LMFDB - Embedded newform 825.6.a.w.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 318 x^{11} + 776 x^{10} + 37929 x^{9} - 75673 x^{8} - 2114192 x^{7} + \cdots + 1037920000 $ x^13 - 3x^12 - 318x^11 + 776x^10 + 37929x^9 - 75673x^8 - 2114192x^7 + 3527024x^6 + 55736416x^5 - 75128864x^4 - 615221248x^3 + 450708480x^2 + 2256236800x + 1037920000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.570908$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.570908 q^{2} +9.00000 q^{3} -31.6741 q^{4} +5.13817 q^{6} -245.179 q^{7} -36.3520 q^{8} +81.0000 q^{9} -121.000 q^{11} -285.067 q^{12} +837.661 q^{13} -139.974 q^{14} +992.816 q^{16} -432.186 q^{17} +46.2435 q^{18} +2875.36 q^{19} -2206.61 q^{21} -69.0798 q^{22} -4417.30 q^{23} -327.168 q^{24} +478.227 q^{26} +729.000 q^{27} +7765.80 q^{28} +1149.20 q^{29} +3722.83 q^{31} +1730.07 q^{32} -1089.00 q^{33} -246.738 q^{34} -2565.60 q^{36} +5219.43 q^{37} +1641.57 q^{38} +7538.95 q^{39} -216.088 q^{41} -1259.77 q^{42} +10915.6 q^{43} +3832.56 q^{44} -2521.87 q^{46} -17830.8 q^{47} +8935.35 q^{48} +43305.6 q^{49} -3889.67 q^{51} -26532.1 q^{52} +25901.8 q^{53} +416.192 q^{54} +8912.74 q^{56} +25878.3 q^{57} +656.089 q^{58} -40288.5 q^{59} +28372.3 q^{61} +2125.39 q^{62} -19859.5 q^{63} -30782.4 q^{64} -621.718 q^{66} +541.795 q^{67} +13689.1 q^{68} -39755.7 q^{69} -50514.3 q^{71} -2944.51 q^{72} -29780.2 q^{73} +2979.81 q^{74} -91074.4 q^{76} +29666.6 q^{77} +4304.05 q^{78} -50850.0 q^{79} +6561.00 q^{81} -123.366 q^{82} -30190.9 q^{83} +69892.2 q^{84} +6231.82 q^{86} +10342.8 q^{87} +4398.59 q^{88} -105116. q^{89} -205377. q^{91} +139914. q^{92} +33505.4 q^{93} -10179.7 q^{94} +15570.6 q^{96} -14653.5 q^{97} +24723.5 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9} - 1573 q^{11} + 2061 q^{12} - 366 q^{13} - 2758 q^{14} + 4141 q^{16} - 2056 q^{17} - 243 q^{18} - 310 q^{19} - 2556 q^{21}+ \cdots - 127413 q^{99}+O(q^{100}) $ 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9 - 1573 * q^11 + 2061 * q^12 - 366 * q^13 - 2758 * q^14 + 4141 * q^16 - 2056 * q^17 - 243 * q^18 - 310 * q^19 - 2556 * q^21 + 363 * q^22 - 3612 * q^23 - 3321 * q^24 + 2280 * q^26 + 9477 * q^27 - 7896 * q^28 - 4848 * q^29 - 24 * q^31 - 38111 * q^32 - 14157 * q^33 + 5518 * q^34 + 18549 * q^36 + 8420 * q^37 - 474 * q^38 - 3294 * q^39 - 15120 * q^41 - 24822 * q^42 - 35492 * q^43 - 27709 * q^44 - 20280 * q^46 - 46544 * q^47 + 37269 * q^48 + 81837 * q^49 - 18504 * q^51 - 107194 * q^52 - 42256 * q^53 - 2187 * q^54 - 196602 * q^56 - 2790 * q^57 - 114160 * q^58 - 65592 * q^59 - 52042 * q^61 - 94972 * q^62 - 23004 * q^63 + 185977 * q^64 + 3267 * q^66 - 80580 * q^67 - 61108 * q^68 - 32508 * q^69 - 77820 * q^71 - 29889 * q^72 - 103050 * q^73 - 240028 * q^74 - 271174 * q^76 + 34364 * q^77 + 20520 * q^78 - 112258 * q^79 + 85293 * q^81 - 64060 * q^82 - 292150 * q^83 - 71064 * q^84 - 319250 * q^86 - 43632 * q^87 + 44649 * q^88 - 295810 * q^89 - 24200 * q^91 - 121328 * q^92 - 216 * q^93 - 358144 * q^94 - 342999 * q^96 - 49072 * q^97 - 101815 * q^98 - 127413 * q^99

Coefficient data

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

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

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

$2$	0.570908	0.100923	0.0504616	−	0.998726i	$-0.483931\pi$
$2$	0.570908	0.100923	0.0504616	+	0.998726i	$0.483931\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−31.6741	−0.989815
$5$	0	0
$5$	0	0
$6$	5.13817	0.0582680
$7$	−245.179	−1.89120	−0.945600	−	0.325331i	$-0.894524\pi$
$7$	−245.179	−1.89120	−0.945600	+	0.325331i	$0.894524\pi$
$8$	−36.3520	−0.200818
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−285.067	−0.571470
$13$	837.661	1.37471	0.687353	−	0.726323i	$-0.258772\pi$
$13$	837.661	1.37471	0.687353	+	0.726323i	$0.258772\pi$
$14$	−139.974	−0.190866
$15$	0	0
$16$	992.816	0.969547
$17$	−432.186	−0.362701	−0.181350	−	0.983419i	$-0.558047\pi$
$17$	−432.186	−0.362701	−0.181350	+	0.983419i	$0.558047\pi$
$18$	46.2435	0.0336411
$19$	2875.36	1.82729	0.913647	−	0.406508i	$-0.133254\pi$
$19$	2875.36	1.82729	0.913647	+	0.406508i	$0.133254\pi$
$20$	0	0
$21$	−2206.61	−1.09189
$22$	−69.0798	−0.0304295
$23$	−4417.30	−1.74115	−0.870577	−	0.492032i	$-0.836254\pi$
$23$	−4417.30	−1.74115	−0.870577	+	0.492032i	$0.836254\pi$
$24$	−327.168	−0.115943
$25$	0	0
$26$	478.227	0.138740
$27$	729.000	0.192450
$28$	7765.80	1.87194
$29$	1149.20	0.253748	0.126874	−	0.991919i	$-0.459506\pi$
$29$	1149.20	0.253748	0.126874	+	0.991919i	$0.459506\pi$
$30$	0	0
$31$	3722.83	0.695775	0.347887	−	0.937536i	$-0.386899\pi$
$31$	3722.83	0.695775	0.347887	+	0.937536i	$0.386899\pi$
$32$	1730.07	0.298668
$33$	−1089.00	−0.174078
$34$	−246.738	−0.0366049
$35$	0	0
$36$	−2565.60	−0.329938
$37$	5219.43	0.626785	0.313393	−	0.949624i	$-0.398534\pi$
$37$	5219.43	0.626785	0.313393	+	0.949624i	$0.398534\pi$
$38$	1641.57	0.184416
$39$	7538.95	0.793687
$40$	0	0
$41$	−216.088	−0.0200757	−0.0100379	−	0.999950i	$-0.503195\pi$
$41$	−216.088	−0.0200757	−0.0100379	+	0.999950i	$0.503195\pi$
$42$	−1259.77	−0.110197
$43$	10915.6	0.900281	0.450140	−	0.892958i	$-0.351374\pi$
$43$	10915.6	0.900281	0.450140	+	0.892958i	$0.351374\pi$
$44$	3832.56	0.298440
$45$	0	0
$46$	−2521.87	−0.175723
$47$	−17830.8	−1.17740	−0.588702	−	0.808350i	$-0.700361\pi$
$47$	−17830.8	−1.17740	−0.588702	+	0.808350i	$0.700361\pi$
$48$	8935.35	0.559768
$49$	43305.6	2.57664
$50$	0	0
$51$	−3889.67	−0.209405
$52$	−26532.1	−1.36070
$53$	25901.8	1.26660	0.633302	−	0.773905i	$-0.281699\pi$
$53$	25901.8	1.26660	0.633302	+	0.773905i	$0.281699\pi$
$54$	416.192	0.0194227
$55$	0	0
$56$	8912.74	0.379788
$57$	25878.3	1.05499
$58$	656.089	0.0256090
$59$	−40288.5	−1.50679	−0.753393	−	0.657571i	$-0.771584\pi$
$59$	−40288.5	−1.50679	−0.753393	+	0.657571i	$0.771584\pi$
$60$	0	0
$61$	28372.3	0.976269	0.488135	−	0.872768i	$-0.337678\pi$
$61$	28372.3	0.976269	0.488135	+	0.872768i	$0.337678\pi$
$62$	2125.39	0.0702198
$63$	−19859.5	−0.630400
$64$	−30782.4	−0.939405
$65$	0	0
$66$	−621.718	−0.0175685
$67$	541.795	0.0147451	0.00737255	−	0.999973i	$-0.497653\pi$
$67$	541.795	0.0147451	0.00737255	+	0.999973i	$0.497653\pi$
$68$	13689.1	0.359006
$69$	−39755.7	−1.00526
$70$	0	0
$71$	−50514.3	−1.18924	−0.594619	−	0.804008i	$-0.702697\pi$
$71$	−50514.3	−1.18924	−0.594619	+	0.804008i	$0.702697\pi$
$72$	−2944.51	−0.0669395
$73$	−29780.2	−0.654064	−0.327032	−	0.945013i	$-0.606049\pi$
$73$	−29780.2	−0.654064	−0.327032	+	0.945013i	$0.606049\pi$
$74$	2979.81	0.0632572
$75$	0	0
$76$	−91074.4	−1.80868
$77$	29666.6	0.570218
$78$	4304.05	0.0801014
$79$	−50850.0	−0.916692	−0.458346	−	0.888774i	$-0.651558\pi$
$79$	−50850.0	−0.916692	−0.458346	+	0.888774i	$0.651558\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−123.366	−0.00202610
$83$	−30190.9	−0.481040	−0.240520	−	0.970644i	$-0.577318\pi$
$83$	−30190.9	−0.481040	−0.240520	+	0.970644i	$0.577318\pi$
$84$	69892.2	1.08076
$85$	0	0
$86$	6231.82	0.0908592
$87$	10342.8	0.146501
$88$	4398.59	0.0605490
$89$	−105116.	−1.40667	−0.703337	−	0.710856i	$-0.748308\pi$
$89$	−105116.	−1.40667	−0.703337	+	0.710856i	$0.748308\pi$
$90$	0	0
$91$	−205377.	−2.59985
$92$	139914.	1.72342
$93$	33505.4	0.401706
$94$	−10179.7	−0.118827
$95$	0	0
$96$	15570.6	0.172436
$97$	−14653.5	−0.158129	−0.0790644	−	0.996870i	$-0.525193\pi$
$97$	−14653.5	−0.158129	−0.0790644	+	0.996870i	$0.525193\pi$
$98$	24723.5	0.260043
$99$	−9801.00	−0.100504
$100$	0	0
$101$	151527.	1.47804	0.739022	−	0.673681i	$-0.235288\pi$
$101$	151527.	1.47804	0.739022	+	0.673681i	$0.235288\pi$
$102$	−2220.64	−0.0211339
$103$	20665.8	0.191937	0.0959686	−	0.995384i	$-0.469405\pi$
$103$	20665.8	0.191937	0.0959686	+	0.995384i	$0.469405\pi$
$104$	−30450.7	−0.276066
$105$	0	0
$106$	14787.5	0.127830
$107$	59110.0	0.499116	0.249558	−	0.968360i	$-0.419715\pi$
$107$	59110.0	0.499116	0.249558	+	0.968360i	$0.419715\pi$
$108$	−23090.4	−0.190490
$109$	−43539.7	−0.351010	−0.175505	−	0.984479i	$-0.556156\pi$
$109$	−43539.7	−0.351010	−0.175505	+	0.984479i	$0.556156\pi$
$110$	0	0
$111$	46974.9	0.361875
$112$	−243417.	−1.83361
$113$	−121844.	−0.897655	−0.448828	−	0.893618i	$-0.648158\pi$
$113$	−121844.	−0.897655	−0.448828	+	0.893618i	$0.648158\pi$
$114$	14774.1	0.106473
$115$	0	0
$116$	−36399.9	−0.251163
$117$	67850.6	0.458236
$118$	−23001.0	−0.152070
$119$	105963.	0.685940
$120$	0	0
$121$	14641.0	0.0909091
$122$	16197.9	0.0985282
$123$	−1944.79	−0.0115907
$124$	−117917.	−0.688688
$125$	0	0
$126$	−11337.9	−0.0636220
$127$	−70642.0	−0.388646	−0.194323	−	0.980938i	$-0.562251\pi$
$127$	−70642.0	−0.388646	−0.194323	+	0.980938i	$0.562251\pi$
$128$	−72936.2	−0.393476
$129$	98240.8	0.519777
$130$	0	0
$131$	142876.	0.727411	0.363706	−	0.931514i	$-0.381511\pi$
$131$	142876.	0.727411	0.363706	+	0.931514i	$0.381511\pi$
$132$	34493.1	0.172305
$133$	−704977.	−3.45578
$134$	309.315	0.00148812
$135$	0	0
$136$	15710.8	0.0728370
$137$	−321212.	−1.46215	−0.731073	−	0.682299i	$-0.760980\pi$
$137$	−321212.	−1.46215	−0.731073	+	0.682299i	$0.760980\pi$
$138$	−22696.8	−0.101454
$139$	168783.	0.740954	0.370477	−	0.928842i	$-0.379194\pi$
$139$	168783.	0.740954	0.370477	+	0.928842i	$0.379194\pi$
$140$	0	0
$141$	−160477.	−0.679774
$142$	−28839.0	−0.120022
$143$	−101357.	−0.414490
$144$	80418.1	0.323182
$145$	0	0
$146$	−17001.7	−0.0660102
$147$	389750.	1.48762
$148$	−165321.	−0.620401
$149$	−262496.	−0.968629	−0.484314	−	0.874894i	$-0.660931\pi$
$149$	−262496.	−0.968629	−0.484314	+	0.874894i	$0.660931\pi$
$150$	0	0
$151$	−247979.	−0.885061	−0.442531	−	0.896753i	$-0.645919\pi$
$151$	−247979.	−0.885061	−0.442531	+	0.896753i	$0.645919\pi$
$152$	−104525.	−0.366954
$153$	−35007.1	−0.120900
$154$	16936.9	0.0575483
$155$	0	0
$156$	−238789.	−0.785603
$157$	−18670.0	−0.0604499	−0.0302250	−	0.999543i	$-0.509622\pi$
$157$	−18670.0	−0.0604499	−0.0302250	+	0.999543i	$0.509622\pi$
$158$	−29030.7	−0.0925155
$159$	233116.	0.731274
$160$	0	0
$161$	1.08303e6	3.29287
$162$	3745.73	0.0112137
$163$	60060.0	0.177058	0.0885291	−	0.996074i	$-0.471783\pi$
$163$	60060.0	0.177058	0.0885291	+	0.996074i	$0.471783\pi$
$164$	6844.38	0.0198712
$165$	0	0
$166$	−17236.2	−0.0485481
$167$	33658.3	0.0933902	0.0466951	−	0.998909i	$-0.485131\pi$
$167$	33658.3	0.0933902	0.0466951	+	0.998909i	$0.485131\pi$
$168$	80214.6	0.219271
$169$	330383.	0.889819
$170$	0	0
$171$	232904.	0.609098
$172$	−345743.	−0.891111
$173$	−368527.	−0.936169	−0.468084	−	0.883684i	$-0.655056\pi$
$173$	−368527.	−0.936169	−0.468084	+	0.883684i	$0.655056\pi$
$174$	5904.80	0.0147854
$175$	0	0
$176$	−120131.	−0.292330
$177$	−362597.	−0.869943
$178$	−60011.5	−0.141966
$179$	−206978.	−0.482826	−0.241413	−	0.970422i	$-0.577611\pi$
$179$	−206978.	−0.482826	−0.241413	+	0.970422i	$0.577611\pi$
$180$	0	0
$181$	688203.	1.56142	0.780711	−	0.624893i	$-0.214857\pi$
$181$	688203.	1.56142	0.780711	+	0.624893i	$0.214857\pi$
$182$	−117251.	−0.262385
$183$	255350.	0.563649
$184$	160578.	0.349656
$185$	0	0
$186$	19128.5	0.0405414
$187$	52294.5	0.109358
$188$	564773.	1.16541
$189$	−178735.	−0.363962
$190$	0	0
$191$	−135209.	−0.268178	−0.134089	−	0.990969i	$-0.542811\pi$
$191$	−135209.	−0.268178	−0.134089	+	0.990969i	$0.542811\pi$
$192$	−277042.	−0.542366
$193$	−699033.	−1.35084	−0.675420	−	0.737433i	$-0.736038\pi$
$193$	−699033.	−1.35084	−0.675420	+	0.737433i	$0.736038\pi$
$194$	−8365.78	−0.0159589
$195$	0	0
$196$	−1.37166e6	−2.55039
$197$	379663.	0.696999	0.348500	−	0.937309i	$-0.386691\pi$
$197$	379663.	0.696999	0.348500	+	0.937309i	$0.386691\pi$
$198$	−5595.47	−0.0101432
$199$	567492.	1.01584	0.507922	−	0.861403i	$-0.330414\pi$
$199$	567492.	1.01584	0.507922	+	0.861403i	$0.330414\pi$
$200$	0	0
$201$	4876.15	0.00851309
$202$	86508.1	0.149169
$203$	−281760.	−0.479888
$204$	123202.	0.207272
$205$	0	0
$206$	11798.3	0.0193709
$207$	−357801.	−0.580385
$208$	831644.	1.33284
$209$	−347919.	−0.550950
$210$	0	0
$211$	−751976.	−1.16278	−0.581391	−	0.813624i	$-0.697491\pi$
$211$	−751976.	−1.16278	−0.581391	+	0.813624i	$0.697491\pi$
$212$	−820416.	−1.25370
$213$	−454629.	−0.686607
$214$	33746.4	0.0503724
$215$	0	0
$216$	−26500.6	−0.0386475
$217$	−912758.	−1.31585
$218$	−24857.1	−0.0354250
$219$	−268022.	−0.377624
$220$	0	0
$221$	−362025.	−0.498607
$222$	26818.3	0.0365215
$223$	−574961.	−0.774241	−0.387120	−	0.922029i	$-0.626530\pi$
$223$	−574961.	−0.774241	−0.387120	+	0.922029i	$0.626530\pi$
$224$	−424176.	−0.564841
$225$	0	0
$226$	−69561.9	−0.0905942
$227$	−894453.	−1.15211	−0.576054	−	0.817412i	$-0.695408\pi$
$227$	−894453.	−1.15211	−0.576054	+	0.817412i	$0.695408\pi$
$228$	−819669.	−1.04424
$229$	156362.	0.197034	0.0985171	−	0.995135i	$-0.468590\pi$
$229$	156362.	0.197034	0.0985171	+	0.995135i	$0.468590\pi$
$230$	0	0
$231$	267000.	0.329216
$232$	−41775.9	−0.0509572
$233$	105270.	0.127033	0.0635164	−	0.997981i	$-0.479768\pi$
$233$	105270.	0.127033	0.0635164	+	0.997981i	$0.479768\pi$
$234$	38736.4	0.0462466
$235$	0	0
$236$	1.27610e6	1.49144
$237$	−457650.	−0.529252
$238$	60495.0	0.0692272
$239$	−66124.7	−0.0748805	−0.0374403	−	0.999299i	$-0.511920\pi$
$239$	−66124.7	−0.0748805	−0.0374403	+	0.999299i	$0.511920\pi$
$240$	0	0
$241$	1.38620e6	1.53738	0.768692	−	0.639619i	$-0.220908\pi$
$241$	1.38620e6	1.53738	0.768692	+	0.639619i	$0.220908\pi$
$242$	8358.66	0.00917483
$243$	59049.0	0.0641500
$244$	−898665.	−0.966325
$245$	0	0
$246$	−1110.30	−0.00116977
$247$	2.40858e6	2.51199
$248$	−135332.	−0.139724
$249$	−271718.	−0.277728
$250$	0	0
$251$	−39118.6	−0.0391922	−0.0195961	−	0.999808i	$-0.506238\pi$
$251$	−39118.6	−0.0391922	−0.0195961	+	0.999808i	$0.506238\pi$
$252$	629030.	0.623979
$253$	534493.	0.524978
$254$	−40330.1	−0.0392234
$255$	0	0
$256$	943397.	0.899694
$257$	115922.	0.109479	0.0547396	−	0.998501i	$-0.482567\pi$
$257$	115922.	0.109479	0.0547396	+	0.998501i	$0.482567\pi$
$258$	56086.4	0.0524576
$259$	−1.27969e6	−1.18538
$260$	0	0
$261$	93085.5	0.0845825
$262$	81568.8	0.0734127
$263$	−2.06203e6	−1.83826	−0.919128	−	0.393959i	$-0.871105\pi$
$263$	−2.06203e6	−1.83826	−0.919128	+	0.393959i	$0.871105\pi$
$264$	39587.3	0.0349580
$265$	0	0
$266$	−402477.	−0.348768
$267$	−946044.	−0.812144
$268$	−17160.8	−0.0145949
$269$	1.86362e6	1.57028	0.785140	−	0.619318i	$-0.212591\pi$
$269$	1.86362e6	1.57028	0.785140	+	0.619318i	$0.212591\pi$
$270$	0	0
$271$	−1.44161e6	−1.19241	−0.596203	−	0.802833i	$-0.703325\pi$
$271$	−1.44161e6	−1.19241	−0.596203	+	0.802833i	$0.703325\pi$
$272$	−429081.	−0.351655
$273$	−1.84839e6	−1.50102
$274$	−183383.	−0.147564
$275$	0	0
$276$	1.25922e6	0.995017
$277$	1.39431e6	1.09184	0.545922	−	0.837836i	$-0.316180\pi$
$277$	1.39431e6	1.09184	0.545922	+	0.837836i	$0.316180\pi$
$278$	96359.5	0.0747795
$279$	301549.	0.231925
$280$	0	0
$281$	−792429.	−0.598680	−0.299340	−	0.954147i	$-0.596766\pi$
$281$	−792429.	−0.598680	−0.299340	+	0.954147i	$0.596766\pi$
$282$	−91617.5	−0.0686050
$283$	1.54580e6	1.14733	0.573663	−	0.819091i	$-0.305522\pi$
$283$	1.54580e6	1.14733	0.573663	+	0.819091i	$0.305522\pi$
$284$	1.59999e6	1.17712
$285$	0	0
$286$	−57865.5	−0.0418316
$287$	52980.1	0.0379672
$288$	140136.	0.0995561
$289$	−1.23307e6	−0.868448
$290$	0	0
$291$	−131881.	−0.0912957
$292$	943260.	0.647402
$293$	−1.77481e6	−1.20777	−0.603883	−	0.797073i	$-0.706381\pi$
$293$	−1.77481e6	−1.20777	−0.603883	+	0.797073i	$0.706381\pi$
$294$	222511.	0.150136
$295$	0	0
$296$	−189737.	−0.125870
$297$	−88209.0	−0.0580259
$298$	−149861.	−0.0977571
$299$	−3.70020e6	−2.39358
$300$	0	0
$301$	−2.67628e6	−1.70261
$302$	−141573.	−0.0893232
$303$	1.36375e6	0.853349
$304$	2.85471e6	1.77165
$305$	0	0
$306$	−19985.8	−0.0122016
$307$	−1.86110e6	−1.12700	−0.563499	−	0.826117i	$-0.690545\pi$
$307$	−1.86110e6	−1.12700	−0.563499	+	0.826117i	$0.690545\pi$
$308$	−939662.	−0.564410
$309$	185992.	0.110815
$310$	0	0
$311$	−972428.	−0.570107	−0.285053	−	0.958512i	$-0.592011\pi$
$311$	−972428.	−0.570107	−0.285053	+	0.958512i	$0.592011\pi$
$312$	−274056.	−0.159387
$313$	−1.78176e6	−1.02799	−0.513994	−	0.857794i	$-0.671835\pi$
$313$	−1.78176e6	−1.02799	−0.513994	+	0.857794i	$0.671835\pi$
$314$	−10658.9	−0.00610080
$315$	0	0
$316$	1.61063e6	0.907355
$317$	539885.	0.301754	0.150877	−	0.988553i	$-0.451790\pi$
$317$	539885.	0.301754	0.150877	+	0.988553i	$0.451790\pi$
$318$	133088.	0.0738025
$319$	−139054.	−0.0765078
$320$	0	0
$321$	531990.	0.288165
$322$	618309.	0.332327
$323$	−1.24269e6	−0.662761
$324$	−207814.	−0.109979
$325$	0	0
$326$	34288.7	0.0178693
$327$	−391857.	−0.202655
$328$	7855.23	0.00403157
$329$	4.37172e6	2.22671
$330$	0	0
$331$	1.25661e6	0.630420	0.315210	−	0.949022i	$-0.397925\pi$
$331$	1.25661e6	0.630420	0.315210	+	0.949022i	$0.397925\pi$
$332$	956269.	0.476140
$333$	422774.	0.208928
$334$	19215.8	0.00942524
$335$	0	0
$336$	−2.19076e6	−1.05863
$337$	642359.	0.308108	0.154054	−	0.988062i	$-0.450767\pi$
$337$	642359.	0.308108	0.154054	+	0.988062i	$0.450767\pi$
$338$	188618.	0.0898033
$339$	−1.09660e6	−0.518261
$340$	0	0
$341$	−450462.	−0.209784
$342$	132967.	0.0614721
$343$	−6.49689e6	−2.98174
$344$	−396805.	−0.180793
$345$	0	0
$346$	−210395.	−0.0944811
$347$	1.94801e6	0.868496	0.434248	−	0.900793i	$-0.357014\pi$
$347$	1.94801e6	0.868496	0.434248	+	0.900793i	$0.357014\pi$
$348$	−327600.	−0.145009
$349$	322025.	0.141523	0.0707613	−	0.997493i	$-0.477457\pi$
$349$	322025.	0.141523	0.0707613	+	0.997493i	$0.477457\pi$
$350$	0	0
$351$	610655.	0.264562
$352$	−209339.	−0.0900518
$353$	3.34158e6	1.42730	0.713649	−	0.700503i	$-0.247041\pi$
$353$	3.34158e6	1.42730	0.713649	+	0.700503i	$0.247041\pi$
$354$	−207009.	−0.0877974
$355$	0	0
$356$	3.32945e6	1.39235
$357$	953665.	0.396027
$358$	−118165.	−0.0487283
$359$	552279.	0.226163	0.113082	−	0.993586i	$-0.463928\pi$
$359$	552279.	0.226163	0.113082	+	0.993586i	$0.463928\pi$
$360$	0	0
$361$	5.79161e6	2.33900
$362$	392900.	0.157584
$363$	131769.	0.0524864
$364$	6.50511e6	2.57337
$365$	0	0
$366$	145782.	0.0568853
$367$	−3.96818e6	−1.53789	−0.768947	−	0.639312i	$-0.779219\pi$
$367$	−3.96818e6	−1.53789	−0.768947	+	0.639312i	$0.779219\pi$
$368$	−4.38557e6	−1.68813
$369$	−17503.1	−0.00669190
$370$	0	0
$371$	−6.35057e6	−2.39540
$372$	−1.06125e6	−0.397614
$373$	−2.11585e6	−0.787433	−0.393717	−	0.919232i	$-0.628811\pi$
$373$	−2.11585e6	−0.787433	−0.393717	+	0.919232i	$0.628811\pi$
$374$	29855.3	0.0110368
$375$	0	0
$376$	648184.	0.236444
$377$	962643.	0.348829
$378$	−102041.	−0.0367322
$379$	820263.	0.293329	0.146665	−	0.989186i	$-0.453146\pi$
$379$	820263.	0.293329	0.146665	+	0.989186i	$0.453146\pi$
$380$	0	0
$381$	−635778.	−0.224385
$382$	−77192.1	−0.0270654
$383$	−3.23496e6	−1.12686	−0.563432	−	0.826162i	$-0.690519\pi$
$383$	−3.23496e6	−1.12686	−0.563432	+	0.826162i	$0.690519\pi$
$384$	−656426.	−0.227173
$385$	0	0
$386$	−399083.	−0.136331
$387$	884167.	0.300094
$388$	464135.	0.156518
$389$	−2.53332e6	−0.848821	−0.424411	−	0.905470i	$-0.639519\pi$
$389$	−2.53332e6	−0.848821	−0.424411	+	0.905470i	$0.639519\pi$
$390$	0	0
$391$	1.90910e6	0.631518
$392$	−1.57424e6	−0.517437
$393$	1.28588e6	0.419971
$394$	216752.	0.0703434
$395$	0	0
$396$	310438.	0.0994801
$397$	1.58602e6	0.505046	0.252523	−	0.967591i	$-0.418740\pi$
$397$	1.58602e6	0.505046	0.252523	+	0.967591i	$0.418740\pi$
$398$	323985.	0.102522
$399$	−6.34480e6	−1.99520
$400$	0	0
$401$	−2.39011e6	−0.742262	−0.371131	−	0.928581i	$-0.621030\pi$
$401$	−2.39011e6	−0.742262	−0.371131	+	0.928581i	$0.621030\pi$
$402$	2783.83	0.000859168 0
$403$	3.11847e6	0.956486
$404$	−4.79949e6	−1.46299
$405$	0	0
$406$	−160859.	−0.0484318
$407$	−631551.	−0.188983
$408$	141397.	0.0420524
$409$	−4.85226e6	−1.43429	−0.717144	−	0.696925i	$-0.754551\pi$
$409$	−4.85226e6	−1.43429	−0.717144	+	0.696925i	$0.754551\pi$
$410$	0	0
$411$	−2.89091e6	−0.844170
$412$	−654570.	−0.189982
$413$	9.87789e6	2.84963
$414$	−204272.	−0.0585743
$415$	0	0
$416$	1.44921e6	0.410581
$417$	1.51905e6	0.427790
$418$	−198629.	−0.0556036
$419$	759538.	0.211356	0.105678	−	0.994400i	$-0.466299\pi$
$419$	759538.	0.211356	0.105678	+	0.994400i	$0.466299\pi$
$420$	0	0
$421$	−3.32838e6	−0.915224	−0.457612	−	0.889152i	$-0.651295\pi$
$421$	−3.32838e6	−0.915224	−0.457612	+	0.889152i	$0.651295\pi$
$422$	−429309.	−0.117352
$423$	−1.44429e6	−0.392468
$424$	−941583.	−0.254357
$425$	0	0
$426$	−259551.	−0.0692945
$427$	−6.95628e6	−1.84632
$428$	−1.87225e6	−0.494032
$429$	−912213.	−0.239306
$430$	0	0
$431$	−5.26651e6	−1.36562	−0.682810	−	0.730596i	$-0.739242\pi$
$431$	−5.26651e6	−1.36562	−0.682810	+	0.730596i	$0.739242\pi$
$432$	723763.	0.186589
$433$	−3.14161e6	−0.805252	−0.402626	−	0.915365i	$-0.631903\pi$
$433$	−3.14161e6	−0.805252	−0.402626	+	0.915365i	$0.631903\pi$
$434$	−521101.	−0.132800
$435$	0	0
$436$	1.37908e6	0.347434
$437$	−1.27013e7	−3.18160
$438$	−153016.	−0.0381110
$439$	774879.	0.191899	0.0959495	−	0.995386i	$-0.469411\pi$
$439$	774879.	0.191899	0.0959495	+	0.995386i	$0.469411\pi$
$440$	0	0
$441$	3.50775e6	0.858880
$442$	−206683.	−0.0503210
$443$	3.37212e6	0.816382	0.408191	−	0.912897i	$-0.366160\pi$
$443$	3.37212e6	0.816382	0.408191	+	0.912897i	$0.366160\pi$
$444$	−1.48789e6	−0.358189
$445$	0	0
$446$	−328249.	−0.0781388
$447$	−2.36247e6	−0.559238
$448$	7.54719e6	1.77660
$449$	4.72484e6	1.10604	0.553020	−	0.833168i	$-0.313475\pi$
$449$	4.72484e6	1.10604	0.553020	+	0.833168i	$0.313475\pi$
$450$	0	0
$451$	26146.6	0.00605305
$452$	3.85931e6	0.888512
$453$	−2.23181e6	−0.510990
$454$	−510650.	−0.116274
$455$	0	0
$456$	−940727.	−0.211861
$457$	−8.20431e6	−1.83760	−0.918802	−	0.394720i	$-0.870842\pi$
$457$	−8.20431e6	−1.83760	−0.918802	+	0.394720i	$0.870842\pi$
$458$	89268.0	0.0198853
$459$	−315064.	−0.0698018
$460$	0	0
$461$	−8.94858e6	−1.96111	−0.980554	−	0.196248i	$-0.937124\pi$
$461$	−8.94858e6	−1.96111	−0.980554	+	0.196248i	$0.937124\pi$
$462$	152432.	0.0332255
$463$	−1.53900e6	−0.333646	−0.166823	−	0.985987i	$-0.553351\pi$
$463$	−1.53900e6	−0.333646	−0.166823	+	0.985987i	$0.553351\pi$
$464$	1.14095e6	0.246020
$465$	0	0
$466$	60099.6	0.0128206
$467$	4.34475e6	0.921875	0.460938	−	0.887433i	$-0.347513\pi$
$467$	4.34475e6	0.921875	0.460938	+	0.887433i	$0.347513\pi$
$468$	−2.14910e6	−0.453568
$469$	−132836.	−0.0278859
$470$	0	0
$471$	−168030.	−0.0349008
$472$	1.46457e6	0.302590
$473$	−1.32079e6	−0.271445
$474$	−261276.	−0.0534138
$475$	0	0
$476$	−3.35627e6	−0.678953
$477$	2.09805e6	0.422201
$478$	−37751.1	−0.00755718
$479$	1.20356e6	0.239678	0.119839	−	0.992793i	$-0.461762\pi$
$479$	1.20356e6	0.239678	0.119839	+	0.992793i	$0.461762\pi$
$480$	0	0
$481$	4.37212e6	0.861646
$482$	791391.	0.155158
$483$	9.74725e6	1.90114
$484$	−463740.	−0.0899831
$485$	0	0
$486$	33711.5	0.00647422
$487$	−7.84121e6	−1.49817	−0.749084	−	0.662475i	$-0.769506\pi$
$487$	−7.84121e6	−1.49817	−0.749084	+	0.662475i	$0.769506\pi$
$488$	−1.03139e6	−0.196053
$489$	540540.	0.102225
$490$	0	0
$491$	6.73581e6	1.26091	0.630457	−	0.776224i	$-0.282867\pi$
$491$	6.73581e6	1.26091	0.630457	+	0.776224i	$0.282867\pi$
$492$	61599.4	0.0114727
$493$	−496670.	−0.0920344
$494$	1.37508e6	0.253518
$495$	0	0
$496$	3.69608e6	0.674587
$497$	1.23850e7	2.24909
$498$	−155126.	−0.0280292
$499$	6.63311e6	1.19252	0.596260	−	0.802791i	$-0.296653\pi$
$499$	6.63311e6	1.19252	0.596260	+	0.802791i	$0.296653\pi$
$500$	0	0
$501$	302925.	0.0539189
$502$	−22333.1	−0.00395540
$503$	−265757.	−0.0468344	−0.0234172	−	0.999726i	$-0.507455\pi$
$503$	−265757.	−0.0468344	−0.0234172	+	0.999726i	$0.507455\pi$
$504$	721932.	0.126596
$505$	0	0
$506$	305146.	0.0529824
$507$	2.97345e6	0.513737
$508$	2.23752e6	0.384687
$509$	−3.17214e6	−0.542697	−0.271349	−	0.962481i	$-0.587470\pi$
$509$	−3.17214e6	−0.542697	−0.271349	+	0.962481i	$0.587470\pi$
$510$	0	0
$511$	7.30147e6	1.23697
$512$	2.87255e6	0.484276
$513$	2.09614e6	0.351663
$514$	66180.5	0.0110490
$515$	0	0
$516$	−3.11168e6	−0.514483
$517$	2.15752e6	0.355001
$518$	−730587.	−0.119632
$519$	−3.31674e6	−0.540497
$520$	0	0
$521$	8.20953e6	1.32502	0.662512	−	0.749051i	$-0.269490\pi$
$521$	8.20953e6	1.32502	0.662512	+	0.749051i	$0.269490\pi$
$522$	53143.2	0.00853634
$523$	4.16785e6	0.666282	0.333141	−	0.942877i	$-0.391891\pi$
$523$	4.16785e6	0.666282	0.333141	+	0.942877i	$0.391891\pi$
$524$	−4.52545e6	−0.720002
$525$	0	0
$526$	−1.17723e6	−0.185523
$527$	−1.60895e6	−0.252358
$528$	−1.08118e6	−0.168777
$529$	1.30762e7	2.03162
$530$	0	0
$531$	−3.26337e6	−0.502262
$532$	2.23295e7	3.42058
$533$	−181008.	−0.0275982
$534$	−540104.	−0.0819642
$535$	0	0
$536$	−19695.3	−0.00296109
$537$	−1.86280e6	−0.278760
$538$	1.06396e6	0.158478
$539$	−5.23997e6	−0.776886
$540$	0	0
$541$	4.42151e6	0.649498	0.324749	−	0.945800i	$-0.394720\pi$
$541$	4.42151e6	0.649498	0.324749	+	0.945800i	$0.394720\pi$
$542$	−823026.	−0.120341
$543$	6.19383e6	0.901487
$544$	−747712.	−0.108327
$545$	0	0
$546$	−1.05526e6	−0.151488
$547$	−8.47388e6	−1.21092	−0.605458	−	0.795877i	$-0.707010\pi$
$547$	−8.47388e6	−1.21092	−0.605458	+	0.795877i	$0.707010\pi$
$548$	1.01741e7	1.44725
$549$	2.29815e6	0.325423
$550$	0	0
$551$	3.30438e6	0.463672
$552$	1.44520e6	0.201874
$553$	1.24673e7	1.73365
$554$	796023.	0.110192
$555$	0	0
$556$	−5.34604e6	−0.733407
$557$	−1.28187e6	−0.175068	−0.0875341	−	0.996162i	$-0.527899\pi$
$557$	−1.28187e6	−0.175068	−0.0875341	+	0.996162i	$0.527899\pi$
$558$	172157.	0.0234066
$559$	9.14361e6	1.23762
$560$	0	0
$561$	470651.	0.0631381
$562$	−452404.	−0.0604207
$563$	−5.28249e6	−0.702372	−0.351186	−	0.936306i	$-0.614222\pi$
$563$	−5.28249e6	−0.702372	−0.351186	+	0.936306i	$0.614222\pi$
$564$	5.08296e6	0.672851
$565$	0	0
$566$	882508.	0.115792
$567$	−1.60862e6	−0.210133
$568$	1.83630e6	0.238821
$569$	133510.	0.0172875	0.00864375	−	0.999963i	$-0.497249\pi$
$569$	133510.	0.0172875	0.00864375	+	0.999963i	$0.497249\pi$
$570$	0	0
$571$	9.41785e6	1.20882	0.604410	−	0.796673i	$-0.293409\pi$
$571$	9.41785e6	1.20882	0.604410	+	0.796673i	$0.293409\pi$
$572$	3.21039e6	0.410268
$573$	−1.21689e6	−0.154833
$574$	30246.8	0.00383177
$575$	0	0
$576$	−2.49338e6	−0.313135
$577$	2.40962e6	0.301306	0.150653	−	0.988587i	$-0.451862\pi$
$577$	2.40962e6	0.301306	0.150653	+	0.988587i	$0.451862\pi$
$578$	−703970.	−0.0876466
$579$	−6.29129e6	−0.779908
$580$	0	0
$581$	7.40217e6	0.909743
$582$	−75292.0	−0.00921385
$583$	−3.13412e6	−0.381895
$584$	1.08257e6	0.131348
$585$	0	0
$586$	−1.01325e6	−0.121892
$587$	2.91474e6	0.349144	0.174572	−	0.984644i	$-0.444146\pi$
$587$	2.91474e6	0.349144	0.174572	+	0.984644i	$0.444146\pi$
$588$	−1.23450e7	−1.47247
$589$	1.07045e7	1.27139
$590$	0	0
$591$	3.41696e6	0.402413
$592$	5.18194e6	0.607698
$593$	8.05223e6	0.940328	0.470164	−	0.882579i	$-0.344195\pi$
$593$	8.05223e6	0.940328	0.470164	+	0.882579i	$0.344195\pi$
$594$	−50359.2	−0.00585616
$595$	0	0
$596$	8.31432e6	0.958763
$597$	5.10742e6	0.586497
$598$	−2.11247e6	−0.241567
$599$	−9.62449e6	−1.09600	−0.548000	−	0.836478i	$-0.684611\pi$
$599$	−9.62449e6	−1.09600	−0.548000	+	0.836478i	$0.684611\pi$
$600$	0	0
$601$	−8.04903e6	−0.908986	−0.454493	−	0.890750i	$-0.650180\pi$
$601$	−8.04903e6	−0.908986	−0.454493	+	0.890750i	$0.650180\pi$
$602$	−1.52791e6	−0.171833
$603$	43885.4	0.00491503
$604$	7.85452e6	0.876046
$605$	0	0
$606$	778573.	0.0861227
$607$	4.68740e6	0.516369	0.258185	−	0.966096i	$-0.416876\pi$
$607$	4.68740e6	0.516369	0.258185	+	0.966096i	$0.416876\pi$
$608$	4.97458e6	0.545755
$609$	−2.53584e6	−0.277063
$610$	0	0
$611$	−1.49361e7	−1.61858
$612$	1.10882e6	0.119669
$613$	−575438.	−0.0618511	−0.0309256	−	0.999522i	$-0.509845\pi$
$613$	−575438.	−0.0618511	−0.0309256	+	0.999522i	$0.509845\pi$
$614$	−1.06252e6	−0.113740
$615$	0	0
$616$	−1.07844e6	−0.114510
$617$	3.01787e6	0.319144	0.159572	−	0.987186i	$-0.448989\pi$
$617$	3.01787e6	0.319144	0.159572	+	0.987186i	$0.448989\pi$
$618$	106184.	0.0111838
$619$	1.08497e6	0.113813	0.0569065	−	0.998380i	$-0.481876\pi$
$619$	1.08497e6	0.113813	0.0569065	+	0.998380i	$0.481876\pi$
$620$	0	0
$621$	−3.22021e6	−0.335085
$622$	−555166.	−0.0575370
$623$	2.57722e7	2.66030
$624$	7.48480e6	0.769517
$625$	0	0
$626$	−1.01722e6	−0.103748
$627$	−3.13127e6	−0.318091
$628$	591355.	0.0598342
$629$	−2.25576e6	−0.227335
$630$	0	0
$631$	7.88956e6	0.788823	0.394411	−	0.918934i	$-0.370949\pi$
$631$	7.88956e6	0.788823	0.394411	+	0.918934i	$0.370949\pi$
$632$	1.84850e6	0.184089
$633$	−6.76779e6	−0.671332
$634$	308225.	0.0304540
$635$	0	0
$636$	−7.38374e6	−0.723825
$637$	3.62754e7	3.54212
$638$	−79386.8	−0.00772141
$639$	−4.09166e6	−0.396412
$640$	0	0
$641$	1.76459e7	1.69628	0.848142	−	0.529769i	$-0.177721\pi$
$641$	1.76459e7	1.69628	0.848142	+	0.529769i	$0.177721\pi$
$642$	303717.	0.0290825
$643$	−1.12792e7	−1.07585	−0.537923	−	0.842994i	$-0.680791\pi$
$643$	−1.12792e7	−1.07585	−0.537923	+	0.842994i	$0.680791\pi$
$644$	−3.43039e7	−3.25933
$645$	0	0
$646$	−709462.	−0.0668879
$647$	−6.63585e6	−0.623213	−0.311606	−	0.950211i	$-0.600867\pi$
$647$	−6.63585e6	−0.623213	−0.311606	+	0.950211i	$0.600867\pi$
$648$	−238506.	−0.0223132
$649$	4.87491e6	0.454313
$650$	0	0
$651$	−8.21482e6	−0.759706
$652$	−1.90234e6	−0.175255
$653$	−1.77241e7	−1.62660	−0.813301	−	0.581843i	$-0.802332\pi$
$653$	−1.77241e7	−1.62660	−0.813301	+	0.581843i	$0.802332\pi$
$654$	−223714.	−0.0204526
$655$	0	0
$656$	−214536.	−0.0194643
$657$	−2.41220e6	−0.218021
$658$	2.49585e6	0.224726
$659$	−6.73897e6	−0.604477	−0.302239	−	0.953232i	$-0.597734\pi$
$659$	−6.73897e6	−0.604477	−0.302239	+	0.953232i	$0.597734\pi$
$660$	0	0
$661$	887181.	0.0789784	0.0394892	−	0.999220i	$-0.487427\pi$
$661$	887181.	0.0789784	0.0394892	+	0.999220i	$0.487427\pi$
$662$	717407.	0.0636240
$663$	−3.25823e6	−0.287871
$664$	1.09750e6	0.0966016
$665$	0	0
$666$	241365.	0.0210857
$667$	−5.07638e6	−0.441814
$668$	−1.06610e6	−0.0924390
$669$	−5.17465e6	−0.447008
$670$	0	0
$671$	−3.43304e6	−0.294356
$672$	−3.81759e6	−0.326111
$673$	7.24951e6	0.616980	0.308490	−	0.951228i	$-0.400176\pi$
$673$	7.24951e6	0.616980	0.308490	+	0.951228i	$0.400176\pi$
$674$	366728.	0.0310952
$675$	0	0
$676$	−1.04646e7	−0.880756
$677$	1.86561e7	1.56440	0.782201	−	0.623026i	$-0.214097\pi$
$677$	1.86561e7	1.56440	0.782201	+	0.623026i	$0.214097\pi$
$678$	−626057.	−0.0523046
$679$	3.59272e6	0.299053
$680$	0	0
$681$	−8.05008e6	−0.665170
$682$	−257172.	−0.0211721
$683$	7.50966e6	0.615982	0.307991	−	0.951389i	$-0.400343\pi$
$683$	7.50966e6	0.615982	0.307991	+	0.951389i	$0.400343\pi$
$684$	−7.37703e6	−0.602894
$685$	0	0
$686$	−3.70912e6	−0.300927
$687$	1.40725e6	0.113758
$688$	1.08372e7	0.872865
$689$	2.16970e7	1.74121
$690$	0	0
$691$	783725.	0.0624408	0.0312204	−	0.999513i	$-0.490061\pi$
$691$	783725.	0.0624408	0.0312204	+	0.999513i	$0.490061\pi$
$692$	1.16728e7	0.926633
$693$	2.40300e6	0.190073
$694$	1.11214e6	0.0876514
$695$	0	0
$696$	−375983.	−0.0294201
$697$	93390.1	0.00728147
$698$	183846.	0.0142829
$699$	947432.	0.0733424
$700$	0	0
$701$	−1.75538e7	−1.34920	−0.674599	−	0.738184i	$-0.735683\pi$
$701$	−1.75538e7	−1.34920	−0.674599	+	0.738184i	$0.735683\pi$
$702$	348628.	0.0267005
$703$	1.50078e7	1.14532
$704$	3.72467e6	0.283241
$705$	0	0
$706$	1.90773e6	0.144048
$707$	−3.71513e7	−2.79528
$708$	1.14849e7	0.861082
$709$	1.54247e7	1.15240	0.576198	−	0.817310i	$-0.304536\pi$
$709$	1.54247e7	1.15240	0.576198	+	0.817310i	$0.304536\pi$
$710$	0	0
$711$	−4.11885e6	−0.305564
$712$	3.82118e6	0.282486
$713$	−1.64448e7	−1.21145
$714$	544455.	0.0399683
$715$	0	0
$716$	6.55582e6	0.477908
$717$	−595122.	−0.0432323
$718$	315300.	0.0228251
$719$	−2.41023e7	−1.73874	−0.869372	−	0.494158i	$-0.835477\pi$
$719$	−2.41023e7	−1.73874	−0.869372	+	0.494158i	$0.835477\pi$
$720$	0	0
$721$	−5.06681e6	−0.362992
$722$	3.30647e6	0.236060
$723$	1.24758e7	0.887609
$724$	−2.17982e7	−1.54552
$725$	0	0
$726$	75227.9	0.00529709
$727$	−8.51257e6	−0.597345	−0.298672	−	0.954356i	$-0.596544\pi$
$727$	−8.51257e6	−0.597345	−0.298672	+	0.954356i	$0.596544\pi$
$728$	7.46586e6	0.522097
$729$	531441.	0.0370370
$730$	0	0
$731$	−4.71759e6	−0.326533
$732$	−8.08799e6	−0.557908
$733$	−1.07249e7	−0.737285	−0.368642	−	0.929571i	$-0.620177\pi$
$733$	−1.07249e7	−0.737285	−0.368642	+	0.929571i	$0.620177\pi$
$734$	−2.26547e6	−0.155209
$735$	0	0
$736$	−7.64224e6	−0.520028
$737$	−65557.2	−0.00444582
$738$	−9992.66	−0.000675368 0
$739$	32829.8	0.00221135	0.00110567	−	0.999999i	$-0.499648\pi$
$739$	32829.8	0.00221135	0.00110567	+	0.999999i	$0.499648\pi$
$740$	0	0
$741$	2.16772e7	1.45030
$742$	−3.62559e6	−0.241751
$743$	1.47610e7	0.980941	0.490470	−	0.871458i	$-0.336825\pi$
$743$	1.47610e7	0.980941	0.490470	+	0.871458i	$0.336825\pi$
$744$	−1.21799e6	−0.0806699
$745$	0	0
$746$	−1.20796e6	−0.0794703
$747$	−2.44546e6	−0.160347
$748$	−1.65638e6	−0.108244
$749$	−1.44925e7	−0.943928
$750$	0	0
$751$	1.54410e7	0.999022	0.499511	−	0.866307i	$-0.333513\pi$
$751$	1.54410e7	0.999022	0.499511	+	0.866307i	$0.333513\pi$
$752$	−1.77027e7	−1.14155
$753$	−352068.	−0.0226276
$754$	549581.	0.0352049
$755$	0	0
$756$	5.66127e6	0.360255
$757$	−3.01507e7	−1.91231	−0.956153	−	0.292868i	$-0.905390\pi$
$757$	−3.01507e7	−1.91231	−0.956153	+	0.292868i	$0.905390\pi$
$758$	468295.	0.0296037
$759$	4.81044e6	0.303096
$760$	0	0
$761$	4.43840e6	0.277821	0.138910	−	0.990305i	$-0.455640\pi$
$761$	4.43840e6	0.277821	0.138910	+	0.990305i	$0.455640\pi$
$762$	−362971.	−0.0226456
$763$	1.06750e7	0.663830
$764$	4.28263e6	0.265447
$765$	0	0
$766$	−1.84686e6	−0.113727
$767$	−3.37482e7	−2.07139
$768$	8.49058e6	0.519439
$769$	−1.48416e7	−0.905034	−0.452517	−	0.891756i	$-0.649474\pi$
$769$	−1.48416e7	−0.905034	−0.452517	+	0.891756i	$0.649474\pi$
$770$	0	0
$771$	1.04329e6	0.0632078
$772$	2.21412e7	1.33708
$773$	4.82603e6	0.290497	0.145248	−	0.989395i	$-0.453602\pi$
$773$	4.82603e6	0.290497	0.145248	+	0.989395i	$0.453602\pi$
$774$	504778.	0.0302864
$775$	0	0
$776$	532683.	0.0317552
$777$	−1.15172e7	−0.684378
$778$	−1.44629e6	−0.0856657
$779$	−621331.	−0.0366842
$780$	0	0
$781$	6.11223e6	0.358569
$782$	1.08992e6	0.0637348
$783$	837769.	0.0488338
$784$	4.29945e7	2.49817
$785$	0	0
$786$	734119.	0.0423848
$787$	−3.04485e6	−0.175239	−0.0876193	−	0.996154i	$-0.527926\pi$
$787$	−3.04485e6	−0.175239	−0.0876193	+	0.996154i	$0.527926\pi$
$788$	−1.20255e7	−0.689900
$789$	−1.85583e7	−1.06132
$790$	0	0
$791$	2.98736e7	1.69765
$792$	356286.	0.0201830
$793$	2.37664e7	1.34208
$794$	905469.	0.0509709
$795$	0	0
$796$	−1.79748e7	−1.00550
$797$	−350546.	−0.0195478	−0.00977392	−	0.999952i	$-0.503111\pi$
$797$	−350546.	−0.0195478	−0.00977392	+	0.999952i	$0.503111\pi$
$798$	−3.62229e6	−0.201361
$799$	7.70621e6	0.427045
$800$	0	0
$801$	−8.51439e6	−0.468892
$802$	−1.36453e6	−0.0749114
$803$	3.60340e6	0.197208
$804$	−154448.	−0.00842638
$805$	0	0
$806$	1.78036e6	0.0965316
$807$	1.67726e7	0.906602
$808$	−5.50832e6	−0.296818
$809$	2.10278e7	1.12959	0.564797	−	0.825230i	$-0.308955\pi$
$809$	2.10278e7	1.12959	0.564797	+	0.825230i	$0.308955\pi$
$810$	0	0
$811$	−2.76340e7	−1.47534	−0.737669	−	0.675163i	$-0.764073\pi$
$811$	−2.76340e7	−1.47534	−0.737669	+	0.675163i	$0.764073\pi$
$812$	8.92449e6	0.475000
$813$	−1.29745e7	−0.688436
$814$	−360557.	−0.0190728
$815$	0	0
$816$	−3.86173e6	−0.203028
$817$	3.13864e7	1.64508
$818$	−2.77020e6	−0.144753
$819$	−1.66355e7	−0.866615
$820$	0	0
$821$	−1.60377e7	−0.830395	−0.415198	−	0.909731i	$-0.636288\pi$
$821$	−1.60377e7	−0.830395	−0.415198	+	0.909731i	$0.636288\pi$
$822$	−1.65044e6	−0.0851964
$823$	−1.06543e7	−0.548309	−0.274155	−	0.961686i	$-0.588398\pi$
$823$	−1.06543e7	−0.548309	−0.274155	+	0.961686i	$0.588398\pi$
$824$	−751243.	−0.0385445
$825$	0	0
$826$	5.63936e6	0.287594
$827$	2.59175e7	1.31774	0.658870	−	0.752256i	$-0.271035\pi$
$827$	2.59175e7	1.31774	0.658870	+	0.752256i	$0.271035\pi$
$828$	1.13330e7	0.574473
$829$	2.00440e7	1.01297	0.506487	−	0.862247i	$-0.330944\pi$
$829$	2.00440e7	1.01297	0.506487	+	0.862247i	$0.330944\pi$
$830$	0	0
$831$	1.25488e7	0.630376
$832$	−2.57852e7	−1.29141
$833$	−1.87161e7	−0.934549
$834$	867235.	0.0431739
$835$	0	0
$836$	1.10200e7	0.545338
$837$	2.71394e6	0.133902
$838$	433626.	0.0213307
$839$	4.53868e6	0.222600	0.111300	−	0.993787i	$-0.464499\pi$
$839$	4.53868e6	0.222600	0.111300	+	0.993787i	$0.464499\pi$
$840$	0	0
$841$	−1.91905e7	−0.935612
$842$	−1.90020e6	−0.0923674
$843$	−7.13186e6	−0.345648
$844$	2.38182e7	1.15094
$845$	0	0
$846$	−824558.	−0.0396091
$847$	−3.58966e6	−0.171927
$848$	2.57158e7	1.22803
$849$	1.39122e7	0.662409
$850$	0	0
$851$	−2.30558e7	−1.09133
$852$	1.43999e7	0.679613
$853$	−2.05582e7	−0.967416	−0.483708	−	0.875230i	$-0.660710\pi$
$853$	−2.05582e7	−0.967416	−0.483708	+	0.875230i	$0.660710\pi$
$854$	−3.97139e6	−0.186337
$855$	0	0
$856$	−2.14877e6	−0.100232
$857$	−1.42934e7	−0.664790	−0.332395	−	0.943140i	$-0.607857\pi$
$857$	−1.42934e7	−0.664790	−0.332395	+	0.943140i	$0.607857\pi$
$858$	−520790.	−0.0241515
$859$	−1.65900e7	−0.767120	−0.383560	−	0.923516i	$-0.625302\pi$
$859$	−1.65900e7	−0.767120	−0.383560	+	0.923516i	$0.625302\pi$
$860$	0	0
$861$	476821.	0.0219204
$862$	−3.00669e6	−0.137823
$863$	3.37000e7	1.54029	0.770147	−	0.637867i	$-0.220183\pi$
$863$	3.37000e7	1.54029	0.770147	+	0.637867i	$0.220183\pi$
$864$	1.26122e6	0.0574787
$865$	0	0
$866$	−1.79357e6	−0.0812686
$867$	−1.10977e7	−0.501399
$868$	2.89108e7	1.30245
$869$	6.15285e6	0.276393
$870$	0	0
$871$	453840.	0.0202702
$872$	1.58276e6	0.0704892
$873$	−1.18693e6	−0.0527096
$874$	−7.25129e6	−0.321097
$875$	0	0
$876$	8.48934e6	0.373778
$877$	−2.66775e7	−1.17124	−0.585621	−	0.810585i	$-0.699149\pi$
$877$	−2.66775e7	−1.17124	−0.585621	+	0.810585i	$0.699149\pi$
$878$	442384.	0.0193671
$879$	−1.59733e7	−0.697304
$880$	0	0
$881$	−1.03891e7	−0.450959	−0.225480	−	0.974248i	$-0.572395\pi$
$881$	−1.03891e7	−0.450959	−0.225480	+	0.974248i	$0.572395\pi$
$882$	2.00260e6	0.0866809
$883$	−1.35919e7	−0.586650	−0.293325	−	0.956013i	$-0.594762\pi$
$883$	−1.35919e7	−0.586650	−0.293325	+	0.956013i	$0.594762\pi$
$884$	1.14668e7	0.493529
$885$	0	0
$886$	1.92517e6	0.0823919
$887$	1.74524e7	0.744812	0.372406	−	0.928070i	$-0.378533\pi$
$887$	1.74524e7	0.744812	0.372406	+	0.928070i	$0.378533\pi$
$888$	−1.70763e6	−0.0726711
$889$	1.73199e7	0.735007
$890$	0	0
$891$	−793881.	−0.0335013
$892$	1.82113e7	0.766355
$893$	−5.12699e7	−2.15146
$894$	−1.34875e6	−0.0564401
$895$	0	0
$896$	1.78824e7	0.744142
$897$	−3.33018e7	−1.38193
$898$	2.69745e6	0.111625
$899$	4.27829e6	0.176551
$900$	0	0
$901$	−1.11944e7	−0.459398
$902$	14927.3	0.000610893 0
$903$	−2.40865e7	−0.983003
$904$	4.42929e6	0.180266
$905$	0	0
$906$	−1.27416e6	−0.0515708
$907$	2.54933e7	1.02898	0.514491	−	0.857496i	$-0.327981\pi$
$907$	2.54933e7	1.02898	0.514491	+	0.857496i	$0.327981\pi$
$908$	2.83310e7	1.14037
$909$	1.22737e7	0.492681
$910$	0	0
$911$	−3.07739e7	−1.22853	−0.614267	−	0.789098i	$-0.710548\pi$
$911$	−3.07739e7	−1.22853	−0.614267	+	0.789098i	$0.710548\pi$
$912$	2.56924e7	1.02286
$913$	3.65310e6	0.145039
$914$	−4.68391e6	−0.185457
$915$	0	0
$916$	−4.95261e6	−0.195027
$917$	−3.50301e7	−1.37568
$918$	−179872.	−0.00704462
$919$	4.48014e7	1.74986	0.874929	−	0.484252i	$-0.160908\pi$
$919$	4.48014e7	1.74986	0.874929	+	0.484252i	$0.160908\pi$
$920$	0	0
$921$	−1.67499e7	−0.650673
$922$	−5.10881e6	−0.197921
$923$	−4.23139e7	−1.63485
$924$	−8.45696e6	−0.325863
$925$	0	0
$926$	−878628.	−0.0336727
$927$	1.67393e6	0.0639791
$928$	1.98820e6	0.0757863
$929$	−1.61239e7	−0.612958	−0.306479	−	0.951877i	$-0.599151\pi$
$929$	−1.61239e7	−0.612958	−0.306479	+	0.951877i	$0.599151\pi$
$930$	0	0
$931$	1.24519e8	4.70828
$932$	−3.33434e6	−0.125739
$933$	−8.75185e6	−0.329151
$934$	2.48045e6	0.0930386
$935$	0	0
$936$	−2.46650e6	−0.0920221
$937$	−1.77870e6	−0.0661839	−0.0330920	−	0.999452i	$-0.510535\pi$
$937$	−1.77870e6	−0.0661839	−0.0330920	+	0.999452i	$0.510535\pi$
$938$	−75837.4	−0.00281434
$939$	−1.60358e7	−0.593509
$940$	0	0
$941$	7.93718e6	0.292208	0.146104	−	0.989269i	$-0.453327\pi$
$941$	7.93718e6	0.292208	0.146104	+	0.989269i	$0.453327\pi$
$942$	−95929.7	−0.00352230
$943$	954525.	0.0349549
$944$	−3.99991e7	−1.46090
$945$	0	0
$946$	−754051.	−0.0273951
$947$	4.35885e6	0.157942	0.0789709	−	0.996877i	$-0.474837\pi$
$947$	4.35885e6	0.157942	0.0789709	+	0.996877i	$0.474837\pi$
$948$	1.44956e7	0.523862
$949$	−2.49457e7	−0.899146
$950$	0	0
$951$	4.85897e6	0.174218
$952$	−3.85196e6	−0.137749
$953$	−2.81338e7	−1.00345	−0.501725	−	0.865027i	$-0.667301\pi$
$953$	−2.81338e7	−1.00345	−0.501725	+	0.865027i	$0.667301\pi$
$954$	1.19779e6	0.0426099
$955$	0	0
$956$	2.09444e6	0.0741178
$957$	−1.25148e6	−0.0441718
$958$	687122.	0.0241891
$959$	7.87544e7	2.76521
$960$	0	0
$961$	−1.47697e7	−0.515897
$962$	2.49607e6	0.0869601
$963$	4.78791e6	0.166372
$964$	−4.39065e7	−1.52173
$965$	0	0
$966$	5.56478e6	0.191869
$967$	−1.89021e7	−0.650046	−0.325023	−	0.945706i	$-0.605372\pi$
$967$	−1.89021e7	−0.650046	−0.325023	+	0.945706i	$0.605372\pi$
$968$	−532230.	−0.0182562
$969$	−1.11842e7	−0.382645
$970$	0	0
$971$	−2.59802e7	−0.884289	−0.442144	−	0.896944i	$-0.645782\pi$
$971$	−2.59802e7	−0.884289	−0.442144	+	0.896944i	$0.645782\pi$
$972$	−1.87032e6	−0.0634966
$973$	−4.13820e7	−1.40129
$974$	−4.47661e6	−0.151200
$975$	0	0
$976$	2.81685e7	0.946539
$977$	4.80735e7	1.61127	0.805637	−	0.592409i	$-0.201823\pi$
$977$	4.80735e7	1.61127	0.805637	+	0.592409i	$0.201823\pi$
$978$	308598.	0.0103168
$979$	1.27190e7	0.424128
$980$	0	0
$981$	−3.52671e6	−0.117003
$982$	3.84552e6	0.127256
$983$	−4.87846e7	−1.61027	−0.805135	−	0.593091i	$-0.797907\pi$
$983$	−4.87846e7	−1.61027	−0.805135	+	0.593091i	$0.797907\pi$
$984$	70697.1	0.00232763
$985$	0	0
$986$	−283553.	−0.00928841
$987$	3.93455e7	1.28559
$988$	−7.62895e7	−2.48641
$989$	−4.82177e7	−1.56753
$990$	0	0
$991$	−1.23808e7	−0.400466	−0.200233	−	0.979748i	$-0.564170\pi$
$991$	−1.23808e7	−0.400466	−0.200233	+	0.979748i	$0.564170\pi$
$992$	6.44076e6	0.207806
$993$	1.13095e7	0.363973
$994$	7.07071e6	0.226985
$995$	0	0
$996$	8.60642e6	0.274900
$997$	−1.02152e7	−0.325468	−0.162734	−	0.986670i	$-0.552031\pi$
$997$	−1.02152e7	−0.325468	−0.162734	+	0.986670i	$0.552031\pi$
$998$	3.78689e6	0.120353
$999$	3.80497e6	0.120625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.w.1.7		13
5.2	odd	4		165.6.c.a.34.14	yes	26
5.3	odd	4		165.6.c.a.34.13	✓	26
5.4	even	2		825.6.a.x.1.7		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.a.34.13	✓	26	5.3	odd	4
165.6.c.a.34.14	yes	26	5.2	odd	4
825.6.a.w.1.7		13	1.1	even	1	trivial
825.6.a.x.1.7		13	5.4	even	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+0.570908 q^{2} +9.00000 q^{3} -31.6741 q^{4} +5.13817 q^{6} -245.179 q^{7} -36.3520 q^{8} +81.0000 q^{9} -121.000 q^{11} -285.067 q^{12} +837.661 q^{13} -139.974 q^{14} +992.816 q^{16} -432.186 q^{17} +46.2435 q^{18} +2875.36 q^{19} -2206.61 q^{21} -69.0798 q^{22} -4417.30 q^{23} -327.168 q^{24} +478.227 q^{26} +729.000 q^{27} +7765.80 q^{28} +1149.20 q^{29} +3722.83 q^{31} +1730.07 q^{32} -1089.00 q^{33} -246.738 q^{34} -2565.60 q^{36} +5219.43 q^{37} +1641.57 q^{38} +7538.95 q^{39} -216.088 q^{41} -1259.77 q^{42} +10915.6 q^{43} +3832.56 q^{44} -2521.87 q^{46} -17830.8 q^{47} +8935.35 q^{48} +43305.6 q^{49} -3889.67 q^{51} -26532.1 q^{52} +25901.8 q^{53} +416.192 q^{54} +8912.74 q^{56} +25878.3 q^{57} +656.089 q^{58} -40288.5 q^{59} +28372.3 q^{61} +2125.39 q^{62} -19859.5 q^{63} -30782.4 q^{64} -621.718 q^{66} +541.795 q^{67} +13689.1 q^{68} -39755.7 q^{69} -50514.3 q^{71} -2944.51 q^{72} -29780.2 q^{73} +2979.81 q^{74} -91074.4 q^{76} +29666.6 q^{77} +4304.05 q^{78} -50850.0 q^{79} +6561.00 q^{81} -123.366 q^{82} -30190.9 q^{83} +69892.2 q^{84} +6231.82 q^{86} +10342.8 q^{87} +4398.59 q^{88} -105116. q^{89} -205377. q^{91} +139914. q^{92} +33505.4 q^{93} -10179.7 q^{94} +15570.6 q^{96} -14653.5 q^{97} +24723.5 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9} - 1573 q^{11} + 2061 q^{12} - 366 q^{13} - 2758 q^{14} + 4141 q^{16} - 2056 q^{17} - 243 q^{18} - 310 q^{19} - 2556 q^{21}+ \cdots - 127413 q^{99}+O(q^{100}) \) 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9 - 1573 * q^11 + 2061 * q^12 - 366 * q^13 - 2758 * q^14 + 4141 * q^16 - 2056 * q^17 - 243 * q^18 - 310 * q^19 - 2556 * q^21 + 363 * q^22 - 3612 * q^23 - 3321 * q^24 + 2280 * q^26 + 9477 * q^27 - 7896 * q^28 - 4848 * q^29 - 24 * q^31 - 38111 * q^32 - 14157 * q^33 + 5518 * q^34 + 18549 * q^36 + 8420 * q^37 - 474 * q^38 - 3294 * q^39 - 15120 * q^41 - 24822 * q^42 - 35492 * q^43 - 27709 * q^44 - 20280 * q^46 - 46544 * q^47 + 37269 * q^48 + 81837 * q^49 - 18504 * q^51 - 107194 * q^52 - 42256 * q^53 - 2187 * q^54 - 196602 * q^56 - 2790 * q^57 - 114160 * q^58 - 65592 * q^59 - 52042 * q^61 - 94972 * q^62 - 23004 * q^63 + 185977 * q^64 + 3267 * q^66 - 80580 * q^67 - 61108 * q^68 - 32508 * q^69 - 77820 * q^71 - 29889 * q^72 - 103050 * q^73 - 240028 * q^74 - 271174 * q^76 + 34364 * q^77 + 20520 * q^78 - 112258 * q^79 + 85293 * q^81 - 64060 * q^82 - 292150 * q^83 - 71064 * q^84 - 319250 * q^86 - 43632 * q^87 + 44649 * q^88 - 295810 * q^89 - 24200 * q^91 - 121328 * q^92 - 216 * q^93 - 358144 * q^94 - 342999 * q^96 - 49072 * q^97 - 101815 * q^98 - 127413 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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