LMFDB - Embedded newform 1305.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, 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("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.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:	$76.9974925575$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 $ x^6 - x^5 - 49x^4 + 27x^3 + 692x^2 - 82x - 2588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.35472$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35472 q^{2} -2.45528 q^{4} +5.00000 q^{5} -3.94392 q^{7} +24.6193 q^{8} +O(q^{10})$	\(q-2.35472 q^{2} -2.45528 q^{4} +5.00000 q^{5} -3.94392 q^{7} +24.6193 q^{8} -11.7736 q^{10} -42.8507 q^{11} +36.4288 q^{13} +9.28685 q^{14} -38.3294 q^{16} -17.8279 q^{17} +84.6809 q^{19} -12.2764 q^{20} +100.902 q^{22} -93.9097 q^{23} +25.0000 q^{25} -85.7797 q^{26} +9.68343 q^{28} -29.0000 q^{29} +122.782 q^{31} -106.699 q^{32} +41.9798 q^{34} -19.7196 q^{35} -61.2323 q^{37} -199.400 q^{38} +123.096 q^{40} -13.8437 q^{41} +397.388 q^{43} +105.210 q^{44} +221.131 q^{46} -235.310 q^{47} -327.445 q^{49} -58.8681 q^{50} -89.4428 q^{52} +50.4554 q^{53} -214.253 q^{55} -97.0966 q^{56} +68.2870 q^{58} +476.712 q^{59} -75.4071 q^{61} -289.119 q^{62} +557.882 q^{64} +182.144 q^{65} -698.609 q^{67} +43.7725 q^{68} +46.4342 q^{70} -44.6799 q^{71} +669.611 q^{73} +144.185 q^{74} -207.915 q^{76} +169.000 q^{77} +269.236 q^{79} -191.647 q^{80} +32.5981 q^{82} -869.721 q^{83} -89.1395 q^{85} -935.739 q^{86} -1054.95 q^{88} +9.85326 q^{89} -143.672 q^{91} +230.574 q^{92} +554.089 q^{94} +423.405 q^{95} -1524.47 q^{97} +771.043 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8}+O(q^{10}) $ 6 * q - q^2 + 51 * q^4 + 30 * q^5 + 47 * q^7 - 51 * q^8	$ 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8} - 5 q^{10} - 81 q^{11} + 169 q^{13} + 30 q^{14} + 131 q^{16} + q^{17} + 116 q^{19} + 255 q^{20} + 90 q^{22} + 52 q^{23} + 150 q^{25} - 294 q^{26} + 344 q^{28} - 174 q^{29} + 340 q^{31} - 499 q^{32} + 920 q^{34} + 235 q^{35} + 332 q^{37} + 378 q^{38} - 255 q^{40} + 616 q^{41} + 334 q^{43} + 52 q^{44} - 158 q^{46} + 85 q^{47} + 879 q^{49} - 25 q^{50} + 2220 q^{52} + 850 q^{53} - 405 q^{55} + 624 q^{56} + 29 q^{58} + 758 q^{59} - 36 q^{61} + 152 q^{62} + 1795 q^{64} + 845 q^{65} + 939 q^{67} + 186 q^{68} + 150 q^{70} + 1388 q^{71} + 1708 q^{73} + 814 q^{74} + 566 q^{76} - 2585 q^{77} + 1250 q^{79} + 655 q^{80} + 1372 q^{82} + 748 q^{83} + 5 q^{85} + 800 q^{86} + 536 q^{88} - 1099 q^{89} + 539 q^{91} - 1698 q^{92} - 4542 q^{94} + 580 q^{95} - 22 q^{97} + 1433 q^{98}+O(q^{100}) $ 6 * q - q^2 + 51 * q^4 + 30 * q^5 + 47 * q^7 - 51 * q^8 - 5 * q^10 - 81 * q^11 + 169 * q^13 + 30 * q^14 + 131 * q^16 + q^17 + 116 * q^19 + 255 * q^20 + 90 * q^22 + 52 * q^23 + 150 * q^25 - 294 * q^26 + 344 * q^28 - 174 * q^29 + 340 * q^31 - 499 * q^32 + 920 * q^34 + 235 * q^35 + 332 * q^37 + 378 * q^38 - 255 * q^40 + 616 * q^41 + 334 * q^43 + 52 * q^44 - 158 * q^46 + 85 * q^47 + 879 * q^49 - 25 * q^50 + 2220 * q^52 + 850 * q^53 - 405 * q^55 + 624 * q^56 + 29 * q^58 + 758 * q^59 - 36 * q^61 + 152 * q^62 + 1795 * q^64 + 845 * q^65 + 939 * q^67 + 186 * q^68 + 150 * q^70 + 1388 * q^71 + 1708 * q^73 + 814 * q^74 + 566 * q^76 - 2585 * q^77 + 1250 * q^79 + 655 * q^80 + 1372 * q^82 + 748 * q^83 + 5 * q^85 + 800 * q^86 + 536 * q^88 - 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 4542 * q^94 + 580 * q^95 - 22 * q^97 + 1433 * 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 $

$2$	−2.35472	−0.832520	−0.416260	−	0.909246i	$-0.636659\pi$
$2$	−2.35472	−0.832520	−0.416260	+	0.909246i	$0.636659\pi$
$3$	0	0
$3$	0	0
$4$	−2.45528	−0.306910
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−3.94392	−0.212952	−0.106476	−	0.994315i	$-0.533957\pi$
$7$	−3.94392	−0.212952	−0.106476	+	0.994315i	$0.533957\pi$
$8$	24.6193	1.08803
$9$	0	0
$10$	−11.7736	−0.372314
$11$	−42.8507	−1.17454	−0.587271	−	0.809390i	$-0.699798\pi$
$11$	−42.8507	−1.17454	−0.587271	+	0.809390i	$0.699798\pi$
$12$	0	0
$13$	36.4288	0.777194	0.388597	−	0.921408i	$-0.372960\pi$
$13$	36.4288	0.777194	0.388597	+	0.921408i	$0.372960\pi$
$14$	9.28685	0.177287
$15$	0	0
$16$	−38.3294	−0.598897
$17$	−17.8279	−0.254347	−0.127174	−	0.991880i	$-0.540591\pi$
$17$	−17.8279	−0.254347	−0.127174	+	0.991880i	$0.540591\pi$
$18$	0	0
$19$	84.6809	1.02248	0.511241	−	0.859438i	$-0.329186\pi$
$19$	84.6809	1.02248	0.511241	+	0.859438i	$0.329186\pi$
$20$	−12.2764	−0.137254
$21$	0	0
$22$	100.902	0.977831
$23$	−93.9097	−0.851371	−0.425685	−	0.904871i	$-0.639967\pi$
$23$	−93.9097	−0.851371	−0.425685	+	0.904871i	$0.639967\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−85.7797	−0.647030
$27$	0	0
$28$	9.68343	0.0653570
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	122.782	0.711367	0.355683	−	0.934607i	$-0.384248\pi$
$31$	122.782	0.711367	0.355683	+	0.934607i	$0.384248\pi$
$32$	−106.699	−0.589435
$33$	0	0
$34$	41.9798	0.211749
$35$	−19.7196	−0.0952350
$36$	0	0
$37$	−61.2323	−0.272069	−0.136034	−	0.990704i	$-0.543436\pi$
$37$	−61.2323	−0.272069	−0.136034	+	0.990704i	$0.543436\pi$
$38$	−199.400	−0.851236
$39$	0	0
$40$	123.096	0.486581
$41$	−13.8437	−0.0527323	−0.0263662	−	0.999652i	$-0.508394\pi$
$41$	−13.8437	−0.0527323	−0.0263662	+	0.999652i	$0.508394\pi$
$42$	0	0
$43$	397.388	1.40933	0.704664	−	0.709541i	$-0.251098\pi$
$43$	397.388	1.40933	0.704664	+	0.709541i	$0.251098\pi$
$44$	105.210	0.360479
$45$	0	0
$46$	221.131	0.708783
$47$	−235.310	−0.730286	−0.365143	−	0.930952i	$-0.618980\pi$
$47$	−235.310	−0.730286	−0.365143	+	0.930952i	$0.618980\pi$
$48$	0	0
$49$	−327.445	−0.954652
$50$	−58.8681	−0.166504
$51$	0	0
$52$	−89.4428	−0.238529
$53$	50.4554	0.130766	0.0653829	−	0.997860i	$-0.479173\pi$
$53$	50.4554	0.130766	0.0653829	+	0.997860i	$0.479173\pi$
$54$	0	0
$55$	−214.253	−0.525271
$56$	−97.0966	−0.231698
$57$	0	0
$58$	68.2870	0.154595
$59$	476.712	1.05191	0.525955	−	0.850513i	$-0.323708\pi$
$59$	476.712	1.05191	0.525955	+	0.850513i	$0.323708\pi$
$60$	0	0
$61$	−75.4071	−0.158277	−0.0791384	−	0.996864i	$-0.525217\pi$
$61$	−75.4071	−0.158277	−0.0791384	+	0.996864i	$0.525217\pi$
$62$	−289.119	−0.592227
$63$	0	0
$64$	557.882	1.08961
$65$	182.144	0.347572
$66$	0	0
$67$	−698.609	−1.27386	−0.636931	−	0.770921i	$-0.719796\pi$
$67$	−698.609	−1.27386	−0.636931	+	0.770921i	$0.719796\pi$
$68$	43.7725	0.0780616
$69$	0	0
$70$	46.4342	0.0792850
$71$	−44.6799	−0.0746835	−0.0373417	−	0.999303i	$-0.511889\pi$
$71$	−44.6799	−0.0746835	−0.0373417	+	0.999303i	$0.511889\pi$
$72$	0	0
$73$	669.611	1.07359	0.536795	−	0.843713i	$-0.319635\pi$
$73$	669.611	1.07359	0.536795	+	0.843713i	$0.319635\pi$
$74$	144.185	0.226503
$75$	0	0
$76$	−207.915	−0.313809
$77$	169.000	0.250121
$78$	0	0
$79$	269.236	0.383435	0.191718	−	0.981450i	$-0.438594\pi$
$79$	269.236	0.383435	0.191718	+	0.981450i	$0.438594\pi$
$80$	−191.647	−0.267835
$81$	0	0
$82$	32.5981	0.0439007
$83$	−869.721	−1.15017	−0.575086	−	0.818093i	$-0.695031\pi$
$83$	−869.721	−1.15017	−0.575086	+	0.818093i	$0.695031\pi$
$84$	0	0
$85$	−89.1395	−0.113747
$86$	−935.739	−1.17329
$87$	0	0
$88$	−1054.95	−1.27794
$89$	9.85326	0.0117353	0.00586766	−	0.999983i	$-0.498132\pi$
$89$	9.85326	0.0117353	0.00586766	+	0.999983i	$0.498132\pi$
$90$	0	0
$91$	−143.672	−0.165505
$92$	230.574	0.261294
$93$	0	0
$94$	554.089	0.607978
$95$	423.405	0.457267
$96$	0	0
$97$	−1524.47	−1.59574	−0.797868	−	0.602832i	$-0.794039\pi$
$97$	−1524.47	−1.59574	−0.797868	+	0.602832i	$0.794039\pi$
$98$	771.043	0.794767
$99$	0	0
$100$	−61.3820	−0.0613820
$101$	−517.330	−0.509666	−0.254833	−	0.966985i	$-0.582020\pi$
$101$	−517.330	−0.509666	−0.254833	+	0.966985i	$0.582020\pi$
$102$	0	0
$103$	−1447.99	−1.38519	−0.692595	−	0.721327i	$-0.743533\pi$
$103$	−1447.99	−1.38519	−0.692595	+	0.721327i	$0.743533\pi$
$104$	896.851	0.845610
$105$	0	0
$106$	−118.809	−0.108865
$107$	332.747	0.300634	0.150317	−	0.988638i	$-0.451971\pi$
$107$	332.747	0.300634	0.150317	+	0.988638i	$0.451971\pi$
$108$	0	0
$109$	890.530	0.782544	0.391272	−	0.920275i	$-0.372035\pi$
$109$	890.530	0.782544	0.391272	+	0.920275i	$0.372035\pi$
$110$	504.508	0.437299
$111$	0	0
$112$	151.168	0.127536
$113$	936.802	0.779884	0.389942	−	0.920839i	$-0.372495\pi$
$113$	936.802	0.779884	0.389942	+	0.920839i	$0.372495\pi$
$114$	0	0
$115$	−469.548	−0.380745
$116$	71.2031	0.0569917
$117$	0	0
$118$	−1122.53	−0.875736
$119$	70.3119	0.0541637
$120$	0	0
$121$	505.182	0.379551
$122$	177.563	0.131769
$123$	0	0
$124$	−301.465	−0.218325
$125$	125.000	0.0894427
$126$	0	0
$127$	2450.72	1.71234	0.856168	−	0.516698i	$-0.172839\pi$
$127$	2450.72	1.71234	0.856168	+	0.516698i	$0.172839\pi$
$128$	−460.065	−0.317690
$129$	0	0
$130$	−428.898	−0.289361
$131$	−546.681	−0.364609	−0.182304	−	0.983242i	$-0.558356\pi$
$131$	−546.681	−0.364609	−0.182304	+	0.983242i	$0.558356\pi$
$132$	0	0
$133$	−333.975	−0.217739
$134$	1645.03	1.06052
$135$	0	0
$136$	−438.910	−0.276737
$137$	618.064	0.385436	0.192718	−	0.981254i	$-0.438270\pi$
$137$	618.064	0.385436	0.192718	+	0.981254i	$0.438270\pi$
$138$	0	0
$139$	1285.51	0.784428	0.392214	−	0.919874i	$-0.371709\pi$
$139$	1285.51	0.784428	0.392214	+	0.919874i	$0.371709\pi$
$140$	48.4171	0.0292285
$141$	0	0
$142$	105.209	0.0621755
$143$	−1561.00	−0.912848
$144$	0	0
$145$	−145.000	−0.0830455
$146$	−1576.75	−0.893786
$147$	0	0
$148$	150.342	0.0835005
$149$	125.043	0.0687512	0.0343756	−	0.999409i	$-0.489056\pi$
$149$	125.043	0.0687512	0.0343756	+	0.999409i	$0.489056\pi$
$150$	0	0
$151$	1968.65	1.06097	0.530485	−	0.847694i	$-0.322010\pi$
$151$	1968.65	1.06097	0.530485	+	0.847694i	$0.322010\pi$
$152$	2084.78	1.11249
$153$	0	0
$154$	−397.948	−0.208231
$155$	613.912	0.318133
$156$	0	0
$157$	1125.84	0.572304	0.286152	−	0.958184i	$-0.407624\pi$
$157$	1125.84	0.572304	0.286152	+	0.958184i	$0.407624\pi$
$158$	−633.976	−0.319218
$159$	0	0
$160$	−533.496	−0.263604
$161$	370.373	0.181301
$162$	0	0
$163$	1773.87	0.852394	0.426197	−	0.904630i	$-0.359853\pi$
$163$	1773.87	0.852394	0.426197	+	0.904630i	$0.359853\pi$
$164$	33.9902	0.0161841
$165$	0	0
$166$	2047.95	0.957542
$167$	1420.91	0.658403	0.329201	−	0.944260i	$-0.393221\pi$
$167$	1420.91	0.658403	0.329201	+	0.944260i	$0.393221\pi$
$168$	0	0
$169$	−869.944	−0.395969
$170$	209.899	0.0946971
$171$	0	0
$172$	−975.698	−0.432536
$173$	339.659	0.149270	0.0746352	−	0.997211i	$-0.476221\pi$
$173$	339.659	0.149270	0.0746352	+	0.997211i	$0.476221\pi$
$174$	0	0
$175$	−98.5981	−0.0425904
$176$	1642.44	0.703430
$177$	0	0
$178$	−23.2017	−0.00976990
$179$	2593.10	1.08278	0.541389	−	0.840772i	$-0.317899\pi$
$179$	2593.10	1.08278	0.541389	+	0.840772i	$0.317899\pi$
$180$	0	0
$181$	−335.225	−0.137663	−0.0688317	−	0.997628i	$-0.521927\pi$
$181$	−335.225	−0.137663	−0.0688317	+	0.997628i	$0.521927\pi$
$182$	338.309	0.137786
$183$	0	0
$184$	−2311.99	−0.926316
$185$	−306.162	−0.121673
$186$	0	0
$187$	763.938	0.298742
$188$	577.750	0.224132
$189$	0	0
$190$	−997.001	−0.380685
$191$	−927.961	−0.351544	−0.175772	−	0.984431i	$-0.556242\pi$
$191$	−927.961	−0.351544	−0.175772	+	0.984431i	$0.556242\pi$
$192$	0	0
$193$	2184.47	0.814725	0.407362	−	0.913267i	$-0.366449\pi$
$193$	2184.47	0.814725	0.407362	+	0.913267i	$0.366449\pi$
$194$	3589.70	1.32848
$195$	0	0
$196$	803.970	0.292992
$197$	3765.55	1.36185	0.680924	−	0.732354i	$-0.261578\pi$
$197$	3765.55	1.36185	0.680924	+	0.732354i	$0.261578\pi$
$198$	0	0
$199$	−60.0727	−0.0213992	−0.0106996	−	0.999943i	$-0.503406\pi$
$199$	−60.0727	−0.0213992	−0.0106996	+	0.999943i	$0.503406\pi$
$200$	615.482	0.217606
$201$	0	0
$202$	1218.17	0.424307
$203$	114.374	0.0395442
$204$	0	0
$205$	−69.2186	−0.0235826
$206$	3409.61	1.15320
$207$	0	0
$208$	−1396.29	−0.465459
$209$	−3628.64	−1.20095
$210$	0	0
$211$	−2276.29	−0.742682	−0.371341	−	0.928497i	$-0.621102\pi$
$211$	−2276.29	−0.742682	−0.371341	+	0.928497i	$0.621102\pi$
$212$	−123.882	−0.0401333
$213$	0	0
$214$	−783.527	−0.250284
$215$	1986.94	0.630271
$216$	0	0
$217$	−484.244	−0.151487
$218$	−2096.95	−0.651484
$219$	0	0
$220$	526.052	0.161211
$221$	−649.449	−0.197677
$222$	0	0
$223$	3889.04	1.16784	0.583922	−	0.811809i	$-0.301517\pi$
$223$	3889.04	1.16784	0.583922	+	0.811809i	$0.301517\pi$
$224$	420.813	0.125521
$225$	0	0
$226$	−2205.91	−0.649269
$227$	5689.80	1.66363	0.831817	−	0.555050i	$-0.187301\pi$
$227$	5689.80	1.66363	0.831817	+	0.555050i	$0.187301\pi$
$228$	0	0
$229$	−3253.91	−0.938972	−0.469486	−	0.882940i	$-0.655561\pi$
$229$	−3253.91	−0.938972	−0.469486	+	0.882940i	$0.655561\pi$
$230$	1105.66	0.316978
$231$	0	0
$232$	−713.959	−0.202042
$233$	−1363.72	−0.383436	−0.191718	−	0.981450i	$-0.561406\pi$
$233$	−1363.72	−0.383436	−0.191718	+	0.981450i	$0.561406\pi$
$234$	0	0
$235$	−1176.55	−0.326594
$236$	−1170.46	−0.322841
$237$	0	0
$238$	−165.565	−0.0450924
$239$	4788.65	1.29603	0.648017	−	0.761626i	$-0.275598\pi$
$239$	4788.65	1.29603	0.648017	+	0.761626i	$0.275598\pi$
$240$	0	0
$241$	−632.898	−0.169164	−0.0845820	−	0.996417i	$-0.526956\pi$
$241$	−632.898	−0.169164	−0.0845820	+	0.996417i	$0.526956\pi$
$242$	−1189.56	−0.315984
$243$	0	0
$244$	185.145	0.0485767
$245$	−1637.23	−0.426933
$246$	0	0
$247$	3084.82	0.794667
$248$	3022.81	0.773988
$249$	0	0
$250$	−294.340	−0.0744629
$251$	7413.17	1.86420	0.932101	−	0.362198i	$-0.117973\pi$
$251$	7413.17	1.86420	0.932101	+	0.362198i	$0.117973\pi$
$252$	0	0
$253$	4024.10	0.999971
$254$	−5770.78	−1.42555
$255$	0	0
$256$	−3379.73	−0.825130
$257$	3920.20	0.951500	0.475750	−	0.879581i	$-0.342177\pi$
$257$	3920.20	0.951500	0.475750	+	0.879581i	$0.342177\pi$
$258$	0	0
$259$	241.496	0.0579375
$260$	−447.214	−0.106673
$261$	0	0
$262$	1287.28	0.303544
$263$	−2505.55	−0.587448	−0.293724	−	0.955890i	$-0.594895\pi$
$263$	−2505.55	−0.587448	−0.293724	+	0.955890i	$0.594895\pi$
$264$	0	0
$265$	252.277	0.0584802
$266$	786.419	0.181272
$267$	0	0
$268$	1715.28	0.390960
$269$	3707.17	0.840261	0.420131	−	0.907464i	$-0.361984\pi$
$269$	3707.17	0.840261	0.420131	+	0.907464i	$0.361984\pi$
$270$	0	0
$271$	4768.39	1.06885	0.534426	−	0.845215i	$-0.320528\pi$
$271$	4768.39	1.06885	0.534426	+	0.845215i	$0.320528\pi$
$272$	683.332	0.152328
$273$	0	0
$274$	−1455.37	−0.320883
$275$	−1071.27	−0.234909
$276$	0	0
$277$	2908.10	0.630798	0.315399	−	0.948959i	$-0.397862\pi$
$277$	2908.10	0.630798	0.315399	+	0.948959i	$0.397862\pi$
$278$	−3027.02	−0.653053
$279$	0	0
$280$	−485.483	−0.103618
$281$	−495.963	−0.105291	−0.0526454	−	0.998613i	$-0.516765\pi$
$281$	−495.963	−0.105291	−0.0526454	+	0.998613i	$0.516765\pi$
$282$	0	0
$283$	7296.12	1.53254	0.766271	−	0.642517i	$-0.222110\pi$
$283$	7296.12	1.53254	0.766271	+	0.642517i	$0.222110\pi$
$284$	109.702	0.0229211
$285$	0	0
$286$	3675.72	0.759964
$287$	54.5986	0.0112294
$288$	0	0
$289$	−4595.17	−0.935308
$290$	341.435	0.0691371
$291$	0	0
$292$	−1644.08	−0.329495
$293$	−853.949	−0.170267	−0.0851335	−	0.996370i	$-0.527132\pi$
$293$	−853.949	−0.170267	−0.0851335	+	0.996370i	$0.527132\pi$
$294$	0	0
$295$	2383.56	0.470428
$296$	−1507.50	−0.296018
$297$	0	0
$298$	−294.442	−0.0572368
$299$	−3421.02	−0.661680
$300$	0	0
$301$	−1567.27	−0.300119
$302$	−4635.63	−0.883280
$303$	0	0
$304$	−3245.77	−0.612361
$305$	−377.035	−0.0707836
$306$	0	0
$307$	−694.287	−0.129072	−0.0645359	−	0.997915i	$-0.520557\pi$
$307$	−694.287	−0.129072	−0.0645359	+	0.997915i	$0.520557\pi$
$308$	−414.942	−0.0767646
$309$	0	0
$310$	−1445.59	−0.264852
$311$	−2587.58	−0.471794	−0.235897	−	0.971778i	$-0.575803\pi$
$311$	−2587.58	−0.471794	−0.235897	+	0.971778i	$0.575803\pi$
$312$	0	0
$313$	−3467.68	−0.626214	−0.313107	−	0.949718i	$-0.601370\pi$
$313$	−3467.68	−0.626214	−0.313107	+	0.949718i	$0.601370\pi$
$314$	−2651.04	−0.476455
$315$	0	0
$316$	−661.048	−0.117680
$317$	5158.66	0.914003	0.457001	−	0.889466i	$-0.348923\pi$
$317$	5158.66	0.914003	0.457001	+	0.889466i	$0.348923\pi$
$318$	0	0
$319$	1242.67	0.218107
$320$	2789.41	0.487290
$321$	0	0
$322$	−872.125	−0.150937
$323$	−1509.68	−0.260065
$324$	0	0
$325$	910.720	0.155439
$326$	−4176.97	−0.709635
$327$	0	0
$328$	−340.822	−0.0573743
$329$	928.043	0.155516
$330$	0	0
$331$	11056.9	1.83609	0.918043	−	0.396480i	$-0.129768\pi$
$331$	11056.9	1.83609	0.918043	+	0.396480i	$0.129768\pi$
$332$	2135.41	0.352999
$333$	0	0
$334$	−3345.85	−0.548134
$335$	−3493.05	−0.569688
$336$	0	0
$337$	6128.18	0.990574	0.495287	−	0.868729i	$-0.335063\pi$
$337$	6128.18	0.990574	0.495287	+	0.868729i	$0.335063\pi$
$338$	2048.48	0.329652
$339$	0	0
$340$	218.862	0.0349102
$341$	−5261.31	−0.835531
$342$	0	0
$343$	2644.19	0.416247
$344$	9783.41	1.53339
$345$	0	0
$346$	−799.803	−0.124271
$347$	9277.85	1.43533	0.717667	−	0.696386i	$-0.245210\pi$
$347$	9277.85	1.43533	0.717667	+	0.696386i	$0.245210\pi$
$348$	0	0
$349$	4375.41	0.671090	0.335545	−	0.942024i	$-0.391079\pi$
$349$	4375.41	0.671090	0.335545	+	0.942024i	$0.391079\pi$
$350$	232.171	0.0354573
$351$	0	0
$352$	4572.13	0.692317
$353$	3256.89	0.491067	0.245534	−	0.969388i	$-0.421037\pi$
$353$	3256.89	0.491067	0.245534	+	0.969388i	$0.421037\pi$
$354$	0	0
$355$	−223.399	−0.0333995
$356$	−24.1925	−0.00360169
$357$	0	0
$358$	−6106.03	−0.901435
$359$	6219.24	0.914315	0.457158	−	0.889386i	$-0.348868\pi$
$359$	6219.24	0.914315	0.457158	+	0.889386i	$0.348868\pi$
$360$	0	0
$361$	311.863	0.0454677
$362$	789.362	0.114608
$363$	0	0
$364$	352.755	0.0507951
$365$	3348.06	0.480124
$366$	0	0
$367$	10388.3	1.47756	0.738780	−	0.673947i	$-0.235402\pi$
$367$	10388.3	1.47756	0.738780	+	0.673947i	$0.235402\pi$
$368$	3599.50	0.509883
$369$	0	0
$370$	720.926	0.101295
$371$	−198.992	−0.0278468
$372$	0	0
$373$	13200.4	1.83241	0.916204	−	0.400712i	$-0.131237\pi$
$373$	13200.4	1.83241	0.916204	+	0.400712i	$0.131237\pi$
$374$	−1798.86	−0.248708
$375$	0	0
$376$	−5793.15	−0.794572
$377$	−1056.43	−0.144321
$378$	0	0
$379$	−5927.59	−0.803376	−0.401688	−	0.915777i	$-0.631576\pi$
$379$	−5927.59	−0.803376	−0.401688	+	0.915777i	$0.631576\pi$
$380$	−1039.58	−0.140340
$381$	0	0
$382$	2185.09	0.292667
$383$	−7463.33	−0.995713	−0.497857	−	0.867259i	$-0.665879\pi$
$383$	−7463.33	−0.995713	−0.497857	+	0.867259i	$0.665879\pi$
$384$	0	0
$385$	844.999	0.111858
$386$	−5143.83	−0.678275
$387$	0	0
$388$	3743.00	0.489747
$389$	409.685	0.0533981	0.0266991	−	0.999644i	$-0.491500\pi$
$389$	409.685	0.0533981	0.0266991	+	0.999644i	$0.491500\pi$
$390$	0	0
$391$	1674.21	0.216544
$392$	−8061.47	−1.03869
$393$	0	0
$394$	−8866.82	−1.13377
$395$	1346.18	0.171477
$396$	0	0
$397$	1112.31	0.140618	0.0703091	−	0.997525i	$-0.477601\pi$
$397$	1112.31	0.140618	0.0703091	+	0.997525i	$0.477601\pi$
$398$	141.455	0.0178153
$399$	0	0
$400$	−958.235	−0.119779
$401$	−14999.2	−1.86789	−0.933947	−	0.357411i	$-0.883660\pi$
$401$	−14999.2	−1.86789	−0.933947	+	0.357411i	$0.883660\pi$
$402$	0	0
$403$	4472.81	0.552870
$404$	1270.19	0.156421
$405$	0	0
$406$	−269.319	−0.0329213
$407$	2623.85	0.319556
$408$	0	0
$409$	7595.21	0.918238	0.459119	−	0.888375i	$-0.348165\pi$
$409$	7595.21	0.918238	0.459119	+	0.888375i	$0.348165\pi$
$410$	162.991	0.0196330
$411$	0	0
$412$	3555.21	0.425128
$413$	−1880.12	−0.224006
$414$	0	0
$415$	−4348.61	−0.514373
$416$	−3886.92	−0.458106
$417$	0	0
$418$	8544.44	0.999814
$419$	6501.56	0.758047	0.379024	−	0.925387i	$-0.376260\pi$
$419$	6501.56	0.758047	0.379024	+	0.925387i	$0.376260\pi$
$420$	0	0
$421$	−4381.60	−0.507235	−0.253617	−	0.967305i	$-0.581620\pi$
$421$	−4381.60	−0.507235	−0.253617	+	0.967305i	$0.581620\pi$
$422$	5360.02	0.618298
$423$	0	0
$424$	1242.18	0.142277
$425$	−445.697	−0.0508694
$426$	0	0
$427$	297.400	0.0337053
$428$	−816.986	−0.0922676
$429$	0	0
$430$	−4678.69	−0.524713
$431$	698.269	0.0780382	0.0390191	−	0.999238i	$-0.487577\pi$
$431$	698.269	0.0780382	0.0390191	+	0.999238i	$0.487577\pi$
$432$	0	0
$433$	−6899.14	−0.765707	−0.382854	−	0.923809i	$-0.625059\pi$
$433$	−6899.14	−0.765707	−0.382854	+	0.923809i	$0.625059\pi$
$434$	1140.26	0.126116
$435$	0	0
$436$	−2186.50	−0.240170
$437$	−7952.36	−0.870510
$438$	0	0
$439$	−12502.8	−1.35929	−0.679644	−	0.733542i	$-0.737866\pi$
$439$	−12502.8	−1.35929	−0.679644	+	0.733542i	$0.737866\pi$
$440$	−5274.77	−0.571511
$441$	0	0
$442$	1529.27	0.164570
$443$	−7684.41	−0.824148	−0.412074	−	0.911150i	$-0.635195\pi$
$443$	−7684.41	−0.824148	−0.412074	+	0.911150i	$0.635195\pi$
$444$	0	0
$445$	49.2663	0.00524820
$446$	−9157.61	−0.972255
$447$	0	0
$448$	−2200.24	−0.232035
$449$	−6931.63	−0.728561	−0.364280	−	0.931289i	$-0.618685\pi$
$449$	−6931.63	−0.728561	−0.364280	+	0.931289i	$0.618685\pi$
$450$	0	0
$451$	593.213	0.0619364
$452$	−2300.11	−0.239354
$453$	0	0
$454$	−13397.9	−1.38501
$455$	−718.362	−0.0740161
$456$	0	0
$457$	−669.132	−0.0684916	−0.0342458	−	0.999413i	$-0.510903\pi$
$457$	−669.132	−0.0684916	−0.0342458	+	0.999413i	$0.510903\pi$
$458$	7662.06	0.781713
$459$	0	0
$460$	1152.87	0.116854
$461$	−1369.33	−0.138343	−0.0691715	−	0.997605i	$-0.522036\pi$
$461$	−1369.33	−0.138343	−0.0691715	+	0.997605i	$0.522036\pi$
$462$	0	0
$463$	−8213.65	−0.824451	−0.412225	−	0.911082i	$-0.635248\pi$
$463$	−8213.65	−0.824451	−0.412225	+	0.911082i	$0.635248\pi$
$464$	1111.55	0.111212
$465$	0	0
$466$	3211.19	0.319218
$467$	−19762.1	−1.95821	−0.979103	−	0.203366i	$-0.934812\pi$
$467$	−19762.1	−1.95821	−0.979103	+	0.203366i	$0.934812\pi$
$468$	0	0
$469$	2755.26	0.271271
$470$	2770.44	0.271896
$471$	0	0
$472$	11736.3	1.14451
$473$	−17028.3	−1.65532
$474$	0	0
$475$	2117.02	0.204496
$476$	−172.635	−0.0166234
$477$	0	0
$478$	−11276.0	−1.07898
$479$	−1437.29	−0.137101	−0.0685506	−	0.997648i	$-0.521837\pi$
$479$	−1437.29	−0.137101	−0.0685506	+	0.997648i	$0.521837\pi$
$480$	0	0
$481$	−2230.62	−0.211450
$482$	1490.30	0.140833
$483$	0	0
$484$	−1240.36	−0.116488
$485$	−7622.35	−0.713635
$486$	0	0
$487$	12731.4	1.18463	0.592313	−	0.805708i	$-0.298215\pi$
$487$	12731.4	1.18463	0.592313	+	0.805708i	$0.298215\pi$
$488$	−1856.47	−0.172210
$489$	0	0
$490$	3855.22	0.355431
$491$	15590.5	1.43298	0.716488	−	0.697600i	$-0.245749\pi$
$491$	15590.5	1.43298	0.716488	+	0.697600i	$0.245749\pi$
$492$	0	0
$493$	517.009	0.0472311
$494$	−7263.91	−0.661576
$495$	0	0
$496$	−4706.17	−0.426035
$497$	176.214	0.0159040
$498$	0	0
$499$	−7309.53	−0.655750	−0.327875	−	0.944721i	$-0.606333\pi$
$499$	−7309.53	−0.655750	−0.327875	+	0.944721i	$0.606333\pi$
$500$	−306.910	−0.0274508
$501$	0	0
$502$	−17456.0	−1.55199
$503$	−11671.8	−1.03463	−0.517315	−	0.855795i	$-0.673068\pi$
$503$	−11671.8	−1.03463	−0.517315	+	0.855795i	$0.673068\pi$
$504$	0	0
$505$	−2586.65	−0.227930
$506$	−9475.63	−0.832496
$507$	0	0
$508$	−6017.21	−0.525532
$509$	9056.40	0.788640	0.394320	−	0.918973i	$-0.370980\pi$
$509$	9056.40	0.788640	0.394320	+	0.918973i	$0.370980\pi$
$510$	0	0
$511$	−2640.90	−0.228623
$512$	11638.9	1.00463
$513$	0	0
$514$	−9230.99	−0.792143
$515$	−7239.94	−0.619476
$516$	0	0
$517$	10083.2	0.857752
$518$	−568.655	−0.0482341
$519$	0	0
$520$	4484.25	0.378168
$521$	13061.2	1.09831	0.549156	−	0.835720i	$-0.314949\pi$
$521$	13061.2	1.09831	0.549156	+	0.835720i	$0.314949\pi$
$522$	0	0
$523$	−12711.3	−1.06277	−0.531383	−	0.847132i	$-0.678328\pi$
$523$	−12711.3	−1.06277	−0.531383	+	0.847132i	$0.678328\pi$
$524$	1342.25	0.111902
$525$	0	0
$526$	5899.88	0.489063
$527$	−2188.95	−0.180934
$528$	0	0
$529$	−3347.97	−0.275168
$530$	−594.043	−0.0486860
$531$	0	0
$532$	820.002	0.0668263
$533$	−504.310	−0.0409833
$534$	0	0
$535$	1663.73	0.134448
$536$	−17199.3	−1.38600
$537$	0	0
$538$	−8729.36	−0.699535
$539$	14031.3	1.12128
$540$	0	0
$541$	−16338.9	−1.29845	−0.649227	−	0.760595i	$-0.724908\pi$
$541$	−16338.9	−1.29845	−0.649227	+	0.760595i	$0.724908\pi$
$542$	−11228.2	−0.889841
$543$	0	0
$544$	1902.22	0.149921
$545$	4452.65	0.349964
$546$	0	0
$547$	−9583.07	−0.749072	−0.374536	−	0.927212i	$-0.622198\pi$
$547$	−9583.07	−0.749072	−0.374536	+	0.927212i	$0.622198\pi$
$548$	−1517.52	−0.118294
$549$	0	0
$550$	2522.54	0.195566
$551$	−2455.75	−0.189870
$552$	0	0
$553$	−1061.84	−0.0816532
$554$	−6847.78	−0.525152
$555$	0	0
$556$	−3156.28	−0.240749
$557$	1288.54	0.0980198	0.0490099	−	0.998798i	$-0.484393\pi$
$557$	1288.54	0.0980198	0.0490099	+	0.998798i	$0.484393\pi$
$558$	0	0
$559$	14476.4	1.09532
$560$	755.841	0.0570359
$561$	0	0
$562$	1167.86	0.0876567
$563$	13901.3	1.04062	0.520312	−	0.853976i	$-0.325816\pi$
$563$	13901.3	1.04062	0.520312	+	0.853976i	$0.325816\pi$
$564$	0	0
$565$	4684.01	0.348775
$566$	−17180.4	−1.27587
$567$	0	0
$568$	−1099.99	−0.0812578
$569$	−12631.6	−0.930654	−0.465327	−	0.885139i	$-0.654063\pi$
$569$	−12631.6	−0.930654	−0.465327	+	0.885139i	$0.654063\pi$
$570$	0	0
$571$	836.256	0.0612894	0.0306447	−	0.999530i	$-0.490244\pi$
$571$	836.256	0.0612894	0.0306447	+	0.999530i	$0.490244\pi$
$572$	3832.69	0.280162
$573$	0	0
$574$	−128.564	−0.00934874
$575$	−2347.74	−0.170274
$576$	0	0
$577$	55.7407	0.00402169	0.00201085	−	0.999998i	$-0.499360\pi$
$577$	55.7407	0.00402169	0.00201085	+	0.999998i	$0.499360\pi$
$578$	10820.3	0.778663
$579$	0	0
$580$	356.015	0.0254875
$581$	3430.11	0.244931
$582$	0	0
$583$	−2162.05	−0.153590
$584$	16485.4	1.16810
$585$	0	0
$586$	2010.81	0.141751
$587$	24044.3	1.69065	0.845327	−	0.534250i	$-0.179406\pi$
$587$	24044.3	1.69065	0.845327	+	0.534250i	$0.179406\pi$
$588$	0	0
$589$	10397.3	0.727359
$590$	−5612.63	−0.391641
$591$	0	0
$592$	2347.00	0.162941
$593$	15066.0	1.04332	0.521659	−	0.853154i	$-0.325313\pi$
$593$	15066.0	1.04332	0.521659	+	0.853154i	$0.325313\pi$
$594$	0	0
$595$	351.559	0.0242227
$596$	−307.016	−0.0211004
$597$	0	0
$598$	8055.55	0.550862
$599$	−8354.77	−0.569894	−0.284947	−	0.958543i	$-0.591976\pi$
$599$	−8354.77	−0.569894	−0.284947	+	0.958543i	$0.591976\pi$
$600$	0	0
$601$	5059.15	0.343372	0.171686	−	0.985152i	$-0.445078\pi$
$601$	5059.15	0.343372	0.171686	+	0.985152i	$0.445078\pi$
$602$	3690.48	0.249855
$603$	0	0
$604$	−4833.59	−0.325622
$605$	2525.91	0.169740
$606$	0	0
$607$	15350.8	1.02647	0.513236	−	0.858248i	$-0.328447\pi$
$607$	15350.8	1.02647	0.513236	+	0.858248i	$0.328447\pi$
$608$	−9035.39	−0.602687
$609$	0	0
$610$	887.814	0.0589288
$611$	−8572.04	−0.567574
$612$	0	0
$613$	−17964.0	−1.18362	−0.591811	−	0.806077i	$-0.701587\pi$
$613$	−17964.0	−1.18362	−0.591811	+	0.806077i	$0.701587\pi$
$614$	1634.85	0.107455
$615$	0	0
$616$	4160.66	0.272139
$617$	−14791.0	−0.965093	−0.482546	−	0.875870i	$-0.660288\pi$
$617$	−14791.0	−0.965093	−0.482546	+	0.875870i	$0.660288\pi$
$618$	0	0
$619$	−12870.4	−0.835708	−0.417854	−	0.908514i	$-0.637218\pi$
$619$	−12870.4	−0.835708	−0.417854	+	0.908514i	$0.637218\pi$
$620$	−1507.32	−0.0976381
$621$	0	0
$622$	6093.03	0.392779
$623$	−38.8605	−0.00249906
$624$	0	0
$625$	625.000	0.0400000
$626$	8165.43	0.521336
$627$	0	0
$628$	−2764.25	−0.175646
$629$	1091.64	0.0691998
$630$	0	0
$631$	−16299.2	−1.02831	−0.514154	−	0.857698i	$-0.671894\pi$
$631$	−16299.2	−1.02831	−0.514154	+	0.857698i	$0.671894\pi$
$632$	6628.39	0.417189
$633$	0	0
$634$	−12147.2	−0.760926
$635$	12253.6	0.765780
$636$	0	0
$637$	−11928.4	−0.741950
$638$	−2926.14	−0.181579
$639$	0	0
$640$	−2300.32	−0.142075
$641$	9154.93	0.564115	0.282058	−	0.959397i	$-0.408983\pi$
$641$	9154.93	0.564115	0.282058	+	0.959397i	$0.408983\pi$
$642$	0	0
$643$	7509.43	0.460564	0.230282	−	0.973124i	$-0.426035\pi$
$643$	7509.43	0.460564	0.230282	+	0.973124i	$0.426035\pi$
$644$	−909.368	−0.0556430
$645$	0	0
$646$	3554.89	0.216510
$647$	19945.4	1.21195	0.605977	−	0.795482i	$-0.292782\pi$
$647$	19945.4	1.21195	0.605977	+	0.795482i	$0.292782\pi$
$648$	0	0
$649$	−20427.5	−1.23551
$650$	−2144.49	−0.129406
$651$	0	0
$652$	−4355.34	−0.261608
$653$	8028.91	0.481157	0.240579	−	0.970630i	$-0.422663\pi$
$653$	8028.91	0.481157	0.240579	+	0.970630i	$0.422663\pi$
$654$	0	0
$655$	−2733.40	−0.163058
$656$	530.621	0.0315812
$657$	0	0
$658$	−2185.28	−0.129470
$659$	2943.00	0.173965	0.0869827	−	0.996210i	$-0.472278\pi$
$659$	2943.00	0.173965	0.0869827	+	0.996210i	$0.472278\pi$
$660$	0	0
$661$	−30286.3	−1.78215	−0.891073	−	0.453859i	$-0.850047\pi$
$661$	−30286.3	−1.78215	−0.891073	+	0.453859i	$0.850047\pi$
$662$	−26036.0	−1.52858
$663$	0	0
$664$	−21411.9	−1.25142
$665$	−1669.88	−0.0973759
$666$	0	0
$667$	2723.38	0.158096
$668$	−3488.73	−0.202070
$669$	0	0
$670$	8225.16	0.474277
$671$	3231.25	0.185903
$672$	0	0
$673$	2201.07	0.126070	0.0630348	−	0.998011i	$-0.479922\pi$
$673$	2201.07	0.126070	0.0630348	+	0.998011i	$0.479922\pi$
$674$	−14430.2	−0.824673
$675$	0	0
$676$	2135.95	0.121527
$677$	−22392.9	−1.27124	−0.635619	−	0.772003i	$-0.719255\pi$
$677$	−22392.9	−1.27124	−0.635619	+	0.772003i	$0.719255\pi$
$678$	0	0
$679$	6012.39	0.339815
$680$	−2194.55	−0.123761
$681$	0	0
$682$	12388.9	0.695596
$683$	31770.8	1.77991	0.889953	−	0.456053i	$-0.150737\pi$
$683$	31770.8	1.77991	0.889953	+	0.456053i	$0.150737\pi$
$684$	0	0
$685$	3090.32	0.172372
$686$	−6226.33	−0.346534
$687$	0	0
$688$	−15231.6	−0.844042
$689$	1838.03	0.101630
$690$	0	0
$691$	−19995.8	−1.10084	−0.550418	−	0.834889i	$-0.685532\pi$
$691$	−19995.8	−1.10084	−0.550418	+	0.834889i	$0.685532\pi$
$692$	−833.957	−0.0458126
$693$	0	0
$694$	−21846.8	−1.19495
$695$	6427.55	0.350807
$696$	0	0
$697$	246.804	0.0134123
$698$	−10302.9	−0.558696
$699$	0	0
$700$	242.086	0.0130714
$701$	−5940.97	−0.320096	−0.160048	−	0.987109i	$-0.551165\pi$
$701$	−5940.97	−0.320096	−0.160048	+	0.987109i	$0.551165\pi$
$702$	0	0
$703$	−5185.21	−0.278185
$704$	−23905.6	−1.27980
$705$	0	0
$706$	−7669.08	−0.408824
$707$	2040.31	0.108534
$708$	0	0
$709$	−17955.3	−0.951094	−0.475547	−	0.879690i	$-0.657750\pi$
$709$	−17955.3	−0.951094	−0.475547	+	0.879690i	$0.657750\pi$
$710$	526.044	0.0278057
$711$	0	0
$712$	242.580	0.0127684
$713$	−11530.5	−0.605637
$714$	0	0
$715$	−7804.99	−0.408238
$716$	−6366.78	−0.332315
$717$	0	0
$718$	−14644.6	−0.761186
$719$	22809.5	1.18310	0.591551	−	0.806268i	$-0.298516\pi$
$719$	22809.5	1.18310	0.591551	+	0.806268i	$0.298516\pi$
$720$	0	0
$721$	5710.75	0.294979
$722$	−734.351	−0.0378528
$723$	0	0
$724$	823.070	0.0422502
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−28267.1	−1.44205	−0.721023	−	0.692911i	$-0.756328\pi$
$727$	−28267.1	−1.44205	−0.721023	+	0.692911i	$0.756328\pi$
$728$	−3537.11	−0.180074
$729$	0	0
$730$	−7883.75	−0.399713
$731$	−7084.59	−0.358458
$732$	0	0
$733$	18679.6	0.941264	0.470632	−	0.882330i	$-0.344026\pi$
$733$	18679.6	0.941264	0.470632	+	0.882330i	$0.344026\pi$
$734$	−24461.6	−1.23010
$735$	0	0
$736$	10020.1	0.501828
$737$	29935.9	1.49620
$738$	0	0
$739$	−8577.65	−0.426974	−0.213487	−	0.976946i	$-0.568482\pi$
$739$	−8577.65	−0.426974	−0.213487	+	0.976946i	$0.568482\pi$
$740$	751.712	0.0373426
$741$	0	0
$742$	468.572	0.0231830
$743$	−12442.0	−0.614337	−0.307169	−	0.951655i	$-0.599382\pi$
$743$	−12442.0	−0.614337	−0.307169	+	0.951655i	$0.599382\pi$
$744$	0	0
$745$	625.216	0.0307465
$746$	−31083.2	−1.52552
$747$	0	0
$748$	−1875.68	−0.0916867
$749$	−1312.33	−0.0640206
$750$	0	0
$751$	−28327.2	−1.37640	−0.688198	−	0.725523i	$-0.741598\pi$
$751$	−28327.2	−1.37640	−0.688198	+	0.725523i	$0.741598\pi$
$752$	9019.27	0.437366
$753$	0	0
$754$	2487.61	0.120150
$755$	9843.26	0.474480
$756$	0	0
$757$	28073.3	1.34787	0.673937	−	0.738789i	$-0.264602\pi$
$757$	28073.3	1.34787	0.673937	+	0.738789i	$0.264602\pi$
$758$	13957.8	0.668827
$759$	0	0
$760$	10423.9	0.497520
$761$	12715.7	0.605708	0.302854	−	0.953037i	$-0.402061\pi$
$761$	12715.7	0.605708	0.302854	+	0.953037i	$0.402061\pi$
$762$	0	0
$763$	−3512.18	−0.166644
$764$	2278.40	0.107892
$765$	0	0
$766$	17574.1	0.828952
$767$	17366.0	0.817538
$768$	0	0
$769$	−2730.97	−0.128064	−0.0640320	−	0.997948i	$-0.520396\pi$
$769$	−2730.97	−0.128064	−0.0640320	+	0.997948i	$0.520396\pi$
$770$	−1989.74	−0.0931237
$771$	0	0
$772$	−5363.49	−0.250047
$773$	−21816.6	−1.01512	−0.507561	−	0.861616i	$-0.669453\pi$
$773$	−21816.6	−1.01512	−0.507561	+	0.861616i	$0.669453\pi$
$774$	0	0
$775$	3069.56	0.142273
$776$	−37531.3	−1.73621
$777$	0	0
$778$	−964.696	−0.0444550
$779$	−1172.30	−0.0539178
$780$	0	0
$781$	1914.56	0.0877189
$782$	−3942.31	−0.180277
$783$	0	0
$784$	12550.8	0.571738
$785$	5629.20	0.255942
$786$	0	0
$787$	16740.5	0.758239	0.379119	−	0.925348i	$-0.376227\pi$
$787$	16740.5	0.758239	0.379119	+	0.925348i	$0.376227\pi$
$788$	−9245.47	−0.417965
$789$	0	0
$790$	−3169.88	−0.142758
$791$	−3694.67	−0.166078
$792$	0	0
$793$	−2746.99	−0.123012
$794$	−2619.19	−0.117068
$795$	0	0
$796$	147.495	0.00656762
$797$	−3731.13	−0.165826	−0.0829131	−	0.996557i	$-0.526422\pi$
$797$	−3731.13	−0.165826	−0.0829131	+	0.996557i	$0.526422\pi$
$798$	0	0
$799$	4195.07	0.185746
$800$	−2667.48	−0.117887
$801$	0	0
$802$	35319.0	1.55506
$803$	−28693.3	−1.26098
$804$	0	0
$805$	1851.86	0.0810802
$806$	−10532.2	−0.460276
$807$	0	0
$808$	−12736.3	−0.554531
$809$	1318.79	0.0573129	0.0286564	−	0.999589i	$-0.490877\pi$
$809$	1318.79	0.0573129	0.0286564	+	0.999589i	$0.490877\pi$
$810$	0	0
$811$	−6890.79	−0.298358	−0.149179	−	0.988810i	$-0.547663\pi$
$811$	−6890.79	−0.298358	−0.149179	+	0.988810i	$0.547663\pi$
$812$	−280.819	−0.0121365
$813$	0	0
$814$	−6178.44	−0.266037
$815$	8869.35	0.381202
$816$	0	0
$817$	33651.2	1.44101
$818$	−17884.6	−0.764452
$819$	0	0
$820$	169.951	0.00723773
$821$	−25055.7	−1.06510	−0.532552	−	0.846397i	$-0.678767\pi$
$821$	−25055.7	−1.06510	−0.532552	+	0.846397i	$0.678767\pi$
$822$	0	0
$823$	−28476.2	−1.20610	−0.603049	−	0.797704i	$-0.706048\pi$
$823$	−28476.2	−1.20610	−0.603049	+	0.797704i	$0.706048\pi$
$824$	−35648.4	−1.50713
$825$	0	0
$826$	4427.15	0.186490
$827$	6404.82	0.269308	0.134654	−	0.990893i	$-0.457008\pi$
$827$	6404.82	0.269308	0.134654	+	0.990893i	$0.457008\pi$
$828$	0	0
$829$	42500.5	1.78058	0.890292	−	0.455390i	$-0.150500\pi$
$829$	42500.5	1.78058	0.890292	+	0.455390i	$0.150500\pi$
$830$	10239.8	0.428226
$831$	0	0
$832$	20323.0	0.846841
$833$	5837.66	0.242813
$834$	0	0
$835$	7104.55	0.294447
$836$	8909.31	0.368583
$837$	0	0
$838$	−15309.4	−0.631090
$839$	−7738.21	−0.318418	−0.159209	−	0.987245i	$-0.550894\pi$
$839$	−7738.21	−0.318418	−0.159209	+	0.987245i	$0.550894\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	10317.4	0.422283
$843$	0	0
$844$	5588.91	0.227936
$845$	−4349.72	−0.177083
$846$	0	0
$847$	−1992.40	−0.0808260
$848$	−1933.93	−0.0783152
$849$	0	0
$850$	1049.49	0.0423498
$851$	5750.31	0.231631
$852$	0	0
$853$	31504.1	1.26457	0.632286	−	0.774735i	$-0.282117\pi$
$853$	31504.1	1.26457	0.632286	+	0.774735i	$0.282117\pi$
$854$	−700.294	−0.0280604
$855$	0	0
$856$	8191.99	0.327099
$857$	33621.2	1.34011	0.670057	−	0.742310i	$-0.266270\pi$
$857$	33621.2	1.34011	0.670057	+	0.742310i	$0.266270\pi$
$858$	0	0
$859$	35655.1	1.41622	0.708111	−	0.706101i	$-0.249547\pi$
$859$	35655.1	1.41622	0.708111	+	0.706101i	$0.249547\pi$
$860$	−4878.49	−0.193436
$861$	0	0
$862$	−1644.23	−0.0649684
$863$	20500.2	0.808615	0.404307	−	0.914623i	$-0.367513\pi$
$863$	20500.2	0.808615	0.404307	+	0.914623i	$0.367513\pi$
$864$	0	0
$865$	1698.29	0.0667558
$866$	16245.6	0.637467
$867$	0	0
$868$	1188.95	0.0464928
$869$	−11536.9	−0.450361
$870$	0	0
$871$	−25449.5	−0.990038
$872$	21924.2	0.851431
$873$	0	0
$874$	18725.6	0.724718
$875$	−492.990	−0.0190470
$876$	0	0
$877$	−15240.7	−0.586822	−0.293411	−	0.955986i	$-0.594790\pi$
$877$	−15240.7	−0.586822	−0.293411	+	0.955986i	$0.594790\pi$
$878$	29440.7	1.13164
$879$	0	0
$880$	8212.20	0.314583
$881$	17749.3	0.678761	0.339381	−	0.940649i	$-0.389783\pi$
$881$	17749.3	0.678761	0.339381	+	0.940649i	$0.389783\pi$
$882$	0	0
$883$	−2290.27	−0.0872861	−0.0436431	−	0.999047i	$-0.513896\pi$
$883$	−2290.27	−0.0872861	−0.0436431	+	0.999047i	$0.513896\pi$
$884$	1594.58	0.0606690
$885$	0	0
$886$	18094.7	0.686120
$887$	45630.9	1.72732	0.863662	−	0.504072i	$-0.168165\pi$
$887$	45630.9	1.72732	0.863662	+	0.504072i	$0.168165\pi$
$888$	0	0
$889$	−9665.47	−0.364645
$890$	−116.009	−0.00436923
$891$	0	0
$892$	−9548.68	−0.358423
$893$	−19926.2	−0.746703
$894$	0	0
$895$	12965.5	0.484233
$896$	1814.46	0.0676527
$897$	0	0
$898$	16322.1	0.606542
$899$	−3560.69	−0.132097
$900$	0	0
$901$	−899.514	−0.0332599
$902$	−1396.85	−0.0515633
$903$	0	0
$904$	23063.4	0.848537
$905$	−1676.12	−0.0615649
$906$	0	0
$907$	38959.7	1.42628	0.713140	−	0.701022i	$-0.247272\pi$
$907$	38959.7	1.42628	0.713140	+	0.701022i	$0.247272\pi$
$908$	−13970.0	−0.510586
$909$	0	0
$910$	1691.54	0.0616199
$911$	−34221.7	−1.24458	−0.622291	−	0.782786i	$-0.713798\pi$
$911$	−34221.7	−1.24458	−0.622291	+	0.782786i	$0.713798\pi$
$912$	0	0
$913$	37268.1	1.35093
$914$	1575.62	0.0570207
$915$	0	0
$916$	7989.26	0.288180
$917$	2156.07	0.0776441
$918$	0	0
$919$	519.522	0.0186479	0.00932396	−	0.999957i	$-0.497032\pi$
$919$	519.522	0.0186479	0.00932396	+	0.999957i	$0.497032\pi$
$920$	−11559.9	−0.414261
$921$	0	0
$922$	3224.40	0.115173
$923$	−1627.63	−0.0580436
$924$	0	0
$925$	−1530.81	−0.0544137
$926$	19340.9	0.686372
$927$	0	0
$928$	3094.28	0.109455
$929$	−36926.5	−1.30411	−0.652056	−	0.758171i	$-0.726093\pi$
$929$	−36926.5	−1.30411	−0.652056	+	0.758171i	$0.726093\pi$
$930$	0	0
$931$	−27728.4	−0.976113
$932$	3348.32	0.117680
$933$	0	0
$934$	46534.3	1.63025
$935$	3819.69	0.133601
$936$	0	0
$937$	24916.0	0.868697	0.434348	−	0.900745i	$-0.356979\pi$
$937$	24916.0	0.868697	0.434348	+	0.900745i	$0.356979\pi$
$938$	−6487.88	−0.225839
$939$	0	0
$940$	2888.75	0.100235
$941$	41585.1	1.44063	0.720317	−	0.693645i	$-0.243996\pi$
$941$	41585.1	1.44063	0.720317	+	0.693645i	$0.243996\pi$
$942$	0	0
$943$	1300.06	0.0448948
$944$	−18272.1	−0.629985
$945$	0	0
$946$	40097.1	1.37808
$947$	−173.666	−0.00595923	−0.00297962	−	0.999996i	$-0.500948\pi$
$947$	−173.666	−0.00595923	−0.00297962	+	0.999996i	$0.500948\pi$
$948$	0	0
$949$	24393.1	0.834388
$950$	−4985.01	−0.170247
$951$	0	0
$952$	1731.03	0.0589317
$953$	22394.5	0.761207	0.380603	−	0.924738i	$-0.375716\pi$
$953$	22394.5	0.761207	0.380603	+	0.924738i	$0.375716\pi$
$954$	0	0
$955$	−4639.80	−0.157215
$956$	−11757.5	−0.397766
$957$	0	0
$958$	3384.42	0.114140
$959$	−2437.60	−0.0820793
$960$	0	0
$961$	−14715.5	−0.493957
$962$	5252.49	0.176037
$963$	0	0
$964$	1553.94	0.0519181
$965$	10922.4	0.364356
$966$	0	0
$967$	35240.6	1.17194	0.585968	−	0.810334i	$-0.300715\pi$
$967$	35240.6	1.17194	0.585968	+	0.810334i	$0.300715\pi$
$968$	12437.2	0.412962
$969$	0	0
$970$	17948.5	0.594116
$971$	23385.4	0.772888	0.386444	−	0.922313i	$-0.373703\pi$
$971$	23385.4	0.772888	0.386444	+	0.922313i	$0.373703\pi$
$972$	0	0
$973$	−5069.95	−0.167045
$974$	−29978.8	−0.986226
$975$	0	0
$976$	2890.31	0.0947915
$977$	−37442.2	−1.22608	−0.613041	−	0.790051i	$-0.710054\pi$
$977$	−37442.2	−1.22608	−0.613041	+	0.790051i	$0.710054\pi$
$978$	0	0
$979$	−422.219	−0.0137836
$980$	4019.85	0.131030
$981$	0	0
$982$	−36711.4	−1.19298
$983$	35213.6	1.14256	0.571282	−	0.820754i	$-0.306446\pi$
$983$	35213.6	1.14256	0.571282	+	0.820754i	$0.306446\pi$
$984$	0	0
$985$	18827.7	0.609037
$986$	−1217.41	−0.0393208
$987$	0	0
$988$	−7574.10	−0.243891
$989$	−37318.6	−1.19986
$990$	0	0
$991$	−43883.0	−1.40665	−0.703324	−	0.710869i	$-0.748302\pi$
$991$	−43883.0	−1.40665	−0.703324	+	0.710869i	$0.748302\pi$
$992$	−13100.8	−0.419305
$993$	0	0
$994$	−414.935	−0.0132404
$995$	−300.363	−0.00957001
$996$	0	0
$997$	−8968.04	−0.284875	−0.142438	−	0.989804i	$-0.545494\pi$
$997$	−8968.04	−0.284875	−0.142438	+	0.989804i	$0.545494\pi$
$998$	17211.9	0.545925
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.h.1.3		6
3.2	odd	2		435.4.a.h.1.4	✓	6
15.14	odd	2		2175.4.a.k.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.h.1.4	✓	6	3.2	odd	2
1305.4.a.h.1.3		6	1.1	even	1	trivial
2175.4.a.k.1.3		6	15.14	odd	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35472 q^{2} -2.45528 q^{4} +5.00000 q^{5} -3.94392 q^{7} +24.6193 q^{8} +O(q^{10})\)	\(q-2.35472 q^{2} -2.45528 q^{4} +5.00000 q^{5} -3.94392 q^{7} +24.6193 q^{8} -11.7736 q^{10} -42.8507 q^{11} +36.4288 q^{13} +9.28685 q^{14} -38.3294 q^{16} -17.8279 q^{17} +84.6809 q^{19} -12.2764 q^{20} +100.902 q^{22} -93.9097 q^{23} +25.0000 q^{25} -85.7797 q^{26} +9.68343 q^{28} -29.0000 q^{29} +122.782 q^{31} -106.699 q^{32} +41.9798 q^{34} -19.7196 q^{35} -61.2323 q^{37} -199.400 q^{38} +123.096 q^{40} -13.8437 q^{41} +397.388 q^{43} +105.210 q^{44} +221.131 q^{46} -235.310 q^{47} -327.445 q^{49} -58.8681 q^{50} -89.4428 q^{52} +50.4554 q^{53} -214.253 q^{55} -97.0966 q^{56} +68.2870 q^{58} +476.712 q^{59} -75.4071 q^{61} -289.119 q^{62} +557.882 q^{64} +182.144 q^{65} -698.609 q^{67} +43.7725 q^{68} +46.4342 q^{70} -44.6799 q^{71} +669.611 q^{73} +144.185 q^{74} -207.915 q^{76} +169.000 q^{77} +269.236 q^{79} -191.647 q^{80} +32.5981 q^{82} -869.721 q^{83} -89.1395 q^{85} -935.739 q^{86} -1054.95 q^{88} +9.85326 q^{89} -143.672 q^{91} +230.574 q^{92} +554.089 q^{94} +423.405 q^{95} -1524.47 q^{97} +771.043 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8}+O(q^{10}) \) 6 * q - q^2 + 51 * q^4 + 30 * q^5 + 47 * q^7 - 51 * q^8	\( 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8} - 5 q^{10} - 81 q^{11} + 169 q^{13} + 30 q^{14} + 131 q^{16} + q^{17} + 116 q^{19} + 255 q^{20} + 90 q^{22} + 52 q^{23} + 150 q^{25} - 294 q^{26} + 344 q^{28} - 174 q^{29} + 340 q^{31} - 499 q^{32} + 920 q^{34} + 235 q^{35} + 332 q^{37} + 378 q^{38} - 255 q^{40} + 616 q^{41} + 334 q^{43} + 52 q^{44} - 158 q^{46} + 85 q^{47} + 879 q^{49} - 25 q^{50} + 2220 q^{52} + 850 q^{53} - 405 q^{55} + 624 q^{56} + 29 q^{58} + 758 q^{59} - 36 q^{61} + 152 q^{62} + 1795 q^{64} + 845 q^{65} + 939 q^{67} + 186 q^{68} + 150 q^{70} + 1388 q^{71} + 1708 q^{73} + 814 q^{74} + 566 q^{76} - 2585 q^{77} + 1250 q^{79} + 655 q^{80} + 1372 q^{82} + 748 q^{83} + 5 q^{85} + 800 q^{86} + 536 q^{88} - 1099 q^{89} + 539 q^{91} - 1698 q^{92} - 4542 q^{94} + 580 q^{95} - 22 q^{97} + 1433 q^{98}+O(q^{100}) \) 6 * q - q^2 + 51 * q^4 + 30 * q^5 + 47 * q^7 - 51 * q^8 - 5 * q^10 - 81 * q^11 + 169 * q^13 + 30 * q^14 + 131 * q^16 + q^17 + 116 * q^19 + 255 * q^20 + 90 * q^22 + 52 * q^23 + 150 * q^25 - 294 * q^26 + 344 * q^28 - 174 * q^29 + 340 * q^31 - 499 * q^32 + 920 * q^34 + 235 * q^35 + 332 * q^37 + 378 * q^38 - 255 * q^40 + 616 * q^41 + 334 * q^43 + 52 * q^44 - 158 * q^46 + 85 * q^47 + 879 * q^49 - 25 * q^50 + 2220 * q^52 + 850 * q^53 - 405 * q^55 + 624 * q^56 + 29 * q^58 + 758 * q^59 - 36 * q^61 + 152 * q^62 + 1795 * q^64 + 845 * q^65 + 939 * q^67 + 186 * q^68 + 150 * q^70 + 1388 * q^71 + 1708 * q^73 + 814 * q^74 + 566 * q^76 - 2585 * q^77 + 1250 * q^79 + 655 * q^80 + 1372 * q^82 + 748 * q^83 + 5 * q^85 + 800 * q^86 + 536 * q^88 - 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 4542 * q^94 + 580 * q^95 - 22 * q^97 + 1433 * q^98

Introduction

L-functions

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Database

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