LMFDB - Embedded newform 2052.3.s.a.901.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(829,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2052.829");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.s (of order $6$, 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:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			901.18
Character	$\chi$	$=$	2052.901
Dual form			2052.3.s.a.829.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.340706 - 0.590120i) q^{5} +(-2.79581 - 4.84248i) q^{7} +O(q^{10})$	$q+(-0.340706 - 0.590120i) q^{5} +(-2.79581 - 4.84248i) q^{7} +(-7.44896 - 12.9020i) q^{11} -2.36780i q^{13} +(8.72908 - 15.1192i) q^{17} +(-17.2199 - 8.02967i) q^{19} +10.4486 q^{23} +(12.2678 - 21.2485i) q^{25} +(37.6328 + 21.7273i) q^{29} +(28.9832 + 16.7334i) q^{31} +(-1.90510 + 3.29973i) q^{35} -30.3219i q^{37} +(-48.7811 + 28.1638i) q^{41} -62.4810 q^{43} +(-43.3232 + 75.0381i) q^{47} +(8.86690 - 15.3579i) q^{49} +(-43.7359 + 25.2509i) q^{53} +(-5.07582 + 8.79157i) q^{55} +(-13.9077 + 8.02962i) q^{59} +(0.219952 - 0.380967i) q^{61} +(-1.39729 + 0.806726i) q^{65} -41.0728i q^{67} +(-62.9245 - 36.3295i) q^{71} +(38.3367 - 66.4012i) q^{73} +(-41.6518 + 72.1430i) q^{77} +120.760i q^{79} +(53.7042 + 93.0184i) q^{83} -11.8962 q^{85} +(122.606 - 70.7865i) q^{89} +(-11.4661 + 6.61993i) q^{91} +(1.12845 + 12.8976i) q^{95} -171.665i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - q^{7}+O(q^{10}) $ 80 * q - q^7	$ 80 q - q^{7} + 6 q^{11} + 21 q^{17} - 20 q^{19} + 48 q^{23} - 200 q^{25} + 27 q^{29} - 24 q^{31} + 54 q^{35} + 18 q^{41} - 152 q^{43} + 12 q^{47} - 267 q^{49} + 36 q^{53} + 135 q^{59} - 7 q^{61} + 288 q^{65} + 81 q^{71} + 55 q^{73} - 30 q^{77} + 93 q^{83} - 216 q^{89} + 96 q^{91} - 288 q^{95}+O(q^{100}) $ 80 * q - q^7 + 6 * q^11 + 21 * q^17 - 20 * q^19 + 48 * q^23 - 200 * q^25 + 27 * q^29 - 24 * q^31 + 54 * q^35 + 18 * q^41 - 152 * q^43 + 12 * q^47 - 267 * q^49 + 36 * q^53 + 135 * q^59 - 7 * q^61 + 288 * q^65 + 81 * q^71 + 55 * q^73 - 30 * q^77 + 93 * q^83 - 216 * q^89 + 96 * q^91 - 288 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−0.340706	−	0.590120i	−0.0681412	−	0.118024i	0.829942	−	0.557850i	$-0.188374\pi$
$5$	−0.340706	−	0.590120i	−0.0681412	−	0.118024i	−0.898083	+	0.439826i	$0.855040\pi$
$6$		0			0
$7$	−2.79581	−	4.84248i	−0.399401	−	0.691783i	0.594251	−	0.804280i	$-0.297449\pi$
$7$	−2.79581	−	4.84248i	−0.399401	−	0.691783i	−0.993652	+	0.112496i	$0.964115\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.44896	−	12.9020i	−0.677179	−	1.17291i	−0.975827	−	0.218545i	$-0.929869\pi$
$11$	−7.44896	−	12.9020i	−0.677179	−	1.17291i	0.298648	−	0.954363i	$-0.403464\pi$
$12$		0			0
$13$		−	2.36780i		−	0.182139i	−0.995845	−	0.0910694i	$-0.970971\pi$
$13$		−	2.36780i		−	0.182139i	0.995845	−	0.0910694i	$-0.0290285\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	8.72908	−	15.1192i	0.513475	−	0.889365i	−0.486403	−	0.873735i	$-0.661691\pi$
$17$	8.72908	−	15.1192i	0.513475	−	0.889365i	0.999878	−	0.0156303i	$-0.00497549\pi$
$18$		0			0
$19$	−17.2199	−	8.02967i	−0.906310	−	0.422614i
$19$	−17.2199	−	8.02967i	−0.906310	−	0.422614i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	10.4486			0.454289			0.227144	−	0.973861i	$-0.427061\pi$
$23$	10.4486			0.454289			0.227144	+	0.973861i	$0.427061\pi$
$24$		0			0
$25$	12.2678	−	21.2485i	0.490714	−	0.849941i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	37.6328	+	21.7273i	1.29768	+	0.749217i	0.980003	−	0.198981i	$-0.0637632\pi$
$29$	37.6328	+	21.7273i	1.29768	+	0.749217i	0.317679	+	0.948198i	$0.397097\pi$
$30$		0			0
$31$	28.9832	+	16.7334i	0.934940	+	0.539788i	0.888371	−	0.459127i	$-0.151838\pi$
$31$	28.9832	+	16.7334i	0.934940	+	0.539788i	0.0465698	+	0.998915i	$0.485171\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.90510	+	3.29973i	−0.0544314	+	0.0942780i
$36$		0			0
$37$		−	30.3219i		−	0.819510i	−0.912196	−	0.409755i	$-0.865614\pi$
$37$		−	30.3219i		−	0.819510i	0.912196	−	0.409755i	$-0.134386\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−48.7811	+	28.1638i	−1.18978	+	0.686921i	−0.958257	−	0.285908i	$-0.907705\pi$
$41$	−48.7811	+	28.1638i	−1.18978	+	0.686921i	−0.231525	+	0.972829i	$0.574372\pi$
$42$		0			0
$43$	−62.4810			−1.45305			−0.726523	−	0.687142i	$-0.758865\pi$
$43$	−62.4810			−1.45305			−0.726523	+	0.687142i	$0.758865\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−43.3232	+	75.0381i	−0.921771	+	1.59655i	−0.125098	+	0.992144i	$0.539925\pi$
$47$	−43.3232	+	75.0381i	−0.921771	+	1.59655i	−0.796673	+	0.604410i	$0.793409\pi$
$48$		0			0
$49$	8.86690	−	15.3579i	0.180957	−	0.313427i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−43.7359	+	25.2509i	−0.825206	+	0.476433i	−0.852208	−	0.523202i	$-0.824737\pi$
$53$	−43.7359	+	25.2509i	−0.825206	+	0.476433i	0.0270024	+	0.999635i	$0.491404\pi$
$54$		0			0
$55$	−5.07582	+	8.79157i	−0.0922876	+	0.159847i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−13.9077	+	8.02962i	−0.235724	+	0.136095i	−0.613210	−	0.789920i	$-0.710122\pi$
$59$	−13.9077	+	8.02962i	−0.235724	+	0.136095i	0.377486	+	0.926015i	$0.376789\pi$
$60$		0			0
$61$	0.219952	−	0.380967i	0.00360576	−	0.00624536i	−0.864217	−	0.503119i	$-0.832186\pi$
$61$	0.219952	−	0.380967i	0.00360576	−	0.00624536i	0.867823	+	0.496874i	$0.165519\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.39729	+	0.806726i	−0.0214968	+	0.0124112i
$66$		0			0
$67$		−	41.0728i		−	0.613027i	−0.951866	−	0.306513i	$-0.900838\pi$
$67$		−	41.0728i		−	0.613027i	0.951866	−	0.306513i	$-0.0991624\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−62.9245	−	36.3295i	−0.886260	−	0.511683i	−0.0135429	−	0.999908i	$-0.504311\pi$
$71$	−62.9245	−	36.3295i	−0.886260	−	0.511683i	−0.872717	+	0.488226i	$0.837644\pi$
$72$		0			0
$73$	38.3367	−	66.4012i	0.525161	−	0.909605i	−0.474410	−	0.880304i	$-0.657339\pi$
$73$	38.3367	−	66.4012i	0.525161	−	0.909605i	0.999571	−	0.0293009i	$-0.00932811\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−41.6518	+	72.1430i	−0.540932	+	0.936922i
$78$		0			0
$79$			120.760i			1.52861i	0.644856	+	0.764304i	$0.276918\pi$
$79$			120.760i			1.52861i	−0.644856	+	0.764304i	$0.723082\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	53.7042	+	93.0184i	0.647038	+	1.12070i	0.983827	+	0.179123i	$0.0573260\pi$
$83$	53.7042	+	93.0184i	0.647038	+	1.12070i	−0.336788	+	0.941580i	$0.609341\pi$
$84$		0			0
$85$	−11.8962			−0.139955
$86$		0			0
$87$		0			0
$88$		0			0
$89$	122.606	−	70.7865i	1.37759	−	0.795354i	0.385725	−	0.922614i	$-0.373951\pi$
$89$	122.606	−	70.7865i	1.37759	−	0.795354i	0.991869	+	0.127259i	$0.0406181\pi$
$90$		0			0
$91$	−11.4661	+	6.61993i	−0.126001	+	0.0727465i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.12845	+	12.8976i	0.0118784	+	0.135764i
$96$		0			0
$97$		−	171.665i		−	1.76974i	−0.465836	−	0.884871i	$-0.654246\pi$
$97$		−	171.665i		−	1.76974i	0.465836	−	0.884871i	$-0.345754\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−95.2105	+	164.910i	−0.942679	+	1.63277i	−0.182345	+	0.983235i	$0.558369\pi$
$101$	−95.2105	+	164.910i	−0.942679	+	1.63277i	−0.760334	+	0.649533i	$0.774965\pi$
$102$		0			0
$103$	−82.2476	−	47.4857i	−0.798520	−	0.461026i	0.0444333	−	0.999012i	$-0.485852\pi$
$103$	−82.2476	−	47.4857i	−0.798520	−	0.461026i	−0.842953	+	0.537987i	$0.819185\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	21.9122i		−	0.204787i	−0.994744	−	0.102393i	$-0.967350\pi$
$107$		−	21.9122i		−	0.204787i	0.994744	−	0.102393i	$-0.0326500\pi$
$108$		0			0
$109$	−170.338	−	98.3447i	−1.56273	−	0.902245i	−0.996979	−	0.0776724i	$-0.975251\pi$
$109$	−170.338	−	98.3447i	−1.56273	−	0.902245i	−0.565756	−	0.824573i	$-0.691415\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.06724	−	2.34822i	−0.0359933	−	0.0207807i	0.481895	−	0.876229i	$-0.339949\pi$
$113$	−4.06724	−	2.34822i	−0.0359933	−	0.0207807i	−0.517889	+	0.855448i	$0.673282\pi$
$114$		0			0
$115$	−3.55992	−	6.16596i	−0.0309558	−	0.0536170i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−97.6193			−0.820331
$120$		0			0
$121$	−50.4742	+	87.4238i	−0.417142	+	0.722511i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−33.7542			−0.270034
$126$		0			0
$127$	−180.051	+	103.952i	−1.41772	+	0.818523i	−0.996099	−	0.0882482i	$-0.971873\pi$
$127$	−180.051	+	103.952i	−1.41772	+	0.818523i	−0.421624	+	0.906771i	$0.638540\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	101.507	+	175.815i	0.774861	+	1.34210i	0.934873	+	0.354983i	$0.115513\pi$
$131$	101.507	+	175.815i	0.774861	+	1.34210i	−0.160012	+	0.987115i	$0.551153\pi$
$132$		0			0
$133$	9.25996	+	105.836i	0.0696238	+	0.795763i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−44.5917	+	77.2351i	−0.325487	+	0.563760i	−0.981611	−	0.190894i	$-0.938861\pi$
$137$	−44.5917	+	77.2351i	−0.325487	+	0.563760i	0.656124	+	0.754653i	$0.272195\pi$
$138$		0			0
$139$	−165.847			−1.19315			−0.596573	−	0.802559i	$-0.703471\pi$
$139$	−165.847			−1.19315			−0.596573	+	0.802559i	$0.703471\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−30.5494	+	17.6377i	−0.213632	+	0.123340i
$144$		0			0
$145$		−	29.6105i		−	0.204210i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−77.9783	−	135.062i	−0.523344	−	0.906459i	−0.999631	−	0.0271686i	$-0.991351\pi$
$149$	−77.9783	−	135.062i	−0.523344	−	0.906459i	0.476287	−	0.879290i	$-0.341982\pi$
$150$		0			0
$151$	255.830	−	147.703i	1.69424	−	0.978168i	0.743212	−	0.669056i	$-0.233302\pi$
$151$	255.830	−	147.703i	1.69424	−	0.978168i	0.951026	−	0.309112i	$-0.100032\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	22.8047i		−	0.147127i
$156$		0			0
$157$	34.2907	+	59.3933i	0.218412	+	0.378301i	0.954323	−	0.298778i	$-0.0965789\pi$
$157$	34.2907	+	59.3933i	0.218412	+	0.378301i	−0.735910	+	0.677079i	$0.763246\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−29.2124	−	50.5974i	−0.181444	−	0.314269i
$162$		0			0
$163$	54.7677			0.335998			0.167999	−	0.985787i	$-0.446269\pi$
$163$	54.7677			0.335998			0.167999	+	0.985787i	$0.446269\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	104.833i		−	0.627745i	−0.949465	−	0.313873i	$-0.898373\pi$
$167$		−	104.833i		−	0.627745i	0.949465	−	0.313873i	$-0.101627\pi$
$168$		0			0
$169$	163.394			0.966825
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	115.208i		−	0.665940i	−0.942937	−	0.332970i	$-0.891949\pi$
$173$		−	115.208i		−	0.665940i	0.942937	−	0.332970i	$-0.108051\pi$
$174$		0			0
$175$	−137.194			−0.783966
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	136.922i		−	0.764928i	−0.923970	−	0.382464i	$-0.875076\pi$
$179$		−	136.922i		−	0.764928i	0.923970	−	0.382464i	$-0.124924\pi$
$180$		0			0
$181$	−80.6052	+	46.5374i	−0.445332	+	0.257113i	−0.705857	−	0.708354i	$-0.749438\pi$
$181$	−80.6052	+	46.5374i	−0.445332	+	0.257113i	0.260524	+	0.965467i	$0.416105\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−17.8936	+	10.3309i	−0.0967220	+	0.0558425i
$186$		0			0
$187$	−260.090			−1.39086
$188$		0			0
$189$		0			0
$190$		0			0
$191$	17.8353	+	30.8917i	0.0933786	+	0.161737i	0.908931	−	0.416947i	$-0.136900\pi$
$191$	17.8353	+	30.8917i	0.0933786	+	0.161737i	−0.815552	+	0.578684i	$0.803567\pi$
$192$		0			0
$193$	−127.581	+	73.6587i	−0.661039	+	0.381651i	−0.792673	−	0.609647i	$-0.791311\pi$
$193$	−127.581	+	73.6587i	−0.661039	+	0.381651i	0.131634	+	0.991298i	$0.457978\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−91.9844			−0.466926			−0.233463	−	0.972366i	$-0.575006\pi$
$197$	−91.9844			−0.466926			−0.233463	+	0.972366i	$0.575006\pi$
$198$		0			0
$199$	79.6590	+	137.973i	0.400296	+	0.693334i	0.993762	−	0.111526i	$-0.0355738\pi$
$199$	79.6590	+	137.973i	0.400296	+	0.693334i	−0.593465	+	0.804860i	$0.702241\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	242.982i		−	1.19695i
$204$		0			0
$205$	33.2400	+	19.1911i	0.162146	+	0.0936153i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.6716	+	281.983i	0.118046	+	1.34920i
$210$		0			0
$211$	−237.914	+	137.360i	−1.12755	+	0.650993i	−0.943318	−	0.331889i	$-0.892314\pi$
$211$	−237.914	+	137.360i	−1.12755	+	0.650993i	−0.184235	+	0.982882i	$0.558981\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	21.2877	+	36.8713i	0.0990124	+	0.171494i
$216$		0			0
$217$		−	187.134i		−	0.862368i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−35.7993	−	20.6687i	−0.161988	−	0.0935237i
$222$		0			0
$223$			128.065i			0.574281i	0.957888	+	0.287141i	$0.0927047\pi$
$223$			128.065i			0.574281i	−0.957888	+	0.287141i	$0.907295\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−79.1787	+	45.7138i	−0.348805	+	0.201383i	−0.664159	−	0.747592i	$-0.731210\pi$
$227$	−79.1787	+	45.7138i	−0.348805	+	0.201383i	0.315354	+	0.948974i	$0.397877\pi$
$228$		0			0
$229$	−17.3912	+	30.1225i	−0.0759442	+	0.131539i	−0.901496	−	0.432786i	$-0.857530\pi$
$229$	−17.3912	+	30.1225i	−0.0759442	+	0.131539i	0.825552	+	0.564326i	$0.190864\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−46.6581	+	80.8142i	−0.200249	+	0.346842i	−0.948609	−	0.316452i	$-0.897509\pi$
$233$	−46.6581	+	80.8142i	−0.200249	+	0.346842i	0.748359	+	0.663293i	$0.230842\pi$
$234$		0			0
$235$	59.0420			0.251243
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−206.209	+	357.165i	−0.862801	+	1.49441i	0.00641358	+	0.999979i	$0.497958\pi$
$239$	−206.209	+	357.165i	−0.862801	+	1.49441i	−0.869214	+	0.494435i	$0.835375\pi$
$240$		0			0
$241$	−148.987	−	86.0178i	−0.618204	−	0.356920i	0.157966	−	0.987445i	$-0.449507\pi$
$241$	−148.987	−	86.0178i	−0.618204	−	0.356920i	−0.776169	+	0.630524i	$0.782840\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−12.0840			−0.0493226
$246$		0			0
$247$	−19.0127	+	40.7733i	−0.0769744	+	0.165074i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−186.771	−	323.497i	−0.744108	−	1.28883i	−0.950610	−	0.310387i	$-0.899541\pi$
$251$	−186.771	−	323.497i	−0.744108	−	1.28883i	0.206502	−	0.978446i	$-0.433792\pi$
$252$		0			0
$253$	−77.8316	−	134.808i	−0.307635	−	0.532839i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			130.571i			0.508058i	0.967197	+	0.254029i	$0.0817559\pi$
$257$			130.571i			0.508058i	−0.967197	+	0.254029i	$0.918244\pi$
$258$		0			0
$259$	−146.833	+	84.7742i	−0.566924	+	0.327314i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	25.3716			0.0964699			0.0482349	−	0.998836i	$-0.484640\pi$
$263$	25.3716			0.0964699			0.0482349	+	0.998836i	$0.484640\pi$
$264$		0			0
$265$	29.8022	+	17.2063i	0.112461	+	0.0649295i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	128.863	+	74.3993i	0.479046	+	0.276577i	0.720019	−	0.693954i	$-0.244133\pi$
$269$	128.863	+	74.3993i	0.479046	+	0.276577i	−0.240973	+	0.970532i	$0.577466\pi$
$270$		0			0
$271$	198.014	−	342.970i	0.730678	−	1.26557i	−0.225916	−	0.974147i	$-0.572537\pi$
$271$	198.014	−	342.970i	0.730678	−	1.26557i	0.956594	−	0.291425i	$-0.0941292\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−365.531			−1.32920
$276$		0			0
$277$	137.542	+	238.230i	0.496541	+	0.860035i	0.999992	−	0.00398925i	$-0.00126982\pi$
$277$	137.542	+	238.230i	0.496541	+	0.860035i	−0.503451	+	0.864024i	$0.667936\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−321.322	−	185.515i	−1.14349	−	0.660197i	−0.196201	−	0.980564i	$-0.562860\pi$
$281$	−321.322	−	185.515i	−1.14349	−	0.660197i	−0.947294	+	0.320367i	$0.896194\pi$
$282$		0			0
$283$	49.0115	+	84.8904i	0.173185	+	0.299966i	0.939532	−	0.342462i	$-0.111261\pi$
$283$	49.0115	+	84.8904i	0.173185	+	0.299966i	−0.766346	+	0.642428i	$0.777927\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	272.765	+	157.481i	0.950401	+	0.548714i
$288$		0			0
$289$	−7.89361	−	13.6721i	−0.0273135	−	0.0473084i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	341.458	+	197.141i	1.16538	+	0.672835i	0.952588	−	0.304262i	$-0.0984097\pi$
$293$	341.458	+	197.141i	1.16538	+	0.672835i	0.212796	+	0.977097i	$0.431743\pi$
$294$		0			0
$295$	9.47689	+	5.47148i	0.0321250	+	0.0185474i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	24.7403i		−	0.0827436i
$300$		0			0
$301$	174.685	+	302.563i	0.580349	+	1.00519i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.299755			−0.000982805
$306$		0			0
$307$	−450.819	−	260.280i	−1.46847	−	0.847819i	−0.469090	−	0.883150i	$-0.655418\pi$
$307$	−450.819	−	260.280i	−1.46847	−	0.847819i	−0.999376	+	0.0353313i	$0.988751\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	89.2653	−	154.612i	0.287027	−	0.497145i	−0.686072	−	0.727534i	$-0.740666\pi$
$311$	89.2653	−	154.612i	0.287027	−	0.497145i	0.973099	+	0.230389i	$0.0739998\pi$
$312$		0			0
$313$	95.6595	−	165.687i	0.305621	−	0.529352i	−0.671778	−	0.740752i	$-0.734469\pi$
$313$	95.6595	−	165.687i	0.305621	−	0.529352i	0.977399	+	0.211401i	$0.0678025\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−142.358	−	82.1907i	−0.449080	−	0.259277i	0.258361	−	0.966048i	$-0.416817\pi$
$317$	−142.358	−	82.1907i	−0.449080	−	0.259277i	−0.707442	+	0.706772i	$0.750151\pi$
$318$		0			0
$319$		−	647.384i		−	2.02942i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−271.716	+	190.259i	−0.841226	+	0.589038i
$324$		0			0
$325$	−50.3123	−	29.0478i	−0.154807	−	0.0893780i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	484.494			1.47263
$330$		0			0
$331$	86.5524	−	49.9711i	0.261488	−	0.150970i	−0.363525	−	0.931584i	$-0.618427\pi$
$331$	86.5524	−	49.9711i	0.261488	−	0.150970i	0.625013	+	0.780614i	$0.285094\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−24.2379	+	13.9938i	−0.0723519	+	0.0417724i
$336$		0			0
$337$	−385.329	+	222.470i	−1.14341	+	0.660148i	−0.947273	−	0.320429i	$-0.896173\pi$
$337$	−385.329	+	222.470i	−1.14341	+	0.660148i	−0.196137	+	0.980576i	$0.562840\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	498.587i		−	1.46213i
$342$		0			0
$343$	−373.150			−1.08790
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−25.7406	−	44.5839i	−0.0741803	−	0.128484i	0.826549	−	0.562864i	$-0.190301\pi$
$347$	−25.7406	−	44.5839i	−0.0741803	−	0.128484i	−0.900730	+	0.434380i	$0.856967\pi$
$348$		0			0
$349$	−46.4841	−	80.5128i	−0.133192	−	0.230696i	0.791713	−	0.610893i	$-0.209189\pi$
$349$	−46.4841	−	80.5128i	−0.133192	−	0.230696i	−0.924905	+	0.380197i	$0.875856\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	332.939	+	576.668i	0.943171	+	1.63362i	0.759373	+	0.650655i	$0.225506\pi$
$353$	332.939	+	576.668i	0.943171	+	1.63362i	0.183797	+	0.982964i	$0.441161\pi$
$354$		0			0
$355$			49.5107i			0.139467i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	17.3255	−	30.0087i	0.0482605	−	0.0835897i	−0.840886	−	0.541212i	$-0.817966\pi$
$359$	17.3255	−	30.0087i	0.0482605	−	0.0835897i	0.889147	+	0.457623i	$0.151299\pi$
$360$		0			0
$361$	232.049	+	276.540i	0.642794	+	0.766039i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−52.2462			−0.143140
$366$		0			0
$367$	−223.471	+	387.063i	−0.608912	+	1.05467i	0.382508	+	0.923952i	$0.375060\pi$
$367$	−223.471	+	387.063i	−0.608912	+	1.05467i	−0.991420	+	0.130715i	$0.958273\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	244.555	+	141.194i	0.659177	+	0.380576i
$372$		0			0
$373$	359.861	+	207.766i	0.964775	+	0.557013i	0.897639	−	0.440731i	$-0.145281\pi$
$373$	359.861	+	207.766i	0.964775	+	0.557013i	0.0671357	+	0.997744i	$0.478614\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	51.4460	−	89.1071i	0.136462	−	0.236358i
$378$		0			0
$379$			485.460i			1.28090i	0.768001	+	0.640449i	$0.221252\pi$
$379$			485.460i			1.28090i	−0.768001	+	0.640449i	$0.778748\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	148.710	−	85.8577i	0.388276	−	0.224172i	−0.293137	−	0.956071i	$-0.594699\pi$
$383$	148.710	−	85.8577i	0.388276	−	0.224172i	0.681413	+	0.731899i	$0.261366\pi$
$384$		0			0
$385$	56.7641			0.147439
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−141.077	+	244.352i	−0.362665	+	0.628154i	−0.988398	−	0.151883i	$-0.951466\pi$
$389$	−141.077	+	244.352i	−0.362665	+	0.628154i	0.625734	+	0.780037i	$0.284800\pi$
$390$		0			0
$391$	91.2070	−	157.975i	0.233266	−	0.404029i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	71.2630	−	41.1437i	0.180413	−	0.104161i
$396$		0			0
$397$	252.435	−	437.231i	0.635857	−	1.10134i	−0.350476	−	0.936572i	$-0.613980\pi$
$397$	252.435	−	437.231i	0.635857	−	1.10134i	0.986333	−	0.164765i	$-0.0526866\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−36.8889	+	21.2978i	−0.0919923	+	0.0531118i	−0.545290	−	0.838247i	$-0.683581\pi$
$401$	−36.8889	+	21.2978i	−0.0919923	+	0.0531118i	0.453298	+	0.891359i	$0.350247\pi$
$402$		0			0
$403$	39.6215	−	68.6264i	0.0983163	−	0.170289i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−391.213	+	225.867i	−0.961210	+	0.554955i
$408$		0			0
$409$		−	404.458i		−	0.988896i	−0.869207	−	0.494448i	$-0.835370\pi$
$409$		−	404.458i		−	0.988896i	0.869207	−	0.494448i	$-0.164630\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	77.7666	+	44.8986i	0.188297	+	0.108713i
$414$		0			0
$415$	36.5947	−	63.3839i	0.0881800	−	0.152732i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	216.308	−	374.657i	0.516249	−	0.894169i	−0.483573	−	0.875304i	$-0.660661\pi$
$419$	216.308	−	374.657i	0.516249	−	0.894169i	0.999822	−	0.0188652i	$-0.00600535\pi$
$420$		0			0
$421$		−	23.5995i		−	0.0560557i	−0.999607	−	0.0280279i	$-0.991077\pi$
$421$		−	23.5995i		−	0.0560557i	0.999607	−	0.0280279i	$-0.00892272\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−214.174	−	370.960i	−0.503938	−	0.872847i
$426$		0			0
$427$	−2.45977			−0.00576059
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−89.7445	+	51.8140i	−0.208224	+	0.120218i	−0.600486	−	0.799636i	$-0.705026\pi$
$431$	−89.7445	+	51.8140i	−0.208224	+	0.120218i	0.392262	+	0.919854i	$0.371693\pi$
$432$		0			0
$433$	−464.495	+	268.176i	−1.07274	+	0.619344i	−0.928928	−	0.370261i	$-0.879268\pi$
$433$	−464.495	+	268.176i	−1.07274	+	0.619344i	−0.143808	+	0.989606i	$0.545935\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−179.924	−	83.8992i	−0.411726	−	0.191989i
$438$		0			0
$439$		−	260.499i		−	0.593392i	−0.954972	−	0.296696i	$-0.904115\pi$
$439$		−	260.499i		−	0.593392i	0.954972	−	0.296696i	$-0.0958848\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−244.994	+	424.342i	−0.553034	+	0.957882i	0.445020	+	0.895521i	$0.353197\pi$
$443$	−244.994	+	424.342i	−0.553034	+	0.957882i	−0.998054	+	0.0623617i	$0.980137\pi$
$444$		0			0
$445$	−83.5452	−	48.2348i	−0.187742	−	0.108393i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	783.722i		−	1.74548i	−0.488182	−	0.872742i	$-0.662340\pi$
$449$		−	783.722i		−	1.74548i	0.488182	−	0.872742i	$-0.337660\pi$
$450$		0			0
$451$	726.737	+	419.582i	1.61139	+	0.930336i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	7.81311	+	4.51090i	0.0171717	+	0.00991407i
$456$		0			0
$457$	−166.574	−	288.514i	−0.364493	−	0.631321i	0.624201	−	0.781264i	$-0.285425\pi$
$457$	−166.574	−	288.514i	−0.364493	−	0.631321i	−0.988695	+	0.149942i	$0.952091\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	473.007			1.02605			0.513023	−	0.858375i	$-0.328526\pi$
$461$	473.007			1.02605			0.513023	+	0.858375i	$0.328526\pi$
$462$		0			0
$463$	151.128	−	261.762i	0.326411	−	0.565360i	−0.655386	−	0.755294i	$-0.727494\pi$
$463$	151.128	−	261.762i	0.326411	−	0.565360i	0.981797	+	0.189934i	$0.0608274\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−420.810			−0.901093			−0.450546	−	0.892753i	$-0.648771\pi$
$467$	−420.810			−0.901093			−0.450546	+	0.892753i	$0.648771\pi$
$468$		0			0
$469$	−198.894	+	114.832i	−0.424082	+	0.244844i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	465.419	+	806.129i	0.983972	+	1.70429i
$474$		0			0
$475$	−381.869	+	267.390i	−0.803936	+	0.562927i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−186.508	+	323.041i	−0.389369	+	0.674407i	−0.992365	−	0.123337i	$-0.960640\pi$
$479$	−186.508	+	323.041i	−0.389369	+	0.674407i	0.602996	+	0.797745i	$0.293974\pi$
$480$		0			0
$481$	−71.7963			−0.149265
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−101.303	+	58.4873i	−0.208872	+	0.120592i
$486$		0			0
$487$		−	666.331i		−	1.36824i	−0.729371	−	0.684118i	$-0.760187\pi$
$487$		−	666.331i		−	1.36824i	0.729371	−	0.684118i	$-0.239813\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	278.509	+	482.392i	0.567228	+	0.982468i	0.996839	+	0.0794534i	$0.0253175\pi$
$491$	278.509	+	482.392i	0.567228	+	0.982468i	−0.429611	+	0.903014i	$0.641349\pi$
$492$		0			0
$493$	656.999	−	379.319i	1.33266	−	0.769409i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			406.281i			0.817467i
$498$		0			0
$499$	312.553	+	541.357i	0.626358	+	1.08488i	0.988277	+	0.152675i	$0.0487886\pi$
$499$	312.553	+	541.357i	0.626358	+	1.08488i	−0.361918	+	0.932210i	$0.617878\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−99.7875	−	172.837i	−0.198385	−	0.343613i	0.749620	−	0.661868i	$-0.230236\pi$
$503$	−99.7875	−	172.837i	−0.198385	−	0.343613i	−0.948005	+	0.318256i	$0.896903\pi$
$504$		0			0
$505$	129.755			0.256941
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	176.640i		−	0.347034i	−0.984831	−	0.173517i	$-0.944487\pi$
$509$		−	176.640i		−	0.347034i	0.984831	−	0.173517i	$-0.0555131\pi$
$510$		0			0
$511$	−428.729			−0.838999
$512$		0			0
$513$		0			0
$514$		0			0
$515$			64.7146i			0.125659i
$516$		0			0
$517$	1290.85			2.49682
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	143.921i		−	0.276240i	−0.990415	−	0.138120i	$-0.955894\pi$
$521$		−	143.921i		−	0.276240i	0.990415	−	0.138120i	$-0.0441060\pi$
$522$		0			0
$523$	22.5109	−	12.9967i	0.0430419	−	0.0248503i	−0.478325	−	0.878183i	$-0.658756\pi$
$523$	22.5109	−	12.9967i	0.0430419	−	0.0248503i	0.521367	+	0.853333i	$0.325422\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	505.992	−	292.135i	0.960137	−	0.554336i
$528$		0			0
$529$	−419.826			−0.793622
$530$		0			0
$531$		0			0
$532$		0			0
$533$	66.6863	+	115.504i	0.125115	+	0.216705i
$534$		0			0
$535$	−12.9308	+	7.46561i	−0.0241698	+	0.0139544i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−264.197			−0.490161
$540$		0			0
$541$	387.168	+	670.595i	0.715653	+	1.23955i	0.962707	+	0.270546i	$0.0872041\pi$
$541$	387.168	+	670.595i	0.715653	+	1.23955i	−0.247054	+	0.969002i	$0.579463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			134.027i			0.245920i
$546$		0			0
$547$	14.4722	+	8.35552i	0.0264574	+	0.0152752i	0.513170	−	0.858287i	$-0.328471\pi$
$547$	14.4722	+	8.35552i	0.0264574	+	0.0152752i	−0.486713	+	0.873562i	$0.661804\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−473.569	−	676.321i	−0.859472	−	1.22744i
$552$		0			0
$553$	584.779	−	337.622i	1.05747	−	0.610528i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−269.959	−	467.583i	−0.484666	−	0.839467i	0.515178	−	0.857083i	$-0.327726\pi$
$557$	−269.959	−	467.583i	−0.484666	−	0.839467i	−0.999845	+	0.0176161i	$0.994392\pi$
$558$		0			0
$559$			147.943i			0.264656i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	504.214	+	291.108i	0.895585	+	0.517066i	0.875765	−	0.482738i	$-0.160358\pi$
$563$	504.214	+	291.108i	0.895585	+	0.517066i	0.0198195	+	0.999804i	$0.493691\pi$
$564$		0			0
$565$			3.20022i			0.00566410i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	520.303	−	300.397i	0.914417	−	0.527939i	0.0325670	−	0.999470i	$-0.489632\pi$
$569$	520.303	−	300.397i	0.914417	−	0.527939i	0.881850	+	0.471531i	$0.156298\pi$
$570$		0			0
$571$	263.244	−	455.953i	0.461023	−	0.798516i	−0.537989	−	0.842952i	$-0.680816\pi$
$571$	263.244	−	455.953i	0.461023	−	0.798516i	0.999012	+	0.0444360i	$0.0141491\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	128.182	−	222.018i	0.222926	−	0.386119i
$576$		0			0
$577$	679.846			1.17824			0.589121	−	0.808045i	$-0.299474\pi$
$577$	679.846			1.17824			0.589121	+	0.808045i	$0.299474\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	300.293	−	520.123i	0.516856	−	0.895221i
$582$		0			0
$583$	651.575	+	376.187i	1.11762	+	0.645260i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−431.522			−0.735130			−0.367565	−	0.929998i	$-0.619809\pi$
$587$	−431.522			−0.735130			−0.367565	+	0.929998i	$0.619809\pi$
$588$		0			0
$589$	−364.723	−	520.873i	−0.619223	−	0.884334i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	184.931	+	320.310i	0.311856	+	0.540151i	0.978764	−	0.204989i	$-0.0657159\pi$
$593$	184.931	+	320.310i	0.311856	+	0.540151i	−0.666908	+	0.745140i	$0.732383\pi$
$594$		0			0
$595$	33.2595	+	57.6072i	0.0558983	+	0.0968188i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			740.472i			1.23618i	0.786107	+	0.618090i	$0.212093\pi$
$599$			740.472i			1.23618i	−0.786107	+	0.618090i	$0.787907\pi$
$600$		0			0
$601$	15.1465	−	8.74483i	0.0252021	−	0.0145505i	−0.487346	−	0.873209i	$-0.662035\pi$
$601$	15.1465	−	8.74483i	0.0252021	−	0.0145505i	0.512548	+	0.858658i	$0.328702\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	68.7874			0.113698
$606$		0			0
$607$	601.906	+	347.510i	0.991607	+	0.572505i	0.905754	−	0.423803i	$-0.139305\pi$
$607$	601.906	+	347.510i	0.991607	+	0.572505i	0.0858530	+	0.996308i	$0.472638\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	177.675	+	102.581i	0.290795	+	0.167890i
$612$		0			0
$613$	140.459	−	243.281i	0.229133	−	0.396870i	−0.728418	−	0.685133i	$-0.759744\pi$
$613$	140.459	−	243.281i	0.229133	−	0.396870i	0.957551	+	0.288263i	$0.0930776\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	425.784			0.690087			0.345044	−	0.938587i	$-0.387864\pi$
$617$	425.784			0.690087			0.345044	+	0.938587i	$0.387864\pi$
$618$		0			0
$619$	−268.153	−	464.455i	−0.433204	−	0.750332i	0.563943	−	0.825814i	$-0.309284\pi$
$619$	−268.153	−	464.455i	−0.433204	−	0.750332i	−0.997147	+	0.0754821i	$0.975950\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−685.565	−	395.811i	−1.10043	−	0.635331i
$624$		0			0
$625$	−295.196	−	511.294i	−0.472313	−	0.818070i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−458.443	−	264.682i	−0.728844	−	0.420798i
$630$		0			0
$631$	42.3648	+	73.3780i	0.0671391	+	0.116288i	0.897641	−	0.440728i	$-0.145280\pi$
$631$	42.3648	+	73.3780i	0.0671391	+	0.116288i	−0.830502	+	0.557016i	$0.811946\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	122.689	+	70.8344i	0.193211	+	0.111550i
$636$		0			0
$637$	−36.3646	−	20.9951i	−0.0570872	−	0.0329593i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1011.89i		−	1.57862i	−0.613996	−	0.789309i	$-0.710439\pi$
$641$		−	1011.89i		−	1.57862i	0.613996	−	0.789309i	$-0.289561\pi$
$642$		0			0
$643$	−620.422	−	1074.60i	−0.964887	−	1.67123i	−0.709919	−	0.704283i	$-0.751269\pi$
$643$	−620.422	−	1074.60i	−0.964887	−	1.67123i	−0.254967	−	0.966950i	$-0.582065\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	491.485			0.759637			0.379819	−	0.925061i	$-0.375986\pi$
$647$	491.485			0.759637			0.379819	+	0.925061i	$0.375986\pi$
$648$		0			0
$649$	207.196	+	119.625i	0.319254	+	0.184322i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	544.371	−	942.878i	0.833646	−	1.44392i	−0.0614817	−	0.998108i	$-0.519583\pi$
$653$	544.371	−	942.878i	0.833646	−	1.44392i	0.895128	−	0.445809i	$-0.147084\pi$
$654$		0			0
$655$	69.1680	−	119.802i	0.105600	−	0.182905i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−38.8040	−	22.4035i	−0.0588831	−	0.0339962i	0.470269	−	0.882523i	$-0.344157\pi$
$659$	−38.8040	−	22.4035i	−0.0588831	−	0.0339962i	−0.529153	+	0.848527i	$0.677490\pi$
$660$		0			0
$661$			626.834i			0.948312i	0.880441	+	0.474156i	$0.157247\pi$
$661$			626.834i			0.948312i	−0.880441	+	0.474156i	$0.842753\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	59.3013	−	41.5236i	0.0891749	−	0.0624415i
$666$		0			0
$667$	393.212	+	227.021i	0.589523	+	0.340361i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.55365			−0.00976698
$672$		0			0
$673$	978.836	−	565.131i	1.45444	−	0.839719i	0.455708	−	0.890129i	$-0.349386\pi$
$673$	978.836	−	565.131i	1.45444	−	0.839719i	0.998729	+	0.0504103i	$0.0160529\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	269.720	−	155.723i	0.398405	−	0.230019i	−0.287390	−	0.957814i	$-0.592788\pi$
$677$	269.720	−	155.723i	0.398405	−	0.230019i	0.685796	+	0.727794i	$0.259454\pi$
$678$		0			0
$679$	−831.285	+	479.942i	−1.22428	+	0.706837i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			30.4855i			0.0446348i	0.999751	+	0.0223174i	$0.00710443\pi$
$683$			30.4855i			0.0446348i	−0.999751	+	0.0223174i	$0.992896\pi$
$684$		0			0
$685$	60.7706			0.0887163
$686$		0			0
$687$		0			0
$688$		0			0
$689$	59.7893	+	103.558i	0.0867769	+	0.150302i
$690$		0			0
$691$	−231.091	−	400.262i	−0.334431	−	0.579251i	0.648945	−	0.760836i	$-0.275211\pi$
$691$	−231.091	−	400.262i	−0.334431	−	0.579251i	−0.983375	+	0.181585i	$0.941877\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	56.5052	+	97.8699i	0.0813024	+	0.140820i
$696$		0			0
$697$			983.375i			1.41087i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	640.659	−	1109.65i	0.913922	−	1.58296i	0.105449	−	0.994425i	$-0.466372\pi$
$701$	640.659	−	1109.65i	0.913922	−	1.58296i	0.808473	−	0.588534i	$-0.200295\pi$
$702$		0			0
$703$	−243.475	+	522.139i	−0.346337	+	0.742730i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1064.76			1.50603
$708$		0			0
$709$	59.5253	−	103.101i	0.0839567	−	0.145417i	−0.820989	−	0.570943i	$-0.806578\pi$
$709$	59.5253	−	103.101i	0.0839567	−	0.145417i	0.904946	+	0.425526i	$0.139911\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	302.835	+	174.842i	0.424733	+	0.245220i
$714$		0			0
$715$	20.8167	+	12.0185i	0.0291143	+	0.0168091i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−23.4411	+	40.6012i	−0.0326024	+	0.0564690i	−0.881866	−	0.471500i	$-0.843713\pi$
$719$	−23.4411	+	40.6012i	−0.0326024	+	0.0564690i	0.849264	+	0.527969i	$0.177046\pi$
$720$		0			0
$721$			531.043i			0.736537i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	923.346	−	533.094i	1.27358	−	0.735302i
$726$		0			0
$727$	−922.694			−1.26918			−0.634590	−	0.772849i	$-0.718831\pi$
$727$	−922.694			−1.26918			−0.634590	+	0.772849i	$0.718831\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−545.401	+	944.663i	−0.746103	+	1.29229i
$732$		0			0
$733$	−379.867	+	657.948i	−0.518236	+	0.897610i	0.481540	+	0.876424i	$0.340078\pi$
$733$	−379.867	+	657.948i	−0.518236	+	0.897610i	−0.999776	+	0.0211862i	$0.993256\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−529.920	+	305.950i	−0.719024	+	0.415128i
$738$		0			0
$739$	−455.753	+	789.387i	−0.616716	+	1.06818i	0.373365	+	0.927684i	$0.378204\pi$
$739$	−455.753	+	789.387i	−0.616716	+	1.06818i	−0.990081	+	0.140498i	$0.955130\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	624.251	−	360.411i	0.840176	−	0.485076i	−0.0171482	−	0.999853i	$-0.505459\pi$
$743$	624.251	−	360.411i	0.840176	−	0.485076i	0.857324	+	0.514777i	$0.172125\pi$
$744$		0			0
$745$	−53.1354	+	92.0332i	−0.0713226	+	0.123534i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−106.109	+	61.2623i	−0.141668	+	0.0817921i
$750$		0			0
$751$		−	204.996i		−	0.272964i	−0.990643	−	0.136482i	$-0.956420\pi$
$751$		−	204.996i		−	0.272964i	0.990643	−	0.136482i	$-0.0435796\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−174.326	−	100.647i	−0.230895	−	0.133307i
$756$		0			0
$757$	−144.571	+	250.405i	−0.190979	+	0.330785i	−0.945575	−	0.325404i	$-0.894500\pi$
$757$	−144.571	+	250.405i	−0.190979	+	0.330785i	0.754596	+	0.656190i	$0.227833\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−312.556	+	541.362i	−0.410717	+	0.711383i	−0.994968	−	0.100190i	$-0.968055\pi$
$761$	−312.556	+	541.362i	−0.410717	+	0.711383i	0.584251	+	0.811573i	$0.301388\pi$
$762$		0			0
$763$			1099.81i			1.44143i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	19.0126	+	32.9307i	0.0247882	+	0.0429345i
$768$		0			0
$769$	891.808			1.15970			0.579849	−	0.814724i	$-0.303112\pi$
$769$	891.808			1.15970			0.579849	+	0.814724i	$0.303112\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	508.443	−	293.550i	0.657753	−	0.379754i	−0.133667	−	0.991026i	$-0.542675\pi$
$773$	508.443	−	293.550i	0.657753	−	0.379754i	0.791420	+	0.611272i	$0.209342\pi$
$774$		0			0
$775$	711.121	−	410.566i	0.917576	−	0.529763i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1066.15	−	93.2808i	1.36861	−	0.119744i
$780$		0			0
$781$			1082.47i			1.38600i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.3661	−	40.4713i	0.0297658	−	0.0515558i
$786$		0			0
$787$	−735.073	−	424.395i	−0.934020	−	0.539257i	−0.0459391	−	0.998944i	$-0.514628\pi$
$787$	−735.073	−	424.395i	−0.934020	−	0.539257i	−0.888081	+	0.459688i	$0.847961\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			26.2607i			0.0331994i
$792$		0			0
$793$	−0.902056	−	0.520802i	−0.00113752	−	0.000656749i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−976.928	−	564.030i	−1.22576	−	0.707691i	−0.259617	−	0.965712i	$-0.583596\pi$
$797$	−976.928	−	564.030i	−1.22576	−	0.707691i	−0.966139	+	0.258021i	$0.916930\pi$
$798$		0			0
$799$	756.344	+	1310.03i	0.946613	+	1.63958i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1142.28			−1.42251
$804$		0			0
$805$	−19.9057	+	34.4777i	−0.0247276	+	0.0428294i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−909.119			−1.12376			−0.561878	−	0.827220i	$-0.689921\pi$
$809$	−909.119			−1.12376			−0.561878	+	0.827220i	$0.689921\pi$
$810$		0			0
$811$	645.717	−	372.805i	0.796199	−	0.459686i	−0.0459414	−	0.998944i	$-0.514629\pi$
$811$	645.717	−	372.805i	0.796199	−	0.459686i	0.842140	+	0.539258i	$0.181295\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.6597	−	32.3195i	−0.0228953	−	0.0396559i
$816$		0			0
$817$	1075.92	+	501.702i	1.31691	+	0.614078i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−58.9196	+	102.052i	−0.0717656	+	0.124302i	−0.899675	−	0.436560i	$-0.856197\pi$
$821$	−58.9196	+	102.052i	−0.0717656	+	0.124302i	0.827910	+	0.560862i	$0.189530\pi$
$822$		0			0
$823$	−446.415			−0.542424			−0.271212	−	0.962520i	$-0.587424\pi$
$823$	−446.415			−0.542424			−0.271212	+	0.962520i	$0.587424\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	198.286	−	114.480i	0.239765	−	0.138428i	−0.375304	−	0.926902i	$-0.622462\pi$
$827$	198.286	−	114.480i	0.239765	−	0.138428i	0.615069	+	0.788474i	$0.289128\pi$
$828$		0			0
$829$			705.074i			0.850511i	0.905073	+	0.425256i	$0.139816\pi$
$829$			705.074i			0.850511i	−0.905073	+	0.425256i	$0.860184\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−154.800	−	268.121i	−0.185834	−	0.321874i
$834$		0			0
$835$	−61.8644	+	35.7174i	−0.0740891	+	0.0427754i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	762.069i		−	0.908306i	−0.890924	−	0.454153i	$-0.849942\pi$
$839$		−	762.069i		−	0.908306i	0.890924	−	0.454153i	$-0.150058\pi$
$840$		0			0
$841$	523.651	+	906.991i	0.622653	+	1.07847i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−55.6692	−	96.4219i	−0.0658807	−	0.114109i
$846$		0			0
$847$	564.464			0.666428
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	316.823i		−	0.372295i
$852$		0			0
$853$	−545.586			−0.639608			−0.319804	−	0.947484i	$-0.603617\pi$
$853$	−545.586			−0.639608			−0.319804	+	0.947484i	$0.603617\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	107.063i		−	0.124927i	−0.998047	−	0.0624636i	$-0.980104\pi$
$857$		−	107.063i		−	0.124927i	0.998047	−	0.0624636i	$-0.0198957\pi$
$858$		0			0
$859$	844.828			0.983502			0.491751	−	0.870736i	$-0.336357\pi$
$859$	844.828			0.983502			0.491751	+	0.870736i	$0.336357\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	140.169i		−	0.162421i	−0.996697	−	0.0812104i	$-0.974121\pi$
$863$		−	140.169i		−	0.162421i	0.996697	−	0.0812104i	$-0.0258786\pi$
$864$		0			0
$865$	−67.9864	+	39.2520i	−0.0785970	+	0.0453780i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1558.05	−	899.538i	1.79292	−	1.03514i
$870$		0			0
$871$	−97.2523			−0.111656
$872$		0			0
$873$		0			0
$874$		0			0
$875$	94.3704	+	163.454i	0.107852	+	0.186805i
$876$		0			0
$877$	621.612	−	358.888i	0.708794	−	0.409222i	−0.101820	−	0.994803i	$-0.532467\pi$
$877$	621.612	−	358.888i	0.708794	−	0.409222i	0.810614	+	0.585581i	$0.199133\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−973.635			−1.10515			−0.552574	−	0.833464i	$-0.686354\pi$
$881$	−973.635			−1.10515			−0.552574	+	0.833464i	$0.686354\pi$
$882$		0			0
$883$	−810.466	−	1403.77i	−0.917855	−	1.58977i	−0.802667	−	0.596427i	$-0.796586\pi$
$883$	−810.466	−	1403.77i	−0.917855	−	1.58977i	−0.115188	−	0.993344i	$-0.536747\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			93.3587i			0.105252i	0.998614	+	0.0526261i	$0.0167591\pi$
$887$			93.3587i			0.105252i	−0.998614	+	0.0526261i	$0.983241\pi$
$888$		0			0
$889$	1006.78	+	581.262i	1.13248	+	0.653838i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1348.55	−	944.275i	1.51014	−	1.05742i
$894$		0			0
$895$	−80.8006	+	46.6502i	−0.0902800	+	0.0521232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	727.145	+	1259.45i	0.808837	+	1.40095i
$900$		0			0
$901$			881.670i			0.978546i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	54.9254	+	31.7112i	0.0606910	+	0.0350400i
$906$		0			0
$907$		−	1442.46i		−	1.59036i	−0.606372	−	0.795181i	$-0.707376\pi$
$907$		−	1442.46i		−	1.59036i	0.606372	−	0.795181i	$-0.292624\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−967.610	+	558.650i	−1.06214	+	0.613227i	−0.926024	−	0.377466i	$-0.876796\pi$
$911$	−967.610	+	558.650i	−1.06214	+	0.613227i	−0.136117	+	0.990693i	$0.543462\pi$
$912$		0			0
$913$	800.081	−	1385.78i	0.876321	−	1.51783i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	567.587	−	983.090i	0.618961	−	1.07207i
$918$		0			0
$919$	−150.532			−0.163800			−0.0819002	−	0.996641i	$-0.526099\pi$
$919$	−150.532			−0.163800			−0.0819002	+	0.996641i	$0.526099\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−86.0211	+	148.993i	−0.0931972	+	0.161422i
$924$		0			0
$925$	−644.295	−	371.984i	−0.696535	−	0.402145i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1219.21			1.31239			0.656193	−	0.754593i	$-0.272166\pi$
$929$	1219.21			1.31239			0.656193	+	0.754593i	$0.272166\pi$
$930$		0			0
$931$	−276.006	+	193.263i	−0.296462	+	0.207587i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	88.6144	+	153.485i	0.0947748	+	0.164155i
$936$		0			0
$937$	−584.150	−	1011.78i	−0.623426	−	1.07981i	−0.988843	−	0.148962i	$-0.952407\pi$
$937$	−584.150	−	1011.78i	−0.623426	−	1.07981i	0.365417	−	0.930844i	$-0.380926\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			757.255i			0.804735i	0.915478	+	0.402367i	$0.131813\pi$
$941$			757.255i			0.804735i	−0.915478	+	0.402367i	$0.868187\pi$
$942$		0			0
$943$	−509.696	+	294.273i	−0.540505	+	0.312061i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−717.348			−0.757495			−0.378748	−	0.925500i	$-0.623645\pi$
$947$	−717.348			−0.757495			−0.378748	+	0.925500i	$0.623645\pi$
$948$		0			0
$949$	−157.225	−	90.7739i	−0.165674	−	0.0956521i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−143.573	−	82.8922i	−0.150654	−	0.0869802i	0.422778	−	0.906233i	$-0.361055\pi$
$953$	−143.573	−	82.8922i	−0.150654	−	0.0869802i	−0.573432	+	0.819253i	$0.694388\pi$
$954$		0			0
$955$	12.1532	−	21.0500i	0.0127259	−	0.0220419i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	498.679			0.519999
$960$		0			0
$961$	79.5155	+	137.725i	0.0827424	+	0.143314i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	86.9350	+	50.1919i	0.0900881	+	0.0520124i
$966$		0			0
$967$	−111.196	−	192.596i	−0.114990	−	0.199169i	0.802786	−	0.596268i	$-0.203350\pi$
$967$	−111.196	−	192.596i	−0.114990	−	0.199169i	−0.917776	+	0.397099i	$0.870017\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1374.21	−	793.401i	−1.41525	−	0.817097i	−0.419377	−	0.907812i	$-0.637751\pi$
$971$	−1374.21	−	793.401i	−1.41525	−	0.817097i	−0.995877	+	0.0907152i	$0.971085\pi$
$972$		0			0
$973$	463.677	+	803.113i	0.476544	+	0.825398i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−484.548	−	279.754i	−0.495955	−	0.286340i	0.231086	−	0.972933i	$-0.425772\pi$
$977$	−484.548	−	279.754i	−0.495955	−	0.286340i	−0.727042	+	0.686593i	$0.759105\pi$
$978$		0			0
$979$	−1826.57	−	1054.57i	−1.86575	−	1.07719i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	879.650i		−	0.894863i	−0.894318	−	0.447431i	$-0.852339\pi$
$983$		−	879.650i		−	0.894863i	0.894318	−	0.447431i	$-0.147661\pi$
$984$		0			0
$985$	31.3396	+	54.2819i	0.0318169	+	0.0551085i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−652.842			−0.660103
$990$		0			0
$991$	−470.061	−	271.390i	−0.474329	−	0.273854i	0.243721	−	0.969845i	$-0.421632\pi$
$991$	−470.061	−	271.390i	−0.474329	−	0.273854i	−0.718050	+	0.695991i	$0.754965\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	54.2806	−	94.0168i	0.0545534	−	0.0944892i
$996$		0			0
$997$	381.603	−	660.955i	0.382751	−	0.662944i	−0.608703	−	0.793398i	$-0.708310\pi$
$997$	381.603	−	660.955i	0.382751	−	0.662944i	0.991454	+	0.130454i	$0.0416434\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.s.a.901.18		80
3.2	odd	2		684.3.s.a.445.5	✓	80
9.2	odd	6		684.3.bl.a.673.21	yes	80
9.7	even	3		2052.3.bl.a.1585.23		80
19.12	odd	6		2052.3.bl.a.145.23		80
57.50	even	6		684.3.bl.a.373.21	yes	80
171.88	odd	6	inner	2052.3.s.a.829.18		80
171.164	even	6		684.3.s.a.601.5	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.5	✓	80	3.2	odd	2
684.3.s.a.601.5	yes	80	171.164	even	6
684.3.bl.a.373.21	yes	80	57.50	even	6
684.3.bl.a.673.21	yes	80	9.2	odd	6
2052.3.s.a.829.18		80	171.88	odd	6	inner
2052.3.s.a.901.18		80	1.1	even	1	trivial
2052.3.bl.a.145.23		80	19.12	odd	6
2052.3.bl.a.1585.23		80	9.7	even	3

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:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.340706 - 0.590120i) q^{5} +(-2.79581 - 4.84248i) q^{7} +O(q^{10})\)	\(q+(-0.340706 - 0.590120i) q^{5} +(-2.79581 - 4.84248i) q^{7} +(-7.44896 - 12.9020i) q^{11} -2.36780i q^{13} +(8.72908 - 15.1192i) q^{17} +(-17.2199 - 8.02967i) q^{19} +10.4486 q^{23} +(12.2678 - 21.2485i) q^{25} +(37.6328 + 21.7273i) q^{29} +(28.9832 + 16.7334i) q^{31} +(-1.90510 + 3.29973i) q^{35} -30.3219i q^{37} +(-48.7811 + 28.1638i) q^{41} -62.4810 q^{43} +(-43.3232 + 75.0381i) q^{47} +(8.86690 - 15.3579i) q^{49} +(-43.7359 + 25.2509i) q^{53} +(-5.07582 + 8.79157i) q^{55} +(-13.9077 + 8.02962i) q^{59} +(0.219952 - 0.380967i) q^{61} +(-1.39729 + 0.806726i) q^{65} -41.0728i q^{67} +(-62.9245 - 36.3295i) q^{71} +(38.3367 - 66.4012i) q^{73} +(-41.6518 + 72.1430i) q^{77} +120.760i q^{79} +(53.7042 + 93.0184i) q^{83} -11.8962 q^{85} +(122.606 - 70.7865i) q^{89} +(-11.4661 + 6.61993i) q^{91} +(1.12845 + 12.8976i) q^{95} -171.665i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7}+O(q^{10}) \) 80 * q - q^7	\( 80 q - q^{7} + 6 q^{11} + 21 q^{17} - 20 q^{19} + 48 q^{23} - 200 q^{25} + 27 q^{29} - 24 q^{31} + 54 q^{35} + 18 q^{41} - 152 q^{43} + 12 q^{47} - 267 q^{49} + 36 q^{53} + 135 q^{59} - 7 q^{61} + 288 q^{65} + 81 q^{71} + 55 q^{73} - 30 q^{77} + 93 q^{83} - 216 q^{89} + 96 q^{91} - 288 q^{95}+O(q^{100}) \) 80 * q - q^7 + 6 * q^11 + 21 * q^17 - 20 * q^19 + 48 * q^23 - 200 * q^25 + 27 * q^29 - 24 * q^31 + 54 * q^35 + 18 * q^41 - 152 * q^43 + 12 * q^47 - 267 * q^49 + 36 * q^53 + 135 * q^59 - 7 * q^61 + 288 * q^65 + 81 * q^71 + 55 * q^73 - 30 * q^77 + 93 * q^83 - 216 * q^89 + 96 * q^91 - 288 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more