LMFDB - Embedded newform 825.6.a.y.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [825,6,Mod(1,825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("825.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.a (trivial)

Newform invariants

comment:select newform

sage:traces = [13,13,117,209] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + \cdots + 522579400 $ x^13 - 306x^11 - 206x^10 + 34574x^9 + 39928x^8 - 1788312x^7 - 2591628x^6 + 42852537x^5 + 63733360x^4 - 448113518x^3 - 549984598x^2 + 1518551280*x + 522579400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{7} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.317411$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.31741 q^{2} +9.00000 q^{3} -30.2644 q^{4} +11.8567 q^{6} -87.4786 q^{7} -82.0279 q^{8} +81.0000 q^{9} +121.000 q^{11} -272.380 q^{12} +521.292 q^{13} -115.245 q^{14} +860.397 q^{16} -68.1523 q^{17} +106.710 q^{18} -1588.76 q^{19} -787.308 q^{21} +159.407 q^{22} +191.904 q^{23} -738.251 q^{24} +686.756 q^{26} +729.000 q^{27} +2647.49 q^{28} -4252.98 q^{29} -83.6771 q^{31} +3758.39 q^{32} +1089.00 q^{33} -89.7846 q^{34} -2451.42 q^{36} -14445.4 q^{37} -2093.06 q^{38} +4691.63 q^{39} +6151.94 q^{41} -1037.21 q^{42} +1408.62 q^{43} -3662.00 q^{44} +252.816 q^{46} -14797.5 q^{47} +7743.57 q^{48} -9154.49 q^{49} -613.371 q^{51} -15776.6 q^{52} +10938.4 q^{53} +960.393 q^{54} +7175.69 q^{56} -14298.9 q^{57} -5602.92 q^{58} -16099.1 q^{59} +41095.0 q^{61} -110.237 q^{62} -7085.77 q^{63} -22581.4 q^{64} +1434.66 q^{66} -1254.02 q^{67} +2062.59 q^{68} +1727.14 q^{69} +47638.4 q^{71} -6644.26 q^{72} -7358.01 q^{73} -19030.5 q^{74} +48083.1 q^{76} -10584.9 q^{77} +6180.81 q^{78} +90184.5 q^{79} +6561.00 q^{81} +8104.64 q^{82} +10857.0 q^{83} +23827.4 q^{84} +1855.73 q^{86} -38276.8 q^{87} -9925.37 q^{88} +112869. q^{89} -45601.9 q^{91} -5807.86 q^{92} -753.094 q^{93} -19494.4 q^{94} +33825.5 q^{96} +96993.9 q^{97} -12060.2 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21}+ \cdots + 127413 q^{99}+O(q^{100}) $ 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * 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 $

$2$	1.31741	0.232888	0.116444	−	0.993197i	$-0.462851\pi$
$2$	1.31741	0.232888	0.116444	+	0.993197i	$0.462851\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−30.2644	−0.945763
$5$	0	0
$5$	0	0
$6$	11.8567	0.134458
$7$	−87.4786	−0.674772	−0.337386	−	0.941366i	$-0.609543\pi$
$7$	−87.4786	−0.674772	−0.337386	+	0.941366i	$0.609543\pi$
$8$	−82.0279	−0.453144
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−272.380	−0.546037
$13$	521.292	0.855506	0.427753	−	0.903896i	$-0.359305\pi$
$13$	521.292	0.855506	0.427753	+	0.903896i	$0.359305\pi$
$14$	−115.245	−0.157146
$15$	0	0
$16$	860.397	0.840232
$17$	−68.1523	−0.0571950	−0.0285975	−	0.999591i	$-0.509104\pi$
$17$	−68.1523	−0.0571950	−0.0285975	+	0.999591i	$0.509104\pi$
$18$	106.710	0.0776292
$19$	−1588.76	−1.00966	−0.504830	−	0.863219i	$-0.668445\pi$
$19$	−1588.76	−1.00966	−0.504830	+	0.863219i	$0.668445\pi$
$20$	0	0
$21$	−787.308	−0.389580
$22$	159.407	0.0702183
$23$	191.904	0.0756422	0.0378211	−	0.999285i	$-0.487958\pi$
$23$	191.904	0.0756422	0.0378211	+	0.999285i	$0.487958\pi$
$24$	−738.251	−0.261623
$25$	0	0
$26$	686.756	0.199237
$27$	729.000	0.192450
$28$	2647.49	0.638174
$29$	−4252.98	−0.939071	−0.469535	−	0.882914i	$-0.655579\pi$
$29$	−4252.98	−0.939071	−0.469535	+	0.882914i	$0.655579\pi$
$30$	0	0
$31$	−83.6771	−0.0156388	−0.00781938	−	0.999969i	$-0.502489\pi$
$31$	−83.6771	−0.0156388	−0.00781938	+	0.999969i	$0.502489\pi$
$32$	3758.39	0.648824
$33$	1089.00	0.174078
$34$	−89.7846	−0.0133200
$35$	0	0
$36$	−2451.42	−0.315254
$37$	−14445.4	−1.73470	−0.867352	−	0.497695i	$-0.834180\pi$
$37$	−14445.4	−1.73470	−0.867352	+	0.497695i	$0.834180\pi$
$38$	−2093.06	−0.235138
$39$	4691.63	0.493926
$40$	0	0
$41$	6151.94	0.571548	0.285774	−	0.958297i	$-0.407749\pi$
$41$	6151.94	0.571548	0.285774	+	0.958297i	$0.407749\pi$
$42$	−1037.21	−0.0907283
$43$	1408.62	0.116177	0.0580886	−	0.998311i	$-0.481499\pi$
$43$	1408.62	0.116177	0.0580886	+	0.998311i	$0.481499\pi$
$44$	−3662.00	−0.285158
$45$	0	0
$46$	252.816	0.0176161
$47$	−14797.5	−0.977110	−0.488555	−	0.872533i	$-0.662476\pi$
$47$	−14797.5	−0.977110	−0.488555	+	0.872533i	$0.662476\pi$
$48$	7743.57	0.485108
$49$	−9154.49	−0.544683
$50$	0	0
$51$	−613.371	−0.0330216
$52$	−15776.6	−0.809106
$53$	10938.4	0.534891	0.267445	−	0.963573i	$-0.413820\pi$
$53$	10938.4	0.534891	0.267445	+	0.963573i	$0.413820\pi$
$54$	960.393	0.0448192
$55$	0	0
$56$	7175.69	0.305769
$57$	−14298.9	−0.582928
$58$	−5602.92	−0.218698
$59$	−16099.1	−0.602103	−0.301052	−	0.953608i	$-0.597338\pi$
$59$	−16099.1	−0.602103	−0.301052	+	0.953608i	$0.597338\pi$
$60$	0	0
$61$	41095.0	1.41405	0.707025	−	0.707189i	$-0.250037\pi$
$61$	41095.0	1.41405	0.707025	+	0.707189i	$0.250037\pi$
$62$	−110.237	−0.00364208
$63$	−7085.77	−0.224924
$64$	−22581.4	−0.689129
$65$	0	0
$66$	1434.66	0.0405405
$67$	−1254.02	−0.0341286	−0.0170643	−	0.999854i	$-0.505432\pi$
$67$	−1254.02	−0.0341286	−0.0170643	+	0.999854i	$0.505432\pi$
$68$	2062.59	0.0540930
$69$	1727.14	0.0436721
$70$	0	0
$71$	47638.4	1.12153	0.560765	−	0.827975i	$-0.310507\pi$
$71$	47638.4	1.12153	0.560765	+	0.827975i	$0.310507\pi$
$72$	−6644.26	−0.151048
$73$	−7358.01	−0.161604	−0.0808022	−	0.996730i	$-0.525748\pi$
$73$	−7358.01	−0.161604	−0.0808022	+	0.996730i	$0.525748\pi$
$74$	−19030.5	−0.403991
$75$	0	0
$76$	48083.1	0.954900
$77$	−10584.9	−0.203451
$78$	6180.81	0.115029
$79$	90184.5	1.62579	0.812895	−	0.582411i	$-0.197890\pi$
$79$	90184.5	1.62579	0.812895	+	0.582411i	$0.197890\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	8104.64	0.133106
$83$	10857.0	0.172987	0.0864935	−	0.996252i	$-0.472434\pi$
$83$	10857.0	0.172987	0.0864935	+	0.996252i	$0.472434\pi$
$84$	23827.4	0.368450
$85$	0	0
$86$	1855.73	0.0270563
$87$	−38276.8	−0.542173
$88$	−9925.37	−0.136628
$89$	112869.	1.51043	0.755216	−	0.655476i	$-0.227532\pi$
$89$	112869.	1.51043	0.755216	+	0.655476i	$0.227532\pi$
$90$	0	0
$91$	−45601.9	−0.577271
$92$	−5807.86	−0.0715396
$93$	−753.094	−0.00902905
$94$	−19494.4	−0.227557
$95$	0	0
$96$	33825.5	0.374599
$97$	96993.9	1.04668	0.523341	−	0.852123i	$-0.324685\pi$
$97$	96993.9	1.04668	0.523341	+	0.852123i	$0.324685\pi$
$98$	−12060.2	−0.126850
$99$	9801.00	0.100504
$100$	0	0
$101$	−55750.8	−0.543810	−0.271905	−	0.962324i	$-0.587654\pi$
$101$	−55750.8	−0.543810	−0.271905	+	0.962324i	$0.587654\pi$
$102$	−808.062	−0.00769031
$103$	107469.	0.998139	0.499069	−	0.866562i	$-0.333675\pi$
$103$	107469.	0.998139	0.499069	+	0.866562i	$0.333675\pi$
$104$	−42760.5	−0.387667
$105$	0	0
$106$	14410.4	0.124569
$107$	25673.1	0.216780	0.108390	−	0.994108i	$-0.465430\pi$
$107$	25673.1	0.216780	0.108390	+	0.994108i	$0.465430\pi$
$108$	−22062.8	−0.182012
$109$	98829.6	0.796748	0.398374	−	0.917223i	$-0.369575\pi$
$109$	98829.6	0.796748	0.398374	+	0.917223i	$0.369575\pi$
$110$	0	0
$111$	−130009.	−1.00153
$112$	−75266.4	−0.566965
$113$	−12055.0	−0.0888120	−0.0444060	−	0.999014i	$-0.514140\pi$
$113$	−12055.0	−0.0888120	−0.0444060	+	0.999014i	$0.514140\pi$
$114$	−18837.5	−0.135757
$115$	0	0
$116$	128714.	0.888139
$117$	42224.7	0.285169
$118$	−21209.1	−0.140222
$119$	5961.87	0.0385936
$120$	0	0
$121$	14641.0	0.0909091
$122$	54139.1	0.329315
$123$	55367.5	0.329983
$124$	2532.44	0.0147906
$125$	0	0
$126$	−9334.87	−0.0523820
$127$	57953.4	0.318838	0.159419	−	0.987211i	$-0.449038\pi$
$127$	57953.4	0.318838	0.159419	+	0.987211i	$0.449038\pi$
$128$	−150017.	−0.809313
$129$	12677.5	0.0670750
$130$	0	0
$131$	−124350.	−0.633095	−0.316548	−	0.948577i	$-0.602524\pi$
$131$	−124350.	−0.633095	−0.316548	+	0.948577i	$0.602524\pi$
$132$	−32958.0	−0.164636
$133$	138983.	0.681291
$134$	−1652.06	−0.00794812
$135$	0	0
$136$	5590.39	0.0259176
$137$	408683.	1.86031	0.930154	−	0.367170i	$-0.119673\pi$
$137$	408683.	1.86031	0.930154	+	0.367170i	$0.119673\pi$
$138$	2275.35	0.0101707
$139$	378463.	1.66145	0.830723	−	0.556686i	$-0.187927\pi$
$139$	378463.	1.66145	0.830723	+	0.556686i	$0.187927\pi$
$140$	0	0
$141$	−133177.	−0.564135
$142$	62759.4	0.261191
$143$	63076.4	0.257945
$144$	69692.2	0.280077
$145$	0	0
$146$	−9693.52	−0.0376357
$147$	−82390.4	−0.314473
$148$	437182.	1.64062
$149$	−271317.	−1.00118	−0.500588	−	0.865686i	$-0.666883\pi$
$149$	−271317.	−1.00118	−0.500588	+	0.865686i	$0.666883\pi$
$150$	0	0
$151$	−146748.	−0.523758	−0.261879	−	0.965101i	$-0.584342\pi$
$151$	−146748.	−0.523758	−0.261879	+	0.965101i	$0.584342\pi$
$152$	130323.	0.457522
$153$	−5520.34	−0.0190650
$154$	−13944.7	−0.0473813
$155$	0	0
$156$	−141990.	−0.467138
$157$	365562.	1.18362	0.591810	−	0.806077i	$-0.298414\pi$
$157$	365562.	1.18362	0.591810	+	0.806077i	$0.298414\pi$
$158$	118810.	0.378626
$159$	98445.8	0.308819
$160$	0	0
$161$	−16787.5	−0.0510412
$162$	8643.54	0.0258764
$163$	−16695.2	−0.0492180	−0.0246090	−	0.999697i	$-0.507834\pi$
$163$	−16695.2	−0.0492180	−0.0246090	+	0.999697i	$0.507834\pi$
$164$	−186185.	−0.540549
$165$	0	0
$166$	14303.1	0.0402865
$167$	568590.	1.57764	0.788821	−	0.614623i	$-0.210692\pi$
$167$	568590.	1.57764	0.788821	+	0.614623i	$0.210692\pi$
$168$	64581.2	0.176536
$169$	−99547.4	−0.268110
$170$	0	0
$171$	−128690.	−0.336554
$172$	−42630.9	−0.109876
$173$	−552203.	−1.40276	−0.701380	−	0.712788i	$-0.747432\pi$
$173$	−552203.	−1.40276	−0.701380	+	0.712788i	$0.747432\pi$
$174$	−50426.3	−0.126265
$175$	0	0
$176$	104108.	0.253339
$177$	−144892.	−0.347624
$178$	148695.	0.351761
$179$	−179962.	−0.419807	−0.209903	−	0.977722i	$-0.567315\pi$
$179$	−179962.	−0.419807	−0.209903	+	0.977722i	$0.567315\pi$
$180$	0	0
$181$	612439.	1.38953	0.694763	−	0.719239i	$-0.255509\pi$
$181$	612439.	1.38953	0.694763	+	0.719239i	$0.255509\pi$
$182$	−60076.5	−0.134439
$183$	369855.	0.816402
$184$	−15741.5	−0.0342768
$185$	0	0
$186$	−992.135	−0.00210275
$187$	−8246.43	−0.0172449
$188$	447838.	0.924115
$189$	−63771.9	−0.129860
$190$	0	0
$191$	65407.8	0.129732	0.0648659	−	0.997894i	$-0.479338\pi$
$191$	65407.8	0.129732	0.0648659	+	0.997894i	$0.479338\pi$
$192$	−203232.	−0.397869
$193$	661050.	1.27744	0.638721	−	0.769439i	$-0.279464\pi$
$193$	661050.	1.27744	0.638721	+	0.769439i	$0.279464\pi$
$194$	127781.	0.243759
$195$	0	0
$196$	277055.	0.515141
$197$	−590186.	−1.08349	−0.541743	−	0.840544i	$-0.682235\pi$
$197$	−590186.	−1.08349	−0.541743	+	0.840544i	$0.682235\pi$
$198$	12911.9	0.0234061
$199$	−342750.	−0.613543	−0.306771	−	0.951783i	$-0.599249\pi$
$199$	−342750.	−0.613543	−0.306771	+	0.951783i	$0.599249\pi$
$200$	0	0
$201$	−11286.2	−0.0197041
$202$	−73446.7	−0.126647
$203$	372045.	0.633658
$204$	18563.3	0.0312306
$205$	0	0
$206$	141581.	0.232454
$207$	15544.2	0.0252141
$208$	448518.	0.718823
$209$	−192241.	−0.304424
$210$	0	0
$211$	−214655.	−0.331921	−0.165960	−	0.986132i	$-0.553072\pi$
$211$	−214655.	−0.331921	−0.165960	+	0.986132i	$0.553072\pi$
$212$	−331045.	−0.505880
$213$	428746.	0.647516
$214$	33822.1	0.0504854
$215$	0	0
$216$	−59798.3	−0.0872076
$217$	7319.96	0.0105526
$218$	130199.	0.185553
$219$	−66222.1	−0.0933023
$220$	0	0
$221$	−35527.3	−0.0489307
$222$	−171275.	−0.233244
$223$	−118747.	−0.159904	−0.0799521	−	0.996799i	$-0.525477\pi$
$223$	−118747.	−0.159904	−0.0799521	+	0.996799i	$0.525477\pi$
$224$	−328779.	−0.437808
$225$	0	0
$226$	−15881.4	−0.0206832
$227$	2183.16	0.00281204	0.00140602	−	0.999999i	$-0.499552\pi$
$227$	2183.16	0.00281204	0.00140602	+	0.999999i	$0.499552\pi$
$228$	432747.	0.551312
$229$	123704.	0.155881	0.0779407	−	0.996958i	$-0.475166\pi$
$229$	123704.	0.155881	0.0779407	+	0.996958i	$0.475166\pi$
$230$	0	0
$231$	−95264.2	−0.117463
$232$	348863.	0.425534
$233$	1.16587e6	1.40689	0.703446	−	0.710748i	$-0.251644\pi$
$233$	1.16587e6	1.40689	0.703446	+	0.710748i	$0.251644\pi$
$234$	55627.3	0.0664122
$235$	0	0
$236$	487229.	0.569447
$237$	811661.	0.938650
$238$	7854.24	0.00898797
$239$	770619.	0.872659	0.436330	−	0.899787i	$-0.356278\pi$
$239$	770619.	0.872659	0.436330	+	0.899787i	$0.356278\pi$
$240$	0	0
$241$	−1.42541e6	−1.58087	−0.790434	−	0.612547i	$-0.790145\pi$
$241$	−1.42541e6	−1.58087	−0.790434	+	0.612547i	$0.790145\pi$
$242$	19288.2	0.0211716
$243$	59049.0	0.0641500
$244$	−1.24372e6	−1.33736
$245$	0	0
$246$	72941.7	0.0768490
$247$	−828211.	−0.863771
$248$	6863.86	0.00708662
$249$	97712.7	0.0998741
$250$	0	0
$251$	−668745.	−0.670002	−0.335001	−	0.942218i	$-0.608737\pi$
$251$	−668745.	−0.670002	−0.335001	+	0.942218i	$0.608737\pi$
$252$	214447.	0.212725
$253$	23220.4	0.0228070
$254$	76348.5	0.0742534
$255$	0	0
$256$	524969.	0.500650
$257$	1.08696e6	1.02655	0.513276	−	0.858223i	$-0.328432\pi$
$257$	1.08696e6	1.02655	0.513276	+	0.858223i	$0.328432\pi$
$258$	16701.5	0.0156209
$259$	1.26366e6	1.17053
$260$	0	0
$261$	−344491.	−0.313024
$262$	−163821.	−0.147440
$263$	865881.	0.771914	0.385957	−	0.922517i	$-0.373871\pi$
$263$	865881.	0.771914	0.385957	+	0.922517i	$0.373871\pi$
$264$	−89328.3	−0.0788823
$265$	0	0
$266$	183098.	0.158664
$267$	1.01582e6	0.872049
$268$	37952.2	0.0322775
$269$	455847.	0.384094	0.192047	−	0.981386i	$-0.438487\pi$
$269$	455847.	0.384094	0.192047	+	0.981386i	$0.438487\pi$
$270$	0	0
$271$	1.93670e6	1.60191	0.800955	−	0.598725i	$-0.204326\pi$
$271$	1.93670e6	1.60191	0.800955	+	0.598725i	$0.204326\pi$
$272$	−58638.1	−0.0480571
$273$	−410417.	−0.333288
$274$	538403.	0.433243
$275$	0	0
$276$	−52270.8	−0.0413034
$277$	−1.67891e6	−1.31470	−0.657352	−	0.753584i	$-0.728323\pi$
$277$	−1.67891e6	−1.31470	−0.657352	+	0.753584i	$0.728323\pi$
$278$	498591.	0.386930
$279$	−6777.85	−0.00521292
$280$	0	0
$281$	−1.66433e6	−1.25740	−0.628700	−	0.777648i	$-0.716413\pi$
$281$	−1.66433e6	−1.25740	−0.628700	+	0.777648i	$0.716413\pi$
$282$	−175449.	−0.131380
$283$	529051.	0.392673	0.196337	−	0.980537i	$-0.437095\pi$
$283$	529051.	0.392673	0.196337	+	0.980537i	$0.437095\pi$
$284$	−1.44175e6	−1.06070
$285$	0	0
$286$	83097.5	0.0600721
$287$	−538163.	−0.385664
$288$	304430.	0.216275
$289$	−1.41521e6	−0.996729
$290$	0	0
$291$	872945.	0.604302
$292$	222686.	0.152839
$293$	−1.17586e6	−0.800178	−0.400089	−	0.916476i	$-0.631021\pi$
$293$	−1.17586e6	−0.800178	−0.400089	+	0.916476i	$0.631021\pi$
$294$	−108542.	−0.0732369
$295$	0	0
$296$	1.18493e6	0.786071
$297$	88209.0	0.0580259
$298$	−357436.	−0.233162
$299$	100038.	0.0647123
$300$	0	0
$301$	−123224.	−0.0783931
$302$	−193328.	−0.121977
$303$	−501757.	−0.313969
$304$	−1.36697e6	−0.848349
$305$	0	0
$306$	−7272.55	−0.00444000
$307$	1.53089e6	0.927037	0.463518	−	0.886087i	$-0.346587\pi$
$307$	1.53089e6	0.927037	0.463518	+	0.886087i	$0.346587\pi$
$308$	320346.	0.192417
$309$	967223.	0.576276
$310$	0	0
$311$	−1.11621e6	−0.654400	−0.327200	−	0.944955i	$-0.606105\pi$
$311$	−1.11621e6	−0.654400	−0.327200	+	0.944955i	$0.606105\pi$
$312$	−384844.	−0.223820
$313$	864587.	0.498824	0.249412	−	0.968397i	$-0.419763\pi$
$313$	864587.	0.498824	0.249412	+	0.968397i	$0.419763\pi$
$314$	481596.	0.275651
$315$	0	0
$316$	−2.72938e6	−1.53761
$317$	−1.80604e6	−1.00943	−0.504717	−	0.863285i	$-0.668403\pi$
$317$	−1.80604e6	−1.00943	−0.504717	+	0.863285i	$0.668403\pi$
$318$	129694.	0.0719202
$319$	−514611.	−0.283140
$320$	0	0
$321$	231058.	0.125158
$322$	−22116.0	−0.0118869
$323$	108278.	0.0577476
$324$	−198565.	−0.105085
$325$	0	0
$326$	−21994.5	−0.0114623
$327$	889466.	0.460003
$328$	−504630.	−0.258994
$329$	1.29446e6	0.659326
$330$	0	0
$331$	1.85701e6	0.931634	0.465817	−	0.884881i	$-0.345761\pi$
$331$	1.85701e6	0.931634	0.465817	+	0.884881i	$0.345761\pi$
$332$	−328580.	−0.163605
$333$	−1.17008e6	−0.578235
$334$	749068.	0.367413
$335$	0	0
$336$	−677397.	−0.327337
$337$	2.18068e6	1.04597	0.522983	−	0.852343i	$-0.324819\pi$
$337$	2.18068e6	1.04597	0.522983	+	0.852343i	$0.324819\pi$
$338$	−131145.	−0.0624395
$339$	−108495.	−0.0512757
$340$	0	0
$341$	−10124.9	−0.00471527
$342$	−169538.	−0.0783792
$343$	2.27108e6	1.04231
$344$	−115546.	−0.0526451
$345$	0	0
$346$	−727478.	−0.326685
$347$	2.59930e6	1.15886	0.579432	−	0.815021i	$-0.303274\pi$
$347$	2.59930e6	1.15886	0.579432	+	0.815021i	$0.303274\pi$
$348$	1.15843e6	0.512767
$349$	1.17216e6	0.515138	0.257569	−	0.966260i	$-0.417079\pi$
$349$	1.17216e6	0.515138	0.257569	+	0.966260i	$0.417079\pi$
$350$	0	0
$351$	380022.	0.164642
$352$	454765.	0.195628
$353$	1.57881e6	0.674364	0.337182	−	0.941440i	$-0.390526\pi$
$353$	1.57881e6	0.674364	0.337182	+	0.941440i	$0.390526\pi$
$354$	−190882.	−0.0809574
$355$	0	0
$356$	−3.41593e6	−1.42851
$357$	53656.8	0.0222820
$358$	−237085.	−0.0977678
$359$	3.19520e6	1.30847	0.654233	−	0.756293i	$-0.272992\pi$
$359$	3.19520e6	1.30847	0.654233	+	0.756293i	$0.272992\pi$
$360$	0	0
$361$	48073.9	0.0194152
$362$	806834.	0.323603
$363$	131769.	0.0524864
$364$	1.38012e6	0.545962
$365$	0	0
$366$	487252.	0.190130
$367$	2.52046e6	0.976819	0.488409	−	0.872615i	$-0.337577\pi$
$367$	2.52046e6	0.976819	0.488409	+	0.872615i	$0.337577\pi$
$368$	165114.	0.0635570
$369$	498307.	0.190516
$370$	0	0
$371$	−956879.	−0.360929
$372$	22792.0	0.00853934
$373$	1.66879e6	0.621053	0.310527	−	0.950565i	$-0.399495\pi$
$373$	1.66879e6	0.621053	0.310527	+	0.950565i	$0.399495\pi$
$374$	−10863.9	−0.00401614
$375$	0	0
$376$	1.21381e6	0.442772
$377$	−2.21705e6	−0.803380
$378$	−84013.9	−0.0302428
$379$	−3.13030e6	−1.11941	−0.559704	−	0.828692i	$-0.689085\pi$
$379$	−3.13030e6	−1.11941	−0.559704	+	0.828692i	$0.689085\pi$
$380$	0	0
$381$	521581.	0.184081
$382$	86169.0	0.0302129
$383$	4.30969e6	1.50124	0.750618	−	0.660737i	$-0.229756\pi$
$383$	4.30969e6	1.50124	0.750618	+	0.660737i	$0.229756\pi$
$384$	−1.35016e6	−0.467257
$385$	0	0
$386$	870875.	0.297500
$387$	114098.	0.0387258
$388$	−2.93546e6	−0.989914
$389$	−2.45755e6	−0.823432	−0.411716	−	0.911312i	$-0.635071\pi$
$389$	−2.45755e6	−0.823432	−0.411716	+	0.911312i	$0.635071\pi$
$390$	0	0
$391$	−13078.7	−0.00432636
$392$	750923.	0.246820
$393$	−1.11915e6	−0.365518
$394$	−777518.	−0.252330
$395$	0	0
$396$	−296622.	−0.0950528
$397$	5.93203e6	1.88898	0.944489	−	0.328544i	$-0.106558\pi$
$397$	5.93203e6	1.88898	0.944489	+	0.328544i	$0.106558\pi$
$398$	−451543.	−0.142887
$399$	1.25085e6	0.393343
$400$	0	0
$401$	4.56758e6	1.41849	0.709243	−	0.704964i	$-0.249037\pi$
$401$	4.56758e6	1.41849	0.709243	+	0.704964i	$0.249037\pi$
$402$	−14868.6	−0.00458885
$403$	−43620.2	−0.0133791
$404$	1.68727e6	0.514316
$405$	0	0
$406$	490136.	0.147571
$407$	−1.74789e6	−0.523033
$408$	50313.5	0.0149635
$409$	−5.87934e6	−1.73788	−0.868941	−	0.494915i	$-0.835199\pi$
$409$	−5.87934e6	−1.73788	−0.868941	+	0.494915i	$0.835199\pi$
$410$	0	0
$411$	3.67814e6	1.07405
$412$	−3.25249e6	−0.944003
$413$	1.40833e6	0.406282
$414$	20478.1	0.00587205
$415$	0	0
$416$	1.95922e6	0.555072
$417$	3.40616e6	0.959236
$418$	−253260.	−0.0708966
$419$	6.12464e6	1.70430	0.852149	−	0.523299i	$-0.175299\pi$
$419$	6.12464e6	1.70430	0.852149	+	0.523299i	$0.175299\pi$
$420$	0	0
$421$	−2.83114e6	−0.778495	−0.389248	−	0.921133i	$-0.627265\pi$
$421$	−2.83114e6	−0.778495	−0.389248	+	0.921133i	$0.627265\pi$
$422$	−282789.	−0.0773003
$423$	−1.19860e6	−0.325703
$424$	−897256.	−0.242383
$425$	0	0
$426$	564834.	0.150799
$427$	−3.59494e6	−0.954161
$428$	−776983.	−0.205023
$429$	567687.	0.148924
$430$	0	0
$431$	6.20538e6	1.60907	0.804536	−	0.593904i	$-0.202414\pi$
$431$	6.20538e6	1.60907	0.804536	+	0.593904i	$0.202414\pi$
$432$	627230.	0.161703
$433$	422736.	0.108355	0.0541775	−	0.998531i	$-0.482746\pi$
$433$	422736.	0.108355	0.0541775	+	0.998531i	$0.482746\pi$
$434$	9643.40	0.00245757
$435$	0	0
$436$	−2.99102e6	−0.753535
$437$	−304890.	−0.0763730
$438$	−87241.7	−0.0217290
$439$	−4.07012e6	−1.00797	−0.503983	−	0.863713i	$-0.668133\pi$
$439$	−4.07012e6	−1.00797	−0.503983	+	0.863713i	$0.668133\pi$
$440$	0	0
$441$	−741514.	−0.181561
$442$	−46804.0	−0.0113953
$443$	4.95245e6	1.19898	0.599488	−	0.800384i	$-0.295371\pi$
$443$	4.95245e6	1.19898	0.599488	+	0.800384i	$0.295371\pi$
$444$	3.93464e6	0.947212
$445$	0	0
$446$	−156438.	−0.0372397
$447$	−2.44185e6	−0.578029
$448$	1.97539e6	0.465004
$449$	7.34037e6	1.71831	0.859157	−	0.511713i	$-0.170989\pi$
$449$	7.34037e6	1.71831	0.859157	+	0.511713i	$0.170989\pi$
$450$	0	0
$451$	744385.	0.172328
$452$	364838.	0.0839952
$453$	−1.32073e6	−0.302392
$454$	2876.12	0.000654888 0
$455$	0	0
$456$	1.17291e6	0.264150
$457$	2.07655e6	0.465106	0.232553	−	0.972584i	$-0.425292\pi$
$457$	2.07655e6	0.465106	0.232553	+	0.972584i	$0.425292\pi$
$458$	162969.	0.0363029
$459$	−49683.0	−0.0110072
$460$	0	0
$461$	−1.68917e6	−0.370186	−0.185093	−	0.982721i	$-0.559259\pi$
$461$	−1.68917e6	−0.370186	−0.185093	+	0.982721i	$0.559259\pi$
$462$	−125502.	−0.0273556
$463$	−1.35953e6	−0.294739	−0.147369	−	0.989082i	$-0.547081\pi$
$463$	−1.35953e6	−0.294739	−0.147369	+	0.989082i	$0.547081\pi$
$464$	−3.65925e6	−0.789037
$465$	0	0
$466$	1.53593e6	0.327648
$467$	−8.47580e6	−1.79841	−0.899205	−	0.437528i	$-0.855854\pi$
$467$	−8.47580e6	−1.79841	−0.899205	+	0.437528i	$0.855854\pi$
$468$	−1.27791e6	−0.269702
$469$	109700.	0.0230290
$470$	0	0
$471$	3.29006e6	0.683363
$472$	1.32057e6	0.272840
$473$	170442.	0.0350288
$474$	1.06929e6	0.218600
$475$	0	0
$476$	−180433.	−0.0365004
$477$	886013.	0.178297
$478$	1.01522e6	0.203232
$479$	−3.52896e6	−0.702762	−0.351381	−	0.936233i	$-0.614288\pi$
$479$	−3.52896e6	−0.702762	−0.351381	+	0.936233i	$0.614288\pi$
$480$	0	0
$481$	−7.53028e6	−1.48405
$482$	−1.87784e6	−0.368165
$483$	−151087.	−0.0294687
$484$	−443101.	−0.0859785
$485$	0	0
$486$	77791.8	0.0149397
$487$	4.66713e6	0.891719	0.445859	−	0.895103i	$-0.352898\pi$
$487$	4.66713e6	0.891719	0.445859	+	0.895103i	$0.352898\pi$
$488$	−3.37094e6	−0.640768
$489$	−150257.	−0.0284160
$490$	0	0
$491$	−7.22516e6	−1.35252	−0.676260	−	0.736663i	$-0.736400\pi$
$491$	−7.22516e6	−1.35252	−0.676260	+	0.736663i	$0.736400\pi$
$492$	−1.67566e6	−0.312086
$493$	289850.	0.0537102
$494$	−1.09109e6	−0.201162
$495$	0	0
$496$	−71995.6	−0.0131402
$497$	−4.16734e6	−0.756777
$498$	128728.	0.0232594
$499$	−1.04815e7	−1.88440	−0.942202	−	0.335046i	$-0.891248\pi$
$499$	−1.04815e7	−1.88440	−0.942202	+	0.335046i	$0.891248\pi$
$500$	0	0
$501$	5.11731e6	0.910852
$502$	−881012.	−0.156035
$503$	−8.37701e6	−1.47628	−0.738140	−	0.674647i	$-0.764296\pi$
$503$	−8.37701e6	−1.47628	−0.738140	+	0.674647i	$0.764296\pi$
$504$	581230.	0.101923
$505$	0	0
$506$	30590.8	0.00531146
$507$	−895926.	−0.154793
$508$	−1.75393e6	−0.301545
$509$	3.17550e6	0.543272	0.271636	−	0.962400i	$-0.412435\pi$
$509$	3.17550e6	0.543272	0.271636	+	0.962400i	$0.412435\pi$
$510$	0	0
$511$	643668.	0.109046
$512$	5.49216e6	0.925908
$513$	−1.15821e6	−0.194309
$514$	1.43198e6	0.239071
$515$	0	0
$516$	−383678.	−0.0634371
$517$	−1.79050e6	−0.294610
$518$	1.66477e6	0.272602
$519$	−4.96982e6	−0.809884
$520$	0	0
$521$	3.72507e6	0.601230	0.300615	−	0.953746i	$-0.402808\pi$
$521$	3.72507e6	0.601230	0.300615	+	0.953746i	$0.402808\pi$
$522$	−453837.	−0.0728993
$523$	−3.77963e6	−0.604219	−0.302110	−	0.953273i	$-0.597691\pi$
$523$	−3.77963e6	−0.604219	−0.302110	+	0.953273i	$0.597691\pi$
$524$	3.76339e6	0.598758
$525$	0	0
$526$	1.14072e6	0.179769
$527$	5702.79	0.000894460 0
$528$	936973.	0.146266
$529$	−6.39952e6	−0.994278
$530$	0	0
$531$	−1.30403e6	−0.200701
$532$	−4.20624e6	−0.644340
$533$	3.20696e6	0.488962
$534$	1.33826e6	0.203089
$535$	0	0
$536$	102865.	0.0154652
$537$	−1.61966e6	−0.242375
$538$	600538.	0.0894509
$539$	−1.10769e6	−0.164228
$540$	0	0
$541$	−6.08511e6	−0.893871	−0.446936	−	0.894566i	$-0.647485\pi$
$541$	−6.08511e6	−0.893871	−0.446936	+	0.894566i	$0.647485\pi$
$542$	2.55142e6	0.373065
$543$	5.51195e6	0.802243
$544$	−256143.	−0.0371095
$545$	0	0
$546$	−540689.	−0.0776186
$547$	−2.49034e6	−0.355870	−0.177935	−	0.984042i	$-0.556942\pi$
$547$	−2.49034e6	−0.355870	−0.177935	+	0.984042i	$0.556942\pi$
$548$	−1.23685e7	−1.75941
$549$	3.32870e6	0.471350
$550$	0	0
$551$	6.75698e6	0.948143
$552$	−141673.	−0.0197897
$553$	−7.88922e6	−1.09704
$554$	−2.21181e6	−0.306178
$555$	0	0
$556$	−1.14540e7	−1.57133
$557$	5.36235e6	0.732347	0.366173	−	0.930547i	$-0.380668\pi$
$557$	5.36235e6	0.732347	0.366173	+	0.930547i	$0.380668\pi$
$558$	−8929.21	−0.00121403
$559$	734300.	0.0993903
$560$	0	0
$561$	−74217.9	−0.00995638
$562$	−2.19261e6	−0.292833
$563$	−479316.	−0.0637310	−0.0318655	−	0.999492i	$-0.510145\pi$
$563$	−479316.	−0.0637310	−0.0318655	+	0.999492i	$0.510145\pi$
$564$	4.03054e6	0.533538
$565$	0	0
$566$	696978.	0.0914488
$567$	−573947.	−0.0749746
$568$	−3.90768e6	−0.508215
$569$	3.08333e6	0.399245	0.199623	−	0.979873i	$-0.436028\pi$
$569$	3.08333e6	0.399245	0.199623	+	0.979873i	$0.436028\pi$
$570$	0	0
$571$	−5.85646e6	−0.751701	−0.375850	−	0.926680i	$-0.622649\pi$
$571$	−5.85646e6	−0.751701	−0.375850	+	0.926680i	$0.622649\pi$
$572$	−1.90897e6	−0.243955
$573$	588670.	0.0749007
$574$	−708982.	−0.0898164
$575$	0	0
$576$	−1.82909e6	−0.229710
$577$	−5.66144e6	−0.707925	−0.353963	−	0.935260i	$-0.615166\pi$
$577$	−5.66144e6	−0.707925	−0.353963	+	0.935260i	$0.615166\pi$
$578$	−1.86442e6	−0.232126
$579$	5.94945e6	0.737531
$580$	0	0
$581$	−949753.	−0.116727
$582$	1.15003e6	0.140735
$583$	1.32355e6	0.161276
$584$	603562.	0.0732301
$585$	0	0
$586$	−1.54909e6	−0.186352
$587$	−1.39978e7	−1.67674	−0.838368	−	0.545104i	$-0.816490\pi$
$587$	−1.39978e7	−1.67674	−0.838368	+	0.545104i	$0.816490\pi$
$588$	2.49350e6	0.297417
$589$	132943.	0.0157899
$590$	0	0
$591$	−5.31167e6	−0.625551
$592$	−1.24288e7	−1.45755
$593$	−1.27936e7	−1.49402	−0.747012	−	0.664810i	$-0.768512\pi$
$593$	−1.27936e7	−1.49402	−0.747012	+	0.664810i	$0.768512\pi$
$594$	116208.	0.0135135
$595$	0	0
$596$	8.21124e6	0.946876
$597$	−3.08475e6	−0.354229
$598$	131791.	0.0150707
$599$	−1.21733e7	−1.38625	−0.693125	−	0.720818i	$-0.743766\pi$
$599$	−1.21733e7	−1.38625	−0.693125	+	0.720818i	$0.743766\pi$
$600$	0	0
$601$	−6.74801e6	−0.762060	−0.381030	−	0.924563i	$-0.624431\pi$
$601$	−6.74801e6	−0.762060	−0.381030	+	0.924563i	$0.624431\pi$
$602$	−162336.	−0.0182568
$603$	−101576.	−0.0113762
$604$	4.44125e6	0.495351
$605$	0	0
$606$	−661020.	−0.0731195
$607$	1.16658e7	1.28512	0.642561	−	0.766234i	$-0.277872\pi$
$607$	1.16658e7	1.28512	0.642561	+	0.766234i	$0.277872\pi$
$608$	−5.97120e6	−0.655092
$609$	3.34840e6	0.365843
$610$	0	0
$611$	−7.71382e6	−0.835923
$612$	167070.	0.0180310
$613$	−1.36615e7	−1.46841	−0.734204	−	0.678928i	$-0.762445\pi$
$613$	−1.36615e7	−1.46841	−0.734204	+	0.678928i	$0.762445\pi$
$614$	2.01681e6	0.215895
$615$	0	0
$616$	868258.	0.0921928
$617$	1.15742e7	1.22399	0.611995	−	0.790862i	$-0.290367\pi$
$617$	1.15742e7	1.22399	0.611995	+	0.790862i	$0.290367\pi$
$618$	1.27423e6	0.134207
$619$	9.22658e6	0.967863	0.483932	−	0.875106i	$-0.339208\pi$
$619$	9.22658e6	0.967863	0.483932	+	0.875106i	$0.339208\pi$
$620$	0	0
$621$	139898.	0.0145574
$622$	−1.47050e6	−0.152402
$623$	−9.87366e6	−1.01920
$624$	4.03667e6	0.415013
$625$	0	0
$626$	1.13902e6	0.116170
$627$	−1.73016e6	−0.175759
$628$	−1.10635e7	−1.11942
$629$	984487.	0.0992164
$630$	0	0
$631$	1.03002e7	1.02985	0.514923	−	0.857237i	$-0.327821\pi$
$631$	1.03002e7	1.02985	0.514923	+	0.857237i	$0.327821\pi$
$632$	−7.39764e6	−0.736717
$633$	−1.93189e6	−0.191635
$634$	−2.37929e6	−0.235085
$635$	0	0
$636$	−2.97941e6	−0.292070
$637$	−4.77217e6	−0.465980
$638$	−677954.	−0.0659399
$639$	3.85871e6	0.373844
$640$	0	0
$641$	−3.06165e6	−0.294313	−0.147157	−	0.989113i	$-0.547012\pi$
$641$	−3.06165e6	−0.294313	−0.147157	+	0.989113i	$0.547012\pi$
$642$	304399.	0.0291478
$643$	−1.62044e6	−0.154563	−0.0772813	−	0.997009i	$-0.524624\pi$
$643$	−1.62044e6	−0.154563	−0.0772813	+	0.997009i	$0.524624\pi$
$644$	508064.	0.0482729
$645$	0	0
$646$	142647.	0.0134487
$647$	1.57992e6	0.148380	0.0741900	−	0.997244i	$-0.476363\pi$
$647$	1.57992e6	0.148380	0.0741900	+	0.997244i	$0.476363\pi$
$648$	−538185.	−0.0503494
$649$	−1.94799e6	−0.181541
$650$	0	0
$651$	65879.6	0.00609254
$652$	505272.	0.0465486
$653$	1.29080e6	0.118462	0.0592308	−	0.998244i	$-0.481135\pi$
$653$	1.29080e6	0.118462	0.0592308	+	0.998244i	$0.481135\pi$
$654$	1.17179e6	0.107129
$655$	0	0
$656$	5.29311e6	0.480232
$657$	−595999.	−0.0538681
$658$	1.70534e6	0.153549
$659$	1.44401e7	1.29525	0.647627	−	0.761957i	$-0.275761\pi$
$659$	1.44401e7	1.29525	0.647627	+	0.761957i	$0.275761\pi$
$660$	0	0
$661$	8.80403e6	0.783751	0.391875	−	0.920018i	$-0.371826\pi$
$661$	8.80403e6	0.783751	0.391875	+	0.920018i	$0.371826\pi$
$662$	2.44645e6	0.216966
$663$	−319745.	−0.0282501
$664$	−890574.	−0.0783880
$665$	0	0
$666$	−1.54147e6	−0.134664
$667$	−816163.	−0.0710334
$668$	−1.72081e7	−1.49208
$669$	−1.06872e6	−0.0923207
$670$	0	0
$671$	4.97250e6	0.426352
$672$	−2.95901e6	−0.252769
$673$	1.43891e7	1.22461	0.612303	−	0.790623i	$-0.290243\pi$
$673$	1.43891e7	1.22461	0.612303	+	0.790623i	$0.290243\pi$
$674$	2.87286e6	0.243593
$675$	0	0
$676$	3.01274e6	0.253569
$677$	1.21710e7	1.02060	0.510299	−	0.859997i	$-0.329535\pi$
$677$	1.21710e7	1.02060	0.510299	+	0.859997i	$0.329535\pi$
$678$	−142933.	−0.0119415
$679$	−8.48489e6	−0.706272
$680$	0	0
$681$	19648.4	0.00162353
$682$	−13338.7	−0.00109813
$683$	−5.96383e6	−0.489186	−0.244593	−	0.969626i	$-0.578654\pi$
$683$	−5.96383e6	−0.489186	−0.244593	+	0.969626i	$0.578654\pi$
$684$	3.89473e6	0.318300
$685$	0	0
$686$	2.99194e6	0.242741
$687$	1.11333e6	0.0899982
$688$	1.21197e6	0.0976158
$689$	5.70212e6	0.457602
$690$	0	0
$691$	−1.10524e7	−0.880565	−0.440283	−	0.897859i	$-0.645122\pi$
$691$	−1.10524e7	−0.880565	−0.440283	+	0.897859i	$0.645122\pi$
$692$	1.67121e7	1.32668
$693$	−857378.	−0.0678171
$694$	3.42434e6	0.269885
$695$	0	0
$696$	3.13977e6	0.245682
$697$	−419269.	−0.0326897
$698$	1.54422e6	0.119969
$699$	1.04928e7	0.812270
$700$	0	0
$701$	−9.86637e6	−0.758337	−0.379168	−	0.925328i	$-0.623790\pi$
$701$	−9.86637e6	−0.758337	−0.379168	+	0.925328i	$0.623790\pi$
$702$	500645.	0.0383431
$703$	2.29503e7	1.75146
$704$	−2.73235e6	−0.207780
$705$	0	0
$706$	2.07995e6	0.157051
$707$	4.87700e6	0.366948
$708$	4.38506e6	0.328770
$709$	−1.65852e7	−1.23910	−0.619550	−	0.784957i	$-0.712685\pi$
$709$	−1.65852e7	−1.23910	−0.619550	+	0.784957i	$0.712685\pi$
$710$	0	0
$711$	7.30495e6	0.541930
$712$	−9.25844e6	−0.684444
$713$	−16058.0	−0.00118295
$714$	70688.1	0.00518921
$715$	0	0
$716$	5.44646e6	0.397038
$717$	6.93557e6	0.503830
$718$	4.20940e6	0.304725
$719$	−1.70621e6	−0.123086	−0.0615430	−	0.998104i	$-0.519602\pi$
$719$	−1.70621e6	−0.123086	−0.0615430	+	0.998104i	$0.519602\pi$
$720$	0	0
$721$	−9.40126e6	−0.673516
$722$	63333.1	0.00452156
$723$	−1.28286e7	−0.912715
$724$	−1.85351e7	−1.31416
$725$	0	0
$726$	173594.	0.0122234
$727$	4.79877e6	0.336739	0.168370	−	0.985724i	$-0.446150\pi$
$727$	4.79877e6	0.336739	0.168370	+	0.985724i	$0.446150\pi$
$728$	3.74063e6	0.261587
$729$	531441.	0.0370370
$730$	0	0
$731$	−96000.4	−0.00664476
$732$	−1.11935e7	−0.772123
$733$	−2.10050e7	−1.44398	−0.721991	−	0.691902i	$-0.756773\pi$
$733$	−2.10050e7	−1.44398	−0.721991	+	0.691902i	$0.756773\pi$
$734$	3.32048e6	0.227489
$735$	0	0
$736$	721250.	0.0490785
$737$	−151737.	−0.0102901
$738$	656475.	0.0443688
$739$	−1.45144e7	−0.977660	−0.488830	−	0.872379i	$-0.662576\pi$
$739$	−1.45144e7	−0.977660	−0.488830	+	0.872379i	$0.662576\pi$
$740$	0	0
$741$	−7.45390e6	−0.498698
$742$	−1.26060e6	−0.0840560
$743$	1.33510e7	0.887245	0.443622	−	0.896214i	$-0.353693\pi$
$743$	1.33510e7	0.887245	0.443622	+	0.896214i	$0.353693\pi$
$744$	61774.7	0.00409146
$745$	0	0
$746$	2.19848e6	0.144636
$747$	879414.	0.0576623
$748$	249573.	0.0163096
$749$	−2.24585e6	−0.146277
$750$	0	0
$751$	1.03902e7	0.672242	0.336121	−	0.941819i	$-0.390885\pi$
$751$	1.03902e7	0.672242	0.336121	+	0.941819i	$0.390885\pi$
$752$	−1.27317e7	−0.820999
$753$	−6.01870e6	−0.386826
$754$	−2.92076e6	−0.187097
$755$	0	0
$756$	1.93002e6	0.122817
$757$	−1.09141e7	−0.692227	−0.346113	−	0.938193i	$-0.612499\pi$
$757$	−1.09141e7	−0.692227	−0.346113	+	0.938193i	$0.612499\pi$
$758$	−4.12390e6	−0.260696
$759$	208983.	0.0131676
$760$	0	0
$761$	1.04145e7	0.651894	0.325947	−	0.945388i	$-0.394317\pi$
$761$	1.04145e7	0.651894	0.325947	+	0.945388i	$0.394317\pi$
$762$	687136.	0.0428702
$763$	−8.64548e6	−0.537623
$764$	−1.97953e6	−0.122696
$765$	0	0
$766$	5.67763e6	0.349619
$767$	−8.39232e6	−0.515103
$768$	4.72472e6	0.289050
$769$	−9.42825e6	−0.574930	−0.287465	−	0.957791i	$-0.592813\pi$
$769$	−9.42825e6	−0.574930	−0.287465	+	0.957791i	$0.592813\pi$
$770$	0	0
$771$	9.78265e6	0.592681
$772$	−2.00063e7	−1.20816
$773$	−1.18529e7	−0.713469	−0.356734	−	0.934206i	$-0.616110\pi$
$773$	−1.18529e7	−0.713469	−0.356734	+	0.934206i	$0.616110\pi$
$774$	150314.	0.00901875
$775$	0	0
$776$	−7.95620e6	−0.474298
$777$	1.13730e7	0.675805
$778$	−3.23760e6	−0.191767
$779$	−9.77398e6	−0.577069
$780$	0	0
$781$	5.76425e6	0.338154
$782$	−17230.0	−0.00100756
$783$	−3.10042e6	−0.180724
$784$	−7.87650e6	−0.457660
$785$	0	0
$786$	−1.47439e6	−0.0851246
$787$	1.84213e7	1.06019	0.530094	−	0.847939i	$-0.322157\pi$
$787$	1.84213e7	1.06019	0.530094	+	0.847939i	$0.322157\pi$
$788$	1.78616e7	1.02472
$789$	7.79293e6	0.445664
$790$	0	0
$791$	1.05456e6	0.0599279
$792$	−803955.	−0.0455427
$793$	2.14225e7	1.20973
$794$	7.81492e6	0.439919
$795$	0	0
$796$	1.03731e7	0.580266
$797$	8.83478e6	0.492663	0.246331	−	0.969186i	$-0.420775\pi$
$797$	8.83478e6	0.492663	0.246331	+	0.969186i	$0.420775\pi$
$798$	1.64788e6	0.0916048
$799$	1.00848e6	0.0558858
$800$	0	0
$801$	9.14242e6	0.503478
$802$	6.01738e6	0.330348
$803$	−890319.	−0.0487255
$804$	341570.	0.0186354
$805$	0	0
$806$	−57465.8	−0.00311582
$807$	4.10262e6	0.221757
$808$	4.57312e6	0.246425
$809$	1.51577e7	0.814259	0.407129	−	0.913370i	$-0.366530\pi$
$809$	1.51577e7	0.814259	0.407129	+	0.913370i	$0.366530\pi$
$810$	0	0
$811$	−1.02018e7	−0.544658	−0.272329	−	0.962204i	$-0.587794\pi$
$811$	−1.02018e7	−0.544658	−0.272329	+	0.962204i	$0.587794\pi$
$812$	−1.12597e7	−0.599291
$813$	1.74303e7	0.924863
$814$	−2.30270e6	−0.121808
$815$	0	0
$816$	−527742.	−0.0277458
$817$	−2.23796e6	−0.117300
$818$	−7.74551e6	−0.404731
$819$	−3.69376e6	−0.192424
$820$	0	0
$821$	2.60152e7	1.34700	0.673501	−	0.739186i	$-0.264790\pi$
$821$	2.60152e7	1.34700	0.673501	+	0.739186i	$0.264790\pi$
$822$	4.84563e6	0.250133
$823$	−1.36038e7	−0.700103	−0.350052	−	0.936730i	$-0.613836\pi$
$823$	−1.36038e7	−0.700103	−0.350052	+	0.936730i	$0.613836\pi$
$824$	−8.81547e6	−0.452301
$825$	0	0
$826$	1.85534e6	0.0946181
$827$	6.94229e6	0.352971	0.176486	−	0.984303i	$-0.443527\pi$
$827$	6.94229e6	0.352971	0.176486	+	0.984303i	$0.443527\pi$
$828$	−470437.	−0.0238465
$829$	−6.04757e6	−0.305629	−0.152814	−	0.988255i	$-0.548834\pi$
$829$	−6.04757e6	−0.305629	−0.152814	+	0.988255i	$0.548834\pi$
$830$	0	0
$831$	−1.51102e7	−0.759044
$832$	−1.17715e7	−0.589553
$833$	623900.	0.0311532
$834$	4.48732e6	0.223394
$835$	0	0
$836$	5.81805e6	0.287913
$837$	−61000.6	−0.00300968
$838$	8.06867e6	0.396910
$839$	4.32449e6	0.212095	0.106047	−	0.994361i	$-0.466180\pi$
$839$	4.32449e6	0.212095	0.106047	+	0.994361i	$0.466180\pi$
$840$	0	0
$841$	−2.42331e6	−0.118146
$842$	−3.72977e6	−0.181302
$843$	−1.49790e7	−0.725961
$844$	6.49641e6	0.313919
$845$	0	0
$846$	−1.57904e6	−0.0758523
$847$	−1.28077e6	−0.0613429
$848$	9.41139e6	0.449432
$849$	4.76146e6	0.226710
$850$	0	0
$851$	−2.77213e6	−0.131217
$852$	−1.29757e7	−0.612397
$853$	−4.01062e7	−1.88729	−0.943645	−	0.330959i	$-0.892628\pi$
$853$	−4.01062e7	−1.88729	−0.943645	+	0.330959i	$0.892628\pi$
$854$	−4.73601e6	−0.222212
$855$	0	0
$856$	−2.10591e6	−0.0982327
$857$	1.21434e7	0.564792	0.282396	−	0.959298i	$-0.408871\pi$
$857$	1.21434e7	0.564792	0.282396	+	0.959298i	$0.408871\pi$
$858$	747878.	0.0346827
$859$	−1.69726e7	−0.784810	−0.392405	−	0.919792i	$-0.628357\pi$
$859$	−1.69726e7	−0.784810	−0.392405	+	0.919792i	$0.628357\pi$
$860$	0	0
$861$	−4.84347e6	−0.222663
$862$	8.17504e6	0.374733
$863$	3.51856e7	1.60819	0.804095	−	0.594501i	$-0.202650\pi$
$863$	3.51856e7	1.60819	0.804095	+	0.594501i	$0.202650\pi$
$864$	2.73987e6	0.124866
$865$	0	0
$866$	556917.	0.0252346
$867$	−1.27369e7	−0.575462
$868$	−221534.	−0.00998026
$869$	1.09123e7	0.490194
$870$	0	0
$871$	−653712.	−0.0291972
$872$	−8.10678e6	−0.361042
$873$	7.85650e6	0.348894
$874$	−401666.	−0.0177863
$875$	0	0
$876$	2.00417e6	0.0882419
$877$	3.15265e7	1.38413	0.692066	−	0.721834i	$-0.256701\pi$
$877$	3.15265e7	1.38413	0.692066	+	0.721834i	$0.256701\pi$
$878$	−5.36203e6	−0.234743
$879$	−1.05827e7	−0.461983
$880$	0	0
$881$	−2.96796e7	−1.28831	−0.644153	−	0.764897i	$-0.722790\pi$
$881$	−2.96796e7	−1.28831	−0.644153	+	0.764897i	$0.722790\pi$
$882$	−976879.	−0.0422833
$883$	1.56983e7	0.677564	0.338782	−	0.940865i	$-0.389985\pi$
$883$	1.56983e7	0.677564	0.338782	+	0.940865i	$0.389985\pi$
$884$	1.07521e6	0.0462768
$885$	0	0
$886$	6.52441e6	0.279227
$887$	532251.	0.0227147	0.0113574	−	0.999936i	$-0.496385\pi$
$887$	532251.	0.0227147	0.0113574	+	0.999936i	$0.496385\pi$
$888$	1.06643e7	0.453838
$889$	−5.06969e6	−0.215143
$890$	0	0
$891$	793881.	0.0335013
$892$	3.59381e6	0.151232
$893$	2.35097e7	0.986550
$894$	−3.21692e6	−0.134616
$895$	0	0
$896$	1.31233e7	0.546102
$897$	900342.	0.0373617
$898$	9.67029e6	0.400174
$899$	355877.	0.0146859
$900$	0	0
$901$	−745479.	−0.0305931
$902$	980661.	0.0401331
$903$	−1.10901e6	−0.0452603
$904$	988848.	0.0402447
$905$	0	0
$906$	−1.73995e6	−0.0704233
$907$	−7.46788e6	−0.301425	−0.150713	−	0.988578i	$-0.548157\pi$
$907$	−7.46788e6	−0.301425	−0.150713	+	0.988578i	$0.548157\pi$
$908$	−66072.1	−0.00265952
$909$	−4.51581e6	−0.181270
$910$	0	0
$911$	4.14937e6	0.165648	0.0828240	−	0.996564i	$-0.473606\pi$
$911$	4.14937e6	0.165648	0.0828240	+	0.996564i	$0.473606\pi$
$912$	−1.23027e7	−0.489795
$913$	1.31369e6	0.0521575
$914$	2.73567e6	0.108318
$915$	0	0
$916$	−3.74383e6	−0.147427
$917$	1.08780e7	0.427195
$918$	−65453.0	−0.00256344
$919$	−2.25977e7	−0.882625	−0.441313	−	0.897353i	$-0.645487\pi$
$919$	−2.25977e7	−0.882625	−0.441313	+	0.897353i	$0.645487\pi$
$920$	0	0
$921$	1.37780e7	0.535225
$922$	−2.22533e6	−0.0862118
$923$	2.48335e7	0.959476
$924$	2.88312e6	0.111092
$925$	0	0
$926$	−1.79106e6	−0.0686410
$927$	8.70500e6	0.332713
$928$	−1.59844e7	−0.609291
$929$	4.69291e7	1.78403	0.892017	−	0.452002i	$-0.149290\pi$
$929$	4.69291e7	1.78403	0.892017	+	0.452002i	$0.149290\pi$
$930$	0	0
$931$	1.45443e7	0.549945
$932$	−3.52844e7	−1.33059
$933$	−1.00458e7	−0.377818
$934$	−1.11661e7	−0.418827
$935$	0	0
$936$	−3.46360e6	−0.129222
$937$	−5.63886e6	−0.209818	−0.104909	−	0.994482i	$-0.533455\pi$
$937$	−5.63886e6	−0.209818	−0.104909	+	0.994482i	$0.533455\pi$
$938$	144520.	0.00536317
$939$	7.78128e6	0.287996
$940$	0	0
$941$	−2.03249e7	−0.748261	−0.374131	−	0.927376i	$-0.622059\pi$
$941$	−2.03249e7	−0.748261	−0.374131	+	0.927376i	$0.622059\pi$
$942$	4.33436e6	0.159147
$943$	1.18058e6	0.0432331
$944$	−1.38516e7	−0.505906
$945$	0	0
$946$	224543.	0.00815777
$947$	1.36521e7	0.494681	0.247340	−	0.968929i	$-0.420443\pi$
$947$	1.36521e7	0.494681	0.247340	+	0.968929i	$0.420443\pi$
$948$	−2.45644e7	−0.887741
$949$	−3.83567e6	−0.138253
$950$	0	0
$951$	−1.62543e7	−0.582797
$952$	−489039.	−0.0174885
$953$	−5.93702e6	−0.211756	−0.105878	−	0.994379i	$-0.533765\pi$
$953$	−5.93702e6	−0.211756	−0.105878	+	0.994379i	$0.533765\pi$
$954$	1.16724e6	0.0415232
$955$	0	0
$956$	−2.33223e7	−0.825329
$957$	−4.63149e6	−0.163471
$958$	−4.64909e6	−0.163664
$959$	−3.57510e7	−1.25528
$960$	0	0
$961$	−2.86221e7	−0.999755
$962$	−9.92047e6	−0.345617
$963$	2.07952e6	0.0722601
$964$	4.31391e7	1.49513
$965$	0	0
$966$	−199044.	−0.00686289
$967$	4.40436e7	1.51467	0.757333	−	0.653029i	$-0.226502\pi$
$967$	4.40436e7	1.51467	0.757333	+	0.653029i	$0.226502\pi$
$968$	−1.20097e6	−0.0411949
$969$	974502.	0.0333406
$970$	0	0
$971$	−1.01824e7	−0.346577	−0.173289	−	0.984871i	$-0.555439\pi$
$971$	−1.01824e7	−0.346577	−0.173289	+	0.984871i	$0.555439\pi$
$972$	−1.78708e6	−0.0606707
$973$	−3.31074e7	−1.12110
$974$	6.14854e6	0.207670
$975$	0	0
$976$	3.53580e7	1.18813
$977$	4.43840e7	1.48761	0.743806	−	0.668395i	$-0.233018\pi$
$977$	4.43840e7	1.48761	0.743806	+	0.668395i	$0.233018\pi$
$978$	−197951.	−0.00661774
$979$	1.36572e7	0.455413
$980$	0	0
$981$	8.00520e6	0.265583
$982$	−9.51850e6	−0.314985
$983$	2.94574e7	0.972324	0.486162	−	0.873869i	$-0.338396\pi$
$983$	2.94574e7	0.972324	0.486162	+	0.873869i	$0.338396\pi$
$984$	−4.54167e6	−0.149530
$985$	0	0
$986$	381852.	0.0125084
$987$	1.16502e7	0.380662
$988$	2.50653e7	0.816923
$989$	270319.	0.00878791
$990$	0	0
$991$	−1.97622e7	−0.639221	−0.319611	−	0.947549i	$-0.603552\pi$
$991$	−1.97622e7	−0.639221	−0.319611	+	0.947549i	$0.603552\pi$
$992$	−314491.	−0.0101468
$993$	1.67131e7	0.537879
$994$	−5.49010e6	−0.176244
$995$	0	0
$996$	−2.95722e6	−0.0944572
$997$	−2.39291e7	−0.762408	−0.381204	−	0.924491i	$-0.624491\pi$
$997$	−2.39291e7	−0.762408	−0.381204	+	0.924491i	$0.624491\pi$
$998$	−1.38085e7	−0.438854
$999$	−1.05307e7	−0.333844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.y.1.7		13
5.2	odd	4		165.6.c.b.34.15	yes	26
5.3	odd	4		165.6.c.b.34.12	✓	26
5.4	even	2		825.6.a.v.1.7		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.12	✓	26	5.3	odd	4
165.6.c.b.34.15	yes	26	5.2	odd	4
825.6.a.v.1.7		13	5.4	even	2
825.6.a.y.1.7		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+1.31741 q^{2} +9.00000 q^{3} -30.2644 q^{4} +11.8567 q^{6} -87.4786 q^{7} -82.0279 q^{8} +81.0000 q^{9} +121.000 q^{11} -272.380 q^{12} +521.292 q^{13} -115.245 q^{14} +860.397 q^{16} -68.1523 q^{17} +106.710 q^{18} -1588.76 q^{19} -787.308 q^{21} +159.407 q^{22} +191.904 q^{23} -738.251 q^{24} +686.756 q^{26} +729.000 q^{27} +2647.49 q^{28} -4252.98 q^{29} -83.6771 q^{31} +3758.39 q^{32} +1089.00 q^{33} -89.7846 q^{34} -2451.42 q^{36} -14445.4 q^{37} -2093.06 q^{38} +4691.63 q^{39} +6151.94 q^{41} -1037.21 q^{42} +1408.62 q^{43} -3662.00 q^{44} +252.816 q^{46} -14797.5 q^{47} +7743.57 q^{48} -9154.49 q^{49} -613.371 q^{51} -15776.6 q^{52} +10938.4 q^{53} +960.393 q^{54} +7175.69 q^{56} -14298.9 q^{57} -5602.92 q^{58} -16099.1 q^{59} +41095.0 q^{61} -110.237 q^{62} -7085.77 q^{63} -22581.4 q^{64} +1434.66 q^{66} -1254.02 q^{67} +2062.59 q^{68} +1727.14 q^{69} +47638.4 q^{71} -6644.26 q^{72} -7358.01 q^{73} -19030.5 q^{74} +48083.1 q^{76} -10584.9 q^{77} +6180.81 q^{78} +90184.5 q^{79} +6561.00 q^{81} +8104.64 q^{82} +10857.0 q^{83} +23827.4 q^{84} +1855.73 q^{86} -38276.8 q^{87} -9925.37 q^{88} +112869. q^{89} -45601.9 q^{91} -5807.86 q^{92} -753.094 q^{93} -19494.4 q^{94} +33825.5 q^{96} +96993.9 q^{97} -12060.2 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21}+ \cdots + 127413 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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