LMFDB - Embedded newform 275.4.b.c.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,4,Mod(199,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("275.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.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:	$16.2255252516$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.4.b.c.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051i q^{2} -5.92820i q^{3} +7.46410 q^{4} -4.33975 q^{6} +16.9282i q^{7} -11.3205i q^{8} -8.14359 q^{9} +O(q^{10})$	\(q-0.732051i q^{2} -5.92820i q^{3} +7.46410 q^{4} -4.33975 q^{6} +16.9282i q^{7} -11.3205i q^{8} -8.14359 q^{9} -11.0000 q^{11} -44.2487i q^{12} -74.6410i q^{13} +12.3923 q^{14} +51.4256 q^{16} -82.7846i q^{17} +5.96152i q^{18} +67.9230 q^{19} +100.354 q^{21} +8.05256i q^{22} -13.3538i q^{23} -67.1103 q^{24} -54.6410 q^{26} -111.785i q^{27} +126.354i q^{28} -168.995 q^{29} -65.4974 q^{31} -128.210i q^{32} +65.2102i q^{33} -60.6025 q^{34} -60.7846 q^{36} +40.8564i q^{37} -49.7231i q^{38} -442.487 q^{39} +274.928 q^{41} -73.4641i q^{42} +2.28719i q^{43} -82.1051 q^{44} -9.77568 q^{46} +71.8461i q^{47} -304.862i q^{48} +56.4359 q^{49} -490.764 q^{51} -557.128i q^{52} +149.005i q^{53} -81.8320 q^{54} +191.636 q^{56} -402.662i q^{57} +123.713i q^{58} -545.631 q^{59} +101.303 q^{61} +47.9474i q^{62} -137.856i q^{63} +317.549 q^{64} +47.7372 q^{66} +411.641i q^{67} -617.913i q^{68} -79.1642 q^{69} -470.636 q^{71} +92.1896i q^{72} -610.600i q^{73} +29.9090 q^{74} +506.985 q^{76} -186.210i q^{77} +323.923i q^{78} +978.225 q^{79} -882.559 q^{81} -201.261i q^{82} -26.1539i q^{83} +749.051 q^{84} +1.67434 q^{86} +1001.84i q^{87} +124.526i q^{88} +352.887 q^{89} +1263.54 q^{91} -99.6743i q^{92} +388.282i q^{93} +52.5950 q^{94} -760.056 q^{96} +847.585i q^{97} -41.3140i q^{98} +89.5795 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9}+O(q^{10}) $ 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9	$ 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9} - 44 q^{11} + 8 q^{14} - 16 q^{16} - 144 q^{19} + 152 q^{21} - 504 q^{24} - 80 q^{26} - 288 q^{29} - 68 q^{31} + 104 q^{34} - 160 q^{36} - 800 q^{39} + 1072 q^{41} - 176 q^{44} - 628 q^{46} + 780 q^{49} - 328 q^{51} + 220 q^{54} + 240 q^{56} - 1268 q^{59} + 1680 q^{61} - 448 q^{64} + 572 q^{66} - 1924 q^{69} - 1356 q^{71} - 12 q^{74} + 864 q^{76} - 632 q^{79} - 2588 q^{81} + 1472 q^{84} - 312 q^{86} + 3684 q^{89} + 2560 q^{91} + 1984 q^{94} - 1904 q^{96} + 968 q^{99}+O(q^{100}) $ 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9 - 44 * q^11 + 8 * q^14 - 16 * q^16 - 144 * q^19 + 152 * q^21 - 504 * q^24 - 80 * q^26 - 288 * q^29 - 68 * q^31 + 104 * q^34 - 160 * q^36 - 800 * q^39 + 1072 * q^41 - 176 * q^44 - 628 * q^46 + 780 * q^49 - 328 * q^51 + 220 * q^54 + 240 * q^56 - 1268 * q^59 + 1680 * q^61 - 448 * q^64 + 572 * q^66 - 1924 * q^69 - 1356 * q^71 - 12 * q^74 + 864 * q^76 - 632 * q^79 - 2588 * q^81 + 1472 * q^84 - 312 * q^86 + 3684 * q^89 + 2560 * q^91 + 1984 * q^94 - 1904 * q^96 + 968 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

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

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 $

$2$	− 0.732051i	− 0.258819i	−0.991591	−	0.129410i	$-0.958692\pi$
$2$	− 0.732051i	− 0.258819i	0.991591	−	0.129410i	$-0.0413082\pi$
$3$	− 5.92820i	− 1.14088i	−0.821338	−	0.570442i	$-0.806772\pi$
$3$	− 5.92820i	− 1.14088i	0.821338	−	0.570442i	$-0.193228\pi$
$4$	7.46410	0.933013
$5$	0	0
$5$	0	0
$6$	−4.33975	−0.295282
$7$	16.9282i	0.914037i	0.889457	+	0.457019i	$0.151083\pi$
$7$	16.9282i	0.914037i	−0.889457	+	0.457019i	$0.848917\pi$
$8$	− 11.3205i	− 0.500301i
$9$	−8.14359	−0.301615
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	− 44.2487i	− 1.06446i
$13$	− 74.6410i	− 1.59244i	−0.605009	−	0.796219i	$-0.706830\pi$
$13$	− 74.6410i	− 1.59244i	0.605009	−	0.796219i	$-0.293170\pi$
$14$	12.3923	0.236570
$15$	0	0
$16$	51.4256	0.803525
$17$	− 82.7846i	− 1.18107i	−0.807011	−	0.590536i	$-0.798916\pi$
$17$	− 82.7846i	− 1.18107i	0.807011	−	0.590536i	$-0.201084\pi$
$18$	5.96152i	0.0780636i
$19$	67.9230	0.820138	0.410069	−	0.912055i	$-0.365505\pi$
$19$	67.9230	0.820138	0.410069	+	0.912055i	$0.365505\pi$
$20$	0	0
$21$	100.354	1.04281
$22$	8.05256i	0.0780369i
$23$	− 13.3538i	− 0.121064i	−0.998166	−	0.0605319i	$-0.980720\pi$
$23$	− 13.3538i	− 0.121064i	0.998166	−	0.0605319i	$-0.0192797\pi$
$24$	−67.1103	−0.570784
$25$	0	0
$26$	−54.6410	−0.412153
$27$	− 111.785i	− 0.796776i
$28$	126.354i	0.852808i
$29$	−168.995	−1.08212	−0.541061	−	0.840983i	$-0.681977\pi$
$29$	−168.995	−1.08212	−0.541061	+	0.840983i	$0.681977\pi$
$30$	0	0
$31$	−65.4974	−0.379474	−0.189737	−	0.981835i	$-0.560763\pi$
$31$	−65.4974	−0.379474	−0.189737	+	0.981835i	$0.560763\pi$
$32$	− 128.210i	− 0.708268i
$33$	65.2102i	0.343989i
$34$	−60.6025	−0.305684
$35$	0	0
$36$	−60.7846	−0.281410
$37$	40.8564i	0.181534i	0.995872	+	0.0907669i	$0.0289318\pi$
$37$	40.8564i	0.181534i	−0.995872	+	0.0907669i	$0.971068\pi$
$38$	− 49.7231i	− 0.212267i
$39$	−442.487	−1.81679
$40$	0	0
$41$	274.928	1.04723	0.523617	−	0.851954i	$-0.324582\pi$
$41$	274.928	1.04723	0.523617	+	0.851954i	$0.324582\pi$
$42$	− 73.4641i	− 0.269899i
$43$	2.28719i	0.00811146i	0.999992	+	0.00405573i	$0.00129098\pi$
$43$	2.28719i	0.00811146i	−0.999992	+	0.00405573i	$0.998709\pi$
$44$	−82.1051	−0.281314
$45$	0	0
$46$	−9.77568	−0.0313336
$47$	71.8461i	0.222975i	0.993766	+	0.111488i	$0.0355615\pi$
$47$	71.8461i	0.222975i	−0.993766	+	0.111488i	$0.964438\pi$
$48$	− 304.862i	− 0.916729i
$49$	56.4359	0.164536
$50$	0	0
$51$	−490.764	−1.34746
$52$	− 557.128i	− 1.48576i
$53$	149.005i	0.386178i	0.981181	+	0.193089i	$0.0618506\pi$
$53$	149.005i	0.386178i	−0.981181	+	0.193089i	$0.938149\pi$
$54$	−81.8320	−0.206221
$55$	0	0
$56$	191.636	0.457293
$57$	− 402.662i	− 0.935681i
$58$	123.713i	0.280074i
$59$	−545.631	−1.20398	−0.601992	−	0.798502i	$-0.705626\pi$
$59$	−545.631	−1.20398	−0.601992	+	0.798502i	$0.705626\pi$
$60$	0	0
$61$	101.303	0.212631	0.106315	−	0.994332i	$-0.466095\pi$
$61$	101.303	0.212631	0.106315	+	0.994332i	$0.466095\pi$
$62$	47.9474i	0.0982150i
$63$	− 137.856i	− 0.275687i
$64$	317.549	0.620212
$65$	0	0
$66$	47.7372	0.0890310
$67$	411.641i	0.750596i	0.926904	+	0.375298i	$0.122460\pi$
$67$	411.641i	0.750596i	−0.926904	+	0.375298i	$0.877540\pi$
$68$	− 617.913i	− 1.10195i
$69$	−79.1642	−0.138120
$70$	0	0
$71$	−470.636	−0.786679	−0.393339	−	0.919393i	$-0.628680\pi$
$71$	−470.636	−0.786679	−0.393339	+	0.919393i	$0.628680\pi$
$72$	92.1896i	0.150898i
$73$	− 610.600i	− 0.978977i	−0.872010	−	0.489488i	$-0.837184\pi$
$73$	− 610.600i	− 0.978977i	0.872010	−	0.489488i	$-0.162816\pi$
$74$	29.9090	0.0469844
$75$	0	0
$76$	506.985	0.765199
$77$	− 186.210i	− 0.275593i
$78$	323.923i	0.470219i
$79$	978.225	1.39315	0.696576	−	0.717483i	$-0.254706\pi$
$79$	978.225	1.39315	0.696576	+	0.717483i	$0.254706\pi$
$80$	0	0
$81$	−882.559	−1.21064
$82$	− 201.261i	− 0.271044i
$83$	− 26.1539i	− 0.0345875i	−0.999850	−	0.0172938i	$-0.994495\pi$
$83$	− 26.1539i	− 0.0345875i	0.999850	−	0.0172938i	$-0.00550505\pi$
$84$	749.051	0.972955
$85$	0	0
$86$	1.67434	0.00209940
$87$	1001.84i	1.23458i
$88$	124.526i	0.150846i
$89$	352.887	0.420292	0.210146	−	0.977670i	$-0.432606\pi$
$89$	352.887	0.420292	0.210146	+	0.977670i	$0.432606\pi$
$90$	0	0
$91$	1263.54	1.45555
$92$	− 99.6743i	− 0.112954i
$93$	388.282i	0.432935i
$94$	52.5950	0.0577102
$95$	0	0
$96$	−760.056	−0.808051
$97$	847.585i	0.887208i	0.896223	+	0.443604i	$0.146300\pi$
$97$	847.585i	0.887208i	−0.896223	+	0.443604i	$0.853700\pi$
$98$	− 41.3140i	− 0.0425851i
$99$	89.5795	0.0909402
$100$	0	0
$101$	1293.46	1.27430	0.637150	−	0.770740i	$-0.280113\pi$
$101$	1293.46	1.27430	0.637150	+	0.770740i	$0.280113\pi$
$102$	359.264i	0.348750i
$103$	1725.24i	1.65042i	0.564828	+	0.825209i	$0.308943\pi$
$103$	1725.24i	1.65042i	−0.564828	+	0.825209i	$0.691057\pi$
$104$	−844.974	−0.796697
$105$	0	0
$106$	109.079	0.0999502
$107$	− 484.179i	− 0.437452i	−0.975786	−	0.218726i	$-0.929810\pi$
$107$	− 484.179i	− 0.437452i	0.975786	−	0.218726i	$-0.0701901\pi$
$108$	− 834.372i	− 0.743402i
$109$	64.2563	0.0564645	0.0282323	−	0.999601i	$-0.491012\pi$
$109$	64.2563	0.0564645	0.0282323	+	0.999601i	$0.491012\pi$
$110$	0	0
$111$	242.205	0.207109
$112$	870.543i	0.734452i
$113$	2005.08i	1.66922i	0.550839	+	0.834612i	$0.314308\pi$
$113$	2005.08i	1.66922i	−0.550839	+	0.834612i	$0.685692\pi$
$114$	−294.769	−0.242172
$115$	0	0
$116$	−1261.39	−1.00963
$117$	607.846i	0.480302i
$118$	399.429i	0.311614i
$119$	1401.39	1.07954
$120$	0	0
$121$	121.000	0.0909091
$122$	− 74.1587i	− 0.0550329i
$123$	− 1629.83i	− 1.19477i
$124$	−488.879	−0.354054
$125$	0	0
$126$	−100.918	−0.0713530
$127$	109.605i	0.0765816i	0.999267	+	0.0382908i	$0.0121913\pi$
$127$	109.605i	0.0765816i	−0.999267	+	0.0382908i	$0.987809\pi$
$128$	− 1258.14i	− 0.868791i
$129$	13.5589	0.00925423
$130$	0	0
$131$	1156.71	0.771469	0.385734	−	0.922610i	$-0.373948\pi$
$131$	1156.71	0.771469	0.385734	+	0.922610i	$0.373948\pi$
$132$	486.736i	0.320946i
$133$	1149.82i	0.749636i
$134$	301.342	0.194269
$135$	0	0
$136$	−937.164	−0.590891
$137$	198.323i	0.123678i	0.998086	+	0.0618391i	$0.0196965\pi$
$137$	198.323i	0.123678i	−0.998086	+	0.0618391i	$0.980303\pi$
$138$	57.9522i	0.0357480i
$139$	2900.14	1.76969	0.884844	−	0.465888i	$-0.154265\pi$
$139$	2900.14	1.76969	0.884844	+	0.465888i	$0.154265\pi$
$140$	0	0
$141$	425.918	0.254389
$142$	344.529i	0.203607i
$143$	821.051i	0.480138i
$144$	−418.789	−0.242355
$145$	0	0
$146$	−446.990	−0.253378
$147$	− 334.564i	− 0.187717i
$148$	304.956i	0.169373i
$149$	−3488.34	−1.91796	−0.958980	−	0.283472i	$-0.908514\pi$
$149$	−3488.34	−1.91796	−0.958980	+	0.283472i	$0.908514\pi$
$150$	0	0
$151$	−1163.32	−0.626953	−0.313477	−	0.949596i	$-0.601494\pi$
$151$	−1163.32	−0.626953	−0.313477	+	0.949596i	$0.601494\pi$
$152$	− 768.923i	− 0.410315i
$153$	674.164i	0.356228i
$154$	−136.315	−0.0713286
$155$	0	0
$156$	−3302.77	−1.69508
$157$	342.057i	0.173880i	0.996214	+	0.0869398i	$0.0277088\pi$
$157$	342.057i	0.173880i	−0.996214	+	0.0869398i	$0.972291\pi$
$158$	− 716.111i	− 0.360574i
$159$	883.333	0.440584
$160$	0	0
$161$	226.056	0.110657
$162$	646.078i	0.313338i
$163$	1394.89i	0.670285i	0.942167	+	0.335142i	$0.108784\pi$
$163$	1394.89i	0.670285i	−0.942167	+	0.335142i	$0.891216\pi$
$164$	2052.09	0.977082
$165$	0	0
$166$	−19.1460	−0.00895191
$167$	478.703i	0.221815i	0.993831	+	0.110908i	$0.0353758\pi$
$167$	478.703i	0.221815i	−0.993831	+	0.110908i	$0.964624\pi$
$168$	− 1136.06i	− 0.521718i
$169$	−3374.28	−1.53586
$170$	0	0
$171$	−553.138	−0.247365
$172$	17.0718i	0.00756809i
$173$	− 1808.58i	− 0.794822i	−0.917641	−	0.397411i	$-0.869909\pi$
$173$	− 1808.58i	− 0.794822i	0.917641	−	0.397411i	$-0.130091\pi$
$174$	733.395	0.319532
$175$	0	0
$176$	−565.682	−0.242272
$177$	3234.61i	1.37361i
$178$	− 258.331i	− 0.108780i
$179$	4429.85	1.84973	0.924867	−	0.380292i	$-0.124176\pi$
$179$	4429.85	1.84973	0.924867	+	0.380292i	$0.124176\pi$
$180$	0	0
$181$	3409.17	1.40001	0.700005	−	0.714138i	$-0.253181\pi$
$181$	3409.17	1.40001	0.700005	+	0.714138i	$0.253181\pi$
$182$	− 924.974i	− 0.376723i
$183$	− 600.543i	− 0.242587i
$184$	−151.172	−0.0605682
$185$	0	0
$186$	284.242	0.112052
$187$	910.631i	0.356106i
$188$	536.267i	0.208039i
$189$	1892.31	0.728283
$190$	0	0
$191$	2923.75	1.10762	0.553810	−	0.832643i	$-0.313173\pi$
$191$	2923.75	1.10762	0.553810	+	0.832643i	$0.313173\pi$
$192$	− 1882.49i	− 0.707590i
$193$	2484.18i	0.926505i	0.886226	+	0.463253i	$0.153318\pi$
$193$	2484.18i	0.926505i	−0.886226	+	0.463253i	$0.846682\pi$
$194$	620.475	0.229626
$195$	0	0
$196$	421.244	0.153514
$197$	− 5125.67i	− 1.85375i	−0.375369	−	0.926876i	$-0.622484\pi$
$197$	− 5125.67i	− 1.85375i	0.375369	−	0.926876i	$-0.377516\pi$
$198$	− 65.5768i	− 0.0235371i
$199$	7.69219	0.00274013	0.00137006	−	0.999999i	$-0.499564\pi$
$199$	7.69219	0.00274013	0.00137006	+	0.999999i	$0.499564\pi$
$200$	0	0
$201$	2440.29	0.856343
$202$	− 946.879i	− 0.329813i
$203$	− 2860.78i	− 0.989100i
$204$	−3663.11	−1.25720
$205$	0	0
$206$	1262.96	0.427160
$207$	108.748i	0.0365146i
$208$	− 3838.46i	− 1.27956i
$209$	−747.154	−0.247281
$210$	0	0
$211$	3107.34	1.01383	0.506915	−	0.861996i	$-0.330786\pi$
$211$	3107.34	1.01383	0.506915	+	0.861996i	$0.330786\pi$
$212$	1112.19i	0.360309i
$213$	2790.03i	0.897509i
$214$	−354.444	−0.113221
$215$	0	0
$216$	−1265.46	−0.398628
$217$	− 1108.75i	− 0.346853i
$218$	− 47.0388i	− 0.0146141i
$219$	−3619.76	−1.11690
$220$	0	0
$221$	−6179.13	−1.88078
$222$	− 177.306i	− 0.0536037i
$223$	12.3185i	0.00369913i	0.999998	+	0.00184957i	$0.000588736\pi$
$223$	12.3185i	0.00369913i	−0.999998	+	0.00184957i	$0.999411\pi$
$224$	2170.37	0.647383
$225$	0	0
$226$	1467.82	0.432027
$227$	4615.90i	1.34964i	0.737983	+	0.674820i	$0.235779\pi$
$227$	4615.90i	1.34964i	−0.737983	+	0.674820i	$0.764221\pi$
$228$	− 3005.51i	− 0.873003i
$229$	−5074.63	−1.46437	−0.732186	−	0.681105i	$-0.761500\pi$
$229$	−5074.63	−1.46437	−0.732186	+	0.681105i	$0.761500\pi$
$230$	0	0
$231$	−1103.89	−0.314419
$232$	1913.11i	0.541386i
$233$	− 211.683i	− 0.0595184i	−0.999557	−	0.0297592i	$-0.990526\pi$
$233$	− 211.683i	− 0.0595184i	0.999557	−	0.0297592i	$-0.00947405\pi$
$234$	444.974	0.124311
$235$	0	0
$236$	−4072.64	−1.12333
$237$	− 5799.12i	− 1.58942i
$238$	− 1025.89i	− 0.279406i
$239$	−4312.49	−1.16716	−0.583581	−	0.812055i	$-0.698349\pi$
$239$	−4312.49	−1.16716	−0.583581	+	0.812055i	$0.698349\pi$
$240$	0	0
$241$	−996.584	−0.266372	−0.133186	−	0.991091i	$-0.542521\pi$
$241$	−996.584	−0.266372	−0.133186	+	0.991091i	$0.542521\pi$
$242$	− 88.5781i	− 0.0235290i
$243$	2213.80i	0.584426i
$244$	756.133	0.198387
$245$	0	0
$246$	−1193.12	−0.309230
$247$	− 5069.85i	− 1.30602i
$248$	741.464i	0.189851i
$249$	−155.046	−0.0394603
$250$	0	0
$251$	−276.892	−0.0696306	−0.0348153	−	0.999394i	$-0.511084\pi$
$251$	−276.892	−0.0696306	−0.0348153	+	0.999394i	$0.511084\pi$
$252$	− 1028.97i	− 0.257219i
$253$	146.892i	0.0365021i
$254$	80.2364	0.0198208
$255$	0	0
$256$	1619.36	0.395352
$257$	− 3235.18i	− 0.785233i	−0.919702	−	0.392617i	$-0.871570\pi$
$257$	− 3235.18i	− 0.785233i	0.919702	−	0.392617i	$-0.128430\pi$
$258$	− 9.92581i	− 0.00239517i
$259$	−691.626	−0.165929
$260$	0	0
$261$	1376.23	0.326384
$262$	− 846.772i	− 0.199671i
$263$	− 207.944i	− 0.0487544i	−0.999703	−	0.0243772i	$-0.992240\pi$
$263$	− 207.944i	− 0.0487544i	0.999703	−	0.0243772i	$-0.00776027\pi$
$264$	738.213	0.172098
$265$	0	0
$266$	841.723	0.194020
$267$	− 2091.99i	− 0.479504i
$268$	3072.53i	0.700316i
$269$	−5033.04	−1.14078	−0.570390	−	0.821374i	$-0.693208\pi$
$269$	−5033.04	−1.14078	−0.570390	+	0.821374i	$0.693208\pi$
$270$	0	0
$271$	1487.01	0.333319	0.166660	−	0.986015i	$-0.446702\pi$
$271$	1487.01	0.333319	0.166660	+	0.986015i	$0.446702\pi$
$272$	− 4257.25i	− 0.949021i
$273$	− 7490.51i	− 1.66061i
$274$	145.183	0.0320102
$275$	0	0
$276$	−590.890	−0.128867
$277$	− 235.836i	− 0.0511552i	−0.999673	−	0.0255776i	$-0.991858\pi$
$277$	− 235.836i	− 0.0511552i	0.999673	−	0.0255776i	$-0.00814250\pi$
$278$	− 2123.05i	− 0.458029i
$279$	533.384	0.114455
$280$	0	0
$281$	−4915.01	−1.04343	−0.521717	−	0.853118i	$-0.674708\pi$
$281$	−4915.01	−1.04343	−0.521717	+	0.853118i	$0.674708\pi$
$282$	− 311.794i	− 0.0658406i
$283$	5199.56i	1.09216i	0.837733	+	0.546081i	$0.183881\pi$
$283$	5199.56i	1.09216i	−0.837733	+	0.546081i	$0.816119\pi$
$284$	−3512.87	−0.733981
$285$	0	0
$286$	601.051	0.124269
$287$	4654.04i	0.957210i
$288$	1044.09i	0.213624i
$289$	−1940.29	−0.394930
$290$	0	0
$291$	5024.65	1.01220
$292$	− 4557.58i	− 0.913398i
$293$	8880.92i	1.77075i	0.464881	+	0.885373i	$0.346097\pi$
$293$	8880.92i	1.77075i	−0.464881	+	0.885373i	$0.653903\pi$
$294$	−244.918	−0.0485846
$295$	0	0
$296$	462.515	0.0908215
$297$	1229.63i	0.240237i
$298$	2553.64i	0.496405i
$299$	−996.743	−0.192786
$300$	0	0
$301$	−38.7180	−0.00741417
$302$	851.612i	0.162267i
$303$	− 7667.90i	− 1.45383i
$304$	3492.99	0.659001
$305$	0	0
$306$	493.522	0.0921987
$307$	− 1497.93i	− 0.278474i	−0.990259	−	0.139237i	$-0.955535\pi$
$307$	− 1497.93i	− 0.278474i	0.990259	−	0.139237i	$-0.0444650\pi$
$308$	− 1389.89i	− 0.257131i
$309$	10227.6	1.88293
$310$	0	0
$311$	−7484.71	−1.36469	−0.682345	−	0.731030i	$-0.739040\pi$
$311$	−7484.71	−1.36469	−0.682345	+	0.731030i	$0.739040\pi$
$312$	5009.18i	0.908939i
$313$	658.363i	0.118891i	0.998232	+	0.0594455i	$0.0189333\pi$
$313$	658.363i	0.118891i	−0.998232	+	0.0594455i	$0.981067\pi$
$314$	250.403	0.0450034
$315$	0	0
$316$	7301.57	1.29983
$317$	233.708i	0.0414080i	0.999786	+	0.0207040i	$0.00659076\pi$
$317$	233.708i	0.0414080i	−0.999786	+	0.0207040i	$0.993409\pi$
$318$	− 646.645i	− 0.114032i
$319$	1858.94	0.326272
$320$	0	0
$321$	−2870.31	−0.499082
$322$	− 165.485i	− 0.0286401i
$323$	− 5622.98i	− 0.968641i
$324$	−6587.51	−1.12955
$325$	0	0
$326$	1021.13	0.173482
$327$	− 380.924i	− 0.0644194i
$328$	− 3112.33i	− 0.523931i
$329$	−1216.23	−0.203808
$330$	0	0
$331$	8532.95	1.41696	0.708480	−	0.705731i	$-0.249381\pi$
$331$	8532.95	1.41696	0.708480	+	0.705731i	$0.249381\pi$
$332$	− 195.215i	− 0.0322706i
$333$	− 332.718i	− 0.0547533i
$334$	350.435	0.0574100
$335$	0	0
$336$	5160.76	0.837924
$337$	11691.2i	1.88979i	0.327373	+	0.944895i	$0.393837\pi$
$337$	11691.2i	1.88979i	−0.327373	+	0.944895i	$0.606163\pi$
$338$	2470.15i	0.397509i
$339$	11886.5	1.90439
$340$	0	0
$341$	720.472	0.114416
$342$	404.925i	0.0640229i
$343$	6761.73i	1.06443i
$344$	25.8921	0.00405817
$345$	0	0
$346$	−1323.98	−0.205715
$347$	4598.79i	0.711459i	0.934589	+	0.355729i	$0.115768\pi$
$347$	4598.79i	0.711459i	−0.934589	+	0.355729i	$0.884232\pi$
$348$	7477.80i	1.15187i
$349$	−6720.27	−1.03074	−0.515369	−	0.856968i	$-0.672345\pi$
$349$	−6720.27	−1.03074	−0.515369	+	0.856968i	$0.672345\pi$
$350$	0	0
$351$	−8343.72	−1.26882
$352$	1410.31i	0.213551i
$353$	− 5738.70i	− 0.865270i	−0.901569	−	0.432635i	$-0.857584\pi$
$353$	− 5738.70i	− 0.865270i	0.901569	−	0.432635i	$-0.142416\pi$
$354$	2367.90	0.355515
$355$	0	0
$356$	2633.99	0.392138
$357$	− 8307.75i	− 1.23163i
$358$	− 3242.87i	− 0.478746i
$359$	4115.27	0.605001	0.302501	−	0.953149i	$-0.402179\pi$
$359$	4115.27	0.605001	0.302501	+	0.953149i	$0.402179\pi$
$360$	0	0
$361$	−2245.46	−0.327374
$362$	− 2495.69i	− 0.362349i
$363$	− 717.313i	− 0.103717i
$364$	9431.18	1.35804
$365$	0	0
$366$	−439.628	−0.0627861
$367$	9662.99i	1.37440i	0.726469	+	0.687199i	$0.241160\pi$
$367$	9662.99i	1.37440i	−0.726469	+	0.687199i	$0.758840\pi$
$368$	− 686.729i	− 0.0972778i
$369$	−2238.90	−0.315861
$370$	0	0
$371$	−2522.39	−0.352981
$372$	2898.18i	0.403934i
$373$	141.780i	0.0196812i	0.999952	+	0.00984062i	$0.00313242\pi$
$373$	141.780i	0.0196812i	−0.999952	+	0.00984062i	$0.996868\pi$
$374$	666.628	0.0921671
$375$	0	0
$376$	813.334	0.111555
$377$	12613.9i	1.72321i
$378$	− 1385.27i	− 0.188494i
$379$	2819.73	0.382163	0.191082	−	0.981574i	$-0.438800\pi$
$379$	2819.73	0.382163	0.191082	+	0.981574i	$0.438800\pi$
$380$	0	0
$381$	649.760	0.0873707
$382$	− 2140.34i	− 0.286673i
$383$	6337.84i	0.845557i	0.906233	+	0.422778i	$0.138945\pi$
$383$	6337.84i	0.845557i	−0.906233	+	0.422778i	$0.861055\pi$
$384$	−7458.53	−0.991189
$385$	0	0
$386$	1818.55	0.239797
$387$	− 18.6259i	− 0.00244653i
$388$	6326.46i	0.827776i
$389$	8805.25	1.14767	0.573836	−	0.818970i	$-0.305455\pi$
$389$	8805.25	1.14767	0.573836	+	0.818970i	$0.305455\pi$
$390$	0	0
$391$	−1105.49	−0.142985
$392$	− 638.883i	− 0.0823176i
$393$	− 6857.23i	− 0.880156i
$394$	−3752.25	−0.479786
$395$	0	0
$396$	668.631	0.0848484
$397$	4315.26i	0.545534i	0.962080	+	0.272767i	$0.0879387\pi$
$397$	4315.26i	0.545534i	−0.962080	+	0.272767i	$0.912061\pi$
$398$	− 5.63108i	0 0.000709197i
$399$	6816.34	0.855247
$400$	0	0
$401$	361.681	0.0450411	0.0225206	−	0.999746i	$-0.492831\pi$
$401$	361.681	0.0450411	0.0225206	+	0.999746i	$0.492831\pi$
$402$	− 1786.42i	− 0.221638i
$403$	4888.79i	0.604288i
$404$	9654.53	1.18894
$405$	0	0
$406$	−2094.24	−0.255998
$407$	− 449.420i	− 0.0547345i
$408$	5555.70i	0.674137i
$409$	−9220.50	−1.11473	−0.557365	−	0.830268i	$-0.688188\pi$
$409$	−9220.50	−1.11473	−0.557365	+	0.830268i	$0.688188\pi$
$410$	0	0
$411$	1175.70	0.141102
$412$	12877.4i	1.53986i
$413$	− 9236.55i	− 1.10049i
$414$	79.6092	0.00945067
$415$	0	0
$416$	−9569.74	−1.12787
$417$	− 17192.6i	− 2.01901i
$418$	546.954i	0.0640010i
$419$	14912.9	1.73876	0.869380	−	0.494144i	$-0.164519\pi$
$419$	14912.9	1.73876	0.869380	+	0.494144i	$0.164519\pi$
$420$	0	0
$421$	−13486.0	−1.56121	−0.780603	−	0.625027i	$-0.785088\pi$
$421$	−13486.0	−1.56121	−0.780603	+	0.625027i	$0.785088\pi$
$422$	− 2274.73i	− 0.262399i
$423$	− 585.085i	− 0.0672525i
$424$	1686.81	0.193205
$425$	0	0
$426$	2042.44	0.232292
$427$	1714.87i	0.194352i
$428$	− 3613.96i	− 0.408148i
$429$	4867.36	0.547782
$430$	0	0
$431$	406.334	0.0454116	0.0227058	−	0.999742i	$-0.492772\pi$
$431$	406.334	0.0454116	0.0227058	+	0.999742i	$0.492772\pi$
$432$	− 5748.59i	− 0.640230i
$433$	1766.69i	0.196078i	0.995183	+	0.0980391i	$0.0312570\pi$
$433$	1766.69i	0.196078i	−0.995183	+	0.0980391i	$0.968743\pi$
$434$	−811.664	−0.0897722
$435$	0	0
$436$	479.615	0.0526821
$437$	− 907.033i	− 0.0992889i
$438$	2649.85i	0.289075i
$439$	−7824.19	−0.850634	−0.425317	−	0.905044i	$-0.639837\pi$
$439$	−7824.19	−0.850634	−0.425317	+	0.905044i	$0.639837\pi$
$440$	0	0
$441$	−459.591	−0.0496265
$442$	4523.44i	0.486783i
$443$	− 11667.9i	− 1.25137i	−0.780075	−	0.625686i	$-0.784819\pi$
$443$	− 11667.9i	− 1.25137i	0.780075	−	0.625686i	$-0.215181\pi$
$444$	1807.84	0.193235
$445$	0	0
$446$	9.01776	0.000957406 0
$447$	20679.6i	2.18817i
$448$	5375.53i	0.566897i
$449$	−16975.3	−1.78421	−0.892107	−	0.451825i	$-0.850773\pi$
$449$	−16975.3	−1.78421	−0.892107	+	0.451825i	$0.850773\pi$
$450$	0	0
$451$	−3024.21	−0.315753
$452$	14966.1i	1.55741i
$453$	6896.42i	0.715280i
$454$	3379.07	0.349312
$455$	0	0
$456$	−4558.33	−0.468122
$457$	− 16192.9i	− 1.65748i	−0.559632	−	0.828741i	$-0.689057\pi$
$457$	− 16192.9i	− 1.65748i	0.559632	−	0.828741i	$-0.310943\pi$
$458$	3714.89i	0.379007i
$459$	−9254.05	−0.941050
$460$	0	0
$461$	8586.04	0.867444	0.433722	−	0.901047i	$-0.357200\pi$
$461$	8586.04	0.867444	0.433722	+	0.901047i	$0.357200\pi$
$462$	808.105i	0.0813776i
$463$	7917.20i	0.794694i	0.917668	+	0.397347i	$0.130069\pi$
$463$	7917.20i	0.794694i	−0.917668	+	0.397347i	$0.869931\pi$
$464$	−8690.67	−0.869513
$465$	0	0
$466$	−154.962	−0.0154045
$467$	− 15155.0i	− 1.50169i	−0.660480	−	0.750844i	$-0.729647\pi$
$467$	− 15155.0i	− 1.50169i	0.660480	−	0.750844i	$-0.270353\pi$
$468$	4537.03i	0.448128i
$469$	−6968.34	−0.686073
$470$	0	0
$471$	2027.78	0.198376
$472$	6176.82i	0.602354i
$473$	− 25.1591i	− 0.00244570i
$474$	−4245.25	−0.411373
$475$	0	0
$476$	10460.2	1.00723
$477$	− 1213.44i	− 0.116477i
$478$	3156.96i	0.302084i
$479$	−10001.1	−0.953993	−0.476996	−	0.878905i	$-0.658275\pi$
$479$	−10001.1	−0.953993	−0.476996	+	0.878905i	$0.658275\pi$
$480$	0	0
$481$	3049.56	0.289081
$482$	729.550i	0.0689421i
$483$	− 1340.11i	− 0.126246i
$484$	903.156	0.0848193
$485$	0	0
$486$	1620.62	0.151261
$487$	7044.54i	0.655480i	0.944768	+	0.327740i	$0.106287\pi$
$487$	7044.54i	0.655480i	−0.944768	+	0.327740i	$0.893713\pi$
$488$	− 1146.80i	− 0.106379i
$489$	8269.21	0.764717
$490$	0	0
$491$	−13326.4	−1.22487	−0.612437	−	0.790520i	$-0.709811\pi$
$491$	−13326.4	−1.22487	−0.612437	+	0.790520i	$0.709811\pi$
$492$	− 12165.2i	− 1.11474i
$493$	13990.2i	1.27806i
$494$	−3711.38	−0.338022
$495$	0	0
$496$	−3368.25	−0.304917
$497$	− 7967.02i	− 0.719054i
$498$	113.501i	0.0102131i
$499$	20069.1	1.80044	0.900218	−	0.435440i	$-0.143407\pi$
$499$	20069.1	1.80044	0.900218	+	0.435440i	$0.143407\pi$
$500$	0	0
$501$	2837.85	0.253065
$502$	202.699i	0.0180217i
$503$	− 7782.35i	− 0.689856i	−0.938629	−	0.344928i	$-0.887903\pi$
$503$	− 7782.35i	− 0.689856i	0.938629	−	0.344928i	$-0.112097\pi$
$504$	−1560.60	−0.137926
$505$	0	0
$506$	107.532	0.00944744
$507$	20003.4i	1.75224i
$508$	818.102i	0.0714516i
$509$	1475.93	0.128526	0.0642628	−	0.997933i	$-0.479530\pi$
$509$	1475.93	0.128526	0.0642628	+	0.997933i	$0.479530\pi$
$510$	0	0
$511$	10336.4	0.894821
$512$	− 11250.6i	− 0.971116i
$513$	− 7592.75i	− 0.653466i
$514$	−2368.32	−0.203233
$515$	0	0
$516$	101.205	0.00863431
$517$	− 790.307i	− 0.0672295i
$518$	506.305i	0.0429455i
$519$	−10721.7	−0.906799
$520$	0	0
$521$	7609.43	0.639875	0.319938	−	0.947439i	$-0.396338\pi$
$521$	7609.43	0.639875	0.319938	+	0.947439i	$0.396338\pi$
$522$	− 1007.47i	− 0.0844744i
$523$	− 12452.9i	− 1.04116i	−0.853812	−	0.520581i	$-0.825715\pi$
$523$	− 12452.9i	− 1.04116i	0.853812	−	0.520581i	$-0.174285\pi$
$524$	8633.82	0.719790
$525$	0	0
$526$	−152.226	−0.0126186
$527$	5422.18i	0.448186i
$528$	3353.48i	0.276404i
$529$	11988.7	0.985344
$530$	0	0
$531$	4443.39	0.363139
$532$	8582.34i	0.699420i
$533$	− 20520.9i	− 1.66765i
$534$	−1531.44	−0.124105
$535$	0	0
$536$	4659.99	0.375524
$537$	− 26261.0i	− 2.11033i
$538$	3684.44i	0.295255i
$539$	−620.795	−0.0496095
$540$	0	0
$541$	9312.17	0.740039	0.370020	−	0.929024i	$-0.379351\pi$
$541$	9312.17	0.740039	0.370020	+	0.929024i	$0.379351\pi$
$542$	− 1088.57i	− 0.0862693i
$543$	− 20210.3i	− 1.59725i
$544$	−10613.8	−0.836515
$545$	0	0
$546$	−5483.44	−0.429797
$547$	− 11018.6i	− 0.861278i	−0.902524	−	0.430639i	$-0.858288\pi$
$547$	− 11018.6i	− 0.861278i	0.902524	−	0.430639i	$-0.141712\pi$
$548$	1480.31i	0.115393i
$549$	−824.968	−0.0641325
$550$	0	0
$551$	−11478.6	−0.887490
$552$	896.179i	0.0691013i
$553$	16559.6i	1.27339i
$554$	−172.644	−0.0132399
$555$	0	0
$556$	21646.9	1.65114
$557$	− 12018.4i	− 0.914250i	−0.889403	−	0.457125i	$-0.848879\pi$
$557$	− 12018.4i	− 0.914250i	0.889403	−	0.457125i	$-0.151121\pi$
$558$	− 390.464i	− 0.0296231i
$559$	170.718	0.0129170
$560$	0	0
$561$	5398.40	0.406276
$562$	3598.04i	0.270061i
$563$	8763.89i	0.656046i	0.944670	+	0.328023i	$0.106382\pi$
$563$	8763.89i	0.656046i	−0.944670	+	0.328023i	$0.893618\pi$
$564$	3179.10	0.237348
$565$	0	0
$566$	3806.34	0.282672
$567$	− 14940.1i	− 1.10657i
$568$	5327.84i	0.393576i
$569$	10273.2	0.756895	0.378447	−	0.925623i	$-0.376458\pi$
$569$	10273.2	0.756895	0.378447	+	0.925623i	$0.376458\pi$
$570$	0	0
$571$	2602.62	0.190747	0.0953734	−	0.995442i	$-0.469596\pi$
$571$	2602.62	0.190747	0.0953734	+	0.995442i	$0.469596\pi$
$572$	6128.41i	0.447975i
$573$	− 17332.6i	− 1.26366i
$574$	3406.99	0.247744
$575$	0	0
$576$	−2585.99	−0.187065
$577$	− 19727.0i	− 1.42331i	−0.702532	−	0.711653i	$-0.747947\pi$
$577$	− 19727.0i	− 1.42331i	0.702532	−	0.711653i	$-0.252053\pi$
$578$	1420.39i	0.102215i
$579$	14726.7	1.05703
$580$	0	0
$581$	442.739	0.0316143
$582$	− 3678.30i	− 0.261977i
$583$	− 1639.06i	− 0.116437i
$584$	−6912.30	−0.489783
$585$	0	0
$586$	6501.28	0.458303
$587$	− 10116.2i	− 0.711309i	−0.934618	−	0.355654i	$-0.884258\pi$
$587$	− 10116.2i	− 0.711309i	0.934618	−	0.355654i	$-0.115742\pi$
$588$	− 2497.22i	− 0.175142i
$589$	−4448.78	−0.311221
$590$	0	0
$591$	−30386.0	−2.11491
$592$	2101.07i	0.145867i
$593$	− 3130.32i	− 0.216774i	−0.994109	−	0.108387i	$-0.965431\pi$
$593$	− 3130.32i	− 0.216774i	0.994109	−	0.108387i	$-0.0345685\pi$
$594$	900.152	0.0621779
$595$	0	0
$596$	−26037.3	−1.78948
$597$	− 45.6009i	− 0.00312616i
$598$	729.667i	0.0498968i
$599$	−10080.1	−0.687581	−0.343790	−	0.939046i	$-0.611711\pi$
$599$	−10080.1	−0.687581	−0.343790	+	0.939046i	$0.611711\pi$
$600$	0	0
$601$	4777.02	0.324224	0.162112	−	0.986772i	$-0.448169\pi$
$601$	4777.02	0.324224	0.162112	+	0.986772i	$0.448169\pi$
$602$	28.3435i	0.00191893i
$603$	− 3352.24i	− 0.226391i
$604$	−8683.16	−0.584955
$605$	0	0
$606$	−5613.29	−0.376278
$607$	− 2571.35i	− 0.171941i	−0.996298	−	0.0859703i	$-0.972601\pi$
$607$	− 2571.35i	− 0.171941i	0.996298	−	0.0859703i	$-0.0273990\pi$
$608$	− 8708.43i	− 0.580877i
$609$	−16959.3	−1.12845
$610$	0	0
$611$	5362.67	0.355074
$612$	5032.03i	0.332366i
$613$	− 12711.9i	− 0.837564i	−0.908087	−	0.418782i	$-0.862457\pi$
$613$	− 12711.9i	− 0.837564i	0.908087	−	0.418782i	$-0.137543\pi$
$614$	−1096.56	−0.0720744
$615$	0	0
$616$	−2107.99	−0.137879
$617$	16236.1i	1.05939i	0.848189	+	0.529693i	$0.177693\pi$
$617$	16236.1i	1.05939i	−0.848189	+	0.529693i	$0.822307\pi$
$618$	− 7487.11i	− 0.487339i
$619$	−12657.3	−0.821874	−0.410937	−	0.911664i	$-0.634798\pi$
$619$	−12657.3	−0.821874	−0.410937	+	0.911664i	$0.634798\pi$
$620$	0	0
$621$	−1492.75	−0.0964607
$622$	5479.19i	0.353208i
$623$	5973.75i	0.384162i
$624$	−22755.2	−1.45983
$625$	0	0
$626$	481.955	0.0307713
$627$	4429.28i	0.282119i
$628$	2553.15i	0.162232i
$629$	3382.28	0.214404
$630$	0	0
$631$	−3949.97	−0.249201	−0.124600	−	0.992207i	$-0.539765\pi$
$631$	−3949.97	−0.249201	−0.124600	+	0.992207i	$0.539765\pi$
$632$	− 11074.0i	− 0.696994i
$633$	− 18421.0i	− 1.15666i
$634$	171.086	0.0107172
$635$	0	0
$636$	6593.29	0.411070
$637$	− 4212.44i	− 0.262014i
$638$	− 1360.84i	− 0.0844455i
$639$	3832.67	0.237274
$640$	0	0
$641$	−7398.27	−0.455872	−0.227936	−	0.973676i	$-0.573198\pi$
$641$	−7398.27	−0.455872	−0.227936	+	0.973676i	$0.573198\pi$
$642$	2101.22i	0.129172i
$643$	12491.7i	0.766134i	0.923721	+	0.383067i	$0.125132\pi$
$643$	12491.7i	0.766134i	−0.923721	+	0.383067i	$0.874868\pi$
$644$	1687.31	0.103244
$645$	0	0
$646$	−4116.31	−0.250703
$647$	− 10472.0i	− 0.636315i	−0.948038	−	0.318158i	$-0.896936\pi$
$647$	− 10472.0i	− 0.636315i	0.948038	−	0.318158i	$-0.103064\pi$
$648$	9991.02i	0.605685i
$649$	6001.94	0.363015
$650$	0	0
$651$	−6572.92	−0.395719
$652$	10411.6i	0.625384i
$653$	− 6337.94i	− 0.379820i	−0.981801	−	0.189910i	$-0.939180\pi$
$653$	− 6337.94i	− 0.379820i	0.981801	−	0.189910i	$-0.0608196\pi$
$654$	−278.856	−0.0166730
$655$	0	0
$656$	14138.4	0.841479
$657$	4972.48i	0.295274i
$658$	890.339i	0.0527493i
$659$	−15196.7	−0.898302	−0.449151	−	0.893456i	$-0.648274\pi$
$659$	−15196.7	−0.898302	−0.449151	+	0.893456i	$0.648274\pi$
$660$	0	0
$661$	2298.17	0.135232	0.0676161	−	0.997711i	$-0.478461\pi$
$661$	2298.17	0.135232	0.0676161	+	0.997711i	$0.478461\pi$
$662$	− 6246.55i	− 0.366736i
$663$	36631.1i	2.14575i
$664$	−296.075	−0.0173042
$665$	0	0
$666$	−243.566	−0.0141712
$667$	2256.73i	0.131006i
$668$	3573.09i	0.206956i
$669$	73.0265	0.00422028
$670$	0	0
$671$	−1114.33	−0.0641106
$672$	− 12866.4i	− 0.738589i
$673$	− 23199.6i	− 1.32880i	−0.747379	−	0.664398i	$-0.768688\pi$
$673$	− 23199.6i	− 1.32880i	0.747379	−	0.664398i	$-0.231312\pi$
$674$	8558.54	0.489114
$675$	0	0
$676$	−25186.0	−1.43298
$677$	− 2145.38i	− 0.121793i	−0.998144	−	0.0608963i	$-0.980604\pi$
$677$	− 2145.38i	− 0.121793i	0.998144	−	0.0608963i	$-0.0193959\pi$
$678$	− 8701.55i	− 0.492892i
$679$	−14348.1	−0.810941
$680$	0	0
$681$	27364.0	1.53978
$682$	− 527.422i	− 0.0296129i
$683$	29544.6i	1.65519i	0.561329	+	0.827593i	$0.310290\pi$
$683$	29544.6i	1.65519i	−0.561329	+	0.827593i	$0.689710\pi$
$684$	−4128.68	−0.230795
$685$	0	0
$686$	4949.93	0.275495
$687$	30083.4i	1.67068i
$688$	117.620i	0.00651776i
$689$	11121.9	0.614964
$690$	0	0
$691$	27803.1	1.53065	0.765325	−	0.643644i	$-0.222578\pi$
$691$	27803.1	1.53065	0.765325	+	0.643644i	$0.222578\pi$
$692$	− 13499.5i	− 0.741579i
$693$	1516.42i	0.0831227i
$694$	3366.55	0.184139
$695$	0	0
$696$	11341.3	0.617659
$697$	− 22759.8i	− 1.23686i
$698$	4919.58i	0.266775i
$699$	−1254.90	−0.0679036
$700$	0	0
$701$	−19697.8	−1.06130	−0.530652	−	0.847590i	$-0.678053\pi$
$701$	−19697.8	−1.06130	−0.530652	+	0.847590i	$0.678053\pi$
$702$	6108.02i	0.328394i
$703$	2775.09i	0.148883i
$704$	−3493.03	−0.187001
$705$	0	0
$706$	−4201.02	−0.223948
$707$	21896.0i	1.16476i
$708$	24143.5i	1.28159i
$709$	−19122.5	−1.01292	−0.506460	−	0.862263i	$-0.669046\pi$
$709$	−19122.5	−1.01292	−0.506460	+	0.862263i	$0.669046\pi$
$710$	0	0
$711$	−7966.27	−0.420195
$712$	− 3994.86i	− 0.210272i
$713$	874.641i	0.0459405i
$714$	−6081.70	−0.318770
$715$	0	0
$716$	33064.8	1.72582
$717$	25565.3i	1.33160i
$718$	− 3012.59i	− 0.156586i
$719$	−1837.44	−0.0953060	−0.0476530	−	0.998864i	$-0.515174\pi$
$719$	−1837.44	−0.0953060	−0.0476530	+	0.998864i	$0.515174\pi$
$720$	0	0
$721$	−29205.2	−1.50854
$722$	1643.79i	0.0847307i
$723$	5907.95i	0.303899i
$724$	25446.4	1.30623
$725$	0	0
$726$	−525.109	−0.0268438
$727$	− 7555.46i	− 0.385442i	−0.981254	−	0.192721i	$-0.938269\pi$
$727$	− 7555.46i	− 0.385442i	0.981254	−	0.192721i	$-0.0617312\pi$
$728$	− 14303.9i	− 0.728211i
$729$	−10705.2	−0.543881
$730$	0	0
$731$	189.344	0.00958021
$732$	− 4482.51i	− 0.226337i
$733$	− 11984.6i	− 0.603905i	−0.953323	−	0.301952i	$-0.902362\pi$
$733$	− 11984.6i	− 0.603905i	0.953323	−	0.301952i	$-0.0976384\pi$
$734$	7073.80	0.355720
$735$	0	0
$736$	−1712.10	−0.0857456
$737$	− 4528.05i	− 0.226313i
$738$	1638.99i	0.0817508i
$739$	27142.5	1.35109	0.675543	−	0.737321i	$-0.263909\pi$
$739$	27142.5	1.35109	0.675543	+	0.737321i	$0.263909\pi$
$740$	0	0
$741$	−30055.1	−1.49001
$742$	1846.52i	0.0913582i
$743$	29222.6i	1.44290i	0.692467	+	0.721450i	$0.256524\pi$
$743$	29222.6i	1.44290i	−0.692467	+	0.721450i	$0.743476\pi$
$744$	4395.55	0.216598
$745$	0	0
$746$	103.790	0.00509388
$747$	212.987i	0.0104321i
$748$	6797.04i	0.332252i
$749$	8196.29	0.399847
$750$	0	0
$751$	−8859.39	−0.430471	−0.215236	−	0.976562i	$-0.569052\pi$
$751$	−8859.39	−0.430471	−0.215236	+	0.976562i	$0.569052\pi$
$752$	3694.73i	0.179166i
$753$	1641.47i	0.0794404i
$754$	9234.05	0.446000
$755$	0	0
$756$	14124.4	0.679497
$757$	35734.4i	1.71571i	0.513896	+	0.857853i	$0.328202\pi$
$757$	35734.4i	1.71571i	−0.513896	+	0.857853i	$0.671798\pi$
$758$	− 2064.19i	− 0.0989112i
$759$	870.806	0.0416446
$760$	0	0
$761$	34394.7	1.63838	0.819189	−	0.573524i	$-0.194424\pi$
$761$	34394.7	1.63838	0.819189	+	0.573524i	$0.194424\pi$
$762$	− 475.658i	− 0.0226132i
$763$	1087.74i	0.0516107i
$764$	21823.2	1.03342
$765$	0	0
$766$	4639.62	0.218846
$767$	40726.4i	1.91727i
$768$	− 9599.92i	− 0.451051i
$769$	11602.7	0.544091	0.272045	−	0.962284i	$-0.412300\pi$
$769$	11602.7	0.544091	0.272045	+	0.962284i	$0.412300\pi$
$770$	0	0
$771$	−19178.8	−0.895859
$772$	18542.2i	0.864441i
$773$	12680.6i	0.590026i	0.955493	+	0.295013i	$0.0953239\pi$
$773$	12680.6i	0.590026i	−0.955493	+	0.295013i	$0.904676\pi$
$774$	−13.6351	−0.000633210 0
$775$	0	0
$776$	9595.09	0.443871
$777$	4100.10i	0.189305i
$778$	− 6445.89i	− 0.297039i
$779$	18674.0	0.858876
$780$	0	0
$781$	5176.99	0.237193
$782$	809.276i	0.0370072i
$783$	18891.0i	0.862210i
$784$	2902.25	0.132209
$785$	0	0
$786$	−5019.84	−0.227801
$787$	− 4417.61i	− 0.200090i	−0.994983	−	0.100045i	$-0.968101\pi$
$787$	− 4417.61i	− 0.200090i	0.994983	−	0.100045i	$-0.0318987\pi$
$788$	− 38258.5i	− 1.72957i
$789$	−1232.74	−0.0556231
$790$	0	0
$791$	−33942.4	−1.52573
$792$	− 1014.09i	− 0.0454974i
$793$	− 7561.33i	− 0.338601i
$794$	3158.99	0.141194
$795$	0	0
$796$	57.4153	0.00255657
$797$	− 27030.1i	− 1.20132i	−0.799504	−	0.600661i	$-0.794904\pi$
$797$	− 27030.1i	− 1.20132i	0.799504	−	0.600661i	$-0.205096\pi$
$798$	− 4989.91i	− 0.221354i
$799$	5947.75	0.263350
$800$	0	0
$801$	−2873.77	−0.126766
$802$	− 264.769i	− 0.0116575i
$803$	6716.60i	0.295173i
$804$	18214.6	0.798979
$805$	0	0
$806$	3578.85	0.156401
$807$	29836.9i	1.30150i
$808$	− 14642.6i	− 0.637533i
$809$	−23647.0	−1.02767	−0.513835	−	0.857889i	$-0.671776\pi$
$809$	−23647.0	−1.02767	−0.513835	+	0.857889i	$0.671776\pi$
$810$	0	0
$811$	33486.1	1.44988	0.724941	−	0.688811i	$-0.241867\pi$
$811$	33486.1	1.44988	0.724941	+	0.688811i	$0.241867\pi$
$812$	− 21353.1i	− 0.922843i
$813$	− 8815.30i	− 0.380278i
$814$	−328.999	−0.0141663
$815$	0	0
$816$	−25237.8	−1.08272
$817$	155.353i	0.00665251i
$818$	6749.88i	0.288513i
$819$	−10289.7	−0.439014
$820$	0	0
$821$	2605.69	0.110766	0.0553832	−	0.998465i	$-0.482362\pi$
$821$	2605.69	0.110766	0.0553832	+	0.998465i	$0.482362\pi$
$822$	− 860.673i	− 0.0365200i
$823$	− 31976.2i	− 1.35434i	−0.735828	−	0.677169i	$-0.763207\pi$
$823$	− 31976.2i	− 1.35434i	0.735828	−	0.677169i	$-0.236793\pi$
$824$	19530.6	0.825705
$825$	0	0
$826$	−6761.62	−0.284827
$827$	− 37759.0i	− 1.58768i	−0.608128	−	0.793839i	$-0.708079\pi$
$827$	− 37759.0i	− 1.58768i	0.608128	−	0.793839i	$-0.291921\pi$
$828$	811.707i	0.0340686i
$829$	1137.55	0.0476584	0.0238292	−	0.999716i	$-0.492414\pi$
$829$	1137.55	0.0476584	0.0238292	+	0.999716i	$0.492414\pi$
$830$	0	0
$831$	−1398.08	−0.0583621
$832$	− 23702.2i	− 0.987649i
$833$	− 4672.03i	− 0.194329i
$834$	−12585.9	−0.522557
$835$	0	0
$836$	−5576.83	−0.230716
$837$	7321.60i	0.302356i
$838$	− 10917.0i	− 0.450024i
$839$	37372.2	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$839$	37372.2	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$840$	0	0
$841$	4170.26	0.170989
$842$	9872.44i	0.404070i
$843$	29137.2i	1.19044i
$844$	23193.5	0.945917
$845$	0	0
$846$	−428.312	−0.0174062
$847$	2048.31i	0.0830943i
$848$	7662.68i	0.310304i
$849$	30824.0	1.24603
$850$	0	0
$851$	545.589	0.0219772
$852$	20825.0i	0.837387i
$853$	22490.8i	0.902780i	0.892327	+	0.451390i	$0.149072\pi$
$853$	22490.8i	0.902780i	−0.892327	+	0.451390i	$0.850928\pi$
$854$	1255.37	0.0503021
$855$	0	0
$856$	−5481.16	−0.218858
$857$	43409.5i	1.73027i	0.501539	+	0.865135i	$0.332767\pi$
$857$	43409.5i	1.73027i	−0.501539	+	0.865135i	$0.667233\pi$
$858$	− 3563.15i	− 0.141776i
$859$	−29533.2	−1.17306	−0.586532	−	0.809926i	$-0.699507\pi$
$859$	−29533.2	−1.17306	−0.586532	+	0.809926i	$0.699507\pi$
$860$	0	0
$861$	27590.1	1.09207
$862$	− 297.457i	− 0.0117534i
$863$	− 14351.6i	− 0.566090i	−0.959107	−	0.283045i	$-0.908655\pi$
$863$	− 14351.6i	− 0.566090i	0.959107	−	0.283045i	$-0.0913446\pi$
$864$	−14331.9	−0.564331
$865$	0	0
$866$	1293.31	0.0507488
$867$	11502.4i	0.450569i
$868$	− 8275.85i	− 0.323618i
$869$	−10760.5	−0.420051
$870$	0	0
$871$	30725.3	1.19528
$872$	− 727.413i	− 0.0282492i
$873$	− 6902.39i	− 0.267595i
$874$	−663.994	−0.0256979
$875$	0	0
$876$	−27018.3	−1.04208
$877$	− 43248.7i	− 1.66523i	−0.553854	−	0.832614i	$-0.686844\pi$
$877$	− 43248.7i	− 1.66523i	0.553854	−	0.832614i	$-0.313156\pi$
$878$	5727.71i	0.220160i
$879$	52647.9	2.02021
$880$	0	0
$881$	3816.13	0.145935	0.0729675	−	0.997334i	$-0.476753\pi$
$881$	3816.13	0.145935	0.0729675	+	0.997334i	$0.476753\pi$
$882$	336.444i	0.0128443i
$883$	− 48787.6i	− 1.85938i	−0.368343	−	0.929690i	$-0.620075\pi$
$883$	− 48787.6i	− 1.85938i	0.368343	−	0.929690i	$-0.379925\pi$
$884$	−46121.6	−1.75479
$885$	0	0
$886$	−8541.49	−0.323879
$887$	41495.1i	1.57077i	0.619009	+	0.785384i	$0.287534\pi$
$887$	41495.1i	1.57077i	−0.619009	+	0.785384i	$0.712466\pi$
$888$	− 2741.88i	− 0.103617i
$889$	−1855.41	−0.0699984
$890$	0	0
$891$	9708.15	0.365023
$892$	91.9464i	0.00345134i
$893$	4880.01i	0.182870i
$894$	15138.5	0.566340
$895$	0	0
$896$	21298.1	0.794107
$897$	5908.90i	0.219947i
$898$	12426.7i	0.461788i
$899$	11068.7	0.410637
$900$	0	0
$901$	12335.3	0.456104
$902$	2213.88i	0.0817228i
$903$	229.528i	0.00845871i
$904$	22698.5	0.835113
$905$	0	0
$906$	5048.53	0.185128
$907$	21615.3i	0.791316i	0.918398	+	0.395658i	$0.129483\pi$
$907$	21615.3i	0.791316i	−0.918398	+	0.395658i	$0.870517\pi$
$908$	34453.6i	1.25923i
$909$	−10533.4	−0.384347
$910$	0	0
$911$	3646.35	0.132611	0.0663057	−	0.997799i	$-0.478879\pi$
$911$	3646.35	0.132611	0.0663057	+	0.997799i	$0.478879\pi$
$912$	− 20707.1i	− 0.751844i
$913$	287.693i	0.0104285i
$914$	−11854.0	−0.428988
$915$	0	0
$916$	−37877.6	−1.36628
$917$	19581.1i	0.705151i
$918$	6774.43i	0.243562i
$919$	−31280.0	−1.12278	−0.561388	−	0.827553i	$-0.689733\pi$
$919$	−31280.0	−1.12278	−0.561388	+	0.827553i	$0.689733\pi$
$920$	0	0
$921$	−8880.05	−0.317707
$922$	− 6285.42i	− 0.224511i
$923$	35128.7i	1.25274i
$924$	−8239.56	−0.293357
$925$	0	0
$926$	5795.79	0.205682
$927$	− 14049.7i	− 0.497790i
$928$	21666.9i	0.766433i
$929$	−6557.92	−0.231602	−0.115801	−	0.993272i	$-0.536944\pi$
$929$	−6557.92	−0.231602	−0.115801	+	0.993272i	$0.536944\pi$
$930$	0	0
$931$	3833.30	0.134942
$932$	− 1580.02i	− 0.0555314i
$933$	44370.9i	1.55695i
$934$	−11094.2	−0.388665
$935$	0	0
$936$	6881.13	0.240296
$937$	− 24473.3i	− 0.853265i	−0.904425	−	0.426632i	$-0.859700\pi$
$937$	− 24473.3i	− 0.853265i	0.904425	−	0.426632i	$-0.140300\pi$
$938$	5101.18i	0.177569i
$939$	3902.91	0.135641
$940$	0	0
$941$	15420.8	0.534224	0.267112	−	0.963665i	$-0.413931\pi$
$941$	15420.8	0.534224	0.267112	+	0.963665i	$0.413931\pi$
$942$	− 1484.44i	− 0.0513436i
$943$	− 3671.34i	− 0.126782i
$944$	−28059.4	−0.967432
$945$	0	0
$946$	−18.4177	−0.000632993 0
$947$	− 33141.2i	− 1.13722i	−0.822609	−	0.568608i	$-0.807482\pi$
$947$	− 33141.2i	− 1.13722i	0.822609	−	0.568608i	$-0.192518\pi$
$948$	− 43285.2i	− 1.48295i
$949$	−45575.8	−1.55896
$950$	0	0
$951$	1385.47	0.0472417
$952$	− 15864.5i	− 0.540096i
$953$	20735.4i	0.704813i	0.935847	+	0.352406i	$0.114637\pi$
$953$	20735.4i	0.704813i	−0.935847	+	0.352406i	$0.885363\pi$
$954$	−888.298	−0.0301464
$955$	0	0
$956$	−32188.9	−1.08898
$957$	− 11020.2i	− 0.372239i
$958$	7321.32i	0.246911i
$959$	−3357.26	−0.113046
$960$	0	0
$961$	−25501.1	−0.856000
$962$	− 2232.44i	− 0.0748198i
$963$	3942.96i	0.131942i
$964$	−7438.60	−0.248528
$965$	0	0
$966$	−981.027	−0.0326750
$967$	− 8178.87i	− 0.271990i	−0.990710	−	0.135995i	$-0.956577\pi$
$967$	− 8178.87i	− 0.271990i	0.990710	−	0.135995i	$-0.0434232\pi$
$968$	− 1369.78i	− 0.0454819i
$969$	−33334.2	−1.10511
$970$	0	0
$971$	−20576.1	−0.680039	−0.340020	−	0.940418i	$-0.610434\pi$
$971$	−20576.1	−0.680039	−0.340020	+	0.940418i	$0.610434\pi$
$972$	16524.1i	0.545277i
$973$	49094.1i	1.61756i
$974$	5156.96	0.169651
$975$	0	0
$976$	5209.55	0.170854
$977$	14541.9i	0.476188i	0.971242	+	0.238094i	$0.0765226\pi$
$977$	14541.9i	0.476188i	−0.971242	+	0.238094i	$0.923477\pi$
$978$	− 6053.48i	− 0.197923i
$979$	−3881.76	−0.126723
$980$	0	0
$981$	−523.277	−0.0170305
$982$	9755.62i	0.317021i
$983$	29285.7i	0.950223i	0.879926	+	0.475111i	$0.157592\pi$
$983$	29285.7i	0.950223i	−0.879926	+	0.475111i	$0.842408\pi$
$984$	−18450.5	−0.597745
$985$	0	0
$986$	10241.5	0.330787
$987$	7210.03i	0.232521i
$988$	− 37841.8i	− 1.21853i
$989$	30.5427	0.000982004 0
$990$	0	0
$991$	−38085.9	−1.22083	−0.610413	−	0.792083i	$-0.708997\pi$
$991$	−38085.9	−1.22083	−0.610413	+	0.792083i	$0.708997\pi$
$992$	8397.44i	0.268769i
$993$	− 50585.1i	− 1.61658i
$994$	−5832.26	−0.186105
$995$	0	0
$996$	−1157.28	−0.0368170
$997$	− 26803.6i	− 0.851434i	−0.904856	−	0.425717i	$-0.860022\pi$
$997$	− 26803.6i	− 0.851434i	0.904856	−	0.425717i	$-0.139978\pi$
$998$	− 14691.6i	− 0.465987i
$999$	4567.12	0.144642

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.b.c.199.2		4
5.2	odd	4		275.4.a.b.1.2		2
5.3	odd	4		11.4.a.a.1.1	✓	2
5.4	even	2	inner	275.4.b.c.199.3		4
15.2	even	4		2475.4.a.q.1.1		2
15.8	even	4		99.4.a.c.1.2		2
20.3	even	4		176.4.a.i.1.1		2
35.13	even	4		539.4.a.e.1.1		2
40.3	even	4		704.4.a.n.1.2		2
40.13	odd	4		704.4.a.p.1.1		2
55.3	odd	20		121.4.c.c.9.1		8
55.8	even	20		121.4.c.f.9.2		8
55.13	even	20		121.4.c.f.81.1		8
55.18	even	20		121.4.c.f.27.2		8
55.28	even	20		121.4.c.f.3.1		8
55.38	odd	20		121.4.c.c.3.2		8
55.43	even	4		121.4.a.c.1.2		2
55.48	odd	20		121.4.c.c.27.1		8
55.53	odd	20		121.4.c.c.81.2		8
60.23	odd	4		1584.4.a.bc.1.2		2
65.38	odd	4		1859.4.a.a.1.2		2
165.98	odd	4		1089.4.a.v.1.1		2
220.43	odd	4		1936.4.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.1	✓	2	5.3	odd	4
99.4.a.c.1.2		2	15.8	even	4
121.4.a.c.1.2		2	55.43	even	4
121.4.c.c.3.2		8	55.38	odd	20
121.4.c.c.9.1		8	55.3	odd	20
121.4.c.c.27.1		8	55.48	odd	20
121.4.c.c.81.2		8	55.53	odd	20
121.4.c.f.3.1		8	55.28	even	20
121.4.c.f.9.2		8	55.8	even	20
121.4.c.f.27.2		8	55.18	even	20
121.4.c.f.81.1		8	55.13	even	20
176.4.a.i.1.1		2	20.3	even	4
275.4.a.b.1.2		2	5.2	odd	4
275.4.b.c.199.2		4	1.1	even	1	trivial
275.4.b.c.199.3		4	5.4	even	2	inner
539.4.a.e.1.1		2	35.13	even	4
704.4.a.n.1.2		2	40.3	even	4
704.4.a.p.1.1		2	40.13	odd	4
1089.4.a.v.1.1		2	165.98	odd	4
1584.4.a.bc.1.2		2	60.23	odd	4
1859.4.a.a.1.2		2	65.38	odd	4
1936.4.a.w.1.1		2	220.43	odd	4
2475.4.a.q.1.1		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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.732051i q^{2} -5.92820i q^{3} +7.46410 q^{4} -4.33975 q^{6} +16.9282i q^{7} -11.3205i q^{8} -8.14359 q^{9} +O(q^{10})\)	\(q-0.732051i q^{2} -5.92820i q^{3} +7.46410 q^{4} -4.33975 q^{6} +16.9282i q^{7} -11.3205i q^{8} -8.14359 q^{9} -11.0000 q^{11} -44.2487i q^{12} -74.6410i q^{13} +12.3923 q^{14} +51.4256 q^{16} -82.7846i q^{17} +5.96152i q^{18} +67.9230 q^{19} +100.354 q^{21} +8.05256i q^{22} -13.3538i q^{23} -67.1103 q^{24} -54.6410 q^{26} -111.785i q^{27} +126.354i q^{28} -168.995 q^{29} -65.4974 q^{31} -128.210i q^{32} +65.2102i q^{33} -60.6025 q^{34} -60.7846 q^{36} +40.8564i q^{37} -49.7231i q^{38} -442.487 q^{39} +274.928 q^{41} -73.4641i q^{42} +2.28719i q^{43} -82.1051 q^{44} -9.77568 q^{46} +71.8461i q^{47} -304.862i q^{48} +56.4359 q^{49} -490.764 q^{51} -557.128i q^{52} +149.005i q^{53} -81.8320 q^{54} +191.636 q^{56} -402.662i q^{57} +123.713i q^{58} -545.631 q^{59} +101.303 q^{61} +47.9474i q^{62} -137.856i q^{63} +317.549 q^{64} +47.7372 q^{66} +411.641i q^{67} -617.913i q^{68} -79.1642 q^{69} -470.636 q^{71} +92.1896i q^{72} -610.600i q^{73} +29.9090 q^{74} +506.985 q^{76} -186.210i q^{77} +323.923i q^{78} +978.225 q^{79} -882.559 q^{81} -201.261i q^{82} -26.1539i q^{83} +749.051 q^{84} +1.67434 q^{86} +1001.84i q^{87} +124.526i q^{88} +352.887 q^{89} +1263.54 q^{91} -99.6743i q^{92} +388.282i q^{93} +52.5950 q^{94} -760.056 q^{96} +847.585i q^{97} -41.3140i q^{98} +89.5795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9}+O(q^{10}) \) 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9	\( 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9} - 44 q^{11} + 8 q^{14} - 16 q^{16} - 144 q^{19} + 152 q^{21} - 504 q^{24} - 80 q^{26} - 288 q^{29} - 68 q^{31} + 104 q^{34} - 160 q^{36} - 800 q^{39} + 1072 q^{41} - 176 q^{44} - 628 q^{46} + 780 q^{49} - 328 q^{51} + 220 q^{54} + 240 q^{56} - 1268 q^{59} + 1680 q^{61} - 448 q^{64} + 572 q^{66} - 1924 q^{69} - 1356 q^{71} - 12 q^{74} + 864 q^{76} - 632 q^{79} - 2588 q^{81} + 1472 q^{84} - 312 q^{86} + 3684 q^{89} + 2560 q^{91} + 1984 q^{94} - 1904 q^{96} + 968 q^{99}+O(q^{100}) \) 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9 - 44 * q^11 + 8 * q^14 - 16 * q^16 - 144 * q^19 + 152 * q^21 - 504 * q^24 - 80 * q^26 - 288 * q^29 - 68 * q^31 + 104 * q^34 - 160 * q^36 - 800 * q^39 + 1072 * q^41 - 176 * q^44 - 628 * q^46 + 780 * q^49 - 328 * q^51 + 220 * q^54 + 240 * q^56 - 1268 * q^59 + 1680 * q^61 - 448 * q^64 + 572 * q^66 - 1924 * q^69 - 1356 * q^71 - 12 * q^74 + 864 * q^76 - 632 * q^79 - 2588 * q^81 + 1472 * q^84 - 312 * q^86 + 3684 * q^89 + 2560 * q^91 + 1984 * q^94 - 1904 * q^96 + 968 * q^99

Introduction

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