LMFDB - Embedded newform 3969.2.a.be.1.2

Newspace parameters

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.59351616.1
Defining polynomial:	$x^{6} - 12 x^{4} + 21 x^{2} - 9$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 441)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16032$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

$f(q)$	$=$	$q-1.69963 q^{2} +0.888736 q^{4} +0.949271 q^{5} +1.88874 q^{8} +O(q^{10})$	$q-1.69963 q^{2} +0.888736 q^{4} +0.949271 q^{5} +1.88874 q^{8} -1.61341 q^{10} -0.588364 q^{11} +5.01974 q^{13} -4.98762 q^{16} +7.58242 q^{17} +4.46122 q^{19} +0.843651 q^{20} +1.00000 q^{22} +2.47710 q^{23} -4.09888 q^{25} -8.53169 q^{26} +5.47710 q^{29} +6.07463 q^{31} +4.69963 q^{32} -12.8873 q^{34} -6.98762 q^{37} -7.58242 q^{38} +1.79292 q^{40} +1.05489 q^{41} +6.98762 q^{43} -0.522900 q^{44} -4.21015 q^{46} +7.47680 q^{47} +6.96658 q^{50} +4.46122 q^{52} +6.92216 q^{53} -0.558517 q^{55} -9.30903 q^{58} -10.4302 q^{59} -11.6529 q^{61} -10.3246 q^{62} +1.98762 q^{64} +4.76509 q^{65} -11.8640 q^{67} +6.73877 q^{68} +4.30037 q^{71} -4.46122 q^{73} +11.8764 q^{74} +3.96485 q^{76} -1.33379 q^{79} -4.73460 q^{80} -1.79292 q^{82} +5.68387 q^{83} +7.19777 q^{85} -11.8764 q^{86} -1.11126 q^{88} -0.843651 q^{89} +2.20149 q^{92} -12.7078 q^{94} +4.23491 q^{95} +3.40633 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6q + 2q^{2} + 6q^{4} + 12q^{8} + O(q^{10}) $	$ 6q + 2q^{2} + 6q^{4} + 12q^{8} + 8q^{11} + 6q^{16} + 6q^{22} + 4q^{23} + 12q^{25} + 22q^{29} + 16q^{32} - 6q^{37} + 6q^{43} - 14q^{44} + 12q^{46} + 56q^{50} + 28q^{53} + 18q^{58} - 24q^{64} - 6q^{65} + 38q^{71} + 36q^{74} - 6q^{79} - 30q^{85} - 36q^{86} - 6q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) $

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 $

$2$	−1.69963	−1.20182	−0.600909	−	0.799317i	$-0.705195\pi$
$2$	−1.69963	−1.20182	−0.600909	+	0.799317i	$0.705195\pi$
$3$	0	0
$3$	0	0
$4$	0.888736	0.444368
$5$	0.949271	0.424527	0.212263	−	0.977212i	$-0.431916\pi$
$5$	0.949271	0.424527	0.212263	+	0.977212i	$0.431916\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.88874	0.667769
$9$	0	0
$10$	−1.61341	−0.510204
$11$	−0.588364	−0.177398	−0.0886992	−	0.996058i	$-0.528271\pi$
$11$	−0.588364	−0.177398	−0.0886992	+	0.996058i	$0.528271\pi$
$12$	0	0
$13$	5.01974	1.39222	0.696112	−	0.717933i	$-0.254912\pi$
$13$	5.01974	1.39222	0.696112	+	0.717933i	$0.254912\pi$
$14$	0	0
$15$	0	0
$16$	−4.98762	−1.24691
$17$	7.58242	1.83901	0.919503	−	0.393083i	$-0.128591\pi$
$17$	7.58242	1.83901	0.919503	+	0.393083i	$0.128591\pi$
$18$	0	0
$19$	4.46122	1.02347	0.511737	−	0.859142i	$-0.329002\pi$
$19$	4.46122	1.02347	0.511737	+	0.859142i	$0.329002\pi$
$20$	0.843651	0.188646
$21$	0	0
$22$	1.00000	0.213201
$23$	2.47710	0.516511	0.258256	−	0.966077i	$-0.416852\pi$
$23$	2.47710	0.516511	0.258256	+	0.966077i	$0.416852\pi$
$24$	0	0
$25$	−4.09888	−0.819777
$26$	−8.53169	−1.67320
$27$	0	0
$28$	0	0
$29$	5.47710	1.01707	0.508536	−	0.861041i	$-0.330187\pi$
$29$	5.47710	1.01707	0.508536	+	0.861041i	$0.330187\pi$
$30$	0	0
$31$	6.07463	1.09104	0.545518	−	0.838099i	$-0.316333\pi$
$31$	6.07463	1.09104	0.545518	+	0.838099i	$0.316333\pi$
$32$	4.69963	0.830785
$33$	0	0
$34$	−12.8873	−2.21015
$35$	0	0
$36$	0	0
$37$	−6.98762	−1.14876	−0.574379	−	0.818590i	$-0.694756\pi$
$37$	−6.98762	−1.14876	−0.574379	+	0.818590i	$0.694756\pi$
$38$	−7.58242	−1.23003
$39$	0	0
$40$	1.79292	0.283486
$41$	1.05489	0.164746	0.0823731	−	0.996602i	$-0.473750\pi$
$41$	1.05489	0.164746	0.0823731	+	0.996602i	$0.473750\pi$
$42$	0	0
$43$	6.98762	1.06560	0.532801	−	0.846241i	$-0.321139\pi$
$43$	6.98762	1.06560	0.532801	+	0.846241i	$0.321139\pi$
$44$	−0.522900	−0.0788302
$45$	0	0
$46$	−4.21015	−0.620753
$47$	7.47680	1.09060	0.545301	−	0.838240i	$-0.316415\pi$
$47$	7.47680	1.09060	0.545301	+	0.838240i	$0.316415\pi$
$48$	0	0
$49$	0	0
$50$	6.96658	0.985223
$51$	0	0
$52$	4.46122	0.618660
$53$	6.92216	0.950831	0.475416	−	0.879761i	$-0.342298\pi$
$53$	6.92216	0.950831	0.475416	+	0.879761i	$0.342298\pi$
$54$	0	0
$55$	−0.558517	−0.0753104
$56$	0	0
$57$	0	0
$58$	−9.30903	−1.22234
$59$	−10.4302	−1.35790	−0.678950	−	0.734184i	$-0.737565\pi$
$59$	−10.4302	−1.35790	−0.678950	+	0.734184i	$0.737565\pi$
$60$	0	0
$61$	−11.6529	−1.49200	−0.745999	−	0.665947i	$-0.768028\pi$
$61$	−11.6529	−1.49200	−0.745999	+	0.665947i	$0.768028\pi$
$62$	−10.3246	−1.31123
$63$	0	0
$64$	1.98762	0.248453
$65$	4.76509	0.591037
$66$	0	0
$67$	−11.8640	−1.44942	−0.724708	−	0.689056i	$-0.758025\pi$
$67$	−11.8640	−1.44942	−0.724708	+	0.689056i	$0.758025\pi$
$68$	6.73877	0.817195
$69$	0	0
$70$	0	0
$71$	4.30037	0.510360	0.255180	−	0.966894i	$-0.417865\pi$
$71$	4.30037	0.510360	0.255180	+	0.966894i	$0.417865\pi$
$72$	0	0
$73$	−4.46122	−0.522146	−0.261073	−	0.965319i	$-0.584076\pi$
$73$	−4.46122	−0.522146	−0.261073	+	0.965319i	$0.584076\pi$
$74$	11.8764	1.38060
$75$	0	0
$76$	3.96485	0.454799
$77$	0	0
$78$	0	0
$79$	−1.33379	−0.150063	−0.0750317	−	0.997181i	$-0.523906\pi$
$79$	−1.33379	−0.150063	−0.0750317	+	0.997181i	$0.523906\pi$
$80$	−4.73460	−0.529345
$81$	0	0
$82$	−1.79292	−0.197995
$83$	5.68387	0.623886	0.311943	−	0.950101i	$-0.399020\pi$
$83$	5.68387	0.623886	0.311943	+	0.950101i	$0.399020\pi$
$84$	0	0
$85$	7.19777	0.780708
$86$	−11.8764	−1.28066
$87$	0	0
$88$	−1.11126	−0.118461
$89$	−0.843651	−0.0894269	−0.0447134	−	0.999000i	$-0.514237\pi$
$89$	−0.843651	−0.0894269	−0.0447134	+	0.999000i	$0.514237\pi$
$90$	0	0
$91$	0	0
$92$	2.20149	0.229521
$93$	0	0
$94$	−12.7078	−1.31071
$95$	4.23491	0.434492
$96$	0	0
$97$	3.40633	0.345860	0.172930	−	0.984934i	$-0.444676\pi$
$97$	3.40633	0.345860	0.172930	+	0.984934i	$0.444676\pi$
$98$	0	0
$99$	0	0
$100$	−3.64283	−0.364283
$101$	9.58658	0.953900	0.476950	−	0.878930i	$-0.341742\pi$
$101$	9.58658	0.953900	0.476950	+	0.878930i	$0.341742\pi$
$102$	0	0
$103$	11.6529	1.14819	0.574096	−	0.818788i	$-0.305353\pi$
$103$	11.6529	1.14819	0.574096	+	0.818788i	$0.305353\pi$
$104$	9.48096	0.929685
$105$	0	0
$106$	−11.7651	−1.14273
$107$	−3.79851	−0.367216	−0.183608	−	0.983000i	$-0.558778\pi$
$107$	−3.79851	−0.367216	−0.183608	+	0.983000i	$0.558778\pi$
$108$	0	0
$109$	−12.8640	−1.23215	−0.616073	−	0.787689i	$-0.711277\pi$
$109$	−12.8640	−1.23215	−0.616073	+	0.787689i	$0.711277\pi$
$110$	0.949271	0.0905094
$111$	0	0
$112$	0	0
$113$	9.02104	0.848628	0.424314	−	0.905515i	$-0.360515\pi$
$113$	9.02104	0.848628	0.424314	+	0.905515i	$0.360515\pi$
$114$	0	0
$115$	2.35144	0.219273
$116$	4.86769	0.451954
$117$	0	0
$118$	17.7275	1.63195
$119$	0	0
$120$	0	0
$121$	−10.6538	−0.968530
$122$	19.8056	1.79311
$123$	0	0
$124$	5.39874	0.484821
$125$	−8.63731	−0.772544
$126$	0	0
$127$	6.43268	0.570808	0.285404	−	0.958407i	$-0.407872\pi$
$127$	6.43268	0.570808	0.285404	+	0.958407i	$0.407872\pi$
$128$	−12.7775	−1.12938
$129$	0	0
$130$	−8.09888	−0.710319
$131$	−6.63315	−0.579541	−0.289770	−	0.957096i	$-0.593579\pi$
$131$	−6.63315	−0.579541	−0.289770	+	0.957096i	$0.593579\pi$
$132$	0	0
$133$	0	0
$134$	20.1643	1.74193
$135$	0	0
$136$	14.3212	1.22803
$137$	14.0334	1.19896	0.599478	−	0.800391i	$-0.295375\pi$
$137$	14.0334	1.19896	0.599478	+	0.800391i	$0.295375\pi$
$138$	0	0
$139$	−8.80507	−0.746836	−0.373418	−	0.927663i	$-0.621814\pi$
$139$	−8.80507	−0.746836	−0.373418	+	0.927663i	$0.621814\pi$
$140$	0	0
$141$	0	0
$142$	−7.30903	−0.613360
$143$	−2.95343	−0.246978
$144$	0	0
$145$	5.19925	0.431774
$146$	7.58242	0.627525
$147$	0	0
$148$	−6.21015	−0.510471
$149$	−4.36584	−0.357663	−0.178832	−	0.983880i	$-0.557232\pi$
$149$	−4.36584	−0.357663	−0.178832	+	0.983880i	$0.557232\pi$
$150$	0	0
$151$	−12.6538	−1.02975	−0.514877	−	0.857264i	$-0.672162\pi$
$151$	−12.6538	−1.02975	−0.514877	+	0.857264i	$0.672162\pi$
$152$	8.42607	0.683444
$153$	0	0
$154$	0	0
$155$	5.76647	0.463174
$156$	0	0
$157$	−11.2739	−0.899754	−0.449877	−	0.893091i	$-0.648532\pi$
$157$	−11.2739	−0.899754	−0.449877	+	0.893091i	$0.648532\pi$
$158$	2.26695	0.180349
$159$	0	0
$160$	4.46122	0.352690
$161$	0	0
$162$	0	0
$163$	−1.66621	−0.130507	−0.0652537	−	0.997869i	$-0.520786\pi$
$163$	−1.66621	−0.130507	−0.0652537	+	0.997869i	$0.520786\pi$
$164$	0.937519	0.0732080
$165$	0	0
$166$	−9.66047	−0.749798
$167$	−3.90270	−0.302000	−0.151000	−	0.988534i	$-0.548249\pi$
$167$	−3.90270	−0.302000	−0.151000	+	0.988534i	$0.548249\pi$
$168$	0	0
$169$	12.1978	0.938290
$170$	−12.2335	−0.938269
$171$	0	0
$172$	6.21015	0.473519
$173$	−16.1141	−1.22513	−0.612566	−	0.790419i	$-0.709863\pi$
$173$	−16.1141	−1.22513	−0.612566	+	0.790419i	$0.709863\pi$
$174$	0	0
$175$	0	0
$176$	2.93454	0.221199
$177$	0	0
$178$	1.43389	0.107475
$179$	14.2880	1.06793	0.533967	−	0.845505i	$-0.320701\pi$
$179$	14.2880	1.06793	0.533967	+	0.845505i	$0.320701\pi$
$180$	0	0
$181$	−12.8873	−0.957905	−0.478952	−	0.877841i	$-0.658983\pi$
$181$	−12.8873	−0.957905	−0.478952	+	0.877841i	$0.658983\pi$
$182$	0	0
$183$	0	0
$184$	4.67859	0.344910
$185$	−6.63315	−0.487679
$186$	0	0
$187$	−4.46122	−0.326237
$188$	6.64490	0.484629
$189$	0	0
$190$	−7.19777	−0.522181
$191$	−2.16435	−0.156607	−0.0783034	−	0.996930i	$-0.524950\pi$
$191$	−2.16435	−0.156607	−0.0783034	+	0.996930i	$0.524950\pi$
$192$	0	0
$193$	10.4313	0.750861	0.375431	−	0.926850i	$-0.377495\pi$
$193$	10.4313	0.750861	0.375431	+	0.926850i	$0.377495\pi$
$194$	−5.78949	−0.415661
$195$	0	0
$196$	0	0
$197$	−18.7848	−1.33836	−0.669179	−	0.743101i	$-0.733354\pi$
$197$	−18.7848	−1.33836	−0.669179	+	0.743101i	$0.733354\pi$
$198$	0	0
$199$	−8.42607	−0.597308	−0.298654	−	0.954361i	$-0.596538\pi$
$199$	−8.42607	−0.597308	−0.298654	+	0.954361i	$0.596538\pi$
$200$	−7.74171	−0.547422
$201$	0	0
$202$	−16.2936	−1.14642
$203$	0	0
$204$	0	0
$205$	1.00138	0.0699392
$206$	−19.8056	−1.37992
$207$	0	0
$208$	−25.0365	−1.73597
$209$	−2.62482	−0.181563
$210$	0	0
$211$	11.2225	0.772591	0.386295	−	0.922375i	$-0.373755\pi$
$211$	11.2225	0.772591	0.386295	+	0.922375i	$0.373755\pi$
$212$	6.15197	0.422519
$213$	0	0
$214$	6.45606	0.441327
$215$	6.63315	0.452377
$216$	0	0
$217$	0	0
$218$	21.8640	1.48082
$219$	0	0
$220$	−0.496374	−0.0334655
$221$	38.0617	2.56031
$222$	0	0
$223$	20.7548	1.38985	0.694923	−	0.719084i	$-0.255438\pi$
$223$	20.7548	1.38985	0.694923	+	0.719084i	$0.255438\pi$
$224$	0	0
$225$	0	0
$226$	−15.3324	−1.01990
$227$	10.4302	0.692279	0.346139	−	0.938183i	$-0.387492\pi$
$227$	10.4302	0.692279	0.346139	+	0.938183i	$0.387492\pi$
$228$	0	0
$229$	−15.0592	−0.995141	−0.497570	−	0.867424i	$-0.665774\pi$
$229$	−15.0592	−0.995141	−0.497570	+	0.867424i	$0.665774\pi$
$230$	−3.99657	−0.263526
$231$	0	0
$232$	10.3448	0.679169
$233$	4.38688	0.287394	0.143697	−	0.989622i	$-0.454101\pi$
$233$	4.38688	0.287394	0.143697	+	0.989622i	$0.454101\pi$
$234$	0	0
$235$	7.09751	0.462990
$236$	−9.26972	−0.603407
$237$	0	0
$238$	0	0
$239$	−9.55122	−0.617817	−0.308909	−	0.951092i	$-0.599964\pi$
$239$	−9.55122	−0.617817	−0.308909	+	0.951092i	$0.599964\pi$
$240$	0	0
$241$	−10.5358	−0.678674	−0.339337	−	0.940665i	$-0.610203\pi$
$241$	−10.5358	−0.678674	−0.339337	+	0.940665i	$0.610203\pi$
$242$	18.1075	1.16400
$243$	0	0
$244$	−10.3563	−0.662996
$245$	0	0
$246$	0	0
$247$	22.3942	1.42491
$248$	11.4734	0.728560
$249$	0	0
$250$	14.6802	0.928458
$251$	24.4346	1.54230	0.771148	−	0.636656i	$-0.219683\pi$
$251$	24.4346	1.54230	0.771148	+	0.636656i	$0.219683\pi$
$252$	0	0
$253$	−1.45744	−0.0916282
$254$	−10.9332	−0.686007
$255$	0	0
$256$	17.7417	1.10886
$257$	−4.00832	−0.250032	−0.125016	−	0.992155i	$-0.539898\pi$
$257$	−4.00832	−0.250032	−0.125016	+	0.992155i	$0.539898\pi$
$258$	0	0
$259$	0	0
$260$	4.23491	0.262638
$261$	0	0
$262$	11.2739	0.696503
$263$	17.6872	1.09064	0.545321	−	0.838227i	$-0.316408\pi$
$263$	17.6872	1.09064	0.545321	+	0.838227i	$0.316408\pi$
$264$	0	0
$265$	6.57100	0.403653
$266$	0	0
$267$	0	0
$268$	−10.5439	−0.644074
$269$	14.2273	0.867455	0.433727	−	0.901044i	$-0.357198\pi$
$269$	14.2273	0.867455	0.433727	+	0.901044i	$0.357198\pi$
$270$	0	0
$271$	5.39874	0.327950	0.163975	−	0.986464i	$-0.447568\pi$
$271$	5.39874	0.327950	0.163975	+	0.986464i	$0.447568\pi$
$272$	−37.8182	−2.29307
$273$	0	0
$274$	−23.8516	−1.44093
$275$	2.41164	0.145427
$276$	0	0
$277$	7.66621	0.460618	0.230309	−	0.973118i	$-0.426026\pi$
$277$	7.66621	0.460618	0.230309	+	0.973118i	$0.426026\pi$
$278$	14.9653	0.897562
$279$	0	0
$280$	0	0
$281$	−22.6625	−1.35193	−0.675965	−	0.736933i	$-0.736273\pi$
$281$	−22.6625	−1.35193	−0.675965	+	0.736933i	$0.736273\pi$
$282$	0	0
$283$	31.8492	1.89324	0.946619	−	0.322353i	$-0.104474\pi$
$283$	31.8492	1.89324	0.946619	+	0.322353i	$0.104474\pi$
$284$	3.82189	0.226788
$285$	0	0
$286$	5.01974	0.296823
$287$	0	0
$288$	0	0
$289$	40.4930	2.38194
$290$	−8.83680	−0.518915
$291$	0	0
$292$	−3.96485	−0.232025
$293$	−27.4936	−1.60619	−0.803097	−	0.595849i	$-0.796816\pi$
$293$	−27.4936	−1.60619	−0.803097	+	0.595849i	$0.796816\pi$
$294$	0	0
$295$	−9.90112	−0.576465
$296$	−13.1978	−0.767105
$297$	0	0
$298$	7.42030	0.429846
$299$	12.4344	0.719099
$300$	0	0
$301$	0	0
$302$	21.5068	1.23758
$303$	0	0
$304$	−22.2509	−1.27618
$305$	−11.0617	−0.633394
$306$	0	0
$307$	−14.8176	−0.845683	−0.422841	−	0.906204i	$-0.638967\pi$
$307$	−14.8176	−0.845683	−0.422841	+	0.906204i	$0.638967\pi$
$308$	0	0
$309$	0	0
$310$	−9.80085	−0.556651
$311$	−29.0635	−1.64804	−0.824021	−	0.566559i	$-0.808274\pi$
$311$	−29.0635	−1.64804	−0.824021	+	0.566559i	$0.808274\pi$
$312$	0	0
$313$	24.4780	1.38358	0.691790	−	0.722099i	$-0.256822\pi$
$313$	24.4780	1.38358	0.691790	+	0.722099i	$0.256822\pi$
$314$	19.1614	1.08134
$315$	0	0
$316$	−1.18539	−0.0666834
$317$	−7.38688	−0.414888	−0.207444	−	0.978247i	$-0.566515\pi$
$317$	−7.38688	−0.414888	−0.207444	+	0.978247i	$0.566515\pi$
$318$	0	0
$319$	−3.22253	−0.180427
$320$	1.88679	0.105475
$321$	0	0
$322$	0	0
$323$	33.8268	1.88218
$324$	0	0
$325$	−20.5753	−1.14131
$326$	2.83193	0.156846
$327$	0	0
$328$	1.99241	0.110012
$329$	0	0
$330$	0	0
$331$	20.0617	1.10269	0.551347	−	0.834276i	$-0.314114\pi$
$331$	20.0617	1.10269	0.551347	+	0.834276i	$0.314114\pi$
$332$	5.05146	0.277235
$333$	0	0
$334$	6.63315	0.362950
$335$	−11.2621	−0.615316
$336$	0	0
$337$	6.40654	0.348986	0.174493	−	0.984658i	$-0.444171\pi$
$337$	6.40654	0.348986	0.174493	+	0.984658i	$0.444171\pi$
$338$	−20.7317	−1.12765
$339$	0	0
$340$	6.39692	0.346921
$341$	−3.57409	−0.193548
$342$	0	0
$343$	0	0
$344$	13.1978	0.711576
$345$	0	0
$346$	27.3880	1.47239
$347$	−29.1927	−1.56714	−0.783572	−	0.621300i	$-0.786605\pi$
$347$	−29.1927	−1.56714	−0.783572	+	0.621300i	$0.786605\pi$
$348$	0	0
$349$	4.34385	0.232521	0.116260	−	0.993219i	$-0.462909\pi$
$349$	4.34385	0.232521	0.116260	+	0.993219i	$0.462909\pi$
$350$	0	0
$351$	0	0
$352$	−2.76509	−0.147380
$353$	−25.7007	−1.36791	−0.683955	−	0.729525i	$-0.739741\pi$
$353$	−25.7007	−1.36791	−0.683955	+	0.729525i	$0.739741\pi$
$354$	0	0
$355$	4.08222	0.216662
$356$	−0.749783	−0.0397384
$357$	0	0
$358$	−24.2843	−1.28346
$359$	20.6872	1.09183	0.545916	−	0.837840i	$-0.316182\pi$
$359$	20.6872	1.09183	0.545916	+	0.837840i	$0.316182\pi$
$360$	0	0
$361$	0.902493	0.0474996
$362$	21.9036	1.15123
$363$	0	0
$364$	0	0
$365$	−4.23491	−0.221665
$366$	0	0
$367$	2.84781	0.148655	0.0743273	−	0.997234i	$-0.476319\pi$
$367$	2.84781	0.148655	0.0743273	+	0.997234i	$0.476319\pi$
$368$	−12.3548	−0.644040
$369$	0	0
$370$	11.2739	0.586101
$371$	0	0
$372$	0	0
$373$	21.4327	1.10974	0.554871	−	0.831936i	$-0.312768\pi$
$373$	21.4327	1.10974	0.554871	+	0.831936i	$0.312768\pi$
$374$	7.58242	0.392077
$375$	0	0
$376$	14.1217	0.728271
$377$	27.4936	1.41599
$378$	0	0
$379$	27.0494	1.38943	0.694716	−	0.719284i	$-0.255530\pi$
$379$	27.0494	1.38943	0.694716	+	0.719284i	$0.255530\pi$
$380$	3.76371	0.193074
$381$	0	0
$382$	3.67859	0.188213
$383$	14.4268	0.737175	0.368588	−	0.929593i	$-0.379841\pi$
$383$	14.4268	0.737175	0.368588	+	0.929593i	$0.379841\pi$
$384$	0	0
$385$	0	0
$386$	−17.7293	−0.902399
$387$	0	0
$388$	3.02733	0.153689
$389$	−6.10755	−0.309665	−0.154832	−	0.987941i	$-0.549484\pi$
$389$	−6.10755	−0.309665	−0.154832	+	0.987941i	$0.549484\pi$
$390$	0	0
$391$	18.7824	0.949867
$392$	0	0
$393$	0	0
$394$	31.9271	1.60846
$395$	−1.26613	−0.0637059
$396$	0	0
$397$	12.8873	0.646794	0.323397	−	0.946263i	$-0.395175\pi$
$397$	12.8873	0.646794	0.323397	+	0.946263i	$0.395175\pi$
$398$	14.3212	0.717856
$399$	0	0
$400$	20.4437	1.02218
$401$	8.39060	0.419006	0.209503	−	0.977808i	$-0.432815\pi$
$401$	8.39060	0.419006	0.209503	+	0.977808i	$0.432815\pi$
$402$	0	0
$403$	30.4930	1.51897
$404$	8.51994	0.423883
$405$	0	0
$406$	0	0
$407$	4.11126	0.203788
$408$	0	0
$409$	−6.81266	−0.336864	−0.168432	−	0.985713i	$-0.553870\pi$
$409$	−6.81266	−0.336864	−0.168432	+	0.985713i	$0.553870\pi$
$410$	−1.70197	−0.0840543
$411$	0	0
$412$	10.3563	0.510220
$413$	0	0
$414$	0	0
$415$	5.39554	0.264857
$416$	23.5909	1.15664
$417$	0	0
$418$	4.46122	0.218205
$419$	10.3246	0.504390	0.252195	−	0.967676i	$-0.418848\pi$
$419$	10.3246	0.504390	0.252195	+	0.967676i	$0.418848\pi$
$420$	0	0
$421$	3.13602	0.152840	0.0764202	−	0.997076i	$-0.475651\pi$
$421$	3.13602	0.152840	0.0764202	+	0.997076i	$0.475651\pi$
$422$	−19.0741	−0.928514
$423$	0	0
$424$	13.0741	0.634936
$425$	−31.0795	−1.50757
$426$	0	0
$427$	0	0
$428$	−3.37587	−0.163179
$429$	0	0
$430$	−11.2739	−0.543675
$431$	31.8726	1.53525	0.767625	−	0.640899i	$-0.221438\pi$
$431$	31.8726	1.53525	0.767625	+	0.640899i	$0.221438\pi$
$432$	0	0
$433$	−7.48855	−0.359877	−0.179938	−	0.983678i	$-0.557590\pi$
$433$	−7.48855	−0.359877	−0.179938	+	0.983678i	$0.557590\pi$
$434$	0	0
$435$	0	0
$436$	−11.4327	−0.547526
$437$	11.0509	0.528636
$438$	0	0
$439$	−2.28930	−0.109262	−0.0546311	−	0.998507i	$-0.517398\pi$
$439$	−2.28930	−0.109262	−0.0546311	+	0.998507i	$0.517398\pi$
$440$	−1.05489	−0.0502900
$441$	0	0
$442$	−64.6908	−3.07703
$443$	37.3497	1.77454	0.887270	−	0.461251i	$-0.152599\pi$
$443$	37.3497	1.77454	0.887270	+	0.461251i	$0.152599\pi$
$444$	0	0
$445$	−0.800854	−0.0379641
$446$	−35.2755	−1.67034
$447$	0	0
$448$	0	0
$449$	−6.20286	−0.292731	−0.146366	−	0.989231i	$-0.546758\pi$
$449$	−6.20286	−0.292731	−0.146366	+	0.989231i	$0.546758\pi$
$450$	0	0
$451$	−0.620660	−0.0292257
$452$	8.01732	0.377103
$453$	0	0
$454$	−17.7275	−0.831993
$455$	0	0
$456$	0	0
$457$	20.1716	0.943589	0.471795	−	0.881709i	$-0.343606\pi$
$457$	20.1716	0.943589	0.471795	+	0.881709i	$0.343606\pi$
$458$	25.5951	1.19598
$459$	0	0
$460$	2.08981	0.0974378
$461$	22.5360	1.04961	0.524803	−	0.851224i	$-0.324139\pi$
$461$	22.5360	1.04961	0.524803	+	0.851224i	$0.324139\pi$
$462$	0	0
$463$	−27.6291	−1.28403	−0.642016	−	0.766691i	$-0.721902\pi$
$463$	−27.6291	−1.28403	−0.642016	+	0.766691i	$0.721902\pi$
$464$	−27.3177	−1.26819
$465$	0	0
$466$	−7.45606	−0.345395
$467$	−20.1224	−0.931155	−0.465577	−	0.885007i	$-0.654153\pi$
$467$	−20.1224	−0.931155	−0.465577	+	0.885007i	$0.654153\pi$
$468$	0	0
$469$	0	0
$470$	−12.0631	−0.556430
$471$	0	0
$472$	−19.6999	−0.906764
$473$	−4.11126	−0.189036
$474$	0	0
$475$	−18.2860	−0.839021
$476$	0	0
$477$	0	0
$478$	16.2335	0.742504
$479$	9.58658	0.438022	0.219011	−	0.975722i	$-0.429717\pi$
$479$	9.58658	0.438022	0.219011	+	0.975722i	$0.429717\pi$
$480$	0	0
$481$	−35.0760	−1.59933
$482$	17.9070	0.815643
$483$	0	0
$484$	−9.46844	−0.430384
$485$	3.23353	0.146827
$486$	0	0
$487$	13.0741	0.592445	0.296223	−	0.955119i	$-0.404273\pi$
$487$	13.0741	0.592445	0.296223	+	0.955119i	$0.404273\pi$
$488$	−22.0092	−0.996311
$489$	0	0
$490$	0	0
$491$	15.3411	0.692333	0.346167	−	0.938173i	$-0.387483\pi$
$491$	15.3411	0.692333	0.346167	+	0.938173i	$0.387483\pi$
$492$	0	0
$493$	41.5297	1.87040
$494$	−38.0617	−1.71248
$495$	0	0
$496$	−30.2979	−1.36042
$497$	0	0
$498$	0	0
$499$	4.86535	0.217803	0.108902	−	0.994053i	$-0.465267\pi$
$499$	4.86535	0.217803	0.108902	+	0.994053i	$0.465267\pi$
$500$	−7.67628	−0.343294
$501$	0	0
$502$	−41.5297	−1.85356
$503$	−16.0085	−0.713783	−0.356892	−	0.934146i	$-0.616163\pi$
$503$	−16.0085	−0.713783	−0.356892	+	0.934146i	$0.616163\pi$
$504$	0	0
$505$	9.10026	0.404956
$506$	2.47710	0.110121
$507$	0	0
$508$	5.71695	0.253649
$509$	31.1851	1.38225	0.691127	−	0.722733i	$-0.257115\pi$
$509$	31.1851	1.38225	0.691127	+	0.722733i	$0.257115\pi$
$510$	0	0
$511$	0	0
$512$	−4.59937	−0.203265
$513$	0	0
$514$	6.81266	0.300494
$515$	11.0617	0.487439
$516$	0	0
$517$	−4.39908	−0.193471
$518$	0	0
$519$	0	0
$520$	9.00000	0.394676
$521$	−20.9661	−0.918541	−0.459270	−	0.888297i	$-0.651889\pi$
$521$	−20.9661	−0.918541	−0.459270	+	0.888297i	$0.651889\pi$
$522$	0	0
$523$	−43.5642	−1.90493	−0.952465	−	0.304647i	$-0.901462\pi$
$523$	−43.5642	−1.90493	−0.952465	+	0.304647i	$0.901462\pi$
$524$	−5.89511	−0.257529
$525$	0	0
$526$	−30.0617	−1.31075
$527$	46.0604	2.00642
$528$	0	0
$529$	−16.8640	−0.733216
$530$	−11.1683	−0.485118
$531$	0	0
$532$	0	0
$533$	5.29528	0.229364
$534$	0	0
$535$	−3.60582	−0.155893
$536$	−22.4079	−0.967875
$537$	0	0
$538$	−24.1811	−1.04252
$539$	0	0
$540$	0	0
$541$	9.86535	0.424145	0.212072	−	0.977254i	$-0.431979\pi$
$541$	9.86535	0.424145	0.212072	+	0.977254i	$0.431979\pi$
$542$	−9.17585	−0.394137
$543$	0	0
$544$	35.6345	1.52782
$545$	−12.2114	−0.523079
$546$	0	0
$547$	0.568701	0.0243159	0.0121579	−	0.999926i	$-0.496130\pi$
$547$	0.568701	0.0243159	0.0121579	+	0.999926i	$0.496130\pi$
$548$	12.4720	0.532778
$549$	0	0
$550$	−4.09888	−0.174777
$551$	24.4346	1.04095
$552$	0	0
$553$	0	0
$554$	−13.0297	−0.553579
$555$	0	0
$556$	−7.82538	−0.331870
$557$	−2.58699	−0.109614	−0.0548071	−	0.998497i	$-0.517454\pi$
$557$	−2.58699	−0.109614	−0.0548071	+	0.998497i	$0.517454\pi$
$558$	0	0
$559$	35.0760	1.48356
$560$	0	0
$561$	0	0
$562$	38.5178	1.62478
$563$	33.2831	1.40272	0.701358	−	0.712809i	$-0.252578\pi$
$563$	33.2831	1.40272	0.701358	+	0.712809i	$0.252578\pi$
$564$	0	0
$565$	8.56341	0.360265
$566$	−54.1318	−2.27533
$567$	0	0
$568$	8.12227	0.340803
$569$	−5.35346	−0.224429	−0.112214	−	0.993684i	$-0.535794\pi$
$569$	−5.35346	−0.224429	−0.112214	+	0.993684i	$0.535794\pi$
$570$	0	0
$571$	4.90112	0.205105	0.102553	−	0.994728i	$-0.467299\pi$
$571$	4.90112	0.205105	0.102553	+	0.994728i	$0.467299\pi$
$572$	−2.62482	−0.109749
$573$	0	0
$574$	0	0
$575$	−10.1533	−0.423424
$576$	0	0
$577$	36.0757	1.50185	0.750925	−	0.660387i	$-0.229608\pi$
$577$	36.0757	1.50185	0.750925	+	0.660387i	$0.229608\pi$
$578$	−68.8231	−2.86266
$579$	0	0
$580$	4.62076	0.191867
$581$	0	0
$582$	0	0
$583$	−4.07275	−0.168676
$584$	−8.42607	−0.348673
$585$	0	0
$586$	46.7289	1.93035
$587$	−1.05489	−0.0435400	−0.0217700	−	0.999763i	$-0.506930\pi$
$587$	−1.05489	−0.0435400	−0.0217700	+	0.999763i	$0.506930\pi$
$588$	0	0
$589$	27.1003	1.11665
$590$	16.8282	0.692807
$591$	0	0
$592$	34.8516	1.43239
$593$	15.0710	0.618890	0.309445	−	0.950917i	$-0.399857\pi$
$593$	15.0710	0.618890	0.309445	+	0.950917i	$0.399857\pi$
$594$	0	0
$595$	0	0
$596$	−3.88007	−0.158934
$597$	0	0
$598$	−21.1338	−0.864227
$599$	42.0566	1.71839	0.859194	−	0.511650i	$-0.170965\pi$
$599$	42.0566	1.71839	0.859194	+	0.511650i	$0.170965\pi$
$600$	0	0
$601$	−18.8998	−0.770938	−0.385469	−	0.922721i	$-0.625960\pi$
$601$	−18.8998	−0.770938	−0.385469	+	0.922721i	$0.625960\pi$
$602$	0	0
$603$	0	0
$604$	−11.2459	−0.457590
$605$	−10.1134	−0.411167
$606$	0	0
$607$	29.4425	1.19504	0.597518	−	0.801856i	$-0.296154\pi$
$607$	29.4425	1.19504	0.597518	+	0.801856i	$0.296154\pi$
$608$	20.9661	0.850287
$609$	0	0
$610$	18.8009	0.761224
$611$	37.5316	1.51836
$612$	0	0
$613$	−11.6676	−0.471249	−0.235625	−	0.971844i	$-0.575714\pi$
$613$	−11.6676	−0.471249	−0.235625	+	0.971844i	$0.575714\pi$
$614$	25.1843	1.01636
$615$	0	0
$616$	0	0
$617$	−32.8109	−1.32092	−0.660458	−	0.750863i	$-0.729638\pi$
$617$	−32.8109	−1.32092	−0.660458	+	0.750863i	$0.729638\pi$
$618$	0	0
$619$	24.1612	0.971119	0.485560	−	0.874204i	$-0.338616\pi$
$619$	24.1612	0.971119	0.485560	+	0.874204i	$0.338616\pi$
$620$	5.12487	0.205820
$621$	0	0
$622$	49.3972	1.98065
$623$	0	0
$624$	0	0
$625$	12.2953	0.491811
$626$	−41.6035	−1.66281
$627$	0	0
$628$	−10.0195	−0.399822
$629$	−52.9830	−2.11257
$630$	0	0
$631$	−11.1003	−0.441894	−0.220947	−	0.975286i	$-0.570915\pi$
$631$	−11.1003	−0.441894	−0.220947	+	0.975286i	$0.570915\pi$
$632$	−2.51918	−0.100208
$633$	0	0
$634$	12.5549	0.498620
$635$	6.10635	0.242323
$636$	0	0
$637$	0	0
$638$	5.47710	0.216840
$639$	0	0
$640$	−12.1293	−0.479452
$641$	7.30037	0.288347	0.144174	−	0.989552i	$-0.453948\pi$
$641$	7.30037	0.288347	0.144174	+	0.989552i	$0.453948\pi$
$642$	0	0
$643$	−21.2512	−0.838066	−0.419033	−	0.907971i	$-0.637631\pi$
$643$	−21.2512	−0.838066	−0.419033	+	0.907971i	$0.637631\pi$
$644$	0	0
$645$	0	0
$646$	−57.4930	−2.26203
$647$	−16.9460	−0.666216	−0.333108	−	0.942889i	$-0.608097\pi$
$647$	−16.9460	−0.666216	−0.333108	+	0.942889i	$0.608097\pi$
$648$	0	0
$649$	6.13677	0.240889
$650$	34.9704	1.37165
$651$	0	0
$652$	−1.48082	−0.0579933
$653$	−3.73305	−0.146085	−0.0730427	−	0.997329i	$-0.523271\pi$
$653$	−3.73305	−0.146085	−0.0730427	+	0.997329i	$0.523271\pi$
$654$	0	0
$655$	−6.29665	−0.246031
$656$	−5.26140	−0.205423
$657$	0	0
$658$	0	0
$659$	−23.5984	−0.919263	−0.459632	−	0.888110i	$-0.652019\pi$
$659$	−23.5984	−0.919263	−0.459632	+	0.888110i	$0.652019\pi$
$660$	0	0
$661$	−34.5175	−1.34258	−0.671288	−	0.741197i	$-0.734258\pi$
$661$	−34.5175	−1.34258	−0.671288	+	0.741197i	$0.734258\pi$
$662$	−34.0975	−1.32524
$663$	0	0
$664$	10.7353	0.416612
$665$	0	0
$666$	0	0
$667$	13.5673	0.525329
$668$	−3.46847	−0.134199
$669$	0	0
$670$	19.1414	0.739498
$671$	6.85614	0.264678
$672$	0	0
$673$	−24.4574	−0.942765	−0.471382	−	0.881929i	$-0.656245\pi$
$673$	−24.4574	−0.942765	−0.471382	+	0.881929i	$0.656245\pi$
$674$	−10.8887	−0.419418
$675$	0	0
$676$	10.8406	0.416946
$677$	8.32045	0.319781	0.159890	−	0.987135i	$-0.448886\pi$
$677$	8.32045	0.319781	0.159890	+	0.987135i	$0.448886\pi$
$678$	0	0
$679$	0	0
$680$	13.5947	0.521332
$681$	0	0
$682$	6.07463	0.232610
$683$	−42.4624	−1.62478	−0.812389	−	0.583116i	$-0.801833\pi$
$683$	−42.4624	−1.62478	−0.812389	+	0.583116i	$0.801833\pi$
$684$	0	0
$685$	13.3215	0.508989
$686$	0	0
$687$	0	0
$688$	−34.8516	−1.32870
$689$	34.7474	1.32377
$690$	0	0
$691$	−35.3929	−1.34641	−0.673204	−	0.739456i	$-0.735083\pi$
$691$	−35.3929	−1.34641	−0.673204	+	0.739456i	$0.735083\pi$
$692$	−14.3212	−0.544410
$693$	0	0
$694$	49.6167	1.88342
$695$	−8.35840	−0.317052
$696$	0	0
$697$	7.99862	0.302969
$698$	−7.38293	−0.279448
$699$	0	0
$700$	0	0
$701$	−7.00372	−0.264527	−0.132263	−	0.991215i	$-0.542224\pi$
$701$	−7.00372	−0.264527	−0.132263	+	0.991215i	$0.542224\pi$
$702$	0	0
$703$	−31.1733	−1.17572
$704$	−1.16944	−0.0440751
$705$	0	0
$706$	43.6816	1.64398
$707$	0	0
$708$	0	0
$709$	−2.22253	−0.0834688	−0.0417344	−	0.999129i	$-0.513288\pi$
$709$	−2.22253	−0.0834688	−0.0417344	+	0.999129i	$0.513288\pi$
$710$	−6.93825	−0.260388
$711$	0	0
$712$	−1.59343	−0.0597165
$713$	15.0475	0.563532
$714$	0	0
$715$	−2.80361	−0.104849
$716$	12.6983	0.474556
$717$	0	0
$718$	−35.1606	−1.31218
$719$	−26.0175	−0.970291	−0.485145	−	0.874434i	$-0.661233\pi$
$719$	−26.0175	−0.970291	−0.485145	+	0.874434i	$0.661233\pi$
$720$	0	0
$721$	0	0
$722$	−1.53390	−0.0570859
$723$	0	0
$724$	−11.4534	−0.425662
$725$	−22.4500	−0.833772
$726$	0	0
$727$	1.37175	0.0508754	0.0254377	−	0.999676i	$-0.491902\pi$
$727$	1.37175	0.0508754	0.0254377	+	0.999676i	$0.491902\pi$
$728$	0	0
$729$	0	0
$730$	7.19777	0.266401
$731$	52.9830	1.95965
$732$	0	0
$733$	0.800174	0.0295551	0.0147776	−	0.999891i	$-0.495296\pi$
$733$	0.800174	0.0295551	0.0147776	+	0.999891i	$0.495296\pi$
$734$	−4.84022	−0.178656
$735$	0	0
$736$	11.6414	0.429109
$737$	6.98034	0.257124
$738$	0	0
$739$	5.37093	0.197573	0.0987865	−	0.995109i	$-0.468504\pi$
$739$	5.37093	0.197573	0.0987865	+	0.995109i	$0.468504\pi$
$740$	−5.89511	−0.216709
$741$	0	0
$742$	0	0
$743$	−13.2632	−0.486581	−0.243290	−	0.969953i	$-0.578227\pi$
$743$	−13.2632	−0.486581	−0.243290	+	0.969953i	$0.578227\pi$
$744$	0	0
$745$	−4.14436	−0.151838
$746$	−36.4276	−1.33371
$747$	0	0
$748$	−3.96485	−0.144969
$749$	0	0
$750$	0	0
$751$	5.55632	0.202753	0.101377	−	0.994848i	$-0.467675\pi$
$751$	5.55632	0.202753	0.101377	+	0.994848i	$0.467675\pi$
$752$	−37.2914	−1.35988
$753$	0	0
$754$	−46.7289	−1.70177
$755$	−12.0119	−0.437158
$756$	0	0
$757$	−13.3942	−0.486819	−0.243410	−	0.969924i	$-0.578266\pi$
$757$	−13.3942	−0.486819	−0.243410	+	0.969924i	$0.578266\pi$
$758$	−45.9739	−1.66985
$759$	0	0
$760$	7.99862	0.290141
$761$	12.8438	0.465588	0.232794	−	0.972526i	$-0.425213\pi$
$761$	12.8438	0.465588	0.232794	+	0.972526i	$0.425213\pi$
$762$	0	0
$763$	0	0
$764$	−1.92353	−0.0695910
$765$	0	0
$766$	−24.5202	−0.885951
$767$	−52.3570	−1.89050
$768$	0	0
$769$	−2.96518	−0.106927	−0.0534636	−	0.998570i	$-0.517026\pi$
$769$	−2.96518	−0.106927	−0.0534636	+	0.998570i	$0.517026\pi$
$770$	0	0
$771$	0	0
$772$	9.27067	0.333659
$773$	19.2788	0.693409	0.346705	−	0.937974i	$-0.387301\pi$
$773$	19.2788	0.693409	0.346705	+	0.937974i	$0.387301\pi$
$774$	0	0
$775$	−24.8992	−0.894406
$776$	6.43366	0.230955
$777$	0	0
$778$	10.3806	0.372161
$779$	4.70610	0.168614
$780$	0	0
$781$	−2.53018	−0.0905371
$782$	−31.9231	−1.14157
$783$	0	0
$784$	0	0
$785$	−10.7020	−0.381970
$786$	0	0
$787$	−13.6453	−0.486402	−0.243201	−	0.969976i	$-0.578197\pi$
$787$	−13.6453	−0.486402	−0.243201	+	0.969976i	$0.578197\pi$
$788$	−16.6947	−0.594724
$789$	0	0
$790$	2.15195	0.0765630
$791$	0	0
$792$	0	0
$793$	−58.4944	−2.07720
$794$	−21.9036	−0.777330
$795$	0	0
$796$	−7.48855	−0.265425
$797$	22.9585	0.813231	0.406616	−	0.913599i	$-0.366709\pi$
$797$	22.9585	0.813231	0.406616	+	0.913599i	$0.366709\pi$
$798$	0	0
$799$	56.6922	2.00563
$800$	−19.2632	−0.681058
$801$	0	0
$802$	−14.2609	−0.503570
$803$	2.62482	0.0926279
$804$	0	0
$805$	0	0
$806$	−51.8268	−1.82552
$807$	0	0
$808$	18.1065	0.636985
$809$	39.4582	1.38728	0.693639	−	0.720323i	$-0.256006\pi$
$809$	39.4582	1.38728	0.693639	+	0.720323i	$0.256006\pi$
$810$	0	0
$811$	0.496374	0.0174300	0.00871502	−	0.999962i	$-0.497226\pi$
$811$	0.496374	0.0174300	0.00871502	+	0.999962i	$0.497226\pi$
$812$	0	0
$813$	0	0
$814$	−6.98762	−0.244916
$815$	−1.58168	−0.0554039
$816$	0	0
$817$	31.1733	1.09062
$818$	11.5790	0.404850
$819$	0	0
$820$	0.889960	0.0310788
$821$	47.1038	1.64393	0.821967	−	0.569534i	$-0.192876\pi$
$821$	47.1038	1.64393	0.821967	+	0.569534i	$0.192876\pi$
$822$	0	0
$823$	−2.19777	−0.0766094	−0.0383047	−	0.999266i	$-0.512196\pi$
$823$	−2.19777	−0.0766094	−0.0383047	+	0.999266i	$0.512196\pi$
$824$	22.0092	0.766727
$825$	0	0
$826$	0	0
$827$	−55.3360	−1.92422	−0.962110	−	0.272661i	$-0.912096\pi$
$827$	−55.3360	−1.92422	−0.962110	+	0.272661i	$0.912096\pi$
$828$	0	0
$829$	20.3206	0.705764	0.352882	−	0.935668i	$-0.385202\pi$
$829$	20.3206	0.705764	0.352882	+	0.935668i	$0.385202\pi$
$830$	−9.17041	−0.318310
$831$	0	0
$832$	9.97733	0.345902
$833$	0	0
$834$	0	0
$835$	−3.70472	−0.128207
$836$	−2.33277	−0.0806806
$837$	0	0
$838$	−17.5480	−0.606186
$839$	−24.5519	−0.847627	−0.423813	−	0.905750i	$-0.639309\pi$
$839$	−24.5519	−0.847627	−0.423813	+	0.905750i	$0.639309\pi$
$840$	0	0
$841$	0.998623	0.0344353
$842$	−5.33007	−0.183686
$843$	0	0
$844$	9.97386	0.343315
$845$	11.5790	0.398329
$846$	0	0
$847$	0	0
$848$	−34.5251	−1.18560
$849$	0	0
$850$	52.8235	1.81183
$851$	−17.3090	−0.593346
$852$	0	0
$853$	−53.5416	−1.83323	−0.916614	−	0.399773i	$-0.869089\pi$
$853$	−53.5416	−1.83323	−0.916614	+	0.399773i	$0.869089\pi$
$854$	0	0
$855$	0	0
$856$	−7.17439	−0.245215
$857$	−54.1553	−1.84991	−0.924955	−	0.380076i	$-0.875898\pi$
$857$	−54.1553	−1.84991	−0.924955	+	0.380076i	$0.875898\pi$
$858$	0	0
$859$	1.79292	0.0611737	0.0305869	−	0.999532i	$-0.490262\pi$
$859$	1.79292	0.0611737	0.0305869	+	0.999532i	$0.490262\pi$
$860$	5.89511	0.201022
$861$	0	0
$862$	−54.1716	−1.84509
$863$	−32.5709	−1.10873	−0.554363	−	0.832275i	$-0.687038\pi$
$863$	−32.5709	−1.10873	−0.554363	+	0.832275i	$0.687038\pi$
$864$	0	0
$865$	−15.2967	−0.520102
$866$	12.7277	0.432506
$867$	0	0
$868$	0	0
$869$	0.784755	0.0266210
$870$	0	0
$871$	−59.5541	−2.01791
$872$	−24.2967	−0.822789
$873$	0	0
$874$	−18.7824	−0.635324
$875$	0	0
$876$	0	0
$877$	−36.7293	−1.24026	−0.620131	−	0.784499i	$-0.712920\pi$
$877$	−36.7293	−1.24026	−0.620131	+	0.784499i	$0.712920\pi$
$878$	3.89095	0.131313
$879$	0	0
$880$	2.78567	0.0939049
$881$	25.3721	0.854807	0.427403	−	0.904061i	$-0.359428\pi$
$881$	25.3721	0.854807	0.427403	+	0.904061i	$0.359428\pi$
$882$	0	0
$883$	−16.9381	−0.570012	−0.285006	−	0.958526i	$-0.591996\pi$
$883$	−16.9381	−0.570012	−0.285006	+	0.958526i	$0.591996\pi$
$884$	33.8268	1.13772
$885$	0	0
$886$	−63.4807	−2.13267
$887$	−48.0137	−1.61214	−0.806071	−	0.591819i	$-0.798410\pi$
$887$	−48.0137	−1.61214	−0.806071	+	0.591819i	$0.798410\pi$
$888$	0	0
$889$	0	0
$890$	1.36115	0.0456260
$891$	0	0
$892$	18.4456	0.617603
$893$	33.3556	1.11620
$894$	0	0
$895$	13.5632	0.453367
$896$	0	0
$897$	0	0
$898$	10.5426	0.351810
$899$	33.2713	1.10966
$900$	0	0
$901$	52.4867	1.74858
$902$	1.05489	0.0351240
$903$	0	0
$904$	17.0384	0.566688
$905$	−12.2335	−0.406656
$906$	0	0
$907$	−24.7775	−0.822722	−0.411361	−	0.911472i	$-0.634947\pi$
$907$	−24.7775	−0.822722	−0.411361	+	0.911472i	$0.634947\pi$
$908$	9.26972	0.307626
$909$	0	0
$910$	0	0
$911$	31.5833	1.04640	0.523200	−	0.852210i	$-0.324738\pi$
$911$	31.5833	1.04640	0.523200	+	0.852210i	$0.324738\pi$
$912$	0	0
$913$	−3.34419	−0.110676
$914$	−34.2843	−1.13402
$915$	0	0
$916$	−13.3837	−0.442209
$917$	0	0
$918$	0	0
$919$	1.59208	0.0525180	0.0262590	−	0.999655i	$-0.491641\pi$
$919$	1.59208	0.0525180	0.0262590	+	0.999655i	$0.491641\pi$
$920$	4.44125	0.146424
$921$	0	0
$922$	−38.3028	−1.26144
$923$	21.5867	0.710536
$924$	0	0
$925$	28.6414	0.941725
$926$	46.9591	1.54317
$927$	0	0
$928$	25.7403	0.844968
$929$	−27.0711	−0.888175	−0.444087	−	0.895983i	$-0.646472\pi$
$929$	−27.0711	−0.888175	−0.444087	+	0.895983i	$0.646472\pi$
$930$	0	0
$931$	0	0
$932$	3.89877	0.127709
$933$	0	0
$934$	34.2006	1.11908
$935$	−4.23491	−0.138496
$936$	0	0
$937$	−32.6624	−1.06704	−0.533518	−	0.845789i	$-0.679130\pi$
$937$	−32.6624	−1.06704	−0.533518	+	0.845789i	$0.679130\pi$
$938$	0	0
$939$	0	0
$940$	6.30781	0.205738
$941$	−4.72285	−0.153961	−0.0769803	−	0.997033i	$-0.524528\pi$
$941$	−4.72285	−0.153961	−0.0769803	+	0.997033i	$0.524528\pi$
$942$	0	0
$943$	2.61307	0.0850933
$944$	52.0220	1.69317
$945$	0	0
$946$	6.98762	0.227187
$947$	56.7810	1.84514	0.922568	−	0.385835i	$-0.126087\pi$
$947$	56.7810	1.84514	0.922568	+	0.385835i	$0.126087\pi$
$948$	0	0
$949$	−22.3942	−0.726945
$950$	31.0795	1.00835
$951$	0	0
$952$	0	0
$953$	−47.1693	−1.52796	−0.763982	−	0.645238i	$-0.776758\pi$
$953$	−47.1693	−1.52796	−0.763982	+	0.645238i	$0.776758\pi$
$954$	0	0
$955$	−2.05455	−0.0664838
$956$	−8.48852	−0.274538
$957$	0	0
$958$	−16.2936	−0.526423
$959$	0	0
$960$	0	0
$961$	5.90112	0.190359
$962$	59.6162	1.92210
$963$	0	0
$964$	−9.36359	−0.301581
$965$	9.90213	0.318761
$966$	0	0
$967$	47.3969	1.52418	0.762091	−	0.647470i	$-0.224173\pi$
$967$	47.3969	1.52418	0.762091	+	0.647470i	$0.224173\pi$
$968$	−20.1223	−0.646754
$969$	0	0
$970$	−5.49580	−0.176459
$971$	−22.7473	−0.729994	−0.364997	−	0.931009i	$-0.618930\pi$
$971$	−22.7473	−0.729994	−0.364997	+	0.931009i	$0.618930\pi$
$972$	0	0
$973$	0	0
$974$	−22.2212	−0.712012
$975$	0	0
$976$	58.1202	1.86038
$977$	35.6849	1.14166	0.570831	−	0.821068i	$-0.306621\pi$
$977$	35.6849	1.14166	0.570831	+	0.821068i	$0.306621\pi$
$978$	0	0
$979$	0.496374	0.0158642
$980$	0	0
$981$	0	0
$982$	−26.0741	−0.832059
$983$	−24.0134	−0.765908	−0.382954	−	0.923767i	$-0.625093\pi$
$983$	−24.0134	−0.765908	−0.382954	+	0.923767i	$0.625093\pi$
$984$	0	0
$985$	−17.8318	−0.568169
$986$	−70.5850	−2.24788
$987$	0	0
$988$	19.9025	0.633183
$989$	17.3090	0.550395
$990$	0	0
$991$	−44.4189	−1.41101	−0.705507	−	0.708703i	$-0.749281\pi$
$991$	−44.4189	−1.41101	−0.705507	+	0.708703i	$0.749281\pi$
$992$	28.5485	0.906416
$993$	0	0
$994$	0	0
$995$	−7.99862	−0.253573
$996$	0	0
$997$	−9.04673	−0.286513	−0.143256	−	0.989686i	$-0.545757\pi$
$997$	−9.04673	−0.286513	−0.143256	+	0.989686i	$0.545757\pi$
$998$	−8.26929	−0.261760
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.be.1.2		6
3.2	odd	2		3969.2.a.bd.1.5		6
7.6	odd	2	inner	3969.2.a.be.1.1		6
9.2	odd	6		1323.2.f.g.442.2		12
9.4	even	3		441.2.f.g.295.5	yes	12
9.5	odd	6		1323.2.f.g.883.2		12
9.7	even	3		441.2.f.g.148.5	✓	12
21.20	even	2		3969.2.a.bd.1.6		6
63.2	odd	6		1323.2.g.g.361.1		12
63.4	even	3		441.2.g.g.79.6		12
63.5	even	6		1323.2.h.g.802.5		12
63.11	odd	6		1323.2.h.g.226.6		12
63.13	odd	6		441.2.f.g.295.6	yes	12
63.16	even	3		441.2.g.g.67.6		12
63.20	even	6		1323.2.f.g.442.1		12
63.23	odd	6		1323.2.h.g.802.6		12
63.25	even	3		441.2.h.g.373.1		12
63.31	odd	6		441.2.g.g.79.5		12
63.32	odd	6		1323.2.g.g.667.1		12
63.34	odd	6		441.2.f.g.148.6	yes	12
63.38	even	6		1323.2.h.g.226.5		12
63.40	odd	6		441.2.h.g.214.2		12
63.41	even	6		1323.2.f.g.883.1		12
63.47	even	6		1323.2.g.g.361.2		12
63.52	odd	6		441.2.h.g.373.2		12
63.58	even	3		441.2.h.g.214.1		12
63.59	even	6		1323.2.g.g.667.2		12
63.61	odd	6		441.2.g.g.67.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.f.g.148.5	✓	12	9.7	even	3
441.2.f.g.148.6	yes	12	63.34	odd	6
441.2.f.g.295.5	yes	12	9.4	even	3
441.2.f.g.295.6	yes	12	63.13	odd	6
441.2.g.g.67.5		12	63.61	odd	6
441.2.g.g.67.6		12	63.16	even	3
441.2.g.g.79.5		12	63.31	odd	6
441.2.g.g.79.6		12	63.4	even	3
441.2.h.g.214.1		12	63.58	even	3
441.2.h.g.214.2		12	63.40	odd	6
441.2.h.g.373.1		12	63.25	even	3
441.2.h.g.373.2		12	63.52	odd	6
1323.2.f.g.442.1		12	63.20	even	6
1323.2.f.g.442.2		12	9.2	odd	6
1323.2.f.g.883.1		12	63.41	even	6
1323.2.f.g.883.2		12	9.5	odd	6
1323.2.g.g.361.1		12	63.2	odd	6
1323.2.g.g.361.2		12	63.47	even	6
1323.2.g.g.667.1		12	63.32	odd	6
1323.2.g.g.667.2		12	63.59	even	6
1323.2.h.g.226.5		12	63.38	even	6
1323.2.h.g.226.6		12	63.11	odd	6
1323.2.h.g.802.5		12	63.5	even	6
1323.2.h.g.802.6		12	63.23	odd	6
3969.2.a.bd.1.5		6	3.2	odd	2
3969.2.a.bd.1.6		6	21.20	even	2
3969.2.a.be.1.1		6	7.6	odd	2	inner
3969.2.a.be.1.2		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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-1.69963 q^{2} +0.888736 q^{4} +0.949271 q^{5} +1.88874 q^{8} +O(q^{10})\)	\(q-1.69963 q^{2} +0.888736 q^{4} +0.949271 q^{5} +1.88874 q^{8} -1.61341 q^{10} -0.588364 q^{11} +5.01974 q^{13} -4.98762 q^{16} +7.58242 q^{17} +4.46122 q^{19} +0.843651 q^{20} +1.00000 q^{22} +2.47710 q^{23} -4.09888 q^{25} -8.53169 q^{26} +5.47710 q^{29} +6.07463 q^{31} +4.69963 q^{32} -12.8873 q^{34} -6.98762 q^{37} -7.58242 q^{38} +1.79292 q^{40} +1.05489 q^{41} +6.98762 q^{43} -0.522900 q^{44} -4.21015 q^{46} +7.47680 q^{47} +6.96658 q^{50} +4.46122 q^{52} +6.92216 q^{53} -0.558517 q^{55} -9.30903 q^{58} -10.4302 q^{59} -11.6529 q^{61} -10.3246 q^{62} +1.98762 q^{64} +4.76509 q^{65} -11.8640 q^{67} +6.73877 q^{68} +4.30037 q^{71} -4.46122 q^{73} +11.8764 q^{74} +3.96485 q^{76} -1.33379 q^{79} -4.73460 q^{80} -1.79292 q^{82} +5.68387 q^{83} +7.19777 q^{85} -11.8764 q^{86} -1.11126 q^{88} -0.843651 q^{89} +2.20149 q^{92} -12.7078 q^{94} +4.23491 q^{95} +3.40633 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6q + 2q^{2} + 6q^{4} + 12q^{8} + O(q^{10}) \)	\( 6q + 2q^{2} + 6q^{4} + 12q^{8} + 8q^{11} + 6q^{16} + 6q^{22} + 4q^{23} + 12q^{25} + 22q^{29} + 16q^{32} - 6q^{37} + 6q^{43} - 14q^{44} + 12q^{46} + 56q^{50} + 28q^{53} + 18q^{58} - 24q^{64} - 6q^{65} + 38q^{71} + 36q^{74} - 6q^{79} - 30q^{85} - 36q^{86} - 6q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about