LMFDB - Embedded newform 2205.4.a.bz.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1163891200.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 $ x^6 - 2x^5 - 23x^4 + 12x^3 + 154x^2 + 152*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.241849$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.65606 q^{2} -5.25746 q^{4} -5.00000 q^{5} +21.9552 q^{8} +O(q^{10})$	$q-1.65606 q^{2} -5.25746 q^{4} -5.00000 q^{5} +21.9552 q^{8} +8.28031 q^{10} -69.5726 q^{11} -68.4326 q^{13} +5.70053 q^{16} +104.332 q^{17} -71.8929 q^{19} +26.2873 q^{20} +115.217 q^{22} +101.031 q^{23} +25.0000 q^{25} +113.329 q^{26} +114.661 q^{29} +73.6505 q^{31} -185.082 q^{32} -172.780 q^{34} -200.933 q^{37} +119.059 q^{38} -109.776 q^{40} +417.308 q^{41} +311.175 q^{43} +365.775 q^{44} -167.313 q^{46} +149.697 q^{47} -41.4016 q^{50} +359.781 q^{52} -271.474 q^{53} +347.863 q^{55} -189.885 q^{58} -518.028 q^{59} +219.926 q^{61} -121.970 q^{62} +260.903 q^{64} +342.163 q^{65} +80.6950 q^{67} -548.520 q^{68} +91.0463 q^{71} -882.282 q^{73} +332.758 q^{74} +377.974 q^{76} +599.877 q^{79} -28.5026 q^{80} -691.087 q^{82} +70.8820 q^{83} -521.659 q^{85} -515.325 q^{86} -1527.48 q^{88} +802.592 q^{89} -531.165 q^{92} -247.908 q^{94} +359.465 q^{95} -145.648 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 14 * q^4 - 30 * q^5 + 66 * q^8	$ 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8} - 10 q^{10} + 16 q^{11} - 168 q^{13} + 298 q^{16} - 4 q^{17} - 308 q^{19} - 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} + 56 q^{26} - 176 q^{29} - 392 q^{31} + 770 q^{32} - 812 q^{34} - 140 q^{37} + 20 q^{38} - 330 q^{40} + 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} + 628 q^{47} + 50 q^{50} - 1520 q^{52} + 676 q^{53} - 80 q^{55} - 2012 q^{58} + 996 q^{59} - 740 q^{61} - 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} - 2940 q^{68} + 224 q^{71} - 2640 q^{73} - 928 q^{74} + 1340 q^{76} + 1636 q^{79} - 1490 q^{80} + 1756 q^{82} + 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 3332 q^{94} + 1540 q^{95} - 516 q^{97}+O(q^{100}) $ 6 * q + 2 * q^2 + 14 * q^4 - 30 * q^5 + 66 * q^8 - 10 * q^10 + 16 * q^11 - 168 * q^13 + 298 * q^16 - 4 * q^17 - 308 * q^19 - 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 + 56 * q^26 - 176 * q^29 - 392 * q^31 + 770 * q^32 - 812 * q^34 - 140 * q^37 + 20 * q^38 - 330 * q^40 + 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 + 628 * q^47 + 50 * q^50 - 1520 * q^52 + 676 * q^53 - 80 * q^55 - 2012 * q^58 + 996 * q^59 - 740 * q^61 - 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 - 2940 * q^68 + 224 * q^71 - 2640 * q^73 - 928 * q^74 + 1340 * q^76 + 1636 * q^79 - 1490 * q^80 + 1756 * q^82 + 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 - 1904 * q^89 + 1952 * q^92 + 3332 * q^94 + 1540 * q^95 - 516 * 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 $

$2$	−1.65606	−0.585506	−0.292753	−	0.956188i	$-0.594571\pi$
$2$	−1.65606	−0.585506	−0.292753	+	0.956188i	$0.594571\pi$
$3$	0	0
$3$	0	0
$4$	−5.25746	−0.657182
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	21.9552	0.970291
$9$	0	0
$10$	8.28031	0.261846
$11$	−69.5726	−1.90699	−0.953497	−	0.301402i	$-0.902545\pi$
$11$	−69.5726	−1.90699	−0.953497	+	0.301402i	$0.902545\pi$
$12$	0	0
$13$	−68.4326	−1.45998	−0.729991	−	0.683456i	$-0.760476\pi$
$13$	−68.4326	−1.45998	−0.729991	+	0.683456i	$0.760476\pi$
$14$	0	0
$15$	0	0
$16$	5.70053	0.0890707
$17$	104.332	1.48848	0.744240	−	0.667912i	$-0.232812\pi$
$17$	104.332	1.48848	0.744240	+	0.667912i	$0.232812\pi$
$18$	0	0
$19$	−71.8929	−0.868072	−0.434036	−	0.900896i	$-0.642911\pi$
$19$	−71.8929	−0.868072	−0.434036	+	0.900896i	$0.642911\pi$
$20$	26.2873	0.293901
$21$	0	0
$22$	115.217	1.11656
$23$	101.031	0.915929	0.457964	−	0.888970i	$-0.348579\pi$
$23$	101.031	0.915929	0.457964	+	0.888970i	$0.348579\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	113.329	0.854829
$27$	0	0
$28$	0	0
$29$	114.661	0.734205	0.367102	−	0.930181i	$-0.380350\pi$
$29$	114.661	0.734205	0.367102	+	0.930181i	$0.380350\pi$
$30$	0	0
$31$	73.6505	0.426710	0.213355	−	0.976975i	$-0.431561\pi$
$31$	73.6505	0.426710	0.213355	+	0.976975i	$0.431561\pi$
$32$	−185.082	−1.02244
$33$	0	0
$34$	−172.780	−0.871514
$35$	0	0
$36$	0	0
$37$	−200.933	−0.892790	−0.446395	−	0.894836i	$-0.647292\pi$
$37$	−200.933	−0.892790	−0.446395	+	0.894836i	$0.647292\pi$
$38$	119.059	0.508262
$39$	0	0
$40$	−109.776	−0.433927
$41$	417.308	1.58957	0.794786	−	0.606889i	$-0.207583\pi$
$41$	417.308	1.58957	0.794786	+	0.606889i	$0.207583\pi$
$42$	0	0
$43$	311.175	1.10357	0.551787	−	0.833985i	$-0.313946\pi$
$43$	311.175	1.10357	0.551787	+	0.833985i	$0.313946\pi$
$44$	365.775	1.25324
$45$	0	0
$46$	−167.313	−0.536282
$47$	149.697	0.464586	0.232293	−	0.972646i	$-0.425377\pi$
$47$	149.697	0.464586	0.232293	+	0.972646i	$0.425377\pi$
$48$	0	0
$49$	0	0
$50$	−41.4016	−0.117101
$51$	0	0
$52$	359.781	0.959475
$53$	−271.474	−0.703582	−0.351791	−	0.936078i	$-0.614427\pi$
$53$	−271.474	−0.703582	−0.351791	+	0.936078i	$0.614427\pi$
$54$	0	0
$55$	347.863	0.852834
$56$	0	0
$57$	0	0
$58$	−189.885	−0.429882
$59$	−518.028	−1.14308	−0.571538	−	0.820575i	$-0.693653\pi$
$59$	−518.028	−1.14308	−0.571538	+	0.820575i	$0.693653\pi$
$60$	0	0
$61$	219.926	0.461616	0.230808	−	0.972999i	$-0.425863\pi$
$61$	219.926	0.461616	0.230808	+	0.972999i	$0.425863\pi$
$62$	−121.970	−0.249842
$63$	0	0
$64$	260.903	0.509576
$65$	342.163	0.652924
$66$	0	0
$67$	80.6950	0.147141	0.0735706	−	0.997290i	$-0.476561\pi$
$67$	80.6950	0.147141	0.0735706	+	0.997290i	$0.476561\pi$
$68$	−548.520	−0.978202
$69$	0	0
$70$	0	0
$71$	91.0463	0.152186	0.0760930	−	0.997101i	$-0.475755\pi$
$71$	91.0463	0.152186	0.0760930	+	0.997101i	$0.475755\pi$
$72$	0	0
$73$	−882.282	−1.41457	−0.707283	−	0.706931i	$-0.750079\pi$
$73$	−882.282	−1.41457	−0.707283	+	0.706931i	$0.750079\pi$
$74$	332.758	0.522734
$75$	0	0
$76$	377.974	0.570481
$77$	0	0
$78$	0	0
$79$	599.877	0.854322	0.427161	−	0.904175i	$-0.359514\pi$
$79$	599.877	0.854322	0.427161	+	0.904175i	$0.359514\pi$
$80$	−28.5026	−0.0398336
$81$	0	0
$82$	−691.087	−0.930705
$83$	70.8820	0.0937387	0.0468694	−	0.998901i	$-0.485076\pi$
$83$	70.8820	0.0937387	0.0468694	+	0.998901i	$0.485076\pi$
$84$	0	0
$85$	−521.659	−0.665668
$86$	−515.325	−0.646150
$87$	0	0
$88$	−1527.48	−1.85034
$89$	802.592	0.955894	0.477947	−	0.878389i	$-0.341381\pi$
$89$	802.592	0.955894	0.477947	+	0.878389i	$0.341381\pi$
$90$	0	0
$91$	0	0
$92$	−531.165	−0.601932
$93$	0	0
$94$	−247.908	−0.272018
$95$	359.465	0.388214
$96$	0	0
$97$	−145.648	−0.152457	−0.0762283	−	0.997090i	$-0.524288\pi$
$97$	−145.648	−0.152457	−0.0762283	+	0.997090i	$0.524288\pi$
$98$	0	0
$99$	0	0
$100$	−131.436	−0.131436
$101$	−619.435	−0.610259	−0.305129	−	0.952311i	$-0.598700\pi$
$101$	−619.435	−0.610259	−0.305129	+	0.952311i	$0.598700\pi$
$102$	0	0
$103$	1822.08	1.74306	0.871528	−	0.490345i	$-0.163129\pi$
$103$	1822.08	1.74306	0.871528	+	0.490345i	$0.163129\pi$
$104$	−1502.45	−1.41661
$105$	0	0
$106$	449.578	0.411952
$107$	−1089.70	−0.984536	−0.492268	−	0.870444i	$-0.663832\pi$
$107$	−1089.70	−0.984536	−0.492268	+	0.870444i	$0.663832\pi$
$108$	0	0
$109$	589.667	0.518164	0.259082	−	0.965855i	$-0.416580\pi$
$109$	589.667	0.518164	0.259082	+	0.965855i	$0.416580\pi$
$110$	−576.083	−0.499340
$111$	0	0
$112$	0	0
$113$	900.358	0.749544	0.374772	−	0.927117i	$-0.377721\pi$
$113$	900.358	0.749544	0.374772	+	0.927117i	$0.377721\pi$
$114$	0	0
$115$	−505.154	−0.409616
$116$	−602.823	−0.482506
$117$	0	0
$118$	857.887	0.669279
$119$	0	0
$120$	0	0
$121$	3509.35	2.63663
$122$	−364.210	−0.270279
$123$	0	0
$124$	−387.214	−0.280426
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1755.75	−1.22676	−0.613378	−	0.789790i	$-0.710190\pi$
$127$	−1755.75	−1.22676	−0.613378	+	0.789790i	$0.710190\pi$
$128$	1048.58	0.724082
$129$	0	0
$130$	−566.643	−0.382291
$131$	1809.10	1.20658	0.603289	−	0.797523i	$-0.293857\pi$
$131$	1809.10	1.20658	0.603289	+	0.797523i	$0.293857\pi$
$132$	0	0
$133$	0	0
$134$	−133.636	−0.0861521
$135$	0	0
$136$	2290.62	1.44426
$137$	18.5134	0.0115453	0.00577265	−	0.999983i	$-0.498162\pi$
$137$	18.5134	0.0115453	0.00577265	+	0.999983i	$0.498162\pi$
$138$	0	0
$139$	625.608	0.381751	0.190875	−	0.981614i	$-0.438867\pi$
$139$	625.608	0.381751	0.190875	+	0.981614i	$0.438867\pi$
$140$	0	0
$141$	0	0
$142$	−150.778	−0.0891059
$143$	4761.03	2.78418
$144$	0	0
$145$	−573.303	−0.328346
$146$	1461.11	0.828237
$147$	0	0
$148$	1056.40	0.586725
$149$	1028.63	0.565563	0.282782	−	0.959184i	$-0.408743\pi$
$149$	1028.63	0.565563	0.282782	+	0.959184i	$0.408743\pi$
$150$	0	0
$151$	71.0073	0.0382682	0.0191341	−	0.999817i	$-0.493909\pi$
$151$	71.0073	0.0382682	0.0191341	+	0.999817i	$0.493909\pi$
$152$	−1578.42	−0.842282
$153$	0	0
$154$	0	0
$155$	−368.252	−0.190831
$156$	0	0
$157$	2061.32	1.04784	0.523922	−	0.851767i	$-0.324468\pi$
$157$	2061.32	1.04784	0.523922	+	0.851767i	$0.324468\pi$
$158$	−993.434	−0.500211
$159$	0	0
$160$	925.409	0.457250
$161$	0	0
$162$	0	0
$163$	−1963.80	−0.943660	−0.471830	−	0.881690i	$-0.656406\pi$
$163$	−1963.80	−0.943660	−0.471830	+	0.881690i	$0.656406\pi$
$164$	−2193.98	−1.04464
$165$	0	0
$166$	−117.385	−0.0548846
$167$	2855.04	1.32293	0.661467	−	0.749974i	$-0.269934\pi$
$167$	2855.04	1.32293	0.661467	+	0.749974i	$0.269934\pi$
$168$	0	0
$169$	2486.01	1.13155
$170$	863.899	0.389753
$171$	0	0
$172$	−1635.99	−0.725249
$173$	−1553.21	−0.682590	−0.341295	−	0.939956i	$-0.610866\pi$
$173$	−1553.21	−0.682590	−0.341295	+	0.939956i	$0.610866\pi$
$174$	0	0
$175$	0	0
$176$	−396.601	−0.169857
$177$	0	0
$178$	−1329.14	−0.559682
$179$	−269.841	−0.112675	−0.0563376	−	0.998412i	$-0.517942\pi$
$179$	−269.841	−0.112675	−0.0563376	+	0.998412i	$0.517942\pi$
$180$	0	0
$181$	−2229.61	−0.915613	−0.457806	−	0.889052i	$-0.651365\pi$
$181$	−2229.61	−0.915613	−0.457806	+	0.889052i	$0.651365\pi$
$182$	0	0
$183$	0	0
$184$	2218.15	0.888717
$185$	1004.67	0.399268
$186$	0	0
$187$	−7258.63	−2.83852
$188$	−787.026	−0.305318
$189$	0	0
$190$	−595.296	−0.227302
$191$	465.920	0.176507	0.0882533	−	0.996098i	$-0.471871\pi$
$191$	465.920	0.176507	0.0882533	+	0.996098i	$0.471871\pi$
$192$	0	0
$193$	−4414.46	−1.64642	−0.823212	−	0.567734i	$-0.807820\pi$
$193$	−4414.46	−1.64642	−0.823212	+	0.567734i	$0.807820\pi$
$194$	241.202	0.0892643
$195$	0	0
$196$	0	0
$197$	289.812	0.104814	0.0524068	−	0.998626i	$-0.483311\pi$
$197$	289.812	0.104814	0.0524068	+	0.998626i	$0.483311\pi$
$198$	0	0
$199$	−4817.73	−1.71618	−0.858091	−	0.513498i	$-0.828349\pi$
$199$	−4817.73	−1.71618	−0.858091	+	0.513498i	$0.828349\pi$
$200$	548.879	0.194058
$201$	0	0
$202$	1025.82	0.357310
$203$	0	0
$204$	0	0
$205$	−2086.54	−0.710879
$206$	−3017.48	−1.02057
$207$	0	0
$208$	−390.102	−0.130042
$209$	5001.78	1.65541
$210$	0	0
$211$	2022.01	0.659719	0.329859	−	0.944030i	$-0.392999\pi$
$211$	2022.01	0.659719	0.329859	+	0.944030i	$0.392999\pi$
$212$	1427.26	0.462382
$213$	0	0
$214$	1804.61	0.576452
$215$	−1555.87	−0.493533
$216$	0	0
$217$	0	0
$218$	−976.525	−0.303388
$219$	0	0
$220$	−1828.88	−0.560467
$221$	−7139.69	−2.17315
$222$	0	0
$223$	−4343.86	−1.30442	−0.652211	−	0.758037i	$-0.726158\pi$
$223$	−4343.86	−1.30442	−0.652211	+	0.758037i	$0.726158\pi$
$224$	0	0
$225$	0	0
$226$	−1491.05	−0.438863
$227$	−2647.59	−0.774127	−0.387064	−	0.922053i	$-0.626511\pi$
$227$	−2647.59	−0.774127	−0.387064	+	0.922053i	$0.626511\pi$
$228$	0	0
$229$	1445.09	0.417006	0.208503	−	0.978022i	$-0.433141\pi$
$229$	1445.09	0.417006	0.208503	+	0.978022i	$0.433141\pi$
$230$	836.566	0.239833
$231$	0	0
$232$	2517.39	0.712392
$233$	6245.91	1.75615	0.878076	−	0.478522i	$-0.158827\pi$
$233$	6245.91	1.75615	0.878076	+	0.478522i	$0.158827\pi$
$234$	0	0
$235$	−748.485	−0.207769
$236$	2723.51	0.751209
$237$	0	0
$238$	0	0
$239$	−1340.24	−0.362731	−0.181366	−	0.983416i	$-0.558052\pi$
$239$	−1340.24	−0.362731	−0.181366	+	0.983416i	$0.558052\pi$
$240$	0	0
$241$	−3369.92	−0.900729	−0.450364	−	0.892845i	$-0.648706\pi$
$241$	−3369.92	−0.900729	−0.450364	+	0.892845i	$0.648706\pi$
$242$	−5811.70	−1.54376
$243$	0	0
$244$	−1156.25	−0.303366
$245$	0	0
$246$	0	0
$247$	4919.82	1.26737
$248$	1617.01	0.414033
$249$	0	0
$250$	207.008	0.0523693
$251$	−3592.64	−0.903449	−0.451724	−	0.892158i	$-0.649191\pi$
$251$	−3592.64	−0.903449	−0.451724	+	0.892158i	$0.649191\pi$
$252$	0	0
$253$	−7028.97	−1.74667
$254$	2907.64	0.718273
$255$	0	0
$256$	−3823.74	−0.933531
$257$	−2.84763	−0.000691167 0	−0.000345584	−	1.00000i	$-0.500110\pi$
$257$	−2.84763	−0.000691167 0	−0.000345584		1.00000i	$0.500110\pi$
$258$	0	0
$259$	0	0
$260$	−1798.91	−0.429090
$261$	0	0
$262$	−2995.98	−0.706459
$263$	2817.07	0.660488	0.330244	−	0.943896i	$-0.392869\pi$
$263$	2817.07	0.660488	0.330244	+	0.943896i	$0.392869\pi$
$264$	0	0
$265$	1357.37	0.314652
$266$	0	0
$267$	0	0
$268$	−424.250	−0.0966986
$269$	−1447.43	−0.328072	−0.164036	−	0.986454i	$-0.552451\pi$
$269$	−1447.43	−0.328072	−0.164036	+	0.986454i	$0.552451\pi$
$270$	0	0
$271$	−8054.50	−1.80545	−0.902724	−	0.430221i	$-0.858436\pi$
$271$	−8054.50	−1.80545	−0.902724	+	0.430221i	$0.858436\pi$
$272$	594.746	0.132580
$273$	0	0
$274$	−30.6593	−0.00675985
$275$	−1739.32	−0.381399
$276$	0	0
$277$	−571.300	−0.123921	−0.0619604	−	0.998079i	$-0.519735\pi$
$277$	−571.300	−0.123921	−0.0619604	+	0.998079i	$0.519735\pi$
$278$	−1036.05	−0.223518
$279$	0	0
$280$	0	0
$281$	1784.48	0.378837	0.189418	−	0.981896i	$-0.439340\pi$
$281$	1784.48	0.378837	0.189418	+	0.981896i	$0.439340\pi$
$282$	0	0
$283$	−3321.05	−0.697582	−0.348791	−	0.937201i	$-0.613408\pi$
$283$	−3321.05	−0.697582	−0.348791	+	0.937201i	$0.613408\pi$
$284$	−478.672	−0.100014
$285$	0	0
$286$	−7884.57	−1.63015
$287$	0	0
$288$	0	0
$289$	5972.11	1.21557
$290$	949.425	0.192249
$291$	0	0
$292$	4638.56	0.929627
$293$	−5049.54	−1.00682	−0.503408	−	0.864049i	$-0.667921\pi$
$293$	−5049.54	−1.00682	−0.503408	+	0.864049i	$0.667921\pi$
$294$	0	0
$295$	2590.14	0.511199
$296$	−4411.52	−0.866266
$297$	0	0
$298$	−1703.48	−0.331141
$299$	−6913.79	−1.33724
$300$	0	0
$301$	0	0
$302$	−117.593	−0.0224063
$303$	0	0
$304$	−409.828	−0.0773198
$305$	−1099.63	−0.206441
$306$	0	0
$307$	1535.73	0.285500	0.142750	−	0.989759i	$-0.454405\pi$
$307$	1535.73	0.285500	0.142750	+	0.989759i	$0.454405\pi$
$308$	0	0
$309$	0	0
$310$	609.849	0.111733
$311$	−9283.05	−1.69258	−0.846291	−	0.532720i	$-0.821170\pi$
$311$	−9283.05	−1.69258	−0.846291	+	0.532720i	$0.821170\pi$
$312$	0	0
$313$	−6025.43	−1.08811	−0.544054	−	0.839050i	$-0.683111\pi$
$313$	−6025.43	−1.08811	−0.544054	+	0.839050i	$0.683111\pi$
$314$	−3413.68	−0.613519
$315$	0	0
$316$	−3153.83	−0.561445
$317$	6977.58	1.23628	0.618139	−	0.786069i	$-0.287887\pi$
$317$	6977.58	1.23628	0.618139	+	0.786069i	$0.287887\pi$
$318$	0	0
$319$	−7977.24	−1.40012
$320$	−1304.51	−0.227889
$321$	0	0
$322$	0	0
$323$	−7500.71	−1.29211
$324$	0	0
$325$	−1710.81	−0.291997
$326$	3252.17	0.552519
$327$	0	0
$328$	9162.06	1.54235
$329$	0	0
$330$	0	0
$331$	984.878	0.163546	0.0817731	−	0.996651i	$-0.473942\pi$
$331$	984.878	0.163546	0.0817731	+	0.996651i	$0.473942\pi$
$332$	−372.659	−0.0616034
$333$	0	0
$334$	−4728.13	−0.774586
$335$	−403.475	−0.0658035
$336$	0	0
$337$	51.9653	0.00839979	0.00419990	−	0.999991i	$-0.498663\pi$
$337$	51.9653	0.00839979	0.00419990	+	0.999991i	$0.498663\pi$
$338$	−4116.99	−0.662529
$339$	0	0
$340$	2742.60	0.437465
$341$	−5124.06	−0.813734
$342$	0	0
$343$	0	0
$344$	6831.89	1.07079
$345$	0	0
$346$	2572.21	0.399661
$347$	11300.5	1.74825	0.874123	−	0.485704i	$-0.161437\pi$
$347$	11300.5	1.74825	0.874123	+	0.485704i	$0.161437\pi$
$348$	0	0
$349$	2016.91	0.309349	0.154674	−	0.987966i	$-0.450567\pi$
$349$	2016.91	0.309349	0.154674	+	0.987966i	$0.450567\pi$
$350$	0	0
$351$	0	0
$352$	12876.6	1.94979
$353$	−7589.41	−1.14432	−0.572158	−	0.820143i	$-0.693894\pi$
$353$	−7589.41	−1.14432	−0.572158	+	0.820143i	$0.693894\pi$
$354$	0	0
$355$	−455.232	−0.0680597
$356$	−4219.59	−0.628196
$357$	0	0
$358$	446.873	0.0659720
$359$	−8734.24	−1.28405	−0.642027	−	0.766682i	$-0.721906\pi$
$359$	−8734.24	−1.28405	−0.642027	+	0.766682i	$0.721906\pi$
$360$	0	0
$361$	−1690.41	−0.246451
$362$	3692.38	0.536097
$363$	0	0
$364$	0	0
$365$	4411.41	0.632613
$366$	0	0
$367$	1890.54	0.268898	0.134449	−	0.990921i	$-0.457074\pi$
$367$	1890.54	0.268898	0.134449	+	0.990921i	$0.457074\pi$
$368$	575.928	0.0815825
$369$	0	0
$370$	−1663.79	−0.233774
$371$	0	0
$372$	0	0
$373$	2713.88	0.376728	0.188364	−	0.982099i	$-0.439682\pi$
$373$	2713.88	0.376728	0.188364	+	0.982099i	$0.439682\pi$
$374$	12020.7	1.66197
$375$	0	0
$376$	3286.63	0.450784
$377$	−7846.52	−1.07193
$378$	0	0
$379$	8941.19	1.21182	0.605908	−	0.795535i	$-0.292810\pi$
$379$	8941.19	1.21182	0.605908	+	0.795535i	$0.292810\pi$
$380$	−1889.87	−0.255127
$381$	0	0
$382$	−771.592	−0.103346
$383$	−9293.88	−1.23994	−0.619968	−	0.784627i	$-0.712854\pi$
$383$	−9293.88	−1.23994	−0.619968	+	0.784627i	$0.712854\pi$
$384$	0	0
$385$	0	0
$386$	7310.62	0.963992
$387$	0	0
$388$	765.737	0.100192
$389$	−10454.8	−1.36267	−0.681333	−	0.731974i	$-0.738599\pi$
$389$	−10454.8	−1.36267	−0.681333	+	0.731974i	$0.738599\pi$
$390$	0	0
$391$	10540.7	1.36334
$392$	0	0
$393$	0	0
$394$	−479.947	−0.0613690
$395$	−2999.39	−0.382065
$396$	0	0
$397$	3626.03	0.458401	0.229200	−	0.973379i	$-0.426389\pi$
$397$	3626.03	0.458401	0.229200	+	0.973379i	$0.426389\pi$
$398$	7978.47	1.00484
$399$	0	0
$400$	142.513	0.0178141
$401$	8422.52	1.04888	0.524440	−	0.851448i	$-0.324275\pi$
$401$	8422.52	1.04888	0.524440	+	0.851448i	$0.324275\pi$
$402$	0	0
$403$	−5040.09	−0.622989
$404$	3256.65	0.401051
$405$	0	0
$406$	0	0
$407$	13979.5	1.70254
$408$	0	0
$409$	14580.7	1.76276	0.881379	−	0.472409i	$-0.156616\pi$
$409$	14580.7	1.76276	0.881379	+	0.472409i	$0.156616\pi$
$410$	3455.44	0.416224
$411$	0	0
$412$	−9579.51	−1.14551
$413$	0	0
$414$	0	0
$415$	−354.410	−0.0419212
$416$	12665.6	1.49275
$417$	0	0
$418$	−8283.26	−0.969252
$419$	−2537.53	−0.295863	−0.147931	−	0.988998i	$-0.547261\pi$
$419$	−2537.53	−0.295863	−0.147931	+	0.988998i	$0.547261\pi$
$420$	0	0
$421$	9649.52	1.11708	0.558538	−	0.829479i	$-0.311363\pi$
$421$	9649.52	1.11708	0.558538	+	0.829479i	$0.311363\pi$
$422$	−3348.57	−0.386269
$423$	0	0
$424$	−5960.27	−0.682680
$425$	2608.29	0.297696
$426$	0	0
$427$	0	0
$428$	5729.06	0.647020
$429$	0	0
$430$	2576.62	0.288967
$431$	7262.56	0.811660	0.405830	−	0.913949i	$-0.366983\pi$
$431$	7262.56	0.811660	0.405830	+	0.913949i	$0.366983\pi$
$432$	0	0
$433$	−11345.0	−1.25914	−0.629570	−	0.776944i	$-0.716769\pi$
$433$	−11345.0	−1.25914	−0.629570	+	0.776944i	$0.716769\pi$
$434$	0	0
$435$	0	0
$436$	−3100.15	−0.340528
$437$	−7263.40	−0.795092
$438$	0	0
$439$	−11705.9	−1.27265	−0.636323	−	0.771423i	$-0.719545\pi$
$439$	−11705.9	−1.27265	−0.636323	+	0.771423i	$0.719545\pi$
$440$	7637.40	0.827497
$441$	0	0
$442$	11823.8	1.27240
$443$	−15078.0	−1.61710	−0.808551	−	0.588426i	$-0.799748\pi$
$443$	−15078.0	−1.61710	−0.808551	+	0.588426i	$0.799748\pi$
$444$	0	0
$445$	−4012.96	−0.427489
$446$	7193.70	0.763747
$447$	0	0
$448$	0	0
$449$	−1075.45	−0.113037	−0.0565185	−	0.998402i	$-0.518000\pi$
$449$	−1075.45	−0.113037	−0.0565185	+	0.998402i	$0.518000\pi$
$450$	0	0
$451$	−29033.2	−3.03131
$452$	−4733.59	−0.492587
$453$	0	0
$454$	4384.58	0.453257
$455$	0	0
$456$	0	0
$457$	−10736.9	−1.09902	−0.549511	−	0.835487i	$-0.685186\pi$
$457$	−10736.9	−1.09902	−0.549511	+	0.835487i	$0.685186\pi$
$458$	−2393.16	−0.244159
$459$	0	0
$460$	2655.82	0.269192
$461$	452.568	0.0457228	0.0228614	−	0.999739i	$-0.492722\pi$
$461$	452.568	0.0457228	0.0228614	+	0.999739i	$0.492722\pi$
$462$	0	0
$463$	7118.15	0.714489	0.357244	−	0.934011i	$-0.383716\pi$
$463$	7118.15	0.714489	0.357244	+	0.934011i	$0.383716\pi$
$464$	653.626	0.0653962
$465$	0	0
$466$	−10343.6	−1.02824
$467$	973.800	0.0964927	0.0482463	−	0.998835i	$-0.484637\pi$
$467$	973.800	0.0964927	0.0482463	+	0.998835i	$0.484637\pi$
$468$	0	0
$469$	0	0
$470$	1239.54	0.121650
$471$	0	0
$472$	−11373.4	−1.10912
$473$	−21649.2	−2.10451
$474$	0	0
$475$	−1797.32	−0.173614
$476$	0	0
$477$	0	0
$478$	2219.52	0.212381
$479$	9714.00	0.926606	0.463303	−	0.886200i	$-0.346664\pi$
$479$	9714.00	0.926606	0.463303	+	0.886200i	$0.346664\pi$
$480$	0	0
$481$	13750.4	1.30346
$482$	5580.80	0.527383
$483$	0	0
$484$	−18450.3	−1.73274
$485$	728.239	0.0681806
$486$	0	0
$487$	923.389	0.0859194	0.0429597	−	0.999077i	$-0.486321\pi$
$487$	923.389	0.0859194	0.0429597	+	0.999077i	$0.486321\pi$
$488$	4828.50	0.447902
$489$	0	0
$490$	0	0
$491$	−1289.11	−0.118486	−0.0592430	−	0.998244i	$-0.518869\pi$
$491$	−1289.11	−0.118486	−0.0592430	+	0.998244i	$0.518869\pi$
$492$	0	0
$493$	11962.7	1.09285
$494$	−8147.52	−0.742053
$495$	0	0
$496$	419.846	0.0380074
$497$	0	0
$498$	0	0
$499$	−19338.3	−1.73487	−0.867436	−	0.497549i	$-0.834234\pi$
$499$	−19338.3	−1.73487	−0.867436	+	0.497549i	$0.834234\pi$
$500$	657.182	0.0587802
$501$	0	0
$502$	5949.64	0.528975
$503$	−1772.84	−0.157151	−0.0785757	−	0.996908i	$-0.525037\pi$
$503$	−1772.84	−0.157151	−0.0785757	+	0.996908i	$0.525037\pi$
$504$	0	0
$505$	3097.18	0.272916
$506$	11640.4	1.02269
$507$	0	0
$508$	9230.80	0.806202
$509$	3151.75	0.274458	0.137229	−	0.990539i	$-0.456180\pi$
$509$	3151.75	0.274458	0.137229	+	0.990539i	$0.456180\pi$
$510$	0	0
$511$	0	0
$512$	−2056.31	−0.177494
$513$	0	0
$514$	4.71585	0.000404683 0
$515$	−9110.40	−0.779519
$516$	0	0
$517$	−10414.8	−0.885964
$518$	0	0
$519$	0	0
$520$	7512.24	0.633526
$521$	−11029.4	−0.927458	−0.463729	−	0.885977i	$-0.653489\pi$
$521$	−11029.4	−0.927458	−0.463729	+	0.885977i	$0.653489\pi$
$522$	0	0
$523$	−14448.4	−1.20800	−0.604002	−	0.796983i	$-0.706428\pi$
$523$	−14448.4	−1.20800	−0.604002	+	0.796983i	$0.706428\pi$
$524$	−9511.26	−0.792941
$525$	0	0
$526$	−4665.25	−0.386720
$527$	7684.08	0.635149
$528$	0	0
$529$	−1959.79	−0.161074
$530$	−2247.89	−0.184231
$531$	0	0
$532$	0	0
$533$	−28557.4	−2.32075
$534$	0	0
$535$	5448.51	0.440298
$536$	1771.67	0.142770
$537$	0	0
$538$	2397.04	0.192089
$539$	0	0
$540$	0	0
$541$	−22274.8	−1.77018	−0.885091	−	0.465418i	$-0.845904\pi$
$541$	−22274.8	−1.77018	−0.885091	+	0.465418i	$0.845904\pi$
$542$	13338.8	1.05710
$543$	0	0
$544$	−19309.9	−1.52188
$545$	−2948.33	−0.231730
$546$	0	0
$547$	18642.6	1.45722	0.728609	−	0.684930i	$-0.240167\pi$
$547$	18642.6	1.45722	0.728609	+	0.684930i	$0.240167\pi$
$548$	−97.3334	−0.00758737
$549$	0	0
$550$	2880.42	0.223311
$551$	−8243.28	−0.637343
$552$	0	0
$553$	0	0
$554$	946.108	0.0725565
$555$	0	0
$556$	−3289.11	−0.250880
$557$	21431.8	1.63033	0.815165	−	0.579229i	$-0.196646\pi$
$557$	21431.8	1.63033	0.815165	+	0.579229i	$0.196646\pi$
$558$	0	0
$559$	−21294.5	−1.61120
$560$	0	0
$561$	0	0
$562$	−2955.21	−0.221811
$563$	−6154.85	−0.460739	−0.230370	−	0.973103i	$-0.573993\pi$
$563$	−6154.85	−0.460739	−0.230370	+	0.973103i	$0.573993\pi$
$564$	0	0
$565$	−4501.79	−0.335206
$566$	5499.86	0.408439
$567$	0	0
$568$	1998.94	0.147665
$569$	8389.99	0.618149	0.309074	−	0.951038i	$-0.399981\pi$
$569$	8389.99	0.618149	0.309074	+	0.951038i	$0.399981\pi$
$570$	0	0
$571$	−600.502	−0.0440109	−0.0220055	−	0.999758i	$-0.507005\pi$
$571$	−600.502	−0.0440109	−0.0220055	+	0.999758i	$0.507005\pi$
$572$	−25030.9	−1.82971
$573$	0	0
$574$	0	0
$575$	2525.77	0.183186
$576$	0	0
$577$	−4062.35	−0.293098	−0.146549	−	0.989203i	$-0.546817\pi$
$577$	−4062.35	−0.293098	−0.146549	+	0.989203i	$0.546817\pi$
$578$	−9890.18	−0.711725
$579$	0	0
$580$	3014.12	0.215783
$581$	0	0
$582$	0	0
$583$	18887.2	1.34173
$584$	−19370.6	−1.37254
$585$	0	0
$586$	8362.35	0.589498
$587$	−8387.50	−0.589760	−0.294880	−	0.955534i	$-0.595280\pi$
$587$	−8387.50	−0.589760	−0.294880	+	0.955534i	$0.595280\pi$
$588$	0	0
$589$	−5294.95	−0.370415
$590$	−4289.43	−0.299310
$591$	0	0
$592$	−1145.43	−0.0795214
$593$	−15342.6	−1.06247	−0.531236	−	0.847224i	$-0.678272\pi$
$593$	−15342.6	−1.06247	−0.531236	+	0.847224i	$0.678272\pi$
$594$	0	0
$595$	0	0
$596$	−5408.00	−0.371678
$597$	0	0
$598$	11449.7	0.782963
$599$	19708.5	1.34435	0.672175	−	0.740392i	$-0.265360\pi$
$599$	19708.5	1.34435	0.672175	+	0.740392i	$0.265360\pi$
$600$	0	0
$601$	19002.5	1.28973	0.644866	−	0.764295i	$-0.276913\pi$
$601$	19002.5	1.28973	0.644866	+	0.764295i	$0.276913\pi$
$602$	0	0
$603$	0	0
$604$	−373.318	−0.0251492
$605$	−17546.8	−1.17914
$606$	0	0
$607$	−29453.1	−1.96946	−0.984732	−	0.174076i	$-0.944306\pi$
$607$	−29453.1	−1.96946	−0.984732	+	0.174076i	$0.944306\pi$
$608$	13306.1	0.887553
$609$	0	0
$610$	1821.05	0.120873
$611$	−10244.2	−0.678288
$612$	0	0
$613$	9987.27	0.658045	0.329023	−	0.944322i	$-0.393281\pi$
$613$	9987.27	0.658045	0.329023	+	0.944322i	$0.393281\pi$
$614$	−2543.26	−0.167162
$615$	0	0
$616$	0	0
$617$	21076.1	1.37519	0.687593	−	0.726096i	$-0.258667\pi$
$617$	21076.1	1.37519	0.687593	+	0.726096i	$0.258667\pi$
$618$	0	0
$619$	314.668	0.0204323	0.0102161	−	0.999948i	$-0.496748\pi$
$619$	314.668	0.0204323	0.0102161	+	0.999948i	$0.496748\pi$
$620$	1936.07	0.125410
$621$	0	0
$622$	15373.3	0.991018
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	9978.49	0.637094
$627$	0	0
$628$	−10837.3	−0.688624
$629$	−20963.7	−1.32890
$630$	0	0
$631$	3314.96	0.209138	0.104569	−	0.994518i	$-0.466654\pi$
$631$	3314.96	0.209138	0.104569	+	0.994518i	$0.466654\pi$
$632$	13170.4	0.828941
$633$	0	0
$634$	−11555.3	−0.723849
$635$	8778.77	0.548622
$636$	0	0
$637$	0	0
$638$	13210.8	0.819782
$639$	0	0
$640$	−5242.92	−0.323820
$641$	−3005.12	−0.185172	−0.0925858	−	0.995705i	$-0.529513\pi$
$641$	−3005.12	−0.185172	−0.0925858	+	0.995705i	$0.529513\pi$
$642$	0	0
$643$	−21225.7	−1.30180	−0.650902	−	0.759162i	$-0.725609\pi$
$643$	−21225.7	−1.30180	−0.650902	+	0.759162i	$0.725609\pi$
$644$	0	0
$645$	0	0
$646$	12421.6	0.756537
$647$	−2740.35	−0.166514	−0.0832568	−	0.996528i	$-0.526532\pi$
$647$	−2740.35	−0.166514	−0.0832568	+	0.996528i	$0.526532\pi$
$648$	0	0
$649$	36040.6	2.17984
$650$	2833.21	0.170966
$651$	0	0
$652$	10324.6	0.620157
$653$	−22790.7	−1.36580	−0.682900	−	0.730511i	$-0.739282\pi$
$653$	−22790.7	−1.36580	−0.682900	+	0.730511i	$0.739282\pi$
$654$	0	0
$655$	−9045.49	−0.539598
$656$	2378.87	0.141584
$657$	0	0
$658$	0	0
$659$	−19405.1	−1.14706	−0.573532	−	0.819183i	$-0.694427\pi$
$659$	−19405.1	−1.14706	−0.573532	+	0.819183i	$0.694427\pi$
$660$	0	0
$661$	−15637.3	−0.920150	−0.460075	−	0.887880i	$-0.652177\pi$
$661$	−15637.3	−0.920150	−0.460075	+	0.887880i	$0.652177\pi$
$662$	−1631.02	−0.0957574
$663$	0	0
$664$	1556.23	0.0909538
$665$	0	0
$666$	0	0
$667$	11584.2	0.672479
$668$	−15010.3	−0.869409
$669$	0	0
$670$	668.179	0.0385284
$671$	−15300.8	−0.880299
$672$	0	0
$673$	−2579.54	−0.147747	−0.0738735	−	0.997268i	$-0.523536\pi$
$673$	−2579.54	−0.147747	−0.0738735	+	0.997268i	$0.523536\pi$
$674$	−86.0578	−0.00491813
$675$	0	0
$676$	−13070.1	−0.743634
$677$	8159.56	0.463216	0.231608	−	0.972809i	$-0.425601\pi$
$677$	8159.56	0.463216	0.231608	+	0.972809i	$0.425601\pi$
$678$	0	0
$679$	0	0
$680$	−11453.1	−0.645892
$681$	0	0
$682$	8485.76	0.476446
$683$	20529.3	1.15012	0.575059	−	0.818112i	$-0.304979\pi$
$683$	20529.3	1.15012	0.575059	+	0.818112i	$0.304979\pi$
$684$	0	0
$685$	−92.5670	−0.00516321
$686$	0	0
$687$	0	0
$688$	1773.86	0.0982961
$689$	18577.7	1.02722
$690$	0	0
$691$	917.295	0.0505001	0.0252500	−	0.999681i	$-0.491962\pi$
$691$	917.295	0.0505001	0.0252500	+	0.999681i	$0.491962\pi$
$692$	8165.92	0.448586
$693$	0	0
$694$	−18714.3	−1.02361
$695$	−3128.04	−0.170724
$696$	0	0
$697$	43538.4	2.36605
$698$	−3340.13	−0.181126
$699$	0	0
$700$	0	0
$701$	10491.3	0.565266	0.282633	−	0.959228i	$-0.408792\pi$
$701$	10491.3	0.565266	0.282633	+	0.959228i	$0.408792\pi$
$702$	0	0
$703$	14445.7	0.775006
$704$	−18151.7	−0.971758
$705$	0	0
$706$	12568.5	0.670005
$707$	0	0
$708$	0	0
$709$	23837.6	1.26268	0.631339	−	0.775507i	$-0.282506\pi$
$709$	23837.6	1.26268	0.631339	+	0.775507i	$0.282506\pi$
$710$	753.892	0.0398494
$711$	0	0
$712$	17621.0	0.927495
$713$	7440.96	0.390836
$714$	0	0
$715$	−23805.2	−1.24512
$716$	1418.68	0.0740481
$717$	0	0
$718$	14464.4	0.751822
$719$	−3926.19	−0.203647	−0.101824	−	0.994802i	$-0.532468\pi$
$719$	−3926.19	−0.203647	−0.101824	+	0.994802i	$0.532468\pi$
$720$	0	0
$721$	0	0
$722$	2799.42	0.144299
$723$	0	0
$724$	11722.1	0.601724
$725$	2866.51	0.146841
$726$	0	0
$727$	21071.2	1.07495	0.537474	−	0.843281i	$-0.319379\pi$
$727$	21071.2	1.07495	0.537474	+	0.843281i	$0.319379\pi$
$728$	0	0
$729$	0	0
$730$	−7305.57	−0.370399
$731$	32465.4	1.64265
$732$	0	0
$733$	22840.7	1.15094	0.575470	−	0.817823i	$-0.304819\pi$
$733$	22840.7	1.15094	0.575470	+	0.817823i	$0.304819\pi$
$734$	−3130.86	−0.157441
$735$	0	0
$736$	−18699.0	−0.936485
$737$	−5614.16	−0.280597
$738$	0	0
$739$	10837.3	0.539456	0.269728	−	0.962937i	$-0.413066\pi$
$739$	10837.3	0.539456	0.269728	+	0.962937i	$0.413066\pi$
$740$	−5281.99	−0.262392
$741$	0	0
$742$	0	0
$743$	21631.9	1.06810	0.534050	−	0.845453i	$-0.320669\pi$
$743$	21631.9	1.06810	0.534050	+	0.845453i	$0.320669\pi$
$744$	0	0
$745$	−5143.17	−0.252928
$746$	−4494.36	−0.220577
$747$	0	0
$748$	38161.9	1.86543
$749$	0	0
$750$	0	0
$751$	−16179.2	−0.786133	−0.393067	−	0.919510i	$-0.628586\pi$
$751$	−16179.2	−0.786133	−0.393067	+	0.919510i	$0.628586\pi$
$752$	853.352	0.0413811
$753$	0	0
$754$	12994.3	0.627620
$755$	−355.037	−0.0171140
$756$	0	0
$757$	−40930.9	−1.96520	−0.982601	−	0.185727i	$-0.940536\pi$
$757$	−40930.9	−1.96520	−0.982601	+	0.185727i	$0.940536\pi$
$758$	−14807.2	−0.709526
$759$	0	0
$760$	7892.11	0.376680
$761$	3183.97	0.151667	0.0758337	−	0.997120i	$-0.475838\pi$
$761$	3183.97	0.151667	0.0758337	+	0.997120i	$0.475838\pi$
$762$	0	0
$763$	0	0
$764$	−2449.55	−0.115997
$765$	0	0
$766$	15391.2	0.725990
$767$	35450.0	1.66887
$768$	0	0
$769$	−33595.8	−1.57542	−0.787708	−	0.616048i	$-0.788733\pi$
$769$	−33595.8	−1.57542	−0.787708	+	0.616048i	$0.788733\pi$
$770$	0	0
$771$	0	0
$772$	23208.8	1.08200
$773$	34386.0	1.59997	0.799986	−	0.600019i	$-0.204840\pi$
$773$	34386.0	1.59997	0.799986	+	0.600019i	$0.204840\pi$
$774$	0	0
$775$	1841.26	0.0853420
$776$	−3197.72	−0.147927
$777$	0	0
$778$	17313.7	0.797849
$779$	−30001.5	−1.37986
$780$	0	0
$781$	−6334.33	−0.290218
$782$	−17456.1	−0.798245
$783$	0	0
$784$	0	0
$785$	−10306.6	−0.468610
$786$	0	0
$787$	8212.43	0.371972	0.185986	−	0.982552i	$-0.440452\pi$
$787$	8212.43	0.371972	0.185986	+	0.982552i	$0.440452\pi$
$788$	−1523.68	−0.0688816
$789$	0	0
$790$	4967.17	0.223701
$791$	0	0
$792$	0	0
$793$	−15050.1	−0.673951
$794$	−6004.92	−0.268396
$795$	0	0
$796$	25329.0	1.12784
$797$	36798.3	1.63546	0.817732	−	0.575600i	$-0.195231\pi$
$797$	36798.3	1.63546	0.817732	+	0.575600i	$0.195231\pi$
$798$	0	0
$799$	15618.2	0.691528
$800$	−4627.05	−0.204488
$801$	0	0
$802$	−13948.2	−0.614125
$803$	61382.7	2.69757
$804$	0	0
$805$	0	0
$806$	8346.70	0.364764
$807$	0	0
$808$	−13599.8	−0.592128
$809$	−10186.2	−0.442678	−0.221339	−	0.975197i	$-0.571043\pi$
$809$	−10186.2	−0.442678	−0.221339	+	0.975197i	$0.571043\pi$
$810$	0	0
$811$	−21196.9	−0.917786	−0.458893	−	0.888492i	$-0.651754\pi$
$811$	−21196.9	−0.917786	−0.458893	+	0.888492i	$0.651754\pi$
$812$	0	0
$813$	0	0
$814$	−23150.8	−0.996851
$815$	9818.99	0.422018
$816$	0	0
$817$	−22371.3	−0.957982
$818$	−24146.5	−1.03211
$819$	0	0
$820$	10969.9	0.467177
$821$	−7563.94	−0.321539	−0.160769	−	0.986992i	$-0.551398\pi$
$821$	−7563.94	−0.321539	−0.160769	+	0.986992i	$0.551398\pi$
$822$	0	0
$823$	8399.80	0.355770	0.177885	−	0.984051i	$-0.443075\pi$
$823$	8399.80	0.355770	0.177885	+	0.984051i	$0.443075\pi$
$824$	40004.1	1.69127
$825$	0	0
$826$	0	0
$827$	−5479.54	−0.230402	−0.115201	−	0.993342i	$-0.536751\pi$
$827$	−5479.54	−0.230402	−0.115201	+	0.993342i	$0.536751\pi$
$828$	0	0
$829$	−9252.56	−0.387641	−0.193821	−	0.981037i	$-0.562088\pi$
$829$	−9252.56	−0.387641	−0.193821	+	0.981037i	$0.562088\pi$
$830$	586.925	0.0245452
$831$	0	0
$832$	−17854.2	−0.743972
$833$	0	0
$834$	0	0
$835$	−14275.2	−0.591634
$836$	−26296.6	−1.08790
$837$	0	0
$838$	4202.31	0.173230
$839$	−34517.6	−1.42036	−0.710178	−	0.704022i	$-0.751386\pi$
$839$	−34517.6	−1.42036	−0.710178	+	0.704022i	$0.751386\pi$
$840$	0	0
$841$	−11241.9	−0.460943
$842$	−15980.2	−0.654055
$843$	0	0
$844$	−10630.6	−0.433555
$845$	−12430.1	−0.506044
$846$	0	0
$847$	0	0
$848$	−1547.55	−0.0626686
$849$	0	0
$850$	−4319.50	−0.174303
$851$	−20300.4	−0.817732
$852$	0	0
$853$	1498.57	0.0601525	0.0300763	−	0.999548i	$-0.490425\pi$
$853$	1498.57	0.0601525	0.0300763	+	0.999548i	$0.490425\pi$
$854$	0	0
$855$	0	0
$856$	−23924.6	−0.955287
$857$	−9357.02	−0.372963	−0.186482	−	0.982458i	$-0.559709\pi$
$857$	−9357.02	−0.372963	−0.186482	+	0.982458i	$0.559709\pi$
$858$	0	0
$859$	29960.9	1.19005	0.595025	−	0.803707i	$-0.297142\pi$
$859$	29960.9	1.19005	0.595025	+	0.803707i	$0.297142\pi$
$860$	8179.94	0.324341
$861$	0	0
$862$	−12027.3	−0.475232
$863$	33941.0	1.33878	0.669389	−	0.742912i	$-0.266556\pi$
$863$	33941.0	1.33878	0.669389	+	0.742912i	$0.266556\pi$
$864$	0	0
$865$	7766.03	0.305264
$866$	18788.1	0.737235
$867$	0	0
$868$	0	0
$869$	−41735.0	−1.62919
$870$	0	0
$871$	−5522.16	−0.214824
$872$	12946.2	0.502769
$873$	0	0
$874$	12028.6	0.465532
$875$	0	0
$876$	0	0
$877$	−39436.6	−1.51845	−0.759224	−	0.650830i	$-0.774421\pi$
$877$	−39436.6	−1.51845	−0.759224	+	0.650830i	$0.774421\pi$
$878$	19385.7	0.745142
$879$	0	0
$880$	1983.00	0.0759625
$881$	32411.4	1.23946	0.619732	−	0.784813i	$-0.287241\pi$
$881$	32411.4	1.23946	0.619732	+	0.784813i	$0.287241\pi$
$882$	0	0
$883$	−17121.4	−0.652526	−0.326263	−	0.945279i	$-0.605790\pi$
$883$	−17121.4	−0.652526	−0.326263	+	0.945279i	$0.605790\pi$
$884$	37536.6	1.42816
$885$	0	0
$886$	24970.1	0.946824
$887$	−687.797	−0.0260360	−0.0130180	−	0.999915i	$-0.504144\pi$
$887$	−687.797	−0.0260360	−0.0130180	+	0.999915i	$0.504144\pi$
$888$	0	0
$889$	0	0
$890$	6645.71	0.250297
$891$	0	0
$892$	22837.6	0.857243
$893$	−10762.2	−0.403294
$894$	0	0
$895$	1349.20	0.0503898
$896$	0	0
$897$	0	0
$898$	1781.01	0.0661839
$899$	8444.81	0.313293
$900$	0	0
$901$	−28323.4	−1.04727
$902$	48080.8	1.77485
$903$	0	0
$904$	19767.5	0.727276
$905$	11148.1	0.409474
$906$	0	0
$907$	19104.2	0.699388	0.349694	−	0.936864i	$-0.386286\pi$
$907$	19104.2	0.699388	0.349694	+	0.936864i	$0.386286\pi$
$908$	13919.6	0.508743
$909$	0	0
$910$	0	0
$911$	−23135.3	−0.841390	−0.420695	−	0.907202i	$-0.638214\pi$
$911$	−23135.3	−0.841390	−0.420695	+	0.907202i	$0.638214\pi$
$912$	0	0
$913$	−4931.45	−0.178759
$914$	17781.0	0.643484
$915$	0	0
$916$	−7597.50	−0.274049
$917$	0	0
$918$	0	0
$919$	47373.9	1.70046	0.850230	−	0.526412i	$-0.176463\pi$
$919$	47373.9	1.70046	0.850230	+	0.526412i	$0.176463\pi$
$920$	−11090.7	−0.397447
$921$	0	0
$922$	−749.481	−0.0267710
$923$	−6230.53	−0.222189
$924$	0	0
$925$	−5023.33	−0.178558
$926$	−11788.1	−0.418338
$927$	0	0
$928$	−21221.6	−0.750682
$929$	−7617.72	−0.269031	−0.134515	−	0.990912i	$-0.542948\pi$
$929$	−7617.72	−0.269031	−0.134515	+	0.990912i	$0.542948\pi$
$930$	0	0
$931$	0	0
$932$	−32837.6	−1.15411
$933$	0	0
$934$	−1612.67	−0.0564971
$935$	36293.2	1.26943
$936$	0	0
$937$	52091.2	1.81616	0.908082	−	0.418792i	$-0.137546\pi$
$937$	52091.2	1.81616	0.908082	+	0.418792i	$0.137546\pi$
$938$	0	0
$939$	0	0
$940$	3935.13	0.136542
$941$	−56765.1	−1.96651	−0.983257	−	0.182225i	$-0.941670\pi$
$941$	−56765.1	−1.96651	−0.983257	+	0.182225i	$0.941670\pi$
$942$	0	0
$943$	42160.9	1.45594
$944$	−2953.03	−0.101815
$945$	0	0
$946$	35852.5	1.23220
$947$	47629.2	1.63436	0.817181	−	0.576381i	$-0.195535\pi$
$947$	47629.2	1.63436	0.817181	+	0.576381i	$0.195535\pi$
$948$	0	0
$949$	60376.8	2.06524
$950$	2976.48	0.101652
$951$	0	0
$952$	0	0
$953$	−12893.5	−0.438259	−0.219129	−	0.975696i	$-0.570322\pi$
$953$	−12893.5	−0.438259	−0.219129	+	0.975696i	$0.570322\pi$
$954$	0	0
$955$	−2329.60	−0.0789362
$956$	7046.24	0.238380
$957$	0	0
$958$	−16087.0	−0.542534
$959$	0	0
$960$	0	0
$961$	−24366.6	−0.817918
$962$	−22771.5	−0.763183
$963$	0	0
$964$	17717.2	0.591943
$965$	22072.3	0.736303
$966$	0	0
$967$	−28420.0	−0.945114	−0.472557	−	0.881300i	$-0.656669\pi$
$967$	−28420.0	−0.945114	−0.472557	+	0.881300i	$0.656669\pi$
$968$	77048.4	2.55830
$969$	0	0
$970$	−1206.01	−0.0399202
$971$	29273.8	0.967497	0.483749	−	0.875207i	$-0.339275\pi$
$971$	29273.8	0.967497	0.483749	+	0.875207i	$0.339275\pi$
$972$	0	0
$973$	0	0
$974$	−1529.19	−0.0503063
$975$	0	0
$976$	1253.69	0.0411165
$977$	−12055.0	−0.394753	−0.197377	−	0.980328i	$-0.563242\pi$
$977$	−12055.0	−0.394753	−0.197377	+	0.980328i	$0.563242\pi$
$978$	0	0
$979$	−55838.4	−1.82288
$980$	0	0
$981$	0	0
$982$	2134.84	0.0693743
$983$	10605.1	0.344099	0.172049	−	0.985088i	$-0.444961\pi$
$983$	10605.1	0.344099	0.172049	+	0.985088i	$0.444961\pi$
$984$	0	0
$985$	−1449.06	−0.0468740
$986$	−19811.0	−0.639870
$987$	0	0
$988$	−25865.7	−0.832893
$989$	31438.2	1.01080
$990$	0	0
$991$	45229.1	1.44980	0.724898	−	0.688856i	$-0.241887\pi$
$991$	45229.1	1.44980	0.724898	+	0.688856i	$0.241887\pi$
$992$	−13631.4	−0.436287
$993$	0	0
$994$	0	0
$995$	24088.7	0.767500
$996$	0	0
$997$	49676.8	1.57802	0.789008	−	0.614383i	$-0.210595\pi$
$997$	49676.8	1.57802	0.789008	+	0.614383i	$0.210595\pi$
$998$	32025.4	1.01578
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bz.1.2		6
3.2	odd	2		245.4.a.o.1.5	✓	6
7.6	odd	2		2205.4.a.ca.1.2		6
15.14	odd	2		1225.4.a.bj.1.2		6
21.2	odd	6		245.4.e.q.116.2		12
21.5	even	6		245.4.e.p.116.2		12
21.11	odd	6		245.4.e.q.226.2		12
21.17	even	6		245.4.e.p.226.2		12
21.20	even	2		245.4.a.p.1.5	yes	6
105.104	even	2		1225.4.a.bi.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.5	✓	6	3.2	odd	2
245.4.a.p.1.5	yes	6	21.20	even	2
245.4.e.p.116.2		12	21.5	even	6
245.4.e.p.226.2		12	21.17	even	6
245.4.e.q.116.2		12	21.2	odd	6
245.4.e.q.226.2		12	21.11	odd	6
1225.4.a.bi.1.2		6	105.104	even	2
1225.4.a.bj.1.2		6	15.14	odd	2
2205.4.a.bz.1.2		6	1.1	even	1	trivial
2205.4.a.ca.1.2		6	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65606 q^{2} -5.25746 q^{4} -5.00000 q^{5} +21.9552 q^{8} +O(q^{10})\)	\(q-1.65606 q^{2} -5.25746 q^{4} -5.00000 q^{5} +21.9552 q^{8} +8.28031 q^{10} -69.5726 q^{11} -68.4326 q^{13} +5.70053 q^{16} +104.332 q^{17} -71.8929 q^{19} +26.2873 q^{20} +115.217 q^{22} +101.031 q^{23} +25.0000 q^{25} +113.329 q^{26} +114.661 q^{29} +73.6505 q^{31} -185.082 q^{32} -172.780 q^{34} -200.933 q^{37} +119.059 q^{38} -109.776 q^{40} +417.308 q^{41} +311.175 q^{43} +365.775 q^{44} -167.313 q^{46} +149.697 q^{47} -41.4016 q^{50} +359.781 q^{52} -271.474 q^{53} +347.863 q^{55} -189.885 q^{58} -518.028 q^{59} +219.926 q^{61} -121.970 q^{62} +260.903 q^{64} +342.163 q^{65} +80.6950 q^{67} -548.520 q^{68} +91.0463 q^{71} -882.282 q^{73} +332.758 q^{74} +377.974 q^{76} +599.877 q^{79} -28.5026 q^{80} -691.087 q^{82} +70.8820 q^{83} -521.659 q^{85} -515.325 q^{86} -1527.48 q^{88} +802.592 q^{89} -531.165 q^{92} -247.908 q^{94} +359.465 q^{95} -145.648 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 14 * q^4 - 30 * q^5 + 66 * q^8	\( 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8} - 10 q^{10} + 16 q^{11} - 168 q^{13} + 298 q^{16} - 4 q^{17} - 308 q^{19} - 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} + 56 q^{26} - 176 q^{29} - 392 q^{31} + 770 q^{32} - 812 q^{34} - 140 q^{37} + 20 q^{38} - 330 q^{40} + 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} + 628 q^{47} + 50 q^{50} - 1520 q^{52} + 676 q^{53} - 80 q^{55} - 2012 q^{58} + 996 q^{59} - 740 q^{61} - 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} - 2940 q^{68} + 224 q^{71} - 2640 q^{73} - 928 q^{74} + 1340 q^{76} + 1636 q^{79} - 1490 q^{80} + 1756 q^{82} + 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 3332 q^{94} + 1540 q^{95} - 516 q^{97}+O(q^{100}) \) 6 * q + 2 * q^2 + 14 * q^4 - 30 * q^5 + 66 * q^8 - 10 * q^10 + 16 * q^11 - 168 * q^13 + 298 * q^16 - 4 * q^17 - 308 * q^19 - 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 + 56 * q^26 - 176 * q^29 - 392 * q^31 + 770 * q^32 - 812 * q^34 - 140 * q^37 + 20 * q^38 - 330 * q^40 + 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 + 628 * q^47 + 50 * q^50 - 1520 * q^52 + 676 * q^53 - 80 * q^55 - 2012 * q^58 + 996 * q^59 - 740 * q^61 - 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 - 2940 * q^68 + 224 * q^71 - 2640 * q^73 - 928 * q^74 + 1340 * q^76 + 1636 * q^79 - 1490 * q^80 + 1756 * q^82 + 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 - 1904 * q^89 + 1952 * q^92 + 3332 * q^94 + 1540 * q^95 - 516 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more