LMFDB - Embedded newform 2151.4.a.f.1.12

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:	$0$
Dimension:	$37$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-2.73040 q^{2} -0.544930 q^{4} -10.5995 q^{5} +15.6616 q^{7} +23.3311 q^{8} +O(q^{10})$	\(q-2.73040 q^{2} -0.544930 q^{4} -10.5995 q^{5} +15.6616 q^{7} +23.3311 q^{8} +28.9408 q^{10} +0.571833 q^{11} -88.1692 q^{13} -42.7624 q^{14} -59.3436 q^{16} +61.0287 q^{17} -111.929 q^{19} +5.77597 q^{20} -1.56133 q^{22} -169.630 q^{23} -12.6510 q^{25} +240.737 q^{26} -8.53448 q^{28} +68.0256 q^{29} +144.016 q^{31} -24.6168 q^{32} -166.633 q^{34} -166.005 q^{35} -21.1263 q^{37} +305.609 q^{38} -247.297 q^{40} +91.9700 q^{41} +234.141 q^{43} -0.311609 q^{44} +463.157 q^{46} -226.822 q^{47} -97.7140 q^{49} +34.5422 q^{50} +48.0460 q^{52} +82.1194 q^{53} -6.06113 q^{55} +365.402 q^{56} -185.737 q^{58} +110.073 q^{59} +16.2105 q^{61} -393.220 q^{62} +541.963 q^{64} +934.548 q^{65} -875.625 q^{67} -33.2564 q^{68} +453.259 q^{70} +53.3295 q^{71} -21.0392 q^{73} +57.6832 q^{74} +60.9932 q^{76} +8.95582 q^{77} +598.027 q^{79} +629.012 q^{80} -251.115 q^{82} -995.800 q^{83} -646.873 q^{85} -639.299 q^{86} +13.3415 q^{88} -1137.76 q^{89} -1380.87 q^{91} +92.4363 q^{92} +619.315 q^{94} +1186.38 q^{95} -92.2384 q^{97} +266.798 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * 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 $

$2$	−2.73040	−0.965341	−0.482671	−	0.875802i	$-0.660333\pi$
$2$	−2.73040	−0.965341	−0.482671	+	0.875802i	$0.660333\pi$
$3$	0	0
$3$	0	0
$4$	−0.544930	−0.0681162
$5$	−10.5995	−0.948046	−0.474023	−	0.880512i	$-0.657199\pi$
$5$	−10.5995	−0.948046	−0.474023	+	0.880512i	$0.657199\pi$
$6$	0	0
$7$	15.6616	0.845647	0.422824	−	0.906212i	$-0.361039\pi$
$7$	15.6616	0.845647	0.422824	+	0.906212i	$0.361039\pi$
$8$	23.3311	1.03110
$9$	0	0
$10$	28.9408	0.915188
$11$	0.571833	0.0156740	0.00783700	−	0.999969i	$-0.497505\pi$
$11$	0.571833	0.0156740	0.00783700	+	0.999969i	$0.497505\pi$
$12$	0	0
$13$	−88.1692	−1.88106	−0.940528	−	0.339715i	$-0.889669\pi$
$13$	−88.1692	−1.88106	−0.940528	+	0.339715i	$0.889669\pi$
$14$	−42.7624	−0.816338
$15$	0	0
$16$	−59.3436	−0.927244
$17$	61.0287	0.870684	0.435342	−	0.900265i	$-0.356627\pi$
$17$	61.0287	0.870684	0.435342	+	0.900265i	$0.356627\pi$
$18$	0	0
$19$	−111.929	−1.35148	−0.675741	−	0.737139i	$-0.736176\pi$
$19$	−111.929	−1.35148	−0.675741	+	0.737139i	$0.736176\pi$
$20$	5.77597	0.0645774
$21$	0	0
$22$	−1.56133	−0.0151308
$23$	−169.630	−1.53784	−0.768919	−	0.639347i	$-0.779205\pi$
$23$	−169.630	−1.53784	−0.768919	+	0.639347i	$0.779205\pi$
$24$	0	0
$25$	−12.6510	−0.101208
$26$	240.737	1.81586
$27$	0	0
$28$	−8.53448	−0.0576023
$29$	68.0256	0.435588	0.217794	−	0.975995i	$-0.430114\pi$
$29$	68.0256	0.435588	0.217794	+	0.975995i	$0.430114\pi$
$30$	0	0
$31$	144.016	0.834386	0.417193	−	0.908818i	$-0.363014\pi$
$31$	144.016	0.834386	0.417193	+	0.908818i	$0.363014\pi$
$32$	−24.6168	−0.135990
$33$	0	0
$34$	−166.633	−0.840507
$35$	−166.005	−0.801713
$36$	0	0
$37$	−21.1263	−0.0938688	−0.0469344	−	0.998898i	$-0.514945\pi$
$37$	−21.1263	−0.0938688	−0.0469344	+	0.998898i	$0.514945\pi$
$38$	305.609	1.30464
$39$	0	0
$40$	−247.297	−0.977528
$41$	91.9700	0.350324	0.175162	−	0.984540i	$-0.443955\pi$
$41$	91.9700	0.350324	0.175162	+	0.984540i	$0.443955\pi$
$42$	0	0
$43$	234.141	0.830377	0.415189	−	0.909735i	$-0.363716\pi$
$43$	234.141	0.830377	0.415189	+	0.909735i	$0.363716\pi$
$44$	−0.311609	−0.00106765
$45$	0	0
$46$	463.157	1.48454
$47$	−226.822	−0.703945	−0.351972	−	0.936010i	$-0.614489\pi$
$47$	−226.822	−0.703945	−0.351972	+	0.936010i	$0.614489\pi$
$48$	0	0
$49$	−97.7140	−0.284880
$50$	34.5422	0.0977002
$51$	0	0
$52$	48.0460	0.128131
$53$	82.1194	0.212830	0.106415	−	0.994322i	$-0.466063\pi$
$53$	82.1194	0.212830	0.106415	+	0.994322i	$0.466063\pi$
$54$	0	0
$55$	−6.06113	−0.0148597
$56$	365.402	0.871944
$57$	0	0
$58$	−185.737	−0.420491
$59$	110.073	0.242887	0.121443	−	0.992598i	$-0.461248\pi$
$59$	110.073	0.242887	0.121443	+	0.992598i	$0.461248\pi$
$60$	0	0
$61$	16.2105	0.0340253	0.0170127	−	0.999855i	$-0.494584\pi$
$61$	16.2105	0.0340253	0.0170127	+	0.999855i	$0.494584\pi$
$62$	−393.220	−0.805467
$63$	0	0
$64$	541.963	1.05852
$65$	934.548	1.78333
$66$	0	0
$67$	−875.625	−1.59664	−0.798318	−	0.602236i	$-0.794276\pi$
$67$	−875.625	−1.59664	−0.798318	+	0.602236i	$0.794276\pi$
$68$	−33.2564	−0.0593077
$69$	0	0
$70$	453.259	0.773927
$71$	53.3295	0.0891415	0.0445708	−	0.999006i	$-0.485808\pi$
$71$	53.3295	0.0891415	0.0445708	+	0.999006i	$0.485808\pi$
$72$	0	0
$73$	−21.0392	−0.0337322	−0.0168661	−	0.999858i	$-0.505369\pi$
$73$	−21.0392	−0.0337322	−0.0168661	+	0.999858i	$0.505369\pi$
$74$	57.6832	0.0906154
$75$	0	0
$76$	60.9932	0.0920579
$77$	8.95582	0.0132547
$78$	0	0
$79$	598.027	0.851687	0.425843	−	0.904797i	$-0.359977\pi$
$79$	598.027	0.851687	0.425843	+	0.904797i	$0.359977\pi$
$80$	629.012	0.879070
$81$	0	0
$82$	−251.115	−0.338183
$83$	−995.800	−1.31691	−0.658453	−	0.752622i	$-0.728789\pi$
$83$	−995.800	−1.31691	−0.658453	+	0.752622i	$0.728789\pi$
$84$	0	0
$85$	−646.873	−0.825449
$86$	−639.299	−0.801597
$87$	0	0
$88$	13.3415	0.0161614
$89$	−1137.76	−1.35508	−0.677542	−	0.735484i	$-0.736955\pi$
$89$	−1137.76	−1.35508	−0.677542	+	0.735484i	$0.736955\pi$
$90$	0	0
$91$	−1380.87	−1.59071
$92$	92.4363	0.104752
$93$	0	0
$94$	619.315	0.679547
$95$	1186.38	1.28127
$96$	0	0
$97$	−92.2384	−0.0965504	−0.0482752	−	0.998834i	$-0.515372\pi$
$97$	−92.2384	−0.0965504	−0.0482752	+	0.998834i	$0.515372\pi$
$98$	266.798	0.275007
$99$	0	0
$100$	6.89391	0.00689391
$101$	544.406	0.536341	0.268171	−	0.963371i	$-0.413581\pi$
$101$	544.406	0.536341	0.268171	+	0.963371i	$0.413581\pi$
$102$	0	0
$103$	−972.828	−0.930637	−0.465318	−	0.885143i	$-0.654060\pi$
$103$	−972.828	−0.930637	−0.465318	+	0.885143i	$0.654060\pi$
$104$	−2057.08	−1.93955
$105$	0	0
$106$	−224.219	−0.205453
$107$	−2034.46	−1.83812	−0.919061	−	0.394116i	$-0.871051\pi$
$107$	−2034.46	−1.83812	−0.919061	+	0.394116i	$0.871051\pi$
$108$	0	0
$109$	−1797.47	−1.57950	−0.789752	−	0.613426i	$-0.789791\pi$
$109$	−1797.47	−1.57950	−0.789752	+	0.613426i	$0.789791\pi$
$110$	16.5493	0.0143447
$111$	0	0
$112$	−929.416	−0.784121
$113$	1375.95	1.14547	0.572736	−	0.819740i	$-0.305882\pi$
$113$	1375.95	1.14547	0.572736	+	0.819740i	$0.305882\pi$
$114$	0	0
$115$	1797.99	1.45794
$116$	−37.0692	−0.0296706
$117$	0	0
$118$	−300.544	−0.234468
$119$	955.808	0.736292
$120$	0	0
$121$	−1330.67	−0.999754
$122$	−44.2612	−0.0328460
$123$	0	0
$124$	−78.4784	−0.0568352
$125$	1459.03	1.04400
$126$	0	0
$127$	−1958.77	−1.36861	−0.684303	−	0.729198i	$-0.739893\pi$
$127$	−1958.77	−1.36861	−0.684303	+	0.729198i	$0.739893\pi$
$128$	−1282.84	−0.885844
$129$	0	0
$130$	−2551.69	−1.72152
$131$	−630.036	−0.420202	−0.210101	−	0.977680i	$-0.567379\pi$
$131$	−630.036	−0.420202	−0.210101	+	0.977680i	$0.567379\pi$
$132$	0	0
$133$	−1752.98	−1.14288
$134$	2390.80	1.54130
$135$	0	0
$136$	1423.86	0.897760
$137$	−2898.33	−1.80745	−0.903725	−	0.428114i	$-0.859178\pi$
$137$	−2898.33	−1.80745	−0.903725	+	0.428114i	$0.859178\pi$
$138$	0	0
$139$	−655.684	−0.400103	−0.200052	−	0.979785i	$-0.564111\pi$
$139$	−655.684	−0.400103	−0.200052	+	0.979785i	$0.564111\pi$
$140$	90.4611	0.0546097
$141$	0	0
$142$	−145.611	−0.0860520
$143$	−50.4180	−0.0294837
$144$	0	0
$145$	−721.036	−0.412957
$146$	57.4453	0.0325631
$147$	0	0
$148$	11.5124	0.00639399
$149$	2485.78	1.36673	0.683367	−	0.730075i	$-0.260515\pi$
$149$	2485.78	1.36673	0.683367	+	0.730075i	$0.260515\pi$
$150$	0	0
$151$	3142.12	1.69339	0.846697	−	0.532076i	$-0.178588\pi$
$151$	3142.12	1.69339	0.846697	+	0.532076i	$0.178588\pi$
$152$	−2611.41	−1.39351
$153$	0	0
$154$	−24.4530	−0.0127953
$155$	−1526.49	−0.791037
$156$	0	0
$157$	2320.78	1.17973	0.589867	−	0.807500i	$-0.299180\pi$
$157$	2320.78	1.17973	0.589867	+	0.807500i	$0.299180\pi$
$158$	−1632.85	−0.822168
$159$	0	0
$160$	260.925	0.128925
$161$	−2656.68	−1.30047
$162$	0	0
$163$	−234.695	−0.112778	−0.0563888	−	0.998409i	$-0.517959\pi$
$163$	−234.695	−0.112778	−0.0563888	+	0.998409i	$0.517959\pi$
$164$	−50.1172	−0.0238628
$165$	0	0
$166$	2718.93	1.27126
$167$	−1851.29	−0.857829	−0.428915	−	0.903345i	$-0.641104\pi$
$167$	−1851.29	−0.857829	−0.428915	+	0.903345i	$0.641104\pi$
$168$	0	0
$169$	5576.81	2.53837
$170$	1766.22	0.796840
$171$	0	0
$172$	−127.591	−0.0565622
$173$	2171.08	0.954130	0.477065	−	0.878868i	$-0.341701\pi$
$173$	2171.08	0.954130	0.477065	+	0.878868i	$0.341701\pi$
$174$	0	0
$175$	−198.135	−0.0855863
$176$	−33.9346	−0.0145336
$177$	0	0
$178$	3106.54	1.30812
$179$	−2.49159	−0.00104039	−0.000520197	−	1.00000i	$-0.500166\pi$
$179$	−2.49159	−0.00104039	−0.000520197		1.00000i	$0.500166\pi$
$180$	0	0
$181$	−389.853	−0.160097	−0.0800484	−	0.996791i	$-0.525508\pi$
$181$	−389.853	−0.160097	−0.0800484	+	0.996791i	$0.525508\pi$
$182$	3770.33	1.53558
$183$	0	0
$184$	−3957.64	−1.58566
$185$	223.928	0.0889919
$186$	0	0
$187$	34.8982	0.0136471
$188$	123.602	0.0479501
$189$	0	0
$190$	−3239.30	−1.23686
$191$	−2780.89	−1.05350	−0.526750	−	0.850020i	$-0.676590\pi$
$191$	−2780.89	−1.05350	−0.526750	+	0.850020i	$0.676590\pi$
$192$	0	0
$193$	3461.12	1.29086	0.645432	−	0.763818i	$-0.276677\pi$
$193$	3461.12	1.29086	0.645432	+	0.763818i	$0.276677\pi$
$194$	251.848	0.0932041
$195$	0	0
$196$	53.2473	0.0194050
$197$	4209.12	1.52227	0.761136	−	0.648592i	$-0.224642\pi$
$197$	4209.12	1.52227	0.761136	+	0.648592i	$0.224642\pi$
$198$	0	0
$199$	2763.58	0.984446	0.492223	−	0.870469i	$-0.336184\pi$
$199$	2763.58	0.984446	0.492223	+	0.870469i	$0.336184\pi$
$200$	−295.161	−0.104355
$201$	0	0
$202$	−1486.45	−0.517752
$203$	1065.39	0.368354
$204$	0	0
$205$	−974.834	−0.332124
$206$	2656.21	0.898382
$207$	0	0
$208$	5232.28	1.74420
$209$	−64.0044	−0.0211831
$210$	0	0
$211$	437.819	0.142847	0.0714235	−	0.997446i	$-0.477246\pi$
$211$	437.819	0.142847	0.0714235	+	0.997446i	$0.477246\pi$
$212$	−44.7493	−0.0144972
$213$	0	0
$214$	5554.89	1.77441
$215$	−2481.78	−0.787236
$216$	0	0
$217$	2255.52	0.705596
$218$	4907.79	1.52476
$219$	0	0
$220$	3.30289	0.00101219
$221$	−5380.85	−1.63781
$222$	0	0
$223$	5122.15	1.53814	0.769069	−	0.639166i	$-0.220720\pi$
$223$	5122.15	1.53814	0.769069	+	0.639166i	$0.220720\pi$
$224$	−385.539	−0.114999
$225$	0	0
$226$	−3756.88	−1.10577
$227$	−3973.20	−1.16172	−0.580860	−	0.814003i	$-0.697284\pi$
$227$	−3973.20	−1.16172	−0.580860	+	0.814003i	$0.697284\pi$
$228$	0	0
$229$	−6014.59	−1.73561	−0.867806	−	0.496904i	$-0.834470\pi$
$229$	−6014.59	−1.73561	−0.867806	+	0.496904i	$0.834470\pi$
$230$	−4909.22	−1.40741
$231$	0	0
$232$	1587.11	0.449133
$233$	1546.85	0.434926	0.217463	−	0.976069i	$-0.430222\pi$
$233$	1546.85	0.434926	0.217463	+	0.976069i	$0.430222\pi$
$234$	0	0
$235$	2404.20	0.667372
$236$	−59.9822	−0.0165445
$237$	0	0
$238$	−2609.73	−0.710773
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	3722.35	0.994928	0.497464	−	0.867485i	$-0.334265\pi$
$241$	3722.35	0.994928	0.497464	+	0.867485i	$0.334265\pi$
$242$	3633.27	0.965104
$243$	0	0
$244$	−8.83359	−0.00231768
$245$	1035.72	0.270080
$246$	0	0
$247$	9868.65	2.54221
$248$	3360.04	0.860333
$249$	0	0
$250$	−3983.73	−1.00781
$251$	193.653	0.0486983	0.0243492	−	0.999704i	$-0.492249\pi$
$251$	193.653	0.0486983	0.0243492	+	0.999704i	$0.492249\pi$
$252$	0	0
$253$	−96.9999	−0.0241041
$254$	5348.22	1.32117
$255$	0	0
$256$	−833.041	−0.203379
$257$	−2741.01	−0.665289	−0.332644	−	0.943052i	$-0.607941\pi$
$257$	−2741.01	−0.665289	−0.332644	+	0.943052i	$0.607941\pi$
$258$	0	0
$259$	−330.872	−0.0793799
$260$	−509.263	−0.121474
$261$	0	0
$262$	1720.25	0.405639
$263$	6162.53	1.44486	0.722429	−	0.691445i	$-0.243026\pi$
$263$	6162.53	1.44486	0.722429	+	0.691445i	$0.243026\pi$
$264$	0	0
$265$	−870.423	−0.201772
$266$	4786.33	1.10327
$267$	0	0
$268$	477.154	0.108757
$269$	−1582.86	−0.358769	−0.179385	−	0.983779i	$-0.557411\pi$
$269$	−1582.86	−0.358769	−0.179385	+	0.983779i	$0.557411\pi$
$270$	0	0
$271$	312.998	0.0701596	0.0350798	−	0.999385i	$-0.488831\pi$
$271$	312.998	0.0701596	0.0350798	+	0.999385i	$0.488831\pi$
$272$	−3621.66	−0.807337
$273$	0	0
$274$	7913.58	1.74481
$275$	−7.23425	−0.00158633
$276$	0	0
$277$	−2388.88	−0.518172	−0.259086	−	0.965854i	$-0.583421\pi$
$277$	−2388.88	−0.518172	−0.259086	+	0.965854i	$0.583421\pi$
$278$	1790.28	0.386236
$279$	0	0
$280$	−3873.07	−0.826644
$281$	2885.40	0.612557	0.306279	−	0.951942i	$-0.400916\pi$
$281$	2885.40	0.612557	0.306279	+	0.951942i	$0.400916\pi$
$282$	0	0
$283$	−1651.31	−0.346855	−0.173427	−	0.984847i	$-0.555484\pi$
$283$	−1651.31	−0.346855	−0.173427	+	0.984847i	$0.555484\pi$
$284$	−29.0608	−0.00607199
$285$	0	0
$286$	137.661	0.0284618
$287$	1440.40	0.296251
$288$	0	0
$289$	−1188.50	−0.241909
$290$	1968.72	0.398645
$291$	0	0
$292$	11.4649	0.00229771
$293$	8090.64	1.61318	0.806588	−	0.591114i	$-0.201312\pi$
$293$	8090.64	1.61318	0.806588	+	0.591114i	$0.201312\pi$
$294$	0	0
$295$	−1166.72	−0.230268
$296$	−492.899	−0.0967878
$297$	0	0
$298$	−6787.18	−1.31936
$299$	14956.1	2.89276
$300$	0	0
$301$	3667.03	0.702206
$302$	−8579.25	−1.63470
$303$	0	0
$304$	6642.24	1.25315
$305$	−171.823	−0.0322576
$306$	0	0
$307$	5908.92	1.09850	0.549251	−	0.835658i	$-0.314913\pi$
$307$	5908.92	1.09850	0.549251	+	0.835658i	$0.314913\pi$
$308$	−4.88029	−0.000902859 0
$309$	0	0
$310$	4167.93	0.763620
$311$	−4420.08	−0.805915	−0.402957	−	0.915219i	$-0.632018\pi$
$311$	−4420.08	−0.805915	−0.402957	+	0.915219i	$0.632018\pi$
$312$	0	0
$313$	−1070.14	−0.193253	−0.0966265	−	0.995321i	$-0.530805\pi$
$313$	−1070.14	−0.193253	−0.0966265	+	0.995321i	$0.530805\pi$
$314$	−6336.65	−1.13885
$315$	0	0
$316$	−325.883	−0.0580137
$317$	−7580.22	−1.34305	−0.671526	−	0.740981i	$-0.734361\pi$
$317$	−7580.22	−1.34305	−0.671526	+	0.740981i	$0.734361\pi$
$318$	0	0
$319$	38.8993	0.00682740
$320$	−5744.52	−1.00353
$321$	0	0
$322$	7253.78	1.25540
$323$	−6830.85	−1.17671
$324$	0	0
$325$	1115.43	0.190378
$326$	640.811	0.108869
$327$	0	0
$328$	2145.76	0.361218
$329$	−3552.40	−0.595289
$330$	0	0
$331$	−3611.44	−0.599705	−0.299853	−	0.953986i	$-0.596938\pi$
$331$	−3611.44	−0.599705	−0.299853	+	0.953986i	$0.596938\pi$
$332$	542.641	0.0897027
$333$	0	0
$334$	5054.77	0.828098
$335$	9281.17	1.51368
$336$	0	0
$337$	−8396.41	−1.35721	−0.678607	−	0.734501i	$-0.737416\pi$
$337$	−8396.41	−1.35721	−0.678607	+	0.734501i	$0.737416\pi$
$338$	−15226.9	−2.45040
$339$	0	0
$340$	352.500	0.0562265
$341$	82.3528	0.0130782
$342$	0	0
$343$	−6902.29	−1.08656
$344$	5462.76	0.856199
$345$	0	0
$346$	−5927.92	−0.921061
$347$	−11983.7	−1.85395	−0.926976	−	0.375121i	$-0.877601\pi$
$347$	−11983.7	−1.85395	−0.926976	+	0.375121i	$0.877601\pi$
$348$	0	0
$349$	10419.3	1.59808	0.799042	−	0.601275i	$-0.205340\pi$
$349$	10419.3	1.59808	0.799042	+	0.601275i	$0.205340\pi$
$350$	540.987	0.0826200
$351$	0	0
$352$	−14.0767	−0.00213151
$353$	4094.87	0.617417	0.308708	−	0.951157i	$-0.400103\pi$
$353$	4094.87	0.617417	0.308708	+	0.951157i	$0.400103\pi$
$354$	0	0
$355$	−565.265	−0.0845103
$356$	620.000	0.0923032
$357$	0	0
$358$	6.80304	0.00100433
$359$	4040.57	0.594020	0.297010	−	0.954874i	$-0.404010\pi$
$359$	4040.57	0.594020	0.297010	+	0.954874i	$0.404010\pi$
$360$	0	0
$361$	5668.99	0.826504
$362$	1064.45	0.154548
$363$	0	0
$364$	752.478	0.108353
$365$	223.004	0.0319797
$366$	0	0
$367$	−1428.02	−0.203111	−0.101556	−	0.994830i	$-0.532382\pi$
$367$	−1428.02	−0.203111	−0.101556	+	0.994830i	$0.532382\pi$
$368$	10066.4	1.42595
$369$	0	0
$370$	−611.412	−0.0859076
$371$	1286.12	0.179979
$372$	0	0
$373$	622.213	0.0863726	0.0431863	−	0.999067i	$-0.486249\pi$
$373$	622.213	0.0863726	0.0431863	+	0.999067i	$0.486249\pi$
$374$	−95.2860	−0.0131741
$375$	0	0
$376$	−5292.00	−0.725835
$377$	−5997.77	−0.819365
$378$	0	0
$379$	−167.005	−0.0226345	−0.0113173	−	0.999936i	$-0.503602\pi$
$379$	−167.005	−0.0226345	−0.0113173	+	0.999936i	$0.503602\pi$
$380$	−646.496	−0.0872751
$381$	0	0
$382$	7592.95	1.01699
$383$	2252.07	0.300458	0.150229	−	0.988651i	$-0.451999\pi$
$383$	2252.07	0.300458	0.150229	+	0.988651i	$0.451999\pi$
$384$	0	0
$385$	−94.9271	−0.0125661
$386$	−9450.22	−1.24612
$387$	0	0
$388$	50.2635	0.00657665
$389$	−10599.7	−1.38156	−0.690778	−	0.723067i	$-0.742732\pi$
$389$	−10599.7	−1.38156	−0.690778	+	0.723067i	$0.742732\pi$
$390$	0	0
$391$	−10352.3	−1.33897
$392$	−2279.77	−0.293739
$393$	0	0
$394$	−11492.6	−1.46951
$395$	−6338.77	−0.807438
$396$	0	0
$397$	−8837.90	−1.11728	−0.558642	−	0.829409i	$-0.688677\pi$
$397$	−8837.90	−1.11728	−0.558642	+	0.829409i	$0.688677\pi$
$398$	−7545.66	−0.950326
$399$	0	0
$400$	750.756	0.0938445
$401$	−4285.74	−0.533715	−0.266857	−	0.963736i	$-0.585985\pi$
$401$	−4285.74	−0.533715	−0.266857	+	0.963736i	$0.585985\pi$
$402$	0	0
$403$	−12697.7	−1.56953
$404$	−296.663	−0.0365335
$405$	0	0
$406$	−2908.94	−0.355587
$407$	−12.0807	−0.00147130
$408$	0	0
$409$	12692.3	1.53446	0.767228	−	0.641374i	$-0.221635\pi$
$409$	12692.3	1.53446	0.767228	+	0.641374i	$0.221635\pi$
$410$	2661.69	0.320613
$411$	0	0
$412$	530.123	0.0633915
$413$	1723.92	0.205396
$414$	0	0
$415$	10555.0	1.24849
$416$	2170.44	0.255805
$417$	0	0
$418$	174.757	0.0204490
$419$	15467.6	1.80344	0.901721	−	0.432319i	$-0.142305\pi$
$419$	15467.6	1.80344	0.901721	+	0.432319i	$0.142305\pi$
$420$	0	0
$421$	1670.43	0.193377	0.0966883	−	0.995315i	$-0.469175\pi$
$421$	1670.43	0.193377	0.0966883	+	0.995315i	$0.469175\pi$
$422$	−1195.42	−0.137896
$423$	0	0
$424$	1915.93	0.219448
$425$	−772.074	−0.0881202
$426$	0	0
$427$	253.883	0.0287734
$428$	1108.64	0.125206
$429$	0	0
$430$	6776.24	0.759952
$431$	3437.31	0.384152	0.192076	−	0.981380i	$-0.438478\pi$
$431$	3437.31	0.384152	0.192076	+	0.981380i	$0.438478\pi$
$432$	0	0
$433$	12771.0	1.41740	0.708700	−	0.705510i	$-0.249282\pi$
$433$	12771.0	1.41740	0.708700	+	0.705510i	$0.249282\pi$
$434$	−6158.46	−0.681141
$435$	0	0
$436$	979.493	0.107590
$437$	18986.4	2.07836
$438$	0	0
$439$	−3741.60	−0.406781	−0.203390	−	0.979098i	$-0.565196\pi$
$439$	−3741.60	−0.406781	−0.203390	+	0.979098i	$0.565196\pi$
$440$	−141.413	−0.0153218
$441$	0	0
$442$	14691.9	1.58104
$443$	13806.5	1.48073	0.740366	−	0.672204i	$-0.234652\pi$
$443$	13806.5	1.48073	0.740366	+	0.672204i	$0.234652\pi$
$444$	0	0
$445$	12059.7	1.28468
$446$	−13985.5	−1.48483
$447$	0	0
$448$	8488.01	0.895135
$449$	14624.3	1.53711	0.768554	−	0.639785i	$-0.220977\pi$
$449$	14624.3	1.53711	0.768554	+	0.639785i	$0.220977\pi$
$450$	0	0
$451$	52.5915	0.00549099
$452$	−749.795	−0.0780252
$453$	0	0
$454$	10848.4	1.12146
$455$	14636.5	1.50807
$456$	0	0
$457$	11260.2	1.15258	0.576292	−	0.817244i	$-0.304499\pi$
$457$	11260.2	1.15258	0.576292	+	0.817244i	$0.304499\pi$
$458$	16422.2	1.67546
$459$	0	0
$460$	−979.777	−0.0993094
$461$	−8092.64	−0.817596	−0.408798	−	0.912625i	$-0.634052\pi$
$461$	−8092.64	−0.817596	−0.408798	+	0.912625i	$0.634052\pi$
$462$	0	0
$463$	1082.38	0.108644	0.0543222	−	0.998523i	$-0.482700\pi$
$463$	1082.38	0.108644	0.0543222	+	0.998523i	$0.482700\pi$
$464$	−4036.89	−0.403896
$465$	0	0
$466$	−4223.53	−0.419852
$467$	8632.21	0.855356	0.427678	−	0.903931i	$-0.359332\pi$
$467$	8632.21	0.855356	0.427678	+	0.903931i	$0.359332\pi$
$468$	0	0
$469$	−13713.7	−1.35019
$470$	−6564.41	−0.644242
$471$	0	0
$472$	2568.12	0.250440
$473$	133.890	0.0130153
$474$	0	0
$475$	1416.01	0.136781
$476$	−520.848	−0.0501534
$477$	0	0
$478$	−652.565	−0.0624427
$479$	16105.8	1.53631	0.768157	−	0.640262i	$-0.221174\pi$
$479$	16105.8	1.53631	0.768157	+	0.640262i	$0.221174\pi$
$480$	0	0
$481$	1862.69	0.176572
$482$	−10163.5	−0.960445
$483$	0	0
$484$	725.124	0.0680995
$485$	977.679	0.0915343
$486$	0	0
$487$	11830.6	1.10082	0.550408	−	0.834896i	$-0.314472\pi$
$487$	11830.6	1.10082	0.550408	+	0.834896i	$0.314472\pi$
$488$	378.208	0.0350834
$489$	0	0
$490$	−2827.92	−0.260719
$491$	12756.1	1.17245	0.586227	−	0.810147i	$-0.300613\pi$
$491$	12756.1	1.17245	0.586227	+	0.810147i	$0.300613\pi$
$492$	0	0
$493$	4151.52	0.379259
$494$	−26945.3	−2.45410
$495$	0	0
$496$	−8546.41	−0.773679
$497$	835.226	0.0753823
$498$	0	0
$499$	−4712.07	−0.422728	−0.211364	−	0.977407i	$-0.567790\pi$
$499$	−4712.07	−0.422728	−0.211364	+	0.977407i	$0.567790\pi$
$500$	−795.069	−0.0711131
$501$	0	0
$502$	−528.750	−0.0470105
$503$	−2404.12	−0.213110	−0.106555	−	0.994307i	$-0.533982\pi$
$503$	−2404.12	−0.213110	−0.106555	+	0.994307i	$0.533982\pi$
$504$	0	0
$505$	−5770.42	−0.508476
$506$	264.848	0.0232686
$507$	0	0
$508$	1067.39	0.0932242
$509$	−4261.43	−0.371090	−0.185545	−	0.982636i	$-0.559405\pi$
$509$	−4261.43	−0.371090	−0.185545	+	0.982636i	$0.559405\pi$
$510$	0	0
$511$	−329.507	−0.0285255
$512$	12537.2	1.08217
$513$	0	0
$514$	7484.04	0.642231
$515$	10311.5	0.882287
$516$	0	0
$517$	−129.704	−0.0110336
$518$	903.412	0.0766287
$519$	0	0
$520$	21804.0	1.83878
$521$	−484.279	−0.0407229	−0.0203615	−	0.999793i	$-0.506482\pi$
$521$	−484.279	−0.0407229	−0.0203615	+	0.999793i	$0.506482\pi$
$522$	0	0
$523$	12235.6	1.02300	0.511498	−	0.859284i	$-0.329091\pi$
$523$	12235.6	1.02300	0.511498	+	0.859284i	$0.329091\pi$
$524$	343.325	0.0286226
$525$	0	0
$526$	−16826.1	−1.39478
$527$	8789.09	0.726487
$528$	0	0
$529$	16607.3	1.36494
$530$	2376.60	0.194779
$531$	0	0
$532$	955.252	0.0778485
$533$	−8108.92	−0.658980
$534$	0	0
$535$	21564.3	1.74262
$536$	−20429.2	−1.64629
$537$	0	0
$538$	4321.85	0.346335
$539$	−55.8761	−0.00446522
$540$	0	0
$541$	−16798.3	−1.33497	−0.667483	−	0.744625i	$-0.732628\pi$
$541$	−16798.3	−1.33497	−0.667483	+	0.744625i	$0.732628\pi$
$542$	−854.608	−0.0677279
$543$	0	0
$544$	−1502.33	−0.118404
$545$	19052.2	1.49744
$546$	0	0
$547$	16021.9	1.25237	0.626185	−	0.779674i	$-0.284615\pi$
$547$	16021.9	1.25237	0.626185	+	0.779674i	$0.284615\pi$
$548$	1579.38	0.123117
$549$	0	0
$550$	19.7524	0.00153135
$551$	−7614.01	−0.588689
$552$	0	0
$553$	9366.06	0.720227
$554$	6522.58	0.500213
$555$	0	0
$556$	357.302	0.0272535
$557$	6332.93	0.481750	0.240875	−	0.970556i	$-0.422566\pi$
$557$	6332.93	0.481750	0.240875	+	0.970556i	$0.422566\pi$
$558$	0	0
$559$	−20644.1	−1.56199
$560$	9851.33	0.743384
$561$	0	0
$562$	−7878.29	−0.591327
$563$	−1564.29	−0.117099	−0.0585496	−	0.998284i	$-0.518648\pi$
$563$	−1564.29	−0.117099	−0.0585496	+	0.998284i	$0.518648\pi$
$564$	0	0
$565$	−14584.3	−1.08596
$566$	4508.72	0.334833
$567$	0	0
$568$	1244.23	0.0919135
$569$	10946.1	0.806478	0.403239	−	0.915095i	$-0.367884\pi$
$569$	10946.1	0.806478	0.403239	+	0.915095i	$0.367884\pi$
$570$	0	0
$571$	24256.4	1.77775	0.888876	−	0.458147i	$-0.151487\pi$
$571$	24256.4	1.77775	0.888876	+	0.458147i	$0.151487\pi$
$572$	27.4743	0.00200832
$573$	0	0
$574$	−3932.86	−0.285983
$575$	2145.99	0.155641
$576$	0	0
$577$	9288.39	0.670157	0.335078	−	0.942190i	$-0.391237\pi$
$577$	9288.39	0.670157	0.335078	+	0.942190i	$0.391237\pi$
$578$	3245.07	0.233525
$579$	0	0
$580$	392.914	0.0281291
$581$	−15595.8	−1.11364
$582$	0	0
$583$	46.9586	0.00333589
$584$	−490.866	−0.0347812
$585$	0	0
$586$	−22090.7	−1.55727
$587$	14563.3	1.02401	0.512005	−	0.858983i	$-0.328903\pi$
$587$	14563.3	1.02401	0.512005	+	0.858983i	$0.328903\pi$
$588$	0	0
$589$	−16119.5	−1.12766
$590$	3185.61	0.222287
$591$	0	0
$592$	1253.71	0.0870392
$593$	−14447.3	−1.00047	−0.500237	−	0.865889i	$-0.666754\pi$
$593$	−14447.3	−1.00047	−0.500237	+	0.865889i	$0.666754\pi$
$594$	0	0
$595$	−10131.1	−0.698039
$596$	−1354.58	−0.0930967
$597$	0	0
$598$	−40836.2	−2.79250
$599$	19615.9	1.33803	0.669017	−	0.743247i	$-0.266715\pi$
$599$	19615.9	1.33803	0.669017	+	0.743247i	$0.266715\pi$
$600$	0	0
$601$	−25557.9	−1.73466	−0.867328	−	0.497736i	$-0.834165\pi$
$601$	−25557.9	−1.73466	−0.867328	+	0.497736i	$0.834165\pi$
$602$	−10012.5	−0.677869
$603$	0	0
$604$	−1712.24	−0.115348
$605$	14104.4	0.947814
$606$	0	0
$607$	309.636	0.0207047	0.0103523	−	0.999946i	$-0.496705\pi$
$607$	309.636	0.0207047	0.0103523	+	0.999946i	$0.496705\pi$
$608$	2755.32	0.183788
$609$	0	0
$610$	469.145	0.0311396
$611$	19998.7	1.32416
$612$	0	0
$613$	−27561.8	−1.81600	−0.908001	−	0.418967i	$-0.862392\pi$
$613$	−27561.8	−1.81600	−0.908001	+	0.418967i	$0.862392\pi$
$614$	−16133.7	−1.06043
$615$	0	0
$616$	208.949	0.0136669
$617$	−1350.58	−0.0881238	−0.0440619	−	0.999029i	$-0.514030\pi$
$617$	−1350.58	−0.0881238	−0.0440619	+	0.999029i	$0.514030\pi$
$618$	0	0
$619$	−1227.49	−0.0797043	−0.0398522	−	0.999206i	$-0.512689\pi$
$619$	−1227.49	−0.0797043	−0.0398522	+	0.999206i	$0.512689\pi$
$620$	831.831	0.0538824
$621$	0	0
$622$	12068.6	0.777983
$623$	−17819.2	−1.14592
$624$	0	0
$625$	−13883.6	−0.888549
$626$	2921.92	0.186555
$627$	0	0
$628$	−1264.66	−0.0803591
$629$	−1289.31	−0.0817301
$630$	0	0
$631$	3734.46	0.235605	0.117802	−	0.993037i	$-0.462415\pi$
$631$	3734.46	0.235605	0.117802	+	0.993037i	$0.462415\pi$
$632$	13952.6	0.878171
$633$	0	0
$634$	20697.0	1.29650
$635$	20762.0	1.29750
$636$	0	0
$637$	8615.36	0.535876
$638$	−106.210	−0.00659077
$639$	0	0
$640$	13597.4	0.839821
$641$	−6135.48	−0.378061	−0.189030	−	0.981971i	$-0.560534\pi$
$641$	−6135.48	−0.378061	−0.189030	+	0.981971i	$0.560534\pi$
$642$	0	0
$643$	−14048.8	−0.861631	−0.430816	−	0.902440i	$-0.641774\pi$
$643$	−14048.8	−0.861631	−0.430816	+	0.902440i	$0.641774\pi$
$644$	1447.70	0.0885830
$645$	0	0
$646$	18650.9	1.13593
$647$	16.2905	0.000989868 0	0.000494934	−	1.00000i	$-0.499842\pi$
$647$	16.2905	0.000989868 0	0.000494934		1.00000i	$0.499842\pi$
$648$	0	0
$649$	62.9434	0.00380701
$650$	−3045.56	−0.183780
$651$	0	0
$652$	127.892	0.00768198
$653$	21875.4	1.31095	0.655476	−	0.755216i	$-0.272468\pi$
$653$	21875.4	1.31095	0.655476	+	0.755216i	$0.272468\pi$
$654$	0	0
$655$	6678.05	0.398371
$656$	−5457.83	−0.324836
$657$	0	0
$658$	9699.46	0.574657
$659$	−8888.44	−0.525409	−0.262704	−	0.964876i	$-0.584614\pi$
$659$	−8888.44	−0.525409	−0.262704	+	0.964876i	$0.584614\pi$
$660$	0	0
$661$	−9008.22	−0.530075	−0.265037	−	0.964238i	$-0.585384\pi$
$661$	−9008.22	−0.530075	−0.265037	+	0.964238i	$0.585384\pi$
$662$	9860.66	0.578920
$663$	0	0
$664$	−23233.1	−1.35786
$665$	18580.7	1.08350
$666$	0	0
$667$	−11539.2	−0.669863
$668$	1008.83	0.0584321
$669$	0	0
$670$	−25341.3	−1.46122
$671$	9.26970	0.000533313 0
$672$	0	0
$673$	16056.0	0.919636	0.459818	−	0.888013i	$-0.347915\pi$
$673$	16056.0	0.919636	0.459818	+	0.888013i	$0.347915\pi$
$674$	22925.5	1.31018
$675$	0	0
$676$	−3038.97	−0.172904
$677$	−23028.7	−1.30733	−0.653667	−	0.756783i	$-0.726770\pi$
$677$	−23028.7	−1.30733	−0.653667	+	0.756783i	$0.726770\pi$
$678$	0	0
$679$	−1444.60	−0.0816476
$680$	−15092.2	−0.851118
$681$	0	0
$682$	−224.856	−0.0126249
$683$	16309.3	0.913704	0.456852	−	0.889543i	$-0.348977\pi$
$683$	16309.3	0.913704	0.456852	+	0.889543i	$0.348977\pi$
$684$	0	0
$685$	30720.7	1.71355
$686$	18846.0	1.04890
$687$	0	0
$688$	−13894.8	−0.769962
$689$	−7240.40	−0.400345
$690$	0	0
$691$	17578.2	0.967739	0.483870	−	0.875140i	$-0.339231\pi$
$691$	17578.2	0.967739	0.483870	+	0.875140i	$0.339231\pi$
$692$	−1183.09	−0.0649917
$693$	0	0
$694$	32720.4	1.78970
$695$	6949.91	0.379317
$696$	0	0
$697$	5612.81	0.305022
$698$	−28448.8	−1.54270
$699$	0	0
$700$	107.970	0.00582981
$701$	−60.1936	−0.00324320	−0.00162160	−	0.999999i	$-0.500516\pi$
$701$	−60.1936	−0.00324320	−0.00162160	+	0.999999i	$0.500516\pi$
$702$	0	0
$703$	2364.64	0.126862
$704$	309.912	0.0165913
$705$	0	0
$706$	−11180.6	−0.596018
$707$	8526.28	0.453556
$708$	0	0
$709$	8848.94	0.468729	0.234364	−	0.972149i	$-0.424699\pi$
$709$	8848.94	0.468729	0.234364	+	0.972149i	$0.424699\pi$
$710$	1543.40	0.0815813
$711$	0	0
$712$	−26545.2	−1.39722
$713$	−24429.3	−1.28315
$714$	0	0
$715$	534.405	0.0279519
$716$	1.35774	7.08677e−5 0
$717$	0	0
$718$	−11032.4	−0.573432
$719$	24968.4	1.29508	0.647542	−	0.762030i	$-0.275797\pi$
$719$	24968.4	1.29508	0.647542	+	0.762030i	$0.275797\pi$
$720$	0	0
$721$	−15236.1	−0.786991
$722$	−15478.6	−0.797859
$723$	0	0
$724$	212.442	0.0109052
$725$	−860.592	−0.0440849
$726$	0	0
$727$	−7652.10	−0.390372	−0.195186	−	0.980766i	$-0.562531\pi$
$727$	−7652.10	−0.390372	−0.195186	+	0.980766i	$0.562531\pi$
$728$	−32217.2	−1.64018
$729$	0	0
$730$	−608.891	−0.0308713
$731$	14289.3	0.722996
$732$	0	0
$733$	−17911.9	−0.902582	−0.451291	−	0.892377i	$-0.649036\pi$
$733$	−17911.9	−0.902582	−0.451291	+	0.892377i	$0.649036\pi$
$734$	3899.05	0.196072
$735$	0	0
$736$	4175.74	0.209130
$737$	−500.711	−0.0250257
$738$	0	0
$739$	−9922.49	−0.493917	−0.246959	−	0.969026i	$-0.579431\pi$
$739$	−9922.49	−0.493917	−0.246959	+	0.969026i	$0.579431\pi$
$740$	−122.025	−0.00606180
$741$	0	0
$742$	−3511.62	−0.173741
$743$	22365.7	1.10433	0.552165	−	0.833735i	$-0.313802\pi$
$743$	22365.7	1.10433	0.552165	+	0.833735i	$0.313802\pi$
$744$	0	0
$745$	−26348.0	−1.29573
$746$	−1698.89	−0.0833790
$747$	0	0
$748$	−19.0171	−0.000929590 0
$749$	−31863.0	−1.55440
$750$	0	0
$751$	−20642.9	−1.00302	−0.501511	−	0.865151i	$-0.667222\pi$
$751$	−20642.9	−1.00302	−0.501511	+	0.865151i	$0.667222\pi$
$752$	13460.4	0.652729
$753$	0	0
$754$	16376.3	0.790967
$755$	−33304.9	−1.60542
$756$	0	0
$757$	−2015.50	−0.0967696	−0.0483848	−	0.998829i	$-0.515407\pi$
$757$	−2015.50	−0.0967696	−0.0483848	+	0.998829i	$0.515407\pi$
$758$	455.991	0.0218500
$759$	0	0
$760$	27679.6	1.32111
$761$	7797.98	0.371454	0.185727	−	0.982601i	$-0.440536\pi$
$761$	7797.98	0.371454	0.185727	+	0.982601i	$0.440536\pi$
$762$	0	0
$763$	−28151.2	−1.33570
$764$	1515.39	0.0717604
$765$	0	0
$766$	−6149.05	−0.290044
$767$	−9705.06	−0.456883
$768$	0	0
$769$	6237.95	0.292518	0.146259	−	0.989246i	$-0.453277\pi$
$769$	6237.95	0.292518	0.146259	+	0.989246i	$0.453277\pi$
$770$	259.189	0.0121305
$771$	0	0
$772$	−1886.07	−0.0879288
$773$	−7860.91	−0.365766	−0.182883	−	0.983135i	$-0.558543\pi$
$773$	−7860.91	−0.365766	−0.182883	+	0.983135i	$0.558543\pi$
$774$	0	0
$775$	−1821.94	−0.0844465
$776$	−2152.02	−0.0995528
$777$	0	0
$778$	28941.3	1.33367
$779$	−10294.1	−0.473457
$780$	0	0
$781$	30.4956	0.00139720
$782$	28265.8	1.29256
$783$	0	0
$784$	5798.70	0.264154
$785$	−24599.1	−1.11844
$786$	0	0
$787$	26804.8	1.21409	0.607044	−	0.794668i	$-0.292355\pi$
$787$	26804.8	1.21409	0.607044	+	0.794668i	$0.292355\pi$
$788$	−2293.68	−0.103691
$789$	0	0
$790$	17307.4	0.779454
$791$	21549.6	0.968665
$792$	0	0
$793$	−1429.27	−0.0640035
$794$	24131.0	1.07856
$795$	0	0
$796$	−1505.96	−0.0670568
$797$	8245.63	0.366468	0.183234	−	0.983069i	$-0.441343\pi$
$797$	8245.63	0.366468	0.183234	+	0.983069i	$0.441343\pi$
$798$	0	0
$799$	−13842.7	−0.612914
$800$	311.427	0.0137633
$801$	0	0
$802$	11701.8	0.515217
$803$	−12.0309	−0.000528719 0
$804$	0	0
$805$	28159.4	1.23290
$806$	34669.9	1.51513
$807$	0	0
$808$	12701.6	0.553020
$809$	8731.00	0.379438	0.189719	−	0.981838i	$-0.439242\pi$
$809$	8731.00	0.379438	0.189719	+	0.981838i	$0.439242\pi$
$810$	0	0
$811$	−16931.3	−0.733094	−0.366547	−	0.930400i	$-0.619460\pi$
$811$	−16931.3	−0.733094	−0.366547	+	0.930400i	$0.619460\pi$
$812$	−580.563	−0.0250909
$813$	0	0
$814$	32.9852	0.00142031
$815$	2487.65	0.106918
$816$	0	0
$817$	−26207.1	−1.12224
$818$	−34655.0	−1.48127
$819$	0	0
$820$	531.216	0.0226230
$821$	12471.0	0.530137	0.265069	−	0.964230i	$-0.414605\pi$
$821$	12471.0	0.530137	0.265069	+	0.964230i	$0.414605\pi$
$822$	0	0
$823$	−18035.4	−0.763880	−0.381940	−	0.924187i	$-0.624744\pi$
$823$	−18035.4	−0.763880	−0.381940	+	0.924187i	$0.624744\pi$
$824$	−22697.1	−0.959576
$825$	0	0
$826$	−4707.00	−0.198278
$827$	6410.22	0.269535	0.134767	−	0.990877i	$-0.456971\pi$
$827$	6410.22	0.269535	0.134767	+	0.990877i	$0.456971\pi$
$828$	0	0
$829$	−45268.6	−1.89655	−0.948277	−	0.317445i	$-0.897175\pi$
$829$	−45268.6	−1.89655	−0.948277	+	0.317445i	$0.897175\pi$
$830$	−28819.2	−1.20522
$831$	0	0
$832$	−47784.4	−1.99114
$833$	−5963.36	−0.248041
$834$	0	0
$835$	19622.8	0.813262
$836$	34.8779	0.00144292
$837$	0	0
$838$	−42232.7	−1.74094
$839$	−3248.96	−0.133691	−0.0668455	−	0.997763i	$-0.521293\pi$
$839$	−3248.96	−0.133691	−0.0668455	+	0.997763i	$0.521293\pi$
$840$	0	0
$841$	−19761.5	−0.810263
$842$	−4560.93	−0.186674
$843$	0	0
$844$	−238.581	−0.00973020
$845$	−59111.3	−2.40650
$846$	0	0
$847$	−20840.5	−0.845440
$848$	−4873.26	−0.197345
$849$	0	0
$850$	2108.07	0.0850661
$851$	3583.65	0.144355
$852$	0	0
$853$	41764.2	1.67641	0.838206	−	0.545354i	$-0.183605\pi$
$853$	41764.2	1.67641	0.838206	+	0.545354i	$0.183605\pi$
$854$	−693.201	−0.0277762
$855$	0	0
$856$	−47466.2	−1.89528
$857$	−16691.6	−0.665314	−0.332657	−	0.943048i	$-0.607945\pi$
$857$	−16691.6	−0.665314	−0.332657	+	0.943048i	$0.607945\pi$
$858$	0	0
$859$	43498.2	1.72775	0.863876	−	0.503704i	$-0.168030\pi$
$859$	43498.2	1.72775	0.863876	+	0.503704i	$0.168030\pi$
$860$	1352.39	0.0536236
$861$	0	0
$862$	−9385.23	−0.370838
$863$	−6832.13	−0.269488	−0.134744	−	0.990880i	$-0.543021\pi$
$863$	−6832.13	−0.269488	−0.134744	+	0.990880i	$0.543021\pi$
$864$	0	0
$865$	−23012.4	−0.904559
$866$	−34869.9	−1.36828
$867$	0	0
$868$	−1229.10	−0.0480626
$869$	341.971	0.0133493
$870$	0	0
$871$	77203.1	3.00336
$872$	−41936.8	−1.62862
$873$	0	0
$874$	−51840.4	−2.00633
$875$	22850.7	0.882853
$876$	0	0
$877$	20263.0	0.780196	0.390098	−	0.920773i	$-0.372441\pi$
$877$	20263.0	0.780196	0.390098	+	0.920773i	$0.372441\pi$
$878$	10216.0	0.392682
$879$	0	0
$880$	359.689	0.0137786
$881$	25105.5	0.960074	0.480037	−	0.877248i	$-0.340623\pi$
$881$	25105.5	0.960074	0.480037	+	0.877248i	$0.340623\pi$
$882$	0	0
$883$	27661.5	1.05423	0.527115	−	0.849794i	$-0.323274\pi$
$883$	27661.5	1.05423	0.527115	+	0.849794i	$0.323274\pi$
$884$	2932.19	0.111561
$885$	0	0
$886$	−37697.1	−1.42941
$887$	−14968.7	−0.566627	−0.283313	−	0.959027i	$-0.591434\pi$
$887$	−14968.7	−0.566627	−0.283313	+	0.959027i	$0.591434\pi$
$888$	0	0
$889$	−30677.5	−1.15736
$890$	−32927.7	−1.24016
$891$	0	0
$892$	−2791.21	−0.104772
$893$	25387.9	0.951369
$894$	0	0
$895$	26.4096	0.000986341 0
$896$	−20091.3	−0.749111
$897$	0	0
$898$	−39930.0	−1.48383
$899$	9796.75	0.363448
$900$	0	0
$901$	5011.64	0.185307
$902$	−143.596	−0.00530068
$903$	0	0
$904$	32102.3	1.18109
$905$	4132.24	0.151779
$906$	0	0
$907$	−909.461	−0.0332945	−0.0166473	−	0.999861i	$-0.505299\pi$
$907$	−909.461	−0.0332945	−0.0166473	+	0.999861i	$0.505299\pi$
$908$	2165.12	0.0791320
$909$	0	0
$910$	−39963.5	−1.45580
$911$	−41510.1	−1.50965	−0.754825	−	0.655927i	$-0.772278\pi$
$911$	−41510.1	−1.50965	−0.754825	+	0.655927i	$0.772278\pi$
$912$	0	0
$913$	−569.431	−0.0206412
$914$	−30744.9	−1.11264
$915$	0	0
$916$	3277.53	0.118223
$917$	−9867.38	−0.355343
$918$	0	0
$919$	−22942.5	−0.823509	−0.411755	−	0.911295i	$-0.635084\pi$
$919$	−22942.5	−0.823509	−0.411755	+	0.911295i	$0.635084\pi$
$920$	41948.9	1.50328
$921$	0	0
$922$	22096.1	0.789260
$923$	−4702.02	−0.167680
$924$	0	0
$925$	267.269	0.00950027
$926$	−2955.32	−0.104879
$927$	0	0
$928$	−1674.57	−0.0592355
$929$	−45780.8	−1.61681	−0.808406	−	0.588626i	$-0.799669\pi$
$929$	−45780.8	−1.61681	−0.808406	+	0.588626i	$0.799669\pi$
$930$	0	0
$931$	10937.0	0.385011
$932$	−842.927	−0.0296255
$933$	0	0
$934$	−23569.4	−0.825710
$935$	−369.903	−0.0129381
$936$	0	0
$937$	1193.11	0.0415980	0.0207990	−	0.999784i	$-0.493379\pi$
$937$	1193.11	0.0415980	0.0207990	+	0.999784i	$0.493379\pi$
$938$	37443.8	1.30339
$939$	0	0
$940$	−1310.12	−0.0454589
$941$	−10760.2	−0.372767	−0.186383	−	0.982477i	$-0.559677\pi$
$941$	−10760.2	−0.372767	−0.186383	+	0.982477i	$0.559677\pi$
$942$	0	0
$943$	−15600.9	−0.538742
$944$	−6532.14	−0.225215
$945$	0	0
$946$	−365.572	−0.0125642
$947$	−8644.46	−0.296628	−0.148314	−	0.988940i	$-0.547385\pi$
$947$	−8644.46	−0.296628	−0.148314	+	0.988940i	$0.547385\pi$
$948$	0	0
$949$	1855.01	0.0634522
$950$	−3866.26	−0.132040
$951$	0	0
$952$	22300.0	0.759188
$953$	27435.4	0.932548	0.466274	−	0.884640i	$-0.345596\pi$
$953$	27435.4	0.932548	0.466274	+	0.884640i	$0.345596\pi$
$954$	0	0
$955$	29476.0	0.998767
$956$	−130.238	−0.00440607
$957$	0	0
$958$	−43975.3	−1.48307
$959$	−45392.4	−1.52847
$960$	0	0
$961$	−9050.50	−0.303800
$962$	−5085.88	−0.170453
$963$	0	0
$964$	−2028.42	−0.0677707
$965$	−36686.0	−1.22380
$966$	0	0
$967$	4087.45	0.135929	0.0679647	−	0.997688i	$-0.478349\pi$
$967$	4087.45	0.135929	0.0679647	+	0.997688i	$0.478349\pi$
$968$	−31046.0	−1.03084
$969$	0	0
$970$	−2669.45	−0.0883618
$971$	27045.3	0.893848	0.446924	−	0.894572i	$-0.352519\pi$
$971$	27045.3	0.893848	0.446924	+	0.894572i	$0.352519\pi$
$972$	0	0
$973$	−10269.1	−0.338346
$974$	−32302.4	−1.06266
$975$	0	0
$976$	−961.991	−0.0315498
$977$	−9587.15	−0.313941	−0.156970	−	0.987603i	$-0.550173\pi$
$977$	−9587.15	−0.313941	−0.156970	+	0.987603i	$0.550173\pi$
$978$	0	0
$979$	−650.609	−0.0212396
$980$	−564.393	−0.0183968
$981$	0	0
$982$	−34829.2	−1.13182
$983$	30674.2	0.995275	0.497638	−	0.867385i	$-0.334201\pi$
$983$	30674.2	0.995275	0.497638	+	0.867385i	$0.334201\pi$
$984$	0	0
$985$	−44614.5	−1.44318
$986$	−11335.3	−0.366115
$987$	0	0
$988$	−5377.72	−0.173166
$989$	−39717.3	−1.27698
$990$	0	0
$991$	51780.0	1.65978	0.829892	−	0.557924i	$-0.188402\pi$
$991$	51780.0	1.65978	0.829892	+	0.557924i	$0.188402\pi$
$992$	−3545.20	−0.113468
$993$	0	0
$994$	−2280.50	−0.0727696
$995$	−29292.5	−0.933301
$996$	0	0
$997$	2886.70	0.0916978	0.0458489	−	0.998948i	$-0.485401\pi$
$997$	2886.70	0.0916978	0.0458489	+	0.998948i	$0.485401\pi$
$998$	12865.8	0.408076
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.f.1.12		37
3.2	odd	2		239.4.a.b.1.26	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.b.1.26	✓	37	3.2	odd	2
2151.4.a.f.1.12		37	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.73040 q^{2} -0.544930 q^{4} -10.5995 q^{5} +15.6616 q^{7} +23.3311 q^{8} +O(q^{10})\)	\(q-2.73040 q^{2} -0.544930 q^{4} -10.5995 q^{5} +15.6616 q^{7} +23.3311 q^{8} +28.9408 q^{10} +0.571833 q^{11} -88.1692 q^{13} -42.7624 q^{14} -59.3436 q^{16} +61.0287 q^{17} -111.929 q^{19} +5.77597 q^{20} -1.56133 q^{22} -169.630 q^{23} -12.6510 q^{25} +240.737 q^{26} -8.53448 q^{28} +68.0256 q^{29} +144.016 q^{31} -24.6168 q^{32} -166.633 q^{34} -166.005 q^{35} -21.1263 q^{37} +305.609 q^{38} -247.297 q^{40} +91.9700 q^{41} +234.141 q^{43} -0.311609 q^{44} +463.157 q^{46} -226.822 q^{47} -97.7140 q^{49} +34.5422 q^{50} +48.0460 q^{52} +82.1194 q^{53} -6.06113 q^{55} +365.402 q^{56} -185.737 q^{58} +110.073 q^{59} +16.2105 q^{61} -393.220 q^{62} +541.963 q^{64} +934.548 q^{65} -875.625 q^{67} -33.2564 q^{68} +453.259 q^{70} +53.3295 q^{71} -21.0392 q^{73} +57.6832 q^{74} +60.9932 q^{76} +8.95582 q^{77} +598.027 q^{79} +629.012 q^{80} -251.115 q^{82} -995.800 q^{83} -646.873 q^{85} -639.299 q^{86} +13.3415 q^{88} -1137.76 q^{89} -1380.87 q^{91} +92.4363 q^{92} +619.315 q^{94} +1186.38 q^{95} -92.2384 q^{97} +266.798 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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