LMFDB - Embedded newform 2160.4.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, 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("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.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:	$127.444125612$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.697.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 5 $ x^3 - 7*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.782816$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} -11.6263 q^{7} -47.4851 q^{11} +84.2727 q^{13} +26.9488 q^{17} +152.060 q^{19} -177.082 q^{23} +25.0000 q^{25} -60.0913 q^{29} +92.1613 q^{31} +58.1315 q^{35} -221.798 q^{37} -115.099 q^{41} -383.827 q^{43} +317.275 q^{47} -207.829 q^{49} +257.738 q^{53} +237.426 q^{55} -642.576 q^{59} -662.092 q^{61} -421.363 q^{65} -597.173 q^{67} -500.152 q^{71} +989.949 q^{73} +552.076 q^{77} +517.554 q^{79} -605.867 q^{83} -134.744 q^{85} +1519.15 q^{89} -979.780 q^{91} -760.301 q^{95} +742.835 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 24 q^{7} + 6 q^{11} + 48 q^{13} + 27 q^{17} + 195 q^{19} - 27 q^{23} + 75 q^{25} - 60 q^{29} + 279 q^{31} - 120 q^{35} - 138 q^{37} - 66 q^{41} - 222 q^{43} - 264 q^{47} - 237 q^{49} + 507 q^{53}+ \cdots - 336 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 24 * q^7 + 6 * q^11 + 48 * q^13 + 27 * q^17 + 195 * q^19 - 27 * q^23 + 75 * q^25 - 60 * q^29 + 279 * q^31 - 120 * q^35 - 138 * q^37 - 66 * q^41 - 222 * q^43 - 264 * q^47 - 237 * q^49 + 507 * q^53 - 30 * q^55 - 960 * q^59 + 543 * q^61 - 240 * q^65 + 1086 * q^67 - 1818 * q^71 + 1362 * q^73 + 1776 * q^77 + 129 * q^79 - 1569 * q^83 - 135 * q^85 + 1770 * q^89 - 1488 * q^91 - 975 * q^95 - 336 * 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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−11.6263	−0.627761	−0.313881	−	0.949462i	$-0.601629\pi$
$7$	−11.6263	−0.627761	−0.313881	+	0.949462i	$0.601629\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−47.4851	−1.30157	−0.650787	−	0.759261i	$-0.725561\pi$
$11$	−47.4851	−1.30157	−0.650787	+	0.759261i	$0.725561\pi$
$12$	0	0
$13$	84.2727	1.79793	0.898963	−	0.438024i	$-0.144322\pi$
$13$	84.2727	1.79793	0.898963	+	0.438024i	$0.144322\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	26.9488	0.384474	0.192237	−	0.981349i	$-0.438426\pi$
$17$	26.9488	0.384474	0.192237	+	0.981349i	$0.438426\pi$
$18$	0	0
$19$	152.060	1.83605	0.918027	−	0.396518i	$-0.129782\pi$
$19$	152.060	1.83605	0.918027	+	0.396518i	$0.129782\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−177.082	−1.60539	−0.802697	−	0.596387i	$-0.796603\pi$
$23$	−177.082	−1.60539	−0.802697	+	0.596387i	$0.796603\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−60.0913	−0.384782	−0.192391	−	0.981318i	$-0.561624\pi$
$29$	−60.0913	−0.384782	−0.192391	+	0.981318i	$0.561624\pi$
$30$	0	0
$31$	92.1613	0.533957	0.266978	−	0.963703i	$-0.413975\pi$
$31$	92.1613	0.533957	0.266978	+	0.963703i	$0.413975\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	58.1315	0.280743
$36$	0	0
$37$	−221.798	−0.985496	−0.492748	−	0.870172i	$-0.664008\pi$
$37$	−221.798	−0.985496	−0.492748	+	0.870172i	$0.664008\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−115.099	−0.438426	−0.219213	−	0.975677i	$-0.570349\pi$
$41$	−115.099	−0.438426	−0.219213	+	0.975677i	$0.570349\pi$
$42$	0	0
$43$	−383.827	−1.36123	−0.680617	−	0.732639i	$-0.738288\pi$
$43$	−383.827	−1.36123	−0.680617	+	0.732639i	$0.738288\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	317.275	0.984666	0.492333	−	0.870407i	$-0.336144\pi$
$47$	317.275	0.984666	0.492333	+	0.870407i	$0.336144\pi$
$48$	0	0
$49$	−207.829	−0.605916
$50$	0	0
$51$	0	0
$52$	0	0
$53$	257.738	0.667981	0.333991	−	0.942576i	$-0.391605\pi$
$53$	257.738	0.667981	0.333991	+	0.942576i	$0.391605\pi$
$54$	0	0
$55$	237.426	0.582081
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−642.576	−1.41790	−0.708951	−	0.705257i	$-0.750831\pi$
$59$	−642.576	−1.41790	−0.708951	+	0.705257i	$0.750831\pi$
$60$	0	0
$61$	−662.092	−1.38971	−0.694854	−	0.719150i	$-0.744531\pi$
$61$	−662.092	−1.38971	−0.694854	+	0.719150i	$0.744531\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−421.363	−0.804057
$66$	0	0
$67$	−597.173	−1.08890	−0.544450	−	0.838794i	$-0.683261\pi$
$67$	−597.173	−1.08890	−0.544450	+	0.838794i	$0.683261\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−500.152	−0.836016	−0.418008	−	0.908443i	$-0.637272\pi$
$71$	−500.152	−0.836016	−0.418008	+	0.908443i	$0.637272\pi$
$72$	0	0
$73$	989.949	1.58719	0.793594	−	0.608447i	$-0.208207\pi$
$73$	989.949	1.58719	0.793594	+	0.608447i	$0.208207\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	552.076	0.817077
$78$	0	0
$79$	517.554	0.737081	0.368540	−	0.929612i	$-0.379858\pi$
$79$	517.554	0.737081	0.368540	+	0.929612i	$0.379858\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−605.867	−0.801235	−0.400617	−	0.916245i	$-0.631204\pi$
$83$	−605.867	−0.801235	−0.400617	+	0.916245i	$0.631204\pi$
$84$	0	0
$85$	−134.744	−0.171942
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1519.15	1.80932	0.904658	−	0.426138i	$-0.140126\pi$
$89$	1519.15	1.80932	0.904658	+	0.426138i	$0.140126\pi$
$90$	0	0
$91$	−979.780	−1.12867
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−760.301	−0.821108
$96$	0	0
$97$	742.835	0.777561	0.388781	−	0.921330i	$-0.372896\pi$
$97$	742.835	0.777561	0.388781	+	0.921330i	$0.372896\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1066.26	1.05047	0.525233	−	0.850959i	$-0.323978\pi$
$101$	1066.26	1.05047	0.525233	+	0.850959i	$0.323978\pi$
$102$	0	0
$103$	471.508	0.451059	0.225529	−	0.974236i	$-0.427589\pi$
$103$	471.508	0.451059	0.225529	+	0.974236i	$0.427589\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1746.59	1.57803	0.789014	−	0.614375i	$-0.210592\pi$
$107$	1746.59	1.57803	0.789014	+	0.614375i	$0.210592\pi$
$108$	0	0
$109$	1596.56	1.40296	0.701480	−	0.712689i	$-0.252523\pi$
$109$	1596.56	1.40296	0.701480	+	0.712689i	$0.252523\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1622.95	−1.35110	−0.675550	−	0.737315i	$-0.736094\pi$
$113$	−1622.95	−1.35110	−0.675550	+	0.737315i	$0.736094\pi$
$114$	0	0
$115$	885.408	0.717954
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−313.315	−0.241358
$120$	0	0
$121$	923.837	0.694092
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1306.32	0.912730	0.456365	−	0.889793i	$-0.349151\pi$
$127$	1306.32	0.912730	0.456365	+	0.889793i	$0.349151\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	539.259	0.359659	0.179829	−	0.983698i	$-0.442445\pi$
$131$	539.259	0.359659	0.179829	+	0.983698i	$0.442445\pi$
$132$	0	0
$133$	−1767.90	−1.15260
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2877.22	1.79429	0.897144	−	0.441738i	$-0.145638\pi$
$137$	2877.22	1.79429	0.897144	+	0.441738i	$0.145638\pi$
$138$	0	0
$139$	1479.89	0.903038	0.451519	−	0.892261i	$-0.350882\pi$
$139$	1479.89	0.903038	0.451519	+	0.892261i	$0.350882\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4001.70	−2.34013
$144$	0	0
$145$	300.457	0.172080
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1398.13	0.768719	0.384359	−	0.923184i	$-0.374422\pi$
$149$	1398.13	0.768719	0.384359	+	0.923184i	$0.374422\pi$
$150$	0	0
$151$	2251.44	1.21337	0.606687	−	0.794941i	$-0.292498\pi$
$151$	2251.44	1.21337	0.606687	+	0.794941i	$0.292498\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−460.806	−0.238793
$156$	0	0
$157$	764.698	0.388723	0.194362	−	0.980930i	$-0.437736\pi$
$157$	764.698	0.388723	0.194362	+	0.980930i	$0.437736\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2058.81	1.00780
$162$	0	0
$163$	1862.15	0.894817	0.447408	−	0.894330i	$-0.352347\pi$
$163$	1862.15	0.894817	0.447408	+	0.894330i	$0.352347\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3324.33	−1.54038	−0.770192	−	0.637812i	$-0.779840\pi$
$167$	−3324.33	−1.54038	−0.770192	+	0.637812i	$0.779840\pi$
$168$	0	0
$169$	4904.89	2.23254
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1344.67	0.590943	0.295471	−	0.955352i	$-0.404523\pi$
$173$	1344.67	0.590943	0.295471	+	0.955352i	$0.404523\pi$
$174$	0	0
$175$	−290.658	−0.125552
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2086.39	−0.871195	−0.435598	−	0.900142i	$-0.643463\pi$
$179$	−2086.39	−0.871195	−0.435598	+	0.900142i	$0.643463\pi$
$180$	0	0
$181$	698.713	0.286933	0.143467	−	0.989655i	$-0.454175\pi$
$181$	698.713	0.286933	0.143467	+	0.989655i	$0.454175\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1108.99	0.440727
$186$	0	0
$187$	−1279.67	−0.500421
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2267.66	−0.859067	−0.429534	−	0.903051i	$-0.641322\pi$
$191$	−2267.66	−0.859067	−0.429534	+	0.903051i	$0.641322\pi$
$192$	0	0
$193$	−3747.64	−1.39772	−0.698862	−	0.715256i	$-0.746310\pi$
$193$	−3747.64	−1.39772	−0.698862	+	0.715256i	$0.746310\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−293.859	−0.106277	−0.0531385	−	0.998587i	$-0.516922\pi$
$197$	−293.859	−0.106277	−0.0531385	+	0.998587i	$0.516922\pi$
$198$	0	0
$199$	−2552.33	−0.909197	−0.454598	−	0.890697i	$-0.650217\pi$
$199$	−2552.33	−0.909197	−0.454598	+	0.890697i	$0.650217\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	698.640	0.241551
$204$	0	0
$205$	575.496	0.196070
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7220.60	−2.38976
$210$	0	0
$211$	5416.57	1.76726	0.883630	−	0.468185i	$-0.155092\pi$
$211$	5416.57	1.76726	0.883630	+	0.468185i	$0.155092\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1919.14	0.608763
$216$	0	0
$217$	−1071.49	−0.335197
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2271.05	0.691256
$222$	0	0
$223$	−4739.25	−1.42316	−0.711578	−	0.702607i	$-0.752019\pi$
$223$	−4739.25	−1.42316	−0.711578	+	0.702607i	$0.752019\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2489.12	−0.727790	−0.363895	−	0.931440i	$-0.618553\pi$
$227$	−2489.12	−0.727790	−0.363895	+	0.931440i	$0.618553\pi$
$228$	0	0
$229$	4239.27	1.22331	0.611657	−	0.791123i	$-0.290503\pi$
$229$	4239.27	1.22331	0.611657	+	0.791123i	$0.290503\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1200.46	0.337531	0.168765	−	0.985656i	$-0.446022\pi$
$233$	1200.46	0.337531	0.168765	+	0.985656i	$0.446022\pi$
$234$	0	0
$235$	−1586.37	−0.440356
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4609.67	1.24759	0.623796	−	0.781587i	$-0.285589\pi$
$239$	4609.67	1.24759	0.623796	+	0.781587i	$0.285589\pi$
$240$	0	0
$241$	−2121.92	−0.567157	−0.283578	−	0.958949i	$-0.591522\pi$
$241$	−2121.92	−0.567157	−0.283578	+	0.958949i	$0.591522\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1039.15	0.270974
$246$	0	0
$247$	12814.5	3.30109
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5032.56	1.26555	0.632773	−	0.774337i	$-0.281916\pi$
$251$	5032.56	1.26555	0.632773	+	0.774337i	$0.281916\pi$
$252$	0	0
$253$	8408.74	2.08954
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5721.18	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$257$	5721.18	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$258$	0	0
$259$	2578.69	0.618657
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3348.24	0.785025	0.392512	−	0.919747i	$-0.371606\pi$
$263$	3348.24	0.785025	0.392512	+	0.919747i	$0.371606\pi$
$264$	0	0
$265$	−1288.69	−0.298730
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1726.71	0.391373	0.195686	−	0.980667i	$-0.437307\pi$
$269$	1726.71	0.391373	0.195686	+	0.980667i	$0.437307\pi$
$270$	0	0
$271$	−6347.21	−1.42275	−0.711375	−	0.702812i	$-0.751927\pi$
$271$	−6347.21	−1.42275	−0.711375	+	0.702812i	$0.751927\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1187.13	−0.260315
$276$	0	0
$277$	−591.832	−0.128374	−0.0641872	−	0.997938i	$-0.520445\pi$
$277$	−591.832	−0.128374	−0.0641872	+	0.997938i	$0.520445\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1405.24	0.298326	0.149163	−	0.988813i	$-0.452342\pi$
$281$	1405.24	0.298326	0.149163	+	0.988813i	$0.452342\pi$
$282$	0	0
$283$	−1417.53	−0.297750	−0.148875	−	0.988856i	$-0.547565\pi$
$283$	−1417.53	−0.297750	−0.148875	+	0.988856i	$0.547565\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1338.18	0.275227
$288$	0	0
$289$	−4186.76	−0.852180
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5127.59	1.02238	0.511189	−	0.859468i	$-0.329205\pi$
$293$	5127.59	1.02238	0.511189	+	0.859468i	$0.329205\pi$
$294$	0	0
$295$	3212.88	0.634105
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14923.2	−2.88638
$300$	0	0
$301$	4462.49	0.854531
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3310.46	0.621497
$306$	0	0
$307$	−2481.35	−0.461296	−0.230648	−	0.973037i	$-0.574085\pi$
$307$	−2481.35	−0.461296	−0.230648	+	0.973037i	$0.574085\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−201.357	−0.0367135	−0.0183567	−	0.999832i	$-0.505843\pi$
$311$	−201.357	−0.0367135	−0.0183567	+	0.999832i	$0.505843\pi$
$312$	0	0
$313$	1850.89	0.334244	0.167122	−	0.985936i	$-0.446553\pi$
$313$	1850.89	0.334244	0.167122	+	0.985936i	$0.446553\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2257.74	−0.400024	−0.200012	−	0.979793i	$-0.564098\pi$
$317$	−2257.74	−0.400024	−0.200012	+	0.979793i	$0.564098\pi$
$318$	0	0
$319$	2853.44	0.500822
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4097.85	0.705915
$324$	0	0
$325$	2106.82	0.359585
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3688.73	−0.618135
$330$	0	0
$331$	4038.32	0.670592	0.335296	−	0.942113i	$-0.391164\pi$
$331$	4038.32	0.670592	0.335296	+	0.942113i	$0.391164\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2985.86	0.486971
$336$	0	0
$337$	−3552.00	−0.574154	−0.287077	−	0.957908i	$-0.592684\pi$
$337$	−3552.00	−0.574154	−0.287077	+	0.957908i	$0.592684\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4376.29	−0.694983
$342$	0	0
$343$	6404.11	1.00813
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−781.347	−0.120879	−0.0604394	−	0.998172i	$-0.519250\pi$
$347$	−781.347	−0.120879	−0.0604394	+	0.998172i	$0.519250\pi$
$348$	0	0
$349$	2492.19	0.382246	0.191123	−	0.981566i	$-0.438787\pi$
$349$	2492.19	0.382246	0.191123	+	0.981566i	$0.438787\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9230.11	−1.39170	−0.695849	−	0.718189i	$-0.744972\pi$
$353$	−9230.11	−1.39170	−0.695849	+	0.718189i	$0.744972\pi$
$354$	0	0
$355$	2500.76	0.373878
$356$	0	0
$357$	0	0
$358$	0	0
$359$	925.350	0.136039	0.0680196	−	0.997684i	$-0.478332\pi$
$359$	925.350	0.136039	0.0680196	+	0.997684i	$0.478332\pi$
$360$	0	0
$361$	16263.3	2.37109
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4949.75	−0.709812
$366$	0	0
$367$	9070.86	1.29018	0.645089	−	0.764108i	$-0.276820\pi$
$367$	9070.86	1.29018	0.645089	+	0.764108i	$0.276820\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2996.54	−0.419333
$372$	0	0
$373$	9388.07	1.30321	0.651603	−	0.758560i	$-0.274097\pi$
$373$	9388.07	1.30321	0.651603	+	0.758560i	$0.274097\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5064.06	−0.691810
$378$	0	0
$379$	−10172.2	−1.37866	−0.689328	−	0.724449i	$-0.742094\pi$
$379$	−10172.2	−1.37866	−0.689328	+	0.724449i	$0.742094\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1505.63	0.200873	0.100436	−	0.994943i	$-0.467976\pi$
$383$	1505.63	0.200873	0.100436	+	0.994943i	$0.467976\pi$
$384$	0	0
$385$	−2760.38	−0.365408
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3179.65	0.414433	0.207216	−	0.978295i	$-0.433560\pi$
$389$	3179.65	0.414433	0.207216	+	0.978295i	$0.433560\pi$
$390$	0	0
$391$	−4772.15	−0.617232
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2587.77	−0.329632
$396$	0	0
$397$	5360.23	0.677638	0.338819	−	0.940852i	$-0.389973\pi$
$397$	5360.23	0.677638	0.338819	+	0.940852i	$0.389973\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12550.4	−1.56294	−0.781468	−	0.623945i	$-0.785529\pi$
$401$	−12550.4	−1.56294	−0.781468	+	0.623945i	$0.785529\pi$
$402$	0	0
$403$	7766.68	0.960015
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10532.1	1.28270
$408$	0	0
$409$	−5395.42	−0.652289	−0.326144	−	0.945320i	$-0.605750\pi$
$409$	−5395.42	−0.652289	−0.326144	+	0.945320i	$0.605750\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7470.78	0.890105
$414$	0	0
$415$	3029.33	0.358323
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4697.84	0.547743	0.273871	−	0.961766i	$-0.411696\pi$
$419$	4697.84	0.547743	0.273871	+	0.961766i	$0.411696\pi$
$420$	0	0
$421$	8845.86	1.02404	0.512020	−	0.858974i	$-0.328897\pi$
$421$	8845.86	1.02404	0.512020	+	0.858974i	$0.328897\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	673.721	0.0768948
$426$	0	0
$427$	7697.69	0.872406
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5676.98	−0.634456	−0.317228	−	0.948349i	$-0.602752\pi$
$431$	−5676.98	−0.634456	−0.317228	+	0.948349i	$0.602752\pi$
$432$	0	0
$433$	17273.3	1.91710	0.958549	−	0.284927i	$-0.0919695\pi$
$433$	17273.3	1.91710	0.958549	+	0.284927i	$0.0919695\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−26927.1	−2.94759
$438$	0	0
$439$	3583.55	0.389598	0.194799	−	0.980843i	$-0.437595\pi$
$439$	3583.55	0.389598	0.194799	+	0.980843i	$0.437595\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6937.40	0.744031	0.372016	−	0.928227i	$-0.378667\pi$
$443$	6937.40	0.744031	0.372016	+	0.928227i	$0.378667\pi$
$444$	0	0
$445$	−7595.73	−0.809151
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12445.1	−1.30806	−0.654032	−	0.756467i	$-0.726924\pi$
$449$	−12445.1	−1.30806	−0.654032	+	0.756467i	$0.726924\pi$
$450$	0	0
$451$	5465.50	0.570644
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4898.90	0.504756
$456$	0	0
$457$	−12112.2	−1.23979	−0.619896	−	0.784684i	$-0.712825\pi$
$457$	−12112.2	−1.23979	−0.619896	+	0.784684i	$0.712825\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4932.76	−0.498355	−0.249178	−	0.968458i	$-0.580160\pi$
$461$	−4932.76	−0.498355	−0.249178	+	0.968458i	$0.580160\pi$
$462$	0	0
$463$	12428.0	1.24747	0.623734	−	0.781637i	$-0.285615\pi$
$463$	12428.0	1.24747	0.623734	+	0.781637i	$0.285615\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4549.10	0.450765	0.225383	−	0.974270i	$-0.427637\pi$
$467$	4549.10	0.450765	0.225383	+	0.974270i	$0.427637\pi$
$468$	0	0
$469$	6942.91	0.683569
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18226.1	1.77175
$474$	0	0
$475$	3801.51	0.367211
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5122.02	0.488583	0.244292	−	0.969702i	$-0.421445\pi$
$479$	5122.02	0.488583	0.244292	+	0.969702i	$0.421445\pi$
$480$	0	0
$481$	−18691.5	−1.77185
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3714.17	−0.347736
$486$	0	0
$487$	8650.25	0.804888	0.402444	−	0.915445i	$-0.368161\pi$
$487$	8650.25	0.804888	0.402444	+	0.915445i	$0.368161\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	985.959	0.0906226	0.0453113	−	0.998973i	$-0.485572\pi$
$491$	985.959	0.0906226	0.0453113	+	0.998973i	$0.485572\pi$
$492$	0	0
$493$	−1619.39	−0.147939
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5814.92	0.524819
$498$	0	0
$499$	−3131.41	−0.280925	−0.140462	−	0.990086i	$-0.544859\pi$
$499$	−3131.41	−0.280925	−0.140462	+	0.990086i	$0.544859\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	13765.8	1.22025	0.610126	−	0.792304i	$-0.291119\pi$
$503$	13765.8	1.22025	0.610126	+	0.792304i	$0.291119\pi$
$504$	0	0
$505$	−5331.31	−0.469783
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8241.51	0.717679	0.358839	−	0.933399i	$-0.383173\pi$
$509$	8241.51	0.717679	0.358839	+	0.933399i	$0.383173\pi$
$510$	0	0
$511$	−11509.4	−0.996376
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2357.54	−0.201720
$516$	0	0
$517$	−15065.8	−1.28161
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2556.69	−0.214991	−0.107496	−	0.994206i	$-0.534283\pi$
$521$	−2556.69	−0.214991	−0.107496	+	0.994206i	$0.534283\pi$
$522$	0	0
$523$	−8627.71	−0.721345	−0.360672	−	0.932693i	$-0.617453\pi$
$523$	−8627.71	−0.721345	−0.360672	+	0.932693i	$0.617453\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2483.64	0.205292
$528$	0	0
$529$	19190.9	1.57729
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9699.72	−0.788258
$534$	0	0
$535$	−8732.94	−0.705716
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9868.79	0.788643
$540$	0	0
$541$	1265.74	0.100589	0.0502943	−	0.998734i	$-0.483984\pi$
$541$	1265.74	0.100589	0.0502943	+	0.998734i	$0.483984\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7982.79	−0.627423
$546$	0	0
$547$	6990.29	0.546405	0.273202	−	0.961957i	$-0.411917\pi$
$547$	6990.29	0.546405	0.273202	+	0.961957i	$0.411917\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9137.50	−0.706481
$552$	0	0
$553$	−6017.24	−0.462711
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19679.0	1.49699	0.748496	−	0.663139i	$-0.230776\pi$
$557$	19679.0	1.49699	0.748496	+	0.663139i	$0.230776\pi$
$558$	0	0
$559$	−32346.2	−2.44740
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12913.2	0.966652	0.483326	−	0.875440i	$-0.339429\pi$
$563$	12913.2	0.966652	0.483326	+	0.875440i	$0.339429\pi$
$564$	0	0
$565$	8114.74	0.604230
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19908.8	−1.46682	−0.733411	−	0.679786i	$-0.762073\pi$
$569$	−19908.8	−1.46682	−0.733411	+	0.679786i	$0.762073\pi$
$570$	0	0
$571$	13278.9	0.973212	0.486606	−	0.873621i	$-0.338235\pi$
$571$	13278.9	0.973212	0.486606	+	0.873621i	$0.338235\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4427.04	−0.321079
$576$	0	0
$577$	−2538.04	−0.183120	−0.0915598	−	0.995800i	$-0.529185\pi$
$577$	−2538.04	−0.183120	−0.0915598	+	0.995800i	$0.529185\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7043.99	0.502984
$582$	0	0
$583$	−12238.7	−0.869426
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15312.4	1.07668	0.538341	−	0.842727i	$-0.319051\pi$
$587$	15312.4	1.07668	0.538341	+	0.842727i	$0.319051\pi$
$588$	0	0
$589$	14014.1	0.980373
$590$	0	0
$591$	0	0
$592$	0	0
$593$	196.207	0.0135873	0.00679363	−	0.999977i	$-0.497838\pi$
$593$	196.207	0.0135873	0.00679363	+	0.999977i	$0.497838\pi$
$594$	0	0
$595$	1566.58	0.107939
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13738.3	−0.937113	−0.468557	−	0.883433i	$-0.655226\pi$
$599$	−13738.3	−0.937113	−0.468557	+	0.883433i	$0.655226\pi$
$600$	0	0
$601$	1702.20	0.115531	0.0577655	−	0.998330i	$-0.481602\pi$
$601$	1702.20	0.115531	0.0577655	+	0.998330i	$0.481602\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4619.18	−0.310408
$606$	0	0
$607$	11802.7	0.789223	0.394611	−	0.918848i	$-0.370879\pi$
$607$	11802.7	0.789223	0.394611	+	0.918848i	$0.370879\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26737.6	1.77036
$612$	0	0
$613$	15365.9	1.01244	0.506218	−	0.862406i	$-0.331043\pi$
$613$	15365.9	1.01244	0.506218	+	0.862406i	$0.331043\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17151.5	1.11911	0.559555	−	0.828793i	$-0.310972\pi$
$617$	17151.5	1.11911	0.559555	+	0.828793i	$0.310972\pi$
$618$	0	0
$619$	−10081.9	−0.654649	−0.327324	−	0.944912i	$-0.606147\pi$
$619$	−10081.9	−0.654649	−0.327324	+	0.944912i	$0.606147\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17662.1	−1.13582
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5977.20	−0.378898
$630$	0	0
$631$	15510.6	0.978554	0.489277	−	0.872128i	$-0.337261\pi$
$631$	15510.6	0.978554	0.489277	+	0.872128i	$0.337261\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6531.58	−0.408185
$636$	0	0
$637$	−17514.3	−1.08939
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10534.8	−0.649143	−0.324571	−	0.945861i	$-0.605220\pi$
$641$	−10534.8	−0.649143	−0.324571	+	0.945861i	$0.605220\pi$
$642$	0	0
$643$	627.807	0.0385044	0.0192522	−	0.999815i	$-0.493871\pi$
$643$	627.807	0.0385044	0.0192522	+	0.999815i	$0.493871\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2880.41	−0.175024	−0.0875121	−	0.996163i	$-0.527892\pi$
$647$	−2880.41	−0.175024	−0.0875121	+	0.996163i	$0.527892\pi$
$648$	0	0
$649$	30512.8	1.84550
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10664.9	−0.639127	−0.319564	−	0.947565i	$-0.603536\pi$
$653$	−10664.9	−0.639127	−0.319564	+	0.947565i	$0.603536\pi$
$654$	0	0
$655$	−2696.30	−0.160844
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4300.38	0.254202	0.127101	−	0.991890i	$-0.459433\pi$
$659$	4300.38	0.254202	0.127101	+	0.991890i	$0.459433\pi$
$660$	0	0
$661$	−10325.3	−0.607578	−0.303789	−	0.952739i	$-0.598252\pi$
$661$	−10325.3	−0.607578	−0.303789	+	0.952739i	$0.598252\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8839.49	0.515460
$666$	0	0
$667$	10641.1	0.617727
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31439.5	1.80881
$672$	0	0
$673$	−13740.9	−0.787030	−0.393515	−	0.919318i	$-0.628741\pi$
$673$	−13740.9	−0.787030	−0.393515	+	0.919318i	$0.628741\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31303.5	−1.77709	−0.888547	−	0.458785i	$-0.848285\pi$
$677$	−31303.5	−1.77709	−0.888547	+	0.458785i	$0.848285\pi$
$678$	0	0
$679$	−8636.42	−0.488123
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8369.31	0.468877	0.234438	−	0.972131i	$-0.424675\pi$
$683$	8369.31	0.468877	0.234438	+	0.972131i	$0.424675\pi$
$684$	0	0
$685$	−14386.1	−0.802430
$686$	0	0
$687$	0	0
$688$	0	0
$689$	21720.3	1.20098
$690$	0	0
$691$	24106.1	1.32712	0.663559	−	0.748124i	$-0.269045\pi$
$691$	24106.1	1.32712	0.663559	+	0.748124i	$0.269045\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7399.43	−0.403851
$696$	0	0
$697$	−3101.79	−0.168563
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30713.6	1.65483	0.827417	−	0.561589i	$-0.189810\pi$
$701$	30713.6	1.65483	0.827417	+	0.561589i	$0.189810\pi$
$702$	0	0
$703$	−33726.7	−1.80942
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12396.7	−0.659442
$708$	0	0
$709$	−518.034	−0.0274403	−0.0137201	−	0.999906i	$-0.504367\pi$
$709$	−518.034	−0.0274403	−0.0137201	+	0.999906i	$0.504367\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−16320.1	−0.857211
$714$	0	0
$715$	20008.5	1.04654
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1692.51	−0.0877885	−0.0438943	−	0.999036i	$-0.513976\pi$
$719$	−1692.51	−0.0877885	−0.0438943	+	0.999036i	$0.513976\pi$
$720$	0	0
$721$	−5481.89	−0.283157
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1502.28	−0.0769564
$726$	0	0
$727$	9191.77	0.468919	0.234459	−	0.972126i	$-0.424668\pi$
$727$	9191.77	0.468919	0.234459	+	0.972126i	$0.424668\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−10343.7	−0.523359
$732$	0	0
$733$	−1047.13	−0.0527649	−0.0263824	−	0.999652i	$-0.508399\pi$
$733$	−1047.13	−0.0527649	−0.0263824	+	0.999652i	$0.508399\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28356.8	1.41728
$738$	0	0
$739$	−21956.9	−1.09296	−0.546479	−	0.837473i	$-0.684032\pi$
$739$	−21956.9	−1.09296	−0.546479	+	0.837473i	$0.684032\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	19304.1	0.953160	0.476580	−	0.879131i	$-0.341876\pi$
$743$	19304.1	0.953160	0.476580	+	0.879131i	$0.341876\pi$
$744$	0	0
$745$	−6990.64	−0.343781
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20306.4	−0.990625
$750$	0	0
$751$	−21883.0	−1.06328	−0.531640	−	0.846970i	$-0.678424\pi$
$751$	−21883.0	−1.06328	−0.531640	+	0.846970i	$0.678424\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11257.2	−0.542637
$756$	0	0
$757$	13738.5	0.659624	0.329812	−	0.944047i	$-0.393015\pi$
$757$	13738.5	0.659624	0.329812	+	0.944047i	$0.393015\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20167.9	0.960689	0.480345	−	0.877080i	$-0.340512\pi$
$761$	20167.9	0.960689	0.480345	+	0.877080i	$0.340512\pi$
$762$	0	0
$763$	−18562.1	−0.880724
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−54151.6	−2.54928
$768$	0	0
$769$	−28655.9	−1.34377	−0.671885	−	0.740655i	$-0.734515\pi$
$769$	−28655.9	−1.34377	−0.671885	+	0.740655i	$0.734515\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16224.0	0.754898	0.377449	−	0.926030i	$-0.376801\pi$
$773$	16224.0	0.754898	0.377449	+	0.926030i	$0.376801\pi$
$774$	0	0
$775$	2304.03	0.106791
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17502.0	−0.804974
$780$	0	0
$781$	23749.8	1.08814
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3823.49	−0.173842
$786$	0	0
$787$	−3401.97	−0.154088	−0.0770440	−	0.997028i	$-0.524548\pi$
$787$	−3401.97	−0.154088	−0.0770440	+	0.997028i	$0.524548\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18868.9	0.848168
$792$	0	0
$793$	−55796.3	−2.49859
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9328.18	−0.414581	−0.207290	−	0.978279i	$-0.566465\pi$
$797$	−9328.18	−0.414581	−0.207290	+	0.978279i	$0.566465\pi$
$798$	0	0
$799$	8550.19	0.378578
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−47007.9	−2.06584
$804$	0	0
$805$	−10294.0	−0.450704
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21366.3	−0.928552	−0.464276	−	0.885690i	$-0.653686\pi$
$809$	−21366.3	−0.928552	−0.464276	+	0.885690i	$0.653686\pi$
$810$	0	0
$811$	−34119.9	−1.47733	−0.738664	−	0.674074i	$-0.764543\pi$
$811$	−34119.9	−1.47733	−0.738664	+	0.674074i	$0.764543\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9310.77	−0.400174
$816$	0	0
$817$	−58364.9	−2.49930
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21760.8	0.925038	0.462519	−	0.886609i	$-0.346946\pi$
$821$	21760.8	0.925038	0.462519	+	0.886609i	$0.346946\pi$
$822$	0	0
$823$	3225.85	0.136629	0.0683147	−	0.997664i	$-0.478238\pi$
$823$	3225.85	0.136629	0.0683147	+	0.997664i	$0.478238\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5208.23	0.218994	0.109497	−	0.993987i	$-0.465076\pi$
$827$	5208.23	0.218994	0.109497	+	0.993987i	$0.465076\pi$
$828$	0	0
$829$	27899.5	1.16886	0.584432	−	0.811442i	$-0.301317\pi$
$829$	27899.5	1.16886	0.584432	+	0.811442i	$0.301317\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5600.75	−0.232959
$834$	0	0
$835$	16621.6	0.688881
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26244.4	−1.07993	−0.539964	−	0.841688i	$-0.681562\pi$
$839$	−26244.4	−1.07993	−0.539964	+	0.841688i	$0.681562\pi$
$840$	0	0
$841$	−20778.0	−0.851943
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−24524.4	−0.998422
$846$	0	0
$847$	−10740.8	−0.435724
$848$	0	0
$849$	0	0
$850$	0	0
$851$	39276.4	1.58211
$852$	0	0
$853$	37048.2	1.48711	0.743556	−	0.668674i	$-0.233138\pi$
$853$	37048.2	1.48711	0.743556	+	0.668674i	$0.233138\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16705.8	−0.665879	−0.332939	−	0.942948i	$-0.608040\pi$
$857$	−16705.8	−0.665879	−0.332939	+	0.942948i	$0.608040\pi$
$858$	0	0
$859$	−13768.5	−0.546887	−0.273443	−	0.961888i	$-0.588163\pi$
$859$	−13768.5	−0.546887	−0.273443	+	0.961888i	$0.588163\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37455.4	1.47740	0.738700	−	0.674034i	$-0.235440\pi$
$863$	37455.4	1.47740	0.738700	+	0.674034i	$0.235440\pi$
$864$	0	0
$865$	−6723.33	−0.264278
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−24576.1	−0.959364
$870$	0	0
$871$	−50325.4	−1.95776
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1453.29	0.0561487
$876$	0	0
$877$	−33351.2	−1.28414	−0.642070	−	0.766646i	$-0.721924\pi$
$877$	−33351.2	−1.28414	−0.642070	+	0.766646i	$0.721924\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	45140.2	1.72623	0.863116	−	0.505005i	$-0.168509\pi$
$881$	45140.2	1.72623	0.863116	+	0.505005i	$0.168509\pi$
$882$	0	0
$883$	−47316.4	−1.80331	−0.901656	−	0.432454i	$-0.857648\pi$
$883$	−47316.4	−1.80331	−0.901656	+	0.432454i	$0.857648\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17513.8	−0.662973	−0.331487	−	0.943460i	$-0.607550\pi$
$887$	−17513.8	−0.662973	−0.331487	+	0.943460i	$0.607550\pi$
$888$	0	0
$889$	−15187.6	−0.572977
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48244.9	1.80790
$894$	0	0
$895$	10431.9	0.389610
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5538.09	−0.205457
$900$	0	0
$901$	6945.74	0.256821
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3493.57	−0.128321
$906$	0	0
$907$	10576.5	0.387196	0.193598	−	0.981081i	$-0.437984\pi$
$907$	10576.5	0.387196	0.193598	+	0.981081i	$0.437984\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−33191.6	−1.20712	−0.603560	−	0.797317i	$-0.706252\pi$
$911$	−33191.6	−1.20712	−0.603560	+	0.797317i	$0.706252\pi$
$912$	0	0
$913$	28769.7	1.04287
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6269.59	−0.225780
$918$	0	0
$919$	51842.8	1.86087	0.930434	−	0.366460i	$-0.119430\pi$
$919$	51842.8	1.86087	0.930434	+	0.366460i	$0.119430\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−42149.2	−1.50310
$924$	0	0
$925$	−5544.95	−0.197099
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−9039.52	−0.319243	−0.159622	−	0.987178i	$-0.551027\pi$
$929$	−9039.52	−0.319243	−0.159622	+	0.987178i	$0.551027\pi$
$930$	0	0
$931$	−31602.5	−1.11249
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6398.35	0.223795
$936$	0	0
$937$	−50041.7	−1.74471	−0.872353	−	0.488876i	$-0.837407\pi$
$937$	−50041.7	−1.74471	−0.872353	+	0.488876i	$0.837407\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12305.6	−0.426304	−0.213152	−	0.977019i	$-0.568373\pi$
$941$	−12305.6	−0.426304	−0.213152	+	0.977019i	$0.568373\pi$
$942$	0	0
$943$	20382.0	0.703847
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12109.2	0.415519	0.207759	−	0.978180i	$-0.433383\pi$
$947$	12109.2	0.415519	0.207759	+	0.978180i	$0.433383\pi$
$948$	0	0
$949$	83425.7	2.85365
$950$	0	0
$951$	0	0
$952$	0	0
$953$	51371.2	1.74615	0.873073	−	0.487589i	$-0.162123\pi$
$953$	51371.2	1.74615	0.873073	+	0.487589i	$0.162123\pi$
$954$	0	0
$955$	11338.3	0.384187
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−33451.4	−1.12638
$960$	0	0
$961$	−21297.3	−0.714890
$962$	0	0
$963$	0	0
$964$	0	0
$965$	18738.2	0.625082
$966$	0	0
$967$	27271.5	0.906921	0.453460	−	0.891276i	$-0.350189\pi$
$967$	27271.5	0.906921	0.453460	+	0.891276i	$0.350189\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41605.1	−1.37505	−0.687524	−	0.726161i	$-0.741303\pi$
$971$	−41605.1	−1.37505	−0.687524	+	0.726161i	$0.741303\pi$
$972$	0	0
$973$	−17205.6	−0.566893
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41821.2	1.36948	0.684738	−	0.728790i	$-0.259917\pi$
$977$	41821.2	1.36948	0.684738	+	0.728790i	$0.259917\pi$
$978$	0	0
$979$	−72136.8	−2.35496
$980$	0	0
$981$	0	0
$982$	0	0
$983$	37312.9	1.21068	0.605340	−	0.795967i	$-0.293037\pi$
$983$	37312.9	1.21068	0.605340	+	0.795967i	$0.293037\pi$
$984$	0	0
$985$	1469.29	0.0475285
$986$	0	0
$987$	0	0
$988$	0	0
$989$	67968.8	2.18532
$990$	0	0
$991$	32598.1	1.04492	0.522459	−	0.852664i	$-0.325015\pi$
$991$	32598.1	1.04492	0.522459	+	0.852664i	$0.325015\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12761.7	0.406605
$996$	0	0
$997$	41856.8	1.32961	0.664804	−	0.747018i	$-0.268515\pi$
$997$	41856.8	1.32961	0.664804	+	0.747018i	$0.268515\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bl.1.1		3
3.2	odd	2		2160.4.a.bt.1.1		3
4.3	odd	2		1080.4.a.c.1.3	✓	3
12.11	even	2		1080.4.a.i.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.c.1.3	✓	3	4.3	odd	2
1080.4.a.i.1.3	yes	3	12.11	even	2
2160.4.a.bl.1.1		3	1.1	even	1	trivial
2160.4.a.bt.1.1		3	3.2	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -11.6263 q^{7} -47.4851 q^{11} +84.2727 q^{13} +26.9488 q^{17} +152.060 q^{19} -177.082 q^{23} +25.0000 q^{25} -60.0913 q^{29} +92.1613 q^{31} +58.1315 q^{35} -221.798 q^{37} -115.099 q^{41} -383.827 q^{43} +317.275 q^{47} -207.829 q^{49} +257.738 q^{53} +237.426 q^{55} -642.576 q^{59} -662.092 q^{61} -421.363 q^{65} -597.173 q^{67} -500.152 q^{71} +989.949 q^{73} +552.076 q^{77} +517.554 q^{79} -605.867 q^{83} -134.744 q^{85} +1519.15 q^{89} -979.780 q^{91} -760.301 q^{95} +742.835 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 24 q^{7} + 6 q^{11} + 48 q^{13} + 27 q^{17} + 195 q^{19} - 27 q^{23} + 75 q^{25} - 60 q^{29} + 279 q^{31} - 120 q^{35} - 138 q^{37} - 66 q^{41} - 222 q^{43} - 264 q^{47} - 237 q^{49} + 507 q^{53}+ \cdots - 336 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 24 * q^7 + 6 * q^11 + 48 * q^13 + 27 * q^17 + 195 * q^19 - 27 * q^23 + 75 * q^25 - 60 * q^29 + 279 * q^31 - 120 * q^35 - 138 * q^37 - 66 * q^41 - 222 * q^43 - 264 * q^47 - 237 * q^49 + 507 * q^53 - 30 * q^55 - 960 * q^59 + 543 * q^61 - 240 * q^65 + 1086 * q^67 - 1818 * q^71 + 1362 * q^73 + 1776 * q^77 + 129 * q^79 - 1569 * q^83 - 135 * q^85 + 1770 * q^89 - 1488 * q^91 - 975 * q^95 - 336 * q^97

Introduction

L-functions

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