LMFDB - Embedded newform 5571.2.a.g.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5571,2,Mod(1,5571)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5571, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5571.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5571 = 3^{2} \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5571.a (trivial)

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:	yes
Analytic conductor:	$44.4846589661$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 619)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	5571.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.37056 q^{2} -0.121559 q^{4} +0.225133 q^{5} +2.47282 q^{7} -2.90773 q^{8} +O(q^{10})$	\(q+1.37056 q^{2} -0.121559 q^{4} +0.225133 q^{5} +2.47282 q^{7} -2.90773 q^{8} +0.308559 q^{10} +4.06256 q^{11} +2.26536 q^{13} +3.38915 q^{14} -3.74211 q^{16} -7.19086 q^{17} -7.47015 q^{19} -0.0273670 q^{20} +5.56800 q^{22} +0.178059 q^{23} -4.94931 q^{25} +3.10482 q^{26} -0.300594 q^{28} -10.5609 q^{29} +1.27650 q^{31} +0.686670 q^{32} -9.85552 q^{34} +0.556714 q^{35} +2.08508 q^{37} -10.2383 q^{38} -0.654627 q^{40} -4.23367 q^{41} +8.52011 q^{43} -0.493842 q^{44} +0.244041 q^{46} -9.96677 q^{47} -0.885174 q^{49} -6.78334 q^{50} -0.275375 q^{52} -0.619984 q^{53} +0.914619 q^{55} -7.19028 q^{56} -14.4744 q^{58} -11.8540 q^{59} +6.30300 q^{61} +1.74952 q^{62} +8.42533 q^{64} +0.510008 q^{65} +12.5388 q^{67} +0.874114 q^{68} +0.763011 q^{70} +9.50401 q^{71} +5.14906 q^{73} +2.85773 q^{74} +0.908065 q^{76} +10.0460 q^{77} -16.3932 q^{79} -0.842473 q^{80} -5.80251 q^{82} -4.79273 q^{83} -1.61890 q^{85} +11.6773 q^{86} -11.8128 q^{88} +1.64933 q^{89} +5.60182 q^{91} -0.0216447 q^{92} -13.6601 q^{94} -1.68178 q^{95} +3.34400 q^{97} -1.21319 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8}+O(q^{10}) $ 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8	$ 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8} + 5 q^{10} - 23 q^{11} + 9 q^{13} - 7 q^{14} + 35 q^{16} - 4 q^{17} - q^{19} - 29 q^{20} - 4 q^{23} + 35 q^{25} - q^{26} - 13 q^{28} - 90 q^{29} + 2 q^{31} - 43 q^{32} - 9 q^{34} - 9 q^{35} + 19 q^{37} - 5 q^{38} - 12 q^{40} - 59 q^{41} - 4 q^{43} - 52 q^{44} - q^{46} - 4 q^{47} + 30 q^{49} - 31 q^{50} - 12 q^{52} - 34 q^{53} - 17 q^{55} - 2 q^{56} + 6 q^{58} - 13 q^{59} + 16 q^{61} - 28 q^{62} + 37 q^{64} - 31 q^{65} - 11 q^{67} + 52 q^{68} - 40 q^{70} - 42 q^{71} - 4 q^{73} - 16 q^{74} - 42 q^{76} - 29 q^{77} + 3 q^{79} - 21 q^{80} - 43 q^{82} + 11 q^{83} + 19 q^{85} + 11 q^{86} - 47 q^{88} - 58 q^{89} - 39 q^{91} + 7 q^{92} - 46 q^{94} - 23 q^{95} - 9 q^{97}+O(q^{100}) $ 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8 + 5 * q^10 - 23 * q^11 + 9 * q^13 - 7 * q^14 + 35 * q^16 - 4 * q^17 - q^19 - 29 * q^20 - 4 * q^23 + 35 * q^25 - q^26 - 13 * q^28 - 90 * q^29 + 2 * q^31 - 43 * q^32 - 9 * q^34 - 9 * q^35 + 19 * q^37 - 5 * q^38 - 12 * q^40 - 59 * q^41 - 4 * q^43 - 52 * q^44 - q^46 - 4 * q^47 + 30 * q^49 - 31 * q^50 - 12 * q^52 - 34 * q^53 - 17 * q^55 - 2 * q^56 + 6 * q^58 - 13 * q^59 + 16 * q^61 - 28 * q^62 + 37 * q^64 - 31 * q^65 - 11 * q^67 + 52 * q^68 - 40 * q^70 - 42 * q^71 - 4 * q^73 - 16 * q^74 - 42 * q^76 - 29 * q^77 + 3 * q^79 - 21 * q^80 - 43 * q^82 + 11 * q^83 + 19 * q^85 + 11 * q^86 - 47 * q^88 - 58 * q^89 - 39 * q^91 + 7 * q^92 - 46 * q^94 - 23 * q^95 - 9 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	1.37056	0.969134	0.484567	−	0.874754i	$-0.338977\pi$
$2$	1.37056	0.969134	0.484567	+	0.874754i	$0.338977\pi$
$3$	0	0
$3$	0	0
$4$	−0.121559	−0.0607796
$5$	0.225133	0.100683	0.0503414	−	0.998732i	$-0.483969\pi$
$5$	0.225133	0.100683	0.0503414	+	0.998732i	$0.483969\pi$
$6$	0	0
$7$	2.47282	0.934637	0.467319	−	0.884089i	$-0.345220\pi$
$7$	2.47282	0.934637	0.467319	+	0.884089i	$0.345220\pi$
$8$	−2.90773	−1.02804
$9$	0	0
$10$	0.308559	0.0975750
$11$	4.06256	1.22491	0.612455	−	0.790506i	$-0.290182\pi$
$11$	4.06256	1.22491	0.612455	+	0.790506i	$0.290182\pi$
$12$	0	0
$13$	2.26536	0.628298	0.314149	−	0.949374i	$-0.398281\pi$
$13$	2.26536	0.628298	0.314149	+	0.949374i	$0.398281\pi$
$14$	3.38915	0.905788
$15$	0	0
$16$	−3.74211	−0.935526
$17$	−7.19086	−1.74404	−0.872019	−	0.489471i	$-0.837190\pi$
$17$	−7.19086	−1.74404	−0.872019	+	0.489471i	$0.837190\pi$
$18$	0	0
$19$	−7.47015	−1.71377	−0.856885	−	0.515507i	$-0.827603\pi$
$19$	−7.47015	−1.71377	−0.856885	+	0.515507i	$0.827603\pi$
$20$	−0.0273670	−0.00611945
$21$	0	0
$22$	5.56800	1.18710
$23$	0.178059	0.0371279	0.0185639	−	0.999828i	$-0.494091\pi$
$23$	0.178059	0.0371279	0.0185639	+	0.999828i	$0.494091\pi$
$24$	0	0
$25$	−4.94931	−0.989863
$26$	3.10482	0.608905
$27$	0	0
$28$	−0.300594	−0.0568068
$29$	−10.5609	−1.96111	−0.980557	−	0.196232i	$-0.937129\pi$
$29$	−10.5609	−1.96111	−0.980557	+	0.196232i	$0.937129\pi$
$30$	0	0
$31$	1.27650	0.229266	0.114633	−	0.993408i	$-0.463431\pi$
$31$	1.27650	0.229266	0.114633	+	0.993408i	$0.463431\pi$
$32$	0.686670	0.121387
$33$	0	0
$34$	−9.85552	−1.69021
$35$	0.556714	0.0941018
$36$	0	0
$37$	2.08508	0.342784	0.171392	−	0.985203i	$-0.445173\pi$
$37$	2.08508	0.342784	0.171392	+	0.985203i	$0.445173\pi$
$38$	−10.2383	−1.66087
$39$	0	0
$40$	−0.654627	−0.103506
$41$	−4.23367	−0.661188	−0.330594	−	0.943773i	$-0.607249\pi$
$41$	−4.23367	−0.661188	−0.330594	+	0.943773i	$0.607249\pi$
$42$	0	0
$43$	8.52011	1.29930	0.649652	−	0.760232i	$-0.274915\pi$
$43$	8.52011	1.29930	0.649652	+	0.760232i	$0.274915\pi$
$44$	−0.493842	−0.0744495
$45$	0	0
$46$	0.244041	0.0359819
$47$	−9.96677	−1.45380	−0.726901	−	0.686742i	$-0.759040\pi$
$47$	−9.96677	−1.45380	−0.726901	+	0.686742i	$0.759040\pi$
$48$	0	0
$49$	−0.885174	−0.126453
$50$	−6.78334	−0.959310
$51$	0	0
$52$	−0.275375	−0.0381877
$53$	−0.619984	−0.0851614	−0.0425807	−	0.999093i	$-0.513558\pi$
$53$	−0.619984	−0.0851614	−0.0425807	+	0.999093i	$0.513558\pi$
$54$	0	0
$55$	0.914619	0.123327
$56$	−7.19028	−0.960842
$57$	0	0
$58$	−14.4744	−1.90058
$59$	−11.8540	−1.54326	−0.771628	−	0.636074i	$-0.780557\pi$
$59$	−11.8540	−1.54326	−0.771628	+	0.636074i	$0.780557\pi$
$60$	0	0
$61$	6.30300	0.807017	0.403508	−	0.914976i	$-0.367791\pi$
$61$	6.30300	0.807017	0.403508	+	0.914976i	$0.367791\pi$
$62$	1.74952	0.222190
$63$	0	0
$64$	8.42533	1.05317
$65$	0.510008	0.0632587
$66$	0	0
$67$	12.5388	1.53186	0.765928	−	0.642927i	$-0.222280\pi$
$67$	12.5388	1.53186	0.765928	+	0.642927i	$0.222280\pi$
$68$	0.874114	0.106002
$69$	0	0
$70$	0.763011	0.0911972
$71$	9.50401	1.12792	0.563959	−	0.825803i	$-0.309277\pi$
$71$	9.50401	1.12792	0.563959	+	0.825803i	$0.309277\pi$
$72$	0	0
$73$	5.14906	0.602652	0.301326	−	0.953521i	$-0.402571\pi$
$73$	5.14906	0.602652	0.301326	+	0.953521i	$0.402571\pi$
$74$	2.85773	0.332204
$75$	0	0
$76$	0.908065	0.104162
$77$	10.0460	1.14485
$78$	0	0
$79$	−16.3932	−1.84438	−0.922190	−	0.386737i	$-0.873602\pi$
$79$	−16.3932	−1.84438	−0.922190	+	0.386737i	$0.873602\pi$
$80$	−0.842473	−0.0941913
$81$	0	0
$82$	−5.80251	−0.640780
$83$	−4.79273	−0.526071	−0.263035	−	0.964786i	$-0.584724\pi$
$83$	−4.79273	−0.526071	−0.263035	+	0.964786i	$0.584724\pi$
$84$	0	0
$85$	−1.61890	−0.175595
$86$	11.6773	1.25920
$87$	0	0
$88$	−11.8128	−1.25925
$89$	1.64933	0.174829	0.0874143	−	0.996172i	$-0.472140\pi$
$89$	1.64933	0.174829	0.0874143	+	0.996172i	$0.472140\pi$
$90$	0	0
$91$	5.60182	0.587231
$92$	−0.0216447	−0.00225662
$93$	0	0
$94$	−13.6601	−1.40893
$95$	−1.68178	−0.172547
$96$	0	0
$97$	3.34400	0.339531	0.169766	−	0.985484i	$-0.445699\pi$
$97$	3.34400	0.339531	0.169766	+	0.985484i	$0.445699\pi$
$98$	−1.21319	−0.122550
$99$	0	0
$100$	0.601634	0.0601634
$101$	12.6806	1.26176	0.630882	−	0.775878i	$-0.282693\pi$
$101$	12.6806	1.26176	0.630882	+	0.775878i	$0.282693\pi$
$102$	0	0
$103$	−4.47808	−0.441238	−0.220619	−	0.975360i	$-0.570808\pi$
$103$	−4.47808	−0.441238	−0.220619	+	0.975360i	$0.570808\pi$
$104$	−6.58705	−0.645914
$105$	0	0
$106$	−0.849727	−0.0825328
$107$	−15.0754	−1.45740	−0.728699	−	0.684834i	$-0.759875\pi$
$107$	−15.0754	−1.45740	−0.728699	+	0.684834i	$0.759875\pi$
$108$	0	0
$109$	3.66599	0.351138	0.175569	−	0.984467i	$-0.443823\pi$
$109$	3.66599	0.351138	0.175569	+	0.984467i	$0.443823\pi$
$110$	1.25354	0.119521
$111$	0	0
$112$	−9.25354	−0.874378
$113$	−6.54316	−0.615528	−0.307764	−	0.951463i	$-0.599581\pi$
$113$	−6.54316	−0.615528	−0.307764	+	0.951463i	$0.599581\pi$
$114$	0	0
$115$	0.0400870	0.00373814
$116$	1.28378	0.119196
$117$	0	0
$118$	−16.2466	−1.49562
$119$	−17.7817	−1.63004
$120$	0	0
$121$	5.50443	0.500403
$122$	8.63866	0.782107
$123$	0	0
$124$	−0.155170	−0.0139347
$125$	−2.23992	−0.200345
$126$	0	0
$127$	−15.0494	−1.33542	−0.667711	−	0.744420i	$-0.732726\pi$
$127$	−15.0494	−1.33542	−0.667711	+	0.744420i	$0.732726\pi$
$128$	10.1741	0.899272
$129$	0	0
$130$	0.698998	0.0613062
$131$	−9.57374	−0.836462	−0.418231	−	0.908341i	$-0.637350\pi$
$131$	−9.57374	−0.836462	−0.418231	+	0.908341i	$0.637350\pi$
$132$	0	0
$133$	−18.4723	−1.60175
$134$	17.1852	1.48457
$135$	0	0
$136$	20.9091	1.79294
$137$	−2.15272	−0.183919	−0.0919595	−	0.995763i	$-0.529313\pi$
$137$	−2.15272	−0.183919	−0.0919595	+	0.995763i	$0.529313\pi$
$138$	0	0
$139$	5.62106	0.476772	0.238386	−	0.971171i	$-0.423382\pi$
$139$	5.62106	0.476772	0.238386	+	0.971171i	$0.423382\pi$
$140$	−0.0676736	−0.00571947
$141$	0	0
$142$	13.0258	1.09310
$143$	9.20317	0.769608
$144$	0	0
$145$	−2.37762	−0.197450
$146$	7.05711	0.584050
$147$	0	0
$148$	−0.253460	−0.0208343
$149$	−15.1382	−1.24017	−0.620086	−	0.784534i	$-0.712902\pi$
$149$	−15.1382	−1.24017	−0.620086	+	0.784534i	$0.712902\pi$
$150$	0	0
$151$	−19.7193	−1.60474	−0.802369	−	0.596828i	$-0.796427\pi$
$151$	−19.7193	−1.60474	−0.802369	+	0.596828i	$0.796427\pi$
$152$	21.7212	1.76182
$153$	0	0
$154$	13.7686	1.10951
$155$	0.287383	0.0230831
$156$	0	0
$157$	15.7111	1.25389	0.626943	−	0.779065i	$-0.284306\pi$
$157$	15.7111	1.25389	0.626943	+	0.779065i	$0.284306\pi$
$158$	−22.4679	−1.78745
$159$	0	0
$160$	0.154592	0.0122216
$161$	0.440308	0.0347011
$162$	0	0
$163$	−11.8717	−0.929864	−0.464932	−	0.885346i	$-0.653921\pi$
$163$	−11.8717	−0.929864	−0.464932	+	0.885346i	$0.653921\pi$
$164$	0.514641	0.0401867
$165$	0	0
$166$	−6.56874	−0.509833
$167$	5.37936	0.416267	0.208134	−	0.978100i	$-0.433261\pi$
$167$	5.37936	0.416267	0.208134	+	0.978100i	$0.433261\pi$
$168$	0	0
$169$	−7.86814	−0.605242
$170$	−2.21881	−0.170175
$171$	0	0
$172$	−1.03570	−0.0789711
$173$	18.6393	1.41712	0.708560	−	0.705651i	$-0.249345\pi$
$173$	18.6393	1.41712	0.708560	+	0.705651i	$0.249345\pi$
$174$	0	0
$175$	−12.2388	−0.925163
$176$	−15.2025	−1.14593
$177$	0	0
$178$	2.26051	0.169432
$179$	22.6264	1.69117	0.845587	−	0.533837i	$-0.179250\pi$
$179$	22.6264	1.69117	0.845587	+	0.533837i	$0.179250\pi$
$180$	0	0
$181$	12.1002	0.899398	0.449699	−	0.893180i	$-0.351531\pi$
$181$	12.1002	0.899398	0.449699	+	0.893180i	$0.351531\pi$
$182$	7.67765	0.569105
$183$	0	0
$184$	−0.517747	−0.0381688
$185$	0.469420	0.0345124
$186$	0	0
$187$	−29.2133	−2.13629
$188$	1.21155	0.0883615
$189$	0	0
$190$	−2.30498	−0.167221
$191$	0.166765	0.0120667	0.00603336	−	0.999982i	$-0.498080\pi$
$191$	0.166765	0.0120667	0.00603336	+	0.999982i	$0.498080\pi$
$192$	0	0
$193$	−16.5260	−1.18956	−0.594782	−	0.803887i	$-0.702762\pi$
$193$	−16.5260	−1.18956	−0.594782	+	0.803887i	$0.702762\pi$
$194$	4.58316	0.329051
$195$	0	0
$196$	0.107601	0.00768578
$197$	−9.82274	−0.699841	−0.349921	−	0.936779i	$-0.613791\pi$
$197$	−9.82274	−0.699841	−0.349921	+	0.936779i	$0.613791\pi$
$198$	0	0
$199$	11.5691	0.820113	0.410056	−	0.912060i	$-0.365509\pi$
$199$	11.5691	0.820113	0.410056	+	0.912060i	$0.365509\pi$
$200$	14.3913	1.01762
$201$	0	0
$202$	17.3795	1.22282
$203$	−26.1152	−1.83293
$204$	0	0
$205$	−0.953140	−0.0665702
$206$	−6.13749	−0.427619
$207$	0	0
$208$	−8.47722	−0.587789
$209$	−30.3480	−2.09921
$210$	0	0
$211$	−19.2316	−1.32396	−0.661978	−	0.749523i	$-0.730283\pi$
$211$	−19.2316	−1.32396	−0.661978	+	0.749523i	$0.730283\pi$
$212$	0.0753647	0.00517607
$213$	0	0
$214$	−20.6618	−1.41241
$215$	1.91816	0.130817
$216$	0	0
$217$	3.15655	0.214281
$218$	5.02447	0.340300
$219$	0	0
$220$	−0.111180	−0.00749577
$221$	−16.2899	−1.09578
$222$	0	0
$223$	−2.26463	−0.151651	−0.0758255	−	0.997121i	$-0.524159\pi$
$223$	−2.26463	−0.151651	−0.0758255	+	0.997121i	$0.524159\pi$
$224$	1.69801	0.113453
$225$	0	0
$226$	−8.96780	−0.596529
$227$	−12.6616	−0.840378	−0.420189	−	0.907437i	$-0.638036\pi$
$227$	−12.6616	−0.840378	−0.420189	+	0.907437i	$0.638036\pi$
$228$	0	0
$229$	27.7524	1.83393	0.916967	−	0.398964i	$-0.130630\pi$
$229$	27.7524	1.83393	0.916967	+	0.398964i	$0.130630\pi$
$230$	0.0549418	0.00362275
$231$	0	0
$232$	30.7083	2.01610
$233$	10.9164	0.715157	0.357579	−	0.933883i	$-0.383602\pi$
$233$	10.9164	0.715157	0.357579	+	0.933883i	$0.383602\pi$
$234$	0	0
$235$	−2.24385	−0.146373
$236$	1.44096	0.0937984
$237$	0	0
$238$	−24.3709	−1.57973
$239$	−6.41476	−0.414936	−0.207468	−	0.978242i	$-0.566522\pi$
$239$	−6.41476	−0.414936	−0.207468	+	0.978242i	$0.566522\pi$
$240$	0	0
$241$	9.85126	0.634576	0.317288	−	0.948329i	$-0.397228\pi$
$241$	9.85126	0.634576	0.317288	+	0.948329i	$0.397228\pi$
$242$	7.54417	0.484957
$243$	0	0
$244$	−0.766188	−0.0490501
$245$	−0.199282	−0.0127317
$246$	0	0
$247$	−16.9226	−1.07676
$248$	−3.71172	−0.235694
$249$	0	0
$250$	−3.06995	−0.194161
$251$	−5.56947	−0.351542	−0.175771	−	0.984431i	$-0.556242\pi$
$251$	−5.56947	−0.351542	−0.175771	+	0.984431i	$0.556242\pi$
$252$	0	0
$253$	0.723376	0.0454783
$254$	−20.6262	−1.29420
$255$	0	0
$256$	−2.90642	−0.181651
$257$	−24.0328	−1.49913	−0.749564	−	0.661932i	$-0.769737\pi$
$257$	−24.0328	−1.49913	−0.749564	+	0.661932i	$0.769737\pi$
$258$	0	0
$259$	5.15601	0.320379
$260$	−0.0619962	−0.00384484
$261$	0	0
$262$	−13.1214	−0.810643
$263$	−14.9598	−0.922463	−0.461232	−	0.887280i	$-0.652592\pi$
$263$	−14.9598	−0.922463	−0.461232	+	0.887280i	$0.652592\pi$
$264$	0	0
$265$	−0.139579	−0.00857428
$266$	−25.3175	−1.55231
$267$	0	0
$268$	−1.52420	−0.0931055
$269$	−13.4385	−0.819357	−0.409679	−	0.912230i	$-0.634359\pi$
$269$	−13.4385	−0.819357	−0.409679	+	0.912230i	$0.634359\pi$
$270$	0	0
$271$	5.72605	0.347833	0.173916	−	0.984760i	$-0.444358\pi$
$271$	5.72605	0.347833	0.173916	+	0.984760i	$0.444358\pi$
$272$	26.9089	1.63159
$273$	0	0
$274$	−2.95043	−0.178242
$275$	−20.1069	−1.21249
$276$	0	0
$277$	12.0981	0.726905	0.363452	−	0.931613i	$-0.381598\pi$
$277$	12.0981	0.726905	0.363452	+	0.931613i	$0.381598\pi$
$278$	7.70401	0.462056
$279$	0	0
$280$	−1.61877	−0.0967402
$281$	6.21778	0.370921	0.185461	−	0.982652i	$-0.440622\pi$
$281$	6.21778	0.370921	0.185461	+	0.982652i	$0.440622\pi$
$282$	0	0
$283$	12.9505	0.769826	0.384913	−	0.922953i	$-0.374231\pi$
$283$	12.9505	0.769826	0.384913	+	0.922953i	$0.374231\pi$
$284$	−1.15530	−0.0685544
$285$	0	0
$286$	12.6135	0.745853
$287$	−10.4691	−0.617971
$288$	0	0
$289$	34.7084	2.04167
$290$	−3.25867	−0.191356
$291$	0	0
$292$	−0.625915	−0.0366289
$293$	−14.3490	−0.838279	−0.419139	−	0.907922i	$-0.637668\pi$
$293$	−14.3490	−0.838279	−0.419139	+	0.907922i	$0.637668\pi$
$294$	0	0
$295$	−2.66873	−0.155379
$296$	−6.06283	−0.352395
$297$	0	0
$298$	−20.7479	−1.20189
$299$	0.403368	0.0233274
$300$	0	0
$301$	21.0687	1.21438
$302$	−27.0266	−1.55521
$303$	0	0
$304$	27.9541	1.60328
$305$	1.41902	0.0812526
$306$	0	0
$307$	−28.7422	−1.64040	−0.820202	−	0.572075i	$-0.806139\pi$
$307$	−28.7422	−1.64040	−0.820202	+	0.572075i	$0.806139\pi$
$308$	−1.22118	−0.0695832
$309$	0	0
$310$	0.393876	0.0223707
$311$	29.9677	1.69931	0.849655	−	0.527339i	$-0.176810\pi$
$311$	29.9677	1.69931	0.849655	+	0.527339i	$0.176810\pi$
$312$	0	0
$313$	−13.6402	−0.770990	−0.385495	−	0.922710i	$-0.625969\pi$
$313$	−13.6402	−0.770990	−0.385495	+	0.922710i	$0.625969\pi$
$314$	21.5331	1.21518
$315$	0	0
$316$	1.99274	0.112101
$317$	−12.9433	−0.726967	−0.363484	−	0.931601i	$-0.618413\pi$
$317$	−12.9433	−0.726967	−0.363484	+	0.931601i	$0.618413\pi$
$318$	0	0
$319$	−42.9044	−2.40219
$320$	1.89682	0.106036
$321$	0	0
$322$	0.603469	0.0336300
$323$	53.7168	2.98888
$324$	0	0
$325$	−11.2120	−0.621929
$326$	−16.2709	−0.901163
$327$	0	0
$328$	12.3104	0.679726
$329$	−24.6460	−1.35878
$330$	0	0
$331$	8.48346	0.466293	0.233147	−	0.972442i	$-0.425098\pi$
$331$	8.48346	0.466293	0.233147	+	0.972442i	$0.425098\pi$
$332$	0.582600	0.0319744
$333$	0	0
$334$	7.37275	0.403419
$335$	2.82290	0.154231
$336$	0	0
$337$	−28.4431	−1.54939	−0.774697	−	0.632332i	$-0.782098\pi$
$337$	−28.4431	−1.54939	−0.774697	+	0.632332i	$0.782098\pi$
$338$	−10.7838	−0.586560
$339$	0	0
$340$	0.196792	0.0106726
$341$	5.18586	0.280830
$342$	0	0
$343$	−19.4986	−1.05283
$344$	−24.7742	−1.33573
$345$	0	0
$346$	25.5463	1.37338
$347$	−6.37057	−0.341990	−0.170995	−	0.985272i	$-0.554698\pi$
$347$	−6.37057	−0.341990	−0.170995	+	0.985272i	$0.554698\pi$
$348$	0	0
$349$	−12.7281	−0.681317	−0.340659	−	0.940187i	$-0.610650\pi$
$349$	−12.7281	−0.681317	−0.340659	+	0.940187i	$0.610650\pi$
$350$	−16.7740	−0.896607
$351$	0	0
$352$	2.78964	0.148688
$353$	−1.48049	−0.0787984	−0.0393992	−	0.999224i	$-0.512544\pi$
$353$	−1.48049	−0.0787984	−0.0393992	+	0.999224i	$0.512544\pi$
$354$	0	0
$355$	2.13967	0.113562
$356$	−0.200491	−0.0106260
$357$	0	0
$358$	31.0109	1.63897
$359$	27.5753	1.45537	0.727685	−	0.685911i	$-0.240596\pi$
$359$	27.5753	1.45537	0.727685	+	0.685911i	$0.240596\pi$
$360$	0	0
$361$	36.8032	1.93701
$362$	16.5840	0.871637
$363$	0	0
$364$	−0.680953	−0.0356916
$365$	1.15923	0.0606766
$366$	0	0
$367$	6.08957	0.317873	0.158936	−	0.987289i	$-0.449193\pi$
$367$	6.08957	0.317873	0.158936	+	0.987289i	$0.449193\pi$
$368$	−0.666316	−0.0347341
$369$	0	0
$370$	0.643369	0.0334472
$371$	−1.53311	−0.0795950
$372$	0	0
$373$	22.4949	1.16474	0.582371	−	0.812923i	$-0.302125\pi$
$373$	22.4949	1.16474	0.582371	+	0.812923i	$0.302125\pi$
$374$	−40.0387	−2.07035
$375$	0	0
$376$	28.9807	1.49456
$377$	−23.9243	−1.23216
$378$	0	0
$379$	−14.1157	−0.725075	−0.362538	−	0.931969i	$-0.618090\pi$
$379$	−14.1157	−0.725075	−0.362538	+	0.931969i	$0.618090\pi$
$380$	0.204436	0.0104873
$381$	0	0
$382$	0.228562	0.0116943
$383$	−24.3617	−1.24482	−0.622412	−	0.782690i	$-0.713847\pi$
$383$	−24.3617	−1.24482	−0.622412	+	0.782690i	$0.713847\pi$
$384$	0	0
$385$	2.26169	0.115266
$386$	−22.6498	−1.15285
$387$	0	0
$388$	−0.406493	−0.0206366
$389$	16.7319	0.848341	0.424170	−	0.905582i	$-0.360566\pi$
$389$	16.7319	0.848341	0.424170	+	0.905582i	$0.360566\pi$
$390$	0	0
$391$	−1.28040	−0.0647525
$392$	2.57385	0.129999
$393$	0	0
$394$	−13.4627	−0.678240
$395$	−3.69066	−0.185697
$396$	0	0
$397$	−21.7689	−1.09255	−0.546274	−	0.837607i	$-0.683954\pi$
$397$	−21.7689	−1.09255	−0.546274	+	0.837607i	$0.683954\pi$
$398$	15.8562	0.794799
$399$	0	0
$400$	18.5209	0.926043
$401$	−20.2269	−1.01008	−0.505041	−	0.863095i	$-0.668523\pi$
$401$	−20.2269	−1.01008	−0.505041	+	0.863095i	$0.668523\pi$
$402$	0	0
$403$	2.89173	0.144047
$404$	−1.54144	−0.0766895
$405$	0	0
$406$	−35.7926	−1.77636
$407$	8.47075	0.419880
$408$	0	0
$409$	28.5461	1.41151	0.705757	−	0.708454i	$-0.250607\pi$
$409$	28.5461	1.41151	0.705757	+	0.708454i	$0.250607\pi$
$410$	−1.30634	−0.0645155
$411$	0	0
$412$	0.544352	0.0268183
$413$	−29.3127	−1.44238
$414$	0	0
$415$	−1.07900	−0.0529662
$416$	1.55555	0.0762673
$417$	0	0
$418$	−41.5938	−2.03442
$419$	36.3207	1.77438	0.887192	−	0.461400i	$-0.152653\pi$
$419$	36.3207	1.77438	0.887192	+	0.461400i	$0.152653\pi$
$420$	0	0
$421$	−20.8986	−1.01853	−0.509267	−	0.860609i	$-0.670083\pi$
$421$	−20.8986	−1.01853	−0.509267	+	0.860609i	$0.670083\pi$
$422$	−26.3581	−1.28309
$423$	0	0
$424$	1.80275	0.0875491
$425$	35.5898	1.72636
$426$	0	0
$427$	15.5862	0.754268
$428$	1.83256	0.0885800
$429$	0	0
$430$	2.62896	0.126780
$431$	−5.65795	−0.272534	−0.136267	−	0.990672i	$-0.543510\pi$
$431$	−5.65795	−0.272534	−0.136267	+	0.990672i	$0.543510\pi$
$432$	0	0
$433$	−0.797593	−0.0383299	−0.0191649	−	0.999816i	$-0.506101\pi$
$433$	−0.797593	−0.0383299	−0.0191649	+	0.999816i	$0.506101\pi$
$434$	4.32625	0.207667
$435$	0	0
$436$	−0.445635	−0.0213420
$437$	−1.33013	−0.0636286
$438$	0	0
$439$	−18.2826	−0.872581	−0.436290	−	0.899806i	$-0.643708\pi$
$439$	−18.2826	−0.872581	−0.436290	+	0.899806i	$0.643708\pi$
$440$	−2.65946	−0.126785
$441$	0	0
$442$	−22.3263	−1.06195
$443$	18.4038	0.874391	0.437195	−	0.899367i	$-0.355972\pi$
$443$	18.4038	0.874391	0.437195	+	0.899367i	$0.355972\pi$
$444$	0	0
$445$	0.371319	0.0176022
$446$	−3.10382	−0.146970
$447$	0	0
$448$	20.8343	0.984329
$449$	−2.95767	−0.139581	−0.0697905	−	0.997562i	$-0.522233\pi$
$449$	−2.95767	−0.139581	−0.0697905	+	0.997562i	$0.522233\pi$
$450$	0	0
$451$	−17.1996	−0.809896
$452$	0.795381	0.0374116
$453$	0	0
$454$	−17.3535	−0.814439
$455$	1.26116	0.0591240
$456$	0	0
$457$	18.1383	0.848472	0.424236	−	0.905552i	$-0.360543\pi$
$457$	18.1383	0.848472	0.424236	+	0.905552i	$0.360543\pi$
$458$	38.0365	1.77733
$459$	0	0
$460$	−0.00487295	−0.000227202 0
$461$	16.5572	0.771145	0.385573	−	0.922677i	$-0.374004\pi$
$461$	16.5572	0.771145	0.385573	+	0.922677i	$0.374004\pi$
$462$	0	0
$463$	4.01151	0.186430	0.0932152	−	0.995646i	$-0.470286\pi$
$463$	4.01151	0.186430	0.0932152	+	0.995646i	$0.470286\pi$
$464$	39.5201	1.83467
$465$	0	0
$466$	14.9616	0.693083
$467$	4.61263	0.213447	0.106724	−	0.994289i	$-0.465964\pi$
$467$	4.61263	0.213447	0.106724	+	0.994289i	$0.465964\pi$
$468$	0	0
$469$	31.0061	1.43173
$470$	−3.07534	−0.141855
$471$	0	0
$472$	34.4681	1.58652
$473$	34.6135	1.59153
$474$	0	0
$475$	36.9721	1.69640
$476$	2.16153	0.0990733
$477$	0	0
$478$	−8.79182	−0.402129
$479$	37.2886	1.70376	0.851879	−	0.523739i	$-0.175463\pi$
$479$	37.2886	1.70376	0.851879	+	0.523739i	$0.175463\pi$
$480$	0	0
$481$	4.72345	0.215371
$482$	13.5018	0.614989
$483$	0	0
$484$	−0.669114	−0.0304143
$485$	0.752845	0.0341849
$486$	0	0
$487$	25.0176	1.13366	0.566828	−	0.823836i	$-0.308170\pi$
$487$	25.0176	1.13366	0.566828	+	0.823836i	$0.308170\pi$
$488$	−18.3274	−0.829644
$489$	0	0
$490$	−0.273129	−0.0123387
$491$	23.7021	1.06966	0.534830	−	0.844960i	$-0.320376\pi$
$491$	23.7021	1.06966	0.534830	+	0.844960i	$0.320376\pi$
$492$	0	0
$493$	75.9421	3.42026
$494$	−23.1935	−1.04352
$495$	0	0
$496$	−4.77680	−0.214485
$497$	23.5017	1.05419
$498$	0	0
$499$	27.1748	1.21651	0.608255	−	0.793742i	$-0.291870\pi$
$499$	27.1748	1.21651	0.608255	+	0.793742i	$0.291870\pi$
$500$	0.272283	0.0121769
$501$	0	0
$502$	−7.63331	−0.340691
$503$	−12.0746	−0.538381	−0.269191	−	0.963087i	$-0.586756\pi$
$503$	−12.0746	−0.538381	−0.269191	+	0.963087i	$0.586756\pi$
$504$	0	0
$505$	2.85482	0.127038
$506$	0.991432	0.0440745
$507$	0	0
$508$	1.82940	0.0811664
$509$	43.7176	1.93775	0.968873	−	0.247557i	$-0.0796279\pi$
$509$	43.7176	1.93775	0.968873	+	0.247557i	$0.0796279\pi$
$510$	0	0
$511$	12.7327	0.563261
$512$	−24.3316	−1.07532
$513$	0	0
$514$	−32.9385	−1.45286
$515$	−1.00817	−0.0444251
$516$	0	0
$517$	−40.4906	−1.78078
$518$	7.06663	0.310490
$519$	0	0
$520$	−1.48297	−0.0650323
$521$	35.3638	1.54932	0.774659	−	0.632379i	$-0.217922\pi$
$521$	35.3638	1.54932	0.774659	+	0.632379i	$0.217922\pi$
$522$	0	0
$523$	19.8938	0.869897	0.434949	−	0.900455i	$-0.356767\pi$
$523$	19.8938	0.869897	0.434949	+	0.900455i	$0.356767\pi$
$524$	1.16378	0.0508398
$525$	0	0
$526$	−20.5034	−0.893990
$527$	−9.17913	−0.399849
$528$	0	0
$529$	−22.9683	−0.998622
$530$	−0.191302	−0.00830962
$531$	0	0
$532$	2.24548	0.0973539
$533$	−9.59079	−0.415423
$534$	0	0
$535$	−3.39399	−0.146735
$536$	−36.4594	−1.57480
$537$	0	0
$538$	−18.4182	−0.794067
$539$	−3.59608	−0.154894
$540$	0	0
$541$	−15.3119	−0.658311	−0.329156	−	0.944276i	$-0.606764\pi$
$541$	−15.3119	−0.658311	−0.329156	+	0.944276i	$0.606764\pi$
$542$	7.84790	0.337096
$543$	0	0
$544$	−4.93774	−0.211704
$545$	0.825337	0.0353535
$546$	0	0
$547$	42.3037	1.80878	0.904388	−	0.426711i	$-0.140328\pi$
$547$	42.3037	1.80878	0.904388	+	0.426711i	$0.140328\pi$
$548$	0.261682	0.0111785
$549$	0	0
$550$	−27.5578	−1.17507
$551$	78.8917	3.36090
$552$	0	0
$553$	−40.5374	−1.72383
$554$	16.5812	0.704468
$555$	0	0
$556$	−0.683291	−0.0289780
$557$	−7.44351	−0.315392	−0.157696	−	0.987488i	$-0.550407\pi$
$557$	−7.44351	−0.315392	−0.157696	+	0.987488i	$0.550407\pi$
$558$	0	0
$559$	19.3011	0.816350
$560$	−2.08328	−0.0880347
$561$	0	0
$562$	8.52185	0.359472
$563$	30.6551	1.29196	0.645978	−	0.763356i	$-0.276450\pi$
$563$	30.6551	1.29196	0.645978	+	0.763356i	$0.276450\pi$
$564$	0	0
$565$	−1.47308	−0.0619731
$566$	17.7494	0.746064
$567$	0	0
$568$	−27.6351	−1.15954
$569$	−45.0445	−1.88836	−0.944182	−	0.329425i	$-0.893145\pi$
$569$	−45.0445	−1.88836	−0.944182	+	0.329425i	$0.893145\pi$
$570$	0	0
$571$	−20.5533	−0.860127	−0.430063	−	0.902799i	$-0.641509\pi$
$571$	−20.5533	−0.860127	−0.430063	+	0.902799i	$0.641509\pi$
$572$	−1.11873	−0.0467764
$573$	0	0
$574$	−14.3485	−0.598897
$575$	−0.881270	−0.0367515
$576$	0	0
$577$	2.43980	0.101570	0.0507851	−	0.998710i	$-0.483828\pi$
$577$	2.43980	0.101570	0.0507851	+	0.998710i	$0.483828\pi$
$578$	47.5700	1.97865
$579$	0	0
$580$	0.289021	0.0120009
$581$	−11.8516	−0.491685
$582$	0	0
$583$	−2.51873	−0.104315
$584$	−14.9721	−0.619549
$585$	0	0
$586$	−19.6662	−0.812404
$587$	−28.0624	−1.15826	−0.579129	−	0.815236i	$-0.696607\pi$
$587$	−28.0624	−1.15826	−0.579129	+	0.815236i	$0.696607\pi$
$588$	0	0
$589$	−9.53565	−0.392910
$590$	−3.65765	−0.150583
$591$	0	0
$592$	−7.80257	−0.320684
$593$	20.1600	0.827873	0.413937	−	0.910306i	$-0.364153\pi$
$593$	20.1600	0.827873	0.413937	+	0.910306i	$0.364153\pi$
$594$	0	0
$595$	−4.00325	−0.164117
$596$	1.84019	0.0753771
$597$	0	0
$598$	0.552841	0.0226073
$599$	−23.5060	−0.960427	−0.480214	−	0.877152i	$-0.659441\pi$
$599$	−23.5060	−0.960427	−0.480214	+	0.877152i	$0.659441\pi$
$600$	0	0
$601$	−20.9839	−0.855952	−0.427976	−	0.903790i	$-0.640773\pi$
$601$	−20.9839	−0.855952	−0.427976	+	0.903790i	$0.640773\pi$
$602$	28.8759	1.17689
$603$	0	0
$604$	2.39707	0.0975353
$605$	1.23923	0.0503819
$606$	0	0
$607$	32.2511	1.30903	0.654516	−	0.756048i	$-0.272872\pi$
$607$	32.2511	1.30903	0.654516	+	0.756048i	$0.272872\pi$
$608$	−5.12953	−0.208030
$609$	0	0
$610$	1.94485	0.0787447
$611$	−22.5783	−0.913421
$612$	0	0
$613$	−37.2120	−1.50298	−0.751489	−	0.659746i	$-0.770664\pi$
$613$	−37.2120	−1.50298	−0.751489	+	0.659746i	$0.770664\pi$
$614$	−39.3930	−1.58977
$615$	0	0
$616$	−29.2110	−1.17694
$617$	−38.3479	−1.54383	−0.771915	−	0.635726i	$-0.780701\pi$
$617$	−38.3479	−1.54383	−0.771915	+	0.635726i	$0.780701\pi$
$618$	0	0
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−0.0349340	−0.00140298
$621$	0	0
$622$	41.0726	1.64686
$623$	4.07849	0.163401
$624$	0	0
$625$	24.2423	0.969692
$626$	−18.6948	−0.747193
$627$	0	0
$628$	−1.90983	−0.0762106
$629$	−14.9935	−0.597829
$630$	0	0
$631$	38.9829	1.55188	0.775942	−	0.630804i	$-0.217275\pi$
$631$	38.9829	1.55188	0.775942	+	0.630804i	$0.217275\pi$
$632$	47.6670	1.89609
$633$	0	0
$634$	−17.7396	−0.704529
$635$	−3.38813	−0.134454
$636$	0	0
$637$	−2.00524	−0.0794504
$638$	−58.8032	−2.32804
$639$	0	0
$640$	2.29053	0.0905412
$641$	0.319675	0.0126264	0.00631321	−	0.999980i	$-0.497990\pi$
$641$	0.319675	0.0126264	0.00631321	+	0.999980i	$0.497990\pi$
$642$	0	0
$643$	0.159017	0.00627101	0.00313550	−	0.999995i	$-0.499002\pi$
$643$	0.159017	0.00627101	0.00313550	+	0.999995i	$0.499002\pi$
$644$	−0.0535234	−0.00210912
$645$	0	0
$646$	73.6222	2.89663
$647$	−24.6688	−0.969829	−0.484915	−	0.874562i	$-0.661149\pi$
$647$	−24.6688	−0.969829	−0.484915	+	0.874562i	$0.661149\pi$
$648$	0	0
$649$	−48.1575	−1.89035
$650$	−15.3667	−0.602732
$651$	0	0
$652$	1.44311	0.0565167
$653$	28.3461	1.10927	0.554635	−	0.832094i	$-0.312858\pi$
$653$	28.3461	1.10927	0.554635	+	0.832094i	$0.312858\pi$
$654$	0	0
$655$	−2.15537	−0.0842172
$656$	15.8428	0.618559
$657$	0	0
$658$	−33.7789	−1.31684
$659$	−39.5061	−1.53894	−0.769470	−	0.638683i	$-0.779479\pi$
$659$	−39.5061	−1.53894	−0.769470	+	0.638683i	$0.779479\pi$
$660$	0	0
$661$	−15.7041	−0.610818	−0.305409	−	0.952221i	$-0.598793\pi$
$661$	−15.7041	−0.610818	−0.305409	+	0.952221i	$0.598793\pi$
$662$	11.6271	0.451900
$663$	0	0
$664$	13.9360	0.540820
$665$	−4.15874	−0.161269
$666$	0	0
$667$	−1.88047	−0.0728120
$668$	−0.653910	−0.0253006
$669$	0	0
$670$	3.86896	0.149471
$671$	25.6064	0.988523
$672$	0	0
$673$	−23.0243	−0.887522	−0.443761	−	0.896145i	$-0.646356\pi$
$673$	−23.0243	−0.887522	−0.443761	+	0.896145i	$0.646356\pi$
$674$	−38.9830	−1.50157
$675$	0	0
$676$	0.956445	0.0367863
$677$	−7.63096	−0.293282	−0.146641	−	0.989190i	$-0.546846\pi$
$677$	−7.63096	−0.293282	−0.146641	+	0.989190i	$0.546846\pi$
$678$	0	0
$679$	8.26909	0.317339
$680$	4.70733	0.180518
$681$	0	0
$682$	7.10755	0.272162
$683$	−0.644724	−0.0246697	−0.0123348	−	0.999924i	$-0.503926\pi$
$683$	−0.644724	−0.0246697	−0.0123348	+	0.999924i	$0.503926\pi$
$684$	0	0
$685$	−0.484648	−0.0185175
$686$	−26.7240	−1.02033
$687$	0	0
$688$	−31.8831	−1.21553
$689$	−1.40449	−0.0535067
$690$	0	0
$691$	−29.5024	−1.12232	−0.561162	−	0.827706i	$-0.689646\pi$
$691$	−29.5024	−1.12232	−0.561162	+	0.827706i	$0.689646\pi$
$692$	−2.26578	−0.0861319
$693$	0	0
$694$	−8.73126	−0.331434
$695$	1.26549	0.0480027
$696$	0	0
$697$	30.4437	1.15314
$698$	−17.4446	−0.660288
$699$	0	0
$700$	1.48773	0.0562310
$701$	30.4332	1.14944	0.574722	−	0.818348i	$-0.305110\pi$
$701$	30.4332	1.14944	0.574722	+	0.818348i	$0.305110\pi$
$702$	0	0
$703$	−15.5758	−0.587453
$704$	34.2285	1.29003
$705$	0	0
$706$	−2.02910	−0.0763662
$707$	31.3568	1.17929
$708$	0	0
$709$	10.4754	0.393413	0.196707	−	0.980462i	$-0.436975\pi$
$709$	10.4754	0.393413	0.196707	+	0.980462i	$0.436975\pi$
$710$	2.93255	0.110057
$711$	0	0
$712$	−4.79580	−0.179730
$713$	0.227292	0.00851217
$714$	0	0
$715$	2.07194	0.0774862
$716$	−2.75044	−0.102789
$717$	0	0
$718$	37.7937	1.41045
$719$	−0.815327	−0.0304066	−0.0152033	−	0.999884i	$-0.504840\pi$
$719$	−0.815327	−0.0304066	−0.0152033	+	0.999884i	$0.504840\pi$
$720$	0	0
$721$	−11.0735	−0.412398
$722$	50.4410	1.87722
$723$	0	0
$724$	−1.47089	−0.0546650
$725$	52.2694	1.94123
$726$	0	0
$727$	5.58478	0.207128	0.103564	−	0.994623i	$-0.466975\pi$
$727$	5.58478	0.207128	0.103564	+	0.994623i	$0.466975\pi$
$728$	−16.2886	−0.603695
$729$	0	0
$730$	1.58879	0.0588038
$731$	−61.2669	−2.26604
$732$	0	0
$733$	27.7414	1.02465	0.512327	−	0.858791i	$-0.328784\pi$
$733$	27.7414	1.02465	0.512327	+	0.858791i	$0.328784\pi$
$734$	8.34613	0.308061
$735$	0	0
$736$	0.122268	0.00450685
$737$	50.9396	1.87638
$738$	0	0
$739$	−10.9980	−0.404569	−0.202285	−	0.979327i	$-0.564837\pi$
$739$	−10.9980	−0.404569	−0.202285	+	0.979327i	$0.564837\pi$
$740$	−0.0570623	−0.00209765
$741$	0	0
$742$	−2.10122	−0.0771382
$743$	−27.5706	−1.01147	−0.505733	−	0.862690i	$-0.668778\pi$
$743$	−27.5706	−1.01147	−0.505733	+	0.862690i	$0.668778\pi$
$744$	0	0
$745$	−3.40812	−0.124864
$746$	30.8307	1.12879
$747$	0	0
$748$	3.55115	0.129843
$749$	−37.2788	−1.36214
$750$	0	0
$751$	2.39041	0.0872274	0.0436137	−	0.999048i	$-0.486113\pi$
$751$	2.39041	0.0872274	0.0436137	+	0.999048i	$0.486113\pi$
$752$	37.2967	1.36007
$753$	0	0
$754$	−32.7897	−1.19413
$755$	−4.43948	−0.161569
$756$	0	0
$757$	22.8502	0.830505	0.415253	−	0.909706i	$-0.363693\pi$
$757$	22.8502	0.830505	0.415253	+	0.909706i	$0.363693\pi$
$758$	−19.3465	−0.702695
$759$	0	0
$760$	4.89016	0.177385
$761$	−35.2537	−1.27795	−0.638974	−	0.769229i	$-0.720641\pi$
$761$	−35.2537	−1.27795	−0.638974	+	0.769229i	$0.720641\pi$
$762$	0	0
$763$	9.06532	0.328187
$764$	−0.0202719	−0.000733410 0
$765$	0	0
$766$	−33.3892	−1.20640
$767$	−26.8535	−0.969625
$768$	0	0
$769$	14.0283	0.505875	0.252937	−	0.967483i	$-0.418603\pi$
$769$	14.0283	0.505875	0.252937	+	0.967483i	$0.418603\pi$
$770$	3.09978	0.111708
$771$	0	0
$772$	2.00888	0.0723012
$773$	−2.05297	−0.0738401	−0.0369200	−	0.999318i	$-0.511755\pi$
$773$	−2.05297	−0.0738401	−0.0369200	+	0.999318i	$0.511755\pi$
$774$	0	0
$775$	−6.31780	−0.226942
$776$	−9.72344	−0.349051
$777$	0	0
$778$	22.9321	0.822156
$779$	31.6262	1.13312
$780$	0	0
$781$	38.6107	1.38160
$782$	−1.75486	−0.0627538
$783$	0	0
$784$	3.31241	0.118300
$785$	3.53710	0.126245
$786$	0	0
$787$	−7.63128	−0.272026	−0.136013	−	0.990707i	$-0.543429\pi$
$787$	−7.63128	−0.272026	−0.136013	+	0.990707i	$0.543429\pi$
$788$	1.19404	0.0425361
$789$	0	0
$790$	−5.05828	−0.179965
$791$	−16.1800	−0.575296
$792$	0	0
$793$	14.2786	0.507047
$794$	−29.8356	−1.05882
$795$	0	0
$796$	−1.40633	−0.0498461
$797$	−23.8432	−0.844570	−0.422285	−	0.906463i	$-0.638772\pi$
$797$	−23.8432	−0.844570	−0.422285	+	0.906463i	$0.638772\pi$
$798$	0	0
$799$	71.6696	2.53549
$800$	−3.39854	−0.120157
$801$	0	0
$802$	−27.7222	−0.978905
$803$	20.9184	0.738194
$804$	0	0
$805$	0.0991279	0.00349380
$806$	3.96330	0.139601
$807$	0	0
$808$	−36.8717	−1.29714
$809$	−19.9428	−0.701152	−0.350576	−	0.936534i	$-0.614014\pi$
$809$	−19.9428	−0.701152	−0.350576	+	0.936534i	$0.614014\pi$
$810$	0	0
$811$	−24.3626	−0.855488	−0.427744	−	0.903900i	$-0.640692\pi$
$811$	−24.3626	−0.855488	−0.427744	+	0.903900i	$0.640692\pi$
$812$	3.17455	0.111405
$813$	0	0
$814$	11.6097	0.406920
$815$	−2.67272	−0.0936212
$816$	0	0
$817$	−63.6465	−2.22671
$818$	39.1242	1.36795
$819$	0	0
$820$	0.115863	0.00404611
$821$	15.6976	0.547850	0.273925	−	0.961751i	$-0.411678\pi$
$821$	15.6976	0.547850	0.273925	+	0.961751i	$0.411678\pi$
$822$	0	0
$823$	−7.38747	−0.257511	−0.128755	−	0.991676i	$-0.541098\pi$
$823$	−7.38747	−0.257511	−0.128755	+	0.991676i	$0.541098\pi$
$824$	13.0210	0.453609
$825$	0	0
$826$	−40.1749	−1.39786
$827$	−11.5262	−0.400806	−0.200403	−	0.979714i	$-0.564225\pi$
$827$	−11.5262	−0.400806	−0.200403	+	0.979714i	$0.564225\pi$
$828$	0	0
$829$	−17.7336	−0.615914	−0.307957	−	0.951400i	$-0.599645\pi$
$829$	−17.7336	−0.615914	−0.307957	+	0.951400i	$0.599645\pi$
$830$	−1.47884	−0.0513314
$831$	0	0
$832$	19.0864	0.661702
$833$	6.36516	0.220540
$834$	0	0
$835$	1.21107	0.0419109
$836$	3.68907	0.127589
$837$	0	0
$838$	49.7798	1.71962
$839$	28.6447	0.988924	0.494462	−	0.869199i	$-0.335365\pi$
$839$	28.6447	0.988924	0.494462	+	0.869199i	$0.335365\pi$
$840$	0	0
$841$	82.5332	2.84597
$842$	−28.6428	−0.987095
$843$	0	0
$844$	2.33777	0.0804695
$845$	−1.77138	−0.0609374
$846$	0	0
$847$	13.6115	0.467695
$848$	2.32005	0.0796707
$849$	0	0
$850$	48.7781	1.67307
$851$	0.371266	0.0127269
$852$	0	0
$853$	41.6536	1.42619	0.713097	−	0.701066i	$-0.247292\pi$
$853$	41.6536	1.42619	0.713097	+	0.701066i	$0.247292\pi$
$854$	21.3618	0.730987
$855$	0	0
$856$	43.8353	1.49826
$857$	9.05282	0.309238	0.154619	−	0.987974i	$-0.450585\pi$
$857$	9.05282	0.309238	0.154619	+	0.987974i	$0.450585\pi$
$858$	0	0
$859$	9.47579	0.323310	0.161655	−	0.986847i	$-0.448317\pi$
$859$	9.47579	0.323310	0.161655	+	0.986847i	$0.448317\pi$
$860$	−0.233170	−0.00795103
$861$	0	0
$862$	−7.75457	−0.264122
$863$	34.6314	1.17887	0.589434	−	0.807817i	$-0.299351\pi$
$863$	34.6314	1.17887	0.589434	+	0.807817i	$0.299351\pi$
$864$	0	0
$865$	4.19633	0.142679
$866$	−1.09315	−0.0371468
$867$	0	0
$868$	−0.383708	−0.0130239
$869$	−66.5985	−2.25920
$870$	0	0
$871$	28.4048	0.962461
$872$	−10.6597	−0.360983
$873$	0	0
$874$	−1.82302	−0.0616647
$875$	−5.53892	−0.187250
$876$	0	0
$877$	15.5441	0.524886	0.262443	−	0.964947i	$-0.415472\pi$
$877$	15.5441	0.524886	0.262443	+	0.964947i	$0.415472\pi$
$878$	−25.0574	−0.845648
$879$	0	0
$880$	−3.42260	−0.115376
$881$	−52.4905	−1.76845	−0.884224	−	0.467063i	$-0.845312\pi$
$881$	−52.4905	−1.76845	−0.884224	+	0.467063i	$0.845312\pi$
$882$	0	0
$883$	−32.8989	−1.10713	−0.553567	−	0.832804i	$-0.686734\pi$
$883$	−32.8989	−1.10713	−0.553567	+	0.832804i	$0.686734\pi$
$884$	1.98018	0.0666008
$885$	0	0
$886$	25.2236	0.847402
$887$	40.9458	1.37483	0.687413	−	0.726267i	$-0.258746\pi$
$887$	40.9458	1.37483	0.687413	+	0.726267i	$0.258746\pi$
$888$	0	0
$889$	−37.2145	−1.24814
$890$	0.508916	0.0170589
$891$	0	0
$892$	0.275287	0.00921729
$893$	74.4533	2.49148
$894$	0	0
$895$	5.09395	0.170272
$896$	25.1587	0.840493
$897$	0	0
$898$	−4.05367	−0.135273
$899$	−13.4810	−0.449617
$900$	0	0
$901$	4.45822	0.148525
$902$	−23.5731	−0.784897
$903$	0	0
$904$	19.0257	0.632786
$905$	2.72415	0.0905539
$906$	0	0
$907$	−22.3371	−0.741692	−0.370846	−	0.928694i	$-0.620932\pi$
$907$	−22.3371	−0.741692	−0.370846	+	0.928694i	$0.620932\pi$
$908$	1.53913	0.0510778
$909$	0	0
$910$	1.72849	0.0572990
$911$	−15.9284	−0.527732	−0.263866	−	0.964559i	$-0.584998\pi$
$911$	−15.9284	−0.527732	−0.263866	+	0.964559i	$0.584998\pi$
$912$	0	0
$913$	−19.4708	−0.644389
$914$	24.8596	0.822283
$915$	0	0
$916$	−3.37356	−0.111466
$917$	−23.6741	−0.781788
$918$	0	0
$919$	−40.6296	−1.34025	−0.670124	−	0.742249i	$-0.733759\pi$
$919$	−40.6296	−1.34025	−0.670124	+	0.742249i	$0.733759\pi$
$920$	−0.116562	−0.00384294
$921$	0	0
$922$	22.6927	0.747343
$923$	21.5300	0.708669
$924$	0	0
$925$	−10.3197	−0.339309
$926$	5.49802	0.180676
$927$	0	0
$928$	−7.25187	−0.238054
$929$	38.1089	1.25031	0.625155	−	0.780500i	$-0.285036\pi$
$929$	38.1089	1.25031	0.625155	+	0.780500i	$0.285036\pi$
$930$	0	0
$931$	6.61238	0.216712
$932$	−1.32699	−0.0434670
$933$	0	0
$934$	6.32190	0.206859
$935$	−6.57689	−0.215087
$936$	0	0
$937$	20.9882	0.685653	0.342827	−	0.939399i	$-0.388616\pi$
$937$	20.9882	0.685653	0.342827	+	0.939399i	$0.388616\pi$
$938$	42.4958	1.38754
$939$	0	0
$940$	0.272761	0.00889648
$941$	−0.271913	−0.00886412	−0.00443206	−	0.999990i	$-0.501411\pi$
$941$	−0.271913	−0.00886412	−0.00443206	+	0.999990i	$0.501411\pi$
$942$	0	0
$943$	−0.753843	−0.0245485
$944$	44.3588	1.44376
$945$	0	0
$946$	47.4399	1.54241
$947$	−18.9489	−0.615756	−0.307878	−	0.951426i	$-0.599619\pi$
$947$	−18.9489	−0.615756	−0.307878	+	0.951426i	$0.599619\pi$
$948$	0	0
$949$	11.6645	0.378645
$950$	50.6726	1.64404
$951$	0	0
$952$	51.7043	1.67575
$953$	2.67357	0.0866055	0.0433027	−	0.999062i	$-0.486212\pi$
$953$	2.67357	0.0866055	0.0433027	+	0.999062i	$0.486212\pi$
$954$	0	0
$955$	0.0375445	0.00121491
$956$	0.779772	0.0252196
$957$	0	0
$958$	51.1063	1.65117
$959$	−5.32327	−0.171897
$960$	0	0
$961$	−29.3705	−0.947437
$962$	6.47378	0.208723
$963$	0	0
$964$	−1.19751	−0.0385692
$965$	−3.72054	−0.119769
$966$	0	0
$967$	20.1196	0.647004	0.323502	−	0.946227i	$-0.395140\pi$
$967$	20.1196	0.647004	0.323502	+	0.946227i	$0.395140\pi$
$968$	−16.0054	−0.514433
$969$	0	0
$970$	1.03182	0.0331298
$971$	−15.9577	−0.512106	−0.256053	−	0.966663i	$-0.582422\pi$
$971$	−15.9577	−0.512106	−0.256053	+	0.966663i	$0.582422\pi$
$972$	0	0
$973$	13.8998	0.445609
$974$	34.2882	1.09866
$975$	0	0
$976$	−23.5865	−0.754985
$977$	−42.1759	−1.34933	−0.674663	−	0.738126i	$-0.735711\pi$
$977$	−42.1759	−1.34933	−0.674663	+	0.738126i	$0.735711\pi$
$978$	0	0
$979$	6.70051	0.214149
$980$	0.0242246	0.000773826 0
$981$	0	0
$982$	32.4852	1.03664
$983$	31.8772	1.01673	0.508363	−	0.861143i	$-0.330251\pi$
$983$	31.8772	1.01673	0.508363	+	0.861143i	$0.330251\pi$
$984$	0	0
$985$	−2.21143	−0.0704619
$986$	104.083	3.31469
$987$	0	0
$988$	2.05709	0.0654449
$989$	1.51708	0.0482404
$990$	0	0
$991$	28.4135	0.902584	0.451292	−	0.892376i	$-0.350963\pi$
$991$	28.4135	0.902584	0.451292	+	0.892376i	$0.350963\pi$
$992$	0.876534	0.0278300
$993$	0	0
$994$	32.2105	1.02166
$995$	2.60459	0.0825712
$996$	0	0
$997$	−38.9058	−1.23216	−0.616079	−	0.787684i	$-0.711280\pi$
$997$	−38.9058	−1.23216	−0.616079	+	0.787684i	$0.711280\pi$
$998$	37.2447	1.17896
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5571.2.a.g.1.23		30
3.2	odd	2		619.2.a.b.1.8	✓	30
12.11	even	2		9904.2.a.n.1.18		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.b.1.8	✓	30	3.2	odd	2
5571.2.a.g.1.23		30	1.1	even	1	trivial
9904.2.a.n.1.18		30	12.11	even	2

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:	\( N \)	\(=\)	\( 5571 = 3^{2} \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5571.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37056 q^{2} -0.121559 q^{4} +0.225133 q^{5} +2.47282 q^{7} -2.90773 q^{8} +O(q^{10})\)	\(q+1.37056 q^{2} -0.121559 q^{4} +0.225133 q^{5} +2.47282 q^{7} -2.90773 q^{8} +0.308559 q^{10} +4.06256 q^{11} +2.26536 q^{13} +3.38915 q^{14} -3.74211 q^{16} -7.19086 q^{17} -7.47015 q^{19} -0.0273670 q^{20} +5.56800 q^{22} +0.178059 q^{23} -4.94931 q^{25} +3.10482 q^{26} -0.300594 q^{28} -10.5609 q^{29} +1.27650 q^{31} +0.686670 q^{32} -9.85552 q^{34} +0.556714 q^{35} +2.08508 q^{37} -10.2383 q^{38} -0.654627 q^{40} -4.23367 q^{41} +8.52011 q^{43} -0.493842 q^{44} +0.244041 q^{46} -9.96677 q^{47} -0.885174 q^{49} -6.78334 q^{50} -0.275375 q^{52} -0.619984 q^{53} +0.914619 q^{55} -7.19028 q^{56} -14.4744 q^{58} -11.8540 q^{59} +6.30300 q^{61} +1.74952 q^{62} +8.42533 q^{64} +0.510008 q^{65} +12.5388 q^{67} +0.874114 q^{68} +0.763011 q^{70} +9.50401 q^{71} +5.14906 q^{73} +2.85773 q^{74} +0.908065 q^{76} +10.0460 q^{77} -16.3932 q^{79} -0.842473 q^{80} -5.80251 q^{82} -4.79273 q^{83} -1.61890 q^{85} +11.6773 q^{86} -11.8128 q^{88} +1.64933 q^{89} +5.60182 q^{91} -0.0216447 q^{92} -13.6601 q^{94} -1.68178 q^{95} +3.34400 q^{97} -1.21319 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8}+O(q^{10}) \) 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8	\( 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8} + 5 q^{10} - 23 q^{11} + 9 q^{13} - 7 q^{14} + 35 q^{16} - 4 q^{17} - q^{19} - 29 q^{20} - 4 q^{23} + 35 q^{25} - q^{26} - 13 q^{28} - 90 q^{29} + 2 q^{31} - 43 q^{32} - 9 q^{34} - 9 q^{35} + 19 q^{37} - 5 q^{38} - 12 q^{40} - 59 q^{41} - 4 q^{43} - 52 q^{44} - q^{46} - 4 q^{47} + 30 q^{49} - 31 q^{50} - 12 q^{52} - 34 q^{53} - 17 q^{55} - 2 q^{56} + 6 q^{58} - 13 q^{59} + 16 q^{61} - 28 q^{62} + 37 q^{64} - 31 q^{65} - 11 q^{67} + 52 q^{68} - 40 q^{70} - 42 q^{71} - 4 q^{73} - 16 q^{74} - 42 q^{76} - 29 q^{77} + 3 q^{79} - 21 q^{80} - 43 q^{82} + 11 q^{83} + 19 q^{85} + 11 q^{86} - 47 q^{88} - 58 q^{89} - 39 q^{91} + 7 q^{92} - 46 q^{94} - 23 q^{95} - 9 q^{97}+O(q^{100}) \) 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8 + 5 * q^10 - 23 * q^11 + 9 * q^13 - 7 * q^14 + 35 * q^16 - 4 * q^17 - q^19 - 29 * q^20 - 4 * q^23 + 35 * q^25 - q^26 - 13 * q^28 - 90 * q^29 + 2 * q^31 - 43 * q^32 - 9 * q^34 - 9 * q^35 + 19 * q^37 - 5 * q^38 - 12 * q^40 - 59 * q^41 - 4 * q^43 - 52 * q^44 - q^46 - 4 * q^47 + 30 * q^49 - 31 * q^50 - 12 * q^52 - 34 * q^53 - 17 * q^55 - 2 * q^56 + 6 * q^58 - 13 * q^59 + 16 * q^61 - 28 * q^62 + 37 * q^64 - 31 * q^65 - 11 * q^67 + 52 * q^68 - 40 * q^70 - 42 * q^71 - 4 * q^73 - 16 * q^74 - 42 * q^76 - 29 * q^77 + 3 * q^79 - 21 * q^80 - 43 * q^82 + 11 * q^83 + 19 * q^85 + 11 * q^86 - 47 * q^88 - 58 * q^89 - 39 * q^91 + 7 * q^92 - 46 * q^94 - 23 * q^95 - 9 * q^97

Introduction

L-functions

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