LMFDB - Embedded newform 7360.2.a.cp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13955077.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 14x^{3} - x^{2} + 32x + 16 $ x^5 - 14x^3 - x^2 + 32x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.36002$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.36002 q^{3} +1.00000 q^{5} +1.90754 q^{7} +8.28974 q^{9} +O(q^{10})$	$q-3.36002 q^{3} +1.00000 q^{5} +1.90754 q^{7} +8.28974 q^{9} -5.48021 q^{11} +1.04937 q^{13} -3.36002 q^{15} -6.74222 q^{17} -1.55049 q^{19} -6.40939 q^{21} +1.00000 q^{23} +1.00000 q^{25} -17.7736 q^{27} +3.38219 q^{29} -10.9327 q^{31} +18.4136 q^{33} +1.90754 q^{35} -5.26201 q^{37} -3.52589 q^{39} +6.09801 q^{41} +8.28974 q^{45} -0.403830 q^{47} -3.36128 q^{49} +22.6540 q^{51} -5.88332 q^{53} -5.48021 q^{55} +5.20968 q^{57} +9.60111 q^{59} +7.09927 q^{61} +15.8130 q^{63} +1.04937 q^{65} +13.7971 q^{67} -3.36002 q^{69} -0.478950 q^{71} +2.40383 q^{73} -3.36002 q^{75} -10.4537 q^{77} +4.24037 q^{79} +34.8505 q^{81} -11.2620 q^{83} -6.74222 q^{85} -11.3642 q^{87} +4.90495 q^{89} +2.00171 q^{91} +36.7340 q^{93} -1.55049 q^{95} -12.3433 q^{97} -45.4295 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 5 * q + 5 * q^5 + 2 * q^7 + 13 * q^9	$ 5 q + 5 q^{5} + 2 q^{7} + 13 q^{9} - q^{11} - 4 q^{13} + 4 q^{17} + 7 q^{19} - 6 q^{21} + 5 q^{23} + 5 q^{25} + 3 q^{27} - 4 q^{29} - 19 q^{31} + 17 q^{33} + 2 q^{35} - 15 q^{37} - 19 q^{39} + 25 q^{41} + 13 q^{45} + 11 q^{47} + 25 q^{49} + 19 q^{51} - 3 q^{53} - q^{55} + 48 q^{57} - q^{59} + 5 q^{61} + 41 q^{63} - 4 q^{65} + 9 q^{67} - q^{71} - q^{73} - 18 q^{77} + 2 q^{79} + 57 q^{81} - 45 q^{83} + 4 q^{85} + 9 q^{87} + 6 q^{89} + 11 q^{91} + 39 q^{93} + 7 q^{95} + 25 q^{97} - 65 q^{99}+O(q^{100}) $ 5 * q + 5 * q^5 + 2 * q^7 + 13 * q^9 - q^11 - 4 * q^13 + 4 * q^17 + 7 * q^19 - 6 * q^21 + 5 * q^23 + 5 * q^25 + 3 * q^27 - 4 * q^29 - 19 * q^31 + 17 * q^33 + 2 * q^35 - 15 * q^37 - 19 * q^39 + 25 * q^41 + 13 * q^45 + 11 * q^47 + 25 * q^49 + 19 * q^51 - 3 * q^53 - q^55 + 48 * q^57 - q^59 + 5 * q^61 + 41 * q^63 - 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 + 2 * q^79 + 57 * q^81 - 45 * q^83 + 4 * q^85 + 9 * q^87 + 6 * q^89 + 11 * q^91 + 39 * q^93 + 7 * q^95 + 25 * q^97 - 65 * 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
$3$	−3.36002	−1.93991	−0.969954	−	0.243287i	$-0.921774\pi$
$3$	−3.36002	−1.93991	−0.969954	+	0.243287i	$0.921774\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.90754	0.720984	0.360492	−	0.932762i	$-0.382609\pi$
$7$	1.90754	0.720984	0.360492	+	0.932762i	$0.382609\pi$
$8$	0	0
$9$	8.28974	2.76325
$10$	0	0
$11$	−5.48021	−1.65234	−0.826172	−	0.563418i	$-0.809486\pi$
$11$	−5.48021	−1.65234	−0.826172	+	0.563418i	$0.809486\pi$
$12$	0	0
$13$	1.04937	0.291041	0.145521	−	0.989355i	$-0.453514\pi$
$13$	1.04937	0.291041	0.145521	+	0.989355i	$0.453514\pi$
$14$	0	0
$15$	−3.36002	−0.867554
$16$	0	0
$17$	−6.74222	−1.63523	−0.817614	−	0.575767i	$-0.804703\pi$
$17$	−6.74222	−1.63523	−0.817614	+	0.575767i	$0.804703\pi$
$18$	0	0
$19$	−1.55049	−0.355707	−0.177853	−	0.984057i	$-0.556915\pi$
$19$	−1.55049	−0.355707	−0.177853	+	0.984057i	$0.556915\pi$
$20$	0	0
$21$	−6.40939	−1.39864
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−17.7736	−3.42054
$28$	0	0
$29$	3.38219	0.628058	0.314029	−	0.949413i	$-0.398321\pi$
$29$	3.38219	0.628058	0.314029	+	0.949413i	$0.398321\pi$
$30$	0	0
$31$	−10.9327	−1.96357	−0.981784	−	0.190000i	$-0.939151\pi$
$31$	−10.9327	−1.96357	−0.981784	+	0.190000i	$0.939151\pi$
$32$	0	0
$33$	18.4136	3.20540
$34$	0	0
$35$	1.90754	0.322434
$36$	0	0
$37$	−5.26201	−0.865069	−0.432534	−	0.901617i	$-0.642381\pi$
$37$	−5.26201	−0.865069	−0.432534	+	0.901617i	$0.642381\pi$
$38$	0	0
$39$	−3.52589	−0.564594
$40$	0	0
$41$	6.09801	0.952350	0.476175	−	0.879351i	$-0.342023\pi$
$41$	6.09801	0.952350	0.476175	+	0.879351i	$0.342023\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	8.28974	1.23576
$46$	0	0
$47$	−0.403830	−0.0589046	−0.0294523	−	0.999566i	$-0.509376\pi$
$47$	−0.403830	−0.0589046	−0.0294523	+	0.999566i	$0.509376\pi$
$48$	0	0
$49$	−3.36128	−0.480183
$50$	0	0
$51$	22.6540	3.17219
$52$	0	0
$53$	−5.88332	−0.808136	−0.404068	−	0.914729i	$-0.632404\pi$
$53$	−5.88332	−0.808136	−0.404068	+	0.914729i	$0.632404\pi$
$54$	0	0
$55$	−5.48021	−0.738951
$56$	0	0
$57$	5.20968	0.690039
$58$	0	0
$59$	9.60111	1.24996	0.624979	−	0.780641i	$-0.285107\pi$
$59$	9.60111	1.24996	0.624979	+	0.780641i	$0.285107\pi$
$60$	0	0
$61$	7.09927	0.908968	0.454484	−	0.890755i	$-0.349824\pi$
$61$	7.09927	0.908968	0.454484	+	0.890755i	$0.349824\pi$
$62$	0	0
$63$	15.8130	1.99226
$64$	0	0
$65$	1.04937	0.130158
$66$	0	0
$67$	13.7971	1.68559	0.842794	−	0.538236i	$-0.180909\pi$
$67$	13.7971	1.68559	0.842794	+	0.538236i	$0.180909\pi$
$68$	0	0
$69$	−3.36002	−0.404499
$70$	0	0
$71$	−0.478950	−0.0568409	−0.0284205	−	0.999596i	$-0.509048\pi$
$71$	−0.478950	−0.0568409	−0.0284205	+	0.999596i	$0.509048\pi$
$72$	0	0
$73$	2.40383	0.281347	0.140673	−	0.990056i	$-0.455073\pi$
$73$	2.40383	0.281347	0.140673	+	0.990056i	$0.455073\pi$
$74$	0	0
$75$	−3.36002	−0.387982
$76$	0	0
$77$	−10.4537	−1.19131
$78$	0	0
$79$	4.24037	0.477079	0.238540	−	0.971133i	$-0.423331\pi$
$79$	4.24037	0.477079	0.238540	+	0.971133i	$0.423331\pi$
$80$	0	0
$81$	34.8505	3.87228
$82$	0	0
$83$	−11.2620	−1.23617	−0.618083	−	0.786113i	$-0.712090\pi$
$83$	−11.2620	−1.23617	−0.618083	+	0.786113i	$0.712090\pi$
$84$	0	0
$85$	−6.74222	−0.731296
$86$	0	0
$87$	−11.3642	−1.21837
$88$	0	0
$89$	4.90495	0.519924	0.259962	−	0.965619i	$-0.416290\pi$
$89$	4.90495	0.519924	0.259962	+	0.965619i	$0.416290\pi$
$90$	0	0
$91$	2.00171	0.209836
$92$	0	0
$93$	36.7340	3.80914
$94$	0	0
$95$	−1.55049	−0.159077
$96$	0	0
$97$	−12.3433	−1.25327	−0.626637	−	0.779311i	$-0.715569\pi$
$97$	−12.3433	−1.25327	−0.626637	+	0.779311i	$0.715569\pi$
$98$	0	0
$99$	−45.4295	−4.56583
$100$	0	0
$101$	1.44692	0.143974	0.0719870	−	0.997406i	$-0.477066\pi$
$101$	1.44692	0.143974	0.0719870	+	0.997406i	$0.477066\pi$
$102$	0	0
$103$	7.76385	0.764995	0.382497	−	0.923957i	$-0.375064\pi$
$103$	7.76385	0.764995	0.382497	+	0.923957i	$0.375064\pi$
$104$	0	0
$105$	−6.40939	−0.625492
$106$	0	0
$107$	−4.54197	−0.439089	−0.219544	−	0.975603i	$-0.570457\pi$
$107$	−4.54197	−0.439089	−0.219544	+	0.975603i	$0.570457\pi$
$108$	0	0
$109$	−12.4136	−1.18901	−0.594504	−	0.804093i	$-0.702652\pi$
$109$	−12.4136	−1.18901	−0.594504	+	0.804093i	$0.702652\pi$
$110$	0	0
$111$	17.6805	1.67815
$112$	0	0
$113$	9.27312	0.872342	0.436171	−	0.899864i	$-0.356334\pi$
$113$	9.27312	0.872342	0.436171	+	0.899864i	$0.356334\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	8.69896	0.804219
$118$	0	0
$119$	−12.8611	−1.17897
$120$	0	0
$121$	19.0327	1.73024
$122$	0	0
$123$	−20.4894	−1.84747
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.4149	−1.10165	−0.550824	−	0.834621i	$-0.685686\pi$
$127$	−12.4149	−1.10165	−0.550824	+	0.834621i	$0.685686\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.55434	0.834766	0.417383	−	0.908731i	$-0.362947\pi$
$131$	9.55434	0.834766	0.417383	+	0.908731i	$0.362947\pi$
$132$	0	0
$133$	−2.95763	−0.256459
$134$	0	0
$135$	−17.7736	−1.52971
$136$	0	0
$137$	−3.36558	−0.287541	−0.143770	−	0.989611i	$-0.545923\pi$
$137$	−3.36558	−0.287541	−0.143770	+	0.989611i	$0.545923\pi$
$138$	0	0
$139$	−20.7361	−1.75881	−0.879406	−	0.476072i	$-0.842060\pi$
$139$	−20.7361	−1.75881	−0.879406	+	0.476072i	$0.842060\pi$
$140$	0	0
$141$	1.35688	0.114270
$142$	0	0
$143$	−5.75074	−0.480901
$144$	0	0
$145$	3.38219	0.280876
$146$	0	0
$147$	11.2940	0.931510
$148$	0	0
$149$	19.9399	1.63354	0.816769	−	0.576965i	$-0.195763\pi$
$149$	19.9399	1.63354	0.816769	+	0.576965i	$0.195763\pi$
$150$	0	0
$151$	−23.7979	−1.93664	−0.968321	−	0.249709i	$-0.919665\pi$
$151$	−23.7979	−1.93664	−0.968321	+	0.249709i	$0.919665\pi$
$152$	0	0
$153$	−55.8912	−4.51854
$154$	0	0
$155$	−10.9327	−0.878134
$156$	0	0
$157$	−13.3175	−1.06285	−0.531425	−	0.847105i	$-0.678343\pi$
$157$	−13.3175	−1.06285	−0.531425	+	0.847105i	$0.678343\pi$
$158$	0	0
$159$	19.7681	1.56771
$160$	0	0
$161$	1.90754	0.150335
$162$	0	0
$163$	16.1529	1.26519	0.632595	−	0.774483i	$-0.281990\pi$
$163$	16.1529	1.26519	0.632595	+	0.774483i	$0.281990\pi$
$164$	0	0
$165$	18.4136	1.43350
$166$	0	0
$167$	20.6782	1.60013	0.800064	−	0.599915i	$-0.204799\pi$
$167$	20.6782	1.60013	0.800064	+	0.599915i	$0.204799\pi$
$168$	0	0
$169$	−11.8988	−0.915295
$170$	0	0
$171$	−12.8532	−0.982905
$172$	0	0
$173$	−7.72058	−0.586985	−0.293492	−	0.955961i	$-0.594818\pi$
$173$	−7.72058	−0.586985	−0.293492	+	0.955961i	$0.594818\pi$
$174$	0	0
$175$	1.90754	0.144197
$176$	0	0
$177$	−32.2599	−2.42480
$178$	0	0
$179$	9.88683	0.738976	0.369488	−	0.929236i	$-0.379533\pi$
$179$	9.88683	0.738976	0.369488	+	0.929236i	$0.379533\pi$
$180$	0	0
$181$	23.9883	1.78304	0.891519	−	0.452983i	$-0.149640\pi$
$181$	23.9883	1.78304	0.891519	+	0.452983i	$0.149640\pi$
$182$	0	0
$183$	−23.8537	−1.76332
$184$	0	0
$185$	−5.26201	−0.386871
$186$	0	0
$187$	36.9487	2.70196
$188$	0	0
$189$	−33.9040	−2.46615
$190$	0	0
$191$	−6.72004	−0.486245	−0.243123	−	0.969996i	$-0.578172\pi$
$191$	−6.72004	−0.486245	−0.243123	+	0.969996i	$0.578172\pi$
$192$	0	0
$193$	8.95007	0.644240	0.322120	−	0.946699i	$-0.395605\pi$
$193$	8.95007	0.644240	0.322120	+	0.946699i	$0.395605\pi$
$194$	0	0
$195$	−3.52589	−0.252494
$196$	0	0
$197$	6.65109	0.473871	0.236935	−	0.971525i	$-0.423857\pi$
$197$	6.65109	0.473871	0.236935	+	0.971525i	$0.423857\pi$
$198$	0	0
$199$	10.9050	0.773032	0.386516	−	0.922283i	$-0.373678\pi$
$199$	10.9050	0.773032	0.386516	+	0.922283i	$0.373678\pi$
$200$	0	0
$201$	−46.3587	−3.26989
$202$	0	0
$203$	6.45168	0.452819
$204$	0	0
$205$	6.09801	0.425904
$206$	0	0
$207$	8.28974	0.576177
$208$	0	0
$209$	8.49700	0.587750
$210$	0	0
$211$	−1.21874	−0.0839014	−0.0419507	−	0.999120i	$-0.513357\pi$
$211$	−1.21874	−0.0839014	−0.0419507	+	0.999120i	$0.513357\pi$
$212$	0	0
$213$	1.60928	0.110266
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.8546	−1.41570
$218$	0	0
$219$	−8.07692	−0.545787
$220$	0	0
$221$	−7.07505	−0.475919
$222$	0	0
$223$	−14.1542	−0.947835	−0.473917	−	0.880569i	$-0.657160\pi$
$223$	−14.1542	−0.947835	−0.473917	+	0.880569i	$0.657160\pi$
$224$	0	0
$225$	8.28974	0.552649
$226$	0	0
$227$	−4.89902	−0.325159	−0.162580	−	0.986695i	$-0.551981\pi$
$227$	−4.89902	−0.325159	−0.162580	+	0.986695i	$0.551981\pi$
$228$	0	0
$229$	−12.3946	−0.819056	−0.409528	−	0.912298i	$-0.634307\pi$
$229$	−12.3946	−0.819056	−0.409528	+	0.912298i	$0.634307\pi$
$230$	0	0
$231$	35.1248	2.31104
$232$	0	0
$233$	0.882062	0.0577858	0.0288929	−	0.999583i	$-0.490802\pi$
$233$	0.882062	0.0577858	0.0288929	+	0.999583i	$0.490802\pi$
$234$	0	0
$235$	−0.403830	−0.0263429
$236$	0	0
$237$	−14.2477	−0.925490
$238$	0	0
$239$	19.9506	1.29049	0.645247	−	0.763974i	$-0.276754\pi$
$239$	19.9506	1.29049	0.645247	+	0.763974i	$0.276754\pi$
$240$	0	0
$241$	−2.81877	−0.181573	−0.0907865	−	0.995870i	$-0.528938\pi$
$241$	−2.81877	−0.181573	−0.0907865	+	0.995870i	$0.528938\pi$
$242$	0	0
$243$	−63.7777	−4.09134
$244$	0	0
$245$	−3.36128	−0.214744
$246$	0	0
$247$	−1.62703	−0.103525
$248$	0	0
$249$	37.8406	2.39805
$250$	0	0
$251$	17.9883	1.13541	0.567706	−	0.823231i	$-0.307831\pi$
$251$	17.9883	1.13541	0.567706	+	0.823231i	$0.307831\pi$
$252$	0	0
$253$	−5.48021	−0.344538
$254$	0	0
$255$	22.6540	1.41865
$256$	0	0
$257$	−14.1705	−0.883930	−0.441965	−	0.897032i	$-0.645718\pi$
$257$	−14.1705	−0.883930	−0.441965	+	0.897032i	$0.645718\pi$
$258$	0	0
$259$	−10.0375	−0.623701
$260$	0	0
$261$	28.0375	1.73548
$262$	0	0
$263$	6.88386	0.424477	0.212238	−	0.977218i	$-0.431925\pi$
$263$	6.88386	0.424477	0.212238	+	0.977218i	$0.431925\pi$
$264$	0	0
$265$	−5.88332	−0.361409
$266$	0	0
$267$	−16.4807	−1.00861
$268$	0	0
$269$	23.3463	1.42345	0.711724	−	0.702459i	$-0.247915\pi$
$269$	23.3463	1.42345	0.711724	+	0.702459i	$0.247915\pi$
$270$	0	0
$271$	−13.9519	−0.847517	−0.423759	−	0.905775i	$-0.639289\pi$
$271$	−13.9519	−0.847517	−0.423759	+	0.905775i	$0.639289\pi$
$272$	0	0
$273$	−6.72579	−0.407063
$274$	0	0
$275$	−5.48021	−0.330469
$276$	0	0
$277$	−3.69490	−0.222005	−0.111003	−	0.993820i	$-0.535406\pi$
$277$	−3.69490	−0.222005	−0.111003	+	0.993820i	$0.535406\pi$
$278$	0	0
$279$	−90.6291	−5.42582
$280$	0	0
$281$	15.9582	0.951984	0.475992	−	0.879450i	$-0.342089\pi$
$281$	15.9582	0.951984	0.475992	+	0.879450i	$0.342089\pi$
$282$	0	0
$283$	−8.21649	−0.488420	−0.244210	−	0.969722i	$-0.578529\pi$
$283$	−8.21649	−0.488420	−0.244210	+	0.969722i	$0.578529\pi$
$284$	0	0
$285$	5.20968	0.308595
$286$	0	0
$287$	11.6322	0.686628
$288$	0	0
$289$	28.4575	1.67397
$290$	0	0
$291$	41.4738	2.43124
$292$	0	0
$293$	−22.0878	−1.29038	−0.645191	−	0.764021i	$-0.723222\pi$
$293$	−22.0878	−1.29038	−0.645191	+	0.764021i	$0.723222\pi$
$294$	0	0
$295$	9.60111	0.558998
$296$	0	0
$297$	97.4032	5.65191
$298$	0	0
$299$	1.04937	0.0606863
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.86168	−0.279296
$304$	0	0
$305$	7.09927	0.406503
$306$	0	0
$307$	15.2091	0.868030	0.434015	−	0.900906i	$-0.357097\pi$
$307$	15.2091	0.868030	0.434015	+	0.900906i	$0.357097\pi$
$308$	0	0
$309$	−26.0867	−1.48402
$310$	0	0
$311$	−0.254818	−0.0144494	−0.00722471	−	0.999974i	$-0.502300\pi$
$311$	−0.254818	−0.0144494	−0.00722471	+	0.999974i	$0.502300\pi$
$312$	0	0
$313$	3.57095	0.201842	0.100921	−	0.994894i	$-0.467821\pi$
$313$	3.57095	0.201842	0.100921	+	0.994894i	$0.467821\pi$
$314$	0	0
$315$	15.8130	0.890964
$316$	0	0
$317$	−12.3235	−0.692155	−0.346078	−	0.938206i	$-0.612487\pi$
$317$	−12.3235	−0.692155	−0.346078	+	0.938206i	$0.612487\pi$
$318$	0	0
$319$	−18.5351	−1.03777
$320$	0	0
$321$	15.2611	0.851792
$322$	0	0
$323$	10.4537	0.581661
$324$	0	0
$325$	1.04937	0.0582083
$326$	0	0
$327$	41.7100	2.30657
$328$	0	0
$329$	−0.770323	−0.0424693
$330$	0	0
$331$	26.1082	1.43503	0.717517	−	0.696541i	$-0.245278\pi$
$331$	26.1082	1.43503	0.717517	+	0.696541i	$0.245278\pi$
$332$	0	0
$333$	−43.6207	−2.39040
$334$	0	0
$335$	13.7971	0.753818
$336$	0	0
$337$	9.68246	0.527437	0.263718	−	0.964600i	$-0.415051\pi$
$337$	9.68246	0.527437	0.263718	+	0.964600i	$0.415051\pi$
$338$	0	0
$339$	−31.1579	−1.69226
$340$	0	0
$341$	59.9134	3.24449
$342$	0	0
$343$	−19.7646	−1.06719
$344$	0	0
$345$	−3.36002	−0.180897
$346$	0	0
$347$	24.9747	1.34071	0.670355	−	0.742040i	$-0.266142\pi$
$347$	24.9747	1.34071	0.670355	+	0.742040i	$0.266142\pi$
$348$	0	0
$349$	35.3019	1.88967	0.944835	−	0.327547i	$-0.106222\pi$
$349$	35.3019	1.88967	0.944835	+	0.327547i	$0.106222\pi$
$350$	0	0
$351$	−18.6510	−0.995518
$352$	0	0
$353$	−10.0326	−0.533980	−0.266990	−	0.963699i	$-0.586029\pi$
$353$	−10.0326	−0.533980	−0.266990	+	0.963699i	$0.586029\pi$
$354$	0	0
$355$	−0.478950	−0.0254200
$356$	0	0
$357$	43.2135	2.28710
$358$	0	0
$359$	30.8691	1.62921	0.814603	−	0.580019i	$-0.196955\pi$
$359$	30.8691	1.62921	0.814603	+	0.580019i	$0.196955\pi$
$360$	0	0
$361$	−16.5960	−0.873473
$362$	0	0
$363$	−63.9502	−3.35651
$364$	0	0
$365$	2.40383	0.125822
$366$	0	0
$367$	−33.3234	−1.73947	−0.869734	−	0.493521i	$-0.835709\pi$
$367$	−33.3234	−1.73947	−0.869734	+	0.493521i	$0.835709\pi$
$368$	0	0
$369$	50.5509	2.63158
$370$	0	0
$371$	−11.2227	−0.582653
$372$	0	0
$373$	−4.62023	−0.239227	−0.119613	−	0.992821i	$-0.538165\pi$
$373$	−4.62023	−0.239227	−0.119613	+	0.992821i	$0.538165\pi$
$374$	0	0
$375$	−3.36002	−0.173511
$376$	0	0
$377$	3.54916	0.182791
$378$	0	0
$379$	15.0020	0.770600	0.385300	−	0.922791i	$-0.374098\pi$
$379$	15.0020	0.770600	0.385300	+	0.922791i	$0.374098\pi$
$380$	0	0
$381$	41.7145	2.13710
$382$	0	0
$383$	−1.87882	−0.0960033	−0.0480016	−	0.998847i	$-0.515285\pi$
$383$	−1.87882	−0.0960033	−0.0480016	+	0.998847i	$0.515285\pi$
$384$	0	0
$385$	−10.4537	−0.532772
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.48539	0.0753121	0.0376560	−	0.999291i	$-0.488011\pi$
$389$	1.48539	0.0753121	0.0376560	+	0.999291i	$0.488011\pi$
$390$	0	0
$391$	−6.74222	−0.340968
$392$	0	0
$393$	−32.1028	−1.61937
$394$	0	0
$395$	4.24037	0.213356
$396$	0	0
$397$	−5.26722	−0.264354	−0.132177	−	0.991226i	$-0.542197\pi$
$397$	−5.26722	−0.264354	−0.132177	+	0.991226i	$0.542197\pi$
$398$	0	0
$399$	9.93769	0.497507
$400$	0	0
$401$	26.1490	1.30582	0.652910	−	0.757436i	$-0.273548\pi$
$401$	26.1490	1.30582	0.652910	+	0.757436i	$0.273548\pi$
$402$	0	0
$403$	−11.4724	−0.571480
$404$	0	0
$405$	34.8505	1.73174
$406$	0	0
$407$	28.8369	1.42939
$408$	0	0
$409$	−14.7503	−0.729355	−0.364677	−	0.931134i	$-0.618821\pi$
$409$	−14.7503	−0.729355	−0.364677	+	0.931134i	$0.618821\pi$
$410$	0	0
$411$	11.3084	0.557803
$412$	0	0
$413$	18.3145	0.901200
$414$	0	0
$415$	−11.2620	−0.552830
$416$	0	0
$417$	69.6737	3.41194
$418$	0	0
$419$	1.10616	0.0540394	0.0270197	−	0.999635i	$-0.491398\pi$
$419$	1.10616	0.0540394	0.0270197	+	0.999635i	$0.491398\pi$
$420$	0	0
$421$	−9.16362	−0.446607	−0.223304	−	0.974749i	$-0.571684\pi$
$421$	−9.16362	−0.446607	−0.223304	+	0.974749i	$0.571684\pi$
$422$	0	0
$423$	−3.34764	−0.162768
$424$	0	0
$425$	−6.74222	−0.327045
$426$	0	0
$427$	13.5422	0.655351
$428$	0	0
$429$	19.3226	0.932904
$430$	0	0
$431$	32.1712	1.54963	0.774817	−	0.632186i	$-0.217842\pi$
$431$	32.1712	1.54963	0.774817	+	0.632186i	$0.217842\pi$
$432$	0	0
$433$	−8.37861	−0.402650	−0.201325	−	0.979524i	$-0.564525\pi$
$433$	−8.37861	−0.402650	−0.201325	+	0.979524i	$0.564525\pi$
$434$	0	0
$435$	−11.3642	−0.544874
$436$	0	0
$437$	−1.55049	−0.0741700
$438$	0	0
$439$	15.3093	0.730673	0.365336	−	0.930876i	$-0.380954\pi$
$439$	15.3093	0.730673	0.365336	+	0.930876i	$0.380954\pi$
$440$	0	0
$441$	−27.8641	−1.32686
$442$	0	0
$443$	−3.24129	−0.153998	−0.0769992	−	0.997031i	$-0.524534\pi$
$443$	−3.24129	−0.153998	−0.0769992	+	0.997031i	$0.524534\pi$
$444$	0	0
$445$	4.90495	0.232517
$446$	0	0
$447$	−66.9984	−3.16892
$448$	0	0
$449$	−9.02215	−0.425782	−0.212891	−	0.977076i	$-0.568288\pi$
$449$	−9.02215	−0.425782	−0.212891	+	0.977076i	$0.568288\pi$
$450$	0	0
$451$	−33.4184	−1.57361
$452$	0	0
$453$	79.9613	3.75691
$454$	0	0
$455$	2.00171	0.0938416
$456$	0	0
$457$	7.03526	0.329096	0.164548	−	0.986369i	$-0.447384\pi$
$457$	7.03526	0.329096	0.164548	+	0.986369i	$0.447384\pi$
$458$	0	0
$459$	119.834	5.59336
$460$	0	0
$461$	1.59617	0.0743411	0.0371705	−	0.999309i	$-0.488166\pi$
$461$	1.59617	0.0743411	0.0371705	+	0.999309i	$0.488166\pi$
$462$	0	0
$463$	−9.61655	−0.446919	−0.223459	−	0.974713i	$-0.571735\pi$
$463$	−9.61655	−0.446919	−0.223459	+	0.974713i	$0.571735\pi$
$464$	0	0
$465$	36.7340	1.70350
$466$	0	0
$467$	24.4764	1.13263	0.566317	−	0.824188i	$-0.308368\pi$
$467$	24.4764	1.13263	0.566317	+	0.824188i	$0.308368\pi$
$468$	0	0
$469$	26.3186	1.21528
$470$	0	0
$471$	44.7470	2.06183
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.55049	−0.0711413
$476$	0	0
$477$	−48.7712	−2.23308
$478$	0	0
$479$	13.2108	0.603618	0.301809	−	0.953368i	$-0.402410\pi$
$479$	13.2108	0.603618	0.301809	+	0.953368i	$0.402410\pi$
$480$	0	0
$481$	−5.52177	−0.251771
$482$	0	0
$483$	−6.40939	−0.291637
$484$	0	0
$485$	−12.3433	−0.560482
$486$	0	0
$487$	23.7078	1.07431	0.537153	−	0.843485i	$-0.319500\pi$
$487$	23.7078	1.07431	0.537153	+	0.843485i	$0.319500\pi$
$488$	0	0
$489$	−54.2739	−2.45435
$490$	0	0
$491$	18.5930	0.839091	0.419546	−	0.907734i	$-0.362189\pi$
$491$	18.5930	0.839091	0.419546	+	0.907734i	$0.362189\pi$
$492$	0	0
$493$	−22.8035	−1.02702
$494$	0	0
$495$	−45.4295	−2.04190
$496$	0	0
$497$	−0.913618	−0.0409814
$498$	0	0
$499$	17.9121	0.801858	0.400929	−	0.916109i	$-0.368687\pi$
$499$	17.9121	0.801858	0.400929	+	0.916109i	$0.368687\pi$
$500$	0	0
$501$	−69.4792	−3.10410
$502$	0	0
$503$	−1.24890	−0.0556855	−0.0278428	−	0.999612i	$-0.508864\pi$
$503$	−1.24890	−0.0556855	−0.0278428	+	0.999612i	$0.508864\pi$
$504$	0	0
$505$	1.44692	0.0643871
$506$	0	0
$507$	39.9803	1.77559
$508$	0	0
$509$	−15.8757	−0.703679	−0.351839	−	0.936060i	$-0.614444\pi$
$509$	−15.8757	−0.703679	−0.351839	+	0.936060i	$0.614444\pi$
$510$	0	0
$511$	4.58541	0.202847
$512$	0	0
$513$	27.5578	1.21671
$514$	0	0
$515$	7.76385	0.342116
$516$	0	0
$517$	2.21307	0.0973307
$518$	0	0
$519$	25.9413	1.13870
$520$	0	0
$521$	26.5488	1.16312	0.581561	−	0.813503i	$-0.302442\pi$
$521$	26.5488	1.16312	0.581561	+	0.813503i	$0.302442\pi$
$522$	0	0
$523$	31.0258	1.35666	0.678331	−	0.734757i	$-0.262704\pi$
$523$	31.0258	1.35666	0.678331	+	0.734757i	$0.262704\pi$
$524$	0	0
$525$	−6.40939	−0.279729
$526$	0	0
$527$	73.7105	3.21088
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	79.5907	3.45394
$532$	0	0
$533$	6.39904	0.277173
$534$	0	0
$535$	−4.54197	−0.196366
$536$	0	0
$537$	−33.2199	−1.43355
$538$	0	0
$539$	18.4205	0.793427
$540$	0	0
$541$	−22.4123	−0.963579	−0.481790	−	0.876287i	$-0.660013\pi$
$541$	−22.4123	−0.963579	−0.481790	+	0.876287i	$0.660013\pi$
$542$	0	0
$543$	−80.6013	−3.45893
$544$	0	0
$545$	−12.4136	−0.531741
$546$	0	0
$547$	−43.3980	−1.85556	−0.927782	−	0.373122i	$-0.878287\pi$
$547$	−43.3980	−1.85556	−0.927782	+	0.373122i	$0.878287\pi$
$548$	0	0
$549$	58.8511	2.51170
$550$	0	0
$551$	−5.24406	−0.223404
$552$	0	0
$553$	8.08870	0.343966
$554$	0	0
$555$	17.6805	0.750494
$556$	0	0
$557$	21.2690	0.901197	0.450599	−	0.892727i	$-0.351211\pi$
$557$	21.2690	0.901197	0.450599	+	0.892727i	$0.351211\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−124.149	−5.24155
$562$	0	0
$563$	10.6629	0.449389	0.224694	−	0.974429i	$-0.427862\pi$
$563$	10.6629	0.449389	0.224694	+	0.974429i	$0.427862\pi$
$564$	0	0
$565$	9.27312	0.390123
$566$	0	0
$567$	66.4789	2.79185
$568$	0	0
$569$	31.8986	1.33726	0.668630	−	0.743596i	$-0.266881\pi$
$569$	31.8986	1.33726	0.668630	+	0.743596i	$0.266881\pi$
$570$	0	0
$571$	−7.87477	−0.329549	−0.164774	−	0.986331i	$-0.552690\pi$
$571$	−7.87477	−0.329549	−0.164774	+	0.986331i	$0.552690\pi$
$572$	0	0
$573$	22.5795	0.943271
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	40.7070	1.69466	0.847328	−	0.531070i	$-0.178210\pi$
$577$	40.7070	1.69466	0.847328	+	0.531070i	$0.178210\pi$
$578$	0	0
$579$	−30.0724	−1.24977
$580$	0	0
$581$	−21.4828	−0.891256
$582$	0	0
$583$	32.2418	1.33532
$584$	0	0
$585$	8.69896	0.359658
$586$	0	0
$587$	−22.9321	−0.946508	−0.473254	−	0.880926i	$-0.656921\pi$
$587$	−22.9321	−0.946508	−0.473254	+	0.880926i	$0.656921\pi$
$588$	0	0
$589$	16.9510	0.698454
$590$	0	0
$591$	−22.3478	−0.919266
$592$	0	0
$593$	27.6250	1.13442	0.567211	−	0.823572i	$-0.308022\pi$
$593$	27.6250	1.13442	0.567211	+	0.823572i	$0.308022\pi$
$594$	0	0
$595$	−12.8611	−0.527252
$596$	0	0
$597$	−36.6409	−1.49961
$598$	0	0
$599$	18.2139	0.744199	0.372099	−	0.928193i	$-0.378638\pi$
$599$	18.2139	0.744199	0.372099	+	0.928193i	$0.378638\pi$
$600$	0	0
$601$	−24.0407	−0.980640	−0.490320	−	0.871543i	$-0.663120\pi$
$601$	−24.0407	−0.980640	−0.490320	+	0.871543i	$0.663120\pi$
$602$	0	0
$603$	114.375	4.65770
$604$	0	0
$605$	19.0327	0.773788
$606$	0	0
$607$	−5.52878	−0.224406	−0.112203	−	0.993685i	$-0.535791\pi$
$607$	−5.52878	−0.224406	−0.112203	+	0.993685i	$0.535791\pi$
$608$	0	0
$609$	−21.6778	−0.878428
$610$	0	0
$611$	−0.423765	−0.0171437
$612$	0	0
$613$	21.1205	0.853050	0.426525	−	0.904476i	$-0.359738\pi$
$613$	21.1205	0.853050	0.426525	+	0.904476i	$0.359738\pi$
$614$	0	0
$615$	−20.4894	−0.826214
$616$	0	0
$617$	19.6523	0.791174	0.395587	−	0.918429i	$-0.370541\pi$
$617$	19.6523	0.791174	0.395587	+	0.918429i	$0.370541\pi$
$618$	0	0
$619$	40.9931	1.64765	0.823826	−	0.566843i	$-0.191836\pi$
$619$	40.9931	1.64765	0.823826	+	0.566843i	$0.191836\pi$
$620$	0	0
$621$	−17.7736	−0.713231
$622$	0	0
$623$	9.35641	0.374857
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−28.5501	−1.14018
$628$	0	0
$629$	35.4776	1.41458
$630$	0	0
$631$	45.5595	1.81369	0.906847	−	0.421461i	$-0.138483\pi$
$631$	45.5595	1.81369	0.906847	+	0.421461i	$0.138483\pi$
$632$	0	0
$633$	4.09499	0.162761
$634$	0	0
$635$	−12.4149	−0.492672
$636$	0	0
$637$	−3.52721	−0.139753
$638$	0	0
$639$	−3.97037	−0.157065
$640$	0	0
$641$	8.97997	0.354688	0.177344	−	0.984149i	$-0.443250\pi$
$641$	8.97997	0.354688	0.177344	+	0.984149i	$0.443250\pi$
$642$	0	0
$643$	2.69616	0.106326	0.0531630	−	0.998586i	$-0.483070\pi$
$643$	2.69616	0.106326	0.0531630	+	0.998586i	$0.483070\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.12638	−0.358795	−0.179398	−	0.983777i	$-0.557415\pi$
$647$	−9.12638	−0.358795	−0.179398	+	0.983777i	$0.557415\pi$
$648$	0	0
$649$	−52.6161	−2.06536
$650$	0	0
$651$	70.0718	2.74633
$652$	0	0
$653$	−41.5497	−1.62596	−0.812982	−	0.582289i	$-0.802157\pi$
$653$	−41.5497	−1.62596	−0.812982	+	0.582289i	$0.802157\pi$
$654$	0	0
$655$	9.55434	0.373319
$656$	0	0
$657$	19.9271	0.777431
$658$	0	0
$659$	20.9256	0.815145	0.407573	−	0.913173i	$-0.366375\pi$
$659$	20.9256	0.815145	0.407573	+	0.913173i	$0.366375\pi$
$660$	0	0
$661$	−5.25688	−0.204469	−0.102234	−	0.994760i	$-0.532599\pi$
$661$	−5.25688	−0.204469	−0.102234	+	0.994760i	$0.532599\pi$
$662$	0	0
$663$	23.7723	0.923240
$664$	0	0
$665$	−2.95763	−0.114692
$666$	0	0
$667$	3.38219	0.130959
$668$	0	0
$669$	47.5584	1.83871
$670$	0	0
$671$	−38.9055	−1.50193
$672$	0	0
$673$	38.1900	1.47212	0.736059	−	0.676918i	$-0.236685\pi$
$673$	38.1900	1.47212	0.736059	+	0.676918i	$0.236685\pi$
$674$	0	0
$675$	−17.7736	−0.684107
$676$	0	0
$677$	4.83529	0.185835	0.0929176	−	0.995674i	$-0.470381\pi$
$677$	4.83529	0.185835	0.0929176	+	0.995674i	$0.470381\pi$
$678$	0	0
$679$	−23.5454	−0.903591
$680$	0	0
$681$	16.4608	0.630780
$682$	0	0
$683$	−50.3044	−1.92484	−0.962422	−	0.271559i	$-0.912461\pi$
$683$	−50.3044	−1.92484	−0.962422	+	0.271559i	$0.912461\pi$
$684$	0	0
$685$	−3.36558	−0.128592
$686$	0	0
$687$	41.6460	1.58889
$688$	0	0
$689$	−6.17375	−0.235201
$690$	0	0
$691$	20.6298	0.784793	0.392396	−	0.919796i	$-0.371646\pi$
$691$	20.6298	0.784793	0.392396	+	0.919796i	$0.371646\pi$
$692$	0	0
$693$	−86.6587	−3.29189
$694$	0	0
$695$	−20.7361	−0.786565
$696$	0	0
$697$	−41.1141	−1.55731
$698$	0	0
$699$	−2.96375	−0.112099
$700$	0	0
$701$	22.9598	0.867182	0.433591	−	0.901110i	$-0.357246\pi$
$701$	22.9598	0.867182	0.433591	+	0.901110i	$0.357246\pi$
$702$	0	0
$703$	8.15869	0.307711
$704$	0	0
$705$	1.35688	0.0511029
$706$	0	0
$707$	2.76006	0.103803
$708$	0	0
$709$	−11.9069	−0.447174	−0.223587	−	0.974684i	$-0.571777\pi$
$709$	−11.9069	−0.447174	−0.223587	+	0.974684i	$0.571777\pi$
$710$	0	0
$711$	35.1516	1.31829
$712$	0	0
$713$	−10.9327	−0.409432
$714$	0	0
$715$	−5.75074	−0.215065
$716$	0	0
$717$	−67.0343	−2.50344
$718$	0	0
$719$	−24.5244	−0.914604	−0.457302	−	0.889311i	$-0.651184\pi$
$719$	−24.5244	−0.914604	−0.457302	+	0.889311i	$0.651184\pi$
$720$	0	0
$721$	14.8099	0.551549
$722$	0	0
$723$	9.47113	0.352235
$724$	0	0
$725$	3.38219	0.125612
$726$	0	0
$727$	−11.4034	−0.422928	−0.211464	−	0.977386i	$-0.567823\pi$
$727$	−11.4034	−0.422928	−0.211464	+	0.977386i	$0.567823\pi$
$728$	0	0
$729$	109.743	4.06454
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−35.8722	−1.32497	−0.662484	−	0.749076i	$-0.730498\pi$
$733$	−35.8722	−1.32497	−0.662484	+	0.749076i	$0.730498\pi$
$734$	0	0
$735$	11.2940	0.416584
$736$	0	0
$737$	−75.6112	−2.78517
$738$	0	0
$739$	−17.1503	−0.630883	−0.315441	−	0.948945i	$-0.602153\pi$
$739$	−17.1503	−0.630883	−0.315441	+	0.948945i	$0.602153\pi$
$740$	0	0
$741$	5.46685	0.200830
$742$	0	0
$743$	−27.7242	−1.01710	−0.508552	−	0.861031i	$-0.669819\pi$
$743$	−27.7242	−1.01710	−0.508552	+	0.861031i	$0.669819\pi$
$744$	0	0
$745$	19.9399	0.730540
$746$	0	0
$747$	−93.3591	−3.41583
$748$	0	0
$749$	−8.66400	−0.316576
$750$	0	0
$751$	2.80173	0.102236	0.0511182	−	0.998693i	$-0.483721\pi$
$751$	2.80173	0.102236	0.0511182	+	0.998693i	$0.483721\pi$
$752$	0	0
$753$	−60.4411	−2.20260
$754$	0	0
$755$	−23.7979	−0.866093
$756$	0	0
$757$	41.6710	1.51456	0.757278	−	0.653092i	$-0.226529\pi$
$757$	41.6710	1.51456	0.757278	+	0.653092i	$0.226529\pi$
$758$	0	0
$759$	18.4136	0.668372
$760$	0	0
$761$	−35.4285	−1.28428	−0.642141	−	0.766587i	$-0.721954\pi$
$761$	−35.4285	−1.28428	−0.642141	+	0.766587i	$0.721954\pi$
$762$	0	0
$763$	−23.6795	−0.857255
$764$	0	0
$765$	−55.8912	−2.02075
$766$	0	0
$767$	10.0751	0.363790
$768$	0	0
$769$	23.8731	0.860885	0.430442	−	0.902618i	$-0.358358\pi$
$769$	23.8731	0.860885	0.430442	+	0.902618i	$0.358358\pi$
$770$	0	0
$771$	47.6131	1.71474
$772$	0	0
$773$	−11.3403	−0.407881	−0.203941	−	0.978983i	$-0.565375\pi$
$773$	−11.3403	−0.407881	−0.203941	+	0.978983i	$0.565375\pi$
$774$	0	0
$775$	−10.9327	−0.392714
$776$	0	0
$777$	33.7262	1.20992
$778$	0	0
$779$	−9.45490	−0.338757
$780$	0	0
$781$	2.62475	0.0939208
$782$	0	0
$783$	−60.1139	−2.14829
$784$	0	0
$785$	−13.3175	−0.475321
$786$	0	0
$787$	−26.1906	−0.933595	−0.466797	−	0.884364i	$-0.654592\pi$
$787$	−26.1906	−0.933595	−0.466797	+	0.884364i	$0.654592\pi$
$788$	0	0
$789$	−23.1299	−0.823446
$790$	0	0
$791$	17.6889	0.628944
$792$	0	0
$793$	7.44973	0.264547
$794$	0	0
$795$	19.7681	0.701101
$796$	0	0
$797$	4.48133	0.158737	0.0793684	−	0.996845i	$-0.474710\pi$
$797$	4.48133	0.158737	0.0793684	+	0.996845i	$0.474710\pi$
$798$	0	0
$799$	2.72271	0.0963224
$800$	0	0
$801$	40.6608	1.43668
$802$	0	0
$803$	−13.1735	−0.464882
$804$	0	0
$805$	1.90754	0.0672321
$806$	0	0
$807$	−78.4440	−2.76136
$808$	0	0
$809$	−32.4608	−1.14126	−0.570630	−	0.821207i	$-0.693301\pi$
$809$	−32.4608	−1.14126	−0.570630	+	0.821207i	$0.693301\pi$
$810$	0	0
$811$	11.3500	0.398554	0.199277	−	0.979943i	$-0.436141\pi$
$811$	11.3500	0.398554	0.199277	+	0.979943i	$0.436141\pi$
$812$	0	0
$813$	46.8786	1.64411
$814$	0	0
$815$	16.1529	0.565810
$816$	0	0
$817$	0	0
$818$	0	0
$819$	16.5936	0.579829
$820$	0	0
$821$	−48.2070	−1.68244	−0.841218	−	0.540697i	$-0.818161\pi$
$821$	−48.2070	−1.68244	−0.841218	+	0.540697i	$0.818161\pi$
$822$	0	0
$823$	37.7179	1.31476	0.657381	−	0.753558i	$-0.271664\pi$
$823$	37.7179	1.31476	0.657381	+	0.753558i	$0.271664\pi$
$824$	0	0
$825$	18.4136	0.641080
$826$	0	0
$827$	−8.89736	−0.309392	−0.154696	−	0.987962i	$-0.549440\pi$
$827$	−8.89736	−0.309392	−0.154696	+	0.987962i	$0.549440\pi$
$828$	0	0
$829$	−30.9058	−1.07340	−0.536702	−	0.843772i	$-0.680330\pi$
$829$	−30.9058	−1.07340	−0.536702	+	0.843772i	$0.680330\pi$
$830$	0	0
$831$	12.4149	0.430670
$832$	0	0
$833$	22.6625	0.785208
$834$	0	0
$835$	20.6782	0.715599
$836$	0	0
$837$	194.313	6.71646
$838$	0	0
$839$	−1.34387	−0.0463954	−0.0231977	−	0.999731i	$-0.507385\pi$
$839$	−1.34387	−0.0463954	−0.0231977	+	0.999731i	$0.507385\pi$
$840$	0	0
$841$	−17.5608	−0.605543
$842$	0	0
$843$	−53.6198	−1.84676
$844$	0	0
$845$	−11.8988	−0.409332
$846$	0	0
$847$	36.3056	1.24748
$848$	0	0
$849$	27.6076	0.947489
$850$	0	0
$851$	−5.26201	−0.180379
$852$	0	0
$853$	26.3940	0.903713	0.451857	−	0.892091i	$-0.350762\pi$
$853$	26.3940	0.903713	0.451857	+	0.892091i	$0.350762\pi$
$854$	0	0
$855$	−12.8532	−0.439569
$856$	0	0
$857$	−17.5048	−0.597953	−0.298976	−	0.954260i	$-0.596645\pi$
$857$	−17.5048	−0.597953	−0.298976	+	0.954260i	$0.596645\pi$
$858$	0	0
$859$	11.7199	0.399877	0.199938	−	0.979808i	$-0.435926\pi$
$859$	11.7199	0.399877	0.199938	+	0.979808i	$0.435926\pi$
$860$	0	0
$861$	−39.0845	−1.33200
$862$	0	0
$863$	−26.6578	−0.907443	−0.453722	−	0.891144i	$-0.649904\pi$
$863$	−26.6578	−0.907443	−0.453722	+	0.891144i	$0.649904\pi$
$864$	0	0
$865$	−7.72058	−0.262508
$866$	0	0
$867$	−95.6177	−3.24735
$868$	0	0
$869$	−23.2381	−0.788299
$870$	0	0
$871$	14.4782	0.490576
$872$	0	0
$873$	−102.323	−3.46311
$874$	0	0
$875$	1.90754	0.0644867
$876$	0	0
$877$	−19.9574	−0.673912	−0.336956	−	0.941520i	$-0.609397\pi$
$877$	−19.9574	−0.673912	−0.336956	+	0.941520i	$0.609397\pi$
$878$	0	0
$879$	74.2154	2.50322
$880$	0	0
$881$	−20.7297	−0.698402	−0.349201	−	0.937048i	$-0.613547\pi$
$881$	−20.7297	−0.698402	−0.349201	+	0.937048i	$0.613547\pi$
$882$	0	0
$883$	2.31093	0.0777690	0.0388845	−	0.999244i	$-0.487620\pi$
$883$	2.31093	0.0777690	0.0388845	+	0.999244i	$0.487620\pi$
$884$	0	0
$885$	−32.2599	−1.08441
$886$	0	0
$887$	11.6924	0.392592	0.196296	−	0.980545i	$-0.437109\pi$
$887$	11.6924	0.392592	0.196296	+	0.980545i	$0.437109\pi$
$888$	0	0
$889$	−23.6820	−0.794270
$890$	0	0
$891$	−190.988	−6.39835
$892$	0	0
$893$	0.626134	0.0209528
$894$	0	0
$895$	9.88683	0.330480
$896$	0	0
$897$	−3.52589	−0.117726
$898$	0	0
$899$	−36.9765	−1.23323
$900$	0	0
$901$	39.6666	1.32149
$902$	0	0
$903$	0	0
$904$	0	0
$905$	23.9883	0.797399
$906$	0	0
$907$	32.5061	1.07935	0.539673	−	0.841875i	$-0.318548\pi$
$907$	32.5061	1.07935	0.539673	+	0.841875i	$0.318548\pi$
$908$	0	0
$909$	11.9946	0.397836
$910$	0	0
$911$	−10.7518	−0.356224	−0.178112	−	0.984010i	$-0.556999\pi$
$911$	−10.7518	−0.356224	−0.178112	+	0.984010i	$0.556999\pi$
$912$	0	0
$913$	61.7181	2.04257
$914$	0	0
$915$	−23.8537	−0.788579
$916$	0	0
$917$	18.2253	0.601853
$918$	0	0
$919$	24.2758	0.800785	0.400392	−	0.916344i	$-0.368874\pi$
$919$	24.2758	0.800785	0.400392	+	0.916344i	$0.368874\pi$
$920$	0	0
$921$	−51.1029	−1.68390
$922$	0	0
$923$	−0.502594	−0.0165431
$924$	0	0
$925$	−5.26201	−0.173014
$926$	0	0
$927$	64.3603	2.11387
$928$	0	0
$929$	8.18842	0.268653	0.134327	−	0.990937i	$-0.457113\pi$
$929$	8.18842	0.268653	0.134327	+	0.990937i	$0.457113\pi$
$930$	0	0
$931$	5.21163	0.170804
$932$	0	0
$933$	0.856194	0.0280305
$934$	0	0
$935$	36.9487	1.20835
$936$	0	0
$937$	26.2312	0.856936	0.428468	−	0.903557i	$-0.359053\pi$
$937$	26.2312	0.856936	0.428468	+	0.903557i	$0.359053\pi$
$938$	0	0
$939$	−11.9985	−0.391556
$940$	0	0
$941$	−23.6292	−0.770291	−0.385145	−	0.922856i	$-0.625849\pi$
$941$	−23.6292	−0.770291	−0.385145	+	0.922856i	$0.625849\pi$
$942$	0	0
$943$	6.09801	0.198579
$944$	0	0
$945$	−33.9040	−1.10290
$946$	0	0
$947$	−15.9026	−0.516765	−0.258383	−	0.966043i	$-0.583190\pi$
$947$	−15.9026	−0.516765	−0.258383	+	0.966043i	$0.583190\pi$
$948$	0	0
$949$	2.52249	0.0818836
$950$	0	0
$951$	41.4071	1.34272
$952$	0	0
$953$	20.9451	0.678478	0.339239	−	0.940700i	$-0.389831\pi$
$953$	20.9451	0.678478	0.339239	+	0.940700i	$0.389831\pi$
$954$	0	0
$955$	−6.72004	−0.217455
$956$	0	0
$957$	62.2784	2.01318
$958$	0	0
$959$	−6.41998	−0.207312
$960$	0	0
$961$	88.5236	2.85560
$962$	0	0
$963$	−37.6517	−1.21331
$964$	0	0
$965$	8.95007	0.288113
$966$	0	0
$967$	14.0732	0.452563	0.226281	−	0.974062i	$-0.427343\pi$
$967$	14.0732	0.452563	0.226281	+	0.974062i	$0.427343\pi$
$968$	0	0
$969$	−35.1248	−1.12837
$970$	0	0
$971$	−11.4404	−0.367139	−0.183569	−	0.983007i	$-0.558765\pi$
$971$	−11.4404	−0.367139	−0.183569	+	0.983007i	$0.558765\pi$
$972$	0	0
$973$	−39.5550	−1.26808
$974$	0	0
$975$	−3.52589	−0.112919
$976$	0	0
$977$	43.9119	1.40487	0.702433	−	0.711750i	$-0.252097\pi$
$977$	43.9119	1.40487	0.702433	+	0.711750i	$0.252097\pi$
$978$	0	0
$979$	−26.8802	−0.859094
$980$	0	0
$981$	−102.906	−3.28552
$982$	0	0
$983$	43.0420	1.37283	0.686413	−	0.727212i	$-0.259184\pi$
$983$	43.0420	1.37283	0.686413	+	0.727212i	$0.259184\pi$
$984$	0	0
$985$	6.65109	0.211921
$986$	0	0
$987$	2.58830	0.0823865
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−43.5288	−1.38274	−0.691369	−	0.722501i	$-0.742992\pi$
$991$	−43.5288	−1.38274	−0.691369	+	0.722501i	$0.742992\pi$
$992$	0	0
$993$	−87.7240	−2.78384
$994$	0	0
$995$	10.9050	0.345710
$996$	0	0
$997$	−26.4448	−0.837517	−0.418758	−	0.908098i	$-0.637535\pi$
$997$	−26.4448	−0.837517	−0.418758	+	0.908098i	$0.637535\pi$
$998$	0	0
$999$	93.5250	2.95900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.cp.1.1		5
4.3	odd	2		7360.2.a.co.1.5		5
8.3	odd	2		920.2.a.j.1.1	✓	5
8.5	even	2		1840.2.a.v.1.5		5
24.11	even	2		8280.2.a.bs.1.3		5
40.3	even	4		4600.2.e.u.4049.1		10
40.19	odd	2		4600.2.a.be.1.5		5
40.27	even	4		4600.2.e.u.4049.10		10
40.29	even	2		9200.2.a.cu.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.1	✓	5	8.3	odd	2
1840.2.a.v.1.5		5	8.5	even	2
4600.2.a.be.1.5		5	40.19	odd	2
4600.2.e.u.4049.1		10	40.3	even	4
4600.2.e.u.4049.10		10	40.27	even	4
7360.2.a.co.1.5		5	4.3	odd	2
7360.2.a.cp.1.1		5	1.1	even	1	trivial
8280.2.a.bs.1.3		5	24.11	even	2
9200.2.a.cu.1.1		5	40.29	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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q-3.36002 q^{3} +1.00000 q^{5} +1.90754 q^{7} +8.28974 q^{9} +O(q^{10})\)	\(q-3.36002 q^{3} +1.00000 q^{5} +1.90754 q^{7} +8.28974 q^{9} -5.48021 q^{11} +1.04937 q^{13} -3.36002 q^{15} -6.74222 q^{17} -1.55049 q^{19} -6.40939 q^{21} +1.00000 q^{23} +1.00000 q^{25} -17.7736 q^{27} +3.38219 q^{29} -10.9327 q^{31} +18.4136 q^{33} +1.90754 q^{35} -5.26201 q^{37} -3.52589 q^{39} +6.09801 q^{41} +8.28974 q^{45} -0.403830 q^{47} -3.36128 q^{49} +22.6540 q^{51} -5.88332 q^{53} -5.48021 q^{55} +5.20968 q^{57} +9.60111 q^{59} +7.09927 q^{61} +15.8130 q^{63} +1.04937 q^{65} +13.7971 q^{67} -3.36002 q^{69} -0.478950 q^{71} +2.40383 q^{73} -3.36002 q^{75} -10.4537 q^{77} +4.24037 q^{79} +34.8505 q^{81} -11.2620 q^{83} -6.74222 q^{85} -11.3642 q^{87} +4.90495 q^{89} +2.00171 q^{91} +36.7340 q^{93} -1.55049 q^{95} -12.3433 q^{97} -45.4295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 5 * q + 5 * q^5 + 2 * q^7 + 13 * q^9	\( 5 q + 5 q^{5} + 2 q^{7} + 13 q^{9} - q^{11} - 4 q^{13} + 4 q^{17} + 7 q^{19} - 6 q^{21} + 5 q^{23} + 5 q^{25} + 3 q^{27} - 4 q^{29} - 19 q^{31} + 17 q^{33} + 2 q^{35} - 15 q^{37} - 19 q^{39} + 25 q^{41} + 13 q^{45} + 11 q^{47} + 25 q^{49} + 19 q^{51} - 3 q^{53} - q^{55} + 48 q^{57} - q^{59} + 5 q^{61} + 41 q^{63} - 4 q^{65} + 9 q^{67} - q^{71} - q^{73} - 18 q^{77} + 2 q^{79} + 57 q^{81} - 45 q^{83} + 4 q^{85} + 9 q^{87} + 6 q^{89} + 11 q^{91} + 39 q^{93} + 7 q^{95} + 25 q^{97} - 65 q^{99}+O(q^{100}) \) 5 * q + 5 * q^5 + 2 * q^7 + 13 * q^9 - q^11 - 4 * q^13 + 4 * q^17 + 7 * q^19 - 6 * q^21 + 5 * q^23 + 5 * q^25 + 3 * q^27 - 4 * q^29 - 19 * q^31 + 17 * q^33 + 2 * q^35 - 15 * q^37 - 19 * q^39 + 25 * q^41 + 13 * q^45 + 11 * q^47 + 25 * q^49 + 19 * q^51 - 3 * q^53 - q^55 + 48 * q^57 - q^59 + 5 * q^61 + 41 * q^63 - 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 + 2 * q^79 + 57 * q^81 - 45 * q^83 + 4 * q^85 + 9 * q^87 + 6 * q^89 + 11 * q^91 + 39 * q^93 + 7 * q^95 + 25 * q^97 - 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more