LMFDB - Embedded newform 825.6.a.k.1.4

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 $ x^5 - x^4 - 143x^3 + 71x^2 + 4216*x - 3740
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.93734$ 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)

$f(q)$	$=$	$q+5.93734 q^{2} +9.00000 q^{3} +3.25206 q^{4} +53.4361 q^{6} +105.553 q^{7} -170.686 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+5.93734 q^{2} +9.00000 q^{3} +3.25206 q^{4} +53.4361 q^{6} +105.553 q^{7} -170.686 q^{8} +81.0000 q^{9} -121.000 q^{11} +29.2685 q^{12} -124.247 q^{13} +626.706 q^{14} -1117.49 q^{16} -1766.45 q^{17} +480.925 q^{18} +1998.70 q^{19} +949.979 q^{21} -718.419 q^{22} +2546.65 q^{23} -1536.18 q^{24} -737.700 q^{26} +729.000 q^{27} +343.265 q^{28} -320.493 q^{29} +8554.39 q^{31} -1172.96 q^{32} -1089.00 q^{33} -10488.0 q^{34} +263.417 q^{36} -4126.28 q^{37} +11867.0 q^{38} -1118.23 q^{39} +4531.80 q^{41} +5640.35 q^{42} +13412.4 q^{43} -393.499 q^{44} +15120.3 q^{46} +24103.2 q^{47} -10057.4 q^{48} -5665.52 q^{49} -15898.1 q^{51} -404.060 q^{52} +28493.8 q^{53} +4328.32 q^{54} -18016.5 q^{56} +17988.3 q^{57} -1902.88 q^{58} -18203.6 q^{59} +4696.58 q^{61} +50790.3 q^{62} +8549.81 q^{63} +28795.4 q^{64} -6465.77 q^{66} -5374.30 q^{67} -5744.60 q^{68} +22919.9 q^{69} -16054.9 q^{71} -13825.6 q^{72} -26176.0 q^{73} -24499.2 q^{74} +6499.88 q^{76} -12771.9 q^{77} -6639.30 q^{78} -21818.8 q^{79} +6561.00 q^{81} +26906.8 q^{82} +65176.7 q^{83} +3089.39 q^{84} +79633.8 q^{86} -2884.44 q^{87} +20653.1 q^{88} +28370.0 q^{89} -13114.7 q^{91} +8281.86 q^{92} +76989.5 q^{93} +143109. q^{94} -10556.6 q^{96} +51795.2 q^{97} -33638.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9}+O(q^{10}) $ 5 * q + q^2 + 45 * q^3 + 127 * q^4 + 9 * q^6 - 116 * q^7 + 153 * q^8 + 405 * q^9	\( 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9} - 605 q^{11} + 1143 q^{12} + 926 q^{13} + 368 q^{14} + 1891 q^{16} + 246 q^{17} + 81 q^{18} + 3420 q^{19} - 1044 q^{21} - 121 q^{22} + 4244 q^{23} + 1377 q^{24} - 8862 q^{26} + 3645 q^{27} + 4904 q^{28} - 2922 q^{29} - 6112 q^{31} + 24757 q^{32} - 5445 q^{33} + 10866 q^{34} + 10287 q^{36} - 6654 q^{37} + 45692 q^{38} + 8334 q^{39} - 14934 q^{41} + 3312 q^{42} - 10804 q^{43} - 15367 q^{44} - 101500 q^{46} + 41460 q^{47} + 17019 q^{48} - 12099 q^{49} + 2214 q^{51} + 97742 q^{52} + 62398 q^{53} + 729 q^{54} - 74368 q^{56} + 30780 q^{57} + 27822 q^{58} + 8524 q^{59} + 59010 q^{61} + 142624 q^{62} - 9396 q^{63} + 13799 q^{64} - 1089 q^{66} + 15772 q^{67} + 83686 q^{68} + 38196 q^{69} + 88124 q^{71} + 12393 q^{72} + 118358 q^{73} + 67194 q^{74} + 100668 q^{76} + 14036 q^{77} - 79758 q^{78} + 57324 q^{79} + 32805 q^{81} - 29102 q^{82} + 7268 q^{83} + 44136 q^{84} - 35288 q^{86} - 26298 q^{87} - 18513 q^{88} + 72978 q^{89} - 1464 q^{91} - 62148 q^{92} - 55008 q^{93} + 344836 q^{94} + 222813 q^{96} + 59174 q^{97} - 272767 q^{98} - 49005 q^{99}+O(q^{100}) \) 5 * q + q^2 + 45 * q^3 + 127 * q^4 + 9 * q^6 - 116 * q^7 + 153 * q^8 + 405 * q^9 - 605 * q^11 + 1143 * q^12 + 926 * q^13 + 368 * q^14 + 1891 * q^16 + 246 * q^17 + 81 * q^18 + 3420 * q^19 - 1044 * q^21 - 121 * q^22 + 4244 * q^23 + 1377 * q^24 - 8862 * q^26 + 3645 * q^27 + 4904 * q^28 - 2922 * q^29 - 6112 * q^31 + 24757 * q^32 - 5445 * q^33 + 10866 * q^34 + 10287 * q^36 - 6654 * q^37 + 45692 * q^38 + 8334 * q^39 - 14934 * q^41 + 3312 * q^42 - 10804 * q^43 - 15367 * q^44 - 101500 * q^46 + 41460 * q^47 + 17019 * q^48 - 12099 * q^49 + 2214 * q^51 + 97742 * q^52 + 62398 * q^53 + 729 * q^54 - 74368 * q^56 + 30780 * q^57 + 27822 * q^58 + 8524 * q^59 + 59010 * q^61 + 142624 * q^62 - 9396 * q^63 + 13799 * q^64 - 1089 * q^66 + 15772 * q^67 + 83686 * q^68 + 38196 * q^69 + 88124 * q^71 + 12393 * q^72 + 118358 * q^73 + 67194 * q^74 + 100668 * q^76 + 14036 * q^77 - 79758 * q^78 + 57324 * q^79 + 32805 * q^81 - 29102 * q^82 + 7268 * q^83 + 44136 * q^84 - 35288 * q^86 - 26298 * q^87 - 18513 * q^88 + 72978 * q^89 - 1464 * q^91 - 62148 * q^92 - 55008 * q^93 + 344836 * q^94 + 222813 * q^96 + 59174 * q^97 - 272767 * q^98 - 49005 * 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 $

$2$	5.93734	1.04958	0.524792	−	0.851230i	$-0.324143\pi$
$2$	5.93734	1.04958	0.524792	+	0.851230i	$0.324143\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	3.25206	0.101627
$5$	0	0
$5$	0	0
$6$	53.4361	0.605978
$7$	105.553	0.814191	0.407096	−	0.913386i	$-0.366542\pi$
$7$	105.553	0.814191	0.407096	+	0.913386i	$0.366542\pi$
$8$	−170.686	−0.942918
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	29.2685	0.0586742
$13$	−124.247	−0.203906	−0.101953	−	0.994789i	$-0.532509\pi$
$13$	−124.247	−0.203906	−0.101953	+	0.994789i	$0.532509\pi$
$14$	626.706	0.854562
$15$	0	0
$16$	−1117.49	−1.09130
$17$	−1766.45	−1.48245	−0.741224	−	0.671257i	$-0.765755\pi$
$17$	−1766.45	−1.48245	−0.741224	+	0.671257i	$0.765755\pi$
$18$	480.925	0.349861
$19$	1998.70	1.27017	0.635087	−	0.772441i	$-0.280964\pi$
$19$	1998.70	1.27017	0.635087	+	0.772441i	$0.280964\pi$
$20$	0	0
$21$	949.979	0.470073
$22$	−718.419	−0.316462
$23$	2546.65	1.00381	0.501903	−	0.864924i	$-0.332633\pi$
$23$	2546.65	1.00381	0.501903	+	0.864924i	$0.332633\pi$
$24$	−1536.18	−0.544394
$25$	0	0
$26$	−737.700	−0.214016
$27$	729.000	0.192450
$28$	343.265	0.0827436
$29$	−320.493	−0.0707659	−0.0353829	−	0.999374i	$-0.511265\pi$
$29$	−320.493	−0.0707659	−0.0353829	+	0.999374i	$0.511265\pi$
$30$	0	0
$31$	8554.39	1.59876	0.799382	−	0.600823i	$-0.205160\pi$
$31$	8554.39	1.59876	0.799382	+	0.600823i	$0.205160\pi$
$32$	−1172.96	−0.202492
$33$	−1089.00	−0.174078
$34$	−10488.0	−1.55595
$35$	0	0
$36$	263.417	0.0338756
$37$	−4126.28	−0.495513	−0.247756	−	0.968822i	$-0.579693\pi$
$37$	−4126.28	−0.495513	−0.247756	+	0.968822i	$0.579693\pi$
$38$	11867.0	1.33315
$39$	−1118.23	−0.117725
$40$	0	0
$41$	4531.80	0.421028	0.210514	−	0.977591i	$-0.432486\pi$
$41$	4531.80	0.421028	0.210514	+	0.977591i	$0.432486\pi$
$42$	5640.35	0.493382
$43$	13412.4	1.10620	0.553100	−	0.833115i	$-0.313445\pi$
$43$	13412.4	1.10620	0.553100	+	0.833115i	$0.313445\pi$
$44$	−393.499	−0.0306416
$45$	0	0
$46$	15120.3	1.05358
$47$	24103.2	1.59159	0.795793	−	0.605569i	$-0.207055\pi$
$47$	24103.2	1.59159	0.795793	+	0.605569i	$0.207055\pi$
$48$	−10057.4	−0.630062
$49$	−5665.52	−0.337093
$50$	0	0
$51$	−15898.1	−0.855892
$52$	−404.060	−0.0207223
$53$	28493.8	1.39335	0.696675	−	0.717387i	$-0.254662\pi$
$53$	28493.8	1.39335	0.696675	+	0.717387i	$0.254662\pi$
$54$	4328.32	0.201993
$55$	0	0
$56$	−18016.5	−0.767716
$57$	17988.3	0.733335
$58$	−1902.88	−0.0742747
$59$	−18203.6	−0.680813	−0.340407	−	0.940278i	$-0.610565\pi$
$59$	−18203.6	−0.680813	−0.340407	+	0.940278i	$0.610565\pi$
$60$	0	0
$61$	4696.58	0.161606	0.0808030	−	0.996730i	$-0.474252\pi$
$61$	4696.58	0.161606	0.0808030	+	0.996730i	$0.474252\pi$
$62$	50790.3	1.67804
$63$	8549.81	0.271397
$64$	28795.4	0.878767
$65$	0	0
$66$	−6465.77	−0.182709
$67$	−5374.30	−0.146263	−0.0731315	−	0.997322i	$-0.523299\pi$
$67$	−5374.30	−0.146263	−0.0731315	+	0.997322i	$0.523299\pi$
$68$	−5744.60	−0.150656
$69$	22919.9	0.579548
$70$	0	0
$71$	−16054.9	−0.377974	−0.188987	−	0.981980i	$-0.560520\pi$
$71$	−16054.9	−0.377974	−0.188987	+	0.981980i	$0.560520\pi$
$72$	−13825.6	−0.314306
$73$	−26176.0	−0.574904	−0.287452	−	0.957795i	$-0.592808\pi$
$73$	−26176.0	−0.574904	−0.287452	+	0.957795i	$0.592808\pi$
$74$	−24499.2	−0.520082
$75$	0	0
$76$	6499.88	0.129084
$77$	−12771.9	−0.245488
$78$	−6639.30	−0.123562
$79$	−21818.8	−0.393336	−0.196668	−	0.980470i	$-0.563012\pi$
$79$	−21818.8	−0.393336	−0.196668	+	0.980470i	$0.563012\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	26906.8	0.441904
$83$	65176.7	1.03848	0.519239	−	0.854629i	$-0.326216\pi$
$83$	65176.7	1.03848	0.519239	+	0.854629i	$0.326216\pi$
$84$	3089.39	0.0477720
$85$	0	0
$86$	79633.8	1.16105
$87$	−2884.44	−0.0408567
$88$	20653.1	0.284301
$89$	28370.0	0.379651	0.189826	−	0.981818i	$-0.439208\pi$
$89$	28370.0	0.379651	0.189826	+	0.981818i	$0.439208\pi$
$90$	0	0
$91$	−13114.7	−0.166018
$92$	8281.86	0.102014
$93$	76989.5	0.923047
$94$	143109.	1.67050
$95$	0	0
$96$	−10556.6	−0.116909
$97$	51795.2	0.558934	0.279467	−	0.960155i	$-0.409842\pi$
$97$	51795.2	0.558934	0.279467	+	0.960155i	$0.409842\pi$
$98$	−33638.1	−0.353807
$99$	−9801.00	−0.100504
$100$	0	0
$101$	152512.	1.48765	0.743823	−	0.668376i	$-0.233010\pi$
$101$	152512.	1.48765	0.743823	+	0.668376i	$0.233010\pi$
$102$	−94392.3	−0.898331
$103$	−190367.	−1.76807	−0.884035	−	0.467420i	$-0.845184\pi$
$103$	−190367.	−1.76807	−0.884035	+	0.467420i	$0.845184\pi$
$104$	21207.4	0.192266
$105$	0	0
$106$	169177.	1.46244
$107$	68674.3	0.579875	0.289938	−	0.957046i	$-0.406365\pi$
$107$	68674.3	0.579875	0.289938	+	0.957046i	$0.406365\pi$
$108$	2370.75	0.0195581
$109$	29463.1	0.237526	0.118763	−	0.992923i	$-0.462107\pi$
$109$	29463.1	0.237526	0.118763	+	0.992923i	$0.462107\pi$
$110$	0	0
$111$	−37136.6	−0.286084
$112$	−117955.	−0.888526
$113$	99307.3	0.731619	0.365810	−	0.930690i	$-0.380792\pi$
$113$	99307.3	0.731619	0.365810	+	0.930690i	$0.380792\pi$
$114$	106803.	0.769697
$115$	0	0
$116$	−1042.26	−0.00719170
$117$	−10064.0	−0.0679685
$118$	−108081.	−0.714571
$119$	−186455.	−1.20700
$120$	0	0
$121$	14641.0	0.0909091
$122$	27885.2	0.169619
$123$	40786.2	0.243080
$124$	27819.3	0.162477
$125$	0	0
$126$	50763.2	0.284854
$127$	−192783.	−1.06062	−0.530308	−	0.847805i	$-0.677924\pi$
$127$	−192783.	−1.06062	−0.530308	+	0.847805i	$0.677924\pi$
$128$	208503.	1.12483
$129$	120711.	0.638665
$130$	0	0
$131$	194663.	0.991074	0.495537	−	0.868587i	$-0.334971\pi$
$131$	194663.	0.991074	0.495537	+	0.868587i	$0.334971\pi$
$132$	−3541.49	−0.0176909
$133$	210969.	1.03416
$134$	−31909.1	−0.153515
$135$	0	0
$136$	301509.	1.39783
$137$	219837.	1.00069	0.500345	−	0.865826i	$-0.333207\pi$
$137$	219837.	1.00069	0.500345	+	0.865826i	$0.333207\pi$
$138$	136083.	0.608284
$139$	−60583.0	−0.265959	−0.132979	−	0.991119i	$-0.542454\pi$
$139$	−60583.0	−0.265959	−0.132979	+	0.991119i	$0.542454\pi$
$140$	0	0
$141$	216929.	0.918902
$142$	−95323.4	−0.396715
$143$	15033.9	0.0614798
$144$	−90516.7	−0.363766
$145$	0	0
$146$	−155416.	−0.603410
$147$	−50989.7	−0.194621
$148$	−13418.9	−0.0503573
$149$	61572.6	0.227207	0.113604	−	0.993526i	$-0.463761\pi$
$149$	61572.6	0.227207	0.113604	+	0.993526i	$0.463761\pi$
$150$	0	0
$151$	−194637.	−0.694676	−0.347338	−	0.937740i	$-0.612914\pi$
$151$	−194637.	−0.694676	−0.347338	+	0.937740i	$0.612914\pi$
$152$	−341151.	−1.19767
$153$	−143083.	−0.494150
$154$	−75831.4	−0.257660
$155$	0	0
$156$	−3636.54	−0.0119640
$157$	−76011.6	−0.246111	−0.123055	−	0.992400i	$-0.539269\pi$
$157$	−76011.6	−0.246111	−0.123055	+	0.992400i	$0.539269\pi$
$158$	−129546.	−0.412839
$159$	256444.	0.804451
$160$	0	0
$161$	268807.	0.817290
$162$	38954.9	0.116620
$163$	498250.	1.46885	0.734426	−	0.678689i	$-0.237452\pi$
$163$	498250.	1.46885	0.734426	+	0.678689i	$0.237452\pi$
$164$	14737.7	0.0427877
$165$	0	0
$166$	386976.	1.08997
$167$	650928.	1.80610	0.903050	−	0.429536i	$-0.141323\pi$
$167$	650928.	1.80610	0.903050	+	0.429536i	$0.141323\pi$
$168$	−162149.	−0.443241
$169$	−355856.	−0.958423
$170$	0	0
$171$	161895.	0.423391
$172$	43617.7	0.112420
$173$	400605.	1.01766	0.508828	−	0.860868i	$-0.330079\pi$
$173$	400605.	1.01766	0.508828	+	0.860868i	$0.330079\pi$
$174$	−17125.9	−0.0428825
$175$	0	0
$176$	135216.	0.329039
$177$	−163833.	−0.393068
$178$	168443.	0.398476
$179$	−820215.	−1.91335	−0.956677	−	0.291151i	$-0.905962\pi$
$179$	−820215.	−1.91335	−0.956677	+	0.291151i	$0.905962\pi$
$180$	0	0
$181$	639863.	1.45175	0.725873	−	0.687828i	$-0.241436\pi$
$181$	639863.	1.45175	0.725873	+	0.687828i	$0.241436\pi$
$182$	−77866.6	−0.174250
$183$	42269.2	0.0933033
$184$	−434679.	−0.946507
$185$	0	0
$186$	457113.	0.968816
$187$	213741.	0.446975
$188$	78384.9	0.161748
$189$	76948.3	0.156691
$190$	0	0
$191$	707635.	1.40354	0.701772	−	0.712402i	$-0.252393\pi$
$191$	707635.	1.40354	0.701772	+	0.712402i	$0.252393\pi$
$192$	259159.	0.507356
$193$	993614.	1.92010	0.960051	−	0.279825i	$-0.0902763\pi$
$193$	993614.	1.92010	0.960051	+	0.279825i	$0.0902763\pi$
$194$	307526.	0.586648
$195$	0	0
$196$	−18424.6	−0.0342576
$197$	661275.	1.21399	0.606997	−	0.794704i	$-0.292374\pi$
$197$	661275.	1.21399	0.606997	+	0.794704i	$0.292374\pi$
$198$	−58191.9	−0.105487
$199$	−414819.	−0.742551	−0.371276	−	0.928523i	$-0.621080\pi$
$199$	−414819.	−0.742551	−0.371276	+	0.928523i	$0.621080\pi$
$200$	0	0
$201$	−48368.7	−0.0844450
$202$	905514.	1.56141
$203$	−33829.1	−0.0576169
$204$	−51701.4	−0.0869815
$205$	0	0
$206$	−1.13028e6	−1.85574
$207$	206279.	0.334602
$208$	138845.	0.222522
$209$	−241842.	−0.382972
$210$	0	0
$211$	−519315.	−0.803017	−0.401509	−	0.915855i	$-0.631514\pi$
$211$	−519315.	−0.803017	−0.401509	+	0.915855i	$0.631514\pi$
$212$	92663.3	0.141602
$213$	−144494.	−0.218223
$214$	407743.	0.608628
$215$	0	0
$216$	−124430.	−0.181465
$217$	902943.	1.30170
$218$	174933.	0.249304
$219$	−235584.	−0.331921
$220$	0	0
$221$	219477.	0.302279
$222$	−220492.	−0.300270
$223$	−1.10323e6	−1.48561	−0.742804	−	0.669509i	$-0.766505\pi$
$223$	−1.10323e6	−1.48561	−0.742804	+	0.669509i	$0.766505\pi$
$224$	−123809.	−0.164867
$225$	0	0
$226$	589622.	0.767896
$227$	−1.07714e6	−1.38742	−0.693712	−	0.720253i	$-0.744026\pi$
$227$	−1.07714e6	−1.38742	−0.693712	+	0.720253i	$0.744026\pi$
$228$	58498.9	0.0745265
$229$	669513.	0.843666	0.421833	−	0.906674i	$-0.361387\pi$
$229$	669513.	0.843666	0.421833	+	0.906674i	$0.361387\pi$
$230$	0	0
$231$	−114947.	−0.141732
$232$	54703.8	0.0667264
$233$	−100605.	−0.121403	−0.0607016	−	0.998156i	$-0.519334\pi$
$233$	−100605.	−0.121403	−0.0607016	+	0.998156i	$0.519334\pi$
$234$	−59753.7	−0.0713387
$235$	0	0
$236$	−59199.2	−0.0691889
$237$	−196370.	−0.227093
$238$	−1.10705e6	−1.26684
$239$	−867105.	−0.981921	−0.490961	−	0.871182i	$-0.663354\pi$
$239$	−867105.	−0.981921	−0.490961	+	0.871182i	$0.663354\pi$
$240$	0	0
$241$	−618174.	−0.685595	−0.342798	−	0.939409i	$-0.611375\pi$
$241$	−618174.	−0.685595	−0.342798	+	0.939409i	$0.611375\pi$
$242$	86928.7	0.0954167
$243$	59049.0	0.0641500
$244$	15273.6	0.0164235
$245$	0	0
$246$	242161.	0.255133
$247$	−248333.	−0.258995
$248$	−1.46012e6	−1.50750
$249$	586590.	0.599565
$250$	0	0
$251$	630505.	0.631691	0.315845	−	0.948811i	$-0.397712\pi$
$251$	630505.	0.631691	0.315845	+	0.948811i	$0.397712\pi$
$252$	27804.5	0.0275812
$253$	−308145.	−0.302659
$254$	−1.14462e6	−1.11321
$255$	0	0
$256$	316500.	0.301838
$257$	−475889.	−0.449441	−0.224720	−	0.974423i	$-0.572147\pi$
$257$	−475889.	−0.449441	−0.224720	+	0.974423i	$0.572147\pi$
$258$	716704.	0.670333
$259$	−435543.	−0.403442
$260$	0	0
$261$	−25959.9	−0.0235886
$262$	1.15578e6	1.04022
$263$	−415568.	−0.370470	−0.185235	−	0.982694i	$-0.559305\pi$
$263$	−415568.	−0.370470	−0.185235	+	0.982694i	$0.559305\pi$
$264$	185878.	0.164141
$265$	0	0
$266$	1.25260e6	1.08544
$267$	255330.	0.219192
$268$	−17477.5	−0.0148642
$269$	−1.86987e6	−1.57554	−0.787772	−	0.615967i	$-0.788765\pi$
$269$	−1.86987e6	−1.57554	−0.787772	+	0.615967i	$0.788765\pi$
$270$	0	0
$271$	−1.23898e6	−1.02481	−0.512404	−	0.858744i	$-0.671245\pi$
$271$	−1.23898e6	−1.02481	−0.512404	+	0.858744i	$0.671245\pi$
$272$	1.97399e6	1.61779
$273$	−118032.	−0.0958506
$274$	1.30525e6	1.05031
$275$	0	0
$276$	74536.7	0.0588976
$277$	−846334.	−0.662739	−0.331370	−	0.943501i	$-0.607511\pi$
$277$	−846334.	−0.662739	−0.331370	+	0.943501i	$0.607511\pi$
$278$	−359702.	−0.279146
$279$	692905.	0.532922
$280$	0	0
$281$	−1.07789e6	−0.814347	−0.407174	−	0.913351i	$-0.633486\pi$
$281$	−1.07789e6	−0.814347	−0.407174	+	0.913351i	$0.633486\pi$
$282$	1.28798e6	0.964465
$283$	2.59466e6	1.92581	0.962906	−	0.269836i	$-0.0869694\pi$
$283$	2.59466e6	1.92581	0.962906	+	0.269836i	$0.0869694\pi$
$284$	−52211.4	−0.0384122
$285$	0	0
$286$	89261.7	0.0645283
$287$	478346.	0.342797
$288$	−95009.5	−0.0674972
$289$	1.70050e6	1.19765
$290$	0	0
$291$	466157.	0.322700
$292$	−85125.7	−0.0584256
$293$	155161.	0.105588	0.0527939	−	0.998605i	$-0.483187\pi$
$293$	155161.	0.105588	0.0527939	+	0.998605i	$0.483187\pi$
$294$	−302743.	−0.204271
$295$	0	0
$296$	704301.	0.467228
$297$	−88209.0	−0.0580259
$298$	365578.	0.238473
$299$	−316415.	−0.204682
$300$	0	0
$301$	1.41572e6	0.900659
$302$	−1.15562e6	−0.729121
$303$	1.37261e6	0.858893
$304$	−2.23352e6	−1.38614
$305$	0	0
$306$	−849531.	−0.518652
$307$	1.94896e6	1.18020	0.590102	−	0.807329i	$-0.299088\pi$
$307$	1.94896e6	1.18020	0.590102	+	0.807329i	$0.299088\pi$
$308$	−41535.1	−0.0249481
$309$	−1.71331e6	−1.02080
$310$	0	0
$311$	853359.	0.500300	0.250150	−	0.968207i	$-0.419520\pi$
$311$	853359.	0.500300	0.250150	+	0.968207i	$0.419520\pi$
$312$	190866.	0.111005
$313$	−594619.	−0.343066	−0.171533	−	0.985178i	$-0.554872\pi$
$313$	−594619.	−0.343066	−0.171533	+	0.985178i	$0.554872\pi$
$314$	−451307.	−0.258314
$315$	0	0
$316$	−70956.1	−0.0399735
$317$	−3.25318e6	−1.81827	−0.909137	−	0.416497i	$-0.863257\pi$
$317$	−3.25318e6	−1.81827	−0.909137	+	0.416497i	$0.863257\pi$
$318$	1.52260e6	0.844339
$319$	38779.7	0.0213367
$320$	0	0
$321$	618069.	0.334791
$322$	1.59600e6	0.857815
$323$	−3.53061e6	−1.88297
$324$	21336.7	0.0112919
$325$	0	0
$326$	2.95828e6	1.54168
$327$	265168.	0.137136
$328$	−773516.	−0.396995
$329$	2.54417e6	1.29585
$330$	0	0
$331$	−2.05862e6	−1.03278	−0.516389	−	0.856354i	$-0.672724\pi$
$331$	−2.05862e6	−1.03278	−0.516389	+	0.856354i	$0.672724\pi$
$332$	211958.	0.105537
$333$	−334229.	−0.165171
$334$	3.86478e6	1.89565
$335$	0	0
$336$	−1.06159e6	−0.512991
$337$	−87610.8	−0.0420226	−0.0210113	−	0.999779i	$-0.506689\pi$
$337$	−87610.8	−0.0420226	−0.0210113	+	0.999779i	$0.506689\pi$
$338$	−2.11284e6	−1.00595
$339$	893766.	0.422400
$340$	0	0
$341$	−1.03508e6	−0.482046
$342$	961224.	0.444385
$343$	−2.37205e6	−1.08865
$344$	−2.28931e6	−1.04306
$345$	0	0
$346$	2.37853e6	1.06812
$347$	3.19161e6	1.42294	0.711470	−	0.702716i	$-0.248030\pi$
$347$	3.19161e6	1.42294	0.711470	+	0.702716i	$0.248030\pi$
$348$	−9380.36	−0.00415213
$349$	−1.36137e6	−0.598293	−0.299146	−	0.954207i	$-0.596702\pi$
$349$	−1.36137e6	−0.598293	−0.299146	+	0.954207i	$0.596702\pi$
$350$	0	0
$351$	−90576.4	−0.0392416
$352$	141928.	0.0610535
$353$	1.84373e6	0.787517	0.393758	−	0.919214i	$-0.371175\pi$
$353$	1.84373e6	0.787517	0.393758	+	0.919214i	$0.371175\pi$
$354$	−972731.	−0.412558
$355$	0	0
$356$	92261.0	0.0385827
$357$	−1.67809e6	−0.696860
$358$	−4.86990e6	−2.00823
$359$	−345148.	−0.141341	−0.0706707	−	0.997500i	$-0.522514\pi$
$359$	−345148.	−0.141341	−0.0706707	+	0.997500i	$0.522514\pi$
$360$	0	0
$361$	1.51869e6	0.613342
$362$	3.79909e6	1.52373
$363$	131769.	0.0524864
$364$	−42649.8	−0.0168719
$365$	0	0
$366$	250967.	0.0979296
$367$	−810586.	−0.314148	−0.157074	−	0.987587i	$-0.550206\pi$
$367$	−810586.	−0.314148	−0.157074	+	0.987587i	$0.550206\pi$
$368$	−2.84586e6	−1.09545
$369$	367075.	0.140343
$370$	0	0
$371$	3.00761e6	1.13445
$372$	250374.	0.0938063
$373$	637008.	0.237068	0.118534	−	0.992950i	$-0.462181\pi$
$373$	637008.	0.237068	0.118534	+	0.992950i	$0.462181\pi$
$374$	1.26905e6	0.469138
$375$	0	0
$376$	−4.11409e6	−1.50073
$377$	39820.4	0.0144295
$378$	456869.	0.164461
$379$	−3.62357e6	−1.29580	−0.647901	−	0.761725i	$-0.724353\pi$
$379$	−3.62357e6	−1.29580	−0.647901	+	0.761725i	$0.724353\pi$
$380$	0	0
$381$	−1.73504e6	−0.612347
$382$	4.20147e6	1.47314
$383$	−419232.	−0.146035	−0.0730175	−	0.997331i	$-0.523263\pi$
$383$	−419232.	−0.146035	−0.0730175	+	0.997331i	$0.523263\pi$
$384$	1.87653e6	0.649422
$385$	0	0
$386$	5.89943e6	2.01531
$387$	1.08640e6	0.368734
$388$	168441.	0.0568026
$389$	−662976.	−0.222139	−0.111069	−	0.993813i	$-0.535428\pi$
$389$	−662976.	−0.222139	−0.111069	+	0.993813i	$0.535428\pi$
$390$	0	0
$391$	−4.49854e6	−1.48809
$392$	967027.	0.317851
$393$	1.75197e6	0.572197
$394$	3.92622e6	1.27419
$395$	0	0
$396$	−31873.4	−0.0102139
$397$	−3.68928e6	−1.17480	−0.587402	−	0.809295i	$-0.699849\pi$
$397$	−3.68928e6	−1.17480	−0.587402	+	0.809295i	$0.699849\pi$
$398$	−2.46293e6	−0.779370
$399$	1.89872e6	0.597075
$400$	0	0
$401$	4.03113e6	1.25189	0.625945	−	0.779867i	$-0.284713\pi$
$401$	4.03113e6	1.25189	0.625945	+	0.779867i	$0.284713\pi$
$402$	−287181.	−0.0886322
$403$	−1.06286e6	−0.325997
$404$	495977.	0.151185
$405$	0	0
$406$	−200855.	−0.0604738
$407$	499280.	0.149403
$408$	2.71359e6	0.807036
$409$	4.07904e6	1.20573	0.602864	−	0.797844i	$-0.294026\pi$
$409$	4.07904e6	1.20573	0.602864	+	0.797844i	$0.294026\pi$
$410$	0	0
$411$	1.97853e6	0.577748
$412$	−619086.	−0.179683
$413$	−1.92145e6	−0.554312
$414$	1.22475e6	0.351193
$415$	0	0
$416$	145737.	0.0412891
$417$	−545247.	−0.153551
$418$	−1.43590e6	−0.401961
$419$	−1.22736e6	−0.341537	−0.170769	−	0.985311i	$-0.554625\pi$
$419$	−1.22736e6	−0.341537	−0.170769	+	0.985311i	$0.554625\pi$
$420$	0	0
$421$	−16159.8	−0.00444357	−0.00222178	−	0.999998i	$-0.500707\pi$
$421$	−16159.8	−0.00444357	−0.00222178	+	0.999998i	$0.500707\pi$
$422$	−3.08335e6	−0.842834
$423$	1.95236e6	0.530528
$424$	−4.86350e6	−1.31381
$425$	0	0
$426$	−857911.	−0.229044
$427$	495739.	0.131578
$428$	223333.	0.0589309
$429$	135305.	0.0354954
$430$	0	0
$431$	185090.	0.0479944	0.0239972	−	0.999712i	$-0.492361\pi$
$431$	185090.	0.0479944	0.0239972	+	0.999712i	$0.492361\pi$
$432$	−814650.	−0.210021
$433$	2.83808e6	0.727453	0.363727	−	0.931506i	$-0.381504\pi$
$433$	2.83808e6	0.727453	0.363727	+	0.931506i	$0.381504\pi$
$434$	5.36108e6	1.36624
$435$	0	0
$436$	95815.6	0.0241390
$437$	5.08999e6	1.27501
$438$	−1.39874e6	−0.348379
$439$	−1.17189e6	−0.290218	−0.145109	−	0.989416i	$-0.546353\pi$
$439$	−1.17189e6	−0.290218	−0.145109	+	0.989416i	$0.546353\pi$
$440$	0	0
$441$	−458907.	−0.112364
$442$	1.30311e6	0.317268
$443$	6.32946e6	1.53235	0.766174	−	0.642634i	$-0.222158\pi$
$443$	6.32946e6	1.53235	0.766174	+	0.642634i	$0.222158\pi$
$444$	−120770.	−0.0290738
$445$	0	0
$446$	−6.55026e6	−1.55927
$447$	554153.	0.131178
$448$	3.03945e6	0.715484
$449$	−2.81992e6	−0.660116	−0.330058	−	0.943961i	$-0.607068\pi$
$449$	−2.81992e6	−0.660116	−0.330058	+	0.943961i	$0.607068\pi$
$450$	0	0
$451$	−548347.	−0.126945
$452$	322953.	0.0743521
$453$	−1.75173e6	−0.401071
$454$	−6.39537e6	−1.45622
$455$	0	0
$456$	−3.07036e6	−0.691475
$457$	−624549.	−0.139887	−0.0699433	−	0.997551i	$-0.522282\pi$
$457$	−624549.	−0.139887	−0.0699433	+	0.997551i	$0.522282\pi$
$458$	3.97513e6	0.885498
$459$	−1.28774e6	−0.285297
$460$	0	0
$461$	3.84656e6	0.842985	0.421492	−	0.906832i	$-0.361506\pi$
$461$	3.84656e6	0.842985	0.421492	+	0.906832i	$0.361506\pi$
$462$	−682483.	−0.148760
$463$	7.90622e6	1.71402	0.857011	−	0.515298i	$-0.172319\pi$
$463$	7.90622e6	1.71402	0.857011	+	0.515298i	$0.172319\pi$
$464$	358148.	0.0772267
$465$	0	0
$466$	−597327.	−0.127423
$467$	−7.13863e6	−1.51469	−0.757343	−	0.653017i	$-0.773503\pi$
$467$	−7.13863e6	−1.51469	−0.757343	+	0.653017i	$0.773503\pi$
$468$	−32728.8	−0.00690742
$469$	−567274.	−0.119086
$470$	0	0
$471$	−684105.	−0.142092
$472$	3.10711e6	0.641951
$473$	−1.62290e6	−0.333532
$474$	−1.16591e6	−0.238353
$475$	0	0
$476$	−606361.	−0.122663
$477$	2.30799e6	0.464450
$478$	−5.14830e6	−1.03061
$479$	−6.55862e6	−1.30609	−0.653046	−	0.757319i	$-0.726509\pi$
$479$	−6.55862e6	−1.30609	−0.653046	+	0.757319i	$0.726509\pi$
$480$	0	0
$481$	512680.	0.101038
$482$	−3.67031e6	−0.719590
$483$	2.41927e6	0.471863
$484$	47613.4	0.00923880
$485$	0	0
$486$	350594.	0.0673309
$487$	6.92845e6	1.32377	0.661887	−	0.749604i	$-0.269756\pi$
$487$	6.92845e6	1.32377	0.661887	+	0.749604i	$0.269756\pi$
$488$	−801643.	−0.152381
$489$	4.48425e6	0.848042
$490$	0	0
$491$	−9.60498e6	−1.79801	−0.899006	−	0.437936i	$-0.855710\pi$
$491$	−9.60498e6	−1.79801	−0.899006	+	0.437936i	$0.855710\pi$
$492$	132639.	0.0247035
$493$	566136.	0.104907
$494$	−1.47444e6	−0.271838
$495$	0	0
$496$	−9.55944e6	−1.74473
$497$	−1.69465e6	−0.307743
$498$	3.48279e6	0.629294
$499$	6.72389e6	1.20884	0.604421	−	0.796665i	$-0.293405\pi$
$499$	6.72389e6	1.20884	0.604421	+	0.796665i	$0.293405\pi$
$500$	0	0
$501$	5.85835e6	1.04275
$502$	3.74353e6	0.663012
$503$	142175.	0.0250555	0.0125277	−	0.999922i	$-0.496012\pi$
$503$	142175.	0.0250555	0.0125277	+	0.999922i	$0.496012\pi$
$504$	−1.45934e6	−0.255905
$505$	0	0
$506$	−1.82956e6	−0.317666
$507$	−3.20270e6	−0.553346
$508$	−626940.	−0.107787
$509$	−1.05255e7	−1.80072	−0.900362	−	0.435143i	$-0.856698\pi$
$509$	−1.05255e7	−1.80072	−0.900362	+	0.435143i	$0.856698\pi$
$510$	0	0
$511$	−2.76296e6	−0.468082
$512$	−4.79293e6	−0.808027
$513$	1.45705e6	0.244445
$514$	−2.82552e6	−0.471726
$515$	0	0
$516$	392560.	0.0649055
$517$	−2.91649e6	−0.479881
$518$	−2.58597e6	−0.423446
$519$	3.60544e6	0.587544
$520$	0	0
$521$	2.99141e6	0.482816	0.241408	−	0.970424i	$-0.422391\pi$
$521$	2.99141e6	0.482816	0.241408	+	0.970424i	$0.422391\pi$
$522$	−154133.	−0.0247582
$523$	5.44567e6	0.870557	0.435279	−	0.900296i	$-0.356650\pi$
$523$	5.44567e6	0.870557	0.435279	+	0.900296i	$0.356650\pi$
$524$	633057.	0.100720
$525$	0	0
$526$	−2.46737e6	−0.388839
$527$	−1.51109e7	−2.37009
$528$	1.21695e6	0.189971
$529$	49092.3	0.00762736
$530$	0	0
$531$	−1.47449e6	−0.226938
$532$	686083.	0.105099
$533$	−563064.	−0.0858499
$534$	1.51598e6	0.230060
$535$	0	0
$536$	917320.	0.137914
$537$	−7.38194e6	−1.10468
$538$	−1.11021e7	−1.65367
$539$	685528.	0.101637
$540$	0	0
$541$	−340079.	−0.0499559	−0.0249780	−	0.999688i	$-0.507952\pi$
$541$	−340079.	−0.0499559	−0.0249780	+	0.999688i	$0.507952\pi$
$542$	−7.35628e6	−1.07562
$543$	5.75877e6	0.838166
$544$	2.07197e6	0.300183
$545$	0	0
$546$	−700799.	−0.100603
$547$	−8.20219e6	−1.17209	−0.586046	−	0.810278i	$-0.699316\pi$
$547$	−8.20219e6	−1.17209	−0.586046	+	0.810278i	$0.699316\pi$
$548$	714922.	0.101697
$549$	380423.	0.0538687
$550$	0	0
$551$	−640569.	−0.0898849
$552$	−3.91211e6	−0.546466
$553$	−2.30305e6	−0.320251
$554$	−5.02498e6	−0.695600
$555$	0	0
$556$	−197019.	−0.0270285
$557$	−6.21617e6	−0.848956	−0.424478	−	0.905438i	$-0.639542\pi$
$557$	−6.21617e6	−0.848956	−0.424478	+	0.905438i	$0.639542\pi$
$558$	4.11402e6	0.559346
$559$	−1.66645e6	−0.225560
$560$	0	0
$561$	1.92367e6	0.258061
$562$	−6.39982e6	−0.854726
$563$	7.83265e6	1.04145	0.520724	−	0.853725i	$-0.325662\pi$
$563$	7.83265e6	1.04145	0.520724	+	0.853725i	$0.325662\pi$
$564$	705464.	0.0933850
$565$	0	0
$566$	1.54054e7	2.02130
$567$	692535.	0.0904657
$568$	2.74035e6	0.356398
$569$	−479265.	−0.0620576	−0.0310288	−	0.999518i	$-0.509878\pi$
$569$	−479265.	−0.0620576	−0.0310288	+	0.999518i	$0.509878\pi$
$570$	0	0
$571$	−5.13344e6	−0.658898	−0.329449	−	0.944173i	$-0.606863\pi$
$571$	−5.13344e6	−0.658898	−0.329449	+	0.944173i	$0.606863\pi$
$572$	48891.2	0.00624800
$573$	6.36871e6	0.810336
$574$	2.84010e6	0.359794
$575$	0	0
$576$	2.33243e6	0.292922
$577$	−7.94103e6	−0.992973	−0.496486	−	0.868045i	$-0.665377\pi$
$577$	−7.94103e6	−0.992973	−0.496486	+	0.868045i	$0.665377\pi$
$578$	1.00964e7	1.25704
$579$	8.94252e6	1.10857
$580$	0	0
$581$	6.87961e6	0.845519
$582$	2.76773e6	0.338701
$583$	−3.44774e6	−0.420111
$584$	4.46788e6	0.542088
$585$	0	0
$586$	921244.	0.110823
$587$	1.01573e7	1.21669	0.608347	−	0.793671i	$-0.291833\pi$
$587$	1.01573e7	1.21669	0.608347	+	0.793671i	$0.291833\pi$
$588$	−165821.	−0.0197787
$589$	1.70976e7	2.03071
$590$	0	0
$591$	5.95147e6	0.700900
$592$	4.61108e6	0.540752
$593$	9.94917e6	1.16185	0.580925	−	0.813957i	$-0.302691\pi$
$593$	9.94917e6	1.16185	0.580925	+	0.813957i	$0.302691\pi$
$594$	−523727.	−0.0609030
$595$	0	0
$596$	200238.	0.0230903
$597$	−3.73338e6	−0.428712
$598$	−1.87866e6	−0.214831
$599$	1.18037e7	1.34416	0.672080	−	0.740478i	$-0.265401\pi$
$599$	1.18037e7	1.34416	0.672080	+	0.740478i	$0.265401\pi$
$600$	0	0
$601$	−8.52084e6	−0.962268	−0.481134	−	0.876647i	$-0.659775\pi$
$601$	−8.52084e6	−0.962268	−0.481134	+	0.876647i	$0.659775\pi$
$602$	8.40560e6	0.945317
$603$	−435318.	−0.0487544
$604$	−632969.	−0.0705977
$605$	0	0
$606$	8.14963e6	0.901480
$607$	−1.48363e7	−1.63438	−0.817191	−	0.576368i	$-0.804470\pi$
$607$	−1.48363e7	−1.63438	−0.817191	+	0.576368i	$0.804470\pi$
$608$	−2.34439e6	−0.257199
$609$	−304462.	−0.0332652
$610$	0	0
$611$	−2.99476e6	−0.324533
$612$	−465313.	−0.0502188
$613$	−1.31231e7	−1.41054	−0.705272	−	0.708936i	$-0.749175\pi$
$613$	−1.31231e7	−1.41054	−0.705272	+	0.708936i	$0.749175\pi$
$614$	1.15716e7	1.23872
$615$	0	0
$616$	2.18000e6	0.231475
$617$	−4.62938e6	−0.489564	−0.244782	−	0.969578i	$-0.578716\pi$
$617$	−4.62938e6	−0.489564	−0.244782	+	0.969578i	$0.578716\pi$
$618$	−1.01725e7	−1.07141
$619$	9.49235e6	0.995743	0.497872	−	0.867251i	$-0.334115\pi$
$619$	9.49235e6	0.995743	0.497872	+	0.867251i	$0.334115\pi$
$620$	0	0
$621$	1.85651e6	0.193183
$622$	5.06669e6	0.525107
$623$	2.99455e6	0.309109
$624$	1.24961e6	0.128473
$625$	0	0
$626$	−3.53045e6	−0.360077
$627$	−2.17658e6	−0.221109
$628$	−247194.	−0.0250115
$629$	7.28888e6	0.734572
$630$	0	0
$631$	1.14798e7	1.14779	0.573893	−	0.818930i	$-0.305432\pi$
$631$	1.14798e7	1.14779	0.573893	+	0.818930i	$0.305432\pi$
$632$	3.72418e6	0.370884
$633$	−4.67384e6	−0.463622
$634$	−1.93152e7	−1.90843
$635$	0	0
$636$	833970.	0.0817537
$637$	703926.	0.0687351
$638$	230248.	0.0223947
$639$	−1.30045e6	−0.125991
$640$	0	0
$641$	−1.85269e7	−1.78097	−0.890486	−	0.455010i	$-0.849636\pi$
$641$	−1.85269e7	−1.78097	−0.890486	+	0.455010i	$0.849636\pi$
$642$	3.66969e6	0.351392
$643$	−1.34056e7	−1.27867	−0.639337	−	0.768927i	$-0.720791\pi$
$643$	−1.34056e7	−1.27867	−0.639337	+	0.768927i	$0.720791\pi$
$644$	874176.	0.0830586
$645$	0	0
$646$	−2.09624e7	−1.97633
$647$	−8.21634e6	−0.771645	−0.385823	−	0.922573i	$-0.626082\pi$
$647$	−8.21634e6	−0.771645	−0.385823	+	0.922573i	$0.626082\pi$
$648$	−1.11987e6	−0.104769
$649$	2.20264e6	0.205273
$650$	0	0
$651$	8.12649e6	0.751537
$652$	1.62034e6	0.149275
$653$	−4.81154e6	−0.441571	−0.220786	−	0.975322i	$-0.570862\pi$
$653$	−4.81154e6	−0.441571	−0.220786	+	0.975322i	$0.570862\pi$
$654$	1.57439e6	0.143936
$655$	0	0
$656$	−5.06424e6	−0.459467
$657$	−2.12025e6	−0.191635
$658$	1.51056e7	1.36011
$659$	−6.57414e6	−0.589693	−0.294846	−	0.955545i	$-0.595268\pi$
$659$	−6.57414e6	−0.589693	−0.294846	+	0.955545i	$0.595268\pi$
$660$	0	0
$661$	1.07312e7	0.955315	0.477657	−	0.878546i	$-0.341486\pi$
$661$	1.07312e7	0.955315	0.477657	+	0.878546i	$0.341486\pi$
$662$	−1.22228e7	−1.08399
$663$	1.97529e6	0.174521
$664$	−1.11248e7	−0.979199
$665$	0	0
$666$	−1.98443e6	−0.173361
$667$	−816184.	−0.0710352
$668$	2.11685e6	0.183548
$669$	−9.92908e6	−0.857716
$670$	0	0
$671$	−568287.	−0.0487260
$672$	−1.11428e6	−0.0951859
$673$	−7.52813e6	−0.640693	−0.320346	−	0.947301i	$-0.603799\pi$
$673$	−7.52813e6	−0.640693	−0.320346	+	0.947301i	$0.603799\pi$
$674$	−520176.	−0.0441063
$675$	0	0
$676$	−1.15726e6	−0.0974014
$677$	1.63689e7	1.37262	0.686308	−	0.727311i	$-0.259230\pi$
$677$	1.63689e7	1.37262	0.686308	+	0.727311i	$0.259230\pi$
$678$	5.30659e6	0.443345
$679$	5.46715e6	0.455079
$680$	0	0
$681$	−9.69429e6	−0.801029
$682$	−6.14563e6	−0.505948
$683$	−2.56590e6	−0.210469	−0.105234	−	0.994447i	$-0.533559\pi$
$683$	−2.56590e6	−0.210469	−0.105234	+	0.994447i	$0.533559\pi$
$684$	526490.	0.0430279
$685$	0	0
$686$	−1.40837e7	−1.14263
$687$	6.02562e6	0.487091
$688$	−1.49882e7	−1.20720
$689$	−3.54028e6	−0.284112
$690$	0	0
$691$	−8.83595e6	−0.703977	−0.351988	−	0.936004i	$-0.614494\pi$
$691$	−8.83595e6	−0.703977	−0.351988	+	0.936004i	$0.614494\pi$
$692$	1.30279e6	0.103421
$693$	−1.03453e6	−0.0818293
$694$	1.89497e7	1.49350
$695$	0	0
$696$	492334.	0.0385245
$697$	−8.00520e6	−0.624152
$698$	−8.08294e6	−0.627959
$699$	−905446.	−0.0700922
$700$	0	0
$701$	43232.3	0.00332287	0.00166143	−	0.999999i	$-0.499471\pi$
$701$	43232.3	0.00332287	0.00166143	+	0.999999i	$0.499471\pi$
$702$	−537783.	−0.0411874
$703$	−8.24719e6	−0.629387
$704$	−3.48425e6	−0.264958
$705$	0	0
$706$	1.09468e7	0.826565
$707$	1.60981e7	1.21123
$708$	−532793.	−0.0399462
$709$	−5.33741e6	−0.398763	−0.199382	−	0.979922i	$-0.563893\pi$
$709$	−5.33741e6	−0.398763	−0.199382	+	0.979922i	$0.563893\pi$
$710$	0	0
$711$	−1.76733e6	−0.131112
$712$	−4.84238e6	−0.357980
$713$	2.17850e7	1.60485
$714$	−9.96341e6	−0.731413
$715$	0	0
$716$	−2.66739e6	−0.194448
$717$	−7.80394e6	−0.566913
$718$	−2.04926e6	−0.148350
$719$	−9.14720e6	−0.659881	−0.329941	−	0.944002i	$-0.607029\pi$
$719$	−9.14720e6	−0.659881	−0.329941	+	0.944002i	$0.607029\pi$
$720$	0	0
$721$	−2.00939e7	−1.43955
$722$	9.01701e6	0.643754
$723$	−5.56356e6	−0.395829
$724$	2.08087e6	0.147536
$725$	0	0
$726$	782358.	0.0550889
$727$	1.61992e6	0.113673	0.0568367	−	0.998383i	$-0.481899\pi$
$727$	1.61992e6	0.113673	0.0568367	+	0.998383i	$0.481899\pi$
$728$	2.23850e6	0.156541
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.36923e7	−1.63989
$732$	137462.	0.00948211
$733$	−2.29112e7	−1.57503	−0.787515	−	0.616296i	$-0.788633\pi$
$733$	−2.29112e7	−1.57503	−0.787515	+	0.616296i	$0.788633\pi$
$734$	−4.81273e6	−0.329724
$735$	0	0
$736$	−2.98711e6	−0.203262
$737$	650290.	0.0441000
$738$	2.17945e6	0.147301
$739$	5.61873e6	0.378466	0.189233	−	0.981932i	$-0.439400\pi$
$739$	5.61873e6	0.378466	0.189233	+	0.981932i	$0.439400\pi$
$740$	0	0
$741$	−2.23500e6	−0.149531
$742$	1.78572e7	1.19070
$743$	2.58649e7	1.71885	0.859426	−	0.511261i	$-0.170821\pi$
$743$	2.58649e7	1.71885	0.859426	+	0.511261i	$0.170821\pi$
$744$	−1.31411e7	−0.870358
$745$	0	0
$746$	3.78214e6	0.248823
$747$	5.27931e6	0.346159
$748$	695097.	0.0454246
$749$	7.24879e6	0.472129
$750$	0	0
$751$	−2.70229e7	−1.74837	−0.874183	−	0.485596i	$-0.838603\pi$
$751$	−2.70229e7	−1.74837	−0.874183	+	0.485596i	$0.838603\pi$
$752$	−2.69351e7	−1.73689
$753$	5.67455e6	0.364707
$754$	236428.	0.0151450
$755$	0	0
$756$	250240.	0.0159240
$757$	1.63080e6	0.103433	0.0517166	−	0.998662i	$-0.483531\pi$
$757$	1.63080e6	0.103433	0.0517166	+	0.998662i	$0.483531\pi$
$758$	−2.15144e7	−1.36005
$759$	−2.77330e6	−0.174740
$760$	0	0
$761$	1.46228e7	0.915311	0.457656	−	0.889130i	$-0.348689\pi$
$761$	1.46228e7	0.915311	0.457656	+	0.889130i	$0.348689\pi$
$762$	−1.03016e7	−0.642710
$763$	3.10992e6	0.193392
$764$	2.30127e6	0.142638
$765$	0	0
$766$	−2.48912e6	−0.153276
$767$	2.26175e6	0.138822
$768$	2.84850e6	0.174266
$769$	2.12198e7	1.29398	0.646988	−	0.762500i	$-0.276028\pi$
$769$	2.12198e7	1.29398	0.646988	+	0.762500i	$0.276028\pi$
$770$	0	0
$771$	−4.28300e6	−0.259485
$772$	3.23129e6	0.195134
$773$	7.33809e6	0.441707	0.220853	−	0.975307i	$-0.429116\pi$
$773$	7.33809e6	0.441707	0.220853	+	0.975307i	$0.429116\pi$
$774$	6.45034e6	0.387017
$775$	0	0
$776$	−8.84074e6	−0.527029
$777$	−3.91988e6	−0.232927
$778$	−3.93632e6	−0.233153
$779$	9.05769e6	0.534778
$780$	0	0
$781$	1.94264e6	0.113963
$782$	−2.67094e7	−1.56188
$783$	−233640.	−0.0136189
$784$	6.33116e6	0.367869
$785$	0	0
$786$	1.04021e7	0.600569
$787$	−6.32394e6	−0.363958	−0.181979	−	0.983302i	$-0.558250\pi$
$787$	−6.32394e6	−0.363958	−0.181979	+	0.983302i	$0.558250\pi$
$788$	2.15050e6	0.123374
$789$	−3.74011e6	−0.213891
$790$	0	0
$791$	1.04822e7	0.595678
$792$	1.67290e6	0.0947669
$793$	−583538.	−0.0329524
$794$	−2.19045e7	−1.23306
$795$	0	0
$796$	−1.34902e6	−0.0754631
$797$	2.37699e7	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$797$	2.37699e7	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$798$	1.12734e7	0.626680
$799$	−4.25771e7	−2.35944
$800$	0	0
$801$	2.29797e6	0.126550
$802$	2.39342e7	1.31396
$803$	3.16729e6	0.173340
$804$	−157298.	−0.00858187
$805$	0	0
$806$	−6.31057e6	−0.342161
$807$	−1.68288e7	−0.909641
$808$	−2.60317e7	−1.40273
$809$	3.06904e7	1.64866	0.824331	−	0.566108i	$-0.191551\pi$
$809$	3.06904e7	1.64866	0.824331	+	0.566108i	$0.191551\pi$
$810$	0	0
$811$	−1.80033e7	−0.961172	−0.480586	−	0.876948i	$-0.659576\pi$
$811$	−1.80033e7	−0.961172	−0.480586	+	0.876948i	$0.659576\pi$
$812$	−110014.	−0.00585542
$813$	−1.11509e7	−0.591673
$814$	2.96440e6	0.156811
$815$	0	0
$816$	1.77659e7	0.934034
$817$	2.68073e7	1.40507
$818$	2.42187e7	1.26551
$819$	−1.06229e6	−0.0553394
$820$	0	0
$821$	−1.63239e7	−0.845213	−0.422607	−	0.906313i	$-0.638885\pi$
$821$	−1.63239e7	−0.845213	−0.422607	+	0.906313i	$0.638885\pi$
$822$	1.17472e7	0.606396
$823$	−1.82069e7	−0.936992	−0.468496	−	0.883466i	$-0.655204\pi$
$823$	−1.82069e7	−0.936992	−0.468496	+	0.883466i	$0.655204\pi$
$824$	3.24931e7	1.66715
$825$	0	0
$826$	−1.14083e7	−0.581797
$827$	3.78345e7	1.92364	0.961820	−	0.273683i	$-0.0882418\pi$
$827$	3.78345e7	1.92364	0.961820	+	0.273683i	$0.0882418\pi$
$828$	670830.	0.0340045
$829$	5.02252e6	0.253826	0.126913	−	0.991914i	$-0.459493\pi$
$829$	5.02252e6	0.253826	0.126913	+	0.991914i	$0.459493\pi$
$830$	0	0
$831$	−7.61701e6	−0.382633
$832$	−3.57776e6	−0.179185
$833$	1.00079e7	0.499723
$834$	−3.23732e6	−0.161165
$835$	0	0
$836$	−786485.	−0.0389202
$837$	6.23615e6	0.307682
$838$	−7.28728e6	−0.358472
$839$	−2.82732e7	−1.38666	−0.693331	−	0.720619i	$-0.743858\pi$
$839$	−2.82732e7	−1.38666	−0.693331	+	0.720619i	$0.743858\pi$
$840$	0	0
$841$	−2.04084e7	−0.994992
$842$	−95946.5	−0.00466390
$843$	−9.70103e6	−0.470164
$844$	−1.68884e6	−0.0816080
$845$	0	0
$846$	1.15918e7	0.556834
$847$	1.54540e6	0.0740174
$848$	−3.18415e7	−1.52056
$849$	2.33519e7	1.11187
$850$	0	0
$851$	−1.05082e7	−0.497399
$852$	−469903.	−0.0221773
$853$	−7.27728e6	−0.342449	−0.171225	−	0.985232i	$-0.554772\pi$
$853$	−7.27728e6	−0.342449	−0.171225	+	0.985232i	$0.554772\pi$
$854$	2.94338e6	0.138102
$855$	0	0
$856$	−1.17218e7	−0.546775
$857$	−2.49180e7	−1.15894	−0.579469	−	0.814994i	$-0.696740\pi$
$857$	−2.49180e7	−1.15894	−0.579469	+	0.814994i	$0.696740\pi$
$858$	803355.	0.0372554
$859$	−3.01693e7	−1.39502	−0.697512	−	0.716573i	$-0.745710\pi$
$859$	−3.01693e7	−1.39502	−0.697512	+	0.716573i	$0.745710\pi$
$860$	0	0
$861$	4.30511e6	0.197914
$862$	1.09894e6	0.0503742
$863$	−3.86426e7	−1.76620	−0.883099	−	0.469187i	$-0.844547\pi$
$863$	−3.86426e7	−1.76620	−0.883099	+	0.469187i	$0.844547\pi$
$864$	−855085.	−0.0389695
$865$	0	0
$866$	1.68507e7	0.763524
$867$	1.53045e7	0.691466
$868$	2.93642e6	0.132288
$869$	2.64008e6	0.118595
$870$	0	0
$871$	667743.	0.0298239
$872$	−5.02895e6	−0.223968
$873$	4.19541e6	0.186311
$874$	3.02210e7	1.33823
$875$	0	0
$876$	−766131.	−0.0337321
$877$	−1.35160e7	−0.593404	−0.296702	−	0.954970i	$-0.595887\pi$
$877$	−1.35160e7	−0.593404	−0.296702	+	0.954970i	$0.595887\pi$
$878$	−6.95790e6	−0.304608
$879$	1.39645e6	0.0609611
$880$	0	0
$881$	1.56537e7	0.679479	0.339739	−	0.940520i	$-0.389661\pi$
$881$	1.56537e7	0.679479	0.339739	+	0.940520i	$0.389661\pi$
$882$	−2.72469e6	−0.117936
$883$	7.31854e6	0.315880	0.157940	−	0.987449i	$-0.449515\pi$
$883$	7.31854e6	0.315880	0.157940	+	0.987449i	$0.449515\pi$
$884$	713752.	0.0307197
$885$	0	0
$886$	3.75802e7	1.60833
$887$	−3.94393e6	−0.168314	−0.0841570	−	0.996453i	$-0.526820\pi$
$887$	−3.94393e6	−0.168314	−0.0841570	+	0.996453i	$0.526820\pi$
$888$	6.33871e6	0.269754
$889$	−2.03488e7	−0.863545
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−3.58777e6	−0.150978
$893$	4.81750e7	2.02159
$894$	3.29020e6	0.137682
$895$	0	0
$896$	2.20082e7	0.915828
$897$	−2.84773e6	−0.118173
$898$	−1.67428e7	−0.692847
$899$	−2.74162e6	−0.113138
$900$	0	0
$901$	−5.03329e7	−2.06557
$902$	−3.25573e6	−0.133239
$903$	1.27415e7	0.519996
$904$	−1.69504e7	−0.689857
$905$	0	0
$906$	−1.04006e7	−0.420958
$907$	2.09119e7	0.844064	0.422032	−	0.906581i	$-0.361317\pi$
$907$	2.09119e7	0.844064	0.422032	+	0.906581i	$0.361317\pi$
$908$	−3.50293e6	−0.140999
$909$	1.23534e7	0.495882
$910$	0	0
$911$	2.63760e7	1.05296	0.526481	−	0.850187i	$-0.323511\pi$
$911$	2.63760e7	1.05296	0.526481	+	0.850187i	$0.323511\pi$
$912$	−2.01017e7	−0.800288
$913$	−7.88638e6	−0.313113
$914$	−3.70816e6	−0.146823
$915$	0	0
$916$	2.17729e6	0.0857390
$917$	2.05474e7	0.806924
$918$	−7.64578e6	−0.299444
$919$	−1.74127e7	−0.680109	−0.340054	−	0.940406i	$-0.610445\pi$
$919$	−1.74127e7	−0.680109	−0.340054	+	0.940406i	$0.610445\pi$
$920$	0	0
$921$	1.75406e7	0.681391
$922$	2.28383e7	0.884783
$923$	1.99478e6	0.0770709
$924$	−373816.	−0.0144038
$925$	0	0
$926$	4.69419e7	1.79901
$927$	−1.54198e7	−0.589357
$928$	375924.	0.0143295
$929$	1.23992e7	0.471360	0.235680	−	0.971831i	$-0.424268\pi$
$929$	1.23992e7	0.471360	0.235680	+	0.971831i	$0.424268\pi$
$930$	0	0
$931$	−1.13237e7	−0.428166
$932$	−327174.	−0.0123378
$933$	7.68023e6	0.288848
$934$	−4.23845e7	−1.58979
$935$	0	0
$936$	1.71780e6	0.0640888
$937$	−6.45952e6	−0.240354	−0.120177	−	0.992752i	$-0.538346\pi$
$937$	−6.45952e6	−0.240354	−0.120177	+	0.992752i	$0.538346\pi$
$938$	−3.36810e6	−0.124991
$939$	−5.35157e6	−0.198069
$940$	0	0
$941$	−4.98442e6	−0.183502	−0.0917510	−	0.995782i	$-0.529246\pi$
$941$	−4.98442e6	−0.183502	−0.0917510	+	0.995782i	$0.529246\pi$
$942$	−4.06176e6	−0.149138
$943$	1.15409e7	0.422630
$944$	2.03424e7	0.742971
$945$	0	0
$946$	−9.63569e6	−0.350070
$947$	−8.49575e6	−0.307841	−0.153921	−	0.988083i	$-0.549190\pi$
$947$	−8.49575e6	−0.307841	−0.153921	+	0.988083i	$0.549190\pi$
$948$	−638605.	−0.0230787
$949$	3.25230e6	0.117226
$950$	0	0
$951$	−2.92786e7	−1.04978
$952$	3.18253e7	1.13810
$953$	−3.18137e7	−1.13470	−0.567351	−	0.823476i	$-0.692032\pi$
$953$	−3.18137e7	−1.13470	−0.567351	+	0.823476i	$0.692032\pi$
$954$	1.37034e7	0.487479
$955$	0	0
$956$	−2.81987e6	−0.0997895
$957$	349017.	0.0123188
$958$	−3.89408e7	−1.37085
$959$	2.32045e7	0.814753
$960$	0	0
$961$	4.45484e7	1.55605
$962$	3.04396e6	0.106048
$963$	5.56262e6	0.193292
$964$	−2.01034e6	−0.0696748
$965$	0	0
$966$	1.43640e7	0.495260
$967$	4.98824e6	0.171546	0.0857732	−	0.996315i	$-0.472664\pi$
$967$	4.98824e6	0.171546	0.0857732	+	0.996315i	$0.472664\pi$
$968$	−2.49902e6	−0.0857198
$969$	−3.17754e7	−1.08713
$970$	0	0
$971$	1.81815e7	0.618845	0.309422	−	0.950925i	$-0.399864\pi$
$971$	1.81815e7	0.618845	0.309422	+	0.950925i	$0.399864\pi$
$972$	192031.	0.00651936
$973$	−6.39473e6	−0.216541
$974$	4.11366e7	1.38941
$975$	0	0
$976$	−5.24838e6	−0.176360
$977$	−1.72113e7	−0.576870	−0.288435	−	0.957499i	$-0.593135\pi$
$977$	−1.72113e7	−0.576870	−0.288435	+	0.957499i	$0.593135\pi$
$978$	2.66245e7	0.890091
$979$	−3.43277e6	−0.114469
$980$	0	0
$981$	2.38651e6	0.0791755
$982$	−5.70281e7	−1.88717
$983$	4.14848e7	1.36932	0.684661	−	0.728861i	$-0.259950\pi$
$983$	4.14848e7	1.36932	0.684661	+	0.728861i	$0.259950\pi$
$984$	−6.96164e6	−0.229205
$985$	0	0
$986$	3.36134e6	0.110108
$987$	2.28975e7	0.748162
$988$	−807593.	−0.0263209
$989$	3.41566e7	1.11041
$990$	0	0
$991$	−1.58584e7	−0.512951	−0.256475	−	0.966551i	$-0.582561\pi$
$991$	−1.58584e7	−0.512951	−0.256475	+	0.966551i	$0.582561\pi$
$992$	−1.00339e7	−0.323736
$993$	−1.85276e7	−0.596274
$994$	−1.00617e7	−0.323002
$995$	0	0
$996$	1.90762e6	0.0609319
$997$	4.08647e7	1.30200	0.651000	−	0.759078i	$-0.274350\pi$
$997$	4.08647e7	1.30200	0.651000	+	0.759078i	$0.274350\pi$
$998$	3.99221e7	1.26878
$999$	−3.00806e6	−0.0953615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.k.1.4		5
5.4	even	2		165.6.a.g.1.2	✓	5
15.14	odd	2		495.6.a.i.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.g.1.2	✓	5	5.4	even	2
495.6.a.i.1.4		5	15.14	odd	2
825.6.a.k.1.4		5	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)

\(f(q)\)	\(=\)	\(q+5.93734 q^{2} +9.00000 q^{3} +3.25206 q^{4} +53.4361 q^{6} +105.553 q^{7} -170.686 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+5.93734 q^{2} +9.00000 q^{3} +3.25206 q^{4} +53.4361 q^{6} +105.553 q^{7} -170.686 q^{8} +81.0000 q^{9} -121.000 q^{11} +29.2685 q^{12} -124.247 q^{13} +626.706 q^{14} -1117.49 q^{16} -1766.45 q^{17} +480.925 q^{18} +1998.70 q^{19} +949.979 q^{21} -718.419 q^{22} +2546.65 q^{23} -1536.18 q^{24} -737.700 q^{26} +729.000 q^{27} +343.265 q^{28} -320.493 q^{29} +8554.39 q^{31} -1172.96 q^{32} -1089.00 q^{33} -10488.0 q^{34} +263.417 q^{36} -4126.28 q^{37} +11867.0 q^{38} -1118.23 q^{39} +4531.80 q^{41} +5640.35 q^{42} +13412.4 q^{43} -393.499 q^{44} +15120.3 q^{46} +24103.2 q^{47} -10057.4 q^{48} -5665.52 q^{49} -15898.1 q^{51} -404.060 q^{52} +28493.8 q^{53} +4328.32 q^{54} -18016.5 q^{56} +17988.3 q^{57} -1902.88 q^{58} -18203.6 q^{59} +4696.58 q^{61} +50790.3 q^{62} +8549.81 q^{63} +28795.4 q^{64} -6465.77 q^{66} -5374.30 q^{67} -5744.60 q^{68} +22919.9 q^{69} -16054.9 q^{71} -13825.6 q^{72} -26176.0 q^{73} -24499.2 q^{74} +6499.88 q^{76} -12771.9 q^{77} -6639.30 q^{78} -21818.8 q^{79} +6561.00 q^{81} +26906.8 q^{82} +65176.7 q^{83} +3089.39 q^{84} +79633.8 q^{86} -2884.44 q^{87} +20653.1 q^{88} +28370.0 q^{89} -13114.7 q^{91} +8281.86 q^{92} +76989.5 q^{93} +143109. q^{94} -10556.6 q^{96} +51795.2 q^{97} -33638.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9}+O(q^{10}) \) 5 * q + q^2 + 45 * q^3 + 127 * q^4 + 9 * q^6 - 116 * q^7 + 153 * q^8 + 405 * q^9	\( 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9} - 605 q^{11} + 1143 q^{12} + 926 q^{13} + 368 q^{14} + 1891 q^{16} + 246 q^{17} + 81 q^{18} + 3420 q^{19} - 1044 q^{21} - 121 q^{22} + 4244 q^{23} + 1377 q^{24} - 8862 q^{26} + 3645 q^{27} + 4904 q^{28} - 2922 q^{29} - 6112 q^{31} + 24757 q^{32} - 5445 q^{33} + 10866 q^{34} + 10287 q^{36} - 6654 q^{37} + 45692 q^{38} + 8334 q^{39} - 14934 q^{41} + 3312 q^{42} - 10804 q^{43} - 15367 q^{44} - 101500 q^{46} + 41460 q^{47} + 17019 q^{48} - 12099 q^{49} + 2214 q^{51} + 97742 q^{52} + 62398 q^{53} + 729 q^{54} - 74368 q^{56} + 30780 q^{57} + 27822 q^{58} + 8524 q^{59} + 59010 q^{61} + 142624 q^{62} - 9396 q^{63} + 13799 q^{64} - 1089 q^{66} + 15772 q^{67} + 83686 q^{68} + 38196 q^{69} + 88124 q^{71} + 12393 q^{72} + 118358 q^{73} + 67194 q^{74} + 100668 q^{76} + 14036 q^{77} - 79758 q^{78} + 57324 q^{79} + 32805 q^{81} - 29102 q^{82} + 7268 q^{83} + 44136 q^{84} - 35288 q^{86} - 26298 q^{87} - 18513 q^{88} + 72978 q^{89} - 1464 q^{91} - 62148 q^{92} - 55008 q^{93} + 344836 q^{94} + 222813 q^{96} + 59174 q^{97} - 272767 q^{98} - 49005 q^{99}+O(q^{100}) \) 5 * q + q^2 + 45 * q^3 + 127 * q^4 + 9 * q^6 - 116 * q^7 + 153 * q^8 + 405 * q^9 - 605 * q^11 + 1143 * q^12 + 926 * q^13 + 368 * q^14 + 1891 * q^16 + 246 * q^17 + 81 * q^18 + 3420 * q^19 - 1044 * q^21 - 121 * q^22 + 4244 * q^23 + 1377 * q^24 - 8862 * q^26 + 3645 * q^27 + 4904 * q^28 - 2922 * q^29 - 6112 * q^31 + 24757 * q^32 - 5445 * q^33 + 10866 * q^34 + 10287 * q^36 - 6654 * q^37 + 45692 * q^38 + 8334 * q^39 - 14934 * q^41 + 3312 * q^42 - 10804 * q^43 - 15367 * q^44 - 101500 * q^46 + 41460 * q^47 + 17019 * q^48 - 12099 * q^49 + 2214 * q^51 + 97742 * q^52 + 62398 * q^53 + 729 * q^54 - 74368 * q^56 + 30780 * q^57 + 27822 * q^58 + 8524 * q^59 + 59010 * q^61 + 142624 * q^62 - 9396 * q^63 + 13799 * q^64 - 1089 * q^66 + 15772 * q^67 + 83686 * q^68 + 38196 * q^69 + 88124 * q^71 + 12393 * q^72 + 118358 * q^73 + 67194 * q^74 + 100668 * q^76 + 14036 * q^77 - 79758 * q^78 + 57324 * q^79 + 32805 * q^81 - 29102 * q^82 + 7268 * q^83 + 44136 * q^84 - 35288 * q^86 - 26298 * q^87 - 18513 * q^88 + 72978 * q^89 - 1464 * q^91 - 62148 * q^92 - 55008 * q^93 + 344836 * q^94 + 222813 * q^96 + 59174 * q^97 - 272767 * q^98 - 49005 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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