LMFDB - Embedded newform 1458.4.a.i.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1458,4,Mod(1,1458)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1458, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1458.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1458.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-30,0,60,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$86.0247847884$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 339 x^{13} - 1151 x^{12} + 35865 x^{11} + 180141 x^{10} - 1644266 x^{9} - 10786662 x^{8} + \cdots + 33185995624 $ x^15 - 339x^13 - 1151x^12 + 35865x^11 + 180141x^10 - 1644266x^9 - 10786662x^8 + 31430970x^7 + 304598356x^6 - 45265230x^5 - 3933959154x^4 - 5634080801x^3 + 16258950054x^2 + 47732969400*x + 33185995624
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{24} $
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-8.61540$ of defining polynomial
Character	$\chi$	$=$	1458.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +18.4078 q^{5} +34.7746 q^{7} -8.00000 q^{8} -36.8156 q^{10} +29.1352 q^{11} -12.2075 q^{13} -69.5493 q^{14} +16.0000 q^{16} +12.2338 q^{17} +77.7365 q^{19} +73.6312 q^{20} -58.2705 q^{22} -6.32515 q^{23} +213.847 q^{25} +24.4151 q^{26} +139.099 q^{28} -133.889 q^{29} -151.052 q^{31} -32.0000 q^{32} -24.4675 q^{34} +640.124 q^{35} -162.050 q^{37} -155.473 q^{38} -147.262 q^{40} +341.655 q^{41} +347.204 q^{43} +116.541 q^{44} +12.6503 q^{46} -151.205 q^{47} +866.275 q^{49} -427.694 q^{50} -48.8302 q^{52} +329.797 q^{53} +536.316 q^{55} -278.197 q^{56} +267.778 q^{58} -229.746 q^{59} +719.548 q^{61} +302.104 q^{62} +64.0000 q^{64} -224.714 q^{65} +385.022 q^{67} +48.9351 q^{68} -1280.25 q^{70} -125.968 q^{71} +369.566 q^{73} +324.100 q^{74} +310.946 q^{76} +1013.17 q^{77} -1009.08 q^{79} +294.525 q^{80} -683.310 q^{82} -1134.08 q^{83} +225.197 q^{85} -694.409 q^{86} -233.082 q^{88} -650.611 q^{89} -424.513 q^{91} -25.3006 q^{92} +302.411 q^{94} +1430.96 q^{95} -720.708 q^{97} -1732.55 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 30 q^{2} + 60 q^{4} - 15 q^{5} + 42 q^{7} - 120 q^{8} + 30 q^{10} - 33 q^{11} + 117 q^{13} - 84 q^{14} + 240 q^{16} - 102 q^{17} + 171 q^{19} - 60 q^{20} + 66 q^{22} - 174 q^{23} + 600 q^{25} - 234 q^{26}+ \cdots - 4002 q^{98}+O(q^{100}) $ 15 * q - 30 * q^2 + 60 * q^4 - 15 * q^5 + 42 * q^7 - 120 * q^8 + 30 * q^10 - 33 * q^11 + 117 * q^13 - 84 * q^14 + 240 * q^16 - 102 * q^17 + 171 * q^19 - 60 * q^20 + 66 * q^22 - 174 * q^23 + 600 * q^25 - 234 * q^26 + 168 * q^28 - 573 * q^29 + 372 * q^31 - 480 * q^32 + 204 * q^34 - 624 * q^35 + 555 * q^37 - 342 * q^38 + 120 * q^40 - 852 * q^41 + 1002 * q^43 - 132 * q^44 + 348 * q^46 - 306 * q^47 + 2001 * q^49 - 1200 * q^50 + 468 * q^52 - 897 * q^53 + 1953 * q^55 - 336 * q^56 + 1146 * q^58 - 795 * q^59 + 1587 * q^61 - 744 * q^62 + 960 * q^64 - 1020 * q^65 + 1884 * q^67 - 408 * q^68 + 1248 * q^70 - 120 * q^71 + 2604 * q^73 - 1110 * q^74 + 684 * q^76 - 984 * q^77 + 1944 * q^79 - 240 * q^80 + 1704 * q^82 - 2079 * q^83 + 2916 * q^85 - 2004 * q^86 + 264 * q^88 - 2604 * q^89 + 3399 * q^91 - 696 * q^92 + 612 * q^94 - 1038 * q^95 + 3690 * q^97 - 4002 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	18.4078	1.64644	0.823222	−	0.567720i	$-0.192174\pi$
$5$	18.4078	1.64644	0.823222	+	0.567720i	$0.192174\pi$
$6$	0	0
$7$	34.7746	1.87765	0.938827	−	0.344390i	$-0.111914\pi$
$7$	34.7746	1.87765	0.938827	+	0.344390i	$0.111914\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−36.8156	−1.16421
$11$	29.1352	0.798601	0.399300	−	0.916820i	$-0.369253\pi$
$11$	29.1352	0.798601	0.399300	+	0.916820i	$0.369253\pi$
$12$	0	0
$13$	−12.2075	−0.260443	−0.130222	−	0.991485i	$-0.541569\pi$
$13$	−12.2075	−0.260443	−0.130222	+	0.991485i	$0.541569\pi$
$14$	−69.5493	−1.32770
$15$	0	0
$16$	16.0000	0.250000
$17$	12.2338	0.174537	0.0872684	−	0.996185i	$-0.472186\pi$
$17$	12.2338	0.174537	0.0872684	+	0.996185i	$0.472186\pi$
$18$	0	0
$19$	77.7365	0.938631	0.469315	−	0.883031i	$-0.344501\pi$
$19$	77.7365	0.938631	0.469315	+	0.883031i	$0.344501\pi$
$20$	73.6312	0.823222
$21$	0	0
$22$	−58.2705	−0.564696
$23$	−6.32515	−0.0573428	−0.0286714	−	0.999589i	$-0.509128\pi$
$23$	−6.32515	−0.0573428	−0.0286714	+	0.999589i	$0.509128\pi$
$24$	0	0
$25$	213.847	1.71078
$26$	24.4151	0.184161
$27$	0	0
$28$	139.099	0.938827
$29$	−133.889	−0.857329	−0.428665	−	0.903464i	$-0.641016\pi$
$29$	−133.889	−0.857329	−0.428665	+	0.903464i	$0.641016\pi$
$30$	0	0
$31$	−151.052	−0.875154	−0.437577	−	0.899181i	$-0.644163\pi$
$31$	−151.052	−0.875154	−0.437577	+	0.899181i	$0.644163\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−24.4675	−0.123416
$35$	640.124	3.09145
$36$	0	0
$37$	−162.050	−0.720024	−0.360012	−	0.932948i	$-0.617227\pi$
$37$	−162.050	−0.720024	−0.360012	+	0.932948i	$0.617227\pi$
$38$	−155.473	−0.663712
$39$	0	0
$40$	−147.262	−0.582106
$41$	341.655	1.30140	0.650702	−	0.759333i	$-0.274475\pi$
$41$	341.655	1.30140	0.650702	+	0.759333i	$0.274475\pi$
$42$	0	0
$43$	347.204	1.23135	0.615677	−	0.787999i	$-0.288883\pi$
$43$	347.204	1.23135	0.615677	+	0.787999i	$0.288883\pi$
$44$	116.541	0.399300
$45$	0	0
$46$	12.6503	0.0405475
$47$	−151.205	−0.469268	−0.234634	−	0.972084i	$-0.575389\pi$
$47$	−151.205	−0.469268	−0.234634	+	0.972084i	$0.575389\pi$
$48$	0	0
$49$	866.275	2.52558
$50$	−427.694	−1.20970
$51$	0	0
$52$	−48.8302	−0.130222
$53$	329.797	0.854738	0.427369	−	0.904077i	$-0.359441\pi$
$53$	329.797	0.854738	0.427369	+	0.904077i	$0.359441\pi$
$54$	0	0
$55$	536.316	1.31485
$56$	−278.197	−0.663851
$57$	0	0
$58$	267.778	0.606223
$59$	−229.746	−0.506956	−0.253478	−	0.967341i	$-0.581574\pi$
$59$	−229.746	−0.506956	−0.253478	+	0.967341i	$0.581574\pi$
$60$	0	0
$61$	719.548	1.51031	0.755153	−	0.655549i	$-0.227563\pi$
$61$	719.548	1.51031	0.755153	+	0.655549i	$0.227563\pi$
$62$	302.104	0.618828
$63$	0	0
$64$	64.0000	0.125000
$65$	−224.714	−0.428805
$66$	0	0
$67$	385.022	0.702058	0.351029	−	0.936365i	$-0.385832\pi$
$67$	385.022	0.702058	0.351029	+	0.936365i	$0.385832\pi$
$68$	48.9351	0.0872684
$69$	0	0
$70$	−1280.25	−2.18599
$71$	−125.968	−0.210559	−0.105280	−	0.994443i	$-0.533574\pi$
$71$	−125.968	−0.210559	−0.105280	+	0.994443i	$0.533574\pi$
$72$	0	0
$73$	369.566	0.592526	0.296263	−	0.955106i	$-0.404260\pi$
$73$	369.566	0.592526	0.296263	+	0.955106i	$0.404260\pi$
$74$	324.100	0.509134
$75$	0	0
$76$	310.946	0.469315
$77$	1013.17	1.49950
$78$	0	0
$79$	−1009.08	−1.43709	−0.718547	−	0.695478i	$-0.755192\pi$
$79$	−1009.08	−1.43709	−0.718547	+	0.695478i	$0.755192\pi$
$80$	294.525	0.411611
$81$	0	0
$82$	−683.310	−0.920232
$83$	−1134.08	−1.49977	−0.749887	−	0.661566i	$-0.769892\pi$
$83$	−1134.08	−1.49977	−0.749887	+	0.661566i	$0.769892\pi$
$84$	0	0
$85$	225.197	0.287365
$86$	−694.409	−0.870698
$87$	0	0
$88$	−233.082	−0.282348
$89$	−650.611	−0.774884	−0.387442	−	0.921894i	$-0.626641\pi$
$89$	−650.611	−0.774884	−0.387442	+	0.921894i	$0.626641\pi$
$90$	0	0
$91$	−424.513	−0.489022
$92$	−25.3006	−0.0286714
$93$	0	0
$94$	302.411	0.331822
$95$	1430.96	1.54540
$96$	0	0
$97$	−720.708	−0.754401	−0.377200	−	0.926132i	$-0.623113\pi$
$97$	−720.708	−0.754401	−0.377200	+	0.926132i	$0.623113\pi$
$98$	−1732.55	−1.78586
$99$	0	0
$100$	855.388	0.855388
$101$	−655.539	−0.645828	−0.322914	−	0.946428i	$-0.604662\pi$
$101$	−655.539	−0.645828	−0.322914	+	0.946428i	$0.604662\pi$
$102$	0	0
$103$	548.083	0.524313	0.262157	−	0.965025i	$-0.415566\pi$
$103$	548.083	0.524313	0.262157	+	0.965025i	$0.415566\pi$
$104$	97.6604	0.0920807
$105$	0	0
$106$	−659.594	−0.604391
$107$	−1872.22	−1.69154	−0.845769	−	0.533549i	$-0.820858\pi$
$107$	−1872.22	−1.69154	−0.845769	+	0.533549i	$0.820858\pi$
$108$	0	0
$109$	−360.413	−0.316709	−0.158354	−	0.987382i	$-0.550619\pi$
$109$	−360.413	−0.316709	−0.158354	+	0.987382i	$0.550619\pi$
$110$	−1072.63	−0.929740
$111$	0	0
$112$	556.394	0.469413
$113$	−1134.20	−0.944219	−0.472109	−	0.881540i	$-0.656507\pi$
$113$	−1134.20	−0.944219	−0.472109	+	0.881540i	$0.656507\pi$
$114$	0	0
$115$	−116.432	−0.0944118
$116$	−535.556	−0.428665
$117$	0	0
$118$	459.492	0.358472
$119$	425.425	0.327720
$120$	0	0
$121$	−482.138	−0.362237
$122$	−1439.10	−1.06795
$123$	0	0
$124$	−604.209	−0.437577
$125$	1635.48	1.17025
$126$	0	0
$127$	−299.617	−0.209344	−0.104672	−	0.994507i	$-0.533379\pi$
$127$	−299.617	−0.209344	−0.104672	+	0.994507i	$0.533379\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	449.428	0.303211
$131$	−1738.16	−1.15927	−0.579634	−	0.814877i	$-0.696804\pi$
$131$	−1738.16	−1.15927	−0.579634	+	0.814877i	$0.696804\pi$
$132$	0	0
$133$	2703.26	1.76242
$134$	−770.043	−0.496430
$135$	0	0
$136$	−97.8702	−0.0617081
$137$	1456.70	0.908425	0.454212	−	0.890893i	$-0.349921\pi$
$137$	1456.70	0.908425	0.454212	+	0.890893i	$0.349921\pi$
$138$	0	0
$139$	1254.74	0.765654	0.382827	−	0.923820i	$-0.374951\pi$
$139$	1254.74	0.765654	0.382827	+	0.923820i	$0.374951\pi$
$140$	2560.50	1.54573
$141$	0	0
$142$	251.937	0.148888
$143$	−355.670	−0.207990
$144$	0	0
$145$	−2464.60	−1.41154
$146$	−739.131	−0.418979
$147$	0	0
$148$	−648.201	−0.360012
$149$	−200.384	−0.110175	−0.0550875	−	0.998482i	$-0.517544\pi$
$149$	−200.384	−0.110175	−0.0550875	+	0.998482i	$0.517544\pi$
$150$	0	0
$151$	1632.11	0.879597	0.439798	−	0.898097i	$-0.355050\pi$
$151$	1632.11	0.879597	0.439798	+	0.898097i	$0.355050\pi$
$152$	−621.892	−0.331856
$153$	0	0
$154$	−2026.33	−1.06030
$155$	−2780.54	−1.44089
$156$	0	0
$157$	650.872	0.330861	0.165431	−	0.986221i	$-0.447099\pi$
$157$	650.872	0.330861	0.165431	+	0.986221i	$0.447099\pi$
$158$	2018.16	1.01618
$159$	0	0
$160$	−589.050	−0.291053
$161$	−219.955	−0.107670
$162$	0	0
$163$	2696.10	1.29555	0.647776	−	0.761831i	$-0.275699\pi$
$163$	2696.10	1.29555	0.647776	+	0.761831i	$0.275699\pi$
$164$	1366.62	0.650702
$165$	0	0
$166$	2268.16	1.06050
$167$	−356.685	−0.165276	−0.0826381	−	0.996580i	$-0.526335\pi$
$167$	−356.685	−0.165276	−0.0826381	+	0.996580i	$0.526335\pi$
$168$	0	0
$169$	−2047.98	−0.932169
$170$	−450.394	−0.203198
$171$	0	0
$172$	1388.82	0.615677
$173$	323.810	0.142305	0.0711526	−	0.997465i	$-0.477332\pi$
$173$	323.810	0.142305	0.0711526	+	0.997465i	$0.477332\pi$
$174$	0	0
$175$	7436.45	3.21224
$176$	466.164	0.199650
$177$	0	0
$178$	1301.22	0.547926
$179$	−82.1470	−0.0343014	−0.0171507	−	0.999853i	$-0.505460\pi$
$179$	−82.1470	−0.0343014	−0.0171507	+	0.999853i	$0.505460\pi$
$180$	0	0
$181$	−354.725	−0.145671	−0.0728356	−	0.997344i	$-0.523205\pi$
$181$	−354.725	−0.145671	−0.0728356	+	0.997344i	$0.523205\pi$
$182$	849.026	0.345791
$183$	0	0
$184$	50.6012	0.0202738
$185$	−2982.99	−1.18548
$186$	0	0
$187$	356.434	0.139385
$188$	−604.822	−0.234634
$189$	0	0
$190$	−2861.92	−1.09276
$191$	−5233.55	−1.98265	−0.991326	−	0.131424i	$-0.958045\pi$
$191$	−5233.55	−1.98265	−0.991326	+	0.131424i	$0.958045\pi$
$192$	0	0
$193$	2029.38	0.756879	0.378440	−	0.925626i	$-0.376461\pi$
$193$	2029.38	0.756879	0.378440	+	0.925626i	$0.376461\pi$
$194$	1441.42	0.533442
$195$	0	0
$196$	3465.10	1.26279
$197$	−564.442	−0.204136	−0.102068	−	0.994777i	$-0.532546\pi$
$197$	−564.442	−0.204136	−0.102068	+	0.994777i	$0.532546\pi$
$198$	0	0
$199$	2977.44	1.06063	0.530314	−	0.847801i	$-0.322074\pi$
$199$	2977.44	1.06063	0.530314	+	0.847801i	$0.322074\pi$
$200$	−1710.78	−0.604851
$201$	0	0
$202$	1311.08	0.456669
$203$	−4655.94	−1.60977
$204$	0	0
$205$	6289.12	2.14269
$206$	−1096.17	−0.370745
$207$	0	0
$208$	−195.321	−0.0651109
$209$	2264.87	0.749591
$210$	0	0
$211$	2954.68	0.964020	0.482010	−	0.876166i	$-0.339907\pi$
$211$	2954.68	0.964020	0.482010	+	0.876166i	$0.339907\pi$
$212$	1319.19	0.427369
$213$	0	0
$214$	3744.45	1.19610
$215$	6391.27	2.02735
$216$	0	0
$217$	−5252.79	−1.64324
$218$	720.825	0.223947
$219$	0	0
$220$	2145.26	0.657425
$221$	−149.344	−0.0454570
$222$	0	0
$223$	−1479.31	−0.444225	−0.222113	−	0.975021i	$-0.571295\pi$
$223$	−1479.31	−0.444225	−0.222113	+	0.975021i	$0.571295\pi$
$224$	−1112.79	−0.331925
$225$	0	0
$226$	2268.40	0.667664
$227$	−4932.53	−1.44222	−0.721109	−	0.692821i	$-0.756367\pi$
$227$	−4932.53	−1.44222	−0.721109	+	0.692821i	$0.756367\pi$
$228$	0	0
$229$	4794.27	1.38347	0.691735	−	0.722152i	$-0.256847\pi$
$229$	4794.27	1.38347	0.691735	+	0.722152i	$0.256847\pi$
$230$	232.864	0.0667592
$231$	0	0
$232$	1071.11	0.303112
$233$	3744.93	1.05296	0.526478	−	0.850189i	$-0.323512\pi$
$233$	3744.93	1.05296	0.526478	+	0.850189i	$0.323512\pi$
$234$	0	0
$235$	−2783.36	−0.772623
$236$	−918.984	−0.253478
$237$	0	0
$238$	−850.850	−0.231733
$239$	5341.60	1.44569	0.722844	−	0.691011i	$-0.242835\pi$
$239$	5341.60	1.44569	0.722844	+	0.691011i	$0.242835\pi$
$240$	0	0
$241$	−6828.24	−1.82509	−0.912543	−	0.408981i	$-0.865884\pi$
$241$	−6828.24	−1.82509	−0.912543	+	0.408981i	$0.865884\pi$
$242$	964.275	0.256140
$243$	0	0
$244$	2878.19	0.755153
$245$	15946.2	4.15823
$246$	0	0
$247$	−948.973	−0.244460
$248$	1208.42	0.309414
$249$	0	0
$250$	−3270.96	−0.827494
$251$	571.003	0.143591	0.0717956	−	0.997419i	$-0.477127\pi$
$251$	571.003	0.143591	0.0717956	+	0.997419i	$0.477127\pi$
$252$	0	0
$253$	−184.285	−0.0457940
$254$	599.234	0.148029
$255$	0	0
$256$	256.000	0.0625000
$257$	2498.43	0.606411	0.303206	−	0.952925i	$-0.401943\pi$
$257$	2498.43	0.606411	0.303206	+	0.952925i	$0.401943\pi$
$258$	0	0
$259$	−5635.24	−1.35196
$260$	−898.856	−0.214403
$261$	0	0
$262$	3476.33	0.819726
$263$	−3141.25	−0.736494	−0.368247	−	0.929728i	$-0.620042\pi$
$263$	−3141.25	−0.736494	−0.368247	+	0.929728i	$0.620042\pi$
$264$	0	0
$265$	6070.84	1.40728
$266$	−5406.52	−1.24622
$267$	0	0
$268$	1540.09	0.351029
$269$	1573.50	0.356647	0.178324	−	0.983972i	$-0.442933\pi$
$269$	1573.50	0.356647	0.178324	+	0.983972i	$0.442933\pi$
$270$	0	0
$271$	5646.06	1.26558	0.632792	−	0.774322i	$-0.281909\pi$
$271$	5646.06	1.26558	0.632792	+	0.774322i	$0.281909\pi$
$272$	195.740	0.0436342
$273$	0	0
$274$	−2913.40	−0.642353
$275$	6230.49	1.36623
$276$	0	0
$277$	−6015.60	−1.30485	−0.652423	−	0.757855i	$-0.726247\pi$
$277$	−6015.60	−1.30485	−0.652423	+	0.757855i	$0.726247\pi$
$278$	−2509.49	−0.541399
$279$	0	0
$280$	−5120.99	−1.09299
$281$	5354.52	1.13674	0.568369	−	0.822773i	$-0.307574\pi$
$281$	5354.52	1.13674	0.568369	+	0.822773i	$0.307574\pi$
$282$	0	0
$283$	−2927.15	−0.614845	−0.307422	−	0.951573i	$-0.599466\pi$
$283$	−2927.15	−0.614845	−0.307422	+	0.951573i	$0.599466\pi$
$284$	−503.874	−0.105280
$285$	0	0
$286$	711.340	0.147071
$287$	11880.9	2.44359
$288$	0	0
$289$	−4763.33	−0.969537
$290$	4929.20	0.998113
$291$	0	0
$292$	1478.26	0.296263
$293$	−3799.54	−0.757583	−0.378791	−	0.925482i	$-0.623660\pi$
$293$	−3799.54	−0.757583	−0.378791	+	0.925482i	$0.623660\pi$
$294$	0	0
$295$	−4229.12	−0.834674
$296$	1296.40	0.254567
$297$	0	0
$298$	400.768	0.0779055
$299$	77.2146	0.0149346
$300$	0	0
$301$	12073.9	2.31205
$302$	−3264.22	−0.621969
$303$	0	0
$304$	1243.78	0.234658
$305$	13245.3	2.48663
$306$	0	0
$307$	8454.09	1.57166	0.785831	−	0.618441i	$-0.212236\pi$
$307$	8454.09	1.57166	0.785831	+	0.618441i	$0.212236\pi$
$308$	4052.67	0.749748
$309$	0	0
$310$	5561.08	1.01886
$311$	−4214.21	−0.768380	−0.384190	−	0.923254i	$-0.625519\pi$
$311$	−4214.21	−0.768380	−0.384190	+	0.923254i	$0.625519\pi$
$312$	0	0
$313$	−8532.69	−1.54088	−0.770441	−	0.637511i	$-0.779964\pi$
$313$	−8532.69	−1.54088	−0.770441	+	0.637511i	$0.779964\pi$
$314$	−1301.74	−0.233954
$315$	0	0
$316$	−4036.32	−0.718547
$317$	1971.11	0.349238	0.174619	−	0.984636i	$-0.444131\pi$
$317$	1971.11	0.349238	0.174619	+	0.984636i	$0.444131\pi$
$318$	0	0
$319$	−3900.89	−0.684664
$320$	1178.10	0.205805
$321$	0	0
$322$	439.910	0.0761342
$323$	951.011	0.163826
$324$	0	0
$325$	−2610.55	−0.445560
$326$	−5392.21	−0.916094
$327$	0	0
$328$	−2733.24	−0.460116
$329$	−5258.11	−0.881122
$330$	0	0
$331$	−2273.39	−0.377513	−0.188757	−	0.982024i	$-0.560446\pi$
$331$	−2273.39	−0.377513	−0.188757	+	0.982024i	$0.560446\pi$
$332$	−4536.31	−0.749887
$333$	0	0
$334$	713.370	0.116868
$335$	7087.40	1.15590
$336$	0	0
$337$	−1411.81	−0.228208	−0.114104	−	0.993469i	$-0.536400\pi$
$337$	−1411.81	−0.228208	−0.114104	+	0.993469i	$0.536400\pi$
$338$	4095.95	0.659143
$339$	0	0
$340$	900.787	0.143683
$341$	−4400.94	−0.698899
$342$	0	0
$343$	18196.7	2.86451
$344$	−2777.64	−0.435349
$345$	0	0
$346$	−647.620	−0.100625
$347$	9661.46	1.49468	0.747340	−	0.664441i	$-0.231330\pi$
$347$	9661.46	1.49468	0.747340	+	0.664441i	$0.231330\pi$
$348$	0	0
$349$	4702.60	0.721273	0.360636	−	0.932707i	$-0.382560\pi$
$349$	4702.60	0.721273	0.360636	+	0.932707i	$0.382560\pi$
$350$	−14872.9	−2.27140
$351$	0	0
$352$	−932.328	−0.141174
$353$	−13070.5	−1.97074	−0.985368	−	0.170439i	$-0.945482\pi$
$353$	−13070.5	−1.97074	−0.985368	+	0.170439i	$0.945482\pi$
$354$	0	0
$355$	−2318.80	−0.346674
$356$	−2602.45	−0.387442
$357$	0	0
$358$	164.294	0.0242548
$359$	−1858.18	−0.273179	−0.136589	−	0.990628i	$-0.543614\pi$
$359$	−1858.18	−0.273179	−0.136589	+	0.990628i	$0.543614\pi$
$360$	0	0
$361$	−816.029	−0.118972
$362$	709.450	0.103005
$363$	0	0
$364$	−1698.05	−0.244511
$365$	6802.89	0.975560
$366$	0	0
$367$	1632.08	0.232136	0.116068	−	0.993241i	$-0.462971\pi$
$367$	1632.08	0.232136	0.116068	+	0.993241i	$0.462971\pi$
$368$	−101.202	−0.0143357
$369$	0	0
$370$	5965.98	0.838260
$371$	11468.6	1.60490
$372$	0	0
$373$	−10567.7	−1.46695	−0.733475	−	0.679716i	$-0.762103\pi$
$373$	−10567.7	−1.46695	−0.733475	+	0.679716i	$0.762103\pi$
$374$	−712.868	−0.0985602
$375$	0	0
$376$	1209.64	0.165911
$377$	1634.46	0.223286
$378$	0	0
$379$	−4209.91	−0.570576	−0.285288	−	0.958442i	$-0.592089\pi$
$379$	−4209.91	−0.570576	−0.285288	+	0.958442i	$0.592089\pi$
$380$	5723.83	0.772701
$381$	0	0
$382$	10467.1	1.40195
$383$	3237.54	0.431934	0.215967	−	0.976401i	$-0.430710\pi$
$383$	3237.54	0.431934	0.215967	+	0.976401i	$0.430710\pi$
$384$	0	0
$385$	18650.2	2.46883
$386$	−4058.75	−0.535195
$387$	0	0
$388$	−2882.83	−0.377200
$389$	3500.44	0.456245	0.228123	−	0.973632i	$-0.426741\pi$
$389$	3500.44	0.456245	0.228123	+	0.973632i	$0.426741\pi$
$390$	0	0
$391$	−77.3805	−0.0100084
$392$	−6930.20	−0.892928
$393$	0	0
$394$	1128.88	0.144346
$395$	−18574.9	−2.36609
$396$	0	0
$397$	−8960.80	−1.13282	−0.566410	−	0.824124i	$-0.691668\pi$
$397$	−8960.80	−1.13282	−0.566410	+	0.824124i	$0.691668\pi$
$398$	−5954.88	−0.749978
$399$	0	0
$400$	3421.55	0.427694
$401$	−1051.51	−0.130948	−0.0654740	−	0.997854i	$-0.520856\pi$
$401$	−1051.51	−0.130948	−0.0654740	+	0.997854i	$0.520856\pi$
$402$	0	0
$403$	1843.98	0.227928
$404$	−2622.16	−0.322914
$405$	0	0
$406$	9311.88	1.13828
$407$	−4721.37	−0.575012
$408$	0	0
$409$	−88.3827	−0.0106852	−0.00534260	−	0.999986i	$-0.501701\pi$
$409$	−88.3827	−0.0106852	−0.00534260	+	0.999986i	$0.501701\pi$
$410$	−12578.2	−1.51511
$411$	0	0
$412$	2192.33	0.262157
$413$	−7989.33	−0.951887
$414$	0	0
$415$	−20875.9	−2.46929
$416$	390.642	0.0460403
$417$	0	0
$418$	−4529.75	−0.530041
$419$	2783.18	0.324504	0.162252	−	0.986749i	$-0.448124\pi$
$419$	2783.18	0.324504	0.162252	+	0.986749i	$0.448124\pi$
$420$	0	0
$421$	−9579.62	−1.10898	−0.554492	−	0.832189i	$-0.687087\pi$
$421$	−9579.62	−1.10898	−0.554492	+	0.832189i	$0.687087\pi$
$422$	−5909.35	−0.681665
$423$	0	0
$424$	−2638.38	−0.302195
$425$	2616.16	0.298593
$426$	0	0
$427$	25022.0	2.83583
$428$	−7488.89	−0.845769
$429$	0	0
$430$	−12782.5	−1.43356
$431$	6818.04	0.761980	0.380990	−	0.924579i	$-0.375583\pi$
$431$	6818.04	0.761980	0.380990	+	0.924579i	$0.375583\pi$
$432$	0	0
$433$	−4810.43	−0.533891	−0.266945	−	0.963712i	$-0.586014\pi$
$433$	−4810.43	−0.533891	−0.266945	+	0.963712i	$0.586014\pi$
$434$	10505.6	1.16194
$435$	0	0
$436$	−1441.65	−0.158354
$437$	−491.696	−0.0538238
$438$	0	0
$439$	12187.0	1.32495	0.662476	−	0.749083i	$-0.269505\pi$
$439$	12187.0	1.32495	0.662476	+	0.749083i	$0.269505\pi$
$440$	−4290.53	−0.464870
$441$	0	0
$442$	298.689	0.0321429
$443$	−9661.34	−1.03617	−0.518086	−	0.855328i	$-0.673355\pi$
$443$	−9661.34	−1.03617	−0.518086	+	0.855328i	$0.673355\pi$
$444$	0	0
$445$	−11976.3	−1.27580
$446$	2958.63	0.314115
$447$	0	0
$448$	2225.58	0.234707
$449$	−506.871	−0.0532756	−0.0266378	−	0.999645i	$-0.508480\pi$
$449$	−506.871	−0.0532756	−0.0266378	+	0.999645i	$0.508480\pi$
$450$	0	0
$451$	9954.21	1.03930
$452$	−4536.81	−0.472109
$453$	0	0
$454$	9865.06	1.01980
$455$	−7814.35	−0.805148
$456$	0	0
$457$	16622.2	1.70143	0.850715	−	0.525627i	$-0.176169\pi$
$457$	16622.2	1.70143	0.850715	+	0.525627i	$0.176169\pi$
$458$	−9588.54	−0.978260
$459$	0	0
$460$	−465.729	−0.0472059
$461$	−2121.25	−0.214309	−0.107155	−	0.994242i	$-0.534174\pi$
$461$	−2121.25	−0.214309	−0.107155	+	0.994242i	$0.534174\pi$
$462$	0	0
$463$	−8307.85	−0.833906	−0.416953	−	0.908928i	$-0.636902\pi$
$463$	−8307.85	−0.833906	−0.416953	+	0.908928i	$0.636902\pi$
$464$	−2142.22	−0.214332
$465$	0	0
$466$	−7489.87	−0.744553
$467$	13797.5	1.36718	0.683591	−	0.729865i	$-0.260417\pi$
$467$	13797.5	1.36718	0.683591	+	0.729865i	$0.260417\pi$
$468$	0	0
$469$	13389.0	1.31822
$470$	5566.72	0.546327
$471$	0	0
$472$	1837.97	0.179236
$473$	10115.9	0.983359
$474$	0	0
$475$	16623.7	1.60579
$476$	1701.70	0.163860
$477$	0	0
$478$	−10683.2	−1.02226
$479$	15032.8	1.43396	0.716980	−	0.697094i	$-0.245524\pi$
$479$	15032.8	1.43396	0.716980	+	0.697094i	$0.245524\pi$
$480$	0	0
$481$	1978.24	0.187526
$482$	13656.5	1.29053
$483$	0	0
$484$	−1928.55	−0.181119
$485$	−13266.7	−1.24208
$486$	0	0
$487$	17512.1	1.62946	0.814732	−	0.579838i	$-0.196884\pi$
$487$	17512.1	1.62946	0.814732	+	0.579838i	$0.196884\pi$
$488$	−5756.38	−0.533974
$489$	0	0
$490$	−31892.4	−2.94031
$491$	4085.95	0.375553	0.187777	−	0.982212i	$-0.439872\pi$
$491$	4085.95	0.375553	0.187777	+	0.982212i	$0.439872\pi$
$492$	0	0
$493$	−1637.97	−0.149636
$494$	1897.95	0.172860
$495$	0	0
$496$	−2416.84	−0.218789
$497$	−4380.51	−0.395357
$498$	0	0
$499$	9986.63	0.895918	0.447959	−	0.894054i	$-0.352151\pi$
$499$	9986.63	0.895918	0.447959	+	0.894054i	$0.352151\pi$
$500$	6541.91	0.585126
$501$	0	0
$502$	−1142.01	−0.101534
$503$	6434.50	0.570378	0.285189	−	0.958471i	$-0.407944\pi$
$503$	6434.50	0.570378	0.285189	+	0.958471i	$0.407944\pi$
$504$	0	0
$505$	−12067.0	−1.06332
$506$	368.570	0.0323813
$507$	0	0
$508$	−1198.47	−0.104672
$509$	−3596.29	−0.313168	−0.156584	−	0.987665i	$-0.550048\pi$
$509$	−3596.29	−0.313168	−0.156584	+	0.987665i	$0.550048\pi$
$510$	0	0
$511$	12851.5	1.11256
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−4996.86	−0.428797
$515$	10089.0	0.863252
$516$	0	0
$517$	−4405.41	−0.374757
$518$	11270.5	0.955977
$519$	0	0
$520$	1797.71	0.151606
$521$	20052.9	1.68625	0.843123	−	0.537721i	$-0.180715\pi$
$521$	20052.9	1.68625	0.843123	+	0.537721i	$0.180715\pi$
$522$	0	0
$523$	395.675	0.0330816	0.0165408	−	0.999863i	$-0.494735\pi$
$523$	395.675	0.0330816	0.0165408	+	0.999863i	$0.494735\pi$
$524$	−6952.65	−0.579634
$525$	0	0
$526$	6282.50	0.520780
$527$	−1847.94	−0.152747
$528$	0	0
$529$	−12127.0	−0.996712
$530$	−12141.7	−0.995096
$531$	0	0
$532$	10813.0	0.881212
$533$	−4170.77	−0.338942
$534$	0	0
$535$	−34463.5	−2.78502
$536$	−3080.17	−0.248215
$537$	0	0
$538$	−3147.01	−0.252188
$539$	25239.1	2.01693
$540$	0	0
$541$	−19391.5	−1.54105	−0.770525	−	0.637410i	$-0.780006\pi$
$541$	−19391.5	−1.54105	−0.770525	+	0.637410i	$0.780006\pi$
$542$	−11292.1	−0.894904
$543$	0	0
$544$	−391.481	−0.0308540
$545$	−6634.40	−0.521443
$546$	0	0
$547$	1400.29	0.109455	0.0547275	−	0.998501i	$-0.482571\pi$
$547$	1400.29	0.109455	0.0547275	+	0.998501i	$0.482571\pi$
$548$	5826.80	0.454212
$549$	0	0
$550$	−12461.0	−0.966068
$551$	−10408.1	−0.804716
$552$	0	0
$553$	−35090.4	−2.69836
$554$	12031.2	0.922665
$555$	0	0
$556$	5018.97	0.382827
$557$	14991.5	1.14041	0.570205	−	0.821502i	$-0.306864\pi$
$557$	14991.5	1.14041	0.570205	+	0.821502i	$0.306864\pi$
$558$	0	0
$559$	−4238.52	−0.320698
$560$	10242.0	0.772863
$561$	0	0
$562$	−10709.0	−0.803796
$563$	−989.723	−0.0740885	−0.0370443	−	0.999314i	$-0.511794\pi$
$563$	−989.723	−0.0740885	−0.0370443	+	0.999314i	$0.511794\pi$
$564$	0	0
$565$	−20878.2	−1.55460
$566$	5854.30	0.434761
$567$	0	0
$568$	1007.75	0.0744439
$569$	−4215.26	−0.310567	−0.155284	−	0.987870i	$-0.549629\pi$
$569$	−4215.26	−0.310567	−0.155284	+	0.987870i	$0.549629\pi$
$570$	0	0
$571$	3101.72	0.227326	0.113663	−	0.993519i	$-0.463742\pi$
$571$	3101.72	0.227326	0.113663	+	0.993519i	$0.463742\pi$
$572$	−1422.68	−0.103995
$573$	0	0
$574$	−23761.9	−1.72788
$575$	−1352.62	−0.0981008
$576$	0	0
$577$	−6727.12	−0.485362	−0.242681	−	0.970106i	$-0.578027\pi$
$577$	−6727.12	−0.485362	−0.242681	+	0.970106i	$0.578027\pi$
$578$	9526.67	0.685566
$579$	0	0
$580$	−9858.40	−0.705772
$581$	−39437.2	−2.81606
$582$	0	0
$583$	9608.72	0.682594
$584$	−2956.52	−0.209490
$585$	0	0
$586$	7599.09	0.535692
$587$	−16448.8	−1.15658	−0.578292	−	0.815830i	$-0.696281\pi$
$587$	−16448.8	−1.15658	−0.578292	+	0.815830i	$0.696281\pi$
$588$	0	0
$589$	−11742.3	−0.821447
$590$	8458.23	0.590203
$591$	0	0
$592$	−2592.80	−0.180006
$593$	18525.6	1.28289	0.641446	−	0.767168i	$-0.278335\pi$
$593$	18525.6	1.28289	0.641446	+	0.767168i	$0.278335\pi$
$594$	0	0
$595$	7831.14	0.539572
$596$	−801.535	−0.0550875
$597$	0	0
$598$	−154.429	−0.0105603
$599$	−4533.51	−0.309239	−0.154620	−	0.987974i	$-0.549415\pi$
$599$	−4533.51	−0.309239	−0.154620	+	0.987974i	$0.549415\pi$
$600$	0	0
$601$	−2891.36	−0.196241	−0.0981207	−	0.995175i	$-0.531283\pi$
$601$	−2891.36	−0.196241	−0.0981207	+	0.995175i	$0.531283\pi$
$602$	−24147.8	−1.63487
$603$	0	0
$604$	6528.43	0.439798
$605$	−8875.09	−0.596403
$606$	0	0
$607$	−11543.4	−0.771883	−0.385942	−	0.922523i	$-0.626123\pi$
$607$	−11543.4	−0.771883	−0.385942	+	0.922523i	$0.626123\pi$
$608$	−2487.57	−0.165928
$609$	0	0
$610$	−26490.6	−1.75831
$611$	1845.85	0.122218
$612$	0	0
$613$	−3626.32	−0.238933	−0.119466	−	0.992838i	$-0.538118\pi$
$613$	−3626.32	−0.238933	−0.119466	+	0.992838i	$0.538118\pi$
$614$	−16908.2	−1.11133
$615$	0	0
$616$	−8105.34	−0.530152
$617$	−5203.72	−0.339536	−0.169768	−	0.985484i	$-0.554302\pi$
$617$	−5203.72	−0.339536	−0.169768	+	0.985484i	$0.554302\pi$
$618$	0	0
$619$	−25335.9	−1.64513	−0.822564	−	0.568673i	$-0.807457\pi$
$619$	−25335.9	−1.64513	−0.822564	+	0.568673i	$0.807457\pi$
$620$	−11122.2	−0.720446
$621$	0	0
$622$	8428.43	0.543327
$623$	−22624.8	−1.45496
$624$	0	0
$625$	3374.67	0.215979
$626$	17065.4	1.08957
$627$	0	0
$628$	2603.49	0.165431
$629$	−1982.49	−0.125671
$630$	0	0
$631$	−1962.66	−0.123823	−0.0619113	−	0.998082i	$-0.519720\pi$
$631$	−1962.66	−0.123823	−0.0619113	+	0.998082i	$0.519720\pi$
$632$	8072.64	0.508089
$633$	0	0
$634$	−3942.22	−0.246949
$635$	−5515.29	−0.344673
$636$	0	0
$637$	−10575.1	−0.657771
$638$	7801.77	0.484130
$639$	0	0
$640$	−2356.20	−0.145526
$641$	−26185.1	−1.61349	−0.806746	−	0.590898i	$-0.798773\pi$
$641$	−26185.1	−1.61349	−0.806746	+	0.590898i	$0.798773\pi$
$642$	0	0
$643$	−15312.1	−0.939111	−0.469556	−	0.882903i	$-0.655586\pi$
$643$	−15312.1	−0.939111	−0.469556	+	0.882903i	$0.655586\pi$
$644$	−879.819	−0.0538350
$645$	0	0
$646$	−1902.02	−0.115842
$647$	2663.26	0.161829	0.0809146	−	0.996721i	$-0.474216\pi$
$647$	2663.26	0.161829	0.0809146	+	0.996721i	$0.474216\pi$
$648$	0	0
$649$	−6693.70	−0.404855
$650$	5221.10	0.315059
$651$	0	0
$652$	10784.4	0.647776
$653$	28453.8	1.70518	0.852592	−	0.522578i	$-0.175030\pi$
$653$	28453.8	1.70518	0.852592	+	0.522578i	$0.175030\pi$
$654$	0	0
$655$	−31995.7	−1.90867
$656$	5466.48	0.325351
$657$	0	0
$658$	10516.2	0.623047
$659$	14067.7	0.831562	0.415781	−	0.909465i	$-0.363508\pi$
$659$	14067.7	0.831562	0.415781	+	0.909465i	$0.363508\pi$
$660$	0	0
$661$	30493.5	1.79434	0.897171	−	0.441684i	$-0.145619\pi$
$661$	30493.5	1.79434	0.897171	+	0.441684i	$0.145619\pi$
$662$	4546.78	0.266942
$663$	0	0
$664$	9072.63	0.530250
$665$	49761.1	2.90173
$666$	0	0
$667$	846.868	0.0491617
$668$	−1426.74	−0.0826381
$669$	0	0
$670$	−14174.8	−0.817344
$671$	20964.2	1.20613
$672$	0	0
$673$	26508.3	1.51830	0.759152	−	0.650913i	$-0.225614\pi$
$673$	26508.3	1.51830	0.759152	+	0.650913i	$0.225614\pi$
$674$	2823.62	0.161367
$675$	0	0
$676$	−8191.90	−0.466085
$677$	19639.0	1.11490	0.557451	−	0.830210i	$-0.311780\pi$
$677$	19639.0	1.11490	0.557451	+	0.830210i	$0.311780\pi$
$678$	0	0
$679$	−25062.4	−1.41650
$680$	−1801.57	−0.101599
$681$	0	0
$682$	8801.89	0.494196
$683$	−5352.51	−0.299865	−0.149933	−	0.988696i	$-0.547906\pi$
$683$	−5352.51	−0.299865	−0.149933	+	0.988696i	$0.547906\pi$
$684$	0	0
$685$	26814.6	1.49567
$686$	−36393.4	−2.02552
$687$	0	0
$688$	5555.27	0.307838
$689$	−4026.01	−0.222611
$690$	0	0
$691$	28496.8	1.56884	0.784421	−	0.620229i	$-0.212960\pi$
$691$	28496.8	1.56884	0.784421	+	0.620229i	$0.212960\pi$
$692$	1295.24	0.0711526
$693$	0	0
$694$	−19322.9	−1.05690
$695$	23097.1	1.26061
$696$	0	0
$697$	4179.73	0.227143
$698$	−9405.19	−0.510017
$699$	0	0
$700$	29745.8	1.60612
$701$	−32930.0	−1.77425	−0.887125	−	0.461530i	$-0.847301\pi$
$701$	−32930.0	−1.77425	−0.887125	+	0.461530i	$0.847301\pi$
$702$	0	0
$703$	−12597.2	−0.675837
$704$	1864.66	0.0998251
$705$	0	0
$706$	26140.9	1.39352
$707$	−22796.1	−1.21264
$708$	0	0
$709$	4396.02	0.232858	0.116429	−	0.993199i	$-0.462855\pi$
$709$	4396.02	0.232858	0.116429	+	0.993199i	$0.462855\pi$
$710$	4637.60	0.245135
$711$	0	0
$712$	5204.89	0.273963
$713$	955.429	0.0501838
$714$	0	0
$715$	−6547.10	−0.342444
$716$	−328.588	−0.0171507
$717$	0	0
$718$	3716.37	0.193167
$719$	−24851.1	−1.28900	−0.644499	−	0.764605i	$-0.722934\pi$
$719$	−24851.1	−1.28900	−0.644499	+	0.764605i	$0.722934\pi$
$720$	0	0
$721$	19059.4	0.984478
$722$	1632.06	0.0841259
$723$	0	0
$724$	−1418.90	−0.0728356
$725$	−28631.8	−1.46670
$726$	0	0
$727$	−3542.45	−0.180718	−0.0903592	−	0.995909i	$-0.528802\pi$
$727$	−3542.45	−0.180718	−0.0903592	+	0.995909i	$0.528802\pi$
$728$	3396.10	0.172896
$729$	0	0
$730$	−13605.8	−0.689825
$731$	4247.62	0.214916
$732$	0	0
$733$	−32250.5	−1.62510	−0.812550	−	0.582892i	$-0.801921\pi$
$733$	−32250.5	−1.62510	−0.812550	+	0.582892i	$0.801921\pi$
$734$	−3264.16	−0.164145
$735$	0	0
$736$	202.405	0.0101369
$737$	11217.7	0.560664
$738$	0	0
$739$	23798.5	1.18463	0.592316	−	0.805706i	$-0.298214\pi$
$739$	23798.5	1.18463	0.592316	+	0.805706i	$0.298214\pi$
$740$	−11932.0	−0.592739
$741$	0	0
$742$	−22937.1	−1.13484
$743$	15745.4	0.777449	0.388724	−	0.921354i	$-0.372916\pi$
$743$	15745.4	0.777449	0.388724	+	0.921354i	$0.372916\pi$
$744$	0	0
$745$	−3688.62	−0.181397
$746$	21135.3	1.03729
$747$	0	0
$748$	1425.74	0.0696926
$749$	−65105.8	−3.17612
$750$	0	0
$751$	−16089.7	−0.781788	−0.390894	−	0.920436i	$-0.627834\pi$
$751$	−16089.7	−0.781788	−0.390894	+	0.920436i	$0.627834\pi$
$752$	−2419.29	−0.117317
$753$	0	0
$754$	−3268.91	−0.157887
$755$	30043.5	1.44821
$756$	0	0
$757$	28643.2	1.37524	0.687620	−	0.726071i	$-0.258656\pi$
$757$	28643.2	1.37524	0.687620	+	0.726071i	$0.258656\pi$
$758$	8419.82	0.403458
$759$	0	0
$760$	−11447.7	−0.546382
$761$	−27884.4	−1.32827	−0.664133	−	0.747615i	$-0.731199\pi$
$761$	−27884.4	−1.32827	−0.664133	+	0.747615i	$0.731199\pi$
$762$	0	0
$763$	−12533.2	−0.594670
$764$	−20934.2	−0.991326
$765$	0	0
$766$	−6475.09	−0.305424
$767$	2804.63	0.132033
$768$	0	0
$769$	8287.73	0.388639	0.194319	−	0.980938i	$-0.437750\pi$
$769$	8287.73	0.388639	0.194319	+	0.980938i	$0.437750\pi$
$770$	−37300.4	−1.74573
$771$	0	0
$772$	8117.51	0.378440
$773$	−13027.0	−0.606143	−0.303072	−	0.952968i	$-0.598012\pi$
$773$	−13027.0	−0.606143	−0.303072	+	0.952968i	$0.598012\pi$
$774$	0	0
$775$	−32302.1	−1.49719
$776$	5765.67	0.266721
$777$	0	0
$778$	−7000.88	−0.322614
$779$	26559.1	1.22154
$780$	0	0
$781$	−3670.12	−0.168153
$782$	154.761	0.00707703
$783$	0	0
$784$	13860.4	0.631395
$785$	11981.1	0.544745
$786$	0	0
$787$	−3445.44	−0.156057	−0.0780283	−	0.996951i	$-0.524862\pi$
$787$	−3445.44	−0.156057	−0.0780283	+	0.996951i	$0.524862\pi$
$788$	−2257.77	−0.102068
$789$	0	0
$790$	37149.9	1.67308
$791$	−39441.4	−1.77292
$792$	0	0
$793$	−8783.91	−0.393349
$794$	17921.6	0.801025
$795$	0	0
$796$	11909.8	0.530314
$797$	34926.9	1.55229	0.776144	−	0.630555i	$-0.217173\pi$
$797$	34926.9	1.55229	0.776144	+	0.630555i	$0.217173\pi$
$798$	0	0
$799$	−1849.81	−0.0819045
$800$	−6843.11	−0.302425
$801$	0	0
$802$	2103.03	0.0925942
$803$	10767.4	0.473191
$804$	0	0
$805$	−4048.88	−0.177273
$806$	−3687.96	−0.161170
$807$	0	0
$808$	5244.31	0.228335
$809$	−2065.17	−0.0897498	−0.0448749	−	0.998993i	$-0.514289\pi$
$809$	−2065.17	−0.0897498	−0.0448749	+	0.998993i	$0.514289\pi$
$810$	0	0
$811$	−26833.1	−1.16182	−0.580911	−	0.813967i	$-0.697304\pi$
$811$	−26833.1	−1.16182	−0.580911	+	0.813967i	$0.697304\pi$
$812$	−18623.8	−0.804884
$813$	0	0
$814$	9442.75	0.406595
$815$	49629.3	2.13305
$816$	0	0
$817$	26990.5	1.15579
$818$	176.765	0.00755557
$819$	0	0
$820$	25156.5	1.07134
$821$	−33959.2	−1.44359	−0.721794	−	0.692108i	$-0.756682\pi$
$821$	−33959.2	−1.44359	−0.721794	+	0.692108i	$0.756682\pi$
$822$	0	0
$823$	−33850.0	−1.43370	−0.716852	−	0.697225i	$-0.754418\pi$
$823$	−33850.0	−1.43370	−0.716852	+	0.697225i	$0.754418\pi$
$824$	−4384.67	−0.185373
$825$	0	0
$826$	15978.7	0.673086
$827$	16913.3	0.711165	0.355583	−	0.934645i	$-0.384283\pi$
$827$	16913.3	0.711165	0.355583	+	0.934645i	$0.384283\pi$
$828$	0	0
$829$	27078.5	1.13447	0.567235	−	0.823556i	$-0.308013\pi$
$829$	27078.5	1.13447	0.567235	+	0.823556i	$0.308013\pi$
$830$	41751.8	1.74605
$831$	0	0
$832$	−781.283	−0.0325554
$833$	10597.8	0.440807
$834$	0	0
$835$	−6565.79	−0.272118
$836$	9059.49	0.374796
$837$	0	0
$838$	−5566.35	−0.229459
$839$	23089.4	0.950099	0.475050	−	0.879959i	$-0.342430\pi$
$839$	23089.4	0.950099	0.475050	+	0.879959i	$0.342430\pi$
$840$	0	0
$841$	−6462.75	−0.264986
$842$	19159.2	0.784170
$843$	0	0
$844$	11818.7	0.482010
$845$	−37698.7	−1.53476
$846$	0	0
$847$	−16766.2	−0.680156
$848$	5276.75	0.213684
$849$	0	0
$850$	−5232.31	−0.211137
$851$	1024.99	0.0412882
$852$	0	0
$853$	17261.4	0.692871	0.346435	−	0.938074i	$-0.387392\pi$
$853$	17261.4	0.692871	0.346435	+	0.938074i	$0.387392\pi$
$854$	−50044.0	−2.00523
$855$	0	0
$856$	14977.8	0.598049
$857$	8581.99	0.342071	0.171036	−	0.985265i	$-0.445289\pi$
$857$	8581.99	0.342071	0.171036	+	0.985265i	$0.445289\pi$
$858$	0	0
$859$	2306.93	0.0916316	0.0458158	−	0.998950i	$-0.485411\pi$
$859$	2306.93	0.0916316	0.0458158	+	0.998950i	$0.485411\pi$
$860$	25565.1	1.01368
$861$	0	0
$862$	−13636.1	−0.538801
$863$	−23551.9	−0.928988	−0.464494	−	0.885576i	$-0.653764\pi$
$863$	−23551.9	−0.928988	−0.464494	+	0.885576i	$0.653764\pi$
$864$	0	0
$865$	5960.63	0.234298
$866$	9620.87	0.377518
$867$	0	0
$868$	−21011.1	−0.821618
$869$	−29399.8	−1.14766
$870$	0	0
$871$	−4700.17	−0.182846
$872$	2883.30	0.111974
$873$	0	0
$874$	983.391	0.0380592
$875$	56873.1	2.19733
$876$	0	0
$877$	−34079.9	−1.31220	−0.656099	−	0.754675i	$-0.727794\pi$
$877$	−34079.9	−1.31220	−0.656099	+	0.754675i	$0.727794\pi$
$878$	−24374.0	−0.936883
$879$	0	0
$880$	8581.05	0.328713
$881$	−36804.0	−1.40744	−0.703721	−	0.710476i	$-0.748480\pi$
$881$	−36804.0	−1.40744	−0.703721	+	0.710476i	$0.748480\pi$
$882$	0	0
$883$	9104.30	0.346981	0.173490	−	0.984836i	$-0.444495\pi$
$883$	9104.30	0.346981	0.173490	+	0.984836i	$0.444495\pi$
$884$	−597.378	−0.0227285
$885$	0	0
$886$	19322.7	0.732684
$887$	6618.40	0.250535	0.125267	−	0.992123i	$-0.460021\pi$
$887$	6618.40	0.250535	0.125267	+	0.992123i	$0.460021\pi$
$888$	0	0
$889$	−10419.1	−0.393076
$890$	23952.6	0.902129
$891$	0	0
$892$	−5917.26	−0.222113
$893$	−11754.2	−0.440469
$894$	0	0
$895$	−1512.15	−0.0564754
$896$	−4451.15	−0.165963
$897$	0	0
$898$	1013.74	0.0376715
$899$	20224.2	0.750296
$900$	0	0
$901$	4034.66	0.149183
$902$	−19908.4	−0.734898
$903$	0	0
$904$	9073.61	0.333832
$905$	−6529.70	−0.239839
$906$	0	0
$907$	−3247.61	−0.118892	−0.0594461	−	0.998232i	$-0.518933\pi$
$907$	−3247.61	−0.118892	−0.0594461	+	0.998232i	$0.518933\pi$
$908$	−19730.1	−0.721109
$909$	0	0
$910$	15628.7	0.569326
$911$	34500.0	1.25470	0.627352	−	0.778736i	$-0.284139\pi$
$911$	34500.0	1.25470	0.627352	+	0.778736i	$0.284139\pi$
$912$	0	0
$913$	−33041.7	−1.19772
$914$	−33244.4	−1.20309
$915$	0	0
$916$	19177.1	0.691735
$917$	−60444.0	−2.17670
$918$	0	0
$919$	42888.3	1.53945	0.769724	−	0.638376i	$-0.220394\pi$
$919$	42888.3	1.53945	0.769724	+	0.638376i	$0.220394\pi$
$920$	931.457	0.0333796
$921$	0	0
$922$	4242.50	0.151540
$923$	1537.77	0.0548388
$924$	0	0
$925$	−34654.0	−1.23180
$926$	16615.7	0.589661
$927$	0	0
$928$	4284.45	0.151556
$929$	−14809.4	−0.523015	−0.261508	−	0.965201i	$-0.584220\pi$
$929$	−14809.4	−0.523015	−0.261508	+	0.965201i	$0.584220\pi$
$930$	0	0
$931$	67341.2	2.37059
$932$	14979.7	0.526478
$933$	0	0
$934$	−27595.1	−0.966744
$935$	6561.17	0.229490
$936$	0	0
$937$	−23513.5	−0.819800	−0.409900	−	0.912130i	$-0.634436\pi$
$937$	−23513.5	−0.819800	−0.409900	+	0.912130i	$0.634436\pi$
$938$	−26778.0	−0.932123
$939$	0	0
$940$	−11133.4	−0.386311
$941$	−56563.8	−1.95954	−0.979770	−	0.200126i	$-0.935865\pi$
$941$	−56563.8	−1.95954	−0.979770	+	0.200126i	$0.935865\pi$
$942$	0	0
$943$	−2161.02	−0.0746262
$944$	−3675.94	−0.126739
$945$	0	0
$946$	−20231.8	−0.695340
$947$	−36813.0	−1.26321	−0.631606	−	0.775290i	$-0.717604\pi$
$947$	−36813.0	−1.26321	−0.631606	+	0.775290i	$0.717604\pi$
$948$	0	0
$949$	−4511.49	−0.154319
$950$	−33247.5	−1.13546
$951$	0	0
$952$	−3403.40	−0.115866
$953$	5800.77	0.197173	0.0985863	−	0.995129i	$-0.468568\pi$
$953$	5800.77	0.197173	0.0985863	+	0.995129i	$0.468568\pi$
$954$	0	0
$955$	−96338.2	−3.26433
$956$	21366.4	0.722844
$957$	0	0
$958$	−30065.6	−1.01396
$959$	50656.2	1.70571
$960$	0	0
$961$	−6974.22	−0.234105
$962$	−3956.47	−0.132601
$963$	0	0
$964$	−27313.0	−0.912543
$965$	37356.4	1.24616
$966$	0	0
$967$	47543.7	1.58108	0.790539	−	0.612411i	$-0.209800\pi$
$967$	47543.7	1.58108	0.790539	+	0.612411i	$0.209800\pi$
$968$	3857.10	0.128070
$969$	0	0
$970$	26533.3	0.878282
$971$	−51826.6	−1.71287	−0.856434	−	0.516257i	$-0.827325\pi$
$971$	−51826.6	−1.71287	−0.856434	+	0.516257i	$0.827325\pi$
$972$	0	0
$973$	43633.2	1.43763
$974$	−35024.2	−1.15221
$975$	0	0
$976$	11512.8	0.377576
$977$	10438.4	0.341816	0.170908	−	0.985287i	$-0.445330\pi$
$977$	10438.4	0.341816	0.170908	+	0.985287i	$0.445330\pi$
$978$	0	0
$979$	−18955.7	−0.618823
$980$	63784.8	2.07911
$981$	0	0
$982$	−8171.91	−0.265556
$983$	3126.83	0.101455	0.0507275	−	0.998713i	$-0.483846\pi$
$983$	3126.83	0.101455	0.0507275	+	0.998713i	$0.483846\pi$
$984$	0	0
$985$	−10390.1	−0.336099
$986$	3275.93	0.105808
$987$	0	0
$988$	−3795.89	−0.122230
$989$	−2196.12	−0.0706093
$990$	0	0
$991$	−26631.4	−0.853657	−0.426829	−	0.904332i	$-0.640369\pi$
$991$	−26631.4	−0.853657	−0.426829	+	0.904332i	$0.640369\pi$
$992$	4833.67	0.154707
$993$	0	0
$994$	8761.01	0.279560
$995$	54808.1	1.74627
$996$	0	0
$997$	−1948.30	−0.0618889	−0.0309445	−	0.999521i	$-0.509851\pi$
$997$	−1948.30	−0.0618889	−0.0309445	+	0.999521i	$0.509851\pi$
$998$	−19973.3	−0.633509
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.4.a.i.1.15		15
3.2	odd	2		1458.4.a.j.1.1		15
27.4	even	9		54.4.e.b.43.3	✓	30
27.7	even	9		54.4.e.b.49.3	yes	30
27.20	odd	18		162.4.e.b.37.5		30
27.23	odd	18		162.4.e.b.127.5		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.4.e.b.43.3	✓	30	27.4	even	9
54.4.e.b.49.3	yes	30	27.7	even	9
162.4.e.b.37.5		30	27.20	odd	18
162.4.e.b.127.5		30	27.23	odd	18
1458.4.a.i.1.15		15	1.1	even	1	trivial
1458.4.a.j.1.1		15	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1458.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +18.4078 q^{5} +34.7746 q^{7} -8.00000 q^{8} -36.8156 q^{10} +29.1352 q^{11} -12.2075 q^{13} -69.5493 q^{14} +16.0000 q^{16} +12.2338 q^{17} +77.7365 q^{19} +73.6312 q^{20} -58.2705 q^{22} -6.32515 q^{23} +213.847 q^{25} +24.4151 q^{26} +139.099 q^{28} -133.889 q^{29} -151.052 q^{31} -32.0000 q^{32} -24.4675 q^{34} +640.124 q^{35} -162.050 q^{37} -155.473 q^{38} -147.262 q^{40} +341.655 q^{41} +347.204 q^{43} +116.541 q^{44} +12.6503 q^{46} -151.205 q^{47} +866.275 q^{49} -427.694 q^{50} -48.8302 q^{52} +329.797 q^{53} +536.316 q^{55} -278.197 q^{56} +267.778 q^{58} -229.746 q^{59} +719.548 q^{61} +302.104 q^{62} +64.0000 q^{64} -224.714 q^{65} +385.022 q^{67} +48.9351 q^{68} -1280.25 q^{70} -125.968 q^{71} +369.566 q^{73} +324.100 q^{74} +310.946 q^{76} +1013.17 q^{77} -1009.08 q^{79} +294.525 q^{80} -683.310 q^{82} -1134.08 q^{83} +225.197 q^{85} -694.409 q^{86} -233.082 q^{88} -650.611 q^{89} -424.513 q^{91} -25.3006 q^{92} +302.411 q^{94} +1430.96 q^{95} -720.708 q^{97} -1732.55 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 30 q^{2} + 60 q^{4} - 15 q^{5} + 42 q^{7} - 120 q^{8} + 30 q^{10} - 33 q^{11} + 117 q^{13} - 84 q^{14} + 240 q^{16} - 102 q^{17} + 171 q^{19} - 60 q^{20} + 66 q^{22} - 174 q^{23} + 600 q^{25} - 234 q^{26}+ \cdots - 4002 q^{98}+O(q^{100}) \) 15 * q - 30 * q^2 + 60 * q^4 - 15 * q^5 + 42 * q^7 - 120 * q^8 + 30 * q^10 - 33 * q^11 + 117 * q^13 - 84 * q^14 + 240 * q^16 - 102 * q^17 + 171 * q^19 - 60 * q^20 + 66 * q^22 - 174 * q^23 + 600 * q^25 - 234 * q^26 + 168 * q^28 - 573 * q^29 + 372 * q^31 - 480 * q^32 + 204 * q^34 - 624 * q^35 + 555 * q^37 - 342 * q^38 + 120 * q^40 - 852 * q^41 + 1002 * q^43 - 132 * q^44 + 348 * q^46 - 306 * q^47 + 2001 * q^49 - 1200 * q^50 + 468 * q^52 - 897 * q^53 + 1953 * q^55 - 336 * q^56 + 1146 * q^58 - 795 * q^59 + 1587 * q^61 - 744 * q^62 + 960 * q^64 - 1020 * q^65 + 1884 * q^67 - 408 * q^68 + 1248 * q^70 - 120 * q^71 + 2604 * q^73 - 1110 * q^74 + 684 * q^76 - 984 * q^77 + 1944 * q^79 - 240 * q^80 + 1704 * q^82 - 2079 * q^83 + 2916 * q^85 - 2004 * q^86 + 264 * q^88 - 2604 * q^89 + 3399 * q^91 - 696 * q^92 + 612 * q^94 - 1038 * q^95 + 3690 * q^97 - 4002 * q^98

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