LMFDB - Embedded newform 1305.4.a.m.1.1

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 $ x^7 - x^6 - 35x^5 + 18x^4 + 329x^3 - 167x^2 - 767*x + 638
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.96612$ 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)

$f(q)$	$=$	$q-3.96612 q^{2} +7.73008 q^{4} +5.00000 q^{5} -32.8717 q^{7} +1.07055 q^{8} +O(q^{10})$	\(q-3.96612 q^{2} +7.73008 q^{4} +5.00000 q^{5} -32.8717 q^{7} +1.07055 q^{8} -19.8306 q^{10} -12.6426 q^{11} -52.2979 q^{13} +130.373 q^{14} -66.0865 q^{16} +17.2978 q^{17} +48.0486 q^{19} +38.6504 q^{20} +50.1421 q^{22} +130.535 q^{23} +25.0000 q^{25} +207.420 q^{26} -254.100 q^{28} +29.0000 q^{29} +196.614 q^{31} +253.542 q^{32} -68.6049 q^{34} -164.358 q^{35} -119.645 q^{37} -190.566 q^{38} +5.35275 q^{40} +134.133 q^{41} -222.109 q^{43} -97.7283 q^{44} -517.716 q^{46} +569.042 q^{47} +737.546 q^{49} -99.1529 q^{50} -404.267 q^{52} -393.464 q^{53} -63.2131 q^{55} -35.1908 q^{56} -115.017 q^{58} -741.684 q^{59} +758.295 q^{61} -779.796 q^{62} -476.886 q^{64} -261.489 q^{65} -74.7083 q^{67} +133.713 q^{68} +651.864 q^{70} +118.322 q^{71} -498.399 q^{73} +474.525 q^{74} +371.419 q^{76} +415.584 q^{77} +106.032 q^{79} -330.433 q^{80} -531.988 q^{82} -42.5010 q^{83} +86.4888 q^{85} +880.911 q^{86} -13.5346 q^{88} +161.806 q^{89} +1719.12 q^{91} +1009.04 q^{92} -2256.88 q^{94} +240.243 q^{95} +128.506 q^{97} -2925.19 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8}+O(q^{10}) $ 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8	$ 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26} - 637 q^{28} + 203 q^{29} - 480 q^{31} + 779 q^{32} - 897 q^{34} - 185 q^{35} - 1032 q^{37} + 194 q^{38} + 180 q^{40} + 638 q^{41} - 512 q^{43} - 625 q^{44} + 16 q^{46} + 111 q^{47} + 178 q^{49} + 25 q^{50} - 1263 q^{52} - 410 q^{53} + 55 q^{55} - 1174 q^{56} + 29 q^{58} + 426 q^{59} - 1192 q^{61} - 460 q^{62} + 390 q^{64} - 665 q^{65} - 1671 q^{67} - 1509 q^{68} + 375 q^{70} + 1324 q^{71} - 852 q^{73} - 1780 q^{74} - 564 q^{76} + 2107 q^{77} + 366 q^{79} - 265 q^{80} - 318 q^{82} - 470 q^{83} - 105 q^{85} + 2196 q^{86} - 2518 q^{88} - 51 q^{89} - 1297 q^{91} + 684 q^{92} - 1837 q^{94} - 850 q^{95} - 3322 q^{97} - 1068 q^{98}+O(q^{100}) $ 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * 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 $

$2$	−3.96612	−1.40223	−0.701117	−	0.713046i	$-0.747315\pi$
$2$	−3.96612	−1.40223	−0.701117	+	0.713046i	$0.747315\pi$
$3$	0	0
$3$	0	0
$4$	7.73008	0.966259
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−32.8717	−1.77490	−0.887451	−	0.460901i	$-0.847526\pi$
$7$	−32.8717	−1.77490	−0.887451	+	0.460901i	$0.847526\pi$
$8$	1.07055	0.0473121
$9$	0	0
$10$	−19.8306	−0.627098
$11$	−12.6426	−0.346536	−0.173268	−	0.984875i	$-0.555433\pi$
$11$	−12.6426	−0.346536	−0.173268	+	0.984875i	$0.555433\pi$
$12$	0	0
$13$	−52.2979	−1.11576	−0.557878	−	0.829923i	$-0.688384\pi$
$13$	−52.2979	−1.11576	−0.557878	+	0.829923i	$0.688384\pi$
$14$	130.373	2.48883
$15$	0	0
$16$	−66.0865	−1.03260
$17$	17.2978	0.246784	0.123392	−	0.992358i	$-0.460623\pi$
$17$	17.2978	0.246784	0.123392	+	0.992358i	$0.460623\pi$
$18$	0	0
$19$	48.0486	0.580163	0.290082	−	0.957002i	$-0.406318\pi$
$19$	48.0486	0.580163	0.290082	+	0.957002i	$0.406318\pi$
$20$	38.6504	0.432124
$21$	0	0
$22$	50.1421	0.485924
$23$	130.535	1.18341	0.591704	−	0.806156i	$-0.298456\pi$
$23$	130.535	1.18341	0.591704	+	0.806156i	$0.298456\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	207.420	1.56455
$27$	0	0
$28$	−254.100	−1.71502
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	196.614	1.13913	0.569564	−	0.821947i	$-0.307112\pi$
$31$	196.614	1.13913	0.569564	+	0.821947i	$0.307112\pi$
$32$	253.542	1.40064
$33$	0	0
$34$	−68.6049	−0.346048
$35$	−164.358	−0.793761
$36$	0	0
$37$	−119.645	−0.531608	−0.265804	−	0.964027i	$-0.585637\pi$
$37$	−119.645	−0.531608	−0.265804	+	0.964027i	$0.585637\pi$
$38$	−190.566	−0.813525
$39$	0	0
$40$	5.35275	0.0211586
$41$	134.133	0.510929	0.255465	−	0.966818i	$-0.417772\pi$
$41$	134.133	0.510929	0.255465	+	0.966818i	$0.417772\pi$
$42$	0	0
$43$	−222.109	−0.787705	−0.393853	−	0.919174i	$-0.628858\pi$
$43$	−222.109	−0.787705	−0.393853	+	0.919174i	$0.628858\pi$
$44$	−97.7283	−0.334843
$45$	0	0
$46$	−517.716	−1.65941
$47$	569.042	1.76603	0.883013	−	0.469348i	$-0.155511\pi$
$47$	569.042	1.76603	0.883013	+	0.469348i	$0.155511\pi$
$48$	0	0
$49$	737.546	2.15028
$50$	−99.1529	−0.280447
$51$	0	0
$52$	−404.267	−1.07811
$53$	−393.464	−1.01974	−0.509872	−	0.860250i	$-0.670307\pi$
$53$	−393.464	−1.01974	−0.509872	+	0.860250i	$0.670307\pi$
$54$	0	0
$55$	−63.2131	−0.154975
$56$	−35.1908	−0.0839744
$57$	0	0
$58$	−115.017	−0.260388
$59$	−741.684	−1.63659	−0.818297	−	0.574795i	$-0.805082\pi$
$59$	−741.684	−1.63659	−0.818297	+	0.574795i	$0.805082\pi$
$60$	0	0
$61$	758.295	1.59163	0.795817	−	0.605537i	$-0.207042\pi$
$61$	758.295	1.59163	0.795817	+	0.605537i	$0.207042\pi$
$62$	−779.796	−1.59732
$63$	0	0
$64$	−476.886	−0.931419
$65$	−261.489	−0.498981
$66$	0	0
$67$	−74.7083	−0.136225	−0.0681125	−	0.997678i	$-0.521698\pi$
$67$	−74.7083	−0.136225	−0.0681125	+	0.997678i	$0.521698\pi$
$68$	133.713	0.238457
$69$	0	0
$70$	651.864	1.11304
$71$	118.322	0.197779	0.0988893	−	0.995098i	$-0.468471\pi$
$71$	118.322	0.197779	0.0988893	+	0.995098i	$0.468471\pi$
$72$	0	0
$73$	−498.399	−0.799084	−0.399542	−	0.916715i	$-0.630831\pi$
$73$	−498.399	−0.799084	−0.399542	+	0.916715i	$0.630831\pi$
$74$	474.525	0.745438
$75$	0	0
$76$	371.419	0.560588
$77$	415.584	0.615067
$78$	0	0
$79$	106.032	0.151007	0.0755037	−	0.997146i	$-0.475944\pi$
$79$	106.032	0.151007	0.0755037	+	0.997146i	$0.475944\pi$
$80$	−330.433	−0.461794
$81$	0	0
$82$	−531.988	−0.716442
$83$	−42.5010	−0.0562059	−0.0281030	−	0.999605i	$-0.508947\pi$
$83$	−42.5010	−0.0562059	−0.0281030	+	0.999605i	$0.508947\pi$
$84$	0	0
$85$	86.4888	0.110365
$86$	880.911	1.10455
$87$	0	0
$88$	−13.5346	−0.0163953
$89$	161.806	0.192713	0.0963564	−	0.995347i	$-0.469281\pi$
$89$	161.806	0.192713	0.0963564	+	0.995347i	$0.469281\pi$
$90$	0	0
$91$	1719.12	1.98036
$92$	1009.04	1.14348
$93$	0	0
$94$	−2256.88	−2.47638
$95$	240.243	0.259457
$96$	0	0
$97$	128.506	0.134514	0.0672569	−	0.997736i	$-0.478575\pi$
$97$	128.506	0.134514	0.0672569	+	0.997736i	$0.478575\pi$
$98$	−2925.19	−3.01519
$99$	0	0
$100$	193.252	0.193252
$101$	584.973	0.576307	0.288153	−	0.957584i	$-0.406959\pi$
$101$	584.973	0.576307	0.288153	+	0.957584i	$0.406959\pi$
$102$	0	0
$103$	−1908.33	−1.82557	−0.912783	−	0.408446i	$-0.866071\pi$
$103$	−1908.33	−1.82557	−0.912783	+	0.408446i	$0.866071\pi$
$104$	−55.9876	−0.0527888
$105$	0	0
$106$	1560.52	1.42992
$107$	−646.934	−0.584500	−0.292250	−	0.956342i	$-0.594404\pi$
$107$	−646.934	−0.584500	−0.292250	+	0.956342i	$0.594404\pi$
$108$	0	0
$109$	−1065.19	−0.936028	−0.468014	−	0.883721i	$-0.655030\pi$
$109$	−1065.19	−0.936028	−0.468014	+	0.883721i	$0.655030\pi$
$110$	250.710	0.217312
$111$	0	0
$112$	2172.37	1.83277
$113$	1368.69	1.13943	0.569713	−	0.821844i	$-0.307054\pi$
$113$	1368.69	1.13943	0.569713	+	0.821844i	$0.307054\pi$
$114$	0	0
$115$	652.673	0.529236
$116$	224.172	0.179430
$117$	0	0
$118$	2941.61	2.29489
$119$	−568.606	−0.438017
$120$	0	0
$121$	−1171.16	−0.879913
$122$	−3007.49	−2.23184
$123$	0	0
$124$	1519.84	1.10069
$125$	125.000	0.0894427
$126$	0	0
$127$	2118.32	1.48008	0.740040	−	0.672562i	$-0.234806\pi$
$127$	2118.32	1.48008	0.740040	+	0.672562i	$0.234806\pi$
$128$	−136.953	−0.0945704
$129$	0	0
$130$	1037.10	0.699688
$131$	2446.56	1.63173	0.815866	−	0.578241i	$-0.196260\pi$
$131$	2446.56	1.63173	0.815866	+	0.578241i	$0.196260\pi$
$132$	0	0
$133$	−1579.44	−1.02973
$134$	296.302	0.191019
$135$	0	0
$136$	18.5181	0.0116759
$137$	−1586.44	−0.989331	−0.494665	−	0.869084i	$-0.664709\pi$
$137$	−1586.44	−0.989331	−0.494665	+	0.869084i	$0.664709\pi$
$138$	0	0
$139$	1793.32	1.09430	0.547150	−	0.837034i	$-0.315713\pi$
$139$	1793.32	1.09430	0.547150	+	0.837034i	$0.315713\pi$
$140$	−1270.50	−0.766979
$141$	0	0
$142$	−469.280	−0.277332
$143$	661.182	0.386649
$144$	0	0
$145$	145.000	0.0830455
$146$	1976.71	1.12050
$147$	0	0
$148$	−924.863	−0.513671
$149$	2640.05	1.45155	0.725777	−	0.687930i	$-0.241480\pi$
$149$	2640.05	1.45155	0.725777	+	0.687930i	$0.241480\pi$
$150$	0	0
$151$	1132.79	0.610498	0.305249	−	0.952273i	$-0.401260\pi$
$151$	1132.79	0.610498	0.305249	+	0.952273i	$0.401260\pi$
$152$	51.4385	0.0274487
$153$	0	0
$154$	−1648.25	−0.862468
$155$	983.072	0.509434
$156$	0	0
$157$	912.652	0.463933	0.231967	−	0.972724i	$-0.425484\pi$
$157$	912.652	0.463933	0.231967	+	0.972724i	$0.425484\pi$
$158$	−420.537	−0.211748
$159$	0	0
$160$	1267.71	0.626384
$161$	−4290.89	−2.10043
$162$	0	0
$163$	−2303.83	−1.10705	−0.553527	−	0.832831i	$-0.686718\pi$
$163$	−2303.83	−1.10705	−0.553527	+	0.832831i	$0.686718\pi$
$164$	1036.86	0.493690
$165$	0	0
$166$	168.564	0.0788138
$167$	−381.558	−0.176801	−0.0884006	−	0.996085i	$-0.528176\pi$
$167$	−381.558	−0.176801	−0.0884006	+	0.996085i	$0.528176\pi$
$168$	0	0
$169$	538.070	0.244911
$170$	−343.025	−0.154758
$171$	0	0
$172$	−1716.92	−0.761128
$173$	2697.02	1.18526	0.592631	−	0.805474i	$-0.298089\pi$
$173$	2697.02	1.18526	0.592631	+	0.805474i	$0.298089\pi$
$174$	0	0
$175$	−821.791	−0.354981
$176$	835.506	0.357833
$177$	0	0
$178$	−641.742	−0.270228
$179$	−30.7370	−0.0128346	−0.00641729	−	0.999979i	$-0.502043\pi$
$179$	−30.7370	−0.0128346	−0.00641729	+	0.999979i	$0.502043\pi$
$180$	0	0
$181$	159.556	0.0655231	0.0327615	−	0.999463i	$-0.489570\pi$
$181$	159.556	0.0655231	0.0327615	+	0.999463i	$0.489570\pi$
$182$	−6818.22	−2.77692
$183$	0	0
$184$	139.744	0.0559895
$185$	−598.224	−0.237742
$186$	0	0
$187$	−218.689	−0.0855193
$188$	4398.73	1.70644
$189$	0	0
$190$	−952.831	−0.363819
$191$	−96.7632	−0.0366573	−0.0183286	−	0.999832i	$-0.505835\pi$
$191$	−96.7632	−0.0366573	−0.0183286	+	0.999832i	$0.505835\pi$
$192$	0	0
$193$	1017.92	0.379646	0.189823	−	0.981818i	$-0.439209\pi$
$193$	1017.92	0.379646	0.189823	+	0.981818i	$0.439209\pi$
$194$	−509.671	−0.188620
$195$	0	0
$196$	5701.29	2.07773
$197$	−5012.65	−1.81288	−0.906438	−	0.422339i	$-0.861209\pi$
$197$	−5012.65	−1.81288	−0.906438	+	0.422339i	$0.861209\pi$
$198$	0	0
$199$	−4339.56	−1.54585	−0.772924	−	0.634499i	$-0.781206\pi$
$199$	−4339.56	−1.54585	−0.772924	+	0.634499i	$0.781206\pi$
$200$	26.7638	0.00946242
$201$	0	0
$202$	−2320.07	−0.808116
$203$	−953.278	−0.329591
$204$	0	0
$205$	670.666	0.228494
$206$	7568.65	2.55987
$207$	0	0
$208$	3456.19	1.15213
$209$	−607.460	−0.201047
$210$	0	0
$211$	−2360.56	−0.770180	−0.385090	−	0.922879i	$-0.625830\pi$
$211$	−2360.56	−0.770180	−0.385090	+	0.922879i	$0.625830\pi$
$212$	−3041.51	−0.985338
$213$	0	0
$214$	2565.82	0.819605
$215$	−1110.55	−0.352273
$216$	0	0
$217$	−6463.04	−2.02184
$218$	4224.68	1.31253
$219$	0	0
$220$	−488.642	−0.149746
$221$	−904.636	−0.275350
$222$	0	0
$223$	−6063.08	−1.82069	−0.910345	−	0.413851i	$-0.864184\pi$
$223$	−6063.08	−1.82069	−0.910345	+	0.413851i	$0.864184\pi$
$224$	−8334.36	−2.48600
$225$	0	0
$226$	−5428.37	−1.59774
$227$	−4428.61	−1.29488	−0.647438	−	0.762118i	$-0.724160\pi$
$227$	−4428.61	−1.29488	−0.647438	+	0.762118i	$0.724160\pi$
$228$	0	0
$229$	−4517.28	−1.30354	−0.651768	−	0.758418i	$-0.725973\pi$
$229$	−4517.28	−1.30354	−0.651768	+	0.758418i	$0.725973\pi$
$230$	−2588.58	−0.742112
$231$	0	0
$232$	31.0460	0.00878564
$233$	−4051.13	−1.13905	−0.569525	−	0.821974i	$-0.692873\pi$
$233$	−4051.13	−1.13905	−0.569525	+	0.821974i	$0.692873\pi$
$234$	0	0
$235$	2845.21	0.789791
$236$	−5733.28	−1.58138
$237$	0	0
$238$	2255.16	0.614202
$239$	−837.357	−0.226628	−0.113314	−	0.993559i	$-0.536147\pi$
$239$	−837.357	−0.226628	−0.113314	+	0.993559i	$0.536147\pi$
$240$	0	0
$241$	−2011.23	−0.537571	−0.268785	−	0.963200i	$-0.586622\pi$
$241$	−2011.23	−0.537571	−0.268785	+	0.963200i	$0.586622\pi$
$242$	4644.97	1.23384
$243$	0	0
$244$	5861.68	1.53793
$245$	3687.73	0.961634
$246$	0	0
$247$	−2512.84	−0.647321
$248$	210.486	0.0538946
$249$	0	0
$250$	−495.764	−0.125420
$251$	−6458.29	−1.62408	−0.812039	−	0.583603i	$-0.801642\pi$
$251$	−6458.29	−1.62408	−0.812039	+	0.583603i	$0.801642\pi$
$252$	0	0
$253$	−1650.30	−0.410093
$254$	−8401.49	−2.07542
$255$	0	0
$256$	4358.26	1.06403
$257$	−1838.15	−0.446150	−0.223075	−	0.974801i	$-0.571610\pi$
$257$	−1838.15	−0.446150	−0.223075	+	0.974801i	$0.571610\pi$
$258$	0	0
$259$	3932.92	0.943552
$260$	−2021.33	−0.482145
$261$	0	0
$262$	−9703.34	−2.28807
$263$	−2263.00	−0.530580	−0.265290	−	0.964169i	$-0.585468\pi$
$263$	−2263.00	−0.530580	−0.265290	+	0.964169i	$0.585468\pi$
$264$	0	0
$265$	−1967.32	−0.456044
$266$	6264.23	1.44393
$267$	0	0
$268$	−577.501	−0.131629
$269$	−632.951	−0.143464	−0.0717318	−	0.997424i	$-0.522853\pi$
$269$	−632.951	−0.143464	−0.0717318	+	0.997424i	$0.522853\pi$
$270$	0	0
$271$	−987.158	−0.221275	−0.110638	−	0.993861i	$-0.535289\pi$
$271$	−987.158	−0.221275	−0.110638	+	0.993861i	$0.535289\pi$
$272$	−1143.15	−0.254829
$273$	0	0
$274$	6291.99	1.38727
$275$	−316.065	−0.0693071
$276$	0	0
$277$	8123.34	1.76204	0.881018	−	0.473083i	$-0.156859\pi$
$277$	8123.34	1.76204	0.881018	+	0.473083i	$0.156859\pi$
$278$	−7112.53	−1.53447
$279$	0	0
$280$	−175.954	−0.0375545
$281$	5075.60	1.07753	0.538764	−	0.842457i	$-0.318892\pi$
$281$	5075.60	1.07753	0.538764	+	0.842457i	$0.318892\pi$
$282$	0	0
$283$	−2077.30	−0.436335	−0.218168	−	0.975911i	$-0.570008\pi$
$283$	−2077.30	−0.436335	−0.218168	+	0.975911i	$0.570008\pi$
$284$	914.641	0.191105
$285$	0	0
$286$	−2622.32	−0.542172
$287$	−4409.18	−0.906849
$288$	0	0
$289$	−4613.79	−0.939098
$290$	−575.087	−0.116449
$291$	0	0
$292$	−3852.66	−0.772123
$293$	4097.60	0.817011	0.408505	−	0.912756i	$-0.366050\pi$
$293$	4097.60	0.817011	0.408505	+	0.912756i	$0.366050\pi$
$294$	0	0
$295$	−3708.42	−0.731907
$296$	−128.086	−0.0251515
$297$	0	0
$298$	−10470.8	−2.03542
$299$	−6826.69	−1.32039
$300$	0	0
$301$	7301.10	1.39810
$302$	−4492.78	−0.856061
$303$	0	0
$304$	−3175.36	−0.599078
$305$	3791.47	0.711801
$306$	0	0
$307$	2800.27	0.520586	0.260293	−	0.965530i	$-0.416181\pi$
$307$	2800.27	0.520586	0.260293	+	0.965530i	$0.416181\pi$
$308$	3212.49	0.594314
$309$	0	0
$310$	−3898.98	−0.714345
$311$	1007.23	0.183649	0.0918247	−	0.995775i	$-0.470730\pi$
$311$	1007.23	0.183649	0.0918247	+	0.995775i	$0.470730\pi$
$312$	0	0
$313$	−8540.20	−1.54224	−0.771119	−	0.636691i	$-0.780303\pi$
$313$	−8540.20	−1.54224	−0.771119	+	0.636691i	$0.780303\pi$
$314$	−3619.68	−0.650543
$315$	0	0
$316$	819.639	0.145912
$317$	6756.75	1.19715	0.598575	−	0.801067i	$-0.295734\pi$
$317$	6756.75	1.19715	0.598575	+	0.801067i	$0.295734\pi$
$318$	0	0
$319$	−366.636	−0.0643500
$320$	−2384.43	−0.416543
$321$	0	0
$322$	17018.2	2.94530
$323$	831.133	0.143175
$324$	0	0
$325$	−1307.45	−0.223151
$326$	9137.25	1.55235
$327$	0	0
$328$	143.596	0.0241731
$329$	−18705.3	−3.13453
$330$	0	0
$331$	−2186.58	−0.363097	−0.181548	−	0.983382i	$-0.558111\pi$
$331$	−2186.58	−0.363097	−0.181548	+	0.983382i	$0.558111\pi$
$332$	−328.536	−0.0543095
$333$	0	0
$334$	1513.30	0.247917
$335$	−373.542	−0.0609217
$336$	0	0
$337$	5592.65	0.904010	0.452005	−	0.892016i	$-0.350709\pi$
$337$	5592.65	0.904010	0.452005	+	0.892016i	$0.350709\pi$
$338$	−2134.05	−0.343423
$339$	0	0
$340$	668.565	0.106641
$341$	−2485.72	−0.394749
$342$	0	0
$343$	−12969.4	−2.04163
$344$	−237.779	−0.0372680
$345$	0	0
$346$	−10696.7	−1.66201
$347$	−3744.53	−0.579300	−0.289650	−	0.957133i	$-0.593539\pi$
$347$	−3744.53	−0.579300	−0.289650	+	0.957133i	$0.593539\pi$
$348$	0	0
$349$	−11200.7	−1.71794	−0.858970	−	0.512025i	$-0.828895\pi$
$349$	−11200.7	−1.71794	−0.858970	+	0.512025i	$0.828895\pi$
$350$	3259.32	0.497766
$351$	0	0
$352$	−3205.44	−0.485371
$353$	−1695.02	−0.255572	−0.127786	−	0.991802i	$-0.540787\pi$
$353$	−1695.02	−0.255572	−0.127786	+	0.991802i	$0.540787\pi$
$354$	0	0
$355$	591.612	0.0884493
$356$	1250.77	0.186210
$357$	0	0
$358$	121.907	0.0179971
$359$	331.341	0.0487117	0.0243559	−	0.999703i	$-0.492247\pi$
$359$	331.341	0.0487117	0.0243559	+	0.999703i	$0.492247\pi$
$360$	0	0
$361$	−4550.33	−0.663411
$362$	−632.816	−0.0918787
$363$	0	0
$364$	13288.9	1.91354
$365$	−2491.99	−0.357361
$366$	0	0
$367$	−11394.1	−1.62061	−0.810306	−	0.586006i	$-0.800699\pi$
$367$	−11394.1	−1.62061	−0.810306	+	0.586006i	$0.800699\pi$
$368$	−8626.58	−1.22199
$369$	0	0
$370$	2372.63	0.333370
$371$	12933.8	1.80995
$372$	0	0
$373$	6042.53	0.838794	0.419397	−	0.907803i	$-0.362241\pi$
$373$	6042.53	0.838794	0.419397	+	0.907803i	$0.362241\pi$
$374$	867.345	0.119918
$375$	0	0
$376$	609.188	0.0835544
$377$	−1516.64	−0.207191
$378$	0	0
$379$	3446.58	0.467121	0.233560	−	0.972342i	$-0.424962\pi$
$379$	3446.58	0.467121	0.233560	+	0.972342i	$0.424962\pi$
$380$	1857.10	0.250703
$381$	0	0
$382$	383.774	0.0514021
$383$	−6068.80	−0.809664	−0.404832	−	0.914391i	$-0.632670\pi$
$383$	−6068.80	−0.809664	−0.404832	+	0.914391i	$0.632670\pi$
$384$	0	0
$385$	2077.92	0.275066
$386$	−4037.20	−0.532352
$387$	0	0
$388$	993.363	0.129975
$389$	4788.36	0.624112	0.312056	−	0.950064i	$-0.398982\pi$
$389$	4788.36	0.624112	0.312056	+	0.950064i	$0.398982\pi$
$390$	0	0
$391$	2257.96	0.292046
$392$	789.580	0.101734
$393$	0	0
$394$	19880.8	2.54208
$395$	530.162	0.0675325
$396$	0	0
$397$	206.388	0.0260914	0.0130457	−	0.999915i	$-0.495847\pi$
$397$	206.388	0.0260914	0.0130457	+	0.999915i	$0.495847\pi$
$398$	17211.2	2.16764
$399$	0	0
$400$	−1652.16	−0.206520
$401$	10080.5	1.25535	0.627675	−	0.778475i	$-0.284007\pi$
$401$	10080.5	1.25535	0.627675	+	0.778475i	$0.284007\pi$
$402$	0	0
$403$	−10282.5	−1.27099
$404$	4521.88	0.556862
$405$	0	0
$406$	3780.81	0.462164
$407$	1512.62	0.184221
$408$	0	0
$409$	15178.0	1.83497	0.917486	−	0.397769i	$-0.130215\pi$
$409$	15178.0	1.83497	0.917486	+	0.397769i	$0.130215\pi$
$410$	−2659.94	−0.320403
$411$	0	0
$412$	−14751.5	−1.76397
$413$	24380.4	2.90480
$414$	0	0
$415$	−212.505	−0.0251360
$416$	−13259.7	−1.56277
$417$	0	0
$418$	2409.26	0.281915
$419$	6104.30	0.711729	0.355864	−	0.934538i	$-0.384186\pi$
$419$	6104.30	0.711729	0.355864	+	0.934538i	$0.384186\pi$
$420$	0	0
$421$	8543.56	0.989045	0.494522	−	0.869165i	$-0.335343\pi$
$421$	8543.56	0.989045	0.494522	+	0.869165i	$0.335343\pi$
$422$	9362.27	1.07997
$423$	0	0
$424$	−421.223	−0.0482463
$425$	432.444	0.0493567
$426$	0	0
$427$	−24926.4	−2.82500
$428$	−5000.85	−0.564778
$429$	0	0
$430$	4404.55	0.493968
$431$	7712.54	0.861949	0.430975	−	0.902364i	$-0.358170\pi$
$431$	7712.54	0.861949	0.430975	+	0.902364i	$0.358170\pi$
$432$	0	0
$433$	−13294.5	−1.47551	−0.737753	−	0.675071i	$-0.764113\pi$
$433$	−13294.5	−1.47551	−0.737753	+	0.675071i	$0.764113\pi$
$434$	25633.2	2.83510
$435$	0	0
$436$	−8234.03	−0.904446
$437$	6272.01	0.686569
$438$	0	0
$439$	−18032.6	−1.96048	−0.980239	−	0.197816i	$-0.936615\pi$
$439$	−18032.6	−1.96048	−0.980239	+	0.197816i	$0.936615\pi$
$440$	−67.6728	−0.00733221
$441$	0	0
$442$	3587.89	0.386106
$443$	−16475.7	−1.76700	−0.883502	−	0.468427i	$-0.844821\pi$
$443$	−16475.7	−1.76700	−0.883502	+	0.468427i	$0.844821\pi$
$444$	0	0
$445$	809.031	0.0861837
$446$	24046.9	2.55303
$447$	0	0
$448$	15676.1	1.65318
$449$	2651.74	0.278716	0.139358	−	0.990242i	$-0.455496\pi$
$449$	2651.74	0.278716	0.139358	+	0.990242i	$0.455496\pi$
$450$	0	0
$451$	−1695.79	−0.177055
$452$	10580.0	1.10098
$453$	0	0
$454$	17564.4	1.81572
$455$	8595.59	0.885643
$456$	0	0
$457$	−582.597	−0.0596340	−0.0298170	−	0.999555i	$-0.509492\pi$
$457$	−582.597	−0.0596340	−0.0298170	+	0.999555i	$0.509492\pi$
$458$	17916.0	1.82786
$459$	0	0
$460$	5045.21	0.511379
$461$	6537.72	0.660504	0.330252	−	0.943893i	$-0.392866\pi$
$461$	6537.72	0.660504	0.330252	+	0.943893i	$0.392866\pi$
$462$	0	0
$463$	−11895.3	−1.19400	−0.596998	−	0.802243i	$-0.703640\pi$
$463$	−11895.3	−1.19400	−0.596998	+	0.802243i	$0.703640\pi$
$464$	−1916.51	−0.191749
$465$	0	0
$466$	16067.3	1.59721
$467$	14742.5	1.46081	0.730406	−	0.683013i	$-0.239331\pi$
$467$	14742.5	1.46081	0.730406	+	0.683013i	$0.239331\pi$
$468$	0	0
$469$	2455.79	0.241786
$470$	−11284.4	−1.10747
$471$	0	0
$472$	−794.011	−0.0774307
$473$	2808.04	0.272968
$474$	0	0
$475$	1201.21	0.116033
$476$	−4395.37	−0.423238
$477$	0	0
$478$	3321.06	0.317786
$479$	−5864.41	−0.559398	−0.279699	−	0.960088i	$-0.590235\pi$
$479$	−5864.41	−0.559398	−0.279699	+	0.960088i	$0.590235\pi$
$480$	0	0
$481$	6257.17	0.593144
$482$	7976.76	0.753800
$483$	0	0
$484$	−9053.19	−0.850224
$485$	642.531	0.0601564
$486$	0	0
$487$	−15495.7	−1.44184	−0.720920	−	0.693018i	$-0.756281\pi$
$487$	−15495.7	−1.44184	−0.720920	+	0.693018i	$0.756281\pi$
$488$	811.793	0.0753036
$489$	0	0
$490$	−14626.0	−1.34844
$491$	−951.715	−0.0874751	−0.0437376	−	0.999043i	$-0.513927\pi$
$491$	−951.715	−0.0874751	−0.0437376	+	0.999043i	$0.513927\pi$
$492$	0	0
$493$	501.635	0.0458266
$494$	9966.22	0.907695
$495$	0	0
$496$	−12993.6	−1.17627
$497$	−3889.45	−0.351038
$498$	0	0
$499$	12667.6	1.13643	0.568216	−	0.822880i	$-0.307634\pi$
$499$	12667.6	1.13643	0.568216	+	0.822880i	$0.307634\pi$
$500$	966.259	0.0864249
$501$	0	0
$502$	25614.3	2.27734
$503$	−17478.7	−1.54938	−0.774690	−	0.632341i	$-0.782094\pi$
$503$	−17478.7	−1.54938	−0.774690	+	0.632341i	$0.782094\pi$
$504$	0	0
$505$	2924.86	0.257732
$506$	6545.28	0.575046
$507$	0	0
$508$	16374.8	1.43014
$509$	−11463.1	−0.998219	−0.499110	−	0.866539i	$-0.666339\pi$
$509$	−11463.1	−0.998219	−0.499110	+	0.866539i	$0.666339\pi$
$510$	0	0
$511$	16383.2	1.41830
$512$	−16189.8	−1.39745
$513$	0	0
$514$	7290.32	0.625607
$515$	−9541.64	−0.816418
$516$	0	0
$517$	−7194.17	−0.611991
$518$	−15598.4	−1.32308
$519$	0	0
$520$	−279.938	−0.0236079
$521$	−13465.1	−1.13228	−0.566141	−	0.824309i	$-0.691564\pi$
$521$	−13465.1	−1.13228	−0.566141	+	0.824309i	$0.691564\pi$
$522$	0	0
$523$	19829.8	1.65793	0.828963	−	0.559303i	$-0.188931\pi$
$523$	19829.8	1.65793	0.828963	+	0.559303i	$0.188931\pi$
$524$	18912.1	1.57668
$525$	0	0
$526$	8975.32	0.743997
$527$	3400.99	0.281118
$528$	0	0
$529$	4872.30	0.400452
$530$	7802.62	0.639480
$531$	0	0
$532$	−12209.2	−0.994990
$533$	−7014.89	−0.570072
$534$	0	0
$535$	−3234.67	−0.261396
$536$	−79.9790	−0.00644509
$537$	0	0
$538$	2510.36	0.201170
$539$	−9324.51	−0.745148
$540$	0	0
$541$	−8464.80	−0.672699	−0.336349	−	0.941737i	$-0.609192\pi$
$541$	−8464.80	−0.672699	−0.336349	+	0.941737i	$0.609192\pi$
$542$	3915.18	0.310280
$543$	0	0
$544$	4385.72	0.345654
$545$	−5325.97	−0.418605
$546$	0	0
$547$	7649.90	0.597964	0.298982	−	0.954259i	$-0.403353\pi$
$547$	7649.90	0.597964	0.298982	+	0.954259i	$0.403353\pi$
$548$	−12263.3	−0.955950
$549$	0	0
$550$	1253.55	0.0971848
$551$	1393.41	0.107734
$552$	0	0
$553$	−3485.46	−0.268023
$554$	−32218.1	−2.47079
$555$	0	0
$556$	13862.5	1.05738
$557$	−6293.95	−0.478785	−0.239393	−	0.970923i	$-0.576948\pi$
$557$	−6293.95	−0.478785	−0.239393	+	0.970923i	$0.576948\pi$
$558$	0	0
$559$	11615.8	0.878887
$560$	10861.9	0.819639
$561$	0	0
$562$	−20130.4	−1.51095
$563$	4688.80	0.350993	0.175497	−	0.984480i	$-0.443847\pi$
$563$	4688.80	0.350993	0.175497	+	0.984480i	$0.443847\pi$
$564$	0	0
$565$	6843.43	0.509567
$566$	8238.83	0.611844
$567$	0	0
$568$	126.670	0.00935732
$569$	20371.9	1.50094	0.750470	−	0.660904i	$-0.229827\pi$
$569$	20371.9	1.50094	0.750470	+	0.660904i	$0.229827\pi$
$570$	0	0
$571$	−5658.25	−0.414694	−0.207347	−	0.978267i	$-0.566483\pi$
$571$	−5658.25	−0.414694	−0.207347	+	0.978267i	$0.566483\pi$
$572$	5110.99	0.373603
$573$	0	0
$574$	17487.3	1.27161
$575$	3263.37	0.236681
$576$	0	0
$577$	809.648	0.0584161	0.0292081	−	0.999573i	$-0.490701\pi$
$577$	809.648	0.0584161	0.0292081	+	0.999573i	$0.490701\pi$
$578$	18298.8	1.31683
$579$	0	0
$580$	1120.86	0.0802435
$581$	1397.08	0.0997600
$582$	0	0
$583$	4974.42	0.353378
$584$	−533.561	−0.0378064
$585$	0	0
$586$	−16251.5	−1.14564
$587$	−9794.78	−0.688712	−0.344356	−	0.938839i	$-0.611903\pi$
$587$	−9794.78	−0.688712	−0.344356	+	0.938839i	$0.611903\pi$
$588$	0	0
$589$	9447.05	0.660881
$590$	14708.0	1.02631
$591$	0	0
$592$	7906.91	0.548939
$593$	19668.3	1.36202	0.681011	−	0.732273i	$-0.261541\pi$
$593$	19668.3	1.36202	0.681011	+	0.732273i	$0.261541\pi$
$594$	0	0
$595$	−2843.03	−0.195887
$596$	20407.8	1.40258
$597$	0	0
$598$	27075.4	1.85150
$599$	−3461.98	−0.236148	−0.118074	−	0.993005i	$-0.537672\pi$
$599$	−3461.98	−0.236148	−0.118074	+	0.993005i	$0.537672\pi$
$600$	0	0
$601$	1598.93	0.108522	0.0542611	−	0.998527i	$-0.482720\pi$
$601$	1598.93	0.108522	0.0542611	+	0.998527i	$0.482720\pi$
$602$	−28957.0	−1.96046
$603$	0	0
$604$	8756.56	0.589900
$605$	−5855.82	−0.393509
$606$	0	0
$607$	−9756.63	−0.652405	−0.326202	−	0.945300i	$-0.605769\pi$
$607$	−9756.63	−0.652405	−0.326202	+	0.945300i	$0.605769\pi$
$608$	12182.4	0.812598
$609$	0	0
$610$	−15037.4	−0.998111
$611$	−29759.7	−1.97045
$612$	0	0
$613$	−1209.23	−0.0796744	−0.0398372	−	0.999206i	$-0.512684\pi$
$613$	−1209.23	−0.0796744	−0.0398372	+	0.999206i	$0.512684\pi$
$614$	−11106.2	−0.729984
$615$	0	0
$616$	444.903	0.0291001
$617$	−12543.6	−0.818456	−0.409228	−	0.912432i	$-0.634202\pi$
$617$	−12543.6	−0.818456	−0.409228	+	0.912432i	$0.634202\pi$
$618$	0	0
$619$	11124.6	0.722353	0.361176	−	0.932498i	$-0.382375\pi$
$619$	11124.6	0.722353	0.361176	+	0.932498i	$0.382375\pi$
$620$	7599.22	0.492245
$621$	0	0
$622$	−3994.80	−0.257519
$623$	−5318.84	−0.342046
$624$	0	0
$625$	625.000	0.0400000
$626$	33871.4	2.16258
$627$	0	0
$628$	7054.87	0.448280
$629$	−2069.59	−0.131192
$630$	0	0
$631$	5229.61	0.329932	0.164966	−	0.986299i	$-0.447249\pi$
$631$	5229.61	0.329932	0.164966	+	0.986299i	$0.447249\pi$
$632$	113.513	0.00714447
$633$	0	0
$634$	−26798.0	−1.67868
$635$	10591.6	0.661912
$636$	0	0
$637$	−38572.1	−2.39919
$638$	1454.12	0.0902338
$639$	0	0
$640$	−684.763	−0.0422931
$641$	−8390.44	−0.517008	−0.258504	−	0.966010i	$-0.583230\pi$
$641$	−8390.44	−0.517008	−0.258504	+	0.966010i	$0.583230\pi$
$642$	0	0
$643$	25647.2	1.57298	0.786490	−	0.617602i	$-0.211896\pi$
$643$	25647.2	1.57298	0.786490	+	0.617602i	$0.211896\pi$
$644$	−33168.9	−2.02956
$645$	0	0
$646$	−3296.37	−0.200765
$647$	−24363.6	−1.48042	−0.740210	−	0.672376i	$-0.765274\pi$
$647$	−24363.6	−1.48042	−0.740210	+	0.672376i	$0.765274\pi$
$648$	0	0
$649$	9376.83	0.567138
$650$	5185.49	0.312910
$651$	0	0
$652$	−17808.8	−1.06970
$653$	20699.7	1.24049	0.620246	−	0.784407i	$-0.287033\pi$
$653$	20699.7	1.24049	0.620246	+	0.784407i	$0.287033\pi$
$654$	0	0
$655$	12232.8	0.729733
$656$	−8864.40	−0.527586
$657$	0	0
$658$	74187.5	4.39534
$659$	−18977.9	−1.12181	−0.560905	−	0.827880i	$-0.689547\pi$
$659$	−18977.9	−1.12181	−0.560905	+	0.827880i	$0.689547\pi$
$660$	0	0
$661$	19769.9	1.16333	0.581665	−	0.813428i	$-0.302401\pi$
$661$	19769.9	1.16333	0.581665	+	0.813428i	$0.302401\pi$
$662$	8672.21	0.509147
$663$	0	0
$664$	−45.4995	−0.00265922
$665$	−7897.18	−0.460511
$666$	0	0
$667$	3785.51	0.219753
$668$	−2949.47	−0.170836
$669$	0	0
$670$	1481.51	0.0854264
$671$	−9586.83	−0.551558
$672$	0	0
$673$	5776.20	0.330841	0.165420	−	0.986223i	$-0.447102\pi$
$673$	5776.20	0.330841	0.165420	+	0.986223i	$0.447102\pi$
$674$	−22181.1	−1.26763
$675$	0	0
$676$	4159.32	0.236648
$677$	−20304.8	−1.15270	−0.576349	−	0.817203i	$-0.695523\pi$
$677$	−20304.8	−1.15270	−0.576349	+	0.817203i	$0.695523\pi$
$678$	0	0
$679$	−4224.21	−0.238749
$680$	92.5907	0.00522160
$681$	0	0
$682$	9858.65	0.553530
$683$	−2014.43	−0.112855	−0.0564276	−	0.998407i	$-0.517971\pi$
$683$	−2014.43	−0.112855	−0.0564276	+	0.998407i	$0.517971\pi$
$684$	0	0
$685$	−7932.18	−0.442442
$686$	51438.1	2.86285
$687$	0	0
$688$	14678.4	0.813386
$689$	20577.4	1.13779
$690$	0	0
$691$	−25073.6	−1.38038	−0.690190	−	0.723628i	$-0.742473\pi$
$691$	−25073.6	−1.38038	−0.690190	+	0.723628i	$0.742473\pi$
$692$	20848.1	1.14527
$693$	0	0
$694$	14851.2	0.812314
$695$	8966.62	0.489386
$696$	0	0
$697$	2320.20	0.126089
$698$	44423.4	2.40895
$699$	0	0
$700$	−6352.51	−0.343003
$701$	507.308	0.0273335	0.0136667	−	0.999907i	$-0.495650\pi$
$701$	507.308	0.0273335	0.0136667	+	0.999907i	$0.495650\pi$
$702$	0	0
$703$	−5748.76	−0.308419
$704$	6029.09	0.322770
$705$	0	0
$706$	6722.65	0.358372
$707$	−19229.0	−1.02289
$708$	0	0
$709$	−24972.7	−1.32280	−0.661402	−	0.750032i	$-0.730038\pi$
$709$	−24972.7	−1.32280	−0.661402	+	0.750032i	$0.730038\pi$
$710$	−2346.40	−0.124027
$711$	0	0
$712$	173.222	0.00911764
$713$	25665.0	1.34805
$714$	0	0
$715$	3305.91	0.172915
$716$	−237.599	−0.0124015
$717$	0	0
$718$	−1314.14	−0.0683052
$719$	−13494.0	−0.699917	−0.349958	−	0.936765i	$-0.613804\pi$
$719$	−13494.0	−0.699917	−0.349958	+	0.936765i	$0.613804\pi$
$720$	0	0
$721$	62729.9	3.24020
$722$	18047.1	0.930257
$723$	0	0
$724$	1233.38	0.0633123
$725$	725.000	0.0371391
$726$	0	0
$727$	5717.08	0.291657	0.145829	−	0.989310i	$-0.453415\pi$
$727$	5717.08	0.291657	0.145829	+	0.989310i	$0.453415\pi$
$728$	1840.40	0.0936949
$729$	0	0
$730$	9883.53	0.501104
$731$	−3841.99	−0.194393
$732$	0	0
$733$	10147.2	0.511318	0.255659	−	0.966767i	$-0.417708\pi$
$733$	10147.2	0.511318	0.255659	+	0.966767i	$0.417708\pi$
$734$	45190.1	2.27248
$735$	0	0
$736$	33096.1	1.65752
$737$	944.508	0.0472068
$738$	0	0
$739$	39280.9	1.95531	0.977654	−	0.210222i	$-0.0674187\pi$
$739$	39280.9	1.95531	0.977654	+	0.210222i	$0.0674187\pi$
$740$	−4624.32	−0.229721
$741$	0	0
$742$	−51297.0	−2.53797
$743$	−27166.1	−1.34136	−0.670679	−	0.741748i	$-0.733997\pi$
$743$	−27166.1	−1.34136	−0.670679	+	0.741748i	$0.733997\pi$
$744$	0	0
$745$	13200.3	0.649155
$746$	−23965.4	−1.17619
$747$	0	0
$748$	−1690.48	−0.0826339
$749$	21265.8	1.03743
$750$	0	0
$751$	−2310.40	−0.112261	−0.0561303	−	0.998423i	$-0.517876\pi$
$751$	−2310.40	−0.112261	−0.0561303	+	0.998423i	$0.517876\pi$
$752$	−37606.0	−1.82360
$753$	0	0
$754$	6015.17	0.290530
$755$	5663.95	0.273023
$756$	0	0
$757$	28647.5	1.37545	0.687723	−	0.725973i	$-0.258610\pi$
$757$	28647.5	1.37545	0.687723	+	0.725973i	$0.258610\pi$
$758$	−13669.5	−0.655012
$759$	0	0
$760$	257.192	0.0122755
$761$	28161.6	1.34147	0.670733	−	0.741699i	$-0.265980\pi$
$761$	28161.6	1.34147	0.670733	+	0.741699i	$0.265980\pi$
$762$	0	0
$763$	35014.7	1.66136
$764$	−747.987	−0.0354205
$765$	0	0
$766$	24069.6	1.13534
$767$	38788.5	1.82604
$768$	0	0
$769$	−12197.5	−0.571981	−0.285990	−	0.958233i	$-0.592323\pi$
$769$	−12197.5	−0.571981	−0.285990	+	0.958233i	$0.592323\pi$
$770$	−8241.26	−0.385707
$771$	0	0
$772$	7868.62	0.366837
$773$	−21367.3	−0.994216	−0.497108	−	0.867689i	$-0.665605\pi$
$773$	−21367.3	−0.994216	−0.497108	+	0.867689i	$0.665605\pi$
$774$	0	0
$775$	4915.36	0.227826
$776$	137.572	0.00636413
$777$	0	0
$778$	−18991.2	−0.875151
$779$	6444.91	0.296422
$780$	0	0
$781$	−1495.90	−0.0685373
$782$	−8955.32	−0.409516
$783$	0	0
$784$	−48741.9	−2.22038
$785$	4563.26	0.207477
$786$	0	0
$787$	−20891.1	−0.946236	−0.473118	−	0.880999i	$-0.656872\pi$
$787$	−20891.1	−0.946236	−0.473118	+	0.880999i	$0.656872\pi$
$788$	−38748.2	−1.75171
$789$	0	0
$790$	−2102.68	−0.0946964
$791$	−44991.0	−2.02237
$792$	0	0
$793$	−39657.2	−1.77588
$794$	−818.557	−0.0365863
$795$	0	0
$796$	−33545.2	−1.49369
$797$	12968.8	0.576386	0.288193	−	0.957572i	$-0.406946\pi$
$797$	12968.8	0.576386	0.288193	+	0.957572i	$0.406946\pi$
$798$	0	0
$799$	9843.14	0.435827
$800$	6338.56	0.280127
$801$	0	0
$802$	−39980.4	−1.76029
$803$	6301.06	0.276911
$804$	0	0
$805$	−21454.5	−0.939342
$806$	40781.7	1.78222
$807$	0	0
$808$	626.243	0.0272663
$809$	13867.9	0.602683	0.301342	−	0.953516i	$-0.402566\pi$
$809$	13867.9	0.602683	0.301342	+	0.953516i	$0.402566\pi$
$810$	0	0
$811$	−32657.3	−1.41400	−0.707000	−	0.707213i	$-0.749952\pi$
$811$	−32657.3	−1.41400	−0.707000	+	0.707213i	$0.749952\pi$
$812$	−7368.91	−0.318471
$813$	0	0
$814$	−5999.24	−0.258321
$815$	−11519.1	−0.495090
$816$	0	0
$817$	−10672.0	−0.456998
$818$	−60197.7	−2.57306
$819$	0	0
$820$	5184.30	0.220785
$821$	−13615.5	−0.578788	−0.289394	−	0.957210i	$-0.593454\pi$
$821$	−13615.5	−0.578788	−0.289394	+	0.957210i	$0.593454\pi$
$822$	0	0
$823$	−13661.8	−0.578640	−0.289320	−	0.957232i	$-0.593429\pi$
$823$	−13661.8	−0.578640	−0.289320	+	0.957232i	$0.593429\pi$
$824$	−2042.96	−0.0863713
$825$	0	0
$826$	−96695.5	−4.07320
$827$	−31693.6	−1.33264	−0.666320	−	0.745666i	$-0.732131\pi$
$827$	−31693.6	−1.33264	−0.666320	+	0.745666i	$0.732131\pi$
$828$	0	0
$829$	−18908.6	−0.792185	−0.396093	−	0.918211i	$-0.629634\pi$
$829$	−18908.6	−0.792185	−0.396093	+	0.918211i	$0.629634\pi$
$830$	842.819	0.0352466
$831$	0	0
$832$	24940.2	1.03924
$833$	12757.9	0.530654
$834$	0	0
$835$	−1907.79	−0.0790679
$836$	−4695.71	−0.194264
$837$	0	0
$838$	−24210.3	−0.998010
$839$	−29382.9	−1.20907	−0.604534	−	0.796579i	$-0.706641\pi$
$839$	−29382.9	−1.20907	−0.604534	+	0.796579i	$0.706641\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−33884.8	−1.38687
$843$	0	0
$844$	−18247.3	−0.744193
$845$	2690.35	0.109528
$846$	0	0
$847$	38498.1	1.56176
$848$	26002.7	1.05299
$849$	0	0
$850$	−1715.12	−0.0692097
$851$	−15617.8	−0.629108
$852$	0	0
$853$	421.011	0.0168994	0.00844968	−	0.999964i	$-0.497310\pi$
$853$	421.011	0.0168994	0.00844968	+	0.999964i	$0.497310\pi$
$854$	98861.0	3.96131
$855$	0	0
$856$	−692.576	−0.0276539
$857$	41115.0	1.63881	0.819405	−	0.573215i	$-0.194304\pi$
$857$	41115.0	1.63881	0.819405	+	0.573215i	$0.194304\pi$
$858$	0	0
$859$	−578.823	−0.0229909	−0.0114954	−	0.999934i	$-0.503659\pi$
$859$	−578.823	−0.0229909	−0.0114954	+	0.999934i	$0.503659\pi$
$860$	−8584.60	−0.340387
$861$	0	0
$862$	−30588.8	−1.20865
$863$	30773.6	1.21384	0.606921	−	0.794762i	$-0.292404\pi$
$863$	30773.6	1.21384	0.606921	+	0.794762i	$0.292404\pi$
$864$	0	0
$865$	13485.1	0.530065
$866$	52727.6	2.06900
$867$	0	0
$868$	−49959.8	−1.95362
$869$	−1340.53	−0.0523294
$870$	0	0
$871$	3907.09	0.151994
$872$	−1140.34	−0.0442855
$873$	0	0
$874$	−24875.5	−0.962731
$875$	−4108.96	−0.158752
$876$	0	0
$877$	−21118.7	−0.813146	−0.406573	−	0.913618i	$-0.633276\pi$
$877$	−21118.7	−0.813146	−0.406573	+	0.913618i	$0.633276\pi$
$878$	71519.5	2.74905
$879$	0	0
$880$	4177.53	0.160028
$881$	42520.5	1.62605	0.813026	−	0.582228i	$-0.197819\pi$
$881$	42520.5	1.62605	0.813026	+	0.582228i	$0.197819\pi$
$882$	0	0
$883$	25473.5	0.970840	0.485420	−	0.874281i	$-0.338667\pi$
$883$	25473.5	0.970840	0.485420	+	0.874281i	$0.338667\pi$
$884$	−6992.91	−0.266060
$885$	0	0
$886$	65344.5	2.47775
$887$	−36451.4	−1.37984	−0.689920	−	0.723885i	$-0.742354\pi$
$887$	−36451.4	−1.37984	−0.689920	+	0.723885i	$0.742354\pi$
$888$	0	0
$889$	−69632.6	−2.62700
$890$	−3208.71	−0.120850
$891$	0	0
$892$	−46868.1	−1.75926
$893$	27341.6	1.02458
$894$	0	0
$895$	−153.685	−0.00573980
$896$	4501.86	0.167853
$897$	0	0
$898$	−10517.1	−0.390825
$899$	5701.82	0.211531
$900$	0	0
$901$	−6806.05	−0.251656
$902$	6725.72	0.248273
$903$	0	0
$904$	1465.25	0.0539086
$905$	797.778	0.0293028
$906$	0	0
$907$	36563.0	1.33854	0.669270	−	0.743020i	$-0.266607\pi$
$907$	36563.0	1.33854	0.669270	+	0.743020i	$0.266607\pi$
$908$	−34233.5	−1.25119
$909$	0	0
$910$	−34091.1	−1.24188
$911$	−25321.6	−0.920902	−0.460451	−	0.887685i	$-0.652312\pi$
$911$	−25321.6	−0.920902	−0.460451	+	0.887685i	$0.652312\pi$
$912$	0	0
$913$	537.324	0.0194773
$914$	2310.65	0.0836208
$915$	0	0
$916$	−34918.9	−1.25955
$917$	−80422.5	−2.89617
$918$	0	0
$919$	16527.1	0.593230	0.296615	−	0.954997i	$-0.404142\pi$
$919$	16527.1	0.593230	0.296615	+	0.954997i	$0.404142\pi$
$920$	698.720	0.0250393
$921$	0	0
$922$	−25929.4	−0.926181
$923$	−6188.01	−0.220673
$924$	0	0
$925$	−2991.12	−0.106322
$926$	47178.0	1.67426
$927$	0	0
$928$	7352.73	0.260092
$929$	32080.4	1.13297	0.566483	−	0.824073i	$-0.308304\pi$
$929$	32080.4	1.13297	0.566483	+	0.824073i	$0.308304\pi$
$930$	0	0
$931$	35438.0	1.24751
$932$	−31315.6	−1.10062
$933$	0	0
$934$	−58470.3	−2.04840
$935$	−1093.44	−0.0382454
$936$	0	0
$937$	31282.2	1.09066	0.545328	−	0.838223i	$-0.316405\pi$
$937$	31282.2	1.09066	0.545328	+	0.838223i	$0.316405\pi$
$938$	−9739.93	−0.339041
$939$	0	0
$940$	21993.7	0.763143
$941$	−23786.8	−0.824047	−0.412024	−	0.911173i	$-0.635178\pi$
$941$	−23786.8	−0.824047	−0.412024	+	0.911173i	$0.635178\pi$
$942$	0	0
$943$	17509.0	0.604637
$944$	49015.4	1.68995
$945$	0	0
$946$	−11137.0	−0.382765
$947$	−38442.2	−1.31912	−0.659558	−	0.751654i	$-0.729256\pi$
$947$	−38442.2	−1.31912	−0.659558	+	0.751654i	$0.729256\pi$
$948$	0	0
$949$	26065.2	0.891583
$950$	−4764.16	−0.162705
$951$	0	0
$952$	−608.722	−0.0207235
$953$	−41175.0	−1.39957	−0.699784	−	0.714355i	$-0.746720\pi$
$953$	−41175.0	−1.39957	−0.699784	+	0.714355i	$0.746720\pi$
$954$	0	0
$955$	−483.816	−0.0163936
$956$	−6472.84	−0.218982
$957$	0	0
$958$	23258.9	0.784407
$959$	52148.8	1.75597
$960$	0	0
$961$	8866.24	0.297615
$962$	−24816.7	−0.831727
$963$	0	0
$964$	−15546.9	−0.519433
$965$	5089.61	0.169783
$966$	0	0
$967$	20553.4	0.683508	0.341754	−	0.939790i	$-0.388979\pi$
$967$	20553.4	0.683508	0.341754	+	0.939790i	$0.388979\pi$
$968$	−1253.79	−0.0416305
$969$	0	0
$970$	−2548.35	−0.0843533
$971$	−7678.14	−0.253762	−0.126881	−	0.991918i	$-0.540497\pi$
$971$	−7678.14	−0.253762	−0.126881	+	0.991918i	$0.540497\pi$
$972$	0	0
$973$	−58949.5	−1.94228
$974$	61457.7	2.02180
$975$	0	0
$976$	−50113.1	−1.64353
$977$	−46149.6	−1.51121	−0.755607	−	0.655025i	$-0.772658\pi$
$977$	−46149.6	−1.51121	−0.755607	+	0.655025i	$0.772658\pi$
$978$	0	0
$979$	−2045.65	−0.0667818
$980$	28506.4	0.929188
$981$	0	0
$982$	3774.61	0.122661
$983$	17189.7	0.557747	0.278874	−	0.960328i	$-0.410039\pi$
$983$	17189.7	0.557747	0.278874	+	0.960328i	$0.410039\pi$
$984$	0	0
$985$	−25063.3	−0.810743
$986$	−1989.54	−0.0642596
$987$	0	0
$988$	−19424.4	−0.625480
$989$	−28992.9	−0.932176
$990$	0	0
$991$	−14818.9	−0.475012	−0.237506	−	0.971386i	$-0.576330\pi$
$991$	−14818.9	−0.475012	−0.237506	+	0.971386i	$0.576330\pi$
$992$	49850.1	1.59551
$993$	0	0
$994$	15426.0	0.492237
$995$	−21697.8	−0.691324
$996$	0	0
$997$	−60460.3	−1.92056	−0.960280	−	0.279038i	$-0.909984\pi$
$997$	−60460.3	−1.92056	−0.960280	+	0.279038i	$0.909984\pi$
$998$	−50241.2	−1.59354
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.m.1.1		7
3.2	odd	2		435.4.a.j.1.7	✓	7
15.14	odd	2		2175.4.a.m.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.j.1.7	✓	7	3.2	odd	2
1305.4.a.m.1.1		7	1.1	even	1	trivial
2175.4.a.m.1.1		7	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)

\(f(q)\)	\(=\)	\(q-3.96612 q^{2} +7.73008 q^{4} +5.00000 q^{5} -32.8717 q^{7} +1.07055 q^{8} +O(q^{10})\)	\(q-3.96612 q^{2} +7.73008 q^{4} +5.00000 q^{5} -32.8717 q^{7} +1.07055 q^{8} -19.8306 q^{10} -12.6426 q^{11} -52.2979 q^{13} +130.373 q^{14} -66.0865 q^{16} +17.2978 q^{17} +48.0486 q^{19} +38.6504 q^{20} +50.1421 q^{22} +130.535 q^{23} +25.0000 q^{25} +207.420 q^{26} -254.100 q^{28} +29.0000 q^{29} +196.614 q^{31} +253.542 q^{32} -68.6049 q^{34} -164.358 q^{35} -119.645 q^{37} -190.566 q^{38} +5.35275 q^{40} +134.133 q^{41} -222.109 q^{43} -97.7283 q^{44} -517.716 q^{46} +569.042 q^{47} +737.546 q^{49} -99.1529 q^{50} -404.267 q^{52} -393.464 q^{53} -63.2131 q^{55} -35.1908 q^{56} -115.017 q^{58} -741.684 q^{59} +758.295 q^{61} -779.796 q^{62} -476.886 q^{64} -261.489 q^{65} -74.7083 q^{67} +133.713 q^{68} +651.864 q^{70} +118.322 q^{71} -498.399 q^{73} +474.525 q^{74} +371.419 q^{76} +415.584 q^{77} +106.032 q^{79} -330.433 q^{80} -531.988 q^{82} -42.5010 q^{83} +86.4888 q^{85} +880.911 q^{86} -13.5346 q^{88} +161.806 q^{89} +1719.12 q^{91} +1009.04 q^{92} -2256.88 q^{94} +240.243 q^{95} +128.506 q^{97} -2925.19 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8}+O(q^{10}) \) 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8	\( 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26} - 637 q^{28} + 203 q^{29} - 480 q^{31} + 779 q^{32} - 897 q^{34} - 185 q^{35} - 1032 q^{37} + 194 q^{38} + 180 q^{40} + 638 q^{41} - 512 q^{43} - 625 q^{44} + 16 q^{46} + 111 q^{47} + 178 q^{49} + 25 q^{50} - 1263 q^{52} - 410 q^{53} + 55 q^{55} - 1174 q^{56} + 29 q^{58} + 426 q^{59} - 1192 q^{61} - 460 q^{62} + 390 q^{64} - 665 q^{65} - 1671 q^{67} - 1509 q^{68} + 375 q^{70} + 1324 q^{71} - 852 q^{73} - 1780 q^{74} - 564 q^{76} + 2107 q^{77} + 366 q^{79} - 265 q^{80} - 318 q^{82} - 470 q^{83} - 105 q^{85} + 2196 q^{86} - 2518 q^{88} - 51 q^{89} - 1297 q^{91} + 684 q^{92} - 1837 q^{94} - 850 q^{95} - 3322 q^{97} - 1068 q^{98}+O(q^{100}) \) 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * q^98

Introduction

L-functions

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Database

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