LMFDB - Embedded newform 1620.3.o.e.701.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(701,1620)]

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, 5, 0]))

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

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

chi := DirichletCharacter("1620.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.o (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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			701.4
Root			$-0.535233 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	1620.701
Dual form			1620.3.o.e.1241.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.93649 + 1.11803i) q^{5} +(0.854102 + 1.47935i) q^{7} +O(q^{10})$	$q+(1.93649 + 1.11803i) q^{5} +(0.854102 + 1.47935i) q^{7} +(3.21140 - 1.85410i) q^{11} +(6.85410 - 11.8717i) q^{13} -15.0000i q^{17} -14.4164 q^{19} +(-6.56758 - 3.79180i) q^{23} +(2.50000 + 4.33013i) q^{25} +(8.40755 - 4.85410i) q^{29} +(10.2082 - 17.6811i) q^{31} +3.81966i q^{35} -56.2492 q^{37} +(-36.8416 - 21.2705i) q^{41} +(8.97871 + 15.5516i) q^{43} +(1.22665 - 0.708204i) q^{47} +(23.0410 - 39.9082i) q^{49} -70.0820i q^{53} +8.29180 q^{55} +(26.7389 + 15.4377i) q^{59} +(40.7492 + 70.5797i) q^{61} +(26.5458 - 15.3262i) q^{65} +(-7.00000 + 12.1244i) q^{67} -1.95743i q^{71} -27.1246 q^{73} +(5.48572 + 3.16718i) q^{77} +(-6.74922 - 11.6900i) q^{79} +(64.3728 - 37.1656i) q^{83} +(16.7705 - 29.0474i) q^{85} -123.039i q^{89} +23.4164 q^{91} +(-27.9173 - 16.1180i) q^{95} +(5.70820 + 9.88690i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{7}+O(q^{10}) $ 8 * q - 20 * q^7	$ 8 q - 20 q^{7} + 28 q^{13} - 8 q^{19} + 20 q^{25} + 28 q^{31} - 128 q^{37} - 116 q^{43} - 84 q^{49} + 120 q^{55} + 4 q^{61} - 56 q^{67} - 56 q^{73} + 268 q^{79} + 80 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q - 20 * q^7 + 28 * q^13 - 8 * q^19 + 20 * q^25 + 28 * q^31 - 128 * q^37 - 116 * q^43 - 84 * q^49 + 120 * q^55 + 4 * q^61 - 56 * q^67 - 56 * q^73 + 268 * q^79 + 80 * q^91 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	1.93649	+	1.11803i	0.387298	+	0.223607i
$5$	1.93649	+	1.11803i	0.387298	+	0.223607i
$6$		0			0
$7$	0.854102	+	1.47935i	0.122015	+	0.211335i	0.920562	−	0.390596i	$-0.127731\pi$
$7$	0.854102	+	1.47935i	0.122015	+	0.211335i	−0.798547	+	0.601932i	$0.794398\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.21140	−	1.85410i	0.291945	−	0.168555i	−0.346873	−	0.937912i	$-0.612757\pi$
$11$	3.21140	−	1.85410i	0.291945	−	0.168555i	0.638819	+	0.769357i	$0.279423\pi$
$12$		0			0
$13$	6.85410	−	11.8717i	0.527239	−	0.913204i	−0.472257	−	0.881461i	$-0.656561\pi$
$13$	6.85410	−	11.8717i	0.527239	−	0.913204i	0.999496	−	0.0317434i	$-0.0101059\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	15.0000i		−	0.882353i	−0.897420	−	0.441176i	$-0.854561\pi$
$17$		−	15.0000i		−	0.882353i	0.897420	−	0.441176i	$-0.145439\pi$
$18$		0			0
$19$	−14.4164			−0.758758			−0.379379	−	0.925241i	$-0.623862\pi$
$19$	−14.4164			−0.758758			−0.379379	+	0.925241i	$0.623862\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.56758	−	3.79180i	−0.285547	−	0.164861i	0.350385	−	0.936606i	$-0.386051\pi$
$23$	−6.56758	−	3.79180i	−0.285547	−	0.164861i	−0.635932	+	0.771745i	$0.719384\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.40755	−	4.85410i	0.289916	−	0.167383i	−0.347988	−	0.937499i	$-0.613135\pi$
$29$	8.40755	−	4.85410i	0.289916	−	0.167383i	0.637904	+	0.770116i	$0.279802\pi$
$30$		0			0
$31$	10.2082	−	17.6811i	0.329297	−	0.570359i	−0.653076	−	0.757293i	$-0.726522\pi$
$31$	10.2082	−	17.6811i	0.329297	−	0.570359i	0.982373	+	0.186934i	$0.0598550\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.81966i			0.109133i
$36$		0			0
$37$	−56.2492			−1.52025			−0.760125	−	0.649777i	$-0.774862\pi$
$37$	−56.2492			−1.52025			−0.760125	+	0.649777i	$0.774862\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−36.8416	−	21.2705i	−0.898576	−	0.518793i	−0.0218379	−	0.999762i	$-0.506952\pi$
$41$	−36.8416	−	21.2705i	−0.898576	−	0.518793i	−0.876738	+	0.480969i	$0.840285\pi$
$42$		0			0
$43$	8.97871	+	15.5516i	0.208807	+	0.361665i	0.951339	−	0.308146i	$-0.0997085\pi$
$43$	8.97871	+	15.5516i	0.208807	+	0.361665i	−0.742532	+	0.669811i	$0.766375\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.22665	−	0.708204i	0.0260988	−	0.0150682i	−0.486894	−	0.873461i	$-0.661870\pi$
$47$	1.22665	−	0.708204i	0.0260988	−	0.0150682i	0.512993	+	0.858393i	$0.328537\pi$
$48$		0			0
$49$	23.0410	−	39.9082i	0.470225	−	0.814453i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	70.0820i		−	1.32230i	−0.750253	−	0.661151i	$-0.770068\pi$
$53$		−	70.0820i		−	1.32230i	0.750253	−	0.661151i	$-0.229932\pi$
$54$		0			0
$55$	8.29180			0.150760
$56$		0			0
$57$		0			0
$58$		0			0
$59$	26.7389	+	15.4377i	0.453201	+	0.261656i	0.709181	−	0.705026i	$-0.249065\pi$
$59$	26.7389	+	15.4377i	0.453201	+	0.261656i	−0.255980	+	0.966682i	$0.582398\pi$
$60$		0			0
$61$	40.7492	+	70.5797i	0.668020	+	1.15704i	0.978457	+	0.206451i	$0.0661913\pi$
$61$	40.7492	+	70.5797i	0.668020	+	1.15704i	−0.310437	+	0.950594i	$0.600475\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	26.5458	−	15.3262i	0.408397	−	0.235788i
$66$		0			0
$67$	−7.00000	+	12.1244i	−0.104478	+	0.180961i	−0.913525	−	0.406783i	$-0.866650\pi$
$67$	−7.00000	+	12.1244i	−0.104478	+	0.180961i	0.809047	+	0.587744i	$0.199984\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.95743i		−	0.0275694i	−0.999905	−	0.0137847i	$-0.995612\pi$
$71$		−	1.95743i		−	0.0275694i	0.999905	−	0.0137847i	$-0.00438795\pi$
$72$		0			0
$73$	−27.1246			−0.371570			−0.185785	−	0.982590i	$-0.559483\pi$
$73$	−27.1246			−0.371570			−0.185785	+	0.982590i	$0.559483\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.48572	+	3.16718i	0.0712432	+	0.0411323i
$78$		0			0
$79$	−6.74922	−	11.6900i	−0.0854332	−	0.147975i	0.820143	−	0.572159i	$-0.193894\pi$
$79$	−6.74922	−	11.6900i	−0.0854332	−	0.147975i	−0.905576	+	0.424185i	$0.860561\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	64.3728	−	37.1656i	0.775575	−	0.447779i	−0.0592845	−	0.998241i	$-0.518882\pi$
$83$	64.3728	−	37.1656i	0.775575	−	0.447779i	0.834860	+	0.550462i	$0.185549\pi$
$84$		0			0
$85$	16.7705	−	29.0474i	0.197300	−	0.341734i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	123.039i		−	1.38247i	−0.722632	−	0.691233i	$-0.757068\pi$
$89$		−	123.039i		−	1.38247i	0.722632	−	0.691233i	$-0.242932\pi$
$90$		0			0
$91$	23.4164			0.257323
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−27.9173	−	16.1180i	−0.293866	−	0.169664i
$96$		0			0
$97$	5.70820	+	9.88690i	0.0588475	+	0.101927i	0.893948	−	0.448170i	$-0.147924\pi$
$97$	5.70820	+	9.88690i	0.0588475	+	0.101927i	−0.835101	+	0.550097i	$0.814591\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	145.492	−	84.0000i	1.44052	−	0.831683i	0.442634	−	0.896703i	$-0.354044\pi$
$101$	145.492	−	84.0000i	1.44052	−	0.831683i	0.997884	+	0.0650194i	$0.0207109\pi$
$102$		0			0
$103$	9.75078	−	16.8888i	0.0946677	−	0.163969i	−0.814802	−	0.579739i	$-0.803154\pi$
$103$	9.75078	−	16.8888i	0.0946677	−	0.163969i	0.909470	+	0.415770i	$0.136488\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	202.997i		−	1.89717i	−0.316527	−	0.948584i	$-0.602517\pi$
$107$		−	202.997i		−	1.89717i	0.316527	−	0.948584i	$-0.397483\pi$
$108$		0			0
$109$	−32.0820			−0.294331			−0.147165	−	0.989112i	$-0.547015\pi$
$109$	−32.0820			−0.294331			−0.147165	+	0.989112i	$0.547015\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.16372	−	1.24922i	−0.0191480	−	0.0110551i	0.490395	−	0.871500i	$-0.336852\pi$
$113$	−2.16372	−	1.24922i	−0.0191480	−	0.0110551i	−0.509543	+	0.860445i	$0.670186\pi$
$114$		0			0
$115$	−8.47871	−	14.6856i	−0.0737279	−	0.127701i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	22.1902	−	12.8115i	0.186472	−	0.107660i
$120$		0			0
$121$	−53.6246	+	92.8806i	−0.443179	+	0.767608i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	104.748			0.824785			0.412392	−	0.911006i	$-0.364693\pi$
$127$	104.748			0.824785			0.412392	+	0.911006i	$0.364693\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−158.627	−	91.5836i	−1.21090	−	0.699111i	−0.247942	−	0.968775i	$-0.579754\pi$
$131$	−158.627	−	91.5836i	−1.21090	−	0.699111i	−0.962955	+	0.269664i	$0.913088\pi$
$132$		0			0
$133$	−12.3131	−	21.3269i	−0.0925796	−	0.160353i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−78.3766	+	45.2508i	−0.572092	+	0.330298i	−0.757985	−	0.652272i	$-0.773816\pi$
$137$	−78.3766	+	45.2508i	−0.572092	+	0.330298i	0.185892	+	0.982570i	$0.440482\pi$
$138$		0			0
$139$	−89.2492	+	154.584i	−0.642081	+	1.11212i	0.342887	+	0.939377i	$0.388595\pi$
$139$	−89.2492	+	154.584i	−0.642081	+	1.11212i	−0.984968	+	0.172740i	$0.944738\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	50.8328i		−	0.355474i
$144$		0			0
$145$	21.7082			0.149712
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−126.403	−	72.9787i	−0.848341	−	0.489790i	0.0117496	−	0.999931i	$-0.496260\pi$
$149$	−126.403	−	72.9787i	−0.848341	−	0.489790i	−0.860091	+	0.510141i	$0.829593\pi$
$150$		0			0
$151$	−75.4984	−	130.767i	−0.499990	−	0.866008i	0.500010	−	0.866019i	$-0.333330\pi$
$151$	−75.4984	−	130.767i	−0.499990	−	0.866008i	−1.00000		1.18743e-5i	$0.999996\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	39.5362	−	22.8262i	0.255072	−	0.147266i
$156$		0			0
$157$	−8.14590	+	14.1091i	−0.0518847	+	0.0898669i	−0.890801	−	0.454393i	$-0.849856\pi$
$157$	−8.14590	+	14.1091i	−0.0518847	+	0.0898669i	0.838917	+	0.544260i	$0.183189\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	12.9543i		−	0.0804616i
$162$		0			0
$163$	297.830			1.82718			0.913588	−	0.406641i	$-0.133300\pi$
$163$	297.830			1.82718			0.913588	+	0.406641i	$0.133300\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	180.204	+	104.041i	1.07907	+	0.623000i	0.930645	−	0.365924i	$-0.119247\pi$
$167$	180.204	+	104.041i	1.07907	+	0.623000i	0.148423	+	0.988924i	$0.452580\pi$
$168$		0			0
$169$	−9.45743	−	16.3807i	−0.0559611	−	0.0969275i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	28.2893	−	16.3328i	0.163522	−	0.0944093i	−0.416006	−	0.909362i	$-0.636570\pi$
$173$	28.2893	−	16.3328i	0.163522	−	0.0944093i	0.579528	+	0.814953i	$0.303237\pi$
$174$		0			0
$175$	−4.27051	+	7.39674i	−0.0244029	+	0.0422671i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	257.666i		−	1.43947i	−0.694247	−	0.719736i	$-0.744263\pi$
$179$		−	257.666i		−	1.43947i	0.694247	−	0.719736i	$-0.255737\pi$
$180$		0			0
$181$	−191.918			−1.06032			−0.530160	−	0.847898i	$-0.677868\pi$
$181$	−191.918			−1.06032			−0.530160	+	0.847898i	$0.677868\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−108.926	−	62.8885i	−0.588790	−	0.339938i
$186$		0			0
$187$	−27.8115	−	48.1710i	−0.148725	−	0.257599i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	282.177	−	162.915i	1.47737	−	0.852957i	0.477692	−	0.878527i	$-0.341473\pi$
$191$	282.177	−	162.915i	1.47737	−	0.852957i	0.999673	+	0.0255698i	$0.00814000\pi$
$192$		0			0
$193$	54.3131	−	94.0730i	0.281415	−	0.487425i	−0.690319	−	0.723506i	$-0.742530\pi$
$193$	54.3131	−	94.0730i	0.281415	−	0.487425i	0.971733	+	0.236081i	$0.0758629\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	308.666i		−	1.56683i	−0.621499	−	0.783415i	$-0.713476\pi$
$197$		−	308.666i		−	1.56683i	0.621499	−	0.783415i	$-0.286524\pi$
$198$		0			0
$199$	134.413			0.675444			0.337722	−	0.941246i	$-0.390344\pi$
$199$	134.413			0.675444			0.337722	+	0.941246i	$0.390344\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	14.3618	+	8.29180i	0.0707478	+	0.0408463i
$204$		0			0
$205$	−47.5623	−	82.3803i	−0.232011	−	0.401855i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−46.2968	+	26.7295i	−0.221516	+	0.127892i
$210$		0			0
$211$	−166.831	+	288.960i	−0.790669	+	1.36948i	0.134883	+	0.990861i	$0.456934\pi$
$211$	−166.831	+	288.960i	−0.790669	+	1.36948i	−0.925553	+	0.378618i	$0.876399\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			40.1540i			0.186763i
$216$		0			0
$217$	34.8754			0.160716
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−178.075	−	102.812i	−0.805768	−	0.465211i
$222$		0			0
$223$	−113.790	−	197.090i	−0.510270	−	0.883814i	−0.999929	−	0.0118998i	$-0.996212\pi$
$223$	−113.790	−	197.090i	−0.510270	−	0.883814i	0.489659	−	0.871914i	$-0.337121\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	57.6604	−	33.2902i	0.254011	−	0.146653i	−0.367589	−	0.929988i	$-0.619817\pi$
$227$	57.6604	−	33.2902i	0.254011	−	0.146653i	0.621599	+	0.783335i	$0.286483\pi$
$228$		0			0
$229$	84.3328	−	146.069i	0.368266	−	0.637855i	−0.621029	−	0.783788i	$-0.713285\pi$
$229$	84.3328	−	146.069i	0.368266	−	0.637855i	0.989294	+	0.145933i	$0.0466184\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	254.912i		−	1.09404i	−0.837119	−	0.547021i	$-0.815762\pi$
$233$		−	254.912i		−	1.09404i	0.837119	−	0.547021i	$-0.184238\pi$
$234$		0			0
$235$	3.16718			0.0134774
$236$		0			0
$237$		0			0
$238$		0			0
$239$	267.457	+	154.416i	1.11907	+	0.646094i	0.941163	−	0.337954i	$-0.109735\pi$
$239$	267.457	+	154.416i	1.11907	+	0.646094i	0.177905	+	0.984048i	$0.443068\pi$
$240$		0			0
$241$	8.45743	+	14.6487i	0.0350931	+	0.0607830i	0.883039	−	0.469301i	$-0.155494\pi$
$241$	8.45743	+	14.6487i	0.0350931	+	0.0607830i	−0.847945	+	0.530084i	$0.822161\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	89.2375	−	51.5213i	0.364235	−	0.210291i
$246$		0			0
$247$	−98.8115	+	171.147i	−0.400047	+	0.692901i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			433.082i			1.72543i	0.505693	+	0.862713i	$0.331237\pi$
$251$			433.082i			1.72543i	−0.505693	+	0.862713i	$0.668763\pi$
$252$		0			0
$253$	−28.1215			−0.111152
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.9978	−	12.1231i	−0.0817033	−	0.0471714i	0.458592	−	0.888647i	$-0.348354\pi$
$257$	−20.9978	−	12.1231i	−0.0817033	−	0.0471714i	−0.540295	+	0.841476i	$0.681687\pi$
$258$		0			0
$259$	−48.0426	−	83.2122i	−0.185493	−	0.321283i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−44.0909	+	25.4559i	−0.167646	+	0.0967904i	−0.581475	−	0.813564i	$-0.697524\pi$
$263$	−44.0909	+	25.4559i	−0.167646	+	0.0967904i	0.413829	+	0.910354i	$0.364191\pi$
$264$		0			0
$265$	78.3541	−	135.713i	0.295676	−	0.512126i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	78.4133i		−	0.291499i	−0.989322	−	0.145750i	$-0.953441\pi$
$269$		−	78.4133i		−	0.291499i	0.989322	−	0.145750i	$-0.0465594\pi$
$270$		0			0
$271$	−208.246			−0.768436			−0.384218	−	0.923242i	$-0.625529\pi$
$271$	−208.246			−0.768436			−0.384218	+	0.923242i	$0.625529\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16.0570	+	9.27051i	0.0583891	+	0.0337109i
$276$		0			0
$277$	93.5197	+	161.981i	0.337616	+	0.584769i	0.983984	−	0.178258i	$-0.0570460\pi$
$277$	93.5197	+	161.981i	0.337616	+	0.584769i	−0.646367	+	0.763026i	$0.723713\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	185.435	−	107.061i	0.659910	−	0.380999i	−0.132333	−	0.991205i	$-0.542247\pi$
$281$	185.435	−	107.061i	0.659910	−	0.380999i	0.792243	+	0.610206i	$0.208913\pi$
$282$		0			0
$283$	213.520	−	369.827i	0.754487	−	1.30681i	−0.191142	−	0.981562i	$-0.561219\pi$
$283$	213.520	−	369.827i	0.754487	−	1.30681i	0.945629	−	0.325247i	$-0.105447\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	72.6687i		−	0.253201i
$288$		0			0
$289$	64.0000			0.221453
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−120.304	−	69.4574i	−0.410593	−	0.237056i	0.280451	−	0.959868i	$-0.409516\pi$
$293$	−120.304	−	69.4574i	−0.410593	−	0.237056i	−0.691045	+	0.722812i	$0.742849\pi$
$294$		0			0
$295$	34.5197	+	59.7899i	0.117016	+	0.202678i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−90.0298	+	51.9787i	−0.301103	+	0.173842i
$300$		0			0
$301$	−15.3375	+	26.5653i	−0.0509551	+	0.0882568i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			182.236i			0.597495i
$306$		0			0
$307$	−169.502			−0.552122			−0.276061	−	0.961140i	$-0.589029\pi$
$307$	−169.502			−0.552122			−0.276061	+	0.961140i	$0.589029\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−152.963	−	88.3131i	−0.491842	−	0.283965i	0.233497	−	0.972358i	$-0.424983\pi$
$311$	−152.963	−	88.3131i	−0.491842	−	0.283965i	−0.725338	+	0.688393i	$0.758317\pi$
$312$		0			0
$313$	42.5836	+	73.7569i	0.136050	+	0.235645i	0.925998	−	0.377528i	$-0.123226\pi$
$313$	42.5836	+	73.7569i	0.136050	+	0.235645i	−0.789948	+	0.613174i	$0.789893\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−414.679	+	239.415i	−1.30813	+	0.755252i	−0.981785	−	0.189997i	$-0.939152\pi$
$317$	−414.679	+	239.415i	−1.30813	+	0.755252i	−0.326350	+	0.945249i	$0.605819\pi$
$318$		0			0
$319$	18.0000	−	31.1769i	0.0564263	−	0.0977333i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			216.246i			0.669493i
$324$		0			0
$325$	68.5410			0.210895
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.09536	+	1.20976i	0.00636888	+	0.00367707i
$330$		0			0
$331$	157.626	+	273.017i	0.476212	+	0.824823i	0.999629	−	0.0272536i	$-0.00867616\pi$
$331$	157.626	+	273.017i	0.476212	+	0.824823i	−0.523417	+	0.852077i	$0.675343\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−27.1109	+	15.6525i	−0.0809280	+	0.0467238i
$336$		0			0
$337$	−45.7295	+	79.2058i	−0.135696	+	0.235032i	−0.925863	−	0.377859i	$-0.876660\pi$
$337$	−45.7295	+	79.2058i	−0.135696	+	0.235032i	0.790167	+	0.612891i	$0.209994\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	75.7082i		−	0.222018i
$342$		0			0
$343$	162.420			0.473526
$344$		0			0
$345$		0			0
$346$		0			0
$347$	365.315	+	210.915i	1.05278	+	0.607824i	0.923427	−	0.383775i	$-0.125376\pi$
$347$	365.315	+	210.915i	1.05278	+	0.607824i	0.129355	+	0.991598i	$0.458709\pi$
$348$		0			0
$349$	209.871	+	363.507i	0.601349	+	1.04157i	0.992617	+	0.121290i	$0.0387031\pi$
$349$	209.871	+	363.507i	0.601349	+	1.04157i	−0.391268	+	0.920277i	$0.627964\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−334.717	+	193.249i	−0.948208	+	0.547448i	−0.892524	−	0.451000i	$-0.851067\pi$
$353$	−334.717	+	193.249i	−0.948208	+	0.547448i	−0.0556843	+	0.998448i	$0.517734\pi$
$354$		0			0
$355$	2.18847	−	3.79054i	0.00616471	−	0.0106776i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			344.991i			0.960977i	0.877001	+	0.480488i	$0.159541\pi$
$359$			344.991i			0.960977i	−0.877001	+	0.480488i	$0.840459\pi$
$360$		0			0
$361$	−153.167			−0.424286
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−52.5266	−	30.3262i	−0.143908	−	0.0830856i
$366$		0			0
$367$	114.185	+	197.775i	0.311132	+	0.538896i	0.978608	−	0.205735i	$-0.0659586\pi$
$367$	114.185	+	197.775i	0.311132	+	0.538896i	−0.667476	+	0.744631i	$0.732625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	103.676	−	59.8572i	0.279449	−	0.161340i
$372$		0			0
$373$	320.079	−	554.393i	0.858120	−	1.48631i	−0.0155991	−	0.999878i	$-0.504966\pi$
$373$	320.079	−	554.393i	0.858120	−	1.48631i	0.873720	−	0.486430i	$-0.161701\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	133.082i		−	0.353003i
$378$		0			0
$379$	−537.164			−1.41732			−0.708660	−	0.705550i	$-0.750700\pi$
$379$	−537.164			−1.41732			−0.708660	+	0.705550i	$0.750700\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	135.671	+	78.3297i	0.354232	+	0.204516i	0.666548	−	0.745462i	$-0.267771\pi$
$383$	135.671	+	78.3297i	0.354232	+	0.204516i	−0.312315	+	0.949978i	$0.601105\pi$
$384$		0			0
$385$	7.08204	+	12.2665i	0.0183949	+	0.0318609i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	164.003	−	94.6869i	0.421600	−	0.243411i	−0.274161	−	0.961684i	$-0.588400\pi$
$389$	164.003	−	94.6869i	0.421600	−	0.243411i	0.695762	+	0.718273i	$0.255067\pi$
$390$		0			0
$391$	−56.8769	+	98.5138i	−0.145465	+	0.251953i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	30.1834i		−	0.0764138i
$396$		0			0
$397$	−708.529			−1.78471			−0.892353	−	0.451337i	$-0.850947\pi$
$397$	−708.529			−1.78471			−0.892353	+	0.451337i	$0.850947\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−354.565	−	204.708i	−0.884202	−	0.510494i	−0.0121604	−	0.999926i	$-0.503871\pi$
$401$	−354.565	−	204.708i	−0.884202	−	0.510494i	−0.872042	+	0.489432i	$0.837204\pi$
$402$		0			0
$403$	−139.936	−	242.377i	−0.347236	−	0.601431i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−180.639	+	104.292i	−0.443830	+	0.256245i
$408$		0			0
$409$	−196.664	+	340.632i	−0.480841	+	0.832842i	−0.999758	−	0.0219831i	$-0.993002\pi$
$409$	−196.664	+	340.632i	−0.480841	+	0.832842i	0.518917	+	0.854825i	$0.326335\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			52.7415i			0.127703i
$414$		0			0
$415$	166.210			0.400505
$416$		0			0
$417$		0			0
$418$		0			0
$419$	409.585	+	236.474i	0.977530	+	0.564377i	0.901524	−	0.432730i	$-0.142450\pi$
$419$	409.585	+	236.474i	0.977530	+	0.564377i	0.0760065	+	0.997107i	$0.475783\pi$
$420$		0			0
$421$	278.123	+	481.723i	0.660625	+	1.14424i	0.980452	+	0.196760i	$0.0630419\pi$
$421$	278.123	+	481.723i	0.660625	+	1.14424i	−0.319827	+	0.947476i	$0.603625\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	64.9519	−	37.5000i	0.152828	−	0.0882353i
$426$		0			0
$427$	−69.6080	+	120.565i	−0.163016	+	0.282353i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			453.866i			1.05305i	0.850159	+	0.526527i	$0.176506\pi$
$431$			453.866i			1.05305i	−0.850159	+	0.526527i	$0.823494\pi$
$432$		0			0
$433$	564.043			1.30264			0.651319	−	0.758804i	$-0.274216\pi$
$433$	564.043			1.30264			0.651319	+	0.758804i	$0.274216\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	94.6810	+	54.6641i	0.216661	+	0.125089i
$438$		0			0
$439$	94.9953	+	164.537i	0.216390	+	0.374799i	0.953702	−	0.300754i	$-0.0972383\pi$
$439$	94.9953	+	164.537i	0.216390	+	0.374799i	−0.737311	+	0.675553i	$0.763905\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	393.673	−	227.287i	0.888652	−	0.513064i	0.0151507	−	0.999885i	$-0.495177\pi$
$443$	393.673	−	227.287i	0.888652	−	0.513064i	0.873501	+	0.486822i	$0.161844\pi$
$444$		0			0
$445$	137.562	−	238.265i	0.309129	−	0.535427i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			246.875i			0.549834i	0.961468	+	0.274917i	$0.0886503\pi$
$449$			246.875i			0.549834i	−0.961468	+	0.274917i	$0.911350\pi$
$450$		0			0
$451$	−157.751			−0.349780
$452$		0			0
$453$		0			0
$454$		0			0
$455$	45.3457	+	26.1803i	0.0996608	+	0.0575392i
$456$		0			0
$457$	−178.620	−	309.379i	−0.390853	−	0.676978i	0.601709	−	0.798715i	$-0.294487\pi$
$457$	−178.620	−	309.379i	−0.390853	−	0.676978i	−0.992562	+	0.121738i	$0.961153\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−506.975	+	292.702i	−1.09973	+	0.634928i	−0.936150	−	0.351602i	$-0.885637\pi$
$461$	−506.975	+	292.702i	−1.09973	+	0.634928i	−0.163579	+	0.986530i	$0.552304\pi$
$462$		0			0
$463$	409.498	−	709.272i	0.884446	−	1.53191i	0.0380985	−	0.999274i	$-0.487870\pi$
$463$	409.498	−	709.272i	0.884446	−	1.53191i	0.846347	−	0.532631i	$-0.178797\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			675.906i			1.44734i	0.690149	+	0.723668i	$0.257545\pi$
$467$			675.906i			1.44734i	−0.690149	+	0.723668i	$0.742455\pi$
$468$		0			0
$469$	−23.9149			−0.0509912
$470$		0			0
$471$		0			0
$472$		0			0
$473$	57.6685	+	33.2949i	0.121921	+	0.0703909i
$474$		0			0
$475$	−36.0410	−	62.4249i	−0.0758758	−	0.131421i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−822.824	+	475.058i	−1.71780	+	0.991770i	−0.794880	+	0.606767i	$0.792466\pi$
$479$	−822.824	+	475.058i	−1.71780	+	0.991770i	−0.922915	+	0.385003i	$0.874200\pi$
$480$		0			0
$481$	−385.538	+	667.771i	−0.801534	+	1.38830i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			25.5279i			0.0526348i
$486$		0			0
$487$	−66.2129			−0.135961			−0.0679804	−	0.997687i	$-0.521656\pi$
$487$	−66.2129			−0.135961			−0.0679804	+	0.997687i	$0.521656\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	723.697	+	417.827i	1.47392	+	0.850971i	0.999569	−	0.0293626i	$-0.00934776\pi$
$491$	723.697	+	417.827i	1.47392	+	0.850971i	0.474356	+	0.880333i	$0.342681\pi$
$492$		0			0
$493$	−72.8115	−	126.113i	−0.147691	−	0.255808i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.89572	−	1.67184i	0.00582639	−	0.00336387i
$498$		0			0
$499$	52.3754	−	90.7168i	0.104961	−	0.181797i	−0.808761	−	0.588137i	$-0.799862\pi$
$499$	52.3754	−	90.7168i	0.104961	−	0.181797i	0.913722	+	0.406340i	$0.133195\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			743.912i			1.47895i	0.673184	+	0.739475i	$0.264926\pi$
$503$			743.912i			1.47895i	−0.673184	+	0.739475i	$0.735074\pi$
$504$		0			0
$505$	375.659			0.743880
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−632.551	−	365.204i	−1.24273	−	0.717492i	−0.273083	−	0.961990i	$-0.588043\pi$
$509$	−632.551	−	365.204i	−1.24273	−	0.717492i	−0.969650	+	0.244498i	$0.921377\pi$
$510$		0			0
$511$	−23.1672	−	40.1267i	−0.0453370	−	0.0785259i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	37.7646	−	21.8034i	0.0733293	−	0.0423367i
$516$		0			0
$517$	2.62616	−	4.54865i	0.00507962	−	0.00879816i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			490.200i			0.940884i	0.882431	+	0.470442i	$0.155905\pi$
$521$			490.200i			0.940884i	−0.882431	+	0.470442i	$0.844095\pi$
$522$		0			0
$523$	8.12772			0.0155406			0.00777028	−	0.999970i	$-0.497527\pi$
$523$	8.12772			0.0155406			0.00777028	+	0.999970i	$0.497527\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−265.217	−	153.123i	−0.503258	−	0.290556i
$528$		0			0
$529$	−235.745	−	408.322i	−0.445642	−	0.771874i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−505.032	+	291.580i	−0.947528	+	0.547055i
$534$		0			0
$535$	226.957	−	393.102i	0.424219	−	0.734770i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	170.882i		−	0.317035i
$540$		0			0
$541$	−98.8390			−0.182697			−0.0913485	−	0.995819i	$-0.529118\pi$
$541$	−98.8390			−0.182697			−0.0913485	+	0.995819i	$0.529118\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−62.1266	−	35.8688i	−0.113994	−	0.0658143i
$546$		0			0
$547$	110.310	+	191.062i	0.201664	+	0.349292i	0.949065	−	0.315082i	$-0.102032\pi$
$547$	110.310	+	191.062i	0.201664	+	0.349292i	−0.747401	+	0.664373i	$0.768699\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−121.207	+	69.9787i	−0.219976	+	0.127003i
$552$		0			0
$553$	11.5291	−	19.9689i	0.0208482	−	0.0361101i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			440.164i			0.790241i	0.918629	+	0.395120i	$0.129297\pi$
$557$			440.164i			0.790241i	−0.918629	+	0.395120i	$0.870703\pi$
$558$		0			0
$559$	246.164			0.440365
$560$		0			0
$561$		0			0
$562$		0			0
$563$	351.311	+	202.830i	0.623999	+	0.360266i	0.778424	−	0.627739i	$-0.216019\pi$
$563$	351.311	+	202.830i	0.623999	+	0.360266i	−0.154425	+	0.988004i	$0.549353\pi$
$564$		0			0
$565$	−2.79335	−	4.83822i	−0.00494398	−	0.00856323i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	40.9640	−	23.6506i	0.0719929	−	0.0415651i	−0.463571	−	0.886060i	$-0.653432\pi$
$569$	40.9640	−	23.6506i	0.0719929	−	0.0415651i	0.535564	+	0.844494i	$0.320099\pi$
$570$		0			0
$571$	−352.959	+	611.343i	−0.618142	+	1.07065i	0.371683	+	0.928360i	$0.378781\pi$
$571$	−352.959	+	611.343i	−0.618142	+	1.07065i	−0.989825	+	0.142293i	$0.954552\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	37.9180i		−	0.0659443i
$576$		0			0
$577$	537.860			0.932166			0.466083	−	0.884741i	$-0.345665\pi$
$577$	537.860			0.932166			0.466083	+	0.884741i	$0.345665\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	109.962	+	63.4865i	0.189263	+	0.109271i
$582$		0			0
$583$	−129.939	−	225.061i	−0.222880	−	0.386040i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−108.395	+	62.5820i	−0.184660	+	0.106613i	−0.589480	−	0.807783i	$-0.700667\pi$
$587$	−108.395	+	62.5820i	−0.184660	+	0.106613i	0.404820	+	0.914396i	$0.367334\pi$
$588$		0			0
$589$	−147.166	+	254.898i	−0.249857	+	0.432765i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	357.413i		−	0.602721i	−0.953510	−	0.301360i	$-0.902559\pi$
$593$		−	357.413i		−	0.602721i	0.953510	−	0.301360i	$-0.0974407\pi$
$594$		0			0
$595$	57.2949			0.0962940
$596$		0			0
$597$		0			0
$598$		0			0
$599$	489.402	+	282.556i	0.817031	+	0.471713i	0.849392	−	0.527763i	$-0.176969\pi$
$599$	489.402	+	282.556i	0.817031	+	0.471713i	−0.0323606	+	0.999476i	$0.510303\pi$
$600$		0			0
$601$	−230.953	−	400.022i	−0.384281	−	0.665594i	0.607388	−	0.794405i	$-0.292217\pi$
$601$	−230.953	−	400.022i	−0.384281	−	0.665594i	−0.991669	+	0.128811i	$0.958884\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−207.687	+	119.908i	−0.343285	+	0.198195i
$606$		0			0
$607$	−364.310	+	631.003i	−0.600181	+	1.03954i	0.392612	+	0.919704i	$0.371572\pi$
$607$	−364.310	+	631.003i	−0.600181	+	1.03954i	−0.992793	+	0.119840i	$0.961762\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	19.4164i		−	0.0317781i
$612$		0			0
$613$	912.741			1.48897			0.744487	−	0.667637i	$-0.232694\pi$
$613$	912.741			1.48897			0.744487	+	0.667637i	$0.232694\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−84.2887	−	48.6641i	−0.136610	−	0.0788721i	0.430137	−	0.902764i	$-0.358465\pi$
$617$	−84.2887	−	48.6641i	−0.136610	−	0.0788721i	−0.566748	+	0.823891i	$0.691799\pi$
$618$		0			0
$619$	−171.371	−	296.823i	−0.276851	−	0.479520i	0.693749	−	0.720216i	$-0.255957\pi$
$619$	−171.371	−	296.823i	−0.276851	−	0.479520i	−0.970600	+	0.240696i	$0.922624\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	182.018	−	105.088i	0.292164	−	0.168681i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			843.738i			1.34140i
$630$		0			0
$631$	−1002.15			−1.58820			−0.794100	−	0.607787i	$-0.792058\pi$
$631$	−1002.15			−1.58820			−0.794100	+	0.607787i	$0.792058\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	202.843	+	117.111i	0.319438	+	0.184427i
$636$		0			0
$637$	−315.851	−	547.070i	−0.495841	−	0.858823i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−868.405	+	501.374i	−1.35477	+	0.782174i	−0.988913	−	0.148498i	$-0.952556\pi$
$641$	−868.405	+	501.374i	−1.35477	+	0.782174i	−0.365853	+	0.930673i	$0.619223\pi$
$642$		0			0
$643$	265.188	−	459.320i	0.412424	−	0.714339i	−0.582730	−	0.812666i	$-0.698016\pi$
$643$	265.188	−	459.320i	0.412424	−	0.714339i	0.995154	+	0.0983266i	$0.0313490\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			450.325i			0.696020i	0.937491	+	0.348010i	$0.113143\pi$
$647$			450.325i			0.696020i	−0.937491	+	0.348010i	$0.886857\pi$
$648$		0			0
$649$	114.492			0.176413
$650$		0			0
$651$		0			0
$652$		0			0
$653$	497.885	+	287.454i	0.762459	+	0.440206i	0.830178	−	0.557499i	$-0.188239\pi$
$653$	497.885	+	287.454i	0.762459	+	0.440206i	−0.0677192	+	0.997704i	$0.521572\pi$
$654$		0			0
$655$	−204.787	−	354.702i	−0.312652	−	0.541529i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	472.007	−	272.514i	0.716248	−	0.413526i	−0.0971223	−	0.995272i	$-0.530964\pi$
$659$	472.007	−	272.514i	0.716248	−	0.413526i	0.813370	+	0.581747i	$0.197630\pi$
$660$		0			0
$661$	−395.249	+	684.592i	−0.597956	+	1.03569i	0.395166	+	0.918610i	$0.370687\pi$
$661$	−395.249	+	684.592i	−0.597956	+	1.03569i	−0.993122	+	0.117081i	$0.962646\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	55.0658i		−	0.0828057i
$666$		0			0
$667$	−73.6231			−0.110379
$668$		0			0
$669$		0			0
$670$		0			0
$671$	261.724	+	151.106i	0.390051	+	0.225196i
$672$		0			0
$673$	−526.684	−	912.243i	−0.782591	−	1.35549i	−0.930428	−	0.366476i	$-0.880564\pi$
$673$	−526.684	−	912.243i	−0.782591	−	1.35549i	0.147837	−	0.989012i	$-0.452769\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	439.220	−	253.584i	0.648774	−	0.374570i	−0.139213	−	0.990263i	$-0.544457\pi$
$677$	439.220	−	253.584i	0.648774	−	0.374570i	0.787986	+	0.615693i	$0.211124\pi$
$678$		0			0
$679$	−9.75078	+	16.8888i	−0.0143605	+	0.0248731i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1322.49i			1.93629i	0.250382	+	0.968147i	$0.419444\pi$
$683$			1322.49i			1.93629i	−0.250382	+	0.968147i	$0.580556\pi$
$684$		0			0
$685$	−202.368			−0.295427
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−831.990	−	480.349i	−1.20753	−	0.697169i
$690$		0			0
$691$	244.749	+	423.918i	0.354196	+	0.613485i	0.986980	−	0.160843i	$-0.0514213\pi$
$691$	244.749	+	423.918i	0.354196	+	0.613485i	−0.632784	+	0.774328i	$0.718088\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−345.661	+	199.567i	−0.497354	+	0.287147i
$696$		0			0
$697$	−319.058	+	552.624i	−0.457758	+	0.792861i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1063.50i		−	1.51711i	−0.651608	−	0.758556i	$-0.725905\pi$
$701$		−	1063.50i		−	1.51711i	0.651608	−	0.758556i	$-0.274095\pi$
$702$		0			0
$703$	810.912			1.15350
$704$		0			0
$705$		0			0
$706$		0			0
$707$	248.530	+	143.489i	0.351528	+	0.202955i
$708$		0			0
$709$	206.836	+	358.250i	0.291729	+	0.505290i	0.974219	−	0.225606i	$-0.0724361\pi$
$709$	206.836	+	358.250i	0.291729	+	0.505290i	−0.682490	+	0.730895i	$0.739103\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−134.086	+	77.4149i	−0.188060	+	0.108576i
$714$		0			0
$715$	56.8328	−	98.4373i	0.0794865	−	0.137675i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			69.0883i			0.0960894i	0.998845	+	0.0480447i	$0.0152990\pi$
$719$			69.0883i			0.0960894i	−0.998845	+	0.0480447i	$0.984701\pi$
$720$		0			0
$721$	33.3126			0.0462034
$722$		0			0
$723$		0			0
$724$		0			0
$725$	42.0378	+	24.2705i	0.0579831	+	0.0334766i
$726$		0			0
$727$	712.498	+	1234.08i	0.980053	+	1.69750i	0.662138	+	0.749382i	$0.269649\pi$
$727$	712.498	+	1234.08i	0.980053	+	1.69750i	0.317915	+	0.948119i	$0.397017\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	233.274	−	134.681i	0.319116	−	0.184242i
$732$		0			0
$733$	648.249	−	1122.80i	0.884378	−	1.53179i	0.0379535	−	0.999280i	$-0.487916\pi$
$733$	648.249	−	1122.80i	0.884378	−	1.53179i	0.846425	−	0.532508i	$-0.178751\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			51.9149i			0.0704408i
$738$		0			0
$739$	356.246			0.482065			0.241033	−	0.970517i	$-0.422514\pi$
$739$	356.246			0.482065			0.241033	+	0.970517i	$0.422514\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−850.584	−	491.085i	−1.14480	−	0.660949i	−0.197183	−	0.980367i	$-0.563179\pi$
$743$	−850.584	−	491.085i	−1.14480	−	0.660949i	−0.947614	+	0.319418i	$0.896513\pi$
$744$		0			0
$745$	−163.185	−	282.645i	−0.219041	−	0.379390i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	300.303	−	173.380i	0.400939	−	0.231482i
$750$		0			0
$751$	−389.245	+	674.191i	−0.518302	+	0.897725i	0.481472	+	0.876461i	$0.340102\pi$
$751$	−389.245	+	674.191i	−0.518302	+	0.897725i	−0.999774	+	0.0212635i	$0.993231\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	337.639i		−	0.447204i
$756$		0			0
$757$	−160.748			−0.212348			−0.106174	−	0.994348i	$-0.533860\pi$
$757$	−160.748			−0.212348			−0.106174	+	0.994348i	$0.533860\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	553.851	+	319.766i	0.727793	+	0.420192i	0.817614	−	0.575766i	$-0.195296\pi$
$761$	553.851	+	319.766i	0.727793	+	0.420192i	−0.0898210	+	0.995958i	$0.528630\pi$
$762$		0			0
$763$	−27.4013	−	47.4605i	−0.0359126	−	0.0622025i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	366.542	−	211.623i	0.477890	−	0.275910i
$768$		0			0
$769$	−276.336	+	478.628i	−0.359345	+	0.622403i	−0.987851	−	0.155401i	$-0.950333\pi$
$769$	−276.336	+	478.628i	−0.359345	+	0.622403i	0.628507	+	0.777804i	$0.283666\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	934.909i		−	1.20945i	−0.796432	−	0.604727i	$-0.793282\pi$
$773$		−	934.909i		−	1.20945i	0.796432	−	0.604727i	$-0.206718\pi$
$774$		0			0
$775$	102.082			0.131719
$776$		0			0
$777$		0			0
$778$		0			0
$779$	531.124	+	306.644i	0.681802	+	0.393638i
$780$		0			0
$781$	−3.62927	−	6.28608i	−0.00464695	−	0.00804876i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−31.5489	+	18.2148i	−0.0401897	+	0.0232035i
$786$		0			0
$787$	−51.5623	+	89.3085i	−0.0655175	+	0.113480i	−0.896923	−	0.442186i	$-0.854203\pi$
$787$	−51.5623	+	89.3085i	−0.0655175	+	0.113480i	0.831406	+	0.555666i	$0.187536\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	4.26786i		−	0.00539552i
$792$		0			0
$793$	1117.20			1.40882
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1289.71	−	744.615i	−1.61821	−	0.934273i	−0.987384	−	0.158347i	$-0.949384\pi$
$797$	−1289.71	−	744.615i	−1.61821	−	0.934273i	−0.630824	−	0.775926i	$-0.717283\pi$
$798$		0			0
$799$	−10.6231	−	18.3997i	−0.0132954	−	0.0230284i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−87.1079	+	50.2918i	−0.108478	+	0.0626299i
$804$		0			0
$805$	14.4834	−	25.0859i	0.0179918	−	0.0311627i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			278.213i			0.343897i	0.985106	+	0.171949i	$0.0550063\pi$
$809$			278.213i			0.343897i	−0.985106	+	0.171949i	$0.944994\pi$
$810$		0			0
$811$	−838.158			−1.03349			−0.516743	−	0.856140i	$-0.672856\pi$
$811$	−838.158			−1.03349			−0.516743	+	0.856140i	$0.672856\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	576.745	+	332.984i	0.707662	+	0.408569i
$816$		0			0
$817$	−129.441	−	224.198i	−0.158434	−	0.274416i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−464.774	+	268.337i	−0.566107	+	0.326842i	−0.755593	−	0.655041i	$-0.772651\pi$
$821$	−464.774	+	268.337i	−0.566107	+	0.326842i	0.189486	+	0.981883i	$0.439318\pi$
$822$		0			0
$823$	586.602	−	1016.02i	0.712760	−	1.23454i	−0.251057	−	0.967972i	$-0.580778\pi$
$823$	586.602	−	1016.02i	0.712760	−	1.23454i	0.963817	−	0.266565i	$-0.0858886\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			677.754i			0.819533i	0.912190	+	0.409767i	$0.134390\pi$
$827$			677.754i			0.819533i	−0.912190	+	0.409767i	$0.865610\pi$
$828$		0			0
$829$	−141.234			−0.170366			−0.0851832	−	0.996365i	$-0.527148\pi$
$829$	−141.234			−0.170366			−0.0851832	+	0.996365i	$0.527148\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−598.623	−	345.615i	−0.718635	−	0.414904i
$834$		0			0
$835$	232.643	+	402.949i	0.278614	+	0.482574i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−351.464	+	202.918i	−0.418908	+	0.241857i	−0.694610	−	0.719386i	$-0.744423\pi$
$839$	−351.464	+	202.918i	−0.418908	+	0.241857i	0.275702	+	0.961243i	$0.411090\pi$
$840$		0			0
$841$	−373.375	+	646.705i	−0.443966	+	0.768972i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	42.2949i		−	0.0500531i
$846$		0			0
$847$	−183.204			−0.216297
$848$		0			0
$849$		0			0
$850$		0			0
$851$	369.421	+	213.286i	0.434103	+	0.250629i
$852$		0			0
$853$	428.492	+	742.170i	0.502336	+	0.870071i	0.999996	+	0.00269898i	$0.000859113\pi$
$853$	428.492	+	742.170i	0.502336	+	0.870071i	−0.497661	+	0.867372i	$0.665808\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	148.159	−	85.5395i	0.172881	−	0.0998127i	−0.411063	−	0.911607i	$-0.634842\pi$
$857$	148.159	−	85.5395i	0.172881	−	0.0998127i	0.583943	+	0.811794i	$0.301509\pi$
$858$		0			0
$859$	668.861	−	1158.50i	0.778651	−	1.34866i	−0.154068	−	0.988060i	$-0.549238\pi$
$859$	668.861	−	1158.50i	0.778651	−	1.34866i	0.932719	−	0.360603i	$-0.117429\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	768.659i		−	0.890683i	−0.895361	−	0.445341i	$-0.853082\pi$
$863$		−	768.659i		−	0.890683i	0.895361	−	0.445341i	$-0.146918\pi$
$864$		0			0
$865$	73.0426			0.0844423
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−43.3489	−	25.0275i	−0.0498837	−	0.0288003i
$870$		0			0
$871$	95.9574	+	166.203i	0.110169	+	0.190819i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−16.5396	+	9.54915i	−0.0189024	+	0.0109133i
$876$		0			0
$877$	−103.526	+	179.312i	−0.118046	+	0.204461i	−0.918993	−	0.394274i	$-0.870996\pi$
$877$	−103.526	+	179.312i	−0.118046	+	0.204461i	0.800948	+	0.598735i	$0.204330\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1384.61i		−	1.57164i	−0.618456	−	0.785819i	$-0.712242\pi$
$881$		−	1384.61i		−	1.57164i	0.618456	−	0.785819i	$-0.287758\pi$
$882$		0			0
$883$	−317.525			−0.359599			−0.179799	−	0.983703i	$-0.557545\pi$
$883$	−317.525			−0.359599			−0.179799	+	0.983703i	$0.557545\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−688.201	−	397.333i	−0.775874	−	0.447951i	0.0590918	−	0.998253i	$-0.481180\pi$
$887$	−688.201	−	397.333i	−0.775874	−	0.447951i	−0.834966	+	0.550301i	$0.814513\pi$
$888$		0			0
$889$	89.4652	+	154.958i	0.100636	+	0.174306i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−17.6838	+	10.2098i	−0.0198027	+	0.0114331i
$894$		0			0
$895$	288.079	−	498.967i	0.321876	−	0.557505i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	198.207i		−	0.220475i
$900$		0			0
$901$	−1051.23			−1.16674
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−371.648	−	214.571i	−0.410660	−	0.237095i
$906$		0			0
$907$	268.371	+	464.832i	0.295888	+	0.512494i	0.975191	−	0.221364i	$-0.0710510\pi$
$907$	268.371	+	464.832i	0.295888	+	0.512494i	−0.679303	+	0.733858i	$0.737718\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−773.384	+	446.514i	−0.848940	+	0.490136i	−0.860293	−	0.509800i	$-0.829719\pi$
$911$	−773.384	+	446.514i	−0.848940	+	0.490136i	0.0113532	+	0.999936i	$0.496386\pi$
$912$		0			0
$913$	137.818	−	238.707i	0.150950	−	0.261454i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	312.887i		−	0.341207i
$918$		0			0
$919$	−965.988			−1.05113			−0.525565	−	0.850754i	$-0.676146\pi$
$919$	−965.988			−1.05113			−0.525565	+	0.850754i	$0.676146\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−23.2379	−	13.4164i	−0.0251765	−	0.0145357i
$924$		0			0
$925$	−140.623	−	243.566i	−0.152025	−	0.263315i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1115.10	−	643.805i	1.20033	−	0.693009i	0.239699	−	0.970847i	$-0.422951\pi$
$929$	1115.10	−	643.805i	1.20033	−	0.693009i	0.960628	+	0.277838i	$0.0896180\pi$
$930$		0			0
$931$	−332.169	+	575.333i	−0.356787	+	0.617973i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	124.377i		−	0.133023i
$936$		0			0
$937$	−932.328			−0.995014			−0.497507	−	0.867460i	$-0.665751\pi$
$937$	−932.328			−0.995014			−0.497507	+	0.867460i	$0.665751\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	615.173	+	355.170i	0.653744	+	0.377439i	0.789889	−	0.613250i	$-0.210138\pi$
$941$	615.173	+	355.170i	0.653744	+	0.377439i	−0.136145	+	0.990689i	$0.543471\pi$
$942$		0			0
$943$	161.307	+	279.392i	0.171057	+	0.296280i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1405.41	−	811.412i	1.48406	−	0.856823i	0.484226	−	0.874943i	$-0.339101\pi$
$947$	1405.41	−	811.412i	1.48406	−	0.856823i	0.999836	+	0.0181197i	$0.00576799\pi$
$948$		0			0
$949$	−185.915	+	322.014i	−0.195906	+	0.339319i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			33.2461i			0.0348857i	0.999848	+	0.0174429i	$0.00555252\pi$
$953$			33.2461i			0.0348857i	−0.999848	+	0.0174429i	$0.994447\pi$
$954$		0			0
$955$	728.577			0.762908
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−133.883	−	77.2976i	−0.139607	−	0.0806022i
$960$		0			0
$961$	272.085	+	471.265i	0.283127	+	0.490391i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	210.354	−	121.448i	0.217983	−	0.125853i
$966$		0			0
$967$	132.240	−	229.046i	0.136753	−	0.236863i	−0.789513	−	0.613734i	$-0.789667\pi$
$967$	132.240	−	229.046i	0.136753	−	0.236863i	0.926266	+	0.376871i	$0.123000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1477.22i		−	1.52134i	−0.649139	−	0.760670i	$-0.724871\pi$
$971$		−	1477.22i		−	1.52134i	0.649139	−	0.760670i	$-0.275129\pi$
$972$		0			0
$973$	−304.912			−0.313373
$974$		0			0
$975$		0			0
$976$		0			0
$977$	332.980	+	192.246i	0.340819	+	0.196772i	0.660634	−	0.750708i	$-0.270288\pi$
$977$	332.980	+	192.246i	0.340819	+	0.196772i	−0.319815	+	0.947480i	$0.603621\pi$
$978$		0			0
$979$	−228.128	−	395.129i	−0.233021	−	0.403604i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	925.912	−	534.576i	0.941925	−	0.543821i	0.0513619	−	0.998680i	$-0.483644\pi$
$983$	925.912	−	534.576i	0.941925	−	0.543821i	0.890563	+	0.454859i	$0.150310\pi$
$984$		0			0
$985$	345.099	−	597.728i	0.350354	−	0.606831i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	136.182i		−	0.137696i
$990$		0			0
$991$	−233.012			−0.235129			−0.117564	−	0.993065i	$-0.537509\pi$
$991$	−233.012			−0.235129			−0.117564	+	0.993065i	$0.537509\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	260.290	+	150.279i	0.261598	+	0.151034i
$996$		0			0
$997$	124.061	+	214.880i	0.124434	+	0.215526i	0.921512	−	0.388351i	$-0.126955\pi$
$997$	124.061	+	214.880i	0.124434	+	0.215526i	−0.797078	+	0.603877i	$0.793622\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.o.e.701.4		8
3.2	odd	2	inner	1620.3.o.e.701.2		8
9.2	odd	6	inner	1620.3.o.e.1241.4		8
9.4	even	3		540.3.g.d.161.1	✓	4
9.5	odd	6		540.3.g.d.161.3	yes	4
9.7	even	3	inner	1620.3.o.e.1241.2		8
36.23	even	6		2160.3.l.e.161.4		4
36.31	odd	6		2160.3.l.e.161.2		4
45.4	even	6		2700.3.g.n.701.4		4
45.13	odd	12		2700.3.b.g.1349.3		4
45.14	odd	6		2700.3.g.n.701.3		4
45.22	odd	12		2700.3.b.l.1349.2		4
45.23	even	12		2700.3.b.l.1349.3		4
45.32	even	12		2700.3.b.g.1349.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.g.d.161.1	✓	4	9.4	even	3
540.3.g.d.161.3	yes	4	9.5	odd	6
1620.3.o.e.701.2		8	3.2	odd	2	inner
1620.3.o.e.701.4		8	1.1	even	1	trivial
1620.3.o.e.1241.2		8	9.7	even	3	inner
1620.3.o.e.1241.4		8	9.2	odd	6	inner
2160.3.l.e.161.2		4	36.31	odd	6
2160.3.l.e.161.4		4	36.23	even	6
2700.3.b.g.1349.2		4	45.32	even	12
2700.3.b.g.1349.3		4	45.13	odd	12
2700.3.b.l.1349.2		4	45.22	odd	12
2700.3.b.l.1349.3		4	45.23	even	12
2700.3.g.n.701.3		4	45.14	odd	6
2700.3.g.n.701.4		4	45.4	even	6

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.93649 + 1.11803i) q^{5} +(0.854102 + 1.47935i) q^{7} +O(q^{10})\)	\(q+(1.93649 + 1.11803i) q^{5} +(0.854102 + 1.47935i) q^{7} +(3.21140 - 1.85410i) q^{11} +(6.85410 - 11.8717i) q^{13} -15.0000i q^{17} -14.4164 q^{19} +(-6.56758 - 3.79180i) q^{23} +(2.50000 + 4.33013i) q^{25} +(8.40755 - 4.85410i) q^{29} +(10.2082 - 17.6811i) q^{31} +3.81966i q^{35} -56.2492 q^{37} +(-36.8416 - 21.2705i) q^{41} +(8.97871 + 15.5516i) q^{43} +(1.22665 - 0.708204i) q^{47} +(23.0410 - 39.9082i) q^{49} -70.0820i q^{53} +8.29180 q^{55} +(26.7389 + 15.4377i) q^{59} +(40.7492 + 70.5797i) q^{61} +(26.5458 - 15.3262i) q^{65} +(-7.00000 + 12.1244i) q^{67} -1.95743i q^{71} -27.1246 q^{73} +(5.48572 + 3.16718i) q^{77} +(-6.74922 - 11.6900i) q^{79} +(64.3728 - 37.1656i) q^{83} +(16.7705 - 29.0474i) q^{85} -123.039i q^{89} +23.4164 q^{91} +(-27.9173 - 16.1180i) q^{95} +(5.70820 + 9.88690i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{7}+O(q^{10}) \) 8 * q - 20 * q^7	\( 8 q - 20 q^{7} + 28 q^{13} - 8 q^{19} + 20 q^{25} + 28 q^{31} - 128 q^{37} - 116 q^{43} - 84 q^{49} + 120 q^{55} + 4 q^{61} - 56 q^{67} - 56 q^{73} + 268 q^{79} + 80 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q - 20 * q^7 + 28 * q^13 - 8 * q^19 + 20 * q^25 + 28 * q^31 - 128 * q^37 - 116 * q^43 - 84 * q^49 + 120 * q^55 + 4 * q^61 - 56 * q^67 - 56 * q^73 + 268 * q^79 + 80 * q^91 - 8 * q^97

Introduction

L-functions

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