LMFDB - Embedded newform 1620.4.i.x.541.5

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:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904 $ x^12 + 306x^10 + 30777x^8 + 1381040x^6 + 28918584x^4 + 243888288*x^2 + 366645904
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.5
Root			$-6.01416i$ of defining polynomial
Character	$\chi$	$=$	1620.541
Dual form			1620.4.i.x.1081.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{5} +(5.24065 - 9.07707i) q^{7} +O(q^{10})$	$q+(2.50000 + 4.33013i) q^{5} +(5.24065 - 9.07707i) q^{7} +(-16.6714 + 28.8758i) q^{11} +(11.9691 + 20.7311i) q^{13} -72.5536 q^{17} -45.9922 q^{19} +(81.5943 + 141.325i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(15.4887 - 26.8271i) q^{29} +(-152.400 - 263.964i) q^{31} +52.4065 q^{35} +150.155 q^{37} +(-204.532 - 354.261i) q^{41} +(124.587 - 215.791i) q^{43} +(-77.9496 + 135.013i) q^{47} +(116.571 + 201.907i) q^{49} -263.062 q^{53} -166.714 q^{55} +(-122.915 - 212.895i) q^{59} +(233.236 - 403.977i) q^{61} +(-59.8456 + 103.656i) q^{65} +(88.5079 + 153.300i) q^{67} -45.2138 q^{71} -949.331 q^{73} +(174.738 + 302.655i) q^{77} +(-248.459 + 430.343i) q^{79} +(177.318 - 307.124i) q^{83} +(-181.384 - 314.166i) q^{85} +1193.20 q^{89} +250.904 q^{91} +(-114.980 - 199.152i) q^{95} +(204.399 - 354.029i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 30 q^{5} - 12 q^{7}+O(q^{10}) $ 12 * q + 30 * q^5 - 12 * q^7	$ 12 q + 30 q^{5} - 12 q^{7} + 84 q^{13} + 24 q^{17} - 228 q^{19} - 30 q^{23} - 150 q^{25} - 168 q^{29} + 324 q^{31} - 120 q^{35} - 984 q^{37} - 312 q^{41} + 156 q^{43} - 462 q^{47} + 588 q^{49} + 2028 q^{53} - 1008 q^{59} - 36 q^{61} - 420 q^{65} - 144 q^{67} + 2424 q^{71} - 1800 q^{73} - 672 q^{77} + 936 q^{79} - 288 q^{83} + 60 q^{85} - 240 q^{89} + 4572 q^{91} - 570 q^{95} + 1188 q^{97}+O(q^{100}) $ 12 * q + 30 * q^5 - 12 * q^7 + 84 * q^13 + 24 * q^17 - 228 * q^19 - 30 * q^23 - 150 * q^25 - 168 * q^29 + 324 * q^31 - 120 * q^35 - 984 * q^37 - 312 * q^41 + 156 * q^43 - 462 * q^47 + 588 * q^49 + 2028 * q^53 - 1008 * q^59 - 36 * q^61 - 420 * q^65 - 144 * q^67 + 2424 * q^71 - 1800 * q^73 - 672 * q^77 + 936 * q^79 - 288 * q^83 + 60 * q^85 - 240 * q^89 + 4572 * q^91 - 570 * q^95 + 1188 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$7$	5.24065	−	9.07707i	0.282968	−	0.490116i	−0.689146	−	0.724622i	$-0.742014\pi$
$7$	5.24065	−	9.07707i	0.282968	−	0.490116i	0.972114	+	0.234507i	$0.0753475\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−16.6714	+	28.8758i	−0.456966	+	0.791488i	−0.998799	−	0.0489980i	$-0.984397\pi$
$11$	−16.6714	+	28.8758i	−0.456966	+	0.791488i	0.541833	+	0.840486i	$0.317731\pi$
$12$		0			0
$13$	11.9691	+	20.7311i	0.255357	+	0.442291i	0.964992	−	0.262278i	$-0.0844737\pi$
$13$	11.9691	+	20.7311i	0.255357	+	0.442291i	−0.709636	+	0.704569i	$0.751140\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−72.5536			−1.03511			−0.517554	−	0.855651i	$-0.673157\pi$
$17$	−72.5536			−1.03511			−0.517554	+	0.855651i	$0.673157\pi$
$18$		0			0
$19$	−45.9922			−0.555333			−0.277667	−	0.960678i	$-0.589561\pi$
$19$	−45.9922			−0.555333			−0.277667	+	0.960678i	$0.589561\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	81.5943	+	141.325i	0.739721	+	1.28123i	0.952621	+	0.304160i	$0.0983759\pi$
$23$	81.5943	+	141.325i	0.739721	+	1.28123i	−0.212900	+	0.977074i	$0.568291\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	15.4887	−	26.8271i	0.0991783	−	0.171782i	−0.812166	−	0.583426i	$-0.801712\pi$
$29$	15.4887	−	26.8271i	0.0991783	−	0.171782i	0.911345	+	0.411644i	$0.135045\pi$
$30$		0			0
$31$	−152.400	−	263.964i	−0.882960	−	1.52933i	−0.848034	−	0.529942i	$-0.822214\pi$
$31$	−152.400	−	263.964i	−0.882960	−	1.52933i	−0.0349265	−	0.999390i	$-0.511120\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	52.4065			0.253095
$36$		0			0
$37$	150.155			0.667171			0.333586	−	0.942720i	$-0.391741\pi$
$37$	150.155			0.667171			0.333586	+	0.942720i	$0.391741\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−204.532	−	354.261i	−0.779088	−	1.34942i	−0.932468	−	0.361253i	$-0.882349\pi$
$41$	−204.532	−	354.261i	−0.779088	−	1.34942i	0.153380	−	0.988167i	$-0.450984\pi$
$42$		0			0
$43$	124.587	−	215.791i	0.441845	−	0.765297i	−0.555982	−	0.831194i	$-0.687658\pi$
$43$	124.587	−	215.791i	0.441845	−	0.765297i	0.997826	+	0.0658972i	$0.0209909\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−77.9496	+	135.013i	−0.241917	+	0.419013i	−0.961260	−	0.275642i	$-0.911110\pi$
$47$	−77.9496	+	135.013i	−0.241917	+	0.419013i	0.719343	+	0.694655i	$0.244443\pi$
$48$		0			0
$49$	116.571	+	201.907i	0.339858	+	0.588651i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−263.062			−0.681780			−0.340890	−	0.940103i	$-0.610728\pi$
$53$	−263.062			−0.681780			−0.340890	+	0.940103i	$0.610728\pi$
$54$		0			0
$55$	−166.714			−0.408723
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−122.915	−	212.895i	−0.271223	−	0.469771i	0.697953	−	0.716144i	$-0.254095\pi$
$59$	−122.915	−	212.895i	−0.271223	−	0.469771i	−0.969175	+	0.246373i	$0.920761\pi$
$60$		0			0
$61$	233.236	−	403.977i	0.489555	−	0.847934i	−0.510373	−	0.859953i	$-0.670493\pi$
$61$	233.236	−	403.977i	0.489555	−	0.847934i	0.999928	+	0.0120190i	$0.00382584\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−59.8456	+	103.656i	−0.114199	+	0.197799i
$66$		0			0
$67$	88.5079	+	153.300i	0.161387	+	0.279531i	0.935367	−	0.353680i	$-0.115070\pi$
$67$	88.5079	+	153.300i	0.161387	+	0.279531i	−0.773979	+	0.633211i	$0.781736\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−45.2138			−0.0755759			−0.0377879	−	0.999286i	$-0.512031\pi$
$71$	−45.2138			−0.0755759			−0.0377879	+	0.999286i	$0.512031\pi$
$72$		0			0
$73$	−949.331			−1.52207			−0.761033	−	0.648713i	$-0.775307\pi$
$73$	−949.331			−1.52207			−0.761033	+	0.648713i	$0.775307\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	174.738	+	302.655i	0.258614	+	0.447932i
$78$		0			0
$79$	−248.459	+	430.343i	−0.353845	+	0.612878i	−0.986920	−	0.161213i	$-0.948459\pi$
$79$	−248.459	+	430.343i	−0.353845	+	0.612878i	0.633074	+	0.774091i	$0.281793\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	177.318	−	307.124i	0.234496	−	0.406159i	−0.724630	−	0.689138i	$-0.757989\pi$
$83$	177.318	−	307.124i	0.234496	−	0.406159i	0.959126	+	0.282979i	$0.0913226\pi$
$84$		0			0
$85$	−181.384	−	314.166i	−0.231457	−	0.400896i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1193.20			1.42111			0.710554	−	0.703643i	$-0.248444\pi$
$89$	1193.20			1.42111			0.710554	+	0.703643i	$0.248444\pi$
$90$		0			0
$91$	250.904			0.289032
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−114.980	−	199.152i	−0.124176	−	0.215080i
$96$		0			0
$97$	204.399	−	354.029i	0.213954	−	0.370580i	−0.738994	−	0.673712i	$-0.764699\pi$
$97$	204.399	−	354.029i	0.213954	−	0.370580i	0.952949	+	0.303132i	$0.0980323\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−445.213	+	771.132i	−0.438618	+	0.759708i	−0.997583	−	0.0694828i	$-0.977865\pi$
$101$	−445.213	+	771.132i	−0.438618	+	0.759708i	0.558965	+	0.829191i	$0.311198\pi$
$102$		0			0
$103$	−765.584	−	1326.03i	−0.732381	−	1.26852i	−0.955863	−	0.293813i	$-0.905076\pi$
$103$	−765.584	−	1326.03i	−0.732381	−	1.26852i	0.223482	−	0.974708i	$-0.428258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	989.489			0.893996			0.446998	−	0.894535i	$-0.352493\pi$
$107$	989.489			0.893996			0.446998	+	0.894535i	$0.352493\pi$
$108$		0			0
$109$	−296.637			−0.260667			−0.130334	−	0.991470i	$-0.541605\pi$
$109$	−296.637			−0.260667			−0.130334	+	0.991470i	$0.541605\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−518.073	−	897.329i	−0.431294	−	0.747023i	0.565691	−	0.824617i	$-0.308609\pi$
$113$	−518.073	−	897.329i	−0.431294	−	0.747023i	−0.996985	+	0.0775942i	$0.975276\pi$
$114$		0			0
$115$	−407.971	+	706.627i	−0.330813	+	0.572985i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−380.228	+	658.574i	−0.292903	+	0.507323i
$120$		0			0
$121$	109.627	+	189.879i	0.0823644	+	0.142659i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	−1040.10			−0.726726			−0.363363	−	0.931648i	$-0.618372\pi$
$127$	−1040.10			−0.726726			−0.363363	+	0.931648i	$0.618372\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−158.497	−	274.525i	−0.105709	−	0.183094i	0.808318	−	0.588746i	$-0.200378\pi$
$131$	−158.497	−	274.525i	−0.105709	−	0.183094i	−0.914028	+	0.405651i	$0.867045\pi$
$132$		0			0
$133$	−241.029	+	417.474i	−0.157142	+	0.272177i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−850.614	+	1473.31i	−0.530459	+	0.918782i	0.468909	+	0.883246i	$0.344647\pi$
$137$	−850.614	+	1473.31i	−0.530459	+	0.918782i	−0.999368	+	0.0355357i	$0.988686\pi$
$138$		0			0
$139$	−1378.18	−	2387.08i	−0.840978	−	1.45662i	−0.889069	−	0.457773i	$-0.848647\pi$
$139$	−1378.18	−	2387.08i	−0.840978	−	1.45662i	0.0480916	−	0.998843i	$-0.484686\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−798.170			−0.466757
$144$		0			0
$145$	154.887			0.0887078
$146$		0			0
$147$		0			0
$148$		0			0
$149$	164.353	+	284.669i	0.0903648	+	0.156516i	0.907665	−	0.419696i	$-0.137863\pi$
$149$	164.353	+	284.669i	0.0903648	+	0.156516i	−0.817300	+	0.576213i	$0.804530\pi$
$150$		0			0
$151$	158.140	−	273.907i	0.0852269	−	0.147617i	−0.820261	−	0.571989i	$-0.806172\pi$
$151$	158.140	−	273.907i	0.0852269	−	0.147617i	0.905488	+	0.424372i	$0.139505\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	761.998	−	1319.82i	0.394872	−	0.683938i
$156$		0			0
$157$	−1760.60	−	3049.44i	−0.894973	−	1.55014i	−0.833838	−	0.552010i	$-0.813861\pi$
$157$	−1760.60	−	3049.44i	−0.894973	−	1.55014i	−0.0611355	−	0.998129i	$-0.519472\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1710.43			0.837271
$162$		0			0
$163$	−1988.65			−0.955602			−0.477801	−	0.878468i	$-0.658566\pi$
$163$	−1988.65			−0.955602			−0.477801	+	0.878468i	$0.658566\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1316.47	−	2280.20i	−0.610011	−	1.05657i	−0.991238	−	0.132089i	$-0.957832\pi$
$167$	−1316.47	−	2280.20i	−0.610011	−	1.05657i	0.381227	−	0.924481i	$-0.375502\pi$
$168$		0			0
$169$	811.980	−	1406.39i	0.369586	−	0.640141i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−841.615	+	1457.72i	−0.369866	+	0.640627i	−0.989544	−	0.144229i	$-0.953930\pi$
$173$	−841.615	+	1457.72i	−0.369866	+	0.640627i	0.619678	+	0.784856i	$0.287263\pi$
$174$		0			0
$175$	131.016	+	226.927i	0.0565937	+	0.0980231i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2341.61			0.977767			0.488884	−	0.872349i	$-0.337404\pi$
$179$	2341.61			0.977767			0.488884	+	0.872349i	$0.337404\pi$
$180$		0			0
$181$	469.689			0.192882			0.0964412	−	0.995339i	$-0.469254\pi$
$181$	469.689			0.192882			0.0964412	+	0.995339i	$0.469254\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	375.387	+	650.190i	0.149184	+	0.258394i
$186$		0			0
$187$	1209.57	−	2095.04i	0.473009	−	0.819276i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−638.359	+	1105.67i	−0.241833	+	0.418867i	−0.961236	−	0.275726i	$-0.911082\pi$
$191$	−638.359	+	1105.67i	−0.241833	+	0.418867i	0.719404	+	0.694592i	$0.244415\pi$
$192$		0			0
$193$	553.438	+	958.583i	0.206411	+	0.357515i	0.950581	−	0.310475i	$-0.100488\pi$
$193$	553.438	+	958.583i	0.206411	+	0.357515i	−0.744170	+	0.667990i	$0.767155\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1945.23			0.703513			0.351757	−	0.936092i	$-0.385584\pi$
$197$	1945.23			0.703513			0.351757	+	0.936092i	$0.385584\pi$
$198$		0			0
$199$	−426.899			−0.152071			−0.0760353	−	0.997105i	$-0.524226\pi$
$199$	−426.899			−0.152071			−0.0760353	+	0.997105i	$0.524226\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−162.341	−	281.183i	−0.0561287	−	0.0972177i
$204$		0			0
$205$	1022.66	−	1771.30i	0.348419	−	0.603479i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	766.755	−	1328.06i	0.253768	−	0.439540i
$210$		0			0
$211$	−2171.61	−	3761.33i	−0.708529	−	1.22721i	−0.965403	−	0.260763i	$-0.916026\pi$
$211$	−2171.61	−	3761.33i	−0.708529	−	1.22721i	0.256874	−	0.966445i	$-0.417307\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1245.87			0.395198
$216$		0			0
$217$	−3194.69			−0.999399
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−868.403	−	1504.12i	−0.264322	−	0.457819i
$222$		0			0
$223$	1427.45	−	2472.42i	0.428651	−	0.742445i	−0.568103	−	0.822958i	$-0.692322\pi$
$223$	1427.45	−	2472.42i	0.428651	−	0.742445i	0.996754	+	0.0805124i	$0.0256557\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	188.917	−	327.214i	0.0552373	−	0.0956738i	−0.837085	−	0.547074i	$-0.815742\pi$
$227$	188.917	−	327.214i	0.0552373	−	0.0956738i	0.892322	+	0.451400i	$0.149075\pi$
$228$		0			0
$229$	−3071.84	−	5320.58i	−0.886432	−	1.53535i	−0.844063	−	0.536244i	$-0.819843\pi$
$229$	−3071.84	−	5320.58i	−0.886432	−	1.53535i	−0.0423689	−	0.999102i	$-0.513490\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4323.86			1.21573			0.607866	−	0.794040i	$-0.292026\pi$
$233$	4323.86			1.21573			0.607866	+	0.794040i	$0.292026\pi$
$234$		0			0
$235$	−779.496			−0.216378
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−913.603	−	1582.41i	−0.247264	−	0.428273i	0.715502	−	0.698611i	$-0.246198\pi$
$239$	−913.603	−	1582.41i	−0.247264	−	0.428273i	−0.962766	+	0.270337i	$0.912865\pi$
$240$		0			0
$241$	−223.874	+	387.761i	−0.0598381	+	0.103643i	−0.894393	−	0.447283i	$-0.852392\pi$
$241$	−223.874	+	387.761i	−0.0598381	+	0.103643i	0.834554	+	0.550925i	$0.185725\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−582.856	+	1009.54i	−0.151989	+	0.263253i
$246$		0			0
$247$	−550.486	−	953.470i	−0.141808	−	0.245619i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4046.98			−1.01770			−0.508851	−	0.860855i	$-0.669929\pi$
$251$	−4046.98			−1.01770			−0.508851	+	0.860855i	$0.669929\pi$
$252$		0			0
$253$	−5441.17			−1.35211
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1141.57	−	1977.25i	−0.277077	−	0.479912i	0.693580	−	0.720380i	$-0.256032\pi$
$257$	−1141.57	−	1977.25i	−0.277077	−	0.479912i	−0.970657	+	0.240468i	$0.922699\pi$
$258$		0			0
$259$	786.909	−	1362.97i	0.188788	−	0.326991i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1711.65	+	2964.67i	−0.401312	+	0.695093i	−0.993885	−	0.110424i	$-0.964779\pi$
$263$	−1711.65	+	2964.67i	−0.401312	+	0.695093i	0.592572	+	0.805517i	$0.298112\pi$
$264$		0			0
$265$	−657.655	−	1139.09i	−0.152451	−	0.264052i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2257.48			−0.511676			−0.255838	−	0.966720i	$-0.582351\pi$
$269$	−2257.48			−0.511676			−0.255838	+	0.966720i	$0.582351\pi$
$270$		0			0
$271$	−8085.45			−1.81239			−0.906193	−	0.422865i	$-0.861024\pi$
$271$	−8085.45			−1.81239			−0.906193	+	0.422865i	$0.861024\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−416.786	−	721.894i	−0.0913932	−	0.158298i
$276$		0			0
$277$	−2420.37	+	4192.21i	−0.525004	+	0.909333i	0.474572	+	0.880217i	$0.342603\pi$
$277$	−2420.37	+	4192.21i	−0.525004	+	0.909333i	−0.999576	+	0.0291166i	$0.990731\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3277.56	+	5676.90i	−0.695811	+	1.20518i	0.274096	+	0.961702i	$0.411621\pi$
$281$	−3277.56	+	5676.90i	−0.695811	+	1.20518i	−0.969907	+	0.243477i	$0.921712\pi$
$282$		0			0
$283$	−2051.50	−	3553.31i	−0.430916	−	0.746368i	0.566037	−	0.824380i	$-0.308476\pi$
$283$	−2051.50	−	3553.31i	−0.430916	−	0.746368i	−0.996952	+	0.0780120i	$0.975143\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4287.53			−0.881829
$288$		0			0
$289$	351.026			0.0714485
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4250.05	−	7361.31i	−0.847408	−	1.46775i	−0.883513	−	0.468406i	$-0.844828\pi$
$293$	−4250.05	−	7361.31i	−0.847408	−	1.46775i	0.0361047	−	0.999348i	$-0.488505\pi$
$294$		0			0
$295$	614.574	−	1064.47i	0.121294	−	0.210088i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1953.22	+	3383.08i	−0.377786	+	0.654344i
$300$		0			0
$301$	−1305.83	−	2261.77i	−0.250056	−	0.433110i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2332.36			0.437871
$306$		0			0
$307$	5386.01			1.00129			0.500645	−	0.865653i	$-0.333096\pi$
$307$	5386.01			1.00129			0.500645	+	0.865653i	$0.333096\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	744.588	+	1289.66i	0.135761	+	0.235145i	0.925888	−	0.377798i	$-0.123319\pi$
$311$	744.588	+	1289.66i	0.135761	+	0.235145i	−0.790127	+	0.612943i	$0.789985\pi$
$312$		0			0
$313$	2212.65	−	3832.41i	0.399572	−	0.692079i	−0.594101	−	0.804391i	$-0.702492\pi$
$313$	2212.65	−	3832.41i	0.399572	−	0.692079i	0.993673	+	0.112311i	$0.0358253\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1617.97	+	2802.41i	−0.286670	+	0.496527i	−0.973013	−	0.230751i	$-0.925882\pi$
$317$	−1617.97	+	2802.41i	−0.286670	+	0.496527i	0.686343	+	0.727278i	$0.259215\pi$
$318$		0			0
$319$	516.436	+	894.494i	0.0906422	+	0.156997i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3336.90			0.574830
$324$		0			0
$325$	−598.456			−0.102143
$326$		0			0
$327$		0			0
$328$		0			0
$329$	817.013	+	1415.11i	0.136910	+	0.237135i
$330$		0			0
$331$	435.810	−	754.845i	0.0723694	−	0.125348i	−0.827570	−	0.561363i	$-0.810277\pi$
$331$	435.810	−	754.845i	0.0723694	−	0.125348i	0.899939	+	0.436015i	$0.143611\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−442.539	+	766.501i	−0.0721747	+	0.125010i
$336$		0			0
$337$	−1557.07	−	2696.93i	−0.251689	−	0.435938i	0.712302	−	0.701873i	$-0.247653\pi$
$337$	−1557.07	−	2696.93i	−0.251689	−	0.435938i	−0.963991	+	0.265935i	$0.914319\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10162.9			1.61393
$342$		0			0
$343$	6038.72			0.950613
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3709.31	+	6424.72i	0.573851	+	0.993940i	0.996165	+	0.0874900i	$0.0278846\pi$
$347$	3709.31	+	6424.72i	0.573851	+	0.993940i	−0.422314	+	0.906450i	$0.638782\pi$
$348$		0			0
$349$	−3008.34	+	5210.60i	−0.461412	+	0.799189i	−0.999032	−	0.0439983i	$-0.985990\pi$
$349$	−3008.34	+	5210.60i	−0.461412	+	0.799189i	0.537619	+	0.843188i	$0.319324\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2872.88	−	4975.97i	0.433167	−	0.750267i	−0.563977	−	0.825790i	$-0.690729\pi$
$353$	2872.88	−	4975.97i	0.433167	−	0.750267i	0.997144	+	0.0755234i	$0.0240628\pi$
$354$		0			0
$355$	−113.034	−	195.781i	−0.0168993	−	0.0292704i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5133.25			0.754659			0.377329	−	0.926079i	$-0.376842\pi$
$359$	5133.25			0.754659			0.377329	+	0.926079i	$0.376842\pi$
$360$		0			0
$361$	−4743.72			−0.691605
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2373.33	−	4110.72i	−0.340344	−	0.589494i
$366$		0			0
$367$	839.740	−	1454.47i	0.119439	−	0.206874i	−0.800107	−	0.599858i	$-0.795224\pi$
$367$	839.740	−	1454.47i	0.119439	−	0.206874i	0.919545	+	0.392984i	$0.128557\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1378.61	+	2387.83i	−0.192922	+	0.334151i
$372$		0			0
$373$	2455.27	+	4252.66i	0.340829	+	0.590333i	0.984587	−	0.174896i	$-0.0559590\pi$
$373$	2455.27	+	4252.66i	0.340829	+	0.590333i	−0.643758	+	0.765229i	$0.722626\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	741.543			0.101303
$378$		0			0
$379$	−9065.83			−1.22871			−0.614354	−	0.789030i	$-0.710583\pi$
$379$	−9065.83			−1.22871			−0.614354	+	0.789030i	$0.710583\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4136.64	−	7164.87i	−0.551886	−	0.955895i	−0.998138	−	0.0609880i	$-0.980575\pi$
$383$	−4136.64	−	7164.87i	−0.551886	−	0.955895i	0.446252	−	0.894907i	$-0.352758\pi$
$384$		0			0
$385$	−873.691	+	1513.28i	−0.115656	+	0.200321i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1504.28	+	2605.49i	−0.196067	+	0.339598i	−0.947250	−	0.320496i	$-0.896150\pi$
$389$	−1504.28	+	2605.49i	−0.196067	+	0.339598i	0.751183	+	0.660094i	$0.229484\pi$
$390$		0			0
$391$	−5919.96	−	10253.7i	−0.765691	−	1.32622i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2484.59			−0.316489
$396$		0			0
$397$	−11495.3			−1.45324			−0.726618	−	0.687042i	$-0.758909\pi$
$397$	−11495.3			−1.45324			−0.726618	+	0.687042i	$0.758909\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5441.74	+	9425.36i	0.677674	+	1.17377i	0.975680	+	0.219202i	$0.0703453\pi$
$401$	5441.74	+	9425.36i	0.677674	+	1.17377i	−0.298005	+	0.954564i	$0.596321\pi$
$402$		0			0
$403$	3648.18	−	6318.83i	0.450940	−	0.781051i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2503.30	+	4335.84i	−0.304874	+	0.528058i
$408$		0			0
$409$	3127.50	+	5416.99i	0.378105	+	0.654897i	0.990786	−	0.135433i	$-0.0432426\pi$
$409$	3127.50	+	5416.99i	0.378105	+	0.654897i	−0.612682	+	0.790330i	$0.709909\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2576.61			−0.306990
$414$		0			0
$415$	1773.18			0.209740
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5065.61	−	8773.89i	−0.590623	−	1.02299i	−0.994149	−	0.108021i	$-0.965549\pi$
$419$	−5065.61	−	8773.89i	−0.590623	−	1.02299i	0.403525	−	0.914968i	$-0.367785\pi$
$420$		0			0
$421$	−1292.56	+	2238.78i	−0.149633	+	0.259172i	−0.931092	−	0.364785i	$-0.881143\pi$
$421$	−1292.56	+	2238.78i	−0.149633	+	0.259172i	0.781459	+	0.623957i	$0.214476\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	906.920	−	1570.83i	0.103511	−	0.179286i
$426$		0			0
$427$	−2444.62	−	4234.21i	−0.277057	−	0.479877i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5184.08			−0.579370			−0.289685	−	0.957122i	$-0.593551\pi$
$431$	−5184.08			−0.579370			−0.289685	+	0.957122i	$0.593551\pi$
$432$		0			0
$433$	4384.88			0.486660			0.243330	−	0.969944i	$-0.421760\pi$
$433$	4384.88			0.486660			0.243330	+	0.969944i	$0.421760\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3752.70	−	6499.86i	−0.410792	−	0.711512i
$438$		0			0
$439$	4324.59	−	7490.40i	0.470162	−	0.814345i	−0.529256	−	0.848462i	$-0.677529\pi$
$439$	4324.59	−	7490.40i	0.470162	−	0.814345i	0.999418	+	0.0341176i	$0.0108621\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8560.43	−	14827.1i	0.918100	−	1.59020i	0.115801	−	0.993272i	$-0.463056\pi$
$443$	8560.43	−	14827.1i	0.918100	−	1.59020i	0.802298	−	0.596923i	$-0.203610\pi$
$444$		0			0
$445$	2982.99	+	5166.69i	0.317769	+	0.550393i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2688.80			−0.282611			−0.141306	−	0.989966i	$-0.545130\pi$
$449$	−2688.80			−0.282611			−0.141306	+	0.989966i	$0.545130\pi$
$450$		0			0
$451$	13639.4			1.42407
$452$		0			0
$453$		0			0
$454$		0			0
$455$	627.260	+	1086.45i	0.0646294	+	0.111941i
$456$		0			0
$457$	−839.545	+	1454.14i	−0.0859350	+	0.148844i	−0.905789	−	0.423729i	$-0.860721\pi$
$457$	−839.545	+	1454.14i	−0.0859350	+	0.148844i	0.819854	+	0.572572i	$0.194054\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1201.75	−	2081.49i	0.121412	−	0.210292i	−0.798912	−	0.601447i	$-0.794591\pi$
$461$	1201.75	−	2081.49i	0.121412	−	0.210292i	0.920325	+	0.391155i	$0.127924\pi$
$462$		0			0
$463$	−1524.28	−	2640.13i	−0.153001	−	0.265005i	0.779328	−	0.626616i	$-0.215560\pi$
$463$	−1524.28	−	2640.13i	−0.153001	−	0.265005i	−0.932329	+	0.361610i	$0.882227\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6256.23			0.619923			0.309961	−	0.950749i	$-0.399684\pi$
$467$	6256.23			0.619923			0.309961	+	0.950749i	$0.399684\pi$
$468$		0			0
$469$	1855.35			0.182670
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4154.08	+	7195.08i	0.403816	+	0.699429i
$474$		0			0
$475$	574.902	−	995.760i	0.0555333	−	0.0961865i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2316.63	−	4012.51i	0.220980	−	0.382748i	−0.734126	−	0.679013i	$-0.762408\pi$
$479$	2316.63	−	4012.51i	0.220980	−	0.382748i	0.955106	+	0.296265i	$0.0957412\pi$
$480$		0			0
$481$	1797.22	+	3112.88i	0.170367	+	0.295084i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2043.99			0.191366
$486$		0			0
$487$	9823.94			0.914097			0.457049	−	0.889442i	$-0.348907\pi$
$487$	9823.94			0.914097			0.457049	+	0.889442i	$0.348907\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1134.95	+	1965.80i	0.104317	+	0.180683i	0.913459	−	0.406931i	$-0.133401\pi$
$491$	1134.95	+	1965.80i	0.104317	+	0.180683i	−0.809142	+	0.587613i	$0.800068\pi$
$492$		0			0
$493$	−1123.76	+	1946.41i	−0.102660	+	0.177813i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−236.949	+	410.408i	−0.0213856	+	0.0370409i
$498$		0			0
$499$	−2326.16	−	4029.02i	−0.208684	−	0.361450i	0.742617	−	0.669717i	$-0.233585\pi$
$499$	−2326.16	−	4029.02i	−0.208684	−	0.361450i	−0.951300	+	0.308266i	$0.900251\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13783.4			1.22181			0.610906	−	0.791703i	$-0.290805\pi$
$503$	13783.4			1.22181			0.610906	+	0.791703i	$0.290805\pi$
$504$		0			0
$505$	−4452.13			−0.392312
$506$		0			0
$507$		0			0
$508$		0			0
$509$	324.639	+	562.291i	0.0282699	+	0.0489649i	0.879814	−	0.475318i	$-0.157667\pi$
$509$	324.639	+	562.291i	0.0282699	+	0.0489649i	−0.851544	+	0.524283i	$0.824334\pi$
$510$		0			0
$511$	−4975.11	+	8617.14i	−0.430696	+	0.745988i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3827.92	−	6630.16i	0.327531	−	0.567300i
$516$		0			0
$517$	−2599.06	−	4501.71i	−0.221096	−	0.382950i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21010.9			−1.76680			−0.883402	−	0.468616i	$-0.844753\pi$
$521$	−21010.9			−1.76680			−0.883402	+	0.468616i	$0.844753\pi$
$522$		0			0
$523$	−16162.4			−1.35130			−0.675652	−	0.737221i	$-0.736138\pi$
$523$	−16162.4			−1.35130			−0.675652	+	0.737221i	$0.736138\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	11057.1	+	19151.5i	0.913959	+	1.58302i
$528$		0			0
$529$	−7231.75	+	12525.8i	−0.594374	+	1.02949i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4896.15	−	8480.38i	0.397891	−	0.689167i
$534$		0			0
$535$	2473.72	+	4284.62i	0.199904	+	0.346243i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7773.64			−0.621214
$540$		0			0
$541$	20812.9			1.65400			0.827002	−	0.562199i	$-0.190044\pi$
$541$	20812.9			1.65400			0.827002	+	0.562199i	$0.190044\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−741.593	−	1284.48i	−0.0582869	−	0.100956i
$546$		0			0
$547$	500.008	−	866.040i	0.0390838	−	0.0676950i	−0.845822	−	0.533465i	$-0.820889\pi$
$547$	500.008	−	866.040i	0.0390838	−	0.0676950i	0.884906	+	0.465770i	$0.154223\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−712.357	+	1233.84i	−0.0550770	+	0.0953962i
$552$		0			0
$553$	2604.17	+	4510.55i	0.200254	+	0.346850i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4720.42			−0.359085			−0.179543	−	0.983750i	$-0.557462\pi$
$557$	−4720.42			−0.359085			−0.179543	+	0.983750i	$0.557462\pi$
$558$		0			0
$559$	5964.78			0.451312
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3931.74	+	6809.97i	0.294321	+	0.509779i	0.974827	−	0.222964i	$-0.0715731\pi$
$563$	3931.74	+	6809.97i	0.294321	+	0.509779i	−0.680506	+	0.732743i	$0.738240\pi$
$564$		0			0
$565$	2590.37	−	4486.64i	0.192881	−	0.334079i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11277.6	+	19533.4i	−0.830900	+	1.43916i	0.0664259	+	0.997791i	$0.478840\pi$
$569$	−11277.6	+	19533.4i	−0.830900	+	1.43916i	−0.897326	+	0.441369i	$0.854493\pi$
$570$		0			0
$571$	6328.40	+	10961.1i	0.463810	+	0.803342i	0.999147	−	0.0412964i	$-0.0131488\pi$
$571$	6328.40	+	10961.1i	0.463810	+	0.803342i	−0.535337	+	0.844638i	$0.679815\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4079.71			−0.295888
$576$		0			0
$577$	11720.0			0.845601			0.422801	−	0.906223i	$-0.361047\pi$
$577$	11720.0			0.845601			0.422801	+	0.906223i	$0.361047\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1858.52	−	3219.06i	−0.132710	−	0.229860i
$582$		0			0
$583$	4385.62	−	7596.11i	0.311550	−	0.539621i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−474.881	+	822.517i	−0.0333908	+	0.0578346i	−0.882238	−	0.470804i	$-0.843964\pi$
$587$	−474.881	+	822.517i	−0.0333908	+	0.0578346i	0.848847	+	0.528638i	$0.177297\pi$
$588$		0			0
$589$	7009.19	+	12140.3i	0.490337	+	0.849289i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11575.2			0.801580			0.400790	−	0.916170i	$-0.368736\pi$
$593$	11575.2			0.801580			0.400790	+	0.916170i	$0.368736\pi$
$594$		0			0
$595$	−3802.28			−0.261980
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13538.1	+	23448.7i	0.923461	+	1.59948i	0.794018	+	0.607894i	$0.207985\pi$
$599$	13538.1	+	23448.7i	0.923461	+	1.59948i	0.129442	+	0.991587i	$0.458681\pi$
$600$		0			0
$601$	−9931.64	+	17202.1i	−0.674077	+	1.16754i	0.302661	+	0.953098i	$0.402125\pi$
$601$	−9931.64	+	17202.1i	−0.674077	+	1.16754i	−0.976738	+	0.214437i	$0.931208\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−548.135	+	949.397i	−0.0368345	+	0.0637992i
$606$		0			0
$607$	7635.22	+	13224.6i	0.510550	+	0.884299i	0.999925	+	0.0122252i	$0.00389151\pi$
$607$	7635.22	+	13224.6i	0.510550	+	0.884299i	−0.489375	+	0.872073i	$0.662775\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3731.95			−0.247101
$612$		0			0
$613$	−13256.0			−0.873418			−0.436709	−	0.899603i	$-0.643856\pi$
$613$	−13256.0			−0.873418			−0.436709	+	0.899603i	$0.643856\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7174.19	−	12426.1i	−0.468107	−	0.810785i	0.531229	−	0.847228i	$-0.321730\pi$
$617$	−7174.19	−	12426.1i	−0.468107	−	0.810785i	−0.999336	+	0.0364433i	$0.988397\pi$
$618$		0			0
$619$	12002.4	−	20788.7i	0.779348	−	1.34987i	−0.152969	−	0.988231i	$-0.548884\pi$
$619$	12002.4	−	20788.7i	0.779348	−	1.34987i	0.932318	−	0.361640i	$-0.117783\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6253.12	−	10830.7i	0.402128	−	0.696507i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−10894.3			−0.690594
$630$		0			0
$631$	28832.3			1.81901			0.909505	−	0.415693i	$-0.136461\pi$
$631$	28832.3			1.81901			0.909505	+	0.415693i	$0.136461\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2600.26	−	4503.78i	−0.162501	−	0.281460i
$636$		0			0
$637$	−2790.51	+	4833.31i	−0.173570	+	0.300632i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12561.8	+	21757.7i	−0.774042	+	1.34068i	0.161289	+	0.986907i	$0.448435\pi$
$641$	−12561.8	+	21757.7i	−0.774042	+	1.34068i	−0.935331	+	0.353773i	$0.884899\pi$
$642$		0			0
$643$	15680.0	+	27158.5i	0.961676	+	1.66567i	0.718291	+	0.695742i	$0.244925\pi$
$643$	15680.0	+	27158.5i	0.961676	+	1.66567i	0.243385	+	0.969930i	$0.421742\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7235.97			−0.439684			−0.219842	−	0.975536i	$-0.570554\pi$
$647$	−7235.97			−0.439684			−0.219842	+	0.975536i	$0.570554\pi$
$648$		0			0
$649$	8196.66			0.495758
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10762.8	−	18641.8i	−0.644995	−	1.11716i	−0.984303	−	0.176489i	$-0.943526\pi$
$653$	−10762.8	−	18641.8i	−0.644995	−	1.11716i	0.339308	−	0.940675i	$-0.389807\pi$
$654$		0			0
$655$	792.484	−	1372.62i	0.0472747	−	0.0818822i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9726.03	−	16846.0i	0.574920	−	0.995791i	−0.421130	−	0.907000i	$-0.638366\pi$
$659$	9726.03	−	16846.0i	0.574920	−	0.995791i	0.996050	−	0.0887907i	$-0.0283002\pi$
$660$		0			0
$661$	358.375	+	620.723i	0.0210880	+	0.0365255i	0.876377	−	0.481626i	$-0.159954\pi$
$661$	358.375	+	620.723i	0.0210880	+	0.0365255i	−0.855289	+	0.518152i	$0.826620\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2410.29			−0.140552
$666$		0			0
$667$	5055.14			0.293457
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7776.77	+	13469.8i	0.447420	+	0.774954i
$672$		0			0
$673$	−11229.8	+	19450.7i	−0.643207	+	1.11407i	0.341505	+	0.939880i	$0.389063\pi$
$673$	−11229.8	+	19450.7i	−0.643207	+	1.11407i	−0.984712	+	0.174188i	$0.944270\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−835.528	+	1447.18i	−0.0474327	+	0.0821559i	−0.888767	−	0.458359i	$-0.848437\pi$
$677$	−835.528	+	1447.18i	−0.0474327	+	0.0821559i	0.841334	+	0.540515i	$0.181771\pi$
$678$		0			0
$679$	−2142.36	−	3710.68i	−0.121085	−	0.209725i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−22202.2			−1.24384			−0.621921	−	0.783080i	$-0.713647\pi$
$683$	−22202.2			−1.24384			−0.621921	+	0.783080i	$0.713647\pi$
$684$		0			0
$685$	−8506.14			−0.474457
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3148.62	−	5453.57i	−0.174097	−	0.301545i
$690$		0			0
$691$	1935.97	−	3353.20i	0.106582	−	0.184605i	−0.807802	−	0.589454i	$-0.799343\pi$
$691$	1935.97	−	3353.20i	0.106582	−	0.184605i	0.914383	+	0.404850i	$0.132676\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6890.91	−	11935.4i	0.376097	−	0.651418i
$696$		0			0
$697$	14839.6	+	25702.9i	0.806440	+	1.39680i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21918.3			1.18095			0.590474	−	0.807057i	$-0.298941\pi$
$701$	21918.3			1.18095			0.590474	+	0.807057i	$0.298941\pi$
$702$		0			0
$703$	−6905.96			−0.370502
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4666.41	+	8082.46i	0.248230	+	0.429947i
$708$		0			0
$709$	−11166.9	+	19341.6i	−0.591512	+	1.02453i	0.402517	+	0.915412i	$0.368135\pi$
$709$	−11166.9	+	19341.6i	−0.591512	+	1.02453i	−0.994029	+	0.109116i	$0.965198\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	24869.9	−	43075.9i	1.30629	−	2.26256i
$714$		0			0
$715$	−1995.42	−	3456.18i	−0.104370	−	0.180774i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−33133.5			−1.71859			−0.859297	−	0.511477i	$-0.829099\pi$
$719$	−33133.5			−1.71859			−0.859297	+	0.511477i	$0.829099\pi$
$720$		0			0
$721$	−16048.6			−0.828963
$722$		0			0
$723$		0			0
$724$		0			0
$725$	387.217	+	670.679i	0.0198357	+	0.0343564i
$726$		0			0
$727$	−5591.06	+	9684.00i	−0.285228	+	0.494030i	−0.972665	−	0.232215i	$-0.925403\pi$
$727$	−5591.06	+	9684.00i	−0.285228	+	0.494030i	0.687436	+	0.726245i	$0.258736\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9039.23	+	15656.4i	−0.457357	+	0.792165i
$732$		0			0
$733$	16155.9	+	27982.8i	0.814093	+	1.41005i	0.909978	+	0.414657i	$0.136099\pi$
$733$	16155.9	+	27982.8i	0.814093	+	1.41005i	−0.0958849	+	0.995392i	$0.530568\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5902.21			−0.294994
$738$		0			0
$739$	16348.4			0.813784			0.406892	−	0.913476i	$-0.366613\pi$
$739$	16348.4			0.813784			0.406892	+	0.913476i	$0.366613\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6992.11	−	12110.7i	−0.345243	−	0.597979i	0.640155	−	0.768246i	$-0.278870\pi$
$743$	−6992.11	−	12110.7i	−0.345243	−	0.597979i	−0.985398	+	0.170267i	$0.945537\pi$
$744$		0			0
$745$	−821.767	+	1423.34i	−0.0404124	+	0.0699963i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5185.57	−	8981.66i	0.252973	−	0.438161i
$750$		0			0
$751$	−12344.4	−	21381.1i	−0.599805	−	1.03889i	−0.992849	−	0.119373i	$-0.961911\pi$
$751$	−12344.4	−	21381.1i	−0.599805	−	1.03889i	0.393044	−	0.919520i	$-0.371422\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1581.40			0.0762293
$756$		0			0
$757$	21515.4			1.03301			0.516507	−	0.856283i	$-0.327232\pi$
$757$	21515.4			1.03301			0.516507	+	0.856283i	$0.327232\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9577.98	+	16589.5i	0.456244	+	0.790237i	0.998759	−	0.0498090i	$-0.0158612\pi$
$761$	9577.98	+	16589.5i	0.456244	+	0.790237i	−0.542515	+	0.840046i	$0.682528\pi$
$762$		0			0
$763$	−1554.57	+	2692.60i	−0.0737605	+	0.127757i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2942.36	−	5096.32i	0.138517	−	0.239919i
$768$		0			0
$769$	19730.0	+	34173.3i	0.925203	+	1.60250i	0.791234	+	0.611513i	$0.209439\pi$
$769$	19730.0	+	34173.3i	0.925203	+	1.60250i	0.133969	+	0.990986i	$0.457228\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2246.03			−0.104507			−0.0522537	−	0.998634i	$-0.516640\pi$
$773$	−2246.03			−0.104507			−0.0522537	+	0.998634i	$0.516640\pi$
$774$		0			0
$775$	7619.98			0.353184
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9406.89	+	16293.2i	0.432653	+	0.749377i
$780$		0			0
$781$	753.778	−	1305.58i	0.0345356	−	0.0598174i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8802.98	−	15247.2i	0.400244	−	0.693243i
$786$		0			0
$787$	2900.56	+	5023.91i	0.131377	+	0.227552i	0.924208	−	0.381890i	$-0.124727\pi$
$787$	2900.56	+	5023.91i	0.131377	+	0.227552i	−0.792831	+	0.609442i	$0.791394\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−10860.2			−0.488170
$792$		0			0
$793$	11166.5			0.500045
$794$		0			0
$795$		0			0
$796$		0			0
$797$	20199.5	+	34986.6i	0.897747	+	1.55494i	0.830368	+	0.557215i	$0.188130\pi$
$797$	20199.5	+	34986.6i	0.897747	+	1.55494i	0.0673783	+	0.997728i	$0.478537\pi$
$798$		0			0
$799$	5655.53	−	9795.66i	0.250411	−	0.433724i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	15826.7	−	27412.7i	0.695532	−	1.20470i
$804$		0			0
$805$	4276.07	+	7406.37i	0.187219	+	0.324273i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30986.8			1.34665			0.673324	−	0.739348i	$-0.264866\pi$
$809$	30986.8			1.34665			0.673324	+	0.739348i	$0.264866\pi$
$810$		0			0
$811$	8658.08			0.374878			0.187439	−	0.982276i	$-0.439981\pi$
$811$	8658.08			0.374878			0.187439	+	0.982276i	$0.439981\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4971.62	−	8611.11i	−0.213679	−	0.370103i
$816$		0			0
$817$	−5730.02	+	9924.69i	−0.245371	+	0.424995i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15231.5	+	26381.8i	−0.647484	+	1.12147i	0.336238	+	0.941777i	$0.390845\pi$
$821$	−15231.5	+	26381.8i	−0.647484	+	1.12147i	−0.983722	+	0.179698i	$0.942488\pi$
$822$		0			0
$823$	5178.95	+	8970.20i	0.219352	+	0.379929i	0.954610	−	0.297858i	$-0.0962723\pi$
$823$	5178.95	+	8970.20i	0.219352	+	0.379929i	−0.735258	+	0.677787i	$0.762939\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30108.4			−1.26599			−0.632993	−	0.774157i	$-0.718174\pi$
$827$	−30108.4			−1.26599			−0.632993	+	0.774157i	$0.718174\pi$
$828$		0			0
$829$	6248.62			0.261789			0.130895	−	0.991396i	$-0.458215\pi$
$829$	6248.62			0.261789			0.130895	+	0.991396i	$0.458215\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8457.66	−	14649.1i	−0.351790	−	0.609317i
$834$		0			0
$835$	6582.37	−	11401.0i	0.272805	−	0.472512i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5289.81	+	9162.22i	−0.217669	+	0.377014i	−0.954095	−	0.299504i	$-0.903179\pi$
$839$	−5289.81	+	9162.22i	−0.217669	+	0.377014i	0.736426	+	0.676519i	$0.236512\pi$
$840$		0			0
$841$	11714.7	+	20290.5i	0.480327	+	0.831951i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8119.80			0.330568
$846$		0			0
$847$	2298.07			0.0932260
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12251.8	+	21220.7i	0.493520	+	0.854802i
$852$		0			0
$853$	10114.4	−	17518.6i	0.405990	−	0.703195i	−0.588446	−	0.808536i	$-0.700260\pi$
$853$	10114.4	−	17518.6i	0.405990	−	0.703195i	0.994436	+	0.105341i	$0.0335935\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21666.7	+	37527.8i	−0.863618	+	1.49583i	0.00479440	+	0.999989i	$0.498474\pi$
$857$	−21666.7	+	37527.8i	−0.863618	+	1.49583i	−0.868413	+	0.495842i	$0.834859\pi$
$858$		0			0
$859$	−8292.01	−	14362.2i	−0.329359	−	0.570467i	0.653026	−	0.757336i	$-0.273499\pi$
$859$	−8292.01	−	14362.2i	−0.329359	−	0.570467i	−0.982385	+	0.186869i	$0.940166\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3281.90			−0.129452			−0.0647261	−	0.997903i	$-0.520617\pi$
$863$	−3281.90			−0.129452			−0.0647261	+	0.997903i	$0.520617\pi$
$864$		0			0
$865$	−8416.15			−0.330818
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8284.32	−	14348.9i	−0.323390	−	0.560129i
$870$		0			0
$871$	−2118.72	+	3669.74i	−0.0824228	+	0.142760i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−655.081	+	1134.63i	−0.0253095	+	0.0438373i
$876$		0			0
$877$	−9107.38	−	15774.4i	−0.350666	−	0.607372i	0.635700	−	0.771936i	$-0.280712\pi$
$877$	−9107.38	−	15774.4i	−0.350666	−	0.607372i	−0.986366	+	0.164564i	$0.947378\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−29408.8			−1.12464			−0.562320	−	0.826920i	$-0.690091\pi$
$881$	−29408.8			−1.12464			−0.562320	+	0.826920i	$0.690091\pi$
$882$		0			0
$883$	−7442.63			−0.283652			−0.141826	−	0.989892i	$-0.545297\pi$
$883$	−7442.63			−0.283652			−0.141826	+	0.989892i	$0.545297\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10153.4	+	17586.2i	0.384348	+	0.665710i	0.991679	−	0.128739i	$-0.0410929\pi$
$887$	10153.4	+	17586.2i	0.384348	+	0.665710i	−0.607330	+	0.794449i	$0.707760\pi$
$888$		0			0
$889$	−5450.81	+	9441.09i	−0.205641	+	0.356180i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3585.07	−	6209.53i	0.134345	−	0.232692i
$894$		0			0
$895$	5854.03	+	10139.5i	0.218635	+	0.378688i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9441.86			−0.350282
$900$		0			0
$901$	19086.1			0.705716
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1174.22	+	2033.81i	0.0431298	+	0.0747031i
$906$		0			0
$907$	−3200.18	+	5542.87i	−0.117156	+	0.202919i	−0.918639	−	0.395097i	$-0.870711\pi$
$907$	−3200.18	+	5542.87i	−0.117156	+	0.202919i	0.801484	+	0.598017i	$0.204044\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4651.66	+	8056.91i	−0.169173	+	0.293016i	−0.938129	−	0.346285i	$-0.887443\pi$
$911$	−4651.66	+	8056.91i	−0.169173	+	0.293016i	0.768957	+	0.639301i	$0.220776\pi$
$912$		0			0
$913$	5912.29	+	10240.4i	0.214314	+	0.371202i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3322.51			−0.119650
$918$		0			0
$919$	15110.9			0.542397			0.271198	−	0.962523i	$-0.412580\pi$
$919$	15110.9			0.542397			0.271198	+	0.962523i	$0.412580\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−541.169	−	937.332i	−0.0192988	−	0.0334265i
$924$		0			0
$925$	−1876.94	+	3250.95i	−0.0667171	+	0.115557i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	24486.1	−	42411.2i	0.864761	−	1.49781i	−0.00252407	−	0.999997i	$-0.500803\pi$
$929$	24486.1	−	42411.2i	0.864761	−	1.49781i	0.867285	−	0.497812i	$-0.165863\pi$
$930$		0			0
$931$	−5361.36	−	9286.16i	−0.188734	−	0.326897i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12095.7			0.423072
$936$		0			0
$937$	41808.1			1.45764			0.728821	−	0.684705i	$-0.240069\pi$
$937$	41808.1			1.45764			0.728821	+	0.684705i	$0.240069\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	27284.4	+	47257.9i	0.945212	+	1.63716i	0.755325	+	0.655350i	$0.227479\pi$
$941$	27284.4	+	47257.9i	0.945212	+	1.63716i	0.189887	+	0.981806i	$0.439188\pi$
$942$		0			0
$943$	33377.4	−	57811.3i	1.15262	−	1.99639i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19399.9	−	33601.7i	0.665695	−	1.15302i	−0.313401	−	0.949621i	$-0.601468\pi$
$947$	19399.9	−	33601.7i	0.665695	−	1.15302i	0.979096	−	0.203397i	$-0.0651982\pi$
$948$		0			0
$949$	−11362.7	−	19680.7i	−0.388670	−	0.673196i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	38007.3			1.29190			0.645948	−	0.763382i	$-0.276462\pi$
$953$	38007.3			1.29190			0.645948	+	0.763382i	$0.276462\pi$
$954$		0			0
$955$	−6383.59			−0.216302
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8915.54	+	15442.2i	0.300206	+	0.519972i
$960$		0			0
$961$	−31555.8	+	54656.2i	−1.05924	+	1.83465i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2767.19	+	4792.92i	−0.0923099	+	0.159885i
$966$		0			0
$967$	−21453.4	−	37158.3i	−0.713437	−	1.23571i	−0.963559	−	0.267495i	$-0.913804\pi$
$967$	−21453.4	−	37158.3i	−0.713437	−	1.23571i	0.250122	−	0.968214i	$-0.419529\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−38855.2			−1.28416			−0.642082	−	0.766636i	$-0.721929\pi$
$971$	−38855.2			−1.28416			−0.642082	+	0.766636i	$0.721929\pi$
$972$		0			0
$973$	−28890.3			−0.951880
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24119.5	−	41776.2i	−0.789817	−	1.36800i	−0.926079	−	0.377330i	$-0.876842\pi$
$977$	−24119.5	−	41776.2i	−0.789817	−	1.36800i	0.136262	−	0.990673i	$-0.456491\pi$
$978$		0			0
$979$	−19892.3	+	34454.5i	−0.649398	+	1.12479i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	21583.7	−	37384.1i	0.700320	−	1.21299i	−0.268034	−	0.963409i	$-0.586374\pi$
$983$	21583.7	−	37384.1i	0.700320	−	1.21299i	0.968354	−	0.249580i	$-0.0802926\pi$
$984$		0			0
$985$	4863.08	+	8423.10i	0.157310	+	0.272469i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	40662.3			1.30737
$990$		0			0
$991$	57081.7			1.82973			0.914863	−	0.403764i	$-0.132298\pi$
$991$	57081.7			1.82973			0.914863	+	0.403764i	$0.132298\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1067.25	−	1848.53i	−0.0340040	−	0.0588967i
$996$		0			0
$997$	−23836.5	+	41286.1i	−0.757182	+	1.31148i	0.187100	+	0.982341i	$0.440091\pi$
$997$	−23836.5	+	41286.1i	−0.757182	+	1.31148i	−0.944282	+	0.329138i	$0.893242\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.x.541.5		12
3.2	odd	2		1620.4.i.w.541.5		12
9.2	odd	6		1620.4.a.j.1.2	yes	6
9.4	even	3	inner	1620.4.i.x.1081.5		12
9.5	odd	6		1620.4.i.w.1081.5		12
9.7	even	3		1620.4.a.i.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.i.1.2	✓	6	9.7	even	3
1620.4.a.j.1.2	yes	6	9.2	odd	6
1620.4.i.w.541.5		12	3.2	odd	2
1620.4.i.w.1081.5		12	9.5	odd	6
1620.4.i.x.541.5		12	1.1	even	1	trivial
1620.4.i.x.1081.5		12	9.4	even	3	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+(2.50000 + 4.33013i) q^{5} +(5.24065 - 9.07707i) q^{7} +O(q^{10})\)	\(q+(2.50000 + 4.33013i) q^{5} +(5.24065 - 9.07707i) q^{7} +(-16.6714 + 28.8758i) q^{11} +(11.9691 + 20.7311i) q^{13} -72.5536 q^{17} -45.9922 q^{19} +(81.5943 + 141.325i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(15.4887 - 26.8271i) q^{29} +(-152.400 - 263.964i) q^{31} +52.4065 q^{35} +150.155 q^{37} +(-204.532 - 354.261i) q^{41} +(124.587 - 215.791i) q^{43} +(-77.9496 + 135.013i) q^{47} +(116.571 + 201.907i) q^{49} -263.062 q^{53} -166.714 q^{55} +(-122.915 - 212.895i) q^{59} +(233.236 - 403.977i) q^{61} +(-59.8456 + 103.656i) q^{65} +(88.5079 + 153.300i) q^{67} -45.2138 q^{71} -949.331 q^{73} +(174.738 + 302.655i) q^{77} +(-248.459 + 430.343i) q^{79} +(177.318 - 307.124i) q^{83} +(-181.384 - 314.166i) q^{85} +1193.20 q^{89} +250.904 q^{91} +(-114.980 - 199.152i) q^{95} +(204.399 - 354.029i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 30 q^{5} - 12 q^{7}+O(q^{10}) \) 12 * q + 30 * q^5 - 12 * q^7	\( 12 q + 30 q^{5} - 12 q^{7} + 84 q^{13} + 24 q^{17} - 228 q^{19} - 30 q^{23} - 150 q^{25} - 168 q^{29} + 324 q^{31} - 120 q^{35} - 984 q^{37} - 312 q^{41} + 156 q^{43} - 462 q^{47} + 588 q^{49} + 2028 q^{53} - 1008 q^{59} - 36 q^{61} - 420 q^{65} - 144 q^{67} + 2424 q^{71} - 1800 q^{73} - 672 q^{77} + 936 q^{79} - 288 q^{83} + 60 q^{85} - 240 q^{89} + 4572 q^{91} - 570 q^{95} + 1188 q^{97}+O(q^{100}) \) 12 * q + 30 * q^5 - 12 * q^7 + 84 * q^13 + 24 * q^17 - 228 * q^19 - 30 * q^23 - 150 * q^25 - 168 * q^29 + 324 * q^31 - 120 * q^35 - 984 * q^37 - 312 * q^41 + 156 * q^43 - 462 * q^47 + 588 * q^49 + 2028 * q^53 - 1008 * q^59 - 36 * q^61 - 420 * q^65 - 144 * q^67 + 2424 * q^71 - 1800 * q^73 - 672 * q^77 + 936 * q^79 - 288 * q^83 + 60 * q^85 - 240 * q^89 + 4572 * q^91 - 570 * q^95 + 1188 * q^97

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