LMFDB - Embedded newform 2205.4.a.bv.1.3

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 = [5,1,0,21,-25,0,0,69,0,-5,33,0,23] 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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 30x^{3} + 2x^{2} + 164x + 84 $ x^5 - x^4 - 30x^3 + 2x^2 + 164*x + 84
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.544516$ 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)

$f(q)$	$=$	$q-0.544516 q^{2} -7.70350 q^{4} -5.00000 q^{5} +8.55081 q^{8} +2.72258 q^{10} -0.365737 q^{11} -72.4570 q^{13} +56.9720 q^{16} -127.759 q^{17} -129.555 q^{19} +38.5175 q^{20} +0.199150 q^{22} +148.982 q^{23} +25.0000 q^{25} +39.4540 q^{26} -21.4889 q^{29} -25.4269 q^{31} -99.4286 q^{32} +69.5670 q^{34} -148.333 q^{37} +70.5446 q^{38} -42.7541 q^{40} -213.313 q^{41} -285.923 q^{43} +2.81745 q^{44} -81.1233 q^{46} -523.711 q^{47} -13.6129 q^{50} +558.172 q^{52} -218.096 q^{53} +1.82868 q^{55} +11.7011 q^{58} +140.369 q^{59} +474.714 q^{61} +13.8453 q^{62} -401.635 q^{64} +362.285 q^{65} -391.407 q^{67} +984.194 q^{68} -226.869 q^{71} +677.064 q^{73} +80.7697 q^{74} +998.025 q^{76} +330.923 q^{79} -284.860 q^{80} +116.152 q^{82} -1273.24 q^{83} +638.796 q^{85} +155.690 q^{86} -3.12734 q^{88} -1047.36 q^{89} -1147.69 q^{92} +285.169 q^{94} +647.773 q^{95} -1545.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 21 q^{4} - 25 q^{5} + 69 q^{8} - 5 q^{10} + 33 q^{11} + 23 q^{13} + 113 q^{16} - 136 q^{17} + 39 q^{19} - 105 q^{20} - 87 q^{22} + 133 q^{23} + 125 q^{25} - 73 q^{26} - 272 q^{29} + 430 q^{31}+ \cdots + 2168 q^{97}+O(q^{100}) $ 5 * q + q^2 + 21 * q^4 - 25 * q^5 + 69 * q^8 - 5 * q^10 + 33 * q^11 + 23 * q^13 + 113 * q^16 - 136 * q^17 + 39 * q^19 - 105 * q^20 - 87 * q^22 + 133 * q^23 + 125 * q^25 - 73 * q^26 - 272 * q^29 + 430 * q^31 + 573 * q^32 + 372 * q^34 + 3 * q^37 - 837 * q^38 - 345 * q^40 - 627 * q^41 + 108 * q^43 + 1809 * q^44 + 1637 * q^46 - 553 * q^47 + 25 * q^50 + 1047 * q^52 + 1135 * q^53 - 165 * q^55 - 2564 * q^58 - 332 * q^59 + 584 * q^61 - 1562 * q^62 + 1137 * q^64 - 115 * q^65 + 412 * q^67 + 1712 * q^68 + 142 * q^71 + 2074 * q^73 - 605 * q^74 - 9 * q^76 + 28 * q^79 - 565 * q^80 - 1515 * q^82 - 840 * q^83 + 680 * q^85 - 40 * q^86 + 4181 * q^88 - 2978 * q^89 - 531 * q^92 + 843 * q^94 - 195 * q^95 + 2168 * 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 $

$2$	−0.544516	−0.192516	−0.0962578	−	0.995356i	$-0.530687\pi$
$2$	−0.544516	−0.192516	−0.0962578	+	0.995356i	$0.530687\pi$
$3$	0	0
$3$	0	0
$4$	−7.70350	−0.962938
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.55081	0.377896
$9$	0	0
$10$	2.72258	0.0860956
$11$	−0.365737	−0.0100249	−0.00501244	−	0.999987i	$-0.501596\pi$
$11$	−0.365737	−0.0100249	−0.00501244	+	0.999987i	$0.501596\pi$
$12$	0	0
$13$	−72.4570	−1.54584	−0.772921	−	0.634502i	$-0.781205\pi$
$13$	−72.4570	−1.54584	−0.772921	+	0.634502i	$0.781205\pi$
$14$	0	0
$15$	0	0
$16$	56.9720	0.890187
$17$	−127.759	−1.82272	−0.911358	−	0.411615i	$-0.864965\pi$
$17$	−127.759	−1.82272	−0.911358	+	0.411615i	$0.864965\pi$
$18$	0	0
$19$	−129.555	−1.56431	−0.782155	−	0.623084i	$-0.785879\pi$
$19$	−129.555	−1.56431	−0.782155	+	0.623084i	$0.785879\pi$
$20$	38.5175	0.430639
$21$	0	0
$22$	0.199150	0.00192995
$23$	148.982	1.35065	0.675325	−	0.737520i	$-0.264003\pi$
$23$	148.982	1.35065	0.675325	+	0.737520i	$0.264003\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	39.4540	0.297599
$27$	0	0
$28$	0	0
$29$	−21.4889	−0.137600	−0.0687999	−	0.997630i	$-0.521917\pi$
$29$	−21.4889	−0.137600	−0.0687999	+	0.997630i	$0.521917\pi$
$30$	0	0
$31$	−25.4269	−0.147316	−0.0736581	−	0.997284i	$-0.523467\pi$
$31$	−25.4269	−0.147316	−0.0736581	+	0.997284i	$0.523467\pi$
$32$	−99.4286	−0.549271
$33$	0	0
$34$	69.5670	0.350901
$35$	0	0
$36$	0	0
$37$	−148.333	−0.659076	−0.329538	−	0.944142i	$-0.606893\pi$
$37$	−148.333	−0.659076	−0.329538	+	0.944142i	$0.606893\pi$
$38$	70.5446	0.301154
$39$	0	0
$40$	−42.7541	−0.169000
$41$	−213.313	−0.812534	−0.406267	−	0.913755i	$-0.633170\pi$
$41$	−213.313	−0.812534	−0.406267	+	0.913755i	$0.633170\pi$
$42$	0	0
$43$	−285.923	−1.01402	−0.507010	−	0.861940i	$-0.669249\pi$
$43$	−285.923	−1.01402	−0.507010	+	0.861940i	$0.669249\pi$
$44$	2.81745	0.00965334
$45$	0	0
$46$	−81.1233	−0.260021
$47$	−523.711	−1.62534	−0.812671	−	0.582723i	$-0.801987\pi$
$47$	−523.711	−1.62534	−0.812671	+	0.582723i	$0.801987\pi$
$48$	0	0
$49$	0	0
$50$	−13.6129	−0.0385031
$51$	0	0
$52$	558.172	1.48855
$53$	−218.096	−0.565241	−0.282621	−	0.959232i	$-0.591204\pi$
$53$	−218.096	−0.565241	−0.282621	+	0.959232i	$0.591204\pi$
$54$	0	0
$55$	1.82868	0.00448327
$56$	0	0
$57$	0	0
$58$	11.7011	0.0264901
$59$	140.369	0.309736	0.154868	−	0.987935i	$-0.450505\pi$
$59$	140.369	0.309736	0.154868	+	0.987935i	$0.450505\pi$
$60$	0	0
$61$	474.714	0.996407	0.498204	−	0.867060i	$-0.333993\pi$
$61$	474.714	0.996407	0.498204	+	0.867060i	$0.333993\pi$
$62$	13.8453	0.0283606
$63$	0	0
$64$	−401.635	−0.784444
$65$	362.285	0.691322
$66$	0	0
$67$	−391.407	−0.713701	−0.356850	−	0.934162i	$-0.616149\pi$
$67$	−391.407	−0.713701	−0.356850	+	0.934162i	$0.616149\pi$
$68$	984.194	1.75516
$69$	0	0
$70$	0	0
$71$	−226.869	−0.379217	−0.189609	−	0.981860i	$-0.560722\pi$
$71$	−226.869	−0.379217	−0.189609	+	0.981860i	$0.560722\pi$
$72$	0	0
$73$	677.064	1.08554	0.542770	−	0.839882i	$-0.317376\pi$
$73$	677.064	1.08554	0.542770	+	0.839882i	$0.317376\pi$
$74$	80.7697	0.126882
$75$	0	0
$76$	998.025	1.50633
$77$	0	0
$78$	0	0
$79$	330.923	0.471288	0.235644	−	0.971839i	$-0.424280\pi$
$79$	330.923	0.471288	0.235644	+	0.971839i	$0.424280\pi$
$80$	−284.860	−0.398104
$81$	0	0
$82$	116.152	0.156425
$83$	−1273.24	−1.68380	−0.841902	−	0.539630i	$-0.818564\pi$
$83$	−1273.24	−1.68380	−0.841902	+	0.539630i	$0.818564\pi$
$84$	0	0
$85$	638.796	0.815143
$86$	155.690	0.195215
$87$	0	0
$88$	−3.12734	−0.00378836
$89$	−1047.36	−1.24741	−0.623706	−	0.781659i	$-0.714374\pi$
$89$	−1047.36	−1.24741	−0.623706	+	0.781659i	$0.714374\pi$
$90$	0	0
$91$	0	0
$92$	−1147.69	−1.30059
$93$	0	0
$94$	285.169	0.312904
$95$	647.773	0.699581
$96$	0	0
$97$	−1545.00	−1.61723	−0.808616	−	0.588337i	$-0.799783\pi$
$97$	−1545.00	−1.61723	−0.808616	+	0.588337i	$0.799783\pi$
$98$	0	0
$99$	0	0
$100$	−192.588	−0.192588
$101$	−1824.30	−1.79728	−0.898639	−	0.438689i	$-0.855443\pi$
$101$	−1824.30	−1.79728	−0.898639	+	0.438689i	$0.855443\pi$
$102$	0	0
$103$	1337.01	1.27903	0.639513	−	0.768780i	$-0.279136\pi$
$103$	1337.01	1.27903	0.639513	+	0.768780i	$0.279136\pi$
$104$	−619.566	−0.584168
$105$	0	0
$106$	118.757	0.108818
$107$	−378.231	−0.341729	−0.170865	−	0.985295i	$-0.554656\pi$
$107$	−378.231	−0.341729	−0.170865	+	0.985295i	$0.554656\pi$
$108$	0	0
$109$	969.429	0.851876	0.425938	−	0.904752i	$-0.359944\pi$
$109$	969.429	0.851876	0.425938	+	0.904752i	$0.359944\pi$
$110$	−0.995748	−0.000863098 0
$111$	0	0
$112$	0	0
$113$	−643.685	−0.535865	−0.267933	−	0.963438i	$-0.586340\pi$
$113$	−643.685	−0.535865	−0.267933	+	0.963438i	$0.586340\pi$
$114$	0	0
$115$	−744.912	−0.604029
$116$	165.540	0.132500
$117$	0	0
$118$	−76.4330	−0.0596290
$119$	0	0
$120$	0	0
$121$	−1330.87	−0.999900
$122$	−258.489	−0.191824
$123$	0	0
$124$	195.876	0.141856
$125$	−125.000	−0.0894427
$126$	0	0
$127$	228.960	0.159976	0.0799879	−	0.996796i	$-0.474512\pi$
$127$	228.960	0.159976	0.0799879	+	0.996796i	$0.474512\pi$
$128$	1014.13	0.700288
$129$	0	0
$130$	−197.270	−0.133090
$131$	941.820	0.628147	0.314073	−	0.949399i	$-0.398306\pi$
$131$	941.820	0.628147	0.314073	+	0.949399i	$0.398306\pi$
$132$	0	0
$133$	0	0
$134$	213.127	0.137399
$135$	0	0
$136$	−1092.45	−0.688797
$137$	−2912.31	−1.81617	−0.908086	−	0.418785i	$-0.862456\pi$
$137$	−2912.31	−1.81617	−0.908086	+	0.418785i	$0.862456\pi$
$138$	0	0
$139$	−430.438	−0.262657	−0.131328	−	0.991339i	$-0.541924\pi$
$139$	−430.438	−0.262657	−0.131328	+	0.991339i	$0.541924\pi$
$140$	0	0
$141$	0	0
$142$	123.534	0.0730052
$143$	26.5002	0.0154969
$144$	0	0
$145$	107.445	0.0615365
$146$	−368.672	−0.208983
$147$	0	0
$148$	1142.68	0.634649
$149$	547.659	0.301114	0.150557	−	0.988601i	$-0.451893\pi$
$149$	547.659	0.301114	0.150557	+	0.988601i	$0.451893\pi$
$150$	0	0
$151$	1540.36	0.830149	0.415075	−	0.909787i	$-0.363755\pi$
$151$	1540.36	0.830149	0.415075	+	0.909787i	$0.363755\pi$
$152$	−1107.80	−0.591146
$153$	0	0
$154$	0	0
$155$	127.134	0.0658818
$156$	0	0
$157$	2416.86	1.22857	0.614287	−	0.789082i	$-0.289444\pi$
$157$	2416.86	1.22857	0.614287	+	0.789082i	$0.289444\pi$
$158$	−180.193	−0.0907303
$159$	0	0
$160$	497.143	0.245641
$161$	0	0
$162$	0	0
$163$	−1265.17	−0.607952	−0.303976	−	0.952680i	$-0.598314\pi$
$163$	−1265.17	−0.607952	−0.303976	+	0.952680i	$0.598314\pi$
$164$	1643.26	0.782419
$165$	0	0
$166$	693.298	0.324159
$167$	−1503.92	−0.696867	−0.348433	−	0.937334i	$-0.613286\pi$
$167$	−1503.92	−0.696867	−0.348433	+	0.937334i	$0.613286\pi$
$168$	0	0
$169$	3053.01	1.38963
$170$	−347.835	−0.156928
$171$	0	0
$172$	2202.61	0.976438
$173$	1707.41	0.750357	0.375178	−	0.926953i	$-0.377581\pi$
$173$	1707.41	0.750357	0.375178	+	0.926953i	$0.377581\pi$
$174$	0	0
$175$	0	0
$176$	−20.8367	−0.00892402
$177$	0	0
$178$	570.303	0.240146
$179$	158.344	0.0661184	0.0330592	−	0.999453i	$-0.489475\pi$
$179$	158.344	0.0661184	0.0330592	+	0.999453i	$0.489475\pi$
$180$	0	0
$181$	−1998.34	−0.820637	−0.410318	−	0.911942i	$-0.634582\pi$
$181$	−1998.34	−0.820637	−0.410318	+	0.911942i	$0.634582\pi$
$182$	0	0
$183$	0	0
$184$	1273.92	0.510405
$185$	741.665	0.294748
$186$	0	0
$187$	46.7262	0.0182725
$188$	4034.41	1.56510
$189$	0	0
$190$	−352.723	−0.134680
$191$	639.821	0.242386	0.121193	−	0.992629i	$-0.461328\pi$
$191$	639.821	0.242386	0.121193	+	0.992629i	$0.461328\pi$
$192$	0	0
$193$	−1182.58	−0.441058	−0.220529	−	0.975380i	$-0.570778\pi$
$193$	−1182.58	−0.441058	−0.220529	+	0.975380i	$0.570778\pi$
$194$	841.280	0.311342
$195$	0	0
$196$	0	0
$197$	4618.92	1.67048	0.835240	−	0.549885i	$-0.185328\pi$
$197$	4618.92	1.67048	0.835240	+	0.549885i	$0.185328\pi$
$198$	0	0
$199$	2137.99	0.761600	0.380800	−	0.924658i	$-0.375649\pi$
$199$	2137.99	0.761600	0.380800	+	0.924658i	$0.375649\pi$
$200$	213.770	0.0755792
$201$	0	0
$202$	993.363	0.346004
$203$	0	0
$204$	0	0
$205$	1066.56	0.363376
$206$	−728.025	−0.246232
$207$	0	0
$208$	−4128.02	−1.37609
$209$	47.3829	0.0156820
$210$	0	0
$211$	−3877.98	−1.26527	−0.632633	−	0.774452i	$-0.718026\pi$
$211$	−3877.98	−1.26527	−0.632633	+	0.774452i	$0.718026\pi$
$212$	1680.10	0.544292
$213$	0	0
$214$	205.953	0.0657882
$215$	1429.61	0.453483
$216$	0	0
$217$	0	0
$218$	−527.870	−0.163999
$219$	0	0
$220$	−14.0873	−0.00431711
$221$	9257.05	2.81763
$222$	0	0
$223$	−1458.05	−0.437839	−0.218919	−	0.975743i	$-0.570253\pi$
$223$	−1458.05	−0.437839	−0.218919	+	0.975743i	$0.570253\pi$
$224$	0	0
$225$	0	0
$226$	350.497	0.103162
$227$	1812.96	0.530090	0.265045	−	0.964236i	$-0.414613\pi$
$227$	1812.96	0.530090	0.265045	+	0.964236i	$0.414613\pi$
$228$	0	0
$229$	3109.05	0.897168	0.448584	−	0.893741i	$-0.351928\pi$
$229$	3109.05	0.897168	0.448584	+	0.893741i	$0.351928\pi$
$230$	405.616	0.116285
$231$	0	0
$232$	−183.748	−0.0519984
$233$	2620.23	0.736725	0.368362	−	0.929682i	$-0.379919\pi$
$233$	2620.23	0.736725	0.368362	+	0.929682i	$0.379919\pi$
$234$	0	0
$235$	2618.55	0.726875
$236$	−1081.33	−0.298257
$237$	0	0
$238$	0	0
$239$	−2060.20	−0.557587	−0.278793	−	0.960351i	$-0.589934\pi$
$239$	−2060.20	−0.557587	−0.278793	+	0.960351i	$0.589934\pi$
$240$	0	0
$241$	5410.96	1.44627	0.723134	−	0.690708i	$-0.242701\pi$
$241$	5410.96	1.44627	0.723134	+	0.690708i	$0.242701\pi$
$242$	724.678	0.192496
$243$	0	0
$244$	−3656.96	−0.959478
$245$	0	0
$246$	0	0
$247$	9387.14	2.41818
$248$	−217.420	−0.0556702
$249$	0	0
$250$	68.0645	0.0172191
$251$	−4208.37	−1.05829	−0.529144	−	0.848532i	$-0.677487\pi$
$251$	−4208.37	−1.05829	−0.529144	+	0.848532i	$0.677487\pi$
$252$	0	0
$253$	−54.4883	−0.0135401
$254$	−124.672	−0.0307978
$255$	0	0
$256$	2660.87	0.649627
$257$	891.225	0.216315	0.108158	−	0.994134i	$-0.465505\pi$
$257$	891.225	0.216315	0.108158	+	0.994134i	$0.465505\pi$
$258$	0	0
$259$	0	0
$260$	−2790.86	−0.665700
$261$	0	0
$262$	−512.836	−0.120928
$263$	−3868.47	−0.906996	−0.453498	−	0.891257i	$-0.649824\pi$
$263$	−3868.47	−0.906996	−0.453498	+	0.891257i	$0.649824\pi$
$264$	0	0
$265$	1090.48	0.252784
$266$	0	0
$267$	0	0
$268$	3015.20	0.687250
$269$	5786.26	1.31150	0.655752	−	0.754977i	$-0.272352\pi$
$269$	5786.26	1.31150	0.655752	+	0.754977i	$0.272352\pi$
$270$	0	0
$271$	265.995	0.0596239	0.0298119	−	0.999556i	$-0.490509\pi$
$271$	265.995	0.0596239	0.0298119	+	0.999556i	$0.490509\pi$
$272$	−7278.69	−1.62256
$273$	0	0
$274$	1585.80	0.349641
$275$	−9.14342	−0.00200498
$276$	0	0
$277$	1822.07	0.395225	0.197613	−	0.980280i	$-0.436681\pi$
$277$	1822.07	0.395225	0.197613	+	0.980280i	$0.436681\pi$
$278$	234.380	0.0505655
$279$	0	0
$280$	0	0
$281$	−936.690	−0.198855	−0.0994275	−	0.995045i	$-0.531701\pi$
$281$	−936.690	−0.198855	−0.0994275	+	0.995045i	$0.531701\pi$
$282$	0	0
$283$	1131.87	0.237747	0.118874	−	0.992909i	$-0.462072\pi$
$283$	1131.87	0.237747	0.118874	+	0.992909i	$0.462072\pi$
$284$	1747.69	0.365163
$285$	0	0
$286$	−14.4298	−0.00298339
$287$	0	0
$288$	0	0
$289$	11409.4	2.32229
$290$	−58.5053	−0.0118467
$291$	0	0
$292$	−5215.77	−1.04531
$293$	7211.78	1.43794	0.718971	−	0.695040i	$-0.244613\pi$
$293$	7211.78	1.43794	0.718971	+	0.695040i	$0.244613\pi$
$294$	0	0
$295$	−701.843	−0.138518
$296$	−1268.37	−0.249062
$297$	0	0
$298$	−298.209	−0.0579691
$299$	−10794.8	−2.08789
$300$	0	0
$301$	0	0
$302$	−838.750	−0.159817
$303$	0	0
$304$	−7380.98	−1.39253
$305$	−2373.57	−0.445607
$306$	0	0
$307$	−3275.47	−0.608928	−0.304464	−	0.952524i	$-0.598477\pi$
$307$	−3275.47	−0.608928	−0.304464	+	0.952524i	$0.598477\pi$
$308$	0	0
$309$	0	0
$310$	−69.2267	−0.0126833
$311$	−3341.81	−0.609314	−0.304657	−	0.952462i	$-0.598542\pi$
$311$	−3341.81	−0.609314	−0.304657	+	0.952462i	$0.598542\pi$
$312$	0	0
$313$	1741.86	0.314556	0.157278	−	0.987554i	$-0.449728\pi$
$313$	1741.86	0.314556	0.157278	+	0.987554i	$0.449728\pi$
$314$	−1316.02	−0.236520
$315$	0	0
$316$	−2549.27	−0.453821
$317$	5984.78	1.06037	0.530187	−	0.847881i	$-0.322122\pi$
$317$	5984.78	1.06037	0.530187	+	0.847881i	$0.322122\pi$
$318$	0	0
$319$	7.85929	0.00137942
$320$	2008.18	0.350814
$321$	0	0
$322$	0	0
$323$	16551.8	2.85129
$324$	0	0
$325$	−1811.42	−0.309168
$326$	688.908	0.117040
$327$	0	0
$328$	−1824.00	−0.307053
$329$	0	0
$330$	0	0
$331$	6791.08	1.12771	0.563855	−	0.825874i	$-0.309318\pi$
$331$	6791.08	1.12771	0.563855	+	0.825874i	$0.309318\pi$
$332$	9808.38	1.62140
$333$	0	0
$334$	818.908	0.134158
$335$	1957.03	0.319177
$336$	0	0
$337$	4396.69	0.710692	0.355346	−	0.934735i	$-0.384363\pi$
$337$	4396.69	0.710692	0.355346	+	0.934735i	$0.384363\pi$
$338$	−1662.41	−0.267525
$339$	0	0
$340$	−4920.97	−0.784932
$341$	9.29954	0.00147683
$342$	0	0
$343$	0	0
$344$	−2444.87	−0.383194
$345$	0	0
$346$	−929.711	−0.144455
$347$	2611.58	0.404025	0.202013	−	0.979383i	$-0.435252\pi$
$347$	2611.58	0.404025	0.202013	+	0.979383i	$0.435252\pi$
$348$	0	0
$349$	398.363	0.0610999	0.0305500	−	0.999533i	$-0.490274\pi$
$349$	398.363	0.0610999	0.0305500	+	0.999533i	$0.490274\pi$
$350$	0	0
$351$	0	0
$352$	36.3647	0.00550638
$353$	5523.96	0.832891	0.416446	−	0.909161i	$-0.363276\pi$
$353$	5523.96	0.832891	0.416446	+	0.909161i	$0.363276\pi$
$354$	0	0
$355$	1134.35	0.169591
$356$	8068.32	1.20118
$357$	0	0
$358$	−86.2209	−0.0127288
$359$	−5881.42	−0.864651	−0.432325	−	0.901718i	$-0.642307\pi$
$359$	−5881.42	−0.864651	−0.432325	+	0.901718i	$0.642307\pi$
$360$	0	0
$361$	9925.42	1.44706
$362$	1088.13	0.157985
$363$	0	0
$364$	0	0
$365$	−3385.32	−0.485468
$366$	0	0
$367$	−5123.82	−0.728777	−0.364389	−	0.931247i	$-0.618722\pi$
$367$	−5123.82	−0.728777	−0.364389	+	0.931247i	$0.618722\pi$
$368$	8487.82	1.20233
$369$	0	0
$370$	−403.849	−0.0567435
$371$	0	0
$372$	0	0
$373$	2052.04	0.284854	0.142427	−	0.989805i	$-0.454509\pi$
$373$	2052.04	0.284854	0.142427	+	0.989805i	$0.454509\pi$
$374$	−25.4432	−0.00351774
$375$	0	0
$376$	−4478.15	−0.614210
$377$	1557.02	0.212707
$378$	0	0
$379$	5522.74	0.748507	0.374254	−	0.927326i	$-0.377899\pi$
$379$	5522.74	0.748507	0.374254	+	0.927326i	$0.377899\pi$
$380$	−4990.12	−0.673653
$381$	0	0
$382$	−348.393	−0.0466632
$383$	4769.47	0.636314	0.318157	−	0.948038i	$-0.396936\pi$
$383$	4769.47	0.636314	0.318157	+	0.948038i	$0.396936\pi$
$384$	0	0
$385$	0	0
$386$	643.936	0.0849105
$387$	0	0
$388$	11901.9	1.55729
$389$	−5558.15	−0.724446	−0.362223	−	0.932092i	$-0.617982\pi$
$389$	−5558.15	−0.724446	−0.362223	+	0.932092i	$0.617982\pi$
$390$	0	0
$391$	−19033.9	−2.46185
$392$	0	0
$393$	0	0
$394$	−2515.08	−0.321594
$395$	−1654.62	−0.210767
$396$	0	0
$397$	7339.12	0.927808	0.463904	−	0.885886i	$-0.346448\pi$
$397$	7339.12	0.927808	0.463904	+	0.885886i	$0.346448\pi$
$398$	−1164.17	−0.146620
$399$	0	0
$400$	1424.30	0.178037
$401$	−10657.7	−1.32723	−0.663617	−	0.748072i	$-0.730980\pi$
$401$	−10657.7	−1.32723	−0.663617	+	0.748072i	$0.730980\pi$
$402$	0	0
$403$	1842.35	0.227727
$404$	14053.5	1.73067
$405$	0	0
$406$	0	0
$407$	54.2508	0.00660716
$408$	0	0
$409$	−10038.0	−1.21357	−0.606784	−	0.794867i	$-0.707541\pi$
$409$	−10038.0	−1.21357	−0.606784	+	0.794867i	$0.707541\pi$
$410$	−580.762	−0.0699555
$411$	0	0
$412$	−10299.7	−1.23162
$413$	0	0
$414$	0	0
$415$	6366.18	0.753020
$416$	7204.30	0.849086
$417$	0	0
$418$	−25.8008	−0.00301903
$419$	−10649.8	−1.24171	−0.620853	−	0.783927i	$-0.713214\pi$
$419$	−10649.8	−1.24171	−0.620853	+	0.783927i	$0.713214\pi$
$420$	0	0
$421$	−2826.11	−0.327164	−0.163582	−	0.986530i	$-0.552305\pi$
$421$	−2826.11	−0.327164	−0.163582	+	0.986530i	$0.552305\pi$
$422$	2111.62	0.243583
$423$	0	0
$424$	−1864.90	−0.213602
$425$	−3193.98	−0.364543
$426$	0	0
$427$	0	0
$428$	2913.71	0.329064
$429$	0	0
$430$	−778.448	−0.0873026
$431$	6398.53	0.715096	0.357548	−	0.933895i	$-0.383613\pi$
$431$	6398.53	0.715096	0.357548	+	0.933895i	$0.383613\pi$
$432$	0	0
$433$	1976.94	0.219413	0.109706	−	0.993964i	$-0.465009\pi$
$433$	1976.94	0.219413	0.109706	+	0.993964i	$0.465009\pi$
$434$	0	0
$435$	0	0
$436$	−7468.00	−0.820303
$437$	−19301.4	−2.11284
$438$	0	0
$439$	495.075	0.0538238	0.0269119	−	0.999638i	$-0.491433\pi$
$439$	495.075	0.0538238	0.0269119	+	0.999638i	$0.491433\pi$
$440$	15.6367	0.00169421
$441$	0	0
$442$	−5040.61	−0.542438
$443$	13224.3	1.41830	0.709148	−	0.705060i	$-0.249080\pi$
$443$	13224.3	1.41830	0.709148	+	0.705060i	$0.249080\pi$
$444$	0	0
$445$	5236.78	0.557859
$446$	793.930	0.0842907
$447$	0	0
$448$	0	0
$449$	−6415.03	−0.674263	−0.337131	−	0.941458i	$-0.609457\pi$
$449$	−6415.03	−0.674263	−0.337131	+	0.941458i	$0.609457\pi$
$450$	0	0
$451$	78.0163	0.00814556
$452$	4958.63	0.516005
$453$	0	0
$454$	−987.187	−0.102051
$455$	0	0
$456$	0	0
$457$	−11484.9	−1.17558	−0.587791	−	0.809013i	$-0.700002\pi$
$457$	−11484.9	−1.17558	−0.587791	+	0.809013i	$0.700002\pi$
$458$	−1692.93	−0.172719
$459$	0	0
$460$	5738.43	0.581643
$461$	−9098.30	−0.919198	−0.459599	−	0.888127i	$-0.652007\pi$
$461$	−9098.30	−0.919198	−0.459599	+	0.888127i	$0.652007\pi$
$462$	0	0
$463$	−19670.7	−1.97446	−0.987229	−	0.159311i	$-0.949073\pi$
$463$	−19670.7	−1.97446	−0.987229	+	0.159311i	$0.949073\pi$
$464$	−1224.27	−0.122490
$465$	0	0
$466$	−1426.76	−0.141831
$467$	−1825.14	−0.180851	−0.0904255	−	0.995903i	$-0.528823\pi$
$467$	−1825.14	−0.180851	−0.0904255	+	0.995903i	$0.528823\pi$
$468$	0	0
$469$	0	0
$470$	−1425.84	−0.139935
$471$	0	0
$472$	1200.27	0.117048
$473$	104.573	0.0101654
$474$	0	0
$475$	−3238.87	−0.312862
$476$	0	0
$477$	0	0
$478$	1121.81	0.107344
$479$	17960.2	1.71320	0.856600	−	0.515982i	$-0.172573\pi$
$479$	17960.2	1.71320	0.856600	+	0.515982i	$0.172573\pi$
$480$	0	0
$481$	10747.8	1.01883
$482$	−2946.35	−0.278429
$483$	0	0
$484$	10252.3	0.962841
$485$	7725.02	0.723248
$486$	0	0
$487$	−7136.22	−0.664011	−0.332005	−	0.943278i	$-0.607725\pi$
$487$	−7136.22	−0.664011	−0.332005	+	0.943278i	$0.607725\pi$
$488$	4059.19	0.376538
$489$	0	0
$490$	0	0
$491$	4509.39	0.414473	0.207236	−	0.978291i	$-0.433553\pi$
$491$	4509.39	0.414473	0.207236	+	0.978291i	$0.433553\pi$
$492$	0	0
$493$	2745.41	0.250805
$494$	−5111.45	−0.465536
$495$	0	0
$496$	−1448.62	−0.131139
$497$	0	0
$498$	0	0
$499$	3721.35	0.333849	0.166925	−	0.985970i	$-0.446616\pi$
$499$	3721.35	0.333849	0.166925	+	0.985970i	$0.446616\pi$
$500$	962.938	0.0861278
$501$	0	0
$502$	2291.53	0.203737
$503$	3786.92	0.335687	0.167844	−	0.985814i	$-0.446320\pi$
$503$	3786.92	0.335687	0.167844	+	0.985814i	$0.446320\pi$
$504$	0	0
$505$	9121.52	0.803767
$506$	29.6698	0.00260668
$507$	0	0
$508$	−1763.79	−0.154047
$509$	−1785.58	−0.155490	−0.0777450	−	0.996973i	$-0.524772\pi$
$509$	−1785.58	−0.155490	−0.0777450	+	0.996973i	$0.524772\pi$
$510$	0	0
$511$	0	0
$512$	−9561.90	−0.825352
$513$	0	0
$514$	−485.286	−0.0416441
$515$	−6685.06	−0.571998
$516$	0	0
$517$	191.540	0.0162939
$518$	0	0
$519$	0	0
$520$	3097.83	0.261248
$521$	−16766.0	−1.40985	−0.704925	−	0.709282i	$-0.749019\pi$
$521$	−16766.0	−1.40985	−0.704925	+	0.709282i	$0.749019\pi$
$522$	0	0
$523$	15624.6	1.30634	0.653169	−	0.757212i	$-0.273439\pi$
$523$	15624.6	1.30634	0.653169	+	0.757212i	$0.273439\pi$
$524$	−7255.32	−0.604866
$525$	0	0
$526$	2106.44	0.174611
$527$	3248.52	0.268515
$528$	0	0
$529$	10028.7	0.824257
$530$	−593.784	−0.0486648
$531$	0	0
$532$	0	0
$533$	15456.0	1.25605
$534$	0	0
$535$	1891.16	0.152826
$536$	−3346.85	−0.269705
$537$	0	0
$538$	−3150.71	−0.252485
$539$	0	0
$540$	0	0
$541$	14735.1	1.17100	0.585498	−	0.810674i	$-0.300899\pi$
$541$	14735.1	1.17100	0.585498	+	0.810674i	$0.300899\pi$
$542$	−144.839	−0.0114785
$543$	0	0
$544$	12702.9	1.00116
$545$	−4847.15	−0.380970
$546$	0	0
$547$	−2836.82	−0.221743	−0.110872	−	0.993835i	$-0.535364\pi$
$547$	−2836.82	−0.221743	−0.110872	+	0.993835i	$0.535364\pi$
$548$	22435.0	1.74886
$549$	0	0
$550$	4.97874	0.000385989 0
$551$	2783.99	0.215249
$552$	0	0
$553$	0	0
$554$	−992.145	−0.0760870
$555$	0	0
$556$	3315.88	0.252922
$557$	2183.73	0.166118	0.0830588	−	0.996545i	$-0.473531\pi$
$557$	2183.73	0.166118	0.0830588	+	0.996545i	$0.473531\pi$
$558$	0	0
$559$	20717.1	1.56751
$560$	0	0
$561$	0	0
$562$	510.043	0.0382827
$563$	−24196.5	−1.81130	−0.905648	−	0.424031i	$-0.860615\pi$
$563$	−24196.5	−1.81130	−0.905648	+	0.424031i	$0.860615\pi$
$564$	0	0
$565$	3218.43	0.239646
$566$	−616.320	−0.0457700
$567$	0	0
$568$	−1939.92	−0.143305
$569$	−25607.0	−1.88665	−0.943325	−	0.331872i	$-0.892320\pi$
$569$	−25607.0	−1.88665	−0.943325	+	0.331872i	$0.892320\pi$
$570$	0	0
$571$	−1240.00	−0.0908796	−0.0454398	−	0.998967i	$-0.514469\pi$
$571$	−1240.00	−0.0908796	−0.0454398	+	0.998967i	$0.514469\pi$
$572$	−204.144	−0.0149225
$573$	0	0
$574$	0	0
$575$	3724.56	0.270130
$576$	0	0
$577$	2971.15	0.214369	0.107184	−	0.994239i	$-0.465816\pi$
$577$	2971.15	0.214369	0.107184	+	0.994239i	$0.465816\pi$
$578$	−6212.62	−0.447077
$579$	0	0
$580$	−827.700	−0.0592558
$581$	0	0
$582$	0	0
$583$	79.7657	0.00566648
$584$	5789.45	0.410221
$585$	0	0
$586$	−3926.93	−0.276826
$587$	−4859.32	−0.341679	−0.170840	−	0.985299i	$-0.554648\pi$
$587$	−4859.32	−0.341679	−0.170840	+	0.985299i	$0.554648\pi$
$588$	0	0
$589$	3294.17	0.230448
$590$	382.165	0.0266669
$591$	0	0
$592$	−8450.82	−0.586700
$593$	11889.3	0.823332	0.411666	−	0.911335i	$-0.364947\pi$
$593$	11889.3	0.823332	0.411666	+	0.911335i	$0.364947\pi$
$594$	0	0
$595$	0	0
$596$	−4218.89	−0.289954
$597$	0	0
$598$	5877.95	0.401952
$599$	15019.0	1.02448	0.512238	−	0.858844i	$-0.328817\pi$
$599$	15019.0	1.02448	0.512238	+	0.858844i	$0.328817\pi$
$600$	0	0
$601$	20364.5	1.38217	0.691087	−	0.722772i	$-0.257132\pi$
$601$	20364.5	1.38217	0.691087	+	0.722772i	$0.257132\pi$
$602$	0	0
$603$	0	0
$604$	−11866.1	−0.799382
$605$	6654.33	0.447169
$606$	0	0
$607$	−27272.7	−1.82366	−0.911832	−	0.410563i	$-0.865332\pi$
$607$	−27272.7	−1.82366	−0.911832	+	0.410563i	$0.865332\pi$
$608$	12881.4	0.859230
$609$	0	0
$610$	1292.45	0.0857863
$611$	37946.5	2.51252
$612$	0	0
$613$	−407.717	−0.0268638	−0.0134319	−	0.999910i	$-0.504276\pi$
$613$	−407.717	−0.0268638	−0.0134319	+	0.999910i	$0.504276\pi$
$614$	1783.55	0.117228
$615$	0	0
$616$	0	0
$617$	−9470.05	−0.617909	−0.308955	−	0.951077i	$-0.599979\pi$
$617$	−9470.05	−0.617909	−0.308955	+	0.951077i	$0.599979\pi$
$618$	0	0
$619$	12405.7	0.805540	0.402770	−	0.915301i	$-0.368048\pi$
$619$	12405.7	0.805540	0.402770	+	0.915301i	$0.368048\pi$
$620$	−979.380	−0.0634400
$621$	0	0
$622$	1819.67	0.117302
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−948.473	−0.0605569
$627$	0	0
$628$	−18618.3	−1.18304
$629$	18950.9	1.20131
$630$	0	0
$631$	15706.7	0.990923	0.495462	−	0.868630i	$-0.334999\pi$
$631$	15706.7	0.990923	0.495462	+	0.868630i	$0.334999\pi$
$632$	2829.66	0.178098
$633$	0	0
$634$	−3258.81	−0.204139
$635$	−1144.80	−0.0715433
$636$	0	0
$637$	0	0
$638$	−4.27951	−0.000265560 0
$639$	0	0
$640$	−5070.63	−0.313179
$641$	−11333.4	−0.698350	−0.349175	−	0.937057i	$-0.613538\pi$
$641$	−11333.4	−0.698350	−0.349175	+	0.937057i	$0.613538\pi$
$642$	0	0
$643$	383.101	0.0234961	0.0117481	−	0.999931i	$-0.496260\pi$
$643$	383.101	0.0234961	0.0117481	+	0.999931i	$0.496260\pi$
$644$	0	0
$645$	0	0
$646$	−9012.73	−0.548918
$647$	−28076.0	−1.70600	−0.852999	−	0.521912i	$-0.825219\pi$
$647$	−28076.0	−1.70600	−0.852999	+	0.521912i	$0.825219\pi$
$648$	0	0
$649$	−51.3379	−0.00310507
$650$	986.350	0.0595197
$651$	0	0
$652$	9746.28	0.585420
$653$	−17093.9	−1.02440	−0.512202	−	0.858865i	$-0.671170\pi$
$653$	−17093.9	−1.02440	−0.512202	+	0.858865i	$0.671170\pi$
$654$	0	0
$655$	−4709.10	−0.280916
$656$	−12152.9	−0.723307
$657$	0	0
$658$	0	0
$659$	−22079.2	−1.30513	−0.652567	−	0.757731i	$-0.726308\pi$
$659$	−22079.2	−1.30513	−0.652567	+	0.757731i	$0.726308\pi$
$660$	0	0
$661$	10592.7	0.623312	0.311656	−	0.950195i	$-0.399116\pi$
$661$	10592.7	0.623312	0.311656	+	0.950195i	$0.399116\pi$
$662$	−3697.85	−0.217102
$663$	0	0
$664$	−10887.2	−0.636303
$665$	0	0
$666$	0	0
$667$	−3201.47	−0.185849
$668$	11585.4	0.671040
$669$	0	0
$670$	−1065.64	−0.0614465
$671$	−173.620	−0.00998887
$672$	0	0
$673$	−1977.10	−0.113242	−0.0566209	−	0.998396i	$-0.518033\pi$
$673$	−1977.10	−0.113242	−0.0566209	+	0.998396i	$0.518033\pi$
$674$	−2394.07	−0.136819
$675$	0	0
$676$	−23518.9	−1.33812
$677$	−14972.7	−0.849996	−0.424998	−	0.905194i	$-0.639725\pi$
$677$	−14972.7	−0.849996	−0.424998	+	0.905194i	$0.639725\pi$
$678$	0	0
$679$	0	0
$680$	5462.23	0.308039
$681$	0	0
$682$	−5.06375	−0.000284312 0
$683$	−10235.2	−0.573411	−0.286705	−	0.958019i	$-0.592560\pi$
$683$	−10235.2	−0.573411	−0.286705	+	0.958019i	$0.592560\pi$
$684$	0	0
$685$	14561.6	0.812216
$686$	0	0
$687$	0	0
$688$	−16289.6	−0.902667
$689$	15802.6	0.873774
$690$	0	0
$691$	−19726.7	−1.08602	−0.543009	−	0.839727i	$-0.682715\pi$
$691$	−19726.7	−1.08602	−0.543009	+	0.839727i	$0.682715\pi$
$692$	−13153.0	−0.722547
$693$	0	0
$694$	−1422.05	−0.0777811
$695$	2152.19	0.117464
$696$	0	0
$697$	27252.7	1.48102
$698$	−216.915	−0.0117627
$699$	0	0
$700$	0	0
$701$	16077.0	0.866220	0.433110	−	0.901341i	$-0.357416\pi$
$701$	16077.0	0.866220	0.433110	+	0.901341i	$0.357416\pi$
$702$	0	0
$703$	19217.2	1.03100
$704$	146.893	0.00786396
$705$	0	0
$706$	−3007.88	−0.160345
$707$	0	0
$708$	0	0
$709$	13971.9	0.740092	0.370046	−	0.929013i	$-0.379342\pi$
$709$	13971.9	0.740092	0.370046	+	0.929013i	$0.379342\pi$
$710$	−617.670	−0.0326489
$711$	0	0
$712$	−8955.75	−0.471392
$713$	−3788.15	−0.198973
$714$	0	0
$715$	−132.501	−0.00693042
$716$	−1219.80	−0.0636679
$717$	0	0
$718$	3202.53	0.166459
$719$	−31910.8	−1.65518	−0.827588	−	0.561337i	$-0.810287\pi$
$719$	−31910.8	−1.65518	−0.827588	+	0.561337i	$0.810287\pi$
$720$	0	0
$721$	0	0
$722$	−5404.55	−0.278582
$723$	0	0
$724$	15394.2	0.790222
$725$	−537.223	−0.0275200
$726$	0	0
$727$	−1070.60	−0.0546165	−0.0273083	−	0.999627i	$-0.508694\pi$
$727$	−1070.60	−0.0546165	−0.0273083	+	0.999627i	$0.508694\pi$
$728$	0	0
$729$	0	0
$730$	1843.36	0.0934601
$731$	36529.3	1.84827
$732$	0	0
$733$	2601.98	0.131114	0.0655569	−	0.997849i	$-0.479118\pi$
$733$	2601.98	0.131114	0.0655569	+	0.997849i	$0.479118\pi$
$734$	2790.00	0.140301
$735$	0	0
$736$	−14813.1	−0.741873
$737$	143.152	0.00715477
$738$	0	0
$739$	24765.7	1.23278	0.616389	−	0.787442i	$-0.288595\pi$
$739$	24765.7	1.23278	0.616389	+	0.787442i	$0.288595\pi$
$740$	−5713.42	−0.283824
$741$	0	0
$742$	0	0
$743$	−3719.23	−0.183641	−0.0918206	−	0.995776i	$-0.529269\pi$
$743$	−3719.23	−0.183641	−0.0918206	+	0.995776i	$0.529269\pi$
$744$	0	0
$745$	−2738.29	−0.134662
$746$	−1117.37	−0.0548388
$747$	0	0
$748$	−359.956	−0.0175953
$749$	0	0
$750$	0	0
$751$	−21145.2	−1.02743	−0.513715	−	0.857961i	$-0.671731\pi$
$751$	−21145.2	−1.02743	−0.513715	+	0.857961i	$0.671731\pi$
$752$	−29836.8	−1.44686
$753$	0	0
$754$	−847.824	−0.0409495
$755$	−7701.79	−0.371254
$756$	0	0
$757$	−35618.8	−1.71015	−0.855077	−	0.518501i	$-0.826490\pi$
$757$	−35618.8	−1.71015	−0.855077	+	0.518501i	$0.826490\pi$
$758$	−3007.22	−0.144099
$759$	0	0
$760$	5538.99	0.264369
$761$	15371.1	0.732198	0.366099	−	0.930576i	$-0.380693\pi$
$761$	15371.1	0.732198	0.366099	+	0.930576i	$0.380693\pi$
$762$	0	0
$763$	0	0
$764$	−4928.86	−0.233403
$765$	0	0
$766$	−2597.05	−0.122500
$767$	−10170.7	−0.478803
$768$	0	0
$769$	30641.6	1.43688	0.718442	−	0.695587i	$-0.244856\pi$
$769$	30641.6	1.43688	0.718442	+	0.695587i	$0.244856\pi$
$770$	0	0
$771$	0	0
$772$	9110.03	0.424712
$773$	−29515.5	−1.37335	−0.686675	−	0.726964i	$-0.740931\pi$
$773$	−29515.5	−1.37335	−0.686675	+	0.726964i	$0.740931\pi$
$774$	0	0
$775$	−635.672	−0.0294632
$776$	−13211.0	−0.611145
$777$	0	0
$778$	3026.50	0.139467
$779$	27635.7	1.27105
$780$	0	0
$781$	82.9744	0.00380161
$782$	10364.2	0.473945
$783$	0	0
$784$	0	0
$785$	−12084.3	−0.549435
$786$	0	0
$787$	−41493.0	−1.87937	−0.939685	−	0.342041i	$-0.888882\pi$
$787$	−41493.0	−1.87937	−0.939685	+	0.342041i	$0.888882\pi$
$788$	−35581.9	−1.60857
$789$	0	0
$790$	900.966	0.0405758
$791$	0	0
$792$	0	0
$793$	−34396.3	−1.54029
$794$	−3996.27	−0.178617
$795$	0	0
$796$	−16470.0	−0.733373
$797$	7349.90	0.326659	0.163329	−	0.986572i	$-0.447777\pi$
$797$	7349.90	0.326659	0.163329	+	0.986572i	$0.447777\pi$
$798$	0	0
$799$	66908.9	2.96254
$800$	−2485.72	−0.109854
$801$	0	0
$802$	5803.30	0.255513
$803$	−247.627	−0.0108824
$804$	0	0
$805$	0	0
$806$	−1003.19	−0.0438411
$807$	0	0
$808$	−15599.3	−0.679184
$809$	35614.1	1.54775	0.773873	−	0.633341i	$-0.218317\pi$
$809$	35614.1	1.54775	0.773873	+	0.633341i	$0.218317\pi$
$810$	0	0
$811$	2717.28	0.117653	0.0588264	−	0.998268i	$-0.481264\pi$
$811$	2717.28	0.117653	0.0588264	+	0.998268i	$0.481264\pi$
$812$	0	0
$813$	0	0
$814$	−29.5404	−0.00127198
$815$	6325.87	0.271884
$816$	0	0
$817$	37042.7	1.58624
$818$	5465.88	0.233631
$819$	0	0
$820$	−8216.28	−0.349909
$821$	2890.43	0.122871	0.0614353	−	0.998111i	$-0.480432\pi$
$821$	2890.43	0.122871	0.0614353	+	0.998111i	$0.480432\pi$
$822$	0	0
$823$	−28350.4	−1.20077	−0.600385	−	0.799711i	$-0.704986\pi$
$823$	−28350.4	−1.20077	−0.600385	+	0.799711i	$0.704986\pi$
$824$	11432.5	0.483339
$825$	0	0
$826$	0	0
$827$	37285.8	1.56778	0.783889	−	0.620901i	$-0.213233\pi$
$827$	37285.8	1.56778	0.783889	+	0.620901i	$0.213233\pi$
$828$	0	0
$829$	−25495.2	−1.06814	−0.534068	−	0.845441i	$-0.679337\pi$
$829$	−25495.2	−1.06814	−0.534068	+	0.845441i	$0.679337\pi$
$830$	−3466.49	−0.144968
$831$	0	0
$832$	29101.3	1.21263
$833$	0	0
$834$	0	0
$835$	7519.60	0.311648
$836$	−365.014	−0.0151008
$837$	0	0
$838$	5798.97	0.239048
$839$	47206.0	1.94247	0.971236	−	0.238120i	$-0.0765310\pi$
$839$	47206.0	1.94247	0.971236	+	0.238120i	$0.0765310\pi$
$840$	0	0
$841$	−23927.2	−0.981066
$842$	1538.86	0.0629842
$843$	0	0
$844$	29874.0	1.21837
$845$	−15265.1	−0.621460
$846$	0	0
$847$	0	0
$848$	−12425.4	−0.503170
$849$	0	0
$850$	1739.17	0.0701802
$851$	−22099.0	−0.890181
$852$	0	0
$853$	25476.1	1.02261	0.511304	−	0.859400i	$-0.329163\pi$
$853$	25476.1	1.02261	0.511304	+	0.859400i	$0.329163\pi$
$854$	0	0
$855$	0	0
$856$	−3234.19	−0.129138
$857$	−9690.35	−0.386250	−0.193125	−	0.981174i	$-0.561862\pi$
$857$	−9690.35	−0.386250	−0.193125	+	0.981174i	$0.561862\pi$
$858$	0	0
$859$	9759.80	0.387660	0.193830	−	0.981035i	$-0.437909\pi$
$859$	9759.80	0.387660	0.193830	+	0.981035i	$0.437909\pi$
$860$	−11013.0	−0.436676
$861$	0	0
$862$	−3484.10	−0.137667
$863$	−38969.9	−1.53714	−0.768570	−	0.639765i	$-0.779032\pi$
$863$	−38969.9	−1.53714	−0.768570	+	0.639765i	$0.779032\pi$
$864$	0	0
$865$	−8537.03	−0.335570
$866$	−1076.48	−0.0422404
$867$	0	0
$868$	0	0
$869$	−121.031	−0.00472461
$870$	0	0
$871$	28360.2	1.10327
$872$	8289.41	0.321921
$873$	0	0
$874$	10509.9	0.406754
$875$	0	0
$876$	0	0
$877$	15765.3	0.607018	0.303509	−	0.952829i	$-0.401842\pi$
$877$	15765.3	0.607018	0.303509	+	0.952829i	$0.401842\pi$
$878$	−269.577	−0.0103619
$879$	0	0
$880$	104.184	0.00399094
$881$	−19599.2	−0.749506	−0.374753	−	0.927125i	$-0.622272\pi$
$881$	−19599.2	−0.749506	−0.374753	+	0.927125i	$0.622272\pi$
$882$	0	0
$883$	19894.7	0.758221	0.379111	−	0.925351i	$-0.376230\pi$
$883$	19894.7	0.758221	0.379111	+	0.925351i	$0.376230\pi$
$884$	−71311.7	−2.71320
$885$	0	0
$886$	−7200.84	−0.273044
$887$	8648.61	0.327387	0.163693	−	0.986511i	$-0.447659\pi$
$887$	8648.61	0.327387	0.163693	+	0.986511i	$0.447659\pi$
$888$	0	0
$889$	0	0
$890$	−2851.51	−0.107397
$891$	0	0
$892$	11232.1	0.421611
$893$	67849.2	2.54254
$894$	0	0
$895$	−791.720	−0.0295690
$896$	0	0
$897$	0	0
$898$	3493.09	0.129806
$899$	546.396	0.0202707
$900$	0	0
$901$	27863.8	1.03027
$902$	−42.4812	−0.00156815
$903$	0	0
$904$	−5504.03	−0.202501
$905$	9991.69	0.367000
$906$	0	0
$907$	−32406.4	−1.18637	−0.593186	−	0.805066i	$-0.702130\pi$
$907$	−32406.4	−1.18637	−0.593186	+	0.805066i	$0.702130\pi$
$908$	−13966.2	−0.510444
$909$	0	0
$910$	0	0
$911$	8606.20	0.312993	0.156496	−	0.987679i	$-0.449980\pi$
$911$	8606.20	0.312993	0.156496	+	0.987679i	$0.449980\pi$
$912$	0	0
$913$	465.669	0.0168800
$914$	6253.71	0.226318
$915$	0	0
$916$	−23950.5	−0.863917
$917$	0	0
$918$	0	0
$919$	−35113.0	−1.26036	−0.630181	−	0.776448i	$-0.717019\pi$
$919$	−35113.0	−1.26036	−0.630181	+	0.776448i	$0.717019\pi$
$920$	−6369.60	−0.228260
$921$	0	0
$922$	4954.17	0.176960
$923$	16438.3	0.586210
$924$	0	0
$925$	−3708.33	−0.131815
$926$	10711.0	0.380114
$927$	0	0
$928$	2136.61	0.0755795
$929$	−14705.6	−0.519349	−0.259674	−	0.965696i	$-0.583615\pi$
$929$	−14705.6	−0.519349	−0.259674	+	0.965696i	$0.583615\pi$
$930$	0	0
$931$	0	0
$932$	−20184.9	−0.709420
$933$	0	0
$934$	993.818	0.0348166
$935$	−233.631	−0.00817172
$936$	0	0
$937$	−45498.0	−1.58629	−0.793146	−	0.609032i	$-0.791558\pi$
$937$	−45498.0	−1.58629	−0.793146	+	0.609032i	$0.791558\pi$
$938$	0	0
$939$	0	0
$940$	−20172.0	−0.699935
$941$	21257.8	0.736434	0.368217	−	0.929740i	$-0.379968\pi$
$941$	21257.8	0.736434	0.368217	+	0.929740i	$0.379968\pi$
$942$	0	0
$943$	−31779.9	−1.09745
$944$	7997.07	0.275723
$945$	0	0
$946$	−56.9414	−0.00195700
$947$	−18357.8	−0.629935	−0.314968	−	0.949102i	$-0.601994\pi$
$947$	−18357.8	−0.629935	−0.314968	+	0.949102i	$0.601994\pi$
$948$	0	0
$949$	−49058.0	−1.67807
$950$	1763.62	0.0602308
$951$	0	0
$952$	0	0
$953$	30424.0	1.03414	0.517068	−	0.855944i	$-0.327023\pi$
$953$	30424.0	1.03414	0.517068	+	0.855944i	$0.327023\pi$
$954$	0	0
$955$	−3199.10	−0.108399
$956$	15870.7	0.536921
$957$	0	0
$958$	−9779.62	−0.329817
$959$	0	0
$960$	0	0
$961$	−29144.5	−0.978298
$962$	−5852.33	−0.196140
$963$	0	0
$964$	−41683.3	−1.39267
$965$	5912.92	0.197247
$966$	0	0
$967$	42389.3	1.40967	0.704834	−	0.709373i	$-0.251022\pi$
$967$	42389.3	1.40967	0.704834	+	0.709373i	$0.251022\pi$
$968$	−11380.0	−0.377858
$969$	0	0
$970$	−4206.40	−0.139236
$971$	14165.5	0.468169	0.234085	−	0.972216i	$-0.424791\pi$
$971$	14165.5	0.468169	0.234085	+	0.972216i	$0.424791\pi$
$972$	0	0
$973$	0	0
$974$	3885.79	0.127832
$975$	0	0
$976$	27045.4	0.886989
$977$	−449.688	−0.0147255	−0.00736274	−	0.999973i	$-0.502344\pi$
$977$	−449.688	−0.0147255	−0.00736274	+	0.999973i	$0.502344\pi$
$978$	0	0
$979$	383.057	0.0125052
$980$	0	0
$981$	0	0
$982$	−2455.44	−0.0797924
$983$	21218.1	0.688457	0.344228	−	0.938886i	$-0.388141\pi$
$983$	21218.1	0.688457	0.344228	+	0.938886i	$0.388141\pi$
$984$	0	0
$985$	−23094.6	−0.747062
$986$	−1494.92	−0.0482839
$987$	0	0
$988$	−72313.8	−2.32855
$989$	−42597.5	−1.36959
$990$	0	0
$991$	14544.8	0.466228	0.233114	−	0.972449i	$-0.425108\pi$
$991$	14544.8	0.466228	0.233114	+	0.972449i	$0.425108\pi$
$992$	2528.16	0.0809164
$993$	0	0
$994$	0	0
$995$	−10690.0	−0.340598
$996$	0	0
$997$	−62001.6	−1.96952	−0.984759	−	0.173926i	$-0.944355\pi$
$997$	−62001.6	−1.96952	−0.984759	+	0.173926i	$0.944355\pi$
$998$	−2026.34	−0.0642711
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bv.1.3		5
3.2	odd	2		735.4.a.x.1.3		5
7.3	odd	6		315.4.j.f.226.3		10
7.5	odd	6		315.4.j.f.46.3		10
7.6	odd	2		2205.4.a.bw.1.3		5
21.5	even	6		105.4.i.e.46.3	yes	10
21.17	even	6		105.4.i.e.16.3	✓	10
21.20	even	2		735.4.a.y.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.e.16.3	✓	10	21.17	even	6
105.4.i.e.46.3	yes	10	21.5	even	6
315.4.j.f.46.3		10	7.5	odd	6
315.4.j.f.226.3		10	7.3	odd	6
735.4.a.x.1.3		5	3.2	odd	2
735.4.a.y.1.3		5	21.20	even	2
2205.4.a.bv.1.3		5	1.1	even	1	trivial
2205.4.a.bw.1.3		5	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)

\(f(q)\)	\(=\)	\(q-0.544516 q^{2} -7.70350 q^{4} -5.00000 q^{5} +8.55081 q^{8} +2.72258 q^{10} -0.365737 q^{11} -72.4570 q^{13} +56.9720 q^{16} -127.759 q^{17} -129.555 q^{19} +38.5175 q^{20} +0.199150 q^{22} +148.982 q^{23} +25.0000 q^{25} +39.4540 q^{26} -21.4889 q^{29} -25.4269 q^{31} -99.4286 q^{32} +69.5670 q^{34} -148.333 q^{37} +70.5446 q^{38} -42.7541 q^{40} -213.313 q^{41} -285.923 q^{43} +2.81745 q^{44} -81.1233 q^{46} -523.711 q^{47} -13.6129 q^{50} +558.172 q^{52} -218.096 q^{53} +1.82868 q^{55} +11.7011 q^{58} +140.369 q^{59} +474.714 q^{61} +13.8453 q^{62} -401.635 q^{64} +362.285 q^{65} -391.407 q^{67} +984.194 q^{68} -226.869 q^{71} +677.064 q^{73} +80.7697 q^{74} +998.025 q^{76} +330.923 q^{79} -284.860 q^{80} +116.152 q^{82} -1273.24 q^{83} +638.796 q^{85} +155.690 q^{86} -3.12734 q^{88} -1047.36 q^{89} -1147.69 q^{92} +285.169 q^{94} +647.773 q^{95} -1545.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 25 q^{5} + 69 q^{8} - 5 q^{10} + 33 q^{11} + 23 q^{13} + 113 q^{16} - 136 q^{17} + 39 q^{19} - 105 q^{20} - 87 q^{22} + 133 q^{23} + 125 q^{25} - 73 q^{26} - 272 q^{29} + 430 q^{31}+ \cdots + 2168 q^{97}+O(q^{100}) \) 5 * q + q^2 + 21 * q^4 - 25 * q^5 + 69 * q^8 - 5 * q^10 + 33 * q^11 + 23 * q^13 + 113 * q^16 - 136 * q^17 + 39 * q^19 - 105 * q^20 - 87 * q^22 + 133 * q^23 + 125 * q^25 - 73 * q^26 - 272 * q^29 + 430 * q^31 + 573 * q^32 + 372 * q^34 + 3 * q^37 - 837 * q^38 - 345 * q^40 - 627 * q^41 + 108 * q^43 + 1809 * q^44 + 1637 * q^46 - 553 * q^47 + 25 * q^50 + 1047 * q^52 + 1135 * q^53 - 165 * q^55 - 2564 * q^58 - 332 * q^59 + 584 * q^61 - 1562 * q^62 + 1137 * q^64 - 115 * q^65 + 412 * q^67 + 1712 * q^68 + 142 * q^71 + 2074 * q^73 - 605 * q^74 - 9 * q^76 + 28 * q^79 - 565 * q^80 - 1515 * q^82 - 840 * q^83 + 680 * q^85 - 40 * q^86 + 4181 * q^88 - 2978 * q^89 - 531 * q^92 + 843 * q^94 - 195 * q^95 + 2168 * q^97

Introduction

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