LMFDB - Embedded newform 495.6.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,6,Mod(1,495)] 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("495.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-2,0,-42] 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:	$79.3899908074$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.18257.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x + 8 $ x^3 - x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.47894$ 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)

$f(q)$	$=$	\(q+4.47894 q^{2} -11.9391 q^{4} -25.0000 q^{5} -168.818 q^{7} -196.801 q^{8} -111.974 q^{10} -121.000 q^{11} +290.259 q^{13} -756.126 q^{14} -499.408 q^{16} -623.964 q^{17} -398.456 q^{19} +298.477 q^{20} -541.952 q^{22} -3788.06 q^{23} +625.000 q^{25} +1300.06 q^{26} +2015.53 q^{28} -4220.43 q^{29} -5594.57 q^{31} +4060.80 q^{32} -2794.70 q^{34} +4220.45 q^{35} +301.266 q^{37} -1784.66 q^{38} +4920.01 q^{40} +14636.2 q^{41} +151.418 q^{43} +1444.63 q^{44} -16966.5 q^{46} +13535.0 q^{47} +11692.5 q^{49} +2799.34 q^{50} -3465.43 q^{52} +18116.1 q^{53} +3025.00 q^{55} +33223.5 q^{56} -18903.0 q^{58} +50582.4 q^{59} +5984.35 q^{61} -25057.8 q^{62} +34169.1 q^{64} -7256.49 q^{65} +22450.7 q^{67} +7449.56 q^{68} +18903.2 q^{70} -10017.6 q^{71} -476.148 q^{73} +1349.35 q^{74} +4757.19 q^{76} +20427.0 q^{77} -85024.7 q^{79} +12485.2 q^{80} +65554.5 q^{82} +25643.4 q^{83} +15599.1 q^{85} +678.194 q^{86} +23812.9 q^{88} -1807.91 q^{89} -49001.0 q^{91} +45226.0 q^{92} +60622.4 q^{94} +9961.39 q^{95} +11688.5 q^{97} +52370.1 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8} + 50 q^{10} - 363 q^{11} + 290 q^{13} - 916 q^{14} - 590 q^{16} - 434 q^{17} - 2856 q^{19} + 1050 q^{20} + 242 q^{22} + 640 q^{23} + 1875 q^{25}+ \cdots + 154654 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8 + 50 * q^10 - 363 * q^11 + 290 * q^13 - 916 * q^14 - 590 * q^16 - 434 * q^17 - 2856 * q^19 + 1050 * q^20 + 242 * q^22 + 640 * q^23 + 1875 * q^25 - 2132 * q^26 - 580 * q^28 + 4538 * q^29 - 14968 * q^31 + 2496 * q^32 - 13704 * q^34 + 1700 * q^35 - 6190 * q^37 + 11668 * q^38 - 600 * q^40 + 8926 * q^41 - 33592 * q^43 + 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 14693 * q^49 - 1250 * q^50 + 18780 * q^52 + 22934 * q^53 + 9075 * q^55 + 40012 * q^56 - 32304 * q^58 + 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 35474 * q^64 - 7250 * q^65 + 16868 * q^67 + 71288 * q^68 + 22900 * q^70 - 4856 * q^71 + 1910 * q^73 - 29404 * q^74 + 6116 * q^76 + 8228 * q^77 - 36844 * q^79 + 14750 * q^80 + 84000 * q^82 + 48796 * q^83 + 10850 * q^85 + 83492 * q^86 - 2904 * q^88 + 188978 * q^89 - 93208 * q^91 + 6976 * q^92 + 70472 * q^94 + 71400 * q^95 + 247526 * q^97 + 154654 * q^98

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 $

$2$	4.47894	0.791773	0.395886	−	0.918300i	$-0.370437\pi$
$2$	4.47894	0.791773	0.395886	+	0.918300i	$0.370437\pi$
$3$	0	0
$3$	0	0
$4$	−11.9391	−0.373096
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−168.818	−1.30219	−0.651094	−	0.758997i	$-0.725690\pi$
$7$	−168.818	−1.30219	−0.651094	+	0.758997i	$0.725690\pi$
$8$	−196.801	−1.08718
$9$	0	0
$10$	−111.974	−0.354091
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	290.259	0.476352	0.238176	−	0.971222i	$-0.423451\pi$
$13$	290.259	0.476352	0.238176	+	0.971222i	$0.423451\pi$
$14$	−756.126	−1.03104
$15$	0	0
$16$	−499.408	−0.487703
$17$	−623.964	−0.523645	−0.261823	−	0.965116i	$-0.584324\pi$
$17$	−623.964	−0.523645	−0.261823	+	0.965116i	$0.584324\pi$
$18$	0	0
$19$	−398.456	−0.253219	−0.126609	−	0.991953i	$-0.540409\pi$
$19$	−398.456	−0.253219	−0.126609	+	0.991953i	$0.540409\pi$
$20$	298.477	0.166854
$21$	0	0
$22$	−541.952	−0.238728
$23$	−3788.06	−1.49313	−0.746565	−	0.665312i	$-0.768298\pi$
$23$	−3788.06	−1.49313	−0.746565	+	0.665312i	$0.768298\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	1300.06	0.377162
$27$	0	0
$28$	2015.53	0.485841
$29$	−4220.43	−0.931883	−0.465941	−	0.884816i	$-0.654284\pi$
$29$	−4220.43	−0.931883	−0.465941	+	0.884816i	$0.654284\pi$
$30$	0	0
$31$	−5594.57	−1.04559	−0.522797	−	0.852457i	$-0.675111\pi$
$31$	−5594.57	−1.04559	−0.522797	+	0.852457i	$0.675111\pi$
$32$	4060.80	0.701030
$33$	0	0
$34$	−2794.70	−0.414608
$35$	4220.45	0.582356
$36$	0	0
$37$	301.266	0.0361781	0.0180890	−	0.999836i	$-0.494242\pi$
$37$	301.266	0.0361781	0.0180890	+	0.999836i	$0.494242\pi$
$38$	−1784.66	−0.200492
$39$	0	0
$40$	4920.01	0.486202
$41$	14636.2	1.35978	0.679889	−	0.733315i	$-0.262028\pi$
$41$	14636.2	1.35978	0.679889	+	0.733315i	$0.262028\pi$
$42$	0	0
$43$	151.418	0.0124884	0.00624421	−	0.999981i	$-0.498012\pi$
$43$	151.418	0.0124884	0.00624421	+	0.999981i	$0.498012\pi$
$44$	1444.63	0.112493
$45$	0	0
$46$	−16966.5	−1.18222
$47$	13535.0	0.893743	0.446872	−	0.894598i	$-0.352538\pi$
$47$	13535.0	0.893743	0.446872	+	0.894598i	$0.352538\pi$
$48$	0	0
$49$	11692.5	0.695694
$50$	2799.34	0.158355
$51$	0	0
$52$	−3465.43	−0.177725
$53$	18116.1	0.885880	0.442940	−	0.896551i	$-0.353935\pi$
$53$	18116.1	0.885880	0.442940	+	0.896551i	$0.353935\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	33223.5	1.41571
$57$	0	0
$58$	−18903.0	−0.737839
$59$	50582.4	1.89177	0.945887	−	0.324495i	$-0.105194\pi$
$59$	50582.4	1.89177	0.945887	+	0.324495i	$0.105194\pi$
$60$	0	0
$61$	5984.35	0.205917	0.102958	−	0.994686i	$-0.467169\pi$
$61$	5984.35	0.205917	0.102958	+	0.994686i	$0.467169\pi$
$62$	−25057.8	−0.827872
$63$	0	0
$64$	34169.1	1.04276
$65$	−7256.49	−0.213031
$66$	0	0
$67$	22450.7	0.611002	0.305501	−	0.952192i	$-0.401176\pi$
$67$	22450.7	0.611002	0.305501	+	0.952192i	$0.401176\pi$
$68$	7449.56	0.195370
$69$	0	0
$70$	18903.2	0.461094
$71$	−10017.6	−0.235841	−0.117921	−	0.993023i	$-0.537623\pi$
$71$	−10017.6	−0.235841	−0.117921	+	0.993023i	$0.537623\pi$
$72$	0	0
$73$	−476.148	−0.0104577	−0.00522883	−	0.999986i	$-0.501664\pi$
$73$	−476.148	−0.0104577	−0.00522883	+	0.999986i	$0.501664\pi$
$74$	1349.35	0.0286448
$75$	0	0
$76$	4757.19	0.0944750
$77$	20427.0	0.392624
$78$	0	0
$79$	−85024.7	−1.53277	−0.766386	−	0.642380i	$-0.777947\pi$
$79$	−85024.7	−1.53277	−0.766386	+	0.642380i	$0.777947\pi$
$80$	12485.2	0.218107
$81$	0	0
$82$	65554.5	1.07663
$83$	25643.4	0.408583	0.204292	−	0.978910i	$-0.434511\pi$
$83$	25643.4	0.408583	0.204292	+	0.978910i	$0.434511\pi$
$84$	0	0
$85$	15599.1	0.234181
$86$	678.194	0.00988799
$87$	0	0
$88$	23812.9	0.327797
$89$	−1807.91	−0.0241937	−0.0120968	−	0.999927i	$-0.503851\pi$
$89$	−1807.91	−0.0241937	−0.0120968	+	0.999927i	$0.503851\pi$
$90$	0	0
$91$	−49001.0	−0.620300
$92$	45226.0	0.557081
$93$	0	0
$94$	60622.4	0.707642
$95$	9961.39	0.113243
$96$	0	0
$97$	11688.5	0.126133	0.0630666	−	0.998009i	$-0.479912\pi$
$97$	11688.5	0.126133	0.0630666	+	0.998009i	$0.479912\pi$
$98$	52370.1	0.550831
$99$	0	0
$100$	−7461.92	−0.0746192
$101$	135573.	1.32242	0.661210	−	0.750200i	$-0.270043\pi$
$101$	135573.	1.32242	0.661210	+	0.750200i	$0.270043\pi$
$102$	0	0
$103$	−88490.5	−0.821871	−0.410935	−	0.911664i	$-0.634798\pi$
$103$	−88490.5	−0.821871	−0.410935	+	0.911664i	$0.634798\pi$
$104$	−57123.2	−0.517880
$105$	0	0
$106$	81140.9	0.701416
$107$	−218742.	−1.84703	−0.923514	−	0.383564i	$-0.874696\pi$
$107$	−218742.	−1.84703	−0.923514	+	0.383564i	$0.874696\pi$
$108$	0	0
$109$	238399.	1.92193	0.960967	−	0.276663i	$-0.0892287\pi$
$109$	238399.	1.92193	0.960967	+	0.276663i	$0.0892287\pi$
$110$	13548.8	0.106763
$111$	0	0
$112$	84309.1	0.635081
$113$	−187225.	−1.37933	−0.689664	−	0.724130i	$-0.742242\pi$
$113$	−187225.	−1.37933	−0.689664	+	0.724130i	$0.742242\pi$
$114$	0	0
$115$	94701.6	0.667748
$116$	50388.0	0.347682
$117$	0	0
$118$	226556.	1.49786
$119$	105336.	0.681885
$120$	0	0
$121$	14641.0	0.0909091
$122$	26803.5	0.163039
$123$	0	0
$124$	66794.1	0.390107
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	27727.6	0.152547	0.0762733	−	0.997087i	$-0.475698\pi$
$127$	27727.6	0.152547	0.0762733	+	0.997087i	$0.475698\pi$
$128$	23096.0	0.124598
$129$	0	0
$130$	−32501.4	−0.168672
$131$	324623.	1.65273	0.826363	−	0.563138i	$-0.190406\pi$
$131$	324623.	1.65273	0.826363	+	0.563138i	$0.190406\pi$
$132$	0	0
$133$	67266.5	0.329739
$134$	100555.	0.483775
$135$	0	0
$136$	122797.	0.569297
$137$	59748.8	0.271974	0.135987	−	0.990711i	$-0.456579\pi$
$137$	59748.8	0.271974	0.135987	+	0.990711i	$0.456579\pi$
$138$	0	0
$139$	−117646.	−0.516462	−0.258231	−	0.966083i	$-0.583140\pi$
$139$	−117646.	−0.516462	−0.258231	+	0.966083i	$0.583140\pi$
$140$	−50388.3	−0.217275
$141$	0	0
$142$	−44868.5	−0.186733
$143$	−35121.4	−0.143626
$144$	0	0
$145$	105511.	0.416751
$146$	−2132.64	−0.00828009
$147$	0	0
$148$	−3596.84	−0.0134979
$149$	−52786.3	−0.194785	−0.0973925	−	0.995246i	$-0.531050\pi$
$149$	−52786.3	−0.194785	−0.0973925	+	0.995246i	$0.531050\pi$
$150$	0	0
$151$	−430119.	−1.53513	−0.767567	−	0.640969i	$-0.778533\pi$
$151$	−430119.	−1.53513	−0.767567	+	0.640969i	$0.778533\pi$
$152$	78416.3	0.275294
$153$	0	0
$154$	91491.3	0.310869
$155$	139864.	0.467604
$156$	0	0
$157$	−150989.	−0.488873	−0.244436	−	0.969665i	$-0.578603\pi$
$157$	−150989.	−0.488873	−0.244436	+	0.969665i	$0.578603\pi$
$158$	−380821.	−1.21361
$159$	0	0
$160$	−101520.	−0.313510
$161$	639493.	1.94434
$162$	0	0
$163$	142404.	0.419812	0.209906	−	0.977722i	$-0.432684\pi$
$163$	142404.	0.419812	0.209906	+	0.977722i	$0.432684\pi$
$164$	−174742.	−0.507328
$165$	0	0
$166$	114855.	0.323505
$167$	−55468.4	−0.153906	−0.0769528	−	0.997035i	$-0.524519\pi$
$167$	−55468.4	−0.153906	−0.0769528	+	0.997035i	$0.524519\pi$
$168$	0	0
$169$	−287042.	−0.773089
$170$	69867.5	0.185418
$171$	0	0
$172$	−1807.80	−0.00465938
$173$	−133612.	−0.339414	−0.169707	−	0.985495i	$-0.554282\pi$
$173$	−133612.	−0.339414	−0.169707	+	0.985495i	$0.554282\pi$
$174$	0	0
$175$	−105511.	−0.260438
$176$	60428.4	0.147048
$177$	0	0
$178$	−8097.52	−0.0191559
$179$	188254.	0.439150	0.219575	−	0.975596i	$-0.429533\pi$
$179$	188254.	0.439150	0.219575	+	0.975596i	$0.429533\pi$
$180$	0	0
$181$	−652252.	−1.47986	−0.739928	−	0.672686i	$-0.765140\pi$
$181$	−652252.	−1.47986	−0.739928	+	0.672686i	$0.765140\pi$
$182$	−219473.	−0.491136
$183$	0	0
$184$	745493.	1.62330
$185$	−7531.65	−0.0161793
$186$	0	0
$187$	75499.7	0.157885
$188$	−161595.	−0.333452
$189$	0	0
$190$	44616.5	0.0896626
$191$	705620.	1.39955	0.699774	−	0.714365i	$-0.253284\pi$
$191$	705620.	1.39955	0.699774	+	0.714365i	$0.253284\pi$
$192$	0	0
$193$	−87143.5	−0.168400	−0.0841999	−	0.996449i	$-0.526833\pi$
$193$	−87143.5	−0.168400	−0.0841999	+	0.996449i	$0.526833\pi$
$194$	52352.1	0.0998688
$195$	0	0
$196$	−139598.	−0.259561
$197$	667596.	1.22560	0.612800	−	0.790238i	$-0.290043\pi$
$197$	667596.	1.22560	0.612800	+	0.790238i	$0.290043\pi$
$198$	0	0
$199$	−698609.	−1.25055	−0.625276	−	0.780404i	$-0.715013\pi$
$199$	−698609.	−1.25055	−0.625276	+	0.780404i	$0.715013\pi$
$200$	−123000.	−0.217436
$201$	0	0
$202$	607224.	1.04706
$203$	712484.	1.21349
$204$	0	0
$205$	−365904.	−0.608111
$206$	−396344.	−0.650735
$207$	0	0
$208$	−144958.	−0.232318
$209$	48213.1	0.0763484
$210$	0	0
$211$	−558238.	−0.863203	−0.431602	−	0.902064i	$-0.642051\pi$
$211$	−558238.	−0.863203	−0.431602	+	0.902064i	$0.642051\pi$
$212$	−216289.	−0.330518
$213$	0	0
$214$	−979735.	−1.46243
$215$	−3785.46	−0.00558499
$216$	0	0
$217$	944465.	1.36156
$218$	1.06778e6	1.52173
$219$	0	0
$220$	−36115.7	−0.0503083
$221$	−181111.	−0.249440
$222$	0	0
$223$	−382897.	−0.515608	−0.257804	−	0.966197i	$-0.582999\pi$
$223$	−382897.	−0.515608	−0.257804	+	0.966197i	$0.582999\pi$
$224$	−685536.	−0.912873
$225$	0	0
$226$	−838569.	−1.09211
$227$	404735.	0.521322	0.260661	−	0.965430i	$-0.416059\pi$
$227$	404735.	0.521322	0.260661	+	0.965430i	$0.416059\pi$
$228$	0	0
$229$	664894.	0.837845	0.418922	−	0.908022i	$-0.362408\pi$
$229$	664894.	0.837845	0.418922	+	0.908022i	$0.362408\pi$
$230$	424163.	0.528705
$231$	0	0
$232$	830582.	1.01312
$233$	−61531.4	−0.0742518	−0.0371259	−	0.999311i	$-0.511820\pi$
$233$	−61531.4	−0.0742518	−0.0371259	+	0.999311i	$0.511820\pi$
$234$	0	0
$235$	−338374.	−0.399694
$236$	−603907.	−0.705814
$237$	0	0
$238$	471796.	0.539898
$239$	−1.71207e6	−1.93877	−0.969384	−	0.245549i	$-0.921032\pi$
$239$	−1.71207e6	−1.93877	−0.969384	+	0.245549i	$0.921032\pi$
$240$	0	0
$241$	1.31915e6	1.46302	0.731512	−	0.681828i	$-0.238815\pi$
$241$	1.31915e6	1.46302	0.731512	+	0.681828i	$0.238815\pi$
$242$	65576.2	0.0719793
$243$	0	0
$244$	−71447.6	−0.0768268
$245$	−292313.	−0.311124
$246$	0	0
$247$	−115656.	−0.120621
$248$	1.10102e6	1.13675
$249$	0	0
$250$	−69983.5	−0.0708183
$251$	237992.	0.238440	0.119220	−	0.992868i	$-0.461961\pi$
$251$	237992.	0.238440	0.119220	+	0.992868i	$0.461961\pi$
$252$	0	0
$253$	458356.	0.450196
$254$	124190.	0.120782
$255$	0	0
$256$	−989967.	−0.944106
$257$	1.90368e6	1.79789	0.898943	−	0.438066i	$-0.144336\pi$
$257$	1.90368e6	1.79789	0.898943	+	0.438066i	$0.144336\pi$
$258$	0	0
$259$	−50859.1	−0.0471107
$260$	86635.7	0.0794811
$261$	0	0
$262$	1.45397e6	1.30858
$263$	907019.	0.808588	0.404294	−	0.914629i	$-0.367517\pi$
$263$	907019.	0.808588	0.404294	+	0.914629i	$0.367517\pi$
$264$	0	0
$265$	−452902.	−0.396178
$266$	301283.	0.261078
$267$	0	0
$268$	−268041.	−0.227963
$269$	−698485.	−0.588541	−0.294270	−	0.955722i	$-0.595077\pi$
$269$	−698485.	−0.588541	−0.294270	+	0.955722i	$0.595077\pi$
$270$	0	0
$271$	1.19097e6	0.985098	0.492549	−	0.870285i	$-0.336065\pi$
$271$	1.19097e6	0.985098	0.492549	+	0.870285i	$0.336065\pi$
$272$	311613.	0.255383
$273$	0	0
$274$	267611.	0.215342
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	−1.15110e6	−0.901392	−0.450696	−	0.892677i	$-0.648824\pi$
$277$	−1.15110e6	−0.901392	−0.450696	+	0.892677i	$0.648824\pi$
$278$	−526928.	−0.408921
$279$	0	0
$280$	−830587.	−0.633126
$281$	1.54201e6	1.16498	0.582492	−	0.812836i	$-0.302078\pi$
$281$	1.54201e6	1.16498	0.582492	+	0.812836i	$0.302078\pi$
$282$	0	0
$283$	1.28590e6	0.954421	0.477211	−	0.878789i	$-0.341648\pi$
$283$	1.28590e6	0.954421	0.477211	+	0.878789i	$0.341648\pi$
$284$	119601.	0.0879915
$285$	0	0
$286$	−157307.	−0.113719
$287$	−2.47085e6	−1.77069
$288$	0	0
$289$	−1.03053e6	−0.725795
$290$	472576.	0.329972
$291$	0	0
$292$	5684.77	0.00390171
$293$	−1.08802e6	−0.740404	−0.370202	−	0.928951i	$-0.620712\pi$
$293$	−1.08802e6	−0.740404	−0.370202	+	0.928951i	$0.620712\pi$
$294$	0	0
$295$	−1.26456e6	−0.846027
$296$	−59289.3	−0.0393321
$297$	0	0
$298$	−236427.	−0.154225
$299$	−1.09952e6	−0.711255
$300$	0	0
$301$	−25562.2	−0.0162623
$302$	−1.92648e6	−1.21548
$303$	0	0
$304$	198992.	0.123496
$305$	−149609.	−0.0920889
$306$	0	0
$307$	−580340.	−0.351428	−0.175714	−	0.984441i	$-0.556223\pi$
$307$	−580340.	−0.351428	−0.175714	+	0.984441i	$0.556223\pi$
$308$	−243879.	−0.146487
$309$	0	0
$310$	626444.	0.370236
$311$	1.68606e6	0.988492	0.494246	−	0.869322i	$-0.335444\pi$
$311$	1.68606e6	0.988492	0.494246	+	0.869322i	$0.335444\pi$
$312$	0	0
$313$	−1.50328e6	−0.867319	−0.433659	−	0.901077i	$-0.642778\pi$
$313$	−1.50328e6	−0.867319	−0.433659	+	0.901077i	$0.642778\pi$
$314$	−676270.	−0.387076
$315$	0	0
$316$	1.01512e6	0.571871
$317$	−1.31016e6	−0.732281	−0.366140	−	0.930560i	$-0.619321\pi$
$317$	−1.31016e6	−0.732281	−0.366140	+	0.930560i	$0.619321\pi$
$318$	0	0
$319$	510671.	0.280973
$320$	−854228.	−0.466336
$321$	0	0
$322$	2.86425e6	1.53947
$323$	248622.	0.132597
$324$	0	0
$325$	181412.	0.0952704
$326$	637821.	0.332395
$327$	0	0
$328$	−2.88041e6	−1.47832
$329$	−2.28495e6	−1.16382
$330$	0	0
$331$	2.91035e6	1.46007	0.730037	−	0.683408i	$-0.239503\pi$
$331$	2.91035e6	1.46007	0.730037	+	0.683408i	$0.239503\pi$
$332$	−306159.	−0.152441
$333$	0	0
$334$	−248440.	−0.121858
$335$	−561267.	−0.273248
$336$	0	0
$337$	2.47133e6	1.18538	0.592688	−	0.805432i	$-0.298067\pi$
$337$	2.47133e6	1.18538	0.592688	+	0.805432i	$0.298067\pi$
$338$	−1.28565e6	−0.612111
$339$	0	0
$340$	−186239.	−0.0873722
$341$	676944.	0.315258
$342$	0	0
$343$	863415.	0.396264
$344$	−29799.2	−0.0135772
$345$	0	0
$346$	−598440.	−0.268739
$347$	−1.75988e6	−0.784619	−0.392309	−	0.919833i	$-0.628324\pi$
$347$	−1.75988e6	−0.784619	−0.392309	+	0.919833i	$0.628324\pi$
$348$	0	0
$349$	−386142.	−0.169700	−0.0848502	−	0.996394i	$-0.527041\pi$
$349$	−386142.	−0.169700	−0.0848502	+	0.996394i	$0.527041\pi$
$350$	−472579.	−0.206207
$351$	0	0
$352$	−491357.	−0.211368
$353$	2.44184e6	1.04299	0.521495	−	0.853254i	$-0.325375\pi$
$353$	2.44184e6	1.04299	0.521495	+	0.853254i	$0.325375\pi$
$354$	0	0
$355$	250441.	0.105471
$356$	21584.8	0.00902657
$357$	0	0
$358$	843180.	0.347707
$359$	−3.77524e6	−1.54600	−0.772998	−	0.634408i	$-0.781244\pi$
$359$	−3.77524e6	−1.54600	−0.772998	+	0.634408i	$0.781244\pi$
$360$	0	0
$361$	−2.31733e6	−0.935880
$362$	−2.92140e6	−1.17171
$363$	0	0
$364$	585027.	0.231431
$365$	11903.7	0.00467681
$366$	0	0
$367$	−3.98351e6	−1.54383	−0.771917	−	0.635723i	$-0.780702\pi$
$367$	−3.98351e6	−1.54383	−0.771917	+	0.635723i	$0.780702\pi$
$368$	1.89179e6	0.728204
$369$	0	0
$370$	−33733.8	−0.0128104
$371$	−3.05832e6	−1.15358
$372$	0	0
$373$	3.13497e6	1.16670	0.583352	−	0.812219i	$-0.301741\pi$
$373$	3.13497e6	1.16670	0.583352	+	0.812219i	$0.301741\pi$
$374$	338159.	0.125009
$375$	0	0
$376$	−2.66369e6	−0.971660
$377$	−1.22502e6	−0.443904
$378$	0	0
$379$	−2.50995e6	−0.897567	−0.448784	−	0.893640i	$-0.648143\pi$
$379$	−2.50995e6	−0.897567	−0.448784	+	0.893640i	$0.648143\pi$
$380$	−118930.	−0.0422505
$381$	0	0
$382$	3.16043e6	1.10812
$383$	4.20287e6	1.46403	0.732014	−	0.681290i	$-0.238581\pi$
$383$	4.20287e6	1.46403	0.732014	+	0.681290i	$0.238581\pi$
$384$	0	0
$385$	−510675.	−0.175587
$386$	−390311.	−0.133334
$387$	0	0
$388$	−139550.	−0.0470598
$389$	1.91171e6	0.640543	0.320271	−	0.947326i	$-0.396226\pi$
$389$	1.91171e6	0.640543	0.320271	+	0.947326i	$0.396226\pi$
$390$	0	0
$391$	2.36362e6	0.781871
$392$	−2.30110e6	−0.756344
$393$	0	0
$394$	2.99013e6	0.970396
$395$	2.12562e6	0.685477
$396$	0	0
$397$	3.62426e6	1.15410	0.577049	−	0.816709i	$-0.304204\pi$
$397$	3.62426e6	1.15410	0.577049	+	0.816709i	$0.304204\pi$
$398$	−3.12903e6	−0.990152
$399$	0	0
$400$	−312130.	−0.0975406
$401$	2.22052e6	0.689593	0.344797	−	0.938677i	$-0.387948\pi$
$401$	2.22052e6	0.689593	0.344797	+	0.938677i	$0.387948\pi$
$402$	0	0
$403$	−1.62388e6	−0.498070
$404$	−1.61862e6	−0.493390
$405$	0	0
$406$	3.19117e6	0.960805
$407$	−36453.2	−0.0109081
$408$	0	0
$409$	20564.7	0.00607876	0.00303938	−	0.999995i	$-0.499033\pi$
$409$	20564.7	0.00607876	0.00303938	+	0.999995i	$0.499033\pi$
$410$	−1.63886e6	−0.481485
$411$	0	0
$412$	1.05649e6	0.306637
$413$	−8.53922e6	−2.46345
$414$	0	0
$415$	−641086.	−0.182724
$416$	1.17869e6	0.333937
$417$	0	0
$418$	215944.	0.0604505
$419$	2.30684e6	0.641923	0.320962	−	0.947092i	$-0.395994\pi$
$419$	2.30684e6	0.641923	0.320962	+	0.947092i	$0.395994\pi$
$420$	0	0
$421$	5.00382e6	1.37593	0.687965	−	0.725744i	$-0.258504\pi$
$421$	5.00382e6	1.37593	0.687965	+	0.725744i	$0.258504\pi$
$422$	−2.50032e6	−0.683461
$423$	0	0
$424$	−3.56526e6	−0.963111
$425$	−389978.	−0.104729
$426$	0	0
$427$	−1.01027e6	−0.268143
$428$	2.61158e6	0.689119
$429$	0	0
$430$	−16954.9	−0.00442204
$431$	2.70202e6	0.700641	0.350320	−	0.936630i	$-0.386073\pi$
$431$	2.70202e6	0.700641	0.350320	+	0.936630i	$0.386073\pi$
$432$	0	0
$433$	−3.92321e6	−1.00559	−0.502796	−	0.864405i	$-0.667695\pi$
$433$	−3.92321e6	−1.00559	−0.502796	+	0.864405i	$0.667695\pi$
$434$	4.23020e6	1.07805
$435$	0	0
$436$	−2.84627e6	−0.717066
$437$	1.50938e6	0.378089
$438$	0	0
$439$	−53261.3	−0.0131902	−0.00659508	−	0.999978i	$-0.502099\pi$
$439$	−53261.3	−0.0131902	−0.00659508	+	0.999978i	$0.502099\pi$
$440$	−595322.	−0.146595
$441$	0	0
$442$	−811188.	−0.197499
$443$	−3.30457e6	−0.800029	−0.400015	−	0.916509i	$-0.630995\pi$
$443$	−3.30457e6	−0.800029	−0.400015	+	0.916509i	$0.630995\pi$
$444$	0	0
$445$	45197.7	0.0108197
$446$	−1.71497e6	−0.408244
$447$	0	0
$448$	−5.76837e6	−1.35787
$449$	−3.04596e6	−0.713031	−0.356516	−	0.934289i	$-0.616035\pi$
$449$	−3.04596e6	−0.713031	−0.356516	+	0.934289i	$0.616035\pi$
$450$	0	0
$451$	−1.77098e6	−0.409988
$452$	2.23529e6	0.514622
$453$	0	0
$454$	1.81279e6	0.412769
$455$	1.22503e6	0.277407
$456$	0	0
$457$	−2.67154e6	−0.598371	−0.299186	−	0.954195i	$-0.596715\pi$
$457$	−2.67154e6	−0.598371	−0.299186	+	0.954195i	$0.596715\pi$
$458$	2.97802e6	0.663382
$459$	0	0
$460$	−1.13065e6	−0.249134
$461$	3.30382e6	0.724042	0.362021	−	0.932170i	$-0.382087\pi$
$461$	3.30382e6	0.724042	0.362021	+	0.932170i	$0.382087\pi$
$462$	0	0
$463$	−7.66989e6	−1.66279	−0.831394	−	0.555684i	$-0.812457\pi$
$463$	−7.66989e6	−1.66279	−0.831394	+	0.555684i	$0.812457\pi$
$464$	2.10771e6	0.454482
$465$	0	0
$466$	−275596.	−0.0587905
$467$	7.91117e6	1.67860	0.839302	−	0.543665i	$-0.182964\pi$
$467$	7.91117e6	1.67860	0.839302	+	0.543665i	$0.182964\pi$
$468$	0	0
$469$	−3.79008e6	−0.795640
$470$	−1.51556e6	−0.316467
$471$	0	0
$472$	−9.95465e6	−2.05670
$473$	−18321.6	−0.00376540
$474$	0	0
$475$	−249035.	−0.0506438
$476$	−1.25762e6	−0.254409
$477$	0	0
$478$	−7.66825e6	−1.53506
$479$	−5.44099e6	−1.08352	−0.541762	−	0.840532i	$-0.682243\pi$
$479$	−5.44099e6	−1.08352	−0.541762	+	0.840532i	$0.682243\pi$
$480$	0	0
$481$	87445.3	0.0172335
$482$	5.90840e6	1.15838
$483$	0	0
$484$	−174800.	−0.0339178
$485$	−292212.	−0.0564085
$486$	0	0
$487$	−8.50740e6	−1.62545	−0.812727	−	0.582645i	$-0.802018\pi$
$487$	−8.50740e6	−1.62545	−0.812727	+	0.582645i	$0.802018\pi$
$488$	−1.17772e6	−0.223869
$489$	0	0
$490$	−1.30925e6	−0.246339
$491$	334168.	0.0625549	0.0312775	−	0.999511i	$-0.490042\pi$
$491$	334168.	0.0625549	0.0312775	+	0.999511i	$0.490042\pi$
$492$	0	0
$493$	2.63339e6	0.487976
$494$	−518014.	−0.0955046
$495$	0	0
$496$	2.79397e6	0.509939
$497$	1.69116e6	0.307110
$498$	0	0
$499$	−3.50055e6	−0.629339	−0.314669	−	0.949201i	$-0.601894\pi$
$499$	−3.50055e6	−0.629339	−0.314669	+	0.949201i	$0.601894\pi$
$500$	186548.	0.0333707
$501$	0	0
$502$	1.06595e6	0.188790
$503$	−6.56301e6	−1.15660	−0.578300	−	0.815824i	$-0.696284\pi$
$503$	−6.56301e6	−1.15660	−0.578300	+	0.815824i	$0.696284\pi$
$504$	0	0
$505$	−3.38932e6	−0.591405
$506$	2.05295e6	0.356453
$507$	0	0
$508$	−331042.	−0.0569146
$509$	−109860.	−0.0187952	−0.00939760	−	0.999956i	$-0.502991\pi$
$509$	−109860.	−0.0187952	−0.00939760	+	0.999956i	$0.502991\pi$
$510$	0	0
$511$	80382.3	0.0136178
$512$	−5.17308e6	−0.872115
$513$	0	0
$514$	8.52649e6	1.42352
$515$	2.21226e6	0.367552
$516$	0	0
$517$	−1.63773e6	−0.269474
$518$	−227795.	−0.0373009
$519$	0	0
$520$	1.42808e6	0.231603
$521$	914334.	0.147574	0.0737872	−	0.997274i	$-0.476491\pi$
$521$	914334.	0.147574	0.0737872	+	0.997274i	$0.476491\pi$
$522$	0	0
$523$	4.00814e6	0.640750	0.320375	−	0.947291i	$-0.396191\pi$
$523$	4.00814e6	0.640750	0.320375	+	0.947291i	$0.396191\pi$
$524$	−3.87570e6	−0.616626
$525$	0	0
$526$	4.06249e6	0.640218
$527$	3.49081e6	0.547520
$528$	0	0
$529$	7.91308e6	1.22944
$530$	−2.02852e6	−0.313683
$531$	0	0
$532$	−803100.	−0.123024
$533$	4.24829e6	0.647732
$534$	0	0
$535$	5.46856e6	0.826016
$536$	−4.41831e6	−0.664269
$537$	0	0
$538$	−3.12847e6	−0.465990
$539$	−1.41480e6	−0.209760
$540$	0	0
$541$	−1.87833e6	−0.275918	−0.137959	−	0.990438i	$-0.544054\pi$
$541$	−1.87833e6	−0.275918	−0.137959	+	0.990438i	$0.544054\pi$
$542$	5.33431e6	0.779974
$543$	0	0
$544$	−2.53379e6	−0.367091
$545$	−5.95998e6	−0.859515
$546$	0	0
$547$	87797.6	0.0125463	0.00627313	−	0.999980i	$-0.498003\pi$
$547$	87797.6	0.0125463	0.00627313	+	0.999980i	$0.498003\pi$
$548$	−713346.	−0.101473
$549$	0	0
$550$	−338720.	−0.0477457
$551$	1.68165e6	0.235970
$552$	0	0
$553$	1.43537e7	1.99596
$554$	−5.15571e6	−0.713698
$555$	0	0
$556$	1.40458e6	0.192690
$557$	3.46548e6	0.473288	0.236644	−	0.971596i	$-0.423952\pi$
$557$	3.46548e6	0.473288	0.236644	+	0.971596i	$0.423952\pi$
$558$	0	0
$559$	43950.6	0.00594888
$560$	−2.10773e6	−0.284017
$561$	0	0
$562$	6.90655e6	0.922403
$563$	2.85737e6	0.379923	0.189961	−	0.981792i	$-0.439164\pi$
$563$	2.85737e6	0.379923	0.189961	+	0.981792i	$0.439164\pi$
$564$	0	0
$565$	4.68062e6	0.616854
$566$	5.75946e6	0.755685
$567$	0	0
$568$	1.97148e6	0.256402
$569$	5.48030e6	0.709617	0.354808	−	0.934939i	$-0.384546\pi$
$569$	5.48030e6	0.709617	0.354808	+	0.934939i	$0.384546\pi$
$570$	0	0
$571$	8.81180e6	1.13103	0.565515	−	0.824738i	$-0.308677\pi$
$571$	8.81180e6	1.13103	0.565515	+	0.824738i	$0.308677\pi$
$572$	419317.	0.0535861
$573$	0	0
$574$	−1.10668e7	−1.40198
$575$	−2.36754e6	−0.298626
$576$	0	0
$577$	502598.	0.0628465	0.0314233	−	0.999506i	$-0.489996\pi$
$577$	502598.	0.0628465	0.0314233	+	0.999506i	$0.489996\pi$
$578$	−4.61566e6	−0.574665
$579$	0	0
$580$	−1.25970e6	−0.155488
$581$	−4.32907e6	−0.532053
$582$	0	0
$583$	−2.19205e6	−0.267103
$584$	93706.2	0.0113694
$585$	0	0
$586$	−4.87319e6	−0.586232
$587$	7.45878e6	0.893455	0.446728	−	0.894670i	$-0.352589\pi$
$587$	7.45878e6	0.893455	0.446728	+	0.894670i	$0.352589\pi$
$588$	0	0
$589$	2.22919e6	0.264764
$590$	−5.66389e6	−0.669861
$591$	0	0
$592$	−150455.	−0.0176442
$593$	−6.69352e6	−0.781660	−0.390830	−	0.920463i	$-0.627812\pi$
$593$	−6.69352e6	−0.781660	−0.390830	+	0.920463i	$0.627812\pi$
$594$	0	0
$595$	−2.63341e6	−0.304948
$596$	630220.	0.0726736
$597$	0	0
$598$	−4.92469e6	−0.563153
$599$	1.64186e7	1.86968	0.934842	−	0.355065i	$-0.115541\pi$
$599$	1.64186e7	1.86968	0.934842	+	0.355065i	$0.115541\pi$
$600$	0	0
$601$	1.42657e7	1.61105	0.805523	−	0.592564i	$-0.201884\pi$
$601$	1.42657e7	1.61105	0.805523	+	0.592564i	$0.201884\pi$
$602$	−114491.	−0.0128760
$603$	0	0
$604$	5.13522e6	0.572752
$605$	−366025.	−0.0406558
$606$	0	0
$607$	2.38734e6	0.262992	0.131496	−	0.991317i	$-0.458022\pi$
$607$	2.38734e6	0.262992	0.131496	+	0.991317i	$0.458022\pi$
$608$	−1.61805e6	−0.177514
$609$	0	0
$610$	−670089.	−0.0729134
$611$	3.92865e6	0.425736
$612$	0	0
$613$	−1.36125e6	−0.146314	−0.0731570	−	0.997320i	$-0.523307\pi$
$613$	−1.36125e6	−0.146314	−0.0731570	+	0.997320i	$0.523307\pi$
$614$	−2.59931e6	−0.278251
$615$	0	0
$616$	−4.02004e6	−0.426853
$617$	1.33759e6	0.141453	0.0707264	−	0.997496i	$-0.477468\pi$
$617$	1.33759e6	0.141453	0.0707264	+	0.997496i	$0.477468\pi$
$618$	0	0
$619$	1.70842e7	1.79212	0.896062	−	0.443930i	$-0.146416\pi$
$619$	1.70842e7	1.79212	0.896062	+	0.443930i	$0.146416\pi$
$620$	−1.66985e6	−0.174461
$621$	0	0
$622$	7.55179e6	0.782661
$623$	305208.	0.0315047
$624$	0	0
$625$	390625.	0.0400000
$626$	−6.73310e6	−0.686719
$627$	0	0
$628$	1.80267e6	0.182397
$629$	−187979.	−0.0189445
$630$	0	0
$631$	1.70136e7	1.70107	0.850534	−	0.525920i	$-0.176279\pi$
$631$	1.70136e7	1.70107	0.850534	+	0.525920i	$0.176279\pi$
$632$	1.67329e7	1.66640
$633$	0	0
$634$	−5.86815e6	−0.579800
$635$	−693189.	−0.0682209
$636$	0	0
$637$	3.39387e6	0.331395
$638$	2.28727e6	0.222467
$639$	0	0
$640$	−577400.	−0.0557220
$641$	−1.04366e7	−1.00326	−0.501632	−	0.865081i	$-0.667267\pi$
$641$	−1.04366e7	−1.00326	−0.501632	+	0.865081i	$0.667267\pi$
$642$	0	0
$643$	1.32098e7	1.25999	0.629996	−	0.776598i	$-0.283056\pi$
$643$	1.32098e7	1.25999	0.629996	+	0.776598i	$0.283056\pi$
$644$	−7.63496e6	−0.725424
$645$	0	0
$646$	1.11356e6	0.104987
$647$	−3.14204e6	−0.295087	−0.147544	−	0.989056i	$-0.547137\pi$
$647$	−3.14204e6	−0.295087	−0.147544	+	0.989056i	$0.547137\pi$
$648$	0	0
$649$	−6.12047e6	−0.570391
$650$	812534.	0.0754325
$651$	0	0
$652$	−1.70018e6	−0.156630
$653$	−9.82598e6	−0.901764	−0.450882	−	0.892583i	$-0.648891\pi$
$653$	−9.82598e6	−0.901764	−0.450882	+	0.892583i	$0.648891\pi$
$654$	0	0
$655$	−8.11557e6	−0.739122
$656$	−7.30942e6	−0.663167
$657$	0	0
$658$	−1.02341e7	−0.921482
$659$	1.78524e7	1.60134	0.800670	−	0.599105i	$-0.204477\pi$
$659$	1.78524e7	1.60134	0.800670	+	0.599105i	$0.204477\pi$
$660$	0	0
$661$	1.02807e7	0.915207	0.457603	−	0.889156i	$-0.348708\pi$
$661$	1.02807e7	0.915207	0.457603	+	0.889156i	$0.348708\pi$
$662$	1.30353e7	1.15605
$663$	0	0
$664$	−5.04664e6	−0.444204
$665$	−1.68166e6	−0.147464
$666$	0	0
$667$	1.59872e7	1.39142
$668$	662242.	0.0574216
$669$	0	0
$670$	−2.51388e6	−0.216351
$671$	−724106.	−0.0620863
$672$	0	0
$673$	−2.22873e7	−1.89679	−0.948394	−	0.317094i	$-0.897293\pi$
$673$	−2.22873e7	−1.89679	−0.948394	+	0.317094i	$0.897293\pi$
$674$	1.10690e7	0.938549
$675$	0	0
$676$	3.42702e6	0.288436
$677$	1.56465e7	1.31204	0.656018	−	0.754745i	$-0.272239\pi$
$677$	1.56465e7	1.31204	0.656018	+	0.754745i	$0.272239\pi$
$678$	0	0
$679$	−1.97323e6	−0.164249
$680$	−3.06991e6	−0.254597
$681$	0	0
$682$	3.03199e6	0.249613
$683$	1.82208e7	1.49457	0.747283	−	0.664506i	$-0.231358\pi$
$683$	1.82208e7	1.49457	0.747283	+	0.664506i	$0.231358\pi$
$684$	0	0
$685$	−1.49372e6	−0.121631
$686$	3.86719e6	0.313751
$687$	0	0
$688$	−75619.5	−0.00609064
$689$	5.25837e6	0.421991
$690$	0	0
$691$	1.07957e7	0.860116	0.430058	−	0.902801i	$-0.358493\pi$
$691$	1.07957e7	0.860116	0.430058	+	0.902801i	$0.358493\pi$
$692$	1.59520e6	0.126634
$693$	0	0
$694$	−7.88239e6	−0.621240
$695$	2.94114e6	0.230969
$696$	0	0
$697$	−9.13244e6	−0.712041
$698$	−1.72951e6	−0.134364
$699$	0	0
$700$	1.25971e6	0.0971683
$701$	7.59781e6	0.583973	0.291987	−	0.956422i	$-0.405684\pi$
$701$	7.59781e6	0.583973	0.291987	+	0.956422i	$0.405684\pi$
$702$	0	0
$703$	−120041.	−0.00916098
$704$	−4.13447e6	−0.314404
$705$	0	0
$706$	1.09369e7	0.825811
$707$	−2.28872e7	−1.72204
$708$	0	0
$709$	1.26638e7	0.946122	0.473061	−	0.881030i	$-0.343149\pi$
$709$	1.26638e7	0.946122	0.473061	+	0.881030i	$0.343149\pi$
$710$	1.12171e6	0.0835094
$711$	0	0
$712$	355798.	0.0263029
$713$	2.11926e7	1.56121
$714$	0	0
$715$	878035.	0.0642313
$716$	−2.24758e6	−0.163845
$717$	0	0
$718$	−1.69091e7	−1.22408
$719$	8.16704e6	0.589172	0.294586	−	0.955625i	$-0.404818\pi$
$719$	8.16704e6	0.589172	0.294586	+	0.955625i	$0.404818\pi$
$720$	0	0
$721$	1.49388e7	1.07023
$722$	−1.03792e7	−0.741004
$723$	0	0
$724$	7.78729e6	0.552128
$725$	−2.63777e6	−0.186377
$726$	0	0
$727$	1.99934e6	0.140298	0.0701488	−	0.997537i	$-0.477653\pi$
$727$	1.99934e6	0.140298	0.0701488	+	0.997537i	$0.477653\pi$
$728$	9.64343e6	0.674377
$729$	0	0
$730$	53316.0	0.00370297
$731$	−94479.6	−0.00653950
$732$	0	0
$733$	2.26768e7	1.55892	0.779458	−	0.626455i	$-0.215495\pi$
$733$	2.26768e7	1.55892	0.779458	+	0.626455i	$0.215495\pi$
$734$	−1.78419e7	−1.22237
$735$	0	0
$736$	−1.53826e7	−1.04673
$737$	−2.71653e6	−0.184224
$738$	0	0
$739$	−1.49990e7	−1.01030	−0.505151	−	0.863031i	$-0.668563\pi$
$739$	−1.49990e7	−1.01030	−0.505151	+	0.863031i	$0.668563\pi$
$740$	89920.9	0.00603645
$741$	0	0
$742$	−1.36981e7	−0.913375
$743$	8.72196e6	0.579618	0.289809	−	0.957084i	$-0.406408\pi$
$743$	8.72196e6	0.579618	0.289809	+	0.957084i	$0.406408\pi$
$744$	0	0
$745$	1.31966e6	0.0871105
$746$	1.40413e7	0.923765
$747$	0	0
$748$	−901396.	−0.0589063
$749$	3.69277e7	2.40518
$750$	0	0
$751$	3.03235e7	1.96191	0.980957	−	0.194227i	$-0.0622198\pi$
$751$	3.03235e7	1.96191	0.980957	+	0.194227i	$0.0622198\pi$
$752$	−6.75947e6	−0.435881
$753$	0	0
$754$	−5.48679e6	−0.351471
$755$	1.07530e7	0.686532
$756$	0	0
$757$	999401.	0.0633870	0.0316935	−	0.999498i	$-0.489910\pi$
$757$	999401.	0.0633870	0.0316935	+	0.999498i	$0.489910\pi$
$758$	−1.12419e7	−0.710669
$759$	0	0
$760$	−1.96041e6	−0.123115
$761$	−1.23318e7	−0.771904	−0.385952	−	0.922519i	$-0.626127\pi$
$761$	−1.23318e7	−0.771904	−0.385952	+	0.922519i	$0.626127\pi$
$762$	0	0
$763$	−4.02461e7	−2.50272
$764$	−8.42445e6	−0.522166
$765$	0	0
$766$	1.88244e7	1.15918
$767$	1.46820e7	0.901150
$768$	0	0
$769$	−3.88179e6	−0.236710	−0.118355	−	0.992971i	$-0.537762\pi$
$769$	−3.88179e6	−0.236710	−0.118355	+	0.992971i	$0.537762\pi$
$770$	−2.28728e6	−0.139025
$771$	0	0
$772$	1.04041e6	0.0628293
$773$	−2.35138e7	−1.41539	−0.707693	−	0.706520i	$-0.750264\pi$
$773$	−2.35138e7	−1.41539	−0.707693	+	0.706520i	$0.750264\pi$
$774$	0	0
$775$	−3.49661e6	−0.209119
$776$	−2.30030e6	−0.137129
$777$	0	0
$778$	8.56244e6	0.507164
$779$	−5.83186e6	−0.344321
$780$	0	0
$781$	1.21214e6	0.0711088
$782$	1.05865e7	0.619064
$783$	0	0
$784$	−5.83934e6	−0.339292
$785$	3.77472e6	0.218631
$786$	0	0
$787$	2.16876e6	0.124817	0.0624086	−	0.998051i	$-0.480122\pi$
$787$	2.16876e6	0.124817	0.0624086	+	0.998051i	$0.480122\pi$
$788$	−7.97049e6	−0.457266
$789$	0	0
$790$	9.52052e6	0.542742
$791$	3.16069e7	1.79614
$792$	0	0
$793$	1.73701e6	0.0980890
$794$	1.62328e7	0.913784
$795$	0	0
$796$	8.34075e6	0.466576
$797$	1.85869e7	1.03648	0.518241	−	0.855235i	$-0.326587\pi$
$797$	1.85869e7	1.03648	0.518241	+	0.855235i	$0.326587\pi$
$798$	0	0
$799$	−8.44534e6	−0.468005
$800$	2.53800e6	0.140206
$801$	0	0
$802$	9.94556e6	0.546001
$803$	57613.9	0.00315310
$804$	0	0
$805$	−1.59873e7	−0.869534
$806$	−7.27326e6	−0.394359
$807$	0	0
$808$	−2.66808e7	−1.43771
$809$	−2.99966e7	−1.61139	−0.805694	−	0.592332i	$-0.798207\pi$
$809$	−2.99966e7	−1.61139	−0.805694	+	0.592332i	$0.798207\pi$
$810$	0	0
$811$	−1.54479e6	−0.0824740	−0.0412370	−	0.999149i	$-0.513130\pi$
$811$	−1.54479e6	−0.0824740	−0.0412370	+	0.999149i	$0.513130\pi$
$812$	−8.50640e6	−0.452747
$813$	0	0
$814$	−163272.	−0.00863674
$815$	−3.56011e6	−0.187746
$816$	0	0
$817$	−60333.5	−0.00316230
$818$	92108.3	0.00481299
$819$	0	0
$820$	4.36856e6	0.226884
$821$	2.60445e7	1.34852	0.674261	−	0.738493i	$-0.264462\pi$
$821$	2.60445e7	1.34852	0.674261	+	0.738493i	$0.264462\pi$
$822$	0	0
$823$	−9.56891e6	−0.492451	−0.246226	−	0.969213i	$-0.579190\pi$
$823$	−9.56891e6	−0.492451	−0.246226	+	0.969213i	$0.579190\pi$
$824$	1.74150e7	0.893521
$825$	0	0
$826$	−3.82467e7	−1.95049
$827$	−1.07164e7	−0.544860	−0.272430	−	0.962176i	$-0.587827\pi$
$827$	−1.07164e7	−0.544860	−0.272430	+	0.962176i	$0.587827\pi$
$828$	0	0
$829$	−1.64337e7	−0.830519	−0.415259	−	0.909703i	$-0.636309\pi$
$829$	−1.64337e7	−0.830519	−0.415259	+	0.909703i	$0.636309\pi$
$830$	−2.87139e6	−0.144676
$831$	0	0
$832$	9.91791e6	0.496720
$833$	−7.29572e6	−0.364297
$834$	0	0
$835$	1.38671e6	0.0688287
$836$	−575620.	−0.0284853
$837$	0	0
$838$	1.03322e7	0.508257
$839$	−9.64356e6	−0.472969	−0.236484	−	0.971635i	$-0.575995\pi$
$839$	−9.64356e6	−0.472969	−0.236484	+	0.971635i	$0.575995\pi$
$840$	0	0
$841$	−2.69916e6	−0.131595
$842$	2.24118e7	1.08942
$843$	0	0
$844$	6.66485e6	0.322058
$845$	7.17606e6	0.345736
$846$	0	0
$847$	−2.47166e6	−0.118381
$848$	−9.04732e6	−0.432046
$849$	0	0
$850$	−1.74669e6	−0.0829216
$851$	−1.14121e6	−0.0540186
$852$	0	0
$853$	−2.52742e7	−1.18934	−0.594669	−	0.803971i	$-0.702717\pi$
$853$	−2.52742e7	−1.18934	−0.594669	+	0.803971i	$0.702717\pi$
$854$	−4.52492e6	−0.212308
$855$	0	0
$856$	4.30486e7	2.00805
$857$	262739.	0.0122200	0.00611002	−	0.999981i	$-0.498055\pi$
$857$	262739.	0.0122200	0.00611002	+	0.999981i	$0.498055\pi$
$858$	0	0
$859$	−9.43849e6	−0.436435	−0.218218	−	0.975900i	$-0.570024\pi$
$859$	−9.43849e6	−0.436435	−0.218218	+	0.975900i	$0.570024\pi$
$860$	45194.9	0.00208374
$861$	0	0
$862$	1.21022e7	0.554748
$863$	−3.78002e7	−1.72769	−0.863847	−	0.503754i	$-0.831952\pi$
$863$	−3.78002e7	−1.72769	−0.863847	+	0.503754i	$0.831952\pi$
$864$	0	0
$865$	3.34030e6	0.151791
$866$	−1.75718e7	−0.796200
$867$	0	0
$868$	−1.12760e7	−0.507993
$869$	1.02880e7	0.462148
$870$	0	0
$871$	6.51653e6	0.291052
$872$	−4.69171e7	−2.08949
$873$	0	0
$874$	6.76041e6	0.299360
$875$	2.63778e6	0.116471
$876$	0	0
$877$	2.42348e7	1.06400	0.531999	−	0.846745i	$-0.321441\pi$
$877$	2.42348e7	1.06400	0.531999	+	0.846745i	$0.321441\pi$
$878$	−238554.	−0.0104436
$879$	0	0
$880$	−1.51071e6	−0.0657619
$881$	−1.97060e7	−0.855377	−0.427689	−	0.903926i	$-0.640672\pi$
$881$	−1.97060e7	−0.855377	−0.427689	+	0.903926i	$0.640672\pi$
$882$	0	0
$883$	−2.09867e7	−0.905822	−0.452911	−	0.891556i	$-0.649614\pi$
$883$	−2.09867e7	−0.905822	−0.452911	+	0.891556i	$0.649614\pi$
$884$	2.16230e6	0.0930649
$885$	0	0
$886$	−1.48010e7	−0.633441
$887$	3.47934e7	1.48487	0.742434	−	0.669919i	$-0.233671\pi$
$887$	3.47934e7	1.48487	0.742434	+	0.669919i	$0.233671\pi$
$888$	0	0
$889$	−4.68091e6	−0.198644
$890$	202438.	0.00856677
$891$	0	0
$892$	4.57143e6	0.192371
$893$	−5.39309e6	−0.226313
$894$	0	0
$895$	−4.70636e6	−0.196394
$896$	−3.89902e6	−0.162250
$897$	0	0
$898$	−1.36427e7	−0.564559
$899$	2.36115e7	0.974370
$900$	0	0
$901$	−1.13038e7	−0.463887
$902$	−7.93210e6	−0.324617
$903$	0	0
$904$	3.68460e7	1.49958
$905$	1.63063e7	0.661811
$906$	0	0
$907$	−4.14396e7	−1.67262	−0.836310	−	0.548257i	$-0.815291\pi$
$907$	−4.14396e7	−1.67262	−0.836310	+	0.548257i	$0.815291\pi$
$908$	−4.83217e6	−0.194503
$909$	0	0
$910$	5.48682e6	0.219643
$911$	4.55462e7	1.81826	0.909131	−	0.416511i	$-0.136747\pi$
$911$	4.55462e7	1.81826	0.909131	+	0.416511i	$0.136747\pi$
$912$	0	0
$913$	−3.10285e6	−0.123193
$914$	−1.19657e7	−0.473774
$915$	0	0
$916$	−7.93822e6	−0.312597
$917$	−5.48022e7	−2.15216
$918$	0	0
$919$	8.41097e6	0.328517	0.164258	−	0.986417i	$-0.447477\pi$
$919$	8.41097e6	0.328517	0.164258	+	0.986417i	$0.447477\pi$
$920$	−1.86373e7	−0.725962
$921$	0	0
$922$	1.47976e7	0.573277
$923$	−2.90772e6	−0.112343
$924$	0	0
$925$	188291.	0.00723562
$926$	−3.43530e7	−1.31655
$927$	0	0
$928$	−1.71383e7	−0.653278
$929$	−1.89074e6	−0.0718773	−0.0359386	−	0.999354i	$-0.511442\pi$
$929$	−1.89074e6	−0.0718773	−0.0359386	+	0.999354i	$0.511442\pi$
$930$	0	0
$931$	−4.65895e6	−0.176163
$932$	734628.	0.0277031
$933$	0	0
$934$	3.54337e7	1.32907
$935$	−1.88749e6	−0.0706083
$936$	0	0
$937$	3.58861e7	1.33530	0.667648	−	0.744477i	$-0.267301\pi$
$937$	3.58861e7	1.33530	0.667648	+	0.744477i	$0.267301\pi$
$938$	−1.69756e7	−0.629966
$939$	0	0
$940$	4.03988e6	0.149124
$941$	−3.25537e7	−1.19847	−0.599235	−	0.800573i	$-0.704528\pi$
$941$	−3.25537e7	−1.19847	−0.599235	+	0.800573i	$0.704528\pi$
$942$	0	0
$943$	−5.54427e7	−2.03032
$944$	−2.52612e7	−0.922624
$945$	0	0
$946$	−82061.5	−0.00298134
$947$	−6.44146e6	−0.233405	−0.116702	−	0.993167i	$-0.537232\pi$
$947$	−6.44146e6	−0.233405	−0.116702	+	0.993167i	$0.537232\pi$
$948$	0	0
$949$	−138206.	−0.00498153
$950$	−1.11541e6	−0.0400984
$951$	0	0
$952$	−2.07303e7	−0.741332
$953$	−2.40023e7	−0.856093	−0.428046	−	0.903757i	$-0.640798\pi$
$953$	−2.40023e7	−0.856093	−0.428046	+	0.903757i	$0.640798\pi$
$954$	0	0
$955$	−1.76405e7	−0.625897
$956$	2.04405e7	0.723347
$957$	0	0
$958$	−2.43699e7	−0.857905
$959$	−1.00867e7	−0.354162
$960$	0	0
$961$	2.67012e6	0.0932656
$962$	391662.	0.0136450
$963$	0	0
$964$	−1.57494e7	−0.545849
$965$	2.17859e6	0.0753107
$966$	0	0
$967$	4.13239e7	1.42113	0.710567	−	0.703630i	$-0.248439\pi$
$967$	4.13239e7	1.42113	0.710567	+	0.703630i	$0.248439\pi$
$968$	−2.88136e6	−0.0988345
$969$	0	0
$970$	−1.30880e6	−0.0446627
$971$	−8.46004e6	−0.287955	−0.143977	−	0.989581i	$-0.545989\pi$
$971$	−8.46004e6	−0.287955	−0.143977	+	0.989581i	$0.545989\pi$
$972$	0	0
$973$	1.98607e7	0.672531
$974$	−3.81042e7	−1.28699
$975$	0	0
$976$	−2.98863e6	−0.100426
$977$	−1.33050e7	−0.445942	−0.222971	−	0.974825i	$-0.571575\pi$
$977$	−1.33050e7	−0.445942	−0.222971	+	0.974825i	$0.571575\pi$
$978$	0	0
$979$	218757.	0.00729467
$980$	3.48995e6	0.116079
$981$	0	0
$982$	1.49672e6	0.0495293
$983$	−2.90239e7	−0.958014	−0.479007	−	0.877811i	$-0.659003\pi$
$983$	−2.90239e7	−0.958014	−0.479007	+	0.877811i	$0.659003\pi$
$984$	0	0
$985$	−1.66899e7	−0.548105
$986$	1.17948e7	0.386366
$987$	0	0
$988$	1.38082e6	0.0450033
$989$	−573582.	−0.0186468
$990$	0	0
$991$	−4.88551e7	−1.58025	−0.790125	−	0.612946i	$-0.789984\pi$
$991$	−4.88551e7	−1.58025	−0.790125	+	0.612946i	$0.789984\pi$
$992$	−2.27184e7	−0.732992
$993$	0	0
$994$	7.57460e6	0.243161
$995$	1.74652e7	0.559264
$996$	0	0
$997$	−2.19218e7	−0.698453	−0.349227	−	0.937038i	$-0.613556\pi$
$997$	−2.19218e7	−0.698453	−0.349227	+	0.937038i	$0.613556\pi$
$998$	−1.56787e7	−0.498293
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.b.1.3		3
3.2	odd	2		165.6.a.c.1.1	✓	3
15.14	odd	2		825.6.a.g.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.c.1.1	✓	3	3.2	odd	2
495.6.a.b.1.3		3	1.1	even	1	trivial
825.6.a.g.1.3		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+4.47894 q^{2} -11.9391 q^{4} -25.0000 q^{5} -168.818 q^{7} -196.801 q^{8} -111.974 q^{10} -121.000 q^{11} +290.259 q^{13} -756.126 q^{14} -499.408 q^{16} -623.964 q^{17} -398.456 q^{19} +298.477 q^{20} -541.952 q^{22} -3788.06 q^{23} +625.000 q^{25} +1300.06 q^{26} +2015.53 q^{28} -4220.43 q^{29} -5594.57 q^{31} +4060.80 q^{32} -2794.70 q^{34} +4220.45 q^{35} +301.266 q^{37} -1784.66 q^{38} +4920.01 q^{40} +14636.2 q^{41} +151.418 q^{43} +1444.63 q^{44} -16966.5 q^{46} +13535.0 q^{47} +11692.5 q^{49} +2799.34 q^{50} -3465.43 q^{52} +18116.1 q^{53} +3025.00 q^{55} +33223.5 q^{56} -18903.0 q^{58} +50582.4 q^{59} +5984.35 q^{61} -25057.8 q^{62} +34169.1 q^{64} -7256.49 q^{65} +22450.7 q^{67} +7449.56 q^{68} +18903.2 q^{70} -10017.6 q^{71} -476.148 q^{73} +1349.35 q^{74} +4757.19 q^{76} +20427.0 q^{77} -85024.7 q^{79} +12485.2 q^{80} +65554.5 q^{82} +25643.4 q^{83} +15599.1 q^{85} +678.194 q^{86} +23812.9 q^{88} -1807.91 q^{89} -49001.0 q^{91} +45226.0 q^{92} +60622.4 q^{94} +9961.39 q^{95} +11688.5 q^{97} +52370.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8} + 50 q^{10} - 363 q^{11} + 290 q^{13} - 916 q^{14} - 590 q^{16} - 434 q^{17} - 2856 q^{19} + 1050 q^{20} + 242 q^{22} + 640 q^{23} + 1875 q^{25}+ \cdots + 154654 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8 + 50 * q^10 - 363 * q^11 + 290 * q^13 - 916 * q^14 - 590 * q^16 - 434 * q^17 - 2856 * q^19 + 1050 * q^20 + 242 * q^22 + 640 * q^23 + 1875 * q^25 - 2132 * q^26 - 580 * q^28 + 4538 * q^29 - 14968 * q^31 + 2496 * q^32 - 13704 * q^34 + 1700 * q^35 - 6190 * q^37 + 11668 * q^38 - 600 * q^40 + 8926 * q^41 - 33592 * q^43 + 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 14693 * q^49 - 1250 * q^50 + 18780 * q^52 + 22934 * q^53 + 9075 * q^55 + 40012 * q^56 - 32304 * q^58 + 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 35474 * q^64 - 7250 * q^65 + 16868 * q^67 + 71288 * q^68 + 22900 * q^70 - 4856 * q^71 + 1910 * q^73 - 29404 * q^74 + 6116 * q^76 + 8228 * q^77 - 36844 * q^79 + 14750 * q^80 + 84000 * q^82 + 48796 * q^83 + 10850 * q^85 + 83492 * q^86 - 2904 * q^88 + 188978 * q^89 - 93208 * q^91 + 6976 * q^92 + 70472 * q^94 + 71400 * q^95 + 247526 * q^97 + 154654 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more