LMFDB - Embedded newform 531.5.b.a.296.53

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			296.53
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.86286i q^{2} +1.07829 q^{4} -17.4312i q^{5} +88.9215 q^{7} +65.9711i q^{8} +O(q^{10})$	\(q+3.86286i q^{2} +1.07829 q^{4} -17.4312i q^{5} +88.9215 q^{7} +65.9711i q^{8} +67.3343 q^{10} -118.419i q^{11} +299.657 q^{13} +343.491i q^{14} -237.585 q^{16} +163.293i q^{17} -57.9759 q^{19} -18.7958i q^{20} +457.438 q^{22} -461.872i q^{23} +321.154 q^{25} +1157.53i q^{26} +95.8827 q^{28} -897.607i q^{29} -1169.06 q^{31} +137.780i q^{32} -630.777 q^{34} -1550.01i q^{35} +1015.72 q^{37} -223.953i q^{38} +1149.95 q^{40} -365.392i q^{41} -3355.08 q^{43} -127.690i q^{44} +1784.15 q^{46} -3496.62i q^{47} +5506.03 q^{49} +1240.57i q^{50} +323.116 q^{52} +1596.46i q^{53} -2064.19 q^{55} +5866.25i q^{56} +3467.33 q^{58} -453.188i q^{59} +2774.08 q^{61} -4515.92i q^{62} -4333.58 q^{64} -5223.38i q^{65} +3013.19 q^{67} +176.076i q^{68} +5987.46 q^{70} +1951.29i q^{71} +6731.06 q^{73} +3923.60i q^{74} -62.5146 q^{76} -10530.0i q^{77} -4875.32 q^{79} +4141.38i q^{80} +1411.46 q^{82} -2892.99i q^{83} +2846.38 q^{85} -12960.2i q^{86} +7812.26 q^{88} +1598.47i q^{89} +26645.9 q^{91} -498.030i q^{92} +13507.0 q^{94} +1010.59i q^{95} -2056.67 q^{97} +21269.0i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.86286i	0.965716i	0.875699	+	0.482858i	$0.160401\pi$
$2$	3.86286i	0.965716i	−0.875699	+	0.482858i	$0.839599\pi$
$3$	0	0
$3$	0	0
$4$	1.07829	0.0673929
$5$	− 17.4312i	− 0.697247i	−0.937263	−	0.348624i	$-0.886649\pi$
$5$	− 17.4312i	− 0.697247i	0.937263	−	0.348624i	$-0.113351\pi$
$6$	0	0
$7$	88.9215	1.81472	0.907362	−	0.420351i	$-0.138093\pi$
$7$	88.9215	1.81472	0.907362	+	0.420351i	$0.138093\pi$
$8$	65.9711i	1.03080i
$9$	0	0
$10$	67.3343	0.673343
$11$	− 118.419i	− 0.978673i	−0.872095	−	0.489337i	$-0.837239\pi$
$11$	− 118.419i	− 0.978673i	0.872095	−	0.489337i	$-0.162761\pi$
$12$	0	0
$13$	299.657	1.77312	0.886559	−	0.462615i	$-0.153089\pi$
$13$	299.657	1.77312	0.886559	+	0.462615i	$0.153089\pi$
$14$	343.491i	1.75251i
$15$	0	0
$16$	−237.585	−0.928065
$17$	163.293i	0.565026i	0.959263	+	0.282513i	$0.0911681\pi$
$17$	163.293i	0.565026i	−0.959263	+	0.282513i	$0.908832\pi$
$18$	0	0
$19$	−57.9759	−0.160598	−0.0802991	−	0.996771i	$-0.525588\pi$
$19$	−57.9759	−0.160598	−0.0802991	+	0.996771i	$0.525588\pi$
$20$	− 18.7958i	− 0.0469895i
$21$	0	0
$22$	457.438	0.945120
$23$	− 461.872i	− 0.873104i	−0.899679	−	0.436552i	$-0.856200\pi$
$23$	− 461.872i	− 0.873104i	0.899679	−	0.436552i	$-0.143800\pi$
$24$	0	0
$25$	321.154	0.513846
$26$	1157.53i	1.71233i
$27$	0	0
$28$	95.8827	0.122299
$29$	− 897.607i	− 1.06731i	−0.845702	−	0.533655i	$-0.820818\pi$
$29$	− 897.607i	− 1.06731i	0.845702	−	0.533655i	$-0.179182\pi$
$30$	0	0
$31$	−1169.06	−1.21650	−0.608252	−	0.793744i	$-0.708129\pi$
$31$	−1169.06	−1.21650	−0.608252	+	0.793744i	$0.708129\pi$
$32$	137.780i	0.134551i
$33$	0	0
$34$	−630.777	−0.545655
$35$	− 1550.01i	− 1.26531i
$36$	0	0
$37$	1015.72	0.741945	0.370973	−	0.928644i	$-0.379024\pi$
$37$	1015.72	0.741945	0.370973	+	0.928644i	$0.379024\pi$
$38$	− 223.953i	− 0.155092i
$39$	0	0
$40$	1149.95	0.718721
$41$	− 365.392i	− 0.217366i	−0.994076	−	0.108683i	$-0.965337\pi$
$41$	− 365.392i	− 0.217366i	0.994076	−	0.108683i	$-0.0346633\pi$
$42$	0	0
$43$	−3355.08	−1.81454	−0.907269	−	0.420551i	$-0.861837\pi$
$43$	−3355.08	−1.81454	−0.907269	+	0.420551i	$0.861837\pi$
$44$	− 127.690i	− 0.0659556i
$45$	0	0
$46$	1784.15	0.843170
$47$	− 3496.62i	− 1.58290i	−0.611237	−	0.791448i	$-0.709328\pi$
$47$	− 3496.62i	− 1.58290i	0.611237	−	0.791448i	$-0.290672\pi$
$48$	0	0
$49$	5506.03	2.29322
$50$	1240.57i	0.496229i
$51$	0	0
$52$	323.116	0.119495
$53$	1596.46i	0.568338i	0.958774	+	0.284169i	$0.0917176\pi$
$53$	1596.46i	0.568338i	−0.958774	+	0.284169i	$0.908282\pi$
$54$	0	0
$55$	−2064.19	−0.682377
$56$	5866.25i	1.87061i
$57$	0	0
$58$	3467.33	1.03072
$59$	− 453.188i	− 0.130189i
$59$	− 453.188i	− 0.130189i
$60$	0	0
$61$	2774.08	0.745519	0.372759	−	0.927928i	$-0.378412\pi$
$61$	2774.08	0.745519	0.372759	+	0.927928i	$0.378412\pi$
$62$	− 4515.92i	− 1.17480i
$63$	0	0
$64$	−4333.58	−1.05800
$65$	− 5223.38i	− 1.23630i
$66$	0	0
$67$	3013.19	0.671239	0.335619	−	0.941998i	$-0.391054\pi$
$67$	3013.19	0.671239	0.335619	+	0.941998i	$0.391054\pi$
$68$	176.076i	0.0380787i
$69$	0	0
$70$	5987.46	1.22193
$71$	1951.29i	0.387084i	0.981092	+	0.193542i	$0.0619977\pi$
$71$	1951.29i	0.387084i	−0.981092	+	0.193542i	$0.938002\pi$
$72$	0	0
$73$	6731.06	1.26310	0.631550	−	0.775336i	$-0.282419\pi$
$73$	6731.06	1.26310	0.631550	+	0.775336i	$0.282419\pi$
$74$	3923.60i	0.716508i
$75$	0	0
$76$	−62.5146	−0.0108232
$77$	− 10530.0i	− 1.77602i
$78$	0	0
$79$	−4875.32	−0.781177	−0.390588	−	0.920565i	$-0.627728\pi$
$79$	−4875.32	−0.781177	−0.390588	+	0.920565i	$0.627728\pi$
$80$	4141.38i	0.647091i
$81$	0	0
$82$	1411.46	0.209913
$83$	− 2892.99i	− 0.419944i	−0.977707	−	0.209972i	$-0.932663\pi$
$83$	− 2892.99i	− 0.419944i	0.977707	−	0.209972i	$-0.0673373\pi$
$84$	0	0
$85$	2846.38	0.393963
$86$	− 12960.2i	− 1.75233i
$87$	0	0
$88$	7812.26	1.00881
$89$	1598.47i	0.201801i	0.994897	+	0.100901i	$0.0321724\pi$
$89$	1598.47i	0.201801i	−0.994897	+	0.100901i	$0.967828\pi$
$90$	0	0
$91$	26645.9	3.21772
$92$	− 498.030i	− 0.0588410i
$93$	0	0
$94$	13507.0	1.52863
$95$	1010.59i	0.111977i
$96$	0	0
$97$	−2056.67	−0.218586	−0.109293	−	0.994010i	$-0.534859\pi$
$97$	−2056.67	−0.218586	−0.109293	+	0.994010i	$0.534859\pi$
$98$	21269.0i	2.21460i
$99$	0	0
$100$	346.296	0.0346296
$101$	12083.0i	1.18449i	0.805759	+	0.592244i	$0.201758\pi$
$101$	12083.0i	1.18449i	−0.805759	+	0.592244i	$0.798242\pi$
$102$	0	0
$103$	1735.08	0.163548	0.0817740	−	0.996651i	$-0.473941\pi$
$103$	1735.08	0.163548	0.0817740	+	0.996651i	$0.473941\pi$
$104$	19768.7i	1.82773i
$105$	0	0
$106$	−6166.91	−0.548853
$107$	13236.6i	1.15614i	0.815989	+	0.578068i	$0.196193\pi$
$107$	13236.6i	1.15614i	−0.815989	+	0.578068i	$0.803807\pi$
$108$	0	0
$109$	−6547.81	−0.551116	−0.275558	−	0.961284i	$-0.588863\pi$
$109$	−6547.81	−0.551116	−0.275558	+	0.961284i	$0.588863\pi$
$110$	− 7973.69i	− 0.658982i
$111$	0	0
$112$	−21126.4	−1.68418
$113$	10232.1i	0.801322i	0.916226	+	0.400661i	$0.131220\pi$
$113$	10232.1i	0.801322i	−0.916226	+	0.400661i	$0.868780\pi$
$114$	0	0
$115$	−8050.97	−0.608769
$116$	− 967.877i	− 0.0719290i
$117$	0	0
$118$	1750.60	0.125725
$119$	14520.2i	1.02537i
$120$	0	0
$121$	617.837	0.0421991
$122$	10715.9i	0.719959i
$123$	0	0
$124$	−1260.58	−0.0819836
$125$	− 16492.6i	− 1.05553i
$126$	0	0
$127$	−16060.8	−0.995769	−0.497884	−	0.867243i	$-0.665890\pi$
$127$	−16060.8	−0.995769	−0.497884	+	0.867243i	$0.665890\pi$
$128$	− 14535.6i	− 0.887180i
$129$	0	0
$130$	20177.2	1.19392
$131$	14849.5i	0.865305i	0.901561	+	0.432652i	$0.142422\pi$
$131$	14849.5i	0.865305i	−0.901561	+	0.432652i	$0.857578\pi$
$132$	0	0
$133$	−5155.30	−0.291441
$134$	11639.5i	0.648226i
$135$	0	0
$136$	−10772.6	−0.582428
$137$	16828.1i	0.896590i	0.893886	+	0.448295i	$0.147969\pi$
$137$	16828.1i	0.896590i	−0.893886	+	0.448295i	$0.852031\pi$
$138$	0	0
$139$	3609.49	0.186817	0.0934084	−	0.995628i	$-0.470224\pi$
$139$	3609.49	0.186817	0.0934084	+	0.995628i	$0.470224\pi$
$140$	− 1671.35i	− 0.0852729i
$141$	0	0
$142$	−7537.58	−0.373814
$143$	− 35485.2i	− 1.73530i
$144$	0	0
$145$	−15646.4	−0.744179
$146$	26001.1i	1.21979i
$147$	0	0
$148$	1095.24	0.0500018
$149$	11929.3i	0.537330i	0.963234	+	0.268665i	$0.0865824\pi$
$149$	11929.3i	0.537330i	−0.963234	+	0.268665i	$0.913418\pi$
$150$	0	0
$151$	30683.3	1.34570	0.672851	−	0.739778i	$-0.265070\pi$
$151$	30683.3	1.34570	0.672851	+	0.739778i	$0.265070\pi$
$152$	− 3824.74i	− 0.165544i
$153$	0	0
$154$	40676.1	1.71513
$155$	20378.1i	0.848204i
$156$	0	0
$157$	−7752.96	−0.314535	−0.157267	−	0.987556i	$-0.550268\pi$
$157$	−7752.96	−0.314535	−0.157267	+	0.987556i	$0.550268\pi$
$158$	− 18832.7i	− 0.754395i
$159$	0	0
$160$	2401.67	0.0938152
$161$	− 41070.3i	− 1.58444i
$162$	0	0
$163$	−36553.1	−1.37578	−0.687889	−	0.725816i	$-0.741462\pi$
$163$	−36553.1	−1.37578	−0.687889	+	0.725816i	$0.741462\pi$
$164$	− 393.996i	− 0.0146489i
$165$	0	0
$166$	11175.2	0.405546
$167$	− 20745.9i	− 0.743873i	−0.928258	−	0.371936i	$-0.878694\pi$
$167$	− 20745.9i	− 0.743873i	0.928258	−	0.371936i	$-0.121306\pi$
$168$	0	0
$169$	61233.3	2.14395
$170$	10995.2i	0.380456i
$171$	0	0
$172$	−3617.74	−0.122287
$173$	26152.0i	0.873803i	0.899509	+	0.436901i	$0.143924\pi$
$173$	26152.0i	0.873803i	−0.899509	+	0.436901i	$0.856076\pi$
$174$	0	0
$175$	28557.5	0.932489
$176$	28134.7i	0.908273i
$177$	0	0
$178$	−6174.66	−0.194883
$179$	7150.70i	0.223173i	0.993755	+	0.111587i	$0.0355933\pi$
$179$	7150.70i	0.223173i	−0.993755	+	0.111587i	$0.964407\pi$
$180$	0	0
$181$	29158.2	0.890029	0.445014	−	0.895523i	$-0.353199\pi$
$181$	29158.2	0.890029	0.445014	+	0.895523i	$0.353199\pi$
$182$	102930.i	3.10740i
$183$	0	0
$184$	30470.2	0.899994
$185$	− 17705.3i	− 0.517319i
$186$	0	0
$187$	19337.0	0.552976
$188$	− 3770.35i	− 0.106676i
$189$	0	0
$190$	−3903.77	−0.108138
$191$	58142.0i	1.59376i	0.604136	+	0.796881i	$0.293518\pi$
$191$	58142.0i	1.59376i	−0.604136	+	0.796881i	$0.706482\pi$
$192$	0	0
$193$	36004.7	0.966594	0.483297	−	0.875456i	$-0.339439\pi$
$193$	36004.7	0.966594	0.483297	+	0.875456i	$0.339439\pi$
$194$	− 7944.65i	− 0.211092i
$195$	0	0
$196$	5937.07	0.154547
$197$	− 62793.3i	− 1.61801i	−0.587802	−	0.809005i	$-0.700007\pi$
$197$	− 62793.3i	− 1.61801i	0.587802	−	0.809005i	$-0.299993\pi$
$198$	0	0
$199$	−23924.4	−0.604136	−0.302068	−	0.953286i	$-0.597677\pi$
$199$	−23924.4	−0.604136	−0.302068	+	0.953286i	$0.597677\pi$
$200$	21186.9i	0.529672i
$201$	0	0
$202$	−46674.8	−1.14388
$203$	− 79816.6i	− 1.93687i
$204$	0	0
$205$	−6369.21	−0.151558
$206$	6702.38i	0.157941i
$207$	0	0
$208$	−71193.9	−1.64557
$209$	6865.48i	0.157173i
$210$	0	0
$211$	−28945.9	−0.650163	−0.325082	−	0.945686i	$-0.605392\pi$
$211$	−28945.9	−0.650163	−0.325082	+	0.945686i	$0.605392\pi$
$212$	1721.44i	0.0383019i
$213$	0	0
$214$	−51131.2	−1.11650
$215$	58483.0i	1.26518i
$216$	0	0
$217$	−103954.	−2.20762
$218$	− 25293.3i	− 0.532222i
$219$	0	0
$220$	−2225.79	−0.0459873
$221$	48931.8i	1.00186i
$222$	0	0
$223$	83450.6	1.67811	0.839054	−	0.544048i	$-0.183109\pi$
$223$	83450.6	1.67811	0.839054	+	0.544048i	$0.183109\pi$
$224$	12251.6i	0.244172i
$225$	0	0
$226$	−39525.1	−0.773850
$227$	2713.10i	0.0526520i	0.999653	+	0.0263260i	$0.00838079\pi$
$227$	2713.10i	0.0526520i	−0.999653	+	0.0263260i	$0.991619\pi$
$228$	0	0
$229$	74442.8	1.41955	0.709776	−	0.704427i	$-0.248796\pi$
$229$	74442.8	1.41955	0.709776	+	0.704427i	$0.248796\pi$
$230$	− 31099.8i	− 0.587898i
$231$	0	0
$232$	59216.1	1.10018
$233$	− 3839.02i	− 0.0707145i	−0.999375	−	0.0353573i	$-0.988743\pi$
$233$	− 3839.02i	− 0.0707145i	0.999375	−	0.0353573i	$-0.0112569\pi$
$234$	0	0
$235$	−60950.2	−1.10367
$236$	− 488.666i	− 0.00877380i
$237$	0	0
$238$	−56089.6	−0.990213
$239$	71737.9i	1.25589i	0.778256	+	0.627947i	$0.216105\pi$
$239$	71737.9i	1.25589i	−0.778256	+	0.627947i	$0.783895\pi$
$240$	0	0
$241$	−64665.6	−1.11337	−0.556685	−	0.830724i	$-0.687927\pi$
$241$	−64665.6	−1.11337	−0.556685	+	0.830724i	$0.687927\pi$
$242$	2386.62i	0.0407523i
$243$	0	0
$244$	2991.25	0.0502426
$245$	− 95976.5i	− 1.59894i
$246$	0	0
$247$	−17372.9	−0.284760
$248$	− 77124.1i	− 1.25397i
$249$	0	0
$250$	63708.6	1.01934
$251$	− 52433.6i	− 0.832266i	−0.909304	−	0.416133i	$-0.863385\pi$
$251$	− 52433.6i	− 0.832266i	0.909304	−	0.416133i	$-0.136615\pi$
$252$	0	0
$253$	−54694.6	−0.854483
$254$	− 62040.5i	− 0.961630i
$255$	0	0
$256$	−13188.4	−0.201240
$257$	− 57505.6i	− 0.870650i	−0.900273	−	0.435325i	$-0.856633\pi$
$257$	− 57505.6i	− 0.870650i	0.900273	−	0.435325i	$-0.143367\pi$
$258$	0	0
$259$	90319.6	1.34643
$260$	− 5632.29i	− 0.0833179i
$261$	0	0
$262$	−57361.6	−0.835638
$263$	− 82832.3i	− 1.19754i	−0.800923	−	0.598768i	$-0.795657\pi$
$263$	− 82832.3i	− 1.19754i	0.800923	−	0.598768i	$-0.204343\pi$
$264$	0	0
$265$	27828.2	0.396272
$266$	− 19914.2i	− 0.281449i
$267$	0	0
$268$	3249.08	0.0452367
$269$	− 81568.3i	− 1.12724i	−0.826034	−	0.563621i	$-0.809408\pi$
$269$	− 81568.3i	− 1.12724i	0.826034	−	0.563621i	$-0.190592\pi$
$270$	0	0
$271$	28805.7	0.392229	0.196115	−	0.980581i	$-0.437168\pi$
$271$	28805.7	0.392229	0.196115	+	0.980581i	$0.437168\pi$
$272$	− 38795.8i	− 0.524381i
$273$	0	0
$274$	−65004.6	−0.865851
$275$	− 38030.9i	− 0.502887i
$276$	0	0
$277$	−116335.	−1.51618	−0.758089	−	0.652151i	$-0.773867\pi$
$277$	−116335.	−1.51618	−0.758089	+	0.652151i	$0.773867\pi$
$278$	13943.0i	0.180412i
$279$	0	0
$280$	102256.	1.30428
$281$	69993.7i	0.886434i	0.896414	+	0.443217i	$0.146163\pi$
$281$	69993.7i	0.886434i	−0.896414	+	0.443217i	$0.853837\pi$
$282$	0	0
$283$	80668.7	1.00724	0.503619	−	0.863926i	$-0.332001\pi$
$283$	80668.7	1.00724	0.503619	+	0.863926i	$0.332001\pi$
$284$	2104.05i	0.0260867i
$285$	0	0
$286$	137075.	1.67581
$287$	− 32491.1i	− 0.394458i
$288$	0	0
$289$	56856.5	0.680745
$290$	− 60439.7i	− 0.718665i
$291$	0	0
$292$	7258.00	0.0851239
$293$	102246.i	1.19100i	0.803355	+	0.595501i	$0.203046\pi$
$293$	102246.i	1.19100i	−0.803355	+	0.595501i	$0.796954\pi$
$294$	0	0
$295$	−7899.60	−0.0907739
$296$	67008.4i	0.764796i
$297$	0	0
$298$	−46081.1	−0.518908
$299$	− 138403.i	− 1.54812i
$300$	0	0
$301$	−298339.	−3.29289
$302$	118526.i	1.29957i
$303$	0	0
$304$	13774.2	0.149046
$305$	− 48355.4i	− 0.519811i
$306$	0	0
$307$	−85569.4	−0.907908	−0.453954	−	0.891025i	$-0.649987\pi$
$307$	−85569.4	−0.907908	−0.453954	+	0.891025i	$0.649987\pi$
$308$	− 11354.4i	− 0.119691i
$309$	0	0
$310$	−78717.8	−0.819124
$311$	− 75528.5i	− 0.780891i	−0.920626	−	0.390445i	$-0.872321\pi$
$311$	− 75528.5i	− 0.780891i	0.920626	−	0.390445i	$-0.127679\pi$
$312$	0	0
$313$	−96701.7	−0.987064	−0.493532	−	0.869728i	$-0.664294\pi$
$313$	−96701.7	−0.987064	−0.493532	+	0.869728i	$0.664294\pi$
$314$	− 29948.6i	− 0.303751i
$315$	0	0
$316$	−5256.99	−0.0526457
$317$	174427.i	1.73578i	0.496754	+	0.867892i	$0.334525\pi$
$317$	174427.i	1.73578i	−0.496754	+	0.867892i	$0.665475\pi$
$318$	0	0
$319$	−106294.	−1.04455
$320$	75539.4i	0.737690i
$321$	0	0
$322$	158649.	1.53012
$323$	− 9467.04i	− 0.0907422i
$324$	0	0
$325$	96236.0	0.911110
$326$	− 141199.i	− 1.32861i
$327$	0	0
$328$	24105.3	0.224060
$329$	− 310924.i	− 2.87252i
$330$	0	0
$331$	110281.	1.00658	0.503288	−	0.864119i	$-0.332123\pi$
$331$	110281.	1.00658	0.503288	+	0.864119i	$0.332123\pi$
$332$	− 3119.47i	− 0.0283012i
$333$	0	0
$334$	80138.4	0.718370
$335$	− 52523.5i	− 0.468019i
$336$	0	0
$337$	−169204.	−1.48988	−0.744939	−	0.667132i	$-0.767522\pi$
$337$	−169204.	−1.48988	−0.744939	+	0.667132i	$0.767522\pi$
$338$	236536.i	2.07044i
$339$	0	0
$340$	3069.21	0.0265503
$341$	138439.i	1.19056i
$342$	0	0
$343$	276103.	2.34684
$344$	− 221338.i	− 1.87042i
$345$	0	0
$346$	−101022.	−0.843845
$347$	− 15057.2i	− 0.125051i	−0.998043	−	0.0625253i	$-0.980085\pi$
$347$	− 15057.2i	− 0.125051i	0.998043	−	0.0625253i	$-0.0199154\pi$
$348$	0	0
$349$	−40768.0	−0.334710	−0.167355	−	0.985897i	$-0.553523\pi$
$349$	−40768.0	−0.334710	−0.167355	+	0.985897i	$0.553523\pi$
$350$	110314.i	0.900519i
$351$	0	0
$352$	16315.8	0.131681
$353$	− 214933.i	− 1.72486i	−0.506179	−	0.862428i	$-0.668943\pi$
$353$	− 214933.i	− 1.72486i	0.506179	−	0.862428i	$-0.331057\pi$
$354$	0	0
$355$	34013.3	0.269894
$356$	1723.60i	0.0136000i
$357$	0	0
$358$	−27622.2	−0.215522
$359$	− 183128.i	− 1.42091i	−0.703742	−	0.710456i	$-0.748489\pi$
$359$	− 183128.i	− 1.42091i	0.703742	−	0.710456i	$-0.251511\pi$
$360$	0	0
$361$	−126960.	−0.974208
$362$	112634.i	0.859515i
$363$	0	0
$364$	28731.9	0.216851
$365$	− 117330.i	− 0.880692i
$366$	0	0
$367$	−100984.	−0.749754	−0.374877	−	0.927075i	$-0.622315\pi$
$367$	−100984.	−0.749754	−0.374877	+	0.927075i	$0.622315\pi$
$368$	109734.i	0.810297i
$369$	0	0
$370$	68393.0	0.499584
$371$	141960.i	1.03138i
$372$	0	0
$373$	59205.9	0.425547	0.212774	−	0.977102i	$-0.431750\pi$
$373$	59205.9	0.425547	0.212774	+	0.977102i	$0.431750\pi$
$374$	74696.3i	0.534018i
$375$	0	0
$376$	230676.	1.63165
$377$	− 268974.i	− 1.89247i
$378$	0	0
$379$	−73363.8	−0.510744	−0.255372	−	0.966843i	$-0.582198\pi$
$379$	−73363.8	−0.510744	−0.255372	+	0.966843i	$0.582198\pi$
$380$	1089.70i	0.00754642i
$381$	0	0
$382$	−224595.	−1.53912
$383$	46694.0i	0.318320i	0.987253	+	0.159160i	$0.0508785\pi$
$383$	46694.0i	0.318320i	−0.987253	+	0.159160i	$0.949122\pi$
$384$	0	0
$385$	−183551.	−1.23833
$386$	139081.i	0.933456i
$387$	0	0
$388$	−2217.68	−0.0147311
$389$	− 37789.1i	− 0.249728i	−0.992174	−	0.124864i	$-0.960151\pi$
$389$	− 37789.1i	− 0.249728i	0.992174	−	0.124864i	$-0.0398494\pi$
$390$	0	0
$391$	75420.3	0.493327
$392$	363238.i	2.36385i
$393$	0	0
$394$	242562.	1.56254
$395$	84982.6i	0.544673i
$396$	0	0
$397$	−202128.	−1.28247	−0.641233	−	0.767346i	$-0.721577\pi$
$397$	−202128.	−1.28247	−0.641233	+	0.767346i	$0.721577\pi$
$398$	− 92416.6i	− 0.583424i
$399$	0	0
$400$	−76301.3	−0.476883
$401$	− 193375.i	− 1.20258i	−0.799033	−	0.601288i	$-0.794655\pi$
$401$	− 193375.i	− 1.20258i	0.799033	−	0.601288i	$-0.205345\pi$
$402$	0	0
$403$	−350317.	−2.15700
$404$	13028.9i	0.0798260i
$405$	0	0
$406$	308320.	1.87047
$407$	− 120281.i	− 0.726122i
$408$	0	0
$409$	−273757.	−1.63651	−0.818256	−	0.574854i	$-0.805059\pi$
$409$	−273757.	−1.63651	−0.818256	+	0.574854i	$0.805059\pi$
$410$	− 24603.4i	− 0.146362i
$411$	0	0
$412$	1870.91	0.0110220
$413$	− 40298.1i	− 0.236257i
$414$	0	0
$415$	−50428.3	−0.292805
$416$	41286.7i	0.238574i
$417$	0	0
$418$	−26520.4	−0.151785
$419$	274968.i	1.56623i	0.621879	+	0.783113i	$0.286369\pi$
$419$	274968.i	1.56623i	−0.621879	+	0.783113i	$0.713631\pi$
$420$	0	0
$421$	38058.7	0.214728	0.107364	−	0.994220i	$-0.465759\pi$
$421$	38058.7	0.214728	0.107364	+	0.994220i	$0.465759\pi$
$422$	− 111814.i	− 0.627873i
$423$	0	0
$424$	−105320.	−0.585842
$425$	52442.1i	0.290337i
$426$	0	0
$427$	246675.	1.35291
$428$	14272.8i	0.0779153i
$429$	0	0
$430$	−225912.	−1.22181
$431$	− 42019.7i	− 0.226203i	−0.993583	−	0.113102i	$-0.963921\pi$
$431$	− 42019.7i	− 0.226203i	0.993583	−	0.113102i	$-0.0360785\pi$
$432$	0	0
$433$	−62748.9	−0.334680	−0.167340	−	0.985899i	$-0.553518\pi$
$433$	−62748.9	−0.334680	−0.167340	+	0.985899i	$0.553518\pi$
$434$	− 401562.i	− 2.13193i
$435$	0	0
$436$	−7060.41	−0.0371413
$437$	26777.5i	0.140219i
$438$	0	0
$439$	−97116.8	−0.503924	−0.251962	−	0.967737i	$-0.581076\pi$
$439$	−97116.8	−0.503924	−0.251962	+	0.967737i	$0.581076\pi$
$440$	− 136177.i	− 0.703393i
$441$	0	0
$442$	−189017.	−0.967511
$443$	− 324624.i	− 1.65414i	−0.562096	−	0.827072i	$-0.690005\pi$
$443$	− 324624.i	− 1.65414i	0.562096	−	0.827072i	$-0.309995\pi$
$444$	0	0
$445$	27863.2	0.140705
$446$	322358.i	1.62058i
$447$	0	0
$448$	−385348.	−1.91998
$449$	383278.i	1.90117i	0.310459	+	0.950587i	$0.399517\pi$
$449$	383278.i	1.90117i	−0.310459	+	0.950587i	$0.600483\pi$
$450$	0	0
$451$	−43269.5	−0.212730
$452$	11033.1i	0.0540034i
$453$	0	0
$454$	−10480.3	−0.0508468
$455$	− 464470.i	− 2.24355i
$456$	0	0
$457$	57824.2	0.276871	0.138435	−	0.990371i	$-0.455793\pi$
$457$	57824.2	0.276871	0.138435	+	0.990371i	$0.455793\pi$
$458$	287562.i	1.37088i
$459$	0	0
$460$	−8681.25	−0.0410267
$461$	283724.i	1.33504i	0.744591	+	0.667520i	$0.232644\pi$
$461$	283724.i	1.33504i	−0.744591	+	0.667520i	$0.767356\pi$
$462$	0	0
$463$	−124220.	−0.579469	−0.289735	−	0.957107i	$-0.593567\pi$
$463$	−124220.	−0.579469	−0.289735	+	0.957107i	$0.593567\pi$
$464$	213258.i	0.990533i
$465$	0	0
$466$	14829.6	0.0682901
$467$	187668.i	0.860513i	0.902707	+	0.430257i	$0.141577\pi$
$467$	187668.i	0.860513i	−0.902707	+	0.430257i	$0.858423\pi$
$468$	0	0
$469$	267937.	1.21811
$470$	− 235442.i	− 1.06583i
$471$	0	0
$472$	29897.3	0.134198
$473$	397307.i	1.77584i
$474$	0	0
$475$	−18619.2	−0.0825228
$476$	15656.9i	0.0691024i
$477$	0	0
$478$	−277114.	−1.21284
$479$	371142.i	1.61759i	0.588089	+	0.808796i	$0.299880\pi$
$479$	371142.i	1.61759i	−0.588089	+	0.808796i	$0.700120\pi$
$480$	0	0
$481$	304369.	1.31556
$482$	− 249795.i	− 1.07520i
$483$	0	0
$484$	666.205	0.00284392
$485$	35850.3i	0.152408i
$486$	0	0
$487$	−34834.9	−0.146878	−0.0734389	−	0.997300i	$-0.523397\pi$
$487$	−34834.9	−0.146878	−0.0734389	+	0.997300i	$0.523397\pi$
$488$	183009.i	0.768479i
$489$	0	0
$490$	370744.	1.54412
$491$	308128.i	1.27811i	0.769162	+	0.639054i	$0.220674\pi$
$491$	308128.i	1.27811i	−0.769162	+	0.639054i	$0.779326\pi$
$492$	0	0
$493$	146573.	0.603058
$494$	− 67109.1i	− 0.274997i
$495$	0	0
$496$	277751.	1.12899
$497$	173512.i	0.702451i
$498$	0	0
$499$	−87871.9	−0.352898	−0.176449	−	0.984310i	$-0.556461\pi$
$499$	−87871.9	−0.352898	−0.176449	+	0.984310i	$0.556461\pi$
$500$	− 17783.7i	− 0.0711349i
$501$	0	0
$502$	202544.	0.803732
$503$	496622.i	1.96286i	0.191810	+	0.981432i	$0.438564\pi$
$503$	496622.i	1.96286i	−0.191810	+	0.981432i	$0.561436\pi$
$504$	0	0
$505$	210620.	0.825881
$506$	− 211278.i	− 0.825188i
$507$	0	0
$508$	−17318.1	−0.0671077
$509$	454078.i	1.75265i	0.481721	+	0.876325i	$0.340012\pi$
$509$	454078.i	1.75265i	−0.481721	+	0.876325i	$0.659988\pi$
$510$	0	0
$511$	598535.	2.29218
$512$	− 283514.i	− 1.08152i
$513$	0	0
$514$	222136.	0.840801
$515$	− 30244.5i	− 0.114033i
$516$	0	0
$517$	−414067.	−1.54914
$518$	348892.i	1.30026i
$519$	0	0
$520$	344592.	1.27438
$521$	− 328676.i	− 1.21086i	−0.795900	−	0.605428i	$-0.793002\pi$
$521$	− 328676.i	− 1.21086i	0.795900	−	0.605428i	$-0.206998\pi$
$522$	0	0
$523$	−5827.36	−0.0213044	−0.0106522	−	0.999943i	$-0.503391\pi$
$523$	−5827.36	−0.0213044	−0.0106522	+	0.999943i	$0.503391\pi$
$524$	16012.0i	0.0583153i
$525$	0	0
$526$	319970.	1.15648
$527$	− 190899.i	− 0.687356i
$528$	0	0
$529$	66515.4	0.237690
$530$	107497.i	0.382686i
$531$	0	0
$532$	−5558.89	−0.0196411
$533$	− 109492.i	− 0.385415i
$534$	0	0
$535$	230730.	0.806113
$536$	198783.i	0.691912i
$537$	0	0
$538$	315087.	1.08859
$539$	− 652020.i	− 2.24431i
$540$	0	0
$541$	28128.2	0.0961052	0.0480526	−	0.998845i	$-0.484698\pi$
$541$	28128.2	0.0961052	0.0480526	+	0.998845i	$0.484698\pi$
$542$	111273.i	0.378782i
$543$	0	0
$544$	−22498.5	−0.0760247
$545$	114136.i	0.384264i
$546$	0	0
$547$	552585.	1.84682	0.923410	−	0.383815i	$-0.125390\pi$
$547$	552585.	1.84682	0.923410	+	0.383815i	$0.125390\pi$
$548$	18145.5i	0.0604238i
$549$	0	0
$550$	146908.	0.485646
$551$	52039.6i	0.171408i
$552$	0	0
$553$	−433521.	−1.41762
$554$	− 449386.i	− 1.46420i
$555$	0	0
$556$	3892.06	0.0125901
$557$	208598.i	0.672355i	0.941799	+	0.336178i	$0.109134\pi$
$557$	208598.i	0.672355i	−0.941799	+	0.336178i	$0.890866\pi$
$558$	0	0
$559$	−1.00537e6	−3.21739
$560$	368258.i	1.17429i
$561$	0	0
$562$	−270376.	−0.856043
$563$	361493.i	1.14047i	0.821483	+	0.570234i	$0.193147\pi$
$563$	361493.i	1.14047i	−0.821483	+	0.570234i	$0.806853\pi$
$564$	0	0
$565$	178357.	0.558720
$566$	311612.i	0.972706i
$567$	0	0
$568$	−128729.	−0.399006
$569$	− 125877.i	− 0.388796i	−0.980923	−	0.194398i	$-0.937725\pi$
$569$	− 125877.i	− 0.388796i	0.980923	−	0.194398i	$-0.0622753\pi$
$570$	0	0
$571$	−618279.	−1.89632	−0.948161	−	0.317791i	$-0.897059\pi$
$571$	−618279.	−1.89632	−0.948161	+	0.317791i	$0.897059\pi$
$572$	− 38263.2i	− 0.116947i
$573$	0	0
$574$	125509.	0.380935
$575$	− 148332.i	− 0.448641i
$576$	0	0
$577$	348162.	1.04575	0.522877	−	0.852408i	$-0.324859\pi$
$577$	348162.	1.04575	0.522877	+	0.852408i	$0.324859\pi$
$578$	219629.i	0.657406i
$579$	0	0
$580$	−16871.2	−0.0501523
$581$	− 257249.i	− 0.762082i
$582$	0	0
$583$	189052.	0.556217
$584$	444055.i	1.30200i
$585$	0	0
$586$	−394964.	−1.15017
$587$	− 84370.1i	− 0.244857i	−0.992477	−	0.122429i	$-0.960932\pi$
$587$	− 84370.1i	− 0.244857i	0.992477	−	0.122429i	$-0.0390682\pi$
$588$	0	0
$589$	67777.3	0.195368
$590$	− 30515.1i	− 0.0876618i
$591$	0	0
$592$	−241320.	−0.688574
$593$	− 63811.1i	− 0.181463i	−0.995875	−	0.0907313i	$-0.971080\pi$
$593$	− 63811.1i	− 0.181463i	0.995875	−	0.0907313i	$-0.0289204\pi$
$594$	0	0
$595$	253105.	0.714934
$596$	12863.1i	0.0362122i
$597$	0	0
$598$	534632.	1.49504
$599$	395923.i	1.10346i	0.834022	+	0.551731i	$0.186033\pi$
$599$	395923.i	1.10346i	−0.834022	+	0.551731i	$0.813967\pi$
$600$	0	0
$601$	−242745.	−0.672049	−0.336024	−	0.941853i	$-0.609082\pi$
$601$	−242745.	−0.672049	−0.336024	+	0.941853i	$0.609082\pi$
$602$	− 1.15244e6i	− 3.17999i
$603$	0	0
$604$	33085.4	0.0906907
$605$	− 10769.6i	− 0.0294232i
$606$	0	0
$607$	330604.	0.897285	0.448643	−	0.893711i	$-0.351908\pi$
$607$	330604.	0.897285	0.448643	+	0.893711i	$0.351908\pi$
$608$	− 7987.92i	− 0.0216086i
$609$	0	0
$610$	186790.	0.501990
$611$	− 1.04779e6i	− 2.80666i
$612$	0	0
$613$	−496091.	−1.32020	−0.660101	−	0.751176i	$-0.729487\pi$
$613$	−496091.	−1.32020	−0.660101	+	0.751176i	$0.729487\pi$
$614$	− 330543.i	− 0.876781i
$615$	0	0
$616$	694677.	1.83072
$617$	− 425086.i	− 1.11662i	−0.829631	−	0.558312i	$-0.811449\pi$
$617$	− 425086.i	− 1.11662i	0.829631	−	0.558312i	$-0.188551\pi$
$618$	0	0
$619$	−648290.	−1.69195	−0.845977	−	0.533220i	$-0.820982\pi$
$619$	−648290.	−1.69195	−0.845977	+	0.533220i	$0.820982\pi$
$620$	21973.4i	0.0571629i
$621$	0	0
$622$	291756.	0.754119
$623$	142138.i	0.366213i
$624$	0	0
$625$	−86764.0	−0.222116
$626$	− 373545.i	− 0.953223i
$627$	0	0
$628$	−8359.91	−0.0211974
$629$	165860.i	0.419219i
$630$	0	0
$631$	662490.	1.66388	0.831938	−	0.554869i	$-0.187232\pi$
$631$	662490.	1.66388	0.831938	+	0.554869i	$0.187232\pi$
$632$	− 321630.i	− 0.805235i
$633$	0	0
$634$	−673788.	−1.67627
$635$	279958.i	0.694297i
$636$	0	0
$637$	1.64992e6	4.06615
$638$	− 410600.i	− 1.00874i
$639$	0	0
$640$	−253372.	−0.618584
$641$	32481.2i	0.0790525i	0.999219	+	0.0395263i	$0.0125849\pi$
$641$	32481.2i	0.0790525i	−0.999219	+	0.0395263i	$0.987415\pi$
$642$	0	0
$643$	−447448.	−1.08223	−0.541117	−	0.840947i	$-0.681998\pi$
$643$	−447448.	−1.08223	−0.541117	+	0.840947i	$0.681998\pi$
$644$	− 44285.5i	− 0.106780i
$645$	0	0
$646$	36569.9	0.0876312
$647$	71871.2i	0.171690i	0.996308	+	0.0858452i	$0.0273591\pi$
$647$	71871.2i	0.171690i	−0.996308	+	0.0858452i	$0.972641\pi$
$648$	0	0
$649$	−53666.2	−0.127412
$650$	371747.i	0.879873i
$651$	0	0
$652$	−39414.6	−0.0927176
$653$	680407.i	1.59567i	0.602878	+	0.797833i	$0.294020\pi$
$653$	680407.i	1.59567i	−0.602878	+	0.797833i	$0.705980\pi$
$654$	0	0
$655$	258844.	0.603331
$656$	86811.4i	0.201729i
$657$	0	0
$658$	1.20106e6	2.77404
$659$	− 537411.i	− 1.23747i	−0.785599	−	0.618736i	$-0.787645\pi$
$659$	− 537411.i	− 1.23747i	0.785599	−	0.618736i	$-0.212355\pi$
$660$	0	0
$661$	668667.	1.53041	0.765203	−	0.643789i	$-0.222638\pi$
$661$	668667.	1.53041	0.765203	+	0.643789i	$0.222638\pi$
$662$	426002.i	0.972067i
$663$	0	0
$664$	190854.	0.432877
$665$	89863.1i	0.203207i
$666$	0	0
$667$	−414580.	−0.931872
$668$	− 22370.0i	− 0.0501317i
$669$	0	0
$670$	202891.	0.451974
$671$	− 328504.i	− 0.729619i
$672$	0	0
$673$	−103902.	−0.229400	−0.114700	−	0.993400i	$-0.536591\pi$
$673$	−103902.	−0.229400	−0.114700	+	0.993400i	$0.536591\pi$
$674$	− 653612.i	− 1.43880i
$675$	0	0
$676$	66027.0	0.144487
$677$	554524.i	1.20988i	0.796270	+	0.604941i	$0.206803\pi$
$677$	554524.i	1.20988i	−0.796270	+	0.604941i	$0.793197\pi$
$678$	0	0
$679$	−182883.	−0.396673
$680$	187779.i	0.406097i
$681$	0	0
$682$	−534772.	−1.14974
$683$	− 589348.i	− 1.26337i	−0.775225	−	0.631685i	$-0.782364\pi$
$683$	− 589348.i	− 1.26337i	0.775225	−	0.631685i	$-0.217636\pi$
$684$	0	0
$685$	293334.	0.625145
$686$	1.06655e6i	2.26638i
$687$	0	0
$688$	797116.	1.68401
$689$	478391.i	1.00773i
$690$	0	0
$691$	−846904.	−1.77369	−0.886845	−	0.462066i	$-0.847108\pi$
$691$	−846904.	−1.77369	−0.886845	+	0.462066i	$0.847108\pi$
$692$	28199.4i	0.0588881i
$693$	0	0
$694$	58164.0	0.120763
$695$	− 62917.6i	− 0.130258i
$696$	0	0
$697$	59665.7	0.122817
$698$	− 157481.i	− 0.323234i
$699$	0	0
$700$	30793.1	0.0628431
$701$	− 371447.i	− 0.755893i	−0.925827	−	0.377947i	$-0.876630\pi$
$701$	− 371447.i	− 0.755893i	0.925827	−	0.377947i	$-0.123370\pi$
$702$	0	0
$703$	−58887.5	−0.119155
$704$	513180.i	1.03544i
$705$	0	0
$706$	830255.	1.66572
$707$	1.07443e6i	2.14952i
$708$	0	0
$709$	−709675.	−1.41178	−0.705890	−	0.708321i	$-0.749453\pi$
$709$	−709675.	−1.41178	−0.705890	+	0.708321i	$0.749453\pi$
$710$	131389.i	0.260641i
$711$	0	0
$712$	−105453.	−0.208016
$713$	539956.i	1.06213i
$714$	0	0
$715$	−618549.	−1.20994
$716$	7710.50i	0.0150403i
$717$	0	0
$718$	707400.	1.37220
$719$	− 679987.i	− 1.31536i	−0.753299	−	0.657678i	$-0.771539\pi$
$719$	− 679987.i	− 1.31536i	0.753299	−	0.657678i	$-0.228461\pi$
$720$	0	0
$721$	154286.	0.296794
$722$	− 490428.i	− 0.940808i
$723$	0	0
$724$	31440.9	0.0599816
$725$	− 288270.i	− 0.548433i
$726$	0	0
$727$	−653222.	−1.23592	−0.617962	−	0.786208i	$-0.712042\pi$
$727$	−653222.	−1.23592	−0.617962	+	0.786208i	$0.712042\pi$
$728$	1.75786e6i	3.31682i
$729$	0	0
$730$	453231.	0.850499
$731$	− 547860.i	− 1.02526i
$732$	0	0
$733$	360915.	0.671734	0.335867	−	0.941909i	$-0.390971\pi$
$733$	360915.	0.671734	0.335867	+	0.941909i	$0.390971\pi$
$734$	− 390086.i	− 0.724049i
$735$	0	0
$736$	63636.7	0.117477
$737$	− 356820.i	− 0.656923i
$738$	0	0
$739$	−1.03189e6	−1.88948	−0.944742	−	0.327814i	$-0.893688\pi$
$739$	−1.03189e6	−1.88948	−0.944742	+	0.327814i	$0.893688\pi$
$740$	− 19091.3i	− 0.0348636i
$741$	0	0
$742$	−548371.	−0.996016
$743$	− 632641.i	− 1.14599i	−0.819560	−	0.572993i	$-0.805782\pi$
$743$	− 632641.i	− 1.14599i	0.819560	−	0.572993i	$-0.194218\pi$
$744$	0	0
$745$	207941.	0.374652
$746$	228704.i	0.410958i
$747$	0	0
$748$	20850.8	0.0372666
$749$	1.17702e6i	2.09807i
$750$	0	0
$751$	−445763.	−0.790358	−0.395179	−	0.918604i	$-0.629317\pi$
$751$	−445763.	−0.790358	−0.395179	+	0.918604i	$0.629317\pi$
$752$	830743.i	1.46903i
$753$	0	0
$754$	1.03901e6	1.82758
$755$	− 534847.i	− 0.938287i
$756$	0	0
$757$	−267146.	−0.466183	−0.233091	−	0.972455i	$-0.574884\pi$
$757$	−267146.	−0.466183	−0.233091	+	0.972455i	$0.574884\pi$
$758$	− 283394.i	− 0.493234i
$759$	0	0
$760$	−66669.7	−0.115425
$761$	486404.i	0.839900i	0.907547	+	0.419950i	$0.137952\pi$
$761$	486404.i	0.839900i	−0.907547	+	0.419950i	$0.862048\pi$
$762$	0	0
$763$	−582241.	−1.00012
$764$	62693.7i	0.107408i
$765$	0	0
$766$	−180372.	−0.307406
$767$	− 135801.i	− 0.230840i
$768$	0	0
$769$	15470.4	0.0261607	0.0130803	−	0.999914i	$-0.495836\pi$
$769$	15470.4	0.0261607	0.0130803	+	0.999914i	$0.495836\pi$
$770$	− 709032.i	− 1.19587i
$771$	0	0
$772$	38823.3	0.0651416
$773$	− 956704.i	− 1.60110i	−0.599265	−	0.800551i	$-0.704541\pi$
$773$	− 956704.i	− 1.60110i	0.599265	−	0.800551i	$-0.295459\pi$
$774$	0	0
$775$	−375448.	−0.625096
$776$	− 135681.i	− 0.225318i
$777$	0	0
$778$	145974.	0.241166
$779$	21183.9i	0.0349085i
$780$	0	0
$781$	231071.	0.378829
$782$	291338.i	0.476413i
$783$	0	0
$784$	−1.30815e6	−2.12826
$785$	135143.i	0.219308i
$786$	0	0
$787$	163343.	0.263725	0.131863	−	0.991268i	$-0.457904\pi$
$787$	163343.	0.263725	0.131863	+	0.991268i	$0.457904\pi$
$788$	− 67709.2i	− 0.109042i
$789$	0	0
$790$	−328276.	−0.526000
$791$	909852.i	1.45418i
$792$	0	0
$793$	831271.	1.32189
$794$	− 780793.i	− 1.23850i
$795$	0	0
$796$	−25797.3	−0.0407144
$797$	− 482697.i	− 0.759903i	−0.925006	−	0.379951i	$-0.875941\pi$
$797$	− 482697.i	− 0.759903i	0.925006	−	0.379951i	$-0.124059\pi$
$798$	0	0
$799$	570972.	0.894378
$800$	44248.6i	0.0691384i
$801$	0	0
$802$	746982.	1.16135
$803$	− 797088.i	− 1.23616i
$804$	0	0
$805$	−715904.	−1.10475
$806$	− 1.35323e6i	− 2.08305i
$807$	0	0
$808$	−797126.	−1.22097
$809$	− 496607.i	− 0.758780i	−0.925237	−	0.379390i	$-0.876134\pi$
$809$	− 496607.i	− 0.758780i	0.925237	−	0.379390i	$-0.123866\pi$
$810$	0	0
$811$	−997951.	−1.51729	−0.758643	−	0.651506i	$-0.774137\pi$
$811$	−997951.	−1.51729	−0.758643	+	0.651506i	$0.774137\pi$
$812$	− 86065.1i	− 0.130531i
$813$	0	0
$814$	464631.	0.701228
$815$	637163.i	0.959258i
$816$	0	0
$817$	194514.	0.291411
$818$	− 1.05749e6i	− 1.58041i
$819$	0	0
$820$	−6867.82	−0.0102139
$821$	560274.i	0.831217i	0.909544	+	0.415608i	$0.136431\pi$
$821$	560274.i	0.831217i	−0.909544	+	0.415608i	$0.863569\pi$
$822$	0	0
$823$	−854263.	−1.26122	−0.630612	−	0.776098i	$-0.717196\pi$
$823$	−854263.	−1.26122	−0.630612	+	0.776098i	$0.717196\pi$
$824$	114465.i	0.168585i
$825$	0	0
$826$	155666.	0.228157
$827$	1.00848e6i	1.47454i	0.675598	+	0.737270i	$0.263885\pi$
$827$	1.00848e6i	1.47454i	−0.675598	+	0.737270i	$0.736115\pi$
$828$	0	0
$829$	−236145.	−0.343613	−0.171806	−	0.985131i	$-0.554960\pi$
$829$	−236145.	−0.343613	−0.171806	+	0.985131i	$0.554960\pi$
$830$	− 194798.i	− 0.282766i
$831$	0	0
$832$	−1.29859e6	−1.87596
$833$	899093.i	1.29573i
$834$	0	0
$835$	−361625.	−0.518663
$836$	7402.95i	0.0105923i
$837$	0	0
$838$	−1.06216e6	−1.51253
$839$	49221.3i	0.0699245i	0.999389	+	0.0349622i	$0.0111311\pi$
$839$	49221.3i	0.0699245i	−0.999389	+	0.0349622i	$0.988869\pi$
$840$	0	0
$841$	−98418.1	−0.139150
$842$	147015.i	0.207367i
$843$	0	0
$844$	−31212.0	−0.0438163
$845$	− 1.06737e6i	− 1.49486i
$846$	0	0
$847$	54939.0	0.0765797
$848$	− 379295.i	− 0.527455i
$849$	0	0
$850$	−202577.	−0.280383
$851$	− 469134.i	− 0.647795i
$852$	0	0
$853$	1.38463e6	1.90298	0.951492	−	0.307674i	$-0.0995507\pi$
$853$	1.38463e6	1.90298	0.951492	+	0.307674i	$0.0995507\pi$
$854$	952871.i	1.30653i
$855$	0	0
$856$	−873233.	−1.19174
$857$	622410.i	0.847451i	0.905791	+	0.423726i	$0.139278\pi$
$857$	622410.i	0.847451i	−0.905791	+	0.423726i	$0.860722\pi$
$858$	0	0
$859$	−789026.	−1.06931	−0.534657	−	0.845069i	$-0.679559\pi$
$859$	−789026.	−1.06931	−0.534657	+	0.845069i	$0.679559\pi$
$860$	63061.4i	0.0852642i
$861$	0	0
$862$	162316.	0.218448
$863$	− 1.00226e6i	− 1.34573i	−0.739764	−	0.672866i	$-0.765063\pi$
$863$	− 1.00226e6i	− 1.34573i	0.739764	−	0.672866i	$-0.234937\pi$
$864$	0	0
$865$	455861.	0.609257
$866$	− 242390.i	− 0.323206i
$867$	0	0
$868$	−112093.	−0.148778
$869$	577333.i	0.764516i
$870$	0	0
$871$	902924.	1.19019
$872$	− 431966.i	− 0.568090i
$873$	0	0
$874$	−103438.	−0.135412
$875$	− 1.46654e6i	− 1.91549i
$876$	0	0
$877$	−661701.	−0.860325	−0.430163	−	0.902751i	$-0.641544\pi$
$877$	−661701.	−0.860325	−0.430163	+	0.902751i	$0.641544\pi$
$878$	− 375149.i	− 0.486648i
$879$	0	0
$880$	490420.	0.633291
$881$	28550.2i	0.0367838i	0.999831	+	0.0183919i	$0.00585466\pi$
$881$	28550.2i	0.0367838i	−0.999831	+	0.0183919i	$0.994145\pi$
$882$	0	0
$883$	71273.5	0.0914128	0.0457064	−	0.998955i	$-0.485446\pi$
$883$	71273.5	0.0914128	0.0457064	+	0.998955i	$0.485446\pi$
$884$	52762.4i	0.0675181i
$885$	0	0
$886$	1.25398e6	1.59743
$887$	931563.i	1.18404i	0.805925	+	0.592018i	$0.201669\pi$
$887$	931563.i	1.18404i	−0.805925	+	0.592018i	$0.798331\pi$
$888$	0	0
$889$	−1.42815e6	−1.80705
$890$	107632.i	0.135881i
$891$	0	0
$892$	89983.6	0.113092
$893$	202720.i	0.254210i
$894$	0	0
$895$	124645.	0.155607
$896$	− 1.29252e6i	− 1.60999i
$897$	0	0
$898$	−1.48055e6	−1.83599
$899$	1.04936e6i	1.29839i
$900$	0	0
$901$	−260690.	−0.321126
$902$	− 167144.i	− 0.205437i
$903$	0	0
$904$	−675022.	−0.826002
$905$	− 508263.i	− 0.620570i
$906$	0	0
$907$	−1.02900e6	−1.25084	−0.625419	−	0.780289i	$-0.715072\pi$
$907$	−1.02900e6	−1.25084	−0.625419	+	0.780289i	$0.715072\pi$
$908$	2925.50i	0.00354837i
$909$	0	0
$910$	1.79418e6	2.16663
$911$	1.57516e6i	1.89796i	0.315339	+	0.948979i	$0.397882\pi$
$911$	1.57516e6i	1.89796i	−0.315339	+	0.948979i	$0.602118\pi$
$912$	0	0
$913$	−342587.	−0.410988
$914$	223367.i	0.267379i
$915$	0	0
$916$	80270.6	0.0956677
$917$	1.32044e6i	1.57029i
$918$	0	0
$919$	1.05363e6	1.24754	0.623772	−	0.781606i	$-0.285599\pi$
$919$	1.05363e6	1.24754	0.623772	+	0.781606i	$0.285599\pi$
$920$	− 531131.i	− 0.627518i
$921$	0	0
$922$	−1.09599e6	−1.28927
$923$	584719.i	0.686347i
$924$	0	0
$925$	326203.	0.381246
$926$	− 479846.i	− 0.559603i
$927$	0	0
$928$	123672.	0.143607
$929$	69072.4i	0.0800338i	0.999199	+	0.0400169i	$0.0127412\pi$
$929$	69072.4i	0.0800338i	−0.999199	+	0.0400169i	$0.987259\pi$
$930$	0	0
$931$	−319217.	−0.368287
$932$	− 4139.56i	− 0.00476565i
$933$	0	0
$934$	−724938.	−0.831011
$935$	− 337067.i	− 0.385561i
$936$	0	0
$937$	152274.	0.173439	0.0867195	−	0.996233i	$-0.472362\pi$
$937$	152274.	0.173439	0.0867195	+	0.996233i	$0.472362\pi$
$938$	1.03501e6i	1.17635i
$939$	0	0
$940$	−65721.7	−0.0743794
$941$	− 1.23415e6i	− 1.39376i	−0.717187	−	0.696881i	$-0.754571\pi$
$941$	− 1.23415e6i	− 1.39376i	0.717187	−	0.696881i	$-0.245429\pi$
$942$	0	0
$943$	−168764.	−0.189783
$944$	107670.i	0.120824i
$945$	0	0
$946$	−1.53474e6	−1.71496
$947$	− 655688.i	− 0.731135i	−0.930785	−	0.365567i	$-0.880875\pi$
$947$	− 655688.i	− 0.731135i	0.930785	−	0.365567i	$-0.119125\pi$
$948$	0	0
$949$	2.01701e6	2.23962
$950$	− 71923.4i	− 0.0796935i
$951$	0	0
$952$	−957915.	−1.05695
$953$	− 950243.i	− 1.04628i	−0.852246	−	0.523141i	$-0.824760\pi$
$953$	− 950243.i	− 1.04628i	0.852246	−	0.523141i	$-0.175240\pi$
$954$	0	0
$955$	1.01348e6	1.11125
$956$	77354.0i	0.0846383i
$957$	0	0
$958$	−1.43367e6	−1.56213
$959$	1.49638e6i	1.62706i
$960$	0	0
$961$	443179.	0.479880
$962$	1.17573e6i	1.27045i
$963$	0	0
$964$	−69728.0	−0.0750332
$965$	− 627604.i	− 0.673955i
$966$	0	0
$967$	1.28673e6	1.37605	0.688026	−	0.725686i	$-0.258478\pi$
$967$	1.28673e6	1.37605	0.688026	+	0.725686i	$0.258478\pi$
$968$	40759.4i	0.0434987i
$969$	0	0
$970$	−138485.	−0.147183
$971$	− 59770.9i	− 0.0633945i	−0.999498	−	0.0316972i	$-0.989909\pi$
$971$	− 59770.9i	− 0.0633945i	0.999498	−	0.0316972i	$-0.0100912\pi$
$972$	0	0
$973$	320961.	0.339021
$974$	− 134562.i	− 0.141842i
$975$	0	0
$976$	−659078.	−0.691890
$977$	557324.i	0.583873i	0.956438	+	0.291937i	$0.0942996\pi$
$977$	557324.i	0.583873i	−0.956438	+	0.291937i	$0.905700\pi$
$978$	0	0
$979$	189290.	0.197497
$980$	− 103490.i	− 0.107757i
$981$	0	0
$982$	−1.19026e6	−1.23429
$983$	− 1.53249e6i	− 1.58596i	−0.609250	−	0.792979i	$-0.708529\pi$
$983$	− 1.53249e6i	− 1.58596i	0.609250	−	0.792979i	$-0.291471\pi$
$984$	0	0
$985$	−1.09456e6	−1.12815
$986$	566190.i	0.582383i
$987$	0	0
$988$	−18732.9	−0.0191908
$989$	1.54962e6i	1.58428i
$990$	0	0
$991$	18903.3	0.0192482	0.00962410	−	0.999954i	$-0.496937\pi$
$991$	18903.3	0.0192482	0.00962410	+	0.999954i	$0.496937\pi$
$992$	− 161073.i	− 0.163681i
$993$	0	0
$994$	−670252.	−0.678368
$995$	417030.i	0.421232i
$996$	0	0
$997$	1.47934e6	1.48825	0.744126	−	0.668039i	$-0.232866\pi$
$997$	1.47934e6	1.48825	0.744126	+	0.668039i	$0.232866\pi$
$998$	− 339437.i	− 0.340799i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.53	yes	76
3.2	odd	2	inner	531.5.b.a.296.24	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.24	✓	76	3.2	odd	2	inner
531.5.b.a.296.53	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)

\(f(q)\)	\(=\)	\(q+3.86286i q^{2} +1.07829 q^{4} -17.4312i q^{5} +88.9215 q^{7} +65.9711i q^{8} +O(q^{10})\)	\(q+3.86286i q^{2} +1.07829 q^{4} -17.4312i q^{5} +88.9215 q^{7} +65.9711i q^{8} +67.3343 q^{10} -118.419i q^{11} +299.657 q^{13} +343.491i q^{14} -237.585 q^{16} +163.293i q^{17} -57.9759 q^{19} -18.7958i q^{20} +457.438 q^{22} -461.872i q^{23} +321.154 q^{25} +1157.53i q^{26} +95.8827 q^{28} -897.607i q^{29} -1169.06 q^{31} +137.780i q^{32} -630.777 q^{34} -1550.01i q^{35} +1015.72 q^{37} -223.953i q^{38} +1149.95 q^{40} -365.392i q^{41} -3355.08 q^{43} -127.690i q^{44} +1784.15 q^{46} -3496.62i q^{47} +5506.03 q^{49} +1240.57i q^{50} +323.116 q^{52} +1596.46i q^{53} -2064.19 q^{55} +5866.25i q^{56} +3467.33 q^{58} -453.188i q^{59} +2774.08 q^{61} -4515.92i q^{62} -4333.58 q^{64} -5223.38i q^{65} +3013.19 q^{67} +176.076i q^{68} +5987.46 q^{70} +1951.29i q^{71} +6731.06 q^{73} +3923.60i q^{74} -62.5146 q^{76} -10530.0i q^{77} -4875.32 q^{79} +4141.38i q^{80} +1411.46 q^{82} -2892.99i q^{83} +2846.38 q^{85} -12960.2i q^{86} +7812.26 q^{88} +1598.47i q^{89} +26645.9 q^{91} -498.030i q^{92} +13507.0 q^{94} +1010.59i q^{95} -2056.67 q^{97} +21269.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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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