LMFDB - Embedded newform 7360.2.a.ch.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:	$4$
Coefficient field:	4.4.21208.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 13x - 3 $ x^4 - x^3 - 8x^2 + 13x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3680)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.07664$ 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)

$f(q)$	$=$	$q-2.58604 q^{3} +1.00000 q^{5} -2.07664 q^{7} +3.68758 q^{9} +O(q^{10})$	$q-2.58604 q^{3} +1.00000 q^{5} -2.07664 q^{7} +3.68758 q^{9} +3.19698 q^{11} -5.95629 q^{13} -2.58604 q^{15} -6.46569 q^{17} +6.07664 q^{19} +5.37026 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.77811 q^{27} -3.87966 q^{29} -7.56233 q^{31} -8.26751 q^{33} -2.07664 q^{35} -2.64387 q^{37} +15.4032 q^{39} -0.979997 q^{41} +5.66267 q^{43} +3.68758 q^{45} -4.86086 q^{47} -2.68758 q^{49} +16.7205 q^{51} +8.53742 q^{53} +3.19698 q^{55} -15.7144 q^{57} -3.13447 q^{59} +3.38785 q^{61} -7.65777 q^{63} -5.95629 q^{65} +1.11544 q^{67} -2.58604 q^{69} +9.85475 q^{71} -12.3303 q^{73} -2.58604 q^{75} -6.63897 q^{77} +0.874749 q^{79} -6.46449 q^{81} -12.5096 q^{83} -6.46569 q^{85} +10.0329 q^{87} +10.4506 q^{89} +12.3691 q^{91} +19.5565 q^{93} +6.07664 q^{95} +5.54353 q^{97} +11.7891 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^5 + 5 * q^7 + 9 * q^9	$ 4 q - q^{3} + 4 q^{5} + 5 q^{7} + 9 q^{9} + 11 q^{11} + q^{13} - q^{15} - 5 q^{17} + 11 q^{19} + 6 q^{21} + 4 q^{23} + 4 q^{25} + 8 q^{27} - 4 q^{29} - 9 q^{31} - 9 q^{33} + 5 q^{35} + 20 q^{37} + 8 q^{39} + 9 q^{41} + 9 q^{45} - 5 q^{49} + 23 q^{51} - 4 q^{53} + 11 q^{55} - 10 q^{57} + 22 q^{59} + q^{61} - 4 q^{63} + q^{65} + 12 q^{67} - q^{69} + 11 q^{71} - 8 q^{73} - q^{75} + 8 q^{77} - 12 q^{79} - 4 q^{83} - 5 q^{85} + 2 q^{87} + 12 q^{89} + 29 q^{91} - 2 q^{93} + 11 q^{95} - 7 q^{97} + 45 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 + 5 * q^7 + 9 * q^9 + 11 * q^11 + q^13 - q^15 - 5 * q^17 + 11 * q^19 + 6 * q^21 + 4 * q^23 + 4 * q^25 + 8 * q^27 - 4 * q^29 - 9 * q^31 - 9 * q^33 + 5 * q^35 + 20 * q^37 + 8 * q^39 + 9 * q^41 + 9 * q^45 - 5 * q^49 + 23 * q^51 - 4 * q^53 + 11 * q^55 - 10 * q^57 + 22 * q^59 + q^61 - 4 * q^63 + q^65 + 12 * q^67 - q^69 + 11 * q^71 - 8 * q^73 - q^75 + 8 * q^77 - 12 * q^79 - 4 * q^83 - 5 * q^85 + 2 * q^87 + 12 * q^89 + 29 * q^91 - 2 * q^93 + 11 * q^95 - 7 * q^97 + 45 * 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
$3$	−2.58604	−1.49305	−0.746524	−	0.665358i	$-0.768279\pi$
$3$	−2.58604	−1.49305	−0.746524	+	0.665358i	$0.768279\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.07664	−0.784895	−0.392447	−	0.919774i	$-0.628371\pi$
$7$	−2.07664	−0.784895	−0.392447	+	0.919774i	$0.628371\pi$
$8$	0	0
$9$	3.68758	1.22919
$10$	0	0
$11$	3.19698	0.963926	0.481963	−	0.876192i	$-0.339924\pi$
$11$	3.19698	0.963926	0.481963	+	0.876192i	$0.339924\pi$
$12$	0	0
$13$	−5.95629	−1.65198	−0.825989	−	0.563686i	$-0.809383\pi$
$13$	−5.95629	−1.65198	−0.825989	+	0.563686i	$0.809383\pi$
$14$	0	0
$15$	−2.58604	−0.667712
$16$	0	0
$17$	−6.46569	−1.56816	−0.784080	−	0.620659i	$-0.786865\pi$
$17$	−6.46569	−1.56816	−0.784080	+	0.620659i	$0.786865\pi$
$18$	0	0
$19$	6.07664	1.39408	0.697038	−	0.717034i	$-0.254501\pi$
$19$	6.07664	1.39408	0.697038	+	0.717034i	$0.254501\pi$
$20$	0	0
$21$	5.37026	1.17189
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.77811	−0.342197
$28$	0	0
$29$	−3.87966	−0.720434	−0.360217	−	0.932869i	$-0.617297\pi$
$29$	−3.87966	−0.720434	−0.360217	+	0.932869i	$0.617297\pi$
$30$	0	0
$31$	−7.56233	−1.35823	−0.679117	−	0.734030i	$-0.737637\pi$
$31$	−7.56233	−1.35823	−0.679117	+	0.734030i	$0.737637\pi$
$32$	0	0
$33$	−8.26751	−1.43919
$34$	0	0
$35$	−2.07664	−0.351016
$36$	0	0
$37$	−2.64387	−0.434650	−0.217325	−	0.976099i	$-0.569733\pi$
$37$	−2.64387	−0.434650	−0.217325	+	0.976099i	$0.569733\pi$
$38$	0	0
$39$	15.4032	2.46648
$40$	0	0
$41$	−0.979997	−0.153050	−0.0765249	−	0.997068i	$-0.524382\pi$
$41$	−0.979997	−0.153050	−0.0765249	+	0.997068i	$0.524382\pi$
$42$	0	0
$43$	5.66267	0.863549	0.431775	−	0.901981i	$-0.357888\pi$
$43$	5.66267	0.863549	0.431775	+	0.901981i	$0.357888\pi$
$44$	0	0
$45$	3.68758	0.549712
$46$	0	0
$47$	−4.86086	−0.709029	−0.354514	−	0.935051i	$-0.615354\pi$
$47$	−4.86086	−0.709029	−0.354514	+	0.935051i	$0.615354\pi$
$48$	0	0
$49$	−2.68758	−0.383940
$50$	0	0
$51$	16.7205	2.34134
$52$	0	0
$53$	8.53742	1.17270	0.586352	−	0.810056i	$-0.300563\pi$
$53$	8.53742	1.17270	0.586352	+	0.810056i	$0.300563\pi$
$54$	0	0
$55$	3.19698	0.431081
$56$	0	0
$57$	−15.7144	−2.08142
$58$	0	0
$59$	−3.13447	−0.408074	−0.204037	−	0.978963i	$-0.565406\pi$
$59$	−3.13447	−0.408074	−0.204037	+	0.978963i	$0.565406\pi$
$60$	0	0
$61$	3.38785	0.433770	0.216885	−	0.976197i	$-0.430410\pi$
$61$	3.38785	0.433770	0.216885	+	0.976197i	$0.430410\pi$
$62$	0	0
$63$	−7.65777	−0.964788
$64$	0	0
$65$	−5.95629	−0.738787
$66$	0	0
$67$	1.11544	0.136272	0.0681362	−	0.997676i	$-0.478295\pi$
$67$	1.11544	0.136272	0.0681362	+	0.997676i	$0.478295\pi$
$68$	0	0
$69$	−2.58604	−0.311322
$70$	0	0
$71$	9.85475	1.16954	0.584772	−	0.811198i	$-0.301184\pi$
$71$	9.85475	1.16954	0.584772	+	0.811198i	$0.301184\pi$
$72$	0	0
$73$	−12.3303	−1.44315	−0.721573	−	0.692338i	$-0.756581\pi$
$73$	−12.3303	−1.44315	−0.721573	+	0.692338i	$0.756581\pi$
$74$	0	0
$75$	−2.58604	−0.298610
$76$	0	0
$77$	−6.63897	−0.756580
$78$	0	0
$79$	0.874749	0.0984169	0.0492085	−	0.998789i	$-0.484330\pi$
$79$	0.874749	0.0984169	0.0492085	+	0.998789i	$0.484330\pi$
$80$	0	0
$81$	−6.46449	−0.718276
$82$	0	0
$83$	−12.5096	−1.37311	−0.686555	−	0.727077i	$-0.740878\pi$
$83$	−12.5096	−1.37311	−0.686555	+	0.727077i	$0.740878\pi$
$84$	0	0
$85$	−6.46569	−0.701303
$86$	0	0
$87$	10.0329	1.07564
$88$	0	0
$89$	10.4506	1.10776	0.553880	−	0.832596i	$-0.313146\pi$
$89$	10.4506	1.10776	0.553880	+	0.832596i	$0.313146\pi$
$90$	0	0
$91$	12.3691	1.29663
$92$	0	0
$93$	19.5565	2.02791
$94$	0	0
$95$	6.07664	0.623450
$96$	0	0
$97$	5.54353	0.562860	0.281430	−	0.959582i	$-0.409191\pi$
$97$	5.54353	0.562860	0.281430	+	0.959582i	$0.409191\pi$
$98$	0	0
$99$	11.7891	1.18485
$100$	0	0
$101$	0.153273	0.0152512	0.00762560	−	0.999971i	$-0.497573\pi$
$101$	0.153273	0.0152512	0.00762560	+	0.999971i	$0.497573\pi$
$102$	0	0
$103$	−3.09053	−0.304519	−0.152259	−	0.988341i	$-0.548655\pi$
$103$	−3.09053	−0.304519	−0.152259	+	0.988341i	$0.548655\pi$
$104$	0	0
$105$	5.37026	0.524083
$106$	0	0
$107$	5.26871	0.509345	0.254673	−	0.967027i	$-0.418032\pi$
$107$	5.26871	0.509345	0.254673	+	0.967027i	$0.418032\pi$
$108$	0	0
$109$	−10.6615	−1.02118	−0.510592	−	0.859823i	$-0.670574\pi$
$109$	−10.6615	−1.02118	−0.510592	+	0.859823i	$0.670574\pi$
$110$	0	0
$111$	6.83715	0.648954
$112$	0	0
$113$	−6.68147	−0.628540	−0.314270	−	0.949334i	$-0.601760\pi$
$113$	−6.68147	−0.628540	−0.314270	+	0.949334i	$0.601760\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−21.9643	−2.03060
$118$	0	0
$119$	13.4269	1.23084
$120$	0	0
$121$	−0.779314	−0.0708467
$122$	0	0
$123$	2.53431	0.228511
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.90827	−0.613010	−0.306505	−	0.951869i	$-0.599160\pi$
$127$	−6.90827	−0.613010	−0.306505	+	0.951869i	$0.599160\pi$
$128$	0	0
$129$	−14.6439	−1.28932
$130$	0	0
$131$	9.35146	0.817041	0.408520	−	0.912749i	$-0.366045\pi$
$131$	9.35146	0.817041	0.408520	+	0.912749i	$0.366045\pi$
$132$	0	0
$133$	−12.6190	−1.09420
$134$	0	0
$135$	−1.77811	−0.153035
$136$	0	0
$137$	−8.92360	−0.762395	−0.381197	−	0.924494i	$-0.624488\pi$
$137$	−8.92360	−0.762395	−0.381197	+	0.924494i	$0.624488\pi$
$138$	0	0
$139$	19.4949	1.65354	0.826769	−	0.562542i	$-0.190177\pi$
$139$	19.4949	1.65354	0.826769	+	0.562542i	$0.190177\pi$
$140$	0	0
$141$	12.5703	1.05861
$142$	0	0
$143$	−19.0422	−1.59238
$144$	0	0
$145$	−3.87966	−0.322188
$146$	0	0
$147$	6.95018	0.573241
$148$	0	0
$149$	−16.9114	−1.38543	−0.692717	−	0.721209i	$-0.743587\pi$
$149$	−16.9114	−1.38543	−0.692717	+	0.721209i	$0.743587\pi$
$150$	0	0
$151$	−17.4232	−1.41788	−0.708939	−	0.705269i	$-0.750826\pi$
$151$	−17.4232	−1.41788	−0.708939	+	0.705269i	$0.750826\pi$
$152$	0	0
$153$	−23.8428	−1.92757
$154$	0	0
$155$	−7.56233	−0.607421
$156$	0	0
$157$	22.5565	1.80020	0.900101	−	0.435682i	$-0.143493\pi$
$157$	22.5565	1.80020	0.900101	+	0.435682i	$0.143493\pi$
$158$	0	0
$159$	−22.0781	−1.75091
$160$	0	0
$161$	−2.07664	−0.163662
$162$	0	0
$163$	9.31242	0.729405	0.364702	−	0.931124i	$-0.381171\pi$
$163$	9.31242	0.729405	0.364702	+	0.931124i	$0.381171\pi$
$164$	0	0
$165$	−8.26751	−0.643625
$166$	0	0
$167$	6.77811	0.524506	0.262253	−	0.964999i	$-0.415535\pi$
$167$	6.77811	0.524506	0.262253	+	0.964999i	$0.415535\pi$
$168$	0	0
$169$	22.4774	1.72903
$170$	0	0
$171$	22.4081	1.71359
$172$	0	0
$173$	−23.7910	−1.80880	−0.904399	−	0.426687i	$-0.859681\pi$
$173$	−23.7910	−1.80880	−0.904399	+	0.426687i	$0.859681\pi$
$174$	0	0
$175$	−2.07664	−0.156979
$176$	0	0
$177$	8.10586	0.609274
$178$	0	0
$179$	2.45767	0.183695	0.0918475	−	0.995773i	$-0.470723\pi$
$179$	2.45767	0.183695	0.0918475	+	0.995773i	$0.470723\pi$
$180$	0	0
$181$	−3.28871	−0.244448	−0.122224	−	0.992503i	$-0.539003\pi$
$181$	−3.28871	−0.244448	−0.122224	+	0.992503i	$0.539003\pi$
$182$	0	0
$183$	−8.76111	−0.647640
$184$	0	0
$185$	−2.64387	−0.194381
$186$	0	0
$187$	−20.6707	−1.51159
$188$	0	0
$189$	3.69249	0.268589
$190$	0	0
$191$	5.36535	0.388223	0.194112	−	0.980979i	$-0.437818\pi$
$191$	5.36535	0.388223	0.194112	+	0.980979i	$0.437818\pi$
$192$	0	0
$193$	19.4671	1.40127	0.700637	−	0.713517i	$-0.252899\pi$
$193$	19.4671	1.40127	0.700637	+	0.713517i	$0.252899\pi$
$194$	0	0
$195$	15.4032	1.10304
$196$	0	0
$197$	−8.05543	−0.573926	−0.286963	−	0.957942i	$-0.592646\pi$
$197$	−8.05543	−0.573926	−0.286963	+	0.957942i	$0.592646\pi$
$198$	0	0
$199$	−21.4596	−1.52123	−0.760615	−	0.649204i	$-0.775102\pi$
$199$	−21.4596	−1.52123	−0.760615	+	0.649204i	$0.775102\pi$
$200$	0	0
$201$	−2.88456	−0.203461
$202$	0	0
$203$	8.05663	0.565465
$204$	0	0
$205$	−0.979997	−0.0684460
$206$	0	0
$207$	3.68758	0.256305
$208$	0	0
$209$	19.4269	1.34379
$210$	0	0
$211$	2.43397	0.167561	0.0837806	−	0.996484i	$-0.473301\pi$
$211$	2.43397	0.167561	0.0837806	+	0.996484i	$0.473301\pi$
$212$	0	0
$213$	−25.4847	−1.74618
$214$	0	0
$215$	5.66267	0.386191
$216$	0	0
$217$	15.7042	1.06607
$218$	0	0
$219$	31.8865	2.15469
$220$	0	0
$221$	38.5115	2.59057
$222$	0	0
$223$	8.33433	0.558108	0.279054	−	0.960275i	$-0.409979\pi$
$223$	8.33433	0.558108	0.279054	+	0.960275i	$0.409979\pi$
$224$	0	0
$225$	3.68758	0.245839
$226$	0	0
$227$	15.1789	1.00746	0.503729	−	0.863862i	$-0.331961\pi$
$227$	15.1789	1.00746	0.503729	+	0.863862i	$0.331961\pi$
$228$	0	0
$229$	14.0590	0.929048	0.464524	−	0.885561i	$-0.346226\pi$
$229$	14.0590	0.929048	0.464524	+	0.885561i	$0.346226\pi$
$230$	0	0
$231$	17.1686	1.12961
$232$	0	0
$233$	9.31145	0.610014	0.305007	−	0.952350i	$-0.401341\pi$
$233$	9.31145	0.610014	0.305007	+	0.952350i	$0.401341\pi$
$234$	0	0
$235$	−4.86086	−0.317087
$236$	0	0
$237$	−2.26213	−0.146941
$238$	0	0
$239$	−5.14837	−0.333020	−0.166510	−	0.986040i	$-0.553250\pi$
$239$	−5.14837	−0.333020	−0.166510	+	0.986040i	$0.553250\pi$
$240$	0	0
$241$	−15.9526	−1.02760	−0.513798	−	0.857911i	$-0.671762\pi$
$241$	−15.9526	−1.02760	−0.513798	+	0.857911i	$0.671762\pi$
$242$	0	0
$243$	22.0517	1.41462
$244$	0	0
$245$	−2.68758	−0.171703
$246$	0	0
$247$	−36.1942	−2.30298
$248$	0	0
$249$	32.3504	2.05012
$250$	0	0
$251$	−25.6021	−1.61599	−0.807994	−	0.589190i	$-0.799447\pi$
$251$	−25.6021	−1.61599	−0.807994	+	0.589190i	$0.799447\pi$
$252$	0	0
$253$	3.19698	0.200992
$254$	0	0
$255$	16.7205	1.04708
$256$	0	0
$257$	9.33924	0.582566	0.291283	−	0.956637i	$-0.405918\pi$
$257$	9.33924	0.582566	0.291283	+	0.956637i	$0.405918\pi$
$258$	0	0
$259$	5.49036	0.341155
$260$	0	0
$261$	−14.3065	−0.885553
$262$	0	0
$263$	0.612149	0.0377467	0.0188733	−	0.999822i	$-0.493992\pi$
$263$	0.612149	0.0377467	0.0188733	+	0.999822i	$0.493992\pi$
$264$	0	0
$265$	8.53742	0.524450
$266$	0	0
$267$	−27.0256	−1.65394
$268$	0	0
$269$	−6.70518	−0.408822	−0.204411	−	0.978885i	$-0.565528\pi$
$269$	−6.70518	−0.408822	−0.204411	+	0.978885i	$0.565528\pi$
$270$	0	0
$271$	31.2277	1.89695	0.948475	−	0.316852i	$-0.102626\pi$
$271$	31.2277	1.89695	0.948475	+	0.316852i	$0.102626\pi$
$272$	0	0
$273$	−31.9868	−1.93593
$274$	0	0
$275$	3.19698	0.192785
$276$	0	0
$277$	29.8513	1.79359	0.896795	−	0.442447i	$-0.145889\pi$
$277$	29.8513	1.79359	0.896795	+	0.442447i	$0.145889\pi$
$278$	0	0
$279$	−27.8867	−1.66953
$280$	0	0
$281$	18.3910	1.09711	0.548556	−	0.836114i	$-0.315178\pi$
$281$	18.3910	1.09711	0.548556	+	0.836114i	$0.315178\pi$
$282$	0	0
$283$	−21.6319	−1.28588	−0.642941	−	0.765916i	$-0.722286\pi$
$283$	−21.6319	−1.28588	−0.642941	+	0.765916i	$0.722286\pi$
$284$	0	0
$285$	−15.7144	−0.930841
$286$	0	0
$287$	2.03510	0.120128
$288$	0	0
$289$	24.8052	1.45913
$290$	0	0
$291$	−14.3358	−0.840378
$292$	0	0
$293$	−9.83451	−0.574538	−0.287269	−	0.957850i	$-0.592747\pi$
$293$	−9.83451	−0.574538	−0.287269	+	0.957850i	$0.592747\pi$
$294$	0	0
$295$	−3.13447	−0.182496
$296$	0	0
$297$	−5.68458	−0.329853
$298$	0	0
$299$	−5.95629	−0.344461
$300$	0	0
$301$	−11.7593	−0.677795
$302$	0	0
$303$	−0.396369	−0.0227708
$304$	0	0
$305$	3.38785	0.193988
$306$	0	0
$307$	−23.0031	−1.31286	−0.656428	−	0.754388i	$-0.727934\pi$
$307$	−23.0031	−1.31286	−0.656428	+	0.754388i	$0.727934\pi$
$308$	0	0
$309$	7.99222	0.454661
$310$	0	0
$311$	−12.7144	−0.720967	−0.360484	−	0.932766i	$-0.617388\pi$
$311$	−12.7144	−0.720967	−0.360484	+	0.932766i	$0.617388\pi$
$312$	0	0
$313$	−17.5583	−0.992451	−0.496226	−	0.868194i	$-0.665281\pi$
$313$	−17.5583	−0.992451	−0.496226	+	0.868194i	$0.665281\pi$
$314$	0	0
$315$	−7.65777	−0.431466
$316$	0	0
$317$	23.0924	1.29700	0.648500	−	0.761214i	$-0.275397\pi$
$317$	23.0924	1.29700	0.648500	+	0.761214i	$0.275397\pi$
$318$	0	0
$319$	−12.4032	−0.694445
$320$	0	0
$321$	−13.6251	−0.760477
$322$	0	0
$323$	−39.2897	−2.18613
$324$	0	0
$325$	−5.95629	−0.330396
$326$	0	0
$327$	27.5709	1.52468
$328$	0	0
$329$	10.0942	0.556513
$330$	0	0
$331$	7.51454	0.413037	0.206518	−	0.978443i	$-0.433787\pi$
$331$	7.51454	0.413037	0.206518	+	0.978443i	$0.433787\pi$
$332$	0	0
$333$	−9.74950	−0.534269
$334$	0	0
$335$	1.11544	0.0609429
$336$	0	0
$337$	25.3996	1.38360	0.691801	−	0.722088i	$-0.256817\pi$
$337$	25.3996	1.38360	0.691801	+	0.722088i	$0.256817\pi$
$338$	0	0
$339$	17.2785	0.938441
$340$	0	0
$341$	−24.1766	−1.30924
$342$	0	0
$343$	20.1176	1.08625
$344$	0	0
$345$	−2.58604	−0.139227
$346$	0	0
$347$	15.1964	0.815785	0.407892	−	0.913030i	$-0.366264\pi$
$347$	15.1964	0.815785	0.407892	+	0.913030i	$0.366264\pi$
$348$	0	0
$349$	2.32103	0.124242	0.0621209	−	0.998069i	$-0.480214\pi$
$349$	2.32103	0.124242	0.0621209	+	0.998069i	$0.480214\pi$
$350$	0	0
$351$	10.5909	0.565303
$352$	0	0
$353$	−10.5113	−0.559460	−0.279730	−	0.960079i	$-0.590245\pi$
$353$	−10.5113	−0.559460	−0.279730	+	0.960079i	$0.590245\pi$
$354$	0	0
$355$	9.85475	0.523036
$356$	0	0
$357$	−34.7224	−1.83771
$358$	0	0
$359$	−11.6153	−0.613030	−0.306515	−	0.951866i	$-0.599163\pi$
$359$	−11.6153	−0.613030	−0.306515	+	0.951866i	$0.599163\pi$
$360$	0	0
$361$	17.9255	0.943448
$362$	0	0
$363$	2.01533	0.105778
$364$	0	0
$365$	−12.3303	−0.645395
$366$	0	0
$367$	28.0381	1.46358	0.731788	−	0.681533i	$-0.238686\pi$
$367$	28.0381	1.46358	0.731788	+	0.681533i	$0.238686\pi$
$368$	0	0
$369$	−3.61382	−0.188128
$370$	0	0
$371$	−17.7291	−0.920450
$372$	0	0
$373$	27.9224	1.44577	0.722883	−	0.690970i	$-0.242816\pi$
$373$	27.9224	1.44577	0.722883	+	0.690970i	$0.242816\pi$
$374$	0	0
$375$	−2.58604	−0.133542
$376$	0	0
$377$	23.1084	1.19014
$378$	0	0
$379$	17.5978	0.903936	0.451968	−	0.892034i	$-0.350722\pi$
$379$	17.5978	0.903936	0.451968	+	0.892034i	$0.350722\pi$
$380$	0	0
$381$	17.8650	0.915253
$382$	0	0
$383$	3.24333	0.165727	0.0828633	−	0.996561i	$-0.473594\pi$
$383$	3.24333	0.165727	0.0828633	+	0.996561i	$0.473594\pi$
$384$	0	0
$385$	−6.63897	−0.338353
$386$	0	0
$387$	20.8816	1.06147
$388$	0	0
$389$	−21.1964	−1.07470	−0.537350	−	0.843359i	$-0.680575\pi$
$389$	−21.1964	−1.07470	−0.537350	+	0.843359i	$0.680575\pi$
$390$	0	0
$391$	−6.46569	−0.326984
$392$	0	0
$393$	−24.1832	−1.21988
$394$	0	0
$395$	0.874749	0.0440134
$396$	0	0
$397$	−27.6263	−1.38652	−0.693262	−	0.720686i	$-0.743827\pi$
$397$	−27.6263	−1.38652	−0.693262	+	0.720686i	$0.743827\pi$
$398$	0	0
$399$	32.6331	1.63370
$400$	0	0
$401$	36.2093	1.80821	0.904103	−	0.427314i	$-0.140540\pi$
$401$	36.2093	1.80821	0.904103	+	0.427314i	$0.140540\pi$
$402$	0	0
$403$	45.0434	2.24377
$404$	0	0
$405$	−6.46449	−0.321223
$406$	0	0
$407$	−8.45241	−0.418971
$408$	0	0
$409$	33.9231	1.67739	0.838695	−	0.544602i	$-0.183319\pi$
$409$	33.9231	1.67739	0.838695	+	0.544602i	$0.183319\pi$
$410$	0	0
$411$	23.0768	1.13829
$412$	0	0
$413$	6.50916	0.320295
$414$	0	0
$415$	−12.5096	−0.614074
$416$	0	0
$417$	−50.4146	−2.46881
$418$	0	0
$419$	−6.40318	−0.312816	−0.156408	−	0.987693i	$-0.549991\pi$
$419$	−6.40318	−0.312816	−0.156408	+	0.987693i	$0.549991\pi$
$420$	0	0
$421$	11.2013	0.545918	0.272959	−	0.962026i	$-0.411998\pi$
$421$	11.2013	0.545918	0.272959	+	0.962026i	$0.411998\pi$
$422$	0	0
$423$	−17.9248	−0.871534
$424$	0	0
$425$	−6.46569	−0.313632
$426$	0	0
$427$	−7.03534	−0.340464
$428$	0	0
$429$	49.2437	2.37751
$430$	0	0
$431$	25.0002	1.20422	0.602110	−	0.798414i	$-0.294327\pi$
$431$	25.0002	1.20422	0.602110	+	0.798414i	$0.294327\pi$
$432$	0	0
$433$	−19.2955	−0.927284	−0.463642	−	0.886023i	$-0.653458\pi$
$433$	−19.2955	−0.927284	−0.463642	+	0.886023i	$0.653458\pi$
$434$	0	0
$435$	10.0329	0.481042
$436$	0	0
$437$	6.07664	0.290685
$438$	0	0
$439$	38.7725	1.85051	0.925255	−	0.379347i	$-0.123851\pi$
$439$	38.7725	1.85051	0.925255	+	0.379347i	$0.123851\pi$
$440$	0	0
$441$	−9.91067	−0.471937
$442$	0	0
$443$	38.7915	1.84304	0.921520	−	0.388331i	$-0.126948\pi$
$443$	38.7915	1.84304	0.921520	+	0.388331i	$0.126948\pi$
$444$	0	0
$445$	10.4506	0.495406
$446$	0	0
$447$	43.7334	2.06852
$448$	0	0
$449$	−37.7398	−1.78105	−0.890525	−	0.454934i	$-0.849663\pi$
$449$	−37.7398	−1.78105	−0.890525	+	0.454934i	$0.849663\pi$
$450$	0	0
$451$	−3.13303	−0.147529
$452$	0	0
$453$	45.0570	2.11696
$454$	0	0
$455$	12.3691	0.579870
$456$	0	0
$457$	27.6973	1.29562	0.647812	−	0.761800i	$-0.275684\pi$
$457$	27.6973	1.29562	0.647812	+	0.761800i	$0.275684\pi$
$458$	0	0
$459$	11.4967	0.536621
$460$	0	0
$461$	33.5590	1.56300	0.781498	−	0.623907i	$-0.214456\pi$
$461$	33.5590	1.56300	0.781498	+	0.623907i	$0.214456\pi$
$462$	0	0
$463$	−16.7259	−0.777318	−0.388659	−	0.921382i	$-0.627062\pi$
$463$	−16.7259	−0.777318	−0.388659	+	0.921382i	$0.627062\pi$
$464$	0	0
$465$	19.5565	0.906909
$466$	0	0
$467$	8.79033	0.406768	0.203384	−	0.979099i	$-0.434806\pi$
$467$	8.79033	0.406768	0.203384	+	0.979099i	$0.434806\pi$
$468$	0	0
$469$	−2.31636	−0.106960
$470$	0	0
$471$	−58.3318	−2.68779
$472$	0	0
$473$	18.1035	0.832398
$474$	0	0
$475$	6.07664	0.278815
$476$	0	0
$477$	31.4824	1.44148
$478$	0	0
$479$	−17.8726	−0.816619	−0.408309	−	0.912844i	$-0.633882\pi$
$479$	−17.8726	−0.816619	−0.408309	+	0.912844i	$0.633882\pi$
$480$	0	0
$481$	15.7477	0.718032
$482$	0	0
$483$	5.37026	0.244355
$484$	0	0
$485$	5.54353	0.251719
$486$	0	0
$487$	14.1481	0.641113	0.320556	−	0.947229i	$-0.396130\pi$
$487$	14.1481	0.641113	0.320556	+	0.947229i	$0.396130\pi$
$488$	0	0
$489$	−24.0822	−1.08904
$490$	0	0
$491$	0.829600	0.0374393	0.0187197	−	0.999825i	$-0.494041\pi$
$491$	0.829600	0.0374393	0.0187197	+	0.999825i	$0.494041\pi$
$492$	0	0
$493$	25.0847	1.12976
$494$	0	0
$495$	11.7891	0.529882
$496$	0	0
$497$	−20.4647	−0.917968
$498$	0	0
$499$	39.0634	1.74872	0.874358	−	0.485281i	$-0.161283\pi$
$499$	39.0634	1.74872	0.874358	+	0.485281i	$0.161283\pi$
$500$	0	0
$501$	−17.5284	−0.783113
$502$	0	0
$503$	24.4202	1.08884	0.544421	−	0.838812i	$-0.316749\pi$
$503$	24.4202	1.08884	0.544421	+	0.838812i	$0.316749\pi$
$504$	0	0
$505$	0.153273	0.00682055
$506$	0	0
$507$	−58.1274	−2.58153
$508$	0	0
$509$	−21.6488	−0.959565	−0.479783	−	0.877387i	$-0.659284\pi$
$509$	−21.6488	−0.959565	−0.479783	+	0.877387i	$0.659284\pi$
$510$	0	0
$511$	25.6054	1.13272
$512$	0	0
$513$	−10.8049	−0.477049
$514$	0	0
$515$	−3.09053	−0.136185
$516$	0	0
$517$	−15.5401	−0.683451
$518$	0	0
$519$	61.5245	2.70062
$520$	0	0
$521$	18.2788	0.800807	0.400404	−	0.916339i	$-0.368870\pi$
$521$	18.2788	0.800807	0.400404	+	0.916339i	$0.368870\pi$
$522$	0	0
$523$	−0.671658	−0.0293695	−0.0146848	−	0.999892i	$-0.504674\pi$
$523$	−0.671658	−0.0293695	−0.0146848	+	0.999892i	$0.504674\pi$
$524$	0	0
$525$	5.37026	0.234377
$526$	0	0
$527$	48.8957	2.12993
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.5586	−0.501602
$532$	0	0
$533$	5.83715	0.252835
$534$	0	0
$535$	5.26871	0.227786
$536$	0	0
$537$	−6.35563	−0.274266
$538$	0	0
$539$	−8.59215	−0.370090
$540$	0	0
$541$	27.2309	1.17075	0.585373	−	0.810764i	$-0.300948\pi$
$541$	27.2309	1.17075	0.585373	+	0.810764i	$0.300948\pi$
$542$	0	0
$543$	8.50473	0.364973
$544$	0	0
$545$	−10.6615	−0.456687
$546$	0	0
$547$	7.12155	0.304495	0.152248	−	0.988342i	$-0.451349\pi$
$547$	7.12155	0.304495	0.152248	+	0.988342i	$0.451349\pi$
$548$	0	0
$549$	12.4930	0.533187
$550$	0	0
$551$	−23.5753	−1.00434
$552$	0	0
$553$	−1.81654	−0.0772469
$554$	0	0
$555$	6.83715	0.290221
$556$	0	0
$557$	5.32834	0.225769	0.112885	−	0.993608i	$-0.463991\pi$
$557$	5.32834	0.225769	0.112885	+	0.993608i	$0.463991\pi$
$558$	0	0
$559$	−33.7285	−1.42656
$560$	0	0
$561$	53.4551	2.25688
$562$	0	0
$563$	36.1317	1.52277	0.761385	−	0.648300i	$-0.224520\pi$
$563$	36.1317	1.52277	0.761385	+	0.648300i	$0.224520\pi$
$564$	0	0
$565$	−6.68147	−0.281092
$566$	0	0
$567$	13.4244	0.563771
$568$	0	0
$569$	35.4566	1.48642	0.743209	−	0.669060i	$-0.233303\pi$
$569$	35.4566	1.48642	0.743209	+	0.669060i	$0.233303\pi$
$570$	0	0
$571$	3.95365	0.165455	0.0827275	−	0.996572i	$-0.473637\pi$
$571$	3.95365	0.165455	0.0827275	+	0.996572i	$0.473637\pi$
$572$	0	0
$573$	−13.8750	−0.579636
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	35.7234	1.48718	0.743592	−	0.668634i	$-0.233121\pi$
$577$	35.7234	1.48718	0.743592	+	0.668634i	$0.233121\pi$
$578$	0	0
$579$	−50.3427	−2.09217
$580$	0	0
$581$	25.9780	1.07775
$582$	0	0
$583$	27.2940	1.13040
$584$	0	0
$585$	−21.9643	−0.908112
$586$	0	0
$587$	42.0407	1.73520	0.867602	−	0.497259i	$-0.165660\pi$
$587$	42.0407	1.73520	0.867602	+	0.497259i	$0.165660\pi$
$588$	0	0
$589$	−45.9535	−1.89348
$590$	0	0
$591$	20.8316	0.856899
$592$	0	0
$593$	4.71848	0.193765	0.0968823	−	0.995296i	$-0.469113\pi$
$593$	4.71848	0.193765	0.0968823	+	0.995296i	$0.469113\pi$
$594$	0	0
$595$	13.4269	0.550449
$596$	0	0
$597$	55.4952	2.27127
$598$	0	0
$599$	−28.0976	−1.14804	−0.574018	−	0.818843i	$-0.694616\pi$
$599$	−28.0976	−1.14804	−0.574018	+	0.818843i	$0.694616\pi$
$600$	0	0
$601$	16.3995	0.668949	0.334474	−	0.942405i	$-0.391441\pi$
$601$	16.3995	0.668949	0.334474	+	0.942405i	$0.391441\pi$
$602$	0	0
$603$	4.11327	0.167505
$604$	0	0
$605$	−0.779314	−0.0316836
$606$	0	0
$607$	41.6973	1.69244	0.846220	−	0.532834i	$-0.178873\pi$
$607$	41.6973	1.69244	0.846220	+	0.532834i	$0.178873\pi$
$608$	0	0
$609$	−20.8347	−0.844266
$610$	0	0
$611$	28.9527	1.17130
$612$	0	0
$613$	−21.5660	−0.871044	−0.435522	−	0.900178i	$-0.643436\pi$
$613$	−21.5660	−0.871044	−0.435522	+	0.900178i	$0.643436\pi$
$614$	0	0
$615$	2.53431	0.102193
$616$	0	0
$617$	11.5035	0.463115	0.231557	−	0.972821i	$-0.425618\pi$
$617$	11.5035	0.463115	0.231557	+	0.972821i	$0.425618\pi$
$618$	0	0
$619$	−36.9470	−1.48503	−0.742514	−	0.669831i	$-0.766367\pi$
$619$	−36.9470	−1.48503	−0.742514	+	0.669831i	$0.766367\pi$
$620$	0	0
$621$	−1.77811	−0.0713531
$622$	0	0
$623$	−21.7021	−0.869476
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−50.2386	−2.00634
$628$	0	0
$629$	17.0945	0.681601
$630$	0	0
$631$	−36.6839	−1.46036	−0.730181	−	0.683254i	$-0.760564\pi$
$631$	−36.6839	−1.46036	−0.730181	+	0.683254i	$0.760564\pi$
$632$	0	0
$633$	−6.29433	−0.250177
$634$	0	0
$635$	−6.90827	−0.274146
$636$	0	0
$637$	16.0080	0.634261
$638$	0	0
$639$	36.3402	1.43760
$640$	0	0
$641$	−31.7187	−1.25281	−0.626407	−	0.779496i	$-0.715475\pi$
$641$	−31.7187	−1.25281	−0.626407	+	0.779496i	$0.715475\pi$
$642$	0	0
$643$	−49.6953	−1.95979	−0.979896	−	0.199507i	$-0.936066\pi$
$643$	−49.6953	−1.95979	−0.979896	+	0.199507i	$0.936066\pi$
$644$	0	0
$645$	−14.6439	−0.576602
$646$	0	0
$647$	17.9197	0.704495	0.352247	−	0.935907i	$-0.385418\pi$
$647$	17.9197	0.704495	0.352247	+	0.935907i	$0.385418\pi$
$648$	0	0
$649$	−10.0209	−0.393353
$650$	0	0
$651$	−40.6117	−1.59170
$652$	0	0
$653$	31.8591	1.24674	0.623371	−	0.781926i	$-0.285763\pi$
$653$	31.8591	1.24674	0.623371	+	0.781926i	$0.285763\pi$
$654$	0	0
$655$	9.35146	0.365392
$656$	0	0
$657$	−45.4688	−1.77391
$658$	0	0
$659$	42.2689	1.64656	0.823282	−	0.567632i	$-0.192140\pi$
$659$	42.2689	1.64656	0.823282	+	0.567632i	$0.192140\pi$
$660$	0	0
$661$	−25.0251	−0.973365	−0.486683	−	0.873579i	$-0.661793\pi$
$661$	−25.0251	−0.973365	−0.486683	+	0.873579i	$0.661793\pi$
$662$	0	0
$663$	−99.5922	−3.86784
$664$	0	0
$665$	−12.6190	−0.489342
$666$	0	0
$667$	−3.87966	−0.150221
$668$	0	0
$669$	−21.5529	−0.833282
$670$	0	0
$671$	10.8309	0.418122
$672$	0	0
$673$	−25.6058	−0.987030	−0.493515	−	0.869737i	$-0.664288\pi$
$673$	−25.6058	−0.987030	−0.493515	+	0.869737i	$0.664288\pi$
$674$	0	0
$675$	−1.77811	−0.0684395
$676$	0	0
$677$	12.2689	0.471534	0.235767	−	0.971810i	$-0.424240\pi$
$677$	12.2689	0.471534	0.235767	+	0.971810i	$0.424240\pi$
$678$	0	0
$679$	−11.5119	−0.441786
$680$	0	0
$681$	−39.2531	−1.50418
$682$	0	0
$683$	−38.0788	−1.45704	−0.728522	−	0.685022i	$-0.759792\pi$
$683$	−38.0788	−1.45704	−0.728522	+	0.685022i	$0.759792\pi$
$684$	0	0
$685$	−8.92360	−0.340953
$686$	0	0
$687$	−36.3572	−1.38711
$688$	0	0
$689$	−50.8514	−1.93728
$690$	0	0
$691$	−18.9788	−0.721987	−0.360994	−	0.932568i	$-0.617562\pi$
$691$	−18.9788	−0.721987	−0.360994	+	0.932568i	$0.617562\pi$
$692$	0	0
$693$	−24.4817	−0.929984
$694$	0	0
$695$	19.4949	0.739484
$696$	0	0
$697$	6.33636	0.240007
$698$	0	0
$699$	−24.0797	−0.910780
$700$	0	0
$701$	−5.75834	−0.217490	−0.108745	−	0.994070i	$-0.534683\pi$
$701$	−5.75834	−0.217490	−0.108745	+	0.994070i	$0.534683\pi$
$702$	0	0
$703$	−16.0659	−0.605935
$704$	0	0
$705$	12.5703	0.473427
$706$	0	0
$707$	−0.318292	−0.0119706
$708$	0	0
$709$	38.6356	1.45099	0.725495	−	0.688227i	$-0.241611\pi$
$709$	38.6356	1.45099	0.725495	+	0.688227i	$0.241611\pi$
$710$	0	0
$711$	3.22571	0.120973
$712$	0	0
$713$	−7.56233	−0.283211
$714$	0	0
$715$	−19.0422	−0.712136
$716$	0	0
$717$	13.3139	0.497215
$718$	0	0
$719$	−3.30293	−0.123179	−0.0615893	−	0.998102i	$-0.519617\pi$
$719$	−3.30293	−0.123179	−0.0615893	+	0.998102i	$0.519617\pi$
$720$	0	0
$721$	6.41790	0.239015
$722$	0	0
$723$	41.2540	1.53425
$724$	0	0
$725$	−3.87966	−0.144087
$726$	0	0
$727$	9.74362	0.361371	0.180686	−	0.983541i	$-0.442168\pi$
$727$	9.74362	0.361371	0.180686	+	0.983541i	$0.442168\pi$
$728$	0	0
$729$	−37.6331	−1.39382
$730$	0	0
$731$	−36.6131	−1.35418
$732$	0	0
$733$	11.4152	0.421629	0.210814	−	0.977526i	$-0.432388\pi$
$733$	11.4152	0.421629	0.210814	+	0.977526i	$0.432388\pi$
$734$	0	0
$735$	6.95018	0.256361
$736$	0	0
$737$	3.56603	0.131357
$738$	0	0
$739$	53.6434	1.97330	0.986652	−	0.162845i	$-0.0520669\pi$
$739$	53.6434	1.97330	0.986652	+	0.162845i	$0.0520669\pi$
$740$	0	0
$741$	93.5996	3.43847
$742$	0	0
$743$	−24.5457	−0.900494	−0.450247	−	0.892904i	$-0.648664\pi$
$743$	−24.5457	−0.900494	−0.450247	+	0.892904i	$0.648664\pi$
$744$	0	0
$745$	−16.9114	−0.619585
$746$	0	0
$747$	−46.1303	−1.68782
$748$	0	0
$749$	−10.9412	−0.399783
$750$	0	0
$751$	−30.1797	−1.10127	−0.550637	−	0.834745i	$-0.685615\pi$
$751$	−30.1797	−1.10127	−0.550637	+	0.834745i	$0.685615\pi$
$752$	0	0
$753$	66.2079	2.41275
$754$	0	0
$755$	−17.4232	−0.634095
$756$	0	0
$757$	−13.1432	−0.477699	−0.238849	−	0.971057i	$-0.576770\pi$
$757$	−13.1432	−0.477699	−0.238849	+	0.971057i	$0.576770\pi$
$758$	0	0
$759$	−8.26751	−0.300091
$760$	0	0
$761$	4.39467	0.159307	0.0796533	−	0.996823i	$-0.474619\pi$
$761$	4.39467	0.159307	0.0796533	+	0.996823i	$0.474619\pi$
$762$	0	0
$763$	22.1400	0.801522
$764$	0	0
$765$	−23.8428	−0.862037
$766$	0	0
$767$	18.6698	0.674129
$768$	0	0
$769$	−27.9380	−1.00747	−0.503734	−	0.863859i	$-0.668041\pi$
$769$	−27.9380	−1.00747	−0.503734	+	0.863859i	$0.668041\pi$
$770$	0	0
$771$	−24.1516	−0.869799
$772$	0	0
$773$	−43.2096	−1.55414	−0.777070	−	0.629414i	$-0.783295\pi$
$773$	−43.2096	−1.55414	−0.777070	+	0.629414i	$0.783295\pi$
$774$	0	0
$775$	−7.56233	−0.271647
$776$	0	0
$777$	−14.1983	−0.509360
$778$	0	0
$779$	−5.95509	−0.213363
$780$	0	0
$781$	31.5054	1.12735
$782$	0	0
$783$	6.89845	0.246531
$784$	0	0
$785$	22.5565	0.805074
$786$	0	0
$787$	51.0255	1.81886	0.909431	−	0.415854i	$-0.136517\pi$
$787$	51.0255	1.81886	0.909431	+	0.415854i	$0.136517\pi$
$788$	0	0
$789$	−1.58304	−0.0563577
$790$	0	0
$791$	13.8750	0.493338
$792$	0	0
$793$	−20.1790	−0.716578
$794$	0	0
$795$	−22.0781	−0.783029
$796$	0	0
$797$	19.6701	0.696750	0.348375	−	0.937355i	$-0.386734\pi$
$797$	19.6701	0.696750	0.348375	+	0.937355i	$0.386734\pi$
$798$	0	0
$799$	31.4288	1.11187
$800$	0	0
$801$	38.5374	1.36165
$802$	0	0
$803$	−39.4196	−1.39109
$804$	0	0
$805$	−2.07664	−0.0731918
$806$	0	0
$807$	17.3398	0.610391
$808$	0	0
$809$	−45.8691	−1.61267	−0.806336	−	0.591458i	$-0.798553\pi$
$809$	−45.8691	−1.61267	−0.806336	+	0.591458i	$0.798553\pi$
$810$	0	0
$811$	1.97629	0.0693971	0.0346985	−	0.999398i	$-0.488953\pi$
$811$	1.97629	0.0693971	0.0346985	+	0.999398i	$0.488953\pi$
$812$	0	0
$813$	−80.7561	−2.83224
$814$	0	0
$815$	9.31242	0.326200
$816$	0	0
$817$	34.4100	1.20385
$818$	0	0
$819$	45.6119	1.59381
$820$	0	0
$821$	−33.7778	−1.17885	−0.589426	−	0.807822i	$-0.700646\pi$
$821$	−33.7778	−1.17885	−0.589426	+	0.807822i	$0.700646\pi$
$822$	0	0
$823$	−1.97189	−0.0687356	−0.0343678	−	0.999409i	$-0.510942\pi$
$823$	−1.97189	−0.0687356	−0.0343678	+	0.999409i	$0.510942\pi$
$824$	0	0
$825$	−8.26751	−0.287838
$826$	0	0
$827$	−2.06779	−0.0719041	−0.0359520	−	0.999354i	$-0.511446\pi$
$827$	−2.06779	−0.0719041	−0.0359520	+	0.999354i	$0.511446\pi$
$828$	0	0
$829$	−30.0659	−1.04423	−0.522115	−	0.852875i	$-0.674857\pi$
$829$	−30.0659	−1.04423	−0.522115	+	0.852875i	$0.674857\pi$
$830$	0	0
$831$	−77.1965	−2.67792
$832$	0	0
$833$	17.3771	0.602080
$834$	0	0
$835$	6.77811	0.234566
$836$	0	0
$837$	13.4467	0.464784
$838$	0	0
$839$	28.4008	0.980504	0.490252	−	0.871581i	$-0.336905\pi$
$839$	28.4008	0.980504	0.490252	+	0.871581i	$0.336905\pi$
$840$	0	0
$841$	−13.9483	−0.480975
$842$	0	0
$843$	−47.5597	−1.63804
$844$	0	0
$845$	22.4774	0.773247
$846$	0	0
$847$	1.61835	0.0556072
$848$	0	0
$849$	55.9408	1.91988
$850$	0	0
$851$	−2.64387	−0.0906308
$852$	0	0
$853$	23.0965	0.790809	0.395405	−	0.918507i	$-0.370604\pi$
$853$	23.0965	0.790809	0.395405	+	0.918507i	$0.370604\pi$
$854$	0	0
$855$	22.4081	0.766341
$856$	0	0
$857$	18.1530	0.620095	0.310048	−	0.950721i	$-0.399655\pi$
$857$	18.1530	0.620095	0.310048	+	0.950721i	$0.399655\pi$
$858$	0	0
$859$	49.6082	1.69261	0.846305	−	0.532699i	$-0.178822\pi$
$859$	49.6082	1.69261	0.846305	+	0.532699i	$0.178822\pi$
$860$	0	0
$861$	−5.26284	−0.179357
$862$	0	0
$863$	24.3893	0.830221	0.415111	−	0.909771i	$-0.363743\pi$
$863$	24.3893	0.830221	0.415111	+	0.909771i	$0.363743\pi$
$864$	0	0
$865$	−23.7910	−0.808919
$866$	0	0
$867$	−64.1470	−2.17855
$868$	0	0
$869$	2.79656	0.0948666
$870$	0	0
$871$	−6.64387	−0.225119
$872$	0	0
$873$	20.4422	0.691864
$874$	0	0
$875$	−2.07664	−0.0702031
$876$	0	0
$877$	3.19639	0.107934	0.0539672	−	0.998543i	$-0.482813\pi$
$877$	3.19639	0.107934	0.0539672	+	0.998543i	$0.482813\pi$
$878$	0	0
$879$	25.4324	0.857813
$880$	0	0
$881$	−18.8440	−0.634869	−0.317435	−	0.948280i	$-0.602821\pi$
$881$	−18.8440	−0.634869	−0.317435	+	0.948280i	$0.602821\pi$
$882$	0	0
$883$	−4.74254	−0.159599	−0.0797996	−	0.996811i	$-0.525428\pi$
$883$	−4.74254	−0.159599	−0.0797996	+	0.996811i	$0.525428\pi$
$884$	0	0
$885$	8.10586	0.272476
$886$	0	0
$887$	−2.69296	−0.0904206	−0.0452103	−	0.998977i	$-0.514396\pi$
$887$	−2.69296	−0.0904206	−0.0452103	+	0.998977i	$0.514396\pi$
$888$	0	0
$889$	14.3460	0.481148
$890$	0	0
$891$	−20.6668	−0.692365
$892$	0	0
$893$	−29.5377	−0.988440
$894$	0	0
$895$	2.45767	0.0821509
$896$	0	0
$897$	15.4032	0.514297
$898$	0	0
$899$	29.3392	0.978518
$900$	0	0
$901$	−55.2003	−1.83899
$902$	0	0
$903$	30.4100	1.01198
$904$	0	0
$905$	−3.28871	−0.109321
$906$	0	0
$907$	−36.1879	−1.20160	−0.600799	−	0.799400i	$-0.705151\pi$
$907$	−36.1879	−1.20160	−0.600799	+	0.799400i	$0.705151\pi$
$908$	0	0
$909$	0.565206	0.0187467
$910$	0	0
$911$	40.2537	1.33367	0.666833	−	0.745207i	$-0.267650\pi$
$911$	40.2537	1.33367	0.666833	+	0.745207i	$0.267650\pi$
$912$	0	0
$913$	−39.9931	−1.32358
$914$	0	0
$915$	−8.76111	−0.289633
$916$	0	0
$917$	−19.4196	−0.641291
$918$	0	0
$919$	9.92181	0.327290	0.163645	−	0.986519i	$-0.447675\pi$
$919$	9.92181	0.327290	0.163645	+	0.986519i	$0.447675\pi$
$920$	0	0
$921$	59.4869	1.96016
$922$	0	0
$923$	−58.6977	−1.93206
$924$	0	0
$925$	−2.64387	−0.0869300
$926$	0	0
$927$	−11.3966	−0.374313
$928$	0	0
$929$	−32.3155	−1.06024	−0.530119	−	0.847923i	$-0.677853\pi$
$929$	−32.3155	−1.06024	−0.530119	+	0.847923i	$0.677853\pi$
$930$	0	0
$931$	−16.3315	−0.535242
$932$	0	0
$933$	32.8799	1.07644
$934$	0	0
$935$	−20.6707	−0.676004
$936$	0	0
$937$	−30.1049	−0.983484	−0.491742	−	0.870741i	$-0.663640\pi$
$937$	−30.1049	−0.983484	−0.491742	+	0.870741i	$0.663640\pi$
$938$	0	0
$939$	45.4063	1.48178
$940$	0	0
$941$	6.37205	0.207723	0.103861	−	0.994592i	$-0.466880\pi$
$941$	6.37205	0.207723	0.103861	+	0.994592i	$0.466880\pi$
$942$	0	0
$943$	−0.979997	−0.0319131
$944$	0	0
$945$	3.69249	0.120117
$946$	0	0
$947$	−38.5664	−1.25324	−0.626620	−	0.779325i	$-0.715562\pi$
$947$	−38.5664	−1.25324	−0.626620	+	0.779325i	$0.715562\pi$
$948$	0	0
$949$	73.4426	2.38405
$950$	0	0
$951$	−59.7179	−1.93648
$952$	0	0
$953$	44.0322	1.42634	0.713171	−	0.700990i	$-0.247258\pi$
$953$	44.0322	1.42634	0.713171	+	0.700990i	$0.247258\pi$
$954$	0	0
$955$	5.36535	0.173619
$956$	0	0
$957$	32.0751	1.03684
$958$	0	0
$959$	18.5311	0.598400
$960$	0	0
$961$	26.1888	0.844801
$962$	0	0
$963$	19.4288	0.626084
$964$	0	0
$965$	19.4671	0.626669
$966$	0	0
$967$	−23.3837	−0.751967	−0.375984	−	0.926626i	$-0.622695\pi$
$967$	−23.3837	−0.751967	−0.375984	+	0.926626i	$0.622695\pi$
$968$	0	0
$969$	101.604	3.26401
$970$	0	0
$971$	47.4327	1.52219	0.761095	−	0.648641i	$-0.224662\pi$
$971$	47.4327	1.52219	0.761095	+	0.648641i	$0.224662\pi$
$972$	0	0
$973$	−40.4839	−1.29785
$974$	0	0
$975$	15.4032	0.493297
$976$	0	0
$977$	−27.2961	−0.873280	−0.436640	−	0.899636i	$-0.643832\pi$
$977$	−27.2961	−0.873280	−0.436640	+	0.899636i	$0.643832\pi$
$978$	0	0
$979$	33.4104	1.06780
$980$	0	0
$981$	−39.3150	−1.25523
$982$	0	0
$983$	−7.17601	−0.228879	−0.114440	−	0.993430i	$-0.536507\pi$
$983$	−7.17601	−0.228879	−0.114440	+	0.993430i	$0.536507\pi$
$984$	0	0
$985$	−8.05543	−0.256667
$986$	0	0
$987$	−26.1040	−0.830901
$988$	0	0
$989$	5.66267	0.180063
$990$	0	0
$991$	−38.6824	−1.22879	−0.614394	−	0.789000i	$-0.710599\pi$
$991$	−38.6824	−1.22879	−0.614394	+	0.789000i	$0.710599\pi$
$992$	0	0
$993$	−19.4329	−0.616684
$994$	0	0
$995$	−21.4596	−0.680314
$996$	0	0
$997$	22.0184	0.697331	0.348666	−	0.937247i	$-0.386635\pi$
$997$	22.0184	0.697331	0.348666	+	0.937247i	$0.386635\pi$
$998$	0	0
$999$	4.70110	0.148736

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.ch.1.1		4
4.3	odd	2		7360.2.a.ci.1.4		4
8.3	odd	2		3680.2.a.s.1.1	✓	4
8.5	even	2		3680.2.a.t.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.s.1.1	✓	4	8.3	odd	2
3680.2.a.t.1.4	yes	4	8.5	even	2
7360.2.a.ch.1.1		4	1.1	even	1	trivial
7360.2.a.ci.1.4		4	4.3	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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58604 q^{3} +1.00000 q^{5} -2.07664 q^{7} +3.68758 q^{9} +O(q^{10})\)	\(q-2.58604 q^{3} +1.00000 q^{5} -2.07664 q^{7} +3.68758 q^{9} +3.19698 q^{11} -5.95629 q^{13} -2.58604 q^{15} -6.46569 q^{17} +6.07664 q^{19} +5.37026 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.77811 q^{27} -3.87966 q^{29} -7.56233 q^{31} -8.26751 q^{33} -2.07664 q^{35} -2.64387 q^{37} +15.4032 q^{39} -0.979997 q^{41} +5.66267 q^{43} +3.68758 q^{45} -4.86086 q^{47} -2.68758 q^{49} +16.7205 q^{51} +8.53742 q^{53} +3.19698 q^{55} -15.7144 q^{57} -3.13447 q^{59} +3.38785 q^{61} -7.65777 q^{63} -5.95629 q^{65} +1.11544 q^{67} -2.58604 q^{69} +9.85475 q^{71} -12.3303 q^{73} -2.58604 q^{75} -6.63897 q^{77} +0.874749 q^{79} -6.46449 q^{81} -12.5096 q^{83} -6.46569 q^{85} +10.0329 q^{87} +10.4506 q^{89} +12.3691 q^{91} +19.5565 q^{93} +6.07664 q^{95} +5.54353 q^{97} +11.7891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^5 + 5 * q^7 + 9 * q^9	\( 4 q - q^{3} + 4 q^{5} + 5 q^{7} + 9 q^{9} + 11 q^{11} + q^{13} - q^{15} - 5 q^{17} + 11 q^{19} + 6 q^{21} + 4 q^{23} + 4 q^{25} + 8 q^{27} - 4 q^{29} - 9 q^{31} - 9 q^{33} + 5 q^{35} + 20 q^{37} + 8 q^{39} + 9 q^{41} + 9 q^{45} - 5 q^{49} + 23 q^{51} - 4 q^{53} + 11 q^{55} - 10 q^{57} + 22 q^{59} + q^{61} - 4 q^{63} + q^{65} + 12 q^{67} - q^{69} + 11 q^{71} - 8 q^{73} - q^{75} + 8 q^{77} - 12 q^{79} - 4 q^{83} - 5 q^{85} + 2 q^{87} + 12 q^{89} + 29 q^{91} - 2 q^{93} + 11 q^{95} - 7 q^{97} + 45 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 + 5 * q^7 + 9 * q^9 + 11 * q^11 + q^13 - q^15 - 5 * q^17 + 11 * q^19 + 6 * q^21 + 4 * q^23 + 4 * q^25 + 8 * q^27 - 4 * q^29 - 9 * q^31 - 9 * q^33 + 5 * q^35 + 20 * q^37 + 8 * q^39 + 9 * q^41 + 9 * q^45 - 5 * q^49 + 23 * q^51 - 4 * q^53 + 11 * q^55 - 10 * q^57 + 22 * q^59 + q^61 - 4 * q^63 + q^65 + 12 * q^67 - q^69 + 11 * q^71 - 8 * q^73 - q^75 + 8 * q^77 - 12 * q^79 - 4 * q^83 - 5 * q^85 + 2 * q^87 + 12 * q^89 + 29 * q^91 - 2 * q^93 + 11 * q^95 - 7 * q^97 + 45 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more