LMFDB - Embedded newform 405.3.d.a.404.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,Mod(404,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.404"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{18} - 19 x^{16} + 66 x^{14} + 109 x^{12} - 813 x^{10} + 981 x^{8} + 5346 x^{6} + \cdots + 59049 $ x^20 - 3x^18 - 19x^16 + 66x^14 + 109x^12 - 813x^10 + 981x^8 + 5346x^6 - 13851x^4 - 19683*x^2 + 59049
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.19
Root			$1.44078 - 0.961330i$ of defining polynomial
Character	$\chi$	$=$	405.404
Dual form			405.3.d.a.404.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.68728 q^{2} +9.59601 q^{4} +(1.59117 - 4.74006i) q^{5} +2.56739i q^{7} +20.6340 q^{8} +(5.86709 - 17.4779i) q^{10} +9.83010i q^{11} -12.0633i q^{13} +9.46669i q^{14} +37.6993 q^{16} -4.28451 q^{17} +7.16698 q^{19} +(15.2689 - 45.4857i) q^{20} +36.2463i q^{22} -0.510539 q^{23} +(-19.9364 - 15.0845i) q^{25} -44.4808i q^{26} +24.6367i q^{28} +30.5521i q^{29} -19.2272 q^{31} +56.4718 q^{32} -15.7982 q^{34} +(12.1696 + 4.08516i) q^{35} -1.31851i q^{37} +26.4267 q^{38} +(32.8323 - 97.8066i) q^{40} -34.6104i q^{41} +51.9467i q^{43} +94.3297i q^{44} -1.88250 q^{46} -50.9313 q^{47} +42.4085 q^{49} +(-73.5108 - 55.6207i) q^{50} -115.760i q^{52} -86.6349 q^{53} +(46.5953 + 15.6414i) q^{55} +52.9757i q^{56} +112.654i q^{58} +105.962i q^{59} +31.3200 q^{61} -70.8961 q^{62} +57.4297 q^{64} +(-57.1809 - 19.1948i) q^{65} -78.0541i q^{67} -41.1142 q^{68} +(44.8727 + 15.0631i) q^{70} +72.6762i q^{71} +30.3097i q^{73} -4.86172i q^{74} +68.7744 q^{76} -25.2377 q^{77} +115.280 q^{79} +(59.9861 - 178.697i) q^{80} -127.618i q^{82} -60.0138 q^{83} +(-6.81739 + 20.3089i) q^{85} +191.542i q^{86} +202.835i q^{88} -71.2992i q^{89} +30.9713 q^{91} -4.89914 q^{92} -187.798 q^{94} +(11.4039 - 33.9719i) q^{95} -127.964i q^{97} +156.372 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 36 q^{4} + 4 q^{10} + 52 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} + 8 q^{34} + 68 q^{40} + 116 q^{46} + 80 q^{49} + 36 q^{55} + 100 q^{61} + 140 q^{64} + 108 q^{70} + 192 q^{76} + 256 q^{79} + 148 q^{85}+ \cdots - 436 q^{94}+O(q^{100}) $ 20 * q + 36 * q^4 + 4 * q^10 + 52 * q^16 - 8 * q^19 - 4 * q^25 - 56 * q^31 + 8 * q^34 + 68 * q^40 + 116 * q^46 + 80 * q^49 + 36 * q^55 + 100 * q^61 + 140 * q^64 + 108 * q^70 + 192 * q^76 + 256 * q^79 + 148 * q^85 - 288 * q^91 - 436 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$	3.68728			1.84364			0.921819	−	0.387620i	$-0.126703\pi$
$2$	3.68728			1.84364			0.921819	+	0.387620i	$0.126703\pi$
$3$		0			0
$3$		0			0
$4$	9.59601			2.39900
$5$	1.59117	−	4.74006i	0.318234	−	0.948012i
$5$	1.59117	−	4.74006i	0.318234	−	0.948012i
$6$		0			0
$7$			2.56739i			0.366771i	0.983041	+	0.183385i	$0.0587056\pi$
$7$			2.56739i			0.366771i	−0.983041	+	0.183385i	$0.941294\pi$
$8$	20.6340			2.57925
$9$		0			0
$10$	5.86709	−	17.4779i	0.586709	−	1.74779i
$11$			9.83010i			0.893645i	0.894623	+	0.446823i	$0.147444\pi$
$11$			9.83010i			0.893645i	−0.894623	+	0.446823i	$0.852556\pi$
$12$		0			0
$13$		−	12.0633i		−	0.927948i	−0.885849	−	0.463974i	$-0.846423\pi$
$13$		−	12.0633i		−	0.927948i	0.885849	−	0.463974i	$-0.153577\pi$
$14$			9.46669i			0.676192i
$15$		0			0
$16$	37.6993			2.35621
$17$	−4.28451			−0.252030			−0.126015	−	0.992028i	$-0.540219\pi$
$17$	−4.28451			−0.252030			−0.126015	+	0.992028i	$0.540219\pi$
$18$		0			0
$19$	7.16698			0.377210			0.188605	−	0.982053i	$-0.439603\pi$
$19$	7.16698			0.377210			0.188605	+	0.982053i	$0.439603\pi$
$20$	15.2689	−	45.4857i	0.763444	−	2.27428i
$21$		0			0
$22$			36.2463i			1.64756i
$23$	−0.510539			−0.0221973			−0.0110987	−	0.999938i	$-0.503533\pi$
$23$	−0.510539			−0.0221973			−0.0110987	+	0.999938i	$0.503533\pi$
$24$		0			0
$25$	−19.9364	−	15.0845i	−0.797454	−	0.603380i
$26$		−	44.4808i		−	1.71080i
$27$		0			0
$28$			24.6367i			0.879884i
$29$			30.5521i			1.05352i	0.850014	+	0.526760i	$0.176594\pi$
$29$			30.5521i			1.05352i	−0.850014	+	0.526760i	$0.823406\pi$
$30$		0			0
$31$	−19.2272			−0.620233			−0.310116	−	0.950699i	$-0.600368\pi$
$31$	−19.2272			−0.620233			−0.310116	+	0.950699i	$0.600368\pi$
$32$	56.4718			1.76474
$33$		0			0
$34$	−15.7982			−0.464653
$35$	12.1696	+	4.08516i	0.347703	+	0.116719i
$36$		0			0
$37$		−	1.31851i		−	0.0356355i	−0.999841	−	0.0178177i	$-0.994328\pi$
$37$		−	1.31851i		−	0.0356355i	0.999841	−	0.0178177i	$-0.00567186\pi$
$38$	26.4267			0.695438
$39$		0			0
$40$	32.8323	−	97.8066i	0.820807	−	2.44516i
$41$		−	34.6104i		−	0.844156i	−0.906559	−	0.422078i	$-0.861301\pi$
$41$		−	34.6104i		−	0.844156i	0.906559	−	0.422078i	$-0.138699\pi$
$42$		0			0
$43$			51.9467i			1.20806i	0.796960	+	0.604032i	$0.206440\pi$
$43$			51.9467i			1.20806i	−0.796960	+	0.604032i	$0.793560\pi$
$44$			94.3297i			2.14386i
$45$		0			0
$46$	−1.88250			−0.0409239
$47$	−50.9313			−1.08364			−0.541822	−	0.840493i	$-0.682265\pi$
$47$	−50.9313			−1.08364			−0.541822	+	0.840493i	$0.682265\pi$
$48$		0			0
$49$	42.4085			0.865479
$50$	−73.5108	−	55.6207i	−1.47022	−	1.11241i
$51$		0			0
$52$		−	115.760i		−	2.22615i
$53$	−86.6349			−1.63462			−0.817311	−	0.576197i	$-0.804536\pi$
$53$	−86.6349			−1.63462			−0.817311	+	0.576197i	$0.804536\pi$
$54$		0			0
$55$	46.5953	+	15.6414i	0.847186	+	0.284388i
$56$			52.9757i			0.945995i
$57$		0			0
$58$			112.654i			1.94231i
$59$			105.962i			1.79596i	0.440033	+	0.897982i	$0.354967\pi$
$59$			105.962i			1.79596i	−0.440033	+	0.897982i	$0.645033\pi$
$60$		0			0
$61$	31.3200			0.513443			0.256721	−	0.966485i	$-0.417358\pi$
$61$	31.3200			0.513443			0.256721	+	0.966485i	$0.417358\pi$
$62$	−70.8961			−1.14348
$63$		0			0
$64$	57.4297			0.897339
$65$	−57.1809	−	19.1948i	−0.879706	−	0.295305i
$66$		0			0
$67$		−	78.0541i		−	1.16499i	−0.812835	−	0.582493i	$-0.802077\pi$
$67$		−	78.0541i		−	1.16499i	0.812835	−	0.582493i	$-0.197923\pi$
$68$	−41.1142			−0.604621
$69$		0			0
$70$	44.8727	+	15.0631i	0.641039	+	0.215187i
$71$			72.6762i			1.02361i	0.859102	+	0.511804i	$0.171023\pi$
$71$			72.6762i			1.02361i	−0.859102	+	0.511804i	$0.828977\pi$
$72$		0			0
$73$			30.3097i			0.415201i	0.978214	+	0.207601i	$0.0665654\pi$
$73$			30.3097i			0.415201i	−0.978214	+	0.207601i	$0.933435\pi$
$74$		−	4.86172i		−	0.0656989i
$75$		0			0
$76$	68.7744			0.904927
$77$	−25.2377			−0.327763
$78$		0			0
$79$	115.280			1.45923			0.729617	−	0.683856i	$-0.239698\pi$
$79$	115.280			1.45923			0.729617	+	0.683856i	$0.239698\pi$
$80$	59.9861	−	178.697i	0.749826	−	2.23371i
$81$		0			0
$82$		−	127.618i		−	1.55632i
$83$	−60.0138			−0.723057			−0.361529	−	0.932361i	$-0.617745\pi$
$83$	−60.0138			−0.723057			−0.361529	+	0.932361i	$0.617745\pi$
$84$		0			0
$85$	−6.81739	+	20.3089i	−0.0802046	+	0.238928i
$86$			191.542i			2.22723i
$87$		0			0
$88$			202.835i			2.30494i
$89$		−	71.2992i		−	0.801115i	−0.916272	−	0.400558i	$-0.868816\pi$
$89$		−	71.2992i		−	0.801115i	0.916272	−	0.400558i	$-0.131184\pi$
$90$		0			0
$91$	30.9713			0.340344
$92$	−4.89914			−0.0532515
$93$		0			0
$94$	−187.798			−1.99785
$95$	11.4039	−	33.9719i	0.120041	−	0.357599i
$96$		0			0
$97$		−	127.964i		−	1.31921i	−0.751611	−	0.659607i	$-0.770723\pi$
$97$		−	127.964i		−	1.31921i	0.751611	−	0.659607i	$-0.229277\pi$
$98$	156.372			1.59563
$99$		0			0
$100$	−191.309	−	144.751i	−1.91309	−	1.44751i
$101$		−	86.0766i		−	0.852244i	−0.904666	−	0.426122i	$-0.859879\pi$
$101$		−	86.0766i		−	0.852244i	0.904666	−	0.426122i	$-0.140121\pi$
$102$		0			0
$103$			40.8833i			0.396925i	0.980108	+	0.198463i	$0.0635948\pi$
$103$			40.8833i			0.396925i	−0.980108	+	0.198463i	$0.936405\pi$
$104$		−	248.915i		−	2.39341i
$105$		0			0
$106$	−319.447			−3.01365
$107$	−1.66026			−0.0155165			−0.00775823	−	0.999970i	$-0.502470\pi$
$107$	−1.66026			−0.0155165			−0.00775823	+	0.999970i	$0.502470\pi$
$108$		0			0
$109$	−148.641			−1.36368			−0.681839	−	0.731502i	$-0.738820\pi$
$109$	−148.641			−1.36368			−0.681839	+	0.731502i	$0.738820\pi$
$110$	171.810	+	57.6740i	1.56191	+	0.524309i
$111$		0			0
$112$			96.7891i			0.864188i
$113$	−131.457			−1.16333			−0.581667	−	0.813427i	$-0.697599\pi$
$113$	−131.457			−1.16333			−0.581667	+	0.813427i	$0.697599\pi$
$114$		0			0
$115$	−0.812355	+	2.41999i	−0.00706395	+	0.0210434i
$116$			293.178i			2.52740i
$117$		0			0
$118$			390.711i			3.31111i
$119$		−	11.0000i		−	0.0924373i
$120$		0			0
$121$	24.3692			0.201398
$122$	115.486			0.946603
$123$		0			0
$124$	−184.505			−1.48794
$125$	−103.224	+	70.4975i	−0.825788	+	0.563980i
$126$		0			0
$127$		−	101.150i		−	0.796460i	−0.917286	−	0.398230i	$-0.869625\pi$
$127$		−	101.150i		−	0.796460i	0.917286	−	0.398230i	$-0.130375\pi$
$128$	−14.1279			−0.110374
$129$		0			0
$130$	−210.842	−	70.7765i	−1.62186	−	0.544435i
$131$		−	85.7969i		−	0.654938i	−0.944862	−	0.327469i	$-0.893804\pi$
$131$		−	85.7969i		−	0.654938i	0.944862	−	0.327469i	$-0.106196\pi$
$132$		0			0
$133$			18.4005i			0.138349i
$134$		−	287.807i		−	2.14781i
$135$		0			0
$136$	−88.4068			−0.650050
$137$	69.7262			0.508951			0.254475	−	0.967079i	$-0.418097\pi$
$137$	69.7262			0.508951			0.254475	+	0.967079i	$0.418097\pi$
$138$		0			0
$139$	138.917			0.999406			0.499703	−	0.866197i	$-0.333442\pi$
$139$	138.917			0.999406			0.499703	+	0.866197i	$0.333442\pi$
$140$	116.780	+	39.2013i	0.834140	+	0.280009i
$141$		0			0
$142$			267.977i			1.88716i
$143$	118.584			0.829256
$144$		0			0
$145$	144.819	+	48.6136i	0.998751	+	0.335266i
$146$			111.760i			0.765481i
$147$		0			0
$148$		−	12.6524i		−	0.0854895i
$149$			47.7568i			0.320515i	0.987075	+	0.160258i	$0.0512325\pi$
$149$			47.7568i			0.320515i	−0.987075	+	0.160258i	$0.948767\pi$
$150$		0			0
$151$	55.4914			0.367493			0.183746	−	0.982974i	$-0.441178\pi$
$151$	55.4914			0.367493			0.183746	+	0.982974i	$0.441178\pi$
$152$	147.884			0.972920
$153$		0			0
$154$	−93.0585			−0.604276
$155$	−30.5938	+	91.1382i	−0.197379	+	0.587988i
$156$		0			0
$157$		−	11.0721i		−	0.0705232i	−0.999378	−	0.0352616i	$-0.988774\pi$
$157$		−	11.0721i		−	0.0705232i	0.999378	−	0.0352616i	$-0.0112264\pi$
$158$	425.068			2.69030
$159$		0			0
$160$	89.8562	−	267.680i	0.561602	−	1.67300i
$161$		−	1.31076i		−	0.00814134i
$162$		0			0
$163$		−	216.230i		−	1.32656i	−0.748370	−	0.663282i	$-0.769163\pi$
$163$		−	216.230i		−	1.32656i	0.748370	−	0.663282i	$-0.230837\pi$
$164$		−	332.122i		−	2.02513i
$165$		0			0
$166$	−221.287			−1.33306
$167$	61.9468			0.370939			0.185469	−	0.982650i	$-0.440619\pi$
$167$	61.9468			0.370939			0.185469	+	0.982650i	$0.440619\pi$
$168$		0			0
$169$	23.4763			0.138913
$170$	−25.1376	+	74.8844i	−0.147868	+	0.440496i
$171$		0			0
$172$			498.481i			2.89815i
$173$	333.471			1.92758			0.963788	−	0.266670i	$-0.0859233\pi$
$173$	333.471			1.92758			0.963788	+	0.266670i	$0.0859233\pi$
$174$		0			0
$175$	38.7278	−	51.1845i	0.221302	−	0.292483i
$176$			370.588i			2.10561i
$177$		0			0
$178$		−	262.900i		−	1.47697i
$179$		−	193.521i		−	1.08112i	−0.841304	−	0.540562i	$-0.818212\pi$
$179$		−	193.521i		−	1.08112i	0.841304	−	0.540562i	$-0.181788\pi$
$180$		0			0
$181$	243.865			1.34732			0.673661	−	0.739041i	$-0.264721\pi$
$181$	243.865			1.34732			0.673661	+	0.739041i	$0.264721\pi$
$182$	114.200			0.627471
$183$		0			0
$184$	−10.5345			−0.0572526
$185$	−6.24983	−	2.09798i	−0.0337828	−	0.0113404i
$186$		0			0
$187$		−	42.1172i		−	0.225226i
$188$	−488.737			−2.59966
$189$		0			0
$190$	42.0493	−	125.264i	0.221312	−	0.659284i
$191$			141.565i			0.741178i	0.928797	+	0.370589i	$0.120844\pi$
$191$			141.565i			0.741178i	−0.928797	+	0.370589i	$0.879156\pi$
$192$		0			0
$193$			221.422i			1.14727i	0.819112	+	0.573633i	$0.194466\pi$
$193$			221.422i			1.14727i	−0.819112	+	0.573633i	$0.805534\pi$
$194$		−	471.838i		−	2.43215i
$195$		0			0
$196$	406.952			2.07629
$197$	28.4424			0.144378			0.0721889	−	0.997391i	$-0.477002\pi$
$197$	28.4424			0.144378			0.0721889	+	0.997391i	$0.477002\pi$
$198$		0			0
$199$	153.875			0.773244			0.386622	−	0.922238i	$-0.373642\pi$
$199$	153.875			0.773244			0.386622	+	0.922238i	$0.373642\pi$
$200$	−411.367	−	311.254i	−2.05684	−	1.55627i
$201$		0			0
$202$		−	317.388i		−	1.57123i
$203$	−78.4393			−0.386401
$204$		0			0
$205$	−164.055	−	55.0711i	−0.800271	−	0.268639i
$206$			150.748i			0.731786i
$207$		0			0
$208$		−	454.779i		−	2.18644i
$209$			70.4521i			0.337092i
$210$		0			0
$211$	180.178			0.853925			0.426962	−	0.904269i	$-0.359584\pi$
$211$	180.178			0.853925			0.426962	+	0.904269i	$0.359584\pi$
$212$	−831.350			−3.92146
$213$		0			0
$214$	−6.12184			−0.0286067
$215$	246.231	+	82.6561i	1.14526	+	0.384447i
$216$		0			0
$217$		−	49.3638i		−	0.227483i
$218$	−548.080			−2.51413
$219$		0			0
$220$	447.128	+	150.095i	2.03240	+	0.682248i
$221$			51.6855i			0.233871i
$222$		0			0
$223$			215.403i			0.965931i	0.875639	+	0.482966i	$0.160440\pi$
$223$			215.403i			0.965931i	−0.875639	+	0.482966i	$0.839560\pi$
$224$			144.985i			0.647256i
$225$		0			0
$226$	−484.717			−2.14477
$227$	−50.4159			−0.222097			−0.111048	−	0.993815i	$-0.535421\pi$
$227$	−50.4159			−0.222097			−0.111048	+	0.993815i	$0.535421\pi$
$228$		0			0
$229$	−6.16704			−0.0269303			−0.0134652	−	0.999909i	$-0.504286\pi$
$229$	−6.16704			−0.0269303			−0.0134652	+	0.999909i	$0.504286\pi$
$230$	−2.99538	+	8.92316i	−0.0130234	+	0.0387963i
$231$		0			0
$232$			630.413i			2.71730i
$233$	−52.5336			−0.225466			−0.112733	−	0.993625i	$-0.535960\pi$
$233$	−52.5336			−0.225466			−0.112733	+	0.993625i	$0.535960\pi$
$234$		0			0
$235$	−81.0403	+	241.417i	−0.344853	+	1.02731i
$236$			1016.81i			4.30852i
$237$		0			0
$238$		−	40.5602i		−	0.170421i
$239$			97.5840i			0.408301i	0.978939	+	0.204151i	$0.0654432\pi$
$239$			97.5840i			0.408301i	−0.978939	+	0.204151i	$0.934557\pi$
$240$		0			0
$241$	−142.729			−0.592238			−0.296119	−	0.955151i	$-0.595693\pi$
$241$	−142.729			−0.592238			−0.296119	+	0.955151i	$0.595693\pi$
$242$	89.8560			0.371306
$243$		0			0
$244$	300.547			1.23175
$245$	67.4791	−	201.019i	0.275425	−	0.820485i
$246$		0			0
$247$		−	86.4576i		−	0.350031i
$248$	−396.735			−1.59974
$249$		0			0
$250$	−380.614	+	259.944i	−1.52245	+	1.03978i
$251$		−	254.631i		−	1.01447i	−0.861809	−	0.507233i	$-0.830668\pi$
$251$		−	254.631i		−	1.01447i	0.861809	−	0.507233i	$-0.169332\pi$
$252$		0			0
$253$		−	5.01865i		−	0.0198366i
$254$		−	372.969i		−	1.46838i
$255$		0			0
$256$	−281.812			−1.10083
$257$	142.736			0.555394			0.277697	−	0.960669i	$-0.410429\pi$
$257$	142.736			0.555394			0.277697	+	0.960669i	$0.410429\pi$
$258$		0			0
$259$	3.38514			0.0130700
$260$	−548.708	−	184.193i	−2.11042	−	0.708436i
$261$		0			0
$262$		−	316.357i		−	1.20747i
$263$	252.469			0.959957			0.479979	−	0.877280i	$-0.340644\pi$
$263$	252.469			0.959957			0.479979	+	0.877280i	$0.340644\pi$
$264$		0			0
$265$	−137.851	+	410.655i	−0.520192	+	1.54964i
$266$			67.8477i			0.255066i
$267$		0			0
$268$		−	749.008i		−	2.79481i
$269$			388.672i			1.44488i	0.691435	+	0.722439i	$0.256979\pi$
$269$			388.672i			1.44488i	−0.691435	+	0.722439i	$0.743021\pi$
$270$		0			0
$271$	−163.253			−0.602410			−0.301205	−	0.953559i	$-0.597389\pi$
$271$	−163.253			−0.602410			−0.301205	+	0.953559i	$0.597389\pi$
$272$	−161.523			−0.593836
$273$		0			0
$274$	257.100			0.938321
$275$	148.282	−	195.976i	0.539207	−	0.712641i
$276$		0			0
$277$		−	483.823i		−	1.74666i	−0.487133	−	0.873328i	$-0.661957\pi$
$277$		−	483.823i		−	1.74666i	0.487133	−	0.873328i	$-0.338043\pi$
$278$	512.227			1.84254
$279$		0			0
$280$	251.108	+	84.2934i	0.896814	+	0.301048i
$281$		−	267.658i		−	0.952519i	−0.879305	−	0.476260i	$-0.841992\pi$
$281$		−	267.658i		−	0.952519i	0.879305	−	0.476260i	$-0.158008\pi$
$282$		0			0
$283$		−	491.796i		−	1.73780i	−0.494991	−	0.868898i	$-0.664828\pi$
$283$		−	491.796i		−	1.73780i	0.494991	−	0.868898i	$-0.335172\pi$
$284$			697.401i			2.45564i
$285$		0			0
$286$	437.250			1.52885
$287$	88.8586			0.309612
$288$		0			0
$289$	−270.643			−0.936481
$290$	533.987	+	179.252i	1.84133	+	0.618110i
$291$		0			0
$292$			290.852i			0.996069i
$293$	322.146			1.09947			0.549737	−	0.835338i	$-0.314728\pi$
$293$	322.146			1.09947			0.549737	+	0.835338i	$0.314728\pi$
$294$		0			0
$295$	502.266	+	168.603i	1.70260	+	0.571537i
$296$		−	27.2062i		−	0.0919129i
$297$		0			0
$298$			176.092i			0.590914i
$299$			6.15879i			0.0205980i
$300$		0			0
$301$	−133.368			−0.443082
$302$	204.612			0.677523
$303$		0			0
$304$	270.191			0.888785
$305$	49.8355	−	148.459i	0.163395	−	0.486750i
$306$		0			0
$307$		−	426.031i		−	1.38772i	−0.720109	−	0.693861i	$-0.755908\pi$
$307$		−	426.031i		−	1.38772i	0.720109	−	0.693861i	$-0.244092\pi$
$308$	−242.182			−0.786304
$309$		0			0
$310$	−112.808	+	336.052i	−0.363896	+	1.08404i
$311$		−	1.16797i		−	0.00375554i	−0.999998	−	0.00187777i	$-0.999402\pi$
$311$		−	1.16797i		−	0.00375554i	0.999998	−	0.00187777i	$-0.000597714\pi$
$312$		0			0
$313$			260.950i			0.833705i	0.908974	+	0.416852i	$0.136867\pi$
$313$			260.950i			0.833705i	−0.908974	+	0.416852i	$0.863133\pi$
$314$		−	40.8260i		−	0.130019i
$315$		0			0
$316$	1106.22			3.50071
$317$	−131.036			−0.413361			−0.206681	−	0.978408i	$-0.566266\pi$
$317$	−131.036			−0.413361			−0.206681	+	0.978408i	$0.566266\pi$
$318$		0			0
$319$	−300.330			−0.941474
$320$	91.3805	−	272.220i	0.285564	−	0.850689i
$321$		0			0
$322$		−	4.83312i		−	0.0150097i
$323$	−30.7071			−0.0950683
$324$		0			0
$325$	−181.969	+	240.499i	−0.559905	+	0.739996i
$326$		−	797.300i		−	2.44570i
$327$		0			0
$328$		−	714.152i		−	2.17729i
$329$		−	130.761i		−	0.397449i
$330$		0			0
$331$	193.484			0.584543			0.292272	−	0.956335i	$-0.405589\pi$
$331$	193.484			0.584543			0.292272	+	0.956335i	$0.405589\pi$
$332$	−575.893			−1.73462
$333$		0			0
$334$	228.415			0.683877
$335$	−369.981	−	124.197i	−1.10442	−	0.370738i
$336$		0			0
$337$		−	463.595i		−	1.37565i	−0.725875	−	0.687827i	$-0.758565\pi$
$337$		−	463.595i		−	1.37565i	0.725875	−	0.687827i	$-0.241435\pi$
$338$	86.5638			0.256106
$339$		0			0
$340$	−65.4198	+	194.884i	−0.192411	+	0.573188i
$341$		−	189.005i		−	0.554268i
$342$		0			0
$343$			234.682i			0.684203i
$344$			1071.87i			3.11590i
$345$		0			0
$346$	1229.60			3.55375
$347$	9.71868			0.0280077			0.0140039	−	0.999902i	$-0.495542\pi$
$347$	9.71868			0.0280077			0.0140039	+	0.999902i	$0.495542\pi$
$348$		0			0
$349$	−49.3358			−0.141363			−0.0706816	−	0.997499i	$-0.522517\pi$
$349$	−49.3358			−0.141363			−0.0706816	+	0.997499i	$0.522517\pi$
$350$	142.800	−	188.731i	0.408001	−	0.539232i
$351$		0			0
$352$			555.123i			1.57705i
$353$	95.3620			0.270147			0.135074	−	0.990836i	$-0.456873\pi$
$353$	95.3620			0.270147			0.135074	+	0.990836i	$0.456873\pi$
$354$		0			0
$355$	344.489	+	115.640i	0.970393	+	0.325747i
$356$		−	684.188i		−	1.92188i
$357$		0			0
$358$		−	713.566i		−	1.99320i
$359$			539.284i			1.50219i	0.660197	+	0.751093i	$0.270473\pi$
$359$			539.284i			1.50219i	−0.660197	+	0.751093i	$0.729527\pi$
$360$		0			0
$361$	−309.634			−0.857713
$362$	899.199			2.48397
$363$		0			0
$364$	297.201			0.816486
$365$	143.670	+	48.2279i	0.393616	+	0.132131i
$366$		0			0
$367$		−	303.089i		−	0.825854i	−0.910764	−	0.412927i	$-0.864506\pi$
$367$		−	303.089i		−	0.825854i	0.910764	−	0.412927i	$-0.135494\pi$
$368$	−19.2470			−0.0523016
$369$		0			0
$370$	−23.0448	−	7.73582i	−0.0622833	−	0.0209076i
$371$		−	222.426i		−	0.599531i
$372$		0			0
$373$			130.665i			0.350308i	0.984541	+	0.175154i	$0.0560424\pi$
$373$			130.665i			0.350308i	−0.984541	+	0.175154i	$0.943958\pi$
$374$		−	155.298i		−	0.415235i
$375$		0			0
$376$	−1050.92			−2.79499
$377$	368.560			0.977612
$378$		0			0
$379$	545.141			1.43837			0.719183	−	0.694821i	$-0.244516\pi$
$379$	545.141			1.43837			0.719183	+	0.694821i	$0.244516\pi$
$380$	109.432	−	325.995i	0.287979	−	0.857882i
$381$		0			0
$382$			521.989i			1.36646i
$383$	−1.24650			−0.00325456			−0.00162728	−	0.999999i	$-0.500518\pi$
$383$	−1.24650			−0.00325456			−0.00162728	+	0.999999i	$0.500518\pi$
$384$		0			0
$385$	−40.1575	+	119.628i	−0.104305	+	0.310723i
$386$			816.445i			2.11514i
$387$		0			0
$388$		−	1227.94i		−	3.16480i
$389$			398.373i			1.02410i	0.858957	+	0.512048i	$0.171113\pi$
$389$			398.373i			1.02410i	−0.858957	+	0.512048i	$0.828887\pi$
$390$		0			0
$391$	2.18741			0.00559440
$392$	875.058			2.23229
$393$		0			0
$394$	104.875			0.266180
$395$	183.429	−	546.432i	0.464378	−	1.38337i
$396$		0			0
$397$			685.998i			1.72795i	0.503531	+	0.863977i	$0.332034\pi$
$397$			685.998i			1.72795i	−0.503531	+	0.863977i	$0.667966\pi$
$398$	567.381			1.42558
$399$		0			0
$400$	−751.587	−	568.675i	−1.87897	−	1.42169i
$401$			51.8705i			0.129353i	0.997906	+	0.0646764i	$0.0206015\pi$
$401$			51.8705i			0.129353i	−0.997906	+	0.0646764i	$0.979398\pi$
$402$		0			0
$403$			231.944i			0.575543i
$404$		−	825.992i		−	2.04453i
$405$		0			0
$406$	−289.227			−0.712383
$407$	12.9611			0.0318454
$408$		0			0
$409$	−271.296			−0.663317			−0.331658	−	0.943400i	$-0.607608\pi$
$409$	−271.296			−0.663317			−0.331658	+	0.943400i	$0.607608\pi$
$410$	−604.918	−	203.062i	−1.47541	−	0.495274i
$411$		0			0
$412$			392.316i			0.952224i
$413$	−272.046			−0.658707
$414$		0			0
$415$	−95.4921	+	284.469i	−0.230101	+	0.685467i
$416$		−	681.237i		−	1.63759i
$417$		0			0
$418$			259.777i			0.621475i
$419$			27.6513i			0.0659935i	0.999455	+	0.0329967i	$0.0105051\pi$
$419$			27.6513i			0.0659935i	−0.999455	+	0.0329967i	$0.989495\pi$
$420$		0			0
$421$	−437.226			−1.03854			−0.519271	−	0.854610i	$-0.673796\pi$
$421$	−437.226			−1.03854			−0.519271	+	0.854610i	$0.673796\pi$
$422$	664.367			1.57433
$423$		0			0
$424$	−1787.63			−4.21610
$425$	85.4176	+	64.6297i	0.200983	+	0.152070i
$426$		0			0
$427$			80.4108i			0.188316i
$428$	−15.9319			−0.0372240
$429$		0			0
$430$	907.920	+	304.776i	2.11144	+	0.708781i
$431$		−	54.3602i		−	0.126126i	−0.998010	−	0.0630629i	$-0.979913\pi$
$431$		−	54.3602i		−	0.126126i	0.998010	−	0.0630629i	$-0.0200869\pi$
$432$		0			0
$433$		−	526.426i		−	1.21576i	−0.794028	−	0.607882i	$-0.792019\pi$
$433$		−	526.426i		−	1.21576i	0.794028	−	0.607882i	$-0.207981\pi$
$434$		−	182.018i		−	0.419397i
$435$		0			0
$436$	−1426.36			−3.27147
$437$	−3.65903			−0.00837306
$438$		0			0
$439$	−416.248			−0.948173			−0.474087	−	0.880478i	$-0.657222\pi$
$439$	−416.248			−0.948173			−0.474087	+	0.880478i	$0.657222\pi$
$440$	961.448	+	322.744i	2.18511	+	0.733510i
$441$		0			0
$442$			190.579i			0.431173i
$443$	−427.165			−0.964254			−0.482127	−	0.876101i	$-0.660136\pi$
$443$	−427.165			−0.964254			−0.482127	+	0.876101i	$0.660136\pi$
$444$		0			0
$445$	−337.963	−	113.449i	−0.759467	−	0.254942i
$446$			794.249i			1.78083i
$447$		0			0
$448$			147.445i			0.329118i
$449$			236.730i			0.527239i	0.964627	+	0.263620i	$0.0849164\pi$
$449$			236.730i			0.527239i	−0.964627	+	0.263620i	$0.915084\pi$
$450$		0			0
$451$	340.224			0.754376
$452$	−1261.46			−2.79084
$453$		0			0
$454$	−185.898			−0.409466
$455$	49.2806	−	146.806i	0.108309	−	0.322650i
$456$		0			0
$457$			697.781i			1.52687i	0.645883	+	0.763437i	$0.276489\pi$
$457$			697.781i			1.52687i	−0.645883	+	0.763437i	$0.723511\pi$
$458$	−22.7396			−0.0496497
$459$		0			0
$460$	−7.79536	+	23.2222i	−0.0169464	+	0.0504831i
$461$			383.752i			0.832435i	0.909265	+	0.416217i	$0.136644\pi$
$461$			383.752i			0.832435i	−0.909265	+	0.416217i	$0.863356\pi$
$462$		0			0
$463$			270.112i			0.583395i	0.956511	+	0.291698i	$0.0942201\pi$
$463$			270.112i			0.583395i	−0.956511	+	0.291698i	$0.905780\pi$
$464$			1151.79i			2.48232i
$465$		0			0
$466$	−193.706			−0.415678
$467$	−777.952			−1.66585			−0.832926	−	0.553385i	$-0.813336\pi$
$467$	−777.952			−1.66585			−0.832926	+	0.553385i	$0.813336\pi$
$468$		0			0
$469$	200.396			0.427283
$470$	−298.818	+	890.173i	−0.635783	+	1.89398i
$471$		0			0
$472$			2186.42i			4.63225i
$473$	−510.641			−1.07958
$474$		0			0
$475$	−142.884	−	108.110i	−0.300807	−	0.227601i
$476$		−	105.556i		−	0.221757i
$477$		0			0
$478$			359.819i			0.752760i
$479$		−	72.5647i		−	0.151492i	−0.997127	−	0.0757460i	$-0.975866\pi$
$479$		−	72.5647i		−	0.151492i	0.997127	−	0.0757460i	$-0.0241338\pi$
$480$		0			0
$481$	−15.9056			−0.0330678
$482$	−526.283			−1.09187
$483$		0			0
$484$	233.847			0.483155
$485$	−606.556	−	203.612i	−1.25063	−	0.419819i
$486$		0			0
$487$		−	729.487i		−	1.49792i	−0.662616	−	0.748959i	$-0.730554\pi$
$487$		−	729.487i		−	1.49792i	0.662616	−	0.748959i	$-0.269446\pi$
$488$	646.258			1.32430
$489$		0			0
$490$	248.814	−	741.212i	0.507784	−	1.51268i
$491$			3.81570i			0.00777127i	0.999992	+	0.00388564i	$0.00123684\pi$
$491$			3.81570i			0.00777127i	−0.999992	+	0.00388564i	$0.998763\pi$
$492$		0			0
$493$		−	130.901i		−	0.265519i
$494$		−	318.793i		−	0.645330i
$495$		0			0
$496$	−724.853			−1.46140
$497$	−186.588			−0.375429
$498$		0			0
$499$	205.303			0.411428			0.205714	−	0.978612i	$-0.434048\pi$
$499$	205.303			0.411428			0.205714	+	0.978612i	$0.434048\pi$
$500$	−990.534	+	676.495i	−1.98107	+	1.35299i
$501$		0			0
$502$		−	938.895i		−	1.87031i
$503$	−224.016			−0.445360			−0.222680	−	0.974892i	$-0.571481\pi$
$503$	−224.016			−0.445360			−0.222680	+	0.974892i	$0.571481\pi$
$504$		0			0
$505$	−408.008	−	136.963i	−0.807937	−	0.271213i
$506$		−	18.5051i		−	0.0365714i
$507$		0			0
$508$		−	970.640i		−	1.91071i
$509$		−	460.810i		−	0.905324i	−0.891682	−	0.452662i	$-0.850474\pi$
$509$		−	460.810i		−	0.905324i	0.891682	−	0.452662i	$-0.149526\pi$
$510$		0			0
$511$	−77.8169			−0.152284
$512$	−982.608			−1.91916
$513$		0			0
$514$	526.309			1.02395
$515$	193.789	+	65.0523i	0.376290	+	0.126315i
$516$		0			0
$517$		−	500.659i		−	0.968393i
$518$	12.4819			0.0240964
$519$		0			0
$520$	−1179.87	−	396.066i	−2.26898	−	0.761665i
$521$		−	182.438i		−	0.350168i	−0.984554	−	0.175084i	$-0.943980\pi$
$521$		−	182.438i		−	0.350168i	0.984554	−	0.175084i	$-0.0560197\pi$
$522$		0			0
$523$			431.339i			0.824741i	0.911016	+	0.412370i	$0.135299\pi$
$523$			431.339i			0.824741i	−0.911016	+	0.412370i	$0.864701\pi$
$524$		−	823.308i		−	1.57120i
$525$		0			0
$526$	930.922			1.76981
$527$	82.3793			0.156317
$528$		0			0
$529$	−528.739			−0.999507
$530$	−508.295	+	1514.20i	−0.959046	+	2.85698i
$531$		0			0
$532$			176.571i			0.331901i
$533$	−417.516			−0.783333
$534$		0			0
$535$	−2.64176	+	7.86974i	−0.00493786	+	0.0147098i
$536$		−	1610.57i		−	3.00480i
$537$		0			0
$538$			1433.14i			2.66383i
$539$			416.879i			0.773431i
$540$		0			0
$541$	−649.924			−1.20134			−0.600669	−	0.799498i	$-0.705099\pi$
$541$	−649.924			−1.20134			−0.600669	+	0.799498i	$0.705099\pi$
$542$	−601.959			−1.11063
$543$		0			0
$544$	−241.954			−0.444769
$545$	−236.513	+	704.567i	−0.433969	+	1.29278i
$546$		0			0
$547$		−	425.316i		−	0.777542i	−0.921334	−	0.388771i	$-0.872900\pi$
$547$		−	425.316i		−	0.777542i	0.921334	−	0.388771i	$-0.127100\pi$
$548$	669.094			1.22097
$549$		0			0
$550$	546.757	−	722.619i	0.994103	−	1.31385i
$551$			218.966i			0.397398i
$552$		0			0
$553$			295.968i			0.535205i
$554$		−	1783.99i		−	3.22020i
$555$		0			0
$556$	1333.05			2.39758
$557$	325.885			0.585073			0.292536	−	0.956254i	$-0.405501\pi$
$557$	325.885			0.585073			0.292536	+	0.956254i	$0.405501\pi$
$558$		0			0
$559$	626.650			1.12102
$560$	458.786	+	154.008i	0.819261	+	0.275014i
$561$		0			0
$562$		−	986.929i		−	1.75610i
$563$	−706.763			−1.25535			−0.627676	−	0.778475i	$-0.715994\pi$
$563$	−706.763			−1.25535			−0.627676	+	0.778475i	$0.715994\pi$
$564$		0			0
$565$	−209.170	+	623.113i	−0.370213	+	1.10285i
$566$		−	1813.39i		−	3.20387i
$567$		0			0
$568$			1499.60i			2.64015i
$569$			560.236i			0.984597i	0.870426	+	0.492299i	$0.163843\pi$
$569$			560.236i			0.984597i	−0.870426	+	0.492299i	$0.836157\pi$
$570$		0			0
$571$	−210.692			−0.368988			−0.184494	−	0.982834i	$-0.559065\pi$
$571$	−210.692			−0.368988			−0.184494	+	0.982834i	$0.559065\pi$
$572$	1137.93			1.98939
$573$		0			0
$574$	327.646			0.570812
$575$	10.1783	+	7.70122i	0.0177014	+	0.0133934i
$576$		0			0
$577$		−	502.258i		−	0.870464i	−0.900318	−	0.435232i	$-0.856666\pi$
$577$		−	502.258i		−	0.870464i	0.900318	−	0.435232i	$-0.143334\pi$
$578$	−997.935			−1.72653
$579$		0			0
$580$	1389.68	+	466.497i	2.39600	+	0.804304i
$581$		−	154.079i		−	0.265196i
$582$		0			0
$583$		−	851.630i		−	1.46077i
$584$			625.411i			1.07091i
$585$		0			0
$586$	1187.84			2.02703
$587$	409.772			0.698079			0.349040	−	0.937108i	$-0.386508\pi$
$587$	409.772			0.698079			0.349040	+	0.937108i	$0.386508\pi$
$588$		0			0
$589$	−137.801			−0.233958
$590$	1851.99	+	621.687i	3.13897	+	1.05371i
$591$		0			0
$592$		−	49.7070i		−	0.0839646i
$593$	944.139			1.59214			0.796070	−	0.605204i	$-0.206908\pi$
$593$	944.139			1.59214			0.796070	+	0.605204i	$0.206908\pi$
$594$		0			0
$595$	−52.1409	−	17.5029i	−0.0876317	−	0.0294167i
$596$			458.275i			0.768917i
$597$		0			0
$598$			22.7092i			0.0379752i
$599$			532.342i			0.888717i	0.895849	+	0.444359i	$0.146568\pi$
$599$			532.342i			0.888717i	−0.895849	+	0.444359i	$0.853432\pi$
$600$		0			0
$601$	515.566			0.857846			0.428923	−	0.903341i	$-0.358893\pi$
$601$	515.566			0.857846			0.428923	+	0.903341i	$0.358893\pi$
$602$	−491.764			−0.816883
$603$		0			0
$604$	532.496			0.881615
$605$	38.7756	−	115.512i	0.0640918	−	0.190928i
$606$		0			0
$607$			171.153i			0.281965i	0.990012	+	0.140983i	$0.0450262\pi$
$607$			171.153i			0.281965i	−0.990012	+	0.140983i	$0.954974\pi$
$608$	404.732			0.665678
$609$		0			0
$610$	183.757	−	547.409i	0.301241	−	0.897391i
$611$			614.400i			1.00557i
$612$		0			0
$613$			875.826i			1.42875i	0.699761	+	0.714377i	$0.253290\pi$
$613$			875.826i			1.42875i	−0.699761	+	0.714377i	$0.746710\pi$
$614$		−	1570.89i		−	2.55846i
$615$		0			0
$616$	−520.756			−0.845384
$617$	−1131.75			−1.83427			−0.917137	−	0.398571i	$-0.869506\pi$
$617$	−1131.75			−1.83427			−0.917137	+	0.398571i	$0.869506\pi$
$618$		0			0
$619$	520.375			0.840670			0.420335	−	0.907369i	$-0.361913\pi$
$619$	520.375			0.840670			0.420335	+	0.907369i	$0.361913\pi$
$620$	−293.578	+	874.563i	−0.473513	+	1.41058i
$621$		0			0
$622$		−	4.30664i		−	0.00692386i
$623$	183.053			0.293826
$624$		0			0
$625$	169.916	+	601.459i	0.271866	+	0.962335i
$626$			962.193i			1.53705i
$627$		0			0
$628$		−	106.248i		−	0.169185i
$629$			5.64918i			0.00898121i
$630$		0			0
$631$	−607.475			−0.962718			−0.481359	−	0.876523i	$-0.659857\pi$
$631$	−607.475			−0.962718			−0.481359	+	0.876523i	$0.659857\pi$
$632$	2378.68			3.76374
$633$		0			0
$634$	−483.164			−0.762089
$635$	−479.459	−	160.947i	−0.755053	−	0.253461i
$636$		0			0
$637$		−	511.587i		−	0.803119i
$638$	−1107.40			−1.73574
$639$		0			0
$640$	−22.4799	+	66.9670i	−0.0351248	+	0.104636i
$641$		−	186.198i		−	0.290481i	−0.989396	−	0.145240i	$-0.953604\pi$
$641$		−	186.198i		−	0.290481i	0.989396	−	0.145240i	$-0.0463955\pi$
$642$		0			0
$643$			797.522i			1.24031i	0.784478	+	0.620157i	$0.212931\pi$
$643$			797.522i			1.24031i	−0.784478	+	0.620157i	$0.787069\pi$
$644$		−	12.5780i		−	0.0195311i
$645$		0			0
$646$	−113.225			−0.175271
$647$	−849.489			−1.31297			−0.656483	−	0.754341i	$-0.727957\pi$
$647$	−849.489			−1.31297			−0.656483	+	0.754341i	$0.727957\pi$
$648$		0			0
$649$	−1041.61			−1.60495
$650$	−670.970	+	886.785i	−1.03226	+	1.36428i
$651$		0			0
$652$		−	2074.94i		−	3.18243i
$653$	1041.19			1.59447			0.797235	−	0.603669i	$-0.206295\pi$
$653$	1041.19			1.59447			0.797235	+	0.603669i	$0.206295\pi$
$654$		0			0
$655$	−406.683	−	136.518i	−0.620889	−	0.208424i
$656$		−	1304.79i		−	1.98901i
$657$		0			0
$658$		−	482.151i		−	0.732752i
$659$		−	642.380i		−	0.974781i	−0.873184	−	0.487390i	$-0.837949\pi$
$659$		−	642.380i		−	0.974781i	0.873184	−	0.487390i	$-0.162051\pi$
$660$		0			0
$661$	−789.517			−1.19443			−0.597214	−	0.802082i	$-0.703726\pi$
$661$	−789.517			−1.19443			−0.597214	+	0.802082i	$0.703726\pi$
$662$	713.429			1.07769
$663$		0			0
$664$	−1238.33			−1.86495
$665$	87.2194	+	29.2783i	0.131157	+	0.0440275i
$666$		0			0
$667$		−	15.5980i		−	0.0233854i
$668$	594.442			0.889883
$669$		0			0
$670$	−1364.22	−	457.950i	−2.03615	−	0.683508i
$671$			307.879i			0.458836i
$672$		0			0
$673$			1015.93i			1.50956i	0.655978	+	0.754780i	$0.272256\pi$
$673$			1015.93i			1.50956i	−0.655978	+	0.754780i	$0.727744\pi$
$674$		−	1709.40i		−	2.53621i
$675$		0			0
$676$	225.279			0.333253
$677$	−743.622			−1.09841			−0.549204	−	0.835688i	$-0.685069\pi$
$677$	−743.622			−1.09841			−0.549204	+	0.835688i	$0.685069\pi$
$678$		0			0
$679$	328.533			0.483849
$680$	−140.670	+	419.054i	−0.206868	+	0.616255i
$681$		0			0
$682$		−	696.915i		−	1.02187i
$683$	−152.482			−0.223254			−0.111627	−	0.993750i	$-0.535606\pi$
$683$	−152.482			−0.223254			−0.111627	+	0.993750i	$0.535606\pi$
$684$		0			0
$685$	110.946	−	330.507i	0.161965	−	0.482491i
$686$			865.336i			1.26142i
$687$		0			0
$688$			1958.36i			2.84645i
$689$			1045.10i			1.51684i
$690$		0			0
$691$	777.171			1.12471			0.562353	−	0.826898i	$-0.309896\pi$
$691$	777.171			1.12471			0.562353	+	0.826898i	$0.309896\pi$
$692$	3199.99			4.62426
$693$		0			0
$694$	35.8355			0.0516361
$695$	221.041	−	658.477i	0.318045	−	0.947449i
$696$		0			0
$697$			148.289i			0.212753i
$698$	−181.915			−0.260623
$699$		0			0
$700$	371.633	−	491.167i	0.530904	−	0.701667i
$701$			461.657i			0.658569i	0.944231	+	0.329284i	$0.106808\pi$
$701$			461.657i			0.658569i	−0.944231	+	0.329284i	$0.893192\pi$
$702$		0			0
$703$		−	9.44975i		−	0.0134420i
$704$			564.540i			0.801903i
$705$		0			0
$706$	351.626			0.498054
$707$	220.993			0.312578
$708$		0			0
$709$	−358.455			−0.505578			−0.252789	−	0.967521i	$-0.581348\pi$
$709$	−358.455			−0.505578			−0.252789	+	0.967521i	$0.581348\pi$
$710$	1270.23	+	426.397i	1.78905	+	0.600560i
$711$		0			0
$712$		−	1471.19i		−	2.06628i
$713$	9.81624			0.0137675
$714$		0			0
$715$	188.687	−	562.093i	0.263897	−	0.786145i
$716$		−	1857.03i		−	2.59362i
$717$		0			0
$718$			1988.49i			2.76949i
$719$		−	283.414i		−	0.394178i	−0.980386	−	0.197089i	$-0.936851\pi$
$719$		−	283.414i		−	0.394178i	0.980386	−	0.197089i	$-0.0631488\pi$
$720$		0			0
$721$	−104.964			−0.145580
$722$	−1141.71			−1.58131
$723$		0			0
$724$	2340.13			3.23223
$725$	460.863	−	609.098i	0.635673	−	0.840135i
$726$		0			0
$727$			155.359i			0.213699i	0.994275	+	0.106850i	$0.0340763\pi$
$727$			155.359i			0.213699i	−0.994275	+	0.106850i	$0.965924\pi$
$728$	639.063			0.877833
$729$		0			0
$730$	529.750	+	177.830i	0.725685	+	0.243602i
$731$		−	222.566i		−	0.304468i
$732$		0			0
$733$		−	104.932i		−	0.143154i	−0.997435	−	0.0715768i	$-0.977197\pi$
$733$		−	104.932i		−	0.143154i	0.997435	−	0.0715768i	$-0.0228031\pi$
$734$		−	1117.57i		−	1.52258i
$735$		0			0
$736$	−28.8311			−0.0391726
$737$	767.279			1.04108
$738$		0			0
$739$	−627.375			−0.848951			−0.424476	−	0.905439i	$-0.639542\pi$
$739$	−627.375			−0.848951			−0.424476	+	0.905439i	$0.639542\pi$
$740$	−59.9734	−	20.1322i	−0.0810451	−	0.0272057i
$741$		0			0
$742$		−	820.146i		−	1.10532i
$743$	386.590			0.520310			0.260155	−	0.965567i	$-0.416226\pi$
$743$	386.590			0.520310			0.260155	+	0.965567i	$0.416226\pi$
$744$		0			0
$745$	226.370	+	75.9892i	0.303852	+	0.101999i
$746$			481.798i			0.645842i
$747$		0			0
$748$		−	404.157i		−	0.540317i
$749$		−	4.26254i		−	0.00569098i
$750$		0			0
$751$	905.203			1.20533			0.602665	−	0.797994i	$-0.294106\pi$
$751$	905.203			1.20533			0.602665	+	0.797994i	$0.294106\pi$
$752$	−1920.08			−2.55329
$753$		0			0
$754$	1358.98			1.80236
$755$	88.2962	−	263.032i	0.116949	−	0.348387i
$756$		0			0
$757$			332.222i			0.438867i	0.975627	+	0.219433i	$0.0704209\pi$
$757$			332.222i			0.438867i	−0.975627	+	0.219433i	$0.929579\pi$
$758$	2010.08			2.65183
$759$		0			0
$760$	235.308	−	700.978i	0.309616	−	0.922340i
$761$		−	458.100i		−	0.601971i	−0.953629	−	0.300986i	$-0.902684\pi$
$761$		−	458.100i		−	0.601971i	0.953629	−	0.300986i	$-0.0973156\pi$
$762$		0			0
$763$		−	381.620i		−	0.500157i
$764$			1358.46i			1.77809i
$765$		0			0
$766$	−4.59618			−0.00600023
$767$	1278.25			1.66656
$768$		0			0
$769$	−916.392			−1.19167			−0.595833	−	0.803108i	$-0.703178\pi$
$769$	−916.392			−1.19167			−0.595833	+	0.803108i	$0.703178\pi$
$770$	−148.072	+	441.103i	−0.192301	+	0.572861i
$771$		0			0
$772$			2124.77i			2.75229i
$773$	−1186.06			−1.53436			−0.767178	−	0.641434i	$-0.778340\pi$
$773$	−1186.06			−1.53436			−0.767178	+	0.641434i	$0.778340\pi$
$774$		0			0
$775$	383.321	+	290.033i	0.494607	+	0.374236i
$776$		−	2640.41i		−	3.40259i
$777$		0			0
$778$			1468.91i			1.88806i
$779$		−	248.052i		−	0.318424i
$780$		0			0
$781$	−714.414			−0.914742
$782$	8.06559			0.0103141
$783$		0			0
$784$	1598.77			2.03925
$785$	−52.4826	−	17.6177i	−0.0668568	−	0.0224429i
$786$		0			0
$787$			882.479i			1.12132i	0.828046	+	0.560660i	$0.189453\pi$
$787$			882.479i			1.12132i	−0.828046	+	0.560660i	$0.810547\pi$
$788$	272.934			0.346363
$789$		0			0
$790$	676.355	−	2014.85i	0.856146	−	2.55044i
$791$		−	337.501i		−	0.426677i
$792$		0			0
$793$		−	377.823i		−	0.476448i
$794$			2529.46i			3.18572i
$795$		0			0
$796$	1476.59			1.85501
$797$	143.803			0.180431			0.0902153	−	0.995922i	$-0.471244\pi$
$797$	143.803			0.180431			0.0902153	+	0.995922i	$0.471244\pi$
$798$		0			0
$799$	218.216			0.273111
$800$	−1125.84	−	851.848i	−1.40730	−	1.06481i
$801$		0			0
$802$			191.261i			0.238480i
$803$	−297.947			−0.371043
$804$		0			0
$805$	−6.21306	−	2.08563i	−0.00771809	−	0.00259085i
$806$			855.242i			1.06109i
$807$		0			0
$808$		−	1776.11i		−	2.19815i
$809$		−	52.3078i		−	0.0646574i	−0.999477	−	0.0323287i	$-0.989708\pi$
$809$		−	52.3078i		−	0.0646574i	0.999477	−	0.0323287i	$-0.0102923\pi$
$810$		0			0
$811$	1518.65			1.87256			0.936281	−	0.351253i	$-0.114244\pi$
$811$	1518.65			1.87256			0.936281	+	0.351253i	$0.114244\pi$
$812$	−752.704			−0.926976
$813$		0			0
$814$	47.7911			0.0587115
$815$	−1024.94	−	344.059i	−1.25760	−	0.422158i
$816$		0			0
$817$			372.301i			0.455693i
$818$	−1000.35			−1.22292
$819$		0			0
$820$	−1574.28	−	528.462i	−1.91985	−	0.644466i
$821$		−	1097.03i		−	1.33622i	−0.744064	−	0.668108i	$-0.767104\pi$
$821$		−	1097.03i		−	1.33622i	0.744064	−	0.668108i	$-0.232896\pi$
$822$		0			0
$823$		−	957.100i		−	1.16294i	−0.813568	−	0.581470i	$-0.802478\pi$
$823$		−	957.100i		−	1.16294i	0.813568	−	0.581470i	$-0.197522\pi$
$824$			843.587i			1.02377i
$825$		0			0
$826$	−1003.11			−1.21442
$827$	171.626			0.207529			0.103764	−	0.994602i	$-0.466911\pi$
$827$	171.626			0.207529			0.103764	+	0.994602i	$0.466911\pi$
$828$		0			0
$829$	203.896			0.245954			0.122977	−	0.992410i	$-0.460756\pi$
$829$	203.896			0.245954			0.122977	+	0.992410i	$0.460756\pi$
$830$	−352.106	+	1048.92i	−0.424224	+	1.26375i
$831$		0			0
$832$		−	692.793i		−	0.832684i
$833$	−181.700			−0.218127
$834$		0			0
$835$	98.5679	−	293.631i	0.118045	−	0.351654i
$836$			676.059i			0.808683i
$837$		0			0
$838$			101.958i			0.121668i
$839$			1556.65i			1.85536i	0.373377	+	0.927680i	$0.378200\pi$
$839$			1556.65i			1.85536i	−0.373377	+	0.927680i	$0.621800\pi$
$840$		0			0
$841$	−92.4311			−0.109906
$842$	−1612.17			−1.91469
$843$		0			0
$844$	1728.99			2.04857
$845$	37.3549	−	111.279i	0.0442069	−	0.131692i
$846$		0			0
$847$			62.5654i			0.0738670i
$848$	−3266.08			−3.85151
$849$		0			0
$850$	314.958	+	238.308i	0.370539	+	0.280362i
$851$			0.673152i			0.000791013i
$852$		0			0
$853$		−	563.929i		−	0.661112i	−0.943786	−	0.330556i	$-0.892764\pi$
$853$		−	563.929i		−	0.661112i	0.943786	−	0.330556i	$-0.107236\pi$
$854$			296.497i			0.347186i
$855$		0			0
$856$	−34.2579			−0.0400209
$857$	−953.906			−1.11308			−0.556538	−	0.830822i	$-0.687871\pi$
$857$	−953.906			−1.11308			−0.556538	+	0.830822i	$0.687871\pi$
$858$		0			0
$859$	845.456			0.984233			0.492116	−	0.870529i	$-0.336223\pi$
$859$	845.456			0.984233			0.492116	+	0.870529i	$0.336223\pi$
$860$	2362.83	+	793.168i	2.74748	+	0.922289i
$861$		0			0
$862$		−	200.441i		−	0.232530i
$863$	1109.42			1.28554			0.642768	−	0.766061i	$-0.277786\pi$
$863$	1109.42			1.28554			0.642768	+	0.766061i	$0.277786\pi$
$864$		0			0
$865$	530.609	−	1580.67i	0.613420	−	1.82737i
$866$		−	1941.08i		−	2.24143i
$867$		0			0
$868$		−	473.696i		−	0.545733i
$869$			1133.21i			1.30404i
$870$		0			0
$871$	−941.592			−1.08105
$872$	−3067.06			−3.51727
$873$		0			0
$874$	−13.4918			−0.0154369
$875$	−180.995	−	265.016i	−0.206851	−	0.302875i
$876$		0			0
$877$		−	1014.66i		−	1.15697i	−0.815693	−	0.578485i	$-0.803644\pi$
$877$		−	1014.66i		−	1.15697i	0.815693	−	0.578485i	$-0.196356\pi$
$878$	−1534.82			−1.74809
$879$		0			0
$880$	1756.61	+	589.669i	1.99615	+	0.670078i
$881$		−	1226.86i		−	1.39257i	−0.717764	−	0.696287i	$-0.754834\pi$
$881$		−	1226.86i		−	1.39257i	0.717764	−	0.696287i	$-0.245166\pi$
$882$		0			0
$883$			659.407i			0.746780i	0.927674	+	0.373390i	$0.121805\pi$
$883$			659.407i			0.746780i	−0.927674	+	0.373390i	$0.878195\pi$
$884$			495.974i			0.561057i
$885$		0			0
$886$	−1575.07			−1.77774
$887$	771.967			0.870313			0.435156	−	0.900355i	$-0.356693\pi$
$887$	771.967			0.870313			0.435156	+	0.900355i	$0.356693\pi$
$888$		0			0
$889$	259.693			0.292118
$890$	−1246.16	−	418.319i	−1.40018	−	0.470021i
$891$		0			0
$892$			2067.01i			2.31727i
$893$	−365.024			−0.408761
$894$		0			0
$895$	−917.302	−	307.925i	−1.02492	−	0.344050i
$896$		−	36.2718i		−	0.0404820i
$897$		0			0
$898$			872.891i			0.972038i
$899$		−	587.432i		−	0.653428i
$900$		0			0
$901$	371.189			0.411974
$902$	1254.50			1.39080
$903$		0			0
$904$	−2712.48			−3.00053
$905$	388.031	−	1155.94i	0.428764	−	1.27728i
$906$		0			0
$907$			121.471i			0.133926i	0.997755	+	0.0669629i	$0.0213309\pi$
$907$			121.471i			0.133926i	−0.997755	+	0.0669629i	$0.978669\pi$
$908$	−483.792			−0.532810
$909$		0			0
$910$	181.711	−	541.314i	0.199683	−	0.594850i
$911$			1155.23i			1.26809i	0.773295	+	0.634046i	$0.218607\pi$
$911$			1155.23i			1.26809i	−0.773295	+	0.634046i	$0.781393\pi$
$912$		0			0
$913$		−	589.941i		−	0.646157i
$914$			2572.91i			2.81500i
$915$		0			0
$916$	−59.1790			−0.0646059
$917$	220.275			0.240212
$918$		0			0
$919$	−994.576			−1.08224			−0.541119	−	0.840946i	$-0.681999\pi$
$919$	−994.576			−1.08224			−0.541119	+	0.840946i	$0.681999\pi$
$920$	−16.7621	+	49.9341i	−0.0182197	+	0.0542762i
$921$		0			0
$922$			1415.00i			1.53471i
$923$	876.716			0.949855
$924$		0			0
$925$	−19.8891	+	26.2863i	−0.0215017	+	0.0284176i
$926$			995.977i			1.07557i
$927$		0			0
$928$			1725.33i			1.85919i
$929$		−	161.931i		−	0.174307i	−0.996195	−	0.0871534i	$-0.972223\pi$
$929$		−	161.931i		−	0.174307i	0.996195	−	0.0871534i	$-0.0277770\pi$
$930$		0			0
$931$	303.941			0.326467
$932$	−504.113			−0.540893
$933$		0			0
$934$	−2868.53			−3.07123
$935$	−199.638	−	67.0156i	−0.213517	−	0.0716745i
$936$		0			0
$937$			660.489i			0.704898i	0.935831	+	0.352449i	$0.114651\pi$
$937$			660.489i			0.704898i	−0.935831	+	0.352449i	$0.885349\pi$
$938$	738.914			0.787755
$939$		0			0
$940$	−777.664	+	2316.64i	−0.827302	+	2.46451i
$941$		−	1591.25i		−	1.69102i	−0.533961	−	0.845509i	$-0.679297\pi$
$941$		−	1591.25i		−	1.69102i	0.533961	−	0.845509i	$-0.320703\pi$
$942$		0			0
$943$			17.6700i			0.0187380i
$944$			3994.69i			4.23167i
$945$		0			0
$946$	−1882.88			−1.99035
$947$	1563.36			1.65085			0.825426	−	0.564511i	$-0.190935\pi$
$947$	1563.36			1.65085			0.825426	+	0.564511i	$0.190935\pi$
$948$		0			0
$949$	365.635			0.385285
$950$	−526.851	−	398.633i	−0.554580	−	0.419613i
$951$		0			0
$952$		−	226.975i		−	0.238419i
$953$	996.612			1.04576			0.522881	−	0.852405i	$-0.324857\pi$
$953$	996.612			1.04576			0.522881	+	0.852405i	$0.324857\pi$
$954$		0			0
$955$	671.027	+	225.254i	0.702646	+	0.235868i
$956$			936.417i			0.979516i
$957$		0			0
$958$		−	267.566i		−	0.279297i
$959$			179.015i			0.186668i
$960$		0			0
$961$	−591.314			−0.615311
$962$	−58.6484			−0.0609651
$963$		0			0
$964$	−1369.63			−1.42078
$965$	1049.56	+	352.321i	1.08762	+	0.365099i
$966$		0			0
$967$			649.999i			0.672181i	0.941830	+	0.336091i	$0.109105\pi$
$967$			649.999i			0.672181i	−0.941830	+	0.336091i	$0.890895\pi$
$968$	502.835			0.519458
$969$		0			0
$970$	−2236.54	−	750.774i	−2.30571	−	0.773994i
$971$			1717.75i			1.76905i	0.466495	+	0.884524i	$0.345517\pi$
$971$			1717.75i			1.76905i	−0.466495	+	0.884524i	$0.654483\pi$
$972$		0			0
$973$			356.656i			0.366553i
$974$		−	2689.82i		−	2.76162i
$975$		0			0
$976$	1180.74			1.20978
$977$	282.121			0.288763			0.144381	−	0.989522i	$-0.453881\pi$
$977$	282.121			0.288763			0.144381	+	0.989522i	$0.453881\pi$
$978$		0			0
$979$	700.878			0.715913
$980$	647.530	−	1928.98i	0.660745	−	1.96834i
$981$		0			0
$982$			14.0695i			0.0143274i
$983$	−384.535			−0.391185			−0.195592	−	0.980685i	$-0.562663\pi$
$983$	−384.535			−0.391185			−0.195592	+	0.980685i	$0.562663\pi$
$984$		0			0
$985$	45.2568	−	134.819i	0.0459459	−	0.136872i
$986$		−	482.668i		−	0.489521i
$987$		0			0
$988$		−	829.648i		−	0.839725i
$989$		−	26.5208i		−	0.0268158i
$990$		0			0
$991$	399.250			0.402876			0.201438	−	0.979501i	$-0.435439\pi$
$991$	399.250			0.402876			0.201438	+	0.979501i	$0.435439\pi$
$992$	−1085.80			−1.09455
$993$		0			0
$994$	−688.003			−0.692156
$995$	244.842	−	729.379i	0.246072	−	0.733044i
$996$		0			0
$997$			1452.17i			1.45654i	0.685288	+	0.728272i	$0.259676\pi$
$997$			1452.17i			1.45654i	−0.685288	+	0.728272i	$0.740324\pi$
$998$	757.008			0.758525
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.a.404.19		20
3.2	odd	2	inner	405.3.d.a.404.2		20
5.4	even	2	inner	405.3.d.a.404.1		20
9.2	odd	6		45.3.h.a.14.10	yes	20
9.4	even	3		45.3.h.a.29.1	yes	20
9.5	odd	6		135.3.h.a.89.10		20
9.7	even	3		135.3.h.a.44.1		20
15.14	odd	2	inner	405.3.d.a.404.20		20
45.2	even	12		225.3.j.e.176.1		20
45.4	even	6		45.3.h.a.29.10	yes	20
45.7	odd	12		675.3.j.e.476.10		20
45.13	odd	12		225.3.j.e.101.10		20
45.14	odd	6		135.3.h.a.89.1		20
45.22	odd	12		225.3.j.e.101.1		20
45.23	even	12		675.3.j.e.251.1		20
45.29	odd	6		45.3.h.a.14.1	✓	20
45.32	even	12		675.3.j.e.251.10		20
45.34	even	6		135.3.h.a.44.10		20
45.38	even	12		225.3.j.e.176.10		20
45.43	odd	12		675.3.j.e.476.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.1	✓	20	45.29	odd	6
45.3.h.a.14.10	yes	20	9.2	odd	6
45.3.h.a.29.1	yes	20	9.4	even	3
45.3.h.a.29.10	yes	20	45.4	even	6
135.3.h.a.44.1		20	9.7	even	3
135.3.h.a.44.10		20	45.34	even	6
135.3.h.a.89.1		20	45.14	odd	6
135.3.h.a.89.10		20	9.5	odd	6
225.3.j.e.101.1		20	45.22	odd	12
225.3.j.e.101.10		20	45.13	odd	12
225.3.j.e.176.1		20	45.2	even	12
225.3.j.e.176.10		20	45.38	even	12
405.3.d.a.404.1		20	5.4	even	2	inner
405.3.d.a.404.2		20	3.2	odd	2	inner
405.3.d.a.404.19		20	1.1	even	1	trivial
405.3.d.a.404.20		20	15.14	odd	2	inner
675.3.j.e.251.1		20	45.23	even	12
675.3.j.e.251.10		20	45.32	even	12
675.3.j.e.476.1		20	45.43	odd	12
675.3.j.e.476.10		20	45.7	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.68728 q^{2} +9.59601 q^{4} +(1.59117 - 4.74006i) q^{5} +2.56739i q^{7} +20.6340 q^{8} +(5.86709 - 17.4779i) q^{10} +9.83010i q^{11} -12.0633i q^{13} +9.46669i q^{14} +37.6993 q^{16} -4.28451 q^{17} +7.16698 q^{19} +(15.2689 - 45.4857i) q^{20} +36.2463i q^{22} -0.510539 q^{23} +(-19.9364 - 15.0845i) q^{25} -44.4808i q^{26} +24.6367i q^{28} +30.5521i q^{29} -19.2272 q^{31} +56.4718 q^{32} -15.7982 q^{34} +(12.1696 + 4.08516i) q^{35} -1.31851i q^{37} +26.4267 q^{38} +(32.8323 - 97.8066i) q^{40} -34.6104i q^{41} +51.9467i q^{43} +94.3297i q^{44} -1.88250 q^{46} -50.9313 q^{47} +42.4085 q^{49} +(-73.5108 - 55.6207i) q^{50} -115.760i q^{52} -86.6349 q^{53} +(46.5953 + 15.6414i) q^{55} +52.9757i q^{56} +112.654i q^{58} +105.962i q^{59} +31.3200 q^{61} -70.8961 q^{62} +57.4297 q^{64} +(-57.1809 - 19.1948i) q^{65} -78.0541i q^{67} -41.1142 q^{68} +(44.8727 + 15.0631i) q^{70} +72.6762i q^{71} +30.3097i q^{73} -4.86172i q^{74} +68.7744 q^{76} -25.2377 q^{77} +115.280 q^{79} +(59.9861 - 178.697i) q^{80} -127.618i q^{82} -60.0138 q^{83} +(-6.81739 + 20.3089i) q^{85} +191.542i q^{86} +202.835i q^{88} -71.2992i q^{89} +30.9713 q^{91} -4.89914 q^{92} -187.798 q^{94} +(11.4039 - 33.9719i) q^{95} -127.964i q^{97} +156.372 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 36 q^{4} + 4 q^{10} + 52 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} + 8 q^{34} + 68 q^{40} + 116 q^{46} + 80 q^{49} + 36 q^{55} + 100 q^{61} + 140 q^{64} + 108 q^{70} + 192 q^{76} + 256 q^{79} + 148 q^{85}+ \cdots - 436 q^{94}+O(q^{100}) \) 20 * q + 36 * q^4 + 4 * q^10 + 52 * q^16 - 8 * q^19 - 4 * q^25 - 56 * q^31 + 8 * q^34 + 68 * q^40 + 116 * q^46 + 80 * q^49 + 36 * q^55 + 100 * q^61 + 140 * q^64 + 108 * q^70 + 192 * q^76 + 256 * q^79 + 148 * q^85 - 288 * q^91 - 436 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more