LMFDB - Embedded newform 1323.4.a.bk.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504 $ x^7 - x^6 - 43x^5 + 10x^4 + 513x^3 + 258x^2 - 936*x - 504
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.541672$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.541672 q^{2} -7.70659 q^{4} +16.7289 q^{5} +8.50782 q^{8} -9.06158 q^{10} +26.1733 q^{11} +34.4888 q^{13} +57.0443 q^{16} +81.5268 q^{17} +96.8698 q^{19} -128.923 q^{20} -14.1773 q^{22} -183.899 q^{23} +154.856 q^{25} -18.6816 q^{26} +42.2582 q^{29} +279.954 q^{31} -98.9619 q^{32} -44.1608 q^{34} -48.6600 q^{37} -52.4717 q^{38} +142.327 q^{40} +4.73071 q^{41} -419.874 q^{43} -201.707 q^{44} +99.6129 q^{46} -39.9962 q^{47} -83.8812 q^{50} -265.791 q^{52} +287.663 q^{53} +437.850 q^{55} -22.8901 q^{58} +465.137 q^{59} +242.752 q^{61} -151.643 q^{62} -402.749 q^{64} +576.960 q^{65} -634.773 q^{67} -628.293 q^{68} -1027.26 q^{71} +403.402 q^{73} +26.3578 q^{74} -746.536 q^{76} +308.406 q^{79} +954.288 q^{80} -2.56249 q^{82} +1419.65 q^{83} +1363.85 q^{85} +227.434 q^{86} +222.678 q^{88} -1202.20 q^{89} +1417.23 q^{92} +21.6648 q^{94} +1620.52 q^{95} +798.496 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 31 q^{4} + q^{5} + 84 q^{8} - 12 q^{10} + 98 q^{11} + 124 q^{13} + 139 q^{16} - 30 q^{17} - 182 q^{19} + 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} + 245 q^{26} + 323 q^{29} - 26 q^{31}+ \cdots + 3730 q^{97}+O(q^{100}) $ 7 * q + q^2 + 31 * q^4 + q^5 + 84 * q^8 - 12 * q^10 + 98 * q^11 + 124 * q^13 + 139 * q^16 - 30 * q^17 - 182 * q^19 + 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 + 245 * q^26 + 323 * q^29 - 26 * q^31 + 398 * q^32 - 114 * q^34 - 112 * q^37 - 1015 * q^38 + 147 * q^40 + 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 - 288 * q^47 + 2576 * q^50 + 1075 * q^52 + 1353 * q^53 - 156 * q^55 - 81 * q^58 - 165 * q^59 - 56 * q^61 + 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 - 2625 * q^68 + 792 * q^71 - 1487 * q^73 + 2736 * q^74 - 1952 * q^76 - 1273 * q^79 + 2501 * q^80 + 2049 * q^82 + 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 - 1058 * q^89 + 3834 * q^92 - 1653 * q^94 + 3260 * q^95 + 3730 * q^97

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 $

$2$	−0.541672	−0.191510	−0.0957550	−	0.995405i	$-0.530527\pi$
$2$	−0.541672	−0.191510	−0.0957550	+	0.995405i	$0.530527\pi$
$3$	0	0
$3$	0	0
$4$	−7.70659	−0.963324
$5$	16.7289	1.49628	0.748139	−	0.663542i	$-0.230948\pi$
$5$	16.7289	1.49628	0.748139	+	0.663542i	$0.230948\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.50782	0.375996
$9$	0	0
$10$	−9.06158	−0.286552
$11$	26.1733	0.717413	0.358706	−	0.933450i	$-0.383218\pi$
$11$	26.1733	0.717413	0.358706	+	0.933450i	$0.383218\pi$
$12$	0	0
$13$	34.4888	0.735806	0.367903	−	0.929864i	$-0.380076\pi$
$13$	34.4888	0.735806	0.367903	+	0.929864i	$0.380076\pi$
$14$	0	0
$15$	0	0
$16$	57.0443	0.891317
$17$	81.5268	1.16313	0.581563	−	0.813501i	$-0.302441\pi$
$17$	81.5268	1.16313	0.581563	+	0.813501i	$0.302441\pi$
$18$	0	0
$19$	96.8698	1.16966	0.584828	−	0.811157i	$-0.301162\pi$
$19$	96.8698	1.16966	0.584828	+	0.811157i	$0.301162\pi$
$20$	−128.923	−1.44140
$21$	0	0
$22$	−14.1773	−0.137392
$23$	−183.899	−1.66720	−0.833599	−	0.552370i	$-0.813723\pi$
$23$	−183.899	−1.66720	−0.833599	+	0.552370i	$0.813723\pi$
$24$	0	0
$25$	154.856	1.23885
$26$	−18.6816	−0.140914
$27$	0	0
$28$	0	0
$29$	42.2582	0.270591	0.135296	−	0.990805i	$-0.456802\pi$
$29$	42.2582	0.270591	0.135296	+	0.990805i	$0.456802\pi$
$30$	0	0
$31$	279.954	1.62197	0.810987	−	0.585064i	$-0.198931\pi$
$31$	279.954	1.62197	0.810987	+	0.585064i	$0.198931\pi$
$32$	−98.9619	−0.546692
$33$	0	0
$34$	−44.1608	−0.222750
$35$	0	0
$36$	0	0
$37$	−48.6600	−0.216207	−0.108103	−	0.994140i	$-0.534478\pi$
$37$	−48.6600	−0.216207	−0.108103	+	0.994140i	$0.534478\pi$
$38$	−52.4717	−0.224001
$39$	0	0
$40$	142.327	0.562595
$41$	4.73071	0.0180198	0.00900991	−	0.999959i	$-0.497132\pi$
$41$	4.73071	0.0180198	0.00900991	+	0.999959i	$0.497132\pi$
$42$	0	0
$43$	−419.874	−1.48907	−0.744537	−	0.667581i	$-0.767330\pi$
$43$	−419.874	−1.48907	−0.744537	+	0.667581i	$0.767330\pi$
$44$	−201.707	−0.691101
$45$	0	0
$46$	99.6129	0.319285
$47$	−39.9962	−0.124129	−0.0620643	−	0.998072i	$-0.519768\pi$
$47$	−39.9962	−0.124129	−0.0620643	+	0.998072i	$0.519768\pi$
$48$	0	0
$49$	0	0
$50$	−83.8812	−0.237252
$51$	0	0
$52$	−265.791	−0.708820
$53$	287.663	0.745538	0.372769	−	0.927924i	$-0.378408\pi$
$53$	287.663	0.745538	0.372769	+	0.927924i	$0.378408\pi$
$54$	0	0
$55$	437.850	1.07345
$56$	0	0
$57$	0	0
$58$	−22.8901	−0.0518210
$59$	465.137	1.02637	0.513184	−	0.858278i	$-0.328466\pi$
$59$	465.137	1.02637	0.513184	+	0.858278i	$0.328466\pi$
$60$	0	0
$61$	242.752	0.509527	0.254764	−	0.967003i	$-0.418002\pi$
$61$	242.752	0.509527	0.254764	+	0.967003i	$0.418002\pi$
$62$	−151.643	−0.310624
$63$	0	0
$64$	−402.749	−0.786620
$65$	576.960	1.10097
$66$	0	0
$67$	−634.773	−1.15746	−0.578730	−	0.815519i	$-0.696452\pi$
$67$	−634.773	−1.15746	−0.578730	+	0.815519i	$0.696452\pi$
$68$	−628.293	−1.12047
$69$	0	0
$70$	0	0
$71$	−1027.26	−1.71708	−0.858541	−	0.512745i	$-0.828628\pi$
$71$	−1027.26	−1.71708	−0.858541	+	0.512745i	$0.828628\pi$
$72$	0	0
$73$	403.402	0.646775	0.323388	−	0.946267i	$-0.395178\pi$
$73$	403.402	0.646775	0.323388	+	0.946267i	$0.395178\pi$
$74$	26.3578	0.0414058
$75$	0	0
$76$	−746.536	−1.12676
$77$	0	0
$78$	0	0
$79$	308.406	0.439221	0.219610	−	0.975588i	$-0.429521\pi$
$79$	308.406	0.439221	0.219610	+	0.975588i	$0.429521\pi$
$80$	954.288	1.33366
$81$	0	0
$82$	−2.56249	−0.00345098
$83$	1419.65	1.87744	0.938718	−	0.344686i	$-0.112014\pi$
$83$	1419.65	1.87744	0.938718	+	0.344686i	$0.112014\pi$
$84$	0	0
$85$	1363.85	1.74036
$86$	227.434	0.285173
$87$	0	0
$88$	222.678	0.269745
$89$	−1202.20	−1.43183	−0.715914	−	0.698188i	$-0.753990\pi$
$89$	−1202.20	−1.43183	−0.715914	+	0.698188i	$0.753990\pi$
$90$	0	0
$91$	0	0
$92$	1417.23	1.60605
$93$	0	0
$94$	21.6648	0.0237719
$95$	1620.52	1.75013
$96$	0	0
$97$	798.496	0.835824	0.417912	−	0.908487i	$-0.362762\pi$
$97$	798.496	0.835824	0.417912	+	0.908487i	$0.362762\pi$
$98$	0	0
$99$	0	0
$100$	−1193.41	−1.19341
$101$	−1298.42	−1.27918	−0.639590	−	0.768716i	$-0.720896\pi$
$101$	−1298.42	−1.27918	−0.639590	+	0.768716i	$0.720896\pi$
$102$	0	0
$103$	−1118.82	−1.07029	−0.535146	−	0.844759i	$-0.679744\pi$
$103$	−1118.82	−1.07029	−0.535146	+	0.844759i	$0.679744\pi$
$104$	293.425	0.276660
$105$	0	0
$106$	−155.819	−0.142778
$107$	−146.564	−0.132420	−0.0662099	−	0.997806i	$-0.521091\pi$
$107$	−146.564	−0.132420	−0.0662099	+	0.997806i	$0.521091\pi$
$108$	0	0
$109$	−195.984	−0.172219	−0.0861096	−	0.996286i	$-0.527444\pi$
$109$	−195.984	−0.172219	−0.0861096	+	0.996286i	$0.527444\pi$
$110$	−237.171	−0.205576
$111$	0	0
$112$	0	0
$113$	846.025	0.704313	0.352156	−	0.935941i	$-0.385448\pi$
$113$	846.025	0.704313	0.352156	+	0.935941i	$0.385448\pi$
$114$	0	0
$115$	−3076.42	−2.49459
$116$	−325.667	−0.260667
$117$	0	0
$118$	−251.952	−0.196560
$119$	0	0
$120$	0	0
$121$	−645.959	−0.485319
$122$	−131.492	−0.0975796
$123$	0	0
$124$	−2157.49	−1.56249
$125$	499.456	0.357382
$126$	0	0
$127$	1259.98	0.880357	0.440178	−	0.897910i	$-0.354915\pi$
$127$	1259.98	0.880357	0.440178	+	0.897910i	$0.354915\pi$
$128$	1009.85	0.697338
$129$	0	0
$130$	−312.523	−0.210847
$131$	−1221.01	−0.814352	−0.407176	−	0.913350i	$-0.633486\pi$
$131$	−1221.01	−0.814352	−0.407176	+	0.913350i	$0.633486\pi$
$132$	0	0
$133$	0	0
$134$	343.839	0.221665
$135$	0	0
$136$	693.615	0.437331
$137$	3164.23	1.97327	0.986637	−	0.162935i	$-0.0520960\pi$
$137$	3164.23	1.97327	0.986637	+	0.162935i	$0.0520960\pi$
$138$	0	0
$139$	69.3404	0.0423120	0.0211560	−	0.999776i	$-0.493265\pi$
$139$	69.3404	0.0423120	0.0211560	+	0.999776i	$0.493265\pi$
$140$	0	0
$141$	0	0
$142$	556.436	0.328838
$143$	902.686	0.527877
$144$	0	0
$145$	706.933	0.404880
$146$	−218.511	−0.123864
$147$	0	0
$148$	375.003	0.208277
$149$	806.621	0.443496	0.221748	−	0.975104i	$-0.428824\pi$
$149$	806.621	0.443496	0.221748	+	0.975104i	$0.428824\pi$
$150$	0	0
$151$	−2174.31	−1.17181	−0.585904	−	0.810380i	$-0.699261\pi$
$151$	−2174.31	−1.17181	−0.585904	+	0.810380i	$0.699261\pi$
$152$	824.151	0.439786
$153$	0	0
$154$	0	0
$155$	4683.32	2.42692
$156$	0	0
$157$	680.859	0.346105	0.173053	−	0.984913i	$-0.444637\pi$
$157$	680.859	0.346105	0.173053	+	0.984913i	$0.444637\pi$
$158$	−167.055	−0.0841152
$159$	0	0
$160$	−1655.52	−0.818004
$161$	0	0
$162$	0	0
$163$	1181.61	0.567799	0.283899	−	0.958854i	$-0.408372\pi$
$163$	1181.61	0.567799	0.283899	+	0.958854i	$0.408372\pi$
$164$	−36.4577	−0.0173589
$165$	0	0
$166$	−768.987	−0.359548
$167$	−228.073	−0.105681	−0.0528407	−	0.998603i	$-0.516828\pi$
$167$	−228.073	−0.105681	−0.0528407	+	0.998603i	$0.516828\pi$
$168$	0	0
$169$	−1007.52	−0.458590
$170$	−738.761	−0.333296
$171$	0	0
$172$	3235.80	1.43446
$173$	1244.81	0.547058	0.273529	−	0.961864i	$-0.411809\pi$
$173$	1244.81	0.547058	0.273529	+	0.961864i	$0.411809\pi$
$174$	0	0
$175$	0	0
$176$	1493.04	0.639442
$177$	0	0
$178$	651.197	0.274209
$179$	4096.33	1.71047	0.855234	−	0.518242i	$-0.173413\pi$
$179$	4096.33	1.71047	0.855234	+	0.518242i	$0.173413\pi$
$180$	0	0
$181$	3259.45	1.33852	0.669262	−	0.743026i	$-0.266610\pi$
$181$	3259.45	1.33852	0.669262	+	0.743026i	$0.266610\pi$
$182$	0	0
$183$	0	0
$184$	−1564.58	−0.626860
$185$	−814.028	−0.323506
$186$	0	0
$187$	2133.82	0.834442
$188$	308.234	0.119576
$189$	0	0
$190$	−877.793	−0.335167
$191$	1268.88	0.480694	0.240347	−	0.970687i	$-0.422739\pi$
$191$	1268.88	0.480694	0.240347	+	0.970687i	$0.422739\pi$
$192$	0	0
$193$	−178.171	−0.0664508	−0.0332254	−	0.999448i	$-0.510578\pi$
$193$	−178.171	−0.0664508	−0.0332254	+	0.999448i	$0.510578\pi$
$194$	−432.523	−0.160069
$195$	0	0
$196$	0	0
$197$	4790.39	1.73249	0.866246	−	0.499618i	$-0.166526\pi$
$197$	4790.39	1.73249	0.866246	+	0.499618i	$0.166526\pi$
$198$	0	0
$199$	4204.86	1.49786	0.748932	−	0.662647i	$-0.230567\pi$
$199$	4204.86	1.49786	0.748932	+	0.662647i	$0.230567\pi$
$200$	1317.49	0.465802
$201$	0	0
$202$	703.316	0.244976
$203$	0	0
$204$	0	0
$205$	79.1396	0.0269627
$206$	606.031	0.204972
$207$	0	0
$208$	1967.39	0.655836
$209$	2535.40	0.839126
$210$	0	0
$211$	−1998.97	−0.652204	−0.326102	−	0.945335i	$-0.605735\pi$
$211$	−1998.97	−0.652204	−0.326102	+	0.945335i	$0.605735\pi$
$212$	−2216.90	−0.718194
$213$	0	0
$214$	79.3899	0.0253597
$215$	−7024.03	−2.22807
$216$	0	0
$217$	0	0
$218$	106.159	0.0329817
$219$	0	0
$220$	−3374.33	−1.03408
$221$	2811.76	0.855835
$222$	0	0
$223$	−3931.51	−1.18060	−0.590299	−	0.807185i	$-0.700990\pi$
$223$	−3931.51	−1.18060	−0.590299	+	0.807185i	$0.700990\pi$
$224$	0	0
$225$	0	0
$226$	−458.268	−0.134883
$227$	3768.42	1.10184	0.550922	−	0.834557i	$-0.314276\pi$
$227$	3768.42	1.10184	0.550922	+	0.834557i	$0.314276\pi$
$228$	0	0
$229$	2189.72	0.631882	0.315941	−	0.948779i	$-0.397680\pi$
$229$	2189.72	0.631882	0.315941	+	0.948779i	$0.397680\pi$
$230$	1666.41	0.477739
$231$	0	0
$232$	359.525	0.101741
$233$	1952.97	0.549114	0.274557	−	0.961571i	$-0.411469\pi$
$233$	1952.97	0.549114	0.274557	+	0.961571i	$0.411469\pi$
$234$	0	0
$235$	−669.092	−0.185731
$236$	−3584.62	−0.988725
$237$	0	0
$238$	0	0
$239$	−2497.90	−0.676048	−0.338024	−	0.941137i	$-0.609759\pi$
$239$	−2497.90	−0.676048	−0.338024	+	0.941137i	$0.609759\pi$
$240$	0	0
$241$	2258.24	0.603593	0.301796	−	0.953372i	$-0.402414\pi$
$241$	2258.24	0.603593	0.301796	+	0.953372i	$0.402414\pi$
$242$	349.898	0.0929435
$243$	0	0
$244$	−1870.79	−0.490840
$245$	0	0
$246$	0	0
$247$	3340.93	0.860640
$248$	2381.80	0.609856
$249$	0	0
$250$	−270.542	−0.0684422
$251$	2538.43	0.638344	0.319172	−	0.947697i	$-0.396595\pi$
$251$	2538.43	0.638344	0.319172	+	0.947697i	$0.396595\pi$
$252$	0	0
$253$	−4813.23	−1.19607
$254$	−682.497	−0.168597
$255$	0	0
$256$	2674.98	0.653072
$257$	−1790.83	−0.434665	−0.217333	−	0.976098i	$-0.569736\pi$
$257$	−1790.83	−0.434665	−0.217333	+	0.976098i	$0.569736\pi$
$258$	0	0
$259$	0	0
$260$	−4446.39	−1.06059
$261$	0	0
$262$	661.387	0.155957
$263$	−2597.17	−0.608929	−0.304465	−	0.952524i	$-0.598478\pi$
$263$	−2597.17	−0.608929	−0.304465	+	0.952524i	$0.598478\pi$
$264$	0	0
$265$	4812.28	1.11553
$266$	0	0
$267$	0	0
$268$	4891.93	1.11501
$269$	1555.76	0.352625	0.176312	−	0.984334i	$-0.443583\pi$
$269$	1555.76	0.352625	0.176312	+	0.984334i	$0.443583\pi$
$270$	0	0
$271$	7787.65	1.74563	0.872816	−	0.488049i	$-0.162291\pi$
$271$	7787.65	1.74563	0.872816	+	0.488049i	$0.162291\pi$
$272$	4650.64	1.03671
$273$	0	0
$274$	−1713.98	−0.377902
$275$	4053.09	0.888765
$276$	0	0
$277$	2498.39	0.541927	0.270964	−	0.962590i	$-0.412658\pi$
$277$	2498.39	0.541927	0.270964	+	0.962590i	$0.412658\pi$
$278$	−37.5598	−0.00810318
$279$	0	0
$280$	0	0
$281$	−8097.10	−1.71898	−0.859488	−	0.511156i	$-0.829217\pi$
$281$	−8097.10	−1.71898	−0.859488	+	0.511156i	$0.829217\pi$
$282$	0	0
$283$	−5874.53	−1.23394	−0.616969	−	0.786987i	$-0.711640\pi$
$283$	−5874.53	−1.23394	−0.616969	+	0.786987i	$0.711640\pi$
$284$	7916.64	1.65411
$285$	0	0
$286$	−488.960	−0.101094
$287$	0	0
$288$	0	0
$289$	1733.61	0.352862
$290$	−382.926	−0.0775385
$291$	0	0
$292$	−3108.85	−0.623054
$293$	−3987.32	−0.795023	−0.397512	−	0.917597i	$-0.630126\pi$
$293$	−3987.32	−0.795023	−0.397512	+	0.917597i	$0.630126\pi$
$294$	0	0
$295$	7781.24	1.53573
$296$	−413.991	−0.0812930
$297$	0	0
$298$	−436.924	−0.0849340
$299$	−6342.45	−1.22673
$300$	0	0
$301$	0	0
$302$	1177.77	0.224413
$303$	0	0
$304$	5525.87	1.04253
$305$	4060.97	0.762394
$306$	0	0
$307$	−716.337	−0.133171	−0.0665855	−	0.997781i	$-0.521211\pi$
$307$	−716.337	−0.133171	−0.0665855	+	0.997781i	$0.521211\pi$
$308$	0	0
$309$	0	0
$310$	−2536.82	−0.464780
$311$	5505.88	1.00389	0.501945	−	0.864900i	$-0.332618\pi$
$311$	5505.88	1.00389	0.501945	+	0.864900i	$0.332618\pi$
$312$	0	0
$313$	−1845.46	−0.333263	−0.166632	−	0.986019i	$-0.553289\pi$
$313$	−1845.46	−0.333263	−0.166632	+	0.986019i	$0.553289\pi$
$314$	−368.803	−0.0662826
$315$	0	0
$316$	−2376.76	−0.423112
$317$	−1294.20	−0.229305	−0.114652	−	0.993406i	$-0.536575\pi$
$317$	−1294.20	−0.229305	−0.114652	+	0.993406i	$0.536575\pi$
$318$	0	0
$319$	1106.04	0.194126
$320$	−6737.55	−1.17700
$321$	0	0
$322$	0	0
$323$	7897.48	1.36046
$324$	0	0
$325$	5340.80	0.911551
$326$	−640.048	−0.108739
$327$	0	0
$328$	40.2481	0.00677539
$329$	0	0
$330$	0	0
$331$	−5232.87	−0.868956	−0.434478	−	0.900682i	$-0.643067\pi$
$331$	−5232.87	−0.868956	−0.434478	+	0.900682i	$0.643067\pi$
$332$	−10940.7	−1.80858
$333$	0	0
$334$	123.541	0.0202391
$335$	−10619.0	−1.73188
$336$	0	0
$337$	9347.94	1.51102	0.755511	−	0.655136i	$-0.227389\pi$
$337$	9347.94	1.51102	0.755511	+	0.655136i	$0.227389\pi$
$338$	545.746	0.0878245
$339$	0	0
$340$	−10510.7	−1.67653
$341$	7327.31	1.16362
$342$	0	0
$343$	0	0
$344$	−3572.21	−0.559886
$345$	0	0
$346$	−674.278	−0.104767
$347$	−4629.81	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$347$	−4629.81	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$348$	0	0
$349$	88.6438	0.0135960	0.00679798	−	0.999977i	$-0.497836\pi$
$349$	88.6438	0.0135960	0.00679798	+	0.999977i	$0.497836\pi$
$350$	0	0
$351$	0	0
$352$	−2590.16	−0.392204
$353$	10143.0	1.52935	0.764673	−	0.644418i	$-0.222900\pi$
$353$	10143.0	1.52935	0.764673	+	0.644418i	$0.222900\pi$
$354$	0	0
$355$	−17184.8	−2.56923
$356$	9264.85	1.37931
$357$	0	0
$358$	−2218.87	−0.327572
$359$	4607.94	0.677431	0.338716	−	0.940889i	$-0.390008\pi$
$359$	4607.94	0.677431	0.338716	+	0.940889i	$0.390008\pi$
$360$	0	0
$361$	2524.76	0.368094
$362$	−1765.55	−0.256341
$363$	0	0
$364$	0	0
$365$	6748.46	0.967755
$366$	0	0
$367$	−7730.99	−1.09960	−0.549801	−	0.835295i	$-0.685297\pi$
$367$	−7730.99	−1.09960	−0.549801	+	0.835295i	$0.685297\pi$
$368$	−10490.4	−1.48600
$369$	0	0
$370$	440.936	0.0619546
$371$	0	0
$372$	0	0
$373$	−6793.44	−0.943032	−0.471516	−	0.881858i	$-0.656293\pi$
$373$	−6793.44	−0.943032	−0.471516	+	0.881858i	$0.656293\pi$
$374$	−1155.83	−0.159804
$375$	0	0
$376$	−340.281	−0.0466719
$377$	1457.43	0.199103
$378$	0	0
$379$	−6426.60	−0.871009	−0.435504	−	0.900187i	$-0.643430\pi$
$379$	−6426.60	−0.871009	−0.435504	+	0.900187i	$0.643430\pi$
$380$	−12488.7	−1.68594
$381$	0	0
$382$	−687.315	−0.0920578
$383$	7852.39	1.04762	0.523810	−	0.851835i	$-0.324510\pi$
$383$	7852.39	1.04762	0.523810	+	0.851835i	$0.324510\pi$
$384$	0	0
$385$	0	0
$386$	96.5101	0.0127260
$387$	0	0
$388$	−6153.68	−0.805169
$389$	4679.81	0.609963	0.304982	−	0.952358i	$-0.401350\pi$
$389$	4679.81	0.609963	0.304982	+	0.952358i	$0.401350\pi$
$390$	0	0
$391$	−14992.7	−1.93916
$392$	0	0
$393$	0	0
$394$	−2594.82	−0.331790
$395$	5159.30	0.657196
$396$	0	0
$397$	−2491.34	−0.314954	−0.157477	−	0.987523i	$-0.550336\pi$
$397$	−2491.34	−0.314954	−0.157477	+	0.987523i	$0.550336\pi$
$398$	−2277.66	−0.286856
$399$	0	0
$400$	8833.64	1.10421
$401$	−13206.5	−1.64465	−0.822324	−	0.569020i	$-0.807323\pi$
$401$	−13206.5	−1.64465	−0.822324	+	0.569020i	$0.807323\pi$
$402$	0	0
$403$	9655.28	1.19346
$404$	10006.4	1.23227
$405$	0	0
$406$	0	0
$407$	−1273.59	−0.155110
$408$	0	0
$409$	−6028.49	−0.728825	−0.364413	−	0.931238i	$-0.618730\pi$
$409$	−6028.49	−0.728825	−0.364413	+	0.931238i	$0.618730\pi$
$410$	−42.8677	−0.00516362
$411$	0	0
$412$	8622.25	1.03104
$413$	0	0
$414$	0	0
$415$	23749.2	2.80917
$416$	−3413.08	−0.402260
$417$	0	0
$418$	−1373.36	−0.160701
$419$	−10628.5	−1.23923	−0.619613	−	0.784908i	$-0.712710\pi$
$419$	−10628.5	−1.23923	−0.619613	+	0.784908i	$0.712710\pi$
$420$	0	0
$421$	4608.37	0.533487	0.266744	−	0.963768i	$-0.414052\pi$
$421$	4608.37	0.533487	0.266744	+	0.963768i	$0.414052\pi$
$422$	1082.79	0.124904
$423$	0	0
$424$	2447.38	0.280319
$425$	12624.9	1.44094
$426$	0	0
$427$	0	0
$428$	1129.51	0.127563
$429$	0	0
$430$	3804.72	0.426697
$431$	6011.39	0.671829	0.335915	−	0.941892i	$-0.390955\pi$
$431$	6011.39	0.671829	0.335915	+	0.941892i	$0.390955\pi$
$432$	0	0
$433$	−6450.48	−0.715912	−0.357956	−	0.933738i	$-0.616526\pi$
$433$	−6450.48	−0.715912	−0.357956	+	0.933738i	$0.616526\pi$
$434$	0	0
$435$	0	0
$436$	1510.37	0.165903
$437$	−17814.2	−1.95005
$438$	0	0
$439$	−5769.03	−0.627200	−0.313600	−	0.949555i	$-0.601535\pi$
$439$	−5769.03	−0.627200	−0.313600	+	0.949555i	$0.601535\pi$
$440$	3725.15	0.403613
$441$	0	0
$442$	−1523.05	−0.163901
$443$	8997.63	0.964989	0.482495	−	0.875899i	$-0.339731\pi$
$443$	8997.63	0.964989	0.482495	+	0.875899i	$0.339731\pi$
$444$	0	0
$445$	−20111.4	−2.14241
$446$	2129.59	0.226096
$447$	0	0
$448$	0	0
$449$	12051.5	1.26670	0.633348	−	0.773867i	$-0.281680\pi$
$449$	12051.5	1.26670	0.633348	+	0.773867i	$0.281680\pi$
$450$	0	0
$451$	123.818	0.0129277
$452$	−6519.97	−0.678481
$453$	0	0
$454$	−2041.25	−0.211014
$455$	0	0
$456$	0	0
$457$	−19121.1	−1.95721	−0.978605	−	0.205748i	$-0.934037\pi$
$457$	−19121.1	−1.95721	−0.978605	+	0.205748i	$0.934037\pi$
$458$	−1186.11	−0.121012
$459$	0	0
$460$	23708.7	2.40310
$461$	−467.211	−0.0472021	−0.0236011	−	0.999721i	$-0.507513\pi$
$461$	−467.211	−0.0472021	−0.0236011	+	0.999721i	$0.507513\pi$
$462$	0	0
$463$	5728.76	0.575028	0.287514	−	0.957776i	$-0.407171\pi$
$463$	5728.76	0.575028	0.287514	+	0.957776i	$0.407171\pi$
$464$	2410.59	0.241183
$465$	0	0
$466$	−1057.87	−0.105161
$467$	−4662.02	−0.461954	−0.230977	−	0.972959i	$-0.574192\pi$
$467$	−4662.02	−0.461954	−0.230977	+	0.972959i	$0.574192\pi$
$468$	0	0
$469$	0	0
$470$	362.429	0.0355693
$471$	0	0
$472$	3957.31	0.385911
$473$	−10989.5	−1.06828
$474$	0	0
$475$	15000.9	1.44902
$476$	0	0
$477$	0	0
$478$	1353.04	0.129470
$479$	−2215.78	−0.211360	−0.105680	−	0.994400i	$-0.533702\pi$
$479$	−2215.78	−0.211360	−0.105680	+	0.994400i	$0.533702\pi$
$480$	0	0
$481$	−1678.23	−0.159086
$482$	−1223.22	−0.115594
$483$	0	0
$484$	4978.15	0.467519
$485$	13357.9	1.25063
$486$	0	0
$487$	8704.06	0.809894	0.404947	−	0.914340i	$-0.367290\pi$
$487$	8704.06	0.809894	0.404947	+	0.914340i	$0.367290\pi$
$488$	2065.29	0.191580
$489$	0	0
$490$	0	0
$491$	−12210.2	−1.12228	−0.561139	−	0.827721i	$-0.689637\pi$
$491$	−12210.2	−1.12228	−0.561139	+	0.827721i	$0.689637\pi$
$492$	0	0
$493$	3445.17	0.314732
$494$	−1809.69	−0.164821
$495$	0	0
$496$	15969.8	1.44569
$497$	0	0
$498$	0	0
$499$	70.1600	0.00629417	0.00314709	−	0.999995i	$-0.498998\pi$
$499$	70.1600	0.00629417	0.00314709	+	0.999995i	$0.498998\pi$
$500$	−3849.11	−0.344275
$501$	0	0
$502$	−1375.00	−0.122249
$503$	−9773.52	−0.866361	−0.433181	−	0.901307i	$-0.642609\pi$
$503$	−9773.52	−0.866361	−0.433181	+	0.901307i	$0.642609\pi$
$504$	0	0
$505$	−21721.1	−1.91401
$506$	2607.20	0.229059
$507$	0	0
$508$	−9710.17	−0.848069
$509$	6752.01	0.587972	0.293986	−	0.955810i	$-0.405018\pi$
$509$	6752.01	0.587972	0.293986	+	0.955810i	$0.405018\pi$
$510$	0	0
$511$	0	0
$512$	−9527.79	−0.822408
$513$	0	0
$514$	970.043	0.0832427
$515$	−18716.5	−1.60146
$516$	0	0
$517$	−1046.83	−0.0890514
$518$	0	0
$519$	0	0
$520$	4908.67	0.413961
$521$	−12360.2	−1.03936	−0.519682	−	0.854360i	$-0.673950\pi$
$521$	−12360.2	−1.03936	−0.519682	+	0.854360i	$0.673950\pi$
$522$	0	0
$523$	13905.3	1.16259	0.581296	−	0.813692i	$-0.302546\pi$
$523$	13905.3	1.16259	0.581296	+	0.813692i	$0.302546\pi$
$524$	9409.82	0.784485
$525$	0	0
$526$	1406.82	0.116616
$527$	22823.7	1.88656
$528$	0	0
$529$	21651.8	1.77955
$530$	−2606.68	−0.213636
$531$	0	0
$532$	0	0
$533$	163.157	0.0132591
$534$	0	0
$535$	−2451.86	−0.198137
$536$	−5400.53	−0.435201
$537$	0	0
$538$	−842.709	−0.0675312
$539$	0	0
$540$	0	0
$541$	−9392.89	−0.746455	−0.373227	−	0.927740i	$-0.621749\pi$
$541$	−9392.89	−0.746455	−0.373227	+	0.927740i	$0.621749\pi$
$542$	−4218.36	−0.334306
$543$	0	0
$544$	−8068.04	−0.635872
$545$	−3278.60	−0.257688
$546$	0	0
$547$	−25372.6	−1.98328	−0.991639	−	0.129041i	$-0.958810\pi$
$547$	−25372.6	−1.98328	−0.991639	+	0.129041i	$0.958810\pi$
$548$	−24385.4	−1.90090
$549$	0	0
$550$	−2195.44	−0.170207
$551$	4093.54	0.316499
$552$	0	0
$553$	0	0
$554$	−1353.31	−0.103784
$555$	0	0
$556$	−534.378	−0.0407602
$557$	8764.14	0.666694	0.333347	−	0.942804i	$-0.391822\pi$
$557$	8764.14	0.666694	0.333347	+	0.942804i	$0.391822\pi$
$558$	0	0
$559$	−14481.0	−1.09567
$560$	0	0
$561$	0	0
$562$	4385.97	0.329201
$563$	7057.73	0.528327	0.264163	−	0.964478i	$-0.414904\pi$
$563$	7057.73	0.528327	0.264163	+	0.964478i	$0.414904\pi$
$564$	0	0
$565$	14153.1	1.05385
$566$	3182.07	0.236312
$567$	0	0
$568$	−8739.71	−0.645616
$569$	−4913.68	−0.362025	−0.181012	−	0.983481i	$-0.557937\pi$
$569$	−4913.68	−0.362025	−0.181012	+	0.983481i	$0.557937\pi$
$570$	0	0
$571$	−3626.97	−0.265822	−0.132911	−	0.991128i	$-0.542432\pi$
$571$	−3626.97	−0.265822	−0.132911	+	0.991128i	$0.542432\pi$
$572$	−6956.63	−0.508516
$573$	0	0
$574$	0	0
$575$	−28477.8	−2.06540
$576$	0	0
$577$	20573.4	1.48437	0.742186	−	0.670194i	$-0.233789\pi$
$577$	20573.4	1.48437	0.742186	+	0.670194i	$0.233789\pi$
$578$	−939.050	−0.0675767
$579$	0	0
$580$	−5448.04	−0.390030
$581$	0	0
$582$	0	0
$583$	7529.07	0.534858
$584$	3432.07	0.243185
$585$	0	0
$586$	2159.82	0.152255
$587$	8483.89	0.596538	0.298269	−	0.954482i	$-0.403591\pi$
$587$	8483.89	0.596538	0.298269	+	0.954482i	$0.403591\pi$
$588$	0	0
$589$	27119.1	1.89715
$590$	−4214.88	−0.294108
$591$	0	0
$592$	−2775.78	−0.192709
$593$	14571.4	1.00907	0.504534	−	0.863392i	$-0.331664\pi$
$593$	14571.4	1.00907	0.504534	+	0.863392i	$0.331664\pi$
$594$	0	0
$595$	0	0
$596$	−6216.30	−0.427231
$597$	0	0
$598$	3435.53	0.234932
$599$	−10164.8	−0.693362	−0.346681	−	0.937983i	$-0.612691\pi$
$599$	−10164.8	−0.693362	−0.346681	+	0.937983i	$0.612691\pi$
$600$	0	0
$601$	−11347.7	−0.770190	−0.385095	−	0.922877i	$-0.625831\pi$
$601$	−11347.7	−0.770190	−0.385095	+	0.922877i	$0.625831\pi$
$602$	0	0
$603$	0	0
$604$	16756.5	1.12883
$605$	−10806.2	−0.726172
$606$	0	0
$607$	−10938.9	−0.731459	−0.365730	−	0.930721i	$-0.619181\pi$
$607$	−10938.9	−0.731459	−0.365730	+	0.930721i	$0.619181\pi$
$608$	−9586.42	−0.639442
$609$	0	0
$610$	−2199.71	−0.146006
$611$	−1379.42	−0.0913346
$612$	0	0
$613$	17550.8	1.15639	0.578196	−	0.815898i	$-0.303757\pi$
$613$	17550.8	1.15639	0.578196	+	0.815898i	$0.303757\pi$
$614$	388.020	0.0255036
$615$	0	0
$616$	0	0
$617$	−20140.9	−1.31417	−0.657084	−	0.753818i	$-0.728210\pi$
$617$	−20140.9	−1.31417	−0.657084	+	0.753818i	$0.728210\pi$
$618$	0	0
$619$	−9924.85	−0.644448	−0.322224	−	0.946663i	$-0.604430\pi$
$619$	−9924.85	−0.644448	−0.322224	+	0.946663i	$0.604430\pi$
$620$	−36092.4	−2.33791
$621$	0	0
$622$	−2982.38	−0.192255
$623$	0	0
$624$	0	0
$625$	−11001.6	−0.704105
$626$	999.632	0.0638232
$627$	0	0
$628$	−5247.11	−0.333411
$629$	−3967.09	−0.251476
$630$	0	0
$631$	−24226.7	−1.52844	−0.764222	−	0.644954i	$-0.776877\pi$
$631$	−24226.7	−1.52844	−0.764222	+	0.644954i	$0.776877\pi$
$632$	2623.87	0.165145
$633$	0	0
$634$	701.032	0.0439141
$635$	21078.1	1.31726
$636$	0	0
$637$	0	0
$638$	−599.109	−0.0371770
$639$	0	0
$640$	16893.7	1.04341
$641$	6585.11	0.405766	0.202883	−	0.979203i	$-0.434969\pi$
$641$	6585.11	0.405766	0.202883	+	0.979203i	$0.434969\pi$
$642$	0	0
$643$	12810.5	0.785690	0.392845	−	0.919605i	$-0.371491\pi$
$643$	12810.5	0.785690	0.392845	+	0.919605i	$0.371491\pi$
$644$	0	0
$645$	0	0
$646$	−4277.85	−0.260541
$647$	16846.0	1.02363	0.511813	−	0.859097i	$-0.328974\pi$
$647$	16846.0	1.02363	0.511813	+	0.859097i	$0.328974\pi$
$648$	0	0
$649$	12174.2	0.736330
$650$	−2892.96	−0.174571
$651$	0	0
$652$	−9106.22	−0.546974
$653$	−10585.1	−0.634347	−0.317173	−	0.948368i	$-0.602734\pi$
$653$	−10585.1	−0.634347	−0.317173	+	0.948368i	$0.602734\pi$
$654$	0	0
$655$	−20426.1	−1.21850
$656$	269.860	0.0160614
$657$	0	0
$658$	0	0
$659$	11682.7	0.690584	0.345292	−	0.938495i	$-0.387780\pi$
$659$	11682.7	0.690584	0.345292	+	0.938495i	$0.387780\pi$
$660$	0	0
$661$	−13385.5	−0.787651	−0.393826	−	0.919185i	$-0.628849\pi$
$661$	−13385.5	−0.787651	−0.393826	+	0.919185i	$0.628849\pi$
$662$	2834.50	0.166414
$663$	0	0
$664$	12078.2	0.705909
$665$	0	0
$666$	0	0
$667$	−7771.23	−0.451129
$668$	1757.66	0.101805
$669$	0	0
$670$	5752.04	0.331673
$671$	6353.61	0.365541
$672$	0	0
$673$	29896.4	1.71236	0.856181	−	0.516676i	$-0.172831\pi$
$673$	29896.4	1.71236	0.856181	+	0.516676i	$0.172831\pi$
$674$	−5063.52	−0.289376
$675$	0	0
$676$	7764.55	0.441770
$677$	17322.5	0.983395	0.491698	−	0.870766i	$-0.336377\pi$
$677$	17322.5	0.983395	0.491698	+	0.870766i	$0.336377\pi$
$678$	0	0
$679$	0	0
$680$	11603.4	0.654369
$681$	0	0
$682$	−3969.00	−0.222846
$683$	−30767.3	−1.72369	−0.861844	−	0.507173i	$-0.830691\pi$
$683$	−30767.3	−1.72369	−0.861844	+	0.507173i	$0.830691\pi$
$684$	0	0
$685$	52934.1	2.95257
$686$	0	0
$687$	0	0
$688$	−23951.4	−1.32724
$689$	9921.14	0.548571
$690$	0	0
$691$	−9805.50	−0.539824	−0.269912	−	0.962885i	$-0.586995\pi$
$691$	−9805.50	−0.539824	−0.269912	+	0.962885i	$0.586995\pi$
$692$	−9593.22	−0.526994
$693$	0	0
$694$	2507.84	0.137171
$695$	1159.99	0.0633106
$696$	0	0
$697$	385.680	0.0209593
$698$	−48.0159	−0.00260376
$699$	0	0
$700$	0	0
$701$	−19855.5	−1.06980	−0.534902	−	0.844914i	$-0.679652\pi$
$701$	−19855.5	−1.06980	−0.534902	+	0.844914i	$0.679652\pi$
$702$	0	0
$703$	−4713.69	−0.252888
$704$	−10541.3	−0.564331
$705$	0	0
$706$	−5494.20	−0.292885
$707$	0	0
$708$	0	0
$709$	12272.8	0.650092	0.325046	−	0.945698i	$-0.394620\pi$
$709$	12272.8	0.650092	0.325046	+	0.945698i	$0.394620\pi$
$710$	9308.55	0.492034
$711$	0	0
$712$	−10228.1	−0.538362
$713$	−51483.2	−2.70415
$714$	0	0
$715$	15100.9	0.789850
$716$	−31568.7	−1.64774
$717$	0	0
$718$	−2495.99	−0.129735
$719$	−7995.30	−0.414707	−0.207353	−	0.978266i	$-0.566485\pi$
$719$	−7995.30	−0.414707	−0.207353	+	0.978266i	$0.566485\pi$
$720$	0	0
$721$	0	0
$722$	−1367.59	−0.0704938
$723$	0	0
$724$	−25119.2	−1.28943
$725$	6543.93	0.335221
$726$	0	0
$727$	−30320.9	−1.54682	−0.773411	−	0.633905i	$-0.781451\pi$
$727$	−30320.9	−1.54682	−0.773411	+	0.633905i	$0.781451\pi$
$728$	0	0
$729$	0	0
$730$	−3655.46	−0.185335
$731$	−34231.0	−1.73198
$732$	0	0
$733$	−33090.1	−1.66741	−0.833705	−	0.552211i	$-0.813784\pi$
$733$	−33090.1	−1.66741	−0.833705	+	0.552211i	$0.813784\pi$
$734$	4187.66	0.210585
$735$	0	0
$736$	18199.0	0.911444
$737$	−16614.1	−0.830377
$738$	0	0
$739$	−21953.9	−1.09281	−0.546406	−	0.837520i	$-0.684004\pi$
$739$	−21953.9	−1.09281	−0.546406	+	0.837520i	$0.684004\pi$
$740$	6273.38	0.311641
$741$	0	0
$742$	0	0
$743$	−10911.2	−0.538752	−0.269376	−	0.963035i	$-0.586817\pi$
$743$	−10911.2	−0.538752	−0.269376	+	0.963035i	$0.586817\pi$
$744$	0	0
$745$	13493.9	0.663594
$746$	3679.82	0.180600
$747$	0	0
$748$	−16444.5	−0.803837
$749$	0	0
$750$	0	0
$751$	10527.6	0.511530	0.255765	−	0.966739i	$-0.417673\pi$
$751$	10527.6	0.511530	0.255765	+	0.966739i	$0.417673\pi$
$752$	−2281.55	−0.110638
$753$	0	0
$754$	−789.452	−0.0381302
$755$	−36373.9	−1.75335
$756$	0	0
$757$	15270.8	0.733194	0.366597	−	0.930380i	$-0.380523\pi$
$757$	15270.8	0.733194	0.366597	+	0.930380i	$0.380523\pi$
$758$	3481.11	0.166807
$759$	0	0
$760$	13787.1	0.658042
$761$	6708.22	0.319544	0.159772	−	0.987154i	$-0.448924\pi$
$761$	6708.22	0.319544	0.159772	+	0.987154i	$0.448924\pi$
$762$	0	0
$763$	0	0
$764$	−9778.70	−0.463064
$765$	0	0
$766$	−4253.42	−0.200630
$767$	16042.0	0.755208
$768$	0	0
$769$	−7067.41	−0.331414	−0.165707	−	0.986175i	$-0.552991\pi$
$769$	−7067.41	−0.331414	−0.165707	+	0.986175i	$0.552991\pi$
$770$	0	0
$771$	0	0
$772$	1373.09	0.0640136
$773$	2381.32	0.110802	0.0554012	−	0.998464i	$-0.482356\pi$
$773$	2381.32	0.110802	0.0554012	+	0.998464i	$0.482356\pi$
$774$	0	0
$775$	43352.5	2.00938
$776$	6793.46	0.314267
$777$	0	0
$778$	−2534.92	−0.116814
$779$	458.263	0.0210770
$780$	0	0
$781$	−26886.6	−1.23186
$782$	8121.11	0.371369
$783$	0	0
$784$	0	0
$785$	11390.0	0.517869
$786$	0	0
$787$	−33540.7	−1.51918	−0.759592	−	0.650400i	$-0.774601\pi$
$787$	−33540.7	−1.51918	−0.759592	+	0.650400i	$0.774601\pi$
$788$	−36917.5	−1.66895
$789$	0	0
$790$	−2794.65	−0.125860
$791$	0	0
$792$	0	0
$793$	8372.22	0.374913
$794$	1349.49	0.0603168
$795$	0	0
$796$	−32405.1	−1.44293
$797$	−16332.6	−0.725886	−0.362943	−	0.931811i	$-0.618228\pi$
$797$	−16332.6	−0.725886	−0.362943	+	0.931811i	$0.618228\pi$
$798$	0	0
$799$	−3260.76	−0.144377
$800$	−15324.8	−0.677268
$801$	0	0
$802$	7153.62	0.314967
$803$	10558.3	0.464005
$804$	0	0
$805$	0	0
$806$	−5230.00	−0.228559
$807$	0	0
$808$	−11046.7	−0.480967
$809$	32769.7	1.42413	0.712064	−	0.702114i	$-0.247761\pi$
$809$	32769.7	1.42413	0.712064	+	0.702114i	$0.247761\pi$
$810$	0	0
$811$	−30212.2	−1.30813	−0.654066	−	0.756437i	$-0.726938\pi$
$811$	−30212.2	−1.30813	−0.654066	+	0.756437i	$0.726938\pi$
$812$	0	0
$813$	0	0
$814$	689.869	0.0297051
$815$	19767.1	0.849585
$816$	0	0
$817$	−40673.1	−1.74170
$818$	3265.47	0.139577
$819$	0	0
$820$	−609.896	−0.0259738
$821$	44113.0	1.87522	0.937609	−	0.347691i	$-0.113034\pi$
$821$	44113.0	1.87522	0.937609	+	0.347691i	$0.113034\pi$
$822$	0	0
$823$	−1817.29	−0.0769706	−0.0384853	−	0.999259i	$-0.512253\pi$
$823$	−1817.29	−0.0769706	−0.0384853	+	0.999259i	$0.512253\pi$
$824$	−9518.69	−0.402426
$825$	0	0
$826$	0	0
$827$	−1390.30	−0.0584589	−0.0292294	−	0.999573i	$-0.509305\pi$
$827$	−1390.30	−0.0584589	−0.0292294	+	0.999573i	$0.509305\pi$
$828$	0	0
$829$	29072.1	1.21799	0.608995	−	0.793174i	$-0.291573\pi$
$829$	29072.1	1.21799	0.608995	+	0.793174i	$0.291573\pi$
$830$	−12864.3	−0.537984
$831$	0	0
$832$	−13890.3	−0.578800
$833$	0	0
$834$	0	0
$835$	−3815.41	−0.158129
$836$	−19539.3	−0.808350
$837$	0	0
$838$	5757.15	0.237324
$839$	45756.4	1.88282	0.941410	−	0.337264i	$-0.109501\pi$
$839$	45756.4	1.88282	0.941410	+	0.337264i	$0.109501\pi$
$840$	0	0
$841$	−22603.2	−0.926780
$842$	−2496.22	−0.102168
$843$	0	0
$844$	15405.3	0.628284
$845$	−16854.7	−0.686177
$846$	0	0
$847$	0	0
$848$	16409.5	0.664510
$849$	0	0
$850$	−6838.56	−0.275954
$851$	8948.52	0.360460
$852$	0	0
$853$	−6646.22	−0.266779	−0.133389	−	0.991064i	$-0.542586\pi$
$853$	−6646.22	−0.266779	−0.133389	+	0.991064i	$0.542586\pi$
$854$	0	0
$855$	0	0
$856$	−1246.94	−0.0497893
$857$	14576.1	0.580993	0.290497	−	0.956876i	$-0.406179\pi$
$857$	14576.1	0.580993	0.290497	+	0.956876i	$0.406179\pi$
$858$	0	0
$859$	−5100.35	−0.202586	−0.101293	−	0.994857i	$-0.532298\pi$
$859$	−5100.35	−0.202586	−0.101293	+	0.994857i	$0.532298\pi$
$860$	54131.3	2.14635
$861$	0	0
$862$	−3256.20	−0.128662
$863$	16493.9	0.650590	0.325295	−	0.945613i	$-0.394536\pi$
$863$	16493.9	0.650590	0.325295	+	0.945613i	$0.394536\pi$
$864$	0	0
$865$	20824.3	0.818550
$866$	3494.04	0.137104
$867$	0	0
$868$	0	0
$869$	8072.01	0.315103
$870$	0	0
$871$	−21892.6	−0.851666
$872$	−1667.40	−0.0647538
$873$	0	0
$874$	9649.48	0.373454
$875$	0	0
$876$	0	0
$877$	−19853.4	−0.764425	−0.382213	−	0.924074i	$-0.624838\pi$
$877$	−19853.4	−0.764425	−0.382213	+	0.924074i	$0.624838\pi$
$878$	3124.93	0.120115
$879$	0	0
$880$	24976.8	0.956783
$881$	24792.8	0.948117	0.474059	−	0.880493i	$-0.342789\pi$
$881$	24792.8	0.948117	0.474059	+	0.880493i	$0.342789\pi$
$882$	0	0
$883$	2770.17	0.105576	0.0527880	−	0.998606i	$-0.483189\pi$
$883$	2770.17	0.105576	0.0527880	+	0.998606i	$0.483189\pi$
$884$	−21669.1	−0.824446
$885$	0	0
$886$	−4873.76	−0.184805
$887$	5502.04	0.208275	0.104138	−	0.994563i	$-0.466792\pi$
$887$	5502.04	0.208275	0.104138	+	0.994563i	$0.466792\pi$
$888$	0	0
$889$	0	0
$890$	10893.8	0.410294
$891$	0	0
$892$	30298.5	1.13730
$893$	−3874.42	−0.145188
$894$	0	0
$895$	68527.0	2.55934
$896$	0	0
$897$	0	0
$898$	−6527.98	−0.242585
$899$	11830.3	0.438892
$900$	0	0
$901$	23452.2	0.867154
$902$	−67.0689	−0.00247578
$903$	0	0
$904$	7197.83	0.264819
$905$	54527.0	2.00280
$906$	0	0
$907$	−43667.1	−1.59861	−0.799306	−	0.600924i	$-0.794799\pi$
$907$	−43667.1	−1.59861	−0.799306	+	0.600924i	$0.794799\pi$
$908$	−29041.7	−1.06143
$909$	0	0
$910$	0	0
$911$	−22419.5	−0.815358	−0.407679	−	0.913125i	$-0.633662\pi$
$911$	−22419.5	−0.815358	−0.407679	+	0.913125i	$0.633662\pi$
$912$	0	0
$913$	37157.0	1.34690
$914$	10357.3	0.374825
$915$	0	0
$916$	−16875.3	−0.608707
$917$	0	0
$918$	0	0
$919$	−19524.5	−0.700822	−0.350411	−	0.936596i	$-0.613958\pi$
$919$	−19524.5	−0.700822	−0.350411	+	0.936596i	$0.613958\pi$
$920$	−26173.7	−0.937957
$921$	0	0
$922$	253.075	0.00903969
$923$	−35428.8	−1.26344
$924$	0	0
$925$	−7535.29	−0.267847
$926$	−3103.11	−0.110124
$927$	0	0
$928$	−4181.95	−0.147930
$929$	−30706.0	−1.08442	−0.542212	−	0.840242i	$-0.682413\pi$
$929$	−30706.0	−1.08442	−0.542212	+	0.840242i	$0.682413\pi$
$930$	0	0
$931$	0	0
$932$	−15050.8	−0.528974
$933$	0	0
$934$	2525.29	0.0884689
$935$	35696.5	1.24856
$936$	0	0
$937$	20313.5	0.708233	0.354117	−	0.935201i	$-0.384782\pi$
$937$	20313.5	0.708233	0.354117	+	0.935201i	$0.384782\pi$
$938$	0	0
$939$	0	0
$940$	5156.42	0.178919
$941$	−37068.3	−1.28416	−0.642079	−	0.766638i	$-0.721928\pi$
$941$	−37068.3	−1.28416	−0.642079	+	0.766638i	$0.721928\pi$
$942$	0	0
$943$	−869.972	−0.0300426
$944$	26533.4	0.914819
$945$	0	0
$946$	5952.69	0.204586
$947$	−35369.2	−1.21367	−0.606835	−	0.794828i	$-0.707561\pi$
$947$	−35369.2	−1.21367	−0.606835	+	0.794828i	$0.707561\pi$
$948$	0	0
$949$	13912.8	0.475901
$950$	−8125.55	−0.277503
$951$	0	0
$952$	0	0
$953$	33615.0	1.14260	0.571300	−	0.820741i	$-0.306439\pi$
$953$	33615.0	1.14260	0.571300	+	0.820741i	$0.306439\pi$
$954$	0	0
$955$	21226.9	0.719252
$956$	19250.3	0.651254
$957$	0	0
$958$	1200.23	0.0404776
$959$	0	0
$960$	0	0
$961$	48583.2	1.63080
$962$	909.049	0.0304666
$963$	0	0
$964$	−17403.3	−0.581455
$965$	−2980.60	−0.0994289
$966$	0	0
$967$	14905.5	0.495685	0.247842	−	0.968800i	$-0.420279\pi$
$967$	14905.5	0.495685	0.247842	+	0.968800i	$0.420279\pi$
$968$	−5495.71	−0.182478
$969$	0	0
$970$	−7235.63	−0.239507
$971$	9213.04	0.304491	0.152245	−	0.988343i	$-0.451350\pi$
$971$	9213.04	0.304491	0.152245	+	0.988343i	$0.451350\pi$
$972$	0	0
$973$	0	0
$974$	−4714.75	−0.155103
$975$	0	0
$976$	13847.6	0.454150
$977$	11453.4	0.375054	0.187527	−	0.982259i	$-0.439953\pi$
$977$	11453.4	0.375054	0.187527	+	0.982259i	$0.439953\pi$
$978$	0	0
$979$	−31465.5	−1.02721
$980$	0	0
$981$	0	0
$982$	6613.93	0.214928
$983$	42889.2	1.39161	0.695805	−	0.718231i	$-0.255048\pi$
$983$	42889.2	1.39161	0.695805	+	0.718231i	$0.255048\pi$
$984$	0	0
$985$	80137.9	2.59229
$986$	−1866.15	−0.0602743
$987$	0	0
$988$	−25747.1	−0.829075
$989$	77214.3	2.48258
$990$	0	0
$991$	50991.9	1.63452	0.817261	−	0.576267i	$-0.195491\pi$
$991$	50991.9	1.63452	0.817261	+	0.576267i	$0.195491\pi$
$992$	−27704.8	−0.886721
$993$	0	0
$994$	0	0
$995$	70342.7	2.24122
$996$	0	0
$997$	−5789.17	−0.183896	−0.0919482	−	0.995764i	$-0.529309\pi$
$997$	−5789.17	−0.183896	−0.0919482	+	0.995764i	$0.529309\pi$
$998$	−38.0037	−0.00120540
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bk.1.4		7
3.2	odd	2		1323.4.a.bh.1.4		7
7.3	odd	6		189.4.e.f.163.4	yes	14
7.5	odd	6		189.4.e.f.109.4	✓	14
7.6	odd	2		1323.4.a.bj.1.4		7
21.5	even	6		189.4.e.g.109.4	yes	14
21.17	even	6		189.4.e.g.163.4	yes	14
21.20	even	2		1323.4.a.bi.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.f.109.4	✓	14	7.5	odd	6
189.4.e.f.163.4	yes	14	7.3	odd	6
189.4.e.g.109.4	yes	14	21.5	even	6
189.4.e.g.163.4	yes	14	21.17	even	6
1323.4.a.bh.1.4		7	3.2	odd	2
1323.4.a.bi.1.4		7	21.20	even	2
1323.4.a.bj.1.4		7	7.6	odd	2
1323.4.a.bk.1.4		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-0.541672 q^{2} -7.70659 q^{4} +16.7289 q^{5} +8.50782 q^{8} -9.06158 q^{10} +26.1733 q^{11} +34.4888 q^{13} +57.0443 q^{16} +81.5268 q^{17} +96.8698 q^{19} -128.923 q^{20} -14.1773 q^{22} -183.899 q^{23} +154.856 q^{25} -18.6816 q^{26} +42.2582 q^{29} +279.954 q^{31} -98.9619 q^{32} -44.1608 q^{34} -48.6600 q^{37} -52.4717 q^{38} +142.327 q^{40} +4.73071 q^{41} -419.874 q^{43} -201.707 q^{44} +99.6129 q^{46} -39.9962 q^{47} -83.8812 q^{50} -265.791 q^{52} +287.663 q^{53} +437.850 q^{55} -22.8901 q^{58} +465.137 q^{59} +242.752 q^{61} -151.643 q^{62} -402.749 q^{64} +576.960 q^{65} -634.773 q^{67} -628.293 q^{68} -1027.26 q^{71} +403.402 q^{73} +26.3578 q^{74} -746.536 q^{76} +308.406 q^{79} +954.288 q^{80} -2.56249 q^{82} +1419.65 q^{83} +1363.85 q^{85} +227.434 q^{86} +222.678 q^{88} -1202.20 q^{89} +1417.23 q^{92} +21.6648 q^{94} +1620.52 q^{95} +798.496 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 31 q^{4} + q^{5} + 84 q^{8} - 12 q^{10} + 98 q^{11} + 124 q^{13} + 139 q^{16} - 30 q^{17} - 182 q^{19} + 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} + 245 q^{26} + 323 q^{29} - 26 q^{31}+ \cdots + 3730 q^{97}+O(q^{100}) \) 7 * q + q^2 + 31 * q^4 + q^5 + 84 * q^8 - 12 * q^10 + 98 * q^11 + 124 * q^13 + 139 * q^16 - 30 * q^17 - 182 * q^19 + 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 + 245 * q^26 + 323 * q^29 - 26 * q^31 + 398 * q^32 - 114 * q^34 - 112 * q^37 - 1015 * q^38 + 147 * q^40 + 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 - 288 * q^47 + 2576 * q^50 + 1075 * q^52 + 1353 * q^53 - 156 * q^55 - 81 * q^58 - 165 * q^59 - 56 * q^61 + 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 - 2625 * q^68 + 792 * q^71 - 1487 * q^73 + 2736 * q^74 - 1952 * q^76 - 1273 * q^79 + 2501 * q^80 + 2049 * q^82 + 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 - 1058 * q^89 + 3834 * q^92 - 1653 * q^94 + 3260 * q^95 + 3730 * q^97

Introduction

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