LMFDB - Embedded newform 528.6.o.b.175.1

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			175.1
Character	$\chi$	$=$	528.175
Dual form			528.6.o.b.175.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} -34.2910 q^{5} +120.607 q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} -34.2910 q^{5} +120.607 q^{7} -81.0000 q^{9} +(257.124 + 308.120i) q^{11} +1159.73i q^{13} +308.619i q^{15} +1405.86i q^{17} -903.654 q^{19} -1085.46i q^{21} -5015.37i q^{23} -1949.13 q^{25} +729.000i q^{27} -2340.38i q^{29} -6180.89i q^{31} +(2773.08 - 2314.12i) q^{33} -4135.73 q^{35} -6259.81 q^{37} +10437.6 q^{39} -6014.74i q^{41} +3330.66 q^{43} +2777.57 q^{45} +25700.8i q^{47} -2261.02 q^{49} +12652.8 q^{51} -10273.9 q^{53} +(-8817.05 - 10565.8i) q^{55} +8132.89i q^{57} +9815.58i q^{59} +1724.28i q^{61} -9769.15 q^{63} -39768.4i q^{65} -25625.0i q^{67} -45138.3 q^{69} +19210.9i q^{71} -61301.4i q^{73} +17542.1i q^{75} +(31010.9 + 37161.4i) q^{77} -72105.2 q^{79} +6561.00 q^{81} -36504.3 q^{83} -48208.4i q^{85} -21063.4 q^{87} -32281.2 q^{89} +139872. i q^{91} -55628.0 q^{93} +30987.2 q^{95} -47514.1 q^{97} +(-20827.1 - 24957.7i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−34.2910			−0.613416			−0.306708	−	0.951804i	$-0.599228\pi$
$5$	−34.2910			−0.613416			−0.306708	+	0.951804i	$0.599228\pi$
$6$		0			0
$7$	120.607			0.930307			0.465154	−	0.885230i	$-0.345999\pi$
$7$	120.607			0.930307			0.465154	+	0.885230i	$0.345999\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	257.124	+	308.120i	0.640710	+	0.767783i
$11$	257.124	+	308.120i	0.640710	+	0.767783i
$12$		0			0
$13$			1159.73i			1.90327i	0.307235	+	0.951634i	$0.400596\pi$
$13$			1159.73i			1.90327i	−0.307235	+	0.951634i	$0.599404\pi$
$14$		0			0
$15$			308.619i			0.354156i
$16$		0			0
$17$			1405.86i			1.17983i	0.807464	+	0.589916i	$0.200839\pi$
$17$			1405.86i			1.17983i	−0.807464	+	0.589916i	$0.799161\pi$
$18$		0			0
$19$	−903.654			−0.574273			−0.287136	−	0.957890i	$-0.592703\pi$
$19$	−903.654			−0.574273			−0.287136	+	0.957890i	$0.592703\pi$
$20$		0			0
$21$		−	1085.46i		−	0.537113i
$22$		0			0
$23$		−	5015.37i		−	1.97689i	−0.151564	−	0.988447i	$-0.548431\pi$
$23$		−	5015.37i		−	1.97689i	0.151564	−	0.988447i	$-0.451569\pi$
$24$		0			0
$25$	−1949.13			−0.623720
$26$		0			0
$27$			729.000i			0.192450i
$28$		0			0
$29$		−	2340.38i		−	0.516762i	−0.966043	−	0.258381i	$-0.916811\pi$
$29$		−	2340.38i		−	0.516762i	0.966043	−	0.258381i	$-0.0831890\pi$
$30$		0			0
$31$		−	6180.89i		−	1.15517i	−0.816330	−	0.577586i	$-0.803995\pi$
$31$		−	6180.89i		−	1.15517i	0.816330	−	0.577586i	$-0.196005\pi$
$32$		0			0
$33$	2773.08	−	2314.12i	0.443280	−	0.369914i
$34$		0			0
$35$	−4135.73			−0.570666
$36$		0			0
$37$	−6259.81			−0.751721			−0.375860	−	0.926676i	$-0.622653\pi$
$37$	−6259.81			−0.751721			−0.375860	+	0.926676i	$0.622653\pi$
$38$		0			0
$39$	10437.6			1.09885
$40$		0			0
$41$		−	6014.74i		−	0.558801i	−0.960175	−	0.279400i	$-0.909864\pi$
$41$		−	6014.74i		−	0.558801i	0.960175	−	0.279400i	$-0.0901357\pi$
$42$		0			0
$43$	3330.66			0.274700			0.137350	−	0.990523i	$-0.456141\pi$
$43$	3330.66			0.274700			0.137350	+	0.990523i	$0.456141\pi$
$44$		0			0
$45$	2777.57			0.204472
$46$		0			0
$47$			25700.8i			1.69708i	0.529134	+	0.848538i	$0.322517\pi$
$47$			25700.8i			1.69708i	−0.529134	+	0.848538i	$0.677483\pi$
$48$		0			0
$49$	−2261.02			−0.134528
$50$		0			0
$51$	12652.8			0.681177
$52$		0			0
$53$	−10273.9			−0.502397			−0.251199	−	0.967936i	$-0.580825\pi$
$53$	−10273.9			−0.502397			−0.251199	+	0.967936i	$0.580825\pi$
$54$		0			0
$55$	−8817.05	−	10565.8i	−0.393022	−	0.470971i
$56$		0			0
$57$			8132.89i			0.331556i
$58$		0			0
$59$			9815.58i			0.367101i	0.983010	+	0.183551i	$0.0587591\pi$
$59$			9815.58i			0.367101i	−0.983010	+	0.183551i	$0.941241\pi$
$60$		0			0
$61$			1724.28i			0.0593311i	0.999560	+	0.0296655i	$0.00944422\pi$
$61$			1724.28i			0.0593311i	−0.999560	+	0.0296655i	$0.990556\pi$
$62$		0			0
$63$	−9769.15			−0.310102
$64$		0			0
$65$		−	39768.4i		−	1.16750i
$66$		0			0
$67$		−	25625.0i		−	0.697393i	−0.937236	−	0.348697i	$-0.886624\pi$
$67$		−	25625.0i		−	0.697393i	0.937236	−	0.348697i	$-0.113376\pi$
$68$		0			0
$69$	−45138.3			−1.14136
$70$		0			0
$71$			19210.9i			0.452275i	0.974095	+	0.226137i	$0.0726098\pi$
$71$			19210.9i			0.452275i	−0.974095	+	0.226137i	$0.927390\pi$
$72$		0			0
$73$		−	61301.4i		−	1.34637i	−0.739476	−	0.673183i	$-0.764926\pi$
$73$		−	61301.4i		−	1.34637i	0.739476	−	0.673183i	$-0.235074\pi$
$74$		0			0
$75$			17542.1i			0.360105i
$76$		0			0
$77$	31010.9	+	37161.4i	0.596057	+	0.714274i
$78$		0			0
$79$	−72105.2			−1.29987			−0.649933	−	0.759991i	$-0.725203\pi$
$79$	−72105.2			−1.29987			−0.649933	+	0.759991i	$0.725203\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$	−36504.3			−0.581633			−0.290817	−	0.956779i	$-0.593927\pi$
$83$	−36504.3			−0.581633			−0.290817	+	0.956779i	$0.593927\pi$
$84$		0			0
$85$		−	48208.4i		−	0.723729i
$86$		0			0
$87$	−21063.4			−0.298353
$88$		0			0
$89$	−32281.2			−0.431992			−0.215996	−	0.976394i	$-0.569300\pi$
$89$	−32281.2			−0.431992			−0.215996	+	0.976394i	$0.569300\pi$
$90$		0			0
$91$			139872.i			1.77062i
$92$		0			0
$93$	−55628.0			−0.666939
$94$		0			0
$95$	30987.2			0.352268
$96$		0			0
$97$	−47514.1			−0.512735			−0.256367	−	0.966579i	$-0.582526\pi$
$97$	−47514.1			−0.512735			−0.256367	+	0.966579i	$0.582526\pi$
$98$		0			0
$99$	−20827.1	−	24957.7i	−0.213570	−	0.255928i
$100$		0			0
$101$			163409.i			1.59394i	0.604019	+	0.796970i	$0.293565\pi$
$101$			163409.i			1.59394i	−0.604019	+	0.796970i	$0.706435\pi$
$102$		0			0
$103$		−	56144.3i		−	0.521450i	−0.965413	−	0.260725i	$-0.916038\pi$
$103$		−	56144.3i		−	0.521450i	0.965413	−	0.260725i	$-0.0839616\pi$
$104$		0			0
$105$			37221.6i			0.329474i
$106$		0			0
$107$	−56594.8			−0.477878			−0.238939	−	0.971035i	$-0.576800\pi$
$107$	−56594.8			−0.477878			−0.238939	+	0.971035i	$0.576800\pi$
$108$		0			0
$109$		−	211807.i		−	1.70755i	−0.520642	−	0.853775i	$-0.674307\pi$
$109$		−	211807.i		−	1.70755i	0.520642	−	0.853775i	$-0.325693\pi$
$110$		0			0
$111$			56338.3i			0.434006i
$112$		0			0
$113$	−159780.			−1.17713			−0.588567	−	0.808449i	$-0.700308\pi$
$113$	−159780.			−1.17713			−0.588567	+	0.808449i	$0.700308\pi$
$114$		0			0
$115$			171982.i			1.21266i
$116$		0			0
$117$		−	93938.4i		−	0.634423i
$118$		0			0
$119$			169556.i			1.09761i
$120$		0			0
$121$	−28825.2	+	158450.i	−0.178982	+	0.983852i
$122$		0			0
$123$	−54132.6			−0.322624
$124$		0			0
$125$	173997.			0.996017
$126$		0			0
$127$	−277610.			−1.52730			−0.763652	−	0.645628i	$-0.776596\pi$
$127$	−277610.			−1.52730			−0.763652	+	0.645628i	$0.776596\pi$
$128$		0			0
$129$		−	29975.9i		−	0.158598i
$130$		0			0
$131$	−109067.			−0.555283			−0.277642	−	0.960685i	$-0.589553\pi$
$131$	−109067.			−0.555283			−0.277642	+	0.960685i	$0.589553\pi$
$132$		0			0
$133$	−108987.			−0.534250
$134$		0			0
$135$		−	24998.2i		−	0.118052i
$136$		0			0
$137$	72879.1			0.331743			0.165872	−	0.986147i	$-0.446956\pi$
$137$	72879.1			0.331743			0.165872	+	0.986147i	$0.446956\pi$
$138$		0			0
$139$	363091.			1.59396			0.796981	−	0.604004i	$-0.206429\pi$
$139$	363091.			1.59396			0.796981	+	0.604004i	$0.206429\pi$
$140$		0			0
$141$	231307.			0.979808
$142$		0			0
$143$	−357337.	+	298196.i	−1.46130	+	1.21944i
$144$		0			0
$145$			80253.9i			0.316990i
$146$		0			0
$147$			20349.1i			0.0776699i
$148$		0			0
$149$			186242.i			0.687246i	0.939108	+	0.343623i	$0.111654\pi$
$149$			186242.i			0.687246i	−0.939108	+	0.343623i	$0.888346\pi$
$150$		0			0
$151$	−135159.			−0.482394			−0.241197	−	0.970476i	$-0.577540\pi$
$151$	−135159.			−0.482394			−0.241197	+	0.970476i	$0.577540\pi$
$152$		0			0
$153$		−	113875.i		−	0.393278i
$154$		0			0
$155$			211949.i			0.708601i
$156$		0			0
$157$	381242.			1.23439			0.617194	−	0.786811i	$-0.288269\pi$
$157$	381242.			1.23439			0.617194	+	0.786811i	$0.288269\pi$
$158$		0			0
$159$			92465.4i			0.290059i
$160$		0			0
$161$		−	604888.i		−	1.83912i
$162$		0			0
$163$			11634.0i			0.0342972i	0.999853	+	0.0171486i	$0.00545883\pi$
$163$			11634.0i			0.0342972i	−0.999853	+	0.0171486i	$0.994541\pi$
$164$		0			0
$165$	−95091.8	+	79353.5i	−0.271915	+	0.226911i
$166$		0			0
$167$	−77807.9			−0.215890			−0.107945	−	0.994157i	$-0.534427\pi$
$167$	−77807.9			−0.215890			−0.107945	+	0.994157i	$0.534427\pi$
$168$		0			0
$169$	−973689.			−2.62243
$170$		0			0
$171$	73196.0			0.191424
$172$		0			0
$173$		−	698705.i		−	1.77492i	−0.460885	−	0.887460i	$-0.652468\pi$
$173$		−	698705.i		−	1.77492i	0.460885	−	0.887460i	$-0.347532\pi$
$174$		0			0
$175$	−235078.			−0.580252
$176$		0			0
$177$	88340.2			0.211946
$178$		0			0
$179$		−	287620.i		−	0.670944i	−0.942050	−	0.335472i	$-0.891104\pi$
$179$		−	287620.i		−	0.670944i	0.942050	−	0.335472i	$-0.108896\pi$
$180$		0			0
$181$	−428332.			−0.971815			−0.485908	−	0.874010i	$-0.661511\pi$
$181$	−428332.			−0.971815			−0.485908	+	0.874010i	$0.661511\pi$
$182$		0			0
$183$	15518.5			0.0342548
$184$		0			0
$185$	214655.			0.461118
$186$		0			0
$187$	−433175.	+	361481.i	−0.905856	+	0.755930i
$188$		0			0
$189$			87922.3i			0.179038i
$190$		0			0
$191$			649706.i			1.28865i	0.764754	+	0.644323i	$0.222861\pi$
$191$			649706.i			1.28865i	−0.764754	+	0.644323i	$0.777139\pi$
$192$		0			0
$193$		−	64621.7i		−	0.124878i	−0.998049	−	0.0624389i	$-0.980112\pi$
$193$		−	64621.7i		−	0.124878i	0.998049	−	0.0624389i	$-0.0198878\pi$
$194$		0			0
$195$	−357916.			−0.674054
$196$		0			0
$197$			643084.i			1.18060i	0.807185	+	0.590299i	$0.200990\pi$
$197$			643084.i			1.18060i	−0.807185	+	0.590299i	$0.799010\pi$
$198$		0			0
$199$			922637.i			1.65158i	0.563981	+	0.825788i	$0.309269\pi$
$199$			922637.i			1.65158i	−0.563981	+	0.825788i	$0.690731\pi$
$200$		0			0
$201$	−230625.			−0.402640
$202$		0			0
$203$		−	282265.i		−	0.480747i
$204$		0			0
$205$			206252.i			0.342778i
$206$		0			0
$207$			406245.i			0.658965i
$208$		0			0
$209$	−232351.	−	278434.i	−0.367942	−	0.440917i
$210$		0			0
$211$	692990.			1.07157			0.535785	−	0.844354i	$-0.320016\pi$
$211$	692990.			1.07157			0.535785	+	0.844354i	$0.320016\pi$
$212$		0			0
$213$	172898.			0.261121
$214$		0			0
$215$	−114212.			−0.168506
$216$		0			0
$217$		−	745456.i		−	1.07466i
$218$		0			0
$219$	−551713.			−0.777325
$220$		0			0
$221$	−1.63043e6			−2.24554
$222$		0			0
$223$			449464.i			0.605247i	0.953110	+	0.302623i	$0.0978625\pi$
$223$			449464.i			0.605247i	−0.953110	+	0.302623i	$0.902138\pi$
$224$		0			0
$225$	157879.			0.207907
$226$		0			0
$227$	−1.19407e6			−1.53804			−0.769018	−	0.639227i	$-0.779254\pi$
$227$	−1.19407e6			−1.53804			−0.769018	+	0.639227i	$0.779254\pi$
$228$		0			0
$229$	−1.19471e6			−1.50547			−0.752735	−	0.658324i	$-0.771266\pi$
$229$	−1.19471e6			−1.50547			−0.752735	+	0.658324i	$0.771266\pi$
$230$		0			0
$231$	334452.	−	279098.i	0.412386	−	0.344134i
$232$		0			0
$233$			29434.7i			0.0355197i	0.999842	+	0.0177599i	$0.00565344\pi$
$233$			29434.7i			0.0355197i	−0.999842	+	0.0177599i	$0.994347\pi$
$234$		0			0
$235$		−	881306.i		−	1.04101i
$236$		0			0
$237$			648947.i			0.750478i
$238$		0			0
$239$	−565471.			−0.640348			−0.320174	−	0.947359i	$-0.603741\pi$
$239$	−565471.			−0.640348			−0.320174	+	0.947359i	$0.603741\pi$
$240$		0			0
$241$		−	696908.i		−	0.772917i	−0.922307	−	0.386458i	$-0.873698\pi$
$241$		−	696908.i		−	0.772917i	0.922307	−	0.386458i	$-0.126302\pi$
$242$		0			0
$243$		−	59049.0i		−	0.0641500i
$244$		0			0
$245$	77532.6			0.0825218
$246$		0			0
$247$		−	1.04800e6i		−	1.09299i
$248$		0			0
$249$			328539.i			0.335806i
$250$		0			0
$251$			902156.i			0.903852i	0.892055	+	0.451926i	$0.149263\pi$
$251$			902156.i			0.903852i	−0.892055	+	0.451926i	$0.850737\pi$
$252$		0			0
$253$	1.54534e6	−	1.28957e6i	1.51783	−	1.26662i
$254$		0			0
$255$	−433876.			−0.417845
$256$		0			0
$257$	−196306.			−0.185396			−0.0926980	−	0.995694i	$-0.529549\pi$
$257$	−196306.			−0.185396			−0.0926980	+	0.995694i	$0.529549\pi$
$258$		0			0
$259$	−754975.			−0.699331
$260$		0			0
$261$			189570.i			0.172254i
$262$		0			0
$263$	848510.			0.756428			0.378214	−	0.925718i	$-0.376538\pi$
$263$	848510.			0.756428			0.378214	+	0.925718i	$0.376538\pi$
$264$		0			0
$265$	352304.			0.308179
$266$		0			0
$267$			290531.i			0.249410i
$268$		0			0
$269$	682844.			0.575361			0.287681	−	0.957726i	$-0.407116\pi$
$269$	682844.			0.575361			0.287681	+	0.957726i	$0.407116\pi$
$270$		0			0
$271$	1.81503e6			1.50128			0.750639	−	0.660712i	$-0.229746\pi$
$271$	1.81503e6			1.50128			0.750639	+	0.660712i	$0.229746\pi$
$272$		0			0
$273$	1.25885e6			1.02227
$274$		0			0
$275$	−501168.	−	600565.i	−0.399624	−	0.478882i
$276$		0			0
$277$		−	879814.i		−	0.688956i	−0.938794	−	0.344478i	$-0.888056\pi$
$277$		−	879814.i		−	0.688956i	0.938794	−	0.344478i	$-0.111944\pi$
$278$		0			0
$279$			500652.i			0.385057i
$280$		0			0
$281$		−	448022.i		−	0.338480i	−0.985575	−	0.169240i	$-0.945869\pi$
$281$		−	448022.i		−	0.338480i	0.985575	−	0.169240i	$-0.0541313\pi$
$282$		0			0
$283$	288346.			0.214017			0.107008	−	0.994258i	$-0.465873\pi$
$283$	288346.			0.214017			0.107008	+	0.994258i	$0.465873\pi$
$284$		0			0
$285$		−	278885.i		−	0.203382i
$286$		0			0
$287$		−	725418.i		−	0.519857i
$288$		0			0
$289$	−556591.			−0.392005
$290$		0			0
$291$			427627.i			0.296028i
$292$		0			0
$293$			2.11319e6i			1.43804i	0.694991	+	0.719018i	$0.255408\pi$
$293$			2.11319e6i			1.43804i	−0.694991	+	0.719018i	$0.744592\pi$
$294$		0			0
$295$		−	336586.i		−	0.225186i
$296$		0			0
$297$	−224620.	+	187444.i	−0.147760	+	0.123305i
$298$		0			0
$299$	5.81650e6			3.76256
$300$		0			0
$301$	401700.			0.255555
$302$		0			0
$303$	1.47068e6			0.920262
$304$		0			0
$305$		−	59127.2i		−	0.0363947i
$306$		0			0
$307$	−1.65643e6			−1.00306			−0.501530	−	0.865140i	$-0.667229\pi$
$307$	−1.65643e6			−1.00306			−0.501530	+	0.865140i	$0.667229\pi$
$308$		0			0
$309$	−505299.			−0.301059
$310$		0			0
$311$			562712.i			0.329902i	0.986302	+	0.164951i	$0.0527466\pi$
$311$			562712.i			0.329902i	−0.986302	+	0.164951i	$0.947253\pi$
$312$		0			0
$313$	−2.57621e6			−1.48635			−0.743173	−	0.669099i	$-0.766680\pi$
$313$	−2.57621e6			−1.48635			−0.743173	+	0.669099i	$0.766680\pi$
$314$		0			0
$315$	334994.			0.190222
$316$		0			0
$317$	2.48907e6			1.39120			0.695598	−	0.718431i	$-0.255139\pi$
$317$	2.48907e6			1.39120			0.695598	+	0.718431i	$0.255139\pi$
$318$		0			0
$319$	721117.	−	601767.i	0.396761	−	0.331095i
$320$		0			0
$321$			509353.i			0.275903i
$322$		0			0
$323$		−	1.27041e6i		−	0.677546i
$324$		0			0
$325$		−	2.26047e6i		−	1.18711i
$326$		0			0
$327$	−1.90626e6			−0.985855
$328$		0			0
$329$			3.09969e6i			1.57880i
$330$		0			0
$331$		−	3.71066e6i		−	1.86158i	−0.365558	−	0.930789i	$-0.619122\pi$
$331$		−	3.71066e6i		−	1.86158i	0.365558	−	0.930789i	$-0.380878\pi$
$332$		0			0
$333$	507044.			0.250574
$334$		0			0
$335$			878709.i			0.427792i
$336$		0			0
$337$			3.01704e6i			1.44712i	0.690259	+	0.723562i	$0.257497\pi$
$337$			3.01704e6i			1.44712i	−0.690259	+	0.723562i	$0.742503\pi$
$338$		0			0
$339$			1.43802e6i			0.679618i
$340$		0			0
$341$	1.90446e6	−	1.58926e6i	0.886921	−	0.740130i
$342$		0			0
$343$	−2.29973e6			−1.05546
$344$		0			0
$345$	1.54784e6			0.700129
$346$		0			0
$347$	−761085.			−0.339320			−0.169660	−	0.985503i	$-0.554267\pi$
$347$	−761085.			−0.339320			−0.169660	+	0.985503i	$0.554267\pi$
$348$		0			0
$349$			3.89518e6i			1.71184i	0.517105	+	0.855922i	$0.327010\pi$
$349$			3.89518e6i			1.71184i	−0.517105	+	0.855922i	$0.672990\pi$
$350$		0			0
$351$	−845446.			−0.366284
$352$		0			0
$353$	2.02878e6			0.866559			0.433279	−	0.901260i	$-0.357356\pi$
$353$	2.02878e6			0.866559			0.433279	+	0.901260i	$0.357356\pi$
$354$		0			0
$355$		−	658762.i		−	0.277433i
$356$		0			0
$357$	1.52601e6			0.633704
$358$		0			0
$359$	−3.93664e6			−1.61209			−0.806046	−	0.591853i	$-0.798397\pi$
$359$	−3.93664e6			−1.61209			−0.806046	+	0.591853i	$0.798397\pi$
$360$		0			0
$361$	−1.65951e6			−0.670211
$362$		0			0
$363$	1.42605e6	+	259427.i	0.568027	+	0.103335i
$364$		0			0
$365$			2.10209e6i			0.825883i
$366$		0			0
$367$		−	3.77165e6i		−	1.46172i	−0.682525	−	0.730862i	$-0.739118\pi$
$367$		−	3.77165e6i		−	1.46172i	0.682525	−	0.730862i	$-0.260882\pi$
$368$		0			0
$369$			487194.i			0.186267i
$370$		0			0
$371$	−1.23911e6			−0.467384
$372$		0			0
$373$		−	2.47719e6i		−	0.921906i	−0.887424	−	0.460953i	$-0.847508\pi$
$373$		−	2.47719e6i		−	0.921906i	0.887424	−	0.460953i	$-0.152492\pi$
$374$		0			0
$375$		−	1.56597e6i		−	0.575050i
$376$		0			0
$377$	2.71421e6			0.983536
$378$		0			0
$379$		−	1.10616e6i		−	0.395567i	−0.980246	−	0.197784i	$-0.936626\pi$
$379$		−	1.10616e6i		−	0.395567i	0.980246	−	0.197784i	$-0.0633744\pi$
$380$		0			0
$381$			2.49849e6i			0.881790i
$382$		0			0
$383$			720683.i			0.251042i	0.992091	+	0.125521i	$0.0400603\pi$
$383$			720683.i			0.251042i	−0.992091	+	0.125521i	$0.959940\pi$
$384$		0			0
$385$	−1.06340e6	−	1.27430e6i	−0.365631	−	0.438148i
$386$		0			0
$387$	−269783.			−0.0915667
$388$		0			0
$389$	3.32826e6			1.11518			0.557588	−	0.830118i	$-0.311727\pi$
$389$	3.32826e6			1.11518			0.557588	+	0.830118i	$0.311727\pi$
$390$		0			0
$391$	7.05092e6			2.33241
$392$		0			0
$393$			981602.i			0.320593i
$394$		0			0
$395$	2.47256e6			0.797359
$396$		0			0
$397$	−4.72928e6			−1.50598			−0.752989	−	0.658033i	$-0.771389\pi$
$397$	−4.72928e6			−1.50598			−0.752989	+	0.658033i	$0.771389\pi$
$398$		0			0
$399$			980881.i			0.308449i
$400$		0			0
$401$	−2.99271e6			−0.929402			−0.464701	−	0.885468i	$-0.653838\pi$
$401$	−2.99271e6			−0.929402			−0.464701	+	0.885468i	$0.653838\pi$
$402$		0			0
$403$	7.16818e6			2.19860
$404$		0			0
$405$	−224983.			−0.0681574
$406$		0			0
$407$	−1.60955e6	−	1.92877e6i	−0.481635	−	0.577158i
$408$		0			0
$409$		−	2.56859e6i		−	0.759254i	−0.925140	−	0.379627i	$-0.876052\pi$
$409$		−	2.56859e6i		−	0.759254i	0.925140	−	0.379627i	$-0.123948\pi$
$410$		0			0
$411$		−	655912.i		−	0.191532i
$412$		0			0
$413$			1.18383e6i			0.341517i
$414$		0			0
$415$	1.25177e6			0.356783
$416$		0			0
$417$		−	3.26782e6i		−	0.920275i
$418$		0			0
$419$			4.11102e6i			1.14397i	0.820264	+	0.571985i	$0.193827\pi$
$419$			4.11102e6i			1.14397i	−0.820264	+	0.571985i	$0.806173\pi$
$420$		0			0
$421$	2.21871e6			0.610092			0.305046	−	0.952338i	$-0.401328\pi$
$421$	2.21871e6			0.610092			0.305046	+	0.952338i	$0.401328\pi$
$422$		0			0
$423$		−	2.08176e6i		−	0.565692i
$424$		0			0
$425$		−	2.74020e6i		−	0.735886i
$426$		0			0
$427$			207959.i			0.0551962i
$428$		0			0
$429$	2.68376e6	+	3.21604e6i	0.704045	+	0.843680i
$430$		0			0
$431$	−1.42760e6			−0.370181			−0.185091	−	0.982721i	$-0.559258\pi$
$431$	−1.42760e6			−0.370181			−0.185091	+	0.982721i	$0.559258\pi$
$432$		0			0
$433$	−1.56938e6			−0.402262			−0.201131	−	0.979564i	$-0.564462\pi$
$433$	−1.56938e6			−0.402262			−0.201131	+	0.979564i	$0.564462\pi$
$434$		0			0
$435$	722285.			0.183014
$436$		0			0
$437$			4.53216e6i			1.13528i
$438$		0			0
$439$	4.61678e6			1.14335			0.571674	−	0.820481i	$-0.306294\pi$
$439$	4.61678e6			1.14335			0.571674	+	0.820481i	$0.306294\pi$
$440$		0			0
$441$	183142.			0.0448427
$442$		0			0
$443$			1.12949e6i			0.273448i	0.990609	+	0.136724i	$0.0436573\pi$
$443$			1.12949e6i			0.273448i	−0.990609	+	0.136724i	$0.956343\pi$
$444$		0			0
$445$	1.10696e6			0.264991
$446$		0			0
$447$	1.67618e6			0.396782
$448$		0			0
$449$	2.89115e6			0.676791			0.338395	−	0.941004i	$-0.390116\pi$
$449$	2.89115e6			0.676791			0.338395	+	0.941004i	$0.390116\pi$
$450$		0			0
$451$	1.85326e6	−	1.54654e6i	0.429038	−	0.358029i
$452$		0			0
$453$			1.21643e6i			0.278510i
$454$		0			0
$455$		−	4.79634e6i		−	1.08613i
$456$		0			0
$457$		−	5.14632e6i		−	1.15267i	−0.817212	−	0.576337i	$-0.804481\pi$
$457$		−	5.14632e6i		−	1.15267i	0.817212	−	0.576337i	$-0.195519\pi$
$458$		0			0
$459$	−1.02487e6			−0.227059
$460$		0			0
$461$			5.99184e6i			1.31313i	0.754269	+	0.656565i	$0.227991\pi$
$461$			5.99184e6i			1.31313i	−0.754269	+	0.656565i	$0.772009\pi$
$462$		0			0
$463$			2.71999e6i			0.589677i	0.955547	+	0.294838i	$0.0952658\pi$
$463$			2.71999e6i			0.589677i	−0.955547	+	0.294838i	$0.904734\pi$
$464$		0			0
$465$	1.90754e6			0.409111
$466$		0			0
$467$		−	8.46104e6i		−	1.79528i	−0.440732	−	0.897639i	$-0.645281\pi$
$467$		−	8.46104e6i		−	1.79528i	0.440732	−	0.897639i	$-0.354719\pi$
$468$		0			0
$469$		−	3.09055e6i		−	0.648790i
$470$		0			0
$471$		−	3.43118e6i		−	0.712674i
$472$		0			0
$473$	856393.	+	1.02624e6i	0.176003	+	0.210910i
$474$		0			0
$475$	1.76134e6			0.358186
$476$		0			0
$477$	832189.			0.167466
$478$		0			0
$479$	334653.			0.0666431			0.0333216	−	0.999445i	$-0.489391\pi$
$479$	334653.			0.0666431			0.0333216	+	0.999445i	$0.489391\pi$
$480$		0			0
$481$		−	7.25971e6i		−	1.43073i
$482$		0			0
$483$	−5.44399e6			−1.06182
$484$		0			0
$485$	1.62931e6			0.314520
$486$		0			0
$487$		−	6.02740e6i		−	1.15162i	−0.817585	−	0.575808i	$-0.804687\pi$
$487$		−	6.02740e6i		−	1.15162i	0.817585	−	0.575808i	$-0.195313\pi$
$488$		0			0
$489$	104706.			0.0198015
$490$		0			0
$491$	−6.04122e6			−1.13089			−0.565445	−	0.824786i	$-0.691296\pi$
$491$	−6.04122e6			−1.13089			−0.565445	+	0.824786i	$0.691296\pi$
$492$		0			0
$493$	3.29025e6			0.609693
$494$		0			0
$495$	714181.	+	855826.i	0.131007	+	0.156990i
$496$		0			0
$497$			2.31697e6i			0.420754i
$498$		0			0
$499$		−	2.43916e6i		−	0.438520i	−0.975666	−	0.219260i	$-0.929636\pi$
$499$		−	2.43916e6i		−	0.438520i	0.975666	−	0.219260i	$-0.0703643\pi$
$500$		0			0
$501$			700271.i			0.124644i
$502$		0			0
$503$	6.42275e6			1.13188			0.565941	−	0.824446i	$-0.308513\pi$
$503$	6.42275e6			1.13188			0.565941	+	0.824446i	$0.308513\pi$
$504$		0			0
$505$		−	5.60346e6i		−	0.977749i
$506$		0			0
$507$			8.76320e6i			1.51406i
$508$		0			0
$509$	6.13050e6			1.04882			0.524410	−	0.851466i	$-0.324286\pi$
$509$	6.13050e6			1.04882			0.524410	+	0.851466i	$0.324286\pi$
$510$		0			0
$511$		−	7.39336e6i		−	1.25253i
$512$		0			0
$513$		−	658764.i		−	0.110519i
$514$		0			0
$515$			1.92524e6i			0.319866i
$516$		0			0
$517$	−7.91893e6	+	6.60829e6i	−1.30299	+	1.08733i
$518$		0			0
$519$	−6.28835e6			−1.02475
$520$		0			0
$521$	−5.04725e6			−0.814630			−0.407315	−	0.913288i	$-0.633535\pi$
$521$	−5.04725e6			−0.814630			−0.407315	+	0.913288i	$0.633535\pi$
$522$		0			0
$523$	−3.78585e6			−0.605214			−0.302607	−	0.953115i	$-0.597857\pi$
$523$	−3.78585e6			−0.605214			−0.302607	+	0.953115i	$0.597857\pi$
$524$		0			0
$525$			2.11570e6i			0.335008i
$526$		0			0
$527$	8.68947e6			1.36291
$528$		0			0
$529$	−1.87176e7			−2.90811
$530$		0			0
$531$		−	795062.i		−	0.122367i
$532$		0			0
$533$	6.97549e6			1.06355
$534$		0			0
$535$	1.94069e6			0.293138
$536$		0			0
$537$	−2.58858e6			−0.387370
$538$		0			0
$539$	−581362.	−	696665.i	−0.0861936	−	0.103289i
$540$		0			0
$541$			3.73325e6i			0.548396i	0.961673	+	0.274198i	$0.0884124\pi$
$541$			3.73325e6i			0.548396i	−0.961673	+	0.274198i	$0.911588\pi$
$542$		0			0
$543$			3.85498e6i			0.561078i
$544$		0			0
$545$			7.26307e6i			1.04744i
$546$		0			0
$547$	5.08543e6			0.726707			0.363354	−	0.931651i	$-0.381632\pi$
$547$	5.08543e6			0.726707			0.363354	+	0.931651i	$0.381632\pi$
$548$		0			0
$549$		−	139666.i		−	0.0197770i
$550$		0			0
$551$			2.11489e6i			0.296762i
$552$		0			0
$553$	−8.69637e6			−1.20928
$554$		0			0
$555$		−	1.93190e6i		−	0.266227i
$556$		0			0
$557$		−	4.82472e6i		−	0.658923i	−0.944169	−	0.329461i	$-0.893133\pi$
$557$		−	4.82472e6i		−	0.658923i	0.944169	−	0.329461i	$-0.106867\pi$
$558$		0			0
$559$			3.86267e6i			0.522828i
$560$		0			0
$561$	3.25333e6	+	3.89857e6i	0.436437	+	0.522996i
$562$		0			0
$563$	6.47772e6			0.861294			0.430647	−	0.902521i	$-0.358285\pi$
$563$	6.47772e6			0.861294			0.430647	+	0.902521i	$0.358285\pi$
$564$		0			0
$565$	5.47901e6			0.722073
$566$		0			0
$567$	791301.			0.103367
$568$		0			0
$569$		−	1.11982e7i		−	1.45000i	−0.688750	−	0.724998i	$-0.741840\pi$
$569$		−	1.11982e7i		−	1.45000i	0.688750	−	0.724998i	$-0.258160\pi$
$570$		0			0
$571$	−3.76253e6			−0.482936			−0.241468	−	0.970409i	$-0.577629\pi$
$571$	−3.76253e6			−0.482936			−0.241468	+	0.970409i	$0.577629\pi$
$572$		0			0
$573$	5.84735e6			0.744000
$574$		0			0
$575$			9.77559e6i			1.23303i
$576$		0			0
$577$	−4.28649e6			−0.535998			−0.267999	−	0.963419i	$-0.586362\pi$
$577$	−4.28649e6			−0.535998			−0.267999	+	0.963419i	$0.586362\pi$
$578$		0			0
$579$	−581595.			−0.0720982
$580$		0			0
$581$	−4.40267e6			−0.541098
$582$		0			0
$583$	−2.64168e6	−	3.16561e6i	−0.321891	−	0.385732i
$584$		0			0
$585$			3.22124e6i			0.389165i
$586$		0			0
$587$			5.98605e6i			0.717043i	0.933521	+	0.358522i	$0.116719\pi$
$587$			5.98605e6i			0.717043i	−0.933521	+	0.358522i	$0.883281\pi$
$588$		0			0
$589$			5.58538e6i			0.663383i
$590$		0			0
$591$	5.78775e6			0.681618
$592$		0			0
$593$			5.46869e6i			0.638626i	0.947649	+	0.319313i	$0.103452\pi$
$593$			5.46869e6i			0.638626i	−0.947649	+	0.319313i	$0.896548\pi$
$594$		0			0
$595$		−	5.81426e6i		−	0.673290i
$596$		0			0
$597$	8.30374e6			0.953537
$598$		0			0
$599$			8.60665e6i			0.980092i	0.871697	+	0.490046i	$0.163020\pi$
$599$			8.60665e6i			0.980092i	−0.871697	+	0.490046i	$0.836980\pi$
$600$		0			0
$601$		−	1.27483e7i		−	1.43968i	−0.694139	−	0.719841i	$-0.744215\pi$
$601$		−	1.27483e7i		−	1.43968i	0.694139	−	0.719841i	$-0.255785\pi$
$602$		0			0
$603$			2.07563e6i			0.232464i
$604$		0			0
$605$	988445.	−	5.43343e6i	0.109790	−	0.603511i
$606$		0			0
$607$	−1.65173e7			−1.81956			−0.909781	−	0.415088i	$-0.863751\pi$
$607$	−1.65173e7			−1.81956			−0.909781	+	0.415088i	$0.863751\pi$
$608$		0			0
$609$	−2.54039e6			−0.277560
$610$		0			0
$611$	−2.98060e7			−3.22999
$612$		0			0
$613$		−	5.24291e6i		−	0.563535i	−0.959483	−	0.281768i	$-0.909079\pi$
$613$		−	5.24291e6i		−	0.563535i	0.959483	−	0.281768i	$-0.0909207\pi$
$614$		0			0
$615$	1.85626e6			0.197903
$616$		0			0
$617$	−8.23243e6			−0.870592			−0.435296	−	0.900287i	$-0.643356\pi$
$617$	−8.23243e6			−0.870592			−0.435296	+	0.900287i	$0.643356\pi$
$618$		0			0
$619$			929633.i			0.0975180i	0.998811	+	0.0487590i	$0.0155266\pi$
$619$			929633.i			0.0975180i	−0.998811	+	0.0487590i	$0.984473\pi$
$620$		0			0
$621$	3.65621e6			0.380454
$622$		0			0
$623$	−3.89334e6			−0.401885
$624$		0			0
$625$	124486.			0.0127473
$626$		0			0
$627$	−2.50591e6	+	2.09116e6i	−0.254563	+	0.212432i
$628$		0			0
$629$		−	8.80042e6i		−	0.886905i
$630$		0			0
$631$		−	4.64085e6i		−	0.464007i	−0.972715	−	0.232003i	$-0.925472\pi$
$631$		−	4.64085e6i		−	0.464007i	0.972715	−	0.232003i	$-0.0745281\pi$
$632$		0			0
$633$		−	6.23691e6i		−	0.618671i
$634$		0			0
$635$	9.51953e6			0.936874
$636$		0			0
$637$		−	2.62218e6i		−	0.256043i
$638$		0			0
$639$		−	1.55608e6i		−	0.150758i
$640$		0			0
$641$	−4.16366e6			−0.400249			−0.200124	−	0.979771i	$-0.564135\pi$
$641$	−4.16366e6			−0.400249			−0.200124	+	0.979771i	$0.564135\pi$
$642$		0			0
$643$			1.62064e7i			1.54582i	0.634517	+	0.772909i	$0.281199\pi$
$643$			1.62064e7i			1.54582i	−0.634517	+	0.772909i	$0.718801\pi$
$644$		0			0
$645$			1.02790e6i			0.0972867i
$646$		0			0
$647$			6.37433e6i			0.598651i	0.954151	+	0.299326i	$0.0967616\pi$
$647$			6.37433e6i			0.598651i	−0.954151	+	0.299326i	$0.903238\pi$
$648$		0			0
$649$	−3.02438e6	+	2.52382e6i	−0.281854	+	0.235205i
$650$		0			0
$651$	−6.70911e6			−0.620458
$652$		0			0
$653$	9.63555e6			0.884288			0.442144	−	0.896944i	$-0.354218\pi$
$653$	9.63555e6			0.884288			0.442144	+	0.896944i	$0.354218\pi$
$654$		0			0
$655$	3.74001e6			0.340620
$656$		0			0
$657$			4.96541e6i			0.448789i
$658$		0			0
$659$	−8.09896e6			−0.726467			−0.363234	−	0.931698i	$-0.618327\pi$
$659$	−8.09896e6			−0.726467			−0.363234	+	0.931698i	$0.618327\pi$
$660$		0			0
$661$	4.16488e6			0.370765			0.185383	−	0.982666i	$-0.440648\pi$
$661$	4.16488e6			0.370765			0.185383	+	0.982666i	$0.440648\pi$
$662$		0			0
$663$			1.46738e7i			1.29646i
$664$		0			0
$665$	3.73727e6			0.327718
$666$		0			0
$667$	−1.17379e7			−1.02158
$668$		0			0
$669$	4.04517e6			0.349439
$670$		0			0
$671$	−531285.	+	443353.i	−0.0455534	+	0.0380140i
$672$		0			0
$673$		−	1.82243e7i		−	1.55100i	−0.631346	−	0.775501i	$-0.717497\pi$
$673$		−	1.82243e7i		−	1.55100i	0.631346	−	0.775501i	$-0.282503\pi$
$674$		0			0
$675$		−	1.42091e6i		−	0.120035i
$676$		0			0
$677$			2.12559e7i			1.78241i	0.453599	+	0.891206i	$0.350140\pi$
$677$			2.12559e7i			1.78241i	−0.453599	+	0.891206i	$0.649860\pi$
$678$		0			0
$679$	−5.73052e6			−0.477001
$680$		0			0
$681$			1.07467e7i			0.887985i
$682$		0			0
$683$		−	612303.i		−	0.0502244i	−0.999685	−	0.0251122i	$-0.992006\pi$
$683$		−	612303.i		−	0.0502244i	0.999685	−	0.0251122i	$-0.00799430\pi$
$684$		0			0
$685$	−2.49910e6			−0.203497
$686$		0			0
$687$			1.07523e7i			0.869183i
$688$		0			0
$689$		−	1.19150e7i		−	0.956196i
$690$		0			0
$691$			1.95016e7i			1.55372i	0.629670	+	0.776862i	$0.283190\pi$
$691$			1.95016e7i			1.55372i	−0.629670	+	0.776862i	$0.716810\pi$
$692$		0			0
$693$	−2.51188e6	−	3.01007e6i	−0.198686	−	0.238091i
$694$		0			0
$695$	−1.24507e7			−0.977763
$696$		0			0
$697$	8.45589e6			0.659292
$698$		0			0
$699$	264912.			0.0205073
$700$		0			0
$701$			1.46232e7i			1.12395i	0.827155	+	0.561974i	$0.189958\pi$
$701$			1.46232e7i			1.12395i	−0.827155	+	0.561974i	$0.810042\pi$
$702$		0			0
$703$	5.65670e6			0.431693
$704$		0			0
$705$	−7.93175e6			−0.601030
$706$		0			0
$707$			1.97082e7i			1.48285i
$708$		0			0
$709$	−5.29554e6			−0.395635			−0.197817	−	0.980239i	$-0.563385\pi$
$709$	−5.29554e6			−0.395635			−0.197817	+	0.980239i	$0.563385\pi$
$710$		0			0
$711$	5.84052e6			0.433289
$712$		0			0
$713$	−3.09994e7			−2.28365
$714$		0			0
$715$	1.22535e7	−	1.02254e7i	0.896383	−	0.748026i
$716$		0			0
$717$			5.08924e6i			0.369705i
$718$		0			0
$719$		−	1.27161e7i		−	0.917340i	−0.888607	−	0.458670i	$-0.848326\pi$
$719$		−	1.27161e7i		−	0.917340i	0.888607	−	0.458670i	$-0.151674\pi$
$720$		0			0
$721$		−	6.77138e6i		−	0.485109i
$722$		0			0
$723$	−6.27217e6			−0.446244
$724$		0			0
$725$			4.56169e6i			0.322315i
$726$		0			0
$727$			2.32321e7i			1.63024i	0.579290	+	0.815122i	$0.303330\pi$
$727$			2.32321e7i			1.63024i	−0.579290	+	0.815122i	$0.696670\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$			4.68244e6i			0.324100i
$732$		0			0
$733$		−	1.06396e7i		−	0.731420i	−0.930729	−	0.365710i	$-0.880826\pi$
$733$		−	1.06396e7i		−	0.731420i	0.930729	−	0.365710i	$-0.119174\pi$
$734$		0			0
$735$		−	697793.i		−	0.0476440i
$736$		0			0
$737$	7.89559e6	−	6.58882e6i	0.535447	−	0.446827i
$738$		0			0
$739$	8.84091e6			0.595505			0.297753	−	0.954643i	$-0.403763\pi$
$739$	8.84091e6			0.595505			0.297753	+	0.954643i	$0.403763\pi$
$740$		0			0
$741$	−9.43198e6			−0.631041
$742$		0			0
$743$	2.38697e6			0.158626			0.0793130	−	0.996850i	$-0.474727\pi$
$743$	2.38697e6			0.158626			0.0793130	+	0.996850i	$0.474727\pi$
$744$		0			0
$745$		−	6.38643e6i		−	0.421568i
$746$		0			0
$747$	2.95685e6			0.193878
$748$		0			0
$749$	−6.82571e6			−0.444573
$750$		0			0
$751$		−	5.21797e6i		−	0.337599i	−0.985650	−	0.168800i	$-0.946011\pi$
$751$		−	5.21797e6i		−	0.337599i	0.985650	−	0.168800i	$-0.0539891\pi$
$752$		0			0
$753$	8.11941e6			0.521839
$754$		0			0
$755$	4.63473e6			0.295908
$756$		0			0
$757$	−1.21994e7			−0.773746			−0.386873	−	0.922133i	$-0.626445\pi$
$757$	−1.21994e7			−0.773746			−0.386873	+	0.922133i	$0.626445\pi$
$758$		0			0
$759$	−1.16062e7	−	1.39080e7i	−0.731281	−	0.876317i
$760$		0			0
$761$		−	2.57962e7i		−	1.61471i	−0.590069	−	0.807353i	$-0.700900\pi$
$761$		−	2.57962e7i		−	1.61471i	0.590069	−	0.807353i	$-0.299100\pi$
$762$		0			0
$763$		−	2.55453e7i		−	1.58855i
$764$		0			0
$765$			3.90488e6i			0.241243i
$766$		0			0
$767$	−1.13835e7			−0.698692
$768$		0			0
$769$		−	2.25672e6i		−	0.137614i	−0.997630	−	0.0688068i	$-0.978081\pi$
$769$		−	2.25672e6i		−	0.137614i	0.997630	−	0.0688068i	$-0.0219192\pi$
$770$		0			0
$771$			1.76675e6i			0.107038i
$772$		0			0
$773$	9.42539e6			0.567349			0.283675	−	0.958921i	$-0.408446\pi$
$773$	9.42539e6			0.567349			0.283675	+	0.958921i	$0.408446\pi$
$774$		0			0
$775$			1.20473e7i			0.720504i
$776$		0			0
$777$			6.79477e6i			0.403759i
$778$		0			0
$779$			5.43524e6i			0.320904i
$780$		0			0
$781$	−5.91927e6	+	4.93959e6i	−0.347249	+	0.289777i
$782$		0			0
$783$	1.70613e6			0.0994509
$784$		0			0
$785$	−1.30732e7			−0.757194
$786$		0			0
$787$	8.37964e6			0.482268			0.241134	−	0.970492i	$-0.422481\pi$
$787$	8.37964e6			0.482268			0.241134	+	0.970492i	$0.422481\pi$
$788$		0			0
$789$		−	7.63659e6i		−	0.436724i
$790$		0			0
$791$	−1.92705e7			−1.09510
$792$		0			0
$793$	−1.99970e6			−0.112923
$794$		0			0
$795$		−	3.17073e6i		−	0.177927i
$796$		0			0
$797$	2.06198e7			1.14984			0.574922	−	0.818208i	$-0.305032\pi$
$797$	2.06198e7			1.14984			0.574922	+	0.818208i	$0.305032\pi$
$798$		0			0
$799$	−3.61317e7			−2.00227
$800$		0			0
$801$	2.61478e6			0.143997
$802$		0			0
$803$	1.88882e7	−	1.57621e7i	1.03372	−	0.862630i
$804$		0			0
$805$			2.07422e7i			1.12815i
$806$		0			0
$807$		−	6.14559e6i		−	0.332185i
$808$		0			0
$809$		−	7.95202e6i		−	0.427175i	−0.976924	−	0.213588i	$-0.931485\pi$
$809$		−	7.95202e6i		−	0.427175i	0.976924	−	0.213588i	$-0.0685149\pi$
$810$		0			0
$811$	3.93076e6			0.209858			0.104929	−	0.994480i	$-0.466539\pi$
$811$	3.93076e6			0.209858			0.104929	+	0.994480i	$0.466539\pi$
$812$		0			0
$813$		−	1.63353e7i		−	0.866763i
$814$		0			0
$815$		−	398940.i		−	0.0210385i
$816$		0			0
$817$	−3.00976e6			−0.157753
$818$		0			0
$819$		−	1.13296e7i		−	0.590208i
$820$		0			0
$821$			3.62908e7i			1.87905i	0.342479	+	0.939525i	$0.388733\pi$
$821$			3.62908e7i			1.87905i	−0.342479	+	0.939525i	$0.611267\pi$
$822$		0			0
$823$		−	3.73518e6i		−	0.192226i	−0.995370	−	0.0961129i	$-0.969359\pi$
$823$		−	3.73518e6i		−	0.192226i	0.995370	−	0.0961129i	$-0.0306410\pi$
$824$		0			0
$825$	−5.40509e6	+	4.51051e6i	−0.276483	+	0.230723i
$826$		0			0
$827$	−8.84646e6			−0.449786			−0.224893	−	0.974383i	$-0.572203\pi$
$827$	−8.84646e6			−0.449786			−0.224893	+	0.974383i	$0.572203\pi$
$828$		0			0
$829$	−2.32800e7			−1.17651			−0.588257	−	0.808674i	$-0.700186\pi$
$829$	−2.32800e7			−1.17651			−0.588257	+	0.808674i	$0.700186\pi$
$830$		0			0
$831$	−7.91832e6			−0.397769
$832$		0			0
$833$		−	3.17868e6i		−	0.158721i
$834$		0			0
$835$	2.66811e6			0.132431
$836$		0			0
$837$	4.50587e6			0.222313
$838$		0			0
$839$		−	1.34787e7i		−	0.661061i	−0.943795	−	0.330531i	$-0.892772\pi$
$839$		−	1.34787e7i		−	0.661061i	0.943795	−	0.330531i	$-0.107228\pi$
$840$		0			0
$841$	1.50338e7			0.732957
$842$		0			0
$843$	−4.03220e6			−0.195422
$844$		0			0
$845$	3.33888e7			1.60864
$846$		0			0
$847$	−3.47651e6	+	1.91102e7i	−0.166508	+	0.915285i
$848$		0			0
$849$		−	2.59512e6i		−	0.123563i
$850$		0			0
$851$			3.13953e7i			1.48607i
$852$		0			0
$853$		−	2.12303e7i		−	0.999043i	−0.866301	−	0.499522i	$-0.833509\pi$
$853$		−	2.12303e7i		−	0.999043i	0.866301	−	0.499522i	$-0.166491\pi$
$854$		0			0
$855$	−2.50996e6			−0.117423
$856$		0			0
$857$		−	4.81462e6i		−	0.223929i	−0.993712	−	0.111964i	$-0.964286\pi$
$857$		−	4.81462e6i		−	0.223929i	0.993712	−	0.111964i	$-0.0357142\pi$
$858$		0			0
$859$		−	2.16845e7i		−	1.00269i	−0.865248	−	0.501344i	$-0.832839\pi$
$859$		−	2.16845e7i		−	1.00269i	0.865248	−	0.501344i	$-0.167161\pi$
$860$		0			0
$861$	−6.52876e6			−0.300139
$862$		0			0
$863$			7.59435e6i			0.347107i	0.984824	+	0.173554i	$0.0555250\pi$
$863$			7.59435e6i			0.347107i	−0.984824	+	0.173554i	$0.944475\pi$
$864$		0			0
$865$			2.39593e7i			1.08876i
$866$		0			0
$867$			5.00932e6i			0.226324i
$868$		0			0
$869$	−1.85400e7	−	2.22171e7i	−0.832837	−	0.998015i
$870$		0			0
$871$	2.97182e7			1.32733
$872$		0			0
$873$	3.84864e6			0.170912
$874$		0			0
$875$	2.09852e7			0.926602
$876$		0			0
$877$			4.14917e7i			1.82164i	0.412804	+	0.910820i	$0.364550\pi$
$877$			4.14917e7i			1.82164i	−0.412804	+	0.910820i	$0.635450\pi$
$878$		0			0
$879$	1.90187e7			0.830251
$880$		0			0
$881$	−3.25913e7			−1.41469			−0.707346	−	0.706867i	$-0.750108\pi$
$881$	−3.25913e7			−1.41469			−0.707346	+	0.706867i	$0.750108\pi$
$882$		0			0
$883$			1.53548e7i			0.662739i	0.943501	+	0.331369i	$0.107511\pi$
$883$			1.53548e7i			0.662739i	−0.943501	+	0.331369i	$0.892489\pi$
$884$		0			0
$885$	−3.02928e6			−0.130011
$886$		0			0
$887$	3.90024e7			1.66450			0.832248	−	0.554404i	$-0.187054\pi$
$887$	3.90024e7			1.66450			0.832248	+	0.554404i	$0.187054\pi$
$888$		0			0
$889$	−3.34816e7			−1.42086
$890$		0			0
$891$	1.68699e6	+	2.02158e6i	0.0711900	+	0.0853092i
$892$		0			0
$893$		−	2.32246e7i		−	0.974585i
$894$		0			0
$895$			9.86278e6i			0.411568i
$896$		0			0
$897$		−	5.23485e7i		−	2.17231i
$898$		0			0
$899$	−1.44656e7			−0.596949
$900$		0			0
$901$		−	1.44437e7i		−	0.592745i
$902$		0			0
$903$		−	3.61530e6i		−	0.147545i
$904$		0			0
$905$	1.46879e7			0.596127
$906$		0			0
$907$			6.44976e6i			0.260331i	0.991492	+	0.130165i	$0.0415508\pi$
$907$			6.44976e6i			0.260331i	−0.991492	+	0.130165i	$0.958449\pi$
$908$		0			0
$909$		−	1.32361e7i		−	0.531313i
$910$		0			0
$911$		−	2.40055e7i		−	0.958329i	−0.877725	−	0.479164i	$-0.840940\pi$
$911$		−	2.40055e7i		−	0.958329i	0.877725	−	0.479164i	$-0.159060\pi$
$912$		0			0
$913$	−9.38615e6	−	1.12477e7i	−0.372658	−	0.446568i
$914$		0			0
$915$	−532145.			−0.0210125
$916$		0			0
$917$	−1.31542e7			−0.516584
$918$		0			0
$919$	1.97748e7			0.772367			0.386184	−	0.922422i	$-0.373793\pi$
$919$	1.97748e7			0.772367			0.386184	+	0.922422i	$0.373793\pi$
$920$		0			0
$921$			1.49079e7i			0.579117i
$922$		0			0
$923$	−2.22795e7			−0.860800
$924$		0			0
$925$	1.22012e7			0.468864
$926$		0			0
$927$			4.54769e6i			0.173817i
$928$		0			0
$929$	4.62398e7			1.75783			0.878915	−	0.476979i	$-0.158268\pi$
$929$	4.62398e7			1.75783			0.878915	+	0.476979i	$0.158268\pi$
$930$		0			0
$931$	2.04318e6			0.0772559
$932$		0			0
$933$	5.06441e6			0.190469
$934$		0			0
$935$	1.48540e7	−	1.23956e7i	0.555667	−	0.463700i
$936$		0			0
$937$			3.11327e7i			1.15842i	0.815177	+	0.579212i	$0.196640\pi$
$937$			3.11327e7i			1.15842i	−0.815177	+	0.579212i	$0.803360\pi$
$938$		0			0
$939$			2.31859e7i			0.858142i
$940$		0			0
$941$			6.78461e6i			0.249776i	0.992171	+	0.124888i	$0.0398572\pi$
$941$			6.78461e6i			0.249776i	−0.992171	+	0.124888i	$0.960143\pi$
$942$		0			0
$943$	−3.01662e7			−1.10469
$944$		0			0
$945$		−	3.01495e6i		−	0.109825i
$946$		0			0
$947$			2.50870e7i			0.909020i	0.890742	+	0.454510i	$0.150186\pi$
$947$			2.50870e7i			0.909020i	−0.890742	+	0.454510i	$0.849814\pi$
$948$		0			0
$949$	7.10933e7			2.56250
$950$		0			0
$951$		−	2.24016e7i		−	0.803208i
$952$		0			0
$953$			4.06241e7i			1.44894i	0.689304	+	0.724472i	$0.257916\pi$
$953$			4.06241e7i			1.44894i	−0.689304	+	0.724472i	$0.742084\pi$
$954$		0			0
$955$		−	2.22791e7i		−	0.790476i
$956$		0			0
$957$	−5.41591e6	−	6.49005e6i	−0.191158	−	0.229070i
$958$		0			0
$959$	8.78972e6			0.308623
$960$		0			0
$961$	−9.57419e6			−0.334421
$962$		0			0
$963$	4.58418e6			0.159293
$964$		0			0
$965$			2.21594e6i			0.0766020i
$966$		0			0
$967$	−6.77736e6			−0.233074			−0.116537	−	0.993186i	$-0.537179\pi$
$967$	−6.77736e6			−0.233074			−0.116537	+	0.993186i	$0.537179\pi$
$968$		0			0
$969$	−1.14337e7			−0.391181
$970$		0			0
$971$		−	4.53087e7i		−	1.54217i	−0.636730	−	0.771087i	$-0.719713\pi$
$971$		−	4.53087e7i		−	1.54217i	0.636730	−	0.771087i	$-0.280287\pi$
$972$		0			0
$973$	4.37912e7			1.48287
$974$		0			0
$975$	−2.03442e7			−0.685376
$976$		0			0
$977$	2.78797e7			0.934439			0.467220	−	0.884141i	$-0.345256\pi$
$977$	2.78797e7			0.934439			0.467220	+	0.884141i	$0.345256\pi$
$978$		0			0
$979$	−8.30029e6	−	9.94650e6i	−0.276781	−	0.331676i
$980$		0			0
$981$			1.71563e7i			0.569184i
$982$		0			0
$983$		−	1.81592e7i		−	0.599394i	−0.954034	−	0.299697i	$-0.903114\pi$
$983$		−	1.81592e7i		−	0.599394i	0.954034	−	0.299697i	$-0.0968856\pi$
$984$		0			0
$985$		−	2.20520e7i		−	0.724198i
$986$		0			0
$987$	2.78972e7			0.911522
$988$		0			0
$989$		−	1.67045e7i		−	0.543053i
$990$		0			0
$991$			1.66510e7i			0.538587i	0.963058	+	0.269294i	$0.0867902\pi$
$991$			1.66510e7i			0.538587i	−0.963058	+	0.269294i	$0.913210\pi$
$992$		0			0
$993$	−3.33959e7			−1.07478
$994$		0			0
$995$		−	3.16382e7i		−	1.01310i
$996$		0			0
$997$		−	9.89016e6i		−	0.315112i	−0.987510	−	0.157556i	$-0.949638\pi$
$997$		−	9.89016e6i		−	0.315112i	0.987510	−	0.157556i	$-0.0503616\pi$
$998$		0			0
$999$		−	4.56340e6i		−	0.144669i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.o.b.175.1	✓	40
4.3	odd	2	inner	528.6.o.b.175.40	yes	40
11.10	odd	2	inner	528.6.o.b.175.39	yes	40
44.43	even	2	inner	528.6.o.b.175.2	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.6.o.b.175.1	✓	40	1.1	even	1	trivial
528.6.o.b.175.2	yes	40	44.43	even	2	inner
528.6.o.b.175.39	yes	40	11.10	odd	2	inner
528.6.o.b.175.40	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)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} -34.2910 q^{5} +120.607 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} -34.2910 q^{5} +120.607 q^{7} -81.0000 q^{9} +(257.124 + 308.120i) q^{11} +1159.73i q^{13} +308.619i q^{15} +1405.86i q^{17} -903.654 q^{19} -1085.46i q^{21} -5015.37i q^{23} -1949.13 q^{25} +729.000i q^{27} -2340.38i q^{29} -6180.89i q^{31} +(2773.08 - 2314.12i) q^{33} -4135.73 q^{35} -6259.81 q^{37} +10437.6 q^{39} -6014.74i q^{41} +3330.66 q^{43} +2777.57 q^{45} +25700.8i q^{47} -2261.02 q^{49} +12652.8 q^{51} -10273.9 q^{53} +(-8817.05 - 10565.8i) q^{55} +8132.89i q^{57} +9815.58i q^{59} +1724.28i q^{61} -9769.15 q^{63} -39768.4i q^{65} -25625.0i q^{67} -45138.3 q^{69} +19210.9i q^{71} -61301.4i q^{73} +17542.1i q^{75} +(31010.9 + 37161.4i) q^{77} -72105.2 q^{79} +6561.00 q^{81} -36504.3 q^{83} -48208.4i q^{85} -21063.4 q^{87} -32281.2 q^{89} +139872. i q^{91} -55628.0 q^{93} +30987.2 q^{95} -47514.1 q^{97} +(-20827.1 - 24957.7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more