LMFDB - Embedded newform 2394.4.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,4,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, 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("2394.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2394.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:	$141.250572554$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 429x^{5} + 1799x^{4} + 59687x^{3} - 308117x^{2} - 2682459x + 15997617 $ x^7 - 3x^6 - 429x^5 + 1799x^4 + 59687x^3 - 308117x^2 - 2682459x + 15997617
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-12.0547$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -15.0547 q^{5} -7.00000 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} -15.0547 q^{5} -7.00000 q^{7} -8.00000 q^{8} +30.1093 q^{10} +9.90092 q^{11} -30.5297 q^{13} +14.0000 q^{14} +16.0000 q^{16} +12.8034 q^{17} +19.0000 q^{19} -60.2187 q^{20} -19.8018 q^{22} -141.562 q^{23} +101.643 q^{25} +61.0593 q^{26} -28.0000 q^{28} +268.525 q^{29} -95.9998 q^{31} -32.0000 q^{32} -25.6068 q^{34} +105.383 q^{35} +242.366 q^{37} -38.0000 q^{38} +120.437 q^{40} -389.506 q^{41} +438.018 q^{43} +39.6037 q^{44} +283.124 q^{46} -44.8542 q^{47} +49.0000 q^{49} -203.286 q^{50} -122.119 q^{52} -723.290 q^{53} -149.055 q^{55} +56.0000 q^{56} -537.049 q^{58} +277.805 q^{59} +701.898 q^{61} +192.000 q^{62} +64.0000 q^{64} +459.614 q^{65} +204.246 q^{67} +51.2135 q^{68} -210.765 q^{70} -156.351 q^{71} +670.450 q^{73} -484.732 q^{74} +76.0000 q^{76} -69.3064 q^{77} -668.727 q^{79} -240.875 q^{80} +779.011 q^{82} +900.208 q^{83} -192.751 q^{85} -876.035 q^{86} -79.2073 q^{88} +899.341 q^{89} +213.708 q^{91} -566.248 q^{92} +89.7083 q^{94} -286.039 q^{95} +1622.88 q^{97} -98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 14 q^{2} + 28 q^{4} - 18 q^{5} - 49 q^{7} - 56 q^{8}+O(q^{10}) $ 7 * q - 14 * q^2 + 28 * q^4 - 18 * q^5 - 49 * q^7 - 56 * q^8	$ 7 q - 14 q^{2} + 28 q^{4} - 18 q^{5} - 49 q^{7} - 56 q^{8} + 36 q^{10} - 28 q^{11} + 44 q^{13} + 98 q^{14} + 112 q^{16} - 26 q^{17} + 133 q^{19} - 72 q^{20} + 56 q^{22} - 56 q^{23} + 37 q^{25} - 88 q^{26} - 196 q^{28} - 270 q^{29} + 64 q^{31} - 224 q^{32} + 52 q^{34} + 126 q^{35} + 458 q^{37} - 266 q^{38} + 144 q^{40} - 110 q^{41} + 296 q^{43} - 112 q^{44} + 112 q^{46} + 142 q^{47} + 343 q^{49} - 74 q^{50} + 176 q^{52} - 330 q^{53} + 596 q^{55} + 392 q^{56} + 540 q^{58} - 236 q^{59} + 882 q^{61} - 128 q^{62} + 448 q^{64} + 180 q^{65} + 1622 q^{67} - 104 q^{68} - 252 q^{70} - 820 q^{71} + 1130 q^{73} - 916 q^{74} + 532 q^{76} + 196 q^{77} + 1694 q^{79} - 288 q^{80} + 220 q^{82} - 890 q^{83} + 1540 q^{85} - 592 q^{86} + 224 q^{88} - 686 q^{89} - 308 q^{91} - 224 q^{92} - 284 q^{94} - 342 q^{95} + 1772 q^{97} - 686 q^{98}+O(q^{100}) $ 7 * q - 14 * q^2 + 28 * q^4 - 18 * q^5 - 49 * q^7 - 56 * q^8 + 36 * q^10 - 28 * q^11 + 44 * q^13 + 98 * q^14 + 112 * q^16 - 26 * q^17 + 133 * q^19 - 72 * q^20 + 56 * q^22 - 56 * q^23 + 37 * q^25 - 88 * q^26 - 196 * q^28 - 270 * q^29 + 64 * q^31 - 224 * q^32 + 52 * q^34 + 126 * q^35 + 458 * q^37 - 266 * q^38 + 144 * q^40 - 110 * q^41 + 296 * q^43 - 112 * q^44 + 112 * q^46 + 142 * q^47 + 343 * q^49 - 74 * q^50 + 176 * q^52 - 330 * q^53 + 596 * q^55 + 392 * q^56 + 540 * q^58 - 236 * q^59 + 882 * q^61 - 128 * q^62 + 448 * q^64 + 180 * q^65 + 1622 * q^67 - 104 * q^68 - 252 * q^70 - 820 * q^71 + 1130 * q^73 - 916 * q^74 + 532 * q^76 + 196 * q^77 + 1694 * q^79 - 288 * q^80 + 220 * q^82 - 890 * q^83 + 1540 * q^85 - 592 * q^86 + 224 * q^88 - 686 * q^89 - 308 * q^91 - 224 * q^92 - 284 * q^94 - 342 * q^95 + 1772 * q^97 - 686 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−15.0547	−1.34653	−0.673265	−	0.739401i	$-0.735109\pi$
$5$	−15.0547	−1.34653	−0.673265	+	0.739401i	$0.735109\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	30.1093	0.952141
$11$	9.90092	0.271385	0.135693	−	0.990751i	$-0.456674\pi$
$11$	9.90092	0.271385	0.135693	+	0.990751i	$0.456674\pi$
$12$	0	0
$13$	−30.5297	−0.651339	−0.325670	−	0.945484i	$-0.605590\pi$
$13$	−30.5297	−0.651339	−0.325670	+	0.945484i	$0.605590\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	12.8034	0.182663	0.0913317	−	0.995821i	$-0.470888\pi$
$17$	12.8034	0.182663	0.0913317	+	0.995821i	$0.470888\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−60.2187	−0.673265
$21$	0	0
$22$	−19.8018	−0.191898
$23$	−141.562	−1.28338	−0.641690	−	0.766964i	$-0.721766\pi$
$23$	−141.562	−1.28338	−0.641690	+	0.766964i	$0.721766\pi$
$24$	0	0
$25$	101.643	0.813145
$26$	61.0593	0.460566
$27$	0	0
$28$	−28.0000	−0.188982
$29$	268.525	1.71944	0.859720	−	0.510765i	$-0.170638\pi$
$29$	268.525	1.71944	0.859720	+	0.510765i	$0.170638\pi$
$30$	0	0
$31$	−95.9998	−0.556196	−0.278098	−	0.960553i	$-0.589704\pi$
$31$	−95.9998	−0.556196	−0.278098	+	0.960553i	$0.589704\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−25.6068	−0.129163
$35$	105.383	0.508941
$36$	0	0
$37$	242.366	1.07688	0.538442	−	0.842663i	$-0.319013\pi$
$37$	242.366	1.07688	0.538442	+	0.842663i	$0.319013\pi$
$38$	−38.0000	−0.162221
$39$	0	0
$40$	120.437	0.476070
$41$	−389.506	−1.48367	−0.741836	−	0.670581i	$-0.766045\pi$
$41$	−389.506	−1.48367	−0.741836	+	0.670581i	$0.766045\pi$
$42$	0	0
$43$	438.018	1.55342	0.776710	−	0.629858i	$-0.216887\pi$
$43$	438.018	1.55342	0.776710	+	0.629858i	$0.216887\pi$
$44$	39.6037	0.135693
$45$	0	0
$46$	283.124	0.907487
$47$	−44.8542	−0.139205	−0.0696027	−	0.997575i	$-0.522173\pi$
$47$	−44.8542	−0.139205	−0.0696027	+	0.997575i	$0.522173\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−203.286	−0.574980
$51$	0	0
$52$	−122.119	−0.325670
$53$	−723.290	−1.87456	−0.937279	−	0.348581i	$-0.886664\pi$
$53$	−723.290	−1.87456	−0.937279	+	0.348581i	$0.886664\pi$
$54$	0	0
$55$	−149.055	−0.365429
$56$	56.0000	0.133631
$57$	0	0
$58$	−537.049	−1.21583
$59$	277.805	0.613002	0.306501	−	0.951870i	$-0.400842\pi$
$59$	277.805	0.613002	0.306501	+	0.951870i	$0.400842\pi$
$60$	0	0
$61$	701.898	1.47326	0.736630	−	0.676296i	$-0.236416\pi$
$61$	701.898	1.47326	0.736630	+	0.676296i	$0.236416\pi$
$62$	192.000	0.393290
$63$	0	0
$64$	64.0000	0.125000
$65$	459.614	0.877048
$66$	0	0
$67$	204.246	0.372428	0.186214	−	0.982509i	$-0.440378\pi$
$67$	204.246	0.372428	0.186214	+	0.982509i	$0.440378\pi$
$68$	51.2135	0.0913317
$69$	0	0
$70$	−210.765	−0.359875
$71$	−156.351	−0.261344	−0.130672	−	0.991426i	$-0.541713\pi$
$71$	−156.351	−0.261344	−0.130672	+	0.991426i	$0.541713\pi$
$72$	0	0
$73$	670.450	1.07493	0.537467	−	0.843285i	$-0.319381\pi$
$73$	670.450	1.07493	0.537467	+	0.843285i	$0.319381\pi$
$74$	−484.732	−0.761472
$75$	0	0
$76$	76.0000	0.114708
$77$	−69.3064	−0.102574
$78$	0	0
$79$	−668.727	−0.952375	−0.476188	−	0.879344i	$-0.657982\pi$
$79$	−668.727	−0.952375	−0.476188	+	0.879344i	$0.657982\pi$
$80$	−240.875	−0.336633
$81$	0	0
$82$	779.011	1.04911
$83$	900.208	1.19049	0.595245	−	0.803545i	$-0.297055\pi$
$83$	900.208	1.19049	0.595245	+	0.803545i	$0.297055\pi$
$84$	0	0
$85$	−192.751	−0.245962
$86$	−876.035	−1.09843
$87$	0	0
$88$	−79.2073	−0.0959492
$89$	899.341	1.07112	0.535561	−	0.844496i	$-0.320100\pi$
$89$	899.341	1.07112	0.535561	+	0.844496i	$0.320100\pi$
$90$	0	0
$91$	213.708	0.246183
$92$	−566.248	−0.641690
$93$	0	0
$94$	89.7083	0.0984331
$95$	−286.039	−0.308915
$96$	0	0
$97$	1622.88	1.69875	0.849376	−	0.527788i	$-0.176978\pi$
$97$	1622.88	1.69875	0.849376	+	0.527788i	$0.176978\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	406.572	0.406572
$101$	74.5738	0.0734691	0.0367345	−	0.999325i	$-0.488304\pi$
$101$	74.5738	0.0734691	0.0367345	+	0.999325i	$0.488304\pi$
$102$	0	0
$103$	1113.19	1.06491	0.532455	−	0.846458i	$-0.321269\pi$
$103$	1113.19	1.06491	0.532455	+	0.846458i	$0.321269\pi$
$104$	244.237	0.230283
$105$	0	0
$106$	1446.58	1.32551
$107$	−1739.63	−1.57175	−0.785873	−	0.618388i	$-0.787786\pi$
$107$	−1739.63	−1.57175	−0.785873	+	0.618388i	$0.787786\pi$
$108$	0	0
$109$	403.123	0.354240	0.177120	−	0.984189i	$-0.443322\pi$
$109$	403.123	0.354240	0.177120	+	0.984189i	$0.443322\pi$
$110$	298.110	0.258397
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−458.493	−0.381694	−0.190847	−	0.981620i	$-0.561123\pi$
$113$	−458.493	−0.381694	−0.190847	+	0.981620i	$0.561123\pi$
$114$	0	0
$115$	2131.17	1.72811
$116$	1074.10	0.859720
$117$	0	0
$118$	−555.610	−0.433458
$119$	−89.6237	−0.0690403
$120$	0	0
$121$	−1232.97	−0.926350
$122$	−1403.80	−1.04175
$123$	0	0
$124$	−383.999	−0.278098
$125$	351.631	0.251606
$126$	0	0
$127$	2091.64	1.46144	0.730722	−	0.682675i	$-0.239184\pi$
$127$	2091.64	1.46144	0.730722	+	0.682675i	$0.239184\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−919.228	−0.620167
$131$	−2402.38	−1.60226	−0.801132	−	0.598488i	$-0.795769\pi$
$131$	−2402.38	−1.60226	−0.801132	+	0.598488i	$0.795769\pi$
$132$	0	0
$133$	−133.000	−0.0867110
$134$	−408.492	−0.263346
$135$	0	0
$136$	−102.427	−0.0645813
$137$	−1575.11	−0.982270	−0.491135	−	0.871084i	$-0.663418\pi$
$137$	−1575.11	−0.982270	−0.491135	+	0.871084i	$0.663418\pi$
$138$	0	0
$139$	−1149.84	−0.701644	−0.350822	−	0.936442i	$-0.614098\pi$
$139$	−1149.84	−0.701644	−0.350822	+	0.936442i	$0.614098\pi$
$140$	421.531	0.254470
$141$	0	0
$142$	312.701	0.184798
$143$	−302.272	−0.176764
$144$	0	0
$145$	−4042.55	−2.31528
$146$	−1340.90	−0.760094
$147$	0	0
$148$	969.463	0.538442
$149$	1984.67	1.09121	0.545606	−	0.838042i	$-0.316299\pi$
$149$	1984.67	1.09121	0.545606	+	0.838042i	$0.316299\pi$
$150$	0	0
$151$	−935.914	−0.504395	−0.252197	−	0.967676i	$-0.581153\pi$
$151$	−935.914	−0.504395	−0.252197	+	0.967676i	$0.581153\pi$
$152$	−152.000	−0.0811107
$153$	0	0
$154$	138.613	0.0725308
$155$	1445.25	0.748935
$156$	0	0
$157$	1631.27	0.829234	0.414617	−	0.909996i	$-0.363916\pi$
$157$	1631.27	0.829234	0.414617	+	0.909996i	$0.363916\pi$
$158$	1337.45	0.673431
$159$	0	0
$160$	481.749	0.238035
$161$	990.935	0.485072
$162$	0	0
$163$	−3110.40	−1.49463	−0.747317	−	0.664468i	$-0.768658\pi$
$163$	−3110.40	−1.49463	−0.747317	+	0.664468i	$0.768658\pi$
$164$	−1558.02	−0.741836
$165$	0	0
$166$	−1800.42	−0.841803
$167$	1105.79	0.512388	0.256194	−	0.966625i	$-0.417531\pi$
$167$	1105.79	0.512388	0.256194	+	0.966625i	$0.417531\pi$
$168$	0	0
$169$	−1264.94	−0.575757
$170$	385.502	0.173921
$171$	0	0
$172$	1752.07	0.776710
$173$	2890.79	1.27042	0.635210	−	0.772340i	$-0.280914\pi$
$173$	2890.79	1.27042	0.635210	+	0.772340i	$0.280914\pi$
$174$	0	0
$175$	−711.502	−0.307340
$176$	158.415	0.0678463
$177$	0	0
$178$	−1798.68	−0.757398
$179$	−2506.92	−1.04679	−0.523397	−	0.852089i	$-0.675336\pi$
$179$	−2506.92	−1.04679	−0.523397	+	0.852089i	$0.675336\pi$
$180$	0	0
$181$	267.375	0.109800	0.0549002	−	0.998492i	$-0.482516\pi$
$181$	267.375	0.109800	0.0549002	+	0.998492i	$0.482516\pi$
$182$	−427.415	−0.174078
$183$	0	0
$184$	1132.50	0.453743
$185$	−3648.74	−1.45006
$186$	0	0
$187$	126.765	0.0495722
$188$	−179.417	−0.0696027
$189$	0	0
$190$	572.077	0.218436
$191$	1204.26	0.456215	0.228107	−	0.973636i	$-0.426746\pi$
$191$	1204.26	0.456215	0.228107	+	0.973636i	$0.426746\pi$
$192$	0	0
$193$	−1955.46	−0.729313	−0.364657	−	0.931142i	$-0.618814\pi$
$193$	−1955.46	−0.729313	−0.364657	+	0.931142i	$0.618814\pi$
$194$	−3245.77	−1.20120
$195$	0	0
$196$	196.000	0.0714286
$197$	2163.14	0.782322	0.391161	−	0.920322i	$-0.372074\pi$
$197$	2163.14	0.782322	0.391161	+	0.920322i	$0.372074\pi$
$198$	0	0
$199$	126.832	0.0451803	0.0225901	−	0.999745i	$-0.492809\pi$
$199$	126.832	0.0451803	0.0225901	+	0.999745i	$0.492809\pi$
$200$	−813.145	−0.287490
$201$	0	0
$202$	−149.148	−0.0519505
$203$	−1879.67	−0.649888
$204$	0	0
$205$	5863.88	1.99781
$206$	−2226.38	−0.753005
$207$	0	0
$208$	−488.475	−0.162835
$209$	188.117	0.0622601
$210$	0	0
$211$	−3416.34	−1.11465	−0.557323	−	0.830295i	$-0.688172\pi$
$211$	−3416.34	−1.11465	−0.557323	+	0.830295i	$0.688172\pi$
$212$	−2893.16	−0.937279
$213$	0	0
$214$	3479.27	1.11139
$215$	−6594.21	−2.09173
$216$	0	0
$217$	671.999	0.210222
$218$	−806.246	−0.250486
$219$	0	0
$220$	−596.220	−0.182714
$221$	−390.883	−0.118976
$222$	0	0
$223$	3016.35	0.905784	0.452892	−	0.891565i	$-0.350392\pi$
$223$	3016.35	0.905784	0.452892	+	0.891565i	$0.350392\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	916.986	0.269898
$227$	233.339	0.0682257	0.0341128	−	0.999418i	$-0.489139\pi$
$227$	233.339	0.0682257	0.0341128	+	0.999418i	$0.489139\pi$
$228$	0	0
$229$	845.701	0.244041	0.122021	−	0.992528i	$-0.461063\pi$
$229$	845.701	0.244041	0.122021	+	0.992528i	$0.461063\pi$
$230$	−4262.34	−1.22196
$231$	0	0
$232$	−2148.20	−0.607914
$233$	−6389.86	−1.79662	−0.898312	−	0.439358i	$-0.855206\pi$
$233$	−6389.86	−1.79662	−0.898312	+	0.439358i	$0.855206\pi$
$234$	0	0
$235$	675.265	0.187444
$236$	1111.22	0.306501
$237$	0	0
$238$	179.247	0.0488188
$239$	5987.14	1.62040	0.810200	−	0.586153i	$-0.199358\pi$
$239$	5987.14	1.62040	0.810200	+	0.586153i	$0.199358\pi$
$240$	0	0
$241$	−7167.78	−1.91584	−0.957919	−	0.287038i	$-0.907329\pi$
$241$	−7167.78	−1.91584	−0.957919	+	0.287038i	$0.907329\pi$
$242$	2465.94	0.655028
$243$	0	0
$244$	2807.59	0.736630
$245$	−737.679	−0.192362
$246$	0	0
$247$	−580.064	−0.149427
$248$	767.998	0.196645
$249$	0	0
$250$	−703.262	−0.177913
$251$	1755.53	0.441466	0.220733	−	0.975334i	$-0.429155\pi$
$251$	1755.53	0.441466	0.220733	+	0.975334i	$0.429155\pi$
$252$	0	0
$253$	−1401.59	−0.348291
$254$	−4183.29	−1.03340
$255$	0	0
$256$	256.000	0.0625000
$257$	3005.66	0.729526	0.364763	−	0.931100i	$-0.381150\pi$
$257$	3005.66	0.729526	0.364763	+	0.931100i	$0.381150\pi$
$258$	0	0
$259$	−1696.56	−0.407024
$260$	1838.46	0.438524
$261$	0	0
$262$	4804.75	1.13297
$263$	2033.24	0.476712	0.238356	−	0.971178i	$-0.423392\pi$
$263$	2033.24	0.476712	0.238356	+	0.971178i	$0.423392\pi$
$264$	0	0
$265$	10888.9	2.52415
$266$	266.000	0.0613139
$267$	0	0
$268$	816.985	0.186214
$269$	−7962.25	−1.80471	−0.902355	−	0.430994i	$-0.858163\pi$
$269$	−7962.25	−1.80471	−0.902355	+	0.430994i	$0.858163\pi$
$270$	0	0
$271$	−1917.17	−0.429741	−0.214871	−	0.976643i	$-0.568933\pi$
$271$	−1917.17	−0.429741	−0.214871	+	0.976643i	$0.568933\pi$
$272$	204.854	0.0456658
$273$	0	0
$274$	3150.23	0.694570
$275$	1006.36	0.220676
$276$	0	0
$277$	2450.13	0.531459	0.265730	−	0.964048i	$-0.414387\pi$
$277$	2450.13	0.531459	0.265730	+	0.964048i	$0.414387\pi$
$278$	2299.69	0.496137
$279$	0	0
$280$	−843.061	−0.179938
$281$	−6162.94	−1.30836	−0.654182	−	0.756337i	$-0.726987\pi$
$281$	−6162.94	−1.30836	−0.654182	+	0.756337i	$0.726987\pi$
$282$	0	0
$283$	−1181.18	−0.248106	−0.124053	−	0.992276i	$-0.539589\pi$
$283$	−1181.18	−0.248106	−0.124053	+	0.992276i	$0.539589\pi$
$284$	−625.402	−0.130672
$285$	0	0
$286$	604.544	0.124991
$287$	2726.54	0.560775
$288$	0	0
$289$	−4749.07	−0.966634
$290$	8085.10	1.63715
$291$	0	0
$292$	2681.80	0.537467
$293$	3150.00	0.628072	0.314036	−	0.949411i	$-0.398319\pi$
$293$	3150.00	0.628072	0.314036	+	0.949411i	$0.398319\pi$
$294$	0	0
$295$	−4182.26	−0.825426
$296$	−1938.93	−0.380736
$297$	0	0
$298$	−3969.34	−0.771604
$299$	4321.84	0.835915
$300$	0	0
$301$	−3066.12	−0.587138
$302$	1871.83	0.356661
$303$	0	0
$304$	304.000	0.0573539
$305$	−10566.8	−1.98379
$306$	0	0
$307$	6877.43	1.27855	0.639276	−	0.768977i	$-0.279234\pi$
$307$	6877.43	1.27855	0.639276	+	0.768977i	$0.279234\pi$
$308$	−277.226	−0.0512870
$309$	0	0
$310$	−2890.49	−0.529577
$311$	−3904.79	−0.711963	−0.355982	−	0.934493i	$-0.615853\pi$
$311$	−3904.79	−0.711963	−0.355982	+	0.934493i	$0.615853\pi$
$312$	0	0
$313$	−8584.23	−1.55019	−0.775095	−	0.631845i	$-0.782298\pi$
$313$	−8584.23	−1.55019	−0.775095	+	0.631845i	$0.782298\pi$
$314$	−3262.54	−0.586357
$315$	0	0
$316$	−2674.91	−0.476188
$317$	−6801.75	−1.20512	−0.602562	−	0.798072i	$-0.705853\pi$
$317$	−6801.75	−1.20512	−0.602562	+	0.798072i	$0.705853\pi$
$318$	0	0
$319$	2658.64	0.466631
$320$	−963.499	−0.168316
$321$	0	0
$322$	−1981.87	−0.342998
$323$	243.264	0.0419059
$324$	0	0
$325$	−3103.13	−0.529633
$326$	6220.80	1.05687
$327$	0	0
$328$	3116.04	0.524557
$329$	313.979	0.0526147
$330$	0	0
$331$	4066.22	0.675226	0.337613	−	0.941285i	$-0.390380\pi$
$331$	4066.22	0.675226	0.337613	+	0.941285i	$0.390380\pi$
$332$	3600.83	0.595245
$333$	0	0
$334$	−2211.59	−0.362313
$335$	−3074.86	−0.501485
$336$	0	0
$337$	−3483.66	−0.563107	−0.281553	−	0.959546i	$-0.590850\pi$
$337$	−3483.66	−0.563107	−0.281553	+	0.959546i	$0.590850\pi$
$338$	2529.88	0.407122
$339$	0	0
$340$	−771.003	−0.122981
$341$	−950.486	−0.150943
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−3504.14	−0.549217
$345$	0	0
$346$	−5781.58	−0.898322
$347$	791.136	0.122393	0.0611965	−	0.998126i	$-0.480508\pi$
$347$	791.136	0.122393	0.0611965	+	0.998126i	$0.480508\pi$
$348$	0	0
$349$	−9825.92	−1.50708	−0.753538	−	0.657404i	$-0.771654\pi$
$349$	−9825.92	−1.50708	−0.753538	+	0.657404i	$0.771654\pi$
$350$	1423.00	0.217322
$351$	0	0
$352$	−316.829	−0.0479746
$353$	−7309.55	−1.10212	−0.551060	−	0.834466i	$-0.685776\pi$
$353$	−7309.55	−1.10212	−0.551060	+	0.834466i	$0.685776\pi$
$354$	0	0
$355$	2353.81	0.351907
$356$	3597.36	0.535561
$357$	0	0
$358$	5013.85	0.740195
$359$	37.9335	0.00557676	0.00278838	−	0.999996i	$-0.499112\pi$
$359$	37.9335	0.00557676	0.00278838	+	0.999996i	$0.499112\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−534.751	−0.0776406
$363$	0	0
$364$	854.831	0.123092
$365$	−10093.4	−1.44743
$366$	0	0
$367$	9435.21	1.34200	0.671000	−	0.741457i	$-0.265865\pi$
$367$	9435.21	1.34200	0.671000	+	0.741457i	$0.265865\pi$
$368$	−2264.99	−0.320845
$369$	0	0
$370$	7297.48	1.02535
$371$	5063.03	0.708516
$372$	0	0
$373$	11319.7	1.57134	0.785672	−	0.618643i	$-0.212317\pi$
$373$	11319.7	1.57134	0.785672	+	0.618643i	$0.212317\pi$
$374$	−253.531	−0.0350528
$375$	0	0
$376$	358.833	0.0492165
$377$	−8197.97	−1.11994
$378$	0	0
$379$	1605.30	0.217569	0.108784	−	0.994065i	$-0.465304\pi$
$379$	1605.30	0.217569	0.108784	+	0.994065i	$0.465304\pi$
$380$	−1144.15	−0.154458
$381$	0	0
$382$	−2408.51	−0.322592
$383$	−8924.55	−1.19066	−0.595331	−	0.803481i	$-0.702979\pi$
$383$	−8924.55	−1.19066	−0.595331	+	0.803481i	$0.702979\pi$
$384$	0	0
$385$	1043.39	0.138119
$386$	3910.93	0.515702
$387$	0	0
$388$	6491.54	0.849376
$389$	10764.8	1.40308	0.701540	−	0.712630i	$-0.252496\pi$
$389$	10764.8	1.40308	0.701540	+	0.712630i	$0.252496\pi$
$390$	0	0
$391$	−1812.47	−0.234427
$392$	−392.000	−0.0505076
$393$	0	0
$394$	−4326.28	−0.553185
$395$	10067.5	1.28240
$396$	0	0
$397$	6179.17	0.781168	0.390584	−	0.920567i	$-0.372273\pi$
$397$	6179.17	0.781168	0.390584	+	0.920567i	$0.372273\pi$
$398$	−253.664	−0.0319473
$399$	0	0
$400$	1626.29	0.203286
$401$	−8784.76	−1.09399	−0.546995	−	0.837136i	$-0.684228\pi$
$401$	−8784.76	−1.09399	−0.546995	+	0.837136i	$0.684228\pi$
$402$	0	0
$403$	2930.84	0.362272
$404$	298.295	0.0367345
$405$	0	0
$406$	3759.35	0.459540
$407$	2399.64	0.292251
$408$	0	0
$409$	2087.32	0.252350	0.126175	−	0.992008i	$-0.459730\pi$
$409$	2087.32	0.252350	0.126175	+	0.992008i	$0.459730\pi$
$410$	−11727.8	−1.41266
$411$	0	0
$412$	4452.76	0.532455
$413$	−1944.64	−0.231693
$414$	0	0
$415$	−13552.3	−1.60303
$416$	976.949	0.115142
$417$	0	0
$418$	−376.235	−0.0440245
$419$	14055.3	1.63877	0.819387	−	0.573240i	$-0.194314\pi$
$419$	14055.3	1.63877	0.819387	+	0.573240i	$0.194314\pi$
$420$	0	0
$421$	3798.37	0.439718	0.219859	−	0.975532i	$-0.429440\pi$
$421$	3798.37	0.439718	0.219859	+	0.975532i	$0.429440\pi$
$422$	6832.68	0.788174
$423$	0	0
$424$	5786.32	0.662756
$425$	1301.38	0.148532
$426$	0	0
$427$	−4913.29	−0.556840
$428$	−6958.54	−0.785873
$429$	0	0
$430$	13188.4	1.47907
$431$	−6307.50	−0.704923	−0.352461	−	0.935826i	$-0.614655\pi$
$431$	−6307.50	−0.704923	−0.352461	+	0.935826i	$0.614655\pi$
$432$	0	0
$433$	8958.28	0.994244	0.497122	−	0.867681i	$-0.334390\pi$
$433$	8958.28	0.994244	0.497122	+	0.867681i	$0.334390\pi$
$434$	−1344.00	−0.148650
$435$	0	0
$436$	1612.49	0.177120
$437$	−2689.68	−0.294428
$438$	0	0
$439$	−11241.6	−1.22217	−0.611086	−	0.791564i	$-0.709267\pi$
$439$	−11241.6	−1.22217	−0.611086	+	0.791564i	$0.709267\pi$
$440$	1192.44	0.129199
$441$	0	0
$442$	781.766	0.0841286
$443$	490.016	0.0525538	0.0262769	−	0.999655i	$-0.491635\pi$
$443$	490.016	0.0525538	0.0262769	+	0.999655i	$0.491635\pi$
$444$	0	0
$445$	−13539.3	−1.44230
$446$	−6032.71	−0.640486
$447$	0	0
$448$	−448.000	−0.0472456
$449$	−16628.6	−1.74778	−0.873889	−	0.486125i	$-0.838410\pi$
$449$	−16628.6	−1.74778	−0.873889	+	0.486125i	$0.838410\pi$
$450$	0	0
$451$	−3856.46	−0.402647
$452$	−1833.97	−0.190847
$453$	0	0
$454$	−466.677	−0.0482428
$455$	−3217.30	−0.331493
$456$	0	0
$457$	−6329.54	−0.647885	−0.323942	−	0.946077i	$-0.605008\pi$
$457$	−6329.54	−0.647885	−0.323942	+	0.946077i	$0.605008\pi$
$458$	−1691.40	−0.172563
$459$	0	0
$460$	8524.68	0.864055
$461$	−14799.4	−1.49518	−0.747588	−	0.664163i	$-0.768788\pi$
$461$	−14799.4	−1.49518	−0.747588	+	0.664163i	$0.768788\pi$
$462$	0	0
$463$	−18693.9	−1.87642	−0.938208	−	0.346072i	$-0.887515\pi$
$463$	−18693.9	−1.87642	−0.938208	+	0.346072i	$0.887515\pi$
$464$	4296.40	0.429860
$465$	0	0
$466$	12779.7	1.27041
$467$	−1433.55	−0.142049	−0.0710243	−	0.997475i	$-0.522627\pi$
$467$	−1433.55	−0.142049	−0.0710243	+	0.997475i	$0.522627\pi$
$468$	0	0
$469$	−1429.72	−0.140764
$470$	−1350.53	−0.132543
$471$	0	0
$472$	−2222.44	−0.216729
$473$	4336.78	0.421575
$474$	0	0
$475$	1931.22	0.186548
$476$	−358.495	−0.0345201
$477$	0	0
$478$	−11974.3	−1.14580
$479$	−14983.4	−1.42925	−0.714623	−	0.699510i	$-0.753402\pi$
$479$	−14983.4	−1.42925	−0.714623	+	0.699510i	$0.753402\pi$
$480$	0	0
$481$	−7399.35	−0.701416
$482$	14335.6	1.35470
$483$	0	0
$484$	−4931.89	−0.463175
$485$	−24432.0	−2.28742
$486$	0	0
$487$	10768.6	1.00200	0.501000	−	0.865447i	$-0.332966\pi$
$487$	10768.6	1.00200	0.501000	+	0.865447i	$0.332966\pi$
$488$	−5615.19	−0.520876
$489$	0	0
$490$	1475.36	0.136020
$491$	8619.80	0.792273	0.396137	−	0.918192i	$-0.370351\pi$
$491$	8619.80	0.792273	0.396137	+	0.918192i	$0.370351\pi$
$492$	0	0
$493$	3438.03	0.314079
$494$	1160.13	0.105661
$495$	0	0
$496$	−1536.00	−0.139049
$497$	1094.45	0.0987786
$498$	0	0
$499$	−8295.25	−0.744181	−0.372091	−	0.928196i	$-0.621359\pi$
$499$	−8295.25	−0.744181	−0.372091	+	0.928196i	$0.621359\pi$
$500$	1406.52	0.125803
$501$	0	0
$502$	−3511.06	−0.312164
$503$	−8540.67	−0.757077	−0.378539	−	0.925586i	$-0.623573\pi$
$503$	−8540.67	−0.757077	−0.378539	+	0.925586i	$0.623573\pi$
$504$	0	0
$505$	−1122.68	−0.0989283
$506$	2803.19	0.246279
$507$	0	0
$508$	8366.57	0.730722
$509$	6562.54	0.571472	0.285736	−	0.958308i	$-0.407762\pi$
$509$	6562.54	0.571472	0.285736	+	0.958308i	$0.407762\pi$
$510$	0	0
$511$	−4693.15	−0.406287
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−6011.33	−0.515853
$515$	−16758.7	−1.43393
$516$	0	0
$517$	−444.097	−0.0377783
$518$	3393.12	0.287809
$519$	0	0
$520$	−3676.91	−0.310083
$521$	1723.89	0.144962	0.0724808	−	0.997370i	$-0.476908\pi$
$521$	1723.89	0.144962	0.0724808	+	0.997370i	$0.476908\pi$
$522$	0	0
$523$	−16020.5	−1.33944	−0.669721	−	0.742613i	$-0.733586\pi$
$523$	−16020.5	−1.33944	−0.669721	+	0.742613i	$0.733586\pi$
$524$	−9609.51	−0.801132
$525$	0	0
$526$	−4066.49	−0.337086
$527$	−1229.12	−0.101597
$528$	0	0
$529$	7872.83	0.647064
$530$	−21777.8	−1.78484
$531$	0	0
$532$	−532.000	−0.0433555
$533$	11891.5	0.966373
$534$	0	0
$535$	26189.6	2.11640
$536$	−1633.97	−0.131673
$537$	0	0
$538$	15924.5	1.27612
$539$	485.145	0.0387693
$540$	0	0
$541$	−8646.43	−0.687133	−0.343567	−	0.939128i	$-0.611635\pi$
$541$	−8646.43	−0.687133	−0.343567	+	0.939128i	$0.611635\pi$
$542$	3834.34	0.303873
$543$	0	0
$544$	−409.708	−0.0322906
$545$	−6068.89	−0.476996
$546$	0	0
$547$	−6705.84	−0.524170	−0.262085	−	0.965045i	$-0.584410\pi$
$547$	−6705.84	−0.524170	−0.262085	+	0.965045i	$0.584410\pi$
$548$	−6300.45	−0.491135
$549$	0	0
$550$	−2012.72	−0.156041
$551$	5101.97	0.394467
$552$	0	0
$553$	4681.09	0.359964
$554$	−4900.26	−0.375798
$555$	0	0
$556$	−4599.38	−0.350822
$557$	−21140.5	−1.60818	−0.804088	−	0.594511i	$-0.797346\pi$
$557$	−21140.5	−1.60818	−0.804088	+	0.594511i	$0.797346\pi$
$558$	0	0
$559$	−13372.5	−1.01180
$560$	1686.12	0.127235
$561$	0	0
$562$	12325.9	0.925153
$563$	−942.646	−0.0705645	−0.0352822	−	0.999377i	$-0.511233\pi$
$563$	−942.646	−0.0705645	−0.0352822	+	0.999377i	$0.511233\pi$
$564$	0	0
$565$	6902.46	0.513962
$566$	2362.37	0.175438
$567$	0	0
$568$	1250.80	0.0923990
$569$	−5299.56	−0.390455	−0.195228	−	0.980758i	$-0.562545\pi$
$569$	−5299.56	−0.390455	−0.195228	+	0.980758i	$0.562545\pi$
$570$	0	0
$571$	−26052.6	−1.90940	−0.954701	−	0.297565i	$-0.903825\pi$
$571$	−26052.6	−1.90940	−0.954701	+	0.297565i	$0.903825\pi$
$572$	−1209.09	−0.0883819
$573$	0	0
$574$	−5453.08	−0.396528
$575$	−14388.8	−1.04357
$576$	0	0
$577$	9190.95	0.663127	0.331563	−	0.943433i	$-0.392424\pi$
$577$	9190.95	0.663127	0.331563	+	0.943433i	$0.392424\pi$
$578$	9498.15	0.683514
$579$	0	0
$580$	−16170.2	−1.15764
$581$	−6301.45	−0.449963
$582$	0	0
$583$	−7161.24	−0.508728
$584$	−5363.60	−0.380047
$585$	0	0
$586$	−6300.00	−0.444114
$587$	−17941.7	−1.26156	−0.630778	−	0.775963i	$-0.717264\pi$
$587$	−17941.7	−1.26156	−0.630778	+	0.775963i	$0.717264\pi$
$588$	0	0
$589$	−1824.00	−0.127600
$590$	8364.53	0.583665
$591$	0	0
$592$	3877.85	0.269221
$593$	−2707.89	−0.187521	−0.0937604	−	0.995595i	$-0.529889\pi$
$593$	−2707.89	−0.187521	−0.0937604	+	0.995595i	$0.529889\pi$
$594$	0	0
$595$	1349.26	0.0929648
$596$	7938.69	0.545606
$597$	0	0
$598$	−8643.69	−0.591081
$599$	−3980.28	−0.271502	−0.135751	−	0.990743i	$-0.543345\pi$
$599$	−3980.28	−0.271502	−0.135751	+	0.990743i	$0.543345\pi$
$600$	0	0
$601$	−11648.2	−0.790581	−0.395290	−	0.918556i	$-0.629356\pi$
$601$	−11648.2	−0.790581	−0.395290	+	0.918556i	$0.629356\pi$
$602$	6132.25	0.415169
$603$	0	0
$604$	−3743.66	−0.252197
$605$	18562.0	1.24736
$606$	0	0
$607$	−29531.7	−1.97472	−0.987360	−	0.158491i	$-0.949337\pi$
$607$	−29531.7	−1.97472	−0.987360	+	0.158491i	$0.949337\pi$
$608$	−608.000	−0.0405554
$609$	0	0
$610$	21133.7	1.40275
$611$	1369.38	0.0906699
$612$	0	0
$613$	8435.11	0.555777	0.277888	−	0.960613i	$-0.410365\pi$
$613$	8435.11	0.555777	0.277888	+	0.960613i	$0.410365\pi$
$614$	−13754.9	−0.904073
$615$	0	0
$616$	554.451	0.0362654
$617$	16710.5	1.09034	0.545170	−	0.838326i	$-0.316465\pi$
$617$	16710.5	1.09034	0.545170	+	0.838326i	$0.316465\pi$
$618$	0	0
$619$	−6918.55	−0.449241	−0.224620	−	0.974446i	$-0.572114\pi$
$619$	−6918.55	−0.449241	−0.224620	+	0.974446i	$0.572114\pi$
$620$	5780.98	0.374467
$621$	0	0
$622$	7809.59	0.503434
$623$	−6295.38	−0.404846
$624$	0	0
$625$	−17999.1	−1.15194
$626$	17168.5	1.09615
$627$	0	0
$628$	6525.09	0.414617
$629$	3103.10	0.196707
$630$	0	0
$631$	19727.5	1.24460	0.622298	−	0.782780i	$-0.286199\pi$
$631$	19727.5	1.24460	0.622298	+	0.782780i	$0.286199\pi$
$632$	5349.81	0.336715
$633$	0	0
$634$	13603.5	0.852151
$635$	−31489.0	−1.96788
$636$	0	0
$637$	−1495.95	−0.0930484
$638$	−5317.28	−0.329958
$639$	0	0
$640$	1927.00	0.119018
$641$	−23304.5	−1.43599	−0.717997	−	0.696046i	$-0.754941\pi$
$641$	−23304.5	−1.43599	−0.717997	+	0.696046i	$0.754941\pi$
$642$	0	0
$643$	−15766.0	−0.966955	−0.483478	−	0.875357i	$-0.660627\pi$
$643$	−15766.0	−0.966955	−0.483478	+	0.875357i	$0.660627\pi$
$644$	3963.74	0.242536
$645$	0	0
$646$	−486.529	−0.0296319
$647$	25782.1	1.56661	0.783307	−	0.621635i	$-0.213531\pi$
$647$	25782.1	1.56661	0.783307	+	0.621635i	$0.213531\pi$
$648$	0	0
$649$	2750.52	0.166360
$650$	6206.26	0.374507
$651$	0	0
$652$	−12441.6	−0.747317
$653$	2495.59	0.149556	0.0747779	−	0.997200i	$-0.476175\pi$
$653$	2495.59	0.149556	0.0747779	+	0.997200i	$0.476175\pi$
$654$	0	0
$655$	36167.0	2.15750
$656$	−6232.09	−0.370918
$657$	0	0
$658$	−627.958	−0.0372042
$659$	9440.29	0.558029	0.279015	−	0.960287i	$-0.409992\pi$
$659$	9440.29	0.558029	0.279015	+	0.960287i	$0.409992\pi$
$660$	0	0
$661$	−17031.2	−1.00217	−0.501086	−	0.865397i	$-0.667066\pi$
$661$	−17031.2	−1.00217	−0.501086	+	0.865397i	$0.667066\pi$
$662$	−8132.44	−0.477457
$663$	0	0
$664$	−7201.66	−0.420902
$665$	2002.27	0.116759
$666$	0	0
$667$	−38012.9	−2.20670
$668$	4423.17	0.256194
$669$	0	0
$670$	6149.72	0.354603
$671$	6949.44	0.399821
$672$	0	0
$673$	22899.9	1.31163	0.655814	−	0.754922i	$-0.272326\pi$
$673$	22899.9	1.31163	0.655814	+	0.754922i	$0.272326\pi$
$674$	6967.32	0.398177
$675$	0	0
$676$	−5059.76	−0.287879
$677$	22485.0	1.27647	0.638234	−	0.769842i	$-0.279665\pi$
$677$	22485.0	1.27647	0.638234	+	0.769842i	$0.279665\pi$
$678$	0	0
$679$	−11360.2	−0.642068
$680$	1542.01	0.0869606
$681$	0	0
$682$	1900.97	0.106733
$683$	25124.6	1.40757	0.703783	−	0.710415i	$-0.251493\pi$
$683$	25124.6	1.40757	0.703783	+	0.710415i	$0.251493\pi$
$684$	0	0
$685$	23712.8	1.32266
$686$	686.000	0.0381802
$687$	0	0
$688$	7008.28	0.388355
$689$	22081.8	1.22097
$690$	0	0
$691$	−13625.1	−0.750109	−0.375054	−	0.927003i	$-0.622376\pi$
$691$	−13625.1	−0.750109	−0.375054	+	0.927003i	$0.622376\pi$
$692$	11563.2	0.635210
$693$	0	0
$694$	−1582.27	−0.0865450
$695$	17310.5	0.944786
$696$	0	0
$697$	−4986.99	−0.271013
$698$	19651.8	1.06566
$699$	0	0
$700$	−2846.01	−0.153670
$701$	−16392.4	−0.883214	−0.441607	−	0.897209i	$-0.645591\pi$
$701$	−16392.4	−0.883214	−0.441607	+	0.897209i	$0.645591\pi$
$702$	0	0
$703$	4604.95	0.247054
$704$	633.659	0.0339232
$705$	0	0
$706$	14619.1	0.779316
$707$	−522.017	−0.0277687
$708$	0	0
$709$	6482.28	0.343367	0.171683	−	0.985152i	$-0.445079\pi$
$709$	6482.28	0.343367	0.171683	+	0.985152i	$0.445079\pi$
$710$	−4707.61	−0.248836
$711$	0	0
$712$	−7194.72	−0.378699
$713$	13589.9	0.713811
$714$	0	0
$715$	4550.60	0.238018
$716$	−10027.7	−0.523397
$717$	0	0
$718$	−75.8671	−0.00394336
$719$	23342.9	1.21077	0.605384	−	0.795933i	$-0.293019\pi$
$719$	23342.9	1.21077	0.605384	+	0.795933i	$0.293019\pi$
$720$	0	0
$721$	−7792.32	−0.402498
$722$	−722.000	−0.0372161
$723$	0	0
$724$	1069.50	0.0549002
$725$	27293.7	1.39815
$726$	0	0
$727$	−22572.9	−1.15156	−0.575779	−	0.817605i	$-0.695301\pi$
$727$	−22572.9	−1.15156	−0.575779	+	0.817605i	$0.695301\pi$
$728$	−1709.66	−0.0870388
$729$	0	0
$730$	20186.8	1.02349
$731$	5608.11	0.283753
$732$	0	0
$733$	−24428.7	−1.23096	−0.615480	−	0.788153i	$-0.711038\pi$
$733$	−24428.7	−1.23096	−0.615480	+	0.788153i	$0.711038\pi$
$734$	−18870.4	−0.948937
$735$	0	0
$736$	4529.99	0.226872
$737$	2022.22	0.101071
$738$	0	0
$739$	−16728.7	−0.832712	−0.416356	−	0.909202i	$-0.636693\pi$
$739$	−16728.7	−0.832712	−0.416356	+	0.909202i	$0.636693\pi$
$740$	−14595.0	−0.725028
$741$	0	0
$742$	−10126.1	−0.500997
$743$	−2658.61	−0.131272	−0.0656360	−	0.997844i	$-0.520908\pi$
$743$	−2658.61	−0.131272	−0.0656360	+	0.997844i	$0.520908\pi$
$744$	0	0
$745$	−29878.6	−1.46935
$746$	−22639.4	−1.11111
$747$	0	0
$748$	507.061	0.0247861
$749$	12177.4	0.594064
$750$	0	0
$751$	9010.49	0.437813	0.218906	−	0.975746i	$-0.429751\pi$
$751$	9010.49	0.437813	0.218906	+	0.975746i	$0.429751\pi$
$752$	−717.667	−0.0348013
$753$	0	0
$754$	16395.9	0.791916
$755$	14089.9	0.679183
$756$	0	0
$757$	−28702.0	−1.37806	−0.689031	−	0.724732i	$-0.741964\pi$
$757$	−28702.0	−1.37806	−0.689031	+	0.724732i	$0.741964\pi$
$758$	−3210.59	−0.153844
$759$	0	0
$760$	2288.31	0.109218
$761$	21886.0	1.04253	0.521265	−	0.853395i	$-0.325460\pi$
$761$	21886.0	1.04253	0.521265	+	0.853395i	$0.325460\pi$
$762$	0	0
$763$	−2821.86	−0.133890
$764$	4817.03	0.228107
$765$	0	0
$766$	17849.1	0.841925
$767$	−8481.30	−0.399272
$768$	0	0
$769$	35844.3	1.68086	0.840429	−	0.541921i	$-0.182303\pi$
$769$	35844.3	1.68086	0.840429	+	0.541921i	$0.182303\pi$
$770$	−2086.77	−0.0976649
$771$	0	0
$772$	−7821.86	−0.364657
$773$	27743.6	1.29090	0.645452	−	0.763801i	$-0.276669\pi$
$773$	27743.6	1.29090	0.645452	+	0.763801i	$0.276669\pi$
$774$	0	0
$775$	−9757.71	−0.452268
$776$	−12983.1	−0.600600
$777$	0	0
$778$	−21529.6	−0.992127
$779$	−7400.61	−0.340378
$780$	0	0
$781$	−1548.01	−0.0709249
$782$	3624.95	0.165765
$783$	0	0
$784$	784.000	0.0357143
$785$	−24558.3	−1.11659
$786$	0	0
$787$	−9136.73	−0.413836	−0.206918	−	0.978358i	$-0.566343\pi$
$787$	−9136.73	−0.413836	−0.206918	+	0.978358i	$0.566343\pi$
$788$	8652.56	0.391161
$789$	0	0
$790$	−20134.9	−0.906795
$791$	3209.45	0.144267
$792$	0	0
$793$	−21428.7	−0.959592
$794$	−12358.3	−0.552369
$795$	0	0
$796$	507.327	0.0225901
$797$	10900.2	0.484448	0.242224	−	0.970220i	$-0.422123\pi$
$797$	10900.2	0.484448	0.242224	+	0.970220i	$0.422123\pi$
$798$	0	0
$799$	−574.285	−0.0254277
$800$	−3252.58	−0.143745
$801$	0	0
$802$	17569.5	0.773568
$803$	6638.07	0.291722
$804$	0	0
$805$	−14918.2	−0.653164
$806$	−5861.68	−0.256165
$807$	0	0
$808$	−596.591	−0.0259752
$809$	1091.12	0.0474187	0.0237094	−	0.999719i	$-0.492452\pi$
$809$	1091.12	0.0474187	0.0237094	+	0.999719i	$0.492452\pi$
$810$	0	0
$811$	−33671.7	−1.45792	−0.728960	−	0.684557i	$-0.759996\pi$
$811$	−33671.7	−1.45792	−0.728960	+	0.684557i	$0.759996\pi$
$812$	−7518.69	−0.324944
$813$	0	0
$814$	−4799.29	−0.206652
$815$	46826.1	2.01257
$816$	0	0
$817$	8322.33	0.356379
$818$	−4174.64	−0.178439
$819$	0	0
$820$	23455.5	0.998905
$821$	−32281.8	−1.37228	−0.686141	−	0.727468i	$-0.740697\pi$
$821$	−32281.8	−1.37228	−0.686141	+	0.727468i	$0.740697\pi$
$822$	0	0
$823$	−7616.00	−0.322572	−0.161286	−	0.986908i	$-0.551564\pi$
$823$	−7616.00	−0.322572	−0.161286	+	0.986908i	$0.551564\pi$
$824$	−8905.51	−0.376503
$825$	0	0
$826$	3889.27	0.163832
$827$	37011.0	1.55622	0.778112	−	0.628125i	$-0.216177\pi$
$827$	37011.0	1.55622	0.778112	+	0.628125i	$0.216177\pi$
$828$	0	0
$829$	43552.0	1.82464	0.912319	−	0.409481i	$-0.134290\pi$
$829$	43552.0	1.82464	0.912319	+	0.409481i	$0.134290\pi$
$830$	27104.7	1.13351
$831$	0	0
$832$	−1953.90	−0.0814174
$833$	627.366	0.0260948
$834$	0	0
$835$	−16647.4	−0.689947
$836$	752.470	0.0311300
$837$	0	0
$838$	−28110.6	−1.15879
$839$	7350.02	0.302445	0.151222	−	0.988500i	$-0.451679\pi$
$839$	7350.02	0.302445	0.151222	+	0.988500i	$0.451679\pi$
$840$	0	0
$841$	47716.5	1.95648
$842$	−7596.75	−0.310928
$843$	0	0
$844$	−13665.4	−0.557323
$845$	19043.2	0.775275
$846$	0	0
$847$	8630.80	0.350127
$848$	−11572.6	−0.468639
$849$	0	0
$850$	−2602.75	−0.105028
$851$	−34309.8	−1.38205
$852$	0	0
$853$	11462.6	0.460106	0.230053	−	0.973178i	$-0.426110\pi$
$853$	11462.6	0.460106	0.230053	+	0.973178i	$0.426110\pi$
$854$	9826.58	0.393745
$855$	0	0
$856$	13917.1	0.555696
$857$	−20910.5	−0.833476	−0.416738	−	0.909027i	$-0.636827\pi$
$857$	−20910.5	−0.833476	−0.416738	+	0.909027i	$0.636827\pi$
$858$	0	0
$859$	23952.2	0.951384	0.475692	−	0.879612i	$-0.342198\pi$
$859$	23952.2	0.951384	0.475692	+	0.879612i	$0.342198\pi$
$860$	−26376.8	−1.04586
$861$	0	0
$862$	12615.0	0.498456
$863$	2858.69	0.112759	0.0563795	−	0.998409i	$-0.482044\pi$
$863$	2858.69	0.112759	0.0563795	+	0.998409i	$0.482044\pi$
$864$	0	0
$865$	−43519.9	−1.71066
$866$	−17916.6	−0.703036
$867$	0	0
$868$	2687.99	0.105111
$869$	−6621.01	−0.258461
$870$	0	0
$871$	−6235.57	−0.242577
$872$	−3224.99	−0.125243
$873$	0	0
$874$	5379.36	0.208192
$875$	−2461.42	−0.0950983
$876$	0	0
$877$	−36231.3	−1.39503	−0.697516	−	0.716569i	$-0.745712\pi$
$877$	−36231.3	−1.39503	−0.697516	+	0.716569i	$0.745712\pi$
$878$	22483.2	0.864206
$879$	0	0
$880$	−2384.88	−0.0913572
$881$	−4418.48	−0.168970	−0.0844848	−	0.996425i	$-0.526924\pi$
$881$	−4418.48	−0.168970	−0.0844848	+	0.996425i	$0.526924\pi$
$882$	0	0
$883$	7650.53	0.291575	0.145788	−	0.989316i	$-0.453428\pi$
$883$	7650.53	0.291575	0.145788	+	0.989316i	$0.453428\pi$
$884$	−1563.53	−0.0594879
$885$	0	0
$886$	−980.031	−0.0371612
$887$	38408.4	1.45392	0.726961	−	0.686679i	$-0.240932\pi$
$887$	38408.4	1.45392	0.726961	+	0.686679i	$0.240932\pi$
$888$	0	0
$889$	−14641.5	−0.552374
$890$	27078.6	1.01986
$891$	0	0
$892$	12065.4	0.452892
$893$	−852.229	−0.0319359
$894$	0	0
$895$	37740.9	1.40954
$896$	896.000	0.0334077
$897$	0	0
$898$	33257.2	1.23587
$899$	−25778.3	−0.956346
$900$	0	0
$901$	−9260.56	−0.342413
$902$	7712.93	0.284714
$903$	0	0
$904$	3667.94	0.134949
$905$	−4025.25	−0.147850
$906$	0	0
$907$	44355.0	1.62380	0.811899	−	0.583798i	$-0.198434\pi$
$907$	44355.0	1.62380	0.811899	+	0.583798i	$0.198434\pi$
$908$	933.355	0.0341128
$909$	0	0
$910$	6434.60	0.234401
$911$	38466.3	1.39895	0.699476	−	0.714656i	$-0.253417\pi$
$911$	38466.3	1.39895	0.699476	+	0.714656i	$0.253417\pi$
$912$	0	0
$913$	8912.88	0.323081
$914$	12659.1	0.458124
$915$	0	0
$916$	3382.80	0.122021
$917$	16816.6	0.605599
$918$	0	0
$919$	44641.2	1.60237	0.801185	−	0.598416i	$-0.204203\pi$
$919$	44641.2	1.60237	0.801185	+	0.598416i	$0.204203\pi$
$920$	−17049.4	−0.610979
$921$	0	0
$922$	29598.8	1.05725
$923$	4773.33	0.170223
$924$	0	0
$925$	24634.8	0.875662
$926$	37387.9	1.32683
$927$	0	0
$928$	−8592.79	−0.303957
$929$	6110.59	0.215804	0.107902	−	0.994162i	$-0.465587\pi$
$929$	6110.59	0.215804	0.107902	+	0.994162i	$0.465587\pi$
$930$	0	0
$931$	931.000	0.0327737
$932$	−25559.4	−0.898312
$933$	0	0
$934$	2867.10	0.100444
$935$	−1908.41	−0.0667504
$936$	0	0
$937$	32051.6	1.11748	0.558741	−	0.829343i	$-0.311285\pi$
$937$	32051.6	1.11748	0.558741	+	0.829343i	$0.311285\pi$
$938$	2859.45	0.0995354
$939$	0	0
$940$	2701.06	0.0937221
$941$	−8144.56	−0.282152	−0.141076	−	0.989999i	$-0.545056\pi$
$941$	−8144.56	−0.282152	−0.141076	+	0.989999i	$0.545056\pi$
$942$	0	0
$943$	55139.2	1.90411
$944$	4444.88	0.153251
$945$	0	0
$946$	−8673.55	−0.298099
$947$	−44089.7	−1.51291	−0.756453	−	0.654048i	$-0.773069\pi$
$947$	−44089.7	−1.51291	−0.756453	+	0.654048i	$0.773069\pi$
$948$	0	0
$949$	−20468.6	−0.700147
$950$	−3862.44	−0.131909
$951$	0	0
$952$	716.990	0.0244094
$953$	−35960.5	−1.22232	−0.611162	−	0.791506i	$-0.709298\pi$
$953$	−35960.5	−1.22232	−0.611162	+	0.791506i	$0.709298\pi$
$954$	0	0
$955$	−18129.7	−0.614307
$956$	23948.6	0.810200
$957$	0	0
$958$	29966.8	1.01063
$959$	11025.8	0.371263
$960$	0	0
$961$	−20575.0	−0.690646
$962$	14798.7	0.495976
$963$	0	0
$964$	−28671.1	−0.957919
$965$	29438.9	0.982042
$966$	0	0
$967$	−5163.93	−0.171728	−0.0858640	−	0.996307i	$-0.527365\pi$
$967$	−5163.93	−0.171728	−0.0858640	+	0.996307i	$0.527365\pi$
$968$	9863.77	0.327514
$969$	0	0
$970$	48864.0	1.61745
$971$	47479.9	1.56921	0.784604	−	0.619997i	$-0.212866\pi$
$971$	47479.9	1.56921	0.784604	+	0.619997i	$0.212866\pi$
$972$	0	0
$973$	8048.91	0.265197
$974$	−21537.3	−0.708520
$975$	0	0
$976$	11230.4	0.368315
$977$	16411.2	0.537402	0.268701	−	0.963224i	$-0.413406\pi$
$977$	16411.2	0.537402	0.268701	+	0.963224i	$0.413406\pi$
$978$	0	0
$979$	8904.30	0.290687
$980$	−2950.72	−0.0961808
$981$	0	0
$982$	−17239.6	−0.560222
$983$	31820.8	1.03248	0.516239	−	0.856445i	$-0.327332\pi$
$983$	31820.8	1.03248	0.516239	+	0.856445i	$0.327332\pi$
$984$	0	0
$985$	−32565.4	−1.05342
$986$	−6876.05	−0.222087
$987$	0	0
$988$	−2320.25	−0.0747137
$989$	−62006.7	−1.99363
$990$	0	0
$991$	−53254.8	−1.70706	−0.853529	−	0.521046i	$-0.825542\pi$
$991$	−53254.8	−1.70706	−0.853529	+	0.521046i	$0.825542\pi$
$992$	3071.99	0.0983225
$993$	0	0
$994$	−2188.91	−0.0698470
$995$	−1909.41	−0.0608366
$996$	0	0
$997$	30172.6	0.958452	0.479226	−	0.877692i	$-0.340917\pi$
$997$	30172.6	0.958452	0.479226	+	0.877692i	$0.340917\pi$
$998$	16590.5	0.526216
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.4.a.bh.1.2	✓	7
3.2	odd	2		2394.4.a.bi.1.6	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.4.a.bh.1.2	✓	7	1.1	even	1	trivial
2394.4.a.bi.1.6	yes	7	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -15.0547 q^{5} -7.00000 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -15.0547 q^{5} -7.00000 q^{7} -8.00000 q^{8} +30.1093 q^{10} +9.90092 q^{11} -30.5297 q^{13} +14.0000 q^{14} +16.0000 q^{16} +12.8034 q^{17} +19.0000 q^{19} -60.2187 q^{20} -19.8018 q^{22} -141.562 q^{23} +101.643 q^{25} +61.0593 q^{26} -28.0000 q^{28} +268.525 q^{29} -95.9998 q^{31} -32.0000 q^{32} -25.6068 q^{34} +105.383 q^{35} +242.366 q^{37} -38.0000 q^{38} +120.437 q^{40} -389.506 q^{41} +438.018 q^{43} +39.6037 q^{44} +283.124 q^{46} -44.8542 q^{47} +49.0000 q^{49} -203.286 q^{50} -122.119 q^{52} -723.290 q^{53} -149.055 q^{55} +56.0000 q^{56} -537.049 q^{58} +277.805 q^{59} +701.898 q^{61} +192.000 q^{62} +64.0000 q^{64} +459.614 q^{65} +204.246 q^{67} +51.2135 q^{68} -210.765 q^{70} -156.351 q^{71} +670.450 q^{73} -484.732 q^{74} +76.0000 q^{76} -69.3064 q^{77} -668.727 q^{79} -240.875 q^{80} +779.011 q^{82} +900.208 q^{83} -192.751 q^{85} -876.035 q^{86} -79.2073 q^{88} +899.341 q^{89} +213.708 q^{91} -566.248 q^{92} +89.7083 q^{94} -286.039 q^{95} +1622.88 q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 14 q^{2} + 28 q^{4} - 18 q^{5} - 49 q^{7} - 56 q^{8}+O(q^{10}) \) 7 * q - 14 * q^2 + 28 * q^4 - 18 * q^5 - 49 * q^7 - 56 * q^8	\( 7 q - 14 q^{2} + 28 q^{4} - 18 q^{5} - 49 q^{7} - 56 q^{8} + 36 q^{10} - 28 q^{11} + 44 q^{13} + 98 q^{14} + 112 q^{16} - 26 q^{17} + 133 q^{19} - 72 q^{20} + 56 q^{22} - 56 q^{23} + 37 q^{25} - 88 q^{26} - 196 q^{28} - 270 q^{29} + 64 q^{31} - 224 q^{32} + 52 q^{34} + 126 q^{35} + 458 q^{37} - 266 q^{38} + 144 q^{40} - 110 q^{41} + 296 q^{43} - 112 q^{44} + 112 q^{46} + 142 q^{47} + 343 q^{49} - 74 q^{50} + 176 q^{52} - 330 q^{53} + 596 q^{55} + 392 q^{56} + 540 q^{58} - 236 q^{59} + 882 q^{61} - 128 q^{62} + 448 q^{64} + 180 q^{65} + 1622 q^{67} - 104 q^{68} - 252 q^{70} - 820 q^{71} + 1130 q^{73} - 916 q^{74} + 532 q^{76} + 196 q^{77} + 1694 q^{79} - 288 q^{80} + 220 q^{82} - 890 q^{83} + 1540 q^{85} - 592 q^{86} + 224 q^{88} - 686 q^{89} - 308 q^{91} - 224 q^{92} - 284 q^{94} - 342 q^{95} + 1772 q^{97} - 686 q^{98}+O(q^{100}) \) 7 * q - 14 * q^2 + 28 * q^4 - 18 * q^5 - 49 * q^7 - 56 * q^8 + 36 * q^10 - 28 * q^11 + 44 * q^13 + 98 * q^14 + 112 * q^16 - 26 * q^17 + 133 * q^19 - 72 * q^20 + 56 * q^22 - 56 * q^23 + 37 * q^25 - 88 * q^26 - 196 * q^28 - 270 * q^29 + 64 * q^31 - 224 * q^32 + 52 * q^34 + 126 * q^35 + 458 * q^37 - 266 * q^38 + 144 * q^40 - 110 * q^41 + 296 * q^43 - 112 * q^44 + 112 * q^46 + 142 * q^47 + 343 * q^49 - 74 * q^50 + 176 * q^52 - 330 * q^53 + 596 * q^55 + 392 * q^56 + 540 * q^58 - 236 * q^59 + 882 * q^61 - 128 * q^62 + 448 * q^64 + 180 * q^65 + 1622 * q^67 - 104 * q^68 - 252 * q^70 - 820 * q^71 + 1130 * q^73 - 916 * q^74 + 532 * q^76 + 196 * q^77 + 1694 * q^79 - 288 * q^80 + 220 * q^82 - 890 * q^83 + 1540 * q^85 - 592 * q^86 + 224 * q^88 - 686 * q^89 - 308 * q^91 - 224 * q^92 - 284 * q^94 - 342 * q^95 + 1772 * q^97 - 686 * q^98

Introduction

L-functions

Modular forms

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