LMFDB - Embedded newform 1150.4.a.q.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(1,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, 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("1150.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.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:	$67.8521965066$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 $ x^5 - 107x^3 - 3x^2 + 2151*x - 2916
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$9.27140$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +8.27140 q^{3} +4.00000 q^{4} -16.5428 q^{6} -22.8621 q^{7} -8.00000 q^{8} +41.4161 q^{9} -0.987996 q^{11} +33.0856 q^{12} +4.22276 q^{13} +45.7241 q^{14} +16.0000 q^{16} -73.8741 q^{17} -82.8322 q^{18} +71.1586 q^{19} -189.101 q^{21} +1.97599 q^{22} -23.0000 q^{23} -66.1712 q^{24} -8.44552 q^{26} +119.241 q^{27} -91.4483 q^{28} +27.8984 q^{29} -136.861 q^{31} -32.0000 q^{32} -8.17211 q^{33} +147.748 q^{34} +165.664 q^{36} +201.718 q^{37} -142.317 q^{38} +34.9281 q^{39} -204.140 q^{41} +378.203 q^{42} +54.1024 q^{43} -3.95198 q^{44} +46.0000 q^{46} +41.4490 q^{47} +132.342 q^{48} +179.674 q^{49} -611.042 q^{51} +16.8910 q^{52} -428.612 q^{53} -238.483 q^{54} +182.897 q^{56} +588.581 q^{57} -55.7969 q^{58} -164.858 q^{59} +188.248 q^{61} +273.722 q^{62} -946.857 q^{63} +64.0000 q^{64} +16.3442 q^{66} -932.167 q^{67} -295.496 q^{68} -190.242 q^{69} -263.336 q^{71} -331.329 q^{72} -900.401 q^{73} -403.437 q^{74} +284.634 q^{76} +22.5876 q^{77} -69.8563 q^{78} -956.182 q^{79} -131.942 q^{81} +408.279 q^{82} -194.877 q^{83} -756.405 q^{84} -108.205 q^{86} +230.759 q^{87} +7.90397 q^{88} -213.751 q^{89} -96.5410 q^{91} -92.0000 q^{92} -1132.03 q^{93} -82.8981 q^{94} -264.685 q^{96} -628.762 q^{97} -359.348 q^{98} -40.9189 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 - 5 * q^3 + 20 * q^4 + 10 * q^6 + 3 * q^7 - 40 * q^8 + 84 * q^9 - 26 * q^11 - 20 * q^12 + 61 * q^13 - 6 * q^14 + 80 * q^16 - 231 * q^17 - 168 * q^18 + 74 * q^19 - 88 * q^21 + 52 * q^22 - 115 * q^23 + 40 * q^24 - 122 * q^26 - 368 * q^27 + 12 * q^28 + 37 * q^29 + 220 * q^31 - 160 * q^32 - 391 * q^33 + 462 * q^34 + 336 * q^36 - 220 * q^37 - 148 * q^38 + 195 * q^39 + 112 * q^41 + 176 * q^42 - 97 * q^43 - 104 * q^44 + 230 * q^46 - 370 * q^47 - 80 * q^48 + 182 * q^49 + 563 * q^51 + 244 * q^52 - 1224 * q^53 + 736 * q^54 - 24 * q^56 + 1009 * q^57 - 74 * q^58 - 181 * q^59 + 1574 * q^61 - 440 * q^62 - 1829 * q^63 + 320 * q^64 + 782 * q^66 + 435 * q^67 - 924 * q^68 + 115 * q^69 - 30 * q^71 - 672 * q^72 - 2180 * q^73 + 440 * q^74 + 296 * q^76 + 1401 * q^77 - 390 * q^78 + 609 * q^79 + 1453 * q^81 - 224 * q^82 - 2018 * q^83 - 352 * q^84 + 194 * q^86 + 1485 * q^87 + 208 * q^88 - 1663 * q^89 + 361 * q^91 - 460 * q^92 - 2741 * q^93 + 740 * q^94 + 160 * q^96 + 1274 * q^97 - 364 * q^98 + 2397 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	8.27140	1.59183	0.795916	−	0.605407i	$-0.206990\pi$
$3$	8.27140	1.59183	0.795916	+	0.605407i	$0.206990\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−16.5428	−1.12560
$7$	−22.8621	−1.23444	−0.617218	−	0.786792i	$-0.711740\pi$
$7$	−22.8621	−1.23444	−0.617218	+	0.786792i	$0.711740\pi$
$8$	−8.00000	−0.353553
$9$	41.4161	1.53393
$10$	0	0
$11$	−0.987996	−0.0270811	−0.0135405	−	0.999908i	$-0.504310\pi$
$11$	−0.987996	−0.0270811	−0.0135405	+	0.999908i	$0.504310\pi$
$12$	33.0856	0.795916
$13$	4.22276	0.0900910	0.0450455	−	0.998985i	$-0.485657\pi$
$13$	4.22276	0.0900910	0.0450455	+	0.998985i	$0.485657\pi$
$14$	45.7241	0.872878
$15$	0	0
$16$	16.0000	0.250000
$17$	−73.8741	−1.05395	−0.526973	−	0.849882i	$-0.676673\pi$
$17$	−73.8741	−1.05395	−0.526973	+	0.849882i	$0.676673\pi$
$18$	−82.8322	−1.08465
$19$	71.1586	0.859205	0.429602	−	0.903018i	$-0.358654\pi$
$19$	71.1586	0.859205	0.429602	+	0.903018i	$0.358654\pi$
$20$	0	0
$21$	−189.101	−1.96501
$22$	1.97599	0.0191492
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−66.1712	−0.562798
$25$	0	0
$26$	−8.44552	−0.0637039
$27$	119.241	0.849926
$28$	−91.4483	−0.617218
$29$	27.8984	0.178642	0.0893209	−	0.996003i	$-0.471530\pi$
$29$	27.8984	0.178642	0.0893209	+	0.996003i	$0.471530\pi$
$30$	0	0
$31$	−136.861	−0.792935	−0.396468	−	0.918049i	$-0.629764\pi$
$31$	−136.861	−0.792935	−0.396468	+	0.918049i	$0.629764\pi$
$32$	−32.0000	−0.176777
$33$	−8.17211	−0.0431085
$34$	147.748	0.745253
$35$	0	0
$36$	165.664	0.766965
$37$	201.718	0.896278	0.448139	−	0.893964i	$-0.352087\pi$
$37$	201.718	0.896278	0.448139	+	0.893964i	$0.352087\pi$
$38$	−142.317	−0.607550
$39$	34.9281	0.143410
$40$	0	0
$41$	−204.140	−0.777592	−0.388796	−	0.921324i	$-0.627109\pi$
$41$	−204.140	−0.777592	−0.388796	+	0.921324i	$0.627109\pi$
$42$	378.203	1.38947
$43$	54.1024	0.191873	0.0959365	−	0.995387i	$-0.469415\pi$
$43$	54.1024	0.191873	0.0959365	+	0.995387i	$0.469415\pi$
$44$	−3.95198	−0.0135405
$45$	0	0
$46$	46.0000	0.147442
$47$	41.4490	0.128638	0.0643188	−	0.997929i	$-0.479513\pi$
$47$	41.4490	0.128638	0.0643188	+	0.997929i	$0.479513\pi$
$48$	132.342	0.397958
$49$	179.674	0.523831
$50$	0	0
$51$	−611.042	−1.67771
$52$	16.8910	0.0450455
$53$	−428.612	−1.11084	−0.555418	−	0.831571i	$-0.687442\pi$
$53$	−428.612	−1.11084	−0.555418	+	0.831571i	$0.687442\pi$
$54$	−238.483	−0.600988
$55$	0	0
$56$	182.897	0.436439
$57$	588.581	1.36771
$58$	−55.7969	−0.126319
$59$	−164.858	−0.363774	−0.181887	−	0.983319i	$-0.558221\pi$
$59$	−164.858	−0.363774	−0.181887	+	0.983319i	$0.558221\pi$
$60$	0	0
$61$	188.248	0.395125	0.197563	−	0.980290i	$-0.436697\pi$
$61$	188.248	0.395125	0.197563	+	0.980290i	$0.436697\pi$
$62$	273.722	0.560690
$63$	−946.857	−1.89354
$64$	64.0000	0.125000
$65$	0	0
$66$	16.3442	0.0304823
$67$	−932.167	−1.69974	−0.849868	−	0.526995i	$-0.823318\pi$
$67$	−932.167	−1.69974	−0.849868	+	0.526995i	$0.823318\pi$
$68$	−295.496	−0.526973
$69$	−190.242	−0.331920
$70$	0	0
$71$	−263.336	−0.440173	−0.220086	−	0.975480i	$-0.570634\pi$
$71$	−263.336	−0.440173	−0.220086	+	0.975480i	$0.570634\pi$
$72$	−331.329	−0.542326
$73$	−900.401	−1.44362	−0.721808	−	0.692094i	$-0.756689\pi$
$73$	−900.401	−1.44362	−0.721808	+	0.692094i	$0.756689\pi$
$74$	−403.437	−0.633764
$75$	0	0
$76$	284.634	0.429602
$77$	22.5876	0.0334299
$78$	−69.8563	−0.101406
$79$	−956.182	−1.36176	−0.680879	−	0.732396i	$-0.738402\pi$
$79$	−956.182	−1.36176	−0.680879	+	0.732396i	$0.738402\pi$
$80$	0	0
$81$	−131.942	−0.180990
$82$	408.279	0.549841
$83$	−194.877	−0.257717	−0.128858	−	0.991663i	$-0.541131\pi$
$83$	−194.877	−0.257717	−0.128858	+	0.991663i	$0.541131\pi$
$84$	−756.405	−0.982507
$85$	0	0
$86$	−108.205	−0.135675
$87$	230.759	0.284368
$88$	7.90397	0.00957461
$89$	−213.751	−0.254580	−0.127290	−	0.991866i	$-0.540628\pi$
$89$	−213.751	−0.254580	−0.127290	+	0.991866i	$0.540628\pi$
$90$	0	0
$91$	−96.5410	−0.111211
$92$	−92.0000	−0.104257
$93$	−1132.03	−1.26222
$94$	−82.8981	−0.0909605
$95$	0	0
$96$	−264.685	−0.281399
$97$	−628.762	−0.658156	−0.329078	−	0.944303i	$-0.606738\pi$
$97$	−628.762	−0.658156	−0.329078	+	0.944303i	$0.606738\pi$
$98$	−359.348	−0.370404
$99$	−40.9189	−0.0415405
$100$	0	0
$101$	−51.1371	−0.0503796	−0.0251898	−	0.999683i	$-0.508019\pi$
$101$	−51.1371	−0.0503796	−0.0251898	+	0.999683i	$0.508019\pi$
$102$	1222.08	1.18632
$103$	−363.089	−0.347342	−0.173671	−	0.984804i	$-0.555563\pi$
$103$	−363.089	−0.347342	−0.173671	+	0.984804i	$0.555563\pi$
$104$	−33.7821	−0.0318520
$105$	0	0
$106$	857.223	0.785480
$107$	−1514.53	−1.36837	−0.684184	−	0.729310i	$-0.739841\pi$
$107$	−1514.53	−1.36837	−0.684184	+	0.729310i	$0.739841\pi$
$108$	476.965	0.424963
$109$	811.356	0.712971	0.356485	−	0.934301i	$-0.383975\pi$
$109$	811.356	0.712971	0.356485	+	0.934301i	$0.383975\pi$
$110$	0	0
$111$	1668.49	1.42672
$112$	−365.793	−0.308609
$113$	−12.4496	−0.0103642	−0.00518211	−	0.999987i	$-0.501650\pi$
$113$	−12.4496	−0.0103642	−0.00518211	+	0.999987i	$0.501650\pi$
$114$	−1177.16	−0.967117
$115$	0	0
$116$	111.594	0.0893209
$117$	174.890	0.138193
$118$	329.716	0.257227
$119$	1688.91	1.30103
$120$	0	0
$121$	−1330.02	−0.999267
$122$	−376.495	−0.279396
$123$	−1688.52	−1.23780
$124$	−547.445	−0.396468
$125$	0	0
$126$	1893.71	1.33893
$127$	632.599	0.442000	0.221000	−	0.975274i	$-0.429068\pi$
$127$	632.599	0.442000	0.221000	+	0.975274i	$0.429068\pi$
$128$	−128.000	−0.0883883
$129$	447.503	0.305430
$130$	0	0
$131$	2243.16	1.49607	0.748036	−	0.663658i	$-0.230997\pi$
$131$	2243.16	1.49607	0.748036	+	0.663658i	$0.230997\pi$
$132$	−32.6884	−0.0215543
$133$	−1626.83	−1.06063
$134$	1864.33	1.20190
$135$	0	0
$136$	590.993	0.372626
$137$	−1252.77	−0.781252	−0.390626	−	0.920550i	$-0.627741\pi$
$137$	−1252.77	−0.781252	−0.390626	+	0.920550i	$0.627741\pi$
$138$	380.484	0.234703
$139$	2815.19	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$139$	2815.19	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$140$	0	0
$141$	342.842	0.204769
$142$	526.673	0.311249
$143$	−4.17207	−0.00243976
$144$	662.657	0.383482
$145$	0	0
$146$	1800.80	1.02079
$147$	1486.16	0.833851
$148$	806.873	0.448139
$149$	2406.46	1.32312	0.661560	−	0.749892i	$-0.269895\pi$
$149$	2406.46	1.32312	0.661560	+	0.749892i	$0.269895\pi$
$150$	0	0
$151$	2185.59	1.17788	0.588942	−	0.808175i	$-0.299545\pi$
$151$	2185.59	1.17788	0.588942	+	0.808175i	$0.299545\pi$
$152$	−569.268	−0.303775
$153$	−3059.58	−1.61668
$154$	−45.1753	−0.0236385
$155$	0	0
$156$	139.713	0.0717048
$157$	−1074.18	−0.546044	−0.273022	−	0.962008i	$-0.588023\pi$
$157$	−1074.18	−0.546044	−0.273022	+	0.962008i	$0.588023\pi$
$158$	1912.36	0.962908
$159$	−3545.22	−1.76827
$160$	0	0
$161$	525.828	0.257398
$162$	263.884	0.127979
$163$	−2706.30	−1.30045	−0.650226	−	0.759741i	$-0.725326\pi$
$163$	−2706.30	−1.30045	−0.650226	+	0.759741i	$0.725326\pi$
$164$	−816.559	−0.388796
$165$	0	0
$166$	389.754	0.182233
$167$	−455.547	−0.211086	−0.105543	−	0.994415i	$-0.533658\pi$
$167$	−455.547	−0.211086	−0.105543	+	0.994415i	$0.533658\pi$
$168$	1512.81	0.694737
$169$	−2179.17	−0.991884
$170$	0	0
$171$	2947.11	1.31796
$172$	216.410	0.0959365
$173$	1342.64	0.590051	0.295026	−	0.955489i	$-0.404672\pi$
$173$	1342.64	0.590051	0.295026	+	0.955489i	$0.404672\pi$
$174$	−461.518	−0.201078
$175$	0	0
$176$	−15.8079	−0.00677027
$177$	−1363.61	−0.579068
$178$	427.503	0.180015
$179$	−4655.64	−1.94401	−0.972007	−	0.234951i	$-0.924507\pi$
$179$	−4655.64	−1.94401	−0.972007	+	0.234951i	$0.924507\pi$
$180$	0	0
$181$	−176.320	−0.0724074	−0.0362037	−	0.999344i	$-0.511527\pi$
$181$	−176.320	−0.0724074	−0.0362037	+	0.999344i	$0.511527\pi$
$182$	193.082	0.0786384
$183$	1557.07	0.628973
$184$	184.000	0.0737210
$185$	0	0
$186$	2264.07	0.892524
$187$	72.9873	0.0285420
$188$	165.796	0.0643188
$189$	−2726.10	−1.04918
$190$	0	0
$191$	1672.03	0.633422	0.316711	−	0.948522i	$-0.397421\pi$
$191$	1672.03	0.633422	0.316711	+	0.948522i	$0.397421\pi$
$192$	529.370	0.198979
$193$	2954.45	1.10189	0.550947	−	0.834540i	$-0.314267\pi$
$193$	2954.45	1.10189	0.550947	+	0.834540i	$0.314267\pi$
$194$	1257.52	0.465386
$195$	0	0
$196$	718.696	0.261915
$197$	−1817.35	−0.657264	−0.328632	−	0.944458i	$-0.606588\pi$
$197$	−1817.35	−0.657264	−0.328632	+	0.944458i	$0.606588\pi$
$198$	81.8379	0.0293735
$199$	4780.99	1.70309	0.851546	−	0.524280i	$-0.175666\pi$
$199$	4780.99	1.70309	0.851546	+	0.524280i	$0.175666\pi$
$200$	0	0
$201$	−7710.33	−2.70569
$202$	102.274	0.0356237
$203$	−637.816	−0.220522
$204$	−2444.17	−0.838853
$205$	0	0
$206$	726.179	0.245608
$207$	−952.570	−0.319846
$208$	67.5641	0.0225227
$209$	−70.3044	−0.0232682
$210$	0	0
$211$	3201.72	1.04462	0.522312	−	0.852755i	$-0.325070\pi$
$211$	3201.72	1.04462	0.522312	+	0.852755i	$0.325070\pi$
$212$	−1714.45	−0.555418
$213$	−2178.16	−0.700681
$214$	3029.06	0.967582
$215$	0	0
$216$	−953.930	−0.300494
$217$	3128.93	0.978828
$218$	−1622.71	−0.504147
$219$	−7447.58	−2.29799
$220$	0	0
$221$	−311.952	−0.0949511
$222$	−3336.99	−1.00885
$223$	1897.23	0.569721	0.284860	−	0.958569i	$-0.408053\pi$
$223$	1897.23	0.569721	0.284860	+	0.958569i	$0.408053\pi$
$224$	731.586	0.218219
$225$	0	0
$226$	24.8991	0.00732861
$227$	−4468.42	−1.30652	−0.653258	−	0.757135i	$-0.726598\pi$
$227$	−4468.42	−1.30652	−0.653258	+	0.757135i	$0.726598\pi$
$228$	2354.32	0.683855
$229$	−5345.88	−1.54264	−0.771322	−	0.636445i	$-0.780404\pi$
$229$	−5345.88	−1.54264	−0.771322	+	0.636445i	$0.780404\pi$
$230$	0	0
$231$	186.831	0.0532147
$232$	−223.187	−0.0631594
$233$	4480.46	1.25976	0.629882	−	0.776691i	$-0.283103\pi$
$233$	4480.46	1.25976	0.629882	+	0.776691i	$0.283103\pi$
$234$	−349.780	−0.0977173
$235$	0	0
$236$	−659.432	−0.181887
$237$	−7908.97	−2.16769
$238$	−3377.83	−0.919967
$239$	779.836	0.211060	0.105530	−	0.994416i	$-0.466346\pi$
$239$	779.836	0.211060	0.105530	+	0.994416i	$0.466346\pi$
$240$	0	0
$241$	−4204.89	−1.12390	−0.561952	−	0.827170i	$-0.689950\pi$
$241$	−4204.89	−1.12390	−0.561952	+	0.827170i	$0.689950\pi$
$242$	2660.05	0.706588
$243$	−4310.86	−1.13803
$244$	752.991	0.197563
$245$	0	0
$246$	3377.04	0.875254
$247$	300.485	0.0774066
$248$	1094.89	0.280345
$249$	−1611.90	−0.410242
$250$	0	0
$251$	2761.48	0.694433	0.347217	−	0.937785i	$-0.387127\pi$
$251$	2761.48	0.694433	0.347217	+	0.937785i	$0.387127\pi$
$252$	−3787.43	−0.946768
$253$	22.7239	0.00564680
$254$	−1265.20	−0.312542
$255$	0	0
$256$	256.000	0.0625000
$257$	586.469	0.142346	0.0711730	−	0.997464i	$-0.477326\pi$
$257$	586.469	0.142346	0.0711730	+	0.997464i	$0.477326\pi$
$258$	−895.006	−0.215971
$259$	−4611.70	−1.10640
$260$	0	0
$261$	1155.44	0.274024
$262$	−4486.31	−1.05788
$263$	−5970.38	−1.39981	−0.699903	−	0.714238i	$-0.746774\pi$
$263$	−5970.38	−1.39981	−0.699903	+	0.714238i	$0.746774\pi$
$264$	65.3769	0.0152412
$265$	0	0
$266$	3253.66	0.749981
$267$	−1768.02	−0.405248
$268$	−3728.67	−0.849868
$269$	3914.34	0.887217	0.443608	−	0.896221i	$-0.353698\pi$
$269$	3914.34	0.887217	0.443608	+	0.896221i	$0.353698\pi$
$270$	0	0
$271$	78.0865	0.0175034	0.00875170	−	0.999962i	$-0.497214\pi$
$271$	78.0865	0.0175034	0.00875170	+	0.999962i	$0.497214\pi$
$272$	−1181.99	−0.263487
$273$	−798.529	−0.177030
$274$	2505.54	0.552428
$275$	0	0
$276$	−760.969	−0.165960
$277$	76.0512	0.0164963	0.00824815	−	0.999966i	$-0.497375\pi$
$277$	76.0512	0.0164963	0.00824815	+	0.999966i	$0.497375\pi$
$278$	−5630.38	−1.21470
$279$	−5668.26	−1.21631
$280$	0	0
$281$	−3852.67	−0.817905	−0.408952	−	0.912556i	$-0.634106\pi$
$281$	−3852.67	−0.817905	−0.408952	+	0.912556i	$0.634106\pi$
$282$	−685.683	−0.144794
$283$	1550.13	0.325603	0.162802	−	0.986659i	$-0.447947\pi$
$283$	1550.13	0.325603	0.162802	+	0.986659i	$0.447947\pi$
$284$	−1053.35	−0.220086
$285$	0	0
$286$	8.34414	0.00172517
$287$	4667.06	0.959887
$288$	−1325.31	−0.271163
$289$	544.378	0.110804
$290$	0	0
$291$	−5200.74	−1.04767
$292$	−3601.60	−0.721808
$293$	−4236.01	−0.844608	−0.422304	−	0.906454i	$-0.638779\pi$
$293$	−4236.01	−0.844608	−0.422304	+	0.906454i	$0.638779\pi$
$294$	−2972.31	−0.589622
$295$	0	0
$296$	−1613.75	−0.316882
$297$	−117.810	−0.0230169
$298$	−4812.92	−0.935587
$299$	−97.1234	−0.0187853
$300$	0	0
$301$	−1236.89	−0.236855
$302$	−4371.17	−0.832890
$303$	−422.976	−0.0801958
$304$	1138.54	0.214801
$305$	0	0
$306$	6119.15	1.14317
$307$	10078.1	1.87357	0.936786	−	0.349903i	$-0.113785\pi$
$307$	10078.1	1.87357	0.936786	+	0.349903i	$0.113785\pi$
$308$	90.3505	0.0167149
$309$	−3003.26	−0.552911
$310$	0	0
$311$	−2652.97	−0.483717	−0.241859	−	0.970312i	$-0.577757\pi$
$311$	−2652.97	−0.483717	−0.241859	+	0.970312i	$0.577757\pi$
$312$	−279.425	−0.0507030
$313$	−6710.68	−1.21185	−0.605926	−	0.795521i	$-0.707197\pi$
$313$	−6710.68	−1.21185	−0.605926	+	0.795521i	$0.707197\pi$
$314$	2148.36	0.386111
$315$	0	0
$316$	−3824.73	−0.680879
$317$	3677.89	0.651642	0.325821	−	0.945431i	$-0.394359\pi$
$317$	3677.89	0.651642	0.325821	+	0.945431i	$0.394359\pi$
$318$	7090.44	1.25035
$319$	−27.5635	−0.00483781
$320$	0	0
$321$	−12527.3	−2.17821
$322$	−1051.66	−0.182008
$323$	−5256.77	−0.905556
$324$	−527.768	−0.0904952
$325$	0	0
$326$	5412.60	0.919558
$327$	6711.05	1.13493
$328$	1633.12	0.274920
$329$	−947.611	−0.158795
$330$	0	0
$331$	−9806.39	−1.62842	−0.814212	−	0.580568i	$-0.802831\pi$
$331$	−9806.39	−1.62842	−0.814212	+	0.580568i	$0.802831\pi$
$332$	−779.507	−0.128858
$333$	8354.38	1.37483
$334$	911.094	0.149260
$335$	0	0
$336$	−3025.62	−0.491253
$337$	7143.41	1.15468	0.577339	−	0.816505i	$-0.304091\pi$
$337$	7143.41	1.15468	0.577339	+	0.816505i	$0.304091\pi$
$338$	4358.34	0.701368
$339$	−102.975	−0.0164981
$340$	0	0
$341$	135.218	0.0214736
$342$	−5894.22	−0.931938
$343$	3733.97	0.587800
$344$	−432.819	−0.0678374
$345$	0	0
$346$	−2685.28	−0.417229
$347$	5160.20	0.798312	0.399156	−	0.916883i	$-0.369303\pi$
$347$	5160.20	0.798312	0.399156	+	0.916883i	$0.369303\pi$
$348$	923.037	0.142184
$349$	8712.04	1.33623	0.668115	−	0.744058i	$-0.267101\pi$
$349$	8712.04	1.33623	0.668115	+	0.744058i	$0.267101\pi$
$350$	0	0
$351$	503.527	0.0765706
$352$	31.6159	0.00478731
$353$	7117.30	1.07313	0.536566	−	0.843858i	$-0.319721\pi$
$353$	7117.30	1.07313	0.536566	+	0.843858i	$0.319721\pi$
$354$	2727.21	0.409463
$355$	0	0
$356$	−855.005	−0.127290
$357$	13969.7	2.07102
$358$	9311.27	1.37463
$359$	3548.73	0.521712	0.260856	−	0.965378i	$-0.415995\pi$
$359$	3548.73	0.521712	0.260856	+	0.965378i	$0.415995\pi$
$360$	0	0
$361$	−1795.46	−0.261767
$362$	352.639	0.0511998
$363$	−11001.2	−1.59066
$364$	−386.164	−0.0556057
$365$	0	0
$366$	−3114.15	−0.444751
$367$	−80.0110	−0.0113802	−0.00569011	−	0.999984i	$-0.501811\pi$
$367$	−80.0110	−0.0113802	−0.00569011	+	0.999984i	$0.501811\pi$
$368$	−368.000	−0.0521286
$369$	−8454.67	−1.19277
$370$	0	0
$371$	9798.95	1.37126
$372$	−4528.14	−0.631110
$373$	−3211.09	−0.445747	−0.222874	−	0.974847i	$-0.571544\pi$
$373$	−3211.09	−0.445747	−0.222874	+	0.974847i	$0.571544\pi$
$374$	−145.975	−0.0201823
$375$	0	0
$376$	−331.592	−0.0454802
$377$	117.808	0.0160940
$378$	5452.20	0.741881
$379$	−8177.57	−1.10832	−0.554160	−	0.832410i	$-0.686961\pi$
$379$	−8177.57	−1.10832	−0.554160	+	0.832410i	$0.686961\pi$
$380$	0	0
$381$	5232.48	0.703591
$382$	−3344.05	−0.447897
$383$	−12926.5	−1.72458	−0.862291	−	0.506413i	$-0.830971\pi$
$383$	−12926.5	−1.72458	−0.862291	+	0.506413i	$0.830971\pi$
$384$	−1058.74	−0.140699
$385$	0	0
$386$	−5908.89	−0.779157
$387$	2240.71	0.294320
$388$	−2515.05	−0.329078
$389$	4734.66	0.617113	0.308556	−	0.951206i	$-0.400154\pi$
$389$	4734.66	0.617113	0.308556	+	0.951206i	$0.400154\pi$
$390$	0	0
$391$	1699.10	0.219763
$392$	−1437.39	−0.185202
$393$	18554.0	2.38150
$394$	3634.71	0.464756
$395$	0	0
$396$	−163.676	−0.0207702
$397$	9938.80	1.25646	0.628229	−	0.778028i	$-0.283780\pi$
$397$	9938.80	1.25646	0.628229	+	0.778028i	$0.283780\pi$
$398$	−9561.98	−1.20427
$399$	−13456.2	−1.68835
$400$	0	0
$401$	−335.296	−0.0417553	−0.0208777	−	0.999782i	$-0.506646\pi$
$401$	−335.296	−0.0417553	−0.0208777	+	0.999782i	$0.506646\pi$
$402$	15420.7	1.91322
$403$	−577.932	−0.0714363
$404$	−204.549	−0.0251898
$405$	0	0
$406$	1275.63	0.155932
$407$	−199.297	−0.0242722
$408$	4888.34	0.593159
$409$	6560.85	0.793186	0.396593	−	0.917995i	$-0.370192\pi$
$409$	6560.85	0.793186	0.396593	+	0.917995i	$0.370192\pi$
$410$	0	0
$411$	−10362.2	−1.24362
$412$	−1452.36	−0.173671
$413$	3769.00	0.449056
$414$	1905.14	0.226166
$415$	0	0
$416$	−135.128	−0.0159260
$417$	23285.6	2.73453
$418$	140.609	0.0164531
$419$	7116.69	0.829768	0.414884	−	0.909874i	$-0.363822\pi$
$419$	7116.69	0.829768	0.414884	+	0.909874i	$0.363822\pi$
$420$	0	0
$421$	9799.43	1.13443	0.567215	−	0.823570i	$-0.308021\pi$
$421$	9799.43	1.13443	0.567215	+	0.823570i	$0.308021\pi$
$422$	−6403.44	−0.738660
$423$	1716.66	0.197321
$424$	3428.89	0.392740
$425$	0	0
$426$	4356.32	0.495456
$427$	−4303.73	−0.487757
$428$	−6058.13	−0.684184
$429$	−34.5089	−0.00388369
$430$	0	0
$431$	−10523.3	−1.17608	−0.588039	−	0.808832i	$-0.700100\pi$
$431$	−10523.3	−1.17608	−0.588039	+	0.808832i	$0.700100\pi$
$432$	1907.86	0.212481
$433$	4770.97	0.529511	0.264755	−	0.964316i	$-0.414709\pi$
$433$	4770.97	0.529511	0.264755	+	0.964316i	$0.414709\pi$
$434$	−6257.86	−0.692136
$435$	0	0
$436$	3245.42	0.356485
$437$	−1636.65	−0.179157
$438$	14895.2	1.62493
$439$	10231.2	1.11232	0.556159	−	0.831076i	$-0.312274\pi$
$439$	10231.2	1.11232	0.556159	+	0.831076i	$0.312274\pi$
$440$	0	0
$441$	7441.40	0.803520
$442$	623.905	0.0671405
$443$	−10508.1	−1.12699	−0.563493	−	0.826121i	$-0.690543\pi$
$443$	−10508.1	−1.12699	−0.563493	+	0.826121i	$0.690543\pi$
$444$	6673.97	0.713362
$445$	0	0
$446$	−3794.46	−0.402854
$447$	19904.8	2.10618
$448$	−1463.17	−0.154304
$449$	7230.71	0.759996	0.379998	−	0.924987i	$-0.375925\pi$
$449$	7230.71	0.759996	0.379998	+	0.924987i	$0.375925\pi$
$450$	0	0
$451$	201.689	0.0210580
$452$	−49.7983	−0.00518211
$453$	18077.9	1.87499
$454$	8936.83	0.923847
$455$	0	0
$456$	−4708.65	−0.483558
$457$	−2418.74	−0.247580	−0.123790	−	0.992308i	$-0.539505\pi$
$457$	−2418.74	−0.247580	−0.123790	+	0.992308i	$0.539505\pi$
$458$	10691.8	1.09081
$459$	−8808.84	−0.895776
$460$	0	0
$461$	−11342.8	−1.14596	−0.572979	−	0.819570i	$-0.694212\pi$
$461$	−11342.8	−1.14596	−0.572979	+	0.819570i	$0.694212\pi$
$462$	−373.663	−0.0376285
$463$	11785.9	1.18302	0.591508	−	0.806299i	$-0.298533\pi$
$463$	11785.9	1.18302	0.591508	+	0.806299i	$0.298533\pi$
$464$	446.375	0.0446604
$465$	0	0
$466$	−8960.93	−0.890787
$467$	−16294.5	−1.61461	−0.807303	−	0.590137i	$-0.799074\pi$
$467$	−16294.5	−1.61461	−0.807303	+	0.590137i	$0.799074\pi$
$468$	699.561	0.0690966
$469$	21311.3	2.09821
$470$	0	0
$471$	−8884.97	−0.869210
$472$	1318.86	0.128614
$473$	−53.4530	−0.00519613
$474$	15817.9	1.53279
$475$	0	0
$476$	6755.66	0.650515
$477$	−17751.4	−1.70394
$478$	−1559.67	−0.149242
$479$	−785.274	−0.0749062	−0.0374531	−	0.999298i	$-0.511924\pi$
$479$	−785.274	−0.0749062	−0.0374531	+	0.999298i	$0.511924\pi$
$480$	0	0
$481$	851.808	0.0807466
$482$	8409.79	0.794721
$483$	4349.33	0.409734
$484$	−5320.10	−0.499633
$485$	0	0
$486$	8621.72	0.804710
$487$	−8714.29	−0.810846	−0.405423	−	0.914129i	$-0.632876\pi$
$487$	−8714.29	−0.810846	−0.405423	+	0.914129i	$0.632876\pi$
$488$	−1505.98	−0.139698
$489$	−22384.9	−2.07010
$490$	0	0
$491$	−10943.1	−1.00582	−0.502909	−	0.864339i	$-0.667737\pi$
$491$	−10943.1	−1.00582	−0.502909	+	0.864339i	$0.667737\pi$
$492$	−6754.09	−0.618898
$493$	−2060.97	−0.188279
$494$	−600.971	−0.0547347
$495$	0	0
$496$	−2189.78	−0.198234
$497$	6020.41	0.543365
$498$	3223.81	0.290085
$499$	−5601.85	−0.502552	−0.251276	−	0.967916i	$-0.580850\pi$
$499$	−5601.85	−0.502552	−0.251276	+	0.967916i	$0.580850\pi$
$500$	0	0
$501$	−3768.01	−0.336013
$502$	−5522.95	−0.491039
$503$	−21689.8	−1.92266	−0.961331	−	0.275395i	$-0.911191\pi$
$503$	−21689.8	−1.92266	−0.961331	+	0.275395i	$0.911191\pi$
$504$	7574.86	0.669466
$505$	0	0
$506$	−45.4478	−0.00399289
$507$	−18024.8	−1.57891
$508$	2530.39	0.221000
$509$	13826.0	1.20399	0.601993	−	0.798501i	$-0.294373\pi$
$509$	13826.0	1.20399	0.601993	+	0.798501i	$0.294373\pi$
$510$	0	0
$511$	20585.0	1.78205
$512$	−512.000	−0.0441942
$513$	8485.04	0.730260
$514$	−1172.94	−0.100654
$515$	0	0
$516$	1790.01	0.152715
$517$	−40.9515	−0.00348364
$518$	9223.40	0.782341
$519$	11105.5	0.939263
$520$	0	0
$521$	−11971.1	−1.00664	−0.503322	−	0.864099i	$-0.667889\pi$
$521$	−11971.1	−1.00664	−0.503322	+	0.864099i	$0.667889\pi$
$522$	−2310.89	−0.193764
$523$	9668.82	0.808390	0.404195	−	0.914673i	$-0.367552\pi$
$523$	9668.82	0.808390	0.404195	+	0.914673i	$0.367552\pi$
$524$	8972.62	0.748036
$525$	0	0
$526$	11940.8	0.989813
$527$	10110.5	0.835712
$528$	−130.754	−0.0107771
$529$	529.000	0.0434783
$530$	0	0
$531$	−6827.78	−0.558004
$532$	−6507.33	−0.530317
$533$	−862.033	−0.0700540
$534$	3536.05	0.286554
$535$	0	0
$536$	7457.34	0.600948
$537$	−38508.6	−3.09454
$538$	−7828.67	−0.627357
$539$	−177.517	−0.0141859
$540$	0	0
$541$	6793.28	0.539863	0.269932	−	0.962880i	$-0.412999\pi$
$541$	6793.28	0.539863	0.269932	+	0.962880i	$0.412999\pi$
$542$	−156.173	−0.0123768
$543$	−1458.41	−0.115260
$544$	2363.97	0.186313
$545$	0	0
$546$	1597.06	0.125179
$547$	5584.91	0.436551	0.218276	−	0.975887i	$-0.429957\pi$
$547$	5584.91	0.436551	0.218276	+	0.975887i	$0.429957\pi$
$548$	−5011.08	−0.390626
$549$	7796.49	0.606095
$550$	0	0
$551$	1985.21	0.153490
$552$	1521.94	0.117351
$553$	21860.3	1.68100
$554$	−152.102	−0.0116646
$555$	0	0
$556$	11260.8	0.858925
$557$	−4172.24	−0.317385	−0.158693	−	0.987328i	$-0.550728\pi$
$557$	−4172.24	−0.317385	−0.158693	+	0.987328i	$0.550728\pi$
$558$	11336.5	0.860059
$559$	228.461	0.0172860
$560$	0	0
$561$	603.707	0.0454341
$562$	7705.34	0.578346
$563$	−7734.17	−0.578963	−0.289482	−	0.957184i	$-0.593483\pi$
$563$	−7734.17	−0.578963	−0.289482	+	0.957184i	$0.593483\pi$
$564$	1371.37	0.102385
$565$	0	0
$566$	−3100.26	−0.230236
$567$	3016.47	0.223421
$568$	2106.69	0.155625
$569$	1392.55	0.102599	0.0512995	−	0.998683i	$-0.483664\pi$
$569$	1392.55	0.102599	0.0512995	+	0.998683i	$0.483664\pi$
$570$	0	0
$571$	16996.1	1.24565	0.622823	−	0.782363i	$-0.285986\pi$
$571$	16996.1	1.24565	0.622823	+	0.782363i	$0.285986\pi$
$572$	−16.6883	−0.00121988
$573$	13830.0	1.00830
$574$	−9334.11	−0.678743
$575$	0	0
$576$	2650.63	0.191741
$577$	6849.23	0.494172	0.247086	−	0.968994i	$-0.420527\pi$
$577$	6849.23	0.494172	0.247086	+	0.968994i	$0.420527\pi$
$578$	−1088.76	−0.0783500
$579$	24437.4	1.75403
$580$	0	0
$581$	4455.29	0.318135
$582$	10401.5	0.740817
$583$	423.467	0.0300827
$584$	7203.21	0.510395
$585$	0	0
$586$	8472.01	0.597228
$587$	−6701.08	−0.471181	−0.235590	−	0.971852i	$-0.575702\pi$
$587$	−6701.08	−0.471181	−0.235590	+	0.971852i	$0.575702\pi$
$588$	5944.62	0.416925
$589$	−9738.85	−0.681294
$590$	0	0
$591$	−15032.1	−1.04625
$592$	3227.49	0.224070
$593$	−9898.21	−0.685448	−0.342724	−	0.939436i	$-0.611350\pi$
$593$	−9898.21	−0.685448	−0.342724	+	0.939436i	$0.611350\pi$
$594$	235.620	0.0162754
$595$	0	0
$596$	9625.84	0.661560
$597$	39545.5	2.71104
$598$	194.247	0.0132832
$599$	13149.2	0.896934	0.448467	−	0.893799i	$-0.351970\pi$
$599$	13149.2	0.896934	0.448467	+	0.893799i	$0.351970\pi$
$600$	0	0
$601$	27142.4	1.84220	0.921100	−	0.389326i	$-0.127292\pi$
$601$	27142.4	1.84220	0.921100	+	0.389326i	$0.127292\pi$
$602$	2473.79	0.167482
$603$	−38606.7	−2.60728
$604$	8742.34	0.588942
$605$	0	0
$606$	845.952	0.0567070
$607$	6708.40	0.448576	0.224288	−	0.974523i	$-0.427994\pi$
$607$	6708.40	0.448576	0.224288	+	0.974523i	$0.427994\pi$
$608$	−2277.07	−0.151887
$609$	−5275.63	−0.351033
$610$	0	0
$611$	175.029	0.0115891
$612$	−12238.3	−0.808340
$613$	17755.6	1.16989	0.584943	−	0.811074i	$-0.301117\pi$
$613$	17755.6	1.16989	0.584943	+	0.811074i	$0.301117\pi$
$614$	−20156.2	−1.32482
$615$	0	0
$616$	−180.701	−0.0118192
$617$	−14555.4	−0.949722	−0.474861	−	0.880061i	$-0.657502\pi$
$617$	−14555.4	−0.949722	−0.474861	+	0.880061i	$0.657502\pi$
$618$	6006.52	0.390967
$619$	29452.5	1.91243	0.956215	−	0.292665i	$-0.0945422\pi$
$619$	29452.5	1.91243	0.956215	+	0.292665i	$0.0945422\pi$
$620$	0	0
$621$	−2742.55	−0.177222
$622$	5305.94	0.342040
$623$	4886.80	0.314262
$624$	558.850	0.0358524
$625$	0	0
$626$	13421.4	0.856909
$627$	−581.516	−0.0370391
$628$	−4296.72	−0.273022
$629$	−14901.8	−0.944629
$630$	0	0
$631$	9911.19	0.625290	0.312645	−	0.949870i	$-0.398785\pi$
$631$	9911.19	0.625290	0.312645	+	0.949870i	$0.398785\pi$
$632$	7649.46	0.481454
$633$	26482.7	1.66286
$634$	−7355.77	−0.460781
$635$	0	0
$636$	−14180.9	−0.884133
$637$	758.720	0.0471924
$638$	55.1271	0.00342085
$639$	−10906.4	−0.675194
$640$	0	0
$641$	−13548.5	−0.834844	−0.417422	−	0.908713i	$-0.637066\pi$
$641$	−13548.5	−0.834844	−0.417422	+	0.908713i	$0.637066\pi$
$642$	25054.6	1.54023
$643$	−2254.31	−0.138260	−0.0691302	−	0.997608i	$-0.522022\pi$
$643$	−2254.31	−0.138260	−0.0691302	+	0.997608i	$0.522022\pi$
$644$	2103.31	0.128699
$645$	0	0
$646$	10513.5	0.640325
$647$	17967.1	1.09175	0.545874	−	0.837868i	$-0.316198\pi$
$647$	17967.1	1.09175	0.545874	+	0.837868i	$0.316198\pi$
$648$	1055.54	0.0639897
$649$	162.879	0.00985141
$650$	0	0
$651$	25880.6	1.55813
$652$	−10825.2	−0.650226
$653$	−365.239	−0.0218881	−0.0109440	−	0.999940i	$-0.503484\pi$
$653$	−365.239	−0.0218881	−0.0109440	+	0.999940i	$0.503484\pi$
$654$	−13422.1	−0.802517
$655$	0	0
$656$	−3266.24	−0.194398
$657$	−37291.1	−2.21440
$658$	1895.22	0.112285
$659$	29229.1	1.72778	0.863889	−	0.503683i	$-0.168022\pi$
$659$	29229.1	1.72778	0.863889	+	0.503683i	$0.168022\pi$
$660$	0	0
$661$	−23681.1	−1.39348	−0.696740	−	0.717324i	$-0.745367\pi$
$661$	−23681.1	−1.39348	−0.696740	+	0.717324i	$0.745367\pi$
$662$	19612.8	1.15147
$663$	−2580.28	−0.151146
$664$	1559.01	0.0911167
$665$	0	0
$666$	−16708.8	−0.972150
$667$	−641.664	−0.0372494
$668$	−1822.19	−0.105543
$669$	15692.7	0.906900
$670$	0	0
$671$	−185.988	−0.0107004
$672$	6051.24	0.347369
$673$	16016.2	0.917351	0.458676	−	0.888604i	$-0.348324\pi$
$673$	16016.2	0.917351	0.458676	+	0.888604i	$0.348324\pi$
$674$	−14286.8	−0.816480
$675$	0	0
$676$	−8716.67	−0.495942
$677$	24082.5	1.36715	0.683577	−	0.729878i	$-0.260423\pi$
$677$	24082.5	1.36715	0.683577	+	0.729878i	$0.260423\pi$
$678$	205.951	0.0116659
$679$	14374.8	0.812451
$680$	0	0
$681$	−36960.1	−2.07975
$682$	−270.437	−0.0151841
$683$	24224.1	1.35711	0.678556	−	0.734549i	$-0.262606\pi$
$683$	24224.1	1.35711	0.678556	+	0.734549i	$0.262606\pi$
$684$	11788.4	0.658980
$685$	0	0
$686$	−7467.94	−0.415637
$687$	−44217.9	−2.45563
$688$	865.639	0.0479683
$689$	−1809.92	−0.100076
$690$	0	0
$691$	13948.7	0.767919	0.383960	−	0.923350i	$-0.374560\pi$
$691$	13948.7	0.767919	0.383960	+	0.923350i	$0.374560\pi$
$692$	5370.55	0.295026
$693$	935.491	0.0512790
$694$	−10320.4	−0.564492
$695$	0	0
$696$	−1846.07	−0.100539
$697$	15080.6	0.819540
$698$	−17424.1	−0.944858
$699$	37059.7	2.00533
$700$	0	0
$701$	−33334.1	−1.79602	−0.898011	−	0.439973i	$-0.854988\pi$
$701$	−33334.1	−1.79602	−0.898011	+	0.439973i	$0.854988\pi$
$702$	−1007.05	−0.0541436
$703$	14354.0	0.770087
$704$	−63.2317	−0.00338514
$705$	0	0
$706$	−14234.6	−0.758819
$707$	1169.10	0.0621903
$708$	−5454.43	−0.289534
$709$	−7107.92	−0.376507	−0.188253	−	0.982120i	$-0.560283\pi$
$709$	−7107.92	−0.376507	−0.188253	+	0.982120i	$0.560283\pi$
$710$	0	0
$711$	−39601.3	−2.08884
$712$	1710.01	0.0900075
$713$	3147.81	0.165338
$714$	−27939.4	−1.46443
$715$	0	0
$716$	−18622.5	−0.972007
$717$	6450.34	0.335972
$718$	−7097.45	−0.368906
$719$	−13003.1	−0.674456	−0.337228	−	0.941423i	$-0.609489\pi$
$719$	−13003.1	−0.674456	−0.337228	+	0.941423i	$0.609489\pi$
$720$	0	0
$721$	8300.98	0.428772
$722$	3590.92	0.185097
$723$	−34780.4	−1.78907
$724$	−705.279	−0.0362037
$725$	0	0
$726$	22002.3	1.12477
$727$	−5524.86	−0.281851	−0.140926	−	0.990020i	$-0.545008\pi$
$727$	−5524.86	−0.281851	−0.140926	+	0.990020i	$0.545008\pi$
$728$	772.328	0.0393192
$729$	−32094.4	−1.63057
$730$	0	0
$731$	−3996.77	−0.202224
$732$	6228.29	0.314487
$733$	−22106.6	−1.11395	−0.556976	−	0.830529i	$-0.688038\pi$
$733$	−22106.6	−1.11395	−0.556976	+	0.830529i	$0.688038\pi$
$734$	160.022	0.00804703
$735$	0	0
$736$	736.000	0.0368605
$737$	920.977	0.0460307
$738$	16909.3	0.843416
$739$	−6619.82	−0.329518	−0.164759	−	0.986334i	$-0.552685\pi$
$739$	−6619.82	−0.329518	−0.164759	+	0.986334i	$0.552685\pi$
$740$	0	0
$741$	2485.44	0.123218
$742$	−19597.9	−0.969625
$743$	−18894.1	−0.932919	−0.466460	−	0.884543i	$-0.654471\pi$
$743$	−18894.1	−0.932919	−0.466460	+	0.884543i	$0.654471\pi$
$744$	9056.27	0.446262
$745$	0	0
$746$	6422.17	0.315191
$747$	−8071.03	−0.395320
$748$	291.949	0.0142710
$749$	34625.3	1.68916
$750$	0	0
$751$	4931.32	0.239609	0.119805	−	0.992797i	$-0.461773\pi$
$751$	4931.32	0.239609	0.119805	+	0.992797i	$0.461773\pi$
$752$	663.185	0.0321594
$753$	22841.3	1.10542
$754$	−235.617	−0.0113802
$755$	0	0
$756$	−10904.4	−0.524589
$757$	−6248.31	−0.299998	−0.149999	−	0.988686i	$-0.547927\pi$
$757$	−6248.31	−0.299998	−0.149999	+	0.988686i	$0.547927\pi$
$758$	16355.1	0.783701
$759$	187.959	0.00898875
$760$	0	0
$761$	−18867.0	−0.898723	−0.449361	−	0.893350i	$-0.648348\pi$
$761$	−18867.0	−0.898723	−0.449361	+	0.893350i	$0.648348\pi$
$762$	−10465.0	−0.497514
$763$	−18549.3	−0.880117
$764$	6688.11	0.316711
$765$	0	0
$766$	25853.1	1.21946
$767$	−696.156	−0.0327728
$768$	2117.48	0.0994895
$769$	−4752.05	−0.222839	−0.111420	−	0.993773i	$-0.535540\pi$
$769$	−4752.05	−0.222839	−0.111420	+	0.993773i	$0.535540\pi$
$770$	0	0
$771$	4850.92	0.226591
$772$	11817.8	0.550947
$773$	−7050.12	−0.328040	−0.164020	−	0.986457i	$-0.552446\pi$
$773$	−7050.12	−0.328040	−0.164020	+	0.986457i	$0.552446\pi$
$774$	−4481.42	−0.208115
$775$	0	0
$776$	5030.10	0.232693
$777$	−38145.2	−1.76120
$778$	−9469.32	−0.436364
$779$	−14526.3	−0.668111
$780$	0	0
$781$	260.175	0.0119204
$782$	−3398.21	−0.155396
$783$	3326.64	0.151832
$784$	2874.78	0.130958
$785$	0	0
$786$	−37108.1	−1.68397
$787$	−19524.7	−0.884345	−0.442173	−	0.896930i	$-0.645792\pi$
$787$	−19524.7	−0.884345	−0.442173	+	0.896930i	$0.645792\pi$
$788$	−7269.41	−0.328632
$789$	−49383.4	−2.22826
$790$	0	0
$791$	284.623	0.0127940
$792$	327.351	0.0146868
$793$	794.925	0.0355972
$794$	−19877.6	−0.888450
$795$	0	0
$796$	19124.0	0.851546
$797$	25078.7	1.11459	0.557297	−	0.830313i	$-0.311838\pi$
$797$	25078.7	1.11459	0.557297	+	0.830313i	$0.311838\pi$
$798$	26912.4	1.19384
$799$	−3062.01	−0.135577
$800$	0	0
$801$	−8852.74	−0.390507
$802$	670.592	0.0295255
$803$	889.592	0.0390947
$804$	−30841.3	−1.35285
$805$	0	0
$806$	1155.86	0.0505131
$807$	32377.0	1.41230
$808$	409.097	0.0178119
$809$	11722.0	0.509425	0.254712	−	0.967017i	$-0.418019\pi$
$809$	11722.0	0.509425	0.254712	+	0.967017i	$0.418019\pi$
$810$	0	0
$811$	−3540.61	−0.153302	−0.0766509	−	0.997058i	$-0.524423\pi$
$811$	−3540.61	−0.153302	−0.0766509	+	0.997058i	$0.524423\pi$
$812$	−2551.26	−0.110261
$813$	645.885	0.0278625
$814$	398.594	0.0171630
$815$	0	0
$816$	−9776.67	−0.419427
$817$	3849.85	0.164858
$818$	−13121.7	−0.560867
$819$	−3998.35	−0.170591
$820$	0	0
$821$	−43675.7	−1.85663	−0.928315	−	0.371794i	$-0.878743\pi$
$821$	−43675.7	−1.85663	−0.928315	+	0.371794i	$0.878743\pi$
$822$	20724.3	0.879373
$823$	28732.5	1.21695	0.608476	−	0.793572i	$-0.291781\pi$
$823$	28732.5	1.21695	0.608476	+	0.793572i	$0.291781\pi$
$824$	2904.72	0.122804
$825$	0	0
$826$	−7537.99	−0.317531
$827$	−33810.2	−1.42164	−0.710821	−	0.703373i	$-0.751676\pi$
$827$	−33810.2	−1.42164	−0.710821	+	0.703373i	$0.751676\pi$
$828$	−3810.28	−0.159923
$829$	−34664.8	−1.45230	−0.726150	−	0.687536i	$-0.758692\pi$
$829$	−34664.8	−1.45230	−0.726150	+	0.687536i	$0.758692\pi$
$830$	0	0
$831$	629.050	0.0262593
$832$	270.257	0.0112614
$833$	−13273.3	−0.552090
$834$	−46571.1	−1.93360
$835$	0	0
$836$	−281.217	−0.0116341
$837$	−16319.5	−0.673936
$838$	−14233.4	−0.586735
$839$	11115.4	0.457384	0.228692	−	0.973499i	$-0.426555\pi$
$839$	11115.4	0.457384	0.228692	+	0.973499i	$0.426555\pi$
$840$	0	0
$841$	−23610.7	−0.968087
$842$	−19598.9	−0.802163
$843$	−31867.0	−1.30197
$844$	12806.9	0.522312
$845$	0	0
$846$	−3433.31	−0.139527
$847$	30407.1	1.23353
$848$	−6857.79	−0.277709
$849$	12821.8	0.518306
$850$	0	0
$851$	−4639.52	−0.186887
$852$	−8712.64	−0.350341
$853$	−23695.9	−0.951153	−0.475577	−	0.879674i	$-0.657761\pi$
$853$	−23695.9	−0.951153	−0.475577	+	0.879674i	$0.657761\pi$
$854$	8607.46	0.344896
$855$	0	0
$856$	12116.3	0.483791
$857$	20605.9	0.821336	0.410668	−	0.911785i	$-0.365295\pi$
$857$	20605.9	0.821336	0.410668	+	0.911785i	$0.365295\pi$
$858$	69.0177	0.00274618
$859$	−43187.1	−1.71540	−0.857698	−	0.514154i	$-0.828106\pi$
$859$	−43187.1	−1.71540	−0.857698	+	0.514154i	$0.828106\pi$
$860$	0	0
$861$	38603.1	1.52798
$862$	21046.6	0.831613
$863$	47551.7	1.87564	0.937822	−	0.347117i	$-0.112839\pi$
$863$	47551.7	1.87564	0.937822	+	0.347117i	$0.112839\pi$
$864$	−3815.72	−0.150247
$865$	0	0
$866$	−9541.94	−0.374421
$867$	4502.77	0.176381
$868$	12515.7	0.489414
$869$	944.704	0.0368779
$870$	0	0
$871$	−3936.32	−0.153131
$872$	−6490.85	−0.252073
$873$	−26040.9	−1.00956
$874$	3273.29	0.126683
$875$	0	0
$876$	−29790.3	−1.14900
$877$	−8116.63	−0.312519	−0.156260	−	0.987716i	$-0.549944\pi$
$877$	−8116.63	−0.312519	−0.156260	+	0.987716i	$0.549944\pi$
$878$	−20462.4	−0.786528
$879$	−35037.7	−1.34447
$880$	0	0
$881$	24256.1	0.927594	0.463797	−	0.885942i	$-0.346487\pi$
$881$	24256.1	0.927594	0.463797	+	0.885942i	$0.346487\pi$
$882$	−14882.8	−0.568174
$883$	46859.3	1.78589	0.892945	−	0.450167i	$-0.148635\pi$
$883$	46859.3	1.78589	0.892945	+	0.450167i	$0.148635\pi$
$884$	−1247.81	−0.0474755
$885$	0	0
$886$	21016.2	0.796899
$887$	14372.4	0.544057	0.272028	−	0.962289i	$-0.412306\pi$
$887$	14372.4	0.544057	0.272028	+	0.962289i	$0.412306\pi$
$888$	−13347.9	−0.504423
$889$	−14462.5	−0.545621
$890$	0	0
$891$	130.358	0.00490142
$892$	7588.91	0.284860
$893$	2949.45	0.110526
$894$	−39809.6	−1.48930
$895$	0	0
$896$	2926.34	0.109110
$897$	−803.347	−0.0299030
$898$	−14461.4	−0.537398
$899$	−3818.21	−0.141651
$900$	0	0
$901$	31663.3	1.17076
$902$	−403.378	−0.0148903
$903$	−10230.8	−0.377033
$904$	99.5965	0.00366431
$905$	0	0
$906$	−36155.7	−1.32582
$907$	17332.1	0.634512	0.317256	−	0.948340i	$-0.397239\pi$
$907$	17332.1	0.634512	0.317256	+	0.948340i	$0.397239\pi$
$908$	−17873.7	−0.653258
$909$	−2117.90	−0.0772787
$910$	0	0
$911$	44304.9	1.61129	0.805646	−	0.592397i	$-0.201818\pi$
$911$	44304.9	1.61129	0.805646	+	0.592397i	$0.201818\pi$
$912$	9417.30	0.341927
$913$	192.537	0.00697926
$914$	4837.49	0.175066
$915$	0	0
$916$	−21383.5	−0.771322
$917$	−51283.2	−1.84680
$918$	17617.7	0.633410
$919$	−4764.32	−0.171013	−0.0855063	−	0.996338i	$-0.527251\pi$
$919$	−4764.32	−0.171013	−0.0855063	+	0.996338i	$0.527251\pi$
$920$	0	0
$921$	83359.9	2.98241
$922$	22685.6	0.810315
$923$	−1112.01	−0.0396556
$924$	747.325	0.0266074
$925$	0	0
$926$	−23571.8	−0.836519
$927$	−15037.7	−0.532799
$928$	−892.750	−0.0315797
$929$	−23226.2	−0.820267	−0.410134	−	0.912025i	$-0.634518\pi$
$929$	−23226.2	−0.820267	−0.410134	+	0.912025i	$0.634518\pi$
$930$	0	0
$931$	12785.3	0.450078
$932$	17921.9	0.629882
$933$	−21943.8	−0.769996
$934$	32589.1	1.14170
$935$	0	0
$936$	−1399.12	−0.0488587
$937$	23752.5	0.828133	0.414066	−	0.910247i	$-0.364108\pi$
$937$	23752.5	0.828133	0.414066	+	0.910247i	$0.364108\pi$
$938$	−42622.5	−1.48366
$939$	−55506.7	−1.92907
$940$	0	0
$941$	−23210.4	−0.804079	−0.402040	−	0.915622i	$-0.631699\pi$
$941$	−23210.4	−0.804079	−0.402040	+	0.915622i	$0.631699\pi$
$942$	17769.9	0.614624
$943$	4695.21	0.162139
$944$	−2637.73	−0.0909436
$945$	0	0
$946$	106.906	0.00367422
$947$	7778.59	0.266917	0.133458	−	0.991054i	$-0.457392\pi$
$947$	7778.59	0.266917	0.133458	+	0.991054i	$0.457392\pi$
$948$	−31635.9	−1.08385
$949$	−3802.18	−0.130057
$950$	0	0
$951$	30421.3	1.03731
$952$	−13511.3	−0.459983
$953$	−3979.52	−0.135267	−0.0676334	−	0.997710i	$-0.521545\pi$
$953$	−3979.52	−0.135267	−0.0676334	+	0.997710i	$0.521545\pi$
$954$	35502.8	1.20487
$955$	0	0
$956$	3119.34	0.105530
$957$	−227.989	−0.00770098
$958$	1570.55	0.0529667
$959$	28640.9	0.964405
$960$	0	0
$961$	−11060.0	−0.371254
$962$	−1703.62	−0.0570964
$963$	−62726.0	−2.09898
$964$	−16819.6	−0.561952
$965$	0	0
$966$	−8698.66	−0.289726
$967$	−17325.9	−0.576178	−0.288089	−	0.957604i	$-0.593020\pi$
$967$	−17325.9	−0.576178	−0.288089	+	0.957604i	$0.593020\pi$
$968$	10640.2	0.353294
$969$	−43480.9	−1.44149
$970$	0	0
$971$	20183.0	0.667048	0.333524	−	0.942742i	$-0.391762\pi$
$971$	20183.0	0.667048	0.333524	+	0.942742i	$0.391762\pi$
$972$	−17243.4	−0.569016
$973$	−64361.0	−2.12058
$974$	17428.6	0.573355
$975$	0	0
$976$	3011.96	0.0987814
$977$	−47185.7	−1.54514	−0.772571	−	0.634928i	$-0.781030\pi$
$977$	−47185.7	−1.54514	−0.772571	+	0.634928i	$0.781030\pi$
$978$	44769.8	1.46378
$979$	211.185	0.00689430
$980$	0	0
$981$	33603.2	1.09365
$982$	21886.3	0.711221
$983$	7347.90	0.238415	0.119207	−	0.992869i	$-0.461965\pi$
$983$	7347.90	0.238415	0.119207	+	0.992869i	$0.461965\pi$
$984$	13508.2	0.437627
$985$	0	0
$986$	4121.94	0.133133
$987$	−7838.07	−0.252775
$988$	1201.94	0.0387033
$989$	−1244.36	−0.0400083
$990$	0	0
$991$	34327.0	1.10034	0.550168	−	0.835054i	$-0.314564\pi$
$991$	34327.0	1.10034	0.550168	+	0.835054i	$0.314564\pi$
$992$	4379.56	0.140172
$993$	−81112.6	−2.59218
$994$	−12040.8	−0.384217
$995$	0	0
$996$	−6447.62	−0.205121
$997$	−37611.0	−1.19474	−0.597368	−	0.801967i	$-0.703787\pi$
$997$	−37611.0	−1.19474	−0.597368	+	0.801967i	$0.703787\pi$
$998$	11203.7	0.355358
$999$	24053.1	0.761770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.q.1.5	✓	5
5.2	odd	4		1150.4.b.r.599.1		10
5.3	odd	4		1150.4.b.r.599.10		10
5.4	even	2		1150.4.a.v.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.q.1.5	✓	5	1.1	even	1	trivial
1150.4.a.v.1.1	yes	5	5.4	even	2
1150.4.b.r.599.1		10	5.2	odd	4
1150.4.b.r.599.10		10	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +8.27140 q^{3} +4.00000 q^{4} -16.5428 q^{6} -22.8621 q^{7} -8.00000 q^{8} +41.4161 q^{9} -0.987996 q^{11} +33.0856 q^{12} +4.22276 q^{13} +45.7241 q^{14} +16.0000 q^{16} -73.8741 q^{17} -82.8322 q^{18} +71.1586 q^{19} -189.101 q^{21} +1.97599 q^{22} -23.0000 q^{23} -66.1712 q^{24} -8.44552 q^{26} +119.241 q^{27} -91.4483 q^{28} +27.8984 q^{29} -136.861 q^{31} -32.0000 q^{32} -8.17211 q^{33} +147.748 q^{34} +165.664 q^{36} +201.718 q^{37} -142.317 q^{38} +34.9281 q^{39} -204.140 q^{41} +378.203 q^{42} +54.1024 q^{43} -3.95198 q^{44} +46.0000 q^{46} +41.4490 q^{47} +132.342 q^{48} +179.674 q^{49} -611.042 q^{51} +16.8910 q^{52} -428.612 q^{53} -238.483 q^{54} +182.897 q^{56} +588.581 q^{57} -55.7969 q^{58} -164.858 q^{59} +188.248 q^{61} +273.722 q^{62} -946.857 q^{63} +64.0000 q^{64} +16.3442 q^{66} -932.167 q^{67} -295.496 q^{68} -190.242 q^{69} -263.336 q^{71} -331.329 q^{72} -900.401 q^{73} -403.437 q^{74} +284.634 q^{76} +22.5876 q^{77} -69.8563 q^{78} -956.182 q^{79} -131.942 q^{81} +408.279 q^{82} -194.877 q^{83} -756.405 q^{84} -108.205 q^{86} +230.759 q^{87} +7.90397 q^{88} -213.751 q^{89} -96.5410 q^{91} -92.0000 q^{92} -1132.03 q^{93} -82.8981 q^{94} -264.685 q^{96} -628.762 q^{97} -359.348 q^{98} -40.9189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 - 5 * q^3 + 20 * q^4 + 10 * q^6 + 3 * q^7 - 40 * q^8 + 84 * q^9 - 26 * q^11 - 20 * q^12 + 61 * q^13 - 6 * q^14 + 80 * q^16 - 231 * q^17 - 168 * q^18 + 74 * q^19 - 88 * q^21 + 52 * q^22 - 115 * q^23 + 40 * q^24 - 122 * q^26 - 368 * q^27 + 12 * q^28 + 37 * q^29 + 220 * q^31 - 160 * q^32 - 391 * q^33 + 462 * q^34 + 336 * q^36 - 220 * q^37 - 148 * q^38 + 195 * q^39 + 112 * q^41 + 176 * q^42 - 97 * q^43 - 104 * q^44 + 230 * q^46 - 370 * q^47 - 80 * q^48 + 182 * q^49 + 563 * q^51 + 244 * q^52 - 1224 * q^53 + 736 * q^54 - 24 * q^56 + 1009 * q^57 - 74 * q^58 - 181 * q^59 + 1574 * q^61 - 440 * q^62 - 1829 * q^63 + 320 * q^64 + 782 * q^66 + 435 * q^67 - 924 * q^68 + 115 * q^69 - 30 * q^71 - 672 * q^72 - 2180 * q^73 + 440 * q^74 + 296 * q^76 + 1401 * q^77 - 390 * q^78 + 609 * q^79 + 1453 * q^81 - 224 * q^82 - 2018 * q^83 - 352 * q^84 + 194 * q^86 + 1485 * q^87 + 208 * q^88 - 1663 * q^89 + 361 * q^91 - 460 * q^92 - 2741 * q^93 + 740 * q^94 + 160 * q^96 + 1274 * q^97 - 364 * q^98 + 2397 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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