LMFDB - Embedded newform 531.4.a.c.1.6

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:	$1$
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} - 41x^{5} - 7x^{4} + 484x^{3} + 63x^{2} - 1736x - 44 $ x^7 - 41x^5 - 7x^4 + 484x^3 + 63x^2 - 1736*x - 44
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			1.6
Root			$3.06012$ 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)

$f(q)$	$=$	$q+3.06012 q^{2} +1.36432 q^{4} -0.675250 q^{5} -3.38755 q^{7} -20.3060 q^{8} +O(q^{10})$	\(q+3.06012 q^{2} +1.36432 q^{4} -0.675250 q^{5} -3.38755 q^{7} -20.3060 q^{8} -2.06635 q^{10} +8.79097 q^{11} +59.4118 q^{13} -10.3663 q^{14} -73.0532 q^{16} -31.2380 q^{17} -77.5189 q^{19} -0.921258 q^{20} +26.9014 q^{22} -73.1225 q^{23} -124.544 q^{25} +181.807 q^{26} -4.62171 q^{28} +61.5678 q^{29} -278.253 q^{31} -61.1037 q^{32} -95.5920 q^{34} +2.28744 q^{35} -326.365 q^{37} -237.217 q^{38} +13.7116 q^{40} -453.551 q^{41} -174.901 q^{43} +11.9937 q^{44} -223.763 q^{46} +480.883 q^{47} -331.525 q^{49} -381.119 q^{50} +81.0568 q^{52} +185.568 q^{53} -5.93610 q^{55} +68.7874 q^{56} +188.405 q^{58} -59.0000 q^{59} +807.823 q^{61} -851.486 q^{62} +397.441 q^{64} -40.1178 q^{65} -129.081 q^{67} -42.6187 q^{68} +6.99985 q^{70} +272.032 q^{71} -610.054 q^{73} -998.716 q^{74} -105.761 q^{76} -29.7798 q^{77} +886.175 q^{79} +49.3292 q^{80} -1387.92 q^{82} -51.4871 q^{83} +21.0935 q^{85} -535.217 q^{86} -178.509 q^{88} -182.350 q^{89} -201.261 q^{91} -99.7625 q^{92} +1471.56 q^{94} +52.3447 q^{95} -1353.44 q^{97} -1014.50 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * 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 $

$2$	3.06012	1.08192	0.540958	−	0.841050i	$-0.318062\pi$
$2$	3.06012	1.08192	0.540958	+	0.841050i	$0.318062\pi$
$3$	0	0
$3$	0	0
$4$	1.36432	0.170540
$5$	−0.675250	−0.0603962	−0.0301981	−	0.999544i	$-0.509614\pi$
$5$	−0.675250	−0.0603962	−0.0301981	+	0.999544i	$0.509614\pi$
$6$	0	0
$7$	−3.38755	−0.182910	−0.0914552	−	0.995809i	$-0.529152\pi$
$7$	−3.38755	−0.182910	−0.0914552	+	0.995809i	$0.529152\pi$
$8$	−20.3060	−0.897405
$9$	0	0
$10$	−2.06635	−0.0653436
$11$	8.79097	0.240962	0.120481	−	0.992716i	$-0.461556\pi$
$11$	8.79097	0.240962	0.120481	+	0.992716i	$0.461556\pi$
$12$	0	0
$13$	59.4118	1.26753	0.633764	−	0.773526i	$-0.281509\pi$
$13$	59.4118	1.26753	0.633764	+	0.773526i	$0.281509\pi$
$14$	−10.3663	−0.197894
$15$	0	0
$16$	−73.0532	−1.14146
$17$	−31.2380	−0.445666	−0.222833	−	0.974857i	$-0.571531\pi$
$17$	−31.2380	−0.445666	−0.222833	+	0.974857i	$0.571531\pi$
$18$	0	0
$19$	−77.5189	−0.936003	−0.468002	−	0.883728i	$-0.655026\pi$
$19$	−77.5189	−0.936003	−0.468002	+	0.883728i	$0.655026\pi$
$20$	−0.921258	−0.0103000
$21$	0	0
$22$	26.9014	0.260700
$23$	−73.1225	−0.662917	−0.331458	−	0.943470i	$-0.607541\pi$
$23$	−73.1225	−0.662917	−0.331458	+	0.943470i	$0.607541\pi$
$24$	0	0
$25$	−124.544	−0.996352
$26$	181.807	1.37136
$27$	0	0
$28$	−4.62171	−0.0311936
$29$	61.5678	0.394236	0.197118	−	0.980380i	$-0.436842\pi$
$29$	61.5678	0.394236	0.197118	+	0.980380i	$0.436842\pi$
$30$	0	0
$31$	−278.253	−1.61212	−0.806059	−	0.591835i	$-0.798404\pi$
$31$	−278.253	−1.61212	−0.806059	+	0.591835i	$0.798404\pi$
$32$	−61.1037	−0.337554
$33$	0	0
$34$	−95.5920	−0.482173
$35$	2.28744	0.0110471
$36$	0	0
$37$	−326.365	−1.45011	−0.725055	−	0.688691i	$-0.758186\pi$
$37$	−326.365	−1.45011	−0.725055	+	0.688691i	$0.758186\pi$
$38$	−237.217	−1.01268
$39$	0	0
$40$	13.7116	0.0541999
$41$	−453.551	−1.72763	−0.863815	−	0.503809i	$-0.831931\pi$
$41$	−453.551	−1.72763	−0.863815	+	0.503809i	$0.831931\pi$
$42$	0	0
$43$	−174.901	−0.620282	−0.310141	−	0.950691i	$-0.600376\pi$
$43$	−174.901	−0.620282	−0.310141	+	0.950691i	$0.600376\pi$
$44$	11.9937	0.0410936
$45$	0	0
$46$	−223.763	−0.717220
$47$	480.883	1.49243	0.746213	−	0.665707i	$-0.231870\pi$
$47$	480.883	1.49243	0.746213	+	0.665707i	$0.231870\pi$
$48$	0	0
$49$	−331.525	−0.966544
$50$	−381.119	−1.07797
$51$	0	0
$52$	81.0568	0.216165
$53$	185.568	0.480938	0.240469	−	0.970657i	$-0.422699\pi$
$53$	185.568	0.480938	0.240469	+	0.970657i	$0.422699\pi$
$54$	0	0
$55$	−5.93610	−0.0145532
$56$	68.7874	0.164145
$57$	0	0
$58$	188.405	0.426530
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	807.823	1.69559	0.847796	−	0.530322i	$-0.177929\pi$
$61$	807.823	1.69559	0.847796	+	0.530322i	$0.177929\pi$
$62$	−851.486	−1.74417
$63$	0	0
$64$	397.441	0.776252
$65$	−40.1178	−0.0765539
$66$	0	0
$67$	−129.081	−0.235369	−0.117685	−	0.993051i	$-0.537547\pi$
$67$	−129.081	−0.235369	−0.117685	+	0.993051i	$0.537547\pi$
$68$	−42.6187	−0.0760040
$69$	0	0
$70$	6.99985	0.0119520
$71$	272.032	0.454708	0.227354	−	0.973812i	$-0.426993\pi$
$71$	272.032	0.454708	0.227354	+	0.973812i	$0.426993\pi$
$72$	0	0
$73$	−610.054	−0.978102	−0.489051	−	0.872255i	$-0.662657\pi$
$73$	−610.054	−0.978102	−0.489051	+	0.872255i	$0.662657\pi$
$74$	−998.716	−1.56890
$75$	0	0
$76$	−105.761	−0.159626
$77$	−29.7798	−0.0440744
$78$	0	0
$79$	886.175	1.26206	0.631028	−	0.775760i	$-0.282633\pi$
$79$	886.175	1.26206	0.631028	+	0.775760i	$0.282633\pi$
$80$	49.3292	0.0689396
$81$	0	0
$82$	−1387.92	−1.86915
$83$	−51.4871	−0.0680896	−0.0340448	−	0.999420i	$-0.510839\pi$
$83$	−51.4871	−0.0680896	−0.0340448	+	0.999420i	$0.510839\pi$
$84$	0	0
$85$	21.0935	0.0269166
$86$	−535.217	−0.671092
$87$	0	0
$88$	−178.509	−0.216240
$89$	−182.350	−0.217180	−0.108590	−	0.994087i	$-0.534634\pi$
$89$	−182.350	−0.217180	−0.108590	+	0.994087i	$0.534634\pi$
$90$	0	0
$91$	−201.261	−0.231844
$92$	−99.7625	−0.113054
$93$	0	0
$94$	1471.56	1.61468
$95$	52.3447	0.0565311
$96$	0	0
$97$	−1353.44	−1.41671	−0.708354	−	0.705857i	$-0.750562\pi$
$97$	−1353.44	−1.41671	−0.708354	+	0.705857i	$0.750562\pi$
$98$	−1014.50	−1.04572
$99$	0	0
$100$	−169.918	−0.169918
$101$	406.406	0.400385	0.200192	−	0.979757i	$-0.435843\pi$
$101$	406.406	0.400385	0.200192	+	0.979757i	$0.435843\pi$
$102$	0	0
$103$	−304.606	−0.291395	−0.145697	−	0.989329i	$-0.546543\pi$
$103$	−304.606	−0.291395	−0.145697	+	0.989329i	$0.546543\pi$
$104$	−1206.41	−1.13749
$105$	0	0
$106$	567.859	0.520334
$107$	1515.70	1.36942	0.684711	−	0.728815i	$-0.259929\pi$
$107$	1515.70	1.36942	0.684711	+	0.728815i	$0.259929\pi$
$108$	0	0
$109$	−1.57624	−0.00138511	−0.000692553	−	1.00000i	$-0.500220\pi$
$109$	−1.57624	−0.00138511	−0.000692553		1.00000i	$0.500220\pi$
$110$	−18.1652	−0.0157453
$111$	0	0
$112$	247.471	0.208784
$113$	577.813	0.481027	0.240514	−	0.970646i	$-0.422684\pi$
$113$	577.813	0.481027	0.240514	+	0.970646i	$0.422684\pi$
$114$	0	0
$115$	49.3760	0.0400377
$116$	83.9983	0.0672331
$117$	0	0
$118$	−180.547	−0.140853
$119$	105.820	0.0815171
$120$	0	0
$121$	−1253.72	−0.941938
$122$	2472.03	1.83449
$123$	0	0
$124$	−379.626	−0.274931
$125$	168.505	0.120572
$126$	0	0
$127$	291.667	0.203789	0.101895	−	0.994795i	$-0.467510\pi$
$127$	291.667	0.203789	0.101895	+	0.994795i	$0.467510\pi$
$128$	1705.05	1.17739
$129$	0	0
$130$	−122.765	−0.0828249
$131$	1160.30	0.773863	0.386932	−	0.922108i	$-0.373535\pi$
$131$	1160.30	0.773863	0.386932	+	0.922108i	$0.373535\pi$
$132$	0	0
$133$	262.599	0.171205
$134$	−395.003	−0.254649
$135$	0	0
$136$	634.318	0.399943
$137$	−571.419	−0.356348	−0.178174	−	0.983999i	$-0.557019\pi$
$137$	−571.419	−0.356348	−0.178174	+	0.983999i	$0.557019\pi$
$138$	0	0
$139$	−576.322	−0.351677	−0.175838	−	0.984419i	$-0.556264\pi$
$139$	−576.322	−0.351677	−0.175838	+	0.984419i	$0.556264\pi$
$140$	3.12081	0.00188397
$141$	0	0
$142$	832.450	0.491956
$143$	522.287	0.305426
$144$	0	0
$145$	−41.5737	−0.0238104
$146$	−1866.84	−1.05822
$147$	0	0
$148$	−445.267	−0.247302
$149$	1845.62	1.01476	0.507380	−	0.861723i	$-0.330614\pi$
$149$	1845.62	1.01476	0.507380	+	0.861723i	$0.330614\pi$
$150$	0	0
$151$	−1957.64	−1.05503	−0.527517	−	0.849545i	$-0.676877\pi$
$151$	−1957.64	−1.05503	−0.527517	+	0.849545i	$0.676877\pi$
$152$	1574.10	0.839974
$153$	0	0
$154$	−91.1298	−0.0476847
$155$	187.890	0.0973658
$156$	0	0
$157$	791.623	0.402410	0.201205	−	0.979549i	$-0.435514\pi$
$157$	791.623	0.402410	0.201205	+	0.979549i	$0.435514\pi$
$158$	2711.80	1.36544
$159$	0	0
$160$	41.2603	0.0203870
$161$	247.706	0.121254
$162$	0	0
$163$	−3364.00	−1.61650	−0.808248	−	0.588843i	$-0.799584\pi$
$163$	−3364.00	−1.61650	−0.808248	+	0.588843i	$0.799584\pi$
$164$	−618.790	−0.294630
$165$	0	0
$166$	−157.556	−0.0736672
$167$	−514.992	−0.238630	−0.119315	−	0.992856i	$-0.538070\pi$
$167$	−514.992	−0.238630	−0.119315	+	0.992856i	$0.538070\pi$
$168$	0	0
$169$	1332.76	0.606629
$170$	64.5485	0.0291214
$171$	0	0
$172$	−238.621	−0.105783
$173$	−2625.47	−1.15382	−0.576910	−	0.816808i	$-0.695742\pi$
$173$	−2625.47	−1.15382	−0.576910	+	0.816808i	$0.695742\pi$
$174$	0	0
$175$	421.899	0.182243
$176$	−642.208	−0.275047
$177$	0	0
$178$	−558.012	−0.234971
$179$	165.711	0.0691947	0.0345973	−	0.999401i	$-0.488985\pi$
$179$	165.711	0.0691947	0.0345973	+	0.999401i	$0.488985\pi$
$180$	0	0
$181$	1680.47	0.690100	0.345050	−	0.938584i	$-0.387862\pi$
$181$	1680.47	0.690100	0.345050	+	0.938584i	$0.387862\pi$
$182$	−615.881	−0.250836
$183$	0	0
$184$	1484.82	0.594905
$185$	220.378	0.0875812
$186$	0	0
$187$	−274.612	−0.107388
$188$	656.079	0.254519
$189$	0	0
$190$	160.181	0.0611618
$191$	825.467	0.312716	0.156358	−	0.987700i	$-0.450025\pi$
$191$	825.467	0.312716	0.156358	+	0.987700i	$0.450025\pi$
$192$	0	0
$193$	3110.12	1.15995	0.579977	−	0.814633i	$-0.303061\pi$
$193$	3110.12	1.15995	0.579977	+	0.814633i	$0.303061\pi$
$194$	−4141.68	−1.53276
$195$	0	0
$196$	−452.306	−0.164835
$197$	1252.31	0.452909	0.226455	−	0.974022i	$-0.427287\pi$
$197$	1252.31	0.452909	0.226455	+	0.974022i	$0.427287\pi$
$198$	0	0
$199$	854.395	0.304354	0.152177	−	0.988353i	$-0.451372\pi$
$199$	854.395	0.304354	0.152177	+	0.988353i	$0.451372\pi$
$200$	2528.99	0.894132
$201$	0	0
$202$	1243.65	0.433182
$203$	−208.564	−0.0721099
$204$	0	0
$205$	306.261	0.104342
$206$	−932.129	−0.315265
$207$	0	0
$208$	−4340.22	−1.44683
$209$	−681.466	−0.225541
$210$	0	0
$211$	5555.73	1.81266	0.906332	−	0.422566i	$-0.138870\pi$
$211$	5555.73	1.81266	0.906332	+	0.422566i	$0.138870\pi$
$212$	253.174	0.0820192
$213$	0	0
$214$	4638.22	1.48160
$215$	118.102	0.0374627
$216$	0	0
$217$	942.595	0.294873
$218$	−4.82349	−0.00149857
$219$	0	0
$220$	−8.09875	−0.00248190
$221$	−1855.91	−0.564895
$222$	0	0
$223$	−2120.53	−0.636776	−0.318388	−	0.947961i	$-0.603141\pi$
$223$	−2120.53	−0.636776	−0.318388	+	0.947961i	$0.603141\pi$
$224$	206.992	0.0617421
$225$	0	0
$226$	1768.18	0.520431
$227$	−3373.06	−0.986247	−0.493123	−	0.869959i	$-0.664145\pi$
$227$	−3373.06	−0.986247	−0.493123	+	0.869959i	$0.664145\pi$
$228$	0	0
$229$	1545.12	0.445872	0.222936	−	0.974833i	$-0.428436\pi$
$229$	1545.12	0.445872	0.222936	+	0.974833i	$0.428436\pi$
$230$	151.096	0.0433173
$231$	0	0
$232$	−1250.19	−0.353790
$233$	−4860.03	−1.36649	−0.683243	−	0.730191i	$-0.739431\pi$
$233$	−4860.03	−1.36649	−0.683243	+	0.730191i	$0.739431\pi$
$234$	0	0
$235$	−324.717	−0.0901369
$236$	−80.4950	−0.0222024
$237$	0	0
$238$	323.823	0.0881945
$239$	−6957.12	−1.88292	−0.941462	−	0.337119i	$-0.890548\pi$
$239$	−6957.12	−1.88292	−0.941462	+	0.337119i	$0.890548\pi$
$240$	0	0
$241$	−463.881	−0.123988	−0.0619942	−	0.998077i	$-0.519746\pi$
$241$	−463.881	−0.123988	−0.0619942	+	0.998077i	$0.519746\pi$
$242$	−3836.53	−1.01910
$243$	0	0
$244$	1102.13	0.289167
$245$	223.862	0.0583756
$246$	0	0
$247$	−4605.54	−1.18641
$248$	5650.19	1.44672
$249$	0	0
$250$	515.644	0.130449
$251$	7550.40	1.89871	0.949356	−	0.314202i	$-0.101737\pi$
$251$	7550.40	1.89871	0.949356	+	0.314202i	$0.101737\pi$
$252$	0	0
$253$	−642.817	−0.159737
$254$	892.534	0.220483
$255$	0	0
$256$	2038.11	0.497586
$257$	−69.0755	−0.0167658	−0.00838290	−	0.999965i	$-0.502668\pi$
$257$	−69.0755	−0.0167658	−0.00838290	+	0.999965i	$0.502668\pi$
$258$	0	0
$259$	1105.58	0.265240
$260$	−54.7336	−0.0130555
$261$	0	0
$262$	3550.66	0.837254
$263$	6387.24	1.49754	0.748772	−	0.662828i	$-0.230644\pi$
$263$	6387.24	1.49754	0.748772	+	0.662828i	$0.230644\pi$
$264$	0	0
$265$	−125.305	−0.0290468
$266$	803.585	0.185229
$267$	0	0
$268$	−176.108	−0.0401399
$269$	1359.76	0.308201	0.154100	−	0.988055i	$-0.450752\pi$
$269$	1359.76	0.308201	0.154100	+	0.988055i	$0.450752\pi$
$270$	0	0
$271$	−3184.33	−0.713780	−0.356890	−	0.934146i	$-0.616163\pi$
$271$	−3184.33	−0.713780	−0.356890	+	0.934146i	$0.616163\pi$
$272$	2282.04	0.508709
$273$	0	0
$274$	−1748.61	−0.385538
$275$	−1094.86	−0.240083
$276$	0	0
$277$	2086.69	0.452626	0.226313	−	0.974055i	$-0.427333\pi$
$277$	2086.69	0.452626	0.226313	+	0.974055i	$0.427333\pi$
$278$	−1763.61	−0.380484
$279$	0	0
$280$	−46.4487	−0.00991372
$281$	−6730.70	−1.42890	−0.714448	−	0.699688i	$-0.753322\pi$
$281$	−6730.70	−1.42890	−0.714448	+	0.699688i	$0.753322\pi$
$282$	0	0
$283$	6977.08	1.46553	0.732764	−	0.680483i	$-0.238230\pi$
$283$	6977.08	1.46553	0.732764	+	0.680483i	$0.238230\pi$
$284$	371.139	0.0775460
$285$	0	0
$286$	1598.26	0.330445
$287$	1536.43	0.316002
$288$	0	0
$289$	−3937.19	−0.801382
$290$	−127.220	−0.0257608
$291$	0	0
$292$	−832.310	−0.166806
$293$	−1173.58	−0.233999	−0.116999	−	0.993132i	$-0.537328\pi$
$293$	−1173.58	−0.233999	−0.116999	+	0.993132i	$0.537328\pi$
$294$	0	0
$295$	39.8398	0.00786292
$296$	6627.16	1.30134
$297$	0	0
$298$	5647.82	1.09788
$299$	−4344.34	−0.840266
$300$	0	0
$301$	592.485	0.113456
$302$	−5990.60	−1.14146
$303$	0	0
$304$	5663.01	1.06841
$305$	−545.483	−0.102407
$306$	0	0
$307$	−6437.77	−1.19682	−0.598409	−	0.801191i	$-0.704200\pi$
$307$	−6437.77	−1.19682	−0.598409	+	0.801191i	$0.704200\pi$
$308$	−40.6293	−0.00751646
$309$	0	0
$310$	574.966	0.105342
$311$	3665.91	0.668407	0.334204	−	0.942501i	$-0.391533\pi$
$311$	3665.91	0.668407	0.334204	+	0.942501i	$0.391533\pi$
$312$	0	0
$313$	−1795.66	−0.324270	−0.162135	−	0.986769i	$-0.551838\pi$
$313$	−1795.66	−0.324270	−0.162135	+	0.986769i	$0.551838\pi$
$314$	2422.46	0.435373
$315$	0	0
$316$	1209.03	0.215231
$317$	−793.810	−0.140646	−0.0703230	−	0.997524i	$-0.522403\pi$
$317$	−793.810	−0.140646	−0.0703230	+	0.997524i	$0.522403\pi$
$318$	0	0
$319$	541.240	0.0949958
$320$	−268.372	−0.0468827
$321$	0	0
$322$	758.009	0.131187
$323$	2421.54	0.417145
$324$	0	0
$325$	−7399.39	−1.26291
$326$	−10294.2	−1.74891
$327$	0	0
$328$	9209.80	1.55038
$329$	−1629.02	−0.272980
$330$	0	0
$331$	3322.57	0.551737	0.275868	−	0.961195i	$-0.411035\pi$
$331$	3322.57	0.551737	0.275868	+	0.961195i	$0.411035\pi$
$332$	−70.2449	−0.0116120
$333$	0	0
$334$	−1575.94	−0.258178
$335$	87.1619	0.0142154
$336$	0	0
$337$	2092.35	0.338213	0.169106	−	0.985598i	$-0.445912\pi$
$337$	2092.35	0.338213	0.169106	+	0.985598i	$0.445912\pi$
$338$	4078.42	0.656322
$339$	0	0
$340$	28.7783	0.00459036
$341$	−2446.11	−0.388458
$342$	0	0
$343$	2284.99	0.359701
$344$	3551.53	0.556644
$345$	0	0
$346$	−8034.25	−1.24834
$347$	9919.25	1.53456	0.767281	−	0.641311i	$-0.221609\pi$
$347$	9919.25	1.53456	0.767281	+	0.641311i	$0.221609\pi$
$348$	0	0
$349$	1359.74	0.208554	0.104277	−	0.994548i	$-0.466747\pi$
$349$	1359.74	0.208554	0.104277	+	0.994548i	$0.466747\pi$
$350$	1291.06	0.197172
$351$	0	0
$352$	−537.161	−0.0813374
$353$	1123.35	0.169376	0.0846880	−	0.996408i	$-0.473011\pi$
$353$	1123.35	0.169376	0.0846880	+	0.996408i	$0.473011\pi$
$354$	0	0
$355$	−183.690	−0.0274626
$356$	−248.784	−0.0370380
$357$	0	0
$358$	507.096	0.0748628
$359$	−4438.64	−0.652541	−0.326271	−	0.945276i	$-0.605792\pi$
$359$	−4438.64	−0.652541	−0.326271	+	0.945276i	$0.605792\pi$
$360$	0	0
$361$	−849.815	−0.123898
$362$	5142.43	0.746630
$363$	0	0
$364$	−274.584	−0.0395388
$365$	411.939	0.0590736
$366$	0	0
$367$	−12343.3	−1.75563	−0.877817	−	0.478996i	$-0.841001\pi$
$367$	−12343.3	−1.75563	−0.877817	+	0.478996i	$0.841001\pi$
$368$	5341.83	0.756690
$369$	0	0
$370$	674.383	0.0947554
$371$	−628.620	−0.0879685
$372$	0	0
$373$	8458.96	1.17423	0.587116	−	0.809503i	$-0.300263\pi$
$373$	8458.96	1.17423	0.587116	+	0.809503i	$0.300263\pi$
$374$	−840.346	−0.116185
$375$	0	0
$376$	−9764.79	−1.33931
$377$	3657.85	0.499706
$378$	0	0
$379$	−7901.32	−1.07088	−0.535440	−	0.844573i	$-0.679854\pi$
$379$	−7901.32	−1.07088	−0.535440	+	0.844573i	$0.679854\pi$
$380$	71.4150	0.00964082
$381$	0	0
$382$	2526.03	0.338332
$383$	−1775.56	−0.236885	−0.118443	−	0.992961i	$-0.537790\pi$
$383$	−1775.56	−0.236885	−0.118443	+	0.992961i	$0.537790\pi$
$384$	0	0
$385$	20.1088	0.00266193
$386$	9517.33	1.25497
$387$	0	0
$388$	−1846.52	−0.241606
$389$	−12845.2	−1.67423	−0.837115	−	0.547027i	$-0.815759\pi$
$389$	−12845.2	−1.67423	−0.837115	+	0.547027i	$0.815759\pi$
$390$	0	0
$391$	2284.20	0.295440
$392$	6731.92	0.867381
$393$	0	0
$394$	3832.20	0.490009
$395$	−598.390	−0.0762234
$396$	0	0
$397$	3824.54	0.483497	0.241749	−	0.970339i	$-0.422279\pi$
$397$	3824.54	0.483497	0.241749	+	0.970339i	$0.422279\pi$
$398$	2614.55	0.329285
$399$	0	0
$400$	9098.34	1.13729
$401$	10107.1	1.25867	0.629334	−	0.777135i	$-0.283328\pi$
$401$	10107.1	1.25867	0.629334	+	0.777135i	$0.283328\pi$
$402$	0	0
$403$	−16531.5	−2.04341
$404$	554.468	0.0682817
$405$	0	0
$406$	−638.230	−0.0780168
$407$	−2869.07	−0.349421
$408$	0	0
$409$	−863.888	−0.104441	−0.0522207	−	0.998636i	$-0.516630\pi$
$409$	−863.888	−0.104441	−0.0522207	+	0.998636i	$0.516630\pi$
$410$	937.194	0.112890
$411$	0	0
$412$	−415.580	−0.0496945
$413$	199.865	0.0238129
$414$	0	0
$415$	34.7666	0.00411235
$416$	−3630.28	−0.427859
$417$	0	0
$418$	−2085.37	−0.244016
$419$	−10373.1	−1.20945	−0.604725	−	0.796434i	$-0.706717\pi$
$419$	−10373.1	−1.20945	−0.604725	+	0.796434i	$0.706717\pi$
$420$	0	0
$421$	4058.69	0.469853	0.234927	−	0.972013i	$-0.424515\pi$
$421$	4058.69	0.469853	0.234927	+	0.972013i	$0.424515\pi$
$422$	17001.2	1.96115
$423$	0	0
$424$	−3768.13	−0.431596
$425$	3890.51	0.444041
$426$	0	0
$427$	−2736.54	−0.310142
$428$	2067.90	0.233541
$429$	0	0
$430$	361.405	0.0405314
$431$	13813.1	1.54375	0.771873	−	0.635777i	$-0.219320\pi$
$431$	13813.1	1.54375	0.771873	+	0.635777i	$0.219320\pi$
$432$	0	0
$433$	−14065.7	−1.56110	−0.780551	−	0.625093i	$-0.785061\pi$
$433$	−14065.7	−1.56110	−0.780551	+	0.625093i	$0.785061\pi$
$434$	2884.45	0.319028
$435$	0	0
$436$	−2.15050	−0.000236216 0
$437$	5668.37	0.620492
$438$	0	0
$439$	15470.0	1.68187	0.840937	−	0.541133i	$-0.182004\pi$
$439$	15470.0	1.68187	0.840937	+	0.541133i	$0.182004\pi$
$440$	120.538	0.0130601
$441$	0	0
$442$	−5679.29	−0.611168
$443$	−805.886	−0.0864307	−0.0432153	−	0.999066i	$-0.513760\pi$
$443$	−805.886	−0.0864307	−0.0432153	+	0.999066i	$0.513760\pi$
$444$	0	0
$445$	123.132	0.0131169
$446$	−6489.06	−0.688937
$447$	0	0
$448$	−1346.35	−0.141985
$449$	−15047.6	−1.58160	−0.790800	−	0.612074i	$-0.790335\pi$
$449$	−15047.6	−1.58160	−0.790800	+	0.612074i	$0.790335\pi$
$450$	0	0
$451$	−3987.16	−0.416292
$452$	788.323	0.0820345
$453$	0	0
$454$	−10322.0	−1.06704
$455$	135.901	0.0140025
$456$	0	0
$457$	2132.43	0.218273	0.109137	−	0.994027i	$-0.465191\pi$
$457$	2132.43	0.218273	0.109137	+	0.994027i	$0.465191\pi$
$458$	4728.26	0.482395
$459$	0	0
$460$	67.3647	0.00682803
$461$	7140.37	0.721389	0.360695	−	0.932684i	$-0.382540\pi$
$461$	7140.37	0.721389	0.360695	+	0.932684i	$0.382540\pi$
$462$	0	0
$463$	10919.2	1.09602	0.548011	−	0.836471i	$-0.315385\pi$
$463$	10919.2	1.09602	0.548011	+	0.836471i	$0.315385\pi$
$464$	−4497.72	−0.450003
$465$	0	0
$466$	−14872.3	−1.47842
$467$	−13946.8	−1.38197	−0.690987	−	0.722867i	$-0.742824\pi$
$467$	−13946.8	−1.38197	−0.690987	+	0.722867i	$0.742824\pi$
$468$	0	0
$469$	437.268	0.0430515
$470$	−993.671	−0.0975205
$471$	0	0
$472$	1198.05	0.116832
$473$	−1537.55	−0.149464
$474$	0	0
$475$	9654.52	0.932589
$476$	144.373	0.0139019
$477$	0	0
$478$	−21289.6	−2.03716
$479$	−18107.2	−1.72723	−0.863613	−	0.504155i	$-0.831804\pi$
$479$	−18107.2	−1.72723	−0.863613	+	0.504155i	$0.831804\pi$
$480$	0	0
$481$	−19389.9	−1.83806
$482$	−1419.53	−0.134145
$483$	0	0
$484$	−1710.48	−0.160638
$485$	913.908	0.0855638
$486$	0	0
$487$	−15930.5	−1.48230	−0.741150	−	0.671340i	$-0.765719\pi$
$487$	−15930.5	−1.48230	−0.741150	+	0.671340i	$0.765719\pi$
$488$	−16403.6	−1.52163
$489$	0	0
$490$	685.044	0.0631574
$491$	−2619.27	−0.240745	−0.120373	−	0.992729i	$-0.538409\pi$
$491$	−2619.27	−0.240745	−0.120373	+	0.992729i	$0.538409\pi$
$492$	0	0
$493$	−1923.25	−0.175698
$494$	−14093.5	−1.28360
$495$	0	0
$496$	20327.2	1.84016
$497$	−921.522	−0.0831709
$498$	0	0
$499$	−9350.73	−0.838870	−0.419435	−	0.907785i	$-0.637772\pi$
$499$	−9350.73	−0.838870	−0.419435	+	0.907785i	$0.637772\pi$
$500$	229.895	0.0205624
$501$	0	0
$502$	23105.1	2.05425
$503$	7300.43	0.647137	0.323568	−	0.946205i	$-0.395117\pi$
$503$	7300.43	0.647137	0.323568	+	0.946205i	$0.395117\pi$
$504$	0	0
$505$	−274.426	−0.0241817
$506$	−1967.10	−0.172822
$507$	0	0
$508$	397.927	0.0347542
$509$	−16241.5	−1.41432	−0.707162	−	0.707052i	$-0.750025\pi$
$509$	−16241.5	−1.41432	−0.707162	+	0.707052i	$0.750025\pi$
$510$	0	0
$511$	2066.59	0.178905
$512$	−7403.50	−0.639046
$513$	0	0
$514$	−211.379	−0.0181392
$515$	205.685	0.0175991
$516$	0	0
$517$	4227.43	0.359617
$518$	3383.20	0.286968
$519$	0	0
$520$	814.631	0.0686999
$521$	−21182.1	−1.78120	−0.890598	−	0.454792i	$-0.849714\pi$
$521$	−21182.1	−1.78120	−0.890598	+	0.454792i	$0.849714\pi$
$522$	0	0
$523$	19761.6	1.65222	0.826112	−	0.563505i	$-0.190548\pi$
$523$	19761.6	1.65222	0.826112	+	0.563505i	$0.190548\pi$
$524$	1583.03	0.131975
$525$	0	0
$526$	19545.7	1.62022
$527$	8692.06	0.718467
$528$	0	0
$529$	−6820.11	−0.560541
$530$	−383.447	−0.0314262
$531$	0	0
$532$	358.270	0.0291973
$533$	−26946.3	−2.18982
$534$	0	0
$535$	−1023.48	−0.0827079
$536$	2621.11	0.211221
$537$	0	0
$538$	4161.02	0.333447
$539$	−2914.42	−0.232900
$540$	0	0
$541$	1805.49	0.143482	0.0717412	−	0.997423i	$-0.477144\pi$
$541$	1805.49	0.143482	0.0717412	+	0.997423i	$0.477144\pi$
$542$	−9744.44	−0.772250
$543$	0	0
$544$	1908.76	0.150436
$545$	1.06436	8.36552e−5 0
$546$	0	0
$547$	−2017.11	−0.157670	−0.0788349	−	0.996888i	$-0.525120\pi$
$547$	−2017.11	−0.157670	−0.0788349	+	0.996888i	$0.525120\pi$
$548$	−779.599	−0.0607716
$549$	0	0
$550$	−3350.41	−0.259749
$551$	−4772.67	−0.369006
$552$	0	0
$553$	−3001.96	−0.230843
$554$	6385.53	0.489703
$555$	0	0
$556$	−786.289	−0.0599750
$557$	−22836.9	−1.73722	−0.868611	−	0.495495i	$-0.834987\pi$
$557$	−22836.9	−1.73722	−0.868611	+	0.495495i	$0.834987\pi$
$558$	0	0
$559$	−10391.2	−0.786225
$560$	−167.105	−0.0126098
$561$	0	0
$562$	−20596.7	−1.54594
$563$	11521.8	0.862494	0.431247	−	0.902234i	$-0.358074\pi$
$563$	11521.8	0.862494	0.431247	+	0.902234i	$0.358074\pi$
$564$	0	0
$565$	−390.168	−0.0290522
$566$	21350.7	1.58558
$567$	0	0
$568$	−5523.87	−0.408057
$569$	431.235	0.0317721	0.0158860	−	0.999874i	$-0.494943\pi$
$569$	431.235	0.0317721	0.0158860	+	0.999874i	$0.494943\pi$
$570$	0	0
$571$	7544.48	0.552937	0.276468	−	0.961023i	$-0.410836\pi$
$571$	7544.48	0.552937	0.276468	+	0.961023i	$0.410836\pi$
$572$	712.568	0.0520874
$573$	0	0
$574$	4701.65	0.341887
$575$	9106.97	0.660499
$576$	0	0
$577$	−12507.5	−0.902416	−0.451208	−	0.892419i	$-0.649007\pi$
$577$	−12507.5	−0.902416	−0.451208	+	0.892419i	$0.649007\pi$
$578$	−12048.3	−0.867027
$579$	0	0
$580$	−56.7198	−0.00406063
$581$	174.415	0.0124543
$582$	0	0
$583$	1631.32	0.115887
$584$	12387.7	0.877753
$585$	0	0
$586$	−3591.31	−0.253167
$587$	−12553.1	−0.882659	−0.441330	−	0.897345i	$-0.645493\pi$
$587$	−12553.1	−0.882659	−0.441330	+	0.897345i	$0.645493\pi$
$588$	0	0
$589$	21569.9	1.50895
$590$	121.914	0.00850701
$591$	0	0
$592$	23842.0	1.65524
$593$	4198.56	0.290749	0.145374	−	0.989377i	$-0.453561\pi$
$593$	4198.56	0.290749	0.145374	+	0.989377i	$0.453561\pi$
$594$	0	0
$595$	−71.4552	−0.00492332
$596$	2518.02	0.173057
$597$	0	0
$598$	−13294.2	−0.909097
$599$	4991.05	0.340449	0.170224	−	0.985405i	$-0.445551\pi$
$599$	4991.05	0.340449	0.170224	+	0.985405i	$0.445551\pi$
$600$	0	0
$601$	−8600.34	−0.583719	−0.291859	−	0.956461i	$-0.594274\pi$
$601$	−8600.34	−0.583719	−0.291859	+	0.956461i	$0.594274\pi$
$602$	1813.07	0.122750
$603$	0	0
$604$	−2670.84	−0.179926
$605$	846.574	0.0568895
$606$	0	0
$607$	−17833.1	−1.19246	−0.596230	−	0.802814i	$-0.703335\pi$
$607$	−17833.1	−1.19246	−0.596230	+	0.802814i	$0.703335\pi$
$608$	4736.70	0.315951
$609$	0	0
$610$	−1669.24	−0.110796
$611$	28570.2	1.89169
$612$	0	0
$613$	−22215.0	−1.46371	−0.731856	−	0.681459i	$-0.761346\pi$
$613$	−22215.0	−1.46371	−0.731856	+	0.681459i	$0.761346\pi$
$614$	−19700.3	−1.29485
$615$	0	0
$616$	604.708	0.0395526
$617$	−18251.1	−1.19086	−0.595432	−	0.803406i	$-0.703019\pi$
$617$	−18251.1	−1.19086	−0.595432	+	0.803406i	$0.703019\pi$
$618$	0	0
$619$	9872.40	0.641042	0.320521	−	0.947241i	$-0.396142\pi$
$619$	9872.40	0.641042	0.320521	+	0.947241i	$0.396142\pi$
$620$	256.343	0.0166048
$621$	0	0
$622$	11218.1	0.723160
$623$	617.719	0.0397245
$624$	0	0
$625$	15454.2	0.989070
$626$	−5494.92	−0.350833
$627$	0	0
$628$	1080.03	0.0686271
$629$	10195.0	0.646265
$630$	0	0
$631$	−9843.54	−0.621023	−0.310511	−	0.950570i	$-0.600500\pi$
$631$	−9843.54	−0.621023	−0.310511	+	0.950570i	$0.600500\pi$
$632$	−17994.6	−1.13258
$633$	0	0
$634$	−2429.15	−0.152167
$635$	−196.948	−0.0123081
$636$	0	0
$637$	−19696.5	−1.22512
$638$	1656.26	0.102777
$639$	0	0
$640$	−1151.33	−0.0711100
$641$	16239.8	1.00068	0.500338	−	0.865830i	$-0.333209\pi$
$641$	16239.8	1.00068	0.500338	+	0.865830i	$0.333209\pi$
$642$	0	0
$643$	−4102.04	−0.251584	−0.125792	−	0.992057i	$-0.540147\pi$
$643$	−4102.04	−0.251584	−0.125792	+	0.992057i	$0.540147\pi$
$644$	337.951	0.0206788
$645$	0	0
$646$	7410.19	0.451316
$647$	29339.0	1.78274	0.891371	−	0.453274i	$-0.149744\pi$
$647$	29339.0	1.78274	0.891371	+	0.453274i	$0.149744\pi$
$648$	0	0
$649$	−518.667	−0.0313705
$650$	−22643.0	−1.36636
$651$	0	0
$652$	−4589.58	−0.275677
$653$	22467.1	1.34641	0.673206	−	0.739455i	$-0.264917\pi$
$653$	22467.1	1.34641	0.673206	+	0.739455i	$0.264917\pi$
$654$	0	0
$655$	−783.494	−0.0467384
$656$	33133.4	1.97201
$657$	0	0
$658$	−4984.98	−0.295342
$659$	−14150.3	−0.836447	−0.418223	−	0.908344i	$-0.637347\pi$
$659$	−14150.3	−0.836447	−0.418223	+	0.908344i	$0.637347\pi$
$660$	0	0
$661$	−10577.7	−0.622428	−0.311214	−	0.950340i	$-0.600736\pi$
$661$	−10577.7	−0.622428	−0.311214	+	0.950340i	$0.600736\pi$
$662$	10167.4	0.596932
$663$	0	0
$664$	1045.49	0.0611040
$665$	−177.320	−0.0103401
$666$	0	0
$667$	−4501.99	−0.261346
$668$	−702.615	−0.0406961
$669$	0	0
$670$	266.726	0.0153799
$671$	7101.55	0.408573
$672$	0	0
$673$	−9785.95	−0.560506	−0.280253	−	0.959926i	$-0.590418\pi$
$673$	−9785.95	−0.560506	−0.280253	+	0.959926i	$0.590418\pi$
$674$	6402.85	0.365918
$675$	0	0
$676$	1818.32	0.103455
$677$	25327.9	1.43786	0.718929	−	0.695084i	$-0.244633\pi$
$677$	25327.9	1.43786	0.718929	+	0.695084i	$0.244633\pi$
$678$	0	0
$679$	4584.83	0.259131
$680$	−428.323	−0.0241551
$681$	0	0
$682$	−7485.39	−0.420279
$683$	−11813.3	−0.661819	−0.330909	−	0.943662i	$-0.607355\pi$
$683$	−11813.3	−0.661819	−0.330909	+	0.943662i	$0.607355\pi$
$684$	0	0
$685$	385.851	0.0215220
$686$	6992.32	0.389166
$687$	0	0
$688$	12777.1	0.708024
$689$	11024.9	0.609602
$690$	0	0
$691$	10486.4	0.577309	0.288655	−	0.957433i	$-0.406792\pi$
$691$	10486.4	0.577309	0.288655	+	0.957433i	$0.406792\pi$
$692$	−3581.99	−0.196773
$693$	0	0
$694$	30354.1	1.66027
$695$	389.162	0.0212399
$696$	0	0
$697$	14168.0	0.769946
$698$	4160.97	0.225637
$699$	0	0
$700$	575.606	0.0310798
$701$	−10841.7	−0.584144	−0.292072	−	0.956396i	$-0.594345\pi$
$701$	−10841.7	−0.584144	−0.292072	+	0.956396i	$0.594345\pi$
$702$	0	0
$703$	25299.5	1.35731
$704$	3493.89	0.187047
$705$	0	0
$706$	3437.57	0.183250
$707$	−1376.72	−0.0732346
$708$	0	0
$709$	−15139.3	−0.801930	−0.400965	−	0.916093i	$-0.631325\pi$
$709$	−15139.3	−0.801930	−0.400965	+	0.916093i	$0.631325\pi$
$710$	−562.112	−0.0297123
$711$	0	0
$712$	3702.79	0.194899
$713$	20346.5	1.06870
$714$	0	0
$715$	−352.675	−0.0184466
$716$	226.084	0.0118005
$717$	0	0
$718$	−13582.8	−0.705994
$719$	26507.9	1.37494	0.687468	−	0.726215i	$-0.258722\pi$
$719$	26507.9	1.37494	0.687468	+	0.726215i	$0.258722\pi$
$720$	0	0
$721$	1031.87	0.0532992
$722$	−2600.53	−0.134047
$723$	0	0
$724$	2292.70	0.117690
$725$	−7667.90	−0.392798
$726$	0	0
$727$	−4196.55	−0.214087	−0.107044	−	0.994254i	$-0.534138\pi$
$727$	−4196.55	−0.214087	−0.107044	+	0.994254i	$0.534138\pi$
$728$	4086.79	0.208058
$729$	0	0
$730$	1260.58	0.0639127
$731$	5463.55	0.276439
$732$	0	0
$733$	19609.3	0.988110	0.494055	−	0.869431i	$-0.335514\pi$
$733$	19609.3	0.988110	0.494055	+	0.869431i	$0.335514\pi$
$734$	−37772.1	−1.89945
$735$	0	0
$736$	4468.05	0.223770
$737$	−1134.75	−0.0567149
$738$	0	0
$739$	2865.95	0.142660	0.0713299	−	0.997453i	$-0.477276\pi$
$739$	2865.95	0.142660	0.0713299	+	0.997453i	$0.477276\pi$
$740$	300.667	0.0149361
$741$	0	0
$742$	−1923.65	−0.0951745
$743$	−6771.12	−0.334332	−0.167166	−	0.985929i	$-0.553462\pi$
$743$	−6771.12	−0.334332	−0.167166	+	0.985929i	$0.553462\pi$
$744$	0	0
$745$	−1246.26	−0.0612876
$746$	25885.4	1.27042
$747$	0	0
$748$	−374.659	−0.0183140
$749$	−5134.50	−0.250482
$750$	0	0
$751$	−15205.4	−0.738819	−0.369409	−	0.929267i	$-0.620440\pi$
$751$	−15205.4	−0.738819	−0.369409	+	0.929267i	$0.620440\pi$
$752$	−35130.1	−1.70354
$753$	0	0
$754$	11193.5	0.540639
$755$	1321.89	0.0637201
$756$	0	0
$757$	11748.5	0.564077	0.282038	−	0.959403i	$-0.408989\pi$
$757$	11748.5	0.564077	0.282038	+	0.959403i	$0.408989\pi$
$758$	−24179.0	−1.15860
$759$	0	0
$760$	−1062.91	−0.0507313
$761$	33071.4	1.57535	0.787673	−	0.616094i	$-0.211286\pi$
$761$	33071.4	1.57535	0.787673	+	0.616094i	$0.211286\pi$
$762$	0	0
$763$	5.33960	0.000253351 0
$764$	1126.20	0.0533306
$765$	0	0
$766$	−5433.43	−0.256290
$767$	−3505.30	−0.165018
$768$	0	0
$769$	32399.7	1.51933	0.759665	−	0.650315i	$-0.225363\pi$
$769$	32399.7	1.51933	0.759665	+	0.650315i	$0.225363\pi$
$770$	61.5354	0.00287998
$771$	0	0
$772$	4243.20	0.197819
$773$	35967.3	1.67355	0.836774	−	0.547548i	$-0.184439\pi$
$773$	35967.3	1.67355	0.836774	+	0.547548i	$0.184439\pi$
$774$	0	0
$775$	34654.7	1.60624
$776$	27482.8	1.27136
$777$	0	0
$778$	−39307.7	−1.81137
$779$	35158.8	1.61707
$780$	0	0
$781$	2391.43	0.109567
$782$	6989.92	0.319641
$783$	0	0
$784$	24218.9	1.10327
$785$	−534.543	−0.0243040
$786$	0	0
$787$	−2130.77	−0.0965106	−0.0482553	−	0.998835i	$-0.515366\pi$
$787$	−2130.77	−0.0965106	−0.0482553	+	0.998835i	$0.515366\pi$
$788$	1708.55	0.0772392
$789$	0	0
$790$	−1831.14	−0.0824673
$791$	−1957.37	−0.0879849
$792$	0	0
$793$	47994.3	2.14921
$794$	11703.6	0.523103
$795$	0	0
$796$	1165.67	0.0519046
$797$	−25863.7	−1.14949	−0.574743	−	0.818334i	$-0.694898\pi$
$797$	−25863.7	−1.14949	−0.574743	+	0.818334i	$0.694898\pi$
$798$	0	0
$799$	−15021.8	−0.665124
$800$	7610.11	0.336322
$801$	0	0
$802$	30929.0	1.36177
$803$	−5362.97	−0.235685
$804$	0	0
$805$	−167.263	−0.00732331
$806$	−50588.3	−2.21079
$807$	0	0
$808$	−8252.46	−0.359307
$809$	−26720.2	−1.16123	−0.580614	−	0.814179i	$-0.697187\pi$
$809$	−26720.2	−1.16123	−0.580614	+	0.814179i	$0.697187\pi$
$810$	0	0
$811$	23248.5	1.00662	0.503308	−	0.864107i	$-0.332116\pi$
$811$	23248.5	1.00662	0.503308	+	0.864107i	$0.332116\pi$
$812$	−284.548	−0.0122976
$813$	0	0
$814$	−8779.68	−0.378044
$815$	2271.54	0.0976302
$816$	0	0
$817$	13558.1	0.580586
$818$	−2643.60	−0.112997
$819$	0	0
$820$	417.838	0.0177946
$821$	−7799.17	−0.331538	−0.165769	−	0.986165i	$-0.553011\pi$
$821$	−7799.17	−0.331538	−0.165769	+	0.986165i	$0.553011\pi$
$822$	0	0
$823$	2499.10	0.105848	0.0529242	−	0.998599i	$-0.483146\pi$
$823$	2499.10	0.105848	0.0529242	+	0.998599i	$0.483146\pi$
$824$	6185.31	0.261499
$825$	0	0
$826$	611.612	0.0257636
$827$	39871.3	1.67649	0.838246	−	0.545292i	$-0.183581\pi$
$827$	39871.3	1.67649	0.838246	+	0.545292i	$0.183581\pi$
$828$	0	0
$829$	8721.76	0.365403	0.182702	−	0.983168i	$-0.441516\pi$
$829$	8721.76	0.365403	0.182702	+	0.983168i	$0.441516\pi$
$830$	106.390	0.00444922
$831$	0	0
$832$	23612.7	0.983922
$833$	10356.2	0.430756
$834$	0	0
$835$	347.749	0.0144124
$836$	−929.739	−0.0384638
$837$	0	0
$838$	−31742.9	−1.30852
$839$	11850.6	0.487637	0.243818	−	0.969821i	$-0.421600\pi$
$839$	11850.6	0.487637	0.243818	+	0.969821i	$0.421600\pi$
$840$	0	0
$841$	−20598.4	−0.844578
$842$	12420.1	0.508341
$843$	0	0
$844$	7579.80	0.309132
$845$	−899.950	−0.0366381
$846$	0	0
$847$	4247.04	0.172290
$848$	−13556.3	−0.548969
$849$	0	0
$850$	11905.4	0.480414
$851$	23864.6	0.961303
$852$	0	0
$853$	−25857.7	−1.03793	−0.518963	−	0.854797i	$-0.673682\pi$
$853$	−25857.7	−1.03793	−0.518963	+	0.854797i	$0.673682\pi$
$854$	−8374.14	−0.335547
$855$	0	0
$856$	−30777.7	−1.22893
$857$	13217.7	0.526847	0.263424	−	0.964680i	$-0.415148\pi$
$857$	13217.7	0.526847	0.263424	+	0.964680i	$0.415148\pi$
$858$	0	0
$859$	45527.8	1.80837	0.904184	−	0.427142i	$-0.140480\pi$
$859$	45527.8	1.80837	0.904184	+	0.427142i	$0.140480\pi$
$860$	161.129	0.00638889
$861$	0	0
$862$	42269.8	1.67020
$863$	−5044.12	−0.198961	−0.0994807	−	0.995039i	$-0.531718\pi$
$863$	−5044.12	−0.198961	−0.0994807	+	0.995039i	$0.531718\pi$
$864$	0	0
$865$	1772.85	0.0696864
$866$	−43042.9	−1.68898
$867$	0	0
$868$	1286.00	0.0502877
$869$	7790.33	0.304107
$870$	0	0
$871$	−7668.93	−0.298337
$872$	32.0071	0.00124300
$873$	0	0
$874$	17345.9	0.671320
$875$	−570.818	−0.0220539
$876$	0	0
$877$	−16427.0	−0.632498	−0.316249	−	0.948676i	$-0.602424\pi$
$877$	−16427.0	−0.632498	−0.316249	+	0.948676i	$0.602424\pi$
$878$	47340.0	1.81965
$879$	0	0
$880$	433.651	0.0166118
$881$	−49555.6	−1.89508	−0.947542	−	0.319631i	$-0.896441\pi$
$881$	−49555.6	−1.89508	−0.947542	+	0.319631i	$0.896441\pi$
$882$	0	0
$883$	−10096.6	−0.384798	−0.192399	−	0.981317i	$-0.561627\pi$
$883$	−10096.6	−0.384798	−0.192399	+	0.981317i	$0.561627\pi$
$884$	−2532.05	−0.0963373
$885$	0	0
$886$	−2466.11	−0.0935106
$887$	−17531.6	−0.663647	−0.331824	−	0.943341i	$-0.607664\pi$
$887$	−17531.6	−0.663647	−0.331824	+	0.943341i	$0.607664\pi$
$888$	0	0
$889$	−988.035	−0.0372752
$890$	376.798	0.0141913
$891$	0	0
$892$	−2893.08	−0.108596
$893$	−37277.6	−1.39692
$894$	0	0
$895$	−111.897	−0.00417910
$896$	−5775.93	−0.215357
$897$	0	0
$898$	−46047.3	−1.71116
$899$	−17131.4	−0.635555
$900$	0	0
$901$	−5796.77	−0.214338
$902$	−12201.2	−0.450393
$903$	0	0
$904$	−11733.0	−0.431676
$905$	−1134.74	−0.0416794
$906$	0	0
$907$	14205.5	0.520052	0.260026	−	0.965602i	$-0.416269\pi$
$907$	14205.5	0.520052	0.260026	+	0.965602i	$0.416269\pi$
$908$	−4601.94	−0.168195
$909$	0	0
$910$	415.874	0.0151495
$911$	6382.81	0.232132	0.116066	−	0.993242i	$-0.462972\pi$
$911$	6382.81	0.232132	0.116066	+	0.993242i	$0.462972\pi$
$912$	0	0
$913$	−452.621	−0.0164070
$914$	6525.48	0.236153
$915$	0	0
$916$	2108.04	0.0760390
$917$	−3930.58	−0.141548
$918$	0	0
$919$	−4515.26	−0.162073	−0.0810363	−	0.996711i	$-0.525823\pi$
$919$	−4515.26	−0.162073	−0.0810363	+	0.996711i	$0.525823\pi$
$920$	−1002.63	−0.0359300
$921$	0	0
$922$	21850.4	0.780482
$923$	16161.9	0.576356
$924$	0	0
$925$	40646.8	1.44482
$926$	33414.0	1.18580
$927$	0	0
$928$	−3762.02	−0.133076
$929$	33860.3	1.19583	0.597913	−	0.801561i	$-0.295997\pi$
$929$	33860.3	1.19583	0.597913	+	0.801561i	$0.295997\pi$
$930$	0	0
$931$	25699.4	0.904688
$932$	−6630.65	−0.233041
$933$	0	0
$934$	−42678.9	−1.49518
$935$	185.432	0.00648586
$936$	0	0
$937$	−22085.0	−0.769996	−0.384998	−	0.922917i	$-0.625798\pi$
$937$	−22085.0	−0.769996	−0.384998	+	0.922917i	$0.625798\pi$
$938$	1338.09	0.0465780
$939$	0	0
$940$	−443.018	−0.0153720
$941$	14767.9	0.511603	0.255801	−	0.966729i	$-0.417661\pi$
$941$	14767.9	0.511603	0.255801	+	0.966729i	$0.417661\pi$
$942$	0	0
$943$	33164.8	1.14527
$944$	4310.14	0.148605
$945$	0	0
$946$	−4705.07	−0.161707
$947$	−39640.1	−1.36022	−0.680111	−	0.733109i	$-0.738068\pi$
$947$	−39640.1	−1.36022	−0.680111	+	0.733109i	$0.738068\pi$
$948$	0	0
$949$	−36244.4	−1.23977
$950$	29544.0	1.00898
$951$	0	0
$952$	−2148.78	−0.0731538
$953$	10159.5	0.345328	0.172664	−	0.984981i	$-0.444763\pi$
$953$	10159.5	0.345328	0.172664	+	0.984981i	$0.444763\pi$
$954$	0	0
$955$	−557.397	−0.0188869
$956$	−9491.75	−0.321114
$957$	0	0
$958$	−55410.3	−1.86871
$959$	1935.71	0.0651797
$960$	0	0
$961$	47633.6	1.59892
$962$	−59335.5	−1.98862
$963$	0	0
$964$	−632.882	−0.0211450
$965$	−2100.11	−0.0700569
$966$	0	0
$967$	2430.67	0.0808325	0.0404163	−	0.999183i	$-0.487132\pi$
$967$	2430.67	0.0808325	0.0404163	+	0.999183i	$0.487132\pi$
$968$	25458.0	0.845300
$969$	0	0
$970$	2796.67	0.0925728
$971$	−7053.09	−0.233105	−0.116552	−	0.993185i	$-0.537184\pi$
$971$	−7053.09	−0.233105	−0.116552	+	0.993185i	$0.537184\pi$
$972$	0	0
$973$	1952.32	0.0643253
$974$	−48749.2	−1.60372
$975$	0	0
$976$	−59014.1	−1.93545
$977$	−36193.6	−1.18519	−0.592597	−	0.805499i	$-0.701897\pi$
$977$	−36193.6	−1.18519	−0.592597	+	0.805499i	$0.701897\pi$
$978$	0	0
$979$	−1603.03	−0.0523321
$980$	305.420	0.00995538
$981$	0	0
$982$	−8015.27	−0.260466
$983$	12172.1	0.394944	0.197472	−	0.980309i	$-0.436727\pi$
$983$	12172.1	0.394944	0.197472	+	0.980309i	$0.436727\pi$
$984$	0	0
$985$	−845.620	−0.0273540
$986$	−5885.39	−0.190090
$987$	0	0
$988$	−6283.44	−0.202331
$989$	12789.2	0.411195
$990$	0	0
$991$	37357.4	1.19747	0.598737	−	0.800946i	$-0.295670\pi$
$991$	37357.4	1.19747	0.598737	+	0.800946i	$0.295670\pi$
$992$	17002.3	0.544176
$993$	0	0
$994$	−2819.97	−0.0899838
$995$	−576.930	−0.0183818
$996$	0	0
$997$	−43827.1	−1.39220	−0.696098	−	0.717947i	$-0.745082\pi$
$997$	−43827.1	−1.39220	−0.696098	+	0.717947i	$0.745082\pi$
$998$	−28614.3	−0.907586
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.c.1.6		7
3.2	odd	2		177.4.a.b.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.b.1.2	✓	7	3.2	odd	2
531.4.a.c.1.6		7	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)

\(f(q)\)	\(=\)	\(q+3.06012 q^{2} +1.36432 q^{4} -0.675250 q^{5} -3.38755 q^{7} -20.3060 q^{8} +O(q^{10})\)	\(q+3.06012 q^{2} +1.36432 q^{4} -0.675250 q^{5} -3.38755 q^{7} -20.3060 q^{8} -2.06635 q^{10} +8.79097 q^{11} +59.4118 q^{13} -10.3663 q^{14} -73.0532 q^{16} -31.2380 q^{17} -77.5189 q^{19} -0.921258 q^{20} +26.9014 q^{22} -73.1225 q^{23} -124.544 q^{25} +181.807 q^{26} -4.62171 q^{28} +61.5678 q^{29} -278.253 q^{31} -61.1037 q^{32} -95.5920 q^{34} +2.28744 q^{35} -326.365 q^{37} -237.217 q^{38} +13.7116 q^{40} -453.551 q^{41} -174.901 q^{43} +11.9937 q^{44} -223.763 q^{46} +480.883 q^{47} -331.525 q^{49} -381.119 q^{50} +81.0568 q^{52} +185.568 q^{53} -5.93610 q^{55} +68.7874 q^{56} +188.405 q^{58} -59.0000 q^{59} +807.823 q^{61} -851.486 q^{62} +397.441 q^{64} -40.1178 q^{65} -129.081 q^{67} -42.6187 q^{68} +6.99985 q^{70} +272.032 q^{71} -610.054 q^{73} -998.716 q^{74} -105.761 q^{76} -29.7798 q^{77} +886.175 q^{79} +49.3292 q^{80} -1387.92 q^{82} -51.4871 q^{83} +21.0935 q^{85} -535.217 q^{86} -178.509 q^{88} -182.350 q^{89} -201.261 q^{91} -99.7625 q^{92} +1471.56 q^{94} +52.3447 q^{95} -1353.44 q^{97} -1014.50 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * q^98

Introduction

L-functions

Modular forms

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Database

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