LMFDB - Embedded newform 528.6.o.b.175.12

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			175.12
Character	$\chi$	$=$	528.175
Dual form			528.6.o.b.175.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +57.6896 q^{5} +131.273 q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} +57.6896 q^{5} +131.273 q^{7} -81.0000 q^{9} +(-230.624 - 328.426i) q^{11} +558.438i q^{13} +519.207i q^{15} -2074.24i q^{17} -347.233 q^{19} +1181.46i q^{21} -3835.83i q^{23} +203.092 q^{25} -729.000i q^{27} -4843.61i q^{29} -852.598i q^{31} +(2955.83 - 2075.62i) q^{33} +7573.12 q^{35} -14604.2 q^{37} -5025.94 q^{39} -14919.5i q^{41} +2148.63 q^{43} -4672.86 q^{45} +8291.73i q^{47} +425.720 q^{49} +18668.2 q^{51} -33753.6 q^{53} +(-13304.6 - 18946.7i) q^{55} -3125.10i q^{57} -741.252i q^{59} +48133.0i q^{61} -10633.1 q^{63} +32216.1i q^{65} -18143.8i q^{67} +34522.4 q^{69} +44082.9i q^{71} -4533.44i q^{73} +1827.83i q^{75} +(-30274.9 - 43113.6i) q^{77} -44723.3 q^{79} +6561.00 q^{81} +81654.4 q^{83} -119662. i q^{85} +43592.5 q^{87} -28268.0 q^{89} +73308.1i q^{91} +7673.38 q^{93} -20031.8 q^{95} +43447.2 q^{97} +(18680.6 + 26602.5i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	57.6896			1.03198			0.515992	−	0.856594i	$-0.327424\pi$
$5$	57.6896			1.03198			0.515992	+	0.856594i	$0.327424\pi$
$6$		0			0
$7$	131.273			1.01259			0.506293	−	0.862362i	$-0.331015\pi$
$7$	131.273			1.01259			0.506293	+	0.862362i	$0.331015\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	−230.624	−	328.426i	−0.574677	−	0.818380i
$11$	−230.624	−	328.426i	−0.574677	−	0.818380i
$12$		0			0
$13$			558.438i			0.916467i	0.888832	+	0.458233i	$0.151518\pi$
$13$			558.438i			0.916467i	−0.888832	+	0.458233i	$0.848482\pi$
$14$		0			0
$15$			519.207i			0.595816i
$16$		0			0
$17$		−	2074.24i		−	1.74075i	−0.492385	−	0.870377i	$-0.663875\pi$
$17$		−	2074.24i		−	1.74075i	0.492385	−	0.870377i	$-0.336125\pi$
$18$		0			0
$19$	−347.233			−0.220667			−0.110334	−	0.993895i	$-0.535192\pi$
$19$	−347.233			−0.220667			−0.110334	+	0.993895i	$0.535192\pi$
$20$		0			0
$21$			1181.46i			0.584617i
$22$		0			0
$23$		−	3835.83i		−	1.51196i	−0.654597	−	0.755978i	$-0.727162\pi$
$23$		−	3835.83i		−	1.51196i	0.654597	−	0.755978i	$-0.272838\pi$
$24$		0			0
$25$	203.092			0.0649894
$26$		0			0
$27$		−	729.000i		−	0.192450i
$28$		0			0
$29$		−	4843.61i		−	1.06948i	−0.845016	−	0.534742i	$-0.820409\pi$
$29$		−	4843.61i		−	1.06948i	0.845016	−	0.534742i	$-0.179591\pi$
$30$		0			0
$31$		−	852.598i		−	0.159346i	−0.996821	−	0.0796728i	$-0.974612\pi$
$31$		−	852.598i		−	0.159346i	0.996821	−	0.0796728i	$-0.0253875\pi$
$32$		0			0
$33$	2955.83	−	2075.62i	0.472492	−	0.331790i
$34$		0			0
$35$	7573.12			1.04497
$36$		0			0
$37$	−14604.2			−1.75378			−0.876888	−	0.480695i	$-0.840384\pi$
$37$	−14604.2			−1.75378			−0.876888	+	0.480695i	$0.840384\pi$
$38$		0			0
$39$	−5025.94			−0.529122
$40$		0			0
$41$		−	14919.5i		−	1.38610i	−0.720889	−	0.693051i	$-0.756266\pi$
$41$		−	14919.5i		−	1.38610i	0.720889	−	0.693051i	$-0.243734\pi$
$42$		0			0
$43$	2148.63			0.177211			0.0886054	−	0.996067i	$-0.471759\pi$
$43$	2148.63			0.177211			0.0886054	+	0.996067i	$0.471759\pi$
$44$		0			0
$45$	−4672.86			−0.343994
$46$		0			0
$47$			8291.73i			0.547521i	0.961798	+	0.273760i	$0.0882675\pi$
$47$			8291.73i			0.547521i	−0.961798	+	0.273760i	$0.911732\pi$
$48$		0			0
$49$	425.720			0.0253299
$50$		0			0
$51$	18668.2			1.00503
$52$		0			0
$53$	−33753.6			−1.65056			−0.825278	−	0.564727i	$-0.808981\pi$
$53$	−33753.6			−1.65056			−0.825278	+	0.564727i	$0.808981\pi$
$54$		0			0
$55$	−13304.6	−	18946.7i	−0.593057	−	0.844555i
$56$		0			0
$57$		−	3125.10i		−	0.127402i
$58$		0			0
$59$		−	741.252i		−	0.0277227i	−0.999904	−	0.0138614i	$-0.995588\pi$
$59$		−	741.252i		−	0.0277227i	0.999904	−	0.0138614i	$-0.00441235\pi$
$60$		0			0
$61$			48133.0i			1.65622i	0.560564	+	0.828111i	$0.310584\pi$
$61$			48133.0i			1.65622i	−0.560564	+	0.828111i	$0.689416\pi$
$62$		0			0
$63$	−10633.1			−0.337529
$64$		0			0
$65$			32216.1i			0.945778i
$66$		0			0
$67$		−	18143.8i		−	0.493788i	−0.969042	−	0.246894i	$-0.920590\pi$
$67$		−	18143.8i		−	0.493788i	0.969042	−	0.246894i	$-0.0794100\pi$
$68$		0			0
$69$	34522.4			0.872928
$70$		0			0
$71$			44082.9i			1.03782i	0.854827	+	0.518912i	$0.173663\pi$
$71$			44082.9i			1.03782i	−0.854827	+	0.518912i	$0.826337\pi$
$72$		0			0
$73$		−	4533.44i		−	0.0995683i	−0.998760	−	0.0497842i	$-0.984147\pi$
$73$		−	4533.44i		−	0.0995683i	0.998760	−	0.0497842i	$-0.0158533\pi$
$74$		0			0
$75$			1827.83i			0.0375216i
$76$		0			0
$77$	−30274.9	−	43113.6i	−0.581910	−	0.828680i
$78$		0			0
$79$	−44723.3			−0.806243			−0.403121	−	0.915147i	$-0.632075\pi$
$79$	−44723.3			−0.806243			−0.403121	+	0.915147i	$0.632075\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$	81654.4			1.30102			0.650511	−	0.759497i	$-0.274555\pi$
$83$	81654.4			1.30102			0.650511	+	0.759497i	$0.274555\pi$
$84$		0			0
$85$		−	119662.i		−	1.79643i
$86$		0			0
$87$	43592.5			0.617466
$88$		0			0
$89$	−28268.0			−0.378285			−0.189143	−	0.981950i	$-0.560571\pi$
$89$	−28268.0			−0.378285			−0.189143	+	0.981950i	$0.560571\pi$
$90$		0			0
$91$			73308.1i			0.928001i
$92$		0			0
$93$	7673.38			0.0919982
$94$		0			0
$95$	−20031.8			−0.227725
$96$		0			0
$97$	43447.2			0.468849			0.234424	−	0.972134i	$-0.424679\pi$
$97$	43447.2			0.468849			0.234424	+	0.972134i	$0.424679\pi$
$98$		0			0
$99$	18680.6	+	26602.5i	0.191559	+	0.272793i
$100$		0			0
$101$		−	131864.i		−	1.28624i	−0.765766	−	0.643119i	$-0.777640\pi$
$101$		−	131864.i		−	1.28624i	0.765766	−	0.643119i	$-0.222360\pi$
$102$		0			0
$103$		−	192553.i		−	1.78837i	−0.447698	−	0.894185i	$-0.647756\pi$
$103$		−	192553.i		−	1.78837i	0.447698	−	0.894185i	$-0.352244\pi$
$104$		0			0
$105$			68158.0i			0.603315i
$106$		0			0
$107$	−159854.			−1.34978			−0.674892	−	0.737917i	$-0.735810\pi$
$107$	−159854.			−1.34978			−0.674892	+	0.737917i	$0.735810\pi$
$108$		0			0
$109$			14673.5i			0.118295i	0.998249	+	0.0591476i	$0.0188383\pi$
$109$			14673.5i			0.118295i	−0.998249	+	0.0591476i	$0.981162\pi$
$110$		0			0
$111$		−	131438.i		−	1.01254i
$112$		0			0
$113$	−53091.1			−0.391134			−0.195567	−	0.980690i	$-0.562655\pi$
$113$	−53091.1			−0.391134			−0.195567	+	0.980690i	$0.562655\pi$
$114$		0			0
$115$		−	221287.i		−	1.56031i
$116$		0			0
$117$		−	45233.5i		−	0.305489i
$118$		0			0
$119$		−	272293.i		−	1.76266i
$120$		0			0
$121$	−54675.7	+	151486.i	−0.339493	+	0.940609i
$122$		0			0
$123$	134276.			0.800266
$124$		0			0
$125$	−168564.			−0.964915
$126$		0			0
$127$	350327.			1.92737			0.963683	−	0.267048i	$-0.0860483\pi$
$127$	350327.			1.92737			0.963683	+	0.267048i	$0.0860483\pi$
$128$		0			0
$129$			19337.7i			0.102313i
$130$		0			0
$131$	71731.6			0.365201			0.182601	−	0.983187i	$-0.441548\pi$
$131$	71731.6			0.365201			0.182601	+	0.983187i	$0.441548\pi$
$132$		0			0
$133$	−45582.5			−0.223444
$134$		0			0
$135$		−	42055.7i		−	0.198605i
$136$		0			0
$137$	165058.			0.751338			0.375669	−	0.926754i	$-0.377413\pi$
$137$	165058.			0.751338			0.375669	+	0.926754i	$0.377413\pi$
$138$		0			0
$139$	330149.			1.44935			0.724674	−	0.689092i	$-0.241990\pi$
$139$	330149.			1.44935			0.724674	+	0.689092i	$0.241990\pi$
$140$		0			0
$141$	−74625.6			−0.316111
$142$		0			0
$143$	183405.	−	128789.i	0.750018	−	0.526672i
$144$		0			0
$145$		−	279426.i		−	1.10369i
$146$		0			0
$147$			3831.48i			0.0146242i
$148$		0			0
$149$			214414.i			0.791203i	0.918422	+	0.395602i	$0.129464\pi$
$149$			214414.i			0.791203i	−0.918422	+	0.395602i	$0.870536\pi$
$150$		0			0
$151$	160843.			0.574064			0.287032	−	0.957921i	$-0.407331\pi$
$151$	160843.			0.574064			0.287032	+	0.957921i	$0.407331\pi$
$152$		0			0
$153$			168014.i			0.580252i
$154$		0			0
$155$		−	49186.0i		−	0.164442i
$156$		0			0
$157$	−524201.			−1.69726			−0.848631	−	0.528986i	$-0.822573\pi$
$157$	−524201.			−1.69726			−0.848631	+	0.528986i	$0.822573\pi$
$158$		0			0
$159$		−	303782.i		−	0.952949i
$160$		0			0
$161$		−	503542.i		−	1.53099i
$162$		0			0
$163$		−	188009.i		−	0.554255i	−0.960833	−	0.277127i	$-0.910618\pi$
$163$		−	188009.i		−	0.554255i	0.960833	−	0.277127i	$-0.0893824\pi$
$164$		0			0
$165$	170521.	−	119742.i	0.487604	−	0.342402i
$166$		0			0
$167$	506492.			1.40534			0.702670	−	0.711516i	$-0.251991\pi$
$167$	506492.			1.40534			0.702670	+	0.711516i	$0.251991\pi$
$168$		0			0
$169$	59439.9			0.160089
$170$		0			0
$171$	28125.9			0.0735557
$172$		0			0
$173$			208471.i			0.529577i	0.964306	+	0.264789i	$0.0853022\pi$
$173$			208471.i			0.529577i	−0.964306	+	0.264789i	$0.914698\pi$
$174$		0			0
$175$	26660.6			0.0658073
$176$		0			0
$177$	6671.27			0.0160057
$178$		0			0
$179$		−	680588.i		−	1.58764i	−0.608154	−	0.793819i	$-0.708090\pi$
$179$		−	680588.i		−	1.58764i	0.608154	−	0.793819i	$-0.291910\pi$
$180$		0			0
$181$	−57192.8			−0.129761			−0.0648806	−	0.997893i	$-0.520667\pi$
$181$	−57192.8			−0.129761			−0.0648806	+	0.997893i	$0.520667\pi$
$182$		0			0
$183$	−433197.			−0.956220
$184$		0			0
$185$	−842512.			−1.80987
$186$		0			0
$187$	−681235.	+	478371.i	−1.42460	+	1.00037i
$188$		0			0
$189$		−	95698.3i		−	0.194872i
$190$		0			0
$191$			26615.7i			0.0527903i	0.999652	+	0.0263951i	$0.00840281\pi$
$191$			26615.7i			0.0527903i	−0.999652	+	0.0263951i	$0.991597\pi$
$192$		0			0
$193$		−	160688.i		−	0.310520i	−0.987874	−	0.155260i	$-0.950378\pi$
$193$		−	160688.i		−	0.310520i	0.987874	−	0.155260i	$-0.0496216\pi$
$194$		0			0
$195$	−289945.			−0.546045
$196$		0			0
$197$			47966.3i			0.0880584i	0.999030	+	0.0440292i	$0.0140195\pi$
$197$			47966.3i			0.0880584i	−0.999030	+	0.0440292i	$0.985981\pi$
$198$		0			0
$199$		−	274388.i		−	0.491170i	−0.969375	−	0.245585i	$-0.921020\pi$
$199$		−	274388.i		−	0.491170i	0.969375	−	0.245585i	$-0.0789801\pi$
$200$		0			0
$201$	163294.			0.285089
$202$		0			0
$203$		−	635837.i		−	1.08294i
$204$		0			0
$205$		−	860701.i		−	1.43043i
$206$		0			0
$207$			310702.i			0.503985i
$208$		0			0
$209$	80080.5	+	114040.i	0.126812	+	0.180590i
$210$		0			0
$211$	472358.			0.730408			0.365204	−	0.930928i	$-0.380999\pi$
$211$	472358.			0.730408			0.365204	+	0.930928i	$0.380999\pi$
$212$		0			0
$213$	−396746.			−0.599188
$214$		0			0
$215$	123954.			0.182879
$216$		0			0
$217$		−	111923.i		−	0.161351i
$218$		0			0
$219$	40801.0			0.0574858
$220$		0			0
$221$	1.15834e6			1.59534
$222$		0			0
$223$			90508.2i			0.121878i	0.998141	+	0.0609391i	$0.0194095\pi$
$223$			90508.2i			0.121878i	−0.998141	+	0.0609391i	$0.980590\pi$
$224$		0			0
$225$	−16450.4			−0.0216631
$226$		0			0
$227$	849915.			1.09474			0.547370	−	0.836891i	$-0.315629\pi$
$227$	849915.			1.09474			0.547370	+	0.836891i	$0.315629\pi$
$228$		0			0
$229$	−316243.			−0.398503			−0.199251	−	0.979948i	$-0.563851\pi$
$229$	−316243.			−0.398503			−0.199251	+	0.979948i	$0.563851\pi$
$230$		0			0
$231$	388022.	−	272474.i	0.478439	−	0.335966i
$232$		0			0
$233$			702192.i			0.847356i	0.905813	+	0.423678i	$0.139261\pi$
$233$			702192.i			0.847356i	−0.905813	+	0.423678i	$0.860739\pi$
$234$		0			0
$235$			478347.i			0.565032i
$236$		0			0
$237$		−	402509.i		−	0.465484i
$238$		0			0
$239$	321089.			0.363605			0.181803	−	0.983335i	$-0.441807\pi$
$239$	321089.			0.363605			0.181803	+	0.983335i	$0.441807\pi$
$240$		0			0
$241$		−	1.73015e6i		−	1.91885i	−0.281962	−	0.959426i	$-0.590985\pi$
$241$		−	1.73015e6i		−	1.91885i	0.281962	−	0.959426i	$-0.409015\pi$
$242$		0			0
$243$			59049.0i			0.0641500i
$244$		0			0
$245$	24559.6			0.0261401
$246$		0			0
$247$		−	193908.i		−	0.202234i
$248$		0			0
$249$			734890.i			0.751145i
$250$		0			0
$251$			575509.i			0.576591i	0.957542	+	0.288295i	$0.0930885\pi$
$251$			575509.i			0.576591i	−0.957542	+	0.288295i	$0.906912\pi$
$252$		0			0
$253$	−1.25978e6	+	884635.i	−1.23736	+	0.868886i
$254$		0			0
$255$	1.07696e6			1.03717
$256$		0			0
$257$	184062.			0.173833			0.0869164	−	0.996216i	$-0.472299\pi$
$257$	184062.			0.173833			0.0869164	+	0.996216i	$0.472299\pi$
$258$		0			0
$259$	−1.91715e6			−1.77585
$260$		0			0
$261$			392332.i			0.356494i
$262$		0			0
$263$	−2.14183e6			−1.90940			−0.954698	−	0.297575i	$-0.903822\pi$
$263$	−2.14183e6			−1.90940			−0.954698	+	0.297575i	$0.903822\pi$
$264$		0			0
$265$	−1.94723e6			−1.70335
$266$		0			0
$267$		−	254412.i		−	0.218403i
$268$		0			0
$269$	1.53257e6			1.29134			0.645668	−	0.763618i	$-0.276579\pi$
$269$	1.53257e6			1.29134			0.645668	+	0.763618i	$0.276579\pi$
$270$		0			0
$271$	1.81757e6			1.50338			0.751690	−	0.659517i	$-0.229239\pi$
$271$	1.81757e6			1.50338			0.751690	+	0.659517i	$0.229239\pi$
$272$		0			0
$273$	−659773.			−0.535782
$274$		0			0
$275$	−46837.9	−	66700.5i	−0.0373479	−	0.0531860i
$276$		0			0
$277$			256680.i			0.200999i	0.994937	+	0.100499i	$0.0320440\pi$
$277$			256680.i			0.200999i	−0.994937	+	0.100499i	$0.967956\pi$
$278$		0			0
$279$			69060.4i			0.0531152i
$280$		0			0
$281$		−	2.46930e6i		−	1.86555i	−0.360457	−	0.932776i	$-0.617379\pi$
$281$		−	2.46930e6i		−	1.86555i	0.360457	−	0.932776i	$-0.382621\pi$
$282$		0			0
$283$	−1.52706e6			−1.13342			−0.566710	−	0.823917i	$-0.691784\pi$
$283$	−1.52706e6			−1.13342			−0.566710	+	0.823917i	$0.691784\pi$
$284$		0			0
$285$		−	180286.i		−	0.131477i
$286$		0			0
$287$		−	1.95854e6i		−	1.40355i
$288$		0			0
$289$	−2.88263e6			−2.03023
$290$		0			0
$291$			391025.i			0.270690i
$292$		0			0
$293$			1.54689e6i			1.05267i	0.850278	+	0.526334i	$0.176434\pi$
$293$			1.54689e6i			1.05267i	−0.850278	+	0.526334i	$0.823566\pi$
$294$		0			0
$295$		−	42762.6i		−	0.0286094i
$296$		0			0
$297$	−239422.	+	168125.i	−0.157497	+	0.110597i
$298$		0			0
$299$	2.14207e6			1.38566
$300$		0			0
$301$	282058.			0.179441
$302$		0			0
$303$	1.18677e6			0.742610
$304$		0			0
$305$			2.77678e6i			1.70919i
$306$		0			0
$307$	827271.			0.500959			0.250479	−	0.968122i	$-0.419412\pi$
$307$	827271.			0.500959			0.250479	+	0.968122i	$0.419412\pi$
$308$		0			0
$309$	1.73298e6			1.03252
$310$		0			0
$311$			1.85230e6i			1.08595i	0.839748	+	0.542977i	$0.182703\pi$
$311$			1.85230e6i			1.08595i	−0.839748	+	0.542977i	$0.817297\pi$
$312$		0			0
$313$	3.35682e6			1.93672			0.968360	−	0.249557i	$-0.0802852\pi$
$313$	3.35682e6			1.93672			0.968360	+	0.249557i	$0.0802852\pi$
$314$		0			0
$315$	−613422.			−0.348324
$316$		0			0
$317$	1.18358e6			0.661532			0.330766	−	0.943713i	$-0.392693\pi$
$317$	1.18358e6			0.661532			0.330766	+	0.943713i	$0.392693\pi$
$318$		0			0
$319$	−1.59076e6	+	1.11705e6i	−0.875244	+	0.614607i
$320$		0			0
$321$		−	1.43869e6i		−	0.779298i
$322$		0			0
$323$			720247.i			0.384127i
$324$		0			0
$325$			113414.i			0.0595606i
$326$		0			0
$327$	−132061.			−0.0682977
$328$		0			0
$329$			1.08848e6i			0.554412i
$330$		0			0
$331$		−	622103.i		−	0.312099i	−0.987749	−	0.156049i	$-0.950124\pi$
$331$		−	622103.i		−	0.312099i	0.987749	−	0.156049i	$-0.0498759\pi$
$332$		0			0
$333$	1.18294e6			0.584592
$334$		0			0
$335$		−	1.04671e6i		−	0.509581i
$336$		0			0
$337$		−	1.49993e6i		−	0.719445i	−0.933059	−	0.359722i	$-0.882871\pi$
$337$		−	1.49993e6i		−	0.719445i	0.933059	−	0.359722i	$-0.117129\pi$
$338$		0			0
$339$		−	477820.i		−	0.225821i
$340$		0			0
$341$	−280015.	+	196630.i	−0.130405	+	0.0915722i
$342$		0			0
$343$	−2.15043e6			−0.986937
$344$		0			0
$345$	1.99159e6			0.900848
$346$		0			0
$347$	2.76435e6			1.23245			0.616225	−	0.787570i	$-0.288661\pi$
$347$	2.76435e6			1.23245			0.616225	+	0.787570i	$0.288661\pi$
$348$		0			0
$349$		−	367905.i		−	0.161686i	−0.996727	−	0.0808429i	$-0.974239\pi$
$349$		−	367905.i		−	0.161686i	0.996727	−	0.0808429i	$-0.0257612\pi$
$350$		0			0
$351$	407101.			0.176374
$352$		0			0
$353$	−800054.			−0.341730			−0.170865	−	0.985294i	$-0.554656\pi$
$353$	−800054.			−0.341730			−0.170865	+	0.985294i	$0.554656\pi$
$354$		0			0
$355$			2.54312e6i			1.07102i
$356$		0			0
$357$	2.45064e6			1.01767
$358$		0			0
$359$	−973760.			−0.398764			−0.199382	−	0.979922i	$-0.563893\pi$
$359$	−973760.			−0.398764			−0.199382	+	0.979922i	$0.563893\pi$
$360$		0			0
$361$	−2.35553e6			−0.951306
$362$		0			0
$363$	−1.36337e6	−	492081.i	−0.543061	−	0.196006i
$364$		0			0
$365$		−	261533.i		−	0.102753i
$366$		0			0
$367$			1.16077e6i			0.449865i	0.974374	+	0.224932i	$0.0722162\pi$
$367$			1.16077e6i			0.449865i	−0.974374	+	0.224932i	$0.927784\pi$
$368$		0			0
$369$			1.20848e6i			0.462034i
$370$		0			0
$371$	−4.43095e6			−1.67133
$372$		0			0
$373$			4.85089e6i			1.80530i	0.430374	+	0.902651i	$0.358382\pi$
$373$			4.85089e6i			1.80530i	−0.430374	+	0.902651i	$0.641618\pi$
$374$		0			0
$375$		−	1.51707e6i		−	0.557094i
$376$		0			0
$377$	2.70485e6			0.980146
$378$		0			0
$379$			4.42566e6i			1.58263i	0.611408	+	0.791316i	$0.290604\pi$
$379$			4.42566e6i			1.58263i	−0.611408	+	0.791316i	$0.709396\pi$
$380$		0			0
$381$			3.15294e6i			1.11277i
$382$		0			0
$383$			3.74037e6i			1.30292i	0.758684	+	0.651459i	$0.225843\pi$
$383$			3.74037e6i			1.30292i	−0.758684	+	0.651459i	$0.774157\pi$
$384$		0			0
$385$	−1.74655e6	−	2.48720e6i	−0.600521	−	0.855184i
$386$		0			0
$387$	−174039.			−0.0590703
$388$		0			0
$389$	3.93446e6			1.31829			0.659145	−	0.752016i	$-0.270918\pi$
$389$	3.93446e6			1.31829			0.659145	+	0.752016i	$0.270918\pi$
$390$		0			0
$391$	−7.95644e6			−2.63195
$392$		0			0
$393$			645584.i			0.210849i
$394$		0			0
$395$	−2.58007e6			−0.832029
$396$		0			0
$397$	4.01277e6			1.27781			0.638907	−	0.769284i	$-0.279387\pi$
$397$	4.01277e6			1.27781			0.638907	+	0.769284i	$0.279387\pi$
$398$		0			0
$399$		−	410243.i		−	0.129006i
$400$		0			0
$401$	2.17753e6			0.676245			0.338122	−	0.941102i	$-0.390208\pi$
$401$	2.17753e6			0.676245			0.338122	+	0.941102i	$0.390208\pi$
$402$		0			0
$403$	476123.			0.146035
$404$		0			0
$405$	378502.			0.114665
$406$		0			0
$407$	3.36809e6	+	4.79640e6i	1.00785	+	1.43526i
$408$		0			0
$409$		−	4.27963e6i		−	1.26502i	−0.774552	−	0.632511i	$-0.782024\pi$
$409$		−	4.27963e6i		−	1.26502i	0.774552	−	0.632511i	$-0.217976\pi$
$410$		0			0
$411$			1.48552e6i			0.433785i
$412$		0			0
$413$		−	97306.7i		−	0.0280716i
$414$		0			0
$415$	4.71061e6			1.34263
$416$		0			0
$417$			2.97134e6i			0.836782i
$418$		0			0
$419$		−	2.20506e6i		−	0.613600i	−0.951774	−	0.306800i	$-0.900742\pi$
$419$		−	2.20506e6i		−	0.613600i	0.951774	−	0.306800i	$-0.0992582\pi$
$420$		0			0
$421$	1.79770e6			0.494324			0.247162	−	0.968974i	$-0.420502\pi$
$421$	1.79770e6			0.494324			0.247162	+	0.968974i	$0.420502\pi$
$422$		0			0
$423$		−	671630.i		−	0.182507i
$424$		0			0
$425$		−	421262.i		−	0.113131i
$426$		0			0
$427$			6.31859e6i			1.67707i
$428$		0			0
$429$	1.15911e6	+	1.65065e6i	0.304074	+	0.433023i
$430$		0			0
$431$	−6.79438e6			−1.76180			−0.880900	−	0.473302i	$-0.843062\pi$
$431$	−6.79438e6			−1.76180			−0.880900	+	0.473302i	$0.843062\pi$
$432$		0			0
$433$	−3.59923e6			−0.922550			−0.461275	−	0.887257i	$-0.652608\pi$
$433$	−3.59923e6			−0.922550			−0.461275	+	0.887257i	$0.652608\pi$
$434$		0			0
$435$	2.51483e6			0.637215
$436$		0			0
$437$			1.33193e6i			0.333639i
$438$		0			0
$439$	1.21662e6			0.301297			0.150649	−	0.988587i	$-0.451864\pi$
$439$	1.21662e6			0.301297			0.150649	+	0.988587i	$0.451864\pi$
$440$		0			0
$441$	−34483.3			−0.00844331
$442$		0			0
$443$		−	67871.6i		−	0.0164316i	−0.999966	−	0.00821578i	$-0.997385\pi$
$443$		−	67871.6i		−	0.0164316i	0.999966	−	0.00821578i	$-0.00261519\pi$
$444$		0			0
$445$	−1.63077e6			−0.390384
$446$		0			0
$447$	−1.92973e6			−0.456801
$448$		0			0
$449$	−6.97133e6			−1.63192			−0.815961	−	0.578106i	$-0.803792\pi$
$449$	−6.97133e6			−1.63192			−0.815961	+	0.578106i	$0.803792\pi$
$450$		0			0
$451$	−4.89995e6	+	3.44081e6i	−1.13436	+	0.796560i
$452$		0			0
$453$			1.44759e6i			0.331436i
$454$		0			0
$455$			4.22912e6i			0.957682i
$456$		0			0
$457$			5.43056e6i			1.21634i	0.793808	+	0.608169i	$0.208096\pi$
$457$			5.43056e6i			1.21634i	−0.793808	+	0.608169i	$0.791904\pi$
$458$		0			0
$459$	−1.51212e6			−0.335008
$460$		0			0
$461$			1.22198e6i			0.267801i	0.990995	+	0.133901i	$0.0427503\pi$
$461$			1.22198e6i			0.267801i	−0.990995	+	0.133901i	$0.957250\pi$
$462$		0			0
$463$			5.15778e6i			1.11818i	0.829108	+	0.559089i	$0.188849\pi$
$463$			5.15778e6i			1.11818i	−0.829108	+	0.559089i	$0.811151\pi$
$464$		0			0
$465$	442674.			0.0949406
$466$		0			0
$467$		−	2.85758e6i		−	0.606326i	−0.952939	−	0.303163i	$-0.901957\pi$
$467$		−	2.85758e6i		−	0.606326i	0.952939	−	0.303163i	$-0.0980426\pi$
$468$		0			0
$469$		−	2.38180e6i		−	0.500003i
$470$		0			0
$471$		−	4.71781e6i		−	0.979914i
$472$		0			0
$473$	−495527.	−	705665.i	−0.101839	−	0.145026i
$474$		0			0
$475$	−70520.3			−0.0143410
$476$		0			0
$477$	2.73404e6			0.550185
$478$		0			0
$479$	−1.05362e6			−0.209820			−0.104910	−	0.994482i	$-0.533455\pi$
$479$	−1.05362e6			−0.209820			−0.104910	+	0.994482i	$0.533455\pi$
$480$		0			0
$481$		−	8.15555e6i		−	1.60728i
$482$		0			0
$483$	4.53188e6			0.883915
$484$		0			0
$485$	2.50645e6			0.483844
$486$		0			0
$487$		−	3.56735e6i		−	0.681591i	−0.940137	−	0.340795i	$-0.889304\pi$
$487$		−	3.56735e6i		−	0.681591i	0.940137	−	0.340795i	$-0.110696\pi$
$488$		0			0
$489$	1.69208e6			0.319999
$490$		0			0
$491$	4.15903e6			0.778553			0.389276	−	0.921121i	$-0.372725\pi$
$491$	4.15903e6			0.778553			0.389276	+	0.921121i	$0.372725\pi$
$492$		0			0
$493$	−1.00468e7			−1.86171
$494$		0			0
$495$	1.07768e6	+	1.53469e6i	0.197686	+	0.281518i
$496$		0			0
$497$			5.78691e6i			1.05089i
$498$		0			0
$499$			2.29474e6i			0.412555i	0.978494	+	0.206277i	$0.0661349\pi$
$499$			2.29474e6i			0.412555i	−0.978494	+	0.206277i	$0.933865\pi$
$500$		0			0
$501$			4.55843e6i			0.811374i
$502$		0			0
$503$	3.63437e6			0.640485			0.320242	−	0.947336i	$-0.396236\pi$
$503$	3.63437e6			0.640485			0.320242	+	0.947336i	$0.396236\pi$
$504$		0			0
$505$		−	7.60716e6i		−	1.32738i
$506$		0			0
$507$			534959.i			0.0924273i
$508$		0			0
$509$	−3.03436e6			−0.519126			−0.259563	−	0.965726i	$-0.583579\pi$
$509$	−3.03436e6			−0.519126			−0.259563	+	0.965726i	$0.583579\pi$
$510$		0			0
$511$		−	595121.i		−	0.100821i
$512$		0			0
$513$			253133.i			0.0424674i
$514$		0			0
$515$		−	1.11083e7i		−	1.84557i
$516$		0			0
$517$	2.72322e6	−	1.91228e6i	0.448080	−	0.314647i
$518$		0			0
$519$	−1.87623e6			−0.305752
$520$		0			0
$521$	2.39982e6			0.387332			0.193666	−	0.981068i	$-0.437962\pi$
$521$	2.39982e6			0.387332			0.193666	+	0.981068i	$0.437962\pi$
$522$		0			0
$523$	1.00001e7			1.59864			0.799322	−	0.600903i	$-0.205192\pi$
$523$	1.00001e7			1.59864			0.799322	+	0.600903i	$0.205192\pi$
$524$		0			0
$525$			239945.i			0.0379939i
$526$		0			0
$527$	−1.76850e6			−0.277382
$528$		0			0
$529$	−8.27722e6			−1.28601
$530$		0			0
$531$			60041.4i			0.00924091i
$532$		0			0
$533$	8.33163e6			1.27032
$534$		0			0
$535$	−9.22192e6			−1.39295
$536$		0			0
$537$	6.12529e6			0.916623
$538$		0			0
$539$	−98181.5	−	139817.i	−0.0145565	−	0.0207295i
$540$		0			0
$541$		−	6.55644e6i		−	0.963107i	−0.876417	−	0.481554i	$-0.840073\pi$
$541$		−	6.55644e6i		−	0.963107i	0.876417	−	0.481554i	$-0.159927\pi$
$542$		0			0
$543$		−	514736.i		−	0.0749177i
$544$		0			0
$545$			846507.i			0.122079i
$546$		0			0
$547$	−6.23861e6			−0.891496			−0.445748	−	0.895158i	$-0.647062\pi$
$547$	−6.23861e6			−0.891496			−0.445748	+	0.895158i	$0.647062\pi$
$548$		0			0
$549$		−	3.89877e6i		−	0.552074i
$550$		0			0
$551$			1.68186e6i			0.236000i
$552$		0			0
$553$	−5.87098e6			−0.816390
$554$		0			0
$555$		−	7.58261e6i		−	1.04493i
$556$		0			0
$557$		−	272405.i		−	0.0372029i	−0.999827	−	0.0186015i	$-0.994079\pi$
$557$		−	272405.i		−	0.0372029i	0.999827	−	0.0186015i	$-0.00592137\pi$
$558$		0			0
$559$			1.19988e6i			0.162408i
$560$		0			0
$561$	−4.30534e6	−	6.13111e6i	−0.577565	−	0.822493i
$562$		0			0
$563$	950292.			0.126353			0.0631766	−	0.998002i	$-0.479877\pi$
$563$	950292.			0.126353			0.0631766	+	0.998002i	$0.479877\pi$
$564$		0			0
$565$	−3.06280e6			−0.403644
$566$		0			0
$567$	861285.			0.112510
$568$		0			0
$569$		−	780029.i		−	0.101002i	−0.998724	−	0.0505010i	$-0.983918\pi$
$569$		−	780029.i		−	0.101002i	0.998724	−	0.0505010i	$-0.0160818\pi$
$570$		0			0
$571$	−7.08627e6			−0.909552			−0.454776	−	0.890606i	$-0.650281\pi$
$571$	−7.08627e6			−0.909552			−0.454776	+	0.890606i	$0.650281\pi$
$572$		0			0
$573$	−239541.			−0.0304785
$574$		0			0
$575$		−	779025.i		−	0.0982611i
$576$		0			0
$577$	−4.37896e6			−0.547560			−0.273780	−	0.961792i	$-0.588274\pi$
$577$	−4.37896e6			−0.547560			−0.273780	+	0.961792i	$0.588274\pi$
$578$		0			0
$579$	1.44619e6			0.179279
$580$		0			0
$581$	1.07191e7			1.31740
$582$		0			0
$583$	7.78440e6	+	1.10855e7i	0.948536	+	1.35078i
$584$		0			0
$585$		−	2.60950e6i		−	0.315259i
$586$		0			0
$587$			267180.i			0.0320043i	0.999872	+	0.0160021i	$0.00509386\pi$
$587$			267180.i			0.0320043i	−0.999872	+	0.0160021i	$0.994906\pi$
$588$		0			0
$589$			296051.i			0.0351623i
$590$		0			0
$591$	−431697.			−0.0508406
$592$		0			0
$593$		−	8.42013e6i		−	0.983291i	−0.870795	−	0.491646i	$-0.836396\pi$
$593$		−	8.42013e6i		−	0.983291i	0.870795	−	0.491646i	$-0.163604\pi$
$594$		0			0
$595$		−	1.57085e7i		−	1.81904i
$596$		0			0
$597$	2.46949e6			0.283577
$598$		0			0
$599$			2.96617e6i			0.337776i	0.985635	+	0.168888i	$0.0540177\pi$
$599$			2.96617e6i			0.337776i	−0.985635	+	0.168888i	$0.945982\pi$
$600$		0			0
$601$			5.18904e6i			0.586004i	0.956112	+	0.293002i	$0.0946543\pi$
$601$			5.18904e6i			0.586004i	−0.956112	+	0.293002i	$0.905346\pi$
$602$		0			0
$603$			1.46965e6i			0.164596i
$604$		0			0
$605$	−3.15422e6	+	8.73917e6i	−0.350351	+	0.970692i
$606$		0			0
$607$	−957403.			−0.105469			−0.0527343	−	0.998609i	$-0.516794\pi$
$607$	−957403.			−0.105469			−0.0527343	+	0.998609i	$0.516794\pi$
$608$		0			0
$609$	5.72253e6			0.625238
$610$		0			0
$611$	−4.63042e6			−0.501784
$612$		0			0
$613$		−	7.95431e6i		−	0.854971i	−0.904022	−	0.427486i	$-0.859399\pi$
$613$		−	7.95431e6i		−	0.854971i	0.904022	−	0.427486i	$-0.140601\pi$
$614$		0			0
$615$	7.74631e6			0.825861
$616$		0			0
$617$	−2.49277e6			−0.263614			−0.131807	−	0.991275i	$-0.542078\pi$
$617$	−2.49277e6			−0.263614			−0.131807	+	0.991275i	$0.542078\pi$
$618$		0			0
$619$		−	8.20193e6i		−	0.860378i	−0.902739	−	0.430189i	$-0.858447\pi$
$619$		−	8.20193e6i		−	0.860378i	0.902739	−	0.430189i	$-0.141553\pi$
$620$		0			0
$621$	−2.79632e6			−0.290976
$622$		0			0
$623$	−3.71083e6			−0.383046
$624$		0			0
$625$	−1.03590e7			−1.06077
$626$		0			0
$627$	−1.02636e6	+	720725.i	−0.104263	+	0.0732151i
$628$		0			0
$629$			3.02927e7i			3.05289i
$630$		0			0
$631$			1.22176e6i			0.122155i	0.998133	+	0.0610776i	$0.0194537\pi$
$631$			1.22176e6i			0.122155i	−0.998133	+	0.0610776i	$0.980546\pi$
$632$		0			0
$633$			4.25122e6i			0.421701i
$634$		0			0
$635$	2.02102e7			1.98901
$636$		0			0
$637$			237738.i			0.0232140i
$638$		0			0
$639$		−	3.57071e6i		−	0.345942i
$640$		0			0
$641$	1.19673e7			1.15041			0.575204	−	0.818010i	$-0.304923\pi$
$641$	1.19673e7			1.15041			0.575204	+	0.818010i	$0.304923\pi$
$642$		0			0
$643$			1.43285e7i			1.36670i	0.730089	+	0.683352i	$0.239478\pi$
$643$			1.43285e7i			1.36670i	−0.730089	+	0.683352i	$0.760522\pi$
$644$		0			0
$645$			1.11558e6i			0.105585i
$646$		0			0
$647$		−	1.98105e7i		−	1.86052i	−0.366904	−	0.930259i	$-0.619582\pi$
$647$		−	1.98105e7i		−	1.86052i	0.366904	−	0.930259i	$-0.380418\pi$
$648$		0			0
$649$	−243446.	+	170951.i	−0.0226877	+	0.0159316i
$650$		0			0
$651$	1.00731e6			0.0931561
$652$		0			0
$653$	1.60712e7			1.47491			0.737455	−	0.675396i	$-0.236027\pi$
$653$	1.60712e7			1.47491			0.737455	+	0.675396i	$0.236027\pi$
$654$		0			0
$655$	4.13817e6			0.376881
$656$		0			0
$657$			367209.i			0.0331894i
$658$		0			0
$659$	1.08585e7			0.973991			0.486996	−	0.873404i	$-0.338093\pi$
$659$	1.08585e7			0.973991			0.486996	+	0.873404i	$0.338093\pi$
$660$		0			0
$661$	−1.88237e7			−1.67572			−0.837860	−	0.545885i	$-0.816194\pi$
$661$	−1.88237e7			−1.67572			−0.837860	+	0.545885i	$0.816194\pi$
$662$		0			0
$663$			1.04250e7i			0.921072i
$664$		0			0
$665$	−2.62964e6			−0.230591
$666$		0			0
$667$	−1.85792e7			−1.61701
$668$		0			0
$669$	−814574.			−0.0703664
$670$		0			0
$671$	1.58081e7	−	1.11007e7i	1.35542	−	0.951792i
$672$		0			0
$673$			8.23447e6i			0.700806i	0.936599	+	0.350403i	$0.113955\pi$
$673$			8.23447e6i			0.700806i	−0.936599	+	0.350403i	$0.886045\pi$
$674$		0			0
$675$		−	148054.i		−	0.0125072i
$676$		0			0
$677$		−	1.99683e7i		−	1.67444i	−0.546866	−	0.837220i	$-0.684179\pi$
$677$		−	1.99683e7i		−	1.67444i	0.546866	−	0.837220i	$-0.315821\pi$
$678$		0			0
$679$	5.70347e6			0.474750
$680$		0			0
$681$			7.64923e6i			0.632048i
$682$		0			0
$683$		−	4.42724e6i		−	0.363146i	−0.983377	−	0.181573i	$-0.941881\pi$
$683$		−	4.42724e6i		−	0.363146i	0.983377	−	0.181573i	$-0.0581189\pi$
$684$		0			0
$685$	9.52214e6			0.775368
$686$		0			0
$687$		−	2.84618e6i		−	0.230076i
$688$		0			0
$689$		−	1.88493e7i		−	1.51268i
$690$		0			0
$691$		−	4.21923e6i		−	0.336154i	−0.985774	−	0.168077i	$-0.946244\pi$
$691$		−	4.21923e6i		−	0.336154i	0.985774	−	0.168077i	$-0.0537557\pi$
$692$		0			0
$693$	2.45226e6	+	3.49220e6i	0.193970	+	0.276227i
$694$		0			0
$695$	1.90462e7			1.49570
$696$		0			0
$697$	−3.09467e7			−2.41286
$698$		0			0
$699$	−6.31973e6			−0.489221
$700$		0			0
$701$			2.17542e6i			0.167204i	0.996499	+	0.0836022i	$0.0266425\pi$
$701$			2.17542e6i			0.167204i	−0.996499	+	0.0836022i	$0.973357\pi$
$702$		0			0
$703$	5.07107e6			0.387001
$704$		0			0
$705$	−4.30512e6			−0.326221
$706$		0			0
$707$		−	1.73102e7i		−	1.30243i
$708$		0			0
$709$	1.18718e7			0.886953			0.443477	−	0.896286i	$-0.353745\pi$
$709$	1.18718e7			0.886953			0.443477	+	0.896286i	$0.353745\pi$
$710$		0			0
$711$	3.62258e6			0.268748
$712$		0			0
$713$	−3.27042e6			−0.240924
$714$		0			0
$715$	1.05806e7	−	7.42982e6i	0.774006	−	0.543517i
$716$		0			0
$717$			2.88980e6i			0.209928i
$718$		0			0
$719$		−	5.72667e6i		−	0.413124i	−0.978434	−	0.206562i	$-0.933773\pi$
$719$		−	5.72667e6i		−	0.413124i	0.978434	−	0.206562i	$-0.0662275\pi$
$720$		0			0
$721$		−	2.52771e7i		−	1.81088i
$722$		0			0
$723$	1.55714e7			1.10785
$724$		0			0
$725$		−	983697.i		−	0.0695050i
$726$		0			0
$727$		−	1.42526e7i		−	1.00014i	−0.865986	−	0.500068i	$-0.833308\pi$
$727$		−	1.42526e7i		−	1.00014i	0.865986	−	0.500068i	$-0.166692\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$		−	4.45678e6i		−	0.308481i
$732$		0			0
$733$			4.25872e6i			0.292765i	0.989228	+	0.146382i	$0.0467630\pi$
$733$			4.25872e6i			0.292765i	−0.989228	+	0.146382i	$0.953237\pi$
$734$		0			0
$735$			221037.i			0.0150920i
$736$		0			0
$737$	−5.95888e6	+	4.18440e6i	−0.404107	+	0.283769i
$738$		0			0
$739$	−1.78965e7			−1.20547			−0.602737	−	0.797940i	$-0.705923\pi$
$739$	−1.78965e7			−1.20547			−0.602737	+	0.797940i	$0.705923\pi$
$740$		0			0
$741$	1.74518e6			0.116760
$742$		0			0
$743$	−1.94031e7			−1.28943			−0.644716	−	0.764422i	$-0.723025\pi$
$743$	−1.94031e7			−1.28943			−0.644716	+	0.764422i	$0.723025\pi$
$744$		0			0
$745$			1.23695e7i			0.816508i
$746$		0			0
$747$	−6.61401e6			−0.433674
$748$		0			0
$749$	−2.09846e7			−1.36677
$750$		0			0
$751$		−	1.95850e7i		−	1.26714i	−0.773687	−	0.633568i	$-0.781590\pi$
$751$		−	1.95850e7i		−	1.26714i	0.773687	−	0.633568i	$-0.218410\pi$
$752$		0			0
$753$	−5.17958e6			−0.332895
$754$		0			0
$755$	9.27898e6			0.592424
$756$		0			0
$757$	1.46642e7			0.930074			0.465037	−	0.885291i	$-0.346041\pi$
$757$	1.46642e7			0.930074			0.465037	+	0.885291i	$0.346041\pi$
$758$		0			0
$759$	−7.96172e6	−	1.13380e7i	−0.501652	−	0.714388i
$760$		0			0
$761$		−	1.24400e7i		−	0.778677i	−0.921095	−	0.389338i	$-0.872704\pi$
$761$		−	1.24400e7i		−	0.778677i	0.921095	−	0.389338i	$-0.127296\pi$
$762$		0			0
$763$			1.92624e6i			0.119784i
$764$		0			0
$765$			9.69265e6i			0.598810i
$766$		0			0
$767$	413944.			0.0254070
$768$		0			0
$769$		−	2.64466e7i		−	1.61270i	−0.591437	−	0.806351i	$-0.701439\pi$
$769$		−	2.64466e7i		−	1.61270i	0.591437	−	0.806351i	$-0.298561\pi$
$770$		0			0
$771$			1.65656e6i			0.100362i
$772$		0			0
$773$	1.12409e7			0.676634			0.338317	−	0.941032i	$-0.390142\pi$
$773$	1.12409e7			0.676634			0.338317	+	0.941032i	$0.390142\pi$
$774$		0			0
$775$		−	173156.i		−	0.0103558i
$776$		0			0
$777$		−	1.72543e7i		−	1.02529i
$778$		0			0
$779$			5.18055e6i			0.305867i
$780$		0			0
$781$	1.44779e7	−	1.01666e7i	0.849336	−	0.596414i
$782$		0			0
$783$	−3.53099e6			−0.205822
$784$		0			0
$785$	−3.02410e7			−1.75155
$786$		0			0
$787$	2.13056e7			1.22619			0.613093	−	0.790011i	$-0.289925\pi$
$787$	2.13056e7			1.22619			0.613093	+	0.790011i	$0.289925\pi$
$788$		0			0
$789$		−	1.92765e7i		−	1.10239i
$790$		0			0
$791$	−6.96945e6			−0.396057
$792$		0			0
$793$	−2.68793e7			−1.51787
$794$		0			0
$795$		−	1.75251e7i		−	0.983427i
$796$		0			0
$797$	−5.28664e6			−0.294805			−0.147402	−	0.989077i	$-0.547091\pi$
$797$	−5.28664e6			−0.294805			−0.147402	+	0.989077i	$0.547091\pi$
$798$		0			0
$799$	1.71991e7			0.953099
$800$		0			0
$801$	2.28970e6			0.126095
$802$		0			0
$803$	−1.48890e6	+	1.04552e6i	−0.0814848	+	0.0572196i
$804$		0			0
$805$		−	2.90492e7i		−	1.57995i
$806$		0			0
$807$			1.37931e7i			0.745554i
$808$		0			0
$809$		−	2.12970e6i		−	0.114406i	−0.998363	−	0.0572028i	$-0.981782\pi$
$809$		−	2.12970e6i		−	0.114406i	0.998363	−	0.0572028i	$-0.0182182\pi$
$810$		0			0
$811$	−7.76499e6			−0.414561			−0.207281	−	0.978282i	$-0.566461\pi$
$811$	−7.76499e6			−0.414561			−0.207281	+	0.978282i	$0.566461\pi$
$812$		0			0
$813$			1.63582e7i			0.867976i
$814$		0			0
$815$		−	1.08462e7i		−	0.571982i
$816$		0			0
$817$	−746076.			−0.0391046
$818$		0			0
$819$		−	5.93796e6i		−	0.309334i
$820$		0			0
$821$			3.48177e7i			1.80278i	0.433008	+	0.901390i	$0.357452\pi$
$821$			3.48177e7i			1.80278i	−0.433008	+	0.901390i	$0.642548\pi$
$822$		0			0
$823$		−	3.01079e6i		−	0.154946i	−0.996994	−	0.0774732i	$-0.975315\pi$
$823$		−	3.01079e6i		−	0.154946i	0.996994	−	0.0774732i	$-0.0246852\pi$
$824$		0			0
$825$	600305.	−	421542.i	0.0307070	−	0.0215628i
$826$		0			0
$827$	−1.19605e7			−0.608114			−0.304057	−	0.952654i	$-0.598341\pi$
$827$	−1.19605e7			−0.608114			−0.304057	+	0.952654i	$0.598341\pi$
$828$		0			0
$829$	7.31948e6			0.369908			0.184954	−	0.982747i	$-0.440786\pi$
$829$	7.31948e6			0.369908			0.184954	+	0.982747i	$0.440786\pi$
$830$		0			0
$831$	−2.31012e6			−0.116047
$832$		0			0
$833$		−	883048.i		−	0.0440932i
$834$		0			0
$835$	2.92193e7			1.45029
$836$		0			0
$837$	−621544.			−0.0306661
$838$		0			0
$839$			1.99588e7i			0.978881i	0.872037	+	0.489441i	$0.162799\pi$
$839$			1.99588e7i			0.978881i	−0.872037	+	0.489441i	$0.837201\pi$
$840$		0			0
$841$	−2.94938e6			−0.143794
$842$		0			0
$843$	2.22237e7			1.07708
$844$		0			0
$845$	3.42906e6			0.165209
$846$		0			0
$847$	−7.17747e6	+	1.98861e7i	−0.343766	+	0.952447i
$848$		0			0
$849$		−	1.37436e7i		−	0.654380i
$850$		0			0
$851$			5.60192e7i			2.65163i
$852$		0			0
$853$			2.31219e7i			1.08805i	0.839068	+	0.544027i	$0.183101\pi$
$853$			2.31219e7i			1.08805i	−0.839068	+	0.544027i	$0.816899\pi$
$854$		0			0
$855$	1.62257e6			0.0759082
$856$		0			0
$857$		−	2.28728e7i		−	1.06382i	−0.846802	−	0.531908i	$-0.821475\pi$
$857$		−	2.28728e7i		−	1.06382i	0.846802	−	0.531908i	$-0.178525\pi$
$858$		0			0
$859$			3.87503e7i			1.79181i	0.444244	+	0.895906i	$0.353473\pi$
$859$			3.87503e7i			1.79181i	−0.444244	+	0.895906i	$0.646527\pi$
$860$		0			0
$861$	1.76268e7			0.810338
$862$		0			0
$863$		−	1.62061e7i		−	0.740714i	−0.928890	−	0.370357i	$-0.879235\pi$
$863$		−	1.62061e7i		−	0.740714i	0.928890	−	0.370357i	$-0.120765\pi$
$864$		0			0
$865$			1.20266e7i			0.546515i
$866$		0			0
$867$		−	2.59437e7i		−	1.17215i
$868$		0			0
$869$	1.03143e7	+	1.46883e7i	0.463329	+	0.659813i
$870$		0			0
$871$	1.01322e7			0.452540
$872$		0			0
$873$	−3.51923e6			−0.156283
$874$		0			0
$875$	−2.21279e7			−0.977059
$876$		0			0
$877$			4.65570e6i			0.204403i	0.994764	+	0.102201i	$0.0325886\pi$
$877$			4.65570e6i			0.204403i	−0.994764	+	0.102201i	$0.967411\pi$
$878$		0			0
$879$	−1.39221e7			−0.607758
$880$		0			0
$881$	−2.83054e7			−1.22866			−0.614328	−	0.789051i	$-0.710573\pi$
$881$	−2.83054e7			−1.22866			−0.614328	+	0.789051i	$0.710573\pi$
$882$		0			0
$883$		−	2.16701e7i		−	0.935318i	−0.883909	−	0.467659i	$-0.845097\pi$
$883$		−	2.16701e7i		−	0.935318i	0.883909	−	0.467659i	$-0.154903\pi$
$884$		0			0
$885$	384863.			0.0165176
$886$		0			0
$887$	−8.18979e6			−0.349513			−0.174757	−	0.984612i	$-0.555914\pi$
$887$	−8.18979e6			−0.349513			−0.174757	+	0.984612i	$0.555914\pi$
$888$		0			0
$889$	4.59886e7			1.95162
$890$		0			0
$891$	−1.51313e6	−	2.15480e6i	−0.0638530	−	0.0909312i
$892$		0			0
$893$		−	2.87917e6i		−	0.120820i
$894$		0			0
$895$		−	3.92628e7i		−	1.63842i
$896$		0			0
$897$			1.92786e7i			0.800010i
$898$		0			0
$899$	−4.12965e6			−0.170417
$900$		0			0
$901$			7.00132e7i			2.87321i
$902$		0			0
$903$			2.53852e6i			0.103600i
$904$		0			0
$905$	−3.29943e6			−0.133911
$906$		0			0
$907$			4.23143e7i			1.70793i	0.520334	+	0.853963i	$0.325808\pi$
$907$			4.23143e7i			1.70793i	−0.520334	+	0.853963i	$0.674192\pi$
$908$		0			0
$909$			1.06810e7i			0.428746i
$910$		0			0
$911$		−	1.51432e7i		−	0.604536i	−0.953223	−	0.302268i	$-0.902256\pi$
$911$		−	1.51432e7i		−	0.604536i	0.953223	−	0.302268i	$-0.0977438\pi$
$912$		0			0
$913$	−1.88315e7	−	2.68174e7i	−0.747667	−	1.06473i
$914$		0			0
$915$	−2.49910e7			−0.986803
$916$		0			0
$917$	9.41645e6			0.369797
$918$		0			0
$919$	2.73709e7			1.06906			0.534529	−	0.845150i	$-0.320489\pi$
$919$	2.73709e7			1.06906			0.534529	+	0.845150i	$0.320489\pi$
$920$		0			0
$921$			7.44544e6i			0.289229i
$922$		0			0
$923$	−2.46176e7			−0.951132
$924$		0			0
$925$	−2.96600e6			−0.113977
$926$		0			0
$927$			1.55968e7i			0.596123i
$928$		0			0
$929$	−3.09122e7			−1.17514			−0.587571	−	0.809173i	$-0.699916\pi$
$929$	−3.09122e7			−1.17514			−0.587571	+	0.809173i	$0.699916\pi$
$930$		0			0
$931$	−147824.			−0.00558948
$932$		0			0
$933$	−1.66707e7			−0.626975
$934$		0			0
$935$	−3.93002e7	+	2.75971e7i	−1.47016	+	1.03237i
$936$		0			0
$937$			2.17151e7i			0.808003i	0.914758	+	0.404001i	$0.132381\pi$
$937$			2.17151e7i			0.808003i	−0.914758	+	0.404001i	$0.867619\pi$
$938$		0			0
$939$			3.02114e7i			1.11817i
$940$		0			0
$941$		−	3.09366e7i		−	1.13893i	−0.822014	−	0.569467i	$-0.807150\pi$
$941$		−	3.09366e7i		−	1.13893i	0.822014	−	0.569467i	$-0.192850\pi$
$942$		0			0
$943$	−5.72287e7			−2.09573
$944$		0			0
$945$		−	5.52080e6i		−	0.201105i
$946$		0			0
$947$			3.97255e7i			1.43944i	0.694263	+	0.719721i	$0.255730\pi$
$947$			3.97255e7i			1.43944i	−0.694263	+	0.719721i	$0.744270\pi$
$948$		0			0
$949$	2.53165e6			0.0912510
$950$		0			0
$951$			1.06523e7i			0.381936i
$952$		0			0
$953$		−	1.42156e7i		−	0.507031i	−0.967331	−	0.253515i	$-0.918413\pi$
$953$		−	1.42156e7i		−	0.507031i	0.967331	−	0.253515i	$-0.0815869\pi$
$954$		0			0
$955$			1.53545e6i			0.0544787i
$956$		0			0
$957$	−1.00535e7	−	1.43169e7i	−0.354844	−	0.505322i
$958$		0			0
$959$	2.16677e7			0.760794
$960$		0			0
$961$	2.79022e7			0.974609
$962$		0			0
$963$	1.29482e7			0.449928
$964$		0			0
$965$		−	9.27002e6i		−	0.320452i
$966$		0			0
$967$	−1.09098e7			−0.375191			−0.187595	−	0.982246i	$-0.560069\pi$
$967$	−1.09098e7			−0.375191			−0.187595	+	0.982246i	$0.560069\pi$
$968$		0			0
$969$	−6.48222e6			−0.221776
$970$		0			0
$971$		−	1.90463e7i		−	0.648281i	−0.946009	−	0.324140i	$-0.894925\pi$
$971$		−	1.90463e7i		−	0.648281i	0.946009	−	0.324140i	$-0.105075\pi$
$972$		0			0
$973$	4.33398e7			1.46759
$974$		0			0
$975$	−1.02073e6			−0.0343873
$976$		0			0
$977$	1.99539e7			0.668792			0.334396	−	0.942433i	$-0.391468\pi$
$977$	1.99539e7			0.668792			0.334396	+	0.942433i	$0.391468\pi$
$978$		0			0
$979$	6.51928e6	+	9.28392e6i	0.217392	+	0.309581i
$980$		0			0
$981$		−	1.18855e6i		−	0.0394317i
$982$		0			0
$983$			1.51367e7i			0.499630i	0.968294	+	0.249815i	$0.0803698\pi$
$983$			1.51367e7i			0.499630i	−0.968294	+	0.249815i	$0.919630\pi$
$984$		0			0
$985$			2.76716e6i			0.0908748i
$986$		0			0
$987$	−9.79636e6			−0.320090
$988$		0			0
$989$		−	8.24177e6i		−	0.267935i
$990$		0			0
$991$			5.21787e7i			1.68776i	0.536536	+	0.843878i	$0.319733\pi$
$991$			5.21787e7i			1.68776i	−0.536536	+	0.843878i	$0.680267\pi$
$992$		0			0
$993$	5.59893e6			0.180190
$994$		0			0
$995$		−	1.58293e7i		−	0.506879i
$996$		0			0
$997$			6.52302e6i			0.207831i	0.994586	+	0.103916i	$0.0331372\pi$
$997$			6.52302e6i			0.207831i	−0.994586	+	0.103916i	$0.966863\pi$
$998$		0			0
$999$			1.06465e7i			0.337514i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.o.b.175.12	yes	40
4.3	odd	2	inner	528.6.o.b.175.33	yes	40
11.10	odd	2	inner	528.6.o.b.175.34	yes	40
44.43	even	2	inner	528.6.o.b.175.11	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.6.o.b.175.11	✓	40	44.43	even	2	inner
528.6.o.b.175.12	yes	40	1.1	even	1	trivial
528.6.o.b.175.33	yes	40	4.3	odd	2	inner
528.6.o.b.175.34	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)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +57.6896 q^{5} +131.273 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} +57.6896 q^{5} +131.273 q^{7} -81.0000 q^{9} +(-230.624 - 328.426i) q^{11} +558.438i q^{13} +519.207i q^{15} -2074.24i q^{17} -347.233 q^{19} +1181.46i q^{21} -3835.83i q^{23} +203.092 q^{25} -729.000i q^{27} -4843.61i q^{29} -852.598i q^{31} +(2955.83 - 2075.62i) q^{33} +7573.12 q^{35} -14604.2 q^{37} -5025.94 q^{39} -14919.5i q^{41} +2148.63 q^{43} -4672.86 q^{45} +8291.73i q^{47} +425.720 q^{49} +18668.2 q^{51} -33753.6 q^{53} +(-13304.6 - 18946.7i) q^{55} -3125.10i q^{57} -741.252i q^{59} +48133.0i q^{61} -10633.1 q^{63} +32216.1i q^{65} -18143.8i q^{67} +34522.4 q^{69} +44082.9i q^{71} -4533.44i q^{73} +1827.83i q^{75} +(-30274.9 - 43113.6i) q^{77} -44723.3 q^{79} +6561.00 q^{81} +81654.4 q^{83} -119662. i q^{85} +43592.5 q^{87} -28268.0 q^{89} +73308.1i q^{91} +7673.38 q^{93} -20031.8 q^{95} +43447.2 q^{97} +(18680.6 + 26602.5i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) \) 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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