LMFDB - Embedded newform 825.6.a.y.1.12

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:	$0$
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} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + \cdots + 522579400 $ x^13 - 306x^11 - 206x^10 + 34574x^9 + 39928x^8 - 1788312x^7 - 2591628x^6 + 42852537x^5 + 63733360x^4 - 448113518x^3 - 549984598x^2 + 1518551280*x + 522579400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{7} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-9.04603$ 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)

$f(q)$	$=$	$q+10.0460 q^{2} +9.00000 q^{3} +68.9227 q^{4} +90.4143 q^{6} -29.0524 q^{7} +370.927 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+10.0460 q^{2} +9.00000 q^{3} +68.9227 q^{4} +90.4143 q^{6} -29.0524 q^{7} +370.927 q^{8} +81.0000 q^{9} +121.000 q^{11} +620.305 q^{12} +1023.51 q^{13} -291.862 q^{14} +1520.82 q^{16} -1509.69 q^{17} +813.728 q^{18} +1643.04 q^{19} -261.472 q^{21} +1215.57 q^{22} +1478.11 q^{23} +3338.34 q^{24} +10282.2 q^{26} +729.000 q^{27} -2002.37 q^{28} -4572.33 q^{29} +7531.62 q^{31} +3408.50 q^{32} +1089.00 q^{33} -15166.4 q^{34} +5582.74 q^{36} +4408.40 q^{37} +16506.0 q^{38} +9211.62 q^{39} +5629.62 q^{41} -2626.75 q^{42} -2283.11 q^{43} +8339.65 q^{44} +14849.1 q^{46} +5980.48 q^{47} +13687.3 q^{48} -15963.0 q^{49} -13587.2 q^{51} +70543.3 q^{52} +28498.0 q^{53} +7323.56 q^{54} -10776.3 q^{56} +14787.3 q^{57} -45933.7 q^{58} +25942.8 q^{59} -51360.7 q^{61} +75662.8 q^{62} -2353.25 q^{63} -14424.2 q^{64} +10940.1 q^{66} -39180.2 q^{67} -104052. q^{68} +13303.0 q^{69} +37560.3 q^{71} +30045.1 q^{72} +58687.6 q^{73} +44286.9 q^{74} +113242. q^{76} -3515.34 q^{77} +92540.2 q^{78} -8274.06 q^{79} +6561.00 q^{81} +56555.3 q^{82} -21355.4 q^{83} -18021.4 q^{84} -22936.2 q^{86} -41150.9 q^{87} +44882.1 q^{88} +78385.8 q^{89} -29735.5 q^{91} +101875. q^{92} +67784.5 q^{93} +60080.1 q^{94} +30676.5 q^{96} +64574.7 q^{97} -160364. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) $ 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	10.0460	1.77590	0.887952	−	0.459936i	$-0.152128\pi$
$2$	10.0460	1.77590	0.887952	+	0.459936i	$0.152128\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	68.9227	2.15384
$5$	0	0
$5$	0	0
$6$	90.4143	1.02532
$7$	−29.0524	−0.224098	−0.112049	−	0.993703i	$-0.535741\pi$
$7$	−29.0524	−0.224098	−0.112049	+	0.993703i	$0.535741\pi$
$8$	370.927	2.04910
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	620.305	1.24352
$13$	1023.51	1.67971	0.839856	−	0.542809i	$-0.182639\pi$
$13$	1023.51	1.67971	0.839856	+	0.542809i	$0.182639\pi$
$14$	−291.862	−0.397976
$15$	0	0
$16$	1520.82	1.48517
$17$	−1509.69	−1.26696	−0.633482	−	0.773757i	$-0.718375\pi$
$17$	−1509.69	−1.26696	−0.633482	+	0.773757i	$0.718375\pi$
$18$	813.728	0.591968
$19$	1643.04	1.04415	0.522075	−	0.852900i	$-0.325158\pi$
$19$	1643.04	1.04415	0.522075	+	0.852900i	$0.325158\pi$
$20$	0	0
$21$	−261.472	−0.129383
$22$	1215.57	0.535455
$23$	1478.11	0.582622	0.291311	−	0.956628i	$-0.405908\pi$
$23$	1478.11	0.582622	0.291311	+	0.956628i	$0.405908\pi$
$24$	3338.34	1.18305
$25$	0	0
$26$	10282.2	2.98301
$27$	729.000	0.192450
$28$	−2002.37	−0.482669
$29$	−4572.33	−1.00958	−0.504792	−	0.863241i	$-0.668431\pi$
$29$	−4572.33	−1.00958	−0.504792	+	0.863241i	$0.668431\pi$
$30$	0	0
$31$	7531.62	1.40762	0.703808	−	0.710391i	$-0.251482\pi$
$31$	7531.62	1.40762	0.703808	+	0.710391i	$0.251482\pi$
$32$	3408.50	0.588421
$33$	1089.00	0.174078
$34$	−15166.4	−2.25001
$35$	0	0
$36$	5582.74	0.717945
$37$	4408.40	0.529391	0.264695	−	0.964332i	$-0.414729\pi$
$37$	4408.40	0.529391	0.264695	+	0.964332i	$0.414729\pi$
$38$	16506.0	1.85431
$39$	9211.62	0.969783
$40$	0	0
$41$	5629.62	0.523021	0.261511	−	0.965201i	$-0.415779\pi$
$41$	5629.62	0.523021	0.261511	+	0.965201i	$0.415779\pi$
$42$	−2626.75	−0.229771
$43$	−2283.11	−0.188302	−0.0941510	−	0.995558i	$-0.530014\pi$
$43$	−2283.11	−0.188302	−0.0941510	+	0.995558i	$0.530014\pi$
$44$	8339.65	0.649406
$45$	0	0
$46$	14849.1	1.03468
$47$	5980.48	0.394904	0.197452	−	0.980313i	$-0.436733\pi$
$47$	5980.48	0.394904	0.197452	+	0.980313i	$0.436733\pi$
$48$	13687.3	0.857464
$49$	−15963.0	−0.949780
$50$	0	0
$51$	−13587.2	−0.731482
$52$	70543.3	3.61782
$53$	28498.0	1.39356	0.696778	−	0.717287i	$-0.254616\pi$
$53$	28498.0	1.39356	0.696778	+	0.717287i	$0.254616\pi$
$54$	7323.56	0.341773
$55$	0	0
$56$	−10776.3	−0.459199
$57$	14787.3	0.602840
$58$	−45933.7	−1.79292
$59$	25942.8	0.970258	0.485129	−	0.874443i	$-0.338773\pi$
$59$	25942.8	0.970258	0.485129	+	0.874443i	$0.338773\pi$
$60$	0	0
$61$	−51360.7	−1.76728	−0.883642	−	0.468163i	$-0.844916\pi$
$61$	−51360.7	−1.76728	−0.883642	+	0.468163i	$0.844916\pi$
$62$	75662.8	2.49979
$63$	−2353.25	−0.0746992
$64$	−14424.2	−0.440193
$65$	0	0
$66$	10940.1	0.309145
$67$	−39180.2	−1.06630	−0.533151	−	0.846020i	$-0.678992\pi$
$67$	−39180.2	−1.06630	−0.533151	+	0.846020i	$0.678992\pi$
$68$	−104052.	−2.72883
$69$	13303.0	0.336377
$70$	0	0
$71$	37560.3	0.884267	0.442133	−	0.896949i	$-0.354222\pi$
$71$	37560.3	0.884267	0.442133	+	0.896949i	$0.354222\pi$
$72$	30045.1	0.683034
$73$	58687.6	1.28896	0.644480	−	0.764621i	$-0.277074\pi$
$73$	58687.6	1.28896	0.644480	+	0.764621i	$0.277074\pi$
$74$	44286.9	0.940147
$75$	0	0
$76$	113242.	2.24893
$77$	−3515.34	−0.0675680
$78$	92540.2	1.72224
$79$	−8274.06	−0.149160	−0.0745798	−	0.997215i	$-0.523762\pi$
$79$	−8274.06	−0.149160	−0.0745798	+	0.997215i	$0.523762\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	56555.3	0.928836
$83$	−21355.4	−0.340261	−0.170130	−	0.985422i	$-0.554419\pi$
$83$	−21355.4	−0.340261	−0.170130	+	0.985422i	$0.554419\pi$
$84$	−18021.4	−0.278669
$85$	0	0
$86$	−22936.2	−0.334406
$87$	−41150.9	−0.582883
$88$	44882.1	0.617827
$89$	78385.8	1.04897	0.524484	−	0.851420i	$-0.324258\pi$
$89$	78385.8	1.04897	0.524484	+	0.851420i	$0.324258\pi$
$90$	0	0
$91$	−29735.5	−0.376420
$92$	101875.	1.25487
$93$	67784.5	0.812687
$94$	60080.1	0.701312
$95$	0	0
$96$	30676.5	0.339725
$97$	64574.7	0.696840	0.348420	−	0.937339i	$-0.386718\pi$
$97$	64574.7	0.696840	0.348420	+	0.937339i	$0.386718\pi$
$98$	−160364.	−1.68672
$99$	9801.00	0.100504
$100$	0	0
$101$	48364.4	0.471762	0.235881	−	0.971782i	$-0.424203\pi$
$101$	48364.4	0.471762	0.235881	+	0.971782i	$0.424203\pi$
$102$	−136497.	−1.29904
$103$	121923.	1.13238	0.566192	−	0.824273i	$-0.308416\pi$
$103$	121923.	1.13238	0.566192	+	0.824273i	$0.308416\pi$
$104$	379648.	3.44190
$105$	0	0
$106$	286291.	2.47482
$107$	−191883.	−1.62023	−0.810117	−	0.586269i	$-0.800596\pi$
$107$	−191883.	−1.62023	−0.810117	+	0.586269i	$0.800596\pi$
$108$	50244.7	0.414506
$109$	−15660.0	−0.126248	−0.0631240	−	0.998006i	$-0.520106\pi$
$109$	−15660.0	−0.126248	−0.0631240	+	0.998006i	$0.520106\pi$
$110$	0	0
$111$	39675.6	0.305644
$112$	−44183.4	−0.332823
$113$	−15119.6	−0.111389	−0.0556947	−	0.998448i	$-0.517737\pi$
$113$	−15119.6	−0.111389	−0.0556947	+	0.998448i	$0.517737\pi$
$114$	148554.	1.07059
$115$	0	0
$116$	−315137.	−2.17448
$117$	82904.6	0.559904
$118$	260622.	1.72309
$119$	43860.0	0.283924
$120$	0	0
$121$	14641.0	0.0909091
$122$	−515971.	−3.13853
$123$	50666.6	0.301966
$124$	519099.	3.03177
$125$	0	0
$126$	−23640.8	−0.132659
$127$	−223751.	−1.23099	−0.615496	−	0.788140i	$-0.711044\pi$
$127$	−223751.	−1.23099	−0.615496	+	0.788140i	$0.711044\pi$
$128$	−253978.	−1.37016
$129$	−20548.0	−0.108716
$130$	0	0
$131$	269255.	1.37084	0.685418	−	0.728149i	$-0.259619\pi$
$131$	269255.	1.37084	0.685418	+	0.728149i	$0.259619\pi$
$132$	75056.9	0.374935
$133$	−47734.1	−0.233991
$134$	−393606.	−1.89365
$135$	0	0
$136$	−559983.	−2.59614
$137$	172187.	0.783790	0.391895	−	0.920010i	$-0.371820\pi$
$137$	172187.	0.783790	0.391895	+	0.920010i	$0.371820\pi$
$138$	133642.	0.597374
$139$	−350209.	−1.53741	−0.768707	−	0.639601i	$-0.779099\pi$
$139$	−350209.	−1.53741	−0.768707	+	0.639601i	$0.779099\pi$
$140$	0	0
$141$	53824.4	0.227998
$142$	377332.	1.57037
$143$	123845.	0.506452
$144$	123186.	0.495057
$145$	0	0
$146$	589578.	2.28907
$147$	−143667.	−0.548356
$148$	303839.	1.14022
$149$	519184.	1.91582	0.957912	−	0.287062i	$-0.0926785\pi$
$149$	519184.	1.91582	0.957912	+	0.287062i	$0.0926785\pi$
$150$	0	0
$151$	−230203.	−0.821617	−0.410808	−	0.911722i	$-0.634753\pi$
$151$	−230203.	−0.821617	−0.410808	+	0.911722i	$0.634753\pi$
$152$	609446.	2.13957
$153$	−122285.	−0.422321
$154$	−35315.2	−0.119994
$155$	0	0
$156$	634890.	2.08875
$157$	−328864.	−1.06480	−0.532399	−	0.846494i	$-0.678709\pi$
$157$	−328864.	−1.06480	−0.532399	+	0.846494i	$0.678709\pi$
$158$	−83121.5	−0.264893
$159$	256482.	0.804569
$160$	0	0
$161$	−42942.7	−0.130564
$162$	65912.0	0.197323
$163$	−246382.	−0.726341	−0.363171	−	0.931723i	$-0.618306\pi$
$163$	−246382.	−0.726341	−0.363171	+	0.931723i	$0.618306\pi$
$164$	388009.	1.12650
$165$	0	0
$166$	−214537.	−0.604270
$167$	−581413.	−1.61322	−0.806610	−	0.591084i	$-0.798700\pi$
$167$	−581413.	−1.61322	−0.806610	+	0.591084i	$0.798700\pi$
$168$	−96986.9	−0.265118
$169$	676286.	1.82143
$170$	0	0
$171$	133086.	0.348050
$172$	−157358.	−0.405572
$173$	−264650.	−0.672290	−0.336145	−	0.941810i	$-0.609123\pi$
$173$	−264650.	−0.672290	−0.336145	+	0.941810i	$0.609123\pi$
$174$	−413404.	−1.03515
$175$	0	0
$176$	184019.	0.447796
$177$	233485.	0.560179
$178$	787466.	1.86287
$179$	−13886.5	−0.0323937	−0.0161969	−	0.999869i	$-0.505156\pi$
$179$	−13886.5	−0.0323937	−0.0161969	+	0.999869i	$0.505156\pi$
$180$	0	0
$181$	−752593.	−1.70751	−0.853756	−	0.520673i	$-0.825681\pi$
$181$	−752593.	−1.70751	−0.853756	+	0.520673i	$0.825681\pi$
$182$	−298724.	−0.668485
$183$	−462246.	−1.02034
$184$	548271.	1.19385
$185$	0	0
$186$	680966.	1.44325
$187$	−182672.	−0.382004
$188$	412191.	0.850558
$189$	−21179.2	−0.0431276
$190$	0	0
$191$	−25416.1	−0.0504110	−0.0252055	−	0.999682i	$-0.508024\pi$
$191$	−25416.1	−0.0504110	−0.0252055	+	0.999682i	$0.508024\pi$
$192$	−129818.	−0.254145
$193$	−489822.	−0.946553	−0.473277	−	0.880914i	$-0.656929\pi$
$193$	−489822.	−0.946553	−0.473277	+	0.880914i	$0.656929\pi$
$194$	648719.	1.23752
$195$	0	0
$196$	−1.10021e6	−2.04567
$197$	−324827.	−0.596329	−0.298164	−	0.954515i	$-0.596374\pi$
$197$	−324827.	−0.596329	−0.298164	+	0.954515i	$0.596374\pi$
$198$	98461.1	0.178485
$199$	−398516.	−0.713368	−0.356684	−	0.934225i	$-0.616093\pi$
$199$	−398516.	−0.713368	−0.356684	+	0.934225i	$0.616093\pi$
$200$	0	0
$201$	−352622.	−0.615629
$202$	485870.	0.837803
$203$	132837.	0.226245
$204$	−936465.	−1.57549
$205$	0	0
$206$	1.22485e6	2.01101
$207$	119727.	0.194207
$208$	1.55657e6	2.49466
$209$	198807.	0.314823
$210$	0	0
$211$	−868645.	−1.34319	−0.671593	−	0.740920i	$-0.734390\pi$
$211$	−868645.	−1.34319	−0.671593	+	0.740920i	$0.734390\pi$
$212$	1.96416e6	3.00149
$213$	338043.	0.510532
$214$	−1.92766e6	−2.87738
$215$	0	0
$216$	270406.	0.394350
$217$	−218812.	−0.315443
$218$	−157320.	−0.224204
$219$	528189.	0.744181
$220$	0	0
$221$	−1.54518e6	−2.12814
$222$	398582.	0.542794
$223$	−48571.9	−0.0654068	−0.0327034	−	0.999465i	$-0.510412\pi$
$223$	−48571.9	−0.0654068	−0.0327034	+	0.999465i	$0.510412\pi$
$224$	−99025.1	−0.131864
$225$	0	0
$226$	−151892.	−0.197817
$227$	78669.6	0.101331	0.0506655	−	0.998716i	$-0.483866\pi$
$227$	78669.6	0.101331	0.0506655	+	0.998716i	$0.483866\pi$
$228$	1.01918e6	1.29842
$229$	−324090.	−0.408391	−0.204196	−	0.978930i	$-0.565458\pi$
$229$	−324090.	−0.408391	−0.204196	+	0.978930i	$0.565458\pi$
$230$	0	0
$231$	−31638.1	−0.0390104
$232$	−1.69600e6	−2.06874
$233$	1.26828e6	1.53048	0.765238	−	0.643747i	$-0.222621\pi$
$233$	1.26828e6	1.53048	0.765238	+	0.643747i	$0.222621\pi$
$234$	832862.	0.994336
$235$	0	0
$236$	1.78805e6	2.08978
$237$	−74466.6	−0.0861173
$238$	440619.	0.504221
$239$	−1.35381e6	−1.53307	−0.766535	−	0.642202i	$-0.778021\pi$
$239$	−1.35381e6	−1.53307	−0.766535	+	0.642202i	$0.778021\pi$
$240$	0	0
$241$	1.14198e6	1.26653	0.633264	−	0.773936i	$-0.281715\pi$
$241$	1.14198e6	1.26653	0.633264	+	0.773936i	$0.281715\pi$
$242$	147084.	0.161446
$243$	59049.0	0.0641500
$244$	−3.53992e6	−3.80644
$245$	0	0
$246$	508998.	0.536264
$247$	1.68167e6	1.75387
$248$	2.79368e6	2.88434
$249$	−192198.	−0.196449
$250$	0	0
$251$	−321578.	−0.322183	−0.161091	−	0.986940i	$-0.551501\pi$
$251$	−321578.	−0.322183	−0.161091	+	0.986940i	$0.551501\pi$
$252$	−162192.	−0.160890
$253$	178851.	0.175667
$254$	−2.24781e6	−2.18612
$255$	0	0
$256$	−2.08990e6	−1.99308
$257$	1.44275e6	1.36257	0.681286	−	0.732018i	$-0.261421\pi$
$257$	1.44275e6	1.36257	0.681286	+	0.732018i	$0.261421\pi$
$258$	−206425.	−0.193070
$259$	−128075.	−0.118635
$260$	0	0
$261$	−370359.	−0.336528
$262$	2.70495e6	2.43447
$263$	838618.	0.747610	0.373805	−	0.927507i	$-0.378053\pi$
$263$	838618.	0.747610	0.373805	+	0.927507i	$0.378053\pi$
$264$	403939.	0.356703
$265$	0	0
$266$	−479539.	−0.415546
$267$	705472.	0.605622
$268$	−2.70041e6	−2.29664
$269$	−415262.	−0.349898	−0.174949	−	0.984577i	$-0.555976\pi$
$269$	−415262.	−0.349898	−0.174949	+	0.984577i	$0.555976\pi$
$270$	0	0
$271$	647209.	0.535330	0.267665	−	0.963512i	$-0.413748\pi$
$271$	647209.	0.535330	0.267665	+	0.963512i	$0.413748\pi$
$272$	−2.29595e6	−1.88166
$273$	−267620.	−0.217326
$274$	1.72980e6	1.39193
$275$	0	0
$276$	916878.	0.724501
$277$	679011.	0.531713	0.265856	−	0.964013i	$-0.414345\pi$
$277$	679011.	0.531713	0.265856	+	0.964013i	$0.414345\pi$
$278$	−3.51821e6	−2.73030
$279$	610061.	0.469205
$280$	0	0
$281$	−1.78685e6	−1.34997	−0.674983	−	0.737834i	$-0.735849\pi$
$281$	−1.78685e6	−1.34997	−0.674983	+	0.737834i	$0.735849\pi$
$282$	540721.	0.404903
$283$	−872555.	−0.647630	−0.323815	−	0.946120i	$-0.604966\pi$
$283$	−872555.	−0.647630	−0.323815	+	0.946120i	$0.604966\pi$
$284$	2.58876e6	1.90457
$285$	0	0
$286$	1.24415e6	0.899411
$287$	−163554.	−0.117208
$288$	276088.	0.196140
$289$	859295.	0.605198
$290$	0	0
$291$	581172.	0.402321
$292$	4.04491e6	2.77621
$293$	653130.	0.444458	0.222229	−	0.974995i	$-0.428667\pi$
$293$	653130.	0.444458	0.222229	+	0.974995i	$0.428667\pi$
$294$	−1.44328e6	−0.973827
$295$	0	0
$296$	1.63519e6	1.08477
$297$	88209.0	0.0580259
$298$	5.21574e6	3.40232
$299$	1.51286e6	0.978638
$300$	0	0
$301$	66329.8	0.0421980
$302$	−2.31263e6	−1.45911
$303$	435280.	0.272372
$304$	2.49875e6	1.55074
$305$	0	0
$306$	−1.22847e6	−0.750002
$307$	−1.87392e6	−1.13476	−0.567381	−	0.823456i	$-0.692043\pi$
$307$	−1.87392e6	−1.13476	−0.567381	+	0.823456i	$0.692043\pi$
$308$	−242287.	−0.145530
$309$	1.09731e6	0.653783
$310$	0	0
$311$	−1.60626e6	−0.941705	−0.470853	−	0.882212i	$-0.656054\pi$
$311$	−1.60626e6	−0.941705	−0.470853	+	0.882212i	$0.656054\pi$
$312$	3.41684e6	1.98718
$313$	−2.58202e6	−1.48970	−0.744849	−	0.667234i	$-0.767478\pi$
$313$	−2.58202e6	−1.48970	−0.744849	+	0.667234i	$0.767478\pi$
$314$	−3.30377e6	−1.89098
$315$	0	0
$316$	−570271.	−0.321265
$317$	−1.34588e6	−0.752242	−0.376121	−	0.926571i	$-0.622742\pi$
$317$	−1.34588e6	−0.752242	−0.376121	+	0.926571i	$0.622742\pi$
$318$	2.57662e6	1.42884
$319$	−553252.	−0.304401
$320$	0	0
$321$	−1.72695e6	−0.935442
$322$	−431403.	−0.231870
$323$	−2.48047e6	−1.32290
$324$	452202.	0.239315
$325$	0	0
$326$	−2.47516e6	−1.28991
$327$	−140940.	−0.0728893
$328$	2.08818e6	1.07172
$329$	−173748.	−0.0884971
$330$	0	0
$331$	2.74357e6	1.37641	0.688203	−	0.725518i	$-0.258400\pi$
$331$	2.74357e6	1.37641	0.688203	+	0.725518i	$0.258400\pi$
$332$	−1.47187e6	−0.732865
$333$	357080.	0.176464
$334$	−5.84089e6	−2.86492
$335$	0	0
$336$	−397650.	−0.192156
$337$	3.70365e6	1.77646	0.888229	−	0.459400i	$-0.151936\pi$
$337$	3.70365e6	1.77646	0.888229	+	0.459400i	$0.151936\pi$
$338$	6.79399e6	3.23469
$339$	−136076.	−0.0643107
$340$	0	0
$341$	911325.	0.424412
$342$	1.33698e6	0.618103
$343$	952047.	0.436941
$344$	−846866.	−0.385850
$345$	0	0
$346$	−2.65868e6	−1.19392
$347$	−1.17221e6	−0.522616	−0.261308	−	0.965255i	$-0.584154\pi$
$347$	−1.17221e6	−0.522616	−0.261308	+	0.965255i	$0.584154\pi$
$348$	−2.83624e6	−1.25543
$349$	−360082.	−0.158248	−0.0791240	−	0.996865i	$-0.525212\pi$
$349$	−360082.	−0.158248	−0.0791240	+	0.996865i	$0.525212\pi$
$350$	0	0
$351$	746141.	0.323261
$352$	412428.	0.177415
$353$	998723.	0.426588	0.213294	−	0.976988i	$-0.431581\pi$
$353$	998723.	0.426588	0.213294	+	0.976988i	$0.431581\pi$
$354$	2.34560e6	0.994824
$355$	0	0
$356$	5.40256e6	2.25930
$357$	394740.	0.163923
$358$	−139504.	−0.0575282
$359$	1.47706e6	0.604871	0.302436	−	0.953170i	$-0.402200\pi$
$359$	1.47706e6	0.604871	0.302436	+	0.953170i	$0.402200\pi$
$360$	0	0
$361$	223465.	0.0902489
$362$	−7.56057e6	−3.03238
$363$	131769.	0.0524864
$364$	−2.04945e6	−0.810746
$365$	0	0
$366$	−4.64374e6	−1.81203
$367$	3.21905e6	1.24756	0.623781	−	0.781599i	$-0.285596\pi$
$367$	3.21905e6	1.24756	0.623781	+	0.781599i	$0.285596\pi$
$368$	2.24793e6	0.865294
$369$	455999.	0.174340
$370$	0	0
$371$	−827935.	−0.312292
$372$	4.67190e6	1.75039
$373$	−4.65790e6	−1.73348	−0.866739	−	0.498762i	$-0.833788\pi$
$373$	−4.65790e6	−1.73348	−0.866739	+	0.498762i	$0.833788\pi$
$374$	−1.83513e6	−0.678403
$375$	0	0
$376$	2.21832e6	0.809198
$377$	−4.67984e6	−1.69581
$378$	−212767.	−0.0765905
$379$	−4.04644e6	−1.44702	−0.723510	−	0.690314i	$-0.757472\pi$
$379$	−4.04644e6	−1.44702	−0.723510	+	0.690314i	$0.757472\pi$
$380$	0	0
$381$	−2.01376e6	−0.710714
$382$	−255331.	−0.0895252
$383$	824148.	0.287084	0.143542	−	0.989644i	$-0.454151\pi$
$383$	824148.	0.287084	0.143542	+	0.989644i	$0.454151\pi$
$384$	−2.28580e6	−0.791062
$385$	0	0
$386$	−4.92077e6	−1.68099
$387$	−184932.	−0.0627674
$388$	4.45066e6	1.50088
$389$	−2.54462e6	−0.852607	−0.426303	−	0.904580i	$-0.640184\pi$
$389$	−2.54462e6	−0.852607	−0.426303	+	0.904580i	$0.640184\pi$
$390$	0	0
$391$	−2.23148e6	−0.738162
$392$	−5.92109e6	−1.94620
$393$	2.42330e6	0.791453
$394$	−3.26322e6	−1.05902
$395$	0	0
$396$	675512.	0.216469
$397$	−5.17776e6	−1.64879	−0.824395	−	0.566015i	$-0.808484\pi$
$397$	−5.17776e6	−1.64879	−0.824395	+	0.566015i	$0.808484\pi$
$398$	−4.00351e6	−1.26687
$399$	−429607.	−0.135095
$400$	0	0
$401$	−4.82239e6	−1.49762	−0.748809	−	0.662785i	$-0.769374\pi$
$401$	−4.82239e6	−1.49762	−0.748809	+	0.662785i	$0.769374\pi$
$402$	−3.54245e6	−1.09330
$403$	7.70871e6	2.36439
$404$	3.33341e6	1.01610
$405$	0	0
$406$	1.33449e6	0.401790
$407$	533416.	0.159617
$408$	−5.03985e6	−1.49888
$409$	−3.68626e6	−1.08963	−0.544813	−	0.838558i	$-0.683399\pi$
$409$	−3.68626e6	−1.08963	−0.544813	+	0.838558i	$0.683399\pi$
$410$	0	0
$411$	1.54968e6	0.452521
$412$	8.40329e6	2.43897
$413$	−753702.	−0.217433
$414$	1.20278e6	0.344894
$415$	0	0
$416$	3.48864e6	0.988377
$417$	−3.15188e6	−0.887626
$418$	1.99722e6	0.559095
$419$	1.31904e6	0.367049	0.183525	−	0.983015i	$-0.441249\pi$
$419$	1.31904e6	0.367049	0.183525	+	0.983015i	$0.441249\pi$
$420$	0	0
$421$	−3.29518e6	−0.906095	−0.453048	−	0.891486i	$-0.649663\pi$
$421$	−3.29518e6	−0.906095	−0.453048	+	0.891486i	$0.649663\pi$
$422$	−8.72643e6	−2.38537
$423$	484419.	0.131635
$424$	1.05707e7	2.85553
$425$	0	0
$426$	3.39599e6	0.906655
$427$	1.49215e6	0.396044
$428$	−1.32251e7	−3.48972
$429$	1.11461e6	0.292400
$430$	0	0
$431$	1.87137e6	0.485251	0.242625	−	0.970120i	$-0.421991\pi$
$431$	1.87137e6	0.485251	0.242625	+	0.970120i	$0.421991\pi$
$432$	1.10867e6	0.285821
$433$	4.31194e6	1.10523	0.552615	−	0.833437i	$-0.313630\pi$
$433$	4.31194e6	1.10523	0.552615	+	0.833437i	$0.313630\pi$
$434$	−2.19819e6	−0.560197
$435$	0	0
$436$	−1.07933e6	−0.271917
$437$	2.42859e6	0.608345
$438$	5.30620e6	1.32159
$439$	1.13612e6	0.281360	0.140680	−	0.990055i	$-0.455071\pi$
$439$	1.13612e6	0.281360	0.140680	+	0.990055i	$0.455071\pi$
$440$	0	0
$441$	−1.29300e6	−0.316593
$442$	−1.55230e7	−3.77936
$443$	−875875.	−0.212047	−0.106024	−	0.994364i	$-0.533812\pi$
$443$	−875875.	−0.212047	−0.106024	+	0.994364i	$0.533812\pi$
$444$	2.73455e6	0.658306
$445$	0	0
$446$	−487955.	−0.116156
$447$	4.67266e6	1.10610
$448$	419059.	0.0986461
$449$	4.08806e6	0.956977	0.478489	−	0.878094i	$-0.341185\pi$
$449$	4.08806e6	0.956977	0.478489	+	0.878094i	$0.341185\pi$
$450$	0	0
$451$	681184.	0.157697
$452$	−1.04208e6	−0.239915
$453$	−2.07183e6	−0.474361
$454$	790317.	0.179954
$455$	0	0
$456$	5.48501e6	1.23528
$457$	2.85279e6	0.638968	0.319484	−	0.947592i	$-0.396490\pi$
$457$	2.85279e6	0.638968	0.319484	+	0.947592i	$0.396490\pi$
$458$	−3.25581e6	−0.725263
$459$	−1.10056e6	−0.243827
$460$	0	0
$461$	6.16387e6	1.35083	0.675416	−	0.737437i	$-0.263964\pi$
$461$	6.16387e6	1.35083	0.675416	+	0.737437i	$0.263964\pi$
$462$	−317837.	−0.0692787
$463$	−4.52412e6	−0.980804	−0.490402	−	0.871496i	$-0.663150\pi$
$463$	−4.52412e6	−0.980804	−0.490402	+	0.871496i	$0.663150\pi$
$464$	−6.95366e6	−1.49940
$465$	0	0
$466$	1.27412e7	2.71798
$467$	2.23084e6	0.473344	0.236672	−	0.971590i	$-0.423943\pi$
$467$	2.23084e6	0.473344	0.236672	+	0.971590i	$0.423943\pi$
$468$	5.71401e6	1.20594
$469$	1.13828e6	0.238956
$470$	0	0
$471$	−2.95977e6	−0.614761
$472$	9.62289e6	1.98816
$473$	−276256.	−0.0567752
$474$	−748093.	−0.152936
$475$	0	0
$476$	3.02295e6	0.611525
$477$	2.30834e6	0.464518
$478$	−1.36004e7	−2.72259
$479$	−6.62215e6	−1.31874	−0.659372	−	0.751817i	$-0.729178\pi$
$479$	−6.62215e6	−1.31874	−0.659372	+	0.751817i	$0.729178\pi$
$480$	0	0
$481$	4.51205e6	0.889224
$482$	1.14723e7	2.24923
$483$	−386484.	−0.0753813
$484$	1.00910e6	0.195803
$485$	0	0
$486$	593208.	0.113924
$487$	2.17583e6	0.415721	0.207860	−	0.978158i	$-0.433350\pi$
$487$	2.17583e6	0.415721	0.207860	+	0.978158i	$0.433350\pi$
$488$	−1.90511e7	−3.62134
$489$	−2.21744e6	−0.419353
$490$	0	0
$491$	7.67198e6	1.43616	0.718081	−	0.695959i	$-0.245021\pi$
$491$	7.67198e6	1.43616	0.718081	+	0.695959i	$0.245021\pi$
$492$	3.49208e6	0.650386
$493$	6.90278e6	1.27911
$494$	1.68941e7	3.11471
$495$	0	0
$496$	1.14542e7	2.09055
$497$	−1.09122e6	−0.198162
$498$	−1.93083e6	−0.348875
$499$	6.65009e6	1.19557	0.597787	−	0.801655i	$-0.296047\pi$
$499$	6.65009e6	1.19557	0.597787	+	0.801655i	$0.296047\pi$
$500$	0	0
$501$	−5.23272e6	−0.931393
$502$	−3.23058e6	−0.572165
$503$	4.18197e6	0.736989	0.368495	−	0.929630i	$-0.379873\pi$
$503$	4.18197e6	0.736989	0.368495	+	0.929630i	$0.379873\pi$
$504$	−872882.	−0.153066
$505$	0	0
$506$	1.79675e6	0.311968
$507$	6.08657e6	1.05161
$508$	−1.54215e7	−2.65135
$509$	−1.01322e7	−1.73345	−0.866725	−	0.498787i	$-0.833779\pi$
$509$	−1.01322e7	−1.73345	−0.866725	+	0.498787i	$0.833779\pi$
$510$	0	0
$511$	−1.70502e6	−0.288853
$512$	−1.28679e7	−2.16936
$513$	1.19777e6	0.200947
$514$	1.44939e7	2.41980
$515$	0	0
$516$	−1.41622e6	−0.234157
$517$	723639.	0.119068
$518$	−1.28664e6	−0.210685
$519$	−2.38185e6	−0.388147
$520$	0	0
$521$	−6.42546e6	−1.03707	−0.518537	−	0.855055i	$-0.673523\pi$
$521$	−6.42546e6	−1.03707	−0.518537	+	0.855055i	$0.673523\pi$
$522$	−3.72063e6	−0.597641
$523$	−1.17398e6	−0.187674	−0.0938372	−	0.995588i	$-0.529913\pi$
$523$	−1.17398e6	−0.187674	−0.0938372	+	0.995588i	$0.529913\pi$
$524$	1.85578e7	2.95256
$525$	0	0
$526$	8.42478e6	1.32768
$527$	−1.13704e7	−1.78340
$528$	1.65617e6	0.258535
$529$	−4.25153e6	−0.660551
$530$	0	0
$531$	2.10137e6	0.323419
$532$	−3.28997e6	−0.503979
$533$	5.76199e6	0.878525
$534$	7.08719e6	1.07553
$535$	0	0
$536$	−1.45330e7	−2.18496
$537$	−124979.	−0.0187025
$538$	−4.17174e6	−0.621386
$539$	−1.93152e6	−0.286370
$540$	0	0
$541$	−5.25283e6	−0.771614	−0.385807	−	0.922580i	$-0.626077\pi$
$541$	−5.25283e6	−0.771614	−0.385807	+	0.922580i	$0.626077\pi$
$542$	6.50188e6	0.950694
$543$	−6.77334e6	−0.985833
$544$	−5.14576e6	−0.745508
$545$	0	0
$546$	−2.68852e6	−0.385950
$547$	1.13849e7	1.62689	0.813447	−	0.581640i	$-0.197589\pi$
$547$	1.13849e7	1.62689	0.813447	+	0.581640i	$0.197589\pi$
$548$	1.18676e7	1.68815
$549$	−4.16022e6	−0.589095
$550$	0	0
$551$	−7.51249e6	−1.05416
$552$	4.93444e6	0.689271
$553$	240382.	0.0334263
$554$	6.82136e6	0.944271
$555$	0	0
$556$	−2.41374e7	−3.31134
$557$	4.54355e6	0.620522	0.310261	−	0.950651i	$-0.399584\pi$
$557$	4.54355e6	0.620522	0.310261	+	0.950651i	$0.399584\pi$
$558$	6.12869e6	0.833263
$559$	−2.33679e6	−0.316293
$560$	0	0
$561$	−1.64405e6	−0.220550
$562$	−1.79508e7	−2.39741
$563$	−1.22077e7	−1.62317	−0.811583	−	0.584237i	$-0.801394\pi$
$563$	−1.22077e7	−1.62317	−0.811583	+	0.584237i	$0.801394\pi$
$564$	3.70972e6	0.491070
$565$	0	0
$566$	−8.76572e6	−1.15013
$567$	−190613.	−0.0248997
$568$	1.39321e7	1.81195
$569$	−1.58948e6	−0.205814	−0.102907	−	0.994691i	$-0.532814\pi$
$569$	−1.58948e6	−0.205814	−0.102907	+	0.994691i	$0.532814\pi$
$570$	0	0
$571$	−692853.	−0.0889306	−0.0444653	−	0.999011i	$-0.514158\pi$
$571$	−692853.	−0.0889306	−0.0444653	+	0.999011i	$0.514158\pi$
$572$	8.53574e6	1.09082
$573$	−228745.	−0.0291048
$574$	−1.64307e6	−0.208150
$575$	0	0
$576$	−1.16836e6	−0.146731
$577$	1.24815e7	1.56073	0.780365	−	0.625324i	$-0.215033\pi$
$577$	1.24815e7	1.56073	0.780365	+	0.625324i	$0.215033\pi$
$578$	8.63250e6	1.07477
$579$	−4.40840e6	−0.546493
$580$	0	0
$581$	620425.	0.0762516
$582$	5.83847e6	0.714483
$583$	3.44825e6	0.420173
$584$	2.17688e7	2.64121
$585$	0	0
$586$	6.56136e6	0.789314
$587$	−9.13291e6	−1.09399	−0.546996	−	0.837135i	$-0.684229\pi$
$587$	−9.13291e6	−1.09399	−0.546996	+	0.837135i	$0.684229\pi$
$588$	−9.90189e6	−1.18107
$589$	1.23747e7	1.46976
$590$	0	0
$591$	−2.92344e6	−0.344291
$592$	6.70436e6	0.786236
$593$	1.08272e7	1.26438	0.632191	−	0.774813i	$-0.282156\pi$
$593$	1.08272e7	1.26438	0.632191	+	0.774813i	$0.282156\pi$
$594$	886150.	0.103048
$595$	0	0
$596$	3.57836e7	4.12637
$597$	−3.58665e6	−0.411863
$598$	1.51983e7	1.73797
$599$	7.47502e6	0.851227	0.425614	−	0.904905i	$-0.360058\pi$
$599$	7.47502e6	0.851227	0.425614	+	0.904905i	$0.360058\pi$
$600$	0	0
$601$	−1.02516e6	−0.115772	−0.0578861	−	0.998323i	$-0.518436\pi$
$601$	−1.02516e6	−0.115772	−0.0578861	+	0.998323i	$0.518436\pi$
$602$	666351.	0.0749397
$603$	−3.17360e6	−0.355434
$604$	−1.58662e7	−1.76963
$605$	0	0
$606$	4.37283e6	0.483706
$607$	1.33264e7	1.46805	0.734024	−	0.679123i	$-0.237640\pi$
$607$	1.33264e7	1.46805	0.734024	+	0.679123i	$0.237640\pi$
$608$	5.60028e6	0.614399
$609$	1.19553e6	0.130623
$610$	0	0
$611$	6.12110e6	0.663325
$612$	−8.42819e6	−0.909611
$613$	−1.57435e7	−1.69220	−0.846098	−	0.533027i	$-0.821054\pi$
$613$	−1.57435e7	−1.69220	−0.846098	+	0.533027i	$0.821054\pi$
$614$	−1.88254e7	−2.01523
$615$	0	0
$616$	−1.30394e6	−0.138454
$617$	−5.87960e6	−0.621777	−0.310888	−	0.950446i	$-0.600627\pi$
$617$	−5.87960e6	−0.621777	−0.310888	+	0.950446i	$0.600627\pi$
$618$	1.10236e7	1.16106
$619$	−1.81138e6	−0.190013	−0.0950063	−	0.995477i	$-0.530287\pi$
$619$	−1.81138e6	−0.190013	−0.0950063	+	0.995477i	$0.530287\pi$
$620$	0	0
$621$	1.07754e6	0.112126
$622$	−1.61365e7	−1.67238
$623$	−2.27730e6	−0.235071
$624$	1.40092e7	1.44029
$625$	0	0
$626$	−2.59390e7	−2.64556
$627$	1.78927e6	0.181763
$628$	−2.26662e7	−2.29340
$629$	−6.65529e6	−0.670719
$630$	0	0
$631$	−1.43835e7	−1.43811	−0.719054	−	0.694954i	$-0.755425\pi$
$631$	−1.43835e7	−1.43811	−0.719054	+	0.694954i	$0.755425\pi$
$632$	−3.06907e6	−0.305643
$633$	−7.81780e6	−0.775489
$634$	−1.35207e7	−1.33591
$635$	0	0
$636$	1.76774e7	1.73291
$637$	−1.63383e7	−1.59536
$638$	−5.55798e6	−0.540587
$639$	3.04239e6	0.294756
$640$	0	0
$641$	−3.80867e6	−0.366124	−0.183062	−	0.983101i	$-0.558601\pi$
$641$	−3.80867e6	−0.366124	−0.183062	+	0.983101i	$0.558601\pi$
$642$	−1.73490e7	−1.66126
$643$	483920.	0.0461579	0.0230789	−	0.999734i	$-0.492653\pi$
$643$	483920.	0.0461579	0.0230789	+	0.999734i	$0.492653\pi$
$644$	−2.95973e6	−0.281214
$645$	0	0
$646$	−2.49188e7	−2.34934
$647$	1.80652e7	1.69661	0.848306	−	0.529506i	$-0.177623\pi$
$647$	1.80652e7	1.69661	0.848306	+	0.529506i	$0.177623\pi$
$648$	2.43365e6	0.227678
$649$	3.13908e6	0.292544
$650$	0	0
$651$	−1.96931e6	−0.182121
$652$	−1.69813e7	−1.56442
$653$	2.03744e7	1.86983	0.934915	−	0.354871i	$-0.115475\pi$
$653$	2.03744e7	1.86983	0.934915	+	0.354871i	$0.115475\pi$
$654$	−1.41588e6	−0.129444
$655$	0	0
$656$	8.56161e6	0.776776
$657$	4.75370e6	0.429653
$658$	−1.74547e6	−0.157162
$659$	−1.37586e7	−1.23413	−0.617066	−	0.786912i	$-0.711679\pi$
$659$	−1.37586e7	−1.23413	−0.617066	+	0.786912i	$0.711679\pi$
$660$	0	0
$661$	−1.34803e7	−1.20004	−0.600019	−	0.799985i	$-0.704840\pi$
$661$	−1.34803e7	−1.20004	−0.600019	+	0.799985i	$0.704840\pi$
$662$	2.75620e7	2.44437
$663$	−1.39066e7	−1.22868
$664$	−7.92127e6	−0.697228
$665$	0	0
$666$	3.58724e6	0.313382
$667$	−6.75840e6	−0.588206
$668$	−4.00726e7	−3.47461
$669$	−437147.	−0.0377627
$670$	0	0
$671$	−6.21465e6	−0.532856
$672$	−891226.	−0.0761315
$673$	−680925.	−0.0579511	−0.0289755	−	0.999580i	$-0.509224\pi$
$673$	−680925.	−0.0579511	−0.0289755	+	0.999580i	$0.509224\pi$
$674$	3.72070e7	3.15482
$675$	0	0
$676$	4.66115e7	3.92307
$677$	1.36705e7	1.14634	0.573171	−	0.819436i	$-0.305713\pi$
$677$	1.36705e7	1.14634	0.573171	+	0.819436i	$0.305713\pi$
$678$	−1.36703e6	−0.114210
$679$	−1.87605e6	−0.156160
$680$	0	0
$681$	708026.	0.0585034
$682$	9.15520e6	0.753715
$683$	6.79648e6	0.557484	0.278742	−	0.960366i	$-0.410083\pi$
$683$	6.79648e6	0.557484	0.278742	+	0.960366i	$0.410083\pi$
$684$	9.17264e6	0.749642
$685$	0	0
$686$	9.56429e6	0.775966
$687$	−2.91681e6	−0.235785
$688$	−3.47218e6	−0.279661
$689$	2.91680e7	2.34077
$690$	0	0
$691$	9.11491e6	0.726202	0.363101	−	0.931750i	$-0.381718\pi$
$691$	9.11491e6	0.726202	0.363101	+	0.931750i	$0.381718\pi$
$692$	−1.82404e7	−1.44800
$693$	−284743.	−0.0225227
$694$	−1.17761e7	−0.928116
$695$	0	0
$696$	−1.52640e7	−1.19439
$697$	−8.49896e6	−0.662649
$698$	−3.61740e6	−0.281033
$699$	1.14146e7	0.883621
$700$	0	0
$701$	−1.03729e7	−0.797269	−0.398634	−	0.917110i	$-0.630516\pi$
$701$	−1.03729e7	−0.797269	−0.398634	+	0.917110i	$0.630516\pi$
$702$	7.49575e6	0.574080
$703$	7.24315e6	0.552763
$704$	−1.74533e6	−0.132723
$705$	0	0
$706$	1.00332e7	0.757579
$707$	−1.40510e6	−0.105721
$708$	1.60925e7	1.20653
$709$	1.78828e7	1.33604	0.668022	−	0.744142i	$-0.267141\pi$
$709$	1.78828e7	1.33604	0.668022	+	0.744142i	$0.267141\pi$
$710$	0	0
$711$	−670199.	−0.0497199
$712$	2.90754e7	2.14944
$713$	1.11326e7	0.820108
$714$	3.96557e6	0.291112
$715$	0	0
$716$	−957097.	−0.0697708
$717$	−1.21843e7	−0.885119
$718$	1.48386e7	1.07419
$719$	−2.62904e7	−1.89660	−0.948298	−	0.317382i	$-0.897196\pi$
$719$	−2.62904e7	−1.89660	−0.948298	+	0.317382i	$0.897196\pi$
$720$	0	0
$721$	−3.54217e6	−0.253765
$722$	2.24494e6	0.160273
$723$	1.02778e7	0.731231
$724$	−5.18708e7	−3.67770
$725$	0	0
$726$	1.32376e6	0.0932108
$727$	1.40763e7	0.987763	0.493882	−	0.869529i	$-0.335578\pi$
$727$	1.40763e7	0.987763	0.493882	+	0.869529i	$0.335578\pi$
$728$	−1.10297e7	−0.771322
$729$	531441.	0.0370370
$730$	0	0
$731$	3.44677e6	0.238572
$732$	−3.18593e7	−2.19765
$733$	−5.60463e6	−0.385290	−0.192645	−	0.981269i	$-0.561707\pi$
$733$	−5.60463e6	−0.385290	−0.192645	+	0.981269i	$0.561707\pi$
$734$	3.23387e7	2.21555
$735$	0	0
$736$	5.03813e6	0.342827
$737$	−4.74081e6	−0.321502
$738$	4.58098e6	0.309612
$739$	−1.92933e6	−0.129956	−0.0649778	−	0.997887i	$-0.520698\pi$
$739$	−1.92933e6	−0.129956	−0.0649778	+	0.997887i	$0.520698\pi$
$740$	0	0
$741$	1.51350e7	1.01260
$742$	−8.31746e6	−0.554601
$743$	−4.43790e6	−0.294921	−0.147460	−	0.989068i	$-0.547110\pi$
$743$	−4.43790e6	−0.294921	−0.147460	+	0.989068i	$0.547110\pi$
$744$	2.51431e7	1.66528
$745$	0	0
$746$	−4.67934e7	−3.07849
$747$	−1.72978e6	−0.113420
$748$	−1.25903e7	−0.822774
$749$	5.57467e6	0.363090
$750$	0	0
$751$	−1.11577e7	−0.721897	−0.360949	−	0.932586i	$-0.617547\pi$
$751$	−1.11577e7	−0.721897	−0.360949	+	0.932586i	$0.617547\pi$
$752$	9.09521e6	0.586500
$753$	−2.89420e6	−0.186012
$754$	−4.70138e7	−3.01160
$755$	0	0
$756$	−1.45973e6	−0.0928898
$757$	1.16077e7	0.736216	0.368108	−	0.929783i	$-0.380006\pi$
$757$	1.16077e7	0.736216	0.368108	+	0.929783i	$0.380006\pi$
$758$	−4.06506e7	−2.56977
$759$	1.60966e6	0.101422
$760$	0	0
$761$	−1.38952e6	−0.0869770	−0.0434885	−	0.999054i	$-0.513847\pi$
$761$	−1.38952e6	−0.0869770	−0.0434885	+	0.999054i	$0.513847\pi$
$762$	−2.02303e7	−1.26216
$763$	454960.	0.0282919
$764$	−1.75175e6	−0.108577
$765$	0	0
$766$	8.27942e6	0.509833
$767$	2.65528e7	1.62975
$768$	−1.88091e7	−1.15071
$769$	−2.19019e7	−1.33557	−0.667784	−	0.744355i	$-0.732757\pi$
$769$	−2.19019e7	−1.33557	−0.667784	+	0.744355i	$0.732757\pi$
$770$	0	0
$771$	1.29848e7	0.786681
$772$	−3.37599e7	−2.03872
$773$	−9.21665e6	−0.554784	−0.277392	−	0.960757i	$-0.589470\pi$
$773$	−9.21665e6	−0.554784	−0.277392	+	0.960757i	$0.589470\pi$
$774$	−1.85783e6	−0.111469
$775$	0	0
$776$	2.39525e7	1.42789
$777$	−1.15267e6	−0.0684941
$778$	−2.55633e7	−1.51415
$779$	9.24966e6	0.546113
$780$	0	0
$781$	4.54480e6	0.266617
$782$	−2.24175e7	−1.31090
$783$	−3.33323e6	−0.194294
$784$	−2.42767e7	−1.41059
$785$	0	0
$786$	2.43445e7	1.40554
$787$	−2.18581e7	−1.25798	−0.628992	−	0.777412i	$-0.716532\pi$
$787$	−2.18581e7	−1.25798	−0.628992	+	0.777412i	$0.716532\pi$
$788$	−2.23879e7	−1.28439
$789$	7.54756e6	0.431633
$790$	0	0
$791$	439261.	0.0249621
$792$	3.63545e6	0.205942
$793$	−5.25684e7	−2.96853
$794$	−5.20159e7	−2.92809
$795$	0	0
$796$	−2.74668e7	−1.53648
$797$	2.79333e7	1.55767	0.778837	−	0.627226i	$-0.215810\pi$
$797$	2.79333e7	1.55767	0.778837	+	0.627226i	$0.215810\pi$
$798$	−4.31585e6	−0.239916
$799$	−9.02865e6	−0.500329
$800$	0	0
$801$	6.34925e6	0.349656
$802$	−4.84459e7	−2.65963
$803$	7.10120e6	0.388636
$804$	−2.43037e7	−1.32596
$805$	0	0
$806$	7.74419e7	4.19893
$807$	−3.73736e6	−0.202014
$808$	1.79397e7	0.966687
$809$	1.07778e7	0.578974	0.289487	−	0.957182i	$-0.406515\pi$
$809$	1.07778e7	0.578974	0.289487	+	0.957182i	$0.406515\pi$
$810$	0	0
$811$	−2.71394e7	−1.44893	−0.724467	−	0.689309i	$-0.757914\pi$
$811$	−2.71394e7	−1.44893	−0.724467	+	0.689309i	$0.757914\pi$
$812$	9.15550e6	0.487295
$813$	5.82488e6	0.309073
$814$	5.35871e6	0.283465
$815$	0	0
$816$	−2.06636e7	−1.08638
$817$	−3.75122e6	−0.196616
$818$	−3.70322e7	−1.93507
$819$	−2.40858e6	−0.125473
$820$	0	0
$821$	6.84095e6	0.354208	0.177104	−	0.984192i	$-0.443327\pi$
$821$	6.84095e6	0.354208	0.177104	+	0.984192i	$0.443327\pi$
$822$	1.55682e7	0.803634
$823$	1.95930e7	1.00833	0.504163	−	0.863609i	$-0.331801\pi$
$823$	1.95930e7	1.00833	0.504163	+	0.863609i	$0.331801\pi$
$824$	4.52247e7	2.32037
$825$	0	0
$826$	−7.57171e6	−0.386139
$827$	−1.86828e7	−0.949902	−0.474951	−	0.880012i	$-0.657534\pi$
$827$	−1.86828e7	−0.949902	−0.474951	+	0.880012i	$0.657534\pi$
$828$	8.25191e6	0.418291
$829$	−1.58057e7	−0.798779	−0.399390	−	0.916781i	$-0.630778\pi$
$829$	−1.58057e7	−0.798779	−0.399390	+	0.916781i	$0.630778\pi$
$830$	0	0
$831$	6.11109e6	0.306985
$832$	−1.47634e7	−0.739397
$833$	2.40991e7	1.20334
$834$	−3.16639e7	−1.57634
$835$	0	0
$836$	1.37023e7	0.678077
$837$	5.49055e6	0.270896
$838$	1.32512e7	0.651844
$839$	−3.69461e7	−1.81202	−0.906011	−	0.423255i	$-0.860888\pi$
$839$	−3.69461e7	−1.81202	−0.906011	+	0.423255i	$0.860888\pi$
$840$	0	0
$841$	395030.	0.0192593
$842$	−3.31035e7	−1.60914
$843$	−1.60817e7	−0.779403
$844$	−5.98694e7	−2.89300
$845$	0	0
$846$	4.86649e6	0.233771
$847$	−425357.	−0.0203725
$848$	4.33401e7	2.06967
$849$	−7.85300e6	−0.373909
$850$	0	0
$851$	6.51609e6	0.308435
$852$	2.32988e7	1.09960
$853$	2.39478e7	1.12692	0.563459	−	0.826144i	$-0.309470\pi$
$853$	2.39478e7	1.12692	0.563459	+	0.826144i	$0.309470\pi$
$854$	1.49902e7	0.703337
$855$	0	0
$856$	−7.11746e7	−3.32002
$857$	−4.09428e6	−0.190426	−0.0952129	−	0.995457i	$-0.530353\pi$
$857$	−4.09428e6	−0.190426	−0.0952129	+	0.995457i	$0.530353\pi$
$858$	1.11974e7	0.519275
$859$	−6.51782e6	−0.301384	−0.150692	−	0.988581i	$-0.548150\pi$
$859$	−6.51782e6	−0.301384	−0.150692	+	0.988581i	$0.548150\pi$
$860$	0	0
$861$	−1.47199e6	−0.0676700
$862$	1.87998e7	0.861759
$863$	8.54609e6	0.390607	0.195304	−	0.980743i	$-0.437431\pi$
$863$	8.54609e6	0.390607	0.195304	+	0.980743i	$0.437431\pi$
$864$	2.48479e6	0.113242
$865$	0	0
$866$	4.33178e7	1.96278
$867$	7.73365e6	0.349411
$868$	−1.50811e7	−0.679413
$869$	−1.00116e6	−0.0449733
$870$	0	0
$871$	−4.01015e7	−1.79108
$872$	−5.80870e6	−0.258695
$873$	5.23055e6	0.232280
$874$	2.43977e7	1.08036
$875$	0	0
$876$	3.64042e7	1.60284
$877$	−3.63188e7	−1.59453	−0.797266	−	0.603629i	$-0.793721\pi$
$877$	−3.63188e7	−1.59453	−0.797266	+	0.603629i	$0.793721\pi$
$878$	1.14135e7	0.499668
$879$	5.87817e6	0.256608
$880$	0	0
$881$	1.57431e7	0.683364	0.341682	−	0.939816i	$-0.389004\pi$
$881$	1.57431e7	0.683364	0.341682	+	0.939816i	$0.389004\pi$
$882$	−1.29895e7	−0.562240
$883$	1.30059e7	0.561354	0.280677	−	0.959802i	$-0.409441\pi$
$883$	1.30059e7	0.561354	0.280677	+	0.959802i	$0.409441\pi$
$884$	−1.06498e8	−4.58365
$885$	0	0
$886$	−8.79907e6	−0.376576
$887$	−3.12186e7	−1.33231	−0.666154	−	0.745814i	$-0.732061\pi$
$887$	−3.12186e7	−1.33231	−0.666154	+	0.745814i	$0.732061\pi$
$888$	1.47167e7	0.626295
$889$	6.50050e6	0.275862
$890$	0	0
$891$	793881.	0.0335013
$892$	−3.34771e6	−0.140876
$893$	9.82614e6	0.412339
$894$	4.69416e7	1.96433
$895$	0	0
$896$	7.37868e6	0.307050
$897$	1.36158e7	0.565017
$898$	4.10688e7	1.69950
$899$	−3.44370e7	−1.42111
$900$	0	0
$901$	−4.30230e7	−1.76558
$902$	6.84319e6	0.280054
$903$	596968.	0.0243631
$904$	−5.60827e6	−0.228248
$905$	0	0
$906$	−2.08137e7	−0.842419
$907$	3.59253e7	1.45005	0.725023	−	0.688725i	$-0.241829\pi$
$907$	3.59253e7	1.45005	0.725023	+	0.688725i	$0.241829\pi$
$908$	5.42212e6	0.218250
$909$	3.91752e6	0.157254
$910$	0	0
$911$	−1.47634e7	−0.589372	−0.294686	−	0.955594i	$-0.595215\pi$
$911$	−1.47634e7	−0.589372	−0.294686	+	0.955594i	$0.595215\pi$
$912$	2.24888e7	0.895321
$913$	−2.58400e6	−0.102592
$914$	2.86592e7	1.13475
$915$	0	0
$916$	−2.23371e7	−0.879607
$917$	−7.82251e6	−0.307201
$918$	−1.10563e7	−0.433014
$919$	−1.26772e7	−0.495146	−0.247573	−	0.968869i	$-0.579633\pi$
$919$	−1.26772e7	−0.495146	−0.247573	+	0.968869i	$0.579633\pi$
$920$	0	0
$921$	−1.68653e7	−0.655155
$922$	6.19224e7	2.39895
$923$	3.84435e7	1.48531
$924$	−2.18058e6	−0.0840220
$925$	0	0
$926$	−4.54495e7	−1.74181
$927$	9.87580e6	0.377462
$928$	−1.55848e7	−0.594060
$929$	1.20341e7	0.457482	0.228741	−	0.973487i	$-0.426539\pi$
$929$	1.20341e7	0.457482	0.228741	+	0.973487i	$0.426539\pi$
$930$	0	0
$931$	−2.62277e7	−0.991713
$932$	8.74136e7	3.29639
$933$	−1.44563e7	−0.543694
$934$	2.24111e7	0.840614
$935$	0	0
$936$	3.07515e7	1.14730
$937$	−4.51344e7	−1.67942	−0.839708	−	0.543038i	$-0.817274\pi$
$937$	−4.51344e7	−1.67942	−0.839708	+	0.543038i	$0.817274\pi$
$938$	1.14352e7	0.424362
$939$	−2.32381e7	−0.860077
$940$	0	0
$941$	8.44095e6	0.310754	0.155377	−	0.987855i	$-0.450341\pi$
$941$	8.44095e6	0.310754	0.155377	+	0.987855i	$0.450341\pi$
$942$	−2.97340e7	−1.09176
$943$	8.32120e6	0.304724
$944$	3.94543e7	1.44100
$945$	0	0
$946$	−2.77528e6	−0.100827
$947$	2.68109e7	0.971485	0.485743	−	0.874102i	$-0.338549\pi$
$947$	2.68109e7	0.971485	0.485743	+	0.874102i	$0.338549\pi$
$948$	−5.13244e6	−0.185483
$949$	6.00675e7	2.16508
$950$	0	0
$951$	−1.21129e7	−0.434307
$952$	1.62689e7	0.581788
$953$	1.42183e7	0.507127	0.253563	−	0.967319i	$-0.418397\pi$
$953$	1.42183e7	0.507127	0.253563	+	0.967319i	$0.418397\pi$
$954$	2.31896e7	0.824940
$955$	0	0
$956$	−9.33081e7	−3.30198
$957$	−4.97926e6	−0.175746
$958$	−6.65263e7	−2.34196
$959$	−5.00246e6	−0.175645
$960$	0	0
$961$	2.80961e7	0.981380
$962$	4.53282e7	1.57918
$963$	−1.55425e7	−0.540078
$964$	7.87082e7	2.72789
$965$	0	0
$966$	−3.88263e6	−0.133870
$967$	277112.	0.00952993	0.00476496	−	0.999989i	$-0.498483\pi$
$967$	277112.	0.00952993	0.00476496	+	0.999989i	$0.498483\pi$
$968$	5.43074e6	0.186282
$969$	−2.23242e7	−0.763777
$970$	0	0
$971$	7.77514e6	0.264643	0.132321	−	0.991207i	$-0.457757\pi$
$971$	7.77514e6	0.264643	0.132321	+	0.991207i	$0.457757\pi$
$972$	4.06982e6	0.138169
$973$	1.01744e7	0.344531
$974$	2.18584e7	0.738280
$975$	0	0
$976$	−7.81102e7	−2.62472
$977$	4.83351e6	0.162004	0.0810020	−	0.996714i	$-0.474188\pi$
$977$	4.83351e6	0.162004	0.0810020	+	0.996714i	$0.474188\pi$
$978$	−2.22765e7	−0.744731
$979$	9.48468e6	0.316276
$980$	0	0
$981$	−1.26846e6	−0.0420827
$982$	7.70729e7	2.55049
$983$	−9.75876e6	−0.322115	−0.161057	−	0.986945i	$-0.551490\pi$
$983$	−9.75876e6	−0.322115	−0.161057	+	0.986945i	$0.551490\pi$
$984$	1.87936e7	0.618760
$985$	0	0
$986$	6.93455e7	2.27157
$987$	−1.56373e6	−0.0510938
$988$	1.15905e8	3.77755
$989$	−3.37468e6	−0.109709
$990$	0	0
$991$	2.72969e7	0.882935	0.441468	−	0.897277i	$-0.354458\pi$
$991$	2.72969e7	0.882935	0.441468	+	0.897277i	$0.354458\pi$
$992$	2.56715e7	0.828270
$993$	2.46922e7	0.794668
$994$	−1.09624e7	−0.351917
$995$	0	0
$996$	−1.32468e7	−0.423120
$997$	4.85951e7	1.54830	0.774149	−	0.633004i	$-0.218178\pi$
$997$	4.85951e7	1.54830	0.774149	+	0.633004i	$0.218178\pi$
$998$	6.68070e7	2.12322
$999$	3.21372e6	0.101881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.y.1.12		13
5.2	odd	4		165.6.c.b.34.24	yes	26
5.3	odd	4		165.6.c.b.34.3	✓	26
5.4	even	2		825.6.a.v.1.2		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.3	✓	26	5.3	odd	4
165.6.c.b.34.24	yes	26	5.2	odd	4
825.6.a.v.1.2		13	5.4	even	2
825.6.a.y.1.12		13	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+10.0460 q^{2} +9.00000 q^{3} +68.9227 q^{4} +90.4143 q^{6} -29.0524 q^{7} +370.927 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+10.0460 q^{2} +9.00000 q^{3} +68.9227 q^{4} +90.4143 q^{6} -29.0524 q^{7} +370.927 q^{8} +81.0000 q^{9} +121.000 q^{11} +620.305 q^{12} +1023.51 q^{13} -291.862 q^{14} +1520.82 q^{16} -1509.69 q^{17} +813.728 q^{18} +1643.04 q^{19} -261.472 q^{21} +1215.57 q^{22} +1478.11 q^{23} +3338.34 q^{24} +10282.2 q^{26} +729.000 q^{27} -2002.37 q^{28} -4572.33 q^{29} +7531.62 q^{31} +3408.50 q^{32} +1089.00 q^{33} -15166.4 q^{34} +5582.74 q^{36} +4408.40 q^{37} +16506.0 q^{38} +9211.62 q^{39} +5629.62 q^{41} -2626.75 q^{42} -2283.11 q^{43} +8339.65 q^{44} +14849.1 q^{46} +5980.48 q^{47} +13687.3 q^{48} -15963.0 q^{49} -13587.2 q^{51} +70543.3 q^{52} +28498.0 q^{53} +7323.56 q^{54} -10776.3 q^{56} +14787.3 q^{57} -45933.7 q^{58} +25942.8 q^{59} -51360.7 q^{61} +75662.8 q^{62} -2353.25 q^{63} -14424.2 q^{64} +10940.1 q^{66} -39180.2 q^{67} -104052. q^{68} +13303.0 q^{69} +37560.3 q^{71} +30045.1 q^{72} +58687.6 q^{73} +44286.9 q^{74} +113242. q^{76} -3515.34 q^{77} +92540.2 q^{78} -8274.06 q^{79} +6561.00 q^{81} +56555.3 q^{82} -21355.4 q^{83} -18021.4 q^{84} -22936.2 q^{86} -41150.9 q^{87} +44882.1 q^{88} +78385.8 q^{89} -29735.5 q^{91} +101875. q^{92} +67784.5 q^{93} +60080.1 q^{94} +30676.5 q^{96} +64574.7 q^{97} -160364. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more