LMFDB - Embedded newform 1305.4.a.h.1.5

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			1.5
Root			$-3.54861$ 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)

$f(q)$	$=$	$q+3.54861 q^{2} +4.59262 q^{4} +5.00000 q^{5} -21.2667 q^{7} -12.0915 q^{8} +O(q^{10})$	\(q+3.54861 q^{2} +4.59262 q^{4} +5.00000 q^{5} -21.2667 q^{7} -12.0915 q^{8} +17.7430 q^{10} +40.9431 q^{11} -8.89857 q^{13} -75.4671 q^{14} -79.6488 q^{16} +7.68473 q^{17} +40.7787 q^{19} +22.9631 q^{20} +145.291 q^{22} +80.7285 q^{23} +25.0000 q^{25} -31.5775 q^{26} -97.6698 q^{28} -29.0000 q^{29} +82.1098 q^{31} -185.911 q^{32} +27.2701 q^{34} -106.333 q^{35} +223.283 q^{37} +144.708 q^{38} -60.4573 q^{40} +274.422 q^{41} +53.7872 q^{43} +188.036 q^{44} +286.474 q^{46} -17.0236 q^{47} +109.272 q^{49} +88.7152 q^{50} -40.8678 q^{52} +23.0475 q^{53} +204.716 q^{55} +257.145 q^{56} -102.910 q^{58} +399.574 q^{59} +388.308 q^{61} +291.376 q^{62} -22.5340 q^{64} -44.4929 q^{65} +399.434 q^{67} +35.2930 q^{68} -377.336 q^{70} -92.2884 q^{71} +979.407 q^{73} +792.343 q^{74} +187.281 q^{76} -870.724 q^{77} +62.2586 q^{79} -398.244 q^{80} +973.817 q^{82} -163.597 q^{83} +38.4236 q^{85} +190.870 q^{86} -495.062 q^{88} -1371.94 q^{89} +189.243 q^{91} +370.755 q^{92} -60.4101 q^{94} +203.894 q^{95} +1263.64 q^{97} +387.763 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.54861	1.25462	0.627311	−	0.778769i	$-0.284155\pi$
$2$	3.54861	1.25462	0.627311	+	0.778769i	$0.284155\pi$
$3$	0	0
$3$	0	0
$4$	4.59262	0.574078
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−21.2667	−1.14829	−0.574146	−	0.818753i	$-0.694666\pi$
$7$	−21.2667	−1.14829	−0.574146	+	0.818753i	$0.694666\pi$
$8$	−12.0915	−0.534372
$9$	0	0
$10$	17.7430	0.561084
$11$	40.9431	1.12226	0.561128	−	0.827729i	$-0.310368\pi$
$11$	40.9431	1.12226	0.561128	+	0.827729i	$0.310368\pi$
$12$	0	0
$13$	−8.89857	−0.189848	−0.0949238	−	0.995485i	$-0.530261\pi$
$13$	−8.89857	−0.189848	−0.0949238	+	0.995485i	$0.530261\pi$
$14$	−75.4671	−1.44067
$15$	0	0
$16$	−79.6488	−1.24451
$17$	7.68473	0.109636	0.0548182	−	0.998496i	$-0.482542\pi$
$17$	7.68473	0.109636	0.0548182	+	0.998496i	$0.482542\pi$
$18$	0	0
$19$	40.7787	0.492383	0.246192	−	0.969221i	$-0.420821\pi$
$19$	40.7787	0.492383	0.246192	+	0.969221i	$0.420821\pi$
$20$	22.9631	0.256735
$21$	0	0
$22$	145.291	1.40801
$23$	80.7285	0.731872	0.365936	−	0.930640i	$-0.380749\pi$
$23$	80.7285	0.731872	0.365936	+	0.930640i	$0.380749\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−31.5775	−0.238187
$27$	0	0
$28$	−97.6698	−0.659209
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	82.1098	0.475721	0.237861	−	0.971299i	$-0.423554\pi$
$31$	82.1098	0.475721	0.237861	+	0.971299i	$0.423554\pi$
$32$	−185.911	−1.02702
$33$	0	0
$34$	27.2701	0.137552
$35$	−106.333	−0.513532
$36$	0	0
$37$	223.283	0.992093	0.496047	−	0.868296i	$-0.334785\pi$
$37$	223.283	0.992093	0.496047	+	0.868296i	$0.334785\pi$
$38$	144.708	0.617755
$39$	0	0
$40$	−60.4573	−0.238978
$41$	274.422	1.04531	0.522653	−	0.852545i	$-0.324942\pi$
$41$	274.422	1.04531	0.522653	+	0.852545i	$0.324942\pi$
$42$	0	0
$43$	53.7872	0.190755	0.0953775	−	0.995441i	$-0.469594\pi$
$43$	53.7872	0.190755	0.0953775	+	0.995441i	$0.469594\pi$
$44$	188.036	0.644262
$45$	0	0
$46$	286.474	0.918223
$47$	−17.0236	−0.0528329	−0.0264165	−	0.999651i	$-0.508410\pi$
$47$	−17.0236	−0.0528329	−0.0264165	+	0.999651i	$0.508410\pi$
$48$	0	0
$49$	109.272	0.318577
$50$	88.7152	0.250924
$51$	0	0
$52$	−40.8678	−0.108987
$53$	23.0475	0.0597325	0.0298662	−	0.999554i	$-0.490492\pi$
$53$	23.0475	0.0597325	0.0298662	+	0.999554i	$0.490492\pi$
$54$	0	0
$55$	204.716	0.501888
$56$	257.145	0.613615
$57$	0	0
$58$	−102.910	−0.232978
$59$	399.574	0.881697	0.440848	−	0.897582i	$-0.354678\pi$
$59$	399.574	0.881697	0.440848	+	0.897582i	$0.354678\pi$
$60$	0	0
$61$	388.308	0.815046	0.407523	−	0.913195i	$-0.366393\pi$
$61$	388.308	0.815046	0.407523	+	0.913195i	$0.366393\pi$
$62$	291.376	0.596851
$63$	0	0
$64$	−22.5340	−0.0440118
$65$	−44.4929	−0.0849025
$66$	0	0
$67$	399.434	0.728337	0.364169	−	0.931333i	$-0.381353\pi$
$67$	399.434	0.728337	0.364169	+	0.931333i	$0.381353\pi$
$68$	35.2930	0.0629398
$69$	0	0
$70$	−377.336	−0.644289
$71$	−92.2884	−0.154262	−0.0771311	−	0.997021i	$-0.524576\pi$
$71$	−92.2884	−0.154262	−0.0771311	+	0.997021i	$0.524576\pi$
$72$	0	0
$73$	979.407	1.57029	0.785144	−	0.619314i	$-0.212589\pi$
$73$	979.407	1.57029	0.785144	+	0.619314i	$0.212589\pi$
$74$	792.343	1.24470
$75$	0	0
$76$	187.281	0.282666
$77$	−870.724	−1.28868
$78$	0	0
$79$	62.2586	0.0886664	0.0443332	−	0.999017i	$-0.485884\pi$
$79$	62.2586	0.0886664	0.0443332	+	0.999017i	$0.485884\pi$
$80$	−398.244	−0.556563
$81$	0	0
$82$	973.817	1.31146
$83$	−163.597	−0.216350	−0.108175	−	0.994132i	$-0.534501\pi$
$83$	−163.597	−0.216350	−0.108175	+	0.994132i	$0.534501\pi$
$84$	0	0
$85$	38.4236	0.0490309
$86$	190.870	0.239326
$87$	0	0
$88$	−495.062	−0.599702
$89$	−1371.94	−1.63399	−0.816994	−	0.576646i	$-0.804361\pi$
$89$	−1371.94	−1.63399	−0.816994	+	0.576646i	$0.804361\pi$
$90$	0	0
$91$	189.243	0.218001
$92$	370.755	0.420151
$93$	0	0
$94$	−60.4101	−0.0662853
$95$	203.894	0.220200
$96$	0	0
$97$	1263.64	1.32271	0.661355	−	0.750073i	$-0.269982\pi$
$97$	1263.64	1.32271	0.661355	+	0.750073i	$0.269982\pi$
$98$	387.763	0.399693
$99$	0	0
$100$	114.816	0.114816
$101$	−148.533	−0.146332	−0.0731661	−	0.997320i	$-0.523310\pi$
$101$	−148.533	−0.146332	−0.0731661	+	0.997320i	$0.523310\pi$
$102$	0	0
$103$	1485.59	1.42116	0.710580	−	0.703616i	$-0.248433\pi$
$103$	1485.59	1.42116	0.710580	+	0.703616i	$0.248433\pi$
$104$	107.597	0.101449
$105$	0	0
$106$	81.7866	0.0749417
$107$	−461.476	−0.416940	−0.208470	−	0.978029i	$-0.566848\pi$
$107$	−461.476	−0.416940	−0.208470	+	0.978029i	$0.566848\pi$
$108$	0	0
$109$	1516.06	1.33222	0.666112	−	0.745851i	$-0.267957\pi$
$109$	1516.06	1.33222	0.666112	+	0.745851i	$0.267957\pi$
$110$	726.455	0.629680
$111$	0	0
$112$	1693.87	1.42906
$113$	−160.406	−0.133537	−0.0667686	−	0.997768i	$-0.521269\pi$
$113$	−160.406	−0.133537	−0.0667686	+	0.997768i	$0.521269\pi$
$114$	0	0
$115$	403.643	0.327303
$116$	−133.186	−0.106604
$117$	0	0
$118$	1417.93	1.10620
$119$	−163.429	−0.125895
$120$	0	0
$121$	345.339	0.259458
$122$	1377.95	1.02257
$123$	0	0
$124$	377.099	0.273101
$125$	125.000	0.0894427
$126$	0	0
$127$	2583.64	1.80520	0.902601	−	0.430477i	$-0.141655\pi$
$127$	2583.64	1.80520	0.902601	+	0.430477i	$0.141655\pi$
$128$	1407.32	0.971803
$129$	0	0
$130$	−157.888	−0.106521
$131$	151.210	0.100850	0.0504249	−	0.998728i	$-0.483942\pi$
$131$	151.210	0.100850	0.0504249	+	0.998728i	$0.483942\pi$
$132$	0	0
$133$	−867.228	−0.565400
$134$	1417.43	0.913788
$135$	0	0
$136$	−92.9195	−0.0585866
$137$	257.407	0.160524	0.0802619	−	0.996774i	$-0.474424\pi$
$137$	257.407	0.160524	0.0802619	+	0.996774i	$0.474424\pi$
$138$	0	0
$139$	−2294.42	−1.40007	−0.700037	−	0.714106i	$-0.746833\pi$
$139$	−2294.42	−1.40007	−0.700037	+	0.714106i	$0.746833\pi$
$140$	−488.349	−0.294807
$141$	0	0
$142$	−327.495	−0.193541
$143$	−364.335	−0.213058
$144$	0	0
$145$	−145.000	−0.0830455
$146$	3475.53	1.97012
$147$	0	0
$148$	1025.45	0.569538
$149$	835.743	0.459508	0.229754	−	0.973249i	$-0.426208\pi$
$149$	835.743	0.459508	0.229754	+	0.973249i	$0.426208\pi$
$150$	0	0
$151$	−3370.60	−1.81653	−0.908263	−	0.418400i	$-0.862591\pi$
$151$	−3370.60	−1.81653	−0.908263	+	0.418400i	$0.862591\pi$
$152$	−493.074	−0.263116
$153$	0	0
$154$	−3089.86	−1.61680
$155$	410.549	0.212749
$156$	0	0
$157$	−137.311	−0.0698002	−0.0349001	−	0.999391i	$-0.511111\pi$
$157$	−137.311	−0.0698002	−0.0349001	+	0.999391i	$0.511111\pi$
$158$	220.932	0.111243
$159$	0	0
$160$	−929.554	−0.459298
$161$	−1716.83	−0.840404
$162$	0	0
$163$	−859.793	−0.413155	−0.206577	−	0.978430i	$-0.566233\pi$
$163$	−859.793	−0.413155	−0.206577	+	0.978430i	$0.566233\pi$
$164$	1260.32	0.600087
$165$	0	0
$166$	−580.541	−0.271438
$167$	900.368	0.417201	0.208601	−	0.978001i	$-0.433109\pi$
$167$	900.368	0.417201	0.208601	+	0.978001i	$0.433109\pi$
$168$	0	0
$169$	−2117.82	−0.963958
$170$	136.350	0.0615153
$171$	0	0
$172$	247.024	0.109508
$173$	465.780	0.204697	0.102348	−	0.994749i	$-0.467364\pi$
$173$	465.780	0.204697	0.102348	+	0.994749i	$0.467364\pi$
$174$	0	0
$175$	−531.667	−0.229659
$176$	−3261.07	−1.39666
$177$	0	0
$178$	−4868.47	−2.05004
$179$	−1143.90	−0.477650	−0.238825	−	0.971063i	$-0.576762\pi$
$179$	−1143.90	−0.477650	−0.238825	+	0.971063i	$0.576762\pi$
$180$	0	0
$181$	−345.618	−0.141931	−0.0709657	−	0.997479i	$-0.522608\pi$
$181$	−345.618	−0.141931	−0.0709657	+	0.997479i	$0.522608\pi$
$182$	671.550	0.273509
$183$	0	0
$184$	−976.125	−0.391092
$185$	1116.41	0.443677
$186$	0	0
$187$	314.637	0.123040
$188$	−78.1829	−0.0303302
$189$	0	0
$190$	723.539	0.276268
$191$	−3080.33	−1.16694	−0.583469	−	0.812135i	$-0.698305\pi$
$191$	−3080.33	−1.16694	−0.583469	+	0.812135i	$0.698305\pi$
$192$	0	0
$193$	4561.76	1.70136	0.850680	−	0.525684i	$-0.176190\pi$
$193$	4561.76	1.70136	0.850680	+	0.525684i	$0.176190\pi$
$194$	4484.15	1.65950
$195$	0	0
$196$	501.844	0.182888
$197$	−2521.13	−0.911794	−0.455897	−	0.890033i	$-0.650681\pi$
$197$	−2521.13	−0.911794	−0.455897	+	0.890033i	$0.650681\pi$
$198$	0	0
$199$	−3000.49	−1.06884	−0.534420	−	0.845219i	$-0.679470\pi$
$199$	−3000.49	−1.06884	−0.534420	+	0.845219i	$0.679470\pi$
$200$	−302.286	−0.106874
$201$	0	0
$202$	−527.084	−0.183592
$203$	616.734	0.213233
$204$	0	0
$205$	1372.11	0.467475
$206$	5271.78	1.78302
$207$	0	0
$208$	708.761	0.236268
$209$	1669.61	0.552580
$210$	0	0
$211$	843.401	0.275176	0.137588	−	0.990490i	$-0.456065\pi$
$211$	843.401	0.275176	0.137588	+	0.990490i	$0.456065\pi$
$212$	105.849	0.0342911
$213$	0	0
$214$	−1637.60	−0.523103
$215$	268.936	0.0853083
$216$	0	0
$217$	−1746.20	−0.546267
$218$	5379.91	1.67144
$219$	0	0
$220$	940.181	0.288123
$221$	−68.3831	−0.0208142
$222$	0	0
$223$	1068.88	0.320975	0.160488	−	0.987038i	$-0.448693\pi$
$223$	1068.88	0.320975	0.160488	+	0.987038i	$0.448693\pi$
$224$	3953.70	1.17932
$225$	0	0
$226$	−569.217	−0.167539
$227$	−1688.90	−0.493817	−0.246909	−	0.969039i	$-0.579415\pi$
$227$	−1688.90	−0.493817	−0.246909	+	0.969039i	$0.579415\pi$
$228$	0	0
$229$	−3085.30	−0.890317	−0.445158	−	0.895452i	$-0.646853\pi$
$229$	−3085.30	−0.890317	−0.445158	+	0.895452i	$0.646853\pi$
$230$	1432.37	0.410642
$231$	0	0
$232$	350.652	0.0992304
$233$	4810.41	1.35254	0.676268	−	0.736656i	$-0.263596\pi$
$233$	4810.41	1.35254	0.676268	+	0.736656i	$0.263596\pi$
$234$	0	0
$235$	−85.1180	−0.0236276
$236$	1835.09	0.506162
$237$	0	0
$238$	−579.944	−0.157950
$239$	1730.66	0.468399	0.234200	−	0.972189i	$-0.424753\pi$
$239$	1730.66	0.468399	0.234200	+	0.972189i	$0.424753\pi$
$240$	0	0
$241$	−1671.50	−0.446767	−0.223383	−	0.974731i	$-0.571710\pi$
$241$	−1671.50	−0.446767	−0.223383	+	0.974731i	$0.571710\pi$
$242$	1225.47	0.325522
$243$	0	0
$244$	1783.35	0.467900
$245$	546.359	0.142472
$246$	0	0
$247$	−362.872	−0.0934778
$248$	−992.827	−0.254212
$249$	0	0
$250$	443.576	0.112217
$251$	5324.14	1.33887	0.669436	−	0.742870i	$-0.266536\pi$
$251$	5324.14	1.33887	0.669436	+	0.742870i	$0.266536\pi$
$252$	0	0
$253$	3305.28	0.821348
$254$	9168.32	2.26485
$255$	0	0
$256$	5174.31	1.26326
$257$	−1994.20	−0.484026	−0.242013	−	0.970273i	$-0.577808\pi$
$257$	−1994.20	−0.484026	−0.242013	+	0.970273i	$0.577808\pi$
$258$	0	0
$259$	−4748.48	−1.13921
$260$	−204.339	−0.0487406
$261$	0	0
$262$	536.586	0.126528
$263$	1671.01	0.391782	0.195891	−	0.980626i	$-0.437240\pi$
$263$	1671.01	0.391782	0.195891	+	0.980626i	$0.437240\pi$
$264$	0	0
$265$	115.238	0.0267132
$266$	−3077.45	−0.709364
$267$	0	0
$268$	1834.45	0.418122
$269$	−890.546	−0.201850	−0.100925	−	0.994894i	$-0.532180\pi$
$269$	−890.546	−0.201850	−0.100925	+	0.994894i	$0.532180\pi$
$270$	0	0
$271$	6649.82	1.49058	0.745292	−	0.666739i	$-0.232310\pi$
$271$	6649.82	1.49058	0.745292	+	0.666739i	$0.232310\pi$
$272$	−612.079	−0.136444
$273$	0	0
$274$	913.436	0.201397
$275$	1023.58	0.224451
$276$	0	0
$277$	6233.58	1.35213	0.676064	−	0.736843i	$-0.263684\pi$
$277$	6233.58	1.35213	0.676064	+	0.736843i	$0.263684\pi$
$278$	−8142.01	−1.75657
$279$	0	0
$280$	1285.73	0.274417
$281$	3089.87	0.655964	0.327982	−	0.944684i	$-0.393631\pi$
$281$	3089.87	0.655964	0.327982	+	0.944684i	$0.393631\pi$
$282$	0	0
$283$	1644.01	0.345323	0.172662	−	0.984981i	$-0.444763\pi$
$283$	1644.01	0.345323	0.172662	+	0.984981i	$0.444763\pi$
$284$	−423.845	−0.0885584
$285$	0	0
$286$	−1292.88	−0.267307
$287$	−5836.05	−1.20032
$288$	0	0
$289$	−4853.94	−0.987980
$290$	−514.548	−0.104191
$291$	0	0
$292$	4498.05	0.901467
$293$	66.3144	0.0132223	0.00661114	−	0.999978i	$-0.497896\pi$
$293$	66.3144	0.0132223	0.00661114	+	0.999978i	$0.497896\pi$
$294$	0	0
$295$	1997.87	0.394307
$296$	−2699.81	−0.530147
$297$	0	0
$298$	2965.72	0.576510
$299$	−718.369	−0.138944
$300$	0	0
$301$	−1143.87	−0.219043
$302$	−11960.9	−2.27905
$303$	0	0
$304$	−3247.98	−0.612777
$305$	1941.54	0.364500
$306$	0	0
$307$	−6736.52	−1.25236	−0.626179	−	0.779679i	$-0.715382\pi$
$307$	−6736.52	−1.25236	−0.626179	+	0.779679i	$0.715382\pi$
$308$	−3998.91	−0.739801
$309$	0	0
$310$	1456.88	0.266920
$311$	5470.56	0.997450	0.498725	−	0.866760i	$-0.333802\pi$
$311$	5470.56	0.997450	0.498725	+	0.866760i	$0.333802\pi$
$312$	0	0
$313$	903.461	0.163152	0.0815761	−	0.996667i	$-0.474005\pi$
$313$	903.461	0.163152	0.0815761	+	0.996667i	$0.474005\pi$
$314$	−487.264	−0.0875729
$315$	0	0
$316$	285.930	0.0509014
$317$	880.773	0.156054	0.0780270	−	0.996951i	$-0.475138\pi$
$317$	880.773	0.156054	0.0780270	+	0.996951i	$0.475138\pi$
$318$	0	0
$319$	−1187.35	−0.208398
$320$	−112.670	−0.0196827
$321$	0	0
$322$	−6092.35	−1.05439
$323$	313.373	0.0539832
$324$	0	0
$325$	−222.464	−0.0379695
$326$	−3051.07	−0.518353
$327$	0	0
$328$	−3318.16	−0.558582
$329$	362.035	0.0606677
$330$	0	0
$331$	609.140	0.101152	0.0505761	−	0.998720i	$-0.483894\pi$
$331$	609.140	0.101152	0.0505761	+	0.998720i	$0.483894\pi$
$332$	−751.338	−0.124202
$333$	0	0
$334$	3195.05	0.523430
$335$	1997.17	0.325722
$336$	0	0
$337$	−9473.47	−1.53131	−0.765657	−	0.643250i	$-0.777586\pi$
$337$	−9473.47	−1.53131	−0.765657	+	0.643250i	$0.777586\pi$
$338$	−7515.30	−1.20940
$339$	0	0
$340$	176.465	0.0281476
$341$	3361.83	0.533881
$342$	0	0
$343$	4970.62	0.782474
$344$	−650.365	−0.101934
$345$	0	0
$346$	1652.87	0.256817
$347$	−3436.86	−0.531702	−0.265851	−	0.964014i	$-0.585653\pi$
$347$	−3436.86	−0.531702	−0.265851	+	0.964014i	$0.585653\pi$
$348$	0	0
$349$	703.930	0.107967	0.0539835	−	0.998542i	$-0.482808\pi$
$349$	703.930	0.107967	0.0539835	+	0.998542i	$0.482808\pi$
$350$	−1886.68	−0.288135
$351$	0	0
$352$	−7611.77	−1.15258
$353$	−2982.17	−0.449646	−0.224823	−	0.974400i	$-0.572180\pi$
$353$	−2982.17	−0.449646	−0.224823	+	0.974400i	$0.572180\pi$
$354$	0	0
$355$	−461.442	−0.0689881
$356$	−6300.78	−0.938036
$357$	0	0
$358$	−4059.27	−0.599271
$359$	−8585.52	−1.26219	−0.631095	−	0.775705i	$-0.717394\pi$
$359$	−8585.52	−1.26219	−0.631095	+	0.775705i	$0.717394\pi$
$360$	0	0
$361$	−5196.10	−0.757559
$362$	−1226.46	−0.178070
$363$	0	0
$364$	869.122	0.125149
$365$	4897.04	0.702254
$366$	0	0
$367$	6989.50	0.994139	0.497070	−	0.867711i	$-0.334409\pi$
$367$	6989.50	0.994139	0.497070	+	0.867711i	$0.334409\pi$
$368$	−6429.93	−0.910824
$369$	0	0
$370$	3961.71	0.556648
$371$	−490.144	−0.0685904
$372$	0	0
$373$	−1100.55	−0.152773	−0.0763865	−	0.997078i	$-0.524338\pi$
$373$	−1100.55	−0.152773	−0.0763865	+	0.997078i	$0.524338\pi$
$374$	1116.52	0.154369
$375$	0	0
$376$	205.840	0.0282324
$377$	258.059	0.0352538
$378$	0	0
$379$	7950.59	1.07756	0.538779	−	0.842447i	$-0.318886\pi$
$379$	7950.59	1.07756	0.538779	+	0.842447i	$0.318886\pi$
$380$	936.406	0.126412
$381$	0	0
$382$	−10930.9	−1.46407
$383$	−10160.2	−1.35551	−0.677755	−	0.735288i	$-0.737047\pi$
$383$	−10160.2	−1.35551	−0.677755	+	0.735288i	$0.737047\pi$
$384$	0	0
$385$	−4353.62	−0.576315
$386$	16187.9	2.13456
$387$	0	0
$388$	5803.40	0.759338
$389$	−9770.47	−1.27348	−0.636738	−	0.771080i	$-0.719717\pi$
$389$	−9770.47	−1.27348	−0.636738	+	0.771080i	$0.719717\pi$
$390$	0	0
$391$	620.377	0.0802399
$392$	−1321.25	−0.170238
$393$	0	0
$394$	−8946.52	−1.14396
$395$	311.293	0.0396528
$396$	0	0
$397$	−3191.11	−0.403418	−0.201709	−	0.979445i	$-0.564650\pi$
$397$	−3191.11	−0.403418	−0.201709	+	0.979445i	$0.564650\pi$
$398$	−10647.6	−1.34099
$399$	0	0
$400$	−1991.22	−0.248903
$401$	−11101.4	−1.38249	−0.691247	−	0.722619i	$-0.742938\pi$
$401$	−11101.4	−1.38249	−0.691247	+	0.722619i	$0.742938\pi$
$402$	0	0
$403$	−730.660	−0.0903146
$404$	−682.154	−0.0840060
$405$	0	0
$406$	2188.55	0.267526
$407$	9141.89	1.11338
$408$	0	0
$409$	−5263.84	−0.636382	−0.318191	−	0.948027i	$-0.603075\pi$
$409$	−5263.84	−0.636382	−0.318191	+	0.948027i	$0.603075\pi$
$410$	4869.09	0.586505
$411$	0	0
$412$	6822.75	0.815856
$413$	−8497.61	−1.01245
$414$	0	0
$415$	−817.984	−0.0967548
$416$	1654.34	0.194978
$417$	0	0
$418$	5924.78	0.693279
$419$	−5365.09	−0.625541	−0.312771	−	0.949829i	$-0.601257\pi$
$419$	−5365.09	−0.625541	−0.312771	+	0.949829i	$0.601257\pi$
$420$	0	0
$421$	4453.87	0.515602	0.257801	−	0.966198i	$-0.417002\pi$
$421$	4453.87	0.515602	0.257801	+	0.966198i	$0.417002\pi$
$422$	2992.90	0.345242
$423$	0	0
$424$	−278.678	−0.0319193
$425$	192.118	0.0219273
$426$	0	0
$427$	−8258.03	−0.935911
$428$	−2119.39	−0.239356
$429$	0	0
$430$	954.348	0.107030
$431$	15164.6	1.69478	0.847392	−	0.530968i	$-0.178172\pi$
$431$	15164.6	1.69478	0.847392	+	0.530968i	$0.178172\pi$
$432$	0	0
$433$	5748.48	0.638001	0.319000	−	0.947755i	$-0.396653\pi$
$433$	5748.48	0.638001	0.319000	+	0.947755i	$0.396653\pi$
$434$	−6196.59	−0.685359
$435$	0	0
$436$	6962.70	0.764800
$437$	3292.01	0.360362
$438$	0	0
$439$	−2413.19	−0.262358	−0.131179	−	0.991359i	$-0.541876\pi$
$439$	−2413.19	−0.262358	−0.131179	+	0.991359i	$0.541876\pi$
$440$	−2475.31	−0.268195
$441$	0	0
$442$	−242.665	−0.0261140
$443$	5598.47	0.600432	0.300216	−	0.953871i	$-0.402941\pi$
$443$	5598.47	0.600432	0.300216	+	0.953871i	$0.402941\pi$
$444$	0	0
$445$	−6859.68	−0.730742
$446$	3793.03	0.402703
$447$	0	0
$448$	479.224	0.0505384
$449$	−1387.50	−0.145836	−0.0729179	−	0.997338i	$-0.523231\pi$
$449$	−1387.50	−0.145836	−0.0729179	+	0.997338i	$0.523231\pi$
$450$	0	0
$451$	11235.7	1.17310
$452$	−736.682	−0.0766607
$453$	0	0
$454$	−5993.26	−0.619554
$455$	946.216	0.0974929
$456$	0	0
$457$	−1208.57	−0.123707	−0.0618537	−	0.998085i	$-0.519701\pi$
$457$	−1208.57	−0.123707	−0.0618537	+	0.998085i	$0.519701\pi$
$458$	−10948.5	−1.11701
$459$	0	0
$460$	1853.78	0.187897
$461$	1838.27	0.185720	0.0928599	−	0.995679i	$-0.470399\pi$
$461$	1838.27	0.185720	0.0928599	+	0.995679i	$0.470399\pi$
$462$	0	0
$463$	−13925.4	−1.39777	−0.698885	−	0.715234i	$-0.746320\pi$
$463$	−13925.4	−1.39777	−0.698885	+	0.715234i	$0.746320\pi$
$464$	2309.82	0.231100
$465$	0	0
$466$	17070.3	1.69692
$467$	8692.90	0.861369	0.430685	−	0.902502i	$-0.358272\pi$
$467$	8692.90	0.861369	0.430685	+	0.902502i	$0.358272\pi$
$468$	0	0
$469$	−8494.63	−0.836344
$470$	−302.050	−0.0296437
$471$	0	0
$472$	−4831.43	−0.471154
$473$	2202.21	0.214076
$474$	0	0
$475$	1019.47	0.0984766
$476$	−750.566	−0.0722734
$477$	0	0
$478$	6141.45	0.587664
$479$	−9915.68	−0.945843	−0.472922	−	0.881105i	$-0.656801\pi$
$479$	−9915.68	−0.945843	−0.472922	+	0.881105i	$0.656801\pi$
$480$	0	0
$481$	−1986.90	−0.188347
$482$	−5931.50	−0.560523
$483$	0	0
$484$	1586.01	0.148949
$485$	6318.18	0.591534
$486$	0	0
$487$	16633.8	1.54774	0.773871	−	0.633344i	$-0.218318\pi$
$487$	16633.8	1.54774	0.773871	+	0.633344i	$0.218318\pi$
$488$	−4695.21	−0.435538
$489$	0	0
$490$	1938.81	0.178748
$491$	291.435	0.0267867	0.0133933	−	0.999910i	$-0.495737\pi$
$491$	291.435	0.0267867	0.0133933	+	0.999910i	$0.495737\pi$
$492$	0	0
$493$	−222.857	−0.0203590
$494$	−1287.69	−0.117279
$495$	0	0
$496$	−6539.95	−0.592041
$497$	1962.67	0.177138
$498$	0	0
$499$	11845.7	1.06270	0.531350	−	0.847152i	$-0.321685\pi$
$499$	11845.7	1.06270	0.531350	+	0.847152i	$0.321685\pi$
$500$	574.078	0.0513471
$501$	0	0
$502$	18893.3	1.67978
$503$	16484.6	1.46126	0.730629	−	0.682775i	$-0.239227\pi$
$503$	16484.6	1.46126	0.730629	+	0.682775i	$0.239227\pi$
$504$	0	0
$505$	−742.663	−0.0654417
$506$	11729.1	1.03048
$507$	0	0
$508$	11865.7	1.03633
$509$	−11521.1	−1.00327	−0.501636	−	0.865079i	$-0.667268\pi$
$509$	−11521.1	−1.00327	−0.501636	+	0.865079i	$0.667268\pi$
$510$	0	0
$511$	−20828.7	−1.80315
$512$	7103.01	0.613109
$513$	0	0
$514$	−7076.63	−0.607270
$515$	7427.95	0.635562
$516$	0	0
$517$	−696.999	−0.0592920
$518$	−16850.5	−1.42928
$519$	0	0
$520$	537.983	0.0453695
$521$	−6623.78	−0.556993	−0.278496	−	0.960437i	$-0.589836\pi$
$521$	−6623.78	−0.556993	−0.278496	+	0.960437i	$0.589836\pi$
$522$	0	0
$523$	−13552.8	−1.13312	−0.566559	−	0.824021i	$-0.691726\pi$
$523$	−13552.8	−1.13312	−0.566559	+	0.824021i	$0.691726\pi$
$524$	694.452	0.0578956
$525$	0	0
$526$	5929.75	0.491538
$527$	630.992	0.0521564
$528$	0	0
$529$	−5649.91	−0.464363
$530$	408.933	0.0335149
$531$	0	0
$532$	−3982.85	−0.324584
$533$	−2441.97	−0.198449
$534$	0	0
$535$	−2307.38	−0.186461
$536$	−4829.73	−0.389203
$537$	0	0
$538$	−3160.20	−0.253245
$539$	4473.93	0.357524
$540$	0	0
$541$	10971.0	0.871866	0.435933	−	0.899979i	$-0.356418\pi$
$541$	10971.0	0.871866	0.435933	+	0.899979i	$0.356418\pi$
$542$	23597.6	1.87012
$543$	0	0
$544$	−1428.67	−0.112599
$545$	7580.32	0.595789
$546$	0	0
$547$	−12428.0	−0.971453	−0.485726	−	0.874111i	$-0.661445\pi$
$547$	−12428.0	−0.971453	−0.485726	+	0.874111i	$0.661445\pi$
$548$	1182.17	0.0921531
$549$	0	0
$550$	3632.28	0.281601
$551$	−1182.58	−0.0914333
$552$	0	0
$553$	−1324.03	−0.101815
$554$	22120.5	1.69641
$555$	0	0
$556$	−10537.4	−0.803751
$557$	−22737.5	−1.72966	−0.864828	−	0.502069i	$-0.832572\pi$
$557$	−22737.5	−1.72966	−0.864828	+	0.502069i	$0.832572\pi$
$558$	0	0
$559$	−478.629	−0.0362144
$560$	8469.33	0.639097
$561$	0	0
$562$	10964.7	0.822988
$563$	20051.1	1.50099	0.750493	−	0.660878i	$-0.229816\pi$
$563$	20051.1	1.50099	0.750493	+	0.660878i	$0.229816\pi$
$564$	0	0
$565$	−802.028	−0.0597196
$566$	5833.96	0.433250
$567$	0	0
$568$	1115.90	0.0824334
$569$	−10035.7	−0.739397	−0.369699	−	0.929152i	$-0.620539\pi$
$569$	−10035.7	−0.739397	−0.369699	+	0.929152i	$0.620539\pi$
$570$	0	0
$571$	22946.3	1.68174	0.840870	−	0.541237i	$-0.182044\pi$
$571$	22946.3	1.68174	0.840870	+	0.541237i	$0.182044\pi$
$572$	−1673.25	−0.122312
$573$	0	0
$574$	−20709.9	−1.50595
$575$	2018.21	0.146374
$576$	0	0
$577$	14230.3	1.02672	0.513358	−	0.858174i	$-0.328401\pi$
$577$	14230.3	1.02672	0.513358	+	0.858174i	$0.328401\pi$
$578$	−17224.7	−1.23954
$579$	0	0
$580$	−665.930	−0.0476745
$581$	3479.16	0.248434
$582$	0	0
$583$	943.637	0.0670351
$584$	−11842.5	−0.839117
$585$	0	0
$586$	235.324	0.0165890
$587$	−12173.9	−0.855999	−0.428000	−	0.903779i	$-0.640782\pi$
$587$	−12173.9	−0.855999	−0.428000	+	0.903779i	$0.640782\pi$
$588$	0	0
$589$	3348.33	0.234237
$590$	7089.66	0.494706
$591$	0	0
$592$	−17784.2	−1.23467
$593$	−21548.0	−1.49219	−0.746097	−	0.665838i	$-0.768074\pi$
$593$	−21548.0	−1.49219	−0.746097	+	0.665838i	$0.768074\pi$
$594$	0	0
$595$	−817.143	−0.0563019
$596$	3838.25	0.263793
$597$	0	0
$598$	−2549.21	−0.174323
$599$	−22162.4	−1.51174	−0.755870	−	0.654722i	$-0.772786\pi$
$599$	−22162.4	−1.51174	−0.755870	+	0.654722i	$0.772786\pi$
$600$	0	0
$601$	2344.90	0.159152	0.0795761	−	0.996829i	$-0.474643\pi$
$601$	2344.90	0.159152	0.0795761	+	0.996829i	$0.474643\pi$
$602$	−4059.16	−0.274816
$603$	0	0
$604$	−15479.9	−1.04283
$605$	1726.69	0.116033
$606$	0	0
$607$	4273.53	0.285761	0.142881	−	0.989740i	$-0.454363\pi$
$607$	4273.53	0.285761	0.142881	+	0.989740i	$0.454363\pi$
$608$	−7581.20	−0.505688
$609$	0	0
$610$	6889.77	0.457309
$611$	151.486	0.0100302
$612$	0	0
$613$	−28945.8	−1.90719	−0.953597	−	0.301085i	$-0.902651\pi$
$613$	−28945.8	−1.90719	−0.953597	+	0.301085i	$0.902651\pi$
$614$	−23905.3	−1.57124
$615$	0	0
$616$	10528.3	0.688633
$617$	−885.511	−0.0577785	−0.0288892	−	0.999583i	$-0.509197\pi$
$617$	−885.511	−0.0577785	−0.0288892	+	0.999583i	$0.509197\pi$
$618$	0	0
$619$	−6120.02	−0.397390	−0.198695	−	0.980061i	$-0.563670\pi$
$619$	−6120.02	−0.397390	−0.198695	+	0.980061i	$0.563670\pi$
$620$	1885.50	0.122134
$621$	0	0
$622$	19412.9	1.25142
$623$	29176.5	1.87630
$624$	0	0
$625$	625.000	0.0400000
$626$	3206.03	0.204694
$627$	0	0
$628$	−630.618	−0.0400707
$629$	1715.87	0.108770
$630$	0	0
$631$	9436.47	0.595341	0.297670	−	0.954669i	$-0.403790\pi$
$631$	9436.47	0.595341	0.297670	+	0.954669i	$0.403790\pi$
$632$	−752.798	−0.0473808
$633$	0	0
$634$	3125.52	0.195789
$635$	12918.2	0.807311
$636$	0	0
$637$	−972.363	−0.0604810
$638$	−4213.44	−0.261460
$639$	0	0
$640$	7036.61	0.434604
$641$	12119.8	0.746807	0.373403	−	0.927669i	$-0.378191\pi$
$641$	12119.8	0.746807	0.373403	+	0.927669i	$0.378191\pi$
$642$	0	0
$643$	−7693.03	−0.471825	−0.235912	−	0.971774i	$-0.575808\pi$
$643$	−7693.03	−0.471825	−0.235912	+	0.971774i	$0.575808\pi$
$644$	−7884.74	−0.482457
$645$	0	0
$646$	1112.04	0.0677285
$647$	−9921.81	−0.602885	−0.301443	−	0.953484i	$-0.597468\pi$
$647$	−9921.81	−0.602885	−0.301443	+	0.953484i	$0.597468\pi$
$648$	0	0
$649$	16359.8	0.989489
$650$	−789.439	−0.0476374
$651$	0	0
$652$	−3948.70	−0.237183
$653$	−9973.03	−0.597665	−0.298832	−	0.954306i	$-0.596597\pi$
$653$	−9973.03	−0.597665	−0.298832	+	0.954306i	$0.596597\pi$
$654$	0	0
$655$	756.052	0.0451014
$656$	−21857.4	−1.30090
$657$	0	0
$658$	1284.72	0.0761150
$659$	24918.9	1.47299	0.736495	−	0.676443i	$-0.236479\pi$
$659$	24918.9	1.47299	0.736495	+	0.676443i	$0.236479\pi$
$660$	0	0
$661$	14562.7	0.856922	0.428461	−	0.903560i	$-0.359056\pi$
$661$	14562.7	0.856922	0.428461	+	0.903560i	$0.359056\pi$
$662$	2161.60	0.126908
$663$	0	0
$664$	1978.12	0.115612
$665$	−4336.14	−0.252855
$666$	0	0
$667$	−2341.13	−0.135905
$668$	4135.05	0.239506
$669$	0	0
$670$	7087.17	0.408658
$671$	15898.6	0.914690
$672$	0	0
$673$	10744.1	0.615388	0.307694	−	0.951485i	$-0.400443\pi$
$673$	10744.1	0.615388	0.307694	+	0.951485i	$0.400443\pi$
$674$	−33617.6	−1.92122
$675$	0	0
$676$	−9726.32	−0.553387
$677$	−842.278	−0.0478159	−0.0239080	−	0.999714i	$-0.507611\pi$
$677$	−842.278	−0.0478159	−0.0239080	+	0.999714i	$0.507611\pi$
$678$	0	0
$679$	−26873.4	−1.51886
$680$	−464.598	−0.0262007
$681$	0	0
$682$	11929.8	0.669819
$683$	8313.36	0.465742	0.232871	−	0.972508i	$-0.425188\pi$
$683$	8313.36	0.465742	0.232871	+	0.972508i	$0.425188\pi$
$684$	0	0
$685$	1287.03	0.0717884
$686$	17638.8	0.981709
$687$	0	0
$688$	−4284.08	−0.237397
$689$	−205.090	−0.0113401
$690$	0	0
$691$	16891.1	0.929909	0.464954	−	0.885335i	$-0.346071\pi$
$691$	16891.1	0.929909	0.464954	+	0.885335i	$0.346071\pi$
$692$	2139.15	0.117512
$693$	0	0
$694$	−12196.1	−0.667085
$695$	−11472.1	−0.626132
$696$	0	0
$697$	2108.86	0.114604
$698$	2497.97	0.135458
$699$	0	0
$700$	−2441.75	−0.131842
$701$	−26296.2	−1.41682	−0.708412	−	0.705799i	$-0.750588\pi$
$701$	−26296.2	−1.41682	−0.708412	+	0.705799i	$0.750588\pi$
$702$	0	0
$703$	9105.18	0.488490
$704$	−922.613	−0.0493925
$705$	0	0
$706$	−10582.6	−0.564136
$707$	3158.80	0.168032
$708$	0	0
$709$	26644.7	1.41137	0.705686	−	0.708525i	$-0.250639\pi$
$709$	26644.7	1.41137	0.705686	+	0.708525i	$0.250639\pi$
$710$	−1637.48	−0.0865541
$711$	0	0
$712$	16588.7	0.873158
$713$	6628.61	0.348167
$714$	0	0
$715$	−1821.68	−0.0952823
$716$	−5253.52	−0.274208
$717$	0	0
$718$	−30466.7	−1.58357
$719$	−5825.10	−0.302141	−0.151071	−	0.988523i	$-0.548272\pi$
$719$	−5825.10	−0.302141	−0.151071	+	0.988523i	$0.548272\pi$
$720$	0	0
$721$	−31593.6	−1.63191
$722$	−18438.9	−0.950450
$723$	0	0
$724$	−1587.29	−0.0814797
$725$	−725.000	−0.0371391
$726$	0	0
$727$	6715.29	0.342581	0.171290	−	0.985221i	$-0.445206\pi$
$727$	6715.29	0.342581	0.171290	+	0.985221i	$0.445206\pi$
$728$	−2288.22	−0.116493
$729$	0	0
$730$	17377.7	0.881063
$731$	413.340	0.0209137
$732$	0	0
$733$	14206.6	0.715870	0.357935	−	0.933747i	$-0.383481\pi$
$733$	14206.6	0.715870	0.357935	+	0.933747i	$0.383481\pi$
$734$	24803.0	1.24727
$735$	0	0
$736$	−15008.3	−0.751649
$737$	16354.1	0.817381
$738$	0	0
$739$	−30931.2	−1.53968	−0.769840	−	0.638237i	$-0.779664\pi$
$739$	−30931.2	−1.53968	−0.769840	+	0.638237i	$0.779664\pi$
$740$	5127.26	0.254705
$741$	0	0
$742$	−1739.33	−0.0860550
$743$	32851.1	1.62206	0.811030	−	0.585005i	$-0.198907\pi$
$743$	32851.1	1.62206	0.811030	+	0.585005i	$0.198907\pi$
$744$	0	0
$745$	4178.72	0.205498
$746$	−3905.42	−0.191672
$747$	0	0
$748$	1445.01	0.0706346
$749$	9814.07	0.478769
$750$	0	0
$751$	36055.8	1.75192	0.875961	−	0.482382i	$-0.160228\pi$
$751$	36055.8	1.75192	0.875961	+	0.482382i	$0.160228\pi$
$752$	1355.91	0.0657512
$753$	0	0
$754$	915.749	0.0442302
$755$	−16853.0	−0.812375
$756$	0	0
$757$	−7094.16	−0.340610	−0.170305	−	0.985391i	$-0.554475\pi$
$757$	−7094.16	−0.340610	−0.170305	+	0.985391i	$0.554475\pi$
$758$	28213.5	1.35193
$759$	0	0
$760$	−2465.37	−0.117669
$761$	−23664.3	−1.12724	−0.563620	−	0.826034i	$-0.690592\pi$
$761$	−23664.3	−1.12724	−0.563620	+	0.826034i	$0.690592\pi$
$762$	0	0
$763$	−32241.6	−1.52978
$764$	−14146.8	−0.669913
$765$	0	0
$766$	−36054.5	−1.70065
$767$	−3555.64	−0.167388
$768$	0	0
$769$	16743.8	0.785172	0.392586	−	0.919715i	$-0.371581\pi$
$769$	16743.8	0.785172	0.392586	+	0.919715i	$0.371581\pi$
$770$	−15449.3	−0.723057
$771$	0	0
$772$	20950.4	0.976713
$773$	2198.53	0.102297	0.0511485	−	0.998691i	$-0.483712\pi$
$773$	2198.53	0.102297	0.0511485	+	0.998691i	$0.483712\pi$
$774$	0	0
$775$	2052.75	0.0951443
$776$	−15279.2	−0.706819
$777$	0	0
$778$	−34671.6	−1.59773
$779$	11190.6	0.514691
$780$	0	0
$781$	−3778.57	−0.173122
$782$	2201.47	0.100671
$783$	0	0
$784$	−8703.37	−0.396473
$785$	−686.556	−0.0312156
$786$	0	0
$787$	−10092.7	−0.457134	−0.228567	−	0.973528i	$-0.573404\pi$
$787$	−10092.7	−0.457134	−0.228567	+	0.973528i	$0.573404\pi$
$788$	−11578.6	−0.523440
$789$	0	0
$790$	1104.66	0.0497493
$791$	3411.30	0.153340
$792$	0	0
$793$	−3455.39	−0.154735
$794$	−11324.0	−0.506138
$795$	0	0
$796$	−13780.1	−0.613597
$797$	35534.7	1.57930	0.789652	−	0.613555i	$-0.210261\pi$
$797$	35534.7	1.57930	0.789652	+	0.613555i	$0.210261\pi$
$798$	0	0
$799$	−130.822	−0.00579241
$800$	−4647.77	−0.205404
$801$	0	0
$802$	−39394.7	−1.73451
$803$	40100.0	1.76226
$804$	0	0
$805$	−8584.14	−0.375840
$806$	−2592.83	−0.113311
$807$	0	0
$808$	1795.98	0.0781958
$809$	−27966.3	−1.21538	−0.607691	−	0.794173i	$-0.707904\pi$
$809$	−27966.3	−1.21538	−0.607691	+	0.794173i	$0.707904\pi$
$810$	0	0
$811$	−19134.4	−0.828481	−0.414241	−	0.910167i	$-0.635953\pi$
$811$	−19134.4	−0.828481	−0.414241	+	0.910167i	$0.635953\pi$
$812$	2832.42	0.122412
$813$	0	0
$814$	32441.0	1.39687
$815$	−4298.97	−0.184768
$816$	0	0
$817$	2193.37	0.0939246
$818$	−18679.3	−0.798420
$819$	0	0
$820$	6301.59	0.268367
$821$	21420.6	0.910577	0.455288	−	0.890344i	$-0.349536\pi$
$821$	21420.6	0.910577	0.455288	+	0.890344i	$0.349536\pi$
$822$	0	0
$823$	−16122.4	−0.682855	−0.341428	−	0.939908i	$-0.610910\pi$
$823$	−16122.4	−0.682855	−0.341428	+	0.939908i	$0.610910\pi$
$824$	−17962.9	−0.759428
$825$	0	0
$826$	−30154.7	−1.27024
$827$	−22131.9	−0.930596	−0.465298	−	0.885154i	$-0.654053\pi$
$827$	−22131.9	−0.930596	−0.465298	+	0.885154i	$0.654053\pi$
$828$	0	0
$829$	135.660	0.00568355	0.00284178	−	0.999996i	$-0.499095\pi$
$829$	135.660	0.00568355	0.00284178	+	0.999996i	$0.499095\pi$
$830$	−2902.70	−0.121391
$831$	0	0
$832$	200.521	0.00835553
$833$	839.724	0.0349276
$834$	0	0
$835$	4501.84	0.186578
$836$	7667.88	0.317224
$837$	0	0
$838$	−19038.6	−0.784818
$839$	32154.4	1.32311	0.661557	−	0.749895i	$-0.269896\pi$
$839$	32154.4	1.32311	0.661557	+	0.749895i	$0.269896\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	15805.0	0.646886
$843$	0	0
$844$	3873.42	0.157972
$845$	−10589.1	−0.431095
$846$	0	0
$847$	−7344.21	−0.297934
$848$	−1835.71	−0.0743378
$849$	0	0
$850$	681.752	0.0275105
$851$	18025.3	0.726085
$852$	0	0
$853$	2797.50	0.112291	0.0561457	−	0.998423i	$-0.482119\pi$
$853$	2797.50	0.112291	0.0561457	+	0.998423i	$0.482119\pi$
$854$	−29304.5	−1.17422
$855$	0	0
$856$	5579.92	0.222801
$857$	−9053.14	−0.360851	−0.180425	−	0.983589i	$-0.557747\pi$
$857$	−9053.14	−0.360851	−0.180425	+	0.983589i	$0.557747\pi$
$858$	0	0
$859$	23178.7	0.920660	0.460330	−	0.887748i	$-0.347731\pi$
$859$	23178.7	0.920660	0.460330	+	0.887748i	$0.347731\pi$
$860$	1235.12	0.0489736
$861$	0	0
$862$	53813.1	2.12631
$863$	7117.61	0.280749	0.140374	−	0.990098i	$-0.455169\pi$
$863$	7117.61	0.280749	0.140374	+	0.990098i	$0.455169\pi$
$864$	0	0
$865$	2328.90	0.0915432
$866$	20399.1	0.800450
$867$	0	0
$868$	−8019.65	−0.313600
$869$	2549.06	0.0995064
$870$	0	0
$871$	−3554.39	−0.138273
$872$	−18331.4	−0.711904
$873$	0	0
$874$	11682.0	0.452118
$875$	−2658.34	−0.102706
$876$	0	0
$877$	44014.6	1.69472	0.847359	−	0.531020i	$-0.178191\pi$
$877$	44014.6	1.69472	0.847359	+	0.531020i	$0.178191\pi$
$878$	−8563.47	−0.329161
$879$	0	0
$880$	−16305.4	−0.624606
$881$	31970.4	1.22260	0.611299	−	0.791399i	$-0.290647\pi$
$881$	31970.4	1.22260	0.611299	+	0.791399i	$0.290647\pi$
$882$	0	0
$883$	13620.9	0.519116	0.259558	−	0.965727i	$-0.416423\pi$
$883$	13620.9	0.519116	0.259558	+	0.965727i	$0.416423\pi$
$884$	−314.058	−0.0119490
$885$	0	0
$886$	19866.8	0.753315
$887$	−23494.2	−0.889357	−0.444678	−	0.895690i	$-0.646682\pi$
$887$	−23494.2	−0.889357	−0.444678	+	0.895690i	$0.646682\pi$
$888$	0	0
$889$	−54945.4	−2.07290
$890$	−24342.3	−0.916805
$891$	0	0
$892$	4908.96	0.184265
$893$	−694.200	−0.0260140
$894$	0	0
$895$	−5719.52	−0.213612
$896$	−29929.1	−1.11591
$897$	0	0
$898$	−4923.70	−0.182969
$899$	−2381.19	−0.0883392
$900$	0	0
$901$	177.114	0.00654886
$902$	39871.1	1.47180
$903$	0	0
$904$	1939.54	0.0713585
$905$	−1728.09	−0.0634737
$906$	0	0
$907$	11695.6	0.428165	0.214083	−	0.976816i	$-0.431324\pi$
$907$	11695.6	0.428165	0.214083	+	0.976816i	$0.431324\pi$
$908$	−7756.49	−0.283489
$909$	0	0
$910$	3357.75	0.122317
$911$	30991.2	1.12710	0.563549	−	0.826083i	$-0.309436\pi$
$911$	30991.2	1.12710	0.563549	+	0.826083i	$0.309436\pi$
$912$	0	0
$913$	−6698.16	−0.242800
$914$	−4288.72	−0.155206
$915$	0	0
$916$	−14169.6	−0.511111
$917$	−3215.74	−0.115805
$918$	0	0
$919$	−16612.6	−0.596300	−0.298150	−	0.954519i	$-0.596370\pi$
$919$	−16612.6	−0.596300	−0.298150	+	0.954519i	$0.596370\pi$
$920$	−4880.63	−0.174902
$921$	0	0
$922$	6523.30	0.233008
$923$	821.235	0.0292863
$924$	0	0
$925$	5582.07	0.198419
$926$	−49415.8	−1.75367
$927$	0	0
$928$	5391.41	0.190713
$929$	−53475.3	−1.88856	−0.944278	−	0.329149i	$-0.893238\pi$
$929$	−53475.3	−1.88856	−0.944278	+	0.329149i	$0.893238\pi$
$930$	0	0
$931$	4455.96	0.156862
$932$	22092.4	0.776460
$933$	0	0
$934$	30847.7	1.08069
$935$	1573.18	0.0550252
$936$	0	0
$937$	21279.8	0.741923	0.370961	−	0.928648i	$-0.379028\pi$
$937$	21279.8	0.741923	0.370961	+	0.928648i	$0.379028\pi$
$938$	−30144.1	−1.04930
$939$	0	0
$940$	−390.915	−0.0135641
$941$	−13268.0	−0.459645	−0.229822	−	0.973233i	$-0.573814\pi$
$941$	−13268.0	−0.459645	−0.229822	+	0.973233i	$0.573814\pi$
$942$	0	0
$943$	22153.7	0.765031
$944$	−31825.6	−1.09728
$945$	0	0
$946$	7814.80	0.268585
$947$	−21016.6	−0.721170	−0.360585	−	0.932726i	$-0.617423\pi$
$947$	−21016.6	−0.721170	−0.360585	+	0.932726i	$0.617423\pi$
$948$	0	0
$949$	−8715.33	−0.298115
$950$	3617.69	0.123551
$951$	0	0
$952$	1976.09	0.0672746
$953$	−42402.9	−1.44131	−0.720654	−	0.693295i	$-0.756158\pi$
$953$	−42402.9	−1.44131	−0.720654	+	0.693295i	$0.756158\pi$
$954$	0	0
$955$	−15401.7	−0.521871
$956$	7948.28	0.268897
$957$	0	0
$958$	−35186.9	−1.18668
$959$	−5474.19	−0.184328
$960$	0	0
$961$	−23049.0	−0.773689
$962$	−7050.72	−0.236304
$963$	0	0
$964$	−7676.56	−0.256479
$965$	22808.8	0.760871
$966$	0	0
$967$	31613.7	1.05132	0.525661	−	0.850694i	$-0.323818\pi$
$967$	31613.7	1.05132	0.525661	+	0.850694i	$0.323818\pi$
$968$	−4175.65	−0.138647
$969$	0	0
$970$	22420.8	0.742152
$971$	42637.4	1.40916	0.704582	−	0.709622i	$-0.251134\pi$
$971$	42637.4	1.40916	0.704582	+	0.709622i	$0.251134\pi$
$972$	0	0
$973$	48794.8	1.60770
$974$	59026.9	1.94183
$975$	0	0
$976$	−30928.3	−1.01433
$977$	7766.81	0.254332	0.127166	−	0.991881i	$-0.459412\pi$
$977$	7766.81	0.254332	0.127166	+	0.991881i	$0.459412\pi$
$978$	0	0
$979$	−56171.4	−1.83375
$980$	2509.22	0.0817899
$981$	0	0
$982$	1034.19	0.0336072
$983$	13603.8	0.441397	0.220698	−	0.975342i	$-0.429166\pi$
$983$	13603.8	0.441397	0.220698	+	0.975342i	$0.429166\pi$
$984$	0	0
$985$	−12605.7	−0.407767
$986$	−790.832	−0.0255428
$987$	0	0
$988$	−1666.54	−0.0536635
$989$	4342.16	0.139608
$990$	0	0
$991$	−13619.6	−0.436569	−0.218284	−	0.975885i	$-0.570046\pi$
$991$	−13619.6	−0.436569	−0.218284	+	0.975885i	$0.570046\pi$
$992$	−15265.1	−0.488576
$993$	0	0
$994$	6964.74	0.222241
$995$	−15002.4	−0.478000
$996$	0	0
$997$	27262.0	0.865993	0.432996	−	0.901396i	$-0.357456\pi$
$997$	27262.0	0.865993	0.432996	+	0.901396i	$0.357456\pi$
$998$	42035.8	1.33329
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.h.1.5		6
3.2	odd	2		435.4.a.h.1.2	✓	6
15.14	odd	2		2175.4.a.k.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.h.1.2	✓	6	3.2	odd	2
1305.4.a.h.1.5		6	1.1	even	1	trivial
2175.4.a.k.1.5		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)

\(f(q)\)	\(=\)	\(q+3.54861 q^{2} +4.59262 q^{4} +5.00000 q^{5} -21.2667 q^{7} -12.0915 q^{8} +O(q^{10})\)	\(q+3.54861 q^{2} +4.59262 q^{4} +5.00000 q^{5} -21.2667 q^{7} -12.0915 q^{8} +17.7430 q^{10} +40.9431 q^{11} -8.89857 q^{13} -75.4671 q^{14} -79.6488 q^{16} +7.68473 q^{17} +40.7787 q^{19} +22.9631 q^{20} +145.291 q^{22} +80.7285 q^{23} +25.0000 q^{25} -31.5775 q^{26} -97.6698 q^{28} -29.0000 q^{29} +82.1098 q^{31} -185.911 q^{32} +27.2701 q^{34} -106.333 q^{35} +223.283 q^{37} +144.708 q^{38} -60.4573 q^{40} +274.422 q^{41} +53.7872 q^{43} +188.036 q^{44} +286.474 q^{46} -17.0236 q^{47} +109.272 q^{49} +88.7152 q^{50} -40.8678 q^{52} +23.0475 q^{53} +204.716 q^{55} +257.145 q^{56} -102.910 q^{58} +399.574 q^{59} +388.308 q^{61} +291.376 q^{62} -22.5340 q^{64} -44.4929 q^{65} +399.434 q^{67} +35.2930 q^{68} -377.336 q^{70} -92.2884 q^{71} +979.407 q^{73} +792.343 q^{74} +187.281 q^{76} -870.724 q^{77} +62.2586 q^{79} -398.244 q^{80} +973.817 q^{82} -163.597 q^{83} +38.4236 q^{85} +190.870 q^{86} -495.062 q^{88} -1371.94 q^{89} +189.243 q^{91} +370.755 q^{92} -60.4101 q^{94} +203.894 q^{95} +1263.64 q^{97} +387.763 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8}+O(q^{10}) \) 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

Modular forms

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Database

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