LMFDB - Embedded newform 2205.4.a.bf.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,4,Mod(1,2205)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,6,0,6,-10,0,0,-6,0,-30,-28,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$130.099211563$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+4.41421 q^{2} +11.4853 q^{4} -5.00000 q^{5} +15.3848 q^{8} -22.0711 q^{10} +0.142136 q^{11} -32.1421 q^{13} -23.9706 q^{16} +114.368 q^{17} +43.2304 q^{19} -57.4264 q^{20} +0.627417 q^{22} -154.463 q^{23} +25.0000 q^{25} -141.882 q^{26} +40.1472 q^{29} +75.4214 q^{31} -228.889 q^{32} +504.843 q^{34} -400.333 q^{37} +190.828 q^{38} -76.9239 q^{40} +95.4264 q^{41} -340.071 q^{43} +1.63247 q^{44} -681.833 q^{46} +7.49033 q^{47} +110.355 q^{50} -369.161 q^{52} +676.818 q^{53} -0.710678 q^{55} +177.218 q^{58} -796.181 q^{59} -757.102 q^{61} +332.926 q^{62} -818.602 q^{64} +160.711 q^{65} -740.581 q^{67} +1313.54 q^{68} -37.0253 q^{71} +80.8772 q^{73} -1767.16 q^{74} +496.514 q^{76} -317.358 q^{79} +119.853 q^{80} +421.233 q^{82} +945.929 q^{83} -571.838 q^{85} -1501.15 q^{86} +2.18672 q^{88} -783.205 q^{89} -1774.05 q^{92} +33.0639 q^{94} -216.152 q^{95} +393.107 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{2} + 6 q^{4} - 10 q^{5} - 6 q^{8} - 30 q^{10} - 28 q^{11} - 36 q^{13} - 14 q^{16} + 76 q^{17} + 160 q^{19} - 30 q^{20} - 44 q^{22} + 22 q^{23} + 50 q^{25} - 148 q^{26} + 250 q^{29} - 132 q^{31}+ \cdots - 628 q^{97}+O(q^{100}) $ 2 * q + 6 * q^2 + 6 * q^4 - 10 * q^5 - 6 * q^8 - 30 * q^10 - 28 * q^11 - 36 * q^13 - 14 * q^16 + 76 * q^17 + 160 * q^19 - 30 * q^20 - 44 * q^22 + 22 * q^23 + 50 * q^25 - 148 * q^26 + 250 * q^29 - 132 * q^31 - 42 * q^32 + 444 * q^34 - 416 * q^37 + 376 * q^38 + 30 * q^40 + 106 * q^41 - 666 * q^43 + 156 * q^44 - 402 * q^46 + 196 * q^47 + 150 * q^50 - 348 * q^52 + 952 * q^53 + 140 * q^55 + 510 * q^58 - 840 * q^59 + 98 * q^61 + 4 * q^62 - 602 * q^64 + 180 * q^65 - 1286 * q^67 + 1524 * q^68 - 1064 * q^71 - 172 * q^73 - 1792 * q^74 - 144 * q^76 - 1240 * q^79 + 70 * q^80 + 438 * q^82 + 1906 * q^83 - 380 * q^85 - 2018 * q^86 + 604 * q^88 - 650 * q^89 - 2742 * q^92 + 332 * q^94 - 800 * q^95 - 628 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.41421	1.56066	0.780330	−	0.625368i	$-0.215051\pi$
$2$	4.41421	1.56066	0.780330	+	0.625368i	$0.215051\pi$
$3$	0	0
$3$	0	0
$4$	11.4853	1.43566
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	15.3848	0.679917
$9$	0	0
$10$	−22.0711	−0.697948
$11$	0.142136	0.00389595	0.00194798	−	0.999998i	$-0.499380\pi$
$11$	0.142136	0.00389595	0.00194798	+	0.999998i	$0.499380\pi$
$12$	0	0
$13$	−32.1421	−0.685740	−0.342870	−	0.939383i	$-0.611399\pi$
$13$	−32.1421	−0.685740	−0.342870	+	0.939383i	$0.611399\pi$
$14$	0	0
$15$	0	0
$16$	−23.9706	−0.374540
$17$	114.368	1.63166	0.815829	−	0.578293i	$-0.196281\pi$
$17$	114.368	1.63166	0.815829	+	0.578293i	$0.196281\pi$
$18$	0	0
$19$	43.2304	0.521987	0.260993	−	0.965341i	$-0.415950\pi$
$19$	43.2304	0.521987	0.260993	+	0.965341i	$0.415950\pi$
$20$	−57.4264	−0.642047
$21$	0	0
$22$	0.627417	0.00608026
$23$	−154.463	−1.40034	−0.700169	−	0.713977i	$-0.746892\pi$
$23$	−154.463	−1.40034	−0.700169	+	0.713977i	$0.746892\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−141.882	−1.07021
$27$	0	0
$28$	0	0
$29$	40.1472	0.257074	0.128537	−	0.991705i	$-0.458972\pi$
$29$	40.1472	0.257074	0.128537	+	0.991705i	$0.458972\pi$
$30$	0	0
$31$	75.4214	0.436970	0.218485	−	0.975840i	$-0.429888\pi$
$31$	75.4214	0.436970	0.218485	+	0.975840i	$0.429888\pi$
$32$	−228.889	−1.26445
$33$	0	0
$34$	504.843	2.54647
$35$	0	0
$36$	0	0
$37$	−400.333	−1.77877	−0.889383	−	0.457163i	$-0.848866\pi$
$37$	−400.333	−1.77877	−0.889383	+	0.457163i	$0.848866\pi$
$38$	190.828	0.814644
$39$	0	0
$40$	−76.9239	−0.304068
$41$	95.4264	0.363490	0.181745	−	0.983346i	$-0.441825\pi$
$41$	95.4264	0.363490	0.181745	+	0.983346i	$0.441825\pi$
$42$	0	0
$43$	−340.071	−1.20605	−0.603027	−	0.797721i	$-0.706039\pi$
$43$	−340.071	−1.20605	−0.603027	+	0.797721i	$0.706039\pi$
$44$	1.63247	0.00559327
$45$	0	0
$46$	−681.833	−2.18545
$47$	7.49033	0.0232463	0.0116232	−	0.999932i	$-0.496300\pi$
$47$	7.49033	0.0232463	0.0116232	+	0.999932i	$0.496300\pi$
$48$	0	0
$49$	0	0
$50$	110.355	0.312132
$51$	0	0
$52$	−369.161	−0.984490
$53$	676.818	1.75412	0.877058	−	0.480385i	$-0.159503\pi$
$53$	676.818	1.75412	0.877058	+	0.480385i	$0.159503\pi$
$54$	0	0
$55$	−0.710678	−0.00174232
$56$	0	0
$57$	0	0
$58$	177.218	0.401205
$59$	−796.181	−1.75685	−0.878423	−	0.477884i	$-0.841404\pi$
$59$	−796.181	−1.75685	−0.878423	+	0.477884i	$0.841404\pi$
$60$	0	0
$61$	−757.102	−1.58913	−0.794565	−	0.607179i	$-0.792301\pi$
$61$	−757.102	−1.58913	−0.794565	+	0.607179i	$0.792301\pi$
$62$	332.926	0.681962
$63$	0	0
$64$	−818.602	−1.59883
$65$	160.711	0.306672
$66$	0	0
$67$	−740.581	−1.35039	−0.675197	−	0.737638i	$-0.735941\pi$
$67$	−740.581	−1.35039	−0.675197	+	0.737638i	$0.735941\pi$
$68$	1313.54	2.34251
$69$	0	0
$70$	0	0
$71$	−37.0253	−0.0618886	−0.0309443	−	0.999521i	$-0.509851\pi$
$71$	−37.0253	−0.0618886	−0.0309443	+	0.999521i	$0.509851\pi$
$72$	0	0
$73$	80.8772	0.129671	0.0648353	−	0.997896i	$-0.479348\pi$
$73$	80.8772	0.129671	0.0648353	+	0.997896i	$0.479348\pi$
$74$	−1767.16	−2.77605
$75$	0	0
$76$	496.514	0.749395
$77$	0	0
$78$	0	0
$79$	−317.358	−0.451970	−0.225985	−	0.974131i	$-0.572560\pi$
$79$	−317.358	−0.451970	−0.225985	+	0.974131i	$0.572560\pi$
$80$	119.853	0.167499
$81$	0	0
$82$	421.233	0.567285
$83$	945.929	1.25095	0.625477	−	0.780243i	$-0.284904\pi$
$83$	945.929	1.25095	0.625477	+	0.780243i	$0.284904\pi$
$84$	0	0
$85$	−571.838	−0.729700
$86$	−1501.15	−1.88224
$87$	0	0
$88$	2.18672	0.00264893
$89$	−783.205	−0.932804	−0.466402	−	0.884573i	$-0.654450\pi$
$89$	−783.205	−0.932804	−0.466402	+	0.884573i	$0.654450\pi$
$90$	0	0
$91$	0	0
$92$	−1774.05	−2.01041
$93$	0	0
$94$	33.0639	0.0362796
$95$	−216.152	−0.233439
$96$	0	0
$97$	393.107	0.411484	0.205742	−	0.978606i	$-0.434039\pi$
$97$	393.107	0.411484	0.205742	+	0.978606i	$0.434039\pi$
$98$	0	0
$99$	0	0
$100$	287.132	0.287132
$101$	−454.442	−0.447709	−0.223855	−	0.974623i	$-0.571864\pi$
$101$	−454.442	−0.447709	−0.223855	+	0.974623i	$0.571864\pi$
$102$	0	0
$103$	−703.699	−0.673180	−0.336590	−	0.941651i	$-0.609274\pi$
$103$	−703.699	−0.673180	−0.336590	+	0.941651i	$0.609274\pi$
$104$	−494.500	−0.466247
$105$	0	0
$106$	2987.62	2.73758
$107$	−954.997	−0.862832	−0.431416	−	0.902153i	$-0.641986\pi$
$107$	−954.997	−0.862832	−0.431416	+	0.902153i	$0.641986\pi$
$108$	0	0
$109$	288.927	0.253891	0.126946	−	0.991910i	$-0.459483\pi$
$109$	288.927	0.253891	0.126946	+	0.991910i	$0.459483\pi$
$110$	−3.13708	−0.00271918
$111$	0	0
$112$	0	0
$113$	−1251.24	−1.04165	−0.520826	−	0.853663i	$-0.674376\pi$
$113$	−1251.24	−1.04165	−0.520826	+	0.853663i	$0.674376\pi$
$114$	0	0
$115$	772.315	0.626250
$116$	461.102	0.369071
$117$	0	0
$118$	−3514.51	−2.74184
$119$	0	0
$120$	0	0
$121$	−1330.98	−0.999985
$122$	−3342.01	−2.48009
$123$	0	0
$124$	866.235	0.627341
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1947.62	−1.36082	−0.680408	−	0.732833i	$-0.738198\pi$
$127$	−1947.62	−1.36082	−0.680408	+	0.732833i	$0.738198\pi$
$128$	−1782.37	−1.23079
$129$	0	0
$130$	709.411	0.478611
$131$	−1061.65	−0.708064	−0.354032	−	0.935233i	$-0.615190\pi$
$131$	−1061.65	−0.708064	−0.354032	+	0.935233i	$0.615190\pi$
$132$	0	0
$133$	0	0
$134$	−3269.08	−2.10750
$135$	0	0
$136$	1759.52	1.10939
$137$	1439.34	0.897601	0.448800	−	0.893632i	$-0.351851\pi$
$137$	1439.34	0.897601	0.448800	+	0.893632i	$0.351851\pi$
$138$	0	0
$139$	662.132	0.404038	0.202019	−	0.979382i	$-0.435250\pi$
$139$	662.132	0.404038	0.202019	+	0.979382i	$0.435250\pi$
$140$	0	0
$141$	0	0
$142$	−163.437	−0.0965870
$143$	−4.56854	−0.00267161
$144$	0	0
$145$	−200.736	−0.114967
$146$	357.009	0.202372
$147$	0	0
$148$	−4597.94	−2.55370
$149$	787.110	0.432769	0.216384	−	0.976308i	$-0.430574\pi$
$149$	787.110	0.432769	0.216384	+	0.976308i	$0.430574\pi$
$150$	0	0
$151$	−848.355	−0.457206	−0.228603	−	0.973520i	$-0.573416\pi$
$151$	−848.355	−0.457206	−0.228603	+	0.973520i	$0.573416\pi$
$152$	665.091	0.354908
$153$	0	0
$154$	0	0
$155$	−377.107	−0.195419
$156$	0	0
$157$	954.922	0.485421	0.242710	−	0.970099i	$-0.421964\pi$
$157$	954.922	0.485421	0.242710	+	0.970099i	$0.421964\pi$
$158$	−1400.89	−0.705371
$159$	0	0
$160$	1144.45	0.565478
$161$	0	0
$162$	0	0
$163$	−7.36421	−0.00353871	−0.00176936	−	0.999998i	$-0.500563\pi$
$163$	−7.36421	−0.00353871	−0.00176936	+	0.999998i	$0.500563\pi$
$164$	1096.00	0.521848
$165$	0	0
$166$	4175.53	1.95231
$167$	−2921.40	−1.35368	−0.676841	−	0.736129i	$-0.736651\pi$
$167$	−2921.40	−1.35368	−0.676841	+	0.736129i	$0.736651\pi$
$168$	0	0
$169$	−1163.88	−0.529760
$170$	−2524.21	−1.13881
$171$	0	0
$172$	−3905.81	−1.73148
$173$	3969.57	1.74451	0.872256	−	0.489049i	$-0.162656\pi$
$173$	3969.57	1.74451	0.872256	+	0.489049i	$0.162656\pi$
$174$	0	0
$175$	0	0
$176$	−3.40707	−0.00145919
$177$	0	0
$178$	−3457.23	−1.45579
$179$	1105.95	0.461803	0.230901	−	0.972977i	$-0.425833\pi$
$179$	1105.95	0.461803	0.230901	+	0.972977i	$0.425833\pi$
$180$	0	0
$181$	−117.214	−0.0481349	−0.0240674	−	0.999710i	$-0.507662\pi$
$181$	−117.214	−0.0481349	−0.0240674	+	0.999710i	$0.507662\pi$
$182$	0	0
$183$	0	0
$184$	−2376.38	−0.952114
$185$	2001.67	0.795488
$186$	0	0
$187$	16.2557	0.00635687
$188$	86.0286	0.0333738
$189$	0	0
$190$	−954.142	−0.364320
$191$	−1002.60	−0.379820	−0.189910	−	0.981802i	$-0.560820\pi$
$191$	−1002.60	−0.379820	−0.189910	+	0.981802i	$0.560820\pi$
$192$	0	0
$193$	3257.76	1.21502	0.607510	−	0.794312i	$-0.292169\pi$
$193$	3257.76	1.21502	0.607510	+	0.794312i	$0.292169\pi$
$194$	1735.26	0.642187
$195$	0	0
$196$	0	0
$197$	41.6955	0.0150796	0.00753981	−	0.999972i	$-0.497600\pi$
$197$	41.6955	0.0150796	0.00753981	+	0.999972i	$0.497600\pi$
$198$	0	0
$199$	2405.61	0.856932	0.428466	−	0.903558i	$-0.359054\pi$
$199$	2405.61	0.856932	0.428466	+	0.903558i	$0.359054\pi$
$200$	384.619	0.135983
$201$	0	0
$202$	−2006.00	−0.698722
$203$	0	0
$204$	0	0
$205$	−477.132	−0.162558
$206$	−3106.28	−1.05061
$207$	0	0
$208$	770.465	0.256837
$209$	6.14459	0.00203364
$210$	0	0
$211$	1211.24	0.395190	0.197595	−	0.980284i	$-0.436687\pi$
$211$	1211.24	0.395190	0.197595	+	0.980284i	$0.436687\pi$
$212$	7773.45	2.51831
$213$	0	0
$214$	−4215.56	−1.34659
$215$	1700.36	0.539364
$216$	0	0
$217$	0	0
$218$	1275.38	0.396238
$219$	0	0
$220$	−8.16234	−0.00250139
$221$	−3676.02	−1.11889
$222$	0	0
$223$	4164.99	1.25071	0.625354	−	0.780341i	$-0.284954\pi$
$223$	4164.99	1.25071	0.625354	+	0.780341i	$0.284954\pi$
$224$	0	0
$225$	0	0
$226$	−5523.24	−1.62566
$227$	−2356.24	−0.688940	−0.344470	−	0.938797i	$-0.611941\pi$
$227$	−2356.24	−0.688940	−0.344470	+	0.938797i	$0.611941\pi$
$228$	0	0
$229$	4345.16	1.25387	0.626935	−	0.779071i	$-0.284309\pi$
$229$	4345.16	1.25387	0.626935	+	0.779071i	$0.284309\pi$
$230$	3409.16	0.977363
$231$	0	0
$232$	617.655	0.174789
$233$	−2458.25	−0.691182	−0.345591	−	0.938385i	$-0.612321\pi$
$233$	−2458.25	−0.691182	−0.345591	+	0.938385i	$0.612321\pi$
$234$	0	0
$235$	−37.4517	−0.0103961
$236$	−9144.36	−2.52223
$237$	0	0
$238$	0	0
$239$	−322.304	−0.0872307	−0.0436154	−	0.999048i	$-0.513888\pi$
$239$	−322.304	−0.0872307	−0.0436154	+	0.999048i	$0.513888\pi$
$240$	0	0
$241$	−5011.24	−1.33943	−0.669714	−	0.742619i	$-0.733583\pi$
$241$	−5011.24	−1.33943	−0.669714	+	0.742619i	$0.733583\pi$
$242$	−5875.23	−1.56064
$243$	0	0
$244$	−8695.53	−2.28145
$245$	0	0
$246$	0	0
$247$	−1389.52	−0.357947
$248$	1160.34	0.297104
$249$	0	0
$250$	−551.777	−0.139590
$251$	−3936.95	−0.990031	−0.495015	−	0.868884i	$-0.664838\pi$
$251$	−3936.95	−0.990031	−0.495015	+	0.868884i	$0.664838\pi$
$252$	0	0
$253$	−21.9547	−0.00545565
$254$	−8597.23	−2.12377
$255$	0	0
$256$	−1318.94	−0.322008
$257$	3626.16	0.880131	0.440066	−	0.897966i	$-0.354955\pi$
$257$	3626.16	0.880131	0.440066	+	0.897966i	$0.354955\pi$
$258$	0	0
$259$	0	0
$260$	1845.81	0.440277
$261$	0	0
$262$	−4686.33	−1.10505
$263$	786.555	0.184415	0.0922074	−	0.995740i	$-0.470608\pi$
$263$	786.555	0.184415	0.0922074	+	0.995740i	$0.470608\pi$
$264$	0	0
$265$	−3384.09	−0.784465
$266$	0	0
$267$	0	0
$268$	−8505.78	−1.93871
$269$	−4057.94	−0.919766	−0.459883	−	0.887979i	$-0.652109\pi$
$269$	−4057.94	−0.919766	−0.459883	+	0.887979i	$0.652109\pi$
$270$	0	0
$271$	2899.42	0.649916	0.324958	−	0.945728i	$-0.394650\pi$
$271$	2899.42	0.649916	0.324958	+	0.945728i	$0.394650\pi$
$272$	−2741.45	−0.611122
$273$	0	0
$274$	6353.56	1.40085
$275$	3.55339	0.000779191 0
$276$	0	0
$277$	5293.96	1.14831	0.574157	−	0.818745i	$-0.305330\pi$
$277$	5293.96	1.14831	0.574157	+	0.818745i	$0.305330\pi$
$278$	2922.79	0.630566
$279$	0	0
$280$	0	0
$281$	−2359.40	−0.500890	−0.250445	−	0.968131i	$-0.580577\pi$
$281$	−2359.40	−0.500890	−0.250445	+	0.968131i	$0.580577\pi$
$282$	0	0
$283$	−2881.35	−0.605224	−0.302612	−	0.953114i	$-0.597859\pi$
$283$	−2881.35	−0.605224	−0.302612	+	0.953114i	$0.597859\pi$
$284$	−425.245	−0.0888510
$285$	0	0
$286$	−20.1665	−0.00416948
$287$	0	0
$288$	0	0
$289$	8166.93	1.66231
$290$	−886.091	−0.179424
$291$	0	0
$292$	928.897	0.186163
$293$	−760.730	−0.151680	−0.0758402	−	0.997120i	$-0.524164\pi$
$293$	−760.730	−0.151680	−0.0758402	+	0.997120i	$0.524164\pi$
$294$	0	0
$295$	3980.90	0.785685
$296$	−6159.03	−1.20941
$297$	0	0
$298$	3474.47	0.675405
$299$	4964.77	0.960268
$300$	0	0
$301$	0	0
$302$	−3744.82	−0.713544
$303$	0	0
$304$	−1036.26	−0.195505
$305$	3785.51	0.710681
$306$	0	0
$307$	−4968.57	−0.923685	−0.461842	−	0.886962i	$-0.652811\pi$
$307$	−4968.57	−0.923685	−0.461842	+	0.886962i	$0.652811\pi$
$308$	0	0
$309$	0	0
$310$	−1664.63	−0.304983
$311$	5033.96	0.917845	0.458922	−	0.888476i	$-0.348236\pi$
$311$	5033.96	0.917845	0.458922	+	0.888476i	$0.348236\pi$
$312$	0	0
$313$	3599.20	0.649965	0.324983	−	0.945720i	$-0.394642\pi$
$313$	3599.20	0.649965	0.324983	+	0.945720i	$0.394642\pi$
$314$	4215.23	0.757577
$315$	0	0
$316$	−3644.95	−0.648875
$317$	−3541.56	−0.627488	−0.313744	−	0.949508i	$-0.601583\pi$
$317$	−3541.56	−0.627488	−0.313744	+	0.949508i	$0.601583\pi$
$318$	0	0
$319$	5.70635	0.00100155
$320$	4093.01	0.715020
$321$	0	0
$322$	0	0
$323$	4944.16	0.851704
$324$	0	0
$325$	−803.553	−0.137148
$326$	−32.5072	−0.00552272
$327$	0	0
$328$	1468.11	0.247143
$329$	0	0
$330$	0	0
$331$	−7357.96	−1.22184	−0.610922	−	0.791691i	$-0.709201\pi$
$331$	−7357.96	−1.22184	−0.610922	+	0.791691i	$0.709201\pi$
$332$	10864.3	1.79594
$333$	0	0
$334$	−12895.7	−2.11264
$335$	3702.90	0.603914
$336$	0	0
$337$	−2323.24	−0.375534	−0.187767	−	0.982214i	$-0.560125\pi$
$337$	−2323.24	−0.375534	−0.187767	+	0.982214i	$0.560125\pi$
$338$	−5137.63	−0.826776
$339$	0	0
$340$	−6567.72	−1.04760
$341$	10.7201	0.00170242
$342$	0	0
$343$	0	0
$344$	−5231.92	−0.820018
$345$	0	0
$346$	17522.5	2.72259
$347$	5447.79	0.842803	0.421401	−	0.906874i	$-0.361538\pi$
$347$	5447.79	0.842803	0.421401	+	0.906874i	$0.361538\pi$
$348$	0	0
$349$	−1227.84	−0.188324	−0.0941618	−	0.995557i	$-0.530017\pi$
$349$	−1227.84	−0.188324	−0.0941618	+	0.995557i	$0.530017\pi$
$350$	0	0
$351$	0	0
$352$	−32.5333	−0.00492623
$353$	−1981.86	−0.298821	−0.149411	−	0.988775i	$-0.547738\pi$
$353$	−1981.86	−0.298821	−0.149411	+	0.988775i	$0.547738\pi$
$354$	0	0
$355$	185.126	0.0276774
$356$	−8995.33	−1.33919
$357$	0	0
$358$	4881.90	0.720717
$359$	12218.1	1.79622	0.898112	−	0.439768i	$-0.144939\pi$
$359$	12218.1	1.79622	0.898112	+	0.439768i	$0.144939\pi$
$360$	0	0
$361$	−4990.13	−0.727530
$362$	−517.406	−0.0751222
$363$	0	0
$364$	0	0
$365$	−404.386	−0.0579905
$366$	0	0
$367$	13647.7	1.94115	0.970577	−	0.240789i	$-0.0774062\pi$
$367$	13647.7	1.94115	0.970577	+	0.240789i	$0.0774062\pi$
$368$	3702.56	0.524482
$369$	0	0
$370$	8835.78	1.24149
$371$	0	0
$372$	0	0
$373$	−5544.03	−0.769594	−0.384797	−	0.923001i	$-0.625729\pi$
$373$	−5544.03	−0.769594	−0.384797	+	0.923001i	$0.625729\pi$
$374$	71.7561	0.00992091
$375$	0	0
$376$	115.237	0.0158056
$377$	−1290.42	−0.176286
$378$	0	0
$379$	−634.243	−0.0859601	−0.0429801	−	0.999076i	$-0.513685\pi$
$379$	−634.243	−0.0859601	−0.0429801	+	0.999076i	$0.513685\pi$
$380$	−2482.57	−0.335140
$381$	0	0
$382$	−4425.69	−0.592769
$383$	−5790.93	−0.772592	−0.386296	−	0.922375i	$-0.626246\pi$
$383$	−5790.93	−0.772592	−0.386296	+	0.922375i	$0.626246\pi$
$384$	0	0
$385$	0	0
$386$	14380.5	1.89623
$387$	0	0
$388$	4514.94	0.590751
$389$	−1516.99	−0.197724	−0.0988619	−	0.995101i	$-0.531520\pi$
$389$	−1516.99	−0.197724	−0.0988619	+	0.995101i	$0.531520\pi$
$390$	0	0
$391$	−17665.6	−2.28487
$392$	0	0
$393$	0	0
$394$	184.053	0.0235341
$395$	1586.79	0.202127
$396$	0	0
$397$	−7874.94	−0.995547	−0.497773	−	0.867307i	$-0.665849\pi$
$397$	−7874.94	−0.995547	−0.497773	+	0.867307i	$0.665849\pi$
$398$	10618.9	1.33738
$399$	0	0
$400$	−599.264	−0.0749080
$401$	11111.3	1.38372	0.691859	−	0.722033i	$-0.256792\pi$
$401$	11111.3	1.38372	0.691859	+	0.722033i	$0.256792\pi$
$402$	0	0
$403$	−2424.20	−0.299648
$404$	−5219.39	−0.642758
$405$	0	0
$406$	0	0
$407$	−56.9016	−0.00692999
$408$	0	0
$409$	−10710.1	−1.29482	−0.647411	−	0.762141i	$-0.724148\pi$
$409$	−10710.1	−1.29482	−0.647411	+	0.762141i	$0.724148\pi$
$410$	−2106.16	−0.253697
$411$	0	0
$412$	−8082.19	−0.966458
$413$	0	0
$414$	0	0
$415$	−4729.64	−0.559444
$416$	7356.99	0.867082
$417$	0	0
$418$	27.1235	0.00317381
$419$	−8615.98	−1.00458	−0.502289	−	0.864700i	$-0.667509\pi$
$419$	−8615.98	−1.00458	−0.502289	+	0.864700i	$0.667509\pi$
$420$	0	0
$421$	6689.72	0.774435	0.387217	−	0.921988i	$-0.373436\pi$
$421$	6689.72	0.774435	0.387217	+	0.921988i	$0.373436\pi$
$422$	5346.67	0.616758
$423$	0	0
$424$	10412.7	1.19265
$425$	2859.19	0.326332
$426$	0	0
$427$	0	0
$428$	−10968.4	−1.23873
$429$	0	0
$430$	7505.73	0.841764
$431$	−6170.64	−0.689627	−0.344814	−	0.938671i	$-0.612058\pi$
$431$	−6170.64	−0.689627	−0.344814	+	0.938671i	$0.612058\pi$
$432$	0	0
$433$	14001.1	1.55392	0.776961	−	0.629548i	$-0.216760\pi$
$433$	14001.1	1.55392	0.776961	+	0.629548i	$0.216760\pi$
$434$	0	0
$435$	0	0
$436$	3318.41	0.364502
$437$	−6677.50	−0.730957
$438$	0	0
$439$	12207.0	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$439$	12207.0	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$440$	−10.9336	−0.00118464
$441$	0	0
$442$	−16226.7	−1.74621
$443$	−1227.76	−0.131677	−0.0658384	−	0.997830i	$-0.520972\pi$
$443$	−1227.76	−0.131677	−0.0658384	+	0.997830i	$0.520972\pi$
$444$	0	0
$445$	3916.03	0.417163
$446$	18385.1	1.95193
$447$	0	0
$448$	0	0
$449$	1151.70	0.121051	0.0605257	−	0.998167i	$-0.480722\pi$
$449$	1151.70	0.121051	0.0605257	+	0.998167i	$0.480722\pi$
$450$	0	0
$451$	13.5635	0.00141614
$452$	−14370.8	−1.49546
$453$	0	0
$454$	−10401.0	−1.07520
$455$	0	0
$456$	0	0
$457$	18713.8	1.91553	0.957763	−	0.287558i	$-0.0928433\pi$
$457$	18713.8	1.91553	0.957763	+	0.287558i	$0.0928433\pi$
$458$	19180.5	1.95687
$459$	0	0
$460$	8870.25	0.899082
$461$	3154.57	0.318705	0.159352	−	0.987222i	$-0.449059\pi$
$461$	3154.57	0.318705	0.159352	+	0.987222i	$0.449059\pi$
$462$	0	0
$463$	4051.11	0.406633	0.203316	−	0.979113i	$-0.434828\pi$
$463$	4051.11	0.406633	0.203316	+	0.979113i	$0.434828\pi$
$464$	−962.351	−0.0962845
$465$	0	0
$466$	−10851.2	−1.07870
$467$	−16341.6	−1.61927	−0.809635	−	0.586934i	$-0.800335\pi$
$467$	−16341.6	−1.61927	−0.809635	+	0.586934i	$0.800335\pi$
$468$	0	0
$469$	0	0
$470$	−165.320	−0.0162247
$471$	0	0
$472$	−12249.1	−1.19451
$473$	−48.3362	−0.00469873
$474$	0	0
$475$	1080.76	0.104397
$476$	0	0
$477$	0	0
$478$	−1422.72	−0.136138
$479$	−2531.45	−0.241472	−0.120736	−	0.992685i	$-0.538525\pi$
$479$	−2531.45	−0.241472	−0.120736	+	0.992685i	$0.538525\pi$
$480$	0	0
$481$	12867.6	1.21977
$482$	−22120.7	−2.09039
$483$	0	0
$484$	−15286.7	−1.43564
$485$	−1965.53	−0.184021
$486$	0	0
$487$	−8442.18	−0.785527	−0.392763	−	0.919640i	$-0.628481\pi$
$487$	−8442.18	−0.785527	−0.392763	+	0.919640i	$0.628481\pi$
$488$	−11647.8	−1.08048
$489$	0	0
$490$	0	0
$491$	15223.9	1.39928	0.699640	−	0.714496i	$-0.253344\pi$
$491$	15223.9	1.39928	0.699640	+	0.714496i	$0.253344\pi$
$492$	0	0
$493$	4591.53	0.419457
$494$	−6133.63	−0.558634
$495$	0	0
$496$	−1807.89	−0.163663
$497$	0	0
$498$	0	0
$499$	−16622.6	−1.49125	−0.745623	−	0.666368i	$-0.767848\pi$
$499$	−16622.6	−1.49125	−0.745623	+	0.666368i	$0.767848\pi$
$500$	−1435.66	−0.128409
$501$	0	0
$502$	−17378.5	−1.54510
$503$	17506.6	1.55185	0.775923	−	0.630828i	$-0.217285\pi$
$503$	17506.6	1.55185	0.775923	+	0.630828i	$0.217285\pi$
$504$	0	0
$505$	2272.21	0.200222
$506$	−96.9127	−0.00851442
$507$	0	0
$508$	−22369.0	−1.95367
$509$	−12383.2	−1.07834	−0.539169	−	0.842198i	$-0.681261\pi$
$509$	−12383.2	−1.07834	−0.539169	+	0.842198i	$0.681261\pi$
$510$	0	0
$511$	0	0
$512$	8436.86	0.728242
$513$	0	0
$514$	16006.7	1.37359
$515$	3518.50	0.301055
$516$	0	0
$517$	1.06464	9.05666e−5 0
$518$	0	0
$519$	0	0
$520$	2472.50	0.208512
$521$	14979.0	1.25958	0.629790	−	0.776765i	$-0.283141\pi$
$521$	14979.0	1.25958	0.629790	+	0.776765i	$0.283141\pi$
$522$	0	0
$523$	3719.11	0.310947	0.155474	−	0.987840i	$-0.450310\pi$
$523$	3719.11	0.310947	0.155474	+	0.987840i	$0.450310\pi$
$524$	−12193.3	−1.01654
$525$	0	0
$526$	3472.02	0.287809
$527$	8625.75	0.712986
$528$	0	0
$529$	11691.8	0.960945
$530$	−14938.1	−1.22428
$531$	0	0
$532$	0	0
$533$	−3067.21	−0.249260
$534$	0	0
$535$	4774.99	0.385870
$536$	−11393.7	−0.918156
$537$	0	0
$538$	−17912.6	−1.43544
$539$	0	0
$540$	0	0
$541$	−244.726	−0.0194484	−0.00972420	−	0.999953i	$-0.503095\pi$
$541$	−244.726	−0.0194484	−0.00972420	+	0.999953i	$0.503095\pi$
$542$	12798.7	1.01430
$543$	0	0
$544$	−26177.5	−2.06315
$545$	−1444.63	−0.113544
$546$	0	0
$547$	−12685.4	−0.991568	−0.495784	−	0.868446i	$-0.665119\pi$
$547$	−12685.4	−0.991568	−0.495784	+	0.868446i	$0.665119\pi$
$548$	16531.3	1.28865
$549$	0	0
$550$	15.6854	0.00121605
$551$	1735.58	0.134189
$552$	0	0
$553$	0	0
$554$	23368.7	1.79213
$555$	0	0
$556$	7604.77	0.580062
$557$	22062.7	1.67833	0.839164	−	0.543879i	$-0.183045\pi$
$557$	22062.7	1.67833	0.839164	+	0.543879i	$0.183045\pi$
$558$	0	0
$559$	10930.6	0.827040
$560$	0	0
$561$	0	0
$562$	−10414.9	−0.781719
$563$	−7862.20	−0.588547	−0.294274	−	0.955721i	$-0.595078\pi$
$563$	−7862.20	−0.588547	−0.294274	+	0.955721i	$0.595078\pi$
$564$	0	0
$565$	6256.19	0.465841
$566$	−12718.9	−0.944549
$567$	0	0
$568$	−569.625	−0.0420791
$569$	−20941.5	−1.54291	−0.771453	−	0.636286i	$-0.780470\pi$
$569$	−20941.5	−1.54291	−0.771453	+	0.636286i	$0.780470\pi$
$570$	0	0
$571$	−7131.66	−0.522681	−0.261341	−	0.965247i	$-0.584165\pi$
$571$	−7131.66	−0.522681	−0.261341	+	0.965247i	$0.584165\pi$
$572$	−52.4710	−0.00383553
$573$	0	0
$574$	0	0
$575$	−3861.57	−0.280067
$576$	0	0
$577$	−15633.0	−1.12792	−0.563961	−	0.825801i	$-0.690723\pi$
$577$	−15633.0	−1.12792	−0.563961	+	0.825801i	$0.690723\pi$
$578$	36050.6	2.59430
$579$	0	0
$580$	−2305.51	−0.165054
$581$	0	0
$582$	0	0
$583$	96.2000	0.00683396
$584$	1244.28	0.0881654
$585$	0	0
$586$	−3358.03	−0.236721
$587$	19406.2	1.36453	0.682266	−	0.731104i	$-0.260995\pi$
$587$	19406.2	1.36453	0.682266	+	0.731104i	$0.260995\pi$
$588$	0	0
$589$	3260.50	0.228093
$590$	17572.6	1.22619
$591$	0	0
$592$	9596.21	0.666219
$593$	14859.6	1.02903	0.514513	−	0.857483i	$-0.327973\pi$
$593$	14859.6	1.02903	0.514513	+	0.857483i	$0.327973\pi$
$594$	0	0
$595$	0	0
$596$	9040.18	0.621309
$597$	0	0
$598$	21915.6	1.49865
$599$	10529.9	0.718261	0.359131	−	0.933287i	$-0.383073\pi$
$599$	10529.9	0.718261	0.359131	+	0.933287i	$0.383073\pi$
$600$	0	0
$601$	11595.2	0.786984	0.393492	−	0.919328i	$-0.371267\pi$
$601$	11595.2	0.786984	0.393492	+	0.919328i	$0.371267\pi$
$602$	0	0
$603$	0	0
$604$	−9743.60	−0.656393
$605$	6654.90	0.447207
$606$	0	0
$607$	−23282.1	−1.55682	−0.778411	−	0.627755i	$-0.783974\pi$
$607$	−23282.1	−1.55682	−0.778411	+	0.627755i	$0.783974\pi$
$608$	−9894.99	−0.660024
$609$	0	0
$610$	16710.0	1.10913
$611$	−240.755	−0.0159409
$612$	0	0
$613$	−7387.98	−0.486782	−0.243391	−	0.969928i	$-0.578260\pi$
$613$	−7387.98	−0.486782	−0.243391	+	0.969928i	$0.578260\pi$
$614$	−21932.3	−1.44156
$615$	0	0
$616$	0	0
$617$	−667.085	−0.0435265	−0.0217632	−	0.999763i	$-0.506928\pi$
$617$	−667.085	−0.0435265	−0.0217632	+	0.999763i	$0.506928\pi$
$618$	0	0
$619$	−9425.78	−0.612042	−0.306021	−	0.952025i	$-0.598998\pi$
$619$	−9425.78	−0.612042	−0.306021	+	0.952025i	$0.598998\pi$
$620$	−4331.18	−0.280555
$621$	0	0
$622$	22221.0	1.43244
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	15887.7	1.01437
$627$	0	0
$628$	10967.5	0.696899
$629$	−45785.1	−2.90234
$630$	0	0
$631$	28047.3	1.76949	0.884744	−	0.466078i	$-0.154333\pi$
$631$	28047.3	1.76949	0.884744	+	0.466078i	$0.154333\pi$
$632$	−4882.49	−0.307302
$633$	0	0
$634$	−15633.2	−0.979295
$635$	9738.12	0.608576
$636$	0	0
$637$	0	0
$638$	25.1890	0.00156308
$639$	0	0
$640$	8911.85	0.550425
$641$	−23208.3	−1.43007	−0.715034	−	0.699090i	$-0.753589\pi$
$641$	−23208.3	−1.43007	−0.715034	+	0.699090i	$0.753589\pi$
$642$	0	0
$643$	−4294.22	−0.263371	−0.131686	−	0.991292i	$-0.542039\pi$
$643$	−4294.22	−0.263371	−0.131686	+	0.991292i	$0.542039\pi$
$644$	0	0
$645$	0	0
$646$	21824.6	1.32922
$647$	17393.4	1.05689	0.528443	−	0.848969i	$-0.322776\pi$
$647$	17393.4	1.05689	0.528443	+	0.848969i	$0.322776\pi$
$648$	0	0
$649$	−113.166	−0.00684459
$650$	−3547.06	−0.214042
$651$	0	0
$652$	−84.5801	−0.00508039
$653$	−14475.4	−0.867486	−0.433743	−	0.901037i	$-0.642807\pi$
$653$	−14475.4	−0.867486	−0.433743	+	0.901037i	$0.642807\pi$
$654$	0	0
$655$	5308.23	0.316656
$656$	−2287.42	−0.136142
$657$	0	0
$658$	0	0
$659$	−620.113	−0.0366558	−0.0183279	−	0.999832i	$-0.505834\pi$
$659$	−620.113	−0.0366558	−0.0183279	+	0.999832i	$0.505834\pi$
$660$	0	0
$661$	6105.50	0.359268	0.179634	−	0.983733i	$-0.442509\pi$
$661$	6105.50	0.359268	0.179634	+	0.983733i	$0.442509\pi$
$662$	−32479.6	−1.90688
$663$	0	0
$664$	14552.9	0.850546
$665$	0	0
$666$	0	0
$667$	−6201.25	−0.359990
$668$	−33553.1	−1.94343
$669$	0	0
$670$	16345.4	0.942505
$671$	−107.611	−0.00619118
$672$	0	0
$673$	−16302.5	−0.933754	−0.466877	−	0.884322i	$-0.654621\pi$
$673$	−16302.5	−0.933754	−0.466877	+	0.884322i	$0.654621\pi$
$674$	−10255.3	−0.586081
$675$	0	0
$676$	−13367.5	−0.760556
$677$	4819.09	0.273579	0.136789	−	0.990600i	$-0.456322\pi$
$677$	4819.09	0.273579	0.136789	+	0.990600i	$0.456322\pi$
$678$	0	0
$679$	0	0
$680$	−8797.59	−0.496136
$681$	0	0
$682$	47.3206	0.00265689
$683$	8190.32	0.458849	0.229425	−	0.973326i	$-0.426316\pi$
$683$	8190.32	0.458849	0.229425	+	0.973326i	$0.426316\pi$
$684$	0	0
$685$	−7196.71	−0.401419
$686$	0	0
$687$	0	0
$688$	8151.69	0.451716
$689$	−21754.4	−1.20287
$690$	0	0
$691$	476.973	0.0262589	0.0131295	−	0.999914i	$-0.495821\pi$
$691$	476.973	0.0262589	0.0131295	+	0.999914i	$0.495821\pi$
$692$	45591.6	2.50453
$693$	0	0
$694$	24047.7	1.31533
$695$	−3310.66	−0.180691
$696$	0	0
$697$	10913.7	0.593092
$698$	−5419.96	−0.293909
$699$	0	0
$700$	0	0
$701$	20366.7	1.09735	0.548673	−	0.836037i	$-0.315133\pi$
$701$	20366.7	1.09735	0.548673	+	0.836037i	$0.315133\pi$
$702$	0	0
$703$	−17306.6	−0.928492
$704$	−116.353	−0.00622898
$705$	0	0
$706$	−8748.36	−0.466358
$707$	0	0
$708$	0	0
$709$	22415.1	1.18733	0.593664	−	0.804713i	$-0.297681\pi$
$709$	22415.1	1.18733	0.593664	+	0.804713i	$0.297681\pi$
$710$	817.187	0.0431950
$711$	0	0
$712$	−12049.4	−0.634230
$713$	−11649.8	−0.611906
$714$	0	0
$715$	22.8427	0.00119478
$716$	12702.2	0.662992
$717$	0	0
$718$	53933.1	2.80329
$719$	26156.0	1.35668	0.678340	−	0.734748i	$-0.262700\pi$
$719$	26156.0	1.35668	0.678340	+	0.734748i	$0.262700\pi$
$720$	0	0
$721$	0	0
$722$	−22027.5	−1.13543
$723$	0	0
$724$	−1346.23	−0.0691053
$725$	1003.68	0.0514148
$726$	0	0
$727$	−13666.2	−0.697184	−0.348592	−	0.937275i	$-0.613340\pi$
$727$	−13666.2	−0.697184	−0.348592	+	0.937275i	$0.613340\pi$
$728$	0	0
$729$	0	0
$730$	−1785.05	−0.0905035
$731$	−38893.1	−1.96787
$732$	0	0
$733$	25038.2	1.26167	0.630836	−	0.775916i	$-0.282712\pi$
$733$	25038.2	1.26167	0.630836	+	0.775916i	$0.282712\pi$
$734$	60243.8	3.02948
$735$	0	0
$736$	35354.9	1.77065
$737$	−105.263	−0.00526107
$738$	0	0
$739$	3255.20	0.162036	0.0810178	−	0.996713i	$-0.474183\pi$
$739$	3255.20	0.162036	0.0810178	+	0.996713i	$0.474183\pi$
$740$	22989.7	1.14205
$741$	0	0
$742$	0	0
$743$	8366.48	0.413104	0.206552	−	0.978436i	$-0.433776\pi$
$743$	8366.48	0.413104	0.206552	+	0.978436i	$0.433776\pi$
$744$	0	0
$745$	−3935.55	−0.193540
$746$	−24472.5	−1.20108
$747$	0	0
$748$	186.701	0.00912630
$749$	0	0
$750$	0	0
$751$	−10164.5	−0.493884	−0.246942	−	0.969030i	$-0.579426\pi$
$751$	−10164.5	−0.493884	−0.246942	+	0.969030i	$0.579426\pi$
$752$	−179.547	−0.00870668
$753$	0	0
$754$	−5696.17	−0.275123
$755$	4241.77	0.204469
$756$	0	0
$757$	9031.23	0.433614	0.216807	−	0.976214i	$-0.430436\pi$
$757$	9031.23	0.433614	0.216807	+	0.976214i	$0.430436\pi$
$758$	−2799.68	−0.134155
$759$	0	0
$760$	−3325.45	−0.158720
$761$	32434.7	1.54502	0.772508	−	0.635005i	$-0.219002\pi$
$761$	32434.7	1.54502	0.772508	+	0.635005i	$0.219002\pi$
$762$	0	0
$763$	0	0
$764$	−11515.1	−0.545292
$765$	0	0
$766$	−25562.4	−1.20575
$767$	25591.0	1.20474
$768$	0	0
$769$	−22629.4	−1.06117	−0.530583	−	0.847633i	$-0.678027\pi$
$769$	−22629.4	−1.06117	−0.530583	+	0.847633i	$0.678027\pi$
$770$	0	0
$771$	0	0
$772$	37416.3	1.74436
$773$	−2707.01	−0.125956	−0.0629782	−	0.998015i	$-0.520060\pi$
$773$	−2707.01	−0.125956	−0.0629782	+	0.998015i	$0.520060\pi$
$774$	0	0
$775$	1885.53	0.0873940
$776$	6047.86	0.279775
$777$	0	0
$778$	−6696.33	−0.308580
$779$	4125.33	0.189737
$780$	0	0
$781$	−5.26261	−0.000241115 0
$782$	−77979.5	−3.56591
$783$	0	0
$784$	0	0
$785$	−4774.61	−0.217087
$786$	0	0
$787$	−25084.6	−1.13618	−0.568088	−	0.822968i	$-0.692317\pi$
$787$	−25084.6	−1.13618	−0.568088	+	0.822968i	$0.692317\pi$
$788$	478.885	0.0216492
$789$	0	0
$790$	7004.44	0.315451
$791$	0	0
$792$	0	0
$793$	24334.9	1.08973
$794$	−34761.7	−1.55371
$795$	0	0
$796$	27629.1	1.23026
$797$	−7374.99	−0.327774	−0.163887	−	0.986479i	$-0.552403\pi$
$797$	−7374.99	−0.327774	−0.163887	+	0.986479i	$0.552403\pi$
$798$	0	0
$799$	856.651	0.0379301
$800$	−5722.23	−0.252889
$801$	0	0
$802$	49047.5	2.15951
$803$	11.4955	0.000505191 0
$804$	0	0
$805$	0	0
$806$	−10701.0	−0.467649
$807$	0	0
$808$	−6991.48	−0.304405
$809$	−2314.37	−0.100580	−0.0502899	−	0.998735i	$-0.516015\pi$
$809$	−2314.37	−0.100580	−0.0502899	+	0.998735i	$0.516015\pi$
$810$	0	0
$811$	38300.1	1.65832	0.829160	−	0.559011i	$-0.188819\pi$
$811$	38300.1	1.65832	0.829160	+	0.559011i	$0.188819\pi$
$812$	0	0
$813$	0	0
$814$	−251.176	−0.0108154
$815$	36.8211	0.00158256
$816$	0	0
$817$	−14701.4	−0.629544
$818$	−47276.8	−2.02078
$819$	0	0
$820$	−5480.00	−0.233378
$821$	−3942.93	−0.167612	−0.0838059	−	0.996482i	$-0.526708\pi$
$821$	−3942.93	−0.167612	−0.0838059	+	0.996482i	$0.526708\pi$
$822$	0	0
$823$	18502.8	0.783679	0.391839	−	0.920034i	$-0.371839\pi$
$823$	18502.8	0.783679	0.391839	+	0.920034i	$0.371839\pi$
$824$	−10826.3	−0.457707
$825$	0	0
$826$	0	0
$827$	11965.2	0.503110	0.251555	−	0.967843i	$-0.419058\pi$
$827$	11965.2	0.503110	0.251555	+	0.967843i	$0.419058\pi$
$828$	0	0
$829$	−35962.9	−1.50669	−0.753344	−	0.657626i	$-0.771561\pi$
$829$	−35962.9	−1.50669	−0.753344	+	0.657626i	$0.771561\pi$
$830$	−20877.7	−0.873101
$831$	0	0
$832$	26311.6	1.09638
$833$	0	0
$834$	0	0
$835$	14607.0	0.605385
$836$	70.5723	0.00291961
$837$	0	0
$838$	−38032.8	−1.56780
$839$	20174.0	0.830137	0.415069	−	0.909790i	$-0.363758\pi$
$839$	20174.0	0.830137	0.415069	+	0.909790i	$0.363758\pi$
$840$	0	0
$841$	−22777.2	−0.933913
$842$	29529.9	1.20863
$843$	0	0
$844$	13911.4	0.567359
$845$	5819.42	0.236916
$846$	0	0
$847$	0	0
$848$	−16223.7	−0.656987
$849$	0	0
$850$	12621.1	0.509293
$851$	61836.6	2.49087
$852$	0	0
$853$	25297.5	1.01544	0.507719	−	0.861523i	$-0.330489\pi$
$853$	25297.5	1.01544	0.507719	+	0.861523i	$0.330489\pi$
$854$	0	0
$855$	0	0
$856$	−14692.4	−0.586655
$857$	−39269.1	−1.56523	−0.782617	−	0.622504i	$-0.786116\pi$
$857$	−39269.1	−1.56523	−0.782617	+	0.622504i	$0.786116\pi$
$858$	0	0
$859$	−14499.1	−0.575905	−0.287953	−	0.957645i	$-0.592975\pi$
$859$	−14499.1	−0.575905	−0.287953	+	0.957645i	$0.592975\pi$
$860$	19529.1	0.774343
$861$	0	0
$862$	−27238.5	−1.07627
$863$	24411.3	0.962884	0.481442	−	0.876478i	$-0.340113\pi$
$863$	24411.3	0.962884	0.481442	+	0.876478i	$0.340113\pi$
$864$	0	0
$865$	−19847.8	−0.780170
$866$	61803.7	2.42515
$867$	0	0
$868$	0	0
$869$	−45.1079	−0.00176085
$870$	0	0
$871$	23803.8	0.926019
$872$	4445.07	0.172625
$873$	0	0
$874$	−29475.9	−1.14078
$875$	0	0
$876$	0	0
$877$	−5056.65	−0.194699	−0.0973494	−	0.995250i	$-0.531036\pi$
$877$	−5056.65	−0.194699	−0.0973494	+	0.995250i	$0.531036\pi$
$878$	53884.2	2.07119
$879$	0	0
$880$	17.0354	0.000652570 0
$881$	−13233.9	−0.506086	−0.253043	−	0.967455i	$-0.581431\pi$
$881$	−13233.9	−0.506086	−0.253043	+	0.967455i	$0.581431\pi$
$882$	0	0
$883$	−13824.2	−0.526866	−0.263433	−	0.964678i	$-0.584855\pi$
$883$	−13824.2	−0.526866	−0.263433	+	0.964678i	$0.584855\pi$
$884$	−42220.1	−1.60635
$885$	0	0
$886$	−5419.61	−0.205503
$887$	37819.3	1.43162	0.715811	−	0.698294i	$-0.246057\pi$
$887$	37819.3	1.43162	0.715811	+	0.698294i	$0.246057\pi$
$888$	0	0
$889$	0	0
$890$	17286.2	0.651049
$891$	0	0
$892$	47836.0	1.79559
$893$	323.810	0.0121343
$894$	0	0
$895$	−5529.76	−0.206524
$896$	0	0
$897$	0	0
$898$	5083.85	0.188920
$899$	3027.96	0.112334
$900$	0	0
$901$	77406.0	2.86212
$902$	59.8721	0.00221012
$903$	0	0
$904$	−19250.0	−0.708237
$905$	586.068	0.0215266
$906$	0	0
$907$	1052.75	0.0385404	0.0192702	−	0.999814i	$-0.493866\pi$
$907$	1052.75	0.0385404	0.0192702	+	0.999814i	$0.493866\pi$
$908$	−27062.1	−0.989083
$909$	0	0
$910$	0	0
$911$	−7854.40	−0.285651	−0.142825	−	0.989748i	$-0.545619\pi$
$911$	−7854.40	−0.285651	−0.142825	+	0.989748i	$0.545619\pi$
$912$	0	0
$913$	134.450	0.00487366
$914$	82606.8	2.98949
$915$	0	0
$916$	49905.4	1.80013
$917$	0	0
$918$	0	0
$919$	4935.85	0.177169	0.0885846	−	0.996069i	$-0.471766\pi$
$919$	4935.85	0.177169	0.0885846	+	0.996069i	$0.471766\pi$
$920$	11881.9	0.425798
$921$	0	0
$922$	13924.9	0.497390
$923$	1190.07	0.0424395
$924$	0	0
$925$	−10008.3	−0.355753
$926$	17882.4	0.634615
$927$	0	0
$928$	−9189.27	−0.325056
$929$	15705.0	0.554645	0.277322	−	0.960777i	$-0.410553\pi$
$929$	15705.0	0.554645	0.277322	+	0.960777i	$0.410553\pi$
$930$	0	0
$931$	0	0
$932$	−28233.7	−0.992302
$933$	0	0
$934$	−72135.3	−2.52713
$935$	−81.2785	−0.00284288
$936$	0	0
$937$	−43806.5	−1.52732	−0.763658	−	0.645621i	$-0.776599\pi$
$937$	−43806.5	−1.52732	−0.763658	+	0.645621i	$0.776599\pi$
$938$	0	0
$939$	0	0
$940$	−430.143	−0.0149252
$941$	−4062.42	−0.140734	−0.0703672	−	0.997521i	$-0.522417\pi$
$941$	−4062.42	−0.140734	−0.0703672	+	0.997521i	$0.522417\pi$
$942$	0	0
$943$	−14739.8	−0.509009
$944$	19084.9	0.658009
$945$	0	0
$946$	−213.366	−0.00733313
$947$	4860.13	0.166772	0.0833859	−	0.996517i	$-0.473427\pi$
$947$	4860.13	0.166772	0.0833859	+	0.996517i	$0.473427\pi$
$948$	0	0
$949$	−2599.57	−0.0889204
$950$	4770.71	0.162929
$951$	0	0
$952$	0	0
$953$	18999.6	0.645811	0.322906	−	0.946431i	$-0.395340\pi$
$953$	18999.6	0.645811	0.322906	+	0.946431i	$0.395340\pi$
$954$	0	0
$955$	5013.00	0.169861
$956$	−3701.76	−0.125234
$957$	0	0
$958$	−11174.4	−0.376855
$959$	0	0
$960$	0	0
$961$	−24102.6	−0.809057
$962$	56800.2	1.90365
$963$	0	0
$964$	−57555.5	−1.92296
$965$	−16288.8	−0.543373
$966$	0	0
$967$	−32695.1	−1.08728	−0.543641	−	0.839318i	$-0.682955\pi$
$967$	−32695.1	−1.08728	−0.543641	+	0.839318i	$0.682955\pi$
$968$	−20476.8	−0.679907
$969$	0	0
$970$	−8676.29	−0.287195
$971$	6189.69	0.204569	0.102285	−	0.994755i	$-0.467385\pi$
$971$	6189.69	0.204569	0.102285	+	0.994755i	$0.467385\pi$
$972$	0	0
$973$	0	0
$974$	−37265.6	−1.22594
$975$	0	0
$976$	18148.2	0.595193
$977$	−2142.44	−0.0701563	−0.0350782	−	0.999385i	$-0.511168\pi$
$977$	−2142.44	−0.0701563	−0.0350782	+	0.999385i	$0.511168\pi$
$978$	0	0
$979$	−111.321	−0.00363416
$980$	0	0
$981$	0	0
$982$	67201.7	2.18380
$983$	−16808.2	−0.545371	−0.272685	−	0.962103i	$-0.587912\pi$
$983$	−16808.2	−0.545371	−0.272685	+	0.962103i	$0.587912\pi$
$984$	0	0
$985$	−208.478	−0.00674381
$986$	20268.0	0.654630
$987$	0	0
$988$	−15959.0	−0.513891
$989$	52528.4	1.68888
$990$	0	0
$991$	21250.8	0.681185	0.340593	−	0.940211i	$-0.389372\pi$
$991$	21250.8	0.681185	0.340593	+	0.940211i	$0.389372\pi$
$992$	−17263.1	−0.552526
$993$	0	0
$994$	0	0
$995$	−12028.1	−0.383231
$996$	0	0
$997$	−37639.9	−1.19566	−0.597828	−	0.801624i	$-0.703969\pi$
$997$	−37639.9	−1.19566	−0.597828	+	0.801624i	$0.703969\pi$
$998$	−73375.9	−2.32733
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bf.1.2		2
3.2	odd	2		245.4.a.g.1.1		2
7.2	even	3		315.4.j.c.46.1		4
7.4	even	3		315.4.j.c.226.1		4
7.6	odd	2		2205.4.a.bg.1.2		2
15.14	odd	2		1225.4.a.x.1.2		2
21.2	odd	6		35.4.e.b.11.2	✓	4
21.5	even	6		245.4.e.l.116.2		4
21.11	odd	6		35.4.e.b.16.2	yes	4
21.17	even	6		245.4.e.l.226.2		4
21.20	even	2		245.4.a.h.1.1		2
84.11	even	6		560.4.q.i.401.2		4
84.23	even	6		560.4.q.i.81.2		4
105.2	even	12		175.4.k.c.74.1		8
105.23	even	12		175.4.k.c.74.4		8
105.32	even	12		175.4.k.c.149.4		8
105.44	odd	6		175.4.e.c.151.1		4
105.53	even	12		175.4.k.c.149.1		8
105.74	odd	6		175.4.e.c.51.1		4
105.104	even	2		1225.4.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.e.b.11.2	✓	4	21.2	odd	6
35.4.e.b.16.2	yes	4	21.11	odd	6
175.4.e.c.51.1		4	105.74	odd	6
175.4.e.c.151.1		4	105.44	odd	6
175.4.k.c.74.1		8	105.2	even	12
175.4.k.c.74.4		8	105.23	even	12
175.4.k.c.149.1		8	105.53	even	12
175.4.k.c.149.4		8	105.32	even	12
245.4.a.g.1.1		2	3.2	odd	2
245.4.a.h.1.1		2	21.20	even	2
245.4.e.l.116.2		4	21.5	even	6
245.4.e.l.226.2		4	21.17	even	6
315.4.j.c.46.1		4	7.2	even	3
315.4.j.c.226.1		4	7.4	even	3
560.4.q.i.81.2		4	84.23	even	6
560.4.q.i.401.2		4	84.11	even	6
1225.4.a.v.1.2		2	105.104	even	2
1225.4.a.x.1.2		2	15.14	odd	2
2205.4.a.bf.1.2		2	1.1	even	1	trivial
2205.4.a.bg.1.2		2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+4.41421 q^{2} +11.4853 q^{4} -5.00000 q^{5} +15.3848 q^{8} -22.0711 q^{10} +0.142136 q^{11} -32.1421 q^{13} -23.9706 q^{16} +114.368 q^{17} +43.2304 q^{19} -57.4264 q^{20} +0.627417 q^{22} -154.463 q^{23} +25.0000 q^{25} -141.882 q^{26} +40.1472 q^{29} +75.4214 q^{31} -228.889 q^{32} +504.843 q^{34} -400.333 q^{37} +190.828 q^{38} -76.9239 q^{40} +95.4264 q^{41} -340.071 q^{43} +1.63247 q^{44} -681.833 q^{46} +7.49033 q^{47} +110.355 q^{50} -369.161 q^{52} +676.818 q^{53} -0.710678 q^{55} +177.218 q^{58} -796.181 q^{59} -757.102 q^{61} +332.926 q^{62} -818.602 q^{64} +160.711 q^{65} -740.581 q^{67} +1313.54 q^{68} -37.0253 q^{71} +80.8772 q^{73} -1767.16 q^{74} +496.514 q^{76} -317.358 q^{79} +119.853 q^{80} +421.233 q^{82} +945.929 q^{83} -571.838 q^{85} -1501.15 q^{86} +2.18672 q^{88} -783.205 q^{89} -1774.05 q^{92} +33.0639 q^{94} -216.152 q^{95} +393.107 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{2} + 6 q^{4} - 10 q^{5} - 6 q^{8} - 30 q^{10} - 28 q^{11} - 36 q^{13} - 14 q^{16} + 76 q^{17} + 160 q^{19} - 30 q^{20} - 44 q^{22} + 22 q^{23} + 50 q^{25} - 148 q^{26} + 250 q^{29} - 132 q^{31}+ \cdots - 628 q^{97}+O(q^{100}) \) 2 * q + 6 * q^2 + 6 * q^4 - 10 * q^5 - 6 * q^8 - 30 * q^10 - 28 * q^11 - 36 * q^13 - 14 * q^16 + 76 * q^17 + 160 * q^19 - 30 * q^20 - 44 * q^22 + 22 * q^23 + 50 * q^25 - 148 * q^26 + 250 * q^29 - 132 * q^31 - 42 * q^32 + 444 * q^34 - 416 * q^37 + 376 * q^38 + 30 * q^40 + 106 * q^41 - 666 * q^43 + 156 * q^44 - 402 * q^46 + 196 * q^47 + 150 * q^50 - 348 * q^52 + 952 * q^53 + 140 * q^55 + 510 * q^58 - 840 * q^59 + 98 * q^61 + 4 * q^62 - 602 * q^64 + 180 * q^65 - 1286 * q^67 + 1524 * q^68 - 1064 * q^71 - 172 * q^73 - 1792 * q^74 - 144 * q^76 - 1240 * q^79 + 70 * q^80 + 438 * q^82 + 1906 * q^83 - 380 * q^85 - 2018 * q^86 + 604 * q^88 - 650 * q^89 - 2742 * q^92 + 332 * q^94 - 800 * q^95 - 628 * q^97

Introduction

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