LMFDB - Embedded newform 1472.4.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.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:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.334189.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} - 5x + 4 $ x^4 - 2x^3 - 16x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.83969$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.43737 q^{3} +17.9704 q^{5} +32.7301 q^{7} -15.1845 q^{9} -26.7049 q^{11} +14.4956 q^{13} -61.7710 q^{15} +24.7016 q^{17} -94.6224 q^{19} -112.505 q^{21} -23.0000 q^{23} +197.935 q^{25} +145.004 q^{27} +57.5965 q^{29} +88.8691 q^{31} +91.7948 q^{33} +588.173 q^{35} +305.467 q^{37} -49.8269 q^{39} -179.205 q^{41} +96.5826 q^{43} -272.871 q^{45} +218.484 q^{47} +728.258 q^{49} -84.9085 q^{51} +519.174 q^{53} -479.898 q^{55} +325.252 q^{57} +37.2884 q^{59} -96.3052 q^{61} -496.989 q^{63} +260.493 q^{65} -497.552 q^{67} +79.0596 q^{69} -19.6235 q^{71} -208.235 q^{73} -680.378 q^{75} -874.054 q^{77} -446.200 q^{79} -88.4513 q^{81} -501.151 q^{83} +443.897 q^{85} -197.981 q^{87} +1102.82 q^{89} +474.444 q^{91} -305.476 q^{93} -1700.40 q^{95} +1814.37 q^{97} +405.500 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 7 q^{3} - 14 q^{5} + 16 q^{7} - 33 q^{9} - 8 q^{11} - 111 q^{13} + 10 q^{15} + 98 q^{17} - 96 q^{19} - 180 q^{21} - 92 q^{23} + 184 q^{25} + 155 q^{27} - 21 q^{29} - 193 q^{31} - 418 q^{33} + 752 q^{35}+ \cdots + 1498 q^{99}+O(q^{100}) $ 4 * q - 7 * q^3 - 14 * q^5 + 16 * q^7 - 33 * q^9 - 8 * q^11 - 111 * q^13 + 10 * q^15 + 98 * q^17 - 96 * q^19 - 180 * q^21 - 92 * q^23 + 184 * q^25 + 155 * q^27 - 21 * q^29 - 193 * q^31 - 418 * q^33 + 752 * q^35 - 170 * q^37 - 291 * q^39 - 125 * q^41 - 2 * q^43 - 168 * q^45 - 677 * q^47 + 1220 * q^49 + 340 * q^51 + 230 * q^53 - 972 * q^55 + 1322 * q^57 + 1140 * q^59 - 754 * q^61 - 1092 * q^63 + 1318 * q^65 - 488 * q^67 + 161 * q^69 - 401 * q^71 + 1509 * q^73 - 1401 * q^75 - 736 * q^77 - 838 * q^79 - 932 * q^81 - 142 * q^83 - 112 * q^85 + 2223 * q^87 + 2342 * q^89 - 292 * q^91 + 509 * q^93 - 956 * q^95 + 1062 * q^97 + 1498 * 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
$3$	−3.43737	−0.661523	−0.330761	−	0.943714i	$-0.607306\pi$
$3$	−3.43737	−0.661523	−0.330761	+	0.943714i	$0.607306\pi$
$4$	0	0
$5$	17.9704	1.60732	0.803661	−	0.595087i	$-0.202883\pi$
$5$	17.9704	1.60732	0.803661	+	0.595087i	$0.202883\pi$
$6$	0	0
$7$	32.7301	1.76726	0.883629	−	0.468187i	$-0.155093\pi$
$7$	32.7301	1.76726	0.883629	+	0.468187i	$0.155093\pi$
$8$	0	0
$9$	−15.1845	−0.562388
$10$	0	0
$11$	−26.7049	−0.731985	−0.365993	−	0.930618i	$-0.619270\pi$
$11$	−26.7049	−0.731985	−0.365993	+	0.930618i	$0.619270\pi$
$12$	0	0
$13$	14.4956	0.309259	0.154630	−	0.987973i	$-0.450582\pi$
$13$	14.4956	0.309259	0.154630	+	0.987973i	$0.450582\pi$
$14$	0	0
$15$	−61.7710	−1.06328
$16$	0	0
$17$	24.7016	0.352412	0.176206	−	0.984353i	$-0.443617\pi$
$17$	24.7016	0.352412	0.176206	+	0.984353i	$0.443617\pi$
$18$	0	0
$19$	−94.6224	−1.14252	−0.571259	−	0.820770i	$-0.693545\pi$
$19$	−94.6224	−1.14252	−0.571259	+	0.820770i	$0.693545\pi$
$20$	0	0
$21$	−112.505	−1.16908
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	197.935	1.58348
$26$	0	0
$27$	145.004	1.03355
$28$	0	0
$29$	57.5965	0.368807	0.184403	−	0.982851i	$-0.440965\pi$
$29$	57.5965	0.368807	0.184403	+	0.982851i	$0.440965\pi$
$30$	0	0
$31$	88.8691	0.514882	0.257441	−	0.966294i	$-0.417121\pi$
$31$	88.8691	0.514882	0.257441	+	0.966294i	$0.417121\pi$
$32$	0	0
$33$	91.7948	0.484225
$34$	0	0
$35$	588.173	2.84055
$36$	0	0
$37$	305.467	1.35726	0.678629	−	0.734482i	$-0.262575\pi$
$37$	305.467	1.35726	0.678629	+	0.734482i	$0.262575\pi$
$38$	0	0
$39$	−49.8269	−0.204582
$40$	0	0
$41$	−179.205	−0.682613	−0.341306	−	0.939952i	$-0.610869\pi$
$41$	−179.205	−0.682613	−0.341306	+	0.939952i	$0.610869\pi$
$42$	0	0
$43$	96.5826	0.342528	0.171264	−	0.985225i	$-0.445215\pi$
$43$	96.5826	0.342528	0.171264	+	0.985225i	$0.445215\pi$
$44$	0	0
$45$	−272.871	−0.903938
$46$	0	0
$47$	218.484	0.678067	0.339034	−	0.940774i	$-0.389900\pi$
$47$	218.484	0.678067	0.339034	+	0.940774i	$0.389900\pi$
$48$	0	0
$49$	728.258	2.12320
$50$	0	0
$51$	−84.9085	−0.233129
$52$	0	0
$53$	519.174	1.34555	0.672774	−	0.739848i	$-0.265103\pi$
$53$	519.174	1.34555	0.672774	+	0.739848i	$0.265103\pi$
$54$	0	0
$55$	−479.898	−1.17654
$56$	0	0
$57$	325.252	0.755802
$58$	0	0
$59$	37.2884	0.0822803	0.0411401	−	0.999153i	$-0.486901\pi$
$59$	37.2884	0.0822803	0.0411401	+	0.999153i	$0.486901\pi$
$60$	0	0
$61$	−96.3052	−0.202141	−0.101071	−	0.994879i	$-0.532227\pi$
$61$	−96.3052	−0.202141	−0.101071	+	0.994879i	$0.532227\pi$
$62$	0	0
$63$	−496.989	−0.993884
$64$	0	0
$65$	260.493	0.497079
$66$	0	0
$67$	−497.552	−0.907248	−0.453624	−	0.891193i	$-0.649869\pi$
$67$	−497.552	−0.907248	−0.453624	+	0.891193i	$0.649869\pi$
$68$	0	0
$69$	79.0596	0.137937
$70$	0	0
$71$	−19.6235	−0.0328011	−0.0164005	−	0.999866i	$-0.505221\pi$
$71$	−19.6235	−0.0328011	−0.0164005	+	0.999866i	$0.505221\pi$
$72$	0	0
$73$	−208.235	−0.333865	−0.166932	−	0.985968i	$-0.553386\pi$
$73$	−208.235	−0.333865	−0.166932	+	0.985968i	$0.553386\pi$
$74$	0	0
$75$	−680.378	−1.04751
$76$	0	0
$77$	−874.054	−1.29361
$78$	0	0
$79$	−446.200	−0.635461	−0.317730	−	0.948181i	$-0.602921\pi$
$79$	−446.200	−0.635461	−0.317730	+	0.948181i	$0.602921\pi$
$80$	0	0
$81$	−88.4513	−0.121332
$82$	0	0
$83$	−501.151	−0.662752	−0.331376	−	0.943499i	$-0.607513\pi$
$83$	−501.151	−0.662752	−0.331376	+	0.943499i	$0.607513\pi$
$84$	0	0
$85$	443.897	0.566440
$86$	0	0
$87$	−197.981	−0.243974
$88$	0	0
$89$	1102.82	1.31347	0.656733	−	0.754124i	$-0.271938\pi$
$89$	1102.82	1.31347	0.656733	+	0.754124i	$0.271938\pi$
$90$	0	0
$91$	474.444	0.546541
$92$	0	0
$93$	−305.476	−0.340606
$94$	0	0
$95$	−1700.40	−1.83640
$96$	0	0
$97$	1814.37	1.89919	0.949595	−	0.313478i	$-0.101494\pi$
$97$	1814.37	1.89919	0.949595	+	0.313478i	$0.101494\pi$
$98$	0	0
$99$	405.500	0.411659
$100$	0	0
$101$	1386.24	1.36570	0.682852	−	0.730557i	$-0.260739\pi$
$101$	1386.24	1.36570	0.682852	+	0.730557i	$0.260739\pi$
$102$	0	0
$103$	1372.33	1.31281	0.656407	−	0.754407i	$-0.272075\pi$
$103$	1372.33	1.31281	0.656407	+	0.754407i	$0.272075\pi$
$104$	0	0
$105$	−2021.77	−1.87909
$106$	0	0
$107$	−1657.83	−1.49784	−0.748919	−	0.662662i	$-0.769427\pi$
$107$	−1657.83	−1.49784	−0.748919	+	0.662662i	$0.769427\pi$
$108$	0	0
$109$	821.369	0.721770	0.360885	−	0.932610i	$-0.382475\pi$
$109$	821.369	0.721770	0.360885	+	0.932610i	$0.382475\pi$
$110$	0	0
$111$	−1050.01	−0.897857
$112$	0	0
$113$	−267.142	−0.222395	−0.111197	−	0.993798i	$-0.535469\pi$
$113$	−267.142	−0.222395	−0.111197	+	0.993798i	$0.535469\pi$
$114$	0	0
$115$	−413.319	−0.335150
$116$	0	0
$117$	−220.109	−0.173924
$118$	0	0
$119$	808.484	0.622803
$120$	0	0
$121$	−617.847	−0.464198
$122$	0	0
$123$	615.995	0.451564
$124$	0	0
$125$	1310.68	0.937846
$126$	0	0
$127$	−1446.21	−1.01048	−0.505239	−	0.862979i	$-0.668596\pi$
$127$	−1446.21	−1.01048	−0.505239	+	0.862979i	$0.668596\pi$
$128$	0	0
$129$	−331.990	−0.226590
$130$	0	0
$131$	−1459.33	−0.973297	−0.486649	−	0.873598i	$-0.661781\pi$
$131$	−1459.33	−0.973297	−0.486649	+	0.873598i	$0.661781\pi$
$132$	0	0
$133$	−3097.00	−2.01913
$134$	0	0
$135$	2605.78	1.66126
$136$	0	0
$137$	−1889.31	−1.17821	−0.589105	−	0.808056i	$-0.700520\pi$
$137$	−1889.31	−1.17821	−0.589105	+	0.808056i	$0.700520\pi$
$138$	0	0
$139$	1506.31	0.919159	0.459580	−	0.888137i	$-0.348000\pi$
$139$	1506.31	0.919159	0.459580	+	0.888137i	$0.348000\pi$
$140$	0	0
$141$	−751.011	−0.448557
$142$	0	0
$143$	−387.105	−0.226373
$144$	0	0
$145$	1035.03	0.592791
$146$	0	0
$147$	−2503.30	−1.40455
$148$	0	0
$149$	2946.38	1.61998	0.809989	−	0.586444i	$-0.199473\pi$
$149$	2946.38	1.61998	0.809989	+	0.586444i	$0.199473\pi$
$150$	0	0
$151$	1979.37	1.06675	0.533374	−	0.845880i	$-0.320924\pi$
$151$	1979.37	1.06675	0.533374	+	0.845880i	$0.320924\pi$
$152$	0	0
$153$	−375.080	−0.198192
$154$	0	0
$155$	1597.01	0.827582
$156$	0	0
$157$	−505.403	−0.256915	−0.128457	−	0.991715i	$-0.541003\pi$
$157$	−505.403	−0.256915	−0.128457	+	0.991715i	$0.541003\pi$
$158$	0	0
$159$	−1784.59	−0.890110
$160$	0	0
$161$	−752.792	−0.368499
$162$	0	0
$163$	2604.26	1.25142	0.625711	−	0.780055i	$-0.284809\pi$
$163$	2604.26	1.25142	0.625711	+	0.780055i	$0.284809\pi$
$164$	0	0
$165$	1649.59	0.778305
$166$	0	0
$167$	−845.304	−0.391686	−0.195843	−	0.980635i	$-0.562744\pi$
$167$	−845.304	−0.391686	−0.195843	+	0.980635i	$0.562744\pi$
$168$	0	0
$169$	−1986.88	−0.904359
$170$	0	0
$171$	1436.79	0.642539
$172$	0	0
$173$	−886.843	−0.389742	−0.194871	−	0.980829i	$-0.562429\pi$
$173$	−886.843	−0.389742	−0.194871	+	0.980829i	$0.562429\pi$
$174$	0	0
$175$	6478.44	2.79842
$176$	0	0
$177$	−128.174	−0.0544303
$178$	0	0
$179$	−1103.01	−0.460575	−0.230288	−	0.973123i	$-0.573967\pi$
$179$	−1103.01	−0.460575	−0.230288	+	0.973123i	$0.573967\pi$
$180$	0	0
$181$	1200.43	0.492967	0.246483	−	0.969147i	$-0.420725\pi$
$181$	1200.43	0.492967	0.246483	+	0.969147i	$0.420725\pi$
$182$	0	0
$183$	331.037	0.133721
$184$	0	0
$185$	5489.37	2.18155
$186$	0	0
$187$	−659.653	−0.257961
$188$	0	0
$189$	4745.98	1.82656
$190$	0	0
$191$	2415.93	0.915237	0.457618	−	0.889149i	$-0.348703\pi$
$191$	2415.93	0.915237	0.457618	+	0.889149i	$0.348703\pi$
$192$	0	0
$193$	232.884	0.0868568	0.0434284	−	0.999057i	$-0.486172\pi$
$193$	232.884	0.0868568	0.0434284	+	0.999057i	$0.486172\pi$
$194$	0	0
$195$	−895.410	−0.328829
$196$	0	0
$197$	1418.48	0.513008	0.256504	−	0.966543i	$-0.417429\pi$
$197$	1418.48	0.513008	0.256504	+	0.966543i	$0.417429\pi$
$198$	0	0
$199$	1068.21	0.380520	0.190260	−	0.981734i	$-0.439067\pi$
$199$	1068.21	0.380520	0.190260	+	0.981734i	$0.439067\pi$
$200$	0	0
$201$	1710.27	0.600165
$202$	0	0
$203$	1885.14	0.651777
$204$	0	0
$205$	−3220.39	−1.09718
$206$	0	0
$207$	349.243	0.117266
$208$	0	0
$209$	2526.88	0.836307
$210$	0	0
$211$	1537.74	0.501717	0.250859	−	0.968024i	$-0.419287\pi$
$211$	1537.74	0.501717	0.250859	+	0.968024i	$0.419287\pi$
$212$	0	0
$213$	67.4532	0.0216987
$214$	0	0
$215$	1735.63	0.550553
$216$	0	0
$217$	2908.69	0.909930
$218$	0	0
$219$	715.783	0.220859
$220$	0	0
$221$	358.065	0.108987
$222$	0	0
$223$	−5359.68	−1.60946	−0.804732	−	0.593638i	$-0.797691\pi$
$223$	−5359.68	−1.60946	−0.804732	+	0.593638i	$0.797691\pi$
$224$	0	0
$225$	−3005.54	−0.890532
$226$	0	0
$227$	−1108.36	−0.324072	−0.162036	−	0.986785i	$-0.551806\pi$
$227$	−1108.36	−0.324072	−0.162036	+	0.986785i	$0.551806\pi$
$228$	0	0
$229$	5046.34	1.45621	0.728104	−	0.685467i	$-0.240402\pi$
$229$	5046.34	1.45621	0.728104	+	0.685467i	$0.240402\pi$
$230$	0	0
$231$	3004.45	0.855750
$232$	0	0
$233$	7102.11	1.99689	0.998444	−	0.0557635i	$-0.0177593\pi$
$233$	7102.11	1.99689	0.998444	+	0.0557635i	$0.0177593\pi$
$234$	0	0
$235$	3926.25	1.08987
$236$	0	0
$237$	1533.75	0.420372
$238$	0	0
$239$	1556.22	0.421187	0.210594	−	0.977574i	$-0.432460\pi$
$239$	1556.22	0.421187	0.210594	+	0.977574i	$0.432460\pi$
$240$	0	0
$241$	3028.88	0.809573	0.404787	−	0.914411i	$-0.367346\pi$
$241$	3028.88	0.809573	0.404787	+	0.914411i	$0.367346\pi$
$242$	0	0
$243$	−3611.06	−0.953291
$244$	0	0
$245$	13087.1	3.41267
$246$	0	0
$247$	−1371.61	−0.353334
$248$	0	0
$249$	1722.64	0.438426
$250$	0	0
$251$	−1449.39	−0.364480	−0.182240	−	0.983254i	$-0.558335\pi$
$251$	−1449.39	−0.364480	−0.182240	+	0.983254i	$0.558335\pi$
$252$	0	0
$253$	614.213	0.152629
$254$	0	0
$255$	−1525.84	−0.374713
$256$	0	0
$257$	−3099.62	−0.752332	−0.376166	−	0.926552i	$-0.622758\pi$
$257$	−3099.62	−0.752332	−0.376166	+	0.926552i	$0.622758\pi$
$258$	0	0
$259$	9997.97	2.39862
$260$	0	0
$261$	−874.572	−0.207412
$262$	0	0
$263$	−1087.92	−0.255073	−0.127537	−	0.991834i	$-0.540707\pi$
$263$	−1087.92	−0.255073	−0.127537	+	0.991834i	$0.540707\pi$
$264$	0	0
$265$	9329.76	2.16273
$266$	0	0
$267$	−3790.79	−0.868887
$268$	0	0
$269$	−4721.01	−1.07006	−0.535028	−	0.844834i	$-0.679699\pi$
$269$	−4721.01	−1.07006	−0.535028	+	0.844834i	$0.679699\pi$
$270$	0	0
$271$	1401.15	0.314074	0.157037	−	0.987593i	$-0.449806\pi$
$271$	1401.15	0.314074	0.157037	+	0.987593i	$0.449806\pi$
$272$	0	0
$273$	−1630.84	−0.361549
$274$	0	0
$275$	−5285.85	−1.15909
$276$	0	0
$277$	−4122.82	−0.894283	−0.447142	−	0.894463i	$-0.647558\pi$
$277$	−4122.82	−0.894283	−0.447142	+	0.894463i	$0.647558\pi$
$278$	0	0
$279$	−1349.43	−0.289564
$280$	0	0
$281$	803.897	0.170664	0.0853318	−	0.996353i	$-0.472805\pi$
$281$	803.897	0.170664	0.0853318	+	0.996353i	$0.472805\pi$
$282$	0	0
$283$	3147.53	0.661134	0.330567	−	0.943782i	$-0.392760\pi$
$283$	3147.53	0.661134	0.330567	+	0.943782i	$0.392760\pi$
$284$	0	0
$285$	5844.92	1.21482
$286$	0	0
$287$	−5865.40	−1.20635
$288$	0	0
$289$	−4302.83	−0.875806
$290$	0	0
$291$	−6236.67	−1.25636
$292$	0	0
$293$	2812.88	0.560854	0.280427	−	0.959875i	$-0.409524\pi$
$293$	2812.88	0.560854	0.280427	+	0.959875i	$0.409524\pi$
$294$	0	0
$295$	670.088	0.132251
$296$	0	0
$297$	−3872.31	−0.756547
$298$	0	0
$299$	−333.400	−0.0644850
$300$	0	0
$301$	3161.15	0.605335
$302$	0	0
$303$	−4765.02	−0.903444
$304$	0	0
$305$	−1730.64	−0.324906
$306$	0	0
$307$	7885.52	1.46596	0.732981	−	0.680249i	$-0.238128\pi$
$307$	7885.52	1.46596	0.732981	+	0.680249i	$0.238128\pi$
$308$	0	0
$309$	−4717.22	−0.868457
$310$	0	0
$311$	−1904.46	−0.347240	−0.173620	−	0.984813i	$-0.555547\pi$
$311$	−1904.46	−0.347240	−0.173620	+	0.984813i	$0.555547\pi$
$312$	0	0
$313$	−3854.89	−0.696137	−0.348069	−	0.937469i	$-0.613162\pi$
$313$	−3854.89	−0.696137	−0.348069	+	0.937469i	$0.613162\pi$
$314$	0	0
$315$	−8931.09	−1.59749
$316$	0	0
$317$	−1878.70	−0.332865	−0.166432	−	0.986053i	$-0.553225\pi$
$317$	−1878.70	−0.332865	−0.166432	+	0.986053i	$0.553225\pi$
$318$	0	0
$319$	−1538.11	−0.269961
$320$	0	0
$321$	5698.58	0.990853
$322$	0	0
$323$	−2337.32	−0.402638
$324$	0	0
$325$	2869.20	0.489707
$326$	0	0
$327$	−2823.35	−0.477467
$328$	0	0
$329$	7151.00	1.19832
$330$	0	0
$331$	−6829.96	−1.13417	−0.567083	−	0.823661i	$-0.691928\pi$
$331$	−6829.96	−1.13417	−0.567083	+	0.823661i	$0.691928\pi$
$332$	0	0
$333$	−4638.36	−0.763305
$334$	0	0
$335$	−8941.21	−1.45824
$336$	0	0
$337$	11048.9	1.78597	0.892985	−	0.450087i	$-0.148607\pi$
$337$	11048.9	1.78597	0.892985	+	0.450087i	$0.148607\pi$
$338$	0	0
$339$	918.267	0.147119
$340$	0	0
$341$	−2373.24	−0.376886
$342$	0	0
$343$	12609.5	1.98499
$344$	0	0
$345$	1420.73	0.221709
$346$	0	0
$347$	7093.14	1.09735	0.548674	−	0.836037i	$-0.315133\pi$
$347$	7093.14	1.09735	0.548674	+	0.836037i	$0.315133\pi$
$348$	0	0
$349$	5363.16	0.822588	0.411294	−	0.911503i	$-0.365077\pi$
$349$	5363.16	0.822588	0.411294	+	0.911503i	$0.365077\pi$
$350$	0	0
$351$	2101.92	0.319636
$352$	0	0
$353$	11789.3	1.77757	0.888787	−	0.458320i	$-0.151549\pi$
$353$	11789.3	1.77757	0.888787	+	0.458320i	$0.151549\pi$
$354$	0	0
$355$	−352.642	−0.0527219
$356$	0	0
$357$	−2779.06	−0.411999
$358$	0	0
$359$	−2645.66	−0.388949	−0.194474	−	0.980908i	$-0.562300\pi$
$359$	−2645.66	−0.388949	−0.194474	+	0.980908i	$0.562300\pi$
$360$	0	0
$361$	2094.39	0.305349
$362$	0	0
$363$	2123.77	0.307077
$364$	0	0
$365$	−3742.07	−0.536628
$366$	0	0
$367$	−9775.68	−1.39043	−0.695213	−	0.718804i	$-0.744690\pi$
$367$	−9775.68	−1.39043	−0.695213	+	0.718804i	$0.744690\pi$
$368$	0	0
$369$	2721.13	0.383893
$370$	0	0
$371$	16992.6	2.37793
$372$	0	0
$373$	−10981.1	−1.52435	−0.762174	−	0.647373i	$-0.775868\pi$
$373$	−10981.1	−1.52435	−0.762174	+	0.647373i	$0.775868\pi$
$374$	0	0
$375$	−4505.30	−0.620407
$376$	0	0
$377$	834.899	0.114057
$378$	0	0
$379$	2313.77	0.313589	0.156795	−	0.987631i	$-0.449884\pi$
$379$	2313.77	0.313589	0.156795	+	0.987631i	$0.449884\pi$
$380$	0	0
$381$	4971.18	0.668455
$382$	0	0
$383$	9219.25	1.22998	0.614989	−	0.788535i	$-0.289160\pi$
$383$	9219.25	1.22998	0.614989	+	0.788535i	$0.289160\pi$
$384$	0	0
$385$	−15707.1	−2.07924
$386$	0	0
$387$	−1466.55	−0.192633
$388$	0	0
$389$	−5876.90	−0.765991	−0.382996	−	0.923750i	$-0.625108\pi$
$389$	−5876.90	−0.765991	−0.382996	+	0.923750i	$0.625108\pi$
$390$	0	0
$391$	−568.136	−0.0734830
$392$	0	0
$393$	5016.25	0.643858
$394$	0	0
$395$	−8018.39	−1.02139
$396$	0	0
$397$	−14268.6	−1.80383	−0.901916	−	0.431911i	$-0.857840\pi$
$397$	−14268.6	−1.80383	−0.901916	+	0.431911i	$0.857840\pi$
$398$	0	0
$399$	10645.5	1.33570
$400$	0	0
$401$	−11556.1	−1.43911	−0.719557	−	0.694434i	$-0.755655\pi$
$401$	−11556.1	−1.43911	−0.719557	+	0.694434i	$0.755655\pi$
$402$	0	0
$403$	1288.21	0.159232
$404$	0	0
$405$	−1589.51	−0.195020
$406$	0	0
$407$	−8157.48	−0.993492
$408$	0	0
$409$	−14148.3	−1.71049	−0.855245	−	0.518223i	$-0.826594\pi$
$409$	−14148.3	−1.71049	−0.855245	+	0.518223i	$0.826594\pi$
$410$	0	0
$411$	6494.27	0.779413
$412$	0	0
$413$	1220.45	0.145410
$414$	0	0
$415$	−9005.88	−1.06526
$416$	0	0
$417$	−5177.73	−0.608045
$418$	0	0
$419$	2222.16	0.259092	0.129546	−	0.991573i	$-0.458648\pi$
$419$	2222.16	0.259092	0.129546	+	0.991573i	$0.458648\pi$
$420$	0	0
$421$	−2518.51	−0.291555	−0.145777	−	0.989317i	$-0.546568\pi$
$421$	−2518.51	−0.291555	−0.145777	+	0.989317i	$0.546568\pi$
$422$	0	0
$423$	−3317.56	−0.381337
$424$	0	0
$425$	4889.31	0.558039
$426$	0	0
$427$	−3152.08	−0.357236
$428$	0	0
$429$	1330.62	0.149751
$430$	0	0
$431$	9935.17	1.11035	0.555174	−	0.831734i	$-0.312652\pi$
$431$	9935.17	1.11035	0.555174	+	0.831734i	$0.312652\pi$
$432$	0	0
$433$	−8427.47	−0.935331	−0.467665	−	0.883906i	$-0.654905\pi$
$433$	−8427.47	−0.935331	−0.467665	+	0.883906i	$0.654905\pi$
$434$	0	0
$435$	−3557.79	−0.392145
$436$	0	0
$437$	2176.31	0.238232
$438$	0	0
$439$	4886.04	0.531203	0.265601	−	0.964083i	$-0.414430\pi$
$439$	4886.04	0.531203	0.265601	+	0.964083i	$0.414430\pi$
$440$	0	0
$441$	−11058.2	−1.19406
$442$	0	0
$443$	−9724.04	−1.04290	−0.521448	−	0.853283i	$-0.674608\pi$
$443$	−9724.04	−1.04290	−0.521448	+	0.853283i	$0.674608\pi$
$444$	0	0
$445$	19818.1	2.11116
$446$	0	0
$447$	−10127.8	−1.07165
$448$	0	0
$449$	−10093.7	−1.06091	−0.530457	−	0.847712i	$-0.677980\pi$
$449$	−10093.7	−1.06091	−0.530457	+	0.847712i	$0.677980\pi$
$450$	0	0
$451$	4785.66	0.499662
$452$	0	0
$453$	−6803.83	−0.705678
$454$	0	0
$455$	8525.95	0.878467
$456$	0	0
$457$	3561.77	0.364578	0.182289	−	0.983245i	$-0.441649\pi$
$457$	3561.77	0.364578	0.182289	+	0.983245i	$0.441649\pi$
$458$	0	0
$459$	3581.82	0.364237
$460$	0	0
$461$	7492.00	0.756914	0.378457	−	0.925619i	$-0.376455\pi$
$461$	7492.00	0.756914	0.378457	+	0.925619i	$0.376455\pi$
$462$	0	0
$463$	15427.0	1.54849	0.774247	−	0.632884i	$-0.218129\pi$
$463$	15427.0	1.54849	0.774247	+	0.632884i	$0.218129\pi$
$464$	0	0
$465$	−5489.53	−0.547464
$466$	0	0
$467$	−11868.7	−1.17606	−0.588028	−	0.808841i	$-0.700096\pi$
$467$	−11868.7	−1.17606	−0.588028	+	0.808841i	$0.700096\pi$
$468$	0	0
$469$	−16284.9	−1.60334
$470$	0	0
$471$	1737.26	0.169955
$472$	0	0
$473$	−2579.23	−0.250725
$474$	0	0
$475$	−18729.1	−1.80916
$476$	0	0
$477$	−7883.38	−0.756719
$478$	0	0
$479$	697.153	0.0665005	0.0332503	−	0.999447i	$-0.489414\pi$
$479$	697.153	0.0665005	0.0332503	+	0.999447i	$0.489414\pi$
$480$	0	0
$481$	4427.95	0.419744
$482$	0	0
$483$	2587.63	0.243770
$484$	0	0
$485$	32605.0	3.05261
$486$	0	0
$487$	14414.0	1.34119	0.670597	−	0.741822i	$-0.266038\pi$
$487$	14414.0	1.34119	0.670597	+	0.741822i	$0.266038\pi$
$488$	0	0
$489$	−8951.83	−0.827844
$490$	0	0
$491$	−17538.9	−1.61206	−0.806029	−	0.591876i	$-0.798388\pi$
$491$	−17538.9	−1.61206	−0.806029	+	0.591876i	$0.798388\pi$
$492$	0	0
$493$	1422.72	0.129972
$494$	0	0
$495$	7287.00	0.661669
$496$	0	0
$497$	−642.278	−0.0579680
$498$	0	0
$499$	18798.6	1.68645	0.843226	−	0.537559i	$-0.180653\pi$
$499$	18798.6	1.68645	0.843226	+	0.537559i	$0.180653\pi$
$500$	0	0
$501$	2905.62	0.259109
$502$	0	0
$503$	4634.11	0.410785	0.205392	−	0.978680i	$-0.434153\pi$
$503$	4634.11	0.410785	0.205392	+	0.978680i	$0.434153\pi$
$504$	0	0
$505$	24911.3	2.19513
$506$	0	0
$507$	6829.63	0.598254
$508$	0	0
$509$	2193.57	0.191018	0.0955092	−	0.995429i	$-0.469552\pi$
$509$	2193.57	0.191018	0.0955092	+	0.995429i	$0.469552\pi$
$510$	0	0
$511$	−6815.56	−0.590025
$512$	0	0
$513$	−13720.6	−1.18086
$514$	0	0
$515$	24661.4	2.11012
$516$	0	0
$517$	−5834.60	−0.496335
$518$	0	0
$519$	3048.41	0.257823
$520$	0	0
$521$	−4401.81	−0.370147	−0.185074	−	0.982725i	$-0.559252\pi$
$521$	−4401.81	−0.370147	−0.185074	+	0.982725i	$0.559252\pi$
$522$	0	0
$523$	9974.09	0.833913	0.416957	−	0.908926i	$-0.363097\pi$
$523$	9974.09	0.833913	0.416957	+	0.908926i	$0.363097\pi$
$524$	0	0
$525$	−22268.8	−1.85122
$526$	0	0
$527$	2195.20	0.181451
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−566.204	−0.0462734
$532$	0	0
$533$	−2597.69	−0.211104
$534$	0	0
$535$	−29791.9	−2.40751
$536$	0	0
$537$	3791.46	0.304681
$538$	0	0
$539$	−19448.1	−1.55415
$540$	0	0
$541$	9161.72	0.728083	0.364042	−	0.931383i	$-0.381397\pi$
$541$	9161.72	0.728083	0.364042	+	0.931383i	$0.381397\pi$
$542$	0	0
$543$	−4126.31	−0.326109
$544$	0	0
$545$	14760.3	1.16012
$546$	0	0
$547$	−1113.51	−0.0870390	−0.0435195	−	0.999053i	$-0.513857\pi$
$547$	−1113.51	−0.0870390	−0.0435195	+	0.999053i	$0.513857\pi$
$548$	0	0
$549$	1462.34	0.113682
$550$	0	0
$551$	−5449.92	−0.421369
$552$	0	0
$553$	−14604.2	−1.12302
$554$	0	0
$555$	−18869.0	−1.44314
$556$	0	0
$557$	7660.96	0.582775	0.291387	−	0.956605i	$-0.405883\pi$
$557$	7660.96	0.582775	0.291387	+	0.956605i	$0.405883\pi$
$558$	0	0
$559$	1400.03	0.105930
$560$	0	0
$561$	2267.47	0.170647
$562$	0	0
$563$	−17217.7	−1.28888	−0.644441	−	0.764654i	$-0.722910\pi$
$563$	−17217.7	−1.28888	−0.644441	+	0.764654i	$0.722910\pi$
$564$	0	0
$565$	−4800.65	−0.357460
$566$	0	0
$567$	−2895.02	−0.214426
$568$	0	0
$569$	−2385.63	−0.175766	−0.0878830	−	0.996131i	$-0.528010\pi$
$569$	−2385.63	−0.175766	−0.0878830	+	0.996131i	$0.528010\pi$
$570$	0	0
$571$	−16927.9	−1.24065	−0.620323	−	0.784347i	$-0.712998\pi$
$571$	−16927.9	−1.24065	−0.620323	+	0.784347i	$0.712998\pi$
$572$	0	0
$573$	−8304.44	−0.605450
$574$	0	0
$575$	−4552.52	−0.330179
$576$	0	0
$577$	2886.69	0.208275	0.104137	−	0.994563i	$-0.466792\pi$
$577$	2886.69	0.208275	0.104137	+	0.994563i	$0.466792\pi$
$578$	0	0
$579$	−800.509	−0.0574578
$580$	0	0
$581$	−16402.7	−1.17125
$582$	0	0
$583$	−13864.5	−0.984920
$584$	0	0
$585$	−3955.44	−0.279551
$586$	0	0
$587$	−503.810	−0.0354250	−0.0177125	−	0.999843i	$-0.505638\pi$
$587$	−503.810	−0.0354250	−0.0177125	+	0.999843i	$0.505638\pi$
$588$	0	0
$589$	−8409.00	−0.588263
$590$	0	0
$591$	−4875.85	−0.339366
$592$	0	0
$593$	−7654.92	−0.530101	−0.265050	−	0.964235i	$-0.585389\pi$
$593$	−7654.92	−0.530101	−0.265050	+	0.964235i	$0.585389\pi$
$594$	0	0
$595$	14528.8	1.00105
$596$	0	0
$597$	−3671.84	−0.251723
$598$	0	0
$599$	17366.5	1.18460	0.592301	−	0.805717i	$-0.298220\pi$
$599$	17366.5	1.18460	0.592301	+	0.805717i	$0.298220\pi$
$600$	0	0
$601$	3872.81	0.262854	0.131427	−	0.991326i	$-0.458044\pi$
$601$	3872.81	0.262854	0.131427	+	0.991326i	$0.458044\pi$
$602$	0	0
$603$	7555.06	0.510225
$604$	0	0
$605$	−11103.0	−0.746115
$606$	0	0
$607$	−12823.3	−0.857464	−0.428732	−	0.903432i	$-0.641039\pi$
$607$	−12823.3	−0.857464	−0.428732	+	0.903432i	$0.641039\pi$
$608$	0	0
$609$	−6479.92	−0.431165
$610$	0	0
$611$	3167.07	0.209699
$612$	0	0
$613$	17226.4	1.13502	0.567509	−	0.823367i	$-0.307907\pi$
$613$	17226.4	1.13502	0.567509	+	0.823367i	$0.307907\pi$
$614$	0	0
$615$	11069.7	0.725809
$616$	0	0
$617$	−5892.12	−0.384454	−0.192227	−	0.981351i	$-0.561571\pi$
$617$	−5892.12	−0.384454	−0.192227	+	0.981351i	$0.561571\pi$
$618$	0	0
$619$	13036.6	0.846503	0.423251	−	0.906012i	$-0.360889\pi$
$619$	13036.6	0.846503	0.423251	+	0.906012i	$0.360889\pi$
$620$	0	0
$621$	−3335.09	−0.215511
$622$	0	0
$623$	36095.3	2.32123
$624$	0	0
$625$	−1188.48	−0.0760630
$626$	0	0
$627$	−8685.84	−0.553236
$628$	0	0
$629$	7545.52	0.478314
$630$	0	0
$631$	−10044.6	−0.633707	−0.316853	−	0.948475i	$-0.602626\pi$
$631$	−10044.6	−0.633707	−0.316853	+	0.948475i	$0.602626\pi$
$632$	0	0
$633$	−5285.79	−0.331898
$634$	0	0
$635$	−25989.1	−1.62416
$636$	0	0
$637$	10556.6	0.656620
$638$	0	0
$639$	297.972	0.0184469
$640$	0	0
$641$	4583.33	0.282419	0.141209	−	0.989980i	$-0.454901\pi$
$641$	4583.33	0.282419	0.141209	+	0.989980i	$0.454901\pi$
$642$	0	0
$643$	−19260.0	−1.18125	−0.590623	−	0.806948i	$-0.701118\pi$
$643$	−19260.0	−1.18125	−0.590623	+	0.806948i	$0.701118\pi$
$644$	0	0
$645$	−5966.00	−0.364203
$646$	0	0
$647$	−1771.91	−0.107667	−0.0538337	−	0.998550i	$-0.517144\pi$
$647$	−1771.91	−0.107667	−0.0538337	+	0.998550i	$0.517144\pi$
$648$	0	0
$649$	−995.784	−0.0602279
$650$	0	0
$651$	−9998.26	−0.601940
$652$	0	0
$653$	28000.8	1.67803	0.839017	−	0.544106i	$-0.183131\pi$
$653$	28000.8	1.67803	0.839017	+	0.544106i	$0.183131\pi$
$654$	0	0
$655$	−26224.7	−1.56440
$656$	0	0
$657$	3161.94	0.187761
$658$	0	0
$659$	27664.0	1.63526	0.817629	−	0.575745i	$-0.195288\pi$
$659$	27664.0	1.63526	0.817629	+	0.575745i	$0.195288\pi$
$660$	0	0
$661$	−23392.1	−1.37647	−0.688234	−	0.725489i	$-0.741614\pi$
$661$	−23392.1	−1.37647	−0.688234	+	0.725489i	$0.741614\pi$
$662$	0	0
$663$	−1230.80	−0.0720972
$664$	0	0
$665$	−55654.3	−3.24539
$666$	0	0
$667$	−1324.72	−0.0769016
$668$	0	0
$669$	18423.2	1.06470
$670$	0	0
$671$	2571.82	0.147964
$672$	0	0
$673$	4318.41	0.247344	0.123672	−	0.992323i	$-0.460533\pi$
$673$	4318.41	0.247344	0.123672	+	0.992323i	$0.460533\pi$
$674$	0	0
$675$	28701.4	1.63662
$676$	0	0
$677$	−18272.8	−1.03734	−0.518671	−	0.854974i	$-0.673573\pi$
$677$	−18272.8	−1.03734	−0.518671	+	0.854974i	$0.673573\pi$
$678$	0	0
$679$	59384.5	3.35636
$680$	0	0
$681$	3809.84	0.214381
$682$	0	0
$683$	7245.38	0.405910	0.202955	−	0.979188i	$-0.434945\pi$
$683$	7245.38	0.405910	0.202955	+	0.979188i	$0.434945\pi$
$684$	0	0
$685$	−33951.7	−1.89376
$686$	0	0
$687$	−17346.2	−0.963314
$688$	0	0
$689$	7525.76	0.416123
$690$	0	0
$691$	12055.2	0.663679	0.331839	−	0.943336i	$-0.392331\pi$
$691$	12055.2	0.663679	0.331839	+	0.943336i	$0.392331\pi$
$692$	0	0
$693$	13272.0	0.727509
$694$	0	0
$695$	27068.9	1.47738
$696$	0	0
$697$	−4426.64	−0.240561
$698$	0	0
$699$	−24412.6	−1.32099
$700$	0	0
$701$	15301.9	0.824455	0.412228	−	0.911081i	$-0.364751\pi$
$701$	15301.9	0.824455	0.412228	+	0.911081i	$0.364751\pi$
$702$	0	0
$703$	−28904.0	−1.55069
$704$	0	0
$705$	−13496.0	−0.720975
$706$	0	0
$707$	45371.8	2.41355
$708$	0	0
$709$	12173.6	0.644835	0.322418	−	0.946597i	$-0.395504\pi$
$709$	12173.6	0.644835	0.322418	+	0.946597i	$0.395504\pi$
$710$	0	0
$711$	6775.30	0.357375
$712$	0	0
$713$	−2043.99	−0.107360
$714$	0	0
$715$	−6956.44	−0.363854
$716$	0	0
$717$	−5349.32	−0.278625
$718$	0	0
$719$	2639.63	0.136914	0.0684572	−	0.997654i	$-0.478192\pi$
$719$	2639.63	0.136914	0.0684572	+	0.997654i	$0.478192\pi$
$720$	0	0
$721$	44916.5	2.32008
$722$	0	0
$723$	−10411.4	−0.535551
$724$	0	0
$725$	11400.4	0.584000
$726$	0	0
$727$	−32759.5	−1.67123	−0.835613	−	0.549319i	$-0.814887\pi$
$727$	−32759.5	−1.67123	−0.835613	+	0.549319i	$0.814887\pi$
$728$	0	0
$729$	14800.7	0.751956
$730$	0	0
$731$	2385.74	0.120711
$732$	0	0
$733$	−29178.0	−1.47028	−0.735139	−	0.677917i	$-0.762883\pi$
$733$	−29178.0	−1.47028	−0.735139	+	0.677917i	$0.762883\pi$
$734$	0	0
$735$	−44985.2	−2.25756
$736$	0	0
$737$	13287.1	0.664092
$738$	0	0
$739$	13704.9	0.682194	0.341097	−	0.940028i	$-0.389202\pi$
$739$	13704.9	0.682194	0.341097	+	0.940028i	$0.389202\pi$
$740$	0	0
$741$	4714.74	0.233739
$742$	0	0
$743$	−991.593	−0.0489610	−0.0244805	−	0.999700i	$-0.507793\pi$
$743$	−991.593	−0.0489610	−0.0244805	+	0.999700i	$0.507793\pi$
$744$	0	0
$745$	52947.6	2.60383
$746$	0	0
$747$	7609.71	0.372724
$748$	0	0
$749$	−54260.9	−2.64706
$750$	0	0
$751$	9440.73	0.458718	0.229359	−	0.973342i	$-0.426337\pi$
$751$	9440.73	0.458718	0.229359	+	0.973342i	$0.426337\pi$
$752$	0	0
$753$	4982.09	0.241112
$754$	0	0
$755$	35570.1	1.71461
$756$	0	0
$757$	−10480.1	−0.503177	−0.251589	−	0.967834i	$-0.580953\pi$
$757$	−10480.1	−0.503177	−0.251589	+	0.967834i	$0.580953\pi$
$758$	0	0
$759$	−2111.28	−0.100968
$760$	0	0
$761$	−31314.9	−1.49168	−0.745838	−	0.666128i	$-0.767951\pi$
$761$	−31314.9	−1.49168	−0.745838	+	0.666128i	$0.767951\pi$
$762$	0	0
$763$	26883.5	1.27555
$764$	0	0
$765$	−6740.34	−0.318559
$766$	0	0
$767$	540.519	0.0254459
$768$	0	0
$769$	−20862.1	−0.978292	−0.489146	−	0.872202i	$-0.662691\pi$
$769$	−20862.1	−0.978292	−0.489146	+	0.872202i	$0.662691\pi$
$770$	0	0
$771$	10654.6	0.497684
$772$	0	0
$773$	20340.2	0.946426	0.473213	−	0.880948i	$-0.343094\pi$
$773$	20340.2	0.946426	0.473213	+	0.880948i	$0.343094\pi$
$774$	0	0
$775$	17590.3	0.815308
$776$	0	0
$777$	−34366.8	−1.58674
$778$	0	0
$779$	16956.8	0.779898
$780$	0	0
$781$	524.043	0.0240099
$782$	0	0
$783$	8351.71	0.381182
$784$	0	0
$785$	−9082.30	−0.412944
$786$	0	0
$787$	−31293.8	−1.41741	−0.708707	−	0.705503i	$-0.750721\pi$
$787$	−31293.8	−1.41741	−0.708707	+	0.705503i	$0.750721\pi$
$788$	0	0
$789$	3739.60	0.168737
$790$	0	0
$791$	−8743.58	−0.393029
$792$	0	0
$793$	−1396.01	−0.0625140
$794$	0	0
$795$	−32069.9	−1.43069
$796$	0	0
$797$	−32015.4	−1.42289	−0.711446	−	0.702741i	$-0.751959\pi$
$797$	−32015.4	−1.42289	−0.711446	+	0.702741i	$0.751959\pi$
$798$	0	0
$799$	5396.89	0.238959
$800$	0	0
$801$	−16745.7	−0.738677
$802$	0	0
$803$	5560.91	0.244384
$804$	0	0
$805$	−13528.0	−0.592296
$806$	0	0
$807$	16227.9	0.707867
$808$	0	0
$809$	5234.16	0.227470	0.113735	−	0.993511i	$-0.463719\pi$
$809$	5234.16	0.227470	0.113735	+	0.993511i	$0.463719\pi$
$810$	0	0
$811$	2377.40	0.102937	0.0514685	−	0.998675i	$-0.483610\pi$
$811$	2377.40	0.102937	0.0514685	+	0.998675i	$0.483610\pi$
$812$	0	0
$813$	−4816.29	−0.207767
$814$	0	0
$815$	46799.7	2.01144
$816$	0	0
$817$	−9138.87	−0.391345
$818$	0	0
$819$	−7204.18	−0.307368
$820$	0	0
$821$	−5174.70	−0.219974	−0.109987	−	0.993933i	$-0.535081\pi$
$821$	−5174.70	−0.219974	−0.109987	+	0.993933i	$0.535081\pi$
$822$	0	0
$823$	44178.3	1.87115	0.935576	−	0.353126i	$-0.114881\pi$
$823$	44178.3	1.87115	0.935576	+	0.353126i	$0.114881\pi$
$824$	0	0
$825$	18169.4	0.766762
$826$	0	0
$827$	−5766.56	−0.242470	−0.121235	−	0.992624i	$-0.538686\pi$
$827$	−5766.56	−0.242470	−0.121235	+	0.992624i	$0.538686\pi$
$828$	0	0
$829$	−38465.6	−1.61154	−0.805770	−	0.592229i	$-0.798248\pi$
$829$	−38465.6	−1.61154	−0.805770	+	0.592229i	$0.798248\pi$
$830$	0	0
$831$	14171.7	0.591589
$832$	0	0
$833$	17989.1	0.748242
$834$	0	0
$835$	−15190.5	−0.629566
$836$	0	0
$837$	12886.3	0.532159
$838$	0	0
$839$	−17261.8	−0.710301	−0.355151	−	0.934809i	$-0.615570\pi$
$839$	−17261.8	−0.710301	−0.355151	+	0.934809i	$0.615570\pi$
$840$	0	0
$841$	−21071.6	−0.863981
$842$	0	0
$843$	−2763.29	−0.112898
$844$	0	0
$845$	−35705.0	−1.45360
$846$	0	0
$847$	−20222.2	−0.820358
$848$	0	0
$849$	−10819.2	−0.437355
$850$	0	0
$851$	−7025.75	−0.283008
$852$	0	0
$853$	−29038.7	−1.16561	−0.582805	−	0.812612i	$-0.698045\pi$
$853$	−29038.7	−1.16561	−0.582805	+	0.812612i	$0.698045\pi$
$854$	0	0
$855$	25819.7	1.03277
$856$	0	0
$857$	−9865.16	−0.393217	−0.196609	−	0.980482i	$-0.562993\pi$
$857$	−9865.16	−0.393217	−0.196609	+	0.980482i	$0.562993\pi$
$858$	0	0
$859$	24476.6	0.972214	0.486107	−	0.873899i	$-0.338416\pi$
$859$	24476.6	0.972214	0.486107	+	0.873899i	$0.338416\pi$
$860$	0	0
$861$	20161.6	0.798030
$862$	0	0
$863$	−37353.6	−1.47339	−0.736693	−	0.676227i	$-0.763614\pi$
$863$	−37353.6	−1.47339	−0.736693	+	0.676227i	$0.763614\pi$
$864$	0	0
$865$	−15936.9	−0.626442
$866$	0	0
$867$	14790.4	0.579365
$868$	0	0
$869$	11915.7	0.465148
$870$	0	0
$871$	−7212.34	−0.280575
$872$	0	0
$873$	−27550.3	−1.06808
$874$	0	0
$875$	42898.7	1.65742
$876$	0	0
$877$	25920.7	0.998040	0.499020	−	0.866590i	$-0.333693\pi$
$877$	25920.7	0.998040	0.499020	+	0.866590i	$0.333693\pi$
$878$	0	0
$879$	−9668.92	−0.371018
$880$	0	0
$881$	−9337.06	−0.357064	−0.178532	−	0.983934i	$-0.557135\pi$
$881$	−9337.06	−0.357064	−0.178532	+	0.983934i	$0.557135\pi$
$882$	0	0
$883$	70.8656	0.00270081	0.00135041	−	0.999999i	$-0.499570\pi$
$883$	70.8656	0.00270081	0.00135041	+	0.999999i	$0.499570\pi$
$884$	0	0
$885$	−2303.34	−0.0874870
$886$	0	0
$887$	−1058.85	−0.0400819	−0.0200409	−	0.999799i	$-0.506380\pi$
$887$	−1058.85	−0.0400819	−0.0200409	+	0.999799i	$0.506380\pi$
$888$	0	0
$889$	−47334.7	−1.78578
$890$	0	0
$891$	2362.08	0.0888135
$892$	0	0
$893$	−20673.5	−0.774705
$894$	0	0
$895$	−19821.6	−0.740293
$896$	0	0
$897$	1146.02	0.0426583
$898$	0	0
$899$	5118.55	0.189892
$900$	0	0
$901$	12824.4	0.474187
$902$	0	0
$903$	−10866.1	−0.400443
$904$	0	0
$905$	21572.1	0.792356
$906$	0	0
$907$	−29050.2	−1.06350	−0.531751	−	0.846900i	$-0.678466\pi$
$907$	−29050.2	−1.06350	−0.531751	+	0.846900i	$0.678466\pi$
$908$	0	0
$909$	−21049.3	−0.768055
$910$	0	0
$911$	16041.6	0.583404	0.291702	−	0.956509i	$-0.405778\pi$
$911$	16041.6	0.583404	0.291702	+	0.956509i	$0.405778\pi$
$912$	0	0
$913$	13383.2	0.485125
$914$	0	0
$915$	5948.87	0.214933
$916$	0	0
$917$	−47763.9	−1.72007
$918$	0	0
$919$	−30151.2	−1.08226	−0.541130	−	0.840939i	$-0.682003\pi$
$919$	−30151.2	−1.08226	−0.541130	+	0.840939i	$0.682003\pi$
$920$	0	0
$921$	−27105.5	−0.969767
$922$	0	0
$923$	−284.455	−0.0101440
$924$	0	0
$925$	60462.8	2.14919
$926$	0	0
$927$	−20838.1	−0.738311
$928$	0	0
$929$	43339.8	1.53061	0.765303	−	0.643670i	$-0.222589\pi$
$929$	43339.8	1.53061	0.765303	+	0.643670i	$0.222589\pi$
$930$	0	0
$931$	−68909.5	−2.42580
$932$	0	0
$933$	6546.32	0.229707
$934$	0	0
$935$	−11854.2	−0.414626
$936$	0	0
$937$	27291.9	0.951533	0.475766	−	0.879572i	$-0.342171\pi$
$937$	27291.9	0.951533	0.475766	+	0.879572i	$0.342171\pi$
$938$	0	0
$939$	13250.7	0.460511
$940$	0	0
$941$	4358.15	0.150980	0.0754898	−	0.997147i	$-0.475948\pi$
$941$	4358.15	0.150980	0.0754898	+	0.997147i	$0.475948\pi$
$942$	0	0
$943$	4121.72	0.142335
$944$	0	0
$945$	85287.3	2.93587
$946$	0	0
$947$	−39067.8	−1.34059	−0.670293	−	0.742097i	$-0.733831\pi$
$947$	−39067.8	−1.34059	−0.670293	+	0.742097i	$0.733831\pi$
$948$	0	0
$949$	−3018.51	−0.103251
$950$	0	0
$951$	6457.78	0.220197
$952$	0	0
$953$	−52563.8	−1.78668	−0.893341	−	0.449379i	$-0.851645\pi$
$953$	−52563.8	−1.78668	−0.893341	+	0.449379i	$0.851645\pi$
$954$	0	0
$955$	43415.2	1.47108
$956$	0	0
$957$	5287.06	0.178585
$958$	0	0
$959$	−61837.4	−2.08220
$960$	0	0
$961$	−21893.3	−0.734896
$962$	0	0
$963$	25173.3	0.842365
$964$	0	0
$965$	4185.02	0.139607
$966$	0	0
$967$	−2440.46	−0.0811580	−0.0405790	−	0.999176i	$-0.512920\pi$
$967$	−2440.46	−0.0811580	−0.0405790	+	0.999176i	$0.512920\pi$
$968$	0	0
$969$	8034.24	0.266354
$970$	0	0
$971$	−45490.7	−1.50347	−0.751733	−	0.659468i	$-0.770782\pi$
$971$	−45490.7	−1.50347	−0.751733	+	0.659468i	$0.770782\pi$
$972$	0	0
$973$	49301.5	1.62439
$974$	0	0
$975$	−9862.52	−0.323952
$976$	0	0
$977$	−33244.4	−1.08862	−0.544311	−	0.838884i	$-0.683209\pi$
$977$	−33244.4	−1.08862	−0.544311	+	0.838884i	$0.683209\pi$
$978$	0	0
$979$	−29450.6	−0.961437
$980$	0	0
$981$	−12472.1	−0.405914
$982$	0	0
$983$	1171.32	0.0380053	0.0190027	−	0.999819i	$-0.493951\pi$
$983$	1171.32	0.0380053	0.0190027	+	0.999819i	$0.493951\pi$
$984$	0	0
$985$	25490.7	0.824569
$986$	0	0
$987$	−24580.6	−0.792716
$988$	0	0
$989$	−2221.40	−0.0714220
$990$	0	0
$991$	−3714.32	−0.119061	−0.0595304	−	0.998226i	$-0.518960\pi$
$991$	−3714.32	−0.119061	−0.0595304	+	0.998226i	$0.518960\pi$
$992$	0	0
$993$	23477.1	0.750276
$994$	0	0
$995$	19196.2	0.611618
$996$	0	0
$997$	−19521.0	−0.620098	−0.310049	−	0.950721i	$-0.600345\pi$
$997$	−19521.0	−0.620098	−0.310049	+	0.950721i	$0.600345\pi$
$998$	0	0
$999$	44293.9	1.40280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.y.1.2		4
4.3	odd	2		1472.4.a.bf.1.3		4
8.3	odd	2		368.4.a.l.1.2		4
8.5	even	2		23.4.a.b.1.3	✓	4
24.5	odd	2		207.4.a.e.1.2		4
40.13	odd	4		575.4.b.g.24.3		8
40.29	even	2		575.4.a.i.1.2		4
40.37	odd	4		575.4.b.g.24.6		8
56.13	odd	2		1127.4.a.c.1.3		4
184.45	odd	2		529.4.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.b.1.3	✓	4	8.5	even	2
207.4.a.e.1.2		4	24.5	odd	2
368.4.a.l.1.2		4	8.3	odd	2
529.4.a.g.1.3		4	184.45	odd	2
575.4.a.i.1.2		4	40.29	even	2
575.4.b.g.24.3		8	40.13	odd	4
575.4.b.g.24.6		8	40.37	odd	4
1127.4.a.c.1.3		4	56.13	odd	2
1472.4.a.y.1.2		4	1.1	even	1	trivial
1472.4.a.bf.1.3		4	4.3	odd	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q-3.43737 q^{3} +17.9704 q^{5} +32.7301 q^{7} -15.1845 q^{9} -26.7049 q^{11} +14.4956 q^{13} -61.7710 q^{15} +24.7016 q^{17} -94.6224 q^{19} -112.505 q^{21} -23.0000 q^{23} +197.935 q^{25} +145.004 q^{27} +57.5965 q^{29} +88.8691 q^{31} +91.7948 q^{33} +588.173 q^{35} +305.467 q^{37} -49.8269 q^{39} -179.205 q^{41} +96.5826 q^{43} -272.871 q^{45} +218.484 q^{47} +728.258 q^{49} -84.9085 q^{51} +519.174 q^{53} -479.898 q^{55} +325.252 q^{57} +37.2884 q^{59} -96.3052 q^{61} -496.989 q^{63} +260.493 q^{65} -497.552 q^{67} +79.0596 q^{69} -19.6235 q^{71} -208.235 q^{73} -680.378 q^{75} -874.054 q^{77} -446.200 q^{79} -88.4513 q^{81} -501.151 q^{83} +443.897 q^{85} -197.981 q^{87} +1102.82 q^{89} +474.444 q^{91} -305.476 q^{93} -1700.40 q^{95} +1814.37 q^{97} +405.500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 7 q^{3} - 14 q^{5} + 16 q^{7} - 33 q^{9} - 8 q^{11} - 111 q^{13} + 10 q^{15} + 98 q^{17} - 96 q^{19} - 180 q^{21} - 92 q^{23} + 184 q^{25} + 155 q^{27} - 21 q^{29} - 193 q^{31} - 418 q^{33} + 752 q^{35}+ \cdots + 1498 q^{99}+O(q^{100}) \) 4 * q - 7 * q^3 - 14 * q^5 + 16 * q^7 - 33 * q^9 - 8 * q^11 - 111 * q^13 + 10 * q^15 + 98 * q^17 - 96 * q^19 - 180 * q^21 - 92 * q^23 + 184 * q^25 + 155 * q^27 - 21 * q^29 - 193 * q^31 - 418 * q^33 + 752 * q^35 - 170 * q^37 - 291 * q^39 - 125 * q^41 - 2 * q^43 - 168 * q^45 - 677 * q^47 + 1220 * q^49 + 340 * q^51 + 230 * q^53 - 972 * q^55 + 1322 * q^57 + 1140 * q^59 - 754 * q^61 - 1092 * q^63 + 1318 * q^65 - 488 * q^67 + 161 * q^69 - 401 * q^71 + 1509 * q^73 - 1401 * q^75 - 736 * q^77 - 838 * q^79 - 932 * q^81 - 142 * q^83 - 112 * q^85 + 2223 * q^87 + 2342 * q^89 - 292 * q^91 + 509 * q^93 - 956 * q^95 + 1062 * q^97 + 1498 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more