LMFDB - Embedded newform 550.6.b.h.199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,6,Mod(199,550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("550.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 550 = 2 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	550.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$88.2111008971$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{889})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 445x^{2} + 49284 $ x^4 + 445*x^2 + 49284
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$-14.4081i$ of defining polynomial
Character	$\chi$	$=$	550.199
Dual form			550.6.b.h.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000i q^{2} -4.40805i q^{3} -16.0000 q^{4} +17.6322 q^{6} -92.0403i q^{7} -64.0000i q^{8} +223.569 q^{9} +121.000 q^{11} +70.5288i q^{12} -64.2415i q^{13} +368.161 q^{14} +256.000 q^{16} +1394.03i q^{17} +894.276i q^{18} -813.224 q^{19} -405.718 q^{21} +484.000i q^{22} +2668.06i q^{23} -282.115 q^{24} +256.966 q^{26} -2056.66i q^{27} +1472.64i q^{28} -1880.55 q^{29} +3679.50 q^{31} +1024.00i q^{32} -533.374i q^{33} -5576.12 q^{34} -3577.11 q^{36} -15375.8i q^{37} -3252.90i q^{38} -283.180 q^{39} +2875.24 q^{41} -1622.87i q^{42} +20405.3i q^{43} -1936.00 q^{44} -10672.2 q^{46} -26141.6i q^{47} -1128.46i q^{48} +8335.59 q^{49} +6144.95 q^{51} +1027.86i q^{52} +491.587i q^{53} +8226.64 q^{54} -5890.58 q^{56} +3584.73i q^{57} -7522.18i q^{58} +30775.7 q^{59} -12688.8 q^{61} +14718.0i q^{62} -20577.4i q^{63} -4096.00 q^{64} +2133.50 q^{66} -3915.60i q^{67} -22304.5i q^{68} +11760.9 q^{69} -23656.8 q^{71} -14308.4i q^{72} +49810.0i q^{73} +61503.2 q^{74} +13011.6 q^{76} -11136.9i q^{77} -1132.72i q^{78} -5846.55 q^{79} +45261.4 q^{81} +11501.0i q^{82} -29846.2i q^{83} +6491.49 q^{84} -81621.4 q^{86} +8289.54i q^{87} -7744.00i q^{88} +110688. q^{89} -5912.81 q^{91} -42688.9i q^{92} -16219.4i q^{93} +104566. q^{94} +4513.84 q^{96} +65866.2i q^{97} +33342.4i q^{98} +27051.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{4} - 168 q^{6} - 358 q^{9} + 484 q^{11} + 280 q^{14} + 1024 q^{16} - 3074 q^{19} - 3710 q^{21} + 2688 q^{24} - 6128 q^{26} - 15334 q^{29} + 9530 q^{31} + 2264 q^{34} + 5728 q^{36} - 42756 q^{39}+ \cdots - 43318 q^{99}+O(q^{100}) $ 4 * q - 64 * q^4 - 168 * q^6 - 358 * q^9 + 484 * q^11 + 280 * q^14 + 1024 * q^16 - 3074 * q^19 - 3710 * q^21 + 2688 * q^24 - 6128 * q^26 - 15334 * q^29 + 9530 * q^31 + 2264 * q^34 + 5728 * q^36 - 42756 * q^39 + 20088 * q^41 - 7744 * q^44 + 6448 * q^46 + 43778 * q^49 + 97510 * q^51 + 48888 * q^54 - 4480 * q^56 + 30196 * q^59 + 8698 * q^61 - 16384 * q^64 - 20328 * q^66 + 200060 * q^69 - 174594 * q^71 + 228600 * q^74 + 49184 * q^76 + 96236 * q^79 + 100900 * q^81 + 59360 * q^84 - 156176 * q^86 + 317346 * q^89 - 106540 * q^91 + 181168 * q^94 - 43008 * q^96 - 43318 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/550\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 4.40805i	− 0.282777i	−0.989954	−	0.141388i	$-0.954843\pi$
$3$	− 4.40805i	− 0.282777i	0.989954	−	0.141388i	$-0.0451566\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	17.6322	0.199953
$7$	− 92.0403i	− 0.709958i	−0.934874	−	0.354979i	$-0.884488\pi$
$7$	− 92.0403i	− 0.709958i	0.934874	−	0.354979i	$-0.115512\pi$
$8$	− 64.0000i	− 0.353553i
$9$	223.569	0.920037
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	70.5288i	0.141388i
$13$	− 64.2415i	− 0.105428i	−0.998610	−	0.0527142i	$-0.983213\pi$
$13$	− 64.2415i	− 0.105428i	0.998610	−	0.0527142i	$-0.0167872\pi$
$14$	368.161	0.502016
$15$	0	0
$16$	256.000	0.250000
$17$	1394.03i	1.16990i	0.811069	+	0.584951i	$0.198886\pi$
$17$	1394.03i	1.16990i	−0.811069	+	0.584951i	$0.801114\pi$
$18$	894.276i	0.650565i
$19$	−813.224	−0.516804	−0.258402	−	0.966037i	$-0.583196\pi$
$19$	−813.224	−0.516804	−0.258402	+	0.966037i	$0.583196\pi$
$20$	0	0
$21$	−405.718	−0.200760
$22$	484.000i	0.213201i
$23$	2668.06i	1.05166i	0.850589	+	0.525831i	$0.176245\pi$
$23$	2668.06i	1.05166i	−0.850589	+	0.525831i	$0.823755\pi$
$24$	−282.115	−0.0999766
$25$	0	0
$26$	256.966	0.0745491
$27$	− 2056.66i	− 0.542942i
$28$	1472.64i	0.354979i
$29$	−1880.55	−0.415230	−0.207615	−	0.978211i	$-0.566570\pi$
$29$	−1880.55	−0.415230	−0.207615	+	0.978211i	$0.566570\pi$
$30$	0	0
$31$	3679.50	0.687677	0.343839	−	0.939029i	$-0.388273\pi$
$31$	3679.50	0.687677	0.343839	+	0.939029i	$0.388273\pi$
$32$	1024.00i	0.176777i
$33$	− 533.374i	− 0.0852604i
$34$	−5576.12	−0.827246
$35$	0	0
$36$	−3577.11	−0.460019
$37$	− 15375.8i	− 1.84643i	−0.384283	−	0.923215i	$-0.625551\pi$
$37$	− 15375.8i	− 1.84643i	0.384283	−	0.923215i	$-0.374449\pi$
$38$	− 3252.90i	− 0.365436i
$39$	−283.180	−0.0298127
$40$	0	0
$41$	2875.24	0.267125	0.133563	−	0.991040i	$-0.457358\pi$
$41$	2875.24	0.267125	0.133563	+	0.991040i	$0.457358\pi$
$42$	− 1622.87i	− 0.141958i
$43$	20405.3i	1.68296i	0.540291	+	0.841478i	$0.318314\pi$
$43$	20405.3i	1.68296i	−0.540291	+	0.841478i	$0.681686\pi$
$44$	−1936.00	−0.150756
$45$	0	0
$46$	−10672.2	−0.743637
$47$	− 26141.6i	− 1.72619i	−0.505045	−	0.863093i	$-0.668524\pi$
$47$	− 26141.6i	− 1.72619i	0.505045	−	0.863093i	$-0.331476\pi$
$48$	− 1128.46i	− 0.0706942i
$49$	8335.59	0.495959
$50$	0	0
$51$	6144.95	0.330821
$52$	1027.86i	0.0527142i
$53$	491.587i	0.0240387i	0.999928	+	0.0120193i	$0.00382597\pi$
$53$	491.587i	0.0240387i	−0.999928	+	0.0120193i	$0.996174\pi$
$54$	8226.64	0.383918
$55$	0	0
$56$	−5890.58	−0.251008
$57$	3584.73i	0.146140i
$58$	− 7522.18i	− 0.293612i
$59$	30775.7	1.15101	0.575504	−	0.817799i	$-0.304806\pi$
$59$	30775.7	1.15101	0.575504	+	0.817799i	$0.304806\pi$
$60$	0	0
$61$	−12688.8	−0.436613	−0.218307	−	0.975880i	$-0.570053\pi$
$61$	−12688.8	−0.436613	−0.218307	+	0.975880i	$0.570053\pi$
$62$	14718.0i	0.486261i
$63$	− 20577.4i	− 0.653188i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	2133.50	0.0602882
$67$	− 3915.60i	− 0.106564i	−0.998579	−	0.0532822i	$-0.983032\pi$
$67$	− 3915.60i	− 0.106564i	0.998579	−	0.0532822i	$-0.0169683\pi$
$68$	− 22304.5i	− 0.584951i
$69$	11760.9	0.297385
$70$	0	0
$71$	−23656.8	−0.556942	−0.278471	−	0.960445i	$-0.589828\pi$
$71$	−23656.8	−0.556942	−0.278471	+	0.960445i	$0.589828\pi$
$72$	− 14308.4i	− 0.325282i
$73$	49810.0i	1.09398i	0.837139	+	0.546990i	$0.184226\pi$
$73$	49810.0i	1.09398i	−0.837139	+	0.546990i	$0.815774\pi$
$74$	61503.2	1.30562
$75$	0	0
$76$	13011.6	0.258402
$77$	− 11136.9i	− 0.214060i
$78$	− 1132.72i	− 0.0210808i
$79$	−5846.55	−0.105398	−0.0526990	−	0.998610i	$-0.516782\pi$
$79$	−5846.55	−0.105398	−0.0526990	+	0.998610i	$0.516782\pi$
$80$	0	0
$81$	45261.4	0.766506
$82$	11501.0i	0.188886i
$83$	− 29846.2i	− 0.475547i	−0.971321	−	0.237774i	$-0.923582\pi$
$83$	− 29846.2i	− 0.475547i	0.971321	−	0.237774i	$-0.0764177\pi$
$84$	6491.49	0.100380
$85$	0	0
$86$	−81621.4	−1.19003
$87$	8289.54i	0.117417i
$88$	− 7744.00i	− 0.106600i
$89$	110688.	1.48124	0.740621	−	0.671923i	$-0.234531\pi$
$89$	110688.	1.48124	0.740621	+	0.671923i	$0.234531\pi$
$90$	0	0
$91$	−5912.81	−0.0748497
$92$	− 42688.9i	− 0.525831i
$93$	− 16219.4i	− 0.194459i
$94$	104566.	1.22060
$95$	0	0
$96$	4513.84	0.0499883
$97$	65866.2i	0.710777i	0.934719	+	0.355389i	$0.115651\pi$
$97$	65866.2i	0.710777i	−0.934719	+	0.355389i	$0.884349\pi$
$98$	33342.4i	0.350696i
$99$	27051.9	0.277402
$100$	0	0
$101$	170816.	1.66619	0.833097	−	0.553127i	$-0.186565\pi$
$101$	170816.	1.66619	0.833097	+	0.553127i	$0.186565\pi$
$102$	24579.8i	0.233926i
$103$	− 3350.44i	− 0.0311178i	−0.999879	−	0.0155589i	$-0.995047\pi$
$103$	− 3350.44i	− 0.0311178i	0.999879	−	0.0155589i	$-0.00495275\pi$
$104$	−4111.46	−0.0372746
$105$	0	0
$106$	−1966.35	−0.0169979
$107$	43688.4i	0.368898i	0.982842	+	0.184449i	$0.0590501\pi$
$107$	43688.4i	0.368898i	−0.982842	+	0.184449i	$0.940950\pi$
$108$	32906.6i	0.271471i
$109$	−49375.4	−0.398056	−0.199028	−	0.979994i	$-0.563778\pi$
$109$	−49375.4	−0.398056	−0.199028	+	0.979994i	$0.563778\pi$
$110$	0	0
$111$	−67777.3	−0.522128
$112$	− 23562.3i	− 0.177490i
$113$	113388.i	0.835356i	0.908595	+	0.417678i	$0.137156\pi$
$113$	113388.i	0.835356i	−0.908595	+	0.417678i	$0.862844\pi$
$114$	−14338.9	−0.103337
$115$	0	0
$116$	30088.7	0.207615
$117$	− 14362.4i	− 0.0969981i
$118$	123103.i	0.813886i
$119$	128307.	0.830582
$120$	0	0
$121$	14641.0	0.0909091
$122$	− 50755.3i	− 0.308732i
$123$	− 12674.2i	− 0.0755367i
$124$	−58872.0	−0.343839
$125$	0	0
$126$	82309.4	0.461874
$127$	− 315137.i	− 1.73377i	−0.498512	−	0.866883i	$-0.666120\pi$
$127$	− 315137.i	− 1.73377i	0.498512	−	0.866883i	$-0.333880\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	89947.8	0.475901
$130$	0	0
$131$	263284.	1.34044	0.670218	−	0.742165i	$-0.266201\pi$
$131$	263284.	1.34044	0.670218	+	0.742165i	$0.266201\pi$
$132$	8533.99i	0.0426302i
$133$	74849.4i	0.366910i
$134$	15662.4	0.0753524
$135$	0	0
$136$	89217.9	0.413623
$137$	− 101284.i	− 0.461039i	−0.973068	−	0.230519i	$-0.925957\pi$
$137$	− 101284.i	− 0.461039i	0.973068	−	0.230519i	$-0.0740425\pi$
$138$	47043.8i	0.210283i
$139$	156014.	0.684897	0.342449	−	0.939537i	$-0.388744\pi$
$139$	156014.	0.684897	0.342449	+	0.939537i	$0.388744\pi$
$140$	0	0
$141$	−115234.	−0.488125
$142$	− 94627.2i	− 0.393818i
$143$	− 7773.23i	− 0.0317879i
$144$	57233.7	0.230009
$145$	0	0
$146$	−199240.	−0.773560
$147$	− 36743.7i	− 0.140246i
$148$	246013.i	0.923215i
$149$	132795.	0.490024	0.245012	−	0.969520i	$-0.421208\pi$
$149$	132795.	0.490024	0.245012	+	0.969520i	$0.421208\pi$
$150$	0	0
$151$	224845.	0.802491	0.401245	−	0.915971i	$-0.368577\pi$
$151$	224845.	0.802491	0.401245	+	0.915971i	$0.368577\pi$
$152$	52046.3i	0.182718i
$153$	311662.i	1.07635i
$154$	44547.5	0.151364
$155$	0	0
$156$	4530.88	0.0149063
$157$	− 200959.i	− 0.650667i	−0.945599	−	0.325333i	$-0.894524\pi$
$157$	− 200959.i	− 0.650667i	0.945599	−	0.325333i	$-0.105476\pi$
$158$	− 23386.2i	− 0.0745276i
$159$	2166.94	0.00679758
$160$	0	0
$161$	245569.	0.746635
$162$	181046.i	0.542002i
$163$	− 315221.i	− 0.929278i	−0.885500	−	0.464639i	$-0.846184\pi$
$163$	− 315221.i	− 0.929278i	0.885500	−	0.464639i	$-0.153816\pi$
$164$	−46003.8	−0.133563
$165$	0	0
$166$	119385.	0.336263
$167$	461946.i	1.28174i	0.767650	+	0.640870i	$0.221426\pi$
$167$	461946.i	1.28174i	−0.767650	+	0.640870i	$0.778574\pi$
$168$	25966.0i	0.0709792i
$169$	367166.	0.988885
$170$	0	0
$171$	−181812.	−0.475479
$172$	− 326486.i	− 0.841478i
$173$	− 284609.i	− 0.722992i	−0.932374	−	0.361496i	$-0.882266\pi$
$173$	− 284609.i	− 0.722992i	0.932374	−	0.361496i	$-0.117734\pi$
$174$	−33158.2	−0.0830266
$175$	0	0
$176$	30976.0	0.0753778
$177$	− 135661.i	− 0.325478i
$178$	442753.i	1.04740i
$179$	616210.	1.43746	0.718731	−	0.695289i	$-0.244723\pi$
$179$	616210.	1.43746	0.718731	+	0.695289i	$0.244723\pi$
$180$	0	0
$181$	156628.	0.355364	0.177682	−	0.984088i	$-0.443140\pi$
$181$	156628.	0.355364	0.177682	+	0.984088i	$0.443140\pi$
$182$	− 23651.2i	− 0.0529268i
$183$	55933.0i	0.123464i
$184$	170756.	0.371818
$185$	0	0
$186$	64877.7	0.137503
$187$	168678.i	0.352739i
$188$	418266.i	0.863093i
$189$	−189296.	−0.385466
$190$	0	0
$191$	95586.9	0.189590	0.0947949	−	0.995497i	$-0.469780\pi$
$191$	95586.9	0.189590	0.0947949	+	0.995497i	$0.469780\pi$
$192$	18055.4i	0.0353471i
$193$	748658.i	1.44674i	0.690461	+	0.723370i	$0.257408\pi$
$193$	748658.i	1.44674i	−0.690461	+	0.723370i	$0.742592\pi$
$194$	−263465.	−0.502595
$195$	0	0
$196$	−133369.	−0.247980
$197$	456491.i	0.838043i	0.907976	+	0.419022i	$0.137627\pi$
$197$	456491.i	0.838043i	−0.907976	+	0.419022i	$0.862373\pi$
$198$	108207.i	0.196153i
$199$	−591303.	−1.05847	−0.529234	−	0.848476i	$-0.677521\pi$
$199$	−591303.	−1.05847	−0.529234	+	0.848476i	$0.677521\pi$
$200$	0	0
$201$	−17260.2	−0.0301339
$202$	683265.i	1.17818i
$203$	173086.i	0.294796i
$204$	−98319.2	−0.165411
$205$	0	0
$206$	13401.7	0.0220036
$207$	596495.i	0.967567i
$208$	− 16445.8i	− 0.0263571i
$209$	−98400.1	−0.155822
$210$	0	0
$211$	−568294.	−0.878754	−0.439377	−	0.898303i	$-0.644801\pi$
$211$	−568294.	−0.878754	−0.439377	+	0.898303i	$0.644801\pi$
$212$	− 7865.39i	− 0.0120193i
$213$	104280.i	0.157490i
$214$	−174754.	−0.260850
$215$	0	0
$216$	−131626.	−0.191959
$217$	− 338662.i	− 0.488222i
$218$	− 197501.i	− 0.281468i
$219$	219565.	0.309352
$220$	0	0
$221$	89554.6	0.123341
$222$	− 271109.i	− 0.369200i
$223$	355316.i	0.478468i	0.970962	+	0.239234i	$0.0768962\pi$
$223$	355316.i	0.478468i	−0.970962	+	0.239234i	$0.923104\pi$
$224$	94249.2	0.125504
$225$	0	0
$226$	−453553.	−0.590686
$227$	− 627531.i	− 0.808297i	−0.914693	−	0.404148i	$-0.867568\pi$
$227$	− 627531.i	− 0.808297i	0.914693	−	0.404148i	$-0.132432\pi$
$228$	− 57355.7i	− 0.0730701i
$229$	987980.	1.24497	0.622486	−	0.782631i	$-0.286123\pi$
$229$	987980.	1.24497	0.622486	+	0.782631i	$0.286123\pi$
$230$	0	0
$231$	−49091.9	−0.0605313
$232$	120355.i	0.146806i
$233$	− 147032.i	− 0.177428i	−0.996057	−	0.0887139i	$-0.971724\pi$
$233$	− 147032.i	− 0.177428i	0.996057	−	0.0887139i	$-0.0282757\pi$
$234$	57449.7	0.0685880
$235$	0	0
$236$	−492412.	−0.575504
$237$	25771.9i	0.0298041i
$238$	513227.i	0.587310i
$239$	1.36202e6	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$239$	1.36202e6	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$240$	0	0
$241$	1.54270e6	1.71096	0.855478	−	0.517839i	$-0.173263\pi$
$241$	1.54270e6	1.71096	0.855478	+	0.517839i	$0.173263\pi$
$242$	58564.0i	0.0642824i
$243$	− 699283.i	− 0.759692i
$244$	203021.	0.218307
$245$	0	0
$246$	50696.8	0.0534125
$247$	52242.8i	0.0544859i
$248$	− 235488.i	− 0.243131i
$249$	−131564.	−0.134474
$250$	0	0
$251$	−551214.	−0.552250	−0.276125	−	0.961122i	$-0.589050\pi$
$251$	−551214.	−0.552250	−0.276125	+	0.961122i	$0.589050\pi$
$252$	329238.i	0.326594i
$253$	322835.i	0.317088i
$254$	1.26055e6	1.22596
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.50798e6i	1.42417i	0.702094	+	0.712085i	$0.252249\pi$
$257$	1.50798e6i	1.42417i	−0.702094	+	0.712085i	$0.747751\pi$
$258$	359791.i	0.336513i
$259$	−1.41519e6	−1.31089
$260$	0	0
$261$	−420432.	−0.382027
$262$	1.05313e6i	0.947831i
$263$	413554.i	0.368675i	0.982863	+	0.184337i	$0.0590139\pi$
$263$	413554.i	0.368675i	−0.982863	+	0.184337i	$0.940986\pi$
$264$	−34136.0	−0.0301441
$265$	0	0
$266$	−299397.	−0.259444
$267$	− 487919.i	− 0.418861i
$268$	62649.7i	0.0532822i
$269$	−1.92394e6	−1.62110	−0.810551	−	0.585668i	$-0.800832\pi$
$269$	−1.92394e6	−1.62110	−0.810551	+	0.585668i	$0.800832\pi$
$270$	0	0
$271$	1.36505e6	1.12908	0.564540	−	0.825405i	$-0.309053\pi$
$271$	1.36505e6	1.12908	0.564540	+	0.825405i	$0.309053\pi$
$272$	356872.i	0.292476i
$273$	26064.0i	0.0211658i
$274$	405134.	0.326004
$275$	0	0
$276$	−188175.	−0.148693
$277$	− 462704.i	− 0.362330i	−0.983453	−	0.181165i	$-0.942013\pi$
$277$	− 462704.i	− 0.362330i	0.983453	−	0.181165i	$-0.0579867\pi$
$278$	624055.i	0.484296i
$279$	822623.	0.632689
$280$	0	0
$281$	1.31617e6	0.994367	0.497183	−	0.867645i	$-0.334368\pi$
$281$	1.31617e6	0.994367	0.497183	+	0.867645i	$0.334368\pi$
$282$	− 460934.i	− 0.345157i
$283$	− 1.19241e6i	− 0.885035i	−0.896760	−	0.442517i	$-0.854086\pi$
$283$	− 1.19241e6i	− 0.885035i	0.896760	−	0.442517i	$-0.145914\pi$
$284$	378509.	0.278471
$285$	0	0
$286$	31092.9	0.0224774
$287$	− 264638.i	− 0.189648i
$288$	228935.i	0.162641i
$289$	−523461.	−0.368671
$290$	0	0
$291$	290342.	0.200991
$292$	− 796960.i	− 0.546990i
$293$	− 644118.i	− 0.438325i	−0.975688	−	0.219163i	$-0.929668\pi$
$293$	− 644118.i	− 0.438325i	0.975688	−	0.219163i	$-0.0703325\pi$
$294$	146975.	0.0991687
$295$	0	0
$296$	−984050.	−0.652812
$297$	− 248856.i	− 0.163703i
$298$	531181.i	0.346499i
$299$	171400.	0.110875
$300$	0	0
$301$	1.87811e6	1.19483
$302$	899379.i	0.567447i
$303$	− 752967.i	− 0.471161i
$304$	−208185.	−0.129201
$305$	0	0
$306$	−1.24665e6	−0.761097
$307$	− 1.19044e6i	− 0.720875i	−0.932783	−	0.360437i	$-0.882627\pi$
$307$	− 1.19044e6i	− 0.720875i	0.932783	−	0.360437i	$-0.117373\pi$
$308$	178190.i	0.107030i
$309$	−14768.9	−0.00879938
$310$	0	0
$311$	1.25938e6	0.738341	0.369170	−	0.929362i	$-0.379642\pi$
$311$	1.25938e6	0.738341	0.369170	+	0.929362i	$0.379642\pi$
$312$	18123.5i	0.0105404i
$313$	2.23658e6i	1.29040i	0.764014	+	0.645200i	$0.223226\pi$
$313$	2.23658e6i	1.29040i	−0.764014	+	0.645200i	$0.776774\pi$
$314$	803836.	0.460091
$315$	0	0
$316$	93544.8	0.0526990
$317$	− 2.36125e6i	− 1.31976i	−0.751373	−	0.659878i	$-0.770608\pi$
$317$	− 2.36125e6i	− 1.31976i	0.751373	−	0.659878i	$-0.229392\pi$
$318$	8667.76i	0.00480661i
$319$	−227546.	−0.125197
$320$	0	0
$321$	192581.	0.104316
$322$	982275.i	0.527951i
$323$	− 1.13366e6i	− 0.604611i
$324$	−724183.	−0.383253
$325$	0	0
$326$	1.26088e6	0.657099
$327$	217649.i	0.112561i
$328$	− 184015.i	− 0.0944430i
$329$	−2.40608e6	−1.22552
$330$	0	0
$331$	−3.06002e6	−1.53516	−0.767580	−	0.640953i	$-0.778539\pi$
$331$	−3.06002e6	−1.53516	−0.767580	+	0.640953i	$0.778539\pi$
$332$	477539.i	0.237774i
$333$	− 3.43755e6i	− 1.69879i
$334$	−1.84778e6	−0.906327
$335$	0	0
$336$	−103864.	−0.0501899
$337$	− 4.08324e6i	− 1.95853i	−0.202588	−	0.979264i	$-0.564935\pi$
$337$	− 4.08324e6i	− 1.95853i	0.202588	−	0.979264i	$-0.435065\pi$
$338$	1.46866e6i	0.699247i
$339$	499821.	0.236219
$340$	0	0
$341$	445220.	0.207342
$342$	− 727247.i	− 0.336215i
$343$	− 2.31413e6i	− 1.06207i
$344$	1.30594e6	0.595015
$345$	0	0
$346$	1.13844e6	0.511232
$347$	− 2.25118e6i	− 1.00366i	−0.864966	−	0.501830i	$-0.832660\pi$
$347$	− 2.25118e6i	− 1.00366i	0.864966	−	0.501830i	$-0.167340\pi$
$348$	− 132633.i	− 0.0587087i
$349$	−1.09341e6	−0.480531	−0.240265	−	0.970707i	$-0.577234\pi$
$349$	−1.09341e6	−0.480531	−0.240265	+	0.970707i	$0.577234\pi$
$350$	0	0
$351$	−132123.	−0.0572415
$352$	123904.i	0.0533002i
$353$	− 1.73879e6i	− 0.742694i	−0.928494	−	0.371347i	$-0.878896\pi$
$353$	− 1.73879e6i	− 0.742694i	0.928494	−	0.371347i	$-0.121104\pi$
$354$	542644.	0.230148
$355$	0	0
$356$	−1.77101e6	−0.740621
$357$	− 565583.i	− 0.234869i
$358$	2.46484e6i	1.01644i
$359$	−1.93636e6	−0.792959	−0.396480	−	0.918044i	$-0.629768\pi$
$359$	−1.93636e6	−0.792959	−0.396480	+	0.918044i	$0.629768\pi$
$360$	0	0
$361$	−1.81477e6	−0.732913
$362$	626513.i	0.251280i
$363$	− 64538.3i	− 0.0257070i
$364$	94604.9	0.0374249
$365$	0	0
$366$	−223732.	−0.0873022
$367$	1.36996e6i	0.530937i	0.964119	+	0.265469i	$0.0855267\pi$
$367$	1.36996e6i	0.530937i	−0.964119	+	0.265469i	$0.914473\pi$
$368$	683023.i	0.262915i
$369$	642815.	0.245765
$370$	0	0
$371$	45245.8	0.0170665
$372$	259511.i	0.0972295i
$373$	1.02270e6i	0.380608i	0.981725	+	0.190304i	$0.0609473\pi$
$373$	1.02270e6i	0.380608i	−0.981725	+	0.190304i	$0.939053\pi$
$374$	−674710.	−0.249424
$375$	0	0
$376$	−1.67306e6	−0.610299
$377$	120809.i	0.0437770i
$378$	− 757182.i	− 0.272566i
$379$	−2.30604e6	−0.824650	−0.412325	−	0.911037i	$-0.635283\pi$
$379$	−2.30604e6	−0.824650	−0.412325	+	0.911037i	$0.635283\pi$
$380$	0	0
$381$	−1.38914e6	−0.490269
$382$	382348.i	0.134060i
$383$	− 4.44660e6i	− 1.54893i	−0.632618	−	0.774464i	$-0.718020\pi$
$383$	− 4.44660e6i	− 1.54893i	0.632618	−	0.774464i	$-0.281980\pi$
$384$	−72221.5	−0.0249942
$385$	0	0
$386$	−2.99463e6	−1.02300
$387$	4.56201e6i	1.54838i
$388$	− 1.05386e6i	− 0.355389i
$389$	3.04391e6	1.01990	0.509951	−	0.860204i	$-0.329664\pi$
$389$	3.04391e6	1.01990	0.509951	+	0.860204i	$0.329664\pi$
$390$	0	0
$391$	−3.71935e6	−1.23034
$392$	− 533478.i	− 0.175348i
$393$	− 1.16057e6i	− 0.379044i
$394$	−1.82596e6	−0.592586
$395$	0	0
$396$	−432830.	−0.138701
$397$	2.97167e6i	0.946289i	0.880985	+	0.473145i	$0.156881\pi$
$397$	2.97167e6i	0.946289i	−0.880985	+	0.473145i	$0.843119\pi$
$398$	− 2.36521e6i	− 0.748450i
$399$	329940.	0.103753
$400$	0	0
$401$	4.34955e6	1.35078	0.675388	−	0.737463i	$-0.263976\pi$
$401$	4.34955e6	1.35078	0.675388	+	0.737463i	$0.263976\pi$
$402$	− 69040.7i	− 0.0213079i
$403$	− 236377.i	− 0.0725007i
$404$	−2.73306e6	−0.833097
$405$	0	0
$406$	−692343.	−0.208452
$407$	− 1.86047e6i	− 0.556720i
$408$	− 393277.i	− 0.116963i
$409$	−89191.8	−0.0263643	−0.0131822	−	0.999913i	$-0.504196\pi$
$409$	−89191.8	−0.0263643	−0.0131822	+	0.999913i	$0.504196\pi$
$410$	0	0
$411$	−446463.	−0.130371
$412$	53607.0i	0.0155589i
$413$	− 2.83261e6i	− 0.817168i
$414$	−2.38598e6	−0.684173
$415$	0	0
$416$	65783.3	0.0186373
$417$	− 687716.i	− 0.193673i
$418$	− 393600.i	− 0.110183i
$419$	4.18567e6	1.16474	0.582371	−	0.812923i	$-0.302125\pi$
$419$	4.18567e6	1.16474	0.582371	+	0.812923i	$0.302125\pi$
$420$	0	0
$421$	1.96388e6	0.540020	0.270010	−	0.962857i	$-0.412973\pi$
$421$	1.96388e6	0.540020	0.270010	+	0.962857i	$0.412973\pi$
$422$	− 2.27318e6i	− 0.621373i
$423$	− 5.84445e6i	− 1.58816i
$424$	31461.6	0.00849895
$425$	0	0
$426$	−417122.	−0.111362
$427$	1.16788e6i	0.309977i
$428$	− 699014.i	− 0.184449i
$429$	−34264.8	−0.00898886
$430$	0	0
$431$	−1.32562e6	−0.343736	−0.171868	−	0.985120i	$-0.554980\pi$
$431$	−1.32562e6	−0.343736	−0.171868	+	0.985120i	$0.554980\pi$
$432$	− 526505.i	− 0.135735i
$433$	− 5.16016e6i	− 1.32265i	−0.750101	−	0.661323i	$-0.769995\pi$
$433$	− 5.16016e6i	− 1.32265i	0.750101	−	0.661323i	$-0.230005\pi$
$434$	1.35465e6	0.345225
$435$	0	0
$436$	790006.	0.199028
$437$	− 2.16973e6i	− 0.543503i
$438$	878260.i	0.218745i
$439$	−3.93148e6	−0.973632	−0.486816	−	0.873504i	$-0.661842\pi$
$439$	−3.93148e6	−0.973632	−0.486816	+	0.873504i	$0.661842\pi$
$440$	0	0
$441$	1.86358e6	0.456301
$442$	358218.i	0.0872152i
$443$	3.06674e6i	0.742452i	0.928543	+	0.371226i	$0.121062\pi$
$443$	3.06674e6i	0.742452i	−0.928543	+	0.371226i	$0.878938\pi$
$444$	1.08444e6	0.261064
$445$	0	0
$446$	−1.42126e6	−0.338328
$447$	− 585369.i	− 0.138567i
$448$	376997.i	0.0887448i
$449$	−4.68097e6	−1.09577	−0.547886	−	0.836553i	$-0.684567\pi$
$449$	−4.68097e6	−1.09577	−0.547886	+	0.836553i	$0.684567\pi$
$450$	0	0
$451$	347904.	0.0805412
$452$	− 1.81421e6i	− 0.417678i
$453$	− 991127.i	− 0.226926i
$454$	2.51013e6	0.571552
$455$	0	0
$456$	229423.	0.0516684
$457$	6.86401e6i	1.53740i	0.639608	+	0.768701i	$0.279097\pi$
$457$	6.86401e6i	1.53740i	−0.639608	+	0.768701i	$0.720903\pi$
$458$	3.95192e6i	0.880328i
$459$	2.86705e6	0.635189
$460$	0	0
$461$	−3.85378e6	−0.844569	−0.422284	−	0.906463i	$-0.638772\pi$
$461$	−3.85378e6	−0.844569	−0.422284	+	0.906463i	$0.638772\pi$
$462$	− 196368.i	− 0.0428021i
$463$	− 3.15825e6i	− 0.684689i	−0.939574	−	0.342345i	$-0.888779\pi$
$463$	− 3.15825e6i	− 0.684689i	0.939574	−	0.342345i	$-0.111221\pi$
$464$	−481420.	−0.103808
$465$	0	0
$466$	588127.	0.125460
$467$	− 196109.i	− 0.0416107i	−0.999784	−	0.0208054i	$-0.993377\pi$
$467$	− 196109.i	− 0.0416107i	0.999784	−	0.0208054i	$-0.00662303\pi$
$468$	229799.i	0.0484990i
$469$	−360393.	−0.0756562
$470$	0	0
$471$	−885838.	−0.183993
$472$	− 1.96965e6i	− 0.406943i
$473$	2.46905e6i	0.507430i
$474$	−103088.	−0.0210747
$475$	0	0
$476$	−2.05291e6	−0.415291
$477$	109904.i	0.0221165i
$478$	5.44808e6i	1.09062i
$479$	−5.13528e6	−1.02265	−0.511323	−	0.859389i	$-0.670844\pi$
$479$	−5.13528e6	−1.02265	−0.511323	+	0.859389i	$0.670844\pi$
$480$	0	0
$481$	−987764.	−0.194666
$482$	6.17080e6i	1.20983i
$483$	− 1.08248e6i	− 0.211131i
$484$	−234256.	−0.0454545
$485$	0	0
$486$	2.79713e6	0.537183
$487$	− 2.43868e6i	− 0.465943i	−0.972484	−	0.232971i	$-0.925155\pi$
$487$	− 2.43868e6i	− 0.465943i	0.972484	−	0.232971i	$-0.0748448\pi$
$488$	812085.i	0.154366i
$489$	−1.38951e6	−0.262778
$490$	0	0
$491$	4.07687e6	0.763173	0.381587	−	0.924333i	$-0.375378\pi$
$491$	4.07687e6	0.763173	0.381587	+	0.924333i	$0.375378\pi$
$492$	202787.i	0.0377684i
$493$	− 2.62154e6i	− 0.485779i
$494$	−208971.	−0.0385273
$495$	0	0
$496$	941952.	0.171919
$497$	2.17738e6i	0.395406i
$498$	− 526254.i	− 0.0950873i
$499$	−878791.	−0.157992	−0.0789958	−	0.996875i	$-0.525171\pi$
$499$	−878791.	−0.157992	−0.0789958	+	0.996875i	$0.525171\pi$
$500$	0	0
$501$	2.03628e6	0.362446
$502$	− 2.20486e6i	− 0.390500i
$503$	7.00187e6i	1.23394i	0.786987	+	0.616970i	$0.211640\pi$
$503$	7.00187e6i	1.23394i	−0.786987	+	0.616970i	$0.788360\pi$
$504$	−1.31695e6	−0.230937
$505$	0	0
$506$	−1.29134e6	−0.224215
$507$	− 1.61849e6i	− 0.279634i
$508$	5.04220e6i	0.866883i
$509$	−5.54203e6	−0.948144	−0.474072	−	0.880486i	$-0.657216\pi$
$509$	−5.54203e6	−0.948144	−0.474072	+	0.880486i	$0.657216\pi$
$510$	0	0
$511$	4.58452e6	0.776680
$512$	262144.i	0.0441942i
$513$	1.67253e6i	0.280595i
$514$	−6.03190e6	−1.00704
$515$	0	0
$516$	−1.43917e6	−0.237950
$517$	− 3.16313e6i	− 0.520465i
$518$	− 5.66077e6i	− 0.926938i
$519$	−1.25457e6	−0.204445
$520$	0	0
$521$	−5.22863e6	−0.843905	−0.421953	−	0.906618i	$-0.638655\pi$
$521$	−5.22863e6	−0.843905	−0.421953	+	0.906618i	$0.638655\pi$
$522$	− 1.68173e6i	− 0.270134i
$523$	2.80925e6i	0.449092i	0.974463	+	0.224546i	$0.0720900\pi$
$523$	2.80925e6i	0.449092i	−0.974463	+	0.224546i	$0.927910\pi$
$524$	−4.21254e6	−0.670218
$525$	0	0
$526$	−1.65422e6	−0.260692
$527$	5.12933e6i	0.804515i
$528$	− 136544.i	− 0.0213151i
$529$	−682194.	−0.105991
$530$	0	0
$531$	6.88050e6	1.05897
$532$	− 1.19759e6i	− 0.183455i
$533$	− 184710.i	− 0.0281626i
$534$	1.95168e6	0.296179
$535$	0	0
$536$	−250599.	−0.0376762
$537$	− 2.71629e6i	− 0.406480i
$538$	− 7.69575e6i	− 1.14629i
$539$	1.00861e6	0.149537
$540$	0	0
$541$	−7.28856e6	−1.07065	−0.535326	−	0.844645i	$-0.679811\pi$
$541$	−7.28856e6	−1.07065	−0.535326	+	0.844645i	$0.679811\pi$
$542$	5.46020e6i	0.798381i
$543$	− 690425.i	− 0.100489i
$544$	−1.42749e6	−0.206811
$545$	0	0
$546$	−104256.	−0.0149665
$547$	9.33502e6i	1.33397i	0.745070	+	0.666986i	$0.232416\pi$
$547$	9.33502e6i	1.33397i	−0.745070	+	0.666986i	$0.767584\pi$
$548$	1.62054e6i	0.230519i
$549$	−2.83683e6	−0.401700
$550$	0	0
$551$	1.52930e6	0.214593
$552$	− 752700.i	− 0.105142i
$553$	538118.i	0.0748281i
$554$	1.85082e6	0.256206
$555$	0	0
$556$	−2.49622e6	−0.342449
$557$	87187.2i	0.0119073i	0.999982	+	0.00595367i	$0.00189512\pi$
$557$	87187.2i	0.0119073i	−0.999982	+	0.00595367i	$0.998105\pi$
$558$	3.29049e6i	0.447378i
$559$	1.31087e6	0.177431
$560$	0	0
$561$	743539.	0.0997463
$562$	5.26468e6i	0.703123i
$563$	1.99082e6i	0.264705i	0.991203	+	0.132352i	$0.0422530\pi$
$563$	1.99082e6i	0.264705i	−0.991203	+	0.132352i	$0.957747\pi$
$564$	1.84374e6	0.244063
$565$	0	0
$566$	4.76965e6	0.625814
$567$	− 4.16587e6i	− 0.544187i
$568$	1.51404e6i	0.196909i
$569$	−955969.	−0.123784	−0.0618918	−	0.998083i	$-0.519713\pi$
$569$	−955969.	−0.123784	−0.0618918	+	0.998083i	$0.519713\pi$
$570$	0	0
$571$	1.17829e7	1.51239	0.756193	−	0.654349i	$-0.227057\pi$
$571$	1.17829e7	1.51239	0.756193	+	0.654349i	$0.227057\pi$
$572$	124372.i	0.0158939i
$573$	− 421352.i	− 0.0536116i
$574$	1.05855e6	0.134101
$575$	0	0
$576$	−915739.	−0.115005
$577$	7.62998e6i	0.954078i	0.878882	+	0.477039i	$0.158290\pi$
$577$	7.62998e6i	0.954078i	−0.878882	+	0.477039i	$0.841710\pi$
$578$	− 2.09384e6i	− 0.260690i
$579$	3.30013e6	0.409104
$580$	0	0
$581$	−2.74705e6	−0.337619
$582$	1.16137e6i	0.142122i
$583$	59482.0i	0.00724793i
$584$	3.18784e6	0.386780
$585$	0	0
$586$	2.57647e6	0.309943
$587$	− 5.31020e6i	− 0.636086i	−0.948076	−	0.318043i	$-0.896974\pi$
$587$	− 5.31020e6i	− 0.636086i	0.948076	−	0.318043i	$-0.103026\pi$
$588$	587899.i	0.0701229i
$589$	−2.99226e6	−0.355395
$590$	0	0
$591$	2.01223e6	0.236979
$592$	− 3.93620e6i	− 0.461608i
$593$	9.44509e6i	1.10298i	0.834180	+	0.551492i	$0.185941\pi$
$593$	9.44509e6i	1.10298i	−0.834180	+	0.551492i	$0.814059\pi$
$594$	995424.	0.115756
$595$	0	0
$596$	−2.12472e6	−0.245012
$597$	2.60649e6i	0.299310i
$598$	685601.i	0.0784004i
$599$	−8.27369e6	−0.942177	−0.471088	−	0.882086i	$-0.656139\pi$
$599$	−8.27369e6	−0.942177	−0.471088	+	0.882086i	$0.656139\pi$
$600$	0	0
$601$	−1.28982e7	−1.45661	−0.728304	−	0.685254i	$-0.759691\pi$
$601$	−1.28982e7	−1.45661	−0.728304	+	0.685254i	$0.759691\pi$
$602$	7.51245e6i	0.844871i
$603$	− 875408.i	− 0.0980432i
$604$	−3.59751e6	−0.401245
$605$	0	0
$606$	3.01187e6	0.333161
$607$	884063.i	0.0973893i	0.998814	+	0.0486947i	$0.0155061\pi$
$607$	884063.i	0.0973893i	−0.998814	+	0.0486947i	$0.984494\pi$
$608$	− 832742.i	− 0.0913590i
$609$	762971.	0.0833614
$610$	0	0
$611$	−1.67938e6	−0.181989
$612$	− 4.98659e6i	− 0.538177i
$613$	7.44165e6i	0.799868i	0.916544	+	0.399934i	$0.130967\pi$
$613$	7.44165e6i	0.799868i	−0.916544	+	0.399934i	$0.869033\pi$
$614$	4.76174e6	0.509735
$615$	0	0
$616$	−712760.	−0.0756818
$617$	− 1.54607e6i	− 0.163499i	−0.996653	−	0.0817496i	$-0.973949\pi$
$617$	− 1.54607e6i	− 0.163499i	0.996653	−	0.0817496i	$-0.0260508\pi$
$618$	− 59075.6i	− 0.00622210i
$619$	337532.	0.0354069	0.0177035	−	0.999843i	$-0.494365\pi$
$619$	337532.	0.0354069	0.0177035	+	0.999843i	$0.494365\pi$
$620$	0	0
$621$	5.48729e6	0.570991
$622$	5.03753e6i	0.522086i
$623$	− 1.01878e7i	− 1.05162i
$624$	−72494.1	−0.00745317
$625$	0	0
$626$	−8.94633e6	−0.912450
$627$	433753.i	0.0440629i
$628$	3.21535e6i	0.325333i
$629$	2.14343e7	2.16014
$630$	0	0
$631$	−2.60561e6	−0.260517	−0.130259	−	0.991480i	$-0.541581\pi$
$631$	−2.60561e6	−0.260517	−0.130259	+	0.991480i	$0.541581\pi$
$632$	374179.i	0.0372638i
$633$	2.50507e6i	0.248491i
$634$	9.44499e6	0.933208
$635$	0	0
$636$	−34671.0	−0.00339879
$637$	− 535491.i	− 0.0522882i
$638$	− 910184.i	− 0.0885273i
$639$	−5.28893e6	−0.512408
$640$	0	0
$641$	1.37962e7	1.32622	0.663108	−	0.748523i	$-0.269237\pi$
$641$	1.37962e7	1.32622	0.663108	+	0.748523i	$0.269237\pi$
$642$	770323.i	0.0737624i
$643$	4.97213e6i	0.474258i	0.971478	+	0.237129i	$0.0762064\pi$
$643$	4.97213e6i	0.474258i	−0.971478	+	0.237129i	$0.923794\pi$
$644$	−3.92910e6	−0.373318
$645$	0	0
$646$	4.53463e6	0.427524
$647$	− 1.75328e7i	− 1.64661i	−0.567601	−	0.823304i	$-0.692128\pi$
$647$	− 1.75328e7i	− 1.64661i	0.567601	−	0.823304i	$-0.307872\pi$
$648$	− 2.89673e6i	− 0.271001i
$649$	3.72387e6	0.347042
$650$	0	0
$651$	−1.49284e6	−0.138058
$652$	5.04353e6i	0.464639i
$653$	− 1.67551e6i	− 0.153768i	−0.997040	−	0.0768838i	$-0.975503\pi$
$653$	− 1.67551e6i	− 0.153768i	0.997040	−	0.0768838i	$-0.0244971\pi$
$654$	−870596.	−0.0795926
$655$	0	0
$656$	736062.	0.0667813
$657$	1.11360e7i	1.00650i
$658$	− 9.62432e6i	− 0.866573i
$659$	4.27468e6	0.383434	0.191717	−	0.981450i	$-0.438594\pi$
$659$	4.27468e6	0.383434	0.191717	+	0.981450i	$0.438594\pi$
$660$	0	0
$661$	−1.01563e7	−0.904131	−0.452066	−	0.891985i	$-0.649313\pi$
$661$	−1.01563e7	−0.904131	−0.452066	+	0.891985i	$0.649313\pi$
$662$	− 1.22401e7i	− 1.08552i
$663$	− 394761.i	− 0.0348779i
$664$	−1.91016e6	−0.168131
$665$	0	0
$666$	1.37502e7	1.20122
$667$	− 5.01740e6i	− 0.436681i
$668$	− 7.39113e6i	− 0.640870i
$669$	1.56625e6	0.135299
$670$	0	0
$671$	−1.53535e6	−0.131644
$672$	− 415455.i	− 0.0354896i
$673$	1.09681e7i	0.933454i	0.884402	+	0.466727i	$0.154567\pi$
$673$	1.09681e7i	0.933454i	−0.884402	+	0.466727i	$0.845433\pi$
$674$	1.63329e7	1.38489
$675$	0	0
$676$	−5.87466e6	−0.494442
$677$	8.84010e6i	0.741286i	0.928775	+	0.370643i	$0.120863\pi$
$677$	8.84010e6i	0.741286i	−0.928775	+	0.370643i	$0.879137\pi$
$678$	1.99928e6i	0.167032i
$679$	6.06235e6	0.504622
$680$	0	0
$681$	−2.76619e6	−0.228567
$682$	1.78088e6i	0.146613i
$683$	5.98047e6i	0.490551i	0.969453	+	0.245275i	$0.0788784\pi$
$683$	5.98047e6i	0.490551i	−0.969453	+	0.245275i	$0.921122\pi$
$684$	2.90899e6	0.237740
$685$	0	0
$686$	9.25652e6	0.750996
$687$	− 4.35507e6i	− 0.352049i
$688$	5.22377e6i	0.420739i
$689$	31580.3	0.00253436
$690$	0	0
$691$	3.64212e6	0.290175	0.145087	−	0.989419i	$-0.453654\pi$
$691$	3.64212e6	0.290175	0.145087	+	0.989419i	$0.453654\pi$
$692$	4.55374e6i	0.361496i
$693$	− 2.48986e6i	− 0.196944i
$694$	9.00473e6	0.709695
$695$	0	0
$696$	530531.	0.0415133
$697$	4.00817e6i	0.312510i
$698$	− 4.37365e6i	− 0.339786i
$699$	−648124.	−0.0501724
$700$	0	0
$701$	−1.44054e6	−0.110721	−0.0553607	−	0.998466i	$-0.517631\pi$
$701$	−1.44054e6	−0.110721	−0.0553607	+	0.998466i	$0.517631\pi$
$702$	− 528492.i	− 0.0404758i
$703$	1.25040e7i	0.954244i
$704$	−495616.	−0.0376889
$705$	0	0
$706$	6.95515e6	0.525164
$707$	− 1.57220e7i	− 1.18293i
$708$	2.17058e6i	0.162739i
$709$	2.44669e7	1.82795	0.913975	−	0.405771i	$-0.132997\pi$
$709$	2.44669e7	1.82795	0.913975	+	0.405771i	$0.132997\pi$
$710$	0	0
$711$	−1.30711e6	−0.0969700
$712$	− 7.08404e6i	− 0.523698i
$713$	9.81712e6i	0.723203i
$714$	2.26233e6	0.166078
$715$	0	0
$716$	−9.85936e6	−0.718731
$717$	− 6.00386e6i	− 0.436147i
$718$	− 7.74545e6i	− 0.560707i
$719$	2.27569e6	0.164169	0.0820846	−	0.996625i	$-0.473842\pi$
$719$	2.27569e6	0.164169	0.0820846	+	0.996625i	$0.473842\pi$
$720$	0	0
$721$	−308375.	−0.0220923
$722$	− 7.25906e6i	− 0.518248i
$723$	− 6.80030e6i	− 0.483818i
$724$	−2.50605e6	−0.177682
$725$	0	0
$726$	258153.	0.0181776
$727$	− 1.87924e7i	− 1.31870i	−0.751835	−	0.659351i	$-0.770831\pi$
$727$	− 1.87924e7i	− 1.31870i	0.751835	−	0.659351i	$-0.229169\pi$
$728$	378420.i	0.0264634i
$729$	7.91605e6	0.551683
$730$	0	0
$731$	−2.84457e7	−1.96889
$732$	− 894928.i	− 0.0617320i
$733$	− 355042.i	− 0.0244073i	−0.999926	−	0.0122036i	$-0.996115\pi$
$733$	− 355042.i	− 0.0244073i	0.999926	−	0.0122036i	$-0.00388463\pi$
$734$	−5.47985e6	−0.375429
$735$	0	0
$736$	−2.73209e6	−0.185909
$737$	− 473788.i	− 0.0321304i
$738$	2.57126e6i	0.173782i
$739$	1.97063e7	1.32738	0.663689	−	0.748008i	$-0.268990\pi$
$739$	1.97063e7	1.32738	0.663689	+	0.748008i	$0.268990\pi$
$740$	0	0
$741$	230289.	0.0154073
$742$	180983.i	0.0120678i
$743$	1.74672e7i	1.16078i	0.814338	+	0.580391i	$0.197100\pi$
$743$	1.74672e7i	1.16078i	−0.814338	+	0.580391i	$0.802900\pi$
$744$	−1.03804e6	−0.0687517
$745$	0	0
$746$	−4.09081e6	−0.269130
$747$	− 6.67269e6i	− 0.437521i
$748$	− 2.69884e6i	− 0.176369i
$749$	4.02109e6	0.261902
$750$	0	0
$751$	−4.94160e6	−0.319718	−0.159859	−	0.987140i	$-0.551104\pi$
$751$	−4.94160e6	−0.319718	−0.159859	+	0.987140i	$0.551104\pi$
$752$	− 6.69225e6i	− 0.431546i
$753$	2.42978e6i	0.156163i
$754$	−483237.	−0.0309550
$755$	0	0
$756$	3.02873e6	0.192733
$757$	2.46846e7i	1.56562i	0.622261	+	0.782810i	$0.286214\pi$
$757$	2.46846e7i	1.56562i	−0.622261	+	0.782810i	$0.713786\pi$
$758$	− 9.22417e6i	− 0.583115i
$759$	1.42307e6	0.0896650
$760$	0	0
$761$	2.49595e7	1.56234	0.781169	−	0.624320i	$-0.214624\pi$
$761$	2.49595e7	1.56234	0.781169	+	0.624320i	$0.214624\pi$
$762$	− 5.55657e6i	− 0.346672i
$763$	4.54452e6i	0.282603i
$764$	−1.52939e6	−0.0947949
$765$	0	0
$766$	1.77864e7	1.09526
$767$	− 1.97708e6i	− 0.121349i
$768$	− 288886.i	− 0.0176735i
$769$	−6.12897e6	−0.373742	−0.186871	−	0.982384i	$-0.559835\pi$
$769$	−6.12897e6	−0.373742	−0.186871	+	0.982384i	$0.559835\pi$
$770$	0	0
$771$	6.64723e6	0.402722
$772$	− 1.19785e7i	− 0.723370i
$773$	1.14936e7i	0.691842i	0.938264	+	0.345921i	$0.112434\pi$
$773$	1.14936e7i	0.691842i	−0.938264	+	0.345921i	$0.887566\pi$
$774$	−1.82480e7	−1.09487
$775$	0	0
$776$	4.21544e6	0.251298
$777$	6.23824e6i	0.370689i
$778$	1.21757e7i	0.721179i
$779$	−2.33822e6	−0.138051
$780$	0	0
$781$	−2.86247e6	−0.167924
$782$	− 1.48774e7i	− 0.869982i
$783$	3.86764e6i	0.225446i
$784$	2.13391e6	0.123990
$785$	0	0
$786$	4.64227e6	0.268024
$787$	1.34777e7i	0.775673i	0.921728	+	0.387837i	$0.126778\pi$
$787$	1.34777e7i	0.775673i	−0.921728	+	0.387837i	$0.873222\pi$
$788$	− 7.30385e6i	− 0.419022i
$789$	1.82297e6	0.104253
$790$	0	0
$791$	1.04363e7	0.593068
$792$	− 1.73132e6i	− 0.0980763i
$793$	815150.i	0.0460314i
$794$	−1.18867e7	−0.669128
$795$	0	0
$796$	9.46085e6	0.529234
$797$	2.33863e7i	1.30411i	0.758171	+	0.652056i	$0.226093\pi$
$797$	2.33863e7i	1.30411i	−0.758171	+	0.652056i	$0.773907\pi$
$798$	1.31976e6i	0.0733648i
$799$	3.64422e7	2.01947
$800$	0	0
$801$	2.47464e7	1.36280
$802$	1.73982e7i	0.955143i
$803$	6.02701e6i	0.329847i
$804$	276163.	0.0150669
$805$	0	0
$806$	945507.	0.0512657
$807$	8.48082e6i	0.458410i
$808$	− 1.09322e7i	− 0.589089i
$809$	−3.43344e7	−1.84441	−0.922207	−	0.386696i	$-0.873616\pi$
$809$	−3.43344e7	−1.84441	−0.922207	+	0.386696i	$0.873616\pi$
$810$	0	0
$811$	−3.01851e7	−1.61154	−0.805769	−	0.592231i	$-0.798247\pi$
$811$	−3.01851e7	−1.61154	−0.805769	+	0.592231i	$0.798247\pi$
$812$	− 2.76937e6i	− 0.147398i
$813$	− 6.01721e6i	− 0.319278i
$814$	7.44188e6	0.393660
$815$	0	0
$816$	1.57311e6	0.0827053
$817$	− 1.65941e7i	− 0.869759i
$818$	− 356767.i	− 0.0186424i
$819$	−1.32192e6	−0.0688646
$820$	0	0
$821$	−1.75961e7	−0.911087	−0.455543	−	0.890214i	$-0.650555\pi$
$821$	−1.75961e7	−0.911087	−0.455543	+	0.890214i	$0.650555\pi$
$822$	− 1.78585e6i	− 0.0921862i
$823$	2.02786e7i	1.04361i	0.853065	+	0.521804i	$0.174741\pi$
$823$	2.02786e7i	1.04361i	−0.853065	+	0.521804i	$0.825259\pi$
$824$	−214428.	−0.0110018
$825$	0	0
$826$	1.13304e7	0.577825
$827$	− 3.28389e7i	− 1.66965i	−0.550516	−	0.834824i	$-0.685569\pi$
$827$	− 3.28389e7i	− 1.66965i	0.550516	−	0.834824i	$-0.314431\pi$
$828$	− 9.54393e6i	− 0.483784i
$829$	2.40343e7	1.21463	0.607317	−	0.794460i	$-0.292246\pi$
$829$	2.40343e7	1.21463	0.607317	+	0.794460i	$0.292246\pi$
$830$	0	0
$831$	−2.03962e6	−0.102458
$832$	263133.i	0.0131785i
$833$	1.16201e7i	0.580224i
$834$	2.75086e6	0.136947
$835$	0	0
$836$	1.57440e6	0.0779112
$837$	− 7.56748e6i	− 0.373369i
$838$	1.67427e7i	0.823597i
$839$	1.85037e7	0.907515	0.453757	−	0.891125i	$-0.350083\pi$
$839$	1.85037e7	0.907515	0.453757	+	0.891125i	$0.350083\pi$
$840$	0	0
$841$	−1.69747e7	−0.827584
$842$	7.85553e6i	0.381852i
$843$	− 5.80175e6i	− 0.281184i
$844$	9.09271e6	0.439377
$845$	0	0
$846$	2.33778e7	1.12300
$847$	− 1.34756e6i	− 0.0645416i
$848$	125846.i	0.00600967i
$849$	−5.25621e6	−0.250267
$850$	0	0
$851$	4.10235e7	1.94182
$852$	− 1.66849e6i	− 0.0787451i
$853$	− 5.67139e6i	− 0.266881i	−0.991057	−	0.133440i	$-0.957398\pi$
$853$	− 5.67139e6i	− 0.266881i	0.991057	−	0.133440i	$-0.0426025\pi$
$854$	−4.67153e6	−0.219187
$855$	0	0
$856$	2.79606e6	0.130425
$857$	69849.7i	0.00324872i	0.999999	+	0.00162436i	$0.000517050\pi$
$857$	69849.7i	0.00324872i	−0.999999	+	0.00162436i	$0.999483\pi$
$858$	− 137059.i	− 0.00635609i
$859$	−1.24652e7	−0.576391	−0.288195	−	0.957572i	$-0.593055\pi$
$859$	−1.24652e7	−0.576391	−0.288195	+	0.957572i	$0.593055\pi$
$860$	0	0
$861$	−1.16654e6	−0.0536279
$862$	− 5.30246e6i	− 0.243058i
$863$	− 3.28985e7i	− 1.50366i	−0.659358	−	0.751829i	$-0.729172\pi$
$863$	− 3.28985e7i	− 1.50366i	0.659358	−	0.751829i	$-0.270828\pi$
$864$	2.10602e6	0.0959794
$865$	0	0
$866$	2.06406e7	0.935252
$867$	2.30744e6i	0.104252i
$868$	5.41859e6i	0.244111i
$869$	−707433.	−0.0317787
$870$	0	0
$871$	−251544.	−0.0112349
$872$	3.16002e6i	0.140734i
$873$	1.47257e7i	0.653942i
$874$	8.67892e6	0.384315
$875$	0	0
$876$	−3.51304e6	−0.154676
$877$	3.07510e7i	1.35008i	0.737779	+	0.675042i	$0.235875\pi$
$877$	3.07510e7i	1.35008i	−0.737779	+	0.675042i	$0.764125\pi$
$878$	− 1.57259e7i	− 0.688462i
$879$	−2.83930e6	−0.123948
$880$	0	0
$881$	−1.53544e7	−0.666489	−0.333245	−	0.942840i	$-0.608143\pi$
$881$	−1.53544e7	−0.666489	−0.333245	+	0.942840i	$0.608143\pi$
$882$	7.45432e6i	0.322654i
$883$	2.65947e7i	1.14787i	0.818900	+	0.573937i	$0.194585\pi$
$883$	2.65947e7i	1.14787i	−0.818900	+	0.573937i	$0.805415\pi$
$884$	−1.43287e6	−0.0616705
$885$	0	0
$886$	−1.22670e7	−0.524993
$887$	4.22486e6i	0.180303i	0.995928	+	0.0901517i	$0.0287352\pi$
$887$	4.22486e6i	0.180303i	−0.995928	+	0.0901517i	$0.971265\pi$
$888$	4.33774e6i	0.184600i
$889$	−2.90053e7	−1.23090
$890$	0	0
$891$	5.47663e6	0.231110
$892$	− 5.68505e6i	− 0.239234i
$893$	2.12590e7i	0.892101i
$894$	2.34147e6	0.0979818
$895$	0	0
$896$	−1.50799e6	−0.0627520
$897$	− 755541.i	− 0.0313528i
$898$	− 1.87239e7i	− 0.774827i
$899$	−6.91947e6	−0.285544
$900$	0	0
$901$	−685286.	−0.0281229
$902$	1.39162e6i	0.0569512i
$903$	− 8.27882e6i	− 0.337870i
$904$	7.25684e6	0.295343
$905$	0	0
$906$	3.96451e6	0.160461
$907$	1.13522e7i	0.458207i	0.973402	+	0.229104i	$0.0735795\pi$
$907$	1.13522e7i	0.458207i	−0.973402	+	0.229104i	$0.926421\pi$
$908$	1.00405e7i	0.404148i
$909$	3.81892e7	1.53296
$910$	0	0
$911$	−3.28560e7	−1.31165	−0.655826	−	0.754912i	$-0.727680\pi$
$911$	−3.28560e7	−1.31165	−0.655826	+	0.754912i	$0.727680\pi$
$912$	917692.i	0.0365351i
$913$	− 3.61139e6i	− 0.143383i
$914$	−2.74560e7	−1.08711
$915$	0	0
$916$	−1.58077e7	−0.622486
$917$	− 2.42327e7i	− 0.951653i
$918$	1.14682e7i	0.449146i
$919$	3.87685e7	1.51423	0.757113	−	0.653284i	$-0.226609\pi$
$919$	3.87685e7	1.51423	0.757113	+	0.653284i	$0.226609\pi$
$920$	0	0
$921$	−5.24750e6	−0.203847
$922$	− 1.54151e7i	− 0.597200i
$923$	1.51975e6i	0.0587175i
$924$	785470.	0.0302656
$925$	0	0
$926$	1.26330e7	0.484148
$927$	− 749054.i	− 0.0286295i
$928$	− 1.92568e6i	− 0.0734030i
$929$	−3.48798e7	−1.32597	−0.662986	−	0.748632i	$-0.730711\pi$
$929$	−3.48798e7	−1.32597	−0.662986	+	0.748632i	$0.730711\pi$
$930$	0	0
$931$	−6.77870e6	−0.256314
$932$	2.35251e6i	0.0887139i
$933$	− 5.55143e6i	− 0.208786i
$934$	784436.	0.0294232
$935$	0	0
$936$	−919195.	−0.0342940
$937$	4.14192e7i	1.54118i	0.637334	+	0.770588i	$0.280037\pi$
$937$	4.14192e7i	1.54118i	−0.637334	+	0.770588i	$0.719963\pi$
$938$	− 1.44157e6i	− 0.0534970i
$939$	9.85897e6	0.364895
$940$	0	0
$941$	−8.65196e6	−0.318523	−0.159261	−	0.987236i	$-0.550911\pi$
$941$	−8.65196e6	−0.318523	−0.159261	+	0.987236i	$0.550911\pi$
$942$	− 3.54335e6i	− 0.130103i
$943$	7.67131e6i	0.280925i
$944$	7.87859e6	0.287752
$945$	0	0
$946$	−9.87619e6	−0.358808
$947$	− 4.83595e7i	− 1.75229i	−0.482044	−	0.876147i	$-0.660106\pi$
$947$	− 4.83595e7i	− 1.75229i	0.482044	−	0.876147i	$-0.339894\pi$
$948$	− 412350.i	− 0.0149020i
$949$	3.19987e6	0.115337
$950$	0	0
$951$	−1.04085e7	−0.373196
$952$	− 8.21164e6i	− 0.293655i
$953$	− 3.85111e7i	− 1.37358i	−0.726857	−	0.686789i	$-0.759020\pi$
$953$	− 3.85111e7i	− 1.37358i	0.726857	−	0.686789i	$-0.240980\pi$
$954$	−439614.	−0.0156387
$955$	0	0
$956$	−2.17923e7	−0.771186
$957$	1.00303e6i	0.0354027i
$958$	− 2.05411e7i	− 0.723120i
$959$	−9.32216e6	−0.327318
$960$	0	0
$961$	−1.50904e7	−0.527100
$962$	− 3.95106e6i	− 0.137650i
$963$	9.76737e6i	0.339400i
$964$	−2.46832e7	−0.855478
$965$	0	0
$966$	4.32992e6	0.149292
$967$	4.49114e7i	1.54451i	0.635314	+	0.772254i	$0.280871\pi$
$967$	4.49114e7i	1.54451i	−0.635314	+	0.772254i	$0.719129\pi$
$968$	− 937024.i	− 0.0321412i
$969$	−4.99722e6	−0.170970
$970$	0	0
$971$	−5.38907e7	−1.83428	−0.917140	−	0.398565i	$-0.869508\pi$
$971$	−5.38907e7	−1.83428	−0.917140	+	0.398565i	$0.869508\pi$
$972$	1.11885e7i	0.379846i
$973$	− 1.43595e7i	− 0.486248i
$974$	9.75472e6	0.329471
$975$	0	0
$976$	−3.24834e6	−0.109153
$977$	− 5.38133e6i	− 0.180365i	−0.995925	−	0.0901827i	$-0.971255\pi$
$977$	− 5.38133e6i	− 0.180365i	0.995925	−	0.0901827i	$-0.0287451\pi$
$978$	− 5.55804e6i	− 0.185812i
$979$	1.33933e7	0.446611
$980$	0	0
$981$	−1.10388e7	−0.366226
$982$	1.63075e7i	0.539645i
$983$	− 4.23163e7i	− 1.39677i	−0.715724	−	0.698384i	$-0.753903\pi$
$983$	− 4.23163e7i	− 1.39677i	0.715724	−	0.698384i	$-0.246097\pi$
$984$	−811149.	−0.0267063
$985$	0	0
$986$	1.04861e7	0.343497
$987$	1.06061e7i	0.346548i
$988$	− 835884.i	− 0.0272429i
$989$	−5.44427e7	−1.76990
$990$	0	0
$991$	−1.67404e7	−0.541480	−0.270740	−	0.962652i	$-0.587268\pi$
$991$	−1.67404e7	−0.541480	−0.270740	+	0.962652i	$0.587268\pi$
$992$	3.76781e6i	0.121565i
$993$	1.34887e7i	0.434108i
$994$	−8.70951e6	−0.279594
$995$	0	0
$996$	2.10502e6	0.0672369
$997$	2.98987e7i	0.952608i	0.879281	+	0.476304i	$0.158024\pi$
$997$	2.98987e7i	0.952608i	−0.879281	+	0.476304i	$0.841976\pi$
$998$	− 3.51516e6i	− 0.111717i
$999$	−3.16228e7	−1.00250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.6.b.h.199.3		4
5.2	odd	4		110.6.a.e.1.1	✓	2
5.3	odd	4		550.6.a.j.1.2		2
5.4	even	2	inner	550.6.b.h.199.2		4
15.2	even	4		990.6.a.s.1.2		2
20.7	even	4		880.6.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.6.a.e.1.1	✓	2	5.2	odd	4
550.6.a.j.1.2		2	5.3	odd	4
550.6.b.h.199.2		4	5.4	even	2	inner
550.6.b.h.199.3		4	1.1	even	1	trivial
880.6.a.f.1.2		2	20.7	even	4
990.6.a.s.1.2		2	15.2	even	4

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:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	550.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -4.40805i q^{3} -16.0000 q^{4} +17.6322 q^{6} -92.0403i q^{7} -64.0000i q^{8} +223.569 q^{9} +121.000 q^{11} +70.5288i q^{12} -64.2415i q^{13} +368.161 q^{14} +256.000 q^{16} +1394.03i q^{17} +894.276i q^{18} -813.224 q^{19} -405.718 q^{21} +484.000i q^{22} +2668.06i q^{23} -282.115 q^{24} +256.966 q^{26} -2056.66i q^{27} +1472.64i q^{28} -1880.55 q^{29} +3679.50 q^{31} +1024.00i q^{32} -533.374i q^{33} -5576.12 q^{34} -3577.11 q^{36} -15375.8i q^{37} -3252.90i q^{38} -283.180 q^{39} +2875.24 q^{41} -1622.87i q^{42} +20405.3i q^{43} -1936.00 q^{44} -10672.2 q^{46} -26141.6i q^{47} -1128.46i q^{48} +8335.59 q^{49} +6144.95 q^{51} +1027.86i q^{52} +491.587i q^{53} +8226.64 q^{54} -5890.58 q^{56} +3584.73i q^{57} -7522.18i q^{58} +30775.7 q^{59} -12688.8 q^{61} +14718.0i q^{62} -20577.4i q^{63} -4096.00 q^{64} +2133.50 q^{66} -3915.60i q^{67} -22304.5i q^{68} +11760.9 q^{69} -23656.8 q^{71} -14308.4i q^{72} +49810.0i q^{73} +61503.2 q^{74} +13011.6 q^{76} -11136.9i q^{77} -1132.72i q^{78} -5846.55 q^{79} +45261.4 q^{81} +11501.0i q^{82} -29846.2i q^{83} +6491.49 q^{84} -81621.4 q^{86} +8289.54i q^{87} -7744.00i q^{88} +110688. q^{89} -5912.81 q^{91} -42688.9i q^{92} -16219.4i q^{93} +104566. q^{94} +4513.84 q^{96} +65866.2i q^{97} +33342.4i q^{98} +27051.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{4} - 168 q^{6} - 358 q^{9} + 484 q^{11} + 280 q^{14} + 1024 q^{16} - 3074 q^{19} - 3710 q^{21} + 2688 q^{24} - 6128 q^{26} - 15334 q^{29} + 9530 q^{31} + 2264 q^{34} + 5728 q^{36} - 42756 q^{39}+ \cdots - 43318 q^{99}+O(q^{100}) \) 4 * q - 64 * q^4 - 168 * q^6 - 358 * q^9 + 484 * q^11 + 280 * q^14 + 1024 * q^16 - 3074 * q^19 - 3710 * q^21 + 2688 * q^24 - 6128 * q^26 - 15334 * q^29 + 9530 * q^31 + 2264 * q^34 + 5728 * q^36 - 42756 * q^39 + 20088 * q^41 - 7744 * q^44 + 6448 * q^46 + 43778 * q^49 + 97510 * q^51 + 48888 * q^54 - 4480 * q^56 + 30196 * q^59 + 8698 * q^61 - 16384 * q^64 - 20328 * q^66 + 200060 * q^69 - 174594 * q^71 + 228600 * q^74 + 49184 * q^76 + 96236 * q^79 + 100900 * q^81 + 59360 * q^84 - 156176 * q^86 + 317346 * q^89 - 106540 * q^91 + 181168 * q^94 - 43008 * q^96 - 43318 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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