LMFDB - Embedded newform 1216.3.g.d.417.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(417,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.417");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.g (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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 34x^{14} + 509x^{12} - 4794x^{10} + 30356x^{8} - 106386x^{6} + 288389x^{4} - 166634x^{2} + 28561 $ x^16 - 34x^14 + 509x^12 - 4794x^10 + 30356x^8 - 106386x^6 + 288389x^4 - 166634*x^2 + 28561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.7
Root			$1.69895 - 1.19682i$ of defining polynomial
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.d.417.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53741 q^{3} +6.47540i q^{5} -1.41956 q^{7} -2.56155 q^{9} +O(q^{10})$	$q-2.53741 q^{3} +6.47540i q^{5} -1.41956 q^{7} -2.56155 q^{9} +4.80776i q^{11} -12.8287 q^{13} -16.4308i q^{15} -14.3693 q^{17} +(16.9617 + 8.56155i) q^{19} +3.60201 q^{21} -22.4401 q^{23} -16.9309 q^{25} +29.3364 q^{27} -29.2595 q^{29} -9.22674i q^{31} -12.1993i q^{33} -9.19224i q^{35} -23.6348 q^{37} +32.5518 q^{39} -7.12445i q^{41} +9.54640i q^{43} -16.5871i q^{45} +10.9088 q^{47} -46.9848 q^{49} +36.4608 q^{51} +94.9825 q^{53} -31.1322 q^{55} +(-43.0388 - 21.7242i) q^{57} +68.3346 q^{59} +51.3556i q^{61} +3.63628 q^{63} -83.0713i q^{65} +6.63667 q^{67} +56.9397 q^{69} +44.1110i q^{71} +19.3542 q^{73} +42.9606 q^{75} -6.82492i q^{77} +63.7003i q^{79} -51.3845 q^{81} -111.970i q^{83} -93.0471i q^{85} +74.2433 q^{87} -60.7224i q^{89} +18.2112 q^{91} +23.4120i q^{93} +(-55.4395 + 109.834i) q^{95} -136.669i q^{97} -12.3153i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} - 32 q^{17} - 40 q^{25} - 224 q^{49} + 136 q^{57} - 416 q^{73} - 1152 q^{81}+O(q^{100}) $ 16 * q - 8 * q^9 - 32 * q^17 - 40 * q^25 - 224 * q^49 + 136 * q^57 - 416 * q^73 - 1152 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−2.53741			−0.845803			−0.422902	−	0.906176i	$-0.638988\pi$
$3$	−2.53741			−0.845803			−0.422902	+	0.906176i	$0.638988\pi$
$4$		0			0
$5$			6.47540i			1.29508i	0.762031	+	0.647540i	$0.224202\pi$
$5$			6.47540i			1.29508i	−0.762031	+	0.647540i	$0.775798\pi$
$6$		0			0
$7$	−1.41956			−0.202795			−0.101397	−	0.994846i	$-0.532331\pi$
$7$	−1.41956			−0.202795			−0.101397	+	0.994846i	$0.532331\pi$
$8$		0			0
$9$	−2.56155			−0.284617
$10$		0			0
$11$			4.80776i			0.437069i	0.975829	+	0.218535i	$0.0701277\pi$
$11$			4.80776i			0.437069i	−0.975829	+	0.218535i	$0.929872\pi$
$12$		0			0
$13$	−12.8287			−0.986827			−0.493413	−	0.869795i	$-0.664251\pi$
$13$	−12.8287			−0.986827			−0.493413	+	0.869795i	$0.664251\pi$
$14$		0			0
$15$		−	16.4308i		−	1.09538i
$16$		0			0
$17$	−14.3693			−0.845254			−0.422627	−	0.906304i	$-0.638892\pi$
$17$	−14.3693			−0.845254			−0.422627	+	0.906304i	$0.638892\pi$
$18$		0			0
$19$	16.9617	+	8.56155i	0.892722	+	0.450608i
$19$	16.9617	+	8.56155i	0.892722	+	0.450608i
$20$		0			0
$21$	3.60201			0.171524
$22$		0			0
$23$	−22.4401			−0.975656			−0.487828	−	0.872940i	$-0.662211\pi$
$23$	−22.4401			−0.975656			−0.487828	+	0.872940i	$0.662211\pi$
$24$		0			0
$25$	−16.9309			−0.677235
$26$		0			0
$27$	29.3364			1.08653
$28$		0			0
$29$	−29.2595			−1.00895			−0.504474	−	0.863427i	$-0.668314\pi$
$29$	−29.2595			−1.00895			−0.504474	+	0.863427i	$0.668314\pi$
$30$		0			0
$31$		−	9.22674i		−	0.297637i	−0.988865	−	0.148818i	$-0.952453\pi$
$31$		−	9.22674i		−	0.297637i	0.988865	−	0.148818i	$-0.0475470\pi$
$32$		0			0
$33$		−	12.1993i		−	0.369675i
$34$		0			0
$35$		−	9.19224i		−	0.262635i
$36$		0			0
$37$	−23.6348			−0.638778			−0.319389	−	0.947624i	$-0.603478\pi$
$37$	−23.6348			−0.638778			−0.319389	+	0.947624i	$0.603478\pi$
$38$		0			0
$39$	32.5518			0.834661
$40$		0			0
$41$		−	7.12445i		−	0.173767i	−0.996218	−	0.0868835i	$-0.972309\pi$
$41$		−	7.12445i		−	0.173767i	0.996218	−	0.0868835i	$-0.0276908\pi$
$42$		0			0
$43$			9.54640i			0.222009i	0.993820	+	0.111005i	$0.0354068\pi$
$43$			9.54640i			0.222009i	−0.993820	+	0.111005i	$0.964593\pi$
$44$		0			0
$45$		−	16.5871i		−	0.368602i
$46$		0			0
$47$	10.9088			0.232103			0.116052	−	0.993243i	$-0.462976\pi$
$47$	10.9088			0.232103			0.116052	+	0.993243i	$0.462976\pi$
$48$		0			0
$49$	−46.9848			−0.958874
$50$		0			0
$51$	36.4608			0.714918
$52$		0			0
$53$	94.9825			1.79212			0.896061	−	0.443930i	$-0.146416\pi$
$53$	94.9825			1.79212			0.896061	+	0.443930i	$0.146416\pi$
$54$		0			0
$55$	−31.1322			−0.566040
$56$		0			0
$57$	−43.0388	−	21.7242i	−0.755067	−	0.381126i
$58$		0			0
$59$	68.3346			1.15821			0.579107	−	0.815251i	$-0.303401\pi$
$59$	68.3346			1.15821			0.579107	+	0.815251i	$0.303401\pi$
$60$		0			0
$61$			51.3556i			0.841895i	0.907085	+	0.420947i	$0.138302\pi$
$61$			51.3556i			0.841895i	−0.907085	+	0.420947i	$0.861698\pi$
$62$		0			0
$63$	3.63628			0.0577188
$64$		0			0
$65$		−	83.0713i		−	1.27802i
$66$		0			0
$67$	6.63667			0.0990547			0.0495274	−	0.998773i	$-0.484228\pi$
$67$	6.63667			0.0990547			0.0495274	+	0.998773i	$0.484228\pi$
$68$		0			0
$69$	56.9397			0.825213
$70$		0			0
$71$			44.1110i			0.621281i	0.950527	+	0.310641i	$0.100544\pi$
$71$			44.1110i			0.621281i	−0.950527	+	0.310641i	$0.899456\pi$
$72$		0			0
$73$	19.3542			0.265126			0.132563	−	0.991175i	$-0.457679\pi$
$73$	19.3542			0.265126			0.132563	+	0.991175i	$0.457679\pi$
$74$		0			0
$75$	42.9606			0.572807
$76$		0			0
$77$		−	6.82492i		−	0.0886353i
$78$		0			0
$79$			63.7003i			0.806333i	0.915127	+	0.403166i	$0.132090\pi$
$79$			63.7003i			0.806333i	−0.915127	+	0.403166i	$0.867910\pi$
$80$		0			0
$81$	−51.3845			−0.634376
$82$		0			0
$83$		−	111.970i		−	1.34903i	−0.738260	−	0.674516i	$-0.764352\pi$
$83$		−	111.970i		−	1.34903i	0.738260	−	0.674516i	$-0.235648\pi$
$84$		0			0
$85$		−	93.0471i		−	1.09467i
$86$		0			0
$87$	74.2433			0.853372
$88$		0			0
$89$		−	60.7224i		−	0.682274i	−0.940014	−	0.341137i	$-0.889188\pi$
$89$		−	60.7224i		−	0.682274i	0.940014	−	0.341137i	$-0.110812\pi$
$90$		0			0
$91$	18.2112			0.200123
$92$		0			0
$93$			23.4120i			0.251742i
$94$		0			0
$95$	−55.4395	+	109.834i	−0.583574	+	1.15615i
$96$		0			0
$97$		−	136.669i		−	1.40896i	−0.709723	−	0.704481i	$-0.751180\pi$
$97$		−	136.669i		−	1.40896i	0.709723	−	0.704481i	$-0.248820\pi$
$98$		0			0
$99$		−	12.3153i		−	0.124397i
$100$		0			0
$101$		−	97.9282i		−	0.969586i	−0.874629	−	0.484793i	$-0.838895\pi$
$101$		−	97.9282i		−	0.969586i	0.874629	−	0.484793i	$-0.161105\pi$
$102$		0			0
$103$		−	72.9270i		−	0.708029i	−0.935240	−	0.354015i	$-0.884816\pi$
$103$		−	72.9270i		−	0.708029i	0.935240	−	0.354015i	$-0.115184\pi$
$104$		0			0
$105$			23.3245i			0.222138i
$106$		0			0
$107$	113.032			1.05638			0.528189	−	0.849127i	$-0.322871\pi$
$107$	113.032			1.05638			0.528189	+	0.849127i	$0.322871\pi$
$108$		0			0
$109$	−80.5745			−0.739215			−0.369608	−	0.929188i	$-0.620508\pi$
$109$	−80.5745			−0.739215			−0.369608	+	0.929188i	$0.620508\pi$
$110$		0			0
$111$	59.9711			0.540280
$112$		0			0
$113$		−	98.1204i		−	0.868322i	−0.900835	−	0.434161i	$-0.857045\pi$
$113$		−	98.1204i		−	0.868322i	0.900835	−	0.434161i	$-0.142955\pi$
$114$		0			0
$115$		−	145.309i		−	1.26355i
$116$		0			0
$117$	32.8615			0.280868
$118$		0			0
$119$	20.3981			0.171413
$120$		0			0
$121$	97.8854			0.808970
$122$		0			0
$123$			18.0776i			0.146973i
$124$		0			0
$125$			52.2509i			0.418007i
$126$		0			0
$127$		−	191.988i		−	1.51171i	−0.654736	−	0.755857i	$-0.727220\pi$
$127$		−	191.988i		−	1.51171i	0.654736	−	0.755857i	$-0.272780\pi$
$128$		0			0
$129$		−	24.2231i		−	0.187776i
$130$		0			0
$131$		−	161.348i		−	1.23166i	−0.787879	−	0.615830i	$-0.788821\pi$
$131$		−	161.348i		−	1.23166i	0.787879	−	0.615830i	$-0.211179\pi$
$132$		0			0
$133$	−24.0782	−	12.1537i	−0.181039	−	0.0913808i
$134$		0			0
$135$			189.965i			1.40715i
$136$		0			0
$137$	−35.5227			−0.259290			−0.129645	−	0.991560i	$-0.541384\pi$
$137$	−35.5227			−0.259290			−0.129645	+	0.991560i	$0.541384\pi$
$138$		0			0
$139$			7.68466i			0.0552853i	0.999618	+	0.0276427i	$0.00880005\pi$
$139$			7.68466i			0.0552853i	−0.999618	+	0.0276427i	$0.991200\pi$
$140$		0			0
$141$	−27.6802			−0.196314
$142$		0			0
$143$		−	61.6776i		−	0.431312i
$144$		0			0
$145$		−	189.467i		−	1.30667i
$146$		0			0
$147$	119.220			0.811019
$148$		0			0
$149$			236.205i			1.58527i	0.609697	+	0.792634i	$0.291291\pi$
$149$			236.205i			1.58527i	−0.609697	+	0.792634i	$0.708709\pi$
$150$		0			0
$151$		−	234.076i		−	1.55017i	−0.631856	−	0.775086i	$-0.717706\pi$
$151$		−	234.076i		−	1.55017i	0.631856	−	0.775086i	$-0.282294\pi$
$152$		0			0
$153$	36.8078			0.240574
$154$		0			0
$155$	59.7469			0.385464
$156$		0			0
$157$			75.7610i			0.482554i	0.970456	+	0.241277i	$0.0775663\pi$
$157$			75.7610i			0.482554i	−0.970456	+	0.241277i	$0.922434\pi$
$158$		0			0
$159$	−241.010			−1.51578
$160$		0			0
$161$	31.8551			0.197858
$162$		0			0
$163$		−	130.047i		−	0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$163$		−	130.047i		−	0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$164$		0			0
$165$	78.9952			0.478759
$166$		0			0
$167$			110.970i			0.664490i	0.943193	+	0.332245i	$0.107806\pi$
$167$			110.970i			0.664490i	−0.943193	+	0.332245i	$0.892194\pi$
$168$		0			0
$169$	−4.42329			−0.0261733
$170$		0			0
$171$	−43.4483	−	21.9309i	−0.254084	−	0.128251i
$172$		0			0
$173$	−117.038			−0.676520			−0.338260	−	0.941053i	$-0.609838\pi$
$173$	−117.038			−0.676520			−0.338260	+	0.941053i	$0.609838\pi$
$174$		0			0
$175$	24.0344			0.137339
$176$		0			0
$177$	−173.393			−0.979621
$178$		0			0
$179$	131.419			0.734185			0.367092	−	0.930185i	$-0.380353\pi$
$179$	131.419			0.734185			0.367092	+	0.930185i	$0.380353\pi$
$180$		0			0
$181$	−252.529			−1.39519			−0.697595	−	0.716492i	$-0.745747\pi$
$181$	−252.529			−1.39519			−0.697595	+	0.716492i	$0.745747\pi$
$182$		0			0
$183$		−	130.310i		−	0.712077i
$184$		0			0
$185$		−	153.045i		−	0.827269i
$186$		0			0
$187$		−	69.0843i		−	0.369435i
$188$		0			0
$189$	−41.6448			−0.220343
$190$		0			0
$191$	112.648			0.589781			0.294890	−	0.955531i	$-0.404717\pi$
$191$	112.648			0.589781			0.294890	+	0.955531i	$0.404717\pi$
$192$		0			0
$193$		−	68.6470i		−	0.355684i	−0.984059	−	0.177842i	$-0.943088\pi$
$193$		−	68.6470i		−	0.355684i	0.984059	−	0.177842i	$-0.0569116\pi$
$194$		0			0
$195$			210.786i			1.08095i
$196$		0			0
$197$		−	171.200i		−	0.869034i	−0.900664	−	0.434517i	$-0.856919\pi$
$197$		−	171.200i		−	0.869034i	0.900664	−	0.434517i	$-0.143081\pi$
$198$		0			0
$199$	−214.660			−1.07869			−0.539347	−	0.842083i	$-0.681329\pi$
$199$	−214.660			−1.07869			−0.539347	+	0.842083i	$0.681329\pi$
$200$		0			0
$201$	−16.8399			−0.0837808
$202$		0			0
$203$	41.5357			0.204609
$204$		0			0
$205$	46.1337			0.225042
$206$		0			0
$207$	57.4815			0.277688
$208$		0			0
$209$	−41.1619	+	81.5479i	−0.196947	+	0.390181i
$210$		0			0
$211$	6.81208			0.0322847			0.0161424	−	0.999870i	$-0.494862\pi$
$211$	6.81208			0.0322847			0.0161424	+	0.999870i	$0.494862\pi$
$212$		0			0
$213$		−	111.928i		−	0.525482i
$214$		0			0
$215$	−61.8168			−0.287520
$216$		0			0
$217$			13.0979i			0.0603591i
$218$		0			0
$219$	−49.1094			−0.224244
$220$		0			0
$221$	184.340			0.834119
$222$		0			0
$223$		−	6.06815i		−	0.0272115i	−0.999907	−	0.0136057i	$-0.995669\pi$
$223$		−	6.06815i		−	0.0272115i	0.999907	−	0.0136057i	$-0.00433097\pi$
$224$		0			0
$225$	43.3693			0.192753
$226$		0			0
$227$	−191.106			−0.841876			−0.420938	−	0.907089i	$-0.638299\pi$
$227$	−191.106			−0.841876			−0.420938	+	0.907089i	$0.638299\pi$
$228$		0			0
$229$			325.812i			1.42276i	0.702807	+	0.711380i	$0.251930\pi$
$229$			325.812i			1.42276i	−0.702807	+	0.711380i	$0.748070\pi$
$230$		0			0
$231$			17.3176i			0.0749680i
$232$		0			0
$233$	11.7926			0.0506121			0.0253060	−	0.999680i	$-0.491944\pi$
$233$	11.7926			0.0506121			0.0253060	+	0.999680i	$0.491944\pi$
$234$		0			0
$235$			70.6392i			0.300592i
$236$		0			0
$237$		−	161.634i		−	0.681999i
$238$		0			0
$239$	224.073			0.937544			0.468772	−	0.883319i	$-0.344697\pi$
$239$	224.073			0.937544			0.468772	+	0.883319i	$0.344697\pi$
$240$		0			0
$241$			1.77571i			0.00736810i	0.999993	+	0.00368405i	$0.00117267\pi$
$241$			1.77571i			0.00736810i	−0.999993	+	0.00368405i	$0.998827\pi$
$242$		0			0
$243$	−133.644			−0.549976
$244$		0			0
$245$		−	304.246i		−	1.24182i
$246$		0			0
$247$	−217.598	−	109.834i	−0.880962	−	0.444672i
$248$		0			0
$249$			284.113i			1.14102i
$250$		0			0
$251$			174.747i			0.696204i	0.937457	+	0.348102i	$0.113174\pi$
$251$			174.747i			0.696204i	−0.937457	+	0.348102i	$0.886826\pi$
$252$		0			0
$253$		−	107.887i		−	0.426430i
$254$		0			0
$255$			236.099i			0.925877i
$256$		0			0
$257$		−	143.618i		−	0.558826i	−0.960171	−	0.279413i	$-0.909860\pi$
$257$		−	143.618i		−	0.558826i	0.960171	−	0.279413i	$-0.0901399\pi$
$258$		0			0
$259$	33.5510			0.129541
$260$		0			0
$261$	74.9498			0.287164
$262$		0			0
$263$	453.726			1.72519			0.862597	−	0.505892i	$-0.168836\pi$
$263$	453.726			1.72519			0.862597	+	0.505892i	$0.168836\pi$
$264$		0			0
$265$			615.050i			2.32094i
$266$		0			0
$267$			154.078i			0.577070i
$268$		0			0
$269$	323.434			1.20236			0.601178	−	0.799115i	$-0.294698\pi$
$269$	323.434			1.20236			0.601178	+	0.799115i	$0.294698\pi$
$270$		0			0
$271$	142.536			0.525961			0.262981	−	0.964801i	$-0.415294\pi$
$271$	142.536			0.525961			0.262981	+	0.964801i	$0.415294\pi$
$272$		0			0
$273$	−46.2093			−0.169265
$274$		0			0
$275$		−	81.3996i		−	0.295999i
$276$		0			0
$277$			515.794i			1.86207i	0.364926	+	0.931036i	$0.381094\pi$
$277$			515.794i			1.86207i	−0.364926	+	0.931036i	$0.618906\pi$
$278$		0			0
$279$			23.6348i			0.0847124i
$280$		0			0
$281$			218.140i			0.776300i	0.921596	+	0.388150i	$0.126886\pi$
$281$			218.140i			0.776300i	−0.921596	+	0.388150i	$0.873114\pi$
$282$		0			0
$283$			244.562i			0.864175i	0.901832	+	0.432088i	$0.142223\pi$
$283$			244.562i			0.864175i	−0.901832	+	0.432088i	$0.857777\pi$
$284$		0			0
$285$	140.673	−	278.694i	0.493589	−	0.977873i
$286$		0			0
$287$			10.1136i			0.0352390i
$288$		0			0
$289$	−82.5227			−0.285546
$290$		0			0
$291$			346.786i			1.19170i
$292$		0			0
$293$	−231.610			−0.790477			−0.395239	−	0.918578i	$-0.629338\pi$
$293$	−231.610			−0.790477			−0.395239	+	0.918578i	$0.629338\pi$
$294$		0			0
$295$			442.495i			1.49998i
$296$		0			0
$297$			141.042i			0.474890i
$298$		0			0
$299$	287.878			0.962804
$300$		0			0
$301$		−	13.5517i		−	0.0450223i
$302$		0			0
$303$			248.484i			0.820079i
$304$		0			0
$305$	−332.548			−1.09032
$306$		0			0
$307$	−100.970			−0.328893			−0.164446	−	0.986386i	$-0.552584\pi$
$307$	−100.970			−0.328893			−0.164446	+	0.986386i	$0.552584\pi$
$308$		0			0
$309$			185.046i			0.598854i
$310$		0			0
$311$	−229.402			−0.737626			−0.368813	−	0.929504i	$-0.620236\pi$
$311$	−229.402			−0.737626			−0.368813	+	0.929504i	$0.620236\pi$
$312$		0			0
$313$	397.363			1.26953			0.634765	−	0.772706i	$-0.281097\pi$
$313$	397.363			1.26953			0.634765	+	0.772706i	$0.281097\pi$
$314$		0			0
$315$			23.5464i			0.0747505i
$316$		0			0
$317$	53.7811			0.169657			0.0848283	−	0.996396i	$-0.472966\pi$
$317$	53.7811			0.169657			0.0848283	+	0.996396i	$0.472966\pi$
$318$		0			0
$319$		−	140.673i		−	0.440980i
$320$		0			0
$321$	−286.810			−0.893488
$322$		0			0
$323$	−243.728	−	123.024i	−0.754577	−	0.380878i
$324$		0			0
$325$	217.202			0.668313
$326$		0			0
$327$	204.450			0.625231
$328$		0			0
$329$	−15.4858			−0.0470692
$330$		0			0
$331$	−82.0573			−0.247907			−0.123954	−	0.992288i	$-0.539557\pi$
$331$	−82.0573			−0.247907			−0.123954	+	0.992288i	$0.539557\pi$
$332$		0			0
$333$	60.5417			0.181807
$334$		0			0
$335$			42.9751i			0.128284i
$336$		0			0
$337$		−	342.863i		−	1.01740i	−0.860945	−	0.508698i	$-0.830127\pi$
$337$		−	342.863i		−	1.01740i	0.860945	−	0.508698i	$-0.169873\pi$
$338$		0			0
$339$			248.972i			0.734429i
$340$		0			0
$341$	44.3600			0.130088
$342$		0			0
$343$	136.256			0.397249
$344$		0			0
$345$			368.708i			1.06872i
$346$		0			0
$347$		−	74.3627i		−	0.214302i	−0.994243	−	0.107151i	$-0.965827\pi$
$347$		−	74.3627i		−	0.214302i	0.994243	−	0.107151i	$-0.0341728\pi$
$348$		0			0
$349$		−	523.416i		−	1.49976i	−0.661574	−	0.749880i	$-0.730111\pi$
$349$		−	523.416i		−	1.49976i	0.661574	−	0.749880i	$-0.269889\pi$
$350$		0			0
$351$	−376.349			−1.07222
$352$		0			0
$353$	355.380			1.00674			0.503371	−	0.864071i	$-0.332093\pi$
$353$	355.380			1.00674			0.503371	+	0.864071i	$0.332093\pi$
$354$		0			0
$355$	−285.636			−0.804609
$356$		0			0
$357$	−51.7584			−0.144982
$358$		0			0
$359$	−143.878			−0.400776			−0.200388	−	0.979717i	$-0.564220\pi$
$359$	−143.878			−0.400776			−0.200388	+	0.979717i	$0.564220\pi$
$360$		0			0
$361$	214.400	+	290.437i	0.593905	+	0.804535i
$362$		0			0
$363$	−248.375			−0.684230
$364$		0			0
$365$			125.326i			0.343359i
$366$		0			0
$367$	−574.422			−1.56518			−0.782592	−	0.622535i	$-0.786103\pi$
$367$	−574.422			−1.56518			−0.782592	+	0.622535i	$0.786103\pi$
$368$		0			0
$369$			18.2496i			0.0494570i
$370$		0			0
$371$	−134.834			−0.363433
$372$		0			0
$373$	−385.555			−1.03366			−0.516829	−	0.856088i	$-0.672888\pi$
$373$	−385.555			−1.03366			−0.516829	+	0.856088i	$0.672888\pi$
$374$		0			0
$375$		−	132.582i		−	0.353552i
$376$		0			0
$377$	375.363			0.995657
$378$		0			0
$379$	−629.962			−1.66217			−0.831085	−	0.556146i	$-0.812280\pi$
$379$	−629.962			−1.66217			−0.831085	+	0.556146i	$0.812280\pi$
$380$		0			0
$381$			487.152i			1.27861i
$382$		0			0
$383$		−	445.404i		−	1.16293i	−0.813570	−	0.581467i	$-0.802479\pi$
$383$		−	445.404i		−	1.16293i	0.813570	−	0.581467i	$-0.197521\pi$
$384$		0			0
$385$	44.1941			0.114790
$386$		0			0
$387$		−	24.4536i		−	0.0631876i
$388$		0			0
$389$		−	247.408i		−	0.636011i	−0.948089	−	0.318006i	$-0.896987\pi$
$389$		−	247.408i		−	0.636011i	0.948089	−	0.318006i	$-0.103013\pi$
$390$		0			0
$391$	322.449			0.824677
$392$		0			0
$393$			409.405i			1.04174i
$394$		0			0
$395$	−412.485			−1.04427
$396$		0			0
$397$			84.3335i			0.212427i	0.994343	+	0.106213i	$0.0338727\pi$
$397$			84.3335i			0.212427i	−0.994343	+	0.106213i	$0.966127\pi$
$398$		0			0
$399$	61.0963	+	30.8388i	0.153123	+	0.0772902i
$400$		0			0
$401$		−	342.336i		−	0.853707i	−0.904321	−	0.426853i	$-0.859622\pi$
$401$		−	342.336i		−	0.853707i	0.904321	−	0.426853i	$-0.140378\pi$
$402$		0			0
$403$			118.367i			0.293716i
$404$		0			0
$405$		−	332.735i		−	0.821569i
$406$		0			0
$407$		−	113.630i		−	0.279190i
$408$		0			0
$409$			799.815i			1.95554i	0.209685	+	0.977769i	$0.432756\pi$
$409$			799.815i			1.95554i	−0.209685	+	0.977769i	$0.567244\pi$
$410$		0			0
$411$	90.1357			0.219308
$412$		0			0
$413$	−97.0052			−0.234880
$414$		0			0
$415$	725.049			1.74711
$416$		0			0
$417$		−	19.4991i		−	0.0467605i
$418$		0			0
$419$			461.110i			1.10050i	0.835000	+	0.550250i	$0.185468\pi$
$419$			461.110i			1.10050i	−0.835000	+	0.550250i	$0.814532\pi$
$420$		0			0
$421$	−763.462			−1.81345			−0.906725	−	0.421723i	$-0.861425\pi$
$421$	−763.462			−1.81345			−0.906725	+	0.421723i	$0.861425\pi$
$422$		0			0
$423$	−27.9436			−0.0660605
$424$		0			0
$425$	243.285			0.572435
$426$		0			0
$427$		−	72.9024i		−	0.170732i
$428$		0			0
$429$			156.501i			0.364805i
$430$		0			0
$431$			343.023i			0.795877i	0.917412	+	0.397939i	$0.130274\pi$
$431$			343.023i			0.795877i	−0.917412	+	0.397939i	$0.869726\pi$
$432$		0			0
$433$			608.452i			1.40520i	0.711585	+	0.702601i	$0.247978\pi$
$433$			608.452i			1.40520i	−0.711585	+	0.702601i	$0.752022\pi$
$434$		0			0
$435$			480.756i			1.10519i
$436$		0			0
$437$	−380.623	−	192.122i	−0.870990	−	0.439639i
$438$		0			0
$439$		−	762.770i		−	1.73752i	−0.495236	−	0.868758i	$-0.664919\pi$
$439$		−	762.770i		−	1.73752i	0.495236	−	0.868758i	$-0.335081\pi$
$440$		0			0
$441$	120.354			0.272912
$442$		0			0
$443$		−	333.965i		−	0.753871i	−0.926239	−	0.376936i	$-0.876978\pi$
$443$		−	333.965i		−	0.753871i	0.926239	−	0.376936i	$-0.123022\pi$
$444$		0			0
$445$	393.202			0.883601
$446$		0			0
$447$		−	599.349i		−	1.34083i
$448$		0			0
$449$		−	542.030i		−	1.20719i	−0.797290	−	0.603597i	$-0.793734\pi$
$449$		−	542.030i		−	1.20719i	0.797290	−	0.603597i	$-0.206266\pi$
$450$		0			0
$451$	34.2527			0.0759483
$452$		0			0
$453$			593.947i			1.31114i
$454$		0			0
$455$			117.925i			0.259176i
$456$		0			0
$457$	394.110			0.862385			0.431192	−	0.902260i	$-0.358093\pi$
$457$	394.110			0.862385			0.431192	+	0.902260i	$0.358093\pi$
$458$		0			0
$459$	−421.544			−0.918396
$460$		0			0
$461$			545.081i			1.18239i	0.806530	+	0.591194i	$0.201343\pi$
$461$			545.081i			1.18239i	−0.806530	+	0.591194i	$0.798657\pi$
$462$		0			0
$463$	−571.179			−1.23365			−0.616824	−	0.787101i	$-0.711581\pi$
$463$	−571.179			−1.23365			−0.616824	+	0.787101i	$0.711581\pi$
$464$		0			0
$465$	−151.602			−0.326026
$466$		0			0
$467$		−	626.194i		−	1.34089i	−0.741961	−	0.670443i	$-0.766104\pi$
$467$		−	626.194i		−	1.34089i	0.741961	−	0.670443i	$-0.233896\pi$
$468$		0			0
$469$	−9.42116			−0.0200878
$470$		0			0
$471$		−	192.237i		−	0.408146i
$472$		0			0
$473$	−45.8968			−0.0970335
$474$		0			0
$475$	−287.177	−	144.955i	−0.604582	−	0.305167i
$476$		0			0
$477$	−243.303			−0.510069
$478$		0			0
$479$	−282.428			−0.589620			−0.294810	−	0.955556i	$-0.595256\pi$
$479$	−282.428			−0.589620			−0.294810	+	0.955556i	$0.595256\pi$
$480$		0			0
$481$	303.204			0.630363
$482$		0			0
$483$	−80.8294			−0.167349
$484$		0			0
$485$	884.989			1.82472
$486$		0			0
$487$			376.771i			0.773658i	0.922151	+	0.386829i	$0.126430\pi$
$487$			376.771i			0.773658i	−0.922151	+	0.386829i	$0.873570\pi$
$488$		0			0
$489$			329.983i			0.674813i
$490$		0			0
$491$		−	778.634i		−	1.58581i	−0.609343	−	0.792907i	$-0.708567\pi$
$491$		−	778.634i		−	1.58581i	0.609343	−	0.792907i	$-0.291433\pi$
$492$		0			0
$493$	420.439			0.852817
$494$		0			0
$495$	79.7468			0.161105
$496$		0			0
$497$		−	62.6182i		−	0.125992i
$498$		0			0
$499$			445.270i			0.892324i	0.894952	+	0.446162i	$0.147210\pi$
$499$			445.270i			0.892324i	−0.894952	+	0.446162i	$0.852790\pi$
$500$		0			0
$501$		−	281.576i		−	0.562028i
$502$		0			0
$503$	−662.359			−1.31682			−0.658408	−	0.752661i	$-0.728770\pi$
$503$	−662.359			−1.31682			−0.658408	+	0.752661i	$0.728770\pi$
$504$		0			0
$505$	634.125			1.25569
$506$		0			0
$507$	11.2237			0.0221375
$508$		0			0
$509$	−825.583			−1.62197			−0.810985	−	0.585066i	$-0.801068\pi$
$509$	−825.583			−1.62197			−0.810985	+	0.585066i	$0.801068\pi$
$510$		0			0
$511$	−27.4744			−0.0537660
$512$		0			0
$513$	497.596	+	251.165i	0.969972	+	0.489601i
$514$		0			0
$515$	472.232			0.916956
$516$		0			0
$517$			52.4472i			0.101445i
$518$		0			0
$519$	296.973			0.572203
$520$		0			0
$521$		−	467.059i		−	0.896466i	−0.893917	−	0.448233i	$-0.852053\pi$
$521$		−	467.059i		−	0.896466i	0.893917	−	0.448233i	$-0.147947\pi$
$522$		0			0
$523$	−710.261			−1.35805			−0.679026	−	0.734115i	$-0.737597\pi$
$523$	−710.261			−1.35805			−0.679026	+	0.734115i	$0.737597\pi$
$524$		0			0
$525$	−60.9851			−0.116162
$526$		0			0
$527$			132.582i			0.251579i
$528$		0			0
$529$	−25.4422			−0.0480949
$530$		0			0
$531$	−175.043			−0.329647
$532$		0			0
$533$			91.3977i			0.171478i
$534$		0			0
$535$			731.931i			1.36810i
$536$		0			0
$537$	−333.464			−0.620976
$538$		0			0
$539$		−	225.892i		−	0.419095i
$540$		0			0
$541$		−	671.553i		−	1.24132i	−0.784080	−	0.620659i	$-0.786865\pi$
$541$		−	671.553i		−	1.24132i	0.784080	−	0.620659i	$-0.213135\pi$
$542$		0			0
$543$	640.771			1.18006
$544$		0			0
$545$		−	521.752i		−	0.957344i
$546$		0			0
$547$	−642.200			−1.17404			−0.587020	−	0.809572i	$-0.699699\pi$
$547$	−642.200			−1.17404			−0.587020	+	0.809572i	$0.699699\pi$
$548$		0			0
$549$		−	131.550i		−	0.239618i
$550$		0			0
$551$	−496.291	−	250.507i	−0.900710	−	0.454640i
$552$		0			0
$553$		−	90.4265i		−	0.163520i
$554$		0			0
$555$			388.337i			0.699707i
$556$		0			0
$557$		−	160.990i		−	0.289030i	−0.989503	−	0.144515i	$-0.953838\pi$
$557$		−	160.990i		−	0.289030i	0.989503	−	0.144515i	$-0.0461622\pi$
$558$		0			0
$559$		−	122.468i		−	0.219085i
$560$		0			0
$561$			175.295i			0.312469i
$562$		0			0
$563$	751.993			1.33569			0.667845	−	0.744300i	$-0.267217\pi$
$563$	751.993			1.33569			0.667845	+	0.744300i	$0.267217\pi$
$564$		0			0
$565$	635.369			1.12455
$566$		0			0
$567$	72.9434			0.128648
$568$		0			0
$569$			649.303i			1.14113i	0.821252	+	0.570565i	$0.193276\pi$
$569$			649.303i			1.14113i	−0.821252	+	0.570565i	$0.806724\pi$
$570$		0			0
$571$		−	298.371i		−	0.522541i	−0.965266	−	0.261271i	$-0.915858\pi$
$571$		−	298.371i		−	0.522541i	0.965266	−	0.261271i	$-0.0841415\pi$
$572$		0			0
$573$	−285.834			−0.498838
$574$		0			0
$575$	379.930			0.660748
$576$		0			0
$577$	233.216			0.404187			0.202094	−	0.979366i	$-0.435226\pi$
$577$	233.216			0.404187			0.202094	+	0.979366i	$0.435226\pi$
$578$		0			0
$579$			174.186i			0.300839i
$580$		0			0
$581$			158.948i			0.273576i
$582$		0			0
$583$			456.654i			0.783282i
$584$		0			0
$585$			212.792i			0.363746i
$586$		0			0
$587$			407.287i			0.693845i	0.937894	+	0.346922i	$0.112773\pi$
$587$			407.287i			0.693845i	−0.937894	+	0.346922i	$0.887227\pi$
$588$		0			0
$589$	78.9952	−	156.501i	0.134117	−	0.265707i
$590$		0			0
$591$			434.404i			0.735032i
$592$		0			0
$593$	−520.091			−0.877050			−0.438525	−	0.898719i	$-0.644499\pi$
$593$	−520.091			−0.877050			−0.438525	+	0.898719i	$0.644499\pi$
$594$		0			0
$595$			132.086i			0.221994i
$596$		0			0
$597$	544.681			0.912364
$598$		0			0
$599$			38.0428i			0.0635105i	0.999496	+	0.0317553i	$0.0101097\pi$
$599$			38.0428i			0.0635105i	−0.999496	+	0.0317553i	$0.989890\pi$
$600$		0			0
$601$		−	194.246i		−	0.323205i	−0.986856	−	0.161603i	$-0.948334\pi$
$601$		−	194.246i		−	0.323205i	0.986856	−	0.161603i	$-0.0516663\pi$
$602$		0			0
$603$	−17.0002			−0.0281927
$604$		0			0
$605$			633.848i			1.04768i
$606$		0			0
$607$			512.263i			0.843926i	0.906613	+	0.421963i	$0.138659\pi$
$607$			512.263i			0.843926i	−0.906613	+	0.421963i	$0.861341\pi$
$608$		0			0
$609$	−105.393			−0.173059
$610$		0			0
$611$	−139.947			−0.229046
$612$		0			0
$613$		−	587.471i		−	0.958354i	−0.877718	−	0.479177i	$-0.840935\pi$
$613$		−	587.471i		−	0.958354i	0.877718	−	0.479177i	$-0.159065\pi$
$614$		0			0
$615$	−117.060			−0.190342
$616$		0			0
$617$	−454.626			−0.736833			−0.368416	−	0.929661i	$-0.620100\pi$
$617$	−454.626			−0.736833			−0.368416	+	0.929661i	$0.620100\pi$
$618$		0			0
$619$			209.373i			0.338244i	0.985595	+	0.169122i	$0.0540932\pi$
$619$			209.373i			0.338244i	−0.985595	+	0.169122i	$0.945907\pi$
$620$		0			0
$621$	−658.311			−1.06008
$622$		0			0
$623$			86.1992i			0.138361i
$624$		0			0
$625$	−761.617			−1.21859
$626$		0			0
$627$	104.445	−	206.920i	0.166578	−	0.330017i
$628$		0			0
$629$	339.616			0.539929
$630$		0			0
$631$	402.818			0.638380			0.319190	−	0.947691i	$-0.396589\pi$
$631$	402.818			0.638380			0.319190	+	0.947691i	$0.396589\pi$
$632$		0			0
$633$	−17.2850			−0.0273065
$634$		0			0
$635$	1243.20			1.95779
$636$		0			0
$637$	602.757			0.946243
$638$		0			0
$639$		−	112.993i		−	0.176827i
$640$		0			0
$641$		−	1104.13i		−	1.72251i	−0.508173	−	0.861255i	$-0.669679\pi$
$641$		−	1104.13i		−	1.72251i	0.508173	−	0.861255i	$-0.330321\pi$
$642$		0			0
$643$			7.79261i			0.0121191i	0.999982	+	0.00605957i	$0.00192883\pi$
$643$			7.79261i			0.0121191i	−0.999982	+	0.00605957i	$0.998071\pi$
$644$		0			0
$645$	156.855			0.243185
$646$		0			0
$647$	985.289			1.52286			0.761429	−	0.648249i	$-0.224498\pi$
$647$	985.289			1.52286			0.761429	+	0.648249i	$0.224498\pi$
$648$		0			0
$649$			328.537i			0.506220i
$650$		0			0
$651$		−	33.2348i		−	0.0510519i
$652$		0			0
$653$		−	232.471i		−	0.356004i	−0.984030	−	0.178002i	$-0.943037\pi$
$653$		−	232.471i		−	0.356004i	0.984030	−	0.178002i	$-0.0569634\pi$
$654$		0			0
$655$	1044.79			1.59510
$656$		0			0
$657$	−49.5767			−0.0754592
$658$		0			0
$659$	688.832			1.04527			0.522634	−	0.852557i	$-0.324949\pi$
$659$	688.832			1.04527			0.522634	+	0.852557i	$0.324949\pi$
$660$		0			0
$661$	491.343			0.743333			0.371667	−	0.928366i	$-0.378786\pi$
$661$	491.343			0.743333			0.371667	+	0.928366i	$0.378786\pi$
$662$		0			0
$663$	−467.747			−0.705501
$664$		0			0
$665$	78.6998	−	155.916i	0.118346	−	0.234460i
$666$		0			0
$667$	656.586			0.984387
$668$		0			0
$669$			15.3974i			0.0230155i
$670$		0			0
$671$	−246.906			−0.367967
$672$		0			0
$673$		−	856.044i		−	1.27198i	−0.771696	−	0.635991i	$-0.780591\pi$
$673$		−	856.044i		−	1.27198i	0.771696	−	0.635991i	$-0.219409\pi$
$674$		0			0
$675$	−496.691			−0.735838
$676$		0			0
$677$	−431.688			−0.637649			−0.318825	−	0.947814i	$-0.603288\pi$
$677$	−431.688			−0.637649			−0.318825	+	0.947814i	$0.603288\pi$
$678$		0			0
$679$			194.010i			0.285730i
$680$		0			0
$681$	484.914			0.712061
$682$		0			0
$683$	−46.1780			−0.0676106			−0.0338053	−	0.999428i	$-0.510763\pi$
$683$	−46.1780			−0.0676106			−0.0338053	+	0.999428i	$0.510763\pi$
$684$		0			0
$685$		−	230.024i		−	0.335802i
$686$		0			0
$687$		−	826.719i		−	1.20338i
$688$		0			0
$689$	−1218.51			−1.76851
$690$		0			0
$691$		−	853.715i		−	1.23548i	−0.786383	−	0.617739i	$-0.788049\pi$
$691$		−	853.715i		−	1.23548i	0.786383	−	0.617739i	$-0.211951\pi$
$692$		0			0
$693$			17.4824i			0.0252271i
$694$		0			0
$695$	−49.7613			−0.0715990
$696$		0			0
$697$			102.373i			0.146877i
$698$		0			0
$699$	−29.9227			−0.0428078
$700$		0			0
$701$		−	114.570i		−	0.163439i	−0.996655	−	0.0817193i	$-0.973959\pi$
$701$		−	114.570i		−	0.163439i	0.996655	−	0.0817193i	$-0.0260411\pi$
$702$		0			0
$703$	−400.886	−	202.350i	−0.570251	−	0.287838i
$704$		0			0
$705$		−	179.241i		−	0.254242i
$706$		0			0
$707$			139.015i			0.196627i
$708$		0			0
$709$		−	300.162i		−	0.423360i	−0.977339	−	0.211680i	$-0.932107\pi$
$709$		−	300.162i		−	0.423360i	0.977339	−	0.211680i	$-0.0678934\pi$
$710$		0			0
$711$		−	163.172i		−	0.229496i
$712$		0			0
$713$			207.049i			0.290391i
$714$		0			0
$715$	399.387			0.558584
$716$		0			0
$717$	−568.565			−0.792977
$718$		0			0
$719$	−489.773			−0.681186			−0.340593	−	0.940211i	$-0.610628\pi$
$719$	−489.773			−0.681186			−0.340593	+	0.940211i	$0.610628\pi$
$720$		0			0
$721$			103.524i			0.143584i
$722$		0			0
$723$		−	4.50571i		−	0.00623197i
$724$		0			0
$725$	495.389			0.683295
$726$		0			0
$727$	−113.740			−0.156451			−0.0782254	−	0.996936i	$-0.524925\pi$
$727$	−113.740			−0.156451			−0.0782254	+	0.996936i	$0.524925\pi$
$728$		0			0
$729$	801.570			1.09955
$730$		0			0
$731$		−	137.175i		−	0.187654i
$732$		0			0
$733$			845.341i			1.15326i	0.817005	+	0.576631i	$0.195633\pi$
$733$			845.341i			1.15326i	−0.817005	+	0.576631i	$0.804367\pi$
$734$		0			0
$735$			771.996i			1.05034i
$736$		0			0
$737$			31.9075i			0.0432938i
$738$		0			0
$739$			1162.20i			1.57266i	0.617805	+	0.786331i	$0.288022\pi$
$739$			1162.20i			1.57266i	−0.617805	+	0.786331i	$0.711978\pi$
$740$		0			0
$741$	552.134	+	278.694i	0.745120	+	0.376105i
$742$		0			0
$743$		−	834.312i		−	1.12290i	−0.827512	−	0.561448i	$-0.810244\pi$
$743$		−	834.312i		−	1.12290i	0.827512	−	0.561448i	$-0.189756\pi$
$744$		0			0
$745$	−1529.52			−2.05305
$746$		0			0
$747$			286.816i			0.383958i
$748$		0			0
$749$	−160.457			−0.214228
$750$		0			0
$751$			46.3827i			0.0617612i	0.999523	+	0.0308806i	$0.00983117\pi$
$751$			46.3827i			0.0617612i	−0.999523	+	0.0308806i	$0.990169\pi$
$752$		0			0
$753$		−	443.405i		−	0.588851i
$754$		0			0
$755$	1515.74			2.00760
$756$		0			0
$757$			36.3077i			0.0479626i	0.999712	+	0.0239813i	$0.00763422\pi$
$757$			36.3077i			0.0479626i	−0.999712	+	0.0239813i	$0.992366\pi$
$758$		0			0
$759$			273.753i			0.360675i
$760$		0			0
$761$	−415.212			−0.545614			−0.272807	−	0.962069i	$-0.587952\pi$
$761$	−415.212			−0.545614			−0.272807	+	0.962069i	$0.587952\pi$
$762$		0			0
$763$	114.380			0.149909
$764$		0			0
$765$			238.345i			0.311562i
$766$		0			0
$767$	−876.648			−1.14296
$768$		0			0
$769$	−153.051			−0.199026			−0.0995130	−	0.995036i	$-0.531729\pi$
$769$	−153.051			−0.199026			−0.0995130	+	0.995036i	$0.531729\pi$
$770$		0			0
$771$			364.419i			0.472657i
$772$		0			0
$773$	187.001			0.241916			0.120958	−	0.992658i	$-0.461403\pi$
$773$	187.001			0.241916			0.120958	+	0.992658i	$0.461403\pi$
$774$		0			0
$775$			156.217i			0.201570i
$776$		0			0
$777$	−85.1327			−0.109566
$778$		0			0
$779$	60.9963	−	120.843i	0.0783008	−	0.155126i
$780$		0			0
$781$	−212.075			−0.271543
$782$		0			0
$783$	−858.368			−1.09626
$784$		0			0
$785$	−490.583			−0.624947
$786$		0			0
$787$	942.496			1.19758			0.598790	−	0.800906i	$-0.295648\pi$
$787$	942.496			1.19758			0.598790	+	0.800906i	$0.295648\pi$
$788$		0			0
$789$	−1151.29			−1.45917
$790$		0			0
$791$			139.288i			0.176091i
$792$		0			0
$793$		−	658.828i		−	0.830804i
$794$		0			0
$795$		−	1560.63i		−	1.96306i
$796$		0			0
$797$	−1285.59			−1.61304			−0.806518	−	0.591210i	$-0.798651\pi$
$797$	−1285.59			−1.61304			−0.806518	+	0.591210i	$0.798651\pi$
$798$		0			0
$799$	−156.753			−0.196186
$800$		0			0
$801$			155.544i			0.194187i
$802$		0			0
$803$			93.0502i			0.115878i
$804$		0			0
$805$			206.275i			0.256242i
$806$		0			0
$807$	−820.684			−1.01696
$808$		0			0
$809$	−690.576			−0.853616			−0.426808	−	0.904342i	$-0.640362\pi$
$809$	−690.576			−0.853616			−0.426808	+	0.904342i	$0.640362\pi$
$810$		0			0
$811$	−30.7829			−0.0379567			−0.0189783	−	0.999820i	$-0.506041\pi$
$811$	−30.7829			−0.0379567			−0.0189783	+	0.999820i	$0.506041\pi$
$812$		0			0
$813$	−361.671			−0.444860
$814$		0			0
$815$	842.109			1.03326
$816$		0			0
$817$	−81.7320	+	161.923i	−0.100039	+	0.198193i
$818$		0			0
$819$	−46.6489			−0.0569584
$820$		0			0
$821$			240.639i			0.293104i	0.989203	+	0.146552i	$0.0468176\pi$
$821$			240.639i			0.293104i	−0.989203	+	0.146552i	$0.953182\pi$
$822$		0			0
$823$	259.890			0.315784			0.157892	−	0.987456i	$-0.449530\pi$
$823$	259.890			0.315784			0.157892	+	0.987456i	$0.449530\pi$
$824$		0			0
$825$			206.544i			0.250357i
$826$		0			0
$827$	1062.36			1.28460			0.642299	−	0.766454i	$-0.277981\pi$
$827$	1062.36			1.28460			0.642299	+	0.766454i	$0.277981\pi$
$828$		0			0
$829$	275.721			0.332594			0.166297	−	0.986076i	$-0.446819\pi$
$829$	275.721			0.332594			0.166297	+	0.986076i	$0.446819\pi$
$830$		0			0
$831$		−	1308.78i		−	1.57495i
$832$		0			0
$833$	675.140			0.810492
$834$		0			0
$835$	−718.575			−0.860568
$836$		0			0
$837$		−	270.679i		−	0.323392i
$838$		0			0
$839$		−	411.407i		−	0.490354i	−0.969478	−	0.245177i	$-0.921154\pi$
$839$		−	411.407i		−	0.490354i	0.969478	−	0.245177i	$-0.0788460\pi$
$840$		0			0
$841$	15.1183			0.0179766
$842$		0			0
$843$		−	553.511i		−	0.656597i
$844$		0			0
$845$		−	28.6426i		−	0.0338966i
$846$		0			0
$847$	−138.954			−0.164055
$848$		0			0
$849$		−	620.553i		−	0.730922i
$850$		0			0
$851$	530.367			0.623227
$852$		0			0
$853$			386.887i			0.453560i	0.973946	+	0.226780i	$0.0728199\pi$
$853$			386.887i			0.453560i	−0.973946	+	0.226780i	$0.927180\pi$
$854$		0			0
$855$	142.011	−	281.346i	0.166095	−	0.329059i
$856$		0			0
$857$		−	1524.48i		−	1.77886i	−0.457071	−	0.889430i	$-0.651102\pi$
$857$		−	1524.48i		−	1.77886i	0.457071	−	0.889430i	$-0.348898\pi$
$858$		0			0
$859$			1392.04i			1.62054i	0.586058	+	0.810269i	$0.300679\pi$
$859$			1392.04i			1.62054i	−0.586058	+	0.810269i	$0.699321\pi$
$860$		0			0
$861$		−	25.6623i		−	0.0298053i
$862$		0			0
$863$			142.446i			0.165060i	0.996589	+	0.0825298i	$0.0263000\pi$
$863$			142.446i			0.165060i	−0.996589	+	0.0825298i	$0.973700\pi$
$864$		0			0
$865$		−	757.868i		−	0.876148i
$866$		0			0
$867$	209.394			0.241516
$868$		0			0
$869$	−306.256			−0.352423
$870$		0			0
$871$	−85.1401			−0.0977498
$872$		0			0
$873$			350.086i			0.401014i
$874$		0			0
$875$		−	74.1733i		−	0.0847695i
$876$		0			0
$877$	937.883			1.06942			0.534711	−	0.845035i	$-0.320420\pi$
$877$	937.883			1.06942			0.534711	+	0.845035i	$0.320420\pi$
$878$		0			0
$879$	587.689			0.668588
$880$		0			0
$881$	109.435			0.124216			0.0621082	−	0.998069i	$-0.480218\pi$
$881$	109.435			0.124216			0.0621082	+	0.998069i	$0.480218\pi$
$882$		0			0
$883$		−	1486.09i		−	1.68301i	−0.540253	−	0.841503i	$-0.681671\pi$
$883$		−	1486.09i		−	1.68301i	0.540253	−	0.841503i	$-0.318329\pi$
$884$		0			0
$885$		−	1122.79i		−	1.26869i
$886$		0			0
$887$		−	1714.62i		−	1.93305i	−0.256568	−	0.966526i	$-0.582592\pi$
$887$		−	1714.62i		−	1.93305i	0.256568	−	0.966526i	$-0.417408\pi$
$888$		0			0
$889$			272.538i			0.306567i
$890$		0			0
$891$		−	247.044i		−	0.277266i
$892$		0			0
$893$	185.033	+	93.3967i	0.207204	+	0.104588i
$894$		0			0
$895$			850.992i			0.950829i
$896$		0			0
$897$	−730.465			−0.814342
$898$		0			0
$899$			269.970i			0.300300i
$900$		0			0
$901$	−1364.83			−1.51480
$902$		0			0
$903$			34.3862i			0.0380800i
$904$		0			0
$905$		−	1635.23i		−	1.80688i
$906$		0			0
$907$	1409.71			1.55425			0.777127	−	0.629344i	$-0.216676\pi$
$907$	1409.71			1.55425			0.777127	+	0.629344i	$0.216676\pi$
$908$		0			0
$909$			250.848i			0.275961i
$910$		0			0
$911$		−	812.560i		−	0.891943i	−0.895047	−	0.445971i	$-0.852858\pi$
$911$		−	812.560i		−	0.891943i	0.895047	−	0.445971i	$-0.147142\pi$
$912$		0			0
$913$	538.324			0.589621
$914$		0			0
$915$	843.811			0.922198
$916$		0			0
$917$			229.043i			0.249774i
$918$		0			0
$919$	1772.39			1.92861			0.964306	−	0.264792i	$-0.0853032\pi$
$919$	1772.39			1.92861			0.964306	+	0.264792i	$0.0853032\pi$
$920$		0			0
$921$	256.203			0.278179
$922$		0			0
$923$		−	565.888i		−	0.613097i
$924$		0			0
$925$	400.157			0.432602
$926$		0			0
$927$			186.806i			0.201517i
$928$		0			0
$929$	−936.486			−1.00806			−0.504029	−	0.863687i	$-0.668150\pi$
$929$	−936.486			−1.00806			−0.504029	+	0.863687i	$0.668150\pi$
$930$		0			0
$931$	−796.944	−	402.263i	−0.856008	−	0.432077i
$932$		0			0
$933$	582.086			0.623886
$934$		0			0
$935$	447.349			0.478448
$936$		0			0
$937$	−1234.01			−1.31698			−0.658488	−	0.752592i	$-0.728803\pi$
$937$	−1234.01			−1.31698			−0.658488	+	0.752592i	$0.728803\pi$
$938$		0			0
$939$	−1008.27			−1.07377
$940$		0			0
$941$	−327.036			−0.347541			−0.173770	−	0.984786i	$-0.555595\pi$
$941$	−327.036			−0.347541			−0.173770	+	0.984786i	$0.555595\pi$
$942$		0			0
$943$			159.873i			0.169537i
$944$		0			0
$945$		−	269.667i		−	0.285362i
$946$		0			0
$947$		−	1750.53i		−	1.84850i	−0.381792	−	0.924248i	$-0.624693\pi$
$947$		−	1750.53i		−	1.84850i	0.381792	−	0.924248i	$-0.375307\pi$
$948$		0			0
$949$	−248.290			−0.261633
$950$		0			0
$951$	−136.465			−0.143496
$952$		0			0
$953$		−	1438.62i		−	1.50957i	−0.655974	−	0.754784i	$-0.727742\pi$
$953$		−	1438.62i		−	1.50957i	0.655974	−	0.754784i	$-0.272258\pi$
$954$		0			0
$955$			729.442i			0.763814i
$956$		0			0
$957$			356.944i			0.372983i
$958$		0			0
$959$	50.4267			0.0525826
$960$		0			0
$961$	875.867			0.911412
$962$		0			0
$963$	−289.539			−0.300663
$964$		0			0
$965$	444.517			0.460640
$966$		0			0
$967$	−1277.65			−1.32126			−0.660628	−	0.750714i	$-0.729710\pi$
$967$	−1277.65			−1.32126			−0.660628	+	0.750714i	$0.729710\pi$
$968$		0			0
$969$	618.438	+	312.161i	0.638223	+	0.322148i
$970$		0			0
$971$	588.097			0.605662			0.302831	−	0.953044i	$-0.402068\pi$
$971$	588.097			0.605662			0.302831	+	0.953044i	$0.402068\pi$
$972$		0			0
$973$		−	10.9088i		−	0.0112116i
$974$		0			0
$975$	−551.130			−0.565261
$976$		0			0
$977$		−	1234.30i		−	1.26336i	−0.775231	−	0.631678i	$-0.782367\pi$
$977$		−	1234.30i		−	1.26336i	0.775231	−	0.631678i	$-0.217633\pi$
$978$		0			0
$979$	291.939			0.298201
$980$		0			0
$981$	206.396			0.210393
$982$		0			0
$983$			1642.58i			1.67098i	0.549502	+	0.835492i	$0.314817\pi$
$983$			1642.58i			1.67098i	−0.549502	+	0.835492i	$0.685183\pi$
$984$		0			0
$985$	1108.59			1.12547
$986$		0			0
$987$	39.2938			0.0398113
$988$		0			0
$989$		−	214.222i		−	0.216605i
$990$		0			0
$991$			1636.51i			1.65137i	0.564130	+	0.825686i	$0.309212\pi$
$991$			1636.51i			1.65137i	−0.564130	+	0.825686i	$0.690788\pi$
$992$		0			0
$993$	208.213			0.209681
$994$		0			0
$995$		−	1390.01i		−	1.39700i
$996$		0			0
$997$			457.307i			0.458683i	0.973346	+	0.229342i	$0.0736573\pi$
$997$			457.307i			0.458683i	−0.973346	+	0.229342i	$0.926343\pi$
$998$		0			0
$999$	−693.359			−0.694053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.d.417.7	yes	16
4.3	odd	2	inner	1216.3.g.d.417.12	yes	16
8.3	odd	2	inner	1216.3.g.d.417.6	yes	16
8.5	even	2	inner	1216.3.g.d.417.9	yes	16
19.18	odd	2	inner	1216.3.g.d.417.11	yes	16
76.75	even	2	inner	1216.3.g.d.417.8	yes	16
152.37	odd	2	inner	1216.3.g.d.417.5	✓	16
152.75	even	2	inner	1216.3.g.d.417.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.d.417.5	✓	16	152.37	odd	2	inner
1216.3.g.d.417.6	yes	16	8.3	odd	2	inner
1216.3.g.d.417.7	yes	16	1.1	even	1	trivial
1216.3.g.d.417.8	yes	16	76.75	even	2	inner
1216.3.g.d.417.9	yes	16	8.5	even	2	inner
1216.3.g.d.417.10	yes	16	152.75	even	2	inner
1216.3.g.d.417.11	yes	16	19.18	odd	2	inner
1216.3.g.d.417.12	yes	16	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.53741 q^{3} +6.47540i q^{5} -1.41956 q^{7} -2.56155 q^{9} +O(q^{10})\)	\(q-2.53741 q^{3} +6.47540i q^{5} -1.41956 q^{7} -2.56155 q^{9} +4.80776i q^{11} -12.8287 q^{13} -16.4308i q^{15} -14.3693 q^{17} +(16.9617 + 8.56155i) q^{19} +3.60201 q^{21} -22.4401 q^{23} -16.9309 q^{25} +29.3364 q^{27} -29.2595 q^{29} -9.22674i q^{31} -12.1993i q^{33} -9.19224i q^{35} -23.6348 q^{37} +32.5518 q^{39} -7.12445i q^{41} +9.54640i q^{43} -16.5871i q^{45} +10.9088 q^{47} -46.9848 q^{49} +36.4608 q^{51} +94.9825 q^{53} -31.1322 q^{55} +(-43.0388 - 21.7242i) q^{57} +68.3346 q^{59} +51.3556i q^{61} +3.63628 q^{63} -83.0713i q^{65} +6.63667 q^{67} +56.9397 q^{69} +44.1110i q^{71} +19.3542 q^{73} +42.9606 q^{75} -6.82492i q^{77} +63.7003i q^{79} -51.3845 q^{81} -111.970i q^{83} -93.0471i q^{85} +74.2433 q^{87} -60.7224i q^{89} +18.2112 q^{91} +23.4120i q^{93} +(-55.4395 + 109.834i) q^{95} -136.669i q^{97} -12.3153i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} - 32 q^{17} - 40 q^{25} - 224 q^{49} + 136 q^{57} - 416 q^{73} - 1152 q^{81}+O(q^{100}) \) 16 * q - 8 * q^9 - 32 * q^17 - 40 * q^25 - 224 * q^49 + 136 * q^57 - 416 * q^73 - 1152 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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