LMFDB - Embedded newform 153.4.a.f.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:	$3$
Coefficient field:	3.3.5912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 10 $ x^3 - x^2 - 14*x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.75985$ 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)

$f(q)$	$=$	$q-4.75985 q^{2} +14.6562 q^{4} +7.65616 q^{5} -31.5852 q^{7} -31.6823 q^{8} +O(q^{10})$	\(q-4.75985 q^{2} +14.6562 q^{4} +7.65616 q^{5} -31.5852 q^{7} -31.6823 q^{8} -36.4422 q^{10} +7.18910 q^{11} +84.3331 q^{13} +150.341 q^{14} +33.5537 q^{16} -17.0000 q^{17} -37.0838 q^{19} +112.210 q^{20} -34.2190 q^{22} -150.218 q^{23} -66.3832 q^{25} -401.413 q^{26} -462.918 q^{28} +11.5846 q^{29} -53.2865 q^{31} +93.7478 q^{32} +80.9174 q^{34} -241.822 q^{35} -99.2134 q^{37} +176.513 q^{38} -242.565 q^{40} -118.249 q^{41} -456.016 q^{43} +105.365 q^{44} +715.014 q^{46} -571.014 q^{47} +654.627 q^{49} +315.974 q^{50} +1236.00 q^{52} -462.867 q^{53} +55.0409 q^{55} +1000.69 q^{56} -55.1407 q^{58} -48.0674 q^{59} +59.5236 q^{61} +253.636 q^{62} -714.655 q^{64} +645.668 q^{65} -740.787 q^{67} -249.155 q^{68} +1151.03 q^{70} +930.437 q^{71} -697.419 q^{73} +472.241 q^{74} -543.506 q^{76} -227.070 q^{77} +1036.04 q^{79} +256.893 q^{80} +562.849 q^{82} +22.2043 q^{83} -130.155 q^{85} +2170.57 q^{86} -227.767 q^{88} +369.726 q^{89} -2663.68 q^{91} -2201.62 q^{92} +2717.94 q^{94} -283.920 q^{95} +1139.56 q^{97} -3115.93 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8}+O(q^{10}) $ 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8	$ 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8} - 38 q^{10} - 34 q^{11} + 36 q^{13} + 104 q^{14} - 79 q^{16} - 51 q^{17} - 142 q^{19} + 126 q^{20} - 248 q^{22} - 110 q^{23} - 193 q^{25} - 154 q^{26} - 472 q^{28} - 90 q^{29} - 148 q^{31} + 151 q^{32} + 85 q^{34} - 416 q^{35} + 110 q^{37} - 80 q^{38} - 202 q^{40} - 720 q^{41} - 146 q^{43} + 192 q^{44} + 748 q^{46} - 500 q^{47} + 379 q^{49} + 385 q^{50} + 1218 q^{52} - 610 q^{53} + 430 q^{55} + 1368 q^{56} + 1006 q^{58} + 216 q^{59} - 18 q^{61} + 904 q^{62} + 377 q^{64} + 966 q^{65} - 1404 q^{67} - 221 q^{68} + 1472 q^{70} + 960 q^{71} - 794 q^{73} + 1874 q^{74} - 392 q^{76} - 48 q^{77} - 276 q^{79} + 1130 q^{80} + 382 q^{82} + 1552 q^{83} + 136 q^{85} - 16 q^{86} + 1724 q^{88} - 1394 q^{89} - 3968 q^{91} - 2244 q^{92} + 3960 q^{94} + 602 q^{95} + 402 q^{97} - 4109 q^{98}+O(q^{100}) $ 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8 - 38 * q^10 - 34 * q^11 + 36 * q^13 + 104 * q^14 - 79 * q^16 - 51 * q^17 - 142 * q^19 + 126 * q^20 - 248 * q^22 - 110 * q^23 - 193 * q^25 - 154 * q^26 - 472 * q^28 - 90 * q^29 - 148 * q^31 + 151 * q^32 + 85 * q^34 - 416 * q^35 + 110 * q^37 - 80 * q^38 - 202 * q^40 - 720 * q^41 - 146 * q^43 + 192 * q^44 + 748 * q^46 - 500 * q^47 + 379 * q^49 + 385 * q^50 + 1218 * q^52 - 610 * q^53 + 430 * q^55 + 1368 * q^56 + 1006 * q^58 + 216 * q^59 - 18 * q^61 + 904 * q^62 + 377 * q^64 + 966 * q^65 - 1404 * q^67 - 221 * q^68 + 1472 * q^70 + 960 * q^71 - 794 * q^73 + 1874 * q^74 - 392 * q^76 - 48 * q^77 - 276 * q^79 + 1130 * q^80 + 382 * q^82 + 1552 * q^83 + 136 * q^85 - 16 * q^86 + 1724 * q^88 - 1394 * q^89 - 3968 * q^91 - 2244 * q^92 + 3960 * q^94 + 602 * q^95 + 402 * q^97 - 4109 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.75985	−1.68286	−0.841430	−	0.540366i	$-0.818286\pi$
$2$	−4.75985	−1.68286	−0.841430	+	0.540366i	$0.818286\pi$
$3$	0	0
$3$	0	0
$4$	14.6562	1.83202
$5$	7.65616	0.684788	0.342394	−	0.939557i	$-0.388762\pi$
$5$	7.65616	0.684788	0.342394	+	0.939557i	$0.388762\pi$
$6$	0	0
$7$	−31.5852	−1.70544	−0.852721	−	0.522366i	$-0.825049\pi$
$7$	−31.5852	−1.70544	−0.852721	+	0.522366i	$0.825049\pi$
$8$	−31.6823	−1.40017
$9$	0	0
$10$	−36.4422	−1.15240
$11$	7.18910	0.197054	0.0985271	−	0.995134i	$-0.468587\pi$
$11$	7.18910	0.197054	0.0985271	+	0.995134i	$0.468587\pi$
$12$	0	0
$13$	84.3331	1.79921	0.899607	−	0.436700i	$-0.143853\pi$
$13$	84.3331	1.79921	0.899607	+	0.436700i	$0.143853\pi$
$14$	150.341	2.87002
$15$	0	0
$16$	33.5537	0.524277
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−37.0838	−0.447769	−0.223885	−	0.974616i	$-0.571874\pi$
$19$	−37.0838	−0.447769	−0.223885	+	0.974616i	$0.571874\pi$
$20$	112.210	1.25454
$21$	0	0
$22$	−34.2190	−0.331615
$23$	−150.218	−1.36185	−0.680925	−	0.732353i	$-0.738422\pi$
$23$	−150.218	−1.36185	−0.680925	+	0.732353i	$0.738422\pi$
$24$	0	0
$25$	−66.3832	−0.531066
$26$	−401.413	−3.02783
$27$	0	0
$28$	−462.918	−3.12440
$29$	11.5846	0.0741792	0.0370896	−	0.999312i	$-0.488191\pi$
$29$	11.5846	0.0741792	0.0370896	+	0.999312i	$0.488191\pi$
$30$	0	0
$31$	−53.2865	−0.308727	−0.154364	−	0.988014i	$-0.549333\pi$
$31$	−53.2865	−0.308727	−0.154364	+	0.988014i	$0.549333\pi$
$32$	93.7478	0.517889
$33$	0	0
$34$	80.9174	0.408154
$35$	−241.822	−1.16787
$36$	0	0
$37$	−99.2134	−0.440827	−0.220413	−	0.975407i	$-0.570741\pi$
$37$	−99.2134	−0.440827	−0.220413	+	0.975407i	$0.570741\pi$
$38$	176.513	0.753533
$39$	0	0
$40$	−242.565	−0.958821
$41$	−118.249	−0.450425	−0.225213	−	0.974310i	$-0.572308\pi$
$41$	−118.249	−0.450425	−0.225213	+	0.974310i	$0.572308\pi$
$42$	0	0
$43$	−456.016	−1.61725	−0.808626	−	0.588323i	$-0.799789\pi$
$43$	−456.016	−1.61725	−0.808626	+	0.588323i	$0.799789\pi$
$44$	105.365	0.361007
$45$	0	0
$46$	715.014	2.29180
$47$	−571.014	−1.77215	−0.886073	−	0.463545i	$-0.846577\pi$
$47$	−571.014	−1.77215	−0.886073	+	0.463545i	$0.846577\pi$
$48$	0	0
$49$	654.627	1.90853
$50$	315.974	0.893710
$51$	0	0
$52$	1236.00	3.29620
$53$	−462.867	−1.19962	−0.599809	−	0.800143i	$-0.704757\pi$
$53$	−462.867	−1.19962	−0.599809	+	0.800143i	$0.704757\pi$
$54$	0	0
$55$	55.0409	0.134940
$56$	1000.69	2.38792
$57$	0	0
$58$	−55.1407	−0.124833
$59$	−48.0674	−0.106065	−0.0530325	−	0.998593i	$-0.516889\pi$
$59$	−48.0674	−0.106065	−0.0530325	+	0.998593i	$0.516889\pi$
$60$	0	0
$61$	59.5236	0.124938	0.0624689	−	0.998047i	$-0.480103\pi$
$61$	59.5236	0.124938	0.0624689	+	0.998047i	$0.480103\pi$
$62$	253.636	0.519545
$63$	0	0
$64$	−714.655	−1.39581
$65$	645.668	1.23208
$66$	0	0
$67$	−740.787	−1.35077	−0.675384	−	0.737466i	$-0.736022\pi$
$67$	−740.787	−1.35077	−0.675384	+	0.737466i	$0.736022\pi$
$68$	−249.155	−0.444330
$69$	0	0
$70$	1151.03	1.96536
$71$	930.437	1.55525	0.777623	−	0.628730i	$-0.216425\pi$
$71$	930.437	1.55525	0.777623	+	0.628730i	$0.216425\pi$
$72$	0	0
$73$	−697.419	−1.11817	−0.559087	−	0.829109i	$-0.688848\pi$
$73$	−697.419	−1.11817	−0.559087	+	0.829109i	$0.688848\pi$
$74$	472.241	0.741850
$75$	0	0
$76$	−543.506	−0.820322
$77$	−227.070	−0.336065
$78$	0	0
$79$	1036.04	1.47549	0.737747	−	0.675077i	$-0.235890\pi$
$79$	1036.04	1.47549	0.737747	+	0.675077i	$0.235890\pi$
$80$	256.893	0.359018
$81$	0	0
$82$	562.849	0.758003
$83$	22.2043	0.0293643	0.0146822	−	0.999892i	$-0.495326\pi$
$83$	22.2043	0.0293643	0.0146822	+	0.999892i	$0.495326\pi$
$84$	0	0
$85$	−130.155	−0.166085
$86$	2170.57	2.72161
$87$	0	0
$88$	−227.767	−0.275910
$89$	369.726	0.440346	0.220173	−	0.975461i	$-0.429338\pi$
$89$	369.726	0.440346	0.220173	+	0.975461i	$0.429338\pi$
$90$	0	0
$91$	−2663.68	−3.06846
$92$	−2201.62	−2.49494
$93$	0	0
$94$	2717.94	2.98228
$95$	−283.920	−0.306627
$96$	0	0
$97$	1139.56	1.19283	0.596415	−	0.802676i	$-0.296591\pi$
$97$	1139.56	1.19283	0.596415	+	0.802676i	$0.296591\pi$
$98$	−3115.93	−3.21180
$99$	0	0
$100$	−972.923	−0.972923
$101$	−703.083	−0.692667	−0.346334	−	0.938111i	$-0.612573\pi$
$101$	−703.083	−0.692667	−0.346334	+	0.938111i	$0.612573\pi$
$102$	0	0
$103$	−897.160	−0.858250	−0.429125	−	0.903245i	$-0.641178\pi$
$103$	−897.160	−0.858250	−0.429125	+	0.903245i	$0.641178\pi$
$104$	−2671.87	−2.51921
$105$	0	0
$106$	2203.18	2.01879
$107$	1901.21	1.71773	0.858864	−	0.512203i	$-0.171171\pi$
$107$	1901.21	1.71773	0.858864	+	0.512203i	$0.171171\pi$
$108$	0	0
$109$	584.555	0.513671	0.256836	−	0.966455i	$-0.417320\pi$
$109$	584.555	0.513671	0.256836	+	0.966455i	$0.417320\pi$
$110$	−261.986	−0.227086
$111$	0	0
$112$	−1059.80	−0.894124
$113$	63.4225	0.0527990	0.0263995	−	0.999651i	$-0.491596\pi$
$113$	63.4225	0.0527990	0.0263995	+	0.999651i	$0.491596\pi$
$114$	0	0
$115$	−1150.09	−0.932578
$116$	169.785	0.135898
$117$	0	0
$118$	228.793	0.178493
$119$	536.949	0.413631
$120$	0	0
$121$	−1279.32	−0.961170
$122$	−283.323	−0.210253
$123$	0	0
$124$	−780.976	−0.565594
$125$	−1465.26	−1.04845
$126$	0	0
$127$	175.543	0.122653	0.0613266	−	0.998118i	$-0.480467\pi$
$127$	175.543	0.122653	0.0613266	+	0.998118i	$0.480467\pi$
$128$	2651.67	1.83107
$129$	0	0
$130$	−3073.28	−2.07342
$131$	−1865.42	−1.24414	−0.622070	−	0.782961i	$-0.713708\pi$
$131$	−1865.42	−1.24414	−0.622070	+	0.782961i	$0.713708\pi$
$132$	0	0
$133$	1171.30	0.763644
$134$	3526.03	2.27315
$135$	0	0
$136$	538.599	0.339592
$137$	−1057.57	−0.659519	−0.329760	−	0.944065i	$-0.606968\pi$
$137$	−1057.57	−0.659519	−0.329760	+	0.944065i	$0.606968\pi$
$138$	0	0
$139$	904.833	0.552136	0.276068	−	0.961138i	$-0.410968\pi$
$139$	904.833	0.552136	0.276068	+	0.961138i	$0.410968\pi$
$140$	−3544.18	−2.13955
$141$	0	0
$142$	−4428.74	−2.61726
$143$	606.279	0.354543
$144$	0	0
$145$	88.6932	0.0507970
$146$	3319.61	1.88173
$147$	0	0
$148$	−1454.09	−0.807603
$149$	−809.001	−0.444805	−0.222402	−	0.974955i	$-0.571390\pi$
$149$	−809.001	−0.444805	−0.222402	+	0.974955i	$0.571390\pi$
$150$	0	0
$151$	−352.121	−0.189769	−0.0948847	−	0.995488i	$-0.530248\pi$
$151$	−352.121	−0.189769	−0.0948847	+	0.995488i	$0.530248\pi$
$152$	1174.90	0.626954
$153$	0	0
$154$	1080.82	0.565550
$155$	−407.970	−0.211413
$156$	0	0
$157$	537.882	0.273424	0.136712	−	0.990611i	$-0.456346\pi$
$157$	537.882	0.273424	0.136712	+	0.990611i	$0.456346\pi$
$158$	−4931.41	−2.48305
$159$	0	0
$160$	717.748	0.354644
$161$	4744.66	2.32256
$162$	0	0
$163$	1922.74	0.923933	0.461966	−	0.886897i	$-0.347144\pi$
$163$	1922.74	0.923933	0.461966	+	0.886897i	$0.347144\pi$
$164$	−1733.08	−0.825188
$165$	0	0
$166$	−105.689	−0.0494160
$167$	2971.76	1.37702	0.688509	−	0.725228i	$-0.258266\pi$
$167$	2971.76	1.37702	0.688509	+	0.725228i	$0.258266\pi$
$168$	0	0
$169$	4915.07	2.23717
$170$	619.517	0.279499
$171$	0	0
$172$	−6683.45	−2.96284
$173$	−988.564	−0.434446	−0.217223	−	0.976122i	$-0.569700\pi$
$173$	−988.564	−0.434446	−0.217223	+	0.976122i	$0.569700\pi$
$174$	0	0
$175$	2096.73	0.905702
$176$	241.221	0.103311
$177$	0	0
$178$	−1759.84	−0.741042
$179$	1937.65	0.809089	0.404545	−	0.914518i	$-0.367430\pi$
$179$	1937.65	0.809089	0.404545	+	0.914518i	$0.367430\pi$
$180$	0	0
$181$	−2180.07	−0.895267	−0.447634	−	0.894217i	$-0.647733\pi$
$181$	−2180.07	−0.895267	−0.447634	+	0.894217i	$0.647733\pi$
$182$	12678.7	5.16379
$183$	0	0
$184$	4759.24	1.90683
$185$	−759.594	−0.301873
$186$	0	0
$187$	−122.215	−0.0477927
$188$	−8368.86	−3.24661
$189$	0	0
$190$	1351.41	0.516010
$191$	−1675.78	−0.634845	−0.317423	−	0.948284i	$-0.602817\pi$
$191$	−1675.78	−0.634845	−0.317423	+	0.948284i	$0.602817\pi$
$192$	0	0
$193$	−257.961	−0.0962094	−0.0481047	−	0.998842i	$-0.515318\pi$
$193$	−257.961	−0.0962094	−0.0481047	+	0.998842i	$0.515318\pi$
$194$	−5424.12	−2.00737
$195$	0	0
$196$	9594.32	3.49647
$197$	693.466	0.250799	0.125399	−	0.992106i	$-0.459979\pi$
$197$	693.466	0.250799	0.125399	+	0.992106i	$0.459979\pi$
$198$	0	0
$199$	−240.295	−0.0855984	−0.0427992	−	0.999084i	$-0.513628\pi$
$199$	−240.295	−0.0855984	−0.0427992	+	0.999084i	$0.513628\pi$
$200$	2103.17	0.743584
$201$	0	0
$202$	3346.57	1.16566
$203$	−365.901	−0.126508
$204$	0	0
$205$	−905.335	−0.308446
$206$	4270.34	1.44432
$207$	0	0
$208$	2829.69	0.943287
$209$	−266.599	−0.0882348
$210$	0	0
$211$	268.114	0.0874774	0.0437387	−	0.999043i	$-0.486073\pi$
$211$	268.114	0.0874774	0.0437387	+	0.999043i	$0.486073\pi$
$212$	−6783.86	−2.19772
$213$	0	0
$214$	−9049.47	−2.89070
$215$	−3491.33	−1.10747
$216$	0	0
$217$	1683.07	0.526516
$218$	−2782.39	−0.864437
$219$	0	0
$220$	806.688	0.247213
$221$	−1433.66	−0.436374
$222$	0	0
$223$	−5524.43	−1.65894	−0.829468	−	0.558554i	$-0.811357\pi$
$223$	−5524.43	−1.65894	−0.829468	+	0.558554i	$0.811357\pi$
$224$	−2961.05	−0.883229
$225$	0	0
$226$	−301.882	−0.0888534
$227$	384.400	0.112394	0.0561972	−	0.998420i	$-0.482102\pi$
$227$	384.400	0.112394	0.0561972	+	0.998420i	$0.482102\pi$
$228$	0	0
$229$	1395.48	0.402690	0.201345	−	0.979520i	$-0.435469\pi$
$229$	1395.48	0.402690	0.201345	+	0.979520i	$0.435469\pi$
$230$	5474.26	1.56940
$231$	0	0
$232$	−367.025	−0.103864
$233$	−3409.39	−0.958613	−0.479307	−	0.877648i	$-0.659112\pi$
$233$	−3409.39	−0.958613	−0.479307	+	0.877648i	$0.659112\pi$
$234$	0	0
$235$	−4371.77	−1.21354
$236$	−704.483	−0.194313
$237$	0	0
$238$	−2555.80	−0.696083
$239$	1509.18	0.408456	0.204228	−	0.978923i	$-0.434532\pi$
$239$	1509.18	0.408456	0.204228	+	0.978923i	$0.434532\pi$
$240$	0	0
$241$	3406.91	0.910615	0.455307	−	0.890334i	$-0.349529\pi$
$241$	3406.91	0.910615	0.455307	+	0.890334i	$0.349529\pi$
$242$	6089.35	1.61751
$243$	0	0
$244$	872.387	0.228889
$245$	5011.93	1.30694
$246$	0	0
$247$	−3127.39	−0.805633
$248$	1688.24	0.432271
$249$	0	0
$250$	6974.42	1.76440
$251$	−3394.43	−0.853605	−0.426802	−	0.904345i	$-0.640360\pi$
$251$	−3394.43	−0.853605	−0.426802	+	0.904345i	$0.640360\pi$
$252$	0	0
$253$	−1079.93	−0.268358
$254$	−835.560	−0.206408
$255$	0	0
$256$	−6904.30	−1.68562
$257$	−1778.91	−0.431772	−0.215886	−	0.976419i	$-0.569264\pi$
$257$	−1778.91	−0.431772	−0.215886	+	0.976419i	$0.569264\pi$
$258$	0	0
$259$	3133.68	0.751804
$260$	9463.01	2.25719
$261$	0	0
$262$	8879.11	2.09372
$263$	4316.88	1.01213	0.506065	−	0.862495i	$-0.331100\pi$
$263$	4316.88	1.01213	0.506065	+	0.862495i	$0.331100\pi$
$264$	0	0
$265$	−3543.79	−0.821483
$266$	−5575.22	−1.28511
$267$	0	0
$268$	−10857.1	−2.47463
$269$	−6546.31	−1.48378	−0.741888	−	0.670524i	$-0.766069\pi$
$269$	−6546.31	−1.48378	−0.741888	+	0.670524i	$0.766069\pi$
$270$	0	0
$271$	3785.20	0.848466	0.424233	−	0.905553i	$-0.360544\pi$
$271$	3785.20	0.848466	0.424233	+	0.905553i	$0.360544\pi$
$272$	−570.413	−0.127156
$273$	0	0
$274$	5033.86	1.10988
$275$	−477.236	−0.104649
$276$	0	0
$277$	−3521.06	−0.763755	−0.381878	−	0.924213i	$-0.624722\pi$
$277$	−3521.06	−0.763755	−0.381878	+	0.924213i	$0.624722\pi$
$278$	−4306.87	−0.929168
$279$	0	0
$280$	7661.47	1.63521
$281$	2922.30	0.620391	0.310195	−	0.950673i	$-0.399606\pi$
$281$	2922.30	0.620391	0.310195	+	0.950673i	$0.399606\pi$
$282$	0	0
$283$	735.075	0.154402	0.0772008	−	0.997016i	$-0.475402\pi$
$283$	735.075	0.154402	0.0772008	+	0.997016i	$0.475402\pi$
$284$	13636.6	2.84924
$285$	0	0
$286$	−2885.80	−0.596646
$287$	3734.93	0.768175
$288$	0	0
$289$	289.000	0.0588235
$290$	−422.166	−0.0854843
$291$	0	0
$292$	−10221.5	−2.04852
$293$	8702.92	1.73526	0.867628	−	0.497213i	$-0.165643\pi$
$293$	8702.92	1.73526	0.867628	+	0.497213i	$0.165643\pi$
$294$	0	0
$295$	−368.011	−0.0726320
$296$	3143.31	0.617234
$297$	0	0
$298$	3850.72	0.748545
$299$	−12668.3	−2.45026
$300$	0	0
$301$	14403.4	2.75813
$302$	1676.04	0.319355
$303$	0	0
$304$	−1244.30	−0.234755
$305$	455.722	0.0855559
$306$	0	0
$307$	2516.95	0.467916	0.233958	−	0.972247i	$-0.424832\pi$
$307$	2516.95	0.467916	0.233958	+	0.972247i	$0.424832\pi$
$308$	−3327.97	−0.615677
$309$	0	0
$310$	1941.88	0.355778
$311$	−6593.31	−1.20216	−0.601081	−	0.799188i	$-0.705263\pi$
$311$	−6593.31	−1.20216	−0.601081	+	0.799188i	$0.705263\pi$
$312$	0	0
$313$	4392.99	0.793312	0.396656	−	0.917967i	$-0.370171\pi$
$313$	4392.99	0.793312	0.396656	+	0.917967i	$0.370171\pi$
$314$	−2560.24	−0.460135
$315$	0	0
$316$	15184.4	2.70314
$317$	−2601.23	−0.460882	−0.230441	−	0.973086i	$-0.574017\pi$
$317$	−2601.23	−0.460882	−0.230441	+	0.973086i	$0.574017\pi$
$318$	0	0
$319$	83.2825	0.0146173
$320$	−5471.51	−0.955834
$321$	0	0
$322$	−22583.9	−3.90854
$323$	630.425	0.108600
$324$	0	0
$325$	−5598.30	−0.955502
$326$	−9151.97	−1.55485
$327$	0	0
$328$	3746.41	0.630674
$329$	18035.6	3.02229
$330$	0	0
$331$	−4670.49	−0.775568	−0.387784	−	0.921750i	$-0.626759\pi$
$331$	−4670.49	−0.775568	−0.387784	+	0.921750i	$0.626759\pi$
$332$	325.430	0.0537960
$333$	0	0
$334$	−14145.1	−2.31733
$335$	−5671.58	−0.924989
$336$	0	0
$337$	1801.67	0.291226	0.145613	−	0.989342i	$-0.453485\pi$
$337$	1801.67	0.291226	0.145613	+	0.989342i	$0.453485\pi$
$338$	−23395.0	−3.76485
$339$	0	0
$340$	−1907.57	−0.304272
$341$	−383.082	−0.0608360
$342$	0	0
$343$	−9842.83	−1.54945
$344$	14447.7	2.26443
$345$	0	0
$346$	4705.41	0.731112
$347$	−168.340	−0.0260431	−0.0130216	−	0.999915i	$-0.504145\pi$
$347$	−168.340	−0.0260431	−0.0130216	+	0.999915i	$0.504145\pi$
$348$	0	0
$349$	−4447.85	−0.682200	−0.341100	−	0.940027i	$-0.610799\pi$
$349$	−4447.85	−0.682200	−0.341100	+	0.940027i	$0.610799\pi$
$350$	−9980.12	−1.52417
$351$	0	0
$352$	673.963	0.102052
$353$	10509.5	1.58460	0.792298	−	0.610135i	$-0.208885\pi$
$353$	10509.5	1.58460	0.792298	+	0.610135i	$0.208885\pi$
$354$	0	0
$355$	7123.57	1.06501
$356$	5418.76	0.806723
$357$	0	0
$358$	−9222.93	−1.36158
$359$	−8342.99	−1.22654	−0.613268	−	0.789875i	$-0.710145\pi$
$359$	−8342.99	−1.22654	−0.613268	+	0.789875i	$0.710145\pi$
$360$	0	0
$361$	−5483.79	−0.799503
$362$	10376.8	1.50661
$363$	0	0
$364$	−39039.3	−5.62147
$365$	−5339.55	−0.765712
$366$	0	0
$367$	352.402	0.0501232	0.0250616	−	0.999686i	$-0.492022\pi$
$367$	352.402	0.0501232	0.0250616	+	0.999686i	$0.492022\pi$
$368$	−5040.36	−0.713987
$369$	0	0
$370$	3615.55	0.508009
$371$	14619.8	2.04588
$372$	0	0
$373$	12563.2	1.74397	0.871983	−	0.489537i	$-0.162834\pi$
$373$	12563.2	1.74397	0.871983	+	0.489537i	$0.162834\pi$
$374$	581.724	0.0804284
$375$	0	0
$376$	18091.0	2.48131
$377$	976.961	0.133464
$378$	0	0
$379$	−1770.57	−0.239969	−0.119984	−	0.992776i	$-0.538284\pi$
$379$	−1770.57	−0.239969	−0.119984	+	0.992776i	$0.538284\pi$
$380$	−4161.17	−0.561746
$381$	0	0
$382$	7976.47	1.06836
$383$	−4330.57	−0.577759	−0.288880	−	0.957365i	$-0.593283\pi$
$383$	−4330.57	−0.577759	−0.288880	+	0.957365i	$0.593283\pi$
$384$	0	0
$385$	−1738.48	−0.230133
$386$	1227.85	0.161907
$387$	0	0
$388$	16701.5	2.18529
$389$	−10295.5	−1.34191	−0.670957	−	0.741496i	$-0.734117\pi$
$389$	−10295.5	−1.34191	−0.670957	+	0.741496i	$0.734117\pi$
$390$	0	0
$391$	2553.70	0.330297
$392$	−20740.1	−2.67228
$393$	0	0
$394$	−3300.79	−0.422060
$395$	7932.12	1.01040
$396$	0	0
$397$	93.1792	0.0117797	0.00588983	−	0.999983i	$-0.498125\pi$
$397$	93.1792	0.0117797	0.00588983	+	0.999983i	$0.498125\pi$
$398$	1143.77	0.144050
$399$	0	0
$400$	−2227.40	−0.278426
$401$	13320.9	1.65889	0.829443	−	0.558591i	$-0.188658\pi$
$401$	13320.9	1.65889	0.829443	+	0.558591i	$0.188658\pi$
$402$	0	0
$403$	−4493.82	−0.555466
$404$	−10304.5	−1.26898
$405$	0	0
$406$	1741.63	0.212896
$407$	−713.255	−0.0868667
$408$	0	0
$409$	9272.21	1.12098	0.560491	−	0.828161i	$-0.310613\pi$
$409$	9272.21	1.12098	0.560491	+	0.828161i	$0.310613\pi$
$410$	4309.26	0.519071
$411$	0	0
$412$	−13148.9	−1.57233
$413$	1518.22	0.180888
$414$	0	0
$415$	170.000	0.0201083
$416$	7906.05	0.931793
$417$	0	0
$418$	1268.97	0.148487
$419$	−11325.4	−1.32049	−0.660244	−	0.751051i	$-0.729547\pi$
$419$	−11325.4	−1.32049	−0.660244	+	0.751051i	$0.729547\pi$
$420$	0	0
$421$	6934.54	0.802776	0.401388	−	0.915908i	$-0.368528\pi$
$421$	6934.54	0.802776	0.401388	+	0.915908i	$0.368528\pi$
$422$	−1276.18	−0.147212
$423$	0	0
$424$	14664.7	1.67967
$425$	1128.52	0.128802
$426$	0	0
$427$	−1880.07	−0.213074
$428$	27864.4	3.14691
$429$	0	0
$430$	16618.2	1.86373
$431$	2776.55	0.310306	0.155153	−	0.987890i	$-0.450413\pi$
$431$	2776.55	0.310306	0.155153	+	0.987890i	$0.450413\pi$
$432$	0	0
$433$	−3252.22	−0.360951	−0.180476	−	0.983579i	$-0.557764\pi$
$433$	−3252.22	−0.360951	−0.180476	+	0.983579i	$0.557764\pi$
$434$	−8011.15	−0.886054
$435$	0	0
$436$	8567.32	0.941056
$437$	5570.65	0.609795
$438$	0	0
$439$	−13345.7	−1.45093	−0.725464	−	0.688260i	$-0.758375\pi$
$439$	−13345.7	−1.45093	−0.725464	+	0.688260i	$0.758375\pi$
$440$	−1743.82	−0.188940
$441$	0	0
$442$	6824.02	0.734356
$443$	−11639.6	−1.24834	−0.624169	−	0.781290i	$-0.714562\pi$
$443$	−11639.6	−1.24834	−0.624169	+	0.781290i	$0.714562\pi$
$444$	0	0
$445$	2830.68	0.301544
$446$	26295.4	2.79176
$447$	0	0
$448$	22572.6	2.38048
$449$	6937.72	0.729201	0.364601	−	0.931164i	$-0.381206\pi$
$449$	6937.72	0.729201	0.364601	+	0.931164i	$0.381206\pi$
$450$	0	0
$451$	−850.106	−0.0887582
$452$	929.531	0.0967289
$453$	0	0
$454$	−1829.69	−0.189144
$455$	−20393.6	−2.10124
$456$	0	0
$457$	−10285.8	−1.05284	−0.526422	−	0.850224i	$-0.676467\pi$
$457$	−10285.8	−1.05284	−0.526422	+	0.850224i	$0.676467\pi$
$458$	−6642.28	−0.677671
$459$	0	0
$460$	−16855.9	−1.70850
$461$	625.833	0.0632277	0.0316138	−	0.999500i	$-0.489935\pi$
$461$	625.833	0.0632277	0.0316138	+	0.999500i	$0.489935\pi$
$462$	0	0
$463$	−6055.97	−0.607872	−0.303936	−	0.952692i	$-0.598301\pi$
$463$	−6055.97	−0.607872	−0.303936	+	0.952692i	$0.598301\pi$
$464$	388.705	0.0388904
$465$	0	0
$466$	16228.2	1.61321
$467$	815.966	0.0808531	0.0404265	−	0.999183i	$-0.487128\pi$
$467$	815.966	0.0808531	0.0404265	+	0.999183i	$0.487128\pi$
$468$	0	0
$469$	23397.9	2.30366
$470$	20809.0	2.04223
$471$	0	0
$472$	1522.88	0.148509
$473$	−3278.35	−0.318686
$474$	0	0
$475$	2461.74	0.237795
$476$	7869.61	0.757779
$477$	0	0
$478$	−7183.49	−0.687375
$479$	16219.1	1.54712	0.773559	−	0.633724i	$-0.218475\pi$
$479$	16219.1	1.54712	0.773559	+	0.633724i	$0.218475\pi$
$480$	0	0
$481$	−8366.97	−0.793142
$482$	−16216.4	−1.53244
$483$	0	0
$484$	−18749.9	−1.76088
$485$	8724.63	0.816835
$486$	0	0
$487$	2725.13	0.253568	0.126784	−	0.991930i	$-0.459535\pi$
$487$	2725.13	0.253568	0.126784	+	0.991930i	$0.459535\pi$
$488$	−1885.84	−0.174935
$489$	0	0
$490$	−23856.0	−2.19940
$491$	8344.13	0.766935	0.383468	−	0.923554i	$-0.374730\pi$
$491$	8344.13	0.766935	0.383468	+	0.923554i	$0.374730\pi$
$492$	0	0
$493$	−196.937	−0.0179911
$494$	14885.9	1.35577
$495$	0	0
$496$	−1787.96	−0.161858
$497$	−29388.1	−2.65238
$498$	0	0
$499$	13762.0	1.23461	0.617306	−	0.786723i	$-0.288224\pi$
$499$	13762.0	1.23461	0.617306	+	0.786723i	$0.288224\pi$
$500$	−21475.1	−1.92079
$501$	0	0
$502$	16157.0	1.43650
$503$	11909.5	1.05570	0.527852	−	0.849336i	$-0.322997\pi$
$503$	11909.5	1.05570	0.527852	+	0.849336i	$0.322997\pi$
$504$	0	0
$505$	−5382.92	−0.474330
$506$	5140.31	0.451610
$507$	0	0
$508$	2572.79	0.224703
$509$	−742.666	−0.0646721	−0.0323360	−	0.999477i	$-0.510295\pi$
$509$	−742.666	−0.0646721	−0.0323360	+	0.999477i	$0.510295\pi$
$510$	0	0
$511$	22028.1	1.90698
$512$	11650.1	1.00560
$513$	0	0
$514$	8467.35	0.726612
$515$	−6868.80	−0.587719
$516$	0	0
$517$	−4105.07	−0.349209
$518$	−14915.8	−1.26518
$519$	0	0
$520$	−20456.2	−1.72513
$521$	4815.73	0.404953	0.202477	−	0.979287i	$-0.435101\pi$
$521$	4815.73	0.404953	0.202477	+	0.979287i	$0.435101\pi$
$522$	0	0
$523$	−16249.5	−1.35858	−0.679291	−	0.733869i	$-0.737713\pi$
$523$	−16249.5	−1.35858	−0.679291	+	0.733869i	$0.737713\pi$
$524$	−27339.9	−2.27929
$525$	0	0
$526$	−20547.7	−1.70327
$527$	905.871	0.0748773
$528$	0	0
$529$	10398.4	0.854637
$530$	16867.9	1.38244
$531$	0	0
$532$	17166.8	1.39901
$533$	−9972.33	−0.810412
$534$	0	0
$535$	14556.0	1.17628
$536$	23469.8	1.89131
$537$	0	0
$538$	31159.4	2.49699
$539$	4706.18	0.376085
$540$	0	0
$541$	−5458.04	−0.433751	−0.216876	−	0.976199i	$-0.569587\pi$
$541$	−5458.04	−0.433751	−0.216876	+	0.976199i	$0.569587\pi$
$542$	−18017.0	−1.42785
$543$	0	0
$544$	−1593.71	−0.125606
$545$	4475.44	0.351756
$546$	0	0
$547$	17237.0	1.34735	0.673677	−	0.739026i	$-0.264714\pi$
$547$	17237.0	1.34735	0.673677	+	0.739026i	$0.264714\pi$
$548$	−15499.9	−1.20825
$549$	0	0
$550$	2271.57	0.176109
$551$	−429.600	−0.0332152
$552$	0	0
$553$	−32723.7	−2.51637
$554$	16759.7	1.28529
$555$	0	0
$556$	13261.4	1.01152
$557$	8452.71	0.643003	0.321502	−	0.946909i	$-0.395812\pi$
$557$	8452.71	0.643003	0.321502	+	0.946909i	$0.395812\pi$
$558$	0	0
$559$	−38457.3	−2.90978
$560$	−8114.01	−0.612285
$561$	0	0
$562$	−13909.7	−1.04403
$563$	8547.32	0.639834	0.319917	−	0.947446i	$-0.396345\pi$
$563$	8547.32	0.639834	0.319917	+	0.947446i	$0.396345\pi$
$564$	0	0
$565$	485.573	0.0361561
$566$	−3498.85	−0.259836
$567$	0	0
$568$	−29478.4	−2.17762
$569$	−19464.8	−1.43411	−0.717055	−	0.697017i	$-0.754510\pi$
$569$	−19464.8	−1.43411	−0.717055	+	0.697017i	$0.754510\pi$
$570$	0	0
$571$	−3839.06	−0.281366	−0.140683	−	0.990055i	$-0.544930\pi$
$571$	−3839.06	−0.281366	−0.140683	+	0.990055i	$0.544930\pi$
$572$	8885.72	0.649529
$573$	0	0
$574$	−17777.7	−1.29273
$575$	9971.94	0.723232
$576$	0	0
$577$	−18797.7	−1.35625	−0.678127	−	0.734945i	$-0.737208\pi$
$577$	−18797.7	−1.35625	−0.678127	+	0.734945i	$0.737208\pi$
$578$	−1375.60	−0.0989918
$579$	0	0
$580$	1299.90	0.0930611
$581$	−701.328	−0.0500791
$582$	0	0
$583$	−3327.60	−0.236390
$584$	22095.8	1.56564
$585$	0	0
$586$	−41424.6	−2.92020
$587$	−4217.92	−0.296580	−0.148290	−	0.988944i	$-0.547377\pi$
$587$	−4217.92	−0.296580	−0.148290	+	0.988944i	$0.547377\pi$
$588$	0	0
$589$	1976.07	0.138238
$590$	1751.68	0.122230
$591$	0	0
$592$	−3328.98	−0.231115
$593$	3011.92	0.208575	0.104287	−	0.994547i	$-0.466744\pi$
$593$	3011.92	0.208575	0.104287	+	0.994547i	$0.466744\pi$
$594$	0	0
$595$	4110.97	0.283249
$596$	−11856.8	−0.814891
$597$	0	0
$598$	60299.3	4.12345
$599$	−15137.1	−1.03253	−0.516266	−	0.856428i	$-0.672678\pi$
$599$	−15137.1	−1.03253	−0.516266	+	0.856428i	$0.672678\pi$
$600$	0	0
$601$	−18980.1	−1.28821	−0.644104	−	0.764938i	$-0.722770\pi$
$601$	−18980.1	−1.28821	−0.644104	+	0.764938i	$0.722770\pi$
$602$	−68558.0	−4.64155
$603$	0	0
$604$	−5160.74	−0.347661
$605$	−9794.65	−0.658197
$606$	0	0
$607$	−4593.63	−0.307166	−0.153583	−	0.988136i	$-0.549081\pi$
$607$	−4593.63	−0.307166	−0.153583	+	0.988136i	$0.549081\pi$
$608$	−3476.53	−0.231894
$609$	0	0
$610$	−2169.17	−0.143979
$611$	−48155.3	−3.18847
$612$	0	0
$613$	−25654.3	−1.69032	−0.845160	−	0.534513i	$-0.820495\pi$
$613$	−25654.3	−1.69032	−0.845160	+	0.534513i	$0.820495\pi$
$614$	−11980.3	−0.787437
$615$	0	0
$616$	7194.09	0.470549
$617$	7170.97	0.467897	0.233948	−	0.972249i	$-0.424835\pi$
$617$	7170.97	0.467897	0.233948	+	0.972249i	$0.424835\pi$
$618$	0	0
$619$	−13560.6	−0.880525	−0.440263	−	0.897869i	$-0.645115\pi$
$619$	−13560.6	−0.880525	−0.440263	+	0.897869i	$0.645115\pi$
$620$	−5979.27	−0.387312
$621$	0	0
$622$	31383.2	2.02307
$623$	−11677.9	−0.750985
$624$	0	0
$625$	−2920.36	−0.186903
$626$	−20910.0	−1.33503
$627$	0	0
$628$	7883.28	0.500919
$629$	1686.63	0.106916
$630$	0	0
$631$	−1414.98	−0.0892700	−0.0446350	−	0.999003i	$-0.514212\pi$
$631$	−1414.98	−0.0892700	−0.0446350	+	0.999003i	$0.514212\pi$
$632$	−32824.3	−2.06595
$633$	0	0
$634$	12381.4	0.775600
$635$	1343.99	0.0839914
$636$	0	0
$637$	55206.7	3.43386
$638$	−396.412	−0.0245989
$639$	0	0
$640$	20301.6	1.25389
$641$	21708.4	1.33764	0.668822	−	0.743422i	$-0.266799\pi$
$641$	21708.4	1.33764	0.668822	+	0.743422i	$0.266799\pi$
$642$	0	0
$643$	7537.23	0.462270	0.231135	−	0.972922i	$-0.425756\pi$
$643$	7537.23	0.462270	0.231135	+	0.972922i	$0.425756\pi$
$644$	69538.5	4.25497
$645$	0	0
$646$	−3000.73	−0.182759
$647$	32667.3	1.98498	0.992491	−	0.122316i	$-0.0390320\pi$
$647$	32667.3	1.98498	0.992491	+	0.122316i	$0.0390320\pi$
$648$	0	0
$649$	−345.561	−0.0209006
$650$	26647.1	1.60798
$651$	0	0
$652$	28180.1	1.69266
$653$	25845.8	1.54889	0.774443	−	0.632644i	$-0.218030\pi$
$653$	25845.8	1.54889	0.774443	+	0.632644i	$0.218030\pi$
$654$	0	0
$655$	−14281.9	−0.851972
$656$	−3967.70	−0.236148
$657$	0	0
$658$	−85846.7	−5.08610
$659$	15741.2	0.930485	0.465243	−	0.885183i	$-0.345967\pi$
$659$	15741.2	0.930485	0.465243	+	0.885183i	$0.345967\pi$
$660$	0	0
$661$	23495.2	1.38254	0.691269	−	0.722598i	$-0.257052\pi$
$661$	23495.2	1.38254	0.691269	+	0.722598i	$0.257052\pi$
$662$	22230.8	1.30517
$663$	0	0
$664$	−703.483	−0.0411151
$665$	8967.67	0.522934
$666$	0	0
$667$	−1740.21	−0.101021
$668$	43554.7	2.52272
$669$	0	0
$670$	26995.9	1.55663
$671$	427.921	0.0246195
$672$	0	0
$673$	−7057.34	−0.404221	−0.202110	−	0.979363i	$-0.564780\pi$
$673$	−7057.34	−0.404221	−0.202110	+	0.979363i	$0.564780\pi$
$674$	−8575.67	−0.490092
$675$	0	0
$676$	72036.0	4.09855
$677$	−20756.4	−1.17833	−0.589167	−	0.808011i	$-0.700544\pi$
$677$	−20756.4	−1.17833	−0.589167	+	0.808011i	$0.700544\pi$
$678$	0	0
$679$	−35993.2	−2.03430
$680$	4123.60	0.232548
$681$	0	0
$682$	1823.41	0.102378
$683$	7013.65	0.392928	0.196464	−	0.980511i	$-0.437054\pi$
$683$	7013.65	0.392928	0.196464	+	0.980511i	$0.437054\pi$
$684$	0	0
$685$	−8096.91	−0.451631
$686$	46850.4	2.60751
$687$	0	0
$688$	−15301.0	−0.847888
$689$	−39035.0	−2.15837
$690$	0	0
$691$	−4897.47	−0.269622	−0.134811	−	0.990871i	$-0.543043\pi$
$691$	−4897.47	−0.269622	−0.134811	+	0.990871i	$0.543043\pi$
$692$	−14488.5	−0.795913
$693$	0	0
$694$	801.273	0.0438269
$695$	6927.54	0.378096
$696$	0	0
$697$	2010.24	0.109244
$698$	21171.1	1.14805
$699$	0	0
$700$	30730.0	1.65926
$701$	−17525.5	−0.944264	−0.472132	−	0.881528i	$-0.656515\pi$
$701$	−17525.5	−0.944264	−0.472132	+	0.881528i	$0.656515\pi$
$702$	0	0
$703$	3679.21	0.197388
$704$	−5137.73	−0.275050
$705$	0	0
$706$	−50023.4	−2.66665
$707$	22207.1	1.18130
$708$	0	0
$709$	32564.3	1.72493	0.862466	−	0.506115i	$-0.168919\pi$
$709$	32564.3	1.72493	0.862466	+	0.506115i	$0.168919\pi$
$710$	−33907.1	−1.79227
$711$	0	0
$712$	−11713.8	−0.616561
$713$	8004.58	0.420440
$714$	0	0
$715$	4641.77	0.242786
$716$	28398.5	1.48227
$717$	0	0
$718$	39711.4	2.06409
$719$	−14498.2	−0.752007	−0.376004	−	0.926618i	$-0.622702\pi$
$719$	−14498.2	−0.752007	−0.376004	+	0.926618i	$0.622702\pi$
$720$	0	0
$721$	28337.0	1.46370
$722$	26102.0	1.34545
$723$	0	0
$724$	−31951.5	−1.64015
$725$	−769.020	−0.0393941
$726$	0	0
$727$	−25787.6	−1.31556	−0.657778	−	0.753212i	$-0.728503\pi$
$727$	−25787.6	−1.31556	−0.657778	+	0.753212i	$0.728503\pi$
$728$	84391.6	4.29637
$729$	0	0
$730$	25415.4	1.28859
$731$	7752.28	0.392241
$732$	0	0
$733$	−4177.45	−0.210502	−0.105251	−	0.994446i	$-0.533565\pi$
$733$	−4177.45	−0.210502	−0.105251	+	0.994446i	$0.533565\pi$
$734$	−1677.38	−0.0843504
$735$	0	0
$736$	−14082.6	−0.705287
$737$	−5325.59	−0.266175
$738$	0	0
$739$	14115.5	0.702636	0.351318	−	0.936256i	$-0.385734\pi$
$739$	14115.5	0.702636	0.351318	+	0.936256i	$0.385734\pi$
$740$	−11132.7	−0.553037
$741$	0	0
$742$	−69587.9	−3.44293
$743$	−17992.0	−0.888376	−0.444188	−	0.895934i	$-0.646508\pi$
$743$	−17992.0	−0.888376	−0.444188	+	0.895934i	$0.646508\pi$
$744$	0	0
$745$	−6193.84	−0.304597
$746$	−59799.0	−2.93485
$747$	0	0
$748$	−1791.20	−0.0875571
$749$	−60050.2	−2.92949
$750$	0	0
$751$	2055.99	0.0998989	0.0499495	−	0.998752i	$-0.484094\pi$
$751$	2055.99	0.0998989	0.0499495	+	0.998752i	$0.484094\pi$
$752$	−19159.6	−0.929095
$753$	0	0
$754$	−4650.19	−0.224602
$755$	−2695.89	−0.129952
$756$	0	0
$757$	12132.4	0.582508	0.291254	−	0.956646i	$-0.405927\pi$
$757$	12132.4	0.582508	0.291254	+	0.956646i	$0.405927\pi$
$758$	8427.65	0.403834
$759$	0	0
$760$	8995.23	0.429331
$761$	−9319.01	−0.443908	−0.221954	−	0.975057i	$-0.571243\pi$
$761$	−9319.01	−0.443908	−0.221954	+	0.975057i	$0.571243\pi$
$762$	0	0
$763$	−18463.3	−0.876037
$764$	−24560.5	−1.16305
$765$	0	0
$766$	20612.9	0.972288
$767$	−4053.67	−0.190834
$768$	0	0
$769$	38790.8	1.81903	0.909514	−	0.415672i	$-0.136454\pi$
$769$	38790.8	1.81903	0.909514	+	0.415672i	$0.136454\pi$
$770$	8274.90	0.387282
$771$	0	0
$772$	−3780.71	−0.176257
$773$	−12857.4	−0.598252	−0.299126	−	0.954214i	$-0.596695\pi$
$773$	−12857.4	−0.598252	−0.299126	+	0.954214i	$0.596695\pi$
$774$	0	0
$775$	3537.33	0.163954
$776$	−36103.8	−1.67017
$777$	0	0
$778$	49005.2	2.25825
$779$	4385.14	0.201687
$780$	0	0
$781$	6689.00	0.306468
$782$	−12155.2	−0.555844
$783$	0	0
$784$	21965.2	1.00060
$785$	4118.11	0.187238
$786$	0	0
$787$	−26862.0	−1.21668	−0.608339	−	0.793677i	$-0.708164\pi$
$787$	−26862.0	−1.21668	−0.608339	+	0.793677i	$0.708164\pi$
$788$	10163.5	0.459468
$789$	0	0
$790$	−37755.7	−1.70036
$791$	−2003.22	−0.0900457
$792$	0	0
$793$	5019.81	0.224790
$794$	−443.519	−0.0198235
$795$	0	0
$796$	−3521.81	−0.156818
$797$	12471.5	0.554285	0.277142	−	0.960829i	$-0.410613\pi$
$797$	12471.5	0.554285	0.277142	+	0.960829i	$0.410613\pi$
$798$	0	0
$799$	9707.23	0.429809
$800$	−6223.29	−0.275033
$801$	0	0
$802$	−63405.4	−2.79168
$803$	−5013.82	−0.220341
$804$	0	0
$805$	36325.9	1.59046
$806$	21389.9	0.934772
$807$	0	0
$808$	22275.3	0.969854
$809$	24760.8	1.07607	0.538037	−	0.842921i	$-0.319166\pi$
$809$	24760.8	1.07607	0.538037	+	0.842921i	$0.319166\pi$
$810$	0	0
$811$	11237.3	0.486556	0.243278	−	0.969957i	$-0.421777\pi$
$811$	11237.3	0.486556	0.243278	+	0.969957i	$0.421777\pi$
$812$	−5362.70	−0.231766
$813$	0	0
$814$	3394.99	0.146185
$815$	14720.8	0.632698
$816$	0	0
$817$	16910.8	0.724156
$818$	−44134.3	−1.88645
$819$	0	0
$820$	−13268.7	−0.565079
$821$	−39976.4	−1.69937	−0.849687	−	0.527287i	$-0.823209\pi$
$821$	−39976.4	−1.69937	−0.849687	+	0.527287i	$0.823209\pi$
$822$	0	0
$823$	36877.0	1.56191	0.780955	−	0.624588i	$-0.214733\pi$
$823$	36877.0	1.56191	0.780955	+	0.624588i	$0.214733\pi$
$824$	28424.1	1.20170
$825$	0	0
$826$	−7226.49	−0.304409
$827$	14311.9	0.601781	0.300890	−	0.953659i	$-0.402716\pi$
$827$	14311.9	0.601781	0.300890	+	0.953659i	$0.402716\pi$
$828$	0	0
$829$	−12629.7	−0.529130	−0.264565	−	0.964368i	$-0.585228\pi$
$829$	−12629.7	−0.529130	−0.264565	+	0.964368i	$0.585228\pi$
$830$	−809.172	−0.0338395
$831$	0	0
$832$	−60269.1	−2.51136
$833$	−11128.7	−0.462888
$834$	0	0
$835$	22752.3	0.942965
$836$	−3907.32	−0.161648
$837$	0	0
$838$	53907.4	2.22220
$839$	18683.1	0.768786	0.384393	−	0.923170i	$-0.374411\pi$
$839$	18683.1	0.768786	0.384393	+	0.923170i	$0.374411\pi$
$840$	0	0
$841$	−24254.8	−0.994497
$842$	−33007.3	−1.35096
$843$	0	0
$844$	3929.52	0.160260
$845$	37630.6	1.53199
$846$	0	0
$847$	40407.5	1.63922
$848$	−15530.9	−0.628932
$849$	0	0
$850$	−5371.56	−0.216756
$851$	14903.6	0.600340
$852$	0	0
$853$	−35648.7	−1.43094	−0.715468	−	0.698646i	$-0.753786\pi$
$853$	−35648.7	−1.43094	−0.715468	+	0.698646i	$0.753786\pi$
$854$	8948.83	0.358575
$855$	0	0
$856$	−60234.7	−2.40512
$857$	49860.8	1.98741	0.993707	−	0.112010i	$-0.0357290\pi$
$857$	49860.8	1.98741	0.993707	+	0.112010i	$0.0357290\pi$
$858$	0	0
$859$	21487.0	0.853466	0.426733	−	0.904378i	$-0.359664\pi$
$859$	21487.0	0.853466	0.426733	+	0.904378i	$0.359664\pi$
$860$	−51169.6	−2.02892
$861$	0	0
$862$	−13216.0	−0.522201
$863$	15067.6	0.594330	0.297165	−	0.954826i	$-0.403959\pi$
$863$	15067.6	0.594330	0.297165	+	0.954826i	$0.403959\pi$
$864$	0	0
$865$	−7568.60	−0.297503
$866$	15480.1	0.607431
$867$	0	0
$868$	24667.3	0.964588
$869$	7448.23	0.290752
$870$	0	0
$871$	−62472.8	−2.43032
$872$	−18520.0	−0.719229
$873$	0	0
$874$	−26515.4	−1.02620
$875$	46280.6	1.78808
$876$	0	0
$877$	−24852.6	−0.956912	−0.478456	−	0.878111i	$-0.658803\pi$
$877$	−24852.6	−0.956912	−0.478456	+	0.878111i	$0.658803\pi$
$878$	63523.7	2.44171
$879$	0	0
$880$	1846.83	0.0707460
$881$	−1840.51	−0.0703842	−0.0351921	−	0.999381i	$-0.511204\pi$
$881$	−1840.51	−0.0703842	−0.0351921	+	0.999381i	$0.511204\pi$
$882$	0	0
$883$	49803.7	1.89811	0.949054	−	0.315115i	$-0.102043\pi$
$883$	49803.7	1.89811	0.949054	+	0.315115i	$0.102043\pi$
$884$	−21012.0	−0.799445
$885$	0	0
$886$	55402.7	2.10078
$887$	−36314.7	−1.37467	−0.687333	−	0.726342i	$-0.741219\pi$
$887$	−36314.7	−1.37467	−0.687333	+	0.726342i	$0.741219\pi$
$888$	0	0
$889$	−5544.58	−0.209178
$890$	−13473.6	−0.507456
$891$	0	0
$892$	−80966.9	−3.03921
$893$	21175.4	0.793512
$894$	0	0
$895$	14835.0	0.554054
$896$	−83753.6	−3.12278
$897$	0	0
$898$	−33022.5	−1.22714
$899$	−617.300	−0.0229011
$900$	0	0
$901$	7868.75	0.290950
$902$	4046.38	0.149368
$903$	0	0
$904$	−2009.37	−0.0739278
$905$	−16691.0	−0.613068
$906$	0	0
$907$	33679.5	1.23297	0.616487	−	0.787365i	$-0.288555\pi$
$907$	33679.5	1.23297	0.616487	+	0.787365i	$0.288555\pi$
$908$	5633.83	0.205909
$909$	0	0
$910$	97070.3	3.53610
$911$	−32437.2	−1.17968	−0.589841	−	0.807519i	$-0.700810\pi$
$911$	−32437.2	−1.17968	−0.589841	+	0.807519i	$0.700810\pi$
$912$	0	0
$913$	159.629	0.00578636
$914$	48958.8	1.77179
$915$	0	0
$916$	20452.4	0.737736
$917$	58919.7	2.12181
$918$	0	0
$919$	−3511.45	−0.126042	−0.0630208	−	0.998012i	$-0.520073\pi$
$919$	−3511.45	−0.126042	−0.0630208	+	0.998012i	$0.520073\pi$
$920$	36437.5	1.30577
$921$	0	0
$922$	−2978.87	−0.106403
$923$	78466.6	2.79822
$924$	0	0
$925$	6586.11	0.234108
$926$	28825.5	1.02296
$927$	0	0
$928$	1086.03	0.0384166
$929$	−13527.2	−0.477733	−0.238866	−	0.971052i	$-0.576776\pi$
$929$	−13527.2	−0.477733	−0.238866	+	0.971052i	$0.576776\pi$
$930$	0	0
$931$	−24276.1	−0.854583
$932$	−49968.6	−1.75620
$933$	0	0
$934$	−3883.87	−0.136064
$935$	−935.695	−0.0327278
$936$	0	0
$937$	−8862.80	−0.309002	−0.154501	−	0.987993i	$-0.549377\pi$
$937$	−8862.80	−0.309002	−0.154501	+	0.987993i	$0.549377\pi$
$938$	−111371.	−3.87674
$939$	0	0
$940$	−64073.4	−2.22324
$941$	−22824.0	−0.790693	−0.395346	−	0.918532i	$-0.629375\pi$
$941$	−22824.0	−0.790693	−0.395346	+	0.918532i	$0.629375\pi$
$942$	0	0
$943$	17763.1	0.613412
$944$	−1612.84	−0.0556074
$945$	0	0
$946$	15604.4	0.536305
$947$	26308.4	0.902754	0.451377	−	0.892333i	$-0.350933\pi$
$947$	26308.4	0.902754	0.451377	+	0.892333i	$0.350933\pi$
$948$	0	0
$949$	−58815.5	−2.01184
$950$	−11717.5	−0.400176
$951$	0	0
$952$	−17011.8	−0.579155
$953$	−11947.5	−0.406106	−0.203053	−	0.979168i	$-0.565086\pi$
$953$	−11947.5	−0.406106	−0.203053	+	0.979168i	$0.565086\pi$
$954$	0	0
$955$	−12830.1	−0.434734
$956$	22118.9	0.748300
$957$	0	0
$958$	−77200.5	−2.60358
$959$	33403.5	1.12477
$960$	0	0
$961$	−26951.5	−0.904688
$962$	39825.5	1.33475
$963$	0	0
$964$	49932.2	1.66826
$965$	−1974.99	−0.0658830
$966$	0	0
$967$	21505.3	0.715163	0.357581	−	0.933882i	$-0.383601\pi$
$967$	21505.3	0.715163	0.357581	+	0.933882i	$0.383601\pi$
$968$	40531.7	1.34580
$969$	0	0
$970$	−41527.9	−1.37462
$971$	−45685.3	−1.50990	−0.754950	−	0.655783i	$-0.772339\pi$
$971$	−45685.3	−1.50990	−0.754950	+	0.655783i	$0.772339\pi$
$972$	0	0
$973$	−28579.4	−0.941636
$974$	−12971.2	−0.426719
$975$	0	0
$976$	1997.24	0.0655020
$977$	−31710.3	−1.03839	−0.519193	−	0.854657i	$-0.673767\pi$
$977$	−31710.3	−1.03839	−0.519193	+	0.854657i	$0.673767\pi$
$978$	0	0
$979$	2657.99	0.0867721
$980$	73455.7	2.39434
$981$	0	0
$982$	−39716.8	−1.29065
$983$	−46038.9	−1.49381	−0.746904	−	0.664932i	$-0.768460\pi$
$983$	−46038.9	−1.49381	−0.746904	+	0.664932i	$0.768460\pi$
$984$	0	0
$985$	5309.28	0.171744
$986$	937.392	0.0302765
$987$	0	0
$988$	−45835.6	−1.47593
$989$	68501.8	2.20246
$990$	0	0
$991$	14394.9	0.461420	0.230710	−	0.973023i	$-0.425895\pi$
$991$	14394.9	0.461420	0.230710	+	0.973023i	$0.425895\pi$
$992$	−4995.50	−0.159886
$993$	0	0
$994$	139883.	4.46359
$995$	−1839.74	−0.0586167
$996$	0	0
$997$	−33473.4	−1.06330	−0.531652	−	0.846963i	$-0.678428\pi$
$997$	−33473.4	−1.06330	−0.531652	+	0.846963i	$0.678428\pi$
$998$	−65505.0	−2.07768
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.f.1.1		3
3.2	odd	2		51.4.a.e.1.3	✓	3
4.3	odd	2		2448.4.a.bd.1.3		3
12.11	even	2		816.4.a.s.1.1		3
15.14	odd	2		1275.4.a.q.1.1		3
21.20	even	2		2499.4.a.n.1.3		3
51.50	odd	2		867.4.a.k.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.e.1.3	✓	3	3.2	odd	2
153.4.a.f.1.1		3	1.1	even	1	trivial
816.4.a.s.1.1		3	12.11	even	2
867.4.a.k.1.3		3	51.50	odd	2
1275.4.a.q.1.1		3	15.14	odd	2
2448.4.a.bd.1.3		3	4.3	odd	2
2499.4.a.n.1.3		3	21.20	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)

\(f(q)\)	\(=\)	\(q-4.75985 q^{2} +14.6562 q^{4} +7.65616 q^{5} -31.5852 q^{7} -31.6823 q^{8} +O(q^{10})\)	\(q-4.75985 q^{2} +14.6562 q^{4} +7.65616 q^{5} -31.5852 q^{7} -31.6823 q^{8} -36.4422 q^{10} +7.18910 q^{11} +84.3331 q^{13} +150.341 q^{14} +33.5537 q^{16} -17.0000 q^{17} -37.0838 q^{19} +112.210 q^{20} -34.2190 q^{22} -150.218 q^{23} -66.3832 q^{25} -401.413 q^{26} -462.918 q^{28} +11.5846 q^{29} -53.2865 q^{31} +93.7478 q^{32} +80.9174 q^{34} -241.822 q^{35} -99.2134 q^{37} +176.513 q^{38} -242.565 q^{40} -118.249 q^{41} -456.016 q^{43} +105.365 q^{44} +715.014 q^{46} -571.014 q^{47} +654.627 q^{49} +315.974 q^{50} +1236.00 q^{52} -462.867 q^{53} +55.0409 q^{55} +1000.69 q^{56} -55.1407 q^{58} -48.0674 q^{59} +59.5236 q^{61} +253.636 q^{62} -714.655 q^{64} +645.668 q^{65} -740.787 q^{67} -249.155 q^{68} +1151.03 q^{70} +930.437 q^{71} -697.419 q^{73} +472.241 q^{74} -543.506 q^{76} -227.070 q^{77} +1036.04 q^{79} +256.893 q^{80} +562.849 q^{82} +22.2043 q^{83} -130.155 q^{85} +2170.57 q^{86} -227.767 q^{88} +369.726 q^{89} -2663.68 q^{91} -2201.62 q^{92} +2717.94 q^{94} -283.920 q^{95} +1139.56 q^{97} -3115.93 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8}+O(q^{10}) \) 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8	\( 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8} - 38 q^{10} - 34 q^{11} + 36 q^{13} + 104 q^{14} - 79 q^{16} - 51 q^{17} - 142 q^{19} + 126 q^{20} - 248 q^{22} - 110 q^{23} - 193 q^{25} - 154 q^{26} - 472 q^{28} - 90 q^{29} - 148 q^{31} + 151 q^{32} + 85 q^{34} - 416 q^{35} + 110 q^{37} - 80 q^{38} - 202 q^{40} - 720 q^{41} - 146 q^{43} + 192 q^{44} + 748 q^{46} - 500 q^{47} + 379 q^{49} + 385 q^{50} + 1218 q^{52} - 610 q^{53} + 430 q^{55} + 1368 q^{56} + 1006 q^{58} + 216 q^{59} - 18 q^{61} + 904 q^{62} + 377 q^{64} + 966 q^{65} - 1404 q^{67} - 221 q^{68} + 1472 q^{70} + 960 q^{71} - 794 q^{73} + 1874 q^{74} - 392 q^{76} - 48 q^{77} - 276 q^{79} + 1130 q^{80} + 382 q^{82} + 1552 q^{83} + 136 q^{85} - 16 q^{86} + 1724 q^{88} - 1394 q^{89} - 3968 q^{91} - 2244 q^{92} + 3960 q^{94} + 602 q^{95} + 402 q^{97} - 4109 q^{98}+O(q^{100}) \) 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8 - 38 * q^10 - 34 * q^11 + 36 * q^13 + 104 * q^14 - 79 * q^16 - 51 * q^17 - 142 * q^19 + 126 * q^20 - 248 * q^22 - 110 * q^23 - 193 * q^25 - 154 * q^26 - 472 * q^28 - 90 * q^29 - 148 * q^31 + 151 * q^32 + 85 * q^34 - 416 * q^35 + 110 * q^37 - 80 * q^38 - 202 * q^40 - 720 * q^41 - 146 * q^43 + 192 * q^44 + 748 * q^46 - 500 * q^47 + 379 * q^49 + 385 * q^50 + 1218 * q^52 - 610 * q^53 + 430 * q^55 + 1368 * q^56 + 1006 * q^58 + 216 * q^59 - 18 * q^61 + 904 * q^62 + 377 * q^64 + 966 * q^65 - 1404 * q^67 - 221 * q^68 + 1472 * q^70 + 960 * q^71 - 794 * q^73 + 1874 * q^74 - 392 * q^76 - 48 * q^77 - 276 * q^79 + 1130 * q^80 + 382 * q^82 + 1552 * q^83 + 136 * q^85 - 16 * q^86 + 1724 * q^88 - 1394 * q^89 - 3968 * q^91 - 2244 * q^92 + 3960 * q^94 + 602 * q^95 + 402 * q^97 - 4109 * q^98

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