LMFDB - Embedded newform 1216.3.d.d.191.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(191,1216)]

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([1, 0, 0]))

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

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

chi := DirichletCharacter("1216.191");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.d (of order $2$, degree $1$, 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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + \cdots + 16384 $ x^14 - 2x^13 + x^12 + 14x^11 - 42x^10 + 28x^9 + 132x^8 - 440x^7 + 528x^6 + 448x^5 - 2688x^4 + 3584x^3 + 1024x^2 - 8192x + 16384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.5
Root			$1.57398 + 1.23393i$ of defining polynomial
Character	$\chi$	$=$	1216.191
Dual form			1216.3.d.d.191.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.90118i q^{3} -3.66290 q^{5} +1.93414i q^{7} +0.583162 q^{9} +O(q^{10})$	$q-2.90118i q^{3} -3.66290 q^{5} +1.93414i q^{7} +0.583162 q^{9} -0.752428i q^{11} +13.0118 q^{13} +10.6267i q^{15} -23.9304 q^{17} +4.35890i q^{19} +5.61130 q^{21} -6.26712i q^{23} -11.5832 q^{25} -27.8025i q^{27} -33.1165 q^{29} +17.5328i q^{31} -2.18293 q^{33} -7.08457i q^{35} -41.5073 q^{37} -37.7495i q^{39} -5.51661 q^{41} +84.5402i q^{43} -2.13607 q^{45} +18.3582i q^{47} +45.2591 q^{49} +69.4264i q^{51} +41.2010 q^{53} +2.75607i q^{55} +12.6459 q^{57} -69.5279i q^{59} -87.6461 q^{61} +1.12792i q^{63} -47.6609 q^{65} +105.121i q^{67} -18.1820 q^{69} +74.9540i q^{71} +48.7353 q^{73} +33.6048i q^{75} +1.45530 q^{77} +95.9473i q^{79} -75.4115 q^{81} +65.8722i q^{83} +87.6548 q^{85} +96.0768i q^{87} +3.38934 q^{89} +25.1667i q^{91} +50.8657 q^{93} -15.9662i q^{95} -23.5177 q^{97} -0.438788i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 68 q^{9}+O(q^{10}) $ 14 * q - 68 * q^9	$ 14 q - 68 q^{9} - 54 q^{13} + 34 q^{17} + 38 q^{21} - 86 q^{25} - 54 q^{29} + 20 q^{33} - 100 q^{37} + 224 q^{41} + 168 q^{45} - 220 q^{49} - 14 q^{53} + 38 q^{57} - 28 q^{61} - 472 q^{65} - 122 q^{69} + 70 q^{73} - 228 q^{77} + 334 q^{81} - 48 q^{85} + 176 q^{93} + 308 q^{97}+O(q^{100}) $ 14 * q - 68 * q^9 - 54 * q^13 + 34 * q^17 + 38 * q^21 - 86 * q^25 - 54 * q^29 + 20 * q^33 - 100 * q^37 + 224 * q^41 + 168 * q^45 - 220 * q^49 - 14 * q^53 + 38 * q^57 - 28 * q^61 - 472 * q^65 - 122 * q^69 + 70 * q^73 - 228 * q^77 + 334 * q^81 - 48 * q^85 + 176 * q^93 + 308 * q^97

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
$3$	− 2.90118i	− 0.967060i	−0.875328	−	0.483530i	$-0.839354\pi$
$3$	− 2.90118i	− 0.967060i	0.875328	−	0.483530i	$-0.160646\pi$
$4$	0	0
$5$	−3.66290	−0.732580	−0.366290	−	0.930501i	$-0.619372\pi$
$5$	−3.66290	−0.732580	−0.366290	+	0.930501i	$0.619372\pi$
$6$	0	0
$7$	1.93414i	0.276306i	0.990411	+	0.138153i	$0.0441166\pi$
$7$	1.93414i	0.276306i	−0.990411	+	0.138153i	$0.955883\pi$
$8$	0	0
$9$	0.583162	0.0647958
$10$	0	0
$11$	− 0.752428i	− 0.0684025i	−0.999415	−	0.0342013i	$-0.989111\pi$
$11$	− 0.752428i	− 0.0684025i	0.999415	−	0.0342013i	$-0.0108887\pi$
$12$	0	0
$13$	13.0118	1.00091	0.500453	−	0.865763i	$-0.333167\pi$
$13$	13.0118	1.00091	0.500453	+	0.865763i	$0.333167\pi$
$14$	0	0
$15$	10.6267i	0.708449i
$16$	0	0
$17$	−23.9304	−1.40767	−0.703836	−	0.710363i	$-0.748531\pi$
$17$	−23.9304	−1.40767	−0.703836	+	0.710363i	$0.748531\pi$
$18$	0	0
$19$	4.35890i	0.229416i
$19$	4.35890i	0.229416i
$20$	0	0
$21$	5.61130	0.267205
$22$	0	0
$23$	− 6.26712i	− 0.272483i	−0.990676	−	0.136242i	$-0.956498\pi$
$23$	− 6.26712i	− 0.272483i	0.990676	−	0.136242i	$-0.0435024\pi$
$24$	0	0
$25$	−11.5832	−0.463326
$26$	0	0
$27$	− 27.8025i	− 1.02972i
$28$	0	0
$29$	−33.1165	−1.14195	−0.570974	−	0.820968i	$-0.693434\pi$
$29$	−33.1165	−1.14195	−0.570974	+	0.820968i	$0.693434\pi$
$30$	0	0
$31$	17.5328i	0.565574i	0.959183	+	0.282787i	$0.0912589\pi$
$31$	17.5328i	0.565574i	−0.959183	+	0.282787i	$0.908741\pi$
$32$	0	0
$33$	−2.18293	−0.0661493
$34$	0	0
$35$	− 7.08457i	− 0.202416i
$36$	0	0
$37$	−41.5073	−1.12182	−0.560910	−	0.827877i	$-0.689549\pi$
$37$	−41.5073	−1.12182	−0.560910	+	0.827877i	$0.689549\pi$
$38$	0	0
$39$	− 37.7495i	− 0.967937i
$40$	0	0
$41$	−5.51661	−0.134551	−0.0672757	−	0.997734i	$-0.521431\pi$
$41$	−5.51661	−0.134551	−0.0672757	+	0.997734i	$0.521431\pi$
$42$	0	0
$43$	84.5402i	1.96605i	0.183468	+	0.983026i	$0.441267\pi$
$43$	84.5402i	1.96605i	−0.183468	+	0.983026i	$0.558733\pi$
$44$	0	0
$45$	−2.13607	−0.0474681
$46$	0	0
$47$	18.3582i	0.390599i	0.980744	+	0.195300i	$0.0625679\pi$
$47$	18.3582i	0.390599i	−0.980744	+	0.195300i	$0.937432\pi$
$48$	0	0
$49$	45.2591	0.923655
$50$	0	0
$51$	69.4264i	1.36130i
$52$	0	0
$53$	41.2010	0.777377	0.388688	−	0.921369i	$-0.372928\pi$
$53$	41.2010	0.777377	0.388688	+	0.921369i	$0.372928\pi$
$54$	0	0
$55$	2.75607i	0.0501103i
$56$	0	0
$57$	12.6459	0.221859
$58$	0	0
$59$	− 69.5279i	− 1.17844i	−0.807973	−	0.589219i	$-0.799435\pi$
$59$	− 69.5279i	− 1.17844i	0.807973	−	0.589219i	$-0.200565\pi$
$60$	0	0
$61$	−87.6461	−1.43682	−0.718411	−	0.695619i	$-0.755130\pi$
$61$	−87.6461	−1.43682	−0.718411	+	0.695619i	$0.755130\pi$
$62$	0	0
$63$	1.12792i	0.0179035i
$64$	0	0
$65$	−47.6609	−0.733244
$66$	0	0
$67$	105.121i	1.56897i	0.620145	+	0.784487i	$0.287074\pi$
$67$	105.121i	1.56897i	−0.620145	+	0.784487i	$0.712926\pi$
$68$	0	0
$69$	−18.1820	−0.263508
$70$	0	0
$71$	74.9540i	1.05569i	0.849341	+	0.527845i	$0.177000\pi$
$71$	74.9540i	1.05569i	−0.849341	+	0.527845i	$0.823000\pi$
$72$	0	0
$73$	48.7353	0.667607	0.333803	−	0.942643i	$-0.391668\pi$
$73$	48.7353	0.667607	0.333803	+	0.942643i	$0.391668\pi$
$74$	0	0
$75$	33.6048i	0.448064i
$76$	0	0
$77$	1.45530	0.0189000
$78$	0	0
$79$	95.9473i	1.21452i	0.794502	+	0.607261i	$0.207732\pi$
$79$	95.9473i	1.21452i	−0.794502	+	0.607261i	$0.792268\pi$
$80$	0	0
$81$	−75.4115	−0.931006
$82$	0	0
$83$	65.8722i	0.793641i	0.917896	+	0.396821i	$0.129886\pi$
$83$	65.8722i	0.793641i	−0.917896	+	0.396821i	$0.870114\pi$
$84$	0	0
$85$	87.6548	1.03123
$86$	0	0
$87$	96.0768i	1.10433i
$88$	0	0
$89$	3.38934	0.0380825	0.0190412	−	0.999819i	$-0.493939\pi$
$89$	3.38934	0.0380825	0.0190412	+	0.999819i	$0.493939\pi$
$90$	0	0
$91$	25.1667i	0.276557i
$92$	0	0
$93$	50.8657	0.546943
$94$	0	0
$95$	− 15.9662i	− 0.168065i
$96$	0	0
$97$	−23.5177	−0.242451	−0.121225	−	0.992625i	$-0.538682\pi$
$97$	−23.5177	−0.242451	−0.121225	+	0.992625i	$0.538682\pi$
$98$	0	0
$99$	− 0.438788i	− 0.00443220i
$100$	0	0
$101$	153.325	1.51807	0.759035	−	0.651050i	$-0.225671\pi$
$101$	153.325	1.51807	0.759035	+	0.651050i	$0.225671\pi$
$102$	0	0
$103$	114.566i	1.11229i	0.831084	+	0.556147i	$0.187721\pi$
$103$	114.566i	1.11229i	−0.831084	+	0.556147i	$0.812279\pi$
$104$	0	0
$105$	−20.5536	−0.195749
$106$	0	0
$107$	− 161.991i	− 1.51393i	−0.653454	−	0.756966i	$-0.726681\pi$
$107$	− 161.991i	− 1.51393i	0.653454	−	0.756966i	$-0.273319\pi$
$108$	0	0
$109$	−120.924	−1.10939	−0.554696	−	0.832053i	$-0.687166\pi$
$109$	−120.924	−1.10939	−0.554696	+	0.832053i	$0.687166\pi$
$110$	0	0
$111$	120.420i	1.08487i
$112$	0	0
$113$	−91.1817	−0.806918	−0.403459	−	0.914998i	$-0.632192\pi$
$113$	−91.1817	−0.806918	−0.403459	+	0.914998i	$0.632192\pi$
$114$	0	0
$115$	22.9558i	0.199616i
$116$	0	0
$117$	7.58799	0.0648546
$118$	0	0
$119$	− 46.2849i	− 0.388948i
$120$	0	0
$121$	120.434	0.995321
$122$	0	0
$123$	16.0047i	0.130119i
$124$	0	0
$125$	134.000	1.07200
$126$	0	0
$127$	− 191.974i	− 1.51160i	−0.654800	−	0.755802i	$-0.727247\pi$
$127$	− 191.974i	− 1.51160i	0.654800	−	0.755802i	$-0.272753\pi$
$128$	0	0
$129$	245.266	1.90129
$130$	0	0
$131$	92.3292i	0.704803i	0.935849	+	0.352401i	$0.114635\pi$
$131$	92.3292i	0.704803i	−0.935849	+	0.352401i	$0.885365\pi$
$132$	0	0
$133$	−8.43073	−0.0633890
$134$	0	0
$135$	101.838i	0.754353i
$136$	0	0
$137$	−2.46732	−0.0180097	−0.00900483	−	0.999959i	$-0.502866\pi$
$137$	−2.46732	−0.0180097	−0.00900483	+	0.999959i	$0.502866\pi$
$138$	0	0
$139$	245.321i	1.76490i	0.470409	+	0.882448i	$0.344106\pi$
$139$	245.321i	1.76490i	−0.470409	+	0.882448i	$0.655894\pi$
$140$	0	0
$141$	53.2603	0.377733
$142$	0	0
$143$	− 9.79043i	− 0.0684646i
$144$	0	0
$145$	121.302	0.836568
$146$	0	0
$147$	− 131.305i	− 0.893229i
$148$	0	0
$149$	−52.3645	−0.351439	−0.175720	−	0.984440i	$-0.556225\pi$
$149$	−52.3645	−0.351439	−0.175720	+	0.984440i	$0.556225\pi$
$150$	0	0
$151$	25.3459i	0.167853i	0.996472	+	0.0839267i	$0.0267461\pi$
$151$	25.3459i	0.167853i	−0.996472	+	0.0839267i	$0.973254\pi$
$152$	0	0
$153$	−13.9553	−0.0912113
$154$	0	0
$155$	− 64.2208i	− 0.414328i
$156$	0	0
$157$	54.8571	0.349408	0.174704	−	0.984621i	$-0.444103\pi$
$157$	54.8571	0.349408	0.174704	+	0.984621i	$0.444103\pi$
$158$	0	0
$159$	− 119.531i	− 0.751770i
$160$	0	0
$161$	12.1215	0.0752888
$162$	0	0
$163$	− 26.8048i	− 0.164446i	−0.996614	−	0.0822232i	$-0.973798\pi$
$163$	− 26.8048i	− 0.164446i	0.996614	−	0.0822232i	$-0.0262020\pi$
$164$	0	0
$165$	7.99585	0.0484597
$166$	0	0
$167$	62.7282i	0.375618i	0.982206	+	0.187809i	$0.0601386\pi$
$167$	62.7282i	0.375618i	−0.982206	+	0.187809i	$0.939861\pi$
$168$	0	0
$169$	0.306683	0.00181469
$170$	0	0
$171$	2.54195i	0.0148652i
$172$	0	0
$173$	−328.974	−1.90159	−0.950793	−	0.309827i	$-0.899729\pi$
$173$	−328.974	−1.90159	−0.950793	+	0.309827i	$0.899729\pi$
$174$	0	0
$175$	− 22.4035i	− 0.128020i
$176$	0	0
$177$	−201.713	−1.13962
$178$	0	0
$179$	335.845i	1.87623i	0.346327	+	0.938114i	$0.387429\pi$
$179$	335.845i	1.87623i	−0.346327	+	0.938114i	$0.612571\pi$
$180$	0	0
$181$	58.9250	0.325552	0.162776	−	0.986663i	$-0.447955\pi$
$181$	58.9250	0.325552	0.162776	+	0.986663i	$0.447955\pi$
$182$	0	0
$183$	254.277i	1.38949i
$184$	0	0
$185$	152.037	0.821823
$186$	0	0
$187$	18.0059i	0.0962883i
$188$	0	0
$189$	53.7740	0.284518
$190$	0	0
$191$	98.6857i	0.516679i	0.966054	+	0.258339i	$0.0831753\pi$
$191$	98.6857i	0.516679i	−0.966054	+	0.258339i	$0.916825\pi$
$192$	0	0
$193$	−278.391	−1.44244	−0.721220	−	0.692706i	$-0.756418\pi$
$193$	−278.391	−1.44244	−0.721220	+	0.692706i	$0.756418\pi$
$194$	0	0
$195$	138.273i	0.709091i
$196$	0	0
$197$	47.3378	0.240293	0.120147	−	0.992756i	$-0.461664\pi$
$197$	47.3378	0.240293	0.120147	+	0.992756i	$0.461664\pi$
$198$	0	0
$199$	− 251.355i	− 1.26309i	−0.775340	−	0.631545i	$-0.782421\pi$
$199$	− 251.355i	− 1.26309i	0.775340	−	0.631545i	$-0.217579\pi$
$200$	0	0
$201$	304.975	1.51729
$202$	0	0
$203$	− 64.0520i	− 0.315527i
$204$	0	0
$205$	20.2068	0.0985697
$206$	0	0
$207$	− 3.65475i	− 0.0176558i
$208$	0	0
$209$	3.27976	0.0156926
$210$	0	0
$211$	38.9936i	0.184804i	0.995722	+	0.0924019i	$0.0294544\pi$
$211$	38.9936i	0.184804i	−0.995722	+	0.0924019i	$0.970546\pi$
$212$	0	0
$213$	217.455	1.02092
$214$	0	0
$215$	− 309.662i	− 1.44029i
$216$	0	0
$217$	−33.9109	−0.156272
$218$	0	0
$219$	− 141.390i	− 0.645615i
$220$	0	0
$221$	−311.378	−1.40895
$222$	0	0
$223$	189.466i	0.849622i	0.905282	+	0.424811i	$0.139660\pi$
$223$	189.466i	0.849622i	−0.905282	+	0.424811i	$0.860340\pi$
$224$	0	0
$225$	−6.75486	−0.0300216
$226$	0	0
$227$	− 167.625i	− 0.738438i	−0.929342	−	0.369219i	$-0.879625\pi$
$227$	− 167.625i	− 0.738438i	0.929342	−	0.369219i	$-0.120375\pi$
$228$	0	0
$229$	−203.355	−0.888013	−0.444006	−	0.896024i	$-0.646443\pi$
$229$	−203.355	−0.888013	−0.444006	+	0.896024i	$0.646443\pi$
$230$	0	0
$231$	− 4.22209i	− 0.0182775i
$232$	0	0
$233$	−200.769	−0.861671	−0.430836	−	0.902430i	$-0.641781\pi$
$233$	−200.769	−0.861671	−0.430836	+	0.902430i	$0.641781\pi$
$234$	0	0
$235$	− 67.2441i	− 0.286145i
$236$	0	0
$237$	278.360	1.17452
$238$	0	0
$239$	− 165.710i	− 0.693349i	−0.937986	−	0.346674i	$-0.887311\pi$
$239$	− 165.710i	− 0.693349i	0.937986	−	0.346674i	$-0.112689\pi$
$240$	0	0
$241$	−396.818	−1.64655	−0.823273	−	0.567646i	$-0.807854\pi$
$241$	−396.818	−1.64655	−0.823273	+	0.567646i	$0.807854\pi$
$242$	0	0
$243$	− 31.4401i	− 0.129383i
$244$	0	0
$245$	−165.780	−0.676651
$246$	0	0
$247$	56.7171i	0.229624i
$248$	0	0
$249$	191.107	0.767498
$250$	0	0
$251$	− 249.643i	− 0.994592i	−0.867581	−	0.497296i	$-0.834326\pi$
$251$	− 249.643i	− 0.994592i	0.867581	−	0.497296i	$-0.165674\pi$
$252$	0	0
$253$	−4.71555	−0.0186385
$254$	0	0
$255$	− 254.302i	− 0.997263i
$256$	0	0
$257$	24.5770	0.0956302	0.0478151	−	0.998856i	$-0.484774\pi$
$257$	24.5770	0.0956302	0.0478151	+	0.998856i	$0.484774\pi$
$258$	0	0
$259$	− 80.2811i	− 0.309966i
$260$	0	0
$261$	−19.3123	−0.0739934
$262$	0	0
$263$	343.809i	1.30726i	0.756816	+	0.653628i	$0.226754\pi$
$263$	343.809i	1.30726i	−0.756816	+	0.653628i	$0.773246\pi$
$264$	0	0
$265$	−150.915	−0.569491
$266$	0	0
$267$	− 9.83308i	− 0.0368280i
$268$	0	0
$269$	−64.7826	−0.240828	−0.120414	−	0.992724i	$-0.538422\pi$
$269$	−64.7826	−0.240828	−0.120414	+	0.992724i	$0.538422\pi$
$270$	0	0
$271$	113.329i	0.418189i	0.977895	+	0.209095i	$0.0670517\pi$
$271$	113.329i	0.418189i	−0.977895	+	0.209095i	$0.932948\pi$
$272$	0	0
$273$	73.0130	0.267447
$274$	0	0
$275$	8.71549i	0.0316927i
$276$	0	0
$277$	42.8037	0.154526	0.0772629	−	0.997011i	$-0.475382\pi$
$277$	42.8037	0.154526	0.0772629	+	0.997011i	$0.475382\pi$
$278$	0	0
$279$	10.2245i	0.0366468i
$280$	0	0
$281$	−202.018	−0.718924	−0.359462	−	0.933160i	$-0.617040\pi$
$281$	−202.018	−0.718924	−0.359462	+	0.933160i	$0.617040\pi$
$282$	0	0
$283$	− 77.5872i	− 0.274160i	−0.990560	−	0.137080i	$-0.956228\pi$
$283$	− 77.5872i	− 0.274160i	0.990560	−	0.137080i	$-0.0437717\pi$
$284$	0	0
$285$	−46.3208	−0.162529
$286$	0	0
$287$	− 10.6699i	− 0.0371774i
$288$	0	0
$289$	283.665	0.981540
$290$	0	0
$291$	68.2291i	0.234464i
$292$	0	0
$293$	−327.131	−1.11649	−0.558243	−	0.829677i	$-0.688524\pi$
$293$	−327.131	−1.11649	−0.558243	+	0.829677i	$0.688524\pi$
$294$	0	0
$295$	254.674i	0.863300i
$296$	0	0
$297$	−20.9193	−0.0704355
$298$	0	0
$299$	− 81.5464i	− 0.272730i
$300$	0	0
$301$	−163.513	−0.543232
$302$	0	0
$303$	− 444.823i	− 1.46806i
$304$	0	0
$305$	321.039	1.05259
$306$	0	0
$307$	268.290i	0.873910i	0.899483	+	0.436955i	$0.143943\pi$
$307$	268.290i	0.873910i	−0.899483	+	0.436955i	$0.856057\pi$
$308$	0	0
$309$	332.377	1.07566
$310$	0	0
$311$	449.497i	1.44533i	0.691199	+	0.722664i	$0.257083\pi$
$311$	449.497i	1.44533i	−0.691199	+	0.722664i	$0.742917\pi$
$312$	0	0
$313$	−283.901	−0.907032	−0.453516	−	0.891248i	$-0.649830\pi$
$313$	−283.901	−0.907032	−0.453516	+	0.891248i	$0.649830\pi$
$314$	0	0
$315$	− 4.13146i	− 0.0131157i
$316$	0	0
$317$	553.100	1.74480	0.872398	−	0.488797i	$-0.162564\pi$
$317$	553.100	1.74480	0.872398	+	0.488797i	$0.162564\pi$
$318$	0	0
$319$	24.9178i	0.0781121i
$320$	0	0
$321$	−469.964	−1.46406
$322$	0	0
$323$	− 104.310i	− 0.322942i
$324$	0	0
$325$	−150.718	−0.463747
$326$	0	0
$327$	350.822i	1.07285i
$328$	0	0
$329$	−35.5073	−0.107925
$330$	0	0
$331$	− 280.828i	− 0.848422i	−0.905563	−	0.424211i	$-0.860552\pi$
$331$	− 280.828i	− 0.848422i	0.905563	−	0.424211i	$-0.139448\pi$
$332$	0	0
$333$	−24.2055	−0.0726892
$334$	0	0
$335$	− 385.049i	− 1.14940i
$336$	0	0
$337$	93.1413	0.276384	0.138192	−	0.990405i	$-0.455871\pi$
$337$	93.1413	0.276384	0.138192	+	0.990405i	$0.455871\pi$
$338$	0	0
$339$	264.534i	0.780337i
$340$	0	0
$341$	13.1922	0.0386867
$342$	0	0
$343$	182.311i	0.531518i
$344$	0	0
$345$	66.5989	0.193040
$346$	0	0
$347$	221.251i	0.637612i	0.947820	+	0.318806i	$0.103282\pi$
$347$	221.251i	0.637612i	−0.947820	+	0.318806i	$0.896718\pi$
$348$	0	0
$349$	227.761	0.652611	0.326306	−	0.945264i	$-0.394196\pi$
$349$	227.761	0.652611	0.326306	+	0.945264i	$0.394196\pi$
$350$	0	0
$351$	− 361.760i	− 1.03065i
$352$	0	0
$353$	−169.824	−0.481089	−0.240545	−	0.970638i	$-0.577326\pi$
$353$	−169.824	−0.481089	−0.240545	+	0.970638i	$0.577326\pi$
$354$	0	0
$355$	− 274.549i	− 0.773378i
$356$	0	0
$357$	−134.281	−0.376136
$358$	0	0
$359$	− 585.928i	− 1.63211i	−0.577972	−	0.816056i	$-0.696156\pi$
$359$	− 585.928i	− 1.63211i	0.577972	−	0.816056i	$-0.303844\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	− 349.400i	− 0.962535i
$364$	0	0
$365$	−178.512	−0.489075
$366$	0	0
$367$	− 428.832i	− 1.16848i	−0.811581	−	0.584239i	$-0.801393\pi$
$367$	− 428.832i	− 1.16848i	0.811581	−	0.584239i	$-0.198607\pi$
$368$	0	0
$369$	−3.21708	−0.00871837
$370$	0	0
$371$	79.6886i	0.214794i
$372$	0	0
$373$	127.502	0.341830	0.170915	−	0.985286i	$-0.445328\pi$
$373$	127.502	0.341830	0.170915	+	0.985286i	$0.445328\pi$
$374$	0	0
$375$	− 388.759i	− 1.03669i
$376$	0	0
$377$	−430.905	−1.14298
$378$	0	0
$379$	− 244.665i	− 0.645555i	−0.946475	−	0.322777i	$-0.895384\pi$
$379$	− 244.665i	− 0.645555i	0.946475	−	0.322777i	$-0.104616\pi$
$380$	0	0
$381$	−556.950	−1.46181
$382$	0	0
$383$	− 685.025i	− 1.78858i	−0.447491	−	0.894288i	$-0.647682\pi$
$383$	− 685.025i	− 1.78858i	0.447491	−	0.894288i	$-0.352318\pi$
$384$	0	0
$385$	−5.33063	−0.0138458
$386$	0	0
$387$	49.3007i	0.127392i
$388$	0	0
$389$	−290.167	−0.745931	−0.372966	−	0.927845i	$-0.621659\pi$
$389$	−290.167	−0.745931	−0.372966	+	0.927845i	$0.621659\pi$
$390$	0	0
$391$	149.975i	0.383567i
$392$	0	0
$393$	267.863	0.681586
$394$	0	0
$395$	− 351.445i	− 0.889735i
$396$	0	0
$397$	−317.752	−0.800384	−0.400192	−	0.916431i	$-0.631057\pi$
$397$	−317.752	−0.800384	−0.400192	+	0.916431i	$0.631057\pi$
$398$	0	0
$399$	24.4591i	0.0613009i
$400$	0	0
$401$	445.424	1.11078	0.555392	−	0.831589i	$-0.312568\pi$
$401$	445.424	1.11078	0.555392	+	0.831589i	$0.312568\pi$
$402$	0	0
$403$	228.133i	0.566087i
$404$	0	0
$405$	276.225	0.682036
$406$	0	0
$407$	31.2313i	0.0767353i
$408$	0	0
$409$	571.233	1.39666	0.698328	−	0.715778i	$-0.253928\pi$
$409$	571.233	1.39666	0.698328	+	0.715778i	$0.253928\pi$
$410$	0	0
$411$	7.15815i	0.0174164i
$412$	0	0
$413$	134.477	0.325610
$414$	0	0
$415$	− 241.283i	− 0.581406i
$416$	0	0
$417$	711.719	1.70676
$418$	0	0
$419$	− 308.595i	− 0.736502i	−0.929726	−	0.368251i	$-0.879957\pi$
$419$	− 308.595i	− 0.736502i	0.929726	−	0.368251i	$-0.120043\pi$
$420$	0	0
$421$	−38.8298	−0.0922324	−0.0461162	−	0.998936i	$-0.514684\pi$
$421$	−38.8298	−0.0922324	−0.0461162	+	0.998936i	$0.514684\pi$
$422$	0	0
$423$	10.7058i	0.0253092i
$424$	0	0
$425$	277.190	0.652212
$426$	0	0
$427$	− 169.520i	− 0.397003i
$428$	0	0
$429$	−28.4038	−0.0662093
$430$	0	0
$431$	− 479.659i	− 1.11290i	−0.830881	−	0.556450i	$-0.812163\pi$
$431$	− 479.659i	− 1.11290i	0.830881	−	0.556450i	$-0.187837\pi$
$432$	0	0
$433$	434.901	1.00439	0.502195	−	0.864754i	$-0.332526\pi$
$433$	434.901	1.00439	0.502195	+	0.864754i	$0.332526\pi$
$434$	0	0
$435$	− 351.920i	− 0.809011i
$436$	0	0
$437$	27.3177	0.0625120
$438$	0	0
$439$	25.4779i	0.0580363i	0.999579	+	0.0290182i	$0.00923807\pi$
$439$	25.4779i	0.0580363i	−0.999579	+	0.0290182i	$0.990762\pi$
$440$	0	0
$441$	26.3934	0.0598490
$442$	0	0
$443$	− 805.631i	− 1.81858i	−0.416164	−	0.909290i	$-0.636626\pi$
$443$	− 805.631i	− 1.81858i	0.416164	−	0.909290i	$-0.363374\pi$
$444$	0	0
$445$	−12.4148	−0.0278985
$446$	0	0
$447$	151.919i	0.339863i
$448$	0	0
$449$	−106.288	−0.236721	−0.118360	−	0.992971i	$-0.537764\pi$
$449$	−106.288	−0.236721	−0.118360	+	0.992971i	$0.537764\pi$
$450$	0	0
$451$	4.15085i	0.00920366i
$452$	0	0
$453$	73.5328	0.162324
$454$	0	0
$455$	− 92.1830i	− 0.202600i
$456$	0	0
$457$	−730.620	−1.59873	−0.799365	−	0.600846i	$-0.794831\pi$
$457$	−730.620	−1.59873	−0.799365	+	0.600846i	$0.794831\pi$
$458$	0	0
$459$	665.325i	1.44951i
$460$	0	0
$461$	854.696	1.85401	0.927003	−	0.375055i	$-0.122376\pi$
$461$	854.696	1.85401	0.927003	+	0.375055i	$0.122376\pi$
$462$	0	0
$463$	769.128i	1.66118i	0.556882	+	0.830591i	$0.311997\pi$
$463$	769.128i	1.66118i	−0.556882	+	0.830591i	$0.688003\pi$
$464$	0	0
$465$	−186.316	−0.400680
$466$	0	0
$467$	− 54.2556i	− 0.116179i	−0.998311	−	0.0580895i	$-0.981499\pi$
$467$	− 54.2556i	− 0.116179i	0.998311	−	0.0580895i	$-0.0185009\pi$
$468$	0	0
$469$	−203.320	−0.433517
$470$	0	0
$471$	− 159.150i	− 0.337899i
$472$	0	0
$473$	63.6104	0.134483
$474$	0	0
$475$	− 50.4898i	− 0.106294i
$476$	0	0
$477$	24.0269	0.0503708
$478$	0	0
$479$	− 512.359i	− 1.06964i	−0.844965	−	0.534821i	$-0.820379\pi$
$479$	− 512.359i	− 1.06964i	0.844965	−	0.534821i	$-0.179621\pi$
$480$	0	0
$481$	−540.085	−1.12284
$482$	0	0
$483$	− 35.1666i	− 0.0728088i
$484$	0	0
$485$	86.1431	0.177615
$486$	0	0
$487$	− 445.189i	− 0.914146i	−0.889429	−	0.457073i	$-0.848898\pi$
$487$	− 445.189i	− 0.914146i	0.889429	−	0.457073i	$-0.151102\pi$
$488$	0	0
$489$	−77.7654	−0.159030
$490$	0	0
$491$	219.288i	0.446615i	0.974748	+	0.223308i	$0.0716855\pi$
$491$	219.288i	0.446615i	−0.974748	+	0.223308i	$0.928315\pi$
$492$	0	0
$493$	792.492	1.60749
$494$	0	0
$495$	1.60724i	0.00324694i
$496$	0	0
$497$	−144.972	−0.291694
$498$	0	0
$499$	277.168i	0.555446i	0.960661	+	0.277723i	$0.0895798\pi$
$499$	277.168i	0.555446i	−0.960661	+	0.277723i	$0.910420\pi$
$500$	0	0
$501$	181.986	0.363245
$502$	0	0
$503$	328.913i	0.653903i	0.945041	+	0.326951i	$0.106021\pi$
$503$	328.913i	0.653903i	−0.945041	+	0.326951i	$0.893979\pi$
$504$	0	0
$505$	−561.614	−1.11211
$506$	0	0
$507$	− 0.889743i	− 0.00175492i
$508$	0	0
$509$	371.573	0.730006	0.365003	−	0.931006i	$-0.381068\pi$
$509$	371.573	0.730006	0.365003	+	0.931006i	$0.381068\pi$
$510$	0	0
$511$	94.2610i	0.184464i
$512$	0	0
$513$	121.188	0.236234
$514$	0	0
$515$	− 419.645i	− 0.814845i
$516$	0	0
$517$	13.8132	0.0267180
$518$	0	0
$519$	954.413i	1.83895i
$520$	0	0
$521$	−607.799	−1.16660	−0.583300	−	0.812256i	$-0.698239\pi$
$521$	−607.799	−1.16660	−0.583300	+	0.812256i	$0.698239\pi$
$522$	0	0
$523$	− 384.474i	− 0.735131i	−0.929998	−	0.367566i	$-0.880191\pi$
$523$	− 384.474i	− 0.735131i	0.929998	−	0.367566i	$-0.119809\pi$
$524$	0	0
$525$	−64.9965	−0.123803
$526$	0	0
$527$	− 419.567i	− 0.796142i
$528$	0	0
$529$	489.723	0.925753
$530$	0	0
$531$	− 40.5460i	− 0.0763579i
$532$	0	0
$533$	−71.7809	−0.134673
$534$	0	0
$535$	593.356i	1.10908i
$536$	0	0
$537$	974.346	1.81442
$538$	0	0
$539$	− 34.0542i	− 0.0631803i
$540$	0	0
$541$	382.676	0.707350	0.353675	−	0.935368i	$-0.384932\pi$
$541$	382.676	0.707350	0.353675	+	0.935368i	$0.384932\pi$
$542$	0	0
$543$	− 170.952i	− 0.314828i
$544$	0	0
$545$	442.932	0.812719
$546$	0	0
$547$	− 179.104i	− 0.327430i	−0.986508	−	0.163715i	$-0.947652\pi$
$547$	− 179.104i	− 0.327430i	0.986508	−	0.163715i	$-0.0523477\pi$
$548$	0	0
$549$	−51.1119	−0.0931000
$550$	0	0
$551$	− 144.351i	− 0.261981i
$552$	0	0
$553$	−185.576	−0.335580
$554$	0	0
$555$	− 441.087i	− 0.794752i
$556$	0	0
$557$	−720.325	−1.29322	−0.646612	−	0.762819i	$-0.723815\pi$
$557$	−720.325	−1.29322	−0.646612	+	0.762819i	$0.723815\pi$
$558$	0	0
$559$	1100.02i	1.96783i
$560$	0	0
$561$	52.2384	0.0931165
$562$	0	0
$563$	175.948i	0.312519i	0.987716	+	0.156260i	$0.0499436\pi$
$563$	175.948i	0.312519i	−0.987716	+	0.156260i	$0.950056\pi$
$564$	0	0
$565$	333.989	0.591132
$566$	0	0
$567$	− 145.857i	− 0.257243i
$568$	0	0
$569$	−547.930	−0.962971	−0.481485	−	0.876454i	$-0.659903\pi$
$569$	−547.930	−0.962971	−0.481485	+	0.876454i	$0.659903\pi$
$570$	0	0
$571$	− 376.007i	− 0.658507i	−0.944242	−	0.329253i	$-0.893203\pi$
$571$	− 376.007i	− 0.658507i	0.944242	−	0.329253i	$-0.106797\pi$
$572$	0	0
$573$	286.305	0.499659
$574$	0	0
$575$	72.5930i	0.126249i
$576$	0	0
$577$	458.453	0.794546	0.397273	−	0.917700i	$-0.369957\pi$
$577$	458.453	0.794546	0.397273	+	0.917700i	$0.369957\pi$
$578$	0	0
$579$	807.662i	1.39493i
$580$	0	0
$581$	−127.406	−0.219288
$582$	0	0
$583$	− 31.0008i	− 0.0531745i
$584$	0	0
$585$	−27.7940	−0.0475112
$586$	0	0
$587$	− 335.174i	− 0.570995i	−0.958379	−	0.285497i	$-0.907841\pi$
$587$	− 335.174i	− 0.570995i	0.958379	−	0.285497i	$-0.0921588\pi$
$588$	0	0
$589$	−76.4236	−0.129752
$590$	0	0
$591$	− 137.335i	− 0.232378i
$592$	0	0
$593$	−280.543	−0.473092	−0.236546	−	0.971620i	$-0.576015\pi$
$593$	−280.543	−0.473092	−0.236546	+	0.971620i	$0.576015\pi$
$594$	0	0
$595$	169.537i	0.284936i
$596$	0	0
$597$	−729.225	−1.22148
$598$	0	0
$599$	− 677.612i	− 1.13124i	−0.824667	−	0.565619i	$-0.808637\pi$
$599$	− 677.612i	− 1.13124i	0.824667	−	0.565619i	$-0.191363\pi$
$600$	0	0
$601$	177.699	0.295672	0.147836	−	0.989012i	$-0.452769\pi$
$601$	177.699	0.295672	0.147836	+	0.989012i	$0.452769\pi$
$602$	0	0
$603$	61.3028i	0.101663i
$604$	0	0
$605$	−441.137	−0.729152
$606$	0	0
$607$	− 826.068i	− 1.36090i	−0.732793	−	0.680451i	$-0.761784\pi$
$607$	− 826.068i	− 1.36090i	0.732793	−	0.680451i	$-0.238216\pi$
$608$	0	0
$609$	−185.826	−0.305134
$610$	0	0
$611$	238.873i	0.390954i
$612$	0	0
$613$	−839.090	−1.36883	−0.684413	−	0.729095i	$-0.739941\pi$
$613$	−839.090	−1.36883	−0.684413	+	0.729095i	$0.739941\pi$
$614$	0	0
$615$	− 58.6235i	− 0.0953227i
$616$	0	0
$617$	615.452	0.997491	0.498746	−	0.866748i	$-0.333794\pi$
$617$	615.452	0.997491	0.498746	+	0.866748i	$0.333794\pi$
$618$	0	0
$619$	− 118.351i	− 0.191197i	−0.995420	−	0.0955984i	$-0.969524\pi$
$619$	− 118.351i	− 0.191197i	0.995420	−	0.0955984i	$-0.0304765\pi$
$620$	0	0
$621$	−174.241	−0.280582
$622$	0	0
$623$	6.55547i	0.0105224i
$624$	0	0
$625$	−201.251	−0.322002
$626$	0	0
$627$	− 9.51516i	− 0.0151757i
$628$	0	0
$629$	993.288	1.57915
$630$	0	0
$631$	− 665.915i	− 1.05533i	−0.849452	−	0.527667i	$-0.823067\pi$
$631$	− 665.915i	− 1.05533i	0.849452	−	0.527667i	$-0.176933\pi$
$632$	0	0
$633$	113.127	0.178716
$634$	0	0
$635$	703.181i	1.10737i
$636$	0	0
$637$	588.902	0.924493
$638$	0	0
$639$	43.7104i	0.0684043i
$640$	0	0
$641$	396.009	0.617798	0.308899	−	0.951095i	$-0.400039\pi$
$641$	396.009	0.617798	0.308899	+	0.951095i	$0.400039\pi$
$642$	0	0
$643$	890.406i	1.38477i	0.721529	+	0.692384i	$0.243440\pi$
$643$	890.406i	1.38477i	−0.721529	+	0.692384i	$0.756560\pi$
$644$	0	0
$645$	−898.386	−1.39285
$646$	0	0
$647$	1022.42i	1.58025i	0.612945	+	0.790126i	$0.289985\pi$
$647$	1022.42i	1.58025i	−0.612945	+	0.790126i	$0.710015\pi$
$648$	0	0
$649$	−52.3147	−0.0806082
$650$	0	0
$651$	98.3816i	0.151124i
$652$	0	0
$653$	−298.828	−0.457623	−0.228811	−	0.973471i	$-0.573484\pi$
$653$	−298.828	−0.457623	−0.228811	+	0.973471i	$0.573484\pi$
$654$	0	0
$655$	− 338.193i	− 0.516324i
$656$	0	0
$657$	28.4206	0.0432581
$658$	0	0
$659$	76.0870i	0.115458i	0.998332	+	0.0577292i	$0.0183860\pi$
$659$	76.0870i	0.115458i	−0.998332	+	0.0577292i	$0.981614\pi$
$660$	0	0
$661$	63.4958	0.0960602	0.0480301	−	0.998846i	$-0.484706\pi$
$661$	63.4958	0.0960602	0.0480301	+	0.998846i	$0.484706\pi$
$662$	0	0
$663$	903.362i	1.36254i
$664$	0	0
$665$	30.8809	0.0464375
$666$	0	0
$667$	207.545i	0.311162i
$668$	0	0
$669$	549.674	0.821636
$670$	0	0
$671$	65.9474i	0.0982822i
$672$	0	0
$673$	−814.516	−1.21028	−0.605138	−	0.796120i	$-0.706882\pi$
$673$	−814.516	−1.21028	−0.605138	+	0.796120i	$0.706882\pi$
$674$	0	0
$675$	322.040i	0.477097i
$676$	0	0
$677$	−127.732	−0.188673	−0.0943365	−	0.995540i	$-0.530073\pi$
$677$	−127.732	−0.188673	−0.0943365	+	0.995540i	$0.530073\pi$
$678$	0	0
$679$	− 45.4866i	− 0.0669906i
$680$	0	0
$681$	−486.311	−0.714113
$682$	0	0
$683$	− 518.648i	− 0.759367i	−0.925116	−	0.379684i	$-0.876033\pi$
$683$	− 518.648i	− 0.759367i	0.925116	−	0.379684i	$-0.123967\pi$
$684$	0	0
$685$	9.03756	0.0131935
$686$	0	0
$687$	589.969i	0.858761i
$688$	0	0
$689$	536.098	0.778082
$690$	0	0
$691$	1279.00i	1.85094i	0.378823	+	0.925469i	$0.376329\pi$
$691$	1279.00i	1.85094i	−0.378823	+	0.925469i	$0.623671\pi$
$692$	0	0
$693$	0.848678	0.00122464
$694$	0	0
$695$	− 898.585i	− 1.29293i
$696$	0	0
$697$	132.015	0.189404
$698$	0	0
$699$	582.468i	0.833287i
$700$	0	0
$701$	−189.067	−0.269710	−0.134855	−	0.990865i	$-0.543057\pi$
$701$	−189.067	−0.269710	−0.134855	+	0.990865i	$0.543057\pi$
$702$	0	0
$703$	− 180.926i	− 0.257363i
$704$	0	0
$705$	−195.087	−0.276719
$706$	0	0
$707$	296.552i	0.419452i
$708$	0	0
$709$	−722.894	−1.01960	−0.509798	−	0.860294i	$-0.670280\pi$
$709$	−722.894	−1.01960	−0.509798	+	0.860294i	$0.670280\pi$
$710$	0	0
$711$	55.9528i	0.0786960i
$712$	0	0
$713$	109.880	0.154109
$714$	0	0
$715$	35.8614i	0.0501558i
$716$	0	0
$717$	−480.755	−0.670509
$718$	0	0
$719$	768.781i	1.06924i	0.845094	+	0.534618i	$0.179545\pi$
$719$	768.781i	1.06924i	−0.845094	+	0.534618i	$0.820455\pi$
$720$	0	0
$721$	−221.588	−0.307334
$722$	0	0
$723$	1151.24i	1.59231i
$724$	0	0
$725$	383.594	0.529095
$726$	0	0
$727$	1092.23i	1.50238i	0.660085	+	0.751191i	$0.270520\pi$
$727$	1092.23i	1.50238i	−0.660085	+	0.751191i	$0.729480\pi$
$728$	0	0
$729$	−769.916	−1.05613
$730$	0	0
$731$	− 2023.08i	− 2.76756i
$732$	0	0
$733$	−571.768	−0.780038	−0.390019	−	0.920807i	$-0.627532\pi$
$733$	−571.768	−0.780038	−0.390019	+	0.920807i	$0.627532\pi$
$734$	0	0
$735$	480.956i	0.654362i
$736$	0	0
$737$	79.0961	0.107322
$738$	0	0
$739$	299.431i	0.405184i	0.979263	+	0.202592i	$0.0649365\pi$
$739$	299.431i	0.405184i	−0.979263	+	0.202592i	$0.935063\pi$
$740$	0	0
$741$	164.546	0.222060
$742$	0	0
$743$	− 16.0943i	− 0.0216612i	−0.999941	−	0.0108306i	$-0.996552\pi$
$743$	− 16.0943i	− 0.0216612i	0.999941	−	0.0108306i	$-0.00344755\pi$
$744$	0	0
$745$	191.806	0.257458
$746$	0	0
$747$	38.4142i	0.0514246i
$748$	0	0
$749$	313.313	0.418309
$750$	0	0
$751$	699.482i	0.931401i	0.884942	+	0.465701i	$0.154198\pi$
$751$	699.482i	0.931401i	−0.884942	+	0.465701i	$0.845802\pi$
$752$	0	0
$753$	−724.258	−0.961830
$754$	0	0
$755$	− 92.8393i	− 0.122966i
$756$	0	0
$757$	−11.5679	−0.0152812	−0.00764062	−	0.999971i	$-0.502432\pi$
$757$	−11.5679	−0.0152812	−0.00764062	+	0.999971i	$0.502432\pi$
$758$	0	0
$759$	13.6807i	0.0180246i
$760$	0	0
$761$	−354.338	−0.465622	−0.232811	−	0.972522i	$-0.574792\pi$
$761$	−354.338	−0.465622	−0.232811	+	0.972522i	$0.574792\pi$
$762$	0	0
$763$	− 233.884i	− 0.306532i
$764$	0	0
$765$	51.1170	0.0668195
$766$	0	0
$767$	− 904.682i	− 1.17951i
$768$	0	0
$769$	847.602	1.10221	0.551106	−	0.834435i	$-0.314206\pi$
$769$	847.602	1.10221	0.551106	+	0.834435i	$0.314206\pi$
$770$	0	0
$771$	− 71.3021i	− 0.0924801i
$772$	0	0
$773$	−712.408	−0.921615	−0.460807	−	0.887500i	$-0.652440\pi$
$773$	−712.408	−0.921615	−0.460807	+	0.887500i	$0.652440\pi$
$774$	0	0
$775$	− 203.085i	− 0.262045i
$776$	0	0
$777$	−232.910	−0.299755
$778$	0	0
$779$	− 24.0463i	− 0.0308682i
$780$	0	0
$781$	56.3975	0.0722119
$782$	0	0
$783$	920.720i	1.17589i
$784$	0	0
$785$	−200.936	−0.255970
$786$	0	0
$787$	− 514.229i	− 0.653404i	−0.945127	−	0.326702i	$-0.894063\pi$
$787$	− 514.229i	− 0.653404i	0.945127	−	0.326702i	$-0.105937\pi$
$788$	0	0
$789$	997.450	1.26420
$790$	0	0
$791$	− 176.358i	− 0.222956i
$792$	0	0
$793$	−1140.43	−1.43812
$794$	0	0
$795$	437.831i	0.550731i
$796$	0	0
$797$	−934.459	−1.17247	−0.586235	−	0.810141i	$-0.699391\pi$
$797$	−934.459	−1.17247	−0.586235	+	0.810141i	$0.699391\pi$
$798$	0	0
$799$	− 439.319i	− 0.549836i
$800$	0	0
$801$	1.97654	0.00246759
$802$	0	0
$803$	− 36.6698i	− 0.0456660i
$804$	0	0
$805$	−44.3998	−0.0551551
$806$	0	0
$807$	187.946i	0.232895i
$808$	0	0
$809$	1126.44	1.39239	0.696195	−	0.717853i	$-0.254875\pi$
$809$	1126.44	1.39239	0.696195	+	0.717853i	$0.254875\pi$
$810$	0	0
$811$	− 938.339i	− 1.15702i	−0.815677	−	0.578508i	$-0.803635\pi$
$811$	− 938.339i	− 1.15702i	0.815677	−	0.578508i	$-0.196365\pi$
$812$	0	0
$813$	328.789	0.404414
$814$	0	0
$815$	98.1832i	0.120470i
$816$	0	0
$817$	−368.502	−0.451043
$818$	0	0
$819$	14.6763i	0.0179197i
$820$	0	0
$821$	−509.877	−0.621044	−0.310522	−	0.950566i	$-0.600504\pi$
$821$	−509.877	−0.621044	−0.310522	+	0.950566i	$0.600504\pi$
$822$	0	0
$823$	− 791.322i	− 0.961509i	−0.876855	−	0.480754i	$-0.840363\pi$
$823$	− 791.322i	− 0.961509i	0.876855	−	0.480754i	$-0.159637\pi$
$824$	0	0
$825$	25.2852	0.0306487
$826$	0	0
$827$	725.787i	0.877615i	0.898581	+	0.438807i	$0.144599\pi$
$827$	725.787i	0.877615i	−0.898581	+	0.438807i	$0.855401\pi$
$828$	0	0
$829$	987.501	1.19120	0.595598	−	0.803283i	$-0.296915\pi$
$829$	987.501	1.19120	0.595598	+	0.803283i	$0.296915\pi$
$830$	0	0
$831$	− 124.181i	− 0.149436i
$832$	0	0
$833$	−1083.07	−1.30020
$834$	0	0
$835$	− 229.767i	− 0.275170i
$836$	0	0
$837$	487.455	0.582383
$838$	0	0
$839$	1549.38i	1.84670i	0.383959	+	0.923350i	$0.374561\pi$
$839$	1549.38i	1.84670i	−0.383959	+	0.923350i	$0.625439\pi$
$840$	0	0
$841$	255.702	0.304045
$842$	0	0
$843$	586.089i	0.695242i
$844$	0	0
$845$	−1.12335	−0.00132941
$846$	0	0
$847$	232.936i	0.275013i
$848$	0	0
$849$	−225.094	−0.265129
$850$	0	0
$851$	260.131i	0.305677i
$852$	0	0
$853$	108.740	0.127480	0.0637399	−	0.997967i	$-0.479697\pi$
$853$	108.740	0.127480	0.0637399	+	0.997967i	$0.479697\pi$
$854$	0	0
$855$	− 9.31089i	− 0.0108899i
$856$	0	0
$857$	−958.909	−1.11891	−0.559457	−	0.828860i	$-0.688990\pi$
$857$	−958.909	−1.11891	−0.559457	+	0.828860i	$0.688990\pi$
$858$	0	0
$859$	1060.62i	1.23471i	0.786684	+	0.617356i	$0.211796\pi$
$859$	1060.62i	1.23471i	−0.786684	+	0.617356i	$0.788204\pi$
$860$	0	0
$861$	−30.9553	−0.0359527
$862$	0	0
$863$	− 838.621i	− 0.971751i	−0.874028	−	0.485875i	$-0.838501\pi$
$863$	− 838.621i	− 0.971751i	0.874028	−	0.485875i	$-0.161499\pi$
$864$	0	0
$865$	1205.00	1.39306
$866$	0	0
$867$	− 822.963i	− 0.949208i
$868$	0	0
$869$	72.1934	0.0830764
$870$	0	0
$871$	1367.82i	1.57040i
$872$	0	0
$873$	−13.7147	−0.0157098
$874$	0	0
$875$	259.176i	0.296201i
$876$	0	0
$877$	−1453.86	−1.65776	−0.828882	−	0.559424i	$-0.811022\pi$
$877$	−1453.86	−1.65776	−0.828882	+	0.559424i	$0.811022\pi$
$878$	0	0
$879$	949.064i	1.07971i
$880$	0	0
$881$	156.229	0.177332	0.0886659	−	0.996061i	$-0.471740\pi$
$881$	156.229	0.177332	0.0886659	+	0.996061i	$0.471740\pi$
$882$	0	0
$883$	37.1369i	0.0420577i	0.999779	+	0.0210288i	$0.00669418\pi$
$883$	37.1369i	0.0420577i	−0.999779	+	0.0210288i	$0.993306\pi$
$884$	0	0
$885$	738.854	0.834863
$886$	0	0
$887$	− 490.932i	− 0.553474i	−0.960946	−	0.276737i	$-0.910747\pi$
$887$	− 490.932i	− 0.553474i	0.960946	−	0.276737i	$-0.0892532\pi$
$888$	0	0
$889$	371.305	0.417666
$890$	0	0
$891$	56.7417i	0.0636831i
$892$	0	0
$893$	−80.0214	−0.0896096
$894$	0	0
$895$	− 1230.17i	− 1.37449i
$896$	0	0
$897$	−236.581	−0.263747
$898$	0	0
$899$	− 580.624i	− 0.645856i
$900$	0	0
$901$	−985.957	−1.09429
$902$	0	0
$903$	474.380i	0.525338i
$904$	0	0
$905$	−215.836	−0.238493
$906$	0	0
$907$	167.806i	0.185012i	0.995712	+	0.0925060i	$0.0294877\pi$
$907$	167.806i	0.185012i	−0.995712	+	0.0925060i	$0.970512\pi$
$908$	0	0
$909$	89.4134	0.0983645
$910$	0	0
$911$	1217.97i	1.33696i	0.743730	+	0.668481i	$0.233055\pi$
$911$	1217.97i	1.33696i	−0.743730	+	0.668481i	$0.766945\pi$
$912$	0	0
$913$	49.5641	0.0542871
$914$	0	0
$915$	− 931.391i	− 1.01791i
$916$	0	0
$917$	−178.578	−0.194741
$918$	0	0
$919$	1037.14i	1.12855i	0.825585	+	0.564277i	$0.190845\pi$
$919$	1037.14i	1.12855i	−0.825585	+	0.564277i	$0.809155\pi$
$920$	0	0
$921$	778.358	0.845123
$922$	0	0
$923$	975.286i	1.05665i
$924$	0	0
$925$	480.786	0.519769
$926$	0	0
$927$	66.8108i	0.0720720i
$928$	0	0
$929$	−98.9780	−0.106543	−0.0532713	−	0.998580i	$-0.516965\pi$
$929$	−98.9780	−0.106543	−0.0532713	+	0.998580i	$0.516965\pi$
$930$	0	0
$931$	197.280i	0.211901i
$932$	0	0
$933$	1304.07	1.39772
$934$	0	0
$935$	− 65.9539i	− 0.0705389i
$936$	0	0
$937$	1744.65	1.86195	0.930974	−	0.365084i	$-0.118960\pi$
$937$	1744.65	1.86195	0.930974	+	0.365084i	$0.118960\pi$
$938$	0	0
$939$	823.647i	0.877154i
$940$	0	0
$941$	723.767	0.769147	0.384574	−	0.923094i	$-0.374348\pi$
$941$	723.767	0.769147	0.384574	+	0.923094i	$0.374348\pi$
$942$	0	0
$943$	34.5732i	0.0366630i
$944$	0	0
$945$	−196.969	−0.208432
$946$	0	0
$947$	− 819.583i	− 0.865452i	−0.901526	−	0.432726i	$-0.857552\pi$
$947$	− 819.583i	− 0.865452i	0.901526	−	0.432726i	$-0.142448\pi$
$948$	0	0
$949$	634.133	0.668212
$950$	0	0
$951$	− 1604.64i	− 1.68732i
$952$	0	0
$953$	1777.86	1.86554	0.932771	−	0.360470i	$-0.117384\pi$
$953$	1777.86	1.86554	0.932771	+	0.360470i	$0.117384\pi$
$954$	0	0
$955$	− 361.476i	− 0.378509i
$956$	0	0
$957$	72.2909	0.0755391
$958$	0	0
$959$	− 4.77216i	− 0.00497618i
$960$	0	0
$961$	653.601	0.680126
$962$	0	0
$963$	− 94.4669i	− 0.0980965i
$964$	0	0
$965$	1019.72	1.05670
$966$	0	0
$967$	− 1306.29i	− 1.35087i	−0.737421	−	0.675433i	$-0.763957\pi$
$967$	− 1306.29i	− 1.35087i	0.737421	−	0.675433i	$-0.236043\pi$
$968$	0	0
$969$	−302.623	−0.312304
$970$	0	0
$971$	940.020i	0.968094i	0.875042	+	0.484047i	$0.160834\pi$
$971$	940.020i	0.968094i	−0.875042	+	0.484047i	$0.839166\pi$
$972$	0	0
$973$	−474.485	−0.487652
$974$	0	0
$975$	437.259i	0.448471i
$976$	0	0
$977$	−549.002	−0.561927	−0.280963	−	0.959719i	$-0.590654\pi$
$977$	−549.002	−0.561927	−0.280963	+	0.959719i	$0.590654\pi$
$978$	0	0
$979$	− 2.55023i	− 0.00260494i
$980$	0	0
$981$	−70.5182	−0.0718840
$982$	0	0
$983$	506.118i	0.514871i	0.966295	+	0.257436i	$0.0828775\pi$
$983$	506.118i	0.514871i	−0.966295	+	0.257436i	$0.917122\pi$
$984$	0	0
$985$	−173.394	−0.176034
$986$	0	0
$987$	103.013i	0.104370i
$988$	0	0
$989$	529.823	0.535716
$990$	0	0
$991$	213.024i	0.214959i	0.994207	+	0.107479i	$0.0342780\pi$
$991$	213.024i	0.214959i	−0.994207	+	0.107479i	$0.965722\pi$
$992$	0	0
$993$	−814.731	−0.820474
$994$	0	0
$995$	920.687i	0.925314i
$996$	0	0
$997$	940.041	0.942870	0.471435	−	0.881901i	$-0.343736\pi$
$997$	940.041	0.942870	0.471435	+	0.881901i	$0.343736\pi$
$998$	0	0
$999$	1154.01i	1.15516i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.d.d.191.5		14
4.3	odd	2	inner	1216.3.d.d.191.10		14
8.3	odd	2		76.3.b.b.39.12	yes	14
8.5	even	2		76.3.b.b.39.11	✓	14
24.5	odd	2		684.3.g.b.343.4		14
24.11	even	2		684.3.g.b.343.3		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.b.b.39.11	✓	14	8.5	even	2
76.3.b.b.39.12	yes	14	8.3	odd	2
684.3.g.b.343.3		14	24.11	even	2
684.3.g.b.343.4		14	24.5	odd	2
1216.3.d.d.191.5		14	1.1	even	1	trivial
1216.3.d.d.191.10		14	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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.90118i q^{3} -3.66290 q^{5} +1.93414i q^{7} +0.583162 q^{9} +O(q^{10})\)	\(q-2.90118i q^{3} -3.66290 q^{5} +1.93414i q^{7} +0.583162 q^{9} -0.752428i q^{11} +13.0118 q^{13} +10.6267i q^{15} -23.9304 q^{17} +4.35890i q^{19} +5.61130 q^{21} -6.26712i q^{23} -11.5832 q^{25} -27.8025i q^{27} -33.1165 q^{29} +17.5328i q^{31} -2.18293 q^{33} -7.08457i q^{35} -41.5073 q^{37} -37.7495i q^{39} -5.51661 q^{41} +84.5402i q^{43} -2.13607 q^{45} +18.3582i q^{47} +45.2591 q^{49} +69.4264i q^{51} +41.2010 q^{53} +2.75607i q^{55} +12.6459 q^{57} -69.5279i q^{59} -87.6461 q^{61} +1.12792i q^{63} -47.6609 q^{65} +105.121i q^{67} -18.1820 q^{69} +74.9540i q^{71} +48.7353 q^{73} +33.6048i q^{75} +1.45530 q^{77} +95.9473i q^{79} -75.4115 q^{81} +65.8722i q^{83} +87.6548 q^{85} +96.0768i q^{87} +3.38934 q^{89} +25.1667i q^{91} +50.8657 q^{93} -15.9662i q^{95} -23.5177 q^{97} -0.438788i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 68 q^{9}+O(q^{10}) \) 14 * q - 68 * q^9	\( 14 q - 68 q^{9} - 54 q^{13} + 34 q^{17} + 38 q^{21} - 86 q^{25} - 54 q^{29} + 20 q^{33} - 100 q^{37} + 224 q^{41} + 168 q^{45} - 220 q^{49} - 14 q^{53} + 38 q^{57} - 28 q^{61} - 472 q^{65} - 122 q^{69} + 70 q^{73} - 228 q^{77} + 334 q^{81} - 48 q^{85} + 176 q^{93} + 308 q^{97}+O(q^{100}) \) 14 * q - 68 * q^9 - 54 * q^13 + 34 * q^17 + 38 * q^21 - 86 * q^25 - 54 * q^29 + 20 * q^33 - 100 * q^37 + 224 * q^41 + 168 * q^45 - 220 * q^49 - 14 * q^53 + 38 * q^57 - 28 * q^61 - 472 * q^65 - 122 * q^69 + 70 * q^73 - 228 * q^77 + 334 * q^81 - 48 * q^85 + 176 * q^93 + 308 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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