LMFDB - Embedded newform 624.2.d.g.287.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,2,Mod(287,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("624.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	624.d (of order $2$, degree $1$, 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:	$4.98266508613$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.3
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	624.287
Dual form			624.2.d.g.287.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.18614 - 1.26217i) q^{3} +0.939764i q^{5} +2.52434i q^{7} +(-0.186141 - 2.99422i) q^{9} +O(q^{10})$	$q+(1.18614 - 1.26217i) q^{3} +0.939764i q^{5} +2.52434i q^{7} +(-0.186141 - 2.99422i) q^{9} +4.00000 q^{11} +1.00000 q^{13} +(1.18614 + 1.11469i) q^{15} -4.10891i q^{17} +3.46410i q^{19} +(3.18614 + 2.99422i) q^{21} +4.74456 q^{23} +4.11684 q^{25} +(-4.00000 - 3.31662i) q^{27} +6.63325i q^{31} +(4.74456 - 5.04868i) q^{33} -2.37228 q^{35} +0.372281 q^{37} +(1.18614 - 1.26217i) q^{39} -5.04868i q^{41} -0.644810i q^{43} +(2.81386 - 0.174928i) q^{45} -6.37228 q^{47} +0.627719 q^{49} +(-5.18614 - 4.87375i) q^{51} -11.9769i q^{53} +3.75906i q^{55} +(4.37228 + 4.10891i) q^{57} -12.0000 q^{59} -11.4891 q^{61} +(7.55842 - 0.469882i) q^{63} +0.939764i q^{65} +3.46410i q^{67} +(5.62772 - 5.98844i) q^{69} +11.1168 q^{71} -10.7446 q^{73} +(4.88316 - 5.19615i) q^{75} +10.0974i q^{77} -1.58457i q^{79} +(-8.93070 + 1.11469i) q^{81} +12.0000 q^{83} +3.86141 q^{85} +5.04868i q^{89} +2.52434i q^{91} +(8.37228 + 7.86797i) q^{93} -3.25544 q^{95} -2.74456 q^{97} +(-0.744563 - 11.9769i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 5 q^{9}+O(q^{10}) $ 4 * q - q^3 + 5 * q^9	$ 4 q - q^{3} + 5 q^{9} + 16 q^{11} + 4 q^{13} - q^{15} + 7 q^{21} - 4 q^{23} - 18 q^{25} - 16 q^{27} - 4 q^{33} + 2 q^{35} - 10 q^{37} - q^{39} + 17 q^{45} - 14 q^{47} + 14 q^{49} - 15 q^{51} + 6 q^{57} - 48 q^{59} + 13 q^{63} + 34 q^{69} + 10 q^{71} - 20 q^{73} + 54 q^{75} - 7 q^{81} + 48 q^{83} - 42 q^{85} + 22 q^{93} - 36 q^{95} + 12 q^{97} + 20 q^{99}+O(q^{100}) $ 4 * q - q^3 + 5 * q^9 + 16 * q^11 + 4 * q^13 - q^15 + 7 * q^21 - 4 * q^23 - 18 * q^25 - 16 * q^27 - 4 * q^33 + 2 * q^35 - 10 * q^37 - q^39 + 17 * q^45 - 14 * q^47 + 14 * q^49 - 15 * q^51 + 6 * q^57 - 48 * q^59 + 13 * q^63 + 34 * q^69 + 10 * q^71 - 20 * q^73 + 54 * q^75 - 7 * q^81 + 48 * q^83 - 42 * q^85 + 22 * q^93 - 36 * q^95 + 12 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.18614	−	1.26217i	0.684819	−	0.728714i
$3$	1.18614	−	1.26217i	0.684819	−	0.728714i
$4$		0			0
$5$			0.939764i			0.420275i	0.977672	+	0.210138i	$0.0673912\pi$
$5$			0.939764i			0.420275i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$			2.52434i			0.954110i	0.878873	+	0.477055i	$0.158296\pi$
$7$			2.52434i			0.954110i	−0.878873	+	0.477055i	$0.841704\pi$
$8$		0			0
$9$	−0.186141	−	2.99422i	−0.0620469	−	0.998073i
$10$		0			0
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$	1.18614	+	1.11469i	0.306260	+	0.287812i
$16$		0			0
$17$		−	4.10891i		−	0.996557i	−0.867017	−	0.498279i	$-0.833966\pi$
$17$		−	4.10891i		−	0.996557i	0.867017	−	0.498279i	$-0.166034\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	3.18614	+	2.99422i	0.695273	+	0.653392i
$22$		0			0
$23$	4.74456			0.989310			0.494655	−	0.869090i	$-0.335294\pi$
$23$	4.74456			0.989310			0.494655	+	0.869090i	$0.335294\pi$
$24$		0			0
$25$	4.11684			0.823369
$26$		0			0
$27$	−4.00000	−	3.31662i	−0.769800	−	0.638285i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			6.63325i			1.19137i	0.803219	+	0.595683i	$0.203119\pi$
$31$			6.63325i			1.19137i	−0.803219	+	0.595683i	$0.796881\pi$
$32$		0			0
$33$	4.74456	−	5.04868i	0.825922	−	0.878862i
$34$		0			0
$35$	−2.37228			−0.400989
$36$		0			0
$37$	0.372281			0.0612027			0.0306013	−	0.999532i	$-0.490258\pi$
$37$	0.372281			0.0612027			0.0306013	+	0.999532i	$0.490258\pi$
$38$		0			0
$39$	1.18614	−	1.26217i	0.189935	−	0.202109i
$40$		0			0
$41$		−	5.04868i		−	0.788471i	−0.919010	−	0.394235i	$-0.871009\pi$
$41$		−	5.04868i		−	0.788471i	0.919010	−	0.394235i	$-0.128991\pi$
$42$		0			0
$43$		−	0.644810i		−	0.0983326i	−0.998791	−	0.0491663i	$-0.984344\pi$
$43$		−	0.644810i		−	0.0983326i	0.998791	−	0.0491663i	$-0.0156564\pi$
$44$		0			0
$45$	2.81386	−	0.174928i	0.419465	−	0.0260768i
$46$		0			0
$47$	−6.37228			−0.929493			−0.464746	−	0.885444i	$-0.653854\pi$
$47$	−6.37228			−0.929493			−0.464746	+	0.885444i	$0.653854\pi$
$48$		0			0
$49$	0.627719			0.0896741
$50$		0			0
$51$	−5.18614	−	4.87375i	−0.726205	−	0.682461i
$52$		0			0
$53$		−	11.9769i		−	1.64515i	−0.568656	−	0.822575i	$-0.692536\pi$
$53$		−	11.9769i		−	1.64515i	0.568656	−	0.822575i	$-0.307464\pi$
$54$		0			0
$55$			3.75906i			0.506871i
$56$		0			0
$57$	4.37228	+	4.10891i	0.579123	+	0.544239i
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$	−11.4891			−1.47103			−0.735516	−	0.677507i	$-0.763060\pi$
$61$	−11.4891			−1.47103			−0.735516	+	0.677507i	$0.763060\pi$
$62$		0			0
$63$	7.55842	−	0.469882i	0.952272	−	0.0591996i
$64$		0			0
$65$			0.939764i			0.116563i
$66$		0			0
$67$			3.46410i			0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$			3.46410i			0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$	5.62772	−	5.98844i	0.677498	−	0.720923i
$70$		0			0
$71$	11.1168			1.31933			0.659663	−	0.751561i	$-0.270699\pi$
$71$	11.1168			1.31933			0.659663	+	0.751561i	$0.270699\pi$
$72$		0			0
$73$	−10.7446			−1.25756			−0.628778	−	0.777585i	$-0.716445\pi$
$73$	−10.7446			−1.25756			−0.628778	+	0.777585i	$0.716445\pi$
$74$		0			0
$75$	4.88316	−	5.19615i	0.563858	−	0.600000i
$76$		0			0
$77$			10.0974i			1.15070i
$78$		0			0
$79$		−	1.58457i		−	0.178279i	−0.996019	−	0.0891393i	$-0.971588\pi$
$79$		−	1.58457i		−	0.178279i	0.996019	−	0.0891393i	$-0.0284116\pi$
$80$		0			0
$81$	−8.93070	+	1.11469i	−0.992300	+	0.123855i
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$	3.86141			0.418828
$86$		0			0
$87$		0			0
$88$		0			0
$89$			5.04868i			0.535159i	0.963536	+	0.267579i	$0.0862237\pi$
$89$			5.04868i			0.535159i	−0.963536	+	0.267579i	$0.913776\pi$
$90$		0			0
$91$			2.52434i			0.264623i
$92$		0			0
$93$	8.37228	+	7.86797i	0.868165	+	0.815870i
$94$		0			0
$95$	−3.25544			−0.334001
$96$		0			0
$97$	−2.74456			−0.278668			−0.139334	−	0.990245i	$-0.544496\pi$
$97$	−2.74456			−0.278668			−0.139334	+	0.990245i	$0.544496\pi$
$98$		0			0
$99$	−0.744563	−	11.9769i	−0.0748314	−	1.20372i
$100$		0			0
$101$			10.0974i			1.00472i	0.864657	+	0.502362i	$0.167536\pi$
$101$			10.0974i			1.00472i	−0.864657	+	0.502362i	$0.832464\pi$
$102$		0			0
$103$			8.51278i			0.838789i	0.907804	+	0.419394i	$0.137758\pi$
$103$			8.51278i			0.838789i	−0.907804	+	0.419394i	$0.862242\pi$
$104$		0			0
$105$	−2.81386	+	2.99422i	−0.274605	+	0.292206i
$106$		0			0
$107$	−17.4891			−1.69074			−0.845369	−	0.534183i	$-0.820619\pi$
$107$	−17.4891			−1.69074			−0.845369	+	0.534183i	$0.820619\pi$
$108$		0			0
$109$	−12.3723			−1.18505			−0.592525	−	0.805552i	$-0.701869\pi$
$109$	−12.3723			−1.18505			−0.592525	+	0.805552i	$0.701869\pi$
$110$		0			0
$111$	0.441578	−	0.469882i	0.0419127	−	0.0445992i
$112$		0			0
$113$			20.1947i			1.89976i	0.312618	+	0.949879i	$0.398794\pi$
$113$			20.1947i			1.89976i	−0.312618	+	0.949879i	$0.601206\pi$
$114$		0			0
$115$			4.45877i			0.415782i
$116$		0			0
$117$	−0.186141	−	2.99422i	−0.0172087	−	0.276816i
$118$		0			0
$119$	10.3723			0.950825
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$	−6.37228	−	5.98844i	−0.574569	−	0.539959i
$124$		0			0
$125$			8.56768i			0.766317i
$126$		0			0
$127$			6.63325i			0.588606i	0.955712	+	0.294303i	$0.0950874\pi$
$127$			6.63325i			0.588606i	−0.955712	+	0.294303i	$0.904913\pi$
$128$		0			0
$129$	−0.813859	−	0.764836i	−0.0716563	−	0.0673400i
$130$		0			0
$131$	−7.11684			−0.621802			−0.310901	−	0.950442i	$-0.600631\pi$
$131$	−7.11684			−0.621802			−0.310901	+	0.950442i	$0.600631\pi$
$132$		0			0
$133$	−8.74456			−0.758250
$134$		0			0
$135$	3.11684	−	3.75906i	0.268255	−	0.323528i
$136$		0			0
$137$		−	15.1460i		−	1.29401i	−0.762485	−	0.647006i	$-0.776021\pi$
$137$		−	15.1460i		−	1.29401i	0.762485	−	0.647006i	$-0.223979\pi$
$138$		0			0
$139$		−	20.8395i		−	1.76758i	−0.467880	−	0.883792i	$-0.654982\pi$
$139$		−	20.8395i		−	1.76758i	0.467880	−	0.883792i	$-0.345018\pi$
$140$		0			0
$141$	−7.55842	+	8.04290i	−0.636534	+	0.677334i
$142$		0			0
$143$	4.00000			0.334497
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.744563	−	0.792287i	0.0614105	−	0.0653467i
$148$		0			0
$149$		−	17.0256i		−	1.39479i	−0.716688	−	0.697394i	$-0.754343\pi$
$149$		−	17.0256i		−	1.39479i	0.716688	−	0.697394i	$-0.245657\pi$
$150$		0			0
$151$		−	5.69349i		−	0.463329i	−0.972796	−	0.231665i	$-0.925583\pi$
$151$		−	5.69349i		−	0.463329i	0.972796	−	0.231665i	$-0.0744172\pi$
$152$		0			0
$153$	−12.3030	+	0.764836i	−0.994637	+	0.0618333i
$154$		0			0
$155$	−6.23369			−0.500702
$156$		0			0
$157$	2.74456			0.219040			0.109520	−	0.993985i	$-0.465069\pi$
$157$	2.74456			0.219040			0.109520	+	0.993985i	$0.465069\pi$
$158$		0			0
$159$	−15.1168	−	14.2063i	−1.19884	−	1.12663i
$160$		0			0
$161$			11.9769i			0.943910i
$162$		0			0
$163$			7.22316i			0.565761i	0.959155	+	0.282881i	$0.0912900\pi$
$163$			7.22316i			0.565761i	−0.959155	+	0.282881i	$0.908710\pi$
$164$		0			0
$165$	4.74456	+	4.45877i	0.369364	+	0.347115i
$166$		0			0
$167$	4.00000			0.309529			0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000			0.309529			0.154765	+	0.987951i	$0.450538\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	10.3723	−	0.644810i	0.793188	−	0.0493099i
$172$		0			0
$173$		−	6.33830i		−	0.481892i	−0.970539	−	0.240946i	$-0.922542\pi$
$173$		−	6.33830i		−	0.481892i	0.970539	−	0.240946i	$-0.0774576\pi$
$174$		0			0
$175$			10.3923i			0.785584i
$176$		0			0
$177$	−14.2337	+	15.1460i	−1.06987	+	1.13845i
$178$		0			0
$179$	−5.62772			−0.420635			−0.210318	−	0.977633i	$-0.567450\pi$
$179$	−5.62772			−0.420635			−0.210318	+	0.977633i	$0.567450\pi$
$180$		0			0
$181$	−24.2337			−1.80128			−0.900638	−	0.434570i	$-0.856900\pi$
$181$	−24.2337			−1.80128			−0.900638	+	0.434570i	$0.856900\pi$
$182$		0			0
$183$	−13.6277	+	14.5012i	−1.00739	+	1.07196i
$184$		0			0
$185$			0.349857i			0.0257220i
$186$		0			0
$187$		−	16.4356i		−	1.20189i
$188$		0			0
$189$	8.37228	−	10.0974i	0.608994	−	0.734474i
$190$		0			0
$191$	−1.48913			−0.107749			−0.0538747	−	0.998548i	$-0.517157\pi$
$191$	−1.48913			−0.107749			−0.0538747	+	0.998548i	$0.517157\pi$
$192$		0			0
$193$	22.7446			1.63719			0.818595	−	0.574372i	$-0.194754\pi$
$193$	22.7446			1.63719			0.818595	+	0.574372i	$0.194754\pi$
$194$		0			0
$195$	1.18614	+	1.11469i	0.0849413	+	0.0798248i
$196$		0			0
$197$			0.939764i			0.0669554i	0.999439	+	0.0334777i	$0.0106583\pi$
$197$			0.939764i			0.0669554i	−0.999439	+	0.0334777i	$0.989342\pi$
$198$		0			0
$199$			16.7306i			1.18600i	0.805202	+	0.593000i	$0.202057\pi$
$199$			16.7306i			1.18600i	−0.805202	+	0.593000i	$0.797943\pi$
$200$		0			0
$201$	4.37228	+	4.10891i	0.308397	+	0.289820i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.74456			0.331375
$206$		0			0
$207$	−0.883156	−	14.2063i	−0.0613836	−	0.987404i
$208$		0			0
$209$			13.8564i			0.958468i
$210$		0			0
$211$		−	6.28339i		−	0.432567i	−0.976331	−	0.216283i	$-0.930607\pi$
$211$		−	6.28339i		−	0.432567i	0.976331	−	0.216283i	$-0.0693935\pi$
$212$		0			0
$213$	13.1861	−	14.0313i	0.903499	−	0.961411i
$214$		0			0
$215$	0.605969			0.0413268
$216$		0			0
$217$	−16.7446			−1.13669
$218$		0			0
$219$	−12.7446	+	13.5615i	−0.861198	+	0.916398i
$220$		0			0
$221$		−	4.10891i		−	0.276395i
$222$		0			0
$223$		−	3.11425i		−	0.208545i	−0.994549	−	0.104273i	$-0.966749\pi$
$223$		−	3.11425i		−	0.208545i	0.994549	−	0.104273i	$-0.0332515\pi$
$224$		0			0
$225$	−0.766312	−	12.3267i	−0.0510875	−	0.821782i
$226$		0			0
$227$	−8.74456			−0.580397			−0.290199	−	0.956966i	$-0.593721\pi$
$227$	−8.74456			−0.580397			−0.290199	+	0.956966i	$0.593721\pi$
$228$		0			0
$229$	−29.8614			−1.97330			−0.986649	−	0.162863i	$-0.947927\pi$
$229$	−29.8614			−1.97330			−0.986649	+	0.162863i	$0.947927\pi$
$230$		0			0
$231$	12.7446	+	11.9769i	0.838531	+	0.788021i
$232$		0			0
$233$		−	16.0858i		−	1.05382i	−0.849923	−	0.526908i	$-0.823351\pi$
$233$		−	16.0858i		−	1.05382i	0.849923	−	0.526908i	$-0.176649\pi$
$234$		0			0
$235$		−	5.98844i		−	0.390643i
$236$		0			0
$237$	−2.00000	−	1.87953i	−0.129914	−	0.122088i
$238$		0			0
$239$	−7.86141			−0.508512			−0.254256	−	0.967137i	$-0.581831\pi$
$239$	−7.86141			−0.508512			−0.254256	+	0.967137i	$0.581831\pi$
$240$		0			0
$241$	18.0000			1.15948			0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000			1.15948			0.579741	+	0.814801i	$0.303154\pi$
$242$		0			0
$243$	−9.18614	+	12.5942i	−0.589291	+	0.807921i
$244$		0			0
$245$			0.589907i			0.0376878i
$246$		0			0
$247$			3.46410i			0.220416i
$248$		0			0
$249$	14.2337	−	15.1460i	0.902023	−	0.959840i
$250$		0			0
$251$	16.0000			1.00991			0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000			1.00991			0.504956	+	0.863145i	$0.331509\pi$
$252$		0			0
$253$	18.9783			1.19315
$254$		0			0
$255$	4.58017	−	4.87375i	0.286821	−	0.305206i
$256$		0			0
$257$			0.349857i			0.0218234i	0.999940	+	0.0109117i	$0.00347338\pi$
$257$			0.349857i			0.0218234i	−0.999940	+	0.0109117i	$0.996527\pi$
$258$		0			0
$259$			0.939764i			0.0583941i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.7446			−0.785863			−0.392932	−	0.919568i	$-0.628539\pi$
$263$	−12.7446			−0.785863			−0.392932	+	0.919568i	$0.628539\pi$
$264$		0			0
$265$	11.2554			0.691416
$266$		0			0
$267$	6.37228	+	5.98844i	0.389977	+	0.366487i
$268$		0			0
$269$			11.9769i			0.730243i	0.930960	+	0.365122i	$0.118973\pi$
$269$			11.9769i			0.730243i	−0.930960	+	0.365122i	$0.881027\pi$
$270$		0			0
$271$		−	15.7908i		−	0.959225i	−0.877480	−	0.479613i	$-0.840777\pi$
$271$		−	15.7908i		−	0.959225i	0.877480	−	0.479613i	$-0.159223\pi$
$272$		0			0
$273$	3.18614	+	2.99422i	0.192834	+	0.181218i
$274$		0			0
$275$	16.4674			0.993020
$276$		0			0
$277$	24.9783			1.50080			0.750399	−	0.660985i	$-0.229861\pi$
$277$	24.9783			1.50080			0.750399	+	0.660985i	$0.229861\pi$
$278$		0			0
$279$	19.8614	−	1.23472i	1.18907	−	0.0739206i
$280$		0			0
$281$			21.4843i			1.28165i	0.767688	+	0.640824i	$0.221407\pi$
$281$			21.4843i			1.28165i	−0.767688	+	0.640824i	$0.778593\pi$
$282$		0			0
$283$		−	14.1514i		−	0.841211i	−0.907244	−	0.420606i	$-0.861818\pi$
$283$		−	14.1514i		−	0.841211i	0.907244	−	0.420606i	$-0.138182\pi$
$284$		0			0
$285$	−3.86141	+	4.10891i	−0.228730	+	0.243391i
$286$		0			0
$287$	12.7446			0.752288
$288$		0			0
$289$	0.116844			0.00687317
$290$		0			0
$291$	−3.25544	+	3.46410i	−0.190837	+	0.203069i
$292$		0			0
$293$		−	25.5932i		−	1.49517i	−0.664165	−	0.747586i	$-0.731213\pi$
$293$		−	25.5932i		−	1.49517i	0.664165	−	0.747586i	$-0.268787\pi$
$294$		0			0
$295$		−	11.2772i		−	0.656582i
$296$		0			0
$297$	−16.0000	−	13.2665i	−0.928414	−	0.769800i
$298$		0			0
$299$	4.74456			0.274385
$300$		0			0
$301$	1.62772			0.0938201
$302$		0			0
$303$	12.7446	+	11.9769i	0.732156	+	0.688054i
$304$		0			0
$305$		−	10.7971i		−	0.618238i
$306$		0			0
$307$			19.2000i			1.09580i	0.836543	+	0.547902i	$0.184573\pi$
$307$			19.2000i			1.09580i	−0.836543	+	0.547902i	$0.815427\pi$
$308$		0			0
$309$	10.7446	+	10.0974i	0.611237	+	0.574418i
$310$		0			0
$311$	20.7446			1.17632			0.588158	−	0.808746i	$-0.299853\pi$
$311$	20.7446			1.17632			0.588158	+	0.808746i	$0.299853\pi$
$312$		0			0
$313$	20.3723			1.15151			0.575755	−	0.817622i	$-0.304708\pi$
$313$	20.3723			1.15151			0.575755	+	0.817622i	$0.304708\pi$
$314$		0			0
$315$	0.441578	+	7.10313i	0.0248801	+	0.400216i
$316$		0			0
$317$		−	17.0256i		−	0.956250i	−0.878292	−	0.478125i	$-0.841317\pi$
$317$		−	17.0256i		−	0.956250i	0.878292	−	0.478125i	$-0.158683\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−20.7446	+	22.0742i	−1.15785	+	1.23206i
$322$		0			0
$323$	14.2337			0.791984
$324$		0			0
$325$	4.11684			0.228361
$326$		0			0
$327$	−14.6753	+	15.6159i	−0.811544	+	0.863562i
$328$		0			0
$329$		−	16.0858i		−	0.886838i
$330$		0			0
$331$		−	2.17448i		−	0.119520i	−0.998213	−	0.0597602i	$-0.980966\pi$
$331$		−	2.17448i		−	0.119520i	0.998213	−	0.0597602i	$-0.0190336\pi$
$332$		0			0
$333$	−0.0692967	−	1.11469i	−0.00379744	−	0.0610847i
$334$		0			0
$335$	−3.25544			−0.177864
$336$		0			0
$337$	2.88316			0.157056			0.0785278	−	0.996912i	$-0.474978\pi$
$337$	2.88316			0.157056			0.0785278	+	0.996912i	$0.474978\pi$
$338$		0			0
$339$	25.4891	+	23.9538i	1.38438	+	1.30099i
$340$		0			0
$341$			26.5330i			1.43684i
$342$		0			0
$343$			19.2549i			1.03967i
$344$		0			0
$345$	5.62772	+	5.28873i	0.302986	+	0.284735i
$346$		0			0
$347$	−16.6060			−0.891455			−0.445728	−	0.895169i	$-0.647055\pi$
$347$	−16.6060			−0.891455			−0.445728	+	0.895169i	$0.647055\pi$
$348$		0			0
$349$	14.8832			0.796677			0.398339	−	0.917238i	$-0.369587\pi$
$349$	14.8832			0.796677			0.398339	+	0.917238i	$0.369587\pi$
$350$		0			0
$351$	−4.00000	−	3.31662i	−0.213504	−	0.177028i
$352$		0			0
$353$		−	33.4612i		−	1.78096i	−0.455022	−	0.890480i	$-0.650369\pi$
$353$		−	33.4612i		−	1.78096i	0.455022	−	0.890480i	$-0.349631\pi$
$354$		0			0
$355$			10.4472i			0.554480i
$356$		0			0
$357$	12.3030	−	13.0916i	0.651143	−	0.692879i
$358$		0			0
$359$	−21.4891			−1.13415			−0.567076	−	0.823665i	$-0.691926\pi$
$359$	−21.4891			−1.13415			−0.567076	+	0.823665i	$0.691926\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	5.93070	−	6.31084i	0.311281	−	0.331233i
$364$		0			0
$365$		−	10.0974i		−	0.528520i
$366$		0			0
$367$		−	33.7562i		−	1.76206i	−0.473063	−	0.881028i	$-0.656852\pi$
$367$		−	33.7562i		−	1.76206i	0.473063	−	0.881028i	$-0.343148\pi$
$368$		0			0
$369$	−15.1168	+	0.939764i	−0.786951	+	0.0489222i
$370$		0			0
$371$	30.2337			1.56965
$372$		0			0
$373$	1.25544			0.0650041			0.0325020	−	0.999472i	$-0.489652\pi$
$373$	1.25544			0.0650041			0.0325020	+	0.999472i	$0.489652\pi$
$374$		0			0
$375$	10.8139	+	10.1625i	0.558425	+	0.524788i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			17.3205i			0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$			17.3205i			0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$		0			0
$381$	8.37228	+	7.86797i	0.428925	+	0.403088i
$382$		0			0
$383$	−4.88316			−0.249518			−0.124759	−	0.992187i	$-0.539816\pi$
$383$	−4.88316			−0.249518			−0.124759	+	0.992187i	$0.539816\pi$
$384$		0			0
$385$	−9.48913			−0.483611
$386$		0			0
$387$	−1.93070	+	0.120025i	−0.0981432	+	0.00610123i
$388$		0			0
$389$			37.8102i			1.91705i	0.285006	+	0.958526i	$0.408004\pi$
$389$			37.8102i			1.91705i	−0.285006	+	0.958526i	$0.591996\pi$
$390$		0			0
$391$		−	19.4950i		−	0.985904i
$392$		0			0
$393$	−8.44158	+	8.98266i	−0.425821	+	0.453115i
$394$		0			0
$395$	1.48913			0.0749260
$396$		0			0
$397$	4.51087			0.226394			0.113197	−	0.993573i	$-0.463891\pi$
$397$	4.51087			0.226394			0.113197	+	0.993573i	$0.463891\pi$
$398$		0			0
$399$	−10.3723	+	11.0371i	−0.519264	+	0.552547i
$400$		0			0
$401$		−	18.9051i		−	0.944075i	−0.881579	−	0.472037i	$-0.843519\pi$
$401$		−	18.9051i		−	0.944075i	0.881579	−	0.472037i	$-0.156481\pi$
$402$		0			0
$403$			6.63325i			0.330426i
$404$		0			0
$405$	−1.04755	−	8.39275i	−0.0520530	−	0.417039i
$406$		0			0
$407$	1.48913			0.0738132
$408$		0			0
$409$	12.9783			0.641733			0.320867	−	0.947124i	$-0.396026\pi$
$409$	12.9783			0.641733			0.320867	+	0.947124i	$0.396026\pi$
$410$		0			0
$411$	−19.1168	−	17.9653i	−0.942964	−	0.886164i
$412$		0			0
$413$		−	30.2921i		−	1.49057i
$414$		0			0
$415$			11.2772i			0.553574i
$416$		0			0
$417$	−26.3030	−	24.7186i	−1.28806	−	1.21047i
$418$		0			0
$419$	27.8614			1.36112			0.680559	−	0.732693i	$-0.261737\pi$
$419$	27.8614			1.36112			0.680559	+	0.732693i	$0.261737\pi$
$420$		0			0
$421$	−10.8832			−0.530413			−0.265206	−	0.964192i	$-0.585440\pi$
$421$	−10.8832			−0.530413			−0.265206	+	0.964192i	$0.585440\pi$
$422$		0			0
$423$	1.18614	+	19.0800i	0.0576721	+	0.927702i
$424$		0			0
$425$		−	16.9157i		−	0.820534i
$426$		0			0
$427$		−	29.0024i		−	1.40353i
$428$		0			0
$429$	4.74456	−	5.04868i	0.229070	−	0.243752i
$430$		0			0
$431$	−33.3505			−1.60644			−0.803219	−	0.595683i	$-0.796881\pi$
$431$	−33.3505			−1.60644			−0.803219	+	0.595683i	$0.796881\pi$
$432$		0			0
$433$	31.3505			1.50661			0.753305	−	0.657671i	$-0.228458\pi$
$433$	31.3505			1.50661			0.753305	+	0.657671i	$0.228458\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			16.4356i			0.786224i
$438$		0			0
$439$		−	28.1176i		−	1.34198i	−0.741467	−	0.670989i	$-0.765870\pi$
$439$		−	28.1176i		−	1.34198i	0.741467	−	0.670989i	$-0.234130\pi$
$440$		0			0
$441$	−0.116844	−	1.87953i	−0.00556400	−	0.0895013i
$442$		0			0
$443$	2.37228			0.112710			0.0563552	−	0.998411i	$-0.482052\pi$
$443$	2.37228			0.112710			0.0563552	+	0.998411i	$0.482052\pi$
$444$		0			0
$445$	−4.74456			−0.224914
$446$		0			0
$447$	−21.4891	−	20.1947i	−1.01640	−	0.955177i
$448$		0			0
$449$		−	6.92820i		−	0.326962i	−0.986546	−	0.163481i	$-0.947728\pi$
$449$		−	6.92820i		−	0.326962i	0.986546	−	0.163481i	$-0.0522723\pi$
$450$		0			0
$451$		−	20.1947i		−	0.950931i
$452$		0			0
$453$	−7.18614	−	6.75327i	−0.337634	−	0.317297i
$454$		0			0
$455$	−2.37228			−0.111214
$456$		0			0
$457$	28.9783			1.35555			0.677773	−	0.735271i	$-0.262945\pi$
$457$	28.9783			1.35555			0.677773	+	0.735271i	$0.262945\pi$
$458$		0			0
$459$	−13.6277	+	16.4356i	−0.636087	+	0.767150i
$460$		0			0
$461$			17.3754i			0.809254i	0.914482	+	0.404627i	$0.132599\pi$
$461$			17.3754i			0.809254i	−0.914482	+	0.404627i	$0.867401\pi$
$462$		0			0
$463$		−	37.5152i		−	1.74348i	−0.489969	−	0.871740i	$-0.662992\pi$
$463$		−	37.5152i		−	1.74348i	0.489969	−	0.871740i	$-0.337008\pi$
$464$		0			0
$465$	−7.39403	+	7.86797i	−0.342890	+	0.364868i
$466$		0			0
$467$	−25.4891			−1.17950			−0.589748	−	0.807587i	$-0.700773\pi$
$467$	−25.4891			−1.17950			−0.589748	+	0.807587i	$0.700773\pi$
$468$		0			0
$469$	−8.74456			−0.403786
$470$		0			0
$471$	3.25544	−	3.46410i	0.150003	−	0.159617i
$472$		0			0
$473$		−	2.57924i		−	0.118594i
$474$		0			0
$475$			14.2612i			0.654347i
$476$		0			0
$477$	−35.8614	+	2.22938i	−1.64198	+	0.102076i
$478$		0			0
$479$	−27.1168			−1.23900			−0.619500	−	0.784997i	$-0.712665\pi$
$479$	−27.1168			−1.23900			−0.619500	+	0.784997i	$0.712665\pi$
$480$		0			0
$481$	0.372281			0.0169746
$482$		0			0
$483$	15.1168	+	14.2063i	0.687840	+	0.646407i
$484$		0			0
$485$		−	2.57924i		−	0.117117i
$486$		0			0
$487$			40.6844i			1.84358i	0.387684	+	0.921792i	$0.373275\pi$
$487$			40.6844i			1.84358i	−0.387684	+	0.921792i	$0.626725\pi$
$488$		0			0
$489$	9.11684	+	8.56768i	0.412278	+	0.387444i
$490$		0			0
$491$	−29.3505			−1.32457			−0.662285	−	0.749252i	$-0.730413\pi$
$491$	−29.3505			−1.32457			−0.662285	+	0.749252i	$0.730413\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	11.2554	−	0.699713i	0.505894	−	0.0314498i
$496$		0			0
$497$			28.0627i			1.25878i
$498$		0			0
$499$			23.6588i			1.05911i	0.848275	+	0.529557i	$0.177642\pi$
$499$			23.6588i			1.05911i	−0.848275	+	0.529557i	$0.822358\pi$
$500$		0			0
$501$	4.74456	−	5.04868i	0.211971	−	0.225558i
$502$		0			0
$503$	−6.23369			−0.277946			−0.138973	−	0.990296i	$-0.544380\pi$
$503$	−6.23369			−0.277946			−0.138973	+	0.990296i	$0.544380\pi$
$504$		0			0
$505$	−9.48913			−0.422261
$506$		0			0
$507$	1.18614	−	1.26217i	0.0526784	−	0.0560549i
$508$		0			0
$509$			30.8820i			1.36882i	0.729098	+	0.684409i	$0.239940\pi$
$509$			30.8820i			1.36882i	−0.729098	+	0.684409i	$0.760060\pi$
$510$		0			0
$511$		−	27.1229i		−	1.19985i
$512$		0			0
$513$	11.4891	−	13.8564i	0.507257	−	0.611775i
$514$		0			0
$515$	−8.00000			−0.352522
$516$		0			0
$517$	−25.4891			−1.12101
$518$		0			0
$519$	−8.00000	−	7.51811i	−0.351161	−	0.330008i
$520$		0			0
$521$			33.7013i			1.47648i	0.674539	+	0.738239i	$0.264342\pi$
$521$			33.7013i			1.47648i	−0.674539	+	0.738239i	$0.735658\pi$
$522$		0			0
$523$			29.9971i			1.31168i	0.754899	+	0.655841i	$0.227686\pi$
$523$			29.9971i			1.31168i	−0.754899	+	0.655841i	$0.772314\pi$
$524$		0			0
$525$	13.1168	+	12.3267i	0.572466	+	0.537983i
$526$		0			0
$527$	27.2554			1.18727
$528$		0			0
$529$	−0.489125			−0.0212663
$530$		0			0
$531$	2.23369	+	35.9306i	0.0969338	+	1.55926i
$532$		0			0
$533$		−	5.04868i		−	0.218682i
$534$		0			0
$535$		−	16.4356i		−	0.710575i
$536$		0			0
$537$	−6.67527	+	7.10313i	−0.288059	+	0.306523i
$538$		0			0
$539$	2.51087			0.108151
$540$		0			0
$541$	−28.3723			−1.21982			−0.609910	−	0.792471i	$-0.708794\pi$
$541$	−28.3723			−1.21982			−0.609910	+	0.792471i	$0.708794\pi$
$542$		0			0
$543$	−28.7446	+	30.5870i	−1.23355	+	1.31261i
$544$		0			0
$545$		−	11.6270i		−	0.498047i
$546$		0			0
$547$		−	33.9962i		−	1.45357i	−0.686864	−	0.726786i	$-0.741013\pi$
$547$		−	33.9962i		−	1.45357i	0.686864	−	0.726786i	$-0.258987\pi$
$548$		0			0
$549$	2.13859	+	34.4010i	0.0912729	+	1.46820i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	4.00000			0.170097
$554$		0			0
$555$	0.441578	+	0.414979i	0.0187439	+	0.0176149i
$556$		0			0
$557$		−	20.4348i		−	0.865848i	−0.901430	−	0.432924i	$-0.857482\pi$
$557$		−	20.4348i		−	0.865848i	0.901430	−	0.432924i	$-0.142518\pi$
$558$		0			0
$559$		−	0.644810i		−	0.0272726i
$560$		0			0
$561$	−20.7446	−	19.4950i	−0.875836	−	0.823079i
$562$		0			0
$563$	34.0951			1.43694			0.718468	−	0.695560i	$-0.244843\pi$
$563$	34.0951			1.43694			0.718468	+	0.695560i	$0.244843\pi$
$564$		0			0
$565$	−18.9783			−0.798421
$566$		0			0
$567$	−2.81386	−	22.5441i	−0.118171	−	0.946764i
$568$		0			0
$569$			0.349857i			0.0146667i	0.999973	+	0.00733337i	$0.00233431\pi$
$569$			0.349857i			0.0146667i	−0.999973	+	0.00733337i	$0.997666\pi$
$570$		0			0
$571$		−	20.8395i		−	0.872106i	−0.899921	−	0.436053i	$-0.856376\pi$
$571$		−	20.8395i		−	0.872106i	0.899921	−	0.436053i	$-0.143624\pi$
$572$		0			0
$573$	−1.76631	+	1.87953i	−0.0737887	+	0.0785184i
$574$		0			0
$575$	19.5326			0.814567
$576$		0			0
$577$	−2.74456			−0.114258			−0.0571288	−	0.998367i	$-0.518195\pi$
$577$	−2.74456			−0.114258			−0.0571288	+	0.998367i	$0.518195\pi$
$578$		0			0
$579$	26.9783	−	28.7075i	1.12118	−	1.19304i
$580$		0			0
$581$			30.2921i			1.25673i
$582$		0			0
$583$		−	47.9075i		−	1.98413i
$584$		0			0
$585$	2.81386	−	0.174928i	0.116339	−	0.00723239i
$586$		0			0
$587$	0.744563			0.0307314			0.0153657	−	0.999882i	$-0.495109\pi$
$587$	0.744563			0.0307314			0.0153657	+	0.999882i	$0.495109\pi$
$588$		0			0
$589$	−22.9783			−0.946802
$590$		0			0
$591$	1.18614	+	1.11469i	0.0487913	+	0.0458523i
$592$		0			0
$593$			37.2203i			1.52845i	0.644948	+	0.764226i	$0.276879\pi$
$593$			37.2203i			1.52845i	−0.644948	+	0.764226i	$0.723121\pi$
$594$		0			0
$595$			9.74749i			0.399608i
$596$		0			0
$597$	21.1168	+	19.8448i	0.864255	+	0.812195i
$598$		0			0
$599$	−1.48913			−0.0608440			−0.0304220	−	0.999537i	$-0.509685\pi$
$599$	−1.48913			−0.0608440			−0.0304220	+	0.999537i	$0.509685\pi$
$600$		0			0
$601$	−44.8397			−1.82905			−0.914524	−	0.404532i	$-0.867434\pi$
$601$	−44.8397			−1.82905			−0.914524	+	0.404532i	$0.867434\pi$
$602$		0			0
$603$	10.3723	−	0.644810i	0.422392	−	0.0262587i
$604$		0			0
$605$			4.69882i			0.191034i
$606$		0			0
$607$		−	25.5383i		−	1.03657i	−0.855208	−	0.518284i	$-0.826571\pi$
$607$		−	25.5383i		−	1.03657i	0.855208	−	0.518284i	$-0.173429\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.37228			−0.257795
$612$		0			0
$613$	15.4891			0.625600			0.312800	−	0.949819i	$-0.398733\pi$
$613$	15.4891			0.625600			0.312800	+	0.949819i	$0.398733\pi$
$614$		0			0
$615$	5.62772	−	5.98844i	0.226932	−	0.241477i
$616$		0			0
$617$			30.8820i			1.24326i	0.783311	+	0.621630i	$0.213529\pi$
$617$			30.8820i			1.24326i	−0.783311	+	0.621630i	$0.786471\pi$
$618$		0			0
$619$			33.7562i			1.35677i	0.734705	+	0.678387i	$0.237321\pi$
$619$			33.7562i			1.35677i	−0.734705	+	0.678387i	$0.762679\pi$
$620$		0			0
$621$	−18.9783	−	15.7359i	−0.761571	−	0.631461i
$622$		0			0
$623$	−12.7446			−0.510600
$624$		0			0
$625$	12.5326			0.501305
$626$		0			0
$627$	17.4891	+	16.4356i	0.698448	+	0.656377i
$628$		0			0
$629$		−	1.52967i		−	0.0609920i
$630$		0			0
$631$			10.7422i			0.427639i	0.976873	+	0.213819i	$0.0685904\pi$
$631$			10.7422i			0.427639i	−0.976873	+	0.213819i	$0.931410\pi$
$632$		0			0
$633$	−7.93070	−	7.45299i	−0.315217	−	0.296230i
$634$		0			0
$635$	−6.23369			−0.247376
$636$		0			0
$637$	0.627719			0.0248711
$638$		0			0
$639$	−2.06930	−	33.2863i	−0.0818601	−	1.31678i
$640$		0			0
$641$		−	37.8102i		−	1.49341i	−0.665154	−	0.746706i	$-0.731634\pi$
$641$		−	37.8102i		−	1.49341i	0.665154	−	0.746706i	$-0.268366\pi$
$642$		0			0
$643$		−	9.21249i		−	0.363305i	−0.983363	−	0.181653i	$-0.941855\pi$
$643$		−	9.21249i		−	0.363305i	0.983363	−	0.181653i	$-0.0581446\pi$
$644$		0			0
$645$	0.718765	−	0.764836i	0.0283013	−	0.0301154i
$646$		0			0
$647$	34.9783			1.37514			0.687568	−	0.726120i	$-0.258678\pi$
$647$	34.9783			1.37514			0.687568	+	0.726120i	$0.258678\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$	−19.8614	+	21.1345i	−0.778430	+	0.828325i
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		−	6.68815i		−	0.261328i
$656$		0			0
$657$	2.00000	+	32.1716i	0.0780274	+	1.25513i
$658$		0			0
$659$	17.4891			0.681280			0.340640	−	0.940194i	$-0.389356\pi$
$659$	17.4891			0.681280			0.340640	+	0.940194i	$0.389356\pi$
$660$		0			0
$661$	−0.510875			−0.0198707			−0.00993536	−	0.999951i	$-0.503163\pi$
$661$	−0.510875			−0.0198707			−0.00993536	+	0.999951i	$0.503163\pi$
$662$		0			0
$663$	−5.18614	−	4.87375i	−0.201413	−	0.189281i
$664$		0			0
$665$		−	8.21782i		−	0.318674i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−3.93070	−	3.69393i	−0.151970	−	0.142816i
$670$		0			0
$671$	−45.9565			−1.77413
$672$		0			0
$673$	33.1168			1.27656			0.638280	−	0.769804i	$-0.279646\pi$
$673$	33.1168			1.27656			0.638280	+	0.769804i	$0.279646\pi$
$674$		0			0
$675$	−16.4674	−	13.6540i	−0.633830	−	0.525544i
$676$		0			0
$677$		−	7.51811i		−	0.288944i	−0.989509	−	0.144472i	$-0.953852\pi$
$677$		−	7.51811i		−	0.288944i	0.989509	−	0.144472i	$-0.0461484\pi$
$678$		0			0
$679$		−	6.92820i		−	0.265880i
$680$		0			0
$681$	−10.3723	+	11.0371i	−0.397467	+	0.422943i
$682$		0			0
$683$	22.9783			0.879238			0.439619	−	0.898184i	$-0.355113\pi$
$683$	22.9783			0.879238			0.439619	+	0.898184i	$0.355113\pi$
$684$		0			0
$685$	14.2337			0.543841
$686$		0			0
$687$	−35.4198	+	37.6901i	−1.35135	+	1.43797i
$688$		0			0
$689$		−	11.9769i		−	0.456283i
$690$		0			0
$691$			15.4410i			0.587403i	0.955897	+	0.293701i	$0.0948871\pi$
$691$			15.4410i			0.587403i	−0.955897	+	0.293701i	$0.905113\pi$
$692$		0			0
$693$	30.2337	−	1.87953i	1.14848	−	0.0713974i
$694$		0			0
$695$	19.5842			0.742872
$696$		0			0
$697$	−20.7446			−0.785756
$698$		0			0
$699$	−20.3030	−	19.0800i	−0.767929	−	0.721672i
$700$		0			0
$701$		−	9.39764i		−	0.354944i	−0.984126	−	0.177472i	$-0.943208\pi$
$701$		−	9.39764i		−	0.354944i	0.984126	−	0.177472i	$-0.0567919\pi$
$702$		0			0
$703$			1.28962i			0.0486390i
$704$		0			0
$705$	−7.55842	−	7.10313i	−0.284667	−	0.267519i
$706$		0			0
$707$	−25.4891			−0.958617
$708$		0			0
$709$	24.9783			0.938078			0.469039	−	0.883177i	$-0.344600\pi$
$709$	24.9783			0.938078			0.469039	+	0.883177i	$0.344600\pi$
$710$		0			0
$711$	−4.74456	+	0.294954i	−0.177935	+	0.0110616i
$712$		0			0
$713$			31.4719i			1.17863i
$714$		0			0
$715$			3.75906i			0.140581i
$716$		0			0
$717$	−9.32473	+	9.92242i	−0.348239	+	0.370560i
$718$		0			0
$719$	−7.72281			−0.288012			−0.144006	−	0.989577i	$-0.545999\pi$
$719$	−7.72281			−0.288012			−0.144006	+	0.989577i	$0.545999\pi$
$720$		0			0
$721$	−21.4891			−0.800297
$722$		0			0
$723$	21.3505	−	22.7190i	0.794035	−	0.844930i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			41.3841i			1.53485i	0.641139	+	0.767425i	$0.278462\pi$
$727$			41.3841i			1.53485i	−0.641139	+	0.767425i	$0.721538\pi$
$728$		0			0
$729$	5.00000	+	26.5330i	0.185185	+	0.982704i
$730$		0			0
$731$	−2.64947			−0.0979941
$732$		0			0
$733$	11.3505			0.419241			0.209621	−	0.977783i	$-0.432777\pi$
$733$	11.3505			0.419241			0.209621	+	0.977783i	$0.432777\pi$
$734$		0			0
$735$	0.744563	+	0.699713i	0.0274636	+	0.0258093i
$736$		0			0
$737$			13.8564i			0.510407i
$738$		0			0
$739$			19.8997i			0.732024i	0.930610	+	0.366012i	$0.119277\pi$
$739$			19.8997i			0.732024i	−0.930610	+	0.366012i	$0.880723\pi$
$740$		0			0
$741$	4.37228	+	4.10891i	0.160620	+	0.150945i
$742$		0			0
$743$	1.62772			0.0597152			0.0298576	−	0.999554i	$-0.490495\pi$
$743$	1.62772			0.0597152			0.0298576	+	0.999554i	$0.490495\pi$
$744$		0			0
$745$	16.0000			0.586195
$746$		0			0
$747$	−2.23369	−	35.9306i	−0.0817264	−	1.31463i
$748$		0			0
$749$		−	44.1485i		−	1.61315i
$750$		0			0
$751$		−	0.884861i		−	0.0322890i	−0.999870	−	0.0161445i	$-0.994861\pi$
$751$		−	0.884861i		−	0.0322890i	0.999870	−	0.0161445i	$-0.00513918\pi$
$752$		0			0
$753$	18.9783	−	20.1947i	0.691606	−	0.735936i
$754$		0			0
$755$	5.35053			0.194726
$756$		0			0
$757$	−28.9783			−1.05323			−0.526616	−	0.850103i	$-0.676539\pi$
$757$	−28.9783			−1.05323			−0.526616	+	0.850103i	$0.676539\pi$
$758$		0			0
$759$	22.5109	−	23.9538i	0.817093	−	0.869466i
$760$		0			0
$761$			40.9793i			1.48550i	0.669569	+	0.742749i	$0.266479\pi$
$761$			40.9793i			1.48550i	−0.669569	+	0.742749i	$0.733521\pi$
$762$		0			0
$763$		−	31.2318i		−	1.13067i
$764$		0			0
$765$	−0.718765	−	11.5619i	−0.0259870	−	0.418021i
$766$		0			0
$767$	−12.0000			−0.433295
$768$		0			0
$769$	9.72281			0.350614			0.175307	−	0.984514i	$-0.443908\pi$
$769$	9.72281			0.350614			0.175307	+	0.984514i	$0.443908\pi$
$770$		0			0
$771$	0.441578	+	0.414979i	0.0159030	+	0.0149451i
$772$		0			0
$773$		−	31.2318i		−	1.12333i	−0.827365	−	0.561665i	$-0.810161\pi$
$773$		−	31.2318i		−	1.12333i	0.827365	−	0.561665i	$-0.189839\pi$
$774$		0			0
$775$			27.3081i			0.980934i
$776$		0			0
$777$	1.18614	+	1.11469i	0.0425526	+	0.0399894i
$778$		0			0
$779$	17.4891			0.626613
$780$		0			0
$781$	44.4674			1.59117
$782$		0			0
$783$		0			0
$784$		0			0
$785$			2.57924i			0.0920570i
$786$		0			0
$787$			27.4179i			0.977341i	0.872468	+	0.488671i	$0.162518\pi$
$787$			27.4179i			0.977341i	−0.872468	+	0.488671i	$0.837482\pi$
$788$		0			0
$789$	−15.1168	+	16.0858i	−0.538174	+	0.572669i
$790$		0			0
$791$	−50.9783			−1.81258
$792$		0			0
$793$	−11.4891			−0.407991
$794$		0			0
$795$	13.3505	−	14.2063i	0.473495	−	0.503844i
$796$		0			0
$797$			25.8333i			0.915062i	0.889194	+	0.457531i	$0.151266\pi$
$797$			25.8333i			0.915062i	−0.889194	+	0.457531i	$0.848734\pi$
$798$		0			0
$799$			26.1831i			0.926293i
$800$		0			0
$801$	15.1168	−	0.939764i	0.534127	−	0.0332049i
$802$		0			0
$803$	−42.9783			−1.51667
$804$		0			0
$805$	−11.2554			−0.396702
$806$		0			0
$807$	15.1168	+	14.2063i	0.532138	+	0.500084i
$808$		0			0
$809$		−	26.1831i		−	0.920550i	−0.887776	−	0.460275i	$-0.847751\pi$
$809$		−	26.1831i		−	0.920550i	0.887776	−	0.460275i	$-0.152249\pi$
$810$		0			0
$811$			33.0564i			1.16077i	0.814343	+	0.580384i	$0.197098\pi$
$811$			33.0564i			1.16077i	−0.814343	+	0.580384i	$0.802902\pi$
$812$		0			0
$813$	−19.9307	−	18.7302i	−0.699000	−	0.656895i
$814$		0			0
$815$	−6.78806			−0.237775
$816$		0			0
$817$	2.23369			0.0781468
$818$		0			0
$819$	7.55842	−	0.469882i	0.264113	−	0.0164190i
$820$		0			0
$821$		−	50.7268i		−	1.77038i	−0.465232	−	0.885189i	$-0.654029\pi$
$821$		−	50.7268i		−	1.77038i	0.465232	−	0.885189i	$-0.345971\pi$
$822$		0			0
$823$			12.2718i			0.427769i	0.976859	+	0.213885i	$0.0686117\pi$
$823$			12.2718i			0.427769i	−0.976859	+	0.213885i	$0.931388\pi$
$824$		0			0
$825$	19.5326	−	20.7846i	0.680039	−	0.723627i
$826$		0			0
$827$	40.7446			1.41683			0.708414	−	0.705798i	$-0.249411\pi$
$827$	40.7446			1.41683			0.708414	+	0.705798i	$0.249411\pi$
$828$		0			0
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$	29.6277	−	31.5268i	1.02777	−	1.09365i
$832$		0			0
$833$		−	2.57924i		−	0.0893654i
$834$		0			0
$835$			3.75906i			0.130087i
$836$		0			0
$837$	22.0000	−	26.5330i	0.760431	−	0.917115i
$838$		0			0
$839$	1.02175			0.0352747			0.0176374	−	0.999844i	$-0.494386\pi$
$839$	1.02175			0.0352747			0.0176374	+	0.999844i	$0.494386\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	27.1168	+	25.4834i	0.933954	+	0.877696i
$844$		0			0
$845$			0.939764i			0.0323289i
$846$		0			0
$847$			12.6217i			0.433686i
$848$		0			0
$849$	−17.8614	−	16.7855i	−0.613002	−	0.576077i
$850$		0			0
$851$	1.76631			0.0605484
$852$		0			0
$853$	14.6060			0.500099			0.250049	−	0.968233i	$-0.419553\pi$
$853$	14.6060			0.500099			0.250049	+	0.968233i	$0.419553\pi$
$854$		0			0
$855$	0.605969	+	9.74749i	0.0207237	+	0.333357i
$856$		0			0
$857$			10.0974i			0.344919i	0.985017	+	0.172459i	$0.0551714\pi$
$857$			10.0974i			0.344919i	−0.985017	+	0.172459i	$0.944829\pi$
$858$		0			0
$859$		−	26.8280i		−	0.915358i	−0.889118	−	0.457679i	$-0.848681\pi$
$859$		−	26.8280i		−	0.915358i	0.889118	−	0.457679i	$-0.151319\pi$
$860$		0			0
$861$	15.1168	−	16.0858i	0.515181	−	0.548202i
$862$		0			0
$863$	9.35053			0.318296			0.159148	−	0.987255i	$-0.449125\pi$
$863$	9.35053			0.318296			0.159148	+	0.987255i	$0.449125\pi$
$864$		0			0
$865$	5.95650			0.202527
$866$		0			0
$867$	0.138593	−	0.147477i	0.00470688	−	0.00500858i
$868$		0			0
$869$		−	6.33830i		−	0.215012i
$870$		0			0
$871$			3.46410i			0.117377i
$872$		0			0
$873$	0.510875	+	8.21782i	0.0172905	+	0.278131i
$874$		0			0
$875$	−21.6277			−0.731150
$876$		0			0
$877$	−1.11684			−0.0377131			−0.0188566	−	0.999822i	$-0.506003\pi$
$877$	−1.11684			−0.0377131			−0.0188566	+	0.999822i	$0.506003\pi$
$878$		0			0
$879$	−32.3030	−	30.3572i	−1.08955	−	1.02392i
$880$		0			0
$881$			11.6270i			0.391724i	0.980631	+	0.195862i	$0.0627505\pi$
$881$			11.6270i			0.391724i	−0.980631	+	0.195862i	$0.937249\pi$
$882$		0			0
$883$			21.4294i			0.721157i	0.932729	+	0.360579i	$0.117421\pi$
$883$			21.4294i			0.721157i	−0.932729	+	0.360579i	$0.882579\pi$
$884$		0			0
$885$	−14.2337	−	13.3763i	−0.478460	−	0.449640i
$886$		0			0
$887$	−38.2337			−1.28376			−0.641881	−	0.766804i	$-0.721846\pi$
$887$	−38.2337			−1.28376			−0.641881	+	0.766804i	$0.721846\pi$
$888$		0			0
$889$	−16.7446			−0.561595
$890$		0			0
$891$	−35.7228	+	4.45877i	−1.19676	+	0.149374i
$892$		0			0
$893$		−	22.0742i		−	0.738686i
$894$		0			0
$895$		−	5.28873i		−	0.176783i
$896$		0			0
$897$	5.62772	−	5.98844i	0.187904	−	0.199948i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−49.2119			−1.63949
$902$		0			0
$903$	1.93070	−	2.05446i	0.0642498	−	0.0683680i
$904$		0			0
$905$		−	22.7739i		−	0.757031i
$906$		0			0
$907$		−	16.3807i		−	0.543914i	−0.962309	−	0.271957i	$-0.912329\pi$
$907$		−	16.3807i		−	0.543914i	0.962309	−	0.271957i	$-0.0876708\pi$
$908$		0			0
$909$	30.2337	−	1.87953i	1.00279	−	0.0623400i
$910$		0			0
$911$	10.9783			0.363726			0.181863	−	0.983324i	$-0.441787\pi$
$911$	10.9783			0.363726			0.181863	+	0.983324i	$0.441787\pi$
$912$		0			0
$913$	48.0000			1.58857
$914$		0			0
$915$	−13.6277	−	12.8068i	−0.450518	−	0.423381i
$916$		0			0
$917$		−	17.9653i		−	0.593267i
$918$		0			0
$919$			56.4203i			1.86113i	0.366122	+	0.930567i	$0.380685\pi$
$919$			56.4203i			1.86113i	−0.366122	+	0.930567i	$0.619315\pi$
$920$		0			0
$921$	24.2337	+	22.7739i	0.798527	+	0.750427i
$922$		0			0
$923$	11.1168			0.365915
$924$		0			0
$925$	1.53262			0.0503924
$926$		0			0
$927$	25.4891	−	1.58457i	0.837173	−	0.0520442i
$928$		0			0
$929$		−	40.9793i		−	1.34449i	−0.740330	−	0.672244i	$-0.765331\pi$
$929$		−	40.9793i		−	1.34449i	0.740330	−	0.672244i	$-0.234669\pi$
$930$		0			0
$931$			2.17448i			0.0712657i
$932$		0			0
$933$	24.6060	−	26.1831i	0.805563	−	0.857198i
$934$		0			0
$935$	15.4456			0.505126
$936$		0			0
$937$	−12.5109			−0.408712			−0.204356	−	0.978897i	$-0.565510\pi$
$937$	−12.5109			−0.408712			−0.204356	+	0.978897i	$0.565510\pi$
$938$		0			0
$939$	24.1644	−	25.7133i	0.788575	−	0.839121i
$940$		0			0
$941$		−	7.27806i		−	0.237258i	−0.992939	−	0.118629i	$-0.962150\pi$
$941$		−	7.27806i		−	0.237258i	0.992939	−	0.118629i	$-0.0378499\pi$
$942$		0			0
$943$		−	23.9538i		−	0.780042i
$944$		0			0
$945$	9.48913	+	7.86797i	0.308681	+	0.255945i
$946$		0			0
$947$	22.9783			0.746693			0.373346	−	0.927692i	$-0.378210\pi$
$947$	22.9783			0.746693			0.373346	+	0.927692i	$0.378210\pi$
$948$		0			0
$949$	−10.7446			−0.348783
$950$		0			0
$951$	−21.4891	−	20.1947i	−0.696833	−	0.654858i
$952$		0			0
$953$			3.40920i			0.110435i	0.998474	+	0.0552174i	$0.0175852\pi$
$953$			3.40920i			0.110435i	−0.998474	+	0.0552174i	$0.982415\pi$
$954$		0			0
$955$		−	1.39943i		−	0.0452844i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	38.2337			1.23463
$960$		0			0
$961$	−13.0000			−0.419355
$962$		0			0
$963$	3.25544	+	52.3663i	0.104905	+	1.68748i
$964$		0			0
$965$			21.3745i			0.688070i
$966$		0			0
$967$			14.5012i			0.466328i	0.972437	+	0.233164i	$0.0749078\pi$
$967$			14.5012i			0.466328i	−0.972437	+	0.233164i	$0.925092\pi$
$968$		0			0
$969$	16.8832	−	17.9653i	0.542365	−	0.577129i
$970$		0			0
$971$	−11.8614			−0.380651			−0.190325	−	0.981721i	$-0.560954\pi$
$971$	−11.8614			−0.380651			−0.190325	+	0.981721i	$0.560954\pi$
$972$		0			0
$973$	52.6060			1.68647
$974$		0			0
$975$	4.88316	−	5.19615i	0.156386	−	0.166410i
$976$		0			0
$977$			22.6641i			0.725090i	0.931966	+	0.362545i	$0.118092\pi$
$977$			22.6641i			0.725090i	−0.931966	+	0.362545i	$0.881908\pi$
$978$		0			0
$979$			20.1947i			0.645425i
$980$		0			0
$981$	2.30298	+	37.0453i	0.0735286	+	1.18277i
$982$		0			0
$983$	51.1168			1.63037			0.815187	−	0.579198i	$-0.196634\pi$
$983$	51.1168			1.63037			0.815187	+	0.579198i	$0.196634\pi$
$984$		0			0
$985$	−0.883156			−0.0281397
$986$		0			0
$987$	−20.3030	−	19.0800i	−0.646251	−	0.607323i
$988$		0			0
$989$		−	3.05934i		−	0.0972814i
$990$		0			0
$991$		−	6.04334i		−	0.191973i	−0.995383	−	0.0959865i	$-0.969399\pi$
$991$		−	6.04334i		−	0.191973i	0.995383	−	0.0959865i	$-0.0306006\pi$
$992$		0			0
$993$	−2.74456	−	2.57924i	−0.0870961	−	0.0818497i
$994$		0			0
$995$	−15.7228			−0.498447
$996$		0			0
$997$	42.7446			1.35373			0.676867	−	0.736105i	$-0.263337\pi$
$997$	42.7446			1.35373			0.676867	+	0.736105i	$0.263337\pi$
$998$		0			0
$999$	−1.48913	−	1.23472i	−0.0471138	−	0.0390647i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.d.g.287.3	✓	4
3.2	odd	2		624.2.d.i.287.1	yes	4
4.3	odd	2		624.2.d.i.287.2	yes	4
8.3	odd	2		2496.2.d.i.1535.3		4
8.5	even	2		2496.2.d.l.1535.2		4
12.11	even	2	inner	624.2.d.g.287.4	yes	4
24.5	odd	2		2496.2.d.i.1535.4		4
24.11	even	2		2496.2.d.l.1535.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.2.d.g.287.3	✓	4	1.1	even	1	trivial
624.2.d.g.287.4	yes	4	12.11	even	2	inner
624.2.d.i.287.1	yes	4	3.2	odd	2
624.2.d.i.287.2	yes	4	4.3	odd	2
2496.2.d.i.1535.3		4	8.3	odd	2
2496.2.d.i.1535.4		4	24.5	odd	2
2496.2.d.l.1535.1		4	24.11	even	2
2496.2.d.l.1535.2		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.18614 - 1.26217i) q^{3} +0.939764i q^{5} +2.52434i q^{7} +(-0.186141 - 2.99422i) q^{9} +O(q^{10})\)	\(q+(1.18614 - 1.26217i) q^{3} +0.939764i q^{5} +2.52434i q^{7} +(-0.186141 - 2.99422i) q^{9} +4.00000 q^{11} +1.00000 q^{13} +(1.18614 + 1.11469i) q^{15} -4.10891i q^{17} +3.46410i q^{19} +(3.18614 + 2.99422i) q^{21} +4.74456 q^{23} +4.11684 q^{25} +(-4.00000 - 3.31662i) q^{27} +6.63325i q^{31} +(4.74456 - 5.04868i) q^{33} -2.37228 q^{35} +0.372281 q^{37} +(1.18614 - 1.26217i) q^{39} -5.04868i q^{41} -0.644810i q^{43} +(2.81386 - 0.174928i) q^{45} -6.37228 q^{47} +0.627719 q^{49} +(-5.18614 - 4.87375i) q^{51} -11.9769i q^{53} +3.75906i q^{55} +(4.37228 + 4.10891i) q^{57} -12.0000 q^{59} -11.4891 q^{61} +(7.55842 - 0.469882i) q^{63} +0.939764i q^{65} +3.46410i q^{67} +(5.62772 - 5.98844i) q^{69} +11.1168 q^{71} -10.7446 q^{73} +(4.88316 - 5.19615i) q^{75} +10.0974i q^{77} -1.58457i q^{79} +(-8.93070 + 1.11469i) q^{81} +12.0000 q^{83} +3.86141 q^{85} +5.04868i q^{89} +2.52434i q^{91} +(8.37228 + 7.86797i) q^{93} -3.25544 q^{95} -2.74456 q^{97} +(-0.744563 - 11.9769i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 5 q^{9}+O(q^{10}) \) 4 * q - q^3 + 5 * q^9	\( 4 q - q^{3} + 5 q^{9} + 16 q^{11} + 4 q^{13} - q^{15} + 7 q^{21} - 4 q^{23} - 18 q^{25} - 16 q^{27} - 4 q^{33} + 2 q^{35} - 10 q^{37} - q^{39} + 17 q^{45} - 14 q^{47} + 14 q^{49} - 15 q^{51} + 6 q^{57} - 48 q^{59} + 13 q^{63} + 34 q^{69} + 10 q^{71} - 20 q^{73} + 54 q^{75} - 7 q^{81} + 48 q^{83} - 42 q^{85} + 22 q^{93} - 36 q^{95} + 12 q^{97} + 20 q^{99}+O(q^{100}) \) 4 * q - q^3 + 5 * q^9 + 16 * q^11 + 4 * q^13 - q^15 + 7 * q^21 - 4 * q^23 - 18 * q^25 - 16 * q^27 - 4 * q^33 + 2 * q^35 - 10 * q^37 - q^39 + 17 * q^45 - 14 * q^47 + 14 * q^49 - 15 * q^51 + 6 * q^57 - 48 * q^59 + 13 * q^63 + 34 * q^69 + 10 * q^71 - 20 * q^73 + 54 * q^75 - 7 * q^81 + 48 * q^83 - 42 * q^85 + 22 * q^93 - 36 * q^95 + 12 * q^97 + 20 * q^99

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