LMFDB - Embedded newform 2205.4.a.be.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+4.23607 q^{2} +9.94427 q^{4} -5.00000 q^{5} +8.23607 q^{8} -21.1803 q^{10} +41.5279 q^{11} -88.9706 q^{13} -44.6656 q^{16} -120.387 q^{17} +112.138 q^{19} -49.7214 q^{20} +175.915 q^{22} +115.279 q^{23} +25.0000 q^{25} -376.885 q^{26} +144.833 q^{29} +258.079 q^{31} -255.095 q^{32} -509.967 q^{34} +48.3344 q^{37} +475.023 q^{38} -41.1803 q^{40} +200.885 q^{41} -218.217 q^{43} +412.964 q^{44} +488.328 q^{46} +575.659 q^{47} +105.902 q^{50} -884.748 q^{52} +184.302 q^{53} -207.639 q^{55} +613.522 q^{58} -151.502 q^{59} +529.830 q^{61} +1093.24 q^{62} -723.276 q^{64} +444.853 q^{65} +1.28485 q^{67} -1197.16 q^{68} +61.4226 q^{71} -484.800 q^{73} +204.748 q^{74} +1115.13 q^{76} +878.257 q^{79} +223.328 q^{80} +850.964 q^{82} +491.830 q^{83} +601.935 q^{85} -924.381 q^{86} +342.026 q^{88} -415.560 q^{89} +1146.36 q^{92} +2438.53 q^{94} -560.689 q^{95} +1031.70 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 2 q^{4} - 10 q^{5} + 12 q^{8} - 20 q^{10} + 92 q^{11} - 8 q^{13} + 18 q^{16} - 44 q^{17} + 108 q^{19} - 10 q^{20} + 164 q^{22} + 320 q^{23} + 50 q^{25} - 396 q^{26} + 236 q^{29} + 60 q^{31}+ \cdots + 2448 q^{97}+O(q^{100}) $ 2 * q + 4 * q^2 + 2 * q^4 - 10 * q^5 + 12 * q^8 - 20 * q^10 + 92 * q^11 - 8 * q^13 + 18 * q^16 - 44 * q^17 + 108 * q^19 - 10 * q^20 + 164 * q^22 + 320 * q^23 + 50 * q^25 - 396 * q^26 + 236 * q^29 + 60 * q^31 - 300 * q^32 - 528 * q^34 + 204 * q^37 + 476 * q^38 - 60 * q^40 + 44 * q^41 + 136 * q^43 + 12 * q^44 + 440 * q^46 + 400 * q^47 + 100 * q^50 - 1528 * q^52 - 16 * q^53 - 460 * q^55 + 592 * q^58 - 464 * q^59 + 684 * q^61 + 1140 * q^62 - 1214 * q^64 + 40 * q^65 + 736 * q^67 - 1804 * q^68 + 740 * q^71 - 424 * q^73 + 168 * q^74 + 1148 * q^76 - 408 * q^79 - 90 * q^80 + 888 * q^82 + 608 * q^83 + 220 * q^85 - 1008 * q^86 + 532 * q^88 - 1332 * q^89 - 480 * q^92 + 2480 * q^94 - 540 * q^95 + 2448 * 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 $

$2$	4.23607	1.49768	0.748838	−	0.662753i	$-0.230612\pi$
$2$	4.23607	1.49768	0.748838	+	0.662753i	$0.230612\pi$
$3$	0	0
$3$	0	0
$4$	9.94427	1.24303
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.23607	0.363986
$9$	0	0
$10$	−21.1803	−0.669781
$11$	41.5279	1.13828	0.569142	−	0.822239i	$-0.307275\pi$
$11$	41.5279	1.13828	0.569142	+	0.822239i	$0.307275\pi$
$12$	0	0
$13$	−88.9706	−1.89815	−0.949077	−	0.315044i	$-0.897981\pi$
$13$	−88.9706	−1.89815	−0.949077	+	0.315044i	$0.897981\pi$
$14$	0	0
$15$	0	0
$16$	−44.6656	−0.697900
$17$	−120.387	−1.71754	−0.858769	−	0.512364i	$-0.828770\pi$
$17$	−120.387	−1.71754	−0.858769	+	0.512364i	$0.828770\pi$
$18$	0	0
$19$	112.138	1.35401	0.677004	−	0.735979i	$-0.263278\pi$
$19$	112.138	1.35401	0.677004	+	0.735979i	$0.263278\pi$
$20$	−49.7214	−0.555902
$21$	0	0
$22$	175.915	1.70478
$23$	115.279	1.04510	0.522549	−	0.852609i	$-0.324981\pi$
$23$	115.279	1.04510	0.522549	+	0.852609i	$0.324981\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−376.885	−2.84282
$27$	0	0
$28$	0	0
$29$	144.833	0.927406	0.463703	−	0.885991i	$-0.346520\pi$
$29$	144.833	0.927406	0.463703	+	0.885991i	$0.346520\pi$
$30$	0	0
$31$	258.079	1.49524	0.747618	−	0.664128i	$-0.231197\pi$
$31$	258.079	1.49524	0.747618	+	0.664128i	$0.231197\pi$
$32$	−255.095	−1.40922
$33$	0	0
$34$	−509.967	−2.57231
$35$	0	0
$36$	0	0
$37$	48.3344	0.214760	0.107380	−	0.994218i	$-0.465754\pi$
$37$	48.3344	0.214760	0.107380	+	0.994218i	$0.465754\pi$
$38$	475.023	2.02787
$39$	0	0
$40$	−41.1803	−0.162780
$41$	200.885	0.765196	0.382598	−	0.923915i	$-0.375029\pi$
$41$	200.885	0.765196	0.382598	+	0.923915i	$0.375029\pi$
$42$	0	0
$43$	−218.217	−0.773901	−0.386950	−	0.922101i	$-0.626472\pi$
$43$	−218.217	−0.773901	−0.386950	+	0.922101i	$0.626472\pi$
$44$	412.964	1.41493
$45$	0	0
$46$	488.328	1.56522
$47$	575.659	1.78657	0.893283	−	0.449496i	$-0.148396\pi$
$47$	575.659	1.78657	0.893283	+	0.449496i	$0.148396\pi$
$48$	0	0
$49$	0	0
$50$	105.902	0.299535
$51$	0	0
$52$	−884.748	−2.35947
$53$	184.302	0.477657	0.238828	−	0.971062i	$-0.423237\pi$
$53$	184.302	0.477657	0.238828	+	0.971062i	$0.423237\pi$
$54$	0	0
$55$	−207.639	−0.509056
$56$	0	0
$57$	0	0
$58$	613.522	1.38895
$59$	−151.502	−0.334302	−0.167151	−	0.985931i	$-0.553457\pi$
$59$	−151.502	−0.334302	−0.167151	+	0.985931i	$0.553457\pi$
$60$	0	0
$61$	529.830	1.11209	0.556047	−	0.831151i	$-0.312317\pi$
$61$	529.830	1.11209	0.556047	+	0.831151i	$0.312317\pi$
$62$	1093.24	2.23938
$63$	0	0
$64$	−723.276	−1.41265
$65$	444.853	0.848880
$66$	0	0
$67$	1.28485	0.00234283	0.00117142	−	0.999999i	$-0.499627\pi$
$67$	1.28485	0.00234283	0.00117142	+	0.999999i	$0.499627\pi$
$68$	−1197.16	−2.13496
$69$	0	0
$70$	0	0
$71$	61.4226	0.102669	0.0513347	−	0.998682i	$-0.483652\pi$
$71$	61.4226	0.102669	0.0513347	+	0.998682i	$0.483652\pi$
$72$	0	0
$73$	−484.800	−0.777282	−0.388641	−	0.921389i	$-0.627055\pi$
$73$	−484.800	−0.777282	−0.388641	+	0.921389i	$0.627055\pi$
$74$	204.748	0.321641
$75$	0	0
$76$	1115.13	1.68308
$77$	0	0
$78$	0	0
$79$	878.257	1.25078	0.625390	−	0.780312i	$-0.284940\pi$
$79$	878.257	1.25078	0.625390	+	0.780312i	$0.284940\pi$
$80$	223.328	0.312111
$81$	0	0
$82$	850.964	1.14602
$83$	491.830	0.650426	0.325213	−	0.945641i	$-0.394564\pi$
$83$	491.830	0.650426	0.325213	+	0.945641i	$0.394564\pi$
$84$	0	0
$85$	601.935	0.768106
$86$	−924.381	−1.15905
$87$	0	0
$88$	342.026	0.414320
$89$	−415.560	−0.494936	−0.247468	−	0.968896i	$-0.579599\pi$
$89$	−415.560	−0.494936	−0.247468	+	0.968896i	$0.579599\pi$
$90$	0	0
$91$	0	0
$92$	1146.36	1.29909
$93$	0	0
$94$	2438.53	2.67570
$95$	−560.689	−0.605531
$96$	0	0
$97$	1031.70	1.07993	0.539964	−	0.841688i	$-0.318438\pi$
$97$	1031.70	1.07993	0.539964	+	0.841688i	$0.318438\pi$
$98$	0	0
$99$	0	0
$100$	248.607	0.248607
$101$	1447.19	1.42576	0.712878	−	0.701288i	$-0.247391\pi$
$101$	1447.19	1.42576	0.712878	+	0.701288i	$0.247391\pi$
$102$	0	0
$103$	163.567	0.156473	0.0782364	−	0.996935i	$-0.475071\pi$
$103$	163.567	0.156473	0.0782364	+	0.996935i	$0.475071\pi$
$104$	−732.768	−0.690902
$105$	0	0
$106$	780.715	0.715375
$107$	129.653	0.117141	0.0585703	−	0.998283i	$-0.481346\pi$
$107$	129.653	0.117141	0.0585703	+	0.998283i	$0.481346\pi$
$108$	0	0
$109$	566.681	0.497965	0.248983	−	0.968508i	$-0.419904\pi$
$109$	566.681	0.497965	0.248983	+	0.968508i	$0.419904\pi$
$110$	−879.574	−0.762401
$111$	0	0
$112$	0	0
$113$	−809.890	−0.674230	−0.337115	−	0.941463i	$-0.609451\pi$
$113$	−809.890	−0.674230	−0.337115	+	0.941463i	$0.609451\pi$
$114$	0	0
$115$	−576.393	−0.467382
$116$	1440.26	1.15280
$117$	0	0
$118$	−641.771	−0.500676
$119$	0	0
$120$	0	0
$121$	393.563	0.295690
$122$	2244.39	1.66556
$123$	0	0
$124$	2566.41	1.85863
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2584.25	−1.80563	−0.902816	−	0.430028i	$-0.858504\pi$
$127$	−2584.25	−1.80563	−0.902816	+	0.430028i	$0.858504\pi$
$128$	−1023.08	−0.706473
$129$	0	0
$130$	1884.43	1.27135
$131$	−1421.10	−0.947804	−0.473902	−	0.880578i	$-0.657155\pi$
$131$	−1421.10	−0.947804	−0.473902	+	0.880578i	$0.657155\pi$
$132$	0	0
$133$	0	0
$134$	5.44272	0.00350880
$135$	0	0
$136$	−991.515	−0.625160
$137$	−104.878	−0.0654037	−0.0327019	−	0.999465i	$-0.510411\pi$
$137$	−104.878	−0.0654037	−0.0327019	+	0.999465i	$0.510411\pi$
$138$	0	0
$139$	913.160	0.557217	0.278609	−	0.960405i	$-0.410127\pi$
$139$	913.160	0.557217	0.278609	+	0.960405i	$0.410127\pi$
$140$	0	0
$141$	0	0
$142$	260.190	0.153765
$143$	−3694.76	−2.16064
$144$	0	0
$145$	−724.164	−0.414749
$146$	−2053.65	−1.16412
$147$	0	0
$148$	480.650	0.266954
$149$	−1781.45	−0.979476	−0.489738	−	0.871870i	$-0.662908\pi$
$149$	−1781.45	−0.979476	−0.489738	+	0.871870i	$0.662908\pi$
$150$	0	0
$151$	1407.53	0.758564	0.379282	−	0.925281i	$-0.376171\pi$
$151$	1407.53	0.758564	0.379282	+	0.925281i	$0.376171\pi$
$152$	923.574	0.492841
$153$	0	0
$154$	0	0
$155$	−1290.39	−0.668690
$156$	0	0
$157$	1598.94	0.812798	0.406399	−	0.913696i	$-0.366784\pi$
$157$	1598.94	0.812798	0.406399	+	0.913696i	$0.366784\pi$
$158$	3720.36	1.87326
$159$	0	0
$160$	1275.48	0.630220
$161$	0	0
$162$	0	0
$163$	−204.892	−0.0984562	−0.0492281	−	0.998788i	$-0.515676\pi$
$163$	−204.892	−0.0984562	−0.0492281	+	0.998788i	$0.515676\pi$
$164$	1997.66	0.951165
$165$	0	0
$166$	2083.42	0.974127
$167$	−1165.94	−0.540259	−0.270129	−	0.962824i	$-0.587067\pi$
$167$	−1165.94	−0.540259	−0.270129	+	0.962824i	$0.587067\pi$
$168$	0	0
$169$	5718.76	2.60299
$170$	2549.84	1.15037
$171$	0	0
$172$	−2170.01	−0.961985
$173$	−2538.00	−1.11538	−0.557690	−	0.830049i	$-0.688312\pi$
$173$	−2538.00	−1.11538	−0.557690	+	0.830049i	$0.688312\pi$
$174$	0	0
$175$	0	0
$176$	−1854.87	−0.794409
$177$	0	0
$178$	−1760.34	−0.741254
$179$	−392.255	−0.163791	−0.0818954	−	0.996641i	$-0.526097\pi$
$179$	−392.255	−0.163791	−0.0818954	+	0.996641i	$0.526097\pi$
$180$	0	0
$181$	2978.08	1.22298	0.611489	−	0.791253i	$-0.290571\pi$
$181$	2978.08	1.22298	0.611489	+	0.791253i	$0.290571\pi$
$182$	0	0
$183$	0	0
$184$	949.443	0.380401
$185$	−241.672	−0.0960436
$186$	0	0
$187$	−4999.41	−1.95504
$188$	5724.51	2.22076
$189$	0	0
$190$	−2375.12	−0.906890
$191$	1097.37	0.415722	0.207861	−	0.978158i	$-0.433350\pi$
$191$	1097.37	0.415722	0.207861	+	0.978158i	$0.433350\pi$
$192$	0	0
$193$	3500.31	1.30548	0.652740	−	0.757582i	$-0.273619\pi$
$193$	3500.31	1.30548	0.652740	+	0.757582i	$0.273619\pi$
$194$	4370.34	1.61738
$195$	0	0
$196$	0	0
$197$	−1573.96	−0.569237	−0.284618	−	0.958641i	$-0.591867\pi$
$197$	−1573.96	−0.569237	−0.284618	+	0.958641i	$0.591867\pi$
$198$	0	0
$199$	3396.62	1.20995	0.604976	−	0.796244i	$-0.293183\pi$
$199$	3396.62	1.20995	0.604976	+	0.796244i	$0.293183\pi$
$200$	205.902	0.0727972
$201$	0	0
$202$	6130.42	2.13532
$203$	0	0
$204$	0	0
$205$	−1004.43	−0.342206
$206$	692.879	0.234346
$207$	0	0
$208$	3973.93	1.32472
$209$	4656.84	1.54125
$210$	0	0
$211$	3337.81	1.08903	0.544513	−	0.838753i	$-0.316715\pi$
$211$	3337.81	1.08903	0.544513	+	0.838753i	$0.316715\pi$
$212$	1832.75	0.593744
$213$	0	0
$214$	549.220	0.175439
$215$	1091.08	0.346099
$216$	0	0
$217$	0	0
$218$	2400.50	0.745791
$219$	0	0
$220$	−2064.82	−0.632774
$221$	10710.9	3.26015
$222$	0	0
$223$	−127.328	−0.0382356	−0.0191178	−	0.999817i	$-0.506086\pi$
$223$	−127.328	−0.0382356	−0.0191178	+	0.999817i	$0.506086\pi$
$224$	0	0
$225$	0	0
$226$	−3430.75	−1.00978
$227$	3844.12	1.12398	0.561990	−	0.827144i	$-0.310036\pi$
$227$	3844.12	1.12398	0.561990	+	0.827144i	$0.310036\pi$
$228$	0	0
$229$	−2536.95	−0.732080	−0.366040	−	0.930599i	$-0.619287\pi$
$229$	−2536.95	−0.732080	−0.366040	+	0.930599i	$0.619287\pi$
$230$	−2441.64	−0.699987
$231$	0	0
$232$	1192.85	0.337563
$233$	−3987.44	−1.12114	−0.560570	−	0.828107i	$-0.689418\pi$
$233$	−3987.44	−1.12114	−0.560570	+	0.828107i	$0.689418\pi$
$234$	0	0
$235$	−2878.30	−0.798976
$236$	−1506.57	−0.415549
$237$	0	0
$238$	0	0
$239$	3367.18	0.911317	0.455659	−	0.890155i	$-0.349404\pi$
$239$	3367.18	0.911317	0.455659	+	0.890155i	$0.349404\pi$
$240$	0	0
$241$	939.551	0.251128	0.125564	−	0.992086i	$-0.459926\pi$
$241$	939.551	0.251128	0.125564	+	0.992086i	$0.459926\pi$
$242$	1667.16	0.442848
$243$	0	0
$244$	5268.77	1.38237
$245$	0	0
$246$	0	0
$247$	−9976.96	−2.57012
$248$	2125.56	0.544246
$249$	0	0
$250$	−529.508	−0.133956
$251$	−1403.96	−0.353056	−0.176528	−	0.984296i	$-0.556487\pi$
$251$	−1403.96	−0.353056	−0.176528	+	0.984296i	$0.556487\pi$
$252$	0	0
$253$	4787.28	1.18962
$254$	−10947.1	−2.70425
$255$	0	0
$256$	1452.36	0.354579
$257$	−1964.86	−0.476905	−0.238453	−	0.971154i	$-0.576640\pi$
$257$	−1964.86	−0.476905	−0.238453	+	0.971154i	$0.576640\pi$
$258$	0	0
$259$	0	0
$260$	4423.74	1.05519
$261$	0	0
$262$	−6019.89	−1.41950
$263$	393.821	0.0923347	0.0461673	−	0.998934i	$-0.485299\pi$
$263$	393.821	0.0923347	0.0461673	+	0.998934i	$0.485299\pi$
$264$	0	0
$265$	−921.509	−0.213615
$266$	0	0
$267$	0	0
$268$	12.7769	0.00291222
$269$	−1877.03	−0.425444	−0.212722	−	0.977113i	$-0.568233\pi$
$269$	−1877.03	−0.425444	−0.212722	+	0.977113i	$0.568233\pi$
$270$	0	0
$271$	689.909	0.154646	0.0773228	−	0.997006i	$-0.475363\pi$
$271$	689.909	0.154646	0.0773228	+	0.997006i	$0.475363\pi$
$272$	5377.16	1.19867
$273$	0	0
$274$	−444.269	−0.0979536
$275$	1038.20	0.227657
$276$	0	0
$277$	6289.13	1.36418	0.682088	−	0.731270i	$-0.261072\pi$
$277$	6289.13	1.36418	0.682088	+	0.731270i	$0.261072\pi$
$278$	3868.21	0.834531
$279$	0	0
$280$	0	0
$281$	1954.87	0.415010	0.207505	−	0.978234i	$-0.433466\pi$
$281$	1954.87	0.415010	0.207505	+	0.978234i	$0.433466\pi$
$282$	0	0
$283$	−5033.96	−1.05738	−0.528688	−	0.848816i	$-0.677316\pi$
$283$	−5033.96	−1.05738	−0.528688	+	0.848816i	$0.677316\pi$
$284$	610.803	0.127621
$285$	0	0
$286$	−15651.2	−3.23594
$287$	0	0
$288$	0	0
$289$	9580.03	1.94993
$290$	−3067.61	−0.621159
$291$	0	0
$292$	−4820.99	−0.966188
$293$	6369.12	1.26993	0.634963	−	0.772543i	$-0.281015\pi$
$293$	6369.12	1.26993	0.634963	+	0.772543i	$0.281015\pi$
$294$	0	0
$295$	757.508	0.149504
$296$	398.085	0.0781697
$297$	0	0
$298$	−7546.34	−1.46694
$299$	−10256.4	−1.98376
$300$	0	0
$301$	0	0
$302$	5962.39	1.13608
$303$	0	0
$304$	−5008.70	−0.944963
$305$	−2649.15	−0.497344
$306$	0	0
$307$	6619.83	1.23066	0.615332	−	0.788268i	$-0.289022\pi$
$307$	6619.83	1.23066	0.615332	+	0.788268i	$0.289022\pi$
$308$	0	0
$309$	0	0
$310$	−5466.20	−1.00148
$311$	−9909.22	−1.80675	−0.903377	−	0.428848i	$-0.858920\pi$
$311$	−9909.22	−1.80675	−0.903377	+	0.428848i	$0.858920\pi$
$312$	0	0
$313$	422.336	0.0762678	0.0381339	−	0.999273i	$-0.487859\pi$
$313$	422.336	0.0762678	0.0381339	+	0.999273i	$0.487859\pi$
$314$	6773.22	1.21731
$315$	0	0
$316$	8733.63	1.55476
$317$	4902.78	0.868668	0.434334	−	0.900752i	$-0.356984\pi$
$317$	4902.78	0.868668	0.434334	+	0.900752i	$0.356984\pi$
$318$	0	0
$319$	6014.60	1.05565
$320$	3616.38	0.631755
$321$	0	0
$322$	0	0
$323$	−13499.9	−2.32556
$324$	0	0
$325$	−2224.26	−0.379631
$326$	−867.935	−0.147455
$327$	0	0
$328$	1654.51	0.278521
$329$	0	0
$330$	0	0
$331$	−5281.74	−0.877071	−0.438535	−	0.898714i	$-0.644503\pi$
$331$	−5281.74	−0.877071	−0.438535	+	0.898714i	$0.644503\pi$
$332$	4890.89	0.808501
$333$	0	0
$334$	−4939.01	−0.809133
$335$	−6.42426	−0.00104775
$336$	0	0
$337$	4459.60	0.720860	0.360430	−	0.932786i	$-0.382630\pi$
$337$	4459.60	0.720860	0.360430	+	0.932786i	$0.382630\pi$
$338$	24225.1	3.89843
$339$	0	0
$340$	5985.80	0.954782
$341$	10717.5	1.70200
$342$	0	0
$343$	0	0
$344$	−1797.25	−0.281689
$345$	0	0
$346$	−10751.2	−1.67048
$347$	−5261.97	−0.814056	−0.407028	−	0.913416i	$-0.633435\pi$
$347$	−5261.97	−0.814056	−0.407028	+	0.913416i	$0.633435\pi$
$348$	0	0
$349$	−960.325	−0.147292	−0.0736461	−	0.997284i	$-0.523464\pi$
$349$	−960.325	−0.147292	−0.0736461	+	0.997284i	$0.523464\pi$
$350$	0	0
$351$	0	0
$352$	−10593.6	−1.60409
$353$	−8925.80	−1.34581	−0.672907	−	0.739727i	$-0.734955\pi$
$353$	−8925.80	−1.34581	−0.672907	+	0.739727i	$0.734955\pi$
$354$	0	0
$355$	−307.113	−0.0459151
$356$	−4132.45	−0.615222
$357$	0	0
$358$	−1661.62	−0.245306
$359$	3056.27	0.449314	0.224657	−	0.974438i	$-0.427874\pi$
$359$	3056.27	0.449314	0.224657	+	0.974438i	$0.427874\pi$
$360$	0	0
$361$	5715.88	0.833340
$362$	12615.4	1.83162
$363$	0	0
$364$	0	0
$365$	2424.00	0.347611
$366$	0	0
$367$	1813.52	0.257943	0.128971	−	0.991648i	$-0.458832\pi$
$367$	1813.52	0.257943	0.128971	+	0.991648i	$0.458832\pi$
$368$	−5148.99	−0.729375
$369$	0	0
$370$	−1023.74	−0.143842
$371$	0	0
$372$	0	0
$373$	−4517.48	−0.627094	−0.313547	−	0.949573i	$-0.601517\pi$
$373$	−4517.48	−0.627094	−0.313547	+	0.949573i	$0.601517\pi$
$374$	−21177.9	−2.92802
$375$	0	0
$376$	4741.17	0.650285
$377$	−12885.9	−1.76036
$378$	0	0
$379$	−4931.24	−0.668340	−0.334170	−	0.942513i	$-0.608456\pi$
$379$	−4931.24	−0.668340	−0.334170	+	0.942513i	$0.608456\pi$
$380$	−5575.64	−0.752696
$381$	0	0
$382$	4648.53	0.622617
$383$	−1482.37	−0.197770	−0.0988849	−	0.995099i	$-0.531528\pi$
$383$	−1482.37	−0.197770	−0.0988849	+	0.995099i	$0.531528\pi$
$384$	0	0
$385$	0	0
$386$	14827.5	1.95519
$387$	0	0
$388$	10259.5	1.34239
$389$	5448.98	0.710217	0.355109	−	0.934825i	$-0.384444\pi$
$389$	5448.98	0.710217	0.355109	+	0.934825i	$0.384444\pi$
$390$	0	0
$391$	−13878.0	−1.79500
$392$	0	0
$393$	0	0
$394$	−6667.38	−0.852532
$395$	−4391.28	−0.559366
$396$	0	0
$397$	13675.9	1.72891	0.864453	−	0.502713i	$-0.167665\pi$
$397$	13675.9	1.72891	0.864453	+	0.502713i	$0.167665\pi$
$398$	14388.3	1.81212
$399$	0	0
$400$	−1116.64	−0.139580
$401$	−14109.9	−1.75714	−0.878570	−	0.477613i	$-0.841502\pi$
$401$	−14109.9	−1.75714	−0.878570	+	0.477613i	$0.841502\pi$
$402$	0	0
$403$	−22961.4	−2.83819
$404$	14391.3	1.77226
$405$	0	0
$406$	0	0
$407$	2007.22	0.244458
$408$	0	0
$409$	13995.6	1.69203	0.846015	−	0.533159i	$-0.178995\pi$
$409$	13995.6	1.69203	0.846015	+	0.533159i	$0.178995\pi$
$410$	−4254.82	−0.512514
$411$	0	0
$412$	1626.55	0.194501
$413$	0	0
$414$	0	0
$415$	−2459.15	−0.290879
$416$	22696.0	2.67491
$417$	0	0
$418$	19726.7	2.30829
$419$	−9840.61	−1.14736	−0.573682	−	0.819078i	$-0.694485\pi$
$419$	−9840.61	−1.14736	−0.573682	+	0.819078i	$0.694485\pi$
$420$	0	0
$421$	−12660.5	−1.46564	−0.732822	−	0.680420i	$-0.761797\pi$
$421$	−12660.5	−1.46564	−0.732822	+	0.680420i	$0.761797\pi$
$422$	14139.2	1.63101
$423$	0	0
$424$	1517.92	0.173860
$425$	−3009.67	−0.343507
$426$	0	0
$427$	0	0
$428$	1289.31	0.145610
$429$	0	0
$430$	4621.90	0.518344
$431$	4578.91	0.511736	0.255868	−	0.966712i	$-0.417639\pi$
$431$	4578.91	0.511736	0.255868	+	0.966712i	$0.417639\pi$
$432$	0	0
$433$	3279.88	0.364020	0.182010	−	0.983297i	$-0.441740\pi$
$433$	3279.88	0.364020	0.182010	+	0.983297i	$0.441740\pi$
$434$	0	0
$435$	0	0
$436$	5635.23	0.618988
$437$	12927.1	1.41507
$438$	0	0
$439$	427.807	0.0465105	0.0232552	−	0.999730i	$-0.492597\pi$
$439$	427.807	0.0465105	0.0232552	+	0.999730i	$0.492597\pi$
$440$	−1710.13	−0.185289
$441$	0	0
$442$	45372.1	4.88265
$443$	−15441.2	−1.65605	−0.828027	−	0.560688i	$-0.810537\pi$
$443$	−15441.2	−1.65605	−0.828027	+	0.560688i	$0.810537\pi$
$444$	0	0
$445$	2077.80	0.221342
$446$	−539.371	−0.0572645
$447$	0	0
$448$	0	0
$449$	−9382.02	−0.986113	−0.493057	−	0.869997i	$-0.664120\pi$
$449$	−9382.02	−0.986113	−0.493057	+	0.869997i	$0.664120\pi$
$450$	0	0
$451$	8342.34	0.871010
$452$	−8053.77	−0.838091
$453$	0	0
$454$	16284.0	1.68336
$455$	0	0
$456$	0	0
$457$	13570.4	1.38905	0.694524	−	0.719469i	$-0.255615\pi$
$457$	13570.4	1.38905	0.694524	+	0.719469i	$0.255615\pi$
$458$	−10746.7	−1.09642
$459$	0	0
$460$	−5731.81	−0.580972
$461$	1251.88	0.126477	0.0632386	−	0.997998i	$-0.479857\pi$
$461$	1251.88	0.126477	0.0632386	+	0.997998i	$0.479857\pi$
$462$	0	0
$463$	7934.36	0.796417	0.398209	−	0.917295i	$-0.369632\pi$
$463$	7934.36	0.796417	0.398209	+	0.917295i	$0.369632\pi$
$464$	−6469.05	−0.647237
$465$	0	0
$466$	−16891.1	−1.67911
$467$	7583.76	0.751466	0.375733	−	0.926728i	$-0.377391\pi$
$467$	7583.76	0.751466	0.375733	+	0.926728i	$0.377391\pi$
$468$	0	0
$469$	0	0
$470$	−12192.7	−1.19661
$471$	0	0
$472$	−1247.78	−0.121681
$473$	−9062.07	−0.880919
$474$	0	0
$475$	2803.44	0.270802
$476$	0	0
$477$	0	0
$478$	14263.6	1.36486
$479$	−5829.34	−0.556053	−0.278027	−	0.960573i	$-0.589680\pi$
$479$	−5829.34	−0.556053	−0.278027	+	0.960573i	$0.589680\pi$
$480$	0	0
$481$	−4300.34	−0.407648
$482$	3980.00	0.376108
$483$	0	0
$484$	3913.70	0.367553
$485$	−5158.49	−0.482959
$486$	0	0
$487$	−19902.1	−1.85185	−0.925925	−	0.377708i	$-0.876712\pi$
$487$	−19902.1	−1.85185	−0.925925	+	0.377708i	$0.876712\pi$
$488$	4363.71	0.404787
$489$	0	0
$490$	0	0
$491$	16821.6	1.54613	0.773065	−	0.634327i	$-0.218723\pi$
$491$	16821.6	1.54613	0.773065	+	0.634327i	$0.218723\pi$
$492$	0	0
$493$	−17436.0	−1.59285
$494$	−42263.1	−3.84920
$495$	0	0
$496$	−11527.3	−1.04353
$497$	0	0
$498$	0	0
$499$	6031.83	0.541126	0.270563	−	0.962702i	$-0.412790\pi$
$499$	6031.83	0.541126	0.270563	+	0.962702i	$0.412790\pi$
$500$	−1243.03	−0.111180
$501$	0	0
$502$	−5947.27	−0.528764
$503$	−17176.4	−1.52258	−0.761290	−	0.648412i	$-0.775434\pi$
$503$	−17176.4	−1.52258	−0.761290	+	0.648412i	$0.775434\pi$
$504$	0	0
$505$	−7235.97	−0.637617
$506$	20279.2	1.78166
$507$	0	0
$508$	−25698.5	−2.24446
$509$	4706.59	0.409854	0.204927	−	0.978777i	$-0.434304\pi$
$509$	4706.59	0.409854	0.204927	+	0.978777i	$0.434304\pi$
$510$	0	0
$511$	0	0
$512$	14336.9	1.23752
$513$	0	0
$514$	−8323.28	−0.714250
$515$	−817.833	−0.0699768
$516$	0	0
$517$	23905.9	2.03362
$518$	0	0
$519$	0	0
$520$	3663.84	0.308981
$521$	−8557.18	−0.719572	−0.359786	−	0.933035i	$-0.617150\pi$
$521$	−8557.18	−0.719572	−0.359786	+	0.933035i	$0.617150\pi$
$522$	0	0
$523$	−18248.5	−1.52572	−0.762858	−	0.646566i	$-0.776204\pi$
$523$	−18248.5	−1.52572	−0.762858	+	0.646566i	$0.776204\pi$
$524$	−14131.8	−1.17815
$525$	0	0
$526$	1668.25	0.138287
$527$	−31069.3	−2.56813
$528$	0	0
$529$	1122.16	0.0922302
$530$	−3903.58	−0.319925
$531$	0	0
$532$	0	0
$533$	−17872.9	−1.45246
$534$	0	0
$535$	−648.266	−0.0523869
$536$	10.5821	0.000852758 0
$537$	0	0
$538$	−7951.22	−0.637177
$539$	0	0
$540$	0	0
$541$	−5734.17	−0.455696	−0.227848	−	0.973697i	$-0.573169\pi$
$541$	−5734.17	−0.455696	−0.227848	+	0.973697i	$0.573169\pi$
$542$	2922.50	0.231609
$543$	0	0
$544$	30710.1	2.42038
$545$	−2833.41	−0.222697
$546$	0	0
$547$	−8002.52	−0.625527	−0.312763	−	0.949831i	$-0.601255\pi$
$547$	−8002.52	−0.625527	−0.312763	+	0.949831i	$0.601255\pi$
$548$	−1042.93	−0.0812991
$549$	0	0
$550$	4397.87	0.340956
$551$	16241.2	1.25572
$552$	0	0
$553$	0	0
$554$	26641.2	2.04310
$555$	0	0
$556$	9080.71	0.692640
$557$	−1276.82	−0.0971289	−0.0485644	−	0.998820i	$-0.515465\pi$
$557$	−1276.82	−0.0971289	−0.0485644	+	0.998820i	$0.515465\pi$
$558$	0	0
$559$	19414.9	1.46898
$560$	0	0
$561$	0	0
$562$	8280.96	0.621550
$563$	11027.7	0.825507	0.412753	−	0.910843i	$-0.364567\pi$
$563$	11027.7	0.825507	0.412753	+	0.910843i	$0.364567\pi$
$564$	0	0
$565$	4049.45	0.301525
$566$	−21324.2	−1.58361
$567$	0	0
$568$	505.881	0.0373702
$569$	4519.03	0.332948	0.166474	−	0.986046i	$-0.446762\pi$
$569$	4519.03	0.332948	0.166474	+	0.986046i	$0.446762\pi$
$570$	0	0
$571$	3598.81	0.263758	0.131879	−	0.991266i	$-0.457899\pi$
$571$	3598.81	0.263758	0.131879	+	0.991266i	$0.457899\pi$
$572$	−36741.7	−2.68575
$573$	0	0
$574$	0	0
$575$	2881.97	0.209020
$576$	0	0
$577$	3439.23	0.248140	0.124070	−	0.992273i	$-0.460405\pi$
$577$	3439.23	0.248140	0.124070	+	0.992273i	$0.460405\pi$
$578$	40581.6	2.92037
$579$	0	0
$580$	−7201.28	−0.515547
$581$	0	0
$582$	0	0
$583$	7653.66	0.543709
$584$	−3992.85	−0.282920
$585$	0	0
$586$	26980.0	1.90194
$587$	21285.2	1.49665	0.748327	−	0.663330i	$-0.230857\pi$
$587$	21285.2	1.49665	0.748327	+	0.663330i	$0.230857\pi$
$588$	0	0
$589$	28940.4	2.02456
$590$	3208.85	0.223909
$591$	0	0
$592$	−2158.89	−0.149881
$593$	−14200.8	−0.983404	−0.491702	−	0.870764i	$-0.663625\pi$
$593$	−14200.8	−0.983404	−0.491702	+	0.870764i	$0.663625\pi$
$594$	0	0
$595$	0	0
$596$	−17715.2	−1.21752
$597$	0	0
$598$	−43446.8	−2.97103
$599$	8885.05	0.606065	0.303033	−	0.952980i	$-0.402001\pi$
$599$	8885.05	0.606065	0.303033	+	0.952980i	$0.402001\pi$
$600$	0	0
$601$	2052.89	0.139333	0.0696664	−	0.997570i	$-0.477807\pi$
$601$	2052.89	0.139333	0.0696664	+	0.997570i	$0.477807\pi$
$602$	0	0
$603$	0	0
$604$	13996.9	0.942920
$605$	−1967.82	−0.132237
$606$	0	0
$607$	−10280.0	−0.687404	−0.343702	−	0.939079i	$-0.611681\pi$
$607$	−10280.0	−0.687404	−0.343702	+	0.939079i	$0.611681\pi$
$608$	−28605.8	−1.90809
$609$	0	0
$610$	−11222.0	−0.744860
$611$	−51216.8	−3.39118
$612$	0	0
$613$	23409.5	1.54242	0.771208	−	0.636584i	$-0.219653\pi$
$613$	23409.5	1.54242	0.771208	+	0.636584i	$0.219653\pi$
$614$	28042.0	1.84314
$615$	0	0
$616$	0	0
$617$	6632.75	0.432779	0.216389	−	0.976307i	$-0.430572\pi$
$617$	6632.75	0.432779	0.216389	+	0.976307i	$0.430572\pi$
$618$	0	0
$619$	−10734.0	−0.696990	−0.348495	−	0.937311i	$-0.613307\pi$
$619$	−10734.0	−0.696990	−0.348495	+	0.937311i	$0.613307\pi$
$620$	−12832.0	−0.831205
$621$	0	0
$622$	−41976.1	−2.70593
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	1789.04	0.114225
$627$	0	0
$628$	15900.3	1.01034
$629$	−5818.83	−0.368858
$630$	0	0
$631$	−17071.0	−1.07700	−0.538499	−	0.842626i	$-0.681008\pi$
$631$	−17071.0	−1.07700	−0.538499	+	0.842626i	$0.681008\pi$
$632$	7233.38	0.455267
$633$	0	0
$634$	20768.5	1.30098
$635$	12921.3	0.807503
$636$	0	0
$637$	0	0
$638$	25478.2	1.58102
$639$	0	0
$640$	5115.41	0.315945
$641$	19389.7	1.19477	0.597386	−	0.801954i	$-0.296206\pi$
$641$	19389.7	1.19477	0.597386	+	0.801954i	$0.296206\pi$
$642$	0	0
$643$	25409.3	1.55839	0.779196	−	0.626780i	$-0.215628\pi$
$643$	25409.3	1.55839	0.779196	+	0.626780i	$0.215628\pi$
$644$	0	0
$645$	0	0
$646$	−57186.6	−3.48294
$647$	−6039.08	−0.366956	−0.183478	−	0.983024i	$-0.558736\pi$
$647$	−6039.08	−0.366956	−0.183478	+	0.983024i	$0.558736\pi$
$648$	0	0
$649$	−6291.54	−0.380531
$650$	−9422.14	−0.568564
$651$	0	0
$652$	−2037.50	−0.122384
$653$	30666.2	1.83776	0.918882	−	0.394532i	$-0.129093\pi$
$653$	30666.2	1.83776	0.918882	+	0.394532i	$0.129093\pi$
$654$	0	0
$655$	7105.51	0.423871
$656$	−8972.67	−0.534031
$657$	0	0
$658$	0	0
$659$	2765.96	0.163500	0.0817500	−	0.996653i	$-0.473949\pi$
$659$	2765.96	0.163500	0.0817500	+	0.996653i	$0.473949\pi$
$660$	0	0
$661$	−27261.8	−1.60418	−0.802089	−	0.597204i	$-0.796278\pi$
$661$	−27261.8	−1.60418	−0.802089	+	0.597204i	$0.796278\pi$
$662$	−22373.8	−1.31357
$663$	0	0
$664$	4050.74	0.236746
$665$	0	0
$666$	0	0
$667$	16696.1	0.969230
$668$	−11594.4	−0.671560
$669$	0	0
$670$	−27.2136	−0.00156918
$671$	22002.7	1.26588
$672$	0	0
$673$	−1048.17	−0.0600356	−0.0300178	−	0.999549i	$-0.509556\pi$
$673$	−1048.17	−0.0600356	−0.0300178	+	0.999549i	$0.509556\pi$
$674$	18891.2	1.07961
$675$	0	0
$676$	56869.0	3.23560
$677$	−34554.7	−1.96166	−0.980831	−	0.194860i	$-0.937575\pi$
$677$	−34554.7	−1.96166	−0.980831	+	0.194860i	$0.937575\pi$
$678$	0	0
$679$	0	0
$680$	4957.58	0.279580
$681$	0	0
$682$	45399.9	2.54905
$683$	−14711.6	−0.824192	−0.412096	−	0.911140i	$-0.635203\pi$
$683$	−14711.6	−0.824192	−0.412096	+	0.911140i	$0.635203\pi$
$684$	0	0
$685$	524.389	0.0292494
$686$	0	0
$687$	0	0
$688$	9746.79	0.540106
$689$	−16397.4	−0.906666
$690$	0	0
$691$	24522.6	1.35005	0.675024	−	0.737796i	$-0.264133\pi$
$691$	24522.6	1.35005	0.675024	+	0.737796i	$0.264133\pi$
$692$	−25238.6	−1.38646
$693$	0	0
$694$	−22290.1	−1.21919
$695$	−4565.80	−0.249195
$696$	0	0
$697$	−24184.0	−1.31425
$698$	−4068.00	−0.220596
$699$	0	0
$700$	0	0
$701$	−19912.2	−1.07286	−0.536429	−	0.843946i	$-0.680227\pi$
$701$	−19912.2	−1.07286	−0.536429	+	0.843946i	$0.680227\pi$
$702$	0	0
$703$	5420.11	0.290787
$704$	−30036.1	−1.60799
$705$	0	0
$706$	−37810.3	−2.01559
$707$	0	0
$708$	0	0
$709$	6208.79	0.328880	0.164440	−	0.986387i	$-0.447418\pi$
$709$	6208.79	0.328880	0.164440	+	0.986387i	$0.447418\pi$
$710$	−1300.95	−0.0687660
$711$	0	0
$712$	−3422.58	−0.180150
$713$	29751.0	1.56267
$714$	0	0
$715$	18473.8	0.966267
$716$	−3900.69	−0.203597
$717$	0	0
$718$	12946.6	0.672927
$719$	13063.6	0.677593	0.338797	−	0.940860i	$-0.389980\pi$
$719$	13063.6	0.677593	0.338797	+	0.940860i	$0.389980\pi$
$720$	0	0
$721$	0	0
$722$	24212.9	1.24807
$723$	0	0
$724$	29614.8	1.52020
$725$	3620.82	0.185481
$726$	0	0
$727$	12897.0	0.657940	0.328970	−	0.944340i	$-0.393298\pi$
$727$	12897.0	0.657940	0.328970	+	0.944340i	$0.393298\pi$
$728$	0	0
$729$	0	0
$730$	10268.2	0.520609
$731$	26270.5	1.32920
$732$	0	0
$733$	−11699.6	−0.589540	−0.294770	−	0.955568i	$-0.595243\pi$
$733$	−11699.6	−0.589540	−0.294770	+	0.955568i	$0.595243\pi$
$734$	7682.19	0.386315
$735$	0	0
$736$	−29407.0	−1.47277
$737$	53.3571	0.00266681
$738$	0	0
$739$	14974.0	0.745368	0.372684	−	0.927958i	$-0.378438\pi$
$739$	14974.0	0.745368	0.372684	+	0.927958i	$0.378438\pi$
$740$	−2403.25	−0.119385
$741$	0	0
$742$	0	0
$743$	−18500.7	−0.913492	−0.456746	−	0.889597i	$-0.650985\pi$
$743$	−18500.7	−0.913492	−0.456746	+	0.889597i	$0.650985\pi$
$744$	0	0
$745$	8907.24	0.438035
$746$	−19136.3	−0.939184
$747$	0	0
$748$	−49715.5	−2.43019
$749$	0	0
$750$	0	0
$751$	−26348.4	−1.28025	−0.640125	−	0.768271i	$-0.721117\pi$
$751$	−26348.4	−1.28025	−0.640125	+	0.768271i	$0.721117\pi$
$752$	−25712.2	−1.24684
$753$	0	0
$754$	−54585.4	−2.63645
$755$	−7037.65	−0.339240
$756$	0	0
$757$	−28061.7	−1.34732	−0.673659	−	0.739042i	$-0.735278\pi$
$757$	−28061.7	−1.34732	−0.673659	+	0.739042i	$0.735278\pi$
$758$	−20889.1	−1.00096
$759$	0	0
$760$	−4617.87	−0.220405
$761$	−3579.22	−0.170495	−0.0852476	−	0.996360i	$-0.527168\pi$
$761$	−3579.22	−0.170495	−0.0852476	+	0.996360i	$0.527168\pi$
$762$	0	0
$763$	0	0
$764$	10912.5	0.516757
$765$	0	0
$766$	−6279.44	−0.296195
$767$	13479.2	0.634557
$768$	0	0
$769$	−4339.61	−0.203499	−0.101749	−	0.994810i	$-0.532444\pi$
$769$	−4339.61	−0.203499	−0.101749	+	0.994810i	$0.532444\pi$
$770$	0	0
$771$	0	0
$772$	34808.0	1.62276
$773$	10005.1	0.465537	0.232769	−	0.972532i	$-0.425222\pi$
$773$	10005.1	0.465537	0.232769	+	0.972532i	$0.425222\pi$
$774$	0	0
$775$	6451.97	0.299047
$776$	8497.14	0.393079
$777$	0	0
$778$	23082.3	1.06368
$779$	22526.8	1.03608
$780$	0	0
$781$	2550.75	0.116867
$782$	−58788.4	−2.68832
$783$	0	0
$784$	0	0
$785$	−7994.70	−0.363494
$786$	0	0
$787$	−17826.8	−0.807443	−0.403721	−	0.914882i	$-0.632284\pi$
$787$	−17826.8	−0.807443	−0.403721	+	0.914882i	$0.632284\pi$
$788$	−15651.8	−0.707581
$789$	0	0
$790$	−18601.8	−0.837749
$791$	0	0
$792$	0	0
$793$	−47139.3	−2.11093
$794$	57932.2	2.58934
$795$	0	0
$796$	33777.0	1.50401
$797$	36723.0	1.63211	0.816057	−	0.577971i	$-0.196155\pi$
$797$	36723.0	1.63211	0.816057	+	0.577971i	$0.196155\pi$
$798$	0	0
$799$	−69301.9	−3.06849
$800$	−6377.38	−0.281843
$801$	0	0
$802$	−59770.4	−2.63163
$803$	−20132.7	−0.884767
$804$	0	0
$805$	0	0
$806$	−97266.2	−4.25069
$807$	0	0
$808$	11919.2	0.518955
$809$	−5657.55	−0.245870	−0.122935	−	0.992415i	$-0.539231\pi$
$809$	−5657.55	−0.245870	−0.122935	+	0.992415i	$0.539231\pi$
$810$	0	0
$811$	−7532.41	−0.326139	−0.163070	−	0.986615i	$-0.552139\pi$
$811$	−7532.41	−0.326139	−0.163070	+	0.986615i	$0.552139\pi$
$812$	0	0
$813$	0	0
$814$	8502.73	0.366119
$815$	1024.46	0.0440309
$816$	0	0
$817$	−24470.3	−1.04787
$818$	59286.5	2.53411
$819$	0	0
$820$	−9988.30	−0.425374
$821$	6489.25	0.275854	0.137927	−	0.990442i	$-0.455956\pi$
$821$	6489.25	0.275854	0.137927	+	0.990442i	$0.455956\pi$
$822$	0	0
$823$	7901.57	0.334668	0.167334	−	0.985900i	$-0.446484\pi$
$823$	7901.57	0.334668	0.167334	+	0.985900i	$0.446484\pi$
$824$	1347.15	0.0569539
$825$	0	0
$826$	0	0
$827$	37815.8	1.59007	0.795033	−	0.606566i	$-0.207453\pi$
$827$	37815.8	1.59007	0.795033	+	0.606566i	$0.207453\pi$
$828$	0	0
$829$	−26073.5	−1.09236	−0.546182	−	0.837667i	$-0.683919\pi$
$829$	−26073.5	−1.09236	−0.546182	+	0.837667i	$0.683919\pi$
$830$	−10417.1	−0.435643
$831$	0	0
$832$	64350.2	2.68142
$833$	0	0
$834$	0	0
$835$	5829.71	0.241611
$836$	46308.9	1.91582
$837$	0	0
$838$	−41685.5	−1.71838
$839$	−15590.3	−0.641523	−0.320762	−	0.947160i	$-0.603939\pi$
$839$	−15590.3	−0.641523	−0.320762	+	0.947160i	$0.603939\pi$
$840$	0	0
$841$	−3412.46	−0.139918
$842$	−53630.8	−2.19506
$843$	0	0
$844$	33192.1	1.35370
$845$	−28593.8	−1.16409
$846$	0	0
$847$	0	0
$848$	−8231.96	−0.333357
$849$	0	0
$850$	−12749.2	−0.514463
$851$	5571.92	0.224445
$852$	0	0
$853$	17476.1	0.701488	0.350744	−	0.936471i	$-0.385929\pi$
$853$	17476.1	0.701488	0.350744	+	0.936471i	$0.385929\pi$
$854$	0	0
$855$	0	0
$856$	1067.83	0.0426376
$857$	−5694.54	−0.226980	−0.113490	−	0.993539i	$-0.536203\pi$
$857$	−5694.54	−0.226980	−0.113490	+	0.993539i	$0.536203\pi$
$858$	0	0
$859$	−27313.6	−1.08490	−0.542448	−	0.840089i	$-0.682503\pi$
$859$	−27313.6	−1.08490	−0.542448	+	0.840089i	$0.682503\pi$
$860$	10850.0	0.430213
$861$	0	0
$862$	19396.6	0.766415
$863$	9046.07	0.356815	0.178408	−	0.983957i	$-0.442905\pi$
$863$	9046.07	0.356815	0.178408	+	0.983957i	$0.442905\pi$
$864$	0	0
$865$	12690.0	0.498813
$866$	13893.8	0.545185
$867$	0	0
$868$	0	0
$869$	36472.1	1.42374
$870$	0	0
$871$	−114.314	−0.00444705
$872$	4667.22	0.181252
$873$	0	0
$874$	54760.0	2.11932
$875$	0	0
$876$	0	0
$877$	−2104.29	−0.0810224	−0.0405112	−	0.999179i	$-0.512899\pi$
$877$	−2104.29	−0.0810224	−0.0405112	+	0.999179i	$0.512899\pi$
$878$	1812.22	0.0696576
$879$	0	0
$880$	9274.34	0.355270
$881$	−22589.6	−0.863861	−0.431931	−	0.901907i	$-0.642167\pi$
$881$	−22589.6	−0.863861	−0.431931	+	0.901907i	$0.642167\pi$
$882$	0	0
$883$	−2419.71	−0.0922193	−0.0461096	−	0.998936i	$-0.514682\pi$
$883$	−2419.71	−0.0922193	−0.0461096	+	0.998936i	$0.514682\pi$
$884$	106512.	4.05248
$885$	0	0
$886$	−65409.8	−2.48023
$887$	13177.0	0.498806	0.249403	−	0.968400i	$-0.419766\pi$
$887$	13177.0	0.498806	0.249403	+	0.968400i	$0.419766\pi$
$888$	0	0
$889$	0	0
$890$	8801.71	0.331499
$891$	0	0
$892$	−1266.19	−0.0475281
$893$	64553.2	2.41902
$894$	0	0
$895$	1961.28	0.0732495
$896$	0	0
$897$	0	0
$898$	−39742.9	−1.47688
$899$	37378.3	1.38669
$900$	0	0
$901$	−22187.5	−0.820393
$902$	35338.7	1.30449
$903$	0	0
$904$	−6670.31	−0.245411
$905$	−14890.4	−0.546932
$906$	0	0
$907$	9189.14	0.336406	0.168203	−	0.985752i	$-0.446204\pi$
$907$	9189.14	0.336406	0.168203	+	0.985752i	$0.446204\pi$
$908$	38227.0	1.39714
$909$	0	0
$910$	0	0
$911$	17045.8	0.619928	0.309964	−	0.950748i	$-0.399683\pi$
$911$	17045.8	0.619928	0.309964	+	0.950748i	$0.399683\pi$
$912$	0	0
$913$	20424.6	0.740369
$914$	57485.0	2.08034
$915$	0	0
$916$	−25228.1	−0.910001
$917$	0	0
$918$	0	0
$919$	−30825.0	−1.10645	−0.553223	−	0.833033i	$-0.686602\pi$
$919$	−30825.0	−1.10645	−0.553223	+	0.833033i	$0.686602\pi$
$920$	−4747.21	−0.170121
$921$	0	0
$922$	5303.06	0.189422
$923$	−5464.81	−0.194882
$924$	0	0
$925$	1208.36	0.0429520
$926$	33610.5	1.19277
$927$	0	0
$928$	−36946.2	−1.30691
$929$	−5785.88	−0.204336	−0.102168	−	0.994767i	$-0.532578\pi$
$929$	−5785.88	−0.204336	−0.102168	+	0.994767i	$0.532578\pi$
$930$	0	0
$931$	0	0
$932$	−39652.2	−1.39362
$933$	0	0
$934$	32125.3	1.12545
$935$	24997.1	0.874323
$936$	0	0
$937$	13680.9	0.476986	0.238493	−	0.971144i	$-0.423347\pi$
$937$	13680.9	0.476986	0.238493	+	0.971144i	$0.423347\pi$
$938$	0	0
$939$	0	0
$940$	−28622.6	−0.993155
$941$	−45448.8	−1.57448	−0.787242	−	0.616644i	$-0.788492\pi$
$941$	−45448.8	−1.57448	−0.787242	+	0.616644i	$0.788492\pi$
$942$	0	0
$943$	23157.8	0.799705
$944$	6766.91	0.233310
$945$	0	0
$946$	−38387.6	−1.31933
$947$	7788.45	0.267255	0.133628	−	0.991032i	$-0.457337\pi$
$947$	7788.45	0.267255	0.133628	+	0.991032i	$0.457337\pi$
$948$	0	0
$949$	43133.0	1.47540
$950$	11875.6	0.405573
$951$	0	0
$952$	0	0
$953$	−6149.43	−0.209024	−0.104512	−	0.994524i	$-0.533328\pi$
$953$	−6149.43	−0.209024	−0.104512	+	0.994524i	$0.533328\pi$
$954$	0	0
$955$	−5486.85	−0.185917
$956$	33484.2	1.13280
$957$	0	0
$958$	−24693.5	−0.832788
$959$	0	0
$960$	0	0
$961$	36813.7	1.23573
$962$	−18216.5	−0.610524
$963$	0	0
$964$	9343.15	0.312160
$965$	−17501.5	−0.583829
$966$	0	0
$967$	23902.9	0.794896	0.397448	−	0.917625i	$-0.369896\pi$
$967$	23902.9	0.794896	0.397448	+	0.917625i	$0.369896\pi$
$968$	3241.42	0.107627
$969$	0	0
$970$	−21851.7	−0.723316
$971$	−8015.06	−0.264898	−0.132449	−	0.991190i	$-0.542284\pi$
$971$	−8015.06	−0.264898	−0.132449	+	0.991190i	$0.542284\pi$
$972$	0	0
$973$	0	0
$974$	−84306.7	−2.77347
$975$	0	0
$976$	−23665.2	−0.776131
$977$	34861.1	1.14156	0.570780	−	0.821103i	$-0.306641\pi$
$977$	34861.1	1.14156	0.570780	+	0.821103i	$0.306641\pi$
$978$	0	0
$979$	−17257.3	−0.563378
$980$	0	0
$981$	0	0
$982$	71257.6	2.31560
$983$	6620.83	0.214824	0.107412	−	0.994215i	$-0.465744\pi$
$983$	6620.83	0.214824	0.107412	+	0.994215i	$0.465744\pi$
$984$	0	0
$985$	7869.78	0.254570
$986$	−73860.0	−2.38558
$987$	0	0
$988$	−99213.6	−3.19474
$989$	−25155.7	−0.808802
$990$	0	0
$991$	10360.1	0.332089	0.166045	−	0.986118i	$-0.446900\pi$
$991$	10360.1	0.332089	0.166045	+	0.986118i	$0.446900\pi$
$992$	−65834.7	−2.10711
$993$	0	0
$994$	0	0
$995$	−16983.1	−0.541107
$996$	0	0
$997$	40309.3	1.28045	0.640225	−	0.768188i	$-0.278841\pi$
$997$	40309.3	1.28045	0.640225	+	0.768188i	$0.278841\pi$
$998$	25551.3	0.810432
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.be.1.2		2
3.2	odd	2		735.4.a.m.1.1		2
7.6	odd	2		315.4.a.l.1.2		2
21.20	even	2		105.4.a.d.1.1	✓	2
35.34	odd	2		1575.4.a.n.1.1		2
84.83	odd	2		1680.4.a.bd.1.1		2
105.62	odd	4		525.4.d.k.274.1		4
105.83	odd	4		525.4.d.k.274.4		4
105.104	even	2		525.4.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.d.1.1	✓	2	21.20	even	2
315.4.a.l.1.2		2	7.6	odd	2
525.4.a.o.1.2		2	105.104	even	2
525.4.d.k.274.1		4	105.62	odd	4
525.4.d.k.274.4		4	105.83	odd	4
735.4.a.m.1.1		2	3.2	odd	2
1575.4.a.n.1.1		2	35.34	odd	2
1680.4.a.bd.1.1		2	84.83	odd	2
2205.4.a.be.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+4.23607 q^{2} +9.94427 q^{4} -5.00000 q^{5} +8.23607 q^{8} -21.1803 q^{10} +41.5279 q^{11} -88.9706 q^{13} -44.6656 q^{16} -120.387 q^{17} +112.138 q^{19} -49.7214 q^{20} +175.915 q^{22} +115.279 q^{23} +25.0000 q^{25} -376.885 q^{26} +144.833 q^{29} +258.079 q^{31} -255.095 q^{32} -509.967 q^{34} +48.3344 q^{37} +475.023 q^{38} -41.1803 q^{40} +200.885 q^{41} -218.217 q^{43} +412.964 q^{44} +488.328 q^{46} +575.659 q^{47} +105.902 q^{50} -884.748 q^{52} +184.302 q^{53} -207.639 q^{55} +613.522 q^{58} -151.502 q^{59} +529.830 q^{61} +1093.24 q^{62} -723.276 q^{64} +444.853 q^{65} +1.28485 q^{67} -1197.16 q^{68} +61.4226 q^{71} -484.800 q^{73} +204.748 q^{74} +1115.13 q^{76} +878.257 q^{79} +223.328 q^{80} +850.964 q^{82} +491.830 q^{83} +601.935 q^{85} -924.381 q^{86} +342.026 q^{88} -415.560 q^{89} +1146.36 q^{92} +2438.53 q^{94} -560.689 q^{95} +1031.70 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{4} - 10 q^{5} + 12 q^{8} - 20 q^{10} + 92 q^{11} - 8 q^{13} + 18 q^{16} - 44 q^{17} + 108 q^{19} - 10 q^{20} + 164 q^{22} + 320 q^{23} + 50 q^{25} - 396 q^{26} + 236 q^{29} + 60 q^{31}+ \cdots + 2448 q^{97}+O(q^{100}) \) 2 * q + 4 * q^2 + 2 * q^4 - 10 * q^5 + 12 * q^8 - 20 * q^10 + 92 * q^11 - 8 * q^13 + 18 * q^16 - 44 * q^17 + 108 * q^19 - 10 * q^20 + 164 * q^22 + 320 * q^23 + 50 * q^25 - 396 * q^26 + 236 * q^29 + 60 * q^31 - 300 * q^32 - 528 * q^34 + 204 * q^37 + 476 * q^38 - 60 * q^40 + 44 * q^41 + 136 * q^43 + 12 * q^44 + 440 * q^46 + 400 * q^47 + 100 * q^50 - 1528 * q^52 - 16 * q^53 - 460 * q^55 + 592 * q^58 - 464 * q^59 + 684 * q^61 + 1140 * q^62 - 1214 * q^64 + 40 * q^65 + 736 * q^67 - 1804 * q^68 + 740 * q^71 - 424 * q^73 + 168 * q^74 + 1148 * q^76 - 408 * q^79 - 90 * q^80 + 888 * q^82 + 608 * q^83 + 220 * q^85 - 1008 * q^86 + 532 * q^88 - 1332 * q^89 - 480 * q^92 + 2480 * q^94 - 540 * q^95 + 2448 * q^97

Introduction

L-functions

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