LMFDB - Embedded newform 1568.3.h.a.881.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(881,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1568.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.10
Character	$\chi$	$=$	1568.881
Dual form			1568.3.h.a.881.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33563 q^{3} -3.10110 q^{5} -3.54484 q^{9} +O(q^{10})$	$q-2.33563 q^{3} -3.10110 q^{5} -3.54484 q^{9} -4.69506i q^{11} +6.88097 q^{13} +7.24301 q^{15} +16.9953i q^{17} +26.2198 q^{19} -25.8805 q^{23} -15.3832 q^{25} +29.3001 q^{27} +42.2701i q^{29} -18.3625i q^{31} +10.9659i q^{33} +49.8827i q^{37} -16.0714 q^{39} -10.7844i q^{41} -24.1791i q^{43} +10.9929 q^{45} +13.6809i q^{47} -39.6947i q^{51} -6.97379i q^{53} +14.5598i q^{55} -61.2396 q^{57} +106.184 q^{59} -93.4609 q^{61} -21.3386 q^{65} -89.2299i q^{67} +60.4473 q^{69} -81.7898 q^{71} -137.956i q^{73} +35.9294 q^{75} +13.1018 q^{79} -36.5306 q^{81} +2.15689 q^{83} -52.7041i q^{85} -98.7272i q^{87} +101.413i q^{89} +42.8880i q^{93} -81.3101 q^{95} -88.9318i q^{97} +16.6432i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 64 q^{9}+O(q^{10}) $ 28 * q + 64 * q^9	$ 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) $ 28 * q + 64 * q^9 - 28 * q^15 + 60 * q^23 + 64 * q^25 - 40 * q^39 + 124 * q^57 + 104 * q^65 + 136 * q^71 - 324 * q^79 + 36 * q^81 + 580 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.33563	−0.778543	−0.389271	−	0.921123i	$-0.627273\pi$
$3$	−2.33563	−0.778543	−0.389271	+	0.921123i	$0.627273\pi$
$4$	0	0
$5$	−3.10110	−0.620220	−0.310110	−	0.950701i	$-0.600366\pi$
$5$	−3.10110	−0.620220	−0.310110	+	0.950701i	$0.600366\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.54484	−0.393871
$10$	0	0
$11$	− 4.69506i	− 0.426824i	−0.976962	−	0.213412i	$-0.931542\pi$
$11$	− 4.69506i	− 0.426824i	0.976962	−	0.213412i	$-0.0684576\pi$
$12$	0	0
$13$	6.88097	0.529305	0.264653	−	0.964344i	$-0.414743\pi$
$13$	6.88097	0.529305	0.264653	+	0.964344i	$0.414743\pi$
$14$	0	0
$15$	7.24301	0.482868
$16$	0	0
$17$	16.9953i	0.999724i	0.866105	+	0.499862i	$0.166616\pi$
$17$	16.9953i	0.999724i	−0.866105	+	0.499862i	$0.833384\pi$
$18$	0	0
$19$	26.2198	1.37999	0.689994	−	0.723815i	$-0.257613\pi$
$19$	26.2198	1.37999	0.689994	+	0.723815i	$0.257613\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−25.8805	−1.12524	−0.562620	−	0.826716i	$-0.690206\pi$
$23$	−25.8805	−1.12524	−0.562620	+	0.826716i	$0.690206\pi$
$24$	0	0
$25$	−15.3832	−0.615328
$26$	0	0
$27$	29.3001	1.08519
$28$	0	0
$29$	42.2701i	1.45759i	0.684732	+	0.728795i	$0.259919\pi$
$29$	42.2701i	1.45759i	−0.684732	+	0.728795i	$0.740081\pi$
$30$	0	0
$31$	− 18.3625i	− 0.592339i	−0.955135	−	0.296170i	$-0.904291\pi$
$31$	− 18.3625i	− 0.592339i	0.955135	−	0.296170i	$-0.0957094\pi$
$32$	0	0
$33$	10.9659i	0.332301i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	49.8827i	1.34818i	0.738649	+	0.674090i	$0.235464\pi$
$37$	49.8827i	1.34818i	−0.738649	+	0.674090i	$0.764536\pi$
$38$	0	0
$39$	−16.0714	−0.412087
$40$	0	0
$41$	− 10.7844i	− 0.263035i	−0.991314	−	0.131517i	$-0.958015\pi$
$41$	− 10.7844i	− 0.263035i	0.991314	−	0.131517i	$-0.0419849\pi$
$42$	0	0
$43$	− 24.1791i	− 0.562304i	−0.959663	−	0.281152i	$-0.909284\pi$
$43$	− 24.1791i	− 0.562304i	0.959663	−	0.281152i	$-0.0907165\pi$
$44$	0	0
$45$	10.9929	0.244287
$46$	0	0
$47$	13.6809i	0.291083i	0.989352	+	0.145542i	$0.0464925\pi$
$47$	13.6809i	0.291083i	−0.989352	+	0.145542i	$0.953508\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 39.6947i	− 0.778328i
$52$	0	0
$53$	− 6.97379i	− 0.131581i	−0.997833	−	0.0657905i	$-0.979043\pi$
$53$	− 6.97379i	− 0.131581i	0.997833	−	0.0657905i	$-0.0209569\pi$
$54$	0	0
$55$	14.5598i	0.264724i
$56$	0	0
$57$	−61.2396	−1.07438
$58$	0	0
$59$	106.184	1.79974	0.899868	−	0.436163i	$-0.143663\pi$
$59$	106.184	1.79974	0.899868	+	0.436163i	$0.143663\pi$
$60$	0	0
$61$	−93.4609	−1.53215	−0.766073	−	0.642754i	$-0.777792\pi$
$61$	−93.4609	−1.53215	−0.766073	+	0.642754i	$0.777792\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−21.3386	−0.328286
$66$	0	0
$67$	− 89.2299i	− 1.33179i	−0.746046	−	0.665894i	$-0.768050\pi$
$67$	− 89.2299i	− 1.33179i	0.746046	−	0.665894i	$-0.231950\pi$
$68$	0	0
$69$	60.4473	0.876047
$70$	0	0
$71$	−81.7898	−1.15197	−0.575984	−	0.817461i	$-0.695381\pi$
$71$	−81.7898	−1.15197	−0.575984	+	0.817461i	$0.695381\pi$
$72$	0	0
$73$	− 137.956i	− 1.88981i	−0.327347	−	0.944904i	$-0.606155\pi$
$73$	− 137.956i	− 1.88981i	0.327347	−	0.944904i	$-0.393845\pi$
$74$	0	0
$75$	35.9294	0.479059
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.1018	0.165846	0.0829228	−	0.996556i	$-0.473574\pi$
$79$	13.1018	0.165846	0.0829228	+	0.996556i	$0.473574\pi$
$80$	0	0
$81$	−36.5306	−0.450995
$82$	0	0
$83$	2.15689	0.0259867	0.0129933	−	0.999916i	$-0.495864\pi$
$83$	2.15689	0.0259867	0.0129933	+	0.999916i	$0.495864\pi$
$84$	0	0
$85$	− 52.7041i	− 0.620048i
$86$	0	0
$87$	− 98.7272i	− 1.13480i
$88$	0	0
$89$	101.413i	1.13947i	0.821828	+	0.569735i	$0.192954\pi$
$89$	101.413i	1.13947i	−0.821828	+	0.569735i	$0.807046\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	42.8880i	0.461162i
$94$	0	0
$95$	−81.3101	−0.855895
$96$	0	0
$97$	− 88.9318i	− 0.916823i	−0.888740	−	0.458412i	$-0.848419\pi$
$97$	− 88.9318i	− 0.916823i	0.888740	−	0.458412i	$-0.151581\pi$
$98$	0	0
$99$	16.6432i	0.168114i
$100$	0	0
$101$	−20.8477	−0.206413	−0.103206	−	0.994660i	$-0.532910\pi$
$101$	−20.8477	−0.206413	−0.103206	+	0.994660i	$0.532910\pi$
$102$	0	0
$103$	3.43487i	0.0333483i	0.999861	+	0.0166741i	$0.00530779\pi$
$103$	3.43487i	0.0333483i	−0.999861	+	0.0166741i	$0.994692\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 67.5041i	− 0.630880i	−0.948946	−	0.315440i	$-0.897848\pi$
$107$	− 67.5041i	− 0.630880i	0.948946	−	0.315440i	$-0.102152\pi$
$108$	0	0
$109$	− 135.055i	− 1.23903i	−0.784984	−	0.619516i	$-0.787329\pi$
$109$	− 135.055i	− 1.23903i	0.784984	−	0.619516i	$-0.212671\pi$
$110$	0	0
$111$	− 116.507i	− 1.04962i
$112$	0	0
$113$	136.328	1.20645	0.603223	−	0.797573i	$-0.293883\pi$
$113$	136.328	1.20645	0.603223	+	0.797573i	$0.293883\pi$
$114$	0	0
$115$	80.2580	0.697896
$116$	0	0
$117$	−24.3919	−0.208478
$118$	0	0
$119$	0	0
$120$	0	0
$121$	98.9564	0.817821
$122$	0	0
$123$	25.1884i	0.204784i
$124$	0	0
$125$	125.232	1.00186
$126$	0	0
$127$	6.39702	0.0503702	0.0251851	−	0.999683i	$-0.491982\pi$
$127$	6.39702	0.0503702	0.0251851	+	0.999683i	$0.491982\pi$
$128$	0	0
$129$	56.4733i	0.437778i
$130$	0	0
$131$	−173.694	−1.32591	−0.662956	−	0.748659i	$-0.730698\pi$
$131$	−173.694	−1.32591	−0.662956	+	0.748659i	$0.730698\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−90.8624	−0.673055
$136$	0	0
$137$	−76.5851	−0.559016	−0.279508	−	0.960143i	$-0.590171\pi$
$137$	−76.5851	−0.559016	−0.279508	+	0.960143i	$0.590171\pi$
$138$	0	0
$139$	−72.4724	−0.521384	−0.260692	−	0.965422i	$-0.583951\pi$
$139$	−72.4724	−0.521384	−0.260692	+	0.965422i	$0.583951\pi$
$140$	0	0
$141$	− 31.9535i	− 0.226621i
$142$	0	0
$143$	− 32.3066i	− 0.225920i
$144$	0	0
$145$	− 131.084i	− 0.904026i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 244.268i	− 1.63938i	−0.572807	−	0.819690i	$-0.694146\pi$
$149$	− 244.268i	− 1.63938i	0.572807	−	0.819690i	$-0.305854\pi$
$150$	0	0
$151$	−206.218	−1.36568	−0.682841	−	0.730567i	$-0.739256\pi$
$151$	−206.218	−1.36568	−0.682841	+	0.730567i	$0.739256\pi$
$152$	0	0
$153$	− 60.2456i	− 0.393762i
$154$	0	0
$155$	56.9440i	0.367380i
$156$	0	0
$157$	74.5428	0.474795	0.237397	−	0.971413i	$-0.423706\pi$
$157$	74.5428	0.474795	0.237397	+	0.971413i	$0.423706\pi$
$158$	0	0
$159$	16.2882i	0.102441i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 98.2845i	− 0.602972i	−0.953471	−	0.301486i	$-0.902517\pi$
$163$	− 98.2845i	− 0.602972i	0.953471	−	0.301486i	$-0.0974827\pi$
$164$	0	0
$165$	− 34.0064i	− 0.206099i
$166$	0	0
$167$	− 252.539i	− 1.51221i	−0.654449	−	0.756106i	$-0.727099\pi$
$167$	− 252.539i	− 1.51221i	0.654449	−	0.756106i	$-0.272901\pi$
$168$	0	0
$169$	−121.652	−0.719836
$170$	0	0
$171$	−92.9449	−0.543537
$172$	0	0
$173$	150.378	0.869236	0.434618	−	0.900615i	$-0.356883\pi$
$173$	150.378	0.869236	0.434618	+	0.900615i	$0.356883\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−248.007	−1.40117
$178$	0	0
$179$	− 103.490i	− 0.578154i	−0.957306	−	0.289077i	$-0.906652\pi$
$179$	− 103.490i	− 0.578154i	0.957306	−	0.289077i	$-0.0933484\pi$
$180$	0	0
$181$	−95.1121	−0.525481	−0.262741	−	0.964867i	$-0.584626\pi$
$181$	−95.1121	−0.525481	−0.262741	+	0.964867i	$0.584626\pi$
$182$	0	0
$183$	218.290	1.19284
$184$	0	0
$185$	− 154.691i	− 0.836168i
$186$	0	0
$187$	79.7940	0.426706
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.94503	−0.0206546	−0.0103273	−	0.999947i	$-0.503287\pi$
$191$	−3.94503	−0.0206546	−0.0103273	+	0.999947i	$0.503287\pi$
$192$	0	0
$193$	−292.182	−1.51390	−0.756949	−	0.653474i	$-0.773311\pi$
$193$	−292.182	−1.51390	−0.756949	+	0.653474i	$0.773311\pi$
$194$	0	0
$195$	49.8390	0.255584
$196$	0	0
$197$	− 160.503i	− 0.814735i	−0.913264	−	0.407367i	$-0.866447\pi$
$197$	− 160.503i	− 0.814735i	0.913264	−	0.407367i	$-0.133553\pi$
$198$	0	0
$199$	− 201.450i	− 1.01231i	−0.862442	−	0.506156i	$-0.831066\pi$
$199$	− 201.450i	− 1.01231i	0.862442	−	0.506156i	$-0.168934\pi$
$200$	0	0
$201$	208.408i	1.03685i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	33.4436i	0.163139i
$206$	0	0
$207$	91.7422	0.443199
$208$	0	0
$209$	− 123.103i	− 0.589012i
$210$	0	0
$211$	− 170.542i	− 0.808256i	−0.914702	−	0.404128i	$-0.867575\pi$
$211$	− 170.542i	− 0.808256i	0.914702	−	0.404128i	$-0.132425\pi$
$212$	0	0
$213$	191.030	0.896857
$214$	0	0
$215$	74.9816i	0.348752i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	322.214i	1.47130i
$220$	0	0
$221$	116.944i	0.529159i
$222$	0	0
$223$	143.446i	0.643255i	0.946866	+	0.321628i	$0.104230\pi$
$223$	143.446i	0.643255i	−0.946866	+	0.321628i	$0.895770\pi$
$224$	0	0
$225$	54.5309	0.242360
$226$	0	0
$227$	23.5193	0.103609	0.0518047	−	0.998657i	$-0.483503\pi$
$227$	23.5193	0.103609	0.0518047	+	0.998657i	$0.483503\pi$
$228$	0	0
$229$	−61.4080	−0.268157	−0.134079	−	0.990971i	$-0.542807\pi$
$229$	−61.4080	−0.268157	−0.134079	+	0.990971i	$0.542807\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	104.198	0.447203	0.223601	−	0.974681i	$-0.428219\pi$
$233$	104.198	0.447203	0.223601	+	0.974681i	$0.428219\pi$
$234$	0	0
$235$	− 42.4259i	− 0.180536i
$236$	0	0
$237$	−30.6010	−0.129118
$238$	0	0
$239$	−104.695	−0.438056	−0.219028	−	0.975719i	$-0.570289\pi$
$239$	−104.695	−0.438056	−0.219028	+	0.975719i	$0.570289\pi$
$240$	0	0
$241$	− 164.718i	− 0.683478i	−0.939795	−	0.341739i	$-0.888984\pi$
$241$	− 164.718i	− 0.683478i	0.939795	−	0.341739i	$-0.111016\pi$
$242$	0	0
$243$	−178.379	−0.734070
$244$	0	0
$245$	0	0
$246$	0	0
$247$	180.417	0.730435
$248$	0	0
$249$	−5.03770	−0.0202317
$250$	0	0
$251$	399.066	1.58990	0.794952	−	0.606672i	$-0.207496\pi$
$251$	399.066	1.58990	0.794952	+	0.606672i	$0.207496\pi$
$252$	0	0
$253$	121.511i	0.480279i
$254$	0	0
$255$	123.097i	0.482734i
$256$	0	0
$257$	− 2.45191i	− 0.00954049i	−0.999989	−	0.00477025i	$-0.998482\pi$
$257$	− 2.45191i	− 0.00954049i	0.999989	−	0.00477025i	$-0.00151842\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 149.841i	− 0.574102i
$262$	0	0
$263$	−57.5596	−0.218858	−0.109429	−	0.993995i	$-0.534902\pi$
$263$	−57.5596	−0.218858	−0.109429	+	0.993995i	$0.534902\pi$
$264$	0	0
$265$	21.6264i	0.0816091i
$266$	0	0
$267$	− 236.863i	− 0.887126i
$268$	0	0
$269$	240.403	0.893690	0.446845	−	0.894611i	$-0.352548\pi$
$269$	240.403	0.893690	0.446845	+	0.894611i	$0.352548\pi$
$270$	0	0
$271$	134.531i	0.496424i	0.968706	+	0.248212i	$0.0798430\pi$
$271$	134.531i	0.496424i	−0.968706	+	0.248212i	$0.920157\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	72.2250i	0.262636i
$276$	0	0
$277$	− 110.401i	− 0.398558i	−0.979943	−	0.199279i	$-0.936140\pi$
$277$	− 110.401i	− 0.398558i	0.979943	−	0.199279i	$-0.0638600\pi$
$278$	0	0
$279$	65.0922i	0.233305i
$280$	0	0
$281$	−154.087	−0.548351	−0.274175	−	0.961680i	$-0.588405\pi$
$281$	−154.087	−0.548351	−0.274175	+	0.961680i	$0.588405\pi$
$282$	0	0
$283$	−30.9428	−0.109338	−0.0546692	−	0.998505i	$-0.517410\pi$
$283$	−30.9428	−0.109338	−0.0546692	+	0.998505i	$0.517410\pi$
$284$	0	0
$285$	189.910	0.666351
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0.159593	0.000552224 0
$290$	0	0
$291$	207.712i	0.713786i
$292$	0	0
$293$	−511.686	−1.74637	−0.873184	−	0.487390i	$-0.837949\pi$
$293$	−511.686	−1.74637	−0.873184	+	0.487390i	$0.837949\pi$
$294$	0	0
$295$	−329.288	−1.11623
$296$	0	0
$297$	− 137.566i	− 0.463184i
$298$	0	0
$299$	−178.083	−0.595595
$300$	0	0
$301$	0	0
$302$	0	0
$303$	48.6925	0.160701
$304$	0	0
$305$	289.831	0.950267
$306$	0	0
$307$	51.2670	0.166993	0.0834967	−	0.996508i	$-0.473391\pi$
$307$	51.2670	0.166993	0.0834967	+	0.996508i	$0.473391\pi$
$308$	0	0
$309$	− 8.02259i	− 0.0259631i
$310$	0	0
$311$	20.5467i	0.0660666i	0.999454	+	0.0330333i	$0.0105167\pi$
$311$	20.5467i	0.0660666i	−0.999454	+	0.0330333i	$0.989483\pi$
$312$	0	0
$313$	− 337.127i	− 1.07708i	−0.842599	−	0.538541i	$-0.818976\pi$
$313$	− 337.127i	− 1.07708i	0.842599	−	0.538541i	$-0.181024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	93.2575i	0.294188i	0.989123	+	0.147094i	$0.0469920\pi$
$317$	93.2575i	0.294188i	−0.989123	+	0.147094i	$0.953008\pi$
$318$	0	0
$319$	198.461	0.622134
$320$	0	0
$321$	157.665i	0.491167i
$322$	0	0
$323$	445.613i	1.37961i
$324$	0	0
$325$	−105.851	−0.325696
$326$	0	0
$327$	315.437i	0.964640i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	75.0485i	0.226732i	0.993553	+	0.113366i	$0.0361633\pi$
$331$	75.0485i	0.226732i	−0.993553	+	0.113366i	$0.963837\pi$
$332$	0	0
$333$	− 176.826i	− 0.531009i
$334$	0	0
$335$	276.711i	0.826002i
$336$	0	0
$337$	−140.105	−0.415743	−0.207872	−	0.978156i	$-0.566654\pi$
$337$	−140.105	−0.415743	−0.207872	+	0.978156i	$0.566654\pi$
$338$	0	0
$339$	−318.412	−0.939269
$340$	0	0
$341$	−86.2131	−0.252824
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−187.453	−0.543342
$346$	0	0
$347$	− 75.0227i	− 0.216204i	−0.994140	−	0.108102i	$-0.965523\pi$
$347$	− 75.0227i	− 0.216204i	0.994140	−	0.108102i	$-0.0344773\pi$
$348$	0	0
$349$	603.618	1.72956	0.864782	−	0.502148i	$-0.167457\pi$
$349$	603.618	1.72956	0.864782	+	0.502148i	$0.167457\pi$
$350$	0	0
$351$	201.613	0.574396
$352$	0	0
$353$	389.728i	1.10405i	0.833829	+	0.552023i	$0.186144\pi$
$353$	389.728i	1.10405i	−0.833829	+	0.552023i	$0.813856\pi$
$354$	0	0
$355$	253.638	0.714473
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−138.443	−0.385635	−0.192817	−	0.981235i	$-0.561762\pi$
$359$	−138.443	−0.385635	−0.192817	+	0.981235i	$0.561762\pi$
$360$	0	0
$361$	326.476	0.904366
$362$	0	0
$363$	−231.125	−0.636709
$364$	0	0
$365$	427.815i	1.17210i
$366$	0	0
$367$	472.069i	1.28629i	0.765744	+	0.643145i	$0.222371\pi$
$367$	472.069i	1.28629i	−0.765744	+	0.643145i	$0.777629\pi$
$368$	0	0
$369$	38.2291i	0.103602i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 35.2668i	− 0.0945490i	−0.998882	−	0.0472745i	$-0.984946\pi$
$373$	− 35.2668i	− 0.0945490i	0.998882	−	0.0472745i	$-0.0150536\pi$
$374$	0	0
$375$	−292.496	−0.779989
$376$	0	0
$377$	290.859i	0.771510i
$378$	0	0
$379$	230.447i	0.608039i	0.952666	+	0.304019i	$0.0983287\pi$
$379$	230.447i	0.608039i	−0.952666	+	0.304019i	$0.901671\pi$
$380$	0	0
$381$	−14.9411	−0.0392154
$382$	0	0
$383$	− 554.279i	− 1.44720i	−0.690217	−	0.723602i	$-0.742485\pi$
$383$	− 554.279i	− 1.44720i	0.690217	−	0.723602i	$-0.257515\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	85.7109i	0.221475i
$388$	0	0
$389$	397.680i	1.02231i	0.859488	+	0.511157i	$0.170783\pi$
$389$	397.680i	1.02231i	−0.859488	+	0.511157i	$0.829217\pi$
$390$	0	0
$391$	− 439.847i	− 1.12493i
$392$	0	0
$393$	405.686	1.03228
$394$	0	0
$395$	−40.6300	−0.102861
$396$	0	0
$397$	−121.909	−0.307075	−0.153538	−	0.988143i	$-0.549067\pi$
$397$	−121.909	−0.307075	−0.153538	+	0.988143i	$0.549067\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	248.673	0.620133	0.310067	−	0.950715i	$-0.399649\pi$
$401$	248.673	0.620133	0.310067	+	0.950715i	$0.399649\pi$
$402$	0	0
$403$	− 126.352i	− 0.313528i
$404$	0	0
$405$	113.285	0.279716
$406$	0	0
$407$	234.202	0.575435
$408$	0	0
$409$	672.869i	1.64516i	0.568653	+	0.822578i	$0.307465\pi$
$409$	672.869i	1.64516i	−0.568653	+	0.822578i	$0.692535\pi$
$410$	0	0
$411$	178.874	0.435218
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.68874	−0.0161174
$416$	0	0
$417$	169.269	0.405920
$418$	0	0
$419$	−178.795	−0.426718	−0.213359	−	0.976974i	$-0.568440\pi$
$419$	−178.795	−0.426718	−0.213359	+	0.976974i	$0.568440\pi$
$420$	0	0
$421$	− 212.470i	− 0.504679i	−0.967639	−	0.252340i	$-0.918800\pi$
$421$	− 212.470i	− 0.504679i	0.967639	−	0.252340i	$-0.0812000\pi$
$422$	0	0
$423$	− 48.4967i	− 0.114649i
$424$	0	0
$425$	− 261.442i	− 0.615158i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	75.4562i	0.175888i
$430$	0	0
$431$	691.464	1.60433	0.802163	−	0.597105i	$-0.203683\pi$
$431$	691.464	1.60433	0.802163	+	0.597105i	$0.203683\pi$
$432$	0	0
$433$	99.8389i	0.230575i	0.993332	+	0.115287i	$0.0367789\pi$
$433$	99.8389i	0.230575i	−0.993332	+	0.115287i	$0.963221\pi$
$434$	0	0
$435$	306.163i	0.703823i
$436$	0	0
$437$	−678.581	−1.55282
$438$	0	0
$439$	− 692.091i	− 1.57652i	−0.615344	−	0.788259i	$-0.710983\pi$
$439$	− 692.091i	− 1.57652i	0.615344	−	0.788259i	$-0.289017\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 269.702i	− 0.608809i	−0.952543	−	0.304405i	$-0.901543\pi$
$443$	− 269.702i	− 0.608809i	0.952543	−	0.304405i	$-0.0984575\pi$
$444$	0	0
$445$	− 314.491i	− 0.706722i
$446$	0	0
$447$	570.519i	1.27633i
$448$	0	0
$449$	−76.6510	−0.170715	−0.0853575	−	0.996350i	$-0.527203\pi$
$449$	−76.6510	−0.170715	−0.0853575	+	0.996350i	$0.527203\pi$
$450$	0	0
$451$	−50.6336	−0.112270
$452$	0	0
$453$	481.649	1.06324
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−351.233	−0.768562	−0.384281	−	0.923216i	$-0.625551\pi$
$457$	−351.233	−0.768562	−0.384281	+	0.923216i	$0.625551\pi$
$458$	0	0
$459$	497.964i	1.08489i
$460$	0	0
$461$	−296.940	−0.644122	−0.322061	−	0.946719i	$-0.604376\pi$
$461$	−296.940	−0.644122	−0.322061	+	0.946719i	$0.604376\pi$
$462$	0	0
$463$	−25.5350	−0.0551513	−0.0275756	−	0.999620i	$-0.508779\pi$
$463$	−25.5350	−0.0551513	−0.0275756	+	0.999620i	$0.508779\pi$
$464$	0	0
$465$	− 133.000i	− 0.286021i
$466$	0	0
$467$	−125.520	−0.268780	−0.134390	−	0.990929i	$-0.542907\pi$
$467$	−125.520	−0.268780	−0.134390	+	0.990929i	$0.542907\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−174.104	−0.369648
$472$	0	0
$473$	−113.522	−0.240005
$474$	0	0
$475$	−403.344	−0.849145
$476$	0	0
$477$	24.7210i	0.0518259i
$478$	0	0
$479$	173.422i	0.362049i	0.983479	+	0.181025i	$0.0579414\pi$
$479$	173.422i	0.362049i	−0.983479	+	0.181025i	$0.942059\pi$
$480$	0	0
$481$	343.241i	0.713599i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	275.786i	0.568632i
$486$	0	0
$487$	681.511	1.39941	0.699704	−	0.714433i	$-0.253315\pi$
$487$	681.511	1.39941	0.699704	+	0.714433i	$0.253315\pi$
$488$	0	0
$489$	229.556i	0.469440i
$490$	0	0
$491$	− 278.104i	− 0.566404i	−0.959060	−	0.283202i	$-0.908603\pi$
$491$	− 278.104i	− 0.566404i	0.959060	−	0.283202i	$-0.0913966\pi$
$492$	0	0
$493$	−718.393	−1.45719
$494$	0	0
$495$	− 51.6123i	− 0.104267i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 410.435i	− 0.822515i	−0.911519	−	0.411258i	$-0.865090\pi$
$499$	− 410.435i	− 0.822515i	0.911519	−	0.411258i	$-0.134910\pi$
$500$	0	0
$501$	589.838i	1.17732i
$502$	0	0
$503$	554.042i	1.10148i	0.834678	+	0.550738i	$0.185654\pi$
$503$	554.042i	1.10148i	−0.834678	+	0.550738i	$0.814346\pi$
$504$	0	0
$505$	64.6508	0.128021
$506$	0	0
$507$	284.134	0.560423
$508$	0	0
$509$	44.1942	0.0868255	0.0434127	−	0.999057i	$-0.486177\pi$
$509$	44.1942	0.0868255	0.0434127	+	0.999057i	$0.486177\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	768.241	1.49755
$514$	0	0
$515$	− 10.6519i	− 0.0206833i
$516$	0	0
$517$	64.2327	0.124241
$518$	0	0
$519$	−351.227	−0.676738
$520$	0	0
$521$	420.152i	0.806434i	0.915104	+	0.403217i	$0.132108\pi$
$521$	420.152i	0.806434i	−0.915104	+	0.403217i	$0.867892\pi$
$522$	0	0
$523$	274.894	0.525609	0.262805	−	0.964849i	$-0.415353\pi$
$523$	274.894	0.525609	0.262805	+	0.964849i	$0.415353\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	312.077	0.592176
$528$	0	0
$529$	140.801	0.266164
$530$	0	0
$531$	−376.407	−0.708864
$532$	0	0
$533$	− 74.2073i	− 0.139226i
$534$	0	0
$535$	209.337i	0.391284i
$536$	0	0
$537$	241.713i	0.450118i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	560.443i	1.03594i	0.855399	+	0.517969i	$0.173312\pi$
$541$	560.443i	1.03594i	−0.855399	+	0.517969i	$0.826688\pi$
$542$	0	0
$543$	222.146	0.409110
$544$	0	0
$545$	418.818i	0.768473i
$546$	0	0
$547$	− 655.564i	− 1.19847i	−0.800573	−	0.599235i	$-0.795471\pi$
$547$	− 655.564i	− 1.19847i	0.800573	−	0.599235i	$-0.204529\pi$
$548$	0	0
$549$	331.304	0.603468
$550$	0	0
$551$	1108.31i	2.01146i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	361.301i	0.650993i
$556$	0	0
$557$	750.754i	1.34785i	0.738798	+	0.673927i	$0.235394\pi$
$557$	750.754i	1.34785i	−0.738798	+	0.673927i	$0.764606\pi$
$558$	0	0
$559$	− 166.375i	− 0.297630i
$560$	0	0
$561$	−186.369	−0.332209
$562$	0	0
$563$	634.382	1.12679	0.563394	−	0.826188i	$-0.309495\pi$
$563$	634.382	1.12679	0.563394	+	0.826188i	$0.309495\pi$
$564$	0	0
$565$	−422.767	−0.748261
$566$	0	0
$567$	0	0
$568$	0	0
$569$	392.965	0.690623	0.345312	−	0.938488i	$-0.387773\pi$
$569$	392.965	0.690623	0.345312	+	0.938488i	$0.387773\pi$
$570$	0	0
$571$	538.986i	0.943934i	0.881616	+	0.471967i	$0.156456\pi$
$571$	538.986i	0.943934i	−0.881616	+	0.471967i	$0.843544\pi$
$572$	0	0
$573$	9.21414	0.0160805
$574$	0	0
$575$	398.125	0.692391
$576$	0	0
$577$	− 347.973i	− 0.603072i	−0.953455	−	0.301536i	$-0.902501\pi$
$577$	− 347.973i	− 0.603072i	0.953455	−	0.301536i	$-0.0974994\pi$
$578$	0	0
$579$	682.430	1.17863
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−32.7424	−0.0561619
$584$	0	0
$585$	75.6418	0.129302
$586$	0	0
$587$	−258.936	−0.441118	−0.220559	−	0.975374i	$-0.570788\pi$
$587$	−258.936	−0.441118	−0.220559	+	0.975374i	$0.570788\pi$
$588$	0	0
$589$	− 481.461i	− 0.817421i
$590$	0	0
$591$	374.875i	0.634306i
$592$	0	0
$593$	− 76.9637i	− 0.129787i	−0.997892	−	0.0648935i	$-0.979329\pi$
$593$	− 76.9637i	− 0.129787i	0.997892	−	0.0648935i	$-0.0206708\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	470.513i	0.788128i
$598$	0	0
$599$	−1158.98	−1.93485	−0.967426	−	0.253154i	$-0.918532\pi$
$599$	−1158.98	−1.93485	−0.967426	+	0.253154i	$0.918532\pi$
$600$	0	0
$601$	− 976.895i	− 1.62545i	−0.582648	−	0.812724i	$-0.697983\pi$
$601$	− 976.895i	− 1.62545i	0.582648	−	0.812724i	$-0.302017\pi$
$602$	0	0
$603$	316.306i	0.524553i
$604$	0	0
$605$	−306.873	−0.507229
$606$	0	0
$607$	− 790.031i	− 1.30153i	−0.759278	−	0.650767i	$-0.774448\pi$
$607$	− 790.031i	− 1.30153i	0.759278	−	0.650767i	$-0.225552\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	94.1380i	0.154072i
$612$	0	0
$613$	269.842i	0.440198i	0.975478	+	0.220099i	$0.0706381\pi$
$613$	269.842i	0.440198i	−0.975478	+	0.220099i	$0.929362\pi$
$614$	0	0
$615$	− 78.1118i	− 0.127011i
$616$	0	0
$617$	701.515	1.13698	0.568489	−	0.822691i	$-0.307528\pi$
$617$	701.515	1.13698	0.568489	+	0.822691i	$0.307528\pi$
$618$	0	0
$619$	−869.520	−1.40472	−0.702358	−	0.711823i	$-0.747870\pi$
$619$	−869.520	−1.40472	−0.702358	+	0.711823i	$0.747870\pi$
$620$	0	0
$621$	−758.301	−1.22110
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.77760	−0.00604415
$626$	0	0
$627$	287.524i	0.458571i
$628$	0	0
$629$	−847.771	−1.34781
$630$	0	0
$631$	−100.362	−0.159052	−0.0795258	−	0.996833i	$-0.525341\pi$
$631$	−100.362	−0.159052	−0.0795258	+	0.996833i	$0.525341\pi$
$632$	0	0
$633$	398.323i	0.629262i
$634$	0	0
$635$	−19.8378	−0.0312406
$636$	0	0
$637$	0	0
$638$	0	0
$639$	289.932	0.453727
$640$	0	0
$641$	1061.14	1.65545	0.827724	−	0.561135i	$-0.189635\pi$
$641$	1061.14	1.65545	0.827724	+	0.561135i	$0.189635\pi$
$642$	0	0
$643$	−132.853	−0.206615	−0.103308	−	0.994649i	$-0.532943\pi$
$643$	−132.853	−0.206615	−0.103308	+	0.994649i	$0.532943\pi$
$644$	0	0
$645$	− 175.129i	− 0.271518i
$646$	0	0
$647$	− 1152.69i	− 1.78159i	−0.454404	−	0.890796i	$-0.650148\pi$
$647$	− 1152.69i	− 1.78159i	0.454404	−	0.890796i	$-0.349852\pi$
$648$	0	0
$649$	− 498.542i	− 0.768170i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.0705i	0.0445184i	0.999752	+	0.0222592i	$0.00708590\pi$
$653$	29.0705i	0.0445184i	−0.999752	+	0.0222592i	$0.992914\pi$
$654$	0	0
$655$	538.643	0.822356
$656$	0	0
$657$	489.032i	0.744341i
$658$	0	0
$659$	− 705.504i	− 1.07057i	−0.844672	−	0.535283i	$-0.820205\pi$
$659$	− 705.504i	− 1.07057i	0.844672	−	0.535283i	$-0.179795\pi$
$660$	0	0
$661$	−127.336	−0.192641	−0.0963204	−	0.995350i	$-0.530707\pi$
$661$	−127.336	−0.192641	−0.0963204	+	0.995350i	$0.530707\pi$
$662$	0	0
$663$	− 273.138i	− 0.411973i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1093.97i	− 1.64014i
$668$	0	0
$669$	− 335.036i	− 0.500802i
$670$	0	0
$671$	438.805i	0.653956i
$672$	0	0
$673$	−463.380	−0.688528	−0.344264	−	0.938873i	$-0.611872\pi$
$673$	−463.380	−0.688528	−0.344264	+	0.938873i	$0.611872\pi$
$674$	0	0
$675$	−450.729	−0.667746
$676$	0	0
$677$	−376.275	−0.555798	−0.277899	−	0.960610i	$-0.589638\pi$
$677$	−376.275	−0.555798	−0.277899	+	0.960610i	$0.589638\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−54.9324	−0.0806643
$682$	0	0
$683$	− 1036.84i	− 1.51807i	−0.651049	−	0.759035i	$-0.725671\pi$
$683$	− 1036.84i	− 1.51807i	0.651049	−	0.759035i	$-0.274329\pi$
$684$	0	0
$685$	237.498	0.346712
$686$	0	0
$687$	143.426	0.208772
$688$	0	0
$689$	− 47.9865i	− 0.0696465i
$690$	0	0
$691$	355.069	0.513849	0.256924	−	0.966432i	$-0.417291\pi$
$691$	355.069	0.513849	0.256924	+	0.966432i	$0.417291\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	224.744	0.323373
$696$	0	0
$697$	183.285	0.262962
$698$	0	0
$699$	−243.368	−0.348167
$700$	0	0
$701$	1278.63i	1.82400i	0.410185	+	0.912002i	$0.365464\pi$
$701$	1278.63i	1.82400i	−0.410185	+	0.912002i	$0.634536\pi$
$702$	0	0
$703$	1307.91i	1.86047i
$704$	0	0
$705$	99.0911i	0.140555i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 1200.93i	− 1.69383i	−0.531725	−	0.846917i	$-0.678456\pi$
$709$	− 1200.93i	− 1.69383i	0.531725	−	0.846917i	$-0.321544\pi$
$710$	0	0
$711$	−46.4438	−0.0653218
$712$	0	0
$713$	475.231i	0.666524i
$714$	0	0
$715$	100.186i	0.140120i
$716$	0	0
$717$	244.529	0.341045
$718$	0	0
$719$	− 7.38955i	− 0.0102775i	−0.999987	−	0.00513877i	$-0.998364\pi$
$719$	− 7.38955i	− 0.0102775i	0.999987	−	0.00513877i	$-0.00163573\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	384.721i	0.532117i
$724$	0	0
$725$	− 650.249i	− 0.896895i
$726$	0	0
$727$	− 1184.67i	− 1.62953i	−0.579788	−	0.814767i	$-0.696865\pi$
$727$	− 1184.67i	− 1.62953i	0.579788	−	0.814767i	$-0.303135\pi$
$728$	0	0
$729$	745.402	1.02250
$730$	0	0
$731$	410.930	0.562148
$732$	0	0
$733$	939.428	1.28162	0.640810	−	0.767699i	$-0.278598\pi$
$733$	939.428	1.28162	0.640810	+	0.767699i	$0.278598\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−418.940	−0.568439
$738$	0	0
$739$	1137.32i	1.53899i	0.638650	+	0.769497i	$0.279493\pi$
$739$	1137.32i	1.53899i	−0.638650	+	0.769497i	$0.720507\pi$
$740$	0	0
$741$	−421.388	−0.568675
$742$	0	0
$743$	455.212	0.612667	0.306333	−	0.951924i	$-0.400898\pi$
$743$	455.212	0.612667	0.306333	+	0.951924i	$0.400898\pi$
$744$	0	0
$745$	757.498i	1.01678i
$746$	0	0
$747$	−7.64584	−0.0102354
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−188.525	−0.251032	−0.125516	−	0.992092i	$-0.540059\pi$
$751$	−188.525	−0.251032	−0.125516	+	0.992092i	$0.540059\pi$
$752$	0	0
$753$	−932.070	−1.23781
$754$	0	0
$755$	639.502	0.847023
$756$	0	0
$757$	199.539i	0.263591i	0.991277	+	0.131796i	$0.0420743\pi$
$757$	199.539i	0.263591i	−0.991277	+	0.131796i	$0.957926\pi$
$758$	0	0
$759$	− 283.804i	− 0.373918i
$760$	0	0
$761$	− 338.020i	− 0.444179i	−0.975026	−	0.222089i	$-0.928712\pi$
$761$	− 338.020i	− 0.444179i	0.975026	−	0.222089i	$-0.0712877\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	186.828i	0.244219i
$766$	0	0
$767$	730.652	0.952610
$768$	0	0
$769$	− 604.446i	− 0.786015i	−0.919535	−	0.393008i	$-0.871435\pi$
$769$	− 604.446i	− 0.786015i	0.919535	−	0.393008i	$-0.128565\pi$
$770$	0	0
$771$	5.72674i	0.00742768i
$772$	0	0
$773$	−780.427	−1.00961	−0.504804	−	0.863234i	$-0.668435\pi$
$773$	−780.427	−1.00961	−0.504804	+	0.863234i	$0.668435\pi$
$774$	0	0
$775$	282.474i	0.364483i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 282.765i	− 0.362985i
$780$	0	0
$781$	384.008i	0.491687i
$782$	0	0
$783$	1238.52i	1.58176i
$784$	0	0
$785$	−231.164	−0.294477
$786$	0	0
$787$	−963.810	−1.22466	−0.612332	−	0.790601i	$-0.709768\pi$
$787$	−963.810	−1.22466	−0.612332	+	0.790601i	$0.709768\pi$
$788$	0	0
$789$	134.438	0.170390
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−643.101	−0.810973
$794$	0	0
$795$	− 50.5113i	− 0.0635362i
$796$	0	0
$797$	−677.191	−0.849675	−0.424837	−	0.905270i	$-0.639669\pi$
$797$	−677.191	−0.849675	−0.424837	+	0.905270i	$0.639669\pi$
$798$	0	0
$799$	−232.511	−0.291003
$800$	0	0
$801$	− 359.492i	− 0.448804i
$802$	0	0
$803$	−647.712	−0.806615
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−561.491	−0.695776
$808$	0	0
$809$	−417.234	−0.515741	−0.257870	−	0.966180i	$-0.583021\pi$
$809$	−417.234	−0.515741	−0.257870	+	0.966180i	$0.583021\pi$
$810$	0	0
$811$	1431.94	1.76564	0.882821	−	0.469709i	$-0.155641\pi$
$811$	1431.94	1.76564	0.882821	+	0.469709i	$0.155641\pi$
$812$	0	0
$813$	− 314.215i	− 0.386488i
$814$	0	0
$815$	304.790i	0.373975i
$816$	0	0
$817$	− 633.969i	− 0.775972i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 18.3555i	− 0.0223575i	−0.999938	−	0.0111787i	$-0.996442\pi$
$821$	− 18.3555i	− 0.0223575i	0.999938	−	0.0111787i	$-0.00355838\pi$
$822$	0	0
$823$	−1440.59	−1.75041	−0.875204	−	0.483753i	$-0.839273\pi$
$823$	−1440.59	−1.75041	−0.875204	+	0.483753i	$0.839273\pi$
$824$	0	0
$825$	− 168.691i	− 0.204474i
$826$	0	0
$827$	240.040i	0.290255i	0.989413	+	0.145127i	$0.0463592\pi$
$827$	240.040i	0.290255i	−0.989413	+	0.145127i	$0.953641\pi$
$828$	0	0
$829$	1464.13	1.76614	0.883070	−	0.469241i	$-0.155473\pi$
$829$	1464.13	1.76614	0.883070	+	0.469241i	$0.155473\pi$
$830$	0	0
$831$	257.855i	0.310294i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	783.149i	0.937903i
$836$	0	0
$837$	− 538.023i	− 0.642800i
$838$	0	0
$839$	− 896.568i	− 1.06861i	−0.845290	−	0.534307i	$-0.820573\pi$
$839$	− 896.568i	− 1.06861i	0.845290	−	0.534307i	$-0.179427\pi$
$840$	0	0
$841$	−945.761	−1.12457
$842$	0	0
$843$	359.889	0.426914
$844$	0	0
$845$	377.256	0.446456
$846$	0	0
$847$	0	0
$848$	0	0
$849$	72.2708	0.0851247
$850$	0	0
$851$	− 1290.99i	− 1.51703i
$852$	0	0
$853$	1376.70	1.61395	0.806973	−	0.590588i	$-0.201104\pi$
$853$	1376.70	1.61395	0.806973	+	0.590588i	$0.201104\pi$
$854$	0	0
$855$	288.231	0.337112
$856$	0	0
$857$	− 1210.80i	− 1.41284i	−0.707795	−	0.706418i	$-0.750310\pi$
$857$	− 1210.80i	− 1.41284i	0.707795	−	0.706418i	$-0.249690\pi$
$858$	0	0
$859$	−574.651	−0.668977	−0.334488	−	0.942400i	$-0.608564\pi$
$859$	−574.651	−0.668977	−0.334488	+	0.942400i	$0.608564\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	486.669	0.563927	0.281964	−	0.959425i	$-0.409014\pi$
$863$	486.669	0.563927	0.281964	+	0.959425i	$0.409014\pi$
$864$	0	0
$865$	−466.336	−0.539117
$866$	0	0
$867$	−0.372749	−0.000429930 0
$868$	0	0
$869$	− 61.5138i	− 0.0707869i
$870$	0	0
$871$	− 613.988i	− 0.704923i
$872$	0	0
$873$	315.249i	0.361110i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1175.77i	− 1.34067i	−0.742057	−	0.670337i	$-0.766150\pi$
$877$	− 1175.77i	− 1.34067i	0.742057	−	0.670337i	$-0.233850\pi$
$878$	0	0
$879$	1195.11	1.35962
$880$	0	0
$881$	− 197.506i	− 0.224183i	−0.993698	−	0.112092i	$-0.964245\pi$
$881$	− 197.506i	− 0.224183i	0.993698	−	0.112092i	$-0.0357550\pi$
$882$	0	0
$883$	739.290i	0.837248i	0.908160	+	0.418624i	$0.137487\pi$
$883$	739.290i	0.837248i	−0.908160	+	0.418624i	$0.862513\pi$
$884$	0	0
$885$	769.095	0.869034
$886$	0	0
$887$	− 438.003i	− 0.493803i	−0.969041	−	0.246902i	$-0.920588\pi$
$887$	− 438.003i	− 0.493803i	0.969041	−	0.246902i	$-0.0794124\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	171.513i	0.192495i
$892$	0	0
$893$	358.710i	0.401691i
$894$	0	0
$895$	320.931i	0.358582i
$896$	0	0
$897$	415.936	0.463696
$898$	0	0
$899$	776.185	0.863388
$900$	0	0
$901$	118.522	0.131545
$902$	0	0
$903$	0	0
$904$	0	0
$905$	294.952	0.325914
$906$	0	0
$907$	− 1319.50i	− 1.45480i	−0.686214	−	0.727400i	$-0.740729\pi$
$907$	− 1319.50i	− 1.45480i	0.686214	−	0.727400i	$-0.259271\pi$
$908$	0	0
$909$	73.9018	0.0813001
$910$	0	0
$911$	−206.561	−0.226741	−0.113371	−	0.993553i	$-0.536165\pi$
$911$	−206.561	−0.226741	−0.113371	+	0.993553i	$0.536165\pi$
$912$	0	0
$913$	− 10.1268i	− 0.0110917i
$914$	0	0
$915$	−676.938	−0.739823
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−563.385	−0.613042	−0.306521	−	0.951864i	$-0.599165\pi$
$919$	−563.385	−0.613042	−0.306521	+	0.951864i	$0.599165\pi$
$920$	0	0
$921$	−119.741	−0.130012
$922$	0	0
$923$	−562.793	−0.609743
$924$	0	0
$925$	− 767.355i	− 0.829573i
$926$	0	0
$927$	− 12.1761i	− 0.0131349i
$928$	0	0
$929$	1213.20i	1.30592i	0.757392	+	0.652960i	$0.226473\pi$
$929$	1213.20i	1.30592i	−0.757392	+	0.652960i	$0.773527\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 47.9895i	− 0.0514357i
$934$	0	0
$935$	−247.449	−0.264651
$936$	0	0
$937$	− 237.201i	− 0.253149i	−0.991957	−	0.126575i	$-0.959602\pi$
$937$	− 237.201i	− 0.253149i	0.991957	−	0.126575i	$-0.0403983\pi$
$938$	0	0
$939$	787.403i	0.838554i
$940$	0	0
$941$	119.204	0.126678	0.0633391	−	0.997992i	$-0.479825\pi$
$941$	119.204	0.126678	0.0633391	+	0.997992i	$0.479825\pi$
$942$	0	0
$943$	279.107i	0.295977i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	972.815i	1.02726i	0.858012	+	0.513630i	$0.171699\pi$
$947$	972.815i	1.02726i	−0.858012	+	0.513630i	$0.828301\pi$
$948$	0	0
$949$	− 949.271i	− 1.00029i
$950$	0	0
$951$	− 217.815i	− 0.229038i
$952$	0	0
$953$	−840.555	−0.882010	−0.441005	−	0.897505i	$-0.645378\pi$
$953$	−840.555	−0.882010	−0.441005	+	0.897505i	$0.645378\pi$
$954$	0	0
$955$	12.2339	0.0128104
$956$	0	0
$957$	−463.530	−0.484358
$958$	0	0
$959$	0	0
$960$	0	0
$961$	623.818	0.649134
$962$	0	0
$963$	239.291i	0.248485i
$964$	0	0
$965$	906.086	0.938949
$966$	0	0
$967$	−1696.40	−1.75429	−0.877147	−	0.480222i	$-0.840556\pi$
$967$	−1696.40	−1.75429	−0.877147	+	0.480222i	$0.840556\pi$
$968$	0	0
$969$	− 1040.79i	− 1.07408i
$970$	0	0
$971$	−1288.11	−1.32658	−0.663291	−	0.748361i	$-0.730841\pi$
$971$	−1288.11	−1.32658	−0.663291	+	0.748361i	$0.730841\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	247.229	0.253568
$976$	0	0
$977$	−123.909	−0.126826	−0.0634130	−	0.997987i	$-0.520199\pi$
$977$	−123.909	−0.126826	−0.0634130	+	0.997987i	$0.520199\pi$
$978$	0	0
$979$	476.139	0.486353
$980$	0	0
$981$	478.747i	0.488019i
$982$	0	0
$983$	− 1645.78i	− 1.67424i	−0.547016	−	0.837122i	$-0.684236\pi$
$983$	− 1645.78i	− 1.67424i	0.547016	−	0.837122i	$-0.315764\pi$
$984$	0	0
$985$	497.735i	0.505314i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	625.766i	0.632726i
$990$	0	0
$991$	−453.836	−0.457958	−0.228979	−	0.973431i	$-0.573539\pi$
$991$	−453.836	−0.457958	−0.228979	+	0.973431i	$0.573539\pi$
$992$	0	0
$993$	− 175.285i	− 0.176521i
$994$	0	0
$995$	624.717i	0.627856i
$996$	0	0
$997$	906.826	0.909554	0.454777	−	0.890605i	$-0.349719\pi$
$997$	906.826	0.909554	0.454777	+	0.890605i	$0.349719\pi$
$998$	0	0
$999$	1461.57i	1.46303i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.h.a.881.10		28
4.3	odd	2		392.3.h.a.293.8		28
7.4	even	3		224.3.n.a.145.10		28
7.5	odd	6		224.3.n.a.17.5		28
7.6	odd	2	inner	1568.3.h.a.881.20		28
8.3	odd	2		392.3.h.a.293.5		28
8.5	even	2	inner	1568.3.h.a.881.19		28
28.3	even	6		392.3.j.e.117.7		28
28.11	odd	6		56.3.j.a.5.7	✓	28
28.19	even	6		56.3.j.a.45.13	yes	28
28.23	odd	6		392.3.j.e.325.13		28
28.27	even	2		392.3.h.a.293.7		28
56.3	even	6		392.3.j.e.117.13		28
56.5	odd	6		224.3.n.a.17.10		28
56.11	odd	6		56.3.j.a.5.13	yes	28
56.13	odd	2	inner	1568.3.h.a.881.9		28
56.19	even	6		56.3.j.a.45.7	yes	28
56.27	even	2		392.3.h.a.293.6		28
56.51	odd	6		392.3.j.e.325.7		28
56.53	even	6		224.3.n.a.145.5		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.j.a.5.7	✓	28	28.11	odd	6
56.3.j.a.5.13	yes	28	56.11	odd	6
56.3.j.a.45.7	yes	28	56.19	even	6
56.3.j.a.45.13	yes	28	28.19	even	6
224.3.n.a.17.5		28	7.5	odd	6
224.3.n.a.17.10		28	56.5	odd	6
224.3.n.a.145.5		28	56.53	even	6
224.3.n.a.145.10		28	7.4	even	3
392.3.h.a.293.5		28	8.3	odd	2
392.3.h.a.293.6		28	56.27	even	2
392.3.h.a.293.7		28	28.27	even	2
392.3.h.a.293.8		28	4.3	odd	2
392.3.j.e.117.7		28	28.3	even	6
392.3.j.e.117.13		28	56.3	even	6
392.3.j.e.325.7		28	56.51	odd	6
392.3.j.e.325.13		28	28.23	odd	6
1568.3.h.a.881.9		28	56.13	odd	2	inner
1568.3.h.a.881.10		28	1.1	even	1	trivial
1568.3.h.a.881.19		28	8.5	even	2	inner
1568.3.h.a.881.20		28	7.6	odd	2	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.33563 q^{3} -3.10110 q^{5} -3.54484 q^{9} +O(q^{10})\)	\(q-2.33563 q^{3} -3.10110 q^{5} -3.54484 q^{9} -4.69506i q^{11} +6.88097 q^{13} +7.24301 q^{15} +16.9953i q^{17} +26.2198 q^{19} -25.8805 q^{23} -15.3832 q^{25} +29.3001 q^{27} +42.2701i q^{29} -18.3625i q^{31} +10.9659i q^{33} +49.8827i q^{37} -16.0714 q^{39} -10.7844i q^{41} -24.1791i q^{43} +10.9929 q^{45} +13.6809i q^{47} -39.6947i q^{51} -6.97379i q^{53} +14.5598i q^{55} -61.2396 q^{57} +106.184 q^{59} -93.4609 q^{61} -21.3386 q^{65} -89.2299i q^{67} +60.4473 q^{69} -81.7898 q^{71} -137.956i q^{73} +35.9294 q^{75} +13.1018 q^{79} -36.5306 q^{81} +2.15689 q^{83} -52.7041i q^{85} -98.7272i q^{87} +101.413i q^{89} +42.8880i q^{93} -81.3101 q^{95} -88.9318i q^{97} +16.6432i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 64 q^{9}+O(q^{10}) \) 28 * q + 64 * q^9	\( 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) \) 28 * q + 64 * q^9 - 28 * q^15 + 60 * q^23 + 64 * q^25 - 40 * q^39 + 124 * q^57 + 104 * q^65 + 136 * q^71 - 324 * q^79 + 36 * q^81 + 580 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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