LMFDB - Embedded newform 414.4.a.n.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [414,4,Mod(1,414)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(414, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("414.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	414.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,8,0,16,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.4267907424$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 219x^{2} - 468x + 3240 $ x^4 - 2x^3 - 219x^2 - 468*x + 3240
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$16.3784$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +4.00000 q^{4} +13.0601 q^{5} +18.7067 q^{7} +8.00000 q^{8} +26.1202 q^{10} +41.1103 q^{11} +32.6366 q^{13} +37.4133 q^{14} +16.0000 q^{16} -76.8570 q^{17} -164.965 q^{19} +52.2405 q^{20} +82.2206 q^{22} +23.0000 q^{23} +45.5668 q^{25} +65.2732 q^{26} +74.8267 q^{28} -42.5314 q^{29} +246.351 q^{31} +32.0000 q^{32} -153.714 q^{34} +244.311 q^{35} +281.671 q^{37} -329.929 q^{38} +104.481 q^{40} +330.365 q^{41} -160.644 q^{43} +164.441 q^{44} +46.0000 q^{46} -578.490 q^{47} +6.93931 q^{49} +91.1336 q^{50} +130.546 q^{52} -621.131 q^{53} +536.906 q^{55} +149.653 q^{56} -85.0628 q^{58} +347.432 q^{59} +372.119 q^{61} +492.702 q^{62} +64.0000 q^{64} +426.238 q^{65} +884.432 q^{67} -307.428 q^{68} +488.623 q^{70} -777.672 q^{71} -188.099 q^{73} +563.343 q^{74} -659.859 q^{76} +769.037 q^{77} +200.062 q^{79} +208.962 q^{80} +660.730 q^{82} +49.9822 q^{83} -1003.76 q^{85} -321.288 q^{86} +328.882 q^{88} -767.598 q^{89} +610.522 q^{91} +92.0000 q^{92} -1156.98 q^{94} -2154.46 q^{95} +1626.31 q^{97} +13.8786 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} + 18 q^{7} + 32 q^{8} + 42 q^{11} + 108 q^{13} + 36 q^{14} + 64 q^{16} + 26 q^{17} + 132 q^{19} + 84 q^{22} + 92 q^{23} + 260 q^{25} + 216 q^{26} + 72 q^{28} + 252 q^{29} + 428 q^{31}+ \cdots + 2152 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 + 18 * q^7 + 32 * q^8 + 42 * q^11 + 108 * q^13 + 36 * q^14 + 64 * q^16 + 26 * q^17 + 132 * q^19 + 84 * q^22 + 92 * q^23 + 260 * q^25 + 216 * q^26 + 72 * q^28 + 252 * q^29 + 428 * q^31 + 128 * q^32 + 52 * q^34 + 348 * q^35 + 278 * q^37 + 264 * q^38 - 4 * q^41 + 216 * q^43 + 168 * q^44 + 184 * q^46 - 4 * q^47 + 1076 * q^49 + 520 * q^50 + 432 * q^52 - 460 * q^53 + 512 * q^55 + 144 * q^56 + 504 * q^58 - 504 * q^59 + 1086 * q^61 + 856 * q^62 + 256 * q^64 - 1420 * q^65 + 1484 * q^67 + 104 * q^68 + 696 * q^70 - 876 * q^71 + 1072 * q^73 + 556 * q^74 + 528 * q^76 - 1580 * q^77 + 1634 * q^79 - 8 * q^82 - 2246 * q^83 + 872 * q^85 + 432 * q^86 + 336 * q^88 - 474 * q^89 + 40 * q^91 + 368 * q^92 - 8 * q^94 - 2780 * q^95 + 1044 * q^97 + 2152 * q^98

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	13.0601	1.16813	0.584066	−	0.811706i	$-0.301461\pi$
$5$	13.0601	1.16813	0.584066	+	0.811706i	$0.301461\pi$
$6$	0	0
$7$	18.7067	1.01006	0.505032	−	0.863100i	$-0.331481\pi$
$7$	18.7067	1.01006	0.505032	+	0.863100i	$0.331481\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	26.1202	0.825995
$11$	41.1103	1.12684	0.563419	−	0.826171i	$-0.309486\pi$
$11$	41.1103	1.12684	0.563419	+	0.826171i	$0.309486\pi$
$12$	0	0
$13$	32.6366	0.696290	0.348145	−	0.937441i	$-0.386812\pi$
$13$	32.6366	0.696290	0.348145	+	0.937441i	$0.386812\pi$
$14$	37.4133	0.714224
$15$	0	0
$16$	16.0000	0.250000
$17$	−76.8570	−1.09650	−0.548252	−	0.836313i	$-0.684706\pi$
$17$	−76.8570	−1.09650	−0.548252	+	0.836313i	$0.684706\pi$
$18$	0	0
$19$	−164.965	−1.99187	−0.995934	−	0.0900884i	$-0.971285\pi$
$19$	−164.965	−1.99187	−0.995934	+	0.0900884i	$0.971285\pi$
$20$	52.2405	0.584066
$21$	0	0
$22$	82.2206	0.796795
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	45.5668	0.364534
$26$	65.2732	0.492351
$27$	0	0
$28$	74.8267	0.505032
$29$	−42.5314	−0.272341	−0.136170	−	0.990685i	$-0.543479\pi$
$29$	−42.5314	−0.272341	−0.136170	+	0.990685i	$0.543479\pi$
$30$	0	0
$31$	246.351	1.42729	0.713644	−	0.700508i	$-0.247043\pi$
$31$	246.351	1.42729	0.713644	+	0.700508i	$0.247043\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−153.714	−0.775345
$35$	244.311	1.17989
$36$	0	0
$37$	281.671	1.25153	0.625763	−	0.780013i	$-0.284788\pi$
$37$	281.671	1.25153	0.625763	+	0.780013i	$0.284788\pi$
$38$	−329.929	−1.40846
$39$	0	0
$40$	104.481	0.412997
$41$	330.365	1.25840	0.629200	−	0.777244i	$-0.283383\pi$
$41$	330.365	1.25840	0.629200	+	0.777244i	$0.283383\pi$
$42$	0	0
$43$	−160.644	−0.569721	−0.284861	−	0.958569i	$-0.591947\pi$
$43$	−160.644	−0.569721	−0.284861	+	0.958569i	$0.591947\pi$
$44$	164.441	0.563419
$45$	0	0
$46$	46.0000	0.147442
$47$	−578.490	−1.79535	−0.897675	−	0.440658i	$-0.854745\pi$
$47$	−578.490	−1.79535	−0.897675	+	0.440658i	$0.854745\pi$
$48$	0	0
$49$	6.93931	0.0202312
$50$	91.1336	0.257765
$51$	0	0
$52$	130.546	0.348145
$53$	−621.131	−1.60979	−0.804895	−	0.593418i	$-0.797778\pi$
$53$	−621.131	−1.60979	−0.804895	+	0.593418i	$0.797778\pi$
$54$	0	0
$55$	536.906	1.31630
$56$	149.653	0.357112
$57$	0	0
$58$	−85.0628	−0.192574
$59$	347.432	0.766640	0.383320	−	0.923616i	$-0.374781\pi$
$59$	347.432	0.766640	0.383320	+	0.923616i	$0.374781\pi$
$60$	0	0
$61$	372.119	0.781065	0.390532	−	0.920589i	$-0.372291\pi$
$61$	372.119	0.781065	0.390532	+	0.920589i	$0.372291\pi$
$62$	492.702	1.00925
$63$	0	0
$64$	64.0000	0.125000
$65$	426.238	0.813359
$66$	0	0
$67$	884.432	1.61270	0.806348	−	0.591442i	$-0.201441\pi$
$67$	884.432	1.61270	0.806348	+	0.591442i	$0.201441\pi$
$68$	−307.428	−0.548252
$69$	0	0
$70$	488.623	0.834308
$71$	−777.672	−1.29990	−0.649948	−	0.759978i	$-0.725209\pi$
$71$	−777.672	−1.29990	−0.649948	+	0.759978i	$0.725209\pi$
$72$	0	0
$73$	−188.099	−0.301581	−0.150790	−	0.988566i	$-0.548182\pi$
$73$	−188.099	−0.301581	−0.150790	+	0.988566i	$0.548182\pi$
$74$	563.343	0.884963
$75$	0	0
$76$	−659.859	−0.995934
$77$	769.037	1.13818
$78$	0	0
$79$	200.062	0.284921	0.142461	−	0.989800i	$-0.454499\pi$
$79$	200.062	0.284921	0.142461	+	0.989800i	$0.454499\pi$
$80$	208.962	0.292033
$81$	0	0
$82$	660.730	0.889823
$83$	49.9822	0.0660995	0.0330497	−	0.999454i	$-0.489478\pi$
$83$	49.9822	0.0660995	0.0330497	+	0.999454i	$0.489478\pi$
$84$	0	0
$85$	−1003.76	−1.28086
$86$	−321.288	−0.402854
$87$	0	0
$88$	328.882	0.398398
$89$	−767.598	−0.914216	−0.457108	−	0.889411i	$-0.651115\pi$
$89$	−767.598	−0.914216	−0.457108	+	0.889411i	$0.651115\pi$
$90$	0	0
$91$	610.522	0.703298
$92$	92.0000	0.104257
$93$	0	0
$94$	−1156.98	−1.26950
$95$	−2154.46	−2.32677
$96$	0	0
$97$	1626.31	1.70234	0.851171	−	0.524888i	$-0.175893\pi$
$97$	1626.31	1.70234	0.851171	+	0.524888i	$0.175893\pi$
$98$	13.8786	0.0143056
$99$	0	0
$100$	182.267	0.182267
$101$	−366.127	−0.360703	−0.180352	−	0.983602i	$-0.557724\pi$
$101$	−366.127	−0.360703	−0.180352	+	0.983602i	$0.557724\pi$
$102$	0	0
$103$	1828.41	1.74911	0.874554	−	0.484928i	$-0.161154\pi$
$103$	1828.41	1.74911	0.874554	+	0.484928i	$0.161154\pi$
$104$	261.093	0.246176
$105$	0	0
$106$	−1242.26	−1.13829
$107$	−555.660	−0.502034	−0.251017	−	0.967983i	$-0.580765\pi$
$107$	−555.660	−0.502034	−0.251017	+	0.967983i	$0.580765\pi$
$108$	0	0
$109$	−952.852	−0.837309	−0.418654	−	0.908146i	$-0.637498\pi$
$109$	−952.852	−0.837309	−0.418654	+	0.908146i	$0.637498\pi$
$110$	1073.81	0.930763
$111$	0	0
$112$	299.307	0.252516
$113$	−255.634	−0.212815	−0.106407	−	0.994323i	$-0.533935\pi$
$113$	−255.634	−0.212815	−0.106407	+	0.994323i	$0.533935\pi$
$114$	0	0
$115$	300.383	0.243573
$116$	−170.126	−0.136170
$117$	0	0
$118$	694.864	0.542096
$119$	−1437.74	−1.10754
$120$	0	0
$121$	359.058	0.269765
$122$	744.238	0.552296
$123$	0	0
$124$	985.404	0.713644
$125$	−1037.41	−0.742308
$126$	0	0
$127$	−1543.83	−1.07868	−0.539341	−	0.842088i	$-0.681327\pi$
$127$	−1543.83	−1.07868	−0.539341	+	0.842088i	$0.681327\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	852.476	0.575132
$131$	−116.906	−0.0779702	−0.0389851	−	0.999240i	$-0.512412\pi$
$131$	−116.906	−0.0779702	−0.0389851	+	0.999240i	$0.512412\pi$
$132$	0	0
$133$	−3085.94	−2.01192
$134$	1768.86	1.14035
$135$	0	0
$136$	−614.856	−0.387672
$137$	−2770.00	−1.72742	−0.863712	−	0.503986i	$-0.831866\pi$
$137$	−2770.00	−1.72742	−0.863712	+	0.503986i	$0.831866\pi$
$138$	0	0
$139$	−1296.27	−0.790992	−0.395496	−	0.918468i	$-0.629427\pi$
$139$	−1296.27	−0.790992	−0.395496	+	0.918468i	$0.629427\pi$
$140$	977.245	0.589945
$141$	0	0
$142$	−1555.34	−0.919166
$143$	1341.70	0.784606
$144$	0	0
$145$	−555.466	−0.318130
$146$	−376.199	−0.213250
$147$	0	0
$148$	1126.69	0.625763
$149$	−200.699	−0.110348	−0.0551742	−	0.998477i	$-0.517571\pi$
$149$	−200.699	−0.110348	−0.0551742	+	0.998477i	$0.517571\pi$
$150$	0	0
$151$	−109.209	−0.0588564	−0.0294282	−	0.999567i	$-0.509369\pi$
$151$	−109.209	−0.0588564	−0.0294282	+	0.999567i	$0.509369\pi$
$152$	−1319.72	−0.704232
$153$	0	0
$154$	1538.07	0.804815
$155$	3217.38	1.66726
$156$	0	0
$157$	−3299.65	−1.67733	−0.838665	−	0.544647i	$-0.816664\pi$
$157$	−3299.65	−1.67733	−0.838665	+	0.544647i	$0.816664\pi$
$158$	400.125	0.201470
$159$	0	0
$160$	417.924	0.206499
$161$	430.253	0.210613
$162$	0	0
$163$	238.673	0.114689	0.0573446	−	0.998354i	$-0.481737\pi$
$163$	238.673	0.114689	0.0573446	+	0.998354i	$0.481737\pi$
$164$	1321.46	0.629200
$165$	0	0
$166$	99.9644	0.0467394
$167$	3124.43	1.44776	0.723880	−	0.689926i	$-0.242357\pi$
$167$	3124.43	1.44776	0.723880	+	0.689926i	$0.242357\pi$
$168$	0	0
$169$	−1131.85	−0.515181
$170$	−2007.52	−0.905706
$171$	0	0
$172$	−642.577	−0.284861
$173$	−2732.43	−1.20083	−0.600413	−	0.799690i	$-0.704997\pi$
$173$	−2732.43	−1.20083	−0.600413	+	0.799690i	$0.704997\pi$
$174$	0	0
$175$	852.403	0.368203
$176$	657.765	0.281710
$177$	0	0
$178$	−1535.20	−0.646449
$179$	2350.95	0.981665	0.490833	−	0.871254i	$-0.336693\pi$
$179$	2350.95	0.981665	0.490833	+	0.871254i	$0.336693\pi$
$180$	0	0
$181$	−2224.83	−0.913650	−0.456825	−	0.889557i	$-0.651013\pi$
$181$	−2224.83	−0.913650	−0.456825	+	0.889557i	$0.651013\pi$
$182$	1221.04	0.497307
$183$	0	0
$184$	184.000	0.0737210
$185$	3678.66	1.46195
$186$	0	0
$187$	−3159.61	−1.23558
$188$	−2313.96	−0.897675
$189$	0	0
$190$	−4308.92	−1.64527
$191$	3099.04	1.17403	0.587013	−	0.809577i	$-0.300304\pi$
$191$	3099.04	1.17403	0.587013	+	0.809577i	$0.300304\pi$
$192$	0	0
$193$	−679.935	−0.253590	−0.126795	−	0.991929i	$-0.540469\pi$
$193$	−679.935	−0.253590	−0.126795	+	0.991929i	$0.540469\pi$
$194$	3252.63	1.20374
$195$	0	0
$196$	27.7573	0.0101156
$197$	−49.9118	−0.0180511	−0.00902555	−	0.999959i	$-0.502873\pi$
$197$	−49.9118	−0.0180511	−0.00902555	+	0.999959i	$0.502873\pi$
$198$	0	0
$199$	−4052.80	−1.44369	−0.721847	−	0.692052i	$-0.756707\pi$
$199$	−4052.80	−1.44369	−0.721847	+	0.692052i	$0.756707\pi$
$200$	364.534	0.128882
$201$	0	0
$202$	−732.255	−0.255056
$203$	−795.621	−0.275082
$204$	0	0
$205$	4314.61	1.46998
$206$	3656.81	1.23681
$207$	0	0
$208$	522.186	0.174072
$209$	−6781.75	−2.24451
$210$	0	0
$211$	−864.141	−0.281943	−0.140971	−	0.990014i	$-0.545023\pi$
$211$	−864.141	−0.281943	−0.140971	+	0.990014i	$0.545023\pi$
$212$	−2484.52	−0.804895
$213$	0	0
$214$	−1111.32	−0.354992
$215$	−2098.03	−0.665510
$216$	0	0
$217$	4608.41	1.44165
$218$	−1905.70	−0.592067
$219$	0	0
$220$	2147.62	0.658149
$221$	−2508.35	−0.763484
$222$	0	0
$223$	5682.45	1.70639	0.853196	−	0.521591i	$-0.174661\pi$
$223$	5682.45	1.70639	0.853196	+	0.521591i	$0.174661\pi$
$224$	598.613	0.178556
$225$	0	0
$226$	−511.269	−0.150483
$227$	−158.099	−0.0462263	−0.0231132	−	0.999733i	$-0.507358\pi$
$227$	−158.099	−0.0462263	−0.0231132	+	0.999733i	$0.507358\pi$
$228$	0	0
$229$	2744.94	0.792100	0.396050	−	0.918229i	$-0.370381\pi$
$229$	2744.94	0.792100	0.396050	+	0.918229i	$0.370381\pi$
$230$	600.766	0.172232
$231$	0	0
$232$	−340.251	−0.0962871
$233$	3647.63	1.02560	0.512798	−	0.858509i	$-0.328609\pi$
$233$	3647.63	1.02560	0.512798	+	0.858509i	$0.328609\pi$
$234$	0	0
$235$	−7555.15	−2.09721
$236$	1389.73	0.383320
$237$	0	0
$238$	−2875.48	−0.783149
$239$	−2580.14	−0.698306	−0.349153	−	0.937066i	$-0.613531\pi$
$239$	−2580.14	−0.698306	−0.349153	+	0.937066i	$0.613531\pi$
$240$	0	0
$241$	4498.59	1.20240	0.601202	−	0.799097i	$-0.294689\pi$
$241$	4498.59	1.20240	0.601202	+	0.799097i	$0.294689\pi$
$242$	718.115	0.190753
$243$	0	0
$244$	1488.48	0.390532
$245$	90.6283	0.0236328
$246$	0	0
$247$	−5383.89	−1.38692
$248$	1970.81	0.504623
$249$	0	0
$250$	−2074.81	−0.524891
$251$	1285.33	0.323223	0.161612	−	0.986854i	$-0.448331\pi$
$251$	1285.33	0.323223	0.161612	+	0.986854i	$0.448331\pi$
$252$	0	0
$253$	945.537	0.234962
$254$	−3087.66	−0.762743
$255$	0	0
$256$	256.000	0.0625000
$257$	768.793	0.186599	0.0932996	−	0.995638i	$-0.470259\pi$
$257$	768.793	0.186599	0.0932996	+	0.995638i	$0.470259\pi$
$258$	0	0
$259$	5269.13	1.26412
$260$	1704.95	0.406679
$261$	0	0
$262$	−233.811	−0.0551333
$263$	−3454.27	−0.809883	−0.404942	−	0.914342i	$-0.632708\pi$
$263$	−3454.27	−0.809883	−0.404942	+	0.914342i	$0.632708\pi$
$264$	0	0
$265$	−8112.04	−1.88045
$266$	−6171.88	−1.42264
$267$	0	0
$268$	3537.73	0.806348
$269$	−4348.89	−0.985712	−0.492856	−	0.870111i	$-0.664047\pi$
$269$	−4348.89	−0.985712	−0.492856	+	0.870111i	$0.664047\pi$
$270$	0	0
$271$	4876.35	1.09305	0.546527	−	0.837442i	$-0.315950\pi$
$271$	4876.35	1.09305	0.546527	+	0.837442i	$0.315950\pi$
$272$	−1229.71	−0.274126
$273$	0	0
$274$	−5540.00	−1.22147
$275$	1873.27	0.410771
$276$	0	0
$277$	−757.557	−0.164322	−0.0821610	−	0.996619i	$-0.526182\pi$
$277$	−757.557	−0.164322	−0.0821610	+	0.996619i	$0.526182\pi$
$278$	−2592.53	−0.559316
$279$	0	0
$280$	1954.49	0.417154
$281$	−8407.74	−1.78492	−0.892462	−	0.451122i	$-0.851024\pi$
$281$	−8407.74	−1.78492	−0.892462	+	0.451122i	$0.851024\pi$
$282$	0	0
$283$	−2522.06	−0.529757	−0.264878	−	0.964282i	$-0.585332\pi$
$283$	−2522.06	−0.529757	−0.264878	+	0.964282i	$0.585332\pi$
$284$	−3110.69	−0.649948
$285$	0	0
$286$	2683.40	0.554800
$287$	6180.03	1.27106
$288$	0	0
$289$	993.996	0.202319
$290$	−1110.93	−0.224952
$291$	0	0
$292$	−752.398	−0.150790
$293$	3357.13	0.669372	0.334686	−	0.942330i	$-0.391370\pi$
$293$	3357.13	0.669372	0.334686	+	0.942330i	$0.391370\pi$
$294$	0	0
$295$	4537.50	0.895537
$296$	2253.37	0.442481
$297$	0	0
$298$	−401.399	−0.0780282
$299$	750.642	0.145186
$300$	0	0
$301$	−3005.12	−0.575455
$302$	−218.418	−0.0416177
$303$	0	0
$304$	−2639.43	−0.497967
$305$	4859.92	0.912387
$306$	0	0
$307$	−7660.19	−1.42407	−0.712036	−	0.702143i	$-0.752227\pi$
$307$	−7660.19	−1.42407	−0.712036	+	0.702143i	$0.752227\pi$
$308$	3076.15	0.569090
$309$	0	0
$310$	6434.75	1.17893
$311$	344.406	0.0627958	0.0313979	−	0.999507i	$-0.490004\pi$
$311$	344.406	0.0627958	0.0313979	+	0.999507i	$0.490004\pi$
$312$	0	0
$313$	3403.84	0.614685	0.307343	−	0.951599i	$-0.400560\pi$
$313$	3403.84	0.614685	0.307343	+	0.951599i	$0.400560\pi$
$314$	−6599.30	−1.18605
$315$	0	0
$316$	800.249	0.142461
$317$	−5801.37	−1.02788	−0.513939	−	0.857827i	$-0.671814\pi$
$317$	−5801.37	−1.02788	−0.513939	+	0.857827i	$0.671814\pi$
$318$	0	0
$319$	−1748.48	−0.306884
$320$	835.848	0.146017
$321$	0	0
$322$	860.507	0.148926
$323$	12678.7	2.18409
$324$	0	0
$325$	1487.15	0.253822
$326$	477.347	0.0810975
$327$	0	0
$328$	2642.92	0.444911
$329$	−10821.6	−1.81342
$330$	0	0
$331$	8400.66	1.39499	0.697495	−	0.716589i	$-0.254298\pi$
$331$	8400.66	1.39499	0.697495	+	0.716589i	$0.254298\pi$
$332$	199.929	0.0330497
$333$	0	0
$334$	6248.87	1.02372
$335$	11550.8	1.88384
$336$	0	0
$337$	4144.17	0.669873	0.334937	−	0.942241i	$-0.391285\pi$
$337$	4144.17	0.669873	0.334937	+	0.942241i	$0.391285\pi$
$338$	−2263.70	−0.364288
$339$	0	0
$340$	−4015.05	−0.640431
$341$	10127.6	1.60832
$342$	0	0
$343$	−6286.57	−0.989630
$344$	−1285.15	−0.201427
$345$	0	0
$346$	−5464.87	−0.849113
$347$	3851.70	0.595880	0.297940	−	0.954585i	$-0.403701\pi$
$347$	3851.70	0.595880	0.297940	+	0.954585i	$0.403701\pi$
$348$	0	0
$349$	2581.17	0.395893	0.197947	−	0.980213i	$-0.436573\pi$
$349$	2581.17	0.395893	0.197947	+	0.980213i	$0.436573\pi$
$350$	1704.81	0.260359
$351$	0	0
$352$	1315.53	0.199199
$353$	7748.76	1.16834	0.584171	−	0.811630i	$-0.301420\pi$
$353$	7748.76	1.16834	0.584171	+	0.811630i	$0.301420\pi$
$354$	0	0
$355$	−10156.5	−1.51845
$356$	−3070.39	−0.457108
$357$	0	0
$358$	4701.90	0.694142
$359$	1957.02	0.287709	0.143855	−	0.989599i	$-0.454050\pi$
$359$	1957.02	0.287709	0.143855	+	0.989599i	$0.454050\pi$
$360$	0	0
$361$	20354.3	2.96754
$362$	−4449.67	−0.646048
$363$	0	0
$364$	2442.09	0.351649
$365$	−2456.60	−0.352286
$366$	0	0
$367$	5825.76	0.828616	0.414308	−	0.910137i	$-0.364024\pi$
$367$	5825.76	0.828616	0.414308	+	0.910137i	$0.364024\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	7357.32	1.03375
$371$	−11619.3	−1.62599
$372$	0	0
$373$	2339.75	0.324793	0.162396	−	0.986726i	$-0.448078\pi$
$373$	2339.75	0.324793	0.162396	+	0.986726i	$0.448078\pi$
$374$	−6319.23	−0.873689
$375$	0	0
$376$	−4627.92	−0.634752
$377$	−1388.08	−0.189628
$378$	0	0
$379$	10381.9	1.40707	0.703537	−	0.710658i	$-0.251603\pi$
$379$	10381.9	1.40707	0.703537	+	0.710658i	$0.251603\pi$
$380$	−8617.83	−1.16338
$381$	0	0
$382$	6198.09	0.830162
$383$	−4681.97	−0.624641	−0.312320	−	0.949977i	$-0.601106\pi$
$383$	−4681.97	−0.624641	−0.312320	+	0.949977i	$0.601106\pi$
$384$	0	0
$385$	10043.7	1.32955
$386$	−1359.87	−0.179315
$387$	0	0
$388$	6505.26	0.851171
$389$	−2896.26	−0.377497	−0.188748	−	0.982025i	$-0.560443\pi$
$389$	−2896.26	−0.377497	−0.188748	+	0.982025i	$0.560443\pi$
$390$	0	0
$391$	−1767.71	−0.228637
$392$	55.5145	0.00715282
$393$	0	0
$394$	−99.8235	−0.0127641
$395$	2612.84	0.332826
$396$	0	0
$397$	−1599.60	−0.202221	−0.101111	−	0.994875i	$-0.532240\pi$
$397$	−1599.60	−0.202221	−0.101111	+	0.994875i	$0.532240\pi$
$398$	−8105.60	−1.02085
$399$	0	0
$400$	729.069	0.0911336
$401$	−2132.80	−0.265603	−0.132802	−	0.991143i	$-0.542397\pi$
$401$	−2132.80	−0.265603	−0.132802	+	0.991143i	$0.542397\pi$
$402$	0	0
$403$	8040.06	0.993807
$404$	−1464.51	−0.180352
$405$	0	0
$406$	−1591.24	−0.194512
$407$	11579.6	1.41027
$408$	0	0
$409$	−4181.61	−0.505544	−0.252772	−	0.967526i	$-0.581342\pi$
$409$	−4181.61	−0.505544	−0.252772	+	0.967526i	$0.581342\pi$
$410$	8629.22	1.03943
$411$	0	0
$412$	7313.62	0.874554
$413$	6499.29	0.774356
$414$	0	0
$415$	652.773	0.0772130
$416$	1044.37	0.123088
$417$	0	0
$418$	−13563.5	−1.58711
$419$	3866.07	0.450763	0.225381	−	0.974271i	$-0.427637\pi$
$419$	3866.07	0.450763	0.225381	+	0.974271i	$0.427637\pi$
$420$	0	0
$421$	−6888.69	−0.797469	−0.398734	−	0.917066i	$-0.630550\pi$
$421$	−6888.69	−0.797469	−0.398734	+	0.917066i	$0.630550\pi$
$422$	−1728.28	−0.199364
$423$	0	0
$424$	−4969.04	−0.569147
$425$	−3502.13	−0.399713
$426$	0	0
$427$	6961.10	0.788926
$428$	−2222.64	−0.251017
$429$	0	0
$430$	−4196.07	−0.470587
$431$	−12349.8	−1.38021	−0.690105	−	0.723709i	$-0.742436\pi$
$431$	−12349.8	−1.38021	−0.690105	+	0.723709i	$0.742436\pi$
$432$	0	0
$433$	−12531.1	−1.39078	−0.695391	−	0.718632i	$-0.744769\pi$
$433$	−12531.1	−1.39078	−0.695391	+	0.718632i	$0.744769\pi$
$434$	9216.81	1.01940
$435$	0	0
$436$	−3811.41	−0.418654
$437$	−3794.19	−0.415333
$438$	0	0
$439$	−11534.2	−1.25398	−0.626988	−	0.779029i	$-0.715713\pi$
$439$	−11534.2	−1.25398	−0.626988	+	0.779029i	$0.715713\pi$
$440$	4295.25	0.465381
$441$	0	0
$442$	−5016.70	−0.539865
$443$	2982.50	0.319871	0.159935	−	0.987127i	$-0.448871\pi$
$443$	2982.50	0.319871	0.159935	+	0.987127i	$0.448871\pi$
$444$	0	0
$445$	−10024.9	−1.06793
$446$	11364.9	1.20660
$447$	0	0
$448$	1197.23	0.126258
$449$	12630.8	1.32759	0.663793	−	0.747917i	$-0.268946\pi$
$449$	12630.8	1.32759	0.663793	+	0.747917i	$0.268946\pi$
$450$	0	0
$451$	13581.4	1.41801
$452$	−1022.54	−0.106407
$453$	0	0
$454$	−316.197	−0.0326869
$455$	7973.49	0.821545
$456$	0	0
$457$	−13693.1	−1.40161	−0.700804	−	0.713354i	$-0.747175\pi$
$457$	−13693.1	−1.40161	−0.700804	+	0.713354i	$0.747175\pi$
$458$	5489.88	0.560099
$459$	0	0
$460$	1201.53	0.121786
$461$	4588.64	0.463588	0.231794	−	0.972765i	$-0.425540\pi$
$461$	4588.64	0.463588	0.231794	+	0.972765i	$0.425540\pi$
$462$	0	0
$463$	6110.84	0.613380	0.306690	−	0.951809i	$-0.400778\pi$
$463$	6110.84	0.613380	0.306690	+	0.951809i	$0.400778\pi$
$464$	−680.503	−0.0680852
$465$	0	0
$466$	7295.25	0.725206
$467$	2078.48	0.205954	0.102977	−	0.994684i	$-0.467163\pi$
$467$	2078.48	0.205954	0.102977	+	0.994684i	$0.467163\pi$
$468$	0	0
$469$	16544.8	1.62893
$470$	−15110.3	−1.48295
$471$	0	0
$472$	2779.45	0.271048
$473$	−6604.13	−0.641984
$474$	0	0
$475$	−7516.91	−0.726104
$476$	−5750.95	−0.553770
$477$	0	0
$478$	−5160.27	−0.493777
$479$	13245.4	1.26346	0.631730	−	0.775189i	$-0.282345\pi$
$479$	13245.4	1.26346	0.631730	+	0.775189i	$0.282345\pi$
$480$	0	0
$481$	9192.80	0.871425
$482$	8997.17	0.850228
$483$	0	0
$484$	1436.23	0.134883
$485$	21239.9	1.98856
$486$	0	0
$487$	8327.36	0.774843	0.387421	−	0.921903i	$-0.373366\pi$
$487$	8327.36	0.774843	0.387421	+	0.921903i	$0.373366\pi$
$488$	2976.95	0.276148
$489$	0	0
$490$	181.257	0.0167109
$491$	10585.9	0.972984	0.486492	−	0.873685i	$-0.338276\pi$
$491$	10585.9	0.972984	0.486492	+	0.873685i	$0.338276\pi$
$492$	0	0
$493$	3268.84	0.298623
$494$	−10767.8	−0.980698
$495$	0	0
$496$	3941.62	0.356822
$497$	−14547.6	−1.31298
$498$	0	0
$499$	−2630.88	−0.236021	−0.118011	−	0.993012i	$-0.537652\pi$
$499$	−2630.88	−0.236021	−0.118011	+	0.993012i	$0.537652\pi$
$500$	−4149.63	−0.371154
$501$	0	0
$502$	2570.65	0.228553
$503$	−8978.08	−0.795851	−0.397925	−	0.917418i	$-0.630270\pi$
$503$	−8978.08	−0.795851	−0.397925	+	0.917418i	$0.630270\pi$
$504$	0	0
$505$	−4781.67	−0.421349
$506$	1891.07	0.166143
$507$	0	0
$508$	−6175.31	−0.539341
$509$	−10656.1	−0.927942	−0.463971	−	0.885850i	$-0.653576\pi$
$509$	−10656.1	−0.927942	−0.463971	+	0.885850i	$0.653576\pi$
$510$	0	0
$511$	−3518.71	−0.304616
$512$	512.000	0.0441942
$513$	0	0
$514$	1537.59	0.131946
$515$	23879.2	2.04319
$516$	0	0
$517$	−23781.9	−2.02307
$518$	10538.3	0.893870
$519$	0	0
$520$	3409.90	0.287566
$521$	−15732.0	−1.32290	−0.661449	−	0.749990i	$-0.730058\pi$
$521$	−15732.0	−1.32290	−0.661449	+	0.749990i	$0.730058\pi$
$522$	0	0
$523$	3937.97	0.329245	0.164623	−	0.986357i	$-0.447359\pi$
$523$	3937.97	0.329245	0.164623	+	0.986357i	$0.447359\pi$
$524$	−467.623	−0.0389851
$525$	0	0
$526$	−6908.54	−0.572674
$527$	−18933.8	−1.56503
$528$	0	0
$529$	529.000	0.0434783
$530$	−16224.1	−1.32968
$531$	0	0
$532$	−12343.8	−1.00596
$533$	10782.0	0.876210
$534$	0	0
$535$	−7256.98	−0.586442
$536$	7075.46	0.570174
$537$	0	0
$538$	−8697.78	−0.697003
$539$	285.277	0.0227973
$540$	0	0
$541$	−7886.04	−0.626704	−0.313352	−	0.949637i	$-0.601452\pi$
$541$	−7886.04	−0.626704	−0.313352	+	0.949637i	$0.601452\pi$
$542$	9752.71	0.772905
$543$	0	0
$544$	−2459.42	−0.193836
$545$	−12444.4	−0.978088
$546$	0	0
$547$	20947.1	1.63735	0.818677	−	0.574255i	$-0.194708\pi$
$547$	20947.1	1.63735	0.818677	+	0.574255i	$0.194708\pi$
$548$	−11080.0	−0.863712
$549$	0	0
$550$	3746.53	0.290459
$551$	7016.18	0.542467
$552$	0	0
$553$	3742.50	0.287789
$554$	−1515.11	−0.116193
$555$	0	0
$556$	−5185.07	−0.395496
$557$	−14249.9	−1.08400	−0.542001	−	0.840378i	$-0.682333\pi$
$557$	−14249.9	−1.08400	−0.542001	+	0.840378i	$0.682333\pi$
$558$	0	0
$559$	−5242.88	−0.396691
$560$	3908.98	0.294973
$561$	0	0
$562$	−16815.5	−1.26213
$563$	20886.4	1.56351	0.781755	−	0.623586i	$-0.214325\pi$
$563$	20886.4	1.56351	0.781755	+	0.623586i	$0.214325\pi$
$564$	0	0
$565$	−3338.62	−0.248596
$566$	−5044.13	−0.374594
$567$	0	0
$568$	−6221.37	−0.459583
$569$	−18602.8	−1.37060	−0.685299	−	0.728262i	$-0.740328\pi$
$569$	−18602.8	−1.37060	−0.685299	+	0.728262i	$0.740328\pi$
$570$	0	0
$571$	−13118.5	−0.961456	−0.480728	−	0.876870i	$-0.659628\pi$
$571$	−13118.5	−0.961456	−0.480728	+	0.876870i	$0.659628\pi$
$572$	5366.80	0.392303
$573$	0	0
$574$	12360.1	0.898779
$575$	1048.04	0.0760107
$576$	0	0
$577$	15596.4	1.12528	0.562640	−	0.826702i	$-0.309786\pi$
$577$	15596.4	1.12528	0.562640	+	0.826702i	$0.309786\pi$
$578$	1987.99	0.143061
$579$	0	0
$580$	−2221.86	−0.159065
$581$	935.000	0.0667648
$582$	0	0
$583$	−25534.9	−1.81397
$584$	−1504.80	−0.106625
$585$	0	0
$586$	6714.27	0.473317
$587$	−15569.7	−1.09477	−0.547385	−	0.836881i	$-0.684377\pi$
$587$	−15569.7	−1.09477	−0.547385	+	0.836881i	$0.684377\pi$
$588$	0	0
$589$	−40639.2	−2.84297
$590$	9075.00	0.633241
$591$	0	0
$592$	4506.74	0.312882
$593$	−1616.91	−0.111971	−0.0559854	−	0.998432i	$-0.517830\pi$
$593$	−1616.91	−0.111971	−0.0559854	+	0.998432i	$0.517830\pi$
$594$	0	0
$595$	−18777.0	−1.29375
$596$	−802.797	−0.0551742
$597$	0	0
$598$	1501.28	0.102662
$599$	10595.7	0.722752	0.361376	−	0.932420i	$-0.382307\pi$
$599$	10595.7	0.722752	0.361376	+	0.932420i	$0.382307\pi$
$600$	0	0
$601$	−5699.09	−0.386806	−0.193403	−	0.981119i	$-0.561953\pi$
$601$	−5699.09	−0.386806	−0.193403	+	0.981119i	$0.561953\pi$
$602$	−6010.23	−0.406908
$603$	0	0
$604$	−436.836	−0.0294282
$605$	4689.34	0.315122
$606$	0	0
$607$	15513.1	1.03733	0.518664	−	0.854978i	$-0.326430\pi$
$607$	15513.1	1.03733	0.518664	+	0.854978i	$0.326430\pi$
$608$	−5278.87	−0.352116
$609$	0	0
$610$	9719.84	0.645155
$611$	−18880.0	−1.25008
$612$	0	0
$613$	5817.04	0.383276	0.191638	−	0.981466i	$-0.438620\pi$
$613$	5817.04	0.383276	0.191638	+	0.981466i	$0.438620\pi$
$614$	−15320.4	−1.00697
$615$	0	0
$616$	6152.29	0.402407
$617$	−18218.6	−1.18874	−0.594370	−	0.804191i	$-0.702599\pi$
$617$	−18218.6	−1.18874	−0.594370	+	0.804191i	$0.702599\pi$
$618$	0	0
$619$	23581.8	1.53123	0.765617	−	0.643296i	$-0.222434\pi$
$619$	23581.8	1.53123	0.765617	+	0.643296i	$0.222434\pi$
$620$	12869.5	0.833632
$621$	0	0
$622$	688.812	0.0444033
$623$	−14359.2	−0.923418
$624$	0	0
$625$	−19244.5	−1.23165
$626$	6807.68	0.434648
$627$	0	0
$628$	−13198.6	−0.838665
$629$	−21648.4	−1.37230
$630$	0	0
$631$	−14989.0	−0.945649	−0.472825	−	0.881157i	$-0.656766\pi$
$631$	−14989.0	−0.945649	−0.472825	+	0.881157i	$0.656766\pi$
$632$	1600.50	0.100735
$633$	0	0
$634$	−11602.7	−0.726820
$635$	−20162.6	−1.26004
$636$	0	0
$637$	226.476	0.0140868
$638$	−3496.96	−0.217000
$639$	0	0
$640$	1671.70	0.103249
$641$	16769.0	1.03328	0.516642	−	0.856201i	$-0.327182\pi$
$641$	16769.0	1.03328	0.516642	+	0.856201i	$0.327182\pi$
$642$	0	0
$643$	−15863.5	−0.972933	−0.486467	−	0.873699i	$-0.661714\pi$
$643$	−15863.5	−0.972933	−0.486467	+	0.873699i	$0.661714\pi$
$644$	1721.01	0.105307
$645$	0	0
$646$	25357.4	1.54438
$647$	25100.0	1.52516	0.762582	−	0.646891i	$-0.223931\pi$
$647$	25100.0	1.52516	0.762582	+	0.646891i	$0.223931\pi$
$648$	0	0
$649$	14283.0	0.863880
$650$	2974.29	0.179479
$651$	0	0
$652$	954.693	0.0573446
$653$	−7127.01	−0.427108	−0.213554	−	0.976931i	$-0.568504\pi$
$653$	−7127.01	−0.427108	−0.213554	+	0.976931i	$0.568504\pi$
$654$	0	0
$655$	−1526.80	−0.0910795
$656$	5285.84	0.314600
$657$	0	0
$658$	−21643.2	−1.28228
$659$	18985.3	1.12225	0.561124	−	0.827731i	$-0.310369\pi$
$659$	18985.3	1.12225	0.561124	+	0.827731i	$0.310369\pi$
$660$	0	0
$661$	2780.35	0.163605	0.0818026	−	0.996649i	$-0.473932\pi$
$661$	2780.35	0.163605	0.0818026	+	0.996649i	$0.473932\pi$
$662$	16801.3	0.986407
$663$	0	0
$664$	399.857	0.0233697
$665$	−40302.7	−2.35018
$666$	0	0
$667$	−978.223	−0.0567870
$668$	12497.7	0.723880
$669$	0	0
$670$	23101.6	1.33208
$671$	15297.9	0.880134
$672$	0	0
$673$	−9328.01	−0.534277	−0.267138	−	0.963658i	$-0.586078\pi$
$673$	−9328.01	−0.534277	−0.267138	+	0.963658i	$0.586078\pi$
$674$	8288.34	0.473672
$675$	0	0
$676$	−4527.41	−0.257590
$677$	−33535.6	−1.90381	−0.951905	−	0.306393i	$-0.900878\pi$
$677$	−33535.6	−1.90381	−0.951905	+	0.306393i	$0.900878\pi$
$678$	0	0
$679$	30422.9	1.71948
$680$	−8030.09	−0.452853
$681$	0	0
$682$	20255.1	1.13726
$683$	25507.4	1.42901	0.714506	−	0.699629i	$-0.246652\pi$
$683$	25507.4	1.42901	0.714506	+	0.699629i	$0.246652\pi$
$684$	0	0
$685$	−36176.5	−2.01786
$686$	−12573.1	−0.699774
$687$	0	0
$688$	−2570.31	−0.142430
$689$	−20271.6	−1.12088
$690$	0	0
$691$	−3801.07	−0.209261	−0.104631	−	0.994511i	$-0.533366\pi$
$691$	−3801.07	−0.209261	−0.104631	+	0.994511i	$0.533366\pi$
$692$	−10929.7	−0.600413
$693$	0	0
$694$	7703.41	0.421351
$695$	−16929.4	−0.923984
$696$	0	0
$697$	−25390.9	−1.37984
$698$	5162.34	0.279939
$699$	0	0
$700$	3409.61	0.184102
$701$	24101.5	1.29858	0.649289	−	0.760542i	$-0.275067\pi$
$701$	24101.5	1.29858	0.649289	+	0.760542i	$0.275067\pi$
$702$	0	0
$703$	−46465.8	−2.49287
$704$	2631.06	0.140855
$705$	0	0
$706$	15497.5	0.826143
$707$	−6849.02	−0.364334
$708$	0	0
$709$	27015.2	1.43100	0.715498	−	0.698615i	$-0.246200\pi$
$709$	27015.2	1.43100	0.715498	+	0.698615i	$0.246200\pi$
$710$	−20313.0	−1.07371
$711$	0	0
$712$	−6140.79	−0.323224
$713$	5666.07	0.297610
$714$	0	0
$715$	17522.8	0.916524
$716$	9403.79	0.490833
$717$	0	0
$718$	3914.04	0.203441
$719$	−12022.5	−0.623593	−0.311797	−	0.950149i	$-0.600931\pi$
$719$	−12022.5	−0.623593	−0.311797	+	0.950149i	$0.600931\pi$
$720$	0	0
$721$	34203.4	1.76671
$722$	40708.7	2.09837
$723$	0	0
$724$	−8899.33	−0.456825
$725$	−1938.02	−0.0992776
$726$	0	0
$727$	20108.7	1.02585	0.512923	−	0.858435i	$-0.328563\pi$
$727$	20108.7	1.02585	0.512923	+	0.858435i	$0.328563\pi$
$728$	4884.18	0.248653
$729$	0	0
$730$	−4913.20	−0.249104
$731$	12346.6	0.624701
$732$	0	0
$733$	16026.4	0.807568	0.403784	−	0.914854i	$-0.367695\pi$
$733$	16026.4	0.807568	0.403784	+	0.914854i	$0.367695\pi$
$734$	11651.5	0.585920
$735$	0	0
$736$	736.000	0.0368605
$737$	36359.3	1.81725
$738$	0	0
$739$	−121.045	−0.00602532	−0.00301266	−	0.999995i	$-0.500959\pi$
$739$	−121.045	−0.00602532	−0.00301266	+	0.999995i	$0.500959\pi$
$740$	14714.6	0.730975
$741$	0	0
$742$	−23238.6	−1.14975
$743$	32902.8	1.62461	0.812306	−	0.583231i	$-0.198212\pi$
$743$	32902.8	1.62461	0.812306	+	0.583231i	$0.198212\pi$
$744$	0	0
$745$	−2621.16	−0.128902
$746$	4679.50	0.229663
$747$	0	0
$748$	−12638.5	−0.617791
$749$	−10394.5	−0.507087
$750$	0	0
$751$	−8822.39	−0.428673	−0.214337	−	0.976760i	$-0.568759\pi$
$751$	−8822.39	−0.428673	−0.214337	+	0.976760i	$0.568759\pi$
$752$	−9255.84	−0.448838
$753$	0	0
$754$	−2776.16	−0.134087
$755$	−1426.28	−0.0687520
$756$	0	0
$757$	17256.9	0.828552	0.414276	−	0.910151i	$-0.364035\pi$
$757$	17256.9	0.828552	0.414276	+	0.910151i	$0.364035\pi$
$758$	20763.8	0.994952
$759$	0	0
$760$	−17235.7	−0.822636
$761$	28563.8	1.36063	0.680313	−	0.732921i	$-0.261844\pi$
$761$	28563.8	1.36063	0.680313	+	0.732921i	$0.261844\pi$
$762$	0	0
$763$	−17824.7	−0.845736
$764$	12396.2	0.587013
$765$	0	0
$766$	−9363.94	−0.441688
$767$	11339.0	0.533804
$768$	0	0
$769$	36869.9	1.72895	0.864475	−	0.502676i	$-0.167651\pi$
$769$	36869.9	1.72895	0.864475	+	0.502676i	$0.167651\pi$
$770$	20087.4	0.940131
$771$	0	0
$772$	−2719.74	−0.126795
$773$	−31615.1	−1.47104	−0.735520	−	0.677503i	$-0.763062\pi$
$773$	−31615.1	−1.47104	−0.735520	+	0.677503i	$0.763062\pi$
$774$	0	0
$775$	11225.4	0.520296
$776$	13010.5	0.601869
$777$	0	0
$778$	−5792.52	−0.266931
$779$	−54498.6	−2.50656
$780$	0	0
$781$	−31970.3	−1.46477
$782$	−3535.42	−0.161671
$783$	0	0
$784$	111.029	0.00505781
$785$	−43093.9	−1.95934
$786$	0	0
$787$	−22822.8	−1.03373	−0.516865	−	0.856067i	$-0.672901\pi$
$787$	−22822.8	−1.03373	−0.516865	+	0.856067i	$0.672901\pi$
$788$	−199.647	−0.00902555
$789$	0	0
$790$	5225.68	0.235343
$791$	−4782.07	−0.214957
$792$	0	0
$793$	12144.7	0.543847
$794$	−3199.21	−0.142992
$795$	0	0
$796$	−16211.2	−0.721847
$797$	32080.5	1.42578	0.712892	−	0.701274i	$-0.247385\pi$
$797$	32080.5	1.42578	0.712892	+	0.701274i	$0.247385\pi$
$798$	0	0
$799$	44461.0	1.96861
$800$	1458.14	0.0644412
$801$	0	0
$802$	−4265.60	−0.187810
$803$	−7732.83	−0.339833
$804$	0	0
$805$	5619.16	0.246024
$806$	16080.1	0.702727
$807$	0	0
$808$	−2929.02	−0.127528
$809$	12622.8	0.548570	0.274285	−	0.961649i	$-0.411559\pi$
$809$	12622.8	0.548570	0.274285	+	0.961649i	$0.411559\pi$
$810$	0	0
$811$	14725.4	0.637581	0.318791	−	0.947825i	$-0.396723\pi$
$811$	14725.4	0.637581	0.318791	+	0.947825i	$0.396723\pi$
$812$	−3182.48	−0.137541
$813$	0	0
$814$	23159.2	0.997210
$815$	3117.10	0.133972
$816$	0	0
$817$	26500.6	1.13481
$818$	−8363.23	−0.357474
$819$	0	0
$820$	17258.4	0.734989
$821$	−25304.1	−1.07566	−0.537832	−	0.843052i	$-0.680756\pi$
$821$	−25304.1	−1.07566	−0.537832	+	0.843052i	$0.680756\pi$
$822$	0	0
$823$	−26359.2	−1.11643	−0.558217	−	0.829695i	$-0.688514\pi$
$823$	−26359.2	−1.11643	−0.558217	+	0.829695i	$0.688514\pi$
$824$	14627.2	0.618403
$825$	0	0
$826$	12998.6	0.547553
$827$	23399.7	0.983902	0.491951	−	0.870623i	$-0.336284\pi$
$827$	23399.7	0.983902	0.491951	+	0.870623i	$0.336284\pi$
$828$	0	0
$829$	31965.7	1.33922	0.669612	−	0.742711i	$-0.266461\pi$
$829$	31965.7	1.33922	0.669612	+	0.742711i	$0.266461\pi$
$830$	1305.55	0.0545978
$831$	0	0
$832$	2088.74	0.0870362
$833$	−533.335	−0.0221836
$834$	0	0
$835$	40805.5	1.69118
$836$	−27127.0	−1.12226
$837$	0	0
$838$	7732.13	0.318738
$839$	31922.9	1.31359	0.656793	−	0.754071i	$-0.271912\pi$
$839$	31922.9	1.31359	0.656793	+	0.754071i	$0.271912\pi$
$840$	0	0
$841$	−22580.1	−0.925830
$842$	−13777.4	−0.563896
$843$	0	0
$844$	−3456.57	−0.140971
$845$	−14782.1	−0.601799
$846$	0	0
$847$	6716.77	0.272481
$848$	−9938.09	−0.402447
$849$	0	0
$850$	−7004.25	−0.282640
$851$	6478.44	0.260961
$852$	0	0
$853$	23656.8	0.949583	0.474791	−	0.880098i	$-0.342523\pi$
$853$	23656.8	0.949583	0.474791	+	0.880098i	$0.342523\pi$
$854$	13922.2	0.557855
$855$	0	0
$856$	−4445.28	−0.177496
$857$	−31502.3	−1.25566	−0.627829	−	0.778351i	$-0.716056\pi$
$857$	−31502.3	−1.25566	−0.627829	+	0.778351i	$0.716056\pi$
$858$	0	0
$859$	−4487.44	−0.178242	−0.0891208	−	0.996021i	$-0.528406\pi$
$859$	−4487.44	−0.178242	−0.0891208	+	0.996021i	$0.528406\pi$
$860$	−8392.13	−0.332755
$861$	0	0
$862$	−24699.7	−0.975956
$863$	−21730.6	−0.857147	−0.428574	−	0.903507i	$-0.640984\pi$
$863$	−21730.6	−0.857147	−0.428574	+	0.903507i	$0.640984\pi$
$864$	0	0
$865$	−35685.9	−1.40273
$866$	−25062.3	−0.983431
$867$	0	0
$868$	18433.6	0.720827
$869$	8224.63	0.321060
$870$	0	0
$871$	28864.9	1.12290
$872$	−7622.82	−0.296033
$873$	0	0
$874$	−7588.37	−0.293685
$875$	−19406.4	−0.749780
$876$	0	0
$877$	−9754.59	−0.375586	−0.187793	−	0.982209i	$-0.560133\pi$
$877$	−9754.59	−0.375586	−0.187793	+	0.982209i	$0.560133\pi$
$878$	−23068.3	−0.886696
$879$	0	0
$880$	8590.49	0.329074
$881$	25789.6	0.986235	0.493118	−	0.869963i	$-0.335857\pi$
$881$	25789.6	0.986235	0.493118	+	0.869963i	$0.335857\pi$
$882$	0	0
$883$	−8851.25	−0.337337	−0.168668	−	0.985673i	$-0.553947\pi$
$883$	−8851.25	−0.337337	−0.168668	+	0.985673i	$0.553947\pi$
$884$	−10033.4	−0.381742
$885$	0	0
$886$	5964.99	0.226183
$887$	−13140.9	−0.497439	−0.248720	−	0.968576i	$-0.580010\pi$
$887$	−13140.9	−0.497439	−0.248720	+	0.968576i	$0.580010\pi$
$888$	0	0
$889$	−28879.9	−1.08954
$890$	−20049.9	−0.755138
$891$	0	0
$892$	22729.8	0.853196
$893$	95430.4	3.57610
$894$	0	0
$895$	30703.7	1.14672
$896$	2394.45	0.0892780
$897$	0	0
$898$	25261.7	0.938744
$899$	−10477.7	−0.388709
$900$	0	0
$901$	47738.2	1.76514
$902$	27162.8	1.00269
$903$	0	0
$904$	−2045.07	−0.0752413
$905$	−29056.6	−1.06726
$906$	0	0
$907$	−12195.6	−0.446471	−0.223235	−	0.974765i	$-0.571662\pi$
$907$	−12195.6	−0.446471	−0.223235	+	0.974765i	$0.571662\pi$
$908$	−632.394	−0.0231132
$909$	0	0
$910$	15947.0	0.580920
$911$	23873.1	0.868223	0.434111	−	0.900859i	$-0.357062\pi$
$911$	23873.1	0.868223	0.434111	+	0.900859i	$0.357062\pi$
$912$	0	0
$913$	2054.78	0.0744835
$914$	−27386.1	−0.991087
$915$	0	0
$916$	10979.8	0.396050
$917$	−2186.92	−0.0787550
$918$	0	0
$919$	−25048.2	−0.899092	−0.449546	−	0.893257i	$-0.648414\pi$
$919$	−25048.2	−0.899092	−0.449546	+	0.893257i	$0.648414\pi$
$920$	2403.06	0.0861159
$921$	0	0
$922$	9177.28	0.327806
$923$	−25380.6	−0.905105
$924$	0	0
$925$	12834.9	0.456224
$926$	12221.7	0.433725
$927$	0	0
$928$	−1361.01	−0.0481435
$929$	40641.7	1.43532	0.717660	−	0.696394i	$-0.245213\pi$
$929$	40641.7	1.43532	0.717660	+	0.696394i	$0.245213\pi$
$930$	0	0
$931$	−1144.74	−0.0402979
$932$	14590.5	0.512798
$933$	0	0
$934$	4156.96	0.145632
$935$	−41265.0	−1.44332
$936$	0	0
$937$	39519.4	1.37785	0.688924	−	0.724833i	$-0.258083\pi$
$937$	39519.4	1.37785	0.688924	+	0.724833i	$0.258083\pi$
$938$	33089.6	1.15183
$939$	0	0
$940$	−30220.6	−1.04860
$941$	32189.2	1.11513	0.557566	−	0.830133i	$-0.311735\pi$
$941$	32189.2	1.11513	0.557566	+	0.830133i	$0.311735\pi$
$942$	0	0
$943$	7598.40	0.262394
$944$	5558.91	0.191660
$945$	0	0
$946$	−13208.3	−0.453951
$947$	18703.2	0.641786	0.320893	−	0.947115i	$-0.396017\pi$
$947$	18703.2	0.641786	0.320893	+	0.947115i	$0.396017\pi$
$948$	0	0
$949$	−6138.93	−0.209987
$950$	−15033.8	−0.513433
$951$	0	0
$952$	−11501.9	−0.391574
$953$	−19901.0	−0.676449	−0.338224	−	0.941066i	$-0.609826\pi$
$953$	−19901.0	−0.676449	−0.338224	+	0.941066i	$0.609826\pi$
$954$	0	0
$955$	40473.9	1.37142
$956$	−10320.5	−0.349153
$957$	0	0
$958$	26490.8	0.893401
$959$	−51817.4	−1.74481
$960$	0	0
$961$	30897.9	1.03715
$962$	18385.6	0.616191
$963$	0	0
$964$	17994.3	0.601202
$965$	−8880.03	−0.296226
$966$	0	0
$967$	−1654.58	−0.0550236	−0.0275118	−	0.999621i	$-0.508758\pi$
$967$	−1654.58	−0.0550236	−0.0275118	+	0.999621i	$0.508758\pi$
$968$	2872.46	0.0953765
$969$	0	0
$970$	42479.7	1.40613
$971$	−21532.9	−0.711661	−0.355831	−	0.934550i	$-0.615802\pi$
$971$	−21532.9	−0.711661	−0.355831	+	0.934550i	$0.615802\pi$
$972$	0	0
$973$	−24248.8	−0.798953
$974$	16654.7	0.547897
$975$	0	0
$976$	5953.90	0.195266
$977$	−28732.0	−0.940856	−0.470428	−	0.882438i	$-0.655901\pi$
$977$	−28732.0	−0.940856	−0.470428	+	0.882438i	$0.655901\pi$
$978$	0	0
$979$	−31556.2	−1.03017
$980$	362.513	0.0118164
$981$	0	0
$982$	21171.8	0.688004
$983$	7497.77	0.243277	0.121639	−	0.992574i	$-0.461185\pi$
$983$	7497.77	0.243277	0.121639	+	0.992574i	$0.461185\pi$
$984$	0	0
$985$	−651.854	−0.0210861
$986$	6537.67	0.211158
$987$	0	0
$988$	−21535.5	−0.693458
$989$	−3694.82	−0.118795
$990$	0	0
$991$	32374.5	1.03775	0.518874	−	0.854851i	$-0.326351\pi$
$991$	32374.5	1.03775	0.518874	+	0.854851i	$0.326351\pi$
$992$	7883.23	0.252311
$993$	0	0
$994$	−29095.3	−0.928417
$995$	−52930.0	−1.68643
$996$	0	0
$997$	19368.9	0.615264	0.307632	−	0.951505i	$-0.400463\pi$
$997$	19368.9	0.615264	0.307632	+	0.951505i	$0.400463\pi$
$998$	−5261.77	−0.166892
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.4.a.n.1.3	yes	4
3.2	odd	2		414.4.a.m.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.4.a.m.1.2	✓	4	3.2	odd	2
414.4.a.n.1.3	yes	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +13.0601 q^{5} +18.7067 q^{7} +8.00000 q^{8} +26.1202 q^{10} +41.1103 q^{11} +32.6366 q^{13} +37.4133 q^{14} +16.0000 q^{16} -76.8570 q^{17} -164.965 q^{19} +52.2405 q^{20} +82.2206 q^{22} +23.0000 q^{23} +45.5668 q^{25} +65.2732 q^{26} +74.8267 q^{28} -42.5314 q^{29} +246.351 q^{31} +32.0000 q^{32} -153.714 q^{34} +244.311 q^{35} +281.671 q^{37} -329.929 q^{38} +104.481 q^{40} +330.365 q^{41} -160.644 q^{43} +164.441 q^{44} +46.0000 q^{46} -578.490 q^{47} +6.93931 q^{49} +91.1336 q^{50} +130.546 q^{52} -621.131 q^{53} +536.906 q^{55} +149.653 q^{56} -85.0628 q^{58} +347.432 q^{59} +372.119 q^{61} +492.702 q^{62} +64.0000 q^{64} +426.238 q^{65} +884.432 q^{67} -307.428 q^{68} +488.623 q^{70} -777.672 q^{71} -188.099 q^{73} +563.343 q^{74} -659.859 q^{76} +769.037 q^{77} +200.062 q^{79} +208.962 q^{80} +660.730 q^{82} +49.9822 q^{83} -1003.76 q^{85} -321.288 q^{86} +328.882 q^{88} -767.598 q^{89} +610.522 q^{91} +92.0000 q^{92} -1156.98 q^{94} -2154.46 q^{95} +1626.31 q^{97} +13.8786 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 18 q^{7} + 32 q^{8} + 42 q^{11} + 108 q^{13} + 36 q^{14} + 64 q^{16} + 26 q^{17} + 132 q^{19} + 84 q^{22} + 92 q^{23} + 260 q^{25} + 216 q^{26} + 72 q^{28} + 252 q^{29} + 428 q^{31}+ \cdots + 2152 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 + 18 * q^7 + 32 * q^8 + 42 * q^11 + 108 * q^13 + 36 * q^14 + 64 * q^16 + 26 * q^17 + 132 * q^19 + 84 * q^22 + 92 * q^23 + 260 * q^25 + 216 * q^26 + 72 * q^28 + 252 * q^29 + 428 * q^31 + 128 * q^32 + 52 * q^34 + 348 * q^35 + 278 * q^37 + 264 * q^38 - 4 * q^41 + 216 * q^43 + 168 * q^44 + 184 * q^46 - 4 * q^47 + 1076 * q^49 + 520 * q^50 + 432 * q^52 - 460 * q^53 + 512 * q^55 + 144 * q^56 + 504 * q^58 - 504 * q^59 + 1086 * q^61 + 856 * q^62 + 256 * q^64 - 1420 * q^65 + 1484 * q^67 + 104 * q^68 + 696 * q^70 - 876 * q^71 + 1072 * q^73 + 556 * q^74 + 528 * q^76 - 1580 * q^77 + 1634 * q^79 - 8 * q^82 - 2246 * q^83 + 872 * q^85 + 432 * q^86 + 336 * q^88 - 474 * q^89 + 40 * q^91 + 368 * q^92 - 8 * q^94 - 2780 * q^95 + 1044 * q^97 + 2152 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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