LMFDB - Embedded newform 528.6.o.b.175.16

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			175.16
Character	$\chi$	$=$	528.175
Dual form			528.6.o.b.175.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +1.95371 q^{5} -158.286 q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} +1.95371 q^{5} -158.286 q^{7} -81.0000 q^{9} +(391.215 + 89.4537i) q^{11} +135.638i q^{13} +17.5834i q^{15} -602.426i q^{17} -456.875 q^{19} -1424.58i q^{21} -3406.92i q^{23} -3121.18 q^{25} -729.000i q^{27} +3322.64i q^{29} +3733.95i q^{31} +(-805.083 + 3520.93i) q^{33} -309.245 q^{35} -9308.84 q^{37} -1220.74 q^{39} -13951.4i q^{41} +19806.5 q^{43} -158.250 q^{45} -13946.3i q^{47} +8247.55 q^{49} +5421.83 q^{51} +34164.1 q^{53} +(764.319 + 174.766i) q^{55} -4111.88i q^{57} +24786.9i q^{59} -20536.1i q^{61} +12821.2 q^{63} +264.997i q^{65} +71140.8i q^{67} +30662.3 q^{69} -70020.9i q^{71} +31351.2i q^{73} -28090.6i q^{75} +(-61923.9 - 14159.3i) q^{77} +54269.4 q^{79} +6561.00 q^{81} +74040.1 q^{83} -1176.96i q^{85} -29903.8 q^{87} +8217.42 q^{89} -21469.7i q^{91} -33605.5 q^{93} -892.601 q^{95} -113366. q^{97} +(-31688.4 - 7245.75i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.95371			0.0349490			0.0174745	−	0.999847i	$-0.494437\pi$
$5$	1.95371			0.0349490			0.0174745	+	0.999847i	$0.494437\pi$
$6$		0			0
$7$	−158.286			−1.22095			−0.610475	−	0.792035i	$-0.709022\pi$
$7$	−158.286			−1.22095			−0.610475	+	0.792035i	$0.709022\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	391.215	+	89.4537i	0.974841	+	0.222903i
$11$	391.215	+	89.4537i	0.974841	+	0.222903i
$12$		0			0
$13$			135.638i			0.222599i	0.993787	+	0.111300i	$0.0355013\pi$
$13$			135.638i			0.222599i	−0.993787	+	0.111300i	$0.964499\pi$
$14$		0			0
$15$			17.5834i			0.0201778i
$16$		0			0
$17$		−	602.426i		−	0.505570i	−0.967522	−	0.252785i	$-0.918653\pi$
$17$		−	602.426i		−	0.505570i	0.967522	−	0.252785i	$-0.0813465\pi$
$18$		0			0
$19$	−456.875			−0.290345			−0.145172	−	0.989406i	$-0.546374\pi$
$19$	−456.875			−0.290345			−0.145172	+	0.989406i	$0.546374\pi$
$20$		0			0
$21$		−	1424.58i		−	0.704916i
$22$		0			0
$23$		−	3406.92i		−	1.34289i	−0.741052	−	0.671447i	$-0.765673\pi$
$23$		−	3406.92i		−	1.34289i	0.741052	−	0.671447i	$-0.234327\pi$
$24$		0			0
$25$	−3121.18			−0.998779
$26$		0			0
$27$		−	729.000i		−	0.192450i
$28$		0			0
$29$			3322.64i			0.733649i	0.930290	+	0.366825i	$0.119555\pi$
$29$			3322.64i			0.733649i	−0.930290	+	0.366825i	$0.880445\pi$
$30$		0			0
$31$			3733.95i			0.697853i	0.937150	+	0.348927i	$0.113454\pi$
$31$			3733.95i			0.697853i	−0.937150	+	0.348927i	$0.886546\pi$
$32$		0			0
$33$	−805.083	+	3520.93i	−0.128693	+	0.562824i
$34$		0			0
$35$	−309.245			−0.0426710
$36$		0			0
$37$	−9308.84			−1.11787			−0.558935	−	0.829212i	$-0.688790\pi$
$37$	−9308.84			−1.11787			−0.558935	+	0.829212i	$0.688790\pi$
$38$		0			0
$39$	−1220.74			−0.128518
$40$		0			0
$41$		−	13951.4i		−	1.29616i	−0.761574	−	0.648078i	$-0.775573\pi$
$41$		−	13951.4i		−	1.29616i	0.761574	−	0.648078i	$-0.224427\pi$
$42$		0			0
$43$	19806.5			1.63357			0.816785	−	0.576942i	$-0.195754\pi$
$43$	19806.5			1.63357			0.816785	+	0.576942i	$0.195754\pi$
$44$		0			0
$45$	−158.250			−0.0116497
$46$		0			0
$47$		−	13946.3i		−	0.920903i	−0.887685	−	0.460451i	$-0.847688\pi$
$47$		−	13946.3i		−	0.920903i	0.887685	−	0.460451i	$-0.152312\pi$
$48$		0			0
$49$	8247.55			0.490721
$50$		0			0
$51$	5421.83			0.291891
$52$		0			0
$53$	34164.1			1.67063			0.835315	−	0.549771i	$-0.185285\pi$
$53$	34164.1			1.67063			0.835315	+	0.549771i	$0.185285\pi$
$54$		0			0
$55$	764.319	+	174.766i	0.0340697	+	0.00779024i
$56$		0			0
$57$		−	4111.88i		−	0.167630i
$58$		0			0
$59$			24786.9i			0.927026i	0.886090	+	0.463513i	$0.153411\pi$
$59$			24786.9i			0.927026i	−0.886090	+	0.463513i	$0.846589\pi$
$60$		0			0
$61$		−	20536.1i		−	0.706632i	−0.935504	−	0.353316i	$-0.885054\pi$
$61$		−	20536.1i		−	0.706632i	0.935504	−	0.353316i	$-0.114946\pi$
$62$		0			0
$63$	12821.2			0.406984
$64$		0			0
$65$			264.997i			0.00777962i
$66$		0			0
$67$			71140.8i			1.93612i	0.250726	+	0.968058i	$0.419331\pi$
$67$			71140.8i			1.93612i	−0.250726	+	0.968058i	$0.580669\pi$
$68$		0			0
$69$	30662.3			0.775321
$70$		0			0
$71$		−	70020.9i		−	1.64847i	−0.566245	−	0.824237i	$-0.691604\pi$
$71$		−	70020.9i		−	1.64847i	0.566245	−	0.824237i	$-0.308396\pi$
$72$		0			0
$73$			31351.2i			0.688568i	0.938865	+	0.344284i	$0.111878\pi$
$73$			31351.2i			0.688568i	−0.938865	+	0.344284i	$0.888122\pi$
$74$		0			0
$75$		−	28090.6i		−	0.576645i
$76$		0			0
$77$	−61923.9	−	14159.3i	−1.19023	−	0.272154i
$78$		0			0
$79$	54269.4			0.978335			0.489167	−	0.872190i	$-0.337301\pi$
$79$	54269.4			0.978335			0.489167	+	0.872190i	$0.337301\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$	74040.1			1.17970			0.589850	−	0.807513i	$-0.299187\pi$
$83$	74040.1			1.17970			0.589850	+	0.807513i	$0.299187\pi$
$84$		0			0
$85$		−	1176.96i		−	0.0176692i
$86$		0			0
$87$	−29903.8			−0.423573
$88$		0			0
$89$	8217.42			0.109967			0.0549833	−	0.998487i	$-0.482489\pi$
$89$	8217.42			0.109967			0.0549833	+	0.998487i	$0.482489\pi$
$90$		0			0
$91$		−	21469.7i		−	0.271783i
$92$		0			0
$93$	−33605.5			−0.402906
$94$		0			0
$95$	−892.601			−0.0101472
$96$		0			0
$97$	−113366.			−1.22336			−0.611680	−	0.791106i	$-0.709506\pi$
$97$	−113366.			−1.22336			−0.611680	+	0.791106i	$0.709506\pi$
$98$		0			0
$99$	−31688.4	−	7245.75i	−0.324947	−	0.0743011i
$100$		0			0
$101$			202342.i			1.97370i	0.161631	+	0.986851i	$0.448325\pi$
$101$			202342.i			1.97370i	−0.161631	+	0.986851i	$0.551675\pi$
$102$		0			0
$103$		−	96916.7i		−	0.900131i	−0.892996	−	0.450065i	$-0.851401\pi$
$103$		−	96916.7i		−	0.900131i	0.892996	−	0.450065i	$-0.148599\pi$
$104$		0			0
$105$		−	2783.21i		−	0.0246361i
$106$		0			0
$107$	116209.			0.981252			0.490626	−	0.871370i	$-0.336768\pi$
$107$	116209.			0.981252			0.490626	+	0.871370i	$0.336768\pi$
$108$		0			0
$109$			103450.i			0.833997i	0.908907	+	0.416998i	$0.136918\pi$
$109$			103450.i			0.833997i	−0.908907	+	0.416998i	$0.863082\pi$
$110$		0			0
$111$		−	83779.5i		−	0.645402i
$112$		0			0
$113$	44018.8			0.324296			0.162148	−	0.986766i	$-0.448158\pi$
$113$	44018.8			0.324296			0.162148	+	0.986766i	$0.448158\pi$
$114$		0			0
$115$		−	6656.12i		−	0.0469328i
$116$		0			0
$117$		−	10986.7i		−	0.0741998i
$118$		0			0
$119$			95355.7i			0.617276i
$120$		0			0
$121$	145047.	+	69991.2i	0.900628	+	0.434590i
$122$		0			0
$123$	125562.			0.748336
$124$		0			0
$125$	−12203.2			−0.0698553
$126$		0			0
$127$	212159.			1.16722			0.583609	−	0.812035i	$-0.301640\pi$
$127$	212159.			1.16722			0.583609	+	0.812035i	$0.301640\pi$
$128$		0			0
$129$			178259.i			0.943142i
$130$		0			0
$131$	59902.6			0.304977			0.152489	−	0.988305i	$-0.451271\pi$
$131$	59902.6			0.304977			0.152489	+	0.988305i	$0.451271\pi$
$132$		0			0
$133$	72317.1			0.354496
$134$		0			0
$135$		−	1424.25i		−	0.00672594i
$136$		0			0
$137$	119739.			0.545048			0.272524	−	0.962149i	$-0.412142\pi$
$137$	119739.			0.545048			0.272524	+	0.962149i	$0.412142\pi$
$138$		0			0
$139$	404876.			1.77740			0.888699	−	0.458490i	$-0.151610\pi$
$139$	404876.			1.77740			0.888699	+	0.458490i	$0.151610\pi$
$140$		0			0
$141$	125517.			0.531683
$142$		0			0
$143$	−12133.3	+	53063.7i	−0.0496181	+	0.216999i
$144$		0			0
$145$			6491.47i			0.0256403i
$146$		0			0
$147$			74227.9i			0.283318i
$148$		0			0
$149$			35281.7i			0.130192i	0.997879	+	0.0650959i	$0.0207353\pi$
$149$			35281.7i			0.130192i	−0.997879	+	0.0650959i	$0.979265\pi$
$150$		0			0
$151$	−344527.			−1.22965			−0.614825	−	0.788664i	$-0.710773\pi$
$151$	−344527.			−1.22965			−0.614825	+	0.788664i	$0.710773\pi$
$152$		0			0
$153$			48796.5i			0.168523i
$154$		0			0
$155$			7295.04i			0.0243893i
$156$		0			0
$157$	179355.			0.580716			0.290358	−	0.956918i	$-0.406226\pi$
$157$	179355.			0.580716			0.290358	+	0.956918i	$0.406226\pi$
$158$		0			0
$159$			307477.i			0.964539i
$160$		0			0
$161$			539268.i			1.63961i
$162$		0			0
$163$		−	379337.i		−	1.11829i	−0.829068	−	0.559147i	$-0.811129\pi$
$163$		−	379337.i		−	1.11829i	0.829068	−	0.559147i	$-0.188871\pi$
$164$		0			0
$165$	−1572.90	+	6878.87i	−0.00449770	+	0.0196701i
$166$		0			0
$167$	595756.			1.65302			0.826508	−	0.562924i	$-0.190324\pi$
$167$	595756.			1.65302			0.826508	+	0.562924i	$0.190324\pi$
$168$		0			0
$169$	352895.			0.950450
$170$		0			0
$171$	37006.9			0.0967815
$172$		0			0
$173$			76224.9i			0.193634i	0.995302	+	0.0968170i	$0.0308662\pi$
$173$			76224.9i			0.193634i	−0.995302	+	0.0968170i	$0.969134\pi$
$174$		0			0
$175$	494040.			1.21946
$176$		0			0
$177$	−223082.			−0.535219
$178$		0			0
$179$		−	83638.6i		−	0.195108i	−0.995230	−	0.0975538i	$-0.968898\pi$
$179$		−	83638.6i		−	0.195108i	0.995230	−	0.0975538i	$-0.0311018\pi$
$180$		0			0
$181$	−463551.			−1.05172			−0.525862	−	0.850570i	$-0.676257\pi$
$181$	−463551.			−1.05172			−0.525862	+	0.850570i	$0.676257\pi$
$182$		0			0
$183$	184825.			0.407974
$184$		0			0
$185$	−18186.7			−0.0390684
$186$		0			0
$187$	53889.2	−	235678.i	0.112693	−	0.492850i
$188$		0			0
$189$			115391.i			0.234972i
$190$		0			0
$191$		−	100226.i		−	0.198791i	−0.995048	−	0.0993957i	$-0.968309\pi$
$191$		−	100226.i		−	0.198791i	0.995048	−	0.0993957i	$-0.0316910\pi$
$192$		0			0
$193$			437537.i			0.845515i	0.906243	+	0.422758i	$0.138938\pi$
$193$			437537.i			0.845515i	−0.906243	+	0.422758i	$0.861062\pi$
$194$		0			0
$195$	−2384.98			−0.00449156
$196$		0			0
$197$		−	121092.i		−	0.222305i	−0.993803	−	0.111152i	$-0.964546\pi$
$197$		−	121092.i		−	0.222305i	0.993803	−	0.111152i	$-0.0354542\pi$
$198$		0			0
$199$		−	579375.i		−	1.03712i	−0.855043	−	0.518558i	$-0.826469\pi$
$199$		−	579375.i		−	1.03712i	0.855043	−	0.518558i	$-0.173531\pi$
$200$		0			0
$201$	−640267.			−1.11782
$202$		0			0
$203$		−	525929.i		−	0.895750i
$204$		0			0
$205$		−	27256.9i		−	0.0452993i
$206$		0			0
$207$			275960.i			0.447632i
$208$		0			0
$209$	−178736.	−	40869.2i	−0.283040	−	0.0647187i
$210$		0			0
$211$	1.08322e6			1.67498			0.837489	−	0.546454i	$-0.184023\pi$
$211$	1.08322e6			1.67498			0.837489	+	0.546454i	$0.184023\pi$
$212$		0			0
$213$	630188.			0.951747
$214$		0			0
$215$	38696.2			0.0570916
$216$		0			0
$217$		−	591033.i		−	0.852044i
$218$		0			0
$219$	−282161.			−0.397545
$220$		0			0
$221$	81712.0			0.112540
$222$		0			0
$223$			869343.i			1.17066i	0.810797	+	0.585328i	$0.199034\pi$
$223$			869343.i			1.17066i	−0.810797	+	0.585328i	$0.800966\pi$
$224$		0			0
$225$	252816.			0.332926
$226$		0			0
$227$	1.25730e6			1.61947			0.809736	−	0.586794i	$-0.199610\pi$
$227$	1.25730e6			1.61947			0.809736	+	0.586794i	$0.199610\pi$
$228$		0			0
$229$	−598638.			−0.754354			−0.377177	−	0.926141i	$-0.623105\pi$
$229$	−598638.			−0.754354			−0.377177	+	0.926141i	$0.623105\pi$
$230$		0			0
$231$	127434.	−	557315.i	0.157128	−	0.687181i
$232$		0			0
$233$			848868.i			1.02435i	0.858880	+	0.512177i	$0.171161\pi$
$233$			848868.i			1.02435i	−0.858880	+	0.512177i	$0.828839\pi$
$234$		0			0
$235$		−	27247.0i		−	0.0321846i
$236$		0			0
$237$			488425.i			0.564842i
$238$		0			0
$239$	−521020.			−0.590010			−0.295005	−	0.955496i	$-0.595321\pi$
$239$	−521020.			−0.590010			−0.295005	+	0.955496i	$0.595321\pi$
$240$		0			0
$241$		−	803874.i		−	0.891549i	−0.895145	−	0.445775i	$-0.852928\pi$
$241$		−	803874.i		−	0.891549i	0.895145	−	0.445775i	$-0.147072\pi$
$242$		0			0
$243$			59049.0i			0.0641500i
$244$		0			0
$245$	16113.3			0.0171502
$246$		0			0
$247$		−	61969.8i		−	0.0646305i
$248$		0			0
$249$			666360.i			0.681100i
$250$		0			0
$251$			360446.i			0.361123i	0.983564	+	0.180562i	$0.0577915\pi$
$251$			360446.i			0.361123i	−0.983564	+	0.180562i	$0.942208\pi$
$252$		0			0
$253$	304761.	−	1.33284e6i	0.299336	−	1.30911i
$254$		0			0
$255$	10592.7			0.0102013
$256$		0			0
$257$	595859.			0.562744			0.281372	−	0.959599i	$-0.409211\pi$
$257$	595859.			0.562744			0.281372	+	0.959599i	$0.409211\pi$
$258$		0			0
$259$	1.47346e6			1.36486
$260$		0			0
$261$		−	269134.i		−	0.244550i
$262$		0			0
$263$	−721518.			−0.643217			−0.321609	−	0.946873i	$-0.604224\pi$
$263$	−721518.			−0.643217			−0.321609	+	0.946873i	$0.604224\pi$
$264$		0			0
$265$	66746.7			0.0583868
$266$		0			0
$267$			73956.8i			0.0634892i
$268$		0			0
$269$	−147411.			−0.124208			−0.0621038	−	0.998070i	$-0.519781\pi$
$269$	−147411.			−0.124208			−0.0621038	+	0.998070i	$0.519781\pi$
$270$		0			0
$271$	266072.			0.220077			0.110039	−	0.993927i	$-0.464903\pi$
$271$	266072.			0.220077			0.110039	+	0.993927i	$0.464903\pi$
$272$		0			0
$273$	193227.			0.156914
$274$		0			0
$275$	−1.22105e6	−	279201.i	−0.973650	−	0.222631i
$276$		0			0
$277$			45010.0i			0.0352459i	0.999845	+	0.0176230i	$0.00560986\pi$
$277$			45010.0i			0.0352459i	−0.999845	+	0.0176230i	$0.994390\pi$
$278$		0			0
$279$		−	302450.i		−	0.232618i
$280$		0			0
$281$		−	239278.i		−	0.180774i	−0.995907	−	0.0903871i	$-0.971190\pi$
$281$		−	239278.i		−	0.180774i	0.995907	−	0.0903871i	$-0.0288104\pi$
$282$		0			0
$283$	11915.3			0.00884377			0.00442188	−	0.999990i	$-0.498592\pi$
$283$	11915.3			0.00884377			0.00442188	+	0.999990i	$0.498592\pi$
$284$		0			0
$285$		−	8033.41i		−	0.00585851i
$286$		0			0
$287$			2.20831e6i			1.58254i
$288$		0			0
$289$	1.05694e6			0.744399
$290$		0			0
$291$		−	1.02030e6i		−	0.706307i
$292$		0			0
$293$		−	439907.i		−	0.299359i	−0.988735	−	0.149679i	$-0.952176\pi$
$293$		−	439907.i		−	0.299359i	0.988735	−	0.149679i	$-0.0478242\pi$
$294$		0			0
$295$			48426.3i			0.0323986i
$296$		0			0
$297$	65211.7	−	285196.i	0.0428978	−	0.187608i
$298$		0			0
$299$	462108.			0.298927
$300$		0			0
$301$	−3.13510e6			−1.99451
$302$		0			0
$303$	−1.82107e6			−1.13952
$304$		0			0
$305$		−	40121.5i		−	0.0246961i
$306$		0			0
$307$	−2.29235e6			−1.38815			−0.694073	−	0.719904i	$-0.744186\pi$
$307$	−2.29235e6			−1.38815			−0.694073	+	0.719904i	$0.744186\pi$
$308$		0			0
$309$	872250.			0.519691
$310$		0			0
$311$		−	2.10786e6i		−	1.23578i	−0.786266	−	0.617888i	$-0.787988\pi$
$311$		−	2.10786e6i		−	1.23578i	0.786266	−	0.617888i	$-0.212012\pi$
$312$		0			0
$313$	516607.			0.298057			0.149029	−	0.988833i	$-0.452385\pi$
$313$	516607.			0.298057			0.149029	+	0.988833i	$0.452385\pi$
$314$		0			0
$315$	25048.9			0.0142237
$316$		0			0
$317$	1.37598e6			0.769065			0.384533	−	0.923111i	$-0.374363\pi$
$317$	1.37598e6			0.769065			0.384533	+	0.923111i	$0.374363\pi$
$318$		0			0
$319$	−297222.	+	1.29987e6i	−0.163533	+	0.715191i
$320$		0			0
$321$			1.04588e6i			0.566526i
$322$		0			0
$323$			275233.i			0.146789i
$324$		0			0
$325$		−	423352.i		−	0.222327i
$326$		0			0
$327$	−931050.			−0.481508
$328$		0			0
$329$			2.20750e6i			1.12438i
$330$		0			0
$331$			2.10600e6i			1.05655i	0.849074	+	0.528274i	$0.177161\pi$
$331$			2.10600e6i			1.05655i	−0.849074	+	0.528274i	$0.822839\pi$
$332$		0			0
$333$	754016.			0.372623
$334$		0			0
$335$			138988.i			0.0676653i
$336$		0			0
$337$		−	930469.i		−	0.446300i	−0.974784	−	0.223150i	$-0.928366\pi$
$337$		−	930469.i		−	0.446300i	0.974784	−	0.223150i	$-0.0716340\pi$
$338$		0			0
$339$			396169.i			0.187233i
$340$		0			0
$341$	−334015.	+	1.46078e6i	−0.155554	+	0.680296i
$342$		0			0
$343$	1.35484e6			0.621805
$344$		0			0
$345$	59905.1			0.0270967
$346$		0			0
$347$	−1.13686e6			−0.506856			−0.253428	−	0.967354i	$-0.581558\pi$
$347$	−1.13686e6			−0.506856			−0.253428	+	0.967354i	$0.581558\pi$
$348$		0			0
$349$		−	4.07167e6i		−	1.78941i	−0.446662	−	0.894703i	$-0.647387\pi$
$349$		−	4.07167e6i		−	1.78941i	0.446662	−	0.894703i	$-0.352613\pi$
$350$		0			0
$351$	98880.3			0.0428393
$352$		0			0
$353$	−2.12742e6			−0.908692			−0.454346	−	0.890825i	$-0.650127\pi$
$353$	−2.12742e6			−0.908692			−0.454346	+	0.890825i	$0.650127\pi$
$354$		0			0
$355$		−	136800.i		−	0.0576125i
$356$		0			0
$357$	−858202.			−0.356385
$358$		0			0
$359$	−2.06805e6			−0.846886			−0.423443	−	0.905923i	$-0.639179\pi$
$359$	−2.06805e6			−0.846886			−0.423443	+	0.905923i	$0.639179\pi$
$360$		0			0
$361$	−2.26736e6			−0.915700
$362$		0			0
$363$	−629921.	+	1.30542e6i	−0.250911	+	0.519978i
$364$		0			0
$365$			61251.1i			0.0240648i
$366$		0			0
$367$		−	2.59155e6i		−	1.00437i	−0.864759	−	0.502186i	$-0.832529\pi$
$367$		−	2.59155e6i		−	1.00437i	0.864759	−	0.502186i	$-0.167471\pi$
$368$		0			0
$369$			1.13006e6i			0.432052i
$370$		0			0
$371$	−5.40771e6			−2.03976
$372$		0			0
$373$		−	3.81521e6i		−	1.41986i	−0.704272	−	0.709931i	$-0.748726\pi$
$373$		−	3.81521e6i		−	1.41986i	0.704272	−	0.709931i	$-0.251274\pi$
$374$		0			0
$375$		−	109829.i		−	0.0403310i
$376$		0			0
$377$	−450677.			−0.163310
$378$		0			0
$379$		−	1.39733e6i		−	0.499690i	−0.968286	−	0.249845i	$-0.919620\pi$
$379$		−	1.39733e6i		−	0.499690i	0.968286	−	0.249845i	$-0.0803797\pi$
$380$		0			0
$381$			1.90943e6i			0.673894i
$382$		0			0
$383$			560457.i			0.195229i	0.995224	+	0.0976147i	$0.0311213\pi$
$383$			560457.i			0.195229i	−0.995224	+	0.0976147i	$0.968879\pi$
$384$		0			0
$385$	−120981.	−	27663.1i	−0.0415974	−	0.00951150i
$386$		0			0
$387$	−1.60433e6			−0.544523
$388$		0			0
$389$	2.63961e6			0.884434			0.442217	−	0.896908i	$-0.354192\pi$
$389$	2.63961e6			0.884434			0.442217	+	0.896908i	$0.354192\pi$
$390$		0			0
$391$	−2.05241e6			−0.678927
$392$		0			0
$393$			539124.i			0.176079i
$394$		0			0
$395$	106027.			0.0341918
$396$		0			0
$397$	−742881.			−0.236561			−0.118280	−	0.992980i	$-0.537738\pi$
$397$	−742881.			−0.236561			−0.118280	+	0.992980i	$0.537738\pi$
$398$		0			0
$399$			650854.i			0.204669i
$400$		0			0
$401$	−225859.			−0.0701416			−0.0350708	−	0.999385i	$-0.511166\pi$
$401$	−225859.			−0.0701416			−0.0350708	+	0.999385i	$0.511166\pi$
$402$		0			0
$403$	−506466.			−0.155342
$404$		0			0
$405$	12818.3			0.00388322
$406$		0			0
$407$	−3.64175e6	−	832710.i	−1.08974	−	0.249177i
$408$		0			0
$409$		−	3.93871e6i		−	1.16425i	−0.813099	−	0.582125i	$-0.802222\pi$
$409$		−	3.93871e6i		−	1.16425i	0.813099	−	0.582125i	$-0.197778\pi$
$410$		0			0
$411$			1.07765e6i			0.314684i
$412$		0			0
$413$		−	3.92342e6i		−	1.13185i
$414$		0			0
$415$	144653.			0.0412293
$416$		0			0
$417$			3.64388e6i			1.02618i
$418$		0			0
$419$		−	813728.i		−	0.226435i	−0.993570	−	0.113218i	$-0.963884\pi$
$419$		−	813728.i		−	0.226435i	0.993570	−	0.113218i	$-0.0361157\pi$
$420$		0			0
$421$	−4.53570e6			−1.24721			−0.623605	−	0.781740i	$-0.714333\pi$
$421$	−4.53570e6			−1.24721			−0.623605	+	0.781740i	$0.714333\pi$
$422$		0			0
$423$			1.12965e6i			0.306968i
$424$		0			0
$425$			1.88028e6i			0.504953i
$426$		0			0
$427$			3.25058e6i			0.862763i
$428$		0			0
$429$	−477573.	−	109200.i	−0.125284	−	0.0286470i
$430$		0			0
$431$	−4.47230e6			−1.15968			−0.579840	−	0.814731i	$-0.696885\pi$
$431$	−4.47230e6			−1.15968			−0.579840	+	0.814731i	$0.696885\pi$
$432$		0			0
$433$	7.31528e6			1.87504			0.937522	−	0.347926i	$-0.113114\pi$
$433$	7.31528e6			1.87504			0.937522	+	0.347926i	$0.113114\pi$
$434$		0			0
$435$	−58423.2			−0.0148034
$436$		0			0
$437$			1.55654e6i			0.389902i
$438$		0			0
$439$	543075.			0.134493			0.0672463	−	0.997736i	$-0.478579\pi$
$439$	543075.			0.134493			0.0672463	+	0.997736i	$0.478579\pi$
$440$		0			0
$441$	−668051.			−0.163574
$442$		0			0
$443$			4.50829e6i			1.09145i	0.837966	+	0.545723i	$0.183745\pi$
$443$			4.50829e6i			1.09145i	−0.837966	+	0.545723i	$0.816255\pi$
$444$		0			0
$445$	16054.4			0.00384322
$446$		0			0
$447$	−317535.			−0.0751662
$448$		0			0
$449$	−2.50982e6			−0.587525			−0.293763	−	0.955878i	$-0.594907\pi$
$449$	−2.50982e6			−0.587525			−0.293763	+	0.955878i	$0.594907\pi$
$450$		0			0
$451$	1.24800e6	−	5.45798e6i	0.288917	−	1.26354i
$452$		0			0
$453$		−	3.10075e6i		−	0.709938i
$454$		0			0
$455$		−	41945.5i		−	0.00949853i
$456$		0			0
$457$		−	5.03587e6i		−	1.12793i	−0.825797	−	0.563967i	$-0.809274\pi$
$457$		−	5.03587e6i		−	1.12793i	0.825797	−	0.563967i	$-0.190726\pi$
$458$		0			0
$459$	−439168.			−0.0972970
$460$		0			0
$461$		−	7.43886e6i		−	1.63025i	−0.579286	−	0.815124i	$-0.696669\pi$
$461$		−	7.43886e6i		−	1.63025i	0.579286	−	0.815124i	$-0.303331\pi$
$462$		0			0
$463$			3.56349e6i			0.772543i	0.922385	+	0.386272i	$0.126237\pi$
$463$			3.56349e6i			0.772543i	−0.922385	+	0.386272i	$0.873763\pi$
$464$		0			0
$465$	−65655.4			−0.0140811
$466$		0			0
$467$			4.93567e6i			1.04726i	0.851946	+	0.523629i	$0.175422\pi$
$467$			4.93567e6i			1.04726i	−0.851946	+	0.523629i	$0.824578\pi$
$468$		0			0
$469$		−	1.12606e7i		−	2.36390i
$470$		0			0
$471$			1.61419e6i			0.335276i
$472$		0			0
$473$	7.74862e6	+	1.77177e6i	1.59247	+	0.364128i
$474$		0			0
$475$	1.42599e6			0.289990
$476$		0			0
$477$	−2.76729e6			−0.556877
$478$		0			0
$479$	1.84757e6			0.367928			0.183964	−	0.982933i	$-0.441107\pi$
$479$	1.84757e6			0.367928			0.183964	+	0.982933i	$0.441107\pi$
$480$		0			0
$481$		−	1.26263e6i		−	0.248837i
$482$		0			0
$483$	−4.85341e6			−0.946628
$484$		0			0
$485$	−221484.			−0.0427552
$486$		0			0
$487$		−	5.19128e6i		−	0.991863i	−0.868362	−	0.495932i	$-0.834827\pi$
$487$		−	5.19128e6i		−	0.991863i	0.868362	−	0.495932i	$-0.165173\pi$
$488$		0			0
$489$	3.41403e6			0.645648
$490$		0			0
$491$	−3.92274e6			−0.734321			−0.367160	−	0.930158i	$-0.619670\pi$
$491$	−3.92274e6			−0.734321			−0.367160	+	0.930158i	$0.619670\pi$
$492$		0			0
$493$	2.00165e6			0.370911
$494$		0			0
$495$	−61909.9	−	14156.1i	−0.0113566	−	0.00259675i
$496$		0			0
$497$			1.10834e7i			2.01271i
$498$		0			0
$499$		−	2.14716e6i		−	0.386023i	−0.981196	−	0.193011i	$-0.938175\pi$
$499$		−	2.14716e6i		−	0.386023i	0.981196	−	0.193011i	$-0.0618255\pi$
$500$		0			0
$501$			5.36180e6i			0.954370i
$502$		0			0
$503$	7.72824e6			1.36195			0.680974	−	0.732307i	$-0.261556\pi$
$503$	7.72824e6			1.36195			0.680974	+	0.732307i	$0.261556\pi$
$504$		0			0
$505$			395316.i			0.0689789i
$506$		0			0
$507$			3.17606e6i			0.548742i
$508$		0			0
$509$	−3.79608e6			−0.649443			−0.324722	−	0.945810i	$-0.605271\pi$
$509$	−3.79608e6			−0.649443			−0.324722	+	0.945810i	$0.605271\pi$
$510$		0			0
$511$		−	4.96247e6i		−	0.840708i
$512$		0			0
$513$			333062.i			0.0558768i
$514$		0			0
$515$		−	189347.i		−	0.0314586i
$516$		0			0
$517$	1.24755e6	−	5.45599e6i	0.205272	−	0.897733i
$518$		0			0
$519$	−686024.			−0.111795
$520$		0			0
$521$	−6.49540e6			−1.04836			−0.524182	−	0.851606i	$-0.675629\pi$
$521$	−6.49540e6			−1.04836			−0.524182	+	0.851606i	$0.675629\pi$
$522$		0			0
$523$	256923.			0.0410723			0.0205362	−	0.999789i	$-0.493463\pi$
$523$	256923.			0.0410723			0.0205362	+	0.999789i	$0.493463\pi$
$524$		0			0
$525$			4.44636e6i			0.704055i
$526$		0			0
$527$	2.24943e6			0.352814
$528$		0			0
$529$	−5.17074e6			−0.803366
$530$		0			0
$531$		−	2.00774e6i		−	0.309009i
$532$		0			0
$533$	1.89234e6			0.288523
$534$		0			0
$535$	227039.			0.0342938
$536$		0			0
$537$	752747.			0.112645
$538$		0			0
$539$	3.22656e6	+	737773.i	0.478375	+	0.109383i
$540$		0			0
$541$		−	1.09419e7i		−	1.60731i	−0.595098	−	0.803653i	$-0.702887\pi$
$541$		−	1.09419e7i		−	1.60731i	0.595098	−	0.803653i	$-0.297113\pi$
$542$		0			0
$543$		−	4.17196e6i		−	0.607213i
$544$		0			0
$545$			202111.i			0.0291473i
$546$		0			0
$547$	3.99242e6			0.570516			0.285258	−	0.958451i	$-0.407921\pi$
$547$	3.99242e6			0.570516			0.285258	+	0.958451i	$0.407921\pi$
$548$		0			0
$549$			1.66342e6i			0.235544i
$550$		0			0
$551$		−	1.51803e6i		−	0.213011i
$552$		0			0
$553$	−8.59011e6			−1.19450
$554$		0			0
$555$		−	163681.i		−	0.0225562i
$556$		0			0
$557$		−	1.36724e6i		−	0.186727i	−0.995632	−	0.0933637i	$-0.970238\pi$
$557$		−	1.36724e6i		−	0.186727i	0.995632	−	0.0933637i	$-0.0297619\pi$
$558$		0			0
$559$			2.68653e6i			0.363631i
$560$		0			0
$561$	2.12110e6	+	485003.i	0.284547	+	0.0650635i
$562$		0			0
$563$	−1.43843e7			−1.91257			−0.956286	−	0.292433i	$-0.905535\pi$
$563$	−1.43843e7			−1.91257			−0.956286	+	0.292433i	$0.905535\pi$
$564$		0			0
$565$	85999.9			0.0113338
$566$		0			0
$567$	−1.03852e6			−0.135661
$568$		0			0
$569$			9.26965e6i			1.20028i	0.799895	+	0.600140i	$0.204888\pi$
$569$			9.26965e6i			1.20028i	−0.799895	+	0.600140i	$0.795112\pi$
$570$		0			0
$571$	8.19537e6			1.05191			0.525955	−	0.850513i	$-0.323708\pi$
$571$	8.19537e6			1.05191			0.525955	+	0.850513i	$0.323708\pi$
$572$		0			0
$573$	902036.			0.114772
$574$		0			0
$575$			1.06336e7i			1.34125i
$576$		0			0
$577$	−6.02825e6			−0.753792			−0.376896	−	0.926256i	$-0.623009\pi$
$577$	−6.02825e6			−0.753792			−0.376896	+	0.926256i	$0.623009\pi$
$578$		0			0
$579$	−3.93783e6			−0.488159
$580$		0			0
$581$	−1.17195e7			−1.44036
$582$		0			0
$583$	1.33655e7	+	3.05610e6i	1.62860	+	0.372389i
$584$		0			0
$585$		−	21464.8i		−	0.00259321i
$586$		0			0
$587$		−	1.30121e7i		−	1.55866i	−0.626614	−	0.779330i	$-0.715560\pi$
$587$		−	1.30121e7i		−	1.55866i	0.626614	−	0.779330i	$-0.284440\pi$
$588$		0			0
$589$		−	1.70595e6i		−	0.202618i
$590$		0			0
$591$	1.08983e6			0.128348
$592$		0			0
$593$			6.48640e6i			0.757472i	0.925505	+	0.378736i	$0.123641\pi$
$593$			6.48640e6i			0.757472i	−0.925505	+	0.378736i	$0.876359\pi$
$594$		0			0
$595$			186297.i			0.0215732i
$596$		0			0
$597$	5.21437e6			0.598779
$598$		0			0
$599$		−	1.19359e7i		−	1.35922i	−0.733574	−	0.679609i	$-0.762149\pi$
$599$		−	1.19359e7i		−	1.35922i	0.733574	−	0.679609i	$-0.237851\pi$
$600$		0			0
$601$			1.18698e7i			1.34047i	0.742149	+	0.670235i	$0.233807\pi$
$601$			1.18698e7i			1.34047i	−0.742149	+	0.670235i	$0.766193\pi$
$602$		0			0
$603$		−	5.76240e6i		−	0.645372i
$604$		0			0
$605$	283380.	+	136742.i	0.0314760	+	0.0151885i
$606$		0			0
$607$	1.09591e6			0.120727			0.0603636	−	0.998176i	$-0.480774\pi$
$607$	1.09591e6			0.120727			0.0603636	+	0.998176i	$0.480774\pi$
$608$		0			0
$609$	4.73336e6			0.517161
$610$		0			0
$611$	1.89165e6			0.204992
$612$		0			0
$613$		−	3.48697e6i		−	0.374798i	−0.982284	−	0.187399i	$-0.939994\pi$
$613$		−	3.48697e6i		−	0.374798i	0.982284	−	0.187399i	$-0.0600058\pi$
$614$		0			0
$615$	245312.			0.0261536
$616$		0			0
$617$	1.77093e7			1.87278			0.936391	−	0.350957i	$-0.114144\pi$
$617$	1.77093e7			1.87278			0.936391	+	0.350957i	$0.114144\pi$
$618$		0			0
$619$		−	1.00174e6i		−	0.105082i	−0.998619	−	0.0525412i	$-0.983268\pi$
$619$		−	1.00174e6i		−	0.105082i	0.998619	−	0.0525412i	$-0.0167321\pi$
$620$		0			0
$621$	−2.48364e6			−0.258440
$622$		0			0
$623$	−1.30070e6			−0.134264
$624$		0			0
$625$	9.72986e6			0.996337
$626$		0			0
$627$	367822.	−	1.60863e6i	0.0373654	−	0.163413i
$628$		0			0
$629$			5.60788e6i			0.565161i
$630$		0			0
$631$			8.36417e6i			0.836275i	0.908384	+	0.418138i	$0.137317\pi$
$631$			8.36417e6i			0.836275i	−0.908384	+	0.418138i	$0.862683\pi$
$632$		0			0
$633$			9.74895e6i			0.967049i
$634$		0			0
$635$	414496.			0.0407931
$636$		0			0
$637$			1.11868e6i			0.109234i
$638$		0			0
$639$			5.67170e6i			0.549491i
$640$		0			0
$641$	5.11119e6			0.491334			0.245667	−	0.969354i	$-0.420993\pi$
$641$	5.11119e6			0.491334			0.245667	+	0.969354i	$0.420993\pi$
$642$		0			0
$643$			4.58534e6i			0.437365i	0.975796	+	0.218683i	$0.0701759\pi$
$643$			4.58534e6i			0.437365i	−0.975796	+	0.218683i	$0.929824\pi$
$644$		0			0
$645$			348266.i			0.0329619i
$646$		0			0
$647$			1.16776e7i			1.09672i	0.836244	+	0.548358i	$0.184747\pi$
$647$			1.16776e7i			1.09672i	−0.836244	+	0.548358i	$0.815253\pi$
$648$		0			0
$649$	−2.21728e6	+	9.69700e6i	−0.206637	+	0.903703i
$650$		0			0
$651$	5.31929e6			0.491928
$652$		0			0
$653$	1.87685e7			1.72245			0.861223	−	0.508227i	$-0.169699\pi$
$653$	1.87685e7			1.72245			0.861223	+	0.508227i	$0.169699\pi$
$654$		0			0
$655$	117032.			0.0106587
$656$		0			0
$657$		−	2.53945e6i		−	0.229523i
$658$		0			0
$659$	1.67325e7			1.50089			0.750443	−	0.660935i	$-0.229840\pi$
$659$	1.67325e7			1.50089			0.750443	+	0.660935i	$0.229840\pi$
$660$		0			0
$661$	−480270.			−0.0427545			−0.0213772	−	0.999771i	$-0.506805\pi$
$661$	−480270.			−0.0427545			−0.0213772	+	0.999771i	$0.506805\pi$
$662$		0			0
$663$			735408.i			0.0649747i
$664$		0			0
$665$	141286.			0.0123893
$666$		0			0
$667$	1.13200e7			0.985214
$668$		0			0
$669$	−7.82409e6			−0.675878
$670$		0			0
$671$	1.83703e6	−	8.03402e6i	0.157511	−	0.688853i
$672$		0			0
$673$		−	1.47901e7i		−	1.25873i	−0.777109	−	0.629366i	$-0.783315\pi$
$673$		−	1.47901e7i		−	1.25873i	0.777109	−	0.629366i	$-0.216685\pi$
$674$		0			0
$675$			2.27534e6i			0.192215i
$676$		0			0
$677$		−	4.61603e6i		−	0.387076i	−0.981093	−	0.193538i	$-0.938004\pi$
$677$		−	4.61603e6i		−	0.387076i	0.981093	−	0.193538i	$-0.0619963\pi$
$678$		0			0
$679$	1.79443e7			1.49366
$680$		0			0
$681$			1.13157e7i			0.935003i
$682$		0			0
$683$			2.25824e7i			1.85233i	0.377114	+	0.926167i	$0.376916\pi$
$683$			2.25824e7i			1.85233i	−0.377114	+	0.926167i	$0.623084\pi$
$684$		0			0
$685$	233935.			0.0190489
$686$		0			0
$687$		−	5.38774e6i		−	0.435527i
$688$		0			0
$689$			4.63396e6i			0.371881i
$690$		0			0
$691$			424882.i			0.0338512i	0.999857	+	0.0169256i	$0.00538784\pi$
$691$			424882.i			0.0338512i	−0.999857	+	0.0169256i	$0.994612\pi$
$692$		0			0
$693$	5.01584e6	+	1.14690e6i	0.396744	+	0.0907180i
$694$		0			0
$695$	791009.			0.0621183
$696$		0			0
$697$	−8.40466e6			−0.655297
$698$		0			0
$699$	−7.63981e6			−0.591411
$700$		0			0
$701$			2.12009e7i			1.62952i	0.579799	+	0.814759i	$0.303131\pi$
$701$			2.12009e7i			1.62952i	−0.579799	+	0.814759i	$0.696869\pi$
$702$		0			0
$703$	4.25298e6			0.324567
$704$		0			0
$705$	245223.			0.0185818
$706$		0			0
$707$		−	3.20279e7i		−	2.40979i
$708$		0			0
$709$	4.49913e6			0.336134			0.168067	−	0.985776i	$-0.446247\pi$
$709$	4.49913e6			0.336134			0.168067	+	0.985776i	$0.446247\pi$
$710$		0			0
$711$	−4.39582e6			−0.326112
$712$		0			0
$713$	1.27212e7			0.937143
$714$		0			0
$715$	−23705.0	+	103671.i	−0.00173410	+	0.00758389i
$716$		0			0
$717$		−	4.68918e6i		−	0.340642i
$718$		0			0
$719$		−	5.62859e6i		−	0.406048i	−0.979174	−	0.203024i	$-0.934923\pi$
$719$		−	5.62859e6i		−	0.406048i	0.979174	−	0.203024i	$-0.0650769\pi$
$720$		0			0
$721$			1.53406e7i			1.09902i
$722$		0			0
$723$	7.23486e6			0.514736
$724$		0			0
$725$		−	1.03706e7i		−	0.732753i
$726$		0			0
$727$			1.39344e7i			0.977802i	0.872339	+	0.488901i	$0.162602\pi$
$727$			1.39344e7i			0.977802i	−0.872339	+	0.488901i	$0.837398\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$		−	1.19320e7i		−	0.825884i
$732$		0			0
$733$		−	9.73937e6i		−	0.669531i	−0.942301	−	0.334766i	$-0.891343\pi$
$733$		−	9.73937e6i		−	0.669531i	0.942301	−	0.334766i	$-0.108657\pi$
$734$		0			0
$735$			145020.i			0.00990167i
$736$		0			0
$737$	−6.36380e6	+	2.78313e7i	−0.431567	+	1.88740i
$738$		0			0
$739$	1.35974e6			0.0915890			0.0457945	−	0.998951i	$-0.485418\pi$
$739$	1.35974e6			0.0915890			0.0457945	+	0.998951i	$0.485418\pi$
$740$		0			0
$741$	557728.			0.0373144
$742$		0			0
$743$	2.17353e7			1.44442			0.722210	−	0.691673i	$-0.243126\pi$
$743$	2.17353e7			1.44442			0.722210	+	0.691673i	$0.243126\pi$
$744$		0			0
$745$			68930.1i			0.00455007i
$746$		0			0
$747$	−5.99724e6			−0.393233
$748$		0			0
$749$	−1.83943e7			−1.19806
$750$		0			0
$751$		−	1.74521e7i		−	1.12914i	−0.825384	−	0.564571i	$-0.809042\pi$
$751$		−	1.74521e7i		−	1.12914i	0.825384	−	0.564571i	$-0.190958\pi$
$752$		0			0
$753$	−3.24401e6			−0.208495
$754$		0			0
$755$	−673106.			−0.0429750
$756$		0			0
$757$	−1.20511e7			−0.764343			−0.382172	−	0.924091i	$-0.624824\pi$
$757$	−1.20511e7			−0.764343			−0.382172	+	0.924091i	$0.624824\pi$
$758$		0			0
$759$	1.19955e7	+	2.74285e6i	0.755814	+	0.172821i
$760$		0			0
$761$			6.25129e6i			0.391299i	0.980674	+	0.195649i	$0.0626814\pi$
$761$			6.25129e6i			0.391299i	−0.980674	+	0.195649i	$0.937319\pi$
$762$		0			0
$763$		−	1.63747e7i		−	1.01827i
$764$		0			0
$765$			95334.1i			0.00588972i
$766$		0			0
$767$	−3.36205e6			−0.206355
$768$		0			0
$769$			2.39307e7i			1.45928i	0.683830	+	0.729641i	$0.260313\pi$
$769$			2.39307e7i			1.45928i	−0.683830	+	0.729641i	$0.739687\pi$
$770$		0			0
$771$			5.36273e6i			0.324900i
$772$		0			0
$773$	−1.97320e7			−1.18774			−0.593871	−	0.804560i	$-0.702401\pi$
$773$	−1.97320e7			−1.18774			−0.593871	+	0.804560i	$0.702401\pi$
$774$		0			0
$775$		−	1.16543e7i		−	0.697001i
$776$		0			0
$777$			1.32611e7i			0.788004i
$778$		0			0
$779$			6.37403e6i			0.376332i
$780$		0			0
$781$	6.26363e6	−	2.73932e7i	0.367450	−	1.60700i
$782$		0			0
$783$	2.42221e6			0.141191
$784$		0			0
$785$	350407.			0.0202954
$786$		0			0
$787$	6.72911e6			0.387276			0.193638	−	0.981073i	$-0.437971\pi$
$787$	6.72911e6			0.387276			0.193638	+	0.981073i	$0.437971\pi$
$788$		0			0
$789$		−	6.49366e6i		−	0.371362i
$790$		0			0
$791$	−6.96757e6			−0.395950
$792$		0			0
$793$	2.78548e6			0.157296
$794$		0			0
$795$			600720.i			0.0337097i
$796$		0			0
$797$	−1.45091e7			−0.809086			−0.404543	−	0.914519i	$-0.632569\pi$
$797$	−1.45091e7			−0.809086			−0.404543	+	0.914519i	$0.632569\pi$
$798$		0			0
$799$	−8.40160e6			−0.465581
$800$		0			0
$801$	−665611.			−0.0366555
$802$		0			0
$803$	−2.80448e6	+	1.22651e7i	−0.153484	+	0.671244i
$804$		0			0
$805$			1.05357e6i			0.0573026i
$806$		0			0
$807$		−	1.32670e6i		−	0.0717113i
$808$		0			0
$809$		−	2.33598e7i		−	1.25487i	−0.778671	−	0.627433i	$-0.784106\pi$
$809$		−	2.33598e7i		−	1.25487i	0.778671	−	0.627433i	$-0.215894\pi$
$810$		0			0
$811$	2.87241e7			1.53354			0.766769	−	0.641923i	$-0.221863\pi$
$811$	2.87241e7			1.53354			0.766769	+	0.641923i	$0.221863\pi$
$812$		0			0
$813$			2.39464e6i			0.127062i
$814$		0			0
$815$		−	741114.i		−	0.0390833i
$816$		0			0
$817$	−9.04912e6			−0.474298
$818$		0			0
$819$			1.73904e6i			0.0905943i
$820$		0			0
$821$		−	2.45048e7i		−	1.26880i	−0.773005	−	0.634399i	$-0.781247\pi$
$821$		−	2.45048e7i		−	1.26880i	0.773005	−	0.634399i	$-0.218753\pi$
$822$		0			0
$823$		−	1.13447e7i		−	0.583838i	−0.956443	−	0.291919i	$-0.905706\pi$
$823$		−	1.13447e7i		−	0.583838i	0.956443	−	0.291919i	$-0.0942938\pi$
$824$		0			0
$825$	2.51281e6	−	1.09895e7i	0.128536	−	0.562137i
$826$		0			0
$827$	1.15273e7			0.586088			0.293044	−	0.956099i	$-0.405332\pi$
$827$	1.15273e7			0.586088			0.293044	+	0.956099i	$0.405332\pi$
$828$		0			0
$829$	3.67971e7			1.85963			0.929816	−	0.368024i	$-0.119966\pi$
$829$	3.67971e7			1.85963			0.929816	+	0.368024i	$0.119966\pi$
$830$		0			0
$831$	−405090.			−0.0203493
$832$		0			0
$833$		−	4.96853e6i		−	0.248094i
$834$		0			0
$835$	1.16393e6			0.0577713
$836$		0			0
$837$	2.72205e6			0.134302
$838$		0			0
$839$			1.41544e7i			0.694204i	0.937827	+	0.347102i	$0.112834\pi$
$839$			1.41544e7i			0.694204i	−0.937827	+	0.347102i	$0.887166\pi$
$840$		0			0
$841$	9.47120e6			0.461759
$842$		0			0
$843$	2.15350e6			0.104370
$844$		0			0
$845$	689454.			0.0332172
$846$		0			0
$847$	−2.29590e7	−	1.10786e7i	−1.09962	−	0.530613i
$848$		0			0
$849$			107237.i			0.00510595i
$850$		0			0
$851$			3.17144e7i			1.50118i
$852$		0			0
$853$			7.77725e6i			0.365977i	0.983115	+	0.182988i	$0.0585770\pi$
$853$			7.77725e6i			0.365977i	−0.983115	+	0.182988i	$0.941423\pi$
$854$		0			0
$855$	72300.6			0.00338242
$856$		0			0
$857$		−	595387.i		−	0.0276915i	−0.999904	−	0.0138458i	$-0.995593\pi$
$857$		−	595387.i		−	0.0276915i	0.999904	−	0.0138458i	$-0.00440738\pi$
$858$		0			0
$859$		−	1.15641e7i		−	0.534721i	−0.963597	−	0.267361i	$-0.913848\pi$
$859$		−	1.15641e7i		−	0.534721i	0.963597	−	0.267361i	$-0.0861515\pi$
$860$		0			0
$861$	−1.98748e7			−0.913681
$862$		0			0
$863$			8.98186e6i			0.410525i	0.978707	+	0.205262i	$0.0658047\pi$
$863$			8.98186e6i			0.410525i	−0.978707	+	0.205262i	$0.934195\pi$
$864$		0			0
$865$			148921.i			0.00676731i
$866$		0			0
$867$			9.51246e6i			0.429779i
$868$		0			0
$869$	2.12310e7	+	4.85460e6i	0.953720	+	0.218074i
$870$		0			0
$871$	−9.64941e6			−0.430978
$872$		0			0
$873$	9.18266e6			0.407786
$874$		0			0
$875$	1.93160e6			0.0852899
$876$		0			0
$877$		−	2.69891e6i		−	0.118492i	−0.998243	−	0.0592460i	$-0.981130\pi$
$877$		−	2.69891e6i		−	0.118492i	0.998243	−	0.0592460i	$-0.0188696\pi$
$878$		0			0
$879$	3.95917e6			0.172835
$880$		0			0
$881$	2.49371e7			1.08245			0.541224	−	0.840879i	$-0.317961\pi$
$881$	2.49371e7			1.08245			0.541224	+	0.840879i	$0.317961\pi$
$882$		0			0
$883$			3.63504e7i			1.56895i	0.620164	+	0.784473i	$0.287066\pi$
$883$			3.63504e7i			1.56895i	−0.620164	+	0.784473i	$0.712934\pi$
$884$		0			0
$885$	−435837.			−0.0187054
$886$		0			0
$887$	2.24755e7			0.959182			0.479591	−	0.877492i	$-0.340785\pi$
$887$	2.24755e7			0.959182			0.479591	+	0.877492i	$0.340785\pi$
$888$		0			0
$889$	−3.35818e7			−1.42512
$890$		0			0
$891$	2.56676e6	+	586906.i	0.108316	+	0.0247670i
$892$		0			0
$893$			6.37171e6i			0.267379i
$894$		0			0
$895$		−	163405.i		−	0.00681881i
$896$		0			0
$897$			4.15897e6i			0.172586i
$898$		0			0
$899$	−1.24066e7			−0.511979
$900$		0			0
$901$		−	2.05813e7i		−	0.844621i
$902$		0			0
$903$		−	2.82159e7i		−	1.15153i
$904$		0			0
$905$	−905644.			−0.0367567
$906$		0			0
$907$		−	1.63738e7i		−	0.660891i	−0.943825	−	0.330446i	$-0.892801\pi$
$907$		−	1.63738e7i		−	0.660891i	0.943825	−	0.330446i	$-0.107199\pi$
$908$		0			0
$909$		−	1.63897e7i		−	0.657901i
$910$		0			0
$911$		−	2.69110e7i		−	1.07432i	−0.843481	−	0.537160i	$-0.819497\pi$
$911$		−	2.69110e7i		−	1.07432i	0.843481	−	0.537160i	$-0.180503\pi$
$912$		0			0
$913$	2.89656e7	+	6.62315e6i	1.15002	+	0.262959i
$914$		0			0
$915$	361094.			0.0142583
$916$		0			0
$917$	−9.48177e6			−0.372362
$918$		0			0
$919$	−2.06200e7			−0.805376			−0.402688	−	0.915337i	$-0.631924\pi$
$919$	−2.06200e7			−0.805376			−0.402688	+	0.915337i	$0.631924\pi$
$920$		0			0
$921$		−	2.06312e7i		−	0.801447i
$922$		0			0
$923$	9.49752e6			0.366949
$924$		0			0
$925$	2.90546e7			1.11650
$926$		0			0
$927$			7.85025e6i			0.300044i
$928$		0			0
$929$	−2.24458e7			−0.853289			−0.426644	−	0.904419i	$-0.640304\pi$
$929$	−2.24458e7			−0.853289			−0.426644	+	0.904419i	$0.640304\pi$
$930$		0			0
$931$	−3.76810e6			−0.142478
$932$		0			0
$933$	1.89707e7			0.713476
$934$		0			0
$935$	105284.	−	460446.i	0.00393851	−	0.0172246i
$936$		0			0
$937$			2.24706e7i			0.836114i	0.908421	+	0.418057i	$0.137289\pi$
$937$			2.24706e7i			0.836114i	−0.908421	+	0.418057i	$0.862711\pi$
$938$		0			0
$939$			4.64946e6i			0.172083i
$940$		0			0
$941$		−	2.21462e7i		−	0.815314i	−0.913135	−	0.407657i	$-0.866346\pi$
$941$		−	2.21462e7i		−	0.815314i	0.913135	−	0.407657i	$-0.133654\pi$
$942$		0			0
$943$	−4.75311e7			−1.74060
$944$		0			0
$945$			225440.i			0.00821204i
$946$		0			0
$947$			2.17396e7i			0.787727i	0.919169	+	0.393864i	$0.128862\pi$
$947$			2.17396e7i			0.787727i	−0.919169	+	0.393864i	$0.871138\pi$
$948$		0			0
$949$	−4.25242e6			−0.153275
$950$		0			0
$951$			1.23838e7i			0.444020i
$952$		0			0
$953$			7.47807e6i			0.266721i	0.991068	+	0.133361i	$0.0425768\pi$
$953$			7.47807e6i			0.266721i	−0.991068	+	0.133361i	$0.957423\pi$
$954$		0			0
$955$		−	195813.i		−	0.00694756i
$956$		0			0
$957$	−1.16988e7	−	2.67500e6i	−0.412916	−	0.0944157i
$958$		0			0
$959$	−1.89531e7			−0.665477
$960$		0			0
$961$	1.46868e7			0.513001
$962$		0			0
$963$	−9.41293e6			−0.327084
$964$		0			0
$965$			854819.i			0.0295499i
$966$		0			0
$967$	−3.99274e7			−1.37311			−0.686553	−	0.727079i	$-0.740877\pi$
$967$	−3.99274e7			−1.37311			−0.686553	+	0.727079i	$0.740877\pi$
$968$		0			0
$969$	−2.47710e6			−0.0847489
$970$		0			0
$971$		−	3.20453e7i		−	1.09073i	−0.838199	−	0.545364i	$-0.816391\pi$
$971$		−	3.20453e7i		−	1.09073i	0.838199	−	0.545364i	$-0.183609\pi$
$972$		0			0
$973$	−6.40863e7			−2.17012
$974$		0			0
$975$	3.81017e6			0.128361
$976$		0			0
$977$	−2.33839e7			−0.783755			−0.391877	−	0.920017i	$-0.628174\pi$
$977$	−2.33839e7			−0.783755			−0.391877	+	0.920017i	$0.628174\pi$
$978$		0			0
$979$	3.21478e6	+	735078.i	0.107200	+	0.0245119i
$980$		0			0
$981$		−	8.37945e6i		−	0.277999i
$982$		0			0
$983$			1.47319e6i			0.0486268i	0.999704	+	0.0243134i	$0.00773996\pi$
$983$			1.47319e6i			0.0486268i	−0.999704	+	0.0243134i	$0.992260\pi$
$984$		0			0
$985$		−	236578.i		−	0.00776933i
$986$		0			0
$987$	−1.98675e7			−0.649159
$988$		0			0
$989$		−	6.74793e7i		−	2.19371i
$990$		0			0
$991$			4.76802e7i			1.54225i	0.636686	+	0.771123i	$0.280305\pi$
$991$			4.76802e7i			1.54225i	−0.636686	+	0.771123i	$0.719695\pi$
$992$		0			0
$993$	−1.89540e7			−0.609998
$994$		0			0
$995$		−	1.13193e6i		−	0.0362461i
$996$		0			0
$997$			4.90650e7i			1.56327i	0.623736	+	0.781635i	$0.285614\pi$
$997$			4.90650e7i			1.56327i	−0.623736	+	0.781635i	$0.714386\pi$
$998$		0			0
$999$			6.78614e6i			0.215134i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.o.b.175.16	yes	40
4.3	odd	2	inner	528.6.o.b.175.19	yes	40
11.10	odd	2	inner	528.6.o.b.175.20	yes	40
44.43	even	2	inner	528.6.o.b.175.15	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.6.o.b.175.15	✓	40	44.43	even	2	inner
528.6.o.b.175.16	yes	40	1.1	even	1	trivial
528.6.o.b.175.19	yes	40	4.3	odd	2	inner
528.6.o.b.175.20	yes	40	11.10	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)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +1.95371 q^{5} -158.286 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} +1.95371 q^{5} -158.286 q^{7} -81.0000 q^{9} +(391.215 + 89.4537i) q^{11} +135.638i q^{13} +17.5834i q^{15} -602.426i q^{17} -456.875 q^{19} -1424.58i q^{21} -3406.92i q^{23} -3121.18 q^{25} -729.000i q^{27} +3322.64i q^{29} +3733.95i q^{31} +(-805.083 + 3520.93i) q^{33} -309.245 q^{35} -9308.84 q^{37} -1220.74 q^{39} -13951.4i q^{41} +19806.5 q^{43} -158.250 q^{45} -13946.3i q^{47} +8247.55 q^{49} +5421.83 q^{51} +34164.1 q^{53} +(764.319 + 174.766i) q^{55} -4111.88i q^{57} +24786.9i q^{59} -20536.1i q^{61} +12821.2 q^{63} +264.997i q^{65} +71140.8i q^{67} +30662.3 q^{69} -70020.9i q^{71} +31351.2i q^{73} -28090.6i q^{75} +(-61923.9 - 14159.3i) q^{77} +54269.4 q^{79} +6561.00 q^{81} +74040.1 q^{83} -1176.96i q^{85} -29903.8 q^{87} +8217.42 q^{89} -21469.7i q^{91} -33605.5 q^{93} -892.601 q^{95} -113366. q^{97} +(-31688.4 - 7245.75i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) \) 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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