LMFDB - Embedded newform 693.4.a.p.1.4

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 30x^{3} + 11x^{2} + 185x + 162 $ x^5 - x^4 - 30x^3 + 11x^2 + 185*x + 162
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.63074$ 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)

$f(q)$	$=$	$q+3.63074 q^{2} +5.18226 q^{4} -21.1113 q^{5} +7.00000 q^{7} -10.2305 q^{8} +O(q^{10})$	\(q+3.63074 q^{2} +5.18226 q^{4} -21.1113 q^{5} +7.00000 q^{7} -10.2305 q^{8} -76.6494 q^{10} -11.0000 q^{11} +87.3602 q^{13} +25.4152 q^{14} -78.6023 q^{16} +28.2691 q^{17} -97.9284 q^{19} -109.404 q^{20} -39.9381 q^{22} +112.065 q^{23} +320.685 q^{25} +317.182 q^{26} +36.2758 q^{28} -14.9213 q^{29} +138.440 q^{31} -203.540 q^{32} +102.638 q^{34} -147.779 q^{35} +206.944 q^{37} -355.552 q^{38} +215.978 q^{40} -321.063 q^{41} +285.198 q^{43} -57.0048 q^{44} +406.877 q^{46} +303.300 q^{47} +49.0000 q^{49} +1164.32 q^{50} +452.723 q^{52} +554.639 q^{53} +232.224 q^{55} -71.6134 q^{56} -54.1752 q^{58} +693.110 q^{59} +156.761 q^{61} +502.638 q^{62} -110.184 q^{64} -1844.28 q^{65} +584.667 q^{67} +146.498 q^{68} -536.546 q^{70} -363.745 q^{71} -747.424 q^{73} +751.360 q^{74} -507.490 q^{76} -77.0000 q^{77} +419.344 q^{79} +1659.39 q^{80} -1165.70 q^{82} -1178.20 q^{83} -596.796 q^{85} +1035.48 q^{86} +112.535 q^{88} +397.908 q^{89} +611.521 q^{91} +580.748 q^{92} +1101.20 q^{94} +2067.39 q^{95} -1333.21 q^{97} +177.906 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8}+O(q^{10}) $ 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8	$ 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8} - 23 q^{10} - 55 q^{11} + 101 q^{13} + 7 q^{14} - 7 q^{16} + 20 q^{17} + 237 q^{19} - 85 q^{20} - 11 q^{22} + 80 q^{23} + 486 q^{25} - 165 q^{26} + 147 q^{28} + 11 q^{29} + 316 q^{31} - 453 q^{32} + 936 q^{34} - 147 q^{35} + 319 q^{37} - 89 q^{38} + 624 q^{40} - 1190 q^{41} + 88 q^{43} - 231 q^{44} + 1000 q^{46} - 377 q^{47} + 245 q^{49} + 644 q^{50} + 1001 q^{52} + 992 q^{53} + 231 q^{55} + 294 q^{56} + 721 q^{58} - 71 q^{59} - 574 q^{61} - 272 q^{62} - 1380 q^{64} - 589 q^{65} - 527 q^{67} + 2974 q^{68} - 161 q^{70} + 1156 q^{71} + 1061 q^{73} + 1609 q^{74} + 2399 q^{76} - 385 q^{77} + 588 q^{79} + 1643 q^{80} - 2602 q^{82} + 212 q^{83} + 1918 q^{85} + 4760 q^{86} - 462 q^{88} - 1030 q^{89} + 707 q^{91} + 1174 q^{92} - 1799 q^{94} + 3593 q^{95} + 2488 q^{97} + 49 q^{98}+O(q^{100}) $ 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8 - 23 * q^10 - 55 * q^11 + 101 * q^13 + 7 * q^14 - 7 * q^16 + 20 * q^17 + 237 * q^19 - 85 * q^20 - 11 * q^22 + 80 * q^23 + 486 * q^25 - 165 * q^26 + 147 * q^28 + 11 * q^29 + 316 * q^31 - 453 * q^32 + 936 * q^34 - 147 * q^35 + 319 * q^37 - 89 * q^38 + 624 * q^40 - 1190 * q^41 + 88 * q^43 - 231 * q^44 + 1000 * q^46 - 377 * q^47 + 245 * q^49 + 644 * q^50 + 1001 * q^52 + 992 * q^53 + 231 * q^55 + 294 * q^56 + 721 * q^58 - 71 * q^59 - 574 * q^61 - 272 * q^62 - 1380 * q^64 - 589 * q^65 - 527 * q^67 + 2974 * q^68 - 161 * q^70 + 1156 * q^71 + 1061 * q^73 + 1609 * q^74 + 2399 * q^76 - 385 * q^77 + 588 * q^79 + 1643 * q^80 - 2602 * q^82 + 212 * q^83 + 1918 * q^85 + 4760 * q^86 - 462 * q^88 - 1030 * q^89 + 707 * q^91 + 1174 * q^92 - 1799 * q^94 + 3593 * q^95 + 2488 * q^97 + 49 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	3.63074	1.28366	0.641830	−	0.766847i	$-0.278176\pi$
$2$	3.63074	1.28366	0.641830	+	0.766847i	$0.278176\pi$
$3$	0	0
$3$	0	0
$4$	5.18226	0.647782
$5$	−21.1113	−1.88825	−0.944124	−	0.329590i	$-0.893089\pi$
$5$	−21.1113	−1.88825	−0.944124	+	0.329590i	$0.893089\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−10.2305	−0.452128
$9$	0	0
$10$	−76.6494	−2.42387
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	87.3602	1.86380	0.931898	−	0.362719i	$-0.118152\pi$
$13$	87.3602	1.86380	0.931898	+	0.362719i	$0.118152\pi$
$14$	25.4152	0.485178
$15$	0	0
$16$	−78.6023	−1.22816
$17$	28.2691	0.403309	0.201655	−	0.979457i	$-0.435368\pi$
$17$	28.2691	0.403309	0.201655	+	0.979457i	$0.435368\pi$
$18$	0	0
$19$	−97.9284	−1.18244	−0.591219	−	0.806511i	$-0.701353\pi$
$19$	−97.9284	−1.18244	−0.591219	+	0.806511i	$0.701353\pi$
$20$	−109.404	−1.22317
$21$	0	0
$22$	−39.9381	−0.387038
$23$	112.065	1.01596	0.507980	−	0.861369i	$-0.330392\pi$
$23$	112.065	1.01596	0.507980	+	0.861369i	$0.330392\pi$
$24$	0	0
$25$	320.685	2.56548
$26$	317.182	2.39248
$27$	0	0
$28$	36.2758	0.244839
$29$	−14.9213	−0.0955451	−0.0477725	−	0.998858i	$-0.515212\pi$
$29$	−14.9213	−0.0955451	−0.0477725	+	0.998858i	$0.515212\pi$
$30$	0	0
$31$	138.440	0.802081	0.401040	−	0.916060i	$-0.368649\pi$
$31$	138.440	0.802081	0.401040	+	0.916060i	$0.368649\pi$
$32$	−203.540	−1.12441
$33$	0	0
$34$	102.638	0.517712
$35$	−147.779	−0.713691
$36$	0	0
$37$	206.944	0.919498	0.459749	−	0.888049i	$-0.347939\pi$
$37$	206.944	0.919498	0.459749	+	0.888049i	$0.347939\pi$
$38$	−355.552	−1.51785
$39$	0	0
$40$	215.978	0.853729
$41$	−321.063	−1.22297	−0.611483	−	0.791257i	$-0.709427\pi$
$41$	−321.063	−1.22297	−0.611483	+	0.791257i	$0.709427\pi$
$42$	0	0
$43$	285.198	1.01145	0.505724	−	0.862696i	$-0.331226\pi$
$43$	285.198	1.01145	0.505724	+	0.862696i	$0.331226\pi$
$44$	−57.0048	−0.195314
$45$	0	0
$46$	406.877	1.30415
$47$	303.300	0.941293	0.470647	−	0.882322i	$-0.344021\pi$
$47$	303.300	0.941293	0.470647	+	0.882322i	$0.344021\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	1164.32	3.29320
$51$	0	0
$52$	452.723	1.20733
$53$	554.639	1.43746	0.718732	−	0.695287i	$-0.244723\pi$
$53$	554.639	1.43746	0.718732	+	0.695287i	$0.244723\pi$
$54$	0	0
$55$	232.224	0.569328
$56$	−71.6134	−0.170888
$57$	0	0
$58$	−54.1752	−0.122647
$59$	693.110	1.52941	0.764705	−	0.644380i	$-0.222885\pi$
$59$	693.110	1.52941	0.764705	+	0.644380i	$0.222885\pi$
$60$	0	0
$61$	156.761	0.329037	0.164518	−	0.986374i	$-0.447393\pi$
$61$	156.761	0.329037	0.164518	+	0.986374i	$0.447393\pi$
$62$	502.638	1.02960
$63$	0	0
$64$	−110.184	−0.215203
$65$	−1844.28	−3.51931
$66$	0	0
$67$	584.667	1.06610	0.533048	−	0.846085i	$-0.321047\pi$
$67$	584.667	1.06610	0.533048	+	0.846085i	$0.321047\pi$
$68$	146.498	0.261257
$69$	0	0
$70$	−536.546	−0.916136
$71$	−363.745	−0.608008	−0.304004	−	0.952671i	$-0.598324\pi$
$71$	−363.745	−0.608008	−0.304004	+	0.952671i	$0.598324\pi$
$72$	0	0
$73$	−747.424	−1.19835	−0.599173	−	0.800619i	$-0.704504\pi$
$73$	−747.424	−1.19835	−0.599173	+	0.800619i	$0.704504\pi$
$74$	751.360	1.18032
$75$	0	0
$76$	−507.490	−0.765962
$77$	−77.0000	−0.113961
$78$	0	0
$79$	419.344	0.597213	0.298607	−	0.954376i	$-0.403478\pi$
$79$	419.344	0.597213	0.298607	+	0.954376i	$0.403478\pi$
$80$	1659.39	2.31907
$81$	0	0
$82$	−1165.70	−1.56987
$83$	−1178.20	−1.55812	−0.779061	−	0.626948i	$-0.784304\pi$
$83$	−1178.20	−1.55812	−0.779061	+	0.626948i	$0.784304\pi$
$84$	0	0
$85$	−596.796	−0.761548
$86$	1035.48	1.29835
$87$	0	0
$88$	112.535	0.136322
$89$	397.908	0.473911	0.236956	−	0.971520i	$-0.423850\pi$
$89$	397.908	0.473911	0.236956	+	0.971520i	$0.423850\pi$
$90$	0	0
$91$	611.521	0.704449
$92$	580.748	0.658121
$93$	0	0
$94$	1101.20	1.20830
$95$	2067.39	2.23274
$96$	0	0
$97$	−1333.21	−1.39553	−0.697766	−	0.716325i	$-0.745823\pi$
$97$	−1333.21	−1.39553	−0.697766	+	0.716325i	$0.745823\pi$
$98$	177.906	0.183380
$99$	0	0
$100$	1661.87	1.66187
$101$	−747.399	−0.736326	−0.368163	−	0.929761i	$-0.620013\pi$
$101$	−747.399	−0.736326	−0.368163	+	0.929761i	$0.620013\pi$
$102$	0	0
$103$	548.716	0.524918	0.262459	−	0.964943i	$-0.415467\pi$
$103$	548.716	0.524918	0.262459	+	0.964943i	$0.415467\pi$
$104$	−893.737	−0.842674
$105$	0	0
$106$	2013.75	1.84521
$107$	−273.875	−0.247444	−0.123722	−	0.992317i	$-0.539483\pi$
$107$	−273.875	−0.247444	−0.123722	+	0.992317i	$0.539483\pi$
$108$	0	0
$109$	2026.77	1.78100	0.890500	−	0.454983i	$-0.150355\pi$
$109$	2026.77	1.78100	0.890500	+	0.454983i	$0.150355\pi$
$110$	843.144	0.730824
$111$	0	0
$112$	−550.216	−0.464201
$113$	1048.04	0.872490	0.436245	−	0.899828i	$-0.356308\pi$
$113$	1048.04	0.872490	0.436245	+	0.899828i	$0.356308\pi$
$114$	0	0
$115$	−2365.83	−1.91839
$116$	−77.3258	−0.0618924
$117$	0	0
$118$	2516.50	1.96324
$119$	197.884	0.152437
$120$	0	0
$121$	121.000	0.0909091
$122$	569.160	0.422371
$123$	0	0
$124$	717.431	0.519574
$125$	−4131.16	−2.95602
$126$	0	0
$127$	−905.819	−0.632901	−0.316451	−	0.948609i	$-0.602491\pi$
$127$	−905.819	−0.632901	−0.316451	+	0.948609i	$0.602491\pi$
$128$	1228.27	0.848166
$129$	0	0
$130$	−6696.11	−4.51760
$131$	−1887.45	−1.25883	−0.629416	−	0.777069i	$-0.716706\pi$
$131$	−1887.45	−1.25883	−0.629416	+	0.777069i	$0.716706\pi$
$132$	0	0
$133$	−685.499	−0.446919
$134$	2122.77	1.36850
$135$	0	0
$136$	−289.206	−0.182347
$137$	−690.753	−0.430767	−0.215383	−	0.976530i	$-0.569100\pi$
$137$	−690.753	−0.430767	−0.215383	+	0.976530i	$0.569100\pi$
$138$	0	0
$139$	−1466.05	−0.894592	−0.447296	−	0.894386i	$-0.647613\pi$
$139$	−1466.05	−0.894592	−0.447296	+	0.894386i	$0.647613\pi$
$140$	−765.828	−0.462316
$141$	0	0
$142$	−1320.66	−0.780476
$143$	−960.962	−0.561956
$144$	0	0
$145$	315.006	0.180413
$146$	−2713.70	−1.53827
$147$	0	0
$148$	1072.44	0.595634
$149$	−1367.88	−0.752088	−0.376044	−	0.926602i	$-0.622716\pi$
$149$	−1367.88	−0.752088	−0.376044	+	0.926602i	$0.622716\pi$
$150$	0	0
$151$	93.9161	0.0506145	0.0253072	−	0.999680i	$-0.491944\pi$
$151$	93.9161	0.0506145	0.0253072	+	0.999680i	$0.491944\pi$
$152$	1001.85	0.534613
$153$	0	0
$154$	−279.567	−0.146287
$155$	−2922.64	−1.51453
$156$	0	0
$157$	−72.3063	−0.0367558	−0.0183779	−	0.999831i	$-0.505850\pi$
$157$	−72.3063	−0.0367558	−0.0183779	+	0.999831i	$0.505850\pi$
$158$	1522.53	0.766619
$159$	0	0
$160$	4296.99	2.12317
$161$	784.453	0.383997
$162$	0	0
$163$	34.3598	0.0165108	0.00825542	−	0.999966i	$-0.497372\pi$
$163$	34.3598	0.0165108	0.00825542	+	0.999966i	$0.497372\pi$
$164$	−1663.83	−0.792216
$165$	0	0
$166$	−4277.73	−2.00010
$167$	−379.079	−0.175653	−0.0878263	−	0.996136i	$-0.527992\pi$
$167$	−379.079	−0.175653	−0.0878263	+	0.996136i	$0.527992\pi$
$168$	0	0
$169$	5434.80	2.47374
$170$	−2166.81	−0.977569
$171$	0	0
$172$	1477.97	0.655198
$173$	1398.29	0.614507	0.307253	−	0.951628i	$-0.400590\pi$
$173$	1398.29	0.614507	0.307253	+	0.951628i	$0.400590\pi$
$174$	0	0
$175$	2244.80	0.969661
$176$	864.625	0.370304
$177$	0	0
$178$	1444.70	0.608341
$179$	2591.03	1.08191	0.540956	−	0.841051i	$-0.318062\pi$
$179$	2591.03	1.08191	0.540956	+	0.841051i	$0.318062\pi$
$180$	0	0
$181$	3408.35	1.39967	0.699837	−	0.714303i	$-0.253256\pi$
$181$	3408.35	1.39967	0.699837	+	0.714303i	$0.253256\pi$
$182$	2220.27	0.904273
$183$	0	0
$184$	−1146.48	−0.459344
$185$	−4368.85	−1.73624
$186$	0	0
$187$	−310.960	−0.121602
$188$	1571.78	0.609753
$189$	0	0
$190$	7506.16	2.86607
$191$	2785.74	1.05533	0.527667	−	0.849451i	$-0.323067\pi$
$191$	2785.74	1.05533	0.527667	+	0.849451i	$0.323067\pi$
$192$	0	0
$193$	−4931.75	−1.83935	−0.919677	−	0.392676i	$-0.871549\pi$
$193$	−4931.75	−1.83935	−0.919677	+	0.392676i	$0.871549\pi$
$194$	−4840.53	−1.79139
$195$	0	0
$196$	253.931	0.0925403
$197$	−28.9860	−0.0104831	−0.00524154	−	0.999986i	$-0.501668\pi$
$197$	−28.9860	−0.0104831	−0.00524154	+	0.999986i	$0.501668\pi$
$198$	0	0
$199$	−499.588	−0.177964	−0.0889821	−	0.996033i	$-0.528361\pi$
$199$	−499.588	−0.177964	−0.0889821	+	0.996033i	$0.528361\pi$
$200$	−3280.76	−1.15992
$201$	0	0
$202$	−2713.61	−0.945192
$203$	−104.449	−0.0361126
$204$	0	0
$205$	6778.04	2.30926
$206$	1992.24	0.673816
$207$	0	0
$208$	−6866.71	−2.28904
$209$	1077.21	0.356518
$210$	0	0
$211$	3435.50	1.12090	0.560450	−	0.828189i	$-0.310628\pi$
$211$	3435.50	1.12090	0.560450	+	0.828189i	$0.310628\pi$
$212$	2874.29	0.931164
$213$	0	0
$214$	−994.369	−0.317634
$215$	−6020.88	−1.90986
$216$	0	0
$217$	969.078	0.303158
$218$	7358.66	2.28620
$219$	0	0
$220$	1203.44	0.368801
$221$	2469.59	0.751687
$222$	0	0
$223$	−707.522	−0.212463	−0.106231	−	0.994341i	$-0.533878\pi$
$223$	−707.522	−0.212463	−0.106231	+	0.994341i	$0.533878\pi$
$224$	−1424.78	−0.424988
$225$	0	0
$226$	3805.16	1.11998
$227$	500.427	0.146319	0.0731596	−	0.997320i	$-0.476692\pi$
$227$	500.427	0.146319	0.0731596	+	0.997320i	$0.476692\pi$
$228$	0	0
$229$	3362.25	0.970236	0.485118	−	0.874449i	$-0.338777\pi$
$229$	3362.25	0.970236	0.485118	+	0.874449i	$0.338777\pi$
$230$	−8589.69	−2.46255
$231$	0	0
$232$	152.652	0.0431986
$233$	2963.07	0.833122	0.416561	−	0.909108i	$-0.363235\pi$
$233$	2963.07	0.833122	0.416561	+	0.909108i	$0.363235\pi$
$234$	0	0
$235$	−6403.03	−1.77740
$236$	3591.88	0.990725
$237$	0	0
$238$	718.463	0.195677
$239$	844.179	0.228474	0.114237	−	0.993454i	$-0.463558\pi$
$239$	844.179	0.228474	0.114237	+	0.993454i	$0.463558\pi$
$240$	0	0
$241$	3616.63	0.966672	0.483336	−	0.875435i	$-0.339425\pi$
$241$	3616.63	0.966672	0.483336	+	0.875435i	$0.339425\pi$
$242$	439.319	0.116696
$243$	0	0
$244$	812.378	0.213144
$245$	−1034.45	−0.269750
$246$	0	0
$247$	−8555.04	−2.20382
$248$	−1416.30	−0.362643
$249$	0	0
$250$	−14999.2	−3.79452
$251$	6190.73	1.55679	0.778397	−	0.627772i	$-0.216033\pi$
$251$	6190.73	1.55679	0.778397	+	0.627772i	$0.216033\pi$
$252$	0	0
$253$	−1232.71	−0.306324
$254$	−3288.79	−0.812430
$255$	0	0
$256$	5341.01	1.30396
$257$	4990.16	1.21120	0.605598	−	0.795771i	$-0.292934\pi$
$257$	4990.16	1.21120	0.605598	+	0.795771i	$0.292934\pi$
$258$	0	0
$259$	1448.61	0.347537
$260$	−9557.55	−2.27975
$261$	0	0
$262$	−6852.82	−1.61591
$263$	672.993	0.157789	0.0788946	−	0.996883i	$-0.474861\pi$
$263$	672.993	0.157789	0.0788946	+	0.996883i	$0.474861\pi$
$264$	0	0
$265$	−11709.1	−2.71429
$266$	−2488.87	−0.573692
$267$	0	0
$268$	3029.89	0.690598
$269$	3558.01	0.806453	0.403226	−	0.915100i	$-0.367889\pi$
$269$	3558.01	0.806453	0.403226	+	0.915100i	$0.367889\pi$
$270$	0	0
$271$	3305.52	0.740944	0.370472	−	0.928844i	$-0.379196\pi$
$271$	3305.52	0.740944	0.370472	+	0.928844i	$0.379196\pi$
$272$	−2222.01	−0.495328
$273$	0	0
$274$	−2507.94	−0.552958
$275$	−3527.54	−0.773522
$276$	0	0
$277$	1428.48	0.309851	0.154926	−	0.987926i	$-0.450486\pi$
$277$	1428.48	0.309851	0.154926	+	0.987926i	$0.450486\pi$
$278$	−5322.83	−1.14835
$279$	0	0
$280$	1511.85	0.322679
$281$	7156.92	1.51938	0.759691	−	0.650284i	$-0.225350\pi$
$281$	7156.92	1.51938	0.759691	+	0.650284i	$0.225350\pi$
$282$	0	0
$283$	−2470.58	−0.518942	−0.259471	−	0.965751i	$-0.583548\pi$
$283$	−2470.58	−0.518942	−0.259471	+	0.965751i	$0.583548\pi$
$284$	−1885.02	−0.393857
$285$	0	0
$286$	−3489.00	−0.721360
$287$	−2247.44	−0.462238
$288$	0	0
$289$	−4113.86	−0.837342
$290$	1143.71	0.231589
$291$	0	0
$292$	−3873.34	−0.776268
$293$	−4544.57	−0.906131	−0.453066	−	0.891477i	$-0.649670\pi$
$293$	−4544.57	−0.906131	−0.453066	+	0.891477i	$0.649670\pi$
$294$	0	0
$295$	−14632.4	−2.88791
$296$	−2117.14	−0.415730
$297$	0	0
$298$	−4966.41	−0.965425
$299$	9789.99	1.89354
$300$	0	0
$301$	1996.38	0.382291
$302$	340.985	0.0649717
$303$	0	0
$304$	7697.39	1.45222
$305$	−3309.43	−0.621303
$306$	0	0
$307$	6711.77	1.24776	0.623878	−	0.781522i	$-0.285556\pi$
$307$	6711.77	1.24776	0.623878	+	0.781522i	$0.285556\pi$
$308$	−399.034	−0.0738217
$309$	0	0
$310$	−10611.3	−1.94414
$311$	3627.08	0.661328	0.330664	−	0.943749i	$-0.392727\pi$
$311$	3627.08	0.661328	0.330664	+	0.943749i	$0.392727\pi$
$312$	0	0
$313$	4355.14	0.786475	0.393238	−	0.919437i	$-0.371355\pi$
$313$	4355.14	0.786475	0.393238	+	0.919437i	$0.371355\pi$
$314$	−262.525	−0.0471820
$315$	0	0
$316$	2173.15	0.386864
$317$	9227.18	1.63486	0.817429	−	0.576029i	$-0.195398\pi$
$317$	9227.18	1.63486	0.817429	+	0.576029i	$0.195398\pi$
$318$	0	0
$319$	164.134	0.0288079
$320$	2326.12	0.406356
$321$	0	0
$322$	2848.14	0.492922
$323$	−2768.35	−0.476888
$324$	0	0
$325$	28015.1	4.78154
$326$	124.751	0.0211943
$327$	0	0
$328$	3284.63	0.552937
$329$	2123.10	0.355775
$330$	0	0
$331$	837.223	0.139027	0.0695135	−	0.997581i	$-0.477855\pi$
$331$	837.223	0.139027	0.0695135	+	0.997581i	$0.477855\pi$
$332$	−6105.73	−1.00932
$333$	0	0
$334$	−1376.34	−0.225478
$335$	−12343.0	−2.01305
$336$	0	0
$337$	−9563.00	−1.54579	−0.772893	−	0.634536i	$-0.781191\pi$
$337$	−9563.00	−1.54579	−0.772893	+	0.634536i	$0.781191\pi$
$338$	19732.3	3.17544
$339$	0	0
$340$	−3092.75	−0.493317
$341$	−1522.84	−0.241837
$342$	0	0
$343$	343.000	0.0539949
$344$	−2917.71	−0.457303
$345$	0	0
$346$	5076.81	0.788818
$347$	7584.04	1.17329	0.586646	−	0.809843i	$-0.300448\pi$
$347$	7584.04	1.17329	0.586646	+	0.809843i	$0.300448\pi$
$348$	0	0
$349$	−1800.32	−0.276128	−0.138064	−	0.990423i	$-0.544088\pi$
$349$	−1800.32	−0.276128	−0.138064	+	0.990423i	$0.544088\pi$
$350$	8150.27	1.24471
$351$	0	0
$352$	2238.94	0.339023
$353$	−9042.84	−1.36346	−0.681731	−	0.731603i	$-0.738772\pi$
$353$	−9042.84	−1.36346	−0.681731	+	0.731603i	$0.738772\pi$
$354$	0	0
$355$	7679.11	1.14807
$356$	2062.06	0.306991
$357$	0	0
$358$	9407.33	1.38881
$359$	−8915.34	−1.31068	−0.655339	−	0.755335i	$-0.727474\pi$
$359$	−8915.34	−1.31068	−0.655339	+	0.755335i	$0.727474\pi$
$360$	0	0
$361$	2730.97	0.398159
$362$	12374.8	1.79671
$363$	0	0
$364$	3169.06	0.456330
$365$	15779.1	2.26278
$366$	0	0
$367$	−11721.9	−1.66724	−0.833622	−	0.552335i	$-0.813737\pi$
$367$	−11721.9	−1.66724	−0.833622	+	0.552335i	$0.813737\pi$
$368$	−8808.53	−1.24776
$369$	0	0
$370$	−15862.2	−2.22874
$371$	3882.48	0.543310
$372$	0	0
$373$	−8089.21	−1.12291	−0.561453	−	0.827509i	$-0.689757\pi$
$373$	−8089.21	−1.12291	−0.561453	+	0.827509i	$0.689757\pi$
$374$	−1129.01	−0.156096
$375$	0	0
$376$	−3102.90	−0.425585
$377$	−1303.52	−0.178077
$378$	0	0
$379$	−10580.4	−1.43398	−0.716989	−	0.697084i	$-0.754480\pi$
$379$	−10580.4	−1.43398	−0.716989	+	0.697084i	$0.754480\pi$
$380$	10713.8	1.44633
$381$	0	0
$382$	10114.3	1.35469
$383$	−10512.0	−1.40245	−0.701224	−	0.712941i	$-0.747363\pi$
$383$	−10512.0	−1.40245	−0.701224	+	0.712941i	$0.747363\pi$
$384$	0	0
$385$	1625.57	0.215186
$386$	−17905.9	−2.36110
$387$	0	0
$388$	−6909.02	−0.904001
$389$	−1019.47	−0.132878	−0.0664388	−	0.997790i	$-0.521164\pi$
$389$	−1019.47	−0.132878	−0.0664388	+	0.997790i	$0.521164\pi$
$390$	0	0
$391$	3167.96	0.409746
$392$	−501.294	−0.0645897
$393$	0	0
$394$	−105.241	−0.0134567
$395$	−8852.87	−1.12769
$396$	0	0
$397$	11793.5	1.49093	0.745467	−	0.666543i	$-0.232227\pi$
$397$	11793.5	1.49093	0.745467	+	0.666543i	$0.232227\pi$
$398$	−1813.87	−0.228445
$399$	0	0
$400$	−25206.6	−3.15082
$401$	5342.21	0.665280	0.332640	−	0.943054i	$-0.392061\pi$
$401$	5342.21	0.665280	0.332640	+	0.943054i	$0.392061\pi$
$402$	0	0
$403$	12094.1	1.49492
$404$	−3873.21	−0.476979
$405$	0	0
$406$	−379.226	−0.0463563
$407$	−2276.39	−0.277239
$408$	0	0
$409$	−8535.63	−1.03193	−0.515965	−	0.856610i	$-0.672567\pi$
$409$	−8535.63	−1.03193	−0.515965	+	0.856610i	$0.672567\pi$
$410$	24609.3	2.96431
$411$	0	0
$412$	2843.59	0.340033
$413$	4851.77	0.578063
$414$	0	0
$415$	24873.3	2.94212
$416$	−17781.3	−2.09568
$417$	0	0
$418$	3911.08	0.457648
$419$	−4300.54	−0.501420	−0.250710	−	0.968062i	$-0.580664\pi$
$419$	−4300.54	−0.501420	−0.250710	+	0.968062i	$0.580664\pi$
$420$	0	0
$421$	−5917.77	−0.685070	−0.342535	−	0.939505i	$-0.611285\pi$
$421$	−5917.77	−0.685070	−0.342535	+	0.939505i	$0.611285\pi$
$422$	12473.4	1.43885
$423$	0	0
$424$	−5674.23	−0.649917
$425$	9065.47	1.03468
$426$	0	0
$427$	1097.33	0.124364
$428$	−1419.29	−0.160290
$429$	0	0
$430$	−21860.2	−2.45161
$431$	−8110.20	−0.906391	−0.453196	−	0.891411i	$-0.649716\pi$
$431$	−8110.20	−0.906391	−0.453196	+	0.891411i	$0.649716\pi$
$432$	0	0
$433$	−4242.54	−0.470862	−0.235431	−	0.971891i	$-0.575650\pi$
$433$	−4242.54	−0.470862	−0.235431	+	0.971891i	$0.575650\pi$
$434$	3518.47	0.389152
$435$	0	0
$436$	10503.2	1.15370
$437$	−10974.3	−1.20131
$438$	0	0
$439$	−595.983	−0.0647944	−0.0323972	−	0.999475i	$-0.510314\pi$
$439$	−595.983	−0.0647944	−0.0323972	+	0.999475i	$0.510314\pi$
$440$	−2375.76	−0.257409
$441$	0	0
$442$	8966.44	0.964910
$443$	4284.60	0.459521	0.229760	−	0.973247i	$-0.426206\pi$
$443$	4284.60	0.459521	0.229760	+	0.973247i	$0.426206\pi$
$444$	0	0
$445$	−8400.33	−0.894862
$446$	−2568.83	−0.272730
$447$	0	0
$448$	−771.286	−0.0813389
$449$	704.286	0.0740252	0.0370126	−	0.999315i	$-0.488216\pi$
$449$	704.286	0.0740252	0.0370126	+	0.999315i	$0.488216\pi$
$450$	0	0
$451$	3531.69	0.368738
$452$	5431.22	0.565184
$453$	0	0
$454$	1816.92	0.187824
$455$	−12910.0	−1.33017
$456$	0	0
$457$	−7013.21	−0.717865	−0.358932	−	0.933364i	$-0.616859\pi$
$457$	−7013.21	−0.717865	−0.358932	+	0.933364i	$0.616859\pi$
$458$	12207.5	1.24545
$459$	0	0
$460$	−12260.3	−1.24270
$461$	−1032.99	−0.104363	−0.0521813	−	0.998638i	$-0.516617\pi$
$461$	−1032.99	−0.104363	−0.0521813	+	0.998638i	$0.516617\pi$
$462$	0	0
$463$	12593.3	1.26406	0.632029	−	0.774945i	$-0.282222\pi$
$463$	12593.3	1.26406	0.632029	+	0.774945i	$0.282222\pi$
$464$	1172.84	0.117345
$465$	0	0
$466$	10758.1	1.06944
$467$	18527.1	1.83583	0.917914	−	0.396779i	$-0.129872\pi$
$467$	18527.1	1.83583	0.917914	+	0.396779i	$0.129872\pi$
$468$	0	0
$469$	4092.67	0.402946
$470$	−23247.7	−2.28157
$471$	0	0
$472$	−7090.85	−0.691489
$473$	−3137.17	−0.304963
$474$	0	0
$475$	−31404.2	−3.03352
$476$	1025.48	0.0987457
$477$	0	0
$478$	3064.99	0.293283
$479$	15402.2	1.46920	0.734599	−	0.678501i	$-0.237370\pi$
$479$	15402.2	1.46920	0.734599	+	0.678501i	$0.237370\pi$
$480$	0	0
$481$	18078.7	1.71376
$482$	13131.0	1.24088
$483$	0	0
$484$	627.053	0.0588893
$485$	28145.7	2.63511
$486$	0	0
$487$	6816.82	0.634291	0.317146	−	0.948377i	$-0.397276\pi$
$487$	6816.82	0.634291	0.317146	+	0.948377i	$0.397276\pi$
$488$	−1603.74	−0.148767
$489$	0	0
$490$	−3755.82	−0.346267
$491$	−6232.66	−0.572864	−0.286432	−	0.958101i	$-0.592469\pi$
$491$	−6232.66	−0.572864	−0.286432	+	0.958101i	$0.592469\pi$
$492$	0	0
$493$	−421.810	−0.0385342
$494$	−31061.1	−2.82896
$495$	0	0
$496$	−10881.7	−0.985084
$497$	−2546.21	−0.229806
$498$	0	0
$499$	−19244.6	−1.72646	−0.863232	−	0.504807i	$-0.831564\pi$
$499$	−19244.6	−1.72646	−0.863232	+	0.504807i	$0.831564\pi$
$500$	−21408.7	−1.91486
$501$	0	0
$502$	22476.9	1.99839
$503$	−7177.94	−0.636280	−0.318140	−	0.948044i	$-0.603058\pi$
$503$	−7177.94	−0.636280	−0.318140	+	0.948044i	$0.603058\pi$
$504$	0	0
$505$	15778.5	1.39037
$506$	−4475.65	−0.393215
$507$	0	0
$508$	−4694.19	−0.409982
$509$	−15314.2	−1.33357	−0.666787	−	0.745248i	$-0.732331\pi$
$509$	−15314.2	−1.33357	−0.666787	+	0.745248i	$0.732331\pi$
$510$	0	0
$511$	−5231.97	−0.452933
$512$	9565.62	0.825674
$513$	0	0
$514$	18118.0	1.55476
$515$	−11584.1	−0.991176
$516$	0	0
$517$	−3336.29	−0.283811
$518$	5259.52	0.446120
$519$	0	0
$520$	18867.9	1.59118
$521$	−9650.62	−0.811519	−0.405760	−	0.913980i	$-0.632993\pi$
$521$	−9650.62	−0.811519	−0.405760	+	0.913980i	$0.632993\pi$
$522$	0	0
$523$	−13116.1	−1.09661	−0.548305	−	0.836278i	$-0.684727\pi$
$523$	−13116.1	−1.09661	−0.548305	+	0.836278i	$0.684727\pi$
$524$	−9781.23	−0.815449
$525$	0	0
$526$	2443.46	0.202548
$527$	3913.56	0.323487
$528$	0	0
$529$	391.484	0.0321759
$530$	−42512.8	−3.48422
$531$	0	0
$532$	−3552.43	−0.289506
$533$	−28048.1	−2.27936
$534$	0	0
$535$	5781.85	0.467236
$536$	−5981.42	−0.482011
$537$	0	0
$538$	12918.2	1.03521
$539$	−539.000	−0.0430730
$540$	0	0
$541$	18625.5	1.48017	0.740086	−	0.672512i	$-0.234785\pi$
$541$	18625.5	1.48017	0.740086	+	0.672512i	$0.234785\pi$
$542$	12001.5	0.951121
$543$	0	0
$544$	−5753.90	−0.453486
$545$	−42787.6	−3.36297
$546$	0	0
$547$	−6967.57	−0.544628	−0.272314	−	0.962208i	$-0.587789\pi$
$547$	−6967.57	−0.544628	−0.272314	+	0.962208i	$0.587789\pi$
$548$	−3579.66	−0.279043
$549$	0	0
$550$	−12807.6	−0.992939
$551$	1461.21	0.112976
$552$	0	0
$553$	2935.41	0.225725
$554$	5186.42	0.397744
$555$	0	0
$556$	−7597.43	−0.579501
$557$	17230.7	1.31075	0.655374	−	0.755305i	$-0.272511\pi$
$557$	17230.7	1.31075	0.655374	+	0.755305i	$0.272511\pi$
$558$	0	0
$559$	24914.9	1.88513
$560$	11615.7	0.876527
$561$	0	0
$562$	25984.9	1.95037
$563$	20751.0	1.55338	0.776688	−	0.629885i	$-0.216898\pi$
$563$	20751.0	1.55338	0.776688	+	0.629885i	$0.216898\pi$
$564$	0	0
$565$	−22125.5	−1.64748
$566$	−8970.01	−0.666145
$567$	0	0
$568$	3721.29	0.274897
$569$	15953.4	1.17540	0.587700	−	0.809079i	$-0.300034\pi$
$569$	15953.4	1.17540	0.587700	+	0.809079i	$0.300034\pi$
$570$	0	0
$571$	10837.6	0.794291	0.397145	−	0.917756i	$-0.370001\pi$
$571$	10837.6	0.794291	0.397145	+	0.917756i	$0.370001\pi$
$572$	−4979.95	−0.364025
$573$	0	0
$574$	−8159.87	−0.593356
$575$	35937.5	2.60643
$576$	0	0
$577$	9613.09	0.693584	0.346792	−	0.937942i	$-0.387271\pi$
$577$	9613.09	0.693584	0.346792	+	0.937942i	$0.387271\pi$
$578$	−14936.3	−1.07486
$579$	0	0
$580$	1632.44	0.116868
$581$	−8247.39	−0.588915
$582$	0	0
$583$	−6101.03	−0.433412
$584$	7646.50	0.541806
$585$	0	0
$586$	−16500.1	−1.16316
$587$	−2169.43	−0.152542	−0.0762708	−	0.997087i	$-0.524301\pi$
$587$	−2169.43	−0.152542	−0.0762708	+	0.997087i	$0.524301\pi$
$588$	0	0
$589$	−13557.2	−0.948411
$590$	−53126.5	−3.70709
$591$	0	0
$592$	−16266.3	−1.12929
$593$	−4688.62	−0.324686	−0.162343	−	0.986734i	$-0.551905\pi$
$593$	−4688.62	−0.324686	−0.162343	+	0.986734i	$0.551905\pi$
$594$	0	0
$595$	−4177.57	−0.287838
$596$	−7088.71	−0.487189
$597$	0	0
$598$	35544.9	2.43067
$599$	−941.884	−0.0642476	−0.0321238	−	0.999484i	$-0.510227\pi$
$599$	−941.884	−0.0642476	−0.0321238	+	0.999484i	$0.510227\pi$
$600$	0	0
$601$	−8131.21	−0.551879	−0.275939	−	0.961175i	$-0.588989\pi$
$601$	−8131.21	−0.551879	−0.275939	+	0.961175i	$0.588989\pi$
$602$	7248.34	0.490732
$603$	0	0
$604$	486.698	0.0327872
$605$	−2554.46	−0.171659
$606$	0	0
$607$	−27616.5	−1.84666	−0.923328	−	0.384011i	$-0.874542\pi$
$607$	−27616.5	−1.84666	−0.923328	+	0.384011i	$0.874542\pi$
$608$	19932.4	1.32955
$609$	0	0
$610$	−12015.7	−0.797542
$611$	26496.3	1.75438
$612$	0	0
$613$	7289.82	0.480315	0.240158	−	0.970734i	$-0.422801\pi$
$613$	7289.82	0.480315	0.240158	+	0.970734i	$0.422801\pi$
$614$	24368.7	1.60169
$615$	0	0
$616$	787.747	0.0515247
$617$	13672.2	0.892094	0.446047	−	0.895010i	$-0.352831\pi$
$617$	13672.2	0.892094	0.446047	+	0.895010i	$0.352831\pi$
$618$	0	0
$619$	20157.3	1.30887	0.654435	−	0.756118i	$-0.272906\pi$
$619$	20157.3	1.30887	0.654435	+	0.756118i	$0.272906\pi$
$620$	−15145.9	−0.981084
$621$	0	0
$622$	13169.0	0.848920
$623$	2785.35	0.179122
$624$	0	0
$625$	47128.3	3.01621
$626$	15812.4	1.00957
$627$	0	0
$628$	−374.710	−0.0238098
$629$	5850.12	0.370842
$630$	0	0
$631$	−21955.4	−1.38515	−0.692576	−	0.721345i	$-0.743524\pi$
$631$	−21955.4	−1.38515	−0.692576	+	0.721345i	$0.743524\pi$
$632$	−4290.09	−0.270017
$633$	0	0
$634$	33501.5	2.09860
$635$	19123.0	1.19507
$636$	0	0
$637$	4280.65	0.266257
$638$	595.927	0.0369796
$639$	0	0
$640$	−25930.4	−1.60155
$641$	−3575.87	−0.220340	−0.110170	−	0.993913i	$-0.535140\pi$
$641$	−3575.87	−0.220340	−0.110170	+	0.993913i	$0.535140\pi$
$642$	0	0
$643$	6676.04	0.409452	0.204726	−	0.978819i	$-0.434370\pi$
$643$	6676.04	0.409452	0.204726	+	0.978819i	$0.434370\pi$
$644$	4065.24	0.248746
$645$	0	0
$646$	−10051.1	−0.612162
$647$	12366.4	0.751430	0.375715	−	0.926735i	$-0.377397\pi$
$647$	12366.4	0.751430	0.375715	+	0.926735i	$0.377397\pi$
$648$	0	0
$649$	−7624.21	−0.461135
$650$	101716.	6.13786
$651$	0	0
$652$	178.061	0.0106954
$653$	−19797.3	−1.18642	−0.593208	−	0.805049i	$-0.702139\pi$
$653$	−19797.3	−1.18642	−0.593208	+	0.805049i	$0.702139\pi$
$654$	0	0
$655$	39846.4	2.37699
$656$	25236.3	1.50200
$657$	0	0
$658$	7708.41	0.456695
$659$	893.599	0.0528220	0.0264110	−	0.999651i	$-0.491592\pi$
$659$	893.599	0.0528220	0.0264110	+	0.999651i	$0.491592\pi$
$660$	0	0
$661$	16418.8	0.966137	0.483068	−	0.875583i	$-0.339522\pi$
$661$	16418.8	0.966137	0.483068	+	0.875583i	$0.339522\pi$
$662$	3039.74	0.178463
$663$	0	0
$664$	12053.5	0.704470
$665$	14471.7	0.843895
$666$	0	0
$667$	−1672.14	−0.0970700
$668$	−1964.48	−0.113785
$669$	0	0
$670$	−44814.4	−2.58408
$671$	−1724.38	−0.0992083
$672$	0	0
$673$	−17447.6	−0.999339	−0.499670	−	0.866216i	$-0.666545\pi$
$673$	−17447.6	−0.999339	−0.499670	+	0.866216i	$0.666545\pi$
$674$	−34720.8	−1.98426
$675$	0	0
$676$	28164.6	1.60244
$677$	23669.4	1.34371	0.671853	−	0.740685i	$-0.265499\pi$
$677$	23669.4	1.34371	0.671853	+	0.740685i	$0.265499\pi$
$678$	0	0
$679$	−9332.45	−0.527462
$680$	6105.51	0.344317
$681$	0	0
$682$	−5529.02	−0.310436
$683$	15236.3	0.853586	0.426793	−	0.904349i	$-0.359643\pi$
$683$	15236.3	0.853586	0.426793	+	0.904349i	$0.359643\pi$
$684$	0	0
$685$	14582.7	0.813394
$686$	1245.34	0.0693111
$687$	0	0
$688$	−22417.2	−1.24222
$689$	48453.4	2.67914
$690$	0	0
$691$	−8336.36	−0.458944	−0.229472	−	0.973315i	$-0.573700\pi$
$691$	−8336.36	−0.458944	−0.229472	+	0.973315i	$0.573700\pi$
$692$	7246.28	0.398067
$693$	0	0
$694$	27535.7	1.50611
$695$	30950.1	1.68921
$696$	0	0
$697$	−9076.15	−0.493234
$698$	−6536.48	−0.354455
$699$	0	0
$700$	11633.1	0.628129
$701$	−798.786	−0.0430381	−0.0215191	−	0.999768i	$-0.506850\pi$
$701$	−798.786	−0.0430381	−0.0215191	+	0.999768i	$0.506850\pi$
$702$	0	0
$703$	−20265.7	−1.08725
$704$	1212.02	0.0648860
$705$	0	0
$706$	−32832.2	−1.75022
$707$	−5231.79	−0.278305
$708$	0	0
$709$	−22352.2	−1.18400	−0.591999	−	0.805939i	$-0.701661\pi$
$709$	−22352.2	−1.18400	−0.591999	+	0.805939i	$0.701661\pi$
$710$	27880.9	1.47373
$711$	0	0
$712$	−4070.79	−0.214268
$713$	15514.2	0.814883
$714$	0	0
$715$	20287.1	1.06111
$716$	13427.4	0.700844
$717$	0	0
$718$	−32369.3	−1.68247
$719$	−13763.8	−0.713914	−0.356957	−	0.934121i	$-0.616186\pi$
$719$	−13763.8	−0.713914	−0.356957	+	0.934121i	$0.616186\pi$
$720$	0	0
$721$	3841.01	0.198400
$722$	9915.44	0.511100
$723$	0	0
$724$	17663.0	0.906684
$725$	−4785.02	−0.245119
$726$	0	0
$727$	16014.2	0.816966	0.408483	−	0.912766i	$-0.366058\pi$
$727$	16014.2	0.816966	0.408483	+	0.912766i	$0.366058\pi$
$728$	−6256.16	−0.318501
$729$	0	0
$730$	57289.6	2.90463
$731$	8062.27	0.407926
$732$	0	0
$733$	21073.3	1.06188	0.530942	−	0.847408i	$-0.321838\pi$
$733$	21073.3	1.06188	0.530942	+	0.847408i	$0.321838\pi$
$734$	−42559.2	−2.14017
$735$	0	0
$736$	−22809.7	−1.14236
$737$	−6431.33	−0.321440
$738$	0	0
$739$	4570.10	0.227488	0.113744	−	0.993510i	$-0.463716\pi$
$739$	4570.10	0.227488	0.113744	+	0.993510i	$0.463716\pi$
$740$	−22640.5	−1.12471
$741$	0	0
$742$	14096.3	0.697426
$743$	−7122.27	−0.351670	−0.175835	−	0.984420i	$-0.556263\pi$
$743$	−7122.27	−0.351670	−0.175835	+	0.984420i	$0.556263\pi$
$744$	0	0
$745$	28877.7	1.42013
$746$	−29369.8	−1.44143
$747$	0	0
$748$	−1611.47	−0.0787718
$749$	−1917.13	−0.0935251
$750$	0	0
$751$	31096.8	1.51097	0.755485	−	0.655166i	$-0.227401\pi$
$751$	31096.8	1.51097	0.755485	+	0.655166i	$0.227401\pi$
$752$	−23840.0	−1.15606
$753$	0	0
$754$	−4732.75	−0.228590
$755$	−1982.69	−0.0955727
$756$	0	0
$757$	2783.78	0.133657	0.0668284	−	0.997764i	$-0.478712\pi$
$757$	2783.78	0.133657	0.0668284	+	0.997764i	$0.478712\pi$
$758$	−38414.6	−1.84074
$759$	0	0
$760$	−21150.4	−1.00948
$761$	−22236.2	−1.05922	−0.529608	−	0.848243i	$-0.677661\pi$
$761$	−22236.2	−1.05922	−0.529608	+	0.848243i	$0.677661\pi$
$762$	0	0
$763$	14187.4	0.673155
$764$	14436.4	0.683626
$765$	0	0
$766$	−38166.3	−1.80027
$767$	60550.2	2.85051
$768$	0	0
$769$	−28682.5	−1.34502	−0.672508	−	0.740090i	$-0.734783\pi$
$769$	−28682.5	−1.34502	−0.672508	+	0.740090i	$0.734783\pi$
$770$	5902.01	0.276225
$771$	0	0
$772$	−25557.6	−1.19150
$773$	−27245.4	−1.26772	−0.633861	−	0.773447i	$-0.718531\pi$
$773$	−27245.4	−1.26772	−0.633861	+	0.773447i	$0.718531\pi$
$774$	0	0
$775$	44395.6	2.05772
$776$	13639.3	0.630959
$777$	0	0
$778$	−3701.45	−0.170570
$779$	31441.2	1.44608
$780$	0	0
$781$	4001.19	0.183321
$782$	11502.0	0.525975
$783$	0	0
$784$	−3851.51	−0.175451
$785$	1526.48	0.0694042
$786$	0	0
$787$	−33427.2	−1.51404	−0.757020	−	0.653391i	$-0.773346\pi$
$787$	−33427.2	−1.51404	−0.757020	+	0.653391i	$0.773346\pi$
$788$	−150.213	−0.00679075
$789$	0	0
$790$	−32142.5	−1.44757
$791$	7336.29	0.329770
$792$	0	0
$793$	13694.7	0.613258
$794$	42819.2	1.91385
$795$	0	0
$796$	−2589.00	−0.115282
$797$	6592.94	0.293016	0.146508	−	0.989209i	$-0.453197\pi$
$797$	6592.94	0.293016	0.146508	+	0.989209i	$0.453197\pi$
$798$	0	0
$799$	8574.00	0.379632
$800$	−65272.4	−2.88466
$801$	0	0
$802$	19396.2	0.853993
$803$	8221.66	0.361315
$804$	0	0
$805$	−16560.8	−0.725082
$806$	43910.6	1.91896
$807$	0	0
$808$	7646.25	0.332913
$809$	−22661.3	−0.984830	−0.492415	−	0.870361i	$-0.663886\pi$
$809$	−22661.3	−0.984830	−0.492415	+	0.870361i	$0.663886\pi$
$810$	0	0
$811$	2938.33	0.127224	0.0636121	−	0.997975i	$-0.479738\pi$
$811$	2938.33	0.127224	0.0636121	+	0.997975i	$0.479738\pi$
$812$	−541.281	−0.0233931
$813$	0	0
$814$	−8264.96	−0.355881
$815$	−725.379	−0.0311766
$816$	0	0
$817$	−27928.9	−1.19597
$818$	−30990.6	−1.32465
$819$	0	0
$820$	35125.6	1.49590
$821$	−27511.4	−1.16949	−0.584747	−	0.811216i	$-0.698806\pi$
$821$	−27511.4	−1.16949	−0.584747	+	0.811216i	$0.698806\pi$
$822$	0	0
$823$	25092.9	1.06280	0.531400	−	0.847121i	$-0.321666\pi$
$823$	25092.9	1.06280	0.531400	+	0.847121i	$0.321666\pi$
$824$	−5613.63	−0.237330
$825$	0	0
$826$	17615.5	0.742036
$827$	−10259.3	−0.431382	−0.215691	−	0.976462i	$-0.569200\pi$
$827$	−10259.3	−0.431382	−0.215691	+	0.976462i	$0.569200\pi$
$828$	0	0
$829$	−40456.9	−1.69497	−0.847483	−	0.530823i	$-0.821883\pi$
$829$	−40456.9	−1.69497	−0.847483	+	0.530823i	$0.821883\pi$
$830$	90308.3	3.77668
$831$	0	0
$832$	−9625.67	−0.401094
$833$	1385.18	0.0576156
$834$	0	0
$835$	8002.83	0.331676
$836$	5582.39	0.230946
$837$	0	0
$838$	−15614.1	−0.643652
$839$	−11387.0	−0.468563	−0.234281	−	0.972169i	$-0.575274\pi$
$839$	−11387.0	−0.468563	−0.234281	+	0.972169i	$0.575274\pi$
$840$	0	0
$841$	−24166.4	−0.990871
$842$	−21485.9	−0.879396
$843$	0	0
$844$	17803.7	0.726099
$845$	−114736.	−4.67103
$846$	0	0
$847$	847.000	0.0343604
$848$	−43595.9	−1.76544
$849$	0	0
$850$	32914.4	1.32818
$851$	23191.1	0.934173
$852$	0	0
$853$	8058.93	0.323485	0.161742	−	0.986833i	$-0.448289\pi$
$853$	8058.93	0.323485	0.161742	+	0.986833i	$0.448289\pi$
$854$	3984.12	0.159641
$855$	0	0
$856$	2801.87	0.111876
$857$	−4025.02	−0.160434	−0.0802170	−	0.996777i	$-0.525561\pi$
$857$	−4025.02	−0.160434	−0.0802170	+	0.996777i	$0.525561\pi$
$858$	0	0
$859$	−4319.84	−0.171584	−0.0857922	−	0.996313i	$-0.527342\pi$
$859$	−4319.84	−0.171584	−0.0857922	+	0.996313i	$0.527342\pi$
$860$	−31201.8	−1.23718
$861$	0	0
$862$	−29446.0	−1.16350
$863$	−11967.7	−0.472055	−0.236028	−	0.971746i	$-0.575846\pi$
$863$	−11967.7	−0.472055	−0.236028	+	0.971746i	$0.575846\pi$
$864$	0	0
$865$	−29519.6	−1.16034
$866$	−15403.5	−0.604427
$867$	0	0
$868$	5022.01	0.196380
$869$	−4612.78	−0.180067
$870$	0	0
$871$	51076.6	1.98699
$872$	−20734.8	−0.805239
$873$	0	0
$874$	−39844.8	−1.54207
$875$	−28918.1	−1.11727
$876$	0	0
$877$	−3248.36	−0.125074	−0.0625368	−	0.998043i	$-0.519919\pi$
$877$	−3248.36	−0.125074	−0.0625368	+	0.998043i	$0.519919\pi$
$878$	−2163.86	−0.0831740
$879$	0	0
$880$	−18253.3	−0.699226
$881$	−1264.38	−0.0483518	−0.0241759	−	0.999708i	$-0.507696\pi$
$881$	−1264.38	−0.0483518	−0.0241759	+	0.999708i	$0.507696\pi$
$882$	0	0
$883$	−11524.4	−0.439216	−0.219608	−	0.975588i	$-0.570478\pi$
$883$	−11524.4	−0.439216	−0.219608	+	0.975588i	$0.570478\pi$
$884$	12798.1	0.486929
$885$	0	0
$886$	15556.3	0.589868
$887$	−24949.8	−0.944457	−0.472228	−	0.881476i	$-0.656550\pi$
$887$	−24949.8	−0.944457	−0.472228	+	0.881476i	$0.656550\pi$
$888$	0	0
$889$	−6340.73	−0.239214
$890$	−30499.4	−1.14870
$891$	0	0
$892$	−3666.56	−0.137630
$893$	−29701.6	−1.11302
$894$	0	0
$895$	−54699.8	−2.04292
$896$	8597.92	0.320576
$897$	0	0
$898$	2557.08	0.0950231
$899$	−2065.69	−0.0766349
$900$	0	0
$901$	15679.1	0.579743
$902$	12822.7	0.473334
$903$	0	0
$904$	−10722.0	−0.394477
$905$	−71954.6	−2.64293
$906$	0	0
$907$	40615.8	1.48691	0.743454	−	0.668787i	$-0.233186\pi$
$907$	40615.8	1.48691	0.743454	+	0.668787i	$0.233186\pi$
$908$	2593.34	0.0947830
$909$	0	0
$910$	−46872.8	−1.70749
$911$	14594.0	0.530760	0.265380	−	0.964144i	$-0.414503\pi$
$911$	14594.0	0.530760	0.265380	+	0.964144i	$0.414503\pi$
$912$	0	0
$913$	12960.2	0.469792
$914$	−25463.1	−0.921494
$915$	0	0
$916$	17424.1	0.628502
$917$	−13212.1	−0.475794
$918$	0	0
$919$	28339.1	1.01721	0.508607	−	0.860999i	$-0.330161\pi$
$919$	28339.1	1.01721	0.508607	+	0.860999i	$0.330161\pi$
$920$	24203.5	0.867355
$921$	0	0
$922$	−3750.51	−0.133966
$923$	−31776.8	−1.13320
$924$	0	0
$925$	66363.9	2.35895
$926$	45722.9	1.62262
$927$	0	0
$928$	3037.08	0.107432
$929$	−25916.5	−0.915276	−0.457638	−	0.889139i	$-0.651304\pi$
$929$	−25916.5	−0.915276	−0.457638	+	0.889139i	$0.651304\pi$
$930$	0	0
$931$	−4798.49	−0.168920
$932$	15355.4	0.539682
$933$	0	0
$934$	67267.0	2.35658
$935$	6564.75	0.229615
$936$	0	0
$937$	−2004.67	−0.0698930	−0.0349465	−	0.999389i	$-0.511126\pi$
$937$	−2004.67	−0.0698930	−0.0349465	+	0.999389i	$0.511126\pi$
$938$	14859.4	0.517246
$939$	0	0
$940$	−33182.2	−1.15137
$941$	36486.7	1.26401	0.632004	−	0.774965i	$-0.282233\pi$
$941$	36486.7	1.26401	0.632004	+	0.774965i	$0.282233\pi$
$942$	0	0
$943$	−35979.8	−1.24249
$944$	−54480.0	−1.87836
$945$	0	0
$946$	−11390.3	−0.391468
$947$	26275.1	0.901611	0.450806	−	0.892622i	$-0.351137\pi$
$947$	26275.1	0.901611	0.450806	+	0.892622i	$0.351137\pi$
$948$	0	0
$949$	−65295.1	−2.23347
$950$	−114020.	−3.89401
$951$	0	0
$952$	−2024.44	−0.0689208
$953$	25800.3	0.876971	0.438485	−	0.898738i	$-0.355515\pi$
$953$	25800.3	0.876971	0.438485	+	0.898738i	$0.355515\pi$
$954$	0	0
$955$	−58810.4	−1.99273
$956$	4374.75	0.148002
$957$	0	0
$958$	55921.5	1.88595
$959$	−4835.27	−0.162814
$960$	0	0
$961$	−10625.4	−0.356666
$962$	65639.0	2.19988
$963$	0	0
$964$	18742.3	0.626193
$965$	104115.	3.47316
$966$	0	0
$967$	−24454.9	−0.813254	−0.406627	−	0.913594i	$-0.633295\pi$
$967$	−24454.9	−0.813254	−0.406627	+	0.913594i	$0.633295\pi$
$968$	−1237.89	−0.0411025
$969$	0	0
$970$	102190.	3.38259
$971$	57692.0	1.90672	0.953359	−	0.301839i	$-0.0976004\pi$
$971$	57692.0	1.90672	0.953359	+	0.301839i	$0.0976004\pi$
$972$	0	0
$973$	−10262.3	−0.338124
$974$	24750.1	0.814214
$975$	0	0
$976$	−12321.8	−0.404110
$977$	6855.66	0.224495	0.112248	−	0.993680i	$-0.464195\pi$
$977$	6855.66	0.224495	0.112248	+	0.993680i	$0.464195\pi$
$978$	0	0
$979$	−4376.98	−0.142890
$980$	−5360.80	−0.174739
$981$	0	0
$982$	−22629.2	−0.735362
$983$	8142.33	0.264191	0.132096	−	0.991237i	$-0.457829\pi$
$983$	8142.33	0.264191	0.132096	+	0.991237i	$0.457829\pi$
$984$	0	0
$985$	611.931	0.0197947
$986$	−1531.48	−0.0494648
$987$	0	0
$988$	−44334.4	−1.42760
$989$	31960.6	1.02759
$990$	0	0
$991$	11895.7	0.381312	0.190656	−	0.981657i	$-0.438938\pi$
$991$	11895.7	0.381312	0.190656	+	0.981657i	$0.438938\pi$
$992$	−28178.1	−0.901870
$993$	0	0
$994$	−9244.64	−0.294992
$995$	10546.9	0.336041
$996$	0	0
$997$	−44608.5	−1.41702	−0.708509	−	0.705702i	$-0.750632\pi$
$997$	−44608.5	−1.41702	−0.708509	+	0.705702i	$0.750632\pi$
$998$	−69872.0	−2.21619
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.p.1.4		5
3.2	odd	2		231.4.a.k.1.2	✓	5
21.20	even	2		1617.4.a.n.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.k.1.2	✓	5	3.2	odd	2
693.4.a.p.1.4		5	1.1	even	1	trivial
1617.4.a.n.1.2		5	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q+3.63074 q^{2} +5.18226 q^{4} -21.1113 q^{5} +7.00000 q^{7} -10.2305 q^{8} +O(q^{10})\)	\(q+3.63074 q^{2} +5.18226 q^{4} -21.1113 q^{5} +7.00000 q^{7} -10.2305 q^{8} -76.6494 q^{10} -11.0000 q^{11} +87.3602 q^{13} +25.4152 q^{14} -78.6023 q^{16} +28.2691 q^{17} -97.9284 q^{19} -109.404 q^{20} -39.9381 q^{22} +112.065 q^{23} +320.685 q^{25} +317.182 q^{26} +36.2758 q^{28} -14.9213 q^{29} +138.440 q^{31} -203.540 q^{32} +102.638 q^{34} -147.779 q^{35} +206.944 q^{37} -355.552 q^{38} +215.978 q^{40} -321.063 q^{41} +285.198 q^{43} -57.0048 q^{44} +406.877 q^{46} +303.300 q^{47} +49.0000 q^{49} +1164.32 q^{50} +452.723 q^{52} +554.639 q^{53} +232.224 q^{55} -71.6134 q^{56} -54.1752 q^{58} +693.110 q^{59} +156.761 q^{61} +502.638 q^{62} -110.184 q^{64} -1844.28 q^{65} +584.667 q^{67} +146.498 q^{68} -536.546 q^{70} -363.745 q^{71} -747.424 q^{73} +751.360 q^{74} -507.490 q^{76} -77.0000 q^{77} +419.344 q^{79} +1659.39 q^{80} -1165.70 q^{82} -1178.20 q^{83} -596.796 q^{85} +1035.48 q^{86} +112.535 q^{88} +397.908 q^{89} +611.521 q^{91} +580.748 q^{92} +1101.20 q^{94} +2067.39 q^{95} -1333.21 q^{97} +177.906 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8}+O(q^{10}) \) 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8	\( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8} - 23 q^{10} - 55 q^{11} + 101 q^{13} + 7 q^{14} - 7 q^{16} + 20 q^{17} + 237 q^{19} - 85 q^{20} - 11 q^{22} + 80 q^{23} + 486 q^{25} - 165 q^{26} + 147 q^{28} + 11 q^{29} + 316 q^{31} - 453 q^{32} + 936 q^{34} - 147 q^{35} + 319 q^{37} - 89 q^{38} + 624 q^{40} - 1190 q^{41} + 88 q^{43} - 231 q^{44} + 1000 q^{46} - 377 q^{47} + 245 q^{49} + 644 q^{50} + 1001 q^{52} + 992 q^{53} + 231 q^{55} + 294 q^{56} + 721 q^{58} - 71 q^{59} - 574 q^{61} - 272 q^{62} - 1380 q^{64} - 589 q^{65} - 527 q^{67} + 2974 q^{68} - 161 q^{70} + 1156 q^{71} + 1061 q^{73} + 1609 q^{74} + 2399 q^{76} - 385 q^{77} + 588 q^{79} + 1643 q^{80} - 2602 q^{82} + 212 q^{83} + 1918 q^{85} + 4760 q^{86} - 462 q^{88} - 1030 q^{89} + 707 q^{91} + 1174 q^{92} - 1799 q^{94} + 3593 q^{95} + 2488 q^{97} + 49 q^{98}+O(q^{100}) \) 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8 - 23 * q^10 - 55 * q^11 + 101 * q^13 + 7 * q^14 - 7 * q^16 + 20 * q^17 + 237 * q^19 - 85 * q^20 - 11 * q^22 + 80 * q^23 + 486 * q^25 - 165 * q^26 + 147 * q^28 + 11 * q^29 + 316 * q^31 - 453 * q^32 + 936 * q^34 - 147 * q^35 + 319 * q^37 - 89 * q^38 + 624 * q^40 - 1190 * q^41 + 88 * q^43 - 231 * q^44 + 1000 * q^46 - 377 * q^47 + 245 * q^49 + 644 * q^50 + 1001 * q^52 + 992 * q^53 + 231 * q^55 + 294 * q^56 + 721 * q^58 - 71 * q^59 - 574 * q^61 - 272 * q^62 - 1380 * q^64 - 589 * q^65 - 527 * q^67 + 2974 * q^68 - 161 * q^70 + 1156 * q^71 + 1061 * q^73 + 1609 * q^74 + 2399 * q^76 - 385 * q^77 + 588 * q^79 + 1643 * q^80 - 2602 * q^82 + 212 * q^83 + 1918 * q^85 + 4760 * q^86 - 462 * q^88 - 1030 * q^89 + 707 * q^91 + 1174 * q^92 - 1799 * q^94 + 3593 * q^95 + 2488 * q^97 + 49 * q^98

Introduction

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Database

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