LMFDB - Embedded newform 1148.4.a.d.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.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:	$67.7341926866$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 238 x^{13} + 602 x^{12} + 21013 x^{11} - 44923 x^{10} - 876344 x^{9} + \cdots - 45134496 $ x^15 - 3x^14 - 238x^13 + 602x^12 + 21013x^11 - 44923x^10 - 876344x^9 + 1512470x^8 + 18440602x^7 - 23548938x^6 - 192795335x^5 + 157072885x^4 + 930883767x^3 - 339053445x^2 - 1549251522x - 45134496
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.86399$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.86399 q^{3} -6.90167 q^{5} -7.00000 q^{7} -12.0696 q^{9} +O(q^{10})$	$q+3.86399 q^{3} -6.90167 q^{5} -7.00000 q^{7} -12.0696 q^{9} -30.9082 q^{11} +53.9527 q^{13} -26.6680 q^{15} +116.266 q^{17} -2.79364 q^{19} -27.0479 q^{21} +118.160 q^{23} -77.3669 q^{25} -150.964 q^{27} -112.011 q^{29} -213.969 q^{31} -119.429 q^{33} +48.3117 q^{35} -428.057 q^{37} +208.473 q^{39} +41.0000 q^{41} +457.364 q^{43} +83.3005 q^{45} +520.909 q^{47} +49.0000 q^{49} +449.248 q^{51} +133.670 q^{53} +213.318 q^{55} -10.7946 q^{57} +44.0745 q^{59} +836.476 q^{61} +84.4873 q^{63} -372.364 q^{65} +274.331 q^{67} +456.569 q^{69} +213.215 q^{71} -132.666 q^{73} -298.945 q^{75} +216.357 q^{77} -398.982 q^{79} -257.445 q^{81} +1119.50 q^{83} -802.427 q^{85} -432.807 q^{87} +1212.09 q^{89} -377.669 q^{91} -826.771 q^{93} +19.2808 q^{95} +993.197 q^{97} +373.050 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.86399	0.743624	0.371812	−	0.928308i	$-0.378737\pi$
$3$	3.86399	0.743624	0.371812	+	0.928308i	$0.378737\pi$
$4$	0	0
$5$	−6.90167	−0.617304	−0.308652	−	0.951175i	$-0.599878\pi$
$5$	−6.90167	−0.617304	−0.308652	+	0.951175i	$0.599878\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−12.0696	−0.447023
$10$	0	0
$11$	−30.9082	−0.847197	−0.423599	−	0.905850i	$-0.639233\pi$
$11$	−30.9082	−0.847197	−0.423599	+	0.905850i	$0.639233\pi$
$12$	0	0
$13$	53.9527	1.15106	0.575530	−	0.817780i	$-0.304796\pi$
$13$	53.9527	1.15106	0.575530	+	0.817780i	$0.304796\pi$
$14$	0	0
$15$	−26.6680	−0.459042
$16$	0	0
$17$	116.266	1.65874	0.829369	−	0.558701i	$-0.188700\pi$
$17$	116.266	1.65874	0.829369	+	0.558701i	$0.188700\pi$
$18$	0	0
$19$	−2.79364	−0.0337319	−0.0168659	−	0.999858i	$-0.505369\pi$
$19$	−2.79364	−0.0337319	−0.0168659	+	0.999858i	$0.505369\pi$
$20$	0	0
$21$	−27.0479	−0.281064
$22$	0	0
$23$	118.160	1.07122	0.535611	−	0.844465i	$-0.320081\pi$
$23$	118.160	1.07122	0.535611	+	0.844465i	$0.320081\pi$
$24$	0	0
$25$	−77.3669	−0.618936
$26$	0	0
$27$	−150.964	−1.07604
$28$	0	0
$29$	−112.011	−0.717236	−0.358618	−	0.933484i	$-0.616752\pi$
$29$	−112.011	−0.717236	−0.358618	+	0.933484i	$0.616752\pi$
$30$	0	0
$31$	−213.969	−1.23967	−0.619837	−	0.784731i	$-0.712801\pi$
$31$	−213.969	−1.23967	−0.619837	+	0.784731i	$0.712801\pi$
$32$	0	0
$33$	−119.429	−0.629996
$34$	0	0
$35$	48.3117	0.233319
$36$	0	0
$37$	−428.057	−1.90195	−0.950976	−	0.309265i	$-0.899917\pi$
$37$	−428.057	−1.90195	−0.950976	+	0.309265i	$0.899917\pi$
$38$	0	0
$39$	208.473	0.855957
$40$	0	0
$41$	41.0000	0.156174
$41$	41.0000	0.156174
$42$	0	0
$43$	457.364	1.62203	0.811015	−	0.585025i	$-0.198915\pi$
$43$	457.364	1.62203	0.811015	+	0.585025i	$0.198915\pi$
$44$	0	0
$45$	83.3005	0.275949
$46$	0	0
$47$	520.909	1.61665	0.808323	−	0.588739i	$-0.200375\pi$
$47$	520.909	1.61665	0.808323	+	0.588739i	$0.200375\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	449.248	1.23348
$52$	0	0
$53$	133.670	0.346434	0.173217	−	0.984884i	$-0.444584\pi$
$53$	133.670	0.346434	0.173217	+	0.984884i	$0.444584\pi$
$54$	0	0
$55$	213.318	0.522978
$56$	0	0
$57$	−10.7946	−0.0250838
$58$	0	0
$59$	44.0745	0.0972545	0.0486272	−	0.998817i	$-0.484515\pi$
$59$	44.0745	0.0972545	0.0486272	+	0.998817i	$0.484515\pi$
$60$	0	0
$61$	836.476	1.75573	0.877867	−	0.478905i	$-0.158966\pi$
$61$	836.476	1.75573	0.877867	+	0.478905i	$0.158966\pi$
$62$	0	0
$63$	84.4873	0.168959
$64$	0	0
$65$	−372.364	−0.710555
$66$	0	0
$67$	274.331	0.500221	0.250111	−	0.968217i	$-0.419533\pi$
$67$	274.331	0.500221	0.250111	+	0.968217i	$0.419533\pi$
$68$	0	0
$69$	456.569	0.796586
$70$	0	0
$71$	213.215	0.356395	0.178197	−	0.983995i	$-0.442973\pi$
$71$	213.215	0.356395	0.178197	+	0.983995i	$0.442973\pi$
$72$	0	0
$73$	−132.666	−0.212704	−0.106352	−	0.994329i	$-0.533917\pi$
$73$	−132.666	−0.212704	−0.106352	+	0.994329i	$0.533917\pi$
$74$	0	0
$75$	−298.945	−0.460256
$76$	0	0
$77$	216.357	0.320210
$78$	0	0
$79$	−398.982	−0.568215	−0.284108	−	0.958792i	$-0.591697\pi$
$79$	−398.982	−0.568215	−0.284108	+	0.958792i	$0.591697\pi$
$80$	0	0
$81$	−257.445	−0.353148
$82$	0	0
$83$	1119.50	1.48050	0.740251	−	0.672331i	$-0.234707\pi$
$83$	1119.50	1.48050	0.740251	+	0.672331i	$0.234707\pi$
$84$	0	0
$85$	−802.427	−1.02395
$86$	0	0
$87$	−432.807	−0.533354
$88$	0	0
$89$	1212.09	1.44361	0.721803	−	0.692099i	$-0.243314\pi$
$89$	1212.09	1.44361	0.721803	+	0.692099i	$0.243314\pi$
$90$	0	0
$91$	−377.669	−0.435060
$92$	0	0
$93$	−826.771	−0.921852
$94$	0	0
$95$	19.2808	0.0208228
$96$	0	0
$97$	993.197	1.03963	0.519814	−	0.854280i	$-0.326001\pi$
$97$	993.197	1.03963	0.519814	+	0.854280i	$0.326001\pi$
$98$	0	0
$99$	373.050	0.378716
$100$	0	0
$101$	1706.95	1.68167	0.840833	−	0.541294i	$-0.182065\pi$
$101$	1706.95	1.68167	0.840833	+	0.541294i	$0.182065\pi$
$102$	0	0
$103$	1489.57	1.42497	0.712486	−	0.701687i	$-0.247569\pi$
$103$	1489.57	1.42497	0.712486	+	0.701687i	$0.247569\pi$
$104$	0	0
$105$	186.676	0.173502
$106$	0	0
$107$	−947.268	−0.855849	−0.427925	−	0.903814i	$-0.640755\pi$
$107$	−947.268	−0.855849	−0.427925	+	0.903814i	$0.640755\pi$
$108$	0	0
$109$	1962.39	1.72443	0.862215	−	0.506543i	$-0.169077\pi$
$109$	1962.39	1.72443	0.862215	+	0.506543i	$0.169077\pi$
$110$	0	0
$111$	−1654.01	−1.41434
$112$	0	0
$113$	120.537	0.100347	0.0501733	−	0.998741i	$-0.484023\pi$
$113$	120.537	0.100347	0.0501733	+	0.998741i	$0.484023\pi$
$114$	0	0
$115$	−815.502	−0.661270
$116$	0	0
$117$	−651.189	−0.514551
$118$	0	0
$119$	−813.859	−0.626944
$120$	0	0
$121$	−375.684	−0.282257
$122$	0	0
$123$	158.423	0.116135
$124$	0	0
$125$	1396.67	0.999376
$126$	0	0
$127$	−264.867	−0.185064	−0.0925320	−	0.995710i	$-0.529496\pi$
$127$	−264.867	−0.185064	−0.0925320	+	0.995710i	$0.529496\pi$
$128$	0	0
$129$	1767.25	1.20618
$130$	0	0
$131$	−868.580	−0.579299	−0.289650	−	0.957133i	$-0.593539\pi$
$131$	−868.580	−0.579299	−0.289650	+	0.957133i	$0.593539\pi$
$132$	0	0
$133$	19.5555	0.0127494
$134$	0	0
$135$	1041.91	0.664245
$136$	0	0
$137$	−2045.72	−1.27575	−0.637875	−	0.770140i	$-0.720187\pi$
$137$	−2045.72	−1.27575	−0.637875	+	0.770140i	$0.720187\pi$
$138$	0	0
$139$	2378.08	1.45112	0.725561	−	0.688158i	$-0.241581\pi$
$139$	2378.08	1.45112	0.725561	+	0.688158i	$0.241581\pi$
$140$	0	0
$141$	2012.78	1.20218
$142$	0	0
$143$	−1667.58	−0.975175
$144$	0	0
$145$	773.061	0.442753
$146$	0	0
$147$	189.335	0.106232
$148$	0	0
$149$	−1450.00	−0.797238	−0.398619	−	0.917117i	$-0.630510\pi$
$149$	−1450.00	−0.797238	−0.398619	+	0.917117i	$0.630510\pi$
$150$	0	0
$151$	−39.4244	−0.0212471	−0.0106236	−	0.999944i	$-0.503382\pi$
$151$	−39.4244	−0.0212471	−0.0106236	+	0.999944i	$0.503382\pi$
$152$	0	0
$153$	−1403.28	−0.741494
$154$	0	0
$155$	1476.74	0.765256
$156$	0	0
$157$	1928.80	0.980480	0.490240	−	0.871588i	$-0.336909\pi$
$157$	1928.80	0.980480	0.490240	+	0.871588i	$0.336909\pi$
$158$	0	0
$159$	516.500	0.257617
$160$	0	0
$161$	−827.121	−0.404884
$162$	0	0
$163$	1194.88	0.574174	0.287087	−	0.957905i	$-0.407313\pi$
$163$	1194.88	0.574174	0.287087	+	0.957905i	$0.407313\pi$
$164$	0	0
$165$	824.258	0.388899
$166$	0	0
$167$	−749.281	−0.347192	−0.173596	−	0.984817i	$-0.555539\pi$
$167$	−749.281	−0.347192	−0.173596	+	0.984817i	$0.555539\pi$
$168$	0	0
$169$	713.896	0.324941
$170$	0	0
$171$	33.7182	0.0150789
$172$	0	0
$173$	−1592.62	−0.699912	−0.349956	−	0.936766i	$-0.613804\pi$
$173$	−1592.62	−0.699912	−0.349956	+	0.936766i	$0.613804\pi$
$174$	0	0
$175$	541.569	0.233936
$176$	0	0
$177$	170.303	0.0723208
$178$	0	0
$179$	−2639.23	−1.10204	−0.551021	−	0.834491i	$-0.685762\pi$
$179$	−2639.23	−1.10204	−0.551021	+	0.834491i	$0.685762\pi$
$180$	0	0
$181$	3989.17	1.63819	0.819096	−	0.573657i	$-0.194476\pi$
$181$	3989.17	1.63819	0.819096	+	0.573657i	$0.194476\pi$
$182$	0	0
$183$	3232.13	1.30561
$184$	0	0
$185$	2954.31	1.17408
$186$	0	0
$187$	−3593.56	−1.40528
$188$	0	0
$189$	1056.75	0.406705
$190$	0	0
$191$	−778.283	−0.294841	−0.147420	−	0.989074i	$-0.547097\pi$
$191$	−778.283	−0.294841	−0.147420	+	0.989074i	$0.547097\pi$
$192$	0	0
$193$	655.748	0.244569	0.122284	−	0.992495i	$-0.460978\pi$
$193$	655.748	0.244569	0.122284	+	0.992495i	$0.460978\pi$
$194$	0	0
$195$	−1438.81	−0.528386
$196$	0	0
$197$	−4484.79	−1.62197	−0.810984	−	0.585068i	$-0.801068\pi$
$197$	−4484.79	−1.62197	−0.810984	+	0.585068i	$0.801068\pi$
$198$	0	0
$199$	−2668.46	−0.950565	−0.475282	−	0.879833i	$-0.657654\pi$
$199$	−2668.46	−0.950565	−0.475282	+	0.879833i	$0.657654\pi$
$200$	0	0
$201$	1060.01	0.371977
$202$	0	0
$203$	784.075	0.271090
$204$	0	0
$205$	−282.968	−0.0964067
$206$	0	0
$207$	−1426.15	−0.478861
$208$	0	0
$209$	86.3464	0.0285775
$210$	0	0
$211$	5597.32	1.82623	0.913116	−	0.407699i	$-0.133669\pi$
$211$	5597.32	1.82623	0.913116	+	0.407699i	$0.133669\pi$
$212$	0	0
$213$	823.861	0.265024
$214$	0	0
$215$	−3156.57	−1.00129
$216$	0	0
$217$	1497.78	0.468553
$218$	0	0
$219$	−512.620	−0.158172
$220$	0	0
$221$	6272.84	1.90931
$222$	0	0
$223$	2368.81	0.711333	0.355666	−	0.934613i	$-0.384254\pi$
$223$	2368.81	0.711333	0.355666	+	0.934613i	$0.384254\pi$
$224$	0	0
$225$	933.789	0.276678
$226$	0	0
$227$	−45.5253	−0.0133111	−0.00665555	−	0.999978i	$-0.502119\pi$
$227$	−45.5253	−0.0133111	−0.00665555	+	0.999978i	$0.502119\pi$
$228$	0	0
$229$	−3752.37	−1.08281	−0.541406	−	0.840762i	$-0.682108\pi$
$229$	−3752.37	−1.08281	−0.541406	+	0.840762i	$0.682108\pi$
$230$	0	0
$231$	836.001	0.238116
$232$	0	0
$233$	50.8927	0.0143094	0.00715470	−	0.999974i	$-0.497723\pi$
$233$	50.8927	0.0143094	0.00715470	+	0.999974i	$0.497723\pi$
$234$	0	0
$235$	−3595.14	−0.997963
$236$	0	0
$237$	−1541.66	−0.422539
$238$	0	0
$239$	1225.48	0.331671	0.165836	−	0.986153i	$-0.446968\pi$
$239$	1225.48	0.331671	0.165836	+	0.986153i	$0.446968\pi$
$240$	0	0
$241$	−4010.41	−1.07192	−0.535960	−	0.844243i	$-0.680050\pi$
$241$	−4010.41	−1.07192	−0.535960	+	0.844243i	$0.680050\pi$
$242$	0	0
$243$	3081.28	0.813432
$244$	0	0
$245$	−338.182	−0.0881863
$246$	0	0
$247$	−150.725	−0.0388274
$248$	0	0
$249$	4325.75	1.10094
$250$	0	0
$251$	−3727.03	−0.937243	−0.468622	−	0.883399i	$-0.655249\pi$
$251$	−3727.03	−0.937243	−0.468622	+	0.883399i	$0.655249\pi$
$252$	0	0
$253$	−3652.12	−0.907536
$254$	0	0
$255$	−3100.57	−0.761431
$256$	0	0
$257$	−986.314	−0.239395	−0.119698	−	0.992810i	$-0.538193\pi$
$257$	−986.314	−0.239395	−0.119698	+	0.992810i	$0.538193\pi$
$258$	0	0
$259$	2996.40	0.718870
$260$	0	0
$261$	1351.93	0.320621
$262$	0	0
$263$	3532.68	0.828267	0.414133	−	0.910216i	$-0.364085\pi$
$263$	3532.68	0.828267	0.414133	+	0.910216i	$0.364085\pi$
$264$	0	0
$265$	−922.548	−0.213855
$266$	0	0
$267$	4683.48	1.07350
$268$	0	0
$269$	−430.290	−0.0975288	−0.0487644	−	0.998810i	$-0.515528\pi$
$269$	−430.290	−0.0975288	−0.0487644	+	0.998810i	$0.515528\pi$
$270$	0	0
$271$	6608.70	1.48136	0.740682	−	0.671856i	$-0.234502\pi$
$271$	6608.70	1.48136	0.740682	+	0.671856i	$0.234502\pi$
$272$	0	0
$273$	−1459.31	−0.323521
$274$	0	0
$275$	2391.27	0.524360
$276$	0	0
$277$	−6104.99	−1.32424	−0.662118	−	0.749400i	$-0.730342\pi$
$277$	−6104.99	−1.32424	−0.662118	+	0.749400i	$0.730342\pi$
$278$	0	0
$279$	2582.52	0.554163
$280$	0	0
$281$	3885.65	0.824905	0.412453	−	0.910979i	$-0.364672\pi$
$281$	3885.65	0.824905	0.412453	+	0.910979i	$0.364672\pi$
$282$	0	0
$283$	−7873.27	−1.65377	−0.826886	−	0.562369i	$-0.809890\pi$
$283$	−7873.27	−1.65377	−0.826886	+	0.562369i	$0.809890\pi$
$284$	0	0
$285$	74.5007	0.0154844
$286$	0	0
$287$	−287.000	−0.0590281
$288$	0	0
$289$	8604.69	1.75141
$290$	0	0
$291$	3837.70	0.773092
$292$	0	0
$293$	−7538.63	−1.50311	−0.751556	−	0.659670i	$-0.770696\pi$
$293$	−7538.63	−1.50311	−0.751556	+	0.659670i	$0.770696\pi$
$294$	0	0
$295$	−304.188	−0.0600356
$296$	0	0
$297$	4666.04	0.911619
$298$	0	0
$299$	6375.06	1.23304
$300$	0	0
$301$	−3201.55	−0.613070
$302$	0	0
$303$	6595.65	1.25053
$304$	0	0
$305$	−5773.08	−1.08382
$306$	0	0
$307$	−3228.95	−0.600280	−0.300140	−	0.953895i	$-0.597033\pi$
$307$	−3228.95	−0.600280	−0.300140	+	0.953895i	$0.597033\pi$
$308$	0	0
$309$	5755.69	1.05964
$310$	0	0
$311$	6146.50	1.12070	0.560348	−	0.828258i	$-0.310668\pi$
$311$	6146.50	1.12070	0.560348	+	0.828258i	$0.310668\pi$
$312$	0	0
$313$	3733.06	0.674138	0.337069	−	0.941480i	$-0.390564\pi$
$313$	3733.06	0.674138	0.337069	+	0.941480i	$0.390564\pi$
$314$	0	0
$315$	−583.104	−0.104299
$316$	0	0
$317$	−8760.58	−1.55219	−0.776093	−	0.630618i	$-0.782801\pi$
$317$	−8760.58	−1.55219	−0.776093	+	0.630618i	$0.782801\pi$
$318$	0	0
$319$	3462.05	0.607641
$320$	0	0
$321$	−3660.23	−0.636430
$322$	0	0
$323$	−324.804	−0.0559523
$324$	0	0
$325$	−4174.16	−0.712433
$326$	0	0
$327$	7582.65	1.28233
$328$	0	0
$329$	−3646.36	−0.611035
$330$	0	0
$331$	3293.14	0.546851	0.273425	−	0.961893i	$-0.411843\pi$
$331$	3293.14	0.546851	0.273425	+	0.961893i	$0.411843\pi$
$332$	0	0
$333$	5166.49	0.850216
$334$	0	0
$335$	−1893.34	−0.308789
$336$	0	0
$337$	4167.08	0.673576	0.336788	−	0.941580i	$-0.390659\pi$
$337$	4167.08	0.673576	0.336788	+	0.941580i	$0.390659\pi$
$338$	0	0
$339$	465.753	0.0746202
$340$	0	0
$341$	6613.38	1.05025
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−3151.09	−0.491736
$346$	0	0
$347$	5657.31	0.875217	0.437608	−	0.899166i	$-0.355826\pi$
$347$	5657.31	0.875217	0.437608	+	0.899166i	$0.355826\pi$
$348$	0	0
$349$	6549.92	1.00461	0.502305	−	0.864690i	$-0.332485\pi$
$349$	6549.92	1.00461	0.502305	+	0.864690i	$0.332485\pi$
$350$	0	0
$351$	−8144.94	−1.23859
$352$	0	0
$353$	−1253.67	−0.189027	−0.0945133	−	0.995524i	$-0.530129\pi$
$353$	−1253.67	−0.189027	−0.0945133	+	0.995524i	$0.530129\pi$
$354$	0	0
$355$	−1471.54	−0.220004
$356$	0	0
$357$	−3144.74	−0.466211
$358$	0	0
$359$	−8767.17	−1.28890	−0.644448	−	0.764648i	$-0.722913\pi$
$359$	−8767.17	−1.28890	−0.644448	+	0.764648i	$0.722913\pi$
$360$	0	0
$361$	−6851.20	−0.998862
$362$	0	0
$363$	−1451.64	−0.209893
$364$	0	0
$365$	915.617	0.131303
$366$	0	0
$367$	−2006.48	−0.285388	−0.142694	−	0.989767i	$-0.545576\pi$
$367$	−2006.48	−0.285388	−0.142694	+	0.989767i	$0.545576\pi$
$368$	0	0
$369$	−494.854	−0.0698132
$370$	0	0
$371$	−935.691	−0.130940
$372$	0	0
$373$	−3135.73	−0.435286	−0.217643	−	0.976028i	$-0.569837\pi$
$373$	−3135.73	−0.435286	−0.217643	+	0.976028i	$0.569837\pi$
$374$	0	0
$375$	5396.71	0.743160
$376$	0	0
$377$	−6043.28	−0.825583
$378$	0	0
$379$	8755.32	1.18662	0.593312	−	0.804972i	$-0.297820\pi$
$379$	8755.32	1.18662	0.593312	+	0.804972i	$0.297820\pi$
$380$	0	0
$381$	−1023.44	−0.137618
$382$	0	0
$383$	−5990.44	−0.799210	−0.399605	−	0.916688i	$-0.630853\pi$
$383$	−5990.44	−0.799210	−0.399605	+	0.916688i	$0.630853\pi$
$384$	0	0
$385$	−1493.23	−0.197667
$386$	0	0
$387$	−5520.21	−0.725085
$388$	0	0
$389$	−1273.14	−0.165941	−0.0829704	−	0.996552i	$-0.526441\pi$
$389$	−1273.14	−0.165941	−0.0829704	+	0.996552i	$0.526441\pi$
$390$	0	0
$391$	13738.0	1.77688
$392$	0	0
$393$	−3356.18	−0.430781
$394$	0	0
$395$	2753.64	0.350762
$396$	0	0
$397$	14890.2	1.88242	0.941209	−	0.337825i	$-0.109691\pi$
$397$	14890.2	1.88242	0.941209	+	0.337825i	$0.109691\pi$
$398$	0	0
$399$	75.5621	0.00948080
$400$	0	0
$401$	2925.52	0.364324	0.182162	−	0.983269i	$-0.441691\pi$
$401$	2925.52	0.364324	0.182162	+	0.983269i	$0.441691\pi$
$402$	0	0
$403$	−11544.2	−1.42694
$404$	0	0
$405$	1776.80	0.218000
$406$	0	0
$407$	13230.5	1.61133
$408$	0	0
$409$	−15337.7	−1.85428	−0.927141	−	0.374712i	$-0.877741\pi$
$409$	−15337.7	−1.85428	−0.927141	+	0.374712i	$0.877741\pi$
$410$	0	0
$411$	−7904.64	−0.948679
$412$	0	0
$413$	−308.522	−0.0367587
$414$	0	0
$415$	−7726.45	−0.913920
$416$	0	0
$417$	9188.85	1.07909
$418$	0	0
$419$	4928.09	0.574590	0.287295	−	0.957842i	$-0.407244\pi$
$419$	4928.09	0.574590	0.287295	+	0.957842i	$0.407244\pi$
$420$	0	0
$421$	12220.5	1.41470	0.707352	−	0.706861i	$-0.249889\pi$
$421$	12220.5	1.41470	0.707352	+	0.706861i	$0.249889\pi$
$422$	0	0
$423$	−6287.17	−0.722678
$424$	0	0
$425$	−8995.11	−1.02665
$426$	0	0
$427$	−5855.33	−0.663605
$428$	0	0
$429$	−6443.51	−0.725164
$430$	0	0
$431$	−12494.9	−1.39642	−0.698210	−	0.715893i	$-0.746020\pi$
$431$	−12494.9	−1.39642	−0.698210	+	0.715893i	$0.746020\pi$
$432$	0	0
$433$	−2142.48	−0.237785	−0.118893	−	0.992907i	$-0.537934\pi$
$433$	−2142.48	−0.237785	−0.118893	+	0.992907i	$0.537934\pi$
$434$	0	0
$435$	2987.09	0.329242
$436$	0	0
$437$	−330.097	−0.0361343
$438$	0	0
$439$	2117.08	0.230166	0.115083	−	0.993356i	$-0.463287\pi$
$439$	2117.08	0.230166	0.115083	+	0.993356i	$0.463287\pi$
$440$	0	0
$441$	−591.411	−0.0638604
$442$	0	0
$443$	3714.48	0.398375	0.199187	−	0.979961i	$-0.436170\pi$
$443$	3714.48	0.398375	0.199187	+	0.979961i	$0.436170\pi$
$444$	0	0
$445$	−8365.42	−0.891144
$446$	0	0
$447$	−5602.77	−0.592846
$448$	0	0
$449$	11662.2	1.22578	0.612890	−	0.790168i	$-0.290007\pi$
$449$	11662.2	1.22578	0.612890	+	0.790168i	$0.290007\pi$
$450$	0	0
$451$	−1267.24	−0.132310
$452$	0	0
$453$	−152.335	−0.0157999
$454$	0	0
$455$	2606.55	0.268564
$456$	0	0
$457$	−6557.47	−0.671215	−0.335608	−	0.942002i	$-0.608942\pi$
$457$	−6557.47	−0.671215	−0.335608	+	0.942002i	$0.608942\pi$
$458$	0	0
$459$	−17552.0	−1.78487
$460$	0	0
$461$	2112.58	0.213433	0.106716	−	0.994289i	$-0.465966\pi$
$461$	2112.58	0.213433	0.106716	+	0.994289i	$0.465966\pi$
$462$	0	0
$463$	−1655.01	−0.166123	−0.0830615	−	0.996544i	$-0.526470\pi$
$463$	−1655.01	−0.166123	−0.0830615	+	0.996544i	$0.526470\pi$
$464$	0	0
$465$	5706.10	0.569063
$466$	0	0
$467$	−9939.76	−0.984920	−0.492460	−	0.870335i	$-0.663902\pi$
$467$	−9939.76	−0.984920	−0.492460	+	0.870335i	$0.663902\pi$
$468$	0	0
$469$	−1920.32	−0.189066
$470$	0	0
$471$	7452.87	0.729109
$472$	0	0
$473$	−14136.3	−1.37418
$474$	0	0
$475$	216.136	0.0208779
$476$	0	0
$477$	−1613.35	−0.154864
$478$	0	0
$479$	12292.6	1.17258	0.586288	−	0.810103i	$-0.300589\pi$
$479$	12292.6	1.17258	0.586288	+	0.810103i	$0.300589\pi$
$480$	0	0
$481$	−23094.9	−2.18926
$482$	0	0
$483$	−3195.98	−0.301081
$484$	0	0
$485$	−6854.72	−0.641767
$486$	0	0
$487$	598.928	0.0557290	0.0278645	−	0.999612i	$-0.491129\pi$
$487$	598.928	0.0557290	0.0278645	+	0.999612i	$0.491129\pi$
$488$	0	0
$489$	4617.00	0.426970
$490$	0	0
$491$	−12622.7	−1.16019	−0.580097	−	0.814547i	$-0.696985\pi$
$491$	−12622.7	−1.16019	−0.580097	+	0.814547i	$0.696985\pi$
$492$	0	0
$493$	−13023.0	−1.18971
$494$	0	0
$495$	−2574.67	−0.233783
$496$	0	0
$497$	−1492.51	−0.134705
$498$	0	0
$499$	−15411.3	−1.38258	−0.691288	−	0.722579i	$-0.742956\pi$
$499$	−15411.3	−1.38258	−0.691288	+	0.722579i	$0.742956\pi$
$500$	0	0
$501$	−2895.21	−0.258180
$502$	0	0
$503$	19143.4	1.69694	0.848472	−	0.529240i	$-0.177523\pi$
$503$	19143.4	1.69694	0.848472	+	0.529240i	$0.177523\pi$
$504$	0	0
$505$	−11780.8	−1.03810
$506$	0	0
$507$	2758.48	0.241634
$508$	0	0
$509$	13709.9	1.19387	0.596937	−	0.802288i	$-0.296384\pi$
$509$	13709.9	1.19387	0.596937	+	0.802288i	$0.296384\pi$
$510$	0	0
$511$	928.662	0.0803945
$512$	0	0
$513$	421.741	0.0362969
$514$	0	0
$515$	−10280.5	−0.879641
$516$	0	0
$517$	−16100.3	−1.36962
$518$	0	0
$519$	−6153.87	−0.520472
$520$	0	0
$521$	4454.64	0.374590	0.187295	−	0.982304i	$-0.440028\pi$
$521$	4454.64	0.374590	0.187295	+	0.982304i	$0.440028\pi$
$522$	0	0
$523$	14160.6	1.18394	0.591970	−	0.805960i	$-0.298351\pi$
$523$	14160.6	1.18394	0.591970	+	0.805960i	$0.298351\pi$
$524$	0	0
$525$	2092.61	0.173960
$526$	0	0
$527$	−24877.2	−2.05629
$528$	0	0
$529$	1794.82	0.147516
$530$	0	0
$531$	−531.963	−0.0434750
$532$	0	0
$533$	2212.06	0.179766
$534$	0	0
$535$	6537.73	0.528319
$536$	0	0
$537$	−10198.0	−0.819505
$538$	0	0
$539$	−1514.50	−0.121028
$540$	0	0
$541$	−12630.8	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$541$	−12630.8	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$542$	0	0
$543$	15414.1	1.21820
$544$	0	0
$545$	−13543.8	−1.06450
$546$	0	0
$547$	−2268.23	−0.177299	−0.0886495	−	0.996063i	$-0.528255\pi$
$547$	−2268.23	−0.177299	−0.0886495	+	0.996063i	$0.528255\pi$
$548$	0	0
$549$	−10095.9	−0.784853
$550$	0	0
$551$	312.918	0.0241937
$552$	0	0
$553$	2792.87	0.214765
$554$	0	0
$555$	11415.4	0.873076
$556$	0	0
$557$	9908.38	0.753737	0.376869	−	0.926267i	$-0.377001\pi$
$557$	9908.38	0.753737	0.376869	+	0.926267i	$0.377001\pi$
$558$	0	0
$559$	24676.0	1.86706
$560$	0	0
$561$	−13885.5	−1.04500
$562$	0	0
$563$	4501.51	0.336973	0.168487	−	0.985704i	$-0.446112\pi$
$563$	4501.51	0.336973	0.168487	+	0.985704i	$0.446112\pi$
$564$	0	0
$565$	−831.907	−0.0619444
$566$	0	0
$567$	1802.11	0.133477
$568$	0	0
$569$	6860.81	0.505484	0.252742	−	0.967534i	$-0.418668\pi$
$569$	6860.81	0.505484	0.252742	+	0.967534i	$0.418668\pi$
$570$	0	0
$571$	23744.4	1.74023	0.870117	−	0.492845i	$-0.164043\pi$
$571$	23744.4	1.74023	0.870117	+	0.492845i	$0.164043\pi$
$572$	0	0
$573$	−3007.27	−0.219251
$574$	0	0
$575$	−9141.69	−0.663017
$576$	0	0
$577$	15030.5	1.08445	0.542226	−	0.840232i	$-0.317582\pi$
$577$	15030.5	1.08445	0.542226	+	0.840232i	$0.317582\pi$
$578$	0	0
$579$	2533.80	0.181867
$580$	0	0
$581$	−7836.53	−0.559577
$582$	0	0
$583$	−4131.50	−0.293498
$584$	0	0
$585$	4494.29	0.317634
$586$	0	0
$587$	−4983.49	−0.350410	−0.175205	−	0.984532i	$-0.556059\pi$
$587$	−4983.49	−0.350410	−0.175205	+	0.984532i	$0.556059\pi$
$588$	0	0
$589$	597.752	0.0418165
$590$	0	0
$591$	−17329.1	−1.20614
$592$	0	0
$593$	−14887.1	−1.03093	−0.515464	−	0.856911i	$-0.672380\pi$
$593$	−14887.1	−1.03093	−0.515464	+	0.856911i	$0.672380\pi$
$594$	0	0
$595$	5616.99	0.387015
$596$	0	0
$597$	−10310.9	−0.706863
$598$	0	0
$599$	−9282.02	−0.633144	−0.316572	−	0.948569i	$-0.602532\pi$
$599$	−9282.02	−0.633144	−0.316572	+	0.948569i	$0.602532\pi$
$600$	0	0
$601$	−20330.7	−1.37988	−0.689940	−	0.723867i	$-0.742363\pi$
$601$	−20330.7	−1.37988	−0.689940	+	0.723867i	$0.742363\pi$
$602$	0	0
$603$	−3311.07	−0.223610
$604$	0	0
$605$	2592.85	0.174238
$606$	0	0
$607$	12731.8	0.851344	0.425672	−	0.904878i	$-0.360038\pi$
$607$	12731.8	0.851344	0.425672	+	0.904878i	$0.360038\pi$
$608$	0	0
$609$	3029.65	0.201589
$610$	0	0
$611$	28104.5	1.86086
$612$	0	0
$613$	10772.8	0.709806	0.354903	−	0.934903i	$-0.384514\pi$
$613$	10772.8	0.709806	0.354903	+	0.934903i	$0.384514\pi$
$614$	0	0
$615$	−1093.39	−0.0716904
$616$	0	0
$617$	−10991.6	−0.717187	−0.358594	−	0.933494i	$-0.616744\pi$
$617$	−10991.6	−0.717187	−0.358594	+	0.933494i	$0.616744\pi$
$618$	0	0
$619$	−9324.73	−0.605481	−0.302740	−	0.953073i	$-0.597902\pi$
$619$	−9324.73	−0.605481	−0.302740	+	0.953073i	$0.597902\pi$
$620$	0	0
$621$	−17838.0	−1.15268
$622$	0	0
$623$	−8484.60	−0.545631
$624$	0	0
$625$	31.5122	0.00201678
$626$	0	0
$627$	333.641	0.0212510
$628$	0	0
$629$	−49768.4	−3.15484
$630$	0	0
$631$	27943.3	1.76292	0.881461	−	0.472257i	$-0.156561\pi$
$631$	27943.3	1.76292	0.881461	+	0.472257i	$0.156561\pi$
$632$	0	0
$633$	21627.9	1.35803
$634$	0	0
$635$	1828.02	0.114241
$636$	0	0
$637$	2643.68	0.164437
$638$	0	0
$639$	−2573.43	−0.159317
$640$	0	0
$641$	−27959.5	−1.72283	−0.861414	−	0.507904i	$-0.830420\pi$
$641$	−27959.5	−1.72283	−0.861414	+	0.507904i	$0.830420\pi$
$642$	0	0
$643$	−13538.7	−0.830351	−0.415176	−	0.909741i	$-0.636280\pi$
$643$	−13538.7	−0.830351	−0.415176	+	0.909741i	$0.636280\pi$
$644$	0	0
$645$	−12197.0	−0.744581
$646$	0	0
$647$	1342.52	0.0815764	0.0407882	−	0.999168i	$-0.487013\pi$
$647$	1342.52	0.0815764	0.0407882	+	0.999168i	$0.487013\pi$
$648$	0	0
$649$	−1362.26	−0.0823937
$650$	0	0
$651$	5787.40	0.348427
$652$	0	0
$653$	−327.793	−0.0196440	−0.00982200	−	0.999952i	$-0.503126\pi$
$653$	−327.793	−0.0196440	−0.00982200	+	0.999952i	$0.503126\pi$
$654$	0	0
$655$	5994.66	0.357604
$656$	0	0
$657$	1601.23	0.0950835
$658$	0	0
$659$	11784.1	0.696576	0.348288	−	0.937387i	$-0.386763\pi$
$659$	11784.1	0.696576	0.348288	+	0.937387i	$0.386763\pi$
$660$	0	0
$661$	1940.34	0.114176	0.0570882	−	0.998369i	$-0.481818\pi$
$661$	1940.34	0.114176	0.0570882	+	0.998369i	$0.481818\pi$
$662$	0	0
$663$	24238.2	1.41981
$664$	0	0
$665$	−134.966	−0.00787029
$666$	0	0
$667$	−13235.2	−0.768319
$668$	0	0
$669$	9153.04	0.528964
$670$	0	0
$671$	−25853.9	−1.48745
$672$	0	0
$673$	2299.56	0.131711	0.0658554	−	0.997829i	$-0.479022\pi$
$673$	2299.56	0.131711	0.0658554	+	0.997829i	$0.479022\pi$
$674$	0	0
$675$	11679.7	0.666000
$676$	0	0
$677$	−30943.5	−1.75665	−0.878326	−	0.478062i	$-0.841340\pi$
$677$	−30943.5	−1.75665	−0.878326	+	0.478062i	$0.841340\pi$
$678$	0	0
$679$	−6952.38	−0.392942
$680$	0	0
$681$	−175.909	−0.00989845
$682$	0	0
$683$	19656.4	1.10122	0.550608	−	0.834764i	$-0.314396\pi$
$683$	19656.4	1.10122	0.550608	+	0.834764i	$0.314396\pi$
$684$	0	0
$685$	14118.9	0.787526
$686$	0	0
$687$	−14499.1	−0.805205
$688$	0	0
$689$	7211.87	0.398767
$690$	0	0
$691$	−13416.9	−0.738642	−0.369321	−	0.929302i	$-0.620410\pi$
$691$	−13416.9	−0.738642	−0.369321	+	0.929302i	$0.620410\pi$
$692$	0	0
$693$	−2611.35	−0.143141
$694$	0	0
$695$	−16412.7	−0.895783
$696$	0	0
$697$	4766.89	0.259051
$698$	0	0
$699$	196.648	0.0106408
$700$	0	0
$701$	26628.3	1.43472	0.717358	−	0.696704i	$-0.245351\pi$
$701$	26628.3	1.43472	0.717358	+	0.696704i	$0.245351\pi$
$702$	0	0
$703$	1195.84	0.0641564
$704$	0	0
$705$	−13891.6	−0.742109
$706$	0	0
$707$	−11948.7	−0.635610
$708$	0	0
$709$	−20010.3	−1.05995	−0.529973	−	0.848014i	$-0.677798\pi$
$709$	−20010.3	−1.05995	−0.529973	+	0.848014i	$0.677798\pi$
$710$	0	0
$711$	4815.56	0.254005
$712$	0	0
$713$	−25282.6	−1.32797
$714$	0	0
$715$	11509.1	0.601980
$716$	0	0
$717$	4735.22	0.246639
$718$	0	0
$719$	15025.4	0.779349	0.389674	−	0.920953i	$-0.372588\pi$
$719$	15025.4	0.779349	0.389674	+	0.920953i	$0.372588\pi$
$720$	0	0
$721$	−10427.0	−0.538589
$722$	0	0
$723$	−15496.1	−0.797106
$724$	0	0
$725$	8665.92	0.443923
$726$	0	0
$727$	7820.61	0.398969	0.199484	−	0.979901i	$-0.436073\pi$
$727$	7820.61	0.398969	0.199484	+	0.979901i	$0.436073\pi$
$728$	0	0
$729$	18857.0	0.958036
$730$	0	0
$731$	53175.7	2.69052
$732$	0	0
$733$	9875.16	0.497609	0.248805	−	0.968554i	$-0.419962\pi$
$733$	9875.16	0.497609	0.248805	+	0.968554i	$0.419962\pi$
$734$	0	0
$735$	−1306.73	−0.0655775
$736$	0	0
$737$	−8479.07	−0.423786
$738$	0	0
$739$	−13209.0	−0.657509	−0.328755	−	0.944415i	$-0.606629\pi$
$739$	−13209.0	−0.657509	−0.328755	+	0.944415i	$0.606629\pi$
$740$	0	0
$741$	−582.398	−0.0288730
$742$	0	0
$743$	16018.2	0.790918	0.395459	−	0.918484i	$-0.370585\pi$
$743$	16018.2	0.790918	0.395459	+	0.918484i	$0.370585\pi$
$744$	0	0
$745$	10007.4	0.492138
$746$	0	0
$747$	−13512.0	−0.661818
$748$	0	0
$749$	6630.88	0.323481
$750$	0	0
$751$	17159.8	0.833782	0.416891	−	0.908956i	$-0.363120\pi$
$751$	17159.8	0.833782	0.416891	+	0.908956i	$0.363120\pi$
$752$	0	0
$753$	−14401.2	−0.696957
$754$	0	0
$755$	272.094	0.0131159
$756$	0	0
$757$	19219.7	0.922790	0.461395	−	0.887195i	$-0.347349\pi$
$757$	19219.7	0.922790	0.461395	+	0.887195i	$0.347349\pi$
$758$	0	0
$759$	−14111.7	−0.674866
$760$	0	0
$761$	−13680.7	−0.651675	−0.325837	−	0.945426i	$-0.605646\pi$
$761$	−13680.7	−0.651675	−0.325837	+	0.945426i	$0.605646\pi$
$762$	0	0
$763$	−13736.7	−0.651773
$764$	0	0
$765$	9684.98	0.457727
$766$	0	0
$767$	2377.94	0.111946
$768$	0	0
$769$	−8982.63	−0.421225	−0.210613	−	0.977570i	$-0.567546\pi$
$769$	−8982.63	−0.421225	−0.210613	+	0.977570i	$0.567546\pi$
$770$	0	0
$771$	−3811.10	−0.178020
$772$	0	0
$773$	31070.1	1.44568	0.722841	−	0.691014i	$-0.242836\pi$
$773$	31070.1	1.44568	0.722841	+	0.691014i	$0.242836\pi$
$774$	0	0
$775$	16554.1	0.767278
$776$	0	0
$777$	11578.1	0.534569
$778$	0	0
$779$	−114.539	−0.00526803
$780$	0	0
$781$	−6590.10	−0.301936
$782$	0	0
$783$	16909.6	0.771776
$784$	0	0
$785$	−13312.0	−0.605254
$786$	0	0
$787$	34619.2	1.56803	0.784017	−	0.620739i	$-0.213167\pi$
$787$	34619.2	1.56803	0.784017	+	0.620739i	$0.213167\pi$
$788$	0	0
$789$	13650.2	0.615919
$790$	0	0
$791$	−843.759	−0.0379275
$792$	0	0
$793$	45130.1	2.02096
$794$	0	0
$795$	−3564.71	−0.159028
$796$	0	0
$797$	30782.6	1.36810	0.684051	−	0.729434i	$-0.260217\pi$
$797$	30782.6	1.36810	0.684051	+	0.729434i	$0.260217\pi$
$798$	0	0
$799$	60563.8	2.68159
$800$	0	0
$801$	−14629.4	−0.645325
$802$	0	0
$803$	4100.47	0.180202
$804$	0	0
$805$	5708.52	0.249936
$806$	0	0
$807$	−1662.63	−0.0725248
$808$	0	0
$809$	−20664.3	−0.898046	−0.449023	−	0.893520i	$-0.648228\pi$
$809$	−20664.3	−0.898046	−0.449023	+	0.893520i	$0.648228\pi$
$810$	0	0
$811$	−21202.4	−0.918025	−0.459013	−	0.888430i	$-0.651797\pi$
$811$	−21202.4	−0.918025	−0.459013	+	0.888430i	$0.651797\pi$
$812$	0	0
$813$	25535.9	1.10158
$814$	0	0
$815$	−8246.68	−0.354440
$816$	0	0
$817$	−1277.71	−0.0547141
$818$	0	0
$819$	4558.32	0.194482
$820$	0	0
$821$	36052.2	1.53256	0.766280	−	0.642507i	$-0.222106\pi$
$821$	36052.2	1.53256	0.766280	+	0.642507i	$0.222106\pi$
$822$	0	0
$823$	35446.6	1.50132	0.750662	−	0.660686i	$-0.229735\pi$
$823$	35446.6	1.50132	0.750662	+	0.660686i	$0.229735\pi$
$824$	0	0
$825$	9239.84	0.389927
$826$	0	0
$827$	−22379.2	−0.940993	−0.470497	−	0.882402i	$-0.655925\pi$
$827$	−22379.2	−0.940993	−0.470497	+	0.882402i	$0.655925\pi$
$828$	0	0
$829$	36872.7	1.54480	0.772401	−	0.635136i	$-0.219056\pi$
$829$	36872.7	1.54480	0.772401	+	0.635136i	$0.219056\pi$
$830$	0	0
$831$	−23589.6	−0.984734
$832$	0	0
$833$	5697.01	0.236963
$834$	0	0
$835$	5171.29	0.214323
$836$	0	0
$837$	32301.6	1.33394
$838$	0	0
$839$	10706.9	0.440576	0.220288	−	0.975435i	$-0.429300\pi$
$839$	10706.9	0.440576	0.220288	+	0.975435i	$0.429300\pi$
$840$	0	0
$841$	−11842.6	−0.485572
$842$	0	0
$843$	15014.1	0.613420
$844$	0	0
$845$	−4927.07	−0.200588
$846$	0	0
$847$	2629.79	0.106683
$848$	0	0
$849$	−30422.2	−1.22979
$850$	0	0
$851$	−50579.3	−2.03741
$852$	0	0
$853$	−40536.1	−1.62712	−0.813559	−	0.581483i	$-0.802473\pi$
$853$	−40536.1	−1.62712	−0.813559	+	0.581483i	$0.802473\pi$
$854$	0	0
$855$	−232.712	−0.00930828
$856$	0	0
$857$	−39584.6	−1.57781	−0.788906	−	0.614514i	$-0.789352\pi$
$857$	−39584.6	−1.57781	−0.788906	+	0.614514i	$0.789352\pi$
$858$	0	0
$859$	−22580.1	−0.896884	−0.448442	−	0.893812i	$-0.648021\pi$
$859$	−22580.1	−0.896884	−0.448442	+	0.893812i	$0.648021\pi$
$860$	0	0
$861$	−1108.96	−0.0438948
$862$	0	0
$863$	34803.5	1.37280	0.686399	−	0.727225i	$-0.259190\pi$
$863$	34803.5	1.37280	0.686399	+	0.727225i	$0.259190\pi$
$864$	0	0
$865$	10991.8	0.432059
$866$	0	0
$867$	33248.4	1.30239
$868$	0	0
$869$	12331.8	0.481390
$870$	0	0
$871$	14800.9	0.575785
$872$	0	0
$873$	−11987.5	−0.464737
$874$	0	0
$875$	−9776.69	−0.377729
$876$	0	0
$877$	−47372.1	−1.82399	−0.911996	−	0.410199i	$-0.865459\pi$
$877$	−47372.1	−1.82399	−0.911996	+	0.410199i	$0.865459\pi$
$878$	0	0
$879$	−29129.2	−1.11775
$880$	0	0
$881$	−23971.0	−0.916688	−0.458344	−	0.888775i	$-0.651557\pi$
$881$	−23971.0	−0.916688	−0.458344	+	0.888775i	$0.651557\pi$
$882$	0	0
$883$	27665.6	1.05439	0.527193	−	0.849746i	$-0.323245\pi$
$883$	27665.6	1.05439	0.527193	+	0.849746i	$0.323245\pi$
$884$	0	0
$885$	−1175.38	−0.0446439
$886$	0	0
$887$	−34567.9	−1.30854	−0.654271	−	0.756261i	$-0.727024\pi$
$887$	−34567.9	−1.30854	−0.654271	+	0.756261i	$0.727024\pi$
$888$	0	0
$889$	1854.07	0.0699476
$890$	0	0
$891$	7957.15	0.299186
$892$	0	0
$893$	−1455.23	−0.0545325
$894$	0	0
$895$	18215.1	0.680295
$896$	0	0
$897$	24633.1	0.916919
$898$	0	0
$899$	23966.8	0.889139
$900$	0	0
$901$	15541.2	0.574644
$902$	0	0
$903$	−12370.7	−0.455894
$904$	0	0
$905$	−27531.9	−1.01126
$906$	0	0
$907$	32540.0	1.19126	0.595630	−	0.803259i	$-0.296902\pi$
$907$	32540.0	1.19126	0.595630	+	0.803259i	$0.296902\pi$
$908$	0	0
$909$	−20602.3	−0.751744
$910$	0	0
$911$	−11728.0	−0.426526	−0.213263	−	0.976995i	$-0.568409\pi$
$911$	−11728.0	−0.426526	−0.213263	+	0.976995i	$0.568409\pi$
$912$	0	0
$913$	−34601.9	−1.25428
$914$	0	0
$915$	−22307.1	−0.805956
$916$	0	0
$917$	6080.06	0.218955
$918$	0	0
$919$	−7080.36	−0.254145	−0.127073	−	0.991893i	$-0.540558\pi$
$919$	−7080.36	−0.254145	−0.127073	+	0.991893i	$0.540558\pi$
$920$	0	0
$921$	−12476.6	−0.446383
$922$	0	0
$923$	11503.6	0.410232
$924$	0	0
$925$	33117.5	1.17719
$926$	0	0
$927$	−17978.6	−0.636995
$928$	0	0
$929$	−15010.4	−0.530115	−0.265057	−	0.964233i	$-0.585391\pi$
$929$	−15010.4	−0.530115	−0.265057	+	0.964233i	$0.585391\pi$
$930$	0	0
$931$	−136.888	−0.00481884
$932$	0	0
$933$	23750.0	0.833376
$934$	0	0
$935$	24801.6	0.867484
$936$	0	0
$937$	29378.6	1.02429	0.512144	−	0.858899i	$-0.328851\pi$
$937$	29378.6	1.02429	0.512144	+	0.858899i	$0.328851\pi$
$938$	0	0
$939$	14424.5	0.501305
$940$	0	0
$941$	−56146.2	−1.94507	−0.972536	−	0.232751i	$-0.925227\pi$
$941$	−56146.2	−1.94507	−0.972536	+	0.232751i	$0.925227\pi$
$942$	0	0
$943$	4844.57	0.167297
$944$	0	0
$945$	−7293.35	−0.251061
$946$	0	0
$947$	−28000.4	−0.960814	−0.480407	−	0.877046i	$-0.659511\pi$
$947$	−28000.4	−0.960814	−0.480407	+	0.877046i	$0.659511\pi$
$948$	0	0
$949$	−7157.69	−0.244835
$950$	0	0
$951$	−33850.7	−1.15424
$952$	0	0
$953$	−7821.43	−0.265856	−0.132928	−	0.991126i	$-0.542438\pi$
$953$	−7821.43	−0.265856	−0.132928	+	0.991126i	$0.542438\pi$
$954$	0	0
$955$	5371.45	0.182006
$956$	0	0
$957$	13377.3	0.451856
$958$	0	0
$959$	14320.1	0.482188
$960$	0	0
$961$	15991.5	0.536791
$962$	0	0
$963$	11433.2	0.382584
$964$	0	0
$965$	−4525.76	−0.150973
$966$	0	0
$967$	17182.5	0.571408	0.285704	−	0.958318i	$-0.407773\pi$
$967$	17182.5	0.571408	0.285704	+	0.958318i	$0.407773\pi$
$968$	0	0
$969$	−1255.04	−0.0416075
$970$	0	0
$971$	3292.20	0.108807	0.0544036	−	0.998519i	$-0.482674\pi$
$971$	3292.20	0.108807	0.0544036	+	0.998519i	$0.482674\pi$
$972$	0	0
$973$	−16646.5	−0.548472
$974$	0	0
$975$	−16128.9	−0.529782
$976$	0	0
$977$	−36759.3	−1.20372	−0.601861	−	0.798601i	$-0.705574\pi$
$977$	−36759.3	−1.20372	−0.601861	+	0.798601i	$0.705574\pi$
$978$	0	0
$979$	−37463.4	−1.22302
$980$	0	0
$981$	−23685.3	−0.770860
$982$	0	0
$983$	22188.6	0.719947	0.359973	−	0.932963i	$-0.382786\pi$
$983$	22188.6	0.719947	0.359973	+	0.932963i	$0.382786\pi$
$984$	0	0
$985$	30952.5	1.00125
$986$	0	0
$987$	−14089.5	−0.454380
$988$	0	0
$989$	54042.2	1.73755
$990$	0	0
$991$	41061.8	1.31622	0.658108	−	0.752924i	$-0.271357\pi$
$991$	41061.8	1.31622	0.658108	+	0.752924i	$0.271357\pi$
$992$	0	0
$993$	12724.7	0.406651
$994$	0	0
$995$	18416.9	0.586788
$996$	0	0
$997$	40801.3	1.29608	0.648039	−	0.761607i	$-0.275590\pi$
$997$	40801.3	1.29608	0.648039	+	0.761607i	$0.275590\pi$
$998$	0	0
$999$	64621.5	2.04658

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.d.1.11	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.d.1.11	✓	15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+3.86399 q^{3} -6.90167 q^{5} -7.00000 q^{7} -12.0696 q^{9} +O(q^{10})\)	\(q+3.86399 q^{3} -6.90167 q^{5} -7.00000 q^{7} -12.0696 q^{9} -30.9082 q^{11} +53.9527 q^{13} -26.6680 q^{15} +116.266 q^{17} -2.79364 q^{19} -27.0479 q^{21} +118.160 q^{23} -77.3669 q^{25} -150.964 q^{27} -112.011 q^{29} -213.969 q^{31} -119.429 q^{33} +48.3117 q^{35} -428.057 q^{37} +208.473 q^{39} +41.0000 q^{41} +457.364 q^{43} +83.3005 q^{45} +520.909 q^{47} +49.0000 q^{49} +449.248 q^{51} +133.670 q^{53} +213.318 q^{55} -10.7946 q^{57} +44.0745 q^{59} +836.476 q^{61} +84.4873 q^{63} -372.364 q^{65} +274.331 q^{67} +456.569 q^{69} +213.215 q^{71} -132.666 q^{73} -298.945 q^{75} +216.357 q^{77} -398.982 q^{79} -257.445 q^{81} +1119.50 q^{83} -802.427 q^{85} -432.807 q^{87} +1212.09 q^{89} -377.669 q^{91} -826.771 q^{93} +19.2808 q^{95} +993.197 q^{97} +373.050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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