LMFDB - Embedded newform 8048.2.a.u.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74581 q^{3} +1.36534 q^{5} +0.0430977 q^{7} +0.0478420 q^{9} +O(q^{10})$	$q-1.74581 q^{3} +1.36534 q^{5} +0.0430977 q^{7} +0.0478420 q^{9} +1.16028 q^{11} +6.53757 q^{13} -2.38363 q^{15} +5.57441 q^{17} -1.21935 q^{19} -0.0752402 q^{21} -0.469846 q^{23} -3.13583 q^{25} +5.15390 q^{27} +8.44786 q^{29} +3.81096 q^{31} -2.02562 q^{33} +0.0588432 q^{35} +3.80432 q^{37} -11.4133 q^{39} +1.03209 q^{41} -4.43596 q^{43} +0.0653208 q^{45} +9.22994 q^{47} -6.99814 q^{49} -9.73184 q^{51} -4.85375 q^{53} +1.58418 q^{55} +2.12875 q^{57} -6.32625 q^{59} +13.0539 q^{61} +0.00206188 q^{63} +8.92604 q^{65} +8.90136 q^{67} +0.820260 q^{69} +16.0137 q^{71} -9.01057 q^{73} +5.47456 q^{75} +0.0500052 q^{77} -9.40313 q^{79} -9.14124 q^{81} -5.75930 q^{83} +7.61099 q^{85} -14.7483 q^{87} +10.7440 q^{89} +0.281754 q^{91} -6.65319 q^{93} -1.66484 q^{95} -4.36039 q^{97} +0.0555100 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.74581	−1.00794	−0.503971	−	0.863721i	$-0.668128\pi$
$3$	−1.74581	−1.00794	−0.503971	+	0.863721i	$0.668128\pi$
$4$	0	0
$5$	1.36534	0.610601	0.305300	−	0.952256i	$-0.401243\pi$
$5$	1.36534	0.610601	0.305300	+	0.952256i	$0.401243\pi$
$6$	0	0
$7$	0.0430977	0.0162894	0.00814469	−	0.999967i	$-0.497407\pi$
$7$	0.0430977	0.0162894	0.00814469	+	0.999967i	$0.497407\pi$
$8$	0	0
$9$	0.0478420	0.0159473
$10$	0	0
$11$	1.16028	0.349837	0.174918	−	0.984583i	$-0.444034\pi$
$11$	1.16028	0.349837	0.174918	+	0.984583i	$0.444034\pi$
$12$	0	0
$13$	6.53757	1.81320	0.906598	−	0.421994i	$-0.138670\pi$
$13$	6.53757	1.81320	0.906598	+	0.421994i	$0.138670\pi$
$14$	0	0
$15$	−2.38363	−0.615450
$16$	0	0
$17$	5.57441	1.35199	0.675997	−	0.736905i	$-0.263713\pi$
$17$	5.57441	1.35199	0.675997	+	0.736905i	$0.263713\pi$
$18$	0	0
$19$	−1.21935	−0.279738	−0.139869	−	0.990170i	$-0.544668\pi$
$19$	−1.21935	−0.279738	−0.139869	+	0.990170i	$0.544668\pi$
$20$	0	0
$21$	−0.0752402	−0.0164188
$22$	0	0
$23$	−0.469846	−0.0979696	−0.0489848	−	0.998800i	$-0.515599\pi$
$23$	−0.469846	−0.0979696	−0.0489848	+	0.998800i	$0.515599\pi$
$24$	0	0
$25$	−3.13583	−0.627167
$26$	0	0
$27$	5.15390	0.991868
$28$	0	0
$29$	8.44786	1.56873	0.784364	−	0.620300i	$-0.212989\pi$
$29$	8.44786	1.56873	0.784364	+	0.620300i	$0.212989\pi$
$30$	0	0
$31$	3.81096	0.684468	0.342234	−	0.939615i	$-0.388816\pi$
$31$	3.81096	0.684468	0.342234	+	0.939615i	$0.388816\pi$
$32$	0	0
$33$	−2.02562	−0.352615
$34$	0	0
$35$	0.0588432	0.00994631
$36$	0	0
$37$	3.80432	0.625426	0.312713	−	0.949848i	$-0.398762\pi$
$37$	3.80432	0.625426	0.312713	+	0.949848i	$0.398762\pi$
$38$	0	0
$39$	−11.4133	−1.82760
$40$	0	0
$41$	1.03209	0.161185	0.0805926	−	0.996747i	$-0.474319\pi$
$41$	1.03209	0.161185	0.0805926	+	0.996747i	$0.474319\pi$
$42$	0	0
$43$	−4.43596	−0.676478	−0.338239	−	0.941060i	$-0.609831\pi$
$43$	−4.43596	−0.676478	−0.338239	+	0.941060i	$0.609831\pi$
$44$	0	0
$45$	0.0653208	0.00973745
$46$	0	0
$47$	9.22994	1.34633	0.673163	−	0.739494i	$-0.264935\pi$
$47$	9.22994	1.34633	0.673163	+	0.739494i	$0.264935\pi$
$48$	0	0
$49$	−6.99814	−0.999735
$50$	0	0
$51$	−9.73184	−1.36273
$52$	0	0
$53$	−4.85375	−0.666714	−0.333357	−	0.942801i	$-0.608181\pi$
$53$	−4.85375	−0.666714	−0.333357	+	0.942801i	$0.608181\pi$
$54$	0	0
$55$	1.58418	0.213611
$56$	0	0
$57$	2.12875	0.281960
$58$	0	0
$59$	−6.32625	−0.823607	−0.411804	−	0.911273i	$-0.635101\pi$
$59$	−6.32625	−0.823607	−0.411804	+	0.911273i	$0.635101\pi$
$60$	0	0
$61$	13.0539	1.67138	0.835690	−	0.549202i	$-0.185068\pi$
$61$	13.0539	1.67138	0.835690	+	0.549202i	$0.185068\pi$
$62$	0	0
$63$	0.00206188	0.000259772 0
$64$	0	0
$65$	8.92604	1.10714
$66$	0	0
$67$	8.90136	1.08747	0.543737	−	0.839256i	$-0.317009\pi$
$67$	8.90136	1.08747	0.543737	+	0.839256i	$0.317009\pi$
$68$	0	0
$69$	0.820260	0.0987477
$70$	0	0
$71$	16.0137	1.90047	0.950237	−	0.311529i	$-0.100841\pi$
$71$	16.0137	1.90047	0.950237	+	0.311529i	$0.100841\pi$
$72$	0	0
$73$	−9.01057	−1.05461	−0.527304	−	0.849677i	$-0.676797\pi$
$73$	−9.01057	−1.05461	−0.527304	+	0.849677i	$0.676797\pi$
$74$	0	0
$75$	5.47456	0.632148
$76$	0	0
$77$	0.0500052	0.00569862
$78$	0	0
$79$	−9.40313	−1.05794	−0.528968	−	0.848642i	$-0.677421\pi$
$79$	−9.40313	−1.05794	−0.528968	+	0.848642i	$0.677421\pi$
$80$	0	0
$81$	−9.14124	−1.01569
$82$	0	0
$83$	−5.75930	−0.632165	−0.316083	−	0.948732i	$-0.602368\pi$
$83$	−5.75930	−0.632165	−0.316083	+	0.948732i	$0.602368\pi$
$84$	0	0
$85$	7.61099	0.825528
$86$	0	0
$87$	−14.7483	−1.58119
$88$	0	0
$89$	10.7440	1.13887	0.569433	−	0.822038i	$-0.307163\pi$
$89$	10.7440	1.13887	0.569433	+	0.822038i	$0.307163\pi$
$90$	0	0
$91$	0.281754	0.0295359
$92$	0	0
$93$	−6.65319	−0.689904
$94$	0	0
$95$	−1.66484	−0.170809
$96$	0	0
$97$	−4.36039	−0.442731	−0.221365	−	0.975191i	$-0.571051\pi$
$97$	−4.36039	−0.442731	−0.221365	+	0.975191i	$0.571051\pi$
$98$	0	0
$99$	0.0555100	0.00557896
$100$	0	0
$101$	−9.83152	−0.978273	−0.489136	−	0.872207i	$-0.662688\pi$
$101$	−9.83152	−0.978273	−0.489136	+	0.872207i	$0.662688\pi$
$102$	0	0
$103$	−3.85766	−0.380107	−0.190053	−	0.981774i	$-0.560866\pi$
$103$	−3.85766	−0.380107	−0.190053	+	0.981774i	$0.560866\pi$
$104$	0	0
$105$	−0.102729	−0.0100253
$106$	0	0
$107$	4.35489	0.421003	0.210501	−	0.977594i	$-0.432490\pi$
$107$	4.35489	0.421003	0.210501	+	0.977594i	$0.432490\pi$
$108$	0	0
$109$	−9.27257	−0.888151	−0.444076	−	0.895989i	$-0.646468\pi$
$109$	−9.27257	−0.888151	−0.444076	+	0.895989i	$0.646468\pi$
$110$	0	0
$111$	−6.64161	−0.630393
$112$	0	0
$113$	−11.5688	−1.08830	−0.544151	−	0.838987i	$-0.683148\pi$
$113$	−11.5688	−1.08830	−0.544151	+	0.838987i	$0.683148\pi$
$114$	0	0
$115$	−0.641501	−0.0598203
$116$	0	0
$117$	0.312771	0.0289157
$118$	0	0
$119$	0.240244	0.0220231
$120$	0	0
$121$	−9.65376	−0.877614
$122$	0	0
$123$	−1.80183	−0.162465
$124$	0	0
$125$	−11.1082	−0.993549
$126$	0	0
$127$	6.48261	0.575238	0.287619	−	0.957745i	$-0.407136\pi$
$127$	6.48261	0.575238	0.287619	+	0.957745i	$0.407136\pi$
$128$	0	0
$129$	7.74433	0.681850
$130$	0	0
$131$	14.8537	1.29778	0.648888	−	0.760884i	$-0.275234\pi$
$131$	14.8537	1.29778	0.648888	+	0.760884i	$0.275234\pi$
$132$	0	0
$133$	−0.0525512	−0.00455677
$134$	0	0
$135$	7.03685	0.605635
$136$	0	0
$137$	−7.54147	−0.644312	−0.322156	−	0.946687i	$-0.604407\pi$
$137$	−7.54147	−0.644312	−0.322156	+	0.946687i	$0.604407\pi$
$138$	0	0
$139$	−0.849294	−0.0720362	−0.0360181	−	0.999351i	$-0.511467\pi$
$139$	−0.849294	−0.0720362	−0.0360181	+	0.999351i	$0.511467\pi$
$140$	0	0
$141$	−16.1137	−1.35702
$142$	0	0
$143$	7.58540	0.634323
$144$	0	0
$145$	11.5342	0.957867
$146$	0	0
$147$	12.2174	1.00767
$148$	0	0
$149$	−6.50261	−0.532714	−0.266357	−	0.963874i	$-0.585820\pi$
$149$	−6.50261	−0.532714	−0.266357	+	0.963874i	$0.585820\pi$
$150$	0	0
$151$	−11.4911	−0.935130	−0.467565	−	0.883959i	$-0.654869\pi$
$151$	−11.4911	−0.935130	−0.467565	+	0.883959i	$0.654869\pi$
$152$	0	0
$153$	0.266691	0.0215607
$154$	0	0
$155$	5.20327	0.417937
$156$	0	0
$157$	19.6416	1.56757	0.783786	−	0.621031i	$-0.213286\pi$
$157$	19.6416	1.56757	0.783786	+	0.621031i	$0.213286\pi$
$158$	0	0
$159$	8.47371	0.672009
$160$	0	0
$161$	−0.0202492	−0.00159586
$162$	0	0
$163$	−10.1847	−0.797729	−0.398865	−	0.917010i	$-0.630596\pi$
$163$	−10.1847	−0.797729	−0.398865	+	0.917010i	$0.630596\pi$
$164$	0	0
$165$	−2.76567	−0.215307
$166$	0	0
$167$	11.1947	0.866271	0.433136	−	0.901329i	$-0.357407\pi$
$167$	11.1947	0.866271	0.433136	+	0.901329i	$0.357407\pi$
$168$	0	0
$169$	29.7399	2.28768
$170$	0	0
$171$	−0.0583362	−0.00446108
$172$	0	0
$173$	3.70191	0.281451	0.140726	−	0.990049i	$-0.455056\pi$
$173$	3.70191	0.281451	0.140726	+	0.990049i	$0.455056\pi$
$174$	0	0
$175$	−0.135147	−0.0102162
$176$	0	0
$177$	11.0444	0.830149
$178$	0	0
$179$	12.5403	0.937304	0.468652	−	0.883383i	$-0.344740\pi$
$179$	12.5403	0.937304	0.468652	+	0.883383i	$0.344740\pi$
$180$	0	0
$181$	23.1494	1.72068	0.860340	−	0.509720i	$-0.170251\pi$
$181$	23.1494	1.72068	0.860340	+	0.509720i	$0.170251\pi$
$182$	0	0
$183$	−22.7896	−1.68465
$184$	0	0
$185$	5.19421	0.381886
$186$	0	0
$187$	6.46786	0.472977
$188$	0	0
$189$	0.222121	0.0161569
$190$	0	0
$191$	26.2971	1.90279	0.951397	−	0.307968i	$-0.0996491\pi$
$191$	26.2971	1.90279	0.951397	+	0.307968i	$0.0996491\pi$
$192$	0	0
$193$	−21.6664	−1.55958	−0.779791	−	0.626040i	$-0.784675\pi$
$193$	−21.6664	−1.55958	−0.779791	+	0.626040i	$0.784675\pi$
$194$	0	0
$195$	−15.5831	−1.11593
$196$	0	0
$197$	−21.3818	−1.52339	−0.761695	−	0.647936i	$-0.775632\pi$
$197$	−21.3818	−1.52339	−0.761695	+	0.647936i	$0.775632\pi$
$198$	0	0
$199$	−5.04042	−0.357306	−0.178653	−	0.983912i	$-0.557174\pi$
$199$	−5.04042	−0.357306	−0.178653	+	0.983912i	$0.557174\pi$
$200$	0	0
$201$	−15.5401	−1.09611
$202$	0	0
$203$	0.364083	0.0255536
$204$	0	0
$205$	1.40916	0.0984198
$206$	0	0
$207$	−0.0224784	−0.00156235
$208$	0	0
$209$	−1.41479	−0.0978628
$210$	0	0
$211$	−21.4818	−1.47887	−0.739435	−	0.673228i	$-0.764907\pi$
$211$	−21.4818	−1.47887	−0.739435	+	0.673228i	$0.764907\pi$
$212$	0	0
$213$	−27.9568	−1.91557
$214$	0	0
$215$	−6.05662	−0.413058
$216$	0	0
$217$	0.164243	0.0111496
$218$	0	0
$219$	15.7307	1.06298
$220$	0	0
$221$	36.4431	2.45143
$222$	0	0
$223$	5.86814	0.392959	0.196480	−	0.980508i	$-0.437049\pi$
$223$	5.86814	0.392959	0.196480	+	0.980508i	$0.437049\pi$
$224$	0	0
$225$	−0.150025	−0.0100016
$226$	0	0
$227$	−18.3502	−1.21795	−0.608974	−	0.793190i	$-0.708419\pi$
$227$	−18.3502	−1.21795	−0.608974	+	0.793190i	$0.708419\pi$
$228$	0	0
$229$	22.2147	1.46799	0.733993	−	0.679157i	$-0.237654\pi$
$229$	22.2147	1.46799	0.733993	+	0.679157i	$0.237654\pi$
$230$	0	0
$231$	−0.0872995	−0.00574388
$232$	0	0
$233$	8.02761	0.525906	0.262953	−	0.964809i	$-0.415304\pi$
$233$	8.02761	0.525906	0.262953	+	0.964809i	$0.415304\pi$
$234$	0	0
$235$	12.6021	0.822067
$236$	0	0
$237$	16.4161	1.06634
$238$	0	0
$239$	−3.41967	−0.221200	−0.110600	−	0.993865i	$-0.535277\pi$
$239$	−3.41967	−0.221200	−0.110600	+	0.993865i	$0.535277\pi$
$240$	0	0
$241$	11.5356	0.743075	0.371537	−	0.928418i	$-0.378831\pi$
$241$	11.5356	0.743075	0.371537	+	0.928418i	$0.378831\pi$
$242$	0	0
$243$	0.497141	0.0318916
$244$	0	0
$245$	−9.55488	−0.610439
$246$	0	0
$247$	−7.97160	−0.507221
$248$	0	0
$249$	10.0546	0.637186
$250$	0	0
$251$	−28.6268	−1.80691	−0.903453	−	0.428686i	$-0.858977\pi$
$251$	−28.6268	−1.80691	−0.903453	+	0.428686i	$0.858977\pi$
$252$	0	0
$253$	−0.545151	−0.0342734
$254$	0	0
$255$	−13.2873	−0.832084
$256$	0	0
$257$	−1.34056	−0.0836221	−0.0418110	−	0.999126i	$-0.513313\pi$
$257$	−1.34056	−0.0836221	−0.0418110	+	0.999126i	$0.513313\pi$
$258$	0	0
$259$	0.163957	0.0101878
$260$	0	0
$261$	0.404163	0.0250170
$262$	0	0
$263$	−3.81522	−0.235256	−0.117628	−	0.993058i	$-0.537529\pi$
$263$	−3.81522	−0.235256	−0.117628	+	0.993058i	$0.537529\pi$
$264$	0	0
$265$	−6.62704	−0.407096
$266$	0	0
$267$	−18.7570	−1.14791
$268$	0	0
$269$	−7.30135	−0.445171	−0.222586	−	0.974913i	$-0.571450\pi$
$269$	−7.30135	−0.445171	−0.222586	+	0.974913i	$0.571450\pi$
$270$	0	0
$271$	29.0222	1.76297	0.881486	−	0.472209i	$-0.156543\pi$
$271$	29.0222	1.76297	0.881486	+	0.472209i	$0.156543\pi$
$272$	0	0
$273$	−0.491888	−0.0297704
$274$	0	0
$275$	−3.63844	−0.219406
$276$	0	0
$277$	−4.01016	−0.240947	−0.120474	−	0.992717i	$-0.538441\pi$
$277$	−4.01016	−0.240947	−0.120474	+	0.992717i	$0.538441\pi$
$278$	0	0
$279$	0.182324	0.0109154
$280$	0	0
$281$	−3.12524	−0.186436	−0.0932182	−	0.995646i	$-0.529715\pi$
$281$	−3.12524	−0.186436	−0.0932182	+	0.995646i	$0.529715\pi$
$282$	0	0
$283$	−15.7536	−0.936457	−0.468229	−	0.883607i	$-0.655108\pi$
$283$	−15.7536	−0.936457	−0.468229	+	0.883607i	$0.655108\pi$
$284$	0	0
$285$	2.90648	0.172165
$286$	0	0
$287$	0.0444806	0.00262561
$288$	0	0
$289$	14.0741	0.827885
$290$	0	0
$291$	7.61240	0.446247
$292$	0	0
$293$	−8.35748	−0.488249	−0.244125	−	0.969744i	$-0.578501\pi$
$293$	−8.35748	−0.488249	−0.244125	+	0.969744i	$0.578501\pi$
$294$	0	0
$295$	−8.63751	−0.502895
$296$	0	0
$297$	5.97995	0.346992
$298$	0	0
$299$	−3.07165	−0.177638
$300$	0	0
$301$	−0.191180	−0.0110194
$302$	0	0
$303$	17.1639	0.986042
$304$	0	0
$305$	17.8231	1.02055
$306$	0	0
$307$	−26.8628	−1.53314	−0.766570	−	0.642161i	$-0.778038\pi$
$307$	−26.8628	−1.53314	−0.766570	+	0.642161i	$0.778038\pi$
$308$	0	0
$309$	6.73474	0.383126
$310$	0	0
$311$	19.9455	1.13101	0.565503	−	0.824747i	$-0.308682\pi$
$311$	19.9455	1.13101	0.565503	+	0.824747i	$0.308682\pi$
$312$	0	0
$313$	22.3377	1.26260	0.631300	−	0.775539i	$-0.282522\pi$
$313$	22.3377	1.26260	0.631300	+	0.775539i	$0.282522\pi$
$314$	0	0
$315$	0.00281518	0.000158617 0
$316$	0	0
$317$	2.52098	0.141593	0.0707963	−	0.997491i	$-0.477446\pi$
$317$	2.52098	0.141593	0.0707963	+	0.997491i	$0.477446\pi$
$318$	0	0
$319$	9.80186	0.548799
$320$	0	0
$321$	−7.60279	−0.424346
$322$	0	0
$323$	−6.79717	−0.378205
$324$	0	0
$325$	−20.5007	−1.13718
$326$	0	0
$327$	16.1881	0.895205
$328$	0	0
$329$	0.397789	0.0219308
$330$	0	0
$331$	−2.86380	−0.157408	−0.0787042	−	0.996898i	$-0.525078\pi$
$331$	−2.86380	−0.157408	−0.0787042	+	0.996898i	$0.525078\pi$
$332$	0	0
$333$	0.182006	0.00997388
$334$	0	0
$335$	12.1534	0.664013
$336$	0	0
$337$	−22.8158	−1.24286	−0.621428	−	0.783471i	$-0.713447\pi$
$337$	−22.8158	−1.24286	−0.621428	+	0.783471i	$0.713447\pi$
$338$	0	0
$339$	20.1969	1.09695
$340$	0	0
$341$	4.42176	0.239452
$342$	0	0
$343$	−0.603287	−0.0325745
$344$	0	0
$345$	1.11994	0.0602954
$346$	0	0
$347$	17.1397	0.920108	0.460054	−	0.887891i	$-0.347830\pi$
$347$	17.1397	0.920108	0.460054	+	0.887891i	$0.347830\pi$
$348$	0	0
$349$	22.4517	1.20181	0.600906	−	0.799320i	$-0.294807\pi$
$349$	22.4517	1.20181	0.600906	+	0.799320i	$0.294807\pi$
$350$	0	0
$351$	33.6940	1.79845
$352$	0	0
$353$	26.5202	1.41153	0.705765	−	0.708446i	$-0.250604\pi$
$353$	26.5202	1.41153	0.705765	+	0.708446i	$0.250604\pi$
$354$	0	0
$355$	21.8642	1.16043
$356$	0	0
$357$	−0.419420	−0.0221981
$358$	0	0
$359$	−6.78218	−0.357950	−0.178975	−	0.983854i	$-0.557278\pi$
$359$	−6.78218	−0.357950	−0.178975	+	0.983854i	$0.557278\pi$
$360$	0	0
$361$	−17.5132	−0.921746
$362$	0	0
$363$	16.8536	0.884584
$364$	0	0
$365$	−12.3025	−0.643944
$366$	0	0
$367$	30.7423	1.60473	0.802366	−	0.596832i	$-0.203574\pi$
$367$	30.7423	1.60473	0.802366	+	0.596832i	$0.203574\pi$
$368$	0	0
$369$	0.0493772	0.00257047
$370$	0	0
$371$	−0.209185	−0.0108604
$372$	0	0
$373$	17.2561	0.893487	0.446743	−	0.894662i	$-0.352584\pi$
$373$	17.2561	0.893487	0.446743	+	0.894662i	$0.352584\pi$
$374$	0	0
$375$	19.3928	1.00144
$376$	0	0
$377$	55.2285	2.84441
$378$	0	0
$379$	2.51056	0.128959	0.0644795	−	0.997919i	$-0.479461\pi$
$379$	2.51056	0.128959	0.0644795	+	0.997919i	$0.479461\pi$
$380$	0	0
$381$	−11.3174	−0.579807
$382$	0	0
$383$	19.4025	0.991419	0.495710	−	0.868488i	$-0.334908\pi$
$383$	19.4025	0.991419	0.495710	+	0.868488i	$0.334908\pi$
$384$	0	0
$385$	0.0682744	0.00347958
$386$	0	0
$387$	−0.212225	−0.0107880
$388$	0	0
$389$	−1.55769	−0.0789781	−0.0394890	−	0.999220i	$-0.512573\pi$
$389$	−1.55769	−0.0789781	−0.0394890	+	0.999220i	$0.512573\pi$
$390$	0	0
$391$	−2.61911	−0.132454
$392$	0	0
$393$	−25.9317	−1.30808
$394$	0	0
$395$	−12.8385	−0.645976
$396$	0	0
$397$	10.1639	0.510113	0.255057	−	0.966926i	$-0.417906\pi$
$397$	10.1639	0.510113	0.255057	+	0.966926i	$0.417906\pi$
$398$	0	0
$399$	0.0917443	0.00459296
$400$	0	0
$401$	−6.51836	−0.325512	−0.162756	−	0.986666i	$-0.552038\pi$
$401$	−6.51836	−0.325512	−0.162756	+	0.986666i	$0.552038\pi$
$402$	0	0
$403$	24.9144	1.24108
$404$	0	0
$405$	−12.4809	−0.620183
$406$	0	0
$407$	4.41406	0.218797
$408$	0	0
$409$	8.81298	0.435774	0.217887	−	0.975974i	$-0.430084\pi$
$409$	8.81298	0.435774	0.217887	+	0.975974i	$0.430084\pi$
$410$	0	0
$411$	13.1660	0.649429
$412$	0	0
$413$	−0.272647	−0.0134161
$414$	0	0
$415$	−7.86343	−0.386001
$416$	0	0
$417$	1.48270	0.0726083
$418$	0	0
$419$	−0.258184	−0.0126131	−0.00630655	−	0.999980i	$-0.502007\pi$
$419$	−0.258184	−0.0126131	−0.00630655	+	0.999980i	$0.502007\pi$
$420$	0	0
$421$	−24.4816	−1.19316	−0.596581	−	0.802553i	$-0.703475\pi$
$421$	−24.4816	−1.19316	−0.596581	+	0.802553i	$0.703475\pi$
$422$	0	0
$423$	0.441579	0.0214703
$424$	0	0
$425$	−17.4804	−0.847925
$426$	0	0
$427$	0.562592	0.0272257
$428$	0	0
$429$	−13.2426	−0.639361
$430$	0	0
$431$	−23.0134	−1.10852	−0.554259	−	0.832345i	$-0.686998\pi$
$431$	−23.0134	−1.10852	−0.554259	+	0.832345i	$0.686998\pi$
$432$	0	0
$433$	−20.4069	−0.980693	−0.490347	−	0.871527i	$-0.663130\pi$
$433$	−20.4069	−0.980693	−0.490347	+	0.871527i	$0.663130\pi$
$434$	0	0
$435$	−20.1366	−0.965474
$436$	0	0
$437$	0.572907	0.0274059
$438$	0	0
$439$	25.7213	1.22761	0.613804	−	0.789458i	$-0.289638\pi$
$439$	25.7213	1.22761	0.613804	+	0.789458i	$0.289638\pi$
$440$	0	0
$441$	−0.334805	−0.0159431
$442$	0	0
$443$	−14.4477	−0.686430	−0.343215	−	0.939257i	$-0.611516\pi$
$443$	−14.4477	−0.686430	−0.343215	+	0.939257i	$0.611516\pi$
$444$	0	0
$445$	14.6693	0.695392
$446$	0	0
$447$	11.3523	0.536945
$448$	0	0
$449$	−6.56432	−0.309790	−0.154895	−	0.987931i	$-0.549504\pi$
$449$	−6.56432	−0.309790	−0.154895	+	0.987931i	$0.549504\pi$
$450$	0	0
$451$	1.19751	0.0563885
$452$	0	0
$453$	20.0612	0.942557
$454$	0	0
$455$	0.384692	0.0180346
$456$	0	0
$457$	21.8351	1.02140	0.510702	−	0.859758i	$-0.329385\pi$
$457$	21.8351	1.02140	0.510702	+	0.859758i	$0.329385\pi$
$458$	0	0
$459$	28.7299	1.34100
$460$	0	0
$461$	−0.916152	−0.0426695	−0.0213347	−	0.999772i	$-0.506792\pi$
$461$	−0.916152	−0.0426695	−0.0213347	+	0.999772i	$0.506792\pi$
$462$	0	0
$463$	28.1207	1.30688	0.653440	−	0.756979i	$-0.273325\pi$
$463$	28.1207	1.30688	0.653440	+	0.756979i	$0.273325\pi$
$464$	0	0
$465$	−9.08390	−0.421256
$466$	0	0
$467$	17.3242	0.801667	0.400833	−	0.916151i	$-0.368721\pi$
$467$	17.3242	0.801667	0.400833	+	0.916151i	$0.368721\pi$
$468$	0	0
$469$	0.383628	0.0177143
$470$	0	0
$471$	−34.2905	−1.58002
$472$	0	0
$473$	−5.14694	−0.236657
$474$	0	0
$475$	3.82368	0.175443
$476$	0	0
$477$	−0.232213	−0.0106323
$478$	0	0
$479$	−24.4510	−1.11719	−0.558597	−	0.829439i	$-0.688660\pi$
$479$	−24.4510	−1.11719	−0.558597	+	0.829439i	$0.688660\pi$
$480$	0	0
$481$	24.8710	1.13402
$482$	0	0
$483$	0.0353513	0.00160854
$484$	0	0
$485$	−5.95344	−0.270332
$486$	0	0
$487$	14.1625	0.641762	0.320881	−	0.947120i	$-0.396021\pi$
$487$	14.1625	0.641762	0.320881	+	0.947120i	$0.396021\pi$
$488$	0	0
$489$	17.7806	0.804065
$490$	0	0
$491$	20.5245	0.926260	0.463130	−	0.886290i	$-0.346726\pi$
$491$	20.5245	0.926260	0.463130	+	0.886290i	$0.346726\pi$
$492$	0	0
$493$	47.0919	2.12091
$494$	0	0
$495$	0.0757902	0.00340652
$496$	0	0
$497$	0.690152	0.0309575
$498$	0	0
$499$	2.52796	0.113167	0.0565834	−	0.998398i	$-0.481979\pi$
$499$	2.52796	0.113167	0.0565834	+	0.998398i	$0.481979\pi$
$500$	0	0
$501$	−19.5438	−0.873151
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−13.4234	−0.597334
$506$	0	0
$507$	−51.9201	−2.30585
$508$	0	0
$509$	−29.8255	−1.32199	−0.660996	−	0.750390i	$-0.729866\pi$
$509$	−29.8255	−1.32199	−0.660996	+	0.750390i	$0.729866\pi$
$510$	0	0
$511$	−0.388335	−0.0171789
$512$	0	0
$513$	−6.28441	−0.277464
$514$	0	0
$515$	−5.26704	−0.232094
$516$	0	0
$517$	10.7093	0.470994
$518$	0	0
$519$	−6.46283	−0.283687
$520$	0	0
$521$	41.7955	1.83110	0.915548	−	0.402209i	$-0.131758\pi$
$521$	41.7955	1.83110	0.915548	+	0.402209i	$0.131758\pi$
$522$	0	0
$523$	−22.1212	−0.967292	−0.483646	−	0.875264i	$-0.660688\pi$
$523$	−22.1212	−0.967292	−0.483646	+	0.875264i	$0.660688\pi$
$524$	0	0
$525$	0.235941	0.0102973
$526$	0	0
$527$	21.2438	0.925396
$528$	0	0
$529$	−22.7792	−0.990402
$530$	0	0
$531$	−0.302660	−0.0131343
$532$	0	0
$533$	6.74735	0.292260
$534$	0	0
$535$	5.94592	0.257065
$536$	0	0
$537$	−21.8929	−0.944748
$538$	0	0
$539$	−8.11978	−0.349744
$540$	0	0
$541$	−10.6824	−0.459270	−0.229635	−	0.973277i	$-0.573753\pi$
$541$	−10.6824	−0.459270	−0.229635	+	0.973277i	$0.573753\pi$
$542$	0	0
$543$	−40.4144	−1.73435
$544$	0	0
$545$	−12.6603	−0.542306
$546$	0	0
$547$	1.05384	0.0450587	0.0225294	−	0.999746i	$-0.492828\pi$
$547$	1.05384	0.0450587	0.0225294	+	0.999746i	$0.492828\pi$
$548$	0	0
$549$	0.624524	0.0266540
$550$	0	0
$551$	−10.3009	−0.438834
$552$	0	0
$553$	−0.405253	−0.0172331
$554$	0	0
$555$	−9.06808	−0.384919
$556$	0	0
$557$	7.86146	0.333101	0.166550	−	0.986033i	$-0.446737\pi$
$557$	7.86146	0.333101	0.166550	+	0.986033i	$0.446737\pi$
$558$	0	0
$559$	−29.0004	−1.22659
$560$	0	0
$561$	−11.2916	−0.476733
$562$	0	0
$563$	40.7108	1.71575	0.857877	−	0.513855i	$-0.171783\pi$
$563$	40.7108	1.71575	0.857877	+	0.513855i	$0.171783\pi$
$564$	0	0
$565$	−15.7954	−0.664519
$566$	0	0
$567$	−0.393966	−0.0165450
$568$	0	0
$569$	−13.8851	−0.582095	−0.291047	−	0.956709i	$-0.594004\pi$
$569$	−13.8851	−0.582095	−0.291047	+	0.956709i	$0.594004\pi$
$570$	0	0
$571$	−9.88969	−0.413870	−0.206935	−	0.978355i	$-0.566349\pi$
$571$	−9.88969	−0.413870	−0.206935	+	0.978355i	$0.566349\pi$
$572$	0	0
$573$	−45.9097	−1.91791
$574$	0	0
$575$	1.47336	0.0614433
$576$	0	0
$577$	29.3324	1.22112	0.610562	−	0.791968i	$-0.290944\pi$
$577$	29.3324	1.22112	0.610562	+	0.791968i	$0.290944\pi$
$578$	0	0
$579$	37.8254	1.57197
$580$	0	0
$581$	−0.248212	−0.0102976
$582$	0	0
$583$	−5.63169	−0.233241
$584$	0	0
$585$	0.427040	0.0176559
$586$	0	0
$587$	−19.1824	−0.791744	−0.395872	−	0.918306i	$-0.629558\pi$
$587$	−19.1824	−0.791744	−0.395872	+	0.918306i	$0.629558\pi$
$588$	0	0
$589$	−4.64690	−0.191472
$590$	0	0
$591$	37.3285	1.53549
$592$	0	0
$593$	33.6816	1.38314	0.691570	−	0.722310i	$-0.256920\pi$
$593$	33.6816	1.38314	0.691570	+	0.722310i	$0.256920\pi$
$594$	0	0
$595$	0.328016	0.0134473
$596$	0	0
$597$	8.79961	0.360144
$598$	0	0
$599$	45.4775	1.85816	0.929081	−	0.369876i	$-0.120600\pi$
$599$	45.4775	1.85816	0.929081	+	0.369876i	$0.120600\pi$
$600$	0	0
$601$	25.5187	1.04093	0.520464	−	0.853884i	$-0.325759\pi$
$601$	25.5187	1.04093	0.520464	+	0.853884i	$0.325759\pi$
$602$	0	0
$603$	0.425859	0.0173423
$604$	0	0
$605$	−13.1807	−0.535872
$606$	0	0
$607$	−0.336455	−0.0136563	−0.00682814	−	0.999977i	$-0.502173\pi$
$607$	−0.336455	−0.0136563	−0.00682814	+	0.999977i	$0.502173\pi$
$608$	0	0
$609$	−0.635619	−0.0257566
$610$	0	0
$611$	60.3414	2.44115
$612$	0	0
$613$	−18.8476	−0.761247	−0.380623	−	0.924730i	$-0.624291\pi$
$613$	−18.8476	−0.761247	−0.380623	+	0.924730i	$0.624291\pi$
$614$	0	0
$615$	−2.46012	−0.0992014
$616$	0	0
$617$	−22.8183	−0.918632	−0.459316	−	0.888273i	$-0.651905\pi$
$617$	−22.8183	−0.918632	−0.459316	+	0.888273i	$0.651905\pi$
$618$	0	0
$619$	31.9207	1.28300	0.641500	−	0.767123i	$-0.278312\pi$
$619$	31.9207	1.28300	0.641500	+	0.767123i	$0.278312\pi$
$620$	0	0
$621$	−2.42154	−0.0971729
$622$	0	0
$623$	0.463043	0.0185514
$624$	0	0
$625$	0.512620	0.0205048
$626$	0	0
$627$	2.46994	0.0986400
$628$	0	0
$629$	21.2068	0.845572
$630$	0	0
$631$	43.3606	1.72616	0.863079	−	0.505070i	$-0.168533\pi$
$631$	43.3606	1.72616	0.863079	+	0.505070i	$0.168533\pi$
$632$	0	0
$633$	37.5031	1.49062
$634$	0	0
$635$	8.85099	0.351241
$636$	0	0
$637$	−45.7509	−1.81272
$638$	0	0
$639$	0.766126	0.0303075
$640$	0	0
$641$	−49.7944	−1.96676	−0.983381	−	0.181555i	$-0.941887\pi$
$641$	−49.7944	−1.96676	−0.983381	+	0.181555i	$0.941887\pi$
$642$	0	0
$643$	1.81963	0.0717591	0.0358796	−	0.999356i	$-0.488577\pi$
$643$	1.81963	0.0717591	0.0358796	+	0.999356i	$0.488577\pi$
$644$	0	0
$645$	10.5737	0.416338
$646$	0	0
$647$	17.8627	0.702256	0.351128	−	0.936327i	$-0.385798\pi$
$647$	17.8627	0.702256	0.351128	+	0.936327i	$0.385798\pi$
$648$	0	0
$649$	−7.34020	−0.288128
$650$	0	0
$651$	−0.286737	−0.0112381
$652$	0	0
$653$	26.5095	1.03740	0.518699	−	0.854957i	$-0.326416\pi$
$653$	26.5095	1.03740	0.518699	+	0.854957i	$0.326416\pi$
$654$	0	0
$655$	20.2804	0.792423
$656$	0	0
$657$	−0.431084	−0.0168182
$658$	0	0
$659$	−20.4126	−0.795162	−0.397581	−	0.917567i	$-0.630150\pi$
$659$	−20.4126	−0.795162	−0.397581	+	0.917567i	$0.630150\pi$
$660$	0	0
$661$	38.5482	1.49935	0.749676	−	0.661805i	$-0.230209\pi$
$661$	38.5482	1.49935	0.749676	+	0.661805i	$0.230209\pi$
$662$	0	0
$663$	−63.6227	−2.47090
$664$	0	0
$665$	−0.0717505	−0.00278237
$666$	0	0
$667$	−3.96919	−0.153688
$668$	0	0
$669$	−10.2446	−0.396080
$670$	0	0
$671$	15.1461	0.584710
$672$	0	0
$673$	30.8228	1.18813	0.594066	−	0.804416i	$-0.297522\pi$
$673$	30.8228	1.18813	0.594066	+	0.804416i	$0.297522\pi$
$674$	0	0
$675$	−16.1618	−0.622067
$676$	0	0
$677$	44.5143	1.71082	0.855411	−	0.517950i	$-0.173305\pi$
$677$	44.5143	1.71082	0.855411	+	0.517950i	$0.173305\pi$
$678$	0	0
$679$	−0.187923	−0.00721181
$680$	0	0
$681$	32.0360	1.22762
$682$	0	0
$683$	−22.4522	−0.859111	−0.429555	−	0.903041i	$-0.641330\pi$
$683$	−22.4522	−0.859111	−0.429555	+	0.903041i	$0.641330\pi$
$684$	0	0
$685$	−10.2967	−0.393417
$686$	0	0
$687$	−38.7825	−1.47965
$688$	0	0
$689$	−31.7317	−1.20888
$690$	0	0
$691$	3.84302	0.146195	0.0730977	−	0.997325i	$-0.476712\pi$
$691$	3.84302	0.146195	0.0730977	+	0.997325i	$0.476712\pi$
$692$	0	0
$693$	0.00239235	9.08779e−5 0
$694$	0	0
$695$	−1.15958	−0.0439854
$696$	0	0
$697$	5.75328	0.217921
$698$	0	0
$699$	−14.0147	−0.530083
$700$	0	0
$701$	−1.69359	−0.0639660	−0.0319830	−	0.999488i	$-0.510182\pi$
$701$	−1.69359	−0.0639660	−0.0319830	+	0.999488i	$0.510182\pi$
$702$	0	0
$703$	−4.63880	−0.174956
$704$	0	0
$705$	−22.0008	−0.828596
$706$	0	0
$707$	−0.423716	−0.0159355
$708$	0	0
$709$	−16.6139	−0.623947	−0.311973	−	0.950091i	$-0.600990\pi$
$709$	−16.6139	−0.623947	−0.311973	+	0.950091i	$0.600990\pi$
$710$	0	0
$711$	−0.449865	−0.0168712
$712$	0	0
$713$	−1.79056	−0.0670570
$714$	0	0
$715$	10.3567	0.387318
$716$	0	0
$717$	5.97008	0.222957
$718$	0	0
$719$	−9.05406	−0.337659	−0.168830	−	0.985645i	$-0.553999\pi$
$719$	−9.05406	−0.337659	−0.168830	+	0.985645i	$0.553999\pi$
$720$	0	0
$721$	−0.166256	−0.00619171
$722$	0	0
$723$	−20.1390	−0.748976
$724$	0	0
$725$	−26.4911	−0.983854
$726$	0	0
$727$	−7.19120	−0.266707	−0.133353	−	0.991069i	$-0.542575\pi$
$727$	−7.19120	−0.266707	−0.133353	+	0.991069i	$0.542575\pi$
$728$	0	0
$729$	26.5558	0.983548
$730$	0	0
$731$	−24.7279	−0.914593
$732$	0	0
$733$	−11.7306	−0.433279	−0.216640	−	0.976252i	$-0.569510\pi$
$733$	−11.7306	−0.433279	−0.216640	+	0.976252i	$0.569510\pi$
$734$	0	0
$735$	16.6810	0.615287
$736$	0	0
$737$	10.3280	0.380438
$738$	0	0
$739$	−12.4945	−0.459618	−0.229809	−	0.973236i	$-0.573810\pi$
$739$	−12.4945	−0.459618	−0.229809	+	0.973236i	$0.573810\pi$
$740$	0	0
$741$	13.9169	0.511249
$742$	0	0
$743$	8.15329	0.299115	0.149558	−	0.988753i	$-0.452215\pi$
$743$	8.15329	0.299115	0.149558	+	0.988753i	$0.452215\pi$
$744$	0	0
$745$	−8.87830	−0.325276
$746$	0	0
$747$	−0.275536	−0.0100813
$748$	0	0
$749$	0.187685	0.00685788
$750$	0	0
$751$	18.0205	0.657579	0.328789	−	0.944403i	$-0.393359\pi$
$751$	18.0205	0.657579	0.328789	+	0.944403i	$0.393359\pi$
$752$	0	0
$753$	49.9768	1.82126
$754$	0	0
$755$	−15.6893	−0.570991
$756$	0	0
$757$	5.83101	0.211932	0.105966	−	0.994370i	$-0.466207\pi$
$757$	5.83101	0.211932	0.105966	+	0.994370i	$0.466207\pi$
$758$	0	0
$759$	0.951728	0.0345456
$760$	0	0
$761$	13.3164	0.482719	0.241359	−	0.970436i	$-0.422407\pi$
$761$	13.3164	0.482719	0.241359	+	0.970436i	$0.422407\pi$
$762$	0	0
$763$	−0.399626	−0.0144674
$764$	0	0
$765$	0.364125	0.0131650
$766$	0	0
$767$	−41.3583	−1.49336
$768$	0	0
$769$	−31.4202	−1.13304	−0.566522	−	0.824047i	$-0.691711\pi$
$769$	−31.4202	−1.13304	−0.566522	+	0.824047i	$0.691711\pi$
$770$	0	0
$771$	2.34037	0.0842862
$772$	0	0
$773$	−12.1365	−0.436520	−0.218260	−	0.975891i	$-0.570038\pi$
$773$	−12.1365	−0.436520	−0.218260	+	0.975891i	$0.570038\pi$
$774$	0	0
$775$	−11.9505	−0.429275
$776$	0	0
$777$	−0.286238	−0.0102687
$778$	0	0
$779$	−1.25848	−0.0450897
$780$	0	0
$781$	18.5803	0.664855
$782$	0	0
$783$	43.5394	1.55597
$784$	0	0
$785$	26.8176	0.957160
$786$	0	0
$787$	−8.38949	−0.299053	−0.149527	−	0.988758i	$-0.547775\pi$
$787$	−8.38949	−0.299053	−0.149527	+	0.988758i	$0.547775\pi$
$788$	0	0
$789$	6.66063	0.237125
$790$	0	0
$791$	−0.498589	−0.0177278
$792$	0	0
$793$	85.3408	3.03054
$794$	0	0
$795$	11.5695	0.410329
$796$	0	0
$797$	7.94141	0.281299	0.140650	−	0.990059i	$-0.455081\pi$
$797$	7.94141	0.281299	0.140650	+	0.990059i	$0.455081\pi$
$798$	0	0
$799$	51.4515	1.82022
$800$	0	0
$801$	0.514016	0.0181619
$802$	0	0
$803$	−10.4548	−0.368940
$804$	0	0
$805$	−0.0276472	−0.000974436 0
$806$	0	0
$807$	12.7467	0.448707
$808$	0	0
$809$	35.0663	1.23287	0.616433	−	0.787407i	$-0.288577\pi$
$809$	35.0663	1.23287	0.616433	+	0.787407i	$0.288577\pi$
$810$	0	0
$811$	0.490097	0.0172096	0.00860481	−	0.999963i	$-0.497261\pi$
$811$	0.490097	0.0172096	0.00860481	+	0.999963i	$0.497261\pi$
$812$	0	0
$813$	−50.6672	−1.77697
$814$	0	0
$815$	−13.9057	−0.487094
$816$	0	0
$817$	5.40900	0.189237
$818$	0	0
$819$	0.0134797	0.000471018 0
$820$	0	0
$821$	−17.2349	−0.601503	−0.300751	−	0.953703i	$-0.597237\pi$
$821$	−17.2349	−0.601503	−0.300751	+	0.953703i	$0.597237\pi$
$822$	0	0
$823$	10.8365	0.377736	0.188868	−	0.982002i	$-0.439518\pi$
$823$	10.8365	0.377736	0.188868	+	0.982002i	$0.439518\pi$
$824$	0	0
$825$	6.35201	0.221148
$826$	0	0
$827$	1.62913	0.0566504	0.0283252	−	0.999599i	$-0.490983\pi$
$827$	1.62913	0.0566504	0.0283252	+	0.999599i	$0.490983\pi$
$828$	0	0
$829$	21.4942	0.746526	0.373263	−	0.927726i	$-0.378239\pi$
$829$	21.4942	0.746526	0.373263	+	0.927726i	$0.378239\pi$
$830$	0	0
$831$	7.00096	0.242861
$832$	0	0
$833$	−39.0105	−1.35163
$834$	0	0
$835$	15.2846	0.528946
$836$	0	0
$837$	19.6413	0.678902
$838$	0	0
$839$	1.89869	0.0655502	0.0327751	−	0.999463i	$-0.489565\pi$
$839$	1.89869	0.0655502	0.0327751	+	0.999463i	$0.489565\pi$
$840$	0	0
$841$	42.3664	1.46091
$842$	0	0
$843$	5.45607	0.187917
$844$	0	0
$845$	40.6052	1.39686
$846$	0	0
$847$	−0.416054	−0.0142958
$848$	0	0
$849$	27.5028	0.943895
$850$	0	0
$851$	−1.78744	−0.0612727
$852$	0	0
$853$	19.9653	0.683597	0.341799	−	0.939773i	$-0.388964\pi$
$853$	19.9653	0.683597	0.341799	+	0.939773i	$0.388964\pi$
$854$	0	0
$855$	−0.0796491	−0.00272394
$856$	0	0
$857$	−22.1238	−0.755734	−0.377867	−	0.925860i	$-0.623342\pi$
$857$	−22.1238	−0.755734	−0.377867	+	0.925860i	$0.623342\pi$
$858$	0	0
$859$	−44.1857	−1.50760	−0.753798	−	0.657107i	$-0.771780\pi$
$859$	−44.1857	−1.50760	−0.753798	+	0.657107i	$0.771780\pi$
$860$	0	0
$861$	−0.0776545	−0.00264646
$862$	0	0
$863$	−4.74537	−0.161534	−0.0807672	−	0.996733i	$-0.525737\pi$
$863$	−4.74537	−0.161534	−0.0807672	+	0.996733i	$0.525737\pi$
$864$	0	0
$865$	5.05439	0.171854
$866$	0	0
$867$	−24.5706	−0.834461
$868$	0	0
$869$	−10.9102	−0.370105
$870$	0	0
$871$	58.1933	1.97180
$872$	0	0
$873$	−0.208610	−0.00706038
$874$	0	0
$875$	−0.478738	−0.0161843
$876$	0	0
$877$	23.5831	0.796344	0.398172	−	0.917311i	$-0.369645\pi$
$877$	23.5831	0.796344	0.398172	+	0.917311i	$0.369645\pi$
$878$	0	0
$879$	14.5905	0.492127
$880$	0	0
$881$	1.64627	0.0554643	0.0277321	−	0.999615i	$-0.491171\pi$
$881$	1.64627	0.0554643	0.0277321	+	0.999615i	$0.491171\pi$
$882$	0	0
$883$	−20.7614	−0.698678	−0.349339	−	0.936996i	$-0.613594\pi$
$883$	−20.7614	−0.698678	−0.349339	+	0.936996i	$0.613594\pi$
$884$	0	0
$885$	15.0794	0.506889
$886$	0	0
$887$	−39.5989	−1.32960	−0.664801	−	0.747021i	$-0.731484\pi$
$887$	−39.5989	−1.32960	−0.664801	+	0.747021i	$0.731484\pi$
$888$	0	0
$889$	0.279385	0.00937028
$890$	0	0
$891$	−10.6064	−0.355327
$892$	0	0
$893$	−11.2545	−0.376619
$894$	0	0
$895$	17.1218	0.572318
$896$	0	0
$897$	5.36251	0.179049
$898$	0	0
$899$	32.1944	1.07374
$900$	0	0
$901$	−27.0568	−0.901392
$902$	0	0
$903$	0.333763	0.0111069
$904$	0	0
$905$	31.6069	1.05065
$906$	0	0
$907$	1.17397	0.0389810	0.0194905	−	0.999810i	$-0.493796\pi$
$907$	1.17397	0.0389810	0.0194905	+	0.999810i	$0.493796\pi$
$908$	0	0
$909$	−0.470360	−0.0156008
$910$	0	0
$911$	−10.1851	−0.337446	−0.168723	−	0.985664i	$-0.553964\pi$
$911$	−10.1851	−0.337446	−0.168723	+	0.985664i	$0.553964\pi$
$912$	0	0
$913$	−6.68238	−0.221155
$914$	0	0
$915$	−31.1156	−1.02865
$916$	0	0
$917$	0.640161	0.0211400
$918$	0	0
$919$	32.9624	1.08733	0.543664	−	0.839303i	$-0.317037\pi$
$919$	32.9624	1.08733	0.543664	+	0.839303i	$0.317037\pi$
$920$	0	0
$921$	46.8972	1.54532
$922$	0	0
$923$	104.691	3.44593
$924$	0	0
$925$	−11.9297	−0.392247
$926$	0	0
$927$	−0.184558	−0.00606169
$928$	0	0
$929$	39.8065	1.30601	0.653004	−	0.757354i	$-0.273508\pi$
$929$	39.8065	1.30601	0.653004	+	0.757354i	$0.273508\pi$
$930$	0	0
$931$	8.53320	0.279664
$932$	0	0
$933$	−34.8210	−1.13999
$934$	0	0
$935$	8.83086	0.288800
$936$	0	0
$937$	13.2727	0.433602	0.216801	−	0.976216i	$-0.430438\pi$
$937$	13.2727	0.433602	0.216801	+	0.976216i	$0.430438\pi$
$938$	0	0
$939$	−38.9973	−1.27263
$940$	0	0
$941$	−57.5266	−1.87532	−0.937658	−	0.347561i	$-0.887010\pi$
$941$	−57.5266	−1.87532	−0.937658	+	0.347561i	$0.887010\pi$
$942$	0	0
$943$	−0.484922	−0.0157912
$944$	0	0
$945$	0.303272	0.00986543
$946$	0	0
$947$	4.90129	0.159271	0.0796353	−	0.996824i	$-0.474624\pi$
$947$	4.90129	0.159271	0.0796353	+	0.996824i	$0.474624\pi$
$948$	0	0
$949$	−58.9073	−1.91221
$950$	0	0
$951$	−4.40115	−0.142717
$952$	0	0
$953$	39.9887	1.29536	0.647680	−	0.761913i	$-0.275739\pi$
$953$	39.9887	1.29536	0.647680	+	0.761913i	$0.275739\pi$
$954$	0	0
$955$	35.9046	1.16185
$956$	0	0
$957$	−17.1122	−0.553157
$958$	0	0
$959$	−0.325020	−0.0104954
$960$	0	0
$961$	−16.4766	−0.531504
$962$	0	0
$963$	0.208347	0.00671387
$964$	0	0
$965$	−29.5821	−0.952282
$966$	0	0
$967$	−54.0355	−1.73767	−0.868833	−	0.495105i	$-0.835129\pi$
$967$	−54.0355	−1.73767	−0.868833	+	0.495105i	$0.835129\pi$
$968$	0	0
$969$	11.8665	0.381208
$970$	0	0
$971$	−27.2287	−0.873810	−0.436905	−	0.899508i	$-0.643925\pi$
$971$	−27.2287	−0.873810	−0.436905	+	0.899508i	$0.643925\pi$
$972$	0	0
$973$	−0.0366026	−0.00117343
$974$	0	0
$975$	35.7903	1.14621
$976$	0	0
$977$	−31.8268	−1.01823	−0.509116	−	0.860698i	$-0.670027\pi$
$977$	−31.8268	−1.01823	−0.509116	+	0.860698i	$0.670027\pi$
$978$	0	0
$979$	12.4661	0.398417
$980$	0	0
$981$	−0.443618	−0.0141636
$982$	0	0
$983$	24.1825	0.771302	0.385651	−	0.922645i	$-0.373977\pi$
$983$	24.1825	0.771302	0.385651	+	0.922645i	$0.373977\pi$
$984$	0	0
$985$	−29.1935	−0.930183
$986$	0	0
$987$	−0.694463	−0.0221050
$988$	0	0
$989$	2.08422	0.0662742
$990$	0	0
$991$	−34.7807	−1.10484	−0.552422	−	0.833565i	$-0.686296\pi$
$991$	−34.7807	−1.10484	−0.552422	+	0.833565i	$0.686296\pi$
$992$	0	0
$993$	4.99963	0.158659
$994$	0	0
$995$	−6.88192	−0.218171
$996$	0	0
$997$	18.4046	0.582881	0.291440	−	0.956589i	$-0.405866\pi$
$997$	18.4046	0.582881	0.291440	+	0.956589i	$0.405866\pi$
$998$	0	0
$999$	19.6071	0.620340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.8		26
4.3	odd	2		503.2.a.f.1.12	✓	26
12.11	even	2		4527.2.a.o.1.15		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.12	✓	26	4.3	odd	2
4527.2.a.o.1.15		26	12.11	even	2
8048.2.a.u.1.8		26	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74581 q^{3} +1.36534 q^{5} +0.0430977 q^{7} +0.0478420 q^{9} +O(q^{10})\)	\(q-1.74581 q^{3} +1.36534 q^{5} +0.0430977 q^{7} +0.0478420 q^{9} +1.16028 q^{11} +6.53757 q^{13} -2.38363 q^{15} +5.57441 q^{17} -1.21935 q^{19} -0.0752402 q^{21} -0.469846 q^{23} -3.13583 q^{25} +5.15390 q^{27} +8.44786 q^{29} +3.81096 q^{31} -2.02562 q^{33} +0.0588432 q^{35} +3.80432 q^{37} -11.4133 q^{39} +1.03209 q^{41} -4.43596 q^{43} +0.0653208 q^{45} +9.22994 q^{47} -6.99814 q^{49} -9.73184 q^{51} -4.85375 q^{53} +1.58418 q^{55} +2.12875 q^{57} -6.32625 q^{59} +13.0539 q^{61} +0.00206188 q^{63} +8.92604 q^{65} +8.90136 q^{67} +0.820260 q^{69} +16.0137 q^{71} -9.01057 q^{73} +5.47456 q^{75} +0.0500052 q^{77} -9.40313 q^{79} -9.14124 q^{81} -5.75930 q^{83} +7.61099 q^{85} -14.7483 q^{87} +10.7440 q^{89} +0.281754 q^{91} -6.65319 q^{93} -1.66484 q^{95} -4.36039 q^{97} +0.0555100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

Introduction

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