LMFDB - Embedded newform 153.4.a.h.1.1

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.1506848.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 18 $ x^4 - 17*x^2 + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.06515$ 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)

$f(q)$	$=$	$q-5.56787 q^{2} +23.0012 q^{4} -5.10214 q^{5} -6.75787 q^{7} -83.5248 q^{8} +O(q^{10})$	\(q-5.56787 q^{2} +23.0012 q^{4} -5.10214 q^{5} -6.75787 q^{7} -83.5248 q^{8} +28.4081 q^{10} +53.8648 q^{11} -45.8539 q^{13} +37.6269 q^{14} +281.046 q^{16} +17.0000 q^{17} +93.8563 q^{19} -117.355 q^{20} -299.913 q^{22} -40.2877 q^{23} -98.9681 q^{25} +255.309 q^{26} -155.439 q^{28} -224.013 q^{29} -235.723 q^{31} -896.629 q^{32} -94.6538 q^{34} +34.4796 q^{35} -203.863 q^{37} -522.580 q^{38} +426.156 q^{40} +245.951 q^{41} +120.165 q^{43} +1238.96 q^{44} +224.317 q^{46} -56.5390 q^{47} -297.331 q^{49} +551.042 q^{50} -1054.70 q^{52} -177.379 q^{53} -274.826 q^{55} +564.449 q^{56} +1247.27 q^{58} -739.853 q^{59} -895.536 q^{61} +1312.48 q^{62} +2743.95 q^{64} +233.953 q^{65} -182.400 q^{67} +391.020 q^{68} -191.978 q^{70} -796.900 q^{71} +764.726 q^{73} +1135.08 q^{74} +2158.81 q^{76} -364.011 q^{77} +613.228 q^{79} -1433.94 q^{80} -1369.42 q^{82} -289.957 q^{83} -86.7364 q^{85} -669.063 q^{86} -4499.05 q^{88} +516.348 q^{89} +309.875 q^{91} -926.665 q^{92} +314.802 q^{94} -478.869 q^{95} +643.270 q^{97} +1655.50 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8} - 2 q^{10} - 50 q^{11} + 26 q^{13} - 80 q^{14} + 138 q^{16} + 68 q^{17} + 34 q^{19} - 312 q^{20} - 254 q^{22} - 382 q^{23} + 138 q^{25} + 22 q^{26} + 52 q^{28} - 540 q^{29} - 356 q^{31} - 730 q^{32} - 68 q^{34} - 304 q^{35} - 404 q^{37} - 298 q^{38} + 332 q^{40} + 114 q^{41} + 570 q^{43} + 1368 q^{44} - 290 q^{46} - 496 q^{47} - 224 q^{49} + 1862 q^{50} - 1012 q^{52} - 92 q^{53} - 482 q^{55} + 1428 q^{56} + 1324 q^{58} + 48 q^{59} - 1036 q^{61} + 2564 q^{62} + 2898 q^{64} + 342 q^{65} + 812 q^{67} + 442 q^{68} + 152 q^{70} - 1044 q^{71} - 1212 q^{73} + 1444 q^{74} + 2268 q^{76} + 564 q^{77} + 488 q^{79} + 1000 q^{80} - 938 q^{82} - 1708 q^{83} - 374 q^{85} + 2446 q^{86} - 3868 q^{88} + 8 q^{89} + 716 q^{91} - 1356 q^{92} - 1224 q^{94} - 1010 q^{95} - 76 q^{97} + 1472 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8 - 2 * q^10 - 50 * q^11 + 26 * q^13 - 80 * q^14 + 138 * q^16 + 68 * q^17 + 34 * q^19 - 312 * q^20 - 254 * q^22 - 382 * q^23 + 138 * q^25 + 22 * q^26 + 52 * q^28 - 540 * q^29 - 356 * q^31 - 730 * q^32 - 68 * q^34 - 304 * q^35 - 404 * q^37 - 298 * q^38 + 332 * q^40 + 114 * q^41 + 570 * q^43 + 1368 * q^44 - 290 * q^46 - 496 * q^47 - 224 * q^49 + 1862 * q^50 - 1012 * q^52 - 92 * q^53 - 482 * q^55 + 1428 * q^56 + 1324 * q^58 + 48 * q^59 - 1036 * q^61 + 2564 * q^62 + 2898 * q^64 + 342 * q^65 + 812 * q^67 + 442 * q^68 + 152 * q^70 - 1044 * q^71 - 1212 * q^73 + 1444 * q^74 + 2268 * q^76 + 564 * q^77 + 488 * q^79 + 1000 * q^80 - 938 * q^82 - 1708 * q^83 - 374 * q^85 + 2446 * q^86 - 3868 * q^88 + 8 * q^89 + 716 * q^91 - 1356 * q^92 - 1224 * q^94 - 1010 * q^95 - 76 * q^97 + 1472 * 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 $

$2$	−5.56787	−1.96854	−0.984270	−	0.176670i	$-0.943467\pi$
$2$	−5.56787	−1.96854	−0.984270	+	0.176670i	$0.943467\pi$
$3$	0	0
$3$	0	0
$4$	23.0012	2.87515
$5$	−5.10214	−0.456350	−0.228175	−	0.973620i	$-0.573276\pi$
$5$	−5.10214	−0.456350	−0.228175	+	0.973620i	$0.573276\pi$
$6$	0	0
$7$	−6.75787	−0.364891	−0.182445	−	0.983216i	$-0.558401\pi$
$7$	−6.75787	−0.364891	−0.182445	+	0.983216i	$0.558401\pi$
$8$	−83.5248	−3.69131
$9$	0	0
$10$	28.4081	0.898343
$11$	53.8648	1.47644	0.738221	−	0.674559i	$-0.235666\pi$
$11$	53.8648	1.47644	0.738221	+	0.674559i	$0.235666\pi$
$12$	0	0
$13$	−45.8539	−0.978276	−0.489138	−	0.872206i	$-0.662689\pi$
$13$	−45.8539	−0.978276	−0.489138	+	0.872206i	$0.662689\pi$
$14$	37.6269	0.718302
$15$	0	0
$16$	281.046	4.39134
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	93.8563	1.13327	0.566635	−	0.823969i	$-0.308245\pi$
$19$	93.8563	1.13327	0.566635	+	0.823969i	$0.308245\pi$
$20$	−117.355	−1.31207
$21$	0	0
$22$	−299.913	−2.90643
$23$	−40.2877	−0.365242	−0.182621	−	0.983183i	$-0.558458\pi$
$23$	−40.2877	−0.365242	−0.182621	+	0.983183i	$0.558458\pi$
$24$	0	0
$25$	−98.9681	−0.791745
$26$	255.309	1.92578
$27$	0	0
$28$	−155.439	−1.04912
$29$	−224.013	−1.43442	−0.717209	−	0.696858i	$-0.754581\pi$
$29$	−224.013	−1.43442	−0.717209	+	0.696858i	$0.754581\pi$
$30$	0	0
$31$	−235.723	−1.36572	−0.682858	−	0.730551i	$-0.739263\pi$
$31$	−235.723	−1.36572	−0.682858	+	0.730551i	$0.739263\pi$
$32$	−896.629	−4.95322
$33$	0	0
$34$	−94.6538	−0.477441
$35$	34.4796	0.166518
$36$	0	0
$37$	−203.863	−0.905808	−0.452904	−	0.891559i	$-0.649612\pi$
$37$	−203.863	−0.905808	−0.452904	+	0.891559i	$0.649612\pi$
$38$	−522.580	−2.23089
$39$	0	0
$40$	426.156	1.68453
$41$	245.951	0.936855	0.468428	−	0.883502i	$-0.344821\pi$
$41$	245.951	0.936855	0.468428	+	0.883502i	$0.344821\pi$
$42$	0	0
$43$	120.165	0.426162	0.213081	−	0.977034i	$-0.431650\pi$
$43$	120.165	0.426162	0.213081	+	0.977034i	$0.431650\pi$
$44$	1238.96	4.24499
$45$	0	0
$46$	224.317	0.718993
$47$	−56.5390	−0.175469	−0.0877347	−	0.996144i	$-0.527963\pi$
$47$	−56.5390	−0.175469	−0.0877347	+	0.996144i	$0.527963\pi$
$48$	0	0
$49$	−297.331	−0.866855
$50$	551.042	1.55858
$51$	0	0
$52$	−1054.70	−2.81269
$53$	−177.379	−0.459714	−0.229857	−	0.973224i	$-0.573826\pi$
$53$	−177.379	−0.459714	−0.229857	+	0.973224i	$0.573826\pi$
$54$	0	0
$55$	−274.826	−0.673774
$56$	564.449	1.34692
$57$	0	0
$58$	1247.27	2.82371
$59$	−739.853	−1.63255	−0.816277	−	0.577661i	$-0.803965\pi$
$59$	−739.853	−1.63255	−0.816277	+	0.577661i	$0.803965\pi$
$60$	0	0
$61$	−895.536	−1.87970	−0.939849	−	0.341590i	$-0.889035\pi$
$61$	−895.536	−1.87970	−0.939849	+	0.341590i	$0.889035\pi$
$62$	1312.48	2.68847
$63$	0	0
$64$	2743.95	5.35927
$65$	233.953	0.446436
$66$	0	0
$67$	−182.400	−0.332593	−0.166297	−	0.986076i	$-0.553181\pi$
$67$	−182.400	−0.332593	−0.166297	+	0.986076i	$0.553181\pi$
$68$	391.020	0.697326
$69$	0	0
$70$	−191.978	−0.327797
$71$	−796.900	−1.33204	−0.666019	−	0.745935i	$-0.732003\pi$
$71$	−796.900	−1.33204	−0.666019	+	0.745935i	$0.732003\pi$
$72$	0	0
$73$	764.726	1.22609	0.613044	−	0.790049i	$-0.289945\pi$
$73$	764.726	1.22609	0.613044	+	0.790049i	$0.289945\pi$
$74$	1135.08	1.78312
$75$	0	0
$76$	2158.81	3.25832
$77$	−364.011	−0.538740
$78$	0	0
$79$	613.228	0.873336	0.436668	−	0.899623i	$-0.356158\pi$
$79$	613.228	0.873336	0.436668	+	0.899623i	$0.356158\pi$
$80$	−1433.94	−2.00399
$81$	0	0
$82$	−1369.42	−1.84424
$83$	−289.957	−0.383457	−0.191729	−	0.981448i	$-0.561409\pi$
$83$	−289.957	−0.383457	−0.191729	+	0.981448i	$0.561409\pi$
$84$	0	0
$85$	−86.7364	−0.110681
$86$	−669.063	−0.838918
$87$	0	0
$88$	−4499.05	−5.45000
$89$	516.348	0.614976	0.307488	−	0.951552i	$-0.400512\pi$
$89$	516.348	0.614976	0.307488	+	0.951552i	$0.400512\pi$
$90$	0	0
$91$	309.875	0.356964
$92$	−926.665	−1.05013
$93$	0	0
$94$	314.802	0.345418
$95$	−478.869	−0.517167
$96$	0	0
$97$	643.270	0.673342	0.336671	−	0.941622i	$-0.390699\pi$
$97$	643.270	0.673342	0.336671	+	0.941622i	$0.390699\pi$
$98$	1655.50	1.70644
$99$	0	0
$100$	−2276.39	−2.27639
$101$	−571.974	−0.563500	−0.281750	−	0.959488i	$-0.590915\pi$
$101$	−571.974	−0.563500	−0.281750	+	0.959488i	$0.590915\pi$
$102$	0	0
$103$	−1144.95	−1.09529	−0.547646	−	0.836710i	$-0.684476\pi$
$103$	−1144.95	−1.09529	−0.547646	+	0.836710i	$0.684476\pi$
$104$	3829.94	3.61112
$105$	0	0
$106$	987.622	0.904966
$107$	611.920	0.552864	0.276432	−	0.961033i	$-0.410848\pi$
$107$	611.920	0.552864	0.276432	+	0.961033i	$0.410848\pi$
$108$	0	0
$109$	777.358	0.683095	0.341547	−	0.939865i	$-0.389049\pi$
$109$	777.358	0.683095	0.341547	+	0.939865i	$0.389049\pi$
$110$	1530.20	1.32635
$111$	0	0
$112$	−1899.27	−1.60236
$113$	−1431.52	−1.19174	−0.595869	−	0.803082i	$-0.703192\pi$
$113$	−1431.52	−1.19174	−0.595869	+	0.803082i	$0.703192\pi$
$114$	0	0
$115$	205.553	0.166678
$116$	−5152.56	−4.12417
$117$	0	0
$118$	4119.41	3.21375
$119$	−114.884	−0.0884989
$120$	0	0
$121$	1570.42	1.17988
$122$	4986.23	3.70026
$123$	0	0
$124$	−5421.92	−3.92664
$125$	1142.72	0.817662
$126$	0	0
$127$	840.643	0.587362	0.293681	−	0.955903i	$-0.405120\pi$
$127$	840.643	0.587362	0.293681	+	0.955903i	$0.405120\pi$
$128$	−8104.93	−5.59673
$129$	0	0
$130$	−1302.62	−0.878827
$131$	689.180	0.459648	0.229824	−	0.973232i	$-0.426185\pi$
$131$	689.180	0.459648	0.229824	+	0.973232i	$0.426185\pi$
$132$	0	0
$133$	−634.269	−0.413519
$134$	1015.58	0.654723
$135$	0	0
$136$	−1419.92	−0.895274
$137$	326.172	0.203407	0.101703	−	0.994815i	$-0.467571\pi$
$137$	326.172	0.203407	0.101703	+	0.994815i	$0.467571\pi$
$138$	0	0
$139$	−982.101	−0.599286	−0.299643	−	0.954051i	$-0.596867\pi$
$139$	−982.101	−0.599286	−0.299643	+	0.954051i	$0.596867\pi$
$140$	793.073	0.478763
$141$	0	0
$142$	4437.04	2.62217
$143$	−2469.91	−1.44437
$144$	0	0
$145$	1142.95	0.654596
$146$	−4257.90	−2.41360
$147$	0	0
$148$	−4689.10	−2.60433
$149$	134.589	0.0739999	0.0370000	−	0.999315i	$-0.488220\pi$
$149$	134.589	0.0739999	0.0370000	+	0.999315i	$0.488220\pi$
$150$	0	0
$151$	2026.89	1.09236	0.546178	−	0.837669i	$-0.316082\pi$
$151$	2026.89	1.09236	0.546178	+	0.837669i	$0.316082\pi$
$152$	−7839.33	−4.18325
$153$	0	0
$154$	2026.77	1.06053
$155$	1202.69	0.623244
$156$	0	0
$157$	1620.41	0.823711	0.411856	−	0.911249i	$-0.364881\pi$
$157$	1620.41	0.823711	0.411856	+	0.911249i	$0.364881\pi$
$158$	−3414.38	−1.71920
$159$	0	0
$160$	4574.73	2.26040
$161$	272.259	0.133273
$162$	0	0
$163$	2475.25	1.18942	0.594712	−	0.803939i	$-0.297266\pi$
$163$	2475.25	1.18942	0.594712	+	0.803939i	$0.297266\pi$
$164$	5657.17	2.69360
$165$	0	0
$166$	1614.44	0.754851
$167$	2.13158	0.000987705 0	0.000493852	−	1.00000i	$-0.499843\pi$
$167$	2.13158	0.000987705 0	0.000493852		1.00000i	$0.499843\pi$
$168$	0	0
$169$	−94.4169	−0.0429754
$170$	482.937	0.217880
$171$	0	0
$172$	2763.94	1.22528
$173$	949.133	0.417117	0.208559	−	0.978010i	$-0.433123\pi$
$173$	949.133	0.417117	0.208559	+	0.978010i	$0.433123\pi$
$174$	0	0
$175$	668.813	0.288900
$176$	15138.5	6.48356
$177$	0	0
$178$	−2874.96	−1.21060
$179$	−2855.87	−1.19250	−0.596250	−	0.802799i	$-0.703343\pi$
$179$	−2855.87	−1.19250	−0.596250	+	0.802799i	$0.703343\pi$
$180$	0	0
$181$	−3017.20	−1.23904	−0.619522	−	0.784980i	$-0.712673\pi$
$181$	−3017.20	−1.23904	−0.619522	+	0.784980i	$0.712673\pi$
$182$	−1725.34	−0.702698
$183$	0	0
$184$	3365.02	1.34822
$185$	1040.14	0.413365
$186$	0	0
$187$	915.702	0.358090
$188$	−1300.46	−0.504501
$189$	0	0
$190$	2666.28	1.01806
$191$	−559.454	−0.211941	−0.105970	−	0.994369i	$-0.533795\pi$
$191$	−559.454	−0.211941	−0.105970	+	0.994369i	$0.533795\pi$
$192$	0	0
$193$	−2942.15	−1.09731	−0.548654	−	0.836049i	$-0.684860\pi$
$193$	−2942.15	−1.09731	−0.548654	+	0.836049i	$0.684860\pi$
$194$	−3581.64	−1.32550
$195$	0	0
$196$	−6838.98	−2.49234
$197$	626.615	0.226622	0.113311	−	0.993560i	$-0.463854\pi$
$197$	626.615	0.226622	0.113311	+	0.993560i	$0.463854\pi$
$198$	0	0
$199$	1007.16	0.358771	0.179386	−	0.983779i	$-0.442589\pi$
$199$	1007.16	0.358771	0.179386	+	0.983779i	$0.442589\pi$
$200$	8266.29	2.92258
$201$	0	0
$202$	3184.68	1.10927
$203$	1513.85	0.523406
$204$	0	0
$205$	−1254.88	−0.427534
$206$	6374.92	2.15612
$207$	0	0
$208$	−12887.1	−4.29594
$209$	5055.56	1.67321
$210$	0	0
$211$	−2049.32	−0.668630	−0.334315	−	0.942461i	$-0.608505\pi$
$211$	−2049.32	−0.668630	−0.334315	+	0.942461i	$0.608505\pi$
$212$	−4079.93	−1.32175
$213$	0	0
$214$	−3407.09	−1.08834
$215$	−613.099	−0.194479
$216$	0	0
$217$	1592.99	0.498337
$218$	−4328.23	−1.34470
$219$	0	0
$220$	−6321.33	−1.93720
$221$	−779.517	−0.237267
$222$	0	0
$223$	−6073.88	−1.82393	−0.911966	−	0.410266i	$-0.865436\pi$
$223$	−6073.88	−1.82393	−0.911966	+	0.410266i	$0.865436\pi$
$224$	6059.30	1.80738
$225$	0	0
$226$	7970.54	2.34598
$227$	−3261.14	−0.953522	−0.476761	−	0.879033i	$-0.658189\pi$
$227$	−3261.14	−0.953522	−0.476761	+	0.879033i	$0.658189\pi$
$228$	0	0
$229$	1470.71	0.424399	0.212200	−	0.977226i	$-0.431937\pi$
$229$	1470.71	0.424399	0.212200	+	0.977226i	$0.431937\pi$
$230$	−1144.50	−0.328112
$231$	0	0
$232$	18710.6	5.29488
$233$	−4300.23	−1.20909	−0.604543	−	0.796572i	$-0.706644\pi$
$233$	−4300.23	−1.20909	−0.604543	+	0.796572i	$0.706644\pi$
$234$	0	0
$235$	288.470	0.0800754
$236$	−17017.5	−4.69384
$237$	0	0
$238$	639.658	0.174214
$239$	1140.66	0.308715	0.154358	−	0.988015i	$-0.450669\pi$
$239$	1140.66	0.308715	0.154358	+	0.988015i	$0.450669\pi$
$240$	0	0
$241$	724.579	0.193669	0.0968345	−	0.995300i	$-0.469128\pi$
$241$	724.579	0.193669	0.0968345	+	0.995300i	$0.469128\pi$
$242$	−8743.90	−2.32264
$243$	0	0
$244$	−20598.4	−5.40442
$245$	1517.03	0.395589
$246$	0	0
$247$	−4303.68	−1.10865
$248$	19688.8	5.04128
$249$	0	0
$250$	−6362.51	−1.60960
$251$	3882.55	0.976352	0.488176	−	0.872745i	$-0.337662\pi$
$251$	3882.55	0.976352	0.488176	+	0.872745i	$0.337662\pi$
$252$	0	0
$253$	−2170.09	−0.539258
$254$	−4680.59	−1.15625
$255$	0	0
$256$	23175.6	5.65811
$257$	2834.39	0.687954	0.343977	−	0.938978i	$-0.388226\pi$
$257$	2834.39	0.687954	0.343977	+	0.938978i	$0.388226\pi$
$258$	0	0
$259$	1377.68	0.330521
$260$	5381.21	1.28357
$261$	0	0
$262$	−3837.26	−0.904836
$263$	−3013.16	−0.706462	−0.353231	−	0.935536i	$-0.614917\pi$
$263$	−3013.16	−0.706462	−0.353231	+	0.935536i	$0.614917\pi$
$264$	0	0
$265$	905.012	0.209790
$266$	3531.53	0.814029
$267$	0	0
$268$	−4195.42	−0.956255
$269$	6292.44	1.42623	0.713117	−	0.701045i	$-0.247283\pi$
$269$	6292.44	1.42623	0.713117	+	0.701045i	$0.247283\pi$
$270$	0	0
$271$	−1349.68	−0.302536	−0.151268	−	0.988493i	$-0.548336\pi$
$271$	−1349.68	−0.302536	−0.151268	+	0.988493i	$0.548336\pi$
$272$	4777.78	1.06506
$273$	0	0
$274$	−1816.08	−0.400414
$275$	−5330.90	−1.16897
$276$	0	0
$277$	−722.744	−0.156771	−0.0783854	−	0.996923i	$-0.524976\pi$
$277$	−722.744	−0.156771	−0.0783854	+	0.996923i	$0.524976\pi$
$278$	5468.21	1.17972
$279$	0	0
$280$	−2879.90	−0.614668
$281$	5792.27	1.22967	0.614836	−	0.788655i	$-0.289222\pi$
$281$	5792.27	1.22967	0.614836	+	0.788655i	$0.289222\pi$
$282$	0	0
$283$	−2389.23	−0.501856	−0.250928	−	0.968006i	$-0.580736\pi$
$283$	−2389.23	−0.501856	−0.250928	+	0.968006i	$0.580736\pi$
$284$	−18329.7	−3.82981
$285$	0	0
$286$	13752.2	2.84330
$287$	−1662.10	−0.341850
$288$	0	0
$289$	289.000	0.0588235
$290$	−6363.77	−1.28860
$291$	0	0
$292$	17589.6	3.52519
$293$	6003.90	1.19711	0.598553	−	0.801084i	$-0.295743\pi$
$293$	6003.90	1.19711	0.598553	+	0.801084i	$0.295743\pi$
$294$	0	0
$295$	3774.84	0.745015
$296$	17027.6	3.34362
$297$	0	0
$298$	−749.376	−0.145672
$299$	1847.35	0.357307
$300$	0	0
$301$	−812.059	−0.155503
$302$	−11285.4	−2.15035
$303$	0	0
$304$	26377.9	4.97657
$305$	4569.15	0.857800
$306$	0	0
$307$	4120.28	0.765983	0.382991	−	0.923752i	$-0.374894\pi$
$307$	4120.28	0.765983	0.382991	+	0.923752i	$0.374894\pi$
$308$	−8372.70	−1.54896
$309$	0	0
$310$	−6696.45	−1.22688
$311$	−6224.78	−1.13497	−0.567484	−	0.823384i	$-0.692083\pi$
$311$	−6224.78	−1.13497	−0.567484	+	0.823384i	$0.692083\pi$
$312$	0	0
$313$	−5451.10	−0.984391	−0.492196	−	0.870485i	$-0.663805\pi$
$313$	−5451.10	−0.984391	−0.492196	+	0.870485i	$0.663805\pi$
$314$	−9022.23	−1.62151
$315$	0	0
$316$	14105.0	2.51097
$317$	9906.47	1.75521	0.877607	−	0.479381i	$-0.159139\pi$
$317$	9906.47	1.75521	0.877607	+	0.479381i	$0.159139\pi$
$318$	0	0
$319$	−12066.4	−2.11783
$320$	−14000.0	−2.44570
$321$	0	0
$322$	−1515.90	−0.262354
$323$	1595.56	0.274858
$324$	0	0
$325$	4538.08	0.774545
$326$	−13781.9	−2.34143
$327$	0	0
$328$	−20543.0	−3.45822
$329$	382.083	0.0640271
$330$	0	0
$331$	−9684.59	−1.60820	−0.804099	−	0.594496i	$-0.797352\pi$
$331$	−9684.59	−1.60820	−0.804099	+	0.594496i	$0.797352\pi$
$332$	−6669.37	−1.10250
$333$	0	0
$334$	−11.8684	−0.00194434
$335$	930.632	0.151779
$336$	0	0
$337$	−7474.88	−1.20826	−0.604128	−	0.796887i	$-0.706479\pi$
$337$	−7474.88	−1.20826	−0.604128	+	0.796887i	$0.706479\pi$
$338$	525.701	0.0845988
$339$	0	0
$340$	−1995.04	−0.318225
$341$	−12697.2	−2.01640
$342$	0	0
$343$	4327.27	0.681198
$344$	−10036.8	−1.57310
$345$	0	0
$346$	−5284.65	−0.821112
$347$	−10849.0	−1.67841	−0.839203	−	0.543818i	$-0.816978\pi$
$347$	−10849.0	−1.67841	−0.839203	+	0.543818i	$0.816978\pi$
$348$	0	0
$349$	8697.09	1.33394	0.666969	−	0.745085i	$-0.267591\pi$
$349$	8697.09	1.33394	0.666969	+	0.745085i	$0.267591\pi$
$350$	−3723.87	−0.568712
$351$	0	0
$352$	−48296.8	−7.31314
$353$	7811.69	1.17783	0.588916	−	0.808194i	$-0.299555\pi$
$353$	7811.69	1.17783	0.588916	+	0.808194i	$0.299555\pi$
$354$	0	0
$355$	4065.90	0.607875
$356$	11876.6	1.76815
$357$	0	0
$358$	15901.1	2.34748
$359$	6585.02	0.968089	0.484045	−	0.875043i	$-0.339167\pi$
$359$	6585.02	0.968089	0.484045	+	0.875043i	$0.339167\pi$
$360$	0	0
$361$	1950.01	0.284300
$362$	16799.4	2.43911
$363$	0	0
$364$	7127.49	1.02632
$365$	−3901.74	−0.559525
$366$	0	0
$367$	3808.98	0.541763	0.270881	−	0.962613i	$-0.412685\pi$
$367$	3808.98	0.541763	0.270881	+	0.962613i	$0.412685\pi$
$368$	−11322.7	−1.60390
$369$	0	0
$370$	−5791.36	−0.813726
$371$	1198.70	0.167745
$372$	0	0
$373$	−2964.87	−0.411568	−0.205784	−	0.978597i	$-0.565974\pi$
$373$	−2964.87	−0.411568	−0.205784	+	0.978597i	$0.565974\pi$
$374$	−5098.51	−0.704914
$375$	0	0
$376$	4722.41	0.647712
$377$	10271.9	1.40326
$378$	0	0
$379$	−186.019	−0.0252115	−0.0126058	−	0.999921i	$-0.504013\pi$
$379$	−186.019	−0.0252115	−0.0126058	+	0.999921i	$0.504013\pi$
$380$	−11014.6	−1.48693
$381$	0	0
$382$	3114.97	0.417214
$383$	9369.11	1.24997	0.624986	−	0.780636i	$-0.285105\pi$
$383$	9369.11	1.24997	0.624986	+	0.780636i	$0.285105\pi$
$384$	0	0
$385$	1857.24	0.245854
$386$	16381.5	2.16010
$387$	0	0
$388$	14796.0	1.93596
$389$	6790.35	0.885050	0.442525	−	0.896756i	$-0.354083\pi$
$389$	6790.35	0.885050	0.442525	+	0.896756i	$0.354083\pi$
$390$	0	0
$391$	−684.890	−0.0885841
$392$	24834.5	3.19983
$393$	0	0
$394$	−3488.91	−0.446114
$395$	−3128.78	−0.398547
$396$	0	0
$397$	8932.61	1.12926	0.564628	−	0.825345i	$-0.309020\pi$
$397$	8932.61	1.12926	0.564628	+	0.825345i	$0.309020\pi$
$398$	−5607.72	−0.706256
$399$	0	0
$400$	−27814.6	−3.47682
$401$	−6223.71	−0.775055	−0.387528	−	0.921858i	$-0.626671\pi$
$401$	−6223.71	−0.775055	−0.387528	+	0.921858i	$0.626671\pi$
$402$	0	0
$403$	10808.8	1.33605
$404$	−13156.1	−1.62015
$405$	0	0
$406$	−8428.91	−1.03034
$407$	−10981.1	−1.33737
$408$	0	0
$409$	−2127.04	−0.257153	−0.128576	−	0.991700i	$-0.541041\pi$
$409$	−2127.04	−0.257153	−0.128576	+	0.991700i	$0.541041\pi$
$410$	6986.99	0.841617
$411$	0	0
$412$	−26335.2	−3.14913
$413$	4999.83	0.595703
$414$	0	0
$415$	1479.40	0.174990
$416$	41114.0	4.84562
$417$	0	0
$418$	−28148.7	−3.29377
$419$	−6473.83	−0.754814	−0.377407	−	0.926047i	$-0.623184\pi$
$419$	−6473.83	−0.754814	−0.377407	+	0.926047i	$0.623184\pi$
$420$	0	0
$421$	3559.22	0.412032	0.206016	−	0.978549i	$-0.433950\pi$
$421$	3559.22	0.412032	0.206016	+	0.978549i	$0.433950\pi$
$422$	11410.3	1.31622
$423$	0	0
$424$	14815.5	1.69695
$425$	−1682.46	−0.192026
$426$	0	0
$427$	6051.91	0.685884
$428$	14074.9	1.58957
$429$	0	0
$430$	3413.66	0.382840
$431$	−3667.29	−0.409854	−0.204927	−	0.978777i	$-0.565696\pi$
$431$	−3667.29	−0.409854	−0.204927	+	0.978777i	$0.565696\pi$
$432$	0	0
$433$	5571.44	0.618352	0.309176	−	0.951005i	$-0.399947\pi$
$433$	5571.44	0.618352	0.309176	+	0.951005i	$0.399947\pi$
$434$	−8869.55	−0.980996
$435$	0	0
$436$	17880.2	1.96400
$437$	−3781.25	−0.413917
$438$	0	0
$439$	2492.30	0.270959	0.135479	−	0.990780i	$-0.456743\pi$
$439$	2492.30	0.270959	0.135479	+	0.990780i	$0.456743\pi$
$440$	22954.8	2.48711
$441$	0	0
$442$	4340.25	0.467069
$443$	−7142.61	−0.766040	−0.383020	−	0.923740i	$-0.625116\pi$
$443$	−7142.61	−0.766040	−0.383020	+	0.923740i	$0.625116\pi$
$444$	0	0
$445$	−2634.48	−0.280644
$446$	33818.6	3.59048
$447$	0	0
$448$	−18543.2	−1.95555
$449$	−16339.1	−1.71735	−0.858676	−	0.512519i	$-0.828712\pi$
$449$	−16339.1	−1.71735	−0.858676	+	0.512519i	$0.828712\pi$
$450$	0	0
$451$	13248.1	1.38321
$452$	−32926.8	−3.42643
$453$	0	0
$454$	18157.6	1.87705
$455$	−1581.03	−0.162900
$456$	0	0
$457$	−17360.0	−1.77695	−0.888474	−	0.458926i	$-0.848234\pi$
$457$	−17360.0	−1.77695	−0.888474	+	0.458926i	$0.848234\pi$
$458$	−8188.74	−0.835447
$459$	0	0
$460$	4727.98	0.479224
$461$	−1379.11	−0.139331	−0.0696653	−	0.997570i	$-0.522193\pi$
$461$	−1379.11	−0.139331	−0.0696653	+	0.997570i	$0.522193\pi$
$462$	0	0
$463$	8719.36	0.875212	0.437606	−	0.899167i	$-0.355826\pi$
$463$	8719.36	0.875212	0.437606	+	0.899167i	$0.355826\pi$
$464$	−62957.8	−6.29902
$465$	0	0
$466$	23943.1	2.38014
$467$	−6893.92	−0.683110	−0.341555	−	0.939862i	$-0.610954\pi$
$467$	−6893.92	−0.683110	−0.341555	+	0.939862i	$0.610954\pi$
$468$	0	0
$469$	1232.64	0.121360
$470$	−1606.16	−0.157632
$471$	0	0
$472$	61796.1	6.02626
$473$	6472.67	0.629204
$474$	0	0
$475$	−9288.79	−0.897261
$476$	−2642.46	−0.254448
$477$	0	0
$478$	−6351.03	−0.607718
$479$	−9847.31	−0.939321	−0.469661	−	0.882847i	$-0.655624\pi$
$479$	−9847.31	−0.939321	−0.469661	+	0.882847i	$0.655624\pi$
$480$	0	0
$481$	9347.92	0.886130
$482$	−4034.36	−0.381245
$483$	0	0
$484$	36121.6	3.39233
$485$	−3282.05	−0.307279
$486$	0	0
$487$	19579.8	1.82186	0.910929	−	0.412563i	$-0.135366\pi$
$487$	19579.8	1.82186	0.910929	+	0.412563i	$0.135366\pi$
$488$	74799.4	6.93855
$489$	0	0
$490$	−8446.61	−0.778733
$491$	14634.7	1.34513	0.672563	−	0.740040i	$-0.265193\pi$
$491$	14634.7	1.34513	0.672563	+	0.740040i	$0.265193\pi$
$492$	0	0
$493$	−3808.22	−0.347897
$494$	23962.4	2.18242
$495$	0	0
$496$	−66249.1	−5.99732
$497$	5385.35	0.486048
$498$	0	0
$499$	−8677.37	−0.778462	−0.389231	−	0.921140i	$-0.627259\pi$
$499$	−8677.37	−0.778462	−0.389231	+	0.921140i	$0.627259\pi$
$500$	26283.9	2.35090
$501$	0	0
$502$	−21617.5	−1.92199
$503$	−15938.1	−1.41281	−0.706404	−	0.707809i	$-0.749684\pi$
$503$	−15938.1	−1.41281	−0.706404	+	0.707809i	$0.749684\pi$
$504$	0	0
$505$	2918.29	0.257153
$506$	12082.8	1.06155
$507$	0	0
$508$	19335.8	1.68875
$509$	16219.5	1.41241	0.706204	−	0.708009i	$-0.250406\pi$
$509$	16219.5	1.41241	0.706204	+	0.708009i	$0.250406\pi$
$510$	0	0
$511$	−5167.92	−0.447388
$512$	−64199.4	−5.54148
$513$	0	0
$514$	−15781.5	−1.35426
$515$	5841.69	0.499836
$516$	0	0
$517$	−3045.46	−0.259070
$518$	−7670.74	−0.650643
$519$	0	0
$520$	−19540.9	−1.64793
$521$	3774.70	0.317414	0.158707	−	0.987326i	$-0.449267\pi$
$521$	3774.70	0.317414	0.158707	+	0.987326i	$0.449267\pi$
$522$	0	0
$523$	5396.29	0.451172	0.225586	−	0.974223i	$-0.427570\pi$
$523$	5396.29	0.451172	0.225586	+	0.974223i	$0.427570\pi$
$524$	15852.0	1.32156
$525$	0	0
$526$	16776.9	1.39070
$527$	−4007.30	−0.331235
$528$	0	0
$529$	−10543.9	−0.866598
$530$	−5038.99	−0.412981
$531$	0	0
$532$	−14588.9	−1.18893
$533$	−11277.8	−0.916503
$534$	0	0
$535$	−3122.10	−0.252299
$536$	15234.9	1.22770
$537$	0	0
$538$	−35035.5	−2.80760
$539$	−16015.7	−1.27986
$540$	0	0
$541$	2144.39	0.170415	0.0852076	−	0.996363i	$-0.472845\pi$
$541$	2144.39	0.170415	0.0852076	+	0.996363i	$0.472845\pi$
$542$	7514.86	0.595555
$543$	0	0
$544$	−15242.7	−1.20133
$545$	−3966.19	−0.311730
$546$	0	0
$547$	7207.70	0.563399	0.281699	−	0.959503i	$-0.409102\pi$
$547$	7207.70	0.563399	0.281699	+	0.959503i	$0.409102\pi$
$548$	7502.34	0.584825
$549$	0	0
$550$	29681.8	2.30116
$551$	−21025.0	−1.62558
$552$	0	0
$553$	−4144.11	−0.318672
$554$	4024.15	0.308610
$555$	0	0
$556$	−22589.5	−1.72304
$557$	9270.18	0.705189	0.352594	−	0.935776i	$-0.385300\pi$
$557$	9270.18	0.705189	0.352594	+	0.935776i	$0.385300\pi$
$558$	0	0
$559$	−5510.04	−0.416905
$560$	9690.35	0.731236
$561$	0	0
$562$	−32250.6	−2.42066
$563$	24102.0	1.80422	0.902112	−	0.431502i	$-0.142016\pi$
$563$	24102.0	1.80422	0.902112	+	0.431502i	$0.142016\pi$
$564$	0	0
$565$	7303.84	0.543849
$566$	13302.9	0.987923
$567$	0	0
$568$	66560.9	4.91696
$569$	6999.68	0.515715	0.257857	−	0.966183i	$-0.416983\pi$
$569$	6999.68	0.515715	0.257857	+	0.966183i	$0.416983\pi$
$570$	0	0
$571$	−7444.31	−0.545595	−0.272798	−	0.962071i	$-0.587949\pi$
$571$	−7444.31	−0.545595	−0.272798	+	0.962071i	$0.587949\pi$
$572$	−56811.0	−4.15278
$573$	0	0
$574$	9254.38	0.672945
$575$	3987.20	0.289178
$576$	0	0
$577$	12243.8	0.883392	0.441696	−	0.897165i	$-0.354377\pi$
$577$	12243.8	0.883392	0.441696	+	0.897165i	$0.354377\pi$
$578$	−1609.12	−0.115796
$579$	0	0
$580$	26289.1	1.88206
$581$	1959.49	0.139920
$582$	0	0
$583$	−9554.48	−0.678741
$584$	−63873.6	−4.52587
$585$	0	0
$586$	−33429.0	−2.35655
$587$	23540.0	1.65520	0.827598	−	0.561322i	$-0.189707\pi$
$587$	23540.0	1.65520	0.827598	+	0.561322i	$0.189707\pi$
$588$	0	0
$589$	−22124.1	−1.54772
$590$	−21017.8	−1.46659
$591$	0	0
$592$	−57294.9	−3.97771
$593$	−17039.8	−1.18000	−0.589999	−	0.807404i	$-0.700872\pi$
$593$	−17039.8	−1.18000	−0.589999	+	0.807404i	$0.700872\pi$
$594$	0	0
$595$	586.153	0.0403865
$596$	3095.72	0.212761
$597$	0	0
$598$	−10285.8	−0.703374
$599$	−23670.3	−1.61459	−0.807297	−	0.590146i	$-0.799070\pi$
$599$	−23670.3	−1.61459	−0.807297	+	0.590146i	$0.799070\pi$
$600$	0	0
$601$	8978.86	0.609410	0.304705	−	0.952447i	$-0.401442\pi$
$601$	8978.86	0.609410	0.304705	+	0.952447i	$0.401442\pi$
$602$	4521.44	0.306113
$603$	0	0
$604$	46620.8	3.14069
$605$	−8012.51	−0.538438
$606$	0	0
$607$	−9594.38	−0.641555	−0.320778	−	0.947155i	$-0.603944\pi$
$607$	−9594.38	−0.641555	−0.320778	+	0.947155i	$0.603944\pi$
$608$	−84154.3	−5.61333
$609$	0	0
$610$	−25440.5	−1.68861
$611$	2592.53	0.171657
$612$	0	0
$613$	−7735.93	−0.509708	−0.254854	−	0.966979i	$-0.582027\pi$
$613$	−7735.93	−0.509708	−0.254854	+	0.966979i	$0.582027\pi$
$614$	−22941.2	−1.50787
$615$	0	0
$616$	30404.0	1.98865
$617$	16107.2	1.05097	0.525487	−	0.850802i	$-0.323883\pi$
$617$	16107.2	1.05097	0.525487	+	0.850802i	$0.323883\pi$
$618$	0	0
$619$	−20401.9	−1.32475	−0.662375	−	0.749172i	$-0.730451\pi$
$619$	−20401.9	−1.32475	−0.662375	+	0.749172i	$0.730451\pi$
$620$	27663.4	1.79192
$621$	0	0
$622$	34658.8	2.23423
$623$	−3489.41	−0.224399
$624$	0	0
$625$	6540.71	0.418605
$626$	30351.0	1.93781
$627$	0	0
$628$	37271.3	2.36829
$629$	−3465.67	−0.219691
$630$	0	0
$631$	−16139.6	−1.01824	−0.509118	−	0.860697i	$-0.670028\pi$
$631$	−16139.6	−1.01824	−0.509118	+	0.860697i	$0.670028\pi$
$632$	−51219.8	−3.22375
$633$	0	0
$634$	−55158.0	−3.45521
$635$	−4289.08	−0.268042
$636$	0	0
$637$	13633.8	0.848024
$638$	67184.2	4.16904
$639$	0	0
$640$	41352.5	2.55406
$641$	−14628.7	−0.901402	−0.450701	−	0.892675i	$-0.648826\pi$
$641$	−14628.7	−0.901402	−0.450701	+	0.892675i	$0.648826\pi$
$642$	0	0
$643$	−22167.5	−1.35957	−0.679784	−	0.733413i	$-0.737926\pi$
$643$	−22167.5	−1.35957	−0.679784	+	0.733413i	$0.737926\pi$
$644$	6262.28	0.383181
$645$	0	0
$646$	−8883.86	−0.541070
$647$	−4522.66	−0.274813	−0.137407	−	0.990515i	$-0.543877\pi$
$647$	−4522.66	−0.274813	−0.137407	+	0.990515i	$0.543877\pi$
$648$	0	0
$649$	−39852.0	−2.41037
$650$	−25267.4	−1.52472
$651$	0	0
$652$	56933.6	3.41978
$653$	−12357.1	−0.740537	−0.370269	−	0.928925i	$-0.620734\pi$
$653$	−12357.1	−0.740537	−0.370269	+	0.928925i	$0.620734\pi$
$654$	0	0
$655$	−3516.29	−0.209760
$656$	69123.4	4.11405
$657$	0	0
$658$	−2127.39	−0.126040
$659$	26975.4	1.59456	0.797279	−	0.603611i	$-0.206272\pi$
$659$	26975.4	1.59456	0.797279	+	0.603611i	$0.206272\pi$
$660$	0	0
$661$	23302.2	1.37118	0.685591	−	0.727987i	$-0.259544\pi$
$661$	23302.2	1.37118	0.685591	+	0.727987i	$0.259544\pi$
$662$	53922.6	3.16580
$663$	0	0
$664$	24218.6	1.41546
$665$	3236.13	0.188709
$666$	0	0
$667$	9024.95	0.523909
$668$	49.0289	0.00283980
$669$	0	0
$670$	−5181.64	−0.298782
$671$	−48237.9	−2.77526
$672$	0	0
$673$	10960.6	0.627786	0.313893	−	0.949458i	$-0.398367\pi$
$673$	10960.6	0.627786	0.313893	+	0.949458i	$0.398367\pi$
$674$	41619.2	2.37850
$675$	0	0
$676$	−2171.70	−0.123561
$677$	9374.04	0.532162	0.266081	−	0.963951i	$-0.414271\pi$
$677$	9374.04	0.532162	0.266081	+	0.963951i	$0.414271\pi$
$678$	0	0
$679$	−4347.13	−0.245696
$680$	7244.64	0.408558
$681$	0	0
$682$	70696.4	3.96936
$683$	−18017.0	−1.00937	−0.504687	−	0.863302i	$-0.668392\pi$
$683$	−18017.0	−1.00937	−0.504687	+	0.863302i	$0.668392\pi$
$684$	0	0
$685$	−1664.18	−0.0928246
$686$	−24093.7	−1.34096
$687$	0	0
$688$	33771.9	1.87142
$689$	8133.51	0.449727
$690$	0	0
$691$	−15504.9	−0.853596	−0.426798	−	0.904347i	$-0.640359\pi$
$691$	−15504.9	−0.853596	−0.426798	+	0.904347i	$0.640359\pi$
$692$	21831.2	1.19927
$693$	0	0
$694$	60406.1	3.30401
$695$	5010.82	0.273484
$696$	0	0
$697$	4181.16	0.227221
$698$	−48424.3	−2.62591
$699$	0	0
$700$	15383.5	0.830632
$701$	18519.8	0.997838	0.498919	−	0.866649i	$-0.333731\pi$
$701$	18519.8	0.997838	0.498919	+	0.866649i	$0.333731\pi$
$702$	0	0
$703$	−19133.8	−1.02652
$704$	147802.	7.91266
$705$	0	0
$706$	−43494.5	−2.31861
$707$	3865.32	0.205616
$708$	0	0
$709$	4941.88	0.261772	0.130886	−	0.991397i	$-0.458218\pi$
$709$	4941.88	0.261772	0.130886	+	0.991397i	$0.458218\pi$
$710$	−22638.4	−1.19663
$711$	0	0
$712$	−43127.9	−2.27007
$713$	9496.75	0.498816
$714$	0	0
$715$	12601.9	0.659137
$716$	−65688.4	−3.42862
$717$	0	0
$718$	−36664.6	−1.90572
$719$	18524.2	0.960827	0.480414	−	0.877042i	$-0.340487\pi$
$719$	18524.2	0.960827	0.480414	+	0.877042i	$0.340487\pi$
$720$	0	0
$721$	7737.40	0.399661
$722$	−10857.4	−0.559656
$723$	0	0
$724$	−69399.3	−3.56244
$725$	22170.1	1.13569
$726$	0	0
$727$	23425.9	1.19508	0.597538	−	0.801841i	$-0.296146\pi$
$727$	23425.9	1.19508	0.597538	+	0.801841i	$0.296146\pi$
$728$	−25882.2	−1.31766
$729$	0	0
$730$	21724.4	1.10145
$731$	2042.80	0.103360
$732$	0	0
$733$	−27185.9	−1.36990	−0.684949	−	0.728591i	$-0.740176\pi$
$733$	−27185.9	−1.36990	−0.684949	+	0.728591i	$0.740176\pi$
$734$	−21207.9	−1.06648
$735$	0	0
$736$	36123.1	1.80912
$737$	−9824.96	−0.491054
$738$	0	0
$739$	−4268.93	−0.212497	−0.106248	−	0.994340i	$-0.533884\pi$
$739$	−4268.93	−0.212497	−0.106248	+	0.994340i	$0.533884\pi$
$740$	23924.4	1.18849
$741$	0	0
$742$	−6674.22	−0.330213
$743$	13590.9	0.671065	0.335532	−	0.942029i	$-0.391084\pi$
$743$	13590.9	0.671065	0.335532	+	0.942029i	$0.391084\pi$
$744$	0	0
$745$	−686.694	−0.0337698
$746$	16508.0	0.810189
$747$	0	0
$748$	21062.3	1.02956
$749$	−4135.27	−0.201735
$750$	0	0
$751$	7011.20	0.340669	0.170334	−	0.985386i	$-0.445515\pi$
$751$	7011.20	0.340669	0.170334	+	0.985386i	$0.445515\pi$
$752$	−15890.0	−0.770546
$753$	0	0
$754$	−57192.4	−2.76237
$755$	−10341.5	−0.498496
$756$	0	0
$757$	−6380.93	−0.306366	−0.153183	−	0.988198i	$-0.548952\pi$
$757$	−6380.93	−0.306366	−0.153183	+	0.988198i	$0.548952\pi$
$758$	1035.73	0.0496299
$759$	0	0
$760$	39997.4	1.90902
$761$	−23805.2	−1.13395	−0.566975	−	0.823735i	$-0.691886\pi$
$761$	−23805.2	−1.13395	−0.566975	+	0.823735i	$0.691886\pi$
$762$	0	0
$763$	−5253.28	−0.249255
$764$	−12868.1	−0.609362
$765$	0	0
$766$	−52166.0	−2.46062
$767$	33925.2	1.59709
$768$	0	0
$769$	7076.30	0.331831	0.165916	−	0.986140i	$-0.446942\pi$
$769$	7076.30	0.331831	0.165916	+	0.986140i	$0.446942\pi$
$770$	−10340.9	−0.483973
$771$	0	0
$772$	−67673.0	−3.15493
$773$	−34178.9	−1.59034	−0.795168	−	0.606390i	$-0.792617\pi$
$773$	−34178.9	−1.59034	−0.795168	+	0.606390i	$0.792617\pi$
$774$	0	0
$775$	23329.1	1.08130
$776$	−53729.0	−2.48551
$777$	0	0
$778$	−37807.8	−1.74226
$779$	23084.0	1.06171
$780$	0	0
$781$	−42924.9	−1.96668
$782$	3813.38	0.174381
$783$	0	0
$784$	−83563.7	−3.80666
$785$	−8267.55	−0.375900
$786$	0	0
$787$	43993.1	1.99261	0.996305	−	0.0858809i	$-0.0273705\pi$
$787$	43993.1	1.99261	0.996305	+	0.0858809i	$0.0273705\pi$
$788$	14412.9	0.651572
$789$	0	0
$790$	17420.6	0.784555
$791$	9674.04	0.434854
$792$	0	0
$793$	41063.8	1.83886
$794$	−49735.6	−2.22299
$795$	0	0
$796$	23165.8	1.03152
$797$	−15592.5	−0.692993	−0.346497	−	0.938051i	$-0.612629\pi$
$797$	−15592.5	−0.692993	−0.346497	+	0.938051i	$0.612629\pi$
$798$	0	0
$799$	−961.163	−0.0425576
$800$	88737.7	3.92169
$801$	0	0
$802$	34652.8	1.52573
$803$	41191.8	1.81025
$804$	0	0
$805$	−1389.10	−0.0608192
$806$	−60182.3	−2.63006
$807$	0	0
$808$	47774.0	2.08005
$809$	−24548.0	−1.06683	−0.533413	−	0.845855i	$-0.679091\pi$
$809$	−24548.0	−1.06683	−0.533413	+	0.845855i	$0.679091\pi$
$810$	0	0
$811$	−22145.4	−0.958852	−0.479426	−	0.877582i	$-0.659155\pi$
$811$	−22145.4	−0.958852	−0.479426	+	0.877582i	$0.659155\pi$
$812$	34820.3	1.50487
$813$	0	0
$814$	61141.1	2.63267
$815$	−12629.1	−0.542794
$816$	0	0
$817$	11278.2	0.482957
$818$	11843.1	0.506216
$819$	0	0
$820$	−28863.7	−1.22922
$821$	26330.9	1.11931	0.559655	−	0.828726i	$-0.310934\pi$
$821$	26330.9	1.11931	0.559655	+	0.828726i	$0.310934\pi$
$822$	0	0
$823$	13534.7	0.573257	0.286628	−	0.958042i	$-0.407466\pi$
$823$	13534.7	0.573257	0.286628	+	0.958042i	$0.407466\pi$
$824$	95631.5	4.04306
$825$	0	0
$826$	−27838.4	−1.17267
$827$	−15037.1	−0.632275	−0.316137	−	0.948713i	$-0.602386\pi$
$827$	−15037.1	−0.632275	−0.316137	+	0.948713i	$0.602386\pi$
$828$	0	0
$829$	−7612.99	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$829$	−7612.99	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$830$	−8237.13	−0.344476
$831$	0	0
$832$	−125821.	−5.24285
$833$	−5054.63	−0.210243
$834$	0	0
$835$	−10.8756	−0.000450739 0
$836$	116284.	4.81072
$837$	0	0
$838$	36045.5	1.48588
$839$	10208.3	0.420058	0.210029	−	0.977695i	$-0.432644\pi$
$839$	10208.3	0.420058	0.210029	+	0.977695i	$0.432644\pi$
$840$	0	0
$841$	25792.7	1.05755
$842$	−19817.3	−0.811102
$843$	0	0
$844$	−47136.8	−1.92241
$845$	481.729	0.0196118
$846$	0	0
$847$	−10612.7	−0.430527
$848$	−49851.6	−2.01876
$849$	0	0
$850$	9367.71	0.378012
$851$	8213.17	0.330839
$852$	0	0
$853$	25197.2	1.01141	0.505707	−	0.862706i	$-0.331232\pi$
$853$	25197.2	1.01141	0.505707	+	0.862706i	$0.331232\pi$
$854$	−33696.3	−1.35019
$855$	0	0
$856$	−51110.5	−2.04079
$857$	−26090.3	−1.03994	−0.519970	−	0.854184i	$-0.674057\pi$
$857$	−26090.3	−1.03994	−0.519970	+	0.854184i	$0.674057\pi$
$858$	0	0
$859$	2004.36	0.0796132	0.0398066	−	0.999207i	$-0.487326\pi$
$859$	2004.36	0.0796132	0.0398066	+	0.999207i	$0.487326\pi$
$860$	−14102.0	−0.559157
$861$	0	0
$862$	20419.0	0.806814
$863$	1735.17	0.0684426	0.0342213	−	0.999414i	$-0.489105\pi$
$863$	1735.17	0.0684426	0.0342213	+	0.999414i	$0.489105\pi$
$864$	0	0
$865$	−4842.61	−0.190351
$866$	−31021.1	−1.21725
$867$	0	0
$868$	36640.6	1.43279
$869$	33031.4	1.28943
$870$	0	0
$871$	8363.77	0.325368
$872$	−64928.6	−2.52151
$873$	0	0
$874$	21053.5	0.814813
$875$	−7722.33	−0.298357
$876$	0	0
$877$	−18021.6	−0.693895	−0.346948	−	0.937885i	$-0.612782\pi$
$877$	−18021.6	−0.693895	−0.346948	+	0.937885i	$0.612782\pi$
$878$	−13876.8	−0.533394
$879$	0	0
$880$	−77238.7	−2.95877
$881$	47830.0	1.82910	0.914548	−	0.404477i	$-0.132546\pi$
$881$	47830.0	1.82910	0.914548	+	0.404477i	$0.132546\pi$
$882$	0	0
$883$	12664.7	0.482676	0.241338	−	0.970441i	$-0.422414\pi$
$883$	12664.7	0.482676	0.241338	+	0.970441i	$0.422414\pi$
$884$	−17929.8	−0.682178
$885$	0	0
$886$	39769.1	1.50798
$887$	−9359.43	−0.354294	−0.177147	−	0.984184i	$-0.556687\pi$
$887$	−9359.43	−0.354294	−0.177147	+	0.984184i	$0.556687\pi$
$888$	0	0
$889$	−5680.95	−0.214323
$890$	14668.5	0.552459
$891$	0	0
$892$	−139706.	−5.24408
$893$	−5306.54	−0.198854
$894$	0	0
$895$	14571.0	0.544197
$896$	54772.0	2.04219
$897$	0	0
$898$	90974.2	3.38068
$899$	52805.1	1.95901
$900$	0	0
$901$	−3015.44	−0.111497
$902$	−73763.7	−2.72291
$903$	0	0
$904$	119568.	4.39907
$905$	15394.2	0.565437
$906$	0	0
$907$	13862.4	0.507488	0.253744	−	0.967271i	$-0.418338\pi$
$907$	13862.4	0.507488	0.253744	+	0.967271i	$0.418338\pi$
$908$	−75010.1	−2.74152
$909$	0	0
$910$	8802.95	0.320676
$911$	−29338.6	−1.06699	−0.533497	−	0.845802i	$-0.679123\pi$
$911$	−29338.6	−1.06699	−0.533497	+	0.845802i	$0.679123\pi$
$912$	0	0
$913$	−15618.5	−0.566152
$914$	96658.2	3.49800
$915$	0	0
$916$	33828.2	1.22021
$917$	−4657.38	−0.167721
$918$	0	0
$919$	26183.6	0.939844	0.469922	−	0.882708i	$-0.344282\pi$
$919$	26183.6	0.939844	0.469922	+	0.882708i	$0.344282\pi$
$920$	−17168.8	−0.615260
$921$	0	0
$922$	7678.69	0.274278
$923$	36541.0	1.30310
$924$	0	0
$925$	20175.9	0.717169
$926$	−48548.3	−1.72289
$927$	0	0
$928$	200856.	7.10499
$929$	32423.2	1.14507	0.572535	−	0.819880i	$-0.305960\pi$
$929$	32423.2	1.14507	0.572535	+	0.819880i	$0.305960\pi$
$930$	0	0
$931$	−27906.4	−0.982380
$932$	−98910.4	−3.47631
$933$	0	0
$934$	38384.4	1.34473
$935$	−4672.04	−0.163414
$936$	0	0
$937$	20162.3	0.702960	0.351480	−	0.936195i	$-0.385679\pi$
$937$	20162.3	0.702960	0.351480	+	0.936195i	$0.385679\pi$
$938$	−6863.16	−0.238902
$939$	0	0
$940$	6635.16	0.230229
$941$	−680.451	−0.0235729	−0.0117864	−	0.999931i	$-0.503752\pi$
$941$	−680.451	−0.0235729	−0.0117864	+	0.999931i	$0.503752\pi$
$942$	0	0
$943$	−9908.79	−0.342179
$944$	−207933.	−7.16910
$945$	0	0
$946$	−36039.0	−1.23861
$947$	8821.76	0.302713	0.151356	−	0.988479i	$-0.451636\pi$
$947$	8821.76	0.302713	0.151356	+	0.988479i	$0.451636\pi$
$948$	0	0
$949$	−35065.7	−1.19945
$950$	51718.8	1.76629
$951$	0	0
$952$	9595.64	0.326677
$953$	−47742.5	−1.62280	−0.811402	−	0.584489i	$-0.801295\pi$
$953$	−47742.5	−1.62280	−0.811402	+	0.584489i	$0.801295\pi$
$954$	0	0
$955$	2854.42	0.0967191
$956$	26236.5	0.887603
$957$	0	0
$958$	54828.5	1.84909
$959$	−2204.23	−0.0742212
$960$	0	0
$961$	25774.5	0.865179
$962$	−52048.1	−1.74438
$963$	0	0
$964$	16666.2	0.556828
$965$	15011.3	0.500756
$966$	0	0
$967$	37776.8	1.25628	0.628138	−	0.778102i	$-0.283817\pi$
$967$	37776.8	1.25628	0.628138	+	0.778102i	$0.283817\pi$
$968$	−131169.	−4.35530
$969$	0	0
$970$	18274.1	0.604891
$971$	14779.3	0.488456	0.244228	−	0.969718i	$-0.421465\pi$
$971$	14779.3	0.488456	0.244228	+	0.969718i	$0.421465\pi$
$972$	0	0
$973$	6636.91	0.218674
$974$	−109018.	−3.58640
$975$	0	0
$976$	−251687.	−8.25440
$977$	8147.95	0.266813	0.133406	−	0.991061i	$-0.457408\pi$
$977$	8147.95	0.266813	0.133406	+	0.991061i	$0.457408\pi$
$978$	0	0
$979$	27813.0	0.907976
$980$	34893.4	1.13738
$981$	0	0
$982$	−81484.4	−2.64793
$983$	−14417.1	−0.467787	−0.233894	−	0.972262i	$-0.575147\pi$
$983$	−14417.1	−0.467787	−0.233894	+	0.972262i	$0.575147\pi$
$984$	0	0
$985$	−3197.08	−0.103419
$986$	21203.7	0.684850
$987$	0	0
$988$	−98989.9	−3.18754
$989$	−4841.17	−0.155652
$990$	0	0
$991$	−9951.72	−0.318998	−0.159499	−	0.987198i	$-0.550988\pi$
$991$	−9951.72	−0.318998	−0.159499	+	0.987198i	$0.550988\pi$
$992$	211356.	6.76469
$993$	0	0
$994$	−29984.9	−0.956805
$995$	−5138.66	−0.163725
$996$	0	0
$997$	55653.4	1.76786	0.883932	−	0.467616i	$-0.154887\pi$
$997$	55653.4	1.76786	0.883932	+	0.467616i	$0.154887\pi$
$998$	48314.5	1.53243
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.h.1.1	✓	4
3.2	odd	2		153.4.a.i.1.4	yes	4
4.3	odd	2		2448.4.a.bo.1.3		4
12.11	even	2		2448.4.a.bs.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.a.h.1.1	✓	4	1.1	even	1	trivial
153.4.a.i.1.4	yes	4	3.2	odd	2
2448.4.a.bo.1.3		4	4.3	odd	2
2448.4.a.bs.1.2		4	12.11	even	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)

\(f(q)\)	\(=\)	\(q-5.56787 q^{2} +23.0012 q^{4} -5.10214 q^{5} -6.75787 q^{7} -83.5248 q^{8} +O(q^{10})\)	\(q-5.56787 q^{2} +23.0012 q^{4} -5.10214 q^{5} -6.75787 q^{7} -83.5248 q^{8} +28.4081 q^{10} +53.8648 q^{11} -45.8539 q^{13} +37.6269 q^{14} +281.046 q^{16} +17.0000 q^{17} +93.8563 q^{19} -117.355 q^{20} -299.913 q^{22} -40.2877 q^{23} -98.9681 q^{25} +255.309 q^{26} -155.439 q^{28} -224.013 q^{29} -235.723 q^{31} -896.629 q^{32} -94.6538 q^{34} +34.4796 q^{35} -203.863 q^{37} -522.580 q^{38} +426.156 q^{40} +245.951 q^{41} +120.165 q^{43} +1238.96 q^{44} +224.317 q^{46} -56.5390 q^{47} -297.331 q^{49} +551.042 q^{50} -1054.70 q^{52} -177.379 q^{53} -274.826 q^{55} +564.449 q^{56} +1247.27 q^{58} -739.853 q^{59} -895.536 q^{61} +1312.48 q^{62} +2743.95 q^{64} +233.953 q^{65} -182.400 q^{67} +391.020 q^{68} -191.978 q^{70} -796.900 q^{71} +764.726 q^{73} +1135.08 q^{74} +2158.81 q^{76} -364.011 q^{77} +613.228 q^{79} -1433.94 q^{80} -1369.42 q^{82} -289.957 q^{83} -86.7364 q^{85} -669.063 q^{86} -4499.05 q^{88} +516.348 q^{89} +309.875 q^{91} -926.665 q^{92} +314.802 q^{94} -478.869 q^{95} +643.270 q^{97} +1655.50 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8} - 2 q^{10} - 50 q^{11} + 26 q^{13} - 80 q^{14} + 138 q^{16} + 68 q^{17} + 34 q^{19} - 312 q^{20} - 254 q^{22} - 382 q^{23} + 138 q^{25} + 22 q^{26} + 52 q^{28} - 540 q^{29} - 356 q^{31} - 730 q^{32} - 68 q^{34} - 304 q^{35} - 404 q^{37} - 298 q^{38} + 332 q^{40} + 114 q^{41} + 570 q^{43} + 1368 q^{44} - 290 q^{46} - 496 q^{47} - 224 q^{49} + 1862 q^{50} - 1012 q^{52} - 92 q^{53} - 482 q^{55} + 1428 q^{56} + 1324 q^{58} + 48 q^{59} - 1036 q^{61} + 2564 q^{62} + 2898 q^{64} + 342 q^{65} + 812 q^{67} + 442 q^{68} + 152 q^{70} - 1044 q^{71} - 1212 q^{73} + 1444 q^{74} + 2268 q^{76} + 564 q^{77} + 488 q^{79} + 1000 q^{80} - 938 q^{82} - 1708 q^{83} - 374 q^{85} + 2446 q^{86} - 3868 q^{88} + 8 q^{89} + 716 q^{91} - 1356 q^{92} - 1224 q^{94} - 1010 q^{95} - 76 q^{97} + 1472 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8 - 2 * q^10 - 50 * q^11 + 26 * q^13 - 80 * q^14 + 138 * q^16 + 68 * q^17 + 34 * q^19 - 312 * q^20 - 254 * q^22 - 382 * q^23 + 138 * q^25 + 22 * q^26 + 52 * q^28 - 540 * q^29 - 356 * q^31 - 730 * q^32 - 68 * q^34 - 304 * q^35 - 404 * q^37 - 298 * q^38 + 332 * q^40 + 114 * q^41 + 570 * q^43 + 1368 * q^44 - 290 * q^46 - 496 * q^47 - 224 * q^49 + 1862 * q^50 - 1012 * q^52 - 92 * q^53 - 482 * q^55 + 1428 * q^56 + 1324 * q^58 + 48 * q^59 - 1036 * q^61 + 2564 * q^62 + 2898 * q^64 + 342 * q^65 + 812 * q^67 + 442 * q^68 + 152 * q^70 - 1044 * q^71 - 1212 * q^73 + 1444 * q^74 + 2268 * q^76 + 564 * q^77 + 488 * q^79 + 1000 * q^80 - 938 * q^82 - 1708 * q^83 - 374 * q^85 + 2446 * q^86 - 3868 * q^88 + 8 * q^89 + 716 * q^91 - 1356 * q^92 - 1224 * q^94 - 1010 * q^95 - 76 * q^97 + 1472 * q^98

Introduction

L-functions

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