LMFDB - Embedded newform 2151.4.a.g.1.44

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.44
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.96328 q^{2} +0.781040 q^{4} -9.17370 q^{5} +2.35613 q^{7} -21.3918 q^{8} +O(q^{10})$	\(q+2.96328 q^{2} +0.781040 q^{4} -9.17370 q^{5} +2.35613 q^{7} -21.3918 q^{8} -27.1843 q^{10} +14.3796 q^{11} +15.1470 q^{13} +6.98188 q^{14} -69.6383 q^{16} +57.8861 q^{17} +103.148 q^{19} -7.16503 q^{20} +42.6109 q^{22} -82.7139 q^{23} -40.8433 q^{25} +44.8848 q^{26} +1.84023 q^{28} +82.2294 q^{29} -156.009 q^{31} -35.2234 q^{32} +171.533 q^{34} -21.6144 q^{35} +202.052 q^{37} +305.656 q^{38} +196.242 q^{40} +3.54593 q^{41} +357.565 q^{43} +11.2311 q^{44} -245.105 q^{46} -147.712 q^{47} -337.449 q^{49} -121.030 q^{50} +11.8304 q^{52} -160.081 q^{53} -131.915 q^{55} -50.4019 q^{56} +243.669 q^{58} +45.9350 q^{59} -743.891 q^{61} -462.299 q^{62} +452.730 q^{64} -138.954 q^{65} +165.554 q^{67} +45.2114 q^{68} -64.0497 q^{70} -747.149 q^{71} +962.660 q^{73} +598.738 q^{74} +80.5627 q^{76} +33.8803 q^{77} +803.730 q^{79} +638.841 q^{80} +10.5076 q^{82} -588.954 q^{83} -531.030 q^{85} +1059.57 q^{86} -307.607 q^{88} +764.364 q^{89} +35.6883 q^{91} -64.6029 q^{92} -437.713 q^{94} -946.248 q^{95} -135.843 q^{97} -999.955 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * 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 $

$2$	2.96328	1.04768	0.523839	−	0.851817i	$-0.324499\pi$
$2$	2.96328	1.04768	0.523839	+	0.851817i	$0.324499\pi$
$3$	0	0
$3$	0	0
$4$	0.781040	0.0976300
$5$	−9.17370	−0.820521	−0.410260	−	0.911968i	$-0.634562\pi$
$5$	−9.17370	−0.820521	−0.410260	+	0.911968i	$0.634562\pi$
$6$	0	0
$7$	2.35613	0.127219	0.0636096	−	0.997975i	$-0.479739\pi$
$7$	2.35613	0.127219	0.0636096	+	0.997975i	$0.479739\pi$
$8$	−21.3918	−0.945394
$9$	0	0
$10$	−27.1843	−0.859642
$11$	14.3796	0.394148	0.197074	−	0.980389i	$-0.436856\pi$
$11$	14.3796	0.394148	0.197074	+	0.980389i	$0.436856\pi$
$12$	0	0
$13$	15.1470	0.323155	0.161578	−	0.986860i	$-0.448342\pi$
$13$	15.1470	0.323155	0.161578	+	0.986860i	$0.448342\pi$
$14$	6.98188	0.133285
$15$	0	0
$16$	−69.6383	−1.08810
$17$	57.8861	0.825849	0.412925	−	0.910765i	$-0.364507\pi$
$17$	57.8861	0.825849	0.412925	+	0.910765i	$0.364507\pi$
$18$	0	0
$19$	103.148	1.24546	0.622730	−	0.782436i	$-0.286023\pi$
$19$	103.148	1.24546	0.622730	+	0.782436i	$0.286023\pi$
$20$	−7.16503	−0.0801074
$21$	0	0
$22$	42.6109	0.412940
$23$	−82.7139	−0.749872	−0.374936	−	0.927051i	$-0.622335\pi$
$23$	−82.7139	−0.749872	−0.374936	+	0.927051i	$0.622335\pi$
$24$	0	0
$25$	−40.8433	−0.326746
$26$	44.8848	0.338563
$27$	0	0
$28$	1.84023	0.0124204
$29$	82.2294	0.526538	0.263269	−	0.964722i	$-0.415199\pi$
$29$	82.2294	0.526538	0.263269	+	0.964722i	$0.415199\pi$
$30$	0	0
$31$	−156.009	−0.903874	−0.451937	−	0.892050i	$-0.649267\pi$
$31$	−156.009	−0.903874	−0.451937	+	0.892050i	$0.649267\pi$
$32$	−35.2234	−0.194584
$33$	0	0
$34$	171.533	0.865225
$35$	−21.6144	−0.104386
$36$	0	0
$37$	202.052	0.897762	0.448881	−	0.893591i	$-0.351823\pi$
$37$	202.052	0.897762	0.448881	+	0.893591i	$0.351823\pi$
$38$	305.656	1.30484
$39$	0	0
$40$	196.242	0.775715
$41$	3.54593	0.0135069	0.00675344	−	0.999977i	$-0.497850\pi$
$41$	3.54593	0.0135069	0.00675344	+	0.999977i	$0.497850\pi$
$42$	0	0
$43$	357.565	1.26810	0.634048	−	0.773293i	$-0.281392\pi$
$43$	357.565	1.26810	0.634048	+	0.773293i	$0.281392\pi$
$44$	11.2311	0.0384807
$45$	0	0
$46$	−245.105	−0.785624
$47$	−147.712	−0.458426	−0.229213	−	0.973376i	$-0.573615\pi$
$47$	−147.712	−0.458426	−0.229213	+	0.973376i	$0.573615\pi$
$48$	0	0
$49$	−337.449	−0.983815
$50$	−121.030	−0.342325
$51$	0	0
$52$	11.8304	0.0315496
$53$	−160.081	−0.414885	−0.207442	−	0.978247i	$-0.566514\pi$
$53$	−160.081	−0.414885	−0.207442	+	0.978247i	$0.566514\pi$
$54$	0	0
$55$	−131.915	−0.323406
$56$	−50.4019	−0.120272
$57$	0	0
$58$	243.669	0.551643
$59$	45.9350	0.101360	0.0506799	−	0.998715i	$-0.483861\pi$
$59$	45.9350	0.101360	0.0506799	+	0.998715i	$0.483861\pi$
$60$	0	0
$61$	−743.891	−1.56140	−0.780701	−	0.624905i	$-0.785138\pi$
$61$	−743.891	−1.56140	−0.780701	+	0.624905i	$0.785138\pi$
$62$	−462.299	−0.946969
$63$	0	0
$64$	452.730	0.884237
$65$	−138.954	−0.265155
$66$	0	0
$67$	165.554	0.301876	0.150938	−	0.988543i	$-0.451771\pi$
$67$	165.554	0.301876	0.150938	+	0.988543i	$0.451771\pi$
$68$	45.2114	0.0806277
$69$	0	0
$70$	−64.0497	−0.109363
$71$	−747.149	−1.24888	−0.624439	−	0.781074i	$-0.714672\pi$
$71$	−747.149	−1.24888	−0.624439	+	0.781074i	$0.714672\pi$
$72$	0	0
$73$	962.660	1.54344	0.771718	−	0.635965i	$-0.219398\pi$
$73$	962.660	1.54344	0.771718	+	0.635965i	$0.219398\pi$
$74$	598.738	0.940566
$75$	0	0
$76$	80.5627	0.121594
$77$	33.8803	0.0501432
$78$	0	0
$79$	803.730	1.14464	0.572321	−	0.820030i	$-0.306043\pi$
$79$	803.730	1.14464	0.572321	+	0.820030i	$0.306043\pi$
$80$	638.841	0.892807
$81$	0	0
$82$	10.5076	0.0141509
$83$	−588.954	−0.778869	−0.389435	−	0.921054i	$-0.627330\pi$
$83$	−588.954	−0.778869	−0.389435	+	0.921054i	$0.627330\pi$
$84$	0	0
$85$	−531.030	−0.677626
$86$	1059.57	1.32856
$87$	0	0
$88$	−307.607	−0.372625
$89$	764.364	0.910365	0.455182	−	0.890398i	$-0.349574\pi$
$89$	764.364	0.910365	0.455182	+	0.890398i	$0.349574\pi$
$90$	0	0
$91$	35.6883	0.0411115
$92$	−64.6029	−0.0732100
$93$	0	0
$94$	−437.713	−0.480283
$95$	−946.248	−1.02193
$96$	0	0
$97$	−135.843	−0.142193	−0.0710966	−	0.997469i	$-0.522650\pi$
$97$	−135.843	−0.142193	−0.0710966	+	0.997469i	$0.522650\pi$
$98$	−999.955	−1.03072
$99$	0	0
$100$	−31.9002	−0.0319002
$101$	−1641.69	−1.61737	−0.808685	−	0.588241i	$-0.799821\pi$
$101$	−1641.69	−1.61737	−0.808685	+	0.588241i	$0.799821\pi$
$102$	0	0
$103$	−1733.25	−1.65808	−0.829041	−	0.559188i	$-0.811113\pi$
$103$	−1733.25	−1.65808	−0.829041	+	0.559188i	$0.811113\pi$
$104$	−324.021	−0.305509
$105$	0	0
$106$	−474.367	−0.434666
$107$	−1549.33	−1.39981	−0.699903	−	0.714238i	$-0.746773\pi$
$107$	−1549.33	−1.39981	−0.699903	+	0.714238i	$0.746773\pi$
$108$	0	0
$109$	−1886.61	−1.65784	−0.828920	−	0.559368i	$-0.811044\pi$
$109$	−1886.61	−1.65784	−0.828920	+	0.559368i	$0.811044\pi$
$110$	−390.900	−0.338826
$111$	0	0
$112$	−164.077	−0.138427
$113$	−177.382	−0.147670	−0.0738348	−	0.997270i	$-0.523524\pi$
$113$	−177.382	−0.147670	−0.0738348	+	0.997270i	$0.523524\pi$
$114$	0	0
$115$	758.793	0.615285
$116$	64.2244	0.0514059
$117$	0	0
$118$	136.118	0.106192
$119$	136.387	0.105064
$120$	0	0
$121$	−1124.23	−0.844648
$122$	−2204.36	−1.63585
$123$	0	0
$124$	−121.849	−0.0882452
$125$	1521.40	1.08862
$126$	0	0
$127$	465.031	0.324920	0.162460	−	0.986715i	$-0.448057\pi$
$127$	465.031	0.324920	0.162460	+	0.986715i	$0.448057\pi$
$128$	1623.35	1.12098
$129$	0	0
$130$	−411.759	−0.277798
$131$	−91.2048	−0.0608290	−0.0304145	−	0.999537i	$-0.509683\pi$
$131$	−91.2048	−0.0608290	−0.0304145	+	0.999537i	$0.509683\pi$
$132$	0	0
$133$	243.030	0.158446
$134$	490.584	0.316268
$135$	0	0
$136$	−1238.29	−0.780753
$137$	−2813.92	−1.75481	−0.877406	−	0.479748i	$-0.840728\pi$
$137$	−2813.92	−1.75481	−0.877406	+	0.479748i	$0.840728\pi$
$138$	0	0
$139$	−2029.26	−1.23827	−0.619136	−	0.785284i	$-0.712517\pi$
$139$	−2029.26	−1.23827	−0.619136	+	0.785284i	$0.712517\pi$
$140$	−16.8817	−0.0101912
$141$	0	0
$142$	−2214.01	−1.30842
$143$	217.808	0.127371
$144$	0	0
$145$	−754.347	−0.432035
$146$	2852.63	1.61703
$147$	0	0
$148$	157.811	0.0876485
$149$	833.802	0.458441	0.229221	−	0.973375i	$-0.426382\pi$
$149$	833.802	0.458441	0.229221	+	0.973375i	$0.426382\pi$
$150$	0	0
$151$	1231.99	0.663957	0.331979	−	0.943287i	$-0.392284\pi$
$151$	1231.99	0.663957	0.331979	+	0.943287i	$0.392284\pi$
$152$	−2206.52	−1.17745
$153$	0	0
$154$	100.397	0.0525339
$155$	1431.18	0.741647
$156$	0	0
$157$	−1856.39	−0.943670	−0.471835	−	0.881687i	$-0.656408\pi$
$157$	−1856.39	−0.943670	−0.471835	+	0.881687i	$0.656408\pi$
$158$	2381.68	1.19922
$159$	0	0
$160$	323.129	0.159660
$161$	−194.885	−0.0953981
$162$	0	0
$163$	−3384.37	−1.62629	−0.813143	−	0.582064i	$-0.802245\pi$
$163$	−3384.37	−1.62629	−0.813143	+	0.582064i	$0.802245\pi$
$164$	2.76952	0.00131868
$165$	0	0
$166$	−1745.24	−0.816005
$167$	1594.39	0.738789	0.369394	−	0.929273i	$-0.379565\pi$
$167$	1594.39	0.738789	0.369394	+	0.929273i	$0.379565\pi$
$168$	0	0
$169$	−1967.57	−0.895571
$170$	−1573.59	−0.709934
$171$	0	0
$172$	279.273	0.123804
$173$	−1538.64	−0.676191	−0.338095	−	0.941112i	$-0.609783\pi$
$173$	−1538.64	−0.676191	−0.338095	+	0.941112i	$0.609783\pi$
$174$	0	0
$175$	−96.2321	−0.0415684
$176$	−1001.37	−0.428872
$177$	0	0
$178$	2265.03	0.953769
$179$	418.587	0.174786	0.0873929	−	0.996174i	$-0.472146\pi$
$179$	418.587	0.174786	0.0873929	+	0.996174i	$0.472146\pi$
$180$	0	0
$181$	1821.82	0.748149	0.374074	−	0.927399i	$-0.377960\pi$
$181$	1821.82	0.748149	0.374074	+	0.927399i	$0.377960\pi$
$182$	105.754	0.0430717
$183$	0	0
$184$	1769.40	0.708924
$185$	−1853.57	−0.736632
$186$	0	0
$187$	832.381	0.325507
$188$	−115.369	−0.0447561
$189$	0	0
$190$	−2804.00	−1.07065
$191$	−1876.74	−0.710974	−0.355487	−	0.934681i	$-0.615685\pi$
$191$	−1876.74	−0.710974	−0.355487	+	0.934681i	$0.615685\pi$
$192$	0	0
$193$	3743.99	1.39637	0.698183	−	0.715919i	$-0.253992\pi$
$193$	3743.99	1.39637	0.698183	+	0.715919i	$0.253992\pi$
$194$	−402.540	−0.148973
$195$	0	0
$196$	−263.561	−0.0960499
$197$	22.9050	0.00828384	0.00414192	−	0.999991i	$-0.498682\pi$
$197$	22.9050	0.00828384	0.00414192	+	0.999991i	$0.498682\pi$
$198$	0	0
$199$	1922.67	0.684897	0.342448	−	0.939537i	$-0.388744\pi$
$199$	1922.67	0.684897	0.342448	+	0.939537i	$0.388744\pi$
$200$	873.711	0.308904
$201$	0	0
$202$	−4864.80	−1.69448
$203$	193.743	0.0669858
$204$	0	0
$205$	−32.5293	−0.0110827
$206$	−5136.11	−1.73714
$207$	0	0
$208$	−1054.81	−0.351624
$209$	1483.23	0.490896
$210$	0	0
$211$	−1483.21	−0.483924	−0.241962	−	0.970286i	$-0.577791\pi$
$211$	−1483.21	−0.483924	−0.241962	+	0.970286i	$0.577791\pi$
$212$	−125.030	−0.0405052
$213$	0	0
$214$	−4591.10	−1.46655
$215$	−3280.19	−1.04050
$216$	0	0
$217$	−367.578	−0.114990
$218$	−5590.56	−1.73688
$219$	0	0
$220$	−103.031	−0.0315742
$221$	876.799	0.266877
$222$	0	0
$223$	−2208.44	−0.663176	−0.331588	−	0.943424i	$-0.607584\pi$
$223$	−2208.44	−0.663176	−0.331588	+	0.943424i	$0.607584\pi$
$224$	−82.9910	−0.0247548
$225$	0	0
$226$	−525.632	−0.154710
$227$	−5649.03	−1.65172	−0.825858	−	0.563879i	$-0.809309\pi$
$227$	−5649.03	−1.65172	−0.825858	+	0.563879i	$0.809309\pi$
$228$	0	0
$229$	5161.69	1.48949	0.744747	−	0.667347i	$-0.232570\pi$
$229$	5161.69	1.48949	0.744747	+	0.667347i	$0.232570\pi$
$230$	2248.52	0.644621
$231$	0	0
$232$	−1759.04	−0.497786
$233$	−1290.84	−0.362943	−0.181472	−	0.983396i	$-0.558086\pi$
$233$	−1290.84	−0.362943	−0.181472	+	0.983396i	$0.558086\pi$
$234$	0	0
$235$	1355.07	0.376148
$236$	35.8771	0.00989575
$237$	0	0
$238$	404.154	0.110073
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−2953.77	−0.789497	−0.394749	−	0.918789i	$-0.629168\pi$
$241$	−2953.77	−0.789497	−0.394749	+	0.918789i	$0.629168\pi$
$242$	−3331.40	−0.884919
$243$	0	0
$244$	−581.009	−0.152440
$245$	3095.65	0.807241
$246$	0	0
$247$	1562.38	0.402477
$248$	3337.32	0.854516
$249$	0	0
$250$	4508.33	1.14053
$251$	−3409.90	−0.857495	−0.428747	−	0.903424i	$-0.641045\pi$
$251$	−3409.90	−0.857495	−0.428747	+	0.903424i	$0.641045\pi$
$252$	0	0
$253$	−1189.40	−0.295560
$254$	1378.02	0.340412
$255$	0	0
$256$	1188.61	0.290189
$257$	−6809.80	−1.65286	−0.826428	−	0.563043i	$-0.809631\pi$
$257$	−6809.80	−1.65286	−0.826428	+	0.563043i	$0.809631\pi$
$258$	0	0
$259$	476.062	0.114213
$260$	−108.528	−0.0258871
$261$	0	0
$262$	−270.265	−0.0637292
$263$	6686.40	1.56768	0.783842	−	0.620960i	$-0.213257\pi$
$263$	6686.40	1.56768	0.783842	+	0.620960i	$0.213257\pi$
$264$	0	0
$265$	1468.54	0.340421
$266$	720.167	0.166001
$267$	0	0
$268$	129.304	0.0294721
$269$	7028.75	1.59312	0.796562	−	0.604557i	$-0.206650\pi$
$269$	7028.75	1.59312	0.796562	+	0.604557i	$0.206650\pi$
$270$	0	0
$271$	6307.26	1.41380	0.706898	−	0.707316i	$-0.250094\pi$
$271$	6307.26	1.41380	0.706898	+	0.707316i	$0.250094\pi$
$272$	−4031.09	−0.898605
$273$	0	0
$274$	−8338.43	−1.83848
$275$	−587.311	−0.128786
$276$	0	0
$277$	997.811	0.216436	0.108218	−	0.994127i	$-0.465486\pi$
$277$	997.811	0.216436	0.108218	+	0.994127i	$0.465486\pi$
$278$	−6013.28	−1.29731
$279$	0	0
$280$	462.372	0.0986858
$281$	−4367.88	−0.927281	−0.463640	−	0.886024i	$-0.653457\pi$
$281$	−4367.88	−0.927281	−0.463640	+	0.886024i	$0.653457\pi$
$282$	0	0
$283$	−4659.41	−0.978705	−0.489352	−	0.872086i	$-0.662767\pi$
$283$	−4659.41	−0.978705	−0.489352	+	0.872086i	$0.662767\pi$
$284$	−583.553	−0.121928
$285$	0	0
$286$	645.427	0.133444
$287$	8.35469	0.00171833
$288$	0	0
$289$	−1562.20	−0.317973
$290$	−2235.34	−0.452634
$291$	0	0
$292$	751.876	0.150686
$293$	1873.11	0.373476	0.186738	−	0.982410i	$-0.440209\pi$
$293$	1873.11	0.373476	0.186738	+	0.982410i	$0.440209\pi$
$294$	0	0
$295$	−421.394	−0.0831677
$296$	−4322.27	−0.848739
$297$	0	0
$298$	2470.79	0.480299
$299$	−1252.87	−0.242325
$300$	0	0
$301$	842.471	0.161326
$302$	3650.72	0.695614
$303$	0	0
$304$	−7183.05	−1.35518
$305$	6824.23	1.28116
$306$	0	0
$307$	9380.18	1.74383	0.871914	−	0.489659i	$-0.162879\pi$
$307$	9380.18	1.74383	0.871914	+	0.489659i	$0.162879\pi$
$308$	26.4619	0.00489548
$309$	0	0
$310$	4241.00	0.777008
$311$	9990.35	1.82155	0.910773	−	0.412908i	$-0.135487\pi$
$311$	9990.35	1.82155	0.910773	+	0.412908i	$0.135487\pi$
$312$	0	0
$313$	−8829.71	−1.59452	−0.797260	−	0.603636i	$-0.793718\pi$
$313$	−8829.71	−1.59452	−0.797260	+	0.603636i	$0.793718\pi$
$314$	−5501.01	−0.988663
$315$	0	0
$316$	627.745	0.111751
$317$	−4211.98	−0.746273	−0.373136	−	0.927776i	$-0.621718\pi$
$317$	−4211.98	−0.746273	−0.373136	+	0.927776i	$0.621718\pi$
$318$	0	0
$319$	1182.43	0.207534
$320$	−4153.20	−0.725535
$321$	0	0
$322$	−577.499	−0.0999465
$323$	5970.83	1.02856
$324$	0	0
$325$	−618.652	−0.105590
$326$	−10028.8	−1.70382
$327$	0	0
$328$	−75.8539	−0.0127693
$329$	−348.029	−0.0583206
$330$	0	0
$331$	6313.47	1.04840	0.524199	−	0.851596i	$-0.324365\pi$
$331$	6313.47	1.04840	0.524199	+	0.851596i	$0.324365\pi$
$332$	−459.997	−0.0760410
$333$	0	0
$334$	4724.63	0.774013
$335$	−1518.74	−0.247695
$336$	0	0
$337$	−7036.97	−1.13747	−0.568736	−	0.822520i	$-0.692567\pi$
$337$	−7036.97	−1.13747	−0.568736	+	0.822520i	$0.692567\pi$
$338$	−5830.46	−0.938270
$339$	0	0
$340$	−414.755	−0.0661567
$341$	−2243.36	−0.356260
$342$	0	0
$343$	−1603.23	−0.252379
$344$	−7648.96	−1.19885
$345$	0	0
$346$	−4559.44	−0.708430
$347$	−10927.9	−1.69061	−0.845306	−	0.534282i	$-0.820582\pi$
$347$	−10927.9	−1.69061	−0.845306	+	0.534282i	$0.820582\pi$
$348$	0	0
$349$	4966.40	0.761734	0.380867	−	0.924630i	$-0.375626\pi$
$349$	4966.40	0.761734	0.380867	+	0.924630i	$0.375626\pi$
$350$	−285.163	−0.0435503
$351$	0	0
$352$	−506.500	−0.0766947
$353$	−7853.14	−1.18408	−0.592040	−	0.805908i	$-0.701677\pi$
$353$	−7853.14	−1.18408	−0.592040	+	0.805908i	$0.701677\pi$
$354$	0	0
$355$	6854.12	1.02473
$356$	596.999	0.0888789
$357$	0	0
$358$	1240.39	0.183119
$359$	−4725.95	−0.694781	−0.347390	−	0.937721i	$-0.612932\pi$
$359$	−4725.95	−0.694781	−0.347390	+	0.937721i	$0.612932\pi$
$360$	0	0
$361$	3780.49	0.551173
$362$	5398.57	0.783819
$363$	0	0
$364$	27.8740	0.00401372
$365$	−8831.16	−1.26642
$366$	0	0
$367$	11340.8	1.61304	0.806522	−	0.591204i	$-0.201347\pi$
$367$	11340.8	1.61304	0.806522	+	0.591204i	$0.201347\pi$
$368$	5760.06	0.815934
$369$	0	0
$370$	−5492.64	−0.771754
$371$	−377.173	−0.0527813
$372$	0	0
$373$	9963.63	1.38310	0.691551	−	0.722328i	$-0.256928\pi$
$373$	9963.63	1.38310	0.691551	+	0.722328i	$0.256928\pi$
$374$	2466.58	0.341026
$375$	0	0
$376$	3159.83	0.433393
$377$	1245.53	0.170154
$378$	0	0
$379$	−3903.68	−0.529073	−0.264536	−	0.964376i	$-0.585219\pi$
$379$	−3903.68	−0.529073	−0.264536	+	0.964376i	$0.585219\pi$
$380$	−739.058	−0.0997707
$381$	0	0
$382$	−5561.31	−0.744872
$383$	−7011.53	−0.935438	−0.467719	−	0.883877i	$-0.654924\pi$
$383$	−7011.53	−0.935438	−0.467719	+	0.883877i	$0.654924\pi$
$384$	0	0
$385$	−310.808	−0.0411435
$386$	11094.5	1.46294
$387$	0	0
$388$	−106.099	−0.0138823
$389$	−6380.67	−0.831652	−0.415826	−	0.909444i	$-0.636507\pi$
$389$	−6380.67	−0.831652	−0.415826	+	0.909444i	$0.636507\pi$
$390$	0	0
$391$	−4787.99	−0.619281
$392$	7218.64	0.930093
$393$	0	0
$394$	67.8741	0.00867880
$395$	−7373.18	−0.939202
$396$	0	0
$397$	13753.7	1.73874	0.869370	−	0.494162i	$-0.164525\pi$
$397$	13753.7	1.73874	0.869370	+	0.494162i	$0.164525\pi$
$398$	5697.41	0.717552
$399$	0	0
$400$	2844.25	0.355532
$401$	2550.97	0.317679	0.158840	−	0.987304i	$-0.449225\pi$
$401$	2550.97	0.317679	0.158840	+	0.987304i	$0.449225\pi$
$402$	0	0
$403$	−2363.07	−0.292091
$404$	−1282.23	−0.157904
$405$	0	0
$406$	574.116	0.0701795
$407$	2905.44	0.353851
$408$	0	0
$409$	10406.3	1.25809	0.629047	−	0.777367i	$-0.283445\pi$
$409$	10406.3	1.25809	0.629047	+	0.777367i	$0.283445\pi$
$410$	−96.3936	−0.0116111
$411$	0	0
$412$	−1353.74	−0.161879
$413$	108.229	0.0128949
$414$	0	0
$415$	5402.89	0.639078
$416$	−533.528	−0.0628807
$417$	0	0
$418$	4395.23	0.514301
$419$	−15119.5	−1.76286	−0.881429	−	0.472317i	$-0.843418\pi$
$419$	−15119.5	−1.76286	−0.881429	+	0.472317i	$0.843418\pi$
$420$	0	0
$421$	−8421.52	−0.974917	−0.487458	−	0.873146i	$-0.662076\pi$
$421$	−8421.52	−0.974917	−0.487458	+	0.873146i	$0.662076\pi$
$422$	−4395.16	−0.506997
$423$	0	0
$424$	3424.43	0.392229
$425$	−2364.26	−0.269843
$426$	0	0
$427$	−1752.71	−0.198640
$428$	−1210.09	−0.136663
$429$	0	0
$430$	−9720.14	−1.09011
$431$	14275.6	1.59544	0.797719	−	0.603030i	$-0.206040\pi$
$431$	14275.6	1.59544	0.797719	+	0.603030i	$0.206040\pi$
$432$	0	0
$433$	13380.4	1.48504	0.742518	−	0.669826i	$-0.233632\pi$
$433$	13380.4	1.48504	0.742518	+	0.669826i	$0.233632\pi$
$434$	−1089.24	−0.120473
$435$	0	0
$436$	−1473.52	−0.161855
$437$	−8531.77	−0.933936
$438$	0	0
$439$	5354.50	0.582133	0.291067	−	0.956703i	$-0.405990\pi$
$439$	5354.50	0.582133	0.291067	+	0.956703i	$0.405990\pi$
$440$	2821.89	0.305746
$441$	0	0
$442$	2598.20	0.279602
$443$	−17239.1	−1.84889	−0.924443	−	0.381321i	$-0.875469\pi$
$443$	−17239.1	−1.84889	−0.924443	+	0.381321i	$0.875469\pi$
$444$	0	0
$445$	−7012.05	−0.746973
$446$	−6544.24	−0.694795
$447$	0	0
$448$	1066.69	0.112492
$449$	12372.6	1.30044	0.650220	−	0.759746i	$-0.274677\pi$
$449$	12372.6	1.30044	0.650220	+	0.759746i	$0.274677\pi$
$450$	0	0
$451$	50.9893	0.00532370
$452$	−138.542	−0.0144170
$453$	0	0
$454$	−16739.7	−1.73047
$455$	−327.394	−0.0337328
$456$	0	0
$457$	10218.6	1.04596	0.522982	−	0.852343i	$-0.324819\pi$
$457$	10218.6	1.04596	0.522982	+	0.852343i	$0.324819\pi$
$458$	15295.5	1.56051
$459$	0	0
$460$	592.648	0.0600703
$461$	−10295.4	−1.04014	−0.520069	−	0.854124i	$-0.674094\pi$
$461$	−10295.4	−1.04014	−0.520069	+	0.854124i	$0.674094\pi$
$462$	0	0
$463$	624.733	0.0627080	0.0313540	−	0.999508i	$-0.490018\pi$
$463$	624.733	0.0627080	0.0313540	+	0.999508i	$0.490018\pi$
$464$	−5726.31	−0.572925
$465$	0	0
$466$	−3825.12	−0.380248
$467$	−3096.27	−0.306805	−0.153403	−	0.988164i	$-0.549023\pi$
$467$	−3096.27	−0.306805	−0.153403	+	0.988164i	$0.549023\pi$
$468$	0	0
$469$	390.067	0.0384044
$470$	4015.44	0.394082
$471$	0	0
$472$	−982.632	−0.0958248
$473$	5141.66	0.499817
$474$	0	0
$475$	−4212.90	−0.406949
$476$	106.524	0.0102574
$477$	0	0
$478$	708.224	0.0677687
$479$	−3505.65	−0.334399	−0.167200	−	0.985923i	$-0.553472\pi$
$479$	−3505.65	−0.334399	−0.167200	+	0.985923i	$0.553472\pi$
$480$	0	0
$481$	3060.48	0.290116
$482$	−8752.84	−0.827139
$483$	0	0
$484$	−878.065	−0.0824629
$485$	1246.18	0.116672
$486$	0	0
$487$	−5020.73	−0.467168	−0.233584	−	0.972337i	$-0.575045\pi$
$487$	−5020.73	−0.467168	−0.233584	+	0.972337i	$0.575045\pi$
$488$	15913.2	1.47614
$489$	0	0
$490$	9173.29	0.845729
$491$	−7985.83	−0.734003	−0.367002	−	0.930220i	$-0.619616\pi$
$491$	−7985.83	−0.734003	−0.367002	+	0.930220i	$0.619616\pi$
$492$	0	0
$493$	4759.94	0.434841
$494$	4629.77	0.421666
$495$	0	0
$496$	10864.2	0.983504
$497$	−1760.38	−0.158881
$498$	0	0
$499$	−19646.0	−1.76248	−0.881240	−	0.472669i	$-0.843291\pi$
$499$	−19646.0	−1.76248	−0.881240	+	0.472669i	$0.843291\pi$
$500$	1188.27	0.106282
$501$	0	0
$502$	−10104.5	−0.898379
$503$	8830.82	0.782797	0.391398	−	0.920221i	$-0.371991\pi$
$503$	8830.82	0.782797	0.391398	+	0.920221i	$0.371991\pi$
$504$	0	0
$505$	15060.4	1.32709
$506$	−3524.52	−0.309652
$507$	0	0
$508$	363.208	0.0317220
$509$	11339.0	0.987413	0.493707	−	0.869628i	$-0.335642\pi$
$509$	11339.0	0.987413	0.493707	+	0.869628i	$0.335642\pi$
$510$	0	0
$511$	2268.16	0.196355
$512$	−9464.62	−0.816955
$513$	0	0
$514$	−20179.4	−1.73166
$515$	15900.3	1.36049
$516$	0	0
$517$	−2124.05	−0.180688
$518$	1410.71	0.119658
$519$	0	0
$520$	2972.47	0.250676
$521$	−10393.5	−0.873985	−0.436992	−	0.899465i	$-0.643956\pi$
$521$	−10393.5	−0.873985	−0.436992	+	0.899465i	$0.643956\pi$
$522$	0	0
$523$	4010.67	0.335324	0.167662	−	0.985845i	$-0.446378\pi$
$523$	4010.67	0.335324	0.167662	+	0.985845i	$0.446378\pi$
$524$	−71.2346	−0.00593874
$525$	0	0
$526$	19813.7	1.64243
$527$	−9030.77	−0.746464
$528$	0	0
$529$	−5325.40	−0.437692
$530$	4351.70	0.356652
$531$	0	0
$532$	189.816	0.0154691
$533$	53.7102	0.00436481
$534$	0	0
$535$	14213.1	1.14857
$536$	−3541.50	−0.285391
$537$	0	0
$538$	20828.2	1.66908
$539$	−4852.39	−0.387769
$540$	0	0
$541$	23915.0	1.90053	0.950264	−	0.311445i	$-0.100813\pi$
$541$	23915.0	1.90053	0.950264	+	0.311445i	$0.100813\pi$
$542$	18690.2	1.48120
$543$	0	0
$544$	−2038.94	−0.160697
$545$	17307.2	1.36029
$546$	0	0
$547$	569.608	0.0445241	0.0222621	−	0.999752i	$-0.492913\pi$
$547$	569.608	0.0445241	0.0222621	+	0.999752i	$0.492913\pi$
$548$	−2197.78	−0.171322
$549$	0	0
$550$	−1740.37	−0.134927
$551$	8481.79	0.655783
$552$	0	0
$553$	1893.69	0.145620
$554$	2956.79	0.226755
$555$	0	0
$556$	−1584.94	−0.120893
$557$	−4270.83	−0.324885	−0.162442	−	0.986718i	$-0.551937\pi$
$557$	−4270.83	−0.324885	−0.162442	+	0.986718i	$0.551937\pi$
$558$	0	0
$559$	5416.03	0.409792
$560$	1505.19	0.113582
$561$	0	0
$562$	−12943.3	−0.971492
$563$	13696.2	1.02527	0.512633	−	0.858608i	$-0.328670\pi$
$563$	13696.2	1.02527	0.512633	+	0.858608i	$0.328670\pi$
$564$	0	0
$565$	1627.25	0.121166
$566$	−13807.2	−1.02537
$567$	0	0
$568$	15982.9	1.18068
$569$	−5737.72	−0.422738	−0.211369	−	0.977406i	$-0.567792\pi$
$569$	−5737.72	−0.422738	−0.211369	+	0.977406i	$0.567792\pi$
$570$	0	0
$571$	9931.62	0.727890	0.363945	−	0.931420i	$-0.381430\pi$
$571$	9931.62	0.727890	0.363945	+	0.931420i	$0.381430\pi$
$572$	170.117	0.0124352
$573$	0	0
$574$	24.7573	0.00180026
$575$	3378.31	0.245018
$576$	0	0
$577$	1313.36	0.0947591	0.0473796	−	0.998877i	$-0.484913\pi$
$577$	1313.36	0.0947591	0.0473796	+	0.998877i	$0.484913\pi$
$578$	−4629.24	−0.333133
$579$	0	0
$580$	−589.176	−0.0421796
$581$	−1387.65	−0.0990871
$582$	0	0
$583$	−2301.91	−0.163526
$584$	−20593.1	−1.45915
$585$	0	0
$586$	5550.56	0.391282
$587$	−14304.2	−1.00579	−0.502894	−	0.864348i	$-0.667732\pi$
$587$	−14304.2	−1.00579	−0.502894	+	0.864348i	$0.667732\pi$
$588$	0	0
$589$	−16092.0	−1.12574
$590$	−1248.71	−0.0871331
$591$	0	0
$592$	−14070.6	−0.976854
$593$	−4557.38	−0.315597	−0.157799	−	0.987471i	$-0.550440\pi$
$593$	−4557.38	−0.315597	−0.157799	+	0.987471i	$0.550440\pi$
$594$	0	0
$595$	−1251.18	−0.0862071
$596$	651.233	0.0447576
$597$	0	0
$598$	−3712.60	−0.253878
$599$	−26296.5	−1.79373	−0.896867	−	0.442300i	$-0.854163\pi$
$599$	−26296.5	−1.79373	−0.896867	+	0.442300i	$0.854163\pi$
$600$	0	0
$601$	19846.9	1.34704	0.673520	−	0.739169i	$-0.264781\pi$
$601$	19846.9	1.34704	0.673520	+	0.739169i	$0.264781\pi$
$602$	2496.48	0.169018
$603$	0	0
$604$	962.230	0.0648222
$605$	10313.3	0.693051
$606$	0	0
$607$	−18072.3	−1.20846	−0.604228	−	0.796812i	$-0.706518\pi$
$607$	−18072.3	−1.20846	−0.604228	+	0.796812i	$0.706518\pi$
$608$	−3633.22	−0.242346
$609$	0	0
$610$	20222.1	1.34225
$611$	−2237.39	−0.148143
$612$	0	0
$613$	12103.1	0.797457	0.398729	−	0.917069i	$-0.369451\pi$
$613$	12103.1	0.797457	0.398729	+	0.917069i	$0.369451\pi$
$614$	27796.1	1.82697
$615$	0	0
$616$	−724.762	−0.0474050
$617$	20613.6	1.34502	0.672508	−	0.740090i	$-0.265217\pi$
$617$	20613.6	1.34502	0.672508	+	0.740090i	$0.265217\pi$
$618$	0	0
$619$	−1056.49	−0.0686007	−0.0343004	−	0.999412i	$-0.510920\pi$
$619$	−1056.49	−0.0686007	−0.0343004	+	0.999412i	$0.510920\pi$
$620$	1117.81	0.0724070
$621$	0	0
$622$	29604.2	1.90839
$623$	1800.94	0.115816
$624$	0	0
$625$	−8851.42	−0.566491
$626$	−26164.9	−1.67054
$627$	0	0
$628$	−1449.92	−0.0921305
$629$	11696.0	0.741416
$630$	0	0
$631$	−8013.19	−0.505547	−0.252773	−	0.967526i	$-0.581343\pi$
$631$	−8013.19	−0.505547	−0.252773	+	0.967526i	$0.581343\pi$
$632$	−17193.2	−1.08214
$633$	0	0
$634$	−12481.3	−0.781854
$635$	−4266.06	−0.266604
$636$	0	0
$637$	−5111.33	−0.317925
$638$	3503.87	0.217429
$639$	0	0
$640$	−14892.1	−0.919787
$641$	−21822.2	−1.34466	−0.672328	−	0.740254i	$-0.734705\pi$
$641$	−21822.2	−1.34466	−0.672328	+	0.740254i	$0.734705\pi$
$642$	0	0
$643$	−6406.87	−0.392943	−0.196472	−	0.980510i	$-0.562948\pi$
$643$	−6406.87	−0.392943	−0.196472	+	0.980510i	$0.562948\pi$
$644$	−152.213	−0.00931371
$645$	0	0
$646$	17693.3	1.07760
$647$	13195.9	0.801830	0.400915	−	0.916115i	$-0.368692\pi$
$647$	13195.9	0.801830	0.400915	+	0.916115i	$0.368692\pi$
$648$	0	0
$649$	660.529	0.0399507
$650$	−1833.24	−0.110624
$651$	0	0
$652$	−2643.33	−0.158774
$653$	−21736.7	−1.30264	−0.651319	−	0.758804i	$-0.725784\pi$
$653$	−21736.7	−1.30264	−0.651319	+	0.758804i	$0.725784\pi$
$654$	0	0
$655$	836.685	0.0499114
$656$	−246.933	−0.0146968
$657$	0	0
$658$	−1031.31	−0.0611012
$659$	16114.2	0.952534	0.476267	−	0.879301i	$-0.341990\pi$
$659$	16114.2	0.952534	0.476267	+	0.879301i	$0.341990\pi$
$660$	0	0
$661$	3118.43	0.183499	0.0917496	−	0.995782i	$-0.470754\pi$
$661$	3118.43	0.183499	0.0917496	+	0.995782i	$0.470754\pi$
$662$	18708.6	1.09838
$663$	0	0
$664$	12598.8	0.736338
$665$	−2229.49	−0.130009
$666$	0	0
$667$	−6801.51	−0.394836
$668$	1245.28	0.0721279
$669$	0	0
$670$	−4500.47	−0.259505
$671$	−10696.9	−0.615423
$672$	0	0
$673$	−7726.86	−0.442569	−0.221284	−	0.975209i	$-0.571025\pi$
$673$	−7726.86	−0.442569	−0.221284	+	0.975209i	$0.571025\pi$
$674$	−20852.5	−1.19171
$675$	0	0
$676$	−1536.75	−0.0874346
$677$	−32338.1	−1.83583	−0.917914	−	0.396780i	$-0.870127\pi$
$677$	−32338.1	−1.83583	−0.917914	+	0.396780i	$0.870127\pi$
$678$	0	0
$679$	−320.063	−0.0180897
$680$	11359.7	0.640624
$681$	0	0
$682$	−6647.70	−0.373246
$683$	−31880.4	−1.78604	−0.893022	−	0.450013i	$-0.851419\pi$
$683$	−31880.4	−1.78604	−0.893022	+	0.450013i	$0.851419\pi$
$684$	0	0
$685$	25814.0	1.43986
$686$	−4750.81	−0.264412
$687$	0	0
$688$	−24900.2	−1.37981
$689$	−2424.75	−0.134072
$690$	0	0
$691$	−11724.0	−0.645447	−0.322723	−	0.946493i	$-0.604598\pi$
$691$	−11724.0	−0.645447	−0.322723	+	0.946493i	$0.604598\pi$
$692$	−1201.74	−0.0660165
$693$	0	0
$694$	−32382.6	−1.77122
$695$	18615.8	1.01603
$696$	0	0
$697$	205.260	0.0111546
$698$	14716.8	0.798052
$699$	0	0
$700$	−75.1611	−0.00405832
$701$	14029.1	0.755880	0.377940	−	0.925830i	$-0.376633\pi$
$701$	14029.1	0.755880	0.377940	+	0.925830i	$0.376633\pi$
$702$	0	0
$703$	20841.3	1.11813
$704$	6510.09	0.348520
$705$	0	0
$706$	−23271.1	−1.24054
$707$	−3868.04	−0.205761
$708$	0	0
$709$	−15232.8	−0.806881	−0.403440	−	0.915006i	$-0.632186\pi$
$709$	−15232.8	−0.806881	−0.403440	+	0.915006i	$0.632186\pi$
$710$	20310.7	1.07359
$711$	0	0
$712$	−16351.1	−0.860653
$713$	12904.1	0.677789
$714$	0	0
$715$	−1998.11	−0.104510
$716$	326.933	0.0170643
$717$	0	0
$718$	−14004.3	−0.727907
$719$	−33270.0	−1.72568	−0.862838	−	0.505481i	$-0.831315\pi$
$719$	−33270.0	−1.72568	−0.862838	+	0.505481i	$0.831315\pi$
$720$	0	0
$721$	−4083.77	−0.210940
$722$	11202.7	0.577452
$723$	0	0
$724$	1422.92	0.0730418
$725$	−3358.51	−0.172044
$726$	0	0
$727$	−16319.6	−0.832543	−0.416272	−	0.909240i	$-0.636663\pi$
$727$	−16319.6	−0.832543	−0.416272	+	0.909240i	$0.636663\pi$
$728$	−763.437	−0.0388666
$729$	0	0
$730$	−26169.2	−1.32680
$731$	20698.0	1.04726
$732$	0	0
$733$	32024.7	1.61372	0.806862	−	0.590741i	$-0.201164\pi$
$733$	32024.7	1.61372	0.806862	+	0.590741i	$0.201164\pi$
$734$	33606.1	1.68995
$735$	0	0
$736$	2913.47	0.145913
$737$	2380.61	0.118984
$738$	0	0
$739$	−9196.88	−0.457798	−0.228899	−	0.973450i	$-0.573513\pi$
$739$	−9196.88	−0.457798	−0.228899	+	0.973450i	$0.573513\pi$
$740$	−1447.71	−0.0719174
$741$	0	0
$742$	−1117.67	−0.0552978
$743$	33744.1	1.66615	0.833075	−	0.553160i	$-0.186578\pi$
$743$	33744.1	1.66615	0.833075	+	0.553160i	$0.186578\pi$
$744$	0	0
$745$	−7649.05	−0.376160
$746$	29525.0	1.44905
$747$	0	0
$748$	650.123	0.0317792
$749$	−3650.42	−0.178082
$750$	0	0
$751$	11732.9	0.570091	0.285046	−	0.958514i	$-0.407991\pi$
$751$	11732.9	0.570091	0.285046	+	0.958514i	$0.407991\pi$
$752$	10286.4	0.498813
$753$	0	0
$754$	3690.85	0.178266
$755$	−11301.9	−0.544791
$756$	0	0
$757$	15986.7	0.767563	0.383781	−	0.923424i	$-0.374622\pi$
$757$	15986.7	0.767563	0.383781	+	0.923424i	$0.374622\pi$
$758$	−11567.7	−0.554298
$759$	0	0
$760$	20242.0	0.966122
$761$	19860.3	0.946039	0.473019	−	0.881052i	$-0.343164\pi$
$761$	19860.3	0.946039	0.473019	+	0.881052i	$0.343164\pi$
$762$	0	0
$763$	−4445.10	−0.210909
$764$	−1465.81	−0.0694124
$765$	0	0
$766$	−20777.2	−0.980038
$767$	695.776	0.0327549
$768$	0	0
$769$	11902.2	0.558133	0.279067	−	0.960272i	$-0.409975\pi$
$769$	11902.2	0.558133	0.279067	+	0.960272i	$0.409975\pi$
$770$	−921.012	−0.0431051
$771$	0	0
$772$	2924.21	0.136327
$773$	−36404.0	−1.69387	−0.846935	−	0.531696i	$-0.821555\pi$
$773$	−36404.0	−1.69387	−0.846935	+	0.531696i	$0.821555\pi$
$774$	0	0
$775$	6371.93	0.295337
$776$	2905.92	0.134428
$777$	0	0
$778$	−18907.7	−0.871304
$779$	365.756	0.0168223
$780$	0	0
$781$	−10743.7	−0.492242
$782$	−14188.2	−0.648807
$783$	0	0
$784$	23499.3	1.07049
$785$	17030.0	0.774301
$786$	0	0
$787$	−14540.7	−0.658603	−0.329301	−	0.944225i	$-0.606813\pi$
$787$	−14540.7	−0.658603	−0.329301	+	0.944225i	$0.606813\pi$
$788$	17.8898	0.000808752 0
$789$	0	0
$790$	−21848.8	−0.983982
$791$	−417.935	−0.0187864
$792$	0	0
$793$	−11267.7	−0.504575
$794$	40756.2	1.82164
$795$	0	0
$796$	1501.68	0.0668665
$797$	−25877.1	−1.15008	−0.575040	−	0.818125i	$-0.695014\pi$
$797$	−25877.1	−1.15008	−0.575040	+	0.818125i	$0.695014\pi$
$798$	0	0
$799$	−8550.48	−0.378591
$800$	1438.64	0.0635794
$801$	0	0
$802$	7559.25	0.332826
$803$	13842.7	0.608342
$804$	0	0
$805$	1787.82	0.0782761
$806$	−7002.44	−0.306018
$807$	0	0
$808$	35118.8	1.52905
$809$	7017.33	0.304965	0.152482	−	0.988306i	$-0.451273\pi$
$809$	7017.33	0.304965	0.152482	+	0.988306i	$0.451273\pi$
$810$	0	0
$811$	−9683.99	−0.419298	−0.209649	−	0.977777i	$-0.567232\pi$
$811$	−9683.99	−0.419298	−0.209649	+	0.977777i	$0.567232\pi$
$812$	151.321	0.00653982
$813$	0	0
$814$	8609.64	0.370722
$815$	31047.2	1.33440
$816$	0	0
$817$	36882.1	1.57936
$818$	30836.9	1.31808
$819$	0	0
$820$	−25.4067	−0.00108200
$821$	−27788.1	−1.18125	−0.590627	−	0.806944i	$-0.701120\pi$
$821$	−27788.1	−1.18125	−0.590627	+	0.806944i	$0.701120\pi$
$822$	0	0
$823$	−33251.1	−1.40834	−0.704168	−	0.710034i	$-0.748680\pi$
$823$	−33251.1	−1.40834	−0.704168	+	0.710034i	$0.748680\pi$
$824$	37077.4	1.56754
$825$	0	0
$826$	320.713	0.0135097
$827$	44335.1	1.86419	0.932093	−	0.362218i	$-0.117980\pi$
$827$	44335.1	1.86419	0.932093	+	0.362218i	$0.117980\pi$
$828$	0	0
$829$	−37480.4	−1.57026	−0.785132	−	0.619328i	$-0.787405\pi$
$829$	−37480.4	−1.57026	−0.785132	+	0.619328i	$0.787405\pi$
$830$	16010.3	0.669548
$831$	0	0
$832$	6857.48	0.285746
$833$	−19533.6	−0.812483
$834$	0	0
$835$	−14626.5	−0.606191
$836$	1158.46	0.0479261
$837$	0	0
$838$	−44803.4	−1.84691
$839$	28545.7	1.17462	0.587310	−	0.809362i	$-0.300187\pi$
$839$	28545.7	1.17462	0.587310	+	0.809362i	$0.300187\pi$
$840$	0	0
$841$	−17627.3	−0.722757
$842$	−24955.3	−1.02140
$843$	0	0
$844$	−1158.44	−0.0472456
$845$	18049.9	0.734834
$846$	0	0
$847$	−2648.82	−0.107455
$848$	11147.8	0.451435
$849$	0	0
$850$	−7005.96	−0.282709
$851$	−16712.5	−0.673206
$852$	0	0
$853$	10701.2	0.429546	0.214773	−	0.976664i	$-0.431099\pi$
$853$	10701.2	0.429546	0.214773	+	0.976664i	$0.431099\pi$
$854$	−5193.76	−0.208111
$855$	0	0
$856$	33142.9	1.32337
$857$	31267.6	1.24630	0.623152	−	0.782101i	$-0.285852\pi$
$857$	31267.6	1.24630	0.623152	+	0.782101i	$0.285852\pi$
$858$	0	0
$859$	−35.6688	−0.00141677	−0.000708385	−	1.00000i	$-0.500225\pi$
$859$	−35.6688	−0.00141677	−0.000708385		1.00000i	$0.500225\pi$
$860$	−2561.96	−0.101584
$861$	0	0
$862$	42302.8	1.67151
$863$	6916.71	0.272825	0.136412	−	0.990652i	$-0.456443\pi$
$863$	6916.71	0.272825	0.136412	+	0.990652i	$0.456443\pi$
$864$	0	0
$865$	14115.1	0.554828
$866$	39649.9	1.55584
$867$	0	0
$868$	−287.093	−0.0112265
$869$	11557.4	0.451158
$870$	0	0
$871$	2507.65	0.0975526
$872$	40358.0	1.56731
$873$	0	0
$874$	−25282.0	−0.978464
$875$	3584.61	0.138494
$876$	0	0
$877$	2327.68	0.0896239	0.0448120	−	0.998995i	$-0.485731\pi$
$877$	2327.68	0.0896239	0.0448120	+	0.998995i	$0.485731\pi$
$878$	15866.9	0.609888
$879$	0	0
$880$	9186.30	0.351898
$881$	2343.54	0.0896205	0.0448103	−	0.998996i	$-0.485732\pi$
$881$	2343.54	0.0896205	0.0448103	+	0.998996i	$0.485732\pi$
$882$	0	0
$883$	−19558.9	−0.745424	−0.372712	−	0.927947i	$-0.621572\pi$
$883$	−19558.9	−0.745424	−0.372712	+	0.927947i	$0.621572\pi$
$884$	684.815	0.0260552
$885$	0	0
$886$	−51084.4	−1.93704
$887$	10331.5	0.391091	0.195546	−	0.980695i	$-0.437352\pi$
$887$	10331.5	0.391091	0.195546	+	0.980695i	$0.437352\pi$
$888$	0	0
$889$	1095.68	0.0413361
$890$	−20778.7	−0.782587
$891$	0	0
$892$	−1724.88	−0.0647459
$893$	−15236.2	−0.570952
$894$	0	0
$895$	−3839.99	−0.143415
$896$	3824.83	0.142610
$897$	0	0
$898$	36663.4	1.36244
$899$	−12828.5	−0.475924
$900$	0	0
$901$	−9266.49	−0.342632
$902$	151.096	0.00557753
$903$	0	0
$904$	3794.52	0.139606
$905$	−16712.8	−0.613871
$906$	0	0
$907$	52052.2	1.90559	0.952793	−	0.303621i	$-0.0981957\pi$
$907$	52052.2	1.90559	0.952793	+	0.303621i	$0.0981957\pi$
$908$	−4412.12	−0.161257
$909$	0	0
$910$	−970.159	−0.0353412
$911$	10108.8	0.367639	0.183820	−	0.982960i	$-0.441154\pi$
$911$	10108.8	0.367639	0.183820	+	0.982960i	$0.441154\pi$
$912$	0	0
$913$	−8468.95	−0.306990
$914$	30280.6	1.09583
$915$	0	0
$916$	4031.48	0.145419
$917$	−214.890	−0.00773861
$918$	0	0
$919$	34840.7	1.25059	0.625293	−	0.780390i	$-0.284979\pi$
$919$	34840.7	1.25059	0.625293	+	0.780390i	$0.284979\pi$
$920$	−16232.0	−0.581687
$921$	0	0
$922$	−30508.1	−1.08973
$923$	−11317.0	−0.403581
$924$	0	0
$925$	−8252.48	−0.293340
$926$	1851.26	0.0656978
$927$	0	0
$928$	−2896.40	−0.102456
$929$	39014.1	1.37784	0.688919	−	0.724838i	$-0.258085\pi$
$929$	39014.1	1.37784	0.688919	+	0.724838i	$0.258085\pi$
$930$	0	0
$931$	−34807.1	−1.22530
$932$	−1008.20	−0.0354342
$933$	0	0
$934$	−9175.11	−0.321433
$935$	−7636.02	−0.267085
$936$	0	0
$937$	−23368.1	−0.814729	−0.407364	−	0.913266i	$-0.633552\pi$
$937$	−23368.1	−0.814729	−0.407364	+	0.913266i	$0.633552\pi$
$938$	1155.88	0.0402354
$939$	0	0
$940$	1058.36	0.0367233
$941$	6990.35	0.242167	0.121083	−	0.992642i	$-0.461363\pi$
$941$	6990.35	0.242167	0.121083	+	0.992642i	$0.461363\pi$
$942$	0	0
$943$	−293.298	−0.0101284
$944$	−3198.83	−0.110289
$945$	0	0
$946$	15236.2	0.523648
$947$	−34940.9	−1.19897	−0.599486	−	0.800385i	$-0.704628\pi$
$947$	−34940.9	−1.19897	−0.599486	+	0.800385i	$0.704628\pi$
$948$	0	0
$949$	14581.4	0.498769
$950$	−12484.0	−0.426352
$951$	0	0
$952$	−2917.57	−0.0993267
$953$	−49498.4	−1.68249	−0.841244	−	0.540655i	$-0.818176\pi$
$953$	−49498.4	−1.68249	−0.841244	+	0.540655i	$0.818176\pi$
$954$	0	0
$955$	17216.6	0.583369
$956$	186.669	0.00631516
$957$	0	0
$958$	−10388.2	−0.350343
$959$	−6629.96	−0.223246
$960$	0	0
$961$	−5452.12	−0.183012
$962$	9069.07	0.303949
$963$	0	0
$964$	−2307.01	−0.0770786
$965$	−34346.3	−1.14575
$966$	0	0
$967$	−22271.8	−0.740655	−0.370328	−	0.928901i	$-0.620755\pi$
$967$	−22271.8	−0.740655	−0.370328	+	0.928901i	$0.620755\pi$
$968$	24049.2	0.798524
$969$	0	0
$970$	3692.78	0.122235
$971$	16361.7	0.540753	0.270376	−	0.962755i	$-0.412852\pi$
$971$	16361.7	0.540753	0.270376	+	0.962755i	$0.412852\pi$
$972$	0	0
$973$	−4781.21	−0.157532
$974$	−14877.8	−0.489442
$975$	0	0
$976$	51803.3	1.69896
$977$	13731.6	0.449654	0.224827	−	0.974399i	$-0.427818\pi$
$977$	13731.6	0.449654	0.224827	+	0.974399i	$0.427818\pi$
$978$	0	0
$979$	10991.3	0.358818
$980$	2417.83	0.0788109
$981$	0	0
$982$	−23664.3	−0.768999
$983$	41619.1	1.35040	0.675200	−	0.737634i	$-0.264057\pi$
$983$	41619.1	1.35040	0.675200	+	0.737634i	$0.264057\pi$
$984$	0	0
$985$	−210.124	−0.00679706
$986$	14105.0	0.455574
$987$	0	0
$988$	1220.28	0.0392938
$989$	−29575.6	−0.950910
$990$	0	0
$991$	15749.0	0.504827	0.252413	−	0.967619i	$-0.418776\pi$
$991$	15749.0	0.504827	0.252413	+	0.967619i	$0.418776\pi$
$992$	5495.18	0.175879
$993$	0	0
$994$	−5216.51	−0.166456
$995$	−17638.0	−0.561972
$996$	0	0
$997$	−16722.9	−0.531212	−0.265606	−	0.964082i	$-0.585572\pi$
$997$	−16722.9	−0.531212	−0.265606	+	0.964082i	$0.585572\pi$
$998$	−58216.8	−1.84651
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.44	✓	59
3.2	odd	2		2151.4.a.h.1.16	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.44	✓	59	1.1	even	1	trivial
2151.4.a.h.1.16	yes	59	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+2.96328 q^{2} +0.781040 q^{4} -9.17370 q^{5} +2.35613 q^{7} -21.3918 q^{8} +O(q^{10})\)	\(q+2.96328 q^{2} +0.781040 q^{4} -9.17370 q^{5} +2.35613 q^{7} -21.3918 q^{8} -27.1843 q^{10} +14.3796 q^{11} +15.1470 q^{13} +6.98188 q^{14} -69.6383 q^{16} +57.8861 q^{17} +103.148 q^{19} -7.16503 q^{20} +42.6109 q^{22} -82.7139 q^{23} -40.8433 q^{25} +44.8848 q^{26} +1.84023 q^{28} +82.2294 q^{29} -156.009 q^{31} -35.2234 q^{32} +171.533 q^{34} -21.6144 q^{35} +202.052 q^{37} +305.656 q^{38} +196.242 q^{40} +3.54593 q^{41} +357.565 q^{43} +11.2311 q^{44} -245.105 q^{46} -147.712 q^{47} -337.449 q^{49} -121.030 q^{50} +11.8304 q^{52} -160.081 q^{53} -131.915 q^{55} -50.4019 q^{56} +243.669 q^{58} +45.9350 q^{59} -743.891 q^{61} -462.299 q^{62} +452.730 q^{64} -138.954 q^{65} +165.554 q^{67} +45.2114 q^{68} -64.0497 q^{70} -747.149 q^{71} +962.660 q^{73} +598.738 q^{74} +80.5627 q^{76} +33.8803 q^{77} +803.730 q^{79} +638.841 q^{80} +10.5076 q^{82} -588.954 q^{83} -531.030 q^{85} +1059.57 q^{86} -307.607 q^{88} +764.364 q^{89} +35.6883 q^{91} -64.6029 q^{92} -437.713 q^{94} -946.248 q^{95} -135.843 q^{97} -999.955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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