LMFDB - Embedded newform 693.4.a.p.1.1

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			1.1
Root			$-4.54345$ 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)

$f(q)$	$=$	$q-4.54345 q^{2} +12.6430 q^{4} -6.53625 q^{5} +7.00000 q^{7} -21.0952 q^{8} +O(q^{10})$	\(q-4.54345 q^{2} +12.6430 q^{4} -6.53625 q^{5} +7.00000 q^{7} -21.0952 q^{8} +29.6972 q^{10} -11.0000 q^{11} +71.3370 q^{13} -31.8042 q^{14} -5.29887 q^{16} -2.45049 q^{17} +80.0091 q^{19} -82.6377 q^{20} +49.9780 q^{22} -61.8164 q^{23} -82.2774 q^{25} -324.117 q^{26} +88.5009 q^{28} +156.839 q^{29} +77.7131 q^{31} +192.837 q^{32} +11.1337 q^{34} -45.7538 q^{35} +84.7185 q^{37} -363.518 q^{38} +137.883 q^{40} -28.8114 q^{41} -352.389 q^{43} -139.073 q^{44} +280.860 q^{46} +256.310 q^{47} +49.0000 q^{49} +373.824 q^{50} +901.913 q^{52} -492.147 q^{53} +71.8988 q^{55} -147.666 q^{56} -712.590 q^{58} +3.13000 q^{59} -159.772 q^{61} -353.086 q^{62} -833.753 q^{64} -466.277 q^{65} -521.200 q^{67} -30.9815 q^{68} +207.880 q^{70} +885.588 q^{71} -375.499 q^{73} -384.914 q^{74} +1011.55 q^{76} -77.0000 q^{77} -1346.03 q^{79} +34.6348 q^{80} +130.903 q^{82} +1379.43 q^{83} +16.0170 q^{85} +1601.07 q^{86} +232.047 q^{88} +111.045 q^{89} +499.359 q^{91} -781.543 q^{92} -1164.53 q^{94} -522.959 q^{95} +472.385 q^{97} -222.629 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.54345	−1.60635	−0.803177	−	0.595741i	$-0.796859\pi$
$2$	−4.54345	−1.60635	−0.803177	+	0.595741i	$0.796859\pi$
$3$	0	0
$3$	0	0
$4$	12.6430	1.58037
$5$	−6.53625	−0.584620	−0.292310	−	0.956324i	$-0.594424\pi$
$5$	−6.53625	−0.584620	−0.292310	+	0.956324i	$0.594424\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−21.0952	−0.932284
$9$	0	0
$10$	29.6972	0.939107
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	71.3370	1.52195	0.760974	−	0.648782i	$-0.224721\pi$
$13$	71.3370	1.52195	0.760974	+	0.648782i	$0.224721\pi$
$14$	−31.8042	−0.607145
$15$	0	0
$16$	−5.29887	−0.0827949
$17$	−2.45049	−0.0349607	−0.0174803	−	0.999847i	$-0.505564\pi$
$17$	−2.45049	−0.0349607	−0.0174803	+	0.999847i	$0.505564\pi$
$18$	0	0
$19$	80.0091	0.966071	0.483035	−	0.875601i	$-0.339534\pi$
$19$	80.0091	0.966071	0.483035	+	0.875601i	$0.339534\pi$
$20$	−82.6377	−0.923917
$21$	0	0
$22$	49.9780	0.484334
$23$	−61.8164	−0.560418	−0.280209	−	0.959939i	$-0.590404\pi$
$23$	−61.8164	−0.560418	−0.280209	+	0.959939i	$0.590404\pi$
$24$	0	0
$25$	−82.2774	−0.658219
$26$	−324.117	−2.44479
$27$	0	0
$28$	88.5009	0.597325
$29$	156.839	1.00428	0.502142	−	0.864785i	$-0.332546\pi$
$29$	156.839	1.00428	0.502142	+	0.864785i	$0.332546\pi$
$30$	0	0
$31$	77.7131	0.450248	0.225124	−	0.974330i	$-0.427721\pi$
$31$	77.7131	0.450248	0.225124	+	0.974330i	$0.427721\pi$
$32$	192.837	1.06528
$33$	0	0
$34$	11.1337	0.0561592
$35$	−45.7538	−0.220966
$36$	0	0
$37$	84.7185	0.376422	0.188211	−	0.982129i	$-0.439731\pi$
$37$	84.7185	0.376422	0.188211	+	0.982129i	$0.439731\pi$
$38$	−363.518	−1.55185
$39$	0	0
$40$	137.883	0.545032
$41$	−28.8114	−0.109746	−0.0548729	−	0.998493i	$-0.517475\pi$
$41$	−28.8114	−0.109746	−0.0548729	+	0.998493i	$0.517475\pi$
$42$	0	0
$43$	−352.389	−1.24974	−0.624871	−	0.780728i	$-0.714848\pi$
$43$	−352.389	−1.24974	−0.624871	+	0.780728i	$0.714848\pi$
$44$	−139.073	−0.476500
$45$	0	0
$46$	280.860	0.900229
$47$	256.310	0.795459	0.397730	−	0.917503i	$-0.369798\pi$
$47$	256.310	0.795459	0.397730	+	0.917503i	$0.369798\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	373.824	1.05733
$51$	0	0
$52$	901.913	2.40525
$53$	−492.147	−1.27550	−0.637751	−	0.770243i	$-0.720135\pi$
$53$	−492.147	−1.27550	−0.637751	+	0.770243i	$0.720135\pi$
$54$	0	0
$55$	71.8988	0.176270
$56$	−147.666	−0.352370
$57$	0	0
$58$	−712.590	−1.61323
$59$	3.13000	0.00690663	0.00345331	−	0.999994i	$-0.498901\pi$
$59$	3.13000	0.00690663	0.00345331	+	0.999994i	$0.498901\pi$
$60$	0	0
$61$	−159.772	−0.335356	−0.167678	−	0.985842i	$-0.553627\pi$
$61$	−159.772	−0.335356	−0.167678	+	0.985842i	$0.553627\pi$
$62$	−353.086	−0.723257
$63$	0	0
$64$	−833.753	−1.62842
$65$	−466.277	−0.889761
$66$	0	0
$67$	−521.200	−0.950370	−0.475185	−	0.879886i	$-0.657619\pi$
$67$	−521.200	−0.950370	−0.475185	+	0.879886i	$0.657619\pi$
$68$	−30.9815	−0.0552509
$69$	0	0
$70$	207.880	0.354949
$71$	885.588	1.48028	0.740141	−	0.672452i	$-0.234759\pi$
$71$	885.588	1.48028	0.740141	+	0.672452i	$0.234759\pi$
$72$	0	0
$73$	−375.499	−0.602039	−0.301020	−	0.953618i	$-0.597327\pi$
$73$	−375.499	−0.602039	−0.301020	+	0.953618i	$0.597327\pi$
$74$	−384.914	−0.604668
$75$	0	0
$76$	1011.55	1.52675
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−1346.03	−1.91697	−0.958484	−	0.285145i	$-0.907958\pi$
$79$	−1346.03	−1.91697	−0.958484	+	0.285145i	$0.907958\pi$
$80$	34.6348	0.0484036
$81$	0	0
$82$	130.903	0.176291
$83$	1379.43	1.82424	0.912121	−	0.409920i	$-0.134443\pi$
$83$	1379.43	1.82424	0.912121	+	0.409920i	$0.134443\pi$
$84$	0	0
$85$	16.0170	0.0204387
$86$	1601.07	2.00753
$87$	0	0
$88$	232.047	0.281094
$89$	111.045	0.132255	0.0661276	−	0.997811i	$-0.478936\pi$
$89$	111.045	0.132255	0.0661276	+	0.997811i	$0.478936\pi$
$90$	0	0
$91$	499.359	0.575242
$92$	−781.543	−0.885669
$93$	0	0
$94$	−1164.53	−1.27779
$95$	−522.959	−0.564784
$96$	0	0
$97$	472.385	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$97$	472.385	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$98$	−222.629	−0.229479
$99$	0	0
$100$	−1040.23	−1.04023
$101$	−1046.51	−1.03101	−0.515505	−	0.856887i	$-0.672396\pi$
$101$	−1046.51	−1.03101	−0.515505	+	0.856887i	$0.672396\pi$
$102$	0	0
$103$	1116.84	1.06840	0.534200	−	0.845358i	$-0.320613\pi$
$103$	1116.84	1.06840	0.534200	+	0.845358i	$0.320613\pi$
$104$	−1504.87	−1.41889
$105$	0	0
$106$	2236.05	2.04891
$107$	−1484.22	−1.34098	−0.670492	−	0.741917i	$-0.733917\pi$
$107$	−1484.22	−1.34098	−0.670492	+	0.741917i	$0.733917\pi$
$108$	0	0
$109$	1064.65	0.935547	0.467774	−	0.883848i	$-0.345056\pi$
$109$	1064.65	0.935547	0.467774	+	0.883848i	$0.345056\pi$
$110$	−326.669	−0.283151
$111$	0	0
$112$	−37.0921	−0.0312935
$113$	438.172	0.364776	0.182388	−	0.983227i	$-0.441617\pi$
$113$	438.172	0.364776	0.182388	+	0.983227i	$0.441617\pi$
$114$	0	0
$115$	404.047	0.327631
$116$	1982.91	1.58714
$117$	0	0
$118$	−14.2210	−0.0110945
$119$	−17.1534	−0.0132139
$120$	0	0
$121$	121.000	0.0909091
$122$	725.917	0.538700
$123$	0	0
$124$	982.525	0.711559
$125$	1354.82	0.969428
$126$	0	0
$127$	1926.01	1.34571	0.672856	−	0.739774i	$-0.265068\pi$
$127$	1926.01	1.34571	0.672856	+	0.739774i	$0.265068\pi$
$128$	2245.43	1.55054
$129$	0	0
$130$	2118.51	1.42927
$131$	703.646	0.469296	0.234648	−	0.972080i	$-0.424606\pi$
$131$	703.646	0.469296	0.234648	+	0.972080i	$0.424606\pi$
$132$	0	0
$133$	560.064	0.365140
$134$	2368.05	1.52663
$135$	0	0
$136$	51.6935	0.0325933
$137$	944.945	0.589285	0.294643	−	0.955608i	$-0.404799\pi$
$137$	944.945	0.589285	0.294643	+	0.955608i	$0.404799\pi$
$138$	0	0
$139$	1222.86	0.746200	0.373100	−	0.927791i	$-0.378295\pi$
$139$	1222.86	0.746200	0.373100	+	0.927791i	$0.378295\pi$
$140$	−578.464	−0.349208
$141$	0	0
$142$	−4023.63	−2.37786
$143$	−784.707	−0.458885
$144$	0	0
$145$	−1025.14	−0.587124
$146$	1706.06	0.967088
$147$	0	0
$148$	1071.09	0.594888
$149$	2723.95	1.49768	0.748840	−	0.662750i	$-0.230611\pi$
$149$	2723.95	1.49768	0.748840	+	0.662750i	$0.230611\pi$
$150$	0	0
$151$	1133.68	0.610980	0.305490	−	0.952195i	$-0.401180\pi$
$151$	1133.68	0.610980	0.305490	+	0.952195i	$0.401180\pi$
$152$	−1687.81	−0.900652
$153$	0	0
$154$	349.846	0.183061
$155$	−507.952	−0.263224
$156$	0	0
$157$	3449.91	1.75371	0.876856	−	0.480753i	$-0.159636\pi$
$157$	3449.91	1.75371	0.876856	+	0.480753i	$0.159636\pi$
$158$	6115.64	3.07933
$159$	0	0
$160$	−1260.43	−0.622785
$161$	−432.715	−0.211818
$162$	0	0
$163$	917.605	0.440935	0.220467	−	0.975394i	$-0.429242\pi$
$163$	917.605	0.440935	0.220467	+	0.975394i	$0.429242\pi$
$164$	−364.261	−0.173439
$165$	0	0
$166$	−6267.38	−2.93038
$167$	3405.63	1.57806	0.789029	−	0.614355i	$-0.210584\pi$
$167$	3405.63	1.57806	0.789029	+	0.614355i	$0.210584\pi$
$168$	0	0
$169$	2891.97	1.31633
$170$	−72.7726	−0.0328318
$171$	0	0
$172$	−4455.25	−1.97506
$173$	−1874.17	−0.823644	−0.411822	−	0.911264i	$-0.635108\pi$
$173$	−1874.17	−0.823644	−0.411822	+	0.911264i	$0.635108\pi$
$174$	0	0
$175$	−575.942	−0.248784
$176$	58.2876	0.0249636
$177$	0	0
$178$	−504.527	−0.212449
$179$	546.539	0.228214	0.114107	−	0.993468i	$-0.463599\pi$
$179$	546.539	0.228214	0.114107	+	0.993468i	$0.463599\pi$
$180$	0	0
$181$	2927.32	1.20213	0.601066	−	0.799200i	$-0.294743\pi$
$181$	2927.32	1.20213	0.601066	+	0.799200i	$0.294743\pi$
$182$	−2268.82	−0.924043
$183$	0	0
$184$	1304.03	0.522468
$185$	−553.741	−0.220064
$186$	0	0
$187$	26.9554	0.0105410
$188$	3240.52	1.25712
$189$	0	0
$190$	2376.04	0.907243
$191$	2232.58	0.845780	0.422890	−	0.906181i	$-0.361016\pi$
$191$	2232.58	0.845780	0.422890	+	0.906181i	$0.361016\pi$
$192$	0	0
$193$	1440.00	0.537065	0.268533	−	0.963271i	$-0.413461\pi$
$193$	1440.00	0.537065	0.268533	+	0.963271i	$0.413461\pi$
$194$	−2146.26	−0.794291
$195$	0	0
$196$	619.506	0.225768
$197$	−1026.53	−0.371256	−0.185628	−	0.982620i	$-0.559432\pi$
$197$	−1026.53	−0.371256	−0.185628	+	0.982620i	$0.559432\pi$
$198$	0	0
$199$	1273.27	0.453565	0.226783	−	0.973945i	$-0.427179\pi$
$199$	1273.27	0.453565	0.226783	+	0.973945i	$0.427179\pi$
$200$	1735.66	0.613647
$201$	0	0
$202$	4754.79	1.65617
$203$	1097.87	0.379584
$204$	0	0
$205$	188.318	0.0641596
$206$	−5074.29	−1.71623
$207$	0	0
$208$	−378.006	−0.126010
$209$	−880.100	−0.291281
$210$	0	0
$211$	−1087.89	−0.354944	−0.177472	−	0.984126i	$-0.556792\pi$
$211$	−1087.89	−0.354944	−0.177472	+	0.984126i	$0.556792\pi$
$212$	−6222.21	−2.01577
$213$	0	0
$214$	6743.50	2.15409
$215$	2303.31	0.730624
$216$	0	0
$217$	543.992	0.170178
$218$	−4837.17	−1.50282
$219$	0	0
$220$	909.015	0.278572
$221$	−174.811	−0.0532083
$222$	0	0
$223$	−1826.52	−0.548487	−0.274244	−	0.961660i	$-0.588427\pi$
$223$	−1826.52	−0.548487	−0.274244	+	0.961660i	$0.588427\pi$
$224$	1349.86	0.402639
$225$	0	0
$226$	−1990.81	−0.585960
$227$	−1821.81	−0.532678	−0.266339	−	0.963879i	$-0.585814\pi$
$227$	−1821.81	−0.532678	−0.266339	+	0.963879i	$0.585814\pi$
$228$	0	0
$229$	−1369.86	−0.395296	−0.197648	−	0.980273i	$-0.563330\pi$
$229$	−1369.86	−0.395296	−0.197648	+	0.980273i	$0.563330\pi$
$230$	−1835.77	−0.526292
$231$	0	0
$232$	−3308.54	−0.936277
$233$	2116.08	0.594975	0.297488	−	0.954726i	$-0.403851\pi$
$233$	2116.08	0.594975	0.297488	+	0.954726i	$0.403851\pi$
$234$	0	0
$235$	−1675.30	−0.465041
$236$	39.5725	0.0109150
$237$	0	0
$238$	77.9358	0.0212262
$239$	−5148.64	−1.39346	−0.696732	−	0.717331i	$-0.745364\pi$
$239$	−5148.64	−1.39346	−0.696732	+	0.717331i	$0.745364\pi$
$240$	0	0
$241$	−3756.11	−1.00395	−0.501975	−	0.864882i	$-0.667393\pi$
$241$	−3756.11	−1.00395	−0.501975	+	0.864882i	$0.667393\pi$
$242$	−549.758	−0.146032
$243$	0	0
$244$	−2019.99	−0.529987
$245$	−320.276	−0.0835171
$246$	0	0
$247$	5707.61	1.47031
$248$	−1639.37	−0.419759
$249$	0	0
$250$	−6155.55	−1.55724
$251$	1906.43	0.479413	0.239707	−	0.970845i	$-0.422949\pi$
$251$	1906.43	0.479413	0.239707	+	0.970845i	$0.422949\pi$
$252$	0	0
$253$	679.980	0.168972
$254$	−8750.72	−2.16169
$255$	0	0
$256$	−3531.97	−0.862298
$257$	143.978	0.0349459	0.0174729	−	0.999847i	$-0.494438\pi$
$257$	143.978	0.0349459	0.0174729	+	0.999847i	$0.494438\pi$
$258$	0	0
$259$	593.029	0.142274
$260$	−5895.13	−1.40615
$261$	0	0
$262$	−3196.98	−0.753856
$263$	6827.81	1.60084	0.800420	−	0.599440i	$-0.204610\pi$
$263$	6827.81	1.60084	0.800420	+	0.599440i	$0.204610\pi$
$264$	0	0
$265$	3216.80	0.745684
$266$	−2544.62	−0.586545
$267$	0	0
$268$	−6589.53	−1.50194
$269$	−8049.95	−1.82459	−0.912294	−	0.409537i	$-0.865690\pi$
$269$	−8049.95	−1.82459	−0.912294	+	0.409537i	$0.865690\pi$
$270$	0	0
$271$	1420.76	0.318469	0.159234	−	0.987241i	$-0.449097\pi$
$271$	1420.76	0.318469	0.159234	+	0.987241i	$0.449097\pi$
$272$	12.9848	0.00289456
$273$	0	0
$274$	−4293.31	−0.946601
$275$	905.052	0.198461
$276$	0	0
$277$	8044.20	1.74487	0.872435	−	0.488730i	$-0.162540\pi$
$277$	8044.20	1.74487	0.872435	+	0.488730i	$0.162540\pi$
$278$	−5556.02	−1.19866
$279$	0	0
$280$	965.183	0.206003
$281$	−3089.28	−0.655839	−0.327919	−	0.944706i	$-0.606347\pi$
$281$	−3089.28	−0.655839	−0.327919	+	0.944706i	$0.606347\pi$
$282$	0	0
$283$	−8123.26	−1.70628	−0.853141	−	0.521681i	$-0.825305\pi$
$283$	−8123.26	−1.70628	−0.853141	+	0.521681i	$0.825305\pi$
$284$	11196.5	2.33940
$285$	0	0
$286$	3565.28	0.737131
$287$	−201.680	−0.0414800
$288$	0	0
$289$	−4907.00	−0.998778
$290$	4657.66	0.943129
$291$	0	0
$292$	−4747.43	−0.951447
$293$	8383.65	1.67160	0.835799	−	0.549036i	$-0.185005\pi$
$293$	8383.65	1.67160	0.835799	+	0.549036i	$0.185005\pi$
$294$	0	0
$295$	−20.4584	−0.00403775
$296$	−1787.15	−0.350933
$297$	0	0
$298$	−12376.1	−2.40581
$299$	−4409.80	−0.852927
$300$	0	0
$301$	−2466.73	−0.472358
$302$	−5150.84	−0.981450
$303$	0	0
$304$	−423.958	−0.0799857
$305$	1044.31	0.196056
$306$	0	0
$307$	−6514.20	−1.21103	−0.605513	−	0.795835i	$-0.707032\pi$
$307$	−6514.20	−1.21103	−0.605513	+	0.795835i	$0.707032\pi$
$308$	−973.510	−0.180100
$309$	0	0
$310$	2307.86	0.422831
$311$	3822.91	0.697034	0.348517	−	0.937303i	$-0.386685\pi$
$311$	3822.91	0.697034	0.348517	+	0.937303i	$0.386685\pi$
$312$	0	0
$313$	1239.73	0.223878	0.111939	−	0.993715i	$-0.464294\pi$
$313$	1239.73	0.223878	0.111939	+	0.993715i	$0.464294\pi$
$314$	−15674.5	−2.81708
$315$	0	0
$316$	−17017.9	−3.02953
$317$	−5377.28	−0.952739	−0.476369	−	0.879245i	$-0.658047\pi$
$317$	−5377.28	−0.952739	−0.476369	+	0.879245i	$0.658047\pi$
$318$	0	0
$319$	−1725.23	−0.302803
$320$	5449.62	0.952010
$321$	0	0
$322$	1966.02	0.340255
$323$	−196.061	−0.0337745
$324$	0	0
$325$	−5869.43	−1.00178
$326$	−4169.10	−0.708297
$327$	0	0
$328$	607.781	0.102314
$329$	1794.17	0.300655
$330$	0	0
$331$	2321.44	0.385493	0.192746	−	0.981249i	$-0.438261\pi$
$331$	2321.44	0.385493	0.192746	+	0.981249i	$0.438261\pi$
$332$	17440.1	2.88298
$333$	0	0
$334$	−15473.3	−2.53492
$335$	3406.70	0.555605
$336$	0	0
$337$	7265.72	1.17445	0.587224	−	0.809425i	$-0.300221\pi$
$337$	7265.72	1.17445	0.587224	+	0.809425i	$0.300221\pi$
$338$	−13139.5	−2.11449
$339$	0	0
$340$	202.503	0.0323008
$341$	−854.844	−0.135755
$342$	0	0
$343$	343.000	0.0539949
$344$	7433.72	1.16511
$345$	0	0
$346$	8515.21	1.32306
$347$	8322.54	1.28754	0.643771	−	0.765218i	$-0.277369\pi$
$347$	8322.54	1.28754	0.643771	+	0.765218i	$0.277369\pi$
$348$	0	0
$349$	4604.41	0.706213	0.353107	−	0.935583i	$-0.385125\pi$
$349$	4604.41	0.706213	0.353107	+	0.935583i	$0.385125\pi$
$350$	2616.77	0.399634
$351$	0	0
$352$	−2121.20	−0.321195
$353$	3257.91	0.491221	0.245611	−	0.969369i	$-0.421012\pi$
$353$	3257.91	0.491221	0.245611	+	0.969369i	$0.421012\pi$
$354$	0	0
$355$	−5788.43	−0.865402
$356$	1403.94	0.209013
$357$	0	0
$358$	−2483.18	−0.366592
$359$	−11269.4	−1.65676	−0.828380	−	0.560167i	$-0.810737\pi$
$359$	−11269.4	−1.65676	−0.828380	+	0.560167i	$0.810737\pi$
$360$	0	0
$361$	−457.546	−0.0667075
$362$	−13300.1	−1.93105
$363$	0	0
$364$	6313.39	0.909097
$365$	2454.36	0.351964
$366$	0	0
$367$	9729.11	1.38380	0.691901	−	0.721992i	$-0.256773\pi$
$367$	9729.11	1.38380	0.691901	+	0.721992i	$0.256773\pi$
$368$	327.557	0.0463997
$369$	0	0
$370$	2515.90	0.353501
$371$	−3445.03	−0.482094
$372$	0	0
$373$	6722.55	0.933191	0.466596	−	0.884471i	$-0.345480\pi$
$373$	6722.55	0.933191	0.466596	+	0.884471i	$0.345480\pi$
$374$	−122.471	−0.0169326
$375$	0	0
$376$	−5406.89	−0.741594
$377$	11188.4	1.52847
$378$	0	0
$379$	−5966.99	−0.808717	−0.404358	−	0.914601i	$-0.632505\pi$
$379$	−5966.99	−0.808717	−0.404358	+	0.914601i	$0.632505\pi$
$380$	−6611.77	−0.892570
$381$	0	0
$382$	−10143.6	−1.35862
$383$	10029.0	1.33801	0.669005	−	0.743258i	$-0.266721\pi$
$383$	10029.0	1.33801	0.669005	+	0.743258i	$0.266721\pi$
$384$	0	0
$385$	503.291	0.0666236
$386$	−6542.58	−0.862717
$387$	0	0
$388$	5972.35	0.781444
$389$	13234.3	1.72495	0.862477	−	0.506096i	$-0.168912\pi$
$389$	13234.3	1.72495	0.862477	+	0.506096i	$0.168912\pi$
$390$	0	0
$391$	151.480	0.0195926
$392$	−1033.66	−0.133183
$393$	0	0
$394$	4664.00	0.596368
$395$	8798.01	1.12070
$396$	0	0
$397$	−9951.99	−1.25813	−0.629063	−	0.777354i	$-0.716561\pi$
$397$	−9951.99	−1.25813	−0.629063	+	0.777354i	$0.716561\pi$
$398$	−5785.03	−0.728586
$399$	0	0
$400$	435.978	0.0544972
$401$	4709.60	0.586499	0.293250	−	0.956036i	$-0.405263\pi$
$401$	4709.60	0.586499	0.293250	+	0.956036i	$0.405263\pi$
$402$	0	0
$403$	5543.82	0.685254
$404$	−13231.1	−1.62938
$405$	0	0
$406$	−4988.13	−0.609745
$407$	−931.903	−0.113496
$408$	0	0
$409$	15044.8	1.81887	0.909434	−	0.415848i	$-0.136515\pi$
$409$	15044.8	1.81887	0.909434	+	0.415848i	$0.136515\pi$
$410$	−855.615	−0.103063
$411$	0	0
$412$	14120.1	1.68847
$413$	21.9100	0.00261046
$414$	0	0
$415$	−9016.30	−1.06649
$416$	13756.4	1.62130
$417$	0	0
$418$	3998.69	0.467901
$419$	−5987.57	−0.698119	−0.349059	−	0.937101i	$-0.613499\pi$
$419$	−5987.57	−0.698119	−0.349059	+	0.937101i	$0.613499\pi$
$420$	0	0
$421$	−6743.22	−0.780628	−0.390314	−	0.920682i	$-0.627634\pi$
$421$	−6743.22	−0.780628	−0.390314	+	0.920682i	$0.627634\pi$
$422$	4942.76	0.570166
$423$	0	0
$424$	10381.9	1.18913
$425$	201.620	0.0230118
$426$	0	0
$427$	−1118.40	−0.126753
$428$	−18765.0	−2.11925
$429$	0	0
$430$	−10465.0	−1.17364
$431$	−1953.86	−0.218363	−0.109181	−	0.994022i	$-0.534823\pi$
$431$	−1953.86	−0.218363	−0.109181	+	0.994022i	$0.534823\pi$
$432$	0	0
$433$	11962.7	1.32769	0.663844	−	0.747871i	$-0.268924\pi$
$433$	11962.7	1.32769	0.663844	+	0.747871i	$0.268924\pi$
$434$	−2471.60	−0.273366
$435$	0	0
$436$	13460.3	1.47851
$437$	−4945.87	−0.541403
$438$	0	0
$439$	5515.13	0.599597	0.299798	−	0.954003i	$-0.403081\pi$
$439$	5515.13	0.599597	0.299798	+	0.954003i	$0.403081\pi$
$440$	−1516.72	−0.164333
$441$	0	0
$442$	794.244	0.0854714
$443$	13131.2	1.40831	0.704156	−	0.710046i	$-0.251326\pi$
$443$	13131.2	1.40831	0.704156	+	0.710046i	$0.251326\pi$
$444$	0	0
$445$	−725.816	−0.0773191
$446$	8298.70	0.881064
$447$	0	0
$448$	−5836.27	−0.615487
$449$	−5101.57	−0.536209	−0.268104	−	0.963390i	$-0.586397\pi$
$449$	−5101.57	−0.536209	−0.268104	+	0.963390i	$0.586397\pi$
$450$	0	0
$451$	316.925	0.0330896
$452$	5539.80	0.576483
$453$	0	0
$454$	8277.31	0.855669
$455$	−3263.94	−0.336298
$456$	0	0
$457$	−1671.55	−0.171098	−0.0855490	−	0.996334i	$-0.527264\pi$
$457$	−1671.55	−0.171098	−0.0855490	+	0.996334i	$0.527264\pi$
$458$	6223.89	0.634985
$459$	0	0
$460$	5108.36	0.517780
$461$	3298.51	0.333247	0.166623	−	0.986021i	$-0.446714\pi$
$461$	3298.51	0.333247	0.166623	+	0.986021i	$0.446714\pi$
$462$	0	0
$463$	7364.93	0.739260	0.369630	−	0.929179i	$-0.379484\pi$
$463$	7364.93	0.739260	0.369630	+	0.929179i	$0.379484\pi$
$464$	−831.069	−0.0831496
$465$	0	0
$466$	−9614.33	−0.955740
$467$	9694.12	0.960579	0.480290	−	0.877110i	$-0.340532\pi$
$467$	9694.12	0.960579	0.480290	+	0.877110i	$0.340532\pi$
$468$	0	0
$469$	−3648.40	−0.359206
$470$	7611.66	0.747021
$471$	0	0
$472$	−66.0279	−0.00643894
$473$	3876.28	0.376811
$474$	0	0
$475$	−6582.94	−0.635887
$476$	−216.871	−0.0208829
$477$	0	0
$478$	23392.6	2.23840
$479$	−11673.5	−1.11352	−0.556760	−	0.830673i	$-0.687956\pi$
$479$	−11673.5	−1.11352	−0.556760	+	0.830673i	$0.687956\pi$
$480$	0	0
$481$	6043.56	0.572895
$482$	17065.7	1.61270
$483$	0	0
$484$	1529.80	0.143670
$485$	−3087.63	−0.289076
$486$	0	0
$487$	−5032.39	−0.468253	−0.234127	−	0.972206i	$-0.575223\pi$
$487$	−5032.39	−0.468253	−0.234127	+	0.972206i	$0.575223\pi$
$488$	3370.42	0.312647
$489$	0	0
$490$	1455.16	0.134158
$491$	−10559.5	−0.970557	−0.485278	−	0.874360i	$-0.661282\pi$
$491$	−10559.5	−0.970557	−0.485278	+	0.874360i	$0.661282\pi$
$492$	0	0
$493$	−384.332	−0.0351104
$494$	−25932.3	−2.36184
$495$	0	0
$496$	−411.792	−0.0372782
$497$	6199.12	0.559494
$498$	0	0
$499$	−7367.48	−0.660949	−0.330475	−	0.943815i	$-0.607209\pi$
$499$	−7367.48	−0.660949	−0.330475	+	0.943815i	$0.607209\pi$
$500$	17128.9	1.53206
$501$	0	0
$502$	−8661.78	−0.770107
$503$	−4656.82	−0.412798	−0.206399	−	0.978468i	$-0.566174\pi$
$503$	−4656.82	−0.412798	−0.206399	+	0.978468i	$0.566174\pi$
$504$	0	0
$505$	6840.28	0.602749
$506$	−3089.46	−0.271429
$507$	0	0
$508$	24350.4	2.12673
$509$	2210.78	0.192517	0.0962583	−	0.995356i	$-0.469313\pi$
$509$	2210.78	0.192517	0.0962583	+	0.995356i	$0.469313\pi$
$510$	0	0
$511$	−2628.50	−0.227550
$512$	−1916.06	−0.165388
$513$	0	0
$514$	−654.157	−0.0561354
$515$	−7299.92	−0.624608
$516$	0	0
$517$	−2819.40	−0.239840
$518$	−2694.40	−0.228543
$519$	0	0
$520$	9836.19	0.829510
$521$	−18348.3	−1.54291	−0.771453	−	0.636286i	$-0.780470\pi$
$521$	−18348.3	−1.54291	−0.771453	+	0.636286i	$0.780470\pi$
$522$	0	0
$523$	−15196.1	−1.27051	−0.635256	−	0.772301i	$-0.719106\pi$
$523$	−15196.1	−1.27051	−0.635256	+	0.772301i	$0.719106\pi$
$524$	8896.18	0.741663
$525$	0	0
$526$	−31021.8	−2.57151
$527$	−190.435	−0.0157410
$528$	0	0
$529$	−8345.74	−0.685932
$530$	−14615.4	−1.19783
$531$	0	0
$532$	7080.87	0.577058
$533$	−2055.32	−0.167027
$534$	0	0
$535$	9701.25	0.783966
$536$	10994.8	0.886014
$537$	0	0
$538$	36574.6	2.93093
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−5051.03	−0.401406	−0.200703	−	0.979652i	$-0.564323\pi$
$541$	−5051.03	−0.401406	−0.200703	+	0.979652i	$0.564323\pi$
$542$	−6455.16	−0.511573
$543$	0	0
$544$	−472.544	−0.0372430
$545$	−6958.80	−0.546940
$546$	0	0
$547$	−14873.9	−1.16263	−0.581317	−	0.813677i	$-0.697462\pi$
$547$	−14873.9	−1.16263	−0.581317	+	0.813677i	$0.697462\pi$
$548$	11946.9	0.931290
$549$	0	0
$550$	−4112.06	−0.318798
$551$	12548.5	0.970209
$552$	0	0
$553$	−9422.23	−0.724546
$554$	−36548.4	−2.80288
$555$	0	0
$556$	15460.6	1.17927
$557$	−20254.8	−1.54079	−0.770396	−	0.637565i	$-0.779942\pi$
$557$	−20254.8	−1.54079	−0.770396	+	0.637565i	$0.779942\pi$
$558$	0	0
$559$	−25138.4	−1.90204
$560$	242.443	0.0182948
$561$	0	0
$562$	14036.0	1.05351
$563$	2659.45	0.199081	0.0995404	−	0.995034i	$-0.468263\pi$
$563$	2659.45	0.199081	0.0995404	+	0.995034i	$0.468263\pi$
$564$	0	0
$565$	−2864.00	−0.213256
$566$	36907.7	2.74089
$567$	0	0
$568$	−18681.6	−1.38004
$569$	14712.3	1.08395	0.541977	−	0.840393i	$-0.317676\pi$
$569$	14712.3	1.08395	0.541977	+	0.840393i	$0.317676\pi$
$570$	0	0
$571$	26158.2	1.91714	0.958570	−	0.284856i	$-0.0919456\pi$
$571$	26158.2	1.91714	0.958570	+	0.284856i	$0.0919456\pi$
$572$	−9921.04	−0.725209
$573$	0	0
$574$	916.322	0.0666316
$575$	5086.09	0.368878
$576$	0	0
$577$	24028.7	1.73367	0.866835	−	0.498595i	$-0.166151\pi$
$577$	24028.7	1.73367	0.866835	+	0.498595i	$0.166151\pi$
$578$	22294.7	1.60439
$579$	0	0
$580$	−12960.8	−0.927875
$581$	9656.01	0.689499
$582$	0	0
$583$	5413.62	0.384578
$584$	7921.23	0.561272
$585$	0	0
$586$	−38090.7	−2.68518
$587$	18554.9	1.30468	0.652338	−	0.757928i	$-0.273788\pi$
$587$	18554.9	1.30468	0.652338	+	0.757928i	$0.273788\pi$
$588$	0	0
$589$	6217.75	0.434971
$590$	92.9520	0.00648606
$591$	0	0
$592$	−448.913	−0.0311659
$593$	−24535.9	−1.69911	−0.849553	−	0.527504i	$-0.823128\pi$
$593$	−24535.9	−1.69911	−0.849553	+	0.527504i	$0.823128\pi$
$594$	0	0
$595$	112.119	0.00772510
$596$	34438.8	2.36689
$597$	0	0
$598$	20035.7	1.37010
$599$	88.4090	0.00603054	0.00301527	−	0.999995i	$-0.499040\pi$
$599$	88.4090	0.00603054	0.00301527	+	0.999995i	$0.499040\pi$
$600$	0	0
$601$	−26290.0	−1.78434	−0.892172	−	0.451695i	$-0.850819\pi$
$601$	−26290.0	−1.78434	−0.892172	+	0.451695i	$0.850819\pi$
$602$	11207.5	0.758774
$603$	0	0
$604$	14333.2	0.965576
$605$	−790.886	−0.0531473
$606$	0	0
$607$	28411.4	1.89981	0.949905	−	0.312540i	$-0.101180\pi$
$607$	28411.4	1.89981	0.949905	+	0.312540i	$0.101180\pi$
$608$	15428.7	1.02914
$609$	0	0
$610$	−4744.77	−0.314935
$611$	18284.4	1.21065
$612$	0	0
$613$	−3295.49	−0.217135	−0.108567	−	0.994089i	$-0.534626\pi$
$613$	−3295.49	−0.217135	−0.108567	+	0.994089i	$0.534626\pi$
$614$	29597.0	1.94534
$615$	0	0
$616$	1624.33	0.106244
$617$	3409.29	0.222452	0.111226	−	0.993795i	$-0.464522\pi$
$617$	3409.29	0.222452	0.111226	+	0.993795i	$0.464522\pi$
$618$	0	0
$619$	−10865.7	−0.705542	−0.352771	−	0.935710i	$-0.614760\pi$
$619$	−10865.7	−0.705542	−0.352771	+	0.935710i	$0.614760\pi$
$620$	−6422.03	−0.415992
$621$	0	0
$622$	−17369.2	−1.11968
$623$	777.313	0.0499878
$624$	0	0
$625$	1429.26	0.0914724
$626$	−5632.66	−0.359627
$627$	0	0
$628$	43617.2	2.77152
$629$	−207.602	−0.0131600
$630$	0	0
$631$	23851.7	1.50479	0.752394	−	0.658713i	$-0.228899\pi$
$631$	23851.7	1.50479	0.752394	+	0.658713i	$0.228899\pi$
$632$	28394.8	1.78716
$633$	0	0
$634$	24431.4	1.53044
$635$	−12588.9	−0.786730
$636$	0	0
$637$	3495.51	0.217421
$638$	7838.49	0.486409
$639$	0	0
$640$	−14676.7	−0.906479
$641$	−13710.2	−0.844809	−0.422404	−	0.906408i	$-0.638814\pi$
$641$	−13710.2	−0.844809	−0.422404	+	0.906408i	$0.638814\pi$
$642$	0	0
$643$	4879.27	0.299253	0.149627	−	0.988743i	$-0.452193\pi$
$643$	4879.27	0.299253	0.149627	+	0.988743i	$0.452193\pi$
$644$	−5470.80	−0.334751
$645$	0	0
$646$	890.796	0.0542537
$647$	−13856.7	−0.841982	−0.420991	−	0.907065i	$-0.638318\pi$
$647$	−13856.7	−0.841982	−0.420991	+	0.907065i	$0.638318\pi$
$648$	0	0
$649$	−34.4300	−0.00208243
$650$	26667.5	1.60921
$651$	0	0
$652$	11601.3	0.696841
$653$	−9861.70	−0.590993	−0.295496	−	0.955344i	$-0.595485\pi$
$653$	−9861.70	−0.590993	−0.295496	+	0.955344i	$0.595485\pi$
$654$	0	0
$655$	−4599.20	−0.274360
$656$	152.668	0.00908640
$657$	0	0
$658$	−8151.72	−0.482959
$659$	−6849.60	−0.404890	−0.202445	−	0.979294i	$-0.564889\pi$
$659$	−6849.60	−0.404890	−0.202445	+	0.979294i	$0.564889\pi$
$660$	0	0
$661$	24156.9	1.42147	0.710737	−	0.703458i	$-0.248361\pi$
$661$	24156.9	1.42147	0.710737	+	0.703458i	$0.248361\pi$
$662$	−10547.4	−0.619238
$663$	0	0
$664$	−29099.3	−1.70071
$665$	−3660.72	−0.213468
$666$	0	0
$667$	−9695.20	−0.562818
$668$	43057.4	2.49392
$669$	0	0
$670$	−15478.2	−0.892498
$671$	1757.49	0.101114
$672$	0	0
$673$	10039.3	0.575015	0.287508	−	0.957778i	$-0.407173\pi$
$673$	10039.3	0.575015	0.287508	+	0.957778i	$0.407173\pi$
$674$	−33011.5	−1.88658
$675$	0	0
$676$	36563.1	2.08029
$677$	−17849.5	−1.01331	−0.506655	−	0.862149i	$-0.669118\pi$
$677$	−17849.5	−1.01331	−0.506655	+	0.862149i	$0.669118\pi$
$678$	0	0
$679$	3306.69	0.186891
$680$	−337.882	−0.0190547
$681$	0	0
$682$	3883.95	0.218070
$683$	8199.58	0.459368	0.229684	−	0.973265i	$-0.426231\pi$
$683$	8199.58	0.459368	0.229684	+	0.973265i	$0.426231\pi$
$684$	0	0
$685$	−6176.39	−0.344508
$686$	−1558.40	−0.0867350
$687$	0	0
$688$	1867.27	0.103472
$689$	−35108.3	−1.94125
$690$	0	0
$691$	6179.89	0.340223	0.170111	−	0.985425i	$-0.445587\pi$
$691$	6179.89	0.340223	0.170111	+	0.985425i	$0.445587\pi$
$692$	−23695.1	−1.30167
$693$	0	0
$694$	−37813.1	−2.06825
$695$	−7992.94	−0.436244
$696$	0	0
$697$	70.6020	0.00383679
$698$	−20919.9	−1.13443
$699$	0	0
$700$	−7281.62	−0.393171
$701$	13389.9	0.721437	0.360719	−	0.932675i	$-0.382531\pi$
$701$	13389.9	0.721437	0.360719	+	0.932675i	$0.382531\pi$
$702$	0	0
$703$	6778.25	0.363651
$704$	9171.29	0.490988
$705$	0	0
$706$	−14802.2	−0.789075
$707$	−7325.60	−0.389685
$708$	0	0
$709$	−4773.49	−0.252852	−0.126426	−	0.991976i	$-0.540351\pi$
$709$	−4773.49	−0.252852	−0.126426	+	0.991976i	$0.540351\pi$
$710$	26299.4	1.39014
$711$	0	0
$712$	−2342.51	−0.123299
$713$	−4803.94	−0.252327
$714$	0	0
$715$	5129.04	0.268273
$716$	6909.88	0.360663
$717$	0	0
$718$	51202.1	2.66134
$719$	8643.93	0.448351	0.224175	−	0.974549i	$-0.428031\pi$
$719$	8643.93	0.448351	0.224175	+	0.974549i	$0.428031\pi$
$720$	0	0
$721$	7817.85	0.403817
$722$	2078.84	0.107156
$723$	0	0
$724$	37010.0	1.89982
$725$	−12904.3	−0.661039
$726$	0	0
$727$	6255.16	0.319108	0.159554	−	0.987189i	$-0.448994\pi$
$727$	6255.16	0.319108	0.159554	+	0.987189i	$0.448994\pi$
$728$	−10534.1	−0.536289
$729$	0	0
$730$	−11151.3	−0.565379
$731$	863.527	0.0436918
$732$	0	0
$733$	8881.40	0.447534	0.223767	−	0.974643i	$-0.428165\pi$
$733$	8881.40	0.447534	0.223767	+	0.974643i	$0.428165\pi$
$734$	−44203.8	−2.22288
$735$	0	0
$736$	−11920.5	−0.597003
$737$	5733.20	0.286547
$738$	0	0
$739$	30833.0	1.53479	0.767395	−	0.641175i	$-0.221553\pi$
$739$	30833.0	1.53479	0.767395	+	0.641175i	$0.221553\pi$
$740$	−7000.94	−0.347783
$741$	0	0
$742$	15652.3	0.774414
$743$	−9598.45	−0.473934	−0.236967	−	0.971518i	$-0.576153\pi$
$743$	−9598.45	−0.473934	−0.236967	+	0.971518i	$0.576153\pi$
$744$	0	0
$745$	−17804.4	−0.875574
$746$	−30543.6	−1.49904
$747$	0	0
$748$	340.797	0.0166588
$749$	−10389.6	−0.506844
$750$	0	0
$751$	−40107.6	−1.94880	−0.974399	−	0.224826i	$-0.927819\pi$
$751$	−40107.6	−1.94880	−0.974399	+	0.224826i	$0.927819\pi$
$752$	−1358.15	−0.0658600
$753$	0	0
$754$	−50834.0	−2.45526
$755$	−7410.05	−0.357191
$756$	0	0
$757$	3714.41	0.178339	0.0891696	−	0.996016i	$-0.471579\pi$
$757$	3714.41	0.178339	0.0891696	+	0.996016i	$0.471579\pi$
$758$	27110.7	1.29908
$759$	0	0
$760$	11031.9	0.526539
$761$	−29623.8	−1.41112	−0.705559	−	0.708651i	$-0.749304\pi$
$761$	−29623.8	−1.41112	−0.705559	+	0.708651i	$0.749304\pi$
$762$	0	0
$763$	7452.53	0.353604
$764$	28226.5	1.33665
$765$	0	0
$766$	−45566.3	−2.14932
$767$	223.285	0.0105115
$768$	0	0
$769$	−39173.6	−1.83698	−0.918489	−	0.395446i	$-0.870590\pi$
$769$	−39173.6	−1.83698	−0.918489	+	0.395446i	$0.870590\pi$
$770$	−2286.68	−0.107021
$771$	0	0
$772$	18205.9	0.848763
$773$	9500.27	0.442045	0.221023	−	0.975269i	$-0.429061\pi$
$773$	9500.27	0.442045	0.221023	+	0.975269i	$0.429061\pi$
$774$	0	0
$775$	−6394.03	−0.296362
$776$	−9965.04	−0.460985
$777$	0	0
$778$	−60129.6	−2.77089
$779$	−2305.17	−0.106022
$780$	0	0
$781$	−9741.47	−0.446322
$782$	−688.244	−0.0314726
$783$	0	0
$784$	−259.645	−0.0118278
$785$	−22549.5	−1.02526
$786$	0	0
$787$	21640.9	0.980195	0.490097	−	0.871668i	$-0.336961\pi$
$787$	21640.9	0.980195	0.490097	+	0.871668i	$0.336961\pi$
$788$	−12978.4	−0.586723
$789$	0	0
$790$	−39973.4	−1.80024
$791$	3067.20	0.137873
$792$	0	0
$793$	−11397.7	−0.510394
$794$	45216.4	2.02100
$795$	0	0
$796$	16097.9	0.716802
$797$	9843.72	0.437494	0.218747	−	0.975782i	$-0.429803\pi$
$797$	9843.72	0.437494	0.218747	+	0.975782i	$0.429803\pi$
$798$	0	0
$799$	−628.084	−0.0278098
$800$	−15866.1	−0.701189
$801$	0	0
$802$	−21397.9	−0.942125
$803$	4130.49	0.181522
$804$	0	0
$805$	2828.33	0.123833
$806$	−25188.1	−1.10076
$807$	0	0
$808$	22076.4	0.961194
$809$	1679.53	0.0729902	0.0364951	−	0.999334i	$-0.488381\pi$
$809$	1679.53	0.0729902	0.0364951	+	0.999334i	$0.488381\pi$
$810$	0	0
$811$	19807.2	0.857612	0.428806	−	0.903397i	$-0.358934\pi$
$811$	19807.2	0.857612	0.428806	+	0.903397i	$0.358934\pi$
$812$	13880.4	0.599883
$813$	0	0
$814$	4234.06	0.182314
$815$	−5997.69	−0.257779
$816$	0	0
$817$	−28194.4	−1.20734
$818$	−68355.3	−2.92175
$819$	0	0
$820$	2380.90	0.101396
$821$	4861.54	0.206661	0.103331	−	0.994647i	$-0.467050\pi$
$821$	4861.54	0.206661	0.103331	+	0.994647i	$0.467050\pi$
$822$	0	0
$823$	15188.4	0.643297	0.321648	−	0.946859i	$-0.395763\pi$
$823$	15188.4	0.643297	0.321648	+	0.946859i	$0.395763\pi$
$824$	−23559.9	−0.996051
$825$	0	0
$826$	−99.5470	−0.00419332
$827$	29620.8	1.24548	0.622742	−	0.782428i	$-0.286019\pi$
$827$	29620.8	1.24548	0.622742	+	0.782428i	$0.286019\pi$
$828$	0	0
$829$	−1614.70	−0.0676487	−0.0338243	−	0.999428i	$-0.510769\pi$
$829$	−1614.70	−0.0676487	−0.0338243	+	0.999428i	$0.510769\pi$
$830$	40965.2	1.71316
$831$	0	0
$832$	−59477.5	−2.47838
$833$	−120.074	−0.00499438
$834$	0	0
$835$	−22260.1	−0.922565
$836$	−11127.1	−0.460333
$837$	0	0
$838$	27204.2	1.12143
$839$	−10645.8	−0.438063	−0.219031	−	0.975718i	$-0.570290\pi$
$839$	−10645.8	−0.438063	−0.219031	+	0.975718i	$0.570290\pi$
$840$	0	0
$841$	209.391	0.00858547
$842$	30637.5	1.25397
$843$	0	0
$844$	−13754.1	−0.560944
$845$	−18902.6	−0.769551
$846$	0	0
$847$	847.000	0.0343604
$848$	2607.83	0.105605
$849$	0	0
$850$	−916.052	−0.0369651
$851$	−5236.99	−0.210954
$852$	0	0
$853$	30300.5	1.21626	0.608130	−	0.793837i	$-0.291920\pi$
$853$	30300.5	1.21626	0.608130	+	0.793837i	$0.291920\pi$
$854$	5081.42	0.203609
$855$	0	0
$856$	31310.0	1.25018
$857$	14243.2	0.567721	0.283861	−	0.958866i	$-0.408385\pi$
$857$	14243.2	0.567721	0.283861	+	0.958866i	$0.408385\pi$
$858$	0	0
$859$	21732.6	0.863221	0.431610	−	0.902060i	$-0.357946\pi$
$859$	21732.6	0.863221	0.431610	+	0.902060i	$0.357946\pi$
$860$	29120.7	1.15466
$861$	0	0
$862$	8877.29	0.350768
$863$	36687.7	1.44712	0.723559	−	0.690263i	$-0.242505\pi$
$863$	36687.7	1.44712	0.723559	+	0.690263i	$0.242505\pi$
$864$	0	0
$865$	12250.0	0.481519
$866$	−54351.8	−2.13274
$867$	0	0
$868$	6877.68	0.268944
$869$	14806.4	0.577988
$870$	0	0
$871$	−37180.9	−1.44641
$872$	−22458.9	−0.872195
$873$	0	0
$874$	22471.3	0.869685
$875$	9483.72	0.366409
$876$	0	0
$877$	2430.02	0.0935643	0.0467822	−	0.998905i	$-0.485103\pi$
$877$	2430.02	0.0935643	0.0467822	+	0.998905i	$0.485103\pi$
$878$	−25057.8	−0.963165
$879$	0	0
$880$	−380.982	−0.0145942
$881$	−31492.8	−1.20434	−0.602168	−	0.798369i	$-0.705696\pi$
$881$	−31492.8	−1.20434	−0.602168	+	0.798369i	$0.705696\pi$
$882$	0	0
$883$	9175.59	0.349698	0.174849	−	0.984595i	$-0.444056\pi$
$883$	9175.59	0.349698	0.174849	+	0.984595i	$0.444056\pi$
$884$	−2210.13	−0.0840890
$885$	0	0
$886$	−59661.0	−2.26225
$887$	−23904.6	−0.904890	−0.452445	−	0.891792i	$-0.649448\pi$
$887$	−23904.6	−0.904890	−0.452445	+	0.891792i	$0.649448\pi$
$888$	0	0
$889$	13482.0	0.508631
$890$	3297.71	0.124202
$891$	0	0
$892$	−23092.6	−0.866814
$893$	20507.1	0.768470
$894$	0	0
$895$	−3572.32	−0.133418
$896$	15718.0	0.586051
$897$	0	0
$898$	23178.7	0.861341
$899$	12188.4	0.452177
$900$	0	0
$901$	1206.00	0.0445924
$902$	−1439.93	−0.0531536
$903$	0	0
$904$	−9243.31	−0.340075
$905$	−19133.7	−0.702790
$906$	0	0
$907$	30088.7	1.10152	0.550760	−	0.834663i	$-0.314338\pi$
$907$	30088.7	1.10152	0.550760	+	0.834663i	$0.314338\pi$
$908$	−23033.1	−0.841829
$909$	0	0
$910$	14829.5	0.540214
$911$	30837.0	1.12149	0.560745	−	0.827989i	$-0.310515\pi$
$911$	30837.0	1.12149	0.560745	+	0.827989i	$0.310515\pi$
$912$	0	0
$913$	−15173.7	−0.550030
$914$	7594.61	0.274844
$915$	0	0
$916$	−17319.1	−0.624715
$917$	4925.52	0.177377
$918$	0	0
$919$	52840.0	1.89666	0.948330	−	0.317285i	$-0.102771\pi$
$919$	52840.0	1.89666	0.948330	+	0.317285i	$0.102771\pi$
$920$	−8523.45	−0.305445
$921$	0	0
$922$	−14986.6	−0.535312
$923$	63175.2	2.25291
$924$	0	0
$925$	−6970.42	−0.247769
$926$	−33462.2	−1.18751
$927$	0	0
$928$	30244.3	1.06984
$929$	−12341.6	−0.435861	−0.217930	−	0.975964i	$-0.569931\pi$
$929$	−12341.6	−0.435861	−0.217930	+	0.975964i	$0.569931\pi$
$930$	0	0
$931$	3920.45	0.138010
$932$	26753.6	0.940282
$933$	0	0
$934$	−44044.8	−1.54303
$935$	−176.187	−0.00616250
$936$	0	0
$937$	3939.78	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$937$	3939.78	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$938$	16576.4	0.577012
$939$	0	0
$940$	−21180.8	−0.734939
$941$	−35201.0	−1.21947	−0.609734	−	0.792606i	$-0.708724\pi$
$941$	−35201.0	−1.21947	−0.609734	+	0.792606i	$0.708724\pi$
$942$	0	0
$943$	1781.01	0.0615035
$944$	−16.5855	−0.000571834 0
$945$	0	0
$946$	−17611.7	−0.605292
$947$	17958.1	0.616218	0.308109	−	0.951351i	$-0.400304\pi$
$947$	17958.1	0.616218	0.308109	+	0.951351i	$0.400304\pi$
$948$	0	0
$949$	−26787.0	−0.916273
$950$	29909.3	1.02146
$951$	0	0
$952$	361.855	0.0123191
$953$	−9631.60	−0.327385	−0.163693	−	0.986511i	$-0.552341\pi$
$953$	−9631.60	−0.327385	−0.163693	+	0.986511i	$0.552341\pi$
$954$	0	0
$955$	−14592.7	−0.494460
$956$	−65094.2	−2.20219
$957$	0	0
$958$	53038.1	1.78871
$959$	6614.61	0.222729
$960$	0	0
$961$	−23751.7	−0.797277
$962$	−27458.7	−0.920273
$963$	0	0
$964$	−47488.4	−1.58662
$965$	−9412.21	−0.313979
$966$	0	0
$967$	−46034.2	−1.53088	−0.765439	−	0.643508i	$-0.777478\pi$
$967$	−46034.2	−1.53088	−0.765439	+	0.643508i	$0.777478\pi$
$968$	−2552.52	−0.0847531
$969$	0	0
$970$	14028.5	0.464358
$971$	17330.8	0.572782	0.286391	−	0.958113i	$-0.407544\pi$
$971$	17330.8	0.572782	0.286391	+	0.958113i	$0.407544\pi$
$972$	0	0
$973$	8560.04	0.282037
$974$	22864.4	0.752180
$975$	0	0
$976$	846.612	0.0277658
$977$	−8836.93	−0.289374	−0.144687	−	0.989477i	$-0.546218\pi$
$977$	−8836.93	−0.289374	−0.144687	+	0.989477i	$0.546218\pi$
$978$	0	0
$979$	−1221.49	−0.0398765
$980$	−4049.25	−0.131988
$981$	0	0
$982$	47976.6	1.55906
$983$	−27933.6	−0.906352	−0.453176	−	0.891421i	$-0.649709\pi$
$983$	−27933.6	−0.906352	−0.453176	+	0.891421i	$0.649709\pi$
$984$	0	0
$985$	6709.67	0.217044
$986$	1746.19	0.0563997
$987$	0	0
$988$	72161.2	2.32364
$989$	21783.4	0.700377
$990$	0	0
$991$	57618.7	1.84694	0.923471	−	0.383668i	$-0.125339\pi$
$991$	57618.7	1.84694	0.923471	+	0.383668i	$0.125339\pi$
$992$	14985.9	0.479641
$993$	0	0
$994$	−28165.4	−0.898745
$995$	−8322.39	−0.265163
$996$	0	0
$997$	−8399.40	−0.266812	−0.133406	−	0.991061i	$-0.542591\pi$
$997$	−8399.40	−0.266812	−0.133406	+	0.991061i	$0.542591\pi$
$998$	33473.8	1.06172
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.p.1.1		5
3.2	odd	2		231.4.a.k.1.5	✓	5
21.20	even	2		1617.4.a.n.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.k.1.5	✓	5	3.2	odd	2
693.4.a.p.1.1		5	1.1	even	1	trivial
1617.4.a.n.1.5		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)

\(f(q)\)	\(=\)	\(q-4.54345 q^{2} +12.6430 q^{4} -6.53625 q^{5} +7.00000 q^{7} -21.0952 q^{8} +O(q^{10})\)	\(q-4.54345 q^{2} +12.6430 q^{4} -6.53625 q^{5} +7.00000 q^{7} -21.0952 q^{8} +29.6972 q^{10} -11.0000 q^{11} +71.3370 q^{13} -31.8042 q^{14} -5.29887 q^{16} -2.45049 q^{17} +80.0091 q^{19} -82.6377 q^{20} +49.9780 q^{22} -61.8164 q^{23} -82.2774 q^{25} -324.117 q^{26} +88.5009 q^{28} +156.839 q^{29} +77.7131 q^{31} +192.837 q^{32} +11.1337 q^{34} -45.7538 q^{35} +84.7185 q^{37} -363.518 q^{38} +137.883 q^{40} -28.8114 q^{41} -352.389 q^{43} -139.073 q^{44} +280.860 q^{46} +256.310 q^{47} +49.0000 q^{49} +373.824 q^{50} +901.913 q^{52} -492.147 q^{53} +71.8988 q^{55} -147.666 q^{56} -712.590 q^{58} +3.13000 q^{59} -159.772 q^{61} -353.086 q^{62} -833.753 q^{64} -466.277 q^{65} -521.200 q^{67} -30.9815 q^{68} +207.880 q^{70} +885.588 q^{71} -375.499 q^{73} -384.914 q^{74} +1011.55 q^{76} -77.0000 q^{77} -1346.03 q^{79} +34.6348 q^{80} +130.903 q^{82} +1379.43 q^{83} +16.0170 q^{85} +1601.07 q^{86} +232.047 q^{88} +111.045 q^{89} +499.359 q^{91} -781.543 q^{92} -1164.53 q^{94} -522.959 q^{95} +472.385 q^{97} -222.629 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8}+O(q^{10}) \) 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

L-functions

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