LMFDB - Embedded newform 153.4.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	153.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:	$9.02729223088$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2636.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 14x - 4 $ x^3 - 14*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.58966$ of defining polynomial
Character	$\chi$	$=$	153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.03251 q^{2} +17.3261 q^{4} -0.885690 q^{5} +3.81828 q^{7} +46.9339 q^{8} +O(q^{10})$	\(q+5.03251 q^{2} +17.3261 q^{4} -0.885690 q^{5} +3.81828 q^{7} +46.9339 q^{8} -4.45724 q^{10} +52.3720 q^{11} -8.06025 q^{13} +19.2156 q^{14} +97.5862 q^{16} +17.0000 q^{17} -66.5154 q^{19} -15.3456 q^{20} +263.563 q^{22} -180.226 q^{23} -124.216 q^{25} -40.5633 q^{26} +66.1562 q^{28} +41.2800 q^{29} -34.9114 q^{31} +115.632 q^{32} +85.5527 q^{34} -3.38182 q^{35} +130.368 q^{37} -334.739 q^{38} -41.5689 q^{40} +17.9081 q^{41} +277.620 q^{43} +907.405 q^{44} -906.987 q^{46} -463.789 q^{47} -328.421 q^{49} -625.116 q^{50} -139.653 q^{52} +329.944 q^{53} -46.3853 q^{55} +179.207 q^{56} +207.742 q^{58} -678.656 q^{59} +340.280 q^{61} -175.692 q^{62} -198.770 q^{64} +7.13888 q^{65} +15.3925 q^{67} +294.545 q^{68} -17.0190 q^{70} +670.203 q^{71} +193.480 q^{73} +656.080 q^{74} -1152.46 q^{76} +199.971 q^{77} +1080.15 q^{79} -86.4311 q^{80} +90.1229 q^{82} +865.668 q^{83} -15.0567 q^{85} +1397.13 q^{86} +2458.02 q^{88} -1129.46 q^{89} -30.7763 q^{91} -3122.61 q^{92} -2334.02 q^{94} +58.9120 q^{95} -379.412 q^{97} -1652.78 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10}) $ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8	$ 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100}) $ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 - 56 * q^10 + 28 * q^11 + 30 * q^13 - 92 * q^14 + 137 * q^16 + 51 * q^17 + 80 * q^19 + 168 * q^20 + 286 * q^22 - 142 * q^23 - 223 * q^25 - 26 * q^26 + 476 * q^28 + 456 * q^29 + 230 * q^31 + 71 * q^32 - 17 * q^34 + 332 * q^35 + 356 * q^37 - 724 * q^38 - 424 * q^40 + 294 * q^41 + 556 * q^43 + 1122 * q^44 - 704 * q^46 - 640 * q^47 - 269 * q^49 - 547 * q^50 - 774 * q^52 - 302 * q^53 + 76 * q^55 - 684 * q^56 - 1304 * q^58 - 636 * q^59 - 84 * q^61 - 508 * q^62 - 919 * q^64 - 408 * q^65 + 1008 * q^67 + 425 * q^68 - 1504 * q^70 + 402 * q^71 + 838 * q^73 - 836 * q^74 - 908 * q^76 + 504 * q^77 - 594 * q^79 + 40 * q^80 + 358 * q^82 + 2396 * q^83 + 136 * q^85 + 1264 * q^86 + 1838 * q^88 + 170 * q^89 - 1016 * q^91 - 4896 * q^92 - 2016 * q^94 + 472 * q^95 - 270 * q^97 - 2857 * 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 $

$2$	5.03251	1.77926	0.889630	−	0.456681i	$-0.150962\pi$
$2$	5.03251	1.77926	0.889630	+	0.456681i	$0.150962\pi$
$3$	0	0
$3$	0	0
$4$	17.3261	2.16577
$5$	−0.885690	−0.0792185	−0.0396092	−	0.999215i	$-0.512611\pi$
$5$	−0.885690	−0.0792185	−0.0396092	+	0.999215i	$0.512611\pi$
$6$	0	0
$7$	3.81828	0.206168	0.103084	−	0.994673i	$-0.467129\pi$
$7$	3.81828	0.206168	0.103084	+	0.994673i	$0.467129\pi$
$8$	46.9339	2.07421
$9$	0	0
$10$	−4.45724	−0.140950
$11$	52.3720	1.43552	0.717761	−	0.696289i	$-0.245167\pi$
$11$	52.3720	1.43552	0.717761	+	0.696289i	$0.245167\pi$
$12$	0	0
$13$	−8.06025	−0.171962	−0.0859811	−	0.996297i	$-0.527402\pi$
$13$	−8.06025	−0.171962	−0.0859811	+	0.996297i	$0.527402\pi$
$14$	19.2156	0.366827
$15$	0	0
$16$	97.5862	1.52478
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−66.5154	−0.803141	−0.401570	−	0.915828i	$-0.631535\pi$
$19$	−66.5154	−0.803141	−0.401570	+	0.915828i	$0.631535\pi$
$20$	−15.3456	−0.171569
$21$	0	0
$22$	263.563	2.55417
$23$	−180.226	−1.63390	−0.816948	−	0.576711i	$-0.804336\pi$
$23$	−180.226	−1.63390	−0.816948	+	0.576711i	$0.804336\pi$
$24$	0	0
$25$	−124.216	−0.993724
$26$	−40.5633	−0.305966
$27$	0	0
$28$	66.1562	0.446512
$29$	41.2800	0.264328	0.132164	−	0.991228i	$-0.457807\pi$
$29$	41.2800	0.264328	0.132164	+	0.991228i	$0.457807\pi$
$30$	0	0
$31$	−34.9114	−0.202267	−0.101133	−	0.994873i	$-0.532247\pi$
$31$	−34.9114	−0.202267	−0.101133	+	0.994873i	$0.532247\pi$
$32$	115.632	0.638783
$33$	0	0
$34$	85.5527	0.431534
$35$	−3.38182	−0.0163323
$36$	0	0
$37$	130.368	0.579255	0.289627	−	0.957139i	$-0.406469\pi$
$37$	130.368	0.579255	0.289627	+	0.957139i	$0.406469\pi$
$38$	−334.739	−1.42900
$39$	0	0
$40$	−41.5689	−0.164315
$41$	17.9081	0.0682142	0.0341071	−	0.999418i	$-0.489141\pi$
$41$	17.9081	0.0682142	0.0341071	+	0.999418i	$0.489141\pi$
$42$	0	0
$43$	277.620	0.984573	0.492287	−	0.870433i	$-0.336161\pi$
$43$	277.620	0.984573	0.492287	+	0.870433i	$0.336161\pi$
$44$	907.405	3.10901
$45$	0	0
$46$	−906.987	−2.90713
$47$	−463.789	−1.43937	−0.719687	−	0.694299i	$-0.755715\pi$
$47$	−463.789	−1.43937	−0.719687	+	0.694299i	$0.755715\pi$
$48$	0	0
$49$	−328.421	−0.957495
$50$	−625.116	−1.76809
$51$	0	0
$52$	−139.653	−0.372431
$53$	329.944	0.855118	0.427559	−	0.903987i	$-0.359374\pi$
$53$	329.944	0.855118	0.427559	+	0.903987i	$0.359374\pi$
$54$	0	0
$55$	−46.3853	−0.113720
$56$	179.207	0.427635
$57$	0	0
$58$	207.742	0.470308
$59$	−678.656	−1.49752	−0.748759	−	0.662843i	$-0.769350\pi$
$59$	−678.656	−1.49752	−0.748759	+	0.662843i	$0.769350\pi$
$60$	0	0
$61$	340.280	0.714237	0.357118	−	0.934059i	$-0.383759\pi$
$61$	340.280	0.714237	0.357118	+	0.934059i	$0.383759\pi$
$62$	−175.692	−0.359885
$63$	0	0
$64$	−198.770	−0.388223
$65$	7.13888	0.0136226
$66$	0	0
$67$	15.3925	0.0280671	0.0140336	−	0.999902i	$-0.495533\pi$
$67$	15.3925	0.0280671	0.0140336	+	0.999902i	$0.495533\pi$
$68$	294.545	0.525276
$69$	0	0
$70$	−17.0190	−0.0290595
$71$	670.203	1.12026	0.560130	−	0.828405i	$-0.310751\pi$
$71$	670.203	1.12026	0.560130	+	0.828405i	$0.310751\pi$
$72$	0	0
$73$	193.480	0.310207	0.155103	−	0.987898i	$-0.450429\pi$
$73$	193.480	0.310207	0.155103	+	0.987898i	$0.450429\pi$
$74$	656.080	1.03065
$75$	0	0
$76$	−1152.46	−1.73942
$77$	199.971	0.295959
$78$	0	0
$79$	1080.15	1.53831	0.769156	−	0.639061i	$-0.220677\pi$
$79$	1080.15	1.53831	0.769156	+	0.639061i	$0.220677\pi$
$80$	−86.4311	−0.120791
$81$	0	0
$82$	90.1229	0.121371
$83$	865.668	1.14481	0.572406	−	0.819970i	$-0.306010\pi$
$83$	865.668	1.14481	0.572406	+	0.819970i	$0.306010\pi$
$84$	0	0
$85$	−15.0567	−0.0192133
$86$	1397.13	1.75181
$87$	0	0
$88$	2458.02	2.97757
$89$	−1129.46	−1.34520	−0.672599	−	0.740008i	$-0.734822\pi$
$89$	−1129.46	−1.34520	−0.672599	+	0.740008i	$0.734822\pi$
$90$	0	0
$91$	−30.7763	−0.0354531
$92$	−3122.61	−3.53864
$93$	0	0
$94$	−2334.02	−2.56102
$95$	58.9120	0.0636236
$96$	0	0
$97$	−379.412	−0.397149	−0.198574	−	0.980086i	$-0.563631\pi$
$97$	−379.412	−0.397149	−0.198574	+	0.980086i	$0.563631\pi$
$98$	−1652.78	−1.70363
$99$	0	0
$100$	−2152.18	−2.15218
$101$	−131.732	−0.129780	−0.0648902	−	0.997892i	$-0.520670\pi$
$101$	−131.732	−0.129780	−0.0648902	+	0.997892i	$0.520670\pi$
$102$	0	0
$103$	195.988	0.187488	0.0937442	−	0.995596i	$-0.470116\pi$
$103$	195.988	0.187488	0.0937442	+	0.995596i	$0.470116\pi$
$104$	−378.299	−0.356685
$105$	0	0
$106$	1660.45	1.52148
$107$	485.147	0.438326	0.219163	−	0.975688i	$-0.429667\pi$
$107$	485.147	0.438326	0.219163	+	0.975688i	$0.429667\pi$
$108$	0	0
$109$	−1255.12	−1.10292	−0.551460	−	0.834201i	$-0.685929\pi$
$109$	−1255.12	−1.10292	−0.551460	+	0.834201i	$0.685929\pi$
$110$	−233.435	−0.202337
$111$	0	0
$112$	372.612	0.314362
$113$	1013.35	0.843612	0.421806	−	0.906686i	$-0.361396\pi$
$113$	1013.35	0.843612	0.421806	+	0.906686i	$0.361396\pi$
$114$	0	0
$115$	159.624	0.129435
$116$	715.224	0.572473
$117$	0	0
$118$	−3415.34	−2.66447
$119$	64.9108	0.0500031
$120$	0	0
$121$	1411.83	1.06073
$122$	1712.46	1.27081
$123$	0	0
$124$	−604.880	−0.438063
$125$	220.728	0.157940
$126$	0	0
$127$	1927.72	1.34691	0.673456	−	0.739227i	$-0.264809\pi$
$127$	1927.72	1.34691	0.673456	+	0.739227i	$0.264809\pi$
$128$	−1925.37	−1.32953
$129$	0	0
$130$	35.9265	0.0242381
$131$	406.738	0.271274	0.135637	−	0.990759i	$-0.456692\pi$
$131$	406.738	0.271274	0.135637	+	0.990759i	$0.456692\pi$
$132$	0	0
$133$	−253.975	−0.165582
$134$	77.4631	0.0499387
$135$	0	0
$136$	797.877	0.503069
$137$	130.552	0.0814149	0.0407074	−	0.999171i	$-0.487039\pi$
$137$	130.552	0.0814149	0.0407074	+	0.999171i	$0.487039\pi$
$138$	0	0
$139$	2073.54	1.26529	0.632644	−	0.774443i	$-0.281970\pi$
$139$	2073.54	1.26529	0.632644	+	0.774443i	$0.281970\pi$
$140$	−58.5938	−0.0353720
$141$	0	0
$142$	3372.80	1.99323
$143$	−422.131	−0.246856
$144$	0	0
$145$	−36.5613	−0.0209397
$146$	973.689	0.551939
$147$	0	0
$148$	2258.78	1.25453
$149$	1852.73	1.01867	0.509334	−	0.860569i	$-0.329892\pi$
$149$	1852.73	1.01867	0.509334	+	0.860569i	$0.329892\pi$
$150$	0	0
$151$	2050.86	1.10527	0.552637	−	0.833422i	$-0.313622\pi$
$151$	2050.86	1.10527	0.552637	+	0.833422i	$0.313622\pi$
$152$	−3121.83	−1.66588
$153$	0	0
$154$	1006.36	0.526588
$155$	30.9207	0.0160233
$156$	0	0
$157$	−262.991	−0.133688	−0.0668438	−	0.997763i	$-0.521293\pi$
$157$	−262.991	−0.133688	−0.0668438	+	0.997763i	$0.521293\pi$
$158$	5435.88	2.73706
$159$	0	0
$160$	−102.414	−0.0506035
$161$	−688.152	−0.336857
$162$	0	0
$163$	−1444.98	−0.694354	−0.347177	−	0.937800i	$-0.612860\pi$
$163$	−1444.98	−0.694354	−0.347177	+	0.937800i	$0.612860\pi$
$164$	310.279	0.147736
$165$	0	0
$166$	4356.48	2.03692
$167$	501.565	0.232409	0.116204	−	0.993225i	$-0.462927\pi$
$167$	501.565	0.232409	0.116204	+	0.993225i	$0.462927\pi$
$168$	0	0
$169$	−2132.03	−0.970429
$170$	−75.7731	−0.0341855
$171$	0	0
$172$	4810.08	2.13236
$173$	2590.14	1.13829	0.569146	−	0.822237i	$-0.307274\pi$
$173$	2590.14	1.13829	0.569146	+	0.822237i	$0.307274\pi$
$174$	0	0
$175$	−474.290	−0.204874
$176$	5110.79	2.18886
$177$	0	0
$178$	−5684.02	−2.39346
$179$	−2165.65	−0.904294	−0.452147	−	0.891943i	$-0.649342\pi$
$179$	−2165.65	−0.904294	−0.452147	+	0.891943i	$0.649342\pi$
$180$	0	0
$181$	−1925.56	−0.790750	−0.395375	−	0.918520i	$-0.629385\pi$
$181$	−1925.56	−0.790750	−0.395375	+	0.918520i	$0.629385\pi$
$182$	−154.882	−0.0630803
$183$	0	0
$184$	−8458.69	−3.38904
$185$	−115.466	−0.0458877
$186$	0	0
$187$	890.324	0.348165
$188$	−8035.68	−3.11735
$189$	0	0
$190$	296.475	0.113203
$191$	2783.52	1.05449	0.527247	−	0.849712i	$-0.323224\pi$
$191$	2783.52	1.05449	0.527247	+	0.849712i	$0.323224\pi$
$192$	0	0
$193$	2258.27	0.842246	0.421123	−	0.907004i	$-0.361636\pi$
$193$	2258.27	0.842246	0.421123	+	0.907004i	$0.361636\pi$
$194$	−1909.39	−0.706631
$195$	0	0
$196$	−5690.27	−2.07371
$197$	1270.70	0.459560	0.229780	−	0.973243i	$-0.426199\pi$
$197$	1270.70	0.459560	0.229780	+	0.973243i	$0.426199\pi$
$198$	0	0
$199$	−4794.36	−1.70786	−0.853928	−	0.520392i	$-0.825786\pi$
$199$	−4794.36	−1.70786	−0.853928	+	0.520392i	$0.825786\pi$
$200$	−5829.92	−2.06119
$201$	0	0
$202$	−662.942	−0.230913
$203$	157.619	0.0544960
$204$	0	0
$205$	−15.8611	−0.00540383
$206$	986.313	0.333591
$207$	0	0
$208$	−786.569	−0.262205
$209$	−3483.54	−1.15293
$210$	0	0
$211$	−2807.00	−0.915837	−0.457918	−	0.888994i	$-0.651405\pi$
$211$	−2807.00	−0.915837	−0.457918	+	0.888994i	$0.651405\pi$
$212$	5716.66	1.85199
$213$	0	0
$214$	2441.50	0.779896
$215$	−245.885	−0.0779964
$216$	0	0
$217$	−133.302	−0.0417009
$218$	−6316.38	−1.96238
$219$	0	0
$220$	−803.679	−0.246291
$221$	−137.024	−0.0417070
$222$	0	0
$223$	4684.30	1.40665	0.703327	−	0.710866i	$-0.251697\pi$
$223$	4684.30	1.40665	0.703327	+	0.710866i	$0.251697\pi$
$224$	441.516	0.131697
$225$	0	0
$226$	5099.70	1.50101
$227$	1395.72	0.408095	0.204047	−	0.978961i	$-0.434590\pi$
$227$	1395.72	0.408095	0.204047	+	0.978961i	$0.434590\pi$
$228$	0	0
$229$	894.638	0.258163	0.129082	−	0.991634i	$-0.458797\pi$
$229$	894.638	0.258163	0.129082	+	0.991634i	$0.458797\pi$
$230$	803.309	0.230298
$231$	0	0
$232$	1937.43	0.548270
$233$	−1196.13	−0.336313	−0.168156	−	0.985760i	$-0.553781\pi$
$233$	−1196.13	−0.336313	−0.168156	+	0.985760i	$0.553781\pi$
$234$	0	0
$235$	410.773	0.114025
$236$	−11758.5	−3.24328
$237$	0	0
$238$	326.664	0.0889685
$239$	−4948.82	−1.33938	−0.669691	−	0.742639i	$-0.733574\pi$
$239$	−4948.82	−1.33938	−0.669691	+	0.742639i	$0.733574\pi$
$240$	0	0
$241$	−6702.73	−1.79154	−0.895770	−	0.444518i	$-0.853375\pi$
$241$	−6702.73	−1.79154	−0.895770	+	0.444518i	$0.853375\pi$
$242$	7105.03	1.88731
$243$	0	0
$244$	5895.75	1.54687
$245$	290.879	0.0758513
$246$	0	0
$247$	536.130	0.138110
$248$	−1638.53	−0.419543
$249$	0	0
$250$	1110.81	0.281016
$251$	4756.08	1.19602	0.598010	−	0.801489i	$-0.295958\pi$
$251$	4756.08	1.19602	0.598010	+	0.801489i	$0.295958\pi$
$252$	0	0
$253$	−9438.77	−2.34550
$254$	9701.29	2.39651
$255$	0	0
$256$	−8099.28	−1.97736
$257$	−2892.84	−0.702143	−0.351071	−	0.936349i	$-0.614183\pi$
$257$	−2892.84	−0.702143	−0.351071	+	0.936349i	$0.614183\pi$
$258$	0	0
$259$	497.784	0.119424
$260$	123.689	0.0295034
$261$	0	0
$262$	2046.92	0.482667
$263$	−5415.48	−1.26971	−0.634853	−	0.772633i	$-0.718939\pi$
$263$	−5415.48	−1.26971	−0.634853	+	0.772633i	$0.718939\pi$
$264$	0	0
$265$	−292.228	−0.0677412
$266$	−1278.13	−0.294613
$267$	0	0
$268$	266.693	0.0607869
$269$	−5787.00	−1.31167	−0.655835	−	0.754904i	$-0.727683\pi$
$269$	−5787.00	−1.31167	−0.655835	+	0.754904i	$0.727683\pi$
$270$	0	0
$271$	5465.13	1.22503	0.612515	−	0.790459i	$-0.290158\pi$
$271$	5465.13	1.22503	0.612515	+	0.790459i	$0.290158\pi$
$272$	1658.97	0.369815
$273$	0	0
$274$	657.006	0.144858
$275$	−6505.42	−1.42651
$276$	0	0
$277$	−1207.65	−0.261952	−0.130976	−	0.991386i	$-0.541811\pi$
$277$	−1207.65	−0.261952	−0.130976	+	0.991386i	$0.541811\pi$
$278$	10435.1	2.25128
$279$	0	0
$280$	−158.722	−0.0338766
$281$	1197.18	0.254155	0.127077	−	0.991893i	$-0.459440\pi$
$281$	1197.18	0.254155	0.127077	+	0.991893i	$0.459440\pi$
$282$	0	0
$283$	3164.73	0.664748	0.332374	−	0.943148i	$-0.392150\pi$
$283$	3164.73	0.664748	0.332374	+	0.943148i	$0.392150\pi$
$284$	11612.0	2.42622
$285$	0	0
$286$	−2124.38	−0.439221
$287$	68.3784	0.0140636
$288$	0	0
$289$	289.000	0.0588235
$290$	−183.995	−0.0372571
$291$	0	0
$292$	3352.26	0.671836
$293$	−7456.21	−1.48668	−0.743339	−	0.668915i	$-0.766759\pi$
$293$	−7456.21	−1.48668	−0.743339	+	0.668915i	$0.766759\pi$
$294$	0	0
$295$	601.079	0.118631
$296$	6118.70	1.20149
$297$	0	0
$298$	9323.89	1.81248
$299$	1452.66	0.280969
$300$	0	0
$301$	1060.03	0.202988
$302$	10321.0	1.96657
$303$	0	0
$304$	−6490.98	−1.22462
$305$	−301.383	−0.0565808
$306$	0	0
$307$	−6535.48	−1.21498	−0.607491	−	0.794327i	$-0.707824\pi$
$307$	−6535.48	−1.21498	−0.607491	+	0.794327i	$0.707824\pi$
$308$	3464.73	0.640978
$309$	0	0
$310$	155.608	0.0285096
$311$	8935.89	1.62928	0.814642	−	0.579963i	$-0.196933\pi$
$311$	8935.89	1.62928	0.814642	+	0.579963i	$0.196933\pi$
$312$	0	0
$313$	−2628.71	−0.474707	−0.237353	−	0.971423i	$-0.576280\pi$
$313$	−2628.71	−0.474707	−0.237353	+	0.971423i	$0.576280\pi$
$314$	−1323.50	−0.237865
$315$	0	0
$316$	18714.9	3.33163
$317$	−4268.54	−0.756293	−0.378147	−	0.925746i	$-0.623438\pi$
$317$	−4268.54	−0.756293	−0.378147	+	0.925746i	$0.623438\pi$
$318$	0	0
$319$	2161.92	0.379449
$320$	176.048	0.0307544
$321$	0	0
$322$	−3463.13	−0.599357
$323$	−1130.76	−0.194790
$324$	0	0
$325$	1001.21	0.170883
$326$	−7271.89	−1.23544
$327$	0	0
$328$	840.500	0.141490
$329$	−1770.88	−0.296753
$330$	0	0
$331$	992.298	0.164778	0.0823892	−	0.996600i	$-0.473745\pi$
$331$	992.298	0.164778	0.0823892	+	0.996600i	$0.473745\pi$
$332$	14998.7	2.47940
$333$	0	0
$334$	2524.13	0.413516
$335$	−13.6330	−0.00222344
$336$	0	0
$337$	8042.26	1.29997	0.649985	−	0.759947i	$-0.274775\pi$
$337$	8042.26	1.29997	0.649985	+	0.759947i	$0.274775\pi$
$338$	−10729.5	−1.72665
$339$	0	0
$340$	−260.875	−0.0416116
$341$	−1828.38	−0.290359
$342$	0	0
$343$	−2563.68	−0.403573
$344$	13029.8	2.04221
$345$	0	0
$346$	13034.9	2.02532
$347$	7414.16	1.14701	0.573506	−	0.819202i	$-0.305583\pi$
$347$	7414.16	1.14701	0.573506	+	0.819202i	$0.305583\pi$
$348$	0	0
$349$	−859.194	−0.131781	−0.0658905	−	0.997827i	$-0.520989\pi$
$349$	−859.194	−0.131781	−0.0658905	+	0.997827i	$0.520989\pi$
$350$	−2386.87	−0.364525
$351$	0	0
$352$	6055.89	0.916988
$353$	−569.084	−0.0858053	−0.0429027	−	0.999079i	$-0.513661\pi$
$353$	−569.084	−0.0858053	−0.0429027	+	0.999079i	$0.513661\pi$
$354$	0	0
$355$	−593.592	−0.0887453
$356$	−19569.2	−2.91339
$357$	0	0
$358$	−10898.7	−1.60897
$359$	5005.21	0.735835	0.367918	−	0.929858i	$-0.380071\pi$
$359$	5005.21	0.735835	0.367918	+	0.929858i	$0.380071\pi$
$360$	0	0
$361$	−2434.71	−0.354965
$362$	−9690.40	−1.40695
$363$	0	0
$364$	−533.235	−0.0767833
$365$	−171.363	−0.0245741
$366$	0	0
$367$	−10975.3	−1.56105	−0.780523	−	0.625127i	$-0.785047\pi$
$367$	−10975.3	−1.56105	−0.780523	+	0.625127i	$0.785047\pi$
$368$	−17587.5	−2.49134
$369$	0	0
$370$	−581.083	−0.0816462
$371$	1259.82	0.176298
$372$	0	0
$373$	−3211.72	−0.445835	−0.222918	−	0.974837i	$-0.571558\pi$
$373$	−3211.72	−0.445835	−0.222918	+	0.974837i	$0.571558\pi$
$374$	4480.56	0.619477
$375$	0	0
$376$	−21767.4	−2.98556
$377$	−332.727	−0.0454544
$378$	0	0
$379$	8051.48	1.09123	0.545616	−	0.838035i	$-0.316296\pi$
$379$	8051.48	1.09123	0.545616	+	0.838035i	$0.316296\pi$
$380$	1020.72	0.137794
$381$	0	0
$382$	14008.1	1.87622
$383$	2584.16	0.344763	0.172382	−	0.985030i	$-0.444854\pi$
$383$	2584.16	0.344763	0.172382	+	0.985030i	$0.444854\pi$
$384$	0	0
$385$	−177.112	−0.0234454
$386$	11364.7	1.49858
$387$	0	0
$388$	−6573.74	−0.860132
$389$	5174.31	0.674417	0.337208	−	0.941430i	$-0.390517\pi$
$389$	5174.31	0.674417	0.337208	+	0.941430i	$0.390517\pi$
$390$	0	0
$391$	−3063.83	−0.396278
$392$	−15414.1	−1.98604
$393$	0	0
$394$	6394.79	0.817677
$395$	−956.680	−0.121863
$396$	0	0
$397$	−5149.36	−0.650980	−0.325490	−	0.945545i	$-0.605529\pi$
$397$	−5149.36	−0.650980	−0.325490	+	0.945545i	$0.605529\pi$
$398$	−24127.7	−3.03872
$399$	0	0
$400$	−12121.7	−1.51522
$401$	−8700.49	−1.08350	−0.541748	−	0.840541i	$-0.682237\pi$
$401$	−8700.49	−1.08350	−0.541748	+	0.840541i	$0.682237\pi$
$402$	0	0
$403$	281.394	0.0347823
$404$	−2282.41	−0.281074
$405$	0	0
$406$	793.219	0.0969625
$407$	6827.65	0.831533
$408$	0	0
$409$	12346.0	1.49260	0.746299	−	0.665611i	$-0.231829\pi$
$409$	12346.0	1.49260	0.746299	+	0.665611i	$0.231829\pi$
$410$	−79.8209	−0.00961482
$411$	0	0
$412$	3395.72	0.406056
$413$	−2591.30	−0.308740
$414$	0	0
$415$	−766.713	−0.0906903
$416$	−932.023	−0.109847
$417$	0	0
$418$	−17531.0	−2.05136
$419$	5763.33	0.671974	0.335987	−	0.941867i	$-0.390930\pi$
$419$	5763.33	0.671974	0.335987	+	0.941867i	$0.390930\pi$
$420$	0	0
$421$	−1876.12	−0.217188	−0.108594	−	0.994086i	$-0.534635\pi$
$421$	−1876.12	−0.217188	−0.108594	+	0.994086i	$0.534635\pi$
$422$	−14126.2	−1.62951
$423$	0	0
$424$	15485.6	1.77369
$425$	−2111.66	−0.241014
$426$	0	0
$427$	1299.29	0.147253
$428$	8405.72	0.949313
$429$	0	0
$430$	−1237.42	−0.138776
$431$	−83.9299	−0.00937996	−0.00468998	−	0.999989i	$-0.501493\pi$
$431$	−83.9299	−0.00937996	−0.00468998	+	0.999989i	$0.501493\pi$
$432$	0	0
$433$	−15345.0	−1.70308	−0.851539	−	0.524291i	$-0.824331\pi$
$433$	−15345.0	−1.70308	−0.851539	+	0.524291i	$0.824331\pi$
$434$	−670.842	−0.0741968
$435$	0	0
$436$	−21746.3	−2.38867
$437$	11987.8	1.31225
$438$	0	0
$439$	3064.74	0.333194	0.166597	−	0.986025i	$-0.446722\pi$
$439$	3064.74	0.333194	0.166597	+	0.986025i	$0.446722\pi$
$440$	−2177.05	−0.235879
$441$	0	0
$442$	−689.575	−0.0742076
$443$	1792.97	0.192295	0.0961474	−	0.995367i	$-0.469348\pi$
$443$	1792.97	0.192295	0.0961474	+	0.995367i	$0.469348\pi$
$444$	0	0
$445$	1000.35	0.106564
$446$	23573.8	2.50281
$447$	0	0
$448$	−758.960	−0.0800391
$449$	−2499.19	−0.262681	−0.131341	−	0.991337i	$-0.541928\pi$
$449$	−2499.19	−0.262681	−0.131341	+	0.991337i	$0.541928\pi$
$450$	0	0
$451$	937.885	0.0979231
$452$	17557.5	1.82707
$453$	0	0
$454$	7024.00	0.726107
$455$	27.2583	0.00280854
$456$	0	0
$457$	14784.4	1.51331	0.756656	−	0.653813i	$-0.226832\pi$
$457$	14784.4	1.51331	0.756656	+	0.653813i	$0.226832\pi$
$458$	4502.28	0.459340
$459$	0	0
$460$	2765.67	0.280326
$461$	17746.9	1.79297	0.896483	−	0.443078i	$-0.146113\pi$
$461$	17746.9	1.79297	0.896483	+	0.443078i	$0.146113\pi$
$462$	0	0
$463$	18486.4	1.85559	0.927793	−	0.373096i	$-0.121704\pi$
$463$	18486.4	1.85559	0.927793	+	0.373096i	$0.121704\pi$
$464$	4028.36	0.403043
$465$	0	0
$466$	−6019.52	−0.598388
$467$	−7406.57	−0.733908	−0.366954	−	0.930239i	$-0.619599\pi$
$467$	−7406.57	−0.733908	−0.366954	+	0.930239i	$0.619599\pi$
$468$	0	0
$469$	58.7731	0.00578655
$470$	2067.22	0.202880
$471$	0	0
$472$	−31852.0	−3.10616
$473$	14539.5	1.41338
$474$	0	0
$475$	8262.24	0.798101
$476$	1124.65	0.108295
$477$	0	0
$478$	−24905.0	−2.38311
$479$	18550.9	1.76955	0.884775	−	0.466019i	$-0.154312\pi$
$479$	18550.9	1.76955	0.884775	+	0.466019i	$0.154312\pi$
$480$	0	0
$481$	−1050.80	−0.0996100
$482$	−33731.6	−3.18762
$483$	0	0
$484$	24461.5	2.29729
$485$	336.041	0.0314615
$486$	0	0
$487$	10203.4	0.949406	0.474703	−	0.880146i	$-0.342556\pi$
$487$	10203.4	0.949406	0.474703	+	0.880146i	$0.342556\pi$
$488$	15970.7	1.48147
$489$	0	0
$490$	1463.85	0.134959
$491$	1247.46	0.114658	0.0573290	−	0.998355i	$-0.481742\pi$
$491$	1247.46	0.114658	0.0573290	+	0.998355i	$0.481742\pi$
$492$	0	0
$493$	701.760	0.0641089
$494$	2698.08	0.245734
$495$	0	0
$496$	−3406.87	−0.308413
$497$	2559.03	0.230962
$498$	0	0
$499$	70.0303	0.00628254	0.00314127	−	0.999995i	$-0.499000\pi$
$499$	70.0303	0.00628254	0.00314127	+	0.999995i	$0.499000\pi$
$500$	3824.36	0.342061
$501$	0	0
$502$	23935.0	2.12803
$503$	−1444.29	−0.128028	−0.0640138	−	0.997949i	$-0.520390\pi$
$503$	−1444.29	−0.128028	−0.0640138	+	0.997949i	$0.520390\pi$
$504$	0	0
$505$	116.674	0.0102810
$506$	−47500.7	−4.17325
$507$	0	0
$508$	33400.0	2.91710
$509$	−14272.8	−1.24289	−0.621445	−	0.783458i	$-0.713454\pi$
$509$	−14272.8	−1.24289	−0.621445	+	0.783458i	$0.713454\pi$
$510$	0	0
$511$	738.761	0.0639547
$512$	−25356.7	−2.18871
$513$	0	0
$514$	−14558.3	−1.24929
$515$	−173.585	−0.0148525
$516$	0	0
$517$	−24289.6	−2.06625
$518$	2505.10	0.212486
$519$	0	0
$520$	335.055	0.0282561
$521$	−14874.0	−1.25075	−0.625376	−	0.780324i	$-0.715054\pi$
$521$	−14874.0	−1.25075	−0.625376	+	0.780324i	$0.715054\pi$
$522$	0	0
$523$	−8142.90	−0.680811	−0.340406	−	0.940279i	$-0.610564\pi$
$523$	−8142.90	−0.680811	−0.340406	+	0.940279i	$0.610564\pi$
$524$	7047.21	0.587517
$525$	0	0
$526$	−27253.4	−2.25914
$527$	−593.494	−0.0490569
$528$	0	0
$529$	20314.2	1.66962
$530$	−1470.64	−0.120529
$531$	0	0
$532$	−4400.40	−0.358612
$533$	−144.344	−0.0117303
$534$	0	0
$535$	−429.689	−0.0347235
$536$	722.432	0.0582170
$537$	0	0
$538$	−29123.1	−2.33380
$539$	−17200.0	−1.37451
$540$	0	0
$541$	3179.67	0.252689	0.126344	−	0.991986i	$-0.459676\pi$
$541$	3179.67	0.252689	0.126344	+	0.991986i	$0.459676\pi$
$542$	27503.3	2.17965
$543$	0	0
$544$	1965.75	0.154928
$545$	1111.64	0.0873716
$546$	0	0
$547$	2107.07	0.164702	0.0823509	−	0.996603i	$-0.473757\pi$
$547$	2107.07	0.164702	0.0823509	+	0.996603i	$0.473757\pi$
$548$	2261.97	0.176326
$549$	0	0
$550$	−32738.6	−2.53814
$551$	−2745.76	−0.212292
$552$	0	0
$553$	4124.33	0.317151
$554$	−6077.51	−0.466081
$555$	0	0
$556$	35926.4	2.74032
$557$	−467.382	−0.0355540	−0.0177770	−	0.999842i	$-0.505659\pi$
$557$	−467.382	−0.0355540	−0.0177770	+	0.999842i	$0.505659\pi$
$558$	0	0
$559$	−2237.69	−0.169309
$560$	−330.019	−0.0249033
$561$	0	0
$562$	6024.80	0.452208
$563$	−14612.6	−1.09387	−0.546935	−	0.837175i	$-0.684206\pi$
$563$	−14612.6	−1.09387	−0.546935	+	0.837175i	$0.684206\pi$
$564$	0	0
$565$	−897.515	−0.0668297
$566$	15926.5	1.18276
$567$	0	0
$568$	31455.3	2.32365
$569$	−11602.3	−0.854821	−0.427410	−	0.904058i	$-0.640574\pi$
$569$	−11602.3	−0.854821	−0.427410	+	0.904058i	$0.640574\pi$
$570$	0	0
$571$	−10534.9	−0.772104	−0.386052	−	0.922477i	$-0.626161\pi$
$571$	−10534.9	−0.772104	−0.386052	+	0.922477i	$0.626161\pi$
$572$	−7313.91	−0.534633
$573$	0	0
$574$	344.115	0.0250228
$575$	22386.8	1.62364
$576$	0	0
$577$	14404.7	1.03930	0.519650	−	0.854379i	$-0.326062\pi$
$577$	14404.7	1.03930	0.519650	+	0.854379i	$0.326062\pi$
$578$	1454.40	0.104662
$579$	0	0
$580$	−633.466	−0.0453504
$581$	3305.37	0.236024
$582$	0	0
$583$	17279.8	1.22754
$584$	9080.77	0.643433
$585$	0	0
$586$	−37523.5	−2.64519
$587$	11004.9	0.773799	0.386900	−	0.922122i	$-0.373546\pi$
$587$	11004.9	0.773799	0.386900	+	0.922122i	$0.373546\pi$
$588$	0	0
$589$	2322.14	0.162449
$590$	3024.94	0.211076
$591$	0	0
$592$	12722.2	0.883239
$593$	−1853.59	−0.128361	−0.0641804	−	0.997938i	$-0.520443\pi$
$593$	−1853.59	−0.128361	−0.0641804	+	0.997938i	$0.520443\pi$
$594$	0	0
$595$	−57.4909	−0.00396117
$596$	32100.7	2.20620
$597$	0	0
$598$	7310.53	0.499916
$599$	−19074.7	−1.30112	−0.650559	−	0.759456i	$-0.725465\pi$
$599$	−19074.7	−1.30112	−0.650559	+	0.759456i	$0.725465\pi$
$600$	0	0
$601$	−27776.0	−1.88520	−0.942600	−	0.333923i	$-0.891627\pi$
$601$	−27776.0	−1.88520	−0.942600	+	0.333923i	$0.891627\pi$
$602$	5334.62	0.361168
$603$	0	0
$604$	35533.4	2.39377
$605$	−1250.44	−0.0840291
$606$	0	0
$607$	18728.3	1.25232	0.626159	−	0.779695i	$-0.284626\pi$
$607$	18728.3	1.25232	0.626159	+	0.779695i	$0.284626\pi$
$608$	−7691.32	−0.513033
$609$	0	0
$610$	−1516.71	−0.100672
$611$	3738.25	0.247518
$612$	0	0
$613$	−24405.3	−1.60802	−0.804012	−	0.594613i	$-0.797305\pi$
$613$	−24405.3	−1.60802	−0.804012	+	0.594613i	$0.797305\pi$
$614$	−32889.8	−2.16177
$615$	0	0
$616$	9385.43	0.613880
$617$	22516.4	1.46917	0.734584	−	0.678518i	$-0.237377\pi$
$617$	22516.4	1.46917	0.734584	+	0.678518i	$0.237377\pi$
$618$	0	0
$619$	−5146.53	−0.334179	−0.167089	−	0.985942i	$-0.553437\pi$
$619$	−5146.53	−0.334179	−0.167089	+	0.985942i	$0.553437\pi$
$620$	535.736	0.0347027
$621$	0	0
$622$	44969.9	2.89892
$623$	−4312.60	−0.277337
$624$	0	0
$625$	15331.4	0.981213
$626$	−13229.0	−0.844627
$627$	0	0
$628$	−4556.62	−0.289536
$629$	2216.26	0.140490
$630$	0	0
$631$	−3858.77	−0.243447	−0.121724	−	0.992564i	$-0.538842\pi$
$631$	−3858.77	−0.243447	−0.121724	+	0.992564i	$0.538842\pi$
$632$	50695.8	3.19078
$633$	0	0
$634$	−21481.5	−1.34564
$635$	−1707.36	−0.106700
$636$	0	0
$637$	2647.15	0.164653
$638$	10879.9	0.675138
$639$	0	0
$640$	1705.28	0.105324
$641$	−18689.3	−1.15161	−0.575805	−	0.817587i	$-0.695311\pi$
$641$	−18689.3	−1.15161	−0.575805	+	0.817587i	$0.695311\pi$
$642$	0	0
$643$	26473.5	1.62366	0.811831	−	0.583893i	$-0.198471\pi$
$643$	26473.5	1.62366	0.811831	+	0.583893i	$0.198471\pi$
$644$	−11923.0	−0.729555
$645$	0	0
$646$	−5690.57	−0.346583
$647$	−14397.7	−0.874855	−0.437427	−	0.899254i	$-0.644110\pi$
$647$	−14397.7	−0.874855	−0.437427	+	0.899254i	$0.644110\pi$
$648$	0	0
$649$	−35542.6	−2.14972
$650$	5038.59	0.304046
$651$	0	0
$652$	−25036.0	−1.50381
$653$	−20939.5	−1.25486	−0.627431	−	0.778672i	$-0.715893\pi$
$653$	−20939.5	−1.25486	−0.627431	+	0.778672i	$0.715893\pi$
$654$	0	0
$655$	−360.244	−0.0214899
$656$	1747.59	0.104012
$657$	0	0
$658$	−8911.96	−0.528001
$659$	−4031.76	−0.238323	−0.119162	−	0.992875i	$-0.538021\pi$
$659$	−4031.76	−0.238323	−0.119162	+	0.992875i	$0.538021\pi$
$660$	0	0
$661$	6691.52	0.393752	0.196876	−	0.980428i	$-0.436920\pi$
$661$	6691.52	0.393752	0.196876	+	0.980428i	$0.436920\pi$
$662$	4993.75	0.293184
$663$	0	0
$664$	40629.2	2.37458
$665$	224.943	0.0131171
$666$	0	0
$667$	−7439.71	−0.431884
$668$	8690.19	0.503344
$669$	0	0
$670$	−68.6083	−0.00395607
$671$	17821.2	1.02530
$672$	0	0
$673$	10319.2	0.591048	0.295524	−	0.955335i	$-0.404506\pi$
$673$	10319.2	0.591048	0.295524	+	0.955335i	$0.404506\pi$
$674$	40472.7	2.31298
$675$	0	0
$676$	−36939.9	−2.10172
$677$	19813.3	1.12480	0.562398	−	0.826866i	$-0.309879\pi$
$677$	19813.3	1.12480	0.562398	+	0.826866i	$0.309879\pi$
$678$	0	0
$679$	−1448.70	−0.0818793
$680$	−706.671	−0.0398524
$681$	0	0
$682$	−9201.33	−0.516624
$683$	−5924.61	−0.331916	−0.165958	−	0.986133i	$-0.553072\pi$
$683$	−5924.61	−0.331916	−0.165958	+	0.986133i	$0.553072\pi$
$684$	0	0
$685$	−115.629	−0.00644957
$686$	−12901.7	−0.718061
$687$	0	0
$688$	27091.9	1.50126
$689$	−2659.43	−0.147048
$690$	0	0
$691$	1973.16	0.108629	0.0543143	−	0.998524i	$-0.482703\pi$
$691$	1973.16	0.108629	0.0543143	+	0.998524i	$0.482703\pi$
$692$	44877.1	2.46528
$693$	0	0
$694$	37311.8	2.04083
$695$	−1836.51	−0.100234
$696$	0	0
$697$	304.439	0.0165444
$698$	−4323.90	−0.234473
$699$	0	0
$700$	−8217.63	−0.443710
$701$	12840.1	0.691815	0.345907	−	0.938269i	$-0.387571\pi$
$701$	12840.1	0.691815	0.345907	+	0.938269i	$0.387571\pi$
$702$	0	0
$703$	−8671.50	−0.465223
$704$	−10410.0	−0.557302
$705$	0	0
$706$	−2863.92	−0.152670
$707$	−502.990	−0.0267566
$708$	0	0
$709$	−27749.7	−1.46990	−0.734952	−	0.678119i	$-0.762796\pi$
$709$	−27749.7	−1.46990	−0.734952	+	0.678119i	$0.762796\pi$
$710$	−2987.26	−0.157901
$711$	0	0
$712$	−53010.0	−2.79022
$713$	6291.92	0.330483
$714$	0	0
$715$	373.877	0.0195555
$716$	−37522.4	−1.95849
$717$	0	0
$718$	25188.8	1.30924
$719$	−16888.3	−0.875979	−0.437989	−	0.898980i	$-0.644309\pi$
$719$	−16888.3	−0.875979	−0.437989	+	0.898980i	$0.644309\pi$
$720$	0	0
$721$	748.339	0.0386541
$722$	−12252.7	−0.631575
$723$	0	0
$724$	−33362.6	−1.71258
$725$	−5127.62	−0.262669
$726$	0	0
$727$	2135.25	0.108930	0.0544649	−	0.998516i	$-0.482655\pi$
$727$	2135.25	0.108930	0.0544649	+	0.998516i	$0.482655\pi$
$728$	−1444.45	−0.0735371
$729$	0	0
$730$	−862.386	−0.0437238
$731$	4719.54	0.238794
$732$	0	0
$733$	4795.27	0.241633	0.120817	−	0.992675i	$-0.461449\pi$
$733$	4795.27	0.241633	0.120817	+	0.992675i	$0.461449\pi$
$734$	−55233.1	−2.77751
$735$	0	0
$736$	−20839.9	−1.04371
$737$	806.138	0.0402910
$738$	0	0
$739$	−32747.6	−1.63010	−0.815048	−	0.579393i	$-0.803290\pi$
$739$	−32747.6	−1.63010	−0.815048	+	0.579393i	$0.803290\pi$
$740$	−2000.58	−0.0993821
$741$	0	0
$742$	6340.05	0.313680
$743$	−12299.4	−0.607298	−0.303649	−	0.952784i	$-0.598205\pi$
$743$	−12299.4	−0.607298	−0.303649	+	0.952784i	$0.598205\pi$
$744$	0	0
$745$	−1640.94	−0.0806974
$746$	−16163.0	−0.793257
$747$	0	0
$748$	15425.9	0.754046
$749$	1852.43	0.0903688
$750$	0	0
$751$	30102.6	1.46266	0.731332	−	0.682021i	$-0.238899\pi$
$751$	30102.6	1.46266	0.731332	+	0.682021i	$0.238899\pi$
$752$	−45259.4	−2.19474
$753$	0	0
$754$	−1674.45	−0.0808753
$755$	−1816.42	−0.0875581
$756$	0	0
$757$	38826.3	1.86416	0.932078	−	0.362257i	$-0.117994\pi$
$757$	38826.3	1.86416	0.932078	+	0.362257i	$0.117994\pi$
$758$	40519.2	1.94159
$759$	0	0
$760$	2764.97	0.131968
$761$	−19981.6	−0.951815	−0.475907	−	0.879495i	$-0.657880\pi$
$761$	−19981.6	−0.951815	−0.475907	+	0.879495i	$0.657880\pi$
$762$	0	0
$763$	−4792.39	−0.227387
$764$	48227.7	2.28379
$765$	0	0
$766$	13004.8	0.613424
$767$	5470.14	0.257517
$768$	0	0
$769$	−22407.7	−1.05077	−0.525384	−	0.850865i	$-0.676078\pi$
$769$	−22407.7	−1.05077	−0.525384	+	0.850865i	$0.676078\pi$
$770$	−891.320	−0.0417155
$771$	0	0
$772$	39127.0	1.82411
$773$	6902.77	0.321184	0.160592	−	0.987021i	$-0.448660\pi$
$773$	6902.77	0.321184	0.160592	+	0.987021i	$0.448660\pi$
$774$	0	0
$775$	4336.54	0.200997
$776$	−17807.3	−0.823768
$777$	0	0
$778$	26039.8	1.19996
$779$	−1191.17	−0.0547856
$780$	0	0
$781$	35099.9	1.60816
$782$	−15418.8	−0.705082
$783$	0	0
$784$	−32049.3	−1.45997
$785$	232.928	0.0105905
$786$	0	0
$787$	−22185.9	−1.00488	−0.502442	−	0.864611i	$-0.667565\pi$
$787$	−22185.9	−1.00488	−0.502442	+	0.864611i	$0.667565\pi$
$788$	22016.3	0.995301
$789$	0	0
$790$	−4814.50	−0.216826
$791$	3869.27	0.173926
$792$	0	0
$793$	−2742.74	−0.122822
$794$	−25914.2	−1.15826
$795$	0	0
$796$	−83067.8	−3.69882
$797$	16291.1	0.724040	0.362020	−	0.932170i	$-0.382087\pi$
$797$	16291.1	0.724040	0.362020	+	0.932170i	$0.382087\pi$
$798$	0	0
$799$	−7884.41	−0.349100
$800$	−14363.3	−0.634775
$801$	0	0
$802$	−43785.3	−1.92782
$803$	10132.9	0.445309
$804$	0	0
$805$	609.489	0.0266853
$806$	1416.12	0.0618867
$807$	0	0
$808$	−6182.70	−0.269191
$809$	−17696.8	−0.769082	−0.384541	−	0.923108i	$-0.625640\pi$
$809$	−17696.8	−0.769082	−0.384541	+	0.923108i	$0.625640\pi$
$810$	0	0
$811$	−3095.34	−0.134022	−0.0670111	−	0.997752i	$-0.521346\pi$
$811$	−3095.34	−0.134022	−0.0670111	+	0.997752i	$0.521346\pi$
$812$	2730.93	0.118026
$813$	0	0
$814$	34360.2	1.47951
$815$	1279.81	0.0550057
$816$	0	0
$817$	−18466.0	−0.790751
$818$	62131.5	2.65572
$819$	0	0
$820$	−274.811	−0.0117034
$821$	−12323.5	−0.523864	−0.261932	−	0.965086i	$-0.584360\pi$
$821$	−12323.5	−0.523864	−0.261932	+	0.965086i	$0.584360\pi$
$822$	0	0
$823$	−34436.5	−1.45854	−0.729271	−	0.684225i	$-0.760140\pi$
$823$	−34436.5	−1.45854	−0.729271	+	0.684225i	$0.760140\pi$
$824$	9198.50	0.388889
$825$	0	0
$826$	−13040.8	−0.549329
$827$	−18761.6	−0.788880	−0.394440	−	0.918922i	$-0.629061\pi$
$827$	−18761.6	−0.788880	−0.394440	+	0.918922i	$0.629061\pi$
$828$	0	0
$829$	22423.8	0.939457	0.469728	−	0.882811i	$-0.344352\pi$
$829$	22423.8	0.939457	0.469728	+	0.882811i	$0.344352\pi$
$830$	−3858.49	−0.161362
$831$	0	0
$832$	1602.13	0.0667596
$833$	−5583.15	−0.232227
$834$	0	0
$835$	−444.231	−0.0184111
$836$	−60356.4	−2.49697
$837$	0	0
$838$	29004.0	1.19562
$839$	−9128.63	−0.375632	−0.187816	−	0.982204i	$-0.560141\pi$
$839$	−9128.63	−0.375632	−0.187816	+	0.982204i	$0.560141\pi$
$840$	0	0
$841$	−22685.0	−0.930131
$842$	−9441.58	−0.386435
$843$	0	0
$844$	−48634.4	−1.98349
$845$	1888.32	0.0768759
$846$	0	0
$847$	5390.75	0.218688
$848$	32198.0	1.30387
$849$	0	0
$850$	−10627.0	−0.428826
$851$	−23495.7	−0.946442
$852$	0	0
$853$	27204.8	1.09200	0.545999	−	0.837786i	$-0.316150\pi$
$853$	27204.8	1.09200	0.545999	+	0.837786i	$0.316150\pi$
$854$	6538.68	0.262001
$855$	0	0
$856$	22769.8	0.909179
$857$	38060.0	1.51704	0.758520	−	0.651649i	$-0.225923\pi$
$857$	38060.0	1.51704	0.758520	+	0.651649i	$0.225923\pi$
$858$	0	0
$859$	−33326.2	−1.32372	−0.661860	−	0.749627i	$-0.730233\pi$
$859$	−33326.2	−1.32372	−0.661860	+	0.749627i	$0.730233\pi$
$860$	−4260.24	−0.168922
$861$	0	0
$862$	−422.378	−0.0166894
$863$	41724.2	1.64578	0.822890	−	0.568201i	$-0.192360\pi$
$863$	41724.2	1.64578	0.822890	+	0.568201i	$0.192360\pi$
$864$	0	0
$865$	−2294.06	−0.0901737
$866$	−77223.8	−3.03022
$867$	0	0
$868$	−2309.60	−0.0903146
$869$	56569.7	2.20828
$870$	0	0
$871$	−124.068	−0.00482649
$872$	−58907.5	−2.28768
$873$	0	0
$874$	60328.6	2.33483
$875$	842.801	0.0325621
$876$	0	0
$877$	−49337.3	−1.89966	−0.949830	−	0.312767i	$-0.898744\pi$
$877$	−49337.3	−1.89966	−0.949830	+	0.312767i	$0.898744\pi$
$878$	15423.3	0.592838
$879$	0	0
$880$	−4526.57	−0.173398
$881$	−8845.46	−0.338265	−0.169132	−	0.985593i	$-0.554097\pi$
$881$	−8845.46	−0.338265	−0.169132	+	0.985593i	$0.554097\pi$
$882$	0	0
$883$	14724.2	0.561165	0.280582	−	0.959830i	$-0.409472\pi$
$883$	14724.2	0.561165	0.280582	+	0.959830i	$0.409472\pi$
$884$	−2374.10	−0.0903277
$885$	0	0
$886$	9023.14	0.342143
$887$	−3864.38	−0.146283	−0.0731415	−	0.997322i	$-0.523302\pi$
$887$	−3864.38	−0.146283	−0.0731415	+	0.997322i	$0.523302\pi$
$888$	0	0
$889$	7360.60	0.277690
$890$	5034.28	0.189606
$891$	0	0
$892$	81160.9	3.04649
$893$	30849.1	1.15602
$894$	0	0
$895$	1918.10	0.0716368
$896$	−7351.61	−0.274107
$897$	0	0
$898$	−12577.2	−0.467379
$899$	−1441.14	−0.0534648
$900$	0	0
$901$	5609.04	0.207397
$902$	4719.92	0.174231
$903$	0	0
$904$	47560.6	1.74982
$905$	1705.45	0.0626421
$906$	0	0
$907$	−743.409	−0.0272155	−0.0136078	−	0.999907i	$-0.504332\pi$
$907$	−743.409	−0.0272155	−0.0136078	+	0.999907i	$0.504332\pi$
$908$	24182.5	0.883839
$909$	0	0
$910$	137.177	0.00499713
$911$	−16291.0	−0.592475	−0.296238	−	0.955114i	$-0.595732\pi$
$911$	−16291.0	−0.592475	−0.296238	+	0.955114i	$0.595732\pi$
$912$	0	0
$913$	45336.8	1.64340
$914$	74402.5	2.69258
$915$	0	0
$916$	15500.6	0.559122
$917$	1553.04	0.0559280
$918$	0	0
$919$	−6188.99	−0.222150	−0.111075	−	0.993812i	$-0.535429\pi$
$919$	−6188.99	−0.222150	−0.111075	+	0.993812i	$0.535429\pi$
$920$	7491.78	0.268474
$921$	0	0
$922$	89311.6	3.19015
$923$	−5402.00	−0.192643
$924$	0	0
$925$	−16193.8	−0.575620
$926$	93033.0	3.30157
$927$	0	0
$928$	4773.30	0.168848
$929$	31661.7	1.11818	0.559089	−	0.829108i	$-0.311151\pi$
$929$	31661.7	1.11818	0.559089	+	0.829108i	$0.311151\pi$
$930$	0	0
$931$	21845.0	0.769003
$932$	−20724.3	−0.728376
$933$	0	0
$934$	−37273.6	−1.30581
$935$	−788.551	−0.0275811
$936$	0	0
$937$	35010.5	1.22064	0.610322	−	0.792153i	$-0.291040\pi$
$937$	35010.5	1.22064	0.610322	+	0.792153i	$0.291040\pi$
$938$	295.776	0.0102958
$939$	0	0
$940$	7117.12	0.246952
$941$	45625.8	1.58061	0.790307	−	0.612711i	$-0.209921\pi$
$941$	45625.8	1.58061	0.790307	+	0.612711i	$0.209921\pi$
$942$	0	0
$943$	−3227.51	−0.111455
$944$	−66227.5	−2.28339
$945$	0	0
$946$	73170.2	2.51477
$947$	21508.4	0.738044	0.369022	−	0.929421i	$-0.379693\pi$
$947$	21508.4	0.738044	0.369022	+	0.929421i	$0.379693\pi$
$948$	0	0
$949$	−1559.50	−0.0533439
$950$	41579.8	1.42003
$951$	0	0
$952$	3046.52	0.103717
$953$	−35686.7	−1.21302	−0.606509	−	0.795076i	$-0.707431\pi$
$953$	−35686.7	−1.21302	−0.606509	+	0.795076i	$0.707431\pi$
$954$	0	0
$955$	−2465.34	−0.0835355
$956$	−85744.0	−2.90079
$957$	0	0
$958$	93357.8	3.14849
$959$	498.486	0.0167851
$960$	0	0
$961$	−28572.2	−0.959088
$962$	−5288.17	−0.177232
$963$	0	0
$964$	−116133.	−3.88006
$965$	−2000.12	−0.0667215
$966$	0	0
$967$	−3731.33	−0.124086	−0.0620432	−	0.998073i	$-0.519762\pi$
$967$	−3731.33	−0.124086	−0.0620432	+	0.998073i	$0.519762\pi$
$968$	66262.5	2.20016
$969$	0	0
$970$	1691.13	0.0559782
$971$	−17645.1	−0.583171	−0.291585	−	0.956545i	$-0.594183\pi$
$971$	−17645.1	−0.583171	−0.291585	+	0.956545i	$0.594183\pi$
$972$	0	0
$973$	7917.35	0.260862
$974$	51348.7	1.68924
$975$	0	0
$976$	33206.7	1.08906
$977$	−24941.2	−0.816723	−0.408362	−	0.912820i	$-0.633900\pi$
$977$	−24941.2	−0.816723	−0.408362	+	0.912820i	$0.633900\pi$
$978$	0	0
$979$	−59152.1	−1.93106
$980$	5039.81	0.164276
$981$	0	0
$982$	6277.85	0.204006
$983$	22506.2	0.730252	0.365126	−	0.930958i	$-0.381026\pi$
$983$	22506.2	0.730252	0.365126	+	0.930958i	$0.381026\pi$
$984$	0	0
$985$	−1125.44	−0.0364057
$986$	3531.62	0.114066
$987$	0	0
$988$	9289.07	0.299114
$989$	−50034.2	−1.60869
$990$	0	0
$991$	32694.1	1.04799	0.523997	−	0.851720i	$-0.324440\pi$
$991$	32694.1	1.04799	0.523997	+	0.851720i	$0.324440\pi$
$992$	−4036.88	−0.129205
$993$	0	0
$994$	12878.3	0.410941
$995$	4246.32	0.135294
$996$	0	0
$997$	18248.8	0.579686	0.289843	−	0.957074i	$-0.406397\pi$
$997$	18248.8	0.579686	0.289843	+	0.957074i	$0.406397\pi$
$998$	352.428	0.0111783
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.g.1.3		3
3.2	odd	2		17.4.a.b.1.1	✓	3
4.3	odd	2		2448.4.a.bi.1.2		3
12.11	even	2		272.4.a.h.1.1		3
15.2	even	4		425.4.b.f.324.1		6
15.8	even	4		425.4.b.f.324.6		6
15.14	odd	2		425.4.a.g.1.3		3
21.20	even	2		833.4.a.d.1.1		3
24.5	odd	2		1088.4.a.v.1.1		3
24.11	even	2		1088.4.a.x.1.3		3
33.32	even	2		2057.4.a.e.1.3		3
51.38	odd	4		289.4.b.b.288.5		6
51.47	odd	4		289.4.b.b.288.6		6
51.50	odd	2		289.4.a.b.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.a.b.1.1	✓	3	3.2	odd	2
153.4.a.g.1.3		3	1.1	even	1	trivial
272.4.a.h.1.1		3	12.11	even	2
289.4.a.b.1.1		3	51.50	odd	2
289.4.b.b.288.5		6	51.38	odd	4
289.4.b.b.288.6		6	51.47	odd	4
425.4.a.g.1.3		3	15.14	odd	2
425.4.b.f.324.1		6	15.2	even	4
425.4.b.f.324.6		6	15.8	even	4
833.4.a.d.1.1		3	21.20	even	2
1088.4.a.v.1.1		3	24.5	odd	2
1088.4.a.x.1.3		3	24.11	even	2
2057.4.a.e.1.3		3	33.32	even	2
2448.4.a.bi.1.2		3	4.3	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:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

\(f(q)\)	\(=\)	\(q+5.03251 q^{2} +17.3261 q^{4} -0.885690 q^{5} +3.81828 q^{7} +46.9339 q^{8} +O(q^{10})\)	\(q+5.03251 q^{2} +17.3261 q^{4} -0.885690 q^{5} +3.81828 q^{7} +46.9339 q^{8} -4.45724 q^{10} +52.3720 q^{11} -8.06025 q^{13} +19.2156 q^{14} +97.5862 q^{16} +17.0000 q^{17} -66.5154 q^{19} -15.3456 q^{20} +263.563 q^{22} -180.226 q^{23} -124.216 q^{25} -40.5633 q^{26} +66.1562 q^{28} +41.2800 q^{29} -34.9114 q^{31} +115.632 q^{32} +85.5527 q^{34} -3.38182 q^{35} +130.368 q^{37} -334.739 q^{38} -41.5689 q^{40} +17.9081 q^{41} +277.620 q^{43} +907.405 q^{44} -906.987 q^{46} -463.789 q^{47} -328.421 q^{49} -625.116 q^{50} -139.653 q^{52} +329.944 q^{53} -46.3853 q^{55} +179.207 q^{56} +207.742 q^{58} -678.656 q^{59} +340.280 q^{61} -175.692 q^{62} -198.770 q^{64} +7.13888 q^{65} +15.3925 q^{67} +294.545 q^{68} -17.0190 q^{70} +670.203 q^{71} +193.480 q^{73} +656.080 q^{74} -1152.46 q^{76} +199.971 q^{77} +1080.15 q^{79} -86.4311 q^{80} +90.1229 q^{82} +865.668 q^{83} -15.0567 q^{85} +1397.13 q^{86} +2458.02 q^{88} -1129.46 q^{89} -30.7763 q^{91} -3122.61 q^{92} -2334.02 q^{94} +58.9120 q^{95} -379.412 q^{97} -1652.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10}) \) 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8	\( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100}) \) 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 - 56 * q^10 + 28 * q^11 + 30 * q^13 - 92 * q^14 + 137 * q^16 + 51 * q^17 + 80 * q^19 + 168 * q^20 + 286 * q^22 - 142 * q^23 - 223 * q^25 - 26 * q^26 + 476 * q^28 + 456 * q^29 + 230 * q^31 + 71 * q^32 - 17 * q^34 + 332 * q^35 + 356 * q^37 - 724 * q^38 - 424 * q^40 + 294 * q^41 + 556 * q^43 + 1122 * q^44 - 704 * q^46 - 640 * q^47 - 269 * q^49 - 547 * q^50 - 774 * q^52 - 302 * q^53 + 76 * q^55 - 684 * q^56 - 1304 * q^58 - 636 * q^59 - 84 * q^61 - 508 * q^62 - 919 * q^64 - 408 * q^65 + 1008 * q^67 + 425 * q^68 - 1504 * q^70 + 402 * q^71 + 838 * q^73 - 836 * q^74 - 908 * q^76 + 504 * q^77 - 594 * q^79 + 40 * q^80 + 358 * q^82 + 2396 * q^83 + 136 * q^85 + 1264 * q^86 + 1838 * q^88 + 170 * q^89 - 1016 * q^91 - 4896 * q^92 - 2016 * q^94 + 472 * q^95 - 270 * q^97 - 2857 * q^98

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