LMFDB - Embedded newform 2205.4.a.bz.1.5

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			1.5
Root			$4.29508$ 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)

$f(q)$	$=$	$q+2.88087 q^{2} +0.299392 q^{4} -5.00000 q^{5} -22.1844 q^{8} +O(q^{10})$	$q+2.88087 q^{2} +0.299392 q^{4} -5.00000 q^{5} -22.1844 q^{8} -14.4043 q^{10} +46.4881 q^{11} +31.0537 q^{13} -66.3055 q^{16} -61.8516 q^{17} -24.6214 q^{19} -1.49696 q^{20} +133.926 q^{22} +154.942 q^{23} +25.0000 q^{25} +89.4616 q^{26} -200.436 q^{29} -129.255 q^{31} -13.5419 q^{32} -178.186 q^{34} -77.9662 q^{37} -70.9309 q^{38} +110.922 q^{40} +235.479 q^{41} -278.388 q^{43} +13.9182 q^{44} +446.366 q^{46} +368.085 q^{47} +72.0217 q^{50} +9.29725 q^{52} +169.584 q^{53} -232.440 q^{55} -577.429 q^{58} +691.490 q^{59} -696.572 q^{61} -372.365 q^{62} +491.432 q^{64} -155.269 q^{65} +2.33311 q^{67} -18.5179 q^{68} +866.599 q^{71} -752.443 q^{73} -224.610 q^{74} -7.37145 q^{76} +842.783 q^{79} +331.528 q^{80} +678.384 q^{82} -1443.44 q^{83} +309.258 q^{85} -801.998 q^{86} -1031.31 q^{88} -1438.06 q^{89} +46.3884 q^{92} +1060.40 q^{94} +123.107 q^{95} -23.4509 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.88087	1.01854	0.509270	−	0.860607i	$-0.329915\pi$
$2$	2.88087	1.01854	0.509270	+	0.860607i	$0.329915\pi$
$3$	0	0
$3$	0	0
$4$	0.299392	0.0374241
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−22.1844	−0.980422
$9$	0	0
$10$	−14.4043	−0.455505
$11$	46.4881	1.27424	0.637122	−	0.770763i	$-0.280125\pi$
$11$	46.4881	1.27424	0.637122	+	0.770763i	$0.280125\pi$
$12$	0	0
$13$	31.0537	0.662519	0.331260	−	0.943540i	$-0.392526\pi$
$13$	31.0537	0.662519	0.331260	+	0.943540i	$0.392526\pi$
$14$	0	0
$15$	0	0
$16$	−66.3055	−1.03602
$17$	−61.8516	−0.882425	−0.441212	−	0.897403i	$-0.645451\pi$
$17$	−61.8516	−0.882425	−0.441212	+	0.897403i	$0.645451\pi$
$18$	0	0
$19$	−24.6214	−0.297291	−0.148645	−	0.988891i	$-0.547491\pi$
$19$	−24.6214	−0.297291	−0.148645	+	0.988891i	$0.547491\pi$
$20$	−1.49696	−0.0167365
$21$	0	0
$22$	133.926	1.29787
$23$	154.942	1.40468	0.702339	−	0.711843i	$-0.252139\pi$
$23$	154.942	1.40468	0.702339	+	0.711843i	$0.252139\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	89.4616	0.674803
$27$	0	0
$28$	0	0
$29$	−200.436	−1.28345	−0.641724	−	0.766936i	$-0.721781\pi$
$29$	−200.436	−1.28345	−0.641724	+	0.766936i	$0.721781\pi$
$30$	0	0
$31$	−129.255	−0.748865	−0.374432	−	0.927254i	$-0.622162\pi$
$31$	−129.255	−0.748865	−0.374432	+	0.927254i	$0.622162\pi$
$32$	−13.5419	−0.0748093
$33$	0	0
$34$	−178.186	−0.898785
$35$	0	0
$36$	0	0
$37$	−77.9662	−0.346421	−0.173210	−	0.984885i	$-0.555414\pi$
$37$	−77.9662	−0.346421	−0.173210	+	0.984885i	$0.555414\pi$
$38$	−70.9309	−0.302803
$39$	0	0
$40$	110.922	0.438458
$41$	235.479	0.896967	0.448483	−	0.893791i	$-0.351964\pi$
$41$	235.479	0.896967	0.448483	+	0.893791i	$0.351964\pi$
$42$	0	0
$43$	−278.388	−0.987296	−0.493648	−	0.869662i	$-0.664337\pi$
$43$	−278.388	−0.987296	−0.493648	+	0.869662i	$0.664337\pi$
$44$	13.9182	0.0476874
$45$	0	0
$46$	446.366	1.43072
$47$	368.085	1.14235	0.571177	−	0.820827i	$-0.306487\pi$
$47$	368.085	1.14235	0.571177	+	0.820827i	$0.306487\pi$
$48$	0	0
$49$	0	0
$50$	72.0217	0.203708
$51$	0	0
$52$	9.29725	0.0247942
$53$	169.584	0.439512	0.219756	−	0.975555i	$-0.429474\pi$
$53$	169.584	0.439512	0.219756	+	0.975555i	$0.429474\pi$
$54$	0	0
$55$	−232.440	−0.569859
$56$	0	0
$57$	0	0
$58$	−577.429	−1.30724
$59$	691.490	1.52584	0.762918	−	0.646496i	$-0.223766\pi$
$59$	691.490	1.52584	0.762918	+	0.646496i	$0.223766\pi$
$60$	0	0
$61$	−696.572	−1.46208	−0.731041	−	0.682334i	$-0.760965\pi$
$61$	−696.572	−1.46208	−0.731041	+	0.682334i	$0.760965\pi$
$62$	−372.365	−0.762749
$63$	0	0
$64$	491.432	0.959827
$65$	−155.269	−0.296288
$66$	0	0
$67$	2.33311	0.00425426	0.00212713	−	0.999998i	$-0.499323\pi$
$67$	2.33311	0.00425426	0.00212713	+	0.999998i	$0.499323\pi$
$68$	−18.5179	−0.0330239
$69$	0	0
$70$	0	0
$71$	866.599	1.44854	0.724270	−	0.689516i	$-0.242177\pi$
$71$	866.599	1.44854	0.724270	+	0.689516i	$0.242177\pi$
$72$	0	0
$73$	−752.443	−1.20639	−0.603197	−	0.797592i	$-0.706107\pi$
$73$	−752.443	−1.20639	−0.603197	+	0.797592i	$0.706107\pi$
$74$	−224.610	−0.352843
$75$	0	0
$76$	−7.37145	−0.0111258
$77$	0	0
$78$	0	0
$79$	842.783	1.20026	0.600130	−	0.799903i	$-0.295116\pi$
$79$	842.783	1.20026	0.600130	+	0.799903i	$0.295116\pi$
$80$	331.528	0.463324
$81$	0	0
$82$	678.384	0.913597
$83$	−1443.44	−1.90890	−0.954450	−	0.298372i	$-0.903556\pi$
$83$	−1443.44	−1.90890	−0.954450	+	0.298372i	$0.903556\pi$
$84$	0	0
$85$	309.258	0.394632
$86$	−801.998	−1.00560
$87$	0	0
$88$	−1031.31	−1.24930
$89$	−1438.06	−1.71274	−0.856372	−	0.516360i	$-0.827287\pi$
$89$	−1438.06	−1.71274	−0.856372	+	0.516360i	$0.827287\pi$
$90$	0	0
$91$	0	0
$92$	46.3884	0.0525687
$93$	0	0
$94$	1060.40	1.16353
$95$	123.107	0.132953
$96$	0	0
$97$	−23.4509	−0.0245472	−0.0122736	−	0.999925i	$-0.503907\pi$
$97$	−23.4509	−0.0245472	−0.0122736	+	0.999925i	$0.503907\pi$
$98$	0	0
$99$	0	0
$100$	7.48481	0.00748481
$101$	−1476.41	−1.45454	−0.727271	−	0.686350i	$-0.759212\pi$
$101$	−1476.41	−1.45454	−0.727271	+	0.686350i	$0.759212\pi$
$102$	0	0
$103$	−1009.44	−0.965663	−0.482831	−	0.875713i	$-0.660392\pi$
$103$	−1009.44	−0.965663	−0.482831	+	0.875713i	$0.660392\pi$
$104$	−688.909	−0.649549
$105$	0	0
$106$	488.549	0.447661
$107$	−1418.51	−1.28161	−0.640805	−	0.767704i	$-0.721399\pi$
$107$	−1418.51	−1.28161	−0.640805	+	0.767704i	$0.721399\pi$
$108$	0	0
$109$	−877.860	−0.771410	−0.385705	−	0.922622i	$-0.626042\pi$
$109$	−877.860	−0.771410	−0.385705	+	0.922622i	$0.626042\pi$
$110$	−669.629	−0.580424
$111$	0	0
$112$	0	0
$113$	1500.11	1.24884	0.624419	−	0.781090i	$-0.285336\pi$
$113$	1500.11	1.24884	0.624419	+	0.781090i	$0.285336\pi$
$114$	0	0
$115$	−774.708	−0.628191
$116$	−60.0090	−0.0480318
$117$	0	0
$118$	1992.09	1.55412
$119$	0	0
$120$	0	0
$121$	830.140	0.623696
$122$	−2006.73	−1.48919
$123$	0	0
$124$	−38.6978	−0.0280255
$125$	−125.000	−0.0894427
$126$	0	0
$127$	416.639	0.291108	0.145554	−	0.989350i	$-0.453503\pi$
$127$	416.639	0.291108	0.145554	+	0.989350i	$0.453503\pi$
$128$	1524.08	1.05243
$129$	0	0
$130$	−447.308	−0.301781
$131$	−410.496	−0.273780	−0.136890	−	0.990586i	$-0.543711\pi$
$131$	−410.496	−0.273780	−0.136890	+	0.990586i	$0.543711\pi$
$132$	0	0
$133$	0	0
$134$	6.72139	0.00433313
$135$	0	0
$136$	1372.14	0.865149
$137$	−1761.97	−1.09880	−0.549400	−	0.835560i	$-0.685144\pi$
$137$	−1761.97	−1.09880	−0.549400	+	0.835560i	$0.685144\pi$
$138$	0	0
$139$	−865.719	−0.528268	−0.264134	−	0.964486i	$-0.585086\pi$
$139$	−865.719	−0.528268	−0.264134	+	0.964486i	$0.585086\pi$
$140$	0	0
$141$	0	0
$142$	2496.56	1.47540
$143$	1443.63	0.844211
$144$	0	0
$145$	1002.18	0.573975
$146$	−2167.69	−1.22876
$147$	0	0
$148$	−23.3425	−0.0129645
$149$	−3019.23	−1.66004	−0.830018	−	0.557737i	$-0.811670\pi$
$149$	−3019.23	−1.66004	−0.830018	+	0.557737i	$0.811670\pi$
$150$	0	0
$151$	541.896	0.292045	0.146023	−	0.989281i	$-0.453353\pi$
$151$	541.896	0.292045	0.146023	+	0.989281i	$0.453353\pi$
$152$	546.211	0.291471
$153$	0	0
$154$	0	0
$155$	646.273	0.334902
$156$	0	0
$157$	−2586.09	−1.31460	−0.657300	−	0.753629i	$-0.728302\pi$
$157$	−2586.09	−1.31460	−0.657300	+	0.753629i	$0.728302\pi$
$158$	2427.94	1.22251
$159$	0	0
$160$	67.7096	0.0334557
$161$	0	0
$162$	0	0
$163$	−2466.14	−1.18505	−0.592525	−	0.805552i	$-0.701869\pi$
$163$	−2466.14	−1.18505	−0.592525	+	0.805552i	$0.701869\pi$
$164$	70.5006	0.0335681
$165$	0	0
$166$	−4158.37	−1.94429
$167$	459.020	0.212695	0.106347	−	0.994329i	$-0.466084\pi$
$167$	459.020	0.212695	0.106347	+	0.994329i	$0.466084\pi$
$168$	0	0
$169$	−1232.67	−0.561068
$170$	890.931	0.401949
$171$	0	0
$172$	−83.3471	−0.0369486
$173$	−4501.46	−1.97826	−0.989132	−	0.147032i	$-0.953028\pi$
$173$	−4501.46	−1.97826	−0.989132	+	0.147032i	$0.953028\pi$
$174$	0	0
$175$	0	0
$176$	−3082.41	−1.32015
$177$	0	0
$178$	−4142.86	−1.74450
$179$	−1960.11	−0.818467	−0.409233	−	0.912430i	$-0.634204\pi$
$179$	−1960.11	−0.818467	−0.409233	+	0.912430i	$0.634204\pi$
$180$	0	0
$181$	−3645.47	−1.49705	−0.748524	−	0.663108i	$-0.769237\pi$
$181$	−3645.47	−1.49705	−0.748524	+	0.663108i	$0.769237\pi$
$182$	0	0
$183$	0	0
$184$	−3437.29	−1.37718
$185$	389.831	0.154924
$186$	0	0
$187$	−2875.36	−1.12442
$188$	110.202	0.0427515
$189$	0	0
$190$	354.654	0.135417
$191$	−2514.45	−0.952560	−0.476280	−	0.879294i	$-0.658015\pi$
$191$	−2514.45	−0.952560	−0.476280	+	0.879294i	$0.658015\pi$
$192$	0	0
$193$	2189.36	0.816549	0.408274	−	0.912859i	$-0.366131\pi$
$193$	2189.36	0.816549	0.408274	+	0.912859i	$0.366131\pi$
$194$	−67.5590	−0.0250023
$195$	0	0
$196$	0	0
$197$	3886.50	1.40559	0.702795	−	0.711392i	$-0.251935\pi$
$197$	3886.50	1.40559	0.702795	+	0.711392i	$0.251935\pi$
$198$	0	0
$199$	−969.361	−0.345307	−0.172654	−	0.984983i	$-0.555234\pi$
$199$	−969.361	−0.345307	−0.172654	+	0.984983i	$0.555234\pi$
$200$	−554.611	−0.196084
$201$	0	0
$202$	−4253.35	−1.48151
$203$	0	0
$204$	0	0
$205$	−1177.39	−0.401136
$206$	−2908.07	−0.983566
$207$	0	0
$208$	−2059.03	−0.686386
$209$	−1144.60	−0.378821
$210$	0	0
$211$	3079.69	1.00481	0.502404	−	0.864633i	$-0.332449\pi$
$211$	3079.69	1.00481	0.502404	+	0.864633i	$0.332449\pi$
$212$	50.7722	0.0164483
$213$	0	0
$214$	−4086.53	−1.30537
$215$	1391.94	0.441532
$216$	0	0
$217$	0	0
$218$	−2529.00	−0.785712
$219$	0	0
$220$	−69.5909	−0.0213264
$221$	−1920.72	−0.584624
$222$	0	0
$223$	162.003	0.0486480	0.0243240	−	0.999704i	$-0.492257\pi$
$223$	162.003	0.0486480	0.0243240	+	0.999704i	$0.492257\pi$
$224$	0	0
$225$	0	0
$226$	4321.62	1.27199
$227$	−379.952	−0.111094	−0.0555470	−	0.998456i	$-0.517690\pi$
$227$	−379.952	−0.111094	−0.0555470	+	0.998456i	$0.517690\pi$
$228$	0	0
$229$	−4781.50	−1.37978	−0.689892	−	0.723912i	$-0.742342\pi$
$229$	−4781.50	−1.37978	−0.689892	+	0.723912i	$0.742342\pi$
$230$	−2231.83	−0.639838
$231$	0	0
$232$	4446.55	1.25832
$233$	2524.54	0.709819	0.354910	−	0.934901i	$-0.384512\pi$
$233$	2524.54	0.709819	0.354910	+	0.934901i	$0.384512\pi$
$234$	0	0
$235$	−1840.42	−0.510876
$236$	207.027	0.0571029
$237$	0	0
$238$	0	0
$239$	−113.452	−0.0307053	−0.0153527	−	0.999882i	$-0.504887\pi$
$239$	−113.452	−0.0307053	−0.0153527	+	0.999882i	$0.504887\pi$
$240$	0	0
$241$	6725.34	1.79758	0.898791	−	0.438377i	$-0.144447\pi$
$241$	6725.34	1.79758	0.898791	+	0.438377i	$0.144447\pi$
$242$	2391.52	0.635260
$243$	0	0
$244$	−208.549	−0.0547170
$245$	0	0
$246$	0	0
$247$	−764.585	−0.196961
$248$	2867.44	0.734203
$249$	0	0
$250$	−360.108	−0.0911010
$251$	3815.00	0.959366	0.479683	−	0.877442i	$-0.340752\pi$
$251$	3815.00	0.959366	0.479683	+	0.877442i	$0.340752\pi$
$252$	0	0
$253$	7202.94	1.78990
$254$	1200.28	0.296506
$255$	0	0
$256$	459.231	0.112117
$257$	−2201.90	−0.534439	−0.267220	−	0.963636i	$-0.586105\pi$
$257$	−2201.90	−0.534439	−0.267220	+	0.963636i	$0.586105\pi$
$258$	0	0
$259$	0	0
$260$	−46.4862	−0.0110883
$261$	0	0
$262$	−1182.58	−0.278856
$263$	1204.18	0.282330	0.141165	−	0.989986i	$-0.454915\pi$
$263$	1204.18	0.282330	0.141165	+	0.989986i	$0.454915\pi$
$264$	0	0
$265$	−847.920	−0.196556
$266$	0	0
$267$	0	0
$268$	0.698517	0.000159212 0
$269$	−5128.98	−1.16253	−0.581263	−	0.813716i	$-0.697441\pi$
$269$	−5128.98	−1.16253	−0.581263	+	0.813716i	$0.697441\pi$
$270$	0	0
$271$	808.391	0.181204	0.0906020	−	0.995887i	$-0.471121\pi$
$271$	808.391	0.181204	0.0906020	+	0.995887i	$0.471121\pi$
$272$	4101.10	0.914213
$273$	0	0
$274$	−5076.01	−1.11917
$275$	1162.20	0.254849
$276$	0	0
$277$	180.265	0.0391014	0.0195507	−	0.999809i	$-0.493776\pi$
$277$	180.265	0.0391014	0.0195507	+	0.999809i	$0.493776\pi$
$278$	−2494.02	−0.538062
$279$	0	0
$280$	0	0
$281$	3068.29	0.651384	0.325692	−	0.945476i	$-0.394403\pi$
$281$	3068.29	0.651384	0.325692	+	0.945476i	$0.394403\pi$
$282$	0	0
$283$	−3868.17	−0.812506	−0.406253	−	0.913761i	$-0.633165\pi$
$283$	−3868.17	−0.812506	−0.406253	+	0.913761i	$0.633165\pi$
$284$	259.453	0.0542103
$285$	0	0
$286$	4158.90	0.859863
$287$	0	0
$288$	0	0
$289$	−1087.38	−0.221326
$290$	2887.14	0.584617
$291$	0	0
$292$	−225.276	−0.0451482
$293$	1967.79	0.392353	0.196176	−	0.980569i	$-0.437147\pi$
$293$	1967.79	0.392353	0.196176	+	0.980569i	$0.437147\pi$
$294$	0	0
$295$	−3457.45	−0.682374
$296$	1729.63	0.339638
$297$	0	0
$298$	−8698.01	−1.69081
$299$	4811.52	0.930626
$300$	0	0
$301$	0	0
$302$	1561.13	0.297460
$303$	0	0
$304$	1632.53	0.308000
$305$	3482.86	0.653863
$306$	0	0
$307$	5487.54	1.02016	0.510082	−	0.860126i	$-0.329615\pi$
$307$	5487.54	1.02016	0.510082	+	0.860126i	$0.329615\pi$
$308$	0	0
$309$	0	0
$310$	1861.83	0.341112
$311$	5172.80	0.943159	0.471579	−	0.881824i	$-0.343684\pi$
$311$	5172.80	0.943159	0.471579	+	0.881824i	$0.343684\pi$
$312$	0	0
$313$	5762.92	1.04070	0.520351	−	0.853953i	$-0.325801\pi$
$313$	5762.92	1.04070	0.520351	+	0.853953i	$0.325801\pi$
$314$	−7450.17	−1.33897
$315$	0	0
$316$	252.323	0.0449186
$317$	−3180.88	−0.563583	−0.281792	−	0.959476i	$-0.590929\pi$
$317$	−3180.88	−0.563583	−0.281792	+	0.959476i	$0.590929\pi$
$318$	0	0
$319$	−9317.87	−1.63543
$320$	−2457.16	−0.429248
$321$	0	0
$322$	0	0
$323$	1522.87	0.262337
$324$	0	0
$325$	776.343	0.132504
$326$	−7104.63	−1.20702
$327$	0	0
$328$	−5223.97	−0.879406
$329$	0	0
$330$	0	0
$331$	9233.02	1.53321	0.766606	−	0.642118i	$-0.221944\pi$
$331$	9233.02	1.53321	0.766606	+	0.642118i	$0.221944\pi$
$332$	−432.156	−0.0714387
$333$	0	0
$334$	1322.38	0.216638
$335$	−11.6656	−0.00190256
$336$	0	0
$337$	−3259.50	−0.526874	−0.263437	−	0.964677i	$-0.584856\pi$
$337$	−3259.50	−0.526874	−0.263437	+	0.964677i	$0.584856\pi$
$338$	−3551.15	−0.571470
$339$	0	0
$340$	92.5895	0.0147687
$341$	−6008.79	−0.954236
$342$	0	0
$343$	0	0
$344$	6175.87	0.967967
$345$	0	0
$346$	−12968.1	−2.01494
$347$	−1851.13	−0.286380	−0.143190	−	0.989695i	$-0.545736\pi$
$347$	−1851.13	−0.286380	−0.143190	+	0.989695i	$0.545736\pi$
$348$	0	0
$349$	1102.14	0.169043	0.0845216	−	0.996422i	$-0.473064\pi$
$349$	1102.14	0.169043	0.0845216	+	0.996422i	$0.473064\pi$
$350$	0	0
$351$	0	0
$352$	−629.538	−0.0953252
$353$	−2670.25	−0.402615	−0.201307	−	0.979528i	$-0.564519\pi$
$353$	−2670.25	−0.402615	−0.201307	+	0.979528i	$0.564519\pi$
$354$	0	0
$355$	−4333.00	−0.647807
$356$	−430.545	−0.0640978
$357$	0	0
$358$	−5646.82	−0.833641
$359$	−435.453	−0.0640176	−0.0320088	−	0.999488i	$-0.510190\pi$
$359$	−435.453	−0.0640176	−0.0320088	+	0.999488i	$0.510190\pi$
$360$	0	0
$361$	−6252.79	−0.911618
$362$	−10502.1	−1.52480
$363$	0	0
$364$	0	0
$365$	3762.22	0.539516
$366$	0	0
$367$	−2474.65	−0.351978	−0.175989	−	0.984392i	$-0.556312\pi$
$367$	−2474.65	−0.351978	−0.175989	+	0.984392i	$0.556312\pi$
$368$	−10273.5	−1.45528
$369$	0	0
$370$	1123.05	0.157796
$371$	0	0
$372$	0	0
$373$	5438.18	0.754902	0.377451	−	0.926030i	$-0.376801\pi$
$373$	5438.18	0.754902	0.377451	+	0.926030i	$0.376801\pi$
$374$	−8283.53	−1.14527
$375$	0	0
$376$	−8165.74	−1.11999
$377$	−6224.28	−0.850309
$378$	0	0
$379$	10597.1	1.43624	0.718122	−	0.695917i	$-0.245002\pi$
$379$	10597.1	1.43624	0.718122	+	0.695917i	$0.245002\pi$
$380$	36.8572	0.00497562
$381$	0	0
$382$	−7243.78	−0.970221
$383$	4671.36	0.623226	0.311613	−	0.950209i	$-0.399131\pi$
$383$	4671.36	0.623226	0.311613	+	0.950209i	$0.399131\pi$
$384$	0	0
$385$	0	0
$386$	6307.27	0.831688
$387$	0	0
$388$	−7.02103	−0.000918656 0
$389$	−3557.69	−0.463707	−0.231853	−	0.972751i	$-0.574479\pi$
$389$	−3557.69	−0.463707	−0.231853	+	0.972751i	$0.574479\pi$
$390$	0	0
$391$	−9583.40	−1.23952
$392$	0	0
$393$	0	0
$394$	11196.5	1.43165
$395$	−4213.91	−0.536772
$396$	0	0
$397$	−9184.57	−1.16111	−0.580554	−	0.814221i	$-0.697164\pi$
$397$	−9184.57	−1.16111	−0.580554	+	0.814221i	$0.697164\pi$
$398$	−2792.60	−0.351709
$399$	0	0
$400$	−1657.64	−0.207205
$401$	9206.52	1.14651	0.573256	−	0.819376i	$-0.305680\pi$
$401$	9206.52	1.14651	0.573256	+	0.819376i	$0.305680\pi$
$402$	0	0
$403$	−4013.83	−0.496137
$404$	−442.027	−0.0544349
$405$	0	0
$406$	0	0
$407$	−3624.50	−0.441424
$408$	0	0
$409$	−7653.66	−0.925303	−0.462652	−	0.886540i	$-0.653102\pi$
$409$	−7653.66	−0.925303	−0.462652	+	0.886540i	$0.653102\pi$
$410$	−3391.92	−0.408573
$411$	0	0
$412$	−302.219	−0.0361390
$413$	0	0
$414$	0	0
$415$	7217.22	0.853686
$416$	−420.527	−0.0495626
$417$	0	0
$418$	−3297.44	−0.385844
$419$	370.864	0.0432408	0.0216204	−	0.999766i	$-0.493117\pi$
$419$	370.864	0.0432408	0.0216204	+	0.999766i	$0.493117\pi$
$420$	0	0
$421$	1221.96	0.141460	0.0707302	−	0.997495i	$-0.477467\pi$
$421$	1221.96	0.141460	0.0707302	+	0.997495i	$0.477467\pi$
$422$	8872.18	1.02344
$423$	0	0
$424$	−3762.12	−0.430908
$425$	−1546.29	−0.176485
$426$	0	0
$427$	0	0
$428$	−424.690	−0.0479630
$429$	0	0
$430$	4009.99	0.449718
$431$	14928.9	1.66845	0.834224	−	0.551426i	$-0.185916\pi$
$431$	14928.9	1.66845	0.834224	+	0.551426i	$0.185916\pi$
$432$	0	0
$433$	4544.42	0.504367	0.252184	−	0.967679i	$-0.418851\pi$
$433$	4544.42	0.504367	0.252184	+	0.967679i	$0.418851\pi$
$434$	0	0
$435$	0	0
$436$	−262.825	−0.0288693
$437$	−3814.88	−0.417598
$438$	0	0
$439$	9431.89	1.02542	0.512710	−	0.858562i	$-0.328642\pi$
$439$	9431.89	1.02542	0.512710	+	0.858562i	$0.328642\pi$
$440$	5156.55	0.558702
$441$	0	0
$442$	−5533.35	−0.595463
$443$	5542.61	0.594441	0.297220	−	0.954809i	$-0.403940\pi$
$443$	5542.61	0.594441	0.297220	+	0.954809i	$0.403940\pi$
$444$	0	0
$445$	7190.31	0.765962
$446$	466.708	0.0495500
$447$	0	0
$448$	0	0
$449$	−16311.6	−1.71446	−0.857229	−	0.514936i	$-0.827816\pi$
$449$	−16311.6	−1.71446	−0.857229	+	0.514936i	$0.827816\pi$
$450$	0	0
$451$	10947.0	1.14295
$452$	449.122	0.0467366
$453$	0	0
$454$	−1094.59	−0.113154
$455$	0	0
$456$	0	0
$457$	−14231.1	−1.45668	−0.728339	−	0.685217i	$-0.759707\pi$
$457$	−14231.1	−1.45668	−0.728339	+	0.685217i	$0.759707\pi$
$458$	−13774.9	−1.40537
$459$	0	0
$460$	−231.942	−0.0235094
$461$	4960.94	0.501202	0.250601	−	0.968090i	$-0.419372\pi$
$461$	4960.94	0.501202	0.250601	+	0.968090i	$0.419372\pi$
$462$	0	0
$463$	15479.5	1.55377	0.776885	−	0.629642i	$-0.216799\pi$
$463$	15479.5	1.55377	0.776885	+	0.629642i	$0.216799\pi$
$464$	13290.0	1.32968
$465$	0	0
$466$	7272.85	0.722979
$467$	14815.6	1.46807	0.734033	−	0.679114i	$-0.237636\pi$
$467$	14815.6	1.46807	0.734033	+	0.679114i	$0.237636\pi$
$468$	0	0
$469$	0	0
$470$	−5302.01	−0.520348
$471$	0	0
$472$	−15340.3	−1.49596
$473$	−12941.7	−1.25805
$474$	0	0
$475$	−615.534	−0.0594582
$476$	0	0
$477$	0	0
$478$	−326.839	−0.0312746
$479$	9098.96	0.867938	0.433969	−	0.900928i	$-0.357113\pi$
$479$	9098.96	0.867938	0.433969	+	0.900928i	$0.357113\pi$
$480$	0	0
$481$	−2421.14	−0.229510
$482$	19374.8	1.83091
$483$	0	0
$484$	248.538	0.0233412
$485$	117.255	0.0109778
$486$	0	0
$487$	−14991.4	−1.39492	−0.697461	−	0.716623i	$-0.745687\pi$
$487$	−14991.4	−1.39492	−0.697461	+	0.716623i	$0.745687\pi$
$488$	15453.1	1.43346
$489$	0	0
$490$	0	0
$491$	−8243.61	−0.757697	−0.378848	−	0.925459i	$-0.623680\pi$
$491$	−8243.61	−0.757697	−0.378848	+	0.925459i	$0.623680\pi$
$492$	0	0
$493$	12397.3	1.13255
$494$	−2202.67	−0.200613
$495$	0	0
$496$	8570.29	0.775841
$497$	0	0
$498$	0	0
$499$	9227.49	0.827814	0.413907	−	0.910319i	$-0.364164\pi$
$499$	9227.49	0.827814	0.413907	+	0.910319i	$0.364164\pi$
$500$	−37.4241	−0.00334731
$501$	0	0
$502$	10990.5	0.977152
$503$	13750.9	1.21893	0.609465	−	0.792813i	$-0.291384\pi$
$503$	13750.9	1.21893	0.609465	+	0.792813i	$0.291384\pi$
$504$	0	0
$505$	7382.07	0.650491
$506$	20750.7	1.82309
$507$	0	0
$508$	124.739	0.0108945
$509$	−15544.0	−1.35358	−0.676792	−	0.736174i	$-0.736630\pi$
$509$	−15544.0	−1.35358	−0.676792	+	0.736174i	$0.736630\pi$
$510$	0	0
$511$	0	0
$512$	−10869.7	−0.938236
$513$	0	0
$514$	−6343.39	−0.544348
$515$	5047.21	0.431858
$516$	0	0
$517$	17111.5	1.45564
$518$	0	0
$519$	0	0
$520$	3444.54	0.290487
$521$	15247.7	1.28218	0.641089	−	0.767466i	$-0.278483\pi$
$521$	15247.7	1.28218	0.641089	+	0.767466i	$0.278483\pi$
$522$	0	0
$523$	−22216.2	−1.85745	−0.928723	−	0.370773i	$-0.879093\pi$
$523$	−22216.2	−1.85745	−0.928723	+	0.370773i	$0.879093\pi$
$524$	−122.899	−0.0102460
$525$	0	0
$526$	3469.08	0.287565
$527$	7994.60	0.660817
$528$	0	0
$529$	11839.9	0.973118
$530$	−2442.74	−0.200200
$531$	0	0
$532$	0	0
$533$	7312.50	0.594258
$534$	0	0
$535$	7092.53	0.573153
$536$	−51.7588	−0.00417097
$537$	0	0
$538$	−14775.9	−1.18408
$539$	0	0
$540$	0	0
$541$	−12985.3	−1.03194	−0.515972	−	0.856605i	$-0.672569\pi$
$541$	−12985.3	−1.03194	−0.515972	+	0.856605i	$0.672569\pi$
$542$	2328.87	0.184564
$543$	0	0
$544$	837.590	0.0660136
$545$	4389.30	0.344985
$546$	0	0
$547$	−8226.94	−0.643069	−0.321534	−	0.946898i	$-0.604199\pi$
$547$	−8226.94	−0.643069	−0.321534	+	0.946898i	$0.604199\pi$
$548$	−527.522	−0.0411215
$549$	0	0
$550$	3348.15	0.259574
$551$	4935.00	0.381557
$552$	0	0
$553$	0	0
$554$	519.320	0.0398263
$555$	0	0
$556$	−259.190	−0.0197699
$557$	−17841.3	−1.35720	−0.678599	−	0.734509i	$-0.737412\pi$
$557$	−17841.3	−1.35720	−0.678599	+	0.734509i	$0.737412\pi$
$558$	0	0
$559$	−8644.97	−0.654103
$560$	0	0
$561$	0	0
$562$	8839.34	0.663461
$563$	−1078.40	−0.0807265	−0.0403633	−	0.999185i	$-0.512852\pi$
$563$	−1078.40	−0.0807265	−0.0403633	+	0.999185i	$0.512852\pi$
$564$	0	0
$565$	−7500.56	−0.558497
$566$	−11143.7	−0.827570
$567$	0	0
$568$	−19225.0	−1.42018
$569$	−16986.5	−1.25151	−0.625756	−	0.780019i	$-0.715209\pi$
$569$	−16986.5	−1.25151	−0.625756	+	0.780019i	$0.715209\pi$
$570$	0	0
$571$	−12263.0	−0.898756	−0.449378	−	0.893342i	$-0.648354\pi$
$571$	−12263.0	−0.898756	−0.449378	+	0.893342i	$0.648354\pi$
$572$	432.211	0.0315938
$573$	0	0
$574$	0	0
$575$	3873.54	0.280935
$576$	0	0
$577$	−7050.51	−0.508694	−0.254347	−	0.967113i	$-0.581861\pi$
$577$	−7050.51	−0.508694	−0.254347	+	0.967113i	$0.581861\pi$
$578$	−3132.59	−0.225430
$579$	0	0
$580$	300.045	0.0214805
$581$	0	0
$582$	0	0
$583$	7883.63	0.560046
$584$	16692.5	1.18278
$585$	0	0
$586$	5668.93	0.399627
$587$	20085.3	1.41228	0.706141	−	0.708071i	$-0.250434\pi$
$587$	20085.3	1.41228	0.706141	+	0.708071i	$0.250434\pi$
$588$	0	0
$589$	3182.42	0.222631
$590$	−9960.45	−0.695026
$591$	0	0
$592$	5169.59	0.358900
$593$	18974.3	1.31397	0.656984	−	0.753905i	$-0.271832\pi$
$593$	18974.3	1.31397	0.656984	+	0.753905i	$0.271832\pi$
$594$	0	0
$595$	0	0
$596$	−903.936	−0.0621252
$597$	0	0
$598$	13861.3	0.947880
$599$	−16795.7	−1.14567	−0.572834	−	0.819671i	$-0.694156\pi$
$599$	−16795.7	−1.14567	−0.572834	+	0.819671i	$0.694156\pi$
$600$	0	0
$601$	−13624.3	−0.924702	−0.462351	−	0.886697i	$-0.652994\pi$
$601$	−13624.3	−0.924702	−0.462351	+	0.886697i	$0.652994\pi$
$602$	0	0
$603$	0	0
$604$	162.240	0.0109295
$605$	−4150.70	−0.278925
$606$	0	0
$607$	−8399.51	−0.561657	−0.280828	−	0.959758i	$-0.590609\pi$
$607$	−8399.51	−0.561657	−0.280828	+	0.959758i	$0.590609\pi$
$608$	333.421	0.0222401
$609$	0	0
$610$	10033.7	0.665985
$611$	11430.4	0.756832
$612$	0	0
$613$	15520.4	1.02262	0.511308	−	0.859398i	$-0.329161\pi$
$613$	15520.4	1.02262	0.511308	+	0.859398i	$0.329161\pi$
$614$	15808.9	1.03908
$615$	0	0
$616$	0	0
$617$	26665.1	1.73987	0.869933	−	0.493169i	$-0.164162\pi$
$617$	26665.1	1.73987	0.869933	+	0.493169i	$0.164162\pi$
$618$	0	0
$619$	−12231.6	−0.794229	−0.397114	−	0.917769i	$-0.629988\pi$
$619$	−12231.6	−0.794229	−0.397114	+	0.917769i	$0.629988\pi$
$620$	193.489	0.0125334
$621$	0	0
$622$	14902.1	0.960645
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	16602.2	1.06000
$627$	0	0
$628$	−774.255	−0.0491977
$629$	4822.34	0.305690
$630$	0	0
$631$	−7237.27	−0.456595	−0.228297	−	0.973591i	$-0.573316\pi$
$631$	−7237.27	−0.456595	−0.228297	+	0.973591i	$0.573316\pi$
$632$	−18696.7	−1.17676
$633$	0	0
$634$	−9163.68	−0.574032
$635$	−2083.20	−0.130188
$636$	0	0
$637$	0	0
$638$	−26843.5	−1.66575
$639$	0	0
$640$	−7620.42	−0.470662
$641$	−4904.83	−0.302230	−0.151115	−	0.988516i	$-0.548286\pi$
$641$	−4904.83	−0.302230	−0.151115	+	0.988516i	$0.548286\pi$
$642$	0	0
$643$	−8394.46	−0.514845	−0.257422	−	0.966299i	$-0.582873\pi$
$643$	−8394.46	−0.514845	−0.257422	+	0.966299i	$0.582873\pi$
$644$	0	0
$645$	0	0
$646$	4387.19	0.267201
$647$	−9526.45	−0.578861	−0.289431	−	0.957199i	$-0.593466\pi$
$647$	−9526.45	−0.578861	−0.289431	+	0.957199i	$0.593466\pi$
$648$	0	0
$649$	32146.0	1.94429
$650$	2236.54	0.134961
$651$	0	0
$652$	−738.345	−0.0443494
$653$	−15950.1	−0.955856	−0.477928	−	0.878399i	$-0.658612\pi$
$653$	−15950.1	−0.955856	−0.477928	+	0.878399i	$0.658612\pi$
$654$	0	0
$655$	2052.48	0.122438
$656$	−15613.6	−0.929279
$657$	0	0
$658$	0	0
$659$	3370.65	0.199244	0.0996221	−	0.995025i	$-0.468237\pi$
$659$	3370.65	0.199244	0.0996221	+	0.995025i	$0.468237\pi$
$660$	0	0
$661$	24254.0	1.42719	0.713594	−	0.700560i	$-0.247066\pi$
$661$	24254.0	1.42719	0.713594	+	0.700560i	$0.247066\pi$
$662$	26599.1	1.56164
$663$	0	0
$664$	32022.0	1.87153
$665$	0	0
$666$	0	0
$667$	−31055.9	−1.80283
$668$	137.427	0.00795991
$669$	0	0
$670$	−33.6070	−0.00193784
$671$	−32382.3	−1.86305
$672$	0	0
$673$	−3510.41	−0.201064	−0.100532	−	0.994934i	$-0.532055\pi$
$673$	−3510.41	−0.201064	−0.100532	+	0.994934i	$0.532055\pi$
$674$	−9390.20	−0.536642
$675$	0	0
$676$	−369.051	−0.0209974
$677$	−9051.19	−0.513834	−0.256917	−	0.966434i	$-0.582707\pi$
$677$	−9051.19	−0.513834	−0.256917	+	0.966434i	$0.582707\pi$
$678$	0	0
$679$	0	0
$680$	−6860.71	−0.386906
$681$	0	0
$682$	−17310.5	−0.971927
$683$	−7105.73	−0.398087	−0.199043	−	0.979991i	$-0.563783\pi$
$683$	−7105.73	−0.398087	−0.199043	+	0.979991i	$0.563783\pi$
$684$	0	0
$685$	8809.87	0.491398
$686$	0	0
$687$	0	0
$688$	18458.6	1.02286
$689$	5266.21	0.291185
$690$	0	0
$691$	−12629.4	−0.695289	−0.347645	−	0.937626i	$-0.613018\pi$
$691$	−12629.4	−0.695289	−0.347645	+	0.937626i	$0.613018\pi$
$692$	−1347.70	−0.0740346
$693$	0	0
$694$	−5332.86	−0.291690
$695$	4328.59	0.236249
$696$	0	0
$697$	−14564.8	−0.791506
$698$	3175.11	0.172177
$699$	0	0
$700$	0	0
$701$	912.952	0.0491893	0.0245947	−	0.999698i	$-0.492170\pi$
$701$	912.952	0.0491893	0.0245947	+	0.999698i	$0.492170\pi$
$702$	0	0
$703$	1919.63	0.102988
$704$	22845.7	1.22305
$705$	0	0
$706$	−7692.63	−0.410079
$707$	0	0
$708$	0	0
$709$	−20710.5	−1.09704	−0.548518	−	0.836139i	$-0.684808\pi$
$709$	−20710.5	−1.09704	−0.548518	+	0.836139i	$0.684808\pi$
$710$	−12482.8	−0.659818
$711$	0	0
$712$	31902.6	1.67921
$713$	−20026.9	−1.05191
$714$	0	0
$715$	−7218.14	−0.377543
$716$	−586.842	−0.0306303
$717$	0	0
$718$	−1254.48	−0.0652045
$719$	−2469.76	−0.128104	−0.0640518	−	0.997947i	$-0.520402\pi$
$719$	−2469.76	−0.128104	−0.0640518	+	0.997947i	$0.520402\pi$
$720$	0	0
$721$	0	0
$722$	−18013.4	−0.928520
$723$	0	0
$724$	−1091.43	−0.0560256
$725$	−5010.89	−0.256690
$726$	0	0
$727$	10893.4	0.555729	0.277865	−	0.960620i	$-0.410373\pi$
$727$	10893.4	0.555729	0.277865	+	0.960620i	$0.410373\pi$
$728$	0	0
$729$	0	0
$730$	10838.4	0.549519
$731$	17218.7	0.871214
$732$	0	0
$733$	26259.9	1.32324	0.661619	−	0.749840i	$-0.269870\pi$
$733$	26259.9	1.32324	0.661619	+	0.749840i	$0.269870\pi$
$734$	−7129.14	−0.358503
$735$	0	0
$736$	−2098.21	−0.105083
$737$	108.462	0.00542096
$738$	0	0
$739$	−1840.83	−0.0916319	−0.0458160	−	0.998950i	$-0.514589\pi$
$739$	−1840.83	−0.0916319	−0.0458160	+	0.998950i	$0.514589\pi$
$740$	116.712	0.00579788
$741$	0	0
$742$	0	0
$743$	−4022.25	−0.198603	−0.0993015	−	0.995057i	$-0.531661\pi$
$743$	−4022.25	−0.198603	−0.0993015	+	0.995057i	$0.531661\pi$
$744$	0	0
$745$	15096.2	0.742390
$746$	15666.7	0.768898
$747$	0	0
$748$	−860.862	−0.0420805
$749$	0	0
$750$	0	0
$751$	25725.3	1.24997	0.624986	−	0.780636i	$-0.285105\pi$
$751$	25725.3	1.24997	0.624986	+	0.780636i	$0.285105\pi$
$752$	−24406.0	−1.18351
$753$	0	0
$754$	−17931.3	−0.866074
$755$	−2709.48	−0.130607
$756$	0	0
$757$	11359.2	0.545385	0.272692	−	0.962101i	$-0.412086\pi$
$757$	11359.2	0.545385	0.272692	+	0.962101i	$0.412086\pi$
$758$	30528.8	1.46287
$759$	0	0
$760$	−2731.05	−0.130350
$761$	−7843.37	−0.373616	−0.186808	−	0.982396i	$-0.559814\pi$
$761$	−7843.37	−0.373616	−0.186808	+	0.982396i	$0.559814\pi$
$762$	0	0
$763$	0	0
$764$	−752.806	−0.0356487
$765$	0	0
$766$	13457.6	0.634781
$767$	21473.3	1.01090
$768$	0	0
$769$	29007.8	1.36027	0.680136	−	0.733086i	$-0.261921\pi$
$769$	29007.8	1.36027	0.680136	+	0.733086i	$0.261921\pi$
$770$	0	0
$771$	0	0
$772$	655.479	0.0305586
$773$	679.160	0.0316012	0.0158006	−	0.999875i	$-0.494970\pi$
$773$	679.160	0.0316012	0.0158006	+	0.999875i	$0.494970\pi$
$774$	0	0
$775$	−3231.36	−0.149773
$776$	520.245	0.0240666
$777$	0	0
$778$	−10249.2	−0.472304
$779$	−5797.81	−0.266660
$780$	0	0
$781$	40286.5	1.84579
$782$	−27608.5	−1.26250
$783$	0	0
$784$	0	0
$785$	12930.4	0.587907
$786$	0	0
$787$	−24329.0	−1.10195	−0.550976	−	0.834521i	$-0.685744\pi$
$787$	−24329.0	−1.10195	−0.550976	+	0.834521i	$0.685744\pi$
$788$	1163.59	0.0526029
$789$	0	0
$790$	−12139.7	−0.546724
$791$	0	0
$792$	0	0
$793$	−21631.2	−0.968657
$794$	−26459.5	−1.18264
$795$	0	0
$796$	−290.219	−0.0129228
$797$	−2791.24	−0.124054	−0.0620269	−	0.998074i	$-0.519756\pi$
$797$	−2791.24	−0.124054	−0.0620269	+	0.998074i	$0.519756\pi$
$798$	0	0
$799$	−22766.6	−1.00804
$800$	−338.548	−0.0149619
$801$	0	0
$802$	26522.7	1.16777
$803$	−34979.6	−1.53724
$804$	0	0
$805$	0	0
$806$	−11563.3	−0.505336
$807$	0	0
$808$	32753.4	1.42607
$809$	−15695.9	−0.682126	−0.341063	−	0.940040i	$-0.610787\pi$
$809$	−15695.9	−0.682126	−0.341063	+	0.940040i	$0.610787\pi$
$810$	0	0
$811$	−9580.84	−0.414832	−0.207416	−	0.978253i	$-0.566505\pi$
$811$	−9580.84	−0.414832	−0.207416	+	0.978253i	$0.566505\pi$
$812$	0	0
$813$	0	0
$814$	−10441.7	−0.449608
$815$	12330.7	0.529971
$816$	0	0
$817$	6854.28	0.293514
$818$	−22049.2	−0.942458
$819$	0	0
$820$	−352.503	−0.0150121
$821$	−27541.0	−1.17075	−0.585376	−	0.810762i	$-0.699053\pi$
$821$	−27541.0	−1.17075	−0.585376	+	0.810762i	$0.699053\pi$
$822$	0	0
$823$	−11746.5	−0.497516	−0.248758	−	0.968566i	$-0.580022\pi$
$823$	−11746.5	−0.497516	−0.248758	+	0.968566i	$0.580022\pi$
$824$	22393.9	0.946757
$825$	0	0
$826$	0	0
$827$	20831.7	0.875924	0.437962	−	0.898994i	$-0.355700\pi$
$827$	20831.7	0.875924	0.437962	+	0.898994i	$0.355700\pi$
$828$	0	0
$829$	−3590.00	−0.150405	−0.0752027	−	0.997168i	$-0.523960\pi$
$829$	−3590.00	−0.150405	−0.0752027	+	0.997168i	$0.523960\pi$
$830$	20791.9	0.869513
$831$	0	0
$832$	15260.8	0.635904
$833$	0	0
$834$	0	0
$835$	−2295.10	−0.0951201
$836$	−342.684	−0.0141770
$837$	0	0
$838$	1068.41	0.0440425
$839$	−10917.2	−0.449229	−0.224614	−	0.974448i	$-0.572112\pi$
$839$	−10917.2	−0.449229	−0.224614	+	0.974448i	$0.572112\pi$
$840$	0	0
$841$	15785.5	0.647239
$842$	3520.31	0.144083
$843$	0	0
$844$	922.036	0.0376040
$845$	6163.33	0.250917
$846$	0	0
$847$	0	0
$848$	−11244.4	−0.455345
$849$	0	0
$850$	−4454.66	−0.179757
$851$	−12080.2	−0.486609
$852$	0	0
$853$	−35912.9	−1.44154	−0.720770	−	0.693175i	$-0.756211\pi$
$853$	−35912.9	−1.44154	−0.720770	+	0.693175i	$0.756211\pi$
$854$	0	0
$855$	0	0
$856$	31468.8	1.25652
$857$	−39353.3	−1.56859	−0.784295	−	0.620388i	$-0.786975\pi$
$857$	−39353.3	−1.56859	−0.784295	+	0.620388i	$0.786975\pi$
$858$	0	0
$859$	−32234.7	−1.28037	−0.640183	−	0.768222i	$-0.721142\pi$
$859$	−32234.7	−1.28037	−0.640183	+	0.768222i	$0.721142\pi$
$860$	416.736	0.0165239
$861$	0	0
$862$	43008.2	1.69938
$863$	−43811.4	−1.72811	−0.864053	−	0.503401i	$-0.832082\pi$
$863$	−43811.4	−1.72811	−0.864053	+	0.503401i	$0.832082\pi$
$864$	0	0
$865$	22507.3	0.884706
$866$	13091.9	0.513718
$867$	0	0
$868$	0	0
$869$	39179.3	1.52942
$870$	0	0
$871$	72.4519	0.00281853
$872$	19474.8	0.756308
$873$	0	0
$874$	−10990.1	−0.425340
$875$	0	0
$876$	0	0
$877$	−9140.68	−0.351948	−0.175974	−	0.984395i	$-0.556308\pi$
$877$	−9140.68	−0.351948	−0.175974	+	0.984395i	$0.556308\pi$
$878$	27172.0	1.04443
$879$	0	0
$880$	15412.1	0.590387
$881$	23013.1	0.880058	0.440029	−	0.897984i	$-0.354968\pi$
$881$	23013.1	0.880058	0.440029	+	0.897984i	$0.354968\pi$
$882$	0	0
$883$	7448.15	0.283862	0.141931	−	0.989877i	$-0.454669\pi$
$883$	7448.15	0.283862	0.141931	+	0.989877i	$0.454669\pi$
$884$	−575.050	−0.0218790
$885$	0	0
$886$	15967.5	0.605462
$887$	46061.1	1.74361	0.871803	−	0.489856i	$-0.162951\pi$
$887$	46061.1	1.74361	0.871803	+	0.489856i	$0.162951\pi$
$888$	0	0
$889$	0	0
$890$	20714.3	0.780163
$891$	0	0
$892$	48.5024	0.00182061
$893$	−9062.74	−0.339612
$894$	0	0
$895$	9800.55	0.366029
$896$	0	0
$897$	0	0
$898$	−46991.5	−1.74624
$899$	25907.2	0.961129
$900$	0	0
$901$	−10489.0	−0.387837
$902$	31536.7	1.16414
$903$	0	0
$904$	−33279.1	−1.22439
$905$	18227.3	0.669500
$906$	0	0
$907$	38496.1	1.40931	0.704653	−	0.709552i	$-0.251103\pi$
$907$	38496.1	1.40931	0.704653	+	0.709552i	$0.251103\pi$
$908$	−113.755	−0.00415758
$909$	0	0
$910$	0	0
$911$	18329.1	0.666596	0.333298	−	0.942822i	$-0.391838\pi$
$911$	18329.1	0.666596	0.333298	+	0.942822i	$0.391838\pi$
$912$	0	0
$913$	−67102.9	−2.43240
$914$	−40997.8	−1.48368
$915$	0	0
$916$	−1431.55	−0.0516371
$917$	0	0
$918$	0	0
$919$	−37448.4	−1.34419	−0.672094	−	0.740466i	$-0.734605\pi$
$919$	−37448.4	−1.34419	−0.672094	+	0.740466i	$0.734605\pi$
$920$	17186.5	0.615892
$921$	0	0
$922$	14291.8	0.510494
$923$	26911.1	0.959686
$924$	0	0
$925$	−1949.15	−0.0692841
$926$	44594.5	1.58258
$927$	0	0
$928$	2714.29	0.0960138
$929$	−2946.34	−0.104054	−0.0520271	−	0.998646i	$-0.516568\pi$
$929$	−2946.34	−0.104054	−0.0520271	+	0.998646i	$0.516568\pi$
$930$	0	0
$931$	0	0
$932$	755.827	0.0265643
$933$	0	0
$934$	42681.9	1.49528
$935$	14376.8	0.502858
$936$	0	0
$937$	−48870.0	−1.70386	−0.851928	−	0.523659i	$-0.824566\pi$
$937$	−48870.0	−1.70386	−0.851928	+	0.523659i	$0.824566\pi$
$938$	0	0
$939$	0	0
$940$	−551.009	−0.0191191
$941$	20661.0	0.715758	0.357879	−	0.933768i	$-0.383500\pi$
$941$	20661.0	0.715758	0.357879	+	0.933768i	$0.383500\pi$
$942$	0	0
$943$	36485.5	1.25995
$944$	−45849.6	−1.58080
$945$	0	0
$946$	−37283.3	−1.28138
$947$	−13130.3	−0.450557	−0.225279	−	0.974294i	$-0.572329\pi$
$947$	−13130.3	−0.450557	−0.225279	+	0.974294i	$0.572329\pi$
$948$	0	0
$949$	−23366.2	−0.799260
$950$	−1773.27	−0.0605605
$951$	0	0
$952$	0	0
$953$	−16098.9	−0.547214	−0.273607	−	0.961842i	$-0.588217\pi$
$953$	−16098.9	−0.547214	−0.273607	+	0.961842i	$0.588217\pi$
$954$	0	0
$955$	12572.2	0.425998
$956$	−33.9665	−0.00114912
$957$	0	0
$958$	26212.9	0.884030
$959$	0	0
$960$	0	0
$961$	−13084.3	−0.439202
$962$	−6974.98	−0.233766
$963$	0	0
$964$	2013.52	0.0672728
$965$	−10946.8	−0.365172
$966$	0	0
$967$	15916.9	0.529320	0.264660	−	0.964342i	$-0.414740\pi$
$967$	15916.9	0.529320	0.264660	+	0.964342i	$0.414740\pi$
$968$	−18416.2	−0.611486
$969$	0	0
$970$	337.795	0.0111814
$971$	−675.909	−0.0223388	−0.0111694	−	0.999938i	$-0.503555\pi$
$971$	−675.909	−0.0223388	−0.0111694	+	0.999938i	$0.503555\pi$
$972$	0	0
$973$	0	0
$974$	−43188.3	−1.42078
$975$	0	0
$976$	46186.6	1.51475
$977$	46169.3	1.51186	0.755931	−	0.654652i	$-0.227185\pi$
$977$	46169.3	1.51186	0.755931	+	0.654652i	$0.227185\pi$
$978$	0	0
$979$	−66852.7	−2.18245
$980$	0	0
$981$	0	0
$982$	−23748.8	−0.771745
$983$	50336.2	1.63324	0.816620	−	0.577175i	$-0.195845\pi$
$983$	50336.2	1.63324	0.816620	+	0.577175i	$0.195845\pi$
$984$	0	0
$985$	−19432.5	−0.628599
$986$	35714.9	1.15354
$987$	0	0
$988$	−228.911	−0.00737108
$989$	−43133.8	−1.38683
$990$	0	0
$991$	40186.9	1.28817	0.644087	−	0.764953i	$-0.277238\pi$
$991$	40186.9	1.28817	0.644087	+	0.764953i	$0.277238\pi$
$992$	1750.36	0.0560220
$993$	0	0
$994$	0	0
$995$	4846.80	0.154426
$996$	0	0
$997$	17750.2	0.563846	0.281923	−	0.959437i	$-0.409028\pi$
$997$	17750.2	0.563846	0.281923	+	0.959437i	$0.409028\pi$
$998$	26583.2	0.843162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bz.1.5		6
3.2	odd	2		245.4.a.o.1.2	✓	6
7.6	odd	2		2205.4.a.ca.1.5		6
15.14	odd	2		1225.4.a.bj.1.5		6
21.2	odd	6		245.4.e.q.116.5		12
21.5	even	6		245.4.e.p.116.5		12
21.11	odd	6		245.4.e.q.226.5		12
21.17	even	6		245.4.e.p.226.5		12
21.20	even	2		245.4.a.p.1.2	yes	6
105.104	even	2		1225.4.a.bi.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.2	✓	6	3.2	odd	2
245.4.a.p.1.2	yes	6	21.20	even	2
245.4.e.p.116.5		12	21.5	even	6
245.4.e.p.226.5		12	21.17	even	6
245.4.e.q.116.5		12	21.2	odd	6
245.4.e.q.226.5		12	21.11	odd	6
1225.4.a.bi.1.5		6	105.104	even	2
1225.4.a.bj.1.5		6	15.14	odd	2
2205.4.a.bz.1.5		6	1.1	even	1	trivial
2205.4.a.ca.1.5		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)

\(f(q)\)	\(=\)	\(q+2.88087 q^{2} +0.299392 q^{4} -5.00000 q^{5} -22.1844 q^{8} +O(q^{10})\)	\(q+2.88087 q^{2} +0.299392 q^{4} -5.00000 q^{5} -22.1844 q^{8} -14.4043 q^{10} +46.4881 q^{11} +31.0537 q^{13} -66.3055 q^{16} -61.8516 q^{17} -24.6214 q^{19} -1.49696 q^{20} +133.926 q^{22} +154.942 q^{23} +25.0000 q^{25} +89.4616 q^{26} -200.436 q^{29} -129.255 q^{31} -13.5419 q^{32} -178.186 q^{34} -77.9662 q^{37} -70.9309 q^{38} +110.922 q^{40} +235.479 q^{41} -278.388 q^{43} +13.9182 q^{44} +446.366 q^{46} +368.085 q^{47} +72.0217 q^{50} +9.29725 q^{52} +169.584 q^{53} -232.440 q^{55} -577.429 q^{58} +691.490 q^{59} -696.572 q^{61} -372.365 q^{62} +491.432 q^{64} -155.269 q^{65} +2.33311 q^{67} -18.5179 q^{68} +866.599 q^{71} -752.443 q^{73} -224.610 q^{74} -7.37145 q^{76} +842.783 q^{79} +331.528 q^{80} +678.384 q^{82} -1443.44 q^{83} +309.258 q^{85} -801.998 q^{86} -1031.31 q^{88} -1438.06 q^{89} +46.3884 q^{92} +1060.40 q^{94} +123.107 q^{95} -23.4509 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8}+O(q^{10}) \) 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

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