LMFDB - Embedded newform 693.4.a.u.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 43x^{6} + 57x^{5} + 560x^{4} - 439x^{3} - 2246x^{2} + 384x + 1056 $ x^8 - 2x^7 - 43x^6 + 57x^5 + 560x^4 - 439x^3 - 2246x^2 + 384*x + 1056
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-4.31088$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.31088 q^{2} +20.2055 q^{4} -7.25200 q^{5} +7.00000 q^{7} +64.8218 q^{8} -38.5145 q^{10} +11.0000 q^{11} +47.6369 q^{13} +37.1762 q^{14} +182.617 q^{16} -31.5955 q^{17} +18.9406 q^{19} -146.530 q^{20} +58.4197 q^{22} +200.570 q^{23} -72.4085 q^{25} +252.994 q^{26} +141.438 q^{28} +224.264 q^{29} -237.936 q^{31} +451.284 q^{32} -167.800 q^{34} -50.7640 q^{35} +226.700 q^{37} +100.591 q^{38} -470.088 q^{40} -31.1161 q^{41} -176.064 q^{43} +222.260 q^{44} +1065.20 q^{46} +526.826 q^{47} +49.0000 q^{49} -384.553 q^{50} +962.527 q^{52} -342.689 q^{53} -79.7720 q^{55} +453.753 q^{56} +1191.04 q^{58} -283.465 q^{59} -216.971 q^{61} -1263.65 q^{62} +935.778 q^{64} -345.463 q^{65} -180.035 q^{67} -638.402 q^{68} -269.602 q^{70} -166.734 q^{71} +44.8888 q^{73} +1203.98 q^{74} +382.704 q^{76} +77.0000 q^{77} +349.350 q^{79} -1324.34 q^{80} -165.254 q^{82} +722.258 q^{83} +229.131 q^{85} -935.053 q^{86} +713.040 q^{88} +443.808 q^{89} +333.459 q^{91} +4052.61 q^{92} +2797.91 q^{94} -137.357 q^{95} -1804.50 q^{97} +260.233 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{2} + 30 q^{4} + 10 q^{5} + 56 q^{7} + 81 q^{8} + 9 q^{10} + 88 q^{11} + 16 q^{13} + 42 q^{14} + 122 q^{16} + 90 q^{17} - 42 q^{19} + 291 q^{20} + 66 q^{22} + 338 q^{23} + 244 q^{25} + 209 q^{26}+ \cdots + 294 q^{98}+O(q^{100}) $ 8 * q + 6 * q^2 + 30 * q^4 + 10 * q^5 + 56 * q^7 + 81 * q^8 + 9 * q^10 + 88 * q^11 + 16 * q^13 + 42 * q^14 + 122 * q^16 + 90 * q^17 - 42 * q^19 + 291 * q^20 + 66 * q^22 + 338 * q^23 + 244 * q^25 + 209 * q^26 + 210 * q^28 + 496 * q^29 - 8 * q^31 + 524 * q^32 - 302 * q^34 + 70 * q^35 - 360 * q^37 + 45 * q^38 - 6 * q^40 + 242 * q^41 - 66 * q^43 + 330 * q^44 + 344 * q^46 + 540 * q^47 + 392 * q^49 + 1171 * q^50 + 465 * q^52 + 906 * q^53 + 110 * q^55 + 567 * q^56 + 977 * q^58 + 1242 * q^59 - 318 * q^61 - 110 * q^62 + 525 * q^64 + 1258 * q^65 + 522 * q^67 + 678 * q^68 + 63 * q^70 + 858 * q^71 - 78 * q^73 + 1651 * q^74 + 1775 * q^76 + 616 * q^77 + 516 * q^79 + 567 * q^80 - 1212 * q^82 + 3192 * q^83 + 720 * q^85 + 1322 * q^86 + 891 * q^88 + 2356 * q^89 + 112 * q^91 + 4504 * q^92 - 423 * q^94 + 3308 * q^95 - 1556 * q^97 + 294 * 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 $

$2$	5.31088	1.87768	0.938840	−	0.344353i	$-0.111902\pi$
$2$	5.31088	1.87768	0.938840	+	0.344353i	$0.111902\pi$
$3$	0	0
$3$	0	0
$4$	20.2055	2.52568
$5$	−7.25200	−0.648639	−0.324319	−	0.945948i	$-0.605135\pi$
$5$	−7.25200	−0.648639	−0.324319	+	0.945948i	$0.605135\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	64.8218	2.86475
$9$	0	0
$10$	−38.5145	−1.21794
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	47.6369	1.01632	0.508158	−	0.861264i	$-0.330327\pi$
$13$	47.6369	1.01632	0.508158	+	0.861264i	$0.330327\pi$
$14$	37.1762	0.709696
$15$	0	0
$16$	182.617	2.85339
$17$	−31.5955	−0.450766	−0.225383	−	0.974270i	$-0.572363\pi$
$17$	−31.5955	−0.450766	−0.225383	+	0.974270i	$0.572363\pi$
$18$	0	0
$19$	18.9406	0.228699	0.114349	−	0.993441i	$-0.463522\pi$
$19$	18.9406	0.228699	0.114349	+	0.993441i	$0.463522\pi$
$20$	−146.530	−1.63826
$21$	0	0
$22$	58.4197	0.566142
$23$	200.570	1.81833	0.909167	−	0.416431i	$-0.136719\pi$
$23$	200.570	1.81833	0.909167	+	0.416431i	$0.136719\pi$
$24$	0	0
$25$	−72.4085	−0.579268
$26$	252.994	1.90832
$27$	0	0
$28$	141.438	0.954619
$29$	224.264	1.43603	0.718014	−	0.696029i	$-0.245052\pi$
$29$	224.264	1.43603	0.718014	+	0.696029i	$0.245052\pi$
$30$	0	0
$31$	−237.936	−1.37853	−0.689267	−	0.724507i	$-0.742067\pi$
$31$	−237.936	−1.37853	−0.689267	+	0.724507i	$0.742067\pi$
$32$	451.284	2.49302
$33$	0	0
$34$	−167.800	−0.846395
$35$	−50.7640	−0.245162
$36$	0	0
$37$	226.700	1.00728	0.503639	−	0.863914i	$-0.331994\pi$
$37$	226.700	1.00728	0.503639	+	0.863914i	$0.331994\pi$
$38$	100.591	0.429423
$39$	0	0
$40$	−470.088	−1.85819
$41$	−31.1161	−0.118525	−0.0592625	−	0.998242i	$-0.518875\pi$
$41$	−31.1161	−0.118525	−0.0592625	+	0.998242i	$0.518875\pi$
$42$	0	0
$43$	−176.064	−0.624406	−0.312203	−	0.950015i	$-0.601067\pi$
$43$	−176.064	−0.624406	−0.312203	+	0.950015i	$0.601067\pi$
$44$	222.260	0.761522
$45$	0	0
$46$	1065.20	3.41425
$47$	526.826	1.63501	0.817505	−	0.575922i	$-0.195357\pi$
$47$	526.826	1.63501	0.817505	+	0.575922i	$0.195357\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−384.553	−1.08768
$51$	0	0
$52$	962.527	2.56689
$53$	−342.689	−0.888151	−0.444076	−	0.895989i	$-0.646468\pi$
$53$	−342.689	−0.888151	−0.444076	+	0.895989i	$0.646468\pi$
$54$	0	0
$55$	−79.7720	−0.195572
$56$	453.753	1.08277
$57$	0	0
$58$	1191.04	2.69640
$59$	−283.465	−0.625492	−0.312746	−	0.949837i	$-0.601249\pi$
$59$	−283.465	−0.625492	−0.312746	+	0.949837i	$0.601249\pi$
$60$	0	0
$61$	−216.971	−0.455415	−0.227708	−	0.973730i	$-0.573123\pi$
$61$	−216.971	−0.455415	−0.227708	+	0.973730i	$0.573123\pi$
$62$	−1263.65	−2.58845
$63$	0	0
$64$	935.778	1.82769
$65$	−345.463	−0.659222
$66$	0	0
$67$	−180.035	−0.328280	−0.164140	−	0.986437i	$-0.552485\pi$
$67$	−180.035	−0.328280	−0.164140	+	0.986437i	$0.552485\pi$
$68$	−638.402	−1.13849
$69$	0	0
$70$	−269.602	−0.460337
$71$	−166.734	−0.278699	−0.139350	−	0.990243i	$-0.544501\pi$
$71$	−166.734	−0.278699	−0.139350	+	0.990243i	$0.544501\pi$
$72$	0	0
$73$	44.8888	0.0719704	0.0359852	−	0.999352i	$-0.488543\pi$
$73$	44.8888	0.0719704	0.0359852	+	0.999352i	$0.488543\pi$
$74$	1203.98	1.89135
$75$	0	0
$76$	382.704	0.577620
$77$	77.0000	0.113961
$78$	0	0
$79$	349.350	0.497530	0.248765	−	0.968564i	$-0.419975\pi$
$79$	349.350	0.497530	0.248765	+	0.968564i	$0.419975\pi$
$80$	−1324.34	−1.85082
$81$	0	0
$82$	−165.254	−0.222552
$83$	722.258	0.955158	0.477579	−	0.878589i	$-0.341514\pi$
$83$	722.258	0.955158	0.477579	+	0.878589i	$0.341514\pi$
$84$	0	0
$85$	229.131	0.292385
$86$	−935.053	−1.17243
$87$	0	0
$88$	713.040	0.863753
$89$	443.808	0.528579	0.264290	−	0.964443i	$-0.414863\pi$
$89$	443.808	0.528579	0.264290	+	0.964443i	$0.414863\pi$
$90$	0	0
$91$	333.459	0.384131
$92$	4052.61	4.59254
$93$	0	0
$94$	2797.91	3.07002
$95$	−137.357	−0.148343
$96$	0	0
$97$	−1804.50	−1.88886	−0.944431	−	0.328710i	$-0.893386\pi$
$97$	−1804.50	−1.88886	−0.944431	+	0.328710i	$0.893386\pi$
$98$	260.233	0.268240
$99$	0	0
$100$	−1463.05	−1.46305
$101$	−1767.79	−1.74160	−0.870801	−	0.491635i	$-0.836399\pi$
$101$	−1767.79	−1.74160	−0.870801	+	0.491635i	$0.836399\pi$
$102$	0	0
$103$	−1224.69	−1.17157	−0.585786	−	0.810465i	$-0.699214\pi$
$103$	−1224.69	−1.17157	−0.585786	+	0.810465i	$0.699214\pi$
$104$	3087.91	2.91149
$105$	0	0
$106$	−1819.98	−1.66766
$107$	−1720.89	−1.55481	−0.777405	−	0.629000i	$-0.783464\pi$
$107$	−1720.89	−1.55481	−0.777405	+	0.629000i	$0.783464\pi$
$108$	0	0
$109$	1811.71	1.59202	0.796011	−	0.605282i	$-0.206940\pi$
$109$	1811.71	1.59202	0.796011	+	0.605282i	$0.206940\pi$
$110$	−423.660	−0.367222
$111$	0	0
$112$	1278.32	1.07848
$113$	−780.246	−0.649552	−0.324776	−	0.945791i	$-0.605289\pi$
$113$	−780.246	−0.649552	−0.324776	+	0.945791i	$0.605289\pi$
$114$	0	0
$115$	−1454.53	−1.17944
$116$	4531.36	3.62695
$117$	0	0
$118$	−1505.45	−1.17447
$119$	−221.168	−0.170374
$120$	0	0
$121$	121.000	0.0909091
$122$	−1152.31	−0.855124
$123$	0	0
$124$	−4807.61	−3.48174
$125$	1431.61	1.02437
$126$	0	0
$127$	−1086.03	−0.758815	−0.379408	−	0.925230i	$-0.623872\pi$
$127$	−1086.03	−0.758815	−0.379408	+	0.925230i	$0.623872\pi$
$128$	1359.54	0.938806
$129$	0	0
$130$	−1834.71	−1.23781
$131$	−1626.30	−1.08466	−0.542328	−	0.840167i	$-0.682457\pi$
$131$	−1626.30	−1.08466	−0.542328	+	0.840167i	$0.682457\pi$
$132$	0	0
$133$	132.584	0.0864399
$134$	−956.144	−0.616405
$135$	0	0
$136$	−2048.08	−1.29133
$137$	2448.94	1.52721	0.763603	−	0.645686i	$-0.223429\pi$
$137$	2448.94	1.52721	0.763603	+	0.645686i	$0.223429\pi$
$138$	0	0
$139$	−2647.66	−1.61562	−0.807811	−	0.589441i	$-0.799348\pi$
$139$	−2647.66	−1.61562	−0.807811	+	0.589441i	$0.799348\pi$
$140$	−1025.71	−0.619203
$141$	0	0
$142$	−885.503	−0.523308
$143$	524.006	0.306431
$144$	0	0
$145$	−1626.36	−0.931463
$146$	238.399	0.135137
$147$	0	0
$148$	4580.59	2.54407
$149$	741.985	0.407958	0.203979	−	0.978975i	$-0.434613\pi$
$149$	741.985	0.407958	0.203979	+	0.978975i	$0.434613\pi$
$150$	0	0
$151$	516.653	0.278441	0.139221	−	0.990261i	$-0.455540\pi$
$151$	516.653	0.278441	0.139221	+	0.990261i	$0.455540\pi$
$152$	1227.76	0.655163
$153$	0	0
$154$	408.938	0.213982
$155$	1725.51	0.894171
$156$	0	0
$157$	−3189.89	−1.62153	−0.810767	−	0.585368i	$-0.800950\pi$
$157$	−3189.89	−1.62153	−0.810767	+	0.585368i	$0.800950\pi$
$158$	1855.35	0.934203
$159$	0	0
$160$	−3272.71	−1.61707
$161$	1403.99	0.687266
$162$	0	0
$163$	−1350.81	−0.649100	−0.324550	−	0.945868i	$-0.605213\pi$
$163$	−1350.81	−0.649100	−0.324550	+	0.945868i	$0.605213\pi$
$164$	−628.716	−0.299357
$165$	0	0
$166$	3835.83	1.79348
$167$	1538.13	0.712721	0.356360	−	0.934349i	$-0.384018\pi$
$167$	1538.13	0.712721	0.356360	+	0.934349i	$0.384018\pi$
$168$	0	0
$169$	72.2788	0.0328988
$170$	1216.89	0.549005
$171$	0	0
$172$	−3557.45	−1.57705
$173$	−256.648	−0.112789	−0.0563947	−	0.998409i	$-0.517961\pi$
$173$	−256.648	−0.112789	−0.0563947	+	0.998409i	$0.517961\pi$
$174$	0	0
$175$	−506.859	−0.218943
$176$	2008.79	0.860331
$177$	0	0
$178$	2357.01	0.992503
$179$	1362.17	0.568789	0.284394	−	0.958707i	$-0.408207\pi$
$179$	1362.17	0.568789	0.284394	+	0.958707i	$0.408207\pi$
$180$	0	0
$181$	2699.57	1.10860	0.554302	−	0.832315i	$-0.312985\pi$
$181$	2699.57	1.10860	0.554302	+	0.832315i	$0.312985\pi$
$182$	1770.96	0.721276
$183$	0	0
$184$	13001.3	5.20907
$185$	−1644.03	−0.653360
$186$	0	0
$187$	−347.550	−0.135911
$188$	10644.8	4.12952
$189$	0	0
$190$	−729.489	−0.278540
$191$	93.4291	0.0353942	0.0176971	−	0.999843i	$-0.494367\pi$
$191$	93.4291	0.0353942	0.0176971	+	0.999843i	$0.494367\pi$
$192$	0	0
$193$	−1259.65	−0.469803	−0.234901	−	0.972019i	$-0.575477\pi$
$193$	−1259.65	−0.469803	−0.234901	+	0.972019i	$0.575477\pi$
$194$	−9583.51	−3.54668
$195$	0	0
$196$	990.068	0.360812
$197$	−1109.80	−0.401371	−0.200686	−	0.979656i	$-0.564317\pi$
$197$	−1109.80	−0.401371	−0.200686	+	0.979656i	$0.564317\pi$
$198$	0	0
$199$	1842.11	0.656200	0.328100	−	0.944643i	$-0.393592\pi$
$199$	1842.11	0.656200	0.328100	+	0.944643i	$0.393592\pi$
$200$	−4693.65	−1.65945
$201$	0	0
$202$	−9388.53	−3.27017
$203$	1569.85	0.542767
$204$	0	0
$205$	225.654	0.0768799
$206$	−6504.17	−2.19984
$207$	0	0
$208$	8699.33	2.89995
$209$	208.347	0.0689552
$210$	0	0
$211$	−3384.17	−1.10415	−0.552075	−	0.833795i	$-0.686164\pi$
$211$	−3384.17	−1.10415	−0.552075	+	0.833795i	$0.686164\pi$
$212$	−6924.20	−2.24319
$213$	0	0
$214$	−9139.44	−2.91944
$215$	1276.81	0.405014
$216$	0	0
$217$	−1665.55	−0.521037
$218$	9621.78	2.98931
$219$	0	0
$220$	−1611.83	−0.493953
$221$	−1505.11	−0.458121
$222$	0	0
$223$	−2571.75	−0.772274	−0.386137	−	0.922441i	$-0.626191\pi$
$223$	−2571.75	−0.772274	−0.386137	+	0.922441i	$0.626191\pi$
$224$	3158.99	0.942271
$225$	0	0
$226$	−4143.79	−1.21965
$227$	853.596	0.249582	0.124791	−	0.992183i	$-0.460174\pi$
$227$	853.596	0.249582	0.124791	+	0.992183i	$0.460174\pi$
$228$	0	0
$229$	4450.26	1.28420	0.642099	−	0.766622i	$-0.278064\pi$
$229$	4450.26	1.28420	0.642099	+	0.766622i	$0.278064\pi$
$230$	−7724.85	−2.21462
$231$	0	0
$232$	14537.2	4.11385
$233$	−1749.92	−0.492023	−0.246011	−	0.969267i	$-0.579120\pi$
$233$	−1749.92	−0.492023	−0.246011	+	0.969267i	$0.579120\pi$
$234$	0	0
$235$	−3820.54	−1.06053
$236$	−5727.55	−1.57979
$237$	0	0
$238$	−1174.60	−0.319907
$239$	2354.68	0.637287	0.318644	−	0.947875i	$-0.396773\pi$
$239$	2354.68	0.637287	0.318644	+	0.947875i	$0.396773\pi$
$240$	0	0
$241$	−5615.20	−1.50086	−0.750429	−	0.660951i	$-0.770153\pi$
$241$	−5615.20	−1.50086	−0.750429	+	0.660951i	$0.770153\pi$
$242$	642.617	0.170698
$243$	0	0
$244$	−4384.01	−1.15023
$245$	−355.348	−0.0926627
$246$	0	0
$247$	902.273	0.232430
$248$	−15423.4	−3.94915
$249$	0	0
$250$	7603.09	1.92345
$251$	−6267.38	−1.57607	−0.788035	−	0.615630i	$-0.788902\pi$
$251$	−6267.38	−1.57607	−0.788035	+	0.615630i	$0.788902\pi$
$252$	0	0
$253$	2206.27	0.548248
$254$	−5767.77	−1.42481
$255$	0	0
$256$	−265.889	−0.0649142
$257$	3537.51	0.858613	0.429307	−	0.903159i	$-0.358758\pi$
$257$	3537.51	0.858613	0.429307	+	0.903159i	$0.358758\pi$
$258$	0	0
$259$	1586.90	0.380716
$260$	−6980.25	−1.66499
$261$	0	0
$262$	−8637.06	−2.03664
$263$	2708.76	0.635093	0.317547	−	0.948243i	$-0.397141\pi$
$263$	2708.76	0.635093	0.317547	+	0.948243i	$0.397141\pi$
$264$	0	0
$265$	2485.18	0.576089
$266$	704.139	0.162307
$267$	0	0
$268$	−3637.69	−0.829132
$269$	5442.62	1.23362	0.616808	−	0.787114i	$-0.288426\pi$
$269$	5442.62	1.23362	0.616808	+	0.787114i	$0.288426\pi$
$270$	0	0
$271$	1121.27	0.251337	0.125669	−	0.992072i	$-0.459892\pi$
$271$	1121.27	0.251337	0.125669	+	0.992072i	$0.459892\pi$
$272$	−5769.88	−1.28621
$273$	0	0
$274$	13006.0	2.86760
$275$	−796.493	−0.174656
$276$	0	0
$277$	−4569.49	−0.991169	−0.495585	−	0.868560i	$-0.665046\pi$
$277$	−4569.49	−0.991169	−0.495585	+	0.868560i	$0.665046\pi$
$278$	−14061.4	−3.03362
$279$	0	0
$280$	−3290.61	−0.702328
$281$	363.403	0.0771488	0.0385744	−	0.999256i	$-0.487718\pi$
$281$	363.403	0.0771488	0.0385744	+	0.999256i	$0.487718\pi$
$282$	0	0
$283$	6985.88	1.46738	0.733688	−	0.679486i	$-0.237797\pi$
$283$	6985.88	1.46738	0.733688	+	0.679486i	$0.237797\pi$
$284$	−3368.93	−0.703906
$285$	0	0
$286$	2782.94	0.575379
$287$	−217.813	−0.0447982
$288$	0	0
$289$	−3914.73	−0.796810
$290$	−8637.42	−1.74899
$291$	0	0
$292$	907.000	0.181774
$293$	203.016	0.0404788	0.0202394	−	0.999795i	$-0.493557\pi$
$293$	203.016	0.0404788	0.0202394	+	0.999795i	$0.493557\pi$
$294$	0	0
$295$	2055.69	0.405718
$296$	14695.1	2.88560
$297$	0	0
$298$	3940.60	0.766015
$299$	9554.53	1.84800
$300$	0	0
$301$	−1232.45	−0.236003
$302$	2743.88	0.522824
$303$	0	0
$304$	3458.88	0.652567
$305$	1573.48	0.295400
$306$	0	0
$307$	−7669.96	−1.42589	−0.712944	−	0.701221i	$-0.752639\pi$
$307$	−7669.96	−1.42589	−0.712944	+	0.701221i	$0.752639\pi$
$308$	1555.82	0.287828
$309$	0	0
$310$	9163.99	1.67897
$311$	4233.11	0.771825	0.385912	−	0.922535i	$-0.373887\pi$
$311$	4233.11	0.771825	0.385912	+	0.922535i	$0.373887\pi$
$312$	0	0
$313$	4622.10	0.834685	0.417342	−	0.908749i	$-0.362962\pi$
$313$	4622.10	0.834685	0.417342	+	0.908749i	$0.362962\pi$
$314$	−16941.1	−3.04472
$315$	0	0
$316$	7058.77	1.25660
$317$	−998.546	−0.176921	−0.0884605	−	0.996080i	$-0.528195\pi$
$317$	−998.546	−0.176921	−0.0884605	+	0.996080i	$0.528195\pi$
$318$	0	0
$319$	2466.90	0.432978
$320$	−6786.27	−1.18551
$321$	0	0
$322$	7456.42	1.29047
$323$	−598.438	−0.103090
$324$	0	0
$325$	−3449.32	−0.588719
$326$	−7173.98	−1.21880
$327$	0	0
$328$	−2017.00	−0.339544
$329$	3687.78	0.617975
$330$	0	0
$331$	5612.23	0.931952	0.465976	−	0.884797i	$-0.345703\pi$
$331$	5612.23	0.931952	0.465976	+	0.884797i	$0.345703\pi$
$332$	14593.6	2.41243
$333$	0	0
$334$	8168.85	1.33826
$335$	1305.61	0.212935
$336$	0	0
$337$	−4655.51	−0.752527	−0.376263	−	0.926513i	$-0.622791\pi$
$337$	−4655.51	−0.752527	−0.376263	+	0.926513i	$0.622791\pi$
$338$	383.864	0.0617735
$339$	0	0
$340$	4629.69	0.738471
$341$	−2617.30	−0.415644
$342$	0	0
$343$	343.000	0.0539949
$344$	−11412.8	−1.78876
$345$	0	0
$346$	−1363.03	−0.211782
$347$	−5166.96	−0.799357	−0.399678	−	0.916655i	$-0.630878\pi$
$347$	−5166.96	−0.799357	−0.399678	+	0.916655i	$0.630878\pi$
$348$	0	0
$349$	9167.31	1.40606	0.703030	−	0.711161i	$-0.251830\pi$
$349$	9167.31	1.40606	0.703030	+	0.711161i	$0.251830\pi$
$350$	−2691.87	−0.411104
$351$	0	0
$352$	4964.12	0.751672
$353$	−29.1770	−0.00439926	−0.00219963	−	0.999998i	$-0.500700\pi$
$353$	−29.1770	−0.00439926	−0.00219963	+	0.999998i	$0.500700\pi$
$354$	0	0
$355$	1209.15	0.180775
$356$	8967.35	1.33502
$357$	0	0
$358$	7234.31	1.06800
$359$	1668.90	0.245352	0.122676	−	0.992447i	$-0.460852\pi$
$359$	1668.90	0.245352	0.122676	+	0.992447i	$0.460852\pi$
$360$	0	0
$361$	−6500.25	−0.947697
$362$	14337.1	2.08161
$363$	0	0
$364$	6737.69	0.970194
$365$	−325.534	−0.0466828
$366$	0	0
$367$	9479.10	1.34824	0.674121	−	0.738621i	$-0.264522\pi$
$367$	9479.10	1.34824	0.674121	+	0.738621i	$0.264522\pi$
$368$	36627.5	5.18842
$369$	0	0
$370$	−8731.26	−1.22680
$371$	−2398.83	−0.335690
$372$	0	0
$373$	−1626.53	−0.225787	−0.112893	−	0.993607i	$-0.536012\pi$
$373$	−1626.53	−0.225787	−0.112893	+	0.993607i	$0.536012\pi$
$374$	−1845.80	−0.255198
$375$	0	0
$376$	34149.8	4.68389
$377$	10683.3	1.45946
$378$	0	0
$379$	−5377.26	−0.728790	−0.364395	−	0.931244i	$-0.618724\pi$
$379$	−5377.26	−0.728790	−0.364395	+	0.931244i	$0.618724\pi$
$380$	−2775.37	−0.374667
$381$	0	0
$382$	496.191	0.0664590
$383$	13111.9	1.74931	0.874657	−	0.484743i	$-0.161087\pi$
$383$	13111.9	1.74931	0.874657	+	0.484743i	$0.161087\pi$
$384$	0	0
$385$	−558.404	−0.0739193
$386$	−6689.88	−0.882139
$387$	0	0
$388$	−36460.8	−4.77067
$389$	895.272	0.116689	0.0583446	−	0.998297i	$-0.481418\pi$
$389$	895.272	0.116689	0.0583446	+	0.998297i	$0.481418\pi$
$390$	0	0
$391$	−6337.10	−0.819644
$392$	3176.27	0.409249
$393$	0	0
$394$	−5894.03	−0.753647
$395$	−2533.48	−0.322718
$396$	0	0
$397$	−5690.01	−0.719329	−0.359664	−	0.933082i	$-0.617109\pi$
$397$	−5690.01	−0.719329	−0.359664	+	0.933082i	$0.617109\pi$
$398$	9783.24	1.23213
$399$	0	0
$400$	−13223.0	−1.65288
$401$	2358.60	0.293723	0.146861	−	0.989157i	$-0.453083\pi$
$401$	2358.60	0.293723	0.146861	+	0.989157i	$0.453083\pi$
$402$	0	0
$403$	−11334.5	−1.40103
$404$	−35719.1	−4.39874
$405$	0	0
$406$	8337.28	1.01914
$407$	2493.70	0.303706
$408$	0	0
$409$	14655.6	1.77182	0.885908	−	0.463861i	$-0.153536\pi$
$409$	14655.6	1.77182	0.885908	+	0.463861i	$0.153536\pi$
$410$	1198.42	0.144356
$411$	0	0
$412$	−24745.4	−2.95902
$413$	−1984.26	−0.236414
$414$	0	0
$415$	−5237.82	−0.619552
$416$	21497.8	2.53369
$417$	0	0
$418$	1106.50	0.129476
$419$	−11602.7	−1.35281	−0.676407	−	0.736528i	$-0.736464\pi$
$419$	−11602.7	−1.35281	−0.676407	+	0.736528i	$0.736464\pi$
$420$	0	0
$421$	12579.8	1.45630	0.728149	−	0.685419i	$-0.240381\pi$
$421$	12579.8	1.45630	0.728149	+	0.685419i	$0.240381\pi$
$422$	−17972.9	−2.07324
$423$	0	0
$424$	−22213.7	−2.54433
$425$	2287.78	0.261114
$426$	0	0
$427$	−1518.80	−0.172131
$428$	−34771.4	−3.92696
$429$	0	0
$430$	6781.01	0.760486
$431$	10004.7	1.11812	0.559060	−	0.829127i	$-0.311162\pi$
$431$	10004.7	1.11812	0.559060	+	0.829127i	$0.311162\pi$
$432$	0	0
$433$	−17934.2	−1.99045	−0.995223	−	0.0976238i	$-0.968876\pi$
$433$	−17934.2	−1.99045	−0.995223	+	0.0976238i	$0.968876\pi$
$434$	−8845.55	−0.978341
$435$	0	0
$436$	36606.5	4.02094
$437$	3798.91	0.415850
$438$	0	0
$439$	−16225.5	−1.76402	−0.882008	−	0.471234i	$-0.843809\pi$
$439$	−16225.5	−1.76402	−0.882008	+	0.471234i	$0.843809\pi$
$440$	−5170.97	−0.560264
$441$	0	0
$442$	−7993.47	−0.860205
$443$	−10194.9	−1.09340	−0.546699	−	0.837329i	$-0.684116\pi$
$443$	−10194.9	−1.09340	−0.546699	+	0.837329i	$0.684116\pi$
$444$	0	0
$445$	−3218.50	−0.342857
$446$	−13658.3	−1.45008
$447$	0	0
$448$	6550.45	0.690803
$449$	−395.095	−0.0415272	−0.0207636	−	0.999784i	$-0.506610\pi$
$449$	−395.095	−0.0415272	−0.0207636	+	0.999784i	$0.506610\pi$
$450$	0	0
$451$	−342.277	−0.0357366
$452$	−15765.2	−1.64056
$453$	0	0
$454$	4533.35	0.468635
$455$	−2418.24	−0.249163
$456$	0	0
$457$	18408.4	1.88426	0.942130	−	0.335247i	$-0.108820\pi$
$457$	18408.4	1.88426	0.942130	+	0.335247i	$0.108820\pi$
$458$	23634.8	2.41131
$459$	0	0
$460$	−29389.5	−2.97890
$461$	16511.8	1.66818	0.834089	−	0.551630i	$-0.185994\pi$
$461$	16511.8	1.66818	0.834089	+	0.551630i	$0.185994\pi$
$462$	0	0
$463$	−15464.7	−1.55228	−0.776141	−	0.630559i	$-0.782826\pi$
$463$	−15464.7	−1.55228	−0.776141	+	0.630559i	$0.782826\pi$
$464$	40954.5	4.09755
$465$	0	0
$466$	−9293.64	−0.923862
$467$	−5014.38	−0.496869	−0.248435	−	0.968649i	$-0.579916\pi$
$467$	−5014.38	−0.496869	−0.248435	+	0.968649i	$0.579916\pi$
$468$	0	0
$469$	−1260.24	−0.124078
$470$	−20290.4	−1.99134
$471$	0	0
$472$	−18374.7	−1.79188
$473$	−1936.70	−0.188265
$474$	0	0
$475$	−1371.46	−0.132478
$476$	−4468.81	−0.430310
$477$	0	0
$478$	12505.4	1.19662
$479$	−10525.9	−1.00405	−0.502026	−	0.864853i	$-0.667412\pi$
$479$	−10525.9	−1.00405	−0.502026	+	0.864853i	$0.667412\pi$
$480$	0	0
$481$	10799.3	1.02371
$482$	−29821.7	−2.81813
$483$	0	0
$484$	2444.86	0.229608
$485$	13086.3	1.22519
$486$	0	0
$487$	15778.5	1.46815	0.734077	−	0.679066i	$-0.237615\pi$
$487$	15778.5	1.46815	0.734077	+	0.679066i	$0.237615\pi$
$488$	−14064.5	−1.30465
$489$	0	0
$490$	−1887.21	−0.173991
$491$	13536.1	1.24415	0.622073	−	0.782959i	$-0.286291\pi$
$491$	13536.1	1.24415	0.622073	+	0.782959i	$0.286291\pi$
$492$	0	0
$493$	−7085.73	−0.647313
$494$	4791.86	0.436429
$495$	0	0
$496$	−43451.2	−3.93350
$497$	−1167.14	−0.105338
$498$	0	0
$499$	18101.7	1.62393	0.811967	−	0.583704i	$-0.198397\pi$
$499$	18101.7	1.62393	0.811967	+	0.583704i	$0.198397\pi$
$500$	28926.3	2.58725
$501$	0	0
$502$	−33285.3	−2.95936
$503$	−5147.40	−0.456285	−0.228142	−	0.973628i	$-0.573265\pi$
$503$	−5147.40	−0.456285	−0.228142	+	0.973628i	$0.573265\pi$
$504$	0	0
$505$	12820.0	1.12967
$506$	11717.2	1.02944
$507$	0	0
$508$	−21943.7	−1.91653
$509$	−7185.23	−0.625697	−0.312848	−	0.949803i	$-0.601283\pi$
$509$	−7185.23	−0.625697	−0.312848	+	0.949803i	$0.601283\pi$
$510$	0	0
$511$	314.222	0.0272023
$512$	−12288.4	−1.06069
$513$	0	0
$514$	18787.3	1.61220
$515$	8881.43	0.759928
$516$	0	0
$517$	5795.08	0.492974
$518$	8427.85	0.714862
$519$	0	0
$520$	−22393.6	−1.88850
$521$	6620.31	0.556700	0.278350	−	0.960480i	$-0.410212\pi$
$521$	6620.31	0.556700	0.278350	+	0.960480i	$0.410212\pi$
$522$	0	0
$523$	8525.60	0.712808	0.356404	−	0.934332i	$-0.384003\pi$
$523$	8525.60	0.712808	0.356404	+	0.934332i	$0.384003\pi$
$524$	−32860.1	−2.73950
$525$	0	0
$526$	14385.9	1.19250
$527$	7517.70	0.621397
$528$	0	0
$529$	28061.2	2.30634
$530$	13198.5	1.08171
$531$	0	0
$532$	2678.93	0.218320
$533$	−1482.28	−0.120459
$534$	0	0
$535$	12479.9	1.00851
$536$	−11670.2	−0.940439
$537$	0	0
$538$	28905.1	2.31634
$539$	539.000	0.0430730
$540$	0	0
$541$	−6872.24	−0.546138	−0.273069	−	0.961994i	$-0.588039\pi$
$541$	−6872.24	−0.546138	−0.273069	+	0.961994i	$0.588039\pi$
$542$	5954.94	0.471931
$543$	0	0
$544$	−14258.5	−1.12377
$545$	−13138.5	−1.03265
$546$	0	0
$547$	−12050.5	−0.941940	−0.470970	−	0.882149i	$-0.656096\pi$
$547$	−12050.5	−0.941940	−0.470970	+	0.882149i	$0.656096\pi$
$548$	49482.0	3.85724
$549$	0	0
$550$	−4230.08	−0.327948
$551$	4247.70	0.328417
$552$	0	0
$553$	2445.45	0.188049
$554$	−24268.0	−1.86110
$555$	0	0
$556$	−53497.2	−4.08055
$557$	−24457.8	−1.86052	−0.930261	−	0.366899i	$-0.880420\pi$
$557$	−24457.8	−1.86052	−0.930261	+	0.366899i	$0.880420\pi$
$558$	0	0
$559$	−8387.13	−0.634594
$560$	−9270.38	−0.699545
$561$	0	0
$562$	1929.99	0.144861
$563$	14710.5	1.10120	0.550599	−	0.834770i	$-0.314399\pi$
$563$	14710.5	1.10120	0.550599	+	0.834770i	$0.314399\pi$
$564$	0	0
$565$	5658.34	0.421324
$566$	37101.2	2.75526
$567$	0	0
$568$	−10808.0	−0.798403
$569$	−24838.0	−1.82999	−0.914993	−	0.403470i	$-0.867804\pi$
$569$	−24838.0	−1.82999	−0.914993	+	0.403470i	$0.867804\pi$
$570$	0	0
$571$	5512.76	0.404031	0.202016	−	0.979382i	$-0.435251\pi$
$571$	5512.76	0.404031	0.202016	+	0.979382i	$0.435251\pi$
$572$	10587.8	0.773947
$573$	0	0
$574$	−1156.78	−0.0841167
$575$	−14522.9	−1.05330
$576$	0	0
$577$	8281.91	0.597539	0.298770	−	0.954325i	$-0.403424\pi$
$577$	8281.91	0.597539	0.298770	+	0.954325i	$0.403424\pi$
$578$	−20790.6	−1.49615
$579$	0	0
$580$	−32861.4	−2.35258
$581$	5055.80	0.361016
$582$	0	0
$583$	−3769.58	−0.267788
$584$	2909.77	0.206177
$585$	0	0
$586$	1078.19	0.0760063
$587$	8424.68	0.592374	0.296187	−	0.955130i	$-0.404285\pi$
$587$	8424.68	0.592374	0.296187	+	0.955130i	$0.404285\pi$
$588$	0	0
$589$	−4506.65	−0.315269
$590$	10917.5	0.761809
$591$	0	0
$592$	41399.4	2.87416
$593$	2199.00	0.152280	0.0761402	−	0.997097i	$-0.475740\pi$
$593$	2199.00	0.152280	0.0761402	+	0.997097i	$0.475740\pi$
$594$	0	0
$595$	1603.91	0.110511
$596$	14992.2	1.03037
$597$	0	0
$598$	50743.0	3.46996
$599$	−8765.83	−0.597934	−0.298967	−	0.954264i	$-0.596642\pi$
$599$	−8765.83	−0.597934	−0.298967	+	0.954264i	$0.596642\pi$
$600$	0	0
$601$	−15362.1	−1.04265	−0.521326	−	0.853357i	$-0.674563\pi$
$601$	−15362.1	−1.04265	−0.521326	+	0.853357i	$0.674563\pi$
$602$	−6545.37	−0.443139
$603$	0	0
$604$	10439.2	0.703255
$605$	−877.492	−0.0589672
$606$	0	0
$607$	1674.58	0.111976	0.0559878	−	0.998431i	$-0.482169\pi$
$607$	1674.58	0.111976	0.0559878	+	0.998431i	$0.482169\pi$
$608$	8547.59	0.570149
$609$	0	0
$610$	8356.55	0.554667
$611$	25096.4	1.66169
$612$	0	0
$613$	−10407.8	−0.685756	−0.342878	−	0.939380i	$-0.611402\pi$
$613$	−10407.8	−0.685756	−0.342878	+	0.939380i	$0.611402\pi$
$614$	−40734.2	−2.67736
$615$	0	0
$616$	4991.28	0.326468
$617$	−8306.01	−0.541957	−0.270978	−	0.962585i	$-0.587347\pi$
$617$	−8306.01	−0.541957	−0.270978	+	0.962585i	$0.587347\pi$
$618$	0	0
$619$	−20339.2	−1.32068	−0.660342	−	0.750965i	$-0.729588\pi$
$619$	−20339.2	−1.32068	−0.660342	+	0.750965i	$0.729588\pi$
$620$	34864.8	2.25839
$621$	0	0
$622$	22481.5	1.44924
$623$	3106.66	0.199784
$624$	0	0
$625$	−1330.96	−0.0851813
$626$	24547.4	1.56727
$627$	0	0
$628$	−64453.2	−4.09548
$629$	−7162.71	−0.454047
$630$	0	0
$631$	3457.35	0.218122	0.109061	−	0.994035i	$-0.465216\pi$
$631$	3457.35	0.218122	0.109061	+	0.994035i	$0.465216\pi$
$632$	22645.5	1.42530
$633$	0	0
$634$	−5303.16	−0.332201
$635$	7875.89	0.492197
$636$	0	0
$637$	2334.21	0.145188
$638$	13101.4	0.812995
$639$	0	0
$640$	−9859.36	−0.608946
$641$	12098.8	0.745516	0.372758	−	0.927929i	$-0.378412\pi$
$641$	12098.8	0.745516	0.372758	+	0.927929i	$0.378412\pi$
$642$	0	0
$643$	5887.31	0.361077	0.180539	−	0.983568i	$-0.442216\pi$
$643$	5887.31	0.361077	0.180539	+	0.983568i	$0.442216\pi$
$644$	28368.2	1.73582
$645$	0	0
$646$	−3178.23	−0.193569
$647$	21800.3	1.32467	0.662333	−	0.749210i	$-0.269566\pi$
$647$	21800.3	1.32467	0.662333	+	0.749210i	$0.269566\pi$
$648$	0	0
$649$	−3118.12	−0.188593
$650$	−18318.9	−1.10543
$651$	0	0
$652$	−27293.7	−1.63942
$653$	28636.2	1.71611	0.858055	−	0.513559i	$-0.171673\pi$
$653$	28636.2	1.71611	0.858055	+	0.513559i	$0.171673\pi$
$654$	0	0
$655$	11793.9	0.703551
$656$	−5682.34	−0.338198
$657$	0	0
$658$	19585.4	1.16036
$659$	23309.7	1.37787	0.688934	−	0.724824i	$-0.258079\pi$
$659$	23309.7	1.37787	0.688934	+	0.724824i	$0.258079\pi$
$660$	0	0
$661$	19540.2	1.14981	0.574906	−	0.818220i	$-0.305039\pi$
$661$	19540.2	1.14981	0.574906	+	0.818220i	$0.305039\pi$
$662$	29805.9	1.74991
$663$	0	0
$664$	46818.0	2.73628
$665$	−961.501	−0.0560683
$666$	0	0
$667$	44980.6	2.61118
$668$	31078.7	1.80011
$669$	0	0
$670$	6933.96	0.399824
$671$	−2386.68	−0.137313
$672$	0	0
$673$	−17079.1	−0.978236	−0.489118	−	0.872218i	$-0.662681\pi$
$673$	−17079.1	−0.978236	−0.489118	+	0.872218i	$0.662681\pi$
$674$	−24724.8	−1.41300
$675$	0	0
$676$	1460.43	0.0830921
$677$	22077.4	1.25333	0.626663	−	0.779290i	$-0.284420\pi$
$677$	22077.4	1.25333	0.626663	+	0.779290i	$0.284420\pi$
$678$	0	0
$679$	−12631.5	−0.713923
$680$	14852.7	0.837608
$681$	0	0
$682$	−13900.1	−0.780446
$683$	−2423.70	−0.135784	−0.0678919	−	0.997693i	$-0.521627\pi$
$683$	−2423.70	−0.135784	−0.0678919	+	0.997693i	$0.521627\pi$
$684$	0	0
$685$	−17759.7	−0.990605
$686$	1821.63	0.101385
$687$	0	0
$688$	−32152.2	−1.78168
$689$	−16324.7	−0.902642
$690$	0	0
$691$	34446.1	1.89637	0.948185	−	0.317720i	$-0.102917\pi$
$691$	34446.1	1.89637	0.948185	+	0.317720i	$0.102917\pi$
$692$	−5185.69	−0.284870
$693$	0	0
$694$	−27441.1	−1.50094
$695$	19200.8	1.04796
$696$	0	0
$697$	983.129	0.0534271
$698$	48686.5	2.64013
$699$	0	0
$700$	−10241.3	−0.552980
$701$	−12732.3	−0.686012	−0.343006	−	0.939333i	$-0.611445\pi$
$701$	−12732.3	−0.686012	−0.343006	+	0.939333i	$0.611445\pi$
$702$	0	0
$703$	4293.84	0.230363
$704$	10293.6	0.551070
$705$	0	0
$706$	−154.956	−0.00826040
$707$	−12374.5	−0.658264
$708$	0	0
$709$	−2619.52	−0.138756	−0.0693782	−	0.997590i	$-0.522102\pi$
$709$	−2619.52	−0.138756	−0.0693782	+	0.997590i	$0.522102\pi$
$710$	6421.67	0.339438
$711$	0	0
$712$	28768.4	1.51425
$713$	−47722.8	−2.50664
$714$	0	0
$715$	−3800.10	−0.198763
$716$	27523.2	1.43658
$717$	0	0
$718$	8863.34	0.460692
$719$	−20757.7	−1.07668	−0.538338	−	0.842729i	$-0.680948\pi$
$719$	−20757.7	−1.07668	−0.538338	+	0.842729i	$0.680948\pi$
$720$	0	0
$721$	−8572.81	−0.442813
$722$	−34522.1	−1.77947
$723$	0	0
$724$	54546.1	2.79999
$725$	−16238.6	−0.831844
$726$	0	0
$727$	11099.5	0.566241	0.283120	−	0.959084i	$-0.408630\pi$
$727$	11099.5	0.566241	0.283120	+	0.959084i	$0.408630\pi$
$728$	21615.4	1.10044
$729$	0	0
$730$	−1728.87	−0.0876554
$731$	5562.81	0.281461
$732$	0	0
$733$	28379.9	1.43006	0.715030	−	0.699094i	$-0.246413\pi$
$733$	28379.9	1.43006	0.715030	+	0.699094i	$0.246413\pi$
$734$	50342.4	2.53157
$735$	0	0
$736$	90513.9	4.53314
$737$	−1980.38	−0.0989802
$738$	0	0
$739$	2611.45	0.129991	0.0649957	−	0.997886i	$-0.479297\pi$
$739$	2611.45	0.129991	0.0649957	+	0.997886i	$0.479297\pi$
$740$	−33218.4	−1.65018
$741$	0	0
$742$	−12739.9	−0.630318
$743$	35458.7	1.75081	0.875407	−	0.483386i	$-0.160593\pi$
$743$	35458.7	1.75081	0.875407	+	0.483386i	$0.160593\pi$
$744$	0	0
$745$	−5380.88	−0.264618
$746$	−8638.29	−0.423955
$747$	0	0
$748$	−7022.42	−0.343269
$749$	−12046.2	−0.587663
$750$	0	0
$751$	15082.5	0.732845	0.366422	−	0.930449i	$-0.380582\pi$
$751$	15082.5	0.732845	0.366422	+	0.930449i	$0.380582\pi$
$752$	96207.4	4.66532
$753$	0	0
$754$	56737.5	2.74040
$755$	−3746.77	−0.180608
$756$	0	0
$757$	11843.4	0.568633	0.284316	−	0.958731i	$-0.408233\pi$
$757$	11843.4	0.568633	0.284316	+	0.958731i	$0.408233\pi$
$758$	−28558.0	−1.36843
$759$	0	0
$760$	−8903.75	−0.424964
$761$	−19043.4	−0.907128	−0.453564	−	0.891224i	$-0.649848\pi$
$761$	−19043.4	−0.907128	−0.453564	+	0.891224i	$0.649848\pi$
$762$	0	0
$763$	12682.0	0.601728
$764$	1887.78	0.0893945
$765$	0	0
$766$	69635.8	3.28465
$767$	−13503.4	−0.635698
$768$	0	0
$769$	19957.0	0.935850	0.467925	−	0.883768i	$-0.345002\pi$
$769$	19957.0	0.935850	0.467925	+	0.883768i	$0.345002\pi$
$770$	−2965.62	−0.138797
$771$	0	0
$772$	−25451.9	−1.18657
$773$	2643.80	0.123015	0.0615076	−	0.998107i	$-0.480409\pi$
$773$	2643.80	0.123015	0.0615076	+	0.998107i	$0.480409\pi$
$774$	0	0
$775$	17228.6	0.798540
$776$	−116971.	−5.41111
$777$	0	0
$778$	4754.69	0.219105
$779$	−589.358	−0.0271065
$780$	0	0
$781$	−1834.07	−0.0840310
$782$	−33655.6	−1.53903
$783$	0	0
$784$	8948.24	0.407628
$785$	23133.1	1.05179
$786$	0	0
$787$	38850.2	1.75967	0.879835	−	0.475279i	$-0.157653\pi$
$787$	38850.2	1.75967	0.879835	+	0.475279i	$0.157653\pi$
$788$	−22424.1	−1.01374
$789$	0	0
$790$	−13455.0	−0.605960
$791$	−5461.72	−0.245507
$792$	0	0
$793$	−10335.9	−0.462846
$794$	−30219.0	−1.35067
$795$	0	0
$796$	37220.7	1.65735
$797$	18821.9	0.836520	0.418260	−	0.908327i	$-0.362640\pi$
$797$	18821.9	0.836520	0.418260	+	0.908327i	$0.362640\pi$
$798$	0	0
$799$	−16645.3	−0.737007
$800$	−32676.8	−1.44412
$801$	0	0
$802$	12526.2	0.551518
$803$	493.777	0.0216999
$804$	0	0
$805$	−10181.7	−0.445787
$806$	−60196.4	−2.63068
$807$	0	0
$808$	−114591.	−4.98925
$809$	−38807.6	−1.68653	−0.843265	−	0.537498i	$-0.819370\pi$
$809$	−38807.6	−1.68653	−0.843265	+	0.537498i	$0.819370\pi$
$810$	0	0
$811$	17348.5	0.751157	0.375579	−	0.926791i	$-0.377444\pi$
$811$	17348.5	0.751157	0.375579	+	0.926791i	$0.377444\pi$
$812$	31719.5	1.37086
$813$	0	0
$814$	13243.8	0.570263
$815$	9796.06	0.421032
$816$	0	0
$817$	−3334.75	−0.142801
$818$	77834.2	3.32690
$819$	0	0
$820$	4559.45	0.194174
$821$	−44722.9	−1.90115	−0.950573	−	0.310502i	$-0.899503\pi$
$821$	−44722.9	−1.90115	−0.950573	+	0.310502i	$0.899503\pi$
$822$	0	0
$823$	−8808.05	−0.373061	−0.186531	−	0.982449i	$-0.559724\pi$
$823$	−8808.05	−0.373061	−0.186531	+	0.982449i	$0.559724\pi$
$824$	−79386.4	−3.35626
$825$	0	0
$826$	−10538.2	−0.443909
$827$	−23454.1	−0.986191	−0.493096	−	0.869975i	$-0.664135\pi$
$827$	−23454.1	−0.986191	−0.493096	+	0.869975i	$0.664135\pi$
$828$	0	0
$829$	−7484.57	−0.313570	−0.156785	−	0.987633i	$-0.550113\pi$
$829$	−7484.57	−0.313570	−0.156785	+	0.987633i	$0.550113\pi$
$830$	−27817.4	−1.16332
$831$	0	0
$832$	44577.6	1.85751
$833$	−1548.18	−0.0643952
$834$	0	0
$835$	−11154.6	−0.462298
$836$	4209.74	0.174159
$837$	0	0
$838$	−61620.6	−2.54015
$839$	39660.2	1.63197	0.815985	−	0.578073i	$-0.196195\pi$
$839$	39660.2	1.63197	0.815985	+	0.578073i	$0.196195\pi$
$840$	0	0
$841$	25905.4	1.06217
$842$	66809.8	2.73446
$843$	0	0
$844$	−68378.7	−2.78873
$845$	−524.166	−0.0213395
$846$	0	0
$847$	847.000	0.0343604
$848$	−62581.0	−2.53424
$849$	0	0
$850$	12150.1	0.490289
$851$	45469.2	1.83157
$852$	0	0
$853$	−1477.78	−0.0593180	−0.0296590	−	0.999560i	$-0.509442\pi$
$853$	−1477.78	−0.0593180	−0.0296590	+	0.999560i	$0.509442\pi$
$854$	−8066.16	−0.323207
$855$	0	0
$856$	−111551.	−4.45414
$857$	−371.368	−0.0148024	−0.00740122	−	0.999973i	$-0.502356\pi$
$857$	−371.368	−0.0148024	−0.00740122	+	0.999973i	$0.502356\pi$
$858$	0	0
$859$	−33342.7	−1.32437	−0.662187	−	0.749339i	$-0.730372\pi$
$859$	−33342.7	−1.32437	−0.662187	+	0.749339i	$0.730372\pi$
$860$	25798.6	1.02294
$861$	0	0
$862$	53133.8	2.09947
$863$	−23208.8	−0.915452	−0.457726	−	0.889093i	$-0.651336\pi$
$863$	−23208.8	−0.915452	−0.457726	+	0.889093i	$0.651336\pi$
$864$	0	0
$865$	1861.21	0.0731596
$866$	−95246.5	−3.73742
$867$	0	0
$868$	−33653.2	−1.31597
$869$	3842.85	0.150011
$870$	0	0
$871$	−8576.31	−0.333636
$872$	117438.	4.56074
$873$	0	0
$874$	20175.6	0.780834
$875$	10021.2	0.387177
$876$	0	0
$877$	−20778.9	−0.800059	−0.400030	−	0.916502i	$-0.631000\pi$
$877$	−20778.9	−0.800059	−0.400030	+	0.916502i	$0.631000\pi$
$878$	−86172.0	−3.31226
$879$	0	0
$880$	−14567.7	−0.558044
$881$	29688.6	1.13534	0.567669	−	0.823257i	$-0.307845\pi$
$881$	29688.6	1.13534	0.567669	+	0.823257i	$0.307845\pi$
$882$	0	0
$883$	36836.6	1.40391	0.701953	−	0.712223i	$-0.252312\pi$
$883$	36836.6	1.40391	0.701953	+	0.712223i	$0.252312\pi$
$884$	−30411.5	−1.15707
$885$	0	0
$886$	−54144.0	−2.05305
$887$	37539.3	1.42102	0.710511	−	0.703686i	$-0.248464\pi$
$887$	37539.3	1.42102	0.710511	+	0.703686i	$0.248464\pi$
$888$	0	0
$889$	−7602.21	−0.286805
$890$	−17093.1	−0.643776
$891$	0	0
$892$	−51963.4	−1.95052
$893$	9978.40	0.373924
$894$	0	0
$895$	−9878.45	−0.368939
$896$	9516.75	0.354835
$897$	0	0
$898$	−2098.30	−0.0779747
$899$	−53360.5	−1.97961
$900$	0	0
$901$	10827.4	0.400349
$902$	−1817.80	−0.0671020
$903$	0	0
$904$	−50576.9	−1.86080
$905$	−19577.3	−0.719084
$906$	0	0
$907$	−31373.5	−1.14856	−0.574278	−	0.818660i	$-0.694717\pi$
$907$	−31373.5	−1.14856	−0.574278	+	0.818660i	$0.694717\pi$
$908$	17247.3	0.630365
$909$	0	0
$910$	−12843.0	−0.467848
$911$	−6292.83	−0.228859	−0.114430	−	0.993431i	$-0.536504\pi$
$911$	−6292.83	−0.228859	−0.114430	+	0.993431i	$0.536504\pi$
$912$	0	0
$913$	7944.84	0.287991
$914$	97764.7	3.53804
$915$	0	0
$916$	89919.5	3.24348
$917$	−11384.1	−0.409962
$918$	0	0
$919$	−36983.0	−1.32748	−0.663742	−	0.747961i	$-0.731033\pi$
$919$	−36983.0	−1.32748	−0.663742	+	0.747961i	$0.731033\pi$
$920$	−94285.4	−3.37880
$921$	0	0
$922$	87692.1	3.13230
$923$	−7942.68	−0.283247
$924$	0	0
$925$	−16415.0	−0.583484
$926$	−82131.4	−2.91469
$927$	0	0
$928$	101207.	3.58004
$929$	−20223.9	−0.714234	−0.357117	−	0.934060i	$-0.616240\pi$
$929$	−20223.9	−0.714234	−0.357117	+	0.934060i	$0.616240\pi$
$930$	0	0
$931$	928.090	0.0326712
$932$	−35358.0	−1.24269
$933$	0	0
$934$	−26630.8	−0.932961
$935$	2520.44	0.0881573
$936$	0	0
$937$	−1301.80	−0.0453873	−0.0226936	−	0.999742i	$-0.507224\pi$
$937$	−1301.80	−0.0453873	−0.0226936	+	0.999742i	$0.507224\pi$
$938$	−6693.01	−0.232979
$939$	0	0
$940$	−77195.8	−2.67856
$941$	−19149.5	−0.663395	−0.331697	−	0.943386i	$-0.607621\pi$
$941$	−19149.5	−0.663395	−0.331697	+	0.943386i	$0.607621\pi$
$942$	0	0
$943$	−6240.96	−0.215518
$944$	−51765.6	−1.78477
$945$	0	0
$946$	−10285.6	−0.353502
$947$	32816.0	1.12606	0.563030	−	0.826437i	$-0.309636\pi$
$947$	32816.0	1.12606	0.563030	+	0.826437i	$0.309636\pi$
$948$	0	0
$949$	2138.37	0.0731447
$950$	−7283.66	−0.248751
$951$	0	0
$952$	−14336.5	−0.488077
$953$	46677.1	1.58659	0.793294	−	0.608838i	$-0.208364\pi$
$953$	46677.1	1.58659	0.793294	+	0.608838i	$0.208364\pi$
$954$	0	0
$955$	−677.548	−0.0229581
$956$	47577.4	1.60959
$957$	0	0
$958$	−55901.8	−1.88529
$959$	17142.6	0.577229
$960$	0	0
$961$	26822.5	0.900356
$962$	57353.9	1.92221
$963$	0	0
$964$	−113458.	−3.79069
$965$	9135.02	0.304732
$966$	0	0
$967$	−24701.2	−0.821445	−0.410723	−	0.911760i	$-0.634724\pi$
$967$	−24701.2	−0.821445	−0.410723	+	0.911760i	$0.634724\pi$
$968$	7843.44	0.260431
$969$	0	0
$970$	69499.6	2.30051
$971$	6025.41	0.199140	0.0995699	−	0.995031i	$-0.468253\pi$
$971$	6025.41	0.199140	0.0995699	+	0.995031i	$0.468253\pi$
$972$	0	0
$973$	−18533.6	−0.610648
$974$	83797.6	2.75672
$975$	0	0
$976$	−39622.7	−1.29948
$977$	−53291.7	−1.74509	−0.872545	−	0.488533i	$-0.837532\pi$
$977$	−53291.7	−1.74509	−0.872545	+	0.488533i	$0.837532\pi$
$978$	0	0
$979$	4881.89	0.159373
$980$	−7179.98	−0.234037
$981$	0	0
$982$	71888.7	2.33611
$983$	−30172.7	−0.979003	−0.489502	−	0.872002i	$-0.662821\pi$
$983$	−30172.7	−0.979003	−0.489502	+	0.872002i	$0.662821\pi$
$984$	0	0
$985$	8048.29	0.260345
$986$	−37631.5	−1.21545
$987$	0	0
$988$	18230.8	0.587045
$989$	−35313.0	−1.13538
$990$	0	0
$991$	−37208.5	−1.19270	−0.596351	−	0.802724i	$-0.703383\pi$
$991$	−37208.5	−1.19270	−0.596351	+	0.802724i	$0.703383\pi$
$992$	−107377.	−3.43671
$993$	0	0
$994$	−6198.52	−0.197792
$995$	−13359.0	−0.425637
$996$	0	0
$997$	−4163.25	−0.132248	−0.0661241	−	0.997811i	$-0.521063\pi$
$997$	−4163.25	−0.132248	−0.0661241	+	0.997811i	$0.521063\pi$
$998$	96135.9	3.04923
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.u.1.8	yes	8
3.2	odd	2		693.4.a.r.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.r.1.1	✓	8	3.2	odd	2
693.4.a.u.1.8	yes	8	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q+5.31088 q^{2} +20.2055 q^{4} -7.25200 q^{5} +7.00000 q^{7} +64.8218 q^{8} -38.5145 q^{10} +11.0000 q^{11} +47.6369 q^{13} +37.1762 q^{14} +182.617 q^{16} -31.5955 q^{17} +18.9406 q^{19} -146.530 q^{20} +58.4197 q^{22} +200.570 q^{23} -72.4085 q^{25} +252.994 q^{26} +141.438 q^{28} +224.264 q^{29} -237.936 q^{31} +451.284 q^{32} -167.800 q^{34} -50.7640 q^{35} +226.700 q^{37} +100.591 q^{38} -470.088 q^{40} -31.1161 q^{41} -176.064 q^{43} +222.260 q^{44} +1065.20 q^{46} +526.826 q^{47} +49.0000 q^{49} -384.553 q^{50} +962.527 q^{52} -342.689 q^{53} -79.7720 q^{55} +453.753 q^{56} +1191.04 q^{58} -283.465 q^{59} -216.971 q^{61} -1263.65 q^{62} +935.778 q^{64} -345.463 q^{65} -180.035 q^{67} -638.402 q^{68} -269.602 q^{70} -166.734 q^{71} +44.8888 q^{73} +1203.98 q^{74} +382.704 q^{76} +77.0000 q^{77} +349.350 q^{79} -1324.34 q^{80} -165.254 q^{82} +722.258 q^{83} +229.131 q^{85} -935.053 q^{86} +713.040 q^{88} +443.808 q^{89} +333.459 q^{91} +4052.61 q^{92} +2797.91 q^{94} -137.357 q^{95} -1804.50 q^{97} +260.233 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 30 q^{4} + 10 q^{5} + 56 q^{7} + 81 q^{8} + 9 q^{10} + 88 q^{11} + 16 q^{13} + 42 q^{14} + 122 q^{16} + 90 q^{17} - 42 q^{19} + 291 q^{20} + 66 q^{22} + 338 q^{23} + 244 q^{25} + 209 q^{26}+ \cdots + 294 q^{98}+O(q^{100}) \) 8 * q + 6 * q^2 + 30 * q^4 + 10 * q^5 + 56 * q^7 + 81 * q^8 + 9 * q^10 + 88 * q^11 + 16 * q^13 + 42 * q^14 + 122 * q^16 + 90 * q^17 - 42 * q^19 + 291 * q^20 + 66 * q^22 + 338 * q^23 + 244 * q^25 + 209 * q^26 + 210 * q^28 + 496 * q^29 - 8 * q^31 + 524 * q^32 - 302 * q^34 + 70 * q^35 - 360 * q^37 + 45 * q^38 - 6 * q^40 + 242 * q^41 - 66 * q^43 + 330 * q^44 + 344 * q^46 + 540 * q^47 + 392 * q^49 + 1171 * q^50 + 465 * q^52 + 906 * q^53 + 110 * q^55 + 567 * q^56 + 977 * q^58 + 1242 * q^59 - 318 * q^61 - 110 * q^62 + 525 * q^64 + 1258 * q^65 + 522 * q^67 + 678 * q^68 + 63 * q^70 + 858 * q^71 - 78 * q^73 + 1651 * q^74 + 1775 * q^76 + 616 * q^77 + 516 * q^79 + 567 * q^80 - 1212 * q^82 + 3192 * q^83 + 720 * q^85 + 1322 * q^86 + 891 * q^88 + 2356 * q^89 + 112 * q^91 + 4504 * q^92 - 423 * q^94 + 3308 * q^95 - 1556 * q^97 + 294 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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