LMFDB - Embedded newform 495.6.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,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, 6, names="a")

//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);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
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:	$79.3899908074$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 $ x^7 - x^6 - 209x^5 + 137x^4 + 12724x^3 - 1040x^2 - 218208*x - 8784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$10.4220$ 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)

$f(q)$	$=$	$q-10.4220 q^{2} +76.6176 q^{4} -25.0000 q^{5} -101.242 q^{7} -465.003 q^{8} +O(q^{10})$	\(q-10.4220 q^{2} +76.6176 q^{4} -25.0000 q^{5} -101.242 q^{7} -465.003 q^{8} +260.549 q^{10} -121.000 q^{11} +463.289 q^{13} +1055.14 q^{14} +2394.49 q^{16} -1571.95 q^{17} -2430.09 q^{19} -1915.44 q^{20} +1261.06 q^{22} -2768.12 q^{23} +625.000 q^{25} -4828.38 q^{26} -7756.92 q^{28} +5699.44 q^{29} -5755.76 q^{31} -10075.2 q^{32} +16382.8 q^{34} +2531.05 q^{35} -2753.45 q^{37} +25326.4 q^{38} +11625.1 q^{40} +19265.1 q^{41} -2080.20 q^{43} -9270.73 q^{44} +28849.3 q^{46} -23142.2 q^{47} -6557.04 q^{49} -6513.73 q^{50} +35496.0 q^{52} +12836.7 q^{53} +3025.00 q^{55} +47077.9 q^{56} -59399.4 q^{58} -31548.2 q^{59} +19918.8 q^{61} +59986.4 q^{62} +28380.0 q^{64} -11582.2 q^{65} -60988.6 q^{67} -120439. q^{68} -26378.6 q^{70} -16291.6 q^{71} -29882.0 q^{73} +28696.4 q^{74} -186188. q^{76} +12250.3 q^{77} -8600.69 q^{79} -59862.3 q^{80} -200781. q^{82} -41891.9 q^{83} +39298.8 q^{85} +21679.8 q^{86} +56265.4 q^{88} +30646.2 q^{89} -46904.3 q^{91} -212086. q^{92} +241188. q^{94} +60752.4 q^{95} -118282. q^{97} +68337.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 195 q^{4} - 175 q^{5} - 153 q^{8}+O(q^{10}) $ 7 * q - q^2 + 195 * q^4 - 175 * q^5 - 153 * q^8	$ 7 q - q^{2} + 195 q^{4} - 175 q^{5} - 153 q^{8} + 25 q^{10} - 847 q^{11} + 1418 q^{13} - 2548 q^{14} + 3699 q^{16} - 630 q^{17} + 2572 q^{19} - 4875 q^{20} + 121 q^{22} - 536 q^{23} + 4375 q^{25} + 7626 q^{26} - 11368 q^{28} + 1038 q^{29} + 1872 q^{31} + 7523 q^{32} + 20790 q^{34} + 24298 q^{37} + 18952 q^{38} + 3825 q^{40} + 17658 q^{41} + 7244 q^{43} - 23595 q^{44} + 31016 q^{46} - 34560 q^{47} + 78735 q^{49} - 625 q^{50} + 110222 q^{52} + 10214 q^{53} + 21175 q^{55} - 81124 q^{56} - 5718 q^{58} - 94676 q^{59} + 69538 q^{61} + 4208 q^{62} + 112339 q^{64} - 35450 q^{65} + 64908 q^{67} + 136010 q^{68} + 63700 q^{70} - 61816 q^{71} - 11890 q^{73} + 124050 q^{74} - 47216 q^{76} + 18928 q^{79} - 92475 q^{80} + 36398 q^{82} - 17492 q^{83} + 15750 q^{85} + 216688 q^{86} + 18513 q^{88} - 25302 q^{89} + 3392 q^{91} + 27408 q^{92} - 30800 q^{94} - 64300 q^{95} - 172546 q^{97} + 615271 q^{98}+O(q^{100}) $ 7 * q - q^2 + 195 * q^4 - 175 * q^5 - 153 * q^8 + 25 * q^10 - 847 * q^11 + 1418 * q^13 - 2548 * q^14 + 3699 * q^16 - 630 * q^17 + 2572 * q^19 - 4875 * q^20 + 121 * q^22 - 536 * q^23 + 4375 * q^25 + 7626 * q^26 - 11368 * q^28 + 1038 * q^29 + 1872 * q^31 + 7523 * q^32 + 20790 * q^34 + 24298 * q^37 + 18952 * q^38 + 3825 * q^40 + 17658 * q^41 + 7244 * q^43 - 23595 * q^44 + 31016 * q^46 - 34560 * q^47 + 78735 * q^49 - 625 * q^50 + 110222 * q^52 + 10214 * q^53 + 21175 * q^55 - 81124 * q^56 - 5718 * q^58 - 94676 * q^59 + 69538 * q^61 + 4208 * q^62 + 112339 * q^64 - 35450 * q^65 + 64908 * q^67 + 136010 * q^68 + 63700 * q^70 - 61816 * q^71 - 11890 * q^73 + 124050 * q^74 - 47216 * q^76 + 18928 * q^79 - 92475 * q^80 + 36398 * q^82 - 17492 * q^83 + 15750 * q^85 + 216688 * q^86 + 18513 * q^88 - 25302 * q^89 + 3392 * q^91 + 27408 * q^92 - 30800 * q^94 - 64300 * q^95 - 172546 * q^97 + 615271 * 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 $

$2$	−10.4220	−1.84236	−0.921181	−	0.389134i	$-0.872774\pi$
$2$	−10.4220	−1.84236	−0.921181	+	0.389134i	$0.872774\pi$
$3$	0	0
$3$	0	0
$4$	76.6176	2.39430
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−101.242	−0.780937	−0.390468	−	0.920616i	$-0.627687\pi$
$7$	−101.242	−0.780937	−0.390468	+	0.920616i	$0.627687\pi$
$8$	−465.003	−2.56880
$9$	0	0
$10$	260.549	0.823930
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	463.289	0.760314	0.380157	−	0.924922i	$-0.375870\pi$
$13$	463.289	0.760314	0.380157	+	0.924922i	$0.375870\pi$
$14$	1055.14	1.43877
$15$	0	0
$16$	2394.49	2.33837
$17$	−1571.95	−1.31922	−0.659610	−	0.751608i	$-0.729278\pi$
$17$	−1571.95	−1.31922	−0.659610	+	0.751608i	$0.729278\pi$
$18$	0	0
$19$	−2430.09	−1.54433	−0.772163	−	0.635424i	$-0.780825\pi$
$19$	−2430.09	−1.54433	−0.772163	+	0.635424i	$0.780825\pi$
$20$	−1915.44	−1.07076
$21$	0	0
$22$	1261.06	0.555493
$23$	−2768.12	−1.09110	−0.545551	−	0.838078i	$-0.683679\pi$
$23$	−2768.12	−1.09110	−0.545551	+	0.838078i	$0.683679\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−4828.38	−1.40077
$27$	0	0
$28$	−7756.92	−1.86980
$29$	5699.44	1.25845	0.629227	−	0.777222i	$-0.283372\pi$
$29$	5699.44	1.25845	0.629227	+	0.777222i	$0.283372\pi$
$30$	0	0
$31$	−5755.76	−1.07572	−0.537859	−	0.843035i	$-0.680767\pi$
$31$	−5755.76	−1.07572	−0.537859	+	0.843035i	$0.680767\pi$
$32$	−10075.2	−1.73932
$33$	0	0
$34$	16382.8	2.43048
$35$	2531.05	0.349246
$36$	0	0
$37$	−2753.45	−0.330653	−0.165327	−	0.986239i	$-0.552868\pi$
$37$	−2753.45	−0.330653	−0.165327	+	0.986239i	$0.552868\pi$
$38$	25326.4	2.84521
$39$	0	0
$40$	11625.1	1.14880
$41$	19265.1	1.78983	0.894915	−	0.446236i	$-0.147236\pi$
$41$	19265.1	1.78983	0.894915	+	0.446236i	$0.147236\pi$
$42$	0	0
$43$	−2080.20	−0.171567	−0.0857837	−	0.996314i	$-0.527339\pi$
$43$	−2080.20	−0.171567	−0.0857837	+	0.996314i	$0.527339\pi$
$44$	−9270.73	−0.721908
$45$	0	0
$46$	28849.3	2.01020
$47$	−23142.2	−1.52813	−0.764066	−	0.645138i	$-0.776800\pi$
$47$	−23142.2	−1.52813	−0.764066	+	0.645138i	$0.776800\pi$
$48$	0	0
$49$	−6557.04	−0.390137
$50$	−6513.73	−0.368472
$51$	0	0
$52$	35496.0	1.82042
$53$	12836.7	0.627715	0.313858	−	0.949470i	$-0.398379\pi$
$53$	12836.7	0.627715	0.313858	+	0.949470i	$0.398379\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	47077.9	2.00607
$57$	0	0
$58$	−59399.4	−2.31853
$59$	−31548.2	−1.17990	−0.589950	−	0.807440i	$-0.700852\pi$
$59$	−31548.2	−1.17990	−0.589950	+	0.807440i	$0.700852\pi$
$60$	0	0
$61$	19918.8	0.685392	0.342696	−	0.939446i	$-0.388660\pi$
$61$	19918.8	0.685392	0.342696	+	0.939446i	$0.388660\pi$
$62$	59986.4	1.98186
$63$	0	0
$64$	28380.0	0.866089
$65$	−11582.2	−0.340023
$66$	0	0
$67$	−60988.6	−1.65982	−0.829912	−	0.557895i	$-0.811609\pi$
$67$	−60988.6	−1.65982	−0.829912	+	0.557895i	$0.811609\pi$
$68$	−120439.	−3.15861
$69$	0	0
$70$	−26378.6	−0.643437
$71$	−16291.6	−0.383546	−0.191773	−	0.981439i	$-0.561424\pi$
$71$	−16291.6	−0.383546	−0.191773	+	0.981439i	$0.561424\pi$
$72$	0	0
$73$	−29882.0	−0.656300	−0.328150	−	0.944626i	$-0.606425\pi$
$73$	−29882.0	−0.656300	−0.328150	+	0.944626i	$0.606425\pi$
$74$	28696.4	0.609183
$75$	0	0
$76$	−186188.	−3.69758
$77$	12250.3	0.235461
$78$	0	0
$79$	−8600.69	−0.155048	−0.0775239	−	0.996990i	$-0.524701\pi$
$79$	−8600.69	−0.155048	−0.0775239	+	0.996990i	$0.524701\pi$
$80$	−59862.3	−1.04575
$81$	0	0
$82$	−200781.	−3.29752
$83$	−41891.9	−0.667474	−0.333737	−	0.942666i	$-0.608310\pi$
$83$	−41891.9	−0.667474	−0.333737	+	0.942666i	$0.608310\pi$
$84$	0	0
$85$	39298.8	0.589973
$86$	21679.8	0.316089
$87$	0	0
$88$	56265.4	0.774524
$89$	30646.2	0.410111	0.205056	−	0.978750i	$-0.434262\pi$
$89$	30646.2	0.410111	0.205056	+	0.978750i	$0.434262\pi$
$90$	0	0
$91$	−46904.3	−0.593758
$92$	−212086.	−2.61242
$93$	0	0
$94$	241188.	2.81537
$95$	60752.4	0.690644
$96$	0	0
$97$	−118282.	−1.27641	−0.638206	−	0.769865i	$-0.720323\pi$
$97$	−118282.	−1.27641	−0.638206	+	0.769865i	$0.720323\pi$
$98$	68337.3	0.718775
$99$	0	0
$100$	47886.0	0.478860
$101$	118856.	1.15936	0.579679	−	0.814845i	$-0.303178\pi$
$101$	118856.	1.15936	0.579679	+	0.814845i	$0.303178\pi$
$102$	0	0
$103$	132043.	1.22637	0.613184	−	0.789940i	$-0.289888\pi$
$103$	132043.	1.22637	0.613184	+	0.789940i	$0.289888\pi$
$104$	−215431.	−1.95310
$105$	0	0
$106$	−133783.	−1.15648
$107$	−193247.	−1.63174	−0.815872	−	0.578232i	$-0.803743\pi$
$107$	−193247.	−1.63174	−0.815872	+	0.578232i	$0.803743\pi$
$108$	0	0
$109$	41338.7	0.333266	0.166633	−	0.986019i	$-0.446711\pi$
$109$	41338.7	0.333266	0.166633	+	0.986019i	$0.446711\pi$
$110$	−31526.5	−0.248424
$111$	0	0
$112$	−242423.	−1.82612
$113$	−112853.	−0.831411	−0.415705	−	0.909499i	$-0.636465\pi$
$113$	−112853.	−0.831411	−0.415705	+	0.909499i	$0.636465\pi$
$114$	0	0
$115$	69202.9	0.487955
$116$	436677.	3.01312
$117$	0	0
$118$	328795.	2.17380
$119$	159148.	1.03023
$120$	0	0
$121$	14641.0	0.0909091
$122$	−207594.	−1.26274
$123$	0	0
$124$	−440992.	−2.57559
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−150866.	−0.830010	−0.415005	−	0.909819i	$-0.636220\pi$
$127$	−150866.	−0.830010	−0.415005	+	0.909819i	$0.636220\pi$
$128$	26631.5	0.143671
$129$	0	0
$130$	120710.	0.626445
$131$	−180602.	−0.919485	−0.459742	−	0.888052i	$-0.652058\pi$
$131$	−180602.	−0.919485	−0.459742	+	0.888052i	$0.652058\pi$
$132$	0	0
$133$	246028.	1.20602
$134$	635622.	3.05800
$135$	0	0
$136$	730963.	3.38882
$137$	75306.6	0.342793	0.171396	−	0.985202i	$-0.445172\pi$
$137$	75306.6	0.342793	0.171396	+	0.985202i	$0.445172\pi$
$138$	0	0
$139$	57578.9	0.252771	0.126385	−	0.991981i	$-0.459662\pi$
$139$	57578.9	0.252771	0.126385	+	0.991981i	$0.459662\pi$
$140$	193923.	0.836199
$141$	0	0
$142$	169790.	0.706630
$143$	−56057.9	−0.229243
$144$	0	0
$145$	−142486.	−0.562798
$146$	311429.	1.20914
$147$	0	0
$148$	−210963.	−0.791683
$149$	157538.	0.581327	0.290663	−	0.956825i	$-0.406124\pi$
$149$	157538.	0.581327	0.290663	+	0.956825i	$0.406124\pi$
$150$	0	0
$151$	−401031.	−1.43132	−0.715658	−	0.698451i	$-0.753873\pi$
$151$	−401031.	−1.43132	−0.715658	+	0.698451i	$0.753873\pi$
$152$	1.13000e6	3.96707
$153$	0	0
$154$	−127672.	−0.433805
$155$	143894.	0.481076
$156$	0	0
$157$	−261690.	−0.847303	−0.423651	−	0.905825i	$-0.639252\pi$
$157$	−261690.	−0.847303	−0.423651	+	0.905825i	$0.639252\pi$
$158$	89636.2	0.285654
$159$	0	0
$160$	251880.	0.777848
$161$	280250.	0.852081
$162$	0	0
$163$	450617.	1.32843	0.664215	−	0.747541i	$-0.268766\pi$
$163$	450617.	1.32843	0.664215	+	0.747541i	$0.268766\pi$
$164$	1.47605e6	4.28539
$165$	0	0
$166$	436596.	1.22973
$167$	236789.	0.657008	0.328504	−	0.944503i	$-0.393455\pi$
$167$	236789.	0.657008	0.328504	+	0.944503i	$0.393455\pi$
$168$	0	0
$169$	−156657.	−0.421922
$170$	−409571.	−1.08694
$171$	0	0
$172$	−159380.	−0.410784
$173$	232600.	0.590872	0.295436	−	0.955362i	$-0.404535\pi$
$173$	232600.	0.590872	0.295436	+	0.955362i	$0.404535\pi$
$174$	0	0
$175$	−63276.3	−0.156187
$176$	−289733.	−0.705045
$177$	0	0
$178$	−319394.	−0.755574
$179$	−287284.	−0.670160	−0.335080	−	0.942190i	$-0.608763\pi$
$179$	−287284.	−0.670160	−0.335080	+	0.942190i	$0.608763\pi$
$180$	0	0
$181$	646197.	1.46612	0.733058	−	0.680166i	$-0.238092\pi$
$181$	646197.	1.46612	0.733058	+	0.680166i	$0.238092\pi$
$182$	488835.	1.09392
$183$	0	0
$184$	1.28718e6	2.80283
$185$	68836.2	0.147873
$186$	0	0
$187$	190206.	0.397760
$188$	−1.77310e6	−3.65880
$189$	0	0
$190$	−633160.	−1.27242
$191$	−94906.9	−0.188241	−0.0941205	−	0.995561i	$-0.530004\pi$
$191$	−94906.9	−0.188241	−0.0941205	+	0.995561i	$0.530004\pi$
$192$	0	0
$193$	567496.	1.09665	0.548327	−	0.836264i	$-0.315265\pi$
$193$	567496.	1.09665	0.548327	+	0.836264i	$0.315265\pi$
$194$	1.23274e6	2.35161
$195$	0	0
$196$	−502384.	−0.934106
$197$	−307151.	−0.563880	−0.281940	−	0.959432i	$-0.590978\pi$
$197$	−307151.	−0.563880	−0.281940	+	0.959432i	$0.590978\pi$
$198$	0	0
$199$	992525.	1.77668	0.888339	−	0.459187i	$-0.151859\pi$
$199$	992525.	1.77668	0.888339	+	0.459187i	$0.151859\pi$
$200$	−290627.	−0.513761
$201$	0	0
$202$	−1.23871e6	−2.13596
$203$	−577023.	−0.982773
$204$	0	0
$205$	−481628.	−0.800436
$206$	−1.37614e6	−2.25942
$207$	0	0
$208$	1.10934e6	1.77790
$209$	294041.	0.465632
$210$	0	0
$211$	1.01044e6	1.56244	0.781219	−	0.624256i	$-0.214598\pi$
$211$	1.01044e6	1.56244	0.781219	+	0.624256i	$0.214598\pi$
$212$	983514.	1.50294
$213$	0	0
$214$	2.01401e6	3.00627
$215$	52005.1	0.0767273
$216$	0	0
$217$	582725.	0.840068
$218$	−430831.	−0.613996
$219$	0	0
$220$	231768.	0.322847
$221$	−728267.	−1.00302
$222$	0	0
$223$	890984.	1.19980	0.599899	−	0.800076i	$-0.295208\pi$
$223$	890984.	1.19980	0.599899	+	0.800076i	$0.295208\pi$
$224$	1.02004e6	1.35830
$225$	0	0
$226$	1.17615e6	1.53176
$227$	485648.	0.625542	0.312771	−	0.949828i	$-0.398743\pi$
$227$	485648.	0.625542	0.312771	+	0.949828i	$0.398743\pi$
$228$	0	0
$229$	755216.	0.951661	0.475831	−	0.879537i	$-0.342148\pi$
$229$	755216.	0.951661	0.475831	+	0.879537i	$0.342148\pi$
$230$	−721231.	−0.898990
$231$	0	0
$232$	−2.65026e6	−3.23272
$233$	885853.	1.06899	0.534493	−	0.845173i	$-0.320503\pi$
$233$	885853.	1.06899	0.534493	+	0.845173i	$0.320503\pi$
$234$	0	0
$235$	578556.	0.683401
$236$	−2.41715e6	−2.82503
$237$	0	0
$238$	−1.65863e6	−1.89805
$239$	−1.12491e6	−1.27386	−0.636930	−	0.770921i	$-0.719796\pi$
$239$	−1.12491e6	−1.27386	−0.636930	+	0.770921i	$0.719796\pi$
$240$	0	0
$241$	865147.	0.959506	0.479753	−	0.877404i	$-0.340726\pi$
$241$	865147.	0.959506	0.479753	+	0.877404i	$0.340726\pi$
$242$	−152588.	−0.167487
$243$	0	0
$244$	1.52613e6	1.64103
$245$	163926.	0.174475
$246$	0	0
$247$	−1.12583e6	−1.17417
$248$	2.67645e6	2.76331
$249$	0	0
$250$	162843.	0.164786
$251$	1.72537e6	1.72861	0.864307	−	0.502964i	$-0.167757\pi$
$251$	1.72537e6	1.72861	0.864307	+	0.502964i	$0.167757\pi$
$252$	0	0
$253$	334942.	0.328979
$254$	1.57233e6	1.52918
$255$	0	0
$256$	−1.18571e6	−1.13078
$257$	−1.19261e6	−1.12633	−0.563164	−	0.826345i	$-0.690416\pi$
$257$	−1.19261e6	−1.12633	−0.563164	+	0.826345i	$0.690416\pi$
$258$	0	0
$259$	278765.	0.258219
$260$	−887401.	−0.814117
$261$	0	0
$262$	1.88223e6	1.69402
$263$	442628.	0.394593	0.197297	−	0.980344i	$-0.436784\pi$
$263$	442628.	0.394593	0.197297	+	0.980344i	$0.436784\pi$
$264$	0	0
$265$	−320917.	−0.280723
$266$	−2.56410e6	−2.22193
$267$	0	0
$268$	−4.67280e6	−3.97411
$269$	−380384.	−0.320510	−0.160255	−	0.987076i	$-0.551232\pi$
$269$	−380384.	−0.320510	−0.160255	+	0.987076i	$0.551232\pi$
$270$	0	0
$271$	−1.28189e6	−1.06029	−0.530147	−	0.847906i	$-0.677863\pi$
$271$	−1.28189e6	−1.06029	−0.530147	+	0.847906i	$0.677863\pi$
$272$	−3.76403e6	−3.08482
$273$	0	0
$274$	−784843.	−0.631548
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	991944.	0.776762	0.388381	−	0.921499i	$-0.373034\pi$
$277$	991944.	0.776762	0.388381	+	0.921499i	$0.373034\pi$
$278$	−600086.	−0.465695
$279$	0	0
$280$	−1.17695e6	−0.897144
$281$	−1.80045e6	−1.36024	−0.680120	−	0.733101i	$-0.738072\pi$
$281$	−1.80045e6	−1.36024	−0.680120	+	0.733101i	$0.738072\pi$
$282$	0	0
$283$	16392.7	0.0121670	0.00608351	−	0.999981i	$-0.498064\pi$
$283$	16392.7	0.0121670	0.00608351	+	0.999981i	$0.498064\pi$
$284$	−1.24822e6	−0.918323
$285$	0	0
$286$	584234.	0.422349
$287$	−1.95044e6	−1.39774
$288$	0	0
$289$	1.05118e6	0.740340
$290$	1.48499e6	1.03688
$291$	0	0
$292$	−2.28949e6	−1.57138
$293$	−1.09513e6	−0.745244	−0.372622	−	0.927983i	$-0.621541\pi$
$293$	−1.09513e6	−0.745244	−0.372622	+	0.927983i	$0.621541\pi$
$294$	0	0
$295$	788706.	0.527667
$296$	1.28036e6	0.849384
$297$	0	0
$298$	−1.64186e6	−1.07101
$299$	−1.28244e6	−0.829580
$300$	0	0
$301$	210604.	0.133983
$302$	4.17953e6	2.63700
$303$	0	0
$304$	−5.81884e6	−3.61121
$305$	−497971.	−0.306517
$306$	0	0
$307$	1.75898e6	1.06516	0.532582	−	0.846379i	$-0.321222\pi$
$307$	1.75898e6	1.06516	0.532582	+	0.846379i	$0.321222\pi$
$308$	938588.	0.563765
$309$	0	0
$310$	−1.49966e6	−0.886316
$311$	−991121.	−0.581066	−0.290533	−	0.956865i	$-0.593833\pi$
$311$	−991121.	−0.581066	−0.290533	+	0.956865i	$0.593833\pi$
$312$	0	0
$313$	103222.	0.0595539	0.0297770	−	0.999557i	$-0.490520\pi$
$313$	103222.	0.0595539	0.0297770	+	0.999557i	$0.490520\pi$
$314$	2.72733e6	1.56104
$315$	0	0
$316$	−658964.	−0.371231
$317$	3.43326e6	1.91893	0.959464	−	0.281832i	$-0.0909422\pi$
$317$	3.43326e6	1.91893	0.959464	+	0.281832i	$0.0909422\pi$
$318$	0	0
$319$	−689632.	−0.379438
$320$	−709500.	−0.387327
$321$	0	0
$322$	−2.92076e6	−1.56984
$323$	3.81999e6	2.03731
$324$	0	0
$325$	289555.	0.152063
$326$	−4.69632e6	−2.44745
$327$	0	0
$328$	−8.95834e6	−4.59772
$329$	2.34297e6	1.19337
$330$	0	0
$331$	−2.61402e6	−1.31141	−0.655706	−	0.755017i	$-0.727629\pi$
$331$	−2.61402e6	−1.31141	−0.655706	+	0.755017i	$0.727629\pi$
$332$	−3.20965e6	−1.59813
$333$	0	0
$334$	−2.46781e6	−1.21045
$335$	1.52472e6	0.742296
$336$	0	0
$337$	214721.	0.102991	0.0514955	−	0.998673i	$-0.483601\pi$
$337$	214721.	0.102991	0.0514955	+	0.998673i	$0.483601\pi$
$338$	1.63267e6	0.777333
$339$	0	0
$340$	3.01098e6	1.41257
$341$	696447.	0.324341
$342$	0	0
$343$	2.36542e6	1.08561
$344$	967302.	0.440723
$345$	0	0
$346$	−2.42415e6	−1.08860
$347$	4.41557e6	1.96862	0.984312	−	0.176435i	$-0.0564564\pi$
$347$	4.41557e6	1.96862	0.984312	+	0.176435i	$0.0564564\pi$
$348$	0	0
$349$	3.80874e6	1.67386	0.836928	−	0.547313i	$-0.184349\pi$
$349$	3.80874e6	1.67386	0.836928	+	0.547313i	$0.184349\pi$
$350$	659464.	0.287754
$351$	0	0
$352$	1.21910e6	0.524425
$353$	−3.03953e6	−1.29829	−0.649143	−	0.760666i	$-0.724872\pi$
$353$	−3.03953e6	−1.29829	−0.649143	+	0.760666i	$0.724872\pi$
$354$	0	0
$355$	407289.	0.171527
$356$	2.34804e6	0.981929
$357$	0	0
$358$	2.99406e6	1.23468
$359$	2.24731e6	0.920293	0.460147	−	0.887843i	$-0.347797\pi$
$359$	2.24731e6	0.920293	0.460147	+	0.887843i	$0.347797\pi$
$360$	0	0
$361$	3.42926e6	1.38494
$362$	−6.73464e6	−2.70112
$363$	0	0
$364$	−3.59369e6	−1.42163
$365$	747050.	0.293506
$366$	0	0
$367$	−694347.	−0.269098	−0.134549	−	0.990907i	$-0.542959\pi$
$367$	−694347.	−0.269098	−0.134549	+	0.990907i	$0.542959\pi$
$368$	−6.62823e6	−2.55140
$369$	0	0
$370$	−717410.	−0.272435
$371$	−1.29961e6	−0.490206
$372$	0	0
$373$	−785602.	−0.292368	−0.146184	−	0.989257i	$-0.546699\pi$
$373$	−785602.	−0.292368	−0.146184	+	0.989257i	$0.546699\pi$
$374$	−1.98232e6	−0.732817
$375$	0	0
$376$	1.07612e7	3.92547
$377$	2.64049e6	0.956821
$378$	0	0
$379$	3.55628e6	1.27174	0.635870	−	0.771797i	$-0.280642\pi$
$379$	3.55628e6	1.27174	0.635870	+	0.771797i	$0.280642\pi$
$380$	4.65470e6	1.65361
$381$	0	0
$382$	989118.	0.346808
$383$	−4.80866e6	−1.67505	−0.837523	−	0.546402i	$-0.815997\pi$
$383$	−4.80866e6	−1.67505	−0.837523	+	0.546402i	$0.815997\pi$
$384$	0	0
$385$	−306257.	−0.105302
$386$	−5.91443e6	−2.02043
$387$	0	0
$388$	−9.06252e6	−3.05611
$389$	2.38186e6	0.798074	0.399037	−	0.916935i	$-0.369345\pi$
$389$	2.38186e6	0.798074	0.399037	+	0.916935i	$0.369345\pi$
$390$	0	0
$391$	4.35135e6	1.43940
$392$	3.04905e6	1.00219
$393$	0	0
$394$	3.20112e6	1.03887
$395$	215017.	0.0693395
$396$	0	0
$397$	−5.40097e6	−1.71987	−0.859934	−	0.510405i	$-0.829496\pi$
$397$	−5.40097e6	−1.71987	−0.859934	+	0.510405i	$0.829496\pi$
$398$	−1.03441e7	−3.27329
$399$	0	0
$400$	1.49656e6	0.467674
$401$	3.50477e6	1.08842	0.544212	−	0.838947i	$-0.316829\pi$
$401$	3.50477e6	1.08842	0.544212	+	0.838947i	$0.316829\pi$
$402$	0	0
$403$	−2.66658e6	−0.817884
$404$	9.10645e6	2.77585
$405$	0	0
$406$	6.01372e6	1.81062
$407$	333167.	0.0996957
$408$	0	0
$409$	−1.39023e6	−0.410939	−0.205470	−	0.978664i	$-0.565872\pi$
$409$	−1.39023e6	−0.410939	−0.205470	+	0.978664i	$0.565872\pi$
$410$	5.01951e6	1.47469
$411$	0	0
$412$	1.01168e7	2.93629
$413$	3.19401e6	0.921427
$414$	0	0
$415$	1.04730e6	0.298504
$416$	−4.66773e6	−1.32243
$417$	0	0
$418$	−3.06449e6	−0.857863
$419$	−5.07746e6	−1.41290	−0.706450	−	0.707763i	$-0.749705\pi$
$419$	−5.07746e6	−1.41290	−0.706450	+	0.707763i	$0.749705\pi$
$420$	0	0
$421$	−2.49864e6	−0.687065	−0.343532	−	0.939141i	$-0.611624\pi$
$421$	−2.49864e6	−0.687065	−0.343532	+	0.939141i	$0.611624\pi$
$422$	−1.05307e7	−2.87858
$423$	0	0
$424$	−5.96909e6	−1.61248
$425$	−982470.	−0.263844
$426$	0	0
$427$	−2.01662e6	−0.535248
$428$	−1.48061e7	−3.90689
$429$	0	0
$430$	−541996.	−0.141359
$431$	6.56337e6	1.70190	0.850949	−	0.525248i	$-0.176027\pi$
$431$	6.56337e6	1.70190	0.850949	+	0.525248i	$0.176027\pi$
$432$	0	0
$433$	682000.	0.174809	0.0874047	−	0.996173i	$-0.472143\pi$
$433$	682000.	0.174809	0.0874047	+	0.996173i	$0.472143\pi$
$434$	−6.07315e6	−1.54771
$435$	0	0
$436$	3.16727e6	0.797938
$437$	6.72679e6	1.68502
$438$	0	0
$439$	−6.01759e6	−1.49026	−0.745129	−	0.666920i	$-0.767612\pi$
$439$	−6.01759e6	−1.49026	−0.745129	+	0.666920i	$0.767612\pi$
$440$	−1.40664e6	−0.346378
$441$	0	0
$442$	7.58999e6	1.84793
$443$	−2.47898e6	−0.600154	−0.300077	−	0.953915i	$-0.597012\pi$
$443$	−2.47898e6	−0.600154	−0.300077	+	0.953915i	$0.597012\pi$
$444$	0	0
$445$	−766155.	−0.183407
$446$	−9.28582e6	−2.21046
$447$	0	0
$448$	−2.87325e6	−0.676361
$449$	−3.30403e6	−0.773444	−0.386722	−	0.922196i	$-0.626393\pi$
$449$	−3.30403e6	−0.773444	−0.386722	+	0.922196i	$0.626393\pi$
$450$	0	0
$451$	−2.33108e6	−0.539654
$452$	−8.64650e6	−1.99065
$453$	0	0
$454$	−5.06141e6	−1.15248
$455$	1.17261e6	0.265536
$456$	0	0
$457$	−2.24767e6	−0.503434	−0.251717	−	0.967801i	$-0.580995\pi$
$457$	−2.24767e6	−0.503434	−0.251717	+	0.967801i	$0.580995\pi$
$458$	−7.87084e6	−1.75331
$459$	0	0
$460$	5.30216e6	1.16831
$461$	4.68058e6	1.02576	0.512882	−	0.858459i	$-0.328578\pi$
$461$	4.68058e6	1.02576	0.512882	+	0.858459i	$0.328578\pi$
$462$	0	0
$463$	−1.44106e6	−0.312413	−0.156206	−	0.987724i	$-0.549927\pi$
$463$	−1.44106e6	−0.312413	−0.156206	+	0.987724i	$0.549927\pi$
$464$	1.36473e7	2.94273
$465$	0	0
$466$	−9.23234e6	−1.96946
$467$	−2.33003e6	−0.494390	−0.247195	−	0.968966i	$-0.579509\pi$
$467$	−2.33003e6	−0.494390	−0.247195	+	0.968966i	$0.579509\pi$
$468$	0	0
$469$	6.17462e6	1.29622
$470$	−6.02970e6	−1.25907
$471$	0	0
$472$	1.46700e7	3.03093
$473$	251705.	0.0517295
$474$	0	0
$475$	−1.51881e6	−0.308865
$476$	1.21935e7	2.46667
$477$	0	0
$478$	1.17238e7	2.34691
$479$	−1.06772e6	−0.212627	−0.106313	−	0.994333i	$-0.533905\pi$
$479$	−1.06772e6	−0.212627	−0.106313	+	0.994333i	$0.533905\pi$
$480$	0	0
$481$	−1.27564e6	−0.251400
$482$	−9.01655e6	−1.76776
$483$	0	0
$484$	1.12176e6	0.217664
$485$	2.95706e6	0.570829
$486$	0	0
$487$	4.64634e6	0.887745	0.443872	−	0.896090i	$-0.353604\pi$
$487$	4.64634e6	0.887745	0.443872	+	0.896090i	$0.353604\pi$
$488$	−9.26232e6	−1.76064
$489$	0	0
$490$	−1.70843e6	−0.321446
$491$	9.98992e6	1.87007	0.935036	−	0.354553i	$-0.115367\pi$
$491$	9.98992e6	1.87007	0.935036	+	0.354553i	$0.115367\pi$
$492$	0	0
$493$	−8.95925e6	−1.66018
$494$	1.17334e7	2.16325
$495$	0	0
$496$	−1.37821e7	−2.51543
$497$	1.64939e6	0.299525
$498$	0	0
$499$	−1.31428e6	−0.236286	−0.118143	−	0.992997i	$-0.537694\pi$
$499$	−1.31428e6	−0.236286	−0.118143	+	0.992997i	$0.537694\pi$
$500$	−1.19715e6	−0.214153
$501$	0	0
$502$	−1.79818e7	−3.18473
$503$	1.07177e7	1.88879	0.944393	−	0.328818i	$-0.106650\pi$
$503$	1.07177e7	1.88879	0.944393	+	0.328818i	$0.106650\pi$
$504$	0	0
$505$	−2.97140e6	−0.518481
$506$	−3.49076e6	−0.606099
$507$	0	0
$508$	−1.15590e7	−1.98729
$509$	5.54239e6	0.948205	0.474103	−	0.880470i	$-0.342773\pi$
$509$	5.54239e6	0.948205	0.474103	+	0.880470i	$0.342773\pi$
$510$	0	0
$511$	3.02531e6	0.512529
$512$	1.15053e7	1.93964
$513$	0	0
$514$	1.24293e7	2.07511
$515$	−3.30107e6	−0.548449
$516$	0	0
$517$	2.80021e6	0.460749
$518$	−2.90528e6	−0.475734
$519$	0	0
$520$	5.38577e6	0.873452
$521$	−5.92275e6	−0.955937	−0.477969	−	0.878377i	$-0.658627\pi$
$521$	−5.92275e6	−0.955937	−0.477969	+	0.878377i	$0.658627\pi$
$522$	0	0
$523$	8.68952e6	1.38913	0.694563	−	0.719432i	$-0.255598\pi$
$523$	8.68952e6	1.38913	0.694563	+	0.719432i	$0.255598\pi$
$524$	−1.38373e7	−2.20152
$525$	0	0
$526$	−4.61306e6	−0.726983
$527$	9.04778e6	1.41911
$528$	0	0
$529$	1.22613e6	0.190501
$530$	3.34459e6	0.517193
$531$	0	0
$532$	1.88501e7	2.88758
$533$	8.92530e6	1.36083
$534$	0	0
$535$	4.83116e6	0.729738
$536$	2.83599e7	4.26376
$537$	0	0
$538$	3.96435e6	0.590496
$539$	793402.	0.117631
$540$	0	0
$541$	−5.42578e6	−0.797020	−0.398510	−	0.917164i	$-0.630472\pi$
$541$	−5.42578e6	−0.797020	−0.398510	+	0.917164i	$0.630472\pi$
$542$	1.33598e7	1.95345
$543$	0	0
$544$	1.58378e7	2.29455
$545$	−1.03347e6	−0.149041
$546$	0	0
$547$	4.94660e6	0.706868	0.353434	−	0.935459i	$-0.385014\pi$
$547$	4.94660e6	0.706868	0.353434	+	0.935459i	$0.385014\pi$
$548$	5.76981e6	0.820748
$549$	0	0
$550$	788162.	0.111099
$551$	−1.38502e7	−1.94346
$552$	0	0
$553$	870752.	0.121083
$554$	−1.03380e7	−1.43108
$555$	0	0
$556$	4.41156e6	0.605209
$557$	−1.48942e6	−0.203413	−0.101706	−	0.994814i	$-0.532430\pi$
$557$	−1.48942e6	−0.203413	−0.101706	+	0.994814i	$0.532430\pi$
$558$	0	0
$559$	−963734.	−0.130445
$560$	6.06058e6	0.816666
$561$	0	0
$562$	1.87642e7	2.50605
$563$	−1.08095e7	−1.43725	−0.718627	−	0.695395i	$-0.755229\pi$
$563$	−1.08095e7	−1.43725	−0.718627	+	0.695395i	$0.755229\pi$
$564$	0	0
$565$	2.82132e6	0.371818
$566$	−170844.	−0.0224161
$567$	0	0
$568$	7.57564e6	0.985254
$569$	3.54847e6	0.459473	0.229737	−	0.973253i	$-0.426214\pi$
$569$	3.54847e6	0.459473	0.229737	+	0.973253i	$0.426214\pi$
$570$	0	0
$571$	287930.	0.0369570	0.0184785	−	0.999829i	$-0.494118\pi$
$571$	287930.	0.0369570	0.0184785	+	0.999829i	$0.494118\pi$
$572$	−4.29502e6	−0.548877
$573$	0	0
$574$	2.03274e7	2.57515
$575$	−1.73007e6	−0.218220
$576$	0	0
$577$	−1.16438e7	−1.45599	−0.727993	−	0.685585i	$-0.759546\pi$
$577$	−1.16438e7	−1.45599	−0.727993	+	0.685585i	$0.759546\pi$
$578$	−1.09553e7	−1.36397
$579$	0	0
$580$	−1.09169e7	−1.34751
$581$	4.24122e6	0.521255
$582$	0	0
$583$	−1.55324e6	−0.189263
$584$	1.38952e7	1.68591
$585$	0	0
$586$	1.14135e7	1.37301
$587$	1.08095e7	1.29482	0.647409	−	0.762143i	$-0.275853\pi$
$587$	1.08095e7	1.29482	0.647409	+	0.762143i	$0.275853\pi$
$588$	0	0
$589$	1.39870e7	1.66126
$590$	−8.21987e6	−0.972154
$591$	0	0
$592$	−6.59311e6	−0.773190
$593$	−1.33217e7	−1.55569	−0.777846	−	0.628455i	$-0.783688\pi$
$593$	−1.33217e7	−1.55569	−0.777846	+	0.628455i	$0.783688\pi$
$594$	0	0
$595$	−3.97869e6	−0.460732
$596$	1.20702e7	1.39187
$597$	0	0
$598$	1.33655e7	1.52839
$599$	−1.41822e7	−1.61501	−0.807506	−	0.589859i	$-0.799183\pi$
$599$	−1.41822e7	−1.61501	−0.807506	+	0.589859i	$0.799183\pi$
$600$	0	0
$601$	−1.28912e7	−1.45582	−0.727909	−	0.685674i	$-0.759508\pi$
$601$	−1.28912e7	−1.45582	−0.727909	+	0.685674i	$0.759508\pi$
$602$	−2.19491e6	−0.246846
$603$	0	0
$604$	−3.07260e7	−3.42700
$605$	−366025.	−0.0406558
$606$	0	0
$607$	−2.94467e6	−0.324388	−0.162194	−	0.986759i	$-0.551857\pi$
$607$	−2.94467e6	−0.324388	−0.162194	+	0.986759i	$0.551857\pi$
$608$	2.44837e7	2.68608
$609$	0	0
$610$	5.18984e6	0.564715
$611$	−1.07215e7	−1.16186
$612$	0	0
$613$	−1.43701e7	−1.54458	−0.772289	−	0.635271i	$-0.780888\pi$
$613$	−1.43701e7	−1.54458	−0.772289	+	0.635271i	$0.780888\pi$
$614$	−1.83321e7	−1.96242
$615$	0	0
$616$	−5.69643e6	−0.604854
$617$	1.65923e7	1.75466	0.877330	−	0.479887i	$-0.159322\pi$
$617$	1.65923e7	1.75466	0.877330	+	0.479887i	$0.159322\pi$
$618$	0	0
$619$	1.21251e7	1.27192	0.635960	−	0.771722i	$-0.280604\pi$
$619$	1.21251e7	1.27192	0.635960	+	0.771722i	$0.280604\pi$
$620$	1.10248e7	1.15184
$621$	0	0
$622$	1.03294e7	1.07053
$623$	−3.10269e6	−0.320271
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.07577e6	−0.109720
$627$	0	0
$628$	−2.00501e7	−2.02870
$629$	4.32829e6	0.436204
$630$	0	0
$631$	1.13432e7	1.13413	0.567066	−	0.823673i	$-0.308079\pi$
$631$	1.13432e7	1.13413	0.567066	+	0.823673i	$0.308079\pi$
$632$	3.99935e6	0.398288
$633$	0	0
$634$	−3.57814e7	−3.53536
$635$	3.77166e6	0.371192
$636$	0	0
$637$	−3.03780e6	−0.296627
$638$	7.18733e6	0.699063
$639$	0	0
$640$	−665787.	−0.0642518
$641$	−5.38122e6	−0.517292	−0.258646	−	0.965972i	$-0.583276\pi$
$641$	−5.38122e6	−0.517292	−0.258646	+	0.965972i	$0.583276\pi$
$642$	0	0
$643$	1.32043e7	1.25947	0.629736	−	0.776809i	$-0.283163\pi$
$643$	1.32043e7	1.25947	0.629736	+	0.776809i	$0.283163\pi$
$644$	2.14721e7	2.04014
$645$	0	0
$646$	−3.98119e7	−3.75346
$647$	1.31287e7	1.23300	0.616499	−	0.787356i	$-0.288550\pi$
$647$	1.31287e7	1.23300	0.616499	+	0.787356i	$0.288550\pi$
$648$	0	0
$649$	3.81733e6	0.355753
$650$	−3.01774e6	−0.280155
$651$	0	0
$652$	3.45252e7	3.18066
$653$	2.34573e6	0.215275	0.107638	−	0.994190i	$-0.465671\pi$
$653$	2.34573e6	0.215275	0.107638	+	0.994190i	$0.465671\pi$
$654$	0	0
$655$	4.51505e6	0.411206
$656$	4.61301e7	4.18529
$657$	0	0
$658$	−2.44184e7	−2.19863
$659$	−7.14853e6	−0.641215	−0.320607	−	0.947212i	$-0.603887\pi$
$659$	−7.14853e6	−0.641215	−0.320607	+	0.947212i	$0.603887\pi$
$660$	0	0
$661$	−2.49201e6	−0.221843	−0.110922	−	0.993829i	$-0.535380\pi$
$661$	−2.49201e6	−0.221843	−0.110922	+	0.993829i	$0.535380\pi$
$662$	2.72432e7	2.41609
$663$	0	0
$664$	1.94799e7	1.71461
$665$	−6.15070e6	−0.539349
$666$	0	0
$667$	−1.57767e7	−1.37310
$668$	1.81422e7	1.57307
$669$	0	0
$670$	−1.58906e7	−1.36758
$671$	−2.41018e6	−0.206654
$672$	0	0
$673$	−1.14320e6	−0.0972938	−0.0486469	−	0.998816i	$-0.515491\pi$
$673$	−1.14320e6	−0.0972938	−0.0486469	+	0.998816i	$0.515491\pi$
$674$	−2.23781e6	−0.189747
$675$	0	0
$676$	−1.20027e7	−1.01021
$677$	−2.03792e7	−1.70889	−0.854447	−	0.519539i	$-0.826104\pi$
$677$	−2.03792e7	−1.70889	−0.854447	+	0.519539i	$0.826104\pi$
$678$	0	0
$679$	1.19752e7	0.996798
$680$	−1.82741e7	−1.51553
$681$	0	0
$682$	−7.25835e6	−0.597554
$683$	−1.02514e7	−0.840874	−0.420437	−	0.907322i	$-0.638123\pi$
$683$	−1.02514e7	−0.840874	−0.420437	+	0.907322i	$0.638123\pi$
$684$	0	0
$685$	−1.88266e6	−0.153301
$686$	−2.46524e7	−2.00009
$687$	0	0
$688$	−4.98103e6	−0.401188
$689$	5.94708e6	0.477261
$690$	0	0
$691$	1.60534e7	1.27901	0.639503	−	0.768788i	$-0.279140\pi$
$691$	1.60534e7	1.27901	0.639503	+	0.768788i	$0.279140\pi$
$692$	1.78212e7	1.41473
$693$	0	0
$694$	−4.60189e7	−3.62692
$695$	−1.43947e6	−0.113042
$696$	0	0
$697$	−3.02838e7	−2.36118
$698$	−3.96946e7	−3.08385
$699$	0	0
$700$	−4.84808e6	−0.373959
$701$	−1.67341e6	−0.128620	−0.0643100	−	0.997930i	$-0.520485\pi$
$701$	−1.67341e6	−0.128620	−0.0643100	+	0.997930i	$0.520485\pi$
$702$	0	0
$703$	6.69114e6	0.510637
$704$	−3.43398e6	−0.261136
$705$	0	0
$706$	3.16780e7	2.39191
$707$	−1.20332e7	−0.905385
$708$	0	0
$709$	1.30861e6	0.0977674	0.0488837	−	0.998804i	$-0.484434\pi$
$709$	1.30861e6	0.0977674	0.0488837	+	0.998804i	$0.484434\pi$
$710$	−4.24476e6	−0.316015
$711$	0	0
$712$	−1.42506e7	−1.05350
$713$	1.59326e7	1.17372
$714$	0	0
$715$	1.40145e6	0.102521
$716$	−2.20110e7	−1.60456
$717$	0	0
$718$	−2.34214e7	−1.69551
$719$	−283809.	−0.0204740	−0.0102370	−	0.999948i	$-0.503259\pi$
$719$	−283809.	−0.0204740	−0.0102370	+	0.999948i	$0.503259\pi$
$720$	0	0
$721$	−1.33683e7	−0.957717
$722$	−3.57396e7	−2.55157
$723$	0	0
$724$	4.95100e7	3.51032
$725$	3.56215e6	0.251691
$726$	0	0
$727$	−1.55937e6	−0.109424	−0.0547121	−	0.998502i	$-0.517424\pi$
$727$	−1.55937e6	−0.109424	−0.0547121	+	0.998502i	$0.517424\pi$
$728$	2.18107e7	1.52525
$729$	0	0
$730$	−7.78573e6	−0.540745
$731$	3.26998e6	0.226335
$732$	0	0
$733$	8.61557e6	0.592276	0.296138	−	0.955145i	$-0.404301\pi$
$733$	8.61557e6	0.592276	0.296138	+	0.955145i	$0.404301\pi$
$734$	7.23646e6	0.495777
$735$	0	0
$736$	2.78894e7	1.89777
$737$	7.37963e6	0.500456
$738$	0	0
$739$	−1.93631e7	−1.30426	−0.652128	−	0.758109i	$-0.726124\pi$
$739$	−1.93631e7	−1.30426	−0.652128	+	0.758109i	$0.726124\pi$
$740$	5.27407e6	0.354051
$741$	0	0
$742$	1.35445e7	0.903137
$743$	9.31274e6	0.618878	0.309439	−	0.950919i	$-0.399859\pi$
$743$	9.31274e6	0.618878	0.309439	+	0.950919i	$0.399859\pi$
$744$	0	0
$745$	−3.93846e6	−0.259977
$746$	8.18752e6	0.538649
$747$	0	0
$748$	1.45731e7	0.952356
$749$	1.95647e7	1.27429
$750$	0	0
$751$	5.82007e6	0.376555	0.188277	−	0.982116i	$-0.439710\pi$
$751$	5.82007e6	0.376555	0.188277	+	0.982116i	$0.439710\pi$
$752$	−5.54139e7	−3.57334
$753$	0	0
$754$	−2.75191e7	−1.76281
$755$	1.00258e7	0.640104
$756$	0	0
$757$	1.66702e7	1.05730	0.528652	−	0.848839i	$-0.322698\pi$
$757$	1.66702e7	1.05730	0.528652	+	0.848839i	$0.322698\pi$
$758$	−3.70635e7	−2.34300
$759$	0	0
$760$	−2.82500e7	−1.77413
$761$	−2.58124e7	−1.61573	−0.807863	−	0.589371i	$-0.799376\pi$
$761$	−2.58124e7	−1.61573	−0.807863	+	0.589371i	$0.799376\pi$
$762$	0	0
$763$	−4.18522e6	−0.260259
$764$	−7.27154e6	−0.450706
$765$	0	0
$766$	5.01157e7	3.08604
$767$	−1.46159e7	−0.897094
$768$	0	0
$769$	−1.08680e7	−0.662725	−0.331362	−	0.943504i	$-0.607508\pi$
$769$	−1.08680e7	−0.662725	−0.331362	+	0.943504i	$0.607508\pi$
$770$	3.19181e6	0.194004
$771$	0	0
$772$	4.34802e7	2.62572
$773$	2.19994e6	0.132422	0.0662112	−	0.997806i	$-0.478909\pi$
$773$	2.19994e6	0.132422	0.0662112	+	0.997806i	$0.478909\pi$
$774$	0	0
$775$	−3.59735e6	−0.215144
$776$	5.50018e7	3.27885
$777$	0	0
$778$	−2.48237e7	−1.47034
$779$	−4.68160e7	−2.76408
$780$	0	0
$781$	1.97128e6	0.115643
$782$	−4.53497e7	−2.65190
$783$	0	0
$784$	−1.57008e7	−0.912286
$785$	6.54226e6	0.378925
$786$	0	0
$787$	−6.16690e6	−0.354920	−0.177460	−	0.984128i	$-0.556788\pi$
$787$	−6.16690e6	−0.354920	−0.177460	+	0.984128i	$0.556788\pi$
$788$	−2.35332e7	−1.35010
$789$	0	0
$790$	−2.24090e6	−0.127748
$791$	1.14254e7	0.649279
$792$	0	0
$793$	9.22817e6	0.521114
$794$	5.62887e7	3.16862
$795$	0	0
$796$	7.60449e7	4.25390
$797$	−1.77670e7	−0.990757	−0.495379	−	0.868677i	$-0.664971\pi$
$797$	−1.77670e7	−0.990757	−0.495379	+	0.868677i	$0.664971\pi$
$798$	0	0
$799$	3.63785e7	2.01594
$800$	−6.29701e6	−0.347864
$801$	0	0
$802$	−3.65266e7	−2.00527
$803$	3.61572e6	0.197882
$804$	0	0
$805$	−7.00625e6	−0.381062
$806$	2.77910e7	1.50684
$807$	0	0
$808$	−5.52684e7	−2.97816
$809$	1.48896e7	0.799855	0.399928	−	0.916547i	$-0.369035\pi$
$809$	1.48896e7	0.799855	0.399928	+	0.916547i	$0.369035\pi$
$810$	0	0
$811$	−3.38022e7	−1.80465	−0.902326	−	0.431055i	$-0.858142\pi$
$811$	−3.38022e7	−1.80465	−0.902326	+	0.431055i	$0.858142\pi$
$812$	−4.42101e7	−2.35305
$813$	0	0
$814$	−3.47226e6	−0.183676
$815$	−1.12654e7	−0.594092
$816$	0	0
$817$	5.05509e6	0.264956
$818$	1.44889e7	0.757099
$819$	0	0
$820$	−3.69012e7	−1.91648
$821$	1.74801e6	0.0905076	0.0452538	−	0.998976i	$-0.485590\pi$
$821$	1.74801e6	0.0905076	0.0452538	+	0.998976i	$0.485590\pi$
$822$	0	0
$823$	−4.28201e6	−0.220368	−0.110184	−	0.993911i	$-0.535144\pi$
$823$	−4.28201e6	−0.220368	−0.110184	+	0.993911i	$0.535144\pi$
$824$	−6.14003e7	−3.15030
$825$	0	0
$826$	−3.32879e7	−1.69760
$827$	2.32294e7	1.18107	0.590534	−	0.807013i	$-0.298917\pi$
$827$	2.32294e7	1.18107	0.590534	+	0.807013i	$0.298917\pi$
$828$	0	0
$829$	−3.07927e7	−1.55618	−0.778092	−	0.628150i	$-0.783812\pi$
$829$	−3.07927e7	−1.55618	−0.778092	+	0.628150i	$0.783812\pi$
$830$	−1.09149e7	−0.549952
$831$	0	0
$832$	1.31481e7	0.658500
$833$	1.03074e7	0.514677
$834$	0	0
$835$	−5.91973e6	−0.293823
$836$	2.25287e7	1.11486
$837$	0	0
$838$	5.29172e7	2.60307
$839$	6.70936e6	0.329061	0.164530	−	0.986372i	$-0.447389\pi$
$839$	6.70936e6	0.329061	0.164530	+	0.986372i	$0.447389\pi$
$840$	0	0
$841$	1.19725e7	0.583706
$842$	2.60407e7	1.26582
$843$	0	0
$844$	7.74172e7	3.74095
$845$	3.91642e6	0.188689
$846$	0	0
$847$	−1.48229e6	−0.0709943
$848$	3.07373e7	1.46783
$849$	0	0
$850$	1.02393e7	0.486096
$851$	7.62187e6	0.360776
$852$	0	0
$853$	2.27734e7	1.07166	0.535828	−	0.844327i	$-0.319999\pi$
$853$	2.27734e7	1.07166	0.535828	+	0.844327i	$0.319999\pi$
$854$	2.10172e7	0.986121
$855$	0	0
$856$	8.98603e7	4.19163
$857$	−4.12568e6	−0.191886	−0.0959431	−	0.995387i	$-0.530587\pi$
$857$	−4.12568e6	−0.191886	−0.0959431	+	0.995387i	$0.530587\pi$
$858$	0	0
$859$	2.66483e7	1.23221	0.616107	−	0.787663i	$-0.288709\pi$
$859$	2.66483e7	1.23221	0.616107	+	0.787663i	$0.288709\pi$
$860$	3.98450e6	0.183708
$861$	0	0
$862$	−6.84033e7	−3.13551
$863$	2.11753e7	0.967836	0.483918	−	0.875113i	$-0.339213\pi$
$863$	2.11753e7	0.967836	0.483918	+	0.875113i	$0.339213\pi$
$864$	0	0
$865$	−5.81499e6	−0.264246
$866$	−7.10779e6	−0.322062
$867$	0	0
$868$	4.46470e7	2.01137
$869$	1.04068e6	0.0467487
$870$	0	0
$871$	−2.82553e7	−1.26199
$872$	−1.92226e7	−0.856094
$873$	0	0
$874$	−7.01064e7	−3.10441
$875$	1.58191e6	0.0698491
$876$	0	0
$877$	−1.39029e7	−0.610388	−0.305194	−	0.952290i	$-0.598721\pi$
$877$	−1.39029e7	−0.610388	−0.305194	+	0.952290i	$0.598721\pi$
$878$	6.27152e7	2.74560
$879$	0	0
$880$	7.24334e6	0.315306
$881$	−2.04219e7	−0.886455	−0.443227	−	0.896409i	$-0.646167\pi$
$881$	−2.04219e7	−0.886455	−0.443227	+	0.896409i	$0.646167\pi$
$882$	0	0
$883$	−1.99749e7	−0.862151	−0.431075	−	0.902316i	$-0.641866\pi$
$883$	−1.99749e7	−0.862151	−0.431075	+	0.902316i	$0.641866\pi$
$884$	−5.57981e7	−2.40153
$885$	0	0
$886$	2.58358e7	1.10570
$887$	−1.78506e7	−0.761805	−0.380903	−	0.924615i	$-0.624387\pi$
$887$	−1.78506e7	−0.761805	−0.380903	+	0.924615i	$0.624387\pi$
$888$	0	0
$889$	1.52740e7	0.648186
$890$	7.98485e6	0.337903
$891$	0	0
$892$	6.82651e7	2.87267
$893$	5.62378e7	2.35993
$894$	0	0
$895$	7.18209e6	0.299705
$896$	−2.69623e6	−0.112198
$897$	0	0
$898$	3.44346e7	1.42496
$899$	−3.28046e7	−1.35374
$900$	0	0
$901$	−2.01786e7	−0.828094
$902$	2.42944e7	0.994238
$903$	0	0
$904$	5.24769e7	2.13573
$905$	−1.61549e7	−0.655667
$906$	0	0
$907$	3.08062e7	1.24343	0.621714	−	0.783245i	$-0.286437\pi$
$907$	3.08062e7	1.24343	0.621714	+	0.783245i	$0.286437\pi$
$908$	3.72092e7	1.49774
$909$	0	0
$910$	−1.22209e7	−0.489214
$911$	−2.08179e7	−0.831078	−0.415539	−	0.909575i	$-0.636407\pi$
$911$	−2.08179e7	−0.831078	−0.415539	+	0.909575i	$0.636407\pi$
$912$	0	0
$913$	5.06892e6	0.201251
$914$	2.34252e7	0.927508
$915$	0	0
$916$	5.78628e7	2.27856
$917$	1.82845e7	0.718060
$918$	0	0
$919$	−9.01768e6	−0.352214	−0.176107	−	0.984371i	$-0.556350\pi$
$919$	−9.01768e6	−0.352214	−0.176107	+	0.984371i	$0.556350\pi$
$920$	−3.21796e7	−1.25346
$921$	0	0
$922$	−4.87809e7	−1.88983
$923$	−7.54770e6	−0.291615
$924$	0	0
$925$	−1.72091e6	−0.0661307
$926$	1.50187e7	0.575578
$927$	0	0
$928$	−5.74231e7	−2.18885
$929$	−3.83351e7	−1.45733	−0.728665	−	0.684871i	$-0.759859\pi$
$929$	−3.83351e7	−1.45733	−0.728665	+	0.684871i	$0.759859\pi$
$930$	0	0
$931$	1.59342e7	0.602499
$932$	6.78719e7	2.55947
$933$	0	0
$934$	2.42835e7	0.910845
$935$	−4.75516e6	−0.177884
$936$	0	0
$937$	4.56705e7	1.69936	0.849682	−	0.527295i	$-0.176794\pi$
$937$	4.56705e7	1.69936	0.849682	+	0.527295i	$0.176794\pi$
$938$	−6.43517e7	−2.38810
$939$	0	0
$940$	4.43276e7	1.63627
$941$	3.18573e7	1.17283	0.586414	−	0.810011i	$-0.300539\pi$
$941$	3.18573e7	1.17283	0.586414	+	0.810011i	$0.300539\pi$
$942$	0	0
$943$	−5.33281e7	−1.95289
$944$	−7.55419e7	−2.75904
$945$	0	0
$946$	−2.62326e6	−0.0953045
$947$	4.42081e6	0.160187	0.0800933	−	0.996787i	$-0.474478\pi$
$947$	4.42081e6	0.160187	0.0800933	+	0.996787i	$0.474478\pi$
$948$	0	0
$949$	−1.38440e7	−0.498994
$950$	1.58290e7	0.569042
$951$	0	0
$952$	−7.40042e7	−2.64645
$953$	2.52087e7	0.899123	0.449561	−	0.893249i	$-0.351580\pi$
$953$	2.52087e7	0.899123	0.449561	+	0.893249i	$0.351580\pi$
$954$	0	0
$955$	2.37267e6	0.0841840
$956$	−8.61877e7	−3.05000
$957$	0	0
$958$	1.11277e7	0.391735
$959$	−7.62419e6	−0.267699
$960$	0	0
$961$	4.49961e6	0.157169
$962$	1.32947e7	0.463171
$963$	0	0
$964$	6.62855e7	2.29734
$965$	−1.41874e7	−0.490439
$966$	0	0
$967$	−3.09081e7	−1.06293	−0.531467	−	0.847079i	$-0.678359\pi$
$967$	−3.09081e7	−1.06293	−0.531467	+	0.847079i	$0.678359\pi$
$968$	−6.80811e6	−0.233528
$969$	0	0
$970$	−3.08184e7	−1.05167
$971$	5.14432e7	1.75098	0.875488	−	0.483240i	$-0.160540\pi$
$971$	5.14432e7	1.75098	0.875488	+	0.483240i	$0.160540\pi$
$972$	0	0
$973$	−5.82941e6	−0.197398
$974$	−4.84240e7	−1.63555
$975$	0	0
$976$	4.76955e7	1.60270
$977$	2.79815e7	0.937854	0.468927	−	0.883237i	$-0.344641\pi$
$977$	2.79815e7	0.937854	0.468927	+	0.883237i	$0.344641\pi$
$978$	0	0
$979$	−3.70819e6	−0.123653
$980$	1.25596e7	0.417745
$981$	0	0
$982$	−1.04115e8	−3.44535
$983$	3.94905e7	1.30349	0.651746	−	0.758437i	$-0.274037\pi$
$983$	3.94905e7	1.30349	0.651746	+	0.758437i	$0.274037\pi$
$984$	0	0
$985$	7.67878e6	0.252175
$986$	9.33731e7	3.05865
$987$	0	0
$988$	−8.62587e7	−2.81132
$989$	5.75825e6	0.187197
$990$	0	0
$991$	1.83336e7	0.593011	0.296506	−	0.955031i	$-0.404179\pi$
$991$	1.83336e7	0.593011	0.296506	+	0.955031i	$0.404179\pi$
$992$	5.79905e7	1.87102
$993$	0	0
$994$	−1.71899e7	−0.551834
$995$	−2.48131e7	−0.794555
$996$	0	0
$997$	3.33685e7	1.06316	0.531580	−	0.847008i	$-0.321599\pi$
$997$	3.33685e7	1.06316	0.531580	+	0.847008i	$0.321599\pi$
$998$	1.36974e7	0.435324
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.n.1.1		7
3.2	odd	2		165.6.a.h.1.7	✓	7
15.14	odd	2		825.6.a.n.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.h.1.7	✓	7	3.2	odd	2
495.6.a.n.1.1		7	1.1	even	1	trivial
825.6.a.n.1.1		7	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-10.4220 q^{2} +76.6176 q^{4} -25.0000 q^{5} -101.242 q^{7} -465.003 q^{8} +O(q^{10})\)	\(q-10.4220 q^{2} +76.6176 q^{4} -25.0000 q^{5} -101.242 q^{7} -465.003 q^{8} +260.549 q^{10} -121.000 q^{11} +463.289 q^{13} +1055.14 q^{14} +2394.49 q^{16} -1571.95 q^{17} -2430.09 q^{19} -1915.44 q^{20} +1261.06 q^{22} -2768.12 q^{23} +625.000 q^{25} -4828.38 q^{26} -7756.92 q^{28} +5699.44 q^{29} -5755.76 q^{31} -10075.2 q^{32} +16382.8 q^{34} +2531.05 q^{35} -2753.45 q^{37} +25326.4 q^{38} +11625.1 q^{40} +19265.1 q^{41} -2080.20 q^{43} -9270.73 q^{44} +28849.3 q^{46} -23142.2 q^{47} -6557.04 q^{49} -6513.73 q^{50} +35496.0 q^{52} +12836.7 q^{53} +3025.00 q^{55} +47077.9 q^{56} -59399.4 q^{58} -31548.2 q^{59} +19918.8 q^{61} +59986.4 q^{62} +28380.0 q^{64} -11582.2 q^{65} -60988.6 q^{67} -120439. q^{68} -26378.6 q^{70} -16291.6 q^{71} -29882.0 q^{73} +28696.4 q^{74} -186188. q^{76} +12250.3 q^{77} -8600.69 q^{79} -59862.3 q^{80} -200781. q^{82} -41891.9 q^{83} +39298.8 q^{85} +21679.8 q^{86} +56265.4 q^{88} +30646.2 q^{89} -46904.3 q^{91} -212086. q^{92} +241188. q^{94} +60752.4 q^{95} -118282. q^{97} +68337.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 195 q^{4} - 175 q^{5} - 153 q^{8}+O(q^{10}) \) 7 * q - q^2 + 195 * q^4 - 175 * q^5 - 153 * q^8	\( 7 q - q^{2} + 195 q^{4} - 175 q^{5} - 153 q^{8} + 25 q^{10} - 847 q^{11} + 1418 q^{13} - 2548 q^{14} + 3699 q^{16} - 630 q^{17} + 2572 q^{19} - 4875 q^{20} + 121 q^{22} - 536 q^{23} + 4375 q^{25} + 7626 q^{26} - 11368 q^{28} + 1038 q^{29} + 1872 q^{31} + 7523 q^{32} + 20790 q^{34} + 24298 q^{37} + 18952 q^{38} + 3825 q^{40} + 17658 q^{41} + 7244 q^{43} - 23595 q^{44} + 31016 q^{46} - 34560 q^{47} + 78735 q^{49} - 625 q^{50} + 110222 q^{52} + 10214 q^{53} + 21175 q^{55} - 81124 q^{56} - 5718 q^{58} - 94676 q^{59} + 69538 q^{61} + 4208 q^{62} + 112339 q^{64} - 35450 q^{65} + 64908 q^{67} + 136010 q^{68} + 63700 q^{70} - 61816 q^{71} - 11890 q^{73} + 124050 q^{74} - 47216 q^{76} + 18928 q^{79} - 92475 q^{80} + 36398 q^{82} - 17492 q^{83} + 15750 q^{85} + 216688 q^{86} + 18513 q^{88} - 25302 q^{89} + 3392 q^{91} + 27408 q^{92} - 30800 q^{94} - 64300 q^{95} - 172546 q^{97} + 615271 q^{98}+O(q^{100}) \) 7 * q - q^2 + 195 * q^4 - 175 * q^5 - 153 * q^8 + 25 * q^10 - 847 * q^11 + 1418 * q^13 - 2548 * q^14 + 3699 * q^16 - 630 * q^17 + 2572 * q^19 - 4875 * q^20 + 121 * q^22 - 536 * q^23 + 4375 * q^25 + 7626 * q^26 - 11368 * q^28 + 1038 * q^29 + 1872 * q^31 + 7523 * q^32 + 20790 * q^34 + 24298 * q^37 + 18952 * q^38 + 3825 * q^40 + 17658 * q^41 + 7244 * q^43 - 23595 * q^44 + 31016 * q^46 - 34560 * q^47 + 78735 * q^49 - 625 * q^50 + 110222 * q^52 + 10214 * q^53 + 21175 * q^55 - 81124 * q^56 - 5718 * q^58 - 94676 * q^59 + 69538 * q^61 + 4208 * q^62 + 112339 * q^64 - 35450 * q^65 + 64908 * q^67 + 136010 * q^68 + 63700 * q^70 - 61816 * q^71 - 11890 * q^73 + 124050 * q^74 - 47216 * q^76 + 18928 * q^79 - 92475 * q^80 + 36398 * q^82 - 17492 * q^83 + 15750 * q^85 + 216688 * q^86 + 18513 * q^88 - 25302 * q^89 + 3392 * q^91 + 27408 * q^92 - 30800 * q^94 - 64300 * q^95 - 172546 * q^97 + 615271 * q^98

Introduction

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