LMFDB - Embedded newform 8280.2.a.bs.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
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:	$\Z[a_1, \ldots, a_{17}]$
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			1.3
Root			$-3.36002$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.90754 q^{7} +5.48021 q^{11} -1.04937 q^{13} +6.74222 q^{17} -1.55049 q^{19} +1.00000 q^{23} +1.00000 q^{25} +3.38219 q^{29} +10.9327 q^{31} -1.90754 q^{35} +5.26201 q^{37} -6.09801 q^{41} -0.403830 q^{47} -3.36128 q^{49} -5.88332 q^{53} +5.48021 q^{55} -9.60111 q^{59} -7.09927 q^{61} -1.04937 q^{65} +13.7971 q^{67} -0.478950 q^{71} +2.40383 q^{73} -10.4537 q^{77} -4.24037 q^{79} +11.2620 q^{83} +6.74222 q^{85} -4.90495 q^{89} +2.00171 q^{91} -1.55049 q^{95} -12.3433 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} - 2 q^{7} + q^{11} + 4 q^{13} - 4 q^{17} + 7 q^{19} + 5 q^{23} + 5 q^{25} - 4 q^{29} + 19 q^{31} - 2 q^{35} + 15 q^{37} - 25 q^{41} + 11 q^{47} + 25 q^{49} - 3 q^{53} + q^{55} + q^{59}+ \cdots + 25 q^{97}+O(q^{100}) $ 5 * q + 5 * q^5 - 2 * q^7 + q^11 + 4 * q^13 - 4 * q^17 + 7 * q^19 + 5 * q^23 + 5 * q^25 - 4 * q^29 + 19 * q^31 - 2 * q^35 + 15 * q^37 - 25 * q^41 + 11 * q^47 + 25 * q^49 - 3 * q^53 + q^55 + q^59 - 5 * q^61 + 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 - 2 * q^79 + 45 * q^83 - 4 * q^85 - 6 * q^89 + 11 * q^91 + 7 * q^95 + 25 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.90754	−0.720984	−0.360492	−	0.932762i	$-0.617391\pi$
$7$	−1.90754	−0.720984	−0.360492	+	0.932762i	$0.617391\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.48021	1.65234	0.826172	−	0.563418i	$-0.190514\pi$
$11$	5.48021	1.65234	0.826172	+	0.563418i	$0.190514\pi$
$12$	0	0
$13$	−1.04937	−0.291041	−0.145521	−	0.989355i	$-0.546486\pi$
$13$	−1.04937	−0.291041	−0.145521	+	0.989355i	$0.546486\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.74222	1.63523	0.817614	−	0.575767i	$-0.195297\pi$
$17$	6.74222	1.63523	0.817614	+	0.575767i	$0.195297\pi$
$18$	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
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	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	$-0.0608489\pi$
$31$	10.9327	1.96357	0.981784	+	0.190000i	$0.0608489\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.90754	−0.322434
$36$	0	0
$37$	5.26201	0.865069	0.432534	−	0.901617i	$-0.357619\pi$
$37$	5.26201	0.865069	0.432534	+	0.901617i	$0.357619\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.09801	−0.952350	−0.476175	−	0.879351i	$-0.657977\pi$
$41$	−6.09801	−0.952350	−0.476175	+	0.879351i	$0.657977\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	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
$57$	0	0
$58$	0	0
$59$	−9.60111	−1.24996	−0.624979	−	0.780641i	$-0.714893\pi$
$59$	−9.60111	−1.24996	−0.624979	+	0.780641i	$0.714893\pi$
$60$	0	0
$61$	−7.09927	−0.908968	−0.454484	−	0.890755i	$-0.650176\pi$
$61$	−7.09927	−0.908968	−0.454484	+	0.890755i	$0.650176\pi$
$62$	0	0
$63$	0	0
$64$	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
$69$	0	0
$70$	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
$75$	0	0
$76$	0	0
$77$	−10.4537	−1.19131
$78$	0	0
$79$	−4.24037	−0.477079	−0.238540	−	0.971133i	$-0.576669\pi$
$79$	−4.24037	−0.477079	−0.238540	+	0.971133i	$0.576669\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.2620	1.23617	0.618083	−	0.786113i	$-0.287910\pi$
$83$	11.2620	1.23617	0.618083	+	0.786113i	$0.287910\pi$
$84$	0	0
$85$	6.74222	0.731296
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.90495	−0.519924	−0.259962	−	0.965619i	$-0.583710\pi$
$89$	−4.90495	−0.519924	−0.259962	+	0.965619i	$0.583710\pi$
$90$	0	0
$91$	2.00171	0.209836
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	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	$-0.624936\pi$
$103$	−7.76385	−0.764995	−0.382497	+	0.923957i	$0.624936\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.54197	0.439089	0.219544	−	0.975603i	$-0.429543\pi$
$107$	4.54197	0.439089	0.219544	+	0.975603i	$0.429543\pi$
$108$	0	0
$109$	12.4136	1.18901	0.594504	−	0.804093i	$-0.297348\pi$
$109$	12.4136	1.18901	0.594504	+	0.804093i	$0.297348\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.27312	−0.872342	−0.436171	−	0.899864i	$-0.643666\pi$
$113$	−9.27312	−0.872342	−0.436171	+	0.899864i	$0.643666\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.8611	−1.17897
$120$	0	0
$121$	19.0327	1.73024
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.4149	1.10165	0.550824	−	0.834621i	$-0.314314\pi$
$127$	12.4149	1.10165	0.550824	+	0.834621i	$0.314314\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.55434	−0.834766	−0.417383	−	0.908731i	$-0.637053\pi$
$131$	−9.55434	−0.834766	−0.417383	+	0.908731i	$0.637053\pi$
$132$	0	0
$133$	2.95763	0.256459
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.36558	0.287541	0.143770	−	0.989611i	$-0.454077\pi$
$137$	3.36558	0.287541	0.143770	+	0.989611i	$0.454077\pi$
$138$	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
$141$	0	0
$142$	0	0
$143$	−5.75074	−0.480901
$144$	0	0
$145$	3.38219	0.280876
$146$	0	0
$147$	0	0
$148$	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	$-0.0803349\pi$
$151$	23.7979	1.93664	0.968321	+	0.249709i	$0.0803349\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.9327	0.878134
$156$	0	0
$157$	13.3175	1.06285	0.531425	−	0.847105i	$-0.321657\pi$
$157$	13.3175	1.06285	0.531425	+	0.847105i	$0.321657\pi$
$158$	0	0
$159$	0	0
$160$	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
$165$	0	0
$166$	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
$171$	0	0
$172$	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
$177$	0	0
$178$	0	0
$179$	−9.88683	−0.738976	−0.369488	−	0.929236i	$-0.620467\pi$
$179$	−9.88683	−0.738976	−0.369488	+	0.929236i	$0.620467\pi$
$180$	0	0
$181$	−23.9883	−1.78304	−0.891519	−	0.452983i	$-0.850360\pi$
$181$	−23.9883	−1.78304	−0.891519	+	0.452983i	$0.850360\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.26201	0.386871
$186$	0	0
$187$	36.9487	2.70196
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	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	$-0.626322\pi$
$199$	−10.9050	−0.773032	−0.386516	+	0.922283i	$0.626322\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.45168	−0.452819
$204$	0	0
$205$	−6.09801	−0.425904
$206$	0	0
$207$	0	0
$208$	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
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.8546	−1.41570
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.07505	−0.475919
$222$	0	0
$223$	14.1542	0.947835	0.473917	−	0.880569i	$-0.342840\pi$
$223$	14.1542	0.947835	0.473917	+	0.880569i	$0.342840\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.89902	0.325159	0.162580	−	0.986695i	$-0.448019\pi$
$227$	4.89902	0.325159	0.162580	+	0.986695i	$0.448019\pi$
$228$	0	0
$229$	12.3946	0.819056	0.409528	−	0.912298i	$-0.365693\pi$
$229$	12.3946	0.819056	0.409528	+	0.912298i	$0.365693\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.882062	−0.0577858	−0.0288929	−	0.999583i	$-0.509198\pi$
$233$	−0.882062	−0.0577858	−0.0288929	+	0.999583i	$0.509198\pi$
$234$	0	0
$235$	−0.403830	−0.0263429
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	−3.36128	−0.214744
$246$	0	0
$247$	1.62703	0.103525
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−17.9883	−1.13541	−0.567706	−	0.823231i	$-0.692169\pi$
$251$	−17.9883	−1.13541	−0.567706	+	0.823231i	$0.692169\pi$
$252$	0	0
$253$	5.48021	0.344538
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.1705	0.883930	0.441965	−	0.897032i	$-0.354282\pi$
$257$	14.1705	0.883930	0.441965	+	0.897032i	$0.354282\pi$
$258$	0	0
$259$	−10.0375	−0.623701
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	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	$-0.360711\pi$
$271$	13.9519	0.847517	0.423759	+	0.905775i	$0.360711\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.48021	0.330469
$276$	0	0
$277$	3.69490	0.222005	0.111003	−	0.993820i	$-0.464594\pi$
$277$	3.69490	0.222005	0.111003	+	0.993820i	$0.464594\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.9582	−0.951984	−0.475992	−	0.879450i	$-0.657911\pi$
$281$	−15.9582	−0.951984	−0.475992	+	0.879450i	$0.657911\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	11.6322	0.686628
$288$	0	0
$289$	28.4575	1.67397
$290$	0	0
$291$	0	0
$292$	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
$297$	0	0
$298$	0	0
$299$	−1.04937	−0.0606863
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	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
$309$	0	0
$310$	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
$315$	0	0
$316$	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
$321$	0	0
$322$	0	0
$323$	−10.4537	−0.581661
$324$	0	0
$325$	−1.04937	−0.0582083
$326$	0	0
$327$	0	0
$328$	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
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	59.9134	3.24449
$342$	0	0
$343$	19.7646	1.06719
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.9747	−1.34071	−0.670355	−	0.742040i	$-0.733858\pi$
$347$	−24.9747	−1.34071	−0.670355	+	0.742040i	$0.733858\pi$
$348$	0	0
$349$	−35.3019	−1.88967	−0.944835	−	0.327547i	$-0.893778\pi$
$349$	−35.3019	−1.88967	−0.944835	+	0.327547i	$0.893778\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.0326	0.533980	0.266990	−	0.963699i	$-0.413971\pi$
$353$	10.0326	0.533980	0.266990	+	0.963699i	$0.413971\pi$
$354$	0	0
$355$	−0.478950	−0.0254200
$356$	0	0
$357$	0	0
$358$	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
$363$	0	0
$364$	0	0
$365$	2.40383	0.125822
$366$	0	0
$367$	33.3234	1.73947	0.869734	−	0.493521i	$-0.164291\pi$
$367$	33.3234	1.73947	0.869734	+	0.493521i	$0.164291\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.2227	0.582653
$372$	0	0
$373$	4.62023	0.239227	0.119613	−	0.992821i	$-0.461835\pi$
$373$	4.62023	0.239227	0.119613	+	0.992821i	$0.461835\pi$
$374$	0	0
$375$	0	0
$376$	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
$381$	0	0
$382$	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
$393$	0	0
$394$	0	0
$395$	−4.24037	−0.213356
$396$	0	0
$397$	5.26722	0.264354	0.132177	−	0.991226i	$-0.457803\pi$
$397$	5.26722	0.264354	0.132177	+	0.991226i	$0.457803\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.1490	−1.30582	−0.652910	−	0.757436i	$-0.726452\pi$
$401$	−26.1490	−1.30582	−0.652910	+	0.757436i	$0.726452\pi$
$402$	0	0
$403$	−11.4724	−0.571480
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	18.3145	0.901200
$414$	0	0
$415$	11.2620	0.552830
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.10616	−0.0540394	−0.0270197	−	0.999635i	$-0.508602\pi$
$419$	−1.10616	−0.0540394	−0.0270197	+	0.999635i	$0.508602\pi$
$420$	0	0
$421$	9.16362	0.446607	0.223304	−	0.974749i	$-0.428316\pi$
$421$	9.16362	0.446607	0.223304	+	0.974749i	$0.428316\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.74222	0.327045
$426$	0	0
$427$	13.5422	0.655351
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	0	0
$437$	−1.55049	−0.0741700
$438$	0	0
$439$	−15.3093	−0.730673	−0.365336	−	0.930876i	$-0.619046\pi$
$439$	−15.3093	−0.730673	−0.365336	+	0.930876i	$0.619046\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.24129	0.153998	0.0769992	−	0.997031i	$-0.475466\pi$
$443$	3.24129	0.153998	0.0769992	+	0.997031i	$0.475466\pi$
$444$	0	0
$445$	−4.90495	−0.232517
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.02215	0.425782	0.212891	−	0.977076i	$-0.431712\pi$
$449$	9.02215	0.425782	0.212891	+	0.977076i	$0.431712\pi$
$450$	0	0
$451$	−33.4184	−1.57361
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	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	$-0.428265\pi$
$463$	9.61655	0.446919	0.223459	+	0.974713i	$0.428265\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.4764	−1.13263	−0.566317	−	0.824188i	$-0.691632\pi$
$467$	−24.4764	−1.13263	−0.566317	+	0.824188i	$0.691632\pi$
$468$	0	0
$469$	−26.3186	−1.21528
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.55049	−0.0711413
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	−12.3433	−0.560482
$486$	0	0
$487$	−23.7078	−1.07431	−0.537153	−	0.843485i	$-0.680500\pi$
$487$	−23.7078	−1.07431	−0.537153	+	0.843485i	$0.680500\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18.5930	−0.839091	−0.419546	−	0.907734i	$-0.637811\pi$
$491$	−18.5930	−0.839091	−0.419546	+	0.907734i	$0.637811\pi$
$492$	0	0
$493$	22.8035	1.02702
$494$	0	0
$495$	0	0
$496$	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
$501$	0	0
$502$	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
$507$	0	0
$508$	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
$513$	0	0
$514$	0	0
$515$	−7.76385	−0.342116
$516$	0	0
$517$	−2.21307	−0.0973307
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.5488	−1.16312	−0.581561	−	0.813503i	$-0.697558\pi$
$521$	−26.5488	−1.16312	−0.581561	+	0.813503i	$0.697558\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	73.7105	3.21088
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.39904	0.277173
$534$	0	0
$535$	4.54197	0.196366
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18.4205	−0.793427
$540$	0	0
$541$	22.4123	0.963579	0.481790	−	0.876287i	$-0.339987\pi$
$541$	22.4123	0.963579	0.481790	+	0.876287i	$0.339987\pi$
$542$	0	0
$543$	0	0
$544$	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
$549$	0	0
$550$	0	0
$551$	−5.24406	−0.223404
$552$	0	0
$553$	8.08870	0.343966
$554$	0	0
$555$	0	0
$556$	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
$561$	0	0
$562$	0	0
$563$	−10.6629	−0.449389	−0.224694	−	0.974429i	$-0.572138\pi$
$563$	−10.6629	−0.449389	−0.224694	+	0.974429i	$0.572138\pi$
$564$	0	0
$565$	−9.27312	−0.390123
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.8986	−1.33726	−0.668630	−	0.743596i	$-0.733119\pi$
$569$	−31.8986	−1.33726	−0.668630	+	0.743596i	$0.733119\pi$
$570$	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
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	−21.4828	−0.891256
$582$	0	0
$583$	−32.2418	−1.33532
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.9321	0.946508	0.473254	−	0.880926i	$-0.343079\pi$
$587$	22.9321	0.946508	0.473254	+	0.880926i	$0.343079\pi$
$588$	0	0
$589$	−16.9510	−0.698454
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−27.6250	−1.13442	−0.567211	−	0.823572i	$-0.691978\pi$
$593$	−27.6250	−1.13442	−0.567211	+	0.823572i	$0.691978\pi$
$594$	0	0
$595$	−12.8611	−0.527252
$596$	0	0
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	19.0327	0.773788
$606$	0	0
$607$	5.52878	0.224406	0.112203	−	0.993685i	$-0.464209\pi$
$607$	5.52878	0.224406	0.112203	+	0.993685i	$0.464209\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.423765	0.0171437
$612$	0	0
$613$	−21.1205	−0.853050	−0.426525	−	0.904476i	$-0.640262\pi$
$613$	−21.1205	−0.853050	−0.426525	+	0.904476i	$0.640262\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.6523	−0.791174	−0.395587	−	0.918429i	$-0.629459\pi$
$617$	−19.6523	−0.791174	−0.395587	+	0.918429i	$0.629459\pi$
$618$	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
$621$	0	0
$622$	0	0
$623$	9.35641	0.374857
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	35.4776	1.41458
$630$	0	0
$631$	−45.5595	−1.81369	−0.906847	−	0.421461i	$-0.861517\pi$
$631$	−45.5595	−1.81369	−0.906847	+	0.421461i	$0.861517\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.4149	0.492672
$636$	0	0
$637$	3.52721	0.139753
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.97997	−0.354688	−0.177344	−	0.984149i	$-0.556750\pi$
$641$	−8.97997	−0.354688	−0.177344	+	0.984149i	$0.556750\pi$
$642$	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
$651$	0	0
$652$	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
$657$	0	0
$658$	0	0
$659$	−20.9256	−0.815145	−0.407573	−	0.913173i	$-0.633625\pi$
$659$	−20.9256	−0.815145	−0.407573	+	0.913173i	$0.633625\pi$
$660$	0	0
$661$	5.25688	0.204469	0.102234	−	0.994760i	$-0.467401\pi$
$661$	5.25688	0.204469	0.102234	+	0.994760i	$0.467401\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.95763	0.114692
$666$	0	0
$667$	3.38219	0.130959
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	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
$681$	0	0
$682$	0	0
$683$	50.3044	1.92484	0.962422	−	0.271559i	$-0.0875391\pi$
$683$	50.3044	1.92484	0.962422	+	0.271559i	$0.0875391\pi$
$684$	0	0
$685$	3.36558	0.128592
$686$	0	0
$687$	0	0
$688$	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
$693$	0	0
$694$	0	0
$695$	−20.7361	−0.786565
$696$	0	0
$697$	−41.1141	−1.55731
$698$	0	0
$699$	0	0
$700$	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
$705$	0	0
$706$	0	0
$707$	−2.76006	−0.103803
$708$	0	0
$709$	11.9069	0.447174	0.223587	−	0.974684i	$-0.428223\pi$
$709$	11.9069	0.447174	0.223587	+	0.974684i	$0.428223\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.9327	0.409432
$714$	0	0
$715$	−5.75074	−0.215065
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	3.38219	0.125612
$726$	0	0
$727$	11.4034	0.422928	0.211464	−	0.977386i	$-0.432177\pi$
$727$	11.4034	0.422928	0.211464	+	0.977386i	$0.432177\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	35.8722	1.32497	0.662484	−	0.749076i	$-0.269502\pi$
$733$	35.8722	1.32497	0.662484	+	0.749076i	$0.269502\pi$
$734$	0	0
$735$	0	0
$736$	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
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	−8.66400	−0.316576
$750$	0	0
$751$	−2.80173	−0.102236	−0.0511182	−	0.998693i	$-0.516279\pi$
$751$	−2.80173	−0.102236	−0.0511182	+	0.998693i	$0.516279\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23.7979	0.866093
$756$	0	0
$757$	−41.6710	−1.51456	−0.757278	−	0.653092i	$-0.773471\pi$
$757$	−41.6710	−1.51456	−0.757278	+	0.653092i	$0.773471\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.4285	1.28428	0.642141	−	0.766587i	$-0.278046\pi$
$761$	35.4285	1.28428	0.642141	+	0.766587i	$0.278046\pi$
$762$	0	0
$763$	−23.6795	−0.857255
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	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
$777$	0	0
$778$	0	0
$779$	9.45490	0.338757
$780$	0	0
$781$	−2.62475	−0.0939208
$782$	0	0
$783$	0	0
$784$	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
$789$	0	0
$790$	0	0
$791$	17.6889	0.628944
$792$	0	0
$793$	7.44973	0.264547
$794$	0	0
$795$	0	0
$796$	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
$801$	0	0
$802$	0	0
$803$	13.1735	0.464882
$804$	0	0
$805$	−1.90754	−0.0672321
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.4608	1.14126	0.570630	−	0.821207i	$-0.306699\pi$
$809$	32.4608	1.14126	0.570630	+	0.821207i	$0.306699\pi$
$810$	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
$813$	0	0
$814$	0	0
$815$	16.1529	0.565810
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	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	$-0.728336\pi$
$823$	−37.7179	−1.31476	−0.657381	+	0.753558i	$0.728336\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.89736	0.309392	0.154696	−	0.987962i	$-0.450560\pi$
$827$	8.89736	0.309392	0.154696	+	0.987962i	$0.450560\pi$
$828$	0	0
$829$	30.9058	1.07340	0.536702	−	0.843772i	$-0.319670\pi$
$829$	30.9058	1.07340	0.536702	+	0.843772i	$0.319670\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−22.6625	−0.785208
$834$	0	0
$835$	20.6782	0.715599
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	0	0
$845$	−11.8988	−0.409332
$846$	0	0
$847$	−36.3056	−1.24748
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.26201	0.180379
$852$	0	0
$853$	−26.3940	−0.903713	−0.451857	−	0.892091i	$-0.649238\pi$
$853$	−26.3940	−0.903713	−0.451857	+	0.892091i	$0.649238\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.5048	0.597953	0.298976	−	0.954260i	$-0.403355\pi$
$857$	17.5048	0.597953	0.298976	+	0.954260i	$0.403355\pi$
$858$	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
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	−23.2381	−0.788299
$870$	0	0
$871$	−14.4782	−0.490576
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.90754	−0.0644867
$876$	0	0
$877$	19.9574	0.673912	0.336956	−	0.941520i	$-0.390603\pi$
$877$	19.9574	0.673912	0.336956	+	0.941520i	$0.390603\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.7297	0.698402	0.349201	−	0.937048i	$-0.386453\pi$
$881$	20.7297	0.698402	0.349201	+	0.937048i	$0.386453\pi$
$882$	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
$885$	0	0
$886$	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
$891$	0	0
$892$	0	0
$893$	0.626134	0.0209528
$894$	0	0
$895$	−9.88683	−0.330480
$896$	0	0
$897$	0	0
$898$	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
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	18.2253	0.601853
$918$	0	0
$919$	−24.2758	−0.800785	−0.400392	−	0.916344i	$-0.631126\pi$
$919$	−24.2758	−0.800785	−0.400392	+	0.916344i	$0.631126\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.502594	0.0165431
$924$	0	0
$925$	5.26201	0.173014
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.18842	−0.268653	−0.134327	−	0.990937i	$-0.542887\pi$
$929$	−8.18842	−0.268653	−0.134327	+	0.990937i	$0.542887\pi$
$930$	0	0
$931$	5.21163	0.170804
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	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
$945$	0	0
$946$	0	0
$947$	15.9026	0.516765	0.258383	−	0.966043i	$-0.416810\pi$
$947$	15.9026	0.516765	0.258383	+	0.966043i	$0.416810\pi$
$948$	0	0
$949$	−2.52249	−0.0818836
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−20.9451	−0.678478	−0.339239	−	0.940700i	$-0.610169\pi$
$953$	−20.9451	−0.678478	−0.339239	+	0.940700i	$0.610169\pi$
$954$	0	0
$955$	−6.72004	−0.217455
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.41998	−0.207312
$960$	0	0
$961$	88.5236	2.85560
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.95007	0.288113
$966$	0	0
$967$	−14.0732	−0.452563	−0.226281	−	0.974062i	$-0.572657\pi$
$967$	−14.0732	−0.452563	−0.226281	+	0.974062i	$0.572657\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.4404	0.367139	0.183569	−	0.983007i	$-0.441235\pi$
$971$	11.4404	0.367139	0.183569	+	0.983007i	$0.441235\pi$
$972$	0	0
$973$	39.5550	1.26808
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.9119	−1.40487	−0.702433	−	0.711750i	$-0.747903\pi$
$977$	−43.9119	−1.40487	−0.702433	+	0.711750i	$0.747903\pi$
$978$	0	0
$979$	−26.8802	−0.859094
$980$	0	0
$981$	0	0
$982$	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
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	43.5288	1.38274	0.691369	−	0.722501i	$-0.257008\pi$
$991$	43.5288	1.38274	0.691369	+	0.722501i	$0.257008\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.9050	−0.345710
$996$	0	0
$997$	26.4448	0.837517	0.418758	−	0.908098i	$-0.362465\pi$
$997$	26.4448	0.837517	0.418758	+	0.908098i	$0.362465\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bs.1.3		5
3.2	odd	2		920.2.a.j.1.1	✓	5
12.11	even	2		1840.2.a.v.1.5		5
15.2	even	4		4600.2.e.u.4049.10		10
15.8	even	4		4600.2.e.u.4049.1		10
15.14	odd	2		4600.2.a.be.1.5		5
24.5	odd	2		7360.2.a.co.1.5		5
24.11	even	2		7360.2.a.cp.1.1		5
60.59	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	3.2	odd	2
1840.2.a.v.1.5		5	12.11	even	2
4600.2.a.be.1.5		5	15.14	odd	2
4600.2.e.u.4049.1		10	15.8	even	4
4600.2.e.u.4049.10		10	15.2	even	4
7360.2.a.co.1.5		5	24.5	odd	2
7360.2.a.cp.1.1		5	24.11	even	2
8280.2.a.bs.1.3		5	1.1	even	1	trivial
9200.2.a.cu.1.1		5	60.59	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.90754 q^{7} +5.48021 q^{11} -1.04937 q^{13} +6.74222 q^{17} -1.55049 q^{19} +1.00000 q^{23} +1.00000 q^{25} +3.38219 q^{29} +10.9327 q^{31} -1.90754 q^{35} +5.26201 q^{37} -6.09801 q^{41} -0.403830 q^{47} -3.36128 q^{49} -5.88332 q^{53} +5.48021 q^{55} -9.60111 q^{59} -7.09927 q^{61} -1.04937 q^{65} +13.7971 q^{67} -0.478950 q^{71} +2.40383 q^{73} -10.4537 q^{77} -4.24037 q^{79} +11.2620 q^{83} +6.74222 q^{85} -4.90495 q^{89} +2.00171 q^{91} -1.55049 q^{95} -12.3433 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} - 2 q^{7} + q^{11} + 4 q^{13} - 4 q^{17} + 7 q^{19} + 5 q^{23} + 5 q^{25} - 4 q^{29} + 19 q^{31} - 2 q^{35} + 15 q^{37} - 25 q^{41} + 11 q^{47} + 25 q^{49} - 3 q^{53} + q^{55} + q^{59}+ \cdots + 25 q^{97}+O(q^{100}) \) 5 * q + 5 * q^5 - 2 * q^7 + q^11 + 4 * q^13 - 4 * q^17 + 7 * q^19 + 5 * q^23 + 5 * q^25 - 4 * q^29 + 19 * q^31 - 2 * q^35 + 15 * q^37 - 25 * q^41 + 11 * q^47 + 25 * q^49 - 3 * q^53 + q^55 + q^59 - 5 * q^61 + 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 - 2 * q^79 + 45 * q^83 - 4 * q^85 - 6 * q^89 + 11 * q^91 + 7 * q^95 + 25 * q^97

Introduction

L-functions

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