LMFDB - Embedded newform 528.6.o.b.175.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(175,528)]

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([1, 0, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("528.175");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.o (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:	$84.6826568613$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			175.13
Character	$\chi$	$=$	528.175
Dual form			528.6.o.b.175.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} +4.87261 q^{5} -228.204 q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} +4.87261 q^{5} -228.204 q^{7} -81.0000 q^{9} +(-241.684 - 320.374i) q^{11} +402.842i q^{13} -43.8535i q^{15} +1224.26i q^{17} -984.004 q^{19} +2053.84i q^{21} +1685.14i q^{23} -3101.26 q^{25} +729.000i q^{27} +534.393i q^{29} -6041.67i q^{31} +(-2883.37 + 2175.16i) q^{33} -1111.95 q^{35} +7458.20 q^{37} +3625.58 q^{39} -3608.08i q^{41} -2909.71 q^{43} -394.681 q^{45} -322.333i q^{47} +35270.2 q^{49} +11018.3 q^{51} +5190.98 q^{53} +(-1177.63 - 1561.06i) q^{55} +8856.04i q^{57} +43008.7i q^{59} -23779.9i q^{61} +18484.6 q^{63} +1962.89i q^{65} -16932.4i q^{67} +15166.3 q^{69} -54343.7i q^{71} +5847.83i q^{73} +27911.3i q^{75} +(55153.5 + 73110.8i) q^{77} -66707.4 q^{79} +6561.00 q^{81} +72755.4 q^{83} +5965.32i q^{85} +4809.54 q^{87} -78866.7 q^{89} -91930.3i q^{91} -54375.1 q^{93} -4794.67 q^{95} +15255.1 q^{97} +(19576.4 + 25950.3i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) $ 40 * q + 88 * q^5 - 3240 * q^9	$ 40 q + 88 q^{5} - 3240 q^{9} + 5464 q^{25} + 6840 q^{33} + 4576 q^{37} - 7128 q^{45} + 58760 q^{49} - 165336 q^{53} + 16632 q^{69} + 28136 q^{77} + 262440 q^{81} - 304608 q^{89} + 102672 q^{93} + 15664 q^{97}+O(q^{100}) $ 40 * q + 88 * q^5 - 3240 * q^9 + 5464 * q^25 + 6840 * q^33 + 4576 * q^37 - 7128 * q^45 + 58760 * q^49 - 165336 * q^53 + 16632 * q^69 + 28136 * q^77 + 262440 * q^81 - 304608 * q^89 + 102672 * q^93 + 15664 * q^97

Display 100 coefficients 10 coefficients

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
$3$		−	9.00000i		−	0.577350i
$3$		−	9.00000i		−	0.577350i
$4$		0			0
$5$	4.87261			0.0871639			0.0435819	−	0.999050i	$-0.486123\pi$
$5$	4.87261			0.0871639			0.0435819	+	0.999050i	$0.486123\pi$
$6$		0			0
$7$	−228.204			−1.76027			−0.880134	−	0.474725i	$-0.842548\pi$
$7$	−228.204			−1.76027			−0.880134	+	0.474725i	$0.842548\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	−241.684	−	320.374i	−0.602236	−	0.798318i
$11$	−241.684	−	320.374i	−0.602236	−	0.798318i
$12$		0			0
$13$			402.842i			0.661114i	0.943786	+	0.330557i	$0.107237\pi$
$13$			402.842i			0.661114i	−0.943786	+	0.330557i	$0.892763\pi$
$14$		0			0
$15$		−	43.8535i		−	0.0503241i
$16$		0			0
$17$			1224.26i			1.02743i	0.857962	+	0.513713i	$0.171730\pi$
$17$			1224.26i			1.02743i	−0.857962	+	0.513713i	$0.828270\pi$
$18$		0			0
$19$	−984.004			−0.625335			−0.312668	−	0.949863i	$-0.601223\pi$
$19$	−984.004			−0.625335			−0.312668	+	0.949863i	$0.601223\pi$
$20$		0			0
$21$			2053.84i			1.01629i
$22$		0			0
$23$			1685.14i			0.664228i	0.943239	+	0.332114i	$0.107762\pi$
$23$			1685.14i			0.664228i	−0.943239	+	0.332114i	$0.892238\pi$
$24$		0			0
$25$	−3101.26			−0.992402
$26$		0			0
$27$			729.000i			0.192450i
$28$		0			0
$29$			534.393i			0.117996i	0.998258	+	0.0589978i	$0.0187905\pi$
$29$			534.393i			0.117996i	−0.998258	+	0.0589978i	$0.981209\pi$
$30$		0			0
$31$		−	6041.67i		−	1.12915i	−0.825381	−	0.564577i	$-0.809039\pi$
$31$		−	6041.67i		−	1.12915i	0.825381	−	0.564577i	$-0.190961\pi$
$32$		0			0
$33$	−2883.37	+	2175.16i	−0.460909	+	0.347701i
$34$		0			0
$35$	−1111.95			−0.153432
$36$		0			0
$37$	7458.20			0.895632			0.447816	−	0.894126i	$-0.352202\pi$
$37$	7458.20			0.895632			0.447816	+	0.894126i	$0.352202\pi$
$38$		0			0
$39$	3625.58			0.381694
$40$		0			0
$41$		−	3608.08i		−	0.335210i	−0.985854	−	0.167605i	$-0.946397\pi$
$41$		−	3608.08i		−	0.335210i	0.985854	−	0.167605i	$-0.0536033\pi$
$42$		0			0
$43$	−2909.71			−0.239982			−0.119991	−	0.992775i	$-0.538287\pi$
$43$	−2909.71			−0.239982			−0.119991	+	0.992775i	$0.538287\pi$
$44$		0			0
$45$	−394.681			−0.0290546
$46$		0			0
$47$		−	322.333i		−	0.0212843i	−0.999943	−	0.0106422i	$-0.996612\pi$
$47$		−	322.333i		−	0.0212843i	0.999943	−	0.0106422i	$-0.00338757\pi$
$48$		0			0
$49$	35270.2			2.09854
$50$		0			0
$51$	11018.3			0.593184
$52$		0			0
$53$	5190.98			0.253840			0.126920	−	0.991913i	$-0.459491\pi$
$53$	5190.98			0.253840			0.126920	+	0.991913i	$0.459491\pi$
$54$		0			0
$55$	−1177.63	−	1561.06i	−0.0524933	−	0.0695845i
$56$		0			0
$57$			8856.04i			0.361037i
$58$		0			0
$59$			43008.7i			1.60852i	0.594278	+	0.804260i	$0.297438\pi$
$59$			43008.7i			1.60852i	−0.594278	+	0.804260i	$0.702562\pi$
$60$		0			0
$61$		−	23779.9i		−	0.818249i	−0.912479	−	0.409124i	$-0.865834\pi$
$61$		−	23779.9i		−	0.818249i	0.912479	−	0.409124i	$-0.134166\pi$
$62$		0			0
$63$	18484.6			0.586756
$64$		0			0
$65$			1962.89i			0.0576252i
$66$		0			0
$67$		−	16932.4i		−	0.460820i	−0.973094	−	0.230410i	$-0.925993\pi$
$67$		−	16932.4i		−	0.460820i	0.973094	−	0.230410i	$-0.0740068\pi$
$68$		0			0
$69$	15166.3			0.383492
$70$		0			0
$71$		−	54343.7i		−	1.27939i	−0.768629	−	0.639695i	$-0.779061\pi$
$71$		−	54343.7i		−	1.27939i	0.768629	−	0.639695i	$-0.220939\pi$
$72$		0			0
$73$			5847.83i			0.128436i	0.997936	+	0.0642181i	$0.0204553\pi$
$73$			5847.83i			0.128436i	−0.997936	+	0.0642181i	$0.979545\pi$
$74$		0			0
$75$			27911.3i			0.572964i
$76$		0			0
$77$	55153.5	+	73110.8i	1.06010	+	1.40525i
$78$		0			0
$79$	−66707.4			−1.20256			−0.601280	−	0.799039i	$-0.705342\pi$
$79$	−66707.4			−1.20256			−0.601280	+	0.799039i	$0.705342\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$	72755.4			1.15923			0.579616	−	0.814890i	$-0.303203\pi$
$83$	72755.4			1.15923			0.579616	+	0.814890i	$0.303203\pi$
$84$		0			0
$85$			5965.32i			0.0895543i
$86$		0			0
$87$	4809.54			0.0681248
$88$		0			0
$89$	−78866.7			−1.05540			−0.527702	−	0.849430i	$-0.676946\pi$
$89$	−78866.7			−1.05540			−0.527702	+	0.849430i	$0.676946\pi$
$90$		0			0
$91$		−	91930.3i		−	1.16374i
$92$		0			0
$93$	−54375.1			−0.651917
$94$		0			0
$95$	−4794.67			−0.0545066
$96$		0			0
$97$	15255.1			0.164622			0.0823108	−	0.996607i	$-0.473770\pi$
$97$	15255.1			0.164622			0.0823108	+	0.996607i	$0.473770\pi$
$98$		0			0
$99$	19576.4	+	25950.3i	0.200745	+	0.266106i
$100$		0			0
$101$			38239.9i			0.373004i	0.982455	+	0.186502i	$0.0597151\pi$
$101$			38239.9i			0.373004i	−0.982455	+	0.186502i	$0.940285\pi$
$102$		0			0
$103$		−	150735.i		−	1.39998i	−0.714153	−	0.699990i	$-0.753188\pi$
$103$		−	150735.i		−	1.39998i	0.714153	−	0.699990i	$-0.246812\pi$
$104$		0			0
$105$			10007.6i			0.0885839i
$106$		0			0
$107$	142247.			1.20112			0.600558	−	0.799581i	$-0.294945\pi$
$107$	142247.			1.20112			0.600558	+	0.799581i	$0.294945\pi$
$108$		0			0
$109$			12564.0i			0.101289i	0.998717	+	0.0506446i	$0.0161276\pi$
$109$			12564.0i			0.101289i	−0.998717	+	0.0506446i	$0.983872\pi$
$110$		0			0
$111$		−	67123.8i		−	0.517093i
$112$		0			0
$113$	66561.8			0.490376			0.245188	−	0.969476i	$-0.421150\pi$
$113$	66561.8			0.490376			0.245188	+	0.969476i	$0.421150\pi$
$114$		0			0
$115$			8211.04i			0.0578967i
$116$		0			0
$117$		−	32630.2i		−	0.220371i
$118$		0			0
$119$		−	279381.i		−	1.80854i
$120$		0			0
$121$	−44228.2	+	154859.i	−0.274622	+	0.961552i
$122$		0			0
$123$	−32472.7			−0.193534
$124$		0			0
$125$	−30338.1			−0.173665
$126$		0			0
$127$	−12372.1			−0.0680666			−0.0340333	−	0.999421i	$-0.510835\pi$
$127$	−12372.1			−0.0680666			−0.0340333	+	0.999421i	$0.510835\pi$
$128$		0			0
$129$			26187.4i			0.138553i
$130$		0			0
$131$	138717.			0.706237			0.353119	−	0.935579i	$-0.385121\pi$
$131$	138717.			0.706237			0.353119	+	0.935579i	$0.385121\pi$
$132$		0			0
$133$	224554.			1.10076
$134$		0			0
$135$			3552.13i			0.0167747i
$136$		0			0
$137$	235737.			1.07307			0.536533	−	0.843879i	$-0.319734\pi$
$137$	235737.			1.07307			0.536533	+	0.843879i	$0.319734\pi$
$138$		0			0
$139$	184093.			0.808167			0.404084	−	0.914722i	$-0.367590\pi$
$139$	184093.			0.808167			0.404084	+	0.914722i	$0.367590\pi$
$140$		0			0
$141$	−2900.99			−0.0122885
$142$		0			0
$143$	129060.	−	97360.6i	0.527779	−	0.398147i
$144$		0			0
$145$			2603.89i			0.0102850i
$146$		0			0
$147$		−	317432.i		−	1.21160i
$148$		0			0
$149$		−	87841.1i		−	0.324140i	−0.986779	−	0.162070i	$-0.948183\pi$
$149$		−	87841.1i		−	0.324140i	0.986779	−	0.162070i	$-0.0518170\pi$
$150$		0			0
$151$	514025.			1.83460			0.917301	−	0.398194i	$-0.130363\pi$
$151$	514025.			1.83460			0.917301	+	0.398194i	$0.130363\pi$
$152$		0			0
$153$		−	99164.8i		−	0.342475i
$154$		0			0
$155$		−	29438.7i		−	0.0984214i
$156$		0			0
$157$	−17991.7			−0.0582536			−0.0291268	−	0.999576i	$-0.509273\pi$
$157$	−17991.7			−0.0582536			−0.0291268	+	0.999576i	$0.509273\pi$
$158$		0			0
$159$		−	46718.8i		−	0.146554i
$160$		0			0
$161$		−	384557.i		−	1.16922i
$162$		0			0
$163$			254495.i			0.750256i	0.926973	+	0.375128i	$0.122401\pi$
$163$			254495.i			0.750256i	−0.926973	+	0.375128i	$0.877599\pi$
$164$		0			0
$165$	−14049.5	+	10598.7i	−0.0401746	+	0.0303070i
$166$		0			0
$167$	−315052.			−0.874161			−0.437081	−	0.899422i	$-0.643988\pi$
$167$	−315052.			−0.874161			−0.437081	+	0.899422i	$0.643988\pi$
$168$		0			0
$169$	209011.			0.562929
$170$		0			0
$171$	79704.3			0.208445
$172$		0			0
$173$			232439.i			0.590464i	0.955426	+	0.295232i	$0.0953970\pi$
$173$			232439.i			0.590464i	−0.955426	+	0.295232i	$0.904603\pi$
$174$		0			0
$175$	707721.			1.74689
$176$		0			0
$177$	387078.			0.928679
$178$		0			0
$179$		−	667770.i		−	1.55774i	−0.627186	−	0.778869i	$-0.715794\pi$
$179$		−	667770.i		−	1.55774i	0.627186	−	0.778869i	$-0.284206\pi$
$180$		0			0
$181$	15934.0			0.0361516			0.0180758	−	0.999837i	$-0.494246\pi$
$181$	15934.0			0.0361516			0.0180758	+	0.999837i	$0.494246\pi$
$182$		0			0
$183$	−214019.			−0.472416
$184$		0			0
$185$	36340.9			0.0780667
$186$		0			0
$187$	392220.	−	295884.i	0.820212	−	0.618753i
$188$		0			0
$189$		−	166361.i		−	0.338764i
$190$		0			0
$191$		−	329369.i		−	0.653280i	−0.945149	−	0.326640i	$-0.894084\pi$
$191$		−	329369.i		−	0.653280i	0.945149	−	0.326640i	$-0.105916\pi$
$192$		0			0
$193$			662466.i			1.28018i	0.768301	+	0.640089i	$0.221103\pi$
$193$			662466.i			1.28018i	−0.768301	+	0.640089i	$0.778897\pi$
$194$		0			0
$195$	17666.0			0.0332699
$196$		0			0
$197$			472488.i			0.867411i	0.901055	+	0.433705i	$0.142794\pi$
$197$			472488.i			0.867411i	−0.901055	+	0.433705i	$0.857206\pi$
$198$		0			0
$199$		−	344660.i		−	0.616961i	−0.951231	−	0.308481i	$-0.900180\pi$
$199$		−	344660.i		−	0.616961i	0.951231	−	0.308481i	$-0.0998205\pi$
$200$		0			0
$201$	−152392.			−0.266055
$202$		0			0
$203$		−	121951.i		−	0.207704i
$204$		0			0
$205$		−	17580.8i		−	0.0292182i
$206$		0			0
$207$		−	136497.i		−	0.221409i
$208$		0			0
$209$	237819.	+	315250.i	0.376600	+	0.499216i
$210$		0			0
$211$	−894976.			−1.38390			−0.691951	−	0.721945i	$-0.743248\pi$
$211$	−894976.			−1.38390			−0.691951	+	0.721945i	$0.743248\pi$
$212$		0			0
$213$	−489093.			−0.738657
$214$		0			0
$215$	−14177.9			−0.0209177
$216$		0			0
$217$			1.37874e6i			1.98761i
$218$		0			0
$219$	52630.5			0.0741527
$220$		0			0
$221$	−493182.			−0.679245
$222$		0			0
$223$			587042.i			0.790509i	0.918572	+	0.395255i	$0.129344\pi$
$223$			587042.i			0.790509i	−0.918572	+	0.395255i	$0.870656\pi$
$224$		0			0
$225$	251202.			0.330801
$226$		0			0
$227$	−1.14147e6			−1.47028			−0.735141	−	0.677915i	$-0.762884\pi$
$227$	−1.14147e6			−1.47028			−0.735141	+	0.677915i	$0.762884\pi$
$228$		0			0
$229$	1.44915e6			1.82610			0.913049	−	0.407851i	$-0.133722\pi$
$229$	1.44915e6			1.82610			0.913049	+	0.407851i	$0.133722\pi$
$230$		0			0
$231$	657997.	−	496381.i	0.811323	−	0.612048i
$232$		0			0
$233$		−	486985.i		−	0.587660i	−0.955858	−	0.293830i	$-0.905070\pi$
$233$		−	486985.i		−	0.587660i	0.955858	−	0.293830i	$-0.0949299\pi$
$234$		0			0
$235$		−	1570.60i		−	0.00185522i
$236$		0			0
$237$			600367.i			0.694298i
$238$		0			0
$239$	1.63479e6			1.85126			0.925631	−	0.378427i	$-0.123535\pi$
$239$	1.63479e6			1.85126			0.925631	+	0.378427i	$0.123535\pi$
$240$		0			0
$241$			852633.i			0.945626i	0.881163	+	0.472813i	$0.156761\pi$
$241$			852633.i			0.945626i	−0.881163	+	0.472813i	$0.843239\pi$
$242$		0			0
$243$		−	59049.0i		−	0.0641500i
$244$		0			0
$245$	171858.			0.182917
$246$		0			0
$247$		−	396398.i		−	0.413418i
$248$		0			0
$249$		−	654799.i		−	0.669282i
$250$		0			0
$251$		−	18386.4i		−	0.0184210i	−0.999958	−	0.00921048i	$-0.997068\pi$
$251$		−	18386.4i		−	0.0184210i	0.999958	−	0.00921048i	$-0.00293183\pi$
$252$		0			0
$253$	539876.	−	407273.i	0.530265	−	0.400022i
$254$		0			0
$255$	53687.9			0.0517042
$256$		0			0
$257$	417084.			0.393904			0.196952	−	0.980413i	$-0.436896\pi$
$257$	417084.			0.393904			0.196952	+	0.980413i	$0.436896\pi$
$258$		0			0
$259$	−1.70199e6			−1.57655
$260$		0			0
$261$		−	43285.9i		−	0.0393319i
$262$		0			0
$263$	1.77628e6			1.58352			0.791758	−	0.610835i	$-0.209166\pi$
$263$	1.77628e6			1.58352			0.791758	+	0.610835i	$0.209166\pi$
$264$		0			0
$265$	25293.6			0.0221256
$266$		0			0
$267$			709800.i			0.609337i
$268$		0			0
$269$	178764.			0.150625			0.0753127	−	0.997160i	$-0.476004\pi$
$269$	178764.			0.150625			0.0753127	+	0.997160i	$0.476004\pi$
$270$		0			0
$271$	−1.19292e6			−0.986708			−0.493354	−	0.869829i	$-0.664229\pi$
$271$	−1.19292e6			−0.986708			−0.493354	+	0.869829i	$0.664229\pi$
$272$		0			0
$273$	−827373.			−0.671884
$274$		0			0
$275$	749526.	+	993563.i	0.597661	+	0.792252i
$276$		0			0
$277$		−	2.16777e6i		−	1.69752i	−0.528780	−	0.848759i	$-0.677351\pi$
$277$		−	2.16777e6i		−	1.69752i	0.528780	−	0.848759i	$-0.322649\pi$
$278$		0			0
$279$			489375.i			0.376384i
$280$		0			0
$281$		−	841253.i		−	0.635566i	−0.948163	−	0.317783i	$-0.897062\pi$
$281$		−	841253.i		−	0.635566i	0.948163	−	0.317783i	$-0.102938\pi$
$282$		0			0
$283$	1.40090e6			1.03978			0.519889	−	0.854234i	$-0.325973\pi$
$283$	1.40090e6			1.03978			0.519889	+	0.854234i	$0.325973\pi$
$284$		0			0
$285$			43152.0i			0.0314694i
$286$		0			0
$287$			823380.i			0.590059i
$288$		0			0
$289$	−78947.5			−0.0556024
$290$		0			0
$291$		−	137296.i		−	0.0950443i
$292$		0			0
$293$		−	638675.i		−	0.434621i	−0.976103	−	0.217310i	$-0.930272\pi$
$293$		−	638675.i		−	0.434621i	0.976103	−	0.217310i	$-0.0697284\pi$
$294$		0			0
$295$			209565.i			0.140205i
$296$		0			0
$297$	233553.	−	176188.i	0.153636	−	0.115900i
$298$		0			0
$299$	−678846.			−0.439130
$300$		0			0
$301$	664008.			0.422432
$302$		0			0
$303$	344159.			0.215354
$304$		0			0
$305$		−	115870.i		−	0.0713217i
$306$		0			0
$307$	−1.95550e6			−1.18417			−0.592083	−	0.805877i	$-0.701694\pi$
$307$	−1.95550e6			−1.18417			−0.592083	+	0.805877i	$0.701694\pi$
$308$		0			0
$309$	−1.35662e6			−0.808278
$310$		0			0
$311$		−	456596.i		−	0.267689i	−0.991002	−	0.133845i	$-0.957268\pi$
$311$		−	456596.i		−	0.267689i	0.991002	−	0.133845i	$-0.0427323\pi$
$312$		0			0
$313$	2.70232e6			1.55911			0.779554	−	0.626335i	$-0.215446\pi$
$313$	2.70232e6			1.55911			0.779554	+	0.626335i	$0.215446\pi$
$314$		0			0
$315$	90068.0			0.0511439
$316$		0			0
$317$	−3.43523e6			−1.92003			−0.960014	−	0.279951i	$-0.909682\pi$
$317$	−3.43523e6			−1.92003			−0.960014	+	0.279951i	$0.909682\pi$
$318$		0			0
$319$	171206.	−	129155.i	0.0941980	−	0.0710613i
$320$		0			0
$321$		−	1.28023e6i		−	0.693465i
$322$		0			0
$323$		−	1.20467e6i		−	0.642485i
$324$		0			0
$325$		−	1.24932e6i		−	0.656091i
$326$		0			0
$327$	113076.			0.0584793
$328$		0			0
$329$			73557.7i			0.0374661i
$330$		0			0
$331$			2.24099e6i			1.12427i	0.827047	+	0.562133i	$0.190019\pi$
$331$			2.24099e6i			1.12427i	−0.827047	+	0.562133i	$0.809981\pi$
$332$		0			0
$333$	−604114.			−0.298544
$334$		0			0
$335$		−	82505.0i		−	0.0401669i
$336$		0			0
$337$			1.91165e6i			0.916924i	0.888714	+	0.458462i	$0.151599\pi$
$337$			1.91165e6i			0.916924i	−0.888714	+	0.458462i	$0.848401\pi$
$338$		0			0
$339$		−	599056.i		−	0.283118i
$340$		0			0
$341$	−1.93560e6	+	1.46018e6i	−0.901423	+	0.680017i
$342$		0			0
$343$	−4.21339e6			−1.93373
$344$		0			0
$345$	73899.4			0.0334267
$346$		0			0
$347$	2.19277e6			0.977619			0.488810	−	0.872390i	$-0.337431\pi$
$347$	2.19277e6			0.977619			0.488810	+	0.872390i	$0.337431\pi$
$348$		0			0
$349$			981912.i			0.431528i	0.976446	+	0.215764i	$0.0692242\pi$
$349$			981912.i			0.431528i	−0.976446	+	0.215764i	$0.930776\pi$
$350$		0			0
$351$	−293672.			−0.127231
$352$		0			0
$353$	−1.19718e6			−0.511356			−0.255678	−	0.966762i	$-0.582299\pi$
$353$	−1.19718e6			−0.511356			−0.255678	+	0.966762i	$0.582299\pi$
$354$		0			0
$355$		−	264795.i		−	0.111517i
$356$		0			0
$357$	−2.51443e6			−1.04416
$358$		0			0
$359$	−4.54143e6			−1.85976			−0.929879	−	0.367866i	$-0.880088\pi$
$359$	−4.54143e6			−1.85976			−0.929879	+	0.367866i	$0.880088\pi$
$360$		0			0
$361$	−1.50783e6			−0.608956
$362$		0			0
$363$	1.39373e6	+	398054.i	0.555152	+	0.158553i
$364$		0			0
$365$			28494.2i			0.0111950i
$366$		0			0
$367$			1.59412e6i			0.617812i	0.951093	+	0.308906i	$0.0999628\pi$
$367$			1.59412e6i			0.617812i	−0.951093	+	0.308906i	$0.900037\pi$
$368$		0			0
$369$			292255.i			0.111737i
$370$		0			0
$371$	−1.18460e6			−0.446826
$372$		0			0
$373$			1.42370e6i			0.529842i	0.964270	+	0.264921i	$0.0853458\pi$
$373$			1.42370e6i			0.529842i	−0.964270	+	0.264921i	$0.914654\pi$
$374$		0			0
$375$			273043.i			0.100266i
$376$		0			0
$377$	−215276.			−0.0780086
$378$		0			0
$379$			4.59635e6i			1.64367i	0.569725	+	0.821835i	$0.307050\pi$
$379$			4.59635e6i			1.64367i	−0.569725	+	0.821835i	$0.692950\pi$
$380$		0			0
$381$			111349.i			0.0392983i
$382$		0			0
$383$			3.49701e6i			1.21815i	0.793114	+	0.609074i	$0.208459\pi$
$383$			3.49701e6i			1.21815i	−0.793114	+	0.609074i	$0.791541\pi$
$384$		0			0
$385$	268741.	+	356240.i	0.0924022	+	0.122487i
$386$		0			0
$387$	235686.			0.0799939
$388$		0			0
$389$	−3.55827e6			−1.19224			−0.596122	−	0.802894i	$-0.703292\pi$
$389$	−3.55827e6			−1.19224			−0.596122	+	0.802894i	$0.703292\pi$
$390$		0			0
$391$	−2.06305e6			−0.682444
$392$		0			0
$393$		−	1.24845e6i		−	0.407746i
$394$		0			0
$395$	−325039.			−0.104820
$396$		0			0
$397$	−825843.			−0.262979			−0.131490	−	0.991318i	$-0.541976\pi$
$397$	−825843.			−0.262979			−0.131490	+	0.991318i	$0.541976\pi$
$398$		0			0
$399$		−	2.02099e6i		−	0.635523i
$400$		0			0
$401$	−2.65200e6			−0.823594			−0.411797	−	0.911276i	$-0.635099\pi$
$401$	−2.65200e6			−0.823594			−0.411797	+	0.911276i	$0.635099\pi$
$402$		0			0
$403$	2.43384e6			0.746499
$404$		0			0
$405$	31969.2			0.00968487
$406$		0			0
$407$	−1.80253e6	−	2.38941e6i	−0.539382	−	0.714999i
$408$		0			0
$409$			692397.i			0.204667i	0.994750	+	0.102333i	$0.0326308\pi$
$409$			692397.i			0.204667i	−0.994750	+	0.102333i	$0.967369\pi$
$410$		0			0
$411$		−	2.12163e6i		−	0.619535i
$412$		0			0
$413$		−	9.81478e6i		−	2.83143i
$414$		0			0
$415$	354509.			0.101043
$416$		0			0
$417$		−	1.65684e6i		−	0.466596i
$418$		0			0
$419$			1.50005e6i			0.417417i	0.977978	+	0.208709i	$0.0669260\pi$
$419$			1.50005e6i			0.417417i	−0.977978	+	0.208709i	$0.933074\pi$
$420$		0			0
$421$	2.18029e6			0.599526			0.299763	−	0.954014i	$-0.403092\pi$
$421$	2.18029e6			0.599526			0.299763	+	0.954014i	$0.403092\pi$
$422$		0			0
$423$			26108.9i			0.00709477i
$424$		0			0
$425$		−	3.79674e6i		−	1.01962i
$426$		0			0
$427$			5.42668e6i			1.44034i
$428$		0			0
$429$	−876246.	−	1.16154e6i	−0.229870	−	0.304713i
$430$		0			0
$431$	−3.66222e6			−0.949624			−0.474812	−	0.880087i	$-0.657484\pi$
$431$	−3.66222e6			−0.949624			−0.474812	+	0.880087i	$0.657484\pi$
$432$		0			0
$433$	5.50788e6			1.41177			0.705886	−	0.708325i	$-0.250549\pi$
$433$	5.50788e6			1.41177			0.705886	+	0.708325i	$0.250549\pi$
$434$		0			0
$435$	23435.0			0.00593802
$436$		0			0
$437$		−	1.65819e6i		−	0.415365i
$438$		0			0
$439$	−1.38840e6			−0.343839			−0.171919	−	0.985111i	$-0.554997\pi$
$439$	−1.38840e6			−0.343839			−0.171919	+	0.985111i	$0.554997\pi$
$440$		0			0
$441$	−2.85689e6			−0.699515
$442$		0			0
$443$			7.80815e6i			1.89033i	0.326586	+	0.945167i	$0.394102\pi$
$443$			7.80815e6i			1.89033i	−0.326586	+	0.945167i	$0.605898\pi$
$444$		0			0
$445$	−384286.			−0.0919930
$446$		0			0
$447$	−790570.			−0.187142
$448$		0			0
$449$	6.42482e6			1.50399			0.751995	−	0.659169i	$-0.229092\pi$
$449$	6.42482e6			1.50399			0.751995	+	0.659169i	$0.229092\pi$
$450$		0			0
$451$	−1.15594e6	+	872018.i	−0.267604	+	0.201876i
$452$		0			0
$453$		−	4.62623e6i		−	1.05921i
$454$		0			0
$455$		−	447940.i		−	0.101436i
$456$		0			0
$457$			7.14672e6i			1.60072i	0.599517	+	0.800362i	$0.295359\pi$
$457$			7.14672e6i			1.60072i	−0.599517	+	0.800362i	$0.704641\pi$
$458$		0			0
$459$	−892483.			−0.197728
$460$		0			0
$461$		−	5.65408e6i		−	1.23911i	−0.784954	−	0.619554i	$-0.787313\pi$
$461$		−	5.65408e6i		−	1.23911i	0.784954	−	0.619554i	$-0.212687\pi$
$462$		0			0
$463$			7.65380e6i			1.65930i	0.558285	+	0.829649i	$0.311460\pi$
$463$			7.65380e6i			1.65930i	−0.558285	+	0.829649i	$0.688540\pi$
$464$		0			0
$465$	−264948.			−0.0568236
$466$		0			0
$467$		−	5.16391e6i		−	1.09569i	−0.836581	−	0.547844i	$-0.815449\pi$
$467$		−	5.16391e6i		−	1.09569i	0.836581	−	0.547844i	$-0.184551\pi$
$468$		0			0
$469$			3.86405e6i			0.811167i
$470$		0			0
$471$			161925.i			0.0336327i
$472$		0			0
$473$	703231.	+	932195.i	0.144526	+	0.191582i
$474$		0			0
$475$	3.05165e6			0.620584
$476$		0			0
$477$	−420469.			−0.0846132
$478$		0			0
$479$	3.60472e6			0.717848			0.358924	−	0.933367i	$-0.383144\pi$
$479$	3.60472e6			0.717848			0.358924	+	0.933367i	$0.383144\pi$
$480$		0			0
$481$			3.00447e6i			0.592115i
$482$		0			0
$483$	−3.46101e6			−0.675049
$484$		0			0
$485$	74332.3			0.0143490
$486$		0			0
$487$			8.28903e6i			1.58373i	0.610696	+	0.791865i	$0.290890\pi$
$487$			8.28903e6i			1.58373i	−0.610696	+	0.791865i	$0.709110\pi$
$488$		0			0
$489$	2.29045e6			0.433161
$490$		0			0
$491$	4.14379e6			0.775701			0.387850	−	0.921722i	$-0.373218\pi$
$491$	4.14379e6			0.775701			0.387850	+	0.921722i	$0.373218\pi$
$492$		0			0
$493$	−654235.			−0.121232
$494$		0			0
$495$	95388.3	+	126446.i	0.0174978	+	0.0231948i
$496$		0			0
$497$			1.24015e7i			2.25207i
$498$		0			0
$499$		−	39066.7i		−	0.00702353i	−0.999994	−	0.00351176i	$-0.998882\pi$
$499$		−	39066.7i		−	0.00702353i	0.999994	−	0.00351176i	$-0.00111783\pi$
$500$		0			0
$501$			2.83547e6i			0.504697i
$502$		0			0
$503$	−6.80581e6			−1.19939			−0.599694	−	0.800229i	$-0.704711\pi$
$503$	−6.80581e6			−1.19939			−0.599694	+	0.800229i	$0.704711\pi$
$504$		0			0
$505$			186328.i			0.0325125i
$506$		0			0
$507$		−	1.88110e6i		−	0.325007i
$508$		0			0
$509$	−7.92404e6			−1.35566			−0.677832	−	0.735217i	$-0.737080\pi$
$509$	−7.92404e6			−1.35566			−0.677832	+	0.735217i	$0.737080\pi$
$510$		0			0
$511$		−	1.33450e6i		−	0.226082i
$512$		0			0
$513$		−	717339.i		−	0.120346i
$514$		0			0
$515$		−	734473.i		−	0.122028i
$516$		0			0
$517$	−103267.	+	77902.8i	−0.0169916	+	0.0128182i
$518$		0			0
$519$	2.09195e6			0.340905
$520$		0			0
$521$	−649709.			−0.104864			−0.0524318	−	0.998625i	$-0.516697\pi$
$521$	−649709.			−0.104864			−0.0524318	+	0.998625i	$0.516697\pi$
$522$		0			0
$523$	−4.06953e6			−0.650563			−0.325282	−	0.945617i	$-0.605459\pi$
$523$	−4.06953e6			−0.650563			−0.325282	+	0.945617i	$0.605459\pi$
$524$		0			0
$525$		−	6.36949e6i		−	1.00857i
$526$		0			0
$527$	7.39656e6			1.16012
$528$		0			0
$529$	3.59664e6			0.558801
$530$		0			0
$531$		−	3.48371e6i		−	0.536173i
$532$		0			0
$533$	1.45349e6			0.221612
$534$		0			0
$535$	693116.			0.104694
$536$		0			0
$537$	−6.00993e6			−0.899361
$538$		0			0
$539$	−8.52427e6	−	1.12997e7i	−1.26382	−	1.67531i
$540$		0			0
$541$			5.15406e6i			0.757106i	0.925580	+	0.378553i	$0.123578\pi$
$541$			5.15406e6i			0.757106i	−0.925580	+	0.378553i	$0.876422\pi$
$542$		0			0
$543$		−	143406.i		−	0.0208722i
$544$		0			0
$545$			61219.7i			0.00882876i
$546$		0			0
$547$	−1.13609e7			−1.62347			−0.811734	−	0.584027i	$-0.801476\pi$
$547$	−1.13609e7			−1.62347			−0.811734	+	0.584027i	$0.801476\pi$
$548$		0			0
$549$			1.92617e6i			0.272750i
$550$		0			0
$551$		−	525845.i		−	0.0737868i
$552$		0			0
$553$	1.52229e7			2.11683
$554$		0			0
$555$		−	327068.i		−	0.0450719i
$556$		0			0
$557$		−	2.52330e6i		−	0.344612i	−0.985043	−	0.172306i	$-0.944878\pi$
$557$		−	2.52330e6i		−	0.344612i	0.985043	−	0.172306i	$-0.0551219\pi$
$558$		0			0
$559$		−	1.17215e6i		−	0.158655i
$560$		0			0
$561$	−2.66295e6	−	3.52998e6i	−0.357237	−	0.473549i
$562$		0			0
$563$	1.13510e7			1.50925			0.754627	−	0.656154i	$-0.227818\pi$
$563$	1.13510e7			1.50925			0.754627	+	0.656154i	$0.227818\pi$
$564$		0			0
$565$	324329.			0.0427430
$566$		0			0
$567$	−1.49725e6			−0.195585
$568$		0			0
$569$			1.07523e7i			1.39226i	0.717917	+	0.696129i	$0.245096\pi$
$569$			1.07523e7i			1.39226i	−0.717917	+	0.696129i	$0.754904\pi$
$570$		0			0
$571$	−8.86062e6			−1.13730			−0.568648	−	0.822581i	$-0.692533\pi$
$571$	−8.86062e6			−1.13730			−0.568648	+	0.822581i	$0.692533\pi$
$572$		0			0
$573$	−2.96432e6			−0.377171
$574$		0			0
$575$		−	5.22606e6i		−	0.659181i
$576$		0			0
$577$	9.61211e6			1.20193			0.600965	−	0.799275i	$-0.294783\pi$
$577$	9.61211e6			1.20193			0.600965	+	0.799275i	$0.294783\pi$
$578$		0			0
$579$	5.96220e6			0.739111
$580$		0			0
$581$	−1.66031e7			−2.04056
$582$		0			0
$583$	−1.25458e6	−	1.66305e6i	−0.152871	−	0.202645i
$584$		0			0
$585$		−	158994.i		−	0.0192084i
$586$		0			0
$587$		−	4.89458e6i		−	0.586300i	−0.956066	−	0.293150i	$-0.905296\pi$
$587$		−	4.89458e6i		−	0.586300i	0.956066	−	0.293150i	$-0.0947035\pi$
$588$		0			0
$589$			5.94503e6i			0.706099i
$590$		0			0
$591$	4.25239e6			0.500800
$592$		0			0
$593$		−	1.57877e7i		−	1.84367i	−0.387585	−	0.921834i	$-0.626691\pi$
$593$		−	1.57877e7i		−	1.84367i	0.387585	−	0.921834i	$-0.373309\pi$
$594$		0			0
$595$		−	1.36131e6i		−	0.157640i
$596$		0			0
$597$	−3.10194e6			−0.356203
$598$		0			0
$599$			1.12714e7i			1.28355i	0.766894	+	0.641774i	$0.221801\pi$
$599$			1.12714e7i			1.28355i	−0.766894	+	0.641774i	$0.778199\pi$
$600$		0			0
$601$			3.51007e6i			0.396396i	0.980162	+	0.198198i	$0.0635090\pi$
$601$			3.51007e6i			0.396396i	−0.980162	+	0.198198i	$0.936491\pi$
$602$		0			0
$603$			1.37152e6i			0.153607i
$604$		0			0
$605$	−215507.	+	754567.i	−0.0239372	+	0.0838126i
$606$		0			0
$607$	1.31768e7			1.45157			0.725786	−	0.687921i	$-0.241476\pi$
$607$	1.31768e7			1.45157			0.725786	+	0.687921i	$0.241476\pi$
$608$		0			0
$609$	−1.09756e6			−0.119918
$610$		0			0
$611$	129849.			0.0140714
$612$		0			0
$613$		−	1.77384e7i		−	1.90661i	−0.302007	−	0.953306i	$-0.597657\pi$
$613$		−	1.77384e7i		−	1.90661i	0.302007	−	0.953306i	$-0.402343\pi$
$614$		0			0
$615$	−158227.			−0.0168691
$616$		0			0
$617$	−8.23926e6			−0.871315			−0.435658	−	0.900113i	$-0.643484\pi$
$617$	−8.23926e6			−0.871315			−0.435658	+	0.900113i	$0.643484\pi$
$618$		0			0
$619$		−	430573.i		−	0.0451668i	−0.999745	−	0.0225834i	$-0.992811\pi$
$619$		−	430573.i		−	0.0451668i	0.999745	−	0.0225834i	$-0.00718914\pi$
$620$		0			0
$621$	−1.22847e6			−0.127831
$622$		0			0
$623$	1.79977e7			1.85779
$624$		0			0
$625$	9.54360e6			0.977265
$626$		0			0
$627$	2.83725e6	−	2.14037e6i	0.288223	−	0.217430i
$628$		0			0
$629$			9.13075e6i			0.920195i
$630$		0			0
$631$		−	1.03675e7i		−	1.03658i	−0.855205	−	0.518289i	$-0.826569\pi$
$631$		−	1.03675e7i		−	1.03658i	0.855205	−	0.518289i	$-0.173431\pi$
$632$		0			0
$633$			8.05478e6i			0.798996i
$634$		0			0
$635$	−60284.4			−0.00593295
$636$		0			0
$637$			1.42083e7i			1.38738i
$638$		0			0
$639$			4.40184e6i			0.426464i
$640$		0			0
$641$	9.46830e6			0.910179			0.455090	−	0.890446i	$-0.349607\pi$
$641$	9.46830e6			0.910179			0.455090	+	0.890446i	$0.349607\pi$
$642$		0			0
$643$		−	1.69765e7i		−	1.61928i	−0.586928	−	0.809639i	$-0.699663\pi$
$643$		−	1.69765e7i		−	1.61928i	0.586928	−	0.809639i	$-0.300337\pi$
$644$		0			0
$645$			127601.i			0.0120769i
$646$		0			0
$647$		−	2.51172e6i		−	0.235891i	−0.993020	−	0.117945i	$-0.962369\pi$
$647$		−	2.51172e6i		−	0.235891i	0.993020	−	0.117945i	$-0.0376308\pi$
$648$		0			0
$649$	1.37789e7	−	1.03945e7i	1.28411	−	0.968709i
$650$		0			0
$651$	1.24086e7			1.14755
$652$		0			0
$653$	1.44877e7			1.32959			0.664795	−	0.747026i	$-0.268519\pi$
$653$	1.44877e7			1.32959			0.664795	+	0.747026i	$0.268519\pi$
$654$		0			0
$655$	675912.			0.0615584
$656$		0			0
$657$		−	473674.i		−	0.0428121i
$658$		0			0
$659$	1.26956e7			1.13878			0.569390	−	0.822068i	$-0.307179\pi$
$659$	1.26956e7			1.13878			0.569390	+	0.822068i	$0.307179\pi$
$660$		0			0
$661$	−9.78535e6			−0.871109			−0.435555	−	0.900162i	$-0.643448\pi$
$661$	−9.78535e6			−0.871109			−0.435555	+	0.900162i	$0.643448\pi$
$662$		0			0
$663$			4.43864e6i			0.392162i
$664$		0			0
$665$	1.09416e6			0.0959463
$666$		0			0
$667$	−900529.			−0.0783760
$668$		0			0
$669$	5.28338e6			0.456401
$670$		0			0
$671$	−7.61847e6	+	5.74723e6i	−0.653223	+	0.492779i
$672$		0			0
$673$		−	4.25243e6i		−	0.361909i	−0.983491	−	0.180955i	$-0.942081\pi$
$673$		−	4.25243e6i		−	0.361909i	0.983491	−	0.180955i	$-0.0579187\pi$
$674$		0			0
$675$		−	2.26082e6i		−	0.190988i
$676$		0			0
$677$		−	1.63891e7i		−	1.37431i	−0.726512	−	0.687153i	$-0.758860\pi$
$677$		−	1.63891e7i		−	1.37431i	0.726512	−	0.687153i	$-0.241140\pi$
$678$		0			0
$679$	−3.48129e6			−0.289778
$680$		0			0
$681$			1.02732e7i			0.848867i
$682$		0			0
$683$		−	7.04609e6i		−	0.577958i	−0.957335	−	0.288979i	$-0.906684\pi$
$683$		−	7.04609e6i		−	0.577958i	0.957335	−	0.288979i	$-0.0933158\pi$
$684$		0			0
$685$	1.14865e6			0.0935326
$686$		0			0
$687$		−	1.30423e7i		−	1.05430i
$688$		0			0
$689$			2.09114e6i			0.167817i
$690$		0			0
$691$			3.22252e6i			0.256744i	0.991726	+	0.128372i	$0.0409751\pi$
$691$			3.22252e6i			0.256744i	−0.991726	+	0.128372i	$0.959025\pi$
$692$		0			0
$693$	−4.46743e6	−	5.92197e6i	−0.353366	−	0.468418i
$694$		0			0
$695$	897015.			0.0704430
$696$		0			0
$697$	4.41722e6			0.344403
$698$		0			0
$699$	−4.38287e6			−0.339286
$700$		0			0
$701$		−	1.35369e7i		−	1.04046i	−0.854028	−	0.520228i	$-0.825847\pi$
$701$		−	1.35369e7i		−	1.04046i	0.854028	−	0.520228i	$-0.174153\pi$
$702$		0			0
$703$	−7.33890e6			−0.560070
$704$		0			0
$705$	−14135.4			−0.00107111
$706$		0			0
$707$		−	8.72652e6i		−	0.656588i
$708$		0			0
$709$	1.10608e7			0.826362			0.413181	−	0.910649i	$-0.364418\pi$
$709$	1.10608e7			0.826362			0.413181	+	0.910649i	$0.364418\pi$
$710$		0			0
$711$	5.40330e6			0.400853
$712$		0			0
$713$	1.01811e7			0.750015
$714$		0			0
$715$	628859.	−	474400.i	0.0460032	−	0.0347040i
$716$		0			0
$717$		−	1.47131e7i		−	1.06883i
$718$		0			0
$719$		−	2.67319e7i		−	1.92844i	−0.265097	−	0.964222i	$-0.585404\pi$
$719$		−	2.67319e7i		−	1.92844i	0.265097	−	0.964222i	$-0.414596\pi$
$720$		0			0
$721$			3.43984e7i			2.46434i
$722$		0			0
$723$	7.67370e6			0.545958
$724$		0			0
$725$		−	1.65729e6i		−	0.117099i
$726$		0			0
$727$		−	7.31883e6i		−	0.513577i	−0.966468	−	0.256788i	$-0.917336\pi$
$727$		−	7.31883e6i		−	0.513577i	0.966468	−	0.256788i	$-0.0826644\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$		−	3.56223e6i		−	0.246563i
$732$		0			0
$733$			6.09801e6i			0.419207i	0.977786	+	0.209603i	$0.0672173\pi$
$733$			6.09801e6i			0.419207i	−0.977786	+	0.209603i	$0.932783\pi$
$734$		0			0
$735$		−	1.54672e6i		−	0.105607i
$736$		0			0
$737$	−5.42471e6	+	4.09230e6i	−0.367881	+	0.277523i
$738$		0			0
$739$	2.24749e7			1.51386			0.756930	−	0.653496i	$-0.226698\pi$
$739$	2.24749e7			1.51386			0.756930	+	0.653496i	$0.226698\pi$
$740$		0			0
$741$	−3.56758e6			−0.238687
$742$		0			0
$743$	2.53511e7			1.68471			0.842356	−	0.538922i	$-0.181168\pi$
$743$	2.53511e7			1.68471			0.842356	+	0.538922i	$0.181168\pi$
$744$		0			0
$745$		−	428015.i		−	0.0282533i
$746$		0			0
$747$	−5.89319e6			−0.386410
$748$		0			0
$749$	−3.24615e7			−2.11429
$750$		0			0
$751$		−	7.73389e6i		−	0.500378i	−0.968197	−	0.250189i	$-0.919507\pi$
$751$		−	7.73389e6i		−	0.500378i	0.968197	−	0.250189i	$-0.0804927\pi$
$752$		0			0
$753$	−165478.			−0.0106353
$754$		0			0
$755$	2.50464e6			0.159911
$756$		0			0
$757$	−1.88657e6			−0.119656			−0.0598279	−	0.998209i	$-0.519055\pi$
$757$	−1.88657e6			−0.119656			−0.0598279	+	0.998209i	$0.519055\pi$
$758$		0			0
$759$	−3.66546e6	−	4.85889e6i	−0.230953	−	0.306149i
$760$		0			0
$761$			3.10483e7i			1.94346i	0.236094	+	0.971730i	$0.424133\pi$
$761$			3.10483e7i			1.94346i	−0.236094	+	0.971730i	$0.575867\pi$
$762$		0			0
$763$		−	2.86717e6i		−	0.178296i
$764$		0			0
$765$		−	483191.i		−	0.0298514i
$766$		0			0
$767$	−1.73257e7			−1.06341
$768$		0			0
$769$			7.32459e6i			0.446650i	0.974744	+	0.223325i	$0.0716912\pi$
$769$			7.32459e6i			0.446650i	−0.974744	+	0.223325i	$0.928309\pi$
$770$		0			0
$771$		−	3.75375e6i		−	0.227421i
$772$		0			0
$773$	2.15385e7			1.29649			0.648243	−	0.761434i	$-0.275504\pi$
$773$	2.15385e7			1.29649			0.648243	+	0.761434i	$0.275504\pi$
$774$		0			0
$775$			1.87368e7i			1.12057i
$776$		0			0
$777$			1.53179e7i			0.910223i
$778$		0			0
$779$			3.55037e6i			0.209619i
$780$		0			0
$781$	−1.74103e7	+	1.31340e7i	−1.02136	+	0.770496i
$782$		0			0
$783$	−389573.			−0.0227083
$784$		0			0
$785$	−87666.4			−0.00507761
$786$		0			0
$787$	7.59115e6			0.436889			0.218444	−	0.975849i	$-0.429902\pi$
$787$	7.59115e6			0.436889			0.218444	+	0.975849i	$0.429902\pi$
$788$		0			0
$789$		−	1.59865e7i		−	0.914243i
$790$		0			0
$791$	−1.51897e7			−0.863193
$792$		0			0
$793$	9.57954e6			0.540956
$794$		0			0
$795$		−	227642.i		−	0.0127742i
$796$		0			0
$797$	6.95075e6			0.387602			0.193801	−	0.981041i	$-0.437918\pi$
$797$	6.95075e6			0.387602			0.193801	+	0.981041i	$0.437918\pi$
$798$		0			0
$799$	394618.			0.0218680
$800$		0			0
$801$	6.38820e6			0.351801
$802$		0			0
$803$	1.87349e6	−	1.41333e6i	0.102533	−	0.0773490i
$804$		0			0
$805$		−	1.87380e6i		−	0.101914i
$806$		0			0
$807$		−	1.60887e6i		−	0.0869637i
$808$		0			0
$809$		−	2.27994e7i		−	1.22477i	−0.790561	−	0.612383i	$-0.790211\pi$
$809$		−	2.27994e7i		−	1.22477i	0.790561	−	0.612383i	$-0.209789\pi$
$810$		0			0
$811$	9.33998e6			0.498648			0.249324	−	0.968420i	$-0.419792\pi$
$811$	9.33998e6			0.498648			0.249324	+	0.968420i	$0.419792\pi$
$812$		0			0
$813$			1.07363e7i			0.569676i
$814$		0			0
$815$			1.24005e6i			0.0653952i
$816$		0			0
$817$	2.86316e6			0.150069
$818$		0			0
$819$			7.44635e6i			0.387913i
$820$		0			0
$821$			1.41165e7i			0.730916i	0.930828	+	0.365458i	$0.119088\pi$
$821$			1.41165e7i			0.730916i	−0.930828	+	0.365458i	$0.880912\pi$
$822$		0			0
$823$			4.37655e6i			0.225233i	0.993639	+	0.112617i	$0.0359232\pi$
$823$			4.37655e6i			0.225233i	−0.993639	+	0.112617i	$0.964077\pi$
$824$		0			0
$825$	8.94207e6	−	6.74573e6i	0.457407	−	0.345060i
$826$		0			0
$827$	−2.57585e7			−1.30966			−0.654828	−	0.755778i	$-0.727259\pi$
$827$	−2.57585e7			−1.30966			−0.654828	+	0.755778i	$0.727259\pi$
$828$		0			0
$829$	−1.82549e7			−0.922556			−0.461278	−	0.887256i	$-0.652609\pi$
$829$	−1.82549e7			−0.922556			−0.461278	+	0.887256i	$0.652609\pi$
$830$		0			0
$831$	−1.95099e7			−0.980062
$832$		0			0
$833$			4.31798e7i			2.15610i
$834$		0			0
$835$	−1.53513e6			−0.0761953
$836$		0			0
$837$	4.40438e6			0.217306
$838$		0			0
$839$		−	444998.i		−	0.0218249i	−0.999940	−	0.0109125i	$-0.996526\pi$
$839$		−	444998.i		−	0.0218249i	0.999940	−	0.0109125i	$-0.00347362\pi$
$840$		0			0
$841$	2.02256e7			0.986077
$842$		0			0
$843$	−7.57127e6			−0.366944
$844$		0			0
$845$	1.01843e6			0.0490670
$846$		0			0
$847$	1.00931e7	−	3.53395e7i	0.483409	−	1.69259i
$848$		0			0
$849$		−	1.26081e7i		−	0.600316i
$850$		0			0
$851$			1.25681e7i			0.594904i
$852$		0			0
$853$		−	1.63016e7i		−	0.767112i	−0.923518	−	0.383556i	$-0.874699\pi$
$853$		−	1.63016e7i		−	0.767112i	0.923518	−	0.383556i	$-0.125301\pi$
$854$		0			0
$855$	388368.			0.0181689
$856$		0			0
$857$		−	3.64004e7i		−	1.69299i	−0.532398	−	0.846494i	$-0.678709\pi$
$857$		−	3.64004e7i		−	1.69299i	0.532398	−	0.846494i	$-0.321291\pi$
$858$		0			0
$859$			3.23024e7i			1.49366i	0.665014	+	0.746831i	$0.268426\pi$
$859$			3.23024e7i			1.49366i	−0.665014	+	0.746831i	$0.731574\pi$
$860$		0			0
$861$	7.41042e6			0.340671
$862$		0			0
$863$		−	1.36957e7i		−	0.625977i	−0.949757	−	0.312989i	$-0.898670\pi$
$863$		−	1.36957e7i		−	0.625977i	0.949757	−	0.312989i	$-0.101330\pi$
$864$		0			0
$865$			1.13258e6i			0.0514672i
$866$		0			0
$867$			710528.i			0.0321021i
$868$		0			0
$869$	1.61221e7	+	2.13713e7i	0.724225	+	0.960024i
$870$		0			0
$871$	6.82108e6			0.304655
$872$		0			0
$873$	−1.23567e6			−0.0548738
$874$		0			0
$875$	6.92329e6			0.305698
$876$		0			0
$877$		−	1.62339e7i		−	0.712727i	−0.934347	−	0.356364i	$-0.884016\pi$
$877$		−	1.62339e7i		−	0.712727i	0.934347	−	0.356364i	$-0.115984\pi$
$878$		0			0
$879$	−5.74807e6			−0.250929
$880$		0			0
$881$	2.28225e7			0.990658			0.495329	−	0.868705i	$-0.335047\pi$
$881$	2.28225e7			0.990658			0.495329	+	0.868705i	$0.335047\pi$
$882$		0			0
$883$			4.47322e7i			1.93071i	0.260931	+	0.965357i	$0.415970\pi$
$883$			4.47322e7i			1.93071i	−0.260931	+	0.965357i	$0.584030\pi$
$884$		0			0
$885$	1.88608e6			0.0809473
$886$		0			0
$887$	1.31892e7			0.562870			0.281435	−	0.959580i	$-0.409190\pi$
$887$	1.31892e7			0.562870			0.281435	+	0.959580i	$0.409190\pi$
$888$		0			0
$889$	2.82337e6			0.119816
$890$		0			0
$891$	−1.58569e6	−	2.10197e6i	−0.0669152	−	0.0887020i
$892$		0			0
$893$			317177.i			0.0133098i
$894$		0			0
$895$		−	3.25378e6i		−	0.135778i
$896$		0			0
$897$			6.10961e6i			0.253532i
$898$		0			0
$899$	3.22863e6			0.133235
$900$		0			0
$901$			6.35509e6i			0.260801i
$902$		0			0
$903$		−	5.97607e6i		−	0.243891i
$904$		0			0
$905$	77640.1			0.00315112
$906$		0			0
$907$			1.00049e7i			0.403825i	0.979404	+	0.201913i	$0.0647157\pi$
$907$			1.00049e7i			0.403825i	−0.979404	+	0.201913i	$0.935284\pi$
$908$		0			0
$909$		−	3.09744e6i		−	0.124335i
$910$		0			0
$911$			1.70099e7i			0.679055i	0.940596	+	0.339528i	$0.110267\pi$
$911$			1.70099e7i			0.679055i	−0.940596	+	0.339528i	$0.889733\pi$
$912$		0			0
$913$	−1.75839e7	−	2.33090e7i	−0.698131	−	0.925435i
$914$		0			0
$915$	−1.04283e6			−0.0411776
$916$		0			0
$917$	−3.16558e7			−1.24317
$918$		0			0
$919$	1.69386e7			0.661592			0.330796	−	0.943702i	$-0.392683\pi$
$919$	1.69386e7			0.661592			0.330796	+	0.943702i	$0.392683\pi$
$920$		0			0
$921$			1.75995e7i			0.683678i
$922$		0			0
$923$	2.18919e7			0.845823
$924$		0			0
$925$	−2.31298e7			−0.888827
$926$		0			0
$927$			1.22095e7i			0.466660i
$928$		0			0
$929$	4.67479e7			1.77714			0.888572	−	0.458737i	$-0.151698\pi$
$929$	4.67479e7			1.77714			0.888572	+	0.458737i	$0.151698\pi$
$930$		0			0
$931$	−3.47061e7			−1.31229
$932$		0			0
$933$	−4.10936e6			−0.154551
$934$		0			0
$935$	1.91114e6	−	1.44173e6i	0.0714928	−	0.0539329i
$936$		0			0
$937$		−	2.97723e7i		−	1.10780i	−0.832582	−	0.553902i	$-0.813138\pi$
$937$		−	2.97723e7i		−	1.10780i	0.832582	−	0.553902i	$-0.186862\pi$
$938$		0			0
$939$		−	2.43209e7i		−	0.900151i
$940$		0			0
$941$		−	4.73475e7i		−	1.74310i	−0.490303	−	0.871552i	$-0.663114\pi$
$941$		−	4.73475e7i		−	1.74310i	0.490303	−	0.871552i	$-0.336886\pi$
$942$		0			0
$943$	6.08013e6			0.222656
$944$		0			0
$945$		−	810612.i		−	0.0295280i
$946$		0			0
$947$		−	1.36837e6i		−	0.0495826i	−0.999693	−	0.0247913i	$-0.992108\pi$
$947$		−	1.36837e6i		−	0.0495826i	0.999693	−	0.0247913i	$-0.00789213\pi$
$948$		0			0
$949$	−2.35575e6			−0.0849110
$950$		0			0
$951$			3.09171e7i			1.10853i
$952$		0			0
$953$		−	1.78467e7i		−	0.636539i	−0.948000	−	0.318269i	$-0.896898\pi$
$953$		−	1.78467e7i		−	0.636539i	0.948000	−	0.318269i	$-0.103102\pi$
$954$		0			0
$955$		−	1.60489e6i		−	0.0569424i
$956$		0			0
$957$	−1.16239e6	−	1.54085e6i	−0.0410273	−	0.0543853i
$958$		0			0
$959$	−5.37962e7			−1.88888
$960$		0			0
$961$	−7.87266e6			−0.274987
$962$		0			0
$963$	−1.15220e7			−0.400372
$964$		0			0
$965$			3.22794e6i			0.111585i
$966$		0			0
$967$	1.58483e7			0.545025			0.272512	−	0.962152i	$-0.412145\pi$
$967$	1.58483e7			0.545025			0.272512	+	0.962152i	$0.412145\pi$
$968$		0			0
$969$	−1.08421e7			−0.370939
$970$		0			0
$971$			4.06818e7i			1.38469i	0.721568	+	0.692344i	$0.243422\pi$
$971$			4.06818e7i			1.38469i	−0.721568	+	0.692344i	$0.756578\pi$
$972$		0			0
$973$	−4.20109e7			−1.42259
$974$		0			0
$975$	−1.12438e7			−0.378794
$976$		0			0
$977$	−6.35637e6			−0.213046			−0.106523	−	0.994310i	$-0.533972\pi$
$977$	−6.35637e6			−0.213046			−0.106523	+	0.994310i	$0.533972\pi$
$978$		0			0
$979$	1.90608e7	+	2.52668e7i	0.635602	+	0.842547i
$980$		0			0
$981$		−	1.01769e6i		−	0.0337631i
$982$		0			0
$983$			2.94380e7i			0.971683i	0.874047	+	0.485841i	$0.161487\pi$
$983$			2.94380e7i			0.971683i	−0.874047	+	0.485841i	$0.838513\pi$
$984$		0			0
$985$			2.30225e6i			0.0756069i
$986$		0			0
$987$	662019.			0.0216311
$988$		0			0
$989$		−	4.90327e6i		−	0.159402i
$990$		0			0
$991$		−	6.08830e7i		−	1.96930i	−0.174539	−	0.984650i	$-0.555844\pi$
$991$		−	6.08830e7i		−	1.96930i	0.174539	−	0.984650i	$-0.444156\pi$
$992$		0			0
$993$	2.01689e7			0.649095
$994$		0			0
$995$		−	1.67939e6i		−	0.0537767i
$996$		0			0
$997$		−	3.53900e7i		−	1.12757i	−0.825922	−	0.563785i	$-0.809345\pi$
$997$		−	3.53900e7i		−	1.12757i	0.825922	−	0.563785i	$-0.190655\pi$
$998$		0			0
$999$			5.43703e6i			0.172364i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.o.b.175.13	✓	40
4.3	odd	2	inner	528.6.o.b.175.32	yes	40
11.10	odd	2	inner	528.6.o.b.175.31	yes	40
44.43	even	2	inner	528.6.o.b.175.14	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.6.o.b.175.13	✓	40	1.1	even	1	trivial
528.6.o.b.175.14	yes	40	44.43	even	2	inner
528.6.o.b.175.31	yes	40	11.10	odd	2	inner
528.6.o.b.175.32	yes	40	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} +4.87261 q^{5} -228.204 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} +4.87261 q^{5} -228.204 q^{7} -81.0000 q^{9} +(-241.684 - 320.374i) q^{11} +402.842i q^{13} -43.8535i q^{15} +1224.26i q^{17} -984.004 q^{19} +2053.84i q^{21} +1685.14i q^{23} -3101.26 q^{25} +729.000i q^{27} +534.393i q^{29} -6041.67i q^{31} +(-2883.37 + 2175.16i) q^{33} -1111.95 q^{35} +7458.20 q^{37} +3625.58 q^{39} -3608.08i q^{41} -2909.71 q^{43} -394.681 q^{45} -322.333i q^{47} +35270.2 q^{49} +11018.3 q^{51} +5190.98 q^{53} +(-1177.63 - 1561.06i) q^{55} +8856.04i q^{57} +43008.7i q^{59} -23779.9i q^{61} +18484.6 q^{63} +1962.89i q^{65} -16932.4i q^{67} +15166.3 q^{69} -54343.7i q^{71} +5847.83i q^{73} +27911.3i q^{75} +(55153.5 + 73110.8i) q^{77} -66707.4 q^{79} +6561.00 q^{81} +72755.4 q^{83} +5965.32i q^{85} +4809.54 q^{87} -78866.7 q^{89} -91930.3i q^{91} -54375.1 q^{93} -4794.67 q^{95} +15255.1 q^{97} +(19576.4 + 25950.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) \) 40 * q + 88 * q^5 - 3240 * q^9	\( 40 q + 88 q^{5} - 3240 q^{9} + 5464 q^{25} + 6840 q^{33} + 4576 q^{37} - 7128 q^{45} + 58760 q^{49} - 165336 q^{53} + 16632 q^{69} + 28136 q^{77} + 262440 q^{81} - 304608 q^{89} + 102672 q^{93} + 15664 q^{97}+O(q^{100}) \) 40 * q + 88 * q^5 - 3240 * q^9 + 5464 * q^25 + 6840 * q^33 + 4576 * q^37 - 7128 * q^45 + 58760 * q^49 - 165336 * q^53 + 16632 * q^69 + 28136 * q^77 + 262440 * q^81 - 304608 * q^89 + 102672 * q^93 + 15664 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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