	+	0.542858i	−0.890041	−	0.455880i	$-0.849325\pi$
$311$	−292.420	+	168.829i	−0.940258	+	0.542858i	−0.0502167	+	0.998738i	$0.515991\pi$
$312$		0			0
$313$	−134.866	+	233.594i	−0.430881	+	0.746308i	−0.996949	−	0.0780502i	$-0.975131\pi$
$313$	−134.866	+	233.594i	−0.430881	+	0.746308i	0.566068	+	0.824358i	$0.308464\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−119.647	−	69.0783i	−0.377436	−	0.217913i	0.299266	−	0.954170i	$-0.403258\pi$
$317$	−119.647	−	69.0783i	−0.377436	−	0.217913i	−0.676702	+	0.736257i	$0.736591\pi$
$318$		0			0
$319$	117.466	+	203.457i	0.368231	+	0.637795i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	201.150i		−	0.622755i
$324$		0			0
$325$	202.527	−	350.787i	0.623160	−	1.07935i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	98.9423	−	366.747i	0.300737	−	1.11473i
$330$		0			0
$331$	58.4473	−	101.234i	0.176578	−	0.305842i	−0.764128	−	0.645064i	$-0.776831\pi$
$331$	58.4473	−	101.234i	0.176578	−	0.305842i	0.940706	+	0.339222i	$0.110164\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	724.340	−	418.198i	2.16221	−	1.24835i
$336$		0			0
$337$	−232.080	+	401.974i	−0.688664	+	1.19280i	0.283606	+	0.958941i	$0.408469\pi$
$337$	−232.080	+	401.974i	−0.688664	+	1.19280i	−0.972270	+	0.233860i	$0.924864\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−215.782	+	124.582i	−0.632791	+	0.365342i
$342$		0			0
$343$	241.290	+	243.779i	0.703469	+	0.710726i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	471.687	−	272.329i	1.35933	−	0.784809i	0.369796	−	0.929113i	$-0.379428\pi$
$347$	471.687	−	272.329i	1.35933	−	0.784809i	0.989533	+	0.144304i	$0.0460942\pi$
$348$		0			0
$349$	−209.129	−	362.222i	−0.599224	−	1.03789i	−0.992936	−	0.118653i	$-0.962142\pi$
$349$	−209.129	−	362.222i	−0.599224	−	1.03789i	0.393712	−	0.919234i	$-0.371191\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	331.665i		−	0.939561i	−0.882783	−	0.469781i	$-0.844333\pi$
$353$		−	331.665i		−	0.939561i	0.882783	−	0.469781i	$-0.155667\pi$
$354$		0			0
$355$	213.749			0.602109
$356$		0			0
$357$		0			0
$358$		0			0
$359$	364.336	−	210.349i	1.01486	−	0.585932i	0.102251	−	0.994759i	$-0.467395\pi$
$359$	364.336	−	210.349i	1.01486	−	0.585932i	0.912612	+	0.408827i	$0.134062\pi$
$360$		0			0
$361$	76.9174	−	133.225i	0.213068	−	0.369044i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.30343	−	3.63928i	−0.0172697	−	0.00997064i
$366$		0			0
$367$	48.7156			0.132740			0.0663701	−	0.997795i	$-0.478858\pi$
$367$	48.7156			0.132740			0.0663701	+	0.997795i	$0.478858\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−99.7077	+	369.584i	−0.268754	+	0.996183i
$372$		0			0
$373$	−352.803			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$373$	−352.803			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	265.913i		−	0.705339i
$378$		0			0
$379$	132.331			0.349158			0.174579	−	0.984643i	$-0.444144\pi$
$379$	132.331			0.349158			0.174579	+	0.984643i	$0.444144\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	73.8011i		−	0.192692i	−0.995348	−	0.0963461i	$-0.969284\pi$
$383$		−	73.8011i		−	0.192692i	0.995348	−	0.0963461i	$-0.0307155\pi$
$384$		0			0
$385$	−139.448	−	524.010i	−0.362202	−	1.36107i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			366.404i			0.941911i	0.882157	+	0.470956i	$0.156091\pi$
$389$			366.404i			0.941911i	−0.882157	+	0.470956i	$0.843909\pi$
$390$		0			0
$391$	267.035	−	462.519i	0.682955	−	1.18291i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−638.593	−	368.692i	−1.61669	−	0.933397i
$396$		0			0
$397$	−266.425	−	461.462i	−0.671097	−	1.16237i	−0.977593	−	0.210502i	$-0.932490\pi$
$397$	−266.425	−	461.462i	−0.671097	−	1.16237i	0.306497	−	0.951872i	$-0.400843\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	607.654i		−	1.51535i	−0.652634	−	0.757673i	$-0.726336\pi$
$401$		−	607.654i		−	1.51535i	0.652634	−	0.757673i	$-0.273664\pi$
$402$		0			0
$403$	282.021			0.699804
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−337.773	+	195.013i	−0.829909	+	0.479148i
$408$		0			0
$409$	−195.462	−	338.550i	−0.477902	−	0.827750i	0.521777	−	0.853082i	$-0.325269\pi$
$409$	−195.462	−	338.550i	−0.477902	−	0.827750i	−0.999679	+	0.0253316i	$0.991936\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	193.118	+	193.780i	0.467598	+	0.469200i
$414$		0			0
$415$	−367.943	−	637.295i	−0.886609	−	1.53565i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	361.229	+	208.556i	0.862122	+	0.497746i	0.864722	−	0.502250i	$-0.167494\pi$
$419$	361.229	+	208.556i	0.862122	+	0.497746i	−0.00260040	+	0.999997i	$0.500828\pi$
$420$		0			0
$421$	35.2932	+	61.1296i	0.0838318	+	0.145201i	0.904893	−	0.425639i	$-0.139951\pi$
$421$	35.2932	+	61.1296i	0.0838318	+	0.145201i	−0.821061	+	0.570840i	$0.806617\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	437.102	+	252.361i	1.02848	+	0.593791i
$426$		0			0
$427$	−440.069	−	118.723i	−1.03061	−	0.278041i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.3061	−	11.7237i	−0.0471139	−	0.0272012i	0.476258	−	0.879306i	$-0.341993\pi$
$431$	−20.3061	−	11.7237i	−0.0471139	−	0.0272012i	−0.523372	+	0.852104i	$0.675326\pi$
$432$		0			0
$433$	−354.517			−0.818746			−0.409373	−	0.912367i	$-0.634253\pi$
$433$	−354.517			−0.818746			−0.409373	+	0.912367i	$0.634253\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−476.350	+	275.021i	−1.09005	+	0.629338i
$438$		0			0
$439$	−254.998	+	441.669i	−0.580861	+	1.00608i	0.414517	+	0.910042i	$0.363951\pi$
$439$	−254.998	+	441.669i	−0.580861	+	1.00608i	−0.995378	+	0.0960389i	$0.969383\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	261.097	+	150.744i	0.589383	+	0.340281i	0.764854	−	0.644204i	$-0.222811\pi$
$443$	261.097	+	150.744i	0.589383	+	0.340281i	−0.175470	+	0.984485i	$0.556145\pi$
$444$		0			0
$445$	81.0384	+	140.363i	0.182109	+	0.315422i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			73.5476i			0.163803i	0.996640	+	0.0819015i	$0.0260993\pi$
$449$			73.5476i			0.163803i	−0.996640	+	0.0819015i	$0.973901\pi$
$450$		0			0
$451$	−260.429	+	451.077i	−0.577449	+	1.00017i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−159.865	+	592.569i	−0.351353	+	1.30235i
$456$		0			0
$457$	−355.247	+	615.306i	−0.777346	+	1.34640i	0.156121	+	0.987738i	$0.450101\pi$
$457$	−355.247	+	615.306i	−0.777346	+	1.34640i	−0.933467	+	0.358664i	$0.883232\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	472.697	−	272.912i	1.02537	−	0.591999i	0.109717	−	0.993963i	$-0.465005\pi$
$461$	472.697	−	272.912i	1.02537	−	0.591999i	0.915656	+	0.401964i	$0.131672\pi$
$462$		0			0
$463$	337.645	−	584.819i	0.729255	−	1.26311i	−0.227943	−	0.973675i	$-0.573200\pi$
$463$	337.645	−	584.819i	0.729255	−	1.26311i	0.957198	−	0.289433i	$-0.0934667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−95.1678	+	54.9452i	−0.203785	+	0.117656i	−0.598420	−	0.801183i	$-0.704205\pi$
$467$	−95.1678	+	54.9452i	−0.203785	+	0.117656i	0.394635	+	0.918838i	$0.370871\pi$
$468$		0			0
$469$	−530.471	+	528.659i	−1.13107	+	1.12720i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−611.350	+	352.963i	−1.29250	+	0.746223i
$474$		0			0
$475$	−259.908	−	450.173i	−0.547174	−	0.947734i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			71.6163i			0.149512i	0.997202	+	0.0747560i	$0.0238178\pi$
$479$			71.6163i			0.149512i	−0.997202	+	0.0747560i	$0.976182\pi$
$480$		0			0
$481$	441.460			0.917797
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1042.99	+	602.173i	−2.15050	+	1.24159i
$486$		0			0
$487$	−294.873	+	510.736i	−0.605490	+	1.04874i	0.386484	+	0.922296i	$0.373689\pi$
$487$	−294.873	+	510.736i	−0.605490	+	1.04874i	−0.991974	+	0.126443i	$0.959644\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	154.271	+	89.0686i	0.314198	+	0.181403i	0.648804	−	0.760956i	$-0.275270\pi$
$491$	154.271	+	89.0686i	0.314198	+	0.181403i	−0.334605	+	0.942358i	$0.608603\pi$
$492$		0			0
$493$	331.343			0.672096
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−184.956	+	49.2199i	−0.372145	+	0.0990341i
$498$		0			0
$499$	−596.724			−1.19584			−0.597920	−	0.801556i	$-0.704006\pi$
$499$	−596.724			−1.19584			−0.597920	+	0.801556i	$0.704006\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	135.851i		−	0.270081i	−0.990840	−	0.135041i	$-0.956884\pi$
$503$		−	135.851i		−	0.270081i	0.990840	−	0.135041i	$-0.0431165\pi$
$504$		0			0
$505$	287.725			0.569753
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	307.717i		−	0.604552i	−0.953220	−	0.302276i	$-0.902254\pi$
$509$		−	307.717i		−	0.604552i	0.953220	−	0.302276i	$-0.0977464\pi$
$510$		0			0
$511$	6.29236	+	1.69757i	0.0123138	+	0.00332206i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			589.709i			1.14507i
$516$		0			0
$517$	268.856	−	465.672i	0.520031	−	0.900720i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−359.480	−	207.546i	−0.689981	−	0.398361i	0.113624	−	0.993524i	$-0.463754\pi$
$521$	−359.480	−	207.546i	−0.689981	−	0.398361i	−0.803605	+	0.595163i	$0.797087\pi$
$522$		0			0
$523$	−32.5093	−	56.3077i	−0.0621592	−	0.107663i	0.833271	−	0.552865i	$-0.186465\pi$
$523$	−32.5093	−	56.3077i	−0.0621592	−	0.107663i	−0.895430	+	0.445202i	$0.853132\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			351.415i			0.666822i
$528$		0			0
$529$	−931.410			−1.76070
$530$		0			0
$531$		0			0
$532$		0			0
$533$	510.562	−	294.773i	0.957902	−	0.553045i
$534$		0			0
$535$	−328.433	−	568.863i	−0.613893	−	1.06329i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	241.328	+	421.314i	0.447732	+	0.781659i
$540$		0			0
$541$	44.9103	+	77.7869i	0.0830135	+	0.143784i	0.904543	−	0.426382i	$-0.140212\pi$
$541$	44.9103	+	77.7869i	0.0830135	+	0.143784i	−0.821530	+	0.570166i	$0.806879\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1097.48	−	633.631i	−2.01373	−	1.16263i
$546$		0			0
$547$	14.9248	+	25.8506i	0.0272849	+	0.0472588i	0.879345	−	0.476184i	$-0.157981\pi$
$547$	14.9248	+	25.8506i	0.0272849	+	0.0472588i	−0.852061	+	0.523443i	$0.824647\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−295.533	−	170.626i	−0.536357	−	0.309666i
$552$		0			0
$553$	637.472	+	171.979i	1.15275	+	0.310993i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−314.117	−	181.355i	−0.563944	−	0.325593i	0.190783	−	0.981632i	$-0.438897\pi$
$557$	−314.117	−	181.355i	−0.563944	−	0.325593i	−0.754727	+	0.656039i	$0.772231\pi$
$558$		0			0
$559$	799.019			1.42937
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−472.574	+	272.840i	−0.839385	+	0.484619i	−0.857055	−	0.515225i	$-0.827708\pi$
$563$	−472.574	+	272.840i	−0.839385	+	0.484619i	0.0176704	+	0.999844i	$0.494375\pi$
$564$		0			0
$565$	−83.8286	+	145.195i	−0.148369	+	0.256983i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−225.698	−	130.307i	−0.396657	−	0.229010i	0.288383	−	0.957515i	$-0.406882\pi$
$569$	−225.698	−	130.307i	−0.396657	−	0.229010i	−0.685041	+	0.728505i	$0.740216\pi$
$570$		0			0
$571$	−325.279	−	563.401i	−0.569666	−	0.986691i	−0.996599	−	0.0824074i	$-0.973739\pi$
$571$	−325.279	−	563.401i	−0.569666	−	0.986691i	0.426932	−	0.904284i	$-0.359594\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1380.16i		−	2.40027i
$576$		0			0
$577$	515.854	−	893.485i	0.894027	−	1.54850i	0.0590234	−	0.998257i	$-0.481201\pi$
$577$	515.854	−	893.485i	0.894027	−	1.54850i	0.835004	−	0.550244i	$-0.185465\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	465.130	+	466.724i	0.800567	+	0.803311i
$582$		0			0
$583$	−270.936	+	469.274i	−0.464726	+	0.804930i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−715.056	+	412.838i	−1.21815	+	0.703302i	−0.964523	−	0.263999i	$-0.914958\pi$
$587$	−715.056	+	412.838i	−1.21815	+	0.703302i	−0.253631	+	0.967301i	$0.581625\pi$
$588$		0			0
$589$	180.962	−	313.435i	0.307236	−	0.532148i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	620.271	−	358.114i	1.04599	−	0.603902i	0.124465	−	0.992224i	$-0.460279\pi$
$593$	620.271	−	358.114i	1.04599	−	0.603902i	0.921524	+	0.388322i	$0.126945\pi$
$594$		0			0
$595$	−738.377	−	199.202i	−1.24097	−	0.334793i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	850.895	−	491.264i	1.42053	−	0.820141i	0.424182	−	0.905577i	$-0.360562\pi$
$599$	850.895	−	491.264i	1.42053	−	0.820141i	0.996344	+	0.0854363i	$0.0272284\pi$
$600$		0			0
$601$	280.080	+	485.113i	0.466024	+	0.807177i	0.999247	−	0.0387975i	$-0.0123527\pi$
$601$	280.080	+	485.113i	0.466024	+	0.807177i	−0.533223	+	0.845975i	$0.679019\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			178.351i			0.294796i
$606$		0			0
$607$	155.174			0.255640			0.127820	−	0.991797i	$-0.459202\pi$
$607$	155.174			0.255640			0.127820	+	0.991797i	$0.459202\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−527.081	+	304.311i	−0.862654	+	0.498053i
$612$		0			0
$613$	−197.990	+	342.929i	−0.322986	+	0.559428i	−0.981103	−	0.193488i	$-0.938020\pi$
$613$	−197.990	+	342.929i	−0.322986	+	0.559428i	0.658117	+	0.752916i	$0.271353\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−346.378	−	199.982i	−0.561391	−	0.324119i	0.192313	−	0.981334i	$-0.438401\pi$
$617$	−346.378	−	199.982i	−0.561391	−	0.324119i	−0.753704	+	0.657215i	$0.771735\pi$
$618$		0			0
$619$	907.561			1.46617			0.733087	−	0.680135i	$-0.238079\pi$
$619$	907.561			1.46617			0.733087	+	0.680135i	$0.238079\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−102.444	−	102.795i	−0.164436	−	0.165000i
$624$		0			0
$625$	−223.569			−0.357710
$626$		0			0
$627$		0			0
$628$		0			0
$629$			550.086i			0.874541i
$630$		0			0
$631$	−105.596			−0.167347			−0.0836736	−	0.996493i	$-0.526665\pi$
$631$	−105.596			−0.167347			−0.0836736	+	0.996493i	$0.526665\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	420.011i		−	0.661434i
$636$		0			0
$637$	1.88021	−	549.561i	0.00295166	−	0.862733i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			382.825i			0.597230i	0.954374	+	0.298615i	$0.0965247\pi$
$641$			382.825i			0.597230i	−0.954374	+	0.298615i	$0.903475\pi$
$642$		0			0
$643$	−245.677	+	425.525i	−0.382080	+	0.661781i	−0.991359	−	0.131174i	$-0.958125\pi$
$643$	−245.677	+	425.525i	−0.382080	+	0.661781i	0.609280	+	0.792955i	$0.291459\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−741.786	−	428.270i	−1.14650	−	0.661933i	−0.198469	−	0.980107i	$-0.563597\pi$
$647$	−741.786	−	428.270i	−1.14650	−	0.661933i	−0.948032	+	0.318175i	$0.896930\pi$
$648$		0			0
$649$	193.633	+	335.381i	0.298355	+	0.516767i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	750.342i		−	1.14907i	−0.818481	−	0.574534i	$-0.805183\pi$
$653$		−	750.342i		−	1.14907i	0.818481	−	0.574534i	$-0.194817\pi$
$654$		0			0
$655$	−477.617			−0.729187
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−390.298	+	225.339i	−0.592258	+	0.341940i	−0.765990	−	0.642853i	$-0.777751\pi$
$659$	−390.298	+	225.339i	−0.592258	+	0.341940i	0.173732	+	0.984793i	$0.444417\pi$
$660$		0			0
$661$	−14.1942	−	24.5851i	−0.0214738	−	0.0371937i	0.855089	−	0.518482i	$-0.173503\pi$
$661$	−14.1942	−	24.5851i	−0.0214738	−	0.0371937i	−0.876563	+	0.481288i	$0.840169\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	555.997	+	557.902i	0.836085	+	0.838950i
$666$		0			0
$667$	−453.027	−	784.666i	−0.679201	−	1.17641i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−558.772	−	322.607i	−0.832745	−	0.480786i
$672$		0			0
$673$	211.002	+	365.466i	0.313524	+	0.543040i	0.979123	−	0.203270i	$-0.0651570\pi$
$673$	211.002	+	365.466i	0.313524	+	0.543040i	−0.665599	+	0.746310i	$0.731824\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	765.661	+	442.055i	1.13096	+	0.652961i	0.944177	−	0.329440i	$-0.106860\pi$
$677$	765.661	+	442.055i	1.13096	+	0.652961i	0.186785	+	0.982401i	$0.440193\pi$
$678$		0			0
$679$	763.838	−	761.229i	1.12495	−	1.12110i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	99.1010	+	57.2160i	0.145097	+	0.0837716i	0.570791	−	0.821096i	$-0.306637\pi$
$683$	99.1010	+	57.2160i	0.145097	+	0.0837716i	−0.425694	+	0.904867i	$0.639970\pi$
$684$		0			0
$685$	961.321			1.40339
$686$		0			0
$687$		0			0
$688$		0			0
$689$	531.158	−	306.664i	0.770912	−	0.445086i
$690$		0			0
$691$	244.279	−	423.104i	0.353515	−	0.612306i	−0.633347	−	0.773868i	$-0.718320\pi$
$691$	244.279	−	423.104i	0.353515	−	0.612306i	0.986863	+	0.161561i	$0.0516529\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	51.1332	+	29.5218i	0.0735730	+	0.0424774i
$696$		0			0
$697$	367.305	+	636.191i	0.526980	+	0.912756i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			57.6991i			0.0823098i	0.999153	+	0.0411549i	$0.0131037\pi$
$701$			57.6991i			0.0823098i	−0.999153	+	0.0411549i	$0.986896\pi$
$702$		0			0
$703$	283.268	−	490.635i	0.402942	−	0.697916i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−248.968	+	66.2545i	−0.352147	+	0.0937122i
$708$		0			0
$709$	312.295	−	540.912i	0.440473	−	0.762922i	−0.557251	−	0.830344i	$-0.688144\pi$
$709$	312.295	−	540.912i	0.440473	−	0.762922i	0.997725	+	0.0674220i	$0.0214774\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	832.199	−	480.470i	1.16718	−	0.673871i
$714$		0			0
$715$	−434.402	+	752.406i	−0.607555	+	1.05232i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	785.622	−	453.579i	1.09266	−	0.630847i	0.158376	−	0.987379i	$-0.449374\pi$
$719$	785.622	−	453.579i	1.09266	−	0.630847i	0.934283	+	0.356531i	$0.116041\pi$
$720$		0			0
$721$	−135.792	−	510.273i	−0.188339	−	0.707730i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	741.546	−	428.132i	1.02282	−	0.590527i
$726$		0			0
$727$	−43.9060	−	76.0475i	−0.0603934	−	0.104605i	0.834248	−	0.551390i	$-0.185902\pi$
$727$	−43.9060	−	76.0475i	−0.0603934	−	0.104605i	−0.894641	+	0.446785i	$0.852569\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			995.626i			1.36201i
$732$		0			0
$733$	−1141.34			−1.55708			−0.778540	−	0.627595i	$-0.784039\pi$
$733$	−1141.34			−1.55708			−0.778540	+	0.627595i	$0.784039\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−918.105	+	530.068i	−1.24573	+	0.719224i
$738$		0			0
$739$	−403.086	+	698.166i	−0.545448	+	0.944744i	0.453130	+	0.891444i	$0.350307\pi$
$739$	−403.086	+	698.166i	−0.545448	+	0.944744i	−0.998579	+	0.0532999i	$0.983026\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−858.109	−	495.429i	−1.15492	−	0.666796i	−0.204842	−	0.978795i	$-0.565668\pi$
$743$	−858.109	−	495.429i	−1.15492	−	0.666796i	−0.950082	+	0.311999i	$0.899001\pi$
$744$		0			0
$745$	−1702.78			−2.28561
$746$		0			0
$747$		0			0
$748$		0			0
$749$	415.184	+	416.607i	0.554318	+	0.556218i
$750$		0			0
$751$	−404.444			−0.538540			−0.269270	−	0.963065i	$-0.586782\pi$
$751$	−404.444			−0.538540			−0.269270	+	0.963065i	$0.586782\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1762.25i			2.33411i
$756$		0			0
$757$	322.682			0.426264			0.213132	−	0.977023i	$-0.431634\pi$
$757$	322.682			0.426264			0.213132	+	0.977023i	$0.431634\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			933.261i			1.22636i	0.789943	+	0.613181i	$0.210110\pi$
$761$			933.261i			1.22636i	−0.789943	+	0.613181i	$0.789890\pi$
$762$		0			0
$763$	1095.55	+	295.562i	1.43585	+	0.387369i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	438.335i		−	0.571493i
$768$		0			0
$769$	109.357	−	189.411i	0.142206	−	0.246309i	−0.786121	−	0.618073i	$-0.787914\pi$
$769$	109.357	−	189.411i	0.142206	−	0.246309i	0.928327	+	0.371764i	$0.121247\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1184.55	−	683.901i	−1.53241	−	0.884736i	−0.999250	−	0.0387193i	$-0.987672\pi$
$773$	−1184.55	−	683.901i	−1.53241	−	0.884736i	−0.533157	−	0.846016i	$-0.678994\pi$
$774$		0			0
$775$	454.067	+	786.467i	0.585893	+	1.01480i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	756.578i		−	0.971217i
$780$		0			0
$781$	−270.928			−0.346899
$782$		0			0
$783$		0			0
$784$		0			0
$785$	724.768	−	418.445i	0.923271	−	0.533051i
$786$		0			0
$787$	290.720	+	503.541i	0.369403	+	0.639824i	0.989472	−	0.144723i	$-0.0462290\pi$
$787$	290.720	+	503.541i	0.369403	+	0.639824i	−0.620070	+	0.784547i	$0.712896\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	39.1025	−	144.940i	0.0494343	−	0.183237i
$792$		0			0
$793$	365.150	+	632.459i	0.460467	+	0.797552i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	209.389	+	120.891i	0.262721	+	0.151682i	0.625575	−	0.780164i	$-0.284864\pi$
$797$	209.389	+	120.891i	0.262721	+	0.151682i	−0.362854	+	0.931846i	$0.618198\pi$
$798$		0			0
$799$	−379.189	−	656.775i	−0.474580	−	0.821996i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.98963	+	4.61281i	0.00994972	+	0.00574448i
$804$		0			0
$805$	537.804	+	2020.94i	0.668080	+	2.51048i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−609.778	−	352.056i	−0.753743	−	0.435174i	0.0733017	−	0.997310i	$-0.476646\pi$
$809$	−609.778	−	352.056i	−0.753743	−	0.435174i	−0.827045	+	0.562136i	$0.809980\pi$
$810$		0			0
$811$	421.013			0.519128			0.259564	−	0.965726i	$-0.416421\pi$
$811$	421.013			0.519128			0.259564	+	0.965726i	$0.416421\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1647.21	−	951.016i	2.02111	−	1.16689i
$816$		0			0
$817$	512.700	−	888.022i	0.627540	−	1.08693i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	602.776	+	348.013i	0.734198	+	0.423889i	0.819956	−	0.572427i	$-0.193998\pi$
$821$	602.776	+	348.013i	0.734198	+	0.423889i	−0.0857582	+	0.996316i	$0.527331\pi$
$822$		0			0
$823$	80.8958	+	140.116i	0.0982938	+	0.170250i	0.910979	−	0.412454i	$-0.135328\pi$
$823$	80.8958	+	140.116i	0.0982938	+	0.170250i	−0.812685	+	0.582704i	$0.801995\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			785.578i			0.949913i	0.880009	+	0.474957i	$0.157536\pi$
$827$			785.578i			0.949913i	−0.880009	+	0.474957i	$0.842464\pi$
$828$		0			0
$829$	−814.931	+	1411.50i	−0.983029	+	1.70266i	−0.332644	+	0.943052i	$0.607941\pi$
$829$	−814.931	+	1411.50i	−0.983029	+	1.70266i	−0.650385	+	0.759604i	$0.725393\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	684.786	+	2.34285i	0.822072	+	0.00281254i
$834$		0			0
$835$	−33.8370	+	58.6074i	−0.0405233	+	0.0701885i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1013.79	−	585.314i	1.20834	−	0.697633i	0.245940	−	0.969285i	$-0.420903\pi$
$839$	1013.79	−	585.314i	1.20834	−	0.697633i	0.962396	+	0.271652i	$0.0875700\pi$
$840$		0			0
$841$	−139.437	+	241.512i	−0.165799	+	0.287173i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−292.546	+	168.902i	−0.346208	+	0.199884i
$846$		0			0
$847$	−41.0690	−	154.327i	−0.0484876	−	0.182204i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1302.68	−	752.102i	1.53076	−	0.883786i
$852$		0			0
$853$	247.678	+	428.991i	0.290361	+	0.502920i	0.973895	−	0.226999i	$-0.0728913\pi$
$853$	247.678	+	428.991i	0.290361	+	0.502920i	−0.683534	+	0.729919i	$0.739558\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	738.976i		−	0.862283i	−0.902284	−	0.431141i	$-0.858111\pi$
$857$		−	738.976i		−	0.862283i	0.902284	−	0.431141i	$-0.141889\pi$
$858$		0			0
$859$	721.520			0.839954			0.419977	−	0.907535i	$-0.362038\pi$
$859$	721.520			0.839954			0.419977	+	0.907535i	$0.362038\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−756.958	+	437.030i	−0.877124	+	0.506408i	−0.869709	−	0.493565i	$-0.835694\pi$
$863$	−756.958	+	437.030i	−0.877124	+	0.506408i	−0.00741492	+	0.999973i	$0.502360\pi$
$864$		0			0
$865$	678.939	−	1175.96i	0.784900	−	1.35949i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	809.421	+	467.319i	0.931439	+	0.537767i
$870$		0			0
$871$	1199.94			1.37766
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−587.807	+	156.425i	−0.671780	+	0.178772i
$876$		0			0
$877$	−552.139			−0.629577			−0.314789	−	0.949162i	$-0.601934\pi$
$877$	−552.139			−0.629577			−0.314789	+	0.949162i	$0.601934\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			665.872i			0.755813i	0.925844	+	0.377907i	$0.123356\pi$
$881$			665.872i			0.755813i	−0.925844	+	0.377907i	$0.876644\pi$
$882$		0			0
$883$	−1122.90			−1.27169			−0.635843	−	0.771819i	$-0.719347\pi$
$883$	−1122.90			−1.27169			−0.635843	+	0.771819i	$0.719347\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	9.45659i		−	0.0106613i	−0.999986	−	0.00533066i	$-0.998303\pi$
$887$		−	9.45659i		−	0.0106613i	0.999986	−	0.00533066i	$-0.00169681\pi$
$888$		0			0
$889$	96.7159	+	363.434i	0.108792	+	0.408812i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			781.058i			0.874645i
$894$		0			0
$895$	−649.618	+	1125.17i	−0.725830	+	1.25718i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	516.305	+	298.089i	0.574310	+	0.331578i
$900$		0			0
$901$	382.122	+	661.855i	0.424109	+	0.734579i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	363.074i		−	0.401187i
$906$		0			0
$907$	−255.906			−0.282146			−0.141073	−	0.989999i	$-0.545055\pi$
$907$	−255.906			−0.282146			−0.141073	+	0.989999i	$0.545055\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	762.662	−	440.323i	0.837170	−	0.483340i	−0.0191314	−	0.999817i	$-0.506090\pi$
$911$	762.662	−	440.323i	0.837170	−	0.483340i	0.856301	+	0.516477i	$0.172757\pi$
$912$		0			0
$913$	466.369	+	807.775i	0.510810	+	0.884748i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	413.281	−	109.981i	0.450688	−	0.119936i
$918$		0			0
$919$	156.040	+	270.269i	0.169793	+	0.294090i	0.938347	−	0.345695i	$-0.112357\pi$
$919$	156.040	+	270.269i	0.169793	+	0.294090i	−0.768554	+	0.639785i	$0.779023\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	265.572	+	153.328i	0.287727	+	0.166119i
$924$		0			0
$925$	710.772	+	1231.09i	0.768402	+	1.33091i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	712.896	+	411.590i	0.767379	+	0.443047i	0.831939	−	0.554867i	$-0.187231\pi$
$929$	712.896	+	411.590i	0.767379	+	0.443047i	−0.0645595	+	0.997914i	$0.520564\pi$
$930$		0			0
$931$	−609.570	−	354.722i	−0.654748	−	0.381012i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−937.544	−	541.291i	−1.00272	−	0.578921i
$936$		0			0
$937$	−1596.78			−1.70414			−0.852070	−	0.523428i	$-0.824653\pi$
$937$	−1596.78			−1.70414			−0.852070	+	0.523428i	$0.824653\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−557.598	+	321.930i	−0.592559	+	0.342114i	−0.766109	−	0.642711i	$-0.777810\pi$
$941$	−557.598	+	321.930i	−0.592559	+	0.342114i	0.173549	+	0.984825i	$0.444476\pi$
$942$		0			0
$943$	1004.39	−	1739.66i	1.06510	−	1.84481i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−852.136	−	491.981i	−0.899827	−	0.519515i	−0.0226826	−	0.999743i	$-0.507221\pi$
$947$	−852.136	−	491.981i	−0.899827	−	0.519515i	−0.877144	+	0.480228i	$0.840554\pi$
$948$		0			0
$949$	−5.22112	−	9.04324i	−0.00550170	−	0.00952923i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			950.170i			0.997030i	0.866881	+	0.498515i	$0.166121\pi$
$953$			950.170i			0.997030i	−0.866881	+	0.498515i	$0.833879\pi$
$954$		0			0
$955$	963.223	−	1668.35i	1.00861	−	1.74697i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−831.829	+	221.363i	−0.867392	+	0.230827i
$960$		0			0
$961$	164.354	−	284.669i	0.171024	−	0.296222i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−412.988	+	238.439i	−0.427967	+	0.247087i
$966$		0			0
$967$	567.995	−	983.796i	0.587378	−	1.01737i	−0.407196	−	0.913341i	$-0.633493\pi$
$967$	567.995	−	983.796i	0.587378	−	1.01737i	0.994574	−	0.104028i	$-0.0331733\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	46.3049	−	26.7342i	0.0476879	−	0.0275326i	−0.475967	−	0.879463i	$-0.657902\pi$
$971$	46.3049	−	26.7342i	0.0476879	−	0.0275326i	0.523654	+	0.851931i	$0.324568\pi$
$972$		0			0
$973$	−51.0434	−	13.7707i	−0.0524598	−	0.0141528i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	49.6031	−	28.6383i	0.0507708	−	0.0293125i	−0.474400	−	0.880310i	$-0.657335\pi$
$977$	49.6031	−	28.6383i	0.0507708	−	0.0293125i	0.525171	+	0.850997i	$0.324002\pi$
$978$		0			0
$979$	−102.717	−	177.910i	−0.104920	−	0.181727i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	210.844i		−	0.214490i	−0.994233	−	0.107245i	$-0.965797\pi$
$983$		−	210.844i		−	0.214490i	0.994233	−	0.107245i	$-0.0342029\pi$
$984$		0			0
$985$	−495.161			−0.502701
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2357.78	−	1361.26i	2.38400	−	1.37640i
$990$		0			0
$991$	472.405	−	818.229i	0.476695	−	0.825660i	−0.522948	−	0.852364i	$-0.675168\pi$
$991$	472.405	−	818.229i	0.476695	−	0.825660i	0.999643	+	0.0267044i	$0.00850128\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	940.416	+	542.949i	0.945142	+	0.545678i
$996$		0			0
$997$	214.100			0.214744			0.107372	−	0.994219i	$-0.465756\pi$
$997$	214.100			0.214744			0.107372	+	0.994219i	$0.465756\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.m.a.557.16		32
3.2	odd	2		252.3.m.a.221.9	yes	32
7.2	even	3		756.3.bh.a.233.16		32
9.2	odd	6		756.3.bh.a.305.16		32
9.7	even	3		252.3.bh.a.137.14	yes	32
21.2	odd	6		252.3.bh.a.149.14	yes	32
63.2	odd	6	inner	756.3.m.a.737.1		32
63.16	even	3		252.3.m.a.65.9	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.m.a.65.9	✓	32	63.16	even	3
252.3.m.a.221.9	yes	32	3.2	odd	2
252.3.bh.a.137.14	yes	32	9.7	even	3
252.3.bh.a.149.14	yes	32	21.2	odd	6
756.3.m.a.557.16		32	1.1	even	1	trivial
756.3.m.a.737.1		32	63.2	odd	6	inner
756.3.bh.a.233.16		32	7.2	even	3
756.3.bh.a.305.16		32	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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	756.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+7.81763i q^{5} +(-1.80016 - 6.76457i) q^{7} +O(q^{10})\)	\(q+7.81763i q^{5} +(-1.80016 - 6.76457i) q^{7} -9.90888i q^{11} +(-5.60780 + 9.71299i) q^{13} +(-12.1030 - 6.98765i) q^{17} +(7.19662 + 12.4649i) q^{19} +38.2153i q^{23} -36.1153 q^{25} +(-20.5328 + 11.8546i) q^{29} +(-12.5727 - 21.7766i) q^{31} +(52.8829 - 14.0730i) q^{35} +(-19.6806 - 34.0879i) q^{37} +(-45.5225 - 26.2824i) q^{41} +(-35.6209 - 61.6972i) q^{43} +(46.9954 + 27.1328i) q^{47} +(-42.5188 + 24.3547i) q^{49} +(-47.3589 - 27.3427i) q^{53} +77.4639 q^{55} +(-33.8465 + 19.5413i) q^{59} +(32.5574 - 56.3910i) q^{61} +(-75.9325 - 43.8397i) q^{65} +(-53.4942 - 92.6547i) q^{67} -27.3419i q^{71} +(-0.465523 + 0.806310i) q^{73} +(-67.0293 + 17.8376i) q^{77} +(-47.1616 + 81.6863i) q^{79} +(-81.5203 + 47.0658i) q^{83} +(54.6269 - 94.6165i) q^{85} +(17.9546 - 10.3661i) q^{89} +(75.7992 + 20.4494i) q^{91} +(-97.4460 + 56.2605i) q^{95} +(77.0276 + 133.416i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - q^{7}+O(q^{10}) \) 32 * q - q^7	\( 32 q - q^{7} - 5 q^{13} + 27 q^{17} - 14 q^{19} - 160 q^{25} - 36 q^{29} - 8 q^{31} + 45 q^{35} - 11 q^{37} - 72 q^{41} + 16 q^{43} + 108 q^{47} + 35 q^{49} - 180 q^{53} - 24 q^{55} - 45 q^{59} - 41 q^{61} + 81 q^{65} - 35 q^{67} - 98 q^{73} - 225 q^{77} - 71 q^{79} - 30 q^{85} + 189 q^{89} + 109 q^{91} + 288 q^{95} + 19 q^{97}+O(q^{100}) \) 32 * q - q^7 - 5 * q^13 + 27 * q^17 - 14 * q^19 - 160 * q^25 - 36 * q^29 - 8 * q^31 + 45 * q^35 - 11 * q^37 - 72 * q^41 + 16 * q^43 + 108 * q^47 + 35 * q^49 - 180 * q^53 - 24 * q^55 - 45 * q^59 - 41 * q^61 + 81 * q^65 - 35 * q^67 - 98 * q^73 - 225 * q^77 - 71 * q^79 - 30 * q^85 + 189 * q^89 + 109 * q^91 + 288 * q^95 + 19 * q^97

Introduction

L-functions

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