LMFDB - Embedded newform 1224.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,4,Mod(1,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1224.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1224.a (trivial)

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:	yes
Analytic conductor:	$72.2183378470$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.9282 q^{5} -28.3923 q^{7} +55.0333 q^{11} -0.430781 q^{13} -17.0000 q^{17} +147.636 q^{19} -108.603 q^{23} +42.1384 q^{25} +107.933 q^{29} -70.1718 q^{31} -367.061 q^{35} -381.769 q^{37} +16.1436 q^{41} +382.354 q^{43} +455.846 q^{47} +463.123 q^{49} -21.0052 q^{53} +711.482 q^{55} +9.91274 q^{59} -679.759 q^{61} -5.56922 q^{65} +708.574 q^{67} -85.0793 q^{71} -37.5795 q^{73} -1562.52 q^{77} +685.290 q^{79} +1294.89 q^{83} -219.779 q^{85} +1572.08 q^{89} +12.2309 q^{91} +1908.67 q^{95} -175.990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{5} - 36 q^{7} + 20 q^{11} - 84 q^{13} - 34 q^{17} + 32 q^{19} - 44 q^{23} - 82 q^{25} + 396 q^{29} + 116 q^{31} - 360 q^{35} - 140 q^{37} + 60 q^{41} + 640 q^{43} + 496 q^{47} + 178 q^{49} - 236 q^{53}+ \cdots + 36 q^{97}+O(q^{100}) $ 2 * q + 12 * q^5 - 36 * q^7 + 20 * q^11 - 84 * q^13 - 34 * q^17 + 32 * q^19 - 44 * q^23 - 82 * q^25 + 396 * q^29 + 116 * q^31 - 360 * q^35 - 140 * q^37 + 60 * q^41 + 640 * q^43 + 496 * q^47 + 178 * q^49 - 236 * q^53 + 744 * q^55 - 576 * q^59 - 348 * q^61 + 72 * q^65 + 1528 * q^67 + 876 * q^71 - 380 * q^73 - 1296 * q^77 + 172 * q^79 + 1024 * q^83 - 204 * q^85 + 844 * q^89 + 648 * q^91 + 2016 * q^95 + 36 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	12.9282	1.15633	0.578167	−	0.815919i	$-0.303768\pi$
$5$	12.9282	1.15633	0.578167	+	0.815919i	$0.303768\pi$
$6$	0	0
$7$	−28.3923	−1.53304	−0.766520	−	0.642220i	$-0.778013\pi$
$7$	−28.3923	−1.53304	−0.766520	+	0.642220i	$0.778013\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	55.0333	1.50847	0.754235	−	0.656605i	$-0.228008\pi$
$11$	55.0333	1.50847	0.754235	+	0.656605i	$0.228008\pi$
$12$	0	0
$13$	−0.430781	−0.00919054	−0.00459527	−	0.999989i	$-0.501463\pi$
$13$	−0.430781	−0.00919054	−0.00459527	+	0.999989i	$0.501463\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	147.636	1.78263	0.891316	−	0.453384i	$-0.149783\pi$
$19$	147.636	1.78263	0.891316	+	0.453384i	$0.149783\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−108.603	−0.984574	−0.492287	−	0.870433i	$-0.663839\pi$
$23$	−108.603	−0.984574	−0.492287	+	0.870433i	$0.663839\pi$
$24$	0	0
$25$	42.1384	0.337108
$26$	0	0
$27$	0	0
$28$	0	0
$29$	107.933	0.691128	0.345564	−	0.938395i	$-0.387688\pi$
$29$	107.933	0.691128	0.345564	+	0.938395i	$0.387688\pi$
$30$	0	0
$31$	−70.1718	−0.406555	−0.203278	−	0.979121i	$-0.565159\pi$
$31$	−70.1718	−0.406555	−0.203278	+	0.979121i	$0.565159\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−367.061	−1.77271
$36$	0	0
$37$	−381.769	−1.69628	−0.848141	−	0.529770i	$-0.822278\pi$
$37$	−381.769	−1.69628	−0.848141	+	0.529770i	$0.822278\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	16.1436	0.0614928	0.0307464	−	0.999527i	$-0.490212\pi$
$41$	16.1436	0.0614928	0.0307464	+	0.999527i	$0.490212\pi$
$42$	0	0
$43$	382.354	1.35601	0.678005	−	0.735057i	$-0.262845\pi$
$43$	382.354	1.35601	0.678005	+	0.735057i	$0.262845\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	455.846	1.41472	0.707362	−	0.706852i	$-0.249885\pi$
$47$	455.846	1.41472	0.707362	+	0.706852i	$0.249885\pi$
$48$	0	0
$49$	463.123	1.35021
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−21.0052	−0.0544392	−0.0272196	−	0.999629i	$-0.508665\pi$
$53$	−21.0052	−0.0544392	−0.0272196	+	0.999629i	$0.508665\pi$
$54$	0	0
$55$	711.482	1.74429
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.91274	0.0218734	0.0109367	−	0.999940i	$-0.496519\pi$
$59$	9.91274	0.0218734	0.0109367	+	0.999940i	$0.496519\pi$
$60$	0	0
$61$	−679.759	−1.42679	−0.713395	−	0.700762i	$-0.752843\pi$
$61$	−679.759	−1.42679	−0.713395	+	0.700762i	$0.752843\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.56922	−0.0106273
$66$	0	0
$67$	708.574	1.29203	0.646016	−	0.763324i	$-0.276434\pi$
$67$	708.574	1.29203	0.646016	+	0.763324i	$0.276434\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−85.0793	−0.142212	−0.0711061	−	0.997469i	$-0.522653\pi$
$71$	−85.0793	−0.142212	−0.0711061	+	0.997469i	$0.522653\pi$
$72$	0	0
$73$	−37.5795	−0.0602514	−0.0301257	−	0.999546i	$-0.509591\pi$
$73$	−37.5795	−0.0602514	−0.0301257	+	0.999546i	$0.509591\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1562.52	−2.31255
$78$	0	0
$79$	685.290	0.975963	0.487982	−	0.872854i	$-0.337733\pi$
$79$	685.290	0.975963	0.487982	+	0.872854i	$0.337733\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1294.89	1.71244	0.856219	−	0.516613i	$-0.172808\pi$
$83$	1294.89	1.71244	0.856219	+	0.516613i	$0.172808\pi$
$84$	0	0
$85$	−219.779	−0.280452
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1572.08	1.87236	0.936182	−	0.351517i	$-0.114334\pi$
$89$	1572.08	1.87236	0.936182	+	0.351517i	$0.114334\pi$
$90$	0	0
$91$	12.2309	0.0140895
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1908.67	2.06132
$96$	0	0
$97$	−175.990	−0.184217	−0.0921085	−	0.995749i	$-0.529361\pi$
$97$	−175.990	−0.184217	−0.0921085	+	0.995749i	$0.529361\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−381.538	−0.375886	−0.187943	−	0.982180i	$-0.560182\pi$
$101$	−381.538	−0.375886	−0.187943	+	0.982180i	$0.560182\pi$
$102$	0	0
$103$	724.246	0.692836	0.346418	−	0.938080i	$-0.387398\pi$
$103$	724.246	0.692836	0.346418	+	0.938080i	$0.387398\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1132.23	−1.02296	−0.511482	−	0.859294i	$-0.670903\pi$
$107$	−1132.23	−1.02296	−0.511482	+	0.859294i	$0.670903\pi$
$108$	0	0
$109$	555.913	0.488503	0.244251	−	0.969712i	$-0.421458\pi$
$109$	555.913	0.488503	0.244251	+	0.969712i	$0.421458\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1156.28	0.962596	0.481298	−	0.876557i	$-0.340165\pi$
$113$	1156.28	0.962596	0.481298	+	0.876557i	$0.340165\pi$
$114$	0	0
$115$	−1404.04	−1.13850
$116$	0	0
$117$	0	0
$118$	0	0
$119$	482.669	0.371817
$120$	0	0
$121$	1697.67	1.27548
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1071.25	−0.766525
$126$	0	0
$127$	−2071.22	−1.44717	−0.723585	−	0.690235i	$-0.757507\pi$
$127$	−2071.22	−1.44717	−0.723585	+	0.690235i	$0.757507\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	498.690	0.332601	0.166300	−	0.986075i	$-0.446818\pi$
$131$	498.690	0.332601	0.166300	+	0.986075i	$0.446818\pi$
$132$	0	0
$133$	−4191.72	−2.73285
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1500.66	0.935837	0.467919	−	0.883772i	$-0.345004\pi$
$137$	1500.66	0.935837	0.467919	+	0.883772i	$0.345004\pi$
$138$	0	0
$139$	−272.043	−0.166003	−0.0830015	−	0.996549i	$-0.526451\pi$
$139$	−272.043	−0.166003	−0.0830015	+	0.996549i	$0.526451\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−23.7073	−0.0138637
$144$	0	0
$145$	1395.38	0.799175
$146$	0	0
$147$	0	0
$148$	0	0
$149$	122.349	0.0672700	0.0336350	−	0.999434i	$-0.489292\pi$
$149$	122.349	0.0672700	0.0336350	+	0.999434i	$0.489292\pi$
$150$	0	0
$151$	−703.944	−0.379378	−0.189689	−	0.981844i	$-0.560748\pi$
$151$	−703.944	−0.379378	−0.189689	+	0.981844i	$0.560748\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−907.195	−0.470114
$156$	0	0
$157$	1785.67	0.907720	0.453860	−	0.891073i	$-0.350047\pi$
$157$	1785.67	0.907720	0.453860	+	0.891073i	$0.350047\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3083.48	1.50939
$162$	0	0
$163$	3588.11	1.72418	0.862092	−	0.506751i	$-0.169154\pi$
$163$	3588.11	1.72418	0.862092	+	0.506751i	$0.169154\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1659.15	0.768794	0.384397	−	0.923168i	$-0.374409\pi$
$167$	1659.15	0.768794	0.384397	+	0.923168i	$0.374409\pi$
$168$	0	0
$169$	−2196.81	−0.999916
$170$	0	0
$171$	0	0
$172$	0	0
$173$	142.148	0.0624702	0.0312351	−	0.999512i	$-0.490056\pi$
$173$	142.148	0.0624702	0.0312351	+	0.999512i	$0.490056\pi$
$174$	0	0
$175$	−1196.41	−0.516799
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−387.749	−0.161909	−0.0809544	−	0.996718i	$-0.525797\pi$
$179$	−387.749	−0.161909	−0.0809544	+	0.996718i	$0.525797\pi$
$180$	0	0
$181$	3683.64	1.51272	0.756360	−	0.654155i	$-0.226976\pi$
$181$	3683.64	1.51272	0.756360	+	0.654155i	$0.226976\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4935.59	−1.96147
$186$	0	0
$187$	−935.566	−0.365858
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1308.05	0.495535	0.247768	−	0.968819i	$-0.420303\pi$
$191$	1308.05	0.495535	0.247768	+	0.968819i	$0.420303\pi$
$192$	0	0
$193$	−3585.28	−1.33717	−0.668586	−	0.743635i	$-0.733100\pi$
$193$	−3585.28	−1.33717	−0.668586	+	0.743635i	$0.733100\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1434.13	0.518667	0.259333	−	0.965788i	$-0.416497\pi$
$197$	1434.13	0.518667	0.259333	+	0.965788i	$0.416497\pi$
$198$	0	0
$199$	1018.55	0.362828	0.181414	−	0.983407i	$-0.441933\pi$
$199$	1018.55	0.362828	0.181414	+	0.983407i	$0.441933\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3064.48	−1.05953
$204$	0	0
$205$	208.708	0.0711062
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8124.89	2.68905
$210$	0	0
$211$	2888.47	0.942421	0.471210	−	0.882021i	$-0.343817\pi$
$211$	2888.47	0.942421	0.471210	+	0.882021i	$0.343817\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4943.15	1.56800
$216$	0	0
$217$	1992.34	0.623266
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.32327	0.00222903
$222$	0	0
$223$	3641.74	1.09358	0.546791	−	0.837269i	$-0.315849\pi$
$223$	3641.74	1.09358	0.546791	+	0.837269i	$0.315849\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5221.59	−1.52674	−0.763368	−	0.645963i	$-0.776456\pi$
$227$	−5221.59	−1.52674	−0.763368	+	0.645963i	$0.776456\pi$
$228$	0	0
$229$	4186.66	1.20813	0.604065	−	0.796935i	$-0.293547\pi$
$229$	4186.66	1.20813	0.604065	+	0.796935i	$0.293547\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3420.08	0.961618	0.480809	−	0.876825i	$-0.340343\pi$
$233$	3420.08	0.961618	0.480809	+	0.876825i	$0.340343\pi$
$234$	0	0
$235$	5893.27	1.63589
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2456.35	0.664803	0.332402	−	0.943138i	$-0.392141\pi$
$239$	2456.35	0.664803	0.332402	+	0.943138i	$0.392141\pi$
$240$	0	0
$241$	2967.19	0.793085	0.396542	−	0.918016i	$-0.370210\pi$
$241$	2967.19	0.793085	0.396542	+	0.918016i	$0.370210\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5987.35	1.56130
$246$	0	0
$247$	−63.5987	−0.0163833
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5652.07	1.42134	0.710668	−	0.703527i	$-0.248393\pi$
$251$	5652.07	1.42134	0.710668	+	0.703527i	$0.248393\pi$
$252$	0	0
$253$	−5976.76	−1.48520
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1480.86	0.359430	0.179715	−	0.983719i	$-0.442482\pi$
$257$	1480.86	0.359430	0.179715	+	0.983719i	$0.442482\pi$
$258$	0	0
$259$	10839.3	2.60047
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1460.73	0.342482	0.171241	−	0.985229i	$-0.445222\pi$
$263$	1460.73	0.342482	0.171241	+	0.985229i	$0.445222\pi$
$264$	0	0
$265$	−271.559	−0.0629499
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6260.73	1.41905	0.709523	−	0.704682i	$-0.248910\pi$
$269$	6260.73	1.41905	0.709523	+	0.704682i	$0.248910\pi$
$270$	0	0
$271$	−269.538	−0.0604179	−0.0302089	−	0.999544i	$-0.509617\pi$
$271$	−269.538	−0.0604179	−0.0302089	+	0.999544i	$0.509617\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2319.02	0.508517
$276$	0	0
$277$	−5097.56	−1.10571	−0.552857	−	0.833276i	$-0.686463\pi$
$277$	−5097.56	−1.10571	−0.552857	+	0.833276i	$0.686463\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1516.13	0.321868	0.160934	−	0.986965i	$-0.448549\pi$
$281$	1516.13	0.321868	0.160934	+	0.986965i	$0.448549\pi$
$282$	0	0
$283$	−7668.88	−1.61084	−0.805420	−	0.592705i	$-0.798060\pi$
$283$	−7668.88	−1.61084	−0.805420	+	0.592705i	$0.798060\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−458.354	−0.0942710
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2580.58	−0.514537	−0.257269	−	0.966340i	$-0.582822\pi$
$293$	−2580.58	−0.514537	−0.257269	+	0.966340i	$0.582822\pi$
$294$	0	0
$295$	128.154	0.0252929
$296$	0	0
$297$	0	0
$298$	0	0
$299$	46.7839	0.00904877
$300$	0	0
$301$	−10855.9	−2.07882
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8788.06	−1.64985
$306$	0	0
$307$	−583.507	−0.108477	−0.0542386	−	0.998528i	$-0.517273\pi$
$307$	−583.507	−0.108477	−0.0542386	+	0.998528i	$0.517273\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	154.973	0.0282563	0.0141282	−	0.999900i	$-0.495503\pi$
$311$	154.973	0.0282563	0.0141282	+	0.999900i	$0.495503\pi$
$312$	0	0
$313$	−5835.86	−1.05387	−0.526936	−	0.849905i	$-0.676659\pi$
$313$	−5835.86	−1.05387	−0.526936	+	0.849905i	$0.676659\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10586.1	−1.87563	−0.937817	−	0.347130i	$-0.887156\pi$
$317$	−10586.1	−1.87563	−0.937817	+	0.347130i	$0.887156\pi$
$318$	0	0
$319$	5939.93	1.04255
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2509.81	−0.432352
$324$	0	0
$325$	−18.1524	−0.00309820
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12942.5	−2.16883
$330$	0	0
$331$	−7094.38	−1.17807	−0.589037	−	0.808106i	$-0.700493\pi$
$331$	−7094.38	−1.17807	−0.589037	+	0.808106i	$0.700493\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9160.59	1.49402
$336$	0	0
$337$	−1579.48	−0.255310	−0.127655	−	0.991819i	$-0.540745\pi$
$337$	−1579.48	−0.255310	−0.127655	+	0.991819i	$0.540745\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3861.78	−0.613277
$342$	0	0
$343$	−3410.57	−0.536890
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1717.38	0.265688	0.132844	−	0.991137i	$-0.457589\pi$
$347$	1717.38	0.265688	0.132844	+	0.991137i	$0.457589\pi$
$348$	0	0
$349$	−6237.70	−0.956724	−0.478362	−	0.878163i	$-0.658769\pi$
$349$	−6237.70	−0.956724	−0.478362	+	0.878163i	$0.658769\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9900.99	−1.49285	−0.746426	−	0.665469i	$-0.768232\pi$
$353$	−9900.99	−1.49285	−0.746426	+	0.665469i	$0.768232\pi$
$354$	0	0
$355$	−1099.92	−0.164445
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7524.52	1.10621	0.553104	−	0.833112i	$-0.313443\pi$
$359$	7524.52	1.10621	0.553104	+	0.833112i	$0.313443\pi$
$360$	0	0
$361$	14937.3	2.17777
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−485.836	−0.0696707
$366$	0	0
$367$	10169.2	1.44640	0.723199	−	0.690640i	$-0.242671\pi$
$367$	10169.2	1.44640	0.723199	+	0.690640i	$0.242671\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	596.385	0.0834576
$372$	0	0
$373$	14115.4	1.95943	0.979716	−	0.200393i	$-0.0642219\pi$
$373$	14115.4	1.95943	0.979716	+	0.200393i	$0.0642219\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−46.4956	−0.00635184
$378$	0	0
$379$	−10480.9	−1.42050	−0.710249	−	0.703951i	$-0.751417\pi$
$379$	−10480.9	−1.42050	−0.710249	+	0.703951i	$0.751417\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9726.14	−1.29760	−0.648802	−	0.760957i	$-0.724730\pi$
$383$	−9726.14	−1.29760	−0.648802	+	0.760957i	$0.724730\pi$
$384$	0	0
$385$	−20200.6	−2.67407
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13217.9	−1.72282	−0.861409	−	0.507912i	$-0.830417\pi$
$389$	−13217.9	−1.72282	−0.861409	+	0.507912i	$0.830417\pi$
$390$	0	0
$391$	1846.24	0.238794
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8859.56	1.12854
$396$	0	0
$397$	5628.14	0.711507	0.355753	−	0.934580i	$-0.384224\pi$
$397$	5628.14	0.711507	0.355753	+	0.934580i	$0.384224\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8890.10	−1.10711	−0.553554	−	0.832813i	$-0.686729\pi$
$401$	−8890.10	−1.10711	−0.553554	+	0.832813i	$0.686729\pi$
$402$	0	0
$403$	30.2286	0.00373647
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21010.0	−2.55879
$408$	0	0
$409$	721.057	0.0871736	0.0435868	−	0.999050i	$-0.486121\pi$
$409$	721.057	0.0871736	0.0435868	+	0.999050i	$0.486121\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−281.446	−0.0335328
$414$	0	0
$415$	16740.6	1.98015
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16306.9	1.90129	0.950647	−	0.310275i	$-0.100421\pi$
$419$	16306.9	1.90129	0.950647	+	0.310275i	$0.100421\pi$
$420$	0	0
$421$	−11568.8	−1.33926	−0.669628	−	0.742697i	$-0.733546\pi$
$421$	−11568.8	−1.33926	−0.669628	+	0.742697i	$0.733546\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−716.353	−0.0817606
$426$	0	0
$427$	19299.9	2.18733
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11733.2	1.31129	0.655647	−	0.755067i	$-0.272396\pi$
$431$	11733.2	1.31129	0.655647	+	0.755067i	$0.272396\pi$
$432$	0	0
$433$	−8900.96	−0.987882	−0.493941	−	0.869495i	$-0.664444\pi$
$433$	−8900.96	−0.987882	−0.493941	+	0.869495i	$0.664444\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16033.6	−1.75513
$438$	0	0
$439$	−8857.07	−0.962927	−0.481464	−	0.876466i	$-0.659895\pi$
$439$	−8857.07	−0.962927	−0.481464	+	0.876466i	$0.659895\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13328.1	−1.42943	−0.714716	−	0.699415i	$-0.753444\pi$
$443$	−13328.1	−1.42943	−0.714716	+	0.699415i	$0.753444\pi$
$444$	0	0
$445$	20324.2	2.16508
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8432.61	0.886323	0.443162	−	0.896442i	$-0.353857\pi$
$449$	8432.61	0.886323	0.443162	+	0.896442i	$0.353857\pi$
$450$	0	0
$451$	888.436	0.0927601
$452$	0	0
$453$	0	0
$454$	0	0
$455$	158.123	0.0162921
$456$	0	0
$457$	6628.94	0.678531	0.339266	−	0.940691i	$-0.389821\pi$
$457$	6628.94	0.678531	0.339266	+	0.940691i	$0.389821\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7327.69	0.740314	0.370157	−	0.928969i	$-0.379304\pi$
$461$	7327.69	0.740314	0.370157	+	0.928969i	$0.379304\pi$
$462$	0	0
$463$	−8996.83	−0.903063	−0.451531	−	0.892255i	$-0.649122\pi$
$463$	−8996.83	−0.903063	−0.451531	+	0.892255i	$0.649122\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12138.4	−1.20278	−0.601388	−	0.798957i	$-0.705386\pi$
$467$	−12138.4	−1.20278	−0.601388	+	0.798957i	$0.705386\pi$
$468$	0	0
$469$	−20118.1	−1.98074
$470$	0	0
$471$	0	0
$472$	0	0
$473$	21042.2	2.04550
$474$	0	0
$475$	6221.14	0.600938
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1644.47	−0.156864	−0.0784320	−	0.996919i	$-0.524991\pi$
$479$	−1644.47	−0.156864	−0.0784320	+	0.996919i	$0.524991\pi$
$480$	0	0
$481$	164.459	0.0155898
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2275.23	−0.213016
$486$	0	0
$487$	−1761.78	−0.163930	−0.0819648	−	0.996635i	$-0.526119\pi$
$487$	−1761.78	−0.163930	−0.0819648	+	0.996635i	$0.526119\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2471.08	−0.227125	−0.113562	−	0.993531i	$-0.536226\pi$
$491$	−2471.08	−0.227125	−0.113562	+	0.993531i	$0.536226\pi$
$492$	0	0
$493$	−1834.87	−0.167623
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2415.60	0.218017
$498$	0	0
$499$	−16582.5	−1.48764	−0.743821	−	0.668379i	$-0.766988\pi$
$499$	−16582.5	−1.48764	−0.743821	+	0.668379i	$0.766988\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19067.9	−1.69025	−0.845126	−	0.534566i	$-0.820475\pi$
$503$	−19067.9	−1.69025	−0.845126	+	0.534566i	$0.820475\pi$
$504$	0	0
$505$	−4932.60	−0.434650
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9295.25	0.809439	0.404720	−	0.914441i	$-0.367369\pi$
$509$	9295.25	0.809439	0.404720	+	0.914441i	$0.367369\pi$
$510$	0	0
$511$	1066.97	0.0923678
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9363.20	0.801149
$516$	0	0
$517$	25086.7	2.13407
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4907.38	−0.412661	−0.206330	−	0.978482i	$-0.566152\pi$
$521$	−4907.38	−0.412661	−0.206330	+	0.978482i	$0.566152\pi$
$522$	0	0
$523$	−11247.5	−0.940381	−0.470191	−	0.882565i	$-0.655815\pi$
$523$	−11247.5	−0.940381	−0.470191	+	0.882565i	$0.655815\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1192.92	0.0986042
$528$	0	0
$529$	−372.488	−0.0306146
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.95435	−0.000565152 0
$534$	0	0
$535$	−14637.7	−1.18289
$536$	0	0
$537$	0	0
$538$	0	0
$539$	25487.2	2.03676
$540$	0	0
$541$	−3992.37	−0.317274	−0.158637	−	0.987337i	$-0.550710\pi$
$541$	−3992.37	−0.317274	−0.158637	+	0.987337i	$0.550710\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7186.95	0.564872
$546$	0	0
$547$	−3014.39	−0.235624	−0.117812	−	0.993036i	$-0.537588\pi$
$547$	−3014.39	−0.235624	−0.117812	+	0.993036i	$0.537588\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15934.8	1.23203
$552$	0	0
$553$	−19457.0	−1.49619
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16174.5	1.23041	0.615203	−	0.788369i	$-0.289074\pi$
$557$	16174.5	1.23041	0.615203	+	0.788369i	$0.289074\pi$
$558$	0	0
$559$	−164.711	−0.0124625
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6626.51	0.496047	0.248023	−	0.968754i	$-0.420219\pi$
$563$	6626.51	0.496047	0.248023	+	0.968754i	$0.420219\pi$
$564$	0	0
$565$	14948.6	1.11308
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2651.54	−0.195357	−0.0976786	−	0.995218i	$-0.531142\pi$
$569$	−2651.54	−0.195357	−0.0976786	+	0.995218i	$0.531142\pi$
$570$	0	0
$571$	7663.21	0.561638	0.280819	−	0.959761i	$-0.409394\pi$
$571$	7663.21	0.561638	0.280819	+	0.959761i	$0.409394\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4576.34	−0.331907
$576$	0	0
$577$	−22136.0	−1.59711	−0.798557	−	0.601919i	$-0.794403\pi$
$577$	−22136.0	−1.59711	−0.798557	+	0.601919i	$0.794403\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−36764.8	−2.62524
$582$	0	0
$583$	−1155.98	−0.0821200
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24994.8	−1.75749	−0.878744	−	0.477294i	$-0.841618\pi$
$587$	−24994.8	−1.75749	−0.878744	+	0.477294i	$0.841618\pi$
$588$	0	0
$589$	−10359.9	−0.724738
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11985.7	−0.830007	−0.415004	−	0.909820i	$-0.636220\pi$
$593$	−11985.7	−0.830007	−0.415004	+	0.909820i	$0.636220\pi$
$594$	0	0
$595$	6240.05	0.429944
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2943.68	−0.200794	−0.100397	−	0.994947i	$-0.532011\pi$
$599$	−2943.68	−0.200794	−0.100397	+	0.994947i	$0.532011\pi$
$600$	0	0
$601$	−4398.52	−0.298535	−0.149267	−	0.988797i	$-0.547692\pi$
$601$	−4398.52	−0.298535	−0.149267	+	0.988797i	$0.547692\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	21947.8	1.47488
$606$	0	0
$607$	19859.6	1.32797	0.663985	−	0.747746i	$-0.268864\pi$
$607$	19859.6	1.32797	0.663985	+	0.747746i	$0.268864\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−196.370	−0.0130021
$612$	0	0
$613$	5581.75	0.367773	0.183887	−	0.982947i	$-0.441132\pi$
$613$	5581.75	0.367773	0.183887	+	0.982947i	$0.441132\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9888.18	0.645192	0.322596	−	0.946537i	$-0.395445\pi$
$617$	9888.18	0.645192	0.322596	+	0.946537i	$0.395445\pi$
$618$	0	0
$619$	9547.62	0.619953	0.309977	−	0.950744i	$-0.399679\pi$
$619$	9547.62	0.619953	0.309977	+	0.950744i	$0.399679\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−44635.0	−2.87041
$624$	0	0
$625$	−19116.7	−1.22347
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6490.08	0.411409
$630$	0	0
$631$	−198.776	−0.0125406	−0.00627032	−	0.999980i	$-0.501996\pi$
$631$	−198.776	−0.0125406	−0.00627032	+	0.999980i	$0.501996\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−26777.1	−1.67341
$636$	0	0
$637$	−199.504	−0.0124092
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15596.6	−0.961043	−0.480522	−	0.876983i	$-0.659553\pi$
$641$	−15596.6	−0.961043	−0.480522	+	0.876983i	$0.659553\pi$
$642$	0	0
$643$	18359.9	1.12604	0.563020	−	0.826443i	$-0.309639\pi$
$643$	18359.9	1.12604	0.563020	+	0.826443i	$0.309639\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23713.7	1.44093	0.720465	−	0.693491i	$-0.243928\pi$
$647$	23713.7	1.44093	0.720465	+	0.693491i	$0.243928\pi$
$648$	0	0
$649$	545.531	0.0329953
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22829.6	−1.36814	−0.684068	−	0.729418i	$-0.739791\pi$
$653$	−22829.6	−1.36814	−0.684068	+	0.729418i	$0.739791\pi$
$654$	0	0
$655$	6447.16	0.384598
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16979.9	1.00371	0.501853	−	0.864953i	$-0.332652\pi$
$659$	16979.9	1.00371	0.501853	+	0.864953i	$0.332652\pi$
$660$	0	0
$661$	−15036.2	−0.884782	−0.442391	−	0.896822i	$-0.645870\pi$
$661$	−15036.2	−0.884782	−0.442391	+	0.896822i	$0.645870\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−54191.4	−3.16008
$666$	0	0
$667$	−11721.8	−0.680467
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−37409.4	−2.15227
$672$	0	0
$673$	−16935.7	−0.970022	−0.485011	−	0.874508i	$-0.661184\pi$
$673$	−16935.7	−0.970022	−0.485011	+	0.874508i	$0.661184\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6514.79	−0.369843	−0.184921	−	0.982753i	$-0.559203\pi$
$677$	−6514.79	−0.369843	−0.184921	+	0.982753i	$0.559203\pi$
$678$	0	0
$679$	4996.75	0.282412
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6261.11	0.350768	0.175384	−	0.984500i	$-0.443883\pi$
$683$	6261.11	0.350768	0.175384	+	0.984500i	$0.443883\pi$
$684$	0	0
$685$	19400.8	1.08214
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.04861	0.000500326 0
$690$	0	0
$691$	12775.3	0.703320	0.351660	−	0.936128i	$-0.385617\pi$
$691$	12775.3	0.703320	0.351660	+	0.936128i	$0.385617\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3517.03	−0.191955
$696$	0	0
$697$	−274.441	−0.0149142
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5939.18	0.320000	0.160000	−	0.987117i	$-0.448851\pi$
$701$	5939.18	0.320000	0.160000	+	0.987117i	$0.448851\pi$
$702$	0	0
$703$	−56362.8	−3.02385
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10832.8	0.576248
$708$	0	0
$709$	−22029.4	−1.16690	−0.583450	−	0.812149i	$-0.698298\pi$
$709$	−22029.4	−1.16690	−0.583450	+	0.812149i	$0.698298\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7620.83	0.400284
$714$	0	0
$715$	−306.493	−0.0160310
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4477.22	−0.232228	−0.116114	−	0.993236i	$-0.537044\pi$
$719$	−4477.22	−0.232228	−0.116114	+	0.993236i	$0.537044\pi$
$720$	0	0
$721$	−20563.0	−1.06215
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4548.14	0.232985
$726$	0	0
$727$	−7878.08	−0.401901	−0.200950	−	0.979601i	$-0.564403\pi$
$727$	−7878.08	−0.401901	−0.200950	+	0.979601i	$0.564403\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6500.02	−0.328881
$732$	0	0
$733$	11332.8	0.571060	0.285530	−	0.958370i	$-0.407830\pi$
$733$	11332.8	0.571060	0.285530	+	0.958370i	$0.407830\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38995.2	1.94899
$738$	0	0
$739$	20027.3	0.996908	0.498454	−	0.866916i	$-0.333901\pi$
$739$	20027.3	0.996908	0.498454	+	0.866916i	$0.333901\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8839.83	0.436477	0.218238	−	0.975896i	$-0.429969\pi$
$743$	8839.83	0.436477	0.218238	+	0.975896i	$0.429969\pi$
$744$	0	0
$745$	1581.75	0.0777865
$746$	0	0
$747$	0	0
$748$	0	0
$749$	32146.7	1.56824
$750$	0	0
$751$	1533.58	0.0745156	0.0372578	−	0.999306i	$-0.488138\pi$
$751$	1533.58	0.0745156	0.0372578	+	0.999306i	$0.488138\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9100.73	−0.438688
$756$	0	0
$757$	16470.7	0.790801	0.395401	−	0.918509i	$-0.370606\pi$
$757$	16470.7	0.790801	0.395401	+	0.918509i	$0.370606\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18629.7	0.887418	0.443709	−	0.896171i	$-0.353662\pi$
$761$	18629.7	0.887418	0.443709	+	0.896171i	$0.353662\pi$
$762$	0	0
$763$	−15783.6	−0.748894
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.27022	−0.000201028 0
$768$	0	0
$769$	−6086.38	−0.285410	−0.142705	−	0.989765i	$-0.545580\pi$
$769$	−6086.38	−0.285410	−0.142705	+	0.989765i	$0.545580\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7472.68	−0.347702	−0.173851	−	0.984772i	$-0.555621\pi$
$773$	−7472.68	−0.347702	−0.173851	+	0.984772i	$0.555621\pi$
$774$	0	0
$775$	−2956.93	−0.137053
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2383.37	0.109619
$780$	0	0
$781$	−4682.20	−0.214523
$782$	0	0
$783$	0	0
$784$	0	0
$785$	23085.5	1.04963
$786$	0	0
$787$	28318.2	1.28263	0.641317	−	0.767276i	$-0.278388\pi$
$787$	28318.2	1.28263	0.641317	+	0.767276i	$0.278388\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−32829.4	−1.47570
$792$	0	0
$793$	292.827	0.0131130
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29551.4	−1.31338	−0.656691	−	0.754160i	$-0.728044\pi$
$797$	−29551.4	−1.31338	−0.656691	+	0.754160i	$0.728044\pi$
$798$	0	0
$799$	−7749.38	−0.343121
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2068.13	−0.0908874
$804$	0	0
$805$	39863.8	1.74536
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−28511.0	−1.23905	−0.619527	−	0.784975i	$-0.712676\pi$
$809$	−28511.0	−1.23905	−0.619527	+	0.784975i	$0.712676\pi$
$810$	0	0
$811$	−29641.1	−1.28340	−0.641700	−	0.766955i	$-0.721771\pi$
$811$	−29641.1	−1.28340	−0.641700	+	0.766955i	$0.721771\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	46387.8	1.99373
$816$	0	0
$817$	56449.1	2.41726
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8825.39	−0.375162	−0.187581	−	0.982249i	$-0.560065\pi$
$821$	−8825.39	−0.375162	−0.187581	+	0.982249i	$0.560065\pi$
$822$	0	0
$823$	−19790.8	−0.838233	−0.419116	−	0.907933i	$-0.637660\pi$
$823$	−19790.8	−0.838233	−0.419116	+	0.907933i	$0.637660\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38948.7	−1.63770	−0.818850	−	0.574008i	$-0.805388\pi$
$827$	−38948.7	−1.63770	−0.818850	+	0.574008i	$0.805388\pi$
$828$	0	0
$829$	34825.0	1.45901	0.729506	−	0.683974i	$-0.239750\pi$
$829$	34825.0	1.45901	0.729506	+	0.683974i	$0.239750\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7873.09	−0.327475
$834$	0	0
$835$	21449.8	0.888982
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17082.9	0.702941	0.351470	−	0.936199i	$-0.385682\pi$
$839$	17082.9	0.702941	0.351470	+	0.936199i	$0.385682\pi$
$840$	0	0
$841$	−12739.4	−0.522342
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−28400.9	−1.15624
$846$	0	0
$847$	−48200.7	−1.95537
$848$	0	0
$849$	0	0
$850$	0	0
$851$	41461.1	1.67012
$852$	0	0
$853$	−38982.3	−1.56475	−0.782373	−	0.622810i	$-0.785991\pi$
$853$	−38982.3	−1.56475	−0.782373	+	0.622810i	$0.785991\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6648.98	0.265023	0.132512	−	0.991181i	$-0.457696\pi$
$857$	6648.98	0.265023	0.132512	+	0.991181i	$0.457696\pi$
$858$	0	0
$859$	−1644.93	−0.0653368	−0.0326684	−	0.999466i	$-0.510401\pi$
$859$	−1644.93	−0.0653368	−0.0326684	+	0.999466i	$0.510401\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24272.3	−0.957402	−0.478701	−	0.877978i	$-0.658892\pi$
$863$	−24272.3	−0.957402	−0.478701	+	0.877978i	$0.658892\pi$
$864$	0	0
$865$	1837.72	0.0722364
$866$	0	0
$867$	0	0
$868$	0	0
$869$	37713.8	1.47221
$870$	0	0
$871$	−305.240	−0.0118745
$872$	0	0
$873$	0	0
$874$	0	0
$875$	30415.3	1.17511
$876$	0	0
$877$	−51065.8	−1.96621	−0.983107	−	0.183030i	$-0.941410\pi$
$877$	−51065.8	−1.96621	−0.983107	+	0.183030i	$0.941410\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10283.5	0.393258	0.196629	−	0.980478i	$-0.437000\pi$
$881$	10283.5	0.393258	0.196629	+	0.980478i	$0.437000\pi$
$882$	0	0
$883$	−5412.08	−0.206264	−0.103132	−	0.994668i	$-0.532886\pi$
$883$	−5412.08	−0.206264	−0.103132	+	0.994668i	$0.532886\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11306.0	−0.427981	−0.213991	−	0.976836i	$-0.568646\pi$
$887$	−11306.0	−0.427981	−0.213991	+	0.976836i	$0.568646\pi$
$888$	0	0
$889$	58806.6	2.21857
$890$	0	0
$891$	0	0
$892$	0	0
$893$	67299.2	2.52193
$894$	0	0
$895$	−5012.89	−0.187221
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7573.87	−0.280982
$900$	0	0
$901$	357.088	0.0132035
$902$	0	0
$903$	0	0
$904$	0	0
$905$	47622.8	1.74921
$906$	0	0
$907$	13122.6	0.480408	0.240204	−	0.970722i	$-0.422786\pi$
$907$	13122.6	0.480408	0.240204	+	0.970722i	$0.422786\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24127.8	0.877485	0.438743	−	0.898613i	$-0.355424\pi$
$911$	24127.8	0.877485	0.438743	+	0.898613i	$0.355424\pi$
$912$	0	0
$913$	71261.9	2.58316
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14159.0	−0.509891
$918$	0	0
$919$	−10597.4	−0.380387	−0.190194	−	0.981747i	$-0.560912\pi$
$919$	−10597.4	−0.380387	−0.190194	+	0.981747i	$0.560912\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	36.6505	0.00130701
$924$	0	0
$925$	−16087.2	−0.571830
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16159.9	−0.570711	−0.285356	−	0.958422i	$-0.592112\pi$
$929$	−16159.9	−0.570711	−0.285356	+	0.958422i	$0.592112\pi$
$930$	0	0
$931$	68373.6	2.40693
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12095.2	−0.423054
$936$	0	0
$937$	12857.2	0.448267	0.224134	−	0.974558i	$-0.428045\pi$
$937$	12857.2	0.448267	0.224134	+	0.974558i	$0.428045\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31043.6	1.07544	0.537722	−	0.843122i	$-0.319285\pi$
$941$	31043.6	1.07544	0.537722	+	0.843122i	$0.319285\pi$
$942$	0	0
$943$	−1753.24	−0.0605442
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39841.0	1.36712	0.683558	−	0.729896i	$-0.260432\pi$
$947$	39841.0	1.36712	0.683558	+	0.729896i	$0.260432\pi$
$948$	0	0
$949$	16.1885	0.000553743 0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26601.9	0.904217	0.452109	−	0.891963i	$-0.350672\pi$
$953$	26601.9	0.904217	0.452109	+	0.891963i	$0.350672\pi$
$954$	0	0
$955$	16910.7	0.573004
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−42607.1	−1.43468
$960$	0	0
$961$	−24866.9	−0.834713
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−46351.2	−1.54622
$966$	0	0
$967$	15998.2	0.532024	0.266012	−	0.963970i	$-0.414294\pi$
$967$	15998.2	0.532024	0.266012	+	0.963970i	$0.414294\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21950.4	0.725461	0.362731	−	0.931894i	$-0.381845\pi$
$971$	21950.4	0.725461	0.362731	+	0.931894i	$0.381845\pi$
$972$	0	0
$973$	7723.94	0.254489
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24955.7	0.817198	0.408599	−	0.912714i	$-0.366017\pi$
$977$	24955.7	0.817198	0.408599	+	0.912714i	$0.366017\pi$
$978$	0	0
$979$	86516.9	2.82440
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−35207.3	−1.14236	−0.571180	−	0.820825i	$-0.693514\pi$
$983$	−35207.3	−1.14236	−0.571180	+	0.820825i	$0.693514\pi$
$984$	0	0
$985$	18540.7	0.599752
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−41524.6	−1.33509
$990$	0	0
$991$	47633.5	1.52687	0.763435	−	0.645885i	$-0.223511\pi$
$991$	47633.5	1.52687	0.763435	+	0.645885i	$0.223511\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13168.0	0.419550
$996$	0	0
$997$	6114.40	0.194228	0.0971139	−	0.995273i	$-0.469039\pi$
$997$	6114.40	0.194228	0.0971139	+	0.995273i	$0.469039\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.4.a.d.1.2		2
3.2	odd	2		136.4.a.a.1.2	✓	2
4.3	odd	2		2448.4.a.z.1.2		2
12.11	even	2		272.4.a.f.1.1		2
24.5	odd	2		1088.4.a.n.1.1		2
24.11	even	2		1088.4.a.p.1.2		2
51.50	odd	2		2312.4.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.a.1.2	✓	2	3.2	odd	2
272.4.a.f.1.1		2	12.11	even	2
1088.4.a.n.1.1		2	24.5	odd	2
1088.4.a.p.1.2		2	24.11	even	2
1224.4.a.d.1.2		2	1.1	even	1	trivial
2312.4.a.b.1.1		2	51.50	odd	2
2448.4.a.z.1.2		2	4.3	odd	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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1224.a (trivial)

\(f(q)\)	\(=\)	\(q+12.9282 q^{5} -28.3923 q^{7} +55.0333 q^{11} -0.430781 q^{13} -17.0000 q^{17} +147.636 q^{19} -108.603 q^{23} +42.1384 q^{25} +107.933 q^{29} -70.1718 q^{31} -367.061 q^{35} -381.769 q^{37} +16.1436 q^{41} +382.354 q^{43} +455.846 q^{47} +463.123 q^{49} -21.0052 q^{53} +711.482 q^{55} +9.91274 q^{59} -679.759 q^{61} -5.56922 q^{65} +708.574 q^{67} -85.0793 q^{71} -37.5795 q^{73} -1562.52 q^{77} +685.290 q^{79} +1294.89 q^{83} -219.779 q^{85} +1572.08 q^{89} +12.2309 q^{91} +1908.67 q^{95} -175.990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{5} - 36 q^{7} + 20 q^{11} - 84 q^{13} - 34 q^{17} + 32 q^{19} - 44 q^{23} - 82 q^{25} + 396 q^{29} + 116 q^{31} - 360 q^{35} - 140 q^{37} + 60 q^{41} + 640 q^{43} + 496 q^{47} + 178 q^{49} - 236 q^{53}+ \cdots + 36 q^{97}+O(q^{100}) \) 2 * q + 12 * q^5 - 36 * q^7 + 20 * q^11 - 84 * q^13 - 34 * q^17 + 32 * q^19 - 44 * q^23 - 82 * q^25 + 396 * q^29 + 116 * q^31 - 360 * q^35 - 140 * q^37 + 60 * q^41 + 640 * q^43 + 496 * q^47 + 178 * q^49 - 236 * q^53 + 744 * q^55 - 576 * q^59 - 348 * q^61 + 72 * q^65 + 1528 * q^67 + 876 * q^71 - 380 * q^73 - 1296 * q^77 + 172 * q^79 + 1024 * q^83 - 204 * q^85 + 844 * q^89 + 648 * q^91 + 2016 * q^95 + 36 * q^97

Introduction

L-functions

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