LMFDB - Embedded newform 528.4.b.d.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,4,Mod(65,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("528.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	528.b (of order $2$, degree $1$, not 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:	$31.1530084830$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 21x^{4} + 114x^{3} - 567x^{2} - 729x + 19683 $ x^6 - x^5 - 21x^4 + 114x^3 - 567x^2 - 729x + 19683
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 66)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Root			$0.832570 + 5.12902i$ of defining polynomial
Character	$\chi$	$=$	528.65
Dual form			528.4.b.d.65.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.832570 + 5.12902i) q^{3} -6.09929i q^{5} -1.16981i q^{7} +(-25.6137 - 8.54053i) q^{9} +(30.4508 + 20.0935i) q^{11} -37.5697i q^{13} +(31.2834 + 5.07809i) q^{15} -40.8925 q^{17} +84.3569i q^{19} +(6.00000 + 0.973952i) q^{21} -101.207i q^{23} +87.7986 q^{25} +(65.1297 - 124.262i) q^{27} +251.561 q^{29} -19.9245 q^{31} +(-128.412 + 139.454i) q^{33} -7.13504 q^{35} +45.9062 q^{37} +(192.696 + 31.2794i) q^{39} +175.597 q^{41} +332.498i q^{43} +(-52.0912 + 156.225i) q^{45} +186.815i q^{47} +341.632 q^{49} +(34.0458 - 209.738i) q^{51} -554.462i q^{53} +(122.556 - 185.728i) q^{55} +(-432.668 - 70.2330i) q^{57} +185.177i q^{59} +754.209i q^{61} +(-9.99084 + 29.9632i) q^{63} -229.149 q^{65} +381.334 q^{67} +(519.094 + 84.2622i) q^{69} -401.820i q^{71} +1034.57i q^{73} +(-73.0985 + 450.321i) q^{75} +(23.5057 - 35.6218i) q^{77} +1044.32i q^{79} +(583.119 + 437.508i) q^{81} -847.048 q^{83} +249.415i q^{85} +(-209.442 + 1290.26i) q^{87} +675.083i q^{89} -43.9496 q^{91} +(16.5886 - 102.193i) q^{93} +514.517 q^{95} -378.080 q^{97} +(-608.347 - 774.734i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 43 q^{9} + 6 q^{11} + 7 q^{15} + 156 q^{17} + 36 q^{21} - 204 q^{25} + 278 q^{27} - 144 q^{29} + 114 q^{31} + 157 q^{33} - 588 q^{35} - 54 q^{37} + 48 q^{39} - 408 q^{41} + 551 q^{45} + 606 q^{49}+ \cdots - 1753 q^{99}+O(q^{100}) $ 6 * q - q^3 + 43 * q^9 + 6 * q^11 + 7 * q^15 + 156 * q^17 + 36 * q^21 - 204 * q^25 + 278 * q^27 - 144 * q^29 + 114 * q^31 + 157 * q^33 - 588 * q^35 - 54 * q^37 + 48 * q^39 - 408 * q^41 + 551 * q^45 + 606 * q^49 + 444 * q^51 - 942 * q^55 - 1344 * q^57 - 12 * q^63 - 1668 * q^65 - 66 * q^67 + 1007 * q^69 + 936 * q^75 - 1800 * q^77 + 2779 * q^81 - 1392 * q^83 - 3084 * q^87 + 3780 * q^91 - 3847 * q^93 - 2088 * q^95 + 1254 * q^97 - 1753 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$.

$n$	$133$	$145$	$353$	$463$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.832570	+	5.12902i	−0.160228	+	0.987080i
$3$	−0.832570	+	5.12902i	−0.160228	+	0.987080i
$4$		0			0
$5$		−	6.09929i		−	0.545537i	−0.962080	−	0.272769i	$-0.912061\pi$
$5$		−	6.09929i		−	0.545537i	0.962080	−	0.272769i	$-0.0879393\pi$
$6$		0			0
$7$		−	1.16981i		−	0.0631640i	−0.999501	−	0.0315820i	$-0.989945\pi$
$7$		−	1.16981i		−	0.0631640i	0.999501	−	0.0315820i	$-0.0100545\pi$
$8$		0			0
$9$	−25.6137	−	8.54053i	−0.948654	−	0.316316i
$10$		0			0
$11$	30.4508	+	20.0935i	0.834660	+	0.550765i
$11$	30.4508	+	20.0935i	0.834660	+	0.550765i
$12$		0			0
$13$		−	37.5697i		−	0.801535i	−0.916180	−	0.400768i	$-0.868743\pi$
$13$		−	37.5697i		−	0.801535i	0.916180	−	0.400768i	$-0.131257\pi$
$14$		0			0
$15$	31.2834	+	5.07809i	0.538489	+	0.0874104i
$16$		0			0
$17$	−40.8925			−0.583404			−0.291702	−	0.956509i	$-0.594222\pi$
$17$	−40.8925			−0.583404			−0.291702	+	0.956509i	$0.594222\pi$
$18$		0			0
$19$			84.3569i			1.01857i	0.860598	+	0.509284i	$0.170090\pi$
$19$			84.3569i			1.01857i	−0.860598	+	0.509284i	$0.829910\pi$
$20$		0			0
$21$	6.00000	+	0.973952i	0.0623480	+	0.0101207i
$22$		0			0
$23$		−	101.207i		−	0.917530i	−0.888558	−	0.458765i	$-0.848292\pi$
$23$		−	101.207i		−	0.917530i	0.888558	−	0.458765i	$-0.151708\pi$
$24$		0			0
$25$	87.7986			0.702389
$26$		0			0
$27$	65.1297	−	124.262i	0.464230	−	0.885715i
$28$		0			0
$29$	251.561			1.61082			0.805408	−	0.592721i	$-0.201946\pi$
$29$	251.561			1.61082			0.805408	+	0.592721i	$0.201946\pi$
$30$		0			0
$31$	−19.9245			−0.115437			−0.0577185	−	0.998333i	$-0.518383\pi$
$31$	−19.9245			−0.115437			−0.0577185	+	0.998333i	$0.518383\pi$
$32$		0			0
$33$	−128.412	+	139.454i	−0.677385	+	0.735628i
$34$		0			0
$35$	−7.13504			−0.0344583
$36$		0			0
$37$	45.9062			0.203971			0.101986	−	0.994786i	$-0.467480\pi$
$37$	45.9062			0.203971			0.101986	+	0.994786i	$0.467480\pi$
$38$		0			0
$39$	192.696	+	31.2794i	0.791180	+	0.128429i
$40$		0			0
$41$	175.597			0.668870			0.334435	−	0.942419i	$-0.391454\pi$
$41$	175.597			0.668870			0.334435	+	0.942419i	$0.391454\pi$
$42$		0			0
$43$			332.498i			1.17920i	0.807697	+	0.589598i	$0.200714\pi$
$43$			332.498i			1.17920i	−0.807697	+	0.589598i	$0.799286\pi$
$44$		0			0
$45$	−52.0912	+	156.225i	−0.172562	+	0.517526i
$46$		0			0
$47$			186.815i			0.579783i	0.957059	+	0.289892i	$0.0936193\pi$
$47$			186.815i			0.579783i	−0.957059	+	0.289892i	$0.906381\pi$
$48$		0			0
$49$	341.632			0.996010
$50$		0			0
$51$	34.0458	−	209.738i	0.0934778	−	0.575867i
$52$		0			0
$53$		−	554.462i		−	1.43700i	−0.695524	−	0.718502i	$-0.744828\pi$
$53$		−	554.462i		−	1.43700i	0.695524	−	0.718502i	$-0.255172\pi$
$54$		0			0
$55$	122.556	−	185.728i	0.300463	−	0.455338i
$56$		0			0
$57$	−432.668	−	70.2330i	−1.00541	−	0.163203i
$58$		0			0
$59$			185.177i			0.408610i	0.978907	+	0.204305i	$0.0654934\pi$
$59$			185.177i			0.408610i	−0.978907	+	0.204305i	$0.934507\pi$
$60$		0			0
$61$			754.209i			1.58306i	0.611131	+	0.791529i	$0.290715\pi$
$61$			754.209i			1.58306i	−0.611131	+	0.791529i	$0.709285\pi$
$62$		0			0
$63$	−9.99084	+	29.9632i	−0.0199798	+	0.0599208i
$64$		0			0
$65$	−229.149			−0.437267
$66$		0			0
$67$	381.334			0.695334			0.347667	−	0.937618i	$-0.386974\pi$
$67$	381.334			0.695334			0.347667	+	0.937618i	$0.386974\pi$
$68$		0			0
$69$	519.094	+	84.2622i	0.905675	+	0.147014i
$70$		0			0
$71$		−	401.820i		−	0.671651i	−0.941924	−	0.335826i	$-0.890985\pi$
$71$		−	401.820i		−	0.671651i	0.941924	−	0.335826i	$-0.109015\pi$
$72$		0			0
$73$			1034.57i			1.65872i	0.558711	+	0.829362i	$0.311296\pi$
$73$			1034.57i			1.65872i	−0.558711	+	0.829362i	$0.688704\pi$
$74$		0			0
$75$	−73.0985	+	450.321i	−0.112542	+	0.693314i
$76$		0			0
$77$	23.5057	−	35.6218i	0.0347885	−	0.0527205i
$78$		0			0
$79$			1044.32i			1.48728i	0.668578	+	0.743642i	$0.266903\pi$
$79$			1044.32i			1.48728i	−0.668578	+	0.743642i	$0.733097\pi$
$80$		0			0
$81$	583.119	+	437.508i	0.799888	+	0.600149i
$82$		0			0
$83$	−847.048			−1.12019			−0.560094	−	0.828429i	$-0.689235\pi$
$83$	−847.048			−1.12019			−0.560094	+	0.828429i	$0.689235\pi$
$84$		0			0
$85$			249.415i			0.318269i
$86$		0			0
$87$	−209.442	+	1290.26i	−0.258098	+	1.59000i
$88$		0			0
$89$			675.083i			0.804030i	0.915633	+	0.402015i	$0.131690\pi$
$89$			675.083i			0.804030i	−0.915633	+	0.402015i	$0.868310\pi$
$90$		0			0
$91$	−43.9496			−0.0506282
$92$		0			0
$93$	16.5886	−	102.193i	0.0184963	−	0.113946i
$94$		0			0
$95$	514.517			0.555667
$96$		0			0
$97$	−378.080			−0.395755			−0.197877	−	0.980227i	$-0.563405\pi$
$97$	−378.080			−0.395755			−0.197877	+	0.980227i	$0.563405\pi$
$98$		0			0
$99$	−608.347	−	774.734i	−0.617588	−	0.786502i
$100$		0			0
$101$	870.938			0.858036			0.429018	−	0.903296i	$-0.358860\pi$
$101$	870.938			0.858036			0.429018	+	0.903296i	$0.358860\pi$
$102$		0			0
$103$	1940.06			1.85592			0.927962	−	0.372675i	$-0.121559\pi$
$103$	1940.06			1.85592			0.927962	+	0.372675i	$0.121559\pi$
$104$		0			0
$105$	5.94042	−	36.5957i	0.00552119	−	0.0340131i
$106$		0			0
$107$	503.199			0.454636			0.227318	−	0.973821i	$-0.427004\pi$
$107$	503.199			0.454636			0.227318	+	0.973821i	$0.427004\pi$
$108$		0			0
$109$		−	1952.64i		−	1.71586i	−0.513766	−	0.857930i	$-0.671750\pi$
$109$		−	1952.64i		−	1.71586i	0.513766	−	0.857930i	$-0.328250\pi$
$110$		0			0
$111$	−38.2201	+	235.454i	−0.0326819	+	0.201336i
$112$		0			0
$113$			1009.29i			0.840227i	0.907472	+	0.420113i	$0.138010\pi$
$113$			1009.29i			0.840227i	−0.907472	+	0.420113i	$0.861990\pi$
$114$		0			0
$115$	−617.293			−0.500547
$116$		0			0
$117$	−320.865	+	962.297i	−0.253538	+	0.760380i
$118$		0			0
$119$			47.8366i			0.0368502i
$120$		0			0
$121$	523.503	+	1223.73i	0.393316	+	0.919403i
$122$		0			0
$123$	−146.197	+	900.642i	−0.107172	+	0.660229i
$124$		0			0
$125$		−	1297.92i		−	0.928717i
$126$		0			0
$127$			84.3828i			0.0589588i	0.999565	+	0.0294794i	$0.00938494\pi$
$127$			84.3828i			0.0589588i	−0.999565	+	0.0294794i	$0.990615\pi$
$128$		0			0
$129$	−1705.39	−	276.828i	−1.16396	−	0.188940i
$130$		0			0
$131$	1786.96			1.19181			0.595905	−	0.803055i	$-0.296793\pi$
$131$	1786.96			1.19181			0.595905	+	0.803055i	$0.296793\pi$
$132$		0			0
$133$	98.6820			0.0643369
$134$		0			0
$135$	−757.912	−	397.245i	−0.483190	−	0.253255i
$136$		0			0
$137$		−	2371.76i		−	1.47907i	−0.673117	−	0.739536i	$-0.735045\pi$
$137$		−	2371.76i		−	1.47907i	0.673117	−	0.739536i	$-0.264955\pi$
$138$		0			0
$139$		−	1760.85i		−	1.07449i	−0.843428	−	0.537243i	$-0.819466\pi$
$139$		−	1760.85i		−	1.07449i	0.843428	−	0.537243i	$-0.180534\pi$
$140$		0			0
$141$	−958.179	−	155.537i	−0.572293	−	0.0928976i
$142$		0			0
$143$	754.906	−	1144.03i	0.441458	−	0.669010i
$144$		0			0
$145$		−	1534.34i		−	0.878760i
$146$		0			0
$147$	−284.432	+	1752.23i	−0.159589	+	0.983142i
$148$		0			0
$149$	970.169			0.533419			0.266709	−	0.963777i	$-0.414064\pi$
$149$	970.169			0.533419			0.266709	+	0.963777i	$0.414064\pi$
$150$		0			0
$151$			1433.20i			0.772399i	0.922415	+	0.386199i	$0.126212\pi$
$151$			1433.20i			0.772399i	−0.922415	+	0.386199i	$0.873788\pi$
$152$		0			0
$153$	1047.41	+	349.243i	0.553449	+	0.184540i
$154$		0			0
$155$			121.525i			0.0629752i
$156$		0			0
$157$	−1787.64			−0.908719			−0.454360	−	0.890818i	$-0.650132\pi$
$157$	−1787.64			−0.908719			−0.454360	+	0.890818i	$0.650132\pi$
$158$		0			0
$159$	2843.85	+	461.629i	1.41844	+	0.230249i
$160$		0			0
$161$	−118.394			−0.0579549
$162$		0			0
$163$	−2607.09			−1.25278			−0.626390	−	0.779510i	$-0.715468\pi$
$163$	−2607.09			−1.25278			−0.626390	+	0.779510i	$0.715468\pi$
$164$		0			0
$165$	850.568	+	783.224i	0.401313	+	0.369539i
$166$		0			0
$167$	1615.67			0.748649			0.374324	−	0.927298i	$-0.377875\pi$
$167$	1615.67			0.748649			0.374324	+	0.927298i	$0.377875\pi$
$168$		0			0
$169$	785.518			0.357541
$170$		0			0
$171$	720.453	−	2160.69i	0.322190	−	0.966269i
$172$		0			0
$173$	3145.99			1.38257			0.691286	−	0.722581i	$-0.257045\pi$
$173$	3145.99			1.38257			0.691286	+	0.722581i	$0.257045\pi$
$174$		0			0
$175$		−	102.708i		−	0.0443657i
$176$		0			0
$177$	−949.776	−	154.173i	−0.403331	−	0.0654708i
$178$		0			0
$179$		−	4440.78i		−	1.85430i	−0.374693	−	0.927149i	$-0.622252\pi$
$179$		−	4440.78i		−	1.85430i	0.374693	−	0.927149i	$-0.377748\pi$
$180$		0			0
$181$	2744.08			1.12688			0.563442	−	0.826156i	$-0.309477\pi$
$181$	2744.08			1.12688			0.563442	+	0.826156i	$0.309477\pi$
$182$		0			0
$183$	−3868.35	−	627.932i	−1.56261	−	0.253651i
$184$		0			0
$185$		−	279.995i		−	0.111274i
$186$		0			0
$187$	−1245.21	−	821.672i	−0.486945	−	0.321319i
$188$		0			0
$189$	−145.364	−	76.1897i	−0.0559453	−	0.0293227i
$190$		0			0
$191$			529.261i			0.200502i	0.994962	+	0.100251i	$0.0319646\pi$
$191$			529.261i			0.200502i	−0.994962	+	0.100251i	$0.968035\pi$
$192$		0			0
$193$		−	688.329i		−	0.256720i	−0.991728	−	0.128360i	$-0.959029\pi$
$193$		−	688.329i		−	0.256720i	0.991728	−	0.128360i	$-0.0409714\pi$
$194$		0			0
$195$	190.782	−	1175.31i	0.0700625	−	0.431618i
$196$		0			0
$197$	−1562.71			−0.565170			−0.282585	−	0.959242i	$-0.591192\pi$
$197$	−1562.71			−0.565170			−0.282585	+	0.959242i	$0.591192\pi$
$198$		0			0
$199$	−3580.49			−1.27545			−0.637725	−	0.770264i	$-0.720124\pi$
$199$	−3580.49			−1.27545			−0.637725	+	0.770264i	$0.720124\pi$
$200$		0			0
$201$	−317.487	+	1955.87i	−0.111412	+	0.686350i
$202$		0			0
$203$		−	294.279i		−	0.101746i
$204$		0			0
$205$		−	1071.02i		−	0.364894i
$206$		0			0
$207$	−864.364	+	2592.29i	−0.290229	+	0.870418i
$208$		0			0
$209$	−1695.03	+	2568.74i	−0.560992	+	0.850159i
$210$		0			0
$211$		−	3783.06i		−	1.23430i	−0.786847	−	0.617149i	$-0.788288\pi$
$211$		−	3783.06i		−	1.23430i	0.786847	−	0.617149i	$-0.211712\pi$
$212$		0			0
$213$	2060.94	+	334.543i	0.662973	+	0.107617i
$214$		0			0
$215$	2028.00			0.643295
$216$		0			0
$217$			23.3080i			0.00729147i
$218$		0			0
$219$	−5306.31	−	861.349i	−1.63729	−	0.265774i
$220$		0			0
$221$			1536.32i			0.467619i
$222$		0			0
$223$	642.113			0.192821			0.0964104	−	0.995342i	$-0.469264\pi$
$223$	642.113			0.192821			0.0964104	+	0.995342i	$0.469264\pi$
$224$		0			0
$225$	−2248.84	−	749.847i	−0.666324	−	0.222177i
$226$		0			0
$227$	2331.21			0.681620			0.340810	−	0.940132i	$-0.389299\pi$
$227$	2331.21			0.681620			0.340810	+	0.940132i	$0.389299\pi$
$228$		0			0
$229$	−6167.24			−1.77966			−0.889831	−	0.456290i	$-0.849178\pi$
$229$	−6167.24			−1.77966			−0.889831	+	0.456290i	$0.849178\pi$
$230$		0			0
$231$	163.135	+	150.219i	0.0464653	+	0.0427864i
$232$		0			0
$233$	−1475.11			−0.414755			−0.207377	−	0.978261i	$-0.566493\pi$
$233$	−1475.11			−0.414755			−0.207377	+	0.978261i	$0.566493\pi$
$234$		0			0
$235$	1139.44			0.316293
$236$		0			0
$237$	−5356.35	−	869.471i	−1.46807	−	0.238305i
$238$		0			0
$239$	−5448.65			−1.47466			−0.737330	−	0.675532i	$-0.763914\pi$
$239$	−5448.65			−1.47466			−0.737330	+	0.675532i	$0.763914\pi$
$240$		0			0
$241$		−	188.475i		−	0.0503766i	−0.999683	−	0.0251883i	$-0.991981\pi$
$241$		−	188.475i		−	0.0503766i	0.999683	−	0.0251883i	$-0.00801853\pi$
$242$		0			0
$243$	−2729.48	+	2626.57i	−0.720559	+	0.693393i
$244$		0			0
$245$		−	2083.71i		−	0.543361i
$246$		0			0
$247$	3169.26			0.816419
$248$		0			0
$249$	705.226	−	4344.52i	0.179486	−	1.10572i
$250$		0			0
$251$			4726.64i			1.18862i	0.804237	+	0.594309i	$0.202574\pi$
$251$			4726.64i			1.18862i	−0.804237	+	0.594309i	$0.797426\pi$
$252$		0			0
$253$	2033.61	−	3081.84i	0.505343	−	0.765826i
$254$		0			0
$255$	−1279.25	−	207.655i	−0.314157	−	0.0509956i
$256$		0			0
$257$		−	4898.69i		−	1.18900i	−0.804097	−	0.594498i	$-0.797351\pi$
$257$		−	4898.69i		−	1.18900i	0.804097	−	0.594498i	$-0.202649\pi$
$258$		0			0
$259$		−	53.7017i		−	0.0128836i
$260$		0			0
$261$	−6443.39	−	2148.46i	−1.52811	−	0.509527i
$262$		0			0
$263$	3627.90			0.850593			0.425296	−	0.905054i	$-0.360170\pi$
$263$	3627.90			0.850593			0.425296	+	0.905054i	$0.360170\pi$
$264$		0			0
$265$	−3381.83			−0.783940
$266$		0			0
$267$	−3462.51	−	562.054i	−0.793642	−	0.128828i
$268$		0			0
$269$			3349.61i			0.759218i	0.925147	+	0.379609i	$0.123941\pi$
$269$			3349.61i			0.759218i	−0.925147	+	0.379609i	$0.876059\pi$
$270$		0			0
$271$			6472.32i			1.45079i	0.688330	+	0.725397i	$0.258344\pi$
$271$			6472.32i			1.45079i	−0.688330	+	0.725397i	$0.741656\pi$
$272$		0			0
$273$	36.5911	−	225.418i	0.00811206	−	0.0499741i
$274$		0			0
$275$	2673.54	+	1764.18i	0.586256	+	0.386851i
$276$		0			0
$277$			2365.72i			0.513149i	0.966525	+	0.256574i	$0.0825938\pi$
$277$			2365.72i			0.513149i	−0.966525	+	0.256574i	$0.917406\pi$
$278$		0			0
$279$	510.340	+	170.166i	0.109510	+	0.0365146i
$280$		0			0
$281$	7527.53			1.59806			0.799030	−	0.601291i	$-0.205347\pi$
$281$	7527.53			1.59806			0.799030	+	0.601291i	$0.205347\pi$
$282$		0			0
$283$		−	5881.45i		−	1.23539i	−0.786417	−	0.617696i	$-0.788066\pi$
$283$		−	5881.45i		−	1.23539i	0.786417	−	0.617696i	$-0.211934\pi$
$284$		0			0
$285$	−428.372	+	2638.97i	−0.0890335	+	0.548488i
$286$		0			0
$287$		−	205.416i		−	0.0422486i
$288$		0			0
$289$	−3240.81			−0.659639
$290$		0			0
$291$	314.778	−	1939.18i	0.0634110	−	0.390642i
$292$		0			0
$293$	−1110.23			−0.221367			−0.110683	−	0.993856i	$-0.535304\pi$
$293$	−1110.23			−0.221367			−0.110683	+	0.993856i	$0.535304\pi$
$294$		0			0
$295$	1129.45			0.222912
$296$		0			0
$297$	4480.12	−	2475.20i	0.875295	−	0.483589i
$298$		0			0
$299$	−3802.33			−0.735433
$300$		0			0
$301$	388.961			0.0744828
$302$		0			0
$303$	−725.117	+	4467.06i	−0.137481	+	0.846950i
$304$		0			0
$305$	4600.14			0.863618
$306$		0			0
$307$		−	9223.13i		−	1.71463i	−0.514792	−	0.857315i	$-0.672131\pi$
$307$		−	9223.13i		−	1.71463i	0.514792	−	0.857315i	$-0.327869\pi$
$308$		0			0
$309$	−1615.24	+	9950.62i	−0.297371	+	1.83195i
$310$		0			0
$311$		−	6601.30i		−	1.20362i	−0.798640	−	0.601809i	$-0.794447\pi$
$311$		−	6601.30i		−	1.20362i	0.798640	−	0.601809i	$-0.205553\pi$
$312$		0			0
$313$	3325.08			0.600463			0.300231	−	0.953866i	$-0.402936\pi$
$313$	3325.08			0.600463			0.300231	+	0.953866i	$0.402936\pi$
$314$		0			0
$315$	182.754	+	60.9370i	0.0326890	+	0.0108997i
$316$		0			0
$317$		−	6364.80i		−	1.12771i	−0.825875	−	0.563853i	$-0.809318\pi$
$317$		−	6364.80i		−	1.12771i	0.825875	−	0.563853i	$-0.190682\pi$
$318$		0			0
$319$	7660.22	+	5054.73i	1.34448	+	0.887181i
$320$		0			0
$321$	−418.948	+	2580.92i	−0.0728455	+	0.448763i
$322$		0			0
$323$		−	3449.56i		−	0.594238i
$324$		0			0
$325$		−	3298.57i		−	0.562990i
$326$		0			0
$327$	10015.1	+	1625.71i	1.69369	+	0.274929i
$328$		0			0
$329$	218.539			0.0366215
$330$		0			0
$331$	−6316.60			−1.04892			−0.524459	−	0.851436i	$-0.675732\pi$
$331$	−6316.60			−1.04892			−0.524459	+	0.851436i	$0.675732\pi$
$332$		0			0
$333$	−1175.83	−	392.063i	−0.193498	−	0.0645193i
$334$		0			0
$335$		−	2325.87i		−	0.379331i
$336$		0			0
$337$		−	3332.25i		−	0.538633i	−0.963052	−	0.269317i	$-0.913202\pi$
$337$		−	3332.25i		−	0.538633i	0.963052	−	0.269317i	$-0.0867978\pi$
$338$		0			0
$339$	−5176.64	−	840.301i	−0.829371	−	0.134628i
$340$		0			0
$341$	−606.718	−	400.353i	−0.0963508	−	0.0635787i
$342$		0			0
$343$		−	800.892i		−	0.126076i
$344$		0			0
$345$	513.939	−	3166.11i	0.0802017	−	0.494080i
$346$		0			0
$347$	−6024.71			−0.932055			−0.466028	−	0.884770i	$-0.654315\pi$
$347$	−6024.71			−0.932055			−0.466028	+	0.884770i	$0.654315\pi$
$348$		0			0
$349$		−	1442.33i		−	0.221222i	−0.993864	−	0.110611i	$-0.964719\pi$
$349$		−	1442.33i		−	0.221222i	0.993864	−	0.110611i	$-0.0352807\pi$
$350$		0			0
$351$	−4668.50	−	2446.90i	−0.709932	−	0.372097i
$352$		0			0
$353$			5171.99i			0.779822i	0.920853	+	0.389911i	$0.127494\pi$
$353$			5171.99i			0.779822i	−0.920853	+	0.389911i	$0.872506\pi$
$354$		0			0
$355$	−2450.82			−0.366411
$356$		0			0
$357$	−245.355	−	39.8273i	−0.0363741	−	0.00590444i
$358$		0			0
$359$	8749.85			1.28635			0.643175	−	0.765720i	$-0.277617\pi$
$359$	8749.85			1.28635			0.643175	+	0.765720i	$0.277617\pi$
$360$		0			0
$361$	−257.090			−0.0374822
$362$		0			0
$363$	−6712.37	+	1666.22i	−0.970545	+	0.240920i
$364$		0			0
$365$	6310.12			0.904896
$366$		0			0
$367$	334.322			0.0475517			0.0237759	−	0.999717i	$-0.492431\pi$
$367$	334.322			0.0475517			0.0237759	+	0.999717i	$0.492431\pi$
$368$		0			0
$369$	−4497.69	−	1499.69i	−0.634527	−	0.211574i
$370$		0			0
$371$	−648.618			−0.0907670
$372$		0			0
$373$			12485.2i			1.73313i	0.499062	+	0.866566i	$0.333678\pi$
$373$			12485.2i			1.73313i	−0.499062	+	0.866566i	$0.666322\pi$
$374$		0			0
$375$	6657.06	+	1080.61i	0.916718	+	0.148807i
$376$		0			0
$377$		−	9451.06i		−	1.29113i
$378$		0			0
$379$	−7043.92			−0.954676			−0.477338	−	0.878720i	$-0.658398\pi$
$379$	−7043.92			−0.954676			−0.477338	+	0.878720i	$0.658398\pi$
$380$		0			0
$381$	−432.801	−	70.2546i	−0.0581970	−	0.00944685i
$382$		0			0
$383$			3641.30i			0.485801i	0.970051	+	0.242900i	$0.0780988\pi$
$383$			3641.30i			0.485801i	−0.970051	+	0.242900i	$0.921901\pi$
$384$		0			0
$385$	−217.268	−	143.368i	−0.0287610	−	0.0189784i
$386$		0			0
$387$	2839.71	−	8516.48i	0.372998	−	1.11865i
$388$		0			0
$389$			305.012i			0.0397551i	0.999802	+	0.0198775i	$0.00632763\pi$
$389$			305.012i			0.0397551i	−0.999802	+	0.0198775i	$0.993672\pi$
$390$		0			0
$391$			4138.62i			0.535291i
$392$		0			0
$393$	−1487.77	+	9165.33i	−0.190961	+	1.17641i
$394$		0			0
$395$	6369.63			0.811369
$396$		0			0
$397$	1097.19			0.138707			0.0693534	−	0.997592i	$-0.477906\pi$
$397$	1097.19			0.138707			0.0693534	+	0.997592i	$0.477906\pi$
$398$		0			0
$399$	−82.1596	+	506.142i	−0.0103086	+	0.0635057i
$400$		0			0
$401$		−	7833.32i		−	0.975504i	−0.872982	−	0.487752i	$-0.837817\pi$
$401$		−	7833.32i		−	0.975504i	0.872982	−	0.487752i	$-0.162183\pi$
$402$		0			0
$403$			748.558i			0.0925269i
$404$		0			0
$405$	2668.49	−	3556.61i	0.327403	−	0.436369i
$406$		0			0
$407$	1397.88	+	922.416i	0.170247	+	0.112340i
$408$		0			0
$409$			4130.56i			0.499371i	0.968327	+	0.249686i	$0.0803273\pi$
$409$			4130.56i			0.499371i	−0.968327	+	0.249686i	$0.919673\pi$
$410$		0			0
$411$	12164.8	+	1974.65i	1.45996	+	0.236989i
$412$		0			0
$413$	216.623			0.0258094
$414$		0			0
$415$			5166.39i			0.611104i
$416$		0			0
$417$	9031.44	+	1466.03i	1.06060	+	0.172163i
$418$		0			0
$419$			8613.76i			1.00432i	0.864775	+	0.502160i	$0.167461\pi$
$419$			8613.76i			1.00432i	−0.864775	+	0.502160i	$0.832539\pi$
$420$		0			0
$421$	8896.13			1.02986			0.514929	−	0.857233i	$-0.327818\pi$
$421$	8896.13			1.02986			0.514929	+	0.857233i	$0.327818\pi$
$422$		0			0
$423$	1595.50	−	4785.02i	0.183395	−	0.550014i
$424$		0			0
$425$	−3590.30			−0.409777
$426$		0			0
$427$	882.285			0.0999924
$428$		0			0
$429$	5239.23	+	4824.41i	0.589632	+	0.542948i
$430$		0			0
$431$	939.369			0.104983			0.0524917	−	0.998621i	$-0.483284\pi$
$431$	939.369			0.104983			0.0524917	+	0.998621i	$0.483284\pi$
$432$		0			0
$433$	9426.38			1.04620			0.523098	−	0.852272i	$-0.324776\pi$
$433$	9426.38			1.04620			0.523098	+	0.852272i	$0.324776\pi$
$434$		0			0
$435$	7869.67	+	1277.45i	0.867406	+	0.140802i
$436$		0			0
$437$	8537.54			0.934567
$438$		0			0
$439$			8187.39i			0.890121i	0.895501	+	0.445060i	$0.146818\pi$
$439$			8187.39i			0.890121i	−0.895501	+	0.445060i	$0.853182\pi$
$440$		0			0
$441$	−8750.43	−	2917.71i	−0.944869	−	0.315054i
$442$		0			0
$443$		−	4287.18i		−	0.459797i	−0.973215	−	0.229898i	$-0.926161\pi$
$443$		−	4287.18i		−	0.459797i	0.973215	−	0.229898i	$-0.0738394\pi$
$444$		0			0
$445$	4117.53			0.438628
$446$		0			0
$447$	−807.734	+	4976.02i	−0.0854687	+	0.526527i
$448$		0			0
$449$			12593.5i			1.32366i	0.749652	+	0.661832i	$0.230221\pi$
$449$			12593.5i			1.32366i	−0.749652	+	0.661832i	$0.769779\pi$
$450$		0			0
$451$	5347.08	+	3528.36i	0.558280	+	0.368390i
$452$		0			0
$453$	−7350.91	−	1193.24i	−0.762419	−	0.123760i
$454$		0			0
$455$			268.061i			0.0276196i
$456$		0			0
$457$			4093.90i			0.419047i	0.977804	+	0.209523i	$0.0671913\pi$
$457$			4093.90i			0.419047i	−0.977804	+	0.209523i	$0.932809\pi$
$458$		0			0
$459$	−2663.31	+	5081.39i	−0.270834	+	0.516730i
$460$		0			0
$461$	−3374.90			−0.340965			−0.170482	−	0.985361i	$-0.554533\pi$
$461$	−3374.90			−0.340965			−0.170482	+	0.985361i	$0.554533\pi$
$462$		0			0
$463$	2113.75			0.212169			0.106084	−	0.994357i	$-0.466169\pi$
$463$	2113.75			0.212169			0.106084	+	0.994357i	$0.466169\pi$
$464$		0			0
$465$	−623.306	−	101.178i	−0.0621616	−	0.0100904i
$466$		0			0
$467$		−	3549.00i		−	0.351666i	−0.984420	−	0.175833i	$-0.943738\pi$
$467$		−	3549.00i		−	0.351666i	0.984420	−	0.175833i	$-0.0562619\pi$
$468$		0			0
$469$		−	446.090i		−	0.0439201i
$470$		0			0
$471$	1488.33	−	9168.82i	0.145602	−	0.896979i
$472$		0			0
$473$	−6681.04	+	10124.8i	−0.649460	+	0.984228i
$474$		0			0
$475$			7406.42i			0.715432i
$476$		0			0
$477$	−4735.40	+	14201.8i	−0.454548	+	1.36322i
$478$		0			0
$479$	−4651.83			−0.443731			−0.221866	−	0.975077i	$-0.571215\pi$
$479$	−4651.83			−0.443731			−0.221866	+	0.975077i	$0.571215\pi$
$480$		0			0
$481$		−	1724.68i		−	0.163490i
$482$		0			0
$483$	98.5711	−	607.244i	0.00928600	−	0.0572061i
$484$		0			0
$485$			2306.02i			0.215899i
$486$		0			0
$487$	575.427			0.0535422			0.0267711	−	0.999642i	$-0.491477\pi$
$487$	575.427			0.0535422			0.0267711	+	0.999642i	$0.491477\pi$
$488$		0			0
$489$	2170.58	−	13371.8i	0.200730	−	1.23659i
$490$		0			0
$491$	−15977.7			−1.46856			−0.734279	−	0.678848i	$-0.762479\pi$
$491$	−15977.7			−1.46856			−0.734279	+	0.678848i	$0.762479\pi$
$492$		0			0
$493$	−10286.9			−0.939757
$494$		0			0
$495$	−4725.33	+	3710.49i	−0.429066	+	0.336917i
$496$		0			0
$497$	−470.055			−0.0424242
$498$		0			0
$499$	4327.29			0.388209			0.194104	−	0.980981i	$-0.437820\pi$
$499$	4327.29			0.388209			0.194104	+	0.980981i	$0.437820\pi$
$500$		0			0
$501$	−1345.16	+	8286.80i	−0.119955	+	0.738976i
$502$		0			0
$503$	−5390.13			−0.477801			−0.238901	−	0.971044i	$-0.576787\pi$
$503$	−5390.13			−0.477801			−0.238901	+	0.971044i	$0.576787\pi$
$504$		0			0
$505$		−	5312.11i		−	0.468090i
$506$		0			0
$507$	−653.998	+	4028.93i	−0.0572881	+	0.352922i
$508$		0			0
$509$		−	13340.3i		−	1.16168i	−0.814016	−	0.580842i	$-0.802723\pi$
$509$		−	13340.3i		−	1.16168i	0.814016	−	0.580842i	$-0.197277\pi$
$510$		0			0
$511$	1210.25			0.104772
$512$		0			0
$513$	10482.4	+	5494.14i	0.902161	+	0.472850i
$514$		0			0
$515$		−	11833.0i		−	1.01248i
$516$		0			0
$517$	−3753.77	+	5688.68i	−0.319324	+	0.483922i
$518$		0			0
$519$	−2619.25	+	16135.8i	−0.221527	+	1.36471i
$520$		0			0
$521$			14671.7i			1.23374i	0.787066	+	0.616869i	$0.211599\pi$
$521$			14671.7i			1.23374i	−0.787066	+	0.616869i	$0.788401\pi$
$522$		0			0
$523$		−	7638.50i		−	0.638639i	−0.947647	−	0.319320i	$-0.896546\pi$
$523$		−	7638.50i		−	0.638639i	0.947647	−	0.319320i	$-0.103454\pi$
$524$		0			0
$525$	526.792	+	85.5117i	0.0437925	+	0.00710864i
$526$		0			0
$527$	814.762			0.0673465
$528$		0			0
$529$	1924.08			0.158139
$530$		0			0
$531$	1581.51	−	4743.06i	0.129250	−	0.387629i
$532$		0			0
$533$		−	6597.14i		−	0.536123i
$534$		0			0
$535$		−	3069.16i		−	0.248021i
$536$		0			0
$537$	22776.8	+	3697.26i	1.83034	+	0.297111i
$538$		0			0
$539$	10403.0	+	6864.57i	0.831330	+	0.548568i
$540$		0			0
$541$			625.465i			0.0497058i	0.999691	+	0.0248529i	$0.00791173\pi$
$541$			625.465i			0.0497058i	−0.999691	+	0.0248529i	$0.992088\pi$
$542$		0			0
$543$	−2284.64	+	14074.4i	−0.180558	+	1.11232i
$544$		0			0
$545$	−11909.7			−0.936066
$546$		0			0
$547$			1099.96i			0.0859797i	0.999076	+	0.0429899i	$0.0136883\pi$
$547$			1099.96i			0.0859797i	−0.999076	+	0.0429899i	$0.986312\pi$
$548$		0			0
$549$	6441.35	−	19318.1i	0.500747	−	1.50177i
$550$		0			0
$551$			21220.9i			1.64073i
$552$		0			0
$553$	1221.66			0.0939429
$554$		0			0
$555$	1436.10	+	233.116i	0.109836	+	0.0178292i
$556$		0			0
$557$	3439.75			0.261664			0.130832	−	0.991405i	$-0.458235\pi$
$557$	3439.75			0.261664			0.130832	+	0.991405i	$0.458235\pi$
$558$		0			0
$559$	12491.8			0.945167
$560$		0			0
$561$	5251.09	−	5702.60i	0.395190	−	0.429169i
$562$		0			0
$563$	7959.85			0.595857			0.297929	−	0.954588i	$-0.403704\pi$
$563$	7959.85			0.595857			0.297929	+	0.954588i	$0.403704\pi$
$564$		0			0
$565$	6155.93			0.458375
$566$		0			0
$567$	511.804	−	682.141i	0.0379078	−	0.0505242i
$568$		0			0
$569$	−12864.2			−0.947791			−0.473896	−	0.880581i	$-0.657153\pi$
$569$	−12864.2			−0.947791			−0.473896	+	0.880581i	$0.657153\pi$
$570$		0			0
$571$			17952.2i			1.31572i	0.753141	+	0.657859i	$0.228538\pi$
$571$			17952.2i			1.31572i	−0.753141	+	0.657859i	$0.771462\pi$
$572$		0			0
$573$	−2714.59	−	440.646i	−0.197912	−	0.0321261i
$574$		0			0
$575$		−	8885.87i		−	0.644463i
$576$		0			0
$577$	−17586.2			−1.26884			−0.634421	−	0.772988i	$-0.718761\pi$
$577$	−17586.2			−1.26884			−0.634421	+	0.772988i	$0.718761\pi$
$578$		0			0
$579$	3530.45	+	573.082i	0.253404	+	0.0411338i
$580$		0			0
$581$			990.889i			0.0707556i
$582$		0			0
$583$	11141.1	−	16883.8i	0.791452	−	1.19941i
$584$		0			0
$585$	5869.33	+	1957.05i	0.414815	+	0.138315i
$586$		0			0
$587$			6329.62i			0.445062i	0.974926	+	0.222531i	$0.0714318\pi$
$587$			6329.62i			0.445062i	−0.974926	+	0.222531i	$0.928568\pi$
$588$		0			0
$589$		−	1680.77i		−	0.117581i
$590$		0			0
$591$	1301.06	−	8015.17i	0.0905561	−	0.557868i
$592$		0			0
$593$	−24403.9			−1.68997			−0.844983	−	0.534794i	$-0.820389\pi$
$593$	−24403.9			−1.68997			−0.844983	+	0.534794i	$0.820389\pi$
$594$		0			0
$595$	291.769			0.0201031
$596$		0			0
$597$	2981.01	−	18364.4i	0.204363	−	1.25897i
$598$		0			0
$599$		−	6281.88i		−	0.428498i	−0.976779	−	0.214249i	$-0.931270\pi$
$599$		−	6281.88i		−	0.428498i	0.976779	−	0.214249i	$-0.0687304\pi$
$600$		0			0
$601$			20299.7i			1.37777i	0.724869	+	0.688887i	$0.241900\pi$
$601$			20299.7i			1.37777i	−0.724869	+	0.688887i	$0.758100\pi$
$602$		0			0
$603$	−9767.36	−	3256.80i	−0.659631	−	0.219945i
$604$		0			0
$605$	7463.86	−	3193.00i	0.501569	−	0.214568i
$606$		0			0
$607$			1642.65i			0.109840i	0.998491	+	0.0549202i	$0.0174904\pi$
$607$			1642.65i			0.109840i	−0.998491	+	0.0549202i	$0.982510\pi$
$608$		0			0
$609$	1509.36	+	245.008i	0.100431	+	0.0163025i
$610$		0			0
$611$	7018.60			0.464717
$612$		0			0
$613$			9464.85i			0.623624i	0.950144	+	0.311812i	$0.100936\pi$
$613$			9464.85i			0.623624i	−0.950144	+	0.311812i	$0.899064\pi$
$614$		0			0
$615$	5493.28	+	891.698i	0.360179	+	0.0584662i
$616$		0			0
$617$			20313.6i			1.32544i	0.748869	+	0.662718i	$0.230597\pi$
$617$			20313.6i			1.32544i	−0.748869	+	0.662718i	$0.769403\pi$
$618$		0			0
$619$	21020.9			1.36494			0.682471	−	0.730912i	$-0.260905\pi$
$619$	21020.9			1.36494			0.682471	+	0.730912i	$0.260905\pi$
$620$		0			0
$621$	−12576.3	−	6591.60i	−0.812670	−	0.425945i
$622$		0			0
$623$	789.722			0.0507858
$624$		0			0
$625$	3058.43			0.195740
$626$		0			0
$627$	−11763.9	−	10832.5i	−0.749288	−	0.689963i
$628$		0			0
$629$	−1877.22			−0.118998
$630$		0			0
$631$	−24742.9			−1.56101			−0.780507	−	0.625148i	$-0.785039\pi$
$631$	−24742.9			−1.56101			−0.780507	+	0.625148i	$0.785039\pi$
$632$		0			0
$633$	19403.4	+	3149.66i	1.21835	+	0.197769i
$634$		0			0
$635$	514.675			0.0321642
$636$		0			0
$637$		−	12835.0i		−	0.798338i
$638$		0			0
$639$	−3431.75	+	10292.1i	−0.212454	+	0.637165i
$640$		0			0
$641$		−	8846.29i		−	0.545098i	−0.962142	−	0.272549i	$-0.912133\pi$
$641$		−	8846.29i		−	0.545098i	0.962142	−	0.272549i	$-0.0878666\pi$
$642$		0			0
$643$	−28124.9			−1.72494			−0.862471	−	0.506106i	$-0.831084\pi$
$643$	−28124.9			−1.72494			−0.862471	+	0.506106i	$0.831084\pi$
$644$		0			0
$645$	−1688.45	+	10401.7i	−0.103074	+	0.634984i
$646$		0			0
$647$		−	22919.1i		−	1.39265i	−0.717728	−	0.696323i	$-0.754818\pi$
$647$		−	22919.1i		−	1.39265i	0.717728	−	0.696323i	$-0.245182\pi$
$648$		0			0
$649$	−3720.85	+	5638.78i	−0.225048	+	0.341050i
$650$		0			0
$651$	−119.547	−	19.4055i	−0.00719727	−	0.00116830i
$652$		0			0
$653$			12310.5i			0.737747i	0.929480	+	0.368873i	$0.120256\pi$
$653$			12310.5i			0.737747i	−0.929480	+	0.368873i	$0.879744\pi$
$654$		0			0
$655$		−	10899.2i		−	0.650177i
$656$		0			0
$657$	8835.75	−	26499.0i	0.524681	−	1.57356i
$658$		0			0
$659$	−21638.9			−1.27911			−0.639554	−	0.768746i	$-0.720881\pi$
$659$	−21638.9			−1.27911			−0.639554	+	0.768746i	$0.720881\pi$
$660$		0			0
$661$	24343.1			1.43243			0.716217	−	0.697878i	$-0.245872\pi$
$661$	24343.1			1.43243			0.716217	+	0.697878i	$0.245872\pi$
$662$		0			0
$663$	−7879.80	−	1279.09i	−0.461578	−	0.0749258i
$664$		0			0
$665$		−	601.890i		−	0.0350982i
$666$		0			0
$667$		−	25459.8i		−	1.47797i
$668$		0			0
$669$	−534.604	+	3293.41i	−0.0308953	+	0.190330i
$670$		0			0
$671$	−15154.7	+	22966.3i	−0.871893	+	1.32132i
$672$		0			0
$673$		−	20487.4i		−	1.17345i	−0.809787	−	0.586724i	$-0.800417\pi$
$673$		−	20487.4i		−	1.17345i	0.809787	−	0.586724i	$-0.199583\pi$
$674$		0			0
$675$	5718.30	−	10910.1i	0.326070	−	0.622116i
$676$		0			0
$677$	23872.9			1.35526			0.677628	−	0.735404i	$-0.263008\pi$
$677$	23872.9			1.35526			0.677628	+	0.735404i	$0.263008\pi$
$678$		0			0
$679$			442.284i			0.0249975i
$680$		0			0
$681$	−1940.89	+	11956.8i	−0.109215	+	0.672813i
$682$		0			0
$683$			8085.75i			0.452991i	0.974012	+	0.226495i	$0.0727268\pi$
$683$			8085.75i			0.452991i	−0.974012	+	0.226495i	$0.927273\pi$
$684$		0			0
$685$	−14466.0			−0.806888
$686$		0			0
$687$	5134.66	−	31631.9i	0.285152	−	1.75667i
$688$		0			0
$689$	−20831.0			−1.15181
$690$		0			0
$691$	3896.37			0.214508			0.107254	−	0.994232i	$-0.465794\pi$
$691$	3896.37			0.214508			0.107254	+	0.994232i	$0.465794\pi$
$692$		0			0
$693$	−906.295	+	711.654i	−0.0496786	+	0.0390094i
$694$		0			0
$695$	−10739.9			−0.586172
$696$		0			0
$697$	−7180.60			−0.390222
$698$		0			0
$699$	1228.13	−	7565.88i	0.0664554	−	0.409396i
$700$		0			0
$701$	−9607.99			−0.517673			−0.258837	−	0.965921i	$-0.583339\pi$
$701$	−9607.99			−0.517673			−0.258837	+	0.965921i	$0.583339\pi$
$702$		0			0
$703$			3872.51i			0.207759i
$704$		0			0
$705$	−948.664	+	5844.21i	−0.0506791	+	0.312207i
$706$		0			0
$707$		−	1018.84i		−	0.0541970i
$708$		0			0
$709$	−3758.71			−0.199099			−0.0995495	−	0.995033i	$-0.531740\pi$
$709$	−3758.71			−0.199099			−0.0995495	+	0.995033i	$0.531740\pi$
$710$		0			0
$711$	8919.07	−	26748.9i	0.470452	−	1.41092i
$712$		0			0
$713$			2016.51i			0.105917i
$714$		0			0
$715$	−6977.76	−	4604.39i	−0.364970	−	0.240832i
$716$		0			0
$717$	4536.38	−	27946.2i	0.236282	−	1.45561i
$718$		0			0
$719$			8115.64i			0.420949i	0.977599	+	0.210474i	$0.0675008\pi$
$719$			8115.64i			0.420949i	−0.977599	+	0.210474i	$0.932499\pi$
$720$		0			0
$721$		−	2269.52i		−	0.117228i
$722$		0			0
$723$	966.692	+	156.919i	0.0497257	+	0.00807174i
$724$		0			0
$725$	22086.7			1.13142
$726$		0			0
$727$	−4061.52			−0.207199			−0.103599	−	0.994619i	$-0.533036\pi$
$727$	−4061.52			−0.207199			−0.103599	+	0.994619i	$0.533036\pi$
$728$		0			0
$729$	−11199.2	−	16186.3i	−0.568981	−	0.822351i
$730$		0			0
$731$		−	13596.6i		−	0.687948i
$732$		0			0
$733$		−	16076.9i		−	0.810112i	−0.914292	−	0.405056i	$-0.867252\pi$
$733$		−	16076.9i		−	0.810112i	0.914292	−	0.405056i	$-0.132748\pi$
$734$		0			0
$735$	10687.4	+	1734.83i	0.536340	+	0.0870617i
$736$		0			0
$737$	11611.9	+	7662.34i	0.580368	+	0.382966i
$738$		0			0
$739$			9518.02i			0.473783i	0.971536	+	0.236892i	$0.0761287\pi$
$739$			9518.02i			0.473783i	−0.971536	+	0.236892i	$0.923871\pi$
$740$		0			0
$741$	−2638.63	+	16255.2i	−0.130813	+	0.805871i
$742$		0			0
$743$	−6327.27			−0.312416			−0.156208	−	0.987724i	$-0.549927\pi$
$743$	−6327.27			−0.312416			−0.156208	+	0.987724i	$0.549927\pi$
$744$		0			0
$745$		−	5917.35i		−	0.291000i
$746$		0			0
$747$	21696.0	+	7234.24i	1.06267	+	0.354333i
$748$		0			0
$749$		−	588.650i		−	0.0287167i
$750$		0			0
$751$	−26753.1			−1.29991			−0.649956	−	0.759972i	$-0.725213\pi$
$751$	−26753.1			−1.29991			−0.649956	+	0.759972i	$0.725213\pi$
$752$		0			0
$753$	−24243.0	−	3935.26i	−1.17326	−	0.190450i
$754$		0			0
$755$	8741.51			0.421372
$756$		0			0
$757$	−14244.7			−0.683927			−0.341964	−	0.939713i	$-0.611092\pi$
$757$	−14244.7			−0.683927			−0.341964	+	0.939713i	$0.611092\pi$
$758$		0			0
$759$	14113.7	+	12996.3i	0.674961	+	0.621521i
$760$		0			0
$761$	−18619.7			−0.886944			−0.443472	−	0.896288i	$-0.646254\pi$
$761$	−18619.7			−0.886944			−0.443472	+	0.896288i	$0.646254\pi$
$762$		0			0
$763$	−2284.22			−0.108381
$764$		0			0
$765$	2130.14	−	6388.43i	0.100674	−	0.301927i
$766$		0			0
$767$	6957.04			0.327515
$768$		0			0
$769$		−	20303.7i		−	0.952109i	−0.879416	−	0.476054i	$-0.842067\pi$
$769$		−	20303.7i		−	0.952109i	0.879416	−	0.476054i	$-0.157933\pi$
$770$		0			0
$771$	25125.5	+	4078.50i	1.17363	+	0.190510i
$772$		0			0
$773$		−	34707.7i		−	1.61494i	−0.589907	−	0.807471i	$-0.700836\pi$
$773$		−	34707.7i		−	1.61494i	0.589907	−	0.807471i	$-0.299164\pi$
$774$		0			0
$775$	−1749.35			−0.0810818
$776$		0			0
$777$	275.437	+	44.7104i	0.0127172	+	0.00206432i
$778$		0			0
$779$			14812.8i			0.681291i
$780$		0			0
$781$	8073.96	−	12235.7i	0.369922	−	0.560601i
$782$		0			0
$783$	16384.1	−	31259.5i	0.747789	−	1.42672i
$784$		0			0
$785$			10903.3i			0.495740i
$786$		0			0
$787$			20177.4i			0.913910i	0.889490	+	0.456955i	$0.151060\pi$
$787$			20177.4i			0.913910i	−0.889490	+	0.456955i	$0.848940\pi$
$788$		0			0
$789$	−3020.48	+	18607.6i	−0.136289	+	0.839603i
$790$		0			0
$791$	1180.68			0.0530721
$792$		0			0
$793$	28335.4			1.26888
$794$		0			0
$795$	2815.61	−	17345.5i	0.125609	−	0.773811i
$796$		0			0
$797$			7076.83i			0.314522i	0.987557	+	0.157261i	$0.0502664\pi$
$797$			7076.83i			0.314522i	−0.987557	+	0.157261i	$0.949734\pi$
$798$		0			0
$799$		−	7639.34i		−	0.338248i
$800$		0			0
$801$	5765.57	−	17291.3i	0.254327	−	0.762746i
$802$		0			0
$803$	−20788.1	+	31503.4i	−0.913567	+	1.38447i
$804$		0			0
$805$			722.118i			0.0316166i
$806$		0			0
$807$	−17180.2	−	2788.79i	−0.749409	−	0.121648i
$808$		0			0
$809$	−42531.5			−1.84837			−0.924183	−	0.381951i	$-0.875252\pi$
$809$	−42531.5			−1.84837			−0.924183	+	0.381951i	$0.875252\pi$
$810$		0			0
$811$			2213.84i			0.0958552i	0.998851	+	0.0479276i	$0.0152617\pi$
$811$			2213.84i			0.0958552i	−0.998851	+	0.0479276i	$0.984738\pi$
$812$		0			0
$813$	−33196.6	−	5388.66i	−1.43205	−	0.232458i
$814$		0			0
$815$			15901.4i			0.683438i
$816$		0			0
$817$	−28048.5			−1.20109
$818$		0			0
$819$	1125.71	+	375.353i	0.0480287	+	0.0160145i
$820$		0			0
$821$	19530.6			0.830234			0.415117	−	0.909768i	$-0.363741\pi$
$821$	19530.6			0.830234			0.415117	+	0.909768i	$0.363741\pi$
$822$		0			0
$823$	−12872.0			−0.545186			−0.272593	−	0.962129i	$-0.587881\pi$
$823$	−12872.0			−0.545186			−0.272593	+	0.962129i	$0.587881\pi$
$824$		0			0
$825$	−11274.4	+	12243.8i	−0.475788	+	0.516697i
$826$		0			0
$827$	−10319.7			−0.433920			−0.216960	−	0.976181i	$-0.569614\pi$
$827$	−10319.7			−0.433920			−0.216960	+	0.976181i	$0.569614\pi$
$828$		0			0
$829$	−40067.5			−1.67865			−0.839326	−	0.543629i	$-0.817050\pi$
$829$	−40067.5			−1.67865			−0.839326	+	0.543629i	$0.817050\pi$
$830$		0			0
$831$	−12133.8	−	1969.62i	−0.506519	−	0.0822208i
$832$		0			0
$833$	−13970.2			−0.581077
$834$		0			0
$835$		−	9854.45i		−	0.408416i
$836$		0			0
$837$	−1297.68	+	2475.87i	−0.0535894	+	0.102244i
$838$		0			0
$839$		−	1122.81i		−	0.0462022i	−0.999733	−	0.0231011i	$-0.992646\pi$
$839$		−	1122.81i		−	0.0462022i	0.999733	−	0.0231011i	$-0.00735397\pi$
$840$		0			0
$841$	38893.8			1.59473
$842$		0			0
$843$	−6267.19	+	38608.8i	−0.256054	+	1.57741i
$844$		0			0
$845$		−	4791.10i		−	0.195052i
$846$		0			0
$847$	1431.53	−	612.402i	0.0580732	−	0.0248434i
$848$		0			0
$849$	30166.1	+	4896.72i	1.21943	+	0.197945i
$850$		0			0
$851$		−	4646.04i		−	0.187150i
$852$		0			0
$853$			10578.2i			0.424609i	0.977204	+	0.212305i	$0.0680969\pi$
$853$			10578.2i			0.424609i	−0.977204	+	0.212305i	$0.931903\pi$
$854$		0			0
$855$	−13178.7	−	4394.25i	−0.527136	−	0.175766i
$856$		0			0
$857$	21583.1			0.860287			0.430143	−	0.902761i	$-0.358463\pi$
$857$	21583.1			0.860287			0.430143	+	0.902761i	$0.358463\pi$
$858$		0			0
$859$	35493.3			1.40980			0.704898	−	0.709309i	$-0.250993\pi$
$859$	35493.3			1.40980			0.704898	+	0.709309i	$0.250993\pi$
$860$		0			0
$861$	1053.58	+	171.023i	0.0417027	+	0.00676941i
$862$		0			0
$863$		−	6119.91i		−	0.241395i	−0.992689	−	0.120698i	$-0.961487\pi$
$863$		−	6119.91i		−	0.241395i	0.992689	−	0.120698i	$-0.0385131\pi$
$864$		0			0
$865$		−	19188.3i		−	0.754244i
$866$		0			0
$867$	2698.20	−	16622.2i	0.105693	−	0.651117i
$868$		0			0
$869$	−20984.1	+	31800.5i	−0.819144	+	1.24138i
$870$		0			0
$871$		−	14326.6i		−	0.557335i
$872$		0			0
$873$	9684.01	+	3229.00i	0.375434	+	0.125184i
$874$		0			0
$875$	−1518.33			−0.0586615
$876$		0			0
$877$		−	14460.8i		−	0.556793i	−0.960466	−	0.278397i	$-0.910197\pi$
$877$		−	14460.8i		−	0.556793i	0.960466	−	0.278397i	$-0.0898030\pi$
$878$		0			0
$879$	924.346	−	5694.40i	0.0354692	−	0.218507i
$880$		0			0
$881$			36160.3i			1.38283i	0.722458	+	0.691415i	$0.243012\pi$
$881$			36160.3i			1.38283i	−0.722458	+	0.691415i	$0.756988\pi$
$882$		0			0
$883$	46776.3			1.78273			0.891364	−	0.453288i	$-0.149749\pi$
$883$	46776.3			1.78273			0.891364	+	0.453288i	$0.149749\pi$
$884$		0			0
$885$	−940.344	+	5792.96i	−0.0357167	+	0.220032i
$886$		0			0
$887$	−45438.0			−1.72002			−0.860011	−	0.510276i	$-0.829543\pi$
$887$	−45438.0			−1.72002			−0.860011	+	0.510276i	$0.829543\pi$
$888$		0			0
$889$	98.7123			0.00372407
$890$		0			0
$891$	8965.36	+	25039.4i	0.337094	+	0.941471i
$892$		0			0
$893$	−15759.2			−0.590549
$894$		0			0
$895$	−27085.6			−1.01159
$896$		0			0
$897$	3165.70	−	19502.2i	0.117837	−	0.725931i
$898$		0			0
$899$	−5012.23			−0.185948
$900$		0			0
$901$			22673.3i			0.838355i
$902$		0			0
$903$	−323.837	+	1994.99i	−0.0119342	+	0.0735205i
$904$		0			0
$905$		−	16736.9i		−	0.614757i
$906$		0			0
$907$	−24873.9			−0.910612			−0.455306	−	0.890335i	$-0.650470\pi$
$907$	−24873.9			−0.910612			−0.455306	+	0.890335i	$0.650470\pi$
$908$		0			0
$909$	−22307.9	−	7438.28i	−0.813979	−	0.271410i
$910$		0			0
$911$		−	5633.58i		−	0.204884i	−0.994739	−	0.102442i	$-0.967334\pi$
$911$		−	5633.58i		−	0.204884i	0.994739	−	0.102442i	$-0.0326655\pi$
$912$		0			0
$913$	−25793.3	−	17020.2i	−0.934976	−	0.616960i
$914$		0			0
$915$	−3829.94	+	23594.2i	−0.138376	+	0.852460i
$916$		0			0
$917$		−	2090.41i		−	0.0752795i
$918$		0			0
$919$		−	27624.4i		−	0.991561i	−0.868448	−	0.495780i	$-0.834882\pi$
$919$		−	27624.4i		−	0.991561i	0.868448	−	0.495780i	$-0.165118\pi$
$920$		0			0
$921$	47305.6	+	7678.89i	1.69248	+	0.274732i
$922$		0			0
$923$	−15096.2			−0.538352
$924$		0			0
$925$	4030.50			0.143267
$926$		0			0
$927$	−49692.1	−	16569.2i	−1.76063	−	0.587058i
$928$		0			0
$929$		−	11931.4i		−	0.421376i	−0.977553	−	0.210688i	$-0.932430\pi$
$929$		−	11931.4i		−	0.421376i	0.977553	−	0.210688i	$-0.0675703\pi$
$930$		0			0
$931$			28819.0i			1.01450i
$932$		0			0
$933$	33858.2	+	5496.04i	1.18807	+	0.192853i
$934$		0			0
$935$	−5011.62	+	7594.89i	−0.175291	+	0.265646i
$936$		0			0
$937$			29994.5i			1.04576i	0.852407	+	0.522879i	$0.175142\pi$
$937$			29994.5i			1.04576i	−0.852407	+	0.522879i	$0.824858\pi$
$938$		0			0
$939$	−2768.36	+	17054.4i	−0.0962110	+	0.592705i
$940$		0			0
$941$	−21952.3			−0.760493			−0.380247	−	0.924885i	$-0.624161\pi$
$941$	−21952.3			−0.760493			−0.380247	+	0.924885i	$0.624161\pi$
$942$		0			0
$943$		−	17771.7i		−	0.613709i
$944$		0			0
$945$	−464.703	+	886.617i	−0.0159966	+	0.0305203i
$946$		0			0
$947$			4114.14i			0.141174i	0.997506	+	0.0705870i	$0.0224872\pi$
$947$			4114.14i			0.141174i	−0.997506	+	0.0705870i	$0.977513\pi$
$948$		0			0
$949$	38868.4			1.32953
$950$		0			0
$951$	32645.2	+	5299.14i	1.11314	+	0.180690i
$952$		0			0
$953$	30162.8			1.02525			0.512627	−	0.858611i	$-0.328672\pi$
$953$	30162.8			1.02525			0.512627	+	0.858611i	$0.328672\pi$
$954$		0			0
$955$	3228.12			0.109382
$956$		0			0
$957$	−32303.5	+	35081.0i	−1.09114	+	1.18496i
$958$		0			0
$959$	−2774.51			−0.0934241
$960$		0			0
$961$	−29394.0			−0.986674
$962$		0			0
$963$	−12888.8	−	4297.59i	−0.431293	−	0.143809i
$964$		0			0
$965$	−4198.32			−0.140051
$966$		0			0
$967$		−	58764.2i		−	1.95422i	−0.212742	−	0.977109i	$-0.568239\pi$
$967$		−	58764.2i		−	1.95422i	0.212742	−	0.977109i	$-0.431761\pi$
$968$		0			0
$969$	17692.9	+	2872.00i	0.586560	+	0.0952136i
$970$		0			0
$971$		−	57412.1i		−	1.89747i	−0.316076	−	0.948734i	$-0.602365\pi$
$971$		−	57412.1i		−	1.89747i	0.316076	−	0.948734i	$-0.397635\pi$
$972$		0			0
$973$	−2059.87			−0.0678689
$974$		0			0
$975$	16918.4	+	2746.29i	0.555716	+	0.0902068i
$976$		0			0
$977$			13185.5i			0.431772i	0.976419	+	0.215886i	$0.0692639\pi$
$977$			13185.5i			0.431772i	−0.976419	+	0.215886i	$0.930736\pi$
$978$		0			0
$979$	−13564.8	+	20556.8i	−0.442832	+	0.671092i
$980$		0			0
$981$	−16676.6	+	50014.2i	−0.542754	+	1.62776i
$982$		0			0
$983$			26690.9i			0.866032i	0.901386	+	0.433016i	$0.142551\pi$
$983$			26690.9i			0.866032i	−0.901386	+	0.433016i	$0.857449\pi$
$984$		0			0
$985$			9531.42i			0.308321i
$986$		0			0
$987$	−181.949	+	1120.89i	−0.00586779	+	0.0361483i
$988$		0			0
$989$	33651.2			1.08195
$990$		0			0
$991$	9295.87			0.297975			0.148987	−	0.988839i	$-0.452399\pi$
$991$	9295.87			0.297975			0.148987	+	0.988839i	$0.452399\pi$
$992$		0			0
$993$	5259.01	−	32398.0i	0.168066	−	1.03537i
$994$		0			0
$995$			21838.5i			0.695805i
$996$		0			0
$997$			11713.6i			0.372089i	0.982541	+	0.186045i	$0.0595669\pi$
$997$			11713.6i			0.372089i	−0.982541	+	0.186045i	$0.940433\pi$
$998$		0			0
$999$	2989.86	−	5704.41i	0.0946896	−	0.180660i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.4.b.d.65.4		6
3.2	odd	2		528.4.b.c.65.3		6
4.3	odd	2		66.4.b.a.65.3	✓	6
11.10	odd	2		528.4.b.c.65.4		6
12.11	even	2		66.4.b.b.65.4	yes	6
33.32	even	2	inner	528.4.b.d.65.3		6
44.43	even	2		66.4.b.b.65.3	yes	6
132.131	odd	2		66.4.b.a.65.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.4.b.a.65.3	✓	6	4.3	odd	2
66.4.b.a.65.4	yes	6	132.131	odd	2
66.4.b.b.65.3	yes	6	44.43	even	2
66.4.b.b.65.4	yes	6	12.11	even	2
528.4.b.c.65.3		6	3.2	odd	2
528.4.b.c.65.4		6	11.10	odd	2
528.4.b.d.65.3		6	33.32	even	2	inner
528.4.b.d.65.4		6	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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	528.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.832570 + 5.12902i) q^{3} -6.09929i q^{5} -1.16981i q^{7} +(-25.6137 - 8.54053i) q^{9} +(30.4508 + 20.0935i) q^{11} -37.5697i q^{13} +(31.2834 + 5.07809i) q^{15} -40.8925 q^{17} +84.3569i q^{19} +(6.00000 + 0.973952i) q^{21} -101.207i q^{23} +87.7986 q^{25} +(65.1297 - 124.262i) q^{27} +251.561 q^{29} -19.9245 q^{31} +(-128.412 + 139.454i) q^{33} -7.13504 q^{35} +45.9062 q^{37} +(192.696 + 31.2794i) q^{39} +175.597 q^{41} +332.498i q^{43} +(-52.0912 + 156.225i) q^{45} +186.815i q^{47} +341.632 q^{49} +(34.0458 - 209.738i) q^{51} -554.462i q^{53} +(122.556 - 185.728i) q^{55} +(-432.668 - 70.2330i) q^{57} +185.177i q^{59} +754.209i q^{61} +(-9.99084 + 29.9632i) q^{63} -229.149 q^{65} +381.334 q^{67} +(519.094 + 84.2622i) q^{69} -401.820i q^{71} +1034.57i q^{73} +(-73.0985 + 450.321i) q^{75} +(23.5057 - 35.6218i) q^{77} +1044.32i q^{79} +(583.119 + 437.508i) q^{81} -847.048 q^{83} +249.415i q^{85} +(-209.442 + 1290.26i) q^{87} +675.083i q^{89} -43.9496 q^{91} +(16.5886 - 102.193i) q^{93} +514.517 q^{95} -378.080 q^{97} +(-608.347 - 774.734i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 43 q^{9} + 6 q^{11} + 7 q^{15} + 156 q^{17} + 36 q^{21} - 204 q^{25} + 278 q^{27} - 144 q^{29} + 114 q^{31} + 157 q^{33} - 588 q^{35} - 54 q^{37} + 48 q^{39} - 408 q^{41} + 551 q^{45} + 606 q^{49}+ \cdots - 1753 q^{99}+O(q^{100}) \) 6 * q - q^3 + 43 * q^9 + 6 * q^11 + 7 * q^15 + 156 * q^17 + 36 * q^21 - 204 * q^25 + 278 * q^27 - 144 * q^29 + 114 * q^31 + 157 * q^33 - 588 * q^35 - 54 * q^37 + 48 * q^39 - 408 * q^41 + 551 * q^45 + 606 * q^49 + 444 * q^51 - 942 * q^55 - 1344 * q^57 - 12 * q^63 - 1668 * q^65 - 66 * q^67 + 1007 * q^69 + 936 * q^75 - 1800 * q^77 + 2779 * q^81 - 1392 * q^83 - 3084 * q^87 + 3780 * q^91 - 3847 * q^93 - 2088 * q^95 + 1254 * q^97 - 1753 * q^99

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