LMFDB - Embedded newform 450.6.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.a (trivial)

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:	yes
Analytic conductor:	$72.1727189158$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1249}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 312 $ x^2 - x - 312
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$18.1706$ of defining polynomial
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} -119.706 q^{7} -64.0000 q^{8} +O(q^{10})$	$q-4.00000 q^{2} +16.0000 q^{4} -119.706 q^{7} -64.0000 q^{8} -263.706 q^{11} +851.118 q^{13} +478.824 q^{14} +256.000 q^{16} -1287.12 q^{17} +2060.47 q^{19} +1054.82 q^{22} -55.5284 q^{23} -3404.47 q^{26} -1915.30 q^{28} +5986.06 q^{29} +4781.76 q^{31} -1024.00 q^{32} +5148.47 q^{34} -12150.4 q^{37} -8241.89 q^{38} -18500.0 q^{41} -2188.47 q^{43} -4219.30 q^{44} +222.113 q^{46} +5597.76 q^{47} -2477.48 q^{49} +13617.9 q^{52} +26463.4 q^{53} +7661.18 q^{56} -23944.2 q^{58} -20825.6 q^{59} +45525.8 q^{61} -19127.1 q^{62} +4096.00 q^{64} -34354.7 q^{67} -20593.9 q^{68} +57489.6 q^{71} -26956.8 q^{73} +48601.7 q^{74} +32967.5 q^{76} +31567.2 q^{77} +42097.8 q^{79} +74000.0 q^{82} -101733. q^{83} +8753.86 q^{86} +16877.2 q^{88} +65551.2 q^{89} -101884. q^{91} -888.454 q^{92} -22391.1 q^{94} -82780.7 q^{97} +9909.92 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} + 114 q^{7} - 128 q^{8}+O(q^{10}) $ 2 * q - 8 * q^2 + 32 * q^4 + 114 * q^7 - 128 * q^8	$ 2 q - 8 q^{2} + 32 q^{4} + 114 q^{7} - 128 q^{8} - 174 q^{11} + 642 q^{13} - 456 q^{14} + 512 q^{16} - 1514 q^{17} - 120 q^{19} + 696 q^{22} - 4352 q^{23} - 2568 q^{26} + 1824 q^{28} + 2430 q^{29} + 11684 q^{31} - 2048 q^{32} + 6056 q^{34} - 17586 q^{37} + 480 q^{38} - 24984 q^{41} - 24168 q^{43} - 2784 q^{44} + 17408 q^{46} + 13316 q^{47} + 35334 q^{49} + 10272 q^{52} + 13698 q^{53} - 7296 q^{56} - 9720 q^{58} + 23730 q^{59} + 57124 q^{61} - 46736 q^{62} + 8192 q^{64} - 38316 q^{67} - 24224 q^{68} + 11076 q^{71} - 88548 q^{73} + 70344 q^{74} - 1920 q^{76} + 52532 q^{77} + 14220 q^{79} + 99936 q^{82} - 50792 q^{83} + 96672 q^{86} + 11136 q^{88} + 60420 q^{89} - 150756 q^{91} - 69632 q^{92} - 53264 q^{94} - 171216 q^{97} - 141336 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 + 114 * q^7 - 128 * q^8 - 174 * q^11 + 642 * q^13 - 456 * q^14 + 512 * q^16 - 1514 * q^17 - 120 * q^19 + 696 * q^22 - 4352 * q^23 - 2568 * q^26 + 1824 * q^28 + 2430 * q^29 + 11684 * q^31 - 2048 * q^32 + 6056 * q^34 - 17586 * q^37 + 480 * q^38 - 24984 * q^41 - 24168 * q^43 - 2784 * q^44 + 17408 * q^46 + 13316 * q^47 + 35334 * q^49 + 10272 * q^52 + 13698 * q^53 - 7296 * q^56 - 9720 * q^58 + 23730 * q^59 + 57124 * q^61 - 46736 * q^62 + 8192 * q^64 - 38316 * q^67 - 24224 * q^68 + 11076 * q^71 - 88548 * q^73 + 70344 * q^74 - 1920 * q^76 + 52532 * q^77 + 14220 * q^79 + 99936 * q^82 - 50792 * q^83 + 96672 * q^86 + 11136 * q^88 + 60420 * q^89 - 150756 * q^91 - 69632 * q^92 - 53264 * q^94 - 171216 * q^97 - 141336 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−119.706	−0.923359	−0.461680	−	0.887047i	$-0.652753\pi$
$7$	−119.706	−0.923359	−0.461680	+	0.887047i	$0.652753\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	0	0
$11$	−263.706	−0.657110	−0.328555	−	0.944485i	$-0.606562\pi$
$11$	−263.706	−0.657110	−0.328555	+	0.944485i	$0.606562\pi$
$12$	0	0
$13$	851.118	1.39679	0.698395	−	0.715712i	$-0.253898\pi$
$13$	851.118	1.39679	0.698395	+	0.715712i	$0.253898\pi$
$14$	478.824	0.652914
$15$	0	0
$16$	256.000	0.250000
$17$	−1287.12	−1.08018	−0.540090	−	0.841607i	$-0.681610\pi$
$17$	−1287.12	−1.08018	−0.540090	+	0.841607i	$0.681610\pi$
$18$	0	0
$19$	2060.47	1.30943	0.654716	−	0.755875i	$-0.272788\pi$
$19$	2060.47	1.30943	0.654716	+	0.755875i	$0.272788\pi$
$20$	0	0
$21$	0	0
$22$	1054.82	0.464647
$23$	−55.5284	−0.0218875	−0.0109437	−	0.999940i	$-0.503484\pi$
$23$	−55.5284	−0.0218875	−0.0109437	+	0.999940i	$0.503484\pi$
$24$	0	0
$25$	0	0
$26$	−3404.47	−0.987680
$27$	0	0
$28$	−1915.30	−0.461680
$29$	5986.06	1.32174	0.660870	−	0.750500i	$-0.270187\pi$
$29$	5986.06	1.32174	0.660870	+	0.750500i	$0.270187\pi$
$30$	0	0
$31$	4781.76	0.893684	0.446842	−	0.894613i	$-0.352549\pi$
$31$	4781.76	0.893684	0.446842	+	0.894613i	$0.352549\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	5148.47	0.763802
$35$	0	0
$36$	0	0
$37$	−12150.4	−1.45911	−0.729553	−	0.683924i	$-0.760272\pi$
$37$	−12150.4	−1.45911	−0.729553	+	0.683924i	$0.760272\pi$
$38$	−8241.89	−0.925908
$39$	0	0
$40$	0	0
$41$	−18500.0	−1.71875	−0.859374	−	0.511348i	$-0.829146\pi$
$41$	−18500.0	−1.71875	−0.859374	+	0.511348i	$0.829146\pi$
$42$	0	0
$43$	−2188.47	−0.180496	−0.0902482	−	0.995919i	$-0.528766\pi$
$43$	−2188.47	−0.180496	−0.0902482	+	0.995919i	$0.528766\pi$
$44$	−4219.30	−0.328555
$45$	0	0
$46$	222.113	0.0154768
$47$	5597.76	0.369632	0.184816	−	0.982773i	$-0.440831\pi$
$47$	5597.76	0.369632	0.184816	+	0.982773i	$0.440831\pi$
$48$	0	0
$49$	−2477.48	−0.147408
$50$	0	0
$51$	0	0
$52$	13617.9	0.698395
$53$	26463.4	1.29406	0.647031	−	0.762463i	$-0.276010\pi$
$53$	26463.4	1.29406	0.647031	+	0.762463i	$0.276010\pi$
$54$	0	0
$55$	0	0
$56$	7661.18	0.326457
$57$	0	0
$58$	−23944.2	−0.934612
$59$	−20825.6	−0.778875	−0.389437	−	0.921053i	$-0.627331\pi$
$59$	−20825.6	−0.778875	−0.389437	+	0.921053i	$0.627331\pi$
$60$	0	0
$61$	45525.8	1.56651	0.783254	−	0.621702i	$-0.213558\pi$
$61$	45525.8	1.56651	0.783254	+	0.621702i	$0.213558\pi$
$62$	−19127.1	−0.631930
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	−34354.7	−0.934974	−0.467487	−	0.884000i	$-0.654840\pi$
$67$	−34354.7	−0.934974	−0.467487	+	0.884000i	$0.654840\pi$
$68$	−20593.9	−0.540090
$69$	0	0
$70$	0	0
$71$	57489.6	1.35345	0.676726	−	0.736235i	$-0.263398\pi$
$71$	57489.6	1.35345	0.676726	+	0.736235i	$0.263398\pi$
$72$	0	0
$73$	−26956.8	−0.592054	−0.296027	−	0.955180i	$-0.595662\pi$
$73$	−26956.8	−0.592054	−0.296027	+	0.955180i	$0.595662\pi$
$74$	48601.7	1.03174
$75$	0	0
$76$	32967.5	0.654716
$77$	31567.2	0.606749
$78$	0	0
$79$	42097.8	0.758912	0.379456	−	0.925210i	$-0.376111\pi$
$79$	42097.8	0.758912	0.379456	+	0.925210i	$0.376111\pi$
$80$	0	0
$81$	0	0
$82$	74000.0	1.21534
$83$	−101733.	−1.62094	−0.810469	−	0.585781i	$-0.800788\pi$
$83$	−101733.	−1.62094	−0.810469	+	0.585781i	$0.800788\pi$
$84$	0	0
$85$	0	0
$86$	8753.86	0.127630
$87$	0	0
$88$	16877.2	0.232324
$89$	65551.2	0.877214	0.438607	−	0.898679i	$-0.355472\pi$
$89$	65551.2	0.877214	0.438607	+	0.898679i	$0.355472\pi$
$90$	0	0
$91$	−101884.	−1.28974
$92$	−888.454	−0.0109437
$93$	0	0
$94$	−22391.1	−0.261370
$95$	0	0
$96$	0	0
$97$	−82780.7	−0.893305	−0.446653	−	0.894708i	$-0.647384\pi$
$97$	−82780.7	−0.893305	−0.446653	+	0.894708i	$0.647384\pi$
$98$	9909.92	0.104233
$99$	0	0
$100$	0	0
$101$	−14644.9	−0.142851	−0.0714255	−	0.997446i	$-0.522755\pi$
$101$	−14644.9	−0.142851	−0.0714255	+	0.997446i	$0.522755\pi$
$102$	0	0
$103$	−199927.	−1.85686	−0.928429	−	0.371511i	$-0.878840\pi$
$103$	−199927.	−1.85686	−0.928429	+	0.371511i	$0.878840\pi$
$104$	−54471.5	−0.493840
$105$	0	0
$106$	−105853.	−0.915041
$107$	−34770.4	−0.293596	−0.146798	−	0.989166i	$-0.546897\pi$
$107$	−34770.4	−0.293596	−0.146798	+	0.989166i	$0.546897\pi$
$108$	0	0
$109$	−19636.5	−0.158306	−0.0791529	−	0.996862i	$-0.525222\pi$
$109$	−19636.5	−0.158306	−0.0791529	+	0.996862i	$0.525222\pi$
$110$	0	0
$111$	0	0
$112$	−30644.7	−0.230840
$113$	19716.6	0.145256	0.0726282	−	0.997359i	$-0.476861\pi$
$113$	19716.6	0.145256	0.0726282	+	0.997359i	$0.476861\pi$
$114$	0	0
$115$	0	0
$116$	95777.0	0.660870
$117$	0	0
$118$	83302.4	0.550748
$119$	154076.	0.997394
$120$	0	0
$121$	−91510.2	−0.568206
$122$	−182103.	−1.10769
$123$	0	0
$124$	76508.2	0.446842
$125$	0	0
$126$	0	0
$127$	−55823.7	−0.307121	−0.153560	−	0.988139i	$-0.549074\pi$
$127$	−55823.7	−0.307121	−0.153560	+	0.988139i	$0.549074\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	0	0
$131$	−136377.	−0.694326	−0.347163	−	0.937805i	$-0.612855\pi$
$131$	−136377.	−0.694326	−0.347163	+	0.937805i	$0.612855\pi$
$132$	0	0
$133$	−246651.	−1.20908
$134$	137419.	0.661126
$135$	0	0
$136$	82375.5	0.381901
$137$	134387.	0.611722	0.305861	−	0.952076i	$-0.401056\pi$
$137$	134387.	0.611722	0.305861	+	0.952076i	$0.401056\pi$
$138$	0	0
$139$	−305523.	−1.34124	−0.670620	−	0.741801i	$-0.733972\pi$
$139$	−305523.	−1.34124	−0.670620	+	0.741801i	$0.733972\pi$
$140$	0	0
$141$	0	0
$142$	−229958.	−0.957036
$143$	−224445.	−0.917846
$144$	0	0
$145$	0	0
$146$	107827.	0.418646
$147$	0	0
$148$	−194407.	−0.729553
$149$	−349387.	−1.28926	−0.644630	−	0.764494i	$-0.722989\pi$
$149$	−349387.	−1.28926	−0.644630	+	0.764494i	$0.722989\pi$
$150$	0	0
$151$	−314525.	−1.12257	−0.561284	−	0.827623i	$-0.689692\pi$
$151$	−314525.	−1.12257	−0.561284	+	0.827623i	$0.689692\pi$
$152$	−131870.	−0.462954
$153$	0	0
$154$	−126269.	−0.429036
$155$	0	0
$156$	0	0
$157$	−131302.	−0.425132	−0.212566	−	0.977147i	$-0.568182\pi$
$157$	−131302.	−0.425132	−0.212566	+	0.977147i	$0.568182\pi$
$158$	−168391.	−0.536632
$159$	0	0
$160$	0	0
$161$	6647.08	0.0202100
$162$	0	0
$163$	−633204.	−1.86670	−0.933351	−	0.358966i	$-0.883129\pi$
$163$	−633204.	−1.86670	−0.933351	+	0.358966i	$0.883129\pi$
$164$	−296000.	−0.859374
$165$	0	0
$166$	406932.	1.14618
$167$	−137690.	−0.382042	−0.191021	−	0.981586i	$-0.561180\pi$
$167$	−137690.	−0.382042	−0.191021	+	0.981586i	$0.561180\pi$
$168$	0	0
$169$	353109.	0.951024
$170$	0	0
$171$	0	0
$172$	−35015.5	−0.0902482
$173$	90278.1	0.229333	0.114667	−	0.993404i	$-0.463420\pi$
$173$	90278.1	0.229333	0.114667	+	0.993404i	$0.463420\pi$
$174$	0	0
$175$	0	0
$176$	−67508.7	−0.164278
$177$	0	0
$178$	−262205.	−0.620284
$179$	−601579.	−1.40333	−0.701665	−	0.712507i	$-0.747560\pi$
$179$	−601579.	−1.40333	−0.701665	+	0.712507i	$0.747560\pi$
$180$	0	0
$181$	447746.	1.01586	0.507931	−	0.861398i	$-0.330410\pi$
$181$	447746.	1.01586	0.507931	+	0.861398i	$0.330410\pi$
$182$	407536.	0.911984
$183$	0	0
$184$	3553.81	0.00773838
$185$	0	0
$186$	0	0
$187$	339421.	0.709797
$188$	89564.2	0.184816
$189$	0	0
$190$	0	0
$191$	−673846.	−1.33653	−0.668263	−	0.743925i	$-0.732962\pi$
$191$	−673846.	−1.33653	−0.668263	+	0.743925i	$0.732962\pi$
$192$	0	0
$193$	255962.	0.494632	0.247316	−	0.968935i	$-0.420451\pi$
$193$	255962.	0.494632	0.247316	+	0.968935i	$0.420451\pi$
$194$	331123.	0.631662
$195$	0	0
$196$	−39639.7	−0.0737038
$197$	−196202.	−0.360195	−0.180097	−	0.983649i	$-0.557641\pi$
$197$	−196202.	−0.360195	−0.180097	+	0.983649i	$0.557641\pi$
$198$	0	0
$199$	118265.	0.211702	0.105851	−	0.994382i	$-0.466243\pi$
$199$	118265.	0.211702	0.105851	+	0.994382i	$0.466243\pi$
$200$	0	0
$201$	0	0
$202$	58579.7	0.101011
$203$	−716567.	−1.22044
$204$	0	0
$205$	0	0
$206$	799708.	1.31300
$207$	0	0
$208$	217886.	0.349198
$209$	−543359.	−0.860441
$210$	0	0
$211$	910278.	1.40756	0.703781	−	0.710417i	$-0.251493\pi$
$211$	910278.	1.40756	0.703781	+	0.710417i	$0.251493\pi$
$212$	423414.	0.647031
$213$	0	0
$214$	139082.	0.207604
$215$	0	0
$216$	0	0
$217$	−572406.	−0.825191
$218$	78545.9	0.111939
$219$	0	0
$220$	0	0
$221$	−1.09549e6	−1.50879
$222$	0	0
$223$	1.04513e6	1.40737	0.703683	−	0.710514i	$-0.251538\pi$
$223$	1.04513e6	1.40737	0.703683	+	0.710514i	$0.251538\pi$
$224$	122579.	0.163228
$225$	0	0
$226$	−78866.3	−0.102712
$227$	−869120.	−1.11948	−0.559739	−	0.828669i	$-0.689098\pi$
$227$	−869120.	−1.11948	−0.559739	+	0.828669i	$0.689098\pi$
$228$	0	0
$229$	−1.39907e6	−1.76299	−0.881494	−	0.472195i	$-0.843462\pi$
$229$	−1.39907e6	−1.76299	−0.881494	+	0.472195i	$0.843462\pi$
$230$	0	0
$231$	0	0
$232$	−383108.	−0.467306
$233$	403494.	0.486908	0.243454	−	0.969912i	$-0.421720\pi$
$233$	403494.	0.486908	0.243454	+	0.969912i	$0.421720\pi$
$234$	0	0
$235$	0	0
$236$	−333210.	−0.389437
$237$	0	0
$238$	−616303.	−0.705264
$239$	611076.	0.691991	0.345995	−	0.938236i	$-0.387541\pi$
$239$	611076.	0.691991	0.345995	+	0.938236i	$0.387541\pi$
$240$	0	0
$241$	−1.50639e6	−1.67068	−0.835342	−	0.549730i	$-0.814730\pi$
$241$	−1.50639e6	−1.67068	−0.835342	+	0.549730i	$0.814730\pi$
$242$	366041.	0.401782
$243$	0	0
$244$	728412.	0.783254
$245$	0	0
$246$	0	0
$247$	1.75370e6	1.82900
$248$	−306033.	−0.315965
$249$	0	0
$250$	0	0
$251$	−558727.	−0.559777	−0.279889	−	0.960032i	$-0.590298\pi$
$251$	−558727.	−0.559777	−0.279889	+	0.960032i	$0.590298\pi$
$252$	0	0
$253$	14643.2	0.0143825
$254$	223295.	0.217167
$255$	0	0
$256$	65536.0	0.0625000
$257$	−680108.	−0.642310	−0.321155	−	0.947027i	$-0.604071\pi$
$257$	−680108.	−0.642310	−0.321155	+	0.947027i	$0.604071\pi$
$258$	0	0
$259$	1.45448e6	1.34728
$260$	0	0
$261$	0	0
$262$	545508.	0.490962
$263$	428668.	0.382148	0.191074	−	0.981576i	$-0.438803\pi$
$263$	428668.	0.382148	0.191074	+	0.981576i	$0.438803\pi$
$264$	0	0
$265$	0	0
$266$	986603.	0.854945
$267$	0	0
$268$	−549675.	−0.467487
$269$	1.51936e6	1.28020	0.640101	−	0.768291i	$-0.278892\pi$
$269$	1.51936e6	1.28020	0.640101	+	0.768291i	$0.278892\pi$
$270$	0	0
$271$	−1.34595e6	−1.11329	−0.556643	−	0.830752i	$-0.687911\pi$
$271$	−1.34595e6	−1.11329	−0.556643	+	0.830752i	$0.687911\pi$
$272$	−329502.	−0.270045
$273$	0	0
$274$	−537546.	−0.432553
$275$	0	0
$276$	0	0
$277$	−885541.	−0.693440	−0.346720	−	0.937969i	$-0.612705\pi$
$277$	−885541.	−0.693440	−0.346720	+	0.937969i	$0.612705\pi$
$278$	1.22209e6	0.948400
$279$	0	0
$280$	0	0
$281$	1.26803e6	0.957999	0.478999	−	0.877815i	$-0.341000\pi$
$281$	1.26803e6	0.957999	0.478999	+	0.877815i	$0.341000\pi$
$282$	0	0
$283$	685833.	0.509040	0.254520	−	0.967067i	$-0.418082\pi$
$283$	685833.	0.509040	0.254520	+	0.967067i	$0.418082\pi$
$284$	919833.	0.676726
$285$	0	0
$286$	897779.	0.649015
$287$	2.21456e6	1.58702
$288$	0	0
$289$	236816.	0.166788
$290$	0	0
$291$	0	0
$292$	−431309.	−0.296027
$293$	1.66857e6	1.13547	0.567736	−	0.823211i	$-0.307819\pi$
$293$	1.66857e6	1.13547	0.567736	+	0.823211i	$0.307819\pi$
$294$	0	0
$295$	0	0
$296$	777626.	0.515872
$297$	0	0
$298$	1.39755e6	0.911645
$299$	−47261.2	−0.0305722
$300$	0	0
$301$	261972.	0.166663
$302$	1.25810e6	0.793776
$303$	0	0
$304$	527481.	0.327358
$305$	0	0
$306$	0	0
$307$	−560776.	−0.339581	−0.169790	−	0.985480i	$-0.554309\pi$
$307$	−560776.	−0.339581	−0.169790	+	0.985480i	$0.554309\pi$
$308$	505075.	0.303374
$309$	0	0
$310$	0	0
$311$	1.33943e6	0.785268	0.392634	−	0.919695i	$-0.371564\pi$
$311$	1.33943e6	0.785268	0.392634	+	0.919695i	$0.371564\pi$
$312$	0	0
$313$	621225.	0.358417	0.179208	−	0.983811i	$-0.442646\pi$
$313$	621225.	0.358417	0.179208	+	0.983811i	$0.442646\pi$
$314$	525210.	0.300614
$315$	0	0
$316$	673565.	0.379456
$317$	−300282.	−0.167834	−0.0839172	−	0.996473i	$-0.526743\pi$
$317$	−300282.	−0.167834	−0.0839172	+	0.996473i	$0.526743\pi$
$318$	0	0
$319$	−1.57856e6	−0.868529
$320$	0	0
$321$	0	0
$322$	−26588.3	−0.0142906
$323$	−2.65207e6	−1.41442
$324$	0	0
$325$	0	0
$326$	2.53282e6	1.31996
$327$	0	0
$328$	1.18400e6	0.607669
$329$	−670086.	−0.341303
$330$	0	0
$331$	−1.94914e6	−0.977851	−0.488925	−	0.872326i	$-0.662611\pi$
$331$	−1.94914e6	−0.977851	−0.488925	+	0.872326i	$0.662611\pi$
$332$	−1.62773e6	−0.810469
$333$	0	0
$334$	550760.	0.270145
$335$	0	0
$336$	0	0
$337$	253373.	0.121530	0.0607652	−	0.998152i	$-0.480646\pi$
$337$	253373.	0.121530	0.0607652	+	0.998152i	$0.480646\pi$
$338$	−1.41243e6	−0.672476
$339$	0	0
$340$	0	0
$341$	−1.26098e6	−0.587249
$342$	0	0
$343$	2.30847e6	1.05947
$344$	140062.	0.0638151
$345$	0	0
$346$	−361112.	−0.162163
$347$	−1.60688e6	−0.716406	−0.358203	−	0.933644i	$-0.616610\pi$
$347$	−1.60688e6	−0.716406	−0.358203	+	0.933644i	$0.616610\pi$
$348$	0	0
$349$	−608374.	−0.267367	−0.133683	−	0.991024i	$-0.542680\pi$
$349$	−608374.	−0.267367	−0.133683	+	0.991024i	$0.542680\pi$
$350$	0	0
$351$	0	0
$352$	270035.	0.116162
$353$	3.83773e6	1.63922	0.819612	−	0.572919i	$-0.194189\pi$
$353$	3.83773e6	1.63922	0.819612	+	0.572919i	$0.194189\pi$
$354$	0	0
$355$	0	0
$356$	1.04882e6	0.438607
$357$	0	0
$358$	2.40632e6	0.992305
$359$	−2.13850e6	−0.875738	−0.437869	−	0.899039i	$-0.644267\pi$
$359$	−2.13850e6	−0.875738	−0.437869	+	0.899039i	$0.644267\pi$
$360$	0	0
$361$	1.76944e6	0.714610
$362$	−1.79098e6	−0.718323
$363$	0	0
$364$	−1.63014e6	−0.644870
$365$	0	0
$366$	0	0
$367$	4.50396e6	1.74554	0.872769	−	0.488133i	$-0.162322\pi$
$367$	4.50396e6	1.74554	0.872769	+	0.488133i	$0.162322\pi$
$368$	−14215.3	−0.00547186
$369$	0	0
$370$	0	0
$371$	−3.16782e6	−1.19488
$372$	0	0
$373$	−1.15223e6	−0.428811	−0.214406	−	0.976745i	$-0.568781\pi$
$373$	−1.15223e6	−0.428811	−0.214406	+	0.976745i	$0.568781\pi$
$374$	−1.35768e6	−0.501902
$375$	0	0
$376$	−358257.	−0.130685
$377$	5.09484e6	1.84619
$378$	0	0
$379$	797740.	0.285275	0.142637	−	0.989775i	$-0.454442\pi$
$379$	797740.	0.285275	0.142637	+	0.989775i	$0.454442\pi$
$380$	0	0
$381$	0	0
$382$	2.69538e6	0.945066
$383$	−1.02124e6	−0.355737	−0.177869	−	0.984054i	$-0.556920\pi$
$383$	−1.02124e6	−0.355737	−0.177869	+	0.984054i	$0.556920\pi$
$384$	0	0
$385$	0	0
$386$	−1.02385e6	−0.349757
$387$	0	0
$388$	−1.32449e6	−0.446653
$389$	1.06820e6	0.357914	0.178957	−	0.983857i	$-0.442728\pi$
$389$	1.06820e6	0.357914	0.178957	+	0.983857i	$0.442728\pi$
$390$	0	0
$391$	71471.5	0.0236424
$392$	158559.	0.0521165
$393$	0	0
$394$	784807.	0.254696
$395$	0	0
$396$	0	0
$397$	109330.	0.0348148	0.0174074	−	0.999848i	$-0.494459\pi$
$397$	109330.	0.0348148	0.0174074	+	0.999848i	$0.494459\pi$
$398$	−473061.	−0.149696
$399$	0	0
$400$	0	0
$401$	−1.42131e6	−0.441395	−0.220697	−	0.975342i	$-0.570833\pi$
$401$	−1.42131e6	−0.441395	−0.220697	+	0.975342i	$0.570833\pi$
$402$	0	0
$403$	4.06985e6	1.24829
$404$	−234319.	−0.0714255
$405$	0	0
$406$	2.86627e6	0.862982
$407$	3.20414e6	0.958793
$408$	0	0
$409$	3.07511e6	0.908975	0.454487	−	0.890753i	$-0.349823\pi$
$409$	3.07511e6	0.908975	0.454487	+	0.890753i	$0.349823\pi$
$410$	0	0
$411$	0	0
$412$	−3.19883e6	−0.928429
$413$	2.49295e6	0.719181
$414$	0	0
$415$	0	0
$416$	−871545.	−0.246920
$417$	0	0
$418$	2.17343e6	0.608423
$419$	−3.06462e6	−0.852789	−0.426395	−	0.904537i	$-0.640217\pi$
$419$	−3.06462e6	−0.852789	−0.426395	+	0.904537i	$0.640217\pi$
$420$	0	0
$421$	837782.	0.230370	0.115185	−	0.993344i	$-0.463254\pi$
$421$	837782.	0.230370	0.115185	+	0.993344i	$0.463254\pi$
$422$	−3.64111e6	−0.995297
$423$	0	0
$424$	−1.69366e6	−0.457520
$425$	0	0
$426$	0	0
$427$	−5.44971e6	−1.44645
$428$	−556326.	−0.146798
$429$	0	0
$430$	0	0
$431$	−5.90720e6	−1.53175	−0.765876	−	0.642988i	$-0.777694\pi$
$431$	−5.90720e6	−1.53175	−0.765876	+	0.642988i	$0.777694\pi$
$432$	0	0
$433$	−1.84995e6	−0.474177	−0.237089	−	0.971488i	$-0.576193\pi$
$433$	−1.84995e6	−0.474177	−0.237089	+	0.971488i	$0.576193\pi$
$434$	2.28962e6	0.583498
$435$	0	0
$436$	−314183.	−0.0791529
$437$	−114415.	−0.0286601
$438$	0	0
$439$	−2.81172e6	−0.696323	−0.348161	−	0.937435i	$-0.613194\pi$
$439$	−2.81172e6	−0.696323	−0.348161	+	0.937435i	$0.613194\pi$
$440$	0	0
$441$	0	0
$442$	4.38196e6	1.06687
$443$	2.41386e6	0.584391	0.292196	−	0.956359i	$-0.405614\pi$
$443$	2.41386e6	0.584391	0.292196	+	0.956359i	$0.405614\pi$
$444$	0	0
$445$	0	0
$446$	−4.18051e6	−0.995158
$447$	0	0
$448$	−490316.	−0.115420
$449$	507124.	0.118713	0.0593565	−	0.998237i	$-0.481095\pi$
$449$	507124.	0.118713	0.0593565	+	0.998237i	$0.481095\pi$
$450$	0	0
$451$	4.87856e6	1.12941
$452$	315465.	0.0726282
$453$	0	0
$454$	3.47648e6	0.791590
$455$	0	0
$456$	0	0
$457$	−1.09879e6	−0.246106	−0.123053	−	0.992400i	$-0.539269\pi$
$457$	−1.09879e6	−0.246106	−0.123053	+	0.992400i	$0.539269\pi$
$458$	5.59626e6	1.24662
$459$	0	0
$460$	0	0
$461$	−106066.	−0.0232446	−0.0116223	−	0.999932i	$-0.503700\pi$
$461$	−106066.	−0.0232446	−0.0116223	+	0.999932i	$0.503700\pi$
$462$	0	0
$463$	−1.36176e6	−0.295222	−0.147611	−	0.989045i	$-0.547158\pi$
$463$	−1.36176e6	−0.295222	−0.147611	+	0.989045i	$0.547158\pi$
$464$	1.53243e6	0.330435
$465$	0	0
$466$	−1.61397e6	−0.344296
$467$	−2.18710e6	−0.464062	−0.232031	−	0.972708i	$-0.574537\pi$
$467$	−2.18710e6	−0.464062	−0.232031	+	0.972708i	$0.574537\pi$
$468$	0	0
$469$	4.11246e6	0.863316
$470$	0	0
$471$	0	0
$472$	1.33284e6	0.275374
$473$	577111.	0.118606
$474$	0	0
$475$	0	0
$476$	2.46521e6	0.498697
$477$	0	0
$478$	−2.44430e6	−0.489311
$479$	−1.01595e6	−0.202318	−0.101159	−	0.994870i	$-0.532255\pi$
$479$	−1.01595e6	−0.202318	−0.101159	+	0.994870i	$0.532255\pi$
$480$	0	0
$481$	−1.03414e7	−2.03807
$482$	6.02555e6	1.18135
$483$	0	0
$484$	−1.46416e6	−0.284103
$485$	0	0
$486$	0	0
$487$	−2.02339e6	−0.386596	−0.193298	−	0.981140i	$-0.561918\pi$
$487$	−2.02339e6	−0.386596	−0.193298	+	0.981140i	$0.561918\pi$
$488$	−2.91365e6	−0.553844
$489$	0	0
$490$	0	0
$491$	4.22487e6	0.790878	0.395439	−	0.918492i	$-0.370592\pi$
$491$	4.22487e6	0.790878	0.395439	+	0.918492i	$0.370592\pi$
$492$	0	0
$493$	−7.70477e6	−1.42772
$494$	−7.01482e6	−1.29330
$495$	0	0
$496$	1.22413e6	0.223421
$497$	−6.88184e6	−1.24972
$498$	0	0
$499$	−6.71381e6	−1.20703	−0.603515	−	0.797352i	$-0.706233\pi$
$499$	−6.71381e6	−1.20703	−0.603515	+	0.797352i	$0.706233\pi$
$500$	0	0
$501$	0	0
$502$	2.23491e6	0.395822
$503$	−5.36486e6	−0.945450	−0.472725	−	0.881210i	$-0.656730\pi$
$503$	−5.36486e6	−0.945450	−0.472725	+	0.881210i	$0.656730\pi$
$504$	0	0
$505$	0	0
$506$	−58572.6	−0.0101699
$507$	0	0
$508$	−893179.	−0.153560
$509$	−1.10627e6	−0.189264	−0.0946318	−	0.995512i	$-0.530167\pi$
$509$	−1.10627e6	−0.189264	−0.0946318	+	0.995512i	$0.530167\pi$
$510$	0	0
$511$	3.22689e6	0.546679
$512$	−262144.	−0.0441942
$513$	0	0
$514$	2.72043e6	0.454182
$515$	0	0
$516$	0	0
$517$	−1.47616e6	−0.242889
$518$	−5.81791e6	−0.952670
$519$	0	0
$520$	0	0
$521$	3.11641e6	0.502991	0.251495	−	0.967859i	$-0.419078\pi$
$521$	3.11641e6	0.502991	0.251495	+	0.967859i	$0.419078\pi$
$522$	0	0
$523$	9.29324e6	1.48564	0.742819	−	0.669492i	$-0.233488\pi$
$523$	9.29324e6	1.48564	0.742819	+	0.669492i	$0.233488\pi$
$524$	−2.18203e6	−0.347163
$525$	0	0
$526$	−1.71467e6	−0.270219
$527$	−6.15469e6	−0.965339
$528$	0	0
$529$	−6.43326e6	−0.999521
$530$	0	0
$531$	0	0
$532$	−3.94641e6	−0.604538
$533$	−1.57457e7	−2.40073
$534$	0	0
$535$	0	0
$536$	2.19870e6	0.330563
$537$	0	0
$538$	−6.07742e6	−0.905240
$539$	653326.	0.0968631
$540$	0	0
$541$	1.00152e7	1.47118	0.735592	−	0.677425i	$-0.236904\pi$
$541$	1.00152e7	1.47118	0.735592	+	0.677425i	$0.236904\pi$
$542$	5.38382e6	0.787213
$543$	0	0
$544$	1.31801e6	0.190951
$545$	0	0
$546$	0	0
$547$	−605416.	−0.0865138	−0.0432569	−	0.999064i	$-0.513773\pi$
$547$	−605416.	−0.0865138	−0.0432569	+	0.999064i	$0.513773\pi$
$548$	2.15018e6	0.305861
$549$	0	0
$550$	0	0
$551$	1.23341e7	1.73073
$552$	0	0
$553$	−5.03936e6	−0.700749
$554$	3.54216e6	0.490336
$555$	0	0
$556$	−4.88836e6	−0.670620
$557$	9.21693e6	1.25878	0.629388	−	0.777091i	$-0.283306\pi$
$557$	9.21693e6	1.25878	0.629388	+	0.777091i	$0.283306\pi$
$558$	0	0
$559$	−1.86264e6	−0.252116
$560$	0	0
$561$	0	0
$562$	−5.07213e6	−0.677407
$563$	9.44213e6	1.25545	0.627724	−	0.778436i	$-0.283986\pi$
$563$	9.44213e6	1.25545	0.627724	+	0.778436i	$0.283986\pi$
$564$	0	0
$565$	0	0
$566$	−2.74333e6	−0.359946
$567$	0	0
$568$	−3.67933e6	−0.478518
$569$	1.24216e6	0.160841	0.0804206	−	0.996761i	$-0.474374\pi$
$569$	1.24216e6	0.160841	0.0804206	+	0.996761i	$0.474374\pi$
$570$	0	0
$571$	−1.35633e7	−1.74090	−0.870450	−	0.492256i	$-0.836172\pi$
$571$	−1.35633e7	−1.74090	−0.870450	+	0.492256i	$0.836172\pi$
$572$	−3.59112e6	−0.458923
$573$	0	0
$574$	−8.85824e6	−1.12219
$575$	0	0
$576$	0	0
$577$	6.75669e6	0.844879	0.422440	−	0.906391i	$-0.361174\pi$
$577$	6.75669e6	0.844879	0.422440	+	0.906391i	$0.361174\pi$
$578$	−947262.	−0.117937
$579$	0	0
$580$	0	0
$581$	1.21780e7	1.49671
$582$	0	0
$583$	−6.97855e6	−0.850342
$584$	1.72524e6	0.209323
$585$	0	0
$586$	−6.67430e6	−0.802900
$587$	−1.08655e7	−1.30153	−0.650764	−	0.759280i	$-0.725551\pi$
$587$	−1.08655e7	−1.30153	−0.650764	+	0.759280i	$0.725551\pi$
$588$	0	0
$589$	9.85269e6	1.17022
$590$	0	0
$591$	0	0
$592$	−3.11051e6	−0.364776
$593$	1.20061e6	0.140205	0.0701027	−	0.997540i	$-0.477667\pi$
$593$	1.20061e6	0.140205	0.0701027	+	0.997540i	$0.477667\pi$
$594$	0	0
$595$	0	0
$596$	−5.59019e6	−0.644630
$597$	0	0
$598$	189045.	0.0216178
$599$	6.77826e6	0.771882	0.385941	−	0.922523i	$-0.373877\pi$
$599$	6.77826e6	0.771882	0.385941	+	0.922523i	$0.373877\pi$
$600$	0	0
$601$	6.32426e6	0.714207	0.357103	−	0.934065i	$-0.383764\pi$
$601$	6.32426e6	0.714207	0.357103	+	0.934065i	$0.383764\pi$
$602$	−1.04789e6	−0.117849
$603$	0	0
$604$	−5.03240e6	−0.561284
$605$	0	0
$606$	0	0
$607$	−2.63732e6	−0.290530	−0.145265	−	0.989393i	$-0.546404\pi$
$607$	−2.63732e6	−0.290530	−0.145265	+	0.989393i	$0.546404\pi$
$608$	−2.10992e6	−0.231477
$609$	0	0
$610$	0	0
$611$	4.76436e6	0.516299
$612$	0	0
$613$	−1.43305e7	−1.54031	−0.770157	−	0.637854i	$-0.779822\pi$
$613$	−1.43305e7	−1.54031	−0.770157	+	0.637854i	$0.779822\pi$
$614$	2.24310e6	0.240120
$615$	0	0
$616$	−2.02030e6	−0.214518
$617$	29265.6	0.00309489	0.00154744	−	0.999999i	$-0.499507\pi$
$617$	29265.6	0.00309489	0.00154744	+	0.999999i	$0.499507\pi$
$618$	0	0
$619$	7.84138e6	0.822557	0.411279	−	0.911510i	$-0.365082\pi$
$619$	7.84138e6	0.822557	0.411279	+	0.911510i	$0.365082\pi$
$620$	0	0
$621$	0	0
$622$	−5.35771e6	−0.555269
$623$	−7.84687e6	−0.809984
$624$	0	0
$625$	0	0
$626$	−2.48490e6	−0.253439
$627$	0	0
$628$	−2.10084e6	−0.212566
$629$	1.56390e7	1.57610
$630$	0	0
$631$	−668942.	−0.0668829	−0.0334414	−	0.999441i	$-0.510647\pi$
$631$	−668942.	−0.0668829	−0.0334414	+	0.999441i	$0.510647\pi$
$632$	−2.69426e6	−0.268316
$633$	0	0
$634$	1.20113e6	0.118677
$635$	0	0
$636$	0	0
$637$	−2.10863e6	−0.205898
$638$	6.31424e6	0.614143
$639$	0	0
$640$	0	0
$641$	−1.26732e7	−1.21826	−0.609130	−	0.793070i	$-0.708481\pi$
$641$	−1.26732e7	−1.21826	−0.609130	+	0.793070i	$0.708481\pi$
$642$	0	0
$643$	6.26256e6	0.597344	0.298672	−	0.954356i	$-0.403456\pi$
$643$	6.26256e6	0.597344	0.298672	+	0.954356i	$0.403456\pi$
$644$	106353.	0.0101050
$645$	0	0
$646$	1.06083e7	1.00015
$647$	8.69452e6	0.816554	0.408277	−	0.912858i	$-0.366130\pi$
$647$	8.69452e6	0.816554	0.408277	+	0.912858i	$0.366130\pi$
$648$	0	0
$649$	5.49184e6	0.511807
$650$	0	0
$651$	0	0
$652$	−1.01313e7	−0.933351
$653$	4.56532e6	0.418976	0.209488	−	0.977811i	$-0.432820\pi$
$653$	4.56532e6	0.418976	0.209488	+	0.977811i	$0.432820\pi$
$654$	0	0
$655$	0	0
$656$	−4.73600e6	−0.429687
$657$	0	0
$658$	2.68034e6	0.241338
$659$	−1.00273e7	−0.899437	−0.449718	−	0.893170i	$-0.648476\pi$
$659$	−1.00273e7	−0.899437	−0.449718	+	0.893170i	$0.648476\pi$
$660$	0	0
$661$	−8.14715e6	−0.725274	−0.362637	−	0.931930i	$-0.618123\pi$
$661$	−8.14715e6	−0.725274	−0.362637	+	0.931930i	$0.618123\pi$
$662$	7.79655e6	0.691445
$663$	0	0
$664$	6.51091e6	0.573088
$665$	0	0
$666$	0	0
$667$	−332396.	−0.0289295
$668$	−2.20304e6	−0.191021
$669$	0	0
$670$	0	0
$671$	−1.20054e7	−1.02937
$672$	0	0
$673$	−1.16421e6	−0.0990818	−0.0495409	−	0.998772i	$-0.515776\pi$
$673$	−1.16421e6	−0.0990818	−0.0495409	+	0.998772i	$0.515776\pi$
$674$	−1.01349e6	−0.0859350
$675$	0	0
$676$	5.64974e6	0.475512
$677$	−1.20869e7	−1.01355	−0.506775	−	0.862079i	$-0.669162\pi$
$677$	−1.20869e7	−1.01355	−0.506775	+	0.862079i	$0.669162\pi$
$678$	0	0
$679$	9.90934e6	0.824841
$680$	0	0
$681$	0	0
$682$	5.04392e6	0.415248
$683$	−7.57524e6	−0.621362	−0.310681	−	0.950514i	$-0.600557\pi$
$683$	−7.57524e6	−0.621362	−0.310681	+	0.950514i	$0.600557\pi$
$684$	0	0
$685$	0	0
$686$	−9.23387e6	−0.749158
$687$	0	0
$688$	−560247.	−0.0451241
$689$	2.25234e7	1.80753
$690$	0	0
$691$	2.32589e7	1.85308	0.926539	−	0.376198i	$-0.122769\pi$
$691$	2.32589e7	1.85308	0.926539	+	0.376198i	$0.122769\pi$
$692$	1.44445e6	0.114667
$693$	0	0
$694$	6.42751e6	0.506575
$695$	0	0
$696$	0	0
$697$	2.38117e7	1.85656
$698$	2.43350e6	0.189057
$699$	0	0
$700$	0	0
$701$	1.11461e7	0.856700	0.428350	−	0.903613i	$-0.359095\pi$
$701$	1.11461e7	0.856700	0.428350	+	0.903613i	$0.359095\pi$
$702$	0	0
$703$	−2.50356e7	−1.91060
$704$	−1.08014e6	−0.0821388
$705$	0	0
$706$	−1.53509e7	−1.15911
$707$	1.75308e6	0.131903
$708$	0	0
$709$	264391.	0.0197529	0.00987646	−	0.999951i	$-0.496856\pi$
$709$	264391.	0.0197529	0.00987646	+	0.999951i	$0.496856\pi$
$710$	0	0
$711$	0	0
$712$	−4.19528e6	−0.310142
$713$	−265523.	−0.0195605
$714$	0	0
$715$	0	0
$716$	−9.62526e6	−0.701665
$717$	0	0
$718$	8.55402e6	0.619240
$719$	−7.09616e6	−0.511919	−0.255959	−	0.966688i	$-0.582391\pi$
$719$	−7.09616e6	−0.511919	−0.255959	+	0.966688i	$0.582391\pi$
$720$	0	0
$721$	2.39325e7	1.71455
$722$	−7.07778e6	−0.505305
$723$	0	0
$724$	7.16393e6	0.507931
$725$	0	0
$726$	0	0
$727$	3.83492e6	0.269104	0.134552	−	0.990907i	$-0.457040\pi$
$727$	3.83492e6	0.269104	0.134552	+	0.990907i	$0.457040\pi$
$728$	6.52057e6	0.455992
$729$	0	0
$730$	0	0
$731$	2.81681e6	0.194969
$732$	0	0
$733$	1.25850e7	0.865151	0.432575	−	0.901598i	$-0.357605\pi$
$733$	1.25850e7	0.865151	0.432575	+	0.901598i	$0.357605\pi$
$734$	−1.80158e7	−1.23428
$735$	0	0
$736$	56861.0	0.00386919
$737$	9.05954e6	0.614381
$738$	0	0
$739$	−1.48983e7	−1.00352	−0.501761	−	0.865006i	$-0.667314\pi$
$739$	−1.48983e7	−1.00352	−0.501761	+	0.865006i	$0.667314\pi$
$740$	0	0
$741$	0	0
$742$	1.26713e7	0.844911
$743$	921340.	0.0612277	0.0306139	−	0.999531i	$-0.490254\pi$
$743$	921340.	0.0612277	0.0306139	+	0.999531i	$0.490254\pi$
$744$	0	0
$745$	0	0
$746$	4.60891e6	0.303215
$747$	0	0
$748$	5.43073e6	0.354899
$749$	4.16222e6	0.271095
$750$	0	0
$751$	−1.19124e7	−0.770726	−0.385363	−	0.922765i	$-0.625924\pi$
$751$	−1.19124e7	−0.770726	−0.385363	+	0.922765i	$0.625924\pi$
$752$	1.43303e6	0.0924081
$753$	0	0
$754$	−2.03794e7	−1.30546
$755$	0	0
$756$	0	0
$757$	−1.43092e7	−0.907558	−0.453779	−	0.891114i	$-0.649924\pi$
$757$	−1.43092e7	−0.907558	−0.453779	+	0.891114i	$0.649924\pi$
$758$	−3.19096e6	−0.201720
$759$	0	0
$760$	0	0
$761$	1.78549e6	0.111762	0.0558812	−	0.998437i	$-0.482203\pi$
$761$	1.78549e6	0.111762	0.0558812	+	0.998437i	$0.482203\pi$
$762$	0	0
$763$	2.35060e6	0.146173
$764$	−1.07815e7	−0.668263
$765$	0	0
$766$	4.08494e6	0.251544
$767$	−1.77250e7	−1.08792
$768$	0	0
$769$	−1.03207e7	−0.629350	−0.314675	−	0.949200i	$-0.601895\pi$
$769$	−1.03207e7	−0.629350	−0.314675	+	0.949200i	$0.601895\pi$
$770$	0	0
$771$	0	0
$772$	4.09539e6	0.247316
$773$	−6.93308e6	−0.417328	−0.208664	−	0.977987i	$-0.566911\pi$
$773$	−6.93308e6	−0.417328	−0.208664	+	0.977987i	$0.566911\pi$
$774$	0	0
$775$	0	0
$776$	5.29797e6	0.315831
$777$	0	0
$778$	−4.27281e6	−0.253084
$779$	−3.81187e7	−2.25058
$780$	0	0
$781$	−1.51603e7	−0.889368
$782$	−285886.	−0.0167177
$783$	0	0
$784$	−634235.	−0.0368519
$785$	0	0
$786$	0	0
$787$	−3.05516e7	−1.75831	−0.879157	−	0.476532i	$-0.841894\pi$
$787$	−3.05516e7	−1.75831	−0.879157	+	0.476532i	$0.841894\pi$
$788$	−3.13923e6	−0.180097
$789$	0	0
$790$	0	0
$791$	−2.36019e6	−0.134124
$792$	0	0
$793$	3.87478e7	2.18808
$794$	−437321.	−0.0246178
$795$	0	0
$796$	1.89225e6	0.105851
$797$	2.76107e7	1.53968	0.769841	−	0.638236i	$-0.220336\pi$
$797$	2.76107e7	1.53968	0.769841	+	0.638236i	$0.220336\pi$
$798$	0	0
$799$	−7.20498e6	−0.399269
$800$	0	0
$801$	0	0
$802$	5.68523e6	0.312113
$803$	7.10867e6	0.389045
$804$	0	0
$805$	0	0
$806$	−1.62794e7	−0.882674
$807$	0	0
$808$	937275.	0.0505055
$809$	−9.60377e6	−0.515906	−0.257953	−	0.966157i	$-0.583048\pi$
$809$	−9.60377e6	−0.515906	−0.257953	+	0.966157i	$0.583048\pi$
$810$	0	0
$811$	1.77352e7	0.946856	0.473428	−	0.880832i	$-0.343016\pi$
$811$	1.77352e7	0.946856	0.473428	+	0.880832i	$0.343016\pi$
$812$	−1.14651e7	−0.610221
$813$	0	0
$814$	−1.28165e7	−0.677969
$815$	0	0
$816$	0	0
$817$	−4.50927e6	−0.236348
$818$	−1.23004e7	−0.642742
$819$	0	0
$820$	0	0
$821$	2.66186e7	1.37825	0.689125	−	0.724643i	$-0.257995\pi$
$821$	2.66186e7	1.37825	0.689125	+	0.724643i	$0.257995\pi$
$822$	0	0
$823$	−1.38059e7	−0.710504	−0.355252	−	0.934771i	$-0.615605\pi$
$823$	−1.38059e7	−0.710504	−0.355252	+	0.934771i	$0.615605\pi$
$824$	1.27953e7	0.656498
$825$	0	0
$826$	−9.97180e6	−0.508538
$827$	3.25738e7	1.65617	0.828084	−	0.560604i	$-0.189431\pi$
$827$	3.25738e7	1.65617	0.828084	+	0.560604i	$0.189431\pi$
$828$	0	0
$829$	1.67628e7	0.847150	0.423575	−	0.905861i	$-0.360775\pi$
$829$	1.67628e7	0.847150	0.423575	+	0.905861i	$0.360775\pi$
$830$	0	0
$831$	0	0
$832$	3.48618e6	0.174599
$833$	3.18881e6	0.159227
$834$	0	0
$835$	0	0
$836$	−8.69374e6	−0.430220
$837$	0	0
$838$	1.22585e7	0.603013
$839$	1.54475e6	0.0757625	0.0378812	−	0.999282i	$-0.487939\pi$
$839$	1.54475e6	0.0757625	0.0378812	+	0.999282i	$0.487939\pi$
$840$	0	0
$841$	1.53218e7	0.746998
$842$	−3.35113e6	−0.162896
$843$	0	0
$844$	1.45644e7	0.703781
$845$	0	0
$846$	0	0
$847$	1.09543e7	0.524658
$848$	6.77462e6	0.323516
$849$	0	0
$850$	0	0
$851$	674692.	0.0319361
$852$	0	0
$853$	2.39801e7	1.12844	0.564220	−	0.825625i	$-0.309177\pi$
$853$	2.39801e7	1.12844	0.564220	+	0.825625i	$0.309177\pi$
$854$	2.17988e7	1.02279
$855$	0	0
$856$	2.22531e6	0.103802
$857$	−3.48306e7	−1.61998	−0.809989	−	0.586445i	$-0.800527\pi$
$857$	−3.48306e7	−1.61998	−0.809989	+	0.586445i	$0.800527\pi$
$858$	0	0
$859$	−1.01532e7	−0.469482	−0.234741	−	0.972058i	$-0.575424\pi$
$859$	−1.01532e7	−0.469482	−0.234741	+	0.972058i	$0.575424\pi$
$860$	0	0
$861$	0	0
$862$	2.36288e7	1.08311
$863$	2.71085e7	1.23902	0.619510	−	0.784989i	$-0.287331\pi$
$863$	2.71085e7	1.23902	0.619510	+	0.784989i	$0.287331\pi$
$864$	0	0
$865$	0	0
$866$	7.39981e6	0.335294
$867$	0	0
$868$	−9.15849e6	−0.412596
$869$	−1.11014e7	−0.498689
$870$	0	0
$871$	−2.92399e7	−1.30596
$872$	1.25673e6	0.0559696
$873$	0	0
$874$	457658.	0.0202658
$875$	0	0
$876$	0	0
$877$	−5.76852e6	−0.253259	−0.126630	−	0.991950i	$-0.540416\pi$
$877$	−5.76852e6	−0.253259	−0.126630	+	0.991950i	$0.540416\pi$
$878$	1.12469e7	0.492374
$879$	0	0
$880$	0	0
$881$	3.56949e7	1.54941	0.774705	−	0.632322i	$-0.217898\pi$
$881$	3.56949e7	1.54941	0.774705	+	0.632322i	$0.217898\pi$
$882$	0	0
$883$	−4.55843e6	−0.196749	−0.0983747	−	0.995149i	$-0.531364\pi$
$883$	−4.55843e6	−0.196749	−0.0983747	+	0.995149i	$0.531364\pi$
$884$	−1.75278e7	−0.754393
$885$	0	0
$886$	−9.65546e6	−0.413227
$887$	3.23885e6	0.138223	0.0691117	−	0.997609i	$-0.477984\pi$
$887$	3.23885e6	0.138223	0.0691117	+	0.997609i	$0.477984\pi$
$888$	0	0
$889$	6.68243e6	0.283583
$890$	0	0
$891$	0	0
$892$	1.67220e7	0.703683
$893$	1.15340e7	0.484008
$894$	0	0
$895$	0	0
$896$	1.96126e6	0.0816142
$897$	0	0
$898$	−2.02850e6	−0.0839428
$899$	2.86239e7	1.18122
$900$	0	0
$901$	−3.40615e7	−1.39782
$902$	−1.95142e7	−0.798611
$903$	0	0
$904$	−1.26186e6	−0.0513559
$905$	0	0
$906$	0	0
$907$	168518.	0.00680186	0.00340093	−	0.999994i	$-0.498917\pi$
$907$	168518.	0.00680186	0.00340093	+	0.999994i	$0.498917\pi$
$908$	−1.39059e7	−0.559739
$909$	0	0
$910$	0	0
$911$	−3.36451e7	−1.34315	−0.671577	−	0.740935i	$-0.734383\pi$
$911$	−3.36451e7	−1.34315	−0.671577	+	0.740935i	$0.734383\pi$
$912$	0	0
$913$	2.68276e7	1.06514
$914$	4.39514e6	0.174023
$915$	0	0
$916$	−2.23851e7	−0.881494
$917$	1.63252e7	0.641112
$918$	0	0
$919$	6.60944e6	0.258152	0.129076	−	0.991635i	$-0.458799\pi$
$919$	6.60944e6	0.258152	0.129076	+	0.991635i	$0.458799\pi$
$920$	0	0
$921$	0	0
$922$	424262.	0.0164364
$923$	4.89304e7	1.89049
$924$	0	0
$925$	0	0
$926$	5.44706e6	0.208754
$927$	0	0
$928$	−6.12973e6	−0.233653
$929$	4.55946e7	1.73330	0.866651	−	0.498915i	$-0.166268\pi$
$929$	4.55946e7	1.73330	0.866651	+	0.498915i	$0.166268\pi$
$930$	0	0
$931$	−5.10478e6	−0.193020
$932$	6.45590e6	0.243454
$933$	0	0
$934$	8.74839e6	0.328141
$935$	0	0
$936$	0	0
$937$	3.05380e6	0.113630	0.0568148	−	0.998385i	$-0.481906\pi$
$937$	3.05380e6	0.113630	0.0568148	+	0.998385i	$0.481906\pi$
$938$	−1.64499e7	−0.610457
$939$	0	0
$940$	0	0
$941$	−3.36840e7	−1.24008	−0.620040	−	0.784570i	$-0.712884\pi$
$941$	−3.36840e7	−1.24008	−0.620040	+	0.784570i	$0.712884\pi$
$942$	0	0
$943$	1.02727e6	0.0376190
$944$	−5.33135e6	−0.194719
$945$	0	0
$946$	−2.30845e6	−0.0838671
$947$	5.69161e6	0.206234	0.103117	−	0.994669i	$-0.467118\pi$
$947$	5.69161e6	0.206234	0.103117	+	0.994669i	$0.467118\pi$
$948$	0	0
$949$	−2.29434e7	−0.826976
$950$	0	0
$951$	0	0
$952$	−9.86084e6	−0.352632
$953$	1.73627e7	0.619278	0.309639	−	0.950854i	$-0.399792\pi$
$953$	1.73627e7	0.619278	0.309639	+	0.950854i	$0.399792\pi$
$954$	0	0
$955$	0	0
$956$	9.77721e6	0.345995
$957$	0	0
$958$	4.06380e6	0.143060
$959$	−1.60869e7	−0.564839
$960$	0	0
$961$	−5.76388e6	−0.201329
$962$	4.13657e7	1.44113
$963$	0	0
$964$	−2.41022e7	−0.835342
$965$	0	0
$966$	0	0
$967$	4.93082e7	1.69572	0.847858	−	0.530224i	$-0.177892\pi$
$967$	4.93082e7	1.69572	0.847858	+	0.530224i	$0.177892\pi$
$968$	5.85665e6	0.200891
$969$	0	0
$970$	0	0
$971$	−3.23520e7	−1.10117	−0.550584	−	0.834780i	$-0.685595\pi$
$971$	−3.23520e7	−1.10117	−0.550584	+	0.834780i	$0.685595\pi$
$972$	0	0
$973$	3.65729e7	1.23845
$974$	8.09357e6	0.273365
$975$	0	0
$976$	1.16546e7	0.391627
$977$	−2.61290e7	−0.875763	−0.437882	−	0.899033i	$-0.644271\pi$
$977$	−2.61290e7	−0.875763	−0.437882	+	0.899033i	$0.644271\pi$
$978$	0	0
$979$	−1.72862e7	−0.576426
$980$	0	0
$981$	0	0
$982$	−1.68995e7	−0.559235
$983$	−3.19496e7	−1.05458	−0.527292	−	0.849684i	$-0.676793\pi$
$983$	−3.19496e7	−1.05458	−0.527292	+	0.849684i	$0.676793\pi$
$984$	0	0
$985$	0	0
$986$	3.08191e7	1.00955
$987$	0	0
$988$	2.80593e7	0.914501
$989$	121522.	0.00395061
$990$	0	0
$991$	1.14783e7	0.371272	0.185636	−	0.982619i	$-0.440565\pi$
$991$	1.14783e7	0.371272	0.185636	+	0.982619i	$0.440565\pi$
$992$	−4.89653e6	−0.157982
$993$	0	0
$994$	2.75274e7	0.883688
$995$	0	0
$996$	0	0
$997$	−5.48929e7	−1.74895	−0.874476	−	0.485068i	$-0.838795\pi$
$997$	−5.48929e7	−1.74895	−0.874476	+	0.485068i	$0.838795\pi$
$998$	2.68552e7	0.853499
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.bb.1.1		2
3.2	odd	2		150.6.a.o.1.1		2
5.2	odd	4		90.6.c.c.19.2		4
5.3	odd	4		90.6.c.c.19.4		4
5.4	even	2		450.6.a.bc.1.2		2
15.2	even	4		30.6.c.b.19.3	yes	4
15.8	even	4		30.6.c.b.19.1	✓	4
15.14	odd	2		150.6.a.n.1.2		2
20.3	even	4		720.6.f.i.289.3		4
20.7	even	4		720.6.f.i.289.4		4
60.23	odd	4		240.6.f.b.49.1		4
60.47	odd	4		240.6.f.b.49.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.6.c.b.19.1	✓	4	15.8	even	4
30.6.c.b.19.3	yes	4	15.2	even	4
90.6.c.c.19.2		4	5.2	odd	4
90.6.c.c.19.4		4	5.3	odd	4
150.6.a.n.1.2		2	15.14	odd	2
150.6.a.o.1.1		2	3.2	odd	2
240.6.f.b.49.1		4	60.23	odd	4
240.6.f.b.49.3		4	60.47	odd	4
450.6.a.bb.1.1		2	1.1	even	1	trivial
450.6.a.bc.1.2		2	5.4	even	2
720.6.f.i.289.3		4	20.3	even	4
720.6.f.i.289.4		4	20.7	even	4

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} -119.706 q^{7} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} -119.706 q^{7} -64.0000 q^{8} -263.706 q^{11} +851.118 q^{13} +478.824 q^{14} +256.000 q^{16} -1287.12 q^{17} +2060.47 q^{19} +1054.82 q^{22} -55.5284 q^{23} -3404.47 q^{26} -1915.30 q^{28} +5986.06 q^{29} +4781.76 q^{31} -1024.00 q^{32} +5148.47 q^{34} -12150.4 q^{37} -8241.89 q^{38} -18500.0 q^{41} -2188.47 q^{43} -4219.30 q^{44} +222.113 q^{46} +5597.76 q^{47} -2477.48 q^{49} +13617.9 q^{52} +26463.4 q^{53} +7661.18 q^{56} -23944.2 q^{58} -20825.6 q^{59} +45525.8 q^{61} -19127.1 q^{62} +4096.00 q^{64} -34354.7 q^{67} -20593.9 q^{68} +57489.6 q^{71} -26956.8 q^{73} +48601.7 q^{74} +32967.5 q^{76} +31567.2 q^{77} +42097.8 q^{79} +74000.0 q^{82} -101733. q^{83} +8753.86 q^{86} +16877.2 q^{88} +65551.2 q^{89} -101884. q^{91} -888.454 q^{92} -22391.1 q^{94} -82780.7 q^{97} +9909.92 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} + 114 q^{7} - 128 q^{8}+O(q^{10}) \) 2 * q - 8 * q^2 + 32 * q^4 + 114 * q^7 - 128 * q^8	\( 2 q - 8 q^{2} + 32 q^{4} + 114 q^{7} - 128 q^{8} - 174 q^{11} + 642 q^{13} - 456 q^{14} + 512 q^{16} - 1514 q^{17} - 120 q^{19} + 696 q^{22} - 4352 q^{23} - 2568 q^{26} + 1824 q^{28} + 2430 q^{29} + 11684 q^{31} - 2048 q^{32} + 6056 q^{34} - 17586 q^{37} + 480 q^{38} - 24984 q^{41} - 24168 q^{43} - 2784 q^{44} + 17408 q^{46} + 13316 q^{47} + 35334 q^{49} + 10272 q^{52} + 13698 q^{53} - 7296 q^{56} - 9720 q^{58} + 23730 q^{59} + 57124 q^{61} - 46736 q^{62} + 8192 q^{64} - 38316 q^{67} - 24224 q^{68} + 11076 q^{71} - 88548 q^{73} + 70344 q^{74} - 1920 q^{76} + 52532 q^{77} + 14220 q^{79} + 99936 q^{82} - 50792 q^{83} + 96672 q^{86} + 11136 q^{88} + 60420 q^{89} - 150756 q^{91} - 69632 q^{92} - 53264 q^{94} - 171216 q^{97} - 141336 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 + 114 * q^7 - 128 * q^8 - 174 * q^11 + 642 * q^13 - 456 * q^14 + 512 * q^16 - 1514 * q^17 - 120 * q^19 + 696 * q^22 - 4352 * q^23 - 2568 * q^26 + 1824 * q^28 + 2430 * q^29 + 11684 * q^31 - 2048 * q^32 + 6056 * q^34 - 17586 * q^37 + 480 * q^38 - 24984 * q^41 - 24168 * q^43 - 2784 * q^44 + 17408 * q^46 + 13316 * q^47 + 35334 * q^49 + 10272 * q^52 + 13698 * q^53 - 7296 * q^56 - 9720 * q^58 + 23730 * q^59 + 57124 * q^61 - 46736 * q^62 + 8192 * q^64 - 38316 * q^67 - 24224 * q^68 + 11076 * q^71 - 88548 * q^73 + 70344 * q^74 - 1920 * q^76 + 52532 * q^77 + 14220 * q^79 + 99936 * q^82 - 50792 * q^83 + 96672 * q^86 + 11136 * q^88 + 60420 * q^89 - 150756 * q^91 - 69632 * q^92 - 53264 * q^94 - 171216 * q^97 - 141336 * q^98

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