LMFDB - Embedded newform 495.6.a.o.1.5

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3 x^{9} - 246 x^{8} + 640 x^{7} + 20433 x^{6} - 44595 x^{5} - 667026 x^{4} + 1173648 x^{3} + \cdots - 30445728 $ x^10 - 3x^9 - 246x^8 + 640x^7 + 20433x^6 - 44595x^5 - 667026x^4 + 1173648x^3 + 7949136x^2 - 8226800*x - 30445728
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.25634$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.25634 q^{2} -21.3963 q^{4} -25.0000 q^{5} -3.69124 q^{7} +173.876 q^{8} +O(q^{10})$	\(q-3.25634 q^{2} -21.3963 q^{4} -25.0000 q^{5} -3.69124 q^{7} +173.876 q^{8} +81.4085 q^{10} +121.000 q^{11} -806.671 q^{13} +12.0199 q^{14} +118.480 q^{16} +989.950 q^{17} +116.088 q^{19} +534.906 q^{20} -394.017 q^{22} -4457.08 q^{23} +625.000 q^{25} +2626.80 q^{26} +78.9788 q^{28} -109.688 q^{29} +4444.57 q^{31} -5949.85 q^{32} -3223.61 q^{34} +92.2811 q^{35} -9616.97 q^{37} -378.021 q^{38} -4346.91 q^{40} -2760.73 q^{41} -2030.26 q^{43} -2588.95 q^{44} +14513.8 q^{46} -13475.5 q^{47} -16793.4 q^{49} -2035.21 q^{50} +17259.7 q^{52} +12268.5 q^{53} -3025.00 q^{55} -641.820 q^{56} +357.182 q^{58} -50349.4 q^{59} -50077.2 q^{61} -14473.0 q^{62} +15583.4 q^{64} +20166.8 q^{65} +55560.1 q^{67} -21181.2 q^{68} -300.499 q^{70} +57968.7 q^{71} +72008.0 q^{73} +31316.1 q^{74} -2483.84 q^{76} -446.641 q^{77} +92740.7 q^{79} -2962.00 q^{80} +8989.86 q^{82} -66553.0 q^{83} -24748.7 q^{85} +6611.22 q^{86} +21039.0 q^{88} +42949.4 q^{89} +2977.62 q^{91} +95364.7 q^{92} +43880.9 q^{94} -2902.19 q^{95} +125017. q^{97} +54684.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{2} + 181 q^{4} - 250 q^{5} + 116 q^{7} - 129 q^{8}+O(q^{10}) $ 10 * q - 3 * q^2 + 181 * q^4 - 250 * q^5 + 116 * q^7 - 129 * q^8	$ 10 q - 3 q^{2} + 181 q^{4} - 250 q^{5} + 116 q^{7} - 129 q^{8} + 75 q^{10} + 1210 q^{11} + 932 q^{13} + 1332 q^{14} + 2701 q^{16} - 96 q^{17} + 1664 q^{19} - 4525 q^{20} - 363 q^{22} - 6288 q^{23} + 6250 q^{25} - 13380 q^{26} + 13868 q^{28} - 11208 q^{29} + 9032 q^{31} - 9801 q^{32} + 14610 q^{34} - 2900 q^{35} + 21572 q^{37} - 15870 q^{38} + 3225 q^{40} - 10800 q^{41} + 21128 q^{43} + 21901 q^{44} + 83982 q^{46} + 17400 q^{47} + 71610 q^{49} - 1875 q^{50} + 40640 q^{52} - 5004 q^{53} - 30250 q^{55} + 54012 q^{56} - 9786 q^{58} + 25272 q^{59} + 52004 q^{61} - 34740 q^{62} + 56953 q^{64} - 23300 q^{65} + 4160 q^{67} + 87978 q^{68} - 33300 q^{70} + 65232 q^{71} + 44252 q^{73} + 49842 q^{74} + 233246 q^{76} + 14036 q^{77} + 112604 q^{79} - 67525 q^{80} + 167910 q^{82} - 70032 q^{83} + 2400 q^{85} + 72978 q^{86} - 15609 q^{88} + 46848 q^{89} + 130672 q^{91} - 121302 q^{92} + 252294 q^{94} - 41600 q^{95} + 129932 q^{97} + 316137 q^{98}+O(q^{100}) $ 10 * q - 3 * q^2 + 181 * q^4 - 250 * q^5 + 116 * q^7 - 129 * q^8 + 75 * q^10 + 1210 * q^11 + 932 * q^13 + 1332 * q^14 + 2701 * q^16 - 96 * q^17 + 1664 * q^19 - 4525 * q^20 - 363 * q^22 - 6288 * q^23 + 6250 * q^25 - 13380 * q^26 + 13868 * q^28 - 11208 * q^29 + 9032 * q^31 - 9801 * q^32 + 14610 * q^34 - 2900 * q^35 + 21572 * q^37 - 15870 * q^38 + 3225 * q^40 - 10800 * q^41 + 21128 * q^43 + 21901 * q^44 + 83982 * q^46 + 17400 * q^47 + 71610 * q^49 - 1875 * q^50 + 40640 * q^52 - 5004 * q^53 - 30250 * q^55 + 54012 * q^56 - 9786 * q^58 + 25272 * q^59 + 52004 * q^61 - 34740 * q^62 + 56953 * q^64 - 23300 * q^65 + 4160 * q^67 + 87978 * q^68 - 33300 * q^70 + 65232 * q^71 + 44252 * q^73 + 49842 * q^74 + 233246 * q^76 + 14036 * q^77 + 112604 * q^79 - 67525 * q^80 + 167910 * q^82 - 70032 * q^83 + 2400 * q^85 + 72978 * q^86 - 15609 * q^88 + 46848 * q^89 + 130672 * q^91 - 121302 * q^92 + 252294 * q^94 - 41600 * q^95 + 129932 * q^97 + 316137 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−3.25634	−0.575645	−0.287822	−	0.957684i	$-0.592931\pi$
$2$	−3.25634	−0.575645	−0.287822	+	0.957684i	$0.592931\pi$
$3$	0	0
$3$	0	0
$4$	−21.3963	−0.668633
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−3.69124	−0.0284726	−0.0142363	−	0.999899i	$-0.504532\pi$
$7$	−3.69124	−0.0284726	−0.0142363	+	0.999899i	$0.504532\pi$
$8$	173.876	0.960540
$9$	0	0
$10$	81.4085	0.257436
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−806.671	−1.32385	−0.661924	−	0.749571i	$-0.730260\pi$
$13$	−806.671	−1.32385	−0.661924	+	0.749571i	$0.730260\pi$
$14$	12.0199	0.0163901
$15$	0	0
$16$	118.480	0.115703
$17$	989.950	0.830789	0.415395	−	0.909641i	$-0.363643\pi$
$17$	989.950	0.830789	0.415395	+	0.909641i	$0.363643\pi$
$18$	0	0
$19$	116.088	0.0737739	0.0368869	−	0.999319i	$-0.488256\pi$
$19$	116.088	0.0737739	0.0368869	+	0.999319i	$0.488256\pi$
$20$	534.906	0.299022
$21$	0	0
$22$	−394.017	−0.173563
$23$	−4457.08	−1.75683	−0.878416	−	0.477896i	$-0.841400\pi$
$23$	−4457.08	−1.75683	−0.878416	+	0.477896i	$0.841400\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	2626.80	0.762067
$27$	0	0
$28$	78.9788	0.0190377
$29$	−109.688	−0.0242195	−0.0121098	−	0.999927i	$-0.503855\pi$
$29$	−109.688	−0.0242195	−0.0121098	+	0.999927i	$0.503855\pi$
$30$	0	0
$31$	4444.57	0.830664	0.415332	−	0.909670i	$-0.363665\pi$
$31$	4444.57	0.830664	0.415332	+	0.909670i	$0.363665\pi$
$32$	−5949.85	−1.02714
$33$	0	0
$34$	−3223.61	−0.478240
$35$	92.2811	0.0127334
$36$	0	0
$37$	−9616.97	−1.15487	−0.577436	−	0.816436i	$-0.695947\pi$
$37$	−9616.97	−1.15487	−0.577436	+	0.816436i	$0.695947\pi$
$38$	−378.021	−0.0424675
$39$	0	0
$40$	−4346.91	−0.429567
$41$	−2760.73	−0.256486	−0.128243	−	0.991743i	$-0.540934\pi$
$41$	−2760.73	−0.256486	−0.128243	+	0.991743i	$0.540934\pi$
$42$	0	0
$43$	−2030.26	−0.167448	−0.0837242	−	0.996489i	$-0.526681\pi$
$43$	−2030.26	−0.167448	−0.0837242	+	0.996489i	$0.526681\pi$
$44$	−2588.95	−0.201600
$45$	0	0
$46$	14513.8	1.01131
$47$	−13475.5	−0.889817	−0.444909	−	0.895576i	$-0.646764\pi$
$47$	−13475.5	−0.889817	−0.444909	+	0.895576i	$0.646764\pi$
$48$	0	0
$49$	−16793.4	−0.999189
$50$	−2035.21	−0.115129
$51$	0	0
$52$	17259.7	0.885169
$53$	12268.5	0.599930	0.299965	−	0.953950i	$-0.403025\pi$
$53$	12268.5	0.599930	0.299965	+	0.953950i	$0.403025\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−641.820	−0.0273491
$57$	0	0
$58$	357.182	0.0139418
$59$	−50349.4	−1.88306	−0.941530	−	0.336929i	$-0.890612\pi$
$59$	−50349.4	−1.88306	−0.941530	+	0.336929i	$0.890612\pi$
$60$	0	0
$61$	−50077.2	−1.72312	−0.861560	−	0.507656i	$-0.830512\pi$
$61$	−50077.2	−1.72312	−0.861560	+	0.507656i	$0.830512\pi$
$62$	−14473.0	−0.478168
$63$	0	0
$64$	15583.4	0.475567
$65$	20166.8	0.592043
$66$	0	0
$67$	55560.1	1.51208	0.756042	−	0.654523i	$-0.227130\pi$
$67$	55560.1	1.51208	0.756042	+	0.654523i	$0.227130\pi$
$68$	−21181.2	−0.555493
$69$	0	0
$70$	−300.499	−0.00732989
$71$	57968.7	1.36473	0.682366	−	0.731011i	$-0.260951\pi$
$71$	57968.7	1.36473	0.682366	+	0.731011i	$0.260951\pi$
$72$	0	0
$73$	72008.0	1.58152	0.790758	−	0.612129i	$-0.209687\pi$
$73$	72008.0	1.58152	0.790758	+	0.612129i	$0.209687\pi$
$74$	31316.1	0.664796
$75$	0	0
$76$	−2483.84	−0.0493276
$77$	−446.641	−0.00858482
$78$	0	0
$79$	92740.7	1.67187	0.835935	−	0.548828i	$-0.184926\pi$
$79$	92740.7	1.67187	0.835935	+	0.548828i	$0.184926\pi$
$80$	−2962.00	−0.0517440
$81$	0	0
$82$	8989.86	0.147645
$83$	−66553.0	−1.06041	−0.530203	−	0.847871i	$-0.677884\pi$
$83$	−66553.0	−1.06041	−0.530203	+	0.847871i	$0.677884\pi$
$84$	0	0
$85$	−24748.7	−0.371540
$86$	6611.22	0.0963908
$87$	0	0
$88$	21039.0	0.289614
$89$	42949.4	0.574753	0.287377	−	0.957818i	$-0.407217\pi$
$89$	42949.4	0.574753	0.287377	+	0.957818i	$0.407217\pi$
$90$	0	0
$91$	2977.62	0.0376935
$92$	95364.7	1.17468
$93$	0	0
$94$	43880.9	0.512219
$95$	−2902.19	−0.0329927
$96$	0	0
$97$	125017.	1.34909	0.674544	−	0.738235i	$-0.264340\pi$
$97$	125017.	1.34909	0.674544	+	0.738235i	$0.264340\pi$
$98$	54684.9	0.575178
$99$	0	0
$100$	−13372.7	−0.133727
$101$	−154798.	−1.50995	−0.754973	−	0.655756i	$-0.772350\pi$
$101$	−154798.	−1.50995	−0.754973	+	0.655756i	$0.772350\pi$
$102$	0	0
$103$	−170675.	−1.58517	−0.792586	−	0.609760i	$-0.791266\pi$
$103$	−170675.	−1.58517	−0.792586	+	0.609760i	$0.791266\pi$
$104$	−140261.	−1.27161
$105$	0	0
$106$	−39950.3	−0.345347
$107$	−175479.	−1.48171	−0.740857	−	0.671663i	$-0.765580\pi$
$107$	−175479.	−1.48171	−0.740857	+	0.671663i	$0.765580\pi$
$108$	0	0
$109$	187795.	1.51397	0.756987	−	0.653430i	$-0.226671\pi$
$109$	187795.	1.51397	0.756987	+	0.653430i	$0.226671\pi$
$110$	9850.43	0.0776199
$111$	0	0
$112$	−437.338	−0.00329437
$113$	−112671.	−0.830074	−0.415037	−	0.909805i	$-0.636231\pi$
$113$	−112671.	−0.830074	−0.415037	+	0.909805i	$0.636231\pi$
$114$	0	0
$115$	111427.	0.785680
$116$	2346.92	0.0161940
$117$	0	0
$118$	163955.	1.08397
$119$	−3654.15	−0.0236548
$120$	0	0
$121$	14641.0	0.0909091
$122$	163068.	0.991905
$123$	0	0
$124$	−95097.2	−0.555410
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	303331.	1.66881	0.834407	−	0.551149i	$-0.185810\pi$
$127$	303331.	1.66881	0.834407	+	0.551149i	$0.185810\pi$
$128$	139651.	0.753386
$129$	0	0
$130$	−65669.9	−0.340807
$131$	33081.2	0.168424	0.0842118	−	0.996448i	$-0.473163\pi$
$131$	33081.2	0.168424	0.0842118	+	0.996448i	$0.473163\pi$
$132$	0	0
$133$	−428.508	−0.00210054
$134$	−180923.	−0.870424
$135$	0	0
$136$	172129.	0.798006
$137$	324831.	1.47862	0.739309	−	0.673367i	$-0.235152\pi$
$137$	324831.	1.47862	0.739309	+	0.673367i	$0.235152\pi$
$138$	0	0
$139$	238982.	1.04913	0.524564	−	0.851371i	$-0.324228\pi$
$139$	238982.	1.04913	0.524564	+	0.851371i	$0.324228\pi$
$140$	−1974.47	−0.00851394
$141$	0	0
$142$	−188766.	−0.785601
$143$	−97607.3	−0.399155
$144$	0	0
$145$	2742.21	0.0108313
$146$	−234482.	−0.910391
$147$	0	0
$148$	205767.	0.772186
$149$	185393.	0.684113	0.342056	−	0.939679i	$-0.388877\pi$
$149$	185393.	0.684113	0.342056	+	0.939679i	$0.388877\pi$
$150$	0	0
$151$	318132.	1.13544	0.567722	−	0.823221i	$-0.307825\pi$
$151$	318132.	1.13544	0.567722	+	0.823221i	$0.307825\pi$
$152$	20184.9	0.0708627
$153$	0	0
$154$	1454.41	0.00494181
$155$	−111114.	−0.371484
$156$	0	0
$157$	−191305.	−0.619410	−0.309705	−	0.950833i	$-0.600230\pi$
$157$	−191305.	−0.619410	−0.309705	+	0.950833i	$0.600230\pi$
$158$	−301995.	−0.962404
$159$	0	0
$160$	148746.	0.459353
$161$	16452.2	0.0500217
$162$	0	0
$163$	31135.6	0.0917884	0.0458942	−	0.998946i	$-0.485386\pi$
$163$	31135.6	0.0917884	0.0458942	+	0.998946i	$0.485386\pi$
$164$	59069.2	0.171495
$165$	0	0
$166$	216719.	0.610417
$167$	300093.	0.832655	0.416328	−	0.909215i	$-0.363317\pi$
$167$	300093.	0.832655	0.416328	+	0.909215i	$0.363317\pi$
$168$	0	0
$169$	279426.	0.752575
$170$	80590.3	0.213875
$171$	0	0
$172$	43440.0	0.111962
$173$	336129.	0.853867	0.426934	−	0.904283i	$-0.359594\pi$
$173$	336129.	0.853867	0.426934	+	0.904283i	$0.359594\pi$
$174$	0	0
$175$	−2307.03	−0.00569453
$176$	14336.1	0.0348858
$177$	0	0
$178$	−139858.	−0.330854
$179$	319924.	0.746301	0.373150	−	0.927771i	$-0.378278\pi$
$179$	319924.	0.746301	0.373150	+	0.927771i	$0.378278\pi$
$180$	0	0
$181$	−731009.	−1.65854	−0.829271	−	0.558847i	$-0.811244\pi$
$181$	−731009.	−1.65854	−0.829271	+	0.558847i	$0.811244\pi$
$182$	−9696.15	−0.0216980
$183$	0	0
$184$	−774980.	−1.68751
$185$	240424.	0.516475
$186$	0	0
$187$	119784.	0.250492
$188$	288326.	0.594961
$189$	0	0
$190$	9450.53	0.0189921
$191$	271429.	0.538361	0.269180	−	0.963090i	$-0.413247\pi$
$191$	271429.	0.538361	0.269180	+	0.963090i	$0.413247\pi$
$192$	0	0
$193$	868150.	1.67765	0.838825	−	0.544401i	$-0.183243\pi$
$193$	868150.	1.67765	0.838825	+	0.544401i	$0.183243\pi$
$194$	−407098.	−0.776596
$195$	0	0
$196$	359315.	0.668091
$197$	700407.	1.28583	0.642917	−	0.765936i	$-0.277724\pi$
$197$	700407.	1.28583	0.642917	+	0.765936i	$0.277724\pi$
$198$	0	0
$199$	350224.	0.626921	0.313461	−	0.949601i	$-0.398512\pi$
$199$	350224.	0.626921	0.313461	+	0.949601i	$0.398512\pi$
$200$	108673.	0.192108
$201$	0	0
$202$	504074.	0.869192
$203$	404.887	0.000689594 0
$204$	0	0
$205$	69018.2	0.114704
$206$	555775.	0.912496
$207$	0	0
$208$	−95574.4	−0.153173
$209$	14046.6	0.0222437
$210$	0	0
$211$	65807.2	0.101758	0.0508788	−	0.998705i	$-0.483798\pi$
$211$	65807.2	0.101758	0.0508788	+	0.998705i	$0.483798\pi$
$212$	−262499.	−0.401133
$213$	0	0
$214$	571418.	0.852941
$215$	50756.6	0.0748852
$216$	0	0
$217$	−16406.0	−0.0236512
$218$	−611525.	−0.871511
$219$	0	0
$220$	64723.7	0.0901585
$221$	−798564.	−1.09984
$222$	0	0
$223$	−382404.	−0.514945	−0.257472	−	0.966286i	$-0.582890\pi$
$223$	−382404.	−0.514945	−0.257472	+	0.966286i	$0.582890\pi$
$224$	21962.4	0.0292455
$225$	0	0
$226$	366895.	0.477828
$227$	260346.	0.335340	0.167670	−	0.985843i	$-0.446376\pi$
$227$	260346.	0.335340	0.167670	+	0.985843i	$0.446376\pi$
$228$	0	0
$229$	733690.	0.924536	0.462268	−	0.886740i	$-0.347036\pi$
$229$	733690.	0.924536	0.462268	+	0.886740i	$0.347036\pi$
$230$	−362844.	−0.452272
$231$	0	0
$232$	−19072.2	−0.0232638
$233$	−1.31459e6	−1.58635	−0.793177	−	0.608991i	$-0.791575\pi$
$233$	−1.31459e6	−1.58635	−0.793177	+	0.608991i	$0.791575\pi$
$234$	0	0
$235$	336888.	0.397938
$236$	1.07729e6	1.25908
$237$	0	0
$238$	11899.1	0.0136167
$239$	−774428.	−0.876973	−0.438487	−	0.898738i	$-0.644485\pi$
$239$	−774428.	−0.876973	−0.438487	+	0.898738i	$0.644485\pi$
$240$	0	0
$241$	397106.	0.440417	0.220209	−	0.975453i	$-0.429326\pi$
$241$	397106.	0.440417	0.220209	+	0.975453i	$0.429326\pi$
$242$	−47676.1	−0.0523314
$243$	0	0
$244$	1.07146e6	1.15213
$245$	419834.	0.446851
$246$	0	0
$247$	−93644.7	−0.0976654
$248$	772806.	0.797886
$249$	0	0
$250$	50880.3	0.0514872
$251$	1.69788e6	1.70107	0.850535	−	0.525919i	$-0.176278\pi$
$251$	1.69788e6	1.70107	0.850535	+	0.525919i	$0.176278\pi$
$252$	0	0
$253$	−539306.	−0.529705
$254$	−987749.	−0.960644
$255$	0	0
$256$	−953418.	−0.909250
$257$	−74448.9	−0.0703114	−0.0351557	−	0.999382i	$-0.511193\pi$
$257$	−74448.9	−0.0703114	−0.0351557	+	0.999382i	$0.511193\pi$
$258$	0	0
$259$	35498.6	0.0328823
$260$	−431494.	−0.395860
$261$	0	0
$262$	−107724.	−0.0969522
$263$	−137773.	−0.122822	−0.0614108	−	0.998113i	$-0.519560\pi$
$263$	−137773.	−0.122822	−0.0614108	+	0.998113i	$0.519560\pi$
$264$	0	0
$265$	−306712.	−0.268297
$266$	1395.37	0.00120916
$267$	0	0
$268$	−1.18878e6	−1.01103
$269$	−532267.	−0.448486	−0.224243	−	0.974533i	$-0.571991\pi$
$269$	−532267.	−0.448486	−0.224243	+	0.974533i	$0.571991\pi$
$270$	0	0
$271$	2.07179e6	1.71365	0.856825	−	0.515607i	$-0.172433\pi$
$271$	2.07179e6	1.71365	0.856825	+	0.515607i	$0.172433\pi$
$272$	117289.	0.0961249
$273$	0	0
$274$	−1.05776e6	−0.851158
$275$	75625.0	0.0603023
$276$	0	0
$277$	−895067.	−0.700900	−0.350450	−	0.936581i	$-0.613971\pi$
$277$	−895067.	−0.700900	−0.350450	+	0.936581i	$0.613971\pi$
$278$	−778207.	−0.603925
$279$	0	0
$280$	16045.5	0.0122309
$281$	−669119.	−0.505519	−0.252760	−	0.967529i	$-0.581338\pi$
$281$	−669119.	−0.505519	−0.252760	+	0.967529i	$0.581338\pi$
$282$	0	0
$283$	−357606.	−0.265423	−0.132712	−	0.991155i	$-0.542368\pi$
$283$	−357606.	−0.265423	−0.132712	+	0.991155i	$0.542368\pi$
$284$	−1.24031e6	−0.912505
$285$	0	0
$286$	317842.	0.229772
$287$	10190.5	0.00730283
$288$	0	0
$289$	−439856.	−0.309789
$290$	−8929.56	−0.00623498
$291$	0	0
$292$	−1.54070e6	−1.05745
$293$	−105022.	−0.0714676	−0.0357338	−	0.999361i	$-0.511377\pi$
$293$	−105022.	−0.0714676	−0.0357338	+	0.999361i	$0.511377\pi$
$294$	0	0
$295$	1.25873e6	0.842130
$296$	−1.67216e6	−1.10930
$297$	0	0
$298$	−603702.	−0.393806
$299$	3.59540e6	2.32578
$300$	0	0
$301$	7494.20	0.00476770
$302$	−1.03595e6	−0.653612
$303$	0	0
$304$	13754.1	0.00853586
$305$	1.25193e6	0.770603
$306$	0	0
$307$	−766406.	−0.464102	−0.232051	−	0.972704i	$-0.574544\pi$
$307$	−766406.	−0.464102	−0.232051	+	0.972704i	$0.574544\pi$
$308$	9556.44	0.00574010
$309$	0	0
$310$	361826.	0.213843
$311$	1.01564e6	0.595441	0.297720	−	0.954653i	$-0.403774\pi$
$311$	1.01564e6	0.595441	0.297720	+	0.954653i	$0.403774\pi$
$312$	0	0
$313$	−2.04181e6	−1.17803	−0.589013	−	0.808124i	$-0.700483\pi$
$313$	−2.04181e6	−1.17803	−0.589013	+	0.808124i	$0.700483\pi$
$314$	622955.	0.356560
$315$	0	0
$316$	−1.98430e6	−1.11787
$317$	−2.88929e6	−1.61489	−0.807444	−	0.589944i	$-0.799150\pi$
$317$	−2.88929e6	−1.61489	−0.807444	+	0.589944i	$0.799150\pi$
$318$	0	0
$319$	−13272.3	−0.00730246
$320$	−389585.	−0.212680
$321$	0	0
$322$	−53573.8	−0.0287947
$323$	114921.	0.0612905
$324$	0	0
$325$	−504170.	−0.264770
$326$	−101388.	−0.0528375
$327$	0	0
$328$	−480025.	−0.246365
$329$	49741.4	0.0253354
$330$	0	0
$331$	−1.14449e6	−0.574171	−0.287085	−	0.957905i	$-0.592686\pi$
$331$	−1.14449e6	−0.574171	−0.287085	+	0.957905i	$0.592686\pi$
$332$	1.42398e6	0.709023
$333$	0	0
$334$	−977206.	−0.479314
$335$	−1.38900e6	−0.676225
$336$	0	0
$337$	1.84439e6	0.884663	0.442331	−	0.896852i	$-0.354152\pi$
$337$	1.84439e6	0.884663	0.442331	+	0.896852i	$0.354152\pi$
$338$	−909906.	−0.433216
$339$	0	0
$340$	529531.	0.248424
$341$	537793.	0.250455
$342$	0	0
$343$	124027.	0.0569222
$344$	−353015.	−0.160841
$345$	0	0
$346$	−1.09455e6	−0.491524
$347$	4.28131e6	1.90877	0.954383	−	0.298585i	$-0.0965146\pi$
$347$	4.28131e6	1.90877	0.954383	+	0.298585i	$0.0965146\pi$
$348$	0	0
$349$	2.85114e6	1.25301	0.626505	−	0.779417i	$-0.284485\pi$
$349$	2.85114e6	1.25301	0.626505	+	0.779417i	$0.284485\pi$
$350$	7512.46	0.00327803
$351$	0	0
$352$	−719932.	−0.309696
$353$	−2.95909e6	−1.26392	−0.631962	−	0.774999i	$-0.717750\pi$
$353$	−2.95909e6	−1.26392	−0.631962	+	0.774999i	$0.717750\pi$
$354$	0	0
$355$	−1.44922e6	−0.610327
$356$	−918955.	−0.384299
$357$	0	0
$358$	−1.04178e6	−0.429604
$359$	−3.62259e6	−1.48348	−0.741742	−	0.670685i	$-0.766000\pi$
$359$	−3.62259e6	−1.48348	−0.741742	+	0.670685i	$0.766000\pi$
$360$	0	0
$361$	−2.46262e6	−0.994557
$362$	2.38041e6	0.954731
$363$	0	0
$364$	−63710.0	−0.0252031
$365$	−1.80020e6	−0.707275
$366$	0	0
$367$	−3.97582e6	−1.54085	−0.770426	−	0.637529i	$-0.779957\pi$
$367$	−3.97582e6	−1.54085	−0.770426	+	0.637529i	$0.779957\pi$
$368$	−528074.	−0.203271
$369$	0	0
$370$	−782903.	−0.297306
$371$	−45285.9	−0.0170816
$372$	0	0
$373$	−634639.	−0.236186	−0.118093	−	0.993003i	$-0.537678\pi$
$373$	−634639.	−0.236186	−0.118093	+	0.993003i	$0.537678\pi$
$374$	−390057.	−0.144195
$375$	0	0
$376$	−2.34307e6	−0.854705
$377$	88482.5	0.0320630
$378$	0	0
$379$	2.71571e6	0.971149	0.485575	−	0.874195i	$-0.338610\pi$
$379$	2.71571e6	0.971149	0.485575	+	0.874195i	$0.338610\pi$
$380$	62096.1	0.0220600
$381$	0	0
$382$	−883866.	−0.309904
$383$	−1.27015e6	−0.442444	−0.221222	−	0.975224i	$-0.571004\pi$
$383$	−1.27015e6	−0.442444	−0.221222	+	0.975224i	$0.571004\pi$
$384$	0	0
$385$	11166.0	0.00383925
$386$	−2.82699e6	−0.965731
$387$	0	0
$388$	−2.67490e6	−0.902045
$389$	2.91899e6	0.978045	0.489023	−	0.872271i	$-0.337354\pi$
$389$	2.91899e6	0.978045	0.489023	+	0.872271i	$0.337354\pi$
$390$	0	0
$391$	−4.41228e6	−1.45956
$392$	−2.91997e6	−0.959761
$393$	0	0
$394$	−2.28076e6	−0.740184
$395$	−2.31852e6	−0.747683
$396$	0	0
$397$	2.21451e6	0.705183	0.352592	−	0.935777i	$-0.385300\pi$
$397$	2.21451e6	0.705183	0.352592	+	0.935777i	$0.385300\pi$
$398$	−1.14045e6	−0.360884
$399$	0	0
$400$	74050.0	0.0231406
$401$	5.68327e6	1.76497	0.882485	−	0.470341i	$-0.155869\pi$
$401$	5.68327e6	1.76497	0.882485	+	0.470341i	$0.155869\pi$
$402$	0	0
$403$	−3.58531e6	−1.09967
$404$	3.31209e6	1.00960
$405$	0	0
$406$	−1318.45	−0.000396961 0
$407$	−1.16365e6	−0.348207
$408$	0	0
$409$	−3.03027e6	−0.895722	−0.447861	−	0.894103i	$-0.647814\pi$
$409$	−3.03027e6	−0.895722	−0.447861	+	0.894103i	$0.647814\pi$
$410$	−224747.	−0.0660288
$411$	0	0
$412$	3.65180e6	1.05990
$413$	185852.	0.0536157
$414$	0	0
$415$	1.66382e6	0.474228
$416$	4.79958e6	1.35978
$417$	0	0
$418$	−45740.6	−0.0128044
$419$	1.55269e6	0.432066	0.216033	−	0.976386i	$-0.430688\pi$
$419$	1.55269e6	0.432066	0.216033	+	0.976386i	$0.430688\pi$
$420$	0	0
$421$	−2.99983e6	−0.824881	−0.412441	−	0.910985i	$-0.635324\pi$
$421$	−2.99983e6	−0.824881	−0.412441	+	0.910985i	$0.635324\pi$
$422$	−214290.	−0.0585763
$423$	0	0
$424$	2.13320e6	0.576257
$425$	618719.	0.166158
$426$	0	0
$427$	184847.	0.0490618
$428$	3.75458e6	0.990723
$429$	0	0
$430$	−165281.	−0.0431073
$431$	−2.84237e6	−0.737034	−0.368517	−	0.929621i	$-0.620134\pi$
$431$	−2.84237e6	−0.737034	−0.368517	+	0.929621i	$0.620134\pi$
$432$	0	0
$433$	5.55596e6	1.42410	0.712048	−	0.702130i	$-0.247768\pi$
$433$	5.55596e6	1.42410	0.712048	+	0.702130i	$0.247768\pi$
$434$	53423.5	0.0136147
$435$	0	0
$436$	−4.01811e6	−1.01229
$437$	−517412.	−0.129608
$438$	0	0
$439$	2.16907e6	0.537170	0.268585	−	0.963256i	$-0.413444\pi$
$439$	2.16907e6	0.537170	0.268585	+	0.963256i	$0.413444\pi$
$440$	−525976.	−0.129519
$441$	0	0
$442$	2.60040e6	0.633117
$443$	−376551.	−0.0911622	−0.0455811	−	0.998961i	$-0.514514\pi$
$443$	−376551.	−0.0911622	−0.0455811	+	0.998961i	$0.514514\pi$
$444$	0	0
$445$	−1.07373e6	−0.257038
$446$	1.24524e6	0.296425
$447$	0	0
$448$	−57522.1	−0.0135406
$449$	3.37874e6	0.790932	0.395466	−	0.918481i	$-0.370583\pi$
$449$	3.37874e6	0.790932	0.395466	+	0.918481i	$0.370583\pi$
$450$	0	0
$451$	−334048.	−0.0773335
$452$	2.41074e6	0.555015
$453$	0	0
$454$	−847774.	−0.193037
$455$	−74440.5	−0.0168570
$456$	0	0
$457$	4.42600e6	0.991337	0.495668	−	0.868512i	$-0.334923\pi$
$457$	4.42600e6	0.991337	0.495668	+	0.868512i	$0.334923\pi$
$458$	−2.38914e6	−0.532205
$459$	0	0
$460$	−2.38412e6	−0.525331
$461$	701092.	0.153646	0.0768232	−	0.997045i	$-0.475522\pi$
$461$	701092.	0.153646	0.0768232	+	0.997045i	$0.475522\pi$
$462$	0	0
$463$	−1.22959e6	−0.266568	−0.133284	−	0.991078i	$-0.542552\pi$
$463$	−1.22959e6	−0.266568	−0.133284	+	0.991078i	$0.542552\pi$
$464$	−12995.9	−0.00280227
$465$	0	0
$466$	4.28075e6	0.913177
$467$	20871.5	0.00442855	0.00221427	−	0.999998i	$-0.499295\pi$
$467$	20871.5	0.00442855	0.00221427	+	0.999998i	$0.499295\pi$
$468$	0	0
$469$	−205086.	−0.0430530
$470$	−1.09702e6	−0.229071
$471$	0	0
$472$	−8.75457e6	−1.80875
$473$	−245662.	−0.0504876
$474$	0	0
$475$	72554.9	0.0147548
$476$	78185.1	0.0158164
$477$	0	0
$478$	2.52180e6	0.504825
$479$	4.15915e6	0.828258	0.414129	−	0.910218i	$-0.364086\pi$
$479$	4.15915e6	0.828258	0.414129	+	0.910218i	$0.364086\pi$
$480$	0	0
$481$	7.75774e6	1.52888
$482$	−1.29311e6	−0.253524
$483$	0	0
$484$	−313263.	−0.0607848
$485$	−3.12543e6	−0.603331
$486$	0	0
$487$	1.74097e6	0.332635	0.166317	−	0.986072i	$-0.446812\pi$
$487$	1.74097e6	0.332635	0.166317	+	0.986072i	$0.446812\pi$
$488$	−8.70724e6	−1.65513
$489$	0	0
$490$	−1.36712e6	−0.257228
$491$	3.92481e6	0.734708	0.367354	−	0.930081i	$-0.380264\pi$
$491$	3.92481e6	0.734708	0.367354	+	0.930081i	$0.380264\pi$
$492$	0	0
$493$	−108586.	−0.0201213
$494$	304939.	0.0562206
$495$	0	0
$496$	526592.	0.0961104
$497$	−213976.	−0.0388575
$498$	0	0
$499$	2.95071e6	0.530487	0.265243	−	0.964181i	$-0.414548\pi$
$499$	2.95071e6	0.530487	0.265243	+	0.964181i	$0.414548\pi$
$500$	334316.	0.0598044
$501$	0	0
$502$	−5.52886e6	−0.979212
$503$	−134861.	−0.0237665	−0.0118832	−	0.999929i	$-0.503783\pi$
$503$	−134861.	−0.0237665	−0.0118832	+	0.999929i	$0.503783\pi$
$504$	0	0
$505$	3.86994e6	0.675268
$506$	1.75616e6	0.304922
$507$	0	0
$508$	−6.49015e6	−1.11582
$509$	−4.60116e6	−0.787179	−0.393589	−	0.919286i	$-0.628767\pi$
$509$	−4.60116e6	−0.787179	−0.393589	+	0.919286i	$0.628767\pi$
$510$	0	0
$511$	−265799.	−0.0450299
$512$	−1.36417e6	−0.229981
$513$	0	0
$514$	242431.	0.0404744
$515$	4.26687e6	0.708911
$516$	0	0
$517$	−1.63054e6	−0.268290
$518$	−115595.	−0.0189285
$519$	0	0
$520$	3.50653e6	0.568681
$521$	1.21848e7	1.96664	0.983319	−	0.181889i	$-0.0582213\pi$
$521$	1.21848e7	1.96664	0.983319	+	0.181889i	$0.0582213\pi$
$522$	0	0
$523$	−8.26146e6	−1.32069	−0.660347	−	0.750960i	$-0.729591\pi$
$523$	−8.26146e6	−1.32069	−0.660347	+	0.750960i	$0.729591\pi$
$524$	−707814.	−0.112614
$525$	0	0
$526$	448636.	0.0707016
$527$	4.39990e6	0.690107
$528$	0	0
$529$	1.34292e7	2.08646
$530$	998757.	0.154444
$531$	0	0
$532$	9168.47	0.00140449
$533$	2.22700e6	0.339549
$534$	0	0
$535$	4.38696e6	0.662643
$536$	9.66059e6	1.45242
$537$	0	0
$538$	1.73324e6	0.258169
$539$	−2.03200e6	−0.301267
$540$	0	0
$541$	−4.17508e6	−0.613298	−0.306649	−	0.951823i	$-0.599208\pi$
$541$	−4.17508e6	−0.613298	−0.306649	+	0.951823i	$0.599208\pi$
$542$	−6.74645e6	−0.986454
$543$	0	0
$544$	−5.89006e6	−0.853340
$545$	−4.69488e6	−0.677069
$546$	0	0
$547$	−7.74600e6	−1.10690	−0.553451	−	0.832882i	$-0.686689\pi$
$547$	−7.74600e6	−1.10690	−0.553451	+	0.832882i	$0.686689\pi$
$548$	−6.95016e6	−0.988652
$549$	0	0
$550$	−246261.	−0.0347127
$551$	−12733.5	−0.00178677
$552$	0	0
$553$	−342329.	−0.0476026
$554$	2.91464e6	0.403470
$555$	0	0
$556$	−5.11333e6	−0.701482
$557$	−3.48809e6	−0.476376	−0.238188	−	0.971219i	$-0.576553\pi$
$557$	−3.48809e6	−0.476376	−0.238188	+	0.971219i	$0.576553\pi$
$558$	0	0
$559$	1.63776e6	0.221676
$560$	10933.5	0.00147329
$561$	0	0
$562$	2.17888e6	0.291000
$563$	−9.49420e6	−1.26237	−0.631186	−	0.775632i	$-0.717431\pi$
$563$	−9.49420e6	−1.26237	−0.631186	+	0.775632i	$0.717431\pi$
$564$	0	0
$565$	2.81678e6	0.371220
$566$	1.16449e6	0.152790
$567$	0	0
$568$	1.00794e7	1.31088
$569$	−953163.	−0.123420	−0.0617102	−	0.998094i	$-0.519655\pi$
$569$	−953163.	−0.123420	−0.0617102	+	0.998094i	$0.519655\pi$
$570$	0	0
$571$	1.22946e7	1.57806	0.789031	−	0.614354i	$-0.210583\pi$
$571$	1.22946e7	1.57806	0.789031	+	0.614354i	$0.210583\pi$
$572$	2.08843e6	0.266888
$573$	0	0
$574$	−33183.8	−0.00420384
$575$	−2.78567e6	−0.351367
$576$	0	0
$577$	1.20281e7	1.50403	0.752015	−	0.659146i	$-0.229082\pi$
$577$	1.20281e7	1.50403	0.752015	+	0.659146i	$0.229082\pi$
$578$	1.43232e6	0.178328
$579$	0	0
$580$	−58673.0	−0.00724216
$581$	245663.	0.0301926
$582$	0	0
$583$	1.48448e6	0.180886
$584$	1.25205e7	1.51911
$585$	0	0
$586$	341986.	0.0411400
$587$	1.63146e6	0.195426	0.0977129	−	0.995215i	$-0.468847\pi$
$587$	1.63146e6	0.195426	0.0977129	+	0.995215i	$0.468847\pi$
$588$	0	0
$589$	515960.	0.0612813
$590$	−4.09887e6	−0.484768
$591$	0	0
$592$	−1.13942e6	−0.133622
$593$	−1.08848e7	−1.27111	−0.635557	−	0.772054i	$-0.719229\pi$
$593$	−1.08848e7	−1.27111	−0.635557	+	0.772054i	$0.719229\pi$
$594$	0	0
$595$	91353.7	0.0105787
$596$	−3.96671e6	−0.457420
$597$	0	0
$598$	−1.17078e7	−1.33882
$599$	−1.72974e6	−0.196976	−0.0984879	−	0.995138i	$-0.531401\pi$
$599$	−1.72974e6	−0.196976	−0.0984879	+	0.995138i	$0.531401\pi$
$600$	0	0
$601$	569573.	0.0643225	0.0321612	−	0.999483i	$-0.489761\pi$
$601$	569573.	0.0643225	0.0321612	+	0.999483i	$0.489761\pi$
$602$	−24403.6	−0.00274450
$603$	0	0
$604$	−6.80684e6	−0.759195
$605$	−366025.	−0.0406558
$606$	0	0
$607$	−813603.	−0.0896274	−0.0448137	−	0.998995i	$-0.514269\pi$
$607$	−813603.	−0.0896274	−0.0448137	+	0.998995i	$0.514269\pi$
$608$	−690705.	−0.0757764
$609$	0	0
$610$	−4.07671e6	−0.443593
$611$	1.08703e7	1.17798
$612$	0	0
$613$	−788143.	−0.0847137	−0.0423569	−	0.999103i	$-0.513487\pi$
$613$	−788143.	−0.0847137	−0.0423569	+	0.999103i	$0.513487\pi$
$614$	2.49568e6	0.267158
$615$	0	0
$616$	−77660.2	−0.00824607
$617$	3.95246e6	0.417980	0.208990	−	0.977918i	$-0.432982\pi$
$617$	3.95246e6	0.417980	0.208990	+	0.977918i	$0.432982\pi$
$618$	0	0
$619$	1.25914e7	1.32083	0.660415	−	0.750901i	$-0.270380\pi$
$619$	1.25914e7	1.32083	0.660415	+	0.750901i	$0.270380\pi$
$620$	2.37743e6	0.248387
$621$	0	0
$622$	−3.30727e6	−0.342763
$623$	−158537.	−0.0163647
$624$	0	0
$625$	390625.	0.0400000
$626$	6.64883e6	0.678124
$627$	0	0
$628$	4.09322e6	0.414158
$629$	−9.52032e6	−0.959456
$630$	0	0
$631$	1.63441e7	1.63414	0.817069	−	0.576540i	$-0.195597\pi$
$631$	1.63441e7	1.63414	0.817069	+	0.576540i	$0.195597\pi$
$632$	1.61254e7	1.60590
$633$	0	0
$634$	9.40849e6	0.929602
$635$	−7.58328e6	−0.746316
$636$	0	0
$637$	1.35467e7	1.32278
$638$	43219.1	0.00420362
$639$	0	0
$640$	−3.49126e6	−0.336925
$641$	1.29025e7	1.24031	0.620155	−	0.784480i	$-0.287070\pi$
$641$	1.29025e7	1.24031	0.620155	+	0.784480i	$0.287070\pi$
$642$	0	0
$643$	1.75490e7	1.67389	0.836944	−	0.547289i	$-0.184340\pi$
$643$	1.75490e7	1.67389	0.836944	+	0.547289i	$0.184340\pi$
$644$	−352015.	−0.0334461
$645$	0	0
$646$	−374222.	−0.0352816
$647$	−1.76105e6	−0.165391	−0.0826954	−	0.996575i	$-0.526353\pi$
$647$	−1.76105e6	−0.165391	−0.0826954	+	0.996575i	$0.526353\pi$
$648$	0	0
$649$	−6.09228e6	−0.567764
$650$	1.64175e6	0.152413
$651$	0	0
$652$	−666184.	−0.0613727
$653$	4.06442e6	0.373006	0.186503	−	0.982454i	$-0.440285\pi$
$653$	4.06442e6	0.373006	0.186503	+	0.982454i	$0.440285\pi$
$654$	0	0
$655$	−827030.	−0.0753214
$656$	−327091.	−0.0296762
$657$	0	0
$658$	−161975.	−0.0145842
$659$	9.29074e6	0.833368	0.416684	−	0.909051i	$-0.363192\pi$
$659$	9.29074e6	0.833368	0.416684	+	0.909051i	$0.363192\pi$
$660$	0	0
$661$	−8.18019e6	−0.728216	−0.364108	−	0.931357i	$-0.618626\pi$
$661$	−8.18019e6	−0.728216	−0.364108	+	0.931357i	$0.618626\pi$
$662$	3.72684e6	0.330518
$663$	0	0
$664$	−1.15720e7	−1.01856
$665$	10712.7	0.000939388 0
$666$	0	0
$667$	488889.	0.0425497
$668$	−6.42088e6	−0.556741
$669$	0	0
$670$	4.52306e6	0.389265
$671$	−6.05934e6	−0.519540
$672$	0	0
$673$	8.50408e6	0.723751	0.361876	−	0.932226i	$-0.382136\pi$
$673$	8.50408e6	0.723751	0.361876	+	0.932226i	$0.382136\pi$
$674$	−6.00595e6	−0.509251
$675$	0	0
$676$	−5.97867e6	−0.503197
$677$	3.16671e6	0.265544	0.132772	−	0.991147i	$-0.457612\pi$
$677$	3.16671e6	0.265544	0.132772	+	0.991147i	$0.457612\pi$
$678$	0	0
$679$	−461469.	−0.0384121
$680$	−4.30322e6	−0.356879
$681$	0	0
$682$	−1.75124e6	−0.144173
$683$	−1.97934e7	−1.62356	−0.811779	−	0.583964i	$-0.801501\pi$
$683$	−1.97934e7	−1.62356	−0.811779	+	0.583964i	$0.801501\pi$
$684$	0	0
$685$	−8.12077e6	−0.661258
$686$	−403875.	−0.0327670
$687$	0	0
$688$	−240545.	−0.0193743
$689$	−9.89662e6	−0.794216
$690$	0	0
$691$	2.33939e7	1.86384	0.931919	−	0.362666i	$-0.118134\pi$
$691$	2.33939e7	1.86384	0.931919	+	0.362666i	$0.118134\pi$
$692$	−7.19190e6	−0.570924
$693$	0	0
$694$	−1.39414e7	−1.09877
$695$	−5.97456e6	−0.469184
$696$	0	0
$697$	−2.73298e6	−0.213086
$698$	−9.28427e6	−0.721289
$699$	0	0
$700$	49361.8	0.00380755
$701$	−1.97814e7	−1.52041	−0.760207	−	0.649681i	$-0.774903\pi$
$701$	−1.97814e7	−1.52041	−0.760207	+	0.649681i	$0.774903\pi$
$702$	0	0
$703$	−1.11641e6	−0.0851994
$704$	1.88559e6	0.143389
$705$	0	0
$706$	9.63580e6	0.727572
$707$	571396.	0.0429921
$708$	0	0
$709$	1.68612e7	1.25971	0.629857	−	0.776711i	$-0.283114\pi$
$709$	1.68612e7	1.25971	0.629857	+	0.776711i	$0.283114\pi$
$710$	4.71914e6	0.351331
$711$	0	0
$712$	7.46787e6	0.552074
$713$	−1.98098e7	−1.45934
$714$	0	0
$715$	2.44018e6	0.178508
$716$	−6.84517e6	−0.499001
$717$	0	0
$718$	1.17964e7	0.853960
$719$	−8.24004e6	−0.594439	−0.297219	−	0.954809i	$-0.596059\pi$
$719$	−8.24004e6	−0.594439	−0.297219	+	0.954809i	$0.596059\pi$
$720$	0	0
$721$	630003.	0.0451340
$722$	8.01913e6	0.572512
$723$	0	0
$724$	1.56409e7	1.10896
$725$	−68555.2	−0.00484390
$726$	0	0
$727$	1.70476e7	1.19627	0.598133	−	0.801397i	$-0.295909\pi$
$727$	1.70476e7	1.19627	0.598133	+	0.801397i	$0.295909\pi$
$728$	517738.	0.0362061
$729$	0	0
$730$	5.86206e6	0.407139
$731$	−2.00986e6	−0.139114
$732$	0	0
$733$	−1.45811e6	−0.100238	−0.0501188	−	0.998743i	$-0.515960\pi$
$733$	−1.45811e6	−0.100238	−0.0501188	+	0.998743i	$0.515960\pi$
$734$	1.29466e7	0.886984
$735$	0	0
$736$	2.65189e7	1.80452
$737$	6.72277e6	0.455911
$738$	0	0
$739$	−1.40591e7	−0.946993	−0.473497	−	0.880796i	$-0.657008\pi$
$739$	−1.40591e7	−0.946993	−0.473497	+	0.880796i	$0.657008\pi$
$740$	−5.14418e6	−0.345332
$741$	0	0
$742$	147466.	0.00983293
$743$	−9.40934e6	−0.625298	−0.312649	−	0.949869i	$-0.601216\pi$
$743$	−9.40934e6	−0.625298	−0.312649	+	0.949869i	$0.601216\pi$
$744$	0	0
$745$	−4.63482e6	−0.305944
$746$	2.06660e6	0.135959
$747$	0	0
$748$	−2.56293e6	−0.167487
$749$	647734.	0.0421883
$750$	0	0
$751$	−2.34850e7	−1.51946	−0.759731	−	0.650237i	$-0.774670\pi$
$751$	−2.34850e7	−1.51946	−0.759731	+	0.650237i	$0.774670\pi$
$752$	−1.59658e6	−0.102955
$753$	0	0
$754$	−288129.	−0.0184569
$755$	−7.95331e6	−0.507786
$756$	0	0
$757$	7.94803e6	0.504104	0.252052	−	0.967714i	$-0.418895\pi$
$757$	7.94803e6	0.504104	0.252052	+	0.967714i	$0.418895\pi$
$758$	−8.84329e6	−0.559037
$759$	0	0
$760$	−504623.	−0.0316908
$761$	1.14046e7	0.713866	0.356933	−	0.934130i	$-0.383822\pi$
$761$	1.14046e7	0.713866	0.356933	+	0.934130i	$0.383822\pi$
$762$	0	0
$763$	−693198.	−0.0431068
$764$	−5.80757e6	−0.359966
$765$	0	0
$766$	4.13604e6	0.254690
$767$	4.06154e7	2.49289
$768$	0	0
$769$	3.65643e6	0.222967	0.111484	−	0.993766i	$-0.464440\pi$
$769$	3.65643e6	0.222967	0.111484	+	0.993766i	$0.464440\pi$
$770$	−36360.3	−0.00221004
$771$	0	0
$772$	−1.85752e7	−1.12173
$773$	−2.51934e7	−1.51649	−0.758244	−	0.651971i	$-0.773942\pi$
$773$	−2.51934e7	−1.51649	−0.758244	+	0.651971i	$0.773942\pi$
$774$	0	0
$775$	2.77786e6	0.166133
$776$	2.17375e7	1.29585
$777$	0	0
$778$	−9.50523e6	−0.563007
$779$	−320487.	−0.0189220
$780$	0	0
$781$	7.01421e6	0.411482
$782$	1.43679e7	0.840187
$783$	0	0
$784$	−1.98968e6	−0.115609
$785$	4.78263e6	0.277008
$786$	0	0
$787$	1.33416e7	0.767840	0.383920	−	0.923366i	$-0.374574\pi$
$787$	1.33416e7	0.767840	0.383920	+	0.923366i	$0.374574\pi$
$788$	−1.49861e7	−0.859751
$789$	0	0
$790$	7.54988e6	0.430400
$791$	415897.	0.0236344
$792$	0	0
$793$	4.03959e7	2.28115
$794$	−7.21121e6	−0.405935
$795$	0	0
$796$	−7.49348e6	−0.419180
$797$	−9.12714e6	−0.508966	−0.254483	−	0.967077i	$-0.581905\pi$
$797$	−9.12714e6	−0.508966	−0.254483	+	0.967077i	$0.581905\pi$
$798$	0	0
$799$	−1.33401e7	−0.739251
$800$	−3.71866e6	−0.205429
$801$	0	0
$802$	−1.85067e7	−1.01600
$803$	8.71296e6	0.476845
$804$	0	0
$805$	−411304.	−0.0223704
$806$	1.16750e7	0.633022
$807$	0	0
$808$	−2.69157e7	−1.45036
$809$	−5.55824e6	−0.298584	−0.149292	−	0.988793i	$-0.547699\pi$
$809$	−5.55824e6	−0.298584	−0.149292	+	0.988793i	$0.547699\pi$
$810$	0	0
$811$	8.03735e6	0.429102	0.214551	−	0.976713i	$-0.431171\pi$
$811$	8.03735e6	0.429102	0.214551	+	0.976713i	$0.431171\pi$
$812$	−8663.06	−0.000461085 0
$813$	0	0
$814$	3.78925e6	0.200444
$815$	−778389.	−0.0410490
$816$	0	0
$817$	−235689.	−0.0123533
$818$	9.86760e6	0.515618
$819$	0	0
$820$	−1.47673e6	−0.0766949
$821$	3.79134e7	1.96306	0.981532	−	0.191298i	$-0.0612698\pi$
$821$	3.79134e7	1.96306	0.981532	+	0.191298i	$0.0612698\pi$
$822$	0	0
$823$	1.85141e7	0.952801	0.476401	−	0.879228i	$-0.341941\pi$
$823$	1.85141e7	0.952801	0.476401	+	0.879228i	$0.341941\pi$
$824$	−2.96763e7	−1.52262
$825$	0	0
$826$	−605197.	−0.0308636
$827$	−2.93172e6	−0.149059	−0.0745295	−	0.997219i	$-0.523745\pi$
$827$	−2.93172e6	−0.149059	−0.0745295	+	0.997219i	$0.523745\pi$
$828$	0	0
$829$	−1.36785e7	−0.691278	−0.345639	−	0.938368i	$-0.612338\pi$
$829$	−1.36785e7	−0.691278	−0.345639	+	0.938368i	$0.612338\pi$
$830$	−5.41798e6	−0.272987
$831$	0	0
$832$	−1.25707e7	−0.629579
$833$	−1.66246e7	−0.830116
$834$	0	0
$835$	−7.50233e6	−0.372375
$836$	−300545.	−0.0148728
$837$	0	0
$838$	−5.05609e6	−0.248717
$839$	2.61080e7	1.28047	0.640234	−	0.768180i	$-0.278838\pi$
$839$	2.61080e7	1.28047	0.640234	+	0.768180i	$0.278838\pi$
$840$	0	0
$841$	−2.04991e7	−0.999413
$842$	9.76846e6	0.474839
$843$	0	0
$844$	−1.40803e6	−0.0680385
$845$	−6.98565e6	−0.336562
$846$	0	0
$847$	−54043.5	−0.00258842
$848$	1.45357e6	0.0694137
$849$	0	0
$850$	−2.01476e6	−0.0956479
$851$	4.28636e7	2.02892
$852$	0	0
$853$	1.74350e7	0.820445	0.410223	−	0.911985i	$-0.365451\pi$
$853$	1.74350e7	0.820445	0.410223	+	0.911985i	$0.365451\pi$
$854$	−601925.	−0.0282422
$855$	0	0
$856$	−3.05116e7	−1.42325
$857$	−1.61674e7	−0.751951	−0.375975	−	0.926630i	$-0.622692\pi$
$857$	−1.61674e7	−0.751951	−0.375975	+	0.926630i	$0.622692\pi$
$858$	0	0
$859$	4.88789e6	0.226016	0.113008	−	0.993594i	$-0.463951\pi$
$859$	4.88789e6	0.226016	0.113008	+	0.993594i	$0.463951\pi$
$860$	−1.08600e6	−0.0500707
$861$	0	0
$862$	9.25572e6	0.424270
$863$	6.16637e6	0.281840	0.140920	−	0.990021i	$-0.454994\pi$
$863$	6.16637e6	0.281840	0.140920	+	0.990021i	$0.454994\pi$
$864$	0	0
$865$	−8.40322e6	−0.381861
$866$	−1.80921e7	−0.819774
$867$	0	0
$868$	351027.	0.0158140
$869$	1.12216e7	0.504088
$870$	0	0
$871$	−4.48188e7	−2.00177
$872$	3.26531e7	1.45423
$873$	0	0
$874$	1.68487e6	0.0746084
$875$	57675.7	0.00254667
$876$	0	0
$877$	−2.48055e7	−1.08905	−0.544526	−	0.838744i	$-0.683291\pi$
$877$	−2.48055e7	−1.08905	−0.544526	+	0.838744i	$0.683291\pi$
$878$	−7.06322e6	−0.309219
$879$	0	0
$880$	−358402.	−0.0156014
$881$	2.66799e7	1.15810	0.579048	−	0.815293i	$-0.303424\pi$
$881$	2.66799e7	1.15810	0.579048	+	0.815293i	$0.303424\pi$
$882$	0	0
$883$	−1.66063e7	−0.716755	−0.358377	−	0.933577i	$-0.616670\pi$
$883$	−1.66063e7	−0.716755	−0.358377	+	0.933577i	$0.616670\pi$
$884$	1.70863e7	0.735389
$885$	0	0
$886$	1.22618e6	0.0524771
$887$	−1.31661e7	−0.561886	−0.280943	−	0.959725i	$-0.590647\pi$
$887$	−1.31661e7	−0.561886	−0.280943	+	0.959725i	$0.590647\pi$
$888$	0	0
$889$	−1.11967e6	−0.0475155
$890$	3.49644e6	0.147962
$891$	0	0
$892$	8.18202e6	0.344309
$893$	−1.56434e6	−0.0656452
$894$	0	0
$895$	−7.99809e6	−0.333756
$896$	−515484.	−0.0214509
$897$	0	0
$898$	−1.10023e7	−0.455296
$899$	−487518.	−0.0201183
$900$	0	0
$901$	1.21452e7	0.498415
$902$	1.08777e6	0.0445166
$903$	0	0
$904$	−1.95908e7	−0.797319
$905$	1.82752e7	0.741722
$906$	0	0
$907$	3.53803e7	1.42805	0.714025	−	0.700121i	$-0.246870\pi$
$907$	3.53803e7	1.42805	0.714025	+	0.700121i	$0.246870\pi$
$908$	−5.57042e6	−0.224220
$909$	0	0
$910$	242404.	0.00970366
$911$	−3.20552e7	−1.27968	−0.639841	−	0.768507i	$-0.721000\pi$
$911$	−3.20552e7	−1.27968	−0.639841	+	0.768507i	$0.721000\pi$
$912$	0	0
$913$	−8.05291e6	−0.319724
$914$	−1.44126e7	−0.570658
$915$	0	0
$916$	−1.56982e7	−0.618176
$917$	−122111.	−0.00479547
$918$	0	0
$919$	5.92372e6	0.231369	0.115685	−	0.993286i	$-0.463094\pi$
$919$	5.92372e6	0.231369	0.115685	+	0.993286i	$0.463094\pi$
$920$	1.93745e7	0.754677
$921$	0	0
$922$	−2.28299e6	−0.0884458
$923$	−4.67617e7	−1.80670
$924$	0	0
$925$	−6.01061e6	−0.230975
$926$	4.00397e6	0.153449
$927$	0	0
$928$	652630.	0.0248769
$929$	1.68912e7	0.642126	0.321063	−	0.947058i	$-0.395960\pi$
$929$	1.68912e7	0.642126	0.321063	+	0.947058i	$0.395960\pi$
$930$	0	0
$931$	−1.94951e6	−0.0737141
$932$	2.81273e7	1.06069
$933$	0	0
$934$	−67964.7	−0.00254927
$935$	−2.99460e6	−0.112024
$936$	0	0
$937$	1.74118e7	0.647879	0.323940	−	0.946078i	$-0.394993\pi$
$937$	1.74118e7	0.647879	0.323940	+	0.946078i	$0.394993\pi$
$938$	667829.	0.0247833
$939$	0	0
$940$	−7.20814e6	−0.266075
$941$	2.93262e7	1.07965	0.539823	−	0.841779i	$-0.318491\pi$
$941$	2.93262e7	1.07965	0.539823	+	0.841779i	$0.318491\pi$
$942$	0	0
$943$	1.23048e7	0.450603
$944$	−5.96539e6	−0.217876
$945$	0	0
$946$	799958.	0.0290629
$947$	1.23412e7	0.447180	0.223590	−	0.974683i	$-0.428222\pi$
$947$	1.23412e7	0.447180	0.223590	+	0.974683i	$0.428222\pi$
$948$	0	0
$949$	−5.80868e7	−2.09369
$950$	−236263.	−0.00849351
$951$	0	0
$952$	−635370.	−0.0227213
$953$	−3.95996e7	−1.41240	−0.706201	−	0.708012i	$-0.749592\pi$
$953$	−3.95996e7	−1.41240	−0.706201	+	0.708012i	$0.749592\pi$
$954$	0	0
$955$	−6.78573e6	−0.240762
$956$	1.65699e7	0.586373
$957$	0	0
$958$	−1.35436e7	−0.476782
$959$	−1.19903e6	−0.0421001
$960$	0	0
$961$	−8.87494e6	−0.309997
$962$	−2.52618e7	−0.880090
$963$	0	0
$964$	−8.49659e6	−0.294477
$965$	−2.17037e7	−0.750268
$966$	0	0
$967$	−7.99848e6	−0.275069	−0.137534	−	0.990497i	$-0.543918\pi$
$967$	−7.99848e6	−0.275069	−0.137534	+	0.990497i	$0.543918\pi$
$968$	2.54572e6	0.0873218
$969$	0	0
$970$	1.01775e7	0.347304
$971$	2.44875e7	0.833481	0.416740	−	0.909026i	$-0.363172\pi$
$971$	2.44875e7	0.833481	0.416740	+	0.909026i	$0.363172\pi$
$972$	0	0
$973$	−882142.	−0.0298715
$974$	−5.66918e6	−0.191480
$975$	0	0
$976$	−5.93314e6	−0.199370
$977$	3.03008e7	1.01559	0.507794	−	0.861478i	$-0.330461\pi$
$977$	3.03008e7	1.01559	0.507794	+	0.861478i	$0.330461\pi$
$978$	0	0
$979$	5.19687e6	0.173295
$980$	−8.98288e6	−0.298779
$981$	0	0
$982$	−1.27805e7	−0.422931
$983$	−1.42147e7	−0.469197	−0.234599	−	0.972092i	$-0.575378\pi$
$983$	−1.42147e7	−0.469197	−0.234599	+	0.972092i	$0.575378\pi$
$984$	0	0
$985$	−1.75102e7	−0.575043
$986$	353593.	0.0115827
$987$	0	0
$988$	2.00365e6	0.0653023
$989$	9.04904e6	0.294179
$990$	0	0
$991$	9.73392e6	0.314850	0.157425	−	0.987531i	$-0.449681\pi$
$991$	9.73392e6	0.314850	0.157425	+	0.987531i	$0.449681\pi$
$992$	−2.64445e7	−0.853212
$993$	0	0
$994$	696780.	0.0223681
$995$	−8.75560e6	−0.280368
$996$	0	0
$997$	1.60192e7	0.510390	0.255195	−	0.966890i	$-0.417860\pi$
$997$	1.60192e7	0.510390	0.255195	+	0.966890i	$0.417860\pi$
$998$	−9.60850e6	−0.305372
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.o.1.5	✓	10
3.2	odd	2		495.6.a.p.1.6	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.6.a.o.1.5	✓	10	1.1	even	1	trivial
495.6.a.p.1.6	yes	10	3.2	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)

\(f(q)\)	\(=\)	\(q-3.25634 q^{2} -21.3963 q^{4} -25.0000 q^{5} -3.69124 q^{7} +173.876 q^{8} +O(q^{10})\)	\(q-3.25634 q^{2} -21.3963 q^{4} -25.0000 q^{5} -3.69124 q^{7} +173.876 q^{8} +81.4085 q^{10} +121.000 q^{11} -806.671 q^{13} +12.0199 q^{14} +118.480 q^{16} +989.950 q^{17} +116.088 q^{19} +534.906 q^{20} -394.017 q^{22} -4457.08 q^{23} +625.000 q^{25} +2626.80 q^{26} +78.9788 q^{28} -109.688 q^{29} +4444.57 q^{31} -5949.85 q^{32} -3223.61 q^{34} +92.2811 q^{35} -9616.97 q^{37} -378.021 q^{38} -4346.91 q^{40} -2760.73 q^{41} -2030.26 q^{43} -2588.95 q^{44} +14513.8 q^{46} -13475.5 q^{47} -16793.4 q^{49} -2035.21 q^{50} +17259.7 q^{52} +12268.5 q^{53} -3025.00 q^{55} -641.820 q^{56} +357.182 q^{58} -50349.4 q^{59} -50077.2 q^{61} -14473.0 q^{62} +15583.4 q^{64} +20166.8 q^{65} +55560.1 q^{67} -21181.2 q^{68} -300.499 q^{70} +57968.7 q^{71} +72008.0 q^{73} +31316.1 q^{74} -2483.84 q^{76} -446.641 q^{77} +92740.7 q^{79} -2962.00 q^{80} +8989.86 q^{82} -66553.0 q^{83} -24748.7 q^{85} +6611.22 q^{86} +21039.0 q^{88} +42949.4 q^{89} +2977.62 q^{91} +95364.7 q^{92} +43880.9 q^{94} -2902.19 q^{95} +125017. q^{97} +54684.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} + 181 q^{4} - 250 q^{5} + 116 q^{7} - 129 q^{8}+O(q^{10}) \) 10 * q - 3 * q^2 + 181 * q^4 - 250 * q^5 + 116 * q^7 - 129 * q^8	\( 10 q - 3 q^{2} + 181 q^{4} - 250 q^{5} + 116 q^{7} - 129 q^{8} + 75 q^{10} + 1210 q^{11} + 932 q^{13} + 1332 q^{14} + 2701 q^{16} - 96 q^{17} + 1664 q^{19} - 4525 q^{20} - 363 q^{22} - 6288 q^{23} + 6250 q^{25} - 13380 q^{26} + 13868 q^{28} - 11208 q^{29} + 9032 q^{31} - 9801 q^{32} + 14610 q^{34} - 2900 q^{35} + 21572 q^{37} - 15870 q^{38} + 3225 q^{40} - 10800 q^{41} + 21128 q^{43} + 21901 q^{44} + 83982 q^{46} + 17400 q^{47} + 71610 q^{49} - 1875 q^{50} + 40640 q^{52} - 5004 q^{53} - 30250 q^{55} + 54012 q^{56} - 9786 q^{58} + 25272 q^{59} + 52004 q^{61} - 34740 q^{62} + 56953 q^{64} - 23300 q^{65} + 4160 q^{67} + 87978 q^{68} - 33300 q^{70} + 65232 q^{71} + 44252 q^{73} + 49842 q^{74} + 233246 q^{76} + 14036 q^{77} + 112604 q^{79} - 67525 q^{80} + 167910 q^{82} - 70032 q^{83} + 2400 q^{85} + 72978 q^{86} - 15609 q^{88} + 46848 q^{89} + 130672 q^{91} - 121302 q^{92} + 252294 q^{94} - 41600 q^{95} + 129932 q^{97} + 316137 q^{98}+O(q^{100}) \) 10 * q - 3 * q^2 + 181 * q^4 - 250 * q^5 + 116 * q^7 - 129 * q^8 + 75 * q^10 + 1210 * q^11 + 932 * q^13 + 1332 * q^14 + 2701 * q^16 - 96 * q^17 + 1664 * q^19 - 4525 * q^20 - 363 * q^22 - 6288 * q^23 + 6250 * q^25 - 13380 * q^26 + 13868 * q^28 - 11208 * q^29 + 9032 * q^31 - 9801 * q^32 + 14610 * q^34 - 2900 * q^35 + 21572 * q^37 - 15870 * q^38 + 3225 * q^40 - 10800 * q^41 + 21128 * q^43 + 21901 * q^44 + 83982 * q^46 + 17400 * q^47 + 71610 * q^49 - 1875 * q^50 + 40640 * q^52 - 5004 * q^53 - 30250 * q^55 + 54012 * q^56 - 9786 * q^58 + 25272 * q^59 + 52004 * q^61 - 34740 * q^62 + 56953 * q^64 - 23300 * q^65 + 4160 * q^67 + 87978 * q^68 - 33300 * q^70 + 65232 * q^71 + 44252 * q^73 + 49842 * q^74 + 233246 * q^76 + 14036 * q^77 + 112604 * q^79 - 67525 * q^80 + 167910 * q^82 - 70032 * q^83 + 2400 * q^85 + 72978 * q^86 - 15609 * q^88 + 46848 * q^89 + 130672 * q^91 - 121302 * q^92 + 252294 * q^94 - 41600 * q^95 + 129932 * q^97 + 316137 * q^98

Introduction

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