LMFDB - Embedded newform 756.3.bh.a.233.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,3,Mod(233,756)]

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, 2]))

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

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

chi := DirichletCharacter("756.233");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.bh (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			233.16
Character	$\chi$	$=$	756.233
Dual form			756.3.bh.a.305.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.77026 - 3.90881i) q^{5} +(-4.95821 - 4.94127i) q^{7} +O(q^{10})$	$q+(6.77026 - 3.90881i) q^{5} +(-4.95821 - 4.94127i) q^{7} +(8.58135 + 4.95444i) q^{11} +(-5.60780 + 9.71299i) q^{13} +(12.1030 - 6.98765i) q^{17} +(7.19662 - 12.4649i) q^{19} +(33.0954 - 19.1077i) q^{23} +(18.0576 - 31.2767i) q^{25} +(-20.5328 + 11.8546i) q^{29} +25.1454 q^{31} +(-52.8829 - 14.0730i) q^{35} +(-19.6806 + 34.0879i) q^{37} +(-45.5225 - 26.2824i) q^{41} +(-35.6209 - 61.6972i) q^{43} -54.2656i q^{47} +(0.167642 + 48.9997i) q^{49} +(47.3589 - 27.3427i) q^{53} +77.4639 q^{55} -39.0826i q^{59} -65.1147 q^{61} +87.6793i q^{65} +106.988 q^{67} -27.3419i q^{71} +(-0.465523 - 0.806310i) q^{73} +(-18.0668 - 66.9679i) q^{77} +94.3233 q^{79} +(-81.5203 + 47.0658i) q^{83} +(54.6269 - 94.6165i) q^{85} +(-17.9546 - 10.3661i) q^{89} +(75.7992 - 20.4494i) q^{91} -112.521i q^{95} +(77.0276 + 133.416i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - q^{7}+O(q^{10}) $ 32 * q - q^7	$ 32 q - q^{7} - 18 q^{11} - 5 q^{13} - 27 q^{17} - 14 q^{19} + 45 q^{23} + 80 q^{25} - 36 q^{29} + 16 q^{31} - 45 q^{35} - 11 q^{37} - 72 q^{41} + 16 q^{43} - 37 q^{49} + 180 q^{53} - 24 q^{55} + 82 q^{61} + 70 q^{67} - 98 q^{73} + 135 q^{77} + 142 q^{79} - 30 q^{85} - 189 q^{89} + 109 q^{91} + 19 q^{97}+O(q^{100}) $ 32 * q - q^7 - 18 * q^11 - 5 * q^13 - 27 * q^17 - 14 * q^19 + 45 * q^23 + 80 * q^25 - 36 * q^29 + 16 * q^31 - 45 * q^35 - 11 * q^37 - 72 * q^41 + 16 * q^43 - 37 * q^49 + 180 * q^53 - 24 * q^55 + 82 * q^61 + 70 * q^67 - 98 * q^73 + 135 * q^77 + 142 * q^79 - 30 * q^85 - 189 * q^89 + 109 * q^91 + 19 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	6.77026	−	3.90881i	1.35405	−	0.781763i	0.365238	−	0.930914i	$-0.380988\pi$
$5$	6.77026	−	3.90881i	1.35405	−	0.781763i	0.988814	+	0.149152i	$0.0476542\pi$
$6$		0			0
$7$	−4.95821	−	4.94127i	−0.708315	−	0.705896i
$7$	−4.95821	−	4.94127i	−0.708315	−	0.705896i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.58135	+	4.95444i	0.780122	+	0.450404i	0.836474	−	0.548007i	$-0.184613\pi$
$11$	8.58135	+	4.95444i	0.780122	+	0.450404i	−0.0563513	+	0.998411i	$0.517947\pi$
$12$		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	$-0.531833\pi$
$17$	12.1030	−	6.98765i	0.711939	−	0.411038i	0.811779	+	0.583965i	$0.198500\pi$
$18$		0			0
$19$	7.19662	−	12.4649i	0.378769	−	0.656048i	−0.612114	−	0.790769i	$-0.709681\pi$
$19$	7.19662	−	12.4649i	0.378769	−	0.656048i	0.990883	+	0.134722i	$0.0430141\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	33.0954	−	19.1077i	1.43893	−	0.830768i	0.441156	−	0.897430i	$-0.354569\pi$
$23$	33.0954	−	19.1077i	1.43893	−	0.830768i	0.997776	+	0.0666629i	$0.0212352\pi$
$24$		0			0
$25$	18.0576	−	31.2767i	0.722305	−	1.25107i
$26$		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
$31$	25.1454			0.811143			0.405571	−	0.914063i	$-0.367073\pi$
$31$	25.1454			0.811143			0.405571	+	0.914063i	$0.367073\pi$
$32$		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	0.467397	+	0.884048i	$0.345192\pi$
$37$	−19.6806	+	34.0879i	−0.531909	+	0.921294i	−0.999306	+	0.0372462i	$0.988141\pi$
$38$		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
$47$		−	54.2656i		−	1.15459i	−0.816536	−	0.577294i	$-0.804109\pi$
$47$		−	54.2656i		−	1.15459i	0.816536	−	0.577294i	$-0.195891\pi$
$48$		0			0
$49$	0.167642	+	48.9997i	0.00342127	+	0.999994i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	47.3589	−	27.3427i	0.893565	−	0.515900i	0.0184578	−	0.999830i	$-0.494124\pi$
$53$	47.3589	−	27.3427i	0.893565	−	0.515900i	0.875107	+	0.483930i	$0.160791\pi$
$54$		0			0
$55$	77.4639			1.40844
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	39.0826i		−	0.662417i	−0.943558	−	0.331209i	$-0.892544\pi$
$59$		−	39.0826i		−	0.662417i	0.943558	−	0.331209i	$-0.107456\pi$
$60$		0			0
$61$	−65.1147			−1.06745			−0.533727	−	0.845657i	$-0.679209\pi$
$61$	−65.1147			−1.06745			−0.533727	+	0.845657i	$0.679209\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			87.6793i			1.34891i
$66$		0			0
$67$	106.988			1.59684			0.798421	−	0.602100i	$-0.205669\pi$
$67$	106.988			1.59684			0.798421	+	0.602100i	$0.205669\pi$
$68$		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	0.862819	−	0.505513i	$-0.168697\pi$
$73$	−0.465523	−	0.806310i	−0.00637703	−	0.0110453i	−0.869196	+	0.494467i	$0.835363\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−18.0668	−	66.9679i	−0.234634	−	0.869713i
$78$		0			0
$79$	94.3233			1.19397			0.596983	−	0.802254i	$-0.296366\pi$
$79$	94.3233			1.19397			0.596983	+	0.802254i	$0.296366\pi$
$80$		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	$-0.370492\pi$
$89$	−17.9546	−	10.3661i	−0.201738	−	0.116473i	−0.597466	+	0.801894i	$0.703826\pi$
$90$		0			0
$91$	75.7992	−	20.4494i	0.832958	−	0.224718i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	112.521i		−	1.18443i
$96$		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
$101$	31.8738	+	18.4023i	0.315582	+	0.182201i	0.649422	−	0.760428i	$-0.275011\pi$
$101$	31.8738	+	18.4023i	0.315582	+	0.182201i	−0.333840	+	0.942630i	$0.608344\pi$
$102$		0			0
$103$	−37.7166	−	65.3271i	−0.366181	−	0.634244i	0.622784	−	0.782394i	$-0.286001\pi$
$103$	−37.7166	−	65.3271i	−0.366181	−	0.634244i	−0.988965	+	0.148150i	$0.952668\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	72.7667	+	42.0119i	0.680062	+	0.392634i	0.799879	−	0.600162i	$-0.204897\pi$
$107$	72.7667	+	42.0119i	0.680062	+	0.392634i	−0.119816	+	0.992796i	$0.538231\pi$
$108$		0			0
$109$	−81.0516	−	140.385i	−0.743592	−	1.28794i	−0.950850	−	0.309653i	$-0.899787\pi$
$109$	−81.0516	−	140.385i	−0.743592	−	1.28794i	0.207257	−	0.978287i	$-0.433546\pi$
$110$		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
$115$	149.376	−	258.728i	1.29893	−	2.24981i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−94.5369	−	25.1578i	−0.794428	−	0.211410i
$120$		0			0
$121$	−11.4070	−	19.7575i	−0.0942728	−	0.163285i
$122$		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
$131$	52.9098	−	30.5475i	0.403891	−	0.233187i	−0.284270	−	0.958744i	$-0.591751\pi$
$131$	52.9098	−	30.5475i	0.403891	−	0.233187i	0.688162	+	0.725557i	$0.258418\pi$
$132$		0			0
$133$	−97.2748	+	26.2431i	−0.731390	+	0.197317i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	106.494	+	61.4842i	0.777327	+	0.448790i	0.835482	−	0.549518i	$-0.185188\pi$
$137$	106.494	+	61.4842i	0.777327	+	0.448790i	−0.0581551	+	0.998308i	$0.518522\pi$
$138$		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
$149$	188.632	−	108.907i	1.26598	−	0.730917i	0.291759	−	0.956492i	$-0.405759\pi$
$149$	188.632	−	108.907i	1.26598	−	0.730917i	0.974226	+	0.225575i	$0.0724261\pi$
$150$		0			0
$151$	−112.710	+	195.220i	−0.746426	+	1.29285i	0.203100	+	0.979158i	$0.434899\pi$
$151$	−112.710	+	195.220i	−0.746426	+	1.29285i	−0.949526	+	0.313690i	$0.898435\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	170.241	−	98.2888i	1.09833	−	0.634121i
$156$		0			0
$157$	107.052			0.681858			0.340929	−	0.940089i	$-0.389258\pi$
$157$	107.052			0.681858			0.340929	+	0.940089i	$0.389258\pi$
$158$		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	0.203254	+	0.979126i	$0.434848\pi$
$163$	−121.650	+	210.704i	−0.746321	+	1.29267i	−0.949575	+	0.313539i	$0.898485\pi$
$164$		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
$173$			173.694i			1.00401i	0.864864	+	0.502007i	$0.167405\pi$
$173$			173.694i			1.00401i	−0.864864	+	0.502007i	$0.832595\pi$
$174$		0			0
$175$	−244.080	+	65.8489i	−1.39475	+	0.376279i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−143.928	+	83.0966i	−0.804065	+	0.464227i	−0.844890	−	0.534939i	$-0.820334\pi$
$179$	−143.928	+	83.0966i	−0.804065	+	0.464227i	0.0408259	+	0.999166i	$0.487001\pi$
$180$		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
$185$			307.712i			1.66331i
$186$		0			0
$187$	138.480			0.740533
$188$		0			0
$189$		0			0
$190$		0			0
$191$			246.423i			1.29018i	0.764109	+	0.645088i	$0.223179\pi$
$191$			246.423i			1.29018i	−0.764109	+	0.645088i	$0.776821\pi$
$192$		0			0
$193$	−61.0003			−0.316064			−0.158032	−	0.987434i	$-0.550515\pi$
$193$	−61.0003			−0.316064			−0.158032	+	0.987434i	$0.550515\pi$
$194$		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	0.986073	−	0.166313i	$-0.0531863\pi$
$199$	69.4520	+	120.294i	0.349005	+	0.604494i	−0.637068	+	0.770808i	$0.719853\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	160.382	+	42.6804i	0.790061	+	0.210248i
$204$		0			0
$205$	−410.932			−2.00455
$206$		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
$221$			156.741i			0.709237i
$222$		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
$227$	78.5926	+	45.3754i	0.346223	+	0.199892i	0.663020	−	0.748601i	$-0.269274\pi$
$227$	78.5926	+	45.3754i	0.346223	+	0.199892i	−0.316798	+	0.948493i	$0.602608\pi$
$228$		0			0
$229$	−96.9776	−	167.970i	−0.423483	−	0.733494i	0.572795	−	0.819699i	$-0.305859\pi$
$229$	−96.9776	−	167.970i	−0.423483	−	0.733494i	−0.996277	+	0.0862051i	$0.972526\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	342.650	+	197.829i	1.47060	+	0.849053i	0.999455	−	0.0330016i	$-0.0105066\pi$
$233$	342.650	+	197.829i	1.47060	+	0.849053i	0.471147	+	0.882054i	$0.343840\pi$
$234$		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
$241$	−77.6043	+	134.415i	−0.322010	+	0.557737i	−0.980903	−	0.194500i	$-0.937692\pi$
$241$	−77.6043	+	134.415i	−0.322010	+	0.557737i	0.658893	+	0.752237i	$0.271025\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	192.666	+	331.086i	0.786391	+	1.35137i
$246$		0			0
$247$	80.7143	+	139.801i	0.326779	+	0.565997i
$248$		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
$257$	262.148	−	151.351i	1.02003	−	0.588914i	0.105917	−	0.994375i	$-0.466222\pi$
$257$	262.148	−	151.351i	1.02003	−	0.588914i	0.914113	+	0.405461i	$0.132889\pi$
$258$		0			0
$259$	266.018	−	71.7673i	1.02710	−	0.277094i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−297.807	−	171.939i	−1.13235	−	0.653760i	−0.187822	−	0.982203i	$-0.560143\pi$
$263$	−297.807	−	171.939i	−1.13235	−	0.653760i	−0.944524	+	0.328443i	$0.893476\pi$
$264$		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	$-0.831733\pi$
$269$	−230.930	+	133.327i	−0.858474	+	0.495640i	0.00502670	+	0.999987i	$0.498400\pi$
$270$		0			0
$271$	201.627	−	349.227i	0.744009	−	1.28866i	−0.206647	−	0.978416i	$-0.566255\pi$
$271$	201.627	−	349.227i	0.744009	−	1.28866i	0.950656	−	0.310247i	$-0.100412\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	309.918	−	178.931i	1.12697	−	0.650658i
$276$		0			0
$277$	−179.217	+	310.413i	−0.646993	+	1.12062i	0.336845	+	0.941560i	$0.390640\pi$
$277$	−179.217	+	310.413i	−0.646993	+	1.12062i	−0.983837	+	0.179064i	$0.942693\pi$
$278$		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
$283$	−378.349			−1.33692			−0.668461	−	0.743747i	$-0.733047\pi$
$283$	−378.349			−1.33692			−0.668461	+	0.743747i	$0.733047\pi$
$284$		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
$299$			428.607i			1.43347i
$300$		0			0
$301$	−128.247	+	481.920i	−0.426069	+	1.60106i
$302$		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
$311$		−	337.658i		−	1.08572i	−0.839824	−	0.542858i	$-0.817342\pi$
$311$		−	337.658i		−	1.08572i	0.839824	−	0.542858i	$-0.182658\pi$
$312$		0			0
$313$	269.732			0.861763			0.430881	−	0.902409i	$-0.358203\pi$
$313$	269.732			0.861763			0.430881	+	0.902409i	$0.358203\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			138.157i			0.435825i	0.975968	+	0.217913i	$0.0699248\pi$
$317$			138.157i			0.435825i	−0.975968	+	0.217913i	$0.930075\pi$
$318$		0			0
$319$	−234.932			−0.736463
$320$		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
$329$	−268.141	+	269.060i	−0.815019	+	0.817812i
$330$		0			0
$331$	−116.895			−0.353156			−0.176578	−	0.984287i	$-0.556503\pi$
$331$	−116.895			−0.353156			−0.176578	+	0.984287i	$0.556503\pi$
$332$		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
$347$			544.658i			1.56962i	0.619737	+	0.784809i	$0.287239\pi$
$347$			544.658i			1.56962i	−0.619737	+	0.784809i	$0.712761\pi$
$348$		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
$353$	287.230	+	165.833i	0.813684	+	0.469781i	0.848234	−	0.529622i	$-0.177666\pi$
$353$	287.230	+	165.833i	0.813684	+	0.469781i	−0.0345497	+	0.999403i	$0.511000\pi$
$354$		0			0
$355$	−106.874	−	185.112i	−0.301055	−	0.521442i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−364.336	−	210.349i	−1.01486	−	0.585932i	−0.102251	−	0.994759i	$-0.532605\pi$
$359$	−364.336	−	210.349i	−1.01486	−	0.585932i	−0.912612	+	0.408827i	$0.865938\pi$
$360$		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
$367$	−24.3578	+	42.1890i	−0.0663701	+	0.114956i	−0.897301	−	0.441419i	$-0.854475\pi$
$367$	−24.3578	+	42.1890i	−0.0663701	+	0.114956i	0.830931	+	0.556376i	$0.187808\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−369.923	−	98.4426i	−0.997097	−	0.265344i
$372$		0			0
$373$	176.402	+	305.537i	0.472927	+	0.819133i	0.999520	−	0.0309841i	$-0.00986414\pi$
$373$	176.402	+	305.537i	0.472927	+	0.819133i	−0.526593	+	0.850118i	$0.676531\pi$
$374$		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
$383$	−63.9136	+	36.9005i	−0.166876	+	0.0963461i	−0.581112	−	0.813824i	$-0.697382\pi$
$383$	−63.9136	+	36.9005i	−0.166876	+	0.0963461i	0.414236	+	0.910170i	$0.364049\pi$
$384$		0			0
$385$	−384.082	−	382.771i	−0.997616	−	0.994209i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−317.315	−	183.202i	−0.815719	−	0.470956i	0.0332189	−	0.999448i	$-0.489424\pi$
$389$	−317.315	−	183.202i	−0.815719	−	0.470956i	−0.848938	+	0.528492i	$0.822757\pi$
$390$		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	0.306497	+	0.951872i	$0.400843\pi$
$397$	−266.425	+	461.462i	−0.671097	+	1.16237i	−0.977593	+	0.210502i	$0.932490\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−526.244	+	303.827i	−1.31233	+	0.757673i	−0.982481	−	0.186361i	$-0.940330\pi$
$401$	−526.244	+	303.827i	−1.31233	+	0.757673i	−0.329847	+	0.944034i	$0.606997\pi$
$402$		0			0
$403$	−141.010	+	244.237i	−0.349902	+	0.606048i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−337.773	+	195.013i	−0.829909	+	0.479148i
$408$		0			0
$409$	390.924			0.955803			0.477902	−	0.878413i	$-0.341397\pi$
$409$	390.924			0.955803			0.477902	+	0.878413i	$0.341397\pi$
$410$		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
$425$		−	504.722i		−	1.18758i
$426$		0			0
$427$	322.852	+	321.750i	0.756094	+	0.753512i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.3061	−	11.7237i	0.0471139	−	0.0272012i	−0.476258	−	0.879306i	$-0.658007\pi$
$431$	20.3061	−	11.7237i	0.0471139	−	0.0272012i	0.523372	+	0.852104i	$0.324674\pi$
$432$		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
$437$		−	550.042i		−	1.25868i
$438$		0			0
$439$	509.996			1.16172			0.580861	−	0.814003i	$-0.302716\pi$
$439$	509.996			1.16172			0.580861	+	0.814003i	$0.302716\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	301.489i		−	0.680561i	−0.940324	−	0.340281i	$-0.889478\pi$
$443$		−	301.489i		−	0.680561i	0.940324	−	0.340281i	$-0.110522\pi$
$444$		0			0
$445$	−162.077			−0.364218
$446$		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
$455$	433.247	−	434.732i	0.952192	−	0.955455i
$456$		0			0
$457$	710.494			1.55469			0.777346	−	0.629074i	$-0.216566\pi$
$457$	710.494			1.55469			0.777346	+	0.629074i	$0.216566\pi$
$458$		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	$-0.295795\pi$
$467$	95.1678	+	54.9452i	0.203785	+	0.117656i	−0.394635	+	0.918838i	$0.629129\pi$
$468$		0			0
$469$	−530.471	−	528.659i	−1.13107	−	1.12720i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	705.927i		−	1.49245i
$474$		0			0
$475$	−259.908	−	450.173i	−0.547174	−	0.947734i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−62.0215	−	35.8081i	−0.129481	−	0.0747560i	0.433860	−	0.900980i	$-0.357151\pi$
$479$	−62.0215	−	35.8081i	−0.129481	−	0.0747560i	−0.563342	+	0.826224i	$0.690484\pi$
$480$		0			0
$481$	−220.730	−	382.316i	−0.458898	−	0.794835i
$482$		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	−0.991974	−	0.126443i	$-0.959644\pi$
$487$	−294.873	−	510.736i	−0.605490	−	1.04874i	0.386484	−	0.922296i	$-0.373689\pi$
$488$		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
$493$	−165.672	+	286.952i	−0.336048	+	0.582052i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−135.104	+	135.567i	−0.271839	+	0.272770i
$498$		0			0
$499$	298.362	+	516.779i	0.597920	+	1.03563i	0.993128	+	0.117036i	$0.0373394\pi$
$499$	298.362	+	516.779i	0.597920	+	1.03563i	−0.395207	+	0.918592i	$0.629327\pi$
$500$		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
$509$	−266.491	+	153.859i	−0.523558	+	0.302276i	−0.738389	−	0.674375i	$-0.764413\pi$
$509$	−266.491	+	153.859i	−0.523558	+	0.302276i	0.214831	+	0.976651i	$0.431080\pi$
$510$		0			0
$511$	−1.67604	+	6.29813i	−0.00327991	+	0.0123251i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−510.703	−	294.854i	−0.991656	−	0.572533i
$516$		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	$-0.536246\pi$
$521$	359.480	−	207.546i	0.689981	−	0.398361i	0.803605	+	0.595163i	$0.202913\pi$
$522$		0			0
$523$	−32.5093	+	56.3077i	−0.0621592	+	0.107663i	−0.895430	−	0.445202i	$-0.853132\pi$
$523$	−32.5093	+	56.3077i	−0.0621592	+	0.107663i	0.833271	+	0.552865i	$0.186465\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	304.334	−	175.708i	0.577485	−	0.333411i
$528$		0			0
$529$	465.705	−	806.624i	0.880349	−	1.52481i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	510.562	−	294.773i	0.957902	−	0.553045i
$534$		0			0
$535$	656.866			1.22779
$536$		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	−0.821530	−	0.570166i	$-0.806879\pi$
$541$	44.9103	−	77.7869i	0.0830135	−	0.143784i	0.904543	+	0.426382i	$0.140212\pi$
$542$		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
$551$			341.252i			0.619332i
$552$		0			0
$553$	−467.674	−	466.077i	−0.845704	−	0.842816i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	314.117	−	181.355i	0.563944	−	0.325593i	−0.190783	−	0.981632i	$-0.561103\pi$
$557$	314.117	−	181.355i	0.563944	−	0.325593i	0.754727	+	0.656039i	$0.227769\pi$
$558$		0			0
$559$	799.019			1.42937
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	545.681i		−	0.969238i	−0.874725	−	0.484619i	$-0.838958\pi$
$563$		−	545.681i		−	0.969238i	0.874725	−	0.484619i	$-0.161042\pi$
$564$		0			0
$565$	167.657			0.296738
$566$		0			0
$567$		0			0
$568$		0			0
$569$			260.614i			0.458020i	0.973424	+	0.229010i	$0.0735489\pi$
$569$			260.614i			0.458020i	−0.973424	+	0.229010i	$0.926451\pi$
$570$		0			0
$571$	650.559			1.13933			0.569666	−	0.821876i	$-0.307072\pi$
$571$	650.559			1.13933			0.569666	+	0.821876i	$0.307072\pi$
$572$		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	0.835004	+	0.550244i	$0.185465\pi$
$577$	515.854	+	893.485i	0.894027	+	1.54850i	0.0590234	+	0.998257i	$0.481201\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	636.759	+	169.452i	1.09597	+	0.291656i
$582$		0			0
$583$	541.871			0.929453
$584$		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	$-0.539721\pi$
$593$	−620.271	−	358.114i	−1.04599	−	0.603902i	−0.921524	+	0.388322i	$0.873055\pi$
$594$		0			0
$595$	−738.377	+	199.202i	−1.24097	+	0.334793i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			982.529i			1.64028i	0.572162	+	0.820141i	$0.306105\pi$
$599$			982.529i			1.64028i	−0.572162	+	0.820141i	$0.693895\pi$
$600$		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
$605$	−154.457	−	89.1757i	−0.255301	−	0.147398i
$606$		0			0
$607$	−77.5868	−	134.384i	−0.127820	−	0.221391i	0.795012	−	0.606594i	$-0.207465\pi$
$607$	−77.5868	−	134.384i	−0.127820	−	0.221391i	−0.922832	+	0.385203i	$0.874131\pi$
$608$		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	0.658117	−	0.752916i	$-0.271353\pi$
$613$	−197.990	−	342.929i	−0.322986	−	0.559428i	−0.981103	+	0.193488i	$0.938020\pi$
$614$		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
$619$	−453.781	+	785.971i	−0.733087	+	1.26974i	0.222471	+	0.974939i	$0.428588\pi$
$619$	−453.781	+	785.971i	−0.733087	+	1.26974i	−0.955558	+	0.294804i	$0.904746\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	37.8010	+	140.116i	0.0606758	+	0.224906i
$624$		0			0
$625$	111.785	+	193.616i	0.178855	+	0.309786i
$626$		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
$635$	−363.740	+	210.005i	−0.572819	+	0.330717i
$636$		0			0
$637$	−476.874	−	273.152i	−0.748624	−	0.428810i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−331.536	−	191.412i	−0.517217	−	0.298615i	0.218578	−	0.975819i	$-0.429858\pi$
$641$	−331.536	−	191.412i	−0.517217	−	0.298615i	−0.735795	+	0.677204i	$0.763191\pi$
$642$		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	$-0.436403\pi$
$647$	741.786	−	428.270i	1.14650	−	0.661933i	0.948032	+	0.318175i	$0.103070\pi$
$648$		0			0
$649$	193.633	−	335.381i	0.298355	−	0.516767i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−649.815	+	375.171i	−0.995122	+	0.574534i	−0.906801	−	0.421558i	$-0.861483\pi$
$653$	−649.815	+	375.171i	−0.995122	+	0.574534i	−0.0883208	+	0.996092i	$0.528150\pi$
$654$		0			0
$655$	238.809	−	413.629i	0.364593	−	0.631494i
$656$		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
$661$	28.3884			0.0429476			0.0214738	−	0.999769i	$-0.493164\pi$
$661$	28.3884			0.0429476			0.0214738	+	0.999769i	$0.493164\pi$
$662$		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
$677$		−	884.109i		−	1.30592i	−0.757391	−	0.652961i	$-0.773527\pi$
$677$		−	884.109i		−	1.30592i	0.757391	−	0.652961i	$-0.226473\pi$
$678$		0			0
$679$	277.325	−	1042.12i	0.408431	−	1.53478i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−99.1010	+	57.2160i	−0.145097	+	0.0837716i	−0.570791	−	0.821096i	$-0.693363\pi$
$683$	−99.1010	+	57.2160i	−0.145097	+	0.0837716i	0.425694	+	0.904867i	$0.360030\pi$
$684$		0			0
$685$	961.321			1.40339
$686$		0			0
$687$		0			0
$688$		0			0
$689$			613.329i			0.890173i
$690$		0			0
$691$	−488.558			−0.707031			−0.353515	−	0.935429i	$-0.615014\pi$
$691$	−488.558			−0.707031			−0.353515	+	0.935429i	$0.615014\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	59.0435i		−	0.0849547i
$696$		0			0
$697$	−734.610			−1.05396
$698$		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
$707$	−67.1059	−	248.740i	−0.0949164	−	0.351824i
$708$		0			0
$709$	−624.591			−0.880946			−0.440473	−	0.897766i	$-0.645189\pi$
$709$	−624.591			−0.880946			−0.440473	+	0.897766i	$0.645189\pi$
$710$		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	$-0.550626\pi$
$719$	−785.622	−	453.579i	−1.09266	−	0.630847i	−0.934283	+	0.356531i	$0.883959\pi$
$720$		0			0
$721$	−135.792	+	510.273i	−0.188339	+	0.707730i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			856.264i			1.18105i
$726$		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
$731$	−862.237	−	497.813i	−1.17953	−	0.681003i
$732$		0			0
$733$	570.670	+	988.429i	0.778540	+	1.34847i	0.932783	+	0.360438i	$0.117373\pi$
$733$	570.670	+	988.429i	0.778540	+	1.34847i	−0.154243	+	0.988033i	$0.549294\pi$
$734$		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	−0.998579	−	0.0532999i	$-0.983026\pi$
$739$	−403.086	−	698.166i	−0.545448	−	0.944744i	0.453130	−	0.891444i	$-0.350307\pi$
$740$		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
$745$	851.391	−	1474.65i	1.14281	−	1.97940i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−153.200	−	567.863i	−0.204540	−	0.758162i
$750$		0			0
$751$	202.222	+	350.259i	0.269270	+	0.466390i	0.968674	−	0.248338i	$-0.0798842\pi$
$751$	202.222	+	350.259i	0.269270	+	0.466390i	−0.699403	+	0.714727i	$0.746551\pi$
$752$		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
$761$	808.228	−	466.630i	1.06206	−	0.613181i	0.136059	−	0.990701i	$-0.456556\pi$
$761$	808.228	−	466.630i	1.06206	−	0.613181i	0.926001	+	0.377520i	$0.123223\pi$
$762$		0			0
$763$	−291.812	+	1096.56i	−0.382454	+	1.43717i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	379.609	+	219.167i	0.494927	+	0.285746i
$768$		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	0.533157	−	0.846016i	$-0.321006\pi$
$773$	1184.55	−	683.901i	1.53241	−	0.884736i	0.999250	−	0.0387193i	$-0.0123278\pi$
$774$		0			0
$775$	454.067	−	786.467i	0.585893	−	1.01480i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−655.216	+	378.289i	−0.841099	+	0.485608i
$780$		0			0
$781$	135.464	−	234.630i	0.173449	−	0.300423i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	724.768	−	418.445i	0.923271	−	0.533051i
$786$		0			0
$787$	−581.440			−0.738805			−0.369403	−	0.929269i	$-0.620438\pi$
$787$	−581.440			−0.738805			−0.369403	+	0.929269i	$0.620438\pi$
$788$		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
$803$		−	9.22563i		−	0.0114890i
$804$		0			0
$805$	−2019.08	+	544.715i	−2.50818	+	0.676665i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	609.778	−	352.056i	0.753743	−	0.435174i	−0.0733017	−	0.997310i	$-0.523354\pi$
$809$	609.778	−	352.056i	0.753743	−	0.435174i	0.827045	+	0.562136i	$0.190020\pi$
$810$		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
$815$			1902.03i			2.33378i
$816$		0			0
$817$	−1025.40			−1.25508
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	696.026i		−	0.847778i	−0.905714	−	0.423889i	$-0.860665\pi$
$821$		−	696.026i		−	0.847778i	0.905714	−	0.423889i	$-0.139335\pi$
$822$		0			0
$823$	−161.792			−0.196588			−0.0982938	−	0.995157i	$-0.531338\pi$
$823$	−161.792			−0.196588			−0.0982938	+	0.995157i	$0.531338\pi$
$824$		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	−0.650385	−	0.759604i	$-0.725393\pi$
$829$	−814.931	−	1411.50i	−0.983029	−	1.70266i	−0.332644	−	0.943052i	$-0.607941\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	344.422	+	591.871i	0.413472	+	0.710529i
$834$		0			0
$835$	67.6740			0.0810467
$836$		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
$851$			1504.20i			1.76757i
$852$		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
$857$	639.972	+	369.488i	0.746759	+	0.431141i	0.824522	−	0.565831i	$-0.191444\pi$
$857$	639.972	+	369.488i	0.746759	+	0.431141i	−0.0777629	+	0.996972i	$0.524778\pi$
$858$		0			0
$859$	−360.760	−	624.855i	−0.419977	−	0.727421i	0.575960	−	0.817478i	$-0.304628\pi$
$859$	−360.760	−	624.855i	−0.419977	−	0.727421i	−0.995937	+	0.0900568i	$0.971295\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	756.958	+	437.030i	0.877124	+	0.506408i	0.869709	−	0.493565i	$-0.164306\pi$
$863$	756.958	+	437.030i	0.877124	+	0.506408i	0.00741492	+	0.999973i	$0.497640\pi$
$864$		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
$871$	−599.969	+	1039.18i	−0.688828	+	1.19309i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−429.372	+	430.844i	−0.490711	+	0.492393i
$876$		0			0
$877$	276.070	+	478.167i	0.314789	+	0.545230i	0.979393	−	0.201966i	$-0.0647331\pi$
$877$	276.070	+	478.167i	0.314789	+	0.545230i	−0.664604	+	0.747196i	$0.731400\pi$
$878$		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
$887$	−8.18965	+	4.72830i	−0.00923298	+	0.00533066i	−0.504609	−	0.863348i	$-0.668363\pi$
$887$	−8.18965	+	4.72830i	−0.00923298	+	0.00533066i	0.495376	+	0.868678i	$0.335030\pi$
$888$		0			0
$889$	266.385	+	265.475i	0.299646	+	0.298623i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−676.416	−	390.529i	−0.757465	−	0.437322i
$894$		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
$905$	−314.432	+	181.537i	−0.347438	+	0.200594i
$906$		0			0
$907$	127.953	−	221.621i	0.141073	−	0.244345i	−0.786828	−	0.617172i	$-0.788278\pi$
$907$	127.953	−	221.621i	0.141073	−	0.244345i	0.927901	+	0.372827i	$0.121611\pi$
$908$		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
$913$	−932.738			−1.02162
$914$		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	−0.768554	−	0.639785i	$-0.779023\pi$
$919$	156.040	−	270.269i	0.169793	−	0.294090i	0.938347	+	0.345695i	$0.112357\pi$
$920$		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
$929$		−	823.181i		−	0.886094i	−0.896499	−	0.443047i	$-0.853898\pi$
$929$		−	823.181i		−	0.886094i	0.896499	−	0.443047i	$-0.146102\pi$
$930$		0			0
$931$	611.983	+	350.543i	0.657340	+	0.376523i
$932$		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
$941$		−	643.859i		−	0.684229i	−0.939658	−	0.342114i	$-0.888857\pi$
$941$		−	643.859i		−	0.684229i	0.939658	−	0.342114i	$-0.111143\pi$
$942$		0			0
$943$	−2008.78			−2.13020
$944$		0			0
$945$		0			0
$946$		0			0
$947$			983.962i			1.03903i	0.854461	+	0.519515i	$0.173887\pi$
$947$			983.962i			1.03903i	−0.854461	+	0.519515i	$0.826113\pi$
$948$		0			0
$949$	10.4422			0.0110034
$950$		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
$959$	−224.208	−	831.066i	−0.233794	−	0.866597i
$960$		0			0
$961$	−328.707			−0.342047
$962$		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	$-0.342098\pi$
$971$	−46.3049	−	26.7342i	−0.0476879	−	0.0275326i	−0.523654	+	0.851931i	$0.675432\pi$
$972$		0			0
$973$	−51.0434	+	13.7707i	−0.0524598	+	0.0141528i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			57.2767i			0.0586251i	0.999570	+	0.0293125i	$0.00933180\pi$
$977$			57.2767i			0.0586251i	−0.999570	+	0.0293125i	$0.990668\pi$
$978$		0			0
$979$	−102.717	−	177.910i	−0.104920	−	0.181727i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	182.596	+	105.422i	0.185754	+	0.107245i	0.589993	−	0.807408i	$-0.299130\pi$
$983$	182.596	+	105.422i	0.185754	+	0.107245i	−0.404239	+	0.914653i	$0.632464\pi$
$984$		0			0
$985$	247.580	+	428.822i	0.251351	+	0.435352i
$986$		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	0.999643	−	0.0267044i	$-0.00850128\pi$
$991$	472.405	+	818.229i	0.476695	+	0.825660i	−0.522948	+	0.852364i	$0.675168\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	940.416	+	542.949i	0.945142	+	0.545678i
$996$		0			0
$997$	−107.050	+	185.416i	−0.107372	+	0.185974i	−0.914705	−	0.404123i	$-0.867577\pi$
$997$	−107.050	+	185.416i	−0.107372	+	0.185974i	0.807333	+	0.590096i	$0.200910\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.m.a.65.9	✓	32	9.7	even	3
252.3.m.a.221.9	yes	32	21.11	odd	6
252.3.bh.a.137.14	yes	32	63.25	even	3
252.3.bh.a.149.14	yes	32	3.2	odd	2
756.3.m.a.557.16		32	7.4	even	3
756.3.m.a.737.1		32	9.2	odd	6
756.3.bh.a.233.16		32	1.1	even	1	trivial
756.3.bh.a.305.16		32	63.11	odd	6	inner

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.bh (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.77026 - 3.90881i) q^{5} +(-4.95821 - 4.94127i) q^{7} +O(q^{10})\)	\(q+(6.77026 - 3.90881i) q^{5} +(-4.95821 - 4.94127i) q^{7} +(8.58135 + 4.95444i) q^{11} +(-5.60780 + 9.71299i) q^{13} +(12.1030 - 6.98765i) q^{17} +(7.19662 - 12.4649i) q^{19} +(33.0954 - 19.1077i) q^{23} +(18.0576 - 31.2767i) q^{25} +(-20.5328 + 11.8546i) q^{29} +25.1454 q^{31} +(-52.8829 - 14.0730i) q^{35} +(-19.6806 + 34.0879i) q^{37} +(-45.5225 - 26.2824i) q^{41} +(-35.6209 - 61.6972i) q^{43} -54.2656i q^{47} +(0.167642 + 48.9997i) q^{49} +(47.3589 - 27.3427i) q^{53} +77.4639 q^{55} -39.0826i q^{59} -65.1147 q^{61} +87.6793i q^{65} +106.988 q^{67} -27.3419i q^{71} +(-0.465523 - 0.806310i) q^{73} +(-18.0668 - 66.9679i) q^{77} +94.3233 q^{79} +(-81.5203 + 47.0658i) q^{83} +(54.6269 - 94.6165i) q^{85} +(-17.9546 - 10.3661i) q^{89} +(75.7992 - 20.4494i) q^{91} -112.521i q^{95} +(77.0276 + 133.416i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - q^{7}+O(q^{10}) \) 32 * q - q^7	\( 32 q - q^{7} - 18 q^{11} - 5 q^{13} - 27 q^{17} - 14 q^{19} + 45 q^{23} + 80 q^{25} - 36 q^{29} + 16 q^{31} - 45 q^{35} - 11 q^{37} - 72 q^{41} + 16 q^{43} - 37 q^{49} + 180 q^{53} - 24 q^{55} + 82 q^{61} + 70 q^{67} - 98 q^{73} + 135 q^{77} + 142 q^{79} - 30 q^{85} - 189 q^{89} + 109 q^{91} + 19 q^{97}+O(q^{100}) \) 32 * q - q^7 - 18 * q^11 - 5 * q^13 - 27 * q^17 - 14 * q^19 + 45 * q^23 + 80 * q^25 - 36 * q^29 + 16 * q^31 - 45 * q^35 - 11 * q^37 - 72 * q^41 + 16 * q^43 - 37 * q^49 + 180 * q^53 - 24 * q^55 + 82 * q^61 + 70 * q^67 - 98 * q^73 + 135 * q^77 + 142 * q^79 - 30 * q^85 - 189 * q^89 + 109 * q^91 + 19 * q^97

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