LMFDB - Embedded newform 495.4.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	495.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:	$29.2059454528$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.77801 q^{2} +6.27334 q^{4} +5.00000 q^{5} -28.7933 q^{7} +6.52332 q^{8} +O(q^{10})$	\(q-3.77801 q^{2} +6.27334 q^{4} +5.00000 q^{5} -28.7933 q^{7} +6.52332 q^{8} -18.8900 q^{10} +11.0000 q^{11} +47.1426 q^{13} +108.781 q^{14} -74.8319 q^{16} -108.893 q^{17} +121.335 q^{19} +31.3667 q^{20} -41.5581 q^{22} +40.0253 q^{23} +25.0000 q^{25} -178.105 q^{26} -180.630 q^{28} -7.21200 q^{29} -127.903 q^{31} +230.529 q^{32} +411.400 q^{34} -143.966 q^{35} -227.172 q^{37} -458.403 q^{38} +32.6166 q^{40} +317.456 q^{41} +311.391 q^{43} +69.0068 q^{44} -151.216 q^{46} +16.1134 q^{47} +486.053 q^{49} -94.4502 q^{50} +295.742 q^{52} -119.281 q^{53} +55.0000 q^{55} -187.828 q^{56} +27.2470 q^{58} -20.5979 q^{59} +593.927 q^{61} +483.217 q^{62} -272.285 q^{64} +235.713 q^{65} -612.887 q^{67} -683.125 q^{68} +543.906 q^{70} -162.987 q^{71} -942.575 q^{73} +858.257 q^{74} +761.174 q^{76} -316.726 q^{77} -1086.48 q^{79} -374.160 q^{80} -1199.35 q^{82} -132.951 q^{83} -544.467 q^{85} -1176.44 q^{86} +71.7565 q^{88} -1558.43 q^{89} -1357.39 q^{91} +251.092 q^{92} -60.8767 q^{94} +606.673 q^{95} -1062.26 q^{97} -1836.31 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} + 23 q^{4} + 15 q^{5} - 15 q^{7} - 33 q^{8}+O(q^{10}) $ 3 * q - 5 * q^2 + 23 * q^4 + 15 * q^5 - 15 * q^7 - 33 * q^8	$ 3 q - 5 q^{2} + 23 q^{4} + 15 q^{5} - 15 q^{7} - 33 q^{8} - 25 q^{10} + 33 q^{11} - 14 q^{13} - 111 q^{14} - 149 q^{16} - 91 q^{17} - 19 q^{19} + 115 q^{20} - 55 q^{22} - 330 q^{23} + 75 q^{25} + 218 q^{26} + 193 q^{28} - 413 q^{29} - 127 q^{31} + 251 q^{32} + 477 q^{34} - 75 q^{35} - 543 q^{37} - 531 q^{38} - 165 q^{40} - 58 q^{41} + 532 q^{43} + 253 q^{44} + 270 q^{46} - 478 q^{47} + 1292 q^{49} - 125 q^{50} - 654 q^{52} - 261 q^{53} + 165 q^{55} - 695 q^{56} + 195 q^{58} + 630 q^{59} + 1095 q^{61} + 545 q^{62} - 953 q^{64} - 70 q^{65} - 976 q^{67} - 627 q^{68} - 555 q^{70} + 445 q^{71} - 1838 q^{73} + 577 q^{74} - 219 q^{76} - 165 q^{77} + 318 q^{79} - 745 q^{80} - 1738 q^{82} + 330 q^{83} - 455 q^{85} + 1172 q^{86} - 363 q^{88} - 2573 q^{89} - 4150 q^{91} - 3082 q^{92} - 310 q^{94} - 95 q^{95} - 2744 q^{97} - 5244 q^{98}+O(q^{100}) $ 3 * q - 5 * q^2 + 23 * q^4 + 15 * q^5 - 15 * q^7 - 33 * q^8 - 25 * q^10 + 33 * q^11 - 14 * q^13 - 111 * q^14 - 149 * q^16 - 91 * q^17 - 19 * q^19 + 115 * q^20 - 55 * q^22 - 330 * q^23 + 75 * q^25 + 218 * q^26 + 193 * q^28 - 413 * q^29 - 127 * q^31 + 251 * q^32 + 477 * q^34 - 75 * q^35 - 543 * q^37 - 531 * q^38 - 165 * q^40 - 58 * q^41 + 532 * q^43 + 253 * q^44 + 270 * q^46 - 478 * q^47 + 1292 * q^49 - 125 * q^50 - 654 * q^52 - 261 * q^53 + 165 * q^55 - 695 * q^56 + 195 * q^58 + 630 * q^59 + 1095 * q^61 + 545 * q^62 - 953 * q^64 - 70 * q^65 - 976 * q^67 - 627 * q^68 - 555 * q^70 + 445 * q^71 - 1838 * q^73 + 577 * q^74 - 219 * q^76 - 165 * q^77 + 318 * q^79 - 745 * q^80 - 1738 * q^82 + 330 * q^83 - 455 * q^85 + 1172 * q^86 - 363 * q^88 - 2573 * q^89 - 4150 * q^91 - 3082 * q^92 - 310 * q^94 - 95 * q^95 - 2744 * q^97 - 5244 * 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 $

$2$	−3.77801	−1.33573	−0.667864	−	0.744284i	$-0.732791\pi$
$2$	−3.77801	−1.33573	−0.667864	+	0.744284i	$0.732791\pi$
$3$	0	0
$3$	0	0
$4$	6.27334	0.784168
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−28.7933	−1.55469	−0.777345	−	0.629074i	$-0.783434\pi$
$7$	−28.7933	−1.55469	−0.777345	+	0.629074i	$0.783434\pi$
$8$	6.52332	0.288293
$9$	0	0
$10$	−18.8900	−0.597355
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	47.1426	1.00577	0.502884	−	0.864354i	$-0.332272\pi$
$13$	47.1426	1.00577	0.502884	+	0.864354i	$0.332272\pi$
$14$	108.781	2.07664
$15$	0	0
$16$	−74.8319	−1.16925
$17$	−108.893	−1.55356	−0.776780	−	0.629772i	$-0.783148\pi$
$17$	−108.893	−1.55356	−0.776780	+	0.629772i	$0.783148\pi$
$18$	0	0
$19$	121.335	1.46506	0.732529	−	0.680736i	$-0.238340\pi$
$19$	121.335	1.46506	0.732529	+	0.680736i	$0.238340\pi$
$20$	31.3667	0.350691
$21$	0	0
$22$	−41.5581	−0.402737
$23$	40.0253	0.362863	0.181431	−	0.983404i	$-0.441927\pi$
$23$	40.0253	0.362863	0.181431	+	0.983404i	$0.441927\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−178.105	−1.34343
$27$	0	0
$28$	−180.630	−1.21914
$29$	−7.21200	−0.0461805	−0.0230902	−	0.999733i	$-0.507351\pi$
$29$	−7.21200	−0.0461805	−0.0230902	+	0.999733i	$0.507351\pi$
$30$	0	0
$31$	−127.903	−0.741032	−0.370516	−	0.928826i	$-0.620819\pi$
$31$	−127.903	−0.741032	−0.370516	+	0.928826i	$0.620819\pi$
$32$	230.529	1.27350
$33$	0	0
$34$	411.400	2.07513
$35$	−143.966	−0.695279
$36$	0	0
$37$	−227.172	−1.00937	−0.504687	−	0.863302i	$-0.668392\pi$
$37$	−227.172	−1.00937	−0.504687	+	0.863302i	$0.668392\pi$
$38$	−458.403	−1.95692
$39$	0	0
$40$	32.6166	0.128928
$41$	317.456	1.20923	0.604614	−	0.796519i	$-0.293328\pi$
$41$	317.456	1.20923	0.604614	+	0.796519i	$0.293328\pi$
$42$	0	0
$43$	311.391	1.10434	0.552171	−	0.833731i	$-0.313799\pi$
$43$	311.391	1.10434	0.552171	+	0.833731i	$0.313799\pi$
$44$	69.0068	0.236436
$45$	0	0
$46$	−151.216	−0.484686
$47$	16.1134	0.0500083	0.0250041	−	0.999687i	$-0.492040\pi$
$47$	16.1134	0.0500083	0.0250041	+	0.999687i	$0.492040\pi$
$48$	0	0
$49$	486.053	1.41706
$50$	−94.4502	−0.267145
$51$	0	0
$52$	295.742	0.788692
$53$	−119.281	−0.309143	−0.154571	−	0.987982i	$-0.549400\pi$
$53$	−119.281	−0.309143	−0.154571	+	0.987982i	$0.549400\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	−187.828	−0.448206
$57$	0	0
$58$	27.2470	0.0616846
$59$	−20.5979	−0.0454512	−0.0227256	−	0.999742i	$-0.507234\pi$
$59$	−20.5979	−0.0454512	−0.0227256	+	0.999742i	$0.507234\pi$
$60$	0	0
$61$	593.927	1.24663	0.623316	−	0.781970i	$-0.285785\pi$
$61$	593.927	1.24663	0.623316	+	0.781970i	$0.285785\pi$
$62$	483.217	0.989817
$63$	0	0
$64$	−272.285	−0.531807
$65$	235.713	0.449794
$66$	0	0
$67$	−612.887	−1.11755	−0.558776	−	0.829318i	$-0.688729\pi$
$67$	−612.887	−1.11755	−0.558776	+	0.829318i	$0.688729\pi$
$68$	−683.125	−1.21825
$69$	0	0
$70$	543.906	0.928703
$71$	−162.987	−0.272436	−0.136218	−	0.990679i	$-0.543495\pi$
$71$	−162.987	−0.272436	−0.136218	+	0.990679i	$0.543495\pi$
$72$	0	0
$73$	−942.575	−1.51123	−0.755617	−	0.655014i	$-0.772663\pi$
$73$	−942.575	−1.51123	−0.755617	+	0.655014i	$0.772663\pi$
$74$	858.257	1.34825
$75$	0	0
$76$	761.174	1.14885
$77$	−316.726	−0.468757
$78$	0	0
$79$	−1086.48	−1.54733	−0.773664	−	0.633596i	$-0.781578\pi$
$79$	−1086.48	−1.54733	−0.773664	+	0.633596i	$0.781578\pi$
$80$	−374.160	−0.522904
$81$	0	0
$82$	−1199.35	−1.61520
$83$	−132.951	−0.175822	−0.0879111	−	0.996128i	$-0.528019\pi$
$83$	−132.951	−0.175822	−0.0879111	+	0.996128i	$0.528019\pi$
$84$	0	0
$85$	−544.467	−0.694773
$86$	−1176.44	−1.47510
$87$	0	0
$88$	71.7565	0.0869236
$89$	−1558.43	−1.85611	−0.928054	−	0.372445i	$-0.878520\pi$
$89$	−1558.43	−1.85611	−0.928054	+	0.372445i	$0.878520\pi$
$90$	0	0
$91$	−1357.39	−1.56366
$92$	251.092	0.284545
$93$	0	0
$94$	−60.8767	−0.0667974
$95$	606.673	0.655193
$96$	0	0
$97$	−1062.26	−1.11192	−0.555961	−	0.831208i	$-0.687650\pi$
$97$	−1062.26	−1.11192	−0.555961	+	0.831208i	$0.687650\pi$
$98$	−1836.31	−1.89281
$99$	0	0
$100$	156.834	0.156834
$101$	−1247.21	−1.22873	−0.614366	−	0.789021i	$-0.710588\pi$
$101$	−1247.21	−1.22873	−0.614366	+	0.789021i	$0.710588\pi$
$102$	0	0
$103$	−161.494	−0.154491	−0.0772453	−	0.997012i	$-0.524612\pi$
$103$	−161.494	−0.154491	−0.0772453	+	0.997012i	$0.524612\pi$
$104$	307.526	0.289956
$105$	0	0
$106$	450.646	0.412930
$107$	−739.342	−0.667990	−0.333995	−	0.942575i	$-0.608397\pi$
$107$	−739.342	−0.667990	−0.333995	+	0.942575i	$0.608397\pi$
$108$	0	0
$109$	1712.84	1.50514	0.752568	−	0.658514i	$-0.228815\pi$
$109$	1712.84	1.50514	0.752568	+	0.658514i	$0.228815\pi$
$110$	−207.790	−0.180109
$111$	0	0
$112$	2154.66	1.81782
$113$	−503.233	−0.418940	−0.209470	−	0.977815i	$-0.567174\pi$
$113$	−503.233	−0.418940	−0.209470	+	0.977815i	$0.567174\pi$
$114$	0	0
$115$	200.126	0.162277
$116$	−45.2433	−0.0362133
$117$	0	0
$118$	77.8191	0.0607105
$119$	3135.40	2.41530
$120$	0	0
$121$	121.000	0.0909091
$122$	−2243.86	−1.66516
$123$	0	0
$124$	−802.377	−0.581094
$125$	125.000	0.0894427
$126$	0	0
$127$	−1872.90	−1.30861	−0.654303	−	0.756233i	$-0.727038\pi$
$127$	−1872.90	−1.30861	−0.654303	+	0.756233i	$0.727038\pi$
$128$	−815.537	−0.563156
$129$	0	0
$130$	−890.525	−0.600802
$131$	−994.727	−0.663433	−0.331717	−	0.943379i	$-0.607628\pi$
$131$	−994.727	−0.663433	−0.331717	+	0.943379i	$0.607628\pi$
$132$	0	0
$133$	−3493.62	−2.27771
$134$	2315.49	1.49275
$135$	0	0
$136$	−710.346	−0.447880
$137$	−2341.00	−1.45989	−0.729946	−	0.683505i	$-0.760455\pi$
$137$	−2341.00	−1.45989	−0.729946	+	0.683505i	$0.760455\pi$
$138$	0	0
$139$	−498.576	−0.304235	−0.152118	−	0.988362i	$-0.548609\pi$
$139$	−498.576	−0.304235	−0.152118	+	0.988362i	$0.548609\pi$
$140$	−903.151	−0.545215
$141$	0	0
$142$	615.765	0.363901
$143$	518.568	0.303251
$144$	0	0
$145$	−36.0600	−0.0206525
$146$	3561.06	2.01860
$147$	0	0
$148$	−1425.13	−0.791518
$149$	−701.030	−0.385440	−0.192720	−	0.981254i	$-0.561731\pi$
$149$	−701.030	−0.385440	−0.192720	+	0.981254i	$0.561731\pi$
$150$	0	0
$151$	1914.43	1.03175	0.515875	−	0.856664i	$-0.327467\pi$
$151$	1914.43	1.03175	0.515875	+	0.856664i	$0.327467\pi$
$152$	791.505	0.422365
$153$	0	0
$154$	1196.59	0.626131
$155$	−639.513	−0.331400
$156$	0	0
$157$	−2292.19	−1.16520	−0.582601	−	0.812759i	$-0.697965\pi$
$157$	−2292.19	−1.16520	−0.582601	+	0.812759i	$0.697965\pi$
$158$	4104.74	2.06681
$159$	0	0
$160$	1152.64	0.569529
$161$	−1152.46	−0.564139
$162$	0	0
$163$	579.593	0.278511	0.139255	−	0.990257i	$-0.455529\pi$
$163$	579.593	0.278511	0.139255	+	0.990257i	$0.455529\pi$
$164$	1991.51	0.948237
$165$	0	0
$166$	502.289	0.234851
$167$	3351.09	1.55279	0.776393	−	0.630249i	$-0.217047\pi$
$167$	3351.09	1.55279	0.776393	+	0.630249i	$0.217047\pi$
$168$	0	0
$169$	25.4217	0.0115711
$170$	2057.00	0.928027
$171$	0	0
$172$	1953.46	0.865990
$173$	−374.313	−0.164500	−0.0822500	−	0.996612i	$-0.526211\pi$
$173$	−374.313	−0.164500	−0.0822500	+	0.996612i	$0.526211\pi$
$174$	0	0
$175$	−719.832	−0.310938
$176$	−823.151	−0.352542
$177$	0	0
$178$	5887.78	2.47926
$179$	−18.6528	−0.00778870	−0.00389435	−	0.999992i	$-0.501240\pi$
$179$	−18.6528	−0.00778870	−0.00389435	+	0.999992i	$0.501240\pi$
$180$	0	0
$181$	1558.91	0.640182	0.320091	−	0.947387i	$-0.396287\pi$
$181$	1558.91	0.640182	0.320091	+	0.947387i	$0.396287\pi$
$182$	5128.23	2.08862
$183$	0	0
$184$	261.098	0.104611
$185$	−1135.86	−0.451406
$186$	0	0
$187$	−1197.83	−0.468416
$188$	101.085	0.0392149
$189$	0	0
$190$	−2292.02	−0.875160
$191$	564.077	0.213692	0.106846	−	0.994276i	$-0.465925\pi$
$191$	564.077	0.213692	0.106846	+	0.994276i	$0.465925\pi$
$192$	0	0
$193$	−3509.37	−1.30886	−0.654430	−	0.756122i	$-0.727091\pi$
$193$	−3509.37	−1.30886	−0.654430	+	0.756122i	$0.727091\pi$
$194$	4013.24	1.48523
$195$	0	0
$196$	3049.18	1.11122
$197$	−980.441	−0.354587	−0.177293	−	0.984158i	$-0.556734\pi$
$197$	−980.441	−0.354587	−0.177293	+	0.984158i	$0.556734\pi$
$198$	0	0
$199$	−2603.46	−0.927407	−0.463704	−	0.885990i	$-0.653480\pi$
$199$	−2603.46	−0.927407	−0.463704	+	0.885990i	$0.653480\pi$
$200$	163.083	0.0576586
$201$	0	0
$202$	4711.97	1.64125
$203$	207.657	0.0717964
$204$	0	0
$205$	1587.28	0.540783
$206$	610.127	0.206357
$207$	0	0
$208$	−3527.77	−1.17599
$209$	1334.68	0.441731
$210$	0	0
$211$	4735.78	1.54514	0.772570	−	0.634930i	$-0.218971\pi$
$211$	4735.78	1.54514	0.772570	+	0.634930i	$0.218971\pi$
$212$	−748.293	−0.242420
$213$	0	0
$214$	2793.24	0.892252
$215$	1556.96	0.493877
$216$	0	0
$217$	3682.74	1.15208
$218$	−6471.11	−2.01045
$219$	0	0
$220$	345.034	0.105737
$221$	−5133.51	−1.56252
$222$	0	0
$223$	2434.64	0.731102	0.365551	−	0.930791i	$-0.380880\pi$
$223$	2434.64	0.731102	0.365551	+	0.930791i	$0.380880\pi$
$224$	−6637.68	−1.97991
$225$	0	0
$226$	1901.22	0.559590
$227$	1829.18	0.534833	0.267417	−	0.963581i	$-0.413830\pi$
$227$	1829.18	0.534833	0.267417	+	0.963581i	$0.413830\pi$
$228$	0	0
$229$	−1747.73	−0.504339	−0.252169	−	0.967683i	$-0.581144\pi$
$229$	−1747.73	−0.504339	−0.252169	+	0.967683i	$0.581144\pi$
$230$	−756.079	−0.216758
$231$	0	0
$232$	−47.0462	−0.0133135
$233$	−4120.51	−1.15856	−0.579278	−	0.815130i	$-0.696665\pi$
$233$	−4120.51	−1.15856	−0.579278	+	0.815130i	$0.696665\pi$
$234$	0	0
$235$	80.5672	0.0223644
$236$	−129.218	−0.0356414
$237$	0	0
$238$	−11845.6	−3.22619
$239$	1545.91	0.418397	0.209198	−	0.977873i	$-0.432915\pi$
$239$	1545.91	0.418397	0.209198	+	0.977873i	$0.432915\pi$
$240$	0	0
$241$	−482.502	−0.128966	−0.0644828	−	0.997919i	$-0.520540\pi$
$241$	−482.502	−0.128966	−0.0644828	+	0.997919i	$0.520540\pi$
$242$	−457.139	−0.121430
$243$	0	0
$244$	3725.91	0.977569
$245$	2430.26	0.633730
$246$	0	0
$247$	5720.03	1.47351
$248$	−834.350	−0.213634
$249$	0	0
$250$	−472.251	−0.119471
$251$	−2351.43	−0.591319	−0.295660	−	0.955293i	$-0.595539\pi$
$251$	−2351.43	−0.591319	−0.295660	+	0.955293i	$0.595539\pi$
$252$	0	0
$253$	440.278	0.109407
$254$	7075.83	1.74794
$255$	0	0
$256$	5259.38	1.28403
$257$	7300.82	1.77203	0.886017	−	0.463652i	$-0.153461\pi$
$257$	7300.82	1.77203	0.886017	+	0.463652i	$0.153461\pi$
$258$	0	0
$259$	6541.02	1.56926
$260$	1478.71	0.352714
$261$	0	0
$262$	3758.09	0.886166
$263$	4109.74	0.963565	0.481783	−	0.876291i	$-0.339990\pi$
$263$	4109.74	0.963565	0.481783	+	0.876291i	$0.339990\pi$
$264$	0	0
$265$	−596.407	−0.138253
$266$	13198.9	3.04240
$267$	0	0
$268$	−3844.85	−0.876349
$269$	−6102.62	−1.38321	−0.691605	−	0.722276i	$-0.743096\pi$
$269$	−6102.62	−1.38321	−0.691605	+	0.722276i	$0.743096\pi$
$270$	0	0
$271$	2313.02	0.518473	0.259237	−	0.965814i	$-0.416529\pi$
$271$	2313.02	0.518473	0.259237	+	0.965814i	$0.416529\pi$
$272$	8148.70	1.81650
$273$	0	0
$274$	8844.32	1.95002
$275$	275.000	0.0603023
$276$	0	0
$277$	−2612.89	−0.566763	−0.283382	−	0.959007i	$-0.591456\pi$
$277$	−2612.89	−0.566763	−0.283382	+	0.959007i	$0.591456\pi$
$278$	1883.62	0.406375
$279$	0	0
$280$	−939.139	−0.200444
$281$	175.039	0.0371600	0.0185800	−	0.999827i	$-0.494085\pi$
$281$	175.039	0.0371600	0.0185800	+	0.999827i	$0.494085\pi$
$282$	0	0
$283$	−3488.43	−0.732741	−0.366371	−	0.930469i	$-0.619400\pi$
$283$	−3488.43	−0.732741	−0.366371	+	0.930469i	$0.619400\pi$
$284$	−1022.47	−0.213636
$285$	0	0
$286$	−1959.15	−0.405060
$287$	−9140.60	−1.87997
$288$	0	0
$289$	6944.76	1.41355
$290$	136.235	0.0275862
$291$	0	0
$292$	−5913.10	−1.18506
$293$	−7708.87	−1.53706	−0.768528	−	0.639817i	$-0.779010\pi$
$293$	−7708.87	−1.53706	−0.768528	+	0.639817i	$0.779010\pi$
$294$	0	0
$295$	−102.990	−0.0203264
$296$	−1481.92	−0.290995
$297$	0	0
$298$	2648.50	0.514843
$299$	1886.89	0.364956
$300$	0	0
$301$	−8965.98	−1.71691
$302$	−7232.74	−1.37814
$303$	0	0
$304$	−9079.71	−1.71302
$305$	2969.63	0.557511
$306$	0	0
$307$	−7969.78	−1.48163	−0.740813	−	0.671711i	$-0.765560\pi$
$307$	−7969.78	−1.48163	−0.740813	+	0.671711i	$0.765560\pi$
$308$	−1986.93	−0.367584
$309$	0	0
$310$	2416.09	0.442660
$311$	8739.11	1.59341	0.796703	−	0.604371i	$-0.206575\pi$
$311$	8739.11	1.59341	0.796703	+	0.604371i	$0.206575\pi$
$312$	0	0
$313$	557.816	0.100734	0.0503668	−	0.998731i	$-0.483961\pi$
$313$	557.816	0.100734	0.0503668	+	0.998731i	$0.483961\pi$
$314$	8659.91	1.55639
$315$	0	0
$316$	−6815.88	−1.21336
$317$	622.155	0.110232	0.0551162	−	0.998480i	$-0.482447\pi$
$317$	622.155	0.110232	0.0551162	+	0.998480i	$0.482447\pi$
$318$	0	0
$319$	−79.3320	−0.0139239
$320$	−1361.42	−0.237831
$321$	0	0
$322$	4354.00	0.753537
$323$	−13212.5	−2.27605
$324$	0	0
$325$	1178.56	0.201154
$326$	−2189.71	−0.372014
$327$	0	0
$328$	2070.87	0.348612
$329$	−463.959	−0.0777474
$330$	0	0
$331$	−8911.17	−1.47976	−0.739882	−	0.672736i	$-0.765119\pi$
$331$	−8911.17	−1.47976	−0.739882	+	0.672736i	$0.765119\pi$
$332$	−834.046	−0.137874
$333$	0	0
$334$	−12660.5	−2.07410
$335$	−3064.43	−0.499785
$336$	0	0
$337$	11057.4	1.78734	0.893670	−	0.448724i	$-0.148121\pi$
$337$	11057.4	1.78734	0.893670	+	0.448724i	$0.148121\pi$
$338$	−96.0433	−0.0154558
$339$	0	0
$340$	−3415.63	−0.544819
$341$	−1406.93	−0.223430
$342$	0	0
$343$	−4118.96	−0.648405
$344$	2031.31	0.318374
$345$	0	0
$346$	1414.16	0.219727
$347$	1799.40	0.278378	0.139189	−	0.990266i	$-0.455551\pi$
$347$	1799.40	0.278378	0.139189	+	0.990266i	$0.455551\pi$
$348$	0	0
$349$	−2504.00	−0.384057	−0.192028	−	0.981389i	$-0.561507\pi$
$349$	−2504.00	−0.384057	−0.192028	+	0.981389i	$0.561507\pi$
$350$	2719.53	0.415329
$351$	0	0
$352$	2535.82	0.383976
$353$	2418.48	0.364653	0.182326	−	0.983238i	$-0.441637\pi$
$353$	2418.48	0.364653	0.182326	+	0.983238i	$0.441637\pi$
$354$	0	0
$355$	−814.934	−0.121837
$356$	−9776.59	−1.45550
$357$	0	0
$358$	70.4705	0.0104036
$359$	3191.37	0.469176	0.234588	−	0.972095i	$-0.424626\pi$
$359$	3191.37	0.469176	0.234588	+	0.972095i	$0.424626\pi$
$360$	0	0
$361$	7863.11	1.14639
$362$	−5889.58	−0.855108
$363$	0	0
$364$	−8515.37	−1.22617
$365$	−4712.87	−0.675844
$366$	0	0
$367$	1674.44	0.238161	0.119080	−	0.992885i	$-0.462005\pi$
$367$	1674.44	0.238161	0.119080	+	0.992885i	$0.462005\pi$
$368$	−2995.17	−0.424277
$369$	0	0
$370$	4291.29	0.602955
$371$	3434.50	0.480621
$372$	0	0
$373$	−10450.1	−1.45063	−0.725314	−	0.688418i	$-0.758306\pi$
$373$	−10450.1	−1.45063	−0.725314	+	0.688418i	$0.758306\pi$
$374$	4525.40	0.625676
$375$	0	0
$376$	105.113	0.0144170
$377$	−339.992	−0.0464469
$378$	0	0
$379$	1695.73	0.229825	0.114913	−	0.993376i	$-0.463341\pi$
$379$	1695.73	0.229825	0.114913	+	0.993376i	$0.463341\pi$
$380$	3805.87	0.513782
$381$	0	0
$382$	−2131.09	−0.285434
$383$	−11049.0	−1.47409	−0.737044	−	0.675845i	$-0.763779\pi$
$383$	−11049.0	−1.47409	−0.737044	+	0.675845i	$0.763779\pi$
$384$	0	0
$385$	−1583.63	−0.209634
$386$	13258.4	1.74828
$387$	0	0
$388$	−6663.94	−0.871934
$389$	−259.766	−0.0338578	−0.0169289	−	0.999857i	$-0.505389\pi$
$389$	−259.766	−0.0338578	−0.0169289	+	0.999857i	$0.505389\pi$
$390$	0	0
$391$	−4358.48	−0.563729
$392$	3170.68	0.408529
$393$	0	0
$394$	3704.11	0.473631
$395$	−5432.41	−0.691986
$396$	0	0
$397$	709.737	0.0897246	0.0448623	−	0.998993i	$-0.485715\pi$
$397$	709.737	0.0897246	0.0448623	+	0.998993i	$0.485715\pi$
$398$	9835.88	1.23876
$399$	0	0
$400$	−1870.80	−0.233850
$401$	−4115.30	−0.512489	−0.256244	−	0.966612i	$-0.582485\pi$
$401$	−4115.30	−0.512489	−0.256244	+	0.966612i	$0.582485\pi$
$402$	0	0
$403$	−6029.66	−0.745307
$404$	−7824.17	−0.963533
$405$	0	0
$406$	−784.530	−0.0959004
$407$	−2498.89	−0.304338
$408$	0	0
$409$	1314.42	0.158910	0.0794548	−	0.996838i	$-0.474682\pi$
$409$	1314.42	0.158910	0.0794548	+	0.996838i	$0.474682\pi$
$410$	−5996.76	−0.722339
$411$	0	0
$412$	−1013.11	−0.121147
$413$	593.082	0.0706626
$414$	0	0
$415$	−664.754	−0.0786301
$416$	10867.7	1.28085
$417$	0	0
$418$	−5042.44	−0.590033
$419$	−15347.8	−1.78947	−0.894735	−	0.446598i	$-0.852635\pi$
$419$	−15347.8	−1.78947	−0.894735	+	0.446598i	$0.852635\pi$
$420$	0	0
$421$	−4293.60	−0.497048	−0.248524	−	0.968626i	$-0.579946\pi$
$421$	−4293.60	−0.497048	−0.248524	+	0.968626i	$0.579946\pi$
$422$	−17891.8	−2.06389
$423$	0	0
$424$	−778.111	−0.0891236
$425$	−2722.33	−0.310712
$426$	0	0
$427$	−17101.1	−1.93813
$428$	−4638.15	−0.523816
$429$	0	0
$430$	−5882.19	−0.659685
$431$	6429.62	0.718571	0.359285	−	0.933228i	$-0.383020\pi$
$431$	6429.62	0.718571	0.359285	+	0.933228i	$0.383020\pi$
$432$	0	0
$433$	8563.46	0.950424	0.475212	−	0.879871i	$-0.342371\pi$
$433$	8563.46	0.950424	0.475212	+	0.879871i	$0.342371\pi$
$434$	−13913.4	−1.53886
$435$	0	0
$436$	10745.2	1.18028
$437$	4856.45	0.531615
$438$	0	0
$439$	8651.72	0.940601	0.470301	−	0.882506i	$-0.344145\pi$
$439$	8651.72	0.940601	0.470301	+	0.882506i	$0.344145\pi$
$440$	358.783	0.0388734
$441$	0	0
$442$	19394.4	2.08710
$443$	15214.9	1.63179	0.815895	−	0.578200i	$-0.196245\pi$
$443$	15214.9	1.63179	0.815895	+	0.578200i	$0.196245\pi$
$444$	0	0
$445$	−7792.17	−0.830077
$446$	−9198.11	−0.976554
$447$	0	0
$448$	7839.98	0.826795
$449$	−6940.63	−0.729507	−0.364753	−	0.931104i	$-0.618847\pi$
$449$	−6940.63	−0.729507	−0.364753	+	0.931104i	$0.618847\pi$
$450$	0	0
$451$	3492.02	0.364596
$452$	−3156.96	−0.328519
$453$	0	0
$454$	−6910.67	−0.714392
$455$	−6786.94	−0.699290
$456$	0	0
$457$	8762.51	0.896921	0.448460	−	0.893803i	$-0.351972\pi$
$457$	8762.51	0.896921	0.448460	+	0.893803i	$0.351972\pi$
$458$	6602.95	0.673659
$459$	0	0
$460$	1255.46	0.127253
$461$	−3203.21	−0.323619	−0.161810	−	0.986822i	$-0.551733\pi$
$461$	−3203.21	−0.323619	−0.161810	+	0.986822i	$0.551733\pi$
$462$	0	0
$463$	−435.479	−0.0437115	−0.0218558	−	0.999761i	$-0.506957\pi$
$463$	−435.479	−0.0437115	−0.0218558	+	0.999761i	$0.506957\pi$
$464$	539.688	0.0539965
$465$	0	0
$466$	15567.3	1.54751
$467$	9916.33	0.982598	0.491299	−	0.870991i	$-0.336522\pi$
$467$	9916.33	0.982598	0.491299	+	0.870991i	$0.336522\pi$
$468$	0	0
$469$	17647.0	1.73745
$470$	−304.384	−0.0298727
$471$	0	0
$472$	−134.367	−0.0131033
$473$	3425.30	0.332972
$474$	0	0
$475$	3033.37	0.293011
$476$	19669.4	1.89400
$477$	0	0
$478$	−5840.47	−0.558864
$479$	302.136	0.0288204	0.0144102	−	0.999896i	$-0.495413\pi$
$479$	302.136	0.0288204	0.0144102	+	0.999896i	$0.495413\pi$
$480$	0	0
$481$	−10709.5	−1.01520
$482$	1822.90	0.172263
$483$	0	0
$484$	759.075	0.0712880
$485$	−5311.32	−0.497267
$486$	0	0
$487$	8192.73	0.762317	0.381158	−	0.924510i	$-0.375525\pi$
$487$	8192.73	0.762317	0.381158	+	0.924510i	$0.375525\pi$
$488$	3874.38	0.359395
$489$	0	0
$490$	−9181.56	−0.846491
$491$	−5222.81	−0.480045	−0.240023	−	0.970767i	$-0.577155\pi$
$491$	−5222.81	−0.480045	−0.240023	+	0.970767i	$0.577155\pi$
$492$	0	0
$493$	785.339	0.0717442
$494$	−21610.3	−1.96821
$495$	0	0
$496$	9571.20	0.866451
$497$	4692.92	0.423554
$498$	0	0
$499$	−1844.29	−0.165455	−0.0827273	−	0.996572i	$-0.526363\pi$
$499$	−1844.29	−0.165455	−0.0827273	+	0.996572i	$0.526363\pi$
$500$	784.168	0.0701381
$501$	0	0
$502$	8883.73	0.789841
$503$	−3878.67	−0.343820	−0.171910	−	0.985113i	$-0.554994\pi$
$503$	−3878.67	−0.343820	−0.171910	+	0.985113i	$0.554994\pi$
$504$	0	0
$505$	−6236.05	−0.549506
$506$	−1663.37	−0.146138
$507$	0	0
$508$	−11749.3	−1.02617
$509$	−3796.88	−0.330636	−0.165318	−	0.986240i	$-0.552865\pi$
$509$	−3796.88	−0.330636	−0.165318	+	0.986240i	$0.552865\pi$
$510$	0	0
$511$	27139.8	2.34950
$512$	−13345.7	−1.15196
$513$	0	0
$514$	−27582.6	−2.36696
$515$	−807.472	−0.0690903
$516$	0	0
$517$	177.248	0.0150781
$518$	−24712.0	−2.09611
$519$	0	0
$520$	1537.63	0.129672
$521$	8492.81	0.714159	0.357079	−	0.934074i	$-0.383773\pi$
$521$	8492.81	0.714159	0.357079	+	0.934074i	$0.383773\pi$
$522$	0	0
$523$	−955.757	−0.0799089	−0.0399545	−	0.999202i	$-0.512721\pi$
$523$	−955.757	−0.0799089	−0.0399545	+	0.999202i	$0.512721\pi$
$524$	−6240.27	−0.520243
$525$	0	0
$526$	−15526.6	−1.28706
$527$	13927.7	1.15124
$528$	0	0
$529$	−10565.0	−0.868331
$530$	2253.23	0.184668
$531$	0	0
$532$	−21916.7	−1.78611
$533$	14965.7	1.21620
$534$	0	0
$535$	−3696.71	−0.298734
$536$	−3998.06	−0.322182
$537$	0	0
$538$	23055.8	1.84759
$539$	5346.58	0.427261
$540$	0	0
$541$	20686.5	1.64396	0.821980	−	0.569516i	$-0.192869\pi$
$541$	20686.5	1.64396	0.821980	+	0.569516i	$0.192869\pi$
$542$	−8738.62	−0.692539
$543$	0	0
$544$	−25103.1	−1.97847
$545$	8564.18	0.673118
$546$	0	0
$547$	−24.6763	−0.00192885	−0.000964426	−	1.00000i	$-0.500307\pi$
$547$	−24.6763	−0.00192885	−0.000964426		1.00000i	$0.500307\pi$
$548$	−14685.9	−1.14480
$549$	0	0
$550$	−1038.95	−0.0805474
$551$	−875.066	−0.0676571
$552$	0	0
$553$	31283.4	2.40562
$554$	9871.52	0.757041
$555$	0	0
$556$	−3127.74	−0.238571
$557$	−3976.06	−0.302461	−0.151231	−	0.988499i	$-0.548324\pi$
$557$	−3976.06	−0.302461	−0.151231	+	0.988499i	$0.548324\pi$
$558$	0	0
$559$	14679.8	1.11071
$560$	10773.3	0.812954
$561$	0	0
$562$	−661.300	−0.0496357
$563$	20585.3	1.54097	0.770486	−	0.637457i	$-0.220014\pi$
$563$	20585.3	1.54097	0.770486	+	0.637457i	$0.220014\pi$
$564$	0	0
$565$	−2516.17	−0.187356
$566$	13179.3	0.978742
$567$	0	0
$568$	−1063.22	−0.0785414
$569$	−21579.3	−1.58989	−0.794947	−	0.606679i	$-0.792501\pi$
$569$	−21579.3	−1.58989	−0.794947	+	0.606679i	$0.792501\pi$
$570$	0	0
$571$	−10341.6	−0.757937	−0.378969	−	0.925409i	$-0.623721\pi$
$571$	−10341.6	−0.757937	−0.378969	+	0.925409i	$0.623721\pi$
$572$	3253.16	0.237800
$573$	0	0
$574$	34533.3	2.51113
$575$	1000.63	0.0725726
$576$	0	0
$577$	−22739.2	−1.64063	−0.820315	−	0.571912i	$-0.806202\pi$
$577$	−22739.2	−1.64063	−0.820315	+	0.571912i	$0.806202\pi$
$578$	−26237.4	−1.88811
$579$	0	0
$580$	−226.217	−0.0161951
$581$	3828.09	0.273349
$582$	0	0
$583$	−1312.09	−0.0932100
$584$	−6148.72	−0.435678
$585$	0	0
$586$	29124.2	2.05309
$587$	2964.80	0.208468	0.104234	−	0.994553i	$-0.466761\pi$
$587$	2964.80	0.208468	0.104234	+	0.994553i	$0.466761\pi$
$588$	0	0
$589$	−15519.0	−1.08565
$590$	389.096	0.0271505
$591$	0	0
$592$	16999.7	1.18021
$593$	13009.8	0.900925	0.450462	−	0.892795i	$-0.351259\pi$
$593$	13009.8	0.900925	0.450462	+	0.892795i	$0.351259\pi$
$594$	0	0
$595$	15677.0	1.08016
$596$	−4397.80	−0.302250
$597$	0	0
$598$	−7128.70	−0.487482
$599$	2888.39	0.197022	0.0985112	−	0.995136i	$-0.468592\pi$
$599$	2888.39	0.197022	0.0985112	+	0.995136i	$0.468592\pi$
$600$	0	0
$601$	−1621.64	−0.110063	−0.0550315	−	0.998485i	$-0.517526\pi$
$601$	−1621.64	−0.110063	−0.0550315	+	0.998485i	$0.517526\pi$
$602$	33873.5	2.29333
$603$	0	0
$604$	12009.9	0.809065
$605$	605.000	0.0406558
$606$	0	0
$607$	3059.31	0.204569	0.102285	−	0.994755i	$-0.467385\pi$
$607$	3059.31	0.204569	0.102285	+	0.994755i	$0.467385\pi$
$608$	27971.2	1.86576
$609$	0	0
$610$	−11219.3	−0.744683
$611$	759.629	0.0502968
$612$	0	0
$613$	−11123.0	−0.732875	−0.366437	−	0.930443i	$-0.619423\pi$
$613$	−11123.0	−0.732875	−0.366437	+	0.930443i	$0.619423\pi$
$614$	30109.9	1.97905
$615$	0	0
$616$	−2066.11	−0.135139
$617$	−1958.59	−0.127796	−0.0638978	−	0.997956i	$-0.520353\pi$
$617$	−1958.59	−0.127796	−0.0638978	+	0.997956i	$0.520353\pi$
$618$	0	0
$619$	−5993.01	−0.389143	−0.194571	−	0.980888i	$-0.562332\pi$
$619$	−5993.01	−0.389143	−0.194571	+	0.980888i	$0.562332\pi$
$620$	−4011.89	−0.259873
$621$	0	0
$622$	−33016.4	−2.12836
$623$	44872.4	2.88567
$624$	0	0
$625$	625.000	0.0400000
$626$	−2107.43	−0.134553
$627$	0	0
$628$	−14379.7	−0.913713
$629$	24737.5	1.56812
$630$	0	0
$631$	−5695.23	−0.359308	−0.179654	−	0.983730i	$-0.557498\pi$
$631$	−5695.23	−0.359308	−0.179654	+	0.983730i	$0.557498\pi$
$632$	−7087.48	−0.446083
$633$	0	0
$634$	−2350.51	−0.147241
$635$	−9364.50	−0.585226
$636$	0	0
$637$	22913.8	1.42524
$638$	299.717	0.0185986
$639$	0	0
$640$	−4077.68	−0.251851
$641$	−19626.6	−1.20937	−0.604684	−	0.796465i	$-0.706701\pi$
$641$	−19626.6	−1.20937	−0.604684	+	0.796465i	$0.706701\pi$
$642$	0	0
$643$	9521.51	0.583968	0.291984	−	0.956423i	$-0.405685\pi$
$643$	9521.51	0.583968	0.291984	+	0.956423i	$0.405685\pi$
$644$	−7229.77	−0.442380
$645$	0	0
$646$	49917.1	3.04019
$647$	−30129.3	−1.83077	−0.915383	−	0.402585i	$-0.868112\pi$
$647$	−30129.3	−1.83077	−0.915383	+	0.402585i	$0.868112\pi$
$648$	0	0
$649$	−226.577	−0.0137041
$650$	−4452.62	−0.268687
$651$	0	0
$652$	3635.99	0.218399
$653$	−1103.35	−0.0661215	−0.0330608	−	0.999453i	$-0.510525\pi$
$653$	−1103.35	−0.0661215	−0.0330608	+	0.999453i	$0.510525\pi$
$654$	0	0
$655$	−4973.64	−0.296696
$656$	−23755.8	−1.41389
$657$	0	0
$658$	1752.84	0.103849
$659$	8010.95	0.473539	0.236770	−	0.971566i	$-0.423911\pi$
$659$	8010.95	0.473539	0.236770	+	0.971566i	$0.423911\pi$
$660$	0	0
$661$	−29260.8	−1.72181	−0.860903	−	0.508770i	$-0.830100\pi$
$661$	−29260.8	−1.72181	−0.860903	+	0.508770i	$0.830100\pi$
$662$	33666.5	1.97656
$663$	0	0
$664$	−867.281	−0.0506883
$665$	−17468.1	−1.01862
$666$	0	0
$667$	−288.662	−0.0167572
$668$	21022.6	1.21765
$669$	0	0
$670$	11577.5	0.667576
$671$	6533.20	0.375874
$672$	0	0
$673$	29269.0	1.67643	0.838213	−	0.545342i	$-0.183600\pi$
$673$	29269.0	1.67643	0.838213	+	0.545342i	$0.183600\pi$
$674$	−41774.9	−2.38740
$675$	0	0
$676$	159.479	0.00907368
$677$	13482.1	0.765375	0.382687	−	0.923878i	$-0.374999\pi$
$677$	13482.1	0.765375	0.382687	+	0.923878i	$0.374999\pi$
$678$	0	0
$679$	30586.0	1.72870
$680$	−3551.73	−0.200298
$681$	0	0
$682$	5315.39	0.298441
$683$	2722.37	0.152516	0.0762582	−	0.997088i	$-0.475703\pi$
$683$	2722.37	0.152516	0.0762582	+	0.997088i	$0.475703\pi$
$684$	0	0
$685$	−11705.0	−0.652884
$686$	15561.5	0.866092
$687$	0	0
$688$	−23302.0	−1.29125
$689$	−5623.23	−0.310926
$690$	0	0
$691$	−32590.3	−1.79420	−0.897100	−	0.441828i	$-0.854330\pi$
$691$	−32590.3	−1.79420	−0.897100	+	0.441828i	$0.854330\pi$
$692$	−2348.19	−0.128996
$693$	0	0
$694$	−6798.16	−0.371837
$695$	−2492.88	−0.136058
$696$	0	0
$697$	−34568.9	−1.87861
$698$	9460.12	0.512995
$699$	0	0
$700$	−4515.75	−0.243828
$701$	−5893.99	−0.317565	−0.158782	−	0.987314i	$-0.550757\pi$
$701$	−5893.99	−0.317565	−0.158782	+	0.987314i	$0.550757\pi$
$702$	0	0
$703$	−27563.8	−1.47879
$704$	−2995.13	−0.160346
$705$	0	0
$706$	−9137.02	−0.487077
$707$	35911.2	1.91030
$708$	0	0
$709$	−17782.7	−0.941954	−0.470977	−	0.882146i	$-0.656098\pi$
$709$	−17782.7	−0.941954	−0.470977	+	0.882146i	$0.656098\pi$
$710$	3078.83	0.162741
$711$	0	0
$712$	−10166.2	−0.535103
$713$	−5119.34	−0.268893
$714$	0	0
$715$	2592.84	0.135618
$716$	−117.016	−0.00610765
$717$	0	0
$718$	−12057.0	−0.626691
$719$	12589.0	0.652975	0.326488	−	0.945202i	$-0.394135\pi$
$719$	12589.0	0.652975	0.326488	+	0.945202i	$0.394135\pi$
$720$	0	0
$721$	4649.95	0.240185
$722$	−29706.9	−1.53127
$723$	0	0
$724$	9779.58	0.502010
$725$	−180.300	−0.00923610
$726$	0	0
$727$	20424.1	1.04194	0.520969	−	0.853576i	$-0.325571\pi$
$727$	20424.1	1.04194	0.520969	+	0.853576i	$0.325571\pi$
$728$	−8854.68	−0.450792
$729$	0	0
$730$	17805.3	0.902744
$731$	−33908.4	−1.71566
$732$	0	0
$733$	16706.0	0.841816	0.420908	−	0.907103i	$-0.361711\pi$
$733$	16706.0	0.841816	0.420908	+	0.907103i	$0.361711\pi$
$734$	−6326.04	−0.318118
$735$	0	0
$736$	9226.98	0.462107
$737$	−6741.75	−0.336955
$738$	0	0
$739$	−15764.1	−0.784698	−0.392349	−	0.919816i	$-0.628337\pi$
$739$	−15764.1	−0.784698	−0.392349	+	0.919816i	$0.628337\pi$
$740$	−7125.64	−0.353978
$741$	0	0
$742$	−12975.6	−0.641979
$743$	−20371.1	−1.00584	−0.502922	−	0.864332i	$-0.667742\pi$
$743$	−20371.1	−1.00584	−0.502922	+	0.864332i	$0.667742\pi$
$744$	0	0
$745$	−3505.15	−0.172374
$746$	39480.5	1.93764
$747$	0	0
$748$	−7514.38	−0.367317
$749$	21288.1	1.03852
$750$	0	0
$751$	−15718.6	−0.763757	−0.381879	−	0.924213i	$-0.624723\pi$
$751$	−15718.6	−0.763757	−0.381879	+	0.924213i	$0.624723\pi$
$752$	−1205.80	−0.0584721
$753$	0	0
$754$	1284.49	0.0620404
$755$	9572.16	0.461413
$756$	0	0
$757$	6038.61	0.289930	0.144965	−	0.989437i	$-0.453693\pi$
$757$	6038.61	0.289930	0.144965	+	0.989437i	$0.453693\pi$
$758$	−6406.47	−0.306984
$759$	0	0
$760$	3957.53	0.188888
$761$	−20027.8	−0.954015	−0.477008	−	0.878899i	$-0.658279\pi$
$761$	−20027.8	−0.954015	−0.477008	+	0.878899i	$0.658279\pi$
$762$	0	0
$763$	−49318.1	−2.34002
$764$	3538.65	0.167570
$765$	0	0
$766$	41743.1	1.96898
$767$	−971.039	−0.0457134
$768$	0	0
$769$	−4376.71	−0.205238	−0.102619	−	0.994721i	$-0.532722\pi$
$769$	−4376.71	−0.205238	−0.102619	+	0.994721i	$0.532722\pi$
$770$	5982.97	0.280015
$771$	0	0
$772$	−22015.5	−1.02637
$773$	16200.1	0.753788	0.376894	−	0.926256i	$-0.376992\pi$
$773$	16200.1	0.753788	0.376894	+	0.926256i	$0.376992\pi$
$774$	0	0
$775$	−3197.57	−0.148206
$776$	−6929.48	−0.320559
$777$	0	0
$778$	981.400	0.0452248
$779$	38518.4	1.77159
$780$	0	0
$781$	−1792.85	−0.0821426
$782$	16466.4	0.752988
$783$	0	0
$784$	−36372.3	−1.65690
$785$	−11460.9	−0.521094
$786$	0	0
$787$	24529.0	1.11101	0.555506	−	0.831513i	$-0.312525\pi$
$787$	24529.0	1.11101	0.555506	+	0.831513i	$0.312525\pi$
$788$	−6150.64	−0.278055
$789$	0	0
$790$	20523.7	0.924305
$791$	14489.7	0.651322
$792$	0	0
$793$	27999.2	1.25382
$794$	−2681.39	−0.119848
$795$	0	0
$796$	−16332.4	−0.727243
$797$	−36238.8	−1.61059	−0.805297	−	0.592872i	$-0.797994\pi$
$797$	−36238.8	−1.61059	−0.805297	+	0.592872i	$0.797994\pi$
$798$	0	0
$799$	−1754.65	−0.0776908
$800$	5763.22	0.254701
$801$	0	0
$802$	15547.6	0.684545
$803$	−10368.3	−0.455654
$804$	0	0
$805$	−5762.29	−0.252291
$806$	22780.1	0.995527
$807$	0	0
$808$	−8135.95	−0.354235
$809$	−26737.8	−1.16199	−0.580995	−	0.813907i	$-0.697337\pi$
$809$	−26737.8	−1.16199	−0.580995	+	0.813907i	$0.697337\pi$
$810$	0	0
$811$	11905.5	0.515486	0.257743	−	0.966213i	$-0.417021\pi$
$811$	11905.5	0.515486	0.257743	+	0.966213i	$0.417021\pi$
$812$	1302.70	0.0563004
$813$	0	0
$814$	9440.83	0.406512
$815$	2897.97	0.124554
$816$	0	0
$817$	37782.6	1.61793
$818$	−4965.90	−0.212260
$819$	0	0
$820$	9957.56	0.424065
$821$	9816.33	0.417287	0.208643	−	0.977992i	$-0.433095\pi$
$821$	9816.33	0.417287	0.208643	+	0.977992i	$0.433095\pi$
$822$	0	0
$823$	9058.16	0.383655	0.191827	−	0.981429i	$-0.438559\pi$
$823$	9058.16	0.383655	0.191827	+	0.981429i	$0.438559\pi$
$824$	−1053.48	−0.0445385
$825$	0	0
$826$	−2240.67	−0.0943860
$827$	−34337.6	−1.44382	−0.721908	−	0.691989i	$-0.756735\pi$
$827$	−34337.6	−1.44382	−0.721908	+	0.691989i	$0.756735\pi$
$828$	0	0
$829$	32109.5	1.34524	0.672622	−	0.739986i	$-0.265168\pi$
$829$	32109.5	1.34524	0.672622	+	0.739986i	$0.265168\pi$
$830$	2511.45	0.105028
$831$	0	0
$832$	−12836.2	−0.534875
$833$	−52927.9	−2.20149
$834$	0	0
$835$	16755.5	0.694427
$836$	8372.92	0.346392
$837$	0	0
$838$	57984.0	2.39024
$839$	−3197.28	−0.131564	−0.0657820	−	0.997834i	$-0.520954\pi$
$839$	−3197.28	−0.131564	−0.0657820	+	0.997834i	$0.520954\pi$
$840$	0	0
$841$	−24337.0	−0.997867
$842$	16221.3	0.663921
$843$	0	0
$844$	29709.2	1.21165
$845$	127.108	0.00517475
$846$	0	0
$847$	−3483.99	−0.141336
$848$	8926.05	0.361464
$849$	0	0
$850$	10285.0	0.415026
$851$	−9092.61	−0.366264
$852$	0	0
$853$	−11610.9	−0.466060	−0.233030	−	0.972470i	$-0.574864\pi$
$853$	−11610.9	−0.466060	−0.233030	+	0.972470i	$0.574864\pi$
$854$	64608.1	2.58881
$855$	0	0
$856$	−4822.97	−0.192577
$857$	13007.6	0.518471	0.259236	−	0.965814i	$-0.416529\pi$
$857$	13007.6	0.518471	0.259236	+	0.965814i	$0.416529\pi$
$858$	0	0
$859$	−5077.33	−0.201672	−0.100836	−	0.994903i	$-0.532152\pi$
$859$	−5077.33	−0.201672	−0.100836	+	0.994903i	$0.532152\pi$
$860$	9767.32	0.387283
$861$	0	0
$862$	−24291.2	−0.959815
$863$	43798.2	1.72759	0.863793	−	0.503847i	$-0.168082\pi$
$863$	43798.2	1.72759	0.863793	+	0.503847i	$0.168082\pi$
$864$	0	0
$865$	−1871.57	−0.0735666
$866$	−32352.8	−1.26951
$867$	0	0
$868$	23103.1	0.903421
$869$	−11951.3	−0.466537
$870$	0	0
$871$	−28893.1	−1.12400
$872$	11173.4	0.433920
$873$	0	0
$874$	−18347.7	−0.710092
$875$	−3599.16	−0.139056
$876$	0	0
$877$	16022.7	0.616931	0.308466	−	0.951236i	$-0.400185\pi$
$877$	16022.7	0.616931	0.308466	+	0.951236i	$0.400185\pi$
$878$	−32686.3	−1.25639
$879$	0	0
$880$	−4115.76	−0.157661
$881$	−34840.8	−1.33237	−0.666185	−	0.745787i	$-0.732074\pi$
$881$	−34840.8	−1.33237	−0.666185	+	0.745787i	$0.732074\pi$
$882$	0	0
$883$	5327.52	0.203041	0.101520	−	0.994833i	$-0.467629\pi$
$883$	5327.52	0.203041	0.101520	+	0.994833i	$0.467629\pi$
$884$	−32204.3	−1.22528
$885$	0	0
$886$	−57482.1	−2.17963
$887$	29292.6	1.10885	0.554424	−	0.832234i	$-0.312939\pi$
$887$	29292.6	1.10885	0.554424	+	0.832234i	$0.312939\pi$
$888$	0	0
$889$	53926.9	2.03448
$890$	29438.9	1.10876
$891$	0	0
$892$	15273.4	0.573307
$893$	1955.12	0.0732650
$894$	0	0
$895$	−93.2641	−0.00348321
$896$	23482.0	0.875533
$897$	0	0
$898$	26221.8	0.974422
$899$	922.434	0.0342212
$900$	0	0
$901$	12988.9	0.480271
$902$	−13192.9	−0.487001
$903$	0	0
$904$	−3282.75	−0.120777
$905$	7794.55	0.286298
$906$	0	0
$907$	−6957.01	−0.254690	−0.127345	−	0.991858i	$-0.540646\pi$
$907$	−6957.01	−0.254690	−0.127345	+	0.991858i	$0.540646\pi$
$908$	11475.1	0.419399
$909$	0	0
$910$	25641.1	0.934061
$911$	−22002.6	−0.800195	−0.400097	−	0.916473i	$-0.631024\pi$
$911$	−22002.6	−0.800195	−0.400097	+	0.916473i	$0.631024\pi$
$912$	0	0
$913$	−1462.46	−0.0530124
$914$	−33104.8	−1.19804
$915$	0	0
$916$	−10964.1	−0.395486
$917$	28641.5	1.03143
$918$	0	0
$919$	−20551.4	−0.737682	−0.368841	−	0.929493i	$-0.620245\pi$
$919$	−20551.4	−0.737682	−0.368841	+	0.929493i	$0.620245\pi$
$920$	1305.49	0.0467833
$921$	0	0
$922$	12101.8	0.432267
$923$	−7683.61	−0.274008
$924$	0	0
$925$	−5679.30	−0.201875
$926$	1645.24	0.0583867
$927$	0	0
$928$	−1662.57	−0.0588111
$929$	−7047.73	−0.248900	−0.124450	−	0.992226i	$-0.539717\pi$
$929$	−7047.73	−0.248900	−0.124450	+	0.992226i	$0.539717\pi$
$930$	0	0
$931$	58975.1	2.07608
$932$	−25849.4	−0.908502
$933$	0	0
$934$	−37464.0	−1.31248
$935$	−5989.13	−0.209482
$936$	0	0
$937$	−48501.8	−1.69102	−0.845509	−	0.533961i	$-0.820703\pi$
$937$	−48501.8	−1.69102	−0.845509	+	0.533961i	$0.820703\pi$
$938$	−66670.6	−2.32076
$939$	0	0
$940$	505.426	0.0175374
$941$	43970.6	1.52328	0.761638	−	0.648003i	$-0.224396\pi$
$941$	43970.6	1.52328	0.761638	+	0.648003i	$0.224396\pi$
$942$	0	0
$943$	12706.3	0.438784
$944$	1541.38	0.0531438
$945$	0	0
$946$	−12940.8	−0.444760
$947$	34246.3	1.17514	0.587570	−	0.809174i	$-0.300085\pi$
$947$	34246.3	1.17514	0.587570	+	0.809174i	$0.300085\pi$
$948$	0	0
$949$	−44435.4	−1.51995
$950$	−11460.1	−0.391383
$951$	0	0
$952$	20453.2	0.696315
$953$	1262.86	0.0429256	0.0214628	−	0.999770i	$-0.493168\pi$
$953$	1262.86	0.0429256	0.0214628	+	0.999770i	$0.493168\pi$
$954$	0	0
$955$	2820.38	0.0955659
$956$	9698.05	0.328093
$957$	0	0
$958$	−1141.47	−0.0384962
$959$	67405.1	2.26968
$960$	0	0
$961$	−13431.9	−0.450871
$962$	40460.4	1.35603
$963$	0	0
$964$	−3026.90	−0.101131
$965$	−17546.9	−0.585340
$966$	0	0
$967$	28172.3	0.936877	0.468439	−	0.883496i	$-0.344817\pi$
$967$	28172.3	0.936877	0.468439	+	0.883496i	$0.344817\pi$
$968$	789.322	0.0262084
$969$	0	0
$970$	20066.2	0.664213
$971$	−28373.3	−0.937736	−0.468868	−	0.883268i	$-0.655338\pi$
$971$	−28373.3	−0.937736	−0.468868	+	0.883268i	$0.655338\pi$
$972$	0	0
$973$	14355.6	0.472991
$974$	−30952.2	−1.01825
$975$	0	0
$976$	−44444.7	−1.45762
$977$	59709.8	1.95526	0.977628	−	0.210343i	$-0.0674581\pi$
$977$	59709.8	1.95526	0.977628	+	0.210343i	$0.0674581\pi$
$978$	0	0
$979$	−17142.8	−0.559638
$980$	15245.9	0.496951
$981$	0	0
$982$	19731.8	0.641210
$983$	10526.2	0.341539	0.170770	−	0.985311i	$-0.445375\pi$
$983$	10526.2	0.341539	0.170770	+	0.985311i	$0.445375\pi$
$984$	0	0
$985$	−4902.21	−0.158576
$986$	−2967.02	−0.0958306
$987$	0	0
$988$	35883.7	1.15548
$989$	12463.5	0.400725
$990$	0	0
$991$	−32927.7	−1.05548	−0.527741	−	0.849405i	$-0.676961\pi$
$991$	−32927.7	−1.05548	−0.527741	+	0.849405i	$0.676961\pi$
$992$	−29485.3	−0.943708
$993$	0	0
$994$	−17729.9	−0.565753
$995$	−13017.3	−0.414749
$996$	0	0
$997$	−30529.9	−0.969800	−0.484900	−	0.874570i	$-0.661144\pi$
$997$	−30529.9	−0.969800	−0.484900	+	0.874570i	$0.661144\pi$
$998$	6967.75	0.221002
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.f.1.2		3
3.2	odd	2		55.4.a.c.1.2	✓	3
5.4	even	2		2475.4.a.ba.1.2		3
12.11	even	2		880.4.a.x.1.1		3
15.2	even	4		275.4.b.d.199.5		6
15.8	even	4		275.4.b.d.199.2		6
15.14	odd	2		275.4.a.d.1.2		3
33.32	even	2		605.4.a.h.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.a.c.1.2	✓	3	3.2	odd	2
275.4.a.d.1.2		3	15.14	odd	2
275.4.b.d.199.2		6	15.8	even	4
275.4.b.d.199.5		6	15.2	even	4
495.4.a.f.1.2		3	1.1	even	1	trivial
605.4.a.h.1.2		3	33.32	even	2
880.4.a.x.1.1		3	12.11	even	2
2475.4.a.ba.1.2		3	5.4	even	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-3.77801 q^{2} +6.27334 q^{4} +5.00000 q^{5} -28.7933 q^{7} +6.52332 q^{8} +O(q^{10})\)	\(q-3.77801 q^{2} +6.27334 q^{4} +5.00000 q^{5} -28.7933 q^{7} +6.52332 q^{8} -18.8900 q^{10} +11.0000 q^{11} +47.1426 q^{13} +108.781 q^{14} -74.8319 q^{16} -108.893 q^{17} +121.335 q^{19} +31.3667 q^{20} -41.5581 q^{22} +40.0253 q^{23} +25.0000 q^{25} -178.105 q^{26} -180.630 q^{28} -7.21200 q^{29} -127.903 q^{31} +230.529 q^{32} +411.400 q^{34} -143.966 q^{35} -227.172 q^{37} -458.403 q^{38} +32.6166 q^{40} +317.456 q^{41} +311.391 q^{43} +69.0068 q^{44} -151.216 q^{46} +16.1134 q^{47} +486.053 q^{49} -94.4502 q^{50} +295.742 q^{52} -119.281 q^{53} +55.0000 q^{55} -187.828 q^{56} +27.2470 q^{58} -20.5979 q^{59} +593.927 q^{61} +483.217 q^{62} -272.285 q^{64} +235.713 q^{65} -612.887 q^{67} -683.125 q^{68} +543.906 q^{70} -162.987 q^{71} -942.575 q^{73} +858.257 q^{74} +761.174 q^{76} -316.726 q^{77} -1086.48 q^{79} -374.160 q^{80} -1199.35 q^{82} -132.951 q^{83} -544.467 q^{85} -1176.44 q^{86} +71.7565 q^{88} -1558.43 q^{89} -1357.39 q^{91} +251.092 q^{92} -60.8767 q^{94} +606.673 q^{95} -1062.26 q^{97} -1836.31 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} + 23 q^{4} + 15 q^{5} - 15 q^{7} - 33 q^{8}+O(q^{10}) \) 3 * q - 5 * q^2 + 23 * q^4 + 15 * q^5 - 15 * q^7 - 33 * q^8	\( 3 q - 5 q^{2} + 23 q^{4} + 15 q^{5} - 15 q^{7} - 33 q^{8} - 25 q^{10} + 33 q^{11} - 14 q^{13} - 111 q^{14} - 149 q^{16} - 91 q^{17} - 19 q^{19} + 115 q^{20} - 55 q^{22} - 330 q^{23} + 75 q^{25} + 218 q^{26} + 193 q^{28} - 413 q^{29} - 127 q^{31} + 251 q^{32} + 477 q^{34} - 75 q^{35} - 543 q^{37} - 531 q^{38} - 165 q^{40} - 58 q^{41} + 532 q^{43} + 253 q^{44} + 270 q^{46} - 478 q^{47} + 1292 q^{49} - 125 q^{50} - 654 q^{52} - 261 q^{53} + 165 q^{55} - 695 q^{56} + 195 q^{58} + 630 q^{59} + 1095 q^{61} + 545 q^{62} - 953 q^{64} - 70 q^{65} - 976 q^{67} - 627 q^{68} - 555 q^{70} + 445 q^{71} - 1838 q^{73} + 577 q^{74} - 219 q^{76} - 165 q^{77} + 318 q^{79} - 745 q^{80} - 1738 q^{82} + 330 q^{83} - 455 q^{85} + 1172 q^{86} - 363 q^{88} - 2573 q^{89} - 4150 q^{91} - 3082 q^{92} - 310 q^{94} - 95 q^{95} - 2744 q^{97} - 5244 q^{98}+O(q^{100}) \) 3 * q - 5 * q^2 + 23 * q^4 + 15 * q^5 - 15 * q^7 - 33 * q^8 - 25 * q^10 + 33 * q^11 - 14 * q^13 - 111 * q^14 - 149 * q^16 - 91 * q^17 - 19 * q^19 + 115 * q^20 - 55 * q^22 - 330 * q^23 + 75 * q^25 + 218 * q^26 + 193 * q^28 - 413 * q^29 - 127 * q^31 + 251 * q^32 + 477 * q^34 - 75 * q^35 - 543 * q^37 - 531 * q^38 - 165 * q^40 - 58 * q^41 + 532 * q^43 + 253 * q^44 + 270 * q^46 - 478 * q^47 + 1292 * q^49 - 125 * q^50 - 654 * q^52 - 261 * q^53 + 165 * q^55 - 695 * q^56 + 195 * q^58 + 630 * q^59 + 1095 * q^61 + 545 * q^62 - 953 * q^64 - 70 * q^65 - 976 * q^67 - 627 * q^68 - 555 * q^70 + 445 * q^71 - 1838 * q^73 + 577 * q^74 - 219 * q^76 - 165 * q^77 + 318 * q^79 - 745 * q^80 - 1738 * q^82 + 330 * q^83 - 455 * q^85 + 1172 * q^86 - 363 * q^88 - 2573 * q^89 - 4150 * q^91 - 3082 * q^92 - 310 * q^94 - 95 * q^95 - 2744 * q^97 - 5244 * q^98

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