LMFDB - Embedded newform 495.6.a.e.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.34253.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 52x + 48 $ x^3 - x^2 - 52*x + 48
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.17710$ 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)

$f(q)$	$=$	$q-5.17710 q^{2} -5.19759 q^{4} -25.0000 q^{5} -123.437 q^{7} +192.576 q^{8} +O(q^{10})$	\(q-5.17710 q^{2} -5.19759 q^{4} -25.0000 q^{5} -123.437 q^{7} +192.576 q^{8} +129.428 q^{10} +121.000 q^{11} -500.053 q^{13} +639.048 q^{14} -830.662 q^{16} +422.631 q^{17} -932.948 q^{19} +129.940 q^{20} -626.430 q^{22} +1225.18 q^{23} +625.000 q^{25} +2588.83 q^{26} +641.576 q^{28} +2111.62 q^{29} -159.612 q^{31} -1862.00 q^{32} -2188.00 q^{34} +3085.93 q^{35} +5414.46 q^{37} +4829.97 q^{38} -4814.40 q^{40} +18066.7 q^{41} +6815.47 q^{43} -628.908 q^{44} -6342.88 q^{46} +15098.9 q^{47} -1570.23 q^{49} -3235.69 q^{50} +2599.07 q^{52} -15367.7 q^{53} -3025.00 q^{55} -23771.0 q^{56} -10932.1 q^{58} -23400.5 q^{59} +10768.4 q^{61} +826.328 q^{62} +36221.0 q^{64} +12501.3 q^{65} +14507.1 q^{67} -2196.66 q^{68} -15976.2 q^{70} +28114.0 q^{71} -28836.7 q^{73} -28031.2 q^{74} +4849.08 q^{76} -14935.9 q^{77} -8150.52 q^{79} +20766.6 q^{80} -93533.0 q^{82} +109864. q^{83} -10565.8 q^{85} -35284.4 q^{86} +23301.7 q^{88} -69673.6 q^{89} +61725.2 q^{91} -6367.98 q^{92} -78168.5 q^{94} +23323.7 q^{95} +91551.4 q^{97} +8129.23 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8}+O(q^{10}) $ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8	$ 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8} - 175 q^{10} + 363 q^{11} - 654 q^{13} + 728 q^{14} - 415 q^{16} + 2366 q^{17} - 2872 q^{19} - 625 q^{20} + 847 q^{22} - 2272 q^{23} + 1875 q^{25} - 3422 q^{26} + 4592 q^{28} + 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 2506 q^{34} + 4300 q^{35} - 9126 q^{37} - 13076 q^{38} - 5775 q^{40} + 8758 q^{41} - 14672 q^{43} + 3025 q^{44} - 28768 q^{46} + 19392 q^{47} - 26629 q^{49} + 4375 q^{50} - 61506 q^{52} + 4598 q^{53} - 9075 q^{55} - 2688 q^{56} + 8550 q^{58} + 9348 q^{59} - 60078 q^{61} + 14096 q^{62} - 7087 q^{64} + 16350 q^{65} - 38468 q^{67} - 59778 q^{68} - 18200 q^{70} + 74032 q^{71} - 44442 q^{73} - 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 108116 q^{79} + 10375 q^{80} - 92230 q^{82} + 81892 q^{83} - 59150 q^{85} - 126412 q^{86} + 27951 q^{88} - 167342 q^{89} - 31832 q^{91} - 72960 q^{92} + 12728 q^{94} + 71800 q^{95} + 159702 q^{97} - 163121 q^{98}+O(q^{100}) $ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 - 175 * q^10 + 363 * q^11 - 654 * q^13 + 728 * q^14 - 415 * q^16 + 2366 * q^17 - 2872 * q^19 - 625 * q^20 + 847 * q^22 - 2272 * q^23 + 1875 * q^25 - 3422 * q^26 + 4592 * q^28 + 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 2506 * q^34 + 4300 * q^35 - 9126 * q^37 - 13076 * q^38 - 5775 * q^40 + 8758 * q^41 - 14672 * q^43 + 3025 * q^44 - 28768 * q^46 + 19392 * q^47 - 26629 * q^49 + 4375 * q^50 - 61506 * q^52 + 4598 * q^53 - 9075 * q^55 - 2688 * q^56 + 8550 * q^58 + 9348 * q^59 - 60078 * q^61 + 14096 * q^62 - 7087 * q^64 + 16350 * q^65 - 38468 * q^67 - 59778 * q^68 - 18200 * q^70 + 74032 * q^71 - 44442 * q^73 - 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 108116 * q^79 + 10375 * q^80 - 92230 * q^82 + 81892 * q^83 - 59150 * q^85 - 126412 * q^86 + 27951 * q^88 - 167342 * q^89 - 31832 * q^91 - 72960 * q^92 + 12728 * q^94 + 71800 * q^95 + 159702 * q^97 - 163121 * 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 $

$2$	−5.17710	−0.915191	−0.457596	−	0.889160i	$-0.651289\pi$
$2$	−5.17710	−0.915191	−0.457596	+	0.889160i	$0.651289\pi$
$3$	0	0
$3$	0	0
$4$	−5.19759	−0.162425
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−123.437	−0.952141	−0.476071	−	0.879407i	$-0.657939\pi$
$7$	−123.437	−0.952141	−0.476071	+	0.879407i	$0.657939\pi$
$8$	192.576	1.06384
$9$	0	0
$10$	129.428	0.409286
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−500.053	−0.820649	−0.410324	−	0.911940i	$-0.634585\pi$
$13$	−500.053	−0.820649	−0.410324	+	0.911940i	$0.634585\pi$
$14$	639.048	0.871392
$15$	0	0
$16$	−830.662	−0.811194
$17$	422.631	0.354682	0.177341	−	0.984150i	$-0.443251\pi$
$17$	422.631	0.354682	0.177341	+	0.984150i	$0.443251\pi$
$18$	0	0
$19$	−932.948	−0.592889	−0.296445	−	0.955050i	$-0.595801\pi$
$19$	−932.948	−0.592889	−0.296445	+	0.955050i	$0.595801\pi$
$20$	129.940	0.0726385
$21$	0	0
$22$	−626.430	−0.275941
$23$	1225.18	0.482926	0.241463	−	0.970410i	$-0.422373\pi$
$23$	1225.18	0.482926	0.241463	+	0.970410i	$0.422373\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	2588.83	0.751051
$27$	0	0
$28$	641.576	0.154651
$29$	2111.62	0.466251	0.233126	−	0.972447i	$-0.425105\pi$
$29$	2111.62	0.466251	0.233126	+	0.972447i	$0.425105\pi$
$30$	0	0
$31$	−159.612	−0.0298306	−0.0149153	−	0.999889i	$-0.504748\pi$
$31$	−159.612	−0.0298306	−0.0149153	+	0.999889i	$0.504748\pi$
$32$	−1862.00	−0.321444
$33$	0	0
$34$	−2188.00	−0.324601
$35$	3085.93	0.425811
$36$	0	0
$37$	5414.46	0.650205	0.325103	−	0.945679i	$-0.394601\pi$
$37$	5414.46	0.650205	0.325103	+	0.945679i	$0.394601\pi$
$38$	4829.97	0.542607
$39$	0	0
$40$	−4814.40	−0.475764
$41$	18066.7	1.67849	0.839244	−	0.543756i	$-0.182998\pi$
$41$	18066.7	1.67849	0.839244	+	0.543756i	$0.182998\pi$
$42$	0	0
$43$	6815.47	0.562114	0.281057	−	0.959691i	$-0.409315\pi$
$43$	6815.47	0.562114	0.281057	+	0.959691i	$0.409315\pi$
$44$	−628.908	−0.0489729
$45$	0	0
$46$	−6342.88	−0.441969
$47$	15098.9	0.997012	0.498506	−	0.866886i	$-0.333882\pi$
$47$	15098.9	0.997012	0.498506	+	0.866886i	$0.333882\pi$
$48$	0	0
$49$	−1570.23	−0.0934270
$50$	−3235.69	−0.183038
$51$	0	0
$52$	2599.07	0.133294
$53$	−15367.7	−0.751484	−0.375742	−	0.926724i	$-0.622612\pi$
$53$	−15367.7	−0.751484	−0.375742	+	0.926724i	$0.622612\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−23771.0	−1.01293
$57$	0	0
$58$	−10932.1	−0.426709
$59$	−23400.5	−0.875177	−0.437588	−	0.899175i	$-0.644167\pi$
$59$	−23400.5	−0.875177	−0.437588	+	0.899175i	$0.644167\pi$
$60$	0	0
$61$	10768.4	0.370532	0.185266	−	0.982688i	$-0.440685\pi$
$61$	10768.4	0.370532	0.185266	+	0.982688i	$0.440685\pi$
$62$	826.328	0.0273007
$63$	0	0
$64$	36221.0	1.10538
$65$	12501.3	0.367005
$66$	0	0
$67$	14507.1	0.394814	0.197407	−	0.980322i	$-0.436748\pi$
$67$	14507.1	0.394814	0.197407	+	0.980322i	$0.436748\pi$
$68$	−2196.66	−0.0576090
$69$	0	0
$70$	−15976.2	−0.389698
$71$	28114.0	0.661876	0.330938	−	0.943653i	$-0.392635\pi$
$71$	28114.0	0.661876	0.330938	+	0.943653i	$0.392635\pi$
$72$	0	0
$73$	−28836.7	−0.633342	−0.316671	−	0.948536i	$-0.602565\pi$
$73$	−28836.7	−0.633342	−0.316671	+	0.948536i	$0.602565\pi$
$74$	−28031.2	−0.595062
$75$	0	0
$76$	4849.08	0.0962998
$77$	−14935.9	−0.287081
$78$	0	0
$79$	−8150.52	−0.146932	−0.0734662	−	0.997298i	$-0.523406\pi$
$79$	−8150.52	−0.146932	−0.0734662	+	0.997298i	$0.523406\pi$
$80$	20766.6	0.362777
$81$	0	0
$82$	−93533.0	−1.53614
$83$	109864.	1.75049	0.875246	−	0.483679i	$-0.160700\pi$
$83$	109864.	1.75049	0.875246	+	0.483679i	$0.160700\pi$
$84$	0	0
$85$	−10565.8	−0.158618
$86$	−35284.4	−0.514442
$87$	0	0
$88$	23301.7	0.320760
$89$	−69673.6	−0.932380	−0.466190	−	0.884685i	$-0.654374\pi$
$89$	−69673.6	−0.932380	−0.466190	+	0.884685i	$0.654374\pi$
$90$	0	0
$91$	61725.2	0.781374
$92$	−6367.98	−0.0784390
$93$	0	0
$94$	−78168.5	−0.912457
$95$	23323.7	0.265148
$96$	0	0
$97$	91551.4	0.987952	0.493976	−	0.869476i	$-0.335543\pi$
$97$	91551.4	0.987952	0.493976	+	0.869476i	$0.335543\pi$
$98$	8129.23	0.0855036
$99$	0	0
$100$	−3248.49	−0.0324849
$101$	−21299.3	−0.207760	−0.103880	−	0.994590i	$-0.533126\pi$
$101$	−21299.3	−0.207760	−0.103880	+	0.994590i	$0.533126\pi$
$102$	0	0
$103$	−6548.06	−0.0608163	−0.0304081	−	0.999538i	$-0.509681\pi$
$103$	−6548.06	−0.0608163	−0.0304081	+	0.999538i	$0.509681\pi$
$104$	−96298.1	−0.873040
$105$	0	0
$106$	79560.3	0.687752
$107$	−127171.	−1.07381	−0.536905	−	0.843643i	$-0.680407\pi$
$107$	−127171.	−1.07381	−0.536905	+	0.843643i	$0.680407\pi$
$108$	0	0
$109$	57285.2	0.461823	0.230912	−	0.972975i	$-0.425829\pi$
$109$	57285.2	0.461823	0.230912	+	0.972975i	$0.425829\pi$
$110$	15660.7	0.123404
$111$	0	0
$112$	102535.	0.772371
$113$	67774.2	0.499308	0.249654	−	0.968335i	$-0.419683\pi$
$113$	67774.2	0.499308	0.249654	+	0.968335i	$0.419683\pi$
$114$	0	0
$115$	−30629.5	−0.215971
$116$	−10975.3	−0.0757307
$117$	0	0
$118$	121147.	0.800954
$119$	−52168.4	−0.337707
$120$	0	0
$121$	14641.0	0.0909091
$122$	−55749.1	−0.339108
$123$	0	0
$124$	829.598	0.00484522
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−115352.	−0.634622	−0.317311	−	0.948322i	$-0.602780\pi$
$127$	−115352.	−0.634622	−0.317311	+	0.948322i	$0.602780\pi$
$128$	−127936.	−0.690187
$129$	0	0
$130$	−64720.6	−0.335880
$131$	−43276.4	−0.220329	−0.110165	−	0.993913i	$-0.535138\pi$
$131$	−43276.4	−0.220329	−0.110165	+	0.993913i	$0.535138\pi$
$132$	0	0
$133$	115161.	0.564514
$134$	−75104.6	−0.361331
$135$	0	0
$136$	81388.4	0.377325
$137$	−401439.	−1.82734	−0.913668	−	0.406461i	$-0.866763\pi$
$137$	−401439.	−1.82734	−0.913668	+	0.406461i	$0.866763\pi$
$138$	0	0
$139$	−302018.	−1.32586	−0.662928	−	0.748683i	$-0.730686\pi$
$139$	−302018.	−1.32586	−0.662928	+	0.748683i	$0.730686\pi$
$140$	−16039.4	−0.0691621
$141$	0	0
$142$	−145549.	−0.605743
$143$	−60506.4	−0.247435
$144$	0	0
$145$	−52790.4	−0.208514
$146$	149290.	0.579629
$147$	0	0
$148$	−28142.1	−0.105609
$149$	−326942.	−1.20644	−0.603218	−	0.797576i	$-0.706115\pi$
$149$	−326942.	−1.20644	−0.603218	+	0.797576i	$0.706115\pi$
$150$	0	0
$151$	164681.	0.587761	0.293881	−	0.955842i	$-0.405053\pi$
$151$	164681.	0.587761	0.293881	+	0.955842i	$0.405053\pi$
$152$	−179663.	−0.630740
$153$	0	0
$154$	77324.8	0.262734
$155$	3990.30	0.0133406
$156$	0	0
$157$	248620.	0.804982	0.402491	−	0.915424i	$-0.368144\pi$
$157$	248620.	0.804982	0.402491	+	0.915424i	$0.368144\pi$
$158$	42196.1	0.134471
$159$	0	0
$160$	46550.0	0.143754
$161$	−151233.	−0.459813
$162$	0	0
$163$	−417656.	−1.23126	−0.615629	−	0.788036i	$-0.711098\pi$
$163$	−417656.	−1.23126	−0.615629	+	0.788036i	$0.711098\pi$
$164$	−93903.0	−0.272628
$165$	0	0
$166$	−568777.	−1.60203
$167$	−704955.	−1.95601	−0.978004	−	0.208588i	$-0.933113\pi$
$167$	−704955.	−1.95601	−0.978004	+	0.208588i	$0.933113\pi$
$168$	0	0
$169$	−121240.	−0.326535
$170$	54700.1	0.145166
$171$	0	0
$172$	−35424.0	−0.0913012
$173$	171060.	0.434545	0.217272	−	0.976111i	$-0.430284\pi$
$173$	171060.	0.434545	0.217272	+	0.976111i	$0.430284\pi$
$174$	0	0
$175$	−77148.3	−0.190428
$176$	−100510.	−0.244584
$177$	0	0
$178$	360707.	0.853306
$179$	166327.	0.387999	0.193999	−	0.981002i	$-0.437854\pi$
$179$	166327.	0.387999	0.193999	+	0.981002i	$0.437854\pi$
$180$	0	0
$181$	−584391.	−1.32589	−0.662945	−	0.748668i	$-0.730694\pi$
$181$	−584391.	−1.32589	−0.662945	+	0.748668i	$0.730694\pi$
$182$	−319558.	−0.715107
$183$	0	0
$184$	235940.	0.513756
$185$	−135361.	−0.290781
$186$	0	0
$187$	51138.3	0.106941
$188$	−78477.8	−0.161939
$189$	0	0
$190$	−120749.	−0.242661
$191$	715510.	1.41916	0.709581	−	0.704624i	$-0.248884\pi$
$191$	715510.	1.41916	0.709581	+	0.704624i	$0.248884\pi$
$192$	0	0
$193$	−922088.	−1.78188	−0.890941	−	0.454118i	$-0.849954\pi$
$193$	−922088.	−1.78188	−0.890941	+	0.454118i	$0.849954\pi$
$194$	−473971.	−0.904165
$195$	0	0
$196$	8161.40	0.0151749
$197$	613632.	1.12653	0.563265	−	0.826276i	$-0.309545\pi$
$197$	613632.	1.12653	0.563265	+	0.826276i	$0.309545\pi$
$198$	0	0
$199$	−378985.	−0.678406	−0.339203	−	0.940713i	$-0.610157\pi$
$199$	−378985.	−0.678406	−0.339203	+	0.940713i	$0.610157\pi$
$200$	120360.	0.212768
$201$	0	0
$202$	110269.	0.190140
$203$	−260652.	−0.443937
$204$	0	0
$205$	−451666.	−0.750642
$206$	33900.0	0.0556585
$207$	0	0
$208$	415375.	0.665705
$209$	−112887.	−0.178763
$210$	0	0
$211$	−473721.	−0.732515	−0.366257	−	0.930514i	$-0.619361\pi$
$211$	−473721.	−0.732515	−0.366257	+	0.930514i	$0.619361\pi$
$212$	79875.1	0.122060
$213$	0	0
$214$	658376.	0.982741
$215$	−170387.	−0.251385
$216$	0	0
$217$	19702.1	0.0284029
$218$	−296571.	−0.422657
$219$	0	0
$220$	15722.7	0.0219013
$221$	−211338.	−0.291069
$222$	0	0
$223$	−822747.	−1.10791	−0.553954	−	0.832547i	$-0.686882\pi$
$223$	−822747.	−1.10791	−0.553954	+	0.832547i	$0.686882\pi$
$224$	229840.	0.306060
$225$	0	0
$226$	−350874.	−0.456962
$227$	556097.	0.716285	0.358143	−	0.933667i	$-0.383410\pi$
$227$	556097.	0.716285	0.358143	+	0.933667i	$0.383410\pi$
$228$	0	0
$229$	−634919.	−0.800073	−0.400036	−	0.916499i	$-0.631003\pi$
$229$	−634919.	−0.800073	−0.400036	+	0.916499i	$0.631003\pi$
$230$	158572.	0.197655
$231$	0	0
$232$	406646.	0.496017
$233$	906561.	1.09397	0.546987	−	0.837141i	$-0.315775\pi$
$233$	906561.	1.09397	0.546987	+	0.837141i	$0.315775\pi$
$234$	0	0
$235$	−377472.	−0.445877
$236$	121626.	0.142150
$237$	0	0
$238$	270081.	0.309066
$239$	−662586.	−0.750322	−0.375161	−	0.926960i	$-0.622413\pi$
$239$	−662586.	−0.750322	−0.375161	+	0.926960i	$0.622413\pi$
$240$	0	0
$241$	−1.31823e6	−1.46200	−0.731001	−	0.682376i	$-0.760947\pi$
$241$	−1.31823e6	−1.46200	−0.731001	+	0.682376i	$0.760947\pi$
$242$	−75798.0	−0.0831992
$243$	0	0
$244$	−55969.6	−0.0601836
$245$	39255.7	0.0417818
$246$	0	0
$247$	466523.	0.486554
$248$	−30737.4	−0.0317350
$249$	0	0
$250$	80892.3	0.0818572
$251$	11073.9	0.0110947	0.00554737	−	0.999985i	$-0.498234\pi$
$251$	11073.9	0.0110947	0.00554737	+	0.999985i	$0.498234\pi$
$252$	0	0
$253$	148247.	0.145608
$254$	597188.	0.580800
$255$	0	0
$256$	−496734.	−0.473723
$257$	788596.	0.744769	0.372384	−	0.928079i	$-0.378540\pi$
$257$	788596.	0.744769	0.372384	+	0.928079i	$0.378540\pi$
$258$	0	0
$259$	−668346.	−0.619087
$260$	−64976.7	−0.0596107
$261$	0	0
$262$	224046.	0.201644
$263$	−211325.	−0.188391	−0.0941956	−	0.995554i	$-0.530028\pi$
$263$	−211325.	−0.188391	−0.0941956	+	0.995554i	$0.530028\pi$
$264$	0	0
$265$	384193.	0.336074
$266$	−596198.	−0.516639
$267$	0	0
$268$	−75401.8	−0.0641276
$269$	−552342.	−0.465401	−0.232701	−	0.972548i	$-0.574756\pi$
$269$	−552342.	−0.465401	−0.232701	+	0.972548i	$0.574756\pi$
$270$	0	0
$271$	−1.49255e6	−1.23454	−0.617271	−	0.786751i	$-0.711762\pi$
$271$	−1.49255e6	−1.23454	−0.617271	+	0.786751i	$0.711762\pi$
$272$	−351063.	−0.287715
$273$	0	0
$274$	2.07829e6	1.67236
$275$	75625.0	0.0603023
$276$	0	0
$277$	1.01431e6	0.794275	0.397138	−	0.917759i	$-0.370004\pi$
$277$	1.01431e6	0.794275	0.397138	+	0.917759i	$0.370004\pi$
$278$	1.56358e6	1.21341
$279$	0	0
$280$	594276.	0.452995
$281$	658216.	0.497282	0.248641	−	0.968596i	$-0.420016\pi$
$281$	658216.	0.497282	0.248641	+	0.968596i	$0.420016\pi$
$282$	0	0
$283$	1.65981e6	1.23195	0.615974	−	0.787767i	$-0.288763\pi$
$283$	1.65981e6	1.23195	0.615974	+	0.787767i	$0.288763\pi$
$284$	−146125.	−0.107505
$285$	0	0
$286$	313248.	0.226450
$287$	−2.23010e6	−1.59816
$288$	0	0
$289$	−1.24124e6	−0.874201
$290$	273301.	0.190830
$291$	0	0
$292$	149881.	0.102870
$293$	410864.	0.279594	0.139797	−	0.990180i	$-0.455355\pi$
$293$	410864.	0.279594	0.139797	+	0.990180i	$0.455355\pi$
$294$	0	0
$295$	585013.	0.391391
$296$	1.04269e6	0.691715
$297$	0	0
$298$	1.69261e6	1.10412
$299$	−612654.	−0.396312
$300$	0	0
$301$	−841283.	−0.535212
$302$	−852570.	−0.537914
$303$	0	0
$304$	774965.	0.480948
$305$	−269210.	−0.165707
$306$	0	0
$307$	1.34831e6	0.816477	0.408238	−	0.912875i	$-0.366143\pi$
$307$	1.34831e6	0.816477	0.408238	+	0.912875i	$0.366143\pi$
$308$	77630.7	0.0466291
$309$	0	0
$310$	−20658.2	−0.0122092
$311$	2.52585e6	1.48083	0.740417	−	0.672148i	$-0.234628\pi$
$311$	2.52585e6	1.48083	0.740417	+	0.672148i	$0.234628\pi$
$312$	0	0
$313$	2.23061e6	1.28695	0.643476	−	0.765466i	$-0.277491\pi$
$313$	2.23061e6	1.28695	0.643476	+	0.765466i	$0.277491\pi$
$314$	−1.28713e6	−0.736713
$315$	0	0
$316$	42363.0	0.0238654
$317$	120908.	0.0675780	0.0337890	−	0.999429i	$-0.489243\pi$
$317$	120908.	0.0675780	0.0337890	+	0.999429i	$0.489243\pi$
$318$	0	0
$319$	255506.	0.140580
$320$	−905524.	−0.494339
$321$	0	0
$322$	782948.	0.420817
$323$	−394292.	−0.210287
$324$	0	0
$325$	−312533.	−0.164130
$326$	2.16225e6	1.12684
$327$	0	0
$328$	3.47920e6	1.78564
$329$	−1.86377e6	−0.949296
$330$	0	0
$331$	578480.	0.290214	0.145107	−	0.989416i	$-0.453647\pi$
$331$	578480.	0.290214	0.145107	+	0.989416i	$0.453647\pi$
$332$	−571028.	−0.284323
$333$	0	0
$334$	3.64963e6	1.79012
$335$	−362677.	−0.176566
$336$	0	0
$337$	−1.78710e6	−0.857186	−0.428593	−	0.903498i	$-0.640991\pi$
$337$	−1.78710e6	−0.857186	−0.428593	+	0.903498i	$0.640991\pi$
$338$	627674.	0.298842
$339$	0	0
$340$	54916.5	0.0257635
$341$	−19313.1	−0.00899425
$342$	0	0
$343$	2.26844e6	1.04110
$344$	1.31249e6	0.598000
$345$	0	0
$346$	−885598.	−0.397692
$347$	1.46282e6	0.652179	0.326089	−	0.945339i	$-0.394269\pi$
$347$	1.46282e6	0.652179	0.326089	+	0.945339i	$0.394269\pi$
$348$	0	0
$349$	1.48501e6	0.652627	0.326314	−	0.945262i	$-0.394193\pi$
$349$	1.48501e6	0.652627	0.326314	+	0.945262i	$0.394193\pi$
$350$	399405.	0.174278
$351$	0	0
$352$	−225302.	−0.0969189
$353$	−1.34528e6	−0.574616	−0.287308	−	0.957838i	$-0.592760\pi$
$353$	−1.34528e6	−0.574616	−0.287308	+	0.957838i	$0.592760\pi$
$354$	0	0
$355$	−702850.	−0.296000
$356$	362135.	0.151442
$357$	0	0
$358$	−861092.	−0.355093
$359$	1.83956e6	0.753316	0.376658	−	0.926352i	$-0.377073\pi$
$359$	1.83956e6	0.753316	0.376658	+	0.926352i	$0.377073\pi$
$360$	0	0
$361$	−1.60571e6	−0.648483
$362$	3.02546e6	1.21344
$363$	0	0
$364$	−320822.	−0.126914
$365$	720917.	0.283239
$366$	0	0
$367$	−312079.	−0.120948	−0.0604741	−	0.998170i	$-0.519261\pi$
$367$	−312079.	−0.120948	−0.0604741	+	0.998170i	$0.519261\pi$
$368$	−1.01771e6	−0.391746
$369$	0	0
$370$	700780.	0.266120
$371$	1.89695e6	0.715519
$372$	0	0
$373$	3.38658e6	1.26034	0.630172	−	0.776455i	$-0.282984\pi$
$373$	3.38658e6	1.26034	0.630172	+	0.776455i	$0.282984\pi$
$374$	−264748.	−0.0978710
$375$	0	0
$376$	2.90768e6	1.06066
$377$	−1.05592e6	−0.382629
$378$	0	0
$379$	−1.62370e6	−0.580641	−0.290321	−	0.956929i	$-0.593762\pi$
$379$	−1.62370e6	−0.580641	−0.290321	+	0.956929i	$0.593762\pi$
$380$	−121227.	−0.0430666
$381$	0	0
$382$	−3.70427e6	−1.29881
$383$	3.06187e6	1.06657	0.533285	−	0.845935i	$-0.320957\pi$
$383$	3.06187e6	1.06657	0.533285	+	0.845935i	$0.320957\pi$
$384$	0	0
$385$	373398.	0.128387
$386$	4.77375e6	1.63076
$387$	0	0
$388$	−475847.	−0.160468
$389$	−4.21840e6	−1.41343	−0.706715	−	0.707499i	$-0.749824\pi$
$389$	−4.21840e6	−1.41343	−0.706715	+	0.707499i	$0.749824\pi$
$390$	0	0
$391$	517798.	0.171285
$392$	−302388.	−0.0993915
$393$	0	0
$394$	−3.17684e6	−1.03099
$395$	203763.	0.0657101
$396$	0	0
$397$	513185.	0.163417	0.0817085	−	0.996656i	$-0.473962\pi$
$397$	513185.	0.163417	0.0817085	+	0.996656i	$0.473962\pi$
$398$	1.96205e6	0.620871
$399$	0	0
$400$	−519164.	−0.162239
$401$	2.18458e6	0.678432	0.339216	−	0.940709i	$-0.389838\pi$
$401$	2.18458e6	0.678432	0.339216	+	0.940709i	$0.389838\pi$
$402$	0	0
$403$	79814.5	0.0244804
$404$	110705.	0.0337453
$405$	0	0
$406$	1.34942e6	0.406287
$407$	655149.	0.196044
$408$	0	0
$409$	1.22307e6	0.361529	0.180764	−	0.983526i	$-0.442143\pi$
$409$	1.22307e6	0.361529	0.180764	+	0.983526i	$0.442143\pi$
$410$	2.33832e6	0.686981
$411$	0	0
$412$	34034.1	0.00987806
$413$	2.88850e6	0.833292
$414$	0	0
$415$	−2.74660e6	−0.782844
$416$	931098.	0.263792
$417$	0	0
$418$	584426.	0.163602
$419$	−2.42879e6	−0.675858	−0.337929	−	0.941172i	$-0.609726\pi$
$419$	−2.42879e6	−0.675858	−0.337929	+	0.941172i	$0.609726\pi$
$420$	0	0
$421$	−727467.	−0.200036	−0.100018	−	0.994986i	$-0.531890\pi$
$421$	−727467.	−0.200036	−0.100018	+	0.994986i	$0.531890\pi$
$422$	2.45250e6	0.670391
$423$	0	0
$424$	−2.95945e6	−0.799460
$425$	264144.	0.0709363
$426$	0	0
$427$	−1.32922e6	−0.352799
$428$	660981.	0.174413
$429$	0	0
$430$	882110.	0.230066
$431$	−6.18223e6	−1.60307	−0.801534	−	0.597949i	$-0.795982\pi$
$431$	−6.18223e6	−1.60307	−0.801534	+	0.597949i	$0.795982\pi$
$432$	0	0
$433$	2.42337e6	0.621154	0.310577	−	0.950548i	$-0.399478\pi$
$433$	2.42337e6	0.621154	0.310577	+	0.950548i	$0.399478\pi$
$434$	−102000.	−0.0259941
$435$	0	0
$436$	−297745.	−0.0750115
$437$	−1.14303e6	−0.286321
$438$	0	0
$439$	−3.79993e6	−0.941054	−0.470527	−	0.882385i	$-0.655936\pi$
$439$	−3.79993e6	−0.941054	−0.470527	+	0.882385i	$0.655936\pi$
$440$	−582542.	−0.143448
$441$	0	0
$442$	1.09412e6	0.266384
$443$	5.58044e6	1.35101	0.675506	−	0.737354i	$-0.263925\pi$
$443$	5.58044e6	1.35101	0.675506	+	0.737354i	$0.263925\pi$
$444$	0	0
$445$	1.74184e6	0.416973
$446$	4.25945e6	1.01395
$447$	0	0
$448$	−4.47102e6	−1.05247
$449$	−2.69609e6	−0.631130	−0.315565	−	0.948904i	$-0.602194\pi$
$449$	−2.69609e6	−0.631130	−0.315565	+	0.948904i	$0.602194\pi$
$450$	0	0
$451$	2.18607e6	0.506083
$452$	−352263.	−0.0810999
$453$	0	0
$454$	−2.87897e6	−0.655538
$455$	−1.54313e6	−0.349441
$456$	0	0
$457$	1.14257e6	0.255912	0.127956	−	0.991780i	$-0.459158\pi$
$457$	1.14257e6	0.255912	0.127956	+	0.991780i	$0.459158\pi$
$458$	3.28704e6	0.732220
$459$	0	0
$460$	159199.	0.0350790
$461$	−5.37091e6	−1.17705	−0.588526	−	0.808478i	$-0.700292\pi$
$461$	−5.37091e6	−1.17705	−0.588526	+	0.808478i	$0.700292\pi$
$462$	0	0
$463$	−3.80227e6	−0.824311	−0.412155	−	0.911114i	$-0.635224\pi$
$463$	−3.80227e6	−0.824311	−0.412155	+	0.911114i	$0.635224\pi$
$464$	−1.75404e6	−0.378220
$465$	0	0
$466$	−4.69336e6	−1.00120
$467$	−9.41694e6	−1.99810	−0.999051	−	0.0435609i	$-0.986130\pi$
$467$	−9.41694e6	−1.99810	−0.999051	+	0.0435609i	$0.986130\pi$
$468$	0	0
$469$	−1.79071e6	−0.375919
$470$	1.95421e6	0.408063
$471$	0	0
$472$	−4.50638e6	−0.931049
$473$	824672.	0.169484
$474$	0	0
$475$	−583093.	−0.118578
$476$	271150.	0.0548519
$477$	0	0
$478$	3.43028e6	0.686688
$479$	4.69047e6	0.934065	0.467033	−	0.884240i	$-0.345323\pi$
$479$	4.69047e6	0.934065	0.467033	+	0.884240i	$0.345323\pi$
$480$	0	0
$481$	−2.70751e6	−0.533590
$482$	6.82461e6	1.33801
$483$	0	0
$484$	−76097.9	−0.0147659
$485$	−2.28879e6	−0.441825
$486$	0	0
$487$	7.07177e6	1.35116	0.675579	−	0.737288i	$-0.263894\pi$
$487$	7.07177e6	1.35116	0.675579	+	0.737288i	$0.263894\pi$
$488$	2.07373e6	0.394187
$489$	0	0
$490$	−203231.	−0.0382384
$491$	−906051.	−0.169609	−0.0848045	−	0.996398i	$-0.527027\pi$
$491$	−906051.	−0.169609	−0.0848045	+	0.996398i	$0.527027\pi$
$492$	0	0
$493$	892434.	0.165371
$494$	−2.41524e6	−0.445290
$495$	0	0
$496$	132584.	0.0241984
$497$	−3.47031e6	−0.630199
$498$	0	0
$499$	−7.13323e6	−1.28243	−0.641217	−	0.767360i	$-0.721570\pi$
$499$	−7.13323e6	−1.28243	−0.641217	+	0.767360i	$0.721570\pi$
$500$	81212.3	0.0145277
$501$	0	0
$502$	−57330.9	−0.0101538
$503$	2.34927e6	0.414013	0.207006	−	0.978340i	$-0.433628\pi$
$503$	2.34927e6	0.414013	0.207006	+	0.978340i	$0.433628\pi$
$504$	0	0
$505$	532482.	0.0929131
$506$	−767489.	−0.133259
$507$	0	0
$508$	599551.	0.103078
$509$	−1.97148e6	−0.337286	−0.168643	−	0.985677i	$-0.553938\pi$
$509$	−1.97148e6	−0.337286	−0.168643	+	0.985677i	$0.553938\pi$
$510$	0	0
$511$	3.55952e6	0.603031
$512$	6.66559e6	1.12373
$513$	0	0
$514$	−4.08264e6	−0.681606
$515$	163702.	0.0271979
$516$	0	0
$517$	1.82697e6	0.300610
$518$	3.46010e6	0.566583
$519$	0	0
$520$	2.40745e6	0.390435
$521$	−1.20074e7	−1.93801	−0.969004	−	0.247046i	$-0.920540\pi$
$521$	−1.20074e7	−1.93801	−0.969004	+	0.247046i	$0.920540\pi$
$522$	0	0
$523$	4.16772e6	0.666260	0.333130	−	0.942881i	$-0.391895\pi$
$523$	4.16772e6	0.666260	0.333130	+	0.942881i	$0.391895\pi$
$524$	224933.	0.0357869
$525$	0	0
$526$	1.09405e6	0.172414
$527$	−67456.9	−0.0105804
$528$	0	0
$529$	−4.93528e6	−0.766783
$530$	−1.98901e6	−0.307572
$531$	0	0
$532$	−598557.	−0.0916910
$533$	−9.03428e6	−1.37745
$534$	0	0
$535$	3.17927e6	0.480222
$536$	2.79371e6	0.420020
$537$	0	0
$538$	2.85953e6	0.425931
$539$	−189998.	−0.0281693
$540$	0	0
$541$	1.01106e7	1.48519	0.742594	−	0.669741i	$-0.233595\pi$
$541$	1.01106e7	1.48519	0.742594	+	0.669741i	$0.233595\pi$
$542$	7.72709e6	1.12984
$543$	0	0
$544$	−786938.	−0.114010
$545$	−1.43213e6	−0.206534
$546$	0	0
$547$	−7.39087e6	−1.05615	−0.528077	−	0.849196i	$-0.677087\pi$
$547$	−7.39087e6	−1.05615	−0.528077	+	0.849196i	$0.677087\pi$
$548$	2.08652e6	0.296805
$549$	0	0
$550$	−391519.	−0.0551881
$551$	−1.97003e6	−0.276435
$552$	0	0
$553$	1.00608e6	0.139900
$554$	−5.25119e6	−0.726914
$555$	0	0
$556$	1.56977e6	0.215352
$557$	−1.52459e6	−0.208217	−0.104108	−	0.994566i	$-0.533199\pi$
$557$	−1.52459e6	−0.208217	−0.104108	+	0.994566i	$0.533199\pi$
$558$	0	0
$559$	−3.40809e6	−0.461299
$560$	−2.56337e6	−0.345415
$561$	0	0
$562$	−3.40765e6	−0.455108
$563$	1.40845e7	1.87271	0.936353	−	0.351059i	$-0.114178\pi$
$563$	1.40845e7	1.87271	0.936353	+	0.351059i	$0.114178\pi$
$564$	0	0
$565$	−1.69436e6	−0.223297
$566$	−8.59301e6	−1.12747
$567$	0	0
$568$	5.41407e6	0.704131
$569$	1.30473e7	1.68942	0.844712	−	0.535221i	$-0.179772\pi$
$569$	1.30473e7	1.68942	0.844712	+	0.535221i	$0.179772\pi$
$570$	0	0
$571$	−1.19873e7	−1.53862	−0.769312	−	0.638873i	$-0.779401\pi$
$571$	−1.19873e7	−1.53862	−0.769312	+	0.638873i	$0.779401\pi$
$572$	314487.	0.0401895
$573$	0	0
$574$	1.15455e7	1.46262
$575$	765737.	0.0965851
$576$	0	0
$577$	568904.	0.0711376	0.0355688	−	0.999367i	$-0.488676\pi$
$577$	568904.	0.0711376	0.0355688	+	0.999367i	$0.488676\pi$
$578$	6.42603e6	0.800061
$579$	0	0
$580$	274383.	0.0338678
$581$	−1.35613e7	−1.66671
$582$	0	0
$583$	−1.85949e6	−0.226581
$584$	−5.55325e6	−0.673775
$585$	0	0
$586$	−2.12708e6	−0.255882
$587$	−2.93323e6	−0.351358	−0.175679	−	0.984447i	$-0.556212\pi$
$587$	−2.93323e6	−0.351358	−0.175679	+	0.984447i	$0.556212\pi$
$588$	0	0
$589$	148910.	0.0176862
$590$	−3.02868e6	−0.358198
$591$	0	0
$592$	−4.49758e6	−0.527442
$593$	−5.90557e6	−0.689644	−0.344822	−	0.938668i	$-0.612061\pi$
$593$	−5.90557e6	−0.689644	−0.344822	+	0.938668i	$0.612061\pi$
$594$	0	0
$595$	1.30421e6	0.151027
$596$	1.69931e6	0.195955
$597$	0	0
$598$	3.17178e6	0.362702
$599$	−1.07665e7	−1.22605	−0.613025	−	0.790064i	$-0.710047\pi$
$599$	−1.07665e7	−1.22605	−0.613025	+	0.790064i	$0.710047\pi$
$600$	0	0
$601$	−1.19746e7	−1.35231	−0.676154	−	0.736760i	$-0.736355\pi$
$601$	−1.19746e7	−1.35231	−0.676154	+	0.736760i	$0.736355\pi$
$602$	4.35541e6	0.489822
$603$	0	0
$604$	−855944.	−0.0954669
$605$	−366025.	−0.0406558
$606$	0	0
$607$	−3.45506e6	−0.380613	−0.190307	−	0.981725i	$-0.560948\pi$
$607$	−3.45506e6	−0.380613	−0.190307	+	0.981725i	$0.560948\pi$
$608$	1.73715e6	0.190580
$609$	0	0
$610$	1.39373e6	0.151654
$611$	−7.55024e6	−0.818197
$612$	0	0
$613$	−8.22419e6	−0.883979	−0.441990	−	0.897020i	$-0.645727\pi$
$613$	−8.22419e6	−0.883979	−0.441990	+	0.897020i	$0.645727\pi$
$614$	−6.98034e6	−0.747233
$615$	0	0
$616$	−2.87630e6	−0.305409
$617$	−9.83567e6	−1.04014	−0.520069	−	0.854124i	$-0.674094\pi$
$617$	−9.83567e6	−1.04014	−0.520069	+	0.854124i	$0.674094\pi$
$618$	0	0
$619$	267648.	0.0280762	0.0140381	−	0.999901i	$-0.495531\pi$
$619$	267648.	0.0280762	0.0140381	+	0.999901i	$0.495531\pi$
$620$	−20739.9	−0.00216685
$621$	0	0
$622$	−1.30766e7	−1.35525
$623$	8.60032e6	0.887758
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.15481e7	−1.17781
$627$	0	0
$628$	−1.29222e6	−0.130749
$629$	2.28831e6	0.230616
$630$	0	0
$631$	−4.17135e6	−0.417065	−0.208532	−	0.978015i	$-0.566869\pi$
$631$	−4.17135e6	−0.417065	−0.208532	+	0.978015i	$0.566869\pi$
$632$	−1.56959e6	−0.156313
$633$	0	0
$634$	−625951.	−0.0618468
$635$	2.88379e6	0.283811
$636$	0	0
$637$	785197.	0.0766708
$638$	−1.32278e6	−0.128658
$639$	0	0
$640$	3.19839e6	0.308661
$641$	−8.24673e6	−0.792751	−0.396376	−	0.918088i	$-0.629732\pi$
$641$	−8.24673e6	−0.792751	−0.396376	+	0.918088i	$0.629732\pi$
$642$	0	0
$643$	1.07773e7	1.02797	0.513986	−	0.857799i	$-0.328168\pi$
$643$	1.07773e7	1.02797	0.513986	+	0.857799i	$0.328168\pi$
$644$	786046.	0.0746850
$645$	0	0
$646$	2.04129e6	0.192453
$647$	−7.89194e6	−0.741179	−0.370590	−	0.928797i	$-0.620844\pi$
$647$	−7.89194e6	−0.741179	−0.370590	+	0.928797i	$0.620844\pi$
$648$	0	0
$649$	−2.83147e6	−0.263876
$650$	1.61802e6	0.150210
$651$	0	0
$652$	2.17080e6	0.199987
$653$	−1.47858e7	−1.35695	−0.678473	−	0.734625i	$-0.737358\pi$
$653$	−1.47858e7	−1.35695	−0.678473	+	0.734625i	$0.737358\pi$
$654$	0	0
$655$	1.08191e6	0.0985343
$656$	−1.50073e7	−1.36158
$657$	0	0
$658$	9.64891e6	0.868788
$659$	475088.	0.0426148	0.0213074	−	0.999773i	$-0.493217\pi$
$659$	475088.	0.0426148	0.0213074	+	0.999773i	$0.493217\pi$
$660$	0	0
$661$	2.73604e6	0.243567	0.121784	−	0.992557i	$-0.461139\pi$
$661$	2.73604e6	0.243567	0.121784	+	0.992557i	$0.461139\pi$
$662$	−2.99485e6	−0.265601
$663$	0	0
$664$	2.11571e7	1.86224
$665$	−2.87902e6	−0.252458
$666$	0	0
$667$	2.58711e6	0.225165
$668$	3.66407e6	0.317704
$669$	0	0
$670$	1.87762e6	0.161592
$671$	1.30297e6	0.111720
$672$	0	0
$673$	1.52861e7	1.30094	0.650471	−	0.759531i	$-0.274572\pi$
$673$	1.52861e7	1.30094	0.650471	+	0.759531i	$0.274572\pi$
$674$	9.25202e6	0.784489
$675$	0	0
$676$	630157.	0.0530374
$677$	3.93225e6	0.329739	0.164869	−	0.986315i	$-0.447280\pi$
$677$	3.93225e6	0.329739	0.164869	+	0.986315i	$0.447280\pi$
$678$	0	0
$679$	−1.13009e7	−0.940670
$680$	−2.03471e6	−0.168745
$681$	0	0
$682$	99985.7	0.00823146
$683$	−3.42591e6	−0.281011	−0.140506	−	0.990080i	$-0.544873\pi$
$683$	−3.42591e6	−0.281011	−0.140506	+	0.990080i	$0.544873\pi$
$684$	0	0
$685$	1.00360e7	0.817210
$686$	−1.17439e7	−0.952803
$687$	0	0
$688$	−5.66135e6	−0.455984
$689$	7.68467e6	0.616705
$690$	0	0
$691$	1.81896e7	1.44920	0.724599	−	0.689171i	$-0.242025\pi$
$691$	1.81896e7	1.44920	0.724599	+	0.689171i	$0.242025\pi$
$692$	−889102.	−0.0705808
$693$	0	0
$694$	−7.57316e6	−0.596868
$695$	7.55046e6	0.592941
$696$	0	0
$697$	7.63552e6	0.595328
$698$	−7.68804e6	−0.597279
$699$	0	0
$700$	400985.	0.0309302
$701$	1.15598e6	0.0888494	0.0444247	−	0.999013i	$-0.485855\pi$
$701$	1.15598e6	0.0888494	0.0444247	+	0.999013i	$0.485855\pi$
$702$	0	0
$703$	−5.05141e6	−0.385500
$704$	4.38274e6	0.333283
$705$	0	0
$706$	6.96468e6	0.525883
$707$	2.62913e6	0.197817
$708$	0	0
$709$	2.42664e7	1.81296	0.906481	−	0.422246i	$-0.138758\pi$
$709$	2.42664e7	1.81296	0.906481	+	0.422246i	$0.138758\pi$
$710$	3.63873e6	0.270897
$711$	0	0
$712$	−1.34174e7	−0.991904
$713$	−195553.	−0.0144059
$714$	0	0
$715$	1.51266e6	0.110656
$716$	−864499.	−0.0630205
$717$	0	0
$718$	−9.52359e6	−0.689429
$719$	2.25503e7	1.62679	0.813394	−	0.581714i	$-0.197618\pi$
$719$	2.25503e7	1.62679	0.813394	+	0.581714i	$0.197618\pi$
$720$	0	0
$721$	808276.	0.0579057
$722$	8.31291e6	0.593486
$723$	0	0
$724$	3.03743e6	0.215357
$725$	1.31976e6	0.0932503
$726$	0	0
$727$	−2.17941e7	−1.52933	−0.764667	−	0.644425i	$-0.777097\pi$
$727$	−2.17941e7	−1.52933	−0.764667	+	0.644425i	$0.777097\pi$
$728$	1.18868e7	0.831257
$729$	0	0
$730$	−3.73226e6	−0.259218
$731$	2.88043e6	0.199372
$732$	0	0
$733$	1.73565e7	1.19317	0.596583	−	0.802551i	$-0.296524\pi$
$733$	1.73565e7	1.19317	0.596583	+	0.802551i	$0.296524\pi$
$734$	1.61567e6	0.110691
$735$	0	0
$736$	−2.28129e6	−0.155233
$737$	1.75536e6	0.119041
$738$	0	0
$739$	1.16961e6	0.0787825	0.0393912	−	0.999224i	$-0.487458\pi$
$739$	1.16961e6	0.0787825	0.0393912	+	0.999224i	$0.487458\pi$
$740$	703553.	0.0472300
$741$	0	0
$742$	−9.82071e6	−0.654837
$743$	−2.98047e6	−0.198067	−0.0990336	−	0.995084i	$-0.531575\pi$
$743$	−2.98047e6	−0.198067	−0.0990336	+	0.995084i	$0.531575\pi$
$744$	0	0
$745$	8.17354e6	0.539535
$746$	−1.75327e7	−1.15346
$747$	0	0
$748$	−265796.	−0.0173698
$749$	1.56976e7	1.02242
$750$	0	0
$751$	−1.83578e7	−1.18774	−0.593868	−	0.804563i	$-0.702400\pi$
$751$	−1.83578e7	−1.18774	−0.593868	+	0.804563i	$0.702400\pi$
$752$	−1.25421e7	−0.808770
$753$	0	0
$754$	5.46661e6	0.350178
$755$	−4.11702e6	−0.262855
$756$	0	0
$757$	1.60050e7	1.01511	0.507557	−	0.861618i	$-0.330549\pi$
$757$	1.60050e7	1.01511	0.507557	+	0.861618i	$0.330549\pi$
$758$	8.40607e6	0.531398
$759$	0	0
$760$	4.49158e6	0.282075
$761$	1.67436e7	1.04807	0.524033	−	0.851698i	$-0.324427\pi$
$761$	1.67436e7	1.04807	0.524033	+	0.851698i	$0.324427\pi$
$762$	0	0
$763$	−7.07113e6	−0.439721
$764$	−3.71892e6	−0.230507
$765$	0	0
$766$	−1.58516e7	−0.976116
$767$	1.17015e7	0.718213
$768$	0	0
$769$	−1.73863e7	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$769$	−1.73863e7	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$770$	−1.93312e6	−0.117498
$771$	0	0
$772$	4.79263e6	0.289422
$773$	−1.79095e7	−1.07804	−0.539021	−	0.842292i	$-0.681206\pi$
$773$	−1.79095e7	−1.07804	−0.539021	+	0.842292i	$0.681206\pi$
$774$	0	0
$775$	−99757.5	−0.00596611
$776$	1.76306e7	1.05102
$777$	0	0
$778$	2.18391e7	1.29356
$779$	−1.68552e7	−0.995157
$780$	0	0
$781$	3.40179e6	0.199563
$782$	−2.68070e6	−0.156758
$783$	0	0
$784$	1.30433e6	0.0757874
$785$	−6.21549e6	−0.359999
$786$	0	0
$787$	−3.01291e7	−1.73400	−0.867000	−	0.498308i	$-0.833955\pi$
$787$	−3.01291e7	−1.73400	−0.867000	+	0.498308i	$0.833955\pi$
$788$	−3.18941e6	−0.182976
$789$	0	0
$790$	−1.05490e6	−0.0601374
$791$	−8.36587e6	−0.475412
$792$	0	0
$793$	−5.38476e6	−0.304077
$794$	−2.65681e6	−0.149558
$795$	0	0
$796$	1.96981e6	0.110190
$797$	5.51251e6	0.307400	0.153700	−	0.988118i	$-0.450881\pi$
$797$	5.51251e6	0.307400	0.153700	+	0.988118i	$0.450881\pi$
$798$	0	0
$799$	6.38125e6	0.353622
$800$	−1.16375e6	−0.0642887
$801$	0	0
$802$	−1.13098e7	−0.620895
$803$	−3.48924e6	−0.190960
$804$	0	0
$805$	3.78082e6	0.205635
$806$	−413208.	−0.0224043
$807$	0	0
$808$	−4.10173e6	−0.221024
$809$	−3.11564e7	−1.67369	−0.836847	−	0.547436i	$-0.815604\pi$
$809$	−3.11564e7	−1.67369	−0.836847	+	0.547436i	$0.815604\pi$
$810$	0	0
$811$	−1.11336e7	−0.594406	−0.297203	−	0.954814i	$-0.596054\pi$
$811$	−1.11336e7	−0.594406	−0.297203	+	0.954814i	$0.596054\pi$
$812$	1.35476e6	0.0721063
$813$	0	0
$814$	−3.39178e6	−0.179418
$815$	1.04414e7	0.550636
$816$	0	0
$817$	−6.35848e6	−0.333271
$818$	−6.33195e6	−0.330868
$819$	0	0
$820$	2.34758e6	0.121923
$821$	3.44603e7	1.78427	0.892136	−	0.451767i	$-0.149206\pi$
$821$	3.44603e7	1.78427	0.892136	+	0.451767i	$0.149206\pi$
$822$	0	0
$823$	−9.74417e6	−0.501470	−0.250735	−	0.968056i	$-0.580672\pi$
$823$	−9.74417e6	−0.501470	−0.250735	+	0.968056i	$0.580672\pi$
$824$	−1.26100e6	−0.0646989
$825$	0	0
$826$	−1.49541e7	−0.762622
$827$	9.63214e6	0.489733	0.244866	−	0.969557i	$-0.421256\pi$
$827$	9.63214e6	0.489733	0.244866	+	0.969557i	$0.421256\pi$
$828$	0	0
$829$	−2.79310e7	−1.41156	−0.705780	−	0.708431i	$-0.749403\pi$
$829$	−2.79310e7	−1.41156	−0.705780	+	0.708431i	$0.749403\pi$
$830$	1.42194e7	0.716452
$831$	0	0
$832$	−1.81124e7	−0.907126
$833$	−663626.	−0.0331368
$834$	0	0
$835$	1.76239e7	0.874753
$836$	586739.	0.0290355
$837$	0	0
$838$	1.25741e7	0.618540
$839$	3.51475e7	1.72381	0.861906	−	0.507068i	$-0.169271\pi$
$839$	3.51475e7	1.72381	0.861906	+	0.507068i	$0.169271\pi$
$840$	0	0
$841$	−1.60522e7	−0.782610
$842$	3.76617e6	0.183071
$843$	0	0
$844$	2.46221e6	0.118978
$845$	3.03101e6	0.146031
$846$	0	0
$847$	−1.80725e6	−0.0865583
$848$	1.27654e7	0.609599
$849$	0	0
$850$	−1.36750e6	−0.0649203
$851$	6.63368e6	0.314001
$852$	0	0
$853$	2.50498e7	1.17878	0.589389	−	0.807849i	$-0.299369\pi$
$853$	2.50498e7	1.17878	0.589389	+	0.807849i	$0.299369\pi$
$854$	6.88152e6	0.322879
$855$	0	0
$856$	−2.44900e7	−1.14236
$857$	1.89631e7	0.881975	0.440987	−	0.897513i	$-0.354628\pi$
$857$	1.89631e7	0.881975	0.440987	+	0.897513i	$0.354628\pi$
$858$	0	0
$859$	−3.04605e7	−1.40849	−0.704245	−	0.709957i	$-0.748714\pi$
$859$	−3.04605e7	−1.40849	−0.704245	+	0.709957i	$0.748714\pi$
$860$	885600.	0.0408312
$861$	0	0
$862$	3.20060e7	1.46711
$863$	−3.46176e7	−1.58223	−0.791115	−	0.611667i	$-0.790499\pi$
$863$	−3.46176e7	−1.58223	−0.791115	+	0.611667i	$0.790499\pi$
$864$	0	0
$865$	−4.27651e6	−0.194334
$866$	−1.25460e7	−0.568475
$867$	0	0
$868$	−102403.	−0.00461333
$869$	−986213.	−0.0443018
$870$	0	0
$871$	−7.25430e6	−0.324004
$872$	1.10317e7	0.491307
$873$	0	0
$874$	5.91758e6	0.262039
$875$	1.92871e6	0.0851621
$876$	0	0
$877$	−3.61979e7	−1.58922	−0.794611	−	0.607119i	$-0.792325\pi$
$877$	−3.61979e7	−1.58922	−0.794611	+	0.607119i	$0.792325\pi$
$878$	1.96727e7	0.861245
$879$	0	0
$880$	2.51275e6	0.109381
$881$	−2.40371e7	−1.04338	−0.521690	−	0.853135i	$-0.674698\pi$
$881$	−2.40371e7	−1.04338	−0.521690	+	0.853135i	$0.674698\pi$
$882$	0	0
$883$	3.64938e7	1.57513	0.787567	−	0.616229i	$-0.211341\pi$
$883$	3.64938e7	1.57513	0.787567	+	0.616229i	$0.211341\pi$
$884$	1.09845e6	0.0472768
$885$	0	0
$886$	−2.88905e7	−1.23644
$887$	−2.29197e7	−0.978137	−0.489069	−	0.872245i	$-0.662663\pi$
$887$	−2.29197e7	−0.978137	−0.489069	+	0.872245i	$0.662663\pi$
$888$	0	0
$889$	1.42387e7	0.604250
$890$	−9.01768e6	−0.381610
$891$	0	0
$892$	4.27630e6	0.179952
$893$	−1.40865e7	−0.591117
$894$	0	0
$895$	−4.15817e6	−0.173518
$896$	1.57920e7	0.657156
$897$	0	0
$898$	1.39579e7	0.577605
$899$	−337039.	−0.0139085
$900$	0	0
$901$	−6.49487e6	−0.266538
$902$	−1.13175e7	−0.463163
$903$	0	0
$904$	1.30517e7	0.531184
$905$	1.46098e7	0.592956
$906$	0	0
$907$	−2.66664e7	−1.07633	−0.538167	−	0.842838i	$-0.680883\pi$
$907$	−2.66664e7	−1.07633	−0.538167	+	0.842838i	$0.680883\pi$
$908$	−2.89036e6	−0.116342
$909$	0	0
$910$	7.98894e6	0.319805
$911$	−1.73286e7	−0.691778	−0.345889	−	0.938276i	$-0.612423\pi$
$911$	−1.73286e7	−0.691778	−0.345889	+	0.938276i	$0.612423\pi$
$912$	0	0
$913$	1.32935e7	0.527793
$914$	−5.91518e6	−0.234208
$915$	0	0
$916$	3.30005e6	0.129952
$917$	5.34192e6	0.209785
$918$	0	0
$919$	−1.72198e7	−0.672573	−0.336286	−	0.941760i	$-0.609171\pi$
$919$	−1.72198e7	−0.672573	−0.336286	+	0.941760i	$0.609171\pi$
$920$	−5.89850e6	−0.229759
$921$	0	0
$922$	2.78058e7	1.07723
$923$	−1.40585e7	−0.543168
$924$	0	0
$925$	3.38404e6	0.130041
$926$	1.96848e7	0.754402
$927$	0	0
$928$	−3.93183e6	−0.149874
$929$	−5.47137e6	−0.207997	−0.103998	−	0.994577i	$-0.533164\pi$
$929$	−5.47137e6	−0.207997	−0.103998	+	0.994577i	$0.533164\pi$
$930$	0	0
$931$	1.46494e6	0.0553919
$932$	−4.71193e6	−0.177688
$933$	0	0
$934$	4.87525e7	1.82865
$935$	−1.27846e6	−0.0478252
$936$	0	0
$937$	8.04892e6	0.299494	0.149747	−	0.988724i	$-0.452154\pi$
$937$	8.04892e6	0.299494	0.149747	+	0.988724i	$0.452154\pi$
$938$	9.27072e6	0.344038
$939$	0	0
$940$	1.96195e6	0.0724215
$941$	1.13721e7	0.418665	0.209333	−	0.977844i	$-0.432871\pi$
$941$	1.13721e7	0.418665	0.209333	+	0.977844i	$0.432871\pi$
$942$	0	0
$943$	2.21349e7	0.810584
$944$	1.94379e7	0.709938
$945$	0	0
$946$	−4.26941e6	−0.155110
$947$	−4.97017e7	−1.80093	−0.900463	−	0.434932i	$-0.856772\pi$
$947$	−4.97017e7	−1.80093	−0.900463	+	0.434932i	$0.856772\pi$
$948$	0	0
$949$	1.44199e7	0.519751
$950$	3.01873e6	0.108521
$951$	0	0
$952$	−1.00464e7	−0.359266
$953$	−2.95809e7	−1.05507	−0.527533	−	0.849534i	$-0.676883\pi$
$953$	−2.95809e7	−1.05507	−0.527533	+	0.849534i	$0.676883\pi$
$954$	0	0
$955$	−1.78877e7	−0.634669
$956$	3.44385e6	0.121871
$957$	0	0
$958$	−2.42830e7	−0.854849
$959$	4.95526e7	1.73988
$960$	0	0
$961$	−2.86037e7	−0.999110
$962$	1.40171e7	0.488337
$963$	0	0
$964$	6.85161e6	0.237465
$965$	2.30522e7	0.796882
$966$	0	0
$967$	3.10475e7	1.06773	0.533863	−	0.845571i	$-0.320740\pi$
$967$	3.10475e7	1.06773	0.533863	+	0.845571i	$0.320740\pi$
$968$	2.81950e6	0.0967128
$969$	0	0
$970$	1.18493e7	0.404355
$971$	1.89426e7	0.644751	0.322376	−	0.946612i	$-0.395519\pi$
$971$	1.89426e7	0.644751	0.322376	+	0.946612i	$0.395519\pi$
$972$	0	0
$973$	3.72803e7	1.26240
$974$	−3.66113e7	−1.23657
$975$	0	0
$976$	−8.94489e6	−0.300573
$977$	−5.49466e7	−1.84164	−0.920819	−	0.389989i	$-0.872479\pi$
$977$	−5.49466e7	−1.84164	−0.920819	+	0.389989i	$0.872479\pi$
$978$	0	0
$979$	−8.43050e6	−0.281123
$980$	−204035.	−0.00678640
$981$	0	0
$982$	4.69072e6	0.155225
$983$	4.72203e7	1.55864	0.779318	−	0.626628i	$-0.215565\pi$
$983$	4.72203e7	1.55864	0.779318	+	0.626628i	$0.215565\pi$
$984$	0	0
$985$	−1.53408e7	−0.503799
$986$	−4.62022e6	−0.151346
$987$	0	0
$988$	−2.42480e6	−0.0790283
$989$	8.35018e6	0.271459
$990$	0	0
$991$	4.48844e7	1.45182	0.725908	−	0.687792i	$-0.241420\pi$
$991$	4.48844e7	1.45182	0.725908	+	0.687792i	$0.241420\pi$
$992$	297198.	0.00958885
$993$	0	0
$994$	1.79662e7	0.576753
$995$	9.47463e6	0.303392
$996$	0	0
$997$	5.08797e7	1.62109	0.810543	−	0.585679i	$-0.199172\pi$
$997$	5.08797e7	1.62109	0.810543	+	0.585679i	$0.199172\pi$
$998$	3.69295e7	1.17367
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.a.1.3	✓	3	3.2	odd	2
495.6.a.e.1.1		3	1.1	even	1	trivial
825.6.a.j.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-5.17710 q^{2} -5.19759 q^{4} -25.0000 q^{5} -123.437 q^{7} +192.576 q^{8} +O(q^{10})\)	\(q-5.17710 q^{2} -5.19759 q^{4} -25.0000 q^{5} -123.437 q^{7} +192.576 q^{8} +129.428 q^{10} +121.000 q^{11} -500.053 q^{13} +639.048 q^{14} -830.662 q^{16} +422.631 q^{17} -932.948 q^{19} +129.940 q^{20} -626.430 q^{22} +1225.18 q^{23} +625.000 q^{25} +2588.83 q^{26} +641.576 q^{28} +2111.62 q^{29} -159.612 q^{31} -1862.00 q^{32} -2188.00 q^{34} +3085.93 q^{35} +5414.46 q^{37} +4829.97 q^{38} -4814.40 q^{40} +18066.7 q^{41} +6815.47 q^{43} -628.908 q^{44} -6342.88 q^{46} +15098.9 q^{47} -1570.23 q^{49} -3235.69 q^{50} +2599.07 q^{52} -15367.7 q^{53} -3025.00 q^{55} -23771.0 q^{56} -10932.1 q^{58} -23400.5 q^{59} +10768.4 q^{61} +826.328 q^{62} +36221.0 q^{64} +12501.3 q^{65} +14507.1 q^{67} -2196.66 q^{68} -15976.2 q^{70} +28114.0 q^{71} -28836.7 q^{73} -28031.2 q^{74} +4849.08 q^{76} -14935.9 q^{77} -8150.52 q^{79} +20766.6 q^{80} -93533.0 q^{82} +109864. q^{83} -10565.8 q^{85} -35284.4 q^{86} +23301.7 q^{88} -69673.6 q^{89} +61725.2 q^{91} -6367.98 q^{92} -78168.5 q^{94} +23323.7 q^{95} +91551.4 q^{97} +8129.23 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8}+O(q^{10}) \) 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8	\( 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8} - 175 q^{10} + 363 q^{11} - 654 q^{13} + 728 q^{14} - 415 q^{16} + 2366 q^{17} - 2872 q^{19} - 625 q^{20} + 847 q^{22} - 2272 q^{23} + 1875 q^{25} - 3422 q^{26} + 4592 q^{28} + 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 2506 q^{34} + 4300 q^{35} - 9126 q^{37} - 13076 q^{38} - 5775 q^{40} + 8758 q^{41} - 14672 q^{43} + 3025 q^{44} - 28768 q^{46} + 19392 q^{47} - 26629 q^{49} + 4375 q^{50} - 61506 q^{52} + 4598 q^{53} - 9075 q^{55} - 2688 q^{56} + 8550 q^{58} + 9348 q^{59} - 60078 q^{61} + 14096 q^{62} - 7087 q^{64} + 16350 q^{65} - 38468 q^{67} - 59778 q^{68} - 18200 q^{70} + 74032 q^{71} - 44442 q^{73} - 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 108116 q^{79} + 10375 q^{80} - 92230 q^{82} + 81892 q^{83} - 59150 q^{85} - 126412 q^{86} + 27951 q^{88} - 167342 q^{89} - 31832 q^{91} - 72960 q^{92} + 12728 q^{94} + 71800 q^{95} + 159702 q^{97} - 163121 q^{98}+O(q^{100}) \) 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 - 175 * q^10 + 363 * q^11 - 654 * q^13 + 728 * q^14 - 415 * q^16 + 2366 * q^17 - 2872 * q^19 - 625 * q^20 + 847 * q^22 - 2272 * q^23 + 1875 * q^25 - 3422 * q^26 + 4592 * q^28 + 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 2506 * q^34 + 4300 * q^35 - 9126 * q^37 - 13076 * q^38 - 5775 * q^40 + 8758 * q^41 - 14672 * q^43 + 3025 * q^44 - 28768 * q^46 + 19392 * q^47 - 26629 * q^49 + 4375 * q^50 - 61506 * q^52 + 4598 * q^53 - 9075 * q^55 - 2688 * q^56 + 8550 * q^58 + 9348 * q^59 - 60078 * q^61 + 14096 * q^62 - 7087 * q^64 + 16350 * q^65 - 38468 * q^67 - 59778 * q^68 - 18200 * q^70 + 74032 * q^71 - 44442 * q^73 - 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 108116 * q^79 + 10375 * q^80 - 92230 * q^82 + 81892 * q^83 - 59150 * q^85 - 126412 * q^86 + 27951 * q^88 - 167342 * q^89 - 31832 * q^91 - 72960 * q^92 + 12728 * q^94 + 71800 * q^95 + 159702 * q^97 - 163121 * q^98

Introduction

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