LMFDB - Embedded newform 6080.2.a.cc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.552409$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.87834 q^{3} +1.00000 q^{5} +3.10482 q^{7} +5.28487 q^{9} +O(q^{10})$	$q+2.87834 q^{3} +1.00000 q^{5} +3.10482 q^{7} +5.28487 q^{9} +1.10482 q^{11} -1.77353 q^{13} +2.87834 q^{15} +7.75669 q^{17} -1.00000 q^{19} +8.93674 q^{21} -6.65187 q^{23} +1.00000 q^{25} +6.57664 q^{27} -7.75669 q^{29} +6.57664 q^{31} +3.18005 q^{33} +3.10482 q^{35} +1.40652 q^{37} -5.10482 q^{39} +2.81995 q^{41} -3.10482 q^{43} +5.28487 q^{45} +1.46492 q^{47} +2.63990 q^{49} +22.3264 q^{51} +2.59348 q^{53} +1.10482 q^{55} -2.87834 q^{57} +5.38969 q^{59} -4.07523 q^{61} +16.4086 q^{63} -1.77353 q^{65} +15.8151 q^{67} -19.1464 q^{69} +7.14638 q^{71} +0.243310 q^{73} +2.87834 q^{75} +3.43026 q^{77} -9.38969 q^{79} +3.07523 q^{81} -8.86151 q^{83} +7.75669 q^{85} -22.3264 q^{87} +0.813048 q^{89} -5.50648 q^{91} +18.9298 q^{93} -1.00000 q^{95} +3.19689 q^{97} +5.83882 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 8 * q^9	$ 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} - 4 q^{19} + 4 q^{21} - 8 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 8 q^{33} + 4 q^{35} + 6 q^{37} - 12 q^{39} + 16 q^{41} - 4 q^{43} + 8 q^{45} - 12 q^{47} + 20 q^{49} + 36 q^{51} + 10 q^{53} - 4 q^{55} + 2 q^{57} - 20 q^{61} + 20 q^{63} - 2 q^{65} + 18 q^{67} - 28 q^{69} - 20 q^{71} + 28 q^{73} - 2 q^{75} + 40 q^{77} - 16 q^{79} + 16 q^{81} + 4 q^{85} - 36 q^{87} + 4 q^{89} + 36 q^{91} + 40 q^{93} - 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 8 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^17 - 4 * q^19 + 4 * q^21 - 8 * q^23 + 4 * q^25 + 4 * q^27 - 4 * q^29 + 4 * q^31 + 8 * q^33 + 4 * q^35 + 6 * q^37 - 12 * q^39 + 16 * q^41 - 4 * q^43 + 8 * q^45 - 12 * q^47 + 20 * q^49 + 36 * q^51 + 10 * q^53 - 4 * q^55 + 2 * q^57 - 20 * q^61 + 20 * q^63 - 2 * q^65 + 18 * q^67 - 28 * q^69 - 20 * q^71 + 28 * q^73 - 2 * q^75 + 40 * q^77 - 16 * q^79 + 16 * q^81 + 4 * q^85 - 36 * q^87 + 4 * q^89 + 36 * q^91 + 40 * q^93 - 4 * q^95 + 30 * q^97 + 4 * 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
$3$	2.87834	1.66181	0.830907	−	0.556412i	$-0.187822\pi$
$3$	2.87834	1.66181	0.830907	+	0.556412i	$0.187822\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.10482	1.17351	0.586756	−	0.809764i	$-0.300405\pi$
$7$	3.10482	1.17351	0.586756	+	0.809764i	$0.300405\pi$
$8$	0	0
$9$	5.28487	1.76162
$10$	0	0
$11$	1.10482	0.333115	0.166558	−	0.986032i	$-0.446735\pi$
$11$	1.10482	0.333115	0.166558	+	0.986032i	$0.446735\pi$
$12$	0	0
$13$	−1.77353	−0.491888	−0.245944	−	0.969284i	$-0.579098\pi$
$13$	−1.77353	−0.491888	−0.245944	+	0.969284i	$0.579098\pi$
$14$	0	0
$15$	2.87834	0.743185
$16$	0	0
$17$	7.75669	1.88127	0.940637	−	0.339415i	$-0.110229\pi$
$17$	7.75669	1.88127	0.940637	+	0.339415i	$0.110229\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	8.93674	1.95016
$22$	0	0
$23$	−6.65187	−1.38701	−0.693505	−	0.720451i	$-0.743935\pi$
$23$	−6.65187	−1.38701	−0.693505	+	0.720451i	$0.743935\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	6.57664	1.26567
$28$	0	0
$29$	−7.75669	−1.44038	−0.720191	−	0.693776i	$-0.755946\pi$
$29$	−7.75669	−1.44038	−0.720191	+	0.693776i	$0.755946\pi$
$30$	0	0
$31$	6.57664	1.18120	0.590600	−	0.806965i	$-0.298891\pi$
$31$	6.57664	1.18120	0.590600	+	0.806965i	$0.298891\pi$
$32$	0	0
$33$	3.18005	0.553576
$34$	0	0
$35$	3.10482	0.524810
$36$	0	0
$37$	1.40652	0.231231	0.115616	−	0.993294i	$-0.463116\pi$
$37$	1.40652	0.231231	0.115616	+	0.993294i	$0.463116\pi$
$38$	0	0
$39$	−5.10482	−0.817425
$40$	0	0
$41$	2.81995	0.440402	0.220201	−	0.975454i	$-0.429329\pi$
$41$	2.81995	0.440402	0.220201	+	0.975454i	$0.429329\pi$
$42$	0	0
$43$	−3.10482	−0.473480	−0.236740	−	0.971573i	$-0.576079\pi$
$43$	−3.10482	−0.473480	−0.236740	+	0.971573i	$0.576079\pi$
$44$	0	0
$45$	5.28487	0.787822
$46$	0	0
$47$	1.46492	0.213680	0.106840	−	0.994276i	$-0.465927\pi$
$47$	1.46492	0.213680	0.106840	+	0.994276i	$0.465927\pi$
$48$	0	0
$49$	2.63990	0.377129
$50$	0	0
$51$	22.3264	3.12633
$52$	0	0
$53$	2.59348	0.356241	0.178121	−	0.984009i	$-0.442998\pi$
$53$	2.59348	0.356241	0.178121	+	0.984009i	$0.442998\pi$
$54$	0	0
$55$	1.10482	0.148974
$56$	0	0
$57$	−2.87834	−0.381246
$58$	0	0
$59$	5.38969	0.701678	0.350839	−	0.936436i	$-0.385897\pi$
$59$	5.38969	0.701678	0.350839	+	0.936436i	$0.385897\pi$
$60$	0	0
$61$	−4.07523	−0.521780	−0.260890	−	0.965369i	$-0.584016\pi$
$61$	−4.07523	−0.521780	−0.260890	+	0.965369i	$0.584016\pi$
$62$	0	0
$63$	16.4086	2.06728
$64$	0	0
$65$	−1.77353	−0.219979
$66$	0	0
$67$	15.8151	1.93212	0.966060	−	0.258318i	$-0.0831681\pi$
$67$	15.8151	1.93212	0.966060	+	0.258318i	$0.0831681\pi$
$68$	0	0
$69$	−19.1464	−2.30495
$70$	0	0
$71$	7.14638	0.848119	0.424059	−	0.905634i	$-0.360605\pi$
$71$	7.14638	0.848119	0.424059	+	0.905634i	$0.360605\pi$
$72$	0	0
$73$	0.243310	0.0284773	0.0142387	−	0.999899i	$-0.495468\pi$
$73$	0.243310	0.0284773	0.0142387	+	0.999899i	$0.495468\pi$
$74$	0	0
$75$	2.87834	0.332363
$76$	0	0
$77$	3.43026	0.390915
$78$	0	0
$79$	−9.38969	−1.05642	−0.528211	−	0.849113i	$-0.677137\pi$
$79$	−9.38969	−1.05642	−0.528211	+	0.849113i	$0.677137\pi$
$80$	0	0
$81$	3.07523	0.341692
$82$	0	0
$83$	−8.86151	−0.972677	−0.486338	−	0.873771i	$-0.661668\pi$
$83$	−8.86151	−0.972677	−0.486338	+	0.873771i	$0.661668\pi$
$84$	0	0
$85$	7.75669	0.841331
$86$	0	0
$87$	−22.3264	−2.39364
$88$	0	0
$89$	0.813048	0.0861829	0.0430914	−	0.999071i	$-0.486279\pi$
$89$	0.813048	0.0861829	0.0430914	+	0.999071i	$0.486279\pi$
$90$	0	0
$91$	−5.50648	−0.577236
$92$	0	0
$93$	18.9298	1.96293
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	3.19689	0.324595	0.162297	−	0.986742i	$-0.448110\pi$
$97$	3.19689	0.324595	0.162297	+	0.986742i	$0.448110\pi$
$98$	0	0
$99$	5.83882	0.586824
$100$	0	0
$101$	3.01887	0.300389	0.150195	−	0.988656i	$-0.452010\pi$
$101$	3.01887	0.300389	0.150195	+	0.988656i	$0.452010\pi$
$102$	0	0
$103$	6.05839	0.596951	0.298476	−	0.954417i	$-0.403522\pi$
$103$	6.05839	0.596951	0.298476	+	0.954417i	$0.403522\pi$
$104$	0	0
$105$	8.93674	0.872136
$106$	0	0
$107$	−20.3848	−1.97068	−0.985338	−	0.170616i	$-0.945424\pi$
$107$	−20.3848	−1.97068	−0.985338	+	0.170616i	$0.945424\pi$
$108$	0	0
$109$	−4.72710	−0.452774	−0.226387	−	0.974037i	$-0.572691\pi$
$109$	−4.72710	−0.452774	−0.226387	+	0.974037i	$0.572691\pi$
$110$	0	0
$111$	4.04846	0.384263
$112$	0	0
$113$	−7.98316	−0.750993	−0.375496	−	0.926824i	$-0.622528\pi$
$113$	−7.98316	−0.750993	−0.375496	+	0.926824i	$0.622528\pi$
$114$	0	0
$115$	−6.65187	−0.620290
$116$	0	0
$117$	−9.37285	−0.866520
$118$	0	0
$119$	24.0831	2.20770
$120$	0	0
$121$	−9.77938	−0.889034
$122$	0	0
$123$	8.11679	0.731866
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.63503	0.411293	0.205646	−	0.978626i	$-0.434070\pi$
$127$	4.63503	0.411293	0.205646	+	0.978626i	$0.434070\pi$
$128$	0	0
$129$	−8.93674	−0.786836
$130$	0	0
$131$	−15.5134	−1.35541	−0.677705	−	0.735334i	$-0.737025\pi$
$131$	−15.5134	−1.35541	−0.677705	+	0.735334i	$0.737025\pi$
$132$	0	0
$133$	−3.10482	−0.269222
$134$	0	0
$135$	6.57664	0.566027
$136$	0	0
$137$	−0.813048	−0.0694634	−0.0347317	−	0.999397i	$-0.511058\pi$
$137$	−0.813048	−0.0694634	−0.0347317	+	0.999397i	$0.511058\pi$
$138$	0	0
$139$	12.7687	1.08302	0.541512	−	0.840693i	$-0.317852\pi$
$139$	12.7687	1.08302	0.541512	+	0.840693i	$0.317852\pi$
$140$	0	0
$141$	4.21654	0.355097
$142$	0	0
$143$	−1.95942	−0.163855
$144$	0	0
$145$	−7.75669	−0.644158
$146$	0	0
$147$	7.59854	0.626717
$148$	0	0
$149$	13.7982	1.13040	0.565198	−	0.824955i	$-0.308800\pi$
$149$	13.7982	1.13040	0.565198	+	0.824955i	$0.308800\pi$
$150$	0	0
$151$	5.02269	0.408740	0.204370	−	0.978894i	$-0.434485\pi$
$151$	5.02269	0.408740	0.204370	+	0.978894i	$0.434485\pi$
$152$	0	0
$153$	40.9931	3.31409
$154$	0	0
$155$	6.57664	0.528248
$156$	0	0
$157$	−7.18695	−0.573581	−0.286791	−	0.957993i	$-0.592588\pi$
$157$	−7.18695	−0.573581	−0.286791	+	0.957993i	$0.592588\pi$
$158$	0	0
$159$	7.46492	0.592007
$160$	0	0
$161$	−20.6529	−1.62767
$162$	0	0
$163$	−7.25528	−0.568277	−0.284139	−	0.958783i	$-0.591708\pi$
$163$	−7.25528	−0.568277	−0.284139	+	0.958783i	$0.591708\pi$
$164$	0	0
$165$	3.18005	0.247567
$166$	0	0
$167$	−7.33129	−0.567312	−0.283656	−	0.958926i	$-0.591547\pi$
$167$	−7.33129	−0.567312	−0.283656	+	0.958926i	$0.591547\pi$
$168$	0	0
$169$	−9.85461	−0.758047
$170$	0	0
$171$	−5.28487	−0.404144
$172$	0	0
$173$	5.77353	0.438953	0.219477	−	0.975618i	$-0.429565\pi$
$173$	5.77353	0.438953	0.219477	+	0.975618i	$0.429565\pi$
$174$	0	0
$175$	3.10482	0.234702
$176$	0	0
$177$	15.5134	1.16606
$178$	0	0
$179$	4.48379	0.335134	0.167567	−	0.985861i	$-0.446409\pi$
$179$	4.48379	0.335134	0.167567	+	0.985861i	$0.446409\pi$
$180$	0	0
$181$	2.73400	0.203217	0.101608	−	0.994824i	$-0.467601\pi$
$181$	2.73400	0.203217	0.101608	+	0.994824i	$0.467601\pi$
$182$	0	0
$183$	−11.7299	−0.867101
$184$	0	0
$185$	1.40652	0.103410
$186$	0	0
$187$	8.56974	0.626681
$188$	0	0
$189$	20.4193	1.48528
$190$	0	0
$191$	−17.7230	−1.28239	−0.641196	−	0.767377i	$-0.721562\pi$
$191$	−17.7230	−1.28239	−0.641196	+	0.767377i	$0.721562\pi$
$192$	0	0
$193$	−13.5371	−0.974423	−0.487212	−	0.873284i	$-0.661986\pi$
$193$	−13.5371	−0.974423	−0.487212	+	0.873284i	$0.661986\pi$
$194$	0	0
$195$	−5.10482	−0.365564
$196$	0	0
$197$	21.5134	1.53276	0.766382	−	0.642385i	$-0.222055\pi$
$197$	21.5134	1.53276	0.766382	+	0.642385i	$0.222055\pi$
$198$	0	0
$199$	7.09410	0.502888	0.251444	−	0.967872i	$-0.419095\pi$
$199$	7.09410	0.502888	0.251444	+	0.967872i	$0.419095\pi$
$200$	0	0
$201$	45.5213	3.21082
$202$	0	0
$203$	−24.0831	−1.69030
$204$	0	0
$205$	2.81995	0.196954
$206$	0	0
$207$	−35.1543	−2.44339
$208$	0	0
$209$	−1.10482	−0.0764219
$210$	0	0
$211$	−11.0604	−0.761431	−0.380716	−	0.924692i	$-0.624322\pi$
$211$	−11.0604	−0.761431	−0.380716	+	0.924692i	$0.624322\pi$
$212$	0	0
$213$	20.5697	1.40942
$214$	0	0
$215$	−3.10482	−0.211747
$216$	0	0
$217$	20.4193	1.38615
$218$	0	0
$219$	0.700331	0.0473240
$220$	0	0
$221$	−13.7567	−0.925375
$222$	0	0
$223$	−12.6350	−0.846104	−0.423052	−	0.906105i	$-0.639041\pi$
$223$	−12.6350	−0.846104	−0.423052	+	0.906105i	$0.639041\pi$
$224$	0	0
$225$	5.28487	0.352325
$226$	0	0
$227$	6.62813	0.439925	0.219962	−	0.975508i	$-0.429407\pi$
$227$	6.62813	0.439925	0.219962	+	0.975508i	$0.429407\pi$
$228$	0	0
$229$	0.0752308	0.00497139	0.00248570	−	0.999997i	$-0.499209\pi$
$229$	0.0752308	0.00497139	0.00248570	+	0.999997i	$0.499209\pi$
$230$	0	0
$231$	9.87348	0.649627
$232$	0	0
$233$	9.51338	0.623242	0.311621	−	0.950206i	$-0.399128\pi$
$233$	9.51338	0.623242	0.311621	+	0.950206i	$0.399128\pi$
$234$	0	0
$235$	1.46492	0.0955607
$236$	0	0
$237$	−27.0268	−1.75558
$238$	0	0
$239$	−2.92984	−0.189515	−0.0947577	−	0.995500i	$-0.530208\pi$
$239$	−2.92984	−0.189515	−0.0947577	+	0.995500i	$0.530208\pi$
$240$	0	0
$241$	−9.42743	−0.607274	−0.303637	−	0.952788i	$-0.598201\pi$
$241$	−9.42743	−0.607274	−0.303637	+	0.952788i	$0.598201\pi$
$242$	0	0
$243$	−10.8783	−0.697846
$244$	0	0
$245$	2.63990	0.168657
$246$	0	0
$247$	1.77353	0.112847
$248$	0	0
$249$	−25.5065	−1.61641
$250$	0	0
$251$	14.9298	0.942363	0.471181	−	0.882036i	$-0.343828\pi$
$251$	14.9298	0.942363	0.471181	+	0.882036i	$0.343828\pi$
$252$	0	0
$253$	−7.34911	−0.462035
$254$	0	0
$255$	22.3264	1.39814
$256$	0	0
$257$	−15.1295	−0.943755	−0.471877	−	0.881664i	$-0.656424\pi$
$257$	−15.1295	−0.943755	−0.471877	+	0.881664i	$0.656424\pi$
$258$	0	0
$259$	4.36700	0.271352
$260$	0	0
$261$	−40.9931	−2.53741
$262$	0	0
$263$	−15.2216	−0.938605	−0.469302	−	0.883038i	$-0.655495\pi$
$263$	−15.2216	−0.938605	−0.469302	+	0.883038i	$0.655495\pi$
$264$	0	0
$265$	2.59348	0.159316
$266$	0	0
$267$	2.34023	0.143220
$268$	0	0
$269$	−11.5203	−0.702404	−0.351202	−	0.936300i	$-0.614227\pi$
$269$	−11.5203	−0.702404	−0.351202	+	0.936300i	$0.614227\pi$
$270$	0	0
$271$	2.16118	0.131282	0.0656411	−	0.997843i	$-0.479091\pi$
$271$	2.16118	0.131282	0.0656411	+	0.997843i	$0.479091\pi$
$272$	0	0
$273$	−15.8495	−0.959258
$274$	0	0
$275$	1.10482	0.0666231
$276$	0	0
$277$	−23.4205	−1.40720	−0.703602	−	0.710595i	$-0.748426\pi$
$277$	−23.4205	−1.40720	−0.703602	+	0.710595i	$0.748426\pi$
$278$	0	0
$279$	34.7567	2.08083
$280$	0	0
$281$	25.6824	1.53209	0.766043	−	0.642789i	$-0.222223\pi$
$281$	25.6824	1.53209	0.766043	+	0.642789i	$0.222223\pi$
$282$	0	0
$283$	−7.38587	−0.439045	−0.219522	−	0.975607i	$-0.570450\pi$
$283$	−7.38587	−0.439045	−0.219522	+	0.975607i	$0.570450\pi$
$284$	0	0
$285$	−2.87834	−0.170498
$286$	0	0
$287$	8.75543	0.516817
$288$	0	0
$289$	43.1662	2.53919
$290$	0	0
$291$	9.20174	0.539416
$292$	0	0
$293$	24.1522	1.41099	0.705494	−	0.708716i	$-0.250725\pi$
$293$	24.1522	1.41099	0.705494	+	0.708716i	$0.250725\pi$
$294$	0	0
$295$	5.38969	0.313800
$296$	0	0
$297$	7.26600	0.421616
$298$	0	0
$299$	11.7973	0.682253
$300$	0	0
$301$	−9.63990	−0.555635
$302$	0	0
$303$	8.68936	0.499190
$304$	0	0
$305$	−4.07523	−0.233347
$306$	0	0
$307$	−4.38482	−0.250255	−0.125127	−	0.992141i	$-0.539934\pi$
$307$	−4.38482	−0.250255	−0.125127	+	0.992141i	$0.539934\pi$
$308$	0	0
$309$	17.4381	0.992022
$310$	0	0
$311$	−15.7123	−0.890963	−0.445481	−	0.895291i	$-0.646967\pi$
$311$	−15.7123	−0.890963	−0.445481	+	0.895291i	$0.646967\pi$
$312$	0	0
$313$	24.7794	1.40061	0.700307	−	0.713842i	$-0.253047\pi$
$313$	24.7794	1.40061	0.700307	+	0.713842i	$0.253047\pi$
$314$	0	0
$315$	16.4086	0.924518
$316$	0	0
$317$	27.3798	1.53780	0.768900	−	0.639369i	$-0.220804\pi$
$317$	27.3798	1.53780	0.768900	+	0.639369i	$0.220804\pi$
$318$	0	0
$319$	−8.56974	−0.479813
$320$	0	0
$321$	−58.6745	−3.27489
$322$	0	0
$323$	−7.75669	−0.431594
$324$	0	0
$325$	−1.77353	−0.0983775
$326$	0	0
$327$	−13.6062	−0.752426
$328$	0	0
$329$	4.54831	0.250756
$330$	0	0
$331$	−4.70033	−0.258354	−0.129177	−	0.991622i	$-0.541233\pi$
$331$	−4.70033	−0.258354	−0.129177	+	0.991622i	$0.541233\pi$
$332$	0	0
$333$	7.43329	0.407342
$334$	0	0
$335$	15.8151	0.864070
$336$	0	0
$337$	−20.6360	−1.12412	−0.562058	−	0.827098i	$-0.689990\pi$
$337$	−20.6360	−1.12412	−0.562058	+	0.827098i	$0.689990\pi$
$338$	0	0
$339$	−22.9783	−1.24801
$340$	0	0
$341$	7.26600	0.393476
$342$	0	0
$343$	−13.5373	−0.730947
$344$	0	0
$345$	−19.1464	−1.03081
$346$	0	0
$347$	−30.0148	−1.61128	−0.805639	−	0.592407i	$-0.798178\pi$
$347$	−30.0148	−1.61128	−0.805639	+	0.592407i	$0.798178\pi$
$348$	0	0
$349$	14.7340	0.788693	0.394347	−	0.918962i	$-0.370971\pi$
$349$	14.7340	0.788693	0.394347	+	0.918962i	$0.370971\pi$
$350$	0	0
$351$	−11.6638	−0.622570
$352$	0	0
$353$	−29.6302	−1.57705	−0.788527	−	0.615000i	$-0.789156\pi$
$353$	−29.6302	−1.57705	−0.788527	+	0.615000i	$0.789156\pi$
$354$	0	0
$355$	7.14638	0.379290
$356$	0	0
$357$	69.3195	3.66878
$358$	0	0
$359$	21.7577	1.14833	0.574163	−	0.818741i	$-0.305328\pi$
$359$	21.7577	1.14833	0.574163	+	0.818741i	$0.305328\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−28.1484	−1.47741
$364$	0	0
$365$	0.243310	0.0127355
$366$	0	0
$367$	30.0347	1.56780	0.783898	−	0.620889i	$-0.213228\pi$
$367$	30.0347	1.56780	0.783898	+	0.620889i	$0.213228\pi$
$368$	0	0
$369$	14.9031	0.775823
$370$	0	0
$371$	8.05227	0.418053
$372$	0	0
$373$	37.3055	1.93161	0.965803	−	0.259277i	$-0.0834843\pi$
$373$	37.3055	1.93161	0.965803	+	0.259277i	$0.0834843\pi$
$374$	0	0
$375$	2.87834	0.148637
$376$	0	0
$377$	13.7567	0.708506
$378$	0	0
$379$	14.7794	0.759166	0.379583	−	0.925158i	$-0.376068\pi$
$379$	14.7794	0.759166	0.379583	+	0.925158i	$0.376068\pi$
$380$	0	0
$381$	13.3412	0.683492
$382$	0	0
$383$	−29.0020	−1.48193	−0.740967	−	0.671541i	$-0.765633\pi$
$383$	−29.0020	−1.48193	−0.740967	+	0.671541i	$0.765633\pi$
$384$	0	0
$385$	3.43026	0.174822
$386$	0	0
$387$	−16.4086	−0.834094
$388$	0	0
$389$	−21.1987	−1.07481	−0.537407	−	0.843323i	$-0.680596\pi$
$389$	−21.1987	−1.07481	−0.537407	+	0.843323i	$0.680596\pi$
$390$	0	0
$391$	−51.5965	−2.60935
$392$	0	0
$393$	−44.6529	−2.25244
$394$	0	0
$395$	−9.38969	−0.472446
$396$	0	0
$397$	−10.3856	−0.521238	−0.260619	−	0.965442i	$-0.583927\pi$
$397$	−10.3856	−0.521238	−0.260619	+	0.965442i	$0.583927\pi$
$398$	0	0
$399$	−8.93674	−0.447397
$400$	0	0
$401$	25.6638	1.28159	0.640796	−	0.767712i	$-0.278605\pi$
$401$	25.6638	1.28159	0.640796	+	0.767712i	$0.278605\pi$
$402$	0	0
$403$	−11.6638	−0.581017
$404$	0	0
$405$	3.07523	0.152809
$406$	0	0
$407$	1.55395	0.0770266
$408$	0	0
$409$	5.71611	0.282644	0.141322	−	0.989964i	$-0.454865\pi$
$409$	5.71611	0.282644	0.141322	+	0.989964i	$0.454865\pi$
$410$	0	0
$411$	−2.34023	−0.115435
$412$	0	0
$413$	16.7340	0.823427
$414$	0	0
$415$	−8.86151	−0.434994
$416$	0	0
$417$	36.7526	1.79978
$418$	0	0
$419$	−15.4303	−0.753818	−0.376909	−	0.926250i	$-0.623013\pi$
$419$	−15.4303	−0.753818	−0.376909	+	0.926250i	$0.623013\pi$
$420$	0	0
$421$	1.18005	0.0575121	0.0287561	−	0.999586i	$-0.490845\pi$
$421$	1.18005	0.0575121	0.0287561	+	0.999586i	$0.490845\pi$
$422$	0	0
$423$	7.74190	0.376424
$424$	0	0
$425$	7.75669	0.376255
$426$	0	0
$427$	−12.6529	−0.612315
$428$	0	0
$429$	−5.63990	−0.272297
$430$	0	0
$431$	−14.0255	−0.675585	−0.337792	−	0.941221i	$-0.609680\pi$
$431$	−14.0255	−0.675585	−0.337792	+	0.941221i	$0.609680\pi$
$432$	0	0
$433$	25.5894	1.22975	0.614874	−	0.788625i	$-0.289207\pi$
$433$	25.5894	1.22975	0.614874	+	0.788625i	$0.289207\pi$
$434$	0	0
$435$	−22.3264	−1.07047
$436$	0	0
$437$	6.65187	0.318202
$438$	0	0
$439$	−21.9732	−1.04873	−0.524363	−	0.851495i	$-0.675696\pi$
$439$	−21.9732	−1.04873	−0.524363	+	0.851495i	$0.675696\pi$
$440$	0	0
$441$	13.9515	0.664358
$442$	0	0
$443$	19.3721	0.920395	0.460197	−	0.887817i	$-0.347779\pi$
$443$	19.3721	0.920395	0.460197	+	0.887817i	$0.347779\pi$
$444$	0	0
$445$	0.813048	0.0385422
$446$	0	0
$447$	39.7161	1.87851
$448$	0	0
$449$	−19.1841	−0.905355	−0.452677	−	0.891674i	$-0.649531\pi$
$449$	−19.1841	−0.905355	−0.452677	+	0.891674i	$0.649531\pi$
$450$	0	0
$451$	3.11553	0.146705
$452$	0	0
$453$	14.4570	0.679250
$454$	0	0
$455$	−5.50648	−0.258148
$456$	0	0
$457$	8.36010	0.391069	0.195534	−	0.980697i	$-0.437356\pi$
$457$	8.36010	0.391069	0.195534	+	0.980697i	$0.437356\pi$
$458$	0	0
$459$	51.0130	2.38108
$460$	0	0
$461$	25.0268	1.16561	0.582806	−	0.812611i	$-0.301955\pi$
$461$	25.0268	1.16561	0.582806	+	0.812611i	$0.301955\pi$
$462$	0	0
$463$	32.3749	1.50459	0.752294	−	0.658827i	$-0.228947\pi$
$463$	32.3749	1.50459	0.752294	+	0.658827i	$0.228947\pi$
$464$	0	0
$465$	18.9298	0.877850
$466$	0	0
$467$	−15.2216	−0.704372	−0.352186	−	0.935930i	$-0.614562\pi$
$467$	−15.2216	−0.704372	−0.352186	+	0.935930i	$0.614562\pi$
$468$	0	0
$469$	49.1030	2.26736
$470$	0	0
$471$	−20.6865	−0.953185
$472$	0	0
$473$	−3.43026	−0.157724
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	13.7062	0.627563
$478$	0	0
$479$	−20.0485	−0.916038	−0.458019	−	0.888943i	$-0.651441\pi$
$479$	−20.0485	−0.916038	−0.458019	+	0.888943i	$0.651441\pi$
$480$	0	0
$481$	−2.49451	−0.113740
$482$	0	0
$483$	−59.4460	−2.70489
$484$	0	0
$485$	3.19689	0.145163
$486$	0	0
$487$	−31.6508	−1.43424	−0.717118	−	0.696952i	$-0.754539\pi$
$487$	−31.6508	−1.43424	−0.717118	+	0.696952i	$0.754539\pi$
$488$	0	0
$489$	−20.8832	−0.944371
$490$	0	0
$491$	−24.8033	−1.11936	−0.559679	−	0.828710i	$-0.689076\pi$
$491$	−24.8033	−1.11936	−0.559679	+	0.828710i	$0.689076\pi$
$492$	0	0
$493$	−60.1662	−2.70975
$494$	0	0
$495$	5.83882	0.262436
$496$	0	0
$497$	22.1882	0.995277
$498$	0	0
$499$	33.0237	1.47834	0.739171	−	0.673518i	$-0.235217\pi$
$499$	33.0237	1.47834	0.739171	+	0.673518i	$0.235217\pi$
$500$	0	0
$501$	−21.1020	−0.942767
$502$	0	0
$503$	−3.54396	−0.158017	−0.0790087	−	0.996874i	$-0.525175\pi$
$503$	−3.54396	−0.158017	−0.0790087	+	0.996874i	$0.525175\pi$
$504$	0	0
$505$	3.01887	0.134338
$506$	0	0
$507$	−28.3650	−1.25973
$508$	0	0
$509$	2.11679	0.0938250	0.0469125	−	0.998899i	$-0.485062\pi$
$509$	2.11679	0.0938250	0.0469125	+	0.998899i	$0.485062\pi$
$510$	0	0
$511$	0.755435	0.0334185
$512$	0	0
$513$	−6.57664	−0.290366
$514$	0	0
$515$	6.05839	0.266965
$516$	0	0
$517$	1.61847	0.0711802
$518$	0	0
$519$	16.6182	0.729458
$520$	0	0
$521$	−8.60748	−0.377101	−0.188550	−	0.982064i	$-0.560379\pi$
$521$	−8.60748	−0.377101	−0.188550	+	0.982064i	$0.560379\pi$
$522$	0	0
$523$	−36.8349	−1.61068	−0.805340	−	0.592814i	$-0.798017\pi$
$523$	−36.8349	−1.61068	−0.805340	+	0.592814i	$0.798017\pi$
$524$	0	0
$525$	8.93674	0.390031
$526$	0	0
$527$	51.0130	2.22216
$528$	0	0
$529$	21.2474	0.923799
$530$	0	0
$531$	28.4838	1.23609
$532$	0	0
$533$	−5.00125	−0.216628
$534$	0	0
$535$	−20.3848	−0.881313
$536$	0	0
$537$	12.9059	0.556931
$538$	0	0
$539$	2.91661	0.125627
$540$	0	0
$541$	32.0079	1.37613	0.688063	−	0.725651i	$-0.258461\pi$
$541$	32.0079	1.37613	0.688063	+	0.725651i	$0.258461\pi$
$542$	0	0
$543$	7.86941	0.337709
$544$	0	0
$545$	−4.72710	−0.202487
$546$	0	0
$547$	−17.6240	−0.753550	−0.376775	−	0.926305i	$-0.622967\pi$
$547$	−17.6240	−0.753550	−0.376775	+	0.926305i	$0.622967\pi$
$548$	0	0
$549$	−21.5371	−0.919179
$550$	0	0
$551$	7.75669	0.330446
$552$	0	0
$553$	−29.1533	−1.23972
$554$	0	0
$555$	4.04846	0.171848
$556$	0	0
$557$	34.6865	1.46972	0.734858	−	0.678221i	$-0.237249\pi$
$557$	34.6865	1.46972	0.734858	+	0.678221i	$0.237249\pi$
$558$	0	0
$559$	5.50648	0.232899
$560$	0	0
$561$	24.6667	1.04143
$562$	0	0
$563$	−13.5073	−0.569263	−0.284632	−	0.958637i	$-0.591871\pi$
$563$	−13.5073	−0.569263	−0.284632	+	0.958637i	$0.591871\pi$
$564$	0	0
$565$	−7.98316	−0.335854
$566$	0	0
$567$	9.54803	0.400980
$568$	0	0
$569$	32.3588	1.35655	0.678276	−	0.734807i	$-0.262727\pi$
$569$	32.3588	1.35655	0.678276	+	0.734807i	$0.262727\pi$
$570$	0	0
$571$	8.93293	0.373831	0.186916	−	0.982376i	$-0.440151\pi$
$571$	8.93293	0.373831	0.186916	+	0.982376i	$0.440151\pi$
$572$	0	0
$573$	−51.0130	−2.13110
$574$	0	0
$575$	−6.65187	−0.277402
$576$	0	0
$577$	14.9059	0.620541	0.310270	−	0.950648i	$-0.399580\pi$
$577$	14.9059	0.620541	0.310270	+	0.950648i	$0.399580\pi$
$578$	0	0
$579$	−38.9645	−1.61931
$580$	0	0
$581$	−27.5134	−1.14145
$582$	0	0
$583$	2.86532	0.118669
$584$	0	0
$585$	−9.37285	−0.387520
$586$	0	0
$587$	−7.37207	−0.304278	−0.152139	−	0.988359i	$-0.548616\pi$
$587$	−7.37207	−0.304278	−0.152139	+	0.988359i	$0.548616\pi$
$588$	0	0
$589$	−6.57664	−0.270986
$590$	0	0
$591$	61.9229	2.54717
$592$	0	0
$593$	−17.6853	−0.726247	−0.363124	−	0.931741i	$-0.618290\pi$
$593$	−17.6853	−0.726247	−0.363124	+	0.931741i	$0.618290\pi$
$594$	0	0
$595$	24.0831	0.987312
$596$	0	0
$597$	20.4193	0.835705
$598$	0	0
$599$	5.30657	0.216821	0.108410	−	0.994106i	$-0.465424\pi$
$599$	5.30657	0.216821	0.108410	+	0.994106i	$0.465424\pi$
$600$	0	0
$601$	43.7875	1.78613	0.893065	−	0.449927i	$-0.148550\pi$
$601$	43.7875	1.78613	0.893065	+	0.449927i	$0.148550\pi$
$602$	0	0
$603$	83.5806	3.40367
$604$	0	0
$605$	−9.77938	−0.397588
$606$	0	0
$607$	−7.24535	−0.294080	−0.147040	−	0.989131i	$-0.546975\pi$
$607$	−7.24535	−0.294080	−0.147040	+	0.989131i	$0.546975\pi$
$608$	0	0
$609$	−69.3195	−2.80897
$610$	0	0
$611$	−2.59807	−0.105107
$612$	0	0
$613$	−20.8962	−0.843988	−0.421994	−	0.906599i	$-0.638670\pi$
$613$	−20.8962	−0.843988	−0.421994	+	0.906599i	$0.638670\pi$
$614$	0	0
$615$	8.11679	0.327301
$616$	0	0
$617$	−22.0494	−0.887677	−0.443839	−	0.896107i	$-0.646384\pi$
$617$	−22.0494	−0.887677	−0.443839	+	0.896107i	$0.646384\pi$
$618$	0	0
$619$	0.0484607	0.00194780	0.000973900	−	1.00000i	$-0.499690\pi$
$619$	0.0484607	0.00194780	0.000973900		1.00000i	$0.499690\pi$
$620$	0	0
$621$	−43.7470	−1.75550
$622$	0	0
$623$	2.52437	0.101137
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.18005	−0.126999
$628$	0	0
$629$	10.9100	0.435009
$630$	0	0
$631$	−32.6182	−1.29851	−0.649255	−	0.760571i	$-0.724919\pi$
$631$	−32.6182	−1.29851	−0.649255	+	0.760571i	$0.724919\pi$
$632$	0	0
$633$	−31.8357	−1.26536
$634$	0	0
$635$	4.63503	0.183936
$636$	0	0
$637$	−4.68193	−0.185505
$638$	0	0
$639$	37.7677	1.49407
$640$	0	0
$641$	−13.4136	−0.529806	−0.264903	−	0.964275i	$-0.585340\pi$
$641$	−13.4136	−0.529806	−0.264903	+	0.964275i	$0.585340\pi$
$642$	0	0
$643$	−23.8052	−0.938783	−0.469392	−	0.882990i	$-0.655527\pi$
$643$	−23.8052	−0.938783	−0.469392	+	0.882990i	$0.655527\pi$
$644$	0	0
$645$	−8.93674	−0.351884
$646$	0	0
$647$	−48.4580	−1.90508	−0.952540	−	0.304412i	$-0.901540\pi$
$647$	−48.4580	−1.90508	−0.952540	+	0.304412i	$0.901540\pi$
$648$	0	0
$649$	5.95463	0.233740
$650$	0	0
$651$	58.7737	2.30352
$652$	0	0
$653$	−22.2351	−0.870128	−0.435064	−	0.900399i	$-0.643274\pi$
$653$	−22.2351	−0.870128	−0.435064	+	0.900399i	$0.643274\pi$
$654$	0	0
$655$	−15.5134	−0.606158
$656$	0	0
$657$	1.28586	0.0501663
$658$	0	0
$659$	10.7072	0.417095	0.208547	−	0.978012i	$-0.433126\pi$
$659$	10.7072	0.417095	0.208547	+	0.978012i	$0.433126\pi$
$660$	0	0
$661$	44.7980	1.74244	0.871220	−	0.490893i	$-0.163330\pi$
$661$	44.7980	1.74244	0.871220	+	0.490893i	$0.163330\pi$
$662$	0	0
$663$	−39.5965	−1.53780
$664$	0	0
$665$	−3.10482	−0.120400
$666$	0	0
$667$	51.5965	1.99782
$668$	0	0
$669$	−36.3680	−1.40607
$670$	0	0
$671$	−4.50239	−0.173813
$672$	0	0
$673$	−43.2772	−1.66821	−0.834106	−	0.551604i	$-0.814016\pi$
$673$	−43.2772	−1.66821	−0.834106	+	0.551604i	$0.814016\pi$
$674$	0	0
$675$	6.57664	0.253135
$676$	0	0
$677$	−11.5330	−0.443251	−0.221625	−	0.975132i	$-0.571136\pi$
$677$	−11.5330	−0.443251	−0.221625	+	0.975132i	$0.571136\pi$
$678$	0	0
$679$	9.92575	0.380915
$680$	0	0
$681$	19.0780	0.731072
$682$	0	0
$683$	−0.916090	−0.0350532	−0.0175266	−	0.999846i	$-0.505579\pi$
$683$	−0.916090	−0.0350532	−0.0175266	+	0.999846i	$0.505579\pi$
$684$	0	0
$685$	−0.813048	−0.0310650
$686$	0	0
$687$	0.216540	0.00826153
$688$	0	0
$689$	−4.59960	−0.175231
$690$	0	0
$691$	−14.8278	−0.564077	−0.282039	−	0.959403i	$-0.591011\pi$
$691$	−14.8278	−0.564077	−0.282039	+	0.959403i	$0.591011\pi$
$692$	0	0
$693$	18.1285	0.688644
$694$	0	0
$695$	12.7687	0.484343
$696$	0	0
$697$	21.8735	0.828517
$698$	0	0
$699$	27.3828	1.03571
$700$	0	0
$701$	−36.1722	−1.36620	−0.683102	−	0.730323i	$-0.739369\pi$
$701$	−36.1722	−1.36620	−0.683102	+	0.730323i	$0.739369\pi$
$702$	0	0
$703$	−1.40652	−0.0530481
$704$	0	0
$705$	4.21654	0.158804
$706$	0	0
$707$	9.37305	0.352510
$708$	0	0
$709$	−10.4331	−0.391823	−0.195911	−	0.980622i	$-0.562766\pi$
$709$	−10.4331	−0.391823	−0.195911	+	0.980622i	$0.562766\pi$
$710$	0	0
$711$	−49.6233	−1.86102
$712$	0	0
$713$	−43.7470	−1.63834
$714$	0	0
$715$	−1.95942	−0.0732783
$716$	0	0
$717$	−8.43308	−0.314939
$718$	0	0
$719$	7.73373	0.288420	0.144210	−	0.989547i	$-0.453936\pi$
$719$	7.73373	0.288420	0.144210	+	0.989547i	$0.453936\pi$
$720$	0	0
$721$	18.8102	0.700529
$722$	0	0
$723$	−27.1354	−1.00918
$724$	0	0
$725$	−7.75669	−0.288076
$726$	0	0
$727$	−39.5241	−1.46587	−0.732934	−	0.680300i	$-0.761849\pi$
$727$	−39.5241	−1.46587	−0.732934	+	0.680300i	$0.761849\pi$
$728$	0	0
$729$	−40.5373	−1.50138
$730$	0	0
$731$	−24.0831	−0.890746
$732$	0	0
$733$	−50.7498	−1.87449	−0.937243	−	0.348677i	$-0.886631\pi$
$733$	−50.7498	−1.87449	−0.937243	+	0.348677i	$0.886631\pi$
$734$	0	0
$735$	7.59854	0.280277
$736$	0	0
$737$	17.4728	0.643619
$738$	0	0
$739$	−0.419276	−0.0154233	−0.00771165	−	0.999970i	$-0.502455\pi$
$739$	−0.419276	−0.0154233	−0.00771165	+	0.999970i	$0.502455\pi$
$740$	0	0
$741$	5.10482	0.187530
$742$	0	0
$743$	19.5511	0.717259	0.358630	−	0.933480i	$-0.383244\pi$
$743$	19.5511	0.717259	0.358630	+	0.933480i	$0.383244\pi$
$744$	0	0
$745$	13.7982	0.505529
$746$	0	0
$747$	−46.8319	−1.71349
$748$	0	0
$749$	−63.2912	−2.31261
$750$	0	0
$751$	8.76768	0.319937	0.159969	−	0.987122i	$-0.448861\pi$
$751$	8.76768	0.319937	0.159969	+	0.987122i	$0.448861\pi$
$752$	0	0
$753$	42.9732	1.56603
$754$	0	0
$755$	5.02269	0.182794
$756$	0	0
$757$	48.0535	1.74653	0.873267	−	0.487241i	$-0.161997\pi$
$757$	48.0535	1.74653	0.873267	+	0.487241i	$0.161997\pi$
$758$	0	0
$759$	−21.1533	−0.767815
$760$	0	0
$761$	29.3947	1.06556	0.532779	−	0.846254i	$-0.321148\pi$
$761$	29.3947	1.06556	0.532779	+	0.846254i	$0.321148\pi$
$762$	0	0
$763$	−14.6768	−0.531336
$764$	0	0
$765$	40.9931	1.48211
$766$	0	0
$767$	−9.55875	−0.345146
$768$	0	0
$769$	−13.3790	−0.482458	−0.241229	−	0.970468i	$-0.577551\pi$
$769$	−13.3790	−0.482458	−0.241229	+	0.970468i	$0.577551\pi$
$770$	0	0
$771$	−43.5480	−1.56834
$772$	0	0
$773$	−0.133625	−0.00480617	−0.00240309	−	0.999997i	$-0.500765\pi$
$773$	−0.133625	−0.00480617	−0.00240309	+	0.999997i	$0.500765\pi$
$774$	0	0
$775$	6.57664	0.236240
$776$	0	0
$777$	12.5697	0.450937
$778$	0	0
$779$	−2.81995	−0.101035
$780$	0	0
$781$	7.89545	0.282522
$782$	0	0
$783$	−51.0130	−1.82305
$784$	0	0
$785$	−7.18695	−0.256513
$786$	0	0
$787$	4.84060	0.172549	0.0862744	−	0.996271i	$-0.472504\pi$
$787$	4.84060	0.172549	0.0862744	+	0.996271i	$0.472504\pi$
$788$	0	0
$789$	−43.8130	−1.55979
$790$	0	0
$791$	−24.7863	−0.881299
$792$	0	0
$793$	7.22753	0.256657
$794$	0	0
$795$	7.46492	0.264753
$796$	0	0
$797$	−11.6993	−0.414410	−0.207205	−	0.978298i	$-0.566437\pi$
$797$	−11.6993	−0.414410	−0.207205	+	0.978298i	$0.566437\pi$
$798$	0	0
$799$	11.3629	0.401991
$800$	0	0
$801$	4.29685	0.151822
$802$	0	0
$803$	0.268814	0.00948624
$804$	0	0
$805$	−20.6529	−0.727917
$806$	0	0
$807$	−33.1593	−1.16726
$808$	0	0
$809$	−33.1987	−1.16720	−0.583601	−	0.812040i	$-0.698357\pi$
$809$	−33.1987	−1.16720	−0.583601	+	0.812040i	$0.698357\pi$
$810$	0	0
$811$	26.5766	0.933232	0.466616	−	0.884460i	$-0.345473\pi$
$811$	26.5766	0.933232	0.466616	+	0.884460i	$0.345473\pi$
$812$	0	0
$813$	6.22061	0.218166
$814$	0	0
$815$	−7.25528	−0.254141
$816$	0	0
$817$	3.10482	0.108624
$818$	0	0
$819$	−29.1010	−1.01687
$820$	0	0
$821$	−25.5807	−0.892773	−0.446387	−	0.894840i	$-0.647289\pi$
$821$	−25.5807	−0.892773	−0.446387	+	0.894840i	$0.647289\pi$
$822$	0	0
$823$	12.9783	0.452395	0.226198	−	0.974081i	$-0.427371\pi$
$823$	12.9783	0.452395	0.226198	+	0.974081i	$0.427371\pi$
$824$	0	0
$825$	3.18005	0.110715
$826$	0	0
$827$	12.5167	0.435248	0.217624	−	0.976033i	$-0.430169\pi$
$827$	12.5167	0.435248	0.217624	+	0.976033i	$0.430169\pi$
$828$	0	0
$829$	19.4391	0.675149	0.337574	−	0.941299i	$-0.390394\pi$
$829$	19.4391	0.675149	0.337574	+	0.941299i	$0.390394\pi$
$830$	0	0
$831$	−67.4124	−2.33851
$832$	0	0
$833$	20.4769	0.709482
$834$	0	0
$835$	−7.33129	−0.253710
$836$	0	0
$837$	43.2522	1.49501
$838$	0	0
$839$	−43.9972	−1.51895	−0.759475	−	0.650536i	$-0.774544\pi$
$839$	−43.9972	−1.51895	−0.759475	+	0.650536i	$0.774544\pi$
$840$	0	0
$841$	31.1662	1.07470
$842$	0	0
$843$	73.9229	2.54604
$844$	0	0
$845$	−9.85461	−0.339009
$846$	0	0
$847$	−30.3632	−1.04329
$848$	0	0
$849$	−21.2591	−0.729610
$850$	0	0
$851$	−9.35601	−0.320720
$852$	0	0
$853$	−3.30374	−0.113118	−0.0565590	−	0.998399i	$-0.518013\pi$
$853$	−3.30374	−0.113118	−0.0565590	+	0.998399i	$0.518013\pi$
$854$	0	0
$855$	−5.28487	−0.180739
$856$	0	0
$857$	6.02374	0.205767	0.102883	−	0.994693i	$-0.467193\pi$
$857$	6.02374	0.205767	0.102883	+	0.994693i	$0.467193\pi$
$858$	0	0
$859$	36.0158	1.22884	0.614421	−	0.788978i	$-0.289390\pi$
$859$	36.0158	1.22884	0.614421	+	0.788978i	$0.289390\pi$
$860$	0	0
$861$	25.2012	0.858853
$862$	0	0
$863$	29.9250	1.01866	0.509329	−	0.860572i	$-0.329894\pi$
$863$	29.9250	1.01866	0.509329	+	0.860572i	$0.329894\pi$
$864$	0	0
$865$	5.77353	0.196306
$866$	0	0
$867$	124.247	4.21966
$868$	0	0
$869$	−10.3739	−0.351911
$870$	0	0
$871$	−28.0485	−0.950386
$872$	0	0
$873$	16.8951	0.571813
$874$	0	0
$875$	3.10482	0.104962
$876$	0	0
$877$	−0.618980	−0.0209015	−0.0104507	−	0.999945i	$-0.503327\pi$
$877$	−0.618980	−0.0209015	−0.0104507	+	0.999945i	$0.503327\pi$
$878$	0	0
$879$	69.5184	2.34480
$880$	0	0
$881$	−21.2423	−0.715672	−0.357836	−	0.933784i	$-0.616485\pi$
$881$	−21.2423	−0.715672	−0.357836	+	0.933784i	$0.616485\pi$
$882$	0	0
$883$	48.8971	1.64552	0.822760	−	0.568389i	$-0.192433\pi$
$883$	48.8971	1.64552	0.822760	+	0.568389i	$0.192433\pi$
$884$	0	0
$885$	15.5134	0.521477
$886$	0	0
$887$	58.2797	1.95684	0.978421	−	0.206622i	$-0.0662469\pi$
$887$	58.2797	1.95684	0.978421	+	0.206622i	$0.0662469\pi$
$888$	0	0
$889$	14.3909	0.482657
$890$	0	0
$891$	3.39757	0.113823
$892$	0	0
$893$	−1.46492	−0.0490216
$894$	0	0
$895$	4.48379	0.149877
$896$	0	0
$897$	33.9566	1.13378
$898$	0	0
$899$	−51.0130	−1.70138
$900$	0	0
$901$	20.1168	0.670187
$902$	0	0
$903$	−27.7470	−0.923361
$904$	0	0
$905$	2.73400	0.0908814
$906$	0	0
$907$	36.5154	1.21247	0.606237	−	0.795284i	$-0.292678\pi$
$907$	36.5154	1.21247	0.606237	+	0.795284i	$0.292678\pi$
$908$	0	0
$909$	15.9543	0.529172
$910$	0	0
$911$	6.62735	0.219574	0.109787	−	0.993955i	$-0.464983\pi$
$911$	6.62735	0.219574	0.109787	+	0.993955i	$0.464983\pi$
$912$	0	0
$913$	−9.79036	−0.324014
$914$	0	0
$915$	−11.7299	−0.387779
$916$	0	0
$917$	−48.1662	−1.59059
$918$	0	0
$919$	7.91688	0.261154	0.130577	−	0.991438i	$-0.458317\pi$
$919$	7.91688	0.261154	0.130577	+	0.991438i	$0.458317\pi$
$920$	0	0
$921$	−12.6210	−0.415877
$922$	0	0
$923$	−12.6743	−0.417179
$924$	0	0
$925$	1.40652	0.0462462
$926$	0	0
$927$	32.0178	1.05160
$928$	0	0
$929$	35.8597	1.17652	0.588259	−	0.808673i	$-0.299814\pi$
$929$	35.8597	1.17652	0.588259	+	0.808673i	$0.299814\pi$
$930$	0	0
$931$	−2.63990	−0.0865192
$932$	0	0
$933$	−45.2254	−1.48061
$934$	0	0
$935$	8.56974	0.280260
$936$	0	0
$937$	−11.4420	−0.373793	−0.186896	−	0.982380i	$-0.559843\pi$
$937$	−11.4420	−0.373793	−0.186896	+	0.982380i	$0.559843\pi$
$938$	0	0
$939$	71.3236	2.32756
$940$	0	0
$941$	0.360100	0.0117389	0.00586945	−	0.999983i	$-0.498132\pi$
$941$	0.360100	0.0117389	0.00586945	+	0.999983i	$0.498132\pi$
$942$	0	0
$943$	−18.7579	−0.610843
$944$	0	0
$945$	20.4193	0.664239
$946$	0	0
$947$	−2.63807	−0.0857256	−0.0428628	−	0.999081i	$-0.513648\pi$
$947$	−2.63807	−0.0857256	−0.0428628	+	0.999081i	$0.513648\pi$
$948$	0	0
$949$	−0.431517	−0.0140076
$950$	0	0
$951$	78.8084	2.55554
$952$	0	0
$953$	10.4430	0.338282	0.169141	−	0.985592i	$-0.445901\pi$
$953$	10.4430	0.338282	0.169141	+	0.985592i	$0.445901\pi$
$954$	0	0
$955$	−17.7230	−0.573503
$956$	0	0
$957$	−24.6667	−0.797360
$958$	0	0
$959$	−2.52437	−0.0815160
$960$	0	0
$961$	12.2522	0.395232
$962$	0	0
$963$	−107.731	−3.47159
$964$	0	0
$965$	−13.5371	−0.435775
$966$	0	0
$967$	33.2732	1.06999	0.534996	−	0.844854i	$-0.320313\pi$
$967$	33.2732	1.06999	0.534996	+	0.844854i	$0.320313\pi$
$968$	0	0
$969$	−22.3264	−0.717228
$970$	0	0
$971$	−23.5657	−0.756258	−0.378129	−	0.925753i	$-0.623432\pi$
$971$	−23.5657	−0.756258	−0.378129	+	0.925753i	$0.623432\pi$
$972$	0	0
$973$	39.6444	1.27094
$974$	0	0
$975$	−5.10482	−0.163485
$976$	0	0
$977$	0.402439	0.0128752	0.00643759	−	0.999979i	$-0.497951\pi$
$977$	0.402439	0.0128752	0.00643759	+	0.999979i	$0.497951\pi$
$978$	0	0
$979$	0.898271	0.0287089
$980$	0	0
$981$	−24.9821	−0.797617
$982$	0	0
$983$	11.4927	0.366561	0.183281	−	0.983061i	$-0.441328\pi$
$983$	11.4927	0.366561	0.183281	+	0.983061i	$0.441328\pi$
$984$	0	0
$985$	21.5134	0.685473
$986$	0	0
$987$	13.0916	0.416710
$988$	0	0
$989$	20.6529	0.656723
$990$	0	0
$991$	6.05761	0.192426	0.0962132	−	0.995361i	$-0.469327\pi$
$991$	6.05761	0.192426	0.0962132	+	0.995361i	$0.469327\pi$
$992$	0	0
$993$	−13.5292	−0.429335
$994$	0	0
$995$	7.09410	0.224898
$996$	0	0
$997$	48.1086	1.52362	0.761808	−	0.647803i	$-0.224312\pi$
$997$	48.1086	1.52362	0.761808	+	0.647803i	$0.224312\pi$
$998$	0	0
$999$	9.25020	0.292663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.cc.1.4		4
4.3	odd	2		6080.2.a.ch.1.1		4
8.3	odd	2		1520.2.a.t.1.4		4
8.5	even	2		95.2.a.b.1.2	✓	4
24.5	odd	2		855.2.a.m.1.3		4
40.13	odd	4		475.2.b.e.324.7		8
40.19	odd	2		7600.2.a.cf.1.1		4
40.29	even	2		475.2.a.i.1.3		4
40.37	odd	4		475.2.b.e.324.2		8
56.13	odd	2		4655.2.a.y.1.2		4
120.29	odd	2		4275.2.a.bo.1.2		4
152.37	odd	2		1805.2.a.p.1.3		4
760.189	odd	2		9025.2.a.bf.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.2	✓	4	8.5	even	2
475.2.a.i.1.3		4	40.29	even	2
475.2.b.e.324.2		8	40.37	odd	4
475.2.b.e.324.7		8	40.13	odd	4
855.2.a.m.1.3		4	24.5	odd	2
1520.2.a.t.1.4		4	8.3	odd	2
1805.2.a.p.1.3		4	152.37	odd	2
4275.2.a.bo.1.2		4	120.29	odd	2
4655.2.a.y.1.2		4	56.13	odd	2
6080.2.a.cc.1.4		4	1.1	even	1	trivial
6080.2.a.ch.1.1		4	4.3	odd	2
7600.2.a.cf.1.1		4	40.19	odd	2
9025.2.a.bf.1.2		4	760.189	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+2.87834 q^{3} +1.00000 q^{5} +3.10482 q^{7} +5.28487 q^{9} +O(q^{10})\)	\(q+2.87834 q^{3} +1.00000 q^{5} +3.10482 q^{7} +5.28487 q^{9} +1.10482 q^{11} -1.77353 q^{13} +2.87834 q^{15} +7.75669 q^{17} -1.00000 q^{19} +8.93674 q^{21} -6.65187 q^{23} +1.00000 q^{25} +6.57664 q^{27} -7.75669 q^{29} +6.57664 q^{31} +3.18005 q^{33} +3.10482 q^{35} +1.40652 q^{37} -5.10482 q^{39} +2.81995 q^{41} -3.10482 q^{43} +5.28487 q^{45} +1.46492 q^{47} +2.63990 q^{49} +22.3264 q^{51} +2.59348 q^{53} +1.10482 q^{55} -2.87834 q^{57} +5.38969 q^{59} -4.07523 q^{61} +16.4086 q^{63} -1.77353 q^{65} +15.8151 q^{67} -19.1464 q^{69} +7.14638 q^{71} +0.243310 q^{73} +2.87834 q^{75} +3.43026 q^{77} -9.38969 q^{79} +3.07523 q^{81} -8.86151 q^{83} +7.75669 q^{85} -22.3264 q^{87} +0.813048 q^{89} -5.50648 q^{91} +18.9298 q^{93} -1.00000 q^{95} +3.19689 q^{97} +5.83882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 8 * q^9	\( 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} - 4 q^{19} + 4 q^{21} - 8 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 8 q^{33} + 4 q^{35} + 6 q^{37} - 12 q^{39} + 16 q^{41} - 4 q^{43} + 8 q^{45} - 12 q^{47} + 20 q^{49} + 36 q^{51} + 10 q^{53} - 4 q^{55} + 2 q^{57} - 20 q^{61} + 20 q^{63} - 2 q^{65} + 18 q^{67} - 28 q^{69} - 20 q^{71} + 28 q^{73} - 2 q^{75} + 40 q^{77} - 16 q^{79} + 16 q^{81} + 4 q^{85} - 36 q^{87} + 4 q^{89} + 36 q^{91} + 40 q^{93} - 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 8 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^17 - 4 * q^19 + 4 * q^21 - 8 * q^23 + 4 * q^25 + 4 * q^27 - 4 * q^29 + 4 * q^31 + 8 * q^33 + 4 * q^35 + 6 * q^37 - 12 * q^39 + 16 * q^41 - 4 * q^43 + 8 * q^45 - 12 * q^47 + 20 * q^49 + 36 * q^51 + 10 * q^53 - 4 * q^55 + 2 * q^57 - 20 * q^61 + 20 * q^63 - 2 * q^65 + 18 * q^67 - 28 * q^69 - 20 * q^71 + 28 * q^73 - 2 * q^75 + 40 * q^77 - 16 * q^79 + 16 * q^81 + 4 * q^85 - 36 * q^87 + 4 * q^89 + 36 * q^91 + 40 * q^93 - 4 * q^95 + 30 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more