LMFDB - Embedded newform 531.5.b.a.296.55

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(296,531)]

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([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("531.296");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			296.55
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.10252i q^{2} -0.830633 q^{4} -6.63332i q^{5} +3.66343 q^{7} +62.2326i q^{8} +O(q^{10})$	\(q+4.10252i q^{2} -0.830633 q^{4} -6.63332i q^{5} +3.66343 q^{7} +62.2326i q^{8} +27.2133 q^{10} +44.2447i q^{11} +32.0041 q^{13} +15.0293i q^{14} -268.600 q^{16} -340.426i q^{17} -603.386 q^{19} +5.50985i q^{20} -181.515 q^{22} +800.256i q^{23} +580.999 q^{25} +131.297i q^{26} -3.04297 q^{28} +239.753i q^{29} +356.875 q^{31} -106.215i q^{32} +1396.60 q^{34} -24.3007i q^{35} -154.813 q^{37} -2475.40i q^{38} +412.808 q^{40} +383.522i q^{41} -2582.87 q^{43} -36.7511i q^{44} -3283.06 q^{46} +2546.66i q^{47} -2387.58 q^{49} +2383.56i q^{50} -26.5836 q^{52} -502.821i q^{53} +293.489 q^{55} +227.985i q^{56} -983.588 q^{58} +453.188i q^{59} -6410.51 q^{61} +1464.09i q^{62} -3861.85 q^{64} -212.293i q^{65} +1829.26 q^{67} +282.769i q^{68} +99.6940 q^{70} +1832.05i q^{71} -125.909 q^{73} -635.123i q^{74} +501.192 q^{76} +162.088i q^{77} -1526.37 q^{79} +1781.71i q^{80} -1573.41 q^{82} +12144.3i q^{83} -2258.16 q^{85} -10596.3i q^{86} -2753.46 q^{88} -14425.5i q^{89} +117.245 q^{91} -664.718i q^{92} -10447.7 q^{94} +4002.45i q^{95} -14332.3 q^{97} -9795.08i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) $ 76 * q - 632 * q^4 - 48 * q^7	$ 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) $ 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$-1$	$1$

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 $

$2$	4.10252i	1.02563i	0.858499	+	0.512814i	$0.171397\pi$
$2$	4.10252i	1.02563i	−0.858499	+	0.512814i	$0.828603\pi$
$3$	0	0
$3$	0	0
$4$	−0.830633	−0.0519145
$5$	− 6.63332i	− 0.265333i	−0.991161	−	0.132666i	$-0.957646\pi$
$5$	− 6.63332i	− 0.265333i	0.991161	−	0.132666i	$-0.0423539\pi$
$6$	0	0
$7$	3.66343	0.0747639	0.0373820	−	0.999301i	$-0.488098\pi$
$7$	3.66343	0.0747639	0.0373820	+	0.999301i	$0.488098\pi$
$8$	62.2326i	0.972384i
$9$	0	0
$10$	27.2133	0.272133
$11$	44.2447i	0.365659i	0.983145	+	0.182830i	$0.0585256\pi$
$11$	44.2447i	0.365659i	−0.983145	+	0.182830i	$0.941474\pi$
$12$	0	0
$13$	32.0041	0.189373	0.0946865	−	0.995507i	$-0.469815\pi$
$13$	32.0041	0.189373	0.0946865	+	0.995507i	$0.469815\pi$
$14$	15.0293i	0.0766800i
$15$	0	0
$16$	−268.600	−1.04922
$17$	− 340.426i	− 1.17795i	−0.808153	−	0.588973i	$-0.799532\pi$
$17$	− 340.426i	− 1.17795i	0.808153	−	0.588973i	$-0.200468\pi$
$18$	0	0
$19$	−603.386	−1.67143	−0.835714	−	0.549165i	$-0.814946\pi$
$19$	−603.386	−1.67143	−0.835714	+	0.549165i	$0.814946\pi$
$20$	5.50985i	0.0137746i
$21$	0	0
$22$	−181.515	−0.375030
$23$	800.256i	1.51277i	0.654126	+	0.756385i	$0.273036\pi$
$23$	800.256i	1.51277i	−0.654126	+	0.756385i	$0.726964\pi$
$24$	0	0
$25$	580.999	0.929599
$26$	131.297i	0.194226i
$27$	0	0
$28$	−3.04297	−0.00388133
$29$	239.753i	0.285080i	0.989789	+	0.142540i	$0.0455270\pi$
$29$	239.753i	0.285080i	−0.989789	+	0.142540i	$0.954473\pi$
$30$	0	0
$31$	356.875	0.371358	0.185679	−	0.982610i	$-0.440551\pi$
$31$	356.875	0.371358	0.185679	+	0.982610i	$0.440551\pi$
$32$	− 106.215i	− 0.103726i
$33$	0	0
$34$	1396.60	1.20814
$35$	− 24.3007i	− 0.0198373i
$36$	0	0
$37$	−154.813	−0.113085	−0.0565424	−	0.998400i	$-0.518008\pi$
$37$	−154.813	−0.113085	−0.0565424	+	0.998400i	$0.518008\pi$
$38$	− 2475.40i	− 1.71427i
$39$	0	0
$40$	412.808	0.258005
$41$	383.522i	0.228151i	0.993472	+	0.114076i	$0.0363906\pi$
$41$	383.522i	0.228151i	−0.993472	+	0.114076i	$0.963609\pi$
$42$	0	0
$43$	−2582.87	−1.39690	−0.698451	−	0.715658i	$-0.746127\pi$
$43$	−2582.87	−1.39690	−0.698451	+	0.715658i	$0.746127\pi$
$44$	− 36.7511i	− 0.0189830i
$45$	0	0
$46$	−3283.06	−1.55154
$47$	2546.66i	1.15286i	0.817148	+	0.576428i	$0.195554\pi$
$47$	2546.66i	1.15286i	−0.817148	+	0.576428i	$0.804446\pi$
$48$	0	0
$49$	−2387.58	−0.994410
$50$	2383.56i	0.953423i
$51$	0	0
$52$	−26.5836	−0.00983122
$53$	− 502.821i	− 0.179004i	−0.995987	−	0.0895018i	$-0.971473\pi$
$53$	− 502.821i	− 0.179004i	0.995987	−	0.0895018i	$-0.0285275\pi$
$54$	0	0
$55$	293.489	0.0970213
$56$	227.985i	0.0726992i
$57$	0	0
$58$	−983.588	−0.292387
$59$	453.188i	0.130189i
$59$	453.188i	0.130189i
$60$	0	0
$61$	−6410.51	−1.72279	−0.861396	−	0.507934i	$-0.830409\pi$
$61$	−6410.51	−1.72279	−0.861396	+	0.507934i	$0.830409\pi$
$62$	1464.09i	0.380876i
$63$	0	0
$64$	−3861.85	−0.942835
$65$	− 212.293i	− 0.0502469i
$66$	0	0
$67$	1829.26	0.407498	0.203749	−	0.979023i	$-0.434687\pi$
$67$	1829.26	0.407498	0.203749	+	0.979023i	$0.434687\pi$
$68$	282.769i	0.0611525i
$69$	0	0
$70$	99.6940	0.0203457
$71$	1832.05i	0.363430i	0.983351	+	0.181715i	$0.0581649\pi$
$71$	1832.05i	0.363430i	−0.983351	+	0.181715i	$0.941835\pi$
$72$	0	0
$73$	−125.909	−0.0236271	−0.0118135	−	0.999930i	$-0.503760\pi$
$73$	−125.909	−0.0236271	−0.0118135	+	0.999930i	$0.503760\pi$
$74$	− 635.123i	− 0.115983i
$75$	0	0
$76$	501.192	0.0867714
$77$	162.088i	0.0273381i
$78$	0	0
$79$	−1526.37	−0.244571	−0.122286	−	0.992495i	$-0.539022\pi$
$79$	−1526.37	−0.244571	−0.122286	+	0.992495i	$0.539022\pi$
$80$	1781.71i	0.278392i
$81$	0	0
$82$	−1573.41	−0.233998
$83$	12144.3i	1.76286i	0.472317	+	0.881429i	$0.343418\pi$
$83$	12144.3i	1.76286i	−0.472317	+	0.881429i	$0.656582\pi$
$84$	0	0
$85$	−2258.16	−0.312548
$86$	− 10596.3i	− 1.43270i
$87$	0	0
$88$	−2753.46	−0.355561
$89$	− 14425.5i	− 1.82117i	−0.413322	−	0.910585i	$-0.635632\pi$
$89$	− 14425.5i	− 1.82117i	0.413322	−	0.910585i	$-0.364368\pi$
$90$	0	0
$91$	117.245	0.0141583
$92$	− 664.718i	− 0.0785348i
$93$	0	0
$94$	−10447.7	−1.18240
$95$	4002.45i	0.443485i
$96$	0	0
$97$	−14332.3	−1.52325	−0.761625	−	0.648018i	$-0.775598\pi$
$97$	−14332.3	−1.52325	−0.761625	+	0.648018i	$0.775598\pi$
$98$	− 9795.08i	− 1.01990i
$99$	0	0
$100$	−482.597	−0.0482597
$101$	− 8375.15i	− 0.821013i	−0.911858	−	0.410506i	$-0.865352\pi$
$101$	− 8375.15i	− 0.821013i	0.911858	−	0.410506i	$-0.134648\pi$
$102$	0	0
$103$	−11956.1	−1.12698	−0.563490	−	0.826123i	$-0.690542\pi$
$103$	−11956.1	−1.12698	−0.563490	+	0.826123i	$0.690542\pi$
$104$	1991.69i	0.184143i
$105$	0	0
$106$	2062.83	0.183591
$107$	13036.2i	1.13863i	0.822119	+	0.569316i	$0.192792\pi$
$107$	13036.2i	1.13863i	−0.822119	+	0.569316i	$0.807208\pi$
$108$	0	0
$109$	8632.79	0.726604	0.363302	−	0.931671i	$-0.381649\pi$
$109$	8632.79	0.726604	0.363302	+	0.931671i	$0.381649\pi$
$110$	1204.05i	0.0995079i
$111$	0	0
$112$	−983.998	−0.0784437
$113$	− 7900.79i	− 0.618748i	−0.950940	−	0.309374i	$-0.899881\pi$
$113$	− 7900.79i	− 0.618748i	0.950940	−	0.309374i	$-0.100119\pi$
$114$	0	0
$115$	5308.35	0.401387
$116$	− 199.146i	− 0.0147998i
$117$	0	0
$118$	−1859.21	−0.133526
$119$	− 1247.13i	− 0.0880678i
$120$	0	0
$121$	12683.4	0.866293
$122$	− 26299.2i	− 1.76695i
$123$	0	0
$124$	−296.432	−0.0192789
$125$	− 7999.77i	− 0.511986i
$126$	0	0
$127$	−10246.3	−0.635270	−0.317635	−	0.948213i	$-0.602889\pi$
$127$	−10246.3	−0.635270	−0.317635	+	0.948213i	$0.602889\pi$
$128$	− 17542.8i	− 1.07072i
$129$	0	0
$130$	870.935	0.0515346
$131$	− 27809.0i	− 1.62048i	−0.586099	−	0.810240i	$-0.699337\pi$
$131$	− 27809.0i	− 1.62048i	0.586099	−	0.810240i	$-0.300663\pi$
$132$	0	0
$133$	−2210.46	−0.124963
$134$	7504.56i	0.417942i
$135$	0	0
$136$	21185.6	1.14542
$137$	− 340.194i	− 0.0181253i	−0.999959	−	0.00906267i	$-0.997115\pi$
$137$	− 340.194i	− 0.0181253i	0.999959	−	0.00906267i	$-0.00288478\pi$
$138$	0	0
$139$	−16643.2	−0.861403	−0.430702	−	0.902494i	$-0.641734\pi$
$139$	−16643.2	−0.861403	−0.430702	+	0.902494i	$0.641734\pi$
$140$	20.1850i	0.00102984i
$141$	0	0
$142$	−7516.02	−0.372744
$143$	1416.01i	0.0692460i
$144$	0	0
$145$	1590.35	0.0756411
$146$	− 516.543i	− 0.0242326i
$147$	0	0
$148$	128.593	0.00587075
$149$	22687.6i	1.02192i	0.859605	+	0.510959i	$0.170710\pi$
$149$	22687.6i	1.02192i	−0.859605	+	0.510959i	$0.829290\pi$
$150$	0	0
$151$	−17877.4	−0.784062	−0.392031	−	0.919952i	$-0.628227\pi$
$151$	−17877.4	−0.784062	−0.392031	+	0.919952i	$0.628227\pi$
$152$	− 37550.2i	− 1.62527i
$153$	0	0
$154$	−664.967	−0.0280387
$155$	− 2367.27i	− 0.0985335i
$156$	0	0
$157$	−27724.7	−1.12478	−0.562390	−	0.826872i	$-0.690118\pi$
$157$	−27724.7	−1.12478	−0.562390	+	0.826872i	$0.690118\pi$
$158$	− 6261.95i	− 0.250839i
$159$	0	0
$160$	−704.560	−0.0275219
$161$	2931.68i	0.113101i
$162$	0	0
$163$	48961.9	1.84282	0.921411	−	0.388590i	$-0.127038\pi$
$163$	48961.9	1.84282	0.921411	+	0.388590i	$0.127038\pi$
$164$	− 318.566i	− 0.0118444i
$165$	0	0
$166$	−49822.3	−1.80804
$167$	22726.7i	0.814899i	0.913228	+	0.407450i	$0.133582\pi$
$167$	22726.7i	0.814899i	−0.913228	+	0.407450i	$0.866418\pi$
$168$	0	0
$169$	−27536.7	−0.964138
$170$	− 9264.12i	− 0.320558i
$171$	0	0
$172$	2145.42	0.0725195
$173$	− 32275.9i	− 1.07841i	−0.842173	−	0.539207i	$-0.818724\pi$
$173$	− 32275.9i	− 1.07841i	0.842173	−	0.539207i	$-0.181276\pi$
$174$	0	0
$175$	2128.45	0.0695004
$176$	− 11884.1i	− 0.383657i
$177$	0	0
$178$	59180.8	1.86784
$179$	42947.8i	1.34040i	0.742180	+	0.670201i	$0.233792\pi$
$179$	42947.8i	1.34040i	−0.742180	+	0.670201i	$0.766208\pi$
$180$	0	0
$181$	37439.0	1.14279	0.571396	−	0.820675i	$-0.306402\pi$
$181$	37439.0	1.14279	0.571396	+	0.820675i	$0.306402\pi$
$182$	480.998i	0.0145211i
$183$	0	0
$184$	−49802.0	−1.47099
$185$	1026.92i	0.0300051i
$186$	0	0
$187$	15062.1	0.430727
$188$	− 2115.34i	− 0.0598500i
$189$	0	0
$190$	−16420.1	−0.454851
$191$	− 58642.3i	− 1.60747i	−0.594985	−	0.803737i	$-0.702842\pi$
$191$	− 58642.3i	− 1.60747i	0.594985	−	0.803737i	$-0.297158\pi$
$192$	0	0
$193$	45005.1	1.20822	0.604111	−	0.796900i	$-0.293528\pi$
$193$	45005.1	1.20822	0.604111	+	0.796900i	$0.293528\pi$
$194$	− 58798.3i	− 1.56229i
$195$	0	0
$196$	1983.20	0.0516244
$197$	55233.1i	1.42320i	0.702582	+	0.711602i	$0.252030\pi$
$197$	55233.1i	1.42320i	−0.702582	+	0.711602i	$0.747970\pi$
$198$	0	0
$199$	−63861.9	−1.61263	−0.806316	−	0.591485i	$-0.798542\pi$
$199$	−63861.9	−1.61263	−0.806316	+	0.591485i	$0.798542\pi$
$200$	36157.1i	0.903927i
$201$	0	0
$202$	34359.2	0.842055
$203$	878.317i	0.0213137i
$204$	0	0
$205$	2544.02	0.0605360
$206$	− 49050.2i	− 1.15586i
$207$	0	0
$208$	−8596.29	−0.198694
$209$	− 26696.6i	− 0.611173i
$210$	0	0
$211$	29884.3	0.671240	0.335620	−	0.941998i	$-0.391054\pi$
$211$	29884.3	0.671240	0.335620	+	0.941998i	$0.391054\pi$
$212$	417.660i	0.00929289i
$213$	0	0
$214$	−53481.2	−1.16781
$215$	17133.0i	0.370644i
$216$	0	0
$217$	1307.39	0.0277642
$218$	35416.1i	0.745226i
$219$	0	0
$220$	−243.782	−0.00503682
$221$	− 10895.0i	− 0.223071i
$222$	0	0
$223$	−2943.34	−0.0591877	−0.0295938	−	0.999562i	$-0.509421\pi$
$223$	−2943.34	−0.0591877	−0.0295938	+	0.999562i	$0.509421\pi$
$224$	− 389.113i	− 0.00775495i
$225$	0	0
$226$	32413.1	0.634605
$227$	− 30535.6i	− 0.592591i	−0.955096	−	0.296296i	$-0.904249\pi$
$227$	− 30535.6i	− 0.592591i	0.955096	−	0.296296i	$-0.0957514\pi$
$228$	0	0
$229$	86377.9	1.64714	0.823572	−	0.567211i	$-0.191978\pi$
$229$	86377.9	1.64714	0.823572	+	0.567211i	$0.191978\pi$
$230$	21777.6i	0.411675i
$231$	0	0
$232$	−14920.4	−0.277207
$233$	2235.29i	0.0411739i	0.999788	+	0.0205869i	$0.00655349\pi$
$233$	2235.29i	0.0411739i	−0.999788	+	0.0205869i	$0.993447\pi$
$234$	0	0
$235$	16892.8	0.305891
$236$	− 376.432i	− 0.00675870i
$237$	0	0
$238$	5116.36	0.0903249
$239$	78377.4i	1.37213i	0.727540	+	0.686065i	$0.240663\pi$
$239$	78377.4i	1.37213i	−0.727540	+	0.686065i	$0.759337\pi$
$240$	0	0
$241$	−44559.1	−0.767189	−0.383594	−	0.923502i	$-0.625314\pi$
$241$	−44559.1	−0.767189	−0.383594	+	0.923502i	$0.625314\pi$
$242$	52033.9i	0.888496i
$243$	0	0
$244$	5324.78	0.0894380
$245$	15837.6i	0.263850i
$246$	0	0
$247$	−19310.8	−0.316524
$248$	22209.3i	0.361103i
$249$	0	0
$250$	32819.2	0.525107
$251$	− 35862.5i	− 0.569237i	−0.958641	−	0.284619i	$-0.908133\pi$
$251$	− 35862.5i	− 0.569237i	0.958641	−	0.284619i	$-0.0918669\pi$
$252$	0	0
$253$	−35407.1	−0.553158
$254$	− 42035.5i	− 0.651551i
$255$	0	0
$256$	10179.8	0.155331
$257$	29060.1i	0.439978i	0.975502	+	0.219989i	$0.0706021\pi$
$257$	29060.1i	0.439978i	−0.975502	+	0.219989i	$0.929398\pi$
$258$	0	0
$259$	−567.147	−0.00845467
$260$	176.338i	0.00260854i
$261$	0	0
$262$	114087.	1.66201
$263$	− 14630.6i	− 0.211519i	−0.994392	−	0.105760i	$-0.966273\pi$
$263$	− 14630.6i	− 0.211519i	0.994392	−	0.105760i	$-0.0337274\pi$
$264$	0	0
$265$	−3335.37	−0.0474955
$266$	− 9068.45i	− 0.128165i
$267$	0	0
$268$	−1519.44	−0.0211551
$269$	88337.6i	1.22079i	0.792097	+	0.610395i	$0.208989\pi$
$269$	88337.6i	1.22079i	−0.792097	+	0.610395i	$0.791011\pi$
$270$	0	0
$271$	−89165.3	−1.21411	−0.607054	−	0.794661i	$-0.707649\pi$
$271$	−89165.3	−1.21411	−0.607054	+	0.794661i	$0.707649\pi$
$272$	91438.6i	1.23592i
$273$	0	0
$274$	1395.65	0.0185899
$275$	25706.2i	0.339916i
$276$	0	0
$277$	−80041.3	−1.04317	−0.521584	−	0.853200i	$-0.674659\pi$
$277$	−80041.3	−1.04317	−0.521584	+	0.853200i	$0.674659\pi$
$278$	− 68278.9i	− 0.883480i
$279$	0	0
$280$	1512.30	0.0192895
$281$	138701.i	1.75657i	0.478135	+	0.878286i	$0.341313\pi$
$281$	138701.i	1.75657i	−0.478135	+	0.878286i	$0.658687\pi$
$282$	0	0
$283$	116931.	1.46001	0.730007	−	0.683439i	$-0.239517\pi$
$283$	116931.	1.46001	0.730007	+	0.683439i	$0.239517\pi$
$284$	− 1521.76i	− 0.0188673i
$285$	0	0
$286$	−5809.21	−0.0710207
$287$	1405.01i	0.0170575i
$288$	0	0
$289$	−32369.1	−0.387556
$290$	6524.45i	0.0775797i
$291$	0	0
$292$	104.584	0.00122659
$293$	131126.i	1.52741i	0.645567	+	0.763704i	$0.276621\pi$
$293$	131126.i	1.52741i	−0.645567	+	0.763704i	$0.723379\pi$
$294$	0	0
$295$	3006.14	0.0345434
$296$	− 9634.42i	− 0.109962i
$297$	0	0
$298$	−93076.3	−1.04811
$299$	25611.4i	0.286478i
$300$	0	0
$301$	−9462.17	−0.104438
$302$	− 73342.3i	− 0.804157i
$303$	0	0
$304$	162069.	1.75370
$305$	42522.9i	0.457113i
$306$	0	0
$307$	140000.	1.48543	0.742713	−	0.669610i	$-0.233539\pi$
$307$	140000.	1.48543	0.742713	+	0.669610i	$0.233539\pi$
$308$	− 134.635i	− 0.00141924i
$309$	0	0
$310$	9711.75	0.101059
$311$	78392.2i	0.810498i	0.914206	+	0.405249i	$0.132815\pi$
$311$	78392.2i	0.810498i	−0.914206	+	0.405249i	$0.867185\pi$
$312$	0	0
$313$	29293.1	0.299004	0.149502	−	0.988761i	$-0.452233\pi$
$313$	29293.1	0.299004	0.149502	+	0.988761i	$0.452233\pi$
$314$	− 113741.i	− 1.15361i
$315$	0	0
$316$	1267.85	0.0126968
$317$	86378.5i	0.859582i	0.902928	+	0.429791i	$0.141413\pi$
$317$	86378.5i	0.859582i	−0.902928	+	0.429791i	$0.858587\pi$
$318$	0	0
$319$	−10607.8	−0.104242
$320$	25616.9i	0.250165i
$321$	0	0
$322$	−12027.3	−0.115999
$323$	205408.i	1.96885i
$324$	0	0
$325$	18594.3	0.176041
$326$	200867.i	1.89005i
$327$	0	0
$328$	−23867.6	−0.221851
$329$	9329.52i	0.0861921i
$330$	0	0
$331$	−155534.	−1.41961	−0.709806	−	0.704397i	$-0.751217\pi$
$331$	−155534.	−1.41961	−0.709806	+	0.704397i	$0.751217\pi$
$332$	− 10087.5i	− 0.0915179i
$333$	0	0
$334$	−93236.7	−0.835784
$335$	− 12134.1i	− 0.108123i
$336$	0	0
$337$	128245.	1.12923	0.564614	−	0.825355i	$-0.309025\pi$
$337$	128245.	1.12923	0.564614	+	0.825355i	$0.309025\pi$
$338$	− 112970.i	− 0.988848i
$339$	0	0
$340$	1875.70	0.0162258
$341$	15789.9i	0.135791i
$342$	0	0
$343$	−17542.6	−0.149110
$344$	− 160739.i	− 1.35832i
$345$	0	0
$346$	132412.	1.10605
$347$	− 112231.i	− 0.932080i	−0.884764	−	0.466040i	$-0.845680\pi$
$347$	− 112231.i	− 0.932080i	0.884764	−	0.466040i	$-0.154320\pi$
$348$	0	0
$349$	160932.	1.32127	0.660635	−	0.750708i	$-0.270287\pi$
$349$	160932.	1.32127	0.660635	+	0.750708i	$0.270287\pi$
$350$	8732.00i	0.0712816i
$351$	0	0
$352$	4699.47	0.0379283
$353$	22177.4i	0.177976i	0.996033	+	0.0889878i	$0.0283632\pi$
$353$	22177.4i	0.177976i	−0.996033	+	0.0889878i	$0.971637\pi$
$354$	0	0
$355$	12152.6	0.0964299
$356$	11982.3i	0.0945452i
$357$	0	0
$358$	−176194.	−1.37475
$359$	97089.4i	0.753326i	0.926350	+	0.376663i	$0.122929\pi$
$359$	97089.4i	0.753326i	−0.926350	+	0.376663i	$0.877071\pi$
$360$	0	0
$361$	233753.	1.79367
$362$	153594.i	1.17208i
$363$	0	0
$364$	−97.3872	−0.000735020 0
$365$	835.193i	0.00626904i
$366$	0	0
$367$	−147769.	−1.09711	−0.548556	−	0.836114i	$-0.684822\pi$
$367$	−147769.	−1.09711	−0.548556	+	0.836114i	$0.684822\pi$
$368$	− 214949.i	− 1.58723i
$369$	0	0
$370$	−4212.98	−0.0307741
$371$	− 1842.05i	− 0.0133830i
$372$	0	0
$373$	25595.8	0.183971	0.0919857	−	0.995760i	$-0.470679\pi$
$373$	25595.8	0.183971	0.0919857	+	0.995760i	$0.470679\pi$
$374$	61792.4i	0.441766i
$375$	0	0
$376$	−158485.	−1.12102
$377$	7673.05i	0.0539865i
$378$	0	0
$379$	264410.	1.84077	0.920386	−	0.391010i	$-0.127874\pi$
$379$	264410.	1.84077	0.920386	+	0.391010i	$0.127874\pi$
$380$	− 3324.56i	− 0.0230233i
$381$	0	0
$382$	240581.	1.64867
$383$	− 49039.5i	− 0.334309i	−0.985931	−	0.167155i	$-0.946542\pi$
$383$	− 49039.5i	− 0.334309i	0.985931	−	0.167155i	$-0.0534579\pi$
$384$	0	0
$385$	1075.18	0.00725369
$386$	184634.i	1.23919i
$387$	0	0
$388$	11904.8	0.0790788
$389$	103660.i	0.685031i	0.939512	+	0.342515i	$0.111279\pi$
$389$	103660.i	0.685031i	−0.939512	+	0.342515i	$0.888721\pi$
$390$	0	0
$391$	272428.	1.78196
$392$	− 148585.i	− 0.966949i
$393$	0	0
$394$	−226595.	−1.45968
$395$	10124.9i	0.0648928i
$396$	0	0
$397$	−184214.	−1.16880	−0.584402	−	0.811464i	$-0.698671\pi$
$397$	−184214.	−1.16880	−0.584402	+	0.811464i	$0.698671\pi$
$398$	− 261994.i	− 1.65396i
$399$	0	0
$400$	−156056.	−0.975353
$401$	− 179457.i	− 1.11602i	−0.829835	−	0.558010i	$-0.811565\pi$
$401$	− 179457.i	− 1.11602i	0.829835	−	0.558010i	$-0.188435\pi$
$402$	0	0
$403$	11421.5	0.0703253
$404$	6956.68i	0.0426225i
$405$	0	0
$406$	−3603.31	−0.0218600
$407$	− 6849.67i	− 0.0413505i
$408$	0	0
$409$	72402.3	0.432818	0.216409	−	0.976303i	$-0.430565\pi$
$409$	72402.3	0.432818	0.216409	+	0.976303i	$0.430565\pi$
$410$	10436.9i	0.0620874i
$411$	0	0
$412$	9931.16	0.0585067
$413$	1660.22i	0.00973343i
$414$	0	0
$415$	80557.2	0.467744
$416$	− 3399.32i	− 0.0196429i
$417$	0	0
$418$	109523.	0.626837
$419$	293865.i	1.67386i	0.547309	+	0.836931i	$0.315652\pi$
$419$	293865.i	1.67386i	−0.547309	+	0.836931i	$0.684348\pi$
$420$	0	0
$421$	41354.2	0.233322	0.116661	−	0.993172i	$-0.462781\pi$
$421$	41354.2	0.233322	0.116661	+	0.993172i	$0.462781\pi$
$422$	122601.i	0.688443i
$423$	0	0
$424$	31291.9	0.174060
$425$	− 197787.i	− 1.09502i
$426$	0	0
$427$	−23484.5	−0.128803
$428$	− 10828.3i	− 0.0591115i
$429$	0	0
$430$	−70288.4	−0.380143
$431$	− 80255.8i	− 0.432038i	−0.976389	−	0.216019i	$-0.930693\pi$
$431$	− 80255.8i	− 0.432038i	0.976389	−	0.216019i	$-0.0693073\pi$
$432$	0	0
$433$	96385.7	0.514087	0.257044	−	0.966400i	$-0.417252\pi$
$433$	96385.7	0.514087	0.257044	+	0.966400i	$0.417252\pi$
$434$	5363.58i	0.0284758i
$435$	0	0
$436$	−7170.67	−0.0377213
$437$	− 482863.i	− 2.52849i
$438$	0	0
$439$	244788.	1.27017	0.635085	−	0.772442i	$-0.280965\pi$
$439$	244788.	1.27017	0.635085	+	0.772442i	$0.280965\pi$
$440$	18264.6i	0.0943419i
$441$	0	0
$442$	44697.0	0.228788
$443$	41377.4i	0.210841i	0.994428	+	0.105421i	$0.0336189\pi$
$443$	41377.4i	0.210841i	−0.994428	+	0.105421i	$0.966381\pi$
$444$	0	0
$445$	−95688.8	−0.483216
$446$	− 12075.1i	− 0.0607046i
$447$	0	0
$448$	−14147.6	−0.0704900
$449$	− 236114.i	− 1.17119i	−0.810603	−	0.585597i	$-0.800860\pi$
$449$	− 236114.i	− 1.17119i	0.810603	−	0.585597i	$-0.199140\pi$
$450$	0	0
$451$	−16968.8	−0.0834255
$452$	6562.65i	0.0321220i
$453$	0	0
$454$	125273.	0.607779
$455$	− 777.721i	− 0.00375665i
$456$	0	0
$457$	−79310.1	−0.379748	−0.189874	−	0.981808i	$-0.560808\pi$
$457$	−79310.1	−0.379748	−0.189874	+	0.981808i	$0.560808\pi$
$458$	354367.i	1.68936i
$459$	0	0
$460$	−4409.29	−0.0208378
$461$	225969.i	1.06328i	0.846970	+	0.531640i	$0.178424\pi$
$461$	225969.i	1.06328i	−0.846970	+	0.531640i	$0.821576\pi$
$462$	0	0
$463$	−3993.47	−0.0186289	−0.00931447	−	0.999957i	$-0.502965\pi$
$463$	−3993.47	−0.0186289	−0.00931447	+	0.999957i	$0.502965\pi$
$464$	− 64397.6i	− 0.299112i
$465$	0	0
$466$	−9170.31	−0.0422291
$467$	95763.2i	0.439101i	0.975601	+	0.219551i	$0.0704591\pi$
$467$	95763.2i	0.439101i	−0.975601	+	0.219551i	$0.929541\pi$
$468$	0	0
$469$	6701.36	0.0304661
$470$	69303.0i	0.313730i
$471$	0	0
$472$	−28203.0	−0.126594
$473$	− 114278.i	− 0.510790i
$474$	0	0
$475$	−350567.	−1.55376
$476$	1035.91i	0.00457200i
$477$	0	0
$478$	−321545.	−1.40730
$479$	102843.i	0.448234i	0.974562	+	0.224117i	$0.0719499\pi$
$479$	102843.i	0.448234i	−0.974562	+	0.224117i	$0.928050\pi$
$480$	0	0
$481$	−4954.65	−0.0214152
$482$	− 182804.i	− 0.786851i
$483$	0	0
$484$	−10535.2	−0.0449732
$485$	95070.4i	0.404168i
$486$	0	0
$487$	72008.8	0.303618	0.151809	−	0.988410i	$-0.451490\pi$
$487$	72008.8	0.303618	0.151809	+	0.988410i	$0.451490\pi$
$488$	− 398942.i	− 1.67522i
$489$	0	0
$490$	−64973.9	−0.270612
$491$	98779.6i	0.409736i	0.978790	+	0.204868i	$0.0656765\pi$
$491$	98779.6i	0.409736i	−0.978790	+	0.204868i	$0.934323\pi$
$492$	0	0
$493$	81618.1	0.335809
$494$	− 79222.8i	− 0.324636i
$495$	0	0
$496$	−95856.8	−0.389636
$497$	6711.60i	0.0271715i
$498$	0	0
$499$	108301.	0.434942	0.217471	−	0.976067i	$-0.430219\pi$
$499$	108301.	0.434942	0.217471	+	0.976067i	$0.430219\pi$
$500$	6644.87i	0.0265795i
$501$	0	0
$502$	147126.	0.583826
$503$	− 214167.i	− 0.846481i	−0.906017	−	0.423240i	$-0.860893\pi$
$503$	− 214167.i	− 0.846481i	0.906017	−	0.423240i	$-0.139107\pi$
$504$	0	0
$505$	−55555.0	−0.217842
$506$	− 145258.i	− 0.567335i
$507$	0	0
$508$	8510.89	0.0329798
$509$	− 334006.i	− 1.28919i	−0.764522	−	0.644597i	$-0.777025\pi$
$509$	− 334006.i	− 1.28919i	0.764522	−	0.644597i	$-0.222975\pi$
$510$	0	0
$511$	−461.258	−0.00176645
$512$	− 238921.i	− 0.911413i
$513$	0	0
$514$	−119219.	−0.451254
$515$	79308.9i	0.299025i
$516$	0	0
$517$	−112676.	−0.421552
$518$	− 2326.73i	− 0.00867135i
$519$	0	0
$520$	13211.5	0.0488592
$521$	− 462012.i	− 1.70207i	−0.525107	−	0.851036i	$-0.675975\pi$
$521$	− 462012.i	− 1.70207i	0.525107	−	0.851036i	$-0.324025\pi$
$522$	0	0
$523$	323406.	1.18235	0.591174	−	0.806544i	$-0.298665\pi$
$523$	323406.	1.18235	0.591174	+	0.806544i	$0.298665\pi$
$524$	23099.1i	0.0841264i
$525$	0	0
$526$	60022.1	0.216940
$527$	− 121490.i	− 0.437440i
$528$	0	0
$529$	−360568.	−1.28847
$530$	− 13683.4i	− 0.0487128i
$531$	0	0
$532$	1836.08	0.00648737
$533$	12274.3i	0.0432057i
$534$	0	0
$535$	86473.2	0.302116
$536$	113839.i	0.396245i
$537$	0	0
$538$	−362406.	−1.25208
$539$	− 105638.i	− 0.363615i
$540$	0	0
$541$	198892.	0.679552	0.339776	−	0.940506i	$-0.389649\pi$
$541$	198892.	0.679552	0.339776	+	0.940506i	$0.389649\pi$
$542$	− 365802.i	− 1.24522i
$543$	0	0
$544$	−36158.5	−0.122183
$545$	− 57264.0i	− 0.192792i
$546$	0	0
$547$	265497.	0.887329	0.443664	−	0.896193i	$-0.353678\pi$
$547$	265497.	0.887329	0.443664	+	0.896193i	$0.353678\pi$
$548$	282.577i	0 0.000940968i
$549$	0	0
$550$	−105460.	−0.348628
$551$	− 144663.i	− 0.476491i
$552$	0	0
$553$	−5591.75	−0.0182851
$554$	− 328371.i	− 1.06990i
$555$	0	0
$556$	13824.4	0.0447193
$557$	535867.i	1.72722i	0.504164	+	0.863608i	$0.331801\pi$
$557$	535867.i	1.72722i	−0.504164	+	0.863608i	$0.668199\pi$
$558$	0	0
$559$	−82662.3	−0.264536
$560$	6527.17i	0.0208137i
$561$	0	0
$562$	−569022.	−1.80159
$563$	− 178426.i	− 0.562912i	−0.959574	−	0.281456i	$-0.909183\pi$
$563$	− 178426.i	− 0.562912i	0.959574	−	0.281456i	$-0.0908174\pi$
$564$	0	0
$565$	−52408.4	−0.164174
$566$	479712.i	1.49743i
$567$	0	0
$568$	−114013.	−0.353394
$569$	429115.i	1.32541i	0.748881	+	0.662704i	$0.230591\pi$
$569$	429115.i	1.32541i	−0.748881	+	0.662704i	$0.769409\pi$
$570$	0	0
$571$	47827.9	0.146693	0.0733465	−	0.997307i	$-0.476632\pi$
$571$	47827.9	0.146693	0.0733465	+	0.997307i	$0.476632\pi$
$572$	− 1176.19i	− 0.00359487i
$573$	0	0
$574$	−5764.06	−0.0174946
$575$	464948.i	1.40627i
$576$	0	0
$577$	33367.6	0.100225	0.0501123	−	0.998744i	$-0.484042\pi$
$577$	33367.6	0.100225	0.0501123	+	0.998744i	$0.484042\pi$
$578$	− 132795.i	− 0.397489i
$579$	0	0
$580$	−1321.00	−0.00392687
$581$	44489.9i	0.131798i
$582$	0	0
$583$	22247.2	0.0654543
$584$	− 7835.63i	− 0.0229746i
$585$	0	0
$586$	−537948.	−1.56655
$587$	− 369349.i	− 1.07192i	−0.844245	−	0.535958i	$-0.819951\pi$
$587$	− 369349.i	− 1.07192i	0.844245	−	0.535958i	$-0.180049\pi$
$588$	0	0
$589$	−215333.	−0.620699
$590$	12332.7i	0.0354287i
$591$	0	0
$592$	41582.8	0.118651
$593$	159988.i	0.454966i	0.973782	+	0.227483i	$0.0730497\pi$
$593$	159988.i	0.454966i	−0.973782	+	0.227483i	$0.926950\pi$
$594$	0	0
$595$	−8272.60	−0.0233673
$596$	− 18845.1i	− 0.0530524i
$597$	0	0
$598$	−105071.	−0.293820
$599$	506389.i	1.41134i	0.708542	+	0.705669i	$0.249353\pi$
$599$	506389.i	1.41134i	−0.708542	+	0.705669i	$0.750647\pi$
$600$	0	0
$601$	−195258.	−0.540580	−0.270290	−	0.962779i	$-0.587120\pi$
$601$	−195258.	−0.540580	−0.270290	+	0.962779i	$0.587120\pi$
$602$	− 38818.7i	− 0.107114i
$603$	0	0
$604$	14849.6	0.0407042
$605$	− 84133.0i	− 0.229856i
$606$	0	0
$607$	−171318.	−0.464970	−0.232485	−	0.972600i	$-0.574686\pi$
$607$	−171318.	−0.464970	−0.232485	+	0.972600i	$0.574686\pi$
$608$	64088.8i	0.173370i
$609$	0	0
$610$	−174451.	−0.468828
$611$	81503.4i	0.218320i
$612$	0	0
$613$	598207.	1.59196	0.795978	−	0.605326i	$-0.206957\pi$
$613$	598207.	1.59196	0.795978	+	0.605326i	$0.206957\pi$
$614$	574352.i	1.52350i
$615$	0	0
$616$	−10087.1	−0.0265831
$617$	− 82287.7i	− 0.216155i	−0.994142	−	0.108077i	$-0.965531\pi$
$617$	− 82287.7i	− 0.216155i	0.994142	−	0.108077i	$-0.0344694\pi$
$618$	0	0
$619$	−204576.	−0.533916	−0.266958	−	0.963708i	$-0.586018\pi$
$619$	−204576.	−0.533916	−0.266958	+	0.963708i	$0.586018\pi$
$620$	1966.33i	0.00511532i
$621$	0	0
$622$	−321605.	−0.831270
$623$	− 52846.8i	− 0.136158i
$624$	0	0
$625$	310059.	0.793752
$626$	120176.i	0.306667i
$627$	0	0
$628$	23029.0	0.0583924
$629$	52702.5i	0.133208i
$630$	0	0
$631$	−399291.	−1.00284	−0.501418	−	0.865205i	$-0.667188\pi$
$631$	−399291.	−1.00284	−0.501418	+	0.865205i	$0.667188\pi$
$632$	− 94989.9i	− 0.237817i
$633$	0	0
$634$	−354369.	−0.881612
$635$	67966.8i	0.168558i
$636$	0	0
$637$	−76412.2	−0.188315
$638$	− 43518.6i	− 0.106914i
$639$	0	0
$640$	−116367.	−0.284098
$641$	117293.i	0.285468i	0.989761	+	0.142734i	$0.0455894\pi$
$641$	117293.i	0.285468i	−0.989761	+	0.142734i	$0.954411\pi$
$642$	0	0
$643$	381231.	0.922075	0.461037	−	0.887381i	$-0.347477\pi$
$643$	381231.	0.922075	0.461037	+	0.887381i	$0.347477\pi$
$644$	− 2435.15i	− 0.00587157i
$645$	0	0
$646$	−842691.	−2.01931
$647$	− 484544.i	− 1.15751i	−0.815501	−	0.578755i	$-0.803539\pi$
$647$	− 484544.i	− 1.15751i	0.815501	−	0.578755i	$-0.196461\pi$
$648$	0	0
$649$	−20051.2	−0.0476048
$650$	76283.5i	0.180553i
$651$	0	0
$652$	−40669.4	−0.0956693
$653$	170027.i	0.398742i	0.979924	+	0.199371i	$0.0638899\pi$
$653$	170027.i	0.398742i	−0.979924	+	0.199371i	$0.936110\pi$
$654$	0	0
$655$	−184466.	−0.429966
$656$	− 103014.i	− 0.239381i
$657$	0	0
$658$	−38274.5	−0.0884011
$659$	67562.4i	0.155573i	0.996970	+	0.0777865i	$0.0247852\pi$
$659$	67562.4i	0.155573i	−0.996970	+	0.0777865i	$0.975215\pi$
$660$	0	0
$661$	−181271.	−0.414883	−0.207441	−	0.978247i	$-0.566514\pi$
$661$	−181271.	−0.414883	−0.207441	+	0.978247i	$0.566514\pi$
$662$	− 638081.i	− 1.45599i
$663$	0	0
$664$	−755772.	−1.71417
$665$	14662.7i	0.0331566i
$666$	0	0
$667$	−191863.	−0.431261
$668$	− 18877.6i	− 0.0423051i
$669$	0	0
$670$	49780.1	0.110894
$671$	− 283631.i	− 0.629955i
$672$	0	0
$673$	130571.	0.288281	0.144141	−	0.989557i	$-0.453958\pi$
$673$	130571.	0.288281	0.144141	+	0.989557i	$0.453958\pi$
$674$	526128.i	1.15817i
$675$	0	0
$676$	22872.9	0.0500528
$677$	− 173344.i	− 0.378209i	−0.981957	−	0.189105i	$-0.939441\pi$
$677$	− 173344.i	− 0.378209i	0.981957	−	0.189105i	$-0.0605586\pi$
$678$	0	0
$679$	−52505.2	−0.113884
$680$	− 140531.i	− 0.303916i
$681$	0	0
$682$	−64778.1	−0.139271
$683$	− 378528.i	− 0.811439i	−0.913998	−	0.405720i	$-0.867021\pi$
$683$	− 378528.i	− 0.811439i	0.913998	−	0.405720i	$-0.132979\pi$
$684$	0	0
$685$	−2256.62	−0.00480924
$686$	− 71968.9i	− 0.152931i
$687$	0	0
$688$	693760.	1.46566
$689$	− 16092.3i	− 0.0338985i
$690$	0	0
$691$	−232301.	−0.486514	−0.243257	−	0.969962i	$-0.578216\pi$
$691$	−232301.	−0.486514	−0.243257	+	0.969962i	$0.578216\pi$
$692$	26809.4i	0.0559854i
$693$	0	0
$694$	460429.	0.955968
$695$	110399.i	0.228558i
$696$	0	0
$697$	130561.	0.268750
$698$	660226.i	1.35513i
$699$	0	0
$700$	−1767.96	−0.00360808
$701$	− 386733.i	− 0.787001i	−0.919324	−	0.393500i	$-0.871264\pi$
$701$	− 386733.i	− 0.787001i	0.919324	−	0.393500i	$-0.128736\pi$
$702$	0	0
$703$	93412.0	0.189013
$704$	− 170867.i	− 0.344756i
$705$	0	0
$706$	−90983.0	−0.182537
$707$	− 30681.8i	− 0.0613821i
$708$	0	0
$709$	70981.5	0.141206	0.0706030	−	0.997504i	$-0.477508\pi$
$709$	70981.5	0.141206	0.0706030	+	0.997504i	$0.477508\pi$
$710$	49856.1i	0.0989013i
$711$	0	0
$712$	897735.	1.77088
$713$	285591.i	0.561780i
$714$	0	0
$715$	9392.85	0.0183732
$716$	− 35673.9i	− 0.0695863i
$717$	0	0
$718$	−398311.	−0.772633
$719$	279132.i	0.539948i	0.962868	+	0.269974i	$0.0870151\pi$
$719$	279132.i	0.539948i	−0.962868	+	0.269974i	$0.912985\pi$
$720$	0	0
$721$	−43800.5	−0.0842575
$722$	958976.i	1.83964i
$723$	0	0
$724$	−31098.0	−0.0593275
$725$	139296.i	0.265010i
$726$	0	0
$727$	−122003.	−0.230835	−0.115418	−	0.993317i	$-0.536821\pi$
$727$	−122003.	−0.230835	−0.115418	+	0.993317i	$0.536821\pi$
$728$	7296.44i	0.0137673i
$729$	0	0
$730$	−3426.39	−0.00642971
$731$	879277.i	1.64547i
$732$	0	0
$733$	−7614.50	−0.0141721	−0.00708604	−	0.999975i	$-0.502256\pi$
$733$	−7614.50	−0.0141721	−0.00708604	+	0.999975i	$0.502256\pi$
$734$	− 606224.i	− 1.12523i
$735$	0	0
$736$	84999.4	0.156913
$737$	80935.1i	0.149005i
$738$	0	0
$739$	794720.	1.45521	0.727604	−	0.685997i	$-0.240634\pi$
$739$	794720.	1.45521	0.727604	+	0.685997i	$0.240634\pi$
$740$	− 852.997i	− 0.00155770i
$741$	0	0
$742$	7557.04	0.0137260
$743$	− 909419.i	− 1.64735i	−0.567061	−	0.823676i	$-0.691919\pi$
$743$	− 909419.i	− 1.64735i	0.567061	−	0.823676i	$-0.308081\pi$
$744$	0	0
$745$	150494.	0.271148
$746$	105007.i	0.188686i
$747$	0	0
$748$	−12511.1	−0.0223610
$749$	47757.2i	0.0851286i
$750$	0	0
$751$	1.04671e6	1.85587	0.927937	−	0.372737i	$-0.121581\pi$
$751$	1.04671e6	1.85587	0.927937	+	0.372737i	$0.121581\pi$
$752$	− 684033.i	− 1.20960i
$753$	0	0
$754$	−31478.8	−0.0553701
$755$	118587.i	0.208037i
$756$	0	0
$757$	−770041.	−1.34376	−0.671881	−	0.740659i	$-0.734513\pi$
$757$	−770041.	−1.34376	−0.671881	+	0.740659i	$0.734513\pi$
$758$	1.08475e6i	1.88795i
$759$	0	0
$760$	−249083.	−0.431237
$761$	− 134766.i	− 0.232707i	−0.993208	−	0.116354i	$-0.962879\pi$
$761$	− 134766.i	− 0.232707i	0.993208	−	0.116354i	$-0.0371206\pi$
$762$	0	0
$763$	31625.6	0.0543238
$764$	48710.2i	0.0834513i
$765$	0	0
$766$	201185.	0.342877
$767$	14503.8i	0.0246543i
$768$	0	0
$769$	−805016.	−1.36129	−0.680647	−	0.732612i	$-0.738301\pi$
$769$	−805016.	−1.36129	−0.680647	+	0.732612i	$0.738301\pi$
$770$	4410.94i	0.00743960i
$771$	0	0
$772$	−37382.7	−0.0627243
$773$	− 827235.i	− 1.38443i	−0.721693	−	0.692213i	$-0.756636\pi$
$773$	− 827235.i	− 1.38443i	0.721693	−	0.692213i	$-0.243364\pi$
$774$	0	0
$775$	207344.	0.345214
$776$	− 891933.i	− 1.48118i
$777$	0	0
$778$	−425265.	−0.702587
$779$	− 231412.i	− 0.381338i
$780$	0	0
$781$	−81058.7	−0.132892
$782$	1.11764e6i	1.82763i
$783$	0	0
$784$	641304.	1.04335
$785$	183907.i	0.298441i
$786$	0	0
$787$	−487518.	−0.787121	−0.393560	−	0.919299i	$-0.628757\pi$
$787$	−487518.	−0.787121	−0.393560	+	0.919299i	$0.628757\pi$
$788$	− 45878.5i	− 0.0738850i
$789$	0	0
$790$	−41537.5	−0.0665559
$791$	− 28944.0i	− 0.0462600i
$792$	0	0
$793$	−205162.	−0.326250
$794$	− 755741.i	− 1.19876i
$795$	0	0
$796$	53045.7	0.0837191
$797$	− 1.10591e6i	− 1.74101i	−0.492158	−	0.870506i	$-0.663792\pi$
$797$	− 1.10591e6i	− 1.74101i	0.492158	−	0.870506i	$-0.336208\pi$
$798$	0	0
$799$	866950.	1.35800
$800$	− 61711.0i	− 0.0964235i
$801$	0	0
$802$	736225.	1.14462
$803$	− 5570.80i	− 0.00863946i
$804$	0	0
$805$	19446.8	0.0300093
$806$	46856.7i	0.0721276i
$807$	0	0
$808$	521207.	0.798340
$809$	− 380885.i	− 0.581966i	−0.956728	−	0.290983i	$-0.906018\pi$
$809$	− 380885.i	− 0.581966i	0.956728	−	0.290983i	$-0.0939822\pi$
$810$	0	0
$811$	−1.13397e6	−1.72410	−0.862048	−	0.506827i	$-0.830818\pi$
$811$	−1.13397e6	−1.72410	−0.862048	+	0.506827i	$0.830818\pi$
$812$	− 729.559i	− 0.00110649i
$813$	0	0
$814$	28100.9	0.0424103
$815$	− 324780.i	− 0.488961i
$816$	0	0
$817$	1.55847e6	2.33482
$818$	297032.i	0.443911i
$819$	0	0
$820$	−2113.15	−0.00314270
$821$	789018.i	1.17058i	0.810825	+	0.585289i	$0.199019\pi$
$821$	789018.i	1.17058i	−0.810825	+	0.585289i	$0.800981\pi$
$822$	0	0
$823$	749791.	1.10698	0.553491	−	0.832855i	$-0.313295\pi$
$823$	749791.	1.10698	0.553491	+	0.832855i	$0.313295\pi$
$824$	− 744061.i	− 1.09586i
$825$	0	0
$826$	−6811.09	−0.00998289
$827$	950531.i	1.38981i	0.719102	+	0.694905i	$0.244554\pi$
$827$	950531.i	1.38981i	−0.719102	+	0.694905i	$0.755446\pi$
$828$	0	0
$829$	216495.	0.315020	0.157510	−	0.987517i	$-0.449653\pi$
$829$	216495.	0.315020	0.157510	+	0.987517i	$0.449653\pi$
$830$	330487.i	0.479731i
$831$	0	0
$832$	−123595.	−0.178548
$833$	812795.i	1.17136i
$834$	0	0
$835$	150754.	0.216219
$836$	22175.1i	0.0317288i
$837$	0	0
$838$	−1.20558e6	−1.71676
$839$	139527.i	0.198214i	0.995077	+	0.0991070i	$0.0315986\pi$
$839$	139527.i	0.198214i	−0.995077	+	0.0991070i	$0.968401\pi$
$840$	0	0
$841$	649800.	0.918729
$842$	169656.i	0.239302i
$843$	0	0
$844$	−24822.8	−0.0348471
$845$	182660.i	0.255817i
$846$	0	0
$847$	46464.8	0.0647675
$848$	135058.i	0.187814i
$849$	0	0
$850$	811426.	1.12308
$851$	− 123890.i	− 0.171071i
$852$	0	0
$853$	131490.	0.180715	0.0903575	−	0.995909i	$-0.471199\pi$
$853$	131490.	0.180715	0.0903575	+	0.995909i	$0.471199\pi$
$854$	− 96345.4i	− 0.132104i
$855$	0	0
$856$	−811276.	−1.10719
$857$	− 138262.i	− 0.188253i	−0.995560	−	0.0941265i	$-0.969994\pi$
$857$	− 138262.i	− 0.188253i	0.995560	−	0.0941265i	$-0.0300058\pi$
$858$	0	0
$859$	386559.	0.523877	0.261938	−	0.965085i	$-0.415638\pi$
$859$	386559.	0.523877	0.261938	+	0.965085i	$0.415638\pi$
$860$	− 14231.2i	− 0.0192418i
$861$	0	0
$862$	329251.	0.443111
$863$	1.38478e6i	1.85935i	0.368387	+	0.929673i	$0.379910\pi$
$863$	1.38478e6i	1.85935i	−0.368387	+	0.929673i	$0.620090\pi$
$864$	0	0
$865$	−214096.	−0.286139
$866$	395424.i	0.527262i
$867$	0	0
$868$	−1085.96	−0.00144137
$869$	− 67533.8i	− 0.0894297i
$870$	0	0
$871$	58543.7	0.0771692
$872$	537240.i	0.706538i
$873$	0	0
$874$	1.98095e6	2.59329
$875$	− 29306.6i	− 0.0382780i
$876$	0	0
$877$	−247651.	−0.321989	−0.160995	−	0.986955i	$-0.551470\pi$
$877$	−247651.	−0.321989	−0.160995	+	0.986955i	$0.551470\pi$
$878$	1.00425e6i	1.30272i
$879$	0	0
$880$	−78831.3	−0.101797
$881$	1.27133e6i	1.63798i	0.573810	+	0.818989i	$0.305465\pi$
$881$	1.27133e6i	1.63798i	−0.573810	+	0.818989i	$0.694535\pi$
$882$	0	0
$883$	19451.0	0.0249472	0.0124736	−	0.999922i	$-0.496029\pi$
$883$	19451.0	0.0249472	0.0124736	+	0.999922i	$0.496029\pi$
$884$	9049.76i	0.0115806i
$885$	0	0
$886$	−169751.	−0.216245
$887$	− 940689.i	− 1.19564i	−0.801632	−	0.597818i	$-0.796035\pi$
$887$	− 940689.i	− 1.19564i	0.801632	−	0.597818i	$-0.203965\pi$
$888$	0	0
$889$	−37536.5	−0.0474953
$890$	− 392565.i	− 0.495600i
$891$	0	0
$892$	2444.84	0.00307270
$893$	− 1.53662e6i	− 1.92692i
$894$	0	0
$895$	284887.	0.355652
$896$	− 64266.7i	− 0.0800516i
$897$	0	0
$898$	968660.	1.20121
$899$	85561.8i	0.105867i
$900$	0	0
$901$	−171174.	−0.210857
$902$	− 69614.9i	− 0.0855636i
$903$	0	0
$904$	491686.	0.601660
$905$	− 248345.i	− 0.303220i
$906$	0	0
$907$	647481.	0.787068	0.393534	−	0.919310i	$-0.371252\pi$
$907$	647481.	0.787068	0.393534	+	0.919310i	$0.371252\pi$
$908$	25363.9i	0.0307641i
$909$	0	0
$910$	3190.61	0.00385293
$911$	− 616156.i	− 0.742428i	−0.928547	−	0.371214i	$-0.878942\pi$
$911$	− 616156.i	− 0.742428i	0.928547	−	0.371214i	$-0.121058\pi$
$912$	0	0
$913$	−537323.	−0.644605
$914$	− 325371.i	− 0.389481i
$915$	0	0
$916$	−71748.3	−0.0855108
$917$	− 101877.i	− 0.121153i
$918$	0	0
$919$	319420.	0.378209	0.189104	−	0.981957i	$-0.439442\pi$
$919$	319420.	0.378209	0.189104	+	0.981957i	$0.439442\pi$
$920$	330352.i	0.390303i
$921$	0	0
$922$	−927043.	−1.09053
$923$	58633.1i	0.0688239i
$924$	0	0
$925$	−89946.3	−0.105124
$926$	− 16383.3i	− 0.0191064i
$927$	0	0
$928$	25465.4	0.0295702
$929$	− 660110.i	− 0.764865i	−0.923983	−	0.382432i	$-0.875087\pi$
$929$	− 660110.i	− 0.764865i	0.923983	−	0.382432i	$-0.124913\pi$
$930$	0	0
$931$	1.44063e6	1.66209
$932$	− 1856.70i	− 0.00213752i
$933$	0	0
$934$	−392870.	−0.450355
$935$	− 99911.5i	− 0.114286i
$936$	0	0
$937$	1.51083e6	1.72082	0.860409	−	0.509603i	$-0.170208\pi$
$937$	1.51083e6	1.72082	0.860409	+	0.509603i	$0.170208\pi$
$938$	27492.5i	0.0312470i
$939$	0	0
$940$	−14031.7	−0.0158802
$941$	145794.i	0.164650i	0.996606	+	0.0823248i	$0.0262345\pi$
$941$	145794.i	0.164650i	−0.996606	+	0.0823248i	$0.973766\pi$
$942$	0	0
$943$	−306916.	−0.345140
$944$	− 121726.i	− 0.136597i
$945$	0	0
$946$	468829.	0.523881
$947$	− 989306.i	− 1.10314i	−0.834129	−	0.551570i	$-0.814029\pi$
$947$	− 989306.i	− 1.10314i	0.834129	−	0.551570i	$-0.185971\pi$
$948$	0	0
$949$	−4029.59	−0.00447434
$950$	− 1.43820e6i	− 1.59358i
$951$	0	0
$952$	77612.0	0.0856357
$953$	530701.i	0.584337i	0.956367	+	0.292169i	$0.0943769\pi$
$953$	530701.i	0.584337i	−0.956367	+	0.292169i	$0.905623\pi$
$954$	0	0
$955$	−388993.	−0.426515
$956$	− 65102.8i	− 0.0712335i
$957$	0	0
$958$	−421916.	−0.459722
$959$	− 1246.28i	− 0.00135512i
$960$	0	0
$961$	−796161.	−0.862093
$962$	− 20326.5i	− 0.0219641i
$963$	0	0
$964$	37012.2	0.0398283
$965$	− 298533.i	− 0.320581i
$966$	0	0
$967$	511289.	0.546781	0.273390	−	0.961903i	$-0.411855\pi$
$967$	511289.	0.546781	0.273390	+	0.961903i	$0.411855\pi$
$968$	789321.i	0.842370i
$969$	0	0
$970$	−390028.	−0.414526
$971$	897751.i	0.952176i	0.879398	+	0.476088i	$0.157946\pi$
$971$	897751.i	0.952176i	−0.879398	+	0.476088i	$0.842054\pi$
$972$	0	0
$973$	−60971.1	−0.0644019
$974$	295417.i	0.311399i
$975$	0	0
$976$	1.72186e6	1.80759
$977$	− 687566.i	− 0.720319i	−0.932891	−	0.360160i	$-0.882722\pi$
$977$	− 687566.i	− 0.720319i	0.932891	−	0.360160i	$-0.117278\pi$
$978$	0	0
$979$	638252.	0.665927
$980$	− 13155.2i	− 0.0136976i
$981$	0	0
$982$	−405245.	−0.420237
$983$	822846.i	0.851553i	0.904828	+	0.425776i	$0.139999\pi$
$983$	822846.i	0.851553i	−0.904828	+	0.425776i	$0.860001\pi$
$984$	0	0
$985$	366379.	0.377623
$986$	334839.i	0.344416i
$987$	0	0
$988$	16040.2	0.0164322
$989$	− 2.06696e6i	− 2.11319i
$990$	0	0
$991$	32140.8	0.0327272	0.0163636	−	0.999866i	$-0.494791\pi$
$991$	32140.8	0.0327272	0.0163636	+	0.999866i	$0.494791\pi$
$992$	− 37905.6i	− 0.0385195i
$993$	0	0
$994$	−27534.4	−0.0278678
$995$	423616.i	0.427884i
$996$	0	0
$997$	−1.34689e6	−1.35501	−0.677505	−	0.735518i	$-0.736939\pi$
$997$	−1.34689e6	−1.35501	−0.677505	+	0.735518i	$0.736939\pi$
$998$	444306.i	0.446089i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.55	yes	76
3.2	odd	2	inner	531.5.b.a.296.22	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.22	✓	76	3.2	odd	2	inner
531.5.b.a.296.55	yes	76	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.10252i q^{2} -0.830633 q^{4} -6.63332i q^{5} +3.66343 q^{7} +62.2326i q^{8} +O(q^{10})\)	\(q+4.10252i q^{2} -0.830633 q^{4} -6.63332i q^{5} +3.66343 q^{7} +62.2326i q^{8} +27.2133 q^{10} +44.2447i q^{11} +32.0041 q^{13} +15.0293i q^{14} -268.600 q^{16} -340.426i q^{17} -603.386 q^{19} +5.50985i q^{20} -181.515 q^{22} +800.256i q^{23} +580.999 q^{25} +131.297i q^{26} -3.04297 q^{28} +239.753i q^{29} +356.875 q^{31} -106.215i q^{32} +1396.60 q^{34} -24.3007i q^{35} -154.813 q^{37} -2475.40i q^{38} +412.808 q^{40} +383.522i q^{41} -2582.87 q^{43} -36.7511i q^{44} -3283.06 q^{46} +2546.66i q^{47} -2387.58 q^{49} +2383.56i q^{50} -26.5836 q^{52} -502.821i q^{53} +293.489 q^{55} +227.985i q^{56} -983.588 q^{58} +453.188i q^{59} -6410.51 q^{61} +1464.09i q^{62} -3861.85 q^{64} -212.293i q^{65} +1829.26 q^{67} +282.769i q^{68} +99.6940 q^{70} +1832.05i q^{71} -125.909 q^{73} -635.123i q^{74} +501.192 q^{76} +162.088i q^{77} -1526.37 q^{79} +1781.71i q^{80} -1573.41 q^{82} +12144.3i q^{83} -2258.16 q^{85} -10596.3i q^{86} -2753.46 q^{88} -14425.5i q^{89} +117.245 q^{91} -664.718i q^{92} -10447.7 q^{94} +4002.45i q^{95} -14332.3 q^{97} -9795.08i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 76 * q - 632 * q^4 - 48 * q^7	\( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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