LMFDB - Embedded newform 575.4.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, 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("575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.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:	$33.9260982533$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 37x^{5} + 123x^{4} + 304x^{3} - 1196x^{2} + 264x + 864 $ x^7 - 3x^6 - 37x^5 + 123x^4 + 304x^3 - 1196x^2 + 264x + 864
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.711047$ of defining polynomial
Character	$\chi$	$=$	575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.711047 q^{2} -0.689108 q^{3} -7.49441 q^{4} +0.489988 q^{6} -19.0526 q^{7} +11.0173 q^{8} -26.5251 q^{9} +50.7019 q^{11} +5.16446 q^{12} +45.4342 q^{13} +13.5473 q^{14} +52.1215 q^{16} +53.0220 q^{17} +18.8606 q^{18} +58.8778 q^{19} +13.1293 q^{21} -36.0514 q^{22} -23.0000 q^{23} -7.59208 q^{24} -32.3058 q^{26} +36.8846 q^{27} +142.788 q^{28} -70.7345 q^{29} -286.748 q^{31} -125.199 q^{32} -34.9391 q^{33} -37.7011 q^{34} +198.790 q^{36} -175.393 q^{37} -41.8649 q^{38} -31.3091 q^{39} -95.0049 q^{41} -9.33557 q^{42} +148.383 q^{43} -379.981 q^{44} +16.3541 q^{46} +23.5469 q^{47} -35.9174 q^{48} +20.0033 q^{49} -36.5379 q^{51} -340.503 q^{52} +379.219 q^{53} -26.2267 q^{54} -209.908 q^{56} -40.5731 q^{57} +50.2956 q^{58} -741.332 q^{59} +5.17078 q^{61} +203.891 q^{62} +505.374 q^{63} -327.950 q^{64} +24.8433 q^{66} +974.454 q^{67} -397.368 q^{68} +15.8495 q^{69} -336.469 q^{71} -292.234 q^{72} -317.142 q^{73} +124.713 q^{74} -441.254 q^{76} -966.005 q^{77} +22.2622 q^{78} -577.050 q^{79} +690.761 q^{81} +67.5530 q^{82} -225.514 q^{83} -98.3966 q^{84} -105.507 q^{86} +48.7438 q^{87} +558.596 q^{88} -1141.92 q^{89} -865.642 q^{91} +172.371 q^{92} +197.600 q^{93} -16.7429 q^{94} +86.2756 q^{96} -147.766 q^{97} -14.2233 q^{98} -1344.87 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + q^{3} + 27 q^{4} - 41 q^{6} + q^{7} - 57 q^{8} + 10 q^{9} - 52 q^{11} + 65 q^{12} + 45 q^{13} - 42 q^{14} - 85 q^{16} + 85 q^{17} + 18 q^{18} - 10 q^{19} - 202 q^{21} - 71 q^{22} - 161 q^{23}+ \cdots - 2143 q^{99}+O(q^{100}) $ 7 * q + 3 * q^2 + q^3 + 27 * q^4 - 41 * q^6 + q^7 - 57 * q^8 + 10 * q^9 - 52 * q^11 + 65 * q^12 + 45 * q^13 - 42 * q^14 - 85 * q^16 + 85 * q^17 + 18 * q^18 - 10 * q^19 - 202 * q^21 - 71 * q^22 - 161 * q^23 - 303 * q^24 - 26 * q^26 - 242 * q^27 + 186 * q^28 - 455 * q^29 - 690 * q^31 - 233 * q^32 - 203 * q^33 - 657 * q^34 + 204 * q^36 - 892 * q^37 + 421 * q^38 - 277 * q^39 - 230 * q^41 - 888 * q^42 + 217 * q^43 - 1003 * q^44 - 69 * q^46 - 166 * q^47 + 1225 * q^48 - 802 * q^49 - 695 * q^51 - 1640 * q^52 + 1148 * q^53 - 1247 * q^54 - 386 * q^56 - 845 * q^57 + 1454 * q^58 - 551 * q^59 - 1242 * q^61 - 296 * q^62 + 1111 * q^63 - 1501 * q^64 + 987 * q^66 + 201 * q^67 + 1575 * q^68 - 23 * q^69 + 186 * q^71 - 2964 * q^72 + 530 * q^73 - 1530 * q^74 - 1775 * q^76 - 1339 * q^77 + 1364 * q^78 - 2003 * q^79 - 365 * q^81 - 2063 * q^82 + 726 * q^83 - 1352 * q^84 + 1894 * q^86 + 263 * q^87 + 2145 * q^88 - 2629 * q^89 - 1295 * q^91 - 621 * q^92 - 549 * q^93 - 3468 * q^94 - 135 * q^96 + 472 * q^97 + 2389 * q^98 - 2143 * q^99

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 $

$2$	−0.711047	−0.251393	−0.125697	−	0.992069i	$-0.540117\pi$
$2$	−0.711047	−0.251393	−0.125697	+	0.992069i	$0.540117\pi$
$3$	−0.689108	−0.132619	−0.0663095	−	0.997799i	$-0.521122\pi$
$3$	−0.689108	−0.132619	−0.0663095	+	0.997799i	$0.521122\pi$
$4$	−7.49441	−0.936802
$5$	0	0
$5$	0	0
$6$	0.489988	0.0333395
$7$	−19.0526	−1.02875	−0.514373	−	0.857566i	$-0.671975\pi$
$7$	−19.0526	−1.02875	−0.514373	+	0.857566i	$0.671975\pi$
$8$	11.0173	0.486899
$9$	−26.5251	−0.982412
$10$	0	0
$11$	50.7019	1.38974	0.694872	−	0.719133i	$-0.255461\pi$
$11$	50.7019	1.38974	0.694872	+	0.719133i	$0.255461\pi$
$12$	5.16446	0.124238
$13$	45.4342	0.969321	0.484661	−	0.874702i	$-0.338943\pi$
$13$	45.4342	0.969321	0.484661	+	0.874702i	$0.338943\pi$
$14$	13.5473	0.258620
$15$	0	0
$16$	52.1215	0.814399
$17$	53.0220	0.756454	0.378227	−	0.925713i	$-0.376534\pi$
$17$	53.0220	0.756454	0.378227	+	0.925713i	$0.376534\pi$
$18$	18.8606	0.246972
$19$	58.8778	0.710920	0.355460	−	0.934691i	$-0.384324\pi$
$19$	58.8778	0.710920	0.355460	+	0.934691i	$0.384324\pi$
$20$	0	0
$21$	13.1293	0.136431
$22$	−36.0514	−0.349372
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−7.59208	−0.0645720
$25$	0	0
$26$	−32.3058	−0.243681
$27$	36.8846	0.262905
$28$	142.788	0.963731
$29$	−70.7345	−0.452934	−0.226467	−	0.974019i	$-0.572717\pi$
$29$	−70.7345	−0.452934	−0.226467	+	0.974019i	$0.572717\pi$
$30$	0	0
$31$	−286.748	−1.66134	−0.830669	−	0.556766i	$-0.812042\pi$
$31$	−286.748	−1.66134	−0.830669	+	0.556766i	$0.812042\pi$
$32$	−125.199	−0.691633
$33$	−34.9391	−0.184306
$34$	−37.7011	−0.190167
$35$	0	0
$36$	198.790	0.920325
$37$	−175.393	−0.779310	−0.389655	−	0.920961i	$-0.627406\pi$
$37$	−175.393	−0.779310	−0.389655	+	0.920961i	$0.627406\pi$
$38$	−41.8649	−0.178720
$39$	−31.3091	−0.128550
$40$	0	0
$41$	−95.0049	−0.361885	−0.180942	−	0.983494i	$-0.557915\pi$
$41$	−95.0049	−0.361885	−0.180942	+	0.983494i	$0.557915\pi$
$42$	−9.33557	−0.0342979
$43$	148.383	0.526237	0.263118	−	0.964764i	$-0.415249\pi$
$43$	148.383	0.526237	0.263118	+	0.964764i	$0.415249\pi$
$44$	−379.981	−1.30191
$45$	0	0
$46$	16.3541	0.0524191
$47$	23.5469	0.0730779	0.0365390	−	0.999332i	$-0.488367\pi$
$47$	23.5469	0.0730779	0.0365390	+	0.999332i	$0.488367\pi$
$48$	−35.9174	−0.108005
$49$	20.0033	0.0583188
$50$	0	0
$51$	−36.5379	−0.100320
$52$	−340.503	−0.908062
$53$	379.219	0.982825	0.491413	−	0.870927i	$-0.336481\pi$
$53$	379.219	0.982825	0.491413	+	0.870927i	$0.336481\pi$
$54$	−26.2267	−0.0660926
$55$	0	0
$56$	−209.908	−0.500895
$57$	−40.5731	−0.0942815
$58$	50.2956	0.113864
$59$	−741.332	−1.63582	−0.817908	−	0.575349i	$-0.804866\pi$
$59$	−741.332	−1.63582	−0.817908	+	0.575349i	$0.804866\pi$
$60$	0	0
$61$	5.17078	0.0108533	0.00542664	−	0.999985i	$-0.498273\pi$
$61$	5.17078	0.0108533	0.00542664	+	0.999985i	$0.498273\pi$
$62$	203.891	0.417649
$63$	505.374	1.01065
$64$	−327.950	−0.640527
$65$	0	0
$66$	24.8433	0.0463334
$67$	974.454	1.77684	0.888422	−	0.459028i	$-0.151802\pi$
$67$	974.454	1.77684	0.888422	+	0.459028i	$0.151802\pi$
$68$	−397.368	−0.708647
$69$	15.8495	0.0276530
$70$	0	0
$71$	−336.469	−0.562416	−0.281208	−	0.959647i	$-0.590735\pi$
$71$	−336.469	−0.562416	−0.281208	+	0.959647i	$0.590735\pi$
$72$	−292.234	−0.478335
$73$	−317.142	−0.508474	−0.254237	−	0.967142i	$-0.581824\pi$
$73$	−317.142	−0.508474	−0.254237	+	0.967142i	$0.581824\pi$
$74$	124.713	0.195913
$75$	0	0
$76$	−441.254	−0.665991
$77$	−966.005	−1.42969
$78$	22.2622	0.0323167
$79$	−577.050	−0.821812	−0.410906	−	0.911678i	$-0.634788\pi$
$79$	−577.050	−0.821812	−0.410906	+	0.911678i	$0.634788\pi$
$80$	0	0
$81$	690.761	0.947546
$82$	67.5530	0.0909754
$83$	−225.514	−0.298234	−0.149117	−	0.988820i	$-0.547643\pi$
$83$	−225.514	−0.298234	−0.149117	+	0.988820i	$0.547643\pi$
$84$	−98.3966	−0.127809
$85$	0	0
$86$	−105.507	−0.132292
$87$	48.7438	0.0600676
$88$	558.596	0.676665
$89$	−1141.92	−1.36003	−0.680016	−	0.733197i	$-0.738027\pi$
$89$	−1141.92	−1.36003	−0.680016	+	0.733197i	$0.738027\pi$
$90$	0	0
$91$	−865.642	−0.997186
$92$	172.371	0.195337
$93$	197.600	0.220325
$94$	−16.7429	−0.0183713
$95$	0	0
$96$	86.2756	0.0917236
$97$	−147.766	−0.154674	−0.0773370	−	0.997005i	$-0.524642\pi$
$97$	−147.766	−0.154674	−0.0773370	+	0.997005i	$0.524642\pi$
$98$	−14.2233	−0.0146609
$99$	−1344.87	−1.36530
$100$	0	0
$101$	−909.233	−0.895763	−0.447881	−	0.894093i	$-0.647821\pi$
$101$	−909.233	−0.895763	−0.447881	+	0.894093i	$0.647821\pi$
$102$	25.9801	0.0252198
$103$	1330.08	1.27240	0.636198	−	0.771526i	$-0.280506\pi$
$103$	1330.08	1.27240	0.636198	+	0.771526i	$0.280506\pi$
$104$	500.560	0.471961
$105$	0	0
$106$	−269.643	−0.247075
$107$	−1209.82	−1.09306	−0.546529	−	0.837440i	$-0.684051\pi$
$107$	−1209.82	−1.09306	−0.546529	+	0.837440i	$0.684051\pi$
$108$	−276.428	−0.246290
$109$	507.933	0.446341	0.223171	−	0.974779i	$-0.428359\pi$
$109$	507.933	0.446341	0.223171	+	0.974779i	$0.428359\pi$
$110$	0	0
$111$	120.865	0.103351
$112$	−993.053	−0.837809
$113$	2.39609	0.00199474	0.000997369	−	1.00000i	$-0.499683\pi$
$113$	2.39609	0.00199474	0.000997369		1.00000i	$0.499683\pi$
$114$	28.8494	0.0237017
$115$	0	0
$116$	530.114	0.424309
$117$	−1205.15	−0.952273
$118$	527.122	0.411233
$119$	−1010.21	−0.778199
$120$	0	0
$121$	1239.68	0.931390
$122$	−3.67666	−0.00272844
$123$	65.4687	0.0479928
$124$	2149.01	1.55634
$125$	0	0
$126$	−359.345	−0.254071
$127$	−1355.81	−0.947310	−0.473655	−	0.880710i	$-0.657066\pi$
$127$	−1355.81	−0.947310	−0.473655	+	0.880710i	$0.657066\pi$
$128$	1234.78	0.852657
$129$	−102.252	−0.0697890
$130$	0	0
$131$	−1701.47	−1.13479	−0.567397	−	0.823444i	$-0.692050\pi$
$131$	−1701.47	−1.13479	−0.567397	+	0.823444i	$0.692050\pi$
$132$	261.848	0.172659
$133$	−1121.78	−0.731356
$134$	−692.883	−0.446686
$135$	0	0
$136$	584.157	0.368316
$137$	−2505.35	−1.56238	−0.781191	−	0.624292i	$-0.785387\pi$
$137$	−2505.35	−1.56238	−0.781191	+	0.624292i	$0.785387\pi$
$138$	−11.2697	−0.00695176
$139$	−245.374	−0.149729	−0.0748646	−	0.997194i	$-0.523852\pi$
$139$	−245.374	−0.149729	−0.0748646	+	0.997194i	$0.523852\pi$
$140$	0	0
$141$	−16.2263	−0.00969151
$142$	239.245	0.141388
$143$	2303.60	1.34711
$144$	−1382.53	−0.800075
$145$	0	0
$146$	225.503	0.127827
$147$	−13.7845	−0.00773418
$148$	1314.47	0.730059
$149$	2542.90	1.39814	0.699068	−	0.715055i	$-0.253598\pi$
$149$	2542.90	1.39814	0.699068	+	0.715055i	$0.253598\pi$
$150$	0	0
$151$	2976.66	1.60422	0.802110	−	0.597176i	$-0.203711\pi$
$151$	2976.66	1.60422	0.802110	+	0.597176i	$0.203711\pi$
$152$	648.671	0.346146
$153$	−1406.41	−0.743150
$154$	686.875	0.359415
$155$	0	0
$156$	234.643	0.120426
$157$	3622.21	1.84130	0.920648	−	0.390393i	$-0.127661\pi$
$157$	3622.21	1.84130	0.920648	+	0.390393i	$0.127661\pi$
$158$	410.310	0.206598
$159$	−261.323	−0.130341
$160$	0	0
$161$	438.211	0.214508
$162$	−491.164	−0.238207
$163$	−3486.81	−1.67551	−0.837754	−	0.546048i	$-0.816132\pi$
$163$	−3486.81	−1.67551	−0.837754	+	0.546048i	$0.816132\pi$
$164$	712.006	0.339014
$165$	0	0
$166$	160.351	0.0749738
$167$	2818.90	1.30618	0.653092	−	0.757278i	$-0.273471\pi$
$167$	2818.90	1.30618	0.653092	+	0.757278i	$0.273471\pi$
$168$	144.649	0.0664282
$169$	−132.734	−0.0604161
$170$	0	0
$171$	−1561.74	−0.698417
$172$	−1112.04	−0.492980
$173$	−2919.22	−1.28291	−0.641456	−	0.767159i	$-0.721670\pi$
$173$	−2919.22	−1.28291	−0.641456	+	0.767159i	$0.721670\pi$
$174$	−34.6591	−0.0151006
$175$	0	0
$176$	2642.66	1.13181
$177$	510.858	0.216940
$178$	811.956	0.341903
$179$	−2992.42	−1.24952	−0.624761	−	0.780816i	$-0.714803\pi$
$179$	−2992.42	−1.24952	−0.624761	+	0.780816i	$0.714803\pi$
$180$	0	0
$181$	−3665.87	−1.50542	−0.752712	−	0.658350i	$-0.771255\pi$
$181$	−3665.87	−1.50542	−0.752712	+	0.658350i	$0.771255\pi$
$182$	615.512	0.250686
$183$	−3.56322	−0.00143935
$184$	−253.397	−0.101525
$185$	0	0
$186$	−140.503	−0.0553882
$187$	2688.31	1.05128
$188$	−176.470	−0.0684595
$189$	−702.749	−0.270463
$190$	0	0
$191$	−2159.35	−0.818035	−0.409018	−	0.912526i	$-0.634129\pi$
$191$	−2159.35	−0.818035	−0.409018	+	0.912526i	$0.634129\pi$
$192$	225.993	0.0849460
$193$	−1865.76	−0.695856	−0.347928	−	0.937521i	$-0.613115\pi$
$193$	−1865.76	−0.695856	−0.347928	+	0.937521i	$0.613115\pi$
$194$	105.069	0.0388840
$195$	0	0
$196$	−149.913	−0.0546331
$197$	−2638.16	−0.954117	−0.477058	−	0.878872i	$-0.658297\pi$
$197$	−2638.16	−0.954117	−0.477058	+	0.878872i	$0.658297\pi$
$198$	956.269	0.343228
$199$	626.916	0.223321	0.111661	−	0.993746i	$-0.464383\pi$
$199$	626.916	0.223321	0.111661	+	0.993746i	$0.464383\pi$
$200$	0	0
$201$	−671.504	−0.235643
$202$	646.507	0.225189
$203$	1347.68	0.465954
$204$	273.830	0.0939800
$205$	0	0
$206$	−945.751	−0.319872
$207$	610.078	0.204847
$208$	2368.10	0.789414
$209$	2985.21	0.987998
$210$	0	0
$211$	−3090.41	−1.00831	−0.504154	−	0.863614i	$-0.668196\pi$
$211$	−3090.41	−1.00831	−0.504154	+	0.863614i	$0.668196\pi$
$212$	−2842.02	−0.920712
$213$	231.864	0.0745870
$214$	860.236	0.274787
$215$	0	0
$216$	406.367	0.128008
$217$	5463.31	1.70910
$218$	−361.165	−0.112207
$219$	218.545	0.0674333
$220$	0	0
$221$	2409.01	0.733247
$222$	−85.9407	−0.0259818
$223$	−5702.67	−1.71246	−0.856230	−	0.516594i	$-0.827200\pi$
$223$	−5702.67	−1.71246	−0.856230	+	0.516594i	$0.827200\pi$
$224$	2385.37	0.711515
$225$	0	0
$226$	−1.70373	−0.000501463 0
$227$	3203.45	0.936654	0.468327	−	0.883555i	$-0.344857\pi$
$227$	3203.45	0.936654	0.468327	+	0.883555i	$0.344857\pi$
$228$	304.072	0.0883230
$229$	648.466	0.187126	0.0935630	−	0.995613i	$-0.470174\pi$
$229$	648.466	0.187126	0.0935630	+	0.995613i	$0.470174\pi$
$230$	0	0
$231$	665.682	0.189605
$232$	−779.301	−0.220533
$233$	2556.98	0.718941	0.359470	−	0.933156i	$-0.382957\pi$
$233$	2556.98	0.718941	0.359470	+	0.933156i	$0.382957\pi$
$234$	856.917	0.239395
$235$	0	0
$236$	5555.85	1.53244
$237$	397.650	0.108988
$238$	718.306	0.195634
$239$	4424.46	1.19747	0.598733	−	0.800949i	$-0.295671\pi$
$239$	4424.46	1.19747	0.598733	+	0.800949i	$0.295671\pi$
$240$	0	0
$241$	−3335.19	−0.891447	−0.445724	−	0.895171i	$-0.647054\pi$
$241$	−3335.19	−0.891447	−0.445724	+	0.895171i	$0.647054\pi$
$242$	−881.471	−0.234145
$243$	−1471.89	−0.388568
$244$	−38.7519	−0.0101674
$245$	0	0
$246$	−46.5513	−0.0120651
$247$	2675.06	0.689110
$248$	−3159.18	−0.808903
$249$	155.404	0.0395514
$250$	0	0
$251$	1952.47	0.490990	0.245495	−	0.969398i	$-0.421049\pi$
$251$	1952.47	0.490990	0.245495	+	0.969398i	$0.421049\pi$
$252$	−3787.48	−0.946781
$253$	−1166.14	−0.289782
$254$	964.042	0.238147
$255$	0	0
$256$	1745.61	0.426175
$257$	−543.328	−0.131875	−0.0659374	−	0.997824i	$-0.521004\pi$
$257$	−543.328	−0.131875	−0.0659374	+	0.997824i	$0.521004\pi$
$258$	72.7059	0.0175445
$259$	3341.71	0.801713
$260$	0	0
$261$	1876.24	0.444968
$262$	1209.82	0.285279
$263$	−3370.82	−0.790318	−0.395159	−	0.918613i	$-0.629310\pi$
$263$	−3370.82	−0.790318	−0.395159	+	0.918613i	$0.629310\pi$
$264$	−384.933	−0.0897385
$265$	0	0
$266$	797.636	0.183858
$267$	786.904	0.180366
$268$	−7302.96	−1.66455
$269$	−2778.40	−0.629747	−0.314873	−	0.949134i	$-0.601962\pi$
$269$	−2778.40	−0.629747	−0.314873	+	0.949134i	$0.601962\pi$
$270$	0	0
$271$	6476.65	1.45177	0.725883	−	0.687819i	$-0.241432\pi$
$271$	6476.65	1.45177	0.725883	+	0.687819i	$0.241432\pi$
$272$	2763.59	0.616055
$273$	596.521	0.132246
$274$	1781.42	0.392772
$275$	0	0
$276$	−118.783	−0.0259053
$277$	1555.57	0.337420	0.168710	−	0.985666i	$-0.446040\pi$
$277$	1555.57	0.337420	0.168710	+	0.985666i	$0.446040\pi$
$278$	174.473	0.0376409
$279$	7606.03	1.63212
$280$	0	0
$281$	4311.09	0.915225	0.457612	−	0.889152i	$-0.348705\pi$
$281$	4311.09	0.915225	0.457612	+	0.889152i	$0.348705\pi$
$282$	11.5377	0.00243638
$283$	−6866.82	−1.44237	−0.721183	−	0.692744i	$-0.756402\pi$
$283$	−6866.82	−1.44237	−0.721183	+	0.692744i	$0.756402\pi$
$284$	2521.64	0.526872
$285$	0	0
$286$	−1637.97	−0.338654
$287$	1810.10	0.372288
$288$	3320.92	0.679468
$289$	−2101.67	−0.427777
$290$	0	0
$291$	101.827	0.0205127
$292$	2376.79	0.476339
$293$	7215.56	1.43870	0.719348	−	0.694650i	$-0.244441\pi$
$293$	7215.56	1.43870	0.719348	+	0.694650i	$0.244441\pi$
$294$	9.80141	0.00194432
$295$	0	0
$296$	−1932.35	−0.379445
$297$	1870.12	0.365371
$298$	−1808.12	−0.351482
$299$	−1044.99	−0.202117
$300$	0	0
$301$	−2827.09	−0.541364
$302$	−2116.55	−0.403290
$303$	626.560	0.118795
$304$	3068.80	0.578972
$305$	0	0
$306$	1000.03	0.186823
$307$	−3524.91	−0.655300	−0.327650	−	0.944799i	$-0.606257\pi$
$307$	−3524.91	−0.655300	−0.327650	+	0.944799i	$0.606257\pi$
$308$	7239.64	1.33934
$309$	−916.570	−0.168744
$310$	0	0
$311$	3378.27	0.615962	0.307981	−	0.951393i	$-0.400347\pi$
$311$	3378.27	0.615962	0.307981	+	0.951393i	$0.400347\pi$
$312$	−344.940	−0.0625910
$313$	−2962.83	−0.535045	−0.267522	−	0.963552i	$-0.586205\pi$
$313$	−2962.83	−0.535045	−0.267522	+	0.963552i	$0.586205\pi$
$314$	−2575.56	−0.462889
$315$	0	0
$316$	4324.65	0.769875
$317$	−553.945	−0.0981472	−0.0490736	−	0.998795i	$-0.515627\pi$
$317$	−553.945	−0.0981472	−0.0490736	+	0.998795i	$0.515627\pi$
$318$	185.813	0.0327669
$319$	−3586.37	−0.629462
$320$	0	0
$321$	833.694	0.144960
$322$	−311.589	−0.0539259
$323$	3121.81	0.537778
$324$	−5176.85	−0.887663
$325$	0	0
$326$	2479.28	0.421211
$327$	−350.021	−0.0591933
$328$	−1046.69	−0.176201
$329$	−448.630	−0.0751786
$330$	0	0
$331$	−2236.93	−0.371458	−0.185729	−	0.982601i	$-0.559465\pi$
$331$	−2236.93	−0.371458	−0.185729	+	0.982601i	$0.559465\pi$
$332$	1690.10	0.279386
$333$	4652.33	0.765604
$334$	−2004.37	−0.328366
$335$	0	0
$336$	684.321	0.111109
$337$	−586.710	−0.0948372	−0.0474186	−	0.998875i	$-0.515099\pi$
$337$	−586.710	−0.0948372	−0.0474186	+	0.998875i	$0.515099\pi$
$338$	94.3802	0.0151882
$339$	−1.65117	−0.000264540 0
$340$	0	0
$341$	−14538.7	−2.30884
$342$	1110.47	0.175577
$343$	6153.94	0.968751
$344$	1634.77	0.256224
$345$	0	0
$346$	2075.70	0.322515
$347$	9174.08	1.41928	0.709640	−	0.704564i	$-0.248857\pi$
$347$	9174.08	1.41928	0.709640	+	0.704564i	$0.248857\pi$
$348$	−365.306	−0.0562714
$349$	5325.68	0.816840	0.408420	−	0.912794i	$-0.366080\pi$
$349$	5325.68	0.816840	0.408420	+	0.912794i	$0.366080\pi$
$350$	0	0
$351$	1675.82	0.254840
$352$	−6347.82	−0.961193
$353$	−9803.34	−1.47813	−0.739064	−	0.673635i	$-0.764732\pi$
$353$	−9803.34	−1.47813	−0.739064	+	0.673635i	$0.764732\pi$
$354$	−363.244	−0.0545373
$355$	0	0
$356$	8557.99	1.27408
$357$	696.143	0.103204
$358$	2127.75	0.314121
$359$	10089.9	1.48336	0.741681	−	0.670753i	$-0.234029\pi$
$359$	10089.9	1.48336	0.741681	+	0.670753i	$0.234029\pi$
$360$	0	0
$361$	−3392.41	−0.494593
$362$	2606.60	0.378453
$363$	−854.274	−0.123520
$364$	6487.48	0.934165
$365$	0	0
$366$	2.53362	0.000361843 0
$367$	8291.81	1.17937	0.589685	−	0.807633i	$-0.299252\pi$
$367$	8291.81	1.17937	0.589685	+	0.807633i	$0.299252\pi$
$368$	−1198.79	−0.169814
$369$	2520.02	0.355520
$370$	0	0
$371$	−7225.13	−1.01108
$372$	−1480.90	−0.206401
$373$	−7493.58	−1.04022	−0.520111	−	0.854098i	$-0.674110\pi$
$373$	−7493.58	−1.04022	−0.520111	+	0.854098i	$0.674110\pi$
$374$	−1911.52	−0.264284
$375$	0	0
$376$	259.422	0.0355815
$377$	−3213.77	−0.439038
$378$	499.688	0.0679925
$379$	−3630.87	−0.492099	−0.246049	−	0.969257i	$-0.579133\pi$
$379$	−3630.87	−0.492099	−0.246049	+	0.969257i	$0.579133\pi$
$380$	0	0
$381$	934.297	0.125631
$382$	1535.40	0.205648
$383$	−952.474	−0.127074	−0.0635368	−	0.997979i	$-0.520238\pi$
$383$	−952.474	−0.127074	−0.0635368	+	0.997979i	$0.520238\pi$
$384$	−850.896	−0.113078
$385$	0	0
$386$	1326.64	0.174933
$387$	−3935.88	−0.516982
$388$	1107.42	0.144899
$389$	−4541.34	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$389$	−4541.34	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$390$	0	0
$391$	−1219.51	−0.157732
$392$	220.382	0.0283953
$393$	1172.50	0.150495
$394$	1875.86	0.239858
$395$	0	0
$396$	10079.0	1.27902
$397$	−14345.6	−1.81356	−0.906780	−	0.421604i	$-0.861467\pi$
$397$	−14345.6	−1.81356	−0.906780	+	0.421604i	$0.861467\pi$
$398$	−445.767	−0.0561414
$399$	773.026	0.0969917
$400$	0	0
$401$	−6358.53	−0.791845	−0.395923	−	0.918284i	$-0.629575\pi$
$401$	−6358.53	−0.791845	−0.395923	+	0.918284i	$0.629575\pi$
$402$	477.471	0.0592390
$403$	−13028.2	−1.61037
$404$	6814.17	0.839152
$405$	0	0
$406$	−958.264	−0.117138
$407$	−8892.77	−1.08304
$408$	−402.547	−0.0488457
$409$	−11543.2	−1.39553	−0.697767	−	0.716325i	$-0.745823\pi$
$409$	−11543.2	−1.39553	−0.697767	+	0.716325i	$0.745823\pi$
$410$	0	0
$411$	1726.46	0.207201
$412$	−9968.18	−1.19198
$413$	14124.3	1.68284
$414$	−433.794	−0.0514971
$415$	0	0
$416$	−5688.31	−0.670414
$417$	169.089	0.0198569
$418$	−2122.63	−0.248376
$419$	−3691.25	−0.430380	−0.215190	−	0.976572i	$-0.569037\pi$
$419$	−3691.25	−0.430380	−0.215190	+	0.976572i	$0.569037\pi$
$420$	0	0
$421$	−7551.44	−0.874191	−0.437096	−	0.899415i	$-0.643993\pi$
$421$	−7551.44	−0.874191	−0.437096	+	0.899415i	$0.643993\pi$
$422$	2197.43	0.253482
$423$	−624.583	−0.0717926
$424$	4177.95	0.478536
$425$	0	0
$426$	−164.866	−0.0187507
$427$	−98.5170	−0.0111653
$428$	9066.86	1.02398
$429$	−1587.43	−0.178652
$430$	0	0
$431$	−16426.2	−1.83578	−0.917891	−	0.396833i	$-0.870109\pi$
$431$	−16426.2	−1.83578	−0.917891	+	0.396833i	$0.870109\pi$
$432$	1922.48	0.214110
$433$	−6363.88	−0.706301	−0.353151	−	0.935566i	$-0.614890\pi$
$433$	−6363.88	−0.706301	−0.353151	+	0.935566i	$0.614890\pi$
$434$	−3884.67	−0.429655
$435$	0	0
$436$	−3806.66	−0.418133
$437$	−1354.19	−0.148237
$438$	−155.396	−0.0169523
$439$	−7513.99	−0.816910	−0.408455	−	0.912779i	$-0.633932\pi$
$439$	−7513.99	−0.816910	−0.408455	+	0.912779i	$0.633932\pi$
$440$	0	0
$441$	−530.591	−0.0572931
$442$	−1712.92	−0.184333
$443$	−4413.43	−0.473337	−0.236669	−	0.971590i	$-0.576056\pi$
$443$	−4413.43	−0.473337	−0.236669	+	0.971590i	$0.576056\pi$
$444$	−905.812	−0.0968196
$445$	0	0
$446$	4054.86	0.430501
$447$	−1752.33	−0.185419
$448$	6248.31	0.658940
$449$	−10795.9	−1.13472	−0.567361	−	0.823469i	$-0.692036\pi$
$449$	−10795.9	−1.13472	−0.567361	+	0.823469i	$0.692036\pi$
$450$	0	0
$451$	−4816.93	−0.502928
$452$	−17.9573	−0.00186867
$453$	−2051.24	−0.212750
$454$	−2277.80	−0.235468
$455$	0	0
$456$	−447.005	−0.0459055
$457$	15726.1	1.60971	0.804854	−	0.593473i	$-0.202244\pi$
$457$	15726.1	1.60971	0.804854	+	0.593473i	$0.202244\pi$
$458$	−461.090	−0.0470422
$459$	1955.69	0.198876
$460$	0	0
$461$	−997.530	−0.100780	−0.0503900	−	0.998730i	$-0.516046\pi$
$461$	−997.530	−0.100780	−0.0503900	+	0.998730i	$0.516046\pi$
$462$	−473.331	−0.0476653
$463$	13338.6	1.33887	0.669434	−	0.742872i	$-0.266537\pi$
$463$	13338.6	1.33887	0.669434	+	0.742872i	$0.266537\pi$
$464$	−3686.79	−0.368869
$465$	0	0
$466$	−1818.13	−0.180737
$467$	5767.52	0.571496	0.285748	−	0.958305i	$-0.407758\pi$
$467$	5767.52	0.571496	0.285748	+	0.958305i	$0.407758\pi$
$468$	9031.87	0.892091
$469$	−18565.9	−1.82792
$470$	0	0
$471$	−2496.09	−0.244191
$472$	−8167.44	−0.796477
$473$	7523.29	0.731335
$474$	−282.748	−0.0273988
$475$	0	0
$476$	7570.92	0.729018
$477$	−10058.8	−0.965540
$478$	−3146.00	−0.301035
$479$	9581.36	0.913953	0.456977	−	0.889479i	$-0.348932\pi$
$479$	9581.36	0.913953	0.456977	+	0.889479i	$0.348932\pi$
$480$	0	0
$481$	−7968.85	−0.755402
$482$	2371.48	0.224104
$483$	−301.975	−0.0284479
$484$	−9290.68	−0.872528
$485$	0	0
$486$	1046.59	0.0976833
$487$	4204.25	0.391197	0.195598	−	0.980684i	$-0.437335\pi$
$487$	4204.25	0.391197	0.195598	+	0.980684i	$0.437335\pi$
$488$	56.9678	0.00528444
$489$	2402.79	0.222204
$490$	0	0
$491$	5650.56	0.519361	0.259681	−	0.965695i	$-0.416383\pi$
$491$	5650.56	0.519361	0.259681	+	0.965695i	$0.416383\pi$
$492$	−490.649	−0.0449597
$493$	−3750.48	−0.342623
$494$	−1902.10	−0.173238
$495$	0	0
$496$	−14945.7	−1.35299
$497$	6410.63	0.578583
$498$	−110.499	−0.00994295
$499$	−19210.6	−1.72341	−0.861706	−	0.507408i	$-0.830604\pi$
$499$	−19210.6	−1.72341	−0.861706	+	0.507408i	$0.830604\pi$
$500$	0	0
$501$	−1942.53	−0.173225
$502$	−1388.30	−0.123432
$503$	−10429.5	−0.924506	−0.462253	−	0.886748i	$-0.652959\pi$
$503$	−10429.5	−0.924506	−0.462253	+	0.886748i	$0.652959\pi$
$504$	5567.83	0.492085
$505$	0	0
$506$	829.183	0.0728491
$507$	91.4682	0.00801231
$508$	10161.0	0.887442
$509$	−10350.4	−0.901324	−0.450662	−	0.892695i	$-0.648812\pi$
$509$	−10350.4	−0.901324	−0.450662	+	0.892695i	$0.648812\pi$
$510$	0	0
$511$	6042.39	0.523091
$512$	−11119.4	−0.959794
$513$	2171.68	0.186905
$514$	386.331	0.0331524
$515$	0	0
$516$	766.318	0.0653784
$517$	1193.87	0.101560
$518$	−2376.11	−0.201545
$519$	2011.66	0.170139
$520$	0	0
$521$	12477.6	1.04924	0.524620	−	0.851336i	$-0.324207\pi$
$521$	12477.6	1.04924	0.524620	+	0.851336i	$0.324207\pi$
$522$	−1334.10	−0.111862
$523$	11220.6	0.938134	0.469067	−	0.883163i	$-0.344590\pi$
$523$	11220.6	0.938134	0.469067	+	0.883163i	$0.344590\pi$
$524$	12751.5	1.06308
$525$	0	0
$526$	2396.81	0.198680
$527$	−15204.0	−1.25673
$528$	−1821.08	−0.150099
$529$	529.000	0.0434783
$530$	0	0
$531$	19663.9	1.60705
$532$	8407.06	0.685136
$533$	−4316.47	−0.350783
$534$	−559.525	−0.0453428
$535$	0	0
$536$	10735.8	0.865143
$537$	2062.10	0.165710
$538$	1975.57	0.158314
$539$	1014.21	0.0810482
$540$	0	0
$541$	−4220.58	−0.335410	−0.167705	−	0.985837i	$-0.553636\pi$
$541$	−4220.58	−0.335410	−0.167705	+	0.985837i	$0.553636\pi$
$542$	−4605.20	−0.364964
$543$	2526.18	0.199648
$544$	−6638.29	−0.523188
$545$	0	0
$546$	−424.154	−0.0332457
$547$	−298.329	−0.0233193	−0.0116596	−	0.999932i	$-0.503711\pi$
$547$	−298.329	−0.0233193	−0.0116596	+	0.999932i	$0.503711\pi$
$548$	18776.1	1.46364
$549$	−137.155	−0.0106624
$550$	0	0
$551$	−4164.69	−0.322000
$552$	174.618	0.0134642
$553$	10994.3	0.845436
$554$	−1106.09	−0.0848251
$555$	0	0
$556$	1838.93	0.140267
$557$	1097.07	0.0834545	0.0417272	−	0.999129i	$-0.486714\pi$
$557$	1097.07	0.0834545	0.0417272	+	0.999129i	$0.486714\pi$
$558$	−5408.25	−0.410303
$559$	6741.66	0.510093
$560$	0	0
$561$	−1852.54	−0.139419
$562$	−3065.39	−0.230081
$563$	−5409.44	−0.404939	−0.202469	−	0.979289i	$-0.564897\pi$
$563$	−5409.44	−0.404939	−0.202469	+	0.979289i	$0.564897\pi$
$564$	121.607	0.00907902
$565$	0	0
$566$	4882.63	0.362601
$567$	−13160.8	−0.974784
$568$	−3706.97	−0.273840
$569$	10942.8	0.806231	0.403115	−	0.915149i	$-0.367927\pi$
$569$	10942.8	0.806231	0.403115	+	0.915149i	$0.367927\pi$
$570$	0	0
$571$	−1144.71	−0.0838959	−0.0419480	−	0.999120i	$-0.513356\pi$
$571$	−1144.71	−0.0838959	−0.0419480	+	0.999120i	$0.513356\pi$
$572$	−17264.1	−1.26197
$573$	1488.02	0.108487
$574$	−1287.06	−0.0935906
$575$	0	0
$576$	8698.91	0.629261
$577$	9892.27	0.713727	0.356864	−	0.934157i	$-0.383846\pi$
$577$	9892.27	0.713727	0.356864	+	0.934157i	$0.383846\pi$
$578$	1494.39	0.107540
$579$	1285.71	0.0922836
$580$	0	0
$581$	4296.64	0.306807
$582$	−72.4036	−0.00515675
$583$	19227.1	1.36588
$584$	−3494.03	−0.247575
$585$	0	0
$586$	−5130.60	−0.361678
$587$	−12528.8	−0.880950	−0.440475	−	0.897765i	$-0.645190\pi$
$587$	−12528.8	−0.880950	−0.440475	+	0.897765i	$0.645190\pi$
$588$	103.307	0.00724539
$589$	−16883.1	−1.18108
$590$	0	0
$591$	1817.98	0.126534
$592$	−9141.76	−0.634669
$593$	10978.3	0.760241	0.380121	−	0.924937i	$-0.375882\pi$
$593$	10978.3	0.760241	0.380121	+	0.924937i	$0.375882\pi$
$594$	−1329.74	−0.0918518
$595$	0	0
$596$	−19057.5	−1.30978
$597$	−432.013	−0.0296166
$598$	743.035	0.0508109
$599$	−583.723	−0.0398168	−0.0199084	−	0.999802i	$-0.506337\pi$
$599$	−583.723	−0.0398168	−0.0199084	+	0.999802i	$0.506337\pi$
$600$	0	0
$601$	−4111.87	−0.279080	−0.139540	−	0.990216i	$-0.544562\pi$
$601$	−4111.87	−0.279080	−0.139540	+	0.990216i	$0.544562\pi$
$602$	2010.19	0.136095
$603$	−25847.5	−1.74559
$604$	−22308.3	−1.50284
$605$	0	0
$606$	−445.514	−0.0298643
$607$	15367.4	1.02758	0.513792	−	0.857915i	$-0.328240\pi$
$607$	15367.4	1.02758	0.513792	+	0.857915i	$0.328240\pi$
$608$	−7371.43	−0.491696
$609$	−928.697	−0.0617943
$610$	0	0
$611$	1069.83	0.0708360
$612$	10540.3	0.696184
$613$	14127.7	0.930853	0.465426	−	0.885087i	$-0.345901\pi$
$613$	14127.7	0.930853	0.465426	+	0.885087i	$0.345901\pi$
$614$	2506.38	0.164738
$615$	0	0
$616$	−10642.7	−0.696116
$617$	−24427.9	−1.59389	−0.796944	−	0.604053i	$-0.793552\pi$
$617$	−24427.9	−1.59389	−0.796944	+	0.604053i	$0.793552\pi$
$618$	651.724	0.0424210
$619$	23038.8	1.49597	0.747987	−	0.663713i	$-0.231020\pi$
$619$	23038.8	1.49597	0.747987	+	0.663713i	$0.231020\pi$
$620$	0	0
$621$	−848.346	−0.0548196
$622$	−2402.11	−0.154848
$623$	21756.5	1.39913
$624$	−1631.88	−0.104691
$625$	0	0
$626$	2106.71	0.134507
$627$	−2057.13	−0.131027
$628$	−27146.3	−1.72493
$629$	−9299.70	−0.589512
$630$	0	0
$631$	−11336.7	−0.715223	−0.357611	−	0.933871i	$-0.616409\pi$
$631$	−11336.7	−0.715223	−0.357611	+	0.933871i	$0.616409\pi$
$632$	−6357.50	−0.400139
$633$	2129.63	0.133721
$634$	393.881	0.0246735
$635$	0	0
$636$	1958.46	0.122104
$637$	908.836	0.0565297
$638$	2550.08	0.158242
$639$	8924.89	0.552524
$640$	0	0
$641$	27030.1	1.66556	0.832782	−	0.553601i	$-0.186747\pi$
$641$	27030.1	1.66556	0.832782	+	0.553601i	$0.186747\pi$
$642$	−592.795	−0.0364420
$643$	6776.53	0.415615	0.207807	−	0.978170i	$-0.433367\pi$
$643$	6776.53	0.415615	0.207807	+	0.978170i	$0.433367\pi$
$644$	−3284.13	−0.200952
$645$	0	0
$646$	−2219.76	−0.135194
$647$	25310.7	1.53797	0.768984	−	0.639268i	$-0.220763\pi$
$647$	25310.7	1.53797	0.768984	+	0.639268i	$0.220763\pi$
$648$	7610.29	0.461359
$649$	−37586.9	−2.27337
$650$	0	0
$651$	−3764.81	−0.226658
$652$	26131.6	1.56962
$653$	−24830.6	−1.48805	−0.744024	−	0.668153i	$-0.767085\pi$
$653$	−24830.6	−1.48805	−0.744024	+	0.668153i	$0.767085\pi$
$654$	248.881	0.0148808
$655$	0	0
$656$	−4951.80	−0.294719
$657$	8412.22	0.499531
$658$	318.997	0.0188994
$659$	19514.4	1.15353	0.576764	−	0.816911i	$-0.304315\pi$
$659$	19514.4	1.15353	0.576764	+	0.816911i	$0.304315\pi$
$660$	0	0
$661$	−20107.4	−1.18319	−0.591594	−	0.806236i	$-0.701501\pi$
$661$	−20107.4	−1.18319	−0.591594	+	0.806236i	$0.701501\pi$
$662$	1590.56	0.0933820
$663$	−1660.07	−0.0972424
$664$	−2484.55	−0.145209
$665$	0	0
$666$	−3308.03	−0.192468
$667$	1626.89	0.0944432
$668$	−21126.0	−1.22364
$669$	3929.75	0.227105
$670$	0	0
$671$	262.168	0.0150833
$672$	−1643.78	−0.0943603
$673$	−34488.4	−1.97538	−0.987689	−	0.156432i	$-0.950001\pi$
$673$	−34488.4	−1.97538	−0.987689	+	0.156432i	$0.950001\pi$
$674$	417.179	0.0238414
$675$	0	0
$676$	994.764	0.0565979
$677$	−20506.3	−1.16414	−0.582070	−	0.813139i	$-0.697757\pi$
$677$	−20506.3	−1.16414	−0.582070	+	0.813139i	$0.697757\pi$
$678$	1.17406	6.65035e−5 0
$679$	2815.33	0.159120
$680$	0	0
$681$	−2207.52	−0.124218
$682$	10337.7	0.580425
$683$	2269.13	0.127124	0.0635620	−	0.997978i	$-0.479754\pi$
$683$	2269.13	0.127124	0.0635620	+	0.997978i	$0.479754\pi$
$684$	11704.3	0.654278
$685$	0	0
$686$	−4375.74	−0.243537
$687$	−446.863	−0.0248164
$688$	7733.94	0.428567
$689$	17229.5	0.952673
$690$	0	0
$691$	−27891.7	−1.53553	−0.767764	−	0.640733i	$-0.778631\pi$
$691$	−27891.7	−1.53553	−0.767764	+	0.640733i	$0.778631\pi$
$692$	21877.8	1.20183
$693$	25623.4	1.40455
$694$	−6523.20	−0.356797
$695$	0	0
$696$	537.022	0.0292468
$697$	−5037.35	−0.273749
$698$	−3786.81	−0.205348
$699$	−1762.03	−0.0953452
$700$	0	0
$701$	27181.1	1.46450	0.732251	−	0.681034i	$-0.238470\pi$
$701$	27181.1	1.46450	0.732251	+	0.681034i	$0.238470\pi$
$702$	−1191.59	−0.0640650
$703$	−10326.8	−0.554027
$704$	−16627.7	−0.890169
$705$	0	0
$706$	6970.64	0.371591
$707$	17323.3	0.921513
$708$	−3828.58	−0.203230
$709$	2398.23	0.127035	0.0635173	−	0.997981i	$-0.479768\pi$
$709$	2398.23	0.127035	0.0635173	+	0.997981i	$0.479768\pi$
$710$	0	0
$711$	15306.3	0.807358
$712$	−12580.8	−0.662198
$713$	6595.21	0.346413
$714$	−494.991	−0.0259448
$715$	0	0
$716$	22426.5	1.17055
$717$	−3048.93	−0.158807
$718$	−7174.42	−0.372907
$719$	11387.8	0.590674	0.295337	−	0.955393i	$-0.404568\pi$
$719$	11387.8	0.590674	0.295337	+	0.955393i	$0.404568\pi$
$720$	0	0
$721$	−25341.6	−1.30897
$722$	2412.16	0.124337
$723$	2298.31	0.118223
$724$	27473.5	1.41028
$725$	0	0
$726$	607.429	0.0310521
$727$	−2762.84	−0.140946	−0.0704732	−	0.997514i	$-0.522451\pi$
$727$	−2762.84	−0.140946	−0.0704732	+	0.997514i	$0.522451\pi$
$728$	−9537.00	−0.485528
$729$	−17636.3	−0.896015
$730$	0	0
$731$	7867.56	0.398074
$732$	26.7043	0.00134839
$733$	1782.84	0.0898370	0.0449185	−	0.998991i	$-0.485697\pi$
$733$	1782.84	0.0898370	0.0449185	+	0.998991i	$0.485697\pi$
$734$	−5895.87	−0.296486
$735$	0	0
$736$	2879.57	0.144215
$737$	49406.7	2.46936
$738$	−1791.85	−0.0893753
$739$	−987.346	−0.0491476	−0.0245738	−	0.999698i	$-0.507823\pi$
$739$	−987.346	−0.0491476	−0.0245738	+	0.999698i	$0.507823\pi$
$740$	0	0
$741$	−1843.41	−0.0913890
$742$	5137.40	0.254178
$743$	11744.8	0.579910	0.289955	−	0.957040i	$-0.406360\pi$
$743$	11744.8	0.579910	0.289955	+	0.957040i	$0.406360\pi$
$744$	2177.01	0.107276
$745$	0	0
$746$	5328.29	0.261505
$747$	5981.79	0.292988
$748$	−20147.3	−0.984839
$749$	23050.2	1.12448
$750$	0	0
$751$	−14156.0	−0.687828	−0.343914	−	0.939001i	$-0.611753\pi$
$751$	−14156.0	−0.687828	−0.343914	+	0.939001i	$0.611753\pi$
$752$	1227.30	0.0595145
$753$	−1345.46	−0.0651146
$754$	2285.14	0.110371
$755$	0	0
$756$	5266.69	0.253370
$757$	35240.9	1.69201	0.846005	−	0.533174i	$-0.179001\pi$
$757$	35240.9	1.69201	0.846005	+	0.533174i	$0.179001\pi$
$758$	2581.72	0.123710
$759$	803.599	0.0384306
$760$	0	0
$761$	15253.8	0.726609	0.363305	−	0.931670i	$-0.381648\pi$
$761$	15253.8	0.726609	0.363305	+	0.931670i	$0.381648\pi$
$762$	−664.329	−0.0315828
$763$	−9677.48	−0.459172
$764$	16183.0	0.766337
$765$	0	0
$766$	677.254	0.0319454
$767$	−33681.8	−1.58563
$768$	−1202.92	−0.0565189
$769$	33344.6	1.56364	0.781818	−	0.623507i	$-0.214293\pi$
$769$	33344.6	1.56364	0.781818	+	0.623507i	$0.214293\pi$
$770$	0	0
$771$	374.411	0.0174891
$772$	13982.8	0.651879
$773$	14412.0	0.670587	0.335294	−	0.942114i	$-0.391164\pi$
$773$	14412.0	0.670587	0.335294	+	0.942114i	$0.391164\pi$
$774$	2798.59	0.129966
$775$	0	0
$776$	−1627.98	−0.0753105
$777$	−2302.80	−0.106322
$778$	3229.11	0.148803
$779$	−5593.68	−0.257271
$780$	0	0
$781$	−17059.6	−0.781615
$782$	867.126	0.0396526
$783$	−2609.02	−0.119079
$784$	1042.60	0.0474947
$785$	0	0
$786$	−833.700	−0.0378335
$787$	15474.1	0.700880	0.350440	−	0.936585i	$-0.386032\pi$
$787$	15474.1	0.700880	0.350440	+	0.936585i	$0.386032\pi$
$788$	19771.5	0.893818
$789$	2322.86	0.104811
$790$	0	0
$791$	−45.6519	−0.00205208
$792$	−14816.8	−0.664764
$793$	234.930	0.0105203
$794$	10200.4	0.455916
$795$	0	0
$796$	−4698.37	−0.209208
$797$	−15561.2	−0.691603	−0.345802	−	0.938308i	$-0.612393\pi$
$797$	−15561.2	−0.691603	−0.345802	+	0.938308i	$0.612393\pi$
$798$	−549.658	−0.0243830
$799$	1248.50	0.0552801
$800$	0	0
$801$	30289.5	1.33611
$802$	4521.21	0.199064
$803$	−16079.7	−0.706649
$804$	5032.53	0.220751
$805$	0	0
$806$	9263.64	0.404836
$807$	1914.62	0.0835163
$808$	−10017.3	−0.436146
$809$	14327.9	0.622674	0.311337	−	0.950299i	$-0.399223\pi$
$809$	14327.9	0.622674	0.311337	+	0.950299i	$0.399223\pi$
$810$	0	0
$811$	−8539.31	−0.369736	−0.184868	−	0.982763i	$-0.559186\pi$
$811$	−8539.31	−0.369736	−0.184868	+	0.982763i	$0.559186\pi$
$812$	−10100.1	−0.436506
$813$	−4463.11	−0.192532
$814$	6323.18	0.272269
$815$	0	0
$816$	−1904.41	−0.0817005
$817$	8736.45	0.374112
$818$	8207.74	0.350828
$819$	22961.3	0.979647
$820$	0	0
$821$	42664.3	1.81364	0.906818	−	0.421523i	$-0.138504\pi$
$821$	42664.3	1.81364	0.906818	+	0.421523i	$0.138504\pi$
$822$	−1227.59	−0.0520890
$823$	−7687.86	−0.325616	−0.162808	−	0.986658i	$-0.552055\pi$
$823$	−7687.86	−0.325616	−0.162808	+	0.986658i	$0.552055\pi$
$824$	14653.8	0.619528
$825$	0	0
$826$	−10043.1	−0.423054
$827$	−27712.7	−1.16525	−0.582627	−	0.812740i	$-0.697975\pi$
$827$	−27712.7	−1.16525	−0.582627	+	0.812740i	$0.697975\pi$
$828$	−4572.18	−0.191901
$829$	−13667.8	−0.572619	−0.286310	−	0.958137i	$-0.592429\pi$
$829$	−13667.8	−0.572619	−0.286310	+	0.958137i	$0.592429\pi$
$830$	0	0
$831$	−1071.96	−0.0447483
$832$	−14900.1	−0.620876
$833$	1060.62	0.0441155
$834$	−120.230	−0.00499189
$835$	0	0
$836$	−22372.4	−0.925558
$837$	−10576.6	−0.436775
$838$	2624.65	0.108195
$839$	−13429.4	−0.552604	−0.276302	−	0.961071i	$-0.589109\pi$
$839$	−13429.4	−0.552604	−0.276302	+	0.961071i	$0.589109\pi$
$840$	0	0
$841$	−19385.6	−0.794851
$842$	5369.43	0.219766
$843$	−2970.81	−0.121376
$844$	23160.8	0.944584
$845$	0	0
$846$	444.108	0.0180482
$847$	−23619.2	−0.958164
$848$	19765.5	0.800411
$849$	4731.98	0.191285
$850$	0	0
$851$	4034.05	0.162497
$852$	−1737.68	−0.0698732
$853$	1540.70	0.0618434	0.0309217	−	0.999522i	$-0.490156\pi$
$853$	1540.70	0.0618434	0.0309217	+	0.999522i	$0.490156\pi$
$854$	70.0502	0.00280687
$855$	0	0
$856$	−13328.8	−0.532209
$857$	−33588.2	−1.33880	−0.669399	−	0.742903i	$-0.733448\pi$
$857$	−33588.2	−1.33880	−0.669399	+	0.742903i	$0.733448\pi$
$858$	1128.74	0.0449119
$859$	4955.28	0.196824	0.0984122	−	0.995146i	$-0.468624\pi$
$859$	4955.28	0.196824	0.0984122	+	0.995146i	$0.468624\pi$
$860$	0	0
$861$	−1247.35	−0.0493724
$862$	11679.8	0.461503
$863$	−41833.5	−1.65009	−0.825046	−	0.565066i	$-0.808851\pi$
$863$	−41833.5	−1.65009	−0.825046	+	0.565066i	$0.808851\pi$
$864$	−4617.91	−0.181834
$865$	0	0
$866$	4525.02	0.177559
$867$	1448.28	0.0567314
$868$	−40944.3	−1.60108
$869$	−29257.5	−1.14211
$870$	0	0
$871$	44273.5	1.72233
$872$	5596.03	0.217323
$873$	3919.51	0.151954
$874$	962.892	0.0372658
$875$	0	0
$876$	−1637.87	−0.0631716
$877$	−34708.9	−1.33642	−0.668208	−	0.743974i	$-0.732938\pi$
$877$	−34708.9	−1.33642	−0.668208	+	0.743974i	$0.732938\pi$
$878$	5342.80	0.205365
$879$	−4972.30	−0.190798
$880$	0	0
$881$	−6072.43	−0.232219	−0.116110	−	0.993236i	$-0.537042\pi$
$881$	−6072.43	−0.232219	−0.116110	+	0.993236i	$0.537042\pi$
$882$	377.275	0.0144031
$883$	−18884.3	−0.719712	−0.359856	−	0.933008i	$-0.617174\pi$
$883$	−18884.3	−0.719712	−0.359856	+	0.933008i	$0.617174\pi$
$884$	−18054.1	−0.686907
$885$	0	0
$886$	3138.16	0.118994
$887$	46338.3	1.75410	0.877051	−	0.480397i	$-0.159507\pi$
$887$	46338.3	1.75410	0.877051	+	0.480397i	$0.159507\pi$
$888$	1331.60	0.0503216
$889$	25831.7	0.974542
$890$	0	0
$891$	35022.9	1.31685
$892$	42738.1	1.60424
$893$	1386.39	0.0519526
$894$	1245.99	0.0466132
$895$	0	0
$896$	−23525.8	−0.877167
$897$	720.109	0.0268046
$898$	7676.40	0.285261
$899$	20283.0	0.752476
$900$	0	0
$901$	20106.9	0.743462
$902$	3425.06	0.126433
$903$	1948.17	0.0717951
$904$	26.3984	0.000971235 0
$905$	0	0
$906$	1458.53	0.0534839
$907$	22074.9	0.808141	0.404070	−	0.914728i	$-0.367595\pi$
$907$	22074.9	0.808141	0.404070	+	0.914728i	$0.367595\pi$
$908$	−24008.0	−0.877459
$909$	24117.5	0.880008
$910$	0	0
$911$	−16529.7	−0.601156	−0.300578	−	0.953757i	$-0.597180\pi$
$911$	−16529.7	−0.601156	−0.300578	+	0.953757i	$0.597180\pi$
$912$	−2114.73	−0.0767827
$913$	−11434.0	−0.414468
$914$	−11182.0	−0.404670
$915$	0	0
$916$	−4859.87	−0.175300
$917$	32417.5	1.16742
$918$	−1390.59	−0.0499960
$919$	37282.7	1.33824	0.669120	−	0.743154i	$-0.266671\pi$
$919$	37282.7	1.33824	0.669120	+	0.743154i	$0.266671\pi$
$920$	0	0
$921$	2429.04	0.0869052
$922$	709.291	0.0253354
$923$	−15287.2	−0.545162
$924$	−4988.89	−0.177622
$925$	0	0
$926$	−9484.35	−0.336582
$927$	−35280.6	−1.25002
$928$	8855.89	0.313264
$929$	37919.2	1.33917	0.669585	−	0.742736i	$-0.266472\pi$
$929$	37919.2	1.33917	0.669585	+	0.742736i	$0.266472\pi$
$930$	0	0
$931$	1177.75	0.0414600
$932$	−19163.0	−0.673505
$933$	−2327.99	−0.0816882
$934$	−4100.97	−0.143670
$935$	0	0
$936$	−13277.4	−0.463660
$937$	−7139.87	−0.248932	−0.124466	−	0.992224i	$-0.539722\pi$
$937$	−7139.87	−0.248932	−0.124466	+	0.992224i	$0.539722\pi$
$938$	13201.2	0.459527
$939$	2041.71	0.0709570
$940$	0	0
$941$	44548.2	1.54328	0.771642	−	0.636057i	$-0.219436\pi$
$941$	44548.2	1.54328	0.771642	+	0.636057i	$0.219436\pi$
$942$	1774.84	0.0613879
$943$	2185.11	0.0754582
$944$	−38639.3	−1.33221
$945$	0	0
$946$	−5349.42	−0.183853
$947$	11796.3	0.404782	0.202391	−	0.979305i	$-0.435129\pi$
$947$	11796.3	0.404782	0.202391	+	0.979305i	$0.435129\pi$
$948$	−2980.15	−0.102100
$949$	−14409.1	−0.492875
$950$	0	0
$951$	381.728	0.0130162
$952$	−11129.7	−0.378904
$953$	16156.5	0.549171	0.274586	−	0.961563i	$-0.411459\pi$
$953$	16156.5	0.549171	0.274586	+	0.961563i	$0.411459\pi$
$954$	7152.30	0.242730
$955$	0	0
$956$	−33158.7	−1.12179
$957$	2471.40	0.0834786
$958$	−6812.80	−0.229762
$959$	47733.5	1.60729
$960$	0	0
$961$	52433.5	1.76004
$962$	5666.23	0.189903
$963$	32090.5	1.07383
$964$	24995.3	0.835109
$965$	0	0
$966$	214.718	0.00715160
$967$	50550.0	1.68105	0.840527	−	0.541770i	$-0.182246\pi$
$967$	50550.0	1.68105	0.840527	+	0.541770i	$0.182246\pi$
$968$	13657.9	0.453493
$969$	−2151.27	−0.0713196
$970$	0	0
$971$	33282.9	1.10000	0.550000	−	0.835165i	$-0.314628\pi$
$971$	33282.9	1.10000	0.550000	+	0.835165i	$0.314628\pi$
$972$	11031.0	0.364011
$973$	4675.03	0.154033
$974$	−2989.42	−0.0983441
$975$	0	0
$976$	269.509	0.00883889
$977$	−2729.36	−0.0893755	−0.0446877	−	0.999001i	$-0.514229\pi$
$977$	−2729.36	−0.0893755	−0.0446877	+	0.999001i	$0.514229\pi$
$978$	−1708.49	−0.0558606
$979$	−57897.3	−1.89010
$980$	0	0
$981$	−13473.0	−0.438491
$982$	−4017.82	−0.130564
$983$	14904.5	0.483600	0.241800	−	0.970326i	$-0.422262\pi$
$983$	14904.5	0.483600	0.241800	+	0.970326i	$0.422262\pi$
$984$	721.285	0.0233676
$985$	0	0
$986$	2666.77	0.0861332
$987$	309.154	0.00997011
$988$	−20048.0	−0.645559
$989$	−3412.81	−0.109728
$990$	0	0
$991$	39748.4	1.27412	0.637058	−	0.770816i	$-0.280151\pi$
$991$	39748.4	1.27412	0.637058	+	0.770816i	$0.280151\pi$
$992$	35900.5	1.14904
$993$	1541.48	0.0492624
$994$	−4558.26	−0.145452
$995$	0	0
$996$	−1164.66	−0.0370518
$997$	−47343.4	−1.50389	−0.751946	−	0.659225i	$-0.770884\pi$
$997$	−47343.4	−1.50389	−0.751946	+	0.659225i	$0.770884\pi$
$998$	13659.6	0.433254
$999$	−6469.31	−0.204885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.a.m.1.3	yes	7
5.2	odd	4		575.4.b.j.24.7		14
5.3	odd	4		575.4.b.j.24.8		14
5.4	even	2		575.4.a.l.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
575.4.a.l.1.5	✓	7	5.4	even	2
575.4.a.m.1.3	yes	7	1.1	even	1	trivial
575.4.b.j.24.7		14	5.2	odd	4
575.4.b.j.24.8		14	5.3	odd	4

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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.a (trivial)

\(f(q)\)	\(=\)	\(q-0.711047 q^{2} -0.689108 q^{3} -7.49441 q^{4} +0.489988 q^{6} -19.0526 q^{7} +11.0173 q^{8} -26.5251 q^{9} +50.7019 q^{11} +5.16446 q^{12} +45.4342 q^{13} +13.5473 q^{14} +52.1215 q^{16} +53.0220 q^{17} +18.8606 q^{18} +58.8778 q^{19} +13.1293 q^{21} -36.0514 q^{22} -23.0000 q^{23} -7.59208 q^{24} -32.3058 q^{26} +36.8846 q^{27} +142.788 q^{28} -70.7345 q^{29} -286.748 q^{31} -125.199 q^{32} -34.9391 q^{33} -37.7011 q^{34} +198.790 q^{36} -175.393 q^{37} -41.8649 q^{38} -31.3091 q^{39} -95.0049 q^{41} -9.33557 q^{42} +148.383 q^{43} -379.981 q^{44} +16.3541 q^{46} +23.5469 q^{47} -35.9174 q^{48} +20.0033 q^{49} -36.5379 q^{51} -340.503 q^{52} +379.219 q^{53} -26.2267 q^{54} -209.908 q^{56} -40.5731 q^{57} +50.2956 q^{58} -741.332 q^{59} +5.17078 q^{61} +203.891 q^{62} +505.374 q^{63} -327.950 q^{64} +24.8433 q^{66} +974.454 q^{67} -397.368 q^{68} +15.8495 q^{69} -336.469 q^{71} -292.234 q^{72} -317.142 q^{73} +124.713 q^{74} -441.254 q^{76} -966.005 q^{77} +22.2622 q^{78} -577.050 q^{79} +690.761 q^{81} +67.5530 q^{82} -225.514 q^{83} -98.3966 q^{84} -105.507 q^{86} +48.7438 q^{87} +558.596 q^{88} -1141.92 q^{89} -865.642 q^{91} +172.371 q^{92} +197.600 q^{93} -16.7429 q^{94} +86.2756 q^{96} -147.766 q^{97} -14.2233 q^{98} -1344.87 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + q^{3} + 27 q^{4} - 41 q^{6} + q^{7} - 57 q^{8} + 10 q^{9} - 52 q^{11} + 65 q^{12} + 45 q^{13} - 42 q^{14} - 85 q^{16} + 85 q^{17} + 18 q^{18} - 10 q^{19} - 202 q^{21} - 71 q^{22} - 161 q^{23}+ \cdots - 2143 q^{99}+O(q^{100}) \) 7 * q + 3 * q^2 + q^3 + 27 * q^4 - 41 * q^6 + q^7 - 57 * q^8 + 10 * q^9 - 52 * q^11 + 65 * q^12 + 45 * q^13 - 42 * q^14 - 85 * q^16 + 85 * q^17 + 18 * q^18 - 10 * q^19 - 202 * q^21 - 71 * q^22 - 161 * q^23 - 303 * q^24 - 26 * q^26 - 242 * q^27 + 186 * q^28 - 455 * q^29 - 690 * q^31 - 233 * q^32 - 203 * q^33 - 657 * q^34 + 204 * q^36 - 892 * q^37 + 421 * q^38 - 277 * q^39 - 230 * q^41 - 888 * q^42 + 217 * q^43 - 1003 * q^44 - 69 * q^46 - 166 * q^47 + 1225 * q^48 - 802 * q^49 - 695 * q^51 - 1640 * q^52 + 1148 * q^53 - 1247 * q^54 - 386 * q^56 - 845 * q^57 + 1454 * q^58 - 551 * q^59 - 1242 * q^61 - 296 * q^62 + 1111 * q^63 - 1501 * q^64 + 987 * q^66 + 201 * q^67 + 1575 * q^68 - 23 * q^69 + 186 * q^71 - 2964 * q^72 + 530 * q^73 - 1530 * q^74 - 1775 * q^76 - 1339 * q^77 + 1364 * q^78 - 2003 * q^79 - 365 * q^81 - 2063 * q^82 + 726 * q^83 - 1352 * q^84 + 1894 * q^86 + 263 * q^87 + 2145 * q^88 - 2629 * q^89 - 1295 * q^91 - 621 * q^92 - 549 * q^93 - 3468 * q^94 - 135 * q^96 + 472 * q^97 + 2389 * q^98 - 2143 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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