LMFDB - Embedded newform 531.4.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-4.21744$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.21744 q^{2} +9.78684 q^{4} -17.5635 q^{5} +14.8996 q^{7} +7.53588 q^{8} +O(q^{10})$	\(q+4.21744 q^{2} +9.78684 q^{4} -17.5635 q^{5} +14.8996 q^{7} +7.53588 q^{8} -74.0731 q^{10} +18.9001 q^{11} +31.7591 q^{13} +62.8381 q^{14} -46.5125 q^{16} +84.1910 q^{17} +126.537 q^{19} -171.891 q^{20} +79.7102 q^{22} +21.1777 q^{23} +183.477 q^{25} +133.942 q^{26} +145.820 q^{28} +12.6033 q^{29} +215.817 q^{31} -256.451 q^{32} +355.071 q^{34} -261.689 q^{35} -3.62282 q^{37} +533.661 q^{38} -132.357 q^{40} +381.728 q^{41} +168.813 q^{43} +184.972 q^{44} +89.3156 q^{46} -613.793 q^{47} -121.003 q^{49} +773.804 q^{50} +310.821 q^{52} -270.545 q^{53} -331.952 q^{55} +112.281 q^{56} +53.1537 q^{58} +59.0000 q^{59} +311.206 q^{61} +910.195 q^{62} -709.468 q^{64} -557.802 q^{65} +375.157 q^{67} +823.963 q^{68} -1103.66 q^{70} +987.919 q^{71} -359.475 q^{73} -15.2790 q^{74} +1238.39 q^{76} +281.604 q^{77} +933.998 q^{79} +816.923 q^{80} +1609.92 q^{82} -1262.64 q^{83} -1478.69 q^{85} +711.960 q^{86} +142.429 q^{88} -273.294 q^{89} +473.197 q^{91} +207.262 q^{92} -2588.64 q^{94} -2222.43 q^{95} -897.200 q^{97} -510.323 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	4.21744	1.49109	0.745546	−	0.666454i	$-0.232189\pi$
$2$	4.21744	1.49109	0.745546	+	0.666454i	$0.232189\pi$
$3$	0	0
$3$	0	0
$4$	9.78684	1.22335
$5$	−17.5635	−1.57093	−0.785464	−	0.618907i	$-0.787576\pi$
$5$	−17.5635	−1.57093	−0.785464	+	0.618907i	$0.787576\pi$
$6$	0	0
$7$	14.8996	0.804501	0.402251	−	0.915530i	$-0.368228\pi$
$7$	14.8996	0.804501	0.402251	+	0.915530i	$0.368228\pi$
$8$	7.53588	0.333042
$9$	0	0
$10$	−74.0731	−2.34240
$11$	18.9001	0.518054	0.259027	−	0.965870i	$-0.416598\pi$
$11$	18.9001	0.518054	0.259027	+	0.965870i	$0.416598\pi$
$12$	0	0
$13$	31.7591	0.677569	0.338784	−	0.940864i	$-0.389984\pi$
$13$	31.7591	0.677569	0.338784	+	0.940864i	$0.389984\pi$
$14$	62.8381	1.19958
$15$	0	0
$16$	−46.5125	−0.726758
$17$	84.1910	1.20114	0.600568	−	0.799574i	$-0.294941\pi$
$17$	84.1910	1.20114	0.600568	+	0.799574i	$0.294941\pi$
$18$	0	0
$19$	126.537	1.52787	0.763934	−	0.645294i	$-0.223265\pi$
$19$	126.537	1.52787	0.763934	+	0.645294i	$0.223265\pi$
$20$	−171.891	−1.92180
$21$	0	0
$22$	79.7102	0.772467
$23$	21.1777	0.191993	0.0959967	−	0.995382i	$-0.469396\pi$
$23$	21.1777	0.191993	0.0959967	+	0.995382i	$0.469396\pi$
$24$	0	0
$25$	183.477	1.46782
$26$	133.942	1.01032
$27$	0	0
$28$	145.820	0.984190
$29$	12.6033	0.0807025	0.0403512	−	0.999186i	$-0.487152\pi$
$29$	12.6033	0.0807025	0.0403512	+	0.999186i	$0.487152\pi$
$30$	0	0
$31$	215.817	1.25038	0.625191	−	0.780472i	$-0.285021\pi$
$31$	215.817	1.25038	0.625191	+	0.780472i	$0.285021\pi$
$32$	−256.451	−1.41671
$33$	0	0
$34$	355.071	1.79100
$35$	−261.689	−1.26381
$36$	0	0
$37$	−3.62282	−0.0160970	−0.00804848	−	0.999968i	$-0.502562\pi$
$37$	−3.62282	−0.0160970	−0.00804848	+	0.999968i	$0.502562\pi$
$38$	533.661	2.27819
$39$	0	0
$40$	−132.357	−0.523185
$41$	381.728	1.45405	0.727023	−	0.686613i	$-0.240903\pi$
$41$	381.728	1.45405	0.727023	+	0.686613i	$0.240903\pi$
$42$	0	0
$43$	168.813	0.598692	0.299346	−	0.954145i	$-0.403231\pi$
$43$	168.813	0.598692	0.299346	+	0.954145i	$0.403231\pi$
$44$	184.972	0.633764
$45$	0	0
$46$	89.3156	0.286280
$47$	−613.793	−1.90491	−0.952457	−	0.304674i	$-0.901452\pi$
$47$	−613.793	−1.90491	−0.952457	+	0.304674i	$0.901452\pi$
$48$	0	0
$49$	−121.003	−0.352778
$50$	773.804	2.18865
$51$	0	0
$52$	310.821	0.828907
$53$	−270.545	−0.701173	−0.350587	−	0.936530i	$-0.614018\pi$
$53$	−270.545	−0.701173	−0.350587	+	0.936530i	$0.614018\pi$
$54$	0	0
$55$	−331.952	−0.813826
$56$	112.281	0.267933
$57$	0	0
$58$	53.1537	0.120335
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	311.206	0.653211	0.326605	−	0.945161i	$-0.394095\pi$
$61$	311.206	0.653211	0.326605	+	0.945161i	$0.394095\pi$
$62$	910.195	1.86443
$63$	0	0
$64$	−709.468	−1.38568
$65$	−557.802	−1.06441
$66$	0	0
$67$	375.157	0.684071	0.342035	−	0.939687i	$-0.388884\pi$
$67$	375.157	0.684071	0.342035	+	0.939687i	$0.388884\pi$
$68$	823.963	1.46942
$69$	0	0
$70$	−1103.66	−1.88446
$71$	987.919	1.65133	0.825665	−	0.564160i	$-0.190800\pi$
$71$	987.919	1.65133	0.825665	+	0.564160i	$0.190800\pi$
$72$	0	0
$73$	−359.475	−0.576348	−0.288174	−	0.957578i	$-0.593048\pi$
$73$	−359.475	−0.576348	−0.288174	+	0.957578i	$0.593048\pi$
$74$	−15.2790	−0.0240020
$75$	0	0
$76$	1238.39	1.86912
$77$	281.604	0.416775
$78$	0	0
$79$	933.998	1.33016	0.665082	−	0.746770i	$-0.268397\pi$
$79$	933.998	1.33016	0.665082	+	0.746770i	$0.268397\pi$
$80$	816.923	1.14169
$81$	0	0
$82$	1609.92	2.16811
$83$	−1262.64	−1.66979	−0.834893	−	0.550412i	$-0.814471\pi$
$83$	−1262.64	−1.66979	−0.834893	+	0.550412i	$0.814471\pi$
$84$	0	0
$85$	−1478.69	−1.88690
$86$	711.960	0.892705
$87$	0	0
$88$	142.429	0.172534
$89$	−273.294	−0.325496	−0.162748	−	0.986668i	$-0.552036\pi$
$89$	−273.294	−0.325496	−0.162748	+	0.986668i	$0.552036\pi$
$90$	0	0
$91$	473.197	0.545105
$92$	207.262	0.234876
$93$	0	0
$94$	−2588.64	−2.84040
$95$	−2222.43	−2.40017
$96$	0	0
$97$	−897.200	−0.939143	−0.469571	−	0.882895i	$-0.655592\pi$
$97$	−897.200	−0.939143	−0.469571	+	0.882895i	$0.655592\pi$
$98$	−510.323	−0.526024
$99$	0	0
$100$	1795.66	1.79566
$101$	−296.784	−0.292387	−0.146194	−	0.989256i	$-0.546702\pi$
$101$	−296.784	−0.292387	−0.146194	+	0.989256i	$0.546702\pi$
$102$	0	0
$103$	−36.8226	−0.0352257	−0.0176128	−	0.999845i	$-0.505607\pi$
$103$	−36.8226	−0.0352257	−0.0176128	+	0.999845i	$0.505607\pi$
$104$	239.333	0.225659
$105$	0	0
$106$	−1141.01	−1.04551
$107$	479.349	0.433088	0.216544	−	0.976273i	$-0.430521\pi$
$107$	479.349	0.433088	0.216544	+	0.976273i	$0.430521\pi$
$108$	0	0
$109$	946.780	0.831973	0.415987	−	0.909371i	$-0.363436\pi$
$109$	946.780	0.831973	0.415987	+	0.909371i	$0.363436\pi$
$110$	−1399.99	−1.21349
$111$	0	0
$112$	−693.017	−0.584678
$113$	−2131.61	−1.77456	−0.887279	−	0.461234i	$-0.847407\pi$
$113$	−2131.61	−1.77456	−0.887279	+	0.461234i	$0.847407\pi$
$114$	0	0
$115$	−371.954	−0.301608
$116$	123.346	0.0987277
$117$	0	0
$118$	248.829	0.194124
$119$	1254.41	0.966315
$120$	0	0
$121$	−973.786	−0.731620
$122$	1312.49	0.973997
$123$	0	0
$124$	2112.16	1.52966
$125$	−1027.06	−0.734904
$126$	0	0
$127$	2054.51	1.43550	0.717751	−	0.696300i	$-0.245172\pi$
$127$	2054.51	1.43550	0.717751	+	0.696300i	$0.245172\pi$
$128$	−940.532	−0.649469
$129$	0	0
$130$	−2352.50	−1.58714
$131$	602.309	0.401710	0.200855	−	0.979621i	$-0.435628\pi$
$131$	602.309	0.401710	0.200855	+	0.979621i	$0.435628\pi$
$132$	0	0
$133$	1885.34	1.22917
$134$	1582.20	1.02001
$135$	0	0
$136$	634.453	0.400029
$137$	−1919.26	−1.19689	−0.598444	−	0.801164i	$-0.704214\pi$
$137$	−1919.26	−1.19689	−0.598444	+	0.801164i	$0.704214\pi$
$138$	0	0
$139$	−448.518	−0.273689	−0.136845	−	0.990593i	$-0.543696\pi$
$139$	−448.518	−0.273689	−0.136845	+	0.990593i	$0.543696\pi$
$140$	−2561.10	−1.54609
$141$	0	0
$142$	4166.49	2.46228
$143$	600.251	0.351017
$144$	0	0
$145$	−221.358	−0.126778
$146$	−1516.07	−0.859388
$147$	0	0
$148$	−35.4559	−0.0196923
$149$	1613.45	0.887105	0.443553	−	0.896248i	$-0.353718\pi$
$149$	1613.45	0.887105	0.443553	+	0.896248i	$0.353718\pi$
$150$	0	0
$151$	−1522.22	−0.820374	−0.410187	−	0.912002i	$-0.634537\pi$
$151$	−1522.22	−0.820374	−0.410187	+	0.912002i	$0.634537\pi$
$152$	953.565	0.508844
$153$	0	0
$154$	1187.65	0.621450
$155$	−3790.50	−1.96426
$156$	0	0
$157$	−79.4233	−0.0403737	−0.0201868	−	0.999796i	$-0.506426\pi$
$157$	−79.4233	−0.0403737	−0.0201868	+	0.999796i	$0.506426\pi$
$158$	3939.08	1.98340
$159$	0	0
$160$	4504.18	2.22554
$161$	315.538	0.154459
$162$	0	0
$163$	2254.35	1.08328	0.541639	−	0.840611i	$-0.317804\pi$
$163$	2254.35	1.08328	0.541639	+	0.840611i	$0.317804\pi$
$164$	3735.91	1.77881
$165$	0	0
$166$	−5325.10	−2.48981
$167$	382.224	0.177110	0.0885550	−	0.996071i	$-0.471775\pi$
$167$	382.224	0.177110	0.0885550	+	0.996071i	$0.471775\pi$
$168$	0	0
$169$	−1188.36	−0.540901
$170$	−6236.29	−2.81354
$171$	0	0
$172$	1652.15	0.732413
$173$	929.824	0.408631	0.204316	−	0.978905i	$-0.434503\pi$
$173$	929.824	0.408631	0.204316	+	0.978905i	$0.434503\pi$
$174$	0	0
$175$	2733.73	1.18086
$176$	−879.092	−0.376500
$177$	0	0
$178$	−1152.60	−0.485344
$179$	509.106	0.212583	0.106291	−	0.994335i	$-0.466102\pi$
$179$	509.106	0.212583	0.106291	+	0.994335i	$0.466102\pi$
$180$	0	0
$181$	824.865	0.338739	0.169369	−	0.985553i	$-0.445827\pi$
$181$	824.865	0.338739	0.169369	+	0.985553i	$0.445827\pi$
$182$	1995.68	0.812801
$183$	0	0
$184$	159.592	0.0639419
$185$	63.6294	0.0252872
$186$	0	0
$187$	1591.22	0.622254
$188$	−6007.09	−2.33038
$189$	0	0
$190$	−9372.97	−3.57888
$191$	3681.09	1.39453	0.697263	−	0.716815i	$-0.254401\pi$
$191$	3681.09	1.39453	0.697263	+	0.716815i	$0.254401\pi$
$192$	0	0
$193$	−3051.17	−1.13797	−0.568985	−	0.822348i	$-0.692664\pi$
$193$	−3051.17	−1.13797	−0.568985	+	0.822348i	$0.692664\pi$
$194$	−3783.89	−1.40035
$195$	0	0
$196$	−1184.23	−0.431573
$197$	5332.57	1.92858	0.964290	−	0.264849i	$-0.0853221\pi$
$197$	5332.57	1.92858	0.964290	+	0.264849i	$0.0853221\pi$
$198$	0	0
$199$	−5371.72	−1.91352	−0.956761	−	0.290874i	$-0.906054\pi$
$199$	−5371.72	−1.91352	−0.956761	+	0.290874i	$0.906054\pi$
$200$	1382.66	0.488844
$201$	0	0
$202$	−1251.67	−0.435976
$203$	187.784	0.0649252
$204$	0	0
$205$	−6704.48	−2.28420
$206$	−155.297	−0.0525247
$207$	0	0
$208$	−1477.20	−0.492429
$209$	2391.56	0.791519
$210$	0	0
$211$	1896.92	0.618907	0.309454	−	0.950915i	$-0.399854\pi$
$211$	1896.92	0.618907	0.309454	+	0.950915i	$0.399854\pi$
$212$	−2647.78	−0.857784
$213$	0	0
$214$	2021.63	0.645775
$215$	−2964.95	−0.940503
$216$	0	0
$217$	3215.57	1.00593
$218$	3992.99	1.24055
$219$	0	0
$220$	−3248.76	−0.995598
$221$	2673.83	0.813852
$222$	0	0
$223$	−2808.85	−0.843472	−0.421736	−	0.906719i	$-0.638579\pi$
$223$	−2808.85	−0.843472	−0.421736	+	0.906719i	$0.638579\pi$
$224$	−3821.01	−1.13974
$225$	0	0
$226$	−8989.95	−2.64603
$227$	−3302.83	−0.965711	−0.482856	−	0.875700i	$-0.660400\pi$
$227$	−3302.83	−0.965711	−0.482856	+	0.875700i	$0.660400\pi$
$228$	0	0
$229$	−5780.19	−1.66797	−0.833987	−	0.551785i	$-0.813947\pi$
$229$	−5780.19	−1.66797	−0.833987	+	0.551785i	$0.813947\pi$
$230$	−1568.70	−0.449725
$231$	0	0
$232$	94.9769	0.0268773
$233$	−4892.84	−1.37571	−0.687856	−	0.725847i	$-0.741448\pi$
$233$	−4892.84	−1.37571	−0.687856	+	0.725847i	$0.741448\pi$
$234$	0	0
$235$	10780.4	2.99248
$236$	577.423	0.159267
$237$	0	0
$238$	5290.40	1.44086
$239$	6461.39	1.74876	0.874378	−	0.485246i	$-0.161270\pi$
$239$	6461.39	1.74876	0.874378	+	0.485246i	$0.161270\pi$
$240$	0	0
$241$	−60.7729	−0.0162437	−0.00812184	−	0.999967i	$-0.502585\pi$
$241$	−60.7729	−0.0162437	−0.00812184	+	0.999967i	$0.502585\pi$
$242$	−4106.89	−1.09091
$243$	0	0
$244$	3045.72	0.799108
$245$	2125.23	0.554189
$246$	0	0
$247$	4018.69	1.03524
$248$	1626.37	0.416429
$249$	0	0
$250$	−4331.57	−1.09581
$251$	3782.13	0.951100	0.475550	−	0.879689i	$-0.342249\pi$
$251$	3782.13	0.951100	0.475550	+	0.879689i	$0.342249\pi$
$252$	0	0
$253$	400.260	0.0994630
$254$	8664.80	2.14046
$255$	0	0
$256$	1709.10	0.417261
$257$	72.7427	0.0176559	0.00882795	−	0.999961i	$-0.497190\pi$
$257$	72.7427	0.0176559	0.00882795	+	0.999961i	$0.497190\pi$
$258$	0	0
$259$	−53.9784	−0.0129500
$260$	−5459.11	−1.30215
$261$	0	0
$262$	2540.20	0.598986
$263$	4787.49	1.12247	0.561235	−	0.827657i	$-0.310327\pi$
$263$	4787.49	1.12247	0.561235	+	0.827657i	$0.310327\pi$
$264$	0	0
$265$	4751.72	1.10149
$266$	7951.32	1.83281
$267$	0	0
$268$	3671.60	0.836861
$269$	−3320.18	−0.752546	−0.376273	−	0.926509i	$-0.622795\pi$
$269$	−3320.18	−0.752546	−0.376273	+	0.926509i	$0.622795\pi$
$270$	0	0
$271$	7147.95	1.60224	0.801120	−	0.598504i	$-0.204238\pi$
$271$	7147.95	1.60224	0.801120	+	0.598504i	$0.204238\pi$
$272$	−3915.94	−0.872936
$273$	0	0
$274$	−8094.39	−1.78467
$275$	3467.73	0.760408
$276$	0	0
$277$	4611.19	1.00021	0.500107	−	0.865963i	$-0.333294\pi$
$277$	4611.19	1.00021	0.500107	+	0.865963i	$0.333294\pi$
$278$	−1891.60	−0.408096
$279$	0	0
$280$	−1972.05	−0.420903
$281$	−8307.67	−1.76368	−0.881840	−	0.471548i	$-0.843695\pi$
$281$	−8307.67	−1.76368	−0.881840	+	0.471548i	$0.843695\pi$
$282$	0	0
$283$	−8181.70	−1.71856	−0.859279	−	0.511507i	$-0.829087\pi$
$283$	−8181.70	−1.71856	−0.859279	+	0.511507i	$0.829087\pi$
$284$	9668.60	2.02016
$285$	0	0
$286$	2531.52	0.523399
$287$	5687.58	1.16978
$288$	0	0
$289$	2175.12	0.442728
$290$	−933.565	−0.189037
$291$	0	0
$292$	−3518.13	−0.705078
$293$	264.862	0.0528102	0.0264051	−	0.999651i	$-0.491594\pi$
$293$	264.862	0.0528102	0.0264051	+	0.999651i	$0.491594\pi$
$294$	0	0
$295$	−1036.25	−0.204517
$296$	−27.3011	−0.00536096
$297$	0	0
$298$	6804.62	1.32276
$299$	672.584	0.130089
$300$	0	0
$301$	2515.24	0.481649
$302$	−6419.87	−1.22325
$303$	0	0
$304$	−5885.54	−1.11039
$305$	−5465.87	−1.02615
$306$	0	0
$307$	4379.69	0.814209	0.407104	−	0.913382i	$-0.366539\pi$
$307$	4379.69	0.814209	0.407104	+	0.913382i	$0.366539\pi$
$308$	2756.01	0.509864
$309$	0	0
$310$	−15986.2	−2.92889
$311$	−10544.6	−1.92261	−0.961303	−	0.275494i	$-0.911159\pi$
$311$	−10544.6	−1.92261	−0.961303	+	0.275494i	$0.911159\pi$
$312$	0	0
$313$	−6496.71	−1.17321	−0.586607	−	0.809872i	$-0.699537\pi$
$313$	−6496.71	−1.17321	−0.586607	+	0.809872i	$0.699537\pi$
$314$	−334.963	−0.0602009
$315$	0	0
$316$	9140.88	1.62726
$317$	−2628.81	−0.465769	−0.232884	−	0.972504i	$-0.574816\pi$
$317$	−2628.81	−0.465769	−0.232884	+	0.972504i	$0.574816\pi$
$318$	0	0
$319$	238.204	0.0418083
$320$	12460.7	2.17680
$321$	0	0
$322$	1330.76	0.230312
$323$	10653.2	1.83518
$324$	0	0
$325$	5827.06	0.994546
$326$	9507.59	1.61527
$327$	0	0
$328$	2876.65	0.484258
$329$	−9145.25	−1.53250
$330$	0	0
$331$	−3679.05	−0.610933	−0.305466	−	0.952203i	$-0.598812\pi$
$331$	−3679.05	−0.610933	−0.305466	+	0.952203i	$0.598812\pi$
$332$	−12357.2	−2.04274
$333$	0	0
$334$	1612.01	0.264087
$335$	−6589.08	−1.07463
$336$	0	0
$337$	1323.74	0.213972	0.106986	−	0.994261i	$-0.465880\pi$
$337$	1323.74	0.213972	0.106986	+	0.994261i	$0.465880\pi$
$338$	−5011.84	−0.806532
$339$	0	0
$340$	−14471.7	−2.30835
$341$	4078.96	0.647765
$342$	0	0
$343$	−6913.44	−1.08831
$344$	1272.16	0.199390
$345$	0	0
$346$	3921.48	0.609307
$347$	−5541.30	−0.857270	−0.428635	−	0.903478i	$-0.641005\pi$
$347$	−5541.30	−0.857270	−0.428635	+	0.903478i	$0.641005\pi$
$348$	0	0
$349$	−6590.45	−1.01083	−0.505414	−	0.862877i	$-0.668660\pi$
$349$	−6590.45	−1.01083	−0.505414	+	0.862877i	$0.668660\pi$
$350$	11529.3	1.76077
$351$	0	0
$352$	−4846.95	−0.733930
$353$	5633.11	0.849349	0.424674	−	0.905346i	$-0.360389\pi$
$353$	5633.11	0.849349	0.424674	+	0.905346i	$0.360389\pi$
$354$	0	0
$355$	−17351.3	−2.59412
$356$	−2674.69	−0.398197
$357$	0	0
$358$	2147.13	0.316981
$359$	−2672.65	−0.392916	−0.196458	−	0.980512i	$-0.562944\pi$
$359$	−2672.65	−0.392916	−0.196458	+	0.980512i	$0.562944\pi$
$360$	0	0
$361$	9152.52	1.33438
$362$	3478.82	0.505091
$363$	0	0
$364$	4631.10	0.666856
$365$	6313.65	0.905402
$366$	0	0
$367$	−2881.92	−0.409904	−0.204952	−	0.978772i	$-0.565704\pi$
$367$	−2881.92	−0.409904	−0.204952	+	0.978772i	$0.565704\pi$
$368$	−985.027	−0.139533
$369$	0	0
$370$	268.353	0.0377055
$371$	−4031.00	−0.564095
$372$	0	0
$373$	7349.59	1.02023	0.510117	−	0.860105i	$-0.329602\pi$
$373$	7349.59	1.02023	0.510117	+	0.860105i	$0.329602\pi$
$374$	6710.88	0.927837
$375$	0	0
$376$	−4625.47	−0.634416
$377$	400.269	0.0546815
$378$	0	0
$379$	6001.29	0.813366	0.406683	−	0.913569i	$-0.366685\pi$
$379$	6001.29	0.813366	0.406683	+	0.913569i	$0.366685\pi$
$380$	−21750.5	−2.93626
$381$	0	0
$382$	15524.8	2.07937
$383$	7844.21	1.04653	0.523264	−	0.852171i	$-0.324714\pi$
$383$	7844.21	1.04653	0.523264	+	0.852171i	$0.324714\pi$
$384$	0	0
$385$	−4945.95	−0.654724
$386$	−12868.2	−1.69682
$387$	0	0
$388$	−8780.75	−1.14890
$389$	574.603	0.0748934	0.0374467	−	0.999299i	$-0.488078\pi$
$389$	574.603	0.0748934	0.0374467	+	0.999299i	$0.488078\pi$
$390$	0	0
$391$	1782.97	0.230610
$392$	−911.863	−0.117490
$393$	0	0
$394$	22489.8	2.87569
$395$	−16404.3	−2.08959
$396$	0	0
$397$	−14329.0	−1.81147	−0.905735	−	0.423844i	$-0.860680\pi$
$397$	−14329.0	−1.81147	−0.905735	+	0.423844i	$0.860680\pi$
$398$	−22654.9	−2.85324
$399$	0	0
$400$	−8533.98	−1.06675
$401$	−8440.51	−1.05112	−0.525560	−	0.850757i	$-0.676144\pi$
$401$	−8440.51	−1.05112	−0.525560	+	0.850757i	$0.676144\pi$
$402$	0	0
$403$	6854.15	0.847219
$404$	−2904.57	−0.357693
$405$	0	0
$406$	791.967	0.0968095
$407$	−68.4716	−0.00833910
$408$	0	0
$409$	−3995.02	−0.482985	−0.241492	−	0.970403i	$-0.577637\pi$
$409$	−3995.02	−0.482985	−0.241492	+	0.970403i	$0.577637\pi$
$410$	−28275.8	−3.40595
$411$	0	0
$412$	−360.377	−0.0430935
$413$	879.075	0.104737
$414$	0	0
$415$	22176.3	2.62312
$416$	−8144.66	−0.959915
$417$	0	0
$418$	10086.3	1.18023
$419$	1885.71	0.219864	0.109932	−	0.993939i	$-0.464937\pi$
$419$	1885.71	0.219864	0.109932	+	0.993939i	$0.464937\pi$
$420$	0	0
$421$	−9268.38	−1.07295	−0.536477	−	0.843915i	$-0.680245\pi$
$421$	−9268.38	−1.07295	−0.536477	+	0.843915i	$0.680245\pi$
$422$	8000.16	0.922848
$423$	0	0
$424$	−2038.79	−0.233520
$425$	15447.1	1.76305
$426$	0	0
$427$	4636.84	0.525509
$428$	4691.31	0.529821
$429$	0	0
$430$	−12504.5	−1.40238
$431$	3346.11	0.373959	0.186980	−	0.982364i	$-0.440130\pi$
$431$	3346.11	0.373959	0.186980	+	0.982364i	$0.440130\pi$
$432$	0	0
$433$	−927.848	−0.102978	−0.0514891	−	0.998674i	$-0.516397\pi$
$433$	−927.848	−0.102978	−0.0514891	+	0.998674i	$0.516397\pi$
$434$	13561.5	1.49994
$435$	0	0
$436$	9265.98	1.01780
$437$	2679.75	0.293341
$438$	0	0
$439$	10124.5	1.10072	0.550361	−	0.834927i	$-0.314490\pi$
$439$	10124.5	1.10072	0.550361	+	0.834927i	$0.314490\pi$
$440$	−2501.55	−0.271038
$441$	0	0
$442$	11276.7	1.21353
$443$	6492.44	0.696310	0.348155	−	0.937437i	$-0.386808\pi$
$443$	6492.44	0.696310	0.348155	+	0.937437i	$0.386808\pi$
$444$	0	0
$445$	4800.01	0.511331
$446$	−11846.2	−1.25769
$447$	0	0
$448$	−10570.8	−1.11478
$449$	17708.0	1.86123	0.930614	−	0.366003i	$-0.119274\pi$
$449$	17708.0	1.86123	0.930614	+	0.366003i	$0.119274\pi$
$450$	0	0
$451$	7214.70	0.753275
$452$	−20861.7	−2.17091
$453$	0	0
$454$	−13929.5	−1.43996
$455$	−8311.00	−0.856321
$456$	0	0
$457$	10785.3	1.10397	0.551984	−	0.833855i	$-0.313871\pi$
$457$	10785.3	1.10397	0.551984	+	0.833855i	$0.313871\pi$
$458$	−24377.6	−2.48710
$459$	0	0
$460$	−3640.25	−0.368973
$461$	14512.5	1.46619	0.733097	−	0.680124i	$-0.238074\pi$
$461$	14512.5	1.46619	0.733097	+	0.680124i	$0.238074\pi$
$462$	0	0
$463$	−9637.81	−0.967402	−0.483701	−	0.875233i	$-0.660708\pi$
$463$	−9637.81	−0.967402	−0.483701	+	0.875233i	$0.660708\pi$
$464$	−586.211	−0.0586512
$465$	0	0
$466$	−20635.3	−2.05131
$467$	7260.33	0.719417	0.359709	−	0.933065i	$-0.382876\pi$
$467$	7260.33	0.719417	0.359709	+	0.933065i	$0.382876\pi$
$468$	0	0
$469$	5589.68	0.550336
$470$	45465.6	4.46207
$471$	0	0
$472$	444.617	0.0433584
$473$	3190.59	0.310155
$474$	0	0
$475$	23216.6	2.24263
$476$	12276.7	1.18215
$477$	0	0
$478$	27250.6	2.60756
$479$	−12069.4	−1.15128	−0.575641	−	0.817703i	$-0.695247\pi$
$479$	−12069.4	−1.15128	−0.575641	+	0.817703i	$0.695247\pi$
$480$	0	0
$481$	−115.057	−0.0109068
$482$	−256.306	−0.0242208
$483$	0	0
$484$	−9530.28	−0.895030
$485$	15758.0	1.47533
$486$	0	0
$487$	−6015.96	−0.559773	−0.279886	−	0.960033i	$-0.590297\pi$
$487$	−6015.96	−0.559773	−0.279886	+	0.960033i	$0.590297\pi$
$488$	2345.21	0.217547
$489$	0	0
$490$	8963.06	0.826346
$491$	−4022.81	−0.369749	−0.184875	−	0.982762i	$-0.559188\pi$
$491$	−4022.81	−0.369749	−0.184875	+	0.982762i	$0.559188\pi$
$492$	0	0
$493$	1061.08	0.0969347
$494$	16948.6	1.54363
$495$	0	0
$496$	−10038.2	−0.908725
$497$	14719.6	1.32850
$498$	0	0
$499$	−14112.8	−1.26608	−0.633042	−	0.774117i	$-0.718194\pi$
$499$	−14112.8	−1.26608	−0.633042	+	0.774117i	$0.718194\pi$
$500$	−10051.7	−0.899048
$501$	0	0
$502$	15950.9	1.41818
$503$	−689.493	−0.0611192	−0.0305596	−	0.999533i	$-0.509729\pi$
$503$	−689.493	−0.0611192	−0.0305596	+	0.999533i	$0.509729\pi$
$504$	0	0
$505$	5212.56	0.459319
$506$	1688.07	0.148308
$507$	0	0
$508$	20107.2	1.75613
$509$	12620.3	1.09899	0.549495	−	0.835497i	$-0.314820\pi$
$509$	12620.3	1.09899	0.549495	+	0.835497i	$0.314820\pi$
$510$	0	0
$511$	−5356.03	−0.463673
$512$	14732.3	1.27164
$513$	0	0
$514$	306.788	0.0263266
$515$	646.735	0.0553370
$516$	0	0
$517$	−11600.8	−0.986849
$518$	−227.651	−0.0193097
$519$	0	0
$520$	−4203.53	−0.354494
$521$	−2186.17	−0.183835	−0.0919174	−	0.995767i	$-0.529300\pi$
$521$	−2186.17	−0.183835	−0.0919174	+	0.995767i	$0.529300\pi$
$522$	0	0
$523$	−18639.4	−1.55840	−0.779202	−	0.626773i	$-0.784375\pi$
$523$	−18639.4	−1.55840	−0.779202	+	0.626773i	$0.784375\pi$
$524$	5894.70	0.491433
$525$	0	0
$526$	20191.0	1.67370
$527$	18169.8	1.50188
$528$	0	0
$529$	−11718.5	−0.963139
$530$	20040.1	1.64243
$531$	0	0
$532$	18451.5	1.50371
$533$	12123.3	0.985216
$534$	0	0
$535$	−8419.06	−0.680351
$536$	2827.14	0.227824
$537$	0	0
$538$	−14002.7	−1.12211
$539$	−2286.97	−0.182758
$540$	0	0
$541$	−6609.92	−0.525291	−0.262646	−	0.964892i	$-0.584595\pi$
$541$	−6609.92	−0.525291	−0.262646	+	0.964892i	$0.584595\pi$
$542$	30146.1	2.38909
$543$	0	0
$544$	−21590.9	−1.70166
$545$	−16628.8	−1.30697
$546$	0	0
$547$	8883.37	0.694379	0.347190	−	0.937795i	$-0.387136\pi$
$547$	8883.37	0.694379	0.347190	+	0.937795i	$0.387136\pi$
$548$	−18783.5	−1.46422
$549$	0	0
$550$	14625.0	1.13384
$551$	1594.78	0.123303
$552$	0	0
$553$	13916.2	1.07012
$554$	19447.4	1.49141
$555$	0	0
$556$	−4389.58	−0.334819
$557$	−13797.8	−1.04961	−0.524805	−	0.851222i	$-0.675862\pi$
$557$	−13797.8	−1.04961	−0.524805	+	0.851222i	$0.675862\pi$
$558$	0	0
$559$	5361.36	0.405655
$560$	12171.8	0.918487
$561$	0	0
$562$	−35037.1	−2.62981
$563$	666.036	0.0498581	0.0249290	−	0.999689i	$-0.492064\pi$
$563$	666.036	0.0498581	0.0249290	+	0.999689i	$0.492064\pi$
$564$	0	0
$565$	37438.6	2.78770
$566$	−34505.9	−2.56253
$567$	0	0
$568$	7444.84	0.549962
$569$	18945.7	1.39586	0.697932	−	0.716164i	$-0.254104\pi$
$569$	18945.7	1.39586	0.697932	+	0.716164i	$0.254104\pi$
$570$	0	0
$571$	23781.4	1.74295	0.871473	−	0.490443i	$-0.163165\pi$
$571$	23781.4	1.74295	0.871473	+	0.490443i	$0.163165\pi$
$572$	5874.56	0.429419
$573$	0	0
$574$	23987.0	1.74425
$575$	3885.61	0.281811
$576$	0	0
$577$	12773.4	0.921601	0.460800	−	0.887504i	$-0.347562\pi$
$577$	12773.4	0.921601	0.460800	+	0.887504i	$0.347562\pi$
$578$	9173.45	0.660148
$579$	0	0
$580$	−2166.39	−0.155094
$581$	−18812.7	−1.34335
$582$	0	0
$583$	−5113.33	−0.363246
$584$	−2708.96	−0.191948
$585$	0	0
$586$	1117.04	0.0787448
$587$	−11287.3	−0.793658	−0.396829	−	0.917893i	$-0.629889\pi$
$587$	−11287.3	−0.793658	−0.396829	+	0.917893i	$0.629889\pi$
$588$	0	0
$589$	27308.7	1.91042
$590$	−4370.31	−0.304954
$591$	0	0
$592$	168.506	0.0116986
$593$	23122.5	1.60123	0.800614	−	0.599181i	$-0.204507\pi$
$593$	23122.5	1.60123	0.800614	+	0.599181i	$0.204507\pi$
$594$	0	0
$595$	−22031.8	−1.51801
$596$	15790.5	1.08524
$597$	0	0
$598$	2836.58	0.193974
$599$	28372.3	1.93533	0.967665	−	0.252239i	$-0.0811670\pi$
$599$	28372.3	1.93533	0.967665	+	0.252239i	$0.0811670\pi$
$600$	0	0
$601$	−15958.6	−1.08314	−0.541569	−	0.840656i	$-0.682170\pi$
$601$	−15958.6	−1.08314	−0.541569	+	0.840656i	$0.682170\pi$
$602$	10607.9	0.718182
$603$	0	0
$604$	−14897.7	−1.00361
$605$	17103.1	1.14932
$606$	0	0
$607$	12001.9	0.802540	0.401270	−	0.915960i	$-0.368569\pi$
$607$	12001.9	0.802540	0.401270	+	0.915960i	$0.368569\pi$
$608$	−32450.5	−2.16454
$609$	0	0
$610$	−23052.0	−1.53008
$611$	−19493.5	−1.29071
$612$	0	0
$613$	−17040.5	−1.12277	−0.561384	−	0.827555i	$-0.689731\pi$
$613$	−17040.5	−1.12277	−0.561384	+	0.827555i	$0.689731\pi$
$614$	18471.1	1.21406
$615$	0	0
$616$	2122.13	0.138804
$617$	23366.1	1.52461	0.762304	−	0.647219i	$-0.224068\pi$
$617$	23366.1	1.52461	0.762304	+	0.647219i	$0.224068\pi$
$618$	0	0
$619$	3475.30	0.225661	0.112831	−	0.993614i	$-0.464008\pi$
$619$	3475.30	0.225661	0.112831	+	0.993614i	$0.464008\pi$
$620$	−37097.0	−2.40298
$621$	0	0
$622$	−44471.3	−2.86678
$623$	−4071.97	−0.261862
$624$	0	0
$625$	−4895.84	−0.313334
$626$	−27399.5	−1.74937
$627$	0	0
$628$	−777.303	−0.0493913
$629$	−305.008	−0.0193346
$630$	0	0
$631$	16882.7	1.06512	0.532561	−	0.846392i	$-0.321230\pi$
$631$	16882.7	1.06512	0.532561	+	0.846392i	$0.321230\pi$
$632$	7038.49	0.443000
$633$	0	0
$634$	−11086.9	−0.694504
$635$	−36084.5	−2.25507
$636$	0	0
$637$	−3842.94	−0.239031
$638$	1004.61	0.0623400
$639$	0	0
$640$	16519.0	1.02027
$641$	−19928.1	−1.22794	−0.613971	−	0.789328i	$-0.710429\pi$
$641$	−19928.1	−1.22794	−0.613971	+	0.789328i	$0.710429\pi$
$642$	0	0
$643$	−27843.0	−1.70765	−0.853827	−	0.520556i	$-0.825725\pi$
$643$	−27843.0	−1.70765	−0.853827	+	0.520556i	$0.825725\pi$
$644$	3088.12	0.188958
$645$	0	0
$646$	44929.5	2.73642
$647$	−28838.3	−1.75232	−0.876160	−	0.482021i	$-0.839903\pi$
$647$	−28838.3	−1.75232	−0.876160	+	0.482021i	$0.839903\pi$
$648$	0	0
$649$	1115.11	0.0674449
$650$	24575.3	1.48296
$651$	0	0
$652$	22062.9	1.32523
$653$	−17697.2	−1.06056	−0.530280	−	0.847823i	$-0.677913\pi$
$653$	−17697.2	−1.06056	−0.530280	+	0.847823i	$0.677913\pi$
$654$	0	0
$655$	−10578.7	−0.631057
$656$	−17755.1	−1.05674
$657$	0	0
$658$	−38569.6	−2.28511
$659$	−21288.4	−1.25839	−0.629196	−	0.777247i	$-0.716616\pi$
$659$	−21288.4	−1.25839	−0.629196	+	0.777247i	$0.716616\pi$
$660$	0	0
$661$	−3564.16	−0.209727	−0.104864	−	0.994487i	$-0.533441\pi$
$661$	−3564.16	−0.209727	−0.104864	+	0.994487i	$0.533441\pi$
$662$	−15516.2	−0.910957
$663$	0	0
$664$	−9515.07	−0.556109
$665$	−33113.2	−1.93094
$666$	0	0
$667$	266.908	0.0154943
$668$	3740.76	0.216668
$669$	0	0
$670$	−27789.1	−1.60237
$671$	5881.83	0.338399
$672$	0	0
$673$	803.878	0.0460434	0.0230217	−	0.999735i	$-0.492671\pi$
$673$	803.878	0.0460434	0.0230217	+	0.999735i	$0.492671\pi$
$674$	5582.80	0.319053
$675$	0	0
$676$	−11630.3	−0.661713
$677$	14587.1	0.828106	0.414053	−	0.910253i	$-0.364113\pi$
$677$	14587.1	0.828106	0.414053	+	0.910253i	$0.364113\pi$
$678$	0	0
$679$	−13367.9	−0.755541
$680$	−11143.2	−0.628416
$681$	0	0
$682$	17202.8	0.965878
$683$	−19187.3	−1.07494	−0.537469	−	0.843284i	$-0.680619\pi$
$683$	−19187.3	−1.07494	−0.537469	+	0.843284i	$0.680619\pi$
$684$	0	0
$685$	33709.0	1.88023
$686$	−29157.1	−1.62277
$687$	0	0
$688$	−7851.93	−0.435105
$689$	−8592.26	−0.475093
$690$	0	0
$691$	19785.3	1.08924	0.544622	−	0.838682i	$-0.316673\pi$
$691$	19785.3	1.08924	0.544622	+	0.838682i	$0.316673\pi$
$692$	9100.03	0.499901
$693$	0	0
$694$	−23370.1	−1.27827
$695$	7877.56	0.429946
$696$	0	0
$697$	32138.0	1.74651
$698$	−27794.9	−1.50724
$699$	0	0
$700$	26754.5	1.44461
$701$	14311.7	0.771104	0.385552	−	0.922686i	$-0.374011\pi$
$701$	14311.7	0.771104	0.385552	+	0.922686i	$0.374011\pi$
$702$	0	0
$703$	−458.419	−0.0245940
$704$	−13409.0	−0.717857
$705$	0	0
$706$	23757.3	1.26646
$707$	−4421.95	−0.235226
$708$	0	0
$709$	−11747.7	−0.622277	−0.311139	−	0.950365i	$-0.600710\pi$
$709$	−11747.7	−0.622277	−0.311139	+	0.950365i	$0.600710\pi$
$710$	−73178.3	−3.86807
$711$	0	0
$712$	−2059.51	−0.108404
$713$	4570.49	0.240065
$714$	0	0
$715$	−10542.5	−0.551423
$716$	4982.53	0.260064
$717$	0	0
$718$	−11271.7	−0.585874
$719$	−1263.52	−0.0655374	−0.0327687	−	0.999463i	$-0.510432\pi$
$719$	−1263.52	−0.0655374	−0.0327687	+	0.999463i	$0.510432\pi$
$720$	0	0
$721$	−548.642	−0.0283391
$722$	38600.3	1.98969
$723$	0	0
$724$	8072.82	0.414398
$725$	2312.41	0.118456
$726$	0	0
$727$	−15608.6	−0.796275	−0.398138	−	0.917326i	$-0.630343\pi$
$727$	−15608.6	−0.796275	−0.398138	+	0.917326i	$0.630343\pi$
$728$	3565.96	0.181543
$729$	0	0
$730$	26627.5	1.35004
$731$	14212.6	0.719111
$732$	0	0
$733$	5555.30	0.279931	0.139966	−	0.990156i	$-0.455301\pi$
$733$	5555.30	0.279931	0.139966	+	0.990156i	$0.455301\pi$
$734$	−12154.3	−0.611205
$735$	0	0
$736$	−5431.03	−0.271998
$737$	7090.51	0.354386
$738$	0	0
$739$	−6833.98	−0.340179	−0.170089	−	0.985429i	$-0.554406\pi$
$739$	−6833.98	−0.340179	−0.170089	+	0.985429i	$0.554406\pi$
$740$	622.730	0.0309352
$741$	0	0
$742$	−17000.5	−0.841117
$743$	−35475.6	−1.75165	−0.875823	−	0.482633i	$-0.839681\pi$
$743$	−35475.6	−1.75165	−0.875823	+	0.482633i	$0.839681\pi$
$744$	0	0
$745$	−28337.8	−1.39358
$746$	30996.5	1.52126
$747$	0	0
$748$	15573.0	0.761237
$749$	7142.10	0.348420
$750$	0	0
$751$	−33087.6	−1.60770	−0.803850	−	0.594832i	$-0.797218\pi$
$751$	−33087.6	−1.60770	−0.803850	+	0.594832i	$0.797218\pi$
$752$	28549.1	1.38441
$753$	0	0
$754$	1688.11	0.0815351
$755$	26735.5	1.28875
$756$	0	0
$757$	−11497.2	−0.552013	−0.276007	−	0.961156i	$-0.589011\pi$
$757$	−11497.2	−0.552013	−0.276007	+	0.961156i	$0.589011\pi$
$758$	25310.1	1.21280
$759$	0	0
$760$	−16747.9	−0.799358
$761$	−7865.73	−0.374681	−0.187341	−	0.982295i	$-0.559987\pi$
$761$	−7865.73	−0.374681	−0.187341	+	0.982295i	$0.559987\pi$
$762$	0	0
$763$	14106.6	0.669323
$764$	36026.3	1.70600
$765$	0	0
$766$	33082.5	1.56047
$767$	1873.79	0.0882119
$768$	0	0
$769$	8020.94	0.376128	0.188064	−	0.982157i	$-0.439779\pi$
$769$	8020.94	0.376128	0.188064	+	0.982157i	$0.439779\pi$
$770$	−20859.3	−0.976254
$771$	0	0
$772$	−29861.3	−1.39214
$773$	17731.7	0.825053	0.412526	−	0.910946i	$-0.364646\pi$
$773$	17731.7	0.825053	0.412526	+	0.910946i	$0.364646\pi$
$774$	0	0
$775$	39597.4	1.83533
$776$	−6761.19	−0.312774
$777$	0	0
$778$	2423.36	0.111673
$779$	48302.5	2.22159
$780$	0	0
$781$	18671.8	0.855479
$782$	7519.57	0.343861
$783$	0	0
$784$	5628.15	0.256384
$785$	1394.95	0.0634242
$786$	0	0
$787$	−10866.2	−0.492170	−0.246085	−	0.969248i	$-0.579144\pi$
$787$	−10866.2	−0.492170	−0.246085	+	0.969248i	$0.579144\pi$
$788$	52189.0	2.35934
$789$	0	0
$790$	−69184.1	−3.11577
$791$	−31760.1	−1.42763
$792$	0	0
$793$	9883.63	0.442595
$794$	−60431.9	−2.70107
$795$	0	0
$796$	−52572.1	−2.34092
$797$	36334.0	1.61483	0.807414	−	0.589985i	$-0.200866\pi$
$797$	36334.0	1.61483	0.807414	+	0.589985i	$0.200866\pi$
$798$	0	0
$799$	−51675.8	−2.28806
$800$	−47052.8	−2.07946
$801$	0	0
$802$	−35597.4	−1.56731
$803$	−6794.13	−0.298580
$804$	0	0
$805$	−5541.96	−0.242644
$806$	28907.0	1.26328
$807$	0	0
$808$	−2236.53	−0.0973772
$809$	−29433.4	−1.27914	−0.639569	−	0.768734i	$-0.720887\pi$
$809$	−29433.4	−1.27914	−0.639569	+	0.768734i	$0.720887\pi$
$810$	0	0
$811$	−6295.17	−0.272569	−0.136284	−	0.990670i	$-0.543516\pi$
$811$	−6295.17	−0.272569	−0.136284	+	0.990670i	$0.543516\pi$
$812$	1837.81	0.0794266
$813$	0	0
$814$	−288.775	−0.0124344
$815$	−39594.3	−1.70175
$816$	0	0
$817$	21361.1	0.914723
$818$	−16848.8	−0.720175
$819$	0	0
$820$	−65615.6	−2.79439
$821$	−39899.7	−1.69611	−0.848057	−	0.529905i	$-0.822228\pi$
$821$	−39899.7	−1.69611	−0.848057	+	0.529905i	$0.822228\pi$
$822$	0	0
$823$	−1122.81	−0.0475560	−0.0237780	−	0.999717i	$-0.507569\pi$
$823$	−1122.81	−0.0475560	−0.0237780	+	0.999717i	$0.507569\pi$
$824$	−277.491	−0.0117316
$825$	0	0
$826$	3707.45	0.156173
$827$	11050.5	0.464649	0.232325	−	0.972638i	$-0.425367\pi$
$827$	11050.5	0.464649	0.232325	+	0.972638i	$0.425367\pi$
$828$	0	0
$829$	28472.9	1.19289	0.596444	−	0.802654i	$-0.296580\pi$
$829$	28472.9	1.19289	0.596444	+	0.802654i	$0.296580\pi$
$830$	93527.4	3.91131
$831$	0	0
$832$	−22532.1	−0.938893
$833$	−10187.3	−0.423734
$834$	0	0
$835$	−6713.19	−0.278227
$836$	23405.8	0.968308
$837$	0	0
$838$	7952.88	0.327837
$839$	−6906.38	−0.284189	−0.142095	−	0.989853i	$-0.545384\pi$
$839$	−6906.38	−0.284189	−0.142095	+	0.989853i	$0.545384\pi$
$840$	0	0
$841$	−24230.2	−0.993487
$842$	−39088.9	−1.59987
$843$	0	0
$844$	18564.9	0.757143
$845$	20871.7	0.849716
$846$	0	0
$847$	−14509.0	−0.588589
$848$	12583.7	0.509584
$849$	0	0
$850$	65147.3	2.62886
$851$	−76.7228	−0.00309051
$852$	0	0
$853$	−22035.5	−0.884504	−0.442252	−	0.896891i	$-0.645820\pi$
$853$	−22035.5	−0.884504	−0.442252	+	0.896891i	$0.645820\pi$
$854$	19555.6	0.783582
$855$	0	0
$856$	3612.32	0.144237
$857$	−39983.9	−1.59373	−0.796864	−	0.604159i	$-0.793509\pi$
$857$	−39983.9	−1.59373	−0.796864	+	0.604159i	$0.793509\pi$
$858$	0	0
$859$	−17967.4	−0.713668	−0.356834	−	0.934168i	$-0.616144\pi$
$859$	−17967.4	−0.713668	−0.356834	+	0.934168i	$0.616144\pi$
$860$	−29017.5	−1.15057
$861$	0	0
$862$	14112.0	0.557608
$863$	−42533.4	−1.67770	−0.838850	−	0.544363i	$-0.816771\pi$
$863$	−42533.4	−1.67770	−0.838850	+	0.544363i	$0.816771\pi$
$864$	0	0
$865$	−16331.0	−0.641930
$866$	−3913.15	−0.153550
$867$	0	0
$868$	31470.3	1.23061
$869$	17652.7	0.689097
$870$	0	0
$871$	11914.7	0.463505
$872$	7134.82	0.277082
$873$	0	0
$874$	11301.7	0.437398
$875$	−15302.8	−0.591231
$876$	0	0
$877$	48368.0	1.86234	0.931169	−	0.364589i	$-0.118791\pi$
$877$	48368.0	1.86234	0.931169	+	0.364589i	$0.118791\pi$
$878$	42699.6	1.64128
$879$	0	0
$880$	15439.9	0.591455
$881$	−29172.9	−1.11562	−0.557810	−	0.829969i	$-0.688358\pi$
$881$	−29172.9	−1.11562	−0.557810	+	0.829969i	$0.688358\pi$
$882$	0	0
$883$	−17067.5	−0.650471	−0.325235	−	0.945633i	$-0.605444\pi$
$883$	−17067.5	−0.650471	−0.325235	+	0.945633i	$0.605444\pi$
$884$	26168.3	0.995630
$885$	0	0
$886$	27381.5	1.03826
$887$	−19672.0	−0.744671	−0.372335	−	0.928098i	$-0.621443\pi$
$887$	−19672.0	−0.744671	−0.372335	+	0.928098i	$0.621443\pi$
$888$	0	0
$889$	30611.4	1.15486
$890$	20243.8	0.762441
$891$	0	0
$892$	−27489.7	−1.03187
$893$	−77667.3	−2.91046
$894$	0	0
$895$	−8941.68	−0.333953
$896$	−14013.5	−0.522499
$897$	0	0
$898$	74682.4	2.77526
$899$	2720.00	0.100909
$900$	0	0
$901$	−22777.4	−0.842205
$902$	30427.6	1.12320
$903$	0	0
$904$	−16063.6	−0.591002
$905$	−14487.5	−0.532134
$906$	0	0
$907$	−6730.37	−0.246393	−0.123196	−	0.992382i	$-0.539315\pi$
$907$	−6730.37	−0.246393	−0.123196	+	0.992382i	$0.539315\pi$
$908$	−32324.2	−1.18141
$909$	0	0
$910$	−35051.2	−1.27685
$911$	−5094.05	−0.185262	−0.0926308	−	0.995701i	$-0.529528\pi$
$911$	−5094.05	−0.185262	−0.0926308	+	0.995701i	$0.529528\pi$
$912$	0	0
$913$	−23864.0	−0.865040
$914$	45486.2	1.64612
$915$	0	0
$916$	−56569.8	−2.04052
$917$	8974.14	0.323176
$918$	0	0
$919$	−18873.8	−0.677464	−0.338732	−	0.940883i	$-0.609998\pi$
$919$	−18873.8	−0.677464	−0.338732	+	0.940883i	$0.609998\pi$
$920$	−2803.00	−0.100448
$921$	0	0
$922$	61205.7	2.18623
$923$	31375.4	1.11889
$924$	0	0
$925$	−664.703	−0.0236274
$926$	−40646.9	−1.44248
$927$	0	0
$928$	−3232.13	−0.114332
$929$	−42077.8	−1.48603	−0.743017	−	0.669272i	$-0.766606\pi$
$929$	−42077.8	−1.48603	−0.743017	+	0.669272i	$0.766606\pi$
$930$	0	0
$931$	−15311.3	−0.538998
$932$	−47885.4	−1.68298
$933$	0	0
$934$	30620.0	1.07272
$935$	−27947.4	−0.977516
$936$	0	0
$937$	−16791.9	−0.585452	−0.292726	−	0.956196i	$-0.594562\pi$
$937$	−16791.9	−0.585452	−0.292726	+	0.956196i	$0.594562\pi$
$938$	23574.2	0.820601
$939$	0	0
$940$	105506.	3.66087
$941$	21099.3	0.730943	0.365472	−	0.930822i	$-0.380908\pi$
$941$	21099.3	0.730943	0.365472	+	0.930822i	$0.380908\pi$
$942$	0	0
$943$	8084.10	0.279167
$944$	−2744.24	−0.0946159
$945$	0	0
$946$	13456.1	0.462470
$947$	13159.7	0.451567	0.225783	−	0.974178i	$-0.427506\pi$
$947$	13159.7	0.451567	0.225783	+	0.974178i	$0.427506\pi$
$948$	0	0
$949$	−11416.6	−0.390516
$950$	97914.5	3.34396
$951$	0	0
$952$	9453.08	0.321824
$953$	14229.9	0.483685	0.241843	−	0.970315i	$-0.422248\pi$
$953$	14229.9	0.483685	0.241843	+	0.970315i	$0.422248\pi$
$954$	0	0
$955$	−64652.9	−2.19070
$956$	63236.6	2.13935
$957$	0	0
$958$	−50901.9	−1.71667
$959$	−28596.2	−0.962898
$960$	0	0
$961$	16785.8	0.563453
$962$	−485.248	−0.0162630
$963$	0	0
$964$	−594.774	−0.0198718
$965$	53589.3	1.78767
$966$	0	0
$967$	−43103.2	−1.43341	−0.716704	−	0.697377i	$-0.754350\pi$
$967$	−43103.2	−1.43341	−0.716704	+	0.697377i	$0.754350\pi$
$968$	−7338.33	−0.243660
$969$	0	0
$970$	66458.4	2.19985
$971$	24953.4	0.824710	0.412355	−	0.911023i	$-0.364706\pi$
$971$	24953.4	0.824710	0.412355	+	0.911023i	$0.364706\pi$
$972$	0	0
$973$	−6682.73	−0.220183
$974$	−25372.0	−0.834673
$975$	0	0
$976$	−14475.0	−0.474726
$977$	7739.34	0.253432	0.126716	−	0.991939i	$-0.459556\pi$
$977$	7739.34	0.253432	0.126716	+	0.991939i	$0.459556\pi$
$978$	0	0
$979$	−5165.29	−0.168625
$980$	20799.3	0.677969
$981$	0	0
$982$	−16966.0	−0.551330
$983$	−19799.0	−0.642410	−0.321205	−	0.947010i	$-0.604088\pi$
$983$	−19799.0	−0.642410	−0.321205	+	0.947010i	$0.604088\pi$
$984$	0	0
$985$	−93658.7	−3.02966
$986$	4475.06	0.144538
$987$	0	0
$988$	39330.3	1.26646
$989$	3575.07	0.114945
$990$	0	0
$991$	10262.8	0.328968	0.164484	−	0.986380i	$-0.447404\pi$
$991$	10262.8	0.328968	0.164484	+	0.986380i	$0.447404\pi$
$992$	−55346.4	−1.77142
$993$	0	0
$994$	62079.0	1.98091
$995$	94346.2	3.00601
$996$	0	0
$997$	−18748.2	−0.595549	−0.297774	−	0.954636i	$-0.596244\pi$
$997$	−18748.2	−0.595549	−0.297774	+	0.954636i	$0.596244\pi$
$998$	−59520.0	−1.88785
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.f.1.7		8
3.2	odd	2		177.4.a.c.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.2	✓	8	3.2	odd	2
531.4.a.f.1.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+4.21744 q^{2} +9.78684 q^{4} -17.5635 q^{5} +14.8996 q^{7} +7.53588 q^{8} +O(q^{10})\)	\(q+4.21744 q^{2} +9.78684 q^{4} -17.5635 q^{5} +14.8996 q^{7} +7.53588 q^{8} -74.0731 q^{10} +18.9001 q^{11} +31.7591 q^{13} +62.8381 q^{14} -46.5125 q^{16} +84.1910 q^{17} +126.537 q^{19} -171.891 q^{20} +79.7102 q^{22} +21.1777 q^{23} +183.477 q^{25} +133.942 q^{26} +145.820 q^{28} +12.6033 q^{29} +215.817 q^{31} -256.451 q^{32} +355.071 q^{34} -261.689 q^{35} -3.62282 q^{37} +533.661 q^{38} -132.357 q^{40} +381.728 q^{41} +168.813 q^{43} +184.972 q^{44} +89.3156 q^{46} -613.793 q^{47} -121.003 q^{49} +773.804 q^{50} +310.821 q^{52} -270.545 q^{53} -331.952 q^{55} +112.281 q^{56} +53.1537 q^{58} +59.0000 q^{59} +311.206 q^{61} +910.195 q^{62} -709.468 q^{64} -557.802 q^{65} +375.157 q^{67} +823.963 q^{68} -1103.66 q^{70} +987.919 q^{71} -359.475 q^{73} -15.2790 q^{74} +1238.39 q^{76} +281.604 q^{77} +933.998 q^{79} +816.923 q^{80} +1609.92 q^{82} -1262.64 q^{83} -1478.69 q^{85} +711.960 q^{86} +142.429 q^{88} -273.294 q^{89} +473.197 q^{91} +207.262 q^{92} -2588.64 q^{94} -2222.43 q^{95} -897.200 q^{97} -510.323 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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