LMFDB - Embedded newform 1620.4.i.a.1081.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, degree $2$, not minimal)

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:	no
Analytic conductor:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.a.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 4.33013i) q^{5} +(-16.0000 - 27.7128i) q^{7} +(-18.0000 - 31.1769i) q^{11} +(5.00000 - 8.66025i) q^{13} -78.0000 q^{17} +140.000 q^{19} +(96.0000 - 166.277i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-3.00000 - 5.19615i) q^{29} +(8.00000 - 13.8564i) q^{31} +160.000 q^{35} -34.0000 q^{37} +(195.000 - 337.750i) q^{41} +(26.0000 + 45.0333i) q^{43} +(-204.000 - 353.338i) q^{47} +(-340.500 + 589.763i) q^{49} -114.000 q^{53} +180.000 q^{55} +(-258.000 + 446.869i) q^{59} +(29.0000 + 50.2295i) q^{61} +(25.0000 + 43.3013i) q^{65} +(446.000 - 772.495i) q^{67} -120.000 q^{71} -646.000 q^{73} +(-576.000 + 997.661i) q^{77} +(584.000 + 1011.52i) q^{79} +(366.000 + 633.931i) q^{83} +(195.000 - 337.750i) q^{85} -1590.00 q^{89} -320.000 q^{91} +(-350.000 + 606.218i) q^{95} +(-97.0000 - 168.009i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{5} - 32 q^{7} - 36 q^{11} + 10 q^{13} - 156 q^{17} + 280 q^{19} + 192 q^{23} - 25 q^{25} - 6 q^{29} + 16 q^{31} + 320 q^{35} - 68 q^{37} + 390 q^{41} + 52 q^{43} - 408 q^{47} - 681 q^{49} - 228 q^{53}+ \cdots - 194 q^{97}+O(q^{100}) $ 2 * q - 5 * q^5 - 32 * q^7 - 36 * q^11 + 10 * q^13 - 156 * q^17 + 280 * q^19 + 192 * q^23 - 25 * q^25 - 6 * q^29 + 16 * q^31 + 320 * q^35 - 68 * q^37 + 390 * q^41 + 52 * q^43 - 408 * q^47 - 681 * q^49 - 228 * q^53 + 360 * q^55 - 516 * q^59 + 58 * q^61 + 50 * q^65 + 892 * q^67 - 240 * q^71 - 1292 * q^73 - 1152 * q^77 + 1168 * q^79 + 732 * q^83 + 390 * q^85 - 3180 * q^89 - 640 * q^91 - 700 * q^95 - 194 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	−16.0000	−	27.7128i	−0.863919	−	1.49635i	−0.868117	−	0.496360i	$-0.834670\pi$
$7$	−16.0000	−	27.7128i	−0.863919	−	1.49635i	0.00419795	−	0.999991i	$-0.498664\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−18.0000	−	31.1769i	−0.493382	−	0.854563i	0.506589	−	0.862188i	$-0.330906\pi$
$11$	−18.0000	−	31.1769i	−0.493382	−	0.854563i	−0.999971	+	0.00762479i	$0.997573\pi$
$12$		0			0
$13$	5.00000	−	8.66025i	0.106673	−	0.184763i	−0.807747	−	0.589529i	$-0.799313\pi$
$13$	5.00000	−	8.66025i	0.106673	−	0.184763i	0.914421	+	0.404765i	$0.132647\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−78.0000			−1.11281			−0.556405	−	0.830911i	$-0.687820\pi$
$17$	−78.0000			−1.11281			−0.556405	+	0.830911i	$0.687820\pi$
$18$		0			0
$19$	140.000			1.69043			0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000			1.69043			0.845216	+	0.534425i	$0.179472\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	96.0000	−	166.277i	0.870321	−	1.50744i	0.00865615	−	0.999963i	$-0.497245\pi$
$23$	96.0000	−	166.277i	0.870321	−	1.50744i	0.861665	−	0.507478i	$-0.169422\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000	−	5.19615i	−0.0192099	−	0.0332725i	0.856261	−	0.516544i	$-0.172782\pi$
$29$	−3.00000	−	5.19615i	−0.0192099	−	0.0332725i	−0.875471	+	0.483272i	$0.839448\pi$
$30$		0			0
$31$	8.00000	−	13.8564i	0.0463498	−	0.0802801i	−0.841920	−	0.539603i	$-0.818574\pi$
$31$	8.00000	−	13.8564i	0.0463498	−	0.0802801i	0.888270	+	0.459323i	$0.151908\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	160.000			0.772712
$36$		0			0
$37$	−34.0000			−0.151069			−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000			−0.151069			−0.0755347	+	0.997143i	$0.524066\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	195.000	−	337.750i	0.742778	−	1.28653i	−0.208448	−	0.978033i	$-0.566841\pi$
$41$	195.000	−	337.750i	0.742778	−	1.28653i	0.951226	−	0.308495i	$-0.0998254\pi$
$42$		0			0
$43$	26.0000	+	45.0333i	0.0922084	+	0.159710i	0.908440	−	0.418015i	$-0.137274\pi$
$43$	26.0000	+	45.0333i	0.0922084	+	0.159710i	−0.816232	+	0.577725i	$0.803941\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−204.000	−	353.338i	−0.633116	−	1.09659i	−0.986911	−	0.161266i	$-0.948442\pi$
$47$	−204.000	−	353.338i	−0.633116	−	1.09659i	0.353795	−	0.935323i	$-0.384891\pi$
$48$		0			0
$49$	−340.500	+	589.763i	−0.992711	+	1.71943i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−114.000			−0.295455			−0.147727	−	0.989028i	$-0.547196\pi$
$53$	−114.000			−0.295455			−0.147727	+	0.989028i	$0.547196\pi$
$54$		0			0
$55$	180.000			0.441294
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−258.000	+	446.869i	−0.569301	+	0.986058i	0.427335	+	0.904094i	$0.359453\pi$
$59$	−258.000	+	446.869i	−0.569301	+	0.986058i	−0.996635	+	0.0819641i	$0.973881\pi$
$60$		0			0
$61$	29.0000	+	50.2295i	0.0608700	+	0.105430i	0.894855	−	0.446358i	$-0.147279\pi$
$61$	29.0000	+	50.2295i	0.0608700	+	0.105430i	−0.833985	+	0.551788i	$0.813946\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	25.0000	+	43.3013i	0.0477057	+	0.0826286i
$66$		0			0
$67$	446.000	−	772.495i	0.813247	−	1.40859i	−0.0973322	−	0.995252i	$-0.531031\pi$
$67$	446.000	−	772.495i	0.813247	−	1.40859i	0.910580	−	0.413334i	$-0.135636\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−120.000			−0.200583			−0.100291	−	0.994958i	$-0.531978\pi$
$71$	−120.000			−0.200583			−0.100291	+	0.994958i	$0.531978\pi$
$72$		0			0
$73$	−646.000			−1.03573			−0.517867	−	0.855461i	$-0.673274\pi$
$73$	−646.000			−1.03573			−0.517867	+	0.855461i	$0.673274\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−576.000	+	997.661i	−0.852484	+	1.47655i
$78$		0			0
$79$	584.000	+	1011.52i	0.831711	+	1.44056i	0.896681	+	0.442678i	$0.145971\pi$
$79$	584.000	+	1011.52i	0.831711	+	1.44056i	−0.0649702	+	0.997887i	$0.520695\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	366.000	+	633.931i	0.484021	+	0.838348i	0.999832	−	0.0183540i	$-0.00584260\pi$
$83$	366.000	+	633.931i	0.484021	+	0.838348i	−0.515811	+	0.856703i	$0.672509\pi$
$84$		0			0
$85$	195.000	−	337.750i	0.248832	−	0.430990i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1590.00			−1.89370			−0.946852	−	0.321669i	$-0.895756\pi$
$89$	−1590.00			−1.89370			−0.946852	+	0.321669i	$0.895756\pi$
$90$		0			0
$91$	−320.000			−0.368628
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−350.000	+	606.218i	−0.377992	+	0.654701i
$96$		0			0
$97$	−97.0000	−	168.009i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−97.0000	−	168.009i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−399.000	−	691.088i	−0.393089	−	0.680850i	0.599766	−	0.800175i	$-0.295260\pi$
$101$	−399.000	−	691.088i	−0.393089	−	0.680850i	−0.992855	+	0.119325i	$0.961927\pi$
$102$		0			0
$103$	−136.000	+	235.559i	−0.130102	+	0.225343i	−0.923716	−	0.383079i	$-0.874864\pi$
$103$	−136.000	+	235.559i	−0.130102	+	0.225343i	0.793614	+	0.608422i	$0.208197\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	156.000			0.140945			0.0704724	−	0.997514i	$-0.477549\pi$
$107$	156.000			0.140945			0.0704724	+	0.997514i	$0.477549\pi$
$108$		0			0
$109$	1622.00			1.42532			0.712658	−	0.701512i	$-0.247491\pi$
$109$	1622.00			1.42532			0.712658	+	0.701512i	$0.247491\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−537.000	+	930.111i	−0.447051	+	0.774314i	−0.998193	−	0.0600972i	$-0.980859\pi$
$113$	−537.000	+	930.111i	−0.447051	+	0.774314i	0.551142	+	0.834411i	$0.314192\pi$
$114$		0			0
$115$	480.000	+	831.384i	0.389219	+	0.674148i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1248.00	+	2161.60i	0.961378	+	1.66516i
$120$		0			0
$121$	17.5000	−	30.3109i	0.0131480	−	0.0227730i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	−1528.00			−1.06762			−0.533811	−	0.845604i	$-0.679241\pi$
$127$	−1528.00			−1.06762			−0.533811	+	0.845604i	$0.679241\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1206.00	+	2088.85i	−0.804341	+	1.39316i	0.112394	+	0.993664i	$0.464148\pi$
$131$	−1206.00	+	2088.85i	−0.804341	+	1.39316i	−0.916735	+	0.399496i	$0.869185\pi$
$132$		0			0
$133$	−2240.00	−	3879.79i	−1.46040	−	2.52948i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1053.00	−	1823.85i	−0.656671	−	1.13739i	−0.981472	−	0.191605i	$-0.938631\pi$
$137$	−1053.00	−	1823.85i	−0.656671	−	1.13739i	0.324802	−	0.945782i	$-0.394702\pi$
$138$		0			0
$139$	278.000	−	481.510i	0.169638	−	0.293821i	−0.768655	−	0.639664i	$-0.779074\pi$
$139$	278.000	−	481.510i	0.169638	−	0.293821i	0.938293	+	0.345843i	$0.112407\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−360.000			−0.210522
$144$		0			0
$145$	30.0000			0.0171818
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1209.00	−	2094.05i	0.664732	−	1.15135i	−0.314625	−	0.949216i	$-0.601879\pi$
$149$	1209.00	−	2094.05i	0.664732	−	1.15135i	0.979358	−	0.202134i	$-0.0647878\pi$
$150$		0			0
$151$	−1420.00	−	2459.51i	−0.765285	−	1.32551i	−0.940096	−	0.340910i	$-0.889265\pi$
$151$	−1420.00	−	2459.51i	−0.765285	−	1.32551i	0.174812	−	0.984602i	$-0.444068\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	40.0000	+	69.2820i	0.0207282	+	0.0359024i
$156$		0			0
$157$	−1027.00	+	1778.82i	−0.522061	+	0.904236i	0.477610	+	0.878572i	$0.341503\pi$
$157$	−1027.00	+	1778.82i	−0.522061	+	0.904236i	−0.999671	+	0.0256636i	$0.991830\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6144.00			−3.00755
$162$		0			0
$163$	−460.000			−0.221043			−0.110521	−	0.993874i	$-0.535252\pi$
$163$	−460.000			−0.221043			−0.110521	+	0.993874i	$0.535252\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1008.00	+	1745.91i	−0.467074	+	0.808996i	−0.999292	−	0.0376110i	$-0.988025\pi$
$167$	−1008.00	+	1745.91i	−0.467074	+	0.808996i	0.532218	+	0.846607i	$0.321359\pi$
$168$		0			0
$169$	1048.50	+	1816.06i	0.477242	+	0.826607i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	309.000	+	535.204i	0.135797	+	0.235207i	0.925902	−	0.377765i	$-0.123307\pi$
$173$	309.000	+	535.204i	0.135797	+	0.235207i	−0.790105	+	0.612972i	$0.789974\pi$
$174$		0			0
$175$	−400.000	+	692.820i	−0.172784	+	0.299270i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2964.00			−1.23765			−0.618826	−	0.785528i	$-0.712391\pi$
$179$	−2964.00			−1.23765			−0.618826	+	0.785528i	$0.712391\pi$
$180$		0			0
$181$	−370.000			−0.151944			−0.0759721	−	0.997110i	$-0.524206\pi$
$181$	−370.000			−0.151944			−0.0759721	+	0.997110i	$0.524206\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	85.0000	−	147.224i	0.0337801	−	0.0585089i
$186$		0			0
$187$	1404.00	+	2431.80i	0.549041	+	0.950967i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−552.000	−	956.092i	−0.209117	−	0.362201i	0.742320	−	0.670046i	$-0.233726\pi$
$191$	−552.000	−	956.092i	−0.209117	−	0.362201i	−0.951437	+	0.307845i	$0.900392\pi$
$192$		0			0
$193$	1199.00	−	2076.73i	0.447181	−	0.774540i	−0.551020	−	0.834492i	$-0.685761\pi$
$193$	1199.00	−	2076.73i	0.447181	−	0.774540i	0.998201	+	0.0599518i	$0.0190947\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1278.00			0.462202			0.231101	−	0.972930i	$-0.425767\pi$
$197$	1278.00			0.462202			0.231101	+	0.972930i	$0.425767\pi$
$198$		0			0
$199$	4472.00			1.59302			0.796512	−	0.604623i	$-0.206676\pi$
$199$	4472.00			1.59302			0.796512	+	0.604623i	$0.206676\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−96.0000	+	166.277i	−0.0331915	+	0.0574894i
$204$		0			0
$205$	975.000	+	1688.75i	0.332180	+	0.575353i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2520.00	−	4364.77i	−0.834029	−	1.44458i
$210$		0			0
$211$	−670.000	+	1160.47i	−0.218600	+	0.378627i	−0.954380	−	0.298594i	$-0.903482\pi$
$211$	−670.000	+	1160.47i	−0.218600	+	0.378627i	0.735780	+	0.677221i	$0.236816\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−260.000			−0.0824737
$216$		0			0
$217$	−512.000			−0.160170
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−390.000	+	675.500i	−0.118707	+	0.205606i
$222$		0			0
$223$	−1180.00	−	2043.82i	−0.354344	−	0.613741i	0.632662	−	0.774428i	$-0.281962\pi$
$223$	−1180.00	−	2043.82i	−0.354344	−	0.613741i	−0.987005	+	0.160687i	$0.948629\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−690.000	−	1195.12i	−0.201748	−	0.349439i	0.747343	−	0.664438i	$-0.231329\pi$
$227$	−690.000	−	1195.12i	−0.201748	−	0.349439i	−0.949092	+	0.314999i	$0.897996\pi$
$228$		0			0
$229$	−847.000	+	1467.05i	−0.244416	+	0.423341i	−0.961967	−	0.273164i	$-0.911930\pi$
$229$	−847.000	+	1467.05i	−0.244416	+	0.423341i	0.717551	+	0.696506i	$0.245263\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5190.00			−1.45926			−0.729631	−	0.683841i	$-0.760308\pi$
$233$	−5190.00			−1.45926			−0.729631	+	0.683841i	$0.760308\pi$
$234$		0			0
$235$	2040.00			0.566276
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1176.00	+	2036.89i	−0.318281	+	0.551279i	−0.980129	−	0.198359i	$-0.936439\pi$
$239$	−1176.00	+	2036.89i	−0.318281	+	0.551279i	0.661849	+	0.749638i	$0.269772\pi$
$240$		0			0
$241$	1751.00	+	3032.82i	0.468016	+	0.810627i	0.999332	−	0.0365464i	$-0.0116357\pi$
$241$	1751.00	+	3032.82i	0.468016	+	0.810627i	−0.531316	+	0.847174i	$0.678302\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1702.50	−	2948.82i	−0.443954	−	0.768951i
$246$		0			0
$247$	700.000	−	1212.44i	0.180324	−	0.312330i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4788.00			1.20405			0.602024	−	0.798478i	$-0.294361\pi$
$251$	4788.00			1.20405			0.602024	+	0.798478i	$0.294361\pi$
$252$		0			0
$253$	−6912.00			−1.71760
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−753.000	+	1304.23i	−0.182766	+	0.316560i	−0.942821	−	0.333298i	$-0.891838\pi$
$257$	−753.000	+	1304.23i	−0.182766	+	0.316560i	0.760055	+	0.649858i	$0.225172\pi$
$258$		0			0
$259$	544.000	+	942.236i	0.130512	+	0.226053i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	216.000	+	374.123i	0.0506431	+	0.0877164i	0.890236	−	0.455500i	$-0.150540\pi$
$263$	216.000	+	374.123i	0.0506431	+	0.0877164i	−0.839593	+	0.543217i	$0.817206\pi$
$264$		0			0
$265$	285.000	−	493.634i	0.0660657	−	0.114429i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	54.0000			0.0122395			0.00611977	−	0.999981i	$-0.498052\pi$
$269$	54.0000			0.0122395			0.00611977	+	0.999981i	$0.498052\pi$
$270$		0			0
$271$	−6496.00			−1.45610			−0.728051	−	0.685522i	$-0.759574\pi$
$271$	−6496.00			−1.45610			−0.728051	+	0.685522i	$0.759574\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−450.000	+	779.423i	−0.0986764	+	0.170913i
$276$		0			0
$277$	233.000	+	403.568i	0.0505401	+	0.0875381i	0.890189	−	0.455592i	$-0.150572\pi$
$277$	233.000	+	403.568i	0.0505401	+	0.0875381i	−0.839649	+	0.543130i	$0.817239\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2427.00	+	4203.69i	0.515241	+	0.892423i	0.999844	+	0.0176890i	$0.00563087\pi$
$281$	2427.00	+	4203.69i	0.515241	+	0.892423i	−0.484603	+	0.874734i	$0.661036\pi$
$282$		0			0
$283$	2258.00	−	3910.97i	0.474290	−	0.821495i	−0.525276	−	0.850932i	$-0.676038\pi$
$283$	2258.00	−	3910.97i	0.474290	−	0.821495i	0.999567	+	0.0294368i	$0.00937136\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12480.0			−2.56680
$288$		0			0
$289$	1171.00			0.238347
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4287.00	+	7425.30i	−0.854775	+	1.48051i	0.0220777	+	0.999756i	$0.492972\pi$
$293$	−4287.00	+	7425.30i	−0.854775	+	1.48051i	−0.876853	+	0.480758i	$0.840361\pi$
$294$		0			0
$295$	−1290.00	−	2234.35i	−0.254599	−	0.440978i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−960.000	−	1662.77i	−0.185680	−	0.321607i
$300$		0			0
$301$	832.000	−	1441.07i	0.159321	−	0.275952i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−290.000			−0.0544438
$306$		0			0
$307$	3476.00			0.646208			0.323104	−	0.946363i	$-0.395274\pi$
$307$	3476.00			0.646208			0.323104	+	0.946363i	$0.395274\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1212.00	+	2099.25i	−0.220985	+	0.382757i	−0.955107	−	0.296260i	$-0.904260\pi$
$311$	−1212.00	+	2099.25i	−0.220985	+	0.382757i	0.734123	+	0.679017i	$0.237594\pi$
$312$		0			0
$313$	779.000	+	1349.27i	0.140676	+	0.243659i	0.927751	−	0.373199i	$-0.121739\pi$
$313$	779.000	+	1349.27i	0.140676	+	0.243659i	−0.787075	+	0.616857i	$0.788406\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4269.00	+	7394.12i	0.756375	+	1.31008i	0.944688	+	0.327971i	$0.106365\pi$
$317$	4269.00	+	7394.12i	0.756375	+	1.31008i	−0.188313	+	0.982109i	$0.560302\pi$
$318$		0			0
$319$	−108.000	+	187.061i	−0.0189556	+	0.0328321i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10920.0			−1.88113
$324$		0			0
$325$	−250.000			−0.0426692
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6528.00	+	11306.8i	−1.09392	+	1.89473i
$330$		0			0
$331$	494.000	+	855.633i	0.0820323	+	0.142084i	0.904123	−	0.427273i	$-0.140526\pi$
$331$	494.000	+	855.633i	0.0820323	+	0.142084i	−0.822090	+	0.569357i	$0.807192\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2230.00	+	3862.47i	0.363695	+	0.629939i
$336$		0			0
$337$	−1273.00	+	2204.90i	−0.205771	+	0.356405i	−0.950378	−	0.311097i	$-0.899303\pi$
$337$	−1273.00	+	2204.90i	−0.205771	+	0.356405i	0.744607	+	0.667503i	$0.232637\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−576.000			−0.0914726
$342$		0			0
$343$	10816.0			1.70265
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4278.00	+	7409.71i	−0.661830	+	1.14632i	0.318304	+	0.947989i	$0.396887\pi$
$347$	−4278.00	+	7409.71i	−0.661830	+	1.14632i	−0.980134	+	0.198335i	$0.936447\pi$
$348$		0			0
$349$	1853.00	+	3209.49i	0.284209	+	0.492264i	0.972417	−	0.233249i	$-0.0749358\pi$
$349$	1853.00	+	3209.49i	0.284209	+	0.492264i	−0.688208	+	0.725513i	$0.741602\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5697.00	−	9867.49i	−0.858982	−	1.48780i	−0.872901	−	0.487898i	$-0.837764\pi$
$353$	−5697.00	−	9867.49i	−0.858982	−	1.48780i	0.0139186	−	0.999903i	$-0.495569\pi$
$354$		0			0
$355$	300.000	−	519.615i	0.0448517	−	0.0776854i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−264.000			−0.0388117			−0.0194058	−	0.999812i	$-0.506177\pi$
$359$	−264.000			−0.0388117			−0.0194058	+	0.999812i	$0.506177\pi$
$360$		0			0
$361$	12741.0			1.85756
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1615.00	−	2797.26i	0.231597	−	0.401138i
$366$		0			0
$367$	−5116.00	−	8861.17i	−0.727665	−	1.26035i	−0.957868	−	0.287210i	$-0.907272\pi$
$367$	−5116.00	−	8861.17i	−0.727665	−	1.26035i	0.230203	−	0.973143i	$-0.426061\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1824.00	+	3159.26i	0.255249	+	0.442104i
$372$		0			0
$373$	281.000	−	486.706i	0.0390070	−	0.0675622i	−0.845863	−	0.533401i	$-0.820914\pi$
$373$	281.000	−	486.706i	0.0390070	−	0.0675622i	0.884870	+	0.465838i	$0.154247\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−60.0000			−0.00819670
$378$		0			0
$379$	−7228.00			−0.979624			−0.489812	−	0.871828i	$-0.662935\pi$
$379$	−7228.00			−0.979624			−0.489812	+	0.871828i	$0.662935\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2868.00	−	4967.52i	0.382632	−	0.662738i	−0.608806	−	0.793319i	$-0.708351\pi$
$383$	2868.00	−	4967.52i	0.382632	−	0.662738i	0.991438	+	0.130582i	$0.0416845\pi$
$384$		0			0
$385$	−2880.00	−	4988.31i	−0.381243	−	0.660332i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4593.00	+	7955.31i	0.598649	+	1.03689i	0.993021	+	0.117939i	$0.0376288\pi$
$389$	4593.00	+	7955.31i	0.598649	+	1.03689i	−0.394372	+	0.918951i	$0.629038\pi$
$390$		0			0
$391$	−7488.00	+	12969.6i	−0.968502	+	1.67750i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5840.00			−0.743905
$396$		0			0
$397$	−394.000			−0.0498093			−0.0249047	−	0.999690i	$-0.507928\pi$
$397$	−394.000			−0.0498093			−0.0249047	+	0.999690i	$0.507928\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	807.000	−	1397.77i	0.100498	−	0.174067i	−0.811392	−	0.584502i	$-0.801290\pi$
$401$	807.000	−	1397.77i	0.100498	−	0.174067i	0.911890	+	0.410435i	$0.134623\pi$
$402$		0			0
$403$	−80.0000	−	138.564i	−0.00988855	−	0.0171275i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	612.000	+	1060.02i	0.0745349	+	0.129098i
$408$		0			0
$409$	−517.000	+	895.470i	−0.0625037	+	0.108260i	−0.895584	−	0.444893i	$-0.853242\pi$
$409$	−517.000	+	895.470i	−0.0625037	+	0.108260i	0.833080	+	0.553152i	$0.186575\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	16512.0			1.96732
$414$		0			0
$415$	−3660.00			−0.432921
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1854.00	+	3211.22i	−0.216167	+	0.374412i	−0.953633	−	0.300972i	$-0.902689\pi$
$419$	−1854.00	+	3211.22i	−0.216167	+	0.374412i	0.737466	+	0.675384i	$0.236022\pi$
$420$		0			0
$421$	2465.00	+	4269.51i	0.285360	+	0.494259i	0.972697	−	0.232081i	$-0.0745534\pi$
$421$	2465.00	+	4269.51i	0.285360	+	0.494259i	−0.687336	+	0.726340i	$0.741220\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	975.000	+	1688.75i	0.111281	+	0.192744i
$426$		0			0
$427$	928.000	−	1607.34i	0.105173	−	0.182166i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2592.00			−0.289680			−0.144840	−	0.989455i	$-0.546267\pi$
$431$	−2592.00			−0.289680			−0.144840	+	0.989455i	$0.546267\pi$
$432$		0			0
$433$	2162.00			0.239952			0.119976	−	0.992777i	$-0.461718\pi$
$433$	2162.00			0.239952			0.119976	+	0.992777i	$0.461718\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	13440.0	−	23278.8i	1.47122	−	2.54822i
$438$		0			0
$439$	−676.000	−	1170.87i	−0.0734937	−	0.127295i	0.826937	−	0.562295i	$-0.190082\pi$
$439$	−676.000	−	1170.87i	−0.0734937	−	0.127295i	−0.900430	+	0.435000i	$0.856748\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2766.00	−	4790.85i	−0.296652	−	0.513816i	0.678716	−	0.734401i	$-0.262537\pi$
$443$	−2766.00	−	4790.85i	−0.296652	−	0.513816i	−0.975368	+	0.220585i	$0.929203\pi$
$444$		0			0
$445$	3975.00	−	6884.90i	0.423445	−	0.733428i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3198.00			−0.336131			−0.168066	−	0.985776i	$-0.553752\pi$
$449$	−3198.00			−0.336131			−0.168066	+	0.985776i	$0.553752\pi$
$450$		0			0
$451$	−14040.0			−1.46589
$452$		0			0
$453$		0			0
$454$		0			0
$455$	800.000	−	1385.64i	0.0824276	−	0.142769i
$456$		0			0
$457$	755.000	+	1307.70i	0.0772810	+	0.133855i	0.902076	−	0.431577i	$-0.142043\pi$
$457$	755.000	+	1307.70i	0.0772810	+	0.133855i	−0.824795	+	0.565432i	$0.808709\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8043.00	−	13930.9i	−0.812581	−	1.40743i	−0.911052	−	0.412292i	$-0.864728\pi$
$461$	−8043.00	−	13930.9i	−0.812581	−	1.40743i	0.0984709	−	0.995140i	$-0.468605\pi$
$462$		0			0
$463$	−2692.00	+	4662.68i	−0.270211	+	0.468020i	−0.968916	−	0.247391i	$-0.920427\pi$
$463$	−2692.00	+	4662.68i	−0.270211	+	0.468020i	0.698704	+	0.715410i	$0.253760\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2604.00			−0.258027			−0.129014	−	0.991643i	$-0.541181\pi$
$467$	−2604.00			−0.258027			−0.129014	+	0.991643i	$0.541181\pi$
$468$		0			0
$469$	−28544.0			−2.81032
$470$		0			0
$471$		0			0
$472$		0			0
$473$	936.000	−	1621.20i	0.0909880	−	0.157596i
$474$		0			0
$475$	−1750.00	−	3031.09i	−0.169043	−	0.292791i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5568.00	−	9644.06i	−0.531124	−	0.919934i	−0.999340	−	0.0363199i	$-0.988436\pi$
$479$	−5568.00	−	9644.06i	−0.531124	−	0.919934i	0.468216	−	0.883614i	$-0.344897\pi$
$480$		0			0
$481$	−170.000	+	294.449i	−0.0161150	+	0.0279121i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	970.000			0.0908153
$486$		0			0
$487$	14624.0			1.36073			0.680366	−	0.732872i	$-0.261821\pi$
$487$	14624.0			1.36073			0.680366	+	0.732872i	$0.261821\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5922.00	+	10257.2i	−0.544310	+	0.942772i	0.454340	+	0.890828i	$0.349875\pi$
$491$	−5922.00	+	10257.2i	−0.544310	+	0.942772i	−0.998650	+	0.0519440i	$0.983458\pi$
$492$		0			0
$493$	234.000	+	405.300i	0.0213769	+	0.0370259i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1920.00	+	3325.54i	0.173287	+	0.300142i
$498$		0			0
$499$	5642.00	−	9772.23i	0.506154	−	0.876684i	−0.493821	−	0.869564i	$-0.664400\pi$
$499$	5642.00	−	9772.23i	0.506154	−	0.876684i	0.999975	−	0.00712011i	$-0.00226642\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4032.00			0.357412			0.178706	−	0.983903i	$-0.442809\pi$
$503$	4032.00			0.357412			0.178706	+	0.983903i	$0.442809\pi$
$504$		0			0
$505$	3990.00			0.351589
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8781.00	−	15209.1i	0.764658	−	1.32443i	−0.175769	−	0.984431i	$-0.556241\pi$
$509$	8781.00	−	15209.1i	0.764658	−	1.32443i	0.940427	−	0.339995i	$-0.110425\pi$
$510$		0			0
$511$	10336.0	+	17902.5i	0.894790	+	1.54982i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−680.000	−	1177.79i	−0.0581833	−	0.100776i
$516$		0			0
$517$	−7344.00	+	12720.2i	−0.624736	+	1.08208i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3162.00			0.265892			0.132946	−	0.991123i	$-0.457556\pi$
$521$	3162.00			0.265892			0.132946	+	0.991123i	$0.457556\pi$
$522$		0			0
$523$	6764.00			0.565524			0.282762	−	0.959190i	$-0.408749\pi$
$523$	6764.00			0.565524			0.282762	+	0.959190i	$0.408749\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−624.000	+	1080.80i	−0.0515785	+	0.0893366i
$528$		0			0
$529$	−12348.5	−	21388.2i	−1.01492	−	1.75789i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1950.00	−	3377.50i	−0.158469	−	0.274476i
$534$		0			0
$535$	−390.000	+	675.500i	−0.0315162	+	0.0545877i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	24516.0			1.95914
$540$		0			0
$541$	17798.0			1.41441			0.707205	−	0.707009i	$-0.249956\pi$
$541$	17798.0			1.41441			0.707205	+	0.707009i	$0.249956\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4055.00	+	7023.47i	−0.318710	+	0.552022i
$546$		0			0
$547$	9998.00	+	17317.0i	0.781506	+	1.35361i	0.931064	+	0.364855i	$0.118881\pi$
$547$	9998.00	+	17317.0i	0.781506	+	1.35361i	−0.149559	+	0.988753i	$0.547785\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−420.000	−	727.461i	−0.0324730	−	0.0562448i
$552$		0			0
$553$	18688.0	−	32368.6i	1.43706	−	2.48906i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11094.0			0.843928			0.421964	−	0.906613i	$-0.361341\pi$
$557$	11094.0			0.843928			0.421964	+	0.906613i	$0.361341\pi$
$558$		0			0
$559$	520.000			0.0393446
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−450.000	+	779.423i	−0.0336860	+	0.0583459i	−0.882377	−	0.470543i	$-0.844058\pi$
$563$	−450.000	+	779.423i	−0.0336860	+	0.0583459i	0.848691	+	0.528889i	$0.177391\pi$
$564$		0			0
$565$	−2685.00	−	4650.56i	−0.199927	−	0.346284i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3957.00	−	6853.73i	−0.291540	−	0.504962i	0.682634	−	0.730760i	$-0.260834\pi$
$569$	−3957.00	−	6853.73i	−0.291540	−	0.504962i	−0.974174	+	0.225799i	$0.927501\pi$
$570$		0			0
$571$	1190.00	−	2061.14i	0.0872153	−	0.151061i	−0.819118	−	0.573625i	$-0.805537\pi$
$571$	1190.00	−	2061.14i	0.0872153	−	0.151061i	0.906333	+	0.422564i	$0.138870\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4800.00			−0.348128
$576$		0			0
$577$	−25726.0			−1.85613			−0.928065	−	0.372417i	$-0.878529\pi$
$577$	−25726.0			−1.85613			−0.928065	+	0.372417i	$0.878529\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11712.0	−	20285.8i	0.836309	−	1.44853i
$582$		0			0
$583$	2052.00	+	3554.17i	0.145772	+	0.252485i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1806.00	−	3128.08i	−0.126987	−	0.219949i	0.795521	−	0.605926i	$-0.207197\pi$
$587$	−1806.00	−	3128.08i	−0.126987	−	0.219949i	−0.922508	+	0.385978i	$0.873864\pi$
$588$		0			0
$589$	1120.00	−	1939.90i	0.0783511	−	0.135708i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2898.00			0.200686			0.100343	−	0.994953i	$-0.468006\pi$
$593$	2898.00			0.200686			0.100343	+	0.994953i	$0.468006\pi$
$594$		0			0
$595$	−12480.0			−0.859883
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1332.00	+	2307.09i	−0.0908582	+	0.157371i	−0.907872	−	0.419246i	$-0.862294\pi$
$599$	−1332.00	+	2307.09i	−0.0908582	+	0.157371i	0.817014	+	0.576617i	$0.195628\pi$
$600$		0			0
$601$	251.000	+	434.745i	0.0170358	+	0.0295068i	0.874418	−	0.485174i	$-0.161244\pi$
$601$	251.000	+	434.745i	0.0170358	+	0.0295068i	−0.857382	+	0.514681i	$0.827910\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	87.5000	+	151.554i	0.00587997	+	0.0101844i
$606$		0			0
$607$	−3988.00	+	6907.42i	−0.266669	+	0.461884i	−0.967999	−	0.250952i	$-0.919256\pi$
$607$	−3988.00	+	6907.42i	−0.266669	+	0.461884i	0.701331	+	0.712836i	$0.252590\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4080.00			−0.270146
$612$		0			0
$613$	20414.0			1.34505			0.672523	−	0.740076i	$-0.265210\pi$
$613$	20414.0			1.34505			0.672523	+	0.740076i	$0.265210\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3171.00	−	5492.33i	0.206904	−	0.358368i	−0.743834	−	0.668365i	$-0.766995\pi$
$617$	3171.00	−	5492.33i	0.206904	−	0.358368i	0.950738	+	0.309997i	$0.100328\pi$
$618$		0			0
$619$	−11338.0	−	19638.0i	−0.736208	−	1.27515i	−0.954191	−	0.299197i	$-0.903281\pi$
$619$	−11338.0	−	19638.0i	−0.736208	−	1.27515i	0.217983	−	0.975952i	$-0.430052\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	25440.0	+	44063.4i	1.63601	+	2.83365i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2652.00			0.168112
$630$		0			0
$631$	−7048.00			−0.444654			−0.222327	−	0.974972i	$-0.571365\pi$
$631$	−7048.00			−0.444654			−0.222327	+	0.974972i	$0.571365\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3820.00	−	6616.43i	0.238728	−	0.413488i
$636$		0			0
$637$	3405.00	+	5897.63i	0.211791	+	0.366833i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10143.0	+	17568.2i	0.624999	+	1.08253i	0.988541	+	0.150952i	$0.0482340\pi$
$641$	10143.0	+	17568.2i	0.624999	+	1.08253i	−0.363542	+	0.931578i	$0.618433\pi$
$642$		0			0
$643$	8054.00	−	13949.9i	0.493964	−	0.855570i	−0.506012	−	0.862526i	$-0.668881\pi$
$643$	8054.00	−	13949.9i	0.493964	−	0.855570i	0.999976	+	0.00695598i	$0.00221417\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	27456.0			1.66833			0.834163	−	0.551518i	$-0.185951\pi$
$647$	27456.0			1.66833			0.834163	+	0.551518i	$0.185951\pi$
$648$		0			0
$649$	18576.0			1.12353
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6261.00	−	10844.4i	0.375210	−	0.649882i	−0.615149	−	0.788411i	$-0.710904\pi$
$653$	6261.00	−	10844.4i	0.375210	−	0.649882i	0.990358	+	0.138529i	$0.0442374\pi$
$654$		0			0
$655$	−6030.00	−	10444.3i	−0.359712	−	0.623040i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8154.00	+	14123.1i	0.481995	+	0.834840i	0.999786	−	0.0206670i	$-0.00657898\pi$
$659$	8154.00	+	14123.1i	0.481995	+	0.834840i	−0.517791	+	0.855507i	$0.673246\pi$
$660$		0			0
$661$	−16039.0	+	27780.4i	−0.943789	+	1.63469i	−0.185632	+	0.982619i	$0.559433\pi$
$661$	−16039.0	+	27780.4i	−0.943789	+	1.63469i	−0.758157	+	0.652072i	$0.773900\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	22400.0			1.30622
$666$		0			0
$667$	−1152.00			−0.0668750
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1044.00	−	1808.26i	0.0600643	−	0.104034i
$672$		0			0
$673$	−2305.00	−	3992.38i	−0.132023	−	0.228670i	0.792434	−	0.609958i	$-0.208814\pi$
$673$	−2305.00	−	3992.38i	−0.132023	−	0.228670i	−0.924456	+	0.381288i	$0.875480\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5391.00	−	9337.49i	−0.306046	−	0.530087i	0.671448	−	0.741052i	$-0.265673\pi$
$677$	−5391.00	−	9337.49i	−0.306046	−	0.530087i	−0.977494	+	0.210965i	$0.932339\pi$
$678$		0			0
$679$	−3104.00	+	5376.29i	−0.175435	+	0.303863i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	2892.00			0.162019			0.0810097	−	0.996713i	$-0.474186\pi$
$683$	2892.00			0.162019			0.0810097	+	0.996713i	$0.474186\pi$
$684$		0			0
$685$	10530.0			0.587344
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−570.000	+	987.269i	−0.0315171	+	0.0545892i
$690$		0			0
$691$	14786.0	+	25610.1i	0.814017	+	1.40992i	0.910031	+	0.414539i	$0.136057\pi$
$691$	14786.0	+	25610.1i	0.814017	+	1.40992i	−0.0960141	+	0.995380i	$0.530609\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1390.00	+	2407.55i	0.0758643	+	0.131401i
$696$		0			0
$697$	−15210.0	+	26344.5i	−0.826571	+	1.43166i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	5766.00			0.310669			0.155334	−	0.987862i	$-0.450354\pi$
$701$	5766.00			0.310669			0.155334	+	0.987862i	$0.450354\pi$
$702$		0			0
$703$	−4760.00			−0.255372
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12768.0	+	22114.8i	−0.679194	+	1.17640i
$708$		0			0
$709$	−1663.00	−	2880.40i	−0.0880892	−	0.152575i	0.818614	−	0.574344i	$-0.194743\pi$
$709$	−1663.00	−	2880.40i	−0.0880892	−	0.152575i	−0.906703	+	0.421769i	$0.861409\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1536.00	−	2660.43i	−0.0806783	−	0.139739i
$714$		0			0
$715$	900.000	−	1558.85i	0.0470743	−	0.0815350i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7728.00			−0.400843			−0.200421	−	0.979710i	$-0.564231\pi$
$719$	−7728.00			−0.400843			−0.200421	+	0.979710i	$0.564231\pi$
$720$		0			0
$721$	8704.00			0.449589
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−75.0000	+	129.904i	−0.00384197	+	0.00665449i
$726$		0			0
$727$	10808.0	+	18720.0i	0.551371	+	0.955002i	0.998176	+	0.0603709i	$0.0192283\pi$
$727$	10808.0	+	18720.0i	0.551371	+	0.955002i	−0.446805	+	0.894631i	$0.647438\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2028.00	−	3512.60i	−0.102611	−	0.177727i
$732$		0			0
$733$	−5059.00	+	8762.45i	−0.254923	+	0.441539i	−0.964875	−	0.262711i	$-0.915383\pi$
$733$	−5059.00	+	8762.45i	−0.254923	+	0.441539i	0.709952	+	0.704250i	$0.248717\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−32112.0			−1.60497
$738$		0			0
$739$	10460.0			0.520673			0.260336	−	0.965518i	$-0.416167\pi$
$739$	10460.0			0.520673			0.260336	+	0.965518i	$0.416167\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8616.00	+	14923.3i	−0.425424	+	0.736857i	−0.996460	−	0.0840686i	$-0.973209\pi$
$743$	−8616.00	+	14923.3i	−0.425424	+	0.736857i	0.571035	+	0.820925i	$0.306542\pi$
$744$		0			0
$745$	6045.00	+	10470.2i	0.297277	+	0.514900i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2496.00	−	4323.20i	−0.121765	−	0.210903i
$750$		0			0
$751$	−13456.0	+	23306.5i	−0.653817	+	1.13244i	0.328372	+	0.944548i	$0.393500\pi$
$751$	−13456.0	+	23306.5i	−0.653817	+	1.13244i	−0.982189	+	0.187896i	$0.939833\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14200.0			0.684491
$756$		0			0
$757$	13838.0			0.664400			0.332200	−	0.943209i	$-0.392209\pi$
$757$	13838.0			0.664400			0.332200	+	0.943209i	$0.392209\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8619.00	−	14928.5i	0.410563	−	0.711116i	−0.584388	−	0.811474i	$-0.698665\pi$
$761$	8619.00	−	14928.5i	0.410563	−	0.711116i	0.994951	+	0.100358i	$0.0319988\pi$
$762$		0			0
$763$	−25952.0	−	44950.2i	−1.23136	−	2.13277i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2580.00	+	4468.69i	0.121458	+	0.210372i
$768$		0			0
$769$	−10849.0	+	18791.0i	−0.508745	+	0.881172i	0.491204	+	0.871045i	$0.336557\pi$
$769$	−10849.0	+	18791.0i	−0.508745	+	0.881172i	−0.999949	+	0.0101275i	$0.996776\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18366.0			0.854565			0.427283	−	0.904118i	$-0.359471\pi$
$773$	18366.0			0.854565			0.427283	+	0.904118i	$0.359471\pi$
$774$		0			0
$775$	−400.000			−0.0185399
$776$		0			0
$777$		0			0
$778$		0			0
$779$	27300.0	−	47285.0i	1.25561	−	2.17479i
$780$		0			0
$781$	2160.00	+	3741.23i	0.0989640	+	0.171411i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5135.00	−	8894.08i	−0.233473	−	0.404386i
$786$		0			0
$787$	15158.0	−	26254.4i	0.686562	−	1.18916i	−0.286381	−	0.958116i	$-0.592452\pi$
$787$	15158.0	−	26254.4i	0.686562	−	1.18916i	0.972943	−	0.231045i	$-0.0742143\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	34368.0			1.54486
$792$		0			0
$793$	580.000			0.0259728
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3747.00	+	6489.99i	−0.166531	+	0.288441i	−0.937198	−	0.348798i	$-0.886590\pi$
$797$	−3747.00	+	6489.99i	−0.166531	+	0.288441i	0.770667	+	0.637239i	$0.219923\pi$
$798$		0			0
$799$	15912.0	+	27560.4i	0.704538	+	1.22030i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11628.0	+	20140.3i	0.511013	+	0.885100i
$804$		0			0
$805$	15360.0	−	26604.3i	0.672508	−	1.16482i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11526.0			−0.500906			−0.250453	−	0.968129i	$-0.580579\pi$
$809$	−11526.0			−0.500906			−0.250453	+	0.968129i	$0.580579\pi$
$810$		0			0
$811$	−33820.0			−1.46434			−0.732171	−	0.681121i	$-0.761493\pi$
$811$	−33820.0			−1.46434			−0.732171	+	0.681121i	$0.761493\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1150.00	−	1991.86i	0.0494267	−	0.0856095i
$816$		0			0
$817$	3640.00	+	6304.66i	0.155872	+	0.269978i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9783.00	−	16944.7i	−0.415870	−	0.720308i	0.579650	−	0.814866i	$-0.303189\pi$
$821$	−9783.00	−	16944.7i	−0.415870	−	0.720308i	−0.995519	+	0.0945583i	$0.969856\pi$
$822$		0			0
$823$	20048.0	−	34724.2i	0.849124	−	1.47073i	−0.0328664	−	0.999460i	$-0.510464\pi$
$823$	20048.0	−	34724.2i	0.849124	−	1.47073i	0.881991	−	0.471267i	$-0.156203\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31884.0			1.34065			0.670324	−	0.742069i	$-0.266155\pi$
$827$	31884.0			1.34065			0.670324	+	0.742069i	$0.266155\pi$
$828$		0			0
$829$	−24442.0			−1.02401			−0.512006	−	0.858982i	$-0.671097\pi$
$829$	−24442.0			−1.02401			−0.512006	+	0.858982i	$0.671097\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	26559.0	−	46001.5i	1.10470	−	1.91340i
$834$		0			0
$835$	−5040.00	−	8729.54i	−0.208882	−	0.361794i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15972.0	+	27664.3i	0.657228	+	1.13835i	0.981330	+	0.192331i	$0.0616047\pi$
$839$	15972.0	+	27664.3i	0.657228	+	1.13835i	−0.324102	+	0.946022i	$0.605062\pi$
$840$		0			0
$841$	12176.5	−	21090.3i	0.499262	−	0.864747i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10485.0			−0.426858
$846$		0			0
$847$	−1120.00			−0.0454352
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3264.00	+	5653.41i	−0.131479	+	0.227728i
$852$		0			0
$853$	−8743.00	−	15143.3i	−0.350943	−	0.607852i	0.635472	−	0.772124i	$-0.280806\pi$
$853$	−8743.00	−	15143.3i	−0.350943	−	0.607852i	−0.986415	+	0.164273i	$0.947472\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21717.0	−	37614.9i	−0.865623	−	1.49930i	−0.866428	−	0.499303i	$-0.833590\pi$
$857$	−21717.0	−	37614.9i	−0.865623	−	1.49930i	0.000804999	−	1.00000i	$-0.499744\pi$
$858$		0			0
$859$	−5410.00	+	9370.39i	−0.214886	+	0.372193i	−0.953237	−	0.302223i	$-0.902271\pi$
$859$	−5410.00	+	9370.39i	−0.214886	+	0.372193i	0.738352	+	0.674416i	$0.235605\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	29976.0			1.18238			0.591191	−	0.806532i	$-0.298658\pi$
$863$	29976.0			1.18238			0.591191	+	0.806532i	$0.298658\pi$
$864$		0			0
$865$	−3090.00			−0.121460
$866$		0			0
$867$		0			0
$868$		0			0
$869$	21024.0	−	36414.6i	0.820702	−	1.42150i
$870$		0			0
$871$	−4460.00	−	7724.95i	−0.173503	−	0.300516i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2000.00	−	3464.10i	−0.0772712	−	0.133838i
$876$		0			0
$877$	20261.0	−	35093.1i	0.780120	−	1.35121i	−0.151751	−	0.988419i	$-0.548491\pi$
$877$	20261.0	−	35093.1i	0.780120	−	1.35121i	0.931871	−	0.362789i	$-0.118175\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15570.0			0.595422			0.297711	−	0.954656i	$-0.403777\pi$
$881$	15570.0			0.595422			0.297711	+	0.954656i	$0.403777\pi$
$882$		0			0
$883$	−1084.00			−0.0413131			−0.0206566	−	0.999787i	$-0.506576\pi$
$883$	−1084.00			−0.0413131			−0.0206566	+	0.999787i	$0.506576\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4104.00	−	7108.34i	0.155354	−	0.269081i	−0.777834	−	0.628470i	$-0.783682\pi$
$887$	4104.00	−	7108.34i	0.155354	−	0.269081i	0.933188	+	0.359389i	$0.117015\pi$
$888$		0			0
$889$	24448.0	+	42345.2i	0.922339	+	1.59754i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−28560.0	−	49467.4i	−1.07024	−	1.85371i
$894$		0			0
$895$	7410.00	−	12834.5i	0.276747	−	0.479341i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−96.0000			−0.00356149
$900$		0			0
$901$	8892.00			0.328785
$902$		0			0
$903$		0			0
$904$		0			0
$905$	925.000	−	1602.15i	0.0339757	−	0.0588477i
$906$		0			0
$907$	−17038.0	−	29510.7i	−0.623746	−	1.08036i	−0.988782	−	0.149366i	$-0.952277\pi$
$907$	−17038.0	−	29510.7i	−0.623746	−	1.08036i	0.365036	−	0.930993i	$-0.381057\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7536.00	+	13052.7i	0.274071	+	0.474705i	0.969900	−	0.243502i	$-0.0782963\pi$
$911$	7536.00	+	13052.7i	0.274071	+	0.474705i	−0.695829	+	0.718207i	$0.744963\pi$
$912$		0			0
$913$	13176.0	−	22821.5i	0.477614	−	0.827252i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	77184.0			2.77954
$918$		0			0
$919$	24392.0			0.875536			0.437768	−	0.899088i	$-0.355769\pi$
$919$	24392.0			0.875536			0.437768	+	0.899088i	$0.355769\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−600.000	+	1039.23i	−0.0213968	+	0.0370603i
$924$		0			0
$925$	425.000	+	736.122i	0.0151069	+	0.0261660i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6801.00	−	11779.7i	−0.240187	−	0.416016i	0.720581	−	0.693371i	$-0.243875\pi$
$929$	−6801.00	−	11779.7i	−0.240187	−	0.416016i	−0.960767	+	0.277355i	$0.910542\pi$
$930$		0			0
$931$	−47670.0	+	82566.9i	−1.67811	+	2.90657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−14040.0			−0.491077
$936$		0			0
$937$	−47974.0			−1.67262			−0.836309	−	0.548259i	$-0.815291\pi$
$937$	−47974.0			−1.67262			−0.836309	+	0.548259i	$0.815291\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	24165.0	−	41855.0i	0.837148	−	1.44998i	−0.0551211	−	0.998480i	$-0.517554\pi$
$941$	24165.0	−	41855.0i	0.837148	−	1.44998i	0.892269	−	0.451504i	$-0.149112\pi$
$942$		0			0
$943$	−37440.0	−	64848.0i	−1.29291	−	2.23939i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4662.00	+	8074.82i	0.159973	+	0.277082i	0.934859	−	0.355020i	$-0.115526\pi$
$947$	4662.00	+	8074.82i	0.159973	+	0.277082i	−0.774886	+	0.632102i	$0.782193\pi$
$948$		0			0
$949$	−3230.00	+	5594.52i	−0.110485	+	0.191366i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−14838.0			−0.504355			−0.252177	−	0.967681i	$-0.581147\pi$
$953$	−14838.0			−0.504355			−0.252177	+	0.967681i	$0.581147\pi$
$954$		0			0
$955$	5520.00			0.187040
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−33696.0	+	58363.2i	−1.13462	+	1.96522i
$960$		0			0
$961$	14767.5	+	25578.1i	0.495703	+	0.858583i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5995.00	+	10383.6i	0.199985	+	0.346385i
$966$		0			0
$967$	−5680.00	+	9838.05i	−0.188890	+	0.327167i	−0.944880	−	0.327416i	$-0.893822\pi$
$967$	−5680.00	+	9838.05i	−0.188890	+	0.327167i	0.755991	+	0.654583i	$0.227156\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6972.00			−0.230424			−0.115212	−	0.993341i	$-0.536755\pi$
$971$	−6972.00			−0.230424			−0.115212	+	0.993341i	$0.536755\pi$
$972$		0			0
$973$	−17792.0			−0.586213
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20583.0	−	35650.8i	0.674011	−	1.16742i	−0.302746	−	0.953071i	$-0.597903\pi$
$977$	20583.0	−	35650.8i	0.674011	−	1.16742i	0.976757	−	0.214350i	$-0.0687632\pi$
$978$		0			0
$979$	28620.0	+	49571.3i	0.934320	+	1.61829i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−11232.0	−	19454.4i	−0.364441	−	0.631230i	0.624246	−	0.781228i	$-0.285406\pi$
$983$	−11232.0	−	19454.4i	−0.364441	−	0.631230i	−0.988686	+	0.149998i	$0.952073\pi$
$984$		0			0
$985$	−3195.00	+	5533.90i	−0.103351	+	0.179010i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9984.00			0.321004
$990$		0			0
$991$	−10192.0			−0.326700			−0.163350	−	0.986568i	$-0.552230\pi$
$991$	−10192.0			−0.326700			−0.163350	+	0.986568i	$0.552230\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−11180.0	+	19364.3i	−0.356211	+	0.616975i
$996$		0			0
$997$	161.000	+	278.860i	0.00511426	+	0.00885817i	0.868571	−	0.495564i	$-0.165039\pi$
$997$	161.000	+	278.860i	0.00511426	+	0.00885817i	−0.863457	+	0.504423i	$0.831705\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.a.1081.1		2
3.2	odd	2		1620.4.i.g.1081.1		2
9.2	odd	6		1620.4.i.g.541.1		2
9.4	even	3		60.4.a.b.1.1	✓	1
9.5	odd	6		180.4.a.c.1.1		1
9.7	even	3	inner	1620.4.i.a.541.1		2
36.23	even	6		720.4.a.c.1.1		1
36.31	odd	6		240.4.a.j.1.1		1
45.4	even	6		300.4.a.e.1.1		1
45.13	odd	12		300.4.d.d.49.1		2
45.14	odd	6		900.4.a.b.1.1		1
45.22	odd	12		300.4.d.d.49.2		2
45.23	even	12		900.4.d.b.649.1		2
45.32	even	12		900.4.d.b.649.2		2
72.13	even	6		960.4.a.bb.1.1		1
72.67	odd	6		960.4.a.a.1.1		1
180.67	even	12		1200.4.f.e.49.1		2
180.103	even	12		1200.4.f.e.49.2		2
180.139	odd	6		1200.4.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.a.b.1.1	✓	1	9.4	even	3
180.4.a.c.1.1		1	9.5	odd	6
240.4.a.j.1.1		1	36.31	odd	6
300.4.a.e.1.1		1	45.4	even	6
300.4.d.d.49.1		2	45.13	odd	12
300.4.d.d.49.2		2	45.22	odd	12
720.4.a.c.1.1		1	36.23	even	6
900.4.a.b.1.1		1	45.14	odd	6
900.4.d.b.649.1		2	45.23	even	12
900.4.d.b.649.2		2	45.32	even	12
960.4.a.a.1.1		1	72.67	odd	6
960.4.a.bb.1.1		1	72.13	even	6
1200.4.a.s.1.1		1	180.139	odd	6
1200.4.f.e.49.1		2	180.67	even	12
1200.4.f.e.49.2		2	180.103	even	12
1620.4.i.a.541.1		2	9.7	even	3	inner
1620.4.i.a.1081.1		2	1.1	even	1	trivial
1620.4.i.g.541.1		2	9.2	odd	6
1620.4.i.g.1081.1		2	3.2	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 4.33013i) q^{5} +(-16.0000 - 27.7128i) q^{7} +(-18.0000 - 31.1769i) q^{11} +(5.00000 - 8.66025i) q^{13} -78.0000 q^{17} +140.000 q^{19} +(96.0000 - 166.277i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-3.00000 - 5.19615i) q^{29} +(8.00000 - 13.8564i) q^{31} +160.000 q^{35} -34.0000 q^{37} +(195.000 - 337.750i) q^{41} +(26.0000 + 45.0333i) q^{43} +(-204.000 - 353.338i) q^{47} +(-340.500 + 589.763i) q^{49} -114.000 q^{53} +180.000 q^{55} +(-258.000 + 446.869i) q^{59} +(29.0000 + 50.2295i) q^{61} +(25.0000 + 43.3013i) q^{65} +(446.000 - 772.495i) q^{67} -120.000 q^{71} -646.000 q^{73} +(-576.000 + 997.661i) q^{77} +(584.000 + 1011.52i) q^{79} +(366.000 + 633.931i) q^{83} +(195.000 - 337.750i) q^{85} -1590.00 q^{89} -320.000 q^{91} +(-350.000 + 606.218i) q^{95} +(-97.0000 - 168.009i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{5} - 32 q^{7} - 36 q^{11} + 10 q^{13} - 156 q^{17} + 280 q^{19} + 192 q^{23} - 25 q^{25} - 6 q^{29} + 16 q^{31} + 320 q^{35} - 68 q^{37} + 390 q^{41} + 52 q^{43} - 408 q^{47} - 681 q^{49} - 228 q^{53}+ \cdots - 194 q^{97}+O(q^{100}) \) 2 * q - 5 * q^5 - 32 * q^7 - 36 * q^11 + 10 * q^13 - 156 * q^17 + 280 * q^19 + 192 * q^23 - 25 * q^25 - 6 * q^29 + 16 * q^31 + 320 * q^35 - 68 * q^37 + 390 * q^41 + 52 * q^43 - 408 * q^47 - 681 * q^49 - 228 * q^53 + 360 * q^55 - 516 * q^59 + 58 * q^61 + 50 * q^65 + 892 * q^67 - 240 * q^71 - 1292 * q^73 - 1152 * q^77 + 1168 * q^79 + 732 * q^83 + 390 * q^85 - 3180 * q^89 - 640 * q^91 - 700 * q^95 - 194 * q^97

Introduction

L-functions

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