LMFDB - Embedded newform 2888.2.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,2,Mod(1,2888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,0,0,-3,0,-3,0,3,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.0607961037$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	2888.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.347296 q^{3} +1.53209 q^{5} -0.305407 q^{7} -2.87939 q^{9} +0.305407 q^{11} +0.347296 q^{13} -0.532089 q^{15} -1.87939 q^{17} +0.106067 q^{21} -2.83750 q^{23} -2.65270 q^{25} +2.04189 q^{27} +3.50980 q^{29} +5.82295 q^{31} -0.106067 q^{33} -0.467911 q^{35} -8.12836 q^{37} -0.120615 q^{39} +2.47565 q^{41} -1.16250 q^{43} -4.41147 q^{45} -6.24897 q^{47} -6.90673 q^{49} +0.652704 q^{51} -7.20708 q^{53} +0.467911 q^{55} -11.3969 q^{59} -11.3746 q^{61} +0.879385 q^{63} +0.532089 q^{65} +0.361844 q^{67} +0.985452 q^{69} +12.9659 q^{71} +5.99319 q^{73} +0.921274 q^{75} -0.0932736 q^{77} +9.04189 q^{79} +7.92902 q^{81} -16.5621 q^{83} -2.87939 q^{85} -1.21894 q^{87} -1.04189 q^{89} -0.106067 q^{91} -2.02229 q^{93} -9.39693 q^{97} -0.879385 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{7} - 3 q^{9} + 3 q^{11} + 3 q^{15} - 12 q^{21} - 6 q^{23} - 9 q^{25} + 3 q^{27} + 12 q^{29} - 3 q^{31} + 12 q^{33} - 6 q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 6 q^{43} - 3 q^{45} - 6 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^7 - 3 * q^9 + 3 * q^11 + 3 * q^15 - 12 * q^21 - 6 * q^23 - 9 * q^25 + 3 * q^27 + 12 * q^29 - 3 * q^31 + 12 * q^33 - 6 * q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 6 * q^43 - 3 * q^45 - 6 * q^47 + 6 * q^49 + 3 * q^51 - 12 * q^53 + 6 * q^55 - 6 * q^59 - 12 * q^61 - 3 * q^63 - 3 * q^65 + 18 * q^67 - 15 * q^69 + 18 * q^71 - 24 * q^73 - 6 * q^75 - 27 * q^77 + 24 * q^79 - 9 * q^81 - 15 * q^83 - 3 * q^85 - 21 * q^87 + 12 * q^91 + 3 * q^99

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
$3$	−0.347296	−0.200512	−0.100256	−	0.994962i	$-0.531966\pi$
$3$	−0.347296	−0.200512	−0.100256	+	0.994962i	$0.531966\pi$
$4$	0	0
$5$	1.53209	0.685171	0.342585	−	0.939487i	$-0.388697\pi$
$5$	1.53209	0.685171	0.342585	+	0.939487i	$0.388697\pi$
$6$	0	0
$7$	−0.305407	−0.115433	−0.0577166	−	0.998333i	$-0.518382\pi$
$7$	−0.305407	−0.115433	−0.0577166	+	0.998333i	$0.518382\pi$
$8$	0	0
$9$	−2.87939	−0.959795
$10$	0	0
$11$	0.305407	0.0920838	0.0460419	−	0.998940i	$-0.485339\pi$
$11$	0.305407	0.0920838	0.0460419	+	0.998940i	$0.485339\pi$
$12$	0	0
$13$	0.347296	0.0963227	0.0481613	−	0.998840i	$-0.484664\pi$
$13$	0.347296	0.0963227	0.0481613	+	0.998840i	$0.484664\pi$
$14$	0	0
$15$	−0.532089	−0.137385
$16$	0	0
$17$	−1.87939	−0.455818	−0.227909	−	0.973682i	$-0.573189\pi$
$17$	−1.87939	−0.455818	−0.227909	+	0.973682i	$0.573189\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.106067	0.0231457
$22$	0	0
$23$	−2.83750	−0.591659	−0.295829	−	0.955241i	$-0.595596\pi$
$23$	−2.83750	−0.591659	−0.295829	+	0.955241i	$0.595596\pi$
$24$	0	0
$25$	−2.65270	−0.530541
$26$	0	0
$27$	2.04189	0.392962
$28$	0	0
$29$	3.50980	0.651754	0.325877	−	0.945412i	$-0.394341\pi$
$29$	3.50980	0.651754	0.325877	+	0.945412i	$0.394341\pi$
$30$	0	0
$31$	5.82295	1.04583	0.522916	−	0.852384i	$-0.324844\pi$
$31$	5.82295	1.04583	0.522916	+	0.852384i	$0.324844\pi$
$32$	0	0
$33$	−0.106067	−0.0184639
$34$	0	0
$35$	−0.467911	−0.0790914
$36$	0	0
$37$	−8.12836	−1.33629	−0.668147	−	0.744030i	$-0.732912\pi$
$37$	−8.12836	−1.33629	−0.668147	+	0.744030i	$0.732912\pi$
$38$	0	0
$39$	−0.120615	−0.0193138
$40$	0	0
$41$	2.47565	0.386632	0.193316	−	0.981137i	$-0.438076\pi$
$41$	2.47565	0.386632	0.193316	+	0.981137i	$0.438076\pi$
$42$	0	0
$43$	−1.16250	−0.177280	−0.0886401	−	0.996064i	$-0.528252\pi$
$43$	−1.16250	−0.177280	−0.0886401	+	0.996064i	$0.528252\pi$
$44$	0	0
$45$	−4.41147	−0.657624
$46$	0	0
$47$	−6.24897	−0.911506	−0.455753	−	0.890106i	$-0.650630\pi$
$47$	−6.24897	−0.911506	−0.455753	+	0.890106i	$0.650630\pi$
$48$	0	0
$49$	−6.90673	−0.986675
$50$	0	0
$51$	0.652704	0.0913968
$52$	0	0
$53$	−7.20708	−0.989969	−0.494984	−	0.868902i	$-0.664826\pi$
$53$	−7.20708	−0.989969	−0.494984	+	0.868902i	$0.664826\pi$
$54$	0	0
$55$	0.467911	0.0630931
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.3969	−1.48375	−0.741877	−	0.670536i	$-0.766064\pi$
$59$	−11.3969	−1.48375	−0.741877	+	0.670536i	$0.766064\pi$
$60$	0	0
$61$	−11.3746	−1.45637	−0.728187	−	0.685379i	$-0.759637\pi$
$61$	−11.3746	−1.45637	−0.728187	+	0.685379i	$0.759637\pi$
$62$	0	0
$63$	0.879385	0.110792
$64$	0	0
$65$	0.532089	0.0659975
$66$	0	0
$67$	0.361844	0.0442063	0.0221032	−	0.999756i	$-0.492964\pi$
$67$	0.361844	0.0442063	0.0221032	+	0.999756i	$0.492964\pi$
$68$	0	0
$69$	0.985452	0.118634
$70$	0	0
$71$	12.9659	1.53876	0.769382	−	0.638789i	$-0.220564\pi$
$71$	12.9659	1.53876	0.769382	+	0.638789i	$0.220564\pi$
$72$	0	0
$73$	5.99319	0.701450	0.350725	−	0.936478i	$-0.385935\pi$
$73$	5.99319	0.701450	0.350725	+	0.936478i	$0.385935\pi$
$74$	0	0
$75$	0.921274	0.106380
$76$	0	0
$77$	−0.0932736	−0.0106295
$78$	0	0
$79$	9.04189	1.01729	0.508646	−	0.860976i	$-0.330146\pi$
$79$	9.04189	1.01729	0.508646	+	0.860976i	$0.330146\pi$
$80$	0	0
$81$	7.92902	0.881002
$82$	0	0
$83$	−16.5621	−1.81793	−0.908964	−	0.416874i	$-0.863126\pi$
$83$	−16.5621	−1.81793	−0.908964	+	0.416874i	$0.863126\pi$
$84$	0	0
$85$	−2.87939	−0.312313
$86$	0	0
$87$	−1.21894	−0.130684
$88$	0	0
$89$	−1.04189	−0.110440	−0.0552200	−	0.998474i	$-0.517586\pi$
$89$	−1.04189	−0.110440	−0.0552200	+	0.998474i	$0.517586\pi$
$90$	0	0
$91$	−0.106067	−0.0111188
$92$	0	0
$93$	−2.02229	−0.209702
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.39693	−0.954113	−0.477057	−	0.878873i	$-0.658296\pi$
$97$	−9.39693	−0.954113	−0.477057	+	0.878873i	$0.658296\pi$
$98$	0	0
$99$	−0.879385	−0.0883815
$100$	0	0
$101$	2.47565	0.246337	0.123168	−	0.992386i	$-0.460695\pi$
$101$	2.47565	0.246337	0.123168	+	0.992386i	$0.460695\pi$
$102$	0	0
$103$	16.7297	1.64842	0.824212	−	0.566281i	$-0.191618\pi$
$103$	16.7297	1.64842	0.824212	+	0.566281i	$0.191618\pi$
$104$	0	0
$105$	0.162504	0.0158587
$106$	0	0
$107$	7.21213	0.697223	0.348612	−	0.937267i	$-0.386653\pi$
$107$	7.21213	0.697223	0.348612	+	0.937267i	$0.386653\pi$
$108$	0	0
$109$	−6.46791	−0.619514	−0.309757	−	0.950816i	$-0.600248\pi$
$109$	−6.46791	−0.619514	−0.309757	+	0.950816i	$0.600248\pi$
$110$	0	0
$111$	2.82295	0.267942
$112$	0	0
$113$	−13.5175	−1.27162	−0.635812	−	0.771844i	$-0.719334\pi$
$113$	−13.5175	−1.27162	−0.635812	+	0.771844i	$0.719334\pi$
$114$	0	0
$115$	−4.34730	−0.405387
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0.573978	0.0526165
$120$	0	0
$121$	−10.9067	−0.991521
$122$	0	0
$123$	−0.859785	−0.0775242
$124$	0	0
$125$	−11.7246	−1.04868
$126$	0	0
$127$	−9.94862	−0.882797	−0.441398	−	0.897311i	$-0.645517\pi$
$127$	−9.94862	−0.882797	−0.441398	+	0.897311i	$0.645517\pi$
$128$	0	0
$129$	0.403733	0.0355467
$130$	0	0
$131$	10.4165	0.910096	0.455048	−	0.890467i	$-0.349622\pi$
$131$	10.4165	0.910096	0.455048	+	0.890467i	$0.349622\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.12836	0.269246
$136$	0	0
$137$	−18.4679	−1.57782	−0.788910	−	0.614509i	$-0.789354\pi$
$137$	−18.4679	−1.57782	−0.788910	+	0.614509i	$0.789354\pi$
$138$	0	0
$139$	−17.0419	−1.44547	−0.722737	−	0.691123i	$-0.757116\pi$
$139$	−17.0419	−1.44547	−0.722737	+	0.691123i	$0.757116\pi$
$140$	0	0
$141$	2.17024	0.182768
$142$	0	0
$143$	0.106067	0.00886975
$144$	0	0
$145$	5.37733	0.446563
$146$	0	0
$147$	2.39868	0.197840
$148$	0	0
$149$	15.8648	1.29970	0.649849	−	0.760063i	$-0.274832\pi$
$149$	15.8648	1.29970	0.649849	+	0.760063i	$0.274832\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	5.41147	0.437492
$154$	0	0
$155$	8.92127	0.716574
$156$	0	0
$157$	11.6604	0.930605	0.465302	−	0.885152i	$-0.345946\pi$
$157$	11.6604	0.930605	0.465302	+	0.885152i	$0.345946\pi$
$158$	0	0
$159$	2.50299	0.198500
$160$	0	0
$161$	0.866592	0.0682970
$162$	0	0
$163$	16.0797	1.25946	0.629728	−	0.776816i	$-0.283166\pi$
$163$	16.0797	1.25946	0.629728	+	0.776816i	$0.283166\pi$
$164$	0	0
$165$	−0.162504	−0.0126509
$166$	0	0
$167$	−5.87258	−0.454434	−0.227217	−	0.973844i	$-0.572963\pi$
$167$	−5.87258	−0.454434	−0.227217	+	0.973844i	$0.572963\pi$
$168$	0	0
$169$	−12.8794	−0.990722
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.75103	−0.589300	−0.294650	−	0.955605i	$-0.595203\pi$
$173$	−7.75103	−0.589300	−0.294650	+	0.955605i	$0.595203\pi$
$174$	0	0
$175$	0.810155	0.0612420
$176$	0	0
$177$	3.95811	0.297510
$178$	0	0
$179$	15.8229	1.18266	0.591331	−	0.806429i	$-0.298603\pi$
$179$	15.8229	1.18266	0.591331	+	0.806429i	$0.298603\pi$
$180$	0	0
$181$	−3.75103	−0.278812	−0.139406	−	0.990235i	$-0.544519\pi$
$181$	−3.75103	−0.278812	−0.139406	+	0.990235i	$0.544519\pi$
$182$	0	0
$183$	3.95037	0.292020
$184$	0	0
$185$	−12.4534	−0.915589
$186$	0	0
$187$	−0.573978	−0.0419734
$188$	0	0
$189$	−0.623608	−0.0453608
$190$	0	0
$191$	4.73917	0.342914	0.171457	−	0.985192i	$-0.445152\pi$
$191$	4.73917	0.342914	0.171457	+	0.985192i	$0.445152\pi$
$192$	0	0
$193$	−9.94862	−0.716117	−0.358059	−	0.933699i	$-0.616561\pi$
$193$	−9.94862	−0.716117	−0.358059	+	0.933699i	$0.616561\pi$
$194$	0	0
$195$	−0.184793	−0.0132333
$196$	0	0
$197$	−13.7392	−0.978875	−0.489438	−	0.872038i	$-0.662798\pi$
$197$	−13.7392	−0.978875	−0.489438	+	0.872038i	$0.662798\pi$
$198$	0	0
$199$	−15.1557	−1.07436	−0.537179	−	0.843468i	$-0.680510\pi$
$199$	−15.1557	−1.07436	−0.537179	+	0.843468i	$0.680510\pi$
$200$	0	0
$201$	−0.125667	−0.00886388
$202$	0	0
$203$	−1.07192	−0.0752339
$204$	0	0
$205$	3.79292	0.264909
$206$	0	0
$207$	8.17024	0.567871
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.1361	−0.835483	−0.417742	−	0.908566i	$-0.637178\pi$
$211$	−12.1361	−0.835483	−0.417742	+	0.908566i	$0.637178\pi$
$212$	0	0
$213$	−4.50299	−0.308540
$214$	0	0
$215$	−1.78106	−0.121467
$216$	0	0
$217$	−1.77837	−0.120724
$218$	0	0
$219$	−2.08141	−0.140649
$220$	0	0
$221$	−0.652704	−0.0439056
$222$	0	0
$223$	24.0009	1.60722	0.803611	−	0.595155i	$-0.202909\pi$
$223$	24.0009	1.60722	0.803611	+	0.595155i	$0.202909\pi$
$224$	0	0
$225$	7.63816	0.509210
$226$	0	0
$227$	15.3500	1.01881	0.509407	−	0.860526i	$-0.329865\pi$
$227$	15.3500	1.01881	0.509407	+	0.860526i	$0.329865\pi$
$228$	0	0
$229$	6.90673	0.456409	0.228205	−	0.973613i	$-0.426714\pi$
$229$	6.90673	0.456409	0.228205	+	0.973613i	$0.426714\pi$
$230$	0	0
$231$	0.0323936	0.00213134
$232$	0	0
$233$	1.53209	0.100370	0.0501852	−	0.998740i	$-0.484019\pi$
$233$	1.53209	0.100370	0.0501852	+	0.998740i	$0.484019\pi$
$234$	0	0
$235$	−9.57398	−0.624537
$236$	0	0
$237$	−3.14022	−0.203979
$238$	0	0
$239$	−20.6013	−1.33259	−0.666294	−	0.745689i	$-0.732121\pi$
$239$	−20.6013	−1.33259	−0.666294	+	0.745689i	$0.732121\pi$
$240$	0	0
$241$	19.2148	1.23774	0.618868	−	0.785495i	$-0.287592\pi$
$241$	19.2148	1.23774	0.618868	+	0.785495i	$0.287592\pi$
$242$	0	0
$243$	−8.87939	−0.569613
$244$	0	0
$245$	−10.5817	−0.676041
$246$	0	0
$247$	0	0
$248$	0	0
$249$	5.75196	0.364516
$250$	0	0
$251$	13.4192	0.847013	0.423507	−	0.905893i	$-0.360799\pi$
$251$	13.4192	0.847013	0.423507	+	0.905893i	$0.360799\pi$
$252$	0	0
$253$	−0.866592	−0.0544822
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	19.2841	1.20291	0.601453	−	0.798908i	$-0.294589\pi$
$257$	19.2841	1.20291	0.601453	+	0.798908i	$0.294589\pi$
$258$	0	0
$259$	2.48246	0.154252
$260$	0	0
$261$	−10.1061	−0.625550
$262$	0	0
$263$	−27.6391	−1.70430	−0.852150	−	0.523298i	$-0.824701\pi$
$263$	−27.6391	−1.70430	−0.852150	+	0.523298i	$0.824701\pi$
$264$	0	0
$265$	−11.0419	−0.678298
$266$	0	0
$267$	0.361844	0.0221445
$268$	0	0
$269$	7.69728	0.469312	0.234656	−	0.972079i	$-0.424604\pi$
$269$	7.69728	0.469312	0.234656	+	0.972079i	$0.424604\pi$
$270$	0	0
$271$	−4.93676	−0.299887	−0.149943	−	0.988695i	$-0.547909\pi$
$271$	−4.93676	−0.299887	−0.149943	+	0.988695i	$0.547909\pi$
$272$	0	0
$273$	0.0368366	0.00222945
$274$	0	0
$275$	−0.810155	−0.0488542
$276$	0	0
$277$	1.77837	0.106852	0.0534260	−	0.998572i	$-0.482986\pi$
$277$	1.77837	0.106852	0.0534260	+	0.998572i	$0.482986\pi$
$278$	0	0
$279$	−16.7665	−1.00378
$280$	0	0
$281$	−21.0205	−1.25398	−0.626990	−	0.779027i	$-0.715713\pi$
$281$	−21.0205	−1.25398	−0.626990	+	0.779027i	$0.715713\pi$
$282$	0	0
$283$	−5.26857	−0.313184	−0.156592	−	0.987663i	$-0.550051\pi$
$283$	−5.26857	−0.313184	−0.156592	+	0.987663i	$0.550051\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.756082	−0.0446301
$288$	0	0
$289$	−13.4679	−0.792230
$290$	0	0
$291$	3.26352	0.191311
$292$	0	0
$293$	−26.6459	−1.55667	−0.778335	−	0.627849i	$-0.783935\pi$
$293$	−26.6459	−1.55667	−0.778335	+	0.627849i	$0.783935\pi$
$294$	0	0
$295$	−17.4611	−1.01662
$296$	0	0
$297$	0.623608	0.0361854
$298$	0	0
$299$	−0.985452	−0.0569902
$300$	0	0
$301$	0.355037	0.0204640
$302$	0	0
$303$	−0.859785	−0.0493934
$304$	0	0
$305$	−17.4270	−0.997865
$306$	0	0
$307$	−2.21894	−0.126642	−0.0633208	−	0.997993i	$-0.520169\pi$
$307$	−2.21894	−0.126642	−0.0633208	+	0.997993i	$0.520169\pi$
$308$	0	0
$309$	−5.81016	−0.330528
$310$	0	0
$311$	15.6946	0.889959	0.444979	−	0.895541i	$-0.353211\pi$
$311$	15.6946	0.889959	0.444979	+	0.895541i	$0.353211\pi$
$312$	0	0
$313$	8.24897	0.466259	0.233130	−	0.972446i	$-0.425103\pi$
$313$	8.24897	0.466259	0.233130	+	0.972446i	$0.425103\pi$
$314$	0	0
$315$	1.34730	0.0759115
$316$	0	0
$317$	5.99319	0.336611	0.168306	−	0.985735i	$-0.446170\pi$
$317$	5.99319	0.336611	0.168306	+	0.985735i	$0.446170\pi$
$318$	0	0
$319$	1.07192	0.0600159
$320$	0	0
$321$	−2.50475	−0.139801
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.921274	−0.0511031
$326$	0	0
$327$	2.24628	0.124220
$328$	0	0
$329$	1.90848	0.105218
$330$	0	0
$331$	4.95542	0.272375	0.136187	−	0.990683i	$-0.456515\pi$
$331$	4.95542	0.272375	0.136187	+	0.990683i	$0.456515\pi$
$332$	0	0
$333$	23.4047	1.28257
$334$	0	0
$335$	0.554378	0.0302889
$336$	0	0
$337$	−7.46379	−0.406579	−0.203289	−	0.979119i	$-0.565163\pi$
$337$	−7.46379	−0.406579	−0.203289	+	0.979119i	$0.565163\pi$
$338$	0	0
$339$	4.69459	0.254975
$340$	0	0
$341$	1.77837	0.0963042
$342$	0	0
$343$	4.24722	0.229328
$344$	0	0
$345$	1.50980	0.0812849
$346$	0	0
$347$	−26.3259	−1.41325	−0.706625	−	0.707588i	$-0.749783\pi$
$347$	−26.3259	−1.41325	−0.706625	+	0.707588i	$0.749783\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	0.709141	0.0378511
$352$	0	0
$353$	3.16756	0.168592	0.0842960	−	0.996441i	$-0.473136\pi$
$353$	3.16756	0.168592	0.0842960	+	0.996441i	$0.473136\pi$
$354$	0	0
$355$	19.8648	1.05432
$356$	0	0
$357$	−0.199340	−0.0105502
$358$	0	0
$359$	25.1165	1.32560	0.662799	−	0.748797i	$-0.269368\pi$
$359$	25.1165	1.32560	0.662799	+	0.748797i	$0.269368\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	3.78787	0.198811
$364$	0	0
$365$	9.18210	0.480613
$366$	0	0
$367$	−20.8161	−1.08659	−0.543297	−	0.839541i	$-0.682824\pi$
$367$	−20.8161	−1.08659	−0.543297	+	0.839541i	$0.682824\pi$
$368$	0	0
$369$	−7.12836	−0.371087
$370$	0	0
$371$	2.20110	0.114275
$372$	0	0
$373$	−1.90673	−0.0987266	−0.0493633	−	0.998781i	$-0.515719\pi$
$373$	−1.90673	−0.0987266	−0.0493633	+	0.998781i	$0.515719\pi$
$374$	0	0
$375$	4.07192	0.210273
$376$	0	0
$377$	1.21894	0.0627786
$378$	0	0
$379$	23.7743	1.22120	0.610601	−	0.791939i	$-0.290928\pi$
$379$	23.7743	1.22120	0.610601	+	0.791939i	$0.290928\pi$
$380$	0	0
$381$	3.45512	0.177011
$382$	0	0
$383$	−1.99319	−0.101847	−0.0509237	−	0.998703i	$-0.516217\pi$
$383$	−1.99319	−0.101847	−0.0509237	+	0.998703i	$0.516217\pi$
$384$	0	0
$385$	−0.142903	−0.00728303
$386$	0	0
$387$	3.34730	0.170153
$388$	0	0
$389$	1.38144	0.0700420	0.0350210	−	0.999387i	$-0.488850\pi$
$389$	1.38144	0.0700420	0.0350210	+	0.999387i	$0.488850\pi$
$390$	0	0
$391$	5.33275	0.269689
$392$	0	0
$393$	−3.61762	−0.182485
$394$	0	0
$395$	13.8530	0.697019
$396$	0	0
$397$	−16.0077	−0.803405	−0.401703	−	0.915770i	$-0.631581\pi$
$397$	−16.0077	−0.803405	−0.401703	+	0.915770i	$0.631581\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.15982	0.407482	0.203741	−	0.979025i	$-0.434690\pi$
$401$	8.15982	0.407482	0.203741	+	0.979025i	$0.434690\pi$
$402$	0	0
$403$	2.02229	0.100737
$404$	0	0
$405$	12.1480	0.603637
$406$	0	0
$407$	−2.48246	−0.123051
$408$	0	0
$409$	−20.1753	−0.997604	−0.498802	−	0.866716i	$-0.666227\pi$
$409$	−20.1753	−0.997604	−0.498802	+	0.866716i	$0.666227\pi$
$410$	0	0
$411$	6.41384	0.316371
$412$	0	0
$413$	3.48070	0.171274
$414$	0	0
$415$	−25.3746	−1.24559
$416$	0	0
$417$	5.91859	0.289834
$418$	0	0
$419$	−10.7784	−0.526558	−0.263279	−	0.964720i	$-0.584804\pi$
$419$	−10.7784	−0.526558	−0.263279	+	0.964720i	$0.584804\pi$
$420$	0	0
$421$	−20.9837	−1.02268	−0.511341	−	0.859378i	$-0.670851\pi$
$421$	−20.9837	−1.02268	−0.511341	+	0.859378i	$0.670851\pi$
$422$	0	0
$423$	17.9932	0.874859
$424$	0	0
$425$	4.98545	0.241830
$426$	0	0
$427$	3.47390	0.168114
$428$	0	0
$429$	−0.0368366	−0.00177849
$430$	0	0
$431$	−18.2053	−0.876920	−0.438460	−	0.898751i	$-0.644476\pi$
$431$	−18.2053	−0.876920	−0.438460	+	0.898751i	$0.644476\pi$
$432$	0	0
$433$	35.6023	1.71094	0.855468	−	0.517856i	$-0.173270\pi$
$433$	35.6023	1.71094	0.855468	+	0.517856i	$0.173270\pi$
$434$	0	0
$435$	−1.86753	−0.0895410
$436$	0	0
$437$	0	0
$438$	0	0
$439$	12.7706	0.609509	0.304754	−	0.952431i	$-0.401426\pi$
$439$	12.7706	0.609509	0.304754	+	0.952431i	$0.401426\pi$
$440$	0	0
$441$	19.8871	0.947006
$442$	0	0
$443$	25.5107	1.21205	0.606026	−	0.795445i	$-0.292763\pi$
$443$	25.5107	1.21205	0.606026	+	0.795445i	$0.292763\pi$
$444$	0	0
$445$	−1.59627	−0.0756703
$446$	0	0
$447$	−5.50980	−0.260605
$448$	0	0
$449$	−30.4783	−1.43836	−0.719181	−	0.694823i	$-0.755483\pi$
$449$	−30.4783	−1.43836	−0.719181	+	0.694823i	$0.755483\pi$
$450$	0	0
$451$	0.756082	0.0356025
$452$	0	0
$453$	5.55674	0.261078
$454$	0	0
$455$	−0.162504	−0.00761830
$456$	0	0
$457$	17.2608	0.807428	0.403714	−	0.914885i	$-0.367719\pi$
$457$	17.2608	0.807428	0.403714	+	0.914885i	$0.367719\pi$
$458$	0	0
$459$	−3.83750	−0.179119
$460$	0	0
$461$	30.9522	1.44159	0.720795	−	0.693149i	$-0.243777\pi$
$461$	30.9522	1.44159	0.720795	+	0.693149i	$0.243777\pi$
$462$	0	0
$463$	30.2472	1.40571	0.702854	−	0.711334i	$-0.251909\pi$
$463$	30.2472	1.40571	0.702854	+	0.711334i	$0.251909\pi$
$464$	0	0
$465$	−3.09833	−0.143681
$466$	0	0
$467$	22.1771	1.02623	0.513116	−	0.858319i	$-0.328491\pi$
$467$	22.1771	1.02623	0.513116	+	0.858319i	$0.328491\pi$
$468$	0	0
$469$	−0.110510	−0.00510287
$470$	0	0
$471$	−4.04963	−0.186597
$472$	0	0
$473$	−0.355037	−0.0163246
$474$	0	0
$475$	0	0
$476$	0	0
$477$	20.7520	0.950167
$478$	0	0
$479$	19.4584	0.889078	0.444539	−	0.895760i	$-0.353368\pi$
$479$	19.4584	0.889078	0.444539	+	0.895760i	$0.353368\pi$
$480$	0	0
$481$	−2.82295	−0.128715
$482$	0	0
$483$	−0.300964	−0.0136943
$484$	0	0
$485$	−14.3969	−0.653731
$486$	0	0
$487$	−30.2864	−1.37241	−0.686204	−	0.727409i	$-0.740724\pi$
$487$	−30.2864	−1.37241	−0.686204	+	0.727409i	$0.740724\pi$
$488$	0	0
$489$	−5.58441	−0.252536
$490$	0	0
$491$	26.5202	1.19684	0.598421	−	0.801182i	$-0.295795\pi$
$491$	26.5202	1.19684	0.598421	+	0.801182i	$0.295795\pi$
$492$	0	0
$493$	−6.59627	−0.297081
$494$	0	0
$495$	−1.34730	−0.0605565
$496$	0	0
$497$	−3.95987	−0.177624
$498$	0	0
$499$	−22.3259	−0.999446	−0.499723	−	0.866185i	$-0.666565\pi$
$499$	−22.3259	−0.999446	−0.499723	+	0.866185i	$0.666565\pi$
$500$	0	0
$501$	2.03952	0.0911193
$502$	0	0
$503$	15.6536	0.697961	0.348981	−	0.937130i	$-0.386528\pi$
$503$	15.6536	0.697961	0.348981	+	0.937130i	$0.386528\pi$
$504$	0	0
$505$	3.79292	0.168783
$506$	0	0
$507$	4.47296	0.198651
$508$	0	0
$509$	3.91716	0.173625	0.0868124	−	0.996225i	$-0.472332\pi$
$509$	3.91716	0.173625	0.0868124	+	0.996225i	$0.472332\pi$
$510$	0	0
$511$	−1.83036	−0.0809706
$512$	0	0
$513$	0	0
$514$	0	0
$515$	25.6313	1.12945
$516$	0	0
$517$	−1.90848	−0.0839349
$518$	0	0
$519$	2.69190	0.118161
$520$	0	0
$521$	−20.5175	−0.898890	−0.449445	−	0.893308i	$-0.648378\pi$
$521$	−20.5175	−0.898890	−0.449445	+	0.893308i	$0.648378\pi$
$522$	0	0
$523$	−4.85978	−0.212504	−0.106252	−	0.994339i	$-0.533885\pi$
$523$	−4.85978	−0.212504	−0.106252	+	0.994339i	$0.533885\pi$
$524$	0	0
$525$	−0.281364	−0.0122797
$526$	0	0
$527$	−10.9436	−0.476709
$528$	0	0
$529$	−14.9486	−0.649940
$530$	0	0
$531$	32.8161	1.42410
$532$	0	0
$533$	0.859785	0.0372414
$534$	0	0
$535$	11.0496	0.477717
$536$	0	0
$537$	−5.49525	−0.237138
$538$	0	0
$539$	−2.10936	−0.0908568
$540$	0	0
$541$	34.0624	1.46446	0.732229	−	0.681059i	$-0.238480\pi$
$541$	34.0624	1.46446	0.732229	+	0.681059i	$0.238480\pi$
$542$	0	0
$543$	1.30272	0.0559050
$544$	0	0
$545$	−9.90941	−0.424473
$546$	0	0
$547$	40.2285	1.72005	0.860024	−	0.510253i	$-0.170448\pi$
$547$	40.2285	1.72005	0.860024	+	0.510253i	$0.170448\pi$
$548$	0	0
$549$	32.7520	1.39782
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−2.76146	−0.117429
$554$	0	0
$555$	4.32501	0.183586
$556$	0	0
$557$	19.9540	0.845478	0.422739	−	0.906252i	$-0.361069\pi$
$557$	19.9540	0.845478	0.422739	+	0.906252i	$0.361069\pi$
$558$	0	0
$559$	−0.403733	−0.0170761
$560$	0	0
$561$	0.199340	0.00841616
$562$	0	0
$563$	14.3054	0.602901	0.301451	−	0.953482i	$-0.402529\pi$
$563$	14.3054	0.602901	0.301451	+	0.953482i	$0.402529\pi$
$564$	0	0
$565$	−20.7101	−0.871279
$566$	0	0
$567$	−2.42158	−0.101697
$568$	0	0
$569$	20.3851	0.854586	0.427293	−	0.904113i	$-0.359467\pi$
$569$	20.3851	0.854586	0.427293	+	0.904113i	$0.359467\pi$
$570$	0	0
$571$	−32.9377	−1.37840	−0.689200	−	0.724571i	$-0.742038\pi$
$571$	−32.9377	−1.37840	−0.689200	+	0.724571i	$0.742038\pi$
$572$	0	0
$573$	−1.64590	−0.0687583
$574$	0	0
$575$	7.52704	0.313899
$576$	0	0
$577$	44.1052	1.83613	0.918063	−	0.396435i	$-0.129753\pi$
$577$	44.1052	1.83613	0.918063	+	0.396435i	$0.129753\pi$
$578$	0	0
$579$	3.45512	0.143590
$580$	0	0
$581$	5.05819	0.209849
$582$	0	0
$583$	−2.20110	−0.0911600
$584$	0	0
$585$	−1.53209	−0.0633441
$586$	0	0
$587$	6.50568	0.268518	0.134259	−	0.990946i	$-0.457135\pi$
$587$	6.50568	0.268518	0.134259	+	0.990946i	$0.457135\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	4.77156	0.196276
$592$	0	0
$593$	12.7929	0.525342	0.262671	−	0.964885i	$-0.415397\pi$
$593$	12.7929	0.525342	0.262671	+	0.964885i	$0.415397\pi$
$594$	0	0
$595$	0.879385	0.0360513
$596$	0	0
$597$	5.26352	0.215421
$598$	0	0
$599$	18.2645	0.746265	0.373133	−	0.927778i	$-0.378284\pi$
$599$	18.2645	0.746265	0.373133	+	0.927778i	$0.378284\pi$
$600$	0	0
$601$	19.9067	0.812012	0.406006	−	0.913870i	$-0.366921\pi$
$601$	19.9067	0.812012	0.406006	+	0.913870i	$0.366921\pi$
$602$	0	0
$603$	−1.04189	−0.0424290
$604$	0	0
$605$	−16.7101	−0.679361
$606$	0	0
$607$	−22.5526	−0.915383	−0.457691	−	0.889111i	$-0.651324\pi$
$607$	−22.5526	−0.915383	−0.457691	+	0.889111i	$0.651324\pi$
$608$	0	0
$609$	0.372273	0.0150853
$610$	0	0
$611$	−2.17024	−0.0877987
$612$	0	0
$613$	−44.7948	−1.80924	−0.904622	−	0.426214i	$-0.859847\pi$
$613$	−44.7948	−1.80924	−0.904622	+	0.426214i	$0.859847\pi$
$614$	0	0
$615$	−1.31727	−0.0531173
$616$	0	0
$617$	25.2449	1.01632	0.508160	−	0.861263i	$-0.330326\pi$
$617$	25.2449	1.01632	0.508160	+	0.861263i	$0.330326\pi$
$618$	0	0
$619$	35.1147	1.41138	0.705690	−	0.708520i	$-0.250637\pi$
$619$	35.1147	1.41138	0.705690	+	0.708520i	$0.250637\pi$
$620$	0	0
$621$	−5.79385	−0.232499
$622$	0	0
$623$	0.318201	0.0127484
$624$	0	0
$625$	−4.69965	−0.187986
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15.2763	0.609106
$630$	0	0
$631$	33.8435	1.34729	0.673644	−	0.739056i	$-0.264728\pi$
$631$	33.8435	1.34729	0.673644	+	0.739056i	$0.264728\pi$
$632$	0	0
$633$	4.21482	0.167524
$634$	0	0
$635$	−15.2422	−0.604867
$636$	0	0
$637$	−2.39868	−0.0950392
$638$	0	0
$639$	−37.3337	−1.47690
$640$	0	0
$641$	23.1198	0.913177	0.456588	−	0.889678i	$-0.349071\pi$
$641$	23.1198	0.913177	0.456588	+	0.889678i	$0.349071\pi$
$642$	0	0
$643$	15.0811	0.594740	0.297370	−	0.954762i	$-0.403890\pi$
$643$	15.0811	0.594740	0.297370	+	0.954762i	$0.403890\pi$
$644$	0	0
$645$	0.618555	0.0243556
$646$	0	0
$647$	25.4783	1.00166	0.500828	−	0.865547i	$-0.333029\pi$
$647$	25.4783	1.00166	0.500828	+	0.865547i	$0.333029\pi$
$648$	0	0
$649$	−3.48070	−0.136630
$650$	0	0
$651$	0.617622	0.0242065
$652$	0	0
$653$	34.0351	1.33190	0.665948	−	0.745998i	$-0.268027\pi$
$653$	34.0351	1.33190	0.665948	+	0.745998i	$0.268027\pi$
$654$	0	0
$655$	15.9590	0.623571
$656$	0	0
$657$	−17.2567	−0.673248
$658$	0	0
$659$	25.2540	0.983757	0.491879	−	0.870664i	$-0.336310\pi$
$659$	25.2540	0.983757	0.491879	+	0.870664i	$0.336310\pi$
$660$	0	0
$661$	−3.85710	−0.150024	−0.0750118	−	0.997183i	$-0.523899\pi$
$661$	−3.85710	−0.150024	−0.0750118	+	0.997183i	$0.523899\pi$
$662$	0	0
$663$	0.226682	0.00880358
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.95904	−0.385616
$668$	0	0
$669$	−8.33544	−0.322267
$670$	0	0
$671$	−3.47390	−0.134108
$672$	0	0
$673$	−5.25671	−0.202631	−0.101316	−	0.994854i	$-0.532305\pi$
$673$	−5.25671	−0.202631	−0.101316	+	0.994854i	$0.532305\pi$
$674$	0	0
$675$	−5.41653	−0.208482
$676$	0	0
$677$	−41.1985	−1.58339	−0.791694	−	0.610918i	$-0.790800\pi$
$677$	−41.1985	−1.58339	−0.791694	+	0.610918i	$0.790800\pi$
$678$	0	0
$679$	2.86989	0.110136
$680$	0	0
$681$	−5.33099	−0.204284
$682$	0	0
$683$	45.6459	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$683$	45.6459	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$684$	0	0
$685$	−28.2945	−1.08108
$686$	0	0
$687$	−2.39868	−0.0915154
$688$	0	0
$689$	−2.50299	−0.0953564
$690$	0	0
$691$	14.9864	0.570109	0.285054	−	0.958511i	$-0.407988\pi$
$691$	14.9864	0.570109	0.285054	+	0.958511i	$0.407988\pi$
$692$	0	0
$693$	0.268571	0.0102022
$694$	0	0
$695$	−26.1097	−0.990397
$696$	0	0
$697$	−4.65270	−0.176234
$698$	0	0
$699$	−0.532089	−0.0201255
$700$	0	0
$701$	3.34224	0.126235	0.0631174	−	0.998006i	$-0.479896\pi$
$701$	3.34224	0.126235	0.0631174	+	0.998006i	$0.479896\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	3.32501	0.125227
$706$	0	0
$707$	−0.756082	−0.0284354
$708$	0	0
$709$	−18.7769	−0.705183	−0.352591	−	0.935777i	$-0.614699\pi$
$709$	−18.7769	−0.705183	−0.352591	+	0.935777i	$0.614699\pi$
$710$	0	0
$711$	−26.0351	−0.976392
$712$	0	0
$713$	−16.5226	−0.618776
$714$	0	0
$715$	0.162504	0.00607730
$716$	0	0
$717$	7.15476	0.267200
$718$	0	0
$719$	−31.6973	−1.18211	−0.591055	−	0.806632i	$-0.701288\pi$
$719$	−31.6973	−1.18211	−0.591055	+	0.806632i	$0.701288\pi$
$720$	0	0
$721$	−5.10936	−0.190283
$722$	0	0
$723$	−6.67324	−0.248180
$724$	0	0
$725$	−9.31046	−0.345782
$726$	0	0
$727$	−20.2558	−0.751245	−0.375623	−	0.926773i	$-0.622571\pi$
$727$	−20.2558	−0.751245	−0.375623	+	0.926773i	$0.622571\pi$
$728$	0	0
$729$	−20.7033	−0.766788
$730$	0	0
$731$	2.18479	0.0808075
$732$	0	0
$733$	0.813453	0.0300456	0.0150228	−	0.999887i	$-0.495218\pi$
$733$	0.813453	0.0300456	0.0150228	+	0.999887i	$0.495218\pi$
$734$	0	0
$735$	3.67499	0.135554
$736$	0	0
$737$	0.110510	0.00407068
$738$	0	0
$739$	−39.9685	−1.47027	−0.735133	−	0.677923i	$-0.762880\pi$
$739$	−39.9685	−1.47027	−0.735133	+	0.677923i	$0.762880\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.89487	0.289635	0.144817	−	0.989458i	$-0.453741\pi$
$743$	7.89487	0.289635	0.144817	+	0.989458i	$0.453741\pi$
$744$	0	0
$745$	24.3063	0.890515
$746$	0	0
$747$	47.6887	1.74484
$748$	0	0
$749$	−2.20264	−0.0804826
$750$	0	0
$751$	−20.6031	−0.751817	−0.375908	−	0.926657i	$-0.622669\pi$
$751$	−20.6031	−0.751817	−0.375908	+	0.926657i	$0.622669\pi$
$752$	0	0
$753$	−4.66044	−0.169836
$754$	0	0
$755$	−24.5134	−0.892135
$756$	0	0
$757$	−26.9445	−0.979314	−0.489657	−	0.871915i	$-0.662878\pi$
$757$	−26.9445	−0.979314	−0.489657	+	0.871915i	$0.662878\pi$
$758$	0	0
$759$	0.300964	0.0109243
$760$	0	0
$761$	7.38919	0.267858	0.133929	−	0.990991i	$-0.457241\pi$
$761$	7.38919	0.267858	0.133929	+	0.990991i	$0.457241\pi$
$762$	0	0
$763$	1.97535	0.0715124
$764$	0	0
$765$	8.29086	0.299757
$766$	0	0
$767$	−3.95811	−0.142919
$768$	0	0
$769$	−15.7510	−0.567997	−0.283998	−	0.958825i	$-0.591661\pi$
$769$	−15.7510	−0.567997	−0.283998	+	0.958825i	$0.591661\pi$
$770$	0	0
$771$	−6.69728	−0.241197
$772$	0	0
$773$	−1.78106	−0.0640602	−0.0320301	−	0.999487i	$-0.510197\pi$
$773$	−1.78106	−0.0640602	−0.0320301	+	0.999487i	$0.510197\pi$
$774$	0	0
$775$	−15.4466	−0.554857
$776$	0	0
$777$	−0.862149	−0.0309294
$778$	0	0
$779$	0	0
$780$	0	0
$781$	3.95987	0.141695
$782$	0	0
$783$	7.16662	0.256114
$784$	0	0
$785$	17.8648	0.637623
$786$	0	0
$787$	−28.7689	−1.02550	−0.512750	−	0.858538i	$-0.671373\pi$
$787$	−28.7689	−1.02550	−0.512750	+	0.858538i	$0.671373\pi$
$788$	0	0
$789$	9.59896	0.341732
$790$	0	0
$791$	4.12836	0.146787
$792$	0	0
$793$	−3.95037	−0.140282
$794$	0	0
$795$	3.83481	0.136007
$796$	0	0
$797$	44.4944	1.57607	0.788037	−	0.615628i	$-0.211098\pi$
$797$	44.4944	1.57607	0.788037	+	0.615628i	$0.211098\pi$
$798$	0	0
$799$	11.7442	0.415481
$800$	0	0
$801$	3.00000	0.106000
$802$	0	0
$803$	1.83036	0.0645922
$804$	0	0
$805$	1.32770	0.0467951
$806$	0	0
$807$	−2.67324	−0.0941024
$808$	0	0
$809$	34.2336	1.20359	0.601795	−	0.798651i	$-0.294453\pi$
$809$	34.2336	1.20359	0.601795	+	0.798651i	$0.294453\pi$
$810$	0	0
$811$	45.7093	1.60507	0.802534	−	0.596606i	$-0.203484\pi$
$811$	45.7093	1.60507	0.802534	+	0.596606i	$0.203484\pi$
$812$	0	0
$813$	1.71452	0.0601307
$814$	0	0
$815$	24.6355	0.862943
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0.305407	0.0106718
$820$	0	0
$821$	44.5090	1.55337	0.776687	−	0.629887i	$-0.216899\pi$
$821$	44.5090	1.55337	0.776687	+	0.629887i	$0.216899\pi$
$822$	0	0
$823$	−47.4579	−1.65428	−0.827140	−	0.561997i	$-0.810033\pi$
$823$	−47.4579	−1.65428	−0.827140	+	0.561997i	$0.810033\pi$
$824$	0	0
$825$	0.281364	0.00979583
$826$	0	0
$827$	38.9454	1.35427	0.677133	−	0.735861i	$-0.263222\pi$
$827$	38.9454	1.35427	0.677133	+	0.735861i	$0.263222\pi$
$828$	0	0
$829$	19.7392	0.685570	0.342785	−	0.939414i	$-0.388630\pi$
$829$	19.7392	0.685570	0.342785	+	0.939414i	$0.388630\pi$
$830$	0	0
$831$	−0.617622	−0.0214251
$832$	0	0
$833$	12.9804	0.449744
$834$	0	0
$835$	−8.99731	−0.311365
$836$	0	0
$837$	11.8898	0.410972
$838$	0	0
$839$	−51.2458	−1.76920	−0.884600	−	0.466350i	$-0.845569\pi$
$839$	−51.2458	−1.76920	−0.884600	+	0.466350i	$0.845569\pi$
$840$	0	0
$841$	−16.6813	−0.575217
$842$	0	0
$843$	7.30035	0.251438
$844$	0	0
$845$	−19.7324	−0.678814
$846$	0	0
$847$	3.33099	0.114454
$848$	0	0
$849$	1.82976	0.0627970
$850$	0	0
$851$	23.0642	0.790630
$852$	0	0
$853$	3.11650	0.106707	0.0533534	−	0.998576i	$-0.483009\pi$
$853$	3.11650	0.106707	0.0533534	+	0.998576i	$0.483009\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.8206	0.574580	0.287290	−	0.957844i	$-0.407246\pi$
$857$	16.8206	0.574580	0.287290	+	0.957844i	$0.407246\pi$
$858$	0	0
$859$	−40.1685	−1.37053	−0.685266	−	0.728293i	$-0.740314\pi$
$859$	−40.1685	−1.37053	−0.685266	+	0.728293i	$0.740314\pi$
$860$	0	0
$861$	0.262585	0.00894886
$862$	0	0
$863$	−1.48784	−0.0506465	−0.0253233	−	0.999679i	$-0.508062\pi$
$863$	−1.48784	−0.0506465	−0.0253233	+	0.999679i	$0.508062\pi$
$864$	0	0
$865$	−11.8753	−0.403771
$866$	0	0
$867$	4.67736	0.158851
$868$	0	0
$869$	2.76146	0.0936761
$870$	0	0
$871$	0.125667	0.00425807
$872$	0	0
$873$	27.0574	0.915753
$874$	0	0
$875$	3.58079	0.121053
$876$	0	0
$877$	14.1607	0.478175	0.239087	−	0.970998i	$-0.423152\pi$
$877$	14.1607	0.478175	0.239087	+	0.970998i	$0.423152\pi$
$878$	0	0
$879$	9.25402	0.312130
$880$	0	0
$881$	17.7000	0.596327	0.298164	−	0.954515i	$-0.403626\pi$
$881$	17.7000	0.596327	0.298164	+	0.954515i	$0.403626\pi$
$882$	0	0
$883$	16.4712	0.554300	0.277150	−	0.960827i	$-0.410610\pi$
$883$	16.4712	0.554300	0.277150	+	0.960827i	$0.410610\pi$
$884$	0	0
$885$	6.06418	0.203845
$886$	0	0
$887$	−49.1513	−1.65034	−0.825169	−	0.564886i	$-0.808920\pi$
$887$	−49.1513	−1.65034	−0.825169	+	0.564886i	$0.808920\pi$
$888$	0	0
$889$	3.03838	0.101904
$890$	0	0
$891$	2.42158	0.0811259
$892$	0	0
$893$	0	0
$894$	0	0
$895$	24.2422	0.810326
$896$	0	0
$897$	0.342244	0.0114272
$898$	0	0
$899$	20.4374	0.681625
$900$	0	0
$901$	13.5449	0.451245
$902$	0	0
$903$	−0.123303	−0.00410327
$904$	0	0
$905$	−5.74691	−0.191034
$906$	0	0
$907$	−25.6177	−0.850623	−0.425311	−	0.905047i	$-0.639835\pi$
$907$	−25.6177	−0.850623	−0.425311	+	0.905047i	$0.639835\pi$
$908$	0	0
$909$	−7.12836	−0.236433
$910$	0	0
$911$	−39.3809	−1.30475	−0.652375	−	0.757897i	$-0.726227\pi$
$911$	−39.3809	−1.30475	−0.652375	+	0.757897i	$0.726227\pi$
$912$	0	0
$913$	−5.05819	−0.167402
$914$	0	0
$915$	6.05232	0.200083
$916$	0	0
$917$	−3.18128	−0.105055
$918$	0	0
$919$	43.2121	1.42544	0.712718	−	0.701450i	$-0.247464\pi$
$919$	43.2121	1.42544	0.712718	+	0.701450i	$0.247464\pi$
$920$	0	0
$921$	0.770630	0.0253931
$922$	0	0
$923$	4.50299	0.148218
$924$	0	0
$925$	21.5621	0.708958
$926$	0	0
$927$	−48.1712	−1.58215
$928$	0	0
$929$	14.9691	0.491122	0.245561	−	0.969381i	$-0.421028\pi$
$929$	14.9691	0.491122	0.245561	+	0.969381i	$0.421028\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5.45067	−0.178447
$934$	0	0
$935$	−0.879385	−0.0287590
$936$	0	0
$937$	−52.1661	−1.70419	−0.852097	−	0.523385i	$-0.824669\pi$
$937$	−52.1661	−1.70419	−0.852097	+	0.523385i	$0.824669\pi$
$938$	0	0
$939$	−2.86484	−0.0934904
$940$	0	0
$941$	−7.04282	−0.229589	−0.114795	−	0.993389i	$-0.536621\pi$
$941$	−7.04282	−0.229589	−0.114795	+	0.993389i	$0.536621\pi$
$942$	0	0
$943$	−7.02465	−0.228754
$944$	0	0
$945$	−0.955423	−0.0310799
$946$	0	0
$947$	15.8785	0.515980	0.257990	−	0.966148i	$-0.416940\pi$
$947$	15.8785	0.515980	0.257990	+	0.966148i	$0.416940\pi$
$948$	0	0
$949$	2.08141	0.0675656
$950$	0	0
$951$	−2.08141	−0.0674945
$952$	0	0
$953$	7.77568	0.251879	0.125940	−	0.992038i	$-0.459805\pi$
$953$	7.77568	0.251879	0.125940	+	0.992038i	$0.459805\pi$
$954$	0	0
$955$	7.26083	0.234955
$956$	0	0
$957$	−0.372273	−0.0120339
$958$	0	0
$959$	5.64023	0.182133
$960$	0	0
$961$	2.90673	0.0937654
$962$	0	0
$963$	−20.7665	−0.669191
$964$	0	0
$965$	−15.2422	−0.490663
$966$	0	0
$967$	−47.5645	−1.52957	−0.764785	−	0.644285i	$-0.777155\pi$
$967$	−47.5645	−1.52957	−0.764785	+	0.644285i	$0.777155\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.76651	0.185056	0.0925281	−	0.995710i	$-0.470505\pi$
$971$	5.76651	0.185056	0.0925281	+	0.995710i	$0.470505\pi$
$972$	0	0
$973$	5.20472	0.166856
$974$	0	0
$975$	0.319955	0.0102468
$976$	0	0
$977$	19.5716	0.626151	0.313076	−	0.949728i	$-0.398641\pi$
$977$	19.5716	0.626151	0.313076	+	0.949728i	$0.398641\pi$
$978$	0	0
$979$	−0.318201	−0.0101697
$980$	0	0
$981$	18.6236	0.594606
$982$	0	0
$983$	−39.3800	−1.25603	−0.628014	−	0.778202i	$-0.716132\pi$
$983$	−39.3800	−1.25603	−0.628014	+	0.778202i	$0.716132\pi$
$984$	0	0
$985$	−21.0496	−0.670697
$986$	0	0
$987$	−0.662809	−0.0210974
$988$	0	0
$989$	3.29860	0.104889
$990$	0	0
$991$	−32.1999	−1.02287	−0.511433	−	0.859323i	$-0.670885\pi$
$991$	−32.1999	−1.02287	−0.511433	+	0.859323i	$0.670885\pi$
$992$	0	0
$993$	−1.72100	−0.0546143
$994$	0	0
$995$	−23.2199	−0.736120
$996$	0	0
$997$	−12.7159	−0.402718	−0.201359	−	0.979517i	$-0.564536\pi$
$997$	−12.7159	−0.402718	−0.201359	+	0.979517i	$0.564536\pi$
$998$	0	0
$999$	−16.5972	−0.525112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.q.1.2		3
4.3	odd	2		5776.2.a.bl.1.2		3
19.9	even	9		152.2.q.a.81.1	✓	6
19.17	even	9		152.2.q.a.137.1	yes	6
19.18	odd	2		2888.2.a.p.1.2		3
76.47	odd	18		304.2.u.d.81.1		6
76.55	odd	18		304.2.u.d.289.1		6
76.75	even	2		5776.2.a.bm.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.a.81.1	✓	6	19.9	even	9
152.2.q.a.137.1	yes	6	19.17	even	9
304.2.u.d.81.1		6	76.47	odd	18
304.2.u.d.289.1		6	76.55	odd	18
2888.2.a.p.1.2		3	19.18	odd	2
2888.2.a.q.1.2		3	1.1	even	1	trivial
5776.2.a.bl.1.2		3	4.3	odd	2
5776.2.a.bm.1.2		3	76.75	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q-0.347296 q^{3} +1.53209 q^{5} -0.305407 q^{7} -2.87939 q^{9} +0.305407 q^{11} +0.347296 q^{13} -0.532089 q^{15} -1.87939 q^{17} +0.106067 q^{21} -2.83750 q^{23} -2.65270 q^{25} +2.04189 q^{27} +3.50980 q^{29} +5.82295 q^{31} -0.106067 q^{33} -0.467911 q^{35} -8.12836 q^{37} -0.120615 q^{39} +2.47565 q^{41} -1.16250 q^{43} -4.41147 q^{45} -6.24897 q^{47} -6.90673 q^{49} +0.652704 q^{51} -7.20708 q^{53} +0.467911 q^{55} -11.3969 q^{59} -11.3746 q^{61} +0.879385 q^{63} +0.532089 q^{65} +0.361844 q^{67} +0.985452 q^{69} +12.9659 q^{71} +5.99319 q^{73} +0.921274 q^{75} -0.0932736 q^{77} +9.04189 q^{79} +7.92902 q^{81} -16.5621 q^{83} -2.87939 q^{85} -1.21894 q^{87} -1.04189 q^{89} -0.106067 q^{91} -2.02229 q^{93} -9.39693 q^{97} -0.879385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{7} - 3 q^{9} + 3 q^{11} + 3 q^{15} - 12 q^{21} - 6 q^{23} - 9 q^{25} + 3 q^{27} + 12 q^{29} - 3 q^{31} + 12 q^{33} - 6 q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 6 q^{43} - 3 q^{45} - 6 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^7 - 3 * q^9 + 3 * q^11 + 3 * q^15 - 12 * q^21 - 6 * q^23 - 9 * q^25 + 3 * q^27 + 12 * q^29 - 3 * q^31 + 12 * q^33 - 6 * q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 6 * q^43 - 3 * q^45 - 6 * q^47 + 6 * q^49 + 3 * q^51 - 12 * q^53 + 6 * q^55 - 6 * q^59 - 12 * q^61 - 3 * q^63 - 3 * q^65 + 18 * q^67 - 15 * q^69 + 18 * q^71 - 24 * q^73 - 6 * q^75 - 27 * q^77 + 24 * q^79 - 9 * q^81 - 15 * q^83 - 3 * q^85 - 21 * q^87 + 12 * q^91 + 3 * q^99

Introduction

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