LMFDB - Embedded newform 2160.3.bs.a.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(881,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5, 0]))

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

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

chi := DirichletCharacter("2160.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.bs (of order $6$, degree $2$, 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:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 25 $ x^4 - 5*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			881.2
Root			$-1.93649 - 1.11803i$ of defining polynomial
Character	$\chi$	$=$	2160.881
Dual form			2160.3.bs.a.1601.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.93649 + 1.11803i) q^{5} +(1.87298 + 3.24410i) q^{7} +O(q^{10})$	$q+(1.93649 + 1.11803i) q^{5} +(1.87298 + 3.24410i) q^{7} +(-10.1190 + 5.84218i) q^{11} +(1.12702 - 1.95205i) q^{13} -11.6844i q^{17} +26.7460 q^{19} +(-17.2379 - 9.95231i) q^{23} +(2.50000 + 4.33013i) q^{25} +(38.2379 - 22.0767i) q^{29} +(-26.1109 + 45.2254i) q^{31} +8.37624i q^{35} +14.0000 q^{37} +(22.5000 + 12.9904i) q^{41} +(20.9919 + 36.3591i) q^{43} +(39.3810 - 22.7367i) q^{47} +(17.4839 - 30.2829i) q^{49} +10.8323i q^{53} -26.1270 q^{55} +(-31.8810 - 18.4065i) q^{59} +(22.6190 + 39.1772i) q^{61} +(4.36492 - 2.52009i) q^{65} +(-49.9758 + 86.5606i) q^{67} +102.603i q^{71} -13.7621 q^{73} +(-37.9052 - 21.8846i) q^{77} +(28.3810 + 49.1574i) q^{79} +(-78.0000 + 45.0333i) q^{83} +(13.0635 - 22.6267i) q^{85} +95.2349i q^{89} +8.44353 q^{91} +(51.7933 + 29.9029i) q^{95} +(50.8488 + 88.0727i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} + 6 q^{11} + 20 q^{13} + 76 q^{19} + 24 q^{23} + 10 q^{25} + 60 q^{29} + 4 q^{31} + 56 q^{37} + 90 q^{41} + 22 q^{43} + 204 q^{47} - 54 q^{49} - 120 q^{55} - 174 q^{59} + 44 q^{61} - 60 q^{65} - 14 q^{67} - 148 q^{73} - 384 q^{77} + 160 q^{79} - 312 q^{83} + 60 q^{85} - 400 q^{91} + 60 q^{95} + 2 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 + 6 * q^11 + 20 * q^13 + 76 * q^19 + 24 * q^23 + 10 * q^25 + 60 * q^29 + 4 * q^31 + 56 * q^37 + 90 * q^41 + 22 * q^43 + 204 * q^47 - 54 * q^49 - 120 * q^55 - 174 * q^59 + 44 * q^61 - 60 * q^65 - 14 * q^67 - 148 * q^73 - 384 * q^77 + 160 * q^79 - 312 * q^83 + 60 * q^85 - 400 * q^91 + 60 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	1.93649	+	1.11803i	0.387298	+	0.223607i
$5$	1.93649	+	1.11803i	0.387298	+	0.223607i
$6$		0			0
$7$	1.87298	+	3.24410i	0.267569	+	0.463443i	0.968234	−	0.250048i	$-0.0804463\pi$
$7$	1.87298	+	3.24410i	0.267569	+	0.463443i	−0.700664	+	0.713491i	$0.747113\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−10.1190	+	5.84218i	−0.919905	+	0.531107i	−0.883605	−	0.468234i	$-0.844891\pi$
$11$	−10.1190	+	5.84218i	−0.919905	+	0.531107i	−0.0362999	+	0.999341i	$0.511557\pi$
$12$		0			0
$13$	1.12702	−	1.95205i	0.0866936	−	0.150158i	−0.819418	−	0.573196i	$-0.805703\pi$
$13$	1.12702	−	1.95205i	0.0866936	−	0.150158i	0.906112	+	0.423039i	$0.139037\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	11.6844i		−	0.687315i	−0.939095	−	0.343658i	$-0.888334\pi$
$17$		−	11.6844i		−	0.687315i	0.939095	−	0.343658i	$-0.111666\pi$
$18$		0			0
$19$	26.7460			1.40768			0.703841	−	0.710357i	$-0.251467\pi$
$19$	26.7460			1.40768			0.703841	+	0.710357i	$0.251467\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−17.2379	−	9.95231i	−0.749474	−	0.432709i	0.0760299	−	0.997106i	$-0.475776\pi$
$23$	−17.2379	−	9.95231i	−0.749474	−	0.432709i	−0.825504	+	0.564397i	$0.809109\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	38.2379	−	22.0767i	1.31855	−	0.761264i	0.335053	−	0.942199i	$-0.391246\pi$
$29$	38.2379	−	22.0767i	1.31855	−	0.761264i	0.983495	+	0.180935i	$0.0579124\pi$
$30$		0			0
$31$	−26.1109	+	45.2254i	−0.842287	+	1.45888i	0.0456704	+	0.998957i	$0.485458\pi$
$31$	−26.1109	+	45.2254i	−0.842287	+	1.45888i	−0.887957	+	0.459927i	$0.847876\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			8.37624i			0.239321i
$36$		0			0
$37$	14.0000			0.378378			0.189189	−	0.981941i	$-0.439414\pi$
$37$	14.0000			0.378378			0.189189	+	0.981941i	$0.439414\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	22.5000	+	12.9904i	0.548780	+	0.316839i	0.748630	−	0.662988i	$-0.230712\pi$
$41$	22.5000	+	12.9904i	0.548780	+	0.316839i	−0.199849	+	0.979827i	$0.564045\pi$
$42$		0			0
$43$	20.9919	+	36.3591i	0.488184	+	0.845560i	0.999908	−	0.0135900i	$-0.00432598\pi$
$43$	20.9919	+	36.3591i	0.488184	+	0.845560i	−0.511723	+	0.859150i	$0.670993\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	39.3810	−	22.7367i	0.837895	−	0.483759i	−0.0186534	−	0.999826i	$-0.505938\pi$
$47$	39.3810	−	22.7367i	0.837895	−	0.483759i	0.856548	+	0.516067i	$0.172605\pi$
$48$		0			0
$49$	17.4839	−	30.2829i	0.356814	−	0.618019i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			10.8323i			0.204383i	0.994765	+	0.102192i	$0.0325855\pi$
$53$			10.8323i			0.204383i	−0.994765	+	0.102192i	$0.967415\pi$
$54$		0			0
$55$	−26.1270			−0.475037
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−31.8810	−	18.4065i	−0.540357	−	0.311975i	0.204867	−	0.978790i	$-0.434324\pi$
$59$	−31.8810	−	18.4065i	−0.540357	−	0.311975i	−0.745224	+	0.666815i	$0.767657\pi$
$60$		0			0
$61$	22.6190	+	39.1772i	0.370802	+	0.642249i	0.989689	−	0.143232i	$-0.0457494\pi$
$61$	22.6190	+	39.1772i	0.370802	+	0.642249i	−0.618887	+	0.785480i	$0.712416\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.36492	−	2.52009i	0.0671526	−	0.0387706i
$66$		0			0
$67$	−49.9758	+	86.5606i	−0.745907	+	1.29195i	0.203862	+	0.979000i	$0.434651\pi$
$67$	−49.9758	+	86.5606i	−0.745907	+	1.29195i	−0.949770	+	0.312950i	$0.898683\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			102.603i			1.44511i	0.691312	+	0.722557i	$0.257033\pi$
$71$			102.603i			1.44511i	−0.691312	+	0.722557i	$0.742967\pi$
$72$		0			0
$73$	−13.7621			−0.188522			−0.0942610	−	0.995548i	$-0.530049\pi$
$73$	−13.7621			−0.188522			−0.0942610	+	0.995548i	$0.530049\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−37.9052	−	21.8846i	−0.492276	−	0.284216i
$78$		0			0
$79$	28.3810	+	49.1574i	0.359254	+	0.622246i	0.987836	−	0.155497i	$-0.0496979\pi$
$79$	28.3810	+	49.1574i	0.359254	+	0.622246i	−0.628583	+	0.777743i	$0.716365\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−78.0000	+	45.0333i	−0.939759	+	0.542570i	−0.889885	−	0.456185i	$-0.849215\pi$
$83$	−78.0000	+	45.0333i	−0.939759	+	0.542570i	−0.0498743	+	0.998756i	$0.515882\pi$
$84$		0			0
$85$	13.0635	−	22.6267i	0.153688	−	0.266196i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			95.2349i			1.07005i	0.844835	+	0.535027i	$0.179699\pi$
$89$			95.2349i			1.07005i	−0.844835	+	0.535027i	$0.820301\pi$
$90$		0			0
$91$	8.44353			0.0927861
$92$		0			0
$93$		0			0
$94$		0			0
$95$	51.7933	+	29.9029i	0.545193	+	0.314767i
$96$		0			0
$97$	50.8488	+	88.0727i	0.524214	+	0.907966i	0.999603	+	0.0281897i	$0.00897424\pi$
$97$	50.8488	+	88.0727i	0.524214	+	0.907966i	−0.475388	+	0.879776i	$0.657692\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	87.7621	−	50.6695i	0.868932	−	0.501678i	0.00193860	−	0.999998i	$-0.499383\pi$
$101$	87.7621	−	50.6695i	0.868932	−	0.501678i	0.866993	+	0.498320i	$0.166050\pi$
$102$		0			0
$103$	8.14315	−	14.1043i	0.0790597	−	0.136935i	−0.823785	−	0.566903i	$-0.808142\pi$
$103$	8.14315	−	14.1043i	0.0790597	−	0.136935i	0.902844	+	0.429967i	$0.141475\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	84.8705i		−	0.793182i	−0.917995	−	0.396591i	$-0.870193\pi$
$107$		−	84.8705i		−	0.793182i	0.917995	−	0.396591i	$-0.129807\pi$
$108$		0			0
$109$	−161.968			−1.48594			−0.742971	−	0.669323i	$-0.766584\pi$
$109$	−161.968			−1.48594			−0.742971	+	0.669323i	$0.766584\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000	−	3.46410i	−0.0530973	−	0.0306558i	0.473216	−	0.880946i	$-0.343093\pi$
$113$	−6.00000	−	3.46410i	−0.0530973	−	0.0306558i	−0.526314	+	0.850290i	$0.676426\pi$
$114$		0			0
$115$	−22.2540	−	38.5451i	−0.193513	−	0.335175i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	37.9052	−	21.8846i	0.318532	−	0.183904i
$120$		0			0
$121$	7.76210	−	13.4444i	0.0641496	−	0.111110i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	225.903			1.77877			0.889383	−	0.457163i	$-0.151135\pi$
$127$	225.903			1.77877			0.889383	+	0.457163i	$0.151135\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−81.7137	−	47.1774i	−0.623769	−	0.360133i	0.154566	−	0.987982i	$-0.450602\pi$
$131$	−81.7137	−	47.1774i	−0.623769	−	0.360133i	−0.778335	+	0.627849i	$0.783935\pi$
$132$		0			0
$133$	50.0948	+	86.7667i	0.376652	+	0.652381i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−179.595	+	103.689i	−1.31091	+	0.756855i	−0.982247	−	0.187592i	$-0.939932\pi$
$137$	−179.595	+	103.689i	−1.31091	+	0.756855i	−0.328664	+	0.944447i	$0.606598\pi$
$138$		0			0
$139$	16.3569	−	28.3309i	0.117675	−	0.203819i	−0.801171	−	0.598436i	$-0.795789\pi$
$139$	16.3569	−	28.3309i	0.117675	−	0.203819i	0.918846	+	0.394616i	$0.129123\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			26.3369i			0.184174i
$144$		0			0
$145$	98.7298			0.680895
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.3327	−	10.5844i	−0.123038	−	0.0710360i	0.437218	−	0.899356i	$-0.355964\pi$
$149$	−18.3327	−	10.5844i	−0.123038	−	0.0710360i	−0.560256	+	0.828320i	$0.689297\pi$
$150$		0			0
$151$	−137.714	−	238.527i	−0.912011	−	1.57965i	−0.811219	−	0.584742i	$-0.801196\pi$
$151$	−137.714	−	238.527i	−0.912011	−	1.57965i	−0.100792	−	0.994908i	$-0.532138\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−101.127	+	58.3857i	−0.652432	+	0.376682i
$156$		0			0
$157$	−89.9677	+	155.829i	−0.573043	+	0.992539i	0.423208	+	0.906032i	$0.360904\pi$
$157$	−89.9677	+	155.829i	−0.573043	+	0.992539i	−0.996251	+	0.0865070i	$0.972430\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	74.5620i		−	0.463118i
$162$		0			0
$163$	264.411			1.62216			0.811078	−	0.584939i	$-0.198881\pi$
$163$	264.411			1.62216			0.811078	+	0.584939i	$0.198881\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	149.522	+	86.3267i	0.895342	+	0.516926i	0.875686	−	0.482881i	$-0.160410\pi$
$167$	149.522	+	86.3267i	0.895342	+	0.516926i	0.0196561	+	0.999807i	$0.493743\pi$
$168$		0			0
$169$	81.9597	+	141.958i	0.484968	+	0.839990i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−254.903	+	147.168i	−1.47343	+	0.850685i	−0.999553	−	0.0299069i	$-0.990479\pi$
$173$	−254.903	+	147.168i	−1.47343	+	0.850685i	−0.473876	+	0.880591i	$0.657146\pi$
$174$		0			0
$175$	−9.36492	+	16.2205i	−0.0535138	+	0.0926886i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			64.1138i			0.358178i	0.983833	+	0.179089i	$0.0573150\pi$
$179$			64.1138i			0.358178i	−0.983833	+	0.179089i	$0.942685\pi$
$180$		0			0
$181$	291.206			1.60887			0.804435	−	0.594040i	$-0.202468\pi$
$181$	291.206			1.60887			0.804435	+	0.594040i	$0.202468\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	27.1109	+	15.6525i	0.146545	+	0.0846080i
$186$		0			0
$187$	68.2621	+	118.233i	0.365038	+	0.632264i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.8085	−	7.97231i	0.0722956	−	0.0417399i	−0.463416	−	0.886141i	$-0.653377\pi$
$191$	13.8085	−	7.97231i	0.0722956	−	0.0417399i	0.535712	+	0.844401i	$0.320043\pi$
$192$		0			0
$193$	122.357	−	211.928i	0.633973	−	1.09807i	−0.352758	−	0.935714i	$-0.614756\pi$
$193$	122.357	−	211.928i	0.633973	−	1.09807i	0.986732	−	0.162360i	$-0.0519104\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			210.374i			1.06789i	0.845519	+	0.533945i	$0.179291\pi$
$197$			210.374i			1.06789i	−0.845519	+	0.533945i	$0.820709\pi$
$198$		0			0
$199$	102.730			0.516230			0.258115	−	0.966114i	$-0.416899\pi$
$199$	102.730			0.516230			0.258115	+	0.966114i	$0.416899\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	143.238	+	82.6984i	0.705605	+	0.407381i
$204$		0			0
$205$	29.0474	+	50.3115i	0.141695	+	0.245422i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−270.641	+	156.255i	−1.29493	+	0.747630i
$210$		0			0
$211$	−113.984	+	197.426i	−0.540208	+	0.935668i	0.458684	+	0.888600i	$0.348321\pi$
$211$	−113.984	+	197.426i	−0.540208	+	0.935668i	−0.998892	+	0.0470680i	$0.985012\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			93.8788i			0.436645i
$216$		0			0
$217$	−195.621			−0.901479
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−22.8085	−	13.1685i	−0.103206	−	0.0595858i
$222$		0			0
$223$	−139.681	−	241.935i	−0.626374	−	1.08491i	−0.988273	−	0.152695i	$-0.951205\pi$
$223$	−139.681	−	241.935i	−0.626374	−	1.08491i	0.361899	−	0.932217i	$-0.382128\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	215.262	−	124.282i	0.948291	−	0.547496i	0.0557415	−	0.998445i	$-0.482248\pi$
$227$	215.262	−	124.282i	0.948291	−	0.547496i	0.892550	+	0.450949i	$0.148914\pi$
$228$		0			0
$229$	−78.8891	+	136.640i	−0.344494	+	0.596681i	−0.985262	−	0.171054i	$-0.945283\pi$
$229$	−78.8891	+	136.640i	−0.344494	+	0.596681i	0.640768	+	0.767735i	$0.278616\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			332.854i			1.42856i	0.699861	+	0.714279i	$0.253245\pi$
$233$			332.854i			1.42856i	−0.699861	+	0.714279i	$0.746755\pi$
$234$		0			0
$235$	101.681			0.432687
$236$		0			0
$237$		0			0
$238$		0			0
$239$	207.714	+	119.924i	0.869095	+	0.501772i	0.867047	−	0.498226i	$-0.166015\pi$
$239$	207.714	+	119.924i	0.869095	+	0.501772i	0.00204759	+	0.999998i	$0.499348\pi$
$240$		0			0
$241$	94.7540	+	164.119i	0.393170	+	0.680991i	0.992866	−	0.119237i	$-0.0380450\pi$
$241$	94.7540	+	164.119i	0.393170	+	0.680991i	−0.599696	+	0.800228i	$0.704712\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	67.7147	−	39.0951i	0.276387	−	0.159572i
$246$		0			0
$247$	30.1431	−	52.2095i	0.122037	−	0.211374i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			447.114i			1.78133i	0.454661	+	0.890664i	$0.349760\pi$
$251$			447.114i			1.78133i	−0.454661	+	0.890664i	$0.650240\pi$
$252$		0			0
$253$	232.573			0.919259
$254$		0			0
$255$		0			0
$256$		0			0
$257$	387.308	+	223.613i	1.50704	+	0.870088i	0.999967	+	0.00818336i	$0.00260487\pi$
$257$	387.308	+	223.613i	1.50704	+	0.870088i	0.507070	+	0.861905i	$0.330728\pi$
$258$		0			0
$259$	26.2218	+	45.4174i	0.101242	+	0.175357i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−307.046	+	177.273i	−1.16748	+	0.674043i	−0.953084	−	0.302705i	$-0.902110\pi$
$263$	−307.046	+	177.273i	−1.16748	+	0.674043i	−0.214392	+	0.976748i	$0.568777\pi$
$264$		0			0
$265$	−12.1109	+	20.9767i	−0.0457014	+	0.0791572i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			227.367i			0.845229i	0.906310	+	0.422614i	$0.138888\pi$
$269$			227.367i			0.845229i	−0.906310	+	0.422614i	$0.861112\pi$
$270$		0			0
$271$	−228.573			−0.843441			−0.421721	−	0.906726i	$-0.638574\pi$
$271$	−228.573			−0.843441			−0.421721	+	0.906726i	$0.638574\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−50.5948	−	29.2109i	−0.183981	−	0.106221i
$276$		0			0
$277$	−220.222	−	381.435i	−0.795024	−	1.37702i	−0.922824	−	0.385223i	$-0.874125\pi$
$277$	−220.222	−	381.435i	−0.795024	−	1.37702i	0.127799	−	0.991800i	$-0.459209\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	272.190	−	157.149i	0.968646	−	0.559248i	0.0698227	−	0.997559i	$-0.477757\pi$
$281$	272.190	−	157.149i	0.968646	−	0.559248i	0.898823	+	0.438311i	$0.144423\pi$
$282$		0			0
$283$	133.984	−	232.067i	0.473441	−	0.820024i	−0.526097	−	0.850425i	$-0.676345\pi$
$283$	133.984	−	232.067i	0.473441	−	0.820024i	0.999538	+	0.0304006i	$0.00967830\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			97.3231i			0.339105i
$288$		0			0
$289$	152.476			0.527598
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−268.524	−	155.033i	−0.916465	−	0.529121i	−0.0339593	−	0.999423i	$-0.510812\pi$
$293$	−268.524	−	155.033i	−0.916465	−	0.529121i	−0.882506	+	0.470302i	$0.844145\pi$
$294$		0			0
$295$	−41.1583	−	71.2882i	−0.139520	−	0.241655i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−38.8548	+	22.4328i	−0.129949	+	0.0750262i
$300$		0			0
$301$	−78.6351	+	136.200i	−0.261246	+	0.452492i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			101.155i			0.331656i
$306$		0			0
$307$	495.952			1.61548			0.807739	−	0.589541i	$-0.200691\pi$
$307$	495.952			1.61548			0.807739	+	0.589541i	$0.200691\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−195.762	−	113.023i	−0.629460	−	0.363419i	0.151083	−	0.988521i	$-0.451724\pi$
$311$	−195.762	−	113.023i	−0.629460	−	0.363419i	−0.780543	+	0.625102i	$0.785057\pi$
$312$		0			0
$313$	−163.579	−	283.326i	−0.522615	−	0.905196i	−0.999654	−	0.0263139i	$-0.991623\pi$
$313$	−163.579	−	283.326i	−0.522615	−	0.905196i	0.477038	−	0.878882i	$-0.341710\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.09475	−	4.09616i	0.0223809	−	0.0129216i	−0.488768	−	0.872414i	$-0.662553\pi$
$317$	7.09475	−	4.09616i	0.0223809	−	0.0129216i	0.511149	+	0.859492i	$0.329220\pi$
$318$		0			0
$319$	−257.952	+	446.785i	−0.808626	+	1.40058i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	312.509i		−	0.967521i
$324$		0			0
$325$	11.2702			0.0346774
$326$		0			0
$327$		0			0
$328$		0			0
$329$	147.520	+	85.1708i	0.448389	+	0.258878i
$330$		0			0
$331$	138.903	+	240.587i	0.419647	+	0.726850i	0.995904	−	0.0904186i	$-0.0288205\pi$
$331$	138.903	+	240.587i	0.419647	+	0.726850i	−0.576257	+	0.817269i	$0.695487\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−193.555	+	111.749i	−0.577777	+	0.333580i
$336$		0			0
$337$	−27.9133	+	48.3473i	−0.0828288	+	0.143464i	−0.904464	−	0.426550i	$-0.859729\pi$
$337$	−27.9133	+	48.3473i	−0.0828288	+	0.143464i	0.821635	+	0.570014i	$0.193062\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	610.178i		−	1.78938i
$342$		0			0
$343$	314.540			0.917027
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−440.117	−	254.102i	−1.26835	−	0.732281i	−0.293673	−	0.955906i	$-0.594878\pi$
$347$	−440.117	−	254.102i	−1.26835	−	0.732281i	−0.974675	+	0.223625i	$0.928211\pi$
$348$		0			0
$349$	−69.2379	−	119.924i	−0.198389	−	0.343621i	0.749617	−	0.661872i	$-0.230238\pi$
$349$	−69.2379	−	119.924i	−0.198389	−	0.343621i	−0.948006	+	0.318251i	$0.896904\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	498.877	−	288.027i	1.41325	−	0.815940i	0.417556	−	0.908651i	$-0.362887\pi$
$353$	498.877	−	288.027i	1.41325	−	0.815940i	0.995693	+	0.0927114i	$0.0295534\pi$
$354$		0			0
$355$	−114.714	+	198.690i	−0.323137	+	0.559690i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			267.888i			0.746207i	0.927790	+	0.373103i	$0.121706\pi$
$359$			267.888i			0.746207i	−0.927790	+	0.373103i	$0.878294\pi$
$360$		0			0
$361$	354.347			0.981570
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−26.6502	−	15.3865i	−0.0730142	−	0.0421548i
$366$		0			0
$367$	30.2379	+	52.3736i	0.0823921	+	0.142707i	0.904277	−	0.426946i	$-0.140411\pi$
$367$	30.2379	+	52.3736i	0.0823921	+	0.142707i	−0.821885	+	0.569654i	$0.807077\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−35.1411	+	20.2887i	−0.0947199	+	0.0546866i
$372$		0			0
$373$	108.254	−	187.501i	0.290225	−	0.502685i	−0.683638	−	0.729822i	$-0.739603\pi$
$373$	108.254	−	187.501i	0.290225	−	0.502685i	0.973863	+	0.227137i	$0.0729364\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	99.5231i		−	0.263987i
$378$		0			0
$379$	−285.254			−0.752649			−0.376325	−	0.926488i	$-0.622812\pi$
$379$	−285.254			−0.752649			−0.376325	+	0.926488i	$0.622812\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	141.665	+	81.7905i	0.369883	+	0.213552i	0.673407	−	0.739272i	$-0.264830\pi$
$383$	141.665	+	81.7905i	0.369883	+	0.213552i	−0.303524	+	0.952824i	$0.598163\pi$
$384$		0			0
$385$	−48.9355	−	84.7587i	−0.127105	−	0.220153i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−250.760	+	144.776i	−0.644627	+	0.372176i	−0.786395	−	0.617724i	$-0.788055\pi$
$389$	−250.760	+	144.776i	−0.644627	+	0.372176i	0.141767	+	0.989900i	$0.454721\pi$
$390$		0			0
$391$	−116.286	+	201.414i	−0.297407	+	0.515125i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			126.924i			0.321326i
$396$		0			0
$397$	−66.5081			−0.167527			−0.0837633	−	0.996486i	$-0.526694\pi$
$397$	−66.5081			−0.167527			−0.0837633	+	0.996486i	$0.526694\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	392.165	+	226.417i	0.977968	+	0.564630i	0.901656	−	0.432454i	$-0.142352\pi$
$401$	392.165	+	226.417i	0.977968	+	0.564630i	0.0763122	+	0.997084i	$0.475685\pi$
$402$		0			0
$403$	58.8548	+	101.940i	0.146042	+	0.252952i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−141.665	+	81.7905i	−0.348072	+	0.200959i
$408$		0			0
$409$	−150.435	+	260.562i	−0.367813	+	0.637071i	−0.989223	−	0.146415i	$-0.953227\pi$
$409$	−150.435	+	260.562i	−0.367813	+	0.637071i	0.621410	+	0.783485i	$0.286560\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	137.901i		−	0.333900i
$414$		0			0
$415$	−201.395			−0.485289
$416$		0			0
$417$		0			0
$418$		0			0
$419$	103.524	+	59.7697i	0.247074	+	0.142649i	0.618424	−	0.785845i	$-0.287771\pi$
$419$	103.524	+	59.7697i	0.247074	+	0.142649i	−0.371349	+	0.928493i	$0.621105\pi$
$420$		0			0
$421$	81.1431	+	140.544i	0.192739	+	0.333834i	0.946157	−	0.323708i	$-0.104930\pi$
$421$	81.1431	+	140.544i	0.192739	+	0.333834i	−0.753418	+	0.657542i	$0.771596\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	50.5948	−	29.2109i	0.119046	−	0.0687315i
$426$		0			0
$427$	−84.7298	+	146.756i	−0.198431	+	0.343692i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	499.815i		−	1.15966i	−0.814736	−	0.579832i	$-0.803118\pi$
$431$		−	499.815i		−	1.15966i	0.814736	−	0.579832i	$-0.196882\pi$
$432$		0			0
$433$	739.883			1.70874			0.854368	−	0.519668i	$-0.173944\pi$
$433$	739.883			1.70874			0.854368	+	0.519668i	$0.173944\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−461.044	−	266.184i	−1.05502	−	0.609117i
$438$		0			0
$439$	124.333	+	215.350i	0.283218	+	0.490548i	0.972175	−	0.234254i	$-0.0752647\pi$
$439$	124.333	+	215.350i	0.283218	+	0.490548i	−0.688958	+	0.724802i	$0.741931\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−78.8831	+	45.5432i	−0.178066	+	0.102806i	−0.586384	−	0.810034i	$-0.699449\pi$
$443$	−78.8831	+	45.5432i	−0.178066	+	0.102806i	0.408318	+	0.912840i	$0.366116\pi$
$444$		0			0
$445$	−106.476	+	184.421i	−0.239271	+	0.414430i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			93.6705i			0.208620i	0.994545	+	0.104310i	$0.0332634\pi$
$449$			93.6705i			0.208620i	−0.994545	+	0.104310i	$0.966737\pi$
$450$		0			0
$451$	−303.569			−0.673101
$452$		0			0
$453$		0			0
$454$		0			0
$455$	16.3508	+	9.44016i	0.0359359	+	0.0207476i
$456$		0			0
$457$	−201.498	−	349.005i	−0.440915	−	0.763686i	0.556843	−	0.830618i	$-0.312012\pi$
$457$	−201.498	−	349.005i	−0.440915	−	0.763686i	−0.997758	+	0.0669313i	$0.978679\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	57.9496	−	33.4572i	0.125704	−	0.0725752i	−0.435829	−	0.900029i	$-0.643545\pi$
$461$	57.9496	−	33.4572i	0.125704	−	0.0725752i	0.561533	+	0.827454i	$0.310212\pi$
$462$		0			0
$463$	88.8105	−	153.824i	0.191815	−	0.332234i	−0.754037	−	0.656832i	$-0.771896\pi$
$463$	88.8105	−	153.824i	0.191815	−	0.332234i	0.945852	+	0.324599i	$0.105229\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	620.647i		−	1.32901i	−0.747285	−	0.664504i	$-0.768643\pi$
$467$		−	620.647i		−	1.32901i	0.747285	−	0.664504i	$-0.231357\pi$
$468$		0			0
$469$	−374.415			−0.798327
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−424.833	−	245.277i	−0.898166	−	0.518557i
$474$		0			0
$475$	66.8649	+	115.813i	0.140768	+	0.243818i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	146.044	−	84.3187i	0.304894	−	0.176031i	−0.339745	−	0.940517i	$-0.610341\pi$
$479$	146.044	−	84.3187i	0.304894	−	0.176031i	0.644639	+	0.764487i	$0.277008\pi$
$480$		0			0
$481$	15.7782	−	27.3287i	0.0328030	−	0.0568164i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			227.403i			0.468871i
$486$		0			0
$487$	−401.935			−0.825330			−0.412665	−	0.910883i	$-0.635402\pi$
$487$	−401.935			−0.825330			−0.412665	+	0.910883i	$0.635402\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−307.833	−	177.727i	−0.626950	−	0.361970i	0.152620	−	0.988285i	$-0.451229\pi$
$491$	−307.833	−	177.727i	−0.626950	−	0.361970i	−0.779570	+	0.626315i	$0.784562\pi$
$492$		0			0
$493$	−257.952	−	446.785i	−0.523228	−	0.906258i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−332.855	+	192.174i	−0.669728	+	0.386668i
$498$		0			0
$499$	−8.30845	+	14.3907i	−0.0166502	+	0.0288390i	−0.874230	−	0.485511i	$-0.838634\pi$
$499$	−8.30845	+	14.3907i	−0.0166502	+	0.0288390i	0.857580	+	0.514350i	$0.171967\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	362.522i		−	0.720721i	−0.932813	−	0.360360i	$-0.882654\pi$
$503$		−	362.522i		−	0.720721i	0.932813	−	0.360360i	$-0.117346\pi$
$504$		0			0
$505$	226.601			0.448714
$506$		0			0
$507$		0			0
$508$		0			0
$509$	578.286	+	333.874i	1.13612	+	0.655941i	0.945467	−	0.325717i	$-0.105606\pi$
$509$	578.286	+	333.874i	1.13612	+	0.655941i	0.190655	+	0.981657i	$0.438939\pi$
$510$		0			0
$511$	−25.7762	−	44.6457i	−0.0504426	−	0.0873692i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	31.5383	−	18.2086i	0.0612394	−	0.0353566i
$516$		0			0
$517$	−265.663	+	460.142i	−0.513855	+	0.890024i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	676.352i		−	1.29818i	−0.760712	−	0.649090i	$-0.775150\pi$
$521$		−	676.352i		−	1.29818i	0.760712	−	0.649090i	$-0.224850\pi$
$522$		0			0
$523$	−189.427			−0.362194			−0.181097	−	0.983465i	$-0.557965\pi$
$523$	−189.427			−0.362194			−0.181097	+	0.983465i	$0.557965\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	528.429	+	305.089i	1.00271	+	0.578916i
$528$		0			0
$529$	−66.4032	−	115.014i	−0.125526	−	0.217417i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	50.7157	−	29.2808i	0.0951515	−	0.0549357i
$534$		0			0
$535$	94.8881	−	164.351i	0.177361	−	0.307198i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			408.575i			0.758025i
$540$		0			0
$541$	−590.629			−1.09174			−0.545868	−	0.837871i	$-0.683800\pi$
$541$	−590.629			−1.09174			−0.545868	+	0.837871i	$0.683800\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−313.649	−	181.085i	−0.575503	−	0.332267i
$546$		0			0
$547$	−394.782	−	683.783i	−0.721722	−	1.25006i	−0.960309	−	0.278939i	$-0.910017\pi$
$547$	−394.782	−	683.783i	−0.721722	−	1.25006i	0.238586	−	0.971121i	$-0.423316\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1022.71	−	590.462i	1.85610	−	1.07162i
$552$		0			0
$553$	−106.314	+	184.142i	−0.192250	+	0.332987i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	1004.69i		−	1.80376i	−0.431987	−	0.901880i	$-0.642187\pi$
$557$		−	1004.69i		−	1.80376i	0.431987	−	0.901880i	$-0.357813\pi$
$558$		0			0
$559$	94.6330			0.169290
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−272.976	−	157.603i	−0.484859	−	0.279934i	0.237580	−	0.971368i	$-0.423646\pi$
$563$	−272.976	−	157.603i	−0.484859	−	0.279934i	−0.722439	+	0.691434i	$0.756979\pi$
$564$		0			0
$565$	−7.74597	−	13.4164i	−0.0137097	−	0.0237459i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	823.923	−	475.692i	1.44802	−	0.836015i	0.449656	−	0.893202i	$-0.351546\pi$
$569$	823.923	−	475.692i	1.44802	−	0.836015i	0.998363	+	0.0571870i	$0.0182131\pi$
$570$		0			0
$571$	229.183	−	396.957i	0.401372	−	0.695197i	−0.592520	−	0.805556i	$-0.701867\pi$
$571$	229.183	−	396.957i	0.401372	−	0.695197i	0.993892	+	0.110359i	$0.0352001\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	99.5231i		−	0.173084i
$576$		0			0
$577$	991.190			1.71783			0.858916	−	0.512116i	$-0.171138\pi$
$577$	991.190			1.71783			0.858916	+	0.512116i	$0.171138\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−292.185	−	168.693i	−0.502901	−	0.290350i
$582$		0			0
$583$	−63.2843	−	109.612i	−0.108549	−	0.188013i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	144.883	−	83.6483i	0.246820	−	0.142501i	−0.371488	−	0.928438i	$-0.621152\pi$
$587$	144.883	−	83.6483i	0.246820	−	0.142501i	0.618307	+	0.785937i	$0.287819\pi$
$588$		0			0
$589$	−698.361	+	1209.60i	−1.18567	+	2.05364i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			868.330i			1.46430i	0.681143	+	0.732150i	$0.261483\pi$
$593$			868.330i			1.46430i	−0.681143	+	0.732150i	$0.738517\pi$
$594$		0			0
$595$	97.8709			0.164489
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−170.335	−	98.3428i	−0.284365	−	0.164178i	0.351033	−	0.936363i	$-0.385831\pi$
$599$	−170.335	−	98.3428i	−0.284365	−	0.164178i	−0.635398	+	0.772185i	$0.719164\pi$
$600$		0			0
$601$	−332.181	−	575.355i	−0.552715	−	0.957330i	−0.998077	−	0.0619795i	$-0.980259\pi$
$601$	−332.181	−	575.355i	−0.552715	−	0.957330i	0.445363	−	0.895350i	$-0.353075\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	30.0625	−	17.3566i	0.0496901	−	0.0286886i
$606$		0			0
$607$	−37.2379	+	64.4979i	−0.0613474	+	0.106257i	−0.895068	−	0.445930i	$-0.852873\pi$
$607$	−37.2379	+	64.4979i	−0.0613474	+	0.106257i	0.833720	+	0.552187i	$0.186206\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	102.498i		−	0.167755i
$612$		0			0
$613$	−304.569			−0.496849			−0.248425	−	0.968651i	$-0.579913\pi$
$613$	−304.569			−0.496849			−0.248425	+	0.968651i	$0.579913\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	815.353	+	470.744i	1.32148	+	0.762956i	0.983965	−	0.178364i	$-0.0570804\pi$
$617$	815.353	+	470.744i	1.32148	+	0.762956i	0.337515	+	0.941320i	$0.390414\pi$
$618$		0			0
$619$	−117.736	−	203.924i	−0.190203	−	0.329442i	0.755114	−	0.655593i	$-0.227581\pi$
$619$	−117.736	−	203.924i	−0.190203	−	0.329442i	−0.945318	+	0.326151i	$0.894248\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−308.952	+	178.373i	−0.495909	+	0.286313i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	163.581i		−	0.260065i
$630$		0			0
$631$	832.125			1.31874			0.659370	−	0.751819i	$-0.270823\pi$
$631$	832.125			1.31874			0.659370	+	0.751819i	$0.270823\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	437.460	+	252.567i	0.688913	+	0.397744i
$636$		0			0
$637$	−39.4092	−	68.2588i	−0.0618669	−	0.107157i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−913.355	+	527.326i	−1.42489	+	0.822661i	−0.996711	−	0.0810330i	$-0.974178\pi$
$641$	−913.355	+	527.326i	−1.42489	+	0.822661i	−0.428179	+	0.903694i	$0.640845\pi$
$642$		0			0
$643$	493.593	−	854.928i	0.767640	−	1.32959i	−0.171199	−	0.985236i	$-0.554764\pi$
$643$	493.593	−	854.928i	0.767640	−	1.32959i	0.938839	−	0.344356i	$-0.111903\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			75.6097i			0.116862i	0.998291	+	0.0584310i	$0.0186098\pi$
$647$			75.6097i			0.116862i	−0.998291	+	0.0584310i	$0.981390\pi$
$648$		0			0
$649$	430.137			0.662769
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1079.95	−	623.510i	−1.65383	−	0.954840i	−0.975476	−	0.220105i	$-0.929360\pi$
$653$	−1079.95	−	623.510i	−1.65383	−	0.954840i	−0.678355	−	0.734734i	$-0.737307\pi$
$654$		0			0
$655$	−105.492	−	182.717i	−0.161056	−	0.278958i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	544.851	−	314.570i	0.826784	−	0.477344i	−0.0259662	−	0.999663i	$-0.508266\pi$
$659$	544.851	−	314.570i	0.826784	−	0.477344i	0.852750	+	0.522319i	$0.174933\pi$
$660$		0			0
$661$	−323.282	+	559.941i	−0.489080	+	0.847112i	−0.999921	−	0.0125632i	$-0.996001\pi$
$661$	−323.282	+	559.941i	−0.489080	+	0.847112i	0.510841	+	0.859675i	$0.329334\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			224.031i			0.336888i
$666$		0			0
$667$	−878.855			−1.31762
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−457.760	−	264.288i	−0.682206	−	0.393872i
$672$		0			0
$673$	−45.9516	−	79.5905i	−0.0682788	−	0.118262i	0.829865	−	0.557964i	$-0.188417\pi$
$673$	−45.9516	−	79.5905i	−0.0682788	−	0.118262i	−0.898144	+	0.439702i	$0.855084\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−633.520	+	365.763i	−0.935776	+	0.540270i	−0.888634	−	0.458618i	$-0.848345\pi$
$677$	−633.520	+	365.763i	−0.935776	+	0.540270i	−0.0471421	+	0.998888i	$0.515011\pi$
$678$		0			0
$679$	−190.478	+	329.917i	−0.280527	+	0.485887i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	682.177i		−	0.998794i	−0.866373	−	0.499397i	$-0.833555\pi$
$683$		−	682.177i		−	0.998794i	0.866373	−	0.499397i	$-0.166445\pi$
$684$		0			0
$685$	−463.712			−0.676951
$686$		0			0
$687$		0			0
$688$		0			0
$689$	21.1452	+	12.2082i	0.0306897	+	0.0177187i
$690$		0			0
$691$	−210.331	−	364.303i	−0.304386	−	0.527212i	0.672739	−	0.739880i	$-0.265118\pi$
$691$	−210.331	−	364.303i	−0.304386	−	0.527212i	−0.977124	+	0.212669i	$0.931785\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	63.3498	−	36.5750i	0.0911508	−	0.0526259i
$696$		0			0
$697$	151.784	−	262.898i	0.217768	−	0.377185i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	70.8185i		−	0.101025i	−0.998723	−	0.0505125i	$-0.983915\pi$
$701$		−	70.8185i		−	0.101025i	0.998723	−	0.0505125i	$-0.0160855\pi$
$702$		0			0
$703$	374.444			0.532637
$704$		0			0
$705$		0			0
$706$		0			0
$707$	328.754	+	189.806i	0.464998	+	0.268467i
$708$		0			0
$709$	−399.696	−	692.293i	−0.563745	−	0.976436i	−0.997165	−	0.0752436i	$-0.976027\pi$
$709$	−399.696	−	692.293i	−0.563745	−	0.976436i	0.433420	−	0.901192i	$-0.357307\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	900.194	−	519.727i	1.26254	−	0.728930i
$714$		0			0
$715$	−29.4456	+	51.0012i	−0.0411826	+	0.0713304i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	368.675i		−	0.512761i	−0.966576	−	0.256381i	$-0.917470\pi$
$719$		−	368.675i		−	0.512761i	0.966576	−	0.256381i	$-0.0825300\pi$
$720$		0			0
$721$	61.0079			0.0846157
$722$		0			0
$723$		0			0
$724$		0			0
$725$	191.190	+	110.383i	0.263710	+	0.152253i
$726$		0			0
$727$	−453.095	−	784.783i	−0.623239	−	1.07948i	−0.988879	−	0.148725i	$-0.952483\pi$
$727$	−453.095	−	784.783i	−0.623239	−	1.07948i	0.365640	−	0.930757i	$-0.380850\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	424.833	−	245.277i	0.581166	−	0.335537i
$732$		0			0
$733$	−50.7601	+	87.9190i	−0.0692497	+	0.119944i	−0.898571	−	0.438828i	$-0.855394\pi$
$733$	−50.7601	+	87.9190i	−0.0692497	+	0.119944i	0.829322	+	0.558772i	$0.188727\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1167.87i		−	1.58463i
$738$		0			0
$739$	−557.665			−0.754622			−0.377311	−	0.926087i	$-0.623151\pi$
$739$	−557.665			−0.754622			−0.377311	+	0.926087i	$0.623151\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−308.806	−	178.289i	−0.415621	−	0.239959i	0.277581	−	0.960702i	$-0.410467\pi$
$743$	−308.806	−	178.289i	−0.415621	−	0.239959i	−0.693202	+	0.720743i	$0.743801\pi$
$744$		0			0
$745$	−23.6673	−	40.9931i	−0.0317683	−	0.0550242i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	275.329	−	158.961i	0.367595	−	0.212231i
$750$		0			0
$751$	244.810	−	424.024i	0.325979	−	0.564613i	−0.655731	−	0.754995i	$-0.727639\pi$
$751$	244.810	−	424.024i	0.325979	−	0.564613i	0.981710	+	0.190382i	$0.0609727\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	615.874i		−	0.815728i
$756$		0			0
$757$	198.379			0.262059			0.131030	−	0.991378i	$-0.458172\pi$
$757$	198.379			0.262059			0.131030	+	0.991378i	$0.458172\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	30.6653	+	17.7046i	0.0402961	+	0.0232649i	0.520013	−	0.854159i	$-0.325927\pi$
$761$	30.6653	+	17.7046i	0.0402961	+	0.0232649i	−0.479717	+	0.877423i	$0.659261\pi$
$762$		0			0
$763$	−303.363	−	525.440i	−0.397592	−	0.688650i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−71.8609	+	41.4889i	−0.0936909	+	0.0540925i
$768$		0			0
$769$	114.552	−	198.411i	0.148963	−	0.258011i	−0.781882	−	0.623427i	$-0.785740\pi$
$769$	114.552	−	198.411i	0.148963	−	0.258011i	0.930844	+	0.365416i	$0.119073\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	905.388i		−	1.17126i	−0.810577	−	0.585632i	$-0.800846\pi$
$773$		−	905.388i		−	1.17126i	0.810577	−	0.585632i	$-0.199154\pi$
$774$		0			0
$775$	−261.109			−0.336915
$776$		0			0
$777$		0			0
$778$		0			0
$779$	601.784	+	347.440i	0.772509	+	0.446008i
$780$		0			0
$781$	−599.425	−	1038.24i	−0.767510	−	1.32937i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−348.444	+	201.174i	−0.443877	+	0.256273i
$786$		0			0
$787$	−74.1452	+	128.423i	−0.0942125	+	0.163181i	−0.909280	−	0.416186i	$-0.863367\pi$
$787$	−74.1452	+	128.423i	−0.0942125	+	0.163181i	0.815067	+	0.579366i	$0.196700\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	25.9528i		−	0.0328101i
$792$		0			0
$793$	101.968			0.128585
$794$		0			0
$795$

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.bs (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.93649 + 1.11803i) q^{5} +(1.87298 + 3.24410i) q^{7} +O(q^{10})\)	\(q+(1.93649 + 1.11803i) q^{5} +(1.87298 + 3.24410i) q^{7} +(-10.1190 + 5.84218i) q^{11} +(1.12702 - 1.95205i) q^{13} -11.6844i q^{17} +26.7460 q^{19} +(-17.2379 - 9.95231i) q^{23} +(2.50000 + 4.33013i) q^{25} +(38.2379 - 22.0767i) q^{29} +(-26.1109 + 45.2254i) q^{31} +8.37624i q^{35} +14.0000 q^{37} +(22.5000 + 12.9904i) q^{41} +(20.9919 + 36.3591i) q^{43} +(39.3810 - 22.7367i) q^{47} +(17.4839 - 30.2829i) q^{49} +10.8323i q^{53} -26.1270 q^{55} +(-31.8810 - 18.4065i) q^{59} +(22.6190 + 39.1772i) q^{61} +(4.36492 - 2.52009i) q^{65} +(-49.9758 + 86.5606i) q^{67} +102.603i q^{71} -13.7621 q^{73} +(-37.9052 - 21.8846i) q^{77} +(28.3810 + 49.1574i) q^{79} +(-78.0000 + 45.0333i) q^{83} +(13.0635 - 22.6267i) q^{85} +95.2349i q^{89} +8.44353 q^{91} +(51.7933 + 29.9029i) q^{95} +(50.8488 + 88.0727i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} + 6 q^{11} + 20 q^{13} + 76 q^{19} + 24 q^{23} + 10 q^{25} + 60 q^{29} + 4 q^{31} + 56 q^{37} + 90 q^{41} + 22 q^{43} + 204 q^{47} - 54 q^{49} - 120 q^{55} - 174 q^{59} + 44 q^{61} - 60 q^{65} - 14 q^{67} - 148 q^{73} - 384 q^{77} + 160 q^{79} - 312 q^{83} + 60 q^{85} - 400 q^{91} + 60 q^{95} + 2 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 + 6 * q^11 + 20 * q^13 + 76 * q^19 + 24 * q^23 + 10 * q^25 + 60 * q^29 + 4 * q^31 + 56 * q^37 + 90 * q^41 + 22 * q^43 + 204 * q^47 - 54 * q^49 - 120 * q^55 - 174 * q^59 + 44 * q^61 - 60 * q^65 - 14 * q^67 - 148 * q^73 - 384 * q^77 + 160 * q^79 - 312 * q^83 + 60 * q^85 - 400 * q^91 + 60 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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