LMFDB - Embedded newform 6288.2.a.bd.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6288,2,Mod(1,6288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6288, 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("6288.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6288 = 2^{4} \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6288.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:	$50.2099327910$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.81589.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} + 8x - 1 $ x^5 - 6x^3 + 8x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3144)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.29150$ of defining polynomial
Character	$\chi$	$=$	6288.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.720312 q^{5} -1.67979 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.720312 q^{5} -1.67979 q^{7} +1.00000 q^{9} +5.90157 q^{11} -2.29150 q^{13} -0.720312 q^{15} -2.89863 q^{17} +1.44455 q^{19} +1.67979 q^{21} -2.01868 q^{23} -4.48115 q^{25} -1.00000 q^{27} -0.177836 q^{29} +5.80184 q^{31} -5.90157 q^{33} -1.20997 q^{35} -6.65452 q^{37} +2.29150 q^{39} -3.89270 q^{41} +8.38092 q^{43} +0.720312 q^{45} -11.5824 q^{47} -4.17831 q^{49} +2.89863 q^{51} +0.153046 q^{53} +4.25098 q^{55} -1.44455 q^{57} -7.29939 q^{59} +9.29508 q^{61} -1.67979 q^{63} -1.65060 q^{65} +3.04347 q^{67} +2.01868 q^{69} +15.1808 q^{71} -15.1376 q^{73} +4.48115 q^{75} -9.91339 q^{77} +5.91695 q^{79} +1.00000 q^{81} -8.74622 q^{83} -2.08792 q^{85} +0.177836 q^{87} -6.19079 q^{89} +3.84923 q^{91} -5.80184 q^{93} +1.04053 q^{95} -3.83742 q^{97} +5.90157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 3 q^{11} - 5 q^{13} + 5 q^{15} - q^{17} + q^{19} - 3 q^{21} + 7 q^{23} - 8 q^{25} - 5 q^{27} - 8 q^{29} + 14 q^{31} - 3 q^{33} - 6 q^{35} - 27 q^{37} + 5 q^{39} + 5 q^{41} + 3 q^{43} - 5 q^{45} + 4 q^{47} - 16 q^{49} + q^{51} + q^{53} + 12 q^{55} - q^{57} - 4 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} + 5 q^{67} - 7 q^{69} + 22 q^{71} - 18 q^{73} + 8 q^{75} - 8 q^{77} + 24 q^{79} + 5 q^{81} + 9 q^{83} - 14 q^{85} + 8 q^{87} - 12 q^{89} + 5 q^{91} - 14 q^{93} + 8 q^{95} - 20 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 3 * q^11 - 5 * q^13 + 5 * q^15 - q^17 + q^19 - 3 * q^21 + 7 * q^23 - 8 * q^25 - 5 * q^27 - 8 * q^29 + 14 * q^31 - 3 * q^33 - 6 * q^35 - 27 * q^37 + 5 * q^39 + 5 * q^41 + 3 * q^43 - 5 * q^45 + 4 * q^47 - 16 * q^49 + q^51 + q^53 + 12 * q^55 - q^57 - 4 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 + 5 * q^67 - 7 * q^69 + 22 * q^71 - 18 * q^73 + 8 * q^75 - 8 * q^77 + 24 * q^79 + 5 * q^81 + 9 * q^83 - 14 * q^85 + 8 * q^87 - 12 * q^89 + 5 * q^91 - 14 * q^93 + 8 * q^95 - 20 * q^97 + 3 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.720312	0.322134	0.161067	−	0.986944i	$-0.448507\pi$
$5$	0.720312	0.322134	0.161067	+	0.986944i	$0.448507\pi$
$6$	0	0
$7$	−1.67979	−0.634900	−0.317450	−	0.948275i	$-0.602827\pi$
$7$	−1.67979	−0.634900	−0.317450	+	0.948275i	$0.602827\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.90157	1.77939	0.889695	−	0.456555i	$-0.150917\pi$
$11$	5.90157	1.77939	0.889695	+	0.456555i	$0.150917\pi$
$12$	0	0
$13$	−2.29150	−0.635548	−0.317774	−	0.948166i	$-0.602935\pi$
$13$	−2.29150	−0.635548	−0.317774	+	0.948166i	$0.602935\pi$
$14$	0	0
$15$	−0.720312	−0.185984
$16$	0	0
$17$	−2.89863	−0.703020	−0.351510	−	0.936184i	$-0.614332\pi$
$17$	−2.89863	−0.703020	−0.351510	+	0.936184i	$0.614332\pi$
$18$	0	0
$19$	1.44455	0.331402	0.165701	−	0.986176i	$-0.447011\pi$
$19$	1.44455	0.331402	0.165701	+	0.986176i	$0.447011\pi$
$20$	0	0
$21$	1.67979	0.366560
$22$	0	0
$23$	−2.01868	−0.420924	−0.210462	−	0.977602i	$-0.567497\pi$
$23$	−2.01868	−0.420924	−0.210462	+	0.977602i	$0.567497\pi$
$24$	0	0
$25$	−4.48115	−0.896230
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.177836	−0.0330234	−0.0165117	−	0.999864i	$-0.505256\pi$
$29$	−0.177836	−0.0330234	−0.0165117	+	0.999864i	$0.505256\pi$
$30$	0	0
$31$	5.80184	1.04204	0.521021	−	0.853544i	$-0.325551\pi$
$31$	5.80184	1.04204	0.521021	+	0.853544i	$0.325551\pi$
$32$	0	0
$33$	−5.90157	−1.02733
$34$	0	0
$35$	−1.20997	−0.204523
$36$	0	0
$37$	−6.65452	−1.09400	−0.546998	−	0.837134i	$-0.684230\pi$
$37$	−6.65452	−1.09400	−0.546998	+	0.837134i	$0.684230\pi$
$38$	0	0
$39$	2.29150	0.366934
$40$	0	0
$41$	−3.89270	−0.607938	−0.303969	−	0.952682i	$-0.598312\pi$
$41$	−3.89270	−0.607938	−0.303969	+	0.952682i	$0.598312\pi$
$42$	0	0
$43$	8.38092	1.27808	0.639039	−	0.769174i	$-0.279332\pi$
$43$	8.38092	1.27808	0.639039	+	0.769174i	$0.279332\pi$
$44$	0	0
$45$	0.720312	0.107378
$46$	0	0
$47$	−11.5824	−1.68946	−0.844732	−	0.535189i	$-0.820240\pi$
$47$	−11.5824	−1.68946	−0.844732	+	0.535189i	$0.820240\pi$
$48$	0	0
$49$	−4.17831	−0.596902
$50$	0	0
$51$	2.89863	0.405889
$52$	0	0
$53$	0.153046	0.0210225	0.0105113	−	0.999945i	$-0.496654\pi$
$53$	0.153046	0.0210225	0.0105113	+	0.999945i	$0.496654\pi$
$54$	0	0
$55$	4.25098	0.573201
$56$	0	0
$57$	−1.44455	−0.191335
$58$	0	0
$59$	−7.29939	−0.950300	−0.475150	−	0.879905i	$-0.657606\pi$
$59$	−7.29939	−0.950300	−0.475150	+	0.879905i	$0.657606\pi$
$60$	0	0
$61$	9.29508	1.19011	0.595057	−	0.803684i	$-0.297130\pi$
$61$	9.29508	1.19011	0.595057	+	0.803684i	$0.297130\pi$
$62$	0	0
$63$	−1.67979	−0.211633
$64$	0	0
$65$	−1.65060	−0.204731
$66$	0	0
$67$	3.04347	0.371819	0.185910	−	0.982567i	$-0.440477\pi$
$67$	3.04347	0.371819	0.185910	+	0.982567i	$0.440477\pi$
$68$	0	0
$69$	2.01868	0.243021
$70$	0	0
$71$	15.1808	1.80162	0.900812	−	0.434208i	$-0.142972\pi$
$71$	15.1808	1.80162	0.900812	+	0.434208i	$0.142972\pi$
$72$	0	0
$73$	−15.1376	−1.77172	−0.885859	−	0.463954i	$-0.846430\pi$
$73$	−15.1376	−1.77172	−0.885859	+	0.463954i	$0.846430\pi$
$74$	0	0
$75$	4.48115	0.517439
$76$	0	0
$77$	−9.91339	−1.12974
$78$	0	0
$79$	5.91695	0.665709	0.332854	−	0.942978i	$-0.391988\pi$
$79$	5.91695	0.665709	0.332854	+	0.942978i	$0.391988\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.74622	−0.960022	−0.480011	−	0.877262i	$-0.659367\pi$
$83$	−8.74622	−0.960022	−0.480011	+	0.877262i	$0.659367\pi$
$84$	0	0
$85$	−2.08792	−0.226466
$86$	0	0
$87$	0.177836	0.0190660
$88$	0	0
$89$	−6.19079	−0.656223	−0.328111	−	0.944639i	$-0.606412\pi$
$89$	−6.19079	−0.656223	−0.328111	+	0.944639i	$0.606412\pi$
$90$	0	0
$91$	3.84923	0.403509
$92$	0	0
$93$	−5.80184	−0.601623
$94$	0	0
$95$	1.04053	0.106756
$96$	0	0
$97$	−3.83742	−0.389631	−0.194816	−	0.980840i	$-0.562411\pi$
$97$	−3.83742	−0.389631	−0.194816	+	0.980840i	$0.562411\pi$
$98$	0	0
$99$	5.90157	0.593130
$100$	0	0
$101$	14.0920	1.40220	0.701102	−	0.713061i	$-0.252692\pi$
$101$	14.0920	1.40220	0.701102	+	0.713061i	$0.252692\pi$
$102$	0	0
$103$	1.13139	0.111479	0.0557394	−	0.998445i	$-0.482248\pi$
$103$	1.13139	0.111479	0.0557394	+	0.998445i	$0.482248\pi$
$104$	0	0
$105$	1.20997	0.118081
$106$	0	0
$107$	−17.3758	−1.67978	−0.839889	−	0.542759i	$-0.817380\pi$
$107$	−17.3758	−1.67978	−0.839889	+	0.542759i	$0.817380\pi$
$108$	0	0
$109$	12.5763	1.20459	0.602294	−	0.798274i	$-0.294253\pi$
$109$	12.5763	1.20459	0.602294	+	0.798274i	$0.294253\pi$
$110$	0	0
$111$	6.65452	0.631619
$112$	0	0
$113$	5.83198	0.548626	0.274313	−	0.961640i	$-0.411550\pi$
$113$	5.83198	0.548626	0.274313	+	0.961640i	$0.411550\pi$
$114$	0	0
$115$	−1.45408	−0.135594
$116$	0	0
$117$	−2.29150	−0.211849
$118$	0	0
$119$	4.86908	0.446348
$120$	0	0
$121$	23.8286	2.16623
$122$	0	0
$123$	3.89270	0.350993
$124$	0	0
$125$	−6.82939	−0.610839
$126$	0	0
$127$	−8.97718	−0.796596	−0.398298	−	0.917256i	$-0.630399\pi$
$127$	−8.97718	−0.796596	−0.398298	+	0.917256i	$0.630399\pi$
$128$	0	0
$129$	−8.38092	−0.737899
$130$	0	0
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	0	0
$133$	−2.42653	−0.210407
$134$	0	0
$135$	−0.720312	−0.0619946
$136$	0	0
$137$	−2.80376	−0.239542	−0.119771	−	0.992802i	$-0.538216\pi$
$137$	−2.80376	−0.239542	−0.119771	+	0.992802i	$0.538216\pi$
$138$	0	0
$139$	−23.1927	−1.96718	−0.983591	−	0.180413i	$-0.942257\pi$
$139$	−23.1927	−1.96718	−0.983591	+	0.180413i	$0.942257\pi$
$140$	0	0
$141$	11.5824	0.975413
$142$	0	0
$143$	−13.5235	−1.13089
$144$	0	0
$145$	−0.128098	−0.0106379
$146$	0	0
$147$	4.17831	0.344622
$148$	0	0
$149$	−13.8439	−1.13414	−0.567070	−	0.823669i	$-0.691923\pi$
$149$	−13.8439	−1.13414	−0.567070	+	0.823669i	$0.691923\pi$
$150$	0	0
$151$	6.73691	0.548242	0.274121	−	0.961695i	$-0.411613\pi$
$151$	6.73691	0.548242	0.274121	+	0.961695i	$0.411613\pi$
$152$	0	0
$153$	−2.89863	−0.234340
$154$	0	0
$155$	4.17914	0.335676
$156$	0	0
$157$	1.50918	0.120445	0.0602227	−	0.998185i	$-0.480819\pi$
$157$	1.50918	0.120445	0.0602227	+	0.998185i	$0.480819\pi$
$158$	0	0
$159$	−0.153046	−0.0121374
$160$	0	0
$161$	3.39095	0.267245
$162$	0	0
$163$	14.8218	1.16093	0.580467	−	0.814283i	$-0.302870\pi$
$163$	14.8218	1.16093	0.580467	+	0.814283i	$0.302870\pi$
$164$	0	0
$165$	−4.25098	−0.330938
$166$	0	0
$167$	13.7339	1.06276	0.531381	−	0.847133i	$-0.321673\pi$
$167$	13.7339	1.06276	0.531381	+	0.847133i	$0.321673\pi$
$168$	0	0
$169$	−7.74902	−0.596079
$170$	0	0
$171$	1.44455	0.110467
$172$	0	0
$173$	−9.30952	−0.707789	−0.353895	−	0.935285i	$-0.615143\pi$
$173$	−9.30952	−0.707789	−0.353895	+	0.935285i	$0.615143\pi$
$174$	0	0
$175$	7.52738	0.569016
$176$	0	0
$177$	7.29939	0.548656
$178$	0	0
$179$	−3.12586	−0.233638	−0.116819	−	0.993153i	$-0.537270\pi$
$179$	−3.12586	−0.233638	−0.116819	+	0.993153i	$0.537270\pi$
$180$	0	0
$181$	−8.68960	−0.645893	−0.322946	−	0.946417i	$-0.604673\pi$
$181$	−8.68960	−0.645893	−0.322946	+	0.946417i	$0.604673\pi$
$182$	0	0
$183$	−9.29508	−0.687112
$184$	0	0
$185$	−4.79333	−0.352413
$186$	0	0
$187$	−17.1065	−1.25095
$188$	0	0
$189$	1.67979	0.122187
$190$	0	0
$191$	0.289421	0.0209418	0.0104709	−	0.999945i	$-0.496667\pi$
$191$	0.289421	0.0209418	0.0104709	+	0.999945i	$0.496667\pi$
$192$	0	0
$193$	−24.3737	−1.75446	−0.877229	−	0.480072i	$-0.840611\pi$
$193$	−24.3737	−1.75446	−0.877229	+	0.480072i	$0.840611\pi$
$194$	0	0
$195$	1.65060	0.118202
$196$	0	0
$197$	13.1393	0.936135	0.468067	−	0.883693i	$-0.344950\pi$
$197$	13.1393	0.936135	0.468067	+	0.883693i	$0.344950\pi$
$198$	0	0
$199$	−19.9498	−1.41420	−0.707101	−	0.707113i	$-0.749997\pi$
$199$	−19.9498	−1.41420	−0.707101	+	0.707113i	$0.749997\pi$
$200$	0	0
$201$	−3.04347	−0.214670
$202$	0	0
$203$	0.298727	0.0209665
$204$	0	0
$205$	−2.80396	−0.195837
$206$	0	0
$207$	−2.01868	−0.140308
$208$	0	0
$209$	8.52510	0.589693
$210$	0	0
$211$	14.8983	1.02564	0.512820	−	0.858496i	$-0.328601\pi$
$211$	14.8983	1.02564	0.512820	+	0.858496i	$0.328601\pi$
$212$	0	0
$213$	−15.1808	−1.04017
$214$	0	0
$215$	6.03688	0.411712
$216$	0	0
$217$	−9.74586	−0.661592
$218$	0	0
$219$	15.1376	1.02290
$220$	0	0
$221$	6.64221	0.446803
$222$	0	0
$223$	12.4077	0.830883	0.415441	−	0.909620i	$-0.363627\pi$
$223$	12.4077	0.830883	0.415441	+	0.909620i	$0.363627\pi$
$224$	0	0
$225$	−4.48115	−0.298743
$226$	0	0
$227$	−4.59189	−0.304774	−0.152387	−	0.988321i	$-0.548696\pi$
$227$	−4.59189	−0.304774	−0.152387	+	0.988321i	$0.548696\pi$
$228$	0	0
$229$	16.9437	1.11967	0.559834	−	0.828605i	$-0.310865\pi$
$229$	16.9437	1.11967	0.559834	+	0.828605i	$0.310865\pi$
$230$	0	0
$231$	9.91339	0.652253
$232$	0	0
$233$	15.1520	0.992644	0.496322	−	0.868139i	$-0.334684\pi$
$233$	15.1520	0.992644	0.496322	+	0.868139i	$0.334684\pi$
$234$	0	0
$235$	−8.34293	−0.544233
$236$	0	0
$237$	−5.91695	−0.384347
$238$	0	0
$239$	26.7657	1.73133	0.865665	−	0.500624i	$-0.166896\pi$
$239$	26.7657	1.73133	0.865665	+	0.500624i	$0.166896\pi$
$240$	0	0
$241$	−26.6854	−1.71896	−0.859478	−	0.511172i	$-0.829211\pi$
$241$	−26.6854	−1.71896	−0.859478	+	0.511172i	$0.829211\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.00969	−0.192282
$246$	0	0
$247$	−3.31018	−0.210622
$248$	0	0
$249$	8.74622	0.554269
$250$	0	0
$251$	−29.9983	−1.89348	−0.946738	−	0.322005i	$-0.895643\pi$
$251$	−29.9983	−1.89348	−0.946738	+	0.322005i	$0.895643\pi$
$252$	0	0
$253$	−11.9134	−0.748988
$254$	0	0
$255$	2.08792	0.130750
$256$	0	0
$257$	−9.66611	−0.602955	−0.301478	−	0.953473i	$-0.597480\pi$
$257$	−9.66611	−0.602955	−0.301478	+	0.953473i	$0.597480\pi$
$258$	0	0
$259$	11.1782	0.694578
$260$	0	0
$261$	−0.177836	−0.0110078
$262$	0	0
$263$	2.20079	0.135707	0.0678533	−	0.997695i	$-0.478385\pi$
$263$	2.20079	0.135707	0.0678533	+	0.997695i	$0.478385\pi$
$264$	0	0
$265$	0.110241	0.00677206
$266$	0	0
$267$	6.19079	0.378870
$268$	0	0
$269$	−10.5019	−0.640314	−0.320157	−	0.947364i	$-0.603736\pi$
$269$	−10.5019	−0.640314	−0.320157	+	0.947364i	$0.603736\pi$
$270$	0	0
$271$	−27.7026	−1.68281	−0.841407	−	0.540403i	$-0.818272\pi$
$271$	−27.7026	−1.68281	−0.841407	+	0.540403i	$0.818272\pi$
$272$	0	0
$273$	−3.84923	−0.232966
$274$	0	0
$275$	−26.4458	−1.59474
$276$	0	0
$277$	−7.85683	−0.472071	−0.236036	−	0.971744i	$-0.575848\pi$
$277$	−7.85683	−0.472071	−0.236036	+	0.971744i	$0.575848\pi$
$278$	0	0
$279$	5.80184	0.347347
$280$	0	0
$281$	−19.0590	−1.13696	−0.568481	−	0.822696i	$-0.692469\pi$
$281$	−19.0590	−1.13696	−0.568481	+	0.822696i	$0.692469\pi$
$282$	0	0
$283$	−29.9123	−1.77810	−0.889051	−	0.457808i	$-0.848635\pi$
$283$	−29.9123	−1.77810	−0.889051	+	0.457808i	$0.848635\pi$
$284$	0	0
$285$	−1.04053	−0.0616354
$286$	0	0
$287$	6.53891	0.385980
$288$	0	0
$289$	−8.59796	−0.505762
$290$	0	0
$291$	3.83742	0.224954
$292$	0	0
$293$	−24.8179	−1.44988	−0.724938	−	0.688814i	$-0.758132\pi$
$293$	−24.8179	−1.44988	−0.724938	+	0.688814i	$0.758132\pi$
$294$	0	0
$295$	−5.25784	−0.306123
$296$	0	0
$297$	−5.90157	−0.342444
$298$	0	0
$299$	4.62581	0.267517
$300$	0	0
$301$	−14.0782	−0.811452
$302$	0	0
$303$	−14.0920	−0.809563
$304$	0	0
$305$	6.69536	0.383375
$306$	0	0
$307$	6.44362	0.367757	0.183878	−	0.982949i	$-0.441135\pi$
$307$	6.44362	0.367757	0.183878	+	0.982949i	$0.441135\pi$
$308$	0	0
$309$	−1.13139	−0.0643624
$310$	0	0
$311$	0.410769	0.0232926	0.0116463	−	0.999932i	$-0.496293\pi$
$311$	0.410769	0.0232926	0.0116463	+	0.999932i	$0.496293\pi$
$312$	0	0
$313$	6.13337	0.346678	0.173339	−	0.984862i	$-0.444544\pi$
$313$	6.13337	0.346678	0.173339	+	0.984862i	$0.444544\pi$
$314$	0	0
$315$	−1.20997	−0.0681742
$316$	0	0
$317$	−2.42092	−0.135972	−0.0679862	−	0.997686i	$-0.521657\pi$
$317$	−2.42092	−0.135972	−0.0679862	+	0.997686i	$0.521657\pi$
$318$	0	0
$319$	−1.04951	−0.0587615
$320$	0	0
$321$	17.3758	0.969820
$322$	0	0
$323$	−4.18720	−0.232982
$324$	0	0
$325$	10.2686	0.569597
$326$	0	0
$327$	−12.5763	−0.695470
$328$	0	0
$329$	19.4559	1.07264
$330$	0	0
$331$	−7.54345	−0.414626	−0.207313	−	0.978275i	$-0.566472\pi$
$331$	−7.54345	−0.414626	−0.207313	+	0.978275i	$0.566472\pi$
$332$	0	0
$333$	−6.65452	−0.364665
$334$	0	0
$335$	2.19225	0.119775
$336$	0	0
$337$	−19.9000	−1.08402	−0.542011	−	0.840371i	$-0.682337\pi$
$337$	−19.9000	−1.08402	−0.542011	+	0.840371i	$0.682337\pi$
$338$	0	0
$339$	−5.83198	−0.316749
$340$	0	0
$341$	34.2400	1.85420
$342$	0	0
$343$	18.7772	1.01387
$344$	0	0
$345$	1.45408	0.0782850
$346$	0	0
$347$	1.52978	0.0821228	0.0410614	−	0.999157i	$-0.486926\pi$
$347$	1.52978	0.0821228	0.0410614	+	0.999157i	$0.486926\pi$
$348$	0	0
$349$	−3.16728	−0.169541	−0.0847703	−	0.996401i	$-0.527016\pi$
$349$	−3.16728	−0.169541	−0.0847703	+	0.996401i	$0.527016\pi$
$350$	0	0
$351$	2.29150	0.122311
$352$	0	0
$353$	32.7255	1.74180	0.870900	−	0.491460i	$-0.163536\pi$
$353$	32.7255	1.74180	0.870900	+	0.491460i	$0.163536\pi$
$354$	0	0
$355$	10.9349	0.580364
$356$	0	0
$357$	−4.86908	−0.257699
$358$	0	0
$359$	−23.3443	−1.23207	−0.616034	−	0.787720i	$-0.711261\pi$
$359$	−23.3443	−1.23207	−0.616034	+	0.787720i	$0.711261\pi$
$360$	0	0
$361$	−16.9133	−0.890173
$362$	0	0
$363$	−23.8286	−1.25067
$364$	0	0
$365$	−10.9038	−0.570730
$366$	0	0
$367$	1.03670	0.0541151	0.0270575	−	0.999634i	$-0.491386\pi$
$367$	1.03670	0.0541151	0.0270575	+	0.999634i	$0.491386\pi$
$368$	0	0
$369$	−3.89270	−0.202646
$370$	0	0
$371$	−0.257085	−0.0133472
$372$	0	0
$373$	−16.5039	−0.854541	−0.427270	−	0.904124i	$-0.640525\pi$
$373$	−16.5039	−0.854541	−0.427270	+	0.904124i	$0.640525\pi$
$374$	0	0
$375$	6.82939	0.352668
$376$	0	0
$377$	0.407512	0.0209879
$378$	0	0
$379$	−13.5755	−0.697329	−0.348665	−	0.937248i	$-0.613365\pi$
$379$	−13.5755	−0.697329	−0.348665	+	0.937248i	$0.613365\pi$
$380$	0	0
$381$	8.97718	0.459915
$382$	0	0
$383$	17.4042	0.889311	0.444656	−	0.895702i	$-0.353326\pi$
$383$	17.4042	0.889311	0.444656	+	0.895702i	$0.353326\pi$
$384$	0	0
$385$	−7.14073	−0.363926
$386$	0	0
$387$	8.38092	0.426026
$388$	0	0
$389$	−26.3608	−1.33654	−0.668272	−	0.743917i	$-0.732966\pi$
$389$	−26.3608	−1.33654	−0.668272	+	0.743917i	$0.732966\pi$
$390$	0	0
$391$	5.85140	0.295918
$392$	0	0
$393$	1.00000	0.0504433
$394$	0	0
$395$	4.26205	0.214447
$396$	0	0
$397$	−15.2856	−0.767164	−0.383582	−	0.923507i	$-0.625310\pi$
$397$	−15.2856	−0.767164	−0.383582	+	0.923507i	$0.625310\pi$
$398$	0	0
$399$	2.42653	0.121479
$400$	0	0
$401$	2.86151	0.142897	0.0714484	−	0.997444i	$-0.477238\pi$
$401$	2.86151	0.142897	0.0714484	+	0.997444i	$0.477238\pi$
$402$	0	0
$403$	−13.2949	−0.662267
$404$	0	0
$405$	0.720312	0.0357926
$406$	0	0
$407$	−39.2721	−1.94665
$408$	0	0
$409$	−0.270720	−0.0133862	−0.00669312	−	0.999978i	$-0.502131\pi$
$409$	−0.270720	−0.0133862	−0.00669312	+	0.999978i	$0.502131\pi$
$410$	0	0
$411$	2.80376	0.138299
$412$	0	0
$413$	12.2614	0.603345
$414$	0	0
$415$	−6.30001	−0.309255
$416$	0	0
$417$	23.1927	1.13575
$418$	0	0
$419$	1.75186	0.0855838	0.0427919	−	0.999084i	$-0.486375\pi$
$419$	1.75186	0.0855838	0.0427919	+	0.999084i	$0.486375\pi$
$420$	0	0
$421$	−22.3458	−1.08907	−0.544534	−	0.838739i	$-0.683293\pi$
$421$	−22.3458	−1.08907	−0.544534	+	0.838739i	$0.683293\pi$
$422$	0	0
$423$	−11.5824	−0.563155
$424$	0	0
$425$	12.9892	0.630068
$426$	0	0
$427$	−15.6138	−0.755603
$428$	0	0
$429$	13.5235	0.652919
$430$	0	0
$431$	−34.4586	−1.65981	−0.829906	−	0.557903i	$-0.811606\pi$
$431$	−34.4586	−1.65981	−0.829906	+	0.557903i	$0.811606\pi$
$432$	0	0
$433$	−9.66343	−0.464395	−0.232197	−	0.972669i	$-0.574592\pi$
$433$	−9.66343	−0.464395	−0.232197	+	0.972669i	$0.574592\pi$
$434$	0	0
$435$	0.128098	0.00614181
$436$	0	0
$437$	−2.91608	−0.139495
$438$	0	0
$439$	25.5621	1.22001	0.610007	−	0.792396i	$-0.291167\pi$
$439$	25.5621	1.22001	0.610007	+	0.792396i	$0.291167\pi$
$440$	0	0
$441$	−4.17831	−0.198967
$442$	0	0
$443$	−27.9754	−1.32915	−0.664576	−	0.747220i	$-0.731388\pi$
$443$	−27.9754	−1.32915	−0.664576	+	0.747220i	$0.731388\pi$
$444$	0	0
$445$	−4.45930	−0.211391
$446$	0	0
$447$	13.8439	0.654796
$448$	0	0
$449$	22.0216	1.03926	0.519631	−	0.854391i	$-0.326069\pi$
$449$	22.0216	1.03926	0.519631	+	0.854391i	$0.326069\pi$
$450$	0	0
$451$	−22.9731	−1.08176
$452$	0	0
$453$	−6.73691	−0.316528
$454$	0	0
$455$	2.77265	0.129984
$456$	0	0
$457$	29.1102	1.36172	0.680858	−	0.732415i	$-0.261607\pi$
$457$	29.1102	1.36172	0.680858	+	0.732415i	$0.261607\pi$
$458$	0	0
$459$	2.89863	0.135296
$460$	0	0
$461$	18.8634	0.878555	0.439277	−	0.898351i	$-0.355235\pi$
$461$	18.8634	0.878555	0.439277	+	0.898351i	$0.355235\pi$
$462$	0	0
$463$	−8.81241	−0.409547	−0.204774	−	0.978809i	$-0.565646\pi$
$463$	−8.81241	−0.409547	−0.204774	+	0.978809i	$0.565646\pi$
$464$	0	0
$465$	−4.17914	−0.193803
$466$	0	0
$467$	29.3190	1.35672	0.678360	−	0.734729i	$-0.262691\pi$
$467$	29.3190	1.35672	0.678360	+	0.734729i	$0.262691\pi$
$468$	0	0
$469$	−5.11238	−0.236068
$470$	0	0
$471$	−1.50918	−0.0695392
$472$	0	0
$473$	49.4606	2.27420
$474$	0	0
$475$	−6.47323	−0.297012
$476$	0	0
$477$	0.153046	0.00700751
$478$	0	0
$479$	−11.1162	−0.507913	−0.253957	−	0.967216i	$-0.581732\pi$
$479$	−11.1162	−0.507913	−0.253957	+	0.967216i	$0.581732\pi$
$480$	0	0
$481$	15.2488	0.695287
$482$	0	0
$483$	−3.39095	−0.154294
$484$	0	0
$485$	−2.76414	−0.125513
$486$	0	0
$487$	−20.4434	−0.926377	−0.463189	−	0.886260i	$-0.653295\pi$
$487$	−20.4434	−0.926377	−0.463189	+	0.886260i	$0.653295\pi$
$488$	0	0
$489$	−14.8218	−0.670266
$490$	0	0
$491$	19.2783	0.870020	0.435010	−	0.900426i	$-0.356745\pi$
$491$	19.2783	0.870020	0.435010	+	0.900426i	$0.356745\pi$
$492$	0	0
$493$	0.515481	0.0232161
$494$	0	0
$495$	4.25098	0.191067
$496$	0	0
$497$	−25.5004	−1.14385
$498$	0	0
$499$	−27.3278	−1.22336	−0.611681	−	0.791105i	$-0.709506\pi$
$499$	−27.3278	−1.22336	−0.611681	+	0.791105i	$0.709506\pi$
$500$	0	0
$501$	−13.7339	−0.613586
$502$	0	0
$503$	34.6421	1.54462	0.772308	−	0.635249i	$-0.219102\pi$
$503$	34.6421	1.54462	0.772308	+	0.635249i	$0.219102\pi$
$504$	0	0
$505$	10.1506	0.451697
$506$	0	0
$507$	7.74902	0.344146
$508$	0	0
$509$	−20.1949	−0.895124	−0.447562	−	0.894253i	$-0.647708\pi$
$509$	−20.1949	−0.895124	−0.447562	+	0.894253i	$0.647708\pi$
$510$	0	0
$511$	25.4279	1.12486
$512$	0	0
$513$	−1.44455	−0.0637783
$514$	0	0
$515$	0.814952	0.0359111
$516$	0	0
$517$	−68.3543	−3.00622
$518$	0	0
$519$	9.30952	0.408642
$520$	0	0
$521$	37.1787	1.62883	0.814414	−	0.580285i	$-0.197059\pi$
$521$	37.1787	1.62883	0.814414	+	0.580285i	$0.197059\pi$
$522$	0	0
$523$	22.1183	0.967164	0.483582	−	0.875299i	$-0.339336\pi$
$523$	22.1183	0.967164	0.483582	+	0.875299i	$0.339336\pi$
$524$	0	0
$525$	−7.52738	−0.328522
$526$	0	0
$527$	−16.8174	−0.732576
$528$	0	0
$529$	−18.9249	−0.822823
$530$	0	0
$531$	−7.29939	−0.316767
$532$	0	0
$533$	8.92013	0.386374
$534$	0	0
$535$	−12.5160	−0.541113
$536$	0	0
$537$	3.12586	0.134891
$538$	0	0
$539$	−24.6586	−1.06212
$540$	0	0
$541$	−7.18556	−0.308931	−0.154466	−	0.987998i	$-0.549366\pi$
$541$	−7.18556	−0.308931	−0.154466	+	0.987998i	$0.549366\pi$
$542$	0	0
$543$	8.68960	0.372906
$544$	0	0
$545$	9.05885	0.388038
$546$	0	0
$547$	18.2277	0.779359	0.389679	−	0.920951i	$-0.372586\pi$
$547$	18.2277	0.779359	0.389679	+	0.920951i	$0.372586\pi$
$548$	0	0
$549$	9.29508	0.396704
$550$	0	0
$551$	−0.256893	−0.0109440
$552$	0	0
$553$	−9.93921	−0.422658
$554$	0	0
$555$	4.79333	0.203466
$556$	0	0
$557$	1.56752	0.0664181	0.0332091	−	0.999448i	$-0.489427\pi$
$557$	1.56752	0.0664181	0.0332091	+	0.999448i	$0.489427\pi$
$558$	0	0
$559$	−19.2049	−0.812280
$560$	0	0
$561$	17.1065	0.722235
$562$	0	0
$563$	21.2699	0.896420	0.448210	−	0.893928i	$-0.352062\pi$
$563$	21.2699	0.896420	0.448210	+	0.893928i	$0.352062\pi$
$564$	0	0
$565$	4.20084	0.176731
$566$	0	0
$567$	−1.67979	−0.0705444
$568$	0	0
$569$	29.3587	1.23078	0.615389	−	0.788223i	$-0.288999\pi$
$569$	29.3587	1.23078	0.615389	+	0.788223i	$0.288999\pi$
$570$	0	0
$571$	−20.5946	−0.861855	−0.430928	−	0.902386i	$-0.641814\pi$
$571$	−20.5946	−0.861855	−0.430928	+	0.902386i	$0.641814\pi$
$572$	0	0
$573$	−0.289421	−0.0120907
$574$	0	0
$575$	9.04601	0.377245
$576$	0	0
$577$	25.7927	1.07377	0.536883	−	0.843657i	$-0.319602\pi$
$577$	25.7927	1.07377	0.536883	+	0.843657i	$0.319602\pi$
$578$	0	0
$579$	24.3737	1.01294
$580$	0	0
$581$	14.6918	0.609518
$582$	0	0
$583$	0.903214	0.0374073
$584$	0	0
$585$	−1.65060	−0.0682438
$586$	0	0
$587$	−17.5618	−0.724854	−0.362427	−	0.932012i	$-0.618052\pi$
$587$	−17.5618	−0.724854	−0.362427	+	0.932012i	$0.618052\pi$
$588$	0	0
$589$	8.38103	0.345334
$590$	0	0
$591$	−13.1393	−0.540478
$592$	0	0
$593$	−30.8003	−1.26481	−0.632407	−	0.774636i	$-0.717933\pi$
$593$	−30.8003	−1.26481	−0.632407	+	0.774636i	$0.717933\pi$
$594$	0	0
$595$	3.50726	0.143784
$596$	0	0
$597$	19.9498	0.816490
$598$	0	0
$599$	9.56184	0.390686	0.195343	−	0.980735i	$-0.437418\pi$
$599$	9.56184	0.390686	0.195343	+	0.980735i	$0.437418\pi$
$600$	0	0
$601$	1.30987	0.0534306	0.0267153	−	0.999643i	$-0.491495\pi$
$601$	1.30987	0.0534306	0.0267153	+	0.999643i	$0.491495\pi$
$602$	0	0
$603$	3.04347	0.123940
$604$	0	0
$605$	17.1640	0.697816
$606$	0	0
$607$	−40.4947	−1.64363	−0.821815	−	0.569754i	$-0.807039\pi$
$607$	−40.4947	−1.64363	−0.821815	+	0.569754i	$0.807039\pi$
$608$	0	0
$609$	−0.298727	−0.0121050
$610$	0	0
$611$	26.5410	1.07374
$612$	0	0
$613$	−36.2204	−1.46293	−0.731465	−	0.681879i	$-0.761163\pi$
$613$	−36.2204	−1.46293	−0.731465	+	0.681879i	$0.761163\pi$
$614$	0	0
$615$	2.80396	0.113067
$616$	0	0
$617$	33.5134	1.34920	0.674600	−	0.738184i	$-0.264316\pi$
$617$	33.5134	1.34920	0.674600	+	0.738184i	$0.264316\pi$
$618$	0	0
$619$	−20.8126	−0.836528	−0.418264	−	0.908325i	$-0.637361\pi$
$619$	−20.8126	−0.836528	−0.418264	+	0.908325i	$0.637361\pi$
$620$	0	0
$621$	2.01868	0.0810068
$622$	0	0
$623$	10.3992	0.416636
$624$	0	0
$625$	17.4865	0.699458
$626$	0	0
$627$	−8.52510	−0.340460
$628$	0	0
$629$	19.2890	0.769102
$630$	0	0
$631$	44.7672	1.78215	0.891077	−	0.453853i	$-0.149951\pi$
$631$	44.7672	1.78215	0.891077	+	0.453853i	$0.149951\pi$
$632$	0	0
$633$	−14.8983	−0.592154
$634$	0	0
$635$	−6.46637	−0.256610
$636$	0	0
$637$	9.57461	0.379360
$638$	0	0
$639$	15.1808	0.600542
$640$	0	0
$641$	−15.2855	−0.603742	−0.301871	−	0.953349i	$-0.597611\pi$
$641$	−15.2855	−0.603742	−0.301871	+	0.953349i	$0.597611\pi$
$642$	0	0
$643$	−13.3298	−0.525676	−0.262838	−	0.964840i	$-0.584658\pi$
$643$	−13.3298	−0.525676	−0.262838	+	0.964840i	$0.584658\pi$
$644$	0	0
$645$	−6.03688	−0.237702
$646$	0	0
$647$	−19.1794	−0.754021	−0.377010	−	0.926209i	$-0.623048\pi$
$647$	−19.1794	−0.754021	−0.377010	+	0.926209i	$0.623048\pi$
$648$	0	0
$649$	−43.0779	−1.69095
$650$	0	0
$651$	9.74586	0.381970
$652$	0	0
$653$	22.1335	0.866151	0.433076	−	0.901358i	$-0.357428\pi$
$653$	22.1335	0.866151	0.433076	+	0.901358i	$0.357428\pi$
$654$	0	0
$655$	−0.720312	−0.0281449
$656$	0	0
$657$	−15.1376	−0.590573
$658$	0	0
$659$	−16.0228	−0.624161	−0.312081	−	0.950056i	$-0.601026\pi$
$659$	−16.0228	−0.624161	−0.312081	+	0.950056i	$0.601026\pi$
$660$	0	0
$661$	16.3547	0.636123	0.318061	−	0.948070i	$-0.396968\pi$
$661$	16.3547	0.636123	0.318061	+	0.948070i	$0.396968\pi$
$662$	0	0
$663$	−6.64221	−0.257962
$664$	0	0
$665$	−1.74786	−0.0677791
$666$	0	0
$667$	0.358994	0.0139003
$668$	0	0
$669$	−12.4077	−0.479710
$670$	0	0
$671$	54.8556	2.11768
$672$	0	0
$673$	12.7683	0.492182	0.246091	−	0.969247i	$-0.420854\pi$
$673$	12.7683	0.492182	0.246091	+	0.969247i	$0.420854\pi$
$674$	0	0
$675$	4.48115	0.172480
$676$	0	0
$677$	−46.2387	−1.77710	−0.888550	−	0.458781i	$-0.848286\pi$
$677$	−46.2387	−1.77710	−0.888550	+	0.458781i	$0.848286\pi$
$678$	0	0
$679$	6.44605	0.247377
$680$	0	0
$681$	4.59189	0.175962
$682$	0	0
$683$	16.7613	0.641355	0.320677	−	0.947188i	$-0.396089\pi$
$683$	16.7613	0.641355	0.320677	+	0.947188i	$0.396089\pi$
$684$	0	0
$685$	−2.01958	−0.0771644
$686$	0	0
$687$	−16.9437	−0.646441
$688$	0	0
$689$	−0.350706	−0.0133608
$690$	0	0
$691$	29.4107	1.11884	0.559418	−	0.828885i	$-0.311025\pi$
$691$	29.4107	1.11884	0.559418	+	0.828885i	$0.311025\pi$
$692$	0	0
$693$	−9.91339	−0.376578
$694$	0	0
$695$	−16.7060	−0.633695
$696$	0	0
$697$	11.2835	0.427393
$698$	0	0
$699$	−15.1520	−0.573103
$700$	0	0
$701$	−3.81153	−0.143959	−0.0719797	−	0.997406i	$-0.522932\pi$
$701$	−3.81153	−0.143959	−0.0719797	+	0.997406i	$0.522932\pi$
$702$	0	0
$703$	−9.61277	−0.362552
$704$	0	0
$705$	8.34293	0.314213
$706$	0	0
$707$	−23.6715	−0.890259
$708$	0	0
$709$	1.65349	0.0620983	0.0310491	−	0.999518i	$-0.490115\pi$
$709$	1.65349	0.0620983	0.0310491	+	0.999518i	$0.490115\pi$
$710$	0	0
$711$	5.91695	0.221903
$712$	0	0
$713$	−11.7121	−0.438620
$714$	0	0
$715$	−9.74111	−0.364297
$716$	0	0
$717$	−26.7657	−0.999583
$718$	0	0
$719$	−17.1801	−0.640709	−0.320354	−	0.947298i	$-0.603802\pi$
$719$	−17.1801	−0.640709	−0.320354	+	0.947298i	$0.603802\pi$
$720$	0	0
$721$	−1.90049	−0.0707779
$722$	0	0
$723$	26.6854	0.992440
$724$	0	0
$725$	0.796911	0.0295965
$726$	0	0
$727$	40.1742	1.48998	0.744989	−	0.667077i	$-0.232455\pi$
$727$	40.1742	1.48998	0.744989	+	0.667077i	$0.232455\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.2932	−0.898515
$732$	0	0
$733$	9.00055	0.332443	0.166221	−	0.986088i	$-0.446843\pi$
$733$	9.00055	0.332443	0.166221	+	0.986088i	$0.446843\pi$
$734$	0	0
$735$	3.00969	0.111014
$736$	0	0
$737$	17.9613	0.661611
$738$	0	0
$739$	18.9591	0.697421	0.348710	−	0.937231i	$-0.386620\pi$
$739$	18.9591	0.697421	0.348710	+	0.937231i	$0.386620\pi$
$740$	0	0
$741$	3.31018	0.121603
$742$	0	0
$743$	−34.9523	−1.28228	−0.641138	−	0.767425i	$-0.721538\pi$
$743$	−34.9523	−1.28228	−0.641138	+	0.767425i	$0.721538\pi$
$744$	0	0
$745$	−9.97197	−0.365345
$746$	0	0
$747$	−8.74622	−0.320007
$748$	0	0
$749$	29.1876	1.06649
$750$	0	0
$751$	−50.8048	−1.85389	−0.926946	−	0.375194i	$-0.877576\pi$
$751$	−50.8048	−1.85389	−0.926946	+	0.375194i	$0.877576\pi$
$752$	0	0
$753$	29.9983	1.09320
$754$	0	0
$755$	4.85268	0.176607
$756$	0	0
$757$	−18.4395	−0.670196	−0.335098	−	0.942183i	$-0.608769\pi$
$757$	−18.4395	−0.670196	−0.335098	+	0.942183i	$0.608769\pi$
$758$	0	0
$759$	11.9134	0.432428
$760$	0	0
$761$	19.2212	0.696769	0.348384	−	0.937352i	$-0.386730\pi$
$761$	19.2212	0.696769	0.348384	+	0.937352i	$0.386730\pi$
$762$	0	0
$763$	−21.1255	−0.764793
$764$	0	0
$765$	−2.08792	−0.0754888
$766$	0	0
$767$	16.7266	0.603961
$768$	0	0
$769$	−17.0566	−0.615078	−0.307539	−	0.951535i	$-0.599505\pi$
$769$	−17.0566	−0.615078	−0.307539	+	0.951535i	$0.599505\pi$
$770$	0	0
$771$	9.66611	0.348116
$772$	0	0
$773$	47.7805	1.71855	0.859273	−	0.511517i	$-0.170916\pi$
$773$	47.7805	1.71855	0.859273	+	0.511517i	$0.170916\pi$
$774$	0	0
$775$	−25.9989	−0.933909
$776$	0	0
$777$	−11.1782	−0.401015
$778$	0	0
$779$	−5.62319	−0.201472
$780$	0	0
$781$	89.5904	3.20579
$782$	0	0
$783$	0.177836	0.00635535
$784$	0	0
$785$	1.08708	0.0387995
$786$	0	0
$787$	−51.2106	−1.82546	−0.912730	−	0.408563i	$-0.866030\pi$
$787$	−51.2106	−1.82546	−0.912730	+	0.408563i	$0.866030\pi$
$788$	0	0
$789$	−2.20079	−0.0783502
$790$	0	0
$791$	−9.79648	−0.348323
$792$	0	0
$793$	−21.2997	−0.756374
$794$	0	0
$795$	−0.110241	−0.00390985
$796$	0	0
$797$	−6.78525	−0.240346	−0.120173	−	0.992753i	$-0.538345\pi$
$797$	−6.78525	−0.240346	−0.120173	+	0.992753i	$0.538345\pi$
$798$	0	0
$799$	33.5730	1.18773
$800$	0	0
$801$	−6.19079	−0.218741
$802$	0	0
$803$	−89.3355	−3.15258
$804$	0	0
$805$	2.44255	0.0860884
$806$	0	0
$807$	10.5019	0.369685
$808$	0	0
$809$	−15.6247	−0.549334	−0.274667	−	0.961539i	$-0.588568\pi$
$809$	−15.6247	−0.549334	−0.274667	+	0.961539i	$0.588568\pi$
$810$	0	0
$811$	0.0678913	0.00238398	0.00119199	−	0.999999i	$-0.499621\pi$
$811$	0.0678913	0.00238398	0.00119199	+	0.999999i	$0.499621\pi$
$812$	0	0
$813$	27.7026	0.971573
$814$	0	0
$815$	10.6763	0.373976
$816$	0	0
$817$	12.1066	0.423558
$818$	0	0
$819$	3.84923	0.134503
$820$	0	0
$821$	−19.1748	−0.669204	−0.334602	−	0.942360i	$-0.608602\pi$
$821$	−19.1748	−0.669204	−0.334602	+	0.942360i	$0.608602\pi$
$822$	0	0
$823$	27.3524	0.953446	0.476723	−	0.879054i	$-0.341825\pi$
$823$	27.3524	0.953446	0.476723	+	0.879054i	$0.341825\pi$
$824$	0	0
$825$	26.4458	0.920726
$826$	0	0
$827$	−19.6722	−0.684070	−0.342035	−	0.939687i	$-0.611116\pi$
$827$	−19.6722	−0.684070	−0.342035	+	0.939687i	$0.611116\pi$
$828$	0	0
$829$	38.9618	1.35320	0.676600	−	0.736351i	$-0.263453\pi$
$829$	38.9618	1.35320	0.676600	+	0.736351i	$0.263453\pi$
$830$	0	0
$831$	7.85683	0.272550
$832$	0	0
$833$	12.1114	0.419634
$834$	0	0
$835$	9.89270	0.342351
$836$	0	0
$837$	−5.80184	−0.200541
$838$	0	0
$839$	−9.60815	−0.331710	−0.165855	−	0.986150i	$-0.553038\pi$
$839$	−9.60815	−0.331710	−0.165855	+	0.986150i	$0.553038\pi$
$840$	0	0
$841$	−28.9684	−0.998909
$842$	0	0
$843$	19.0590	0.656425
$844$	0	0
$845$	−5.58172	−0.192017
$846$	0	0
$847$	−40.0269	−1.37534
$848$	0	0
$849$	29.9123	1.02659
$850$	0	0
$851$	13.4333	0.460489
$852$	0	0
$853$	−6.56231	−0.224689	−0.112345	−	0.993669i	$-0.535836\pi$
$853$	−6.56231	−0.224689	−0.112345	+	0.993669i	$0.535836\pi$
$854$	0	0
$855$	1.04053	0.0355852
$856$	0	0
$857$	−39.8795	−1.36226	−0.681128	−	0.732164i	$-0.738510\pi$
$857$	−39.8795	−1.36226	−0.681128	+	0.732164i	$0.738510\pi$
$858$	0	0
$859$	−15.8241	−0.539911	−0.269955	−	0.962873i	$-0.587009\pi$
$859$	−15.8241	−0.539911	−0.269955	+	0.962873i	$0.587009\pi$
$860$	0	0
$861$	−6.53891	−0.222846
$862$	0	0
$863$	−25.5088	−0.868328	−0.434164	−	0.900834i	$-0.642956\pi$
$863$	−25.5088	−0.868328	−0.434164	+	0.900834i	$0.642956\pi$
$864$	0	0
$865$	−6.70576	−0.228003
$866$	0	0
$867$	8.59796	0.292002
$868$	0	0
$869$	34.9193	1.18456
$870$	0	0
$871$	−6.97411	−0.236309
$872$	0	0
$873$	−3.83742	−0.129877
$874$	0	0
$875$	11.4719	0.387822
$876$	0	0
$877$	11.5263	0.389217	0.194609	−	0.980881i	$-0.437656\pi$
$877$	11.5263	0.389217	0.194609	+	0.980881i	$0.437656\pi$
$878$	0	0
$879$	24.8179	0.837086
$880$	0	0
$881$	11.1516	0.375708	0.187854	−	0.982197i	$-0.439847\pi$
$881$	11.1516	0.375708	0.187854	+	0.982197i	$0.439847\pi$
$882$	0	0
$883$	−53.8635	−1.81265	−0.906326	−	0.422579i	$-0.861125\pi$
$883$	−53.8635	−1.81265	−0.906326	+	0.422579i	$0.861125\pi$
$884$	0	0
$885$	5.25784	0.176740
$886$	0	0
$887$	54.8797	1.84268	0.921340	−	0.388758i	$-0.127096\pi$
$887$	54.8797	1.84268	0.921340	+	0.388758i	$0.127096\pi$
$888$	0	0
$889$	15.0797	0.505758
$890$	0	0
$891$	5.90157	0.197710
$892$	0	0
$893$	−16.7313	−0.559892
$894$	0	0
$895$	−2.25159	−0.0752625
$896$	0	0
$897$	−4.62581	−0.154451
$898$	0	0
$899$	−1.03178	−0.0344117
$900$	0	0
$901$	−0.443624	−0.0147793
$902$	0	0
$903$	14.0782	0.468492
$904$	0	0
$905$	−6.25923	−0.208064
$906$	0	0
$907$	−17.8533	−0.592810	−0.296405	−	0.955062i	$-0.595788\pi$
$907$	−17.8533	−0.592810	−0.296405	+	0.955062i	$0.595788\pi$
$908$	0	0
$909$	14.0920	0.467401
$910$	0	0
$911$	−15.0892	−0.499926	−0.249963	−	0.968255i	$-0.580419\pi$
$911$	−15.0892	−0.499926	−0.249963	+	0.968255i	$0.580419\pi$
$912$	0	0
$913$	−51.6164	−1.70825
$914$	0	0
$915$	−6.69536	−0.221342
$916$	0	0
$917$	1.67979	0.0554715
$918$	0	0
$919$	−15.2599	−0.503377	−0.251689	−	0.967808i	$-0.580986\pi$
$919$	−15.2599	−0.503377	−0.251689	+	0.967808i	$0.580986\pi$
$920$	0	0
$921$	−6.44362	−0.212324
$922$	0	0
$923$	−34.7867	−1.14502
$924$	0	0
$925$	29.8199	0.980472
$926$	0	0
$927$	1.13139	0.0371596
$928$	0	0
$929$	29.2077	0.958275	0.479137	−	0.877740i	$-0.340950\pi$
$929$	29.2077	0.958275	0.479137	+	0.877740i	$0.340950\pi$
$930$	0	0
$931$	−6.03577	−0.197814
$932$	0	0
$933$	−0.410769	−0.0134480
$934$	0	0
$935$	−12.3220	−0.402972
$936$	0	0
$937$	−11.5419	−0.377057	−0.188528	−	0.982068i	$-0.560372\pi$
$937$	−11.5419	−0.377057	−0.188528	+	0.982068i	$0.560372\pi$
$938$	0	0
$939$	−6.13337	−0.200155
$940$	0	0
$941$	−22.9752	−0.748969	−0.374485	−	0.927233i	$-0.622180\pi$
$941$	−22.9752	−0.748969	−0.374485	+	0.927233i	$0.622180\pi$
$942$	0	0
$943$	7.85812	0.255896
$944$	0	0
$945$	1.20997	0.0393604
$946$	0	0
$947$	−5.38959	−0.175138	−0.0875690	−	0.996158i	$-0.527910\pi$
$947$	−5.38959	−0.175138	−0.0875690	+	0.996158i	$0.527910\pi$
$948$	0	0
$949$	34.6878	1.12601
$950$	0	0
$951$	2.42092	0.0785037
$952$	0	0
$953$	−46.3396	−1.50109	−0.750543	−	0.660822i	$-0.770208\pi$
$953$	−46.3396	−1.50109	−0.750543	+	0.660822i	$0.770208\pi$
$954$	0	0
$955$	0.208474	0.00674605
$956$	0	0
$957$	1.04951	0.0339259
$958$	0	0
$959$	4.70972	0.152085
$960$	0	0
$961$	2.66137	0.0858505
$962$	0	0
$963$	−17.3758	−0.559926
$964$	0	0
$965$	−17.5567	−0.565170
$966$	0	0
$967$	50.8372	1.63481	0.817407	−	0.576061i	$-0.195411\pi$
$967$	50.8372	1.63481	0.817407	+	0.576061i	$0.195411\pi$
$968$	0	0
$969$	4.18720	0.134512
$970$	0	0
$971$	−13.1145	−0.420864	−0.210432	−	0.977609i	$-0.567487\pi$
$971$	−13.1145	−0.420864	−0.210432	+	0.977609i	$0.567487\pi$
$972$	0	0
$973$	38.9589	1.24896
$974$	0	0
$975$	−10.2686	−0.328857
$976$	0	0
$977$	−1.45948	−0.0466930	−0.0233465	−	0.999727i	$-0.507432\pi$
$977$	−1.45948	−0.0466930	−0.0233465	+	0.999727i	$0.507432\pi$
$978$	0	0
$979$	−36.5354	−1.16768
$980$	0	0
$981$	12.5763	0.401530
$982$	0	0
$983$	50.0175	1.59531	0.797655	−	0.603115i	$-0.206074\pi$
$983$	50.0175	1.59531	0.797655	+	0.603115i	$0.206074\pi$
$984$	0	0
$985$	9.46438	0.301560
$986$	0	0
$987$	−19.4559	−0.619289
$988$	0	0
$989$	−16.9184	−0.537974
$990$	0	0
$991$	3.34044	0.106113	0.0530563	−	0.998592i	$-0.483104\pi$
$991$	3.34044	0.106113	0.0530563	+	0.998592i	$0.483104\pi$
$992$	0	0
$993$	7.54345	0.239384
$994$	0	0
$995$	−14.3701	−0.455562
$996$	0	0
$997$	−8.27590	−0.262100	−0.131050	−	0.991376i	$-0.541835\pi$
$997$	−8.27590	−0.262100	−0.131050	+	0.991376i	$0.541835\pi$
$998$	0	0
$999$	6.65452	0.210540

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6288.2.a.bd.1.5		5
4.3	odd	2		3144.2.a.h.1.5	✓	5
12.11	even	2		9432.2.a.p.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3144.2.a.h.1.5	✓	5	4.3	odd	2
6288.2.a.bd.1.5		5	1.1	even	1	trivial
9432.2.a.p.1.1		5	12.11	even	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:	\( N \)	\(=\)	\( 6288 = 2^{4} \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6288.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.720312 q^{5} -1.67979 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.720312 q^{5} -1.67979 q^{7} +1.00000 q^{9} +5.90157 q^{11} -2.29150 q^{13} -0.720312 q^{15} -2.89863 q^{17} +1.44455 q^{19} +1.67979 q^{21} -2.01868 q^{23} -4.48115 q^{25} -1.00000 q^{27} -0.177836 q^{29} +5.80184 q^{31} -5.90157 q^{33} -1.20997 q^{35} -6.65452 q^{37} +2.29150 q^{39} -3.89270 q^{41} +8.38092 q^{43} +0.720312 q^{45} -11.5824 q^{47} -4.17831 q^{49} +2.89863 q^{51} +0.153046 q^{53} +4.25098 q^{55} -1.44455 q^{57} -7.29939 q^{59} +9.29508 q^{61} -1.67979 q^{63} -1.65060 q^{65} +3.04347 q^{67} +2.01868 q^{69} +15.1808 q^{71} -15.1376 q^{73} +4.48115 q^{75} -9.91339 q^{77} +5.91695 q^{79} +1.00000 q^{81} -8.74622 q^{83} -2.08792 q^{85} +0.177836 q^{87} -6.19079 q^{89} +3.84923 q^{91} -5.80184 q^{93} +1.04053 q^{95} -3.83742 q^{97} +5.90157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 3 q^{11} - 5 q^{13} + 5 q^{15} - q^{17} + q^{19} - 3 q^{21} + 7 q^{23} - 8 q^{25} - 5 q^{27} - 8 q^{29} + 14 q^{31} - 3 q^{33} - 6 q^{35} - 27 q^{37} + 5 q^{39} + 5 q^{41} + 3 q^{43} - 5 q^{45} + 4 q^{47} - 16 q^{49} + q^{51} + q^{53} + 12 q^{55} - q^{57} - 4 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} + 5 q^{67} - 7 q^{69} + 22 q^{71} - 18 q^{73} + 8 q^{75} - 8 q^{77} + 24 q^{79} + 5 q^{81} + 9 q^{83} - 14 q^{85} + 8 q^{87} - 12 q^{89} + 5 q^{91} - 14 q^{93} + 8 q^{95} - 20 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 3 * q^11 - 5 * q^13 + 5 * q^15 - q^17 + q^19 - 3 * q^21 + 7 * q^23 - 8 * q^25 - 5 * q^27 - 8 * q^29 + 14 * q^31 - 3 * q^33 - 6 * q^35 - 27 * q^37 + 5 * q^39 + 5 * q^41 + 3 * q^43 - 5 * q^45 + 4 * q^47 - 16 * q^49 + q^51 + q^53 + 12 * q^55 - q^57 - 4 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 + 5 * q^67 - 7 * q^69 + 22 * q^71 - 18 * q^73 + 8 * q^75 - 8 * q^77 + 24 * q^79 + 5 * q^81 + 9 * q^83 - 14 * q^85 + 8 * q^87 - 12 * q^89 + 5 * q^91 - 14 * q^93 + 8 * q^95 - 20 * q^97 + 3 * q^99

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