LMFDB - Embedded newform 495.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	495.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:	$29.2059454528$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +6.24621 q^{7} -24.1771 q^{8} +O(q^{10})$	\(q+2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +6.24621 q^{7} -24.1771 q^{8} +12.8078 q^{10} +11.0000 q^{11} -49.1231 q^{13} +16.0000 q^{14} -50.4233 q^{16} -82.7083 q^{17} -130.354 q^{19} -7.19224 q^{20} +28.1771 q^{22} +185.693 q^{23} +25.0000 q^{25} -125.831 q^{26} -8.98485 q^{28} +8.90720 q^{29} +5.26137 q^{31} +64.2547 q^{32} -211.862 q^{34} +31.2311 q^{35} -416.894 q^{37} -333.909 q^{38} -120.885 q^{40} +298.479 q^{41} -513.633 q^{43} -15.8229 q^{44} +475.663 q^{46} -557.295 q^{47} -303.985 q^{49} +64.0388 q^{50} +70.6610 q^{52} +168.064 q^{53} +55.0000 q^{55} -151.015 q^{56} +22.8163 q^{58} -618.773 q^{59} +786.405 q^{61} +13.4773 q^{62} +567.978 q^{64} -245.616 q^{65} -339.015 q^{67} +118.972 q^{68} +80.0000 q^{70} -1120.71 q^{71} -123.430 q^{73} -1067.90 q^{74} +187.508 q^{76} +68.7083 q^{77} -309.835 q^{79} -252.116 q^{80} +764.570 q^{82} +1021.22 q^{83} -413.542 q^{85} -1315.70 q^{86} -265.948 q^{88} +141.879 q^{89} -306.833 q^{91} -267.110 q^{92} -1427.54 q^{94} -651.771 q^{95} +798.345 q^{97} -778.673 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) $ 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 4 * q^7 - 3 * q^8	$ 2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8} + 5 q^{10} + 22 q^{11} - 90 q^{13} + 32 q^{14} - 39 q^{16} + 16 q^{17} - 170 q^{19} - 35 q^{20} + 11 q^{22} + 124 q^{23} + 50 q^{25} - 62 q^{26} + 48 q^{28} + 158 q^{29} + 60 q^{31} - 123 q^{32} - 366 q^{34} - 20 q^{35} - 372 q^{37} - 272 q^{38} - 15 q^{40} - 38 q^{41} - 516 q^{43} - 77 q^{44} + 572 q^{46} - 224 q^{47} - 542 q^{49} + 25 q^{50} + 298 q^{52} - 472 q^{53} + 110 q^{55} - 368 q^{56} - 210 q^{58} - 248 q^{59} + 72 q^{61} - 72 q^{62} + 769 q^{64} - 450 q^{65} - 744 q^{67} - 430 q^{68} + 160 q^{70} - 2060 q^{71} - 486 q^{73} - 1138 q^{74} + 408 q^{76} - 44 q^{77} + 642 q^{79} - 195 q^{80} + 1290 q^{82} + 286 q^{83} + 80 q^{85} - 1312 q^{86} - 33 q^{88} - 244 q^{89} + 112 q^{91} + 76 q^{92} - 1948 q^{94} - 850 q^{95} - 168 q^{97} - 407 q^{98}+O(q^{100}) $ 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 4 * q^7 - 3 * q^8 + 5 * q^10 + 22 * q^11 - 90 * q^13 + 32 * q^14 - 39 * q^16 + 16 * q^17 - 170 * q^19 - 35 * q^20 + 11 * q^22 + 124 * q^23 + 50 * q^25 - 62 * q^26 + 48 * q^28 + 158 * q^29 + 60 * q^31 - 123 * q^32 - 366 * q^34 - 20 * q^35 - 372 * q^37 - 272 * q^38 - 15 * q^40 - 38 * q^41 - 516 * q^43 - 77 * q^44 + 572 * q^46 - 224 * q^47 - 542 * q^49 + 25 * q^50 + 298 * q^52 - 472 * q^53 + 110 * q^55 - 368 * q^56 - 210 * q^58 - 248 * q^59 + 72 * q^61 - 72 * q^62 + 769 * q^64 - 450 * q^65 - 744 * q^67 - 430 * q^68 + 160 * q^70 - 2060 * q^71 - 486 * q^73 - 1138 * q^74 + 408 * q^76 - 44 * q^77 + 642 * q^79 - 195 * q^80 + 1290 * q^82 + 286 * q^83 + 80 * q^85 - 1312 * q^86 - 33 * q^88 - 244 * q^89 + 112 * q^91 + 76 * q^92 - 1948 * q^94 - 850 * q^95 - 168 * q^97 - 407 * 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 $

$2$	2.56155	0.905646	0.452823	−	0.891601i	$-0.350417\pi$
$2$	2.56155	0.905646	0.452823	+	0.891601i	$0.350417\pi$
$3$	0	0
$3$	0	0
$4$	−1.43845	−0.179806
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	6.24621	0.337264	0.168632	−	0.985679i	$-0.446065\pi$
$7$	6.24621	0.337264	0.168632	+	0.985679i	$0.446065\pi$
$8$	−24.1771	−1.06849
$9$	0	0
$10$	12.8078	0.405017
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−49.1231	−1.04802	−0.524011	−	0.851711i	$-0.675565\pi$
$13$	−49.1231	−1.04802	−0.524011	+	0.851711i	$0.675565\pi$
$14$	16.0000	0.305441
$15$	0	0
$16$	−50.4233	−0.787864
$17$	−82.7083	−1.17998	−0.589992	−	0.807409i	$-0.700869\pi$
$17$	−82.7083	−1.17998	−0.589992	+	0.807409i	$0.700869\pi$
$18$	0	0
$19$	−130.354	−1.57396	−0.786981	−	0.616977i	$-0.788357\pi$
$19$	−130.354	−1.57396	−0.786981	+	0.616977i	$0.788357\pi$
$20$	−7.19224	−0.0804116
$21$	0	0
$22$	28.1771	0.273062
$23$	185.693	1.68347	0.841733	−	0.539895i	$-0.181536\pi$
$23$	185.693	1.68347	0.841733	+	0.539895i	$0.181536\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−125.831	−0.949137
$27$	0	0
$28$	−8.98485	−0.0606420
$29$	8.90720	0.0570354	0.0285177	−	0.999593i	$-0.490921\pi$
$29$	8.90720	0.0570354	0.0285177	+	0.999593i	$0.490921\pi$
$30$	0	0
$31$	5.26137	0.0304829	0.0152414	−	0.999884i	$-0.495148\pi$
$31$	5.26137	0.0304829	0.0152414	+	0.999884i	$0.495148\pi$
$32$	64.2547	0.354961
$33$	0	0
$34$	−211.862	−1.06865
$35$	31.2311	0.150829
$36$	0	0
$37$	−416.894	−1.85235	−0.926175	−	0.377094i	$-0.876923\pi$
$37$	−416.894	−1.85235	−0.926175	+	0.377094i	$0.876923\pi$
$38$	−333.909	−1.42545
$39$	0	0
$40$	−120.885	−0.477842
$41$	298.479	1.13694	0.568471	−	0.822703i	$-0.307535\pi$
$41$	298.479	1.13694	0.568471	+	0.822703i	$0.307535\pi$
$42$	0	0
$43$	−513.633	−1.82159	−0.910793	−	0.412863i	$-0.864529\pi$
$43$	−513.633	−1.82159	−0.910793	+	0.412863i	$0.864529\pi$
$44$	−15.8229	−0.0542135
$45$	0	0
$46$	475.663	1.52462
$47$	−557.295	−1.72957	−0.864786	−	0.502140i	$-0.832546\pi$
$47$	−557.295	−1.72957	−0.864786	+	0.502140i	$0.832546\pi$
$48$	0	0
$49$	−303.985	−0.886253
$50$	64.0388	0.181129
$51$	0	0
$52$	70.6610	0.188441
$53$	168.064	0.435574	0.217787	−	0.975996i	$-0.430116\pi$
$53$	168.064	0.435574	0.217787	+	0.975996i	$0.430116\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	−151.015	−0.360362
$57$	0	0
$58$	22.8163	0.0516539
$59$	−618.773	−1.36538	−0.682689	−	0.730709i	$-0.739190\pi$
$59$	−618.773	−1.36538	−0.682689	+	0.730709i	$0.739190\pi$
$60$	0	0
$61$	786.405	1.65064	0.825319	−	0.564667i	$-0.190996\pi$
$61$	786.405	1.65064	0.825319	+	0.564667i	$0.190996\pi$
$62$	13.4773	0.0276067
$63$	0	0
$64$	567.978	1.10933
$65$	−245.616	−0.468690
$66$	0	0
$67$	−339.015	−0.618169	−0.309084	−	0.951035i	$-0.600023\pi$
$67$	−339.015	−0.618169	−0.309084	+	0.951035i	$0.600023\pi$
$68$	118.972	0.212168
$69$	0	0
$70$	80.0000	0.136598
$71$	−1120.71	−1.87329	−0.936645	−	0.350280i	$-0.886087\pi$
$71$	−1120.71	−1.87329	−0.936645	+	0.350280i	$0.886087\pi$
$72$	0	0
$73$	−123.430	−0.197896	−0.0989478	−	0.995093i	$-0.531548\pi$
$73$	−123.430	−0.197896	−0.0989478	+	0.995093i	$0.531548\pi$
$74$	−1067.90	−1.67757
$75$	0	0
$76$	187.508	0.283008
$77$	68.7083	0.101689
$78$	0	0
$79$	−309.835	−0.441255	−0.220628	−	0.975358i	$-0.570811\pi$
$79$	−309.835	−0.441255	−0.220628	+	0.975358i	$0.570811\pi$
$80$	−252.116	−0.352343
$81$	0	0
$82$	764.570	1.02967
$83$	1021.22	1.35053	0.675263	−	0.737577i	$-0.264030\pi$
$83$	1021.22	1.35053	0.675263	+	0.737577i	$0.264030\pi$
$84$	0	0
$85$	−413.542	−0.527705
$86$	−1315.70	−1.64971
$87$	0	0
$88$	−265.948	−0.322161
$89$	141.879	0.168979	0.0844894	−	0.996424i	$-0.473074\pi$
$89$	141.879	0.168979	0.0844894	+	0.996424i	$0.473074\pi$
$90$	0	0
$91$	−306.833	−0.353460
$92$	−267.110	−0.302697
$93$	0	0
$94$	−1427.54	−1.56638
$95$	−651.771	−0.703898
$96$	0	0
$97$	798.345	0.835666	0.417833	−	0.908524i	$-0.362790\pi$
$97$	798.345	0.835666	0.417833	+	0.908524i	$0.362790\pi$
$98$	−778.673	−0.802631
$99$	0	0
$100$	−35.9612	−0.0359612
$101$	−241.400	−0.237823	−0.118912	−	0.992905i	$-0.537941\pi$
$101$	−241.400	−0.237823	−0.118912	+	0.992905i	$0.537941\pi$
$102$	0	0
$103$	1168.38	1.11771	0.558853	−	0.829267i	$-0.311242\pi$
$103$	1168.38	1.11771	0.558853	+	0.829267i	$0.311242\pi$
$104$	1187.65	1.11980
$105$	0	0
$106$	430.506	0.394476
$107$	2106.82	1.90350	0.951748	−	0.306882i	$-0.0992857\pi$
$107$	2106.82	1.90350	0.951748	+	0.306882i	$0.0992857\pi$
$108$	0	0
$109$	493.792	0.433914	0.216957	−	0.976181i	$-0.430387\pi$
$109$	493.792	0.433914	0.216957	+	0.976181i	$0.430387\pi$
$110$	140.885	0.122117
$111$	0	0
$112$	−314.955	−0.265718
$113$	−170.000	−0.141524	−0.0707622	−	0.997493i	$-0.522543\pi$
$113$	−170.000	−0.141524	−0.0707622	+	0.997493i	$0.522543\pi$
$114$	0	0
$115$	928.466	0.752869
$116$	−12.8125	−0.0102553
$117$	0	0
$118$	−1585.02	−1.23655
$119$	−516.614	−0.397966
$120$	0	0
$121$	121.000	0.0909091
$122$	2014.42	1.49489
$123$	0	0
$124$	−7.56820	−0.00548100
$125$	125.000	0.0894427
$126$	0	0
$127$	−948.182	−0.662500	−0.331250	−	0.943543i	$-0.607470\pi$
$127$	−948.182	−0.662500	−0.331250	+	0.943543i	$0.607470\pi$
$128$	940.868	0.649702
$129$	0	0
$130$	−629.157	−0.424467
$131$	1484.84	0.990312	0.495156	−	0.868804i	$-0.335111\pi$
$131$	1484.84	0.990312	0.495156	+	0.868804i	$0.335111\pi$
$132$	0	0
$133$	−814.220	−0.530841
$134$	−868.405	−0.559842
$135$	0	0
$136$	1999.65	1.26080
$137$	−684.928	−0.427134	−0.213567	−	0.976928i	$-0.568508\pi$
$137$	−684.928	−0.427134	−0.213567	+	0.976928i	$0.568508\pi$
$138$	0	0
$139$	830.483	0.506767	0.253384	−	0.967366i	$-0.418457\pi$
$139$	830.483	0.506767	0.253384	+	0.967366i	$0.418457\pi$
$140$	−44.9242	−0.0271199
$141$	0	0
$142$	−2870.75	−1.69654
$143$	−540.354	−0.315991
$144$	0	0
$145$	44.5360	0.0255070
$146$	−316.172	−0.179223
$147$	0	0
$148$	599.680	0.333063
$149$	1213.64	0.667285	0.333642	−	0.942700i	$-0.391722\pi$
$149$	1213.64	0.667285	0.333642	+	0.942700i	$0.391722\pi$
$150$	0	0
$151$	30.8466	0.0166242	0.00831212	−	0.999965i	$-0.497354\pi$
$151$	30.8466	0.0166242	0.00831212	+	0.999965i	$0.497354\pi$
$152$	3151.58	1.68176
$153$	0	0
$154$	176.000	0.0920941
$155$	26.3068	0.0136324
$156$	0	0
$157$	345.239	0.175497	0.0877485	−	0.996143i	$-0.472033\pi$
$157$	345.239	0.175497	0.0877485	+	0.996143i	$0.472033\pi$
$158$	−793.659	−0.399621
$159$	0	0
$160$	321.274	0.158743
$161$	1159.88	0.567772
$162$	0	0
$163$	1921.49	0.923331	0.461665	−	0.887054i	$-0.347252\pi$
$163$	1921.49	0.923331	0.461665	+	0.887054i	$0.347252\pi$
$164$	−429.346	−0.204429
$165$	0	0
$166$	2615.91	1.22310
$167$	−172.297	−0.0798369	−0.0399185	−	0.999203i	$-0.512710\pi$
$167$	−172.297	−0.0798369	−0.0399185	+	0.999203i	$0.512710\pi$
$168$	0	0
$169$	216.080	0.0983521
$170$	−1059.31	−0.477913
$171$	0	0
$172$	738.833	0.327532
$173$	1025.29	0.450587	0.225293	−	0.974291i	$-0.427666\pi$
$173$	1025.29	0.450587	0.225293	+	0.974291i	$0.427666\pi$
$174$	0	0
$175$	156.155	0.0674527
$176$	−554.656	−0.237550
$177$	0	0
$178$	363.430	0.153035
$179$	−1658.29	−0.692437	−0.346219	−	0.938154i	$-0.612534\pi$
$179$	−1658.29	−0.692437	−0.346219	+	0.938154i	$0.612534\pi$
$180$	0	0
$181$	−2021.81	−0.830275	−0.415137	−	0.909759i	$-0.636266\pi$
$181$	−2021.81	−0.830275	−0.415137	+	0.909759i	$0.636266\pi$
$182$	−785.970	−0.320110
$183$	0	0
$184$	−4489.52	−1.79876
$185$	−2084.47	−0.828396
$186$	0	0
$187$	−909.792	−0.355778
$188$	801.640	0.310987
$189$	0	0
$190$	−1669.55	−0.637482
$191$	−1440.22	−0.545607	−0.272803	−	0.962070i	$-0.587951\pi$
$191$	−1440.22	−0.545607	−0.272803	+	0.962070i	$0.587951\pi$
$192$	0	0
$193$	−2798.05	−1.04356	−0.521782	−	0.853079i	$-0.674733\pi$
$193$	−2798.05	−1.04356	−0.521782	+	0.853079i	$0.674733\pi$
$194$	2045.00	0.756817
$195$	0	0
$196$	437.266	0.159354
$197$	−458.943	−0.165982	−0.0829908	−	0.996550i	$-0.526447\pi$
$197$	−458.943	−0.165982	−0.0829908	+	0.996550i	$0.526447\pi$
$198$	0	0
$199$	−2371.04	−0.844615	−0.422308	−	0.906453i	$-0.638780\pi$
$199$	−2371.04	−0.844615	−0.422308	+	0.906453i	$0.638780\pi$
$200$	−604.427	−0.213697
$201$	0	0
$202$	−618.358	−0.215384
$203$	55.6363	0.0192360
$204$	0	0
$205$	1492.40	0.508456
$206$	2992.86	1.01225
$207$	0	0
$208$	2476.95	0.825699
$209$	−1433.90	−0.474568
$210$	0	0
$211$	4319.87	1.40944	0.704721	−	0.709484i	$-0.251072\pi$
$211$	4319.87	1.40944	0.704721	+	0.709484i	$0.251072\pi$
$212$	−241.752	−0.0783187
$213$	0	0
$214$	5396.73	1.72389
$215$	−2568.16	−0.814638
$216$	0	0
$217$	32.8636	0.0102808
$218$	1264.87	0.392973
$219$	0	0
$220$	−79.1146	−0.0242450
$221$	4062.89	1.23665
$222$	0	0
$223$	−3837.73	−1.15244	−0.576219	−	0.817295i	$-0.695473\pi$
$223$	−3837.73	−1.15244	−0.576219	+	0.817295i	$0.695473\pi$
$224$	401.349	0.119715
$225$	0	0
$226$	−435.464	−0.128171
$227$	5003.71	1.46303	0.731515	−	0.681825i	$-0.238813\pi$
$227$	5003.71	1.46303	0.731515	+	0.681825i	$0.238813\pi$
$228$	0	0
$229$	−277.375	−0.0800412	−0.0400206	−	0.999199i	$-0.512742\pi$
$229$	−277.375	−0.0800412	−0.0400206	+	0.999199i	$0.512742\pi$
$230$	2378.31	0.681832
$231$	0	0
$232$	−215.350	−0.0609415
$233$	−2269.91	−0.638225	−0.319113	−	0.947717i	$-0.603385\pi$
$233$	−2269.91	−0.638225	−0.319113	+	0.947717i	$0.603385\pi$
$234$	0	0
$235$	−2786.48	−0.773488
$236$	890.072	0.245503
$237$	0	0
$238$	−1323.33	−0.360416
$239$	1617.11	0.437665	0.218832	−	0.975762i	$-0.429775\pi$
$239$	1617.11	0.437665	0.218832	+	0.975762i	$0.429775\pi$
$240$	0	0
$241$	5646.63	1.50926	0.754629	−	0.656151i	$-0.227817\pi$
$241$	5646.63	1.50926	0.754629	+	0.656151i	$0.227817\pi$
$242$	309.948	0.0823314
$243$	0	0
$244$	−1131.20	−0.296794
$245$	−1519.92	−0.396344
$246$	0	0
$247$	6403.40	1.64955
$248$	−127.204	−0.0325705
$249$	0	0
$250$	320.194	0.0810034
$251$	6217.61	1.56355	0.781777	−	0.623558i	$-0.214313\pi$
$251$	6217.61	1.56355	0.781777	+	0.623558i	$0.214313\pi$
$252$	0	0
$253$	2042.62	0.507584
$254$	−2428.82	−0.599991
$255$	0	0
$256$	−2133.74	−0.520933
$257$	−7712.75	−1.87202	−0.936008	−	0.351980i	$-0.885509\pi$
$257$	−7712.75	−1.87202	−0.936008	+	0.351980i	$0.885509\pi$
$258$	0	0
$259$	−2604.01	−0.624730
$260$	353.305	0.0842732
$261$	0	0
$262$	3803.49	0.896871
$263$	−206.347	−0.0483798	−0.0241899	−	0.999707i	$-0.507701\pi$
$263$	−206.347	−0.0483798	−0.0241899	+	0.999707i	$0.507701\pi$
$264$	0	0
$265$	840.322	0.194795
$266$	−2085.67	−0.480753
$267$	0	0
$268$	487.655	0.111150
$269$	−1712.47	−0.388146	−0.194073	−	0.980987i	$-0.562170\pi$
$269$	−1712.47	−0.388146	−0.194073	+	0.980987i	$0.562170\pi$
$270$	0	0
$271$	−477.081	−0.106940	−0.0534698	−	0.998569i	$-0.517028\pi$
$271$	−477.081	−0.106940	−0.0534698	+	0.998569i	$0.517028\pi$
$272$	4170.43	0.929666
$273$	0	0
$274$	−1754.48	−0.386832
$275$	275.000	0.0603023
$276$	0	0
$277$	4283.48	0.929130	0.464565	−	0.885539i	$-0.346211\pi$
$277$	4283.48	0.929130	0.464565	+	0.885539i	$0.346211\pi$
$278$	2127.33	0.458952
$279$	0	0
$280$	−755.076	−0.161159
$281$	−3477.79	−0.738319	−0.369160	−	0.929366i	$-0.620354\pi$
$281$	−3477.79	−0.738319	−0.369160	+	0.929366i	$0.620354\pi$
$282$	0	0
$283$	−6568.27	−1.37966	−0.689829	−	0.723973i	$-0.742314\pi$
$283$	−6568.27	−1.37966	−0.689829	+	0.723973i	$0.742314\pi$
$284$	1612.08	0.336829
$285$	0	0
$286$	−1384.15	−0.286176
$287$	1864.36	0.383449
$288$	0	0
$289$	1927.67	0.392360
$290$	114.081	0.0231003
$291$	0	0
$292$	177.547	0.0355828
$293$	−8352.29	−1.66534	−0.832672	−	0.553766i	$-0.813190\pi$
$293$	−8352.29	−1.66534	−0.832672	+	0.553766i	$0.813190\pi$
$294$	0	0
$295$	−3093.86	−0.610616
$296$	10079.3	1.97921
$297$	0	0
$298$	3108.81	0.604324
$299$	−9121.83	−1.76431
$300$	0	0
$301$	−3208.26	−0.614355
$302$	79.0152	0.0150557
$303$	0	0
$304$	6572.89	1.24007
$305$	3932.03	0.738187
$306$	0	0
$307$	5383.89	1.00090	0.500448	−	0.865767i	$-0.333169\pi$
$307$	5383.89	1.00090	0.500448	+	0.865767i	$0.333169\pi$
$308$	−98.8333	−0.0182843
$309$	0	0
$310$	67.3863	0.0123461
$311$	1790.41	0.326447	0.163223	−	0.986589i	$-0.447811\pi$
$311$	1790.41	0.326447	0.163223	+	0.986589i	$0.447811\pi$
$312$	0	0
$313$	−809.076	−0.146108	−0.0730538	−	0.997328i	$-0.523274\pi$
$313$	−809.076	−0.146108	−0.0730538	+	0.997328i	$0.523274\pi$
$314$	884.347	0.158938
$315$	0	0
$316$	445.682	0.0793403
$317$	−10744.5	−1.90370	−0.951849	−	0.306567i	$-0.900820\pi$
$317$	−10744.5	−1.90370	−0.951849	+	0.306567i	$0.900820\pi$
$318$	0	0
$319$	97.9792	0.0171968
$320$	2839.89	0.496109
$321$	0	0
$322$	2971.09	0.514200
$323$	10781.4	1.85725
$324$	0	0
$325$	−1228.08	−0.209605
$326$	4922.00	0.836210
$327$	0	0
$328$	−7216.35	−1.21481
$329$	−3480.98	−0.583322
$330$	0	0
$331$	3399.12	0.564449	0.282224	−	0.959348i	$-0.408928\pi$
$331$	3399.12	0.564449	0.282224	+	0.959348i	$0.408928\pi$
$332$	−1468.97	−0.242832
$333$	0	0
$334$	−441.349	−0.0723039
$335$	−1695.08	−0.276453
$336$	0	0
$337$	−11840.0	−1.91384	−0.956919	−	0.290356i	$-0.906226\pi$
$337$	−11840.0	−1.91384	−0.956919	+	0.290356i	$0.906226\pi$
$338$	553.499	0.0890721
$339$	0	0
$340$	594.858	0.0948844
$341$	57.8750	0.00919093
$342$	0	0
$343$	−4041.20	−0.636165
$344$	12418.1	1.94634
$345$	0	0
$346$	2626.34	0.408072
$347$	2076.67	0.321272	0.160636	−	0.987014i	$-0.448645\pi$
$347$	2076.67	0.321272	0.160636	+	0.987014i	$0.448645\pi$
$348$	0	0
$349$	5837.37	0.895322	0.447661	−	0.894203i	$-0.352257\pi$
$349$	5837.37	0.895322	0.447661	+	0.894203i	$0.352257\pi$
$350$	400.000	0.0610883
$351$	0	0
$352$	706.802	0.107025
$353$	2423.64	0.365431	0.182715	−	0.983166i	$-0.441511\pi$
$353$	2423.64	0.365431	0.182715	+	0.983166i	$0.441511\pi$
$354$	0	0
$355$	−5603.54	−0.837761
$356$	−204.085	−0.0303834
$357$	0	0
$358$	−4247.79	−0.627103
$359$	3882.22	0.570740	0.285370	−	0.958417i	$-0.407884\pi$
$359$	3882.22	0.570740	0.285370	+	0.958417i	$0.407884\pi$
$360$	0	0
$361$	10133.2	1.47736
$362$	−5178.96	−0.751935
$363$	0	0
$364$	441.363	0.0635542
$365$	−617.150	−0.0885016
$366$	0	0
$367$	5666.65	0.805986	0.402993	−	0.915203i	$-0.367970\pi$
$367$	5666.65	0.805986	0.402993	+	0.915203i	$0.367970\pi$
$368$	−9363.26	−1.32634
$369$	0	0
$370$	−5339.48	−0.750233
$371$	1049.77	0.146903
$372$	0	0
$373$	174.771	0.0242608	0.0121304	−	0.999926i	$-0.496139\pi$
$373$	174.771	0.0242608	0.0121304	+	0.999926i	$0.496139\pi$
$374$	−2330.48	−0.322209
$375$	0	0
$376$	13473.8	1.84802
$377$	−437.550	−0.0597744
$378$	0	0
$379$	252.686	0.0342470	0.0171235	−	0.999853i	$-0.494549\pi$
$379$	252.686	0.0342470	0.0171235	+	0.999853i	$0.494549\pi$
$380$	937.538	0.126565
$381$	0	0
$382$	−3689.21	−0.494126
$383$	11014.5	1.46950	0.734748	−	0.678340i	$-0.237300\pi$
$383$	11014.5	1.46950	0.734748	+	0.678340i	$0.237300\pi$
$384$	0	0
$385$	343.542	0.0454766
$386$	−7167.35	−0.945099
$387$	0	0
$388$	−1148.38	−0.150258
$389$	−8099.40	−1.05567	−0.527835	−	0.849347i	$-0.676996\pi$
$389$	−8099.40	−1.05567	−0.527835	+	0.849347i	$0.676996\pi$
$390$	0	0
$391$	−15358.4	−1.98646
$392$	7349.47	0.946949
$393$	0	0
$394$	−1175.61	−0.150320
$395$	−1549.18	−0.197335
$396$	0	0
$397$	−424.353	−0.0536465	−0.0268232	−	0.999640i	$-0.508539\pi$
$397$	−424.353	−0.0536465	−0.0268232	+	0.999640i	$0.508539\pi$
$398$	−6073.54	−0.764922
$399$	0	0
$400$	−1260.58	−0.157573
$401$	5904.18	0.735263	0.367632	−	0.929972i	$-0.380169\pi$
$401$	5904.18	0.735263	0.367632	+	0.929972i	$0.380169\pi$
$402$	0	0
$403$	−258.455	−0.0319468
$404$	347.241	0.0427620
$405$	0	0
$406$	142.515	0.0174210
$407$	−4585.83	−0.558504
$408$	0	0
$409$	1370.47	0.165686	0.0828430	−	0.996563i	$-0.473600\pi$
$409$	1370.47	0.165686	0.0828430	+	0.996563i	$0.473600\pi$
$410$	3822.85	0.460481
$411$	0	0
$412$	−1680.65	−0.200970
$413$	−3864.98	−0.460493
$414$	0	0
$415$	5106.11	0.603973
$416$	−3156.39	−0.372007
$417$	0	0
$418$	−3673.00	−0.429790
$419$	−1268.73	−0.147927	−0.0739635	−	0.997261i	$-0.523565\pi$
$419$	−1268.73	−0.147927	−0.0739635	+	0.997261i	$0.523565\pi$
$420$	0	0
$421$	−12241.9	−1.41719	−0.708594	−	0.705617i	$-0.750670\pi$
$421$	−12241.9	−1.41719	−0.708594	+	0.705617i	$0.750670\pi$
$422$	11065.6	1.27646
$423$	0	0
$424$	−4063.31	−0.465405
$425$	−2067.71	−0.235997
$426$	0	0
$427$	4912.05	0.556700
$428$	−3030.55	−0.342260
$429$	0	0
$430$	−6578.48	−0.737774
$431$	−8050.11	−0.899675	−0.449838	−	0.893110i	$-0.648518\pi$
$431$	−8050.11	−0.899675	−0.449838	+	0.893110i	$0.648518\pi$
$432$	0	0
$433$	−16565.7	−1.83856	−0.919282	−	0.393600i	$-0.871230\pi$
$433$	−16565.7	−1.83856	−0.919282	+	0.393600i	$0.871230\pi$
$434$	84.1819	0.00931073
$435$	0	0
$436$	−710.293	−0.0780203
$437$	−24205.9	−2.64971
$438$	0	0
$439$	4705.80	0.511607	0.255804	−	0.966729i	$-0.417660\pi$
$439$	4705.80	0.511607	0.255804	+	0.966729i	$0.417660\pi$
$440$	−1329.74	−0.144075
$441$	0	0
$442$	10407.3	1.11997
$443$	−15094.0	−1.61882	−0.809408	−	0.587246i	$-0.800212\pi$
$443$	−15094.0	−1.61882	−0.809408	+	0.587246i	$0.800212\pi$
$444$	0	0
$445$	709.394	0.0755696
$446$	−9830.56	−1.04370
$447$	0	0
$448$	3547.71	0.374138
$449$	−973.478	−0.102319	−0.0511595	−	0.998690i	$-0.516292\pi$
$449$	−973.478	−0.102319	−0.0511595	+	0.998690i	$0.516292\pi$
$450$	0	0
$451$	3283.27	0.342801
$452$	244.536	0.0254469
$453$	0	0
$454$	12817.3	1.32499
$455$	−1534.17	−0.158072
$456$	0	0
$457$	62.6577	0.00641358	0.00320679	−	0.999995i	$-0.498979\pi$
$457$	62.6577	0.00641358	0.00320679	+	0.999995i	$0.498979\pi$
$458$	−710.510	−0.0724890
$459$	0	0
$460$	−1335.55	−0.135370
$461$	11866.2	1.19884	0.599419	−	0.800436i	$-0.295398\pi$
$461$	11866.2	1.19884	0.599419	+	0.800436i	$0.295398\pi$
$462$	0	0
$463$	−13144.8	−1.31942	−0.659711	−	0.751519i	$-0.729321\pi$
$463$	−13144.8	−1.31942	−0.659711	+	0.751519i	$0.729321\pi$
$464$	−449.131	−0.0449361
$465$	0	0
$466$	−5814.48	−0.578006
$467$	10176.8	1.00840	0.504201	−	0.863586i	$-0.331787\pi$
$467$	10176.8	1.00840	0.504201	+	0.863586i	$0.331787\pi$
$468$	0	0
$469$	−2117.56	−0.208486
$470$	−7137.71	−0.700506
$471$	0	0
$472$	14960.1	1.45889
$473$	−5649.96	−0.549229
$474$	0	0
$475$	−3258.85	−0.314793
$476$	743.121	0.0715565
$477$	0	0
$478$	4142.30	0.396369
$479$	−3431.25	−0.327302	−0.163651	−	0.986518i	$-0.552327\pi$
$479$	−3431.25	−0.327302	−0.163651	+	0.986518i	$0.552327\pi$
$480$	0	0
$481$	20479.1	1.94130
$482$	14464.1	1.36685
$483$	0	0
$484$	−174.052	−0.0163460
$485$	3991.72	0.373721
$486$	0	0
$487$	−2833.20	−0.263624	−0.131812	−	0.991275i	$-0.542079\pi$
$487$	−2833.20	−0.263624	−0.131812	+	0.991275i	$0.542079\pi$
$488$	−19013.0	−1.76368
$489$	0	0
$490$	−3893.37	−0.358948
$491$	2667.29	0.245159	0.122580	−	0.992459i	$-0.460883\pi$
$491$	2667.29	0.245159	0.122580	+	0.992459i	$0.460883\pi$
$492$	0	0
$493$	−736.700	−0.0673008
$494$	16402.7	1.49391
$495$	0	0
$496$	−265.295	−0.0240164
$497$	−7000.18	−0.631793
$498$	0	0
$499$	−11137.0	−0.999120	−0.499560	−	0.866279i	$-0.666505\pi$
$499$	−11137.0	−0.999120	−0.499560	+	0.866279i	$0.666505\pi$
$500$	−179.806	−0.0160823
$501$	0	0
$502$	15926.7	1.41603
$503$	−8780.30	−0.778319	−0.389159	−	0.921170i	$-0.627234\pi$
$503$	−8780.30	−0.778319	−0.389159	+	0.921170i	$0.627234\pi$
$504$	0	0
$505$	−1207.00	−0.106358
$506$	5232.29	0.459691
$507$	0	0
$508$	1363.91	0.119121
$509$	−13597.4	−1.18408	−0.592039	−	0.805910i	$-0.701677\pi$
$509$	−13597.4	−1.18408	−0.592039	+	0.805910i	$0.701677\pi$
$510$	0	0
$511$	−770.969	−0.0667430
$512$	−12992.6	−1.12148
$513$	0	0
$514$	−19756.6	−1.69538
$515$	5841.89	0.499854
$516$	0	0
$517$	−6130.25	−0.521486
$518$	−6670.30	−0.565784
$519$	0	0
$520$	5938.27	0.500789
$521$	−14001.3	−1.17736	−0.588682	−	0.808364i	$-0.700353\pi$
$521$	−14001.3	−1.17736	−0.588682	+	0.808364i	$0.700353\pi$
$522$	0	0
$523$	−14749.8	−1.23320	−0.616602	−	0.787275i	$-0.711491\pi$
$523$	−14749.8	−1.23320	−0.616602	+	0.787275i	$0.711491\pi$
$524$	−2135.86	−0.178064
$525$	0	0
$526$	−528.568	−0.0438149
$527$	−435.159	−0.0359693
$528$	0	0
$529$	22315.0	1.83406
$530$	2152.53	0.176415
$531$	0	0
$532$	1171.21	0.0954483
$533$	−14662.2	−1.19154
$534$	0	0
$535$	10534.1	0.851269
$536$	8196.40	0.660505
$537$	0	0
$538$	−4386.59	−0.351523
$539$	−3343.83	−0.267215
$540$	0	0
$541$	1484.06	0.117939	0.0589694	−	0.998260i	$-0.481219\pi$
$541$	1484.06	0.117939	0.0589694	+	0.998260i	$0.481219\pi$
$542$	−1222.07	−0.0968493
$543$	0	0
$544$	−5314.40	−0.418847
$545$	2468.96	0.194052
$546$	0	0
$547$	−16562.2	−1.29460	−0.647302	−	0.762234i	$-0.724103\pi$
$547$	−16562.2	−1.29460	−0.647302	+	0.762234i	$0.724103\pi$
$548$	985.233	0.0768012
$549$	0	0
$550$	704.427	0.0546125
$551$	−1161.09	−0.0897716
$552$	0	0
$553$	−1935.30	−0.148819
$554$	10972.3	0.841463
$555$	0	0
$556$	−1194.61	−0.0911197
$557$	8821.52	0.671059	0.335529	−	0.942030i	$-0.391085\pi$
$557$	8821.52	0.671059	0.335529	+	0.942030i	$0.391085\pi$
$558$	0	0
$559$	25231.2	1.90906
$560$	−1574.77	−0.118833
$561$	0	0
$562$	−8908.55	−0.668656
$563$	5985.53	0.448064	0.224032	−	0.974582i	$-0.428078\pi$
$563$	5985.53	0.448064	0.224032	+	0.974582i	$0.428078\pi$
$564$	0	0
$565$	−850.000	−0.0632916
$566$	−16825.0	−1.24948
$567$	0	0
$568$	27095.5	2.00158
$569$	3453.08	0.254413	0.127206	−	0.991876i	$-0.459399\pi$
$569$	3453.08	0.254413	0.127206	+	0.991876i	$0.459399\pi$
$570$	0	0
$571$	−21484.5	−1.57460	−0.787302	−	0.616568i	$-0.788523\pi$
$571$	−21484.5	−1.57460	−0.787302	+	0.616568i	$0.788523\pi$
$572$	777.271	0.0568170
$573$	0	0
$574$	4775.67	0.347269
$575$	4642.33	0.336693
$576$	0	0
$577$	−13294.4	−0.959189	−0.479594	−	0.877490i	$-0.659216\pi$
$577$	−13294.4	−0.959189	−0.479594	+	0.877490i	$0.659216\pi$
$578$	4937.82	0.355340
$579$	0	0
$580$	−64.0627	−0.00458631
$581$	6378.77	0.455483
$582$	0	0
$583$	1848.71	0.131330
$584$	2984.18	0.211449
$585$	0	0
$586$	−21394.8	−1.50821
$587$	−6695.73	−0.470805	−0.235402	−	0.971898i	$-0.575641\pi$
$587$	−6695.73	−0.470805	−0.235402	+	0.971898i	$0.575641\pi$
$588$	0	0
$589$	−685.841	−0.0479789
$590$	−7925.09	−0.553002
$591$	0	0
$592$	21021.2	1.45940
$593$	10239.6	0.709088	0.354544	−	0.935039i	$-0.384636\pi$
$593$	10239.6	0.709088	0.354544	+	0.935039i	$0.384636\pi$
$594$	0	0
$595$	−2583.07	−0.177976
$596$	−1745.76	−0.119982
$597$	0	0
$598$	−23366.0	−1.59784
$599$	23890.8	1.62963	0.814817	−	0.579719i	$-0.196838\pi$
$599$	23890.8	1.62963	0.814817	+	0.579719i	$0.196838\pi$
$600$	0	0
$601$	−11343.8	−0.769920	−0.384960	−	0.922933i	$-0.625785\pi$
$601$	−11343.8	−0.769920	−0.384960	+	0.922933i	$0.625785\pi$
$602$	−8218.12	−0.556388
$603$	0	0
$604$	−44.3712	−0.00298914
$605$	605.000	0.0406558
$606$	0	0
$607$	−26032.5	−1.74074	−0.870369	−	0.492399i	$-0.836120\pi$
$607$	−26032.5	−1.74074	−0.870369	+	0.492399i	$0.836120\pi$
$608$	−8375.87	−0.558695
$609$	0	0
$610$	10072.1	0.668536
$611$	27376.1	1.81263
$612$	0	0
$613$	−4568.13	−0.300987	−0.150493	−	0.988611i	$-0.548086\pi$
$613$	−4568.13	−0.300987	−0.150493	+	0.988611i	$0.548086\pi$
$614$	13791.1	0.906457
$615$	0	0
$616$	−1661.17	−0.108653
$617$	12755.9	0.832308	0.416154	−	0.909294i	$-0.363378\pi$
$617$	12755.9	0.832308	0.416154	+	0.909294i	$0.363378\pi$
$618$	0	0
$619$	−1138.94	−0.0739545	−0.0369772	−	0.999316i	$-0.511773\pi$
$619$	−1138.94	−0.0739545	−0.0369772	+	0.999316i	$0.511773\pi$
$620$	−37.8410	−0.00245118
$621$	0	0
$622$	4586.24	0.295645
$623$	886.205	0.0569904
$624$	0	0
$625$	625.000	0.0400000
$626$	−2072.49	−0.132322
$627$	0	0
$628$	−496.607	−0.0315554
$629$	34480.6	2.18574
$630$	0	0
$631$	7997.36	0.504548	0.252274	−	0.967656i	$-0.418822\pi$
$631$	7997.36	0.504548	0.252274	+	0.967656i	$0.418822\pi$
$632$	7490.91	0.471475
$633$	0	0
$634$	−27522.7	−1.72408
$635$	−4740.91	−0.296279
$636$	0	0
$637$	14932.7	0.928814
$638$	250.979	0.0155742
$639$	0	0
$640$	4704.34	0.290555
$641$	573.115	0.0353146	0.0176573	−	0.999844i	$-0.494379\pi$
$641$	573.115	0.0353146	0.0176573	+	0.999844i	$0.494379\pi$
$642$	0	0
$643$	−16027.8	−0.983009	−0.491504	−	0.870875i	$-0.663553\pi$
$643$	−16027.8	−0.983009	−0.491504	+	0.870875i	$0.663553\pi$
$644$	−1668.42	−0.102089
$645$	0	0
$646$	27617.1	1.68201
$647$	2622.74	0.159367	0.0796837	−	0.996820i	$-0.474609\pi$
$647$	2622.74	0.159367	0.0796837	+	0.996820i	$0.474609\pi$
$648$	0	0
$649$	−6806.50	−0.411677
$650$	−3145.79	−0.189827
$651$	0	0
$652$	−2763.97	−0.166020
$653$	−3102.00	−0.185897	−0.0929484	−	0.995671i	$-0.529629\pi$
$653$	−3102.00	−0.185897	−0.0929484	+	0.995671i	$0.529629\pi$
$654$	0	0
$655$	7424.19	0.442881
$656$	−15050.3	−0.895755
$657$	0	0
$658$	−8916.73	−0.528283
$659$	20840.2	1.23190	0.615948	−	0.787787i	$-0.288773\pi$
$659$	20840.2	1.23190	0.615948	+	0.787787i	$0.288773\pi$
$660$	0	0
$661$	18242.9	1.07348	0.536738	−	0.843749i	$-0.319657\pi$
$661$	18242.9	1.07348	0.536738	+	0.843749i	$0.319657\pi$
$662$	8707.03	0.511191
$663$	0	0
$664$	−24690.2	−1.44302
$665$	−4071.10	−0.237399
$666$	0	0
$667$	1654.01	0.0960171
$668$	247.841	0.0143551
$669$	0	0
$670$	−4342.03	−0.250369
$671$	8650.46	0.497686
$672$	0	0
$673$	−12746.6	−0.730084	−0.365042	−	0.930991i	$-0.618945\pi$
$673$	−12746.6	−0.730084	−0.365042	+	0.930991i	$0.618945\pi$
$674$	−30328.7	−1.73326
$675$	0	0
$676$	−310.819	−0.0176843
$677$	7683.11	0.436168	0.218084	−	0.975930i	$-0.430019\pi$
$677$	7683.11	0.436168	0.218084	+	0.975930i	$0.430019\pi$
$678$	0	0
$679$	4986.63	0.281840
$680$	9998.23	0.563845
$681$	0	0
$682$	148.250	0.00832373
$683$	21397.1	1.19874	0.599368	−	0.800473i	$-0.295418\pi$
$683$	21397.1	1.19874	0.599368	+	0.800473i	$0.295418\pi$
$684$	0	0
$685$	−3424.64	−0.191020
$686$	−10351.8	−0.576140
$687$	0	0
$688$	25899.0	1.43516
$689$	−8255.84	−0.456491
$690$	0	0
$691$	26137.5	1.43895	0.719477	−	0.694516i	$-0.244381\pi$
$691$	26137.5	1.43895	0.719477	+	0.694516i	$0.244381\pi$
$692$	−1474.83	−0.0810181
$693$	0	0
$694$	5319.50	0.290959
$695$	4152.41	0.226633
$696$	0	0
$697$	−24686.7	−1.34157
$698$	14952.7	0.810845
$699$	0	0
$700$	−224.621	−0.0121284
$701$	−13382.4	−0.721036	−0.360518	−	0.932752i	$-0.617400\pi$
$701$	−13382.4	−0.721036	−0.360518	+	0.932752i	$0.617400\pi$
$702$	0	0
$703$	54343.9	2.91553
$704$	6247.76	0.334476
$705$	0	0
$706$	6208.27	0.330951
$707$	−1507.83	−0.0802092
$708$	0	0
$709$	18164.6	0.962179	0.481090	−	0.876671i	$-0.340241\pi$
$709$	18164.6	0.962179	0.481090	+	0.876671i	$0.340241\pi$
$710$	−14353.8	−0.758715
$711$	0	0
$712$	−3430.21	−0.180552
$713$	977.000	0.0513169
$714$	0	0
$715$	−2701.77	−0.141315
$716$	2385.36	0.124504
$717$	0	0
$718$	9944.50	0.516888
$719$	−9665.62	−0.501344	−0.250672	−	0.968072i	$-0.580652\pi$
$719$	−9665.62	−0.501344	−0.250672	+	0.968072i	$0.580652\pi$
$720$	0	0
$721$	7297.94	0.376962
$722$	25956.7	1.33796
$723$	0	0
$724$	2908.26	0.149288
$725$	222.680	0.0114071
$726$	0	0
$727$	−29779.6	−1.51921	−0.759605	−	0.650385i	$-0.774608\pi$
$727$	−29779.6	−1.51921	−0.759605	+	0.650385i	$0.774608\pi$
$728$	7418.33	0.377667
$729$	0	0
$730$	−1580.86	−0.0801511
$731$	42481.7	2.14944
$732$	0	0
$733$	35029.5	1.76513	0.882567	−	0.470187i	$-0.155814\pi$
$733$	35029.5	1.76513	0.882567	+	0.470187i	$0.155814\pi$
$734$	14515.4	0.729937
$735$	0	0
$736$	11931.7	0.597564
$737$	−3729.17	−0.186385
$738$	0	0
$739$	23297.3	1.15968	0.579842	−	0.814729i	$-0.303114\pi$
$739$	23297.3	1.15968	0.579842	+	0.814729i	$0.303114\pi$
$740$	2998.40	0.148950
$741$	0	0
$742$	2689.03	0.133042
$743$	−21570.4	−1.06506	−0.532530	−	0.846411i	$-0.678759\pi$
$743$	−21570.4	−1.06506	−0.532530	+	0.846411i	$0.678759\pi$
$744$	0	0
$745$	6068.21	0.298419
$746$	447.684	0.0219717
$747$	0	0
$748$	1308.69	0.0639710
$749$	13159.6	0.641980
$750$	0	0
$751$	28554.8	1.38746	0.693729	−	0.720236i	$-0.255966\pi$
$751$	28554.8	1.38746	0.693729	+	0.720236i	$0.255966\pi$
$752$	28100.7	1.36267
$753$	0	0
$754$	−1120.81	−0.0541344
$755$	154.233	0.00743458
$756$	0	0
$757$	7812.81	0.375114	0.187557	−	0.982254i	$-0.439943\pi$
$757$	7812.81	0.375114	0.187557	+	0.982254i	$0.439943\pi$
$758$	647.268	0.0310156
$759$	0	0
$760$	15757.9	0.752105
$761$	−2875.13	−0.136956	−0.0684778	−	0.997653i	$-0.521814\pi$
$761$	−2875.13	−0.136956	−0.0684778	+	0.997653i	$0.521814\pi$
$762$	0	0
$763$	3084.33	0.146344
$764$	2071.69	0.0981033
$765$	0	0
$766$	28214.3	1.33084
$767$	30396.0	1.43095
$768$	0	0
$769$	−27657.7	−1.29696	−0.648479	−	0.761233i	$-0.724594\pi$
$769$	−27657.7	−1.29696	−0.648479	+	0.761233i	$0.724594\pi$
$770$	880.000	0.0411857
$771$	0	0
$772$	4024.84	0.187639
$773$	3929.35	0.182832	0.0914160	−	0.995813i	$-0.470861\pi$
$773$	3929.35	0.182832	0.0914160	+	0.995813i	$0.470861\pi$
$774$	0	0
$775$	131.534	0.00609658
$776$	−19301.6	−0.892898
$777$	0	0
$778$	−20747.0	−0.956064
$779$	−38908.0	−1.78950
$780$	0	0
$781$	−12327.8	−0.564818
$782$	−39341.3	−1.79903
$783$	0	0
$784$	15327.9	0.698247
$785$	1726.19	0.0784847
$786$	0	0
$787$	−21125.7	−0.956860	−0.478430	−	0.878126i	$-0.658794\pi$
$787$	−21125.7	−0.956860	−0.478430	+	0.878126i	$0.658794\pi$
$788$	660.166	0.0298445
$789$	0	0
$790$	−3968.30	−0.178716
$791$	−1061.86	−0.0477310
$792$	0	0
$793$	−38630.7	−1.72991
$794$	−1087.00	−0.0485847
$795$	0	0
$796$	3410.61	0.151867
$797$	−11696.3	−0.519828	−0.259914	−	0.965632i	$-0.583694\pi$
$797$	−11696.3	−0.519828	−0.259914	+	0.965632i	$0.583694\pi$
$798$	0	0
$799$	46093.0	2.04087
$800$	1606.37	0.0709921
$801$	0	0
$802$	15123.9	0.665888
$803$	−1357.73	−0.0596678
$804$	0	0
$805$	5799.39	0.253915
$806$	−662.045	−0.0289324
$807$	0	0
$808$	5836.34	0.254111
$809$	−14310.2	−0.621902	−0.310951	−	0.950426i	$-0.600648\pi$
$809$	−14310.2	−0.621902	−0.310951	+	0.950426i	$0.600648\pi$
$810$	0	0
$811$	21697.9	0.939477	0.469739	−	0.882806i	$-0.344348\pi$
$811$	21697.9	0.939477	0.469739	+	0.882806i	$0.344348\pi$
$812$	−80.0299	−0.00345874
$813$	0	0
$814$	−11746.9	−0.505807
$815$	9607.46	0.412926
$816$	0	0
$817$	66954.1	2.86711
$818$	3510.54	0.150053
$819$	0	0
$820$	−2146.73	−0.0914233
$821$	−3613.00	−0.153587	−0.0767934	−	0.997047i	$-0.524468\pi$
$821$	−3613.00	−0.153587	−0.0767934	+	0.997047i	$0.524468\pi$
$822$	0	0
$823$	−4763.98	−0.201776	−0.100888	−	0.994898i	$-0.532168\pi$
$823$	−4763.98	−0.201776	−0.100888	+	0.994898i	$0.532168\pi$
$824$	−28248.0	−1.19425
$825$	0	0
$826$	−9900.36	−0.417043
$827$	33571.7	1.41161	0.705806	−	0.708405i	$-0.250585\pi$
$827$	33571.7	1.41161	0.705806	+	0.708405i	$0.250585\pi$
$828$	0	0
$829$	17980.5	0.753303	0.376652	−	0.926355i	$-0.377075\pi$
$829$	17980.5	0.753303	0.376652	+	0.926355i	$0.377075\pi$
$830$	13079.6	0.546986
$831$	0	0
$832$	−27900.9	−1.16261
$833$	25142.1	1.04576
$834$	0	0
$835$	−861.486	−0.0357041
$836$	2062.58	0.0853301
$837$	0	0
$838$	−3249.91	−0.133969
$839$	40139.5	1.65169	0.825847	−	0.563895i	$-0.190698\pi$
$839$	40139.5	1.65169	0.825847	+	0.563895i	$0.190698\pi$
$840$	0	0
$841$	−24309.7	−0.996747
$842$	−31358.4	−1.28347
$843$	0	0
$844$	−6213.91	−0.253426
$845$	1080.40	0.0439844
$846$	0	0
$847$	755.792	0.0306603
$848$	−8474.36	−0.343173
$849$	0	0
$850$	−5296.54	−0.213729
$851$	−77414.4	−3.11837
$852$	0	0
$853$	−15369.2	−0.616919	−0.308459	−	0.951237i	$-0.599813\pi$
$853$	−15369.2	−0.616919	−0.308459	+	0.951237i	$0.599813\pi$
$854$	12582.5	0.504173
$855$	0	0
$856$	−50936.8	−2.03386
$857$	−10324.9	−0.411541	−0.205770	−	0.978600i	$-0.565970\pi$
$857$	−10324.9	−0.411541	−0.205770	+	0.978600i	$0.565970\pi$
$858$	0	0
$859$	−27112.5	−1.07691	−0.538455	−	0.842655i	$-0.680992\pi$
$859$	−27112.5	−1.07691	−0.538455	+	0.842655i	$0.680992\pi$
$860$	3694.17	0.146477
$861$	0	0
$862$	−20620.8	−0.814787
$863$	−30463.6	−1.20161	−0.600807	−	0.799394i	$-0.705154\pi$
$863$	−30463.6	−1.20161	−0.600807	+	0.799394i	$0.705154\pi$
$864$	0	0
$865$	5126.46	0.201508
$866$	−42434.0	−1.66509
$867$	0	0
$868$	−47.2726	−0.00184854
$869$	−3408.19	−0.133044
$870$	0	0
$871$	16653.5	0.647855
$872$	−11938.4	−0.463631
$873$	0	0
$874$	−62004.6	−2.39970
$875$	780.776	0.0301658
$876$	0	0
$877$	−5086.12	−0.195833	−0.0979167	−	0.995195i	$-0.531218\pi$
$877$	−5086.12	−0.195833	−0.0979167	+	0.995195i	$0.531218\pi$
$878$	12054.2	0.463335
$879$	0	0
$880$	−2773.28	−0.106236
$881$	10625.5	0.406338	0.203169	−	0.979144i	$-0.434876\pi$
$881$	10625.5	0.406338	0.203169	+	0.979144i	$0.434876\pi$
$882$	0	0
$883$	13112.2	0.499728	0.249864	−	0.968281i	$-0.419614\pi$
$883$	13112.2	0.499728	0.249864	+	0.968281i	$0.419614\pi$
$884$	−5844.25	−0.222357
$885$	0	0
$886$	−38664.0	−1.46607
$887$	14442.8	0.546719	0.273360	−	0.961912i	$-0.411865\pi$
$887$	14442.8	0.546719	0.273360	+	0.961912i	$0.411865\pi$
$888$	0	0
$889$	−5922.54	−0.223437
$890$	1817.15	0.0684393
$891$	0	0
$892$	5520.38	0.207215
$893$	72645.8	2.72228
$894$	0	0
$895$	−8291.44	−0.309667
$896$	5876.86	0.219121
$897$	0	0
$898$	−2493.61	−0.0926648
$899$	46.8641	0.00173860
$900$	0	0
$901$	−13900.3	−0.513970
$902$	8410.27	0.310456
$903$	0	0
$904$	4110.10	0.151217
$905$	−10109.0	−0.371310
$906$	0	0
$907$	−44981.9	−1.64675	−0.823374	−	0.567499i	$-0.807911\pi$
$907$	−44981.9	−1.64675	−0.823374	+	0.567499i	$0.807911\pi$
$908$	−7197.57	−0.263062
$909$	0	0
$910$	−3929.85	−0.143157
$911$	−6841.96	−0.248830	−0.124415	−	0.992230i	$-0.539705\pi$
$911$	−6841.96	−0.248830	−0.124415	+	0.992230i	$0.539705\pi$
$912$	0	0
$913$	11233.4	0.407199
$914$	160.501	0.00580843
$915$	0	0
$916$	398.989	0.0143919
$917$	9274.61	0.333996
$918$	0	0
$919$	−4753.54	−0.170625	−0.0853127	−	0.996354i	$-0.527189\pi$
$919$	−4753.54	−0.170625	−0.0853127	+	0.996354i	$0.527189\pi$
$920$	−22447.6	−0.804430
$921$	0	0
$922$	30395.9	1.08572
$923$	55052.7	1.96325
$924$	0	0
$925$	−10422.3	−0.370470
$926$	−33671.2	−1.19493
$927$	0	0
$928$	572.330	0.0202453
$929$	7507.93	0.265153	0.132576	−	0.991173i	$-0.457675\pi$
$929$	7507.93	0.265153	0.132576	+	0.991173i	$0.457675\pi$
$930$	0	0
$931$	39625.7	1.39493
$932$	3265.14	0.114757
$933$	0	0
$934$	26068.3	0.913256
$935$	−4548.96	−0.159109
$936$	0	0
$937$	8540.47	0.297764	0.148882	−	0.988855i	$-0.452433\pi$
$937$	8540.47	0.297764	0.148882	+	0.988855i	$0.452433\pi$
$938$	−5424.24	−0.188814
$939$	0	0
$940$	4008.20	0.139078
$941$	−9101.13	−0.315290	−0.157645	−	0.987496i	$-0.550390\pi$
$941$	−9101.13	−0.315290	−0.157645	+	0.987496i	$0.550390\pi$
$942$	0	0
$943$	55425.5	1.91400
$944$	31200.6	1.07573
$945$	0	0
$946$	−14472.7	−0.497407
$947$	−47540.0	−1.63130	−0.815650	−	0.578546i	$-0.803620\pi$
$947$	−47540.0	−1.63130	−0.815650	+	0.578546i	$0.803620\pi$
$948$	0	0
$949$	6063.26	0.207399
$950$	−8347.73	−0.285091
$951$	0	0
$952$	12490.2	0.425221
$953$	−47370.7	−1.61016	−0.805082	−	0.593164i	$-0.797879\pi$
$953$	−47370.7	−1.61016	−0.805082	+	0.593164i	$0.797879\pi$
$954$	0	0
$955$	−7201.12	−0.244003
$956$	−2326.12	−0.0786947
$957$	0	0
$958$	−8789.33	−0.296420
$959$	−4278.20	−0.144057
$960$	0	0
$961$	−29763.3	−0.999071
$962$	52458.4	1.75813
$963$	0	0
$964$	−8122.38	−0.271374
$965$	−13990.2	−0.466696
$966$	0	0
$967$	−36171.6	−1.20290	−0.601448	−	0.798912i	$-0.705409\pi$
$967$	−36171.6	−1.20290	−0.601448	+	0.798912i	$0.705409\pi$
$968$	−2925.43	−0.0971351
$969$	0	0
$970$	10225.0	0.338459
$971$	31713.2	1.04812	0.524060	−	0.851681i	$-0.324417\pi$
$971$	31713.2	1.04812	0.524060	+	0.851681i	$0.324417\pi$
$972$	0	0
$973$	5187.37	0.170914
$974$	−7257.40	−0.238750
$975$	0	0
$976$	−39653.1	−1.30048
$977$	−22800.5	−0.746626	−0.373313	−	0.927706i	$-0.621778\pi$
$977$	−22800.5	−0.746626	−0.373313	+	0.927706i	$0.621778\pi$
$978$	0	0
$979$	1560.67	0.0509490
$980$	2186.33	0.0712651
$981$	0	0
$982$	6832.41	0.222027
$983$	44597.4	1.44704	0.723518	−	0.690305i	$-0.242524\pi$
$983$	44597.4	1.44704	0.723518	+	0.690305i	$0.242524\pi$
$984$	0	0
$985$	−2294.72	−0.0742292
$986$	−1887.10	−0.0609507
$987$	0	0
$988$	−9210.95	−0.296599
$989$	−95378.1	−3.06658
$990$	0	0
$991$	−34788.1	−1.11512	−0.557558	−	0.830138i	$-0.688261\pi$
$991$	−34788.1	−1.11512	−0.557558	+	0.830138i	$0.688261\pi$
$992$	338.068	0.0108202
$993$	0	0
$994$	−17931.3	−0.572180
$995$	−11855.2	−0.377723
$996$	0	0
$997$	−20360.5	−0.646765	−0.323383	−	0.946268i	$-0.604820\pi$
$997$	−20360.5	−0.646765	−0.323383	+	0.946268i	$0.604820\pi$
$998$	−28528.0	−0.904849
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.d.1.2		2
3.2	odd	2		165.4.a.c.1.1	✓	2
5.4	even	2		2475.4.a.n.1.1		2
15.2	even	4		825.4.c.j.199.1		4
15.8	even	4		825.4.c.j.199.4		4
15.14	odd	2		825.4.a.m.1.2		2
33.32	even	2		1815.4.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.c.1.1	✓	2	3.2	odd	2
495.4.a.d.1.2		2	1.1	even	1	trivial
825.4.a.m.1.2		2	15.14	odd	2
825.4.c.j.199.1		4	15.2	even	4
825.4.c.j.199.4		4	15.8	even	4
1815.4.a.n.1.2		2	33.32	even	2
2475.4.a.n.1.1		2	5.4	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +6.24621 q^{7} -24.1771 q^{8} +O(q^{10})\)	\(q+2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +6.24621 q^{7} -24.1771 q^{8} +12.8078 q^{10} +11.0000 q^{11} -49.1231 q^{13} +16.0000 q^{14} -50.4233 q^{16} -82.7083 q^{17} -130.354 q^{19} -7.19224 q^{20} +28.1771 q^{22} +185.693 q^{23} +25.0000 q^{25} -125.831 q^{26} -8.98485 q^{28} +8.90720 q^{29} +5.26137 q^{31} +64.2547 q^{32} -211.862 q^{34} +31.2311 q^{35} -416.894 q^{37} -333.909 q^{38} -120.885 q^{40} +298.479 q^{41} -513.633 q^{43} -15.8229 q^{44} +475.663 q^{46} -557.295 q^{47} -303.985 q^{49} +64.0388 q^{50} +70.6610 q^{52} +168.064 q^{53} +55.0000 q^{55} -151.015 q^{56} +22.8163 q^{58} -618.773 q^{59} +786.405 q^{61} +13.4773 q^{62} +567.978 q^{64} -245.616 q^{65} -339.015 q^{67} +118.972 q^{68} +80.0000 q^{70} -1120.71 q^{71} -123.430 q^{73} -1067.90 q^{74} +187.508 q^{76} +68.7083 q^{77} -309.835 q^{79} -252.116 q^{80} +764.570 q^{82} +1021.22 q^{83} -413.542 q^{85} -1315.70 q^{86} -265.948 q^{88} +141.879 q^{89} -306.833 q^{91} -267.110 q^{92} -1427.54 q^{94} -651.771 q^{95} +798.345 q^{97} -778.673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) \) 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 4 * q^7 - 3 * q^8	\( 2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8} + 5 q^{10} + 22 q^{11} - 90 q^{13} + 32 q^{14} - 39 q^{16} + 16 q^{17} - 170 q^{19} - 35 q^{20} + 11 q^{22} + 124 q^{23} + 50 q^{25} - 62 q^{26} + 48 q^{28} + 158 q^{29} + 60 q^{31} - 123 q^{32} - 366 q^{34} - 20 q^{35} - 372 q^{37} - 272 q^{38} - 15 q^{40} - 38 q^{41} - 516 q^{43} - 77 q^{44} + 572 q^{46} - 224 q^{47} - 542 q^{49} + 25 q^{50} + 298 q^{52} - 472 q^{53} + 110 q^{55} - 368 q^{56} - 210 q^{58} - 248 q^{59} + 72 q^{61} - 72 q^{62} + 769 q^{64} - 450 q^{65} - 744 q^{67} - 430 q^{68} + 160 q^{70} - 2060 q^{71} - 486 q^{73} - 1138 q^{74} + 408 q^{76} - 44 q^{77} + 642 q^{79} - 195 q^{80} + 1290 q^{82} + 286 q^{83} + 80 q^{85} - 1312 q^{86} - 33 q^{88} - 244 q^{89} + 112 q^{91} + 76 q^{92} - 1948 q^{94} - 850 q^{95} - 168 q^{97} - 407 q^{98}+O(q^{100}) \) 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 4 * q^7 - 3 * q^8 + 5 * q^10 + 22 * q^11 - 90 * q^13 + 32 * q^14 - 39 * q^16 + 16 * q^17 - 170 * q^19 - 35 * q^20 + 11 * q^22 + 124 * q^23 + 50 * q^25 - 62 * q^26 + 48 * q^28 + 158 * q^29 + 60 * q^31 - 123 * q^32 - 366 * q^34 - 20 * q^35 - 372 * q^37 - 272 * q^38 - 15 * q^40 - 38 * q^41 - 516 * q^43 - 77 * q^44 + 572 * q^46 - 224 * q^47 - 542 * q^49 + 25 * q^50 + 298 * q^52 - 472 * q^53 + 110 * q^55 - 368 * q^56 - 210 * q^58 - 248 * q^59 + 72 * q^61 - 72 * q^62 + 769 * q^64 - 450 * q^65 - 744 * q^67 - 430 * q^68 + 160 * q^70 - 2060 * q^71 - 486 * q^73 - 1138 * q^74 + 408 * q^76 - 44 * q^77 + 642 * q^79 - 195 * q^80 + 1290 * q^82 + 286 * q^83 + 80 * q^85 - 1312 * q^86 - 33 * q^88 - 244 * q^89 + 112 * q^91 + 76 * q^92 - 1948 * q^94 - 850 * q^95 - 168 * q^97 - 407 * q^98

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