LMFDB - Embedded newform 275.8.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,8,Mod(1,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,8,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$85.9058820081$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.87298$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.74597 q^{2} +49.4758 q^{3} -113.968 q^{4} -185.335 q^{6} -21.1693 q^{7} +906.403 q^{8} +260.855 q^{9} +1331.00 q^{11} -5638.64 q^{12} -4184.41 q^{13} +79.2994 q^{14} +11192.5 q^{16} +32630.7 q^{17} -977.153 q^{18} -40738.5 q^{19} -1047.37 q^{21} -4985.88 q^{22} -2487.20 q^{23} +44845.0 q^{24} +15674.7 q^{26} -95297.6 q^{27} +2412.61 q^{28} +689.254 q^{29} +127963. q^{31} -157946. q^{32} +65852.3 q^{33} -122234. q^{34} -29729.0 q^{36} +467881. q^{37} +152605. q^{38} -207027. q^{39} +391657. q^{41} +3923.40 q^{42} -236581. q^{43} -151691. q^{44} +9316.96 q^{46} -713115. q^{47} +553759. q^{48} -823095. q^{49} +1.61443e6 q^{51} +476888. q^{52} -1.32248e6 q^{53} +356982. q^{54} -19187.9 q^{56} -2.01557e6 q^{57} -2581.92 q^{58} -2.36633e6 q^{59} -3.23345e6 q^{61} -479345. q^{62} -5522.11 q^{63} -840980. q^{64} -246680. q^{66} +3.12834e6 q^{67} -3.71885e6 q^{68} -123056. q^{69} +2.93234e6 q^{71} +236440. q^{72} +1.06930e6 q^{73} -1.75267e6 q^{74} +4.64287e6 q^{76} -28176.3 q^{77} +775517. q^{78} +4.40962e6 q^{79} -5.28541e6 q^{81} -1.46713e6 q^{82} +2.49753e6 q^{83} +119366. q^{84} +886223. q^{86} +34101.4 q^{87} +1.20642e6 q^{88} -5.17831e6 q^{89} +88580.9 q^{91} +283460. q^{92} +6.33107e6 q^{93} +2.67130e6 q^{94} -7.81452e6 q^{96} -1.32101e6 q^{97} +3.08329e6 q^{98} +347198. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 6 q^{3} - 104 q^{4} - 696 q^{6} + 1228 q^{7} - 480 q^{8} - 36 q^{9} + 2662 q^{11} - 6072 q^{12} - 344 q^{13} + 14752 q^{14} - 6368 q^{16} + 8468 q^{17} - 4464 q^{18} - 35280 q^{19} - 55356 q^{21}+ \cdots - 47916 q^{99}+O(q^{100}) $ 2 * q + 8 * q^2 + 6 * q^3 - 104 * q^4 - 696 * q^6 + 1228 * q^7 - 480 * q^8 - 36 * q^9 + 2662 * q^11 - 6072 * q^12 - 344 * q^13 + 14752 * q^14 - 6368 * q^16 + 8468 * q^17 - 4464 * q^18 - 35280 * q^19 - 55356 * q^21 + 10648 * q^22 + 61486 * q^23 + 105120 * q^24 + 60784 * q^26 + 12690 * q^27 + 14864 * q^28 + 179040 * q^29 - 57166 * q^31 - 186752 * q^32 + 7986 * q^33 - 406048 * q^34 - 32688 * q^36 + 877698 * q^37 + 216720 * q^38 - 373992 * q^39 - 283616 * q^41 - 633984 * q^42 - 275484 * q^43 - 138424 * q^44 + 760744 * q^46 - 1662512 * q^47 + 1317216 * q^48 - 86214 * q^49 + 2664924 * q^51 + 515168 * q^52 - 1616484 * q^53 + 1625400 * q^54 - 1751040 * q^56 - 2252880 * q^57 + 2092320 * q^58 - 2454130 * q^59 - 6019176 * q^61 - 2653864 * q^62 - 376344 * q^63 + 1068416 * q^64 - 926376 * q^66 + 174698 * q^67 - 3959696 * q^68 - 2904342 * q^69 - 1151466 * q^71 + 648000 * q^72 - 885944 * q^73 + 3061032 * q^74 + 4697280 * q^76 + 1634468 * q^77 - 1185648 * q^78 + 3801460 * q^79 - 9331038 * q^81 - 9398864 * q^82 + 2282916 * q^83 - 421968 * q^84 + 429264 * q^86 - 7719840 * q^87 - 638880 * q^88 - 13481970 * q^89 + 4885904 * q^91 + 921128 * q^92 + 14379702 * q^93 - 8480288 * q^94 - 6562176 * q^96 + 68078 * q^97 + 11738664 * q^98 - 47916 * q^99

Coefficient data

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

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

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

$2$	−3.74597	−0.331100	−0.165550	−	0.986201i	$-0.552940\pi$
$2$	−3.74597	−0.331100	−0.165550	+	0.986201i	$0.552940\pi$
$3$	49.4758	1.05796	0.528979	−	0.848635i	$-0.322575\pi$
$3$	49.4758	1.05796	0.528979	+	0.848635i	$0.322575\pi$
$4$	−113.968	−0.890373
$5$	0	0
$5$	0	0
$6$	−185.335	−0.350290
$7$	−21.1693	−0.0233272	−0.0116636	−	0.999932i	$-0.503713\pi$
$7$	−21.1693	−0.0233272	−0.0116636	+	0.999932i	$0.503713\pi$
$8$	906.403	0.625902
$9$	260.855	0.119275
$10$	0	0
$11$	1331.00	0.301511
$11$	1331.00	0.301511
$12$	−5638.64	−0.941977
$13$	−4184.41	−0.528242	−0.264121	−	0.964490i	$-0.585082\pi$
$13$	−4184.41	−0.528242	−0.264121	+	0.964490i	$0.585082\pi$
$14$	79.2994	0.00772363
$15$	0	0
$16$	11192.5	0.683137
$17$	32630.7	1.61085	0.805425	−	0.592697i	$-0.201937\pi$
$17$	32630.7	1.61085	0.805425	+	0.592697i	$0.201937\pi$
$18$	−977.153	−0.0394920
$19$	−40738.5	−1.36260	−0.681298	−	0.732006i	$-0.738584\pi$
$19$	−40738.5	−1.36260	−0.681298	+	0.732006i	$0.738584\pi$
$20$	0	0
$21$	−1047.37	−0.0246792
$22$	−4985.88	−0.0998303
$23$	−2487.20	−0.0426248	−0.0213124	−	0.999773i	$-0.506784\pi$
$23$	−2487.20	−0.0426248	−0.0213124	+	0.999773i	$0.506784\pi$
$24$	44845.0	0.662178
$25$	0	0
$26$	15674.7	0.174901
$27$	−95297.6	−0.931770
$28$	2412.61	0.0207699
$29$	689.254	0.00524791	0.00262395	−	0.999997i	$-0.499165\pi$
$29$	689.254	0.00524791	0.00262395	+	0.999997i	$0.499165\pi$
$30$	0	0
$31$	127963.	0.771469	0.385734	−	0.922610i	$-0.373948\pi$
$31$	127963.	0.771469	0.385734	+	0.922610i	$0.373948\pi$
$32$	−157946.	−0.852089
$33$	65852.3	0.318986
$34$	−122234.	−0.533352
$35$	0	0
$36$	−29729.0	−0.106199
$37$	467881.	1.51855	0.759275	−	0.650770i	$-0.225554\pi$
$37$	467881.	1.51855	0.759275	+	0.650770i	$0.225554\pi$
$38$	152605.	0.451155
$39$	−207027.	−0.558857
$40$	0	0
$41$	391657.	0.887487	0.443744	−	0.896154i	$-0.353650\pi$
$41$	391657.	0.887487	0.443744	+	0.896154i	$0.353650\pi$
$42$	3923.40	0.00817128
$43$	−236581.	−0.453774	−0.226887	−	0.973921i	$-0.572855\pi$
$43$	−236581.	−0.453774	−0.226887	+	0.973921i	$0.572855\pi$
$44$	−151691.	−0.268458
$45$	0	0
$46$	9316.96	0.0141131
$47$	−713115.	−1.00188	−0.500941	−	0.865481i	$-0.667013\pi$
$47$	−713115.	−1.00188	−0.500941	+	0.865481i	$0.667013\pi$
$48$	553759.	0.722730
$49$	−823095.	−0.999456
$50$	0	0
$51$	1.61443e6	1.70421
$52$	476888.	0.470332
$53$	−1.32248e6	−1.22018	−0.610090	−	0.792332i	$-0.708867\pi$
$53$	−1.32248e6	−1.22018	−0.610090	+	0.792332i	$0.708867\pi$
$54$	356982.	0.308509
$55$	0	0
$56$	−19187.9	−0.0146005
$57$	−2.01557e6	−1.44157
$58$	−2581.92	−0.00173758
$59$	−2.36633e6	−1.50000	−0.750002	−	0.661435i	$-0.769948\pi$
$59$	−2.36633e6	−1.50000	−0.750002	+	0.661435i	$0.769948\pi$
$60$	0	0
$61$	−3.23345e6	−1.82394	−0.911971	−	0.410254i	$-0.865440\pi$
$61$	−3.23345e6	−1.82394	−0.911971	+	0.410254i	$0.865440\pi$
$62$	−479345.	−0.255433
$63$	−5522.11	−0.00278236
$64$	−840980.	−0.401010
$65$	0	0
$66$	−246680.	−0.105616
$67$	3.12834e6	1.27073	0.635363	−	0.772214i	$-0.280851\pi$
$67$	3.12834e6	1.27073	0.635363	+	0.772214i	$0.280851\pi$
$68$	−3.71885e6	−1.43426
$69$	−123056.	−0.0450953
$70$	0	0
$71$	2.93234e6	0.972321	0.486161	−	0.873869i	$-0.338397\pi$
$71$	2.93234e6	0.972321	0.486161	+	0.873869i	$0.338397\pi$
$72$	236440.	0.0746546
$73$	1.06930e6	0.321715	0.160857	−	0.986978i	$-0.448574\pi$
$73$	1.06930e6	0.321715	0.160857	+	0.986978i	$0.448574\pi$
$74$	−1.75267e6	−0.502792
$75$	0	0
$76$	4.64287e6	1.21322
$77$	−28176.3	−0.00703342
$78$	775517.	0.185038
$79$	4.40962e6	1.00625	0.503125	−	0.864214i	$-0.332183\pi$
$79$	4.40962e6	1.00625	0.503125	+	0.864214i	$0.332183\pi$
$80$	0	0
$81$	−5.28541e6	−1.10505
$82$	−1.46713e6	−0.293847
$83$	2.49753e6	0.479443	0.239722	−	0.970842i	$-0.422944\pi$
$83$	2.49753e6	0.479443	0.239722	+	0.970842i	$0.422944\pi$
$84$	119366.	0.0219737
$85$	0	0
$86$	886223.	0.150244
$87$	34101.4	0.00555207
$88$	1.20642e6	0.188717
$89$	−5.17831e6	−0.778616	−0.389308	−	0.921108i	$-0.627286\pi$
$89$	−5.17831e6	−0.778616	−0.389308	+	0.921108i	$0.627286\pi$
$90$	0	0
$91$	88580.9	0.0123224
$92$	283460.	0.0379520
$93$	6.33107e6	0.816182
$94$	2.67130e6	0.331723
$95$	0	0
$96$	−7.81452e6	−0.901474
$97$	−1.32101e6	−0.146962	−0.0734810	−	0.997297i	$-0.523411\pi$
$97$	−1.32101e6	−0.146962	−0.0734810	+	0.997297i	$0.523411\pi$
$98$	3.08329e6	0.330920
$99$	347198.	0.0359628
$100$	0	0
$101$	−1.71975e7	−1.66089	−0.830447	−	0.557098i	$-0.811915\pi$
$101$	−1.71975e7	−1.66089	−0.830447	+	0.557098i	$0.811915\pi$
$102$	−6.04760e6	−0.564264
$103$	−4.02516e6	−0.362955	−0.181477	−	0.983395i	$-0.558088\pi$
$103$	−4.02516e6	−0.362955	−0.181477	+	0.983395i	$0.558088\pi$
$104$	−3.79276e6	−0.330628
$105$	0	0
$106$	4.95397e6	0.404001
$107$	−2.17827e7	−1.71897	−0.859486	−	0.511160i	$-0.829216\pi$
$107$	−2.17827e7	−1.71897	−0.859486	+	0.511160i	$0.829216\pi$
$108$	1.08608e7	0.829623
$109$	−2.69177e6	−0.199088	−0.0995439	−	0.995033i	$-0.531738\pi$
$109$	−2.69177e6	−0.199088	−0.0995439	+	0.995033i	$0.531738\pi$
$110$	0	0
$111$	2.31488e7	1.60656
$112$	−236937.	−0.0159357
$113$	598776.	0.0390382	0.0195191	−	0.999809i	$-0.493786\pi$
$113$	598776.	0.0390382	0.0195191	+	0.999809i	$0.493786\pi$
$114$	7.55025e6	0.477303
$115$	0	0
$116$	−78552.7	−0.00467260
$117$	−1.09152e6	−0.0630061
$118$	8.86418e6	0.496651
$119$	−690768.	−0.0375767
$120$	0	0
$121$	1.77156e6	0.0909091
$122$	1.21124e7	0.603907
$123$	1.93775e7	0.938924
$124$	−1.45837e7	−0.686895
$125$	0	0
$126$	20685.6	0.000921238 0
$127$	−3.59136e7	−1.55577	−0.777886	−	0.628405i	$-0.783708\pi$
$127$	−3.59136e7	−1.55577	−0.777886	+	0.628405i	$0.783708\pi$
$128$	2.33674e7	0.984863
$129$	−1.17050e7	−0.480074
$130$	0	0
$131$	−1.65324e7	−0.642521	−0.321261	−	0.946991i	$-0.604107\pi$
$131$	−1.65324e7	−0.642521	−0.321261	+	0.946991i	$0.604107\pi$
$132$	−7.50504e6	−0.284017
$133$	862404.	0.0317856
$134$	−1.17187e7	−0.420737
$135$	0	0
$136$	2.95766e7	1.00823
$137$	−2.77432e7	−0.921797	−0.460899	−	0.887453i	$-0.652473\pi$
$137$	−2.77432e7	−0.921797	−0.460899	+	0.887453i	$0.652473\pi$
$138$	460964.	0.0149310
$139$	−5.04731e7	−1.59407	−0.797037	−	0.603930i	$-0.793601\pi$
$139$	−5.04731e7	−1.59407	−0.797037	+	0.603930i	$0.793601\pi$
$140$	0	0
$141$	−3.52819e7	−1.05995
$142$	−1.09844e7	−0.321935
$143$	−5.56945e6	−0.159271
$144$	2.91962e6	0.0814813
$145$	0	0
$146$	−4.00558e6	−0.106520
$147$	−4.07233e7	−1.05738
$148$	−5.33233e7	−1.35208
$149$	3.09903e7	0.767492	0.383746	−	0.923439i	$-0.374634\pi$
$149$	3.09903e7	0.767492	0.383746	+	0.923439i	$0.374634\pi$
$150$	0	0
$151$	−6.75757e6	−0.159724	−0.0798622	−	0.996806i	$-0.525448\pi$
$151$	−6.75757e6	−0.159724	−0.0798622	+	0.996806i	$0.525448\pi$
$152$	−3.69255e7	−0.852852
$153$	8.51188e6	0.192135
$154$	105547.	0.00232876
$155$	0	0
$156$	2.35944e7	0.497591
$157$	−1.89989e7	−0.391813	−0.195907	−	0.980623i	$-0.562765\pi$
$157$	−1.89989e7	−0.391813	−0.195907	+	0.980623i	$0.562765\pi$
$158$	−1.65183e7	−0.333169
$159$	−6.54308e7	−1.29090
$160$	0	0
$161$	52652.1	0.000994318 0
$162$	1.97990e7	0.365881
$163$	6.97589e7	1.26166	0.630831	−	0.775921i	$-0.282714\pi$
$163$	6.97589e7	1.26166	0.630831	+	0.775921i	$0.282714\pi$
$164$	−4.46362e7	−0.790195
$165$	0	0
$166$	−9.35566e6	−0.158744
$167$	2.42233e7	0.402463	0.201231	−	0.979544i	$-0.435506\pi$
$167$	2.42233e7	0.402463	0.201231	+	0.979544i	$0.435506\pi$
$168$	−949336.	−0.0154468
$169$	−4.52392e7	−0.720961
$170$	0	0
$171$	−1.06268e7	−0.162524
$172$	2.69625e7	0.404028
$173$	−6.00438e7	−0.881671	−0.440836	−	0.897588i	$-0.645318\pi$
$173$	−6.00438e7	−0.881671	−0.440836	+	0.897588i	$0.645318\pi$
$174$	−127743.	−0.00183829
$175$	0	0
$176$	1.48972e7	0.205974
$177$	−1.17076e8	−1.58694
$178$	1.93978e7	0.257800
$179$	1.82943e6	0.0238413	0.0119206	−	0.999929i	$-0.496205\pi$
$179$	1.82943e6	0.0238413	0.0119206	+	0.999929i	$0.496205\pi$
$180$	0	0
$181$	−1.24476e8	−1.56030	−0.780152	−	0.625590i	$-0.784858\pi$
$181$	−1.24476e8	−1.56030	−0.780152	+	0.625590i	$0.784858\pi$
$182$	−331821.	−0.00407994
$183$	−1.59977e8	−1.92966
$184$	−2.25440e6	−0.0266790
$185$	0	0
$186$	−2.37160e7	−0.270238
$187$	4.34315e7	0.485690
$188$	8.12720e7	0.892049
$189$	2.01738e6	0.0217356
$190$	0	0
$191$	−7.76142e7	−0.805980	−0.402990	−	0.915204i	$-0.632029\pi$
$191$	−7.76142e7	−0.805980	−0.402990	+	0.915204i	$0.632029\pi$
$192$	−4.16082e7	−0.424252
$193$	−3.85250e7	−0.385738	−0.192869	−	0.981225i	$-0.561779\pi$
$193$	−3.85250e7	−0.385738	−0.192869	+	0.981225i	$0.561779\pi$
$194$	4.94846e6	0.0486591
$195$	0	0
$196$	9.38063e7	0.889888
$197$	−1.49067e8	−1.38915	−0.694575	−	0.719421i	$-0.744407\pi$
$197$	−1.49067e8	−1.38915	−0.694575	+	0.719421i	$0.744407\pi$
$198$	−1.30059e6	−0.0119073
$199$	−6.45185e7	−0.580362	−0.290181	−	0.956972i	$-0.593715\pi$
$199$	−6.45185e7	−0.580362	−0.290181	+	0.956972i	$0.593715\pi$
$200$	0	0
$201$	1.54777e8	1.34437
$202$	6.44214e7	0.549921
$203$	−14591.0	−0.000122419 0
$204$	−1.83993e8	−1.51738
$205$	0	0
$206$	1.50781e7	0.120174
$207$	−648797.	−0.00508409
$208$	−4.68341e7	−0.360861
$209$	−5.42229e7	−0.410838
$210$	0	0
$211$	3.55799e7	0.260745	0.130373	−	0.991465i	$-0.458383\pi$
$211$	3.55799e7	0.260745	0.130373	+	0.991465i	$0.458383\pi$
$212$	1.50720e8	1.08642
$213$	1.45080e8	1.02868
$214$	8.15973e7	0.569151
$215$	0	0
$216$	−8.63780e7	−0.583197
$217$	−2.70888e6	−0.0179962
$218$	1.00833e7	0.0659180
$219$	5.29047e7	0.340361
$220$	0	0
$221$	−1.36540e8	−0.850918
$222$	−8.67146e7	−0.531933
$223$	1.55599e8	0.939590	0.469795	−	0.882776i	$-0.344328\pi$
$223$	1.55599e8	0.939590	0.469795	+	0.882776i	$0.344328\pi$
$224$	3.34361e6	0.0198768
$225$	0	0
$226$	−2.24300e6	−0.0129255
$227$	−1.39233e8	−0.790046	−0.395023	−	0.918671i	$-0.629263\pi$
$227$	−1.39233e8	−0.790046	−0.395023	+	0.918671i	$0.629263\pi$
$228$	2.29710e8	1.28353
$229$	2.52955e8	1.39193	0.695966	−	0.718074i	$-0.254976\pi$
$229$	2.52955e8	1.39193	0.695966	+	0.718074i	$0.254976\pi$
$230$	0	0
$231$	−1.39404e6	−0.00744106
$232$	624742.	0.00328468
$233$	−1.00338e8	−0.519658	−0.259829	−	0.965655i	$-0.583666\pi$
$233$	−1.00338e8	−0.519658	−0.259829	+	0.965655i	$0.583666\pi$
$234$	4.08881e6	0.0208613
$235$	0	0
$236$	2.69685e8	1.33556
$237$	2.18169e8	1.06457
$238$	2.58760e6	0.0124416
$239$	7.53046e7	0.356803	0.178402	−	0.983958i	$-0.442907\pi$
$239$	7.53046e7	0.356803	0.178402	+	0.983958i	$0.442907\pi$
$240$	0	0
$241$	1.25802e8	0.578934	0.289467	−	0.957188i	$-0.406522\pi$
$241$	1.25802e8	0.578934	0.289467	+	0.957188i	$0.406522\pi$
$242$	−6.63621e6	−0.0301000
$243$	−5.30843e7	−0.237325
$244$	3.68509e8	1.62399
$245$	0	0
$246$	−7.25876e7	−0.310878
$247$	1.70467e8	0.719780
$248$	1.15986e8	0.482864
$249$	1.23567e8	0.507231
$250$	0	0
$251$	−1.64527e8	−0.656718	−0.328359	−	0.944553i	$-0.606496\pi$
$251$	−1.64527e8	−0.656718	−0.328359	+	0.944553i	$0.606496\pi$
$252$	629342.	0.00247733
$253$	−3.31046e6	−0.0128519
$254$	1.34531e8	0.515116
$255$	0	0
$256$	2.01119e7	0.0749225
$257$	1.10435e8	0.405828	0.202914	−	0.979197i	$-0.434959\pi$
$257$	1.10435e8	0.405828	0.202914	+	0.979197i	$0.434959\pi$
$258$	4.38466e7	0.158952
$259$	−9.90470e6	−0.0354235
$260$	0	0
$261$	179795.	0.000625945 0
$262$	6.19300e7	0.212739
$263$	3.98541e8	1.35091	0.675457	−	0.737400i	$-0.263947\pi$
$263$	3.98541e8	1.35091	0.675457	+	0.737400i	$0.263947\pi$
$264$	5.96887e7	0.199654
$265$	0	0
$266$	−3.23054e6	−0.0105242
$267$	−2.56201e8	−0.823743
$268$	−3.56530e8	−1.13142
$269$	−2.99250e8	−0.937350	−0.468675	−	0.883371i	$-0.655268\pi$
$269$	−2.99250e8	−0.937350	−0.468675	+	0.883371i	$0.655268\pi$
$270$	0	0
$271$	−1.37158e7	−0.0418628	−0.0209314	−	0.999781i	$-0.506663\pi$
$271$	−1.37158e7	−0.0418628	−0.0209314	+	0.999781i	$0.506663\pi$
$272$	3.65220e8	1.10043
$273$	4.38261e6	0.0130366
$274$	1.03925e8	0.305207
$275$	0	0
$276$	1.40244e7	0.0401516
$277$	6.73807e8	1.90483	0.952415	−	0.304804i	$-0.0985910\pi$
$277$	6.73807e8	1.90483	0.952415	+	0.304804i	$0.0985910\pi$
$278$	1.89071e8	0.527798
$279$	3.33798e7	0.0920171
$280$	0	0
$281$	5.66275e8	1.52249	0.761246	−	0.648463i	$-0.224588\pi$
$281$	5.66275e8	1.52249	0.761246	+	0.648463i	$0.224588\pi$
$282$	1.32165e8	0.350949
$283$	−6.29800e8	−1.65177	−0.825886	−	0.563837i	$-0.809325\pi$
$283$	−6.29800e8	−1.65177	−0.825886	+	0.563837i	$0.809325\pi$
$284$	−3.34192e8	−0.865729
$285$	0	0
$286$	2.08630e7	0.0527345
$287$	−8.29109e6	−0.0207026
$288$	−4.12011e7	−0.101633
$289$	6.54425e8	1.59484
$290$	0	0
$291$	−6.53580e7	−0.155480
$292$	−1.21866e8	−0.286446
$293$	1.23830e6	0.00287600	0.00143800	−	0.999999i	$-0.499542\pi$
$293$	1.23830e6	0.00287600	0.00143800	+	0.999999i	$0.499542\pi$
$294$	1.52548e8	0.350099
$295$	0	0
$296$	4.24089e8	0.950464
$297$	−1.26841e8	−0.280939
$298$	−1.16089e8	−0.254116
$299$	1.04075e7	0.0225162
$300$	0	0
$301$	5.00824e6	0.0105853
$302$	2.53136e7	0.0528847
$303$	−8.50862e8	−1.75716
$304$	−4.55966e8	−0.930840
$305$	0	0
$306$	−3.18852e7	−0.0636157
$307$	4.88441e8	0.963447	0.481723	−	0.876323i	$-0.340011\pi$
$307$	4.88441e8	0.963447	0.481723	+	0.876323i	$0.340011\pi$
$308$	3.21119e6	0.00626236
$309$	−1.99148e8	−0.383991
$310$	0	0
$311$	−4.81123e8	−0.906974	−0.453487	−	0.891263i	$-0.649820\pi$
$311$	−4.81123e8	−0.906974	−0.453487	+	0.891263i	$0.649820\pi$
$312$	−1.87650e8	−0.349790
$313$	−5.17645e7	−0.0954172	−0.0477086	−	0.998861i	$-0.515192\pi$
$313$	−5.17645e7	−0.0954172	−0.0477086	+	0.998861i	$0.515192\pi$
$314$	7.11691e7	0.129729
$315$	0	0
$316$	−5.02554e8	−0.895938
$317$	4.71412e8	0.831177	0.415588	−	0.909553i	$-0.363576\pi$
$317$	4.71412e8	0.831177	0.415588	+	0.909553i	$0.363576\pi$
$318$	2.45102e8	0.427417
$319$	917397.	0.00158230
$320$	0	0
$321$	−1.07772e9	−1.81860
$322$	−197233.	−0.000329219 0
$323$	−1.32933e9	−2.19494
$324$	6.02367e8	0.983905
$325$	0	0
$326$	−2.61314e8	−0.417736
$327$	−1.33177e8	−0.210627
$328$	3.54999e8	0.555480
$329$	1.50961e7	0.0233711
$330$	0	0
$331$	5.97269e8	0.905258	0.452629	−	0.891699i	$-0.350486\pi$
$331$	5.97269e8	0.905258	0.452629	+	0.891699i	$0.350486\pi$
$332$	−2.84638e8	−0.426883
$333$	1.22049e8	0.181125
$334$	−9.07397e7	−0.133255
$335$	0	0
$336$	−1.17227e7	−0.0168593
$337$	8.11952e8	1.15565	0.577824	−	0.816161i	$-0.303902\pi$
$337$	8.11952e8	1.15565	0.577824	+	0.816161i	$0.303902\pi$
$338$	1.69465e8	0.238710
$339$	2.96249e7	0.0413008
$340$	0	0
$341$	1.70319e8	0.232607
$342$	3.98077e7	0.0538116
$343$	3.48581e7	0.0466417
$344$	−2.14437e8	−0.284018
$345$	0	0
$346$	2.24922e8	0.291921
$347$	4.61385e8	0.592803	0.296401	−	0.955063i	$-0.404213\pi$
$347$	4.61385e8	0.592803	0.296401	+	0.955063i	$0.404213\pi$
$348$	−3.88646e6	−0.00494341
$349$	−9.00068e8	−1.13341	−0.566704	−	0.823922i	$-0.691782\pi$
$349$	−9.00068e8	−1.13341	−0.566704	+	0.823922i	$0.691782\pi$
$350$	0	0
$351$	3.98764e8	0.492200
$352$	−2.10227e8	−0.256914
$353$	−7.73849e8	−0.936364	−0.468182	−	0.883632i	$-0.655091\pi$
$353$	−7.73849e8	−0.936364	−0.468182	+	0.883632i	$0.655091\pi$
$354$	4.38562e8	0.525436
$355$	0	0
$356$	5.90161e8	0.693259
$357$	−3.41763e7	−0.0397545
$358$	−6.85298e6	−0.00789384
$359$	1.71828e9	1.96003	0.980016	−	0.198918i	$-0.0637427\pi$
$359$	1.71828e9	1.96003	0.980016	+	0.198918i	$0.0637427\pi$
$360$	0	0
$361$	7.65751e8	0.856668
$362$	4.66281e8	0.516616
$363$	8.76494e7	0.0961780
$364$	−1.00954e7	−0.0109715
$365$	0	0
$366$	5.99270e8	0.638908
$367$	3.00191e8	0.317005	0.158503	−	0.987359i	$-0.449333\pi$
$367$	3.00191e8	0.317005	0.158503	+	0.987359i	$0.449333\pi$
$368$	−2.78380e7	−0.0291186
$369$	1.02166e8	0.105855
$370$	0	0
$371$	2.79960e7	0.0284634
$372$	−7.21538e8	−0.726706
$373$	9.39634e7	0.0937514	0.0468757	−	0.998901i	$-0.485074\pi$
$373$	9.39634e7	0.0937514	0.0468757	+	0.998901i	$0.485074\pi$
$374$	−1.62693e8	−0.160812
$375$	0	0
$376$	−6.46369e8	−0.627080
$377$	−2.88412e6	−0.00277216
$378$	−7.55704e6	−0.00719665
$379$	−7.85315e8	−0.740980	−0.370490	−	0.928836i	$-0.620810\pi$
$379$	−7.85315e8	−0.740980	−0.370490	+	0.928836i	$0.620810\pi$
$380$	0	0
$381$	−1.77685e9	−1.64594
$382$	2.90740e8	0.266860
$383$	1.56792e9	1.42603	0.713014	−	0.701149i	$-0.247329\pi$
$383$	1.56792e9	1.42603	0.713014	+	0.701149i	$0.247329\pi$
$384$	1.15612e9	1.04194
$385$	0	0
$386$	1.44313e8	0.127718
$387$	−6.17132e7	−0.0541239
$388$	1.50552e8	0.130851
$389$	−1.59542e9	−1.37420	−0.687101	−	0.726562i	$-0.741117\pi$
$389$	−1.59542e9	−1.37420	−0.687101	+	0.726562i	$0.741117\pi$
$390$	0	0
$391$	−8.11590e7	−0.0686623
$392$	−7.46056e8	−0.625562
$393$	−8.17956e8	−0.679761
$394$	5.58399e8	0.459947
$395$	0	0
$396$	−3.95693e7	−0.0320203
$397$	−1.23380e8	−0.0989645	−0.0494822	−	0.998775i	$-0.515757\pi$
$397$	−1.23380e8	−0.0989645	−0.0494822	+	0.998775i	$0.515757\pi$
$398$	2.41684e8	0.192158
$399$	4.26681e7	0.0336278
$400$	0	0
$401$	−7.80784e8	−0.604680	−0.302340	−	0.953200i	$-0.597768\pi$
$401$	−7.80784e8	−0.604680	−0.302340	+	0.953200i	$0.597768\pi$
$402$	−5.79790e8	−0.445122
$403$	−5.35450e8	−0.407522
$404$	1.95997e9	1.47881
$405$	0	0
$406$	54657.4	4.05329e−5 0
$407$	6.22749e8	0.457860
$408$	1.46332e9	1.06667
$409$	1.78733e9	1.29174	0.645868	−	0.763449i	$-0.276495\pi$
$409$	1.78733e9	1.29174	0.645868	+	0.763449i	$0.276495\pi$
$410$	0	0
$411$	−1.37262e9	−0.975223
$412$	4.58738e8	0.323165
$413$	5.00934e7	0.0349909
$414$	2.43037e6	0.00168334
$415$	0	0
$416$	6.60913e8	0.450109
$417$	−2.49720e9	−1.68646
$418$	2.03117e8	0.136028
$419$	−1.51640e9	−1.00708	−0.503542	−	0.863971i	$-0.667970\pi$
$419$	−1.51640e9	−1.00708	−0.503542	+	0.863971i	$0.667970\pi$
$420$	0	0
$421$	2.24522e9	1.46646	0.733231	−	0.679980i	$-0.238011\pi$
$421$	2.24522e9	1.46646	0.733231	+	0.679980i	$0.238011\pi$
$422$	−1.33281e8	−0.0863326
$423$	−1.86019e8	−0.119500
$424$	−1.19870e9	−0.763713
$425$	0	0
$426$	−5.43464e8	−0.340594
$427$	6.84497e7	0.0425475
$428$	2.48253e9	1.53053
$429$	−2.75553e8	−0.168502
$430$	0	0
$431$	2.62807e9	1.58113	0.790564	−	0.612380i	$-0.209788\pi$
$431$	2.62807e9	1.58113	0.790564	+	0.612380i	$0.209788\pi$
$432$	−1.06662e9	−0.636526
$433$	−3.73140e8	−0.220884	−0.110442	−	0.993883i	$-0.535227\pi$
$433$	−3.73140e8	−0.220884	−0.110442	+	0.993883i	$0.535227\pi$
$434$	1.01474e7	0.00595854
$435$	0	0
$436$	3.06775e8	0.177262
$437$	1.01325e8	0.0580804
$438$	−1.98179e8	−0.112693
$439$	2.04384e9	1.15298	0.576488	−	0.817105i	$-0.304423\pi$
$439$	2.04384e9	1.15298	0.576488	+	0.817105i	$0.304423\pi$
$440$	0	0
$441$	−2.14708e8	−0.119210
$442$	5.11475e8	0.281739
$443$	3.44729e9	1.88393	0.941965	−	0.335712i	$-0.108977\pi$
$443$	3.44729e9	1.88393	0.941965	+	0.335712i	$0.108977\pi$
$444$	−2.63821e9	−1.43044
$445$	0	0
$446$	−5.82867e8	−0.311098
$447$	1.53327e9	0.811974
$448$	1.78029e7	0.00935445
$449$	1.53897e9	0.802357	0.401178	−	0.916000i	$-0.368601\pi$
$449$	1.53897e9	0.802357	0.401178	+	0.916000i	$0.368601\pi$
$450$	0	0
$451$	5.21295e8	0.267587
$452$	−6.82412e7	−0.0347586
$453$	−3.34336e8	−0.168982
$454$	5.21563e8	0.261584
$455$	0	0
$456$	−1.82692e9	−0.902281
$457$	1.48389e9	0.727268	0.363634	−	0.931542i	$-0.381536\pi$
$457$	1.48389e9	0.727268	0.363634	+	0.931542i	$0.381536\pi$
$458$	−9.47559e8	−0.460869
$459$	−3.10963e9	−1.50094
$460$	0	0
$461$	−2.44680e9	−1.16317	−0.581587	−	0.813484i	$-0.697568\pi$
$461$	−2.44680e9	−1.16317	−0.581587	+	0.813484i	$0.697568\pi$
$462$	5.22205e6	0.00246373
$463$	−1.45022e9	−0.679049	−0.339524	−	0.940597i	$-0.610266\pi$
$463$	−1.45022e9	−0.679049	−0.339524	+	0.940597i	$0.610266\pi$
$464$	7.71448e6	0.00358504
$465$	0	0
$466$	3.75861e8	0.172059
$467$	−1.48083e9	−0.672814	−0.336407	−	0.941717i	$-0.609212\pi$
$467$	−1.48083e9	−0.672814	−0.336407	+	0.941717i	$0.609212\pi$
$468$	1.24398e8	0.0560989
$469$	−6.62246e7	−0.0296425
$470$	0	0
$471$	−9.39984e8	−0.414522
$472$	−2.14485e9	−0.938856
$473$	−3.14889e8	−0.136818
$474$	−8.17255e8	−0.352479
$475$	0	0
$476$	7.87253e7	0.0334572
$477$	−3.44976e8	−0.145537
$478$	−2.82089e8	−0.118138
$479$	−3.76893e9	−1.56691	−0.783455	−	0.621449i	$-0.786544\pi$
$479$	−3.76893e9	−1.56691	−0.783455	+	0.621449i	$0.786544\pi$
$480$	0	0
$481$	−1.95781e9	−0.802162
$482$	−4.71251e8	−0.191685
$483$	2.60501e6	0.00105195
$484$	−2.01901e8	−0.0809430
$485$	0	0
$486$	1.98852e8	0.0785783
$487$	−9.84963e8	−0.386428	−0.193214	−	0.981157i	$-0.561891\pi$
$487$	−9.84963e8	−0.386428	−0.193214	+	0.981157i	$0.561891\pi$
$488$	−2.93081e9	−1.14161
$489$	3.45138e9	1.33478
$490$	0	0
$491$	−3.56853e8	−0.136052	−0.0680259	−	0.997684i	$-0.521670\pi$
$491$	−3.56853e8	−0.136052	−0.0680259	+	0.997684i	$0.521670\pi$
$492$	−2.20841e9	−0.835993
$493$	2.24908e7	0.00845360
$494$	−6.38562e8	−0.238319
$495$	0	0
$496$	1.43223e9	0.527019
$497$	−6.20754e7	−0.0226815
$498$	−4.62879e8	−0.167944
$499$	−2.25684e9	−0.813111	−0.406555	−	0.913626i	$-0.633270\pi$
$499$	−2.25684e9	−0.813111	−0.406555	+	0.913626i	$0.633270\pi$
$500$	0	0
$501$	1.19847e9	0.425789
$502$	6.16312e8	0.217439
$503$	−1.74836e9	−0.612553	−0.306277	−	0.951943i	$-0.599083\pi$
$503$	−1.74836e9	−0.612553	−0.306277	+	0.951943i	$0.599083\pi$
$504$	−5.00525e6	−0.00174148
$505$	0	0
$506$	1.24009e7	0.00425525
$507$	−2.23825e9	−0.762746
$508$	4.09299e9	1.38522
$509$	−4.30470e9	−1.44687	−0.723437	−	0.690391i	$-0.757439\pi$
$509$	−4.30470e9	−1.44687	−0.723437	+	0.690391i	$0.757439\pi$
$510$	0	0
$511$	−2.26364e7	−0.00750471
$512$	−3.06637e9	−1.00967
$513$	3.88228e9	1.26963
$514$	−4.13687e8	−0.134370
$515$	0	0
$516$	1.33399e9	0.427445
$517$	−9.49155e8	−0.302079
$518$	3.71027e7	0.0117287
$519$	−2.97071e9	−0.932771
$520$	0	0
$521$	3.38119e9	1.04746	0.523730	−	0.851885i	$-0.324540\pi$
$521$	3.38119e9	1.04746	0.523730	+	0.851885i	$0.324540\pi$
$522$	−673507.	−0.000207250 0
$523$	−4.20231e9	−1.28449	−0.642247	−	0.766497i	$-0.721998\pi$
$523$	−4.20231e9	−1.28449	−0.642247	+	0.766497i	$0.721998\pi$
$524$	1.88417e9	0.572084
$525$	0	0
$526$	−1.49292e9	−0.447287
$527$	4.17552e9	1.24272
$528$	7.37053e8	0.217911
$529$	−3.39864e9	−0.998183
$530$	0	0
$531$	−6.17268e8	−0.178913
$532$	−9.82862e7	−0.0283010
$533$	−1.63885e9	−0.468808
$534$	9.59721e8	0.272741
$535$	0	0
$536$	2.83554e9	0.795350
$537$	9.05124e7	0.0252231
$538$	1.12098e9	0.310356
$539$	−1.09554e9	−0.301347
$540$	0	0
$541$	−6.90757e9	−1.87558	−0.937788	−	0.347208i	$-0.887130\pi$
$541$	−6.90757e9	−1.87558	−0.937788	+	0.347208i	$0.887130\pi$
$542$	5.13789e7	0.0138608
$543$	−6.15853e9	−1.65074
$544$	−5.15390e9	−1.37259
$545$	0	0
$546$	−1.64171e7	−0.00431641
$547$	1.87201e9	0.489049	0.244525	−	0.969643i	$-0.421368\pi$
$547$	1.87201e9	0.489049	0.244525	+	0.969643i	$0.421368\pi$
$548$	3.16184e9	0.820743
$549$	−8.43460e8	−0.217551
$550$	0	0
$551$	−2.80792e7	−0.00715078
$552$	−1.11538e8	−0.0282252
$553$	−9.33484e7	−0.0234730
$554$	−2.52406e9	−0.630689
$555$	0	0
$556$	5.75231e9	1.41932
$557$	−2.57944e9	−0.632458	−0.316229	−	0.948683i	$-0.602417\pi$
$557$	−2.57944e9	−0.632458	−0.316229	+	0.948683i	$0.602417\pi$
$558$	−1.25039e8	−0.0304668
$559$	9.89950e8	0.239702
$560$	0	0
$561$	2.14881e9	0.513839
$562$	−2.12125e9	−0.504097
$563$	2.44826e7	0.00578199	0.00289100	−	0.999996i	$-0.499080\pi$
$563$	2.44826e7	0.00578199	0.00289100	+	0.999996i	$0.499080\pi$
$564$	4.02100e9	0.943750
$565$	0	0
$566$	2.35921e9	0.546902
$567$	1.11888e8	0.0257777
$568$	2.65788e9	0.608578
$569$	6.24602e9	1.42138	0.710690	−	0.703505i	$-0.248383\pi$
$569$	6.24602e9	1.42138	0.710690	+	0.703505i	$0.248383\pi$
$570$	0	0
$571$	2.67106e9	0.600423	0.300211	−	0.953873i	$-0.402943\pi$
$571$	2.67106e9	0.600423	0.300211	+	0.953873i	$0.402943\pi$
$572$	6.34738e8	0.141810
$573$	−3.84002e9	−0.852693
$574$	3.10581e7	0.00685463
$575$	0	0
$576$	−2.19374e8	−0.0478306
$577$	−2.92211e9	−0.633258	−0.316629	−	0.948549i	$-0.602551\pi$
$577$	−2.92211e9	−0.633258	−0.316629	+	0.948549i	$0.602551\pi$
$578$	−2.45145e9	−0.528051
$579$	−1.90606e9	−0.408094
$580$	0	0
$581$	−5.28709e7	−0.0111841
$582$	2.44829e8	0.0514792
$583$	−1.76022e9	−0.367898
$584$	9.69220e8	0.201362
$585$	0	0
$586$	−4.63863e6	−0.000952245 0
$587$	−8.18950e9	−1.67118	−0.835592	−	0.549351i	$-0.814875\pi$
$587$	−8.18950e9	−1.67118	−0.835592	+	0.549351i	$0.814875\pi$
$588$	4.64114e9	0.941465
$589$	−5.21302e9	−1.05120
$590$	0	0
$591$	−7.37520e9	−1.46966
$592$	5.23676e9	1.03738
$593$	9.08514e9	1.78912	0.894562	−	0.446944i	$-0.147488\pi$
$593$	9.08514e9	1.78912	0.894562	+	0.446944i	$0.147488\pi$
$594$	4.75142e8	0.0930189
$595$	0	0
$596$	−3.53189e9	−0.683354
$597$	−3.19211e9	−0.613998
$598$	−3.89860e7	−0.00745511
$599$	3.46726e9	0.659162	0.329581	−	0.944127i	$-0.393093\pi$
$599$	3.46726e9	0.659162	0.329581	+	0.944127i	$0.393093\pi$
$600$	0	0
$601$	−2.56544e9	−0.482060	−0.241030	−	0.970518i	$-0.577485\pi$
$601$	−2.56544e9	−0.482060	−0.241030	+	0.970518i	$0.577485\pi$
$602$	−1.87607e7	−0.00350478
$603$	8.16042e8	0.151566
$604$	7.70145e8	0.142214
$605$	0	0
$606$	3.18730e9	0.581794
$607$	−1.86107e9	−0.337755	−0.168877	−	0.985637i	$-0.554014\pi$
$607$	−1.86107e9	−0.337755	−0.168877	+	0.985637i	$0.554014\pi$
$608$	6.43449e9	1.16105
$609$	−721902.	−0.000129514 0
$610$	0	0
$611$	2.98396e9	0.529236
$612$	−9.70079e8	−0.171071
$613$	2.35285e9	0.412556	0.206278	−	0.978493i	$-0.433865\pi$
$613$	2.35285e9	0.412556	0.206278	+	0.978493i	$0.433865\pi$
$614$	−1.82968e9	−0.318997
$615$	0	0
$616$	−2.55391e7	−0.00440223
$617$	5.81579e9	0.996806	0.498403	−	0.866946i	$-0.333920\pi$
$617$	5.81579e9	0.996806	0.498403	+	0.866946i	$0.333920\pi$
$618$	7.46002e8	0.127139
$619$	3.71473e9	0.629520	0.314760	−	0.949171i	$-0.398076\pi$
$619$	3.71473e9	0.629520	0.314760	+	0.949171i	$0.398076\pi$
$620$	0	0
$621$	2.37024e8	0.0397165
$622$	1.80227e9	0.300299
$623$	1.09621e8	0.0181629
$624$	−2.31715e9	−0.381776
$625$	0	0
$626$	1.93908e8	0.0315926
$627$	−2.68272e9	−0.434650
$628$	2.16526e9	0.348860
$629$	1.52673e10	2.44616
$630$	0	0
$631$	−6.61199e9	−1.04768	−0.523841	−	0.851816i	$-0.675501\pi$
$631$	−6.61199e9	−1.04768	−0.523841	+	0.851816i	$0.675501\pi$
$632$	3.99689e9	0.629814
$633$	1.76034e9	0.275857
$634$	−1.76589e9	−0.275202
$635$	0	0
$636$	7.45700e9	1.14938
$637$	3.44417e9	0.527954
$638$	−3.43654e6	−0.000523901 0
$639$	7.64914e8	0.115974
$640$	0	0
$641$	−1.31108e10	−1.96619	−0.983093	−	0.183106i	$-0.941385\pi$
$641$	−1.31108e10	−1.96619	−0.983093	+	0.183106i	$0.941385\pi$
$642$	4.03709e9	0.602138
$643$	−8.73321e9	−1.29549	−0.647747	−	0.761856i	$-0.724289\pi$
$643$	−8.73321e9	−1.29549	−0.647747	+	0.761856i	$0.724289\pi$
$644$	−6.00065e6	−0.000885314 0
$645$	0	0
$646$	4.97961e9	0.726744
$647$	4.05488e9	0.588590	0.294295	−	0.955715i	$-0.404915\pi$
$647$	4.05488e9	0.588590	0.294295	+	0.955715i	$0.404915\pi$
$648$	−4.79071e9	−0.691652
$649$	−3.14958e9	−0.452268
$650$	0	0
$651$	−1.34024e8	−0.0190392
$652$	−7.95026e9	−1.12335
$653$	−2.96057e9	−0.416082	−0.208041	−	0.978120i	$-0.566709\pi$
$653$	−2.96057e9	−0.416082	−0.208041	+	0.978120i	$0.566709\pi$
$654$	4.98878e8	0.0697384
$655$	0	0
$656$	4.38362e9	0.606275
$657$	2.78933e8	0.0383726
$658$	−5.65495e7	−0.00773817
$659$	1.25563e10	1.70908	0.854540	−	0.519386i	$-0.173839\pi$
$659$	1.25563e10	1.70908	0.854540	+	0.519386i	$0.173839\pi$
$660$	0	0
$661$	−8.54786e9	−1.15120	−0.575602	−	0.817730i	$-0.695232\pi$
$661$	−8.54786e9	−1.15120	−0.575602	+	0.817730i	$0.695232\pi$
$662$	−2.23735e9	−0.299731
$663$	−6.75544e9	−0.900236
$664$	2.26377e9	0.300085
$665$	0	0
$666$	−4.57191e8	−0.0599706
$667$	−1.71431e6	−0.000223691 0
$668$	−2.76068e9	−0.358342
$669$	7.69836e9	0.994047
$670$	0	0
$671$	−4.30372e9	−0.549939
$672$	1.65428e8	0.0210289
$673$	7.90222e9	0.999301	0.499650	−	0.866227i	$-0.333462\pi$
$673$	7.90222e9	0.999301	0.499650	+	0.866227i	$0.333462\pi$
$674$	−3.04154e9	−0.382635
$675$	0	0
$676$	5.15581e9	0.641924
$677$	8.15033e9	1.00952	0.504760	−	0.863260i	$-0.331581\pi$
$677$	8.15033e9	1.00952	0.504760	+	0.863260i	$0.331581\pi$
$678$	−1.10974e8	−0.0136747
$679$	2.79648e7	0.00342821
$680$	0	0
$681$	−6.88867e9	−0.835835
$682$	−6.38008e8	−0.0770160
$683$	6.93643e8	0.0833036	0.0416518	−	0.999132i	$-0.486738\pi$
$683$	6.93643e8	0.0833036	0.0416518	+	0.999132i	$0.486738\pi$
$684$	1.21112e9	0.144707
$685$	0	0
$686$	−1.30577e8	−0.0154431
$687$	1.25151e10	1.47261
$688$	−2.64793e9	−0.309990
$689$	5.53380e9	0.644550
$690$	0	0
$691$	3.17516e7	0.00366094	0.00183047	−	0.999998i	$-0.499417\pi$
$691$	3.17516e7	0.00366094	0.00183047	+	0.999998i	$0.499417\pi$
$692$	6.84305e9	0.785016
$693$	−7.34992e6	−0.000838912 0
$694$	−1.72833e9	−0.196277
$695$	0	0
$696$	3.09096e7	0.00347505
$697$	1.27800e10	1.42961
$698$	3.37162e9	0.375271
$699$	−4.96428e9	−0.549777
$700$	0	0
$701$	7.68876e9	0.843030	0.421515	−	0.906821i	$-0.361498\pi$
$701$	7.68876e9	0.843030	0.421515	+	0.906821i	$0.361498\pi$
$702$	−1.49376e9	−0.162967
$703$	−1.90608e10	−2.06917
$704$	−1.11934e9	−0.120909
$705$	0	0
$706$	2.89881e9	0.310030
$707$	3.64059e8	0.0387440
$708$	1.33429e10	1.41297
$709$	−4.65053e9	−0.490051	−0.245025	−	0.969517i	$-0.578796\pi$
$709$	−4.65053e9	−0.490051	−0.245025	+	0.969517i	$0.578796\pi$
$710$	0	0
$711$	1.15027e9	0.120021
$712$	−4.69364e9	−0.487337
$713$	−3.18269e8	−0.0328837
$714$	1.28023e8	0.0131627
$715$	0	0
$716$	−2.08496e8	−0.0212276
$717$	3.72576e9	0.377483
$718$	−6.43662e9	−0.648966
$719$	−9.19022e9	−0.922093	−0.461047	−	0.887376i	$-0.652526\pi$
$719$	−9.19022e9	−0.922093	−0.461047	+	0.887376i	$0.652526\pi$
$720$	0	0
$721$	8.52097e7	0.00846672
$722$	−2.86848e9	−0.283643
$723$	6.22417e9	0.612488
$724$	1.41862e10	1.38925
$725$	0	0
$726$	−3.28332e8	−0.0318445
$727$	−9.36688e9	−0.904117	−0.452059	−	0.891988i	$-0.649310\pi$
$727$	−9.36688e9	−0.904117	−0.452059	+	0.891988i	$0.649310\pi$
$728$	8.02900e7	0.00771262
$729$	8.93281e9	0.853969
$730$	0	0
$731$	−7.71979e9	−0.730962
$732$	1.82323e10	1.71811
$733$	−1.29034e10	−1.21015	−0.605076	−	0.796168i	$-0.706857\pi$
$733$	−1.29034e10	−1.21015	−0.605076	+	0.796168i	$0.706857\pi$
$734$	−1.12451e9	−0.104960
$735$	0	0
$736$	3.92844e8	0.0363201
$737$	4.16382e9	0.383138
$738$	−3.82709e8	−0.0350486
$739$	6.13118e9	0.558841	0.279421	−	0.960169i	$-0.409858\pi$
$739$	6.13118e9	0.558841	0.279421	+	0.960169i	$0.409858\pi$
$740$	0	0
$741$	8.43397e9	0.761497
$742$	−1.04872e8	−0.00942422
$743$	−9.12544e9	−0.816193	−0.408096	−	0.912939i	$-0.633807\pi$
$743$	−9.12544e9	−0.816193	−0.408096	+	0.912939i	$0.633807\pi$
$744$	5.73850e9	0.510850
$745$	0	0
$746$	−3.51984e8	−0.0310411
$747$	6.51492e8	0.0571857
$748$	−4.94979e9	−0.432445
$749$	4.61124e8	0.0400988
$750$	0	0
$751$	−1.80702e9	−0.155676	−0.0778381	−	0.996966i	$-0.524802\pi$
$751$	−1.80702e9	−0.155676	−0.0778381	+	0.996966i	$0.524802\pi$
$752$	−7.98154e9	−0.684423
$753$	−8.14010e9	−0.694780
$754$	1.08038e7	0.000917863 0
$755$	0	0
$756$	−2.29916e8	−0.0193528
$757$	−4.78438e9	−0.400858	−0.200429	−	0.979708i	$-0.564234\pi$
$757$	−4.78438e9	−0.400858	−0.200429	+	0.979708i	$0.564234\pi$
$758$	2.94176e9	0.245338
$759$	−1.63788e8	−0.0135967
$760$	0	0
$761$	8.11100e9	0.667157	0.333578	−	0.942722i	$-0.391744\pi$
$761$	8.11100e9	0.667157	0.333578	+	0.942722i	$0.391744\pi$
$762$	6.65604e9	0.544971
$763$	5.69828e7	0.00464416
$764$	8.84551e9	0.717623
$765$	0	0
$766$	−5.87338e9	−0.472158
$767$	9.90168e9	0.792365
$768$	9.95050e8	0.0792649
$769$	−4.03304e9	−0.319809	−0.159904	−	0.987133i	$-0.551119\pi$
$769$	−4.03304e9	−0.319809	−0.159904	+	0.987133i	$0.551119\pi$
$770$	0	0
$771$	5.46388e9	0.429349
$772$	4.39061e9	0.343451
$773$	−3.46346e9	−0.269701	−0.134850	−	0.990866i	$-0.543055\pi$
$773$	−3.46346e9	−0.269701	−0.134850	+	0.990866i	$0.543055\pi$
$774$	2.31175e8	0.0179204
$775$	0	0
$776$	−1.19737e9	−0.0919838
$777$	−4.90043e8	−0.0374766
$778$	5.97638e9	0.454998
$779$	−1.59555e10	−1.20929
$780$	0	0
$781$	3.90294e9	0.293166
$782$	3.04019e8	0.0227341
$783$	−6.56842e7	−0.00488984
$784$	−9.21250e9	−0.682765
$785$	0	0
$786$	3.06404e9	0.225069
$787$	7.81853e9	0.571760	0.285880	−	0.958265i	$-0.407714\pi$
$787$	7.81853e9	0.571760	0.285880	+	0.958265i	$0.407714\pi$
$788$	1.69888e10	1.23686
$789$	1.97181e10	1.42921
$790$	0	0
$791$	−1.26757e7	−0.000910652 0
$792$	3.14701e8	0.0225092
$793$	1.35301e10	0.963482
$794$	4.62179e8	0.0327671
$795$	0	0
$796$	7.35303e9	0.516738
$797$	−1.19003e10	−0.832635	−0.416318	−	0.909219i	$-0.636680\pi$
$797$	−1.19003e10	−0.832635	−0.416318	+	0.909219i	$0.636680\pi$
$798$	−1.59833e8	−0.0111342
$799$	−2.32694e10	−1.61388
$800$	0	0
$801$	−1.35079e9	−0.0928696
$802$	2.92479e9	0.200209
$803$	1.42324e9	0.0970007
$804$	−1.76396e10	−1.19699
$805$	0	0
$806$	2.00578e9	0.134930
$807$	−1.48057e10	−0.991677
$808$	−1.55879e10	−1.03956
$809$	1.04001e10	0.690584	0.345292	−	0.938495i	$-0.387780\pi$
$809$	1.04001e10	0.690584	0.345292	+	0.938495i	$0.387780\pi$
$810$	0	0
$811$	−1.00469e10	−0.661392	−0.330696	−	0.943737i	$-0.607283\pi$
$811$	−1.00469e10	−0.661392	−0.330696	+	0.943737i	$0.607283\pi$
$812$	1.66290e6	0.000108999 0
$813$	−6.78600e8	−0.0442891
$814$	−2.33280e9	−0.151597
$815$	0	0
$816$	1.80695e10	1.16421
$817$	9.63793e9	0.618310
$818$	−6.69529e9	−0.427694
$819$	2.31068e7	0.00146976
$820$	0	0
$821$	2.01156e10	1.26862	0.634309	−	0.773079i	$-0.281285\pi$
$821$	2.01156e10	1.26862	0.634309	+	0.773079i	$0.281285\pi$
$822$	5.14179e9	0.322896
$823$	4.01271e9	0.250922	0.125461	−	0.992099i	$-0.459959\pi$
$823$	4.01271e9	0.250922	0.125461	+	0.992099i	$0.459959\pi$
$824$	−3.64842e9	−0.227174
$825$	0	0
$826$	−1.87648e8	−0.0115855
$827$	1.18312e10	0.727378	0.363689	−	0.931520i	$-0.381517\pi$
$827$	1.18312e10	0.727378	0.363689	+	0.931520i	$0.381517\pi$
$828$	7.39420e7	0.00452673
$829$	7.67662e9	0.467982	0.233991	−	0.972239i	$-0.424821\pi$
$829$	7.67662e9	0.467982	0.233991	+	0.972239i	$0.424821\pi$
$830$	0	0
$831$	3.33371e10	2.01523
$832$	3.51901e9	0.211830
$833$	−2.68582e10	−1.60997
$834$	9.35443e9	0.558388
$835$	0	0
$836$	6.17966e9	0.365799
$837$	−1.21946e10	−0.718832
$838$	5.68039e9	0.333445
$839$	−9.41603e9	−0.550429	−0.275214	−	0.961383i	$-0.588749\pi$
$839$	−9.41603e9	−0.550429	−0.275214	+	0.961383i	$0.588749\pi$
$840$	0	0
$841$	−1.72494e10	−0.999972
$842$	−8.41050e9	−0.485545
$843$	2.80169e10	1.61073
$844$	−4.05496e9	−0.232160
$845$	0	0
$846$	6.96822e8	0.0395663
$847$	−3.75027e7	−0.00212066
$848$	−1.48019e10	−0.833550
$849$	−3.11599e10	−1.74751
$850$	0	0
$851$	−1.16371e9	−0.0647280
$852$	−1.65344e10	−0.915904
$853$	1.58189e9	0.0872680	0.0436340	−	0.999048i	$-0.486106\pi$
$853$	1.58189e9	0.0872680	0.0436340	+	0.999048i	$0.486106\pi$
$854$	−2.56410e8	−0.0140875
$855$	0	0
$856$	−1.97439e10	−1.07591
$857$	1.73709e10	0.942734	0.471367	−	0.881937i	$-0.343761\pi$
$857$	1.73709e10	0.942734	0.471367	+	0.881937i	$0.343761\pi$
$858$	1.03221e9	0.0557909
$859$	8.52987e9	0.459163	0.229581	−	0.973289i	$-0.426264\pi$
$859$	8.52987e9	0.459163	0.229581	+	0.973289i	$0.426264\pi$
$860$	0	0
$861$	−4.10208e8	−0.0219025
$862$	−9.84467e9	−0.523511
$863$	2.50924e10	1.32894	0.664470	−	0.747315i	$-0.268657\pi$
$863$	2.50924e10	1.32894	0.664470	+	0.747315i	$0.268657\pi$
$864$	1.50519e10	0.793950
$865$	0	0
$866$	1.39777e9	0.0731346
$867$	3.23782e10	1.68727
$868$	3.08725e8	0.0160233
$869$	5.86920e9	0.303396
$870$	0	0
$871$	−1.30903e10	−0.671250
$872$	−2.43983e9	−0.124610
$873$	−3.44592e8	−0.0175289
$874$	−3.79559e8	−0.0192304
$875$	0	0
$876$	−6.02943e9	−0.303048
$877$	−1.68456e10	−0.843314	−0.421657	−	0.906756i	$-0.638551\pi$
$877$	−1.68456e10	−0.843314	−0.421657	+	0.906756i	$0.638551\pi$
$878$	−7.65615e9	−0.381750
$879$	6.12659e7	0.00304269
$880$	0	0
$881$	−1.17288e10	−0.577881	−0.288941	−	0.957347i	$-0.593303\pi$
$881$	−1.17288e10	−0.577881	−0.288941	+	0.957347i	$0.593303\pi$
$882$	8.04290e8	0.0394705
$883$	−3.11743e10	−1.52382	−0.761911	−	0.647682i	$-0.775739\pi$
$883$	−3.11743e10	−1.52382	−0.761911	+	0.647682i	$0.775739\pi$
$884$	1.55612e10	0.757635
$885$	0	0
$886$	−1.29134e10	−0.623769
$887$	−2.88046e10	−1.38589	−0.692944	−	0.720991i	$-0.743687\pi$
$887$	−2.88046e10	−1.38589	−0.692944	+	0.720991i	$0.743687\pi$
$888$	2.09821e10	1.00555
$889$	7.60265e8	0.0362918
$890$	0	0
$891$	−7.03488e9	−0.333185
$892$	−1.77332e10	−0.836586
$893$	2.90512e10	1.36516
$894$	−5.74358e9	−0.268845
$895$	0	0
$896$	−4.94671e8	−0.0229741
$897$	5.14917e8	0.0238212
$898$	−5.76492e9	−0.265660
$899$	8.81990e7	0.00404860
$900$	0	0
$901$	−4.31535e10	−1.96553
$902$	−1.95275e9	−0.0885982
$903$	2.47787e8	0.0111988
$904$	5.42733e8	0.0244341
$905$	0	0
$906$	1.25241e9	0.0559498
$907$	1.47721e10	0.657381	0.328690	−	0.944438i	$-0.393393\pi$
$907$	1.47721e10	0.657381	0.328690	+	0.944438i	$0.393393\pi$
$908$	1.58681e10	0.703435
$909$	−4.48606e9	−0.198103
$910$	0	0
$911$	1.23248e10	0.540089	0.270044	−	0.962848i	$-0.412962\pi$
$911$	1.23248e10	0.540089	0.270044	+	0.962848i	$0.412962\pi$
$912$	−2.25593e10	−0.984789
$913$	3.32421e9	0.144558
$914$	−5.55859e9	−0.240798
$915$	0	0
$916$	−2.88287e10	−1.23934
$917$	3.49980e8	0.0149882
$918$	1.16486e10	0.496962
$919$	2.59401e10	1.10247	0.551235	−	0.834350i	$-0.314157\pi$
$919$	2.59401e10	1.10247	0.551235	+	0.834350i	$0.314157\pi$
$920$	0	0
$921$	2.41660e10	1.01929
$922$	9.16562e9	0.385127
$923$	−1.22701e10	−0.513621
$924$	1.58876e8	0.00662532
$925$	0	0
$926$	5.43248e9	0.224833
$927$	−1.04998e9	−0.0432915
$928$	−1.08865e8	−0.00447168
$929$	2.78900e10	1.14128	0.570642	−	0.821199i	$-0.306694\pi$
$929$	2.78900e10	1.14128	0.570642	+	0.821199i	$0.306694\pi$
$930$	0	0
$931$	3.35316e10	1.36185
$932$	1.14352e10	0.462690
$933$	−2.38039e10	−0.959540
$934$	5.54713e9	0.222769
$935$	0	0
$936$	−9.89360e8	−0.0394357
$937$	−9.94837e9	−0.395060	−0.197530	−	0.980297i	$-0.563292\pi$
$937$	−9.94837e9	−0.395060	−0.197530	+	0.980297i	$0.563292\pi$
$938$	2.48075e8	0.00981462
$939$	−2.56109e9	−0.100947
$940$	0	0
$941$	−2.18049e10	−0.853082	−0.426541	−	0.904468i	$-0.640268\pi$
$941$	−2.18049e10	−0.853082	−0.426541	+	0.904468i	$0.640268\pi$
$942$	3.52115e9	0.137248
$943$	−9.74127e8	−0.0378290
$944$	−2.64851e10	−1.02471
$945$	0	0
$946$	1.17956e9	0.0453004
$947$	−1.74659e10	−0.668290	−0.334145	−	0.942522i	$-0.608448\pi$
$947$	−1.74659e10	−0.668290	−0.334145	+	0.942522i	$0.608448\pi$
$948$	−2.48643e10	−0.947865
$949$	−4.47441e9	−0.169943
$950$	0	0
$951$	2.33235e10	0.879350
$952$	−6.26115e8	−0.0235193
$953$	4.46489e10	1.67104	0.835518	−	0.549464i	$-0.185168\pi$
$953$	4.46489e10	1.67104	0.835518	+	0.549464i	$0.185168\pi$
$954$	1.29227e9	0.0481873
$955$	0	0
$956$	−8.58230e9	−0.317688
$957$	4.53890e7	0.00167401
$958$	1.41183e10	0.518804
$959$	5.87304e8	0.0215030
$960$	0	0
$961$	−1.11381e10	−0.404836
$962$	7.33388e9	0.265596
$963$	−5.68213e9	−0.205031
$964$	−1.43374e10	−0.515467
$965$	0	0
$966$	−9.75827e6	−0.000348300 0
$967$	−2.46485e10	−0.876594	−0.438297	−	0.898830i	$-0.644418\pi$
$967$	−2.46485e10	−0.876594	−0.438297	+	0.898830i	$0.644418\pi$
$968$	1.60575e9	0.0569002
$969$	−6.57694e10	−2.32215
$970$	0	0
$971$	−1.77832e10	−0.623365	−0.311682	−	0.950186i	$-0.600892\pi$
$971$	−1.77832e10	−0.623365	−0.311682	+	0.950186i	$0.600892\pi$
$972$	6.04989e9	0.211308
$973$	1.06848e9	0.0371853
$974$	3.68964e9	0.127946
$975$	0	0
$976$	−3.61904e10	−1.24600
$977$	−4.72834e10	−1.62210	−0.811051	−	0.584976i	$-0.801104\pi$
$977$	−4.72834e10	−1.62210	−0.811051	+	0.584976i	$0.801104\pi$
$978$	−1.29287e10	−0.441947
$979$	−6.89234e9	−0.234762
$980$	0	0
$981$	−7.02161e8	−0.0237462
$982$	1.33676e9	0.0450468
$983$	4.34353e10	1.45850	0.729249	−	0.684249i	$-0.239870\pi$
$983$	4.34353e10	1.45850	0.729249	+	0.684249i	$0.239870\pi$
$984$	1.75639e10	0.587675
$985$	0	0
$986$	−8.42500e7	−0.00279898
$987$	7.46892e8	0.0247257
$988$	−1.94277e10	−0.640873
$989$	5.88422e8	0.0193420
$990$	0	0
$991$	3.94776e9	0.128853	0.0644263	−	0.997922i	$-0.479478\pi$
$991$	3.94776e9	0.128853	0.0644263	+	0.997922i	$0.479478\pi$
$992$	−2.02113e10	−0.657360
$993$	2.95504e10	0.957725
$994$	2.32533e8	0.00750985
$995$	0	0
$996$	−1.40827e10	−0.451625
$997$	−1.90391e10	−0.608434	−0.304217	−	0.952603i	$-0.598395\pi$
$997$	−1.90391e10	−0.608434	−0.304217	+	0.952603i	$0.598395\pi$
$998$	8.45406e9	0.269221
$999$	−4.45879e10	−1.41494

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.8.a.a.1.1		2
5.4	even	2		11.8.a.a.1.2	✓	2
15.14	odd	2		99.8.a.c.1.1		2
20.19	odd	2		176.8.a.d.1.2		2
35.34	odd	2		539.8.a.a.1.2		2
55.54	odd	2		121.8.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.8.a.a.1.2	✓	2	5.4	even	2
99.8.a.c.1.1		2	15.14	odd	2
121.8.a.b.1.1		2	55.54	odd	2
176.8.a.d.1.2		2	20.19	odd	2
275.8.a.a.1.1		2	1.1	even	1	trivial
539.8.a.a.1.2		2	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q-3.74597 q^{2} +49.4758 q^{3} -113.968 q^{4} -185.335 q^{6} -21.1693 q^{7} +906.403 q^{8} +260.855 q^{9} +1331.00 q^{11} -5638.64 q^{12} -4184.41 q^{13} +79.2994 q^{14} +11192.5 q^{16} +32630.7 q^{17} -977.153 q^{18} -40738.5 q^{19} -1047.37 q^{21} -4985.88 q^{22} -2487.20 q^{23} +44845.0 q^{24} +15674.7 q^{26} -95297.6 q^{27} +2412.61 q^{28} +689.254 q^{29} +127963. q^{31} -157946. q^{32} +65852.3 q^{33} -122234. q^{34} -29729.0 q^{36} +467881. q^{37} +152605. q^{38} -207027. q^{39} +391657. q^{41} +3923.40 q^{42} -236581. q^{43} -151691. q^{44} +9316.96 q^{46} -713115. q^{47} +553759. q^{48} -823095. q^{49} +1.61443e6 q^{51} +476888. q^{52} -1.32248e6 q^{53} +356982. q^{54} -19187.9 q^{56} -2.01557e6 q^{57} -2581.92 q^{58} -2.36633e6 q^{59} -3.23345e6 q^{61} -479345. q^{62} -5522.11 q^{63} -840980. q^{64} -246680. q^{66} +3.12834e6 q^{67} -3.71885e6 q^{68} -123056. q^{69} +2.93234e6 q^{71} +236440. q^{72} +1.06930e6 q^{73} -1.75267e6 q^{74} +4.64287e6 q^{76} -28176.3 q^{77} +775517. q^{78} +4.40962e6 q^{79} -5.28541e6 q^{81} -1.46713e6 q^{82} +2.49753e6 q^{83} +119366. q^{84} +886223. q^{86} +34101.4 q^{87} +1.20642e6 q^{88} -5.17831e6 q^{89} +88580.9 q^{91} +283460. q^{92} +6.33107e6 q^{93} +2.67130e6 q^{94} -7.81452e6 q^{96} -1.32101e6 q^{97} +3.08329e6 q^{98} +347198. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 6 q^{3} - 104 q^{4} - 696 q^{6} + 1228 q^{7} - 480 q^{8} - 36 q^{9} + 2662 q^{11} - 6072 q^{12} - 344 q^{13} + 14752 q^{14} - 6368 q^{16} + 8468 q^{17} - 4464 q^{18} - 35280 q^{19} - 55356 q^{21}+ \cdots - 47916 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 + 6 * q^3 - 104 * q^4 - 696 * q^6 + 1228 * q^7 - 480 * q^8 - 36 * q^9 + 2662 * q^11 - 6072 * q^12 - 344 * q^13 + 14752 * q^14 - 6368 * q^16 + 8468 * q^17 - 4464 * q^18 - 35280 * q^19 - 55356 * q^21 + 10648 * q^22 + 61486 * q^23 + 105120 * q^24 + 60784 * q^26 + 12690 * q^27 + 14864 * q^28 + 179040 * q^29 - 57166 * q^31 - 186752 * q^32 + 7986 * q^33 - 406048 * q^34 - 32688 * q^36 + 877698 * q^37 + 216720 * q^38 - 373992 * q^39 - 283616 * q^41 - 633984 * q^42 - 275484 * q^43 - 138424 * q^44 + 760744 * q^46 - 1662512 * q^47 + 1317216 * q^48 - 86214 * q^49 + 2664924 * q^51 + 515168 * q^52 - 1616484 * q^53 + 1625400 * q^54 - 1751040 * q^56 - 2252880 * q^57 + 2092320 * q^58 - 2454130 * q^59 - 6019176 * q^61 - 2653864 * q^62 - 376344 * q^63 + 1068416 * q^64 - 926376 * q^66 + 174698 * q^67 - 3959696 * q^68 - 2904342 * q^69 - 1151466 * q^71 + 648000 * q^72 - 885944 * q^73 + 3061032 * q^74 + 4697280 * q^76 + 1634468 * q^77 - 1185648 * q^78 + 3801460 * q^79 - 9331038 * q^81 - 9398864 * q^82 + 2282916 * q^83 - 421968 * q^84 + 429264 * q^86 - 7719840 * q^87 - 638880 * q^88 - 13481970 * q^89 + 4885904 * q^91 + 921128 * q^92 + 14379702 * q^93 - 8480288 * q^94 - 6562176 * q^96 + 68078 * q^97 + 11738664 * q^98 - 47916 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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