LMFDB - Embedded newform 1521.4.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 25x^{2} + 24x + 78 $ x^4 - 2x^3 - 25x^2 + 24*x + 78
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.33039$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.33039 q^{2} +20.4131 q^{4} +16.4131 q^{5} +9.67968 q^{7} -66.1667 q^{8} -87.4882 q^{10} -27.5882 q^{11} -51.5965 q^{14} +189.390 q^{16} -107.928 q^{17} -2.24723 q^{19} +335.042 q^{20} +147.056 q^{22} -41.8090 q^{23} +144.390 q^{25} +197.592 q^{28} -61.6213 q^{29} +191.932 q^{31} -480.187 q^{32} +575.300 q^{34} +158.874 q^{35} +98.4236 q^{37} +11.9786 q^{38} -1086.00 q^{40} +30.7452 q^{41} +238.325 q^{43} -563.160 q^{44} +222.858 q^{46} +511.482 q^{47} -249.304 q^{49} -769.653 q^{50} -492.825 q^{53} -452.807 q^{55} -640.472 q^{56} +328.466 q^{58} -484.179 q^{59} -444.021 q^{61} -1023.07 q^{62} +1044.47 q^{64} +190.114 q^{67} -2203.15 q^{68} -846.858 q^{70} -484.785 q^{71} -957.780 q^{73} -524.636 q^{74} -45.8729 q^{76} -267.045 q^{77} -375.216 q^{79} +3108.47 q^{80} -163.884 q^{82} +715.765 q^{83} -1771.43 q^{85} -1270.37 q^{86} +1825.42 q^{88} +1038.15 q^{89} -853.451 q^{92} -2726.40 q^{94} -36.8840 q^{95} +65.5636 q^{97} +1328.89 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29}+ \cdots + 1906 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 - 62 * q^10 - 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 + 124 * q^19 + 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 - 144 * q^28 - 194 * q^29 + 26 * q^31 - 654 * q^32 + 1062 * q^34 - 88 * q^35 + 102 * q^37 - 332 * q^38 - 998 * q^40 + 1054 * q^41 + 450 * q^43 + 44 * q^44 - 172 * q^46 + 96 * q^47 + 1070 * q^49 - 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 + 722 * q^58 - 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 - 1134 * q^67 - 1786 * q^68 - 2324 * q^70 - 1064 * q^71 + 952 * q^73 - 1158 * q^74 - 1708 * q^76 - 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 1734 * q^82 + 404 * q^83 - 1394 * q^85 - 3168 * q^86 + 3060 * q^88 - 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 + 2166 * q^97 + 1906 * 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 $

$2$	−5.33039	−1.88458	−0.942289	−	0.334800i	$-0.891331\pi$
$2$	−5.33039	−1.88458	−0.942289	+	0.334800i	$0.891331\pi$
$3$	0	0
$3$	0	0
$4$	20.4131	2.55164
$5$	16.4131	1.46803	0.734016	−	0.679132i	$-0.237644\pi$
$5$	16.4131	1.46803	0.734016	+	0.679132i	$0.237644\pi$
$6$	0	0
$7$	9.67968	0.522654	0.261327	−	0.965250i	$-0.415840\pi$
$7$	9.67968	0.522654	0.261327	+	0.965250i	$0.415840\pi$
$8$	−66.1667	−2.92418
$9$	0	0
$10$	−87.4882	−2.76662
$11$	−27.5882	−0.756195	−0.378098	−	0.925766i	$-0.623422\pi$
$11$	−27.5882	−0.756195	−0.378098	+	0.925766i	$0.623422\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−51.5965	−0.984982
$15$	0	0
$16$	189.390	2.95921
$17$	−107.928	−1.53979	−0.769895	−	0.638171i	$-0.779691\pi$
$17$	−107.928	−1.53979	−0.769895	+	0.638171i	$0.779691\pi$
$18$	0	0
$19$	−2.24723	−0.0271342	−0.0135671	−	0.999908i	$-0.504319\pi$
$19$	−2.24723	−0.0271342	−0.0135671	+	0.999908i	$0.504319\pi$
$20$	335.042	3.74588
$21$	0	0
$22$	147.056	1.42511
$23$	−41.8090	−0.379034	−0.189517	−	0.981877i	$-0.560692\pi$
$23$	−41.8090	−0.379034	−0.189517	+	0.981877i	$0.560692\pi$
$24$	0	0
$25$	144.390	1.15512
$26$	0	0
$27$	0	0
$28$	197.592	1.33362
$29$	−61.6213	−0.394579	−0.197289	−	0.980345i	$-0.563214\pi$
$29$	−61.6213	−0.394579	−0.197289	+	0.980345i	$0.563214\pi$
$30$	0	0
$31$	191.932	1.11200	0.556000	−	0.831182i	$-0.312335\pi$
$31$	191.932	1.11200	0.556000	+	0.831182i	$0.312335\pi$
$32$	−480.187	−2.65269
$33$	0	0
$34$	575.300	2.90185
$35$	158.874	0.767272
$36$	0	0
$37$	98.4236	0.437317	0.218659	−	0.975801i	$-0.429832\pi$
$37$	98.4236	0.437317	0.218659	+	0.975801i	$0.429832\pi$
$38$	11.9786	0.0511366
$39$	0	0
$40$	−1086.00	−4.29279
$41$	30.7452	0.117112	0.0585561	−	0.998284i	$-0.481350\pi$
$41$	30.7452	0.117112	0.0585561	+	0.998284i	$0.481350\pi$
$42$	0	0
$43$	238.325	0.845216	0.422608	−	0.906313i	$-0.361115\pi$
$43$	238.325	0.845216	0.422608	+	0.906313i	$0.361115\pi$
$44$	−563.160	−1.92953
$45$	0	0
$46$	222.858	0.714319
$47$	511.482	1.58739	0.793695	−	0.608316i	$-0.208155\pi$
$47$	511.482	1.58739	0.793695	+	0.608316i	$0.208155\pi$
$48$	0	0
$49$	−249.304	−0.726833
$50$	−769.653	−2.17691
$51$	0	0
$52$	0	0
$53$	−492.825	−1.27726	−0.638630	−	0.769514i	$-0.720498\pi$
$53$	−492.825	−1.27726	−0.638630	+	0.769514i	$0.720498\pi$
$54$	0	0
$55$	−452.807	−1.11012
$56$	−640.472	−1.52833
$57$	0	0
$58$	328.466	0.743615
$59$	−484.179	−1.06838	−0.534192	−	0.845363i	$-0.679384\pi$
$59$	−484.179	−1.06838	−0.534192	+	0.845363i	$0.679384\pi$
$60$	0	0
$61$	−444.021	−0.931985	−0.465993	−	0.884789i	$-0.654303\pi$
$61$	−444.021	−0.931985	−0.465993	+	0.884789i	$0.654303\pi$
$62$	−1023.07	−2.09565
$63$	0	0
$64$	1044.47	2.03998
$65$	0	0
$66$	0	0
$67$	190.114	0.346658	0.173329	−	0.984864i	$-0.444548\pi$
$67$	190.114	0.346658	0.173329	+	0.984864i	$0.444548\pi$
$68$	−2203.15	−3.92898
$69$	0	0
$70$	−846.858	−1.44598
$71$	−484.785	−0.810329	−0.405164	−	0.914244i	$-0.632786\pi$
$71$	−484.785	−0.810329	−0.405164	+	0.914244i	$0.632786\pi$
$72$	0	0
$73$	−957.780	−1.53561	−0.767806	−	0.640683i	$-0.778651\pi$
$73$	−957.780	−1.53561	−0.767806	+	0.640683i	$0.778651\pi$
$74$	−524.636	−0.824159
$75$	0	0
$76$	−45.8729	−0.0692367
$77$	−267.045	−0.395228
$78$	0	0
$79$	−375.216	−0.534368	−0.267184	−	0.963646i	$-0.586093\pi$
$79$	−375.216	−0.534368	−0.267184	+	0.963646i	$0.586093\pi$
$80$	3108.47	4.34422
$81$	0	0
$82$	−163.884	−0.220707
$83$	715.765	0.946571	0.473286	−	0.880909i	$-0.343068\pi$
$83$	715.765	0.946571	0.473286	+	0.880909i	$0.343068\pi$
$84$	0	0
$85$	−1771.43	−2.26046
$86$	−1270.37	−1.59288
$87$	0	0
$88$	1825.42	2.21125
$89$	1038.15	1.23645	0.618224	−	0.786002i	$-0.287852\pi$
$89$	1038.15	1.23645	0.618224	+	0.786002i	$0.287852\pi$
$90$	0	0
$91$	0	0
$92$	−853.451	−0.967157
$93$	0	0
$94$	−2726.40	−2.99156
$95$	−36.8840	−0.0398339
$96$	0	0
$97$	65.5636	0.0686286	0.0343143	−	0.999411i	$-0.489075\pi$
$97$	65.5636	0.0686286	0.0343143	+	0.999411i	$0.489075\pi$
$98$	1328.89	1.36977
$99$	0	0
$100$	2947.44	2.94744
$101$	−531.798	−0.523920	−0.261960	−	0.965079i	$-0.584369\pi$
$101$	−531.798	−0.523920	−0.261960	+	0.965079i	$0.584369\pi$
$102$	0	0
$103$	−735.984	−0.704064	−0.352032	−	0.935988i	$-0.614509\pi$
$103$	−735.984	−0.704064	−0.352032	+	0.935988i	$0.614509\pi$
$104$	0	0
$105$	0	0
$106$	2626.95	2.40710
$107$	783.265	0.707673	0.353837	−	0.935307i	$-0.384877\pi$
$107$	783.265	0.707673	0.353837	+	0.935307i	$0.384877\pi$
$108$	0	0
$109$	−532.339	−0.467788	−0.233894	−	0.972262i	$-0.575147\pi$
$109$	−532.339	−0.467788	−0.233894	+	0.972262i	$0.575147\pi$
$110$	2413.64	2.09210
$111$	0	0
$112$	1833.23	1.54664
$113$	180.589	0.150340	0.0751699	−	0.997171i	$-0.476050\pi$
$113$	180.589	0.150340	0.0751699	+	0.997171i	$0.476050\pi$
$114$	0	0
$115$	−686.215	−0.556434
$116$	−1257.88	−1.00682
$117$	0	0
$118$	2580.86	2.01346
$119$	−1044.71	−0.804777
$120$	0	0
$121$	−569.893	−0.428169
$122$	2366.81	1.75640
$123$	0	0
$124$	3917.92	2.83742
$125$	318.242	0.227716
$126$	0	0
$127$	1431.63	1.00029	0.500146	−	0.865941i	$-0.333280\pi$
$127$	1431.63	1.00029	0.500146	+	0.865941i	$0.333280\pi$
$128$	−1725.94	−1.19182
$129$	0	0
$130$	0	0
$131$	−2067.32	−1.37880	−0.689400	−	0.724381i	$-0.742126\pi$
$131$	−2067.32	−1.37880	−0.689400	+	0.724381i	$0.742126\pi$
$132$	0	0
$133$	−21.7525	−0.0141818
$134$	−1013.38	−0.653304
$135$	0	0
$136$	7141.25	4.50262
$137$	387.512	0.241660	0.120830	−	0.992673i	$-0.461444\pi$
$137$	387.512	0.241660	0.120830	+	0.992673i	$0.461444\pi$
$138$	0	0
$139$	752.568	0.459223	0.229611	−	0.973282i	$-0.426254\pi$
$139$	752.568	0.459223	0.229611	+	0.973282i	$0.426254\pi$
$140$	3243.10	1.95780
$141$	0	0
$142$	2584.09	1.52713
$143$	0	0
$144$	0	0
$145$	−1011.40	−0.579254
$146$	5105.34	2.89398
$147$	0	0
$148$	2009.13	1.11587
$149$	2636.72	1.44972	0.724862	−	0.688895i	$-0.241904\pi$
$149$	2636.72	1.44972	0.724862	+	0.688895i	$0.241904\pi$
$150$	0	0
$151$	−3332.42	−1.79595	−0.897975	−	0.440046i	$-0.854962\pi$
$151$	−3332.42	−1.79595	−0.897975	+	0.440046i	$0.854962\pi$
$152$	148.692	0.0793454
$153$	0	0
$154$	1423.45	0.744839
$155$	3150.20	1.63245
$156$	0	0
$157$	−1625.26	−0.826179	−0.413089	−	0.910690i	$-0.635550\pi$
$157$	−1625.26	−0.826179	−0.413089	+	0.910690i	$0.635550\pi$
$158$	2000.05	1.00706
$159$	0	0
$160$	−7881.36	−3.89423
$161$	−404.698	−0.198104
$162$	0	0
$163$	−1835.37	−0.881944	−0.440972	−	0.897521i	$-0.645366\pi$
$163$	−1835.37	−0.881944	−0.440972	+	0.897521i	$0.645366\pi$
$164$	627.605	0.298828
$165$	0	0
$166$	−3815.31	−1.78389
$167$	−1945.00	−0.901248	−0.450624	−	0.892714i	$-0.648798\pi$
$167$	−1945.00	−0.901248	−0.450624	+	0.892714i	$0.648798\pi$
$168$	0	0
$169$	0	0
$170$	9442.44	4.26001
$171$	0	0
$172$	4864.96	2.15668
$173$	−2531.63	−1.11258	−0.556289	−	0.830989i	$-0.687775\pi$
$173$	−2531.63	−1.11258	−0.556289	+	0.830989i	$0.687775\pi$
$174$	0	0
$175$	1397.65	0.603726
$176$	−5224.91	−2.23774
$177$	0	0
$178$	−5533.76	−2.33018
$179$	−4263.01	−1.78007	−0.890035	−	0.455892i	$-0.849320\pi$
$179$	−4263.01	−1.78007	−0.890035	+	0.455892i	$0.849320\pi$
$180$	0	0
$181$	3944.61	1.61989	0.809946	−	0.586504i	$-0.199496\pi$
$181$	3944.61	1.61989	0.809946	+	0.586504i	$0.199496\pi$
$182$	0	0
$183$	0	0
$184$	2766.36	1.10836
$185$	1615.43	0.641995
$186$	0	0
$187$	2977.54	1.16438
$188$	10440.9	4.05044
$189$	0	0
$190$	196.606	0.0750701
$191$	214.109	0.0811119	0.0405559	−	0.999177i	$-0.487087\pi$
$191$	214.109	0.0811119	0.0405559	+	0.999177i	$0.487087\pi$
$192$	0	0
$193$	−1207.19	−0.450234	−0.225117	−	0.974332i	$-0.572276\pi$
$193$	−1207.19	−0.450234	−0.225117	+	0.974332i	$0.572276\pi$
$194$	−349.480	−0.129336
$195$	0	0
$196$	−5089.06	−1.85461
$197$	927.631	0.335487	0.167744	−	0.985831i	$-0.446352\pi$
$197$	927.631	0.335487	0.167744	+	0.985831i	$0.446352\pi$
$198$	0	0
$199$	478.951	0.170613	0.0853064	−	0.996355i	$-0.472813\pi$
$199$	478.951	0.170613	0.0853064	+	0.996355i	$0.472813\pi$
$200$	−9553.77	−3.37777
$201$	0	0
$202$	2834.69	0.987368
$203$	−596.474	−0.206228
$204$	0	0
$205$	504.624	0.171924
$206$	3923.08	1.32686
$207$	0	0
$208$	0	0
$209$	61.9970	0.0205188
$210$	0	0
$211$	−1450.95	−0.473402	−0.236701	−	0.971583i	$-0.576066\pi$
$211$	−1450.95	−0.473402	−0.236701	+	0.971583i	$0.576066\pi$
$212$	−10060.1	−3.25910
$213$	0	0
$214$	−4175.11	−1.33367
$215$	3911.66	1.24080
$216$	0	0
$217$	1857.84	0.581191
$218$	2837.58	0.881583
$219$	0	0
$220$	−9243.19	−2.83262
$221$	0	0
$222$	0	0
$223$	−2059.79	−0.618536	−0.309268	−	0.950975i	$-0.600084\pi$
$223$	−2059.79	−0.618536	−0.309268	+	0.950975i	$0.600084\pi$
$224$	−4648.06	−1.38644
$225$	0	0
$226$	−962.612	−0.283327
$227$	−4482.46	−1.31062	−0.655311	−	0.755359i	$-0.727463\pi$
$227$	−4482.46	−1.31062	−0.655311	+	0.755359i	$0.727463\pi$
$228$	0	0
$229$	−1630.39	−0.470477	−0.235239	−	0.971938i	$-0.575587\pi$
$229$	−1630.39	−0.470477	−0.235239	+	0.971938i	$0.575587\pi$
$230$	3657.80	1.04864
$231$	0	0
$232$	4077.27	1.15382
$233$	1903.69	0.535258	0.267629	−	0.963522i	$-0.413760\pi$
$233$	1903.69	0.535258	0.267629	+	0.963522i	$0.413760\pi$
$234$	0	0
$235$	8395.00	2.33034
$236$	−9883.58	−2.72613
$237$	0	0
$238$	5568.72	1.51667
$239$	−3763.79	−1.01866	−0.509328	−	0.860572i	$-0.670106\pi$
$239$	−3763.79	−1.01866	−0.509328	+	0.860572i	$0.670106\pi$
$240$	0	0
$241$	−3614.74	−0.966166	−0.483083	−	0.875575i	$-0.660483\pi$
$241$	−3614.74	−0.966166	−0.483083	+	0.875575i	$0.660483\pi$
$242$	3037.75	0.806918
$243$	0	0
$244$	−9063.85	−2.37809
$245$	−4091.84	−1.06701
$246$	0	0
$247$	0	0
$248$	−12699.5	−3.25169
$249$	0	0
$250$	−1696.36	−0.429148
$251$	5729.77	1.44088	0.720438	−	0.693520i	$-0.243941\pi$
$251$	5729.77	1.44088	0.720438	+	0.693520i	$0.243941\pi$
$252$	0	0
$253$	1153.43	0.286624
$254$	−7631.18	−1.88513
$255$	0	0
$256$	844.191	0.206101
$257$	5525.79	1.34120	0.670602	−	0.741818i	$-0.266036\pi$
$257$	5525.79	1.34120	0.670602	+	0.741818i	$0.266036\pi$
$258$	0	0
$259$	952.709	0.228565
$260$	0	0
$261$	0	0
$262$	11019.6	2.59846
$263$	−5223.21	−1.22463	−0.612313	−	0.790615i	$-0.709761\pi$
$263$	−5223.21	−1.22463	−0.612313	+	0.790615i	$0.709761\pi$
$264$	0	0
$265$	−8088.78	−1.87506
$266$	115.949	0.0267267
$267$	0	0
$268$	3880.81	0.884545
$269$	−7203.88	−1.63282	−0.816410	−	0.577473i	$-0.804039\pi$
$269$	−7203.88	−1.63282	−0.816410	+	0.577473i	$0.804039\pi$
$270$	0	0
$271$	−8577.69	−1.92272	−0.961360	−	0.275293i	$-0.911225\pi$
$271$	−8577.69	−1.92272	−0.961360	+	0.275293i	$0.911225\pi$
$272$	−20440.5	−4.55656
$273$	0	0
$274$	−2065.59	−0.455427
$275$	−3983.44	−0.873493
$276$	0	0
$277$	7169.19	1.55507	0.777536	−	0.628838i	$-0.216469\pi$
$277$	7169.19	1.55507	0.777536	+	0.628838i	$0.216469\pi$
$278$	−4011.48	−0.865442
$279$	0	0
$280$	−10512.1	−2.24364
$281$	−849.157	−0.180272	−0.0901360	−	0.995929i	$-0.528730\pi$
$281$	−849.157	−0.180272	−0.0901360	+	0.995929i	$0.528730\pi$
$282$	0	0
$283$	1115.37	0.234283	0.117141	−	0.993115i	$-0.462627\pi$
$283$	1115.37	0.234283	0.117141	+	0.993115i	$0.462627\pi$
$284$	−9895.95	−2.06766
$285$	0	0
$286$	0	0
$287$	297.604	0.0612091
$288$	0	0
$289$	6735.49	1.37095
$290$	5391.14	1.09165
$291$	0	0
$292$	−19551.2	−3.91832
$293$	1863.53	0.371565	0.185782	−	0.982591i	$-0.440518\pi$
$293$	1863.53	0.371565	0.185782	+	0.982591i	$0.440518\pi$
$294$	0	0
$295$	−7946.87	−1.56842
$296$	−6512.36	−1.27879
$297$	0	0
$298$	−14054.8	−2.73212
$299$	0	0
$300$	0	0
$301$	2306.91	0.441755
$302$	17763.1	3.38461
$303$	0	0
$304$	−425.602	−0.0802959
$305$	−7287.76	−1.36818
$306$	0	0
$307$	−6387.50	−1.18747	−0.593736	−	0.804660i	$-0.702348\pi$
$307$	−6387.50	−1.18747	−0.593736	+	0.804660i	$0.702348\pi$
$308$	−5451.21	−1.00848
$309$	0	0
$310$	−16791.8	−3.07648
$311$	3492.59	0.636806	0.318403	−	0.947955i	$-0.396853\pi$
$311$	3492.59	0.636806	0.318403	+	0.947955i	$0.396853\pi$
$312$	0	0
$313$	−5912.01	−1.06762	−0.533812	−	0.845603i	$-0.679241\pi$
$313$	−5912.01	−1.06762	−0.533812	+	0.845603i	$0.679241\pi$
$314$	8663.29	1.55700
$315$	0	0
$316$	−7659.31	−1.36351
$317$	1677.54	0.297224	0.148612	−	0.988896i	$-0.452519\pi$
$317$	1677.54	0.297224	0.148612	+	0.988896i	$0.452519\pi$
$318$	0	0
$319$	1700.02	0.298378
$320$	17143.0	2.99476
$321$	0	0
$322$	2157.20	0.373342
$323$	242.540	0.0417810
$324$	0	0
$325$	0	0
$326$	9783.22	1.66209
$327$	0	0
$328$	−2034.31	−0.342457
$329$	4950.98	0.829655
$330$	0	0
$331$	2010.31	0.333827	0.166913	−	0.985972i	$-0.446620\pi$
$331$	2010.31	0.333827	0.166913	+	0.985972i	$0.446620\pi$
$332$	14611.0	2.41531
$333$	0	0
$334$	10367.6	1.69847
$335$	3120.35	0.508905
$336$	0	0
$337$	7139.24	1.15400	0.577002	−	0.816743i	$-0.304222\pi$
$337$	7139.24	1.15400	0.577002	+	0.816743i	$0.304222\pi$
$338$	0	0
$339$	0	0
$340$	−36160.5	−5.76787
$341$	−5295.05	−0.840889
$342$	0	0
$343$	−5733.31	−0.902536
$344$	−15769.2	−2.47156
$345$	0	0
$346$	13494.6	2.09674
$347$	−1.13990	−0.000176349 0	−8.81743e−5	−	1.00000i	$-0.500028\pi$
$347$	−1.13990	−0.000176349 0	−8.81743e−5		1.00000i	$0.500028\pi$
$348$	0	0
$349$	12199.1	1.87107	0.935535	−	0.353235i	$-0.114918\pi$
$349$	12199.1	1.87107	0.935535	+	0.353235i	$0.114918\pi$
$350$	−7450.00	−1.13777
$351$	0	0
$352$	13247.5	2.00595
$353$	10892.3	1.64232	0.821160	−	0.570698i	$-0.193327\pi$
$353$	10892.3	1.64232	0.821160	+	0.570698i	$0.193327\pi$
$354$	0	0
$355$	−7956.81	−1.18959
$356$	21191.9	3.15497
$357$	0	0
$358$	22723.5	3.35468
$359$	3525.78	0.518339	0.259169	−	0.965832i	$-0.416551\pi$
$359$	3525.78	0.518339	0.259169	+	0.965832i	$0.416551\pi$
$360$	0	0
$361$	−6853.95	−0.999264
$362$	−21026.3	−3.05281
$363$	0	0
$364$	0	0
$365$	−15720.1	−2.25433
$366$	0	0
$367$	2383.75	0.339049	0.169525	−	0.985526i	$-0.445777\pi$
$367$	2383.75	0.339049	0.169525	+	0.985526i	$0.445777\pi$
$368$	−7918.19	−1.12164
$369$	0	0
$370$	−8610.90	−1.20989
$371$	−4770.39	−0.667564
$372$	0	0
$373$	−13282.2	−1.84377	−0.921885	−	0.387463i	$-0.873352\pi$
$373$	−13282.2	−1.84377	−0.921885	+	0.387463i	$0.873352\pi$
$374$	−15871.5	−2.19437
$375$	0	0
$376$	−33843.1	−4.64181
$377$	0	0
$378$	0	0
$379$	−4436.73	−0.601318	−0.300659	−	0.953732i	$-0.597207\pi$
$379$	−4436.73	−0.601318	−0.300659	+	0.953732i	$0.597207\pi$
$380$	−752.917	−0.101642
$381$	0	0
$382$	−1141.28	−0.152862
$383$	810.412	0.108120	0.0540602	−	0.998538i	$-0.482784\pi$
$383$	810.412	0.108120	0.0540602	+	0.998538i	$0.482784\pi$
$384$	0	0
$385$	−4383.03	−0.580207
$386$	6434.78	0.848501
$387$	0	0
$388$	1338.35	0.175115
$389$	−3463.79	−0.451469	−0.225734	−	0.974189i	$-0.572478\pi$
$389$	−3463.79	−0.451469	−0.225734	+	0.974189i	$0.572478\pi$
$390$	0	0
$391$	4512.37	0.583633
$392$	16495.6	2.12539
$393$	0	0
$394$	−4944.64	−0.632252
$395$	−6158.45	−0.784469
$396$	0	0
$397$	425.405	0.0537796	0.0268898	−	0.999638i	$-0.491440\pi$
$397$	425.405	0.0537796	0.0268898	+	0.999638i	$0.491440\pi$
$398$	−2553.00	−0.321533
$399$	0	0
$400$	27345.9	3.41823
$401$	1186.85	0.147801	0.0739007	−	0.997266i	$-0.476455\pi$
$401$	1186.85	0.147801	0.0739007	+	0.997266i	$0.476455\pi$
$402$	0	0
$403$	0	0
$404$	−10855.6	−1.33685
$405$	0	0
$406$	3179.44	0.388653
$407$	−2715.33	−0.330697
$408$	0	0
$409$	8007.42	0.968071	0.484036	−	0.875048i	$-0.339170\pi$
$409$	8007.42	0.968071	0.484036	+	0.875048i	$0.339170\pi$
$410$	−2689.84	−0.324005
$411$	0	0
$412$	−15023.7	−1.79652
$413$	−4686.70	−0.558395
$414$	0	0
$415$	11747.9	1.38960
$416$	0	0
$417$	0	0
$418$	−330.468	−0.0386692
$419$	−6832.46	−0.796629	−0.398314	−	0.917249i	$-0.630405\pi$
$419$	−6832.46	−0.796629	−0.398314	+	0.917249i	$0.630405\pi$
$420$	0	0
$421$	10739.6	1.24326	0.621632	−	0.783309i	$-0.286470\pi$
$421$	10739.6	1.24326	0.621632	+	0.783309i	$0.286470\pi$
$422$	7734.16	0.892163
$423$	0	0
$424$	32608.6	3.73494
$425$	−15583.7	−1.77864
$426$	0	0
$427$	−4297.99	−0.487106
$428$	15988.9	1.80573
$429$	0	0
$430$	−20850.7	−2.33839
$431$	−5214.45	−0.582763	−0.291382	−	0.956607i	$-0.594115\pi$
$431$	−5214.45	−0.582763	−0.291382	+	0.956607i	$0.594115\pi$
$432$	0	0
$433$	8642.24	0.959168	0.479584	−	0.877496i	$-0.340788\pi$
$433$	8642.24	0.959168	0.479584	+	0.877496i	$0.340788\pi$
$434$	−9903.02	−1.09530
$435$	0	0
$436$	−10866.7	−1.19362
$437$	93.9545	0.0102848
$438$	0	0
$439$	−13026.2	−1.41619	−0.708097	−	0.706116i	$-0.750446\pi$
$439$	−13026.2	−1.41619	−0.708097	+	0.706116i	$0.750446\pi$
$440$	29960.7	3.24619
$441$	0	0
$442$	0	0
$443$	11533.0	1.23690	0.618450	−	0.785824i	$-0.287761\pi$
$443$	11533.0	1.23690	0.618450	+	0.785824i	$0.287761\pi$
$444$	0	0
$445$	17039.3	1.81515
$446$	10979.5	1.16568
$447$	0	0
$448$	10110.2	1.06621
$449$	−9882.75	−1.03874	−0.519372	−	0.854548i	$-0.673834\pi$
$449$	−9882.75	−1.03874	−0.519372	+	0.854548i	$0.673834\pi$
$450$	0	0
$451$	−848.204	−0.0885596
$452$	3686.38	0.383613
$453$	0	0
$454$	23893.3	2.46997
$455$	0	0
$456$	0	0
$457$	−15628.1	−1.59967	−0.799836	−	0.600218i	$-0.795080\pi$
$457$	−15628.1	−1.59967	−0.799836	+	0.600218i	$0.795080\pi$
$458$	8690.63	0.886652
$459$	0	0
$460$	−14007.8	−1.41982
$461$	7747.46	0.782723	0.391361	−	0.920237i	$-0.372004\pi$
$461$	7747.46	0.782723	0.391361	+	0.920237i	$0.372004\pi$
$462$	0	0
$463$	−333.422	−0.0334675	−0.0167337	−	0.999860i	$-0.505327\pi$
$463$	−333.422	−0.0334675	−0.0167337	+	0.999860i	$0.505327\pi$
$464$	−11670.4	−1.16764
$465$	0	0
$466$	−10147.4	−1.00874
$467$	−8198.33	−0.812363	−0.406182	−	0.913792i	$-0.633140\pi$
$467$	−8198.33	−0.812363	−0.406182	+	0.913792i	$0.633140\pi$
$468$	0	0
$469$	1840.24	0.181182
$470$	−44748.7	−4.39171
$471$	0	0
$472$	32036.5	3.12415
$473$	−6574.96	−0.639148
$474$	0	0
$475$	−324.477	−0.0313432
$476$	−21325.8	−2.05350
$477$	0	0
$478$	20062.5	1.91974
$479$	6435.88	0.613910	0.306955	−	0.951724i	$-0.400690\pi$
$479$	6435.88	0.613910	0.306955	+	0.951724i	$0.400690\pi$
$480$	0	0
$481$	0	0
$482$	19268.0	1.82082
$483$	0	0
$484$	−11633.3	−1.09253
$485$	1076.10	0.100749
$486$	0	0
$487$	8095.37	0.753257	0.376629	−	0.926364i	$-0.377083\pi$
$487$	8095.37	0.753257	0.376629	+	0.926364i	$0.377083\pi$
$488$	29379.4	2.72529
$489$	0	0
$490$	21811.1	2.01087
$491$	−5116.46	−0.470270	−0.235135	−	0.971963i	$-0.575553\pi$
$491$	−5116.46	−0.470270	−0.235135	+	0.971963i	$0.575553\pi$
$492$	0	0
$493$	6650.67	0.607568
$494$	0	0
$495$	0	0
$496$	36349.9	3.29064
$497$	−4692.56	−0.423521
$498$	0	0
$499$	−18050.7	−1.61936	−0.809682	−	0.586870i	$-0.800360\pi$
$499$	−18050.7	−1.61936	−0.809682	+	0.586870i	$0.800360\pi$
$500$	6496.31	0.581047
$501$	0	0
$502$	−30541.9	−2.71544
$503$	−10531.1	−0.933512	−0.466756	−	0.884386i	$-0.654577\pi$
$503$	−10531.1	−0.933512	−0.466756	+	0.884386i	$0.654577\pi$
$504$	0	0
$505$	−8728.45	−0.769131
$506$	−6148.25	−0.540165
$507$	0	0
$508$	29224.1	2.55238
$509$	1963.31	0.170967	0.0854834	−	0.996340i	$-0.472757\pi$
$509$	1963.31	0.170967	0.0854834	+	0.996340i	$0.472757\pi$
$510$	0	0
$511$	−9271.00	−0.802593
$512$	9307.69	0.803409
$513$	0	0
$514$	−29454.6	−2.52760
$515$	−12079.8	−1.03359
$516$	0	0
$517$	−14110.9	−1.20038
$518$	−5078.31	−0.430750
$519$	0	0
$520$	0	0
$521$	7044.93	0.592407	0.296203	−	0.955125i	$-0.404279\pi$
$521$	7044.93	0.592407	0.296203	+	0.955125i	$0.404279\pi$
$522$	0	0
$523$	−3213.29	−0.268657	−0.134328	−	0.990937i	$-0.542888\pi$
$523$	−3213.29	−0.268657	−0.134328	+	0.990937i	$0.542888\pi$
$524$	−42200.4	−3.51819
$525$	0	0
$526$	27841.7	2.30790
$527$	−20714.9	−1.71225
$528$	0	0
$529$	−10419.0	−0.856333
$530$	43116.4	3.53369
$531$	0	0
$532$	−444.036	−0.0361868
$533$	0	0
$534$	0	0
$535$	12855.8	1.03889
$536$	−12579.2	−1.01369
$537$	0	0
$538$	38399.5	3.07718
$539$	6877.83	0.549627
$540$	0	0
$541$	11251.4	0.894150	0.447075	−	0.894497i	$-0.352466\pi$
$541$	11251.4	0.894150	0.447075	+	0.894497i	$0.352466\pi$
$542$	45722.4	3.62352
$543$	0	0
$544$	51825.8	4.08458
$545$	−8737.33	−0.686727
$546$	0	0
$547$	1533.54	0.119871	0.0599353	−	0.998202i	$-0.480911\pi$
$547$	1533.54	0.119871	0.0599353	+	0.998202i	$0.480911\pi$
$548$	7910.31	0.616628
$549$	0	0
$550$	21233.3	1.64617
$551$	138.477	0.0107066
$552$	0	0
$553$	−3631.97	−0.279289
$554$	−38214.6	−2.93066
$555$	0	0
$556$	15362.2	1.17177
$557$	−16845.7	−1.28146	−0.640731	−	0.767766i	$-0.721369\pi$
$557$	−16845.7	−1.28146	−0.640731	+	0.767766i	$0.721369\pi$
$558$	0	0
$559$	0	0
$560$	30089.0	2.27052
$561$	0	0
$562$	4526.34	0.339737
$563$	20820.1	1.55855	0.779273	−	0.626685i	$-0.215589\pi$
$563$	20820.1	1.55855	0.779273	+	0.626685i	$0.215589\pi$
$564$	0	0
$565$	2964.03	0.220704
$566$	−5945.37	−0.441524
$567$	0	0
$568$	32076.6	2.36955
$569$	−23636.6	−1.74147	−0.870735	−	0.491752i	$-0.836357\pi$
$569$	−23636.6	−1.74147	−0.870735	+	0.491752i	$0.836357\pi$
$570$	0	0
$571$	−26955.1	−1.97554	−0.987771	−	0.155913i	$-0.950168\pi$
$571$	−26955.1	−1.97554	−0.987771	+	0.155913i	$0.950168\pi$
$572$	0	0
$573$	0	0
$574$	−1586.35	−0.115353
$575$	−6036.78	−0.437828
$576$	0	0
$577$	23499.8	1.69551	0.847755	−	0.530388i	$-0.177954\pi$
$577$	23499.8	1.69551	0.847755	+	0.530388i	$0.177954\pi$
$578$	−35902.8	−2.58367
$579$	0	0
$580$	−20645.7	−1.47805
$581$	6928.38	0.494729
$582$	0	0
$583$	13596.1	0.965857
$584$	63373.1	4.49040
$585$	0	0
$586$	−9933.33	−0.700243
$587$	−4637.50	−0.326082	−0.163041	−	0.986619i	$-0.552130\pi$
$587$	−4637.50	−0.326082	−0.163041	+	0.986619i	$0.552130\pi$
$588$	0	0
$589$	−431.316	−0.0301733
$590$	42359.9	2.95582
$591$	0	0
$592$	18640.4	1.29411
$593$	−12633.5	−0.874869	−0.437434	−	0.899250i	$-0.644113\pi$
$593$	−12633.5	−0.874869	−0.437434	+	0.899250i	$0.644113\pi$
$594$	0	0
$595$	−17146.9	−1.18144
$596$	53823.7	3.69917
$597$	0	0
$598$	0	0
$599$	18757.1	1.27946	0.639730	−	0.768600i	$-0.279046\pi$
$599$	18757.1	1.27946	0.639730	+	0.768600i	$0.279046\pi$
$600$	0	0
$601$	−3632.98	−0.246576	−0.123288	−	0.992371i	$-0.539344\pi$
$601$	−3632.98	−0.246576	−0.123288	+	0.992371i	$0.539344\pi$
$602$	−12296.8	−0.832522
$603$	0	0
$604$	−68025.0	−4.58261
$605$	−9353.71	−0.628566
$606$	0	0
$607$	−12700.0	−0.849219	−0.424610	−	0.905377i	$-0.639589\pi$
$607$	−12700.0	−0.849219	−0.424610	+	0.905377i	$0.639589\pi$
$608$	1079.09	0.0719786
$609$	0	0
$610$	38846.6	2.57845
$611$	0	0
$612$	0	0
$613$	21640.1	1.42584	0.712918	−	0.701248i	$-0.247373\pi$
$613$	21640.1	1.42584	0.712918	+	0.701248i	$0.247373\pi$
$614$	34047.9	2.23788
$615$	0	0
$616$	17669.5	1.15572
$617$	16541.7	1.07933	0.539663	−	0.841881i	$-0.318552\pi$
$617$	16541.7	1.07933	0.539663	+	0.841881i	$0.318552\pi$
$618$	0	0
$619$	−21138.9	−1.37261	−0.686303	−	0.727316i	$-0.740767\pi$
$619$	−21138.9	−1.37261	−0.686303	+	0.727316i	$0.740767\pi$
$620$	64305.2	4.16542
$621$	0	0
$622$	−18616.9	−1.20011
$623$	10049.0	0.646235
$624$	0	0
$625$	−12825.4	−0.820823
$626$	31513.3	2.01202
$627$	0	0
$628$	−33176.6	−2.10811
$629$	−10622.7	−0.673377
$630$	0	0
$631$	5489.80	0.346348	0.173174	−	0.984891i	$-0.444598\pi$
$631$	5489.80	0.346348	0.173174	+	0.984891i	$0.444598\pi$
$632$	24826.8	1.56259
$633$	0	0
$634$	−8941.94	−0.560141
$635$	23497.5	1.46846
$636$	0	0
$637$	0	0
$638$	−9061.76	−0.562318
$639$	0	0
$640$	−28328.1	−1.74963
$641$	−4297.04	−0.264778	−0.132389	−	0.991198i	$-0.542265\pi$
$641$	−4297.04	−0.264778	−0.132389	+	0.991198i	$0.542265\pi$
$642$	0	0
$643$	25696.9	1.57603	0.788016	−	0.615655i	$-0.211108\pi$
$643$	25696.9	1.57603	0.788016	+	0.615655i	$0.211108\pi$
$644$	−8261.14	−0.505488
$645$	0	0
$646$	−1292.83	−0.0787396
$647$	−2174.98	−0.132160	−0.0660798	−	0.997814i	$-0.521049\pi$
$647$	−2174.98	−0.132160	−0.0660798	+	0.997814i	$0.521049\pi$
$648$	0	0
$649$	13357.6	0.807907
$650$	0	0
$651$	0	0
$652$	−37465.5	−2.25040
$653$	15454.5	0.926160	0.463080	−	0.886316i	$-0.346744\pi$
$653$	15454.5	0.926160	0.463080	+	0.886316i	$0.346744\pi$
$654$	0	0
$655$	−33931.1	−2.02412
$656$	5822.82	0.346560
$657$	0	0
$658$	−26390.7	−1.56355
$659$	3148.77	0.186129	0.0930643	−	0.995660i	$-0.470334\pi$
$659$	3148.77	0.186129	0.0930643	+	0.995660i	$0.470334\pi$
$660$	0	0
$661$	−2099.70	−0.123553	−0.0617767	−	0.998090i	$-0.519677\pi$
$661$	−2099.70	−0.123553	−0.0617767	+	0.998090i	$0.519677\pi$
$662$	−10715.7	−0.629122
$663$	0	0
$664$	−47359.8	−2.76795
$665$	−357.026	−0.0208193
$666$	0	0
$667$	2576.32	0.149559
$668$	−39703.4	−2.29966
$669$	0	0
$670$	−16632.7	−0.959071
$671$	12249.7	0.704763
$672$	0	0
$673$	30970.8	1.77390	0.886950	−	0.461865i	$-0.152819\pi$
$673$	30970.8	1.77390	0.886950	+	0.461865i	$0.152819\pi$
$674$	−38055.0	−2.17481
$675$	0	0
$676$	0	0
$677$	−14640.6	−0.831141	−0.415570	−	0.909561i	$-0.636418\pi$
$677$	−14640.6	−0.831141	−0.415570	+	0.909561i	$0.636418\pi$
$678$	0	0
$679$	634.635	0.0358690
$680$	117210.	6.60999
$681$	0	0
$682$	28224.7	1.58472
$683$	6685.83	0.374563	0.187281	−	0.982306i	$-0.440032\pi$
$683$	6685.83	0.374563	0.187281	+	0.982306i	$0.440032\pi$
$684$	0	0
$685$	6360.27	0.354764
$686$	30560.8	1.70090
$687$	0	0
$688$	45136.3	2.50117
$689$	0	0
$690$	0	0
$691$	−30194.1	−1.66228	−0.831141	−	0.556062i	$-0.812312\pi$
$691$	−30194.1	−1.66228	−0.831141	+	0.556062i	$0.812312\pi$
$692$	−51678.4	−2.83890
$693$	0	0
$694$	6.07611	0.000332343 0
$695$	12352.0	0.674154
$696$	0	0
$697$	−3318.28	−0.180328
$698$	−65026.0	−3.52618
$699$	0	0
$700$	28530.3	1.54049
$701$	30300.9	1.63260	0.816298	−	0.577631i	$-0.196023\pi$
$701$	30300.9	1.63260	0.816298	+	0.577631i	$0.196023\pi$
$702$	0	0
$703$	−221.181	−0.0118663
$704$	−28815.1	−1.54263
$705$	0	0
$706$	−58060.3	−3.09508
$707$	−5147.64	−0.273829
$708$	0	0
$709$	26123.2	1.38375	0.691875	−	0.722017i	$-0.256785\pi$
$709$	26123.2	1.38375	0.691875	+	0.722017i	$0.256785\pi$
$710$	42412.9	2.24187
$711$	0	0
$712$	−68691.1	−3.61560
$713$	−8024.48	−0.421486
$714$	0	0
$715$	0	0
$716$	−87021.3	−4.54209
$717$	0	0
$718$	−18793.8	−0.976850
$719$	−19325.7	−1.00240	−0.501200	−	0.865331i	$-0.667108\pi$
$719$	−19325.7	−1.00240	−0.501200	+	0.865331i	$0.667108\pi$
$720$	0	0
$721$	−7124.09	−0.367982
$722$	36534.2	1.88319
$723$	0	0
$724$	80521.7	4.13338
$725$	−8897.47	−0.455784
$726$	0	0
$727$	26065.8	1.32975	0.664875	−	0.746954i	$-0.268485\pi$
$727$	26065.8	1.32975	0.664875	+	0.746954i	$0.268485\pi$
$728$	0	0
$729$	0	0
$730$	83794.4	4.24845
$731$	−25722.0	−1.30145
$732$	0	0
$733$	1055.45	0.0531843	0.0265921	−	0.999646i	$-0.491534\pi$
$733$	1055.45	0.0531843	0.0265921	+	0.999646i	$0.491534\pi$
$734$	−12706.4	−0.638965
$735$	0	0
$736$	20076.2	1.00546
$737$	−5244.89	−0.262141
$738$	0	0
$739$	9410.40	0.468426	0.234213	−	0.972185i	$-0.424749\pi$
$739$	9410.40	0.468426	0.234213	+	0.972185i	$0.424749\pi$
$740$	32976.0	1.63814
$741$	0	0
$742$	25428.1	1.25808
$743$	7523.70	0.371491	0.185746	−	0.982598i	$-0.440530\pi$
$743$	7523.70	0.371491	0.185746	+	0.982598i	$0.440530\pi$
$744$	0	0
$745$	43276.8	2.12824
$746$	70799.4	3.47473
$747$	0	0
$748$	60780.8	2.97108
$749$	7581.76	0.369868
$750$	0	0
$751$	−12984.1	−0.630886	−0.315443	−	0.948945i	$-0.602153\pi$
$751$	−12984.1	−0.630886	−0.315443	+	0.948945i	$0.602153\pi$
$752$	96869.3	4.69742
$753$	0	0
$754$	0	0
$755$	−54695.3	−2.63651
$756$	0	0
$757$	−27934.6	−1.34122	−0.670609	−	0.741811i	$-0.733967\pi$
$757$	−27934.6	−1.34122	−0.670609	+	0.741811i	$0.733967\pi$
$758$	23649.5	1.13323
$759$	0	0
$760$	2440.49	0.116482
$761$	15519.3	0.739255	0.369627	−	0.929180i	$-0.379485\pi$
$761$	15519.3	0.739255	0.369627	+	0.929180i	$0.379485\pi$
$762$	0	0
$763$	−5152.88	−0.244491
$764$	4370.62	0.206968
$765$	0	0
$766$	−4319.82	−0.203761
$767$	0	0
$768$	0	0
$769$	12885.2	0.604228	0.302114	−	0.953272i	$-0.402308\pi$
$769$	12885.2	0.604228	0.302114	+	0.953272i	$0.402308\pi$
$770$	23363.3	1.09345
$771$	0	0
$772$	−24642.4	−1.14883
$773$	−5892.04	−0.274155	−0.137078	−	0.990560i	$-0.543771\pi$
$773$	−5892.04	−0.274155	−0.137078	+	0.990560i	$0.543771\pi$
$774$	0	0
$775$	27713.0	1.28449
$776$	−4338.12	−0.200682
$777$	0	0
$778$	18463.4	0.850828
$779$	−69.0916	−0.00317775
$780$	0	0
$781$	13374.3	0.612766
$782$	−24052.7	−1.09990
$783$	0	0
$784$	−47215.5	−2.15085
$785$	−26675.6	−1.21286
$786$	0	0
$787$	−21020.4	−0.952091	−0.476045	−	0.879421i	$-0.657930\pi$
$787$	−21020.4	−0.952091	−0.476045	+	0.879421i	$0.657930\pi$
$788$	18935.8	0.856042
$789$	0	0
$790$	32827.0	1.47839
$791$	1748.05	0.0785757
$792$	0	0
$793$	0	0
$794$	−2267.58	−0.101352
$795$	0	0
$796$	9776.87	0.435342
$797$	−31355.6	−1.39356	−0.696782	−	0.717283i	$-0.745386\pi$
$797$	−31355.6	−1.39356	−0.696782	+	0.717283i	$0.745386\pi$
$798$	0	0
$799$	−55203.3	−2.44425
$800$	−69334.0	−3.06416
$801$	0	0
$802$	−6326.37	−0.278543
$803$	26423.4	1.16122
$804$	0	0
$805$	−6642.34	−0.290822
$806$	0	0
$807$	0	0
$808$	35187.3	1.53204
$809$	18132.5	0.788017	0.394009	−	0.919107i	$-0.371088\pi$
$809$	18132.5	0.788017	0.394009	+	0.919107i	$0.371088\pi$
$810$	0	0
$811$	−24755.3	−1.07186	−0.535928	−	0.844263i	$-0.680038\pi$
$811$	−24755.3	−1.07186	−0.535928	+	0.844263i	$0.680038\pi$
$812$	−12175.9	−0.526219
$813$	0	0
$814$	14473.8	0.623225
$815$	−30124.0	−1.29472
$816$	0	0
$817$	−535.572	−0.0229343
$818$	−42682.7	−1.82441
$819$	0	0
$820$	10300.9	0.438688
$821$	−4082.65	−0.173551	−0.0867755	−	0.996228i	$-0.527656\pi$
$821$	−4082.65	−0.173551	−0.0867755	+	0.996228i	$0.527656\pi$
$822$	0	0
$823$	−34327.0	−1.45391	−0.726954	−	0.686687i	$-0.759065\pi$
$823$	−34327.0	−1.45391	−0.726954	+	0.686687i	$0.759065\pi$
$824$	48697.6	2.05881
$825$	0	0
$826$	24981.9	1.05234
$827$	−3228.87	−0.135767	−0.0678833	−	0.997693i	$-0.521625\pi$
$827$	−3228.87	−0.135767	−0.0678833	+	0.997693i	$0.521625\pi$
$828$	0	0
$829$	−10452.4	−0.437908	−0.218954	−	0.975735i	$-0.570265\pi$
$829$	−10452.4	−0.437908	−0.218954	+	0.975735i	$0.570265\pi$
$830$	−62621.0	−2.61880
$831$	0	0
$832$	0	0
$833$	26906.9	1.11917
$834$	0	0
$835$	−31923.4	−1.32306
$836$	1265.55	0.0523564
$837$	0	0
$838$	36419.7	1.50131
$839$	28289.0	1.16406	0.582028	−	0.813169i	$-0.302259\pi$
$839$	28289.0	1.16406	0.582028	+	0.813169i	$0.302259\pi$
$840$	0	0
$841$	−20591.8	−0.844308
$842$	−57246.1	−2.34303
$843$	0	0
$844$	−29618.5	−1.20795
$845$	0	0
$846$	0	0
$847$	−5516.39	−0.223784
$848$	−93335.9	−3.77968
$849$	0	0
$850$	83067.2	3.35198
$851$	−4114.99	−0.165758
$852$	0	0
$853$	−26631.8	−1.06900	−0.534498	−	0.845170i	$-0.679499\pi$
$853$	−26631.8	−1.06900	−0.534498	+	0.845170i	$0.679499\pi$
$854$	22910.0	0.917989
$855$	0	0
$856$	−51826.0	−2.06936
$857$	−11796.7	−0.470209	−0.235104	−	0.971970i	$-0.575543\pi$
$857$	−11796.7	−0.470209	−0.235104	+	0.971970i	$0.575543\pi$
$858$	0	0
$859$	−22672.8	−0.900567	−0.450283	−	0.892886i	$-0.648677\pi$
$859$	−22672.8	−0.900567	−0.450283	+	0.892886i	$0.648677\pi$
$860$	79849.0	3.16608
$861$	0	0
$862$	27795.0	1.09826
$863$	−21421.1	−0.844940	−0.422470	−	0.906377i	$-0.638837\pi$
$863$	−21421.1	−0.844940	−0.422470	+	0.906377i	$0.638837\pi$
$864$	0	0
$865$	−41551.9	−1.63330
$866$	−46066.6	−1.80763
$867$	0	0
$868$	37924.3	1.48299
$869$	10351.5	0.404086
$870$	0	0
$871$	0	0
$872$	35223.1	1.36790
$873$	0	0
$874$	−500.814	−0.0193825
$875$	3080.48	0.119016
$876$	0	0
$877$	−5155.20	−0.198493	−0.0992466	−	0.995063i	$-0.531643\pi$
$877$	−5155.20	−0.198493	−0.0992466	+	0.995063i	$0.531643\pi$
$878$	69435.0	2.66893
$879$	0	0
$880$	−85756.9	−3.28507
$881$	−23692.2	−0.906027	−0.453013	−	0.891504i	$-0.649651\pi$
$881$	−23692.2	−0.906027	−0.453013	+	0.891504i	$0.649651\pi$
$882$	0	0
$883$	−14591.5	−0.556108	−0.278054	−	0.960565i	$-0.589689\pi$
$883$	−14591.5	−0.556108	−0.278054	+	0.960565i	$0.589689\pi$
$884$	0	0
$885$	0	0
$886$	−61475.2	−2.33104
$887$	9722.26	0.368029	0.184014	−	0.982924i	$-0.441091\pi$
$887$	9722.26	0.368029	0.184014	+	0.982924i	$0.441091\pi$
$888$	0	0
$889$	13857.8	0.522806
$890$	−90826.1	−3.42078
$891$	0	0
$892$	−42046.6	−1.57828
$893$	−1149.42	−0.0430726
$894$	0	0
$895$	−69969.2	−2.61320
$896$	−16706.6	−0.622911
$897$	0	0
$898$	52678.9	1.95759
$899$	−11827.1	−0.438771
$900$	0	0
$901$	53189.7	1.96671
$902$	4521.26	0.166898
$903$	0	0
$904$	−11949.0	−0.439621
$905$	64743.2	2.37805
$906$	0	0
$907$	11799.0	0.431951	0.215975	−	0.976399i	$-0.430707\pi$
$907$	11799.0	0.431951	0.215975	+	0.976399i	$0.430707\pi$
$908$	−91500.9	−3.34423
$909$	0	0
$910$	0	0
$911$	43012.4	1.56429	0.782143	−	0.623099i	$-0.214127\pi$
$911$	43012.4	1.56429	0.782143	+	0.623099i	$0.214127\pi$
$912$	0	0
$913$	−19746.6	−0.715793
$914$	83303.8	3.01471
$915$	0	0
$916$	−33281.3	−1.20049
$917$	−20011.0	−0.720635
$918$	0	0
$919$	−4951.41	−0.177728	−0.0888639	−	0.996044i	$-0.528324\pi$
$919$	−4951.41	−0.177728	−0.0888639	+	0.996044i	$0.528324\pi$
$920$	45404.5	1.62711
$921$	0	0
$922$	−41297.0	−1.47510
$923$	0	0
$924$	0	0
$925$	14211.3	0.505152
$926$	1777.27	0.0630721
$927$	0	0
$928$	29589.8	1.04669
$929$	8934.86	0.315547	0.157774	−	0.987475i	$-0.449568\pi$
$929$	8934.86	0.315547	0.157774	+	0.987475i	$0.449568\pi$
$930$	0	0
$931$	560.243	0.0197221
$932$	38860.2	1.36578
$933$	0	0
$934$	43700.3	1.53096
$935$	48870.6	1.70935
$936$	0	0
$937$	−13182.8	−0.459620	−0.229810	−	0.973235i	$-0.573811\pi$
$937$	−13182.8	−0.459620	−0.229810	+	0.973235i	$0.573811\pi$
$938$	−9809.20	−0.341452
$939$	0	0
$940$	171368.	5.94618
$941$	21693.7	0.751536	0.375768	−	0.926714i	$-0.377379\pi$
$941$	21693.7	0.751536	0.375768	+	0.926714i	$0.377379\pi$
$942$	0	0
$943$	−1285.43	−0.0443895
$944$	−91698.4	−3.16158
$945$	0	0
$946$	35047.1	1.20452
$947$	49790.0	1.70851	0.854254	−	0.519856i	$-0.174014\pi$
$947$	49790.0	1.70851	0.854254	+	0.519856i	$0.174014\pi$
$948$	0	0
$949$	0	0
$950$	1729.59	0.0590687
$951$	0	0
$952$	69125.0	2.35331
$953$	−4217.93	−0.143371	−0.0716853	−	0.997427i	$-0.522838\pi$
$953$	−4217.93	−0.143371	−0.0716853	+	0.997427i	$0.522838\pi$
$954$	0	0
$955$	3514.19	0.119075
$956$	−76830.5	−2.59924
$957$	0	0
$958$	−34305.8	−1.15696
$959$	3750.99	0.126304
$960$	0	0
$961$	7046.87	0.236544
$962$	0	0
$963$	0	0
$964$	−73788.1	−2.46530
$965$	−19813.7	−0.660958
$966$	0	0
$967$	−40927.9	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$967$	−40927.9	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$968$	37707.9	1.25204
$969$	0	0
$970$	−5736.04	−0.189869
$971$	17114.8	0.565645	0.282822	−	0.959172i	$-0.408729\pi$
$971$	17114.8	0.565645	0.282822	+	0.959172i	$0.408729\pi$
$972$	0	0
$973$	7284.62	0.240015
$974$	−43151.5	−1.41957
$975$	0	0
$976$	−84093.0	−2.75794
$977$	118.470	0.00387940	0.00193970	−	0.999998i	$-0.499383\pi$
$977$	118.470	0.00387940	0.00193970	+	0.999998i	$0.499383\pi$
$978$	0	0
$979$	−28640.7	−0.934996
$980$	−83527.2	−2.72263
$981$	0	0
$982$	27272.8	0.886261
$983$	−26002.8	−0.843705	−0.421852	−	0.906665i	$-0.638620\pi$
$983$	−26002.8	−0.843705	−0.421852	+	0.906665i	$0.638620\pi$
$984$	0	0
$985$	15225.3	0.492506
$986$	−35450.7	−1.14501
$987$	0	0
$988$	0	0
$989$	−9964.14	−0.320365
$990$	0	0
$991$	−16062.0	−0.514860	−0.257430	−	0.966297i	$-0.582876\pi$
$991$	−16062.0	−0.514860	−0.257430	+	0.966297i	$0.582876\pi$
$992$	−92163.3	−2.94979
$993$	0	0
$994$	25013.2	0.798159
$995$	7861.07	0.250465
$996$	0	0
$997$	1361.54	0.0432503	0.0216251	−	0.999766i	$-0.493116\pi$
$997$	1361.54	0.0432503	0.0216251	+	0.999766i	$0.493116\pi$
$998$	96217.5	3.05182
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.v.1.1		4
3.2	odd	2		507.4.a.m.1.4		4
13.3	even	3		117.4.g.e.100.4		8
13.9	even	3		117.4.g.e.55.4		8
13.12	even	2		1521.4.a.bb.1.4		4
39.5	even	4		507.4.b.h.337.1		8
39.8	even	4		507.4.b.h.337.8		8
39.29	odd	6		39.4.e.c.22.1	yes	8
39.35	odd	6		39.4.e.c.16.1	✓	8
39.38	odd	2		507.4.a.i.1.1		4
156.35	even	6		624.4.q.i.289.1		8
156.107	even	6		624.4.q.i.529.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.1	✓	8	39.35	odd	6
39.4.e.c.22.1	yes	8	39.29	odd	6
117.4.g.e.55.4		8	13.9	even	3
117.4.g.e.100.4		8	13.3	even	3
507.4.a.i.1.1		4	39.38	odd	2
507.4.a.m.1.4		4	3.2	odd	2
507.4.b.h.337.1		8	39.5	even	4
507.4.b.h.337.8		8	39.8	even	4
624.4.q.i.289.1		8	156.35	even	6
624.4.q.i.529.1		8	156.107	even	6
1521.4.a.v.1.1		4	1.1	even	1	trivial
1521.4.a.bb.1.4		4	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-5.33039 q^{2} +20.4131 q^{4} +16.4131 q^{5} +9.67968 q^{7} -66.1667 q^{8} -87.4882 q^{10} -27.5882 q^{11} -51.5965 q^{14} +189.390 q^{16} -107.928 q^{17} -2.24723 q^{19} +335.042 q^{20} +147.056 q^{22} -41.8090 q^{23} +144.390 q^{25} +197.592 q^{28} -61.6213 q^{29} +191.932 q^{31} -480.187 q^{32} +575.300 q^{34} +158.874 q^{35} +98.4236 q^{37} +11.9786 q^{38} -1086.00 q^{40} +30.7452 q^{41} +238.325 q^{43} -563.160 q^{44} +222.858 q^{46} +511.482 q^{47} -249.304 q^{49} -769.653 q^{50} -492.825 q^{53} -452.807 q^{55} -640.472 q^{56} +328.466 q^{58} -484.179 q^{59} -444.021 q^{61} -1023.07 q^{62} +1044.47 q^{64} +190.114 q^{67} -2203.15 q^{68} -846.858 q^{70} -484.785 q^{71} -957.780 q^{73} -524.636 q^{74} -45.8729 q^{76} -267.045 q^{77} -375.216 q^{79} +3108.47 q^{80} -163.884 q^{82} +715.765 q^{83} -1771.43 q^{85} -1270.37 q^{86} +1825.42 q^{88} +1038.15 q^{89} -853.451 q^{92} -2726.40 q^{94} -36.8840 q^{95} +65.5636 q^{97} +1328.89 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29}+ \cdots + 1906 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 - 62 * q^10 - 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 + 124 * q^19 + 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 - 144 * q^28 - 194 * q^29 + 26 * q^31 - 654 * q^32 + 1062 * q^34 - 88 * q^35 + 102 * q^37 - 332 * q^38 - 998 * q^40 + 1054 * q^41 + 450 * q^43 + 44 * q^44 - 172 * q^46 + 96 * q^47 + 1070 * q^49 - 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 + 722 * q^58 - 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 - 1134 * q^67 - 1786 * q^68 - 2324 * q^70 - 1064 * q^71 + 952 * q^73 - 1158 * q^74 - 1708 * q^76 - 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 1734 * q^82 + 404 * q^83 - 1394 * q^85 - 3168 * q^86 + 3060 * q^88 - 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 + 2166 * q^97 + 1906 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more