LMFDB - Embedded newform 153.4.a.g.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:	$0$
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			1.1
Root			$-0.287410$ 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)

$f(q)$	$=$	$q-4.67129 q^{2} +13.8209 q^{4} +11.9174 q^{5} +26.1222 q^{7} -27.1912 q^{8} +O(q^{10})$	\(q-4.67129 q^{2} +13.8209 q^{4} +11.9174 q^{5} +26.1222 q^{7} -27.1912 q^{8} -55.6696 q^{10} +3.24412 q^{11} -20.0515 q^{13} -122.024 q^{14} +16.4506 q^{16} +17.0000 q^{17} +57.3466 q^{19} +164.709 q^{20} -15.1542 q^{22} -77.0438 q^{23} +17.0243 q^{25} +93.6662 q^{26} +361.033 q^{28} +286.162 q^{29} -8.54816 q^{31} +140.684 q^{32} -79.4119 q^{34} +311.309 q^{35} +357.982 q^{37} -267.882 q^{38} -324.049 q^{40} -194.467 q^{41} -74.2619 q^{43} +44.8367 q^{44} +359.894 q^{46} -23.6130 q^{47} +339.369 q^{49} -79.5255 q^{50} -277.130 q^{52} -104.330 q^{53} +38.6614 q^{55} -710.295 q^{56} -1336.75 q^{58} -249.363 q^{59} -370.384 q^{61} +39.9309 q^{62} -788.781 q^{64} -238.961 q^{65} +939.650 q^{67} +234.956 q^{68} -1454.21 q^{70} +520.197 q^{71} +348.741 q^{73} -1672.24 q^{74} +792.583 q^{76} +84.7434 q^{77} -953.827 q^{79} +196.049 q^{80} +908.412 q^{82} +1414.28 q^{83} +202.596 q^{85} +346.899 q^{86} -88.2115 q^{88} +486.132 q^{89} -523.788 q^{91} -1064.82 q^{92} +110.303 q^{94} +683.422 q^{95} -685.281 q^{97} -1585.29 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.67129	−1.65155	−0.825775	−	0.564000i	$-0.809262\pi$
$2$	−4.67129	−1.65155	−0.825775	+	0.564000i	$0.809262\pi$
$3$	0	0
$3$	0	0
$4$	13.8209	1.72762
$5$	11.9174	1.06592	0.532962	−	0.846139i	$-0.321079\pi$
$5$	11.9174	1.06592	0.532962	+	0.846139i	$0.321079\pi$
$6$	0	0
$7$	26.1222	1.41047	0.705233	−	0.708975i	$-0.250842\pi$
$7$	26.1222	1.41047	0.705233	+	0.708975i	$0.250842\pi$
$8$	−27.1912	−1.20169
$9$	0	0
$10$	−55.6696	−1.76043
$11$	3.24412	0.0889216	0.0444608	−	0.999011i	$-0.485843\pi$
$11$	3.24412	0.0889216	0.0444608	+	0.999011i	$0.485843\pi$
$12$	0	0
$13$	−20.0515	−0.427790	−0.213895	−	0.976857i	$-0.568615\pi$
$13$	−20.0515	−0.427790	−0.213895	+	0.976857i	$0.568615\pi$
$14$	−122.024	−2.32945
$15$	0	0
$16$	16.4506	0.257041
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	57.3466	0.692432	0.346216	−	0.938155i	$-0.387466\pi$
$19$	57.3466	0.692432	0.346216	+	0.938155i	$0.387466\pi$
$20$	164.709	1.84151
$21$	0	0
$22$	−15.1542	−0.146858
$23$	−77.0438	−0.698467	−0.349233	−	0.937036i	$-0.613558\pi$
$23$	−77.0438	−0.698467	−0.349233	+	0.937036i	$0.613558\pi$
$24$	0	0
$25$	17.0243	0.136195
$26$	93.6662	0.706517
$27$	0	0
$28$	361.033	2.43674
$29$	286.162	1.83238	0.916190	−	0.400744i	$-0.131248\pi$
$29$	286.162	1.83238	0.916190	+	0.400744i	$0.131248\pi$
$30$	0	0
$31$	−8.54816	−0.0495256	−0.0247628	−	0.999693i	$-0.507883\pi$
$31$	−8.54816	−0.0495256	−0.0247628	+	0.999693i	$0.507883\pi$
$32$	140.684	0.777178
$33$	0	0
$34$	−79.4119	−0.400560
$35$	311.309	1.50345
$36$	0	0
$37$	357.982	1.59059	0.795296	−	0.606221i	$-0.207315\pi$
$37$	357.982	1.59059	0.795296	+	0.606221i	$0.207315\pi$
$38$	−267.882	−1.14359
$39$	0	0
$40$	−324.049	−1.28091
$41$	−194.467	−0.740748	−0.370374	−	0.928883i	$-0.620770\pi$
$41$	−194.467	−0.740748	−0.370374	+	0.928883i	$0.620770\pi$
$42$	0	0
$43$	−74.2619	−0.263368	−0.131684	−	0.991292i	$-0.542038\pi$
$43$	−74.2619	−0.263368	−0.131684	+	0.991292i	$0.542038\pi$
$44$	44.8367	0.153622
$45$	0	0
$46$	359.894	1.15355
$47$	−23.6130	−0.0732831	−0.0366416	−	0.999328i	$-0.511666\pi$
$47$	−23.6130	−0.0732831	−0.0366416	+	0.999328i	$0.511666\pi$
$48$	0	0
$49$	339.369	0.989415
$50$	−79.5255	−0.224932
$51$	0	0
$52$	−277.130	−0.739058
$53$	−104.330	−0.270393	−0.135197	−	0.990819i	$-0.543167\pi$
$53$	−104.330	−0.270393	−0.135197	+	0.990819i	$0.543167\pi$
$54$	0	0
$55$	38.6614	0.0947837
$56$	−710.295	−1.69495
$57$	0	0
$58$	−1336.75	−3.02627
$59$	−249.363	−0.550243	−0.275122	−	0.961409i	$-0.588718\pi$
$59$	−249.363	−0.550243	−0.275122	+	0.961409i	$0.588718\pi$
$60$	0	0
$61$	−370.384	−0.777424	−0.388712	−	0.921359i	$-0.627080\pi$
$61$	−370.384	−0.777424	−0.388712	+	0.921359i	$0.627080\pi$
$62$	39.9309	0.0817940
$63$	0	0
$64$	−788.781	−1.54059
$65$	−238.961	−0.455992
$66$	0	0
$67$	939.650	1.71338	0.856691	−	0.515830i	$-0.172517\pi$
$67$	939.650	1.71338	0.856691	+	0.515830i	$0.172517\pi$
$68$	234.956	0.419008
$69$	0	0
$70$	−1454.21	−2.48302
$71$	520.197	0.869522	0.434761	−	0.900546i	$-0.356833\pi$
$71$	520.197	0.869522	0.434761	+	0.900546i	$0.356833\pi$
$72$	0	0
$73$	348.741	0.559137	0.279568	−	0.960126i	$-0.409809\pi$
$73$	348.741	0.559137	0.279568	+	0.960126i	$0.409809\pi$
$74$	−1672.24	−2.62694
$75$	0	0
$76$	792.583	1.19626
$77$	84.7434	0.125421
$78$	0	0
$79$	−953.827	−1.35840	−0.679202	−	0.733951i	$-0.737674\pi$
$79$	−953.827	−1.35840	−0.679202	+	0.733951i	$0.737674\pi$
$80$	196.049	0.273986
$81$	0	0
$82$	908.412	1.22338
$83$	1414.28	1.87033	0.935166	−	0.354211i	$-0.115250\pi$
$83$	1414.28	1.87033	0.935166	+	0.354211i	$0.115250\pi$
$84$	0	0
$85$	202.596	0.258525
$86$	346.899	0.434966
$87$	0	0
$88$	−88.2115	−0.106857
$89$	486.132	0.578987	0.289493	−	0.957180i	$-0.406513\pi$
$89$	486.132	0.578987	0.289493	+	0.957180i	$0.406513\pi$
$90$	0	0
$91$	−523.788	−0.603384
$92$	−1064.82	−1.20668
$93$	0	0
$94$	110.303	0.121031
$95$	683.422	0.738080
$96$	0	0
$97$	−685.281	−0.717317	−0.358659	−	0.933469i	$-0.616766\pi$
$97$	−685.281	−0.717317	−0.358659	+	0.933469i	$0.616766\pi$
$98$	−1585.29	−1.63407
$99$	0	0
$100$	235.292	0.235292
$101$	−864.755	−0.851944	−0.425972	−	0.904736i	$-0.640068\pi$
$101$	−864.755	−0.851944	−0.425972	+	0.904736i	$0.640068\pi$
$102$	0	0
$103$	1880.91	1.79933	0.899665	−	0.436580i	$-0.143810\pi$
$103$	1880.91	1.79933	0.899665	+	0.436580i	$0.143810\pi$
$104$	545.224	0.514073
$105$	0	0
$106$	487.355	0.446567
$107$	32.8149	0.0296480	0.0148240	−	0.999890i	$-0.495281\pi$
$107$	32.8149	0.0296480	0.0148240	+	0.999890i	$0.495281\pi$
$108$	0	0
$109$	528.727	0.464613	0.232307	−	0.972643i	$-0.425373\pi$
$109$	528.727	0.464613	0.232307	+	0.972643i	$0.425373\pi$
$110$	−180.599	−0.156540
$111$	0	0
$112$	429.727	0.362548
$113$	−414.691	−0.345229	−0.172614	−	0.984989i	$-0.555221\pi$
$113$	−414.691	−0.345229	−0.172614	+	0.984989i	$0.555221\pi$
$114$	0	0
$115$	−918.161	−0.744513
$116$	3955.03	3.16565
$117$	0	0
$118$	1164.85	0.908754
$119$	444.077	0.342088
$120$	0	0
$121$	−1320.48	−0.992093
$122$	1730.17	1.28395
$123$	0	0
$124$	−118.143	−0.0855613
$125$	−1286.79	−0.920751
$126$	0	0
$127$	596.093	0.416494	0.208247	−	0.978076i	$-0.433224\pi$
$127$	596.093	0.416494	0.208247	+	0.978076i	$0.433224\pi$
$128$	2559.15	1.76718
$129$	0	0
$130$	1116.26	0.753094
$131$	−121.819	−0.0812472	−0.0406236	−	0.999175i	$-0.512934\pi$
$131$	−121.819	−0.0812472	−0.0406236	+	0.999175i	$0.512934\pi$
$132$	0	0
$133$	1498.02	0.976652
$134$	−4389.38	−2.82973
$135$	0	0
$136$	−462.251	−0.291454
$137$	897.365	0.559614	0.279807	−	0.960056i	$-0.409730\pi$
$137$	897.365	0.559614	0.279807	+	0.960056i	$0.409730\pi$
$138$	0	0
$139$	−2113.61	−1.28974	−0.644871	−	0.764292i	$-0.723089\pi$
$139$	−2113.61	−1.28974	−0.644871	+	0.764292i	$0.723089\pi$
$140$	4302.57	2.59738
$141$	0	0
$142$	−2429.99	−1.43606
$143$	−65.0493	−0.0380398
$144$	0	0
$145$	3410.31	1.95318
$146$	−1629.07	−0.923442
$147$	0	0
$148$	4947.65	2.74793
$149$	−2580.76	−1.41895	−0.709476	−	0.704729i	$-0.751068\pi$
$149$	−2580.76	−1.41895	−0.709476	+	0.704729i	$0.751068\pi$
$150$	0	0
$151$	1342.77	0.723662	0.361831	−	0.932244i	$-0.382152\pi$
$151$	1342.77	0.723662	0.361831	+	0.932244i	$0.382152\pi$
$152$	−1559.32	−0.832091
$153$	0	0
$154$	−395.861	−0.207139
$155$	−101.872	−0.0527906
$156$	0	0
$157$	−2495.82	−1.26871	−0.634357	−	0.773041i	$-0.718735\pi$
$157$	−2495.82	−1.26871	−0.634357	+	0.773041i	$0.718735\pi$
$158$	4455.60	2.24347
$159$	0	0
$160$	1676.59	0.828413
$161$	−2012.55	−0.985164
$162$	0	0
$163$	1961.58	0.942595	0.471297	−	0.881974i	$-0.343786\pi$
$163$	1961.58	0.942595	0.471297	+	0.881974i	$0.343786\pi$
$164$	−2687.72	−1.27973
$165$	0	0
$166$	−6606.51	−3.08894
$167$	−2179.24	−1.00979	−0.504894	−	0.863182i	$-0.668468\pi$
$167$	−2179.24	−1.00979	−0.504894	+	0.863182i	$0.668468\pi$
$168$	0	0
$169$	−1794.94	−0.816995
$170$	−946.383	−0.426966
$171$	0	0
$172$	−1026.37	−0.454999
$173$	−3111.45	−1.36739	−0.683697	−	0.729766i	$-0.739629\pi$
$173$	−3111.45	−1.36739	−0.683697	+	0.729766i	$0.739629\pi$
$174$	0	0
$175$	444.713	0.192098
$176$	53.3677	0.0228565
$177$	0	0
$178$	−2270.86	−0.956226
$179$	−810.106	−0.338269	−0.169135	−	0.985593i	$-0.554097\pi$
$179$	−810.106	−0.338269	−0.169135	+	0.985593i	$0.554097\pi$
$180$	0	0
$181$	−3356.23	−1.37827	−0.689134	−	0.724634i	$-0.742009\pi$
$181$	−3356.23	−1.37827	−0.689134	+	0.724634i	$0.742009\pi$
$182$	2446.77	0.996519
$183$	0	0
$184$	2094.92	0.839343
$185$	4266.22	1.69545
$186$	0	0
$187$	55.1500	0.0215667
$188$	−326.353	−0.126605
$189$	0	0
$190$	−3192.46	−1.21898
$191$	−1338.41	−0.507038	−0.253519	−	0.967330i	$-0.581588\pi$
$191$	−1338.41	−0.507038	−0.253519	+	0.967330i	$0.581588\pi$
$192$	0	0
$193$	−227.465	−0.0848358	−0.0424179	−	0.999100i	$-0.513506\pi$
$193$	−227.465	−0.0848358	−0.0424179	+	0.999100i	$0.513506\pi$
$194$	3201.15	1.18468
$195$	0	0
$196$	4690.40	1.70933
$197$	−815.549	−0.294952	−0.147476	−	0.989066i	$-0.547115\pi$
$197$	−815.549	−0.294952	−0.147476	+	0.989066i	$0.547115\pi$
$198$	0	0
$199$	1866.90	0.665030	0.332515	−	0.943098i	$-0.392103\pi$
$199$	1866.90	0.665030	0.332515	+	0.943098i	$0.392103\pi$
$200$	−462.912	−0.163664
$201$	0	0
$202$	4039.52	1.40703
$203$	7475.19	2.58451
$204$	0	0
$205$	−2317.54	−0.789581
$206$	−8786.25	−2.97168
$207$	0	0
$208$	−329.859	−0.109960
$209$	186.039	0.0615721
$210$	0	0
$211$	−1102.88	−0.359836	−0.179918	−	0.983682i	$-0.557583\pi$
$211$	−1102.88	−0.359836	−0.179918	+	0.983682i	$0.557583\pi$
$212$	−1441.94	−0.467135
$213$	0	0
$214$	−153.288	−0.0489651
$215$	−885.008	−0.280731
$216$	0	0
$217$	−223.297	−0.0698542
$218$	−2469.84	−0.767332
$219$	0	0
$220$	534.337	0.163750
$221$	−340.875	−0.103754
$222$	0	0
$223$	−568.848	−0.170820	−0.0854100	−	0.996346i	$-0.527220\pi$
$223$	−568.848	−0.170820	−0.0854100	+	0.996346i	$0.527220\pi$
$224$	3674.98	1.09618
$225$	0	0
$226$	1937.14	0.570163
$227$	−2106.99	−0.616061	−0.308030	−	0.951377i	$-0.599670\pi$
$227$	−2106.99	−0.616061	−0.308030	+	0.951377i	$0.599670\pi$
$228$	0	0
$229$	4336.30	1.25131	0.625656	−	0.780099i	$-0.284831\pi$
$229$	4336.30	1.25131	0.625656	+	0.780099i	$0.284831\pi$
$230$	4289.00	1.22960
$231$	0	0
$232$	−7781.11	−2.20196
$233$	4517.39	1.27014	0.635072	−	0.772453i	$-0.280970\pi$
$233$	4517.39	1.27014	0.635072	+	0.772453i	$0.280970\pi$
$234$	0	0
$235$	−281.405	−0.0781143
$236$	−3446.43	−0.950609
$237$	0	0
$238$	−2074.41	−0.564976
$239$	−5300.88	−1.43467	−0.717333	−	0.696731i	$-0.754637\pi$
$239$	−5300.88	−1.43467	−0.717333	+	0.696731i	$0.754637\pi$
$240$	0	0
$241$	−1368.82	−0.365864	−0.182932	−	0.983126i	$-0.558559\pi$
$241$	−1368.82	−0.365864	−0.182932	+	0.983126i	$0.558559\pi$
$242$	6168.32	1.63849
$243$	0	0
$244$	−5119.05	−1.34309
$245$	4044.40	1.05464
$246$	0	0
$247$	−1149.88	−0.296216
$248$	232.435	0.0595146
$249$	0	0
$250$	6010.96	1.52067
$251$	5547.63	1.39507	0.697536	−	0.716549i	$-0.254280\pi$
$251$	5547.63	1.39507	0.697536	+	0.716549i	$0.254280\pi$
$252$	0	0
$253$	−249.939	−0.0621088
$254$	−2784.52	−0.687860
$255$	0	0
$256$	−5644.28	−1.37800
$257$	193.949	0.0470748	0.0235374	−	0.999723i	$-0.492507\pi$
$257$	193.949	0.0470748	0.0235374	+	0.999723i	$0.492507\pi$
$258$	0	0
$259$	9351.29	2.24348
$260$	−3302.67	−0.787780
$261$	0	0
$262$	569.052	0.134184
$263$	1345.63	0.315494	0.157747	−	0.987480i	$-0.449577\pi$
$263$	1345.63	0.315494	0.157747	+	0.987480i	$0.449577\pi$
$264$	0	0
$265$	−1243.34	−0.288218
$266$	−6997.67	−1.61299
$267$	0	0
$268$	12986.8	2.96007
$269$	3083.04	0.698797	0.349398	−	0.936974i	$-0.386386\pi$
$269$	3083.04	0.698797	0.349398	+	0.936974i	$0.386386\pi$
$270$	0	0
$271$	−422.163	−0.0946294	−0.0473147	−	0.998880i	$-0.515066\pi$
$271$	−422.163	−0.0946294	−0.0473147	+	0.998880i	$0.515066\pi$
$272$	279.661	0.0623416
$273$	0	0
$274$	−4191.85	−0.924230
$275$	55.2288	0.0121106
$276$	0	0
$277$	−8260.00	−1.79168	−0.895840	−	0.444377i	$-0.853425\pi$
$277$	−8260.00	−1.79168	−0.895840	+	0.444377i	$0.853425\pi$
$278$	9873.28	2.13007
$279$	0	0
$280$	−8464.86	−1.80669
$281$	−3321.91	−0.705226	−0.352613	−	0.935769i	$-0.614707\pi$
$281$	−3321.91	−0.705226	−0.352613	+	0.935769i	$0.614707\pi$
$282$	0	0
$283$	−7954.43	−1.67082	−0.835409	−	0.549629i	$-0.814769\pi$
$283$	−7954.43	−1.67082	−0.835409	+	0.549629i	$0.814769\pi$
$284$	7189.61	1.50220
$285$	0	0
$286$	303.864	0.0628246
$287$	−5079.91	−1.04480
$288$	0	0
$289$	289.000	0.0588235
$290$	−15930.5	−3.22577
$291$	0	0
$292$	4819.92	0.965974
$293$	1171.99	0.233681	0.116841	−	0.993151i	$-0.462723\pi$
$293$	1171.99	0.233681	0.116841	+	0.993151i	$0.462723\pi$
$294$	0	0
$295$	−2971.76	−0.586517
$296$	−9733.98	−1.91141
$297$	0	0
$298$	12055.5	2.34347
$299$	1544.84	0.298798
$300$	0	0
$301$	−1939.88	−0.371472
$302$	−6272.46	−1.19516
$303$	0	0
$304$	943.387	0.177983
$305$	−4414.01	−0.828675
$306$	0	0
$307$	865.763	0.160950	0.0804751	−	0.996757i	$-0.474356\pi$
$307$	865.763	0.160950	0.0804751	+	0.996757i	$0.474356\pi$
$308$	1171.23	0.216679
$309$	0	0
$310$	475.872	0.0871862
$311$	−6994.83	−1.27537	−0.637685	−	0.770297i	$-0.720108\pi$
$311$	−6994.83	−1.27537	−0.637685	+	0.770297i	$0.720108\pi$
$312$	0	0
$313$	3442.33	0.621635	0.310818	−	0.950470i	$-0.399397\pi$
$313$	3442.33	0.621635	0.310818	+	0.950470i	$0.399397\pi$
$314$	11658.7	2.09534
$315$	0	0
$316$	−13182.8	−2.34680
$317$	−2066.15	−0.366078	−0.183039	−	0.983106i	$-0.558593\pi$
$317$	−2066.15	−0.366078	−0.183039	+	0.983106i	$0.558593\pi$
$318$	0	0
$319$	928.344	0.162938
$320$	−9400.22	−1.64215
$321$	0	0
$322$	9401.21	1.62705
$323$	974.892	0.167939
$324$	0	0
$325$	−341.362	−0.0582627
$326$	−9163.11	−1.55674
$327$	0	0
$328$	5287.80	0.890152
$329$	−616.823	−0.103363
$330$	0	0
$331$	9027.44	1.49907	0.749536	−	0.661964i	$-0.230277\pi$
$331$	9027.44	1.49907	0.749536	+	0.661964i	$0.230277\pi$
$332$	19546.7	3.23121
$333$	0	0
$334$	10179.8	1.66771
$335$	11198.2	1.82633
$336$	0	0
$337$	204.309	0.0330250	0.0165125	−	0.999864i	$-0.494744\pi$
$337$	204.309	0.0330250	0.0165125	+	0.999864i	$0.494744\pi$
$338$	8384.67	1.34931
$339$	0	0
$340$	2800.06	0.446631
$341$	−27.7312	−0.00440390
$342$	0	0
$343$	−94.8397	−0.0149296
$344$	2019.27	0.316488
$345$	0	0
$346$	14534.5	2.25832
$347$	−143.063	−0.0221326	−0.0110663	−	0.999939i	$-0.503523\pi$
$347$	−143.063	−0.0221326	−0.0110663	+	0.999939i	$0.503523\pi$
$348$	0	0
$349$	−3998.42	−0.613268	−0.306634	−	0.951828i	$-0.599203\pi$
$349$	−3998.42	−0.613268	−0.306634	+	0.951828i	$0.599203\pi$
$350$	−2077.38	−0.317259
$351$	0	0
$352$	456.396	0.0691079
$353$	5809.57	0.875956	0.437978	−	0.898986i	$-0.355695\pi$
$353$	5809.57	0.875956	0.437978	+	0.898986i	$0.355695\pi$
$354$	0	0
$355$	6199.39	0.926844
$356$	6718.79	1.00027
$357$	0	0
$358$	3784.24	0.558668
$359$	4895.37	0.719687	0.359844	−	0.933013i	$-0.382830\pi$
$359$	4895.37	0.719687	0.359844	+	0.933013i	$0.382830\pi$
$360$	0	0
$361$	−3570.37	−0.520538
$362$	15677.9	2.27628
$363$	0	0
$364$	−7239.24	−1.04242
$365$	4156.08	0.595998
$366$	0	0
$367$	528.151	0.0751206	0.0375603	−	0.999294i	$-0.488041\pi$
$367$	528.151	0.0751206	0.0375603	+	0.999294i	$0.488041\pi$
$368$	−1267.42	−0.179535
$369$	0	0
$370$	−19928.7	−2.80012
$371$	−2725.33	−0.381380
$372$	0	0
$373$	−10113.5	−1.40390	−0.701950	−	0.712226i	$-0.747687\pi$
$373$	−10113.5	−1.40390	−0.701950	+	0.712226i	$0.747687\pi$
$374$	−257.621	−0.0356184
$375$	0	0
$376$	642.066	0.0880639
$377$	−5737.98	−0.783875
$378$	0	0
$379$	729.385	0.0988548	0.0494274	−	0.998778i	$-0.484260\pi$
$379$	729.385	0.0988548	0.0494274	+	0.998778i	$0.484260\pi$
$380$	9445.52	1.27512
$381$	0	0
$382$	6252.12	0.837399
$383$	1608.08	0.214540	0.107270	−	0.994230i	$-0.465789\pi$
$383$	1608.08	0.214540	0.107270	+	0.994230i	$0.465789\pi$
$384$	0	0
$385$	1009.92	0.133689
$386$	1062.56	0.140111
$387$	0	0
$388$	−9471.22	−1.23925
$389$	−9824.09	−1.28047	−0.640233	−	0.768181i	$-0.721162\pi$
$389$	−9824.09	−1.28047	−0.640233	+	0.768181i	$0.721162\pi$
$390$	0	0
$391$	−1309.74	−0.169403
$392$	−9227.87	−1.18897
$393$	0	0
$394$	3809.66	0.487127
$395$	−11367.1	−1.44796
$396$	0	0
$397$	2876.88	0.363694	0.181847	−	0.983327i	$-0.441792\pi$
$397$	2876.88	0.363694	0.181847	+	0.983327i	$0.441792\pi$
$398$	−8720.82	−1.09833
$399$	0	0
$400$	280.061	0.0350076
$401$	6515.91	0.811444	0.405722	−	0.913996i	$-0.367020\pi$
$401$	6515.91	0.811444	0.405722	+	0.913996i	$0.367020\pi$
$402$	0	0
$403$	171.403	0.0211866
$404$	−11951.7	−1.47183
$405$	0	0
$406$	−34918.8	−4.26845
$407$	1161.34	0.141438
$408$	0	0
$409$	−8870.10	−1.07237	−0.536183	−	0.844101i	$-0.680134\pi$
$409$	−8870.10	−1.07237	−0.536183	+	0.844101i	$0.680134\pi$
$410$	10825.9	1.30403
$411$	0	0
$412$	25995.9	3.10855
$413$	−6513.92	−0.776099
$414$	0	0
$415$	16854.5	1.99363
$416$	−2820.93	−0.332469
$417$	0	0
$418$	−869.041	−0.101689
$419$	1009.53	0.117706	0.0588531	−	0.998267i	$-0.481256\pi$
$419$	1009.53	0.117706	0.0588531	+	0.998267i	$0.481256\pi$
$420$	0	0
$421$	−3253.60	−0.376652	−0.188326	−	0.982107i	$-0.560306\pi$
$421$	−3253.60	−0.376652	−0.188326	+	0.982107i	$0.560306\pi$
$422$	5151.87	0.594287
$423$	0	0
$424$	2836.86	0.324930
$425$	289.413	0.0330320
$426$	0	0
$427$	−9675.25	−1.09653
$428$	453.532	0.0512203
$429$	0	0
$430$	4134.13	0.463640
$431$	2352.51	0.262915	0.131457	−	0.991322i	$-0.458034\pi$
$431$	2352.51	0.262915	0.131457	+	0.991322i	$0.458034\pi$
$432$	0	0
$433$	−5860.51	−0.650434	−0.325217	−	0.945639i	$-0.605437\pi$
$433$	−5860.51	−0.650434	−0.325217	+	0.945639i	$0.605437\pi$
$434$	1043.08	0.115368
$435$	0	0
$436$	7307.50	0.802674
$437$	−4418.20	−0.483641
$438$	0	0
$439$	2894.17	0.314650	0.157325	−	0.987547i	$-0.449713\pi$
$439$	2894.17	0.314650	0.157325	+	0.987547i	$0.449713\pi$
$440$	−1051.25	−0.113901
$441$	0	0
$442$	1592.32	0.171356
$443$	8256.85	0.885541	0.442771	−	0.896635i	$-0.353996\pi$
$443$	8256.85	0.885541	0.442771	+	0.896635i	$0.353996\pi$
$444$	0	0
$445$	5793.42	0.617156
$446$	2657.25	0.282118
$447$	0	0
$448$	−20604.7	−2.17295
$449$	−15487.1	−1.62779	−0.813897	−	0.581009i	$-0.802658\pi$
$449$	−15487.1	−1.62779	−0.813897	+	0.581009i	$0.802658\pi$
$450$	0	0
$451$	−630.874	−0.0658685
$452$	−5731.42	−0.596423
$453$	0	0
$454$	9842.35	1.01745
$455$	−6242.19	−0.643162
$456$	0	0
$457$	−16055.6	−1.64343	−0.821716	−	0.569897i	$-0.806983\pi$
$457$	−16055.6	−1.64343	−0.821716	+	0.569897i	$0.806983\pi$
$458$	−20256.1	−2.06661
$459$	0	0
$460$	−12689.8	−1.28623
$461$	−14064.0	−1.42088	−0.710440	−	0.703758i	$-0.751504\pi$
$461$	−14064.0	−1.42088	−0.710440	+	0.703758i	$0.751504\pi$
$462$	0	0
$463$	−8071.30	−0.810162	−0.405081	−	0.914281i	$-0.632757\pi$
$463$	−8071.30	−0.810162	−0.405081	+	0.914281i	$0.632757\pi$
$464$	4707.55	0.470997
$465$	0	0
$466$	−21102.0	−2.09771
$467$	−8582.41	−0.850421	−0.425211	−	0.905094i	$-0.639800\pi$
$467$	−8582.41	−0.850421	−0.425211	+	0.905094i	$0.639800\pi$
$468$	0	0
$469$	24545.7	2.41667
$470$	1314.53	0.129010
$471$	0	0
$472$	6780.49	0.661224
$473$	−240.914	−0.0234191
$474$	0	0
$475$	976.286	0.0943054
$476$	6137.56	0.590997
$477$	0	0
$478$	24761.9	2.36942
$479$	6320.96	0.602948	0.301474	−	0.953474i	$-0.402521\pi$
$479$	6320.96	0.602948	0.301474	+	0.953474i	$0.402521\pi$
$480$	0	0
$481$	−7178.07	−0.680441
$482$	6394.13	0.604242
$483$	0	0
$484$	−18250.2	−1.71396
$485$	−8166.77	−0.764606
$486$	0	0
$487$	7336.47	0.682643	0.341321	−	0.939947i	$-0.389126\pi$
$487$	7336.47	0.682643	0.341321	+	0.939947i	$0.389126\pi$
$488$	10071.2	0.934225
$489$	0	0
$490$	−18892.6	−1.74179
$491$	−6672.53	−0.613294	−0.306647	−	0.951823i	$-0.599207\pi$
$491$	−6672.53	−0.613294	−0.306647	+	0.951823i	$0.599207\pi$
$492$	0	0
$493$	4864.76	0.444417
$494$	5371.43	0.489215
$495$	0	0
$496$	−140.623	−0.0127301
$497$	13588.7	1.22643
$498$	0	0
$499$	17920.9	1.60772	0.803858	−	0.594821i	$-0.202777\pi$
$499$	17920.9	1.60772	0.803858	+	0.594821i	$0.202777\pi$
$500$	−17784.6	−1.59070
$501$	0	0
$502$	−25914.6	−2.30403
$503$	11325.3	1.00392	0.501959	−	0.864891i	$-0.332613\pi$
$503$	11325.3	1.00392	0.501959	+	0.864891i	$0.332613\pi$
$504$	0	0
$505$	−10305.6	−0.908108
$506$	1167.54	0.102576
$507$	0	0
$508$	8238.56	0.719541
$509$	8313.78	0.723972	0.361986	−	0.932184i	$-0.382099\pi$
$509$	8313.78	0.723972	0.361986	+	0.932184i	$0.382099\pi$
$510$	0	0
$511$	9109.87	0.788644
$512$	5892.85	0.508651
$513$	0	0
$514$	−905.993	−0.0777464
$515$	22415.5	1.91795
$516$	0	0
$517$	−76.6033	−0.00651646
$518$	−43682.6	−3.70521
$519$	0	0
$520$	6497.65	0.547963
$521$	−5121.64	−0.430677	−0.215339	−	0.976539i	$-0.569086\pi$
$521$	−5121.64	−0.430677	−0.215339	+	0.976539i	$0.569086\pi$
$522$	0	0
$523$	13378.5	1.11855	0.559275	−	0.828982i	$-0.311080\pi$
$523$	13378.5	1.11855	0.559275	+	0.828982i	$0.311080\pi$
$524$	−1683.65	−0.140364
$525$	0	0
$526$	−6285.81	−0.521054
$527$	−145.319	−0.0120117
$528$	0	0
$529$	−6231.26	−0.512144
$530$	5808.01	0.476007
$531$	0	0
$532$	20704.0	1.68728
$533$	3899.35	0.316885
$534$	0	0
$535$	391.068	0.0316025
$536$	−25550.2	−2.05896
$537$	0	0
$538$	−14401.8	−1.15410
$539$	1100.95	0.0879804
$540$	0	0
$541$	9906.81	0.787296	0.393648	−	0.919261i	$-0.371213\pi$
$541$	9906.81	0.787296	0.393648	+	0.919261i	$0.371213\pi$
$542$	1972.04	0.156285
$543$	0	0
$544$	2391.63	0.188493
$545$	6301.05	0.495243
$546$	0	0
$547$	16399.6	1.28189	0.640947	−	0.767585i	$-0.278542\pi$
$547$	16399.6	1.28189	0.640947	+	0.767585i	$0.278542\pi$
$548$	12402.4	0.966798
$549$	0	0
$550$	−257.990	−0.0200013
$551$	16410.4	1.26880
$552$	0	0
$553$	−24916.1	−1.91598
$554$	38584.8	2.95905
$555$	0	0
$556$	−29212.0	−2.22818
$557$	−22044.3	−1.67692	−0.838461	−	0.544962i	$-0.816544\pi$
$557$	−22044.3	−1.67692	−0.838461	+	0.544962i	$0.816544\pi$
$558$	0	0
$559$	1489.06	0.112666
$560$	5121.22	0.386448
$561$	0	0
$562$	15517.6	1.16472
$563$	−12048.8	−0.901947	−0.450973	−	0.892537i	$-0.648923\pi$
$563$	−12048.8	−0.901947	−0.450973	+	0.892537i	$0.648923\pi$
$564$	0	0
$565$	−4942.04	−0.367988
$566$	37157.4	2.75944
$567$	0	0
$568$	−14144.8	−1.04490
$569$	23785.4	1.75243	0.876217	−	0.481916i	$-0.160059\pi$
$569$	23785.4	1.75243	0.876217	+	0.481916i	$0.160059\pi$
$570$	0	0
$571$	−10878.3	−0.797271	−0.398635	−	0.917110i	$-0.630516\pi$
$571$	−10878.3	−0.797271	−0.398635	+	0.917110i	$0.630516\pi$
$572$	−899.041	−0.0657182
$573$	0	0
$574$	23729.7	1.72554
$575$	−1311.62	−0.0951274
$576$	0	0
$577$	6315.86	0.455689	0.227845	−	0.973698i	$-0.426832\pi$
$577$	6315.86	0.455689	0.227845	+	0.973698i	$0.426832\pi$
$578$	−1350.00	−0.0971500
$579$	0	0
$580$	47133.7	3.37434
$581$	36944.1	2.63804
$582$	0	0
$583$	−338.459	−0.0240438
$584$	−9482.69	−0.671912
$585$	0	0
$586$	−5474.72	−0.385936
$587$	−18192.1	−1.27916	−0.639581	−	0.768724i	$-0.720892\pi$
$587$	−18192.1	−1.27916	−0.639581	+	0.768724i	$0.720892\pi$
$588$	0	0
$589$	−490.207	−0.0342931
$590$	13882.0	0.968663
$591$	0	0
$592$	5889.03	0.408848
$593$	9828.72	0.680636	0.340318	−	0.940310i	$-0.389465\pi$
$593$	9828.72	0.680636	0.340318	+	0.940310i	$0.389465\pi$
$594$	0	0
$595$	5292.25	0.364640
$596$	−35668.5	−2.45140
$597$	0	0
$598$	−7216.40	−0.493479
$599$	−4662.57	−0.318043	−0.159021	−	0.987275i	$-0.550834\pi$
$599$	−4662.57	−0.318043	−0.159021	+	0.987275i	$0.550834\pi$
$600$	0	0
$601$	21658.6	1.47000	0.735001	−	0.678066i	$-0.237182\pi$
$601$	21658.6	1.47000	0.735001	+	0.678066i	$0.237182\pi$
$602$	9061.76	0.613504
$603$	0	0
$604$	18558.3	1.25021
$605$	−15736.6	−1.05750
$606$	0	0
$607$	25764.7	1.72283	0.861415	−	0.507902i	$-0.169579\pi$
$607$	25764.7	1.72283	0.861415	+	0.507902i	$0.169579\pi$
$608$	8067.76	0.538143
$609$	0	0
$610$	20619.1	1.36860
$611$	473.475	0.0313498
$612$	0	0
$613$	16018.1	1.05541	0.527705	−	0.849428i	$-0.323053\pi$
$613$	16018.1	1.05541	0.527705	+	0.849428i	$0.323053\pi$
$614$	−4044.23	−0.265817
$615$	0	0
$616$	−2304.28	−0.150718
$617$	−22250.3	−1.45180	−0.725902	−	0.687798i	$-0.758578\pi$
$617$	−22250.3	−1.45180	−0.725902	+	0.687798i	$0.758578\pi$
$618$	0	0
$619$	−3765.95	−0.244534	−0.122267	−	0.992497i	$-0.539016\pi$
$619$	−3765.95	−0.244534	−0.122267	+	0.992497i	$0.539016\pi$
$620$	−1407.96	−0.0912018
$621$	0	0
$622$	32674.9	2.10634
$623$	12698.8	0.816642
$624$	0	0
$625$	−17463.2	−1.11765
$626$	−16080.1	−1.02666
$627$	0	0
$628$	−34494.5	−2.19185
$629$	6085.70	0.385775
$630$	0	0
$631$	−20806.5	−1.31267	−0.656334	−	0.754470i	$-0.727894\pi$
$631$	−20806.5	−1.31267	−0.656334	+	0.754470i	$0.727894\pi$
$632$	25935.7	1.63239
$633$	0	0
$634$	9651.60	0.604596
$635$	7103.88	0.443951
$636$	0	0
$637$	−6804.85	−0.423262
$638$	−4336.56	−0.269100
$639$	0	0
$640$	30498.4	1.88368
$641$	−2439.58	−0.150324	−0.0751620	−	0.997171i	$-0.523947\pi$
$641$	−2439.58	−0.150324	−0.0751620	+	0.997171i	$0.523947\pi$
$642$	0	0
$643$	−19320.1	−1.18493	−0.592466	−	0.805595i	$-0.701846\pi$
$643$	−19320.1	−1.18493	−0.592466	+	0.805595i	$0.701846\pi$
$644$	−27815.4	−1.70199
$645$	0	0
$646$	−4554.00	−0.277360
$647$	14067.1	0.854766	0.427383	−	0.904071i	$-0.359436\pi$
$647$	14067.1	0.854766	0.427383	+	0.904071i	$0.359436\pi$
$648$	0	0
$649$	−808.963	−0.0489285
$650$	1594.60	0.0962238
$651$	0	0
$652$	27110.9	1.62844
$653$	−15893.7	−0.952478	−0.476239	−	0.879316i	$-0.658000\pi$
$653$	−15893.7	−0.952478	−0.476239	+	0.879316i	$0.658000\pi$
$654$	0	0
$655$	−1451.77	−0.0866033
$656$	−3199.11	−0.190403
$657$	0	0
$658$	2881.36	0.170710
$659$	9653.54	0.570635	0.285318	−	0.958433i	$-0.407901\pi$
$659$	9653.54	0.570635	0.285318	+	0.958433i	$0.407901\pi$
$660$	0	0
$661$	5389.06	0.317111	0.158555	−	0.987350i	$-0.449316\pi$
$661$	5389.06	0.317111	0.158555	+	0.987350i	$0.449316\pi$
$662$	−42169.7	−2.47579
$663$	0	0
$664$	−38456.0	−2.24757
$665$	17852.5	1.04104
$666$	0	0
$667$	−22047.0	−1.27986
$668$	−30119.1	−1.74453
$669$	0	0
$670$	−52309.9	−3.01628
$671$	−1201.57	−0.0691297
$672$	0	0
$673$	−3032.18	−0.173673	−0.0868366	−	0.996223i	$-0.527676\pi$
$673$	−3032.18	−0.173673	−0.0868366	+	0.996223i	$0.527676\pi$
$674$	−954.386	−0.0545424
$675$	0	0
$676$	−24807.7	−1.41145
$677$	22029.2	1.25059	0.625295	−	0.780388i	$-0.284979\pi$
$677$	22029.2	1.25059	0.625295	+	0.780388i	$0.284979\pi$
$678$	0	0
$679$	−17901.1	−1.01175
$680$	−5508.83	−0.310667
$681$	0	0
$682$	129.540	0.00727326
$683$	9040.72	0.506491	0.253246	−	0.967402i	$-0.418502\pi$
$683$	9040.72	0.506491	0.253246	+	0.967402i	$0.418502\pi$
$684$	0	0
$685$	10694.3	0.596506
$686$	443.023	0.0246570
$687$	0	0
$688$	−1221.65	−0.0676964
$689$	2091.97	0.115672
$690$	0	0
$691$	−22863.5	−1.25871	−0.629355	−	0.777118i	$-0.716681\pi$
$691$	−22863.5	−1.25871	−0.629355	+	0.777118i	$0.716681\pi$
$692$	−43003.1	−2.36233
$693$	0	0
$694$	668.288	0.0365531
$695$	−25188.7	−1.37477
$696$	0	0
$697$	−3305.94	−0.179658
$698$	18677.8	1.01284
$699$	0	0
$700$	6146.34	0.331871
$701$	−1753.00	−0.0944507	−0.0472253	−	0.998884i	$-0.515038\pi$
$701$	−1753.00	−0.0944507	−0.0472253	+	0.998884i	$0.515038\pi$
$702$	0	0
$703$	20529.1	1.10138
$704$	−2558.90	−0.136992
$705$	0	0
$706$	−27138.2	−1.44668
$707$	−22589.3	−1.20164
$708$	0	0
$709$	11547.0	0.611645	0.305823	−	0.952089i	$-0.401069\pi$
$709$	11547.0	0.611645	0.305823	+	0.952089i	$0.401069\pi$
$710$	−28959.2	−1.53073
$711$	0	0
$712$	−13218.5	−0.695765
$713$	658.582	0.0345920
$714$	0	0
$715$	−775.218	−0.0405476
$716$	−11196.4	−0.584399
$717$	0	0
$718$	−22867.7	−1.18860
$719$	10289.8	0.533720	0.266860	−	0.963735i	$-0.414014\pi$
$719$	10289.8	0.533720	0.266860	+	0.963735i	$0.414014\pi$
$720$	0	0
$721$	49133.4	2.53790
$722$	16678.2	0.859695
$723$	0	0
$724$	−46386.2	−2.38112
$725$	4871.72	0.249560
$726$	0	0
$727$	2950.10	0.150499	0.0752497	−	0.997165i	$-0.476025\pi$
$727$	2950.10	0.150499	0.0752497	+	0.997165i	$0.476025\pi$
$728$	14242.5	0.725083
$729$	0	0
$730$	−19414.2	−0.984320
$731$	−1262.45	−0.0638762
$732$	0	0
$733$	24348.2	1.22691	0.613453	−	0.789731i	$-0.289780\pi$
$733$	24348.2	1.22691	0.613453	+	0.789731i	$0.289780\pi$
$734$	−2467.15	−0.124065
$735$	0	0
$736$	−10838.8	−0.542833
$737$	3048.33	0.152357
$738$	0	0
$739$	−29233.5	−1.45517	−0.727585	−	0.686017i	$-0.759358\pi$
$739$	−29233.5	−1.45517	−0.727585	+	0.686017i	$0.759358\pi$
$740$	58963.1	2.92909
$741$	0	0
$742$	12730.8	0.629868
$743$	−15340.6	−0.757457	−0.378729	−	0.925508i	$-0.623639\pi$
$743$	−15340.6	−0.757457	−0.378729	+	0.925508i	$0.623639\pi$
$744$	0	0
$745$	−30755.9	−1.51250
$746$	47242.9	2.31861
$747$	0	0
$748$	762.224	0.0372589
$749$	857.197	0.0418175
$750$	0	0
$751$	39862.6	1.93689	0.968446	−	0.249223i	$-0.0801752\pi$
$751$	39862.6	1.93689	0.968446	+	0.249223i	$0.0801752\pi$
$752$	−388.448	−0.0188368
$753$	0	0
$754$	26803.7	1.29461
$755$	16002.3	0.771369
$756$	0	0
$757$	26375.1	1.26634	0.633169	−	0.774013i	$-0.281754\pi$
$757$	26375.1	1.26634	0.633169	+	0.774013i	$0.281754\pi$
$758$	−3407.17	−0.163264
$759$	0	0
$760$	−18583.1	−0.886946
$761$	−7848.63	−0.373867	−0.186933	−	0.982373i	$-0.559855\pi$
$761$	−7848.63	−0.373867	−0.186933	+	0.982373i	$0.559855\pi$
$762$	0	0
$763$	13811.5	0.655322
$764$	−18498.1	−0.875967
$765$	0	0
$766$	−7511.78	−0.354323
$767$	5000.10	0.235389
$768$	0	0
$769$	31818.9	1.49209	0.746046	−	0.665895i	$-0.231950\pi$
$769$	31818.9	1.49209	0.746046	+	0.665895i	$0.231950\pi$
$770$	−4717.63	−0.220794
$771$	0	0
$772$	−3143.78	−0.146564
$773$	29559.8	1.37541	0.687706	−	0.725990i	$-0.258618\pi$
$773$	29559.8	1.37541	0.687706	+	0.725990i	$0.258618\pi$
$774$	0	0
$775$	−145.527	−0.00674512
$776$	18633.6	0.861996
$777$	0	0
$778$	45891.2	2.11475
$779$	−11152.0	−0.512917
$780$	0	0
$781$	1687.58	0.0773193
$782$	6118.19	0.279778
$783$	0	0
$784$	5582.84	0.254320
$785$	−29743.6	−1.35235
$786$	0	0
$787$	−28038.7	−1.26998	−0.634989	−	0.772521i	$-0.718995\pi$
$787$	−28038.7	−1.26998	−0.634989	+	0.772521i	$0.718995\pi$
$788$	−11271.6	−0.509563
$789$	0	0
$790$	53099.2	2.39137
$791$	−10832.6	−0.486934
$792$	0	0
$793$	7426.75	0.332574
$794$	−13438.7	−0.600659
$795$	0	0
$796$	25802.3	1.14892
$797$	−5320.45	−0.236462	−0.118231	−	0.992986i	$-0.537722\pi$
$797$	−5320.45	−0.236462	−0.118231	+	0.992986i	$0.537722\pi$
$798$	0	0
$799$	−401.421	−0.0177738
$800$	2395.05	0.105847
$801$	0	0
$802$	−30437.7	−1.34014
$803$	1131.35	0.0497194
$804$	0	0
$805$	−23984.4	−1.05011
$806$	−800.673	−0.0349907
$807$	0	0
$808$	23513.8	1.02378
$809$	−30934.0	−1.34435	−0.672177	−	0.740390i	$-0.734641\pi$
$809$	−30934.0	−1.34435	−0.672177	+	0.740390i	$0.734641\pi$
$810$	0	0
$811$	−40364.5	−1.74771	−0.873854	−	0.486189i	$-0.838387\pi$
$811$	−40364.5	−1.74771	−0.873854	+	0.486189i	$0.838387\pi$
$812$	103314.	4.46504
$813$	0	0
$814$	−5424.94	−0.233592
$815$	23376.9	1.00473
$816$	0	0
$817$	−4258.66	−0.182364
$818$	41434.8	1.77107
$819$	0	0
$820$	−32030.6	−1.36409
$821$	−19799.7	−0.841672	−0.420836	−	0.907137i	$-0.638263\pi$
$821$	−19799.7	−0.841672	−0.420836	+	0.907137i	$0.638263\pi$
$822$	0	0
$823$	18756.4	0.794419	0.397210	−	0.917728i	$-0.369979\pi$
$823$	18756.4	0.794419	0.397210	+	0.917728i	$0.369979\pi$
$824$	−51144.1	−2.16225
$825$	0	0
$826$	30428.4	1.28177
$827$	−20958.0	−0.881234	−0.440617	−	0.897695i	$-0.645240\pi$
$827$	−20958.0	−0.881234	−0.440617	+	0.897695i	$0.645240\pi$
$828$	0	0
$829$	−31320.3	−1.31218	−0.656091	−	0.754682i	$-0.727791\pi$
$829$	−31320.3	−1.31218	−0.656091	+	0.754682i	$0.727791\pi$
$830$	−78732.4	−3.29258
$831$	0	0
$832$	15816.2	0.659049
$833$	5769.28	0.239968
$834$	0	0
$835$	−25970.8	−1.07636
$836$	2571.23	0.106373
$837$	0	0
$838$	−4715.82	−0.194398
$839$	30290.6	1.24642	0.623212	−	0.782053i	$-0.285827\pi$
$839$	30290.6	1.24642	0.623212	+	0.782053i	$0.285827\pi$
$840$	0	0
$841$	57499.9	2.35762
$842$	15198.5	0.622060
$843$	0	0
$844$	−15242.8	−0.621659
$845$	−21391.0	−0.870855
$846$	0	0
$847$	−34493.7	−1.39931
$848$	−1716.29	−0.0695021
$849$	0	0
$850$	−1351.93	−0.0545540
$851$	−27580.3	−1.11098
$852$	0	0
$853$	−21111.8	−0.847425	−0.423712	−	0.905797i	$-0.639273\pi$
$853$	−21111.8	−0.847425	−0.423712	+	0.905797i	$0.639273\pi$
$854$	45195.9	1.81097
$855$	0	0
$856$	−892.277	−0.0356278
$857$	39983.0	1.59369	0.796845	−	0.604184i	$-0.206501\pi$
$857$	39983.0	1.59369	0.796845	+	0.604184i	$0.206501\pi$
$858$	0	0
$859$	−39503.3	−1.56907	−0.784537	−	0.620082i	$-0.787099\pi$
$859$	−39503.3	−1.56907	−0.784537	+	0.620082i	$0.787099\pi$
$860$	−12231.6	−0.484995
$861$	0	0
$862$	−10989.2	−0.434217
$863$	−26019.9	−1.02634	−0.513168	−	0.858288i	$-0.671528\pi$
$863$	−26019.9	−1.02634	−0.513168	+	0.858288i	$0.671528\pi$
$864$	0	0
$865$	−37080.4	−1.45754
$866$	27376.1	1.07422
$867$	0	0
$868$	−3086.17	−0.120681
$869$	−3094.33	−0.120791
$870$	0	0
$871$	−18841.4	−0.732968
$872$	−14376.7	−0.558323
$873$	0	0
$874$	20638.7	0.798757
$875$	−33613.8	−1.29869
$876$	0	0
$877$	15038.3	0.579027	0.289514	−	0.957174i	$-0.406506\pi$
$877$	15038.3	0.579027	0.289514	+	0.957174i	$0.406506\pi$
$878$	−13519.5	−0.519660
$879$	0	0
$880$	636.004	0.0243633
$881$	18334.3	0.701133	0.350567	−	0.936538i	$-0.385989\pi$
$881$	18334.3	0.701133	0.350567	+	0.936538i	$0.385989\pi$
$882$	0	0
$883$	26659.5	1.01604	0.508020	−	0.861345i	$-0.330378\pi$
$883$	26659.5	1.01604	0.508020	+	0.861345i	$0.330378\pi$
$884$	−4711.21	−0.179248
$885$	0	0
$886$	−38570.1	−1.46251
$887$	−11473.7	−0.434327	−0.217163	−	0.976135i	$-0.569680\pi$
$887$	−11473.7	−0.434327	−0.217163	+	0.976135i	$0.569680\pi$
$888$	0	0
$889$	15571.3	0.587450
$890$	−27062.7	−1.01926
$891$	0	0
$892$	−7862.01	−0.295111
$893$	−1354.12	−0.0507436
$894$	0	0
$895$	−9654.36	−0.360569
$896$	66850.7	2.49255
$897$	0	0
$898$	72344.6	2.68838
$899$	−2446.16	−0.0907498
$900$	0	0
$901$	−1773.61	−0.0655799
$902$	2946.99	0.108785
$903$	0	0
$904$	11276.0	0.414860
$905$	−39997.5	−1.46913
$906$	0	0
$907$	−20361.6	−0.745421	−0.372710	−	0.927948i	$-0.621571\pi$
$907$	−20361.6	−0.745421	−0.372710	+	0.927948i	$0.621571\pi$
$908$	−29120.5	−1.06432
$909$	0	0
$910$	29159.1	1.06221
$911$	19261.9	0.700523	0.350262	−	0.936652i	$-0.386093\pi$
$911$	19261.9	0.700523	0.350262	+	0.936652i	$0.386093\pi$
$912$	0	0
$913$	4588.09	0.166313
$914$	75000.3	2.71421
$915$	0	0
$916$	59931.7	2.16179
$917$	−3182.18	−0.114596
$918$	0	0
$919$	−21191.8	−0.760666	−0.380333	−	0.924850i	$-0.624191\pi$
$919$	−21191.8	−0.760666	−0.380333	+	0.924850i	$0.624191\pi$
$920$	24965.9	0.894677
$921$	0	0
$922$	65697.0	2.34665
$923$	−10430.7	−0.371973
$924$	0	0
$925$	6094.40	0.216630
$926$	37703.4	1.33802
$927$	0	0
$928$	40258.5	1.42409
$929$	−40815.0	−1.44144	−0.720719	−	0.693227i	$-0.756188\pi$
$929$	−40815.0	−1.44144	−0.720719	+	0.693227i	$0.756188\pi$
$930$	0	0
$931$	19461.7	0.685102
$932$	62434.5	2.19432
$933$	0	0
$934$	40090.9	1.40451
$935$	657.244	0.0229884
$936$	0	0
$937$	−38439.1	−1.34018	−0.670092	−	0.742278i	$-0.733745\pi$
$937$	−38439.1	−1.34018	−0.670092	+	0.742278i	$0.733745\pi$
$938$	−114660.	−3.99124
$939$	0	0
$940$	−3889.28	−0.134951
$941$	2244.08	0.0777415	0.0388708	−	0.999244i	$-0.487624\pi$
$941$	2244.08	0.0777415	0.0388708	+	0.999244i	$0.487624\pi$
$942$	0	0
$943$	14982.5	0.517388
$944$	−4102.18	−0.141435
$945$	0	0
$946$	1125.38	0.0386778
$947$	42289.0	1.45112	0.725559	−	0.688160i	$-0.241581\pi$
$947$	42289.0	1.45112	0.725559	+	0.688160i	$0.241581\pi$
$948$	0	0
$949$	−6992.76	−0.239193
$950$	−4560.51	−0.155750
$951$	0	0
$952$	−12075.0	−0.411085
$953$	−37426.2	−1.27214	−0.636072	−	0.771629i	$-0.719442\pi$
$953$	−37426.2	−1.27214	−0.636072	+	0.771629i	$0.719442\pi$
$954$	0	0
$955$	−15950.4	−0.540464
$956$	−73263.0	−2.47855
$957$	0	0
$958$	−29527.0	−0.995799
$959$	23441.2	0.789317
$960$	0	0
$961$	−29717.9	−0.997547
$962$	33530.8	1.12378
$963$	0	0
$964$	−18918.3	−0.632072
$965$	−2710.80	−0.0904286
$966$	0	0
$967$	1088.56	0.0362003	0.0181001	−	0.999836i	$-0.494238\pi$
$967$	1088.56	0.0362003	0.0181001	+	0.999836i	$0.494238\pi$
$968$	35905.4	1.19219
$969$	0	0
$970$	38149.3	1.26278
$971$	−39506.5	−1.30569	−0.652845	−	0.757491i	$-0.726425\pi$
$971$	−39506.5	−1.30569	−0.652845	+	0.757491i	$0.726425\pi$
$972$	0	0
$973$	−55212.1	−1.81914
$974$	−34270.8	−1.12742
$975$	0	0
$976$	−6093.05	−0.199830
$977$	43326.8	1.41878	0.709389	−	0.704817i	$-0.248971\pi$
$977$	43326.8	1.41878	0.709389	+	0.704817i	$0.248971\pi$
$978$	0	0
$979$	1577.07	0.0514845
$980$	55897.3	1.82202
$981$	0	0
$982$	31169.3	1.01288
$983$	−10664.1	−0.346014	−0.173007	−	0.984921i	$-0.555348\pi$
$983$	−10664.1	−0.346014	−0.173007	+	0.984921i	$0.555348\pi$
$984$	0	0
$985$	−9719.22	−0.314396
$986$	−22724.7	−0.733977
$987$	0	0
$988$	−15892.4	−0.511747
$989$	5721.42	0.183954
$990$	0	0
$991$	15461.4	0.495609	0.247804	−	0.968810i	$-0.420291\pi$
$991$	15461.4	0.495609	0.247804	+	0.968810i	$0.420291\pi$
$992$	−1202.59	−0.0384902
$993$	0	0
$994$	−63476.7	−2.02551
$995$	22248.6	0.708871
$996$	0	0
$997$	46061.1	1.46316	0.731579	−	0.681756i	$-0.238784\pi$
$997$	46061.1	1.46316	0.731579	+	0.681756i	$0.238784\pi$
$998$	−83713.7	−2.65522
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.g.1.1		3
3.2	odd	2		17.4.a.b.1.3	✓	3
4.3	odd	2		2448.4.a.bi.1.3		3
12.11	even	2		272.4.a.h.1.3		3
15.2	even	4		425.4.b.f.324.5		6
15.8	even	4		425.4.b.f.324.2		6
15.14	odd	2		425.4.a.g.1.1		3
21.20	even	2		833.4.a.d.1.3		3
24.5	odd	2		1088.4.a.v.1.3		3
24.11	even	2		1088.4.a.x.1.1		3
33.32	even	2		2057.4.a.e.1.1		3
51.38	odd	4		289.4.b.b.288.2		6
51.47	odd	4		289.4.b.b.288.1		6
51.50	odd	2		289.4.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.a.b.1.3	✓	3	3.2	odd	2
153.4.a.g.1.1		3	1.1	even	1	trivial
272.4.a.h.1.3		3	12.11	even	2
289.4.a.b.1.3		3	51.50	odd	2
289.4.b.b.288.1		6	51.47	odd	4
289.4.b.b.288.2		6	51.38	odd	4
425.4.a.g.1.1		3	15.14	odd	2
425.4.b.f.324.2		6	15.8	even	4
425.4.b.f.324.5		6	15.2	even	4
833.4.a.d.1.3		3	21.20	even	2
1088.4.a.v.1.3		3	24.5	odd	2
1088.4.a.x.1.1		3	24.11	even	2
2057.4.a.e.1.1		3	33.32	even	2
2448.4.a.bi.1.3		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)

\(f(q)\)	\(=\)	\(q-4.67129 q^{2} +13.8209 q^{4} +11.9174 q^{5} +26.1222 q^{7} -27.1912 q^{8} +O(q^{10})\)	\(q-4.67129 q^{2} +13.8209 q^{4} +11.9174 q^{5} +26.1222 q^{7} -27.1912 q^{8} -55.6696 q^{10} +3.24412 q^{11} -20.0515 q^{13} -122.024 q^{14} +16.4506 q^{16} +17.0000 q^{17} +57.3466 q^{19} +164.709 q^{20} -15.1542 q^{22} -77.0438 q^{23} +17.0243 q^{25} +93.6662 q^{26} +361.033 q^{28} +286.162 q^{29} -8.54816 q^{31} +140.684 q^{32} -79.4119 q^{34} +311.309 q^{35} +357.982 q^{37} -267.882 q^{38} -324.049 q^{40} -194.467 q^{41} -74.2619 q^{43} +44.8367 q^{44} +359.894 q^{46} -23.6130 q^{47} +339.369 q^{49} -79.5255 q^{50} -277.130 q^{52} -104.330 q^{53} +38.6614 q^{55} -710.295 q^{56} -1336.75 q^{58} -249.363 q^{59} -370.384 q^{61} +39.9309 q^{62} -788.781 q^{64} -238.961 q^{65} +939.650 q^{67} +234.956 q^{68} -1454.21 q^{70} +520.197 q^{71} +348.741 q^{73} -1672.24 q^{74} +792.583 q^{76} +84.7434 q^{77} -953.827 q^{79} +196.049 q^{80} +908.412 q^{82} +1414.28 q^{83} +202.596 q^{85} +346.899 q^{86} -88.2115 q^{88} +486.132 q^{89} -523.788 q^{91} -1064.82 q^{92} +110.303 q^{94} +683.422 q^{95} -685.281 q^{97} -1585.29 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10}) \) 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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