LMFDB - Embedded newform 6048.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(1,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.39528.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 2x + 12 $ x^4 - 9x^2 - 2x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48380$ of defining polynomial
Character	$\chi$	$=$	6048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79835 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.79835 q^{5} +1.00000 q^{7} +2.96759 q^{11} +6.95053 q^{13} +6.11977 q^{17} -0.967592 q^{19} +7.91812 q^{23} -1.76594 q^{25} -0.798349 q^{29} -0.798349 q^{31} -1.79835 q^{35} +6.76594 q^{37} -3.79835 q^{41} +9.71647 q^{43} -11.0703 q^{47} +1.00000 q^{49} -13.6670 q^{53} -5.33677 q^{55} +4.59670 q^{59} +8.56429 q^{61} -12.4995 q^{65} -13.5148 q^{67} -11.9181 q^{71} +4.33849 q^{73} +2.96759 q^{77} +17.6346 q^{79} -0.765941 q^{83} -11.0055 q^{85} -3.98294 q^{89} +6.95053 q^{91} +1.74007 q^{95} -6.86865 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^5 + 4 * q^7	$ 4 q + 2 q^{5} + 4 q^{7} + 8 q^{13} + 2 q^{17} + 8 q^{19} + 14 q^{25} + 6 q^{29} + 6 q^{31} + 2 q^{35} + 6 q^{37} - 6 q^{41} - 2 q^{43} - 2 q^{47} + 4 q^{49} + 6 q^{53} - 12 q^{55} + 4 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 2 q^{79} + 18 q^{83} + 22 q^{85} - 8 q^{89} + 8 q^{91} + 16 q^{95} + 24 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 4 * q^7 + 8 * q^13 + 2 * q^17 + 8 * q^19 + 14 * q^25 + 6 * q^29 + 6 * q^31 + 2 * q^35 + 6 * q^37 - 6 * q^41 - 2 * q^43 - 2 * q^47 + 4 * q^49 + 6 * q^53 - 12 * q^55 + 4 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 2 * q^79 + 18 * q^83 + 22 * q^85 - 8 * q^89 + 8 * q^91 + 16 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−1.79835	−0.804246	−0.402123	−	0.915586i	$-0.631728\pi$
$5$	−1.79835	−0.804246	−0.402123	+	0.915586i	$0.631728\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.96759	0.894763	0.447381	−	0.894343i	$-0.352357\pi$
$11$	2.96759	0.894763	0.447381	+	0.894343i	$0.352357\pi$
$12$	0	0
$13$	6.95053	1.92773	0.963865	−	0.266392i	$-0.0858315\pi$
$13$	6.95053	1.92773	0.963865	+	0.266392i	$0.0858315\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.11977	1.48426	0.742131	−	0.670254i	$-0.233815\pi$
$17$	6.11977	1.48426	0.742131	+	0.670254i	$0.233815\pi$
$18$	0	0
$19$	−0.967592	−0.221981	−0.110990	−	0.993821i	$-0.535402\pi$
$19$	−0.967592	−0.221981	−0.110990	+	0.993821i	$0.535402\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.91812	1.65104	0.825521	−	0.564371i	$-0.190881\pi$
$23$	7.91812	1.65104	0.825521	+	0.564371i	$0.190881\pi$
$24$	0	0
$25$	−1.76594	−0.353188
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.798349	−0.148250	−0.0741248	−	0.997249i	$-0.523616\pi$
$29$	−0.798349	−0.148250	−0.0741248	+	0.997249i	$0.523616\pi$
$30$	0	0
$31$	−0.798349	−0.143388	−0.0716938	−	0.997427i	$-0.522840\pi$
$31$	−0.798349	−0.143388	−0.0716938	+	0.997427i	$0.522840\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.79835	−0.303976
$36$	0	0
$37$	6.76594	1.11231	0.556157	−	0.831077i	$-0.312275\pi$
$37$	6.76594	1.11231	0.556157	+	0.831077i	$0.312275\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.79835	−0.593202	−0.296601	−	0.955001i	$-0.595853\pi$
$41$	−3.79835	−0.593202	−0.296601	+	0.955001i	$0.595853\pi$
$42$	0	0
$43$	9.71647	1.48175	0.740874	−	0.671644i	$-0.234412\pi$
$43$	9.71647	1.48175	0.740874	+	0.671644i	$0.234412\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.0703	−1.61477	−0.807385	−	0.590025i	$-0.799118\pi$
$47$	−11.0703	−1.61477	−0.807385	+	0.590025i	$0.799118\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.6670	−1.87731	−0.938653	−	0.344862i	$-0.887926\pi$
$53$	−13.6670	−1.87731	−0.938653	+	0.344862i	$0.887926\pi$
$54$	0	0
$55$	−5.33677	−0.719609
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.59670	0.598439	0.299220	−	0.954184i	$-0.403274\pi$
$59$	4.59670	0.598439	0.299220	+	0.954184i	$0.403274\pi$
$60$	0	0
$61$	8.56429	1.09654	0.548272	−	0.836300i	$-0.315286\pi$
$61$	8.56429	1.09654	0.548272	+	0.836300i	$0.315286\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.4995	−1.55037
$66$	0	0
$67$	−13.5148	−1.65110	−0.825549	−	0.564331i	$-0.809134\pi$
$67$	−13.5148	−1.65110	−0.825549	+	0.564331i	$0.809134\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.9181	−1.41442	−0.707210	−	0.707003i	$-0.750047\pi$
$71$	−11.9181	−1.41442	−0.707210	+	0.707003i	$0.750047\pi$
$72$	0	0
$73$	4.33849	0.507781	0.253891	−	0.967233i	$-0.418290\pi$
$73$	4.33849	0.507781	0.253891	+	0.967233i	$0.418290\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.96759	0.338189
$78$	0	0
$79$	17.6346	1.98405	0.992023	−	0.126055i	$-0.0402317\pi$
$79$	17.6346	1.98405	0.992023	+	0.126055i	$0.0402317\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.765941	−0.0840730	−0.0420365	−	0.999116i	$-0.513385\pi$
$83$	−0.765941	−0.0840730	−0.0420365	+	0.999116i	$0.513385\pi$
$84$	0	0
$85$	−11.0055	−1.19371
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.98294	−0.422190	−0.211095	−	0.977466i	$-0.567703\pi$
$89$	−3.98294	−0.422190	−0.211095	+	0.977466i	$0.567703\pi$
$90$	0	0
$91$	6.95053	0.728613
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.74007	0.178527
$96$	0	0
$97$	−6.86865	−0.697406	−0.348703	−	0.937233i	$-0.613378\pi$
$97$	−6.86865	−0.697406	−0.348703	+	0.937233i	$0.613378\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.93518	0.789580	0.394790	−	0.918771i	$-0.370817\pi$
$101$	7.93518	0.789580	0.394790	+	0.918771i	$0.370817\pi$
$102$	0	0
$103$	12.4736	1.22906	0.614530	−	0.788893i	$-0.289346\pi$
$103$	12.4736	1.22906	0.614530	+	0.788893i	$0.289346\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	12.7659	1.22276	0.611378	−	0.791339i	$-0.290616\pi$
$109$	12.7659	1.22276	0.611378	+	0.791339i	$0.290616\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.96759	−0.655456	−0.327728	−	0.944772i	$-0.606283\pi$
$113$	−6.96759	−0.655456	−0.327728	+	0.944772i	$0.606283\pi$
$114$	0	0
$115$	−14.2395	−1.32784
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.11977	0.560999
$120$	0	0
$121$	−2.19340	−0.199400
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1675	1.08830
$126$	0	0
$127$	−11.7335	−1.04118	−0.520591	−	0.853806i	$-0.674288\pi$
$127$	−11.7335	−1.04118	−0.520591	+	0.853806i	$0.674288\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.507730	−0.0443606	−0.0221803	−	0.999754i	$-0.507061\pi$
$131$	−0.507730	−0.0443606	−0.0221803	+	0.999754i	$0.507061\pi$
$132$	0	0
$133$	−0.967592	−0.0839009
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.03241	−0.259076	−0.129538	−	0.991574i	$-0.541349\pi$
$137$	−3.03241	−0.259076	−0.129538	+	0.991574i	$0.541349\pi$
$138$	0	0
$139$	−18.1269	−1.53750	−0.768750	−	0.639549i	$-0.779121\pi$
$139$	−18.1269	−1.53750	−0.768750	+	0.639549i	$0.779121\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.6263	1.72486
$144$	0	0
$145$	1.43571	0.119229
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.13684	−0.256980	−0.128490	−	0.991711i	$-0.541013\pi$
$149$	−3.13684	−0.256980	−0.128490	+	0.991711i	$0.541013\pi$
$150$	0	0
$151$	−14.4412	−1.17521	−0.587604	−	0.809149i	$-0.699929\pi$
$151$	−14.4412	−1.17521	−0.587604	+	0.809149i	$0.699929\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.43571	0.115319
$156$	0	0
$157$	8.24287	0.657852	0.328926	−	0.944356i	$-0.393313\pi$
$157$	8.24287	0.657852	0.328926	+	0.944356i	$0.393313\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.91812	0.624035
$162$	0	0
$163$	11.3780	0.891192	0.445596	−	0.895234i	$-0.352992\pi$
$163$	11.3780	0.891192	0.445596	+	0.895234i	$0.352992\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.7318	−1.13998	−0.569991	−	0.821651i	$-0.693053\pi$
$167$	−14.7318	−1.13998	−0.569991	+	0.821651i	$0.693053\pi$
$168$	0	0
$169$	35.3098	2.71614
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.5319	1.33292	0.666462	−	0.745539i	$-0.267808\pi$
$173$	17.5319	1.33292	0.666462	+	0.745539i	$0.267808\pi$
$174$	0	0
$175$	−1.76594	−0.133493
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.2258	−0.913799	−0.456900	−	0.889518i	$-0.651040\pi$
$179$	−12.2258	−0.913799	−0.456900	+	0.889518i	$0.651040\pi$
$180$	0	0
$181$	3.67858	0.273426	0.136713	−	0.990611i	$-0.456346\pi$
$181$	3.67858	0.273426	0.136713	+	0.990611i	$0.456346\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.1675	−0.894574
$186$	0	0
$187$	18.1610	1.32806
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.6615	0.988512	0.494256	−	0.869316i	$-0.335440\pi$
$191$	13.6615	0.988512	0.494256	+	0.869316i	$0.335440\pi$
$192$	0	0
$193$	6.66151	0.479506	0.239753	−	0.970834i	$-0.422934\pi$
$193$	6.66151	0.479506	0.239753	+	0.970834i	$0.422934\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.4653	1.45810	0.729048	−	0.684463i	$-0.239963\pi$
$197$	20.4653	1.45810	0.729048	+	0.684463i	$0.239963\pi$
$198$	0	0
$199$	10.0055	0.709270	0.354635	−	0.935005i	$-0.384605\pi$
$199$	10.0055	0.709270	0.354635	+	0.935005i	$0.384605\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.798349	−0.0560331
$204$	0	0
$205$	6.83076	0.477081
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.87142	−0.198620
$210$	0	0
$211$	4.22248	0.290687	0.145344	−	0.989381i	$-0.453571\pi$
$211$	4.22248	0.290687	0.145344	+	0.989381i	$0.453571\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−17.4736	−1.19169
$216$	0	0
$217$	−0.798349	−0.0541955
$218$	0	0
$219$	0	0
$220$	0	0
$221$	42.5356	2.86126
$222$	0	0
$223$	−19.5456	−1.30887	−0.654436	−	0.756118i	$-0.727094\pi$
$223$	−19.5456	−1.30887	−0.654436	+	0.756118i	$0.727094\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.9593	1.39112	0.695560	−	0.718468i	$-0.255156\pi$
$227$	20.9593	1.39112	0.695560	+	0.718468i	$0.255156\pi$
$228$	0	0
$229$	24.2395	1.60179	0.800897	−	0.598802i	$-0.204356\pi$
$229$	24.2395	1.60179	0.800897	+	0.598802i	$0.204356\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.6104	−1.15370	−0.576849	−	0.816851i	$-0.695718\pi$
$233$	−17.6104	−1.15370	−0.576849	+	0.816851i	$0.695718\pi$
$234$	0	0
$235$	19.9083	1.29867
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.7236	−1.27581	−0.637905	−	0.770115i	$-0.720199\pi$
$239$	−19.7236	−1.27581	−0.637905	+	0.770115i	$0.720199\pi$
$240$	0	0
$241$	30.4005	1.95827	0.979135	−	0.203210i	$-0.0651374\pi$
$241$	30.4005	1.95827	0.979135	+	0.203210i	$0.0651374\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.79835	−0.114892
$246$	0	0
$247$	−6.72528	−0.427919
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.9593	−1.25982	−0.629911	−	0.776667i	$-0.716909\pi$
$251$	−19.9593	−1.25982	−0.629911	+	0.776667i	$0.716909\pi$
$252$	0	0
$253$	23.4978	1.47729
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.49775	0.342940	0.171470	−	0.985189i	$-0.445148\pi$
$257$	5.49775	0.342940	0.171470	+	0.985189i	$0.445148\pi$
$258$	0	0
$259$	6.76594	0.420415
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.6923	0.844304	0.422152	−	0.906525i	$-0.361275\pi$
$263$	13.6923	0.844304	0.422152	+	0.906525i	$0.361275\pi$
$264$	0	0
$265$	24.5780	1.50982
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.4599	0.942604	0.471302	−	0.881972i	$-0.343784\pi$
$269$	15.4599	0.942604	0.471302	+	0.881972i	$0.343784\pi$
$270$	0	0
$271$	−10.0055	−0.607790	−0.303895	−	0.952706i	$-0.598287\pi$
$271$	−10.0055	−0.607790	−0.303895	+	0.952706i	$0.598287\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.24059	−0.316020
$276$	0	0
$277$	15.1044	0.907537	0.453769	−	0.891120i	$-0.350079\pi$
$277$	15.1044	0.907537	0.453769	+	0.891120i	$0.350079\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.33849	−0.258812	−0.129406	−	0.991592i	$-0.541307\pi$
$281$	−4.33849	−0.258812	−0.129406	+	0.991592i	$0.541307\pi$
$282$	0	0
$283$	−30.4653	−1.81098	−0.905488	−	0.424371i	$-0.860495\pi$
$283$	−30.4653	−1.81098	−0.905488	+	0.424371i	$0.860495\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.79835	−0.224209
$288$	0	0
$289$	20.4516	1.20304
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.2774	−1.18462	−0.592310	−	0.805710i	$-0.701784\pi$
$293$	−20.2774	−1.18462	−0.592310	+	0.805710i	$0.701784\pi$
$294$	0	0
$295$	−8.26647	−0.481292
$296$	0	0
$297$	0	0
$298$	0	0
$299$	55.0351	3.18276
$300$	0	0
$301$	9.71647	0.560048
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−15.4016	−0.881892
$306$	0	0
$307$	5.14509	0.293646	0.146823	−	0.989163i	$-0.453095\pi$
$307$	5.14509	0.293646	0.146823	+	0.989163i	$0.453095\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.0889674	0.00504488	0.00252244	−	0.999997i	$-0.499197\pi$
$311$	0.0889674	0.00504488	0.00252244	+	0.999997i	$0.499197\pi$
$312$	0	0
$313$	−27.6077	−1.56048	−0.780239	−	0.625482i	$-0.784902\pi$
$313$	−27.6077	−1.56048	−0.780239	+	0.625482i	$0.784902\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.6615	0.654976	0.327488	−	0.944855i	$-0.393798\pi$
$317$	11.6615	0.654976	0.327488	+	0.944855i	$0.393798\pi$
$318$	0	0
$319$	−2.36917	−0.132648
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.92144	−0.329478
$324$	0	0
$325$	−12.2742	−0.680851
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.0703	−0.610325
$330$	0	0
$331$	15.8154	0.869294	0.434647	−	0.900601i	$-0.356873\pi$
$331$	15.8154	0.869294	0.434647	+	0.900601i	$0.356873\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	24.3044	1.32789
$336$	0	0
$337$	−9.60872	−0.523420	−0.261710	−	0.965147i	$-0.584286\pi$
$337$	−9.60872	−0.523420	−0.261710	+	0.965147i	$0.584286\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.36917	−0.128298
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.8345	−1.01109	−0.505545	−	0.862800i	$-0.668709\pi$
$347$	−18.8345	−1.01109	−0.505545	+	0.862800i	$0.668709\pi$
$348$	0	0
$349$	4.04775	0.216671	0.108336	−	0.994114i	$-0.465448\pi$
$349$	4.04775	0.216671	0.108336	+	0.994114i	$0.465448\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.52307	−0.347188	−0.173594	−	0.984817i	$-0.555538\pi$
$353$	−6.52307	−0.347188	−0.173594	+	0.984817i	$0.555538\pi$
$354$	0	0
$355$	21.4329	1.13754
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.6093	1.50994	0.754970	−	0.655760i	$-0.227652\pi$
$359$	28.6093	1.50994	0.754970	+	0.655760i	$0.227652\pi$
$360$	0	0
$361$	−18.0638	−0.950724
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.80211	−0.408381
$366$	0	0
$367$	1.66700	0.0870166	0.0435083	−	0.999053i	$-0.486147\pi$
$367$	1.66700	0.0870166	0.0435083	+	0.999053i	$0.486147\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−13.6670	−0.709555
$372$	0	0
$373$	20.1506	1.04336	0.521679	−	0.853142i	$-0.325306\pi$
$373$	20.1506	1.04336	0.521679	+	0.853142i	$0.325306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.54895	−0.285785
$378$	0	0
$379$	−26.7797	−1.37558	−0.687790	−	0.725910i	$-0.741419\pi$
$379$	−26.7797	−1.37558	−0.687790	+	0.725910i	$0.741419\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.7966	−0.653877	−0.326939	−	0.945046i	$-0.606017\pi$
$383$	−12.7966	−0.653877	−0.326939	+	0.945046i	$0.606017\pi$
$384$	0	0
$385$	−5.33677	−0.271987
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.0620	1.21999	0.609997	−	0.792404i	$-0.291171\pi$
$389$	24.0620	1.21999	0.609997	+	0.792404i	$0.291171\pi$
$390$	0	0
$391$	48.4571	2.45058
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−31.7131	−1.59566
$396$	0	0
$397$	−7.15927	−0.359313	−0.179657	−	0.983729i	$-0.557499\pi$
$397$	−7.15927	−0.359313	−0.179657	+	0.983729i	$0.557499\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.7373	0.586133	0.293066	−	0.956092i	$-0.405324\pi$
$401$	11.7373	0.586133	0.293066	+	0.956092i	$0.405324\pi$
$402$	0	0
$403$	−5.54895	−0.276413
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.0786	0.995257
$408$	0	0
$409$	−2.57803	−0.127475	−0.0637377	−	0.997967i	$-0.520302\pi$
$409$	−2.57803	−0.127475	−0.0637377	+	0.997967i	$0.520302\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.59670	0.226189
$414$	0	0
$415$	1.37743	0.0676154
$416$	0	0
$417$	0	0
$418$	0	0
$419$	18.7373	0.915377	0.457688	−	0.889113i	$-0.348678\pi$
$419$	18.7373	0.915377	0.457688	+	0.889113i	$0.348678\pi$
$420$	0	0
$421$	−18.7560	−0.914110	−0.457055	−	0.889438i	$-0.651096\pi$
$421$	−18.7560	−0.914110	−0.457055	+	0.889438i	$0.651096\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−10.8072	−0.524224
$426$	0	0
$427$	8.56429	0.414455
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.9352	−0.863907	−0.431954	−	0.901896i	$-0.642176\pi$
$431$	−17.9352	−0.863907	−0.431954	+	0.901896i	$0.642176\pi$
$432$	0	0
$433$	25.6587	1.23308	0.616540	−	0.787323i	$-0.288534\pi$
$433$	25.6587	1.23308	0.616540	+	0.787323i	$0.288534\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.66151	−0.366500
$438$	0	0
$439$	20.7473	0.990213	0.495107	−	0.868832i	$-0.335129\pi$
$439$	20.7473	0.990213	0.495107	+	0.868832i	$0.335129\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.35543	0.111910	0.0559550	−	0.998433i	$-0.482180\pi$
$443$	2.35543	0.111910	0.0559550	+	0.998433i	$0.482180\pi$
$444$	0	0
$445$	7.16271	0.339545
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−22.5643	−1.06487	−0.532437	−	0.846469i	$-0.678724\pi$
$449$	−22.5643	−1.06487	−0.532437	+	0.846469i	$0.678724\pi$
$450$	0	0
$451$	−11.2720	−0.530775
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−12.4995	−0.585984
$456$	0	0
$457$	−17.9714	−0.840665	−0.420332	−	0.907370i	$-0.638086\pi$
$457$	−17.9714	−0.840665	−0.420332	+	0.907370i	$0.638086\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.29610	−0.339813	−0.169907	−	0.985460i	$-0.554347\pi$
$461$	−7.29610	−0.339813	−0.169907	+	0.985460i	$0.554347\pi$
$462$	0	0
$463$	−32.8280	−1.52565	−0.762823	−	0.646607i	$-0.776187\pi$
$463$	−32.8280	−1.52565	−0.762823	+	0.646607i	$0.776187\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.5439	0.673012	0.336506	−	0.941681i	$-0.390755\pi$
$467$	14.5439	0.673012	0.336506	+	0.941681i	$0.390755\pi$
$468$	0	0
$469$	−13.5148	−0.624056
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28.8345	1.32581
$474$	0	0
$475$	1.70871	0.0784010
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19.2517	0.879632	0.439816	−	0.898088i	$-0.355044\pi$
$479$	19.2517	0.879632	0.439816	+	0.898088i	$0.355044\pi$
$480$	0	0
$481$	47.0269	2.14424
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.3522	0.560886
$486$	0	0
$487$	2.85491	0.129368	0.0646841	−	0.997906i	$-0.479396\pi$
$487$	2.85491	0.129368	0.0646841	+	0.997906i	$0.479396\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.95557	−0.0882538	−0.0441269	−	0.999026i	$-0.514051\pi$
$491$	−1.95557	−0.0882538	−0.0441269	+	0.999026i	$0.514051\pi$
$492$	0	0
$493$	−4.88571	−0.220041
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11.9181	−0.534601
$498$	0	0
$499$	−7.58629	−0.339609	−0.169804	−	0.985478i	$-0.554314\pi$
$499$	−7.58629	−0.339609	−0.169804	+	0.985478i	$0.554314\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.6670	−1.27820	−0.639099	−	0.769124i	$-0.720693\pi$
$503$	−28.6670	−1.27820	−0.639099	+	0.769124i	$0.720693\pi$
$504$	0	0
$505$	−14.2702	−0.635017
$506$	0	0
$507$	0	0
$508$	0	0
$509$	38.3423	1.69949	0.849745	−	0.527194i	$-0.176756\pi$
$509$	38.3423	1.69949	0.849745	+	0.527194i	$0.176756\pi$
$510$	0	0
$511$	4.33849	0.191923
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−22.4319	−0.988467
$516$	0	0
$517$	−32.8521	−1.44484
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.7285	0.645267	0.322633	−	0.946524i	$-0.395432\pi$
$521$	14.7285	0.645267	0.322633	+	0.946524i	$0.395432\pi$
$522$	0	0
$523$	−20.7049	−0.905362	−0.452681	−	0.891673i	$-0.649532\pi$
$523$	−20.7049	−0.905362	−0.452681	+	0.891673i	$0.649532\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.88571	−0.212825
$528$	0	0
$529$	39.6966	1.72594
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−26.4005	−1.14353
$534$	0	0
$535$	7.19340	0.310998
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.96759	0.127823
$540$	0	0
$541$	−10.5539	−0.453747	−0.226873	−	0.973924i	$-0.572850\pi$
$541$	−10.5539	−0.453747	−0.226873	+	0.973924i	$0.572850\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−22.9576	−0.983396
$546$	0	0
$547$	−17.8290	−0.762315	−0.381157	−	0.924510i	$-0.624474\pi$
$547$	−17.8290	−0.762315	−0.381157	+	0.924510i	$0.624474\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.772476	0.0329086
$552$	0	0
$553$	17.6346	0.749899
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−42.6625	−1.80767	−0.903834	−	0.427883i	$-0.859260\pi$
$557$	−42.6625	−1.80767	−0.903834	+	0.427883i	$0.859260\pi$
$558$	0	0
$559$	67.5346	2.85641
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.2175	0.936358	0.468179	−	0.883634i	$-0.344910\pi$
$563$	22.2175	0.936358	0.468179	+	0.883634i	$0.344910\pi$
$564$	0	0
$565$	12.5302	0.527148
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.0451	1.34340	0.671700	−	0.740823i	$-0.265564\pi$
$569$	32.0451	1.34340	0.671700	+	0.740823i	$0.265564\pi$
$570$	0	0
$571$	−7.44624	−0.311615	−0.155808	−	0.987787i	$-0.549798\pi$
$571$	−7.44624	−0.311615	−0.155808	+	0.987787i	$0.549798\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.9829	−0.583129
$576$	0	0
$577$	−9.49775	−0.395397	−0.197698	−	0.980263i	$-0.563347\pi$
$577$	−9.49775	−0.395397	−0.197698	+	0.980263i	$0.563347\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.765941	−0.0317766
$582$	0	0
$583$	−40.5581	−1.67974
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.03617	−0.0840418	−0.0420209	−	0.999117i	$-0.513380\pi$
$587$	−2.03617	−0.0840418	−0.0420209	+	0.999117i	$0.513380\pi$
$588$	0	0
$589$	0.772476	0.0318293
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.2017	−0.665322	−0.332661	−	0.943046i	$-0.607946\pi$
$593$	−16.2017	−0.665322	−0.332661	+	0.943046i	$0.607946\pi$
$594$	0	0
$595$	−11.0055	−0.451181
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−33.1253	−1.35346	−0.676731	−	0.736231i	$-0.736604\pi$
$599$	−33.1253	−1.35346	−0.676731	+	0.736231i	$0.736604\pi$
$600$	0	0
$601$	10.4347	0.425639	0.212819	−	0.977092i	$-0.431735\pi$
$601$	10.4347	0.425639	0.212819	+	0.977092i	$0.431735\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.94449	0.160366
$606$	0	0
$607$	10.1896	0.413584	0.206792	−	0.978385i	$-0.433698\pi$
$607$	10.1896	0.413584	0.206792	+	0.978385i	$0.433698\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−76.9444	−3.11284
$612$	0	0
$613$	3.78794	0.152993	0.0764967	−	0.997070i	$-0.475627\pi$
$613$	3.78794	0.152993	0.0764967	+	0.997070i	$0.475627\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.2737	0.574636	0.287318	−	0.957835i	$-0.407236\pi$
$617$	14.2737	0.574636	0.287318	+	0.957835i	$0.407236\pi$
$618$	0	0
$619$	31.8159	1.27879	0.639394	−	0.768880i	$-0.279185\pi$
$619$	31.8159	1.27879	0.639394	+	0.768880i	$0.279185\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.98294	−0.159573
$624$	0	0
$625$	−13.0517	−0.522070
$626$	0	0
$627$	0	0
$628$	0	0
$629$	41.4060	1.65097
$630$	0	0
$631$	−40.9917	−1.63186	−0.815928	−	0.578154i	$-0.803773\pi$
$631$	−40.9917	−1.63186	−0.815928	+	0.578154i	$0.803773\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	21.1010	0.837367
$636$	0	0
$637$	6.95053	0.275390
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.0500	−1.14741	−0.573703	−	0.819063i	$-0.694494\pi$
$641$	−29.0500	−1.14741	−0.573703	+	0.819063i	$0.694494\pi$
$642$	0	0
$643$	34.4450	1.35838	0.679188	−	0.733964i	$-0.262332\pi$
$643$	34.4450	1.35838	0.679188	+	0.733964i	$0.262332\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14.4033	−0.566252	−0.283126	−	0.959083i	$-0.591372\pi$
$647$	−14.4033	−0.566252	−0.283126	+	0.959083i	$0.591372\pi$
$648$	0	0
$649$	13.6411	0.535461
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.6005	−0.414828	−0.207414	−	0.978253i	$-0.566505\pi$
$653$	−10.6005	−0.414828	−0.207414	+	0.978253i	$0.566505\pi$
$654$	0	0
$655$	0.913076	0.0356768
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.46363	−0.368651	−0.184325	−	0.982865i	$-0.559010\pi$
$659$	−9.46363	−0.368651	−0.184325	+	0.982865i	$0.559010\pi$
$660$	0	0
$661$	−4.11268	−0.159965	−0.0799824	−	0.996796i	$-0.525486\pi$
$661$	−4.11268	−0.159965	−0.0799824	+	0.996796i	$0.525486\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.74007	0.0674770
$666$	0	0
$667$	−6.32142	−0.244766
$668$	0	0
$669$	0	0
$670$	0	0
$671$	25.4153	0.981148
$672$	0	0
$673$	15.6977	0.605101	0.302551	−	0.953133i	$-0.402162\pi$
$673$	15.6977	0.605101	0.302551	+	0.953133i	$0.402162\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.90450	−0.0731958	−0.0365979	−	0.999330i	$-0.511652\pi$
$677$	−1.90450	−0.0731958	−0.0365979	+	0.999330i	$0.511652\pi$
$678$	0	0
$679$	−6.86865	−0.263595
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.5643	−0.404231	−0.202116	−	0.979362i	$-0.564782\pi$
$683$	−10.5643	−0.404231	−0.202116	+	0.979362i	$0.564782\pi$
$684$	0	0
$685$	5.45333	0.208361
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−94.9929	−3.61894
$690$	0	0
$691$	15.7742	0.600079	0.300039	−	0.953927i	$-0.403000\pi$
$691$	15.7742	0.600079	0.300039	+	0.953927i	$0.403000\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	32.5984	1.23653
$696$	0	0
$697$	−23.2450	−0.880468
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.8686	1.09035	0.545177	−	0.838321i	$-0.316462\pi$
$701$	28.8686	1.09035	0.545177	+	0.838321i	$0.316462\pi$
$702$	0	0
$703$	−6.54667	−0.246912
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.93518	0.298433
$708$	0	0
$709$	2.68351	0.100781	0.0503906	−	0.998730i	$-0.483953\pi$
$709$	2.68351	0.100781	0.0503906	+	0.998730i	$0.483953\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.32142	−0.236739
$714$	0	0
$715$	−37.0933	−1.38721
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.43842	0.314700	0.157350	−	0.987543i	$-0.449705\pi$
$719$	8.43842	0.314700	0.157350	+	0.987543i	$0.449705\pi$
$720$	0	0
$721$	12.4736	0.464541
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.40984	0.0523600
$726$	0	0
$727$	33.5202	1.24319	0.621597	−	0.783337i	$-0.286484\pi$
$727$	33.5202	1.24319	0.621597	+	0.783337i	$0.286484\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	59.4626	2.19930
$732$	0	0
$733$	−18.8995	−0.698067	−0.349034	−	0.937110i	$-0.613490\pi$
$733$	−18.8995	−0.698067	−0.349034	+	0.937110i	$0.613490\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−40.1065	−1.47734
$738$	0	0
$739$	−38.1889	−1.40480	−0.702401	−	0.711782i	$-0.747889\pi$
$739$	−38.1889	−1.40480	−0.702401	+	0.711782i	$0.747889\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.9153	1.31761	0.658803	−	0.752315i	$-0.271063\pi$
$743$	35.9153	1.31761	0.658803	+	0.752315i	$0.271063\pi$
$744$	0	0
$745$	5.64113	0.206675
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	27.0979	0.988816	0.494408	−	0.869230i	$-0.335385\pi$
$751$	27.0979	0.988816	0.494408	+	0.869230i	$0.335385\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	25.9703	0.945156
$756$	0	0
$757$	42.0010	1.52655	0.763276	−	0.646073i	$-0.223590\pi$
$757$	42.0010	1.52655	0.763276	+	0.646073i	$0.223590\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24.6637	−0.894057	−0.447029	−	0.894520i	$-0.647518\pi$
$761$	−24.6637	−0.894057	−0.447029	+	0.894520i	$0.647518\pi$
$762$	0	0
$763$	12.7659	0.462158
$764$	0	0
$765$	0	0
$766$	0	0
$767$	31.9495	1.15363
$768$	0	0
$769$	−17.8362	−0.643191	−0.321596	−	0.946877i	$-0.604219\pi$
$769$	−17.8362	−0.643191	−0.321596	+	0.946877i	$0.604219\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.99174	0.323411	0.161705	−	0.986839i	$-0.448301\pi$
$773$	8.99174	0.323411	0.161705	+	0.986839i	$0.448301\pi$
$774$	0	0
$775$	1.40984	0.0506428
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.67525	0.131680
$780$	0	0
$781$	−35.3681	−1.26557
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−14.8236	−0.529075
$786$	0	0
$787$	13.2692	0.472995	0.236498	−	0.971632i	$-0.424000\pi$
$787$	13.2692	0.472995	0.236498	+	0.971632i	$0.424000\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.96759	−0.247739
$792$	0	0
$793$	59.5263	2.11384
$794$	0	0
$795$	0	0
$796$	0	0
$797$	49.2385	1.74412	0.872058	−	0.489402i	$-0.162785\pi$
$797$	49.2385	1.74412	0.872058	+	0.489402i	$0.162785\pi$
$798$	0	0
$799$	−67.7477	−2.39674
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.8749	0.454344
$804$	0	0
$805$	−14.2395	−0.501878
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30.8176	−1.08349	−0.541744	−	0.840543i	$-0.682236\pi$
$809$	−30.8176	−1.08349	−0.541744	+	0.840543i	$0.682236\pi$
$810$	0	0
$811$	38.8038	1.36259	0.681293	−	0.732010i	$-0.261418\pi$
$811$	38.8038	1.36259	0.681293	+	0.732010i	$0.261418\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.4616	−0.716738
$816$	0	0
$817$	−9.40158	−0.328920
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.9935	0.453475	0.226738	−	0.973956i	$-0.427194\pi$
$821$	12.9935	0.453475	0.226738	+	0.973956i	$0.427194\pi$
$822$	0	0
$823$	47.9224	1.67047	0.835236	−	0.549892i	$-0.185331\pi$
$823$	47.9224	1.67047	0.835236	+	0.549892i	$0.185331\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.19617	−0.180688	−0.0903442	−	0.995911i	$-0.528797\pi$
$827$	−5.19617	−0.180688	−0.0903442	+	0.995911i	$0.528797\pi$
$828$	0	0
$829$	33.3170	1.15715	0.578574	−	0.815630i	$-0.303609\pi$
$829$	33.3170	1.15715	0.578574	+	0.815630i	$0.303609\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.11977	0.212038
$834$	0	0
$835$	26.4929	0.916826
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.6966	−0.818099	−0.409049	−	0.912512i	$-0.634140\pi$
$839$	−23.6966	−0.818099	−0.409049	+	0.912512i	$0.634140\pi$
$840$	0	0
$841$	−28.3626	−0.978022
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−63.4994	−2.18445
$846$	0	0
$847$	−2.19340	−0.0753660
$848$	0	0
$849$	0	0
$850$	0	0
$851$	53.5735	1.83648
$852$	0	0
$853$	3.51482	0.120345	0.0601725	−	0.998188i	$-0.480835\pi$
$853$	3.51482	0.120345	0.0601725	+	0.998188i	$0.480835\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.1374	1.50770	0.753852	−	0.657044i	$-0.228193\pi$
$857$	44.1374	1.50770	0.753852	+	0.657044i	$0.228193\pi$
$858$	0	0
$859$	−32.2846	−1.10154	−0.550769	−	0.834658i	$-0.685665\pi$
$859$	−32.2846	−1.10154	−0.550769	+	0.834658i	$0.685665\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23.3169	−0.793718	−0.396859	−	0.917880i	$-0.629900\pi$
$863$	−23.3169	−0.793718	−0.396859	+	0.917880i	$0.629900\pi$
$864$	0	0
$865$	−31.5284	−1.07200
$866$	0	0
$867$	0	0
$868$	0	0
$869$	52.3323	1.77525
$870$	0	0
$871$	−93.9351	−3.18287
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.1675	0.411337
$876$	0	0
$877$	−54.8252	−1.85132	−0.925658	−	0.378361i	$-0.876488\pi$
$877$	−54.8252	−1.85132	−0.925658	+	0.378361i	$0.876488\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−10.0819	−0.339667	−0.169834	−	0.985473i	$-0.554323\pi$
$881$	−10.0819	−0.339667	−0.169834	+	0.985473i	$0.554323\pi$
$882$	0	0
$883$	−13.9144	−0.468255	−0.234128	−	0.972206i	$-0.575223\pi$
$883$	−13.9144	−0.468255	−0.234128	+	0.972206i	$0.575223\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.3088	1.35344	0.676718	−	0.736242i	$-0.263401\pi$
$887$	40.3088	1.35344	0.676718	+	0.736242i	$0.263401\pi$
$888$	0	0
$889$	−11.7335	−0.393530
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10.7115	0.358448
$894$	0	0
$895$	21.9863	0.734920
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.637361	0.0212572
$900$	0	0
$901$	−83.6389	−2.78642
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.61537	−0.219902
$906$	0	0
$907$	−7.23129	−0.240111	−0.120055	−	0.992767i	$-0.538307\pi$
$907$	−7.23129	−0.240111	−0.120055	+	0.992767i	$0.538307\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.07751	−0.300751	−0.150376	−	0.988629i	$-0.548048\pi$
$911$	−9.07751	−0.300751	−0.150376	+	0.988629i	$0.548048\pi$
$912$	0	0
$913$	−2.27300	−0.0752254
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.507730	−0.0167667
$918$	0	0
$919$	−23.5698	−0.777495	−0.388748	−	0.921344i	$-0.627092\pi$
$919$	−23.5698	−0.777495	−0.388748	+	0.921344i	$0.627092\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−82.8372	−2.72662
$924$	0	0
$925$	−11.9483	−0.392856
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−32.4071	−1.06324	−0.531621	−	0.846983i	$-0.678417\pi$
$929$	−32.4071	−1.06324	−0.531621	+	0.846983i	$0.678417\pi$
$930$	0	0
$931$	−0.967592	−0.0317116
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−32.6598	−1.06809
$936$	0	0
$937$	−7.37089	−0.240797	−0.120398	−	0.992726i	$-0.538417\pi$
$937$	−7.37089	−0.240797	−0.120398	+	0.992726i	$0.538417\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.00826	−0.163264	−0.0816322	−	0.996663i	$-0.526013\pi$
$941$	−5.00826	−0.163264	−0.0816322	+	0.996663i	$0.526013\pi$
$942$	0	0
$943$	−30.0758	−0.979402
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.0110	0.780252	0.390126	−	0.920762i	$-0.372432\pi$
$947$	24.0110	0.780252	0.390126	+	0.920762i	$0.372432\pi$
$948$	0	0
$949$	30.1548	0.978865
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14.3692	0.465463	0.232732	−	0.972541i	$-0.425234\pi$
$953$	14.3692	0.465463	0.232732	+	0.972541i	$0.425234\pi$
$954$	0	0
$955$	−24.5682	−0.795007
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.03241	−0.0979215
$960$	0	0
$961$	−30.3626	−0.979440
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−11.9797	−0.385641
$966$	0	0
$967$	−10.4374	−0.335645	−0.167823	−	0.985817i	$-0.553674\pi$
$967$	−10.4374	−0.335645	−0.167823	+	0.985817i	$0.553674\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.6845	−1.33772	−0.668860	−	0.743389i	$-0.733217\pi$
$971$	−41.6845	−1.33772	−0.668860	+	0.743389i	$0.733217\pi$
$972$	0	0
$973$	−18.1269	−0.581120
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.9813	−0.351324	−0.175662	−	0.984451i	$-0.556207\pi$
$977$	−10.9813	−0.351324	−0.175662	+	0.984451i	$0.556207\pi$
$978$	0	0
$979$	−11.8197	−0.377760
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−21.7956	−0.695171	−0.347585	−	0.937648i	$-0.612998\pi$
$983$	−21.7956	−0.695171	−0.347585	+	0.937648i	$0.612998\pi$
$984$	0	0
$985$	−36.8038	−1.17267
$986$	0	0
$987$	0	0
$988$	0	0
$989$	76.9362	2.44643
$990$	0	0
$991$	29.9389	0.951042	0.475521	−	0.879704i	$-0.342260\pi$
$991$	29.9389	0.951042	0.475521	+	0.879704i	$0.342260\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17.9934	−0.570428
$996$	0	0
$997$	−22.9164	−0.725770	−0.362885	−	0.931834i	$-0.618208\pi$
$997$	−22.9164	−0.725770	−0.362885	+	0.931834i	$0.618208\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.bt.1.2	yes	4
3.2	odd	2		6048.2.a.bp.1.3	yes	4
4.3	odd	2		6048.2.a.bs.1.2	yes	4
12.11	even	2		6048.2.a.bk.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.bk.1.3	✓	4	12.11	even	2
6048.2.a.bp.1.3	yes	4	3.2	odd	2
6048.2.a.bs.1.2	yes	4	4.3	odd	2
6048.2.a.bt.1.2	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79835 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.79835 q^{5} +1.00000 q^{7} +2.96759 q^{11} +6.95053 q^{13} +6.11977 q^{17} -0.967592 q^{19} +7.91812 q^{23} -1.76594 q^{25} -0.798349 q^{29} -0.798349 q^{31} -1.79835 q^{35} +6.76594 q^{37} -3.79835 q^{41} +9.71647 q^{43} -11.0703 q^{47} +1.00000 q^{49} -13.6670 q^{53} -5.33677 q^{55} +4.59670 q^{59} +8.56429 q^{61} -12.4995 q^{65} -13.5148 q^{67} -11.9181 q^{71} +4.33849 q^{73} +2.96759 q^{77} +17.6346 q^{79} -0.765941 q^{83} -11.0055 q^{85} -3.98294 q^{89} +6.95053 q^{91} +1.74007 q^{95} -6.86865 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^5 + 4 * q^7	\( 4 q + 2 q^{5} + 4 q^{7} + 8 q^{13} + 2 q^{17} + 8 q^{19} + 14 q^{25} + 6 q^{29} + 6 q^{31} + 2 q^{35} + 6 q^{37} - 6 q^{41} - 2 q^{43} - 2 q^{47} + 4 q^{49} + 6 q^{53} - 12 q^{55} + 4 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 2 q^{79} + 18 q^{83} + 22 q^{85} - 8 q^{89} + 8 q^{91} + 16 q^{95} + 24 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 4 * q^7 + 8 * q^13 + 2 * q^17 + 8 * q^19 + 14 * q^25 + 6 * q^29 + 6 * q^31 + 2 * q^35 + 6 * q^37 - 6 * q^41 - 2 * q^43 - 2 * q^47 + 4 * q^49 + 6 * q^53 - 12 * q^55 + 4 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 2 * q^79 + 18 * q^83 + 22 * q^85 - 8 * q^89 + 8 * q^91 + 16 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more