LMFDB - Embedded newform 1617.4.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, 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("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1617.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:	$95.4060884793$
Analytic rank:	$1$
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			1.5
Root			$-4.54345$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.54345 q^{2} +3.00000 q^{3} +12.6430 q^{4} -6.53625 q^{5} +13.6304 q^{6} +21.0952 q^{8} +9.00000 q^{9} -29.6972 q^{10} +11.0000 q^{11} +37.9289 q^{12} -71.3370 q^{13} -19.6088 q^{15} -5.29887 q^{16} -2.45049 q^{17} +40.8911 q^{18} -80.0091 q^{19} -82.6377 q^{20} +49.9780 q^{22} +61.8164 q^{23} +63.2855 q^{24} -82.2774 q^{25} -324.117 q^{26} +27.0000 q^{27} -156.839 q^{29} -89.0915 q^{30} -77.7131 q^{31} -192.837 q^{32} +33.0000 q^{33} -11.1337 q^{34} +113.787 q^{36} +84.7185 q^{37} -363.518 q^{38} -214.011 q^{39} -137.883 q^{40} -28.8114 q^{41} -352.389 q^{43} +139.073 q^{44} -58.8263 q^{45} +280.860 q^{46} +256.310 q^{47} -15.8966 q^{48} -373.824 q^{50} -7.35147 q^{51} -901.913 q^{52} +492.147 q^{53} +122.673 q^{54} -71.8988 q^{55} -240.027 q^{57} -712.590 q^{58} +3.13000 q^{59} -247.913 q^{60} +159.772 q^{61} -353.086 q^{62} -833.753 q^{64} +466.277 q^{65} +149.934 q^{66} -521.200 q^{67} -30.9815 q^{68} +185.449 q^{69} -885.588 q^{71} +189.857 q^{72} +375.499 q^{73} +384.914 q^{74} -246.832 q^{75} -1011.55 q^{76} -972.350 q^{78} -1346.03 q^{79} +34.6348 q^{80} +81.0000 q^{81} -130.903 q^{82} +1379.43 q^{83} +16.0170 q^{85} -1601.07 q^{86} -470.516 q^{87} +232.047 q^{88} +111.045 q^{89} -267.274 q^{90} +781.543 q^{92} -233.139 q^{93} +1164.53 q^{94} +522.959 q^{95} -578.510 q^{96} -472.385 q^{97} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 15 q^{3} + 21 q^{4} - 21 q^{5} - 3 q^{6} - 42 q^{8} + 45 q^{9} + 23 q^{10} + 55 q^{11} + 63 q^{12} - 101 q^{13} - 63 q^{15} - 7 q^{16} + 20 q^{17} - 9 q^{18} - 237 q^{19} - 85 q^{20} - 11 q^{22}+ \cdots + 495 q^{99}+O(q^{100}) $ 5 * q - q^2 + 15 * q^3 + 21 * q^4 - 21 * q^5 - 3 * q^6 - 42 * q^8 + 45 * q^9 + 23 * q^10 + 55 * q^11 + 63 * q^12 - 101 * q^13 - 63 * q^15 - 7 * q^16 + 20 * q^17 - 9 * q^18 - 237 * q^19 - 85 * q^20 - 11 * q^22 - 80 * q^23 - 126 * q^24 + 486 * q^25 - 165 * q^26 + 135 * q^27 - 11 * q^29 + 69 * q^30 - 316 * q^31 + 453 * q^32 + 165 * q^33 - 936 * q^34 + 189 * q^36 + 319 * q^37 - 89 * q^38 - 303 * q^39 - 624 * q^40 - 1190 * q^41 + 88 * q^43 + 231 * q^44 - 189 * q^45 + 1000 * q^46 - 377 * q^47 - 21 * q^48 - 644 * q^50 + 60 * q^51 - 1001 * q^52 - 992 * q^53 - 27 * q^54 - 231 * q^55 - 711 * q^57 + 721 * q^58 - 71 * q^59 - 255 * q^60 + 574 * q^61 - 272 * q^62 - 1380 * q^64 + 589 * q^65 - 33 * q^66 - 527 * q^67 + 2974 * q^68 - 240 * q^69 - 1156 * q^71 - 378 * q^72 - 1061 * q^73 - 1609 * q^74 + 1458 * q^75 - 2399 * q^76 - 495 * q^78 + 588 * q^79 + 1643 * q^80 + 405 * q^81 + 2602 * q^82 + 212 * q^83 + 1918 * q^85 - 4760 * q^86 - 33 * q^87 - 462 * q^88 - 1030 * q^89 + 207 * q^90 - 1174 * q^92 - 948 * q^93 + 1799 * q^94 - 3593 * q^95 + 1359 * q^96 - 2488 * q^97 + 495 * q^99

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	$-0.203141\pi$
$2$	4.54345	1.60635	0.803177	+	0.595741i	$0.203141\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	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	$0.594424\pi$
$6$	13.6304	0.927429
$7$	0	0
$7$	0	0
$8$	21.0952	0.932284
$9$	9.00000	0.333333
$10$	−29.6972	−0.939107
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	37.9289	0.912429
$13$	−71.3370	−1.52195	−0.760974	−	0.648782i	$-0.775279\pi$
$13$	−71.3370	−1.52195	−0.760974	+	0.648782i	$0.775279\pi$
$14$	0	0
$15$	−19.6088	−0.337531
$16$	−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	$0.505564\pi$
$18$	40.8911	0.535451
$19$	−80.0091	−0.966071	−0.483035	−	0.875601i	$-0.660466\pi$
$19$	−80.0091	−0.966071	−0.483035	+	0.875601i	$0.660466\pi$
$20$	−82.6377	−0.923917
$21$	0	0
$22$	49.9780	0.484334
$23$	61.8164	0.560418	0.280209	−	0.959939i	$-0.409596\pi$
$23$	61.8164	0.560418	0.280209	+	0.959939i	$0.409596\pi$
$24$	63.2855	0.538254
$25$	−82.2774	−0.658219
$26$	−324.117	−2.44479
$27$	27.0000	0.192450
$28$	0	0
$29$	−156.839	−1.00428	−0.502142	−	0.864785i	$-0.667454\pi$
$29$	−156.839	−1.00428	−0.502142	+	0.864785i	$0.667454\pi$
$30$	−89.0915	−0.542193
$31$	−77.7131	−0.450248	−0.225124	−	0.974330i	$-0.572279\pi$
$31$	−77.7131	−0.450248	−0.225124	+	0.974330i	$0.572279\pi$
$32$	−192.837	−1.06528
$33$	33.0000	0.174078
$34$	−11.1337	−0.0561592
$35$	0	0
$36$	113.787	0.526791
$37$	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
$39$	−214.011	−0.878697
$40$	−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
$45$	−58.8263	−0.194873
$46$	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	$0.369798\pi$
$48$	−15.8966	−0.0478017
$49$	0	0
$50$	−373.824	−1.05733
$51$	−7.35147	−0.0201845
$52$	−901.913	−2.40525
$53$	492.147	1.27550	0.637751	−	0.770243i	$-0.279865\pi$
$53$	492.147	1.27550	0.637751	+	0.770243i	$0.279865\pi$
$54$	122.673	0.309143
$55$	−71.8988	−0.176270
$56$	0	0
$57$	−240.027	−0.557761
$58$	−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	$0.498901\pi$
$60$	−247.913	−0.533424
$61$	159.772	0.335356	0.167678	−	0.985842i	$-0.446373\pi$
$61$	159.772	0.335356	0.167678	+	0.985842i	$0.446373\pi$
$62$	−353.086	−0.723257
$63$	0	0
$64$	−833.753	−1.62842
$65$	466.277	0.889761
$66$	149.934	0.279630
$67$	−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
$69$	185.449	0.323557
$70$	0	0
$71$	−885.588	−1.48028	−0.740141	−	0.672452i	$-0.765241\pi$
$71$	−885.588	−1.48028	−0.740141	+	0.672452i	$0.765241\pi$
$72$	189.857	0.310761
$73$	375.499	0.602039	0.301020	−	0.953618i	$-0.402673\pi$
$73$	375.499	0.602039	0.301020	+	0.953618i	$0.402673\pi$
$74$	384.914	0.604668
$75$	−246.832	−0.380023
$76$	−1011.55	−1.52675
$77$	0	0
$78$	−972.350	−1.41150
$79$	−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
$81$	81.0000	0.111111
$82$	−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
$87$	−470.516	−0.579823
$88$	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	$0.478936\pi$
$90$	−267.274	−0.313036
$91$	0	0
$92$	781.543	0.885669
$93$	−233.139	−0.259951
$94$	1164.53	1.27779
$95$	522.959	0.564784
$96$	−578.510	−0.615041
$97$	−472.385	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$97$	−472.385	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	−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	$0.672396\pi$
$102$	−33.4011	−0.0324235
$103$	−1116.84	−1.06840	−0.534200	−	0.845358i	$-0.679387\pi$
$103$	−1116.84	−1.06840	−0.534200	+	0.845358i	$0.679387\pi$
$104$	−1504.87	−1.41889
$105$	0	0
$106$	2236.05	2.04891
$107$	1484.22	1.34098	0.670492	−	0.741917i	$-0.266083\pi$
$107$	1484.22	1.34098	0.670492	+	0.741917i	$0.266083\pi$
$108$	341.360	0.304143
$109$	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
$111$	254.155	0.217328
$112$	0	0
$113$	−438.172	−0.364776	−0.182388	−	0.983227i	$-0.558383\pi$
$113$	−438.172	−0.364776	−0.182388	+	0.983227i	$0.558383\pi$
$114$	−1090.55	−0.895962
$115$	−404.047	−0.327631
$116$	−1982.91	−1.58714
$117$	−642.033	−0.507316
$118$	14.2210	0.0110945
$119$	0	0
$120$	−413.650	−0.314674
$121$	121.000	0.0909091
$122$	725.917	0.538700
$123$	−86.4341	−0.0633618
$124$	−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
$129$	−1057.17	−0.721539
$130$	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	$0.424606\pi$
$132$	417.218	0.275108
$133$	0	0
$134$	−2368.05	−1.52663
$135$	−176.479	−0.112510
$136$	−51.6935	−0.0325933
$137$	−944.945	−0.589285	−0.294643	−	0.955608i	$-0.595201\pi$
$137$	−944.945	−0.589285	−0.294643	+	0.955608i	$0.595201\pi$
$138$	842.580	0.519747
$139$	−1222.86	−0.746200	−0.373100	−	0.927791i	$-0.621705\pi$
$139$	−1222.86	−0.746200	−0.373100	+	0.927791i	$0.621705\pi$
$140$	0	0
$141$	768.929	0.459259
$142$	−4023.63	−2.37786
$143$	−784.707	−0.458885
$144$	−47.6899	−0.0275983
$145$	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	$-0.769389\pi$
$149$	−2723.95	−1.49768	−0.748840	+	0.662750i	$0.769389\pi$
$150$	−1121.47	−0.610452
$151$	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
$153$	−22.0544	−0.0116536
$154$	0	0
$155$	507.952	0.263224
$156$	−2705.74	−1.38867
$157$	−3449.91	−1.75371	−0.876856	−	0.480753i	$-0.840364\pi$
$157$	−3449.91	−1.75371	−0.876856	+	0.480753i	$0.840364\pi$
$158$	−6115.64	−3.07933
$159$	1476.44	0.736411
$160$	1260.43	0.622785
$161$	0	0
$162$	368.020	0.178484
$163$	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
$165$	−215.696	−0.101769
$166$	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
$171$	−720.082	−0.322024
$172$	−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	$0.635108\pi$
$174$	−2137.77	−0.931402
$175$	0	0
$176$	−58.2876	−0.0249636
$177$	9.38999	0.00398754
$178$	504.527	0.212449
$179$	−546.539	−0.228214	−0.114107	−	0.993468i	$-0.536401\pi$
$179$	−546.539	−0.228214	−0.114107	+	0.993468i	$0.536401\pi$
$180$	−743.739	−0.307972
$181$	−2927.32	−1.20213	−0.601066	−	0.799200i	$-0.705257\pi$
$181$	−2927.32	−1.20213	−0.601066	+	0.799200i	$0.705257\pi$
$182$	0	0
$183$	479.316	0.193618
$184$	1304.03	0.522468
$185$	−553.741	−0.220064
$186$	−1059.26	−0.417573
$187$	−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	$-0.638984\pi$
$191$	−2232.58	−0.845780	−0.422890	+	0.906181i	$0.638984\pi$
$192$	−2501.26	−0.940171
$193$	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
$195$	1398.83	0.513704
$196$	0	0
$197$	1026.53	0.371256	0.185628	−	0.982620i	$-0.440568\pi$
$197$	1026.53	0.371256	0.185628	+	0.982620i	$0.440568\pi$
$198$	449.802	0.161445
$199$	−1273.27	−0.453565	−0.226783	−	0.973945i	$-0.572821\pi$
$199$	−1273.27	−0.453565	−0.226783	+	0.973945i	$0.572821\pi$
$200$	−1735.66	−0.613647
$201$	−1563.60	−0.548696
$202$	−4754.79	−1.65617
$203$	0	0
$204$	−92.9445	−0.0318991
$205$	188.318	0.0641596
$206$	−5074.29	−1.71623
$207$	556.347	0.186806
$208$	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
$213$	−2656.76	−0.854641
$214$	6743.50	2.15409
$215$	2303.31	0.730624
$216$	569.570	0.179418
$217$	0	0
$218$	4837.17	1.50282
$219$	1126.50	0.347588
$220$	−909.015	−0.278572
$221$	174.811	0.0532083
$222$	1154.74	0.349105
$223$	1826.52	0.548487	0.274244	−	0.961660i	$-0.411573\pi$
$223$	1826.52	0.548487	0.274244	+	0.961660i	$0.411573\pi$
$224$	0	0
$225$	−740.497	−0.219406
$226$	−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	$0.585814\pi$
$228$	−3034.66	−0.881470
$229$	1369.86	0.395296	0.197648	−	0.980273i	$-0.436670\pi$
$229$	1369.86	0.395296	0.197648	+	0.980273i	$0.436670\pi$
$230$	−1835.77	−0.526292
$231$	0	0
$232$	−3308.54	−0.936277
$233$	−2116.08	−0.594975	−0.297488	−	0.954726i	$-0.596149\pi$
$233$	−2116.08	−0.594975	−0.297488	+	0.954726i	$0.596149\pi$
$234$	−2917.05	−0.814929
$235$	−1675.30	−0.465041
$236$	39.5725	0.0109150
$237$	−4038.10	−1.10676
$238$	0	0
$239$	5148.64	1.39346	0.696732	−	0.717331i	$-0.254636\pi$
$239$	5148.64	1.39346	0.696732	+	0.717331i	$0.254636\pi$
$240$	103.904	0.0279458
$241$	3756.11	1.00395	0.501975	−	0.864882i	$-0.332607\pi$
$241$	3756.11	1.00395	0.501975	+	0.864882i	$0.332607\pi$
$242$	549.758	0.146032
$243$	243.000	0.0641500
$244$	2019.99	0.529987
$245$	0	0
$246$	−392.709	−0.101781
$247$	5707.61	1.47031
$248$	−1639.37	−0.419759
$249$	4138.29	1.05323
$250$	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
$255$	48.0511	0.0118003
$256$	−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	$0.494438\pi$
$258$	−4803.20	−1.15905
$259$	0	0
$260$	5895.13	1.40615
$261$	−1411.55	−0.334761
$262$	3196.98	0.753856
$263$	−6827.81	−1.60084	−0.800420	−	0.599440i	$-0.795390\pi$
$263$	−6827.81	−1.60084	−0.800420	+	0.599440i	$0.795390\pi$
$264$	696.141	0.162290
$265$	−3216.80	−0.745684
$266$	0	0
$267$	333.134	0.0763576
$268$	−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	$0.865690\pi$
$270$	−801.823	−0.180731
$271$	−1420.76	−0.318469	−0.159234	−	0.987241i	$-0.550903\pi$
$271$	−1420.76	−0.318469	−0.159234	+	0.987241i	$0.550903\pi$
$272$	12.9848	0.00289456
$273$	0	0
$274$	−4293.31	−0.946601
$275$	−905.052	−0.198461
$276$	2344.63	0.511341
$277$	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
$279$	−699.418	−0.150083
$280$	0	0
$281$	3089.28	0.655839	0.327919	−	0.944706i	$-0.393653\pi$
$281$	3089.28	0.655839	0.327919	+	0.944706i	$0.393653\pi$
$282$	3493.59	0.737732
$283$	8123.26	1.70628	0.853141	−	0.521681i	$-0.174695\pi$
$283$	8123.26	1.70628	0.853141	+	0.521681i	$0.174695\pi$
$284$	−11196.5	−2.33940
$285$	1568.88	0.326078
$286$	−3565.28	−0.737131
$287$	0	0
$288$	−1735.53	−0.355094
$289$	−4907.00	−0.998778
$290$	4657.66	0.943129
$291$	−1417.15	−0.285481
$292$	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
$297$	297.000	0.0580259
$298$	−12376.1	−2.40581
$299$	−4409.80	−0.852927
$300$	−3120.70	−0.600578
$301$	0	0
$302$	5150.84	0.981450
$303$	−3139.54	−0.595254
$304$	423.958	0.0799857
$305$	−1044.31	−0.196056
$306$	−100.203	−0.0187197
$307$	6514.20	1.21103	0.605513	−	0.795835i	$-0.292968\pi$
$307$	6514.20	1.21103	0.605513	+	0.795835i	$0.292968\pi$
$308$	0	0
$309$	−3350.51	−0.616841
$310$	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	$0.386685\pi$
$312$	−4514.60	−0.819195
$313$	−1239.73	−0.223878	−0.111939	−	0.993715i	$-0.535706\pi$
$313$	−1239.73	−0.223878	−0.111939	+	0.993715i	$0.535706\pi$
$314$	−15674.5	−2.81708
$315$	0	0
$316$	−17017.9	−3.02953
$317$	5377.28	0.952739	0.476369	−	0.879245i	$-0.341953\pi$
$317$	5377.28	0.952739	0.476369	+	0.879245i	$0.341953\pi$
$318$	6708.14	1.18294
$319$	−1725.23	−0.302803
$320$	5449.62	0.952010
$321$	4452.67	0.774217
$322$	0	0
$323$	196.061	0.0337745
$324$	1024.08	0.175597
$325$	5869.43	1.00178
$326$	4169.10	0.708297
$327$	3193.94	0.540138
$328$	−607.781	−0.102314
$329$	0	0
$330$	−980.006	−0.163477
$331$	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
$333$	762.466	0.125474
$334$	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
$339$	−1314.52	−0.210604
$340$	202.503	0.0323008
$341$	−854.844	−0.135755
$342$	−3271.66	−0.517284
$343$	0	0
$344$	−7433.72	−1.16511
$345$	−1212.14	−0.189158
$346$	−8515.21	−1.32306
$347$	−8322.54	−1.28754	−0.643771	−	0.765218i	$-0.722631\pi$
$347$	−8322.54	−1.28754	−0.643771	+	0.765218i	$0.722631\pi$
$348$	−5948.73	−0.916337
$349$	−4604.41	−0.706213	−0.353107	−	0.935583i	$-0.614875\pi$
$349$	−4604.41	−0.706213	−0.353107	+	0.935583i	$0.614875\pi$
$350$	0	0
$351$	−1926.10	−0.292899
$352$	−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	$0.421012\pi$
$354$	42.6630	0.00640541
$355$	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	$-0.189263\pi$
$359$	11269.4	1.65676	0.828380	+	0.560167i	$0.189263\pi$
$360$	−1240.95	−0.181677
$361$	−457.546	−0.0667075
$362$	−13300.1	−1.93105
$363$	363.000	0.0524864
$364$	0	0
$365$	−2454.36	−0.351964
$366$	2177.75	0.311019
$367$	−9729.11	−1.38380	−0.691901	−	0.721992i	$-0.743227\pi$
$367$	−9729.11	−1.38380	−0.691901	+	0.721992i	$0.743227\pi$
$368$	−327.557	−0.0463997
$369$	−259.302	−0.0365819
$370$	−2515.90	−0.353501
$371$	0	0
$372$	−2947.58	−0.410819
$373$	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
$375$	4064.45	0.559700
$376$	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
$381$	5778.02	0.776947
$382$	−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	$0.266721\pi$
$384$	−6736.28	−0.895207
$385$	0	0
$386$	6542.58	0.862717
$387$	−3171.51	−0.416581
$388$	−5972.35	−0.781444
$389$	−13234.3	−1.72495	−0.862477	−	0.506096i	$-0.831088\pi$
$389$	−13234.3	−1.72495	−0.862477	+	0.506096i	$0.831088\pi$
$390$	6355.52	0.825190
$391$	−151.480	−0.0195926
$392$	0	0
$393$	2110.94	0.270948
$394$	4664.00	0.596368
$395$	8798.01	1.12070
$396$	1251.66	0.158833
$397$	9951.99	1.25813	0.629063	−	0.777354i	$-0.283439\pi$
$397$	9951.99	1.25813	0.629063	+	0.777354i	$0.283439\pi$
$398$	−5785.03	−0.728586
$399$	0	0
$400$	435.978	0.0544972
$401$	−4709.60	−0.586499	−0.293250	−	0.956036i	$-0.594737\pi$
$401$	−4709.60	−0.586499	−0.293250	+	0.956036i	$0.594737\pi$
$402$	−7104.15	−0.881400
$403$	5543.82	0.685254
$404$	−13231.1	−1.62938
$405$	−529.436	−0.0649578
$406$	0	0
$407$	931.903	0.113496
$408$	−155.081	−0.0188177
$409$	−15044.8	−1.81887	−0.909434	−	0.415848i	$-0.863485\pi$
$409$	−15044.8	−1.81887	−0.909434	+	0.415848i	$0.863485\pi$
$410$	855.615	0.103063
$411$	−2834.83	−0.340224
$412$	−14120.1	−1.68847
$413$	0	0
$414$	2527.74	0.300076
$415$	−9016.30	−1.06649
$416$	13756.4	1.62130
$417$	−3668.59	−0.430819
$418$	−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
$423$	2306.79	0.265153
$424$	10381.9	1.18913
$425$	201.620	0.0230118
$426$	−12070.9	−1.37286
$427$	0	0
$428$	18765.0	2.11925
$429$	−2354.12	−0.264937
$430$	10465.0	1.17364
$431$	1953.86	0.218363	0.109181	−	0.994022i	$-0.465177\pi$
$431$	1953.86	0.218363	0.109181	+	0.994022i	$0.465177\pi$
$432$	−143.070	−0.0159339
$433$	−11962.7	−1.32769	−0.663844	−	0.747871i	$-0.731076\pi$
$433$	−11962.7	−1.32769	−0.663844	+	0.747871i	$0.731076\pi$
$434$	0	0
$435$	3075.41	0.338976
$436$	13460.3	1.47851
$437$	−4945.87	−0.541403
$438$	5118.19	0.558349
$439$	−5515.13	−0.599597	−0.299798	−	0.954003i	$-0.596919\pi$
$439$	−5515.13	−0.599597	−0.299798	+	0.954003i	$0.596919\pi$
$440$	−1516.72	−0.164333
$441$	0	0
$442$	794.244	0.0854714
$443$	−13131.2	−1.40831	−0.704156	−	0.710046i	$-0.748674\pi$
$443$	−13131.2	−1.40831	−0.704156	+	0.710046i	$0.748674\pi$
$444$	3213.28	0.343459
$445$	−725.816	−0.0773191
$446$	8298.70	0.881064
$447$	−8171.84	−0.864687
$448$	0	0
$449$	5101.57	0.536209	0.268104	−	0.963390i	$-0.413603\pi$
$449$	5101.57	0.536209	0.268104	+	0.963390i	$0.413603\pi$
$450$	−3364.41	−0.352444
$451$	−316.925	−0.0330896
$452$	−5539.80	−0.576483
$453$	3401.05	0.352749
$454$	−8277.31	−0.855669
$455$	0	0
$456$	−5063.42	−0.519992
$457$	−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
$459$	−66.1632	−0.00672818
$460$	−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
$465$	1523.86	0.151972
$466$	−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	$0.340532\pi$
$468$	−8117.21	−0.801749
$469$	0	0
$470$	−7611.66	−0.747021
$471$	−10349.7	−1.01251
$472$	66.0279	0.00643894
$473$	−3876.28	−0.376811
$474$	−18346.9	−1.77785
$475$	6582.94	0.635887
$476$	0	0
$477$	4429.32	0.425167
$478$	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	$0.687956\pi$
$480$	3781.28	0.359565
$481$	−6043.56	−0.572895
$482$	17065.7	1.61270
$483$	0	0
$484$	1529.80	0.143670
$485$	3087.63	0.289076
$486$	1104.06	0.103048
$487$	−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
$489$	2752.81	0.254574
$490$	0	0
$491$	10559.5	0.970557	0.485278	−	0.874360i	$-0.338718\pi$
$491$	10559.5	0.970557	0.485278	+	0.874360i	$0.338718\pi$
$492$	−1092.78	−0.100135
$493$	384.332	0.0351104
$494$	25932.3	2.36184
$495$	−647.089	−0.0587565
$496$	411.792	0.0372782
$497$	0	0
$498$	18802.1	1.69186
$499$	−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
$501$	10216.9	0.911093
$502$	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
$507$	8675.91	0.759982
$508$	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	$0.469313\pi$
$510$	218.318	0.0189554
$511$	0	0
$512$	1916.06	0.165388
$513$	−2160.25	−0.185920
$514$	654.157	0.0561354
$515$	7299.92	0.624608
$516$	−13365.8	−1.14030
$517$	2819.40	0.239840
$518$	0	0
$519$	−5622.51	−0.475531
$520$	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	$0.780470\pi$
$522$	−6413.31	−0.537745
$523$	15196.1	1.27051	0.635256	−	0.772301i	$-0.280894\pi$
$523$	15196.1	1.27051	0.635256	+	0.772301i	$0.280894\pi$
$524$	8896.18	0.741663
$525$	0	0
$526$	−31021.8	−2.57151
$527$	190.435	0.0157410
$528$	−174.863	−0.0144127
$529$	−8345.74	−0.685932
$530$	−14615.4	−1.19783
$531$	28.1700	0.00230221
$532$	0	0
$533$	2055.32	0.167027
$534$	1513.58	0.122657
$535$	−9701.25	−0.783966
$536$	−10994.8	−0.886014
$537$	−1639.62	−0.131759
$538$	−36574.6	−2.93093
$539$	0	0
$540$	−2231.22	−0.177808
$541$	−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
$543$	−8781.95	−0.694051
$544$	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
$549$	1437.95	0.111785
$550$	−4112.06	−0.318798
$551$	12548.5	0.970209
$552$	3912.08	0.301647
$553$	0	0
$554$	36548.4	2.80288
$555$	−1661.22	−0.127054
$556$	−15460.6	−1.17927
$557$	20254.8	1.54079	0.770396	−	0.637565i	$-0.220058\pi$
$557$	20254.8	1.54079	0.770396	+	0.637565i	$0.220058\pi$
$558$	−3177.77	−0.241086
$559$	25138.4	1.90204
$560$	0	0
$561$	−80.8662	−0.00608587
$562$	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	$0.468263\pi$
$564$	9721.55	0.725800
$565$	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	$-0.682324\pi$
$569$	−14712.3	−1.08395	−0.541977	+	0.840393i	$0.682324\pi$
$570$	7128.13	0.523797
$571$	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
$573$	−6697.75	−0.488311
$574$	0	0
$575$	−5086.09	−0.368878
$576$	−7503.78	−0.542808
$577$	−24028.7	−1.73367	−0.866835	−	0.498595i	$-0.833849\pi$
$577$	−24028.7	−1.73367	−0.866835	+	0.498595i	$0.833849\pi$
$578$	−22294.7	−1.60439
$579$	4320.01	0.310075
$580$	12960.8	0.927875
$581$	0	0
$582$	−6438.78	−0.458584
$583$	5413.62	0.384578
$584$	7921.23	0.561272
$585$	4196.49	0.296587
$586$	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
$591$	3079.60	0.214345
$592$	−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	$0.823128\pi$
$594$	1349.41	0.0932101
$595$	0	0
$596$	−34438.8	−2.36689
$597$	−3819.80	−0.261866
$598$	−20035.7	−1.37010
$599$	−88.4090	−0.00603054	−0.00301527	−	0.999995i	$-0.500960\pi$
$599$	−88.4090	−0.00603054	−0.00301527	+	0.999995i	$0.500960\pi$
$600$	−5206.97	−0.354289
$601$	26290.0	1.78434	0.892172	−	0.451695i	$-0.149181\pi$
$601$	26290.0	1.78434	0.892172	+	0.451695i	$0.149181\pi$
$602$	0	0
$603$	−4690.80	−0.316790
$604$	14333.2	0.965576
$605$	−790.886	−0.0531473
$606$	−14264.4	−0.956188
$607$	−28411.4	−1.89981	−0.949905	−	0.312540i	$-0.898820\pi$
$607$	−28411.4	−1.89981	−0.949905	+	0.312540i	$0.898820\pi$
$608$	15428.7	1.02914
$609$	0	0
$610$	−4744.77	−0.314935
$611$	−18284.4	−1.21065
$612$	−278.834	−0.0184170
$613$	−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
$615$	564.955	0.0370426
$616$	0	0
$617$	−3409.29	−0.222452	−0.111226	−	0.993795i	$-0.535478\pi$
$617$	−3409.29	−0.222452	−0.111226	+	0.993795i	$0.535478\pi$
$618$	−15222.9	−0.990864
$619$	10865.7	0.705542	0.352771	−	0.935710i	$-0.385240\pi$
$619$	10865.7	0.705542	0.352771	+	0.935710i	$0.385240\pi$
$620$	6422.03	0.415992
$621$	1669.04	0.107852
$622$	17369.2	1.11968
$623$	0	0
$624$	1134.02	0.0727517
$625$	1429.26	0.0914724
$626$	−5632.66	−0.359627
$627$	−2640.30	−0.168171
$628$	−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
$633$	−3263.66	−0.204927
$634$	24431.4	1.53044
$635$	−12588.9	−0.786730
$636$	18666.6	1.16380
$637$	0	0
$638$	−7838.49	−0.486409
$639$	−7970.29	−0.493427
$640$	14676.7	0.906479
$641$	13710.2	0.844809	0.422404	−	0.906408i	$-0.361186\pi$
$641$	13710.2	0.844809	0.422404	+	0.906408i	$0.361186\pi$
$642$	20230.5	1.24367
$643$	−4879.27	−0.299253	−0.149627	−	0.988743i	$-0.547807\pi$
$643$	−4879.27	−0.299253	−0.149627	+	0.988743i	$0.547807\pi$
$644$	0	0
$645$	6909.92	0.421826
$646$	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	$0.638318\pi$
$648$	1708.71	0.103587
$649$	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	$-0.404515\pi$
$653$	9861.70	0.590993	0.295496	+	0.955344i	$0.404515\pi$
$654$	14511.5	0.867653
$655$	−4599.20	−0.274360
$656$	152.668	0.00908640
$657$	3379.49	0.200680
$658$	0	0
$659$	6849.60	0.404890	0.202445	−	0.979294i	$-0.435111\pi$
$659$	6849.60	0.404890	0.202445	+	0.979294i	$0.435111\pi$
$660$	−2727.04	−0.160833
$661$	−24156.9	−1.42147	−0.710737	−	0.703458i	$-0.751639\pi$
$661$	−24156.9	−1.42147	−0.710737	+	0.703458i	$0.751639\pi$
$662$	10547.4	0.619238
$663$	524.432	0.0307198
$664$	29099.3	1.70071
$665$	0	0
$666$	3464.23	0.201556
$667$	−9695.20	−0.562818
$668$	43057.4	2.49392
$669$	5479.55	0.316669
$670$	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
$675$	−2221.49	−0.126674
$676$	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	$0.669118\pi$
$678$	−5972.44	−0.338304
$679$	0	0
$680$	337.882	0.0190547
$681$	−5465.43	−0.307542
$682$	−3883.95	−0.218070
$683$	−8199.58	−0.459368	−0.229684	−	0.973265i	$-0.573769\pi$
$683$	−8199.58	−0.459368	−0.229684	+	0.973265i	$0.573769\pi$
$684$	−9103.98	−0.508917
$685$	6176.39	0.344508
$686$	0	0
$687$	4109.58	0.228224
$688$	1867.27	0.103472
$689$	−35108.3	−1.94125
$690$	−5507.31	−0.303855
$691$	−6179.89	−0.340223	−0.170111	−	0.985425i	$-0.554413\pi$
$691$	−6179.89	−0.340223	−0.170111	+	0.985425i	$0.554413\pi$
$692$	−23695.1	−1.30167
$693$	0	0
$694$	−37813.1	−2.06825
$695$	7992.94	0.436244
$696$	−9925.62	−0.540560
$697$	70.6020	0.00383679
$698$	−20919.9	−1.13443
$699$	−6348.25	−0.343509
$700$	0	0
$701$	−13389.9	−0.721437	−0.360719	−	0.932675i	$-0.617469\pi$
$701$	−13389.9	−0.721437	−0.360719	+	0.932675i	$0.617469\pi$
$702$	−8751.15	−0.470500
$703$	−6778.25	−0.363651
$704$	−9171.29	−0.490988
$705$	−5025.91	−0.268492
$706$	14802.2	0.789075
$707$	0	0
$708$	118.718	0.00630180
$709$	−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
$711$	−12114.3	−0.638990
$712$	2342.51	0.123299
$713$	−4803.94	−0.252327
$714$	0	0
$715$	5129.04	0.268273
$716$	−6909.88	−0.360663
$717$	15445.9	0.804517
$718$	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	$0.428031\pi$
$720$	311.713	0.0161345
$721$	0	0
$722$	−2078.84	−0.107156
$723$	11268.3	0.579631
$724$	−37010.0	−1.89982
$725$	12904.3	0.661039
$726$	1649.27	0.0843117
$727$	−6255.16	−0.319108	−0.159554	−	0.987189i	$-0.551006\pi$
$727$	−6255.16	−0.319108	−0.159554	+	0.987189i	$0.551006\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−11151.3	−0.565379
$731$	863.527	0.0436918
$732$	6059.98	0.305988
$733$	−8881.40	−0.447534	−0.223767	−	0.974643i	$-0.571835\pi$
$733$	−8881.40	−0.447534	−0.223767	+	0.974643i	$0.571835\pi$
$734$	−44203.8	−2.22288
$735$	0	0
$736$	−11920.5	−0.597003
$737$	−5733.20	−0.286547
$738$	−1178.13	−0.0587635
$739$	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
$741$	17122.8	0.848884
$742$	0	0
$743$	9598.45	0.473934	0.236967	−	0.971518i	$-0.423847\pi$
$743$	9598.45	0.473934	0.236967	+	0.971518i	$0.423847\pi$
$744$	−4918.11	−0.242348
$745$	17804.4	0.875574
$746$	30543.6	1.49904
$747$	12414.9	0.608081
$748$	−340.797	−0.0166588
$749$	0	0
$750$	18466.7	0.899076
$751$	−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
$753$	5719.29	0.276789
$754$	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
$759$	2039.94	0.0975562
$760$	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	$0.749304\pi$
$762$	26252.2	1.24805
$763$	0	0
$764$	−28226.5	−1.33665
$765$	144.153	0.00681290
$766$	45566.3	2.14932
$767$	−223.285	−0.0105115
$768$	−10595.9	−0.497848
$769$	39173.6	1.83698	0.918489	−	0.395446i	$-0.129410\pi$
$769$	39173.6	1.83698	0.918489	+	0.395446i	$0.129410\pi$
$770$	0	0
$771$	431.933	0.0201760
$772$	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	$0.429061\pi$
$774$	−14409.6	−0.669176
$775$	6394.03	0.296362
$776$	−9965.04	−0.460985
$777$	0	0
$778$	−60129.6	−2.77089
$779$	2305.17	0.106022
$780$	17685.4	0.811844
$781$	−9741.47	−0.446322
$782$	−688.244	−0.0314726
$783$	−4234.65	−0.193274
$784$	0	0
$785$	22549.5	1.02526
$786$	9590.95	0.435239
$787$	−21640.9	−0.980195	−0.490097	−	0.871668i	$-0.663039\pi$
$787$	−21640.9	−0.980195	−0.490097	+	0.871668i	$0.663039\pi$
$788$	12978.4	0.586723
$789$	−20483.4	−0.924245
$790$	39973.4	1.80024
$791$	0	0
$792$	2088.42	0.0936981
$793$	−11397.7	−0.510394
$794$	45216.4	2.02100
$795$	−9650.39	−0.430521
$796$	−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
$801$	999.403	0.0440851
$802$	−21397.9	−0.942125
$803$	4130.49	0.181522
$804$	−19768.6	−0.867144
$805$	0	0
$806$	25188.1	1.10076
$807$	−24149.8	−1.05343
$808$	−22076.4	−0.961194
$809$	−1679.53	−0.0729902	−0.0364951	−	0.999334i	$-0.511619\pi$
$809$	−1679.53	−0.0729902	−0.0364951	+	0.999334i	$0.511619\pi$
$810$	−2405.47	−0.104345
$811$	−19807.2	−0.857612	−0.428806	−	0.903397i	$-0.641066\pi$
$811$	−19807.2	−0.857612	−0.428806	+	0.903397i	$0.641066\pi$
$812$	0	0
$813$	−4262.28	−0.183868
$814$	4234.06	0.182314
$815$	−5997.69	−0.257779
$816$	38.9545	0.00167118
$817$	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	$-0.532950\pi$
$821$	−4861.54	−0.206661	−0.103331	+	0.994647i	$0.532950\pi$
$822$	−12879.9	−0.546520
$823$	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
$825$	−2715.16	−0.114581
$826$	0	0
$827$	−29620.8	−1.24548	−0.622742	−	0.782428i	$-0.713981\pi$
$827$	−29620.8	−1.24548	−0.622742	+	0.782428i	$0.713981\pi$
$828$	7033.89	0.295223
$829$	1614.70	0.0676487	0.0338243	−	0.999428i	$-0.489231\pi$
$829$	1614.70	0.0676487	0.0338243	+	0.999428i	$0.489231\pi$
$830$	−40965.2	−1.71316
$831$	24132.6	1.00740
$832$	59477.5	2.47838
$833$	0	0
$834$	−16668.1	−0.692048
$835$	−22260.1	−0.922565
$836$	−11127.1	−0.460333
$837$	−2098.25	−0.0866502
$838$	−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
$843$	9267.83	0.378649
$844$	−13754.1	−0.560944
$845$	−18902.6	−0.769551
$846$	10480.8	0.425930
$847$	0	0
$848$	−2607.83	−0.105605
$849$	24369.8	0.985122
$850$	916.052	0.0369651
$851$	5236.99	0.210954
$852$	−33589.4	−1.35065
$853$	−30300.5	−1.21626	−0.608130	−	0.793837i	$-0.708080\pi$
$853$	−30300.5	−1.21626	−0.608130	+	0.793837i	$0.708080\pi$
$854$	0	0
$855$	4706.63	0.188261
$856$	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	$0.408385\pi$
$858$	−10695.8	−0.425583
$859$	−21732.6	−0.863221	−0.431610	−	0.902060i	$-0.642054\pi$
$859$	−21732.6	−0.863221	−0.431610	+	0.902060i	$0.642054\pi$
$860$	29120.7	1.15466
$861$	0	0
$862$	8877.29	0.350768
$863$	−36687.7	−1.44712	−0.723559	−	0.690263i	$-0.757495\pi$
$863$	−36687.7	−1.44712	−0.723559	+	0.690263i	$0.757495\pi$
$864$	−5206.59	−0.205014
$865$	12250.0	0.481519
$866$	−54351.8	−2.13274
$867$	−14721.0	−0.576645
$868$	0	0
$869$	−14806.4	−0.577988
$870$	13973.0	0.544516
$871$	37180.9	1.44641
$872$	22458.9	0.872195
$873$	−4251.46	−0.164823
$874$	−22471.3	−0.869685
$875$	0	0
$876$	14242.3	0.549318
$877$	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
$879$	25150.9	0.965097
$880$	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
$885$	−61.3753	−0.00233120
$886$	−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	$0.649448\pi$
$888$	5361.45	0.202611
$889$	0	0
$890$	−3297.71	−0.124202
$891$	891.000	0.0335013
$892$	23092.6	0.866814
$893$	−20507.1	−0.768470
$894$	−37128.4	−1.38899
$895$	3572.32	0.133418
$896$	0	0
$897$	−13229.4	−0.492437
$898$	23178.7	0.861341
$899$	12188.4	0.452177
$900$	−9362.09	−0.346744
$901$	−1206.00	−0.0445924
$902$	−1439.93	−0.0531536
$903$	0	0
$904$	−9243.31	−0.340075
$905$	19133.7	0.702790
$906$	15452.5	0.566640
$907$	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
$909$	−9418.62	−0.343670
$910$	0	0
$911$	−30837.0	−1.12149	−0.560745	−	0.827989i	$-0.689485\pi$
$911$	−30837.0	−1.12149	−0.560745	+	0.827989i	$0.689485\pi$
$912$	1271.87	0.0461798
$913$	15173.7	0.550030
$914$	−7594.61	−0.274844
$915$	−3132.93	−0.113193
$916$	17319.1	0.624715
$917$	0	0
$918$	−300.610	−0.0108078
$919$	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
$921$	19542.6	0.699187
$922$	14986.6	0.535312
$923$	63175.2	2.25291
$924$	0	0
$925$	−6970.42	−0.247769
$926$	33462.2	1.18751
$927$	−10051.5	−0.356133
$928$	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	$0.569931\pi$
$930$	6923.57	0.244121
$931$	0	0
$932$	−26753.6	−0.940282
$933$	11468.7	0.402433
$934$	44044.8	1.54303
$935$	176.187	0.00616250
$936$	−13543.8	−0.472963
$937$	−3939.78	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$937$	−3939.78	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$938$	0	0
$939$	−3719.19	−0.129256
$940$	−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	$0.708724\pi$
$942$	−47023.5	−1.62644
$943$	−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	$-0.599696\pi$
$947$	−17958.1	−0.616218	−0.308109	+	0.951351i	$0.599696\pi$
$948$	−51053.6	−1.74910
$949$	−26787.0	−0.916273
$950$	29909.3	1.02146
$951$	16131.8	0.550064
$952$	0	0
$953$	9631.60	0.327385	0.163693	−	0.986511i	$-0.447659\pi$
$953$	9631.60	0.327385	0.163693	+	0.986511i	$0.447659\pi$
$954$	20124.4	0.682969
$955$	14592.7	0.494460
$956$	65094.2	2.20219
$957$	−5175.68	−0.174823
$958$	−53038.1	−1.78871
$959$	0	0
$960$	16348.9	0.549643
$961$	−23751.7	−0.797277
$962$	−27458.7	−0.920273
$963$	13358.0	0.446995
$964$	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
$969$	588.184	0.0194997
$970$	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	$0.407544\pi$
$972$	3072.24	0.101381
$973$	0	0
$974$	−22864.4	−0.752180
$975$	17608.3	0.578376
$976$	−846.612	−0.0277658
$977$	8836.93	0.289374	0.144687	−	0.989477i	$-0.453782\pi$
$977$	8836.93	0.289374	0.144687	+	0.989477i	$0.453782\pi$
$978$	12507.3	0.408935
$979$	1221.49	0.0398765
$980$	0	0
$981$	9581.82	0.311849
$982$	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	$0.649709\pi$
$984$	−1823.34	−0.0590712
$985$	−6709.67	−0.217044
$986$	1746.19	0.0563997
$987$	0	0
$988$	72161.2	2.32364
$989$	−21783.4	−0.700377
$990$	−2940.02	−0.0943838
$991$	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
$993$	6964.33	0.222564
$994$	0	0
$995$	8322.39	0.265163
$996$	52320.3	1.66449
$997$	8399.40	0.266812	0.133406	−	0.991061i	$-0.457409\pi$
$997$	8399.40	0.266812	0.133406	+	0.991061i	$0.457409\pi$
$998$	−33473.8	−1.06172
$999$	2287.40	0.0724425

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.k.1.5	✓	5	7.6	odd	2
693.4.a.p.1.1		5	21.20	even	2
1617.4.a.n.1.5		5	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:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q+4.54345 q^{2} +3.00000 q^{3} +12.6430 q^{4} -6.53625 q^{5} +13.6304 q^{6} +21.0952 q^{8} +9.00000 q^{9} -29.6972 q^{10} +11.0000 q^{11} +37.9289 q^{12} -71.3370 q^{13} -19.6088 q^{15} -5.29887 q^{16} -2.45049 q^{17} +40.8911 q^{18} -80.0091 q^{19} -82.6377 q^{20} +49.9780 q^{22} +61.8164 q^{23} +63.2855 q^{24} -82.2774 q^{25} -324.117 q^{26} +27.0000 q^{27} -156.839 q^{29} -89.0915 q^{30} -77.7131 q^{31} -192.837 q^{32} +33.0000 q^{33} -11.1337 q^{34} +113.787 q^{36} +84.7185 q^{37} -363.518 q^{38} -214.011 q^{39} -137.883 q^{40} -28.8114 q^{41} -352.389 q^{43} +139.073 q^{44} -58.8263 q^{45} +280.860 q^{46} +256.310 q^{47} -15.8966 q^{48} -373.824 q^{50} -7.35147 q^{51} -901.913 q^{52} +492.147 q^{53} +122.673 q^{54} -71.8988 q^{55} -240.027 q^{57} -712.590 q^{58} +3.13000 q^{59} -247.913 q^{60} +159.772 q^{61} -353.086 q^{62} -833.753 q^{64} +466.277 q^{65} +149.934 q^{66} -521.200 q^{67} -30.9815 q^{68} +185.449 q^{69} -885.588 q^{71} +189.857 q^{72} +375.499 q^{73} +384.914 q^{74} -246.832 q^{75} -1011.55 q^{76} -972.350 q^{78} -1346.03 q^{79} +34.6348 q^{80} +81.0000 q^{81} -130.903 q^{82} +1379.43 q^{83} +16.0170 q^{85} -1601.07 q^{86} -470.516 q^{87} +232.047 q^{88} +111.045 q^{89} -267.274 q^{90} +781.543 q^{92} -233.139 q^{93} +1164.53 q^{94} +522.959 q^{95} -578.510 q^{96} -472.385 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 15 q^{3} + 21 q^{4} - 21 q^{5} - 3 q^{6} - 42 q^{8} + 45 q^{9} + 23 q^{10} + 55 q^{11} + 63 q^{12} - 101 q^{13} - 63 q^{15} - 7 q^{16} + 20 q^{17} - 9 q^{18} - 237 q^{19} - 85 q^{20} - 11 q^{22}+ \cdots + 495 q^{99}+O(q^{100}) \) 5 * q - q^2 + 15 * q^3 + 21 * q^4 - 21 * q^5 - 3 * q^6 - 42 * q^8 + 45 * q^9 + 23 * q^10 + 55 * q^11 + 63 * q^12 - 101 * q^13 - 63 * q^15 - 7 * q^16 + 20 * q^17 - 9 * q^18 - 237 * q^19 - 85 * q^20 - 11 * q^22 - 80 * q^23 - 126 * q^24 + 486 * q^25 - 165 * q^26 + 135 * q^27 - 11 * q^29 + 69 * q^30 - 316 * q^31 + 453 * q^32 + 165 * q^33 - 936 * q^34 + 189 * q^36 + 319 * q^37 - 89 * q^38 - 303 * q^39 - 624 * q^40 - 1190 * q^41 + 88 * q^43 + 231 * q^44 - 189 * q^45 + 1000 * q^46 - 377 * q^47 - 21 * q^48 - 644 * q^50 + 60 * q^51 - 1001 * q^52 - 992 * q^53 - 27 * q^54 - 231 * q^55 - 711 * q^57 + 721 * q^58 - 71 * q^59 - 255 * q^60 + 574 * q^61 - 272 * q^62 - 1380 * q^64 + 589 * q^65 - 33 * q^66 - 527 * q^67 + 2974 * q^68 - 240 * q^69 - 1156 * q^71 - 378 * q^72 - 1061 * q^73 - 1609 * q^74 + 1458 * q^75 - 2399 * q^76 - 495 * q^78 + 588 * q^79 + 1643 * q^80 + 405 * q^81 + 2602 * q^82 + 212 * q^83 + 1918 * q^85 - 4760 * q^86 - 33 * q^87 - 462 * q^88 - 1030 * q^89 + 207 * q^90 - 1174 * q^92 - 948 * q^93 + 1799 * q^94 - 3593 * q^95 + 1359 * q^96 - 2488 * q^97 + 495 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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