LMFDB - Embedded newform 495.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("495.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	495.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:	$3.95259490005$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.48704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 4x + 6 $ x^4 - 2x^3 - 6x^2 + 4*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.28632$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.28632 q^{2} +3.22727 q^{4} +1.00000 q^{5} +2.51962 q^{7} -2.80595 q^{8} +O(q^{10})$	\(q-2.28632 q^{2} +3.22727 q^{4} +1.00000 q^{5} +2.51962 q^{7} -2.80595 q^{8} -2.28632 q^{10} -1.00000 q^{11} +6.05302 q^{13} -5.76067 q^{14} -0.0392472 q^{16} +4.97417 q^{17} -7.02720 q^{19} +3.22727 q^{20} +2.28632 q^{22} -4.45455 q^{23} +1.00000 q^{25} -13.8392 q^{26} +8.13152 q^{28} -0.921150 q^{29} +3.03925 q^{31} +5.70163 q^{32} -11.3726 q^{34} +2.51962 q^{35} -3.49380 q^{37} +16.0664 q^{38} -2.80595 q^{40} +10.0664 q^{41} +1.48038 q^{43} -3.22727 q^{44} +10.1845 q^{46} +8.10605 q^{47} -0.651497 q^{49} -2.28632 q^{50} +19.5348 q^{52} +1.54545 q^{53} -1.00000 q^{55} -7.06993 q^{56} +2.10605 q^{58} +7.59984 q^{59} +10.1060 q^{61} -6.94870 q^{62} -12.9573 q^{64} +6.05302 q^{65} -8.69074 q^{67} +16.0530 q^{68} -5.76067 q^{70} -1.54545 q^{71} +6.05302 q^{73} +7.98795 q^{74} -22.6787 q^{76} -2.51962 q^{77} +11.0272 q^{79} -0.0392472 q^{80} -23.0151 q^{82} +8.50757 q^{83} +4.97417 q^{85} -3.38462 q^{86} +2.80595 q^{88} -15.1453 q^{89} +15.2513 q^{91} -14.3761 q^{92} -18.5330 q^{94} -7.02720 q^{95} -11.5998 q^{97} +1.48953 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8	$ 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8} + 2 q^{10} - 4 q^{11} + 8 q^{13} - 8 q^{14} + 12 q^{16} + 4 q^{17} + 4 q^{19} + 8 q^{20} - 2 q^{22} - 8 q^{23} + 4 q^{25} - 16 q^{26} - 8 q^{28} - 4 q^{29} + 14 q^{32} + 4 q^{34} + 4 q^{35} + 8 q^{37} + 20 q^{38} + 6 q^{40} - 4 q^{41} + 12 q^{43} - 8 q^{44} - 16 q^{46} + 20 q^{49} + 2 q^{50} + 20 q^{52} + 16 q^{53} - 4 q^{55} - 48 q^{56} - 24 q^{58} - 24 q^{59} + 8 q^{61} - 20 q^{62} + 8 q^{65} + 48 q^{68} - 8 q^{70} - 16 q^{71} + 8 q^{73} + 12 q^{74} - 36 q^{76} - 4 q^{77} + 12 q^{79} + 12 q^{80} - 40 q^{82} + 8 q^{83} + 4 q^{85} + 16 q^{86} - 6 q^{88} - 16 q^{89} - 16 q^{91} - 72 q^{92} - 40 q^{94} + 4 q^{95} + 8 q^{97} + 62 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8 + 2 * q^10 - 4 * q^11 + 8 * q^13 - 8 * q^14 + 12 * q^16 + 4 * q^17 + 4 * q^19 + 8 * q^20 - 2 * q^22 - 8 * q^23 + 4 * q^25 - 16 * q^26 - 8 * q^28 - 4 * q^29 + 14 * q^32 + 4 * q^34 + 4 * q^35 + 8 * q^37 + 20 * q^38 + 6 * q^40 - 4 * q^41 + 12 * q^43 - 8 * q^44 - 16 * q^46 + 20 * q^49 + 2 * q^50 + 20 * q^52 + 16 * q^53 - 4 * q^55 - 48 * q^56 - 24 * q^58 - 24 * q^59 + 8 * q^61 - 20 * q^62 + 8 * q^65 + 48 * q^68 - 8 * q^70 - 16 * q^71 + 8 * q^73 + 12 * q^74 - 36 * q^76 - 4 * q^77 + 12 * q^79 + 12 * q^80 - 40 * q^82 + 8 * q^83 + 4 * q^85 + 16 * q^86 - 6 * q^88 - 16 * q^89 - 16 * q^91 - 72 * q^92 - 40 * q^94 + 4 * q^95 + 8 * q^97 + 62 * q^98

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 $

$2$	−2.28632	−1.61667	−0.808337	−	0.588720i	$-0.799632\pi$
$2$	−2.28632	−1.61667	−0.808337	+	0.588720i	$0.799632\pi$
$3$	0	0
$3$	0	0
$4$	3.22727	1.61364
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.51962	0.952328	0.476164	−	0.879356i	$-0.342027\pi$
$7$	2.51962	0.952328	0.476164	+	0.879356i	$0.342027\pi$
$8$	−2.80595	−0.992052
$9$	0	0
$10$	−2.28632	−0.722999
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.05302	1.67881	0.839403	−	0.543509i	$-0.182905\pi$
$13$	6.05302	1.67881	0.839403	+	0.543509i	$0.182905\pi$
$14$	−5.76067	−1.53960
$15$	0	0
$16$	−0.0392472	−0.00981179
$17$	4.97417	1.20641	0.603207	−	0.797585i	$-0.293889\pi$
$17$	4.97417	1.20641	0.603207	+	0.797585i	$0.293889\pi$
$18$	0	0
$19$	−7.02720	−1.61215	−0.806075	−	0.591814i	$-0.798412\pi$
$19$	−7.02720	−1.61215	−0.806075	+	0.591814i	$0.798412\pi$
$20$	3.22727	0.721641
$21$	0	0
$22$	2.28632	0.487446
$23$	−4.45455	−0.928838	−0.464419	−	0.885616i	$-0.653737\pi$
$23$	−4.45455	−0.928838	−0.464419	+	0.885616i	$0.653737\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−13.8392	−2.71408
$27$	0	0
$28$	8.13152	1.53671
$29$	−0.921150	−0.171053	−0.0855266	−	0.996336i	$-0.527257\pi$
$29$	−0.921150	−0.171053	−0.0855266	+	0.996336i	$0.527257\pi$
$30$	0	0
$31$	3.03925	0.545865	0.272932	−	0.962033i	$-0.412006\pi$
$31$	3.03925	0.545865	0.272932	+	0.962033i	$0.412006\pi$
$32$	5.70163	1.00791
$33$	0	0
$34$	−11.3726	−1.95038
$35$	2.51962	0.425894
$36$	0	0
$37$	−3.49380	−0.574377	−0.287188	−	0.957874i	$-0.592721\pi$
$37$	−3.49380	−0.574377	−0.287188	+	0.957874i	$0.592721\pi$
$38$	16.0664	2.60632
$39$	0	0
$40$	−2.80595	−0.443659
$41$	10.0664	1.57211	0.786057	−	0.618154i	$-0.212119\pi$
$41$	10.0664	1.57211	0.786057	+	0.618154i	$0.212119\pi$
$42$	0	0
$43$	1.48038	0.225755	0.112878	−	0.993609i	$-0.463993\pi$
$43$	1.48038	0.225755	0.112878	+	0.993609i	$0.463993\pi$
$44$	−3.22727	−0.486530
$45$	0	0
$46$	10.1845	1.50163
$47$	8.10605	1.18239	0.591194	−	0.806529i	$-0.298657\pi$
$47$	8.10605	1.18239	0.591194	+	0.806529i	$0.298657\pi$
$48$	0	0
$49$	−0.651497	−0.0930710
$50$	−2.28632	−0.323335
$51$	0	0
$52$	19.5348	2.70899
$53$	1.54545	0.212284	0.106142	−	0.994351i	$-0.466150\pi$
$53$	1.54545	0.212284	0.106142	+	0.994351i	$0.466150\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−7.06993	−0.944759
$57$	0	0
$58$	2.10605	0.276537
$59$	7.59984	0.989415	0.494708	−	0.869059i	$-0.335275\pi$
$59$	7.59984	0.989415	0.494708	+	0.869059i	$0.335275\pi$
$60$	0	0
$61$	10.1060	1.29395	0.646973	−	0.762513i	$-0.276034\pi$
$61$	10.1060	1.29395	0.646973	+	0.762513i	$0.276034\pi$
$62$	−6.94870	−0.882486
$63$	0	0
$64$	−12.9573	−1.61966
$65$	6.05302	0.750785
$66$	0	0
$67$	−8.69074	−1.06174	−0.530872	−	0.847452i	$-0.678135\pi$
$67$	−8.69074	−1.06174	−0.530872	+	0.847452i	$0.678135\pi$
$68$	16.0530	1.94672
$69$	0	0
$70$	−5.76067	−0.688532
$71$	−1.54545	−0.183411	−0.0917056	−	0.995786i	$-0.529232\pi$
$71$	−1.54545	−0.183411	−0.0917056	+	0.995786i	$0.529232\pi$
$72$	0	0
$73$	6.05302	0.708453	0.354226	−	0.935160i	$-0.384744\pi$
$73$	6.05302	0.708453	0.354226	+	0.935160i	$0.384744\pi$
$74$	7.98795	0.928580
$75$	0	0
$76$	−22.6787	−2.60142
$77$	−2.51962	−0.287138
$78$	0	0
$79$	11.0272	1.24066	0.620328	−	0.784342i	$-0.286999\pi$
$79$	11.0272	1.24066	0.620328	+	0.784342i	$0.286999\pi$
$80$	−0.0392472	−0.00438797
$81$	0	0
$82$	−23.0151	−2.54160
$83$	8.50757	0.933827	0.466914	−	0.884303i	$-0.345366\pi$
$83$	8.50757	0.933827	0.466914	+	0.884303i	$0.345366\pi$
$84$	0	0
$85$	4.97417	0.539525
$86$	−3.38462	−0.364973
$87$	0	0
$88$	2.80595	0.299115
$89$	−15.1453	−1.60540	−0.802699	−	0.596384i	$-0.796603\pi$
$89$	−15.1453	−1.60540	−0.802699	+	0.596384i	$0.796603\pi$
$90$	0	0
$91$	15.2513	1.59877
$92$	−14.3761	−1.49881
$93$	0	0
$94$	−18.5330	−1.91154
$95$	−7.02720	−0.720975
$96$	0	0
$97$	−11.5998	−1.17779	−0.588893	−	0.808211i	$-0.700436\pi$
$97$	−11.5998	−1.17779	−0.588893	+	0.808211i	$0.700436\pi$
$98$	1.48953	0.150466
$99$	0	0
$100$	3.22727	0.322727
$101$	0.118097	0.0117511	0.00587556	−	0.999983i	$-0.498130\pi$
$101$	0.118097	0.0117511	0.00587556	+	0.999983i	$0.498130\pi$
$102$	0	0
$103$	10.6391	1.04830	0.524150	−	0.851626i	$-0.324383\pi$
$103$	10.6391	1.04830	0.524150	+	0.851626i	$0.324383\pi$
$104$	−16.9845	−1.66546
$105$	0	0
$106$	−3.53340	−0.343194
$107$	−18.6921	−1.80703	−0.903517	−	0.428551i	$-0.859024\pi$
$107$	−18.6921	−1.80703	−0.903517	+	0.428551i	$0.859024\pi$
$108$	0	0
$109$	−7.14529	−0.684395	−0.342198	−	0.939628i	$-0.611171\pi$
$109$	−7.14529	−0.684395	−0.342198	+	0.939628i	$0.611171\pi$
$110$	2.28632	0.217992
$111$	0	0
$112$	−0.0988881	−0.00934405
$113$	2.58470	0.243148	0.121574	−	0.992582i	$-0.461206\pi$
$113$	2.58470	0.243148	0.121574	+	0.992582i	$0.461206\pi$
$114$	0	0
$115$	−4.45455	−0.415389
$116$	−2.97280	−0.276018
$117$	0	0
$118$	−17.3757	−1.59956
$119$	12.5330	1.14890
$120$	0	0
$121$	1.00000	0.0909091
$122$	−23.1057	−2.09189
$123$	0	0
$124$	9.80849	0.880828
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.4287	1.01414	0.507068	−	0.861906i	$-0.330729\pi$
$127$	11.4287	1.01414	0.507068	+	0.861906i	$0.330729\pi$
$128$	18.2212	1.61055
$129$	0	0
$130$	−13.8392	−1.21378
$131$	−8.10605	−0.708229	−0.354114	−	0.935202i	$-0.615218\pi$
$131$	−8.10605	−0.708229	−0.354114	+	0.935202i	$0.615218\pi$
$132$	0	0
$133$	−17.7059	−1.53530
$134$	19.8699	1.71649
$135$	0	0
$136$	−13.9573	−1.19683
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−20.0664	−1.70201	−0.851007	−	0.525155i	$-0.824007\pi$
$139$	−20.0664	−1.70201	−0.851007	+	0.525155i	$0.824007\pi$
$140$	8.13152	0.687239
$141$	0	0
$142$	3.53340	0.296516
$143$	−6.05302	−0.506179
$144$	0	0
$145$	−0.921150	−0.0764973
$146$	−13.8392	−1.14534
$147$	0	0
$148$	−11.2754	−0.926836
$149$	−21.2634	−1.74196	−0.870982	−	0.491314i	$-0.836517\pi$
$149$	−21.2634	−1.74196	−0.870982	+	0.491314i	$0.836517\pi$
$150$	0	0
$151$	−7.02720	−0.571865	−0.285933	−	0.958250i	$-0.592303\pi$
$151$	−7.02720	−0.571865	−0.285933	+	0.958250i	$0.592303\pi$
$152$	19.7179	1.59934
$153$	0	0
$154$	5.76067	0.464208
$155$	3.03925	0.244118
$156$	0	0
$157$	21.0936	1.68346	0.841728	−	0.539902i	$-0.181539\pi$
$157$	21.0936	1.68346	0.841728	+	0.539902i	$0.181539\pi$
$158$	−25.2117	−2.00574
$159$	0	0
$160$	5.70163	0.450753
$161$	−11.2238	−0.884558
$162$	0	0
$163$	−7.65150	−0.599311	−0.299656	−	0.954047i	$-0.596872\pi$
$163$	−7.65150	−0.599311	−0.299656	+	0.954047i	$0.596872\pi$
$164$	32.4872	2.53682
$165$	0	0
$166$	−19.4511	−1.50970
$167$	3.49243	0.270252	0.135126	−	0.990828i	$-0.456856\pi$
$167$	3.49243	0.270252	0.135126	+	0.990828i	$0.456856\pi$
$168$	0	0
$169$	23.6391	1.81839
$170$	−11.3726	−0.872236
$171$	0	0
$172$	4.77758	0.364287
$173$	16.9742	1.29052	0.645261	−	0.763962i	$-0.276749\pi$
$173$	16.9742	1.29052	0.645261	+	0.763962i	$0.276749\pi$
$174$	0	0
$175$	2.51962	0.190466
$176$	0.0392472	0.00295837
$177$	0	0
$178$	34.6270	2.59541
$179$	−9.14529	−0.683551	−0.341776	−	0.939782i	$-0.611028\pi$
$179$	−9.14529	−0.683551	−0.341776	+	0.939782i	$0.611028\pi$
$180$	0	0
$181$	22.6391	1.68275	0.841375	−	0.540452i	$-0.181747\pi$
$181$	22.6391	1.68275	0.841375	+	0.540452i	$0.181747\pi$
$182$	−34.8695	−2.58470
$183$	0	0
$184$	12.4992	0.921455
$185$	−3.49380	−0.256869
$186$	0	0
$187$	−4.97417	−0.363748
$188$	26.1604	1.90795
$189$	0	0
$190$	16.0664	1.16558
$191$	−12.2362	−0.885380	−0.442690	−	0.896675i	$-0.645976\pi$
$191$	−12.2362	−0.885380	−0.442690	+	0.896675i	$0.645976\pi$
$192$	0	0
$193$	1.03788	0.0747081	0.0373540	−	0.999302i	$-0.488107\pi$
$193$	1.03788	0.0747081	0.0373540	+	0.999302i	$0.488107\pi$
$194$	26.5210	1.90410
$195$	0	0
$196$	−2.10256	−0.150183
$197$	8.06507	0.574613	0.287306	−	0.957839i	$-0.407240\pi$
$197$	8.06507	0.574613	0.287306	+	0.957839i	$0.407240\pi$
$198$	0	0
$199$	−13.1453	−0.931845	−0.465923	−	0.884825i	$-0.654277\pi$
$199$	−13.1453	−0.931845	−0.465923	+	0.884825i	$0.654277\pi$
$200$	−2.80595	−0.198410
$201$	0	0
$202$	−0.270009	−0.0189977
$203$	−2.32095	−0.162899
$204$	0	0
$205$	10.0664	0.703071
$206$	−24.3244	−1.69476
$207$	0	0
$208$	−0.237564	−0.0164721
$209$	7.02720	0.486081
$210$	0	0
$211$	−3.05130	−0.210060	−0.105030	−	0.994469i	$-0.533494\pi$
$211$	−3.05130	−0.210060	−0.105030	+	0.994469i	$0.533494\pi$
$212$	4.98759	0.342549
$213$	0	0
$214$	42.7362	2.92139
$215$	1.48038	0.100961
$216$	0	0
$217$	7.65776	0.519843
$218$	16.3365	1.10644
$219$	0	0
$220$	−3.22727	−0.217583
$221$	30.1088	2.02534
$222$	0	0
$223$	−13.9483	−0.934050	−0.467025	−	0.884244i	$-0.654674\pi$
$223$	−13.9483	−0.934050	−0.467025	+	0.884244i	$0.654674\pi$
$224$	14.3660	0.959865
$225$	0	0
$226$	−5.90945	−0.393091
$227$	29.6529	1.96813	0.984065	−	0.177809i	$-0.0569011\pi$
$227$	29.6529	1.96813	0.984065	+	0.177809i	$0.0569011\pi$
$228$	0	0
$229$	18.2121	1.20349	0.601744	−	0.798689i	$-0.294473\pi$
$229$	18.2121	1.20349	0.601744	+	0.798689i	$0.294473\pi$
$230$	10.1845	0.671549
$231$	0	0
$232$	2.58470	0.169694
$233$	−13.0802	−0.856914	−0.428457	−	0.903562i	$-0.640943\pi$
$233$	−13.0802	−0.856914	−0.428457	+	0.903562i	$0.640943\pi$
$234$	0	0
$235$	8.10605	0.528780
$236$	24.5268	1.59656
$237$	0	0
$238$	−28.6546	−1.85740
$239$	−0.803053	−0.0519452	−0.0259726	−	0.999663i	$-0.508268\pi$
$239$	−0.803053	−0.0519452	−0.0259726	+	0.999663i	$0.508268\pi$
$240$	0	0
$241$	3.03925	0.195775	0.0978876	−	0.995197i	$-0.468791\pi$
$241$	3.03925	0.195775	0.0978876	+	0.995197i	$0.468791\pi$
$242$	−2.28632	−0.146970
$243$	0	0
$244$	32.6150	2.08796
$245$	−0.651497	−0.0416226
$246$	0	0
$247$	−42.5358	−2.70649
$248$	−8.52797	−0.541526
$249$	0	0
$250$	−2.28632	−0.144600
$251$	−7.36365	−0.464789	−0.232395	−	0.972622i	$-0.574656\pi$
$251$	−7.36365	−0.464789	−0.232395	+	0.972622i	$0.574656\pi$
$252$	0	0
$253$	4.45455	0.280055
$254$	−26.1298	−1.63953
$255$	0	0
$256$	−15.7451	−0.984071
$257$	3.94835	0.246291	0.123146	−	0.992389i	$-0.460702\pi$
$257$	3.94835	0.246291	0.123146	+	0.992389i	$0.460702\pi$
$258$	0	0
$259$	−8.80305	−0.546995
$260$	19.5348	1.21149
$261$	0	0
$262$	18.5330	1.14498
$263$	−28.6405	−1.76605	−0.883023	−	0.469329i	$-0.844496\pi$
$263$	−28.6405	−1.76605	−0.883023	+	0.469329i	$0.844496\pi$
$264$	0	0
$265$	1.54545	0.0949363
$266$	40.4814	2.48207
$267$	0	0
$268$	−28.0474	−1.71327
$269$	−28.2121	−1.72012	−0.860061	−	0.510191i	$-0.829575\pi$
$269$	−28.2121	−1.72012	−0.860061	+	0.510191i	$0.829575\pi$
$270$	0	0
$271$	11.2634	0.684202	0.342101	−	0.939663i	$-0.388861\pi$
$271$	11.2634	0.684202	0.342101	+	0.939663i	$0.388861\pi$
$272$	−0.195222	−0.0118371
$273$	0	0
$274$	13.7179	0.828731
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−29.0165	−1.74343	−0.871717	−	0.490010i	$-0.836993\pi$
$277$	−29.0165	−1.74343	−0.871717	+	0.490010i	$0.836993\pi$
$278$	45.8784	2.75160
$279$	0	0
$280$	−7.06993	−0.422509
$281$	−30.9755	−1.84785	−0.923923	−	0.382579i	$-0.875036\pi$
$281$	−30.9755	−1.84785	−0.923923	+	0.382579i	$0.875036\pi$
$282$	0	0
$283$	−23.7951	−1.41447	−0.707235	−	0.706979i	$-0.750058\pi$
$283$	−23.7951	−1.41447	−0.707235	+	0.706979i	$0.750058\pi$
$284$	−4.98759	−0.295959
$285$	0	0
$286$	13.8392	0.818327
$287$	25.3636	1.49717
$288$	0	0
$289$	7.74240	0.455435
$290$	2.10605	0.123671
$291$	0	0
$292$	19.5348	1.14319
$293$	0.998275	0.0583198	0.0291599	−	0.999575i	$-0.490717\pi$
$293$	0.998275	0.0583198	0.0291599	+	0.999575i	$0.490717\pi$
$294$	0	0
$295$	7.59984	0.442480
$296$	9.80341	0.569812
$297$	0	0
$298$	48.6150	2.81619
$299$	−26.9635	−1.55934
$300$	0	0
$301$	3.72999	0.214993
$302$	16.0664	0.924520
$303$	0	0
$304$	0.275798	0.0158181
$305$	10.1060	0.578671
$306$	0	0
$307$	−27.7710	−1.58497	−0.792486	−	0.609890i	$-0.791214\pi$
$307$	−27.7710	−1.58497	−0.792486	+	0.609890i	$0.791214\pi$
$308$	−8.13152	−0.463336
$309$	0	0
$310$	−6.94870	−0.394660
$311$	17.5213	0.993545	0.496772	−	0.867881i	$-0.334518\pi$
$311$	17.5213	0.993545	0.496772	+	0.867881i	$0.334518\pi$
$312$	0	0
$313$	5.41530	0.306091	0.153045	−	0.988219i	$-0.451092\pi$
$313$	5.41530	0.306091	0.153045	+	0.988219i	$0.451092\pi$
$314$	−48.2269	−2.72160
$315$	0	0
$316$	35.5878	2.00197
$317$	−6.79679	−0.381746	−0.190873	−	0.981615i	$-0.561132\pi$
$317$	−6.79679	−0.381746	−0.190873	+	0.981615i	$0.561132\pi$
$318$	0	0
$319$	0.921150	0.0515745
$320$	−12.9573	−0.724333
$321$	0	0
$322$	25.6612	1.43004
$323$	−34.9545	−1.94492
$324$	0	0
$325$	6.05302	0.335761
$326$	17.4938	0.968892
$327$	0	0
$328$	−28.2459	−1.55962
$329$	20.4242	1.12602
$330$	0	0
$331$	−7.11845	−0.391266	−0.195633	−	0.980677i	$-0.562676\pi$
$331$	−7.11845	−0.391266	−0.195633	+	0.980677i	$0.562676\pi$
$332$	27.4563	1.50686
$333$	0	0
$334$	−7.98482	−0.436910
$335$	−8.69074	−0.474826
$336$	0	0
$337$	24.9105	1.35696	0.678480	−	0.734619i	$-0.262639\pi$
$337$	24.9105	1.35696	0.678480	+	0.734619i	$0.262639\pi$
$338$	−54.0466	−2.93975
$339$	0	0
$340$	16.0530	0.870597
$341$	−3.03925	−0.164584
$342$	0	0
$343$	−19.2789	−1.04096
$344$	−4.15386	−0.223961
$345$	0	0
$346$	−38.8084	−2.08636
$347$	23.6253	1.26827	0.634137	−	0.773221i	$-0.281356\pi$
$347$	23.6253	1.26827	0.634137	+	0.773221i	$0.281356\pi$
$348$	0	0
$349$	−31.1721	−1.66861	−0.834303	−	0.551306i	$-0.814130\pi$
$349$	−31.1721	−1.66861	−0.834303	+	0.551306i	$0.814130\pi$
$350$	−5.76067	−0.307921
$351$	0	0
$352$	−5.70163	−0.303898
$353$	0.533044	0.0283711	0.0141855	−	0.999899i	$-0.495484\pi$
$353$	0.533044	0.0283711	0.0141855	+	0.999899i	$0.495484\pi$
$354$	0	0
$355$	−1.54545	−0.0820240
$356$	−48.8780	−2.59053
$357$	0	0
$358$	20.9091	1.10508
$359$	12.0268	0.634752	0.317376	−	0.948300i	$-0.397198\pi$
$359$	12.0268	0.634752	0.317376	+	0.948300i	$0.397198\pi$
$360$	0	0
$361$	30.3815	1.59903
$362$	−51.7603	−2.72046
$363$	0	0
$364$	49.2203	2.57984
$365$	6.05302	0.316830
$366$	0	0
$367$	−14.9608	−0.780945	−0.390472	−	0.920615i	$-0.627688\pi$
$367$	−14.9608	−0.780945	−0.390472	+	0.920615i	$0.627688\pi$
$368$	0.174828	0.00911357
$369$	0	0
$370$	7.98795	0.415274
$371$	3.89395	0.202164
$372$	0	0
$373$	20.1315	1.04237	0.521185	−	0.853444i	$-0.325490\pi$
$373$	20.1315	1.04237	0.521185	+	0.853444i	$0.325490\pi$
$374$	11.3726	0.588062
$375$	0	0
$376$	−22.7451	−1.17299
$377$	−5.57574	−0.287165
$378$	0	0
$379$	15.8967	0.816558	0.408279	−	0.912857i	$-0.366129\pi$
$379$	15.8967	0.816558	0.408279	+	0.912857i	$0.366129\pi$
$380$	−22.6787	−1.16339
$381$	0	0
$382$	27.9759	1.43137
$383$	2.63909	0.134851	0.0674256	−	0.997724i	$-0.478521\pi$
$383$	2.63909	0.134851	0.0674256	+	0.997724i	$0.478521\pi$
$384$	0	0
$385$	−2.51962	−0.128412
$386$	−2.37292	−0.120779
$387$	0	0
$388$	−37.4359	−1.90052
$389$	−13.8423	−0.701832	−0.350916	−	0.936407i	$-0.614130\pi$
$389$	−13.8423	−0.701832	−0.350916	+	0.936407i	$0.614130\pi$
$390$	0	0
$391$	−22.1577	−1.12056
$392$	1.82807	0.0923313
$393$	0	0
$394$	−18.4394	−0.928962
$395$	11.0272	0.554838
$396$	0	0
$397$	−25.6783	−1.28876	−0.644379	−	0.764706i	$-0.722884\pi$
$397$	−25.6783	−1.28876	−0.644379	+	0.764706i	$0.722884\pi$
$398$	30.0544	1.50649
$399$	0	0
$400$	−0.0392472	−0.00196236
$401$	−31.1212	−1.55412	−0.777059	−	0.629428i	$-0.783289\pi$
$401$	−31.1212	−1.55412	−0.777059	+	0.629428i	$0.783289\pi$
$402$	0	0
$403$	18.3966	0.916402
$404$	0.381133	0.0189621
$405$	0	0
$406$	5.30644	0.263354
$407$	3.49380	0.173181
$408$	0	0
$409$	12.7514	0.630516	0.315258	−	0.949006i	$-0.397909\pi$
$409$	12.7514	0.630516	0.315258	+	0.949006i	$0.397909\pi$
$410$	−23.0151	−1.13664
$411$	0	0
$412$	34.3353	1.69158
$413$	19.1487	0.942248
$414$	0	0
$415$	8.50757	0.417620
$416$	34.5121	1.69209
$417$	0	0
$418$	−16.0664	−0.785835
$419$	−0.236195	−0.0115389	−0.00576943	−	0.999983i	$-0.501836\pi$
$419$	−0.236195	−0.0115389	−0.00576943	+	0.999983i	$0.501836\pi$
$420$	0	0
$421$	−21.7300	−1.05905	−0.529527	−	0.848293i	$-0.677631\pi$
$421$	−21.7300	−1.05905	−0.529527	+	0.848293i	$0.677631\pi$
$422$	6.97625	0.339599
$423$	0	0
$424$	−4.33645	−0.210597
$425$	4.97417	0.241283
$426$	0	0
$427$	25.4634	1.23226
$428$	−60.3246	−2.91590
$429$	0	0
$430$	−3.38462	−0.163221
$431$	16.2121	0.780909	0.390455	−	0.920622i	$-0.372318\pi$
$431$	16.2121	0.780909	0.390455	+	0.920622i	$0.372318\pi$
$432$	0	0
$433$	13.5213	0.649795	0.324897	−	0.945749i	$-0.394670\pi$
$433$	13.5213	0.649795	0.324897	+	0.945749i	$0.394670\pi$
$434$	−17.5081	−0.840416
$435$	0	0
$436$	−23.0598	−1.10437
$437$	31.3030	1.49743
$438$	0	0
$439$	0.0664435	0.00317118	0.00158559	−	0.999999i	$-0.499495\pi$
$439$	0.0664435	0.00317118	0.00158559	+	0.999999i	$0.499495\pi$
$440$	2.80595	0.133768
$441$	0	0
$442$	−68.8384	−3.27431
$443$	0.560596	0.0266347	0.0133174	−	0.999911i	$-0.495761\pi$
$443$	0.560596	0.0266347	0.0133174	+	0.999911i	$0.495761\pi$
$444$	0	0
$445$	−15.1453	−0.717956
$446$	31.8904	1.51006
$447$	0	0
$448$	−32.6474	−1.54245
$449$	−9.92079	−0.468191	−0.234096	−	0.972214i	$-0.575213\pi$
$449$	−9.92079	−0.468191	−0.234096	+	0.972214i	$0.575213\pi$
$450$	0	0
$451$	−10.0664	−0.474010
$452$	8.34153	0.392353
$453$	0	0
$454$	−67.7960	−3.18183
$455$	15.2513	0.714994
$456$	0	0
$457$	8.93457	0.417942	0.208971	−	0.977922i	$-0.432989\pi$
$457$	8.93457	0.417942	0.208971	+	0.977922i	$0.432989\pi$
$458$	−41.6387	−1.94565
$459$	0	0
$460$	−14.3761	−0.670287
$461$	−34.1484	−1.59045	−0.795225	−	0.606315i	$-0.792647\pi$
$461$	−34.1484	−1.59045	−0.795225	+	0.606315i	$0.792647\pi$
$462$	0	0
$463$	−24.9359	−1.15887	−0.579436	−	0.815018i	$-0.696727\pi$
$463$	−24.9359	−1.15887	−0.579436	+	0.815018i	$0.696727\pi$
$464$	0.0361525	0.00167834
$465$	0	0
$466$	29.9056	1.38535
$467$	2.84844	0.131810	0.0659051	−	0.997826i	$-0.479007\pi$
$467$	2.84844	0.131810	0.0659051	+	0.997826i	$0.479007\pi$
$468$	0	0
$469$	−21.8974	−1.01113
$470$	−18.5330	−0.854866
$471$	0	0
$472$	−21.3248	−0.981552
$473$	−1.48038	−0.0680678
$474$	0	0
$475$	−7.02720	−0.322430
$476$	40.4476	1.85391
$477$	0	0
$478$	1.83604	0.0839784
$479$	15.1177	0.690747	0.345374	−	0.938465i	$-0.387752\pi$
$479$	15.1177	0.690747	0.345374	+	0.938465i	$0.387752\pi$
$480$	0	0
$481$	−21.1480	−0.964267
$482$	−6.94870	−0.316505
$483$	0	0
$484$	3.22727	0.146694
$485$	−11.5998	−0.526722
$486$	0	0
$487$	−4.53376	−0.205444	−0.102722	−	0.994710i	$-0.532755\pi$
$487$	−4.53376	−0.205444	−0.102722	+	0.994710i	$0.532755\pi$
$488$	−28.3570	−1.28366
$489$	0	0
$490$	1.48953	0.0672902
$491$	15.0909	0.681043	0.340521	−	0.940237i	$-0.389396\pi$
$491$	15.0909	0.681043	0.340521	+	0.940237i	$0.389396\pi$
$492$	0	0
$493$	−4.58196	−0.206361
$494$	97.2506	4.37551
$495$	0	0
$496$	−0.119282	−0.00535591
$497$	−3.89395	−0.174668
$498$	0	0
$499$	−8.93320	−0.399905	−0.199952	−	0.979806i	$-0.564079\pi$
$499$	−8.93320	−0.399905	−0.199952	+	0.979806i	$0.564079\pi$
$500$	3.22727	0.144328
$501$	0	0
$502$	16.8357	0.751413
$503$	−26.7162	−1.19122	−0.595609	−	0.803275i	$-0.703089\pi$
$503$	−26.7162	−1.19122	−0.595609	+	0.803275i	$0.703089\pi$
$504$	0	0
$505$	0.118097	0.00525526
$506$	−10.1845	−0.452758
$507$	0	0
$508$	36.8836	1.63645
$509$	10.9876	0.487017	0.243508	−	0.969899i	$-0.421702\pi$
$509$	10.9876	0.487017	0.243508	+	0.969899i	$0.421702\pi$
$510$	0	0
$511$	15.2513	0.674680
$512$	−0.444024	−0.0196233
$513$	0	0
$514$	−9.02720	−0.398173
$515$	10.6391	0.468814
$516$	0	0
$517$	−8.10605	−0.356504
$518$	20.1266	0.884313
$519$	0	0
$520$	−16.9845	−0.744818
$521$	23.2782	1.01984	0.509918	−	0.860223i	$-0.329676\pi$
$521$	23.2782	1.01984	0.509918	+	0.860223i	$0.329676\pi$
$522$	0	0
$523$	−18.4163	−0.805289	−0.402645	−	0.915356i	$-0.631909\pi$
$523$	−18.4163	−0.805289	−0.402645	+	0.915356i	$0.631909\pi$
$524$	−26.1604	−1.14282
$525$	0	0
$526$	65.4814	2.85512
$527$	15.1177	0.658539
$528$	0	0
$529$	−3.15699	−0.137260
$530$	−3.53340	−0.153481
$531$	0	0
$532$	−57.1418	−2.47741
$533$	60.9324	2.63928
$534$	0	0
$535$	−18.6921	−0.808131
$536$	24.3858	1.05330
$537$	0	0
$538$	64.5020	2.78088
$539$	0.651497	0.0280620
$540$	0	0
$541$	−31.3815	−1.34920	−0.674598	−	0.738186i	$-0.735683\pi$
$541$	−31.3815	−1.34920	−0.674598	+	0.738186i	$0.735683\pi$
$542$	−25.7518	−1.10613
$543$	0	0
$544$	28.3609	1.21596
$545$	−7.14529	−0.306071
$546$	0	0
$547$	−3.74412	−0.160087	−0.0800436	−	0.996791i	$-0.525506\pi$
$547$	−3.74412	−0.160087	−0.0800436	+	0.996791i	$0.525506\pi$
$548$	−19.3636	−0.827174
$549$	0	0
$550$	2.28632	0.0974892
$551$	6.47310	0.275763
$552$	0	0
$553$	27.7844	1.18151
$554$	66.3411	2.81856
$555$	0	0
$556$	−64.7599	−2.74643
$557$	25.3164	1.07269	0.536345	−	0.843999i	$-0.319804\pi$
$557$	25.3164	1.07269	0.536345	+	0.843999i	$0.319804\pi$
$558$	0	0
$559$	8.96075	0.378999
$560$	−0.0988881	−0.00417879
$561$	0	0
$562$	70.8201	2.98737
$563$	24.7465	1.04294	0.521470	−	0.853269i	$-0.325384\pi$
$563$	24.7465	1.04294	0.521470	+	0.853269i	$0.325384\pi$
$564$	0	0
$565$	2.58470	0.108739
$566$	54.4032	2.28674
$567$	0	0
$568$	4.33645	0.181953
$569$	44.0148	1.84520	0.922598	−	0.385763i	$-0.126062\pi$
$569$	44.0148	1.84520	0.922598	+	0.385763i	$0.126062\pi$
$570$	0	0
$571$	−2.01205	−0.0842017	−0.0421009	−	0.999113i	$-0.513405\pi$
$571$	−2.01205	−0.0842017	−0.0421009	+	0.999113i	$0.513405\pi$
$572$	−19.5348	−0.816790
$573$	0	0
$574$	−57.9895	−2.42044
$575$	−4.45455	−0.185768
$576$	0	0
$577$	26.3181	1.09564	0.547819	−	0.836597i	$-0.315458\pi$
$577$	26.3181	1.09564	0.547819	+	0.836597i	$0.315458\pi$
$578$	−17.7016	−0.736291
$579$	0	0
$580$	−2.97280	−0.123439
$581$	21.4359	0.889310
$582$	0	0
$583$	−1.54545	−0.0640060
$584$	−16.9845	−0.702822
$585$	0	0
$586$	−2.28238	−0.0942842
$587$	−35.8112	−1.47809	−0.739044	−	0.673657i	$-0.764722\pi$
$587$	−35.8112	−1.47809	−0.739044	+	0.673657i	$0.764722\pi$
$588$	0	0
$589$	−21.3574	−0.880016
$590$	−17.3757	−0.715346
$591$	0	0
$592$	0.137122	0.00563567
$593$	7.61953	0.312896	0.156448	−	0.987686i	$-0.449996\pi$
$593$	7.61953	0.312896	0.156448	+	0.987686i	$0.449996\pi$
$594$	0	0
$595$	12.5330	0.513805
$596$	−68.6228	−2.81090
$597$	0	0
$598$	61.6473	2.52094
$599$	−40.2121	−1.64302	−0.821511	−	0.570193i	$-0.806868\pi$
$599$	−40.2121	−1.64302	−0.821511	+	0.570193i	$0.806868\pi$
$600$	0	0
$601$	26.3181	1.07354	0.536770	−	0.843729i	$-0.319644\pi$
$601$	26.3181	1.07354	0.536770	+	0.843729i	$0.319644\pi$
$602$	−8.52797	−0.347574
$603$	0	0
$604$	−22.6787	−0.922783
$605$	1.00000	0.0406558
$606$	0	0
$607$	11.4287	0.463878	0.231939	−	0.972730i	$-0.425493\pi$
$607$	11.4287	0.463878	0.231939	+	0.972730i	$0.425493\pi$
$608$	−40.0664	−1.62491
$609$	0	0
$610$	−23.1057	−0.935522
$611$	49.0661	1.98500
$612$	0	0
$613$	−21.1198	−0.853022	−0.426511	−	0.904482i	$-0.640257\pi$
$613$	−21.1198	−0.853022	−0.426511	+	0.904482i	$0.640257\pi$
$614$	63.4934	2.56239
$615$	0	0
$616$	7.06993	0.284856
$617$	−31.3905	−1.26373	−0.631867	−	0.775077i	$-0.717711\pi$
$617$	−31.3905	−1.26373	−0.631867	+	0.775077i	$0.717711\pi$
$618$	0	0
$619$	−7.11845	−0.286115	−0.143057	−	0.989714i	$-0.545693\pi$
$619$	−7.11845	−0.286115	−0.143057	+	0.989714i	$0.545693\pi$
$620$	9.80849	0.393918
$621$	0	0
$622$	−40.0595	−1.60624
$623$	−38.1604	−1.52887
$624$	0	0
$625$	1.00000	0.0400000
$626$	−12.3811	−0.494850
$627$	0	0
$628$	68.0750	2.71649
$629$	−17.3787	−0.692936
$630$	0	0
$631$	21.3574	0.850224	0.425112	−	0.905141i	$-0.360235\pi$
$631$	21.3574	0.850224	0.425112	+	0.905141i	$0.360235\pi$
$632$	−30.9417	−1.23080
$633$	0	0
$634$	15.5397	0.617159
$635$	11.4287	0.453535
$636$	0	0
$637$	−3.94353	−0.156248
$638$	−2.10605	−0.0833792
$639$	0	0
$640$	18.2212	0.720258
$641$	26.1329	1.03219	0.516093	−	0.856532i	$-0.327386\pi$
$641$	26.1329	1.03219	0.516093	+	0.856532i	$0.327386\pi$
$642$	0	0
$643$	30.5358	1.20421	0.602107	−	0.798416i	$-0.294328\pi$
$643$	30.5358	1.20421	0.602107	+	0.798416i	$0.294328\pi$
$644$	−36.2222	−1.42736
$645$	0	0
$646$	79.9173	3.14430
$647$	−11.5482	−0.454006	−0.227003	−	0.973894i	$-0.572893\pi$
$647$	−11.5482	−0.454006	−0.227003	+	0.973894i	$0.572893\pi$
$648$	0	0
$649$	−7.59984	−0.298320
$650$	−13.8392	−0.542817
$651$	0	0
$652$	−24.6935	−0.967071
$653$	−22.2389	−0.870277	−0.435138	−	0.900364i	$-0.643301\pi$
$653$	−22.2389	−0.870277	−0.435138	+	0.900364i	$0.643301\pi$
$654$	0	0
$655$	−8.10605	−0.316729
$656$	−0.395079	−0.0154253
$657$	0	0
$658$	−46.6963	−1.82041
$659$	39.4359	1.53620	0.768102	−	0.640328i	$-0.221201\pi$
$659$	39.4359	1.53620	0.768102	+	0.640328i	$0.221201\pi$
$660$	0	0
$661$	21.8967	0.851683	0.425841	−	0.904798i	$-0.359978\pi$
$661$	21.8967	0.851683	0.425841	+	0.904798i	$0.359978\pi$
$662$	16.2751	0.632549
$663$	0	0
$664$	−23.8718	−0.926405
$665$	−17.7059	−0.686605
$666$	0	0
$667$	4.10331	0.158881
$668$	11.2710	0.436089
$669$	0	0
$670$	19.8699	0.767639
$671$	−10.1060	−0.390140
$672$	0	0
$673$	−18.2920	−0.705103	−0.352552	−	0.935792i	$-0.614686\pi$
$673$	−18.2920	−0.705103	−0.352552	+	0.935792i	$0.614686\pi$
$674$	−56.9534	−2.19376
$675$	0	0
$676$	76.2898	2.93422
$677$	0.817184	0.0314069	0.0157035	−	0.999877i	$-0.495001\pi$
$677$	0.817184	0.0314069	0.0157035	+	0.999877i	$0.495001\pi$
$678$	0	0
$679$	−29.2272	−1.12164
$680$	−13.9573	−0.535237
$681$	0	0
$682$	6.94870	0.266080
$683$	−12.8031	−0.489895	−0.244948	−	0.969536i	$-0.578771\pi$
$683$	−12.8031	−0.489895	−0.244948	+	0.969536i	$0.578771\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	44.0778	1.68290
$687$	0	0
$688$	−0.0581006	−0.00221506
$689$	9.35465	0.356384
$690$	0	0
$691$	−22.9394	−0.872656	−0.436328	−	0.899788i	$-0.643721\pi$
$691$	−22.9394	−0.872656	−0.436328	+	0.899788i	$0.643721\pi$
$692$	54.7803	2.08244
$693$	0	0
$694$	−54.0151	−2.05039
$695$	−20.0664	−0.761164
$696$	0	0
$697$	50.0722	1.89662
$698$	71.2696	2.69759
$699$	0	0
$700$	8.13152	0.307342
$701$	−32.3329	−1.22120	−0.610599	−	0.791940i	$-0.709071\pi$
$701$	−32.3329	−1.22120	−0.610599	+	0.791940i	$0.709071\pi$
$702$	0	0
$703$	24.5516	0.925981
$704$	12.9573	0.488345
$705$	0	0
$706$	−1.21871	−0.0458668
$707$	0.297561	0.0111909
$708$	0	0
$709$	−9.25760	−0.347677	−0.173838	−	0.984774i	$-0.555617\pi$
$709$	−9.25760	−0.347677	−0.173838	+	0.984774i	$0.555617\pi$
$710$	3.53340	0.132606
$711$	0	0
$712$	42.4969	1.59264
$713$	−13.5385	−0.507020
$714$	0	0
$715$	−6.05302	−0.226370
$716$	−29.5144	−1.10300
$717$	0	0
$718$	−27.4972	−1.02619
$719$	−7.01170	−0.261492	−0.130746	−	0.991416i	$-0.541737\pi$
$719$	−7.01170	−0.261492	−0.130746	+	0.991416i	$0.541737\pi$
$720$	0	0
$721$	26.8065	0.998326
$722$	−69.4619	−2.58510
$723$	0	0
$724$	73.0626	2.71535
$725$	−0.921150	−0.0342106
$726$	0	0
$727$	18.0303	0.668706	0.334353	−	0.942448i	$-0.391482\pi$
$727$	18.0303	0.668706	0.334353	+	0.942448i	$0.391482\pi$
$728$	−42.7945	−1.58607
$729$	0	0
$730$	−13.8392	−0.512211
$731$	7.36365	0.272354
$732$	0	0
$733$	−22.9890	−0.849117	−0.424558	−	0.905401i	$-0.639571\pi$
$733$	−22.9890	−0.849117	−0.424558	+	0.905401i	$0.639571\pi$
$734$	34.2051	1.26253
$735$	0	0
$736$	−25.3982	−0.936189
$737$	8.69074	0.320128
$738$	0	0
$739$	−30.0967	−1.10713	−0.553563	−	0.832807i	$-0.686732\pi$
$739$	−30.0967	−1.10713	−0.553563	+	0.832807i	$0.686732\pi$
$740$	−11.2754	−0.414493
$741$	0	0
$742$	−8.90284	−0.326833
$743$	32.7706	1.20224	0.601119	−	0.799160i	$-0.294722\pi$
$743$	32.7706	1.20224	0.601119	+	0.799160i	$0.294722\pi$
$744$	0	0
$745$	−21.2634	−0.779030
$746$	−46.0272	−1.68517
$747$	0	0
$748$	−16.0530	−0.586957
$749$	−47.0971	−1.72089
$750$	0	0
$751$	−0.132887	−0.00484912	−0.00242456	−	0.999997i	$-0.500772\pi$
$751$	−0.132887	−0.00484912	−0.00242456	+	0.999997i	$0.500772\pi$
$752$	−0.318139	−0.0116014
$753$	0	0
$754$	12.7479	0.464253
$755$	−7.02720	−0.255746
$756$	0	0
$757$	−40.1363	−1.45878	−0.729390	−	0.684098i	$-0.760196\pi$
$757$	−40.1363	−1.45878	−0.729390	+	0.684098i	$0.760196\pi$
$758$	−36.3450	−1.32011
$759$	0	0
$760$	19.7179	0.715245
$761$	−30.9487	−1.12189	−0.560945	−	0.827853i	$-0.689562\pi$
$761$	−30.9487	−1.12189	−0.560945	+	0.827853i	$0.689562\pi$
$762$	0	0
$763$	−18.0035	−0.651769
$764$	−39.4896	−1.42868
$765$	0	0
$766$	−6.03381	−0.218011
$767$	46.0020	1.66104
$768$	0	0
$769$	−7.17213	−0.258634	−0.129317	−	0.991603i	$-0.541278\pi$
$769$	−7.17213	−0.258634	−0.129317	+	0.991603i	$0.541278\pi$
$770$	5.76067	0.207600
$771$	0	0
$772$	3.34952	0.120552
$773$	15.3546	0.552268	0.276134	−	0.961119i	$-0.410947\pi$
$773$	15.3546	0.552268	0.276134	+	0.961119i	$0.410947\pi$
$774$	0	0
$775$	3.03925	0.109173
$776$	32.5485	1.16842
$777$	0	0
$778$	31.6480	1.13463
$779$	−70.7389	−2.53448
$780$	0	0
$781$	1.54545	0.0553006
$782$	50.6597	1.81159
$783$	0	0
$784$	0.0255694	0.000913193 0
$785$	21.0936	0.752864
$786$	0	0
$787$	10.8619	0.387184	0.193592	−	0.981082i	$-0.437986\pi$
$787$	10.8619	0.387184	0.193592	+	0.981082i	$0.437986\pi$
$788$	26.0282	0.927217
$789$	0	0
$790$	−25.2117	−0.896993
$791$	6.51247	0.231557
$792$	0	0
$793$	61.1721	2.17229
$794$	58.7090	2.08350
$795$	0	0
$796$	−42.4235	−1.50366
$797$	40.7720	1.44422	0.722109	−	0.691780i	$-0.243173\pi$
$797$	40.7720	1.44422	0.722109	+	0.691780i	$0.243173\pi$
$798$	0	0
$799$	40.3209	1.42645
$800$	5.70163	0.201583
$801$	0	0
$802$	71.1531	2.51250
$803$	−6.05302	−0.213607
$804$	0	0
$805$	−11.2238	−0.395587
$806$	−42.0607	−1.48152
$807$	0	0
$808$	−0.331375	−0.0116577
$809$	14.7634	0.519055	0.259528	−	0.965736i	$-0.416433\pi$
$809$	14.7634	0.519055	0.259528	+	0.965736i	$0.416433\pi$
$810$	0	0
$811$	3.93356	0.138126	0.0690629	−	0.997612i	$-0.477999\pi$
$811$	3.93356	0.138126	0.0690629	+	0.997612i	$0.477999\pi$
$812$	−7.49035	−0.262860
$813$	0	0
$814$	−7.98795	−0.279977
$815$	−7.65150	−0.268020
$816$	0	0
$817$	−10.4029	−0.363951
$818$	−29.1538	−1.01934
$819$	0	0
$820$	32.4872	1.13450
$821$	−20.4603	−0.714071	−0.357035	−	0.934091i	$-0.616212\pi$
$821$	−20.4603	−0.714071	−0.357035	+	0.934091i	$0.616212\pi$
$822$	0	0
$823$	23.7603	0.828231	0.414116	−	0.910224i	$-0.364091\pi$
$823$	23.7603	0.828231	0.414116	+	0.910224i	$0.364091\pi$
$824$	−29.8527	−1.03997
$825$	0	0
$826$	−43.7802	−1.52331
$827$	−30.1253	−1.04756	−0.523779	−	0.851854i	$-0.675478\pi$
$827$	−30.1253	−1.04756	−0.523779	+	0.851854i	$0.675478\pi$
$828$	0	0
$829$	37.8388	1.31420	0.657098	−	0.753806i	$-0.271784\pi$
$829$	37.8388	1.31420	0.657098	+	0.753806i	$0.271784\pi$
$830$	−19.4511	−0.675156
$831$	0	0
$832$	−78.4306	−2.71909
$833$	−3.24066	−0.112282
$834$	0	0
$835$	3.49243	0.120860
$836$	22.6787	0.784359
$837$	0	0
$838$	0.540017	0.0186546
$839$	−45.7871	−1.58075	−0.790374	−	0.612625i	$-0.790114\pi$
$839$	−45.7871	−1.58075	−0.790374	+	0.612625i	$0.790114\pi$
$840$	0	0
$841$	−28.1515	−0.970741
$842$	49.6818	1.71215
$843$	0	0
$844$	−9.84738	−0.338961
$845$	23.6391	0.813209
$846$	0	0
$847$	2.51962	0.0865753
$848$	−0.0606546	−0.00208289
$849$	0	0
$850$	−11.3726	−0.390076
$851$	15.5633	0.533503
$852$	0	0
$853$	23.3312	0.798845	0.399423	−	0.916767i	$-0.369211\pi$
$853$	23.3312	0.798845	0.399423	+	0.916767i	$0.369211\pi$
$854$	−58.2176	−1.99217
$855$	0	0
$856$	52.4491	1.79267
$857$	−16.0892	−0.549596	−0.274798	−	0.961502i	$-0.588611\pi$
$857$	−16.0892	−0.549596	−0.274798	+	0.961502i	$0.588611\pi$
$858$	0	0
$859$	13.1969	0.450274	0.225137	−	0.974327i	$-0.427717\pi$
$859$	13.1969	0.450274	0.225137	+	0.974327i	$0.427717\pi$
$860$	4.77758	0.162914
$861$	0	0
$862$	−37.0661	−1.26248
$863$	−48.5606	−1.65302	−0.826511	−	0.562921i	$-0.809678\pi$
$863$	−48.5606	−1.65302	−0.826511	+	0.562921i	$0.809678\pi$
$864$	0	0
$865$	16.9742	0.577139
$866$	−30.9142	−1.05051
$867$	0	0
$868$	24.7137	0.838837
$869$	−11.0272	−0.374072
$870$	0	0
$871$	−52.6053	−1.78246
$872$	20.0493	0.678955
$873$	0	0
$874$	−71.5688	−2.42085
$875$	2.51962	0.0851788
$876$	0	0
$877$	17.4862	0.590466	0.295233	−	0.955425i	$-0.404603\pi$
$877$	17.4862	0.590466	0.295233	+	0.955425i	$0.404603\pi$
$878$	−0.151911	−0.00512676
$879$	0	0
$880$	0.0392472	0.00132302
$881$	−23.2238	−0.782429	−0.391215	−	0.920299i	$-0.627945\pi$
$881$	−23.2238	−0.782429	−0.391215	+	0.920299i	$0.627945\pi$
$882$	0	0
$883$	39.8967	1.34263	0.671315	−	0.741172i	$-0.265730\pi$
$883$	39.8967	1.34263	0.671315	+	0.741172i	$0.265730\pi$
$884$	97.1693	3.26816
$885$	0	0
$886$	−1.28170	−0.0430597
$887$	13.6770	0.459228	0.229614	−	0.973282i	$-0.426254\pi$
$887$	13.6770	0.459228	0.229614	+	0.973282i	$0.426254\pi$
$888$	0	0
$889$	28.7961	0.965789
$890$	34.6270	1.16070
$891$	0	0
$892$	−45.0151	−1.50722
$893$	−56.9628	−1.90619
$894$	0	0
$895$	−9.14529	−0.305693
$896$	45.9107	1.53377
$897$	0	0
$898$	22.6821	0.756913
$899$	−2.79960	−0.0933720
$900$	0	0
$901$	7.68734	0.256102
$902$	23.0151	0.766321
$903$	0	0
$904$	−7.25252	−0.241215
$905$	22.6391	0.752549
$906$	0	0
$907$	1.73625	0.0576513	0.0288257	−	0.999584i	$-0.490823\pi$
$907$	1.73625	0.0576513	0.0288257	+	0.999584i	$0.490823\pi$
$908$	95.6979	3.17585
$909$	0	0
$910$	−34.8695	−1.15591
$911$	49.4662	1.63889	0.819444	−	0.573160i	$-0.194283\pi$
$911$	49.4662	1.63889	0.819444	+	0.573160i	$0.194283\pi$
$912$	0	0
$913$	−8.50757	−0.281560
$914$	−20.4273	−0.675676
$915$	0	0
$916$	58.7754	1.94199
$917$	−20.4242	−0.674466
$918$	0	0
$919$	40.0423	1.32087	0.660437	−	0.750881i	$-0.270371\pi$
$919$	40.0423	1.32087	0.660437	+	0.750881i	$0.270371\pi$
$920$	12.4992	0.412087
$921$	0	0
$922$	78.0743	2.57124
$923$	−9.35465	−0.307912
$924$	0	0
$925$	−3.49380	−0.114875
$926$	57.0116	1.87352
$927$	0	0
$928$	−5.25205	−0.172407
$929$	−29.0420	−0.952837	−0.476418	−	0.879219i	$-0.658065\pi$
$929$	−29.0420	−0.952837	−0.476418	+	0.879219i	$0.658065\pi$
$930$	0	0
$931$	4.57820	0.150044
$932$	−42.2135	−1.38275
$933$	0	0
$934$	−6.51247	−0.213094
$935$	−4.97417	−0.162673
$936$	0	0
$937$	40.0282	1.30766	0.653832	−	0.756639i	$-0.273160\pi$
$937$	40.0282	1.30766	0.653832	+	0.756639i	$0.273160\pi$
$938$	50.0645	1.63467
$939$	0	0
$940$	26.1604	0.853259
$941$	−13.9604	−0.455096	−0.227548	−	0.973767i	$-0.573071\pi$
$941$	−13.9604	−0.455096	−0.227548	+	0.973767i	$0.573071\pi$
$942$	0	0
$943$	−44.8415	−1.46024
$944$	−0.298272	−0.00970794
$945$	0	0
$946$	3.38462	0.110043
$947$	−17.4875	−0.568269	−0.284134	−	0.958785i	$-0.591706\pi$
$947$	−17.4875	−0.568269	−0.284134	+	0.958785i	$0.591706\pi$
$948$	0	0
$949$	36.6391	1.18936
$950$	16.0664	0.521264
$951$	0	0
$952$	−35.1671	−1.13977
$953$	49.1890	1.59339	0.796694	−	0.604383i	$-0.206580\pi$
$953$	49.1890	1.59339	0.796694	+	0.604383i	$0.206580\pi$
$954$	0	0
$955$	−12.2362	−0.395954
$956$	−2.59167	−0.0838206
$957$	0	0
$958$	−34.5640	−1.11671
$959$	−15.1177	−0.488177
$960$	0	0
$961$	−21.7630	−0.702032
$962$	48.3512	1.55891
$963$	0	0
$964$	9.80849	0.315910
$965$	1.03788	0.0334105
$966$	0	0
$967$	−23.5589	−0.757602	−0.378801	−	0.925478i	$-0.623664\pi$
$967$	−23.5589	−0.757602	−0.378801	+	0.925478i	$0.623664\pi$
$968$	−2.80595	−0.0901866
$969$	0	0
$970$	26.5210	0.851538
$971$	5.16940	0.165894	0.0829469	−	0.996554i	$-0.473567\pi$
$971$	5.16940	0.165894	0.0829469	+	0.996554i	$0.473567\pi$
$972$	0	0
$973$	−50.5599	−1.62088
$974$	10.3656	0.332136
$975$	0	0
$976$	−0.396634	−0.0126959
$977$	9.73970	0.311601	0.155800	−	0.987789i	$-0.450204\pi$
$977$	9.73970	0.311601	0.155800	+	0.987789i	$0.450204\pi$
$978$	0	0
$979$	15.1453	0.484046
$980$	−2.10256	−0.0671638
$981$	0	0
$982$	−34.5027	−1.10102
$983$	−2.16670	−0.0691070	−0.0345535	−	0.999403i	$-0.511001\pi$
$983$	−2.16670	−0.0691070	−0.0345535	+	0.999403i	$0.511001\pi$
$984$	0	0
$985$	8.06507	0.256975
$986$	10.4758	0.333619
$987$	0	0
$988$	−137.275	−4.36729
$989$	−6.59441	−0.209690
$990$	0	0
$991$	−25.2265	−0.801347	−0.400674	−	0.916221i	$-0.631224\pi$
$991$	−25.2265	−0.801347	−0.400674	+	0.916221i	$0.631224\pi$
$992$	17.3286	0.550185
$993$	0	0
$994$	8.90284	0.282381
$995$	−13.1453	−0.416734
$996$	0	0
$997$	29.5950	0.937281	0.468641	−	0.883389i	$-0.344744\pi$
$997$	29.5950	0.937281	0.468641	+	0.883389i	$0.344744\pi$
$998$	20.4242	0.646516
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.2.a.g.1.1	yes	4
3.2	odd	2		495.2.a.f.1.4	✓	4
4.3	odd	2		7920.2.a.cn.1.2		4
5.2	odd	4		2475.2.c.s.199.2		8
5.3	odd	4		2475.2.c.s.199.7		8
5.4	even	2		2475.2.a.bf.1.4		4
11.10	odd	2		5445.2.a.bh.1.4		4
12.11	even	2		7920.2.a.cm.1.2		4
15.2	even	4		2475.2.c.t.199.7		8
15.8	even	4		2475.2.c.t.199.2		8
15.14	odd	2		2475.2.a.bj.1.1		4
33.32	even	2		5445.2.a.bs.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.a.f.1.4	✓	4	3.2	odd	2
495.2.a.g.1.1	yes	4	1.1	even	1	trivial
2475.2.a.bf.1.4		4	5.4	even	2
2475.2.a.bj.1.1		4	15.14	odd	2
2475.2.c.s.199.2		8	5.2	odd	4
2475.2.c.s.199.7		8	5.3	odd	4
2475.2.c.t.199.2		8	15.8	even	4
2475.2.c.t.199.7		8	15.2	even	4
5445.2.a.bh.1.4		4	11.10	odd	2
5445.2.a.bs.1.1		4	33.32	even	2
7920.2.a.cm.1.2		4	12.11	even	2
7920.2.a.cn.1.2		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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-2.28632 q^{2} +3.22727 q^{4} +1.00000 q^{5} +2.51962 q^{7} -2.80595 q^{8} +O(q^{10})\)	\(q-2.28632 q^{2} +3.22727 q^{4} +1.00000 q^{5} +2.51962 q^{7} -2.80595 q^{8} -2.28632 q^{10} -1.00000 q^{11} +6.05302 q^{13} -5.76067 q^{14} -0.0392472 q^{16} +4.97417 q^{17} -7.02720 q^{19} +3.22727 q^{20} +2.28632 q^{22} -4.45455 q^{23} +1.00000 q^{25} -13.8392 q^{26} +8.13152 q^{28} -0.921150 q^{29} +3.03925 q^{31} +5.70163 q^{32} -11.3726 q^{34} +2.51962 q^{35} -3.49380 q^{37} +16.0664 q^{38} -2.80595 q^{40} +10.0664 q^{41} +1.48038 q^{43} -3.22727 q^{44} +10.1845 q^{46} +8.10605 q^{47} -0.651497 q^{49} -2.28632 q^{50} +19.5348 q^{52} +1.54545 q^{53} -1.00000 q^{55} -7.06993 q^{56} +2.10605 q^{58} +7.59984 q^{59} +10.1060 q^{61} -6.94870 q^{62} -12.9573 q^{64} +6.05302 q^{65} -8.69074 q^{67} +16.0530 q^{68} -5.76067 q^{70} -1.54545 q^{71} +6.05302 q^{73} +7.98795 q^{74} -22.6787 q^{76} -2.51962 q^{77} +11.0272 q^{79} -0.0392472 q^{80} -23.0151 q^{82} +8.50757 q^{83} +4.97417 q^{85} -3.38462 q^{86} +2.80595 q^{88} -15.1453 q^{89} +15.2513 q^{91} -14.3761 q^{92} -18.5330 q^{94} -7.02720 q^{95} -11.5998 q^{97} +1.48953 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8	\( 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8} + 2 q^{10} - 4 q^{11} + 8 q^{13} - 8 q^{14} + 12 q^{16} + 4 q^{17} + 4 q^{19} + 8 q^{20} - 2 q^{22} - 8 q^{23} + 4 q^{25} - 16 q^{26} - 8 q^{28} - 4 q^{29} + 14 q^{32} + 4 q^{34} + 4 q^{35} + 8 q^{37} + 20 q^{38} + 6 q^{40} - 4 q^{41} + 12 q^{43} - 8 q^{44} - 16 q^{46} + 20 q^{49} + 2 q^{50} + 20 q^{52} + 16 q^{53} - 4 q^{55} - 48 q^{56} - 24 q^{58} - 24 q^{59} + 8 q^{61} - 20 q^{62} + 8 q^{65} + 48 q^{68} - 8 q^{70} - 16 q^{71} + 8 q^{73} + 12 q^{74} - 36 q^{76} - 4 q^{77} + 12 q^{79} + 12 q^{80} - 40 q^{82} + 8 q^{83} + 4 q^{85} + 16 q^{86} - 6 q^{88} - 16 q^{89} - 16 q^{91} - 72 q^{92} - 40 q^{94} + 4 q^{95} + 8 q^{97} + 62 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8 + 2 * q^10 - 4 * q^11 + 8 * q^13 - 8 * q^14 + 12 * q^16 + 4 * q^17 + 4 * q^19 + 8 * q^20 - 2 * q^22 - 8 * q^23 + 4 * q^25 - 16 * q^26 - 8 * q^28 - 4 * q^29 + 14 * q^32 + 4 * q^34 + 4 * q^35 + 8 * q^37 + 20 * q^38 + 6 * q^40 - 4 * q^41 + 12 * q^43 - 8 * q^44 - 16 * q^46 + 20 * q^49 + 2 * q^50 + 20 * q^52 + 16 * q^53 - 4 * q^55 - 48 * q^56 - 24 * q^58 - 24 * q^59 + 8 * q^61 - 20 * q^62 + 8 * q^65 + 48 * q^68 - 8 * q^70 - 16 * q^71 + 8 * q^73 + 12 * q^74 - 36 * q^76 - 4 * q^77 + 12 * q^79 + 12 * q^80 - 40 * q^82 + 8 * q^83 + 4 * q^85 + 16 * q^86 - 6 * q^88 - 16 * q^89 - 16 * q^91 - 72 * q^92 - 40 * q^94 + 4 * q^95 + 8 * q^97 + 62 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more