LMFDB - Embedded newform 1323.4.a.be.1.5

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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 40x^{4} + 453x^{2} - 1278 $ x^6 - 40x^4 + 453x^2 - 1278
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.68757$ 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)

$f(q)$	$=$	$q+3.68757 q^{2} +5.59820 q^{4} +11.8719 q^{5} -8.85681 q^{8} +O(q^{10})$	$q+3.68757 q^{2} +5.59820 q^{4} +11.8719 q^{5} -8.85681 q^{8} +43.7785 q^{10} +22.2104 q^{11} -75.1040 q^{13} -77.4457 q^{16} -74.1097 q^{17} +7.41977 q^{19} +66.4612 q^{20} +81.9023 q^{22} -205.772 q^{23} +15.9417 q^{25} -276.952 q^{26} +148.304 q^{29} +164.438 q^{31} -214.732 q^{32} -273.285 q^{34} -205.239 q^{37} +27.3610 q^{38} -105.147 q^{40} -83.9896 q^{41} -368.677 q^{43} +124.338 q^{44} -758.800 q^{46} -98.3339 q^{47} +58.7864 q^{50} -420.448 q^{52} -293.897 q^{53} +263.679 q^{55} +546.883 q^{58} -509.272 q^{59} +696.384 q^{61} +606.376 q^{62} -172.276 q^{64} -891.627 q^{65} +370.879 q^{67} -414.881 q^{68} +121.723 q^{71} +682.103 q^{73} -756.832 q^{74} +41.5374 q^{76} -669.950 q^{79} -919.427 q^{80} -309.718 q^{82} +1182.06 q^{83} -879.823 q^{85} -1359.52 q^{86} -196.713 q^{88} +598.651 q^{89} -1151.95 q^{92} -362.614 q^{94} +88.0867 q^{95} +1037.20 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 32 q^{4}+O(q^{10}) $ 6 * q + 32 * q^4	$ 6 q + 32 q^{4} + 20 q^{10} - 52 q^{13} - 148 q^{16} + 62 q^{19} - 356 q^{22} - 46 q^{25} + 82 q^{31} - 420 q^{34} - 1132 q^{37} + 444 q^{40} - 1566 q^{43} - 888 q^{46} + 72 q^{52} - 224 q^{55} + 4 q^{58} - 886 q^{61} - 924 q^{64} - 2084 q^{67} + 2398 q^{73} - 3204 q^{76} + 984 q^{79} + 3892 q^{82} - 3600 q^{85} - 5796 q^{88} - 2772 q^{94} + 682 q^{97}+O(q^{100}) $ 6 * q + 32 * q^4 + 20 * q^10 - 52 * q^13 - 148 * q^16 + 62 * q^19 - 356 * q^22 - 46 * q^25 + 82 * q^31 - 420 * q^34 - 1132 * q^37 + 444 * q^40 - 1566 * q^43 - 888 * q^46 + 72 * q^52 - 224 * q^55 + 4 * q^58 - 886 * q^61 - 924 * q^64 - 2084 * q^67 + 2398 * q^73 - 3204 * q^76 + 984 * q^79 + 3892 * q^82 - 3600 * q^85 - 5796 * q^88 - 2772 * q^94 + 682 * 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 $

$2$	3.68757	1.30375	0.651877	−	0.758325i	$-0.273982\pi$
$2$	3.68757	1.30375	0.651877	+	0.758325i	$0.273982\pi$
$3$	0	0
$3$	0	0
$4$	5.59820	0.699775
$5$	11.8719	1.06185	0.530927	−	0.847418i	$-0.321844\pi$
$5$	11.8719	1.06185	0.530927	+	0.847418i	$0.321844\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.85681	−0.391419
$9$	0	0
$10$	43.7785	1.38440
$11$	22.2104	0.608788	0.304394	−	0.952546i	$-0.401546\pi$
$11$	22.2104	0.608788	0.304394	+	0.952546i	$0.401546\pi$
$12$	0	0
$13$	−75.1040	−1.60232	−0.801158	−	0.598453i	$-0.795782\pi$
$13$	−75.1040	−1.60232	−0.801158	+	0.598453i	$0.795782\pi$
$14$	0	0
$15$	0	0
$16$	−77.4457	−1.21009
$17$	−74.1097	−1.05731	−0.528654	−	0.848837i	$-0.677303\pi$
$17$	−74.1097	−1.05731	−0.528654	+	0.848837i	$0.677303\pi$
$18$	0	0
$19$	7.41977	0.0895901	0.0447951	−	0.998996i	$-0.485737\pi$
$19$	7.41977	0.0895901	0.0447951	+	0.998996i	$0.485737\pi$
$20$	66.4612	0.743059
$21$	0	0
$22$	81.9023	0.793711
$23$	−205.772	−1.86550	−0.932749	−	0.360526i	$-0.882597\pi$
$23$	−205.772	−1.86550	−0.932749	+	0.360526i	$0.882597\pi$
$24$	0	0
$25$	15.9417	0.127534
$26$	−276.952	−2.08903
$27$	0	0
$28$	0	0
$29$	148.304	0.949636	0.474818	−	0.880084i	$-0.342514\pi$
$29$	148.304	0.949636	0.474818	+	0.880084i	$0.342514\pi$
$30$	0	0
$31$	164.438	0.952705	0.476353	−	0.879254i	$-0.341959\pi$
$31$	164.438	0.952705	0.476353	+	0.879254i	$0.341959\pi$
$32$	−214.732	−1.18624
$33$	0	0
$34$	−273.285	−1.37847
$35$	0	0
$36$	0	0
$37$	−205.239	−0.911919	−0.455960	−	0.890001i	$-0.650704\pi$
$37$	−205.239	−0.911919	−0.455960	+	0.890001i	$0.650704\pi$
$38$	27.3610	0.116803
$39$	0	0
$40$	−105.147	−0.415630
$41$	−83.9896	−0.319926	−0.159963	−	0.987123i	$-0.551137\pi$
$41$	−83.9896	−0.319926	−0.159963	+	0.987123i	$0.551137\pi$
$42$	0	0
$43$	−368.677	−1.30751	−0.653753	−	0.756708i	$-0.726806\pi$
$43$	−368.677	−1.30751	−0.653753	+	0.756708i	$0.726806\pi$
$44$	124.338	0.426015
$45$	0	0
$46$	−758.800	−2.43215
$47$	−98.3339	−0.305180	−0.152590	−	0.988290i	$-0.548761\pi$
$47$	−98.3339	−0.305180	−0.152590	+	0.988290i	$0.548761\pi$
$48$	0	0
$49$	0	0
$50$	58.7864	0.166273
$51$	0	0
$52$	−420.448	−1.12126
$53$	−293.897	−0.761696	−0.380848	−	0.924638i	$-0.624368\pi$
$53$	−293.897	−0.761696	−0.380848	+	0.924638i	$0.624368\pi$
$54$	0	0
$55$	263.679	0.646444
$56$	0	0
$57$	0	0
$58$	546.883	1.23809
$59$	−509.272	−1.12376	−0.561878	−	0.827220i	$-0.689921\pi$
$59$	−509.272	−1.12376	−0.561878	+	0.827220i	$0.689921\pi$
$60$	0	0
$61$	696.384	1.46169	0.730843	−	0.682546i	$-0.239127\pi$
$61$	696.384	1.46169	0.730843	+	0.682546i	$0.239127\pi$
$62$	606.376	1.24209
$63$	0	0
$64$	−172.276	−0.336477
$65$	−891.627	−1.70143
$66$	0	0
$67$	370.879	0.676270	0.338135	−	0.941098i	$-0.390204\pi$
$67$	370.879	0.676270	0.338135	+	0.941098i	$0.390204\pi$
$68$	−414.881	−0.739879
$69$	0	0
$70$	0	0
$71$	121.723	0.203462	0.101731	−	0.994812i	$-0.467562\pi$
$71$	121.723	0.203462	0.101731	+	0.994812i	$0.467562\pi$
$72$	0	0
$73$	682.103	1.09362	0.546809	−	0.837257i	$-0.315842\pi$
$73$	682.103	1.09362	0.546809	+	0.837257i	$0.315842\pi$
$74$	−756.832	−1.18892
$75$	0	0
$76$	41.5374	0.0626929
$77$	0	0
$78$	0	0
$79$	−669.950	−0.954118	−0.477059	−	0.878871i	$-0.658297\pi$
$79$	−669.950	−0.954118	−0.477059	+	0.878871i	$0.658297\pi$
$80$	−919.427	−1.28494
$81$	0	0
$82$	−309.718	−0.417105
$83$	1182.06	1.56323	0.781616	−	0.623759i	$-0.214396\pi$
$83$	1182.06	1.56323	0.781616	+	0.623759i	$0.214396\pi$
$84$	0	0
$85$	−879.823	−1.12271
$86$	−1359.52	−1.70467
$87$	0	0
$88$	−196.713	−0.238291
$89$	598.651	0.712999	0.356500	−	0.934295i	$-0.383970\pi$
$89$	598.651	0.712999	0.356500	+	0.934295i	$0.383970\pi$
$90$	0	0
$91$	0	0
$92$	−1151.95	−1.30543
$93$	0	0
$94$	−362.614	−0.397880
$95$	88.0867	0.0951316
$96$	0	0
$97$	1037.20	1.08569	0.542846	−	0.839833i	$-0.317347\pi$
$97$	1037.20	1.08569	0.542846	+	0.839833i	$0.317347\pi$
$98$	0	0
$99$	0	0
$100$	89.2451	0.0892451
$101$	137.285	0.135251	0.0676254	−	0.997711i	$-0.478458\pi$
$101$	137.285	0.135251	0.0676254	+	0.997711i	$0.478458\pi$
$102$	0	0
$103$	327.237	0.313045	0.156522	−	0.987674i	$-0.449972\pi$
$103$	327.237	0.313045	0.156522	+	0.987674i	$0.449972\pi$
$104$	665.182	0.627177
$105$	0	0
$106$	−1083.77	−0.993064
$107$	−1051.42	−0.949947	−0.474973	−	0.880000i	$-0.657542\pi$
$107$	−1051.42	−0.949947	−0.474973	+	0.880000i	$0.657542\pi$
$108$	0	0
$109$	255.171	0.224229	0.112114	−	0.993695i	$-0.464238\pi$
$109$	255.171	0.224229	0.112114	+	0.993695i	$0.464238\pi$
$110$	972.335	0.842805
$111$	0	0
$112$	0	0
$113$	−1990.49	−1.65708	−0.828538	−	0.559933i	$-0.810827\pi$
$113$	−1990.49	−1.65708	−0.828538	+	0.559933i	$0.810827\pi$
$114$	0	0
$115$	−2442.90	−1.98089
$116$	830.238	0.664532
$117$	0	0
$118$	−1877.98	−1.46510
$119$	0	0
$120$	0	0
$121$	−837.700	−0.629377
$122$	2567.97	1.90568
$123$	0	0
$124$	920.555	0.666680
$125$	−1294.73	−0.926432
$126$	0	0
$127$	536.451	0.374822	0.187411	−	0.982282i	$-0.439990\pi$
$127$	536.451	0.374822	0.187411	+	0.982282i	$0.439990\pi$
$128$	1082.58	0.747558
$129$	0	0
$130$	−3287.94	−2.21824
$131$	−2560.09	−1.70745	−0.853727	−	0.520721i	$-0.825663\pi$
$131$	−2560.09	−1.70745	−0.853727	+	0.520721i	$0.825663\pi$
$132$	0	0
$133$	0	0
$134$	1367.64	0.881690
$135$	0	0
$136$	656.376	0.413851
$137$	−2266.25	−1.41327	−0.706637	−	0.707577i	$-0.749788\pi$
$137$	−2266.25	−1.41327	−0.706637	+	0.707577i	$0.749788\pi$
$138$	0	0
$139$	−1837.27	−1.12112	−0.560559	−	0.828114i	$-0.689414\pi$
$139$	−1837.27	−1.12112	−0.560559	+	0.828114i	$0.689414\pi$
$140$	0	0
$141$	0	0
$142$	448.861	0.265265
$143$	−1668.09	−0.975472
$144$	0	0
$145$	1760.65	1.00837
$146$	2515.31	1.42581
$147$	0	0
$148$	−1148.97	−0.638138
$149$	626.201	0.344298	0.172149	−	0.985071i	$-0.444929\pi$
$149$	626.201	0.344298	0.172149	+	0.985071i	$0.444929\pi$
$150$	0	0
$151$	−1204.68	−0.649243	−0.324621	−	0.945844i	$-0.605237\pi$
$151$	−1204.68	−0.649243	−0.324621	+	0.945844i	$0.605237\pi$
$152$	−65.7155	−0.0350673
$153$	0	0
$154$	0	0
$155$	1952.19	1.01163
$156$	0	0
$157$	−2976.32	−1.51297	−0.756486	−	0.654010i	$-0.773085\pi$
$157$	−2976.32	−1.51297	−0.756486	+	0.654010i	$0.773085\pi$
$158$	−2470.49	−1.24394
$159$	0	0
$160$	−2549.28	−1.25961
$161$	0	0
$162$	0	0
$163$	816.520	0.392360	0.196180	−	0.980568i	$-0.437146\pi$
$163$	816.520	0.392360	0.196180	+	0.980568i	$0.437146\pi$
$164$	−470.191	−0.223876
$165$	0	0
$166$	4358.95	2.03807
$167$	1597.72	0.740331	0.370166	−	0.928966i	$-0.379301\pi$
$167$	1597.72	0.740331	0.370166	+	0.928966i	$0.379301\pi$
$168$	0	0
$169$	3443.62	1.56742
$170$	−3244.41	−1.46374
$171$	0	0
$172$	−2063.93	−0.914960
$173$	1330.40	0.584675	0.292337	−	0.956315i	$-0.405567\pi$
$173$	1330.40	0.584675	0.292337	+	0.956315i	$0.405567\pi$
$174$	0	0
$175$	0	0
$176$	−1720.10	−0.736689
$177$	0	0
$178$	2207.57	0.929576
$179$	4088.95	1.70739	0.853694	−	0.520775i	$-0.174357\pi$
$179$	4088.95	1.70739	0.853694	+	0.520775i	$0.174357\pi$
$180$	0	0
$181$	1223.86	0.502591	0.251295	−	0.967910i	$-0.419143\pi$
$181$	1223.86	0.502591	0.251295	+	0.967910i	$0.419143\pi$
$182$	0	0
$183$	0	0
$184$	1822.48	0.730192
$185$	−2436.57	−0.968325
$186$	0	0
$187$	−1646.00	−0.643677
$188$	−550.493	−0.213558
$189$	0	0
$190$	324.826	0.124028
$191$	1418.72	0.537460	0.268730	−	0.963216i	$-0.413396\pi$
$191$	1418.72	0.537460	0.268730	+	0.963216i	$0.413396\pi$
$192$	0	0
$193$	−3519.66	−1.31270	−0.656348	−	0.754458i	$-0.727900\pi$
$193$	−3519.66	−1.31270	−0.656348	+	0.754458i	$0.727900\pi$
$194$	3824.76	1.41547
$195$	0	0
$196$	0	0
$197$	−689.885	−0.249504	−0.124752	−	0.992188i	$-0.539813\pi$
$197$	−689.885	−0.249504	−0.124752	+	0.992188i	$0.539813\pi$
$198$	0	0
$199$	511.903	0.182351	0.0911754	−	0.995835i	$-0.470938\pi$
$199$	511.903	0.182351	0.0911754	+	0.995835i	$0.470938\pi$
$200$	−141.193	−0.0499192
$201$	0	0
$202$	506.248	0.176334
$203$	0	0
$204$	0	0
$205$	−997.115	−0.339715
$206$	1206.71	0.408133
$207$	0	0
$208$	5816.49	1.93895
$209$	164.796	0.0545414
$210$	0	0
$211$	−4487.25	−1.46405	−0.732027	−	0.681276i	$-0.761425\pi$
$211$	−4487.25	−1.46405	−0.732027	+	0.681276i	$0.761425\pi$
$212$	−1645.30	−0.533016
$213$	0	0
$214$	−3877.18	−1.23850
$215$	−4376.89	−1.38838
$216$	0	0
$217$	0	0
$218$	940.961	0.292339
$219$	0	0
$220$	1476.13	0.452366
$221$	5565.94	1.69414
$222$	0	0
$223$	−779.975	−0.234220	−0.117110	−	0.993119i	$-0.537363\pi$
$223$	−779.975	−0.234220	−0.117110	+	0.993119i	$0.537363\pi$
$224$	0	0
$225$	0	0
$226$	−7340.08	−2.16042
$227$	733.716	0.214531	0.107265	−	0.994230i	$-0.465791\pi$
$227$	733.716	0.214531	0.107265	+	0.994230i	$0.465791\pi$
$228$	0	0
$229$	4774.12	1.37765	0.688827	−	0.724926i	$-0.258126\pi$
$229$	4774.12	1.37765	0.688827	+	0.724926i	$0.258126\pi$
$230$	−9008.39	−2.58259
$231$	0	0
$232$	−1313.50	−0.371706
$233$	6104.15	1.71629	0.858146	−	0.513406i	$-0.171617\pi$
$233$	6104.15	1.71629	0.858146	+	0.513406i	$0.171617\pi$
$234$	0	0
$235$	−1167.41	−0.324057
$236$	−2851.01	−0.786377
$237$	0	0
$238$	0	0
$239$	6750.43	1.82698	0.913491	−	0.406859i	$-0.133376\pi$
$239$	6750.43	1.82698	0.913491	+	0.406859i	$0.133376\pi$
$240$	0	0
$241$	−1720.38	−0.459833	−0.229916	−	0.973210i	$-0.573845\pi$
$241$	−1720.38	−0.459833	−0.229916	+	0.973210i	$0.573845\pi$
$242$	−3089.08	−0.820552
$243$	0	0
$244$	3898.50	1.02285
$245$	0	0
$246$	0	0
$247$	−557.255	−0.143552
$248$	−1456.39	−0.372907
$249$	0	0
$250$	−4774.40	−1.20784
$251$	−5719.78	−1.43836	−0.719181	−	0.694822i	$-0.755483\pi$
$251$	−5719.78	−1.43836	−0.719181	+	0.694822i	$0.755483\pi$
$252$	0	0
$253$	−4570.27	−1.13569
$254$	1978.20	0.488675
$255$	0	0
$256$	5370.30	1.31111
$257$	812.156	0.197124	0.0985621	−	0.995131i	$-0.468576\pi$
$257$	812.156	0.197124	0.0985621	+	0.995131i	$0.468576\pi$
$258$	0	0
$259$	0	0
$260$	−4991.51	−1.19062
$261$	0	0
$262$	−9440.54	−2.22610
$263$	−6976.18	−1.63563	−0.817813	−	0.575484i	$-0.804814\pi$
$263$	−6976.18	−1.63563	−0.817813	+	0.575484i	$0.804814\pi$
$264$	0	0
$265$	−3489.11	−0.808809
$266$	0	0
$267$	0	0
$268$	2076.26	0.473237
$269$	265.535	0.0601857	0.0300928	−	0.999547i	$-0.490420\pi$
$269$	265.535	0.0601857	0.0300928	+	0.999547i	$0.490420\pi$
$270$	0	0
$271$	−8109.83	−1.81785	−0.908925	−	0.416960i	$-0.863096\pi$
$271$	−8109.83	−1.81785	−0.908925	+	0.416960i	$0.863096\pi$
$272$	5739.48	1.27944
$273$	0	0
$274$	−8356.95	−1.84256
$275$	354.072	0.0776412
$276$	0	0
$277$	−5941.40	−1.28875	−0.644375	−	0.764709i	$-0.722883\pi$
$277$	−5941.40	−1.28875	−0.644375	+	0.764709i	$0.722883\pi$
$278$	−6775.08	−1.46166
$279$	0	0
$280$	0	0
$281$	5686.26	1.20717	0.603584	−	0.797300i	$-0.293739\pi$
$281$	5686.26	1.20717	0.603584	+	0.797300i	$0.293739\pi$
$282$	0	0
$283$	3582.77	0.752558	0.376279	−	0.926506i	$-0.377203\pi$
$283$	3582.77	0.752558	0.376279	+	0.926506i	$0.377203\pi$
$284$	681.428	0.142378
$285$	0	0
$286$	−6151.19	−1.27178
$287$	0	0
$288$	0	0
$289$	579.254	0.117902
$290$	6492.54	1.31467
$291$	0	0
$292$	3818.55	0.765287
$293$	140.120	0.0279382	0.0139691	−	0.999902i	$-0.495553\pi$
$293$	140.120	0.0279382	0.0139691	+	0.999902i	$0.495553\pi$
$294$	0	0
$295$	−6046.03	−1.19327
$296$	1817.76	0.356943
$297$	0	0
$298$	2309.16	0.448880
$299$	15454.3	2.98912
$300$	0	0
$301$	0	0
$302$	−4442.35	−0.846453
$303$	0	0
$304$	−574.630	−0.108412
$305$	8267.39	1.55210
$306$	0	0
$307$	−2318.11	−0.430949	−0.215475	−	0.976509i	$-0.569130\pi$
$307$	−2318.11	−0.430949	−0.215475	+	0.976509i	$0.569130\pi$
$308$	0	0
$309$	0	0
$310$	7198.83	1.31892
$311$	5860.43	1.06853	0.534267	−	0.845316i	$-0.320588\pi$
$311$	5860.43	1.06853	0.534267	+	0.845316i	$0.320588\pi$
$312$	0	0
$313$	−401.620	−0.0725268	−0.0362634	−	0.999342i	$-0.511546\pi$
$313$	−401.620	−0.0725268	−0.0362634	+	0.999342i	$0.511546\pi$
$314$	−10975.4	−1.97254
$315$	0	0
$316$	−3750.52	−0.667668
$317$	9058.74	1.60501	0.802507	−	0.596643i	$-0.203499\pi$
$317$	9058.74	1.60501	0.802507	+	0.596643i	$0.203499\pi$
$318$	0	0
$319$	3293.89	0.578127
$320$	−2045.24	−0.357289
$321$	0	0
$322$	0	0
$323$	−549.877	−0.0947244
$324$	0	0
$325$	−1197.29	−0.204350
$326$	3010.98	0.511542
$327$	0	0
$328$	743.879	0.125225
$329$	0	0
$330$	0	0
$331$	−2357.68	−0.391510	−0.195755	−	0.980653i	$-0.562716\pi$
$331$	−2357.68	−0.391510	−0.195755	+	0.980653i	$0.562716\pi$
$332$	6617.43	1.09391
$333$	0	0
$334$	5891.71	0.965210
$335$	4403.03	0.718100
$336$	0	0
$337$	4774.30	0.771729	0.385865	−	0.922555i	$-0.373903\pi$
$337$	4774.30	0.771729	0.385865	+	0.922555i	$0.373903\pi$
$338$	12698.6	2.04353
$339$	0	0
$340$	−4925.43	−0.785643
$341$	3652.22	0.579996
$342$	0	0
$343$	0	0
$344$	3265.30	0.511783
$345$	0	0
$346$	4905.96	0.762272
$347$	7684.47	1.18883	0.594415	−	0.804159i	$-0.297384\pi$
$347$	7684.47	1.18883	0.594415	+	0.804159i	$0.297384\pi$
$348$	0	0
$349$	3771.66	0.578488	0.289244	−	0.957255i	$-0.406596\pi$
$349$	3771.66	0.578488	0.289244	+	0.957255i	$0.406596\pi$
$350$	0	0
$351$	0	0
$352$	−4769.28	−0.722170
$353$	−7924.16	−1.19479	−0.597394	−	0.801948i	$-0.703797\pi$
$353$	−7924.16	−1.19479	−0.597394	+	0.801948i	$0.703797\pi$
$354$	0	0
$355$	1445.08	0.216047
$356$	3351.37	0.498939
$357$	0	0
$358$	15078.3	2.22601
$359$	−9078.25	−1.33463	−0.667314	−	0.744776i	$-0.732556\pi$
$359$	−9078.25	−1.33463	−0.667314	+	0.744776i	$0.732556\pi$
$360$	0	0
$361$	−6803.95	−0.991974
$362$	4513.08	0.655255
$363$	0	0
$364$	0	0
$365$	8097.85	1.16126
$366$	0	0
$367$	2091.98	0.297549	0.148775	−	0.988871i	$-0.452467\pi$
$367$	2091.98	0.297549	0.148775	+	0.988871i	$0.452467\pi$
$368$	15936.2	2.25742
$369$	0	0
$370$	−8985.03	−1.26246
$371$	0	0
$372$	0	0
$373$	−9157.84	−1.27125	−0.635623	−	0.771999i	$-0.719257\pi$
$373$	−9157.84	−1.27125	−0.635623	+	0.771999i	$0.719257\pi$
$374$	−6069.76	−0.839197
$375$	0	0
$376$	870.925	0.119453
$377$	−11138.3	−1.52162
$378$	0	0
$379$	−604.558	−0.0819368	−0.0409684	−	0.999160i	$-0.513044\pi$
$379$	−604.558	−0.0819368	−0.0409684	+	0.999160i	$0.513044\pi$
$380$	493.127	0.0665708
$381$	0	0
$382$	5231.63	0.700716
$383$	2872.17	0.383188	0.191594	−	0.981474i	$-0.438634\pi$
$383$	2872.17	0.383188	0.191594	+	0.981474i	$0.438634\pi$
$384$	0	0
$385$	0	0
$386$	−12979.0	−1.71143
$387$	0	0
$388$	5806.47	0.759740
$389$	−2832.57	−0.369196	−0.184598	−	0.982814i	$-0.559098\pi$
$389$	−2832.57	−0.369196	−0.184598	+	0.982814i	$0.559098\pi$
$390$	0	0
$391$	15249.7	1.97241
$392$	0	0
$393$	0	0
$394$	−2544.00	−0.325292
$395$	−7953.58	−1.01313
$396$	0	0
$397$	1788.13	0.226055	0.113027	−	0.993592i	$-0.463945\pi$
$397$	1788.13	0.226055	0.113027	+	0.993592i	$0.463945\pi$
$398$	1887.68	0.237741
$399$	0	0
$400$	−1234.62	−0.154328
$401$	6715.05	0.836243	0.418122	−	0.908391i	$-0.362689\pi$
$401$	6715.05	0.836243	0.418122	+	0.908391i	$0.362689\pi$
$402$	0	0
$403$	−12349.9	−1.52654
$404$	768.548	0.0946452
$405$	0	0
$406$	0	0
$407$	−4558.42	−0.555166
$408$	0	0
$409$	14483.0	1.75095	0.875477	−	0.483259i	$-0.160547\pi$
$409$	14483.0	1.75095	0.875477	+	0.483259i	$0.160547\pi$
$410$	−3676.93	−0.442904
$411$	0	0
$412$	1831.94	0.219061
$413$	0	0
$414$	0	0
$415$	14033.3	1.65992
$416$	16127.3	1.90073
$417$	0	0
$418$	607.696	0.0711086
$419$	4513.58	0.526260	0.263130	−	0.964760i	$-0.415245\pi$
$419$	4513.58	0.526260	0.263130	+	0.964760i	$0.415245\pi$
$420$	0	0
$421$	8684.07	1.00531	0.502655	−	0.864487i	$-0.332357\pi$
$421$	8684.07	1.00531	0.502655	+	0.864487i	$0.332357\pi$
$422$	−16547.1	−1.90877
$423$	0	0
$424$	2602.99	0.298142
$425$	−1181.44	−0.134843
$426$	0	0
$427$	0	0
$428$	−5886.04	−0.664749
$429$	0	0
$430$	−16140.1	−1.81011
$431$	−10770.7	−1.20373	−0.601863	−	0.798599i	$-0.705575\pi$
$431$	−10770.7	−1.20373	−0.601863	+	0.798599i	$0.705575\pi$
$432$	0	0
$433$	−1704.52	−0.189178	−0.0945890	−	0.995516i	$-0.530154\pi$
$433$	−1704.52	−0.189178	−0.0945890	+	0.995516i	$0.530154\pi$
$434$	0	0
$435$	0	0
$436$	1428.50	0.156910
$437$	−1526.78	−0.167130
$438$	0	0
$439$	−4516.83	−0.491063	−0.245531	−	0.969389i	$-0.578962\pi$
$439$	−4516.83	−0.491063	−0.245531	+	0.969389i	$0.578962\pi$
$440$	−2335.35	−0.253031
$441$	0	0
$442$	20524.8	2.20875
$443$	13103.6	1.40536	0.702678	−	0.711508i	$-0.251987\pi$
$443$	13103.6	1.40536	0.702678	+	0.711508i	$0.251987\pi$
$444$	0	0
$445$	7107.12	0.757101
$446$	−2876.22	−0.305365
$447$	0	0
$448$	0	0
$449$	1429.10	0.150208	0.0751040	−	0.997176i	$-0.476071\pi$
$449$	1429.10	0.150208	0.0751040	+	0.997176i	$0.476071\pi$
$450$	0	0
$451$	−1865.44	−0.194767
$452$	−11143.2	−1.15958
$453$	0	0
$454$	2705.63	0.279695
$455$	0	0
$456$	0	0
$457$	−16338.3	−1.67237	−0.836186	−	0.548447i	$-0.815219\pi$
$457$	−16338.3	−1.67237	−0.836186	+	0.548447i	$0.815219\pi$
$458$	17604.9	1.79612
$459$	0	0
$460$	−13675.9	−1.38618
$461$	13542.4	1.36818	0.684089	−	0.729398i	$-0.260200\pi$
$461$	13542.4	1.36818	0.684089	+	0.729398i	$0.260200\pi$
$462$	0	0
$463$	1851.49	0.185844	0.0929222	−	0.995673i	$-0.470379\pi$
$463$	1851.49	0.185844	0.0929222	+	0.995673i	$0.470379\pi$
$464$	−11485.5	−1.14914
$465$	0	0
$466$	22509.5	2.23762
$467$	79.3624	0.00786393	0.00393196	−	0.999992i	$-0.498748\pi$
$467$	79.3624	0.00786393	0.00393196	+	0.999992i	$0.498748\pi$
$468$	0	0
$469$	0	0
$470$	−4304.91	−0.422491
$471$	0	0
$472$	4510.53	0.439860
$473$	−8188.45	−0.795994
$474$	0	0
$475$	118.284	0.0114258
$476$	0	0
$477$	0	0
$478$	24892.7	2.38194
$479$	1895.96	0.180853	0.0904263	−	0.995903i	$-0.471177\pi$
$479$	1895.96	0.180853	0.0904263	+	0.995903i	$0.471177\pi$
$480$	0	0
$481$	15414.2	1.46118
$482$	−6344.04	−0.599509
$483$	0	0
$484$	−4689.62	−0.440422
$485$	12313.6	1.15285
$486$	0	0
$487$	3897.51	0.362655	0.181327	−	0.983423i	$-0.441961\pi$
$487$	3897.51	0.362655	0.181327	+	0.983423i	$0.441961\pi$
$488$	−6167.74	−0.572132
$489$	0	0
$490$	0	0
$491$	13409.7	1.23253	0.616266	−	0.787538i	$-0.288645\pi$
$491$	13409.7	1.23253	0.616266	+	0.787538i	$0.288645\pi$
$492$	0	0
$493$	−10990.8	−1.00406
$494$	−2054.92	−0.187156
$495$	0	0
$496$	−12735.0	−1.15286
$497$	0	0
$498$	0	0
$499$	4044.94	0.362878	0.181439	−	0.983402i	$-0.441924\pi$
$499$	4044.94	0.362878	0.181439	+	0.983402i	$0.441924\pi$
$500$	−7248.15	−0.648294
$501$	0	0
$502$	−21092.1	−1.87527
$503$	15977.7	1.41632	0.708160	−	0.706052i	$-0.249526\pi$
$503$	15977.7	1.41632	0.708160	+	0.706052i	$0.249526\pi$
$504$	0	0
$505$	1629.83	0.143617
$506$	−16853.2	−1.48067
$507$	0	0
$508$	3003.16	0.262291
$509$	−3949.42	−0.343919	−0.171960	−	0.985104i	$-0.555010\pi$
$509$	−3949.42	−0.343919	−0.171960	+	0.985104i	$0.555010\pi$
$510$	0	0
$511$	0	0
$512$	11142.7	0.961805
$513$	0	0
$514$	2994.89	0.257002
$515$	3884.92	0.332408
$516$	0	0
$517$	−2184.03	−0.185790
$518$	0	0
$519$	0	0
$520$	7896.97	0.665971
$521$	−19894.6	−1.67294	−0.836468	−	0.548016i	$-0.815383\pi$
$521$	−19894.6	−1.67294	−0.836468	+	0.548016i	$0.815383\pi$
$522$	0	0
$523$	−20668.8	−1.72807	−0.864037	−	0.503428i	$-0.832072\pi$
$523$	−20668.8	−1.72807	−0.864037	+	0.503428i	$0.832072\pi$
$524$	−14331.9	−1.19483
$525$	0	0
$526$	−25725.2	−2.13245
$527$	−12186.4	−1.00730
$528$	0	0
$529$	30175.2	2.48008
$530$	−12866.4	−1.05449
$531$	0	0
$532$	0	0
$533$	6307.95	0.512623
$534$	0	0
$535$	−12482.3	−1.00870
$536$	−3284.80	−0.264705
$537$	0	0
$538$	979.179	0.0784673
$539$	0	0
$540$	0	0
$541$	−9452.74	−0.751211	−0.375605	−	0.926780i	$-0.622565\pi$
$541$	−9452.74	−0.751211	−0.375605	+	0.926780i	$0.622565\pi$
$542$	−29905.6	−2.37003
$543$	0	0
$544$	15913.8	1.25422
$545$	3029.36	0.238098
$546$	0	0
$547$	15550.8	1.21555	0.607773	−	0.794111i	$-0.292063\pi$
$547$	15550.8	1.21555	0.607773	+	0.794111i	$0.292063\pi$
$548$	−12686.9	−0.988974
$549$	0	0
$550$	1305.67	0.101225
$551$	1100.38	0.0850780
$552$	0	0
$553$	0	0
$554$	−21909.3	−1.68021
$555$	0	0
$556$	−10285.4	−0.784531
$557$	−15865.0	−1.20686	−0.603432	−	0.797414i	$-0.706201\pi$
$557$	−15865.0	−1.20686	−0.603432	+	0.797414i	$0.706201\pi$
$558$	0	0
$559$	27689.1	2.09504
$560$	0	0
$561$	0	0
$562$	20968.5	1.57385
$563$	−198.200	−0.0148369	−0.00741843	−	0.999972i	$-0.502361\pi$
$563$	−198.200	−0.0148369	−0.00741843	+	0.999972i	$0.502361\pi$
$564$	0	0
$565$	−23630.9	−1.75957
$566$	13211.7	0.981150
$567$	0	0
$568$	−1078.07	−0.0796390
$569$	−18915.2	−1.39361	−0.696807	−	0.717259i	$-0.745396\pi$
$569$	−18915.2	−1.39361	−0.696807	+	0.717259i	$0.745396\pi$
$570$	0	0
$571$	1678.21	0.122996	0.0614981	−	0.998107i	$-0.480412\pi$
$571$	1678.21	0.122996	0.0614981	+	0.998107i	$0.480412\pi$
$572$	−9338.29	−0.682611
$573$	0	0
$574$	0	0
$575$	−3280.37	−0.237914
$576$	0	0
$577$	−9694.82	−0.699481	−0.349741	−	0.936847i	$-0.613730\pi$
$577$	−9694.82	−0.699481	−0.349741	+	0.936847i	$0.613730\pi$
$578$	2136.04	0.153716
$579$	0	0
$580$	9856.49	0.705636
$581$	0	0
$582$	0	0
$583$	−6527.56	−0.463711
$584$	−6041.25	−0.428063
$585$	0	0
$586$	516.702	0.0364245
$587$	19044.8	1.33912	0.669559	−	0.742759i	$-0.266483\pi$
$587$	19044.8	1.33912	0.669559	+	0.742759i	$0.266483\pi$
$588$	0	0
$589$	1220.09	0.0853530
$590$	−22295.2	−1.55572
$591$	0	0
$592$	15894.9	1.10350
$593$	−6298.22	−0.436150	−0.218075	−	0.975932i	$-0.569978\pi$
$593$	−6298.22	−0.436150	−0.218075	+	0.975932i	$0.569978\pi$
$594$	0	0
$595$	0	0
$596$	3505.60	0.240931
$597$	0	0
$598$	56988.9	3.89708
$599$	987.205	0.0673391	0.0336695	−	0.999433i	$-0.489281\pi$
$599$	987.205	0.0673391	0.0336695	+	0.999433i	$0.489281\pi$
$600$	0	0
$601$	20530.0	1.39341	0.696703	−	0.717360i	$-0.254650\pi$
$601$	20530.0	1.39341	0.696703	+	0.717360i	$0.254650\pi$
$602$	0	0
$603$	0	0
$604$	−6744.05	−0.454324
$605$	−9945.08	−0.668306
$606$	0	0
$607$	−2185.99	−0.146172	−0.0730861	−	0.997326i	$-0.523285\pi$
$607$	−2185.99	−0.146172	−0.0730861	+	0.997326i	$0.523285\pi$
$608$	−1593.27	−0.106275
$609$	0	0
$610$	30486.6	2.02355
$611$	7385.28	0.488996
$612$	0	0
$613$	−14150.3	−0.932343	−0.466172	−	0.884694i	$-0.654367\pi$
$613$	−14150.3	−0.932343	−0.466172	+	0.884694i	$0.654367\pi$
$614$	−8548.19	−0.561852
$615$	0	0
$616$	0	0
$617$	1478.16	0.0964478	0.0482239	−	0.998837i	$-0.484644\pi$
$617$	1478.16	0.0964478	0.0482239	+	0.998837i	$0.484644\pi$
$618$	0	0
$619$	6480.22	0.420779	0.210389	−	0.977618i	$-0.432527\pi$
$619$	6480.22	0.420779	0.210389	+	0.977618i	$0.432527\pi$
$620$	10928.7	0.707917
$621$	0	0
$622$	21610.8	1.39311
$623$	0	0
$624$	0	0
$625$	−17363.6	−1.11127
$626$	−1481.00	−0.0945571
$627$	0	0
$628$	−16662.1	−1.05874
$629$	15210.2	0.964180
$630$	0	0
$631$	−1844.94	−0.116396	−0.0581979	−	0.998305i	$-0.518535\pi$
$631$	−1844.94	−0.116396	−0.0581979	+	0.998305i	$0.518535\pi$
$632$	5933.62	0.373460
$633$	0	0
$634$	33404.8	2.09254
$635$	6368.69	0.398006
$636$	0	0
$637$	0	0
$638$	12146.5	0.753736
$639$	0	0
$640$	12852.3	0.793797
$641$	−13064.2	−0.804997	−0.402498	−	0.915421i	$-0.631858\pi$
$641$	−13064.2	−0.804997	−0.402498	+	0.915421i	$0.631858\pi$
$642$	0	0
$643$	10753.3	0.659518	0.329759	−	0.944065i	$-0.393032\pi$
$643$	10753.3	0.659518	0.329759	+	0.944065i	$0.393032\pi$
$644$	0	0
$645$	0	0
$646$	−2027.71	−0.123497
$647$	1836.66	0.111602	0.0558010	−	0.998442i	$-0.482229\pi$
$647$	1836.66	0.111602	0.0558010	+	0.998442i	$0.482229\pi$
$648$	0	0
$649$	−11311.1	−0.684130
$650$	−4415.09	−0.266422
$651$	0	0
$652$	4571.04	0.274564
$653$	−9412.50	−0.564073	−0.282037	−	0.959404i	$-0.591010\pi$
$653$	−9412.50	−0.564073	−0.282037	+	0.959404i	$0.591010\pi$
$654$	0	0
$655$	−30393.2	−1.81307
$656$	6504.63	0.387139
$657$	0	0
$658$	0	0
$659$	1425.12	0.0842408	0.0421204	−	0.999113i	$-0.486589\pi$
$659$	1425.12	0.0842408	0.0421204	+	0.999113i	$0.486589\pi$
$660$	0	0
$661$	−13381.3	−0.787403	−0.393701	−	0.919238i	$-0.628806\pi$
$661$	−13381.3	−0.787403	−0.393701	+	0.919238i	$0.628806\pi$
$662$	−8694.12	−0.510433
$663$	0	0
$664$	−10469.3	−0.611879
$665$	0	0
$666$	0	0
$667$	−30516.9	−1.77154
$668$	8944.36	0.518066
$669$	0	0
$670$	16236.5	0.936226
$671$	15466.9	0.889857
$672$	0	0
$673$	−14806.5	−0.848069	−0.424035	−	0.905646i	$-0.639387\pi$
$673$	−14806.5	−0.848069	−0.424035	+	0.905646i	$0.639387\pi$
$674$	17605.6	1.00615
$675$	0	0
$676$	19278.1	1.09684
$677$	−15282.0	−0.867555	−0.433777	−	0.901020i	$-0.642820\pi$
$677$	−15282.0	−0.867555	−0.433777	+	0.901020i	$0.642820\pi$
$678$	0	0
$679$	0	0
$680$	7792.42	0.439449
$681$	0	0
$682$	13467.8	0.756172
$683$	−7166.26	−0.401478	−0.200739	−	0.979645i	$-0.564334\pi$
$683$	−7166.26	−0.401478	−0.200739	+	0.979645i	$0.564334\pi$
$684$	0	0
$685$	−26904.6	−1.50069
$686$	0	0
$687$	0	0
$688$	28552.5	1.58220
$689$	22072.9	1.22048
$690$	0	0
$691$	11758.4	0.647339	0.323669	−	0.946170i	$-0.395083\pi$
$691$	11758.4	0.647339	0.323669	+	0.946170i	$0.395083\pi$
$692$	7447.87	0.409141
$693$	0	0
$694$	28337.0	1.54994
$695$	−21811.9	−1.19046
$696$	0	0
$697$	6224.44	0.338261
$698$	13908.3	0.754207
$699$	0	0
$700$	0	0
$701$	−3610.56	−0.194535	−0.0972674	−	0.995258i	$-0.531010\pi$
$701$	−3610.56	−0.194535	−0.0972674	+	0.995258i	$0.531010\pi$
$702$	0	0
$703$	−1522.82	−0.0816989
$704$	−3826.31	−0.204843
$705$	0	0
$706$	−29220.9	−1.55771
$707$	0	0
$708$	0	0
$709$	13816.1	0.731842	0.365921	−	0.930646i	$-0.380754\pi$
$709$	13816.1	0.731842	0.365921	+	0.930646i	$0.380754\pi$
$710$	5328.83	0.281672
$711$	0	0
$712$	−5302.14	−0.279082
$713$	−33836.7	−1.77727
$714$	0	0
$715$	−19803.3	−1.03581
$716$	22890.8	1.19479
$717$	0	0
$718$	−33476.7	−1.74003
$719$	−382.174	−0.0198229	−0.00991147	−	0.999951i	$-0.503155\pi$
$719$	−382.174	−0.0198229	−0.00991147	+	0.999951i	$0.503155\pi$
$720$	0	0
$721$	0	0
$722$	−25090.1	−1.29329
$723$	0	0
$724$	6851.43	0.351701
$725$	2364.23	0.121111
$726$	0	0
$727$	−3909.70	−0.199453	−0.0997267	−	0.995015i	$-0.531797\pi$
$727$	−3909.70	−0.199453	−0.0997267	+	0.995015i	$0.531797\pi$
$728$	0	0
$729$	0	0
$730$	29861.4	1.51400
$731$	27322.6	1.38244
$732$	0	0
$733$	−1365.99	−0.0688320	−0.0344160	−	0.999408i	$-0.510957\pi$
$733$	−1365.99	−0.0688320	−0.0344160	+	0.999408i	$0.510957\pi$
$734$	7714.34	0.387931
$735$	0	0
$736$	44186.0	2.21293
$737$	8237.35	0.411705
$738$	0	0
$739$	17222.5	0.857293	0.428647	−	0.903472i	$-0.358991\pi$
$739$	17222.5	0.857293	0.428647	+	0.903472i	$0.358991\pi$
$740$	−13640.4	−0.677610
$741$	0	0
$742$	0	0
$743$	−524.887	−0.0259168	−0.0129584	−	0.999916i	$-0.504125\pi$
$743$	−524.887	−0.0259168	−0.0129584	+	0.999916i	$0.504125\pi$
$744$	0	0
$745$	7434.19	0.365594
$746$	−33770.2	−1.65739
$747$	0	0
$748$	−9214.66	−0.450430
$749$	0	0
$750$	0	0
$751$	13190.1	0.640898	0.320449	−	0.947266i	$-0.396166\pi$
$751$	13190.1	0.640898	0.320449	+	0.947266i	$0.396166\pi$
$752$	7615.55	0.369296
$753$	0	0
$754$	−41073.2	−1.98381
$755$	−14301.8	−0.689401
$756$	0	0
$757$	−24776.1	−1.18957	−0.594784	−	0.803885i	$-0.702763\pi$
$757$	−24776.1	−1.18957	−0.594784	+	0.803885i	$0.702763\pi$
$758$	−2229.35	−0.106825
$759$	0	0
$760$	−780.167	−0.0372363
$761$	−7116.89	−0.339010	−0.169505	−	0.985529i	$-0.554217\pi$
$761$	−7116.89	−0.339010	−0.169505	+	0.985529i	$0.554217\pi$
$762$	0	0
$763$	0	0
$764$	7942.27	0.376101
$765$	0	0
$766$	10591.3	0.499584
$767$	38248.4	1.80061
$768$	0	0
$769$	17.4695	0.000819200 0	0.000409600	−	1.00000i	$-0.499870\pi$
$769$	17.4695	0.000819200 0	0.000409600		1.00000i	$0.499870\pi$
$770$	0	0
$771$	0	0
$772$	−19703.8	−0.918593
$773$	−34872.0	−1.62258	−0.811292	−	0.584640i	$-0.801236\pi$
$773$	−34872.0	−1.62258	−0.811292	+	0.584640i	$0.801236\pi$
$774$	0	0
$775$	2621.42	0.121502
$776$	−9186.31	−0.424960
$777$	0	0
$778$	−10445.3	−0.481341
$779$	−623.183	−0.0286622
$780$	0	0
$781$	2703.50	0.123865
$782$	56234.5	2.57154
$783$	0	0
$784$	0	0
$785$	−35334.6	−1.60656
$786$	0	0
$787$	−2978.09	−0.134889	−0.0674443	−	0.997723i	$-0.521485\pi$
$787$	−2978.09	−0.134889	−0.0674443	+	0.997723i	$0.521485\pi$
$788$	−3862.11	−0.174597
$789$	0	0
$790$	−29329.4	−1.32088
$791$	0	0
$792$	0	0
$793$	−52301.2	−2.34208
$794$	6593.87	0.294720
$795$	0	0
$796$	2865.74	0.127605
$797$	2764.97	0.122886	0.0614431	−	0.998111i	$-0.480430\pi$
$797$	2764.97	0.122886	0.0614431	+	0.998111i	$0.480430\pi$
$798$	0	0
$799$	7287.50	0.322670
$800$	−3423.21	−0.151286
$801$	0	0
$802$	24762.2	1.09026
$803$	15149.7	0.665782
$804$	0	0
$805$	0	0
$806$	−45541.3	−1.99023
$807$	0	0
$808$	−1215.90	−0.0529398
$809$	18307.8	0.795633	0.397817	−	0.917465i	$-0.369768\pi$
$809$	18307.8	0.795633	0.397817	+	0.917465i	$0.369768\pi$
$810$	0	0
$811$	21748.4	0.941664	0.470832	−	0.882223i	$-0.343954\pi$
$811$	21748.4	0.941664	0.470832	+	0.882223i	$0.343954\pi$
$812$	0	0
$813$	0	0
$814$	−16809.5	−0.723800
$815$	9693.63	0.416630
$816$	0	0
$817$	−2735.50	−0.117140
$818$	53407.3	2.28281
$819$	0	0
$820$	−5582.05	−0.237724
$821$	−10998.9	−0.467557	−0.233778	−	0.972290i	$-0.575109\pi$
$821$	−10998.9	−0.467557	−0.233778	+	0.972290i	$0.575109\pi$
$822$	0	0
$823$	7773.89	0.329260	0.164630	−	0.986355i	$-0.447357\pi$
$823$	7773.89	0.329260	0.164630	+	0.986355i	$0.447357\pi$
$824$	−2898.27	−0.122532
$825$	0	0
$826$	0	0
$827$	29154.8	1.22589	0.612946	−	0.790125i	$-0.289984\pi$
$827$	29154.8	1.22589	0.612946	+	0.790125i	$0.289984\pi$
$828$	0	0
$829$	−22928.6	−0.960607	−0.480303	−	0.877102i	$-0.659473\pi$
$829$	−22928.6	−0.960607	−0.480303	+	0.877102i	$0.659473\pi$
$830$	51748.9	2.16413
$831$	0	0
$832$	12938.6	0.539142
$833$	0	0
$834$	0	0
$835$	18968.0	0.786124
$836$	922.560	0.0381667
$837$	0	0
$838$	16644.2	0.686114
$839$	−37893.1	−1.55925	−0.779627	−	0.626244i	$-0.784591\pi$
$839$	−37893.1	−1.55925	−0.779627	+	0.626244i	$0.784591\pi$
$840$	0	0
$841$	−2394.81	−0.0981922
$842$	32023.2	1.31068
$843$	0	0
$844$	−25120.6	−1.02451
$845$	40882.2	1.66437
$846$	0	0
$847$	0	0
$848$	22761.1	0.921720
$849$	0	0
$850$	−4356.64	−0.175802
$851$	42232.4	1.70118
$852$	0	0
$853$	−11927.9	−0.478783	−0.239392	−	0.970923i	$-0.576948\pi$
$853$	−11927.9	−0.478783	−0.239392	+	0.970923i	$0.576948\pi$
$854$	0	0
$855$	0	0
$856$	9312.19	0.371827
$857$	−34449.4	−1.37312	−0.686562	−	0.727071i	$-0.740881\pi$
$857$	−34449.4	−1.37312	−0.686562	+	0.727071i	$0.740881\pi$
$858$	0	0
$859$	−19600.3	−0.778528	−0.389264	−	0.921126i	$-0.627271\pi$
$859$	−19600.3	−0.778528	−0.389264	+	0.921126i	$0.627271\pi$
$860$	−24502.7	−0.971554
$861$	0	0
$862$	−39717.8	−1.56936
$863$	14606.4	0.576140	0.288070	−	0.957609i	$-0.406986\pi$
$863$	14606.4	0.576140	0.288070	+	0.957609i	$0.406986\pi$
$864$	0	0
$865$	15794.4	0.620839
$866$	−6285.55	−0.246642
$867$	0	0
$868$	0	0
$869$	−14879.8	−0.580856
$870$	0	0
$871$	−27854.5	−1.08360
$872$	−2260.00	−0.0877674
$873$	0	0
$874$	−5630.12	−0.217897
$875$	0	0
$876$	0	0
$877$	1509.83	0.0581337	0.0290669	−	0.999577i	$-0.490746\pi$
$877$	1509.83	0.0581337	0.0290669	+	0.999577i	$0.490746\pi$
$878$	−16656.1	−0.640225
$879$	0	0
$880$	−20420.8	−0.782256
$881$	−38219.1	−1.46156	−0.730779	−	0.682614i	$-0.760843\pi$
$881$	−38219.1	−1.46156	−0.730779	+	0.682614i	$0.760843\pi$
$882$	0	0
$883$	40698.0	1.55107	0.775535	−	0.631304i	$-0.217480\pi$
$883$	40698.0	1.55107	0.775535	+	0.631304i	$0.217480\pi$
$884$	31159.3	1.18552
$885$	0	0
$886$	48320.7	1.83224
$887$	−10228.5	−0.387193	−0.193597	−	0.981081i	$-0.562015\pi$
$887$	−10228.5	−0.387193	−0.193597	+	0.981081i	$0.562015\pi$
$888$	0	0
$889$	0	0
$890$	26208.0	0.987074
$891$	0	0
$892$	−4366.46	−0.163901
$893$	−729.615	−0.0273411
$894$	0	0
$895$	48543.5	1.81300
$896$	0	0
$897$	0	0
$898$	5269.91	0.195834
$899$	24386.8	0.904723
$900$	0	0
$901$	21780.6	0.805347
$902$	−6878.94	−0.253929
$903$	0	0
$904$	17629.4	0.648611
$905$	14529.6	0.533678
$906$	0	0
$907$	−36302.2	−1.32899	−0.664496	−	0.747292i	$-0.731354\pi$
$907$	−36302.2	−1.32899	−0.664496	+	0.747292i	$0.731354\pi$
$908$	4107.49	0.150123
$909$	0	0
$910$	0	0
$911$	23768.0	0.864401	0.432201	−	0.901777i	$-0.357737\pi$
$911$	23768.0	0.864401	0.432201	+	0.901777i	$0.357737\pi$
$912$	0	0
$913$	26254.0	0.951678
$914$	−60248.7	−2.18036
$915$	0	0
$916$	26726.5	0.964048
$917$	0	0
$918$	0	0
$919$	−1364.96	−0.0489944	−0.0244972	−	0.999700i	$-0.507798\pi$
$919$	−1364.96	−0.0489944	−0.0244972	+	0.999700i	$0.507798\pi$
$920$	21636.3	0.775357
$921$	0	0
$922$	49938.4	1.78377
$923$	−9141.86	−0.326011
$924$	0	0
$925$	−3271.86	−0.116301
$926$	6827.50	0.242296
$927$	0	0
$928$	−31845.8	−1.12650
$929$	−18850.3	−0.665725	−0.332862	−	0.942975i	$-0.608014\pi$
$929$	−18850.3	−0.665725	−0.332862	+	0.942975i	$0.608014\pi$
$930$	0	0
$931$	0	0
$932$	34172.2	1.20102
$933$	0	0
$934$	292.655	0.0102526
$935$	−19541.2	−0.683492
$936$	0	0
$937$	31404.8	1.09493	0.547465	−	0.836828i	$-0.315593\pi$
$937$	31404.8	1.09493	0.547465	+	0.836828i	$0.315593\pi$
$938$	0	0
$939$	0	0
$940$	−6535.40	−0.226767
$941$	31589.8	1.09437	0.547184	−	0.837013i	$-0.315700\pi$
$941$	31589.8	1.09437	0.547184	+	0.837013i	$0.315700\pi$
$942$	0	0
$943$	17282.7	0.596821
$944$	39441.0	1.35985
$945$	0	0
$946$	−30195.5	−1.03778
$947$	12062.2	0.413907	0.206954	−	0.978351i	$-0.433645\pi$
$947$	12062.2	0.413907	0.206954	+	0.978351i	$0.433645\pi$
$948$	0	0
$949$	−51228.7	−1.75232
$950$	436.181	0.0148964
$951$	0	0
$952$	0	0
$953$	2655.82	0.0902733	0.0451366	−	0.998981i	$-0.485628\pi$
$953$	2655.82	0.0902733	0.0451366	+	0.998981i	$0.485628\pi$
$954$	0	0
$955$	16842.9	0.570704
$956$	37790.3	1.27848
$957$	0	0
$958$	6991.48	0.235787
$959$	0	0
$960$	0	0
$961$	−2751.27	−0.0923524
$962$	56841.2	1.90502
$963$	0	0
$964$	−9631.06	−0.321779
$965$	−41785.0	−1.39389
$966$	0	0
$967$	−57274.5	−1.90468	−0.952339	−	0.305041i	$-0.901330\pi$
$967$	−57274.5	−1.90468	−0.952339	+	0.305041i	$0.901330\pi$
$968$	7419.35	0.246350
$969$	0	0
$970$	45407.2	1.50303
$971$	−25867.0	−0.854904	−0.427452	−	0.904038i	$-0.640589\pi$
$971$	−25867.0	−0.854904	−0.427452	+	0.904038i	$0.640589\pi$
$972$	0	0
$973$	0	0
$974$	14372.3	0.472813
$975$	0	0
$976$	−53932.0	−1.76877
$977$	−14650.4	−0.479741	−0.239870	−	0.970805i	$-0.577105\pi$
$977$	−14650.4	−0.479741	−0.239870	+	0.970805i	$0.577105\pi$
$978$	0	0
$979$	13296.3	0.434066
$980$	0	0
$981$	0	0
$982$	49449.4	1.60692
$983$	4533.90	0.147110	0.0735548	−	0.997291i	$-0.476566\pi$
$983$	4533.90	0.147110	0.0735548	+	0.997291i	$0.476566\pi$
$984$	0	0
$985$	−8190.23	−0.264937
$986$	−40529.4	−1.30905
$987$	0	0
$988$	−3119.62	−0.100454
$989$	75863.5	2.43915
$990$	0	0
$991$	−4375.14	−0.140243	−0.0701216	−	0.997538i	$-0.522339\pi$
$991$	−4375.14	−0.140243	−0.0701216	+	0.997538i	$0.522339\pi$
$992$	−35310.1	−1.13014
$993$	0	0
$994$	0	0
$995$	6077.25	0.193630
$996$	0	0
$997$	−10801.8	−0.343124	−0.171562	−	0.985173i	$-0.554881\pi$
$997$	−10801.8	−0.343124	−0.171562	+	0.985173i	$0.554881\pi$
$998$	14916.0	0.473104
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.be.1.5		6
3.2	odd	2	inner	1323.4.a.be.1.2		6
7.3	odd	6		189.4.e.e.163.2	yes	12
7.5	odd	6		189.4.e.e.109.2	✓	12
7.6	odd	2		1323.4.a.bd.1.5		6
21.5	even	6		189.4.e.e.109.5	yes	12
21.17	even	6		189.4.e.e.163.5	yes	12
21.20	even	2		1323.4.a.bd.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.e.109.2	✓	12	7.5	odd	6
189.4.e.e.109.5	yes	12	21.5	even	6
189.4.e.e.163.2	yes	12	7.3	odd	6
189.4.e.e.163.5	yes	12	21.17	even	6
1323.4.a.bd.1.2		6	21.20	even	2
1323.4.a.bd.1.5		6	7.6	odd	2
1323.4.a.be.1.2		6	3.2	odd	2	inner
1323.4.a.be.1.5		6	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)

\(f(q)\)	\(=\)	\(q+3.68757 q^{2} +5.59820 q^{4} +11.8719 q^{5} -8.85681 q^{8} +O(q^{10})\)	\(q+3.68757 q^{2} +5.59820 q^{4} +11.8719 q^{5} -8.85681 q^{8} +43.7785 q^{10} +22.2104 q^{11} -75.1040 q^{13} -77.4457 q^{16} -74.1097 q^{17} +7.41977 q^{19} +66.4612 q^{20} +81.9023 q^{22} -205.772 q^{23} +15.9417 q^{25} -276.952 q^{26} +148.304 q^{29} +164.438 q^{31} -214.732 q^{32} -273.285 q^{34} -205.239 q^{37} +27.3610 q^{38} -105.147 q^{40} -83.9896 q^{41} -368.677 q^{43} +124.338 q^{44} -758.800 q^{46} -98.3339 q^{47} +58.7864 q^{50} -420.448 q^{52} -293.897 q^{53} +263.679 q^{55} +546.883 q^{58} -509.272 q^{59} +696.384 q^{61} +606.376 q^{62} -172.276 q^{64} -891.627 q^{65} +370.879 q^{67} -414.881 q^{68} +121.723 q^{71} +682.103 q^{73} -756.832 q^{74} +41.5374 q^{76} -669.950 q^{79} -919.427 q^{80} -309.718 q^{82} +1182.06 q^{83} -879.823 q^{85} -1359.52 q^{86} -196.713 q^{88} +598.651 q^{89} -1151.95 q^{92} -362.614 q^{94} +88.0867 q^{95} +1037.20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 32 q^{4}+O(q^{10}) \) 6 * q + 32 * q^4	\( 6 q + 32 q^{4} + 20 q^{10} - 52 q^{13} - 148 q^{16} + 62 q^{19} - 356 q^{22} - 46 q^{25} + 82 q^{31} - 420 q^{34} - 1132 q^{37} + 444 q^{40} - 1566 q^{43} - 888 q^{46} + 72 q^{52} - 224 q^{55} + 4 q^{58} - 886 q^{61} - 924 q^{64} - 2084 q^{67} + 2398 q^{73} - 3204 q^{76} + 984 q^{79} + 3892 q^{82} - 3600 q^{85} - 5796 q^{88} - 2772 q^{94} + 682 q^{97}+O(q^{100}) \) 6 * q + 32 * q^4 + 20 * q^10 - 52 * q^13 - 148 * q^16 + 62 * q^19 - 356 * q^22 - 46 * q^25 + 82 * q^31 - 420 * q^34 - 1132 * q^37 + 444 * q^40 - 1566 * q^43 - 888 * q^46 + 72 * q^52 - 224 * q^55 + 4 * q^58 - 886 * q^61 - 924 * q^64 - 2084 * q^67 + 2398 * q^73 - 3204 * q^76 + 984 * q^79 + 3892 * q^82 - 3600 * q^85 - 5796 * q^88 - 2772 * q^94 + 682 * q^97

Introduction

L-functions

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