LMFDB - Embedded newform 1305.4.a.h.1.2

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			1.2
Root			$4.39070$ 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)

$f(q)$	$=$	$q-4.39070 q^{2} +11.2783 q^{4} +5.00000 q^{5} +31.5336 q^{7} -14.3939 q^{8} +O(q^{10})$	\(q-4.39070 q^{2} +11.2783 q^{4} +5.00000 q^{5} +31.5336 q^{7} -14.3939 q^{8} -21.9535 q^{10} +5.28853 q^{11} -5.98731 q^{13} -138.455 q^{14} -27.0267 q^{16} -84.6758 q^{17} +57.3480 q^{19} +56.3914 q^{20} -23.2204 q^{22} -31.0898 q^{23} +25.0000 q^{25} +26.2885 q^{26} +355.645 q^{28} -29.0000 q^{29} -14.6873 q^{31} +233.818 q^{32} +371.786 q^{34} +157.668 q^{35} -7.76174 q^{37} -251.798 q^{38} -71.9697 q^{40} +399.867 q^{41} -17.9967 q^{43} +59.6455 q^{44} +136.506 q^{46} +262.798 q^{47} +651.368 q^{49} -109.768 q^{50} -67.5265 q^{52} +64.9925 q^{53} +26.4427 q^{55} -453.893 q^{56} +127.330 q^{58} -122.788 q^{59} +264.384 q^{61} +64.4877 q^{62} -810.411 q^{64} -29.9366 q^{65} +622.712 q^{67} -954.997 q^{68} -692.274 q^{70} -327.717 q^{71} -833.518 q^{73} +34.0795 q^{74} +646.787 q^{76} +166.767 q^{77} -666.457 q^{79} -135.133 q^{80} -1755.70 q^{82} +1320.72 q^{83} -423.379 q^{85} +79.0180 q^{86} -76.1229 q^{88} +189.059 q^{89} -188.801 q^{91} -350.640 q^{92} -1153.87 q^{94} +286.740 q^{95} -137.630 q^{97} -2859.97 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.39070	−1.55235	−0.776174	−	0.630519i	$-0.782842\pi$
$2$	−4.39070	−1.55235	−0.776174	+	0.630519i	$0.782842\pi$
$3$	0	0
$3$	0	0
$4$	11.2783	1.40978
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	31.5336	1.70265	0.851327	−	0.524635i	$-0.175798\pi$
$7$	31.5336	1.70265	0.851327	+	0.524635i	$0.175798\pi$
$8$	−14.3939	−0.636128
$9$	0	0
$10$	−21.9535	−0.694231
$11$	5.28853	0.144959	0.0724797	−	0.997370i	$-0.476909\pi$
$11$	5.28853	0.144959	0.0724797	+	0.997370i	$0.476909\pi$
$12$	0	0
$13$	−5.98731	−0.127737	−0.0638685	−	0.997958i	$-0.520344\pi$
$13$	−5.98731	−0.127737	−0.0638685	+	0.997958i	$0.520344\pi$
$14$	−138.455	−2.64311
$15$	0	0
$16$	−27.0267	−0.422292
$17$	−84.6758	−1.20805	−0.604026	−	0.796965i	$-0.706438\pi$
$17$	−84.6758	−1.20805	−0.604026	+	0.796965i	$0.706438\pi$
$18$	0	0
$19$	57.3480	0.692449	0.346225	−	0.938152i	$-0.387464\pi$
$19$	57.3480	0.692449	0.346225	+	0.938152i	$0.387464\pi$
$20$	56.3914	0.630475
$21$	0	0
$22$	−23.2204	−0.225027
$23$	−31.0898	−0.281855	−0.140928	−	0.990020i	$-0.545009\pi$
$23$	−31.0898	−0.281855	−0.140928	+	0.990020i	$0.545009\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	26.2885	0.198292
$27$	0	0
$28$	355.645	2.40038
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−14.6873	−0.0850942	−0.0425471	−	0.999094i	$-0.513547\pi$
$31$	−14.6873	−0.0850942	−0.0425471	+	0.999094i	$0.513547\pi$
$32$	233.818	1.29167
$33$	0	0
$34$	371.786	1.87532
$35$	157.668	0.761450
$36$	0	0
$37$	−7.76174	−0.0344871	−0.0172435	−	0.999851i	$-0.505489\pi$
$37$	−7.76174	−0.0344871	−0.0172435	+	0.999851i	$0.505489\pi$
$38$	−251.798	−1.07492
$39$	0	0
$40$	−71.9697	−0.284485
$41$	399.867	1.52314	0.761569	−	0.648084i	$-0.224429\pi$
$41$	399.867	1.52314	0.761569	+	0.648084i	$0.224429\pi$
$42$	0	0
$43$	−17.9967	−0.0638247	−0.0319124	−	0.999491i	$-0.510160\pi$
$43$	−17.9967	−0.0638247	−0.0319124	+	0.999491i	$0.510160\pi$
$44$	59.6455	0.204361
$45$	0	0
$46$	136.506	0.437538
$47$	262.798	0.815595	0.407798	−	0.913072i	$-0.366297\pi$
$47$	262.798	0.815595	0.407798	+	0.913072i	$0.366297\pi$
$48$	0	0
$49$	651.368	1.89903
$50$	−109.768	−0.310470
$51$	0	0
$52$	−67.5265	−0.180082
$53$	64.9925	0.168442	0.0842209	−	0.996447i	$-0.473160\pi$
$53$	64.9925	0.168442	0.0842209	+	0.996447i	$0.473160\pi$
$54$	0	0
$55$	26.4427	0.0648278
$56$	−453.893	−1.08311
$57$	0	0
$58$	127.330	0.288264
$59$	−122.788	−0.270942	−0.135471	−	0.990781i	$-0.543255\pi$
$59$	−122.788	−0.270942	−0.135471	+	0.990781i	$0.543255\pi$
$60$	0	0
$61$	264.384	0.554932	0.277466	−	0.960735i	$-0.410505\pi$
$61$	264.384	0.554932	0.277466	+	0.960735i	$0.410505\pi$
$62$	64.4877	0.132096
$63$	0	0
$64$	−810.411	−1.58283
$65$	−29.9366	−0.0571257
$66$	0	0
$67$	622.712	1.13547	0.567734	−	0.823212i	$-0.307820\pi$
$67$	622.712	1.13547	0.567734	+	0.823212i	$0.307820\pi$
$68$	−954.997	−1.70309
$69$	0	0
$70$	−692.274	−1.18204
$71$	−327.717	−0.547786	−0.273893	−	0.961760i	$-0.588311\pi$
$71$	−327.717	−0.547786	−0.273893	+	0.961760i	$0.588311\pi$
$72$	0	0
$73$	−833.518	−1.33638	−0.668191	−	0.743989i	$-0.732931\pi$
$73$	−833.518	−1.33638	−0.668191	+	0.743989i	$0.732931\pi$
$74$	34.0795	0.0535360
$75$	0	0
$76$	646.787	0.976204
$77$	166.767	0.246816
$78$	0	0
$79$	−666.457	−0.949143	−0.474571	−	0.880217i	$-0.657397\pi$
$79$	−666.457	−0.949143	−0.474571	+	0.880217i	$0.657397\pi$
$80$	−135.133	−0.188855
$81$	0	0
$82$	−1755.70	−2.36444
$83$	1320.72	1.74660	0.873301	−	0.487182i	$-0.161975\pi$
$83$	1320.72	1.74660	0.873301	+	0.487182i	$0.161975\pi$
$84$	0	0
$85$	−423.379	−0.540257
$86$	79.0180	0.0990782
$87$	0	0
$88$	−76.1229	−0.0922128
$89$	189.059	0.225172	0.112586	−	0.993642i	$-0.464087\pi$
$89$	189.059	0.225172	0.112586	+	0.993642i	$0.464087\pi$
$90$	0	0
$91$	−188.801	−0.217492
$92$	−350.640	−0.397355
$93$	0	0
$94$	−1153.87	−1.26609
$95$	286.740	0.309673
$96$	0	0
$97$	−137.630	−0.144064	−0.0720321	−	0.997402i	$-0.522948\pi$
$97$	−137.630	−0.144064	−0.0720321	+	0.997402i	$0.522948\pi$
$98$	−2859.97	−2.94796
$99$	0	0
$100$	281.957	0.281957
$101$	1721.15	1.69565	0.847826	−	0.530275i	$-0.177911\pi$
$101$	1721.15	1.69565	0.847826	+	0.530275i	$0.177911\pi$
$102$	0	0
$103$	797.585	0.762994	0.381497	−	0.924370i	$-0.375409\pi$
$103$	797.585	0.762994	0.381497	+	0.924370i	$0.375409\pi$
$104$	86.1810	0.0812571
$105$	0	0
$106$	−285.363	−0.261480
$107$	778.452	0.703325	0.351662	−	0.936127i	$-0.385616\pi$
$107$	778.452	0.703325	0.351662	+	0.936127i	$0.385616\pi$
$108$	0	0
$109$	721.205	0.633752	0.316876	−	0.948467i	$-0.397366\pi$
$109$	721.205	0.633752	0.316876	+	0.948467i	$0.397366\pi$
$110$	−116.102	−0.100635
$111$	0	0
$112$	−852.249	−0.719017
$113$	559.024	0.465386	0.232693	−	0.972550i	$-0.425246\pi$
$113$	559.024	0.465386	0.232693	+	0.972550i	$0.425246\pi$
$114$	0	0
$115$	−155.449	−0.126050
$116$	−327.070	−0.261790
$117$	0	0
$118$	539.124	0.420597
$119$	−2670.13	−2.05690
$120$	0	0
$121$	−1303.03	−0.978987
$122$	−1160.83	−0.861448
$123$	0	0
$124$	−165.648	−0.119965
$125$	125.000	0.0894427
$126$	0	0
$127$	−626.324	−0.437616	−0.218808	−	0.975768i	$-0.570217\pi$
$127$	−626.324	−0.437616	−0.218808	+	0.975768i	$0.570217\pi$
$128$	1687.73	1.16544
$129$	0	0
$130$	131.443	0.0886790
$131$	−1764.12	−1.17658	−0.588289	−	0.808651i	$-0.700198\pi$
$131$	−1764.12	−1.17658	−0.588289	+	0.808651i	$0.700198\pi$
$132$	0	0
$133$	1808.39	1.17900
$134$	−2734.14	−1.76264
$135$	0	0
$136$	1218.82	0.768476
$137$	433.723	0.270478	0.135239	−	0.990813i	$-0.456820\pi$
$137$	433.723	0.270478	0.135239	+	0.990813i	$0.456820\pi$
$138$	0	0
$139$	−504.659	−0.307947	−0.153974	−	0.988075i	$-0.549207\pi$
$139$	−504.659	−0.307947	−0.153974	+	0.988075i	$0.549207\pi$
$140$	1778.22	1.07348
$141$	0	0
$142$	1438.91	0.850355
$143$	−31.6641	−0.0185167
$144$	0	0
$145$	−145.000	−0.0830455
$146$	3659.73	2.07453
$147$	0	0
$148$	−87.5391	−0.0486194
$149$	354.655	0.194997	0.0974983	−	0.995236i	$-0.468916\pi$
$149$	354.655	0.194997	0.0974983	+	0.995236i	$0.468916\pi$
$150$	0	0
$151$	−325.611	−0.175483	−0.0877413	−	0.996143i	$-0.527965\pi$
$151$	−325.611	−0.175483	−0.0877413	+	0.996143i	$0.527965\pi$
$152$	−825.464	−0.440487
$153$	0	0
$154$	−732.222	−0.383144
$155$	−73.4366	−0.0380553
$156$	0	0
$157$	−2690.09	−1.36747	−0.683733	−	0.729732i	$-0.739645\pi$
$157$	−2690.09	−1.36747	−0.683733	+	0.729732i	$0.739645\pi$
$158$	2926.22	1.47340
$159$	0	0
$160$	1169.09	0.577653
$161$	−980.374	−0.479902
$162$	0	0
$163$	1411.51	0.678270	0.339135	−	0.940738i	$-0.389866\pi$
$163$	1411.51	0.678270	0.339135	+	0.940738i	$0.389866\pi$
$164$	4509.81	2.14730
$165$	0	0
$166$	−5798.89	−2.71133
$167$	2538.25	1.17614	0.588072	−	0.808809i	$-0.299887\pi$
$167$	2538.25	1.17614	0.588072	+	0.808809i	$0.299887\pi$
$168$	0	0
$169$	−2161.15	−0.983683
$170$	1858.93	0.838667
$171$	0	0
$172$	−202.971	−0.0899791
$173$	305.030	0.134052	0.0670260	−	0.997751i	$-0.478649\pi$
$173$	305.030	0.134052	0.0670260	+	0.997751i	$0.478649\pi$
$174$	0	0
$175$	788.340	0.340531
$176$	−142.932	−0.0612152
$177$	0	0
$178$	−830.104	−0.349545
$179$	−3093.43	−1.29170	−0.645849	−	0.763465i	$-0.723496\pi$
$179$	−3093.43	−1.29170	−0.645849	+	0.763465i	$0.723496\pi$
$180$	0	0
$181$	2800.38	1.15000	0.575001	−	0.818153i	$-0.305002\pi$
$181$	2800.38	1.15000	0.575001	+	0.818153i	$0.305002\pi$
$182$	828.971	0.337623
$183$	0	0
$184$	447.505	0.179296
$185$	−38.8087	−0.0154231
$186$	0	0
$187$	−447.811	−0.175118
$188$	2963.91	1.14981
$189$	0	0
$190$	−1258.99	−0.480720
$191$	−4208.78	−1.59443	−0.797217	−	0.603693i	$-0.793695\pi$
$191$	−4208.78	−1.59443	−0.797217	+	0.603693i	$0.793695\pi$
$192$	0	0
$193$	−247.563	−0.0923315	−0.0461658	−	0.998934i	$-0.514700\pi$
$193$	−247.563	−0.0923315	−0.0461658	+	0.998934i	$0.514700\pi$
$194$	604.293	0.223638
$195$	0	0
$196$	7346.31	2.67723
$197$	4574.07	1.65426	0.827129	−	0.562013i	$-0.189973\pi$
$197$	4574.07	1.65426	0.827129	+	0.562013i	$0.189973\pi$
$198$	0	0
$199$	2601.05	0.926551	0.463276	−	0.886214i	$-0.346674\pi$
$199$	2601.05	0.926551	0.463276	+	0.886214i	$0.346674\pi$
$200$	−359.849	−0.127226
$201$	0	0
$202$	−7557.06	−2.63224
$203$	−914.475	−0.316175
$204$	0	0
$205$	1999.33	0.681168
$206$	−3501.96	−1.18443
$207$	0	0
$208$	161.817	0.0539423
$209$	303.287	0.100377
$210$	0	0
$211$	244.148	0.0796582	0.0398291	−	0.999207i	$-0.487319\pi$
$211$	244.148	0.0796582	0.0398291	+	0.999207i	$0.487319\pi$
$212$	733.004	0.237467
$213$	0	0
$214$	−3417.95	−1.09181
$215$	−89.9833	−0.0285433
$216$	0	0
$217$	−463.144	−0.144886
$218$	−3166.60	−0.983803
$219$	0	0
$220$	298.228	0.0913932
$221$	506.980	0.154313
$222$	0	0
$223$	4179.26	1.25500	0.627498	−	0.778618i	$-0.284079\pi$
$223$	4179.26	1.25500	0.627498	+	0.778618i	$0.284079\pi$
$224$	7373.12	2.19927
$225$	0	0
$226$	−2454.51	−0.722441
$227$	−4034.64	−1.17968	−0.589842	−	0.807518i	$-0.700810\pi$
$227$	−4034.64	−1.17968	−0.589842	+	0.807518i	$0.700810\pi$
$228$	0	0
$229$	4501.53	1.29899	0.649497	−	0.760364i	$-0.274980\pi$
$229$	4501.53	1.29899	0.649497	+	0.760364i	$0.274980\pi$
$230$	682.531	0.195673
$231$	0	0
$232$	417.424	0.118126
$233$	4710.72	1.32450	0.662252	−	0.749282i	$-0.269601\pi$
$233$	4710.72	1.32450	0.662252	+	0.749282i	$0.269601\pi$
$234$	0	0
$235$	1313.99	0.364745
$236$	−1384.83	−0.381970
$237$	0	0
$238$	11723.8	3.19302
$239$	496.634	0.134412	0.0672062	−	0.997739i	$-0.478591\pi$
$239$	496.634	0.134412	0.0672062	+	0.997739i	$0.478591\pi$
$240$	0	0
$241$	6414.62	1.71453	0.857266	−	0.514874i	$-0.172161\pi$
$241$	6414.62	1.71453	0.857266	+	0.514874i	$0.172161\pi$
$242$	5721.22	1.51973
$243$	0	0
$244$	2981.79	0.782335
$245$	3256.84	0.849274
$246$	0	0
$247$	−343.360	−0.0884514
$248$	211.409	0.0541309
$249$	0	0
$250$	−548.838	−0.138846
$251$	−4741.55	−1.19237	−0.596183	−	0.802848i	$-0.703317\pi$
$251$	−4741.55	−1.19237	−0.596183	+	0.802848i	$0.703317\pi$
$252$	0	0
$253$	−164.420	−0.0408576
$254$	2750.00	0.679333
$255$	0	0
$256$	−927.044	−0.226329
$257$	6110.47	1.48312	0.741558	−	0.670889i	$-0.234087\pi$
$257$	6110.47	1.48312	0.741558	+	0.670889i	$0.234087\pi$
$258$	0	0
$259$	−244.756	−0.0587196
$260$	−337.633	−0.0805350
$261$	0	0
$262$	7745.72	1.82646
$263$	3905.40	0.915654	0.457827	−	0.889041i	$-0.348628\pi$
$263$	3905.40	0.915654	0.457827	+	0.889041i	$0.348628\pi$
$264$	0	0
$265$	324.963	0.0753294
$266$	−7940.10	−1.83022
$267$	0	0
$268$	7023.11	1.60076
$269$	6684.63	1.51513	0.757563	−	0.652762i	$-0.226390\pi$
$269$	6684.63	1.51513	0.757563	+	0.652762i	$0.226390\pi$
$270$	0	0
$271$	−6784.76	−1.52083	−0.760415	−	0.649438i	$-0.775004\pi$
$271$	−6784.76	−1.52083	−0.760415	+	0.649438i	$0.775004\pi$
$272$	2288.50	0.510151
$273$	0	0
$274$	−1904.35	−0.419875
$275$	132.213	0.0289919
$276$	0	0
$277$	3811.10	0.826666	0.413333	−	0.910580i	$-0.364364\pi$
$277$	3811.10	0.826666	0.413333	+	0.910580i	$0.364364\pi$
$278$	2215.81	0.478041
$279$	0	0
$280$	−2269.46	−0.484380
$281$	−1848.27	−0.392380	−0.196190	−	0.980566i	$-0.562857\pi$
$281$	−1848.27	−0.392380	−0.196190	+	0.980566i	$0.562857\pi$
$282$	0	0
$283$	1381.94	0.290275	0.145138	−	0.989411i	$-0.453638\pi$
$283$	1381.94	0.290275	0.145138	+	0.989411i	$0.453638\pi$
$284$	−3696.08	−0.772261
$285$	0	0
$286$	139.028	0.0287443
$287$	12609.2	2.59338
$288$	0	0
$289$	2256.98	0.459390
$290$	636.652	0.128915
$291$	0	0
$292$	−9400.65	−1.88401
$293$	−6221.44	−1.24048	−0.620239	−	0.784413i	$-0.712964\pi$
$293$	−6221.44	−1.24048	−0.620239	+	0.784413i	$0.712964\pi$
$294$	0	0
$295$	−613.938	−0.121169
$296$	111.722	0.0219382
$297$	0	0
$298$	−1557.19	−0.302702
$299$	186.144	0.0360034
$300$	0	0
$301$	−567.499	−0.108671
$302$	1429.66	0.272410
$303$	0	0
$304$	−1549.93	−0.292416
$305$	1321.92	0.248173
$306$	0	0
$307$	8135.50	1.51244	0.756218	−	0.654320i	$-0.227045\pi$
$307$	8135.50	1.51244	0.756218	+	0.654320i	$0.227045\pi$
$308$	1880.84	0.347957
$309$	0	0
$310$	322.438	0.0590751
$311$	−3117.65	−0.568444	−0.284222	−	0.958759i	$-0.591735\pi$
$311$	−3117.65	−0.568444	−0.284222	+	0.958759i	$0.591735\pi$
$312$	0	0
$313$	7536.36	1.36096	0.680480	−	0.732767i	$-0.261771\pi$
$313$	7536.36	1.36096	0.680480	+	0.732767i	$0.261771\pi$
$314$	11811.4	2.12278
$315$	0	0
$316$	−7516.49	−1.33809
$317$	−3579.28	−0.634172	−0.317086	−	0.948397i	$-0.602704\pi$
$317$	−3579.28	−0.634172	−0.317086	+	0.948397i	$0.602704\pi$
$318$	0	0
$319$	−153.367	−0.0269183
$320$	−4052.05	−0.707865
$321$	0	0
$322$	4304.53	0.744976
$323$	−4855.98	−0.836515
$324$	0	0
$325$	−149.683	−0.0255474
$326$	−6197.52	−1.05291
$327$	0	0
$328$	−5755.66	−0.968911
$329$	8286.96	1.38868
$330$	0	0
$331$	−6257.20	−1.03905	−0.519527	−	0.854454i	$-0.673892\pi$
$331$	−6257.20	−1.03905	−0.519527	+	0.854454i	$0.673892\pi$
$332$	14895.5	2.46233
$333$	0	0
$334$	−11144.7	−1.82578
$335$	3113.56	0.507797
$336$	0	0
$337$	−4034.64	−0.652168	−0.326084	−	0.945341i	$-0.605729\pi$
$337$	−4034.64	−0.652168	−0.326084	+	0.945341i	$0.605729\pi$
$338$	9488.98	1.52702
$339$	0	0
$340$	−4774.98	−0.761647
$341$	−77.6744	−0.0123352
$342$	0	0
$343$	9723.97	1.53074
$344$	259.043	0.0406007
$345$	0	0
$346$	−1339.30	−0.208095
$347$	10221.1	1.58126	0.790630	−	0.612294i	$-0.209753\pi$
$347$	10221.1	1.58126	0.790630	+	0.612294i	$0.209753\pi$
$348$	0	0
$349$	−2543.26	−0.390079	−0.195040	−	0.980795i	$-0.562484\pi$
$349$	−2543.26	−0.390079	−0.195040	+	0.980795i	$0.562484\pi$
$350$	−3461.37	−0.528623
$351$	0	0
$352$	1236.55	0.187240
$353$	−11741.3	−1.77034	−0.885168	−	0.465272i	$-0.845956\pi$
$353$	−11741.3	−1.77034	−0.885168	+	0.465272i	$0.845956\pi$
$354$	0	0
$355$	−1638.58	−0.244978
$356$	2132.27	0.317443
$357$	0	0
$358$	13582.3	2.00517
$359$	−5475.18	−0.804927	−0.402463	−	0.915436i	$-0.631846\pi$
$359$	−5475.18	−0.804927	−0.402463	+	0.915436i	$0.631846\pi$
$360$	0	0
$361$	−3570.21	−0.520514
$362$	−12295.6	−1.78520
$363$	0	0
$364$	−2129.36	−0.306617
$365$	−4167.59	−0.597649
$366$	0	0
$367$	11158.0	1.58704	0.793522	−	0.608542i	$-0.208245\pi$
$367$	11158.0	1.58704	0.793522	+	0.608542i	$0.208245\pi$
$368$	840.254	0.119025
$369$	0	0
$370$	170.398	0.0239420
$371$	2049.45	0.286798
$372$	0	0
$373$	5823.05	0.808327	0.404163	−	0.914687i	$-0.367563\pi$
$373$	5823.05	0.808327	0.404163	+	0.914687i	$0.367563\pi$
$374$	1966.20	0.271845
$375$	0	0
$376$	−3782.70	−0.518823
$377$	173.632	0.0237202
$378$	0	0
$379$	2272.79	0.308036	0.154018	−	0.988068i	$-0.450779\pi$
$379$	2272.79	0.308036	0.154018	+	0.988068i	$0.450779\pi$
$380$	3233.93	0.436572
$381$	0	0
$382$	18479.5	2.47512
$383$	−8686.06	−1.15884	−0.579422	−	0.815028i	$-0.696722\pi$
$383$	−8686.06	−1.15884	−0.579422	+	0.815028i	$0.696722\pi$
$384$	0	0
$385$	833.833	0.110379
$386$	1086.98	0.143331
$387$	0	0
$388$	−1552.23	−0.203099
$389$	−5410.18	−0.705159	−0.352580	−	0.935782i	$-0.614695\pi$
$389$	−5410.18	−0.705159	−0.352580	+	0.935782i	$0.614695\pi$
$390$	0	0
$391$	2632.55	0.340496
$392$	−9375.76	−1.20803
$393$	0	0
$394$	−20083.4	−2.56798
$395$	−3332.28	−0.424470
$396$	0	0
$397$	13470.8	1.70297	0.851486	−	0.524377i	$-0.175702\pi$
$397$	13470.8	1.70297	0.851486	+	0.524377i	$0.175702\pi$
$398$	−11420.4	−1.43833
$399$	0	0
$400$	−675.667	−0.0844584
$401$	−1749.26	−0.217840	−0.108920	−	0.994050i	$-0.534739\pi$
$401$	−1749.26	−0.217840	−0.108920	+	0.994050i	$0.534739\pi$
$402$	0	0
$403$	87.9376	0.0108697
$404$	19411.6	2.39050
$405$	0	0
$406$	4015.19	0.490814
$407$	−41.0482	−0.00499923
$408$	0	0
$409$	−4261.47	−0.515198	−0.257599	−	0.966252i	$-0.582931\pi$
$409$	−4261.47	−0.515198	−0.257599	+	0.966252i	$0.582931\pi$
$410$	−8778.48	−1.05741
$411$	0	0
$412$	8995.38	1.07566
$413$	−3871.94	−0.461321
$414$	0	0
$415$	6603.60	0.781104
$416$	−1399.94	−0.164994
$417$	0	0
$418$	−1331.64	−0.155820
$419$	3355.37	0.391218	0.195609	−	0.980682i	$-0.437332\pi$
$419$	3355.37	0.391218	0.195609	+	0.980682i	$0.437332\pi$
$420$	0	0
$421$	−7859.86	−0.909896	−0.454948	−	0.890518i	$-0.650342\pi$
$421$	−7859.86	−0.909896	−0.454948	+	0.890518i	$0.650342\pi$
$422$	−1071.98	−0.123657
$423$	0	0
$424$	−935.499	−0.107151
$425$	−2116.89	−0.241610
$426$	0	0
$427$	8336.98	0.944858
$428$	8779.59	0.991537
$429$	0	0
$430$	395.090	0.0443091
$431$	9083.57	1.01517	0.507587	−	0.861600i	$-0.330537\pi$
$431$	9083.57	1.01517	0.507587	+	0.861600i	$0.330537\pi$
$432$	0	0
$433$	10303.2	1.14351	0.571756	−	0.820424i	$-0.306262\pi$
$433$	10303.2	1.14351	0.571756	+	0.820424i	$0.306262\pi$
$434$	2033.53	0.224914
$435$	0	0
$436$	8133.95	0.893453
$437$	−1782.94	−0.195171
$438$	0	0
$439$	10285.7	1.11824	0.559122	−	0.829086i	$-0.311139\pi$
$439$	10285.7	1.11824	0.559122	+	0.829086i	$0.311139\pi$
$440$	−380.614	−0.0412388
$441$	0	0
$442$	−2226.00	−0.239547
$443$	−11628.0	−1.24710	−0.623549	−	0.781784i	$-0.714310\pi$
$443$	−11628.0	−1.24710	−0.623549	+	0.781784i	$0.714310\pi$
$444$	0	0
$445$	945.297	0.100700
$446$	−18349.9	−1.94819
$447$	0	0
$448$	−25555.2	−2.69502
$449$	13543.0	1.42346	0.711730	−	0.702453i	$-0.247912\pi$
$449$	13543.0	1.42346	0.711730	+	0.702453i	$0.247912\pi$
$450$	0	0
$451$	2114.71	0.220793
$452$	6304.83	0.656094
$453$	0	0
$454$	17714.9	1.83128
$455$	−944.007	−0.0972654
$456$	0	0
$457$	16383.2	1.67696	0.838481	−	0.544930i	$-0.183444\pi$
$457$	16383.2	1.67696	0.838481	+	0.544930i	$0.183444\pi$
$458$	−19764.9	−2.01649
$459$	0	0
$460$	−1753.20	−0.177703
$461$	−13577.7	−1.37175	−0.685877	−	0.727717i	$-0.740581\pi$
$461$	−13577.7	−1.37175	−0.685877	+	0.727717i	$0.740581\pi$
$462$	0	0
$463$	−17986.2	−1.80538	−0.902688	−	0.430297i	$-0.858409\pi$
$463$	−17986.2	−1.80538	−0.902688	+	0.430297i	$0.858409\pi$
$464$	783.774	0.0784176
$465$	0	0
$466$	−20683.4	−2.05609
$467$	−792.472	−0.0785251	−0.0392625	−	0.999229i	$-0.512501\pi$
$467$	−792.472	−0.0785251	−0.0392625	+	0.999229i	$0.512501\pi$
$468$	0	0
$469$	19636.3	1.93331
$470$	−5769.33	−0.566212
$471$	0	0
$472$	1767.40	0.172354
$473$	−95.1759	−0.00925199
$474$	0	0
$475$	1433.70	0.138490
$476$	−30114.5	−2.89978
$477$	0	0
$478$	−2180.57	−0.208655
$479$	6655.02	0.634813	0.317407	−	0.948290i	$-0.397188\pi$
$479$	6655.02	0.634813	0.317407	+	0.948290i	$0.397188\pi$
$480$	0	0
$481$	46.4719	0.00440528
$482$	−28164.7	−2.66155
$483$	0	0
$484$	−14695.9	−1.38016
$485$	−688.151	−0.0644274
$486$	0	0
$487$	−13972.7	−1.30013	−0.650064	−	0.759880i	$-0.725258\pi$
$487$	−13972.7	−1.30013	−0.650064	+	0.759880i	$0.725258\pi$
$488$	−3805.53	−0.353008
$489$	0	0
$490$	−14299.8	−1.31837
$491$	−3638.63	−0.334438	−0.167219	−	0.985920i	$-0.553479\pi$
$491$	−3638.63	−0.334438	−0.167219	+	0.985920i	$0.553479\pi$
$492$	0	0
$493$	2455.60	0.224330
$494$	1507.59	0.137307
$495$	0	0
$496$	396.950	0.0359346
$497$	−10334.1	−0.932691
$498$	0	0
$499$	11313.5	1.01496	0.507478	−	0.861665i	$-0.330578\pi$
$499$	11313.5	1.01496	0.507478	+	0.861665i	$0.330578\pi$
$500$	1409.78	0.126095
$501$	0	0
$502$	20818.7	1.85097
$503$	4598.92	0.407665	0.203833	−	0.979006i	$-0.434660\pi$
$503$	4598.92	0.407665	0.203833	+	0.979006i	$0.434660\pi$
$504$	0	0
$505$	8605.75	0.758318
$506$	721.917	0.0634252
$507$	0	0
$508$	−7063.86	−0.616945
$509$	−6967.41	−0.606729	−0.303364	−	0.952875i	$-0.598110\pi$
$509$	−6967.41	−0.606729	−0.303364	+	0.952875i	$0.598110\pi$
$510$	0	0
$511$	−26283.8	−2.27540
$512$	−9431.48	−0.814095
$513$	0	0
$514$	−26829.3	−2.30231
$515$	3987.92	0.341221
$516$	0	0
$517$	1389.81	0.118228
$518$	1074.65	0.0911533
$519$	0	0
$520$	430.905	0.0363393
$521$	11038.5	0.928228	0.464114	−	0.885776i	$-0.346373\pi$
$521$	11038.5	0.928228	0.464114	+	0.885776i	$0.346373\pi$
$522$	0	0
$523$	−15368.6	−1.28493	−0.642467	−	0.766314i	$-0.722089\pi$
$523$	−15368.6	−1.28493	−0.642467	+	0.766314i	$0.722089\pi$
$524$	−19896.2	−1.65872
$525$	0	0
$526$	−17147.4	−1.42141
$527$	1243.66	0.102798
$528$	0	0
$529$	−11200.4	−0.920558
$530$	−1426.81	−0.116937
$531$	0	0
$532$	20395.5	1.66214
$533$	−2394.12	−0.194561
$534$	0	0
$535$	3892.26	0.314536
$536$	−8963.28	−0.722303
$537$	0	0
$538$	−29350.2	−2.35200
$539$	3444.78	0.275283
$540$	0	0
$541$	−10945.5	−0.869839	−0.434920	−	0.900469i	$-0.643223\pi$
$541$	−10945.5	−0.869839	−0.434920	+	0.900469i	$0.643223\pi$
$542$	29789.9	2.36086
$543$	0	0
$544$	−19798.7	−1.56041
$545$	3606.03	0.283422
$546$	0	0
$547$	10140.1	0.792616	0.396308	−	0.918118i	$-0.370291\pi$
$547$	10140.1	0.792616	0.396308	+	0.918118i	$0.370291\pi$
$548$	4891.64	0.381315
$549$	0	0
$550$	−580.510	−0.0450055
$551$	−1663.09	−0.128585
$552$	0	0
$553$	−21015.8	−1.61606
$554$	−16733.4	−1.28327
$555$	0	0
$556$	−5691.69	−0.434139
$557$	−18329.0	−1.39430	−0.697151	−	0.716925i	$-0.745549\pi$
$557$	−18329.0	−1.39430	−0.697151	+	0.716925i	$0.745549\pi$
$558$	0	0
$559$	107.752	0.00815278
$560$	−4261.24	−0.321554
$561$	0	0
$562$	8115.21	0.609110
$563$	−2868.99	−0.214766	−0.107383	−	0.994218i	$-0.534247\pi$
$563$	−2868.99	−0.214766	−0.107383	+	0.994218i	$0.534247\pi$
$564$	0	0
$565$	2795.12	0.208127
$566$	−6067.69	−0.450608
$567$	0	0
$568$	4717.14	0.348462
$569$	−18562.0	−1.36759	−0.683795	−	0.729674i	$-0.739672\pi$
$569$	−18562.0	−1.36759	−0.683795	+	0.729674i	$0.739672\pi$
$570$	0	0
$571$	−10162.6	−0.744815	−0.372408	−	0.928069i	$-0.621468\pi$
$571$	−10162.6	−0.744815	−0.372408	+	0.928069i	$0.621468\pi$
$572$	−357.116	−0.0261045
$573$	0	0
$574$	−55363.4	−4.02583
$575$	−777.245	−0.0563711
$576$	0	0
$577$	2525.24	0.182196	0.0910981	−	0.995842i	$-0.470962\pi$
$577$	2525.24	0.182196	0.0910981	+	0.995842i	$0.470962\pi$
$578$	−9909.74	−0.713133
$579$	0	0
$580$	−1635.35	−0.117076
$581$	41647.1	2.97386
$582$	0	0
$583$	343.715	0.0244172
$584$	11997.6	0.850111
$585$	0	0
$586$	27316.5	1.92565
$587$	−3327.35	−0.233960	−0.116980	−	0.993134i	$-0.537321\pi$
$587$	−3327.35	−0.233960	−0.116980	+	0.993134i	$0.537321\pi$
$588$	0	0
$589$	−842.289	−0.0589234
$590$	2695.62	0.188097
$591$	0	0
$592$	209.774	0.0145636
$593$	16731.4	1.15864	0.579321	−	0.815099i	$-0.303318\pi$
$593$	16731.4	1.15864	0.579321	+	0.815099i	$0.303318\pi$
$594$	0	0
$595$	−13350.7	−0.919872
$596$	3999.90	0.274903
$597$	0	0
$598$	−817.305	−0.0558898
$599$	12851.1	0.876600	0.438300	−	0.898829i	$-0.355581\pi$
$599$	12851.1	0.876600	0.438300	+	0.898829i	$0.355581\pi$
$600$	0	0
$601$	−8773.05	−0.595441	−0.297720	−	0.954653i	$-0.596226\pi$
$601$	−8773.05	−0.595441	−0.297720	+	0.954653i	$0.596226\pi$
$602$	2491.72	0.168696
$603$	0	0
$604$	−3672.33	−0.247393
$605$	−6515.16	−0.437816
$606$	0	0
$607$	−19083.1	−1.27605	−0.638023	−	0.770017i	$-0.720247\pi$
$607$	−19083.1	−1.27605	−0.638023	+	0.770017i	$0.720247\pi$
$608$	13409.0	0.894417
$609$	0	0
$610$	−5804.15	−0.385251
$611$	−1573.45	−0.104182
$612$	0	0
$613$	20976.8	1.38213	0.691063	−	0.722794i	$-0.257142\pi$
$613$	20976.8	1.38213	0.691063	+	0.722794i	$0.257142\pi$
$614$	−35720.6	−2.34783
$615$	0	0
$616$	−2400.43	−0.157007
$617$	15290.1	0.997658	0.498829	−	0.866700i	$-0.333764\pi$
$617$	15290.1	0.997658	0.498829	+	0.866700i	$0.333764\pi$
$618$	0	0
$619$	18786.8	1.21988	0.609938	−	0.792449i	$-0.291194\pi$
$619$	18786.8	1.21988	0.609938	+	0.792449i	$0.291194\pi$
$620$	−828.239	−0.0536498
$621$	0	0
$622$	13688.7	0.882422
$623$	5961.73	0.383389
$624$	0	0
$625$	625.000	0.0400000
$626$	−33089.9	−2.11268
$627$	0	0
$628$	−30339.5	−1.92783
$629$	657.231	0.0416622
$630$	0	0
$631$	14273.7	0.900516	0.450258	−	0.892898i	$-0.351332\pi$
$631$	14273.7	0.900516	0.450258	+	0.892898i	$0.351332\pi$
$632$	9592.94	0.603777
$633$	0	0
$634$	15715.6	0.984456
$635$	−3131.62	−0.195708
$636$	0	0
$637$	−3899.95	−0.242577
$638$	673.391	0.0417865
$639$	0	0
$640$	8438.66	0.521199
$641$	−10936.9	−0.673919	−0.336959	−	0.941519i	$-0.609398\pi$
$641$	−10936.9	−0.673919	−0.336959	+	0.941519i	$0.609398\pi$
$642$	0	0
$643$	5329.32	0.326855	0.163427	−	0.986555i	$-0.447745\pi$
$643$	5329.32	0.326855	0.163427	+	0.986555i	$0.447745\pi$
$644$	−11056.9	−0.676559
$645$	0	0
$646$	21321.2	1.29856
$647$	15807.3	0.960509	0.480255	−	0.877129i	$-0.340544\pi$
$647$	15807.3	0.960509	0.480255	+	0.877129i	$0.340544\pi$
$648$	0	0
$649$	−649.367	−0.0392756
$650$	657.213	0.0396585
$651$	0	0
$652$	15919.4	0.956215
$653$	11047.2	0.662037	0.331019	−	0.943624i	$-0.392608\pi$
$653$	11047.2	0.662037	0.331019	+	0.943624i	$0.392608\pi$
$654$	0	0
$655$	−8820.59	−0.526182
$656$	−10807.1	−0.643209
$657$	0	0
$658$	−36385.6	−2.15571
$659$	20355.2	1.20322	0.601612	−	0.798789i	$-0.294525\pi$
$659$	20355.2	1.20322	0.601612	+	0.798789i	$0.294525\pi$
$660$	0	0
$661$	−27873.5	−1.64017	−0.820086	−	0.572241i	$-0.806074\pi$
$661$	−27873.5	−1.64017	−0.820086	+	0.572241i	$0.806074\pi$
$662$	27473.5	1.61297
$663$	0	0
$664$	−19010.4	−1.11106
$665$	9041.95	0.527266
$666$	0	0
$667$	901.605	0.0523392
$668$	28627.1	1.65811
$669$	0	0
$670$	−13670.7	−0.788277
$671$	1398.20	0.0804427
$672$	0	0
$673$	−9555.18	−0.547288	−0.273644	−	0.961831i	$-0.588229\pi$
$673$	−9555.18	−0.547288	−0.273644	+	0.961831i	$0.588229\pi$
$674$	17714.9	1.01239
$675$	0	0
$676$	−24374.1	−1.38678
$677$	−17556.1	−0.996658	−0.498329	−	0.866988i	$-0.666053\pi$
$677$	−17556.1	−0.996658	−0.498329	+	0.866988i	$0.666053\pi$
$678$	0	0
$679$	−4339.97	−0.245292
$680$	6094.09	0.343673
$681$	0	0
$682$	341.045	0.0191485
$683$	−11046.7	−0.618875	−0.309437	−	0.950920i	$-0.600141\pi$
$683$	−11046.7	−0.618875	−0.309437	+	0.950920i	$0.600141\pi$
$684$	0	0
$685$	2168.61	0.120961
$686$	−42695.1	−2.37625
$687$	0	0
$688$	486.390	0.0269527
$689$	−389.130	−0.0215162
$690$	0	0
$691$	−7999.45	−0.440396	−0.220198	−	0.975455i	$-0.570670\pi$
$691$	−7999.45	−0.440396	−0.220198	+	0.975455i	$0.570670\pi$
$692$	3440.21	0.188984
$693$	0	0
$694$	−44877.8	−2.45467
$695$	−2523.30	−0.137718
$696$	0	0
$697$	−33859.0	−1.84003
$698$	11166.7	0.605538
$699$	0	0
$700$	8891.12	0.480075
$701$	25401.3	1.36861	0.684303	−	0.729198i	$-0.260107\pi$
$701$	25401.3	1.36861	0.684303	+	0.729198i	$0.260107\pi$
$702$	0	0
$703$	−445.120	−0.0238806
$704$	−4285.88	−0.229447
$705$	0	0
$706$	51552.7	2.74818
$707$	54274.1	2.88711
$708$	0	0
$709$	−9479.34	−0.502122	−0.251061	−	0.967971i	$-0.580779\pi$
$709$	−9479.34	−0.502122	−0.251061	+	0.967971i	$0.580779\pi$
$710$	7194.54	0.380290
$711$	0	0
$712$	−2721.31	−0.143238
$713$	456.626	0.0239843
$714$	0	0
$715$	−158.320	−0.00828091
$716$	−34888.6	−1.82102
$717$	0	0
$718$	24039.9	1.24953
$719$	32420.3	1.68161	0.840803	−	0.541341i	$-0.182083\pi$
$719$	32420.3	1.68161	0.840803	+	0.541341i	$0.182083\pi$
$720$	0	0
$721$	25150.7	1.29912
$722$	15675.7	0.808019
$723$	0	0
$724$	31583.4	1.62126
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−6808.51	−0.347337	−0.173668	−	0.984804i	$-0.555562\pi$
$727$	−6808.51	−0.347337	−0.173668	+	0.984804i	$0.555562\pi$
$728$	2717.60	0.138353
$729$	0	0
$730$	18298.7	0.927759
$731$	1523.88	0.0771036
$732$	0	0
$733$	20190.4	1.01740	0.508698	−	0.860945i	$-0.330127\pi$
$733$	20190.4	1.01740	0.508698	+	0.860945i	$0.330127\pi$
$734$	−48991.6	−2.46364
$735$	0	0
$736$	−7269.35	−0.364065
$737$	3293.23	0.164597
$738$	0	0
$739$	15130.0	0.753134	0.376567	−	0.926389i	$-0.377105\pi$
$739$	15130.0	0.753134	0.376567	+	0.926389i	$0.377105\pi$
$740$	−437.695	−0.0217432
$741$	0	0
$742$	−8998.52	−0.445210
$743$	13315.6	0.657470	0.328735	−	0.944422i	$-0.393378\pi$
$743$	13315.6	0.657470	0.328735	+	0.944422i	$0.393378\pi$
$744$	0	0
$745$	1773.28	0.0872051
$746$	−25567.3	−1.25480
$747$	0	0
$748$	−5050.53	−0.246879
$749$	24547.4	1.19752
$750$	0	0
$751$	−18846.4	−0.915734	−0.457867	−	0.889021i	$-0.651386\pi$
$751$	−18846.4	−0.915734	−0.457867	+	0.889021i	$0.651386\pi$
$752$	−7102.55	−0.344419
$753$	0	0
$754$	−762.367	−0.0368220
$755$	−1628.06	−0.0784782
$756$	0	0
$757$	699.522	0.0335859	0.0167930	−	0.999859i	$-0.494654\pi$
$757$	699.522	0.0335859	0.0167930	+	0.999859i	$0.494654\pi$
$758$	−9979.16	−0.478178
$759$	0	0
$760$	−4127.32	−0.196992
$761$	37430.3	1.78298	0.891489	−	0.453041i	$-0.149661\pi$
$761$	37430.3	1.78298	0.891489	+	0.453041i	$0.149661\pi$
$762$	0	0
$763$	22742.2	1.07906
$764$	−47467.8	−2.24781
$765$	0	0
$766$	38137.9	1.79893
$767$	735.168	0.0346093
$768$	0	0
$769$	30299.7	1.42085	0.710426	−	0.703772i	$-0.248502\pi$
$769$	30299.7	1.42085	0.710426	+	0.703772i	$0.248502\pi$
$770$	−3661.11	−0.171347
$771$	0	0
$772$	−2792.09	−0.130168
$773$	−16926.9	−0.787603	−0.393802	−	0.919195i	$-0.628840\pi$
$773$	−16926.9	−0.787603	−0.393802	+	0.919195i	$0.628840\pi$
$774$	0	0
$775$	−367.183	−0.0170188
$776$	1981.04	0.0916433
$777$	0	0
$778$	23754.5	1.09465
$779$	22931.5	1.05470
$780$	0	0
$781$	−1733.14	−0.0794068
$782$	−11558.8	−0.528568
$783$	0	0
$784$	−17604.3	−0.801946
$785$	−13450.4	−0.611550
$786$	0	0
$787$	−20726.9	−0.938799	−0.469400	−	0.882986i	$-0.655530\pi$
$787$	−20726.9	−0.938799	−0.469400	+	0.882986i	$0.655530\pi$
$788$	51587.6	2.33215
$789$	0	0
$790$	14631.1	0.658924
$791$	17628.1	0.792391
$792$	0	0
$793$	−1582.95	−0.0708854
$794$	−59146.3	−2.64361
$795$	0	0
$796$	29335.4	1.30624
$797$	25560.6	1.13601	0.568006	−	0.823025i	$-0.307715\pi$
$797$	25560.6	1.13601	0.568006	+	0.823025i	$0.307715\pi$
$798$	0	0
$799$	−22252.6	−0.985282
$800$	5845.44	0.258334
$801$	0	0
$802$	7680.49	0.338164
$803$	−4408.09	−0.193721
$804$	0	0
$805$	−4901.87	−0.214619
$806$	−386.108	−0.0168735
$807$	0	0
$808$	−24774.1	−1.07865
$809$	45525.7	1.97849	0.989244	−	0.146273i	$-0.0467279\pi$
$809$	45525.7	1.97849	0.989244	+	0.146273i	$0.0467279\pi$
$810$	0	0
$811$	8515.77	0.368717	0.184358	−	0.982859i	$-0.440979\pi$
$811$	8515.77	0.368717	0.184358	+	0.982859i	$0.440979\pi$
$812$	−10313.7	−0.445739
$813$	0	0
$814$	180.231	0.00776054
$815$	7057.55	0.303332
$816$	0	0
$817$	−1032.07	−0.0441954
$818$	18710.8	0.799767
$819$	0	0
$820$	22549.0	0.960300
$821$	−18922.1	−0.804369	−0.402185	−	0.915559i	$-0.631749\pi$
$821$	−18922.1	−0.804369	−0.402185	+	0.915559i	$0.631749\pi$
$822$	0	0
$823$	−13297.5	−0.563210	−0.281605	−	0.959530i	$-0.590867\pi$
$823$	−13297.5	−0.563210	−0.281605	+	0.959530i	$0.590867\pi$
$824$	−11480.4	−0.485362
$825$	0	0
$826$	17000.5	0.716131
$827$	26020.8	1.09411	0.547056	−	0.837096i	$-0.315748\pi$
$827$	26020.8	1.09411	0.547056	+	0.837096i	$0.315748\pi$
$828$	0	0
$829$	8756.88	0.366874	0.183437	−	0.983031i	$-0.441278\pi$
$829$	8756.88	0.366874	0.183437	+	0.983031i	$0.441278\pi$
$830$	−28994.5	−1.21255
$831$	0	0
$832$	4852.18	0.202186
$833$	−55155.1	−2.29413
$834$	0	0
$835$	12691.3	0.525987
$836$	3420.55	0.141510
$837$	0	0
$838$	−14732.4	−0.607307
$839$	1131.02	0.0465400	0.0232700	−	0.999729i	$-0.492592\pi$
$839$	1131.02	0.0465400	0.0232700	+	0.999729i	$0.492592\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	34510.3	1.41248
$843$	0	0
$844$	2753.57	0.112301
$845$	−10805.8	−0.439917
$846$	0	0
$847$	−41089.3	−1.66688
$848$	−1756.53	−0.0711316
$849$	0	0
$850$	9294.65	0.375063
$851$	241.311	0.00972037
$852$	0	0
$853$	−12199.3	−0.489679	−0.244839	−	0.969564i	$-0.578735\pi$
$853$	−12199.3	−0.489679	−0.244839	+	0.969564i	$0.578735\pi$
$854$	−36605.2	−1.46675
$855$	0	0
$856$	−11205.0	−0.447405
$857$	−37800.4	−1.50670	−0.753348	−	0.657622i	$-0.771562\pi$
$857$	−37800.4	−1.50670	−0.753348	+	0.657622i	$0.771562\pi$
$858$	0	0
$859$	30888.1	1.22688	0.613438	−	0.789743i	$-0.289786\pi$
$859$	30888.1	1.22688	0.613438	+	0.789743i	$0.289786\pi$
$860$	−1014.86	−0.0402399
$861$	0	0
$862$	−39883.3	−1.57590
$863$	−38008.4	−1.49921	−0.749607	−	0.661883i	$-0.769758\pi$
$863$	−38008.4	−1.49921	−0.749607	+	0.661883i	$0.769758\pi$
$864$	0	0
$865$	1525.15	0.0599499
$866$	−45238.3	−1.77513
$867$	0	0
$868$	−5223.47	−0.204258
$869$	−3524.58	−0.137587
$870$	0	0
$871$	−3728.37	−0.145041
$872$	−10381.0	−0.403147
$873$	0	0
$874$	7828.35	0.302973
$875$	3941.70	0.152290
$876$	0	0
$877$	45609.0	1.75611	0.878055	−	0.478561i	$-0.158841\pi$
$877$	45609.0	1.75611	0.878055	+	0.478561i	$0.158841\pi$
$878$	−45161.4	−1.73590
$879$	0	0
$880$	−714.658	−0.0273762
$881$	−15306.9	−0.585360	−0.292680	−	0.956210i	$-0.594547\pi$
$881$	−15306.9	−0.585360	−0.292680	+	0.956210i	$0.594547\pi$
$882$	0	0
$883$	−4293.37	−0.163628	−0.0818140	−	0.996648i	$-0.526071\pi$
$883$	−4293.37	−0.163628	−0.0818140	+	0.996648i	$0.526071\pi$
$884$	5717.86	0.217548
$885$	0	0
$886$	51055.3	1.93593
$887$	−15161.7	−0.573935	−0.286968	−	0.957940i	$-0.592647\pi$
$887$	−15161.7	−0.573935	−0.286968	+	0.957940i	$0.592647\pi$
$888$	0	0
$889$	−19750.3	−0.745110
$890$	−4150.52	−0.156321
$891$	0	0
$892$	47134.9	1.76927
$893$	15070.9	0.564758
$894$	0	0
$895$	−15467.2	−0.577665
$896$	53220.3	1.98434
$897$	0	0
$898$	−59463.3	−2.20971
$899$	425.932	0.0158016
$900$	0	0
$901$	−5503.29	−0.203486
$902$	−9285.05	−0.342748
$903$	0	0
$904$	−8046.57	−0.296045
$905$	14001.9	0.514297
$906$	0	0
$907$	45095.6	1.65091	0.825454	−	0.564469i	$-0.190919\pi$
$907$	45095.6	1.65091	0.825454	+	0.564469i	$0.190919\pi$
$908$	−45503.8	−1.66310
$909$	0	0
$910$	4144.86	0.150990
$911$	−10578.3	−0.384714	−0.192357	−	0.981325i	$-0.561613\pi$
$911$	−10578.3	−0.384714	−0.192357	+	0.981325i	$0.561613\pi$
$912$	0	0
$913$	6984.67	0.253186
$914$	−71933.6	−2.60323
$915$	0	0
$916$	50769.5	1.83130
$917$	−55629.0	−2.00331
$918$	0	0
$919$	50326.0	1.80642	0.903210	−	0.429198i	$-0.141204\pi$
$919$	50326.0	1.80642	0.903210	+	0.429198i	$0.141204\pi$
$920$	2237.53	0.0801837
$921$	0	0
$922$	59615.9	2.12944
$923$	1962.14	0.0699726
$924$	0	0
$925$	−194.044	−0.00689742
$926$	78972.0	2.80257
$927$	0	0
$928$	−6780.71	−0.239858
$929$	−8373.10	−0.295708	−0.147854	−	0.989009i	$-0.547237\pi$
$929$	−8373.10	−0.295708	−0.147854	+	0.989009i	$0.547237\pi$
$930$	0	0
$931$	37354.7	1.31498
$932$	53128.8	1.86726
$933$	0	0
$934$	3479.51	0.121898
$935$	−2239.05	−0.0783154
$936$	0	0
$937$	53048.3	1.84953	0.924766	−	0.380537i	$-0.124261\pi$
$937$	53048.3	1.84953	0.924766	+	0.380537i	$0.124261\pi$
$938$	−86217.4	−3.00117
$939$	0	0
$940$	14819.5	0.514212
$941$	1015.79	0.0351899	0.0175949	−	0.999845i	$-0.494399\pi$
$941$	1015.79	0.0351899	0.0175949	+	0.999845i	$0.494399\pi$
$942$	0	0
$943$	−12431.8	−0.429305
$944$	3318.54	0.114417
$945$	0	0
$946$	417.889	0.0143623
$947$	26038.4	0.893490	0.446745	−	0.894661i	$-0.352583\pi$
$947$	26038.4	0.893490	0.446745	+	0.894661i	$0.352583\pi$
$948$	0	0
$949$	4990.53	0.170706
$950$	−6294.95	−0.214984
$951$	0	0
$952$	38433.7	1.30845
$953$	22396.1	0.761260	0.380630	−	0.924727i	$-0.375707\pi$
$953$	22396.1	0.761260	0.380630	+	0.924727i	$0.375707\pi$
$954$	0	0
$955$	−21043.9	−0.713053
$956$	5601.17	0.189492
$957$	0	0
$958$	−29220.2	−0.985451
$959$	13676.8	0.460530
$960$	0	0
$961$	−29575.3	−0.992759
$962$	−204.045	−0.00683852
$963$	0	0
$964$	72345.9	2.41712
$965$	−1237.82	−0.0412919
$966$	0	0
$967$	−33614.9	−1.11787	−0.558936	−	0.829211i	$-0.688790\pi$
$967$	−33614.9	−1.11787	−0.558936	+	0.829211i	$0.688790\pi$
$968$	18755.8	0.622761
$969$	0	0
$970$	3021.47	0.100014
$971$	30440.6	1.00606	0.503030	−	0.864269i	$-0.332219\pi$
$971$	30440.6	1.00606	0.503030	+	0.864269i	$0.332219\pi$
$972$	0	0
$973$	−15913.7	−0.524328
$974$	61349.8	2.01825
$975$	0	0
$976$	−7145.42	−0.234343
$977$	−18698.6	−0.612305	−0.306153	−	0.951982i	$-0.599042\pi$
$977$	−18698.6	−0.612305	−0.306153	+	0.951982i	$0.599042\pi$
$978$	0	0
$979$	999.847	0.0326407
$980$	36731.6	1.19729
$981$	0	0
$982$	15976.1	0.519164
$983$	10438.4	0.338692	0.169346	−	0.985557i	$-0.445834\pi$
$983$	10438.4	0.338692	0.169346	+	0.985557i	$0.445834\pi$
$984$	0	0
$985$	22870.3	0.739806
$986$	−10781.8	−0.348238
$987$	0	0
$988$	−3872.51	−0.124697
$989$	559.513	0.0179893
$990$	0	0
$991$	20547.5	0.658641	0.329320	−	0.944218i	$-0.393180\pi$
$991$	20547.5	0.658641	0.329320	+	0.944218i	$0.393180\pi$
$992$	−3434.16	−0.109914
$993$	0	0
$994$	45373.9	1.44786
$995$	13005.3	0.414366
$996$	0	0
$997$	6486.05	0.206033	0.103017	−	0.994680i	$-0.467150\pi$
$997$	6486.05	0.206033	0.103017	+	0.994680i	$0.467150\pi$
$998$	−49674.3	−1.57556
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-4.39070 q^{2} +11.2783 q^{4} +5.00000 q^{5} +31.5336 q^{7} -14.3939 q^{8} +O(q^{10})\)	\(q-4.39070 q^{2} +11.2783 q^{4} +5.00000 q^{5} +31.5336 q^{7} -14.3939 q^{8} -21.9535 q^{10} +5.28853 q^{11} -5.98731 q^{13} -138.455 q^{14} -27.0267 q^{16} -84.6758 q^{17} +57.3480 q^{19} +56.3914 q^{20} -23.2204 q^{22} -31.0898 q^{23} +25.0000 q^{25} +26.2885 q^{26} +355.645 q^{28} -29.0000 q^{29} -14.6873 q^{31} +233.818 q^{32} +371.786 q^{34} +157.668 q^{35} -7.76174 q^{37} -251.798 q^{38} -71.9697 q^{40} +399.867 q^{41} -17.9967 q^{43} +59.6455 q^{44} +136.506 q^{46} +262.798 q^{47} +651.368 q^{49} -109.768 q^{50} -67.5265 q^{52} +64.9925 q^{53} +26.4427 q^{55} -453.893 q^{56} +127.330 q^{58} -122.788 q^{59} +264.384 q^{61} +64.4877 q^{62} -810.411 q^{64} -29.9366 q^{65} +622.712 q^{67} -954.997 q^{68} -692.274 q^{70} -327.717 q^{71} -833.518 q^{73} +34.0795 q^{74} +646.787 q^{76} +166.767 q^{77} -666.457 q^{79} -135.133 q^{80} -1755.70 q^{82} +1320.72 q^{83} -423.379 q^{85} +79.0180 q^{86} -76.1229 q^{88} +189.059 q^{89} -188.801 q^{91} -350.640 q^{92} -1153.87 q^{94} +286.740 q^{95} -137.630 q^{97} -2859.97 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8}+O(q^{10}) \) 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

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