LMFDB - Embedded newform 675.4.a.bc.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,30,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 21x^{4} + 5x^{3} + 101x^{2} + 29x + 2 $ x^6 - x^5 - 21x^4 + 5x^3 + 101x^2 + 29x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.52770$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.62699 q^{2} +5.15507 q^{4} -22.4519 q^{7} -10.3185 q^{8} +52.4410 q^{11} +84.8342 q^{13} -81.4328 q^{14} -78.6658 q^{16} +45.4636 q^{17} -85.6658 q^{19} +190.203 q^{22} +55.7673 q^{23} +307.693 q^{26} -115.741 q^{28} +96.5180 q^{29} +107.596 q^{31} -202.772 q^{32} +164.896 q^{34} +81.7139 q^{37} -310.709 q^{38} +248.066 q^{41} +292.163 q^{43} +270.337 q^{44} +202.268 q^{46} +240.137 q^{47} +161.088 q^{49} +437.326 q^{52} +505.234 q^{53} +231.671 q^{56} +350.070 q^{58} +613.029 q^{59} +147.014 q^{61} +390.251 q^{62} -106.125 q^{64} +55.7251 q^{67} +234.368 q^{68} -541.270 q^{71} -73.4510 q^{73} +296.376 q^{74} -441.613 q^{76} -1177.40 q^{77} -1222.46 q^{79} +899.735 q^{82} -718.235 q^{83} +1059.67 q^{86} -541.115 q^{88} -540.674 q^{89} -1904.69 q^{91} +287.484 q^{92} +870.975 q^{94} +1669.58 q^{97} +584.263 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 30 q^{4} + 36 q^{7} + 162 q^{13} + 42 q^{16} + 450 q^{22} + 828 q^{28} + 126 q^{31} - 534 q^{34} + 1008 q^{37} + 558 q^{43} - 834 q^{46} + 1434 q^{49} + 2610 q^{52} - 270 q^{58} + 396 q^{61} - 1134 q^{64}+ \cdots + 4104 q^{97}+O(q^{100}) $ 6 * q + 30 * q^4 + 36 * q^7 + 162 * q^13 + 42 * q^16 + 450 * q^22 + 828 * q^28 + 126 * q^31 - 534 * q^34 + 1008 * q^37 + 558 * q^43 - 834 * q^46 + 1434 * q^49 + 2610 * q^52 - 270 * q^58 + 396 * q^61 - 1134 * q^64 + 2268 * q^67 + 144 * q^73 + 912 * q^76 - 1098 * q^79 + 5544 * q^82 + 702 * q^88 - 1692 * q^91 + 2574 * q^94 + 4104 * q^97

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 $

$2$	3.62699	1.28234	0.641168	−	0.767401i	$-0.278450\pi$
$2$	3.62699	1.28234	0.641168	+	0.767401i	$0.278450\pi$
$3$	0	0
$3$	0	0
$4$	5.15507	0.644383
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−22.4519	−1.21229	−0.606144	−	0.795355i	$-0.707285\pi$
$7$	−22.4519	−1.21229	−0.606144	+	0.795355i	$0.707285\pi$
$8$	−10.3185	−0.456020
$9$	0	0
$10$	0	0
$11$	52.4410	1.43742	0.718708	−	0.695312i	$-0.244734\pi$
$11$	52.4410	1.43742	0.718708	+	0.695312i	$0.244734\pi$
$12$	0	0
$13$	84.8342	1.80991	0.904953	−	0.425512i	$-0.139906\pi$
$13$	84.8342	1.80991	0.904953	+	0.425512i	$0.139906\pi$
$14$	−81.4328	−1.55456
$15$	0	0
$16$	−78.6658	−1.22915
$17$	45.4636	0.648620	0.324310	−	0.945951i	$-0.394868\pi$
$17$	45.4636	0.648620	0.324310	+	0.945951i	$0.394868\pi$
$18$	0	0
$19$	−85.6658	−1.03437	−0.517186	−	0.855873i	$-0.673021\pi$
$19$	−85.6658	−1.03437	−0.517186	+	0.855873i	$0.673021\pi$
$20$	0	0
$21$	0	0
$22$	190.203	1.84325
$23$	55.7673	0.505578	0.252789	−	0.967521i	$-0.418652\pi$
$23$	55.7673	0.505578	0.252789	+	0.967521i	$0.418652\pi$
$24$	0	0
$25$	0	0
$26$	307.693	2.32091
$27$	0	0
$28$	−115.741	−0.781179
$29$	96.5180	0.618033	0.309016	−	0.951057i	$-0.400000\pi$
$29$	96.5180	0.618033	0.309016	+	0.951057i	$0.400000\pi$
$30$	0	0
$31$	107.596	0.623382	0.311691	−	0.950183i	$-0.399105\pi$
$31$	107.596	0.623382	0.311691	+	0.950183i	$0.399105\pi$
$32$	−202.772	−1.12017
$33$	0	0
$34$	164.896	0.831748
$35$	0	0
$36$	0	0
$37$	81.7139	0.363073	0.181536	−	0.983384i	$-0.441893\pi$
$37$	81.7139	0.363073	0.181536	+	0.983384i	$0.441893\pi$
$38$	−310.709	−1.32641
$39$	0	0
$40$	0	0
$41$	248.066	0.944914	0.472457	−	0.881354i	$-0.343367\pi$
$41$	248.066	0.944914	0.472457	+	0.881354i	$0.343367\pi$
$42$	0	0
$43$	292.163	1.03615	0.518075	−	0.855335i	$-0.326649\pi$
$43$	292.163	1.03615	0.518075	+	0.855335i	$0.326649\pi$
$44$	270.337	0.926247
$45$	0	0
$46$	202.268	0.648320
$47$	240.137	0.745268	0.372634	−	0.927978i	$-0.378455\pi$
$47$	240.137	0.745268	0.372634	+	0.927978i	$0.378455\pi$
$48$	0	0
$49$	161.088	0.469643
$50$	0	0
$51$	0	0
$52$	437.326	1.16627
$53$	505.234	1.30942	0.654710	−	0.755881i	$-0.272791\pi$
$53$	505.234	1.30942	0.654710	+	0.755881i	$0.272791\pi$
$54$	0	0
$55$	0	0
$56$	231.671	0.552827
$57$	0	0
$58$	350.070	0.792525
$59$	613.029	1.35270	0.676352	−	0.736579i	$-0.263560\pi$
$59$	613.029	1.35270	0.676352	+	0.736579i	$0.263560\pi$
$60$	0	0
$61$	147.014	0.308577	0.154288	−	0.988026i	$-0.450692\pi$
$61$	147.014	0.308577	0.154288	+	0.988026i	$0.450692\pi$
$62$	390.251	0.799385
$63$	0	0
$64$	−106.125	−0.207276
$65$	0	0
$66$	0	0
$67$	55.7251	0.101611	0.0508053	−	0.998709i	$-0.483821\pi$
$67$	55.7251	0.101611	0.0508053	+	0.998709i	$0.483821\pi$
$68$	234.368	0.417960
$69$	0	0
$70$	0	0
$71$	−541.270	−0.904746	−0.452373	−	0.891829i	$-0.649422\pi$
$71$	−541.270	−0.904746	−0.452373	+	0.891829i	$0.649422\pi$
$72$	0	0
$73$	−73.4510	−0.117764	−0.0588821	−	0.998265i	$-0.518754\pi$
$73$	−73.4510	−0.117764	−0.0588821	+	0.998265i	$0.518754\pi$
$74$	296.376	0.465581
$75$	0	0
$76$	−441.613	−0.666533
$77$	−1177.40	−1.74256
$78$	0	0
$79$	−1222.46	−1.74099	−0.870494	−	0.492179i	$-0.836200\pi$
$79$	−1222.46	−1.74099	−0.870494	+	0.492179i	$0.836200\pi$
$80$	0	0
$81$	0	0
$82$	899.735	1.21170
$83$	−718.235	−0.949838	−0.474919	−	0.880029i	$-0.657523\pi$
$83$	−718.235	−0.949838	−0.474919	+	0.880029i	$0.657523\pi$
$84$	0	0
$85$	0	0
$86$	1059.67	1.32869
$87$	0	0
$88$	−541.115	−0.655489
$89$	−540.674	−0.643948	−0.321974	−	0.946749i	$-0.604346\pi$
$89$	−540.674	−0.643948	−0.321974	+	0.946749i	$0.604346\pi$
$90$	0	0
$91$	−1904.69	−2.19413
$92$	287.484	0.325786
$93$	0	0
$94$	870.975	0.955683
$95$	0	0
$96$	0	0
$97$	1669.58	1.74763	0.873816	−	0.486256i	$-0.161638\pi$
$97$	1669.58	1.74763	0.873816	+	0.486256i	$0.161638\pi$
$98$	584.263	0.602240
$99$	0	0
$100$	0	0
$101$	−261.609	−0.257734	−0.128867	−	0.991662i	$-0.541134\pi$
$101$	−261.609	−0.257734	−0.128867	+	0.991662i	$0.541134\pi$
$102$	0	0
$103$	−557.014	−0.532856	−0.266428	−	0.963855i	$-0.585844\pi$
$103$	−557.014	−0.532856	−0.266428	+	0.963855i	$0.585844\pi$
$104$	−875.366	−0.825352
$105$	0	0
$106$	1832.48	1.67911
$107$	295.633	0.267102	0.133551	−	0.991042i	$-0.457362\pi$
$107$	295.633	0.267102	0.133551	+	0.991042i	$0.457362\pi$
$108$	0	0
$109$	1080.17	0.949190	0.474595	−	0.880204i	$-0.342595\pi$
$109$	1080.17	0.949190	0.474595	+	0.880204i	$0.342595\pi$
$110$	0	0
$111$	0	0
$112$	1766.20	1.49009
$113$	−2191.08	−1.82406	−0.912032	−	0.410119i	$-0.865487\pi$
$113$	−2191.08	−1.82406	−0.912032	+	0.410119i	$0.865487\pi$
$114$	0	0
$115$	0	0
$116$	497.557	0.398250
$117$	0	0
$118$	2223.45	1.73462
$119$	−1020.74	−0.786315
$120$	0	0
$121$	1419.06	1.06616
$122$	533.218	0.395699
$123$	0	0
$124$	554.666	0.401697
$125$	0	0
$126$	0	0
$127$	1029.13	0.719058	0.359529	−	0.933134i	$-0.382937\pi$
$127$	1029.13	0.719058	0.359529	+	0.933134i	$0.382937\pi$
$128$	1237.26	0.854370
$129$	0	0
$130$	0	0
$131$	−2502.10	−1.66878	−0.834388	−	0.551177i	$-0.814179\pi$
$131$	−2502.10	−1.66878	−0.834388	+	0.551177i	$0.814179\pi$
$132$	0	0
$133$	1923.36	1.25396
$134$	202.114	0.130299
$135$	0	0
$136$	−469.118	−0.295784
$137$	539.139	0.336217	0.168109	−	0.985768i	$-0.446234\pi$
$137$	539.139	0.336217	0.168109	+	0.985768i	$0.446234\pi$
$138$	0	0
$139$	−1050.00	−0.640720	−0.320360	−	0.947296i	$-0.603804\pi$
$139$	−1050.00	−0.640720	−0.320360	+	0.947296i	$0.603804\pi$
$140$	0	0
$141$	0	0
$142$	−1963.18	−1.16019
$143$	4448.79	2.60159
$144$	0	0
$145$	0	0
$146$	−266.406	−0.151013
$147$	0	0
$148$	421.241	0.233958
$149$	−2464.13	−1.35483	−0.677415	−	0.735601i	$-0.736900\pi$
$149$	−2464.13	−1.35483	−0.677415	+	0.735601i	$0.736900\pi$
$150$	0	0
$151$	818.338	0.441029	0.220514	−	0.975384i	$-0.429226\pi$
$151$	818.338	0.441029	0.220514	+	0.975384i	$0.429226\pi$
$152$	883.947	0.471694
$153$	0	0
$154$	−4270.42	−2.23455
$155$	0	0
$156$	0	0
$157$	588.328	0.299068	0.149534	−	0.988757i	$-0.452223\pi$
$157$	588.328	0.299068	0.149534	+	0.988757i	$0.452223\pi$
$158$	−4433.87	−2.23253
$159$	0	0
$160$	0	0
$161$	−1252.08	−0.612906
$162$	0	0
$163$	1371.87	0.659220	0.329610	−	0.944117i	$-0.393083\pi$
$163$	1371.87	0.659220	0.329610	+	0.944117i	$0.393083\pi$
$164$	1278.80	0.608887
$165$	0	0
$166$	−2605.03	−1.21801
$167$	482.446	0.223550	0.111775	−	0.993734i	$-0.464346\pi$
$167$	482.446	0.223550	0.111775	+	0.993734i	$0.464346\pi$
$168$	0	0
$169$	4999.84	2.27576
$170$	0	0
$171$	0	0
$172$	1506.12	0.667678
$173$	−1210.98	−0.532192	−0.266096	−	0.963947i	$-0.585734\pi$
$173$	−1210.98	−0.532192	−0.266096	+	0.963947i	$0.585734\pi$
$174$	0	0
$175$	0	0
$176$	−4125.32	−1.76680
$177$	0	0
$178$	−1961.02	−0.825757
$179$	−1372.10	−0.572936	−0.286468	−	0.958090i	$-0.592481\pi$
$179$	−1372.10	−0.572936	−0.286468	+	0.958090i	$0.592481\pi$
$180$	0	0
$181$	1733.63	0.711931	0.355965	−	0.934499i	$-0.384152\pi$
$181$	1733.63	0.711931	0.355965	+	0.934499i	$0.384152\pi$
$182$	−6908.29	−2.81361
$183$	0	0
$184$	−575.438	−0.230553
$185$	0	0
$186$	0	0
$187$	2384.16	0.932336
$188$	1237.92	0.480238
$189$	0	0
$190$	0	0
$191$	−4029.39	−1.52648	−0.763238	−	0.646118i	$-0.776391\pi$
$191$	−4029.39	−1.52648	−0.763238	+	0.646118i	$0.776391\pi$
$192$	0	0
$193$	1095.49	0.408575	0.204288	−	0.978911i	$-0.434512\pi$
$193$	1095.49	0.408575	0.204288	+	0.978911i	$0.434512\pi$
$194$	6055.56	2.24105
$195$	0	0
$196$	830.417	0.302630
$197$	−254.777	−0.0921428	−0.0460714	−	0.998938i	$-0.514670\pi$
$197$	−254.777	−0.0921428	−0.0460714	+	0.998938i	$0.514670\pi$
$198$	0	0
$199$	942.807	0.335848	0.167924	−	0.985800i	$-0.446294\pi$
$199$	942.807	0.335848	0.167924	+	0.985800i	$0.446294\pi$
$200$	0	0
$201$	0	0
$202$	−948.854	−0.330501
$203$	−2167.01	−0.749234
$204$	0	0
$205$	0	0
$206$	−2020.28	−0.683301
$207$	0	0
$208$	−6673.55	−2.22465
$209$	−4492.40	−1.48682
$210$	0	0
$211$	−2624.31	−0.856233	−0.428117	−	0.903723i	$-0.640823\pi$
$211$	−2624.31	−0.856233	−0.428117	+	0.903723i	$0.640823\pi$
$212$	2604.51	0.843768
$213$	0	0
$214$	1072.26	0.342515
$215$	0	0
$216$	0	0
$217$	−2415.74	−0.755719
$218$	3917.77	1.21718
$219$	0	0
$220$	0	0
$221$	3856.87	1.17394
$222$	0	0
$223$	976.561	0.293253	0.146626	−	0.989192i	$-0.453159\pi$
$223$	976.561	0.293253	0.146626	+	0.989192i	$0.453159\pi$
$224$	4552.61	1.35797
$225$	0	0
$226$	−7947.02	−2.33906
$227$	4591.31	1.34245	0.671224	−	0.741255i	$-0.265769\pi$
$227$	4591.31	1.34245	0.671224	+	0.741255i	$0.265769\pi$
$228$	0	0
$229$	−4640.73	−1.33916	−0.669581	−	0.742739i	$-0.733526\pi$
$229$	−4640.73	−1.33916	−0.669581	+	0.742739i	$0.733526\pi$
$230$	0	0
$231$	0	0
$232$	−995.926	−0.281835
$233$	3508.00	0.986339	0.493169	−	0.869933i	$-0.335838\pi$
$233$	3508.00	0.986339	0.493169	+	0.869933i	$0.335838\pi$
$234$	0	0
$235$	0	0
$236$	3160.20	0.871660
$237$	0	0
$238$	−3702.23	−1.00832
$239$	−889.109	−0.240635	−0.120317	−	0.992735i	$-0.538391\pi$
$239$	−889.109	−0.240635	−0.120317	+	0.992735i	$0.538391\pi$
$240$	0	0
$241$	6433.23	1.71951	0.859753	−	0.510710i	$-0.170618\pi$
$241$	6433.23	1.71951	0.859753	+	0.510710i	$0.170618\pi$
$242$	5146.93	1.36718
$243$	0	0
$244$	757.866	0.198842
$245$	0	0
$246$	0	0
$247$	−7267.39	−1.87212
$248$	−1110.24	−0.284275
$249$	0	0
$250$	0	0
$251$	4538.67	1.14135	0.570674	−	0.821177i	$-0.306682\pi$
$251$	4538.67	1.14135	0.570674	+	0.821177i	$0.306682\pi$
$252$	0	0
$253$	2924.50	0.726725
$254$	3732.64	0.922074
$255$	0	0
$256$	5336.53	1.30286
$257$	−4469.84	−1.08491	−0.542453	−	0.840086i	$-0.682505\pi$
$257$	−4469.84	−1.08491	−0.542453	+	0.840086i	$0.682505\pi$
$258$	0	0
$259$	−1834.63	−0.440149
$260$	0	0
$261$	0	0
$262$	−9075.10	−2.13993
$263$	−1738.49	−0.407603	−0.203802	−	0.979012i	$-0.565330\pi$
$263$	−1738.49	−0.407603	−0.203802	+	0.979012i	$0.565330\pi$
$264$	0	0
$265$	0	0
$266$	6976.01	1.60799
$267$	0	0
$268$	287.267	0.0654761
$269$	−7058.43	−1.59985	−0.799926	−	0.600099i	$-0.795128\pi$
$269$	−7058.43	−1.59985	−0.799926	+	0.600099i	$0.795128\pi$
$270$	0	0
$271$	−5909.97	−1.32474	−0.662371	−	0.749176i	$-0.730450\pi$
$271$	−5909.97	−1.32474	−0.662371	+	0.749176i	$0.730450\pi$
$272$	−3576.43	−0.797254
$273$	0	0
$274$	1955.45	0.431143
$275$	0	0
$276$	0	0
$277$	−5374.61	−1.16581	−0.582904	−	0.812541i	$-0.698084\pi$
$277$	−5374.61	−1.16581	−0.582904	+	0.812541i	$0.698084\pi$
$278$	−3808.35	−0.821618
$279$	0	0
$280$	0	0
$281$	8132.88	1.72657	0.863286	−	0.504715i	$-0.168402\pi$
$281$	8132.88	1.72657	0.863286	+	0.504715i	$0.168402\pi$
$282$	0	0
$283$	4016.36	0.843631	0.421816	−	0.906682i	$-0.361393\pi$
$283$	4016.36	0.843631	0.421816	+	0.906682i	$0.361393\pi$
$284$	−2790.28	−0.583003
$285$	0	0
$286$	16135.7	3.33610
$287$	−5569.56	−1.14551
$288$	0	0
$289$	−2846.06	−0.579292
$290$	0	0
$291$	0	0
$292$	−378.645	−0.0758853
$293$	−105.803	−0.0210958	−0.0105479	−	0.999944i	$-0.503358\pi$
$293$	−105.803	−0.0210958	−0.0105479	+	0.999944i	$0.503358\pi$
$294$	0	0
$295$	0	0
$296$	−843.169	−0.165568
$297$	0	0
$298$	−8937.39	−1.73735
$299$	4730.98	0.915048
$300$	0	0
$301$	−6559.62	−1.25611
$302$	2968.10	0.565547
$303$	0	0
$304$	6738.97	1.27140
$305$	0	0
$306$	0	0
$307$	4638.61	0.862344	0.431172	−	0.902270i	$-0.358100\pi$
$307$	4638.61	0.862344	0.431172	+	0.902270i	$0.358100\pi$
$308$	−6069.58	−1.12288
$309$	0	0
$310$	0	0
$311$	2931.33	0.534471	0.267235	−	0.963631i	$-0.413890\pi$
$311$	2931.33	0.534471	0.267235	+	0.963631i	$0.413890\pi$
$312$	0	0
$313$	−5131.09	−0.926601	−0.463301	−	0.886201i	$-0.653335\pi$
$313$	−5131.09	−0.926601	−0.463301	+	0.886201i	$0.653335\pi$
$314$	2133.86	0.383506
$315$	0	0
$316$	−6301.89	−1.12186
$317$	7218.07	1.27889	0.639443	−	0.768838i	$-0.279165\pi$
$317$	7218.07	1.27889	0.639443	+	0.768838i	$0.279165\pi$
$318$	0	0
$319$	5061.51	0.888370
$320$	0	0
$321$	0	0
$322$	−4541.29	−0.785951
$323$	−3894.68	−0.670915
$324$	0	0
$325$	0	0
$326$	4975.75	0.845341
$327$	0	0
$328$	−2559.69	−0.430899
$329$	−5391.53	−0.903480
$330$	0	0
$331$	1540.33	0.255783	0.127892	−	0.991788i	$-0.459179\pi$
$331$	1540.33	0.255783	0.127892	+	0.991788i	$0.459179\pi$
$332$	−3702.55	−0.612060
$333$	0	0
$334$	1749.83	0.286666
$335$	0	0
$336$	0	0
$337$	10254.4	1.65754	0.828772	−	0.559587i	$-0.189040\pi$
$337$	10254.4	1.65754	0.828772	+	0.559587i	$0.189040\pi$
$338$	18134.4	2.91828
$339$	0	0
$340$	0	0
$341$	5642.46	0.896059
$342$	0	0
$343$	4084.28	0.642946
$344$	−3014.70	−0.472505
$345$	0	0
$346$	−4392.22	−0.682448
$347$	−7036.23	−1.08854	−0.544272	−	0.838909i	$-0.683194\pi$
$347$	−7036.23	−1.08854	−0.544272	+	0.838909i	$0.683194\pi$
$348$	0	0
$349$	−2916.17	−0.447275	−0.223637	−	0.974672i	$-0.571793\pi$
$349$	−2916.17	−0.447275	−0.223637	+	0.974672i	$0.571793\pi$
$350$	0	0
$351$	0	0
$352$	−10633.6	−1.61015
$353$	−1203.79	−0.181505	−0.0907525	−	0.995873i	$-0.528927\pi$
$353$	−1203.79	−0.181505	−0.0907525	+	0.995873i	$0.528927\pi$
$354$	0	0
$355$	0	0
$356$	−2787.21	−0.414949
$357$	0	0
$358$	−4976.59	−0.734696
$359$	9039.30	1.32890	0.664451	−	0.747331i	$-0.268665\pi$
$359$	9039.30	1.32890	0.664451	+	0.747331i	$0.268665\pi$
$360$	0	0
$361$	479.632	0.0699275
$362$	6287.85	0.912934
$363$	0	0
$364$	−9818.80	−1.41386
$365$	0	0
$366$	0	0
$367$	−3080.85	−0.438199	−0.219099	−	0.975703i	$-0.570312\pi$
$367$	−3080.85	−0.438199	−0.219099	+	0.975703i	$0.570312\pi$
$368$	−4386.98	−0.621433
$369$	0	0
$370$	0	0
$371$	−11343.5	−1.58739
$372$	0	0
$373$	6715.41	0.932201	0.466100	−	0.884732i	$-0.345659\pi$
$373$	6715.41	0.932201	0.466100	+	0.884732i	$0.345659\pi$
$374$	8647.32	1.19557
$375$	0	0
$376$	−2477.87	−0.339857
$377$	8188.03	1.11858
$378$	0	0
$379$	−5898.85	−0.799482	−0.399741	−	0.916628i	$-0.630900\pi$
$379$	−5898.85	−0.799482	−0.399741	+	0.916628i	$0.630900\pi$
$380$	0	0
$381$	0	0
$382$	−14614.6	−1.95745
$383$	7272.18	0.970212	0.485106	−	0.874455i	$-0.338781\pi$
$383$	7272.18	0.970212	0.485106	+	0.874455i	$0.338781\pi$
$384$	0	0
$385$	0	0
$386$	3973.33	0.523930
$387$	0	0
$388$	8606.81	1.12615
$389$	4756.07	0.619904	0.309952	−	0.950752i	$-0.399687\pi$
$389$	4756.07	0.619904	0.309952	+	0.950752i	$0.399687\pi$
$390$	0	0
$391$	2535.38	0.327928
$392$	−1662.19	−0.214166
$393$	0	0
$394$	−924.075	−0.118158
$395$	0	0
$396$	0	0
$397$	3579.97	0.452578	0.226289	−	0.974060i	$-0.427341\pi$
$397$	3579.97	0.452578	0.226289	+	0.974060i	$0.427341\pi$
$398$	3419.55	0.430670
$399$	0	0
$400$	0	0
$401$	−11282.3	−1.40502	−0.702509	−	0.711674i	$-0.747937\pi$
$401$	−11282.3	−1.40502	−0.702509	+	0.711674i	$0.747937\pi$
$402$	0	0
$403$	9127.84	1.12826
$404$	−1348.61	−0.166079
$405$	0	0
$406$	−7859.74	−0.960769
$407$	4285.16	0.521886
$408$	0	0
$409$	2402.70	0.290479	0.145240	−	0.989397i	$-0.453605\pi$
$409$	2402.70	0.290479	0.145240	+	0.989397i	$0.453605\pi$
$410$	0	0
$411$	0	0
$412$	−2871.44	−0.343364
$413$	−13763.7	−1.63987
$414$	0	0
$415$	0	0
$416$	−17202.0	−2.02740
$417$	0	0
$418$	−16293.9	−1.90661
$419$	3480.08	0.405759	0.202879	−	0.979204i	$-0.434970\pi$
$419$	3480.08	0.405759	0.202879	+	0.979204i	$0.434970\pi$
$420$	0	0
$421$	−5308.38	−0.614524	−0.307262	−	0.951625i	$-0.599413\pi$
$421$	−5308.38	−0.614524	−0.307262	+	0.951625i	$0.599413\pi$
$422$	−9518.36	−1.09798
$423$	0	0
$424$	−5213.28	−0.597121
$425$	0	0
$426$	0	0
$427$	−3300.74	−0.374084
$428$	1524.01	0.172116
$429$	0	0
$430$	0	0
$431$	−10783.8	−1.20519	−0.602595	−	0.798047i	$-0.705867\pi$
$431$	−10783.8	−1.20519	−0.602595	+	0.798047i	$0.705867\pi$
$432$	0	0
$433$	5480.10	0.608215	0.304107	−	0.952638i	$-0.401642\pi$
$433$	5480.10	0.608215	0.304107	+	0.952638i	$0.401642\pi$
$434$	−8761.87	−0.969085
$435$	0	0
$436$	5568.36	0.611642
$437$	−4777.35	−0.522956
$438$	0	0
$439$	8264.22	0.898473	0.449237	−	0.893413i	$-0.351696\pi$
$439$	8264.22	0.898473	0.449237	+	0.893413i	$0.351696\pi$
$440$	0	0
$441$	0	0
$442$	13988.8	1.50539
$443$	1252.96	0.134379	0.0671896	−	0.997740i	$-0.478597\pi$
$443$	1252.96	0.134379	0.0671896	+	0.997740i	$0.478597\pi$
$444$	0	0
$445$	0	0
$446$	3541.98	0.376048
$447$	0	0
$448$	2382.72	0.251278
$449$	1792.10	0.188362	0.0941808	−	0.995555i	$-0.469977\pi$
$449$	1792.10	0.188362	0.0941808	+	0.995555i	$0.469977\pi$
$450$	0	0
$451$	13008.9	1.35823
$452$	−11295.2	−1.17540
$453$	0	0
$454$	16652.6	1.72147
$455$	0	0
$456$	0	0
$457$	11148.3	1.14113	0.570566	−	0.821252i	$-0.306724\pi$
$457$	11148.3	1.14113	0.570566	+	0.821252i	$0.306724\pi$
$458$	−16831.9	−1.71725
$459$	0	0
$460$	0	0
$461$	9826.57	0.992775	0.496387	−	0.868101i	$-0.334660\pi$
$461$	9826.57	0.992775	0.496387	+	0.868101i	$0.334660\pi$
$462$	0	0
$463$	−11379.0	−1.14217	−0.571087	−	0.820889i	$-0.693478\pi$
$463$	−11379.0	−1.14217	−0.571087	+	0.820889i	$0.693478\pi$
$464$	−7592.67	−0.759657
$465$	0	0
$466$	12723.5	1.26482
$467$	−15503.0	−1.53618	−0.768089	−	0.640343i	$-0.778792\pi$
$467$	−15503.0	−1.53618	−0.768089	+	0.640343i	$0.778792\pi$
$468$	0	0
$469$	−1251.13	−0.123181
$470$	0	0
$471$	0	0
$472$	−6325.56	−0.616859
$473$	15321.3	1.48938
$474$	0	0
$475$	0	0
$476$	−5262.00	−0.506688
$477$	0	0
$478$	−3224.79	−0.308574
$479$	668.853	0.0638010	0.0319005	−	0.999491i	$-0.489844\pi$
$479$	668.853	0.0638010	0.0319005	+	0.999491i	$0.489844\pi$
$480$	0	0
$481$	6932.13	0.657127
$482$	23333.3	2.20498
$483$	0	0
$484$	7315.36	0.687017
$485$	0	0
$486$	0	0
$487$	−2418.99	−0.225082	−0.112541	−	0.993647i	$-0.535899\pi$
$487$	−2418.99	−0.225082	−0.112541	+	0.993647i	$0.535899\pi$
$488$	−1516.97	−0.140717
$489$	0	0
$490$	0	0
$491$	−6837.84	−0.628487	−0.314244	−	0.949342i	$-0.601751\pi$
$491$	−6837.84	−0.628487	−0.314244	+	0.949342i	$0.601751\pi$
$492$	0	0
$493$	4388.06	0.400869
$494$	−26358.8	−2.40068
$495$	0	0
$496$	−8464.15	−0.766233
$497$	12152.5	1.09681
$498$	0	0
$499$	437.043	0.0392079	0.0196039	−	0.999808i	$-0.493759\pi$
$499$	437.043	0.0392079	0.0196039	+	0.999808i	$0.493759\pi$
$500$	0	0
$501$	0	0
$502$	16461.7	1.46359
$503$	6970.27	0.617871	0.308935	−	0.951083i	$-0.400027\pi$
$503$	6970.27	0.617871	0.308935	+	0.951083i	$0.400027\pi$
$504$	0	0
$505$	0	0
$506$	10607.1	0.931906
$507$	0	0
$508$	5305.23	0.463349
$509$	−1570.36	−0.136749	−0.0683744	−	0.997660i	$-0.521781\pi$
$509$	−1570.36	−0.136749	−0.0683744	+	0.997660i	$0.521781\pi$
$510$	0	0
$511$	1649.11	0.142764
$512$	9457.48	0.816339
$513$	0	0
$514$	−16212.1	−1.39121
$515$	0	0
$516$	0	0
$517$	12593.0	1.07126
$518$	−6654.20	−0.564418
$519$	0	0
$520$	0	0
$521$	−21269.3	−1.78853	−0.894265	−	0.447538i	$-0.852301\pi$
$521$	−21269.3	−1.78853	−0.894265	+	0.447538i	$0.852301\pi$
$522$	0	0
$523$	−13992.1	−1.16985	−0.584926	−	0.811087i	$-0.698876\pi$
$523$	−13992.1	−1.16985	−0.584926	+	0.811087i	$0.698876\pi$
$524$	−12898.5	−1.07533
$525$	0	0
$526$	−6305.48	−0.522684
$527$	4891.71	0.404338
$528$	0	0
$529$	−9057.00	−0.744391
$530$	0	0
$531$	0	0
$532$	9915.05	0.808030
$533$	21044.5	1.71020
$534$	0	0
$535$	0	0
$536$	−575.002	−0.0463364
$537$	0	0
$538$	−25600.9	−2.05155
$539$	8447.60	0.675072
$540$	0	0
$541$	−16742.7	−1.33054	−0.665272	−	0.746601i	$-0.731684\pi$
$541$	−16742.7	−1.33054	−0.665272	+	0.746601i	$0.731684\pi$
$542$	−21435.4	−1.69876
$543$	0	0
$544$	−9218.74	−0.726563
$545$	0	0
$546$	0	0
$547$	−6283.10	−0.491126	−0.245563	−	0.969381i	$-0.578973\pi$
$547$	−6283.10	−0.491126	−0.245563	+	0.969381i	$0.578973\pi$
$548$	2779.30	0.216653
$549$	0	0
$550$	0	0
$551$	−8268.30	−0.639276
$552$	0	0
$553$	27446.7	2.11058
$554$	−19493.7	−1.49496
$555$	0	0
$556$	−5412.83	−0.412869
$557$	11043.6	0.840093	0.420047	−	0.907503i	$-0.362014\pi$
$557$	11043.6	0.840093	0.420047	+	0.907503i	$0.362014\pi$
$558$	0	0
$559$	24785.4	1.87533
$560$	0	0
$561$	0	0
$562$	29497.9	2.21404
$563$	−22448.3	−1.68043	−0.840216	−	0.542252i	$-0.817572\pi$
$563$	−22448.3	−1.68043	−0.840216	+	0.542252i	$0.817572\pi$
$564$	0	0
$565$	0	0
$566$	14567.3	1.08182
$567$	0	0
$568$	5585.12	0.412582
$569$	7816.69	0.575910	0.287955	−	0.957644i	$-0.407025\pi$
$569$	7816.69	0.575910	0.287955	+	0.957644i	$0.407025\pi$
$570$	0	0
$571$	−11397.1	−0.835294	−0.417647	−	0.908609i	$-0.637145\pi$
$571$	−11397.1	−0.835294	−0.417647	+	0.908609i	$0.637145\pi$
$572$	22933.8	1.67642
$573$	0	0
$574$	−20200.8	−1.46893
$575$	0	0
$576$	0	0
$577$	−10649.9	−0.768393	−0.384196	−	0.923251i	$-0.625521\pi$
$577$	−10649.9	−0.768393	−0.384196	+	0.923251i	$0.625521\pi$
$578$	−10322.6	−0.742846
$579$	0	0
$580$	0	0
$581$	16125.7	1.15148
$582$	0	0
$583$	26495.0	1.88218
$584$	757.907	0.0537028
$585$	0	0
$586$	−383.746	−0.0270519
$587$	−16310.8	−1.14688	−0.573440	−	0.819247i	$-0.694392\pi$
$587$	−16310.8	−1.14688	−0.573440	+	0.819247i	$0.694392\pi$
$588$	0	0
$589$	−9217.32	−0.644810
$590$	0	0
$591$	0	0
$592$	−6428.09	−0.446272
$593$	−24814.4	−1.71839	−0.859195	−	0.511648i	$-0.829035\pi$
$593$	−24814.4	−1.71839	−0.859195	+	0.511648i	$0.829035\pi$
$594$	0	0
$595$	0	0
$596$	−12702.8	−0.873030
$597$	0	0
$598$	17159.2	1.17340
$599$	22999.3	1.56883	0.784414	−	0.620238i	$-0.212964\pi$
$599$	22999.3	1.56883	0.784414	+	0.620238i	$0.212964\pi$
$600$	0	0
$601$	15453.8	1.04887	0.524437	−	0.851449i	$-0.324276\pi$
$601$	15453.8	1.04887	0.524437	+	0.851449i	$0.324276\pi$
$602$	−23791.7	−1.61076
$603$	0	0
$604$	4218.59	0.284192
$605$	0	0
$606$	0	0
$607$	10510.2	0.702793	0.351396	−	0.936227i	$-0.385707\pi$
$607$	10510.2	0.702793	0.351396	+	0.936227i	$0.385707\pi$
$608$	17370.6	1.15867
$609$	0	0
$610$	0	0
$611$	20371.8	1.34886
$612$	0	0
$613$	18146.3	1.19563	0.597817	−	0.801633i	$-0.296035\pi$
$613$	18146.3	1.19563	0.597817	+	0.801633i	$0.296035\pi$
$614$	16824.2	1.10581
$615$	0	0
$616$	12149.1	0.794642
$617$	−20368.1	−1.32899	−0.664495	−	0.747292i	$-0.731353\pi$
$617$	−20368.1	−1.32899	−0.664495	+	0.747292i	$0.731353\pi$
$618$	0	0
$619$	−5632.50	−0.365734	−0.182867	−	0.983138i	$-0.558538\pi$
$619$	−5632.50	−0.365734	−0.182867	+	0.983138i	$0.558538\pi$
$620$	0	0
$621$	0	0
$622$	10631.9	0.685370
$623$	12139.2	0.780650
$624$	0	0
$625$	0	0
$626$	−18610.4	−1.18821
$627$	0	0
$628$	3032.87	0.192715
$629$	3715.01	0.235496
$630$	0	0
$631$	10522.9	0.663882	0.331941	−	0.943300i	$-0.392296\pi$
$631$	10522.9	0.663882	0.331941	+	0.943300i	$0.392296\pi$
$632$	12614.1	0.793925
$633$	0	0
$634$	26179.9	1.63996
$635$	0	0
$636$	0	0
$637$	13665.7	0.850010
$638$	18358.0	1.13919
$639$	0	0
$640$	0	0
$641$	17567.1	1.08246	0.541232	−	0.840874i	$-0.317958\pi$
$641$	17567.1	1.08246	0.541232	+	0.840874i	$0.317958\pi$
$642$	0	0
$643$	11339.8	0.695489	0.347745	−	0.937589i	$-0.386948\pi$
$643$	11339.8	0.695489	0.347745	+	0.937589i	$0.386948\pi$
$644$	−6454.57	−0.394947
$645$	0	0
$646$	−14126.0	−0.860338
$647$	12382.6	0.752411	0.376205	−	0.926536i	$-0.377229\pi$
$647$	12382.6	0.752411	0.376205	+	0.926536i	$0.377229\pi$
$648$	0	0
$649$	32147.9	1.94440
$650$	0	0
$651$	0	0
$652$	7072.06	0.424790
$653$	−5075.83	−0.304185	−0.152092	−	0.988366i	$-0.548601\pi$
$653$	−5075.83	−0.304185	−0.152092	+	0.988366i	$0.548601\pi$
$654$	0	0
$655$	0	0
$656$	−19514.4	−1.16144
$657$	0	0
$658$	−19555.0	−1.15856
$659$	−4410.87	−0.260733	−0.130367	−	0.991466i	$-0.541615\pi$
$659$	−4410.87	−0.260733	−0.130367	+	0.991466i	$0.541615\pi$
$660$	0	0
$661$	−16077.7	−0.946065	−0.473032	−	0.881045i	$-0.656841\pi$
$661$	−16077.7	−0.946065	−0.473032	+	0.881045i	$0.656841\pi$
$662$	5586.77	0.328000
$663$	0	0
$664$	7411.15	0.433145
$665$	0	0
$666$	0	0
$667$	5382.55	0.312464
$668$	2487.04	0.144052
$669$	0	0
$670$	0	0
$671$	7709.56	0.443553
$672$	0	0
$673$	−6603.48	−0.378225	−0.189113	−	0.981955i	$-0.560561\pi$
$673$	−6603.48	−0.378225	−0.189113	+	0.981955i	$0.560561\pi$
$674$	37192.6	2.12553
$675$	0	0
$676$	25774.5	1.46646
$677$	−8956.50	−0.508458	−0.254229	−	0.967144i	$-0.581822\pi$
$677$	−8956.50	−0.508458	−0.254229	+	0.967144i	$0.581822\pi$
$678$	0	0
$679$	−37485.3	−2.11863
$680$	0	0
$681$	0	0
$682$	20465.1	1.14905
$683$	6280.32	0.351844	0.175922	−	0.984404i	$-0.443709\pi$
$683$	6280.32	0.351844	0.175922	+	0.984404i	$0.443709\pi$
$684$	0	0
$685$	0	0
$686$	14813.6	0.824472
$687$	0	0
$688$	−22983.3	−1.27359
$689$	42861.1	2.36992
$690$	0	0
$691$	−16698.1	−0.919284	−0.459642	−	0.888104i	$-0.652022\pi$
$691$	−16698.1	−0.919284	−0.459642	+	0.888104i	$0.652022\pi$
$692$	−6242.69	−0.342936
$693$	0	0
$694$	−25520.3	−1.39588
$695$	0	0
$696$	0	0
$697$	11278.0	0.612890
$698$	−10576.9	−0.573556
$699$	0	0
$700$	0	0
$701$	11420.7	0.615338	0.307669	−	0.951493i	$-0.400451\pi$
$701$	11420.7	0.615338	0.307669	+	0.951493i	$0.400451\pi$
$702$	0	0
$703$	−7000.09	−0.375552
$704$	−5565.33	−0.297942
$705$	0	0
$706$	−4366.14	−0.232750
$707$	5873.62	0.312447
$708$	0	0
$709$	3436.21	0.182016	0.0910081	−	0.995850i	$-0.470991\pi$
$709$	3436.21	0.182016	0.0910081	+	0.995850i	$0.470991\pi$
$710$	0	0
$711$	0	0
$712$	5578.97	0.293653
$713$	6000.36	0.315168
$714$	0	0
$715$	0	0
$716$	−7073.26	−0.369190
$717$	0	0
$718$	32785.5	1.70410
$719$	−20029.0	−1.03888	−0.519441	−	0.854506i	$-0.673860\pi$
$719$	−20029.0	−1.03888	−0.519441	+	0.854506i	$0.673860\pi$
$720$	0	0
$721$	12506.0	0.645976
$722$	1739.62	0.0896704
$723$	0	0
$724$	8936.97	0.458757
$725$	0	0
$726$	0	0
$727$	2639.99	0.134679	0.0673397	−	0.997730i	$-0.478549\pi$
$727$	2639.99	0.134679	0.0673397	+	0.997730i	$0.478549\pi$
$728$	19653.6	1.00057
$729$	0	0
$730$	0	0
$731$	13282.8	0.672068
$732$	0	0
$733$	−19707.5	−0.993060	−0.496530	−	0.868020i	$-0.665393\pi$
$733$	−19707.5	−0.993060	−0.496530	+	0.868020i	$0.665393\pi$
$734$	−11174.2	−0.561918
$735$	0	0
$736$	−11308.0	−0.566332
$737$	2922.28	0.146056
$738$	0	0
$739$	−11755.6	−0.585163	−0.292581	−	0.956241i	$-0.594514\pi$
$739$	−11755.6	−0.585163	−0.292581	+	0.956241i	$0.594514\pi$
$740$	0	0
$741$	0	0
$742$	−41142.6	−2.03557
$743$	−21494.4	−1.06131	−0.530655	−	0.847588i	$-0.678054\pi$
$743$	−21494.4	−1.06131	−0.530655	+	0.847588i	$0.678054\pi$
$744$	0	0
$745$	0	0
$746$	24356.7	1.19539
$747$	0	0
$748$	12290.5	0.600782
$749$	−6637.53	−0.323805
$750$	0	0
$751$	36993.3	1.79748	0.898739	−	0.438483i	$-0.144484\pi$
$751$	36993.3	1.79748	0.898739	+	0.438483i	$0.144484\pi$
$752$	−18890.6	−0.916049
$753$	0	0
$754$	29697.9	1.43440
$755$	0	0
$756$	0	0
$757$	34002.3	1.63254	0.816272	−	0.577668i	$-0.196037\pi$
$757$	34002.3	1.63254	0.816272	+	0.577668i	$0.196037\pi$
$758$	−21395.1	−1.02520
$759$	0	0
$760$	0	0
$761$	20068.2	0.955943	0.477971	−	0.878375i	$-0.341372\pi$
$761$	20068.2	0.955943	0.477971	+	0.878375i	$0.341372\pi$
$762$	0	0
$763$	−24251.9	−1.15069
$764$	−20771.8	−0.983635
$765$	0	0
$766$	26376.2	1.24414
$767$	52005.8	2.44827
$768$	0	0
$769$	−7422.88	−0.348083	−0.174042	−	0.984738i	$-0.555683\pi$
$769$	−7422.88	−0.348083	−0.174042	+	0.984738i	$0.555683\pi$
$770$	0	0
$771$	0	0
$772$	5647.32	0.263279
$773$	2410.40	0.112155	0.0560777	−	0.998426i	$-0.482141\pi$
$773$	2410.40	0.112155	0.0560777	+	0.998426i	$0.482141\pi$
$774$	0	0
$775$	0	0
$776$	−17227.7	−0.796955
$777$	0	0
$778$	17250.2	0.794924
$779$	−21250.8	−0.977393
$780$	0	0
$781$	−28384.8	−1.30050
$782$	9195.82	0.420514
$783$	0	0
$784$	−12672.1	−0.577263
$785$	0	0
$786$	0	0
$787$	−17506.7	−0.792945	−0.396472	−	0.918047i	$-0.629766\pi$
$787$	−17506.7	−0.792945	−0.396472	+	0.918047i	$0.629766\pi$
$788$	−1313.39	−0.0593753
$789$	0	0
$790$	0	0
$791$	49193.8	2.21129
$792$	0	0
$793$	12471.8	0.558495
$794$	12984.5	0.580356
$795$	0	0
$796$	4860.23	0.216415
$797$	−19875.1	−0.883327	−0.441663	−	0.897181i	$-0.645611\pi$
$797$	−19875.1	−0.883327	−0.441663	+	0.897181i	$0.645611\pi$
$798$	0	0
$799$	10917.5	0.483396
$800$	0	0
$801$	0	0
$802$	−40920.9	−1.80170
$803$	−3851.84	−0.169276
$804$	0	0
$805$	0	0
$806$	33106.6	1.44681
$807$	0	0
$808$	2699.43	0.117532
$809$	23451.1	1.01916	0.509578	−	0.860425i	$-0.329802\pi$
$809$	23451.1	1.01916	0.509578	+	0.860425i	$0.329802\pi$
$810$	0	0
$811$	2190.02	0.0948237	0.0474118	−	0.998875i	$-0.484903\pi$
$811$	2190.02	0.0948237	0.0474118	+	0.998875i	$0.484903\pi$
$812$	−11171.1	−0.482794
$813$	0	0
$814$	15542.2	0.669233
$815$	0	0
$816$	0	0
$817$	−25028.4	−1.07177
$818$	8714.58	0.372492
$819$	0	0
$820$	0	0
$821$	−17276.4	−0.734411	−0.367205	−	0.930140i	$-0.619685\pi$
$821$	−17276.4	−0.734411	−0.367205	+	0.930140i	$0.619685\pi$
$822$	0	0
$823$	−17501.7	−0.741277	−0.370639	−	0.928777i	$-0.620861\pi$
$823$	−17501.7	−0.741277	−0.370639	+	0.928777i	$0.620861\pi$
$824$	5747.57	0.242993
$825$	0	0
$826$	−49920.7	−2.10286
$827$	16849.9	0.708499	0.354250	−	0.935151i	$-0.384736\pi$
$827$	16849.9	0.708499	0.354250	+	0.935151i	$0.384736\pi$
$828$	0	0
$829$	25450.3	1.06626	0.533128	−	0.846034i	$-0.321016\pi$
$829$	25450.3	1.06626	0.533128	+	0.846034i	$0.321016\pi$
$830$	0	0
$831$	0	0
$832$	−9003.06	−0.375150
$833$	7323.62	0.304620
$834$	0	0
$835$	0	0
$836$	−23158.6	−0.958084
$837$	0	0
$838$	12622.2	0.520319
$839$	−17571.1	−0.723028	−0.361514	−	0.932367i	$-0.617740\pi$
$839$	−17571.1	−0.723028	−0.361514	+	0.932367i	$0.617740\pi$
$840$	0	0
$841$	−15073.3	−0.618036
$842$	−19253.4	−0.788025
$843$	0	0
$844$	−13528.5	−0.551743
$845$	0	0
$846$	0	0
$847$	−31860.6	−1.29250
$848$	−39744.6	−1.60948
$849$	0	0
$850$	0	0
$851$	4556.97	0.183561
$852$	0	0
$853$	−40906.9	−1.64200	−0.821001	−	0.570927i	$-0.806584\pi$
$853$	−40906.9	−1.64200	−0.821001	+	0.570927i	$0.806584\pi$
$854$	−11971.7	−0.479701
$855$	0	0
$856$	−3050.50	−0.121804
$857$	−32541.9	−1.29709	−0.648547	−	0.761174i	$-0.724623\pi$
$857$	−32541.9	−1.29709	−0.648547	+	0.761174i	$0.724623\pi$
$858$	0	0
$859$	−36358.6	−1.44417	−0.722084	−	0.691806i	$-0.756815\pi$
$859$	−36358.6	−1.44417	−0.722084	+	0.691806i	$0.756815\pi$
$860$	0	0
$861$	0	0
$862$	−39112.7	−1.54546
$863$	36628.0	1.44476	0.722382	−	0.691494i	$-0.243047\pi$
$863$	36628.0	1.44476	0.722382	+	0.691494i	$0.243047\pi$
$864$	0	0
$865$	0	0
$866$	19876.3	0.779935
$867$	0	0
$868$	−12453.3	−0.486973
$869$	−64107.3	−2.50252
$870$	0	0
$871$	4727.39	0.183905
$872$	−11145.8	−0.432849
$873$	0	0
$874$	−17327.4	−0.670605
$875$	0	0
$876$	0	0
$877$	−7837.47	−0.301770	−0.150885	−	0.988551i	$-0.548212\pi$
$877$	−7837.47	−0.301770	−0.150885	+	0.988551i	$0.548212\pi$
$878$	29974.3	1.15214
$879$	0	0
$880$	0	0
$881$	−49066.1	−1.87637	−0.938183	−	0.346140i	$-0.887492\pi$
$881$	−49066.1	−1.87637	−0.938183	+	0.346140i	$0.887492\pi$
$882$	0	0
$883$	−13551.9	−0.516486	−0.258243	−	0.966080i	$-0.583144\pi$
$883$	−13551.9	−0.516486	−0.258243	+	0.966080i	$0.583144\pi$
$884$	19882.4	0.756468
$885$	0	0
$886$	4544.48	0.172319
$887$	−39447.1	−1.49324	−0.746620	−	0.665251i	$-0.768325\pi$
$887$	−39447.1	−1.49324	−0.746620	+	0.665251i	$0.768325\pi$
$888$	0	0
$889$	−23105.9	−0.871706
$890$	0	0
$891$	0	0
$892$	5034.24	0.188967
$893$	−20571.5	−0.770885
$894$	0	0
$895$	0	0
$896$	−27778.8	−1.03574
$897$	0	0
$898$	6499.92	0.241543
$899$	10385.0	0.385271
$900$	0	0
$901$	22969.8	0.849316
$902$	47183.0	1.74171
$903$	0	0
$904$	22608.7	0.831809
$905$	0	0
$906$	0	0
$907$	−22027.4	−0.806403	−0.403202	−	0.915111i	$-0.632103\pi$
$907$	−22027.4	−0.806403	−0.403202	+	0.915111i	$0.632103\pi$
$908$	23668.5	0.865051
$909$	0	0
$910$	0	0
$911$	25475.9	0.926512	0.463256	−	0.886224i	$-0.346681\pi$
$911$	25475.9	0.926512	0.463256	+	0.886224i	$0.346681\pi$
$912$	0	0
$913$	−37665.0	−1.36531
$914$	40435.0	1.46331
$915$	0	0
$916$	−23923.3	−0.862934
$917$	56176.9	2.02304
$918$	0	0
$919$	53379.4	1.91602	0.958011	−	0.286733i	$-0.0925691\pi$
$919$	53379.4	1.91602	0.958011	+	0.286733i	$0.0925691\pi$
$920$	0	0
$921$	0	0
$922$	35640.9	1.27307
$923$	−45918.2	−1.63750
$924$	0	0
$925$	0	0
$926$	−41271.5	−1.46465
$927$	0	0
$928$	−19571.1	−0.692300
$929$	25713.7	0.908114	0.454057	−	0.890973i	$-0.349976\pi$
$929$	25713.7	0.908114	0.454057	+	0.890973i	$0.349976\pi$
$930$	0	0
$931$	−13799.7	−0.485786
$932$	18084.0	0.635580
$933$	0	0
$934$	−56229.4	−1.96990
$935$	0	0
$936$	0	0
$937$	35971.8	1.25416	0.627080	−	0.778955i	$-0.284250\pi$
$937$	35971.8	1.25416	0.627080	+	0.778955i	$0.284250\pi$
$938$	−4537.85	−0.157960
$939$	0	0
$940$	0	0
$941$	−34548.6	−1.19687	−0.598434	−	0.801172i	$-0.704210\pi$
$941$	−34548.6	−1.19687	−0.598434	+	0.801172i	$0.704210\pi$
$942$	0	0
$943$	13834.0	0.477728
$944$	−48224.4	−1.66268
$945$	0	0
$946$	55570.4	1.90988
$947$	−4988.80	−0.171187	−0.0855936	−	0.996330i	$-0.527279\pi$
$947$	−4988.80	−0.171187	−0.0855936	+	0.996330i	$0.527279\pi$
$948$	0	0
$949$	−6231.15	−0.213142
$950$	0	0
$951$	0	0
$952$	10532.6	0.358575
$953$	5720.11	0.194431	0.0972154	−	0.995263i	$-0.469006\pi$
$953$	5720.11	0.194431	0.0972154	+	0.995263i	$0.469006\pi$
$954$	0	0
$955$	0	0
$956$	−4583.42	−0.155061
$957$	0	0
$958$	2425.92	0.0818142
$959$	−12104.7	−0.407592
$960$	0	0
$961$	−18214.1	−0.611394
$962$	25142.8	0.842657
$963$	0	0
$964$	33163.7	1.10802
$965$	0	0
$966$	0	0
$967$	−47265.0	−1.57181	−0.785904	−	0.618348i	$-0.787802\pi$
$967$	−47265.0	−1.57181	−0.785904	+	0.618348i	$0.787802\pi$
$968$	−14642.7	−0.486191
$969$	0	0
$970$	0	0
$971$	−42574.9	−1.40710	−0.703549	−	0.710647i	$-0.748402\pi$
$971$	−42574.9	−1.40710	−0.703549	+	0.710647i	$0.748402\pi$
$972$	0	0
$973$	23574.5	0.776737
$974$	−8773.66	−0.288631
$975$	0	0
$976$	−11565.0	−0.379288
$977$	−40481.5	−1.32561	−0.662804	−	0.748793i	$-0.730634\pi$
$977$	−40481.5	−1.32561	−0.662804	+	0.748793i	$0.730634\pi$
$978$	0	0
$979$	−28353.5	−0.925620
$980$	0	0
$981$	0	0
$982$	−24800.8	−0.805932
$983$	35891.9	1.16457	0.582285	−	0.812985i	$-0.302159\pi$
$983$	35891.9	1.16457	0.582285	+	0.812985i	$0.302159\pi$
$984$	0	0
$985$	0	0
$986$	15915.4	0.514048
$987$	0	0
$988$	−37463.9	−1.20636
$989$	16293.2	0.523855
$990$	0	0
$991$	2150.76	0.0689416	0.0344708	−	0.999406i	$-0.489025\pi$
$991$	2150.76	0.0689416	0.0344708	+	0.999406i	$0.489025\pi$
$992$	−21817.5	−0.698292
$993$	0	0
$994$	44077.2	1.40648
$995$	0	0
$996$	0	0
$997$	18757.6	0.595847	0.297924	−	0.954590i	$-0.403706\pi$
$997$	18757.6	0.595847	0.297924	+	0.954590i	$0.403706\pi$
$998$	1585.15	0.0502776
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.bc.1.5		6
3.2	odd	2	inner	675.4.a.bc.1.2		6
5.2	odd	4		135.4.b.c.109.9	yes	12
5.3	odd	4		135.4.b.c.109.3	✓	12
5.4	even	2		675.4.a.bb.1.2		6
15.2	even	4		135.4.b.c.109.4	yes	12
15.8	even	4		135.4.b.c.109.10	yes	12
15.14	odd	2		675.4.a.bb.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.c.109.3	✓	12	5.3	odd	4
135.4.b.c.109.4	yes	12	15.2	even	4
135.4.b.c.109.9	yes	12	5.2	odd	4
135.4.b.c.109.10	yes	12	15.8	even	4
675.4.a.bb.1.2		6	5.4	even	2
675.4.a.bb.1.5		6	15.14	odd	2
675.4.a.bc.1.2		6	3.2	odd	2	inner
675.4.a.bc.1.5		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q+3.62699 q^{2} +5.15507 q^{4} -22.4519 q^{7} -10.3185 q^{8} +52.4410 q^{11} +84.8342 q^{13} -81.4328 q^{14} -78.6658 q^{16} +45.4636 q^{17} -85.6658 q^{19} +190.203 q^{22} +55.7673 q^{23} +307.693 q^{26} -115.741 q^{28} +96.5180 q^{29} +107.596 q^{31} -202.772 q^{32} +164.896 q^{34} +81.7139 q^{37} -310.709 q^{38} +248.066 q^{41} +292.163 q^{43} +270.337 q^{44} +202.268 q^{46} +240.137 q^{47} +161.088 q^{49} +437.326 q^{52} +505.234 q^{53} +231.671 q^{56} +350.070 q^{58} +613.029 q^{59} +147.014 q^{61} +390.251 q^{62} -106.125 q^{64} +55.7251 q^{67} +234.368 q^{68} -541.270 q^{71} -73.4510 q^{73} +296.376 q^{74} -441.613 q^{76} -1177.40 q^{77} -1222.46 q^{79} +899.735 q^{82} -718.235 q^{83} +1059.67 q^{86} -541.115 q^{88} -540.674 q^{89} -1904.69 q^{91} +287.484 q^{92} +870.975 q^{94} +1669.58 q^{97} +584.263 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 30 q^{4} + 36 q^{7} + 162 q^{13} + 42 q^{16} + 450 q^{22} + 828 q^{28} + 126 q^{31} - 534 q^{34} + 1008 q^{37} + 558 q^{43} - 834 q^{46} + 1434 q^{49} + 2610 q^{52} - 270 q^{58} + 396 q^{61} - 1134 q^{64}+ \cdots + 4104 q^{97}+O(q^{100}) \) 6 * q + 30 * q^4 + 36 * q^7 + 162 * q^13 + 42 * q^16 + 450 * q^22 + 828 * q^28 + 126 * q^31 - 534 * q^34 + 1008 * q^37 + 558 * q^43 - 834 * q^46 + 1434 * q^49 + 2610 * q^52 - 270 * q^58 + 396 * q^61 - 1134 * q^64 + 2268 * q^67 + 144 * q^73 + 912 * q^76 - 1098 * q^79 + 5544 * q^82 + 702 * q^88 - 1692 * q^91 + 2574 * q^94 + 4104 * q^97

Introduction

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