LMFDB - Embedded newform 152.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [152,4,Mod(1,152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("152.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 152 = 2^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	152.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:	$8.96829032087$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.7057.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 22x + 32 $ x^3 - x^2 - 22*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.36181$ of defining polynomial
Character	$\chi$	$=$	152.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33180 q^{3} +18.4427 q^{5} +10.3872 q^{7} -25.2263 q^{9} +50.4519 q^{11} -61.8352 q^{13} +24.5619 q^{15} +68.1070 q^{17} -19.0000 q^{19} +13.8337 q^{21} +145.632 q^{23} +215.132 q^{25} -69.5550 q^{27} +42.6097 q^{29} +91.6582 q^{31} +67.1918 q^{33} +191.568 q^{35} -400.965 q^{37} -82.3522 q^{39} -123.355 q^{41} +449.802 q^{43} -465.240 q^{45} -453.075 q^{47} -235.105 q^{49} +90.7050 q^{51} +437.142 q^{53} +930.466 q^{55} -25.3042 q^{57} -159.352 q^{59} -476.816 q^{61} -262.031 q^{63} -1140.41 q^{65} -629.682 q^{67} +193.953 q^{69} +471.459 q^{71} -725.055 q^{73} +286.512 q^{75} +524.055 q^{77} -1057.66 q^{79} +588.477 q^{81} -726.957 q^{83} +1256.07 q^{85} +56.7476 q^{87} -468.065 q^{89} -642.297 q^{91} +122.071 q^{93} -350.410 q^{95} -891.891 q^{97} -1272.71 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 21 q^{9} + 103 q^{11} + 32 q^{13} + 122 q^{15} + 11 q^{17} - 57 q^{19} + 114 q^{21} + 316 q^{23} + 162 q^{25} + 178 q^{27} - 138 q^{29} + 420 q^{31} - 330 q^{33} + 333 q^{35}+ \cdots - 1305 q^{99}+O(q^{100}) $ 3 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 21 * q^9 + 103 * q^11 + 32 * q^13 + 122 * q^15 + 11 * q^17 - 57 * q^19 + 114 * q^21 + 316 * q^23 + 162 * q^25 + 178 * q^27 - 138 * q^29 + 420 * q^31 - 330 * q^33 + 333 * q^35 + 102 * q^37 + 164 * q^39 - 370 * q^41 + 431 * q^43 - 429 * q^45 - 199 * q^47 - 802 * q^49 + 272 * q^51 - 308 * q^53 + 85 * q^55 - 76 * q^57 - 188 * q^59 - 609 * q^61 - 61 * q^63 - 1536 * q^65 - 246 * q^67 - 892 * q^69 - 954 * q^71 - 629 * q^73 - 1074 * q^75 - 71 * q^77 + 452 * q^79 - 361 * q^81 - 780 * q^83 + 1883 * q^85 - 96 * q^87 - 1356 * q^89 - 670 * q^91 + 1092 * q^93 - 133 * q^95 - 548 * q^97 - 1305 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.33180	0.256305	0.128153	−	0.991754i	$-0.459095\pi$
$3$	1.33180	0.256305	0.128153	+	0.991754i	$0.459095\pi$
$4$	0	0
$5$	18.4427	1.64956	0.824781	−	0.565453i	$-0.191299\pi$
$5$	18.4427	1.64956	0.824781	+	0.565453i	$0.191299\pi$
$6$	0	0
$7$	10.3872	0.560858	0.280429	−	0.959875i	$-0.409523\pi$
$7$	10.3872	0.560858	0.280429	+	0.959875i	$0.409523\pi$
$8$	0	0
$9$	−25.2263	−0.934308
$10$	0	0
$11$	50.4519	1.38289	0.691446	−	0.722428i	$-0.256974\pi$
$11$	50.4519	1.38289	0.691446	+	0.722428i	$0.256974\pi$
$12$	0	0
$13$	−61.8352	−1.31923	−0.659616	−	0.751603i	$-0.729281\pi$
$13$	−61.8352	−1.31923	−0.659616	+	0.751603i	$0.729281\pi$
$14$	0	0
$15$	24.5619	0.422791
$16$	0	0
$17$	68.1070	0.971669	0.485835	−	0.874051i	$-0.338516\pi$
$17$	68.1070	0.971669	0.485835	+	0.874051i	$0.338516\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	13.8337	0.143751
$22$	0	0
$23$	145.632	1.32028	0.660138	−	0.751144i	$-0.270498\pi$
$23$	145.632	1.32028	0.660138	+	0.751144i	$0.270498\pi$
$24$	0	0
$25$	215.132	1.72105
$26$	0	0
$27$	−69.5550	−0.495773
$28$	0	0
$29$	42.6097	0.272842	0.136421	−	0.990651i	$-0.456440\pi$
$29$	42.6097	0.272842	0.136421	+	0.990651i	$0.456440\pi$
$30$	0	0
$31$	91.6582	0.531042	0.265521	−	0.964105i	$-0.414456\pi$
$31$	91.6582	0.531042	0.265521	+	0.964105i	$0.414456\pi$
$32$	0	0
$33$	67.1918	0.354442
$34$	0	0
$35$	191.568	0.925169
$36$	0	0
$37$	−400.965	−1.78158	−0.890788	−	0.454419i	$-0.849847\pi$
$37$	−400.965	−1.78158	−0.890788	+	0.454419i	$0.849847\pi$
$38$	0	0
$39$	−82.3522	−0.338126
$40$	0	0
$41$	−123.355	−0.469873	−0.234936	−	0.972011i	$-0.575488\pi$
$41$	−123.355	−0.469873	−0.234936	+	0.972011i	$0.575488\pi$
$42$	0	0
$43$	449.802	1.59521	0.797607	−	0.603178i	$-0.206099\pi$
$43$	449.802	1.59521	0.797607	+	0.603178i	$0.206099\pi$
$44$	0	0
$45$	−465.240	−1.54120
$46$	0	0
$47$	−453.075	−1.40612	−0.703062	−	0.711129i	$-0.748184\pi$
$47$	−453.075	−1.40612	−0.703062	+	0.711129i	$0.748184\pi$
$48$	0	0
$49$	−235.105	−0.685439
$50$	0	0
$51$	90.7050	0.249044
$52$	0	0
$53$	437.142	1.13294	0.566472	−	0.824081i	$-0.308308\pi$
$53$	437.142	1.13294	0.566472	+	0.824081i	$0.308308\pi$
$54$	0	0
$55$	930.466	2.28116
$56$	0	0
$57$	−25.3042	−0.0588004
$58$	0	0
$59$	−159.352	−0.351624	−0.175812	−	0.984424i	$-0.556255\pi$
$59$	−159.352	−0.351624	−0.175812	+	0.984424i	$0.556255\pi$
$60$	0	0
$61$	−476.816	−1.00082	−0.500410	−	0.865789i	$-0.666817\pi$
$61$	−476.816	−1.00082	−0.500410	+	0.865789i	$0.666817\pi$
$62$	0	0
$63$	−262.031	−0.524014
$64$	0	0
$65$	−1140.41	−2.17615
$66$	0	0
$67$	−629.682	−1.14818	−0.574089	−	0.818793i	$-0.694643\pi$
$67$	−629.682	−1.14818	−0.574089	+	0.818793i	$0.694643\pi$
$68$	0	0
$69$	193.953	0.338394
$70$	0	0
$71$	471.459	0.788056	0.394028	−	0.919099i	$-0.371081\pi$
$71$	471.459	0.788056	0.394028	+	0.919099i	$0.371081\pi$
$72$	0	0
$73$	−725.055	−1.16248	−0.581241	−	0.813731i	$-0.697433\pi$
$73$	−725.055	−1.16248	−0.581241	+	0.813731i	$0.697433\pi$
$74$	0	0
$75$	286.512	0.441115
$76$	0	0
$77$	524.055	0.775605
$78$	0	0
$79$	−1057.66	−1.50628	−0.753139	−	0.657861i	$-0.771461\pi$
$79$	−1057.66	−1.50628	−0.753139	+	0.657861i	$0.771461\pi$
$80$	0	0
$81$	588.477	0.807238
$82$	0	0
$83$	−726.957	−0.961373	−0.480686	−	0.876893i	$-0.659612\pi$
$83$	−726.957	−0.961373	−0.480686	+	0.876893i	$0.659612\pi$
$84$	0	0
$85$	1256.07	1.60283
$86$	0	0
$87$	56.7476	0.0699309
$88$	0	0
$89$	−468.065	−0.557469	−0.278735	−	0.960368i	$-0.589915\pi$
$89$	−468.065	−0.557469	−0.278735	+	0.960368i	$0.589915\pi$
$90$	0	0
$91$	−642.297	−0.739901
$92$	0	0
$93$	122.071	0.136109
$94$	0	0
$95$	−350.410	−0.378435
$96$	0	0
$97$	−891.891	−0.933585	−0.466793	−	0.884367i	$-0.654591\pi$
$97$	−891.891	−0.933585	−0.466793	+	0.884367i	$0.654591\pi$
$98$	0	0
$99$	−1272.71	−1.29205
$100$	0	0
$101$	−16.6636	−0.0164167	−0.00820835	−	0.999966i	$-0.502613\pi$
$101$	−16.6636	−0.0164167	−0.00820835	+	0.999966i	$0.502613\pi$
$102$	0	0
$103$	1292.68	1.23662	0.618309	−	0.785935i	$-0.287818\pi$
$103$	1292.68	1.23662	0.618309	+	0.785935i	$0.287818\pi$
$104$	0	0
$105$	255.130	0.237126
$106$	0	0
$107$	−176.185	−0.159182	−0.0795911	−	0.996828i	$-0.525361\pi$
$107$	−176.185	−0.159182	−0.0795911	+	0.996828i	$0.525361\pi$
$108$	0	0
$109$	1378.74	1.21155	0.605776	−	0.795635i	$-0.292863\pi$
$109$	1378.74	1.21155	0.605776	+	0.795635i	$0.292863\pi$
$110$	0	0
$111$	−534.006	−0.456627
$112$	0	0
$113$	−601.795	−0.500992	−0.250496	−	0.968118i	$-0.580594\pi$
$113$	−601.795	−0.500992	−0.250496	+	0.968118i	$0.580594\pi$
$114$	0	0
$115$	2685.84	2.17788
$116$	0	0
$117$	1559.87	1.23257
$118$	0	0
$119$	707.443	0.544968
$120$	0	0
$121$	1214.39	0.912389
$122$	0	0
$123$	−164.284	−0.120431
$124$	0	0
$125$	1662.26	1.18942
$126$	0	0
$127$	1890.38	1.32082	0.660409	−	0.750906i	$-0.270383\pi$
$127$	1890.38	1.32082	0.660409	+	0.750906i	$0.270383\pi$
$128$	0	0
$129$	599.047	0.408861
$130$	0	0
$131$	331.964	0.221403	0.110701	−	0.993854i	$-0.464690\pi$
$131$	331.964	0.221403	0.110701	+	0.993854i	$0.464690\pi$
$132$	0	0
$133$	−197.357	−0.128670
$134$	0	0
$135$	−1282.78	−0.817808
$136$	0	0
$137$	−1956.20	−1.21992	−0.609962	−	0.792430i	$-0.708815\pi$
$137$	−1956.20	−1.21992	−0.609962	+	0.792430i	$0.708815\pi$
$138$	0	0
$139$	547.821	0.334285	0.167142	−	0.985933i	$-0.446546\pi$
$139$	547.821	0.334285	0.167142	+	0.985933i	$0.446546\pi$
$140$	0	0
$141$	−603.406	−0.360397
$142$	0	0
$143$	−3119.70	−1.82435
$144$	0	0
$145$	785.836	0.450070
$146$	0	0
$147$	−313.114	−0.175682
$148$	0	0
$149$	−992.480	−0.545686	−0.272843	−	0.962059i	$-0.587964\pi$
$149$	−992.480	−0.545686	−0.272843	+	0.962059i	$0.587964\pi$
$150$	0	0
$151$	−148.167	−0.0798520	−0.0399260	−	0.999203i	$-0.512712\pi$
$151$	−148.167	−0.0798520	−0.0399260	+	0.999203i	$0.512712\pi$
$152$	0	0
$153$	−1718.09	−0.907838
$154$	0	0
$155$	1690.42	0.875987
$156$	0	0
$157$	2356.36	1.19782	0.598910	−	0.800817i	$-0.295601\pi$
$157$	2356.36	1.19782	0.598910	+	0.800817i	$0.295601\pi$
$158$	0	0
$159$	582.186	0.290380
$160$	0	0
$161$	1512.71	0.740487
$162$	0	0
$163$	−2974.86	−1.42950	−0.714751	−	0.699379i	$-0.753460\pi$
$163$	−2974.86	−1.42950	−0.714751	+	0.699379i	$0.753460\pi$
$164$	0	0
$165$	1239.20	0.584674
$166$	0	0
$167$	−3133.09	−1.45177	−0.725885	−	0.687816i	$-0.758570\pi$
$167$	−3133.09	−1.45177	−0.725885	+	0.687816i	$0.758570\pi$
$168$	0	0
$169$	1626.60	0.740372
$170$	0	0
$171$	479.300	0.214345
$172$	0	0
$173$	3415.40	1.50097	0.750485	−	0.660887i	$-0.229820\pi$
$173$	3415.40	1.50097	0.750485	+	0.660887i	$0.229820\pi$
$174$	0	0
$175$	2234.62	0.965265
$176$	0	0
$177$	−212.225	−0.0901231
$178$	0	0
$179$	80.0505	0.0334260	0.0167130	−	0.999860i	$-0.494680\pi$
$179$	80.0505	0.0334260	0.0167130	+	0.999860i	$0.494680\pi$
$180$	0	0
$181$	3379.91	1.38799	0.693996	−	0.719978i	$-0.255848\pi$
$181$	3379.91	1.38799	0.693996	+	0.719978i	$0.255848\pi$
$182$	0	0
$183$	−635.023	−0.256515
$184$	0	0
$185$	−7394.87	−2.93882
$186$	0	0
$187$	3436.13	1.34371
$188$	0	0
$189$	−722.484	−0.278058
$190$	0	0
$191$	−3288.78	−1.24590	−0.622952	−	0.782260i	$-0.714067\pi$
$191$	−3288.78	−1.24590	−0.622952	+	0.782260i	$0.714067\pi$
$192$	0	0
$193$	2906.89	1.08416	0.542079	−	0.840327i	$-0.317637\pi$
$193$	2906.89	1.08416	0.542079	+	0.840327i	$0.317637\pi$
$194$	0	0
$195$	−1518.79	−0.557759
$196$	0	0
$197$	−17.1879	−0.00621617	−0.00310808	−	0.999995i	$-0.500989\pi$
$197$	−17.1879	−0.00621617	−0.00310808	+	0.999995i	$0.500989\pi$
$198$	0	0
$199$	−613.599	−0.218577	−0.109289	−	0.994010i	$-0.534857\pi$
$199$	−613.599	−0.218577	−0.109289	+	0.994010i	$0.534857\pi$
$200$	0	0
$201$	−838.611	−0.294284
$202$	0	0
$203$	442.597	0.153026
$204$	0	0
$205$	−2274.99	−0.775084
$206$	0	0
$207$	−3673.76	−1.23354
$208$	0	0
$209$	−958.585	−0.317257
$210$	0	0
$211$	−3991.85	−1.30242	−0.651209	−	0.758898i	$-0.725738\pi$
$211$	−3991.85	−1.30242	−0.651209	+	0.758898i	$0.725738\pi$
$212$	0	0
$213$	627.890	0.201983
$214$	0	0
$215$	8295.54	2.63140
$216$	0	0
$217$	952.075	0.297839
$218$	0	0
$219$	−965.628	−0.297950
$220$	0	0
$221$	−4211.41	−1.28186
$222$	0	0
$223$	−1051.16	−0.315653	−0.157827	−	0.987467i	$-0.550449\pi$
$223$	−1051.16	−0.315653	−0.157827	+	0.987467i	$0.550449\pi$
$224$	0	0
$225$	−5426.97	−1.60799
$226$	0	0
$227$	3430.64	1.00308	0.501541	−	0.865134i	$-0.332767\pi$
$227$	3430.64	1.00308	0.501541	+	0.865134i	$0.332767\pi$
$228$	0	0
$229$	−5907.75	−1.70478	−0.852392	−	0.522904i	$-0.824849\pi$
$229$	−5907.75	−1.70478	−0.852392	+	0.522904i	$0.824849\pi$
$230$	0	0
$231$	697.937	0.198792
$232$	0	0
$233$	−2109.67	−0.593171	−0.296586	−	0.955006i	$-0.595848\pi$
$233$	−2109.67	−0.593171	−0.296586	+	0.955006i	$0.595848\pi$
$234$	0	0
$235$	−8355.91	−2.31949
$236$	0	0
$237$	−1408.59	−0.386067
$238$	0	0
$239$	2898.33	0.784426	0.392213	−	0.919875i	$-0.371710\pi$
$239$	2898.33	0.784426	0.392213	+	0.919875i	$0.371710\pi$
$240$	0	0
$241$	−734.884	−0.196423	−0.0982117	−	0.995166i	$-0.531312\pi$
$241$	−734.884	−0.196423	−0.0982117	+	0.995166i	$0.531312\pi$
$242$	0	0
$243$	2661.72	0.702672
$244$	0	0
$245$	−4335.97	−1.13067
$246$	0	0
$247$	1174.87	0.302652
$248$	0	0
$249$	−968.162	−0.246405
$250$	0	0
$251$	2758.13	0.693593	0.346796	−	0.937940i	$-0.387270\pi$
$251$	2758.13	0.693593	0.346796	+	0.937940i	$0.387270\pi$
$252$	0	0
$253$	7347.40	1.82580
$254$	0	0
$255$	1672.84	0.410813
$256$	0	0
$257$	486.309	0.118036	0.0590178	−	0.998257i	$-0.481203\pi$
$257$	486.309	0.118036	0.0590178	+	0.998257i	$0.481203\pi$
$258$	0	0
$259$	−4164.92	−0.999211
$260$	0	0
$261$	−1074.89	−0.254919
$262$	0	0
$263$	3381.74	0.792878	0.396439	−	0.918061i	$-0.370246\pi$
$263$	3381.74	0.792878	0.396439	+	0.918061i	$0.370246\pi$
$264$	0	0
$265$	8062.06	1.86886
$266$	0	0
$267$	−623.369	−0.142882
$268$	0	0
$269$	−2628.18	−0.595699	−0.297849	−	0.954613i	$-0.596269\pi$
$269$	−2628.18	−0.595699	−0.297849	+	0.954613i	$0.596269\pi$
$270$	0	0
$271$	1104.06	0.247480	0.123740	−	0.992315i	$-0.460511\pi$
$271$	1104.06	0.247480	0.123740	+	0.992315i	$0.460511\pi$
$272$	0	0
$273$	−855.411	−0.189640
$274$	0	0
$275$	10853.8	2.38003
$276$	0	0
$277$	−549.699	−0.119235	−0.0596177	−	0.998221i	$-0.518988\pi$
$277$	−549.699	−0.119235	−0.0596177	+	0.998221i	$0.518988\pi$
$278$	0	0
$279$	−2312.20	−0.496157
$280$	0	0
$281$	−182.930	−0.0388352	−0.0194176	−	0.999811i	$-0.506181\pi$
$281$	−182.930	−0.0388352	−0.0194176	+	0.999811i	$0.506181\pi$
$282$	0	0
$283$	5218.27	1.09609	0.548046	−	0.836448i	$-0.315372\pi$
$283$	5218.27	1.09609	0.548046	+	0.836448i	$0.315372\pi$
$284$	0	0
$285$	−466.677	−0.0969949
$286$	0	0
$287$	−1281.31	−0.263532
$288$	0	0
$289$	−274.434	−0.0558587
$290$	0	0
$291$	−1187.82	−0.239283
$292$	0	0
$293$	6160.17	1.22826	0.614131	−	0.789204i	$-0.289506\pi$
$293$	6160.17	1.22826	0.614131	+	0.789204i	$0.289506\pi$
$294$	0	0
$295$	−2938.87	−0.580026
$296$	0	0
$297$	−3509.18	−0.685600
$298$	0	0
$299$	−9005.19	−1.74175
$300$	0	0
$301$	4672.20	0.894687
$302$	0	0
$303$	−22.1925	−0.00420768
$304$	0	0
$305$	−8793.75	−1.65091
$306$	0	0
$307$	6620.89	1.23086	0.615430	−	0.788192i	$-0.288982\pi$
$307$	6620.89	1.23086	0.615430	+	0.788192i	$0.288982\pi$
$308$	0	0
$309$	1721.59	0.316952
$310$	0	0
$311$	8280.27	1.50975	0.754873	−	0.655870i	$-0.227698\pi$
$311$	8280.27	1.50975	0.754873	+	0.655870i	$0.227698\pi$
$312$	0	0
$313$	3338.77	0.602934	0.301467	−	0.953477i	$-0.402524\pi$
$313$	3338.77	0.602934	0.301467	+	0.953477i	$0.402524\pi$
$314$	0	0
$315$	−4832.55	−0.864392
$316$	0	0
$317$	9587.72	1.69874	0.849369	−	0.527800i	$-0.176983\pi$
$317$	9587.72	1.69874	0.849369	+	0.527800i	$0.176983\pi$
$318$	0	0
$319$	2149.74	0.377311
$320$	0	0
$321$	−234.644	−0.0407992
$322$	0	0
$323$	−1294.03	−0.222916
$324$	0	0
$325$	−13302.7	−2.27047
$326$	0	0
$327$	1836.20	0.310527
$328$	0	0
$329$	−4706.19	−0.788635
$330$	0	0
$331$	11195.2	1.85905	0.929526	−	0.368756i	$-0.120216\pi$
$331$	11195.2	1.85905	0.929526	+	0.368756i	$0.120216\pi$
$332$	0	0
$333$	10114.9	1.66454
$334$	0	0
$335$	−11613.0	−1.89399
$336$	0	0
$337$	5991.21	0.968432	0.484216	−	0.874948i	$-0.339105\pi$
$337$	5991.21	0.968432	0.484216	+	0.874948i	$0.339105\pi$
$338$	0	0
$339$	−801.470	−0.128407
$340$	0	0
$341$	4624.33	0.734374
$342$	0	0
$343$	−6004.91	−0.945291
$344$	0	0
$345$	3577.00	0.558201
$346$	0	0
$347$	6755.35	1.04509	0.522545	−	0.852612i	$-0.324983\pi$
$347$	6755.35	1.04509	0.522545	+	0.852612i	$0.324983\pi$
$348$	0	0
$349$	−2358.14	−0.361685	−0.180843	−	0.983512i	$-0.557883\pi$
$349$	−2358.14	−0.361685	−0.180843	+	0.983512i	$0.557883\pi$
$350$	0	0
$351$	4300.95	0.654039
$352$	0	0
$353$	−5950.00	−0.897129	−0.448565	−	0.893750i	$-0.648065\pi$
$353$	−5950.00	−0.897129	−0.448565	+	0.893750i	$0.648065\pi$
$354$	0	0
$355$	8694.96	1.29995
$356$	0	0
$357$	942.173	0.139678
$358$	0	0
$359$	1892.90	0.278283	0.139141	−	0.990273i	$-0.455566\pi$
$359$	1892.90	0.278283	0.139141	+	0.990273i	$0.455566\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	1617.32	0.233850
$364$	0	0
$365$	−13371.9	−1.91759
$366$	0	0
$367$	3643.63	0.518245	0.259122	−	0.965844i	$-0.416567\pi$
$367$	3643.63	0.518245	0.259122	+	0.965844i	$0.416567\pi$
$368$	0	0
$369$	3111.79	0.439006
$370$	0	0
$371$	4540.69	0.635421
$372$	0	0
$373$	4820.77	0.669195	0.334598	−	0.942361i	$-0.391400\pi$
$373$	4820.77	0.669195	0.334598	+	0.942361i	$0.391400\pi$
$374$	0	0
$375$	2213.81	0.304854
$376$	0	0
$377$	−2634.78	−0.359942
$378$	0	0
$379$	−8138.02	−1.10296	−0.551480	−	0.834188i	$-0.685937\pi$
$379$	−8138.02	−1.10296	−0.551480	+	0.834188i	$0.685937\pi$
$380$	0	0
$381$	2517.61	0.338532
$382$	0	0
$383$	904.971	0.120736	0.0603679	−	0.998176i	$-0.480773\pi$
$383$	904.971	0.120736	0.0603679	+	0.998176i	$0.480773\pi$
$384$	0	0
$385$	9664.96	1.27941
$386$	0	0
$387$	−11346.8	−1.49042
$388$	0	0
$389$	8885.27	1.15810	0.579050	−	0.815292i	$-0.303424\pi$
$389$	8885.27	1.15810	0.579050	+	0.815292i	$0.303424\pi$
$390$	0	0
$391$	9918.56	1.28287
$392$	0	0
$393$	442.109	0.0567467
$394$	0	0
$395$	−19506.1	−2.48470
$396$	0	0
$397$	−981.841	−0.124124	−0.0620619	−	0.998072i	$-0.519768\pi$
$397$	−981.841	−0.124124	−0.0620619	+	0.998072i	$0.519768\pi$
$398$	0	0
$399$	−262.841	−0.0329787
$400$	0	0
$401$	−15512.7	−1.93184	−0.965919	−	0.258845i	$-0.916658\pi$
$401$	−15512.7	−1.93184	−0.965919	+	0.258845i	$0.916658\pi$
$402$	0	0
$403$	−5667.71	−0.700568
$404$	0	0
$405$	10853.1	1.33159
$406$	0	0
$407$	−20229.5	−2.46373
$408$	0	0
$409$	−13347.9	−1.61371	−0.806857	−	0.590747i	$-0.798833\pi$
$409$	−13347.9	−1.61371	−0.806857	+	0.590747i	$0.798833\pi$
$410$	0	0
$411$	−2605.27	−0.312673
$412$	0	0
$413$	−1655.22	−0.197211
$414$	0	0
$415$	−13407.0	−1.58584
$416$	0	0
$417$	729.588	0.0856789
$418$	0	0
$419$	1557.12	0.181552	0.0907758	−	0.995871i	$-0.471065\pi$
$419$	1557.12	0.181552	0.0907758	+	0.995871i	$0.471065\pi$
$420$	0	0
$421$	1281.41	0.148342	0.0741710	−	0.997246i	$-0.476369\pi$
$421$	1281.41	0.148342	0.0741710	+	0.997246i	$0.476369\pi$
$422$	0	0
$423$	11429.4	1.31375
$424$	0	0
$425$	14652.0	1.67229
$426$	0	0
$427$	−4952.79	−0.561317
$428$	0	0
$429$	−4154.82	−0.467591
$430$	0	0
$431$	9742.61	1.08883	0.544414	−	0.838817i	$-0.316752\pi$
$431$	9742.61	1.08883	0.544414	+	0.838817i	$0.316752\pi$
$432$	0	0
$433$	−5262.21	−0.584031	−0.292016	−	0.956414i	$-0.594326\pi$
$433$	−5262.21	−0.584031	−0.292016	+	0.956414i	$0.594326\pi$
$434$	0	0
$435$	1046.58	0.115355
$436$	0	0
$437$	−2767.01	−0.302892
$438$	0	0
$439$	−15015.7	−1.63249	−0.816243	−	0.577708i	$-0.803947\pi$
$439$	−15015.7	−1.63249	−0.816243	+	0.577708i	$0.803947\pi$
$440$	0	0
$441$	5930.84	0.640411
$442$	0	0
$443$	−4296.96	−0.460846	−0.230423	−	0.973091i	$-0.574011\pi$
$443$	−4296.96	−0.460846	−0.230423	+	0.973091i	$0.574011\pi$
$444$	0	0
$445$	−8632.36	−0.919580
$446$	0	0
$447$	−1321.79	−0.139862
$448$	0	0
$449$	6749.23	0.709389	0.354695	−	0.934982i	$-0.384585\pi$
$449$	6749.23	0.709389	0.354695	+	0.934982i	$0.384585\pi$
$450$	0	0
$451$	−6223.48	−0.649783
$452$	0	0
$453$	−197.329	−0.0204665
$454$	0	0
$455$	−11845.7	−1.22051
$456$	0	0
$457$	11399.9	1.16689	0.583443	−	0.812154i	$-0.301705\pi$
$457$	11399.9	1.16689	0.583443	+	0.812154i	$0.301705\pi$
$458$	0	0
$459$	−4737.19	−0.481727
$460$	0	0
$461$	−5799.27	−0.585898	−0.292949	−	0.956128i	$-0.594637\pi$
$461$	−5799.27	−0.585898	−0.292949	+	0.956128i	$0.594637\pi$
$462$	0	0
$463$	620.160	0.0622490	0.0311245	−	0.999516i	$-0.490091\pi$
$463$	620.160	0.0622490	0.0311245	+	0.999516i	$0.490091\pi$
$464$	0	0
$465$	2251.30	0.224520
$466$	0	0
$467$	9426.16	0.934027	0.467014	−	0.884250i	$-0.345330\pi$
$467$	9426.16	0.934027	0.467014	+	0.884250i	$0.345330\pi$
$468$	0	0
$469$	−6540.65	−0.643964
$470$	0	0
$471$	3138.20	0.307007
$472$	0	0
$473$	22693.3	2.20601
$474$	0	0
$475$	−4087.50	−0.394836
$476$	0	0
$477$	−11027.5	−1.05852
$478$	0	0
$479$	15813.6	1.50844	0.754218	−	0.656625i	$-0.228016\pi$
$479$	15813.6	1.50844	0.754218	+	0.656625i	$0.228016\pi$
$480$	0	0
$481$	24793.8	2.35031
$482$	0	0
$483$	2014.63	0.189791
$484$	0	0
$485$	−16448.8	−1.54001
$486$	0	0
$487$	−6832.36	−0.635736	−0.317868	−	0.948135i	$-0.602967\pi$
$487$	−6832.36	−0.635736	−0.317868	+	0.948135i	$0.602967\pi$
$488$	0	0
$489$	−3961.92	−0.366389
$490$	0	0
$491$	−15084.5	−1.38646	−0.693232	−	0.720714i	$-0.743814\pi$
$491$	−15084.5	−1.38646	−0.693232	+	0.720714i	$0.743814\pi$
$492$	0	0
$493$	2902.02	0.265112
$494$	0	0
$495$	−23472.2	−2.13131
$496$	0	0
$497$	4897.16	0.441987
$498$	0	0
$499$	21771.2	1.95313	0.976564	−	0.215226i	$-0.0690490\pi$
$499$	21771.2	1.95313	0.976564	+	0.215226i	$0.0690490\pi$
$500$	0	0
$501$	−4172.65	−0.372096
$502$	0	0
$503$	21300.9	1.88819	0.944097	−	0.329668i	$-0.106937\pi$
$503$	21300.9	1.88819	0.944097	+	0.329668i	$0.106937\pi$
$504$	0	0
$505$	−307.320	−0.0270803
$506$	0	0
$507$	2166.30	0.189761
$508$	0	0
$509$	−15897.9	−1.38440	−0.692202	−	0.721704i	$-0.743359\pi$
$509$	−15897.9	−1.38440	−0.692202	+	0.721704i	$0.743359\pi$
$510$	0	0
$511$	−7531.31	−0.651987
$512$	0	0
$513$	1321.55	0.113738
$514$	0	0
$515$	23840.5	2.03988
$516$	0	0
$517$	−22858.5	−1.94452
$518$	0	0
$519$	4548.63	0.384707
$520$	0	0
$521$	−13864.6	−1.16587	−0.582937	−	0.812517i	$-0.698097\pi$
$521$	−13864.6	−1.16587	−0.582937	+	0.812517i	$0.698097\pi$
$522$	0	0
$523$	−20503.0	−1.71421	−0.857106	−	0.515141i	$-0.827740\pi$
$523$	−20503.0	−1.71421	−0.857106	+	0.515141i	$0.827740\pi$
$524$	0	0
$525$	2976.07	0.247402
$526$	0	0
$527$	6242.57	0.515997
$528$	0	0
$529$	9041.66	0.743130
$530$	0	0
$531$	4019.86	0.328525
$532$	0	0
$533$	7627.67	0.619871
$534$	0	0
$535$	−3249.33	−0.262581
$536$	0	0
$537$	106.611	0.00856726
$538$	0	0
$539$	−11861.5	−0.947887
$540$	0	0
$541$	−15933.8	−1.26626	−0.633131	−	0.774045i	$-0.718231\pi$
$541$	−15933.8	−1.26626	−0.633131	+	0.774045i	$0.718231\pi$
$542$	0	0
$543$	4501.37	0.355750
$544$	0	0
$545$	25427.6	1.99853
$546$	0	0
$547$	20542.0	1.60569	0.802847	−	0.596185i	$-0.203318\pi$
$547$	20542.0	1.60569	0.802847	+	0.596185i	$0.203318\pi$
$548$	0	0
$549$	12028.3	0.935073
$550$	0	0
$551$	−809.584	−0.0625943
$552$	0	0
$553$	−10986.2	−0.844808
$554$	0	0
$555$	−9848.49	−0.753235
$556$	0	0
$557$	12338.5	0.938599	0.469299	−	0.883039i	$-0.344507\pi$
$557$	12338.5	0.938599	0.469299	+	0.883039i	$0.344507\pi$
$558$	0	0
$559$	−27813.6	−2.10446
$560$	0	0
$561$	4576.23	0.344401
$562$	0	0
$563$	7611.22	0.569759	0.284880	−	0.958563i	$-0.408046\pi$
$563$	7611.22	0.569759	0.284880	+	0.958563i	$0.408046\pi$
$564$	0	0
$565$	−11098.7	−0.826417
$566$	0	0
$567$	6112.64	0.452746
$568$	0	0
$569$	9237.35	0.680579	0.340290	−	0.940321i	$-0.389475\pi$
$569$	9237.35	0.680579	0.340290	+	0.940321i	$0.389475\pi$
$570$	0	0
$571$	7061.48	0.517537	0.258769	−	0.965939i	$-0.416683\pi$
$571$	7061.48	0.517537	0.258769	+	0.965939i	$0.416683\pi$
$572$	0	0
$573$	−4380.00	−0.319332
$574$	0	0
$575$	31330.0	2.27226
$576$	0	0
$577$	−12167.1	−0.877856	−0.438928	−	0.898522i	$-0.644642\pi$
$577$	−12167.1	−0.877856	−0.438928	+	0.898522i	$0.644642\pi$
$578$	0	0
$579$	3871.40	0.277876
$580$	0	0
$581$	−7551.07	−0.539193
$582$	0	0
$583$	22054.6	1.56674
$584$	0	0
$585$	28768.2	2.03320
$586$	0	0
$587$	−11396.4	−0.801327	−0.400663	−	0.916225i	$-0.631220\pi$
$587$	−11396.4	−0.801327	−0.400663	+	0.916225i	$0.631220\pi$
$588$	0	0
$589$	−1741.51	−0.121829
$590$	0	0
$591$	−22.8908	−0.00159324
$592$	0	0
$593$	−10401.1	−0.720276	−0.360138	−	0.932899i	$-0.617270\pi$
$593$	−10401.1	−0.720276	−0.360138	+	0.932899i	$0.617270\pi$
$594$	0	0
$595$	13047.1	0.898958
$596$	0	0
$597$	−817.192	−0.0560225
$598$	0	0
$599$	2296.79	0.156668	0.0783340	−	0.996927i	$-0.475040\pi$
$599$	2296.79	0.156668	0.0783340	+	0.996927i	$0.475040\pi$
$600$	0	0
$601$	−7376.50	−0.500655	−0.250328	−	0.968161i	$-0.580538\pi$
$601$	−7376.50	−0.500655	−0.250328	+	0.968161i	$0.580538\pi$
$602$	0	0
$603$	15884.5	1.07275
$604$	0	0
$605$	22396.6	1.50504
$606$	0	0
$607$	−11793.0	−0.788570	−0.394285	−	0.918988i	$-0.629008\pi$
$607$	−11793.0	−0.788570	−0.394285	+	0.918988i	$0.629008\pi$
$608$	0	0
$609$	589.451	0.0392213
$610$	0	0
$611$	28016.0	1.85500
$612$	0	0
$613$	20918.1	1.37826	0.689131	−	0.724637i	$-0.257992\pi$
$613$	20918.1	1.37826	0.689131	+	0.724637i	$0.257992\pi$
$614$	0	0
$615$	−3029.83	−0.198658
$616$	0	0
$617$	−3951.13	−0.257806	−0.128903	−	0.991657i	$-0.541146\pi$
$617$	−3951.13	−0.257806	−0.128903	+	0.991657i	$0.541146\pi$
$618$	0	0
$619$	−22159.8	−1.43890	−0.719450	−	0.694544i	$-0.755606\pi$
$619$	−22159.8	−1.43890	−0.719450	+	0.694544i	$0.755606\pi$
$620$	0	0
$621$	−10129.4	−0.654557
$622$	0	0
$623$	−4861.90	−0.312661
$624$	0	0
$625$	3765.14	0.240969
$626$	0	0
$627$	−1276.64	−0.0813146
$628$	0	0
$629$	−27308.6	−1.73110
$630$	0	0
$631$	11645.2	0.734687	0.367344	−	0.930085i	$-0.380267\pi$
$631$	11645.2	0.734687	0.367344	+	0.930085i	$0.380267\pi$
$632$	0	0
$633$	−5316.35	−0.333817
$634$	0	0
$635$	34863.6	2.17877
$636$	0	0
$637$	14537.8	0.904252
$638$	0	0
$639$	−11893.2	−0.736286
$640$	0	0
$641$	−21970.1	−1.35377	−0.676885	−	0.736089i	$-0.736671\pi$
$641$	−21970.1	−1.35377	−0.676885	+	0.736089i	$0.736671\pi$
$642$	0	0
$643$	−14047.2	−0.861537	−0.430769	−	0.902462i	$-0.641757\pi$
$643$	−14047.2	−0.861537	−0.430769	+	0.902462i	$0.641757\pi$
$644$	0	0
$645$	11048.0	0.674442
$646$	0	0
$647$	20700.0	1.25780	0.628902	−	0.777485i	$-0.283505\pi$
$647$	20700.0	1.25780	0.628902	+	0.777485i	$0.283505\pi$
$648$	0	0
$649$	−8039.59	−0.486258
$650$	0	0
$651$	1267.97	0.0763377
$652$	0	0
$653$	17197.8	1.03063	0.515315	−	0.857001i	$-0.327675\pi$
$653$	17197.8	1.03063	0.515315	+	0.857001i	$0.327675\pi$
$654$	0	0
$655$	6122.29	0.365218
$656$	0	0
$657$	18290.5	1.08612
$658$	0	0
$659$	−23084.0	−1.36453	−0.682264	−	0.731106i	$-0.739005\pi$
$659$	−23084.0	−1.36453	−0.682264	+	0.731106i	$0.739005\pi$
$660$	0	0
$661$	−9283.22	−0.546256	−0.273128	−	0.961978i	$-0.588058\pi$
$661$	−9283.22	−0.546256	−0.273128	+	0.961978i	$0.588058\pi$
$662$	0	0
$663$	−5608.76	−0.328547
$664$	0	0
$665$	−3639.79	−0.212248
$666$	0	0
$667$	6205.33	0.360227
$668$	0	0
$669$	−1399.93	−0.0809036
$670$	0	0
$671$	−24056.2	−1.38402
$672$	0	0
$673$	11565.0	0.662403	0.331202	−	0.943560i	$-0.392546\pi$
$673$	11565.0	0.662403	0.331202	+	0.943560i	$0.392546\pi$
$674$	0	0
$675$	−14963.5	−0.853251
$676$	0	0
$677$	8808.39	0.500050	0.250025	−	0.968239i	$-0.419561\pi$
$677$	8808.39	0.500050	0.250025	+	0.968239i	$0.419561\pi$
$678$	0	0
$679$	−9264.27	−0.523608
$680$	0	0
$681$	4568.93	0.257095
$682$	0	0
$683$	16282.3	0.912186	0.456093	−	0.889932i	$-0.349248\pi$
$683$	16282.3	0.912186	0.456093	+	0.889932i	$0.349248\pi$
$684$	0	0
$685$	−36077.6	−2.01234
$686$	0	0
$687$	−7867.95	−0.436945
$688$	0	0
$689$	−27030.8	−1.49462
$690$	0	0
$691$	−21900.4	−1.20569	−0.602845	−	0.797859i	$-0.705966\pi$
$691$	−21900.4	−1.20569	−0.602845	+	0.797859i	$0.705966\pi$
$692$	0	0
$693$	−13220.0	−0.724654
$694$	0	0
$695$	10103.3	0.551423
$696$	0	0
$697$	−8401.33	−0.456561
$698$	0	0
$699$	−2809.66	−0.152033
$700$	0	0
$701$	−12286.6	−0.661994	−0.330997	−	0.943632i	$-0.607385\pi$
$701$	−12286.6	−0.661994	−0.330997	+	0.943632i	$0.607385\pi$
$702$	0	0
$703$	7618.34	0.408722
$704$	0	0
$705$	−11128.4	−0.594496
$706$	0	0
$707$	−173.088	−0.00920743
$708$	0	0
$709$	−12918.9	−0.684315	−0.342158	−	0.939643i	$-0.611158\pi$
$709$	−12918.9	−0.684315	−0.342158	+	0.939643i	$0.611158\pi$
$710$	0	0
$711$	26680.8	1.40733
$712$	0	0
$713$	13348.4	0.701122
$714$	0	0
$715$	−57535.6	−3.00938
$716$	0	0
$717$	3860.00	0.201052
$718$	0	0
$719$	−15819.9	−0.820558	−0.410279	−	0.911960i	$-0.634569\pi$
$719$	−15819.9	−0.820558	−0.410279	+	0.911960i	$0.634569\pi$
$720$	0	0
$721$	13427.4	0.693567
$722$	0	0
$723$	−978.719	−0.0503443
$724$	0	0
$725$	9166.69	0.469576
$726$	0	0
$727$	1793.81	0.0915112	0.0457556	−	0.998953i	$-0.485430\pi$
$727$	1793.81	0.0915112	0.0457556	+	0.998953i	$0.485430\pi$
$728$	0	0
$729$	−12344.0	−0.627140
$730$	0	0
$731$	30634.7	1.55002
$732$	0	0
$733$	10370.0	0.522541	0.261271	−	0.965266i	$-0.415858\pi$
$733$	10370.0	0.522541	0.261271	+	0.965266i	$0.415858\pi$
$734$	0	0
$735$	−5774.65	−0.289797
$736$	0	0
$737$	−31768.6	−1.58780
$738$	0	0
$739$	29414.1	1.46416	0.732081	−	0.681218i	$-0.238549\pi$
$739$	29414.1	1.46416	0.732081	+	0.681218i	$0.238549\pi$
$740$	0	0
$741$	1564.69	0.0775714
$742$	0	0
$743$	−14932.0	−0.737286	−0.368643	−	0.929571i	$-0.620177\pi$
$743$	−14932.0	−0.737286	−0.368643	+	0.929571i	$0.620177\pi$
$744$	0	0
$745$	−18304.0	−0.900142
$746$	0	0
$747$	18338.4	0.898218
$748$	0	0
$749$	−1830.08	−0.0892785
$750$	0	0
$751$	21916.9	1.06492	0.532462	−	0.846454i	$-0.321267\pi$
$751$	21916.9	1.06492	0.532462	+	0.846454i	$0.321267\pi$
$752$	0	0
$753$	3673.28	0.177771
$754$	0	0
$755$	−2732.59	−0.131721
$756$	0	0
$757$	−26973.4	−1.29506	−0.647532	−	0.762038i	$-0.724199\pi$
$757$	−26973.4	−1.29506	−0.647532	+	0.762038i	$0.724199\pi$
$758$	0	0
$759$	9785.27	0.467962
$760$	0	0
$761$	−16145.3	−0.769076	−0.384538	−	0.923109i	$-0.625639\pi$
$761$	−16145.3	−0.769076	−0.384538	+	0.923109i	$0.625639\pi$
$762$	0	0
$763$	14321.3	0.679508
$764$	0	0
$765$	−31686.1	−1.49753
$766$	0	0
$767$	9853.55	0.463874
$768$	0	0
$769$	−23777.1	−1.11499	−0.557494	−	0.830181i	$-0.688237\pi$
$769$	−23777.1	−1.11499	−0.557494	+	0.830181i	$0.688237\pi$
$770$	0	0
$771$	647.667	0.0302531
$772$	0	0
$773$	27962.2	1.30107	0.650537	−	0.759475i	$-0.274544\pi$
$773$	27962.2	1.30107	0.650537	+	0.759475i	$0.274544\pi$
$774$	0	0
$775$	19718.6	0.913951
$776$	0	0
$777$	−5546.84	−0.256103
$778$	0	0
$779$	2343.74	0.107796
$780$	0	0
$781$	23786.0	1.08980
$782$	0	0
$783$	−2963.72	−0.135268
$784$	0	0
$785$	43457.5	1.97588
$786$	0	0
$787$	19000.4	0.860600	0.430300	−	0.902686i	$-0.358408\pi$
$787$	19000.4	0.860600	0.430300	+	0.902686i	$0.358408\pi$
$788$	0	0
$789$	4503.80	0.203219
$790$	0	0
$791$	−6250.98	−0.280985
$792$	0	0
$793$	29484.0	1.32031
$794$	0	0
$795$	10737.1	0.478999
$796$	0	0
$797$	−19991.0	−0.888480	−0.444240	−	0.895908i	$-0.646526\pi$
$797$	−19991.0	−0.888480	−0.444240	+	0.895908i	$0.646526\pi$
$798$	0	0
$799$	−30857.6	−1.36629
$800$	0	0
$801$	11807.5	0.520848
$802$	0	0
$803$	−36580.3	−1.60759
$804$	0	0
$805$	27898.4	1.22148
$806$	0	0
$807$	−3500.21	−0.152681
$808$	0	0
$809$	−40779.8	−1.77224	−0.886120	−	0.463455i	$-0.846609\pi$
$809$	−40779.8	−1.77224	−0.886120	+	0.463455i	$0.846609\pi$
$810$	0	0
$811$	−23160.4	−1.00280	−0.501400	−	0.865216i	$-0.667181\pi$
$811$	−23160.4	−1.00280	−0.501400	+	0.865216i	$0.667181\pi$
$812$	0	0
$813$	1470.39	0.0634304
$814$	0	0
$815$	−54864.3	−2.35805
$816$	0	0
$817$	−8546.24	−0.365967
$818$	0	0
$819$	16202.8	0.691295
$820$	0	0
$821$	29453.3	1.25204	0.626021	−	0.779806i	$-0.284682\pi$
$821$	29453.3	1.25204	0.626021	+	0.779806i	$0.284682\pi$
$822$	0	0
$823$	−43227.2	−1.83087	−0.915435	−	0.402465i	$-0.868153\pi$
$823$	−43227.2	−1.83087	−0.915435	+	0.402465i	$0.868153\pi$
$824$	0	0
$825$	14455.1	0.610014
$826$	0	0
$827$	−34232.7	−1.43940	−0.719702	−	0.694283i	$-0.755722\pi$
$827$	−34232.7	−1.43940	−0.719702	+	0.694283i	$0.755722\pi$
$828$	0	0
$829$	32410.2	1.35785	0.678923	−	0.734209i	$-0.262447\pi$
$829$	32410.2	1.35785	0.678923	+	0.734209i	$0.262447\pi$
$830$	0	0
$831$	−732.089	−0.0305606
$832$	0	0
$833$	−16012.3	−0.666020
$834$	0	0
$835$	−57782.5	−2.39478
$836$	0	0
$837$	−6375.29	−0.263276
$838$	0	0
$839$	−6363.50	−0.261851	−0.130925	−	0.991392i	$-0.541795\pi$
$839$	−6363.50	−0.261851	−0.130925	+	0.991392i	$0.541795\pi$
$840$	0	0
$841$	−22573.4	−0.925557
$842$	0	0
$843$	−243.626	−0.00995366
$844$	0	0
$845$	29998.8	1.22129
$846$	0	0
$847$	12614.1	0.511720
$848$	0	0
$849$	6949.69	0.280934
$850$	0	0
$851$	−58393.4	−2.35217
$852$	0	0
$853$	22798.2	0.915117	0.457558	−	0.889180i	$-0.348724\pi$
$853$	22798.2	0.915117	0.457558	+	0.889180i	$0.348724\pi$
$854$	0	0
$855$	8839.56	0.353575
$856$	0	0
$857$	26426.8	1.05335	0.526675	−	0.850067i	$-0.323438\pi$
$857$	26426.8	1.05335	0.526675	+	0.850067i	$0.323438\pi$
$858$	0	0
$859$	22484.5	0.893087	0.446544	−	0.894762i	$-0.352655\pi$
$859$	22484.5	0.893087	0.446544	+	0.894762i	$0.352655\pi$
$860$	0	0
$861$	−1706.46	−0.0675445
$862$	0	0
$863$	−22062.6	−0.870242	−0.435121	−	0.900372i	$-0.643294\pi$
$863$	−22062.6	−0.870242	−0.435121	+	0.900372i	$0.643294\pi$
$864$	0	0
$865$	62989.0	2.47594
$866$	0	0
$867$	−365.491	−0.0143169
$868$	0	0
$869$	−53360.9	−2.08302
$870$	0	0
$871$	38936.5	1.51471
$872$	0	0
$873$	22499.1	0.872256
$874$	0	0
$875$	17266.3	0.667095
$876$	0	0
$877$	17.2794	0.000665316 0	0.000332658	−	1.00000i	$-0.499894\pi$
$877$	17.2794	0.000665316 0	0.000332658		1.00000i	$0.499894\pi$
$878$	0	0
$879$	8204.12	0.314810
$880$	0	0
$881$	19338.2	0.739524	0.369762	−	0.929126i	$-0.379439\pi$
$881$	19338.2	0.739524	0.369762	+	0.929126i	$0.379439\pi$
$882$	0	0
$883$	35899.9	1.36821	0.684105	−	0.729384i	$-0.260193\pi$
$883$	35899.9	1.36821	0.684105	+	0.729384i	$0.260193\pi$
$884$	0	0
$885$	−3913.99	−0.148664
$886$	0	0
$887$	51076.9	1.93348	0.966738	−	0.255767i	$-0.0823280\pi$
$887$	51076.9	1.93348	0.966738	+	0.255767i	$0.0823280\pi$
$888$	0	0
$889$	19635.8	0.740791
$890$	0	0
$891$	29689.7	1.11632
$892$	0	0
$893$	8608.43	0.322587
$894$	0	0
$895$	1476.34	0.0551383
$896$	0	0
$897$	−11993.1	−0.446420
$898$	0	0
$899$	3905.53	0.144891
$900$	0	0
$901$	29772.4	1.10085
$902$	0	0
$903$	6222.43	0.229313
$904$	0	0
$905$	62334.5	2.28958
$906$	0	0
$907$	18999.5	0.695555	0.347778	−	0.937577i	$-0.386936\pi$
$907$	18999.5	0.695555	0.347778	+	0.937577i	$0.386936\pi$
$908$	0	0
$909$	420.360	0.0153382
$910$	0	0
$911$	17147.0	0.623605	0.311802	−	0.950147i	$-0.399067\pi$
$911$	17147.0	0.623605	0.311802	+	0.950147i	$0.399067\pi$
$912$	0	0
$913$	−36676.3	−1.32947
$914$	0	0
$915$	−11711.5	−0.423137
$916$	0	0
$917$	3448.18	0.124176
$918$	0	0
$919$	−10013.4	−0.359425	−0.179712	−	0.983719i	$-0.557517\pi$
$919$	−10013.4	−0.359425	−0.179712	+	0.983719i	$0.557517\pi$
$920$	0	0
$921$	8817.70	0.315476
$922$	0	0
$923$	−29152.8	−1.03963
$924$	0	0
$925$	−86260.3	−3.06619
$926$	0	0
$927$	−32609.6	−1.15538
$928$	0	0
$929$	−22893.7	−0.808524	−0.404262	−	0.914643i	$-0.632472\pi$
$929$	−22893.7	−0.808524	−0.404262	+	0.914643i	$0.632472\pi$
$930$	0	0
$931$	4467.00	0.157250
$932$	0	0
$933$	11027.7	0.386956
$934$	0	0
$935$	63371.3	2.21654
$936$	0	0
$937$	1003.65	0.0349922	0.0174961	−	0.999847i	$-0.494431\pi$
$937$	1003.65	0.0349922	0.0174961	+	0.999847i	$0.494431\pi$
$938$	0	0
$939$	4446.58	0.154535
$940$	0	0
$941$	14679.7	0.508550	0.254275	−	0.967132i	$-0.418163\pi$
$941$	14679.7	0.508550	0.254275	+	0.967132i	$0.418163\pi$
$942$	0	0
$943$	−17964.4	−0.620362
$944$	0	0
$945$	−13324.5	−0.458674
$946$	0	0
$947$	38570.8	1.32353	0.661765	−	0.749711i	$-0.269808\pi$
$947$	38570.8	1.32353	0.661765	+	0.749711i	$0.269808\pi$
$948$	0	0
$949$	44833.9	1.53358
$950$	0	0
$951$	12768.9	0.435395
$952$	0	0
$953$	2208.75	0.0750770	0.0375385	−	0.999295i	$-0.488048\pi$
$953$	2208.75	0.0750770	0.0375385	+	0.999295i	$0.488048\pi$
$954$	0	0
$955$	−60653.8	−2.05520
$956$	0	0
$957$	2863.02	0.0967068
$958$	0	0
$959$	−20319.5	−0.684204
$960$	0	0
$961$	−21389.8	−0.717994
$962$	0	0
$963$	4444.51	0.148725
$964$	0	0
$965$	53610.8	1.78839
$966$	0	0
$967$	5440.71	0.180932	0.0904661	−	0.995900i	$-0.471164\pi$
$967$	5440.71	0.180932	0.0904661	+	0.995900i	$0.471164\pi$
$968$	0	0
$969$	−1723.39	−0.0571346
$970$	0	0
$971$	25018.4	0.826857	0.413428	−	0.910537i	$-0.364331\pi$
$971$	25018.4	0.826857	0.413428	+	0.910537i	$0.364331\pi$
$972$	0	0
$973$	5690.34	0.187486
$974$	0	0
$975$	−17716.6	−0.581932
$976$	0	0
$977$	−45055.8	−1.47540	−0.737698	−	0.675131i	$-0.764087\pi$
$977$	−45055.8	−1.47540	−0.737698	+	0.675131i	$0.764087\pi$
$978$	0	0
$979$	−23614.7	−0.770920
$980$	0	0
$981$	−34780.5	−1.13196
$982$	0	0
$983$	−13675.5	−0.443725	−0.221862	−	0.975078i	$-0.571214\pi$
$983$	−13675.5	−0.443725	−0.221862	+	0.975078i	$0.571214\pi$
$984$	0	0
$985$	−316.990	−0.0102539
$986$	0	0
$987$	−6267.71	−0.202131
$988$	0	0
$989$	65505.5	2.10612
$990$	0	0
$991$	23018.0	0.737832	0.368916	−	0.929463i	$-0.379729\pi$
$991$	23018.0	0.737832	0.368916	+	0.929463i	$0.379729\pi$
$992$	0	0
$993$	14909.8	0.476485
$994$	0	0
$995$	−11316.4	−0.360557
$996$	0	0
$997$	−16189.1	−0.514256	−0.257128	−	0.966377i	$-0.582776\pi$
$997$	−16189.1	−0.514256	−0.257128	+	0.966377i	$0.582776\pi$
$998$	0	0
$999$	27889.2	0.883258

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.a.c.1.2	✓	3
3.2	odd	2		1368.4.a.d.1.1		3
4.3	odd	2		304.4.a.g.1.2		3
8.3	odd	2		1216.4.a.w.1.2		3
8.5	even	2		1216.4.a.r.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.c.1.2	✓	3	1.1	even	1	trivial
304.4.a.g.1.2		3	4.3	odd	2
1216.4.a.r.1.2		3	8.5	even	2
1216.4.a.w.1.2		3	8.3	odd	2
1368.4.a.d.1.1		3	3.2	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:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	152.a (trivial)

\(f(q)\)	\(=\)	\(q+1.33180 q^{3} +18.4427 q^{5} +10.3872 q^{7} -25.2263 q^{9} +50.4519 q^{11} -61.8352 q^{13} +24.5619 q^{15} +68.1070 q^{17} -19.0000 q^{19} +13.8337 q^{21} +145.632 q^{23} +215.132 q^{25} -69.5550 q^{27} +42.6097 q^{29} +91.6582 q^{31} +67.1918 q^{33} +191.568 q^{35} -400.965 q^{37} -82.3522 q^{39} -123.355 q^{41} +449.802 q^{43} -465.240 q^{45} -453.075 q^{47} -235.105 q^{49} +90.7050 q^{51} +437.142 q^{53} +930.466 q^{55} -25.3042 q^{57} -159.352 q^{59} -476.816 q^{61} -262.031 q^{63} -1140.41 q^{65} -629.682 q^{67} +193.953 q^{69} +471.459 q^{71} -725.055 q^{73} +286.512 q^{75} +524.055 q^{77} -1057.66 q^{79} +588.477 q^{81} -726.957 q^{83} +1256.07 q^{85} +56.7476 q^{87} -468.065 q^{89} -642.297 q^{91} +122.071 q^{93} -350.410 q^{95} -891.891 q^{97} -1272.71 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 21 q^{9} + 103 q^{11} + 32 q^{13} + 122 q^{15} + 11 q^{17} - 57 q^{19} + 114 q^{21} + 316 q^{23} + 162 q^{25} + 178 q^{27} - 138 q^{29} + 420 q^{31} - 330 q^{33} + 333 q^{35}+ \cdots - 1305 q^{99}+O(q^{100}) \) 3 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 21 * q^9 + 103 * q^11 + 32 * q^13 + 122 * q^15 + 11 * q^17 - 57 * q^19 + 114 * q^21 + 316 * q^23 + 162 * q^25 + 178 * q^27 - 138 * q^29 + 420 * q^31 - 330 * q^33 + 333 * q^35 + 102 * q^37 + 164 * q^39 - 370 * q^41 + 431 * q^43 - 429 * q^45 - 199 * q^47 - 802 * q^49 + 272 * q^51 - 308 * q^53 + 85 * q^55 - 76 * q^57 - 188 * q^59 - 609 * q^61 - 61 * q^63 - 1536 * q^65 - 246 * q^67 - 892 * q^69 - 954 * q^71 - 629 * q^73 - 1074 * q^75 - 71 * q^77 + 452 * q^79 - 361 * q^81 - 780 * q^83 + 1883 * q^85 - 96 * q^87 - 1356 * q^89 - 670 * q^91 + 1092 * q^93 - 133 * q^95 - 548 * q^97 - 1305 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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