LMFDB - Embedded newform 2205.2.b.d.881.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,2,Mod(881,2205)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2205.881"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-16,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$17.6070136457$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 790 x^{12} - 2192 x^{11} + 5036 x^{10} - 9472 x^{9} + \cdots + 17 $ x^16 - 8x^15 + 52x^14 - 224x^13 + 790x^12 - 2192x^11 + 5036x^10 - 9472x^9 + 14669x^8 - 18520x^7 + 18744x^6 - 14872x^5 + 8956x^4 - 3912x^3 + 1160x^2 - 208*x + 17
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$0.500000 - 0.878189i$ of defining polynomial
Character	$\chi$	$=$	2205.881
Dual form			2205.2.b.d.881.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.607761i q^{2} +1.63063 q^{4} +1.00000 q^{5} -2.20655i q^{8} -0.607761i q^{10} -2.07205i q^{11} -5.72027i q^{13} +1.92020 q^{16} +1.54734 q^{17} -7.10901i q^{19} +1.63063 q^{20} -1.25931 q^{22} +5.33347i q^{23} +1.00000 q^{25} -3.47655 q^{26} +7.57079i q^{29} -1.34780i q^{31} -5.58013i q^{32} -0.940414i q^{34} -1.44977 q^{37} -4.32058 q^{38} -2.20655i q^{40} -9.80448 q^{41} -6.75795 q^{43} -3.37874i q^{44} +3.24147 q^{46} -5.15570 q^{47} -0.607761i q^{50} -9.32762i q^{52} -9.34386i q^{53} -2.07205i q^{55} +4.60123 q^{58} +2.89929 q^{59} +4.65202i q^{61} -0.819142 q^{62} +0.449013 q^{64} -5.72027i q^{65} +9.35484 q^{67} +2.52314 q^{68} +4.17724i q^{71} +2.29124i q^{73} +0.881111i q^{74} -11.5921i q^{76} +14.0324 q^{79} +1.92020 q^{80} +5.95878i q^{82} -5.90939 q^{83} +1.54734 q^{85} +4.10722i q^{86} -4.57209 q^{88} +4.13776 q^{89} +8.69690i q^{92} +3.13343i q^{94} -7.10901i q^{95} -12.8624i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{5} - 16 q^{20} - 16 q^{22} + 16 q^{25} + 32 q^{26} - 16 q^{38} - 32 q^{41} + 32 q^{43} - 16 q^{46} + 32 q^{47} + 48 q^{58} - 32 q^{59} - 32 q^{62} + 16 q^{64} - 16 q^{68} + 32 q^{79}+ \cdots + 64 q^{89}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^5 - 16 * q^20 - 16 * q^22 + 16 * q^25 + 32 * q^26 - 16 * q^38 - 32 * q^41 + 32 * q^43 - 16 * q^46 + 32 * q^47 + 48 * q^58 - 32 * q^59 - 32 * q^62 + 16 * q^64 - 16 * q^68 + 32 * q^79 + 16 * q^88 + 64 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times$.

$n$	$442$	$1081$	$1226$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	− 0.607761i	− 0.429752i	−0.976641	−	0.214876i	$-0.931065\pi$
$2$	− 0.607761i	− 0.429752i	0.976641	−	0.214876i	$-0.0689347\pi$
$3$	0	0
$3$	0	0
$4$	1.63063	0.815313
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 2.20655i	− 0.780134i
$9$	0	0
$10$	− 0.607761i	− 0.192191i
$11$	− 2.07205i	− 0.624746i	−0.949959	−	0.312373i	$-0.898876\pi$
$11$	− 2.07205i	− 0.624746i	0.949959	−	0.312373i	$-0.101124\pi$
$12$	0	0
$13$	− 5.72027i	− 1.58652i	−0.608885	−	0.793258i	$-0.708383\pi$
$13$	− 5.72027i	− 1.58652i	0.608885	−	0.793258i	$-0.291617\pi$
$14$	0	0
$15$	0	0
$16$	1.92020	0.480049
$17$	1.54734	0.375286	0.187643	−	0.982237i	$-0.439915\pi$
$17$	1.54734	0.375286	0.187643	+	0.982237i	$0.439915\pi$
$18$	0	0
$19$	− 7.10901i	− 1.63092i	−0.578814	−	0.815460i	$-0.696484\pi$
$19$	− 7.10901i	− 1.63092i	0.578814	−	0.815460i	$-0.303516\pi$
$20$	1.63063	0.364619
$21$	0	0
$22$	−1.25931	−0.268486
$23$	5.33347i	1.11211i	0.831147	+	0.556053i	$0.187685\pi$
$23$	5.33347i	1.11211i	−0.831147	+	0.556053i	$0.812315\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.47655	−0.681808
$27$	0	0
$28$	0	0
$29$	7.57079i	1.40586i	0.711259	+	0.702930i	$0.248125\pi$
$29$	7.57079i	1.40586i	−0.711259	+	0.702930i	$0.751875\pi$
$30$	0	0
$31$	− 1.34780i	− 0.242073i	−0.992648	−	0.121036i	$-0.961378\pi$
$31$	− 1.34780i	− 0.242073i	0.992648	−	0.121036i	$-0.0386217\pi$
$32$	− 5.58013i	− 0.986436i
$33$	0	0
$34$	− 0.940414i	− 0.161280i
$35$	0	0
$36$	0	0
$37$	−1.44977	−0.238340	−0.119170	−	0.992874i	$-0.538023\pi$
$37$	−1.44977	−0.238340	−0.119170	+	0.992874i	$0.538023\pi$
$38$	−4.32058	−0.700891
$39$	0	0
$40$	− 2.20655i	− 0.348887i
$41$	−9.80448	−1.53120	−0.765601	−	0.643315i	$-0.777558\pi$
$41$	−9.80448	−1.53120	−0.765601	+	0.643315i	$0.777558\pi$
$42$	0	0
$43$	−6.75795	−1.03058	−0.515289	−	0.857017i	$-0.672315\pi$
$43$	−6.75795	−1.03058	−0.515289	+	0.857017i	$0.672315\pi$
$44$	− 3.37874i	− 0.509364i
$45$	0	0
$46$	3.24147	0.477929
$47$	−5.15570	−0.752036	−0.376018	−	0.926612i	$-0.622707\pi$
$47$	−5.15570	−0.752036	−0.376018	+	0.926612i	$0.622707\pi$
$48$	0	0
$49$	0	0
$50$	− 0.607761i	− 0.0859504i
$51$	0	0
$52$	− 9.32762i	− 1.29351i
$53$	− 9.34386i	− 1.28348i	−0.766923	−	0.641739i	$-0.778213\pi$
$53$	− 9.34386i	− 1.28348i	0.766923	−	0.641739i	$-0.221787\pi$
$54$	0	0
$55$	− 2.07205i	− 0.279395i
$56$	0	0
$57$	0	0
$58$	4.60123	0.604171
$59$	2.89929	0.377456	0.188728	−	0.982029i	$-0.439564\pi$
$59$	2.89929	0.377456	0.188728	+	0.982029i	$0.439564\pi$
$60$	0	0
$61$	4.65202i	0.595630i	0.954624	+	0.297815i	$0.0962578\pi$
$61$	4.65202i	0.595630i	−0.954624	+	0.297815i	$0.903742\pi$
$62$	−0.819142	−0.104031
$63$	0	0
$64$	0.449013	0.0561266
$65$	− 5.72027i	− 0.709512i
$66$	0	0
$67$	9.35484	1.14287	0.571437	−	0.820646i	$-0.306386\pi$
$67$	9.35484	1.14287	0.571437	+	0.820646i	$0.306386\pi$
$68$	2.52314	0.305976
$69$	0	0
$70$	0	0
$71$	4.17724i	0.495747i	0.968792	+	0.247874i	$0.0797318\pi$
$71$	4.17724i	0.495747i	−0.968792	+	0.247874i	$0.920268\pi$
$72$	0	0
$73$	2.29124i	0.268170i	0.990970	+	0.134085i	$0.0428095\pi$
$73$	2.29124i	0.268170i	−0.990970	+	0.134085i	$0.957190\pi$
$74$	0.881111i	0.102427i
$75$	0	0
$76$	− 11.5921i	− 1.32971i
$77$	0	0
$78$	0	0
$79$	14.0324	1.57877	0.789386	−	0.613898i	$-0.210399\pi$
$79$	14.0324	1.57877	0.789386	+	0.613898i	$0.210399\pi$
$80$	1.92020	0.214685
$81$	0	0
$82$	5.95878i	0.658037i
$83$	−5.90939	−0.648640	−0.324320	−	0.945948i	$-0.605135\pi$
$83$	−5.90939	−0.648640	−0.324320	+	0.945948i	$0.605135\pi$
$84$	0	0
$85$	1.54734	0.167833
$86$	4.10722i	0.442893i
$87$	0	0
$88$	−4.57209	−0.487386
$89$	4.13776	0.438601	0.219301	−	0.975657i	$-0.429622\pi$
$89$	4.13776	0.438601	0.219301	+	0.975657i	$0.429622\pi$
$90$	0	0
$91$	0	0
$92$	8.69690i	0.906715i
$93$	0	0
$94$	3.13343i	0.323189i
$95$	− 7.10901i	− 0.729369i
$96$	0	0
$97$	− 12.8624i	− 1.30598i	−0.757366	−	0.652991i	$-0.773514\pi$
$97$	− 12.8624i	− 1.30598i	0.757366	−	0.652991i	$-0.226486\pi$
$98$	0	0
$99$	0	0
$100$	1.63063	0.163063
$101$	11.5786	1.15212	0.576059	−	0.817408i	$-0.304590\pi$
$101$	11.5786	1.15212	0.576059	+	0.817408i	$0.304590\pi$
$102$	0	0
$103$	− 8.76554i	− 0.863694i	−0.901947	−	0.431847i	$-0.857862\pi$
$103$	− 8.76554i	− 0.863694i	0.901947	−	0.431847i	$-0.142138\pi$
$104$	−12.6221	−1.23770
$105$	0	0
$106$	−5.67883	−0.551577
$107$	− 13.2290i	− 1.27890i	−0.768834	−	0.639449i	$-0.779163\pi$
$107$	− 13.2290i	− 1.27890i	0.768834	−	0.639449i	$-0.220837\pi$
$108$	0	0
$109$	11.3520	1.08733	0.543664	−	0.839303i	$-0.317037\pi$
$109$	11.3520	1.08733	0.543664	+	0.839303i	$0.317037\pi$
$110$	−1.25931	−0.120071
$111$	0	0
$112$	0	0
$113$	13.7120i	1.28992i	0.764218	+	0.644958i	$0.223125\pi$
$113$	13.7120i	1.28992i	−0.764218	+	0.644958i	$0.776875\pi$
$114$	0	0
$115$	5.33347i	0.497349i
$116$	12.3451i	1.14622i
$117$	0	0
$118$	− 1.76208i	− 0.162212i
$119$	0	0
$120$	0	0
$121$	6.70661	0.609692
$122$	2.82731	0.255973
$123$	0	0
$124$	− 2.19776i	− 0.197365i
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.3356	−1.80450	−0.902248	−	0.431217i	$-0.858084\pi$
$127$	−20.3356	−1.80450	−0.902248	+	0.431217i	$0.858084\pi$
$128$	− 11.4331i	− 1.01056i
$129$	0	0
$130$	−3.47655	−0.304914
$131$	1.51493	0.132360	0.0661802	−	0.997808i	$-0.478919\pi$
$131$	1.51493	0.132360	0.0661802	+	0.997808i	$0.478919\pi$
$132$	0	0
$133$	0	0
$134$	− 5.68550i	− 0.491153i
$135$	0	0
$136$	− 3.41429i	− 0.292773i
$137$	6.72571i	0.574616i	0.957838	+	0.287308i	$0.0927604\pi$
$137$	6.72571i	0.574616i	−0.957838	+	0.287308i	$0.907240\pi$
$138$	0	0
$139$	5.33120i	0.452186i	0.974106	+	0.226093i	$0.0725954\pi$
$139$	5.33120i	0.452186i	−0.974106	+	0.226093i	$0.927405\pi$
$140$	0	0
$141$	0	0
$142$	2.53876	0.213048
$143$	−11.8527	−0.991171
$144$	0	0
$145$	7.57079i	0.628720i
$146$	1.39253	0.115246
$147$	0	0
$148$	−2.36403	−0.194322
$149$	20.3546i	1.66751i	0.552134	+	0.833755i	$0.313813\pi$
$149$	20.3546i	1.66751i	−0.552134	+	0.833755i	$0.686187\pi$
$150$	0	0
$151$	23.0060	1.87220	0.936101	−	0.351732i	$-0.114407\pi$
$151$	23.0060	1.87220	0.936101	+	0.351732i	$0.114407\pi$
$152$	−15.6864	−1.27234
$153$	0	0
$154$	0	0
$155$	− 1.34780i	− 0.108258i
$156$	0	0
$157$	− 6.97420i	− 0.556602i	−0.960494	−	0.278301i	$-0.910229\pi$
$157$	− 6.97420i	− 0.556602i	0.960494	−	0.278301i	$-0.0897713\pi$
$158$	− 8.52836i	− 0.678480i
$159$	0	0
$160$	− 5.58013i	− 0.441148i
$161$	0	0
$162$	0	0
$163$	−5.65699	−0.443089	−0.221545	−	0.975150i	$-0.571110\pi$
$163$	−5.65699	−0.443089	−0.221545	+	0.975150i	$0.571110\pi$
$164$	−15.9874	−1.24841
$165$	0	0
$166$	3.59149i	0.278754i
$167$	19.6550	1.52095	0.760475	−	0.649367i	$-0.224966\pi$
$167$	19.6550	1.52095	0.760475	+	0.649367i	$0.224966\pi$
$168$	0	0
$169$	−19.7215	−1.51704
$170$	− 0.940414i	− 0.0721265i
$171$	0	0
$172$	−11.0197	−0.840244
$173$	24.1809	1.83844	0.919219	−	0.393747i	$-0.128821\pi$
$173$	24.1809	1.83844	0.919219	+	0.393747i	$0.128821\pi$
$174$	0	0
$175$	0	0
$176$	− 3.97874i	− 0.299909i
$177$	0	0
$178$	− 2.51477i	− 0.188490i
$179$	− 5.41987i	− 0.405100i	−0.979272	−	0.202550i	$-0.935077\pi$
$179$	− 5.41987i	− 0.405100i	0.979272	−	0.202550i	$-0.0649228\pi$
$180$	0	0
$181$	1.45743i	0.108330i	0.998532	+	0.0541651i	$0.0172497\pi$
$181$	1.45743i	0.108330i	−0.998532	+	0.0541651i	$0.982750\pi$
$182$	0	0
$183$	0	0
$184$	11.7686	0.867592
$185$	−1.44977	−0.106589
$186$	0	0
$187$	− 3.20617i	− 0.234458i
$188$	−8.40703	−0.613145
$189$	0	0
$190$	−4.32058	−0.313448
$191$	− 13.6957i	− 0.990985i	−0.868612	−	0.495492i	$-0.834988\pi$
$191$	− 13.6957i	− 0.990985i	0.868612	−	0.495492i	$-0.165012\pi$
$192$	0	0
$193$	7.48896	0.539067	0.269534	−	0.962991i	$-0.413130\pi$
$193$	7.48896	0.539067	0.269534	+	0.962991i	$0.413130\pi$
$194$	−7.81728	−0.561248
$195$	0	0
$196$	0	0
$197$	2.07356i	0.147735i	0.997268	+	0.0738677i	$0.0235342\pi$
$197$	2.07356i	0.147735i	−0.997268	+	0.0738677i	$0.976466\pi$
$198$	0	0
$199$	1.44061i	0.102122i	0.998696	+	0.0510611i	$0.0162603\pi$
$199$	1.44061i	0.102122i	−0.998696	+	0.0510611i	$0.983740\pi$
$200$	− 2.20655i	− 0.156027i
$201$	0	0
$202$	− 7.03705i	− 0.495125i
$203$	0	0
$204$	0	0
$205$	−9.80448	−0.684775
$206$	−5.32735	−0.371174
$207$	0	0
$208$	− 10.9840i	− 0.761606i
$209$	−14.7302	−1.01891
$210$	0	0
$211$	16.4747	1.13417	0.567083	−	0.823661i	$-0.308072\pi$
$211$	16.4747	1.13417	0.567083	+	0.823661i	$0.308072\pi$
$212$	− 15.2364i	− 1.04644i
$213$	0	0
$214$	−8.04008	−0.549608
$215$	−6.75795	−0.460888
$216$	0	0
$217$	0	0
$218$	− 6.89932i	− 0.467281i
$219$	0	0
$220$	− 3.37874i	− 0.227795i
$221$	− 8.85122i	− 0.595397i
$222$	0	0
$223$	− 2.52216i	− 0.168896i	−0.996428	−	0.0844482i	$-0.973087\pi$
$223$	− 2.52216i	− 0.168896i	0.996428	−	0.0844482i	$-0.0269128\pi$
$224$	0	0
$225$	0	0
$226$	8.33361	0.554344
$227$	15.3723	1.02030	0.510148	−	0.860087i	$-0.329591\pi$
$227$	15.3723	1.02030	0.510148	+	0.860087i	$0.329591\pi$
$228$	0	0
$229$	4.55926i	0.301284i	0.988588	+	0.150642i	$0.0481341\pi$
$229$	4.55926i	0.301284i	−0.988588	+	0.150642i	$0.951866\pi$
$230$	3.24147	0.213737
$231$	0	0
$232$	16.7053	1.09676
$233$	7.72652i	0.506181i	0.967443	+	0.253091i	$0.0814470\pi$
$233$	7.72652i	0.506181i	−0.967443	+	0.253091i	$0.918553\pi$
$234$	0	0
$235$	−5.15570	−0.336321
$236$	4.72766	0.307745
$237$	0	0
$238$	0	0
$239$	0.869688i	0.0562554i	0.999604	+	0.0281277i	$0.00895451\pi$
$239$	0.869688i	0.0562554i	−0.999604	+	0.0281277i	$0.991045\pi$
$240$	0	0
$241$	26.0615i	1.67877i	0.543541	+	0.839383i	$0.317083\pi$
$241$	26.0615i	1.67877i	−0.543541	+	0.839383i	$0.682917\pi$
$242$	− 4.07602i	− 0.262016i
$243$	0	0
$244$	7.58570i	0.485625i
$245$	0	0
$246$	0	0
$247$	−40.6655	−2.58748
$248$	−2.97400	−0.188849
$249$	0	0
$250$	− 0.607761i	− 0.0384382i
$251$	−22.9225	−1.44686	−0.723428	−	0.690400i	$-0.757435\pi$
$251$	−22.9225	−1.44686	−0.723428	+	0.690400i	$0.757435\pi$
$252$	0	0
$253$	11.0512	0.694784
$254$	12.3592i	0.775486i
$255$	0	0
$256$	−6.05059	−0.378162
$257$	−27.3731	−1.70749	−0.853745	−	0.520692i	$-0.825674\pi$
$257$	−27.3731	−1.70749	−0.853745	+	0.520692i	$0.825674\pi$
$258$	0	0
$259$	0	0
$260$	− 9.32762i	− 0.578475i
$261$	0	0
$262$	− 0.920717i	− 0.0568821i
$263$	15.1797i	0.936020i	0.883723	+	0.468010i	$0.155029\pi$
$263$	15.1797i	0.936020i	−0.883723	+	0.468010i	$0.844971\pi$
$264$	0	0
$265$	− 9.34386i	− 0.573989i
$266$	0	0
$267$	0	0
$268$	15.2542	0.931801
$269$	−26.3996	−1.60961	−0.804806	−	0.593538i	$-0.797730\pi$
$269$	−26.3996	−1.60961	−0.804806	+	0.593538i	$0.797730\pi$
$270$	0	0
$271$	− 8.33144i	− 0.506099i	−0.967453	−	0.253050i	$-0.918566\pi$
$271$	− 8.33144i	− 0.506099i	0.967453	−	0.253050i	$-0.0814336\pi$
$272$	2.97120	0.180156
$273$	0	0
$274$	4.08762	0.246942
$275$	− 2.07205i	− 0.124949i
$276$	0	0
$277$	−4.24827	−0.255254	−0.127627	−	0.991822i	$-0.540736\pi$
$277$	−4.24827	−0.255254	−0.127627	+	0.991822i	$0.540736\pi$
$278$	3.24009	0.194328
$279$	0	0
$280$	0	0
$281$	− 4.35141i	− 0.259583i	−0.991541	−	0.129792i	$-0.958569\pi$
$281$	− 4.35141i	− 0.259583i	0.991541	−	0.129792i	$-0.0414309\pi$
$282$	0	0
$283$	27.7304i	1.64840i	0.566297	+	0.824202i	$0.308376\pi$
$283$	27.7304i	1.64840i	−0.566297	+	0.824202i	$0.691624\pi$
$284$	6.81152i	0.404190i
$285$	0	0
$286$	7.20359i	0.425957i
$287$	0	0
$288$	0	0
$289$	−14.6057	−0.859161
$290$	4.60123	0.270193
$291$	0	0
$292$	3.73616i	0.218643i
$293$	20.8950	1.22070	0.610349	−	0.792133i	$-0.291029\pi$
$293$	20.8950	1.22070	0.610349	+	0.792133i	$0.291029\pi$
$294$	0	0
$295$	2.89929	0.168803
$296$	3.19898i	0.185937i
$297$	0	0
$298$	12.3707	0.716615
$299$	30.5089	1.76437
$300$	0	0
$301$	0	0
$302$	− 13.9821i	− 0.804582i
$303$	0	0
$304$	− 13.6507i	− 0.782922i
$305$	4.65202i	0.266374i
$306$	0	0
$307$	10.2546i	0.585259i	0.956226	+	0.292629i	$0.0945302\pi$
$307$	10.2546i	0.585259i	−0.956226	+	0.292629i	$0.905470\pi$
$308$	0	0
$309$	0	0
$310$	−0.819142	−0.0465241
$311$	13.0858	0.742029	0.371015	−	0.928627i	$-0.379010\pi$
$311$	13.0858	0.742029	0.371015	+	0.928627i	$0.379010\pi$
$312$	0	0
$313$	− 9.36063i	− 0.529094i	−0.964373	−	0.264547i	$-0.914778\pi$
$313$	− 9.36063i	− 0.529094i	0.964373	−	0.264547i	$-0.0852224\pi$
$314$	−4.23865	−0.239201
$315$	0	0
$316$	22.8816	1.28719
$317$	− 25.7813i	− 1.44802i	−0.689787	−	0.724012i	$-0.742296\pi$
$317$	− 25.7813i	− 1.44802i	0.689787	−	0.724012i	$-0.257704\pi$
$318$	0	0
$319$	15.6870	0.878306
$320$	0.449013	0.0251006
$321$	0	0
$322$	0	0
$323$	− 11.0001i	− 0.612061i
$324$	0	0
$325$	− 5.72027i	− 0.317303i
$326$	3.43810i	0.190418i
$327$	0	0
$328$	21.6341i	1.19454i
$329$	0	0
$330$	0	0
$331$	−20.9665	−1.15243	−0.576213	−	0.817300i	$-0.695470\pi$
$331$	−20.9665	−1.15243	−0.576213	+	0.817300i	$0.695470\pi$
$332$	−9.63601	−0.528845
$333$	0	0
$334$	− 11.9455i	− 0.653631i
$335$	9.35484	0.511109
$336$	0	0
$337$	−22.7010	−1.23660	−0.618302	−	0.785941i	$-0.712179\pi$
$337$	−22.7010	−1.23660	−0.618302	+	0.785941i	$0.712179\pi$
$338$	11.9859i	0.651949i
$339$	0	0
$340$	2.52314	0.136836
$341$	−2.79271	−0.151234
$342$	0	0
$343$	0	0
$344$	14.9118i	0.803989i
$345$	0	0
$346$	− 14.6962i	− 0.790072i
$347$	− 20.6183i	− 1.10685i	−0.832900	−	0.553424i	$-0.813321\pi$
$347$	− 20.6183i	− 1.10685i	0.832900	−	0.553424i	$-0.186679\pi$
$348$	0	0
$349$	26.9920i	1.44485i	0.691450	+	0.722424i	$0.256972\pi$
$349$	26.9920i	1.44485i	−0.691450	+	0.722424i	$0.743028\pi$
$350$	0	0
$351$	0	0
$352$	−11.5623	−0.616272
$353$	6.22357	0.331247	0.165624	−	0.986189i	$-0.447036\pi$
$353$	6.22357	0.331247	0.165624	+	0.986189i	$0.447036\pi$
$354$	0	0
$355$	4.17724i	0.221705i
$356$	6.74714	0.357598
$357$	0	0
$358$	−3.29398	−0.174092
$359$	33.7134i	1.77933i	0.456618	+	0.889663i	$0.349061\pi$
$359$	33.7134i	1.77933i	−0.456618	+	0.889663i	$0.650939\pi$
$360$	0	0
$361$	−31.5381	−1.65990
$362$	0.885771	0.0465551
$363$	0	0
$364$	0	0
$365$	2.29124i	0.119929i
$366$	0	0
$367$	− 22.0596i	− 1.15150i	−0.817625	−	0.575751i	$-0.804710\pi$
$367$	− 22.0596i	− 1.15150i	0.817625	−	0.575751i	$-0.195290\pi$
$368$	10.2413i	0.533866i
$369$	0	0
$370$	0.881111i	0.0458068i
$371$	0	0
$372$	0	0
$373$	15.4260	0.798728	0.399364	−	0.916792i	$-0.369231\pi$
$373$	15.4260	0.798728	0.399364	+	0.916792i	$0.369231\pi$
$374$	−1.94858	−0.100759
$375$	0	0
$376$	11.3763i	0.586689i
$377$	43.3069	2.23042
$378$	0	0
$379$	33.5254	1.72208	0.861042	−	0.508534i	$-0.169812\pi$
$379$	33.5254	1.72208	0.861042	+	0.508534i	$0.169812\pi$
$380$	− 11.5921i	− 0.594665i
$381$	0	0
$382$	−8.32370	−0.425878
$383$	−37.3377	−1.90787	−0.953933	−	0.300018i	$-0.903007\pi$
$383$	−37.3377	−1.90787	−0.953933	+	0.300018i	$0.903007\pi$
$384$	0	0
$385$	0	0
$386$	− 4.55150i	− 0.231665i
$387$	0	0
$388$	− 20.9738i	− 1.06478i
$389$	30.6366i	1.55334i	0.629909	+	0.776669i	$0.283092\pi$
$389$	30.6366i	1.55334i	−0.629909	+	0.776669i	$0.716908\pi$
$390$	0	0
$391$	8.25271i	0.417357i
$392$	0	0
$393$	0	0
$394$	1.26023	0.0634895
$395$	14.0324	0.706048
$396$	0	0
$397$	26.4280i	1.32639i	0.748449	+	0.663193i	$0.230799\pi$
$397$	26.4280i	1.32639i	−0.748449	+	0.663193i	$0.769201\pi$
$398$	0.875546	0.0438872
$399$	0	0
$400$	1.92020	0.0960099
$401$	18.5578i	0.926732i	0.886167	+	0.463366i	$0.153358\pi$
$401$	18.5578i	0.926732i	−0.886167	+	0.463366i	$0.846642\pi$
$402$	0	0
$403$	−7.70979	−0.384052
$404$	18.8804	0.939337
$405$	0	0
$406$	0	0
$407$	3.00399i	0.148902i
$408$	0	0
$409$	30.1064i	1.48867i	0.667808	+	0.744334i	$0.267233\pi$
$409$	30.1064i	1.48867i	−0.667808	+	0.744334i	$0.732767\pi$
$410$	5.95878i	0.294283i
$411$	0	0
$412$	− 14.2933i	− 0.704181i
$413$	0	0
$414$	0	0
$415$	−5.90939	−0.290080
$416$	−31.9198	−1.56500
$417$	0	0
$418$	8.95245i	0.437879i
$419$	−11.8430	−0.578570	−0.289285	−	0.957243i	$-0.593418\pi$
$419$	−11.8430	−0.578570	−0.289285	+	0.957243i	$0.593418\pi$
$420$	0	0
$421$	−15.0914	−0.735511	−0.367756	−	0.929922i	$-0.619874\pi$
$421$	−15.0914	−0.735511	−0.367756	+	0.929922i	$0.619874\pi$
$422$	− 10.0127i	− 0.487410i
$423$	0	0
$424$	−20.6177	−1.00129
$425$	1.54734	0.0750572
$426$	0	0
$427$	0	0
$428$	− 21.5716i	− 1.04270i
$429$	0	0
$430$	4.10722i	0.198068i
$431$	7.09697i	0.341849i	0.985284	+	0.170924i	$0.0546754\pi$
$431$	7.09697i	0.341849i	−0.985284	+	0.170924i	$0.945325\pi$
$432$	0	0
$433$	36.5702i	1.75745i	0.477326	+	0.878726i	$0.341606\pi$
$433$	36.5702i	1.75745i	−0.477326	+	0.878726i	$0.658394\pi$
$434$	0	0
$435$	0	0
$436$	18.5109	0.886513
$437$	37.9157	1.81375
$438$	0	0
$439$	− 18.7009i	− 0.892547i	−0.894897	−	0.446274i	$-0.852751\pi$
$439$	− 18.7009i	− 0.892547i	0.894897	−	0.446274i	$-0.147249\pi$
$440$	−4.57209	−0.217966
$441$	0	0
$442$	−5.37942	−0.255873
$443$	29.4215i	1.39786i	0.715190	+	0.698930i	$0.246340\pi$
$443$	29.4215i	1.39786i	−0.715190	+	0.698930i	$0.753660\pi$
$444$	0	0
$445$	4.13776	0.196148
$446$	−1.53287	−0.0725836
$447$	0	0
$448$	0	0
$449$	1.50209i	0.0708879i	0.999372	+	0.0354440i	$0.0112845\pi$
$449$	1.50209i	0.0708879i	−0.999372	+	0.0354440i	$0.988715\pi$
$450$	0	0
$451$	20.3154i	0.956613i
$452$	22.3591i	1.05169i
$453$	0	0
$454$	− 9.34269i	− 0.438474i
$455$	0	0
$456$	0	0
$457$	−12.1408	−0.567924	−0.283962	−	0.958836i	$-0.591649\pi$
$457$	−12.1408	−0.567924	−0.283962	+	0.958836i	$0.591649\pi$
$458$	2.77094	0.129477
$459$	0	0
$460$	8.69690i	0.405495i
$461$	28.6537	1.33454	0.667268	−	0.744817i	$-0.267463\pi$
$461$	28.6537	1.33454	0.667268	+	0.744817i	$0.267463\pi$
$462$	0	0
$463$	6.28648	0.292158	0.146079	−	0.989273i	$-0.453335\pi$
$463$	6.28648	0.292158	0.146079	+	0.989273i	$0.453335\pi$
$464$	14.5374i	0.674882i
$465$	0	0
$466$	4.69587	0.217532
$467$	−3.38674	−0.156720	−0.0783599	−	0.996925i	$-0.524968\pi$
$467$	−3.38674	−0.156720	−0.0783599	+	0.996925i	$0.524968\pi$
$468$	0	0
$469$	0	0
$470$	3.13343i	0.144535i
$471$	0	0
$472$	− 6.39744i	− 0.294466i
$473$	14.0028i	0.643850i
$474$	0	0
$475$	− 7.10901i	− 0.326184i
$476$	0	0
$477$	0	0
$478$	0.528562	0.0241759
$479$	18.6355	0.851480	0.425740	−	0.904846i	$-0.360014\pi$
$479$	18.6355	0.851480	0.425740	+	0.904846i	$0.360014\pi$
$480$	0	0
$481$	8.29305i	0.378130i
$482$	15.8391	0.721453
$483$	0	0
$484$	10.9360	0.497090
$485$	− 12.8624i	− 0.584053i
$486$	0	0
$487$	−26.8409	−1.21628	−0.608139	−	0.793831i	$-0.708084\pi$
$487$	−26.8409	−1.21628	−0.608139	+	0.793831i	$0.708084\pi$
$488$	10.2649	0.464671
$489$	0	0
$490$	0	0
$491$	− 41.1046i	− 1.85503i	−0.373791	−	0.927513i	$-0.621942\pi$
$491$	− 41.1046i	− 1.85503i	0.373791	−	0.927513i	$-0.378058\pi$
$492$	0	0
$493$	11.7146i	0.527599i
$494$	24.7149i	1.11197i
$495$	0	0
$496$	− 2.58805i	− 0.116207i
$497$	0	0
$498$	0	0
$499$	5.89202	0.263763	0.131882	−	0.991265i	$-0.457898\pi$
$499$	5.89202	0.263763	0.131882	+	0.991265i	$0.457898\pi$
$500$	1.63063	0.0729238
$501$	0	0
$502$	13.9314i	0.621789i
$503$	−31.4443	−1.40203	−0.701016	−	0.713146i	$-0.747270\pi$
$503$	−31.4443	−1.40203	−0.701016	+	0.713146i	$0.747270\pi$
$504$	0	0
$505$	11.5786	0.515243
$506$	− 6.71650i	− 0.298585i
$507$	0	0
$508$	−33.1598	−1.47123
$509$	35.9489	1.59341	0.796704	−	0.604370i	$-0.206575\pi$
$509$	35.9489	1.59341	0.796704	+	0.604370i	$0.206575\pi$
$510$	0	0
$511$	0	0
$512$	− 19.1890i	− 0.848041i
$513$	0	0
$514$	16.6363i	0.733797i
$515$	− 8.76554i	− 0.386256i
$516$	0	0
$517$	10.6829i	0.469832i
$518$	0	0
$519$	0	0
$520$	−12.6221	−0.553515
$521$	24.3888	1.06849	0.534247	−	0.845329i	$-0.320595\pi$
$521$	24.3888	1.06849	0.534247	+	0.845329i	$0.320595\pi$
$522$	0	0
$523$	41.9347i	1.83367i	0.399261	+	0.916837i	$0.369267\pi$
$523$	41.9347i	1.83367i	−0.399261	+	0.916837i	$0.630733\pi$
$524$	2.47029	0.107915
$525$	0	0
$526$	9.22562	0.402256
$527$	− 2.08551i	− 0.0908464i
$528$	0	0
$529$	−5.44592	−0.236779
$530$	−5.67883	−0.246673
$531$	0	0
$532$	0	0
$533$	56.0842i	2.42928i
$534$	0	0
$535$	− 13.2290i	− 0.571940i
$536$	− 20.6419i	− 0.891596i
$537$	0	0
$538$	16.0446i	0.691733i
$539$	0	0
$540$	0	0
$541$	4.12241	0.177236	0.0886181	−	0.996066i	$-0.471755\pi$
$541$	4.12241	0.177236	0.0886181	+	0.996066i	$0.471755\pi$
$542$	−5.06353	−0.217497
$543$	0	0
$544$	− 8.63437i	− 0.370196i
$545$	11.3520	0.486268
$546$	0	0
$547$	35.3868	1.51303	0.756515	−	0.653976i	$-0.226900\pi$
$547$	35.3868	1.51303	0.756515	+	0.653976i	$0.226900\pi$
$548$	10.9671i	0.468492i
$549$	0	0
$550$	−1.25931	−0.0536972
$551$	53.8208	2.29284
$552$	0	0
$553$	0	0
$554$	2.58193i	0.109696i
$555$	0	0
$556$	8.69319i	0.368674i
$557$	− 38.1922i	− 1.61826i	−0.587633	−	0.809128i	$-0.699940\pi$
$557$	− 38.1922i	− 1.61826i	0.587633	−	0.809128i	$-0.300060\pi$
$558$	0	0
$559$	38.6573i	1.63503i
$560$	0	0
$561$	0	0
$562$	−2.64462	−0.111556
$563$	−13.0661	−0.550669	−0.275335	−	0.961348i	$-0.588789\pi$
$563$	−13.0661	−0.550669	−0.275335	+	0.961348i	$0.588789\pi$
$564$	0	0
$565$	13.7120i	0.576868i
$566$	16.8535	0.708404
$567$	0	0
$568$	9.21730	0.386750
$569$	− 13.8167i	− 0.579227i	−0.957144	−	0.289614i	$-0.906473\pi$
$569$	− 13.8167i	− 0.579227i	0.957144	−	0.289614i	$-0.0935268\pi$
$570$	0	0
$571$	−6.62806	−0.277376	−0.138688	−	0.990336i	$-0.544288\pi$
$571$	−6.62806	−0.277376	−0.138688	+	0.990336i	$0.544288\pi$
$572$	−19.3273	−0.808115
$573$	0	0
$574$	0	0
$575$	5.33347i	0.222421i
$576$	0	0
$577$	− 4.10673i	− 0.170966i	−0.996340	−	0.0854828i	$-0.972757\pi$
$577$	− 4.10673i	− 0.170966i	0.996340	−	0.0854828i	$-0.0272433\pi$
$578$	8.87679i	0.369226i
$579$	0	0
$580$	12.3451i	0.512603i
$581$	0	0
$582$	0	0
$583$	−19.3609	−0.801848
$584$	5.05575	0.209209
$585$	0	0
$586$	− 12.6992i	− 0.524597i
$587$	43.6331	1.80093	0.900464	−	0.434930i	$-0.143227\pi$
$587$	43.6331	1.80093	0.900464	+	0.434930i	$0.143227\pi$
$588$	0	0
$589$	−9.58155	−0.394801
$590$	− 1.76208i	− 0.0725435i
$591$	0	0
$592$	−2.78384	−0.114415
$593$	4.96236	0.203780	0.101890	−	0.994796i	$-0.467511\pi$
$593$	4.96236	0.203780	0.101890	+	0.994796i	$0.467511\pi$
$594$	0	0
$595$	0	0
$596$	33.1907i	1.35954i
$597$	0	0
$598$	− 18.5421i	− 0.758243i
$599$	− 16.8333i	− 0.687788i	−0.939008	−	0.343894i	$-0.888254\pi$
$599$	− 16.8333i	− 0.687788i	0.939008	−	0.343894i	$-0.111746\pi$
$600$	0	0
$601$	11.0127i	0.449217i	0.974449	+	0.224609i	$0.0721103\pi$
$601$	11.0127i	0.449217i	−0.974449	+	0.224609i	$0.927890\pi$
$602$	0	0
$603$	0	0
$604$	37.5142	1.52643
$605$	6.70661	0.272663
$606$	0	0
$607$	6.61806i	0.268619i	0.990939	+	0.134309i	$0.0428816\pi$
$607$	6.61806i	0.268619i	−0.990939	+	0.134309i	$0.957118\pi$
$608$	−39.6692	−1.60880
$609$	0	0
$610$	2.82731	0.114475
$611$	29.4920i	1.19312i
$612$	0	0
$613$	2.87382	0.116072	0.0580362	−	0.998314i	$-0.481516\pi$
$613$	2.87382	0.116072	0.0580362	+	0.998314i	$0.481516\pi$
$614$	6.23232	0.251516
$615$	0	0
$616$	0	0
$617$	− 11.6224i	− 0.467900i	−0.972249	−	0.233950i	$-0.924835\pi$
$617$	− 11.6224i	− 0.467900i	0.972249	−	0.233950i	$-0.0751652\pi$
$618$	0	0
$619$	− 10.2458i	− 0.411813i	−0.978572	−	0.205907i	$-0.933986\pi$
$619$	− 10.2458i	− 0.411813i	0.978572	−	0.205907i	$-0.0660143\pi$
$620$	− 2.19776i	− 0.0882643i
$621$	0	0
$622$	− 7.95305i	− 0.318888i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−5.68902	−0.227379
$627$	0	0
$628$	− 11.3723i	− 0.453805i
$629$	−2.24328	−0.0894456
$630$	0	0
$631$	−20.0096	−0.796571	−0.398285	−	0.917262i	$-0.630395\pi$
$631$	−20.0096	−0.796571	−0.398285	+	0.917262i	$0.630395\pi$
$632$	− 30.9633i	− 1.23165i
$633$	0	0
$634$	−15.6689	−0.622291
$635$	−20.3356	−0.806995
$636$	0	0
$637$	0	0
$638$	− 9.53397i	− 0.377453i
$639$	0	0
$640$	− 11.4331i	− 0.451935i
$641$	7.62239i	0.301066i	0.988605	+	0.150533i	$0.0480990\pi$
$641$	7.62239i	0.301066i	−0.988605	+	0.150533i	$0.951901\pi$
$642$	0	0
$643$	− 11.2819i	− 0.444917i	−0.974942	−	0.222458i	$-0.928592\pi$
$643$	− 11.2819i	− 0.444917i	0.974942	−	0.222458i	$-0.0714081\pi$
$644$	0	0
$645$	0	0
$646$	−6.68542	−0.263034
$647$	1.05563	0.0415012	0.0207506	−	0.999785i	$-0.493394\pi$
$647$	1.05563	0.0415012	0.0207506	+	0.999785i	$0.493394\pi$
$648$	0	0
$649$	− 6.00748i	− 0.235814i
$650$	−3.47655	−0.136362
$651$	0	0
$652$	−9.22443	−0.361257
$653$	16.3928i	0.641501i	0.947164	+	0.320751i	$0.103935\pi$
$653$	16.3928i	0.641501i	−0.947164	+	0.320751i	$0.896065\pi$
$654$	0	0
$655$	1.51493	0.0591933
$656$	−18.8265	−0.735053
$657$	0	0
$658$	0	0
$659$	− 1.28390i	− 0.0500137i	−0.999687	−	0.0250068i	$-0.992039\pi$
$659$	− 1.28390i	− 0.0500137i	0.999687	−	0.0250068i	$-0.00796076\pi$
$660$	0	0
$661$	− 10.1657i	− 0.395398i	−0.980263	−	0.197699i	$-0.936653\pi$
$661$	− 10.1657i	− 0.395398i	0.980263	−	0.197699i	$-0.0633469\pi$
$662$	12.7426i	0.495257i
$663$	0	0
$664$	13.0394i	0.506026i
$665$	0	0
$666$	0	0
$667$	−40.3786	−1.56346
$668$	32.0500	1.24005
$669$	0	0
$670$	− 5.68550i	− 0.219650i
$671$	9.63921	0.372118
$672$	0	0
$673$	8.80255	0.339314	0.169657	−	0.985503i	$-0.445734\pi$
$673$	8.80255	0.339314	0.169657	+	0.985503i	$0.445734\pi$
$674$	13.7968i	0.531433i
$675$	0	0
$676$	−32.1584	−1.23686
$677$	6.38933	0.245562	0.122781	−	0.992434i	$-0.460819\pi$
$677$	6.38933	0.245562	0.122781	+	0.992434i	$0.460819\pi$
$678$	0	0
$679$	0	0
$680$	− 3.41429i	− 0.130932i
$681$	0	0
$682$	1.69730i	0.0649931i
$683$	17.8275i	0.682149i	0.940036	+	0.341074i	$0.110791\pi$
$683$	17.8275i	0.682149i	−0.940036	+	0.341074i	$0.889209\pi$
$684$	0	0
$685$	6.72571i	0.256976i
$686$	0	0
$687$	0	0
$688$	−12.9766	−0.494728
$689$	−53.4494	−2.03626
$690$	0	0
$691$	− 25.1781i	− 0.957818i	−0.877864	−	0.478909i	$-0.841032\pi$
$691$	− 25.1781i	− 0.957818i	0.877864	−	0.478909i	$-0.158968\pi$
$692$	39.4300	1.49890
$693$	0	0
$694$	−12.5310	−0.475670
$695$	5.33120i	0.202224i
$696$	0	0
$697$	−15.1709	−0.574639
$698$	16.4047	0.620926
$699$	0	0
$700$	0	0
$701$	− 8.86413i	− 0.334794i	−0.985890	−	0.167397i	$-0.946464\pi$
$701$	− 8.86413i	− 0.334794i	0.985890	−	0.167397i	$-0.0535361\pi$
$702$	0	0
$703$	10.3064i	0.388713i
$704$	− 0.930376i	− 0.0350649i
$705$	0	0
$706$	− 3.78244i	− 0.142354i
$707$	0	0
$708$	0	0
$709$	25.4921	0.957376	0.478688	−	0.877985i	$-0.341113\pi$
$709$	25.4921	0.957376	0.478688	+	0.877985i	$0.341113\pi$
$710$	2.53876	0.0952781
$711$	0	0
$712$	− 9.13018i	− 0.342168i
$713$	7.18847	0.269210
$714$	0	0
$715$	−11.8527	−0.443265
$716$	− 8.83778i	− 0.330283i
$717$	0	0
$718$	20.4897	0.764668
$719$	−39.0080	−1.45475	−0.727376	−	0.686239i	$-0.759260\pi$
$719$	−39.0080	−1.45475	−0.727376	+	0.686239i	$0.759260\pi$
$720$	0	0
$721$	0	0
$722$	19.1676i	0.713344i
$723$	0	0
$724$	2.37653i	0.0883231i
$725$	7.57079i	0.281172i
$726$	0	0
$727$	− 41.7031i	− 1.54668i	−0.633990	−	0.773342i	$-0.718584\pi$
$727$	− 41.7031i	− 1.54668i	0.633990	−	0.773342i	$-0.281416\pi$
$728$	0	0
$729$	0	0
$730$	1.39253	0.0515398
$731$	−10.4569	−0.386761
$732$	0	0
$733$	− 12.1775i	− 0.449788i	−0.974383	−	0.224894i	$-0.927796\pi$
$733$	− 12.1775i	− 0.449788i	0.974383	−	0.224894i	$-0.0722036\pi$
$734$	−13.4070	−0.494860
$735$	0	0
$736$	29.7614	1.09702
$737$	− 19.3837i	− 0.714007i
$738$	0	0
$739$	14.5460	0.535084	0.267542	−	0.963546i	$-0.413789\pi$
$739$	14.5460	0.535084	0.267542	+	0.963546i	$0.413789\pi$
$740$	−2.36403	−0.0869033
$741$	0	0
$742$	0	0
$743$	− 9.12577i	− 0.334792i	−0.985890	−	0.167396i	$-0.946464\pi$
$743$	− 9.12577i	− 0.334792i	0.985890	−	0.167396i	$-0.0535359\pi$
$744$	0	0
$745$	20.3546i	0.745733i
$746$	− 9.37532i	− 0.343255i
$747$	0	0
$748$	− 5.22807i	− 0.191157i
$749$	0	0
$750$	0	0
$751$	−24.8928	−0.908351	−0.454176	−	0.890912i	$-0.650066\pi$
$751$	−24.8928	−0.908351	−0.454176	+	0.890912i	$0.650066\pi$
$752$	−9.89997	−0.361015
$753$	0	0
$754$	− 26.3203i	− 0.958527i
$755$	23.0060	0.837274
$756$	0	0
$757$	6.56329	0.238547	0.119273	−	0.992861i	$-0.461943\pi$
$757$	6.56329	0.238547	0.119273	+	0.992861i	$0.461943\pi$
$758$	− 20.3754i	− 0.740069i
$759$	0	0
$760$	−15.6864	−0.569006
$761$	13.0172	0.471874	0.235937	−	0.971768i	$-0.424184\pi$
$761$	13.0172	0.471874	0.235937	+	0.971768i	$0.424184\pi$
$762$	0	0
$763$	0	0
$764$	− 22.3326i	− 0.807963i
$765$	0	0
$766$	22.6924i	0.819909i
$767$	− 16.5847i	− 0.598840i
$768$	0	0
$769$	27.2477i	0.982578i	0.870997	+	0.491289i	$0.163474\pi$
$769$	27.2477i	0.982578i	−0.870997	+	0.491289i	$0.836526\pi$
$770$	0	0
$771$	0	0
$772$	12.2117	0.439509
$773$	9.97277	0.358696	0.179348	−	0.983786i	$-0.442601\pi$
$773$	9.97277	0.358696	0.179348	+	0.983786i	$0.442601\pi$
$774$	0	0
$775$	− 1.34780i	− 0.0484145i
$776$	−28.3816	−1.01884
$777$	0	0
$778$	18.6197	0.667550
$779$	69.7002i	2.49727i
$780$	0	0
$781$	8.65545	0.309716
$782$	5.01567	0.179360
$783$	0	0
$784$	0	0
$785$	− 6.97420i	− 0.248920i
$786$	0	0
$787$	− 8.42880i	− 0.300454i	−0.988651	−	0.150227i	$-0.951999\pi$
$787$	− 8.42880i	− 0.300454i	0.988651	−	0.150227i	$-0.0480005\pi$
$788$	3.38121i	0.120451i
$789$	0	0
$790$	− 8.52836i	− 0.303425i
$791$	0	0
$792$	0	0
$793$	26.6108	0.944977
$794$	16.0619	0.570017
$795$	0	0
$796$	2.34910i	0.0832615i
$797$	−54.1574	−1.91835	−0.959176	−	0.282809i	$-0.908734\pi$
$797$	−54.1574	−1.91835	−0.959176	+	0.282809i	$0.908734\pi$
$798$	0	0
$799$	−7.97764	−0.282229
$800$	− 5.58013i	− 0.197287i
$801$	0	0
$802$	11.2787	0.398265
$803$	4.74757	0.167538
$804$	0	0
$805$	0	0
$806$	4.68571i	0.165047i
$807$	0	0
$808$	− 25.5489i	− 0.898807i
$809$	− 39.6101i	− 1.39262i	−0.717742	−	0.696309i	$-0.754824\pi$
$809$	− 39.6101i	− 1.39262i	0.717742	−	0.696309i	$-0.245176\pi$
$810$	0	0
$811$	− 24.1806i	− 0.849096i	−0.905406	−	0.424548i	$-0.860433\pi$
$811$	− 24.1806i	− 0.849096i	0.905406	−	0.424548i	$-0.139567\pi$
$812$	0	0
$813$	0	0
$814$	1.82570	0.0639909
$815$	−5.65699	−0.198156
$816$	0	0
$817$	48.0424i	1.68079i
$818$	18.2975	0.639758
$819$	0	0
$820$	−15.9874	−0.558306
$821$	− 6.69975i	− 0.233823i	−0.993142	−	0.116911i	$-0.962701\pi$
$821$	− 6.69975i	− 0.233823i	0.993142	−	0.116911i	$-0.0372994\pi$
$822$	0	0
$823$	46.9759	1.63748	0.818738	−	0.574167i	$-0.194674\pi$
$823$	46.9759	1.63748	0.818738	+	0.574167i	$0.194674\pi$
$824$	−19.3416	−0.673797
$825$	0	0
$826$	0	0
$827$	− 24.0149i	− 0.835078i	−0.908659	−	0.417539i	$-0.862893\pi$
$827$	− 24.0149i	− 0.835078i	0.908659	−	0.417539i	$-0.137107\pi$
$828$	0	0
$829$	27.0973i	0.941128i	0.882366	+	0.470564i	$0.155950\pi$
$829$	27.0973i	0.941128i	−0.882366	+	0.470564i	$0.844050\pi$
$830$	3.59149i	0.124663i
$831$	0	0
$832$	− 2.56847i	− 0.0890458i
$833$	0	0
$834$	0	0
$835$	19.6550	0.680190
$836$	−24.0195	−0.830732
$837$	0	0
$838$	7.19773i	0.248642i
$839$	−15.3834	−0.531094	−0.265547	−	0.964098i	$-0.585552\pi$
$839$	−15.3834	−0.531094	−0.265547	+	0.964098i	$0.585552\pi$
$840$	0	0
$841$	−28.3168	−0.976441
$842$	9.17198i	0.316087i
$843$	0	0
$844$	26.8641	0.924700
$845$	−19.7215	−0.678439
$846$	0	0
$847$	0	0
$848$	− 17.9421i	− 0.616133i
$849$	0	0
$850$	− 0.940414i	− 0.0322559i
$851$	− 7.73228i	− 0.265059i
$852$	0	0
$853$	− 26.1335i	− 0.894795i	−0.894335	−	0.447398i	$-0.852351\pi$
$853$	− 26.1335i	− 0.894795i	0.894335	−	0.447398i	$-0.147649\pi$
$854$	0	0
$855$	0	0
$856$	−29.1905	−0.997711
$857$	48.2877	1.64948	0.824738	−	0.565515i	$-0.191323\pi$
$857$	48.2877	1.64948	0.824738	+	0.565515i	$0.191323\pi$
$858$	0	0
$859$	3.98550i	0.135984i	0.997686	+	0.0679918i	$0.0216592\pi$
$859$	3.98550i	0.135984i	−0.997686	+	0.0679918i	$0.978341\pi$
$860$	−11.0197	−0.375768
$861$	0	0
$862$	4.31326	0.146910
$863$	20.4965i	0.697710i	0.937177	+	0.348855i	$0.113429\pi$
$863$	20.4965i	0.697710i	−0.937177	+	0.348855i	$0.886571\pi$
$864$	0	0
$865$	24.1809	0.822174
$866$	22.2259	0.755268
$867$	0	0
$868$	0	0
$869$	− 29.0759i	− 0.986332i
$870$	0	0
$871$	− 53.5122i	− 1.81319i
$872$	− 25.0489i	− 0.848262i
$873$	0	0
$874$	− 23.0437i	− 0.779464i
$875$	0	0
$876$	0	0
$877$	−7.07912	−0.239045	−0.119522	−	0.992832i	$-0.538136\pi$
$877$	−7.07912	−0.239045	−0.119522	+	0.992832i	$0.538136\pi$
$878$	−11.3657	−0.383574
$879$	0	0
$880$	− 3.97874i	− 0.134123i
$881$	−29.7131	−1.00106	−0.500529	−	0.865720i	$-0.666861\pi$
$881$	−29.7131	−1.00106	−0.500529	+	0.865720i	$0.666861\pi$
$882$	0	0
$883$	21.6525	0.728666	0.364333	−	0.931269i	$-0.381297\pi$
$883$	21.6525	0.728666	0.364333	+	0.931269i	$0.381297\pi$
$884$	− 14.4330i	− 0.485435i
$885$	0	0
$886$	17.8813	0.600733
$887$	−37.3619	−1.25449	−0.627246	−	0.778821i	$-0.715818\pi$
$887$	−37.3619	−1.25449	−0.627246	+	0.778821i	$0.715818\pi$
$888$	0	0
$889$	0	0
$890$	− 2.51477i	− 0.0842952i
$891$	0	0
$892$	− 4.11271i	− 0.137704i
$893$	36.6520i	1.22651i
$894$	0	0
$895$	− 5.41987i	− 0.181166i
$896$	0	0
$897$	0	0
$898$	0.912911	0.0304642
$899$	10.2039	0.340320
$900$	0	0
$901$	− 14.4582i	− 0.481671i
$902$	12.3469	0.411106
$903$	0	0
$904$	30.2562	1.00631
$905$	1.45743i	0.0484467i
$906$	0	0
$907$	0.0978680	0.00324965	0.00162483	−	0.999999i	$-0.499483\pi$
$907$	0.0978680	0.00324965	0.00162483	+	0.999999i	$0.499483\pi$
$908$	25.0665	0.831861
$909$	0	0
$910$	0	0
$911$	25.9494i	0.859740i	0.902891	+	0.429870i	$0.141441\pi$
$911$	25.9494i	0.859740i	−0.902891	+	0.429870i	$0.858559\pi$
$912$	0	0
$913$	12.2445i	0.405235i
$914$	7.37871i	0.244066i
$915$	0	0
$916$	7.43444i	0.245641i
$917$	0	0
$918$	0	0
$919$	17.6538	0.582344	0.291172	−	0.956671i	$-0.405955\pi$
$919$	17.6538	0.582344	0.291172	+	0.956671i	$0.405955\pi$
$920$	11.7686	0.387999
$921$	0	0
$922$	− 17.4146i	− 0.573520i
$923$	23.8949	0.786512
$924$	0	0
$925$	−1.44977	−0.0476680
$926$	− 3.82068i	− 0.125555i
$927$	0	0
$928$	42.2459	1.38679
$929$	24.8525	0.815385	0.407693	−	0.913119i	$-0.366334\pi$
$929$	24.8525	0.815385	0.407693	+	0.913119i	$0.366334\pi$
$930$	0	0
$931$	0	0
$932$	12.5991i	0.412696i
$933$	0	0
$934$	2.05833i	0.0673506i
$935$	− 3.20617i	− 0.104853i
$936$	0	0
$937$	− 30.1769i	− 0.985837i	−0.870075	−	0.492918i	$-0.835930\pi$
$937$	− 30.1769i	− 0.985837i	0.870075	−	0.492918i	$-0.164070\pi$
$938$	0	0
$939$	0	0
$940$	−8.40703	−0.274207
$941$	−31.4158	−1.02413	−0.512063	−	0.858948i	$-0.671119\pi$
$941$	−31.4158	−1.02413	−0.512063	+	0.858948i	$0.671119\pi$
$942$	0	0
$943$	− 52.2919i	− 1.70286i
$944$	5.56721	0.181197
$945$	0	0
$946$	8.51036	0.276696
$947$	− 50.7620i	− 1.64954i	−0.565466	−	0.824772i	$-0.691304\pi$
$947$	− 50.7620i	− 1.64954i	0.565466	−	0.824772i	$-0.308696\pi$
$948$	0	0
$949$	13.1065	0.425456
$950$	−4.32058	−0.140178
$951$	0	0
$952$	0	0
$953$	34.7241i	1.12482i	0.826858	+	0.562411i	$0.190126\pi$
$953$	34.7241i	1.12482i	−0.826858	+	0.562411i	$0.809874\pi$
$954$	0	0
$955$	− 13.6957i	− 0.443182i
$956$	1.41814i	0.0458658i
$957$	0	0
$958$	− 11.3260i	− 0.365925i
$959$	0	0
$960$	0	0
$961$	29.1834	0.941401
$962$	5.04019	0.162502
$963$	0	0
$964$	42.4965i	1.36872i
$965$	7.48896	0.241078
$966$	0	0
$967$	−34.7915	−1.11882	−0.559410	−	0.828891i	$-0.688972\pi$
$967$	−34.7915	−1.11882	−0.559410	+	0.828891i	$0.688972\pi$
$968$	− 14.7985i	− 0.475642i
$969$	0	0
$970$	−7.81728	−0.250998
$971$	−47.9049	−1.53734	−0.768670	−	0.639646i	$-0.779081\pi$
$971$	−47.9049	−1.53734	−0.768670	+	0.639646i	$0.779081\pi$
$972$	0	0
$973$	0	0
$974$	16.3129i	0.522698i
$975$	0	0
$976$	8.93279i	0.285932i
$977$	29.7314i	0.951192i	0.879664	+	0.475596i	$0.157768\pi$
$977$	29.7314i	0.951192i	−0.879664	+	0.475596i	$0.842232\pi$
$978$	0	0
$979$	− 8.57364i	− 0.274015i
$980$	0	0
$981$	0	0
$982$	−24.9818	−0.797201
$983$	−38.1303	−1.21617	−0.608084	−	0.793873i	$-0.708062\pi$
$983$	−38.1303	−1.21617	−0.608084	+	0.793873i	$0.708062\pi$
$984$	0	0
$985$	2.07356i	0.0660693i
$986$	7.11968	0.226737
$987$	0	0
$988$	−66.3102	−2.10961
$989$	− 36.0433i	− 1.14611i
$990$	0	0
$991$	13.7304	0.436159	0.218080	−	0.975931i	$-0.430021\pi$
$991$	13.7304	0.436159	0.218080	+	0.975931i	$0.430021\pi$
$992$	−7.52091	−0.238789
$993$	0	0
$994$	0	0
$995$	1.44061i	0.0456704i
$996$	0	0
$997$	57.5721i	1.82333i	0.410936	+	0.911664i	$0.365202\pi$
$997$	57.5721i	1.82333i	−0.410936	+	0.911664i	$0.634798\pi$
$998$	− 3.58094i	− 0.113353i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.b.d.881.7	yes	16
3.2	odd	2		2205.2.b.c.881.10	yes	16
7.6	odd	2		2205.2.b.c.881.7	✓	16
21.20	even	2	inner	2205.2.b.d.881.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2205.2.b.c.881.7	✓	16	7.6	odd	2
2205.2.b.c.881.10	yes	16	3.2	odd	2
2205.2.b.d.881.7	yes	16	1.1	even	1	trivial
2205.2.b.d.881.10	yes	16	21.20	even	2	inner

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.607761i q^{2} +1.63063 q^{4} +1.00000 q^{5} -2.20655i q^{8} -0.607761i q^{10} -2.07205i q^{11} -5.72027i q^{13} +1.92020 q^{16} +1.54734 q^{17} -7.10901i q^{19} +1.63063 q^{20} -1.25931 q^{22} +5.33347i q^{23} +1.00000 q^{25} -3.47655 q^{26} +7.57079i q^{29} -1.34780i q^{31} -5.58013i q^{32} -0.940414i q^{34} -1.44977 q^{37} -4.32058 q^{38} -2.20655i q^{40} -9.80448 q^{41} -6.75795 q^{43} -3.37874i q^{44} +3.24147 q^{46} -5.15570 q^{47} -0.607761i q^{50} -9.32762i q^{52} -9.34386i q^{53} -2.07205i q^{55} +4.60123 q^{58} +2.89929 q^{59} +4.65202i q^{61} -0.819142 q^{62} +0.449013 q^{64} -5.72027i q^{65} +9.35484 q^{67} +2.52314 q^{68} +4.17724i q^{71} +2.29124i q^{73} +0.881111i q^{74} -11.5921i q^{76} +14.0324 q^{79} +1.92020 q^{80} +5.95878i q^{82} -5.90939 q^{83} +1.54734 q^{85} +4.10722i q^{86} -4.57209 q^{88} +4.13776 q^{89} +8.69690i q^{92} +3.13343i q^{94} -7.10901i q^{95} -12.8624i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{5} - 16 q^{20} - 16 q^{22} + 16 q^{25} + 32 q^{26} - 16 q^{38} - 32 q^{41} + 32 q^{43} - 16 q^{46} + 32 q^{47} + 48 q^{58} - 32 q^{59} - 32 q^{62} + 16 q^{64} - 16 q^{68} + 32 q^{79}+ \cdots + 64 q^{89}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^5 - 16 * q^20 - 16 * q^22 + 16 * q^25 + 32 * q^26 - 16 * q^38 - 32 * q^41 + 32 * q^43 - 16 * q^46 + 32 * q^47 + 48 * q^58 - 32 * q^59 - 32 * q^62 + 16 * q^64 - 16 * q^68 + 32 * q^79 + 16 * q^88 + 64 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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