LMFDB - Embedded newform 441.4.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 74x^{6} + 1469x^{4} - 8828x^{2} + 2500 $ x^8 - 74x^6 + 1469x^4 - 8828*x^2 + 2500
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.98569$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.39991 q^{2} +21.1590 q^{4} +15.5768 q^{5} -71.0573 q^{8} +O(q^{10})$	$q-5.39991 q^{2} +21.1590 q^{4} +15.5768 q^{5} -71.0573 q^{8} -84.1131 q^{10} -31.9702 q^{11} +72.5746 q^{13} +214.431 q^{16} -29.0288 q^{17} +108.829 q^{19} +329.589 q^{20} +172.636 q^{22} +55.2869 q^{23} +117.636 q^{25} -391.896 q^{26} -17.7363 q^{29} +56.1189 q^{31} -589.448 q^{32} +156.753 q^{34} -295.816 q^{37} -587.668 q^{38} -1106.84 q^{40} +238.605 q^{41} +16.8202 q^{43} -676.457 q^{44} -298.544 q^{46} +511.909 q^{47} -635.223 q^{50} +1535.60 q^{52} -265.205 q^{53} -497.992 q^{55} +95.7742 q^{58} -254.181 q^{59} -72.8352 q^{61} -303.037 q^{62} +1467.52 q^{64} +1130.48 q^{65} -506.360 q^{67} -614.220 q^{68} +827.722 q^{71} -372.577 q^{73} +1597.38 q^{74} +2302.72 q^{76} +1028.45 q^{79} +3340.14 q^{80} -1288.44 q^{82} +453.148 q^{83} -452.175 q^{85} -90.8276 q^{86} +2271.71 q^{88} +332.065 q^{89} +1169.81 q^{92} -2764.26 q^{94} +1695.21 q^{95} -1164.54 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 68 q^{4}+O(q^{10}) $ 8 * q + 68 * q^4	$ 8 q + 68 q^{4} + 804 q^{16} + 976 q^{22} + 536 q^{25} + 64 q^{37} + 2160 q^{43} - 768 q^{46} + 2184 q^{58} + 7588 q^{64} + 5392 q^{79} + 2864 q^{85} + 5616 q^{88}+O(q^{100}) $ 8 * q + 68 * q^4 + 804 * q^16 + 976 * q^22 + 536 * q^25 + 64 * q^37 + 2160 * q^43 - 768 * q^46 + 2184 * q^58 + 7588 * q^64 + 5392 * q^79 + 2864 * q^85 + 5616 * q^88

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 $

$2$	−5.39991	−1.90916	−0.954578	−	0.297962i	$-0.903693\pi$
$2$	−5.39991	−1.90916	−0.954578	+	0.297962i	$0.903693\pi$
$3$	0	0
$3$	0	0
$4$	21.1590	2.64487
$5$	15.5768	1.39323	0.696615	−	0.717446i	$-0.254689\pi$
$5$	15.5768	1.39323	0.696615	+	0.717446i	$0.254689\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−71.0573	−3.14032
$9$	0	0
$10$	−84.1131	−2.65989
$11$	−31.9702	−0.876306	−0.438153	−	0.898900i	$-0.644367\pi$
$11$	−31.9702	−0.876306	−0.438153	+	0.898900i	$0.644367\pi$
$12$	0	0
$13$	72.5746	1.54835	0.774176	−	0.632971i	$-0.218165\pi$
$13$	72.5746	1.54835	0.774176	+	0.632971i	$0.218165\pi$
$14$	0	0
$15$	0	0
$16$	214.431	3.35048
$17$	−29.0288	−0.414148	−0.207074	−	0.978325i	$-0.566394\pi$
$17$	−29.0288	−0.414148	−0.207074	+	0.978325i	$0.566394\pi$
$18$	0	0
$19$	108.829	1.31406	0.657030	−	0.753864i	$-0.271812\pi$
$19$	108.829	1.31406	0.657030	+	0.753864i	$0.271812\pi$
$20$	329.589	3.68492
$21$	0	0
$22$	172.636	1.67300
$23$	55.2869	0.501222	0.250611	−	0.968088i	$-0.419368\pi$
$23$	55.2869	0.501222	0.250611	+	0.968088i	$0.419368\pi$
$24$	0	0
$25$	117.636	0.941088
$26$	−391.896	−2.95604
$27$	0	0
$28$	0	0
$29$	−17.7363	−0.113570	−0.0567852	−	0.998386i	$-0.518085\pi$
$29$	−17.7363	−0.113570	−0.0567852	+	0.998386i	$0.518085\pi$
$30$	0	0
$31$	56.1189	0.325137	0.162568	−	0.986697i	$-0.448022\pi$
$31$	56.1189	0.325137	0.162568	+	0.986697i	$0.448022\pi$
$32$	−589.448	−3.25627
$33$	0	0
$34$	156.753	0.790673
$35$	0	0
$36$	0	0
$37$	−295.816	−1.31437	−0.657187	−	0.753728i	$-0.728254\pi$
$37$	−295.816	−1.31437	−0.657187	+	0.753728i	$0.728254\pi$
$38$	−587.668	−2.50875
$39$	0	0
$40$	−1106.84	−4.37518
$41$	238.605	0.908873	0.454437	−	0.890779i	$-0.349841\pi$
$41$	238.605	0.908873	0.454437	+	0.890779i	$0.349841\pi$
$42$	0	0
$43$	16.8202	0.0596526	0.0298263	−	0.999555i	$-0.490505\pi$
$43$	16.8202	0.0596526	0.0298263	+	0.999555i	$0.490505\pi$
$44$	−676.457	−2.31772
$45$	0	0
$46$	−298.544	−0.956911
$47$	511.909	1.58871	0.794357	−	0.607451i	$-0.207808\pi$
$47$	511.909	1.58871	0.794357	+	0.607451i	$0.207808\pi$
$48$	0	0
$49$	0	0
$50$	−635.223	−1.79668
$51$	0	0
$52$	1535.60	4.09519
$53$	−265.205	−0.687335	−0.343667	−	0.939091i	$-0.611669\pi$
$53$	−265.205	−0.687335	−0.343667	+	0.939091i	$0.611669\pi$
$54$	0	0
$55$	−497.992	−1.22090
$56$	0	0
$57$	0	0
$58$	95.7742	0.216823
$59$	−254.181	−0.560875	−0.280437	−	0.959872i	$-0.590480\pi$
$59$	−254.181	−0.560875	−0.280437	+	0.959872i	$0.590480\pi$
$60$	0	0
$61$	−72.8352	−0.152878	−0.0764392	−	0.997074i	$-0.524355\pi$
$61$	−72.8352	−0.152878	−0.0764392	+	0.997074i	$0.524355\pi$
$62$	−303.037	−0.620737
$63$	0	0
$64$	1467.52	2.86625
$65$	1130.48	2.15721
$66$	0	0
$67$	−506.360	−0.923308	−0.461654	−	0.887060i	$-0.652744\pi$
$67$	−506.360	−0.923308	−0.461654	+	0.887060i	$0.652744\pi$
$68$	−614.220	−1.09537
$69$	0	0
$70$	0	0
$71$	827.722	1.38356	0.691779	−	0.722110i	$-0.256827\pi$
$71$	827.722	1.38356	0.691779	+	0.722110i	$0.256827\pi$
$72$	0	0
$73$	−372.577	−0.597354	−0.298677	−	0.954354i	$-0.596545\pi$
$73$	−372.577	−0.597354	−0.298677	+	0.954354i	$0.596545\pi$
$74$	1597.38	2.50934
$75$	0	0
$76$	2302.72	3.47552
$77$	0	0
$78$	0	0
$79$	1028.45	1.46468	0.732341	−	0.680938i	$-0.238428\pi$
$79$	1028.45	1.46468	0.732341	+	0.680938i	$0.238428\pi$
$80$	3340.14	4.66799
$81$	0	0
$82$	−1288.44	−1.73518
$83$	453.148	0.599270	0.299635	−	0.954054i	$-0.403135\pi$
$83$	453.148	0.599270	0.299635	+	0.954054i	$0.403135\pi$
$84$	0	0
$85$	−452.175	−0.577003
$86$	−90.8276	−0.113886
$87$	0	0
$88$	2271.71	2.75188
$89$	332.065	0.395493	0.197746	−	0.980253i	$-0.436638\pi$
$89$	332.065	0.395493	0.197746	+	0.980253i	$0.436638\pi$
$90$	0	0
$91$	0	0
$92$	1169.81	1.32567
$93$	0	0
$94$	−2764.26	−3.03310
$95$	1695.21	1.83079
$96$	0	0
$97$	−1164.54	−1.21898	−0.609489	−	0.792795i	$-0.708625\pi$
$97$	−1164.54	−1.21898	−0.609489	+	0.792795i	$0.708625\pi$
$98$	0	0
$99$	0	0
$100$	2489.06	2.48906
$101$	−1863.54	−1.83593	−0.917967	−	0.396656i	$-0.870171\pi$
$101$	−1863.54	−1.83593	−0.917967	+	0.396656i	$0.870171\pi$
$102$	0	0
$103$	986.280	0.943506	0.471753	−	0.881731i	$-0.343621\pi$
$103$	986.280	0.943506	0.471753	+	0.881731i	$0.343621\pi$
$104$	−5156.95	−4.86232
$105$	0	0
$106$	1432.08	1.31223
$107$	695.685	0.628546	0.314273	−	0.949333i	$-0.398239\pi$
$107$	695.685	0.628546	0.314273	+	0.949333i	$0.398239\pi$
$108$	0	0
$109$	1715.00	1.50703	0.753517	−	0.657428i	$-0.228356\pi$
$109$	1715.00	1.50703	0.753517	+	0.657428i	$0.228356\pi$
$110$	2689.11	2.33088
$111$	0	0
$112$	0	0
$113$	−877.721	−0.730700	−0.365350	−	0.930870i	$-0.619051\pi$
$113$	−877.721	−0.730700	−0.365350	+	0.930870i	$0.619051\pi$
$114$	0	0
$115$	861.191	0.698317
$116$	−375.281	−0.300379
$117$	0	0
$118$	1372.56	1.07080
$119$	0	0
$120$	0	0
$121$	−308.908	−0.232087
$122$	393.303	0.291869
$123$	0	0
$124$	1187.42	0.859946
$125$	−114.708	−0.0820784
$126$	0	0
$127$	781.088	0.545751	0.272875	−	0.962049i	$-0.412025\pi$
$127$	781.088	0.545751	0.272875	+	0.962049i	$0.412025\pi$
$128$	−3208.88	−2.21584
$129$	0	0
$130$	−6104.48	−4.11845
$131$	1961.24	1.30805	0.654024	−	0.756474i	$-0.273080\pi$
$131$	1961.24	1.30805	0.654024	+	0.756474i	$0.273080\pi$
$132$	0	0
$133$	0	0
$134$	2734.29	1.76274
$135$	0	0
$136$	2062.71	1.30056
$137$	1220.31	0.761009	0.380504	−	0.924779i	$-0.375750\pi$
$137$	1220.31	0.761009	0.380504	+	0.924779i	$0.375750\pi$
$138$	0	0
$139$	1068.10	0.651765	0.325882	−	0.945410i	$-0.394339\pi$
$139$	1068.10	0.651765	0.325882	+	0.945410i	$0.394339\pi$
$140$	0	0
$141$	0	0
$142$	−4469.62	−2.64143
$143$	−2320.22	−1.35683
$144$	0	0
$145$	−276.274	−0.158230
$146$	2011.88	1.14044
$147$	0	0
$148$	−6259.16	−3.47635
$149$	2590.82	1.42449	0.712243	−	0.701933i	$-0.247679\pi$
$149$	2590.82	1.42449	0.712243	+	0.701933i	$0.247679\pi$
$150$	0	0
$151$	1929.54	1.03989	0.519946	−	0.854199i	$-0.325952\pi$
$151$	1929.54	1.03989	0.519946	+	0.854199i	$0.325952\pi$
$152$	−7733.12	−4.12657
$153$	0	0
$154$	0	0
$155$	874.151	0.452990
$156$	0	0
$157$	2625.73	1.33475	0.667377	−	0.744720i	$-0.267417\pi$
$157$	2625.73	1.33475	0.667377	+	0.744720i	$0.267417\pi$
$158$	−5553.54	−2.79630
$159$	0	0
$160$	−9181.70	−4.53673
$161$	0	0
$162$	0	0
$163$	−3100.17	−1.48972	−0.744858	−	0.667223i	$-0.767483\pi$
$163$	−3100.17	−1.48972	−0.744858	+	0.667223i	$0.767483\pi$
$164$	5048.63	2.40385
$165$	0	0
$166$	−2446.95	−1.14410
$167$	3264.73	1.51277	0.756386	−	0.654126i	$-0.226963\pi$
$167$	3264.73	1.51277	0.756386	+	0.654126i	$0.226963\pi$
$168$	0	0
$169$	3070.07	1.39739
$170$	2441.70	1.10159
$171$	0	0
$172$	355.899	0.157773
$173$	−2036.31	−0.894900	−0.447450	−	0.894309i	$-0.647668\pi$
$173$	−2036.31	−0.894900	−0.447450	+	0.894309i	$0.647668\pi$
$174$	0	0
$175$	0	0
$176$	−6855.39	−2.93605
$177$	0	0
$178$	−1793.12	−0.755057
$179$	3582.35	1.49585	0.747925	−	0.663783i	$-0.231050\pi$
$179$	3582.35	1.49585	0.747925	+	0.663783i	$0.231050\pi$
$180$	0	0
$181$	−1637.35	−0.672392	−0.336196	−	0.941792i	$-0.609140\pi$
$181$	−1637.35	−0.672392	−0.336196	+	0.941792i	$0.609140\pi$
$182$	0	0
$183$	0	0
$184$	−3928.53	−1.57400
$185$	−4607.86	−1.83122
$186$	0	0
$187$	928.056	0.362921
$188$	10831.5	4.20195
$189$	0	0
$190$	−9153.97	−3.49526
$191$	−2824.78	−1.07013	−0.535063	−	0.844812i	$-0.679712\pi$
$191$	−2824.78	−1.07013	−0.535063	+	0.844812i	$0.679712\pi$
$192$	0	0
$193$	−1657.53	−0.618195	−0.309098	−	0.951030i	$-0.600027\pi$
$193$	−1657.53	−0.618195	−0.309098	+	0.951030i	$0.600027\pi$
$194$	6288.39	2.32722
$195$	0	0
$196$	0	0
$197$	−1890.78	−0.683819	−0.341909	−	0.939733i	$-0.611074\pi$
$197$	−1890.78	−0.683819	−0.341909	+	0.939733i	$0.611074\pi$
$198$	0	0
$199$	−1392.75	−0.496126	−0.248063	−	0.968744i	$-0.579794\pi$
$199$	−1392.75	−0.496126	−0.248063	+	0.968744i	$0.579794\pi$
$200$	−8358.89	−2.95532
$201$	0	0
$202$	10063.0	3.50508
$203$	0	0
$204$	0	0
$205$	3716.69	1.26627
$206$	−5325.82	−1.80130
$207$	0	0
$208$	15562.2	5.18772
$209$	−3479.29	−1.15152
$210$	0	0
$211$	3314.53	1.08143	0.540714	−	0.841206i	$-0.318154\pi$
$211$	3314.53	1.08143	0.540714	+	0.841206i	$0.318154\pi$
$212$	−5611.47	−1.81791
$213$	0	0
$214$	−3756.64	−1.19999
$215$	262.005	0.0831097
$216$	0	0
$217$	0	0
$218$	−9260.82	−2.87716
$219$	0	0
$220$	−10537.0	−3.22911
$221$	−2106.75	−0.641247
$222$	0	0
$223$	−5576.50	−1.67457	−0.837287	−	0.546764i	$-0.815860\pi$
$223$	−5576.50	−1.67457	−0.837287	+	0.546764i	$0.815860\pi$
$224$	0	0
$225$	0	0
$226$	4739.61	1.39502
$227$	1447.92	0.423357	0.211678	−	0.977339i	$-0.432107\pi$
$227$	1447.92	0.423357	0.211678	+	0.977339i	$0.432107\pi$
$228$	0	0
$229$	1617.55	0.466771	0.233385	−	0.972384i	$-0.425020\pi$
$229$	1617.55	0.466771	0.233385	+	0.972384i	$0.425020\pi$
$230$	−4650.35	−1.33320
$231$	0	0
$232$	1260.29	0.356647
$233$	6092.11	1.71291	0.856454	−	0.516224i	$-0.172663\pi$
$233$	6092.11	1.71291	0.856454	+	0.516224i	$0.172663\pi$
$234$	0	0
$235$	7973.89	2.21344
$236$	−5378.22	−1.48344
$237$	0	0
$238$	0	0
$239$	1595.90	0.431927	0.215963	−	0.976401i	$-0.430711\pi$
$239$	1595.90	0.431927	0.215963	+	0.976401i	$0.430711\pi$
$240$	0	0
$241$	2188.83	0.585041	0.292521	−	0.956259i	$-0.405506\pi$
$241$	2188.83	0.585041	0.292521	+	0.956259i	$0.405506\pi$
$242$	1668.07	0.443090
$243$	0	0
$244$	−1541.12	−0.404344
$245$	0	0
$246$	0	0
$247$	7898.24	2.03463
$248$	−3987.65	−1.02103
$249$	0	0
$250$	619.413	0.156700
$251$	6203.07	1.55990	0.779949	−	0.625843i	$-0.215245\pi$
$251$	6203.07	1.55990	0.779949	+	0.625843i	$0.215245\pi$
$252$	0	0
$253$	−1767.53	−0.439224
$254$	−4217.80	−1.04192
$255$	0	0
$256$	5587.48	1.36413
$257$	268.323	0.0651266	0.0325633	−	0.999470i	$-0.489633\pi$
$257$	268.323	0.0651266	0.0325633	+	0.999470i	$0.489633\pi$
$258$	0	0
$259$	0	0
$260$	23919.8	5.70554
$261$	0	0
$262$	−10590.5	−2.49727
$263$	−3724.96	−0.873349	−0.436675	−	0.899619i	$-0.643844\pi$
$263$	−3724.96	−0.873349	−0.436675	+	0.899619i	$0.643844\pi$
$264$	0	0
$265$	−4131.04	−0.957615
$266$	0	0
$267$	0	0
$268$	−10714.1	−2.44203
$269$	8557.25	1.93957	0.969786	−	0.243958i	$-0.0784458\pi$
$269$	8557.25	1.93957	0.969786	+	0.243958i	$0.0784458\pi$
$270$	0	0
$271$	−5279.66	−1.18346	−0.591728	−	0.806137i	$-0.701554\pi$
$271$	−5279.66	−1.18346	−0.591728	+	0.806137i	$0.701554\pi$
$272$	−6224.67	−1.38760
$273$	0	0
$274$	−6589.56	−1.45288
$275$	−3760.84	−0.824681
$276$	0	0
$277$	−441.548	−0.0957764	−0.0478882	−	0.998853i	$-0.515249\pi$
$277$	−441.548	−0.0957764	−0.0478882	+	0.998853i	$0.515249\pi$
$278$	−5767.66	−1.24432
$279$	0	0
$280$	0	0
$281$	3766.49	0.799609	0.399804	−	0.916601i	$-0.369078\pi$
$281$	3766.49	0.799609	0.399804	+	0.916601i	$0.369078\pi$
$282$	0	0
$283$	1811.02	0.380403	0.190201	−	0.981745i	$-0.439086\pi$
$283$	1811.02	0.380403	0.190201	+	0.981745i	$0.439086\pi$
$284$	17513.8	3.65933
$285$	0	0
$286$	12529.0	2.59040
$287$	0	0
$288$	0	0
$289$	−4070.33	−0.828481
$290$	1491.85	0.302085
$291$	0	0
$292$	−7883.35	−1.57993
$293$	−5815.74	−1.15959	−0.579794	−	0.814763i	$-0.696867\pi$
$293$	−5815.74	−1.15959	−0.579794	+	0.814763i	$0.696867\pi$
$294$	0	0
$295$	−3959.33	−0.781427
$296$	21019.9	4.12755
$297$	0	0
$298$	−13990.2	−2.71957
$299$	4012.42	0.776068
$300$	0	0
$301$	0	0
$302$	−10419.3	−1.98532
$303$	0	0
$304$	23336.4	4.40274
$305$	−1134.54	−0.212995
$306$	0	0
$307$	−1974.93	−0.367150	−0.183575	−	0.983006i	$-0.558767\pi$
$307$	−1974.93	−0.367150	−0.183575	+	0.983006i	$0.558767\pi$
$308$	0	0
$309$	0	0
$310$	−4720.33	−0.864829
$311$	1159.07	0.211334	0.105667	−	0.994402i	$-0.466302\pi$
$311$	1159.07	0.211334	0.105667	+	0.994402i	$0.466302\pi$
$312$	0	0
$313$	4493.67	0.811494	0.405747	−	0.913986i	$-0.367011\pi$
$313$	4493.67	0.811494	0.405747	+	0.913986i	$0.367011\pi$
$314$	−14178.7	−2.54825
$315$	0	0
$316$	21761.0	3.87390
$317$	−4677.50	−0.828752	−0.414376	−	0.910106i	$-0.636000\pi$
$317$	−4677.50	−0.828752	−0.414376	+	0.910106i	$0.636000\pi$
$318$	0	0
$319$	567.031	0.0995225
$320$	22859.2	3.99334
$321$	0	0
$322$	0	0
$323$	−3159.19	−0.544216
$324$	0	0
$325$	8537.38	1.45713
$326$	16740.6	2.84410
$327$	0	0
$328$	−16954.6	−2.85415
$329$	0	0
$330$	0	0
$331$	−2982.71	−0.495301	−0.247650	−	0.968849i	$-0.579658\pi$
$331$	−2982.71	−0.495301	−0.247650	+	0.968849i	$0.579658\pi$
$332$	9588.15	1.58499
$333$	0	0
$334$	−17629.3	−2.88811
$335$	−7887.45	−1.28638
$336$	0	0
$337$	7328.53	1.18460	0.592301	−	0.805717i	$-0.298220\pi$
$337$	7328.53	1.18460	0.592301	+	0.805717i	$0.298220\pi$
$338$	−16578.1	−2.66784
$339$	0	0
$340$	−9567.57	−1.52610
$341$	−1794.13	−0.284920
$342$	0	0
$343$	0	0
$344$	−1195.20	−0.187328
$345$	0	0
$346$	10995.9	1.70850
$347$	−8309.33	−1.28550	−0.642749	−	0.766076i	$-0.722206\pi$
$347$	−8309.33	−1.28550	−0.642749	+	0.766076i	$0.722206\pi$
$348$	0	0
$349$	334.303	0.0512745	0.0256373	−	0.999671i	$-0.491839\pi$
$349$	334.303	0.0512745	0.0256373	+	0.999671i	$0.491839\pi$
$350$	0	0
$351$	0	0
$352$	18844.8	2.85349
$353$	5458.19	0.822975	0.411487	−	0.911415i	$-0.365009\pi$
$353$	5458.19	0.822975	0.411487	+	0.911415i	$0.365009\pi$
$354$	0	0
$355$	12893.2	1.92761
$356$	7026.17	1.04603
$357$	0	0
$358$	−19344.3	−2.85581
$359$	−7196.90	−1.05804	−0.529022	−	0.848608i	$-0.677441\pi$
$359$	−7196.90	−1.05804	−0.529022	+	0.848608i	$0.677441\pi$
$360$	0	0
$361$	4984.82	0.726755
$362$	8841.52	1.28370
$363$	0	0
$364$	0	0
$365$	−5803.55	−0.832251
$366$	0	0
$367$	−6324.43	−0.899544	−0.449772	−	0.893143i	$-0.648495\pi$
$367$	−6324.43	−0.899544	−0.449772	+	0.893143i	$0.648495\pi$
$368$	11855.2	1.67934
$369$	0	0
$370$	24882.0	3.49609
$371$	0	0
$372$	0	0
$373$	−10931.8	−1.51749	−0.758746	−	0.651386i	$-0.774188\pi$
$373$	−10931.8	−1.51749	−0.758746	+	0.651386i	$0.774188\pi$
$374$	−5011.42	−0.692872
$375$	0	0
$376$	−36374.9	−4.98907
$377$	−1287.20	−0.175847
$378$	0	0
$379$	6024.02	0.816446	0.408223	−	0.912882i	$-0.366149\pi$
$379$	6024.02	0.816446	0.408223	+	0.912882i	$0.366149\pi$
$380$	35868.9	4.84220
$381$	0	0
$382$	15253.6	2.04304
$383$	−6848.10	−0.913633	−0.456816	−	0.889561i	$-0.651010\pi$
$383$	−6848.10	−0.913633	−0.456816	+	0.889561i	$0.651010\pi$
$384$	0	0
$385$	0	0
$386$	8950.51	1.18023
$387$	0	0
$388$	−24640.4	−3.22404
$389$	−5161.88	−0.672796	−0.336398	−	0.941720i	$-0.609209\pi$
$389$	−5161.88	−0.672796	−0.336398	+	0.941720i	$0.609209\pi$
$390$	0	0
$391$	−1604.91	−0.207580
$392$	0	0
$393$	0	0
$394$	10210.0	1.30552
$395$	16020.0	2.04064
$396$	0	0
$397$	−344.768	−0.0435854	−0.0217927	−	0.999763i	$-0.506937\pi$
$397$	−344.768	−0.0435854	−0.0217927	+	0.999763i	$0.506937\pi$
$398$	7520.70	0.947182
$399$	0	0
$400$	25224.8	3.15310
$401$	−5098.84	−0.634972	−0.317486	−	0.948263i	$-0.602839\pi$
$401$	−5098.84	−0.634972	−0.317486	+	0.948263i	$0.602839\pi$
$402$	0	0
$403$	4072.80	0.503426
$404$	−39430.7	−4.85582
$405$	0	0
$406$	0	0
$407$	9457.28	1.15179
$408$	0	0
$409$	323.124	0.0390647	0.0195323	−	0.999809i	$-0.493782\pi$
$409$	323.124	0.0390647	0.0195323	+	0.999809i	$0.493782\pi$
$410$	−20069.8	−2.41750
$411$	0	0
$412$	20868.7	2.49545
$413$	0	0
$414$	0	0
$415$	7058.58	0.834921
$416$	−42779.0	−5.04185
$417$	0	0
$418$	18787.8	2.19843
$419$	4415.98	0.514880	0.257440	−	0.966294i	$-0.417121\pi$
$419$	4415.98	0.514880	0.257440	+	0.966294i	$0.417121\pi$
$420$	0	0
$421$	1379.37	0.159683	0.0798415	−	0.996808i	$-0.474559\pi$
$421$	1379.37	0.159683	0.0798415	+	0.996808i	$0.474559\pi$
$422$	−17898.1	−2.06461
$423$	0	0
$424$	18844.8	2.15845
$425$	−3414.83	−0.389750
$426$	0	0
$427$	0	0
$428$	14720.0	1.66243
$429$	0	0
$430$	−1414.80	−0.158669
$431$	−1655.79	−0.185050	−0.0925248	−	0.995710i	$-0.529494\pi$
$431$	−1655.79	−0.185050	−0.0925248	+	0.995710i	$0.529494\pi$
$432$	0	0
$433$	−8612.65	−0.955883	−0.477942	−	0.878392i	$-0.658617\pi$
$433$	−8612.65	−0.955883	−0.477942	+	0.878392i	$0.658617\pi$
$434$	0	0
$435$	0	0
$436$	36287.6	3.98592
$437$	6016.83	0.658636
$438$	0	0
$439$	5975.39	0.649635	0.324817	−	0.945777i	$-0.394697\pi$
$439$	5975.39	0.649635	0.324817	+	0.945777i	$0.394697\pi$
$440$	35386.0	3.83400
$441$	0	0
$442$	11376.3	1.22424
$443$	−806.532	−0.0865000	−0.0432500	−	0.999064i	$-0.513771\pi$
$443$	−806.532	−0.0865000	−0.0432500	+	0.999064i	$0.513771\pi$
$444$	0	0
$445$	5172.51	0.551012
$446$	30112.6	3.19702
$447$	0	0
$448$	0	0
$449$	−6253.04	−0.657237	−0.328618	−	0.944463i	$-0.606583\pi$
$449$	−6253.04	−0.657237	−0.328618	+	0.944463i	$0.606583\pi$
$450$	0	0
$451$	−7628.23	−0.796451
$452$	−18571.7	−1.93261
$453$	0	0
$454$	−7818.65	−0.808254
$455$	0	0
$456$	0	0
$457$	160.288	0.0164069	0.00820344	−	0.999966i	$-0.497389\pi$
$457$	160.288	0.0164069	0.00820344	+	0.999966i	$0.497389\pi$
$458$	−8734.60	−0.891138
$459$	0	0
$460$	18221.9	1.84696
$461$	−2408.80	−0.243360	−0.121680	−	0.992569i	$-0.538828\pi$
$461$	−2408.80	−0.243360	−0.121680	+	0.992569i	$0.538828\pi$
$462$	0	0
$463$	−1092.89	−0.109699	−0.0548496	−	0.998495i	$-0.517468\pi$
$463$	−1092.89	−0.109699	−0.0548496	+	0.998495i	$0.517468\pi$
$464$	−3803.20	−0.380516
$465$	0	0
$466$	−32896.8	−3.27021
$467$	−15054.8	−1.49177	−0.745884	−	0.666076i	$-0.767973\pi$
$467$	−15054.8	−1.49177	−0.745884	+	0.666076i	$0.767973\pi$
$468$	0	0
$469$	0	0
$470$	−43058.3	−4.22581
$471$	0	0
$472$	18061.5	1.76133
$473$	−537.745	−0.0522739
$474$	0	0
$475$	12802.2	1.23665
$476$	0	0
$477$	0	0
$478$	−8617.73	−0.824615
$479$	−10455.6	−0.997346	−0.498673	−	0.866790i	$-0.666179\pi$
$479$	−10455.6	−0.997346	−0.498673	+	0.866790i	$0.666179\pi$
$480$	0	0
$481$	−21468.7	−2.03511
$482$	−11819.5	−1.11693
$483$	0	0
$484$	−6536.18	−0.613841
$485$	−18139.7	−1.69831
$486$	0	0
$487$	−12717.7	−1.18335	−0.591675	−	0.806176i	$-0.701533\pi$
$487$	−12717.7	−1.18335	−0.591675	+	0.806176i	$0.701533\pi$
$488$	5175.47	0.480087
$489$	0	0
$490$	0	0
$491$	−20983.1	−1.92863	−0.964313	−	0.264763i	$-0.914706\pi$
$491$	−20983.1	−1.92863	−0.964313	+	0.264763i	$0.914706\pi$
$492$	0	0
$493$	514.863	0.0470350
$494$	−42649.8	−3.88442
$495$	0	0
$496$	12033.6	1.08937
$497$	0	0
$498$	0	0
$499$	12718.1	1.14096	0.570480	−	0.821311i	$-0.306757\pi$
$499$	12718.1	1.14096	0.570480	+	0.821311i	$0.306757\pi$
$500$	−2427.11	−0.217087
$501$	0	0
$502$	−33496.0	−2.97809
$503$	−15675.9	−1.38957	−0.694785	−	0.719218i	$-0.744500\pi$
$503$	−15675.9	−1.38957	−0.694785	+	0.719218i	$0.744500\pi$
$504$	0	0
$505$	−29028.0	−2.55788
$506$	9544.50	0.838547
$507$	0	0
$508$	16527.0	1.44344
$509$	−10218.3	−0.889819	−0.444910	−	0.895575i	$-0.646764\pi$
$509$	−10218.3	−0.889819	−0.444910	+	0.895575i	$0.646764\pi$
$510$	0	0
$511$	0	0
$512$	−4500.87	−0.388501
$513$	0	0
$514$	−1448.92	−0.124337
$515$	15363.1	1.31452
$516$	0	0
$517$	−16365.8	−1.39220
$518$	0	0
$519$	0	0
$520$	−80328.7	−6.77432
$521$	−4808.13	−0.404315	−0.202157	−	0.979353i	$-0.564795\pi$
$521$	−4808.13	−0.404315	−0.202157	+	0.979353i	$0.564795\pi$
$522$	0	0
$523$	9936.18	0.830744	0.415372	−	0.909652i	$-0.363651\pi$
$523$	9936.18	0.830744	0.415372	+	0.909652i	$0.363651\pi$
$524$	41497.8	3.45962
$525$	0	0
$526$	20114.4	1.66736
$527$	−1629.06	−0.134655
$528$	0	0
$529$	−9110.36	−0.748777
$530$	22307.2	1.82823
$531$	0	0
$532$	0	0
$533$	17316.6	1.40725
$534$	0	0
$535$	10836.5	0.875709
$536$	35980.5	2.89948
$537$	0	0
$538$	−46208.4	−3.70294
$539$	0	0
$540$	0	0
$541$	−2048.66	−0.162808	−0.0814038	−	0.996681i	$-0.525940\pi$
$541$	−2048.66	−0.162808	−0.0814038	+	0.996681i	$0.525940\pi$
$542$	28509.7	2.25940
$543$	0	0
$544$	17111.0	1.34858
$545$	26714.1	2.09964
$546$	0	0
$547$	6154.72	0.481091	0.240546	−	0.970638i	$-0.422674\pi$
$547$	6154.72	0.481091	0.240546	+	0.970638i	$0.422674\pi$
$548$	25820.5	2.01277
$549$	0	0
$550$	20308.2	1.57444
$551$	−1930.22	−0.149238
$552$	0	0
$553$	0	0
$554$	2384.32	0.182852
$555$	0	0
$556$	22600.0	1.72384
$557$	4446.75	0.338267	0.169134	−	0.985593i	$-0.445903\pi$
$557$	4446.75	0.338267	0.169134	+	0.985593i	$0.445903\pi$
$558$	0	0
$559$	1220.72	0.0923631
$560$	0	0
$561$	0	0
$562$	−20338.7	−1.52658
$563$	9686.59	0.725117	0.362559	−	0.931961i	$-0.381903\pi$
$563$	9686.59	0.725117	0.362559	+	0.931961i	$0.381903\pi$
$564$	0	0
$565$	−13672.1	−1.01803
$566$	−9779.35	−0.726248
$567$	0	0
$568$	−58815.7	−4.34481
$569$	−7306.92	−0.538351	−0.269176	−	0.963091i	$-0.586751\pi$
$569$	−7306.92	−0.538351	−0.269176	+	0.963091i	$0.586751\pi$
$570$	0	0
$571$	9109.34	0.667625	0.333813	−	0.942639i	$-0.391665\pi$
$571$	9109.34	0.667625	0.333813	+	0.942639i	$0.391665\pi$
$572$	−49093.6	−3.58864
$573$	0	0
$574$	0	0
$575$	6503.72	0.471694
$576$	0	0
$577$	18707.7	1.34976	0.674880	−	0.737927i	$-0.264195\pi$
$577$	18707.7	1.34976	0.674880	+	0.737927i	$0.264195\pi$
$578$	21979.4	1.58170
$579$	0	0
$580$	−5845.67	−0.418497
$581$	0	0
$582$	0	0
$583$	8478.66	0.602316
$584$	26474.3	1.87588
$585$	0	0
$586$	31404.4	2.21383
$587$	−24610.4	−1.73046	−0.865230	−	0.501375i	$-0.832828\pi$
$587$	−24610.4	−1.73046	−0.865230	+	0.501375i	$0.832828\pi$
$588$	0	0
$589$	6107.38	0.427250
$590$	21380.0	1.49187
$591$	0	0
$592$	−63432.0	−4.40378
$593$	18840.0	1.30466	0.652332	−	0.757933i	$-0.273791\pi$
$593$	18840.0	1.30466	0.652332	+	0.757933i	$0.273791\pi$
$594$	0	0
$595$	0	0
$596$	54819.2	3.76759
$597$	0	0
$598$	−21666.7	−1.48163
$599$	−20647.6	−1.40841	−0.704206	−	0.709996i	$-0.748697\pi$
$599$	−20647.6	−1.40841	−0.704206	+	0.709996i	$0.748697\pi$
$600$	0	0
$601$	15772.8	1.07053	0.535264	−	0.844685i	$-0.320212\pi$
$601$	15772.8	1.07053	0.535264	+	0.844685i	$0.320212\pi$
$602$	0	0
$603$	0	0
$604$	40827.1	2.75038
$605$	−4811.79	−0.323350
$606$	0	0
$607$	−5182.64	−0.346552	−0.173276	−	0.984873i	$-0.555435\pi$
$607$	−5182.64	−0.346552	−0.173276	+	0.984873i	$0.555435\pi$
$608$	−64149.2	−4.27894
$609$	0	0
$610$	6126.39	0.406640
$611$	37151.6	2.45989
$612$	0	0
$613$	−28839.1	−1.90016	−0.950081	−	0.312004i	$-0.899000\pi$
$613$	−28839.1	−1.90016	−0.950081	+	0.312004i	$0.899000\pi$
$614$	10664.4	0.700946
$615$	0	0
$616$	0	0
$617$	−5114.80	−0.333734	−0.166867	−	0.985979i	$-0.553365\pi$
$617$	−5114.80	−0.333734	−0.166867	+	0.985979i	$0.553365\pi$
$618$	0	0
$619$	−29213.9	−1.89694	−0.948471	−	0.316864i	$-0.897370\pi$
$619$	−29213.9	−1.89694	−0.948471	+	0.316864i	$0.897370\pi$
$620$	18496.1	1.19810
$621$	0	0
$622$	−6258.87	−0.403469
$623$	0	0
$624$	0	0
$625$	−16491.3	−1.05544
$626$	−24265.4	−1.54927
$627$	0	0
$628$	55557.8	3.53025
$629$	8587.18	0.544345
$630$	0	0
$631$	19557.5	1.23387	0.616934	−	0.787015i	$-0.288374\pi$
$631$	19557.5	1.23387	0.616934	+	0.787015i	$0.288374\pi$
$632$	−73079.0	−4.59957
$633$	0	0
$634$	25258.1	1.58222
$635$	12166.8	0.760356
$636$	0	0
$637$	0	0
$638$	−3061.92	−0.190004
$639$	0	0
$640$	−49983.9	−3.08717
$641$	14632.3	0.901624	0.450812	−	0.892619i	$-0.351135\pi$
$641$	14632.3	0.901624	0.450812	+	0.892619i	$0.351135\pi$
$642$	0	0
$643$	23808.1	1.46019	0.730094	−	0.683347i	$-0.239476\pi$
$643$	23808.1	1.46019	0.730094	+	0.683347i	$0.239476\pi$
$644$	0	0
$645$	0	0
$646$	17059.3	1.03899
$647$	−17322.2	−1.05256	−0.526279	−	0.850312i	$-0.676413\pi$
$647$	−17322.2	−1.05256	−0.526279	+	0.850312i	$0.676413\pi$
$648$	0	0
$649$	8126.23	0.491498
$650$	−46101.1	−2.78190
$651$	0	0
$652$	−65596.4	−3.94011
$653$	−3241.68	−0.194268	−0.0971338	−	0.995271i	$-0.530967\pi$
$653$	−3241.68	−0.194268	−0.0971338	+	0.995271i	$0.530967\pi$
$654$	0	0
$655$	30549.8	1.82241
$656$	51164.2	3.04516
$657$	0	0
$658$	0	0
$659$	−16358.2	−0.966958	−0.483479	−	0.875356i	$-0.660627\pi$
$659$	−16358.2	−0.966958	−0.483479	+	0.875356i	$0.660627\pi$
$660$	0	0
$661$	−12572.6	−0.739814	−0.369907	−	0.929069i	$-0.620611\pi$
$661$	−12572.6	−0.739814	−0.369907	+	0.929069i	$0.620611\pi$
$662$	16106.4	0.945606
$663$	0	0
$664$	−32199.5	−1.88190
$665$	0	0
$666$	0	0
$667$	−980.582	−0.0569240
$668$	69078.5	4.00109
$669$	0	0
$670$	42591.5	2.45590
$671$	2328.55	0.133968
$672$	0	0
$673$	13130.7	0.752082	0.376041	−	0.926603i	$-0.377285\pi$
$673$	13130.7	0.752082	0.376041	+	0.926603i	$0.377285\pi$
$674$	−39573.4	−2.26159
$675$	0	0
$676$	64959.6	3.69592
$677$	18624.2	1.05729	0.528644	−	0.848843i	$-0.322701\pi$
$677$	18624.2	1.05729	0.528644	+	0.848843i	$0.322701\pi$
$678$	0	0
$679$	0	0
$680$	32130.4	1.81198
$681$	0	0
$682$	9688.13	0.543956
$683$	25377.0	1.42170	0.710852	−	0.703341i	$-0.248309\pi$
$683$	25377.0	1.42170	0.710852	+	0.703341i	$0.248309\pi$
$684$	0	0
$685$	19008.5	1.06026
$686$	0	0
$687$	0	0
$688$	3606.78	0.199865
$689$	−19247.2	−1.06424
$690$	0	0
$691$	−2135.42	−0.117562	−0.0587810	−	0.998271i	$-0.518721\pi$
$691$	−2135.42	−0.117562	−0.0587810	+	0.998271i	$0.518721\pi$
$692$	−43086.2	−2.36690
$693$	0	0
$694$	44869.6	2.45422
$695$	16637.6	0.908058
$696$	0	0
$697$	−6926.41	−0.376408
$698$	−1805.20	−0.0978910
$699$	0	0
$700$	0	0
$701$	9679.27	0.521513	0.260757	−	0.965405i	$-0.416028\pi$
$701$	9679.27	0.521513	0.260757	+	0.965405i	$0.416028\pi$
$702$	0	0
$703$	−32193.4	−1.72717
$704$	−46916.8	−2.51171
$705$	0	0
$706$	−29473.7	−1.57119
$707$	0	0
$708$	0	0
$709$	−25743.2	−1.36362	−0.681809	−	0.731530i	$-0.738807\pi$
$709$	−25743.2	−1.36362	−0.681809	+	0.731530i	$0.738807\pi$
$710$	−69622.3	−3.68011
$711$	0	0
$712$	−23595.7	−1.24197
$713$	3102.63	0.162966
$714$	0	0
$715$	−36141.6	−1.89038
$716$	75798.9	3.95634
$717$	0	0
$718$	38862.6	2.01997
$719$	10508.5	0.545065	0.272532	−	0.962147i	$-0.412139\pi$
$719$	10508.5	0.545065	0.272532	+	0.962147i	$0.412139\pi$
$720$	0	0
$721$	0	0
$722$	−26917.5	−1.38749
$723$	0	0
$724$	−34644.6	−1.77839
$725$	−2086.42	−0.106880
$726$	0	0
$727$	24259.4	1.23759	0.618797	−	0.785551i	$-0.287620\pi$
$727$	24259.4	1.23759	0.618797	+	0.785551i	$0.287620\pi$
$728$	0	0
$729$	0	0
$730$	31338.6	1.58890
$731$	−488.271	−0.0247050
$732$	0	0
$733$	−19632.2	−0.989263	−0.494632	−	0.869103i	$-0.664697\pi$
$733$	−19632.2	−0.989263	−0.494632	+	0.869103i	$0.664697\pi$
$734$	34151.3	1.71737
$735$	0	0
$736$	−32588.7	−1.63212
$737$	16188.4	0.809101
$738$	0	0
$739$	−26353.6	−1.31182	−0.655909	−	0.754840i	$-0.727714\pi$
$739$	−26353.6	−1.31182	−0.655909	+	0.754840i	$0.727714\pi$
$740$	−97497.6	−4.84335
$741$	0	0
$742$	0	0
$743$	31464.6	1.55360	0.776799	−	0.629749i	$-0.216842\pi$
$743$	31464.6	1.55360	0.776799	+	0.629749i	$0.216842\pi$
$744$	0	0
$745$	40356.7	1.98464
$746$	59030.4	2.89713
$747$	0	0
$748$	19636.7	0.959880
$749$	0	0
$750$	0	0
$751$	5411.52	0.262942	0.131471	−	0.991320i	$-0.458030\pi$
$751$	5411.52	0.262942	0.131471	+	0.991320i	$0.458030\pi$
$752$	109769.	5.32296
$753$	0	0
$754$	6950.77	0.335719
$755$	30056.0	1.44881
$756$	0	0
$757$	3607.94	0.173227	0.0866135	−	0.996242i	$-0.472395\pi$
$757$	3607.94	0.173227	0.0866135	+	0.996242i	$0.472395\pi$
$758$	−32529.1	−1.55872
$759$	0	0
$760$	−120457.	−5.74926
$761$	−4663.70	−0.222154	−0.111077	−	0.993812i	$-0.535430\pi$
$761$	−4663.70	−0.222154	−0.111077	+	0.993812i	$0.535430\pi$
$762$	0	0
$763$	0	0
$764$	−59769.5	−2.83035
$765$	0	0
$766$	36979.1	1.74427
$767$	−18447.1	−0.868431
$768$	0	0
$769$	9725.21	0.456047	0.228023	−	0.973656i	$-0.426774\pi$
$769$	9725.21	0.456047	0.228023	+	0.973656i	$0.426774\pi$
$770$	0	0
$771$	0	0
$772$	−35071.7	−1.63505
$773$	3092.80	0.143907	0.0719536	−	0.997408i	$-0.477077\pi$
$773$	3092.80	0.143907	0.0719536	+	0.997408i	$0.477077\pi$
$774$	0	0
$775$	6601.60	0.305982
$776$	82748.8	3.82798
$777$	0	0
$778$	27873.7	1.28447
$779$	25967.2	1.19431
$780$	0	0
$781$	−26462.4	−1.21242
$782$	8666.37	0.396303
$783$	0	0
$784$	0	0
$785$	40900.4	1.85962
$786$	0	0
$787$	−22812.9	−1.03328	−0.516640	−	0.856203i	$-0.672817\pi$
$787$	−22812.9	−1.03328	−0.516640	+	0.856203i	$0.672817\pi$
$788$	−40006.9	−1.80861
$789$	0	0
$790$	−86506.3	−3.89589
$791$	0	0
$792$	0	0
$793$	−5285.98	−0.236710
$794$	1861.71	0.0832113
$795$	0	0
$796$	−29469.1	−1.31219
$797$	34305.4	1.52467	0.762334	−	0.647184i	$-0.224053\pi$
$797$	34305.4	1.52467	0.762334	+	0.647184i	$0.224053\pi$
$798$	0	0
$799$	−14860.1	−0.657963
$800$	−69340.3	−3.06444
$801$	0	0
$802$	27533.3	1.21226
$803$	11911.4	0.523465
$804$	0	0
$805$	0	0
$806$	−21992.8	−0.961119
$807$	0	0
$808$	132418.	5.76542
$809$	−3264.80	−0.141884	−0.0709421	−	0.997480i	$-0.522601\pi$
$809$	−3264.80	−0.141884	−0.0709421	+	0.997480i	$0.522601\pi$
$810$	0	0
$811$	−27264.0	−1.18048	−0.590239	−	0.807228i	$-0.700967\pi$
$811$	−27264.0	−1.18048	−0.590239	+	0.807228i	$0.700967\pi$
$812$	0	0
$813$	0	0
$814$	−51068.4	−2.19895
$815$	−48290.6	−2.07552
$816$	0	0
$817$	1830.53	0.0783871
$818$	−1744.84	−0.0745805
$819$	0	0
$820$	78641.4	3.34912
$821$	36334.2	1.54454	0.772272	−	0.635292i	$-0.219120\pi$
$821$	36334.2	1.54454	0.772272	+	0.635292i	$0.219120\pi$
$822$	0	0
$823$	7791.80	0.330019	0.165009	−	0.986292i	$-0.447235\pi$
$823$	7791.80	0.330019	0.165009	+	0.986292i	$0.447235\pi$
$824$	−70082.4	−2.96291
$825$	0	0
$826$	0	0
$827$	−36082.7	−1.51719	−0.758596	−	0.651561i	$-0.774114\pi$
$827$	−36082.7	−1.51719	−0.758596	+	0.651561i	$0.774114\pi$
$828$	0	0
$829$	42995.0	1.80130	0.900650	−	0.434545i	$-0.143091\pi$
$829$	42995.0	1.80130	0.900650	+	0.434545i	$0.143091\pi$
$830$	−38115.7	−1.59399
$831$	0	0
$832$	106505.	4.43796
$833$	0	0
$834$	0	0
$835$	50854.0	2.10764
$836$	−73618.3	−3.04562
$837$	0	0
$838$	−23845.9	−0.982986
$839$	−28252.8	−1.16257	−0.581283	−	0.813701i	$-0.697449\pi$
$839$	−28252.8	−1.16257	−0.581283	+	0.813701i	$0.697449\pi$
$840$	0	0
$841$	−24074.4	−0.987102
$842$	−7448.49	−0.304860
$843$	0	0
$844$	70132.0	2.86024
$845$	47821.8	1.94689
$846$	0	0
$847$	0	0
$848$	−56868.2	−2.30290
$849$	0	0
$850$	18439.8	0.744093
$851$	−16354.7	−0.658793
$852$	0	0
$853$	−28994.8	−1.16385	−0.581924	−	0.813243i	$-0.697700\pi$
$853$	−28994.8	−1.16385	−0.581924	+	0.813243i	$0.697700\pi$
$854$	0	0
$855$	0	0
$856$	−49433.5	−1.97384
$857$	8632.87	0.344099	0.172050	−	0.985088i	$-0.444961\pi$
$857$	8632.87	0.344099	0.172050	+	0.985088i	$0.444961\pi$
$858$	0	0
$859$	−33947.1	−1.34838	−0.674191	−	0.738557i	$-0.735507\pi$
$859$	−33947.1	−1.34838	−0.674191	+	0.738557i	$0.735507\pi$
$860$	5543.76	0.219815
$861$	0	0
$862$	8941.09	0.353288
$863$	70.6612	0.00278718	0.00139359	−	0.999999i	$-0.499556\pi$
$863$	70.6612	0.00278718	0.00139359	+	0.999999i	$0.499556\pi$
$864$	0	0
$865$	−31719.1	−1.24680
$866$	46507.5	1.82493
$867$	0	0
$868$	0	0
$869$	−32879.8	−1.28351
$870$	0	0
$871$	−36748.8	−1.42961
$872$	−121863.	−4.73257
$873$	0	0
$874$	−32490.3	−1.25744
$875$	0	0
$876$	0	0
$877$	−11175.3	−0.430289	−0.215144	−	0.976582i	$-0.569022\pi$
$877$	−11175.3	−0.430289	−0.215144	+	0.976582i	$0.569022\pi$
$878$	−32266.5	−1.24025
$879$	0	0
$880$	−106785.	−4.09059
$881$	−14341.3	−0.548433	−0.274216	−	0.961668i	$-0.588418\pi$
$881$	−14341.3	−0.548433	−0.274216	+	0.961668i	$0.588418\pi$
$882$	0	0
$883$	−23559.2	−0.897884	−0.448942	−	0.893561i	$-0.648199\pi$
$883$	−23559.2	−0.897884	−0.448942	+	0.893561i	$0.648199\pi$
$884$	−44576.8	−1.69602
$885$	0	0
$886$	4355.20	0.165142
$887$	−31744.4	−1.20166	−0.600831	−	0.799376i	$-0.705163\pi$
$887$	−31744.4	−1.20166	−0.600831	+	0.799376i	$0.705163\pi$
$888$	0	0
$889$	0	0
$890$	−27931.1	−1.05197
$891$	0	0
$892$	−117993.	−4.42904
$893$	55710.7	2.08767
$894$	0	0
$895$	55801.4	2.08406
$896$	0	0
$897$	0	0
$898$	33765.8	1.25477
$899$	−995.339	−0.0369259
$900$	0	0
$901$	7698.59	0.284658
$902$	41191.8	1.52055
$903$	0	0
$904$	62368.5	2.29463
$905$	−25504.6	−0.936796
$906$	0	0
$907$	6062.67	0.221949	0.110974	−	0.993823i	$-0.464603\pi$
$907$	6062.67	0.221949	0.110974	+	0.993823i	$0.464603\pi$
$908$	30636.6	1.11973
$909$	0	0
$910$	0	0
$911$	25862.9	0.940589	0.470295	−	0.882509i	$-0.344148\pi$
$911$	25862.9	0.940589	0.470295	+	0.882509i	$0.344148\pi$
$912$	0	0
$913$	−14487.2	−0.525144
$914$	−865.539	−0.0313233
$915$	0	0
$916$	34225.7	1.23455
$917$	0	0
$918$	0	0
$919$	1455.14	0.0522314	0.0261157	−	0.999659i	$-0.491686\pi$
$919$	1455.14	0.0522314	0.0261157	+	0.999659i	$0.491686\pi$
$920$	−61193.9	−2.19294
$921$	0	0
$922$	13007.3	0.464612
$923$	60071.6	2.14223
$924$	0	0
$925$	−34798.6	−1.23694
$926$	5901.48	0.209433
$927$	0	0
$928$	10454.6	0.369816
$929$	3077.74	0.108695	0.0543474	−	0.998522i	$-0.482692\pi$
$929$	3077.74	0.108695	0.0543474	+	0.998522i	$0.482692\pi$
$930$	0	0
$931$	0	0
$932$	128903.	4.53042
$933$	0	0
$934$	81294.8	2.84802
$935$	14456.1	0.505632
$936$	0	0
$937$	−5354.80	−0.186695	−0.0933477	−	0.995634i	$-0.529757\pi$
$937$	−5354.80	−0.186695	−0.0933477	+	0.995634i	$0.529757\pi$
$938$	0	0
$939$	0	0
$940$	168719.	5.85428
$941$	−47797.8	−1.65586	−0.827930	−	0.560832i	$-0.810482\pi$
$941$	−47797.8	−1.65586	−0.827930	+	0.560832i	$0.810482\pi$
$942$	0	0
$943$	13191.7	0.455547
$944$	−54504.4	−1.87920
$945$	0	0
$946$	2903.78	0.0997990
$947$	−2491.69	−0.0855006	−0.0427503	−	0.999086i	$-0.513612\pi$
$947$	−2491.69	−0.0855006	−0.0427503	+	0.999086i	$0.513612\pi$
$948$	0	0
$949$	−27039.6	−0.924914
$950$	−69130.9	−2.36095
$951$	0	0
$952$	0	0
$953$	−13130.4	−0.446313	−0.223156	−	0.974783i	$-0.571636\pi$
$953$	−13130.4	−0.446313	−0.223156	+	0.974783i	$0.571636\pi$
$954$	0	0
$955$	−44001.0	−1.49093
$956$	33767.7	1.14239
$957$	0	0
$958$	56459.3	1.90409
$959$	0	0
$960$	0	0
$961$	−26641.7	−0.894286
$962$	115929.	3.88534
$963$	0	0
$964$	46313.4	1.54736
$965$	−25819.0	−0.861287
$966$	0	0
$967$	43314.8	1.44044	0.720222	−	0.693743i	$-0.244040\pi$
$967$	43314.8	1.44044	0.720222	+	0.693743i	$0.244040\pi$
$968$	21950.2	0.728827
$969$	0	0
$970$	97952.8	3.24235
$971$	−19755.8	−0.652929	−0.326464	−	0.945210i	$-0.605857\pi$
$971$	−19755.8	−0.652929	−0.326464	+	0.945210i	$0.605857\pi$
$972$	0	0
$973$	0	0
$974$	68674.1	2.25920
$975$	0	0
$976$	−15618.1	−0.512217
$977$	−11140.1	−0.364794	−0.182397	−	0.983225i	$-0.558386\pi$
$977$	−11140.1	−0.364794	−0.182397	+	0.983225i	$0.558386\pi$
$978$	0	0
$979$	−10616.2	−0.346573
$980$	0	0
$981$	0	0
$982$	113307.	3.68205
$983$	−3575.76	−0.116021	−0.0580107	−	0.998316i	$-0.518476\pi$
$983$	−3575.76	−0.116021	−0.0580107	+	0.998316i	$0.518476\pi$
$984$	0	0
$985$	−29452.2	−0.952716
$986$	−2780.21	−0.0897971
$987$	0	0
$988$	167119.	5.38133
$989$	929.937	0.0298992
$990$	0	0
$991$	22391.8	0.717760	0.358880	−	0.933384i	$-0.383159\pi$
$991$	22391.8	0.717760	0.358880	+	0.933384i	$0.383159\pi$
$992$	−33079.2	−1.05873
$993$	0	0
$994$	0	0
$995$	−21694.5	−0.691218
$996$	0	0
$997$	−17466.5	−0.554834	−0.277417	−	0.960750i	$-0.589478\pi$
$997$	−17466.5	−0.554834	−0.277417	+	0.960750i	$0.589478\pi$
$998$	−68676.4	−2.17827
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.x.1.2	yes	8
3.2	odd	2	inner	441.4.a.x.1.7	yes	8
7.2	even	3		441.4.e.z.361.7		16
7.3	odd	6		441.4.e.z.226.8		16
7.4	even	3		441.4.e.z.226.7		16
7.5	odd	6		441.4.e.z.361.8		16
7.6	odd	2	inner	441.4.a.x.1.1	✓	8
21.2	odd	6		441.4.e.z.361.2		16
21.5	even	6		441.4.e.z.361.1		16
21.11	odd	6		441.4.e.z.226.2		16
21.17	even	6		441.4.e.z.226.1		16
21.20	even	2	inner	441.4.a.x.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.4.a.x.1.1	✓	8	7.6	odd	2	inner
441.4.a.x.1.2	yes	8	1.1	even	1	trivial
441.4.a.x.1.7	yes	8	3.2	odd	2	inner
441.4.a.x.1.8	yes	8	21.20	even	2	inner
441.4.e.z.226.1		16	21.17	even	6
441.4.e.z.226.2		16	21.11	odd	6
441.4.e.z.226.7		16	7.4	even	3
441.4.e.z.226.8		16	7.3	odd	6
441.4.e.z.361.1		16	21.5	even	6
441.4.e.z.361.2		16	21.2	odd	6
441.4.e.z.361.7		16	7.2	even	3
441.4.e.z.361.8		16	7.5	odd	6

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-5.39991 q^{2} +21.1590 q^{4} +15.5768 q^{5} -71.0573 q^{8} +O(q^{10})\)	\(q-5.39991 q^{2} +21.1590 q^{4} +15.5768 q^{5} -71.0573 q^{8} -84.1131 q^{10} -31.9702 q^{11} +72.5746 q^{13} +214.431 q^{16} -29.0288 q^{17} +108.829 q^{19} +329.589 q^{20} +172.636 q^{22} +55.2869 q^{23} +117.636 q^{25} -391.896 q^{26} -17.7363 q^{29} +56.1189 q^{31} -589.448 q^{32} +156.753 q^{34} -295.816 q^{37} -587.668 q^{38} -1106.84 q^{40} +238.605 q^{41} +16.8202 q^{43} -676.457 q^{44} -298.544 q^{46} +511.909 q^{47} -635.223 q^{50} +1535.60 q^{52} -265.205 q^{53} -497.992 q^{55} +95.7742 q^{58} -254.181 q^{59} -72.8352 q^{61} -303.037 q^{62} +1467.52 q^{64} +1130.48 q^{65} -506.360 q^{67} -614.220 q^{68} +827.722 q^{71} -372.577 q^{73} +1597.38 q^{74} +2302.72 q^{76} +1028.45 q^{79} +3340.14 q^{80} -1288.44 q^{82} +453.148 q^{83} -452.175 q^{85} -90.8276 q^{86} +2271.71 q^{88} +332.065 q^{89} +1169.81 q^{92} -2764.26 q^{94} +1695.21 q^{95} -1164.54 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 68 q^{4}+O(q^{10}) \) 8 * q + 68 * q^4	\( 8 q + 68 q^{4} + 804 q^{16} + 976 q^{22} + 536 q^{25} + 64 q^{37} + 2160 q^{43} - 768 q^{46} + 2184 q^{58} + 7588 q^{64} + 5392 q^{79} + 2864 q^{85} + 5616 q^{88}+O(q^{100}) \) 8 * q + 68 * q^4 + 804 * q^16 + 976 * q^22 + 536 * q^25 + 64 * q^37 + 2160 * q^43 - 768 * q^46 + 2184 * q^58 + 7588 * q^64 + 5392 * q^79 + 2864 * q^85 + 5616 * q^88

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