LMFDB - Embedded newform 882.4.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.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:	$52.0396846251$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +15.8995 q^{5} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} +15.8995 q^{5} +8.00000 q^{8} +31.7990 q^{10} -57.3970 q^{11} +5.69848 q^{13} +16.0000 q^{16} +51.8995 q^{17} +16.2010 q^{19} +63.5980 q^{20} -114.794 q^{22} +213.397 q^{23} +127.794 q^{25} +11.3970 q^{26} +218.191 q^{29} -251.397 q^{31} +32.0000 q^{32} +103.799 q^{34} +386.794 q^{37} +32.4020 q^{38} +127.196 q^{40} +328.503 q^{41} -37.5879 q^{43} -229.588 q^{44} +426.794 q^{46} +254.995 q^{47} +255.588 q^{50} +22.7939 q^{52} -211.588 q^{53} -912.583 q^{55} +436.382 q^{58} +412.201 q^{59} -836.693 q^{61} -502.794 q^{62} +64.0000 q^{64} +90.6030 q^{65} -165.588 q^{67} +207.598 q^{68} +465.015 q^{71} +449.658 q^{73} +773.588 q^{74} +64.8040 q^{76} -343.558 q^{79} +254.392 q^{80} +657.005 q^{82} -1502.33 q^{83} +825.176 q^{85} -75.1758 q^{86} -459.176 q^{88} +341.085 q^{89} +853.588 q^{92} +509.990 q^{94} +257.588 q^{95} -865.437 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{8} + 24 q^{10} + 4 q^{11} - 48 q^{13} + 32 q^{16} + 84 q^{17} + 72 q^{19} + 48 q^{20} + 8 q^{22} + 308 q^{23} + 18 q^{25} - 96 q^{26} + 80 q^{29} - 384 q^{31} + 64 q^{32} + 168 q^{34} + 536 q^{37} + 144 q^{38} + 96 q^{40} + 756 q^{41} + 400 q^{43} + 16 q^{44} + 616 q^{46} + 312 q^{47} + 36 q^{50} - 192 q^{52} + 52 q^{53} - 1152 q^{55} + 160 q^{58} + 864 q^{59} - 1416 q^{61} - 768 q^{62} + 128 q^{64} + 300 q^{65} + 144 q^{67} + 336 q^{68} + 1524 q^{71} - 744 q^{73} + 1072 q^{74} + 288 q^{76} + 976 q^{79} + 192 q^{80} + 1512 q^{82} - 312 q^{83} + 700 q^{85} + 800 q^{86} + 32 q^{88} + 108 q^{89} + 1232 q^{92} + 624 q^{94} + 40 q^{95} + 744 q^{97}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^8 + 24 * q^10 + 4 * q^11 - 48 * q^13 + 32 * q^16 + 84 * q^17 + 72 * q^19 + 48 * q^20 + 8 * q^22 + 308 * q^23 + 18 * q^25 - 96 * q^26 + 80 * q^29 - 384 * q^31 + 64 * q^32 + 168 * q^34 + 536 * q^37 + 144 * q^38 + 96 * q^40 + 756 * q^41 + 400 * q^43 + 16 * q^44 + 616 * q^46 + 312 * q^47 + 36 * q^50 - 192 * q^52 + 52 * q^53 - 1152 * q^55 + 160 * q^58 + 864 * q^59 - 1416 * q^61 - 768 * q^62 + 128 * q^64 + 300 * q^65 + 144 * q^67 + 336 * q^68 + 1524 * q^71 - 744 * q^73 + 1072 * q^74 + 288 * q^76 + 976 * q^79 + 192 * q^80 + 1512 * q^82 - 312 * q^83 + 700 * q^85 + 800 * q^86 + 32 * q^88 + 108 * q^89 + 1232 * q^92 + 624 * q^94 + 40 * q^95 + 744 * q^97

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	15.8995	1.42209	0.711047	−	0.703144i	$-0.248221\pi$
$5$	15.8995	1.42209	0.711047	+	0.703144i	$0.248221\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	0	0
$10$	31.7990	1.00557
$11$	−57.3970	−1.57326	−0.786629	−	0.617426i	$-0.788176\pi$
$11$	−57.3970	−1.57326	−0.786629	+	0.617426i	$0.788176\pi$
$12$	0	0
$13$	5.69848	0.121575	0.0607875	−	0.998151i	$-0.480639\pi$
$13$	5.69848	0.121575	0.0607875	+	0.998151i	$0.480639\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	51.8995	0.740440	0.370220	−	0.928944i	$-0.379282\pi$
$17$	51.8995	0.740440	0.370220	+	0.928944i	$0.379282\pi$
$18$	0	0
$19$	16.2010	0.195619	0.0978096	−	0.995205i	$-0.468816\pi$
$19$	16.2010	0.195619	0.0978096	+	0.995205i	$0.468816\pi$
$20$	63.5980	0.711047
$21$	0	0
$22$	−114.794	−1.11246
$23$	213.397	1.93462	0.967312	−	0.253590i	$-0.0816114\pi$
$23$	213.397	1.93462	0.967312	+	0.253590i	$0.0816114\pi$
$24$	0	0
$25$	127.794	1.02235
$26$	11.3970	0.0859665
$27$	0	0
$28$	0	0
$29$	218.191	1.39714	0.698570	−	0.715542i	$-0.253820\pi$
$29$	218.191	1.39714	0.698570	+	0.715542i	$0.253820\pi$
$30$	0	0
$31$	−251.397	−1.45652	−0.728262	−	0.685299i	$-0.759671\pi$
$31$	−251.397	−1.45652	−0.728262	+	0.685299i	$0.759671\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	103.799	0.523570
$35$	0	0
$36$	0	0
$37$	386.794	1.71861	0.859304	−	0.511464i	$-0.170897\pi$
$37$	386.794	1.71861	0.859304	+	0.511464i	$0.170897\pi$
$38$	32.4020	0.138324
$39$	0	0
$40$	127.196	0.502786
$41$	328.503	1.25130	0.625652	−	0.780102i	$-0.284833\pi$
$41$	328.503	1.25130	0.625652	+	0.780102i	$0.284833\pi$
$42$	0	0
$43$	−37.5879	−0.133305	−0.0666523	−	0.997776i	$-0.521232\pi$
$43$	−37.5879	−0.133305	−0.0666523	+	0.997776i	$0.521232\pi$
$44$	−229.588	−0.786629
$45$	0	0
$46$	426.794	1.36799
$47$	254.995	0.791379	0.395690	−	0.918384i	$-0.370506\pi$
$47$	254.995	0.791379	0.395690	+	0.918384i	$0.370506\pi$
$48$	0	0
$49$	0	0
$50$	255.588	0.722912
$51$	0	0
$52$	22.7939	0.0607875
$53$	−211.588	−0.548374	−0.274187	−	0.961676i	$-0.588409\pi$
$53$	−211.588	−0.548374	−0.274187	+	0.961676i	$0.588409\pi$
$54$	0	0
$55$	−912.583	−2.23732
$56$	0	0
$57$	0	0
$58$	436.382	0.987927
$59$	412.201	0.909559	0.454780	−	0.890604i	$-0.349718\pi$
$59$	412.201	0.909559	0.454780	+	0.890604i	$0.349718\pi$
$60$	0	0
$61$	−836.693	−1.75619	−0.878095	−	0.478486i	$-0.841186\pi$
$61$	−836.693	−1.75619	−0.878095	+	0.478486i	$0.841186\pi$
$62$	−502.794	−1.02992
$63$	0	0
$64$	64.0000	0.125000
$65$	90.6030	0.172891
$66$	0	0
$67$	−165.588	−0.301937	−0.150969	−	0.988539i	$-0.548239\pi$
$67$	−165.588	−0.301937	−0.150969	+	0.988539i	$0.548239\pi$
$68$	207.598	0.370220
$69$	0	0
$70$	0	0
$71$	465.015	0.777284	0.388642	−	0.921389i	$-0.372944\pi$
$71$	465.015	0.777284	0.388642	+	0.921389i	$0.372944\pi$
$72$	0	0
$73$	449.658	0.720938	0.360469	−	0.932771i	$-0.382617\pi$
$73$	449.658	0.720938	0.360469	+	0.932771i	$0.382617\pi$
$74$	773.588	1.21524
$75$	0	0
$76$	64.8040	0.0978096
$77$	0	0
$78$	0	0
$79$	−343.558	−0.489282	−0.244641	−	0.969614i	$-0.578670\pi$
$79$	−343.558	−0.489282	−0.244641	+	0.969614i	$0.578670\pi$
$80$	254.392	0.355524
$81$	0	0
$82$	657.005	0.884806
$83$	−1502.33	−1.98677	−0.993387	−	0.114812i	$-0.963373\pi$
$83$	−1502.33	−1.98677	−0.993387	+	0.114812i	$0.963373\pi$
$84$	0	0
$85$	825.176	1.05298
$86$	−75.1758	−0.0942606
$87$	0	0
$88$	−459.176	−0.556231
$89$	341.085	0.406236	0.203118	−	0.979154i	$-0.434893\pi$
$89$	341.085	0.406236	0.203118	+	0.979154i	$0.434893\pi$
$90$	0	0
$91$	0	0
$92$	853.588	0.967312
$93$	0	0
$94$	509.990	0.559590
$95$	257.588	0.278189
$96$	0	0
$97$	−865.437	−0.905895	−0.452947	−	0.891537i	$-0.649627\pi$
$97$	−865.437	−0.905895	−0.452947	+	0.891537i	$0.649627\pi$
$98$	0	0
$99$	0	0
$100$	511.176	0.511176
$101$	243.256	0.239652	0.119826	−	0.992795i	$-0.461766\pi$
$101$	243.256	0.239652	0.119826	+	0.992795i	$0.461766\pi$
$102$	0	0
$103$	−953.346	−0.912000	−0.456000	−	0.889980i	$-0.650718\pi$
$103$	−953.346	−0.912000	−0.456000	+	0.889980i	$0.650718\pi$
$104$	45.5879	0.0429833
$105$	0	0
$106$	−423.176	−0.387759
$107$	1344.95	1.21516	0.607578	−	0.794260i	$-0.292141\pi$
$107$	1344.95	1.21516	0.607578	+	0.794260i	$0.292141\pi$
$108$	0	0
$109$	1734.35	1.52404	0.762022	−	0.647551i	$-0.224207\pi$
$109$	1734.35	1.52404	0.762022	+	0.647551i	$0.224207\pi$
$110$	−1825.17	−1.58202
$111$	0	0
$112$	0	0
$113$	−1441.18	−1.19977	−0.599887	−	0.800085i	$-0.704788\pi$
$113$	−1441.18	−1.19977	−0.599887	+	0.800085i	$0.704788\pi$
$114$	0	0
$115$	3392.90	2.75122
$116$	872.764	0.698570
$117$	0	0
$118$	824.402	0.643156
$119$	0	0
$120$	0	0
$121$	1963.41	1.47514
$122$	−1673.39	−1.24181
$123$	0	0
$124$	−1005.59	−0.728262
$125$	44.4222	0.0317860
$126$	0	0
$127$	1184.70	0.827759	0.413880	−	0.910332i	$-0.364173\pi$
$127$	1184.70	0.827759	0.413880	+	0.910332i	$0.364173\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	181.206	0.122252
$131$	−297.588	−0.198476	−0.0992381	−	0.995064i	$-0.531641\pi$
$131$	−297.588	−0.198476	−0.0992381	+	0.995064i	$0.531641\pi$
$132$	0	0
$133$	0	0
$134$	−331.176	−0.213502
$135$	0	0
$136$	415.196	0.261785
$137$	−620.985	−0.387258	−0.193629	−	0.981075i	$-0.562026\pi$
$137$	−620.985	−0.387258	−0.193629	+	0.981075i	$0.562026\pi$
$138$	0	0
$139$	−898.754	−0.548426	−0.274213	−	0.961669i	$-0.588417\pi$
$139$	−898.754	−0.548426	−0.274213	+	0.961669i	$0.588417\pi$
$140$	0	0
$141$	0	0
$142$	930.030	0.549623
$143$	−327.076	−0.191269
$144$	0	0
$145$	3469.13	1.98686
$146$	899.316	0.509780
$147$	0	0
$148$	1547.18	0.859304
$149$	3054.70	1.67954	0.839769	−	0.542945i	$-0.182691\pi$
$149$	3054.70	1.67954	0.839769	+	0.542945i	$0.182691\pi$
$150$	0	0
$151$	−65.1455	−0.0351090	−0.0175545	−	0.999846i	$-0.505588\pi$
$151$	−65.1455	−0.0351090	−0.0175545	+	0.999846i	$0.505588\pi$
$152$	129.608	0.0691619
$153$	0	0
$154$	0	0
$155$	−3997.08	−2.07131
$156$	0	0
$157$	1542.22	0.783966	0.391983	−	0.919973i	$-0.371789\pi$
$157$	1542.22	0.783966	0.391983	+	0.919973i	$0.371789\pi$
$158$	−687.115	−0.345974
$159$	0	0
$160$	508.784	0.251393
$161$	0	0
$162$	0	0
$163$	−2514.73	−1.20840	−0.604200	−	0.796833i	$-0.706507\pi$
$163$	−2514.73	−1.20840	−0.604200	+	0.796833i	$0.706507\pi$
$164$	1314.01	0.625652
$165$	0	0
$166$	−3004.66	−1.40486
$167$	−528.643	−0.244956	−0.122478	−	0.992471i	$-0.539084\pi$
$167$	−528.643	−0.244956	−0.122478	+	0.992471i	$0.539084\pi$
$168$	0	0
$169$	−2164.53	−0.985220
$170$	1650.35	0.744566
$171$	0	0
$172$	−150.352	−0.0666523
$173$	−96.8439	−0.0425602	−0.0212801	−	0.999774i	$-0.506774\pi$
$173$	−96.8439	−0.0425602	−0.0212801	+	0.999774i	$0.506774\pi$
$174$	0	0
$175$	0	0
$176$	−918.352	−0.393314
$177$	0	0
$178$	682.171	0.287252
$179$	534.542	0.223204	0.111602	−	0.993753i	$-0.464402\pi$
$179$	534.542	0.223204	0.111602	+	0.993753i	$0.464402\pi$
$180$	0	0
$181$	2087.00	0.857049	0.428524	−	0.903530i	$-0.359034\pi$
$181$	2087.00	0.857049	0.428524	+	0.903530i	$0.359034\pi$
$182$	0	0
$183$	0	0
$184$	1707.18	0.683993
$185$	6149.83	2.44402
$186$	0	0
$187$	−2978.87	−1.16490
$188$	1019.98	0.395690
$189$	0	0
$190$	515.176	0.196709
$191$	−3387.69	−1.28337	−0.641687	−	0.766966i	$-0.721765\pi$
$191$	−3387.69	−1.28337	−0.641687	+	0.766966i	$0.721765\pi$
$192$	0	0
$193$	−1908.35	−0.711742	−0.355871	−	0.934535i	$-0.615816\pi$
$193$	−1908.35	−0.711742	−0.355871	+	0.934535i	$0.615816\pi$
$194$	−1730.87	−0.640564
$195$	0	0
$196$	0	0
$197$	2061.88	0.745699	0.372850	−	0.927892i	$-0.378381\pi$
$197$	2061.88	0.745699	0.372850	+	0.927892i	$0.378381\pi$
$198$	0	0
$199$	3171.50	1.12976	0.564878	−	0.825174i	$-0.308923\pi$
$199$	3171.50	1.12976	0.564878	+	0.825174i	$0.308923\pi$
$200$	1022.35	0.361456
$201$	0	0
$202$	486.512	0.169460
$203$	0	0
$204$	0	0
$205$	5223.02	1.77947
$206$	−1906.69	−0.644882
$207$	0	0
$208$	91.1758	0.0303938
$209$	−929.889	−0.307760
$210$	0	0
$211$	1349.97	0.440454	0.220227	−	0.975449i	$-0.429320\pi$
$211$	1349.97	0.440454	0.220227	+	0.975449i	$0.429320\pi$
$212$	−846.352	−0.274187
$213$	0	0
$214$	2689.91	0.859245
$215$	−597.628	−0.189572
$216$	0	0
$217$	0	0
$218$	3468.70	1.07766
$219$	0	0
$220$	−3650.33	−1.11866
$221$	295.748	0.0900190
$222$	0	0
$223$	−1361.85	−0.408951	−0.204476	−	0.978872i	$-0.565549\pi$
$223$	−1361.85	−0.408951	−0.204476	+	0.978872i	$0.565549\pi$
$224$	0	0
$225$	0	0
$226$	−2882.35	−0.848368
$227$	−1861.81	−0.544373	−0.272186	−	0.962245i	$-0.587747\pi$
$227$	−1861.81	−0.544373	−0.272186	+	0.962245i	$0.587747\pi$
$228$	0	0
$229$	−5358.78	−1.54637	−0.773184	−	0.634181i	$-0.781337\pi$
$229$	−5358.78	−1.54637	−0.773184	+	0.634181i	$0.781337\pi$
$230$	6785.81	1.94540
$231$	0	0
$232$	1745.53	0.493963
$233$	−5441.12	−1.52987	−0.764935	−	0.644107i	$-0.777229\pi$
$233$	−5441.12	−1.52987	−0.764935	+	0.644107i	$0.777229\pi$
$234$	0	0
$235$	4054.29	1.12542
$236$	1648.80	0.454780
$237$	0	0
$238$	0	0
$239$	1157.28	0.313213	0.156607	−	0.987661i	$-0.449945\pi$
$239$	1157.28	0.313213	0.156607	+	0.987661i	$0.449945\pi$
$240$	0	0
$241$	−3969.38	−1.06095	−0.530477	−	0.847699i	$-0.677987\pi$
$241$	−3969.38	−1.06095	−0.530477	+	0.847699i	$0.677987\pi$
$242$	3926.82	1.04308
$243$	0	0
$244$	−3346.77	−0.878095
$245$	0	0
$246$	0	0
$247$	92.3212	0.0237824
$248$	−2011.18	−0.514959
$249$	0	0
$250$	88.8444	0.0224761
$251$	−5978.75	−1.50349	−0.751744	−	0.659455i	$-0.770787\pi$
$251$	−5978.75	−1.50349	−0.751744	+	0.659455i	$0.770787\pi$
$252$	0	0
$253$	−12248.3	−3.04366
$254$	2369.41	0.585314
$255$	0	0
$256$	256.000	0.0625000
$257$	−4650.15	−1.12867	−0.564335	−	0.825546i	$-0.690868\pi$
$257$	−4650.15	−1.12867	−0.564335	+	0.825546i	$0.690868\pi$
$258$	0	0
$259$	0	0
$260$	362.412	0.0864456
$261$	0	0
$262$	−595.176	−0.140344
$263$	−3695.37	−0.866411	−0.433205	−	0.901295i	$-0.642618\pi$
$263$	−3695.37	−0.866411	−0.433205	+	0.901295i	$0.642618\pi$
$264$	0	0
$265$	−3364.14	−0.779840
$266$	0	0
$267$	0	0
$268$	−662.352	−0.150969
$269$	7157.69	1.62235	0.811175	−	0.584804i	$-0.198829\pi$
$269$	7157.69	1.62235	0.811175	+	0.584804i	$0.198829\pi$
$270$	0	0
$271$	4038.37	0.905216	0.452608	−	0.891710i	$-0.350494\pi$
$271$	4038.37	0.905216	0.452608	+	0.891710i	$0.350494\pi$
$272$	830.392	0.185110
$273$	0	0
$274$	−1241.97	−0.273833
$275$	−7334.98	−1.60842
$276$	0	0
$277$	−2754.82	−0.597550	−0.298775	−	0.954324i	$-0.596578\pi$
$277$	−2754.82	−0.597550	−0.298775	+	0.954324i	$0.596578\pi$
$278$	−1797.51	−0.387796
$279$	0	0
$280$	0	0
$281$	772.742	0.164050	0.0820248	−	0.996630i	$-0.473861\pi$
$281$	772.742	0.164050	0.0820248	+	0.996630i	$0.473861\pi$
$282$	0	0
$283$	−6745.49	−1.41688	−0.708441	−	0.705770i	$-0.750601\pi$
$283$	−6745.49	−1.41688	−0.708441	+	0.705770i	$0.750601\pi$
$284$	1860.06	0.388642
$285$	0	0
$286$	−654.152	−0.135248
$287$	0	0
$288$	0	0
$289$	−2219.44	−0.451749
$290$	6938.25	1.40492
$291$	0	0
$292$	1798.63	0.360469
$293$	−1922.69	−0.383362	−0.191681	−	0.981457i	$-0.561394\pi$
$293$	−1922.69	−0.383362	−0.191681	+	0.981457i	$0.561394\pi$
$294$	0	0
$295$	6553.79	1.29348
$296$	3094.35	0.607620
$297$	0	0
$298$	6109.41	1.18761
$299$	1216.04	0.235202
$300$	0	0
$301$	0	0
$302$	−130.291	−0.0248258
$303$	0	0
$304$	259.216	0.0489048
$305$	−13303.0	−2.49747
$306$	0	0
$307$	2016.68	0.374913	0.187456	−	0.982273i	$-0.439976\pi$
$307$	2016.68	0.374913	0.187456	+	0.982273i	$0.439976\pi$
$308$	0	0
$309$	0	0
$310$	−7994.17	−1.46464
$311$	−7149.99	−1.30366	−0.651831	−	0.758365i	$-0.725999\pi$
$311$	−7149.99	−1.30366	−0.651831	+	0.758365i	$0.725999\pi$
$312$	0	0
$313$	8596.49	1.55240	0.776202	−	0.630484i	$-0.217144\pi$
$313$	8596.49	1.55240	0.776202	+	0.630484i	$0.217144\pi$
$314$	3084.44	0.554347
$315$	0	0
$316$	−1374.23	−0.244641
$317$	2853.24	0.505532	0.252766	−	0.967527i	$-0.418660\pi$
$317$	2853.24	0.505532	0.252766	+	0.967527i	$0.418660\pi$
$318$	0	0
$319$	−12523.5	−2.19806
$320$	1017.57	0.177762
$321$	0	0
$322$	0	0
$323$	840.824	0.144844
$324$	0	0
$325$	728.232	0.124292
$326$	−5029.47	−0.854467
$327$	0	0
$328$	2628.02	0.442403
$329$	0	0
$330$	0	0
$331$	1619.12	0.268866	0.134433	−	0.990923i	$-0.457079\pi$
$331$	1619.12	0.268866	0.134433	+	0.990923i	$0.457079\pi$
$332$	−6009.33	−0.993387
$333$	0	0
$334$	−1057.29	−0.173210
$335$	−2632.76	−0.429383
$336$	0	0
$337$	−3278.67	−0.529972	−0.264986	−	0.964252i	$-0.585367\pi$
$337$	−3278.67	−0.529972	−0.264986	+	0.964252i	$0.585367\pi$
$338$	−4329.05	−0.696655
$339$	0	0
$340$	3300.70	0.526488
$341$	14429.4	2.29149
$342$	0	0
$343$	0	0
$344$	−300.703	−0.0471303
$345$	0	0
$346$	−193.688	−0.0300946
$347$	2850.30	0.440957	0.220479	−	0.975392i	$-0.429238\pi$
$347$	2850.30	0.440957	0.220479	+	0.975392i	$0.429238\pi$
$348$	0	0
$349$	−4725.32	−0.724758	−0.362379	−	0.932031i	$-0.618035\pi$
$349$	−4725.32	−0.724758	−0.362379	+	0.932031i	$0.618035\pi$
$350$	0	0
$351$	0	0
$352$	−1836.70	−0.278115
$353$	−6727.44	−1.01435	−0.507175	−	0.861843i	$-0.669310\pi$
$353$	−6727.44	−1.01435	−0.507175	+	0.861843i	$0.669310\pi$
$354$	0	0
$355$	7393.51	1.10537
$356$	1364.34	0.203118
$357$	0	0
$358$	1069.08	0.157829
$359$	7331.89	1.07789	0.538945	−	0.842341i	$-0.318823\pi$
$359$	7331.89	1.07789	0.538945	+	0.842341i	$0.318823\pi$
$360$	0	0
$361$	−6596.53	−0.961733
$362$	4174.01	0.606025
$363$	0	0
$364$	0	0
$365$	7149.34	1.02524
$366$	0	0
$367$	2774.43	0.394616	0.197308	−	0.980342i	$-0.436780\pi$
$367$	2774.43	0.394616	0.197308	+	0.980342i	$0.436780\pi$
$368$	3414.35	0.483656
$369$	0	0
$370$	12299.7	1.72819
$371$	0	0
$372$	0	0
$373$	2527.35	0.350834	0.175417	−	0.984494i	$-0.443873\pi$
$373$	2527.35	0.350834	0.175417	+	0.984494i	$0.443873\pi$
$374$	−5957.75	−0.823711
$375$	0	0
$376$	2039.96	0.279795
$377$	1243.36	0.169857
$378$	0	0
$379$	3116.40	0.422371	0.211186	−	0.977446i	$-0.432268\pi$
$379$	3116.40	0.422371	0.211186	+	0.977446i	$0.432268\pi$
$380$	1030.35	0.139095
$381$	0	0
$382$	−6775.38	−0.907483
$383$	−1518.07	−0.202532	−0.101266	−	0.994859i	$-0.532289\pi$
$383$	−1518.07	−0.202532	−0.101266	+	0.994859i	$0.532289\pi$
$384$	0	0
$385$	0	0
$386$	−3816.70	−0.503277
$387$	0	0
$388$	−3461.75	−0.452947
$389$	3246.77	0.423182	0.211591	−	0.977358i	$-0.432135\pi$
$389$	3246.77	0.423182	0.211591	+	0.977358i	$0.432135\pi$
$390$	0	0
$391$	11075.2	1.43247
$392$	0	0
$393$	0	0
$394$	4123.76	0.527289
$395$	−5462.39	−0.695804
$396$	0	0
$397$	−1830.62	−0.231427	−0.115713	−	0.993283i	$-0.536915\pi$
$397$	−1830.62	−0.231427	−0.115713	+	0.993283i	$0.536915\pi$
$398$	6342.99	0.798858
$399$	0	0
$400$	2044.70	0.255588
$401$	3385.81	0.421644	0.210822	−	0.977524i	$-0.432386\pi$
$401$	3385.81	0.421644	0.210822	+	0.977524i	$0.432386\pi$
$402$	0	0
$403$	−1432.58	−0.177077
$404$	973.024	0.119826
$405$	0	0
$406$	0	0
$407$	−22200.8	−2.70382
$408$	0	0
$409$	−9253.17	−1.11868	−0.559340	−	0.828938i	$-0.688945\pi$
$409$	−9253.17	−1.11868	−0.559340	+	0.828938i	$0.688945\pi$
$410$	10446.0	1.25828
$411$	0	0
$412$	−3813.39	−0.456000
$413$	0	0
$414$	0	0
$415$	−23886.3	−2.82538
$416$	182.352	0.0214916
$417$	0	0
$418$	−1859.78	−0.217619
$419$	3547.52	0.413622	0.206811	−	0.978381i	$-0.433692\pi$
$419$	3547.52	0.413622	0.206811	+	0.978381i	$0.433692\pi$
$420$	0	0
$421$	7848.87	0.908624	0.454312	−	0.890843i	$-0.349885\pi$
$421$	7848.87	0.908624	0.454312	+	0.890843i	$0.349885\pi$
$422$	2699.94	0.311448
$423$	0	0
$424$	−1692.70	−0.193880
$425$	6632.44	0.756990
$426$	0	0
$427$	0	0
$428$	5379.82	0.607578
$429$	0	0
$430$	−1195.26	−0.134047
$431$	4447.32	0.497030	0.248515	−	0.968628i	$-0.420057\pi$
$431$	4447.32	0.497030	0.248515	+	0.968628i	$0.420057\pi$
$432$	0	0
$433$	6994.82	0.776327	0.388164	−	0.921590i	$-0.373110\pi$
$433$	6994.82	0.776327	0.388164	+	0.921590i	$0.373110\pi$
$434$	0	0
$435$	0	0
$436$	6937.41	0.762022
$437$	3457.25	0.378450
$438$	0	0
$439$	−636.182	−0.0691647	−0.0345823	−	0.999402i	$-0.511010\pi$
$439$	−636.182	−0.0691647	−0.0345823	+	0.999402i	$0.511010\pi$
$440$	−7300.66	−0.791012
$441$	0	0
$442$	591.497	0.0636530
$443$	4474.24	0.479859	0.239929	−	0.970790i	$-0.422876\pi$
$443$	4474.24	0.479859	0.239929	+	0.970790i	$0.422876\pi$
$444$	0	0
$445$	5423.08	0.577705
$446$	−2723.70	−0.289172
$447$	0	0
$448$	0	0
$449$	2389.42	0.251144	0.125572	−	0.992085i	$-0.459923\pi$
$449$	2389.42	0.251144	0.125572	+	0.992085i	$0.459923\pi$
$450$	0	0
$451$	−18855.0	−1.96862
$452$	−5764.70	−0.599887
$453$	0	0
$454$	−3723.62	−0.384930
$455$	0	0
$456$	0	0
$457$	−4438.34	−0.454304	−0.227152	−	0.973859i	$-0.572941\pi$
$457$	−4438.34	−0.454304	−0.227152	+	0.973859i	$0.572941\pi$
$458$	−10717.6	−1.09345
$459$	0	0
$460$	13571.6	1.37561
$461$	−14079.8	−1.42248	−0.711240	−	0.702949i	$-0.751866\pi$
$461$	−14079.8	−1.42248	−0.711240	+	0.702949i	$0.751866\pi$
$462$	0	0
$463$	4687.50	0.470511	0.235255	−	0.971934i	$-0.424407\pi$
$463$	4687.50	0.470511	0.235255	+	0.971934i	$0.424407\pi$
$464$	3491.05	0.349285
$465$	0	0
$466$	−10882.2	−1.08178
$467$	8447.26	0.837029	0.418514	−	0.908210i	$-0.362551\pi$
$467$	8447.26	0.837029	0.418514	+	0.908210i	$0.362551\pi$
$468$	0	0
$469$	0	0
$470$	8108.58	0.795789
$471$	0	0
$472$	3297.61	0.321578
$473$	2157.43	0.209723
$474$	0	0
$475$	2070.39	0.199992
$476$	0	0
$477$	0	0
$478$	2314.55	0.221475
$479$	−4369.41	−0.416792	−0.208396	−	0.978045i	$-0.566824\pi$
$479$	−4369.41	−0.416792	−0.208396	+	0.978045i	$0.566824\pi$
$480$	0	0
$481$	2204.14	0.208940
$482$	−7938.75	−0.750208
$483$	0	0
$484$	7853.65	0.737570
$485$	−13760.0	−1.28827
$486$	0	0
$487$	−14477.7	−1.34712	−0.673561	−	0.739132i	$-0.735236\pi$
$487$	−14477.7	−1.34712	−0.673561	+	0.739132i	$0.735236\pi$
$488$	−6693.55	−0.620907
$489$	0	0
$490$	0	0
$491$	−9306.12	−0.855355	−0.427677	−	0.903931i	$-0.640668\pi$
$491$	−9306.12	−0.855355	−0.427677	+	0.903931i	$0.640668\pi$
$492$	0	0
$493$	11324.0	1.03450
$494$	184.642	0.0168167
$495$	0	0
$496$	−4022.35	−0.364131
$497$	0	0
$498$	0	0
$499$	−12237.5	−1.09785	−0.548926	−	0.835871i	$-0.684963\pi$
$499$	−12237.5	−1.09785	−0.548926	+	0.835871i	$0.684963\pi$
$500$	177.689	0.0158930
$501$	0	0
$502$	−11957.5	−1.06313
$503$	5524.30	0.489694	0.244847	−	0.969562i	$-0.421262\pi$
$503$	5524.30	0.489694	0.244847	+	0.969562i	$0.421262\pi$
$504$	0	0
$505$	3867.65	0.340808
$506$	−24496.7	−2.15219
$507$	0	0
$508$	4738.81	0.413880
$509$	−10079.6	−0.877743	−0.438871	−	0.898550i	$-0.644622\pi$
$509$	−10079.6	−0.877743	−0.438871	+	0.898550i	$0.644622\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	0	0
$514$	−9300.30	−0.798091
$515$	−15157.7	−1.29695
$516$	0	0
$517$	−14635.9	−1.24504
$518$	0	0
$519$	0	0
$520$	724.824	0.0611262
$521$	5706.61	0.479868	0.239934	−	0.970789i	$-0.422874\pi$
$521$	5706.61	0.479868	0.239934	+	0.970789i	$0.422874\pi$
$522$	0	0
$523$	10657.3	0.891032	0.445516	−	0.895274i	$-0.353020\pi$
$523$	10657.3	0.891032	0.445516	+	0.895274i	$0.353020\pi$
$524$	−1190.35	−0.0992381
$525$	0	0
$526$	−7390.73	−0.612645
$527$	−13047.4	−1.07847
$528$	0	0
$529$	33371.3	2.74277
$530$	−6728.28	−0.551430
$531$	0	0
$532$	0	0
$533$	1871.97	0.152127
$534$	0	0
$535$	21384.1	1.72807
$536$	−1324.70	−0.106751
$537$	0	0
$538$	14315.4	1.14717
$539$	0	0
$540$	0	0
$541$	−4010.48	−0.318714	−0.159357	−	0.987221i	$-0.550942\pi$
$541$	−4010.48	−0.318714	−0.159357	+	0.987221i	$0.550942\pi$
$542$	8076.74	0.640084
$543$	0	0
$544$	1660.78	0.130892
$545$	27575.3	2.16733
$546$	0	0
$547$	−17619.8	−1.37728	−0.688638	−	0.725105i	$-0.741791\pi$
$547$	−17619.8	−1.37728	−0.688638	+	0.725105i	$0.741791\pi$
$548$	−2483.94	−0.193629
$549$	0	0
$550$	−14670.0	−1.13733
$551$	3534.91	0.273307
$552$	0	0
$553$	0	0
$554$	−5509.65	−0.422532
$555$	0	0
$556$	−3595.01	−0.274213
$557$	10337.7	0.786395	0.393198	−	0.919454i	$-0.371369\pi$
$557$	10337.7	0.786395	0.393198	+	0.919454i	$0.371369\pi$
$558$	0	0
$559$	−214.194	−0.0162065
$560$	0	0
$561$	0	0
$562$	1545.48	0.116001
$563$	−24023.7	−1.79836	−0.899180	−	0.437580i	$-0.855836\pi$
$563$	−24023.7	−1.79836	−0.899180	+	0.437580i	$0.855836\pi$
$564$	0	0
$565$	−22914.0	−1.70619
$566$	−13491.0	−1.00189
$567$	0	0
$568$	3720.12	0.274811
$569$	13179.2	0.971001	0.485501	−	0.874236i	$-0.338637\pi$
$569$	13179.2	0.971001	0.485501	+	0.874236i	$0.338637\pi$
$570$	0	0
$571$	7776.26	0.569924	0.284962	−	0.958539i	$-0.408019\pi$
$571$	7776.26	0.569924	0.284962	+	0.958539i	$0.408019\pi$
$572$	−1308.30	−0.0956344
$573$	0	0
$574$	0	0
$575$	27270.8	1.97787
$576$	0	0
$577$	−20167.0	−1.45505	−0.727525	−	0.686081i	$-0.759330\pi$
$577$	−20167.0	−1.45505	−0.727525	+	0.686081i	$0.759330\pi$
$578$	−4438.88	−0.319435
$579$	0	0
$580$	13876.5	0.993432
$581$	0	0
$582$	0	0
$583$	12144.5	0.862734
$584$	3597.26	0.254890
$585$	0	0
$586$	−3845.39	−0.271078
$587$	−8365.08	−0.588184	−0.294092	−	0.955777i	$-0.595017\pi$
$587$	−8365.08	−0.588184	−0.294092	+	0.955777i	$0.595017\pi$
$588$	0	0
$589$	−4072.88	−0.284924
$590$	13107.6	0.914628
$591$	0	0
$592$	6188.70	0.429652
$593$	−27621.9	−1.91281	−0.956403	−	0.292050i	$-0.905663\pi$
$593$	−27621.9	−1.91281	−0.956403	+	0.292050i	$0.905663\pi$
$594$	0	0
$595$	0	0
$596$	12218.8	0.839769
$597$	0	0
$598$	2432.08	0.166313
$599$	−538.318	−0.0367197	−0.0183598	−	0.999831i	$-0.505844\pi$
$599$	−538.318	−0.0367197	−0.0183598	+	0.999831i	$0.505844\pi$
$600$	0	0
$601$	−6958.64	−0.472294	−0.236147	−	0.971717i	$-0.575885\pi$
$601$	−6958.64	−0.472294	−0.236147	+	0.971717i	$0.575885\pi$
$602$	0	0
$603$	0	0
$604$	−260.582	−0.0175545
$605$	31217.3	2.09779
$606$	0	0
$607$	17297.6	1.15665	0.578326	−	0.815806i	$-0.303706\pi$
$607$	17297.6	1.15665	0.578326	+	0.815806i	$0.303706\pi$
$608$	518.432	0.0345809
$609$	0	0
$610$	−26606.0	−1.76598
$611$	1453.08	0.0962120
$612$	0	0
$613$	−839.158	−0.0552908	−0.0276454	−	0.999618i	$-0.508801\pi$
$613$	−839.158	−0.0552908	−0.0276454	+	0.999618i	$0.508801\pi$
$614$	4033.37	0.265103
$615$	0	0
$616$	0	0
$617$	16040.0	1.04659	0.523295	−	0.852152i	$-0.324703\pi$
$617$	16040.0	1.04659	0.523295	+	0.852152i	$0.324703\pi$
$618$	0	0
$619$	5429.28	0.352538	0.176269	−	0.984342i	$-0.443597\pi$
$619$	5429.28	0.352538	0.176269	+	0.984342i	$0.443597\pi$
$620$	−15988.3	−1.03566
$621$	0	0
$622$	−14300.0	−0.921828
$623$	0	0
$624$	0	0
$625$	−15268.0	−0.977149
$626$	17193.0	1.09772
$627$	0	0
$628$	6168.88	0.391983
$629$	20074.4	1.27253
$630$	0	0
$631$	−1807.86	−0.114057	−0.0570284	−	0.998373i	$-0.518163\pi$
$631$	−1807.86	−0.114057	−0.0570284	+	0.998373i	$0.518163\pi$
$632$	−2748.46	−0.172987
$633$	0	0
$634$	5706.47	0.357465
$635$	18836.2	1.17715
$636$	0	0
$637$	0	0
$638$	−25047.0	−1.55426
$639$	0	0
$640$	2035.14	0.125697
$641$	5904.56	0.363832	0.181916	−	0.983314i	$-0.441770\pi$
$641$	5904.56	0.363832	0.181916	+	0.983314i	$0.441770\pi$
$642$	0	0
$643$	−8092.42	−0.496320	−0.248160	−	0.968719i	$-0.579826\pi$
$643$	−8092.42	−0.496320	−0.248160	+	0.968719i	$0.579826\pi$
$644$	0	0
$645$	0	0
$646$	1681.65	0.102420
$647$	20192.7	1.22698	0.613490	−	0.789702i	$-0.289765\pi$
$647$	20192.7	1.22698	0.613490	+	0.789702i	$0.289765\pi$
$648$	0	0
$649$	−23659.1	−1.43097
$650$	1456.46	0.0878880
$651$	0	0
$652$	−10058.9	−0.604200
$653$	21180.8	1.26933	0.634664	−	0.772789i	$-0.281139\pi$
$653$	21180.8	1.26933	0.634664	+	0.772789i	$0.281139\pi$
$654$	0	0
$655$	−4731.50	−0.282252
$656$	5256.04	0.312826
$657$	0	0
$658$	0	0
$659$	−28411.3	−1.67944	−0.839718	−	0.543023i	$-0.817280\pi$
$659$	−28411.3	−1.67944	−0.839718	+	0.543023i	$0.817280\pi$
$660$	0	0
$661$	−16704.9	−0.982975	−0.491488	−	0.870885i	$-0.663547\pi$
$661$	−16704.9	−0.982975	−0.491488	+	0.870885i	$0.663547\pi$
$662$	3238.23	0.190117
$663$	0	0
$664$	−12018.7	−0.702431
$665$	0	0
$666$	0	0
$667$	46561.3	2.70294
$668$	−2114.57	−0.122478
$669$	0	0
$670$	−5265.53	−0.303620
$671$	48023.7	2.76294
$672$	0	0
$673$	−9047.09	−0.518187	−0.259093	−	0.965852i	$-0.583424\pi$
$673$	−9047.09	−0.518187	−0.259093	+	0.965852i	$0.583424\pi$
$674$	−6557.35	−0.374747
$675$	0	0
$676$	−8658.11	−0.492610
$677$	−7844.26	−0.445316	−0.222658	−	0.974897i	$-0.571473\pi$
$677$	−7844.26	−0.445316	−0.222658	+	0.974897i	$0.571473\pi$
$678$	0	0
$679$	0	0
$680$	6601.41	0.372283
$681$	0	0
$682$	28858.8	1.62033
$683$	−25766.1	−1.44350	−0.721751	−	0.692153i	$-0.756662\pi$
$683$	−25766.1	−1.44350	−0.721751	+	0.692153i	$0.756662\pi$
$684$	0	0
$685$	−9873.35	−0.550717
$686$	0	0
$687$	0	0
$688$	−601.406	−0.0333261
$689$	−1205.73	−0.0666686
$690$	0	0
$691$	−24674.1	−1.35839	−0.679195	−	0.733958i	$-0.737671\pi$
$691$	−24674.1	−1.35839	−0.679195	+	0.733958i	$0.737671\pi$
$692$	−387.376	−0.0212801
$693$	0	0
$694$	5700.60	0.311804
$695$	−14289.7	−0.779914
$696$	0	0
$697$	17049.1	0.926515
$698$	−9450.63	−0.512481
$699$	0	0
$700$	0	0
$701$	−29377.9	−1.58286	−0.791431	−	0.611258i	$-0.790664\pi$
$701$	−29377.9	−1.58286	−0.791431	+	0.611258i	$0.790664\pi$
$702$	0	0
$703$	6266.45	0.336193
$704$	−3673.41	−0.196657
$705$	0	0
$706$	−13454.9	−0.717253
$707$	0	0
$708$	0	0
$709$	30594.3	1.62058	0.810291	−	0.586028i	$-0.199309\pi$
$709$	30594.3	1.62058	0.810291	+	0.586028i	$0.199309\pi$
$710$	14787.0	0.781615
$711$	0	0
$712$	2728.68	0.143626
$713$	−53647.4	−2.81782
$714$	0	0
$715$	−5200.34	−0.272002
$716$	2138.17	0.111602
$717$	0	0
$718$	14663.8	0.762183
$719$	1946.94	0.100985	0.0504927	−	0.998724i	$-0.483921\pi$
$719$	1946.94	0.100985	0.0504927	+	0.998724i	$0.483921\pi$
$720$	0	0
$721$	0	0
$722$	−13193.1	−0.680048
$723$	0	0
$724$	8348.02	0.428524
$725$	27883.5	1.42837
$726$	0	0
$727$	15750.6	0.803518	0.401759	−	0.915745i	$-0.368399\pi$
$727$	15750.6	0.803518	0.401759	+	0.915745i	$0.368399\pi$
$728$	0	0
$729$	0	0
$730$	14298.7	0.724956
$731$	−1950.79	−0.0987040
$732$	0	0
$733$	14349.6	0.723074	0.361537	−	0.932358i	$-0.382252\pi$
$733$	14349.6	0.723074	0.361537	+	0.932358i	$0.382252\pi$
$734$	5548.86	0.279036
$735$	0	0
$736$	6828.70	0.341996
$737$	9504.24	0.475025
$738$	0	0
$739$	−17758.1	−0.883953	−0.441976	−	0.897027i	$-0.645722\pi$
$739$	−17758.1	−0.883953	−0.441976	+	0.897027i	$0.645722\pi$
$740$	24599.3	1.22201
$741$	0	0
$742$	0	0
$743$	29187.6	1.44117	0.720586	−	0.693366i	$-0.243873\pi$
$743$	29187.6	1.44117	0.720586	+	0.693366i	$0.243873\pi$
$744$	0	0
$745$	48568.2	2.38846
$746$	5054.69	0.248077
$747$	0	0
$748$	−11915.5	−0.582451
$749$	0	0
$750$	0	0
$751$	13781.9	0.669654	0.334827	−	0.942280i	$-0.391322\pi$
$751$	13781.9	0.669654	0.334827	+	0.942280i	$0.391322\pi$
$752$	4079.92	0.197845
$753$	0	0
$754$	2486.72	0.120107
$755$	−1035.78	−0.0499283
$756$	0	0
$757$	36952.7	1.77420	0.887099	−	0.461579i	$-0.152717\pi$
$757$	36952.7	1.77420	0.887099	+	0.461579i	$0.152717\pi$
$758$	6232.80	0.298662
$759$	0	0
$760$	2060.70	0.0983547
$761$	−28816.5	−1.37266	−0.686332	−	0.727288i	$-0.740780\pi$
$761$	−28816.5	−1.37266	−0.686332	+	0.727288i	$0.740780\pi$
$762$	0	0
$763$	0	0
$764$	−13550.8	−0.641687
$765$	0	0
$766$	−3036.14	−0.143212
$767$	2348.92	0.110580
$768$	0	0
$769$	25285.2	1.18571	0.592854	−	0.805310i	$-0.298001\pi$
$769$	25285.2	1.18571	0.592854	+	0.805310i	$0.298001\pi$
$770$	0	0
$771$	0	0
$772$	−7633.41	−0.355871
$773$	18418.2	0.856995	0.428497	−	0.903543i	$-0.359043\pi$
$773$	18418.2	0.856995	0.428497	+	0.903543i	$0.359043\pi$
$774$	0	0
$775$	−32127.0	−1.48908
$776$	−6923.49	−0.320282
$777$	0	0
$778$	6493.55	0.299235
$779$	5322.07	0.244779
$780$	0	0
$781$	−26690.5	−1.22287
$782$	22150.4	1.01291
$783$	0	0
$784$	0	0
$785$	24520.5	1.11487
$786$	0	0
$787$	−11075.8	−0.501664	−0.250832	−	0.968031i	$-0.580704\pi$
$787$	−11075.8	−0.501664	−0.250832	+	0.968031i	$0.580704\pi$
$788$	8247.52	0.372850
$789$	0	0
$790$	−10924.8	−0.492008
$791$	0	0
$792$	0	0
$793$	−4767.88	−0.213509
$794$	−3661.25	−0.163643
$795$	0	0
$796$	12686.0	0.564878
$797$	4838.83	0.215057	0.107528	−	0.994202i	$-0.465706\pi$
$797$	4838.83	0.215057	0.107528	+	0.994202i	$0.465706\pi$
$798$	0	0
$799$	13234.1	0.585969
$800$	4089.41	0.180728
$801$	0	0
$802$	6771.62	0.298147
$803$	−25809.0	−1.13422
$804$	0	0
$805$	0	0
$806$	−2865.16	−0.125212
$807$	0	0
$808$	1946.05	0.0847299
$809$	31509.9	1.36938	0.684690	−	0.728834i	$-0.259938\pi$
$809$	31509.9	1.36938	0.684690	+	0.728834i	$0.259938\pi$
$810$	0	0
$811$	29463.3	1.27570	0.637851	−	0.770160i	$-0.279823\pi$
$811$	29463.3	1.27570	0.637851	+	0.770160i	$0.279823\pi$
$812$	0	0
$813$	0	0
$814$	−44401.6	−1.91189
$815$	−39983.0	−1.71846
$816$	0	0
$817$	−608.962	−0.0260770
$818$	−18506.3	−0.791026
$819$	0	0
$820$	20892.1	0.889736
$821$	−3502.22	−0.148877	−0.0744386	−	0.997226i	$-0.523716\pi$
$821$	−3502.22	−0.148877	−0.0744386	+	0.997226i	$0.523716\pi$
$822$	0	0
$823$	39993.0	1.69389	0.846943	−	0.531684i	$-0.178440\pi$
$823$	39993.0	1.69389	0.846943	+	0.531684i	$0.178440\pi$
$824$	−7626.77	−0.322441
$825$	0	0
$826$	0	0
$827$	10733.6	0.451322	0.225661	−	0.974206i	$-0.427546\pi$
$827$	10733.6	0.451322	0.225661	+	0.974206i	$0.427546\pi$
$828$	0	0
$829$	14537.5	0.609056	0.304528	−	0.952503i	$-0.401501\pi$
$829$	14537.5	0.609056	0.304528	+	0.952503i	$0.401501\pi$
$830$	−47772.6	−1.99785
$831$	0	0
$832$	364.703	0.0151969
$833$	0	0
$834$	0	0
$835$	−8405.16	−0.348351
$836$	−3719.56	−0.153880
$837$	0	0
$838$	7095.03	0.292475
$839$	7353.57	0.302591	0.151295	−	0.988489i	$-0.451656\pi$
$839$	7353.57	0.302591	0.151295	+	0.988489i	$0.451656\pi$
$840$	0	0
$841$	23218.3	0.951998
$842$	15697.7	0.642494
$843$	0	0
$844$	5399.88	0.220227
$845$	−34414.9	−1.40107
$846$	0	0
$847$	0	0
$848$	−3385.41	−0.137094
$849$	0	0
$850$	13264.9	0.535273
$851$	82540.7	3.32486
$852$	0	0
$853$	−19293.6	−0.774442	−0.387221	−	0.921987i	$-0.626565\pi$
$853$	−19293.6	−0.774442	−0.387221	+	0.921987i	$0.626565\pi$
$854$	0	0
$855$	0	0
$856$	10759.6	0.429622
$857$	−14161.6	−0.564468	−0.282234	−	0.959346i	$-0.591076\pi$
$857$	−14161.6	−0.564468	−0.282234	+	0.959346i	$0.591076\pi$
$858$	0	0
$859$	−8219.13	−0.326465	−0.163232	−	0.986588i	$-0.552192\pi$
$859$	−8219.13	−0.326465	−0.163232	+	0.986588i	$0.552192\pi$
$860$	−2390.51	−0.0947858
$861$	0	0
$862$	8894.65	0.351454
$863$	−2574.95	−0.101567	−0.0507835	−	0.998710i	$-0.516172\pi$
$863$	−2574.95	−0.101567	−0.0507835	+	0.998710i	$0.516172\pi$
$864$	0	0
$865$	−1539.77	−0.0605246
$866$	13989.6	0.548946
$867$	0	0
$868$	0	0
$869$	19719.2	0.769766
$870$	0	0
$871$	−943.600	−0.0367080
$872$	13874.8	0.538831
$873$	0	0
$874$	6914.49	0.267604
$875$	0	0
$876$	0	0
$877$	30981.1	1.19288	0.596442	−	0.802656i	$-0.296581\pi$
$877$	30981.1	1.19288	0.596442	+	0.802656i	$0.296581\pi$
$878$	−1272.36	−0.0489068
$879$	0	0
$880$	−14601.3	−0.559330
$881$	−41781.8	−1.59780	−0.798902	−	0.601461i	$-0.794585\pi$
$881$	−41781.8	−1.59780	−0.798902	+	0.601461i	$0.794585\pi$
$882$	0	0
$883$	−39289.6	−1.49740	−0.748699	−	0.662911i	$-0.769321\pi$
$883$	−39289.6	−1.49740	−0.748699	+	0.662911i	$0.769321\pi$
$884$	1182.99	0.0450095
$885$	0	0
$886$	8948.48	0.339312
$887$	5832.44	0.220783	0.110391	−	0.993888i	$-0.464790\pi$
$887$	5832.44	0.220783	0.110391	+	0.993888i	$0.464790\pi$
$888$	0	0
$889$	0	0
$890$	10846.2	0.408499
$891$	0	0
$892$	−5447.39	−0.204476
$893$	4131.18	0.154809
$894$	0	0
$895$	8498.95	0.317418
$896$	0	0
$897$	0	0
$898$	4778.84	0.177586
$899$	−54852.5	−2.03497
$900$	0	0
$901$	−10981.3	−0.406038
$902$	−37710.1	−1.39203
$903$	0	0
$904$	−11529.4	−0.424184
$905$	33182.3	1.21880
$906$	0	0
$907$	42387.9	1.55178	0.775892	−	0.630866i	$-0.217300\pi$
$907$	42387.9	1.55178	0.775892	+	0.630866i	$0.217300\pi$
$908$	−7447.24	−0.272186
$909$	0	0
$910$	0	0
$911$	−2275.12	−0.0827423	−0.0413711	−	0.999144i	$-0.513173\pi$
$911$	−2275.12	−0.0827423	−0.0413711	+	0.999144i	$0.513173\pi$
$912$	0	0
$913$	86229.3	3.12571
$914$	−8876.68	−0.321241
$915$	0	0
$916$	−21435.1	−0.773184
$917$	0	0
$918$	0	0
$919$	−31284.3	−1.12293	−0.561465	−	0.827500i	$-0.689762\pi$
$919$	−31284.3	−1.12293	−0.561465	+	0.827500i	$0.689762\pi$
$920$	27143.2	0.972702
$921$	0	0
$922$	−28159.7	−1.00585
$923$	2649.88	0.0944983
$924$	0	0
$925$	49429.9	1.75702
$926$	9374.99	0.332701
$927$	0	0
$928$	6982.11	0.246982
$929$	32196.6	1.13707	0.568535	−	0.822659i	$-0.307511\pi$
$929$	32196.6	1.13707	0.568535	+	0.822659i	$0.307511\pi$
$930$	0	0
$931$	0	0
$932$	−21764.5	−0.764935
$933$	0	0
$934$	16894.5	0.591869
$935$	−47362.6	−1.65660
$936$	0	0
$937$	−22293.6	−0.777269	−0.388635	−	0.921392i	$-0.627053\pi$
$937$	−22293.6	−0.777269	−0.388635	+	0.921392i	$0.627053\pi$
$938$	0	0
$939$	0	0
$940$	16217.2	0.562708
$941$	−31809.7	−1.10198	−0.550991	−	0.834511i	$-0.685750\pi$
$941$	−31809.7	−1.10198	−0.550991	+	0.834511i	$0.685750\pi$
$942$	0	0
$943$	70101.4	2.42080
$944$	6595.22	0.227390
$945$	0	0
$946$	4314.86	0.148296
$947$	−18982.6	−0.651376	−0.325688	−	0.945477i	$-0.605596\pi$
$947$	−18982.6	−0.651376	−0.325688	+	0.945477i	$0.605596\pi$
$948$	0	0
$949$	2562.37	0.0876481
$950$	4140.78	0.141415
$951$	0	0
$952$	0	0
$953$	−9254.58	−0.314570	−0.157285	−	0.987553i	$-0.550274\pi$
$953$	−9254.58	−0.314570	−0.157285	+	0.987553i	$0.550274\pi$
$954$	0	0
$955$	−53862.5	−1.82508
$956$	4629.10	0.156607
$957$	0	0
$958$	−8738.81	−0.294716
$959$	0	0
$960$	0	0
$961$	33409.4	1.12146
$962$	4408.28	0.147743
$963$	0	0
$964$	−15877.5	−0.530477
$965$	−30341.8	−1.01216
$966$	0	0
$967$	15317.5	0.509387	0.254694	−	0.967022i	$-0.418025\pi$
$967$	15317.5	0.509387	0.254694	+	0.967022i	$0.418025\pi$
$968$	15707.3	0.521541
$969$	0	0
$970$	−27520.0	−0.910943
$971$	22432.6	0.741398	0.370699	−	0.928753i	$-0.379118\pi$
$971$	22432.6	0.741398	0.370699	+	0.928753i	$0.379118\pi$
$972$	0	0
$973$	0	0
$974$	−28955.5	−0.952559
$975$	0	0
$976$	−13387.1	−0.439048
$977$	627.864	0.0205600	0.0102800	−	0.999947i	$-0.496728\pi$
$977$	627.864	0.0205600	0.0102800	+	0.999947i	$0.496728\pi$
$978$	0	0
$979$	−19577.3	−0.639114
$980$	0	0
$981$	0	0
$982$	−18612.2	−0.604827
$983$	45032.6	1.46116	0.730579	−	0.682828i	$-0.239250\pi$
$983$	45032.6	1.46116	0.730579	+	0.682828i	$0.239250\pi$
$984$	0	0
$985$	32782.8	1.06045
$986$	22648.0	0.731500
$987$	0	0
$988$	369.285	0.0118912
$989$	−8021.14	−0.257894
$990$	0	0
$991$	−31981.0	−1.02514	−0.512569	−	0.858646i	$-0.671306\pi$
$991$	−31981.0	−1.02514	−0.512569	+	0.858646i	$0.671306\pi$
$992$	−8044.70	−0.257479
$993$	0	0
$994$	0	0
$995$	50425.2	1.60662
$996$	0	0
$997$	−12378.8	−0.393221	−0.196611	−	0.980482i	$-0.562993\pi$
$997$	−12378.8	−0.393221	−0.196611	+	0.980482i	$0.562993\pi$
$998$	−24475.1	−0.776298
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.bi.1.2		2
3.2	odd	2		294.4.a.k.1.1	yes	2
7.2	even	3		882.4.g.y.361.1		4
7.3	odd	6		882.4.g.bd.667.2		4
7.4	even	3		882.4.g.y.667.1		4
7.5	odd	6		882.4.g.bd.361.2		4
7.6	odd	2		882.4.a.bc.1.1		2
12.11	even	2		2352.4.a.bn.1.1		2
21.2	odd	6		294.4.e.n.67.2		4
21.5	even	6		294.4.e.o.67.1		4
21.11	odd	6		294.4.e.n.79.2		4
21.17	even	6		294.4.e.o.79.1		4
21.20	even	2		294.4.a.j.1.2	✓	2
84.83	odd	2		2352.4.a.cd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.4.a.j.1.2	✓	2	21.20	even	2
294.4.a.k.1.1	yes	2	3.2	odd	2
294.4.e.n.67.2		4	21.2	odd	6
294.4.e.n.79.2		4	21.11	odd	6
294.4.e.o.67.1		4	21.5	even	6
294.4.e.o.79.1		4	21.17	even	6
882.4.a.bc.1.1		2	7.6	odd	2
882.4.a.bi.1.2		2	1.1	even	1	trivial
882.4.g.y.361.1		4	7.2	even	3
882.4.g.y.667.1		4	7.4	even	3
882.4.g.bd.361.2		4	7.5	odd	6
882.4.g.bd.667.2		4	7.3	odd	6
2352.4.a.bn.1.1		2	12.11	even	2
2352.4.a.cd.1.2		2	84.83	odd	2

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +15.8995 q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +15.8995 q^{5} +8.00000 q^{8} +31.7990 q^{10} -57.3970 q^{11} +5.69848 q^{13} +16.0000 q^{16} +51.8995 q^{17} +16.2010 q^{19} +63.5980 q^{20} -114.794 q^{22} +213.397 q^{23} +127.794 q^{25} +11.3970 q^{26} +218.191 q^{29} -251.397 q^{31} +32.0000 q^{32} +103.799 q^{34} +386.794 q^{37} +32.4020 q^{38} +127.196 q^{40} +328.503 q^{41} -37.5879 q^{43} -229.588 q^{44} +426.794 q^{46} +254.995 q^{47} +255.588 q^{50} +22.7939 q^{52} -211.588 q^{53} -912.583 q^{55} +436.382 q^{58} +412.201 q^{59} -836.693 q^{61} -502.794 q^{62} +64.0000 q^{64} +90.6030 q^{65} -165.588 q^{67} +207.598 q^{68} +465.015 q^{71} +449.658 q^{73} +773.588 q^{74} +64.8040 q^{76} -343.558 q^{79} +254.392 q^{80} +657.005 q^{82} -1502.33 q^{83} +825.176 q^{85} -75.1758 q^{86} -459.176 q^{88} +341.085 q^{89} +853.588 q^{92} +509.990 q^{94} +257.588 q^{95} -865.437 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{8} + 24 q^{10} + 4 q^{11} - 48 q^{13} + 32 q^{16} + 84 q^{17} + 72 q^{19} + 48 q^{20} + 8 q^{22} + 308 q^{23} + 18 q^{25} - 96 q^{26} + 80 q^{29} - 384 q^{31} + 64 q^{32} + 168 q^{34} + 536 q^{37} + 144 q^{38} + 96 q^{40} + 756 q^{41} + 400 q^{43} + 16 q^{44} + 616 q^{46} + 312 q^{47} + 36 q^{50} - 192 q^{52} + 52 q^{53} - 1152 q^{55} + 160 q^{58} + 864 q^{59} - 1416 q^{61} - 768 q^{62} + 128 q^{64} + 300 q^{65} + 144 q^{67} + 336 q^{68} + 1524 q^{71} - 744 q^{73} + 1072 q^{74} + 288 q^{76} + 976 q^{79} + 192 q^{80} + 1512 q^{82} - 312 q^{83} + 700 q^{85} + 800 q^{86} + 32 q^{88} + 108 q^{89} + 1232 q^{92} + 624 q^{94} + 40 q^{95} + 744 q^{97}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^8 + 24 * q^10 + 4 * q^11 - 48 * q^13 + 32 * q^16 + 84 * q^17 + 72 * q^19 + 48 * q^20 + 8 * q^22 + 308 * q^23 + 18 * q^25 - 96 * q^26 + 80 * q^29 - 384 * q^31 + 64 * q^32 + 168 * q^34 + 536 * q^37 + 144 * q^38 + 96 * q^40 + 756 * q^41 + 400 * q^43 + 16 * q^44 + 616 * q^46 + 312 * q^47 + 36 * q^50 - 192 * q^52 + 52 * q^53 - 1152 * q^55 + 160 * q^58 + 864 * q^59 - 1416 * q^61 - 768 * q^62 + 128 * q^64 + 300 * q^65 + 144 * q^67 + 336 * q^68 + 1524 * q^71 - 744 * q^73 + 1072 * q^74 + 288 * q^76 + 976 * q^79 + 192 * q^80 + 1512 * q^82 - 312 * q^83 + 700 * q^85 + 800 * q^86 + 32 * q^88 + 108 * q^89 + 1232 * q^92 + 624 * q^94 + 40 * q^95 + 744 * q^97

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