LMFDB - Embedded newform 315.4.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.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:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.22952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 18x + 14 $ x^3 - x^2 - 18*x + 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.15010$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.15010 q^{2} +18.5235 q^{4} +5.00000 q^{5} +7.00000 q^{7} +54.1971 q^{8} +25.7505 q^{10} -14.0073 q^{11} -10.5931 q^{13} +36.0507 q^{14} +130.932 q^{16} +131.801 q^{17} +38.6871 q^{19} +92.6176 q^{20} -72.1391 q^{22} -102.481 q^{23} +25.0000 q^{25} -54.5553 q^{26} +129.665 q^{28} -232.376 q^{29} -165.501 q^{31} +240.738 q^{32} +678.789 q^{34} +35.0000 q^{35} +280.004 q^{37} +199.242 q^{38} +270.986 q^{40} -122.271 q^{41} -431.844 q^{43} -259.465 q^{44} -527.787 q^{46} +295.692 q^{47} +49.0000 q^{49} +128.752 q^{50} -196.221 q^{52} +243.639 q^{53} -70.0366 q^{55} +379.380 q^{56} -1196.76 q^{58} +566.480 q^{59} -188.012 q^{61} -852.346 q^{62} +192.364 q^{64} -52.9653 q^{65} -871.647 q^{67} +2441.42 q^{68} +180.253 q^{70} -176.800 q^{71} -220.524 q^{73} +1442.05 q^{74} +716.621 q^{76} -98.0513 q^{77} -190.494 q^{79} +654.662 q^{80} -629.710 q^{82} +518.241 q^{83} +659.006 q^{85} -2224.04 q^{86} -759.157 q^{88} -598.355 q^{89} -74.1515 q^{91} -1898.31 q^{92} +1522.85 q^{94} +193.436 q^{95} -1882.24 q^{97} +252.355 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 14 q^{4} + 15 q^{5} + 21 q^{7} + 66 q^{8} + 10 q^{10} - 20 q^{11} + 14 q^{14} + 114 q^{16} + 234 q^{17} - 82 q^{19} + 70 q^{20} - 236 q^{22} + 30 q^{23} + 75 q^{25} + 76 q^{26} + 98 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 14 * q^4 + 15 * q^5 + 21 * q^7 + 66 * q^8 + 10 * q^10 - 20 * q^11 + 14 * q^14 + 114 * q^16 + 234 * q^17 - 82 * q^19 + 70 * q^20 - 236 * q^22 + 30 * q^23 + 75 * q^25 + 76 * q^26 + 98 * q^28 - 32 * q^29 - 362 * q^31 + 430 * q^32 + 596 * q^34 + 105 * q^35 + 302 * q^37 + 160 * q^38 + 330 * q^40 + 626 * q^41 - 68 * q^43 + 300 * q^44 - 412 * q^46 + 936 * q^47 + 147 * q^49 + 50 * q^50 - 684 * q^52 + 298 * q^53 - 100 * q^55 + 462 * q^56 - 1184 * q^58 + 428 * q^59 + 1050 * q^61 - 608 * q^62 - 154 * q^64 - 52 * q^67 + 1964 * q^68 + 70 * q^70 + 416 * q^71 - 312 * q^73 + 1668 * q^74 + 1712 * q^76 - 140 * q^77 - 488 * q^79 + 570 * q^80 - 1476 * q^82 + 948 * q^83 + 1170 * q^85 - 1928 * q^86 + 124 * q^88 - 142 * q^89 - 3220 * q^92 + 328 * q^94 - 410 * q^95 - 3052 * q^97 + 98 * 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 $

$2$	5.15010	1.82083	0.910417	−	0.413691i	$-0.135761\pi$
$2$	5.15010	1.82083	0.910417	+	0.413691i	$0.135761\pi$
$3$	0	0
$3$	0	0
$4$	18.5235	2.31544
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	54.1971	2.39520
$9$	0	0
$10$	25.7505	0.814302
$11$	−14.0073	−0.383943	−0.191971	−	0.981401i	$-0.561488\pi$
$11$	−14.0073	−0.383943	−0.191971	+	0.981401i	$0.561488\pi$
$12$	0	0
$13$	−10.5931	−0.225999	−0.113000	−	0.993595i	$-0.536046\pi$
$13$	−10.5931	−0.225999	−0.113000	+	0.993595i	$0.536046\pi$
$14$	36.0507	0.688211
$15$	0	0
$16$	130.932	2.04582
$17$	131.801	1.88038	0.940191	−	0.340649i	$-0.110647\pi$
$17$	131.801	1.88038	0.940191	+	0.340649i	$0.110647\pi$
$18$	0	0
$19$	38.6871	0.467128	0.233564	−	0.972341i	$-0.424961\pi$
$19$	38.6871	0.467128	0.233564	+	0.972341i	$0.424961\pi$
$20$	92.6176	1.03550
$21$	0	0
$22$	−72.1391	−0.699096
$23$	−102.481	−0.929076	−0.464538	−	0.885553i	$-0.653780\pi$
$23$	−102.481	−0.929076	−0.464538	+	0.885553i	$0.653780\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−54.5553	−0.411507
$27$	0	0
$28$	129.665	0.875154
$29$	−232.376	−1.48797	−0.743986	−	0.668195i	$-0.767067\pi$
$29$	−232.376	−1.48797	−0.743986	+	0.668195i	$0.767067\pi$
$30$	0	0
$31$	−165.501	−0.958866	−0.479433	−	0.877578i	$-0.659158\pi$
$31$	−165.501	−0.958866	−0.479433	+	0.877578i	$0.659158\pi$
$32$	240.738	1.32990
$33$	0	0
$34$	678.789	3.42386
$35$	35.0000	0.169031
$36$	0	0
$37$	280.004	1.24412	0.622059	−	0.782970i	$-0.286296\pi$
$37$	280.004	1.24412	0.622059	+	0.782970i	$0.286296\pi$
$38$	199.242	0.850563
$39$	0	0
$40$	270.986	1.07116
$41$	−122.271	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$41$	−122.271	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$42$	0	0
$43$	−431.844	−1.53153	−0.765764	−	0.643122i	$-0.777639\pi$
$43$	−431.844	−1.53153	−0.765764	+	0.643122i	$0.777639\pi$
$44$	−259.465	−0.888996
$45$	0	0
$46$	−527.787	−1.69169
$47$	295.692	0.917685	0.458842	−	0.888518i	$-0.348264\pi$
$47$	295.692	0.917685	0.458842	+	0.888518i	$0.348264\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	128.752	0.364167
$51$	0	0
$52$	−196.221	−0.523287
$53$	243.639	0.631441	0.315721	−	0.948852i	$-0.397754\pi$
$53$	243.639	0.631441	0.315721	+	0.948852i	$0.397754\pi$
$54$	0	0
$55$	−70.0366	−0.171704
$56$	379.380	0.905299
$57$	0	0
$58$	−1196.76	−2.70935
$59$	566.480	1.24999	0.624995	−	0.780628i	$-0.285101\pi$
$59$	566.480	1.24999	0.624995	+	0.780628i	$0.285101\pi$
$60$	0	0
$61$	−188.012	−0.394630	−0.197315	−	0.980340i	$-0.563222\pi$
$61$	−188.012	−0.394630	−0.197315	+	0.980340i	$0.563222\pi$
$62$	−852.346	−1.74594
$63$	0	0
$64$	192.364	0.375711
$65$	−52.9653	−0.101070
$66$	0	0
$67$	−871.647	−1.58938	−0.794692	−	0.607013i	$-0.792367\pi$
$67$	−871.647	−1.58938	−0.794692	+	0.607013i	$0.792367\pi$
$68$	2441.42	4.35391
$69$	0	0
$70$	180.253	0.307777
$71$	−176.800	−0.295525	−0.147762	−	0.989023i	$-0.547207\pi$
$71$	−176.800	−0.295525	−0.147762	+	0.989023i	$0.547207\pi$
$72$	0	0
$73$	−220.524	−0.353566	−0.176783	−	0.984250i	$-0.556569\pi$
$73$	−220.524	−0.353566	−0.176783	+	0.984250i	$0.556569\pi$
$74$	1442.05	2.26533
$75$	0	0
$76$	716.621	1.08161
$77$	−98.0513	−0.145117
$78$	0	0
$79$	−190.494	−0.271294	−0.135647	−	0.990757i	$-0.543311\pi$
$79$	−190.494	−0.271294	−0.135647	+	0.990757i	$0.543311\pi$
$80$	654.662	0.914918
$81$	0	0
$82$	−629.710	−0.848047
$83$	518.241	0.685353	0.342676	−	0.939453i	$-0.388667\pi$
$83$	518.241	0.685353	0.342676	+	0.939453i	$0.388667\pi$
$84$	0	0
$85$	659.006	0.840932
$86$	−2224.04	−2.78866
$87$	0	0
$88$	−759.157	−0.919618
$89$	−598.355	−0.712646	−0.356323	−	0.934363i	$-0.615970\pi$
$89$	−598.355	−0.712646	−0.356323	+	0.934363i	$0.615970\pi$
$90$	0	0
$91$	−74.1515	−0.0854196
$92$	−1898.31	−2.15122
$93$	0	0
$94$	1522.85	1.67095
$95$	193.436	0.208906
$96$	0	0
$97$	−1882.24	−1.97023	−0.985116	−	0.171889i	$-0.945013\pi$
$97$	−1882.24	−1.97023	−0.985116	+	0.171889i	$0.945013\pi$
$98$	252.355	0.260119
$99$	0	0
$100$	463.088	0.463088
$101$	190.791	0.187964	0.0939821	−	0.995574i	$-0.470040\pi$
$101$	190.791	0.187964	0.0939821	+	0.995574i	$0.470040\pi$
$102$	0	0
$103$	−1496.53	−1.43163	−0.715813	−	0.698292i	$-0.753944\pi$
$103$	−1496.53	−1.43163	−0.715813	+	0.698292i	$0.753944\pi$
$104$	−574.114	−0.541312
$105$	0	0
$106$	1254.76	1.14975
$107$	−1965.85	−1.77613	−0.888066	−	0.459716i	$-0.847951\pi$
$107$	−1965.85	−1.77613	−0.888066	+	0.459716i	$0.847951\pi$
$108$	0	0
$109$	−870.482	−0.764927	−0.382464	−	0.923971i	$-0.624924\pi$
$109$	−870.482	−0.764927	−0.382464	+	0.923971i	$0.624924\pi$
$110$	−360.696	−0.312645
$111$	0	0
$112$	916.527	0.773247
$113$	2051.20	1.70762	0.853809	−	0.520587i	$-0.174287\pi$
$113$	2051.20	1.70762	0.853809	+	0.520587i	$0.174287\pi$
$114$	0	0
$115$	−512.405	−0.415496
$116$	−4304.42	−3.44531
$117$	0	0
$118$	2917.43	2.27603
$119$	922.608	0.710717
$120$	0	0
$121$	−1134.79	−0.852588
$122$	−968.279	−0.718556
$123$	0	0
$124$	−3065.66	−2.22020
$125$	125.000	0.0894427
$126$	0	0
$127$	2501.15	1.74757	0.873785	−	0.486313i	$-0.161659\pi$
$127$	2501.15	1.74757	0.873785	+	0.486313i	$0.161659\pi$
$128$	−935.209	−0.645793
$129$	0	0
$130$	−272.777	−0.184032
$131$	534.190	0.356278	0.178139	−	0.984005i	$-0.442992\pi$
$131$	534.190	0.356278	0.178139	+	0.984005i	$0.442992\pi$
$132$	0	0
$133$	270.810	0.176558
$134$	−4489.07	−2.89400
$135$	0	0
$136$	7143.24	4.50388
$137$	503.159	0.313779	0.156890	−	0.987616i	$-0.449853\pi$
$137$	503.159	0.313779	0.156890	+	0.987616i	$0.449853\pi$
$138$	0	0
$139$	1424.87	0.869466	0.434733	−	0.900559i	$-0.356843\pi$
$139$	1424.87	0.869466	0.434733	+	0.900559i	$0.356843\pi$
$140$	648.323	0.391381
$141$	0	0
$142$	−910.536	−0.538102
$143$	148.381	0.0867707
$144$	0	0
$145$	−1161.88	−0.665441
$146$	−1135.72	−0.643786
$147$	0	0
$148$	5186.66	2.88068
$149$	1482.49	0.815105	0.407552	−	0.913182i	$-0.366382\pi$
$149$	1482.49	0.815105	0.407552	+	0.913182i	$0.366382\pi$
$150$	0	0
$151$	−1267.34	−0.683008	−0.341504	−	0.939880i	$-0.610936\pi$
$151$	−1267.34	−0.683008	−0.341504	+	0.939880i	$0.610936\pi$
$152$	2096.73	1.11886
$153$	0	0
$154$	−504.974	−0.264233
$155$	−827.505	−0.428818
$156$	0	0
$157$	3234.00	1.64396	0.821979	−	0.569517i	$-0.192870\pi$
$157$	3234.00	1.64396	0.821979	+	0.569517i	$0.192870\pi$
$158$	−981.063	−0.493982
$159$	0	0
$160$	1203.69	0.594750
$161$	−717.367	−0.351158
$162$	0	0
$163$	1758.52	0.845017	0.422508	−	0.906359i	$-0.361150\pi$
$163$	1758.52	0.845017	0.422508	+	0.906359i	$0.361150\pi$
$164$	−2264.90	−1.07841
$165$	0	0
$166$	2668.99	1.24791
$167$	1682.57	0.779646	0.389823	−	0.920890i	$-0.372536\pi$
$167$	1682.57	0.779646	0.389823	+	0.920890i	$0.372536\pi$
$168$	0	0
$169$	−2084.79	−0.948924
$170$	3393.95	1.53120
$171$	0	0
$172$	−7999.28	−3.54616
$173$	3211.53	1.41138	0.705688	−	0.708523i	$-0.250638\pi$
$173$	3211.53	1.41138	0.705688	+	0.708523i	$0.250638\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	−1834.01	−0.785477
$177$	0	0
$178$	−3081.59	−1.29761
$179$	3311.97	1.38295	0.691476	−	0.722399i	$-0.256961\pi$
$179$	3311.97	1.38295	0.691476	+	0.722399i	$0.256961\pi$
$180$	0	0
$181$	2062.42	0.846954	0.423477	−	0.905907i	$-0.360809\pi$
$181$	2062.42	0.846954	0.423477	+	0.905907i	$0.360809\pi$
$182$	−381.887	−0.155535
$183$	0	0
$184$	−5554.17	−2.22532
$185$	1400.02	0.556386
$186$	0	0
$187$	−1846.18	−0.721958
$188$	5477.26	2.12484
$189$	0	0
$190$	996.212	0.380383
$191$	2783.01	1.05430	0.527150	−	0.849772i	$-0.323261\pi$
$191$	2783.01	1.05430	0.527150	+	0.849772i	$0.323261\pi$
$192$	0	0
$193$	300.293	0.111998	0.0559988	−	0.998431i	$-0.482166\pi$
$193$	300.293	0.111998	0.0559988	+	0.998431i	$0.482166\pi$
$194$	−9693.72	−3.58747
$195$	0	0
$196$	907.652	0.330777
$197$	−1485.64	−0.537295	−0.268648	−	0.963239i	$-0.586577\pi$
$197$	−1485.64	−0.537295	−0.268648	+	0.963239i	$0.586577\pi$
$198$	0	0
$199$	−3040.55	−1.08311	−0.541555	−	0.840665i	$-0.682164\pi$
$199$	−3040.55	−1.08311	−0.541555	+	0.840665i	$0.682164\pi$
$200$	1354.93	0.479039
$201$	0	0
$202$	982.591	0.342252
$203$	−1626.63	−0.562400
$204$	0	0
$205$	−611.357	−0.208288
$206$	−7707.28	−2.60676
$207$	0	0
$208$	−1386.98	−0.462353
$209$	−541.903	−0.179350
$210$	0	0
$211$	−2886.74	−0.941855	−0.470928	−	0.882172i	$-0.656081\pi$
$211$	−2886.74	−0.941855	−0.470928	+	0.882172i	$0.656081\pi$
$212$	4513.05	1.46206
$213$	0	0
$214$	−10124.3	−3.23404
$215$	−2159.22	−0.684920
$216$	0	0
$217$	−1158.51	−0.362417
$218$	−4483.07	−1.39281
$219$	0	0
$220$	−1297.32	−0.397571
$221$	−1396.18	−0.424964
$222$	0	0
$223$	4591.34	1.37874	0.689369	−	0.724410i	$-0.257888\pi$
$223$	4591.34	1.37874	0.689369	+	0.724410i	$0.257888\pi$
$224$	1685.16	0.502655
$225$	0	0
$226$	10563.9	3.10929
$227$	−2547.05	−0.744729	−0.372364	−	0.928087i	$-0.621453\pi$
$227$	−2547.05	−0.744729	−0.372364	+	0.928087i	$0.621453\pi$
$228$	0	0
$229$	2007.40	0.579269	0.289635	−	0.957137i	$-0.406466\pi$
$229$	2007.40	0.579269	0.289635	+	0.957137i	$0.406466\pi$
$230$	−2638.94	−0.756549
$231$	0	0
$232$	−12594.1	−3.56399
$233$	2810.65	0.790266	0.395133	−	0.918624i	$-0.370699\pi$
$233$	2810.65	0.790266	0.395133	+	0.918624i	$0.370699\pi$
$234$	0	0
$235$	1478.46	0.410401
$236$	10493.2	2.89428
$237$	0	0
$238$	4751.52	1.29410
$239$	1848.68	0.500339	0.250170	−	0.968202i	$-0.419514\pi$
$239$	1848.68	0.500339	0.250170	+	0.968202i	$0.419514\pi$
$240$	0	0
$241$	5722.64	1.52958	0.764788	−	0.644283i	$-0.222844\pi$
$241$	5722.64	1.52958	0.764788	+	0.644283i	$0.222844\pi$
$242$	−5844.30	−1.55242
$243$	0	0
$244$	−3482.64	−0.913742
$245$	245.000	0.0638877
$246$	0	0
$247$	−409.815	−0.105571
$248$	−8969.68	−2.29667
$249$	0	0
$250$	643.762	0.162860
$251$	1723.38	0.433381	0.216691	−	0.976240i	$-0.430474\pi$
$251$	1723.38	0.433381	0.216691	+	0.976240i	$0.430474\pi$
$252$	0	0
$253$	1435.48	0.356712
$254$	12881.2	3.18203
$255$	0	0
$256$	−6355.33	−1.55159
$257$	−5754.01	−1.39660	−0.698298	−	0.715807i	$-0.746059\pi$
$257$	−5754.01	−1.39660	−0.698298	+	0.715807i	$0.746059\pi$
$258$	0	0
$259$	1960.03	0.470232
$260$	−981.104	−0.234021
$261$	0	0
$262$	2751.13	0.648723
$263$	−6680.33	−1.56626	−0.783130	−	0.621857i	$-0.786378\pi$
$263$	−6680.33	−1.56626	−0.783130	+	0.621857i	$0.786378\pi$
$264$	0	0
$265$	1218.19	0.282389
$266$	1394.70	0.321483
$267$	0	0
$268$	−16146.0	−3.68012
$269$	7468.96	1.69290	0.846451	−	0.532466i	$-0.178735\pi$
$269$	7468.96	1.69290	0.846451	+	0.532466i	$0.178735\pi$
$270$	0	0
$271$	−757.673	−0.169835	−0.0849176	−	0.996388i	$-0.527063\pi$
$271$	−757.673	−0.169835	−0.0849176	+	0.996388i	$0.527063\pi$
$272$	17257.0	3.84692
$273$	0	0
$274$	2591.32	0.571340
$275$	−350.183	−0.0767885
$276$	0	0
$277$	1169.29	0.253632	0.126816	−	0.991926i	$-0.459524\pi$
$277$	1169.29	0.253632	0.126816	+	0.991926i	$0.459524\pi$
$278$	7338.22	1.58315
$279$	0	0
$280$	1896.90	0.404862
$281$	5735.20	1.21756	0.608778	−	0.793341i	$-0.291660\pi$
$281$	5735.20	1.21756	0.608778	+	0.793341i	$0.291660\pi$
$282$	0	0
$283$	−9010.69	−1.89269	−0.946343	−	0.323164i	$-0.895253\pi$
$283$	−9010.69	−1.89269	−0.946343	+	0.323164i	$0.895253\pi$
$284$	−3274.95	−0.684270
$285$	0	0
$286$	764.174	0.157995
$287$	−855.900	−0.176035
$288$	0	0
$289$	12458.6	2.53583
$290$	−5983.80	−1.21166
$291$	0	0
$292$	−4084.87	−0.818662
$293$	5303.84	1.05752	0.528760	−	0.848771i	$-0.322657\pi$
$293$	5303.84	1.05752	0.528760	+	0.848771i	$0.322657\pi$
$294$	0	0
$295$	2832.40	0.559013
$296$	15175.4	2.97991
$297$	0	0
$298$	7634.99	1.48417
$299$	1085.59	0.209970
$300$	0	0
$301$	−3022.91	−0.578863
$302$	−6526.90	−1.24365
$303$	0	0
$304$	5065.40	0.955659
$305$	−940.059	−0.176484
$306$	0	0
$307$	8783.81	1.63296	0.816480	−	0.577374i	$-0.195922\pi$
$307$	8783.81	1.63296	0.816480	+	0.577374i	$0.195922\pi$
$308$	−1816.25	−0.336009
$309$	0	0
$310$	−4261.73	−0.780807
$311$	−2357.25	−0.429799	−0.214899	−	0.976636i	$-0.568942\pi$
$311$	−2357.25	−0.429799	−0.214899	+	0.976636i	$0.568942\pi$
$312$	0	0
$313$	−2633.84	−0.475635	−0.237817	−	0.971310i	$-0.576432\pi$
$313$	−2633.84	−0.475635	−0.237817	+	0.971310i	$0.576432\pi$
$314$	16655.4	2.99338
$315$	0	0
$316$	−3528.62	−0.628166
$317$	−842.572	−0.149286	−0.0746428	−	0.997210i	$-0.523782\pi$
$317$	−842.572	−0.149286	−0.0746428	+	0.997210i	$0.523782\pi$
$318$	0	0
$319$	3254.97	0.571296
$320$	961.820	0.168023
$321$	0	0
$322$	−3694.51	−0.639401
$323$	5099.01	0.878379
$324$	0	0
$325$	−264.827	−0.0451998
$326$	9056.54	1.53864
$327$	0	0
$328$	−6626.76	−1.11555
$329$	2069.85	0.346852
$330$	0	0
$331$	11058.8	1.83639	0.918197	−	0.396125i	$-0.129645\pi$
$331$	11058.8	1.83639	0.918197	+	0.396125i	$0.129645\pi$
$332$	9599.64	1.58689
$333$	0	0
$334$	8665.38	1.41961
$335$	−4358.24	−0.710794
$336$	0	0
$337$	−1197.28	−0.193531	−0.0967657	−	0.995307i	$-0.530850\pi$
$337$	−1197.28	−0.193531	−0.0967657	+	0.995307i	$0.530850\pi$
$338$	−10736.9	−1.72783
$339$	0	0
$340$	12207.1	1.94713
$341$	2318.23	0.368150
$342$	0	0
$343$	343.000	0.0539949
$344$	−23404.7	−3.66831
$345$	0	0
$346$	16539.7	2.56988
$347$	−518.087	−0.0801509	−0.0400755	−	0.999197i	$-0.512760\pi$
$347$	−518.087	−0.0801509	−0.0400755	+	0.999197i	$0.512760\pi$
$348$	0	0
$349$	−1776.08	−0.272412	−0.136206	−	0.990681i	$-0.543491\pi$
$349$	−1776.08	−0.272412	−0.136206	+	0.990681i	$0.543491\pi$
$350$	901.267	0.137642
$351$	0	0
$352$	−3372.09	−0.510606
$353$	−7507.35	−1.13194	−0.565972	−	0.824425i	$-0.691499\pi$
$353$	−7507.35	−1.13194	−0.565972	+	0.824425i	$0.691499\pi$
$354$	0	0
$355$	−883.999	−0.132163
$356$	−11083.6	−1.65009
$357$	0	0
$358$	17057.0	2.51813
$359$	−4915.58	−0.722658	−0.361329	−	0.932438i	$-0.617677\pi$
$359$	−4915.58	−0.722658	−0.361329	+	0.932438i	$0.617677\pi$
$360$	0	0
$361$	−5362.31	−0.781791
$362$	10621.7	1.54216
$363$	0	0
$364$	−1373.55	−0.197784
$365$	−1102.62	−0.158120
$366$	0	0
$367$	−3628.98	−0.516161	−0.258080	−	0.966123i	$-0.583090\pi$
$367$	−3628.98	−0.516161	−0.258080	+	0.966123i	$0.583090\pi$
$368$	−13418.1	−1.90072
$369$	0	0
$370$	7210.24	1.01309
$371$	1705.47	0.238662
$372$	0	0
$373$	−2836.97	−0.393814	−0.196907	−	0.980422i	$-0.563090\pi$
$373$	−2836.97	−0.393814	−0.196907	+	0.980422i	$0.563090\pi$
$374$	−9508.02	−1.31457
$375$	0	0
$376$	16025.7	2.19804
$377$	2461.58	0.336280
$378$	0	0
$379$	9703.96	1.31519	0.657597	−	0.753370i	$-0.271573\pi$
$379$	9703.96	1.31519	0.657597	+	0.753370i	$0.271573\pi$
$380$	3583.11	0.483709
$381$	0	0
$382$	14332.8	1.91971
$383$	−3855.90	−0.514431	−0.257216	−	0.966354i	$-0.582805\pi$
$383$	−3855.90	−0.514431	−0.257216	+	0.966354i	$0.582805\pi$
$384$	0	0
$385$	−490.256	−0.0648981
$386$	1546.54	0.203929
$387$	0	0
$388$	−34865.7	−4.56195
$389$	7106.46	0.926251	0.463125	−	0.886293i	$-0.346728\pi$
$389$	7106.46	0.926251	0.463125	+	0.886293i	$0.346728\pi$
$390$	0	0
$391$	−13507.1	−1.74702
$392$	2655.66	0.342171
$393$	0	0
$394$	−7651.17	−0.978326
$395$	−952.470	−0.121327
$396$	0	0
$397$	−6775.82	−0.856596	−0.428298	−	0.903637i	$-0.640887\pi$
$397$	−6775.82	−0.856596	−0.428298	+	0.903637i	$0.640887\pi$
$398$	−15659.1	−1.97217
$399$	0	0
$400$	3273.31	0.409164
$401$	6444.62	0.802566	0.401283	−	0.915954i	$-0.368564\pi$
$401$	6444.62	0.802566	0.401283	+	0.915954i	$0.368564\pi$
$402$	0	0
$403$	1753.16	0.216703
$404$	3534.11	0.435220
$405$	0	0
$406$	−8377.32	−1.02404
$407$	−3922.11	−0.477670
$408$	0	0
$409$	4807.99	0.581270	0.290635	−	0.956834i	$-0.406133\pi$
$409$	4807.99	0.581270	0.290635	+	0.956834i	$0.406133\pi$
$410$	−3148.55	−0.379258
$411$	0	0
$412$	−27721.0	−3.31484
$413$	3965.36	0.472452
$414$	0	0
$415$	2591.20	0.306499
$416$	−2550.15	−0.300556
$417$	0	0
$418$	−2790.85	−0.326567
$419$	−4521.51	−0.527184	−0.263592	−	0.964634i	$-0.584907\pi$
$419$	−4521.51	−0.527184	−0.263592	+	0.964634i	$0.584907\pi$
$420$	0	0
$421$	−5821.82	−0.673962	−0.336981	−	0.941511i	$-0.609406\pi$
$421$	−5821.82	−0.673962	−0.336981	+	0.941511i	$0.609406\pi$
$422$	−14867.0	−1.71496
$423$	0	0
$424$	13204.5	1.51243
$425$	3295.03	0.376076
$426$	0	0
$427$	−1316.08	−0.149156
$428$	−36414.5	−4.11252
$429$	0	0
$430$	−11120.2	−1.24713
$431$	−8382.59	−0.936833	−0.468417	−	0.883508i	$-0.655175\pi$
$431$	−8382.59	−0.936833	−0.468417	+	0.883508i	$0.655175\pi$
$432$	0	0
$433$	2628.05	0.291677	0.145838	−	0.989308i	$-0.453412\pi$
$433$	2628.05	0.291677	0.145838	+	0.989308i	$0.453412\pi$
$434$	−5966.42	−0.659902
$435$	0	0
$436$	−16124.4	−1.77114
$437$	−3964.69	−0.433998
$438$	0	0
$439$	−4122.97	−0.448243	−0.224121	−	0.974561i	$-0.571951\pi$
$439$	−4122.97	−0.448243	−0.224121	+	0.974561i	$0.571951\pi$
$440$	−3795.78	−0.411266
$441$	0	0
$442$	−7190.46	−0.773790
$443$	−210.327	−0.0225574	−0.0112787	−	0.999936i	$-0.503590\pi$
$443$	−210.327	−0.0225574	−0.0112787	+	0.999936i	$0.503590\pi$
$444$	0	0
$445$	−2991.77	−0.318705
$446$	23645.8	2.51045
$447$	0	0
$448$	1346.55	0.142005
$449$	4414.35	0.463978	0.231989	−	0.972718i	$-0.425477\pi$
$449$	4414.35	0.463978	0.231989	+	0.972718i	$0.425477\pi$
$450$	0	0
$451$	1712.70	0.178820
$452$	37995.4	3.95388
$453$	0	0
$454$	−13117.5	−1.35603
$455$	−370.757	−0.0382008
$456$	0	0
$457$	14355.4	1.46941	0.734704	−	0.678388i	$-0.237321\pi$
$457$	14355.4	1.46941	0.734704	+	0.678388i	$0.237321\pi$
$458$	10338.3	1.05475
$459$	0	0
$460$	−9491.54	−0.962055
$461$	−14366.9	−1.45149	−0.725744	−	0.687965i	$-0.758504\pi$
$461$	−14366.9	−1.45149	−0.725744	+	0.687965i	$0.758504\pi$
$462$	0	0
$463$	11634.1	1.16778	0.583892	−	0.811832i	$-0.301529\pi$
$463$	11634.1	1.16778	0.583892	+	0.811832i	$0.301529\pi$
$464$	−30425.6	−3.04412
$465$	0	0
$466$	14475.1	1.43894
$467$	−12112.2	−1.20019	−0.600094	−	0.799929i	$-0.704870\pi$
$467$	−12112.2	−1.20019	−0.600094	+	0.799929i	$0.704870\pi$
$468$	0	0
$469$	−6101.53	−0.600730
$470$	7614.23	0.747273
$471$	0	0
$472$	30701.6	2.99397
$473$	6048.99	0.588018
$474$	0	0
$475$	967.178	0.0934256
$476$	17089.9	1.64562
$477$	0	0
$478$	9520.88	0.911035
$479$	11380.8	1.08560	0.542801	−	0.839861i	$-0.317364\pi$
$479$	11380.8	1.08560	0.542801	+	0.839861i	$0.317364\pi$
$480$	0	0
$481$	−2966.10	−0.281170
$482$	29472.2	2.78510
$483$	0	0
$484$	−21020.4	−1.97412
$485$	−9411.20	−0.881115
$486$	0	0
$487$	−11615.5	−1.08080	−0.540401	−	0.841408i	$-0.681727\pi$
$487$	−11615.5	−1.08080	−0.540401	+	0.841408i	$0.681727\pi$
$488$	−10189.7	−0.945217
$489$	0	0
$490$	1261.77	0.116329
$491$	−6045.64	−0.555674	−0.277837	−	0.960628i	$-0.589617\pi$
$491$	−6045.64	−0.555674	−0.277837	+	0.960628i	$0.589617\pi$
$492$	0	0
$493$	−30627.5	−2.79795
$494$	−2110.59	−0.192226
$495$	0	0
$496$	−21669.4	−1.96167
$497$	−1237.60	−0.111698
$498$	0	0
$499$	860.279	0.0771771	0.0385886	−	0.999255i	$-0.487714\pi$
$499$	860.279	0.0771771	0.0385886	+	0.999255i	$0.487714\pi$
$500$	2315.44	0.207099
$501$	0	0
$502$	8875.57	0.789115
$503$	3341.49	0.296202	0.148101	−	0.988972i	$-0.452684\pi$
$503$	3341.49	0.296202	0.148101	+	0.988972i	$0.452684\pi$
$504$	0	0
$505$	953.953	0.0840601
$506$	7392.89	0.649514
$507$	0	0
$508$	46330.1	4.04639
$509$	19754.5	1.72024	0.860122	−	0.510088i	$-0.170387\pi$
$509$	19754.5	1.72024	0.860122	+	0.510088i	$0.170387\pi$
$510$	0	0
$511$	−1543.67	−0.133636
$512$	−25248.9	−2.17940
$513$	0	0
$514$	−29633.7	−2.54297
$515$	−7482.65	−0.640243
$516$	0	0
$517$	−4141.86	−0.352338
$518$	10094.3	0.856215
$519$	0	0
$520$	−2870.57	−0.242082
$521$	20754.2	1.74522	0.872609	−	0.488420i	$-0.162427\pi$
$521$	20754.2	1.74522	0.872609	+	0.488420i	$0.162427\pi$
$522$	0	0
$523$	−9495.26	−0.793879	−0.396940	−	0.917845i	$-0.629928\pi$
$523$	−9495.26	−0.793879	−0.396940	+	0.917845i	$0.629928\pi$
$524$	9895.07	0.824939
$525$	0	0
$526$	−34404.3	−2.85190
$527$	−21813.2	−1.80303
$528$	0	0
$529$	−1664.65	−0.136817
$530$	6273.82	0.514184
$531$	0	0
$532$	5016.35	0.408809
$533$	1295.23	0.105258
$534$	0	0
$535$	−9829.26	−0.794310
$536$	−47240.8	−3.80689
$537$	0	0
$538$	38465.9	3.08250
$539$	−686.359	−0.0548489
$540$	0	0
$541$	15585.9	1.23862	0.619308	−	0.785148i	$-0.287413\pi$
$541$	15585.9	1.23862	0.619308	+	0.785148i	$0.287413\pi$
$542$	−3902.09	−0.309242
$543$	0	0
$544$	31729.5	2.50072
$545$	−4352.41	−0.342086
$546$	0	0
$547$	−23104.6	−1.80600	−0.902998	−	0.429644i	$-0.858639\pi$
$547$	−23104.6	−1.80600	−0.902998	+	0.429644i	$0.858639\pi$
$548$	9320.27	0.726537
$549$	0	0
$550$	−1803.48	−0.139819
$551$	−8989.96	−0.695073
$552$	0	0
$553$	−1333.46	−0.102540
$554$	6021.98	0.461822
$555$	0	0
$556$	26393.6	2.01320
$557$	−21731.2	−1.65311	−0.826553	−	0.562859i	$-0.809701\pi$
$557$	−21731.2	−1.65311	−0.826553	+	0.562859i	$0.809701\pi$
$558$	0	0
$559$	4574.56	0.346124
$560$	4582.63	0.345807
$561$	0	0
$562$	29536.8	2.21697
$563$	13595.1	1.01770	0.508850	−	0.860855i	$-0.330071\pi$
$563$	13595.1	1.01770	0.508850	+	0.860855i	$0.330071\pi$
$564$	0	0
$565$	10256.0	0.763670
$566$	−46405.9	−3.44627
$567$	0	0
$568$	−9582.04	−0.707840
$569$	−10018.2	−0.738107	−0.369054	−	0.929408i	$-0.620318\pi$
$569$	−10018.2	−0.738107	−0.369054	+	0.929408i	$0.620318\pi$
$570$	0	0
$571$	−6658.73	−0.488020	−0.244010	−	0.969773i	$-0.578463\pi$
$571$	−6658.73	−0.488020	−0.244010	+	0.969773i	$0.578463\pi$
$572$	2748.53	0.200912
$573$	0	0
$574$	−4407.97	−0.320531
$575$	−2562.02	−0.185815
$576$	0	0
$577$	−12676.0	−0.914570	−0.457285	−	0.889320i	$-0.651178\pi$
$577$	−12676.0	−0.914570	−0.457285	+	0.889320i	$0.651178\pi$
$578$	64162.8	4.61733
$579$	0	0
$580$	−21522.1	−1.54079
$581$	3627.68	0.259039
$582$	0	0
$583$	−3412.73	−0.242437
$584$	−11951.8	−0.846861
$585$	0	0
$586$	27315.3	1.92557
$587$	−18625.6	−1.30965	−0.654823	−	0.755782i	$-0.727257\pi$
$587$	−18625.6	−1.30965	−0.654823	+	0.755782i	$0.727257\pi$
$588$	0	0
$589$	−6402.76	−0.447913
$590$	14587.1	1.01787
$591$	0	0
$592$	36661.6	2.54524
$593$	−19417.3	−1.34465	−0.672323	−	0.740258i	$-0.734703\pi$
$593$	−19417.3	−1.34465	−0.672323	+	0.740258i	$0.734703\pi$
$594$	0	0
$595$	4613.04	0.317842
$596$	27461.0	1.88733
$597$	0	0
$598$	5590.88	0.382321
$599$	−13376.9	−0.912461	−0.456231	−	0.889862i	$-0.650801\pi$
$599$	−13376.9	−0.912461	−0.456231	+	0.889862i	$0.650801\pi$
$600$	0	0
$601$	14963.9	1.01562	0.507812	−	0.861468i	$-0.330454\pi$
$601$	14963.9	1.01562	0.507812	+	0.861468i	$0.330454\pi$
$602$	−15568.3	−1.05401
$603$	0	0
$604$	−23475.5	−1.58146
$605$	−5673.97	−0.381289
$606$	0	0
$607$	20270.8	1.35546	0.677731	−	0.735310i	$-0.262963\pi$
$607$	20270.8	1.35546	0.677731	+	0.735310i	$0.262963\pi$
$608$	9313.45	0.621234
$609$	0	0
$610$	−4841.40	−0.321348
$611$	−3132.29	−0.207396
$612$	0	0
$613$	13005.1	0.856888	0.428444	−	0.903568i	$-0.359062\pi$
$613$	13005.1	0.856888	0.428444	+	0.903568i	$0.359062\pi$
$614$	45237.5	2.97335
$615$	0	0
$616$	−5314.10	−0.347583
$617$	−11107.4	−0.724745	−0.362372	−	0.932033i	$-0.618033\pi$
$617$	−11107.4	−0.724745	−0.362372	+	0.932033i	$0.618033\pi$
$618$	0	0
$619$	348.077	0.0226016	0.0113008	−	0.999936i	$-0.496403\pi$
$619$	348.077	0.0226016	0.0113008	+	0.999936i	$0.496403\pi$
$620$	−15328.3	−0.992902
$621$	0	0
$622$	−12140.1	−0.782593
$623$	−4188.48	−0.269355
$624$	0	0
$625$	625.000	0.0400000
$626$	−13564.6	−0.866052
$627$	0	0
$628$	59905.1	3.80649
$629$	36904.8	2.33942
$630$	0	0
$631$	7548.88	0.476254	0.238127	−	0.971234i	$-0.423467\pi$
$631$	7548.88	0.476254	0.238127	+	0.971234i	$0.423467\pi$
$632$	−10324.2	−0.649804
$633$	0	0
$634$	−4339.33	−0.271825
$635$	12505.8	0.781537
$636$	0	0
$637$	−519.060	−0.0322856
$638$	16763.4	1.04023
$639$	0	0
$640$	−4676.04	−0.288808
$641$	8720.39	0.537340	0.268670	−	0.963232i	$-0.413416\pi$
$641$	8720.39	0.537340	0.268670	+	0.963232i	$0.413416\pi$
$642$	0	0
$643$	−10517.6	−0.645061	−0.322530	−	0.946559i	$-0.604533\pi$
$643$	−10517.6	−0.645061	−0.322530	+	0.946559i	$0.604533\pi$
$644$	−13288.2	−0.813085
$645$	0	0
$646$	26260.4	1.59938
$647$	−6194.54	−0.376402	−0.188201	−	0.982130i	$-0.560266\pi$
$647$	−6194.54	−0.376402	−0.188201	+	0.982130i	$0.560266\pi$
$648$	0	0
$649$	−7934.87	−0.479925
$650$	−1363.88	−0.0823014
$651$	0	0
$652$	32573.9	1.95658
$653$	16726.9	1.00241	0.501205	−	0.865329i	$-0.332890\pi$
$653$	16726.9	1.00241	0.501205	+	0.865329i	$0.332890\pi$
$654$	0	0
$655$	2670.95	0.159332
$656$	−16009.3	−0.952832
$657$	0	0
$658$	10659.9	0.631561
$659$	−9874.87	−0.583718	−0.291859	−	0.956461i	$-0.594274\pi$
$659$	−9874.87	−0.583718	−0.291859	+	0.956461i	$0.594274\pi$
$660$	0	0
$661$	−26873.6	−1.58134	−0.790668	−	0.612245i	$-0.790267\pi$
$661$	−26873.6	−1.58134	−0.790668	+	0.612245i	$0.790267\pi$
$662$	56953.9	3.34377
$663$	0	0
$664$	28087.1	1.64156
$665$	1354.05	0.0789591
$666$	0	0
$667$	23814.1	1.38244
$668$	31167.0	1.80522
$669$	0	0
$670$	−22445.3	−1.29424
$671$	2633.54	0.151515
$672$	0	0
$673$	270.279	0.0154807	0.00774034	−	0.999970i	$-0.497536\pi$
$673$	270.279	0.0154807	0.00774034	+	0.999970i	$0.497536\pi$
$674$	−6166.12	−0.352389
$675$	0	0
$676$	−38617.6	−2.19718
$677$	−5216.49	−0.296139	−0.148070	−	0.988977i	$-0.547306\pi$
$677$	−5216.49	−0.296139	−0.148070	+	0.988977i	$0.547306\pi$
$678$	0	0
$679$	−13175.7	−0.744678
$680$	35716.2	2.01420
$681$	0	0
$682$	11939.1	0.670339
$683$	597.154	0.0334546	0.0167273	−	0.999860i	$-0.494675\pi$
$683$	597.154	0.0334546	0.0167273	+	0.999860i	$0.494675\pi$
$684$	0	0
$685$	2515.79	0.140326
$686$	1766.48	0.0983158
$687$	0	0
$688$	−56542.4	−3.13323
$689$	−2580.88	−0.142705
$690$	0	0
$691$	−32491.9	−1.78878	−0.894392	−	0.447283i	$-0.852392\pi$
$691$	−32491.9	−1.78878	−0.894392	+	0.447283i	$0.852392\pi$
$692$	59488.8	3.26795
$693$	0	0
$694$	−2668.20	−0.145942
$695$	7124.35	0.388837
$696$	0	0
$697$	−16115.5	−0.875780
$698$	−9147.01	−0.496016
$699$	0	0
$700$	3241.61	0.175031
$701$	25288.3	1.36252	0.681260	−	0.732041i	$-0.261432\pi$
$701$	25288.3	1.36252	0.681260	+	0.732041i	$0.261432\pi$
$702$	0	0
$703$	10832.5	0.581162
$704$	−2694.51	−0.144251
$705$	0	0
$706$	−38663.6	−2.06108
$707$	1335.53	0.0710438
$708$	0	0
$709$	20110.2	1.06524	0.532618	−	0.846355i	$-0.321208\pi$
$709$	20110.2	1.06524	0.532618	+	0.846355i	$0.321208\pi$
$710$	−4552.68	−0.240647
$711$	0	0
$712$	−32429.1	−1.70693
$713$	16960.7	0.890860
$714$	0	0
$715$	741.903	0.0388050
$716$	61349.4	3.20214
$717$	0	0
$718$	−25315.7	−1.31584
$719$	−5650.46	−0.293083	−0.146542	−	0.989205i	$-0.546814\pi$
$719$	−5650.46	−0.293083	−0.146542	+	0.989205i	$0.546814\pi$
$720$	0	0
$721$	−10475.7	−0.541104
$722$	−27616.4	−1.42351
$723$	0	0
$724$	38203.3	1.96107
$725$	−5809.40	−0.297594
$726$	0	0
$727$	6571.94	0.335268	0.167634	−	0.985849i	$-0.446387\pi$
$727$	6571.94	0.335268	0.167634	+	0.985849i	$0.446387\pi$
$728$	−4018.80	−0.204597
$729$	0	0
$730$	−5678.59	−0.287910
$731$	−56917.6	−2.87985
$732$	0	0
$733$	−31444.0	−1.58446	−0.792231	−	0.610222i	$-0.791080\pi$
$733$	−31444.0	−1.58446	−0.792231	+	0.610222i	$0.791080\pi$
$734$	−18689.6	−0.939843
$735$	0	0
$736$	−24671.0	−1.23558
$737$	12209.4	0.610232
$738$	0	0
$739$	8157.80	0.406075	0.203038	−	0.979171i	$-0.434919\pi$
$739$	8157.80	0.406075	0.203038	+	0.979171i	$0.434919\pi$
$740$	25933.3	1.28828
$741$	0	0
$742$	8783.35	0.434565
$743$	−5705.69	−0.281725	−0.140862	−	0.990029i	$-0.544987\pi$
$743$	−5705.69	−0.281725	−0.140862	+	0.990029i	$0.544987\pi$
$744$	0	0
$745$	7412.47	0.364526
$746$	−14610.7	−0.717070
$747$	0	0
$748$	−34197.8	−1.67165
$749$	−13761.0	−0.671315
$750$	0	0
$751$	−7296.31	−0.354522	−0.177261	−	0.984164i	$-0.556724\pi$
$751$	−7296.31	−0.354522	−0.177261	+	0.984164i	$0.556724\pi$
$752$	38715.7	1.87742
$753$	0	0
$754$	12677.4	0.612311
$755$	−6336.68	−0.305451
$756$	0	0
$757$	8240.48	0.395648	0.197824	−	0.980238i	$-0.436613\pi$
$757$	8240.48	0.395648	0.197824	+	0.980238i	$0.436613\pi$
$758$	49976.3	2.39475
$759$	0	0
$760$	10483.7	0.500371
$761$	−30887.8	−1.47133	−0.735665	−	0.677345i	$-0.763130\pi$
$761$	−30887.8	−1.47133	−0.735665	+	0.677345i	$0.763130\pi$
$762$	0	0
$763$	−6093.37	−0.289115
$764$	51551.1	2.44117
$765$	0	0
$766$	−19858.2	−0.936694
$767$	−6000.76	−0.282497
$768$	0	0
$769$	30953.0	1.45149	0.725743	−	0.687966i	$-0.241496\pi$
$769$	30953.0	1.45149	0.725743	+	0.687966i	$0.241496\pi$
$770$	−2524.87	−0.118169
$771$	0	0
$772$	5562.48	0.259324
$773$	−6620.92	−0.308070	−0.154035	−	0.988065i	$-0.549227\pi$
$773$	−6620.92	−0.308070	−0.154035	+	0.988065i	$0.549227\pi$
$774$	0	0
$775$	−4137.52	−0.191773
$776$	−102012.	−4.71910
$777$	0	0
$778$	36598.9	1.68655
$779$	−4730.33	−0.217563
$780$	0	0
$781$	2476.49	0.113465
$782$	−69563.0	−3.18103
$783$	0	0
$784$	6415.69	0.292260
$785$	16170.0	0.735201
$786$	0	0
$787$	−10068.6	−0.456044	−0.228022	−	0.973656i	$-0.573226\pi$
$787$	−10068.6	−0.456044	−0.228022	+	0.973656i	$0.573226\pi$
$788$	−27519.2	−1.24407
$789$	0	0
$790$	−4905.32	−0.220916
$791$	14358.4	0.645419
$792$	0	0
$793$	1991.62	0.0891861
$794$	−34896.1	−1.55972
$795$	0	0
$796$	−56321.7	−2.50788
$797$	42016.5	1.86738	0.933690	−	0.358084i	$-0.116570\pi$
$797$	42016.5	1.86738	0.933690	+	0.358084i	$0.116570\pi$
$798$	0	0
$799$	38972.6	1.72560
$800$	6018.45	0.265980
$801$	0	0
$802$	33190.4	1.46134
$803$	3088.95	0.135749
$804$	0	0
$805$	−3586.83	−0.157043
$806$	9028.96	0.394580
$807$	0	0
$808$	10340.3	0.450211
$809$	−32805.5	−1.42568	−0.712842	−	0.701324i	$-0.752593\pi$
$809$	−32805.5	−1.42568	−0.712842	+	0.701324i	$0.752593\pi$
$810$	0	0
$811$	15395.7	0.666606	0.333303	−	0.942820i	$-0.391837\pi$
$811$	15395.7	0.666606	0.333303	+	0.942820i	$0.391837\pi$
$812$	−30131.0	−1.30220
$813$	0	0
$814$	−20199.2	−0.869758
$815$	8792.59	0.377903
$816$	0	0
$817$	−16706.8	−0.715419
$818$	24761.6	1.05840
$819$	0	0
$820$	−11324.5	−0.482278
$821$	10626.7	0.451734	0.225867	−	0.974158i	$-0.427479\pi$
$821$	10626.7	0.451734	0.225867	+	0.974158i	$0.427479\pi$
$822$	0	0
$823$	−42578.1	−1.80338	−0.901689	−	0.432384i	$-0.857672\pi$
$823$	−42578.1	−1.80338	−0.901689	+	0.432384i	$0.857672\pi$
$824$	−81107.7	−3.42903
$825$	0	0
$826$	20422.0	0.860257
$827$	733.624	0.0308472	0.0154236	−	0.999881i	$-0.495090\pi$
$827$	733.624	0.0308472	0.0154236	+	0.999881i	$0.495090\pi$
$828$	0	0
$829$	15778.0	0.661028	0.330514	−	0.943801i	$-0.392778\pi$
$829$	15778.0	0.661028	0.330514	+	0.943801i	$0.392778\pi$
$830$	13344.9	0.558084
$831$	0	0
$832$	−2037.73	−0.0849104
$833$	6458.26	0.268626
$834$	0	0
$835$	8412.83	0.348668
$836$	−10037.9	−0.415275
$837$	0	0
$838$	−23286.2	−0.959916
$839$	16776.8	0.690343	0.345172	−	0.938540i	$-0.387821\pi$
$839$	16776.8	0.690343	0.345172	+	0.938540i	$0.387821\pi$
$840$	0	0
$841$	29609.7	1.21406
$842$	−29983.0	−1.22717
$843$	0	0
$844$	−53472.6	−2.18081
$845$	−10423.9	−0.424372
$846$	0	0
$847$	−7943.56	−0.322248
$848$	31900.2	1.29181
$849$	0	0
$850$	16969.7	0.684773
$851$	−28695.1	−1.15588
$852$	0	0
$853$	7646.57	0.306933	0.153466	−	0.988154i	$-0.450956\pi$
$853$	7646.57	0.306933	0.153466	+	0.988154i	$0.450956\pi$
$854$	−6777.95	−0.271589
$855$	0	0
$856$	−106544.	−4.25419
$857$	26980.3	1.07541	0.537707	−	0.843132i	$-0.319291\pi$
$857$	26980.3	1.07541	0.537707	+	0.843132i	$0.319291\pi$
$858$	0	0
$859$	3034.02	0.120511	0.0602557	−	0.998183i	$-0.480808\pi$
$859$	3034.02	0.120511	0.0602557	+	0.998183i	$0.480808\pi$
$860$	−39996.4	−1.58589
$861$	0	0
$862$	−43171.2	−1.70582
$863$	−25954.5	−1.02376	−0.511879	−	0.859058i	$-0.671050\pi$
$863$	−25954.5	−1.02376	−0.511879	+	0.859058i	$0.671050\pi$
$864$	0	0
$865$	16057.6	0.631186
$866$	13534.7	0.531095
$867$	0	0
$868$	−21459.6	−0.839155
$869$	2668.31	0.104161
$870$	0	0
$871$	9233.42	0.359199
$872$	−47177.6	−1.83215
$873$	0	0
$874$	−20418.6	−0.790238
$875$	875.000	0.0338062
$876$	0	0
$877$	−11235.7	−0.432616	−0.216308	−	0.976325i	$-0.569402\pi$
$877$	−11235.7	−0.432616	−0.216308	+	0.976325i	$0.569402\pi$
$878$	−21233.7	−0.816176
$879$	0	0
$880$	−9170.07	−0.351276
$881$	−4594.31	−0.175694	−0.0878468	−	0.996134i	$-0.527999\pi$
$881$	−4594.31	−0.175694	−0.0878468	+	0.996134i	$0.527999\pi$
$882$	0	0
$883$	−14162.1	−0.539741	−0.269871	−	0.962897i	$-0.586981\pi$
$883$	−14162.1	−0.539741	−0.269871	+	0.962897i	$0.586981\pi$
$884$	−25862.1	−0.983979
$885$	0	0
$886$	−1083.20	−0.0410733
$887$	−31024.2	−1.17440	−0.587199	−	0.809442i	$-0.699770\pi$
$887$	−31024.2	−1.17440	−0.587199	+	0.809442i	$0.699770\pi$
$888$	0	0
$889$	17508.1	0.660519
$890$	−15407.9	−0.580309
$891$	0	0
$892$	85047.7	3.19238
$893$	11439.5	0.428676
$894$	0	0
$895$	16559.9	0.618475
$896$	−6546.46	−0.244087
$897$	0	0
$898$	22734.4	0.844828
$899$	38458.5	1.42677
$900$	0	0
$901$	32111.9	1.18735
$902$	8820.55	0.325601
$903$	0	0
$904$	111169.	4.09008
$905$	10312.1	0.378770
$906$	0	0
$907$	−10594.5	−0.387855	−0.193928	−	0.981016i	$-0.562123\pi$
$907$	−10594.5	−0.387855	−0.193928	+	0.981016i	$0.562123\pi$
$908$	−47180.2	−1.72437
$909$	0	0
$910$	−1909.44	−0.0695574
$911$	1469.37	0.0534384	0.0267192	−	0.999643i	$-0.491494\pi$
$911$	1469.37	0.0534384	0.0267192	+	0.999643i	$0.491494\pi$
$912$	0	0
$913$	−7259.17	−0.263136
$914$	73931.9	2.67555
$915$	0	0
$916$	37184.1	1.34126
$917$	3739.33	0.134660
$918$	0	0
$919$	41834.6	1.50163	0.750813	−	0.660515i	$-0.229662\pi$
$919$	41834.6	1.50163	0.750813	+	0.660515i	$0.229662\pi$
$920$	−27770.9	−0.995194
$921$	0	0
$922$	−73991.2	−2.64292
$923$	1872.85	0.0667884
$924$	0	0
$925$	7000.10	0.248824
$926$	59916.9	2.12634
$927$	0	0
$928$	−55941.7	−1.97885
$929$	8570.23	0.302670	0.151335	−	0.988483i	$-0.451643\pi$
$929$	8570.23	0.302670	0.151335	+	0.988483i	$0.451643\pi$
$930$	0	0
$931$	1895.67	0.0667326
$932$	52063.1	1.82981
$933$	0	0
$934$	−62379.3	−2.18534
$935$	−9230.91	−0.322870
$936$	0	0
$937$	41260.3	1.43854	0.719271	−	0.694730i	$-0.244476\pi$
$937$	41260.3	1.43854	0.719271	+	0.694730i	$0.244476\pi$
$938$	−31423.5	−1.09383
$939$	0	0
$940$	27386.3	0.950259
$941$	47969.6	1.66181	0.830905	−	0.556414i	$-0.187823\pi$
$941$	47969.6	1.66181	0.830905	+	0.556414i	$0.187823\pi$
$942$	0	0
$943$	12530.5	0.432714
$944$	74170.6	2.55725
$945$	0	0
$946$	31152.9	1.07068
$947$	−15666.3	−0.537577	−0.268789	−	0.963199i	$-0.586623\pi$
$947$	−15666.3	−0.537577	−0.268789	+	0.963199i	$0.586623\pi$
$948$	0	0
$949$	2336.02	0.0799057
$950$	4981.06	0.170113
$951$	0	0
$952$	50002.7	1.70231
$953$	27495.7	0.934600	0.467300	−	0.884099i	$-0.345227\pi$
$953$	27495.7	0.934600	0.467300	+	0.884099i	$0.345227\pi$
$954$	0	0
$955$	13915.0	0.471497
$956$	34244.0	1.15851
$957$	0	0
$958$	58612.4	1.97670
$959$	3522.11	0.118597
$960$	0	0
$961$	−2400.42	−0.0805755
$962$	−15275.7	−0.511963
$963$	0	0
$964$	106003.	3.54164
$965$	1501.46	0.0500869
$966$	0	0
$967$	51246.2	1.70421	0.852103	−	0.523374i	$-0.175327\pi$
$967$	51246.2	1.70421	0.852103	+	0.523374i	$0.175327\pi$
$968$	−61502.6	−2.04212
$969$	0	0
$970$	−48468.6	−1.60436
$971$	40446.0	1.33674	0.668370	−	0.743829i	$-0.266992\pi$
$971$	40446.0	1.33674	0.668370	+	0.743829i	$0.266992\pi$
$972$	0	0
$973$	9974.08	0.328627
$974$	−59821.2	−1.96796
$975$	0	0
$976$	−24616.8	−0.807342
$977$	21457.8	0.702656	0.351328	−	0.936253i	$-0.385730\pi$
$977$	21457.8	0.702656	0.351328	+	0.936253i	$0.385730\pi$
$978$	0	0
$979$	8381.35	0.273615
$980$	4538.26	0.147928
$981$	0	0
$982$	−31135.6	−1.01179
$983$	−8746.68	−0.283801	−0.141900	−	0.989881i	$-0.545321\pi$
$983$	−8746.68	−0.283801	−0.141900	+	0.989881i	$0.545321\pi$
$984$	0	0
$985$	−7428.18	−0.240286
$986$	−157734.	−5.09461
$987$	0	0
$988$	−7591.22	−0.244442
$989$	44255.8	1.42291
$990$	0	0
$991$	−29046.2	−0.931061	−0.465531	−	0.885032i	$-0.654137\pi$
$991$	−29046.2	−0.931061	−0.465531	+	0.885032i	$0.654137\pi$
$992$	−39842.3	−1.27520
$993$	0	0
$994$	−6373.75	−0.203383
$995$	−15202.8	−0.484382
$996$	0	0
$997$	−4656.30	−0.147910	−0.0739552	−	0.997262i	$-0.523562\pi$
$997$	−4656.30	−0.147910	−0.0739552	+	0.997262i	$0.523562\pi$
$998$	4430.52	0.140527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.a.o.1.3	yes	3
3.2	odd	2		315.4.a.n.1.1	✓	3
5.4	even	2		1575.4.a.bb.1.1		3
7.6	odd	2		2205.4.a.bl.1.3		3
15.14	odd	2		1575.4.a.be.1.3		3
21.20	even	2		2205.4.a.bk.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.n.1.1	✓	3	3.2	odd	2
315.4.a.o.1.3	yes	3	1.1	even	1	trivial
1575.4.a.bb.1.1		3	5.4	even	2
1575.4.a.be.1.3		3	15.14	odd	2
2205.4.a.bk.1.1		3	21.20	even	2
2205.4.a.bl.1.3		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+5.15010 q^{2} +18.5235 q^{4} +5.00000 q^{5} +7.00000 q^{7} +54.1971 q^{8} +25.7505 q^{10} -14.0073 q^{11} -10.5931 q^{13} +36.0507 q^{14} +130.932 q^{16} +131.801 q^{17} +38.6871 q^{19} +92.6176 q^{20} -72.1391 q^{22} -102.481 q^{23} +25.0000 q^{25} -54.5553 q^{26} +129.665 q^{28} -232.376 q^{29} -165.501 q^{31} +240.738 q^{32} +678.789 q^{34} +35.0000 q^{35} +280.004 q^{37} +199.242 q^{38} +270.986 q^{40} -122.271 q^{41} -431.844 q^{43} -259.465 q^{44} -527.787 q^{46} +295.692 q^{47} +49.0000 q^{49} +128.752 q^{50} -196.221 q^{52} +243.639 q^{53} -70.0366 q^{55} +379.380 q^{56} -1196.76 q^{58} +566.480 q^{59} -188.012 q^{61} -852.346 q^{62} +192.364 q^{64} -52.9653 q^{65} -871.647 q^{67} +2441.42 q^{68} +180.253 q^{70} -176.800 q^{71} -220.524 q^{73} +1442.05 q^{74} +716.621 q^{76} -98.0513 q^{77} -190.494 q^{79} +654.662 q^{80} -629.710 q^{82} +518.241 q^{83} +659.006 q^{85} -2224.04 q^{86} -759.157 q^{88} -598.355 q^{89} -74.1515 q^{91} -1898.31 q^{92} +1522.85 q^{94} +193.436 q^{95} -1882.24 q^{97} +252.355 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 14 q^{4} + 15 q^{5} + 21 q^{7} + 66 q^{8} + 10 q^{10} - 20 q^{11} + 14 q^{14} + 114 q^{16} + 234 q^{17} - 82 q^{19} + 70 q^{20} - 236 q^{22} + 30 q^{23} + 75 q^{25} + 76 q^{26} + 98 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 14 * q^4 + 15 * q^5 + 21 * q^7 + 66 * q^8 + 10 * q^10 - 20 * q^11 + 14 * q^14 + 114 * q^16 + 234 * q^17 - 82 * q^19 + 70 * q^20 - 236 * q^22 + 30 * q^23 + 75 * q^25 + 76 * q^26 + 98 * q^28 - 32 * q^29 - 362 * q^31 + 430 * q^32 + 596 * q^34 + 105 * q^35 + 302 * q^37 + 160 * q^38 + 330 * q^40 + 626 * q^41 - 68 * q^43 + 300 * q^44 - 412 * q^46 + 936 * q^47 + 147 * q^49 + 50 * q^50 - 684 * q^52 + 298 * q^53 - 100 * q^55 + 462 * q^56 - 1184 * q^58 + 428 * q^59 + 1050 * q^61 - 608 * q^62 - 154 * q^64 - 52 * q^67 + 1964 * q^68 + 70 * q^70 + 416 * q^71 - 312 * q^73 + 1668 * q^74 + 1712 * q^76 - 140 * q^77 - 488 * q^79 + 570 * q^80 - 1476 * q^82 + 948 * q^83 + 1170 * q^85 - 1928 * q^86 + 124 * q^88 - 142 * q^89 - 3220 * q^92 + 328 * q^94 - 410 * q^95 - 3052 * q^97 + 98 * q^98

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