LMFDB - Embedded newform 324.5.k.a.17.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,5,Mod(17,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 11]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("324.17");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.k (of order $18$, degree $6$, 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:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			17.11
Character	$\chi$	$=$	324.17
Dual form			324.5.k.a.305.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(34.9786 + 6.16768i) q^{5} +(32.9047 - 27.6104i) q^{7} +O(q^{10})$	$q+(34.9786 + 6.16768i) q^{5} +(32.9047 - 27.6104i) q^{7} +(225.646 - 39.7875i) q^{11} +(-42.2209 + 15.3671i) q^{13} +(121.583 + 70.1961i) q^{17} +(-213.411 - 369.638i) q^{19} +(-118.681 + 141.438i) q^{23} +(598.158 + 217.712i) q^{25} +(81.9498 - 225.155i) q^{29} +(-733.987 - 615.888i) q^{31} +(1321.26 - 762.827i) q^{35} +(-199.487 + 345.522i) q^{37} +(491.561 + 1350.55i) q^{41} +(545.187 + 3091.91i) q^{43} +(-2046.16 - 2438.52i) q^{47} +(-96.5390 + 547.500i) q^{49} -4424.38i q^{53} +8138.19 q^{55} +(1692.71 + 298.471i) q^{59} +(312.407 - 262.141i) q^{61} +(-1571.61 + 277.117i) q^{65} +(7589.50 - 2762.35i) q^{67} +(6986.58 + 4033.70i) q^{71} +(-2537.15 - 4394.48i) q^{73} +(6326.28 - 7539.37i) q^{77} +(5936.39 + 2160.67i) q^{79} +(2469.90 - 6786.00i) q^{83} +(3819.87 + 3205.25i) q^{85} +(-7529.73 + 4347.29i) q^{89} +(-964.975 + 1671.39i) q^{91} +(-5185.01 - 14245.7i) q^{95} +(-736.704 - 4178.06i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{5}+O(q^{10}) $ 72 * q - 9 * q^5	$ 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) $ 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{11}{18}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	34.9786	+	6.16768i	1.39915	+	0.246707i	0.821792	−	0.569787i	$-0.192974\pi$
$5$	34.9786	+	6.16768i	1.39915	+	0.246707i	0.577354	+	0.816494i	$0.304085\pi$
$6$		0			0
$7$	32.9047	−	27.6104i	0.671525	−	0.563477i	−0.241991	−	0.970278i	$-0.577800\pi$
$7$	32.9047	−	27.6104i	0.671525	−	0.563477i	0.913516	+	0.406802i	$0.133356\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	225.646	−	39.7875i	1.86484	−	0.328822i	0.876541	−	0.481327i	$-0.159845\pi$
$11$	225.646	−	39.7875i	1.86484	−	0.328822i	0.988303	+	0.152505i	$0.0487340\pi$
$12$		0			0
$13$	−42.2209	+	15.3671i	−0.249828	+	0.0909298i	−0.463899	−	0.885888i	$-0.653550\pi$
$13$	−42.2209	+	15.3671i	−0.249828	+	0.0909298i	0.214071	+	0.976818i	$0.431328\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	121.583	+	70.1961i	0.420703	+	0.242893i	0.695378	−	0.718644i	$-0.255237\pi$
$17$	121.583	+	70.1961i	0.420703	+	0.242893i	−0.274675	+	0.961537i	$0.588570\pi$
$18$		0			0
$19$	−213.411	−	369.638i	−0.591166	−	1.02393i	−0.994076	−	0.108689i	$-0.965335\pi$
$19$	−213.411	−	369.638i	−0.591166	−	1.02393i	0.402910	−	0.915239i	$-0.367999\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−118.681	+	141.438i	−0.224350	+	0.267370i	−0.866464	−	0.499239i	$-0.833613\pi$
$23$	−118.681	+	141.438i	−0.224350	+	0.267370i	0.642115	+	0.766609i	$0.278057\pi$
$24$		0			0
$25$	598.158	+	217.712i	0.957052	+	0.348339i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	81.9498	−	225.155i	0.0974433	−	0.267723i	−0.881388	−	0.472394i	$-0.843390\pi$
$29$	81.9498	−	225.155i	0.0974433	−	0.267723i	0.978831	+	0.204670i	$0.0656123\pi$
$30$		0			0
$31$	−733.987	−	615.888i	−0.763774	−	0.640883i	0.175332	−	0.984509i	$-0.443900\pi$
$31$	−733.987	−	615.888i	−0.763774	−	0.640883i	−0.939106	+	0.343627i	$0.888344\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1321.26	−	762.827i	1.07858	−	0.622716i
$36$		0			0
$37$	−199.487	+	345.522i	−0.145718	+	0.252390i	−0.929640	−	0.368468i	$-0.879882\pi$
$37$	−199.487	+	345.522i	−0.145718	+	0.252390i	0.783923	+	0.620858i	$0.213216\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	491.561	+	1350.55i	0.292422	+	0.803422i	0.995711	+	0.0925183i	$0.0294917\pi$
$41$	491.561	+	1350.55i	0.292422	+	0.803422i	−0.703289	+	0.710904i	$0.748286\pi$
$42$		0			0
$43$	545.187	+	3091.91i	0.294855	+	1.67221i	0.667791	+	0.744348i	$0.267240\pi$
$43$	545.187	+	3091.91i	0.294855	+	1.67221i	−0.372936	+	0.927857i	$0.621649\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2046.16	−	2438.52i	−0.926283	−	1.10390i	−0.994343	−	0.106219i	$-0.966125\pi$
$47$	−2046.16	−	2438.52i	−0.926283	−	1.10390i	0.0680601	−	0.997681i	$-0.478319\pi$
$48$		0			0
$49$	−96.5390	+	547.500i	−0.0402078	+	0.228030i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4424.38i		−	1.57507i	−0.616267	−	0.787537i	$-0.711356\pi$
$53$		−	4424.38i		−	1.57507i	0.616267	−	0.787537i	$-0.288644\pi$
$54$		0			0
$55$	8138.19			2.69031
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1692.71	+	298.471i	0.486272	+	0.0857428i	0.411409	−	0.911451i	$-0.365037\pi$
$59$	1692.71	+	298.471i	0.486272	+	0.0857428i	0.0748631	+	0.997194i	$0.476148\pi$
$60$		0			0
$61$	312.407	−	262.141i	0.0839578	−	0.0704490i	−0.599843	−	0.800118i	$-0.704770\pi$
$61$	312.407	−	262.141i	0.0839578	−	0.0704490i	0.683801	+	0.729669i	$0.260326\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1571.61	+	277.117i	−0.371978	+	0.0655898i
$66$		0			0
$67$	7589.50	−	2762.35i	1.69069	−	0.615360i	0.695976	−	0.718065i	$-0.254972\pi$
$67$	7589.50	−	2762.35i	1.69069	−	0.615360i	0.994712	+	0.102705i	$0.0327497\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6986.58	+	4033.70i	1.38595	+	0.800179i	0.992856	−	0.119319i	$-0.0380711\pi$
$71$	6986.58	+	4033.70i	1.38595	+	0.800179i	0.393095	+	0.919498i	$0.371404\pi$
$72$		0			0
$73$	−2537.15	−	4394.48i	−0.476103	−	0.824634i	0.523522	−	0.852012i	$-0.324618\pi$
$73$	−2537.15	−	4394.48i	−0.476103	−	0.824634i	−0.999625	+	0.0273776i	$0.991284\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6326.28	−	7539.37i	1.06701	−	1.27161i
$78$		0			0
$79$	5936.39	+	2160.67i	0.951191	+	0.346205i	0.770576	−	0.637348i	$-0.219969\pi$
$79$	5936.39	+	2160.67i	0.951191	+	0.346205i	0.180616	+	0.983554i	$0.442191\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2469.90	−	6786.00i	0.358528	−	0.985049i	−0.621012	−	0.783801i	$-0.713278\pi$
$83$	2469.90	−	6786.00i	0.358528	−	0.985049i	0.979540	−	0.201248i	$-0.0644996\pi$
$84$		0			0
$85$	3819.87	+	3205.25i	0.528702	+	0.443634i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7529.73	+	4347.29i	−0.950604	+	0.548831i	−0.893268	−	0.449524i	$-0.851594\pi$
$89$	−7529.73	+	4347.29i	−0.950604	+	0.548831i	−0.0573353	+	0.998355i	$0.518260\pi$
$90$		0			0
$91$	−964.975	+	1671.39i	−0.116529	+	0.201834i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5185.01	−	14245.7i	−0.574516	−	1.57847i
$96$		0			0
$97$	−736.704	−	4178.06i	−0.0782978	−	0.444049i	−0.998603	−	0.0528470i	$-0.983170\pi$
$97$	−736.704	−	4178.06i	−0.0782978	−	0.444049i	0.920305	−	0.391202i	$-0.127941\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2786.37	+	3320.67i	0.273147	+	0.325524i	0.885127	−	0.465349i	$-0.154071\pi$
$101$	2786.37	+	3320.67i	0.273147	+	0.325524i	−0.611980	+	0.790873i	$0.709627\pi$
$102$		0			0
$103$	−2972.56	+	16858.2i	−0.280192	+	1.58905i	0.441779	+	0.897124i	$0.354348\pi$
$103$	−2972.56	+	16858.2i	−0.280192	+	1.58905i	−0.721971	+	0.691924i	$0.756763\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			11917.4i			1.04091i	0.853888	+	0.520456i	$0.174238\pi$
$107$			11917.4i			1.04091i	−0.853888	+	0.520456i	$0.825762\pi$
$108$		0			0
$109$	−20310.0			−1.70945			−0.854724	−	0.519082i	$-0.826274\pi$
$109$	−20310.0			−1.70945			−0.854724	+	0.519082i	$0.826274\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10661.7	+	1879.95i	0.834969	+	0.147228i	0.574757	−	0.818324i	$-0.305097\pi$
$113$	10661.7	+	1879.95i	0.834969	+	0.147228i	0.260212	+	0.965552i	$0.416208\pi$
$114$		0			0
$115$	−5023.65	+	4215.34i	−0.379860	+	0.318740i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5938.81	−	1047.17i	0.419378	−	0.0739476i
$120$		0			0
$121$	35575.1	−	12948.3i	2.42983	−	0.884384i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	355.154	+	205.049i	0.0227299	+	0.0131231i
$126$		0			0
$127$	1766.13	+	3059.02i	0.109500	+	0.189660i	0.915568	−	0.402163i	$-0.131742\pi$
$127$	1766.13	+	3059.02i	0.109500	+	0.189660i	−0.806068	+	0.591823i	$0.798408\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9208.36	−	10974.1i	0.536586	−	0.639479i	−0.427833	−	0.903858i	$-0.640723\pi$
$131$	9208.36	−	10974.1i	0.536586	−	0.639479i	0.964419	+	0.264379i	$0.0851671\pi$
$132$		0			0
$133$	−17228.1	−	6270.51i	−0.973943	−	0.354486i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6431.59	+	17670.7i	−0.342671	+	0.941481i	0.641945	+	0.766750i	$0.278128\pi$
$137$	−6431.59	+	17670.7i	−0.342671	+	0.941481i	−0.984616	+	0.174730i	$0.944095\pi$
$138$		0			0
$139$	−20060.3	−	16832.6i	−1.03827	−	0.871208i	−0.0464545	−	0.998920i	$-0.514792\pi$
$139$	−20060.3	−	16832.6i	−1.03827	−	0.871208i	−0.991811	+	0.127712i	$0.959237\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8915.56	+	5147.40i	−0.435990	+	0.251719i
$144$		0			0
$145$	4255.18	−	7370.19i	0.202387	−	0.350544i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5479.45	+	15054.7i	0.246811	+	0.678107i	0.999799	+	0.0200725i	$0.00638969\pi$
$149$	5479.45	+	15054.7i	0.246811	+	0.678107i	−0.752988	+	0.658035i	$0.771388\pi$
$150$		0			0
$151$	4790.26	+	27166.9i	0.210090	+	1.19148i	0.889226	+	0.457467i	$0.151243\pi$
$151$	4790.26	+	27166.9i	0.210090	+	1.19148i	−0.679136	+	0.734012i	$0.737646\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−21875.3	−	26069.9i	−0.910521	−	1.08512i
$156$		0			0
$157$	4287.95	−	24318.1i	0.173960	−	0.986578i	−0.765377	−	0.643582i	$-0.777447\pi$
$157$	4287.95	−	24318.1i	0.173960	−	0.986578i	0.939337	−	0.342995i	$-0.111441\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			7930.82i			0.305961i
$162$		0			0
$163$	−43724.5			−1.64570			−0.822848	−	0.568261i	$-0.807616\pi$
$163$	−43724.5			−1.64570			−0.822848	+	0.568261i	$0.807616\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6662.89	+	1174.85i	0.238907	+	0.0421258i	0.291820	−	0.956473i	$-0.405739\pi$
$167$	6662.89	+	1174.85i	0.238907	+	0.0421258i	−0.0529123	+	0.998599i	$0.516850\pi$
$168$		0			0
$169$	−20332.5	+	17061.0i	−0.711899	+	0.597354i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−41118.4	+	7250.29i	−1.37387	+	0.242249i	−0.811361	−	0.584545i	$-0.801273\pi$
$173$	−41118.4	+	7250.29i	−1.37387	+	0.242249i	−0.562504	+	0.826795i	$0.690162\pi$
$174$		0			0
$175$	25693.3	−	9351.60i	0.838966	−	0.305359i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	19700.7	+	11374.2i	0.614860	+	0.354989i	0.774865	−	0.632127i	$-0.217818\pi$
$179$	19700.7	+	11374.2i	0.614860	+	0.354989i	−0.160005	+	0.987116i	$0.551151\pi$
$180$		0			0
$181$	−11866.6	−	20553.6i	−0.362218	−	0.627381i	0.626107	−	0.779737i	$-0.284647\pi$
$181$	−11866.6	−	20553.6i	−0.362218	−	0.627381i	−0.988326	+	0.152356i	$0.951314\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9108.87	+	10855.5i	−0.266147	+	0.317181i
$186$		0			0
$187$	30227.7	+	11002.0i	0.864415	+	0.314621i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8606.22	+	23645.4i	−0.235910	+	0.648157i	0.764086	+	0.645115i	$0.223190\pi$
$191$	−8606.22	+	23645.4i	−0.235910	+	0.648157i	−0.999995	+	0.00304179i	$0.999032\pi$
$192$		0			0
$193$	−2499.18	−	2097.06i	−0.0670938	−	0.0562984i	0.608624	−	0.793459i	$-0.291722\pi$
$193$	−2499.18	−	2097.06i	−0.0670938	−	0.0562984i	−0.675718	+	0.737160i	$0.736166\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−27617.7	+	15945.1i	−0.711632	+	0.410861i	−0.811665	−	0.584123i	$-0.801438\pi$
$197$	−27617.7	+	15945.1i	−0.711632	+	0.410861i	0.100033	+	0.994984i	$0.468105\pi$
$198$		0			0
$199$	27550.6	−	47719.1i	0.695705	−	1.20500i	−0.274237	−	0.961662i	$-0.588425\pi$
$199$	27550.6	−	47719.1i	0.695705	−	1.20500i	0.969942	−	0.243335i	$-0.0782414\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3520.08	−	9671.34i	−0.0854202	−	0.234690i
$204$		0			0
$205$	8864.36	+	50272.3i	0.210931	+	1.19625i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−62862.3	−	74916.4i	−1.43912	−	1.71508i
$210$		0			0
$211$	−10198.4	+	57837.8i	−0.229069	+	1.29911i	0.625683	+	0.780077i	$0.284820\pi$
$211$	−10198.4	+	57837.8i	−0.229069	+	1.29911i	−0.854752	+	0.519036i	$0.826291\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			111513.i			2.41240i
$216$		0			0
$217$	−41156.5			−0.874016
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6212.07	−	1095.36i	−0.127190	−	0.0224270i
$222$		0			0
$223$	−51268.9	+	43019.7i	−1.03096	+	0.865082i	−0.990965	−	0.134117i	$-0.957180\pi$
$223$	−51268.9	+	43019.7i	−1.03096	+	0.865082i	−0.0399995	+	0.999200i	$0.512736\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−38576.7	+	6802.11i	−0.748641	+	0.132006i	−0.534936	−	0.844893i	$-0.679664\pi$
$227$	−38576.7	+	6802.11i	−0.748641	+	0.132006i	−0.213705	+	0.976898i	$0.568553\pi$
$228$		0			0
$229$	−43918.8	+	15985.2i	−0.837490	+	0.304822i	−0.724929	−	0.688823i	$-0.758128\pi$
$229$	−43918.8	+	15985.2i	−0.837490	+	0.304822i	−0.112561	+	0.993645i	$0.535905\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1081.44	+	624.373i	0.0199201	+	0.0115009i	0.509927	−	0.860218i	$-0.329672\pi$
$233$	1081.44	+	624.373i	0.0199201	+	0.0115009i	−0.490007	+	0.871719i	$0.663006\pi$
$234$		0			0
$235$	−56531.9	−	97916.1i	−1.02366	−	1.77304i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	30616.6	−	36487.5i	0.535996	−	0.638775i	−0.428289	−	0.903642i	$-0.640883\pi$
$239$	30616.6	−	36487.5i	0.535996	−	0.638775i	0.964285	+	0.264867i	$0.0853279\pi$
$240$		0			0
$241$	−63449.2	−	23093.6i	−1.09243	−	0.397611i	−0.267907	−	0.963445i	$-0.586332\pi$
$241$	−63449.2	−	23093.6i	−1.09243	−	0.397611i	−0.824519	+	0.565834i	$0.808554\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6753.60	+	18555.4i	−0.112513	+	0.309127i
$246$		0			0
$247$	14690.7	+	12326.9i	0.240795	+	0.202051i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	59287.2	−	34229.5i	0.941051	−	0.543316i	0.0507616	−	0.998711i	$-0.483835\pi$
$251$	59287.2	−	34229.5i	0.941051	−	0.543316i	0.890290	+	0.455395i	$0.150502\pi$
$252$		0			0
$253$	−21152.4	+	36637.1i	−0.330460	+	0.572374i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2075.12	+	5701.34i	0.0314179	+	0.0863199i	0.954410	−	0.298499i	$-0.0964860\pi$
$257$	2075.12	+	5701.34i	0.0314179	+	0.0863199i	−0.922992	+	0.384819i	$0.874264\pi$
$258$		0			0
$259$	2975.91	+	16877.2i	0.0443630	+	0.251595i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5335.30	+	6358.37i	0.0771343	+	0.0919251i	0.803231	−	0.595667i	$-0.203112\pi$
$263$	5335.30	+	6358.37i	0.0771343	+	0.0919251i	−0.726097	+	0.687592i	$0.758668\pi$
$264$		0			0
$265$	27288.2	−	154759.i	0.388582	−	2.20376i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			55809.9i			0.771271i	0.922651	+	0.385635i	$0.126018\pi$
$269$			55809.9i			0.771271i	−0.922651	+	0.385635i	$0.873982\pi$
$270$		0			0
$271$	87613.4			1.19298			0.596489	−	0.802622i	$-0.296562\pi$
$271$	87613.4			1.19298			0.596489	+	0.802622i	$0.296562\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	143634.	+	25326.6i	1.89929	+	0.334897i
$276$		0			0
$277$	497.459	−	417.418i	0.00648333	−	0.00544016i	−0.639540	−	0.768758i	$-0.720875\pi$
$277$	497.459	−	417.418i	0.00648333	−	0.00544016i	0.646024	+	0.763318i	$0.276431\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−30216.0	+	5327.89i	−0.382670	+	0.0674750i	−0.361674	−	0.932305i	$-0.617795\pi$
$281$	−30216.0	+	5327.89i	−0.382670	+	0.0674750i	−0.0209955	+	0.999780i	$0.506684\pi$
$282$		0			0
$283$	101593.	−	36976.9i	1.26851	−	0.461698i	0.381891	−	0.924207i	$-0.375273\pi$
$283$	101593.	−	36976.9i	1.26851	−	0.461698i	0.886614	+	0.462509i	$0.153051\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	53463.9	+	30867.4i	0.649078	+	0.374746i
$288$		0			0
$289$	−31905.5	−	55262.0i	−0.382006	−	0.661653i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	77038.0	−	91810.3i	0.897366	−	1.06944i	−0.0998598	−	0.995002i	$-0.531839\pi$
$293$	77038.0	−	91810.3i	0.897366	−	1.06944i	0.997226	−	0.0744375i	$-0.0237161\pi$
$294$		0			0
$295$	57367.9	+	20880.2i	0.659212	+	0.239933i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2837.31	−	7795.44i	0.0317369	−	0.0871964i
$300$		0			0
$301$	103308.	+	86685.6i	1.14025	+	0.956785i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12544.4	−	7242.50i	0.134849	−	0.0778554i
$306$		0			0
$307$	−52688.2	+	91258.7i	−0.559032	+	0.968272i	0.438546	+	0.898709i	$0.355494\pi$
$307$	−52688.2	+	91258.7i	−0.559032	+	0.968272i	−0.997578	+	0.0695629i	$0.977840\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4711.44	−	12944.6i	−0.0487116	−	0.133834i	0.912951	−	0.408069i	$-0.133798\pi$
$311$	−4711.44	−	12944.6i	−0.0487116	−	0.133834i	−0.961663	+	0.274235i	$0.911575\pi$
$312$		0			0
$313$	2507.41	+	14220.2i	0.0255939	+	0.145150i	0.994927	−	0.100601i	$-0.0320765\pi$
$313$	2507.41	+	14220.2i	0.0255939	+	0.145150i	−0.969333	+	0.245751i	$0.920965\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−51395.9	−	61251.3i	−0.511458	−	0.609532i	0.447081	−	0.894494i	$-0.352464\pi$
$317$	−51395.9	−	61251.3i	−0.511458	−	0.609532i	−0.958539	+	0.284961i	$0.908019\pi$
$318$		0			0
$319$	9533.29	−	54066.0i	0.0936832	−	0.531304i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	59922.4i		−	0.574360i
$324$		0			0
$325$	−28600.3			−0.270773
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−134657.	−	23743.6i	−1.24404	−	0.219359i
$330$		0			0
$331$	−43127.5	+	36188.2i	−0.393639	+	0.330302i	−0.818029	−	0.575177i	$-0.804933\pi$
$331$	−43127.5	+	36188.2i	−0.393639	+	0.330302i	0.424390	+	0.905480i	$0.360489\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	282508.	−	49813.7i	2.51733	−	0.443874i
$336$		0			0
$337$	−24903.1	+	9064.00i	−0.219278	+	0.0798105i	−0.449323	−	0.893369i	$-0.648335\pi$
$337$	−24903.1	+	9064.00i	−0.219278	+	0.0798105i	0.230045	+	0.973180i	$0.426113\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−190126.	−	109769.i	−1.63506	−	0.944000i
$342$		0			0
$343$	63506.5	+	109996.i	0.539796	+	0.934954i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	25231.5	−	30069.7i	0.209548	−	0.249729i	−0.651026	−	0.759056i	$-0.725661\pi$
$347$	25231.5	−	30069.7i	0.209548	−	0.249729i	0.860573	+	0.509326i	$0.170105\pi$
$348$		0			0
$349$	95322.3	+	34694.5i	0.782607	+	0.284846i	0.702259	−	0.711921i	$-0.252175\pi$
$349$	95322.3	+	34694.5i	0.782607	+	0.284846i	0.0803478	+	0.996767i	$0.474397\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−43472.2	+	119439.i	−0.348869	+	0.958510i	0.633858	+	0.773449i	$0.281470\pi$
$353$	−43472.2	+	119439.i	−0.348869	+	0.958510i	−0.982727	+	0.185061i	$0.940752\pi$
$354$		0			0
$355$	219502.	+	184184.i	1.74174	+	1.46149i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−138882.	+	80183.8i	−1.07760	+	0.622154i	−0.930249	−	0.366930i	$-0.880409\pi$
$359$	−138882.	+	80183.8i	−1.07760	+	0.622154i	−0.147354	+	0.989084i	$0.547076\pi$
$360$		0			0
$361$	−25927.8	+	44908.3i	−0.198953	+	0.344597i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−61642.4	−	169361.i	−0.462694	−	1.27124i
$366$		0			0
$367$	−8005.82	−	45403.3i	−0.0594393	−	0.337097i	0.940557	−	0.339635i	$-0.110303\pi$
$367$	−8005.82	−	45403.3i	−0.0594393	−	0.337097i	−0.999997	+	0.00253745i	$0.999192\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−122159.	−	145583.i	−0.887518	−	1.05770i
$372$		0			0
$373$	6455.45	−	36610.7i	0.0463990	−	0.263142i	−0.952780	−	0.303662i	$-0.901791\pi$
$373$	6455.45	−	36610.7i	0.0463990	−	0.263142i	0.999179	+	0.0405205i	$0.0129016\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			10765.6i			0.0757452i
$378$		0			0
$379$	−1082.06			−0.00753308			−0.00376654	−	0.999993i	$-0.501199\pi$
$379$	−1082.06			−0.00753308			−0.00376654	+	0.999993i	$0.501199\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	67197.3	+	11848.7i	0.458094	+	0.0807743i	0.397933	−	0.917414i	$-0.369728\pi$
$383$	67197.3	+	11848.7i	0.458094	+	0.0807743i	0.0601606	+	0.998189i	$0.480839\pi$
$384$		0			0
$385$	267785.	−	224698.i	1.80661	−	1.51593i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	238825.	−	42111.3i	1.57827	−	0.278291i	0.685248	−	0.728309i	$-0.259694\pi$
$389$	238825.	−	42111.3i	1.57827	−	0.278291i	0.893019	+	0.450018i	$0.148582\pi$
$390$		0			0
$391$	−24358.1	+	8865.61i	−0.159327	+	0.0579903i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	194320.	+	112191.i	1.24544	+	0.719058i
$396$		0			0
$397$	−12499.0	−	21648.8i	−0.0793036	−	0.137358i	0.823646	−	0.567104i	$-0.191936\pi$
$397$	−12499.0	−	21648.8i	−0.0793036	−	0.137358i	−0.902950	+	0.429746i	$0.858603\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−42149.4	+	50231.7i	−0.262121	+	0.312384i	−0.881013	−	0.473093i	$-0.843138\pi$
$401$	−42149.4	+	50231.7i	−0.262121	+	0.312384i	0.618891	+	0.785477i	$0.287582\pi$
$402$		0			0
$403$	40454.0	+	14724.1i	0.249087	+	0.0906604i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−31266.1	+	85902.9i	−0.188749	+	0.518584i
$408$		0			0
$409$	−66508.1	−	55806.9i	−0.397583	−	0.333612i	0.421975	−	0.906607i	$-0.361337\pi$
$409$	−66508.1	−	55806.9i	−0.397583	−	0.333612i	−0.819559	+	0.572995i	$0.805781\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	63939.1	−	36915.3i	0.374858	−	0.216424i
$414$		0			0
$415$	128248.	−	222132.i	0.744652	−	1.28978i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28999.6	−	79675.9i	−0.165183	−	0.453836i	0.829292	−	0.558816i	$-0.188744\pi$
$419$	−28999.6	−	79675.9i	−0.165183	−	0.453836i	−0.994474	+	0.104980i	$0.966522\pi$
$420$		0			0
$421$	19379.7	+	109908.i	0.109341	+	0.620105i	0.989397	+	0.145235i	$0.0463938\pi$
$421$	19379.7	+	109908.i	0.109341	+	0.620105i	−0.880056	+	0.474870i	$0.842495\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	57443.5	+	68458.5i	0.318026	+	0.379009i
$426$		0			0
$427$	3041.88	−	17251.3i	0.0166834	−	0.0946165i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	303821.i		−	1.63555i	−0.575539	−	0.817774i	$-0.695208\pi$
$431$		−	303821.i		−	1.63555i	0.575539	−	0.817774i	$-0.304792\pi$
$432$		0			0
$433$	−2295.55			−0.0122437			−0.00612184	−	0.999981i	$-0.501949\pi$
$433$	−2295.55			−0.0122437			−0.00612184	+	0.999981i	$0.501949\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	77608.9	+	13684.5i	0.406395	+	0.0716584i
$438$		0			0
$439$	−205003.	+	172018.i	−1.06373	+	0.892576i	−0.994470	−	0.105023i	$-0.966508\pi$
$439$	−205003.	+	172018.i	−1.06373	+	0.892576i	−0.0692606	+	0.997599i	$0.522064\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−120916.	+	21320.8i	−0.616137	+	0.108642i	−0.473002	−	0.881062i	$-0.656830\pi$
$443$	−120916.	+	21320.8i	−0.616137	+	0.108642i	−0.143135	+	0.989703i	$0.545718\pi$
$444$		0			0
$445$	−290193.	+	105621.i	−1.46543	+	0.533374i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	266871.	+	154078.i	1.32376	+	0.764271i	0.984326	−	0.176359i	$-0.0564321\pi$
$449$	266871.	+	154078.i	1.32376	+	0.764271i	0.339431	+	0.940631i	$0.389765\pi$
$450$		0			0
$451$	164654.	+	285189.i	0.809504	+	1.40210i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−44062.1	+	52511.2i	−0.212835	+	0.253646i
$456$		0			0
$457$	−216687.	−	78867.7i	−1.03753	−	0.377630i	−0.233587	−	0.972336i	$-0.575046\pi$
$457$	−216687.	−	78867.7i	−1.03753	−	0.377630i	−0.803943	+	0.594706i	$0.797269\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20157.1	+	55381.1i	−0.0948474	+	0.260591i	−0.978039	−	0.208421i	$-0.933168\pi$
$461$	−20157.1	+	55381.1i	−0.0948474	+	0.260591i	0.883192	+	0.469012i	$0.155390\pi$
$462$		0			0
$463$	−84949.4	−	71281.0i	−0.396276	−	0.332515i	0.422776	−	0.906234i	$-0.361056\pi$
$463$	−84949.4	−	71281.0i	−0.396276	−	0.332515i	−0.819052	+	0.573719i	$0.805500\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	219996.	−	127015.i	1.00875	−	0.582399i	0.0979212	−	0.995194i	$-0.468781\pi$
$467$	219996.	−	127015.i	1.00875	−	0.582399i	0.910824	+	0.412795i	$0.135447\pi$
$468$		0			0
$469$	173461.	−	300443.i	0.788599	−	1.36589i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	246039.	+	675985.i	1.09972	+	3.02145i
$474$		0			0
$475$	−47178.7	−	267564.i	−0.209102	−	1.18588i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−261487.	−	311628.i	−1.13967	−	1.35821i	−0.924297	−	0.381675i	$-0.875347\pi$
$479$	−261487.	−	311628.i	−1.13967	−	1.35821i	−0.215374	−	0.976532i	$-0.569097\pi$
$480$		0			0
$481$	3112.84	−	17653.8i	0.0134545	−	0.0763042i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	150687.i		−	0.640606i
$486$		0			0
$487$	81194.7			0.342349			0.171175	−	0.985241i	$-0.445244\pi$
$487$	81194.7			0.342349			0.171175	+	0.985241i	$0.445244\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−62761.6	−	11066.6i	−0.260334	−	0.0459039i	0.0419577	−	0.999119i	$-0.486641\pi$
$491$	−62761.6	−	11066.6i	−0.260334	−	0.0459039i	−0.302292	+	0.953215i	$0.597752\pi$
$492$		0			0
$493$	25768.8	−	21622.6i	0.106023	−	0.0889638i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	341264.	−	60174.0i	1.38158	−	0.243610i
$498$		0			0
$499$	−360229.	+	131113.i	−1.44670	+	0.526555i	−0.941667	−	0.336546i	$-0.890741\pi$
$499$	−360229.	+	131113.i	−1.44670	+	0.526555i	−0.505031	+	0.863101i	$0.668519\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−343606.	−	198381.i	−1.35808	−	0.784086i	−0.368712	−	0.929544i	$-0.620201\pi$
$503$	−343606.	−	198381.i	−1.35808	−	0.784086i	−0.989364	+	0.145458i	$0.953535\pi$
$504$		0			0
$505$	76982.7	+	133338.i	0.301863	+	0.522843i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−320212.	+	381614.i	−1.23595	+	1.47295i	−0.407196	+	0.913341i	$0.633494\pi$
$509$	−320212.	+	381614.i	−1.23595	+	1.47295i	−0.828756	+	0.559610i	$0.810951\pi$
$510$		0			0
$511$	−204817.	−	74547.5i	−0.784378	−	0.285490i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−207952.	+	571344.i	−0.784059	+	2.15418i
$516$		0			0
$517$	−558730.	−	468830.i	−2.09036	−	1.75402i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−190745.	+	110126.i	−0.702711	+	0.405711i	−0.808356	−	0.588693i	$-0.799643\pi$
$521$	−190745.	+	110126.i	−0.702711	+	0.405711i	0.105645	+	0.994404i	$0.466309\pi$
$522$		0			0
$523$	−112297.	+	194503.i	−0.410547	+	0.711088i	−0.994950	−	0.100376i	$-0.967996\pi$
$523$	−112297.	+	194503.i	−0.410547	+	0.711088i	0.584403	+	0.811464i	$0.301329\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−46007.6	−	126405.i	−0.165656	−	0.455137i
$528$		0			0
$529$	42674.2	+	242017.i	0.152494	+	0.864839i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−41508.3	−	49467.6i	−0.146110	−	0.174127i
$534$		0			0
$535$	−73502.7	+	416855.i	−0.256801	+	1.45639i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			127382.i			0.438461i
$540$		0			0
$541$	−104206.			−0.356040			−0.178020	−	0.984027i	$-0.556969\pi$
$541$	−104206.			−0.356040			−0.178020	+	0.984027i	$0.556969\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−710415.	−	125265.i	−2.39177	−	0.421733i
$546$		0			0
$547$	61804.1	−	51859.8i	0.206558	−	0.173323i	−0.533640	−	0.845712i	$-0.679176\pi$
$547$	61804.1	−	51859.8i	0.206558	−	0.173323i	0.740198	+	0.672389i	$0.234732\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−100715.	+	17758.8i	−0.331735	+	0.0584938i
$552$		0			0
$553$	254992.	−	92809.5i	0.833828	−	0.303489i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−301048.	−	173810.i	−0.970342	−	0.560227i	−0.0710018	−	0.997476i	$-0.522620\pi$
$557$	−301048.	−	173810.i	−0.970342	−	0.560227i	−0.899341	+	0.437249i	$0.855953\pi$
$558$		0			0
$559$	−70532.1	−	122165.i	−0.225716	−	0.390952i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−66806.1	+	79616.4i	−0.210765	+	0.251180i	−0.861062	−	0.508500i	$-0.830200\pi$
$563$	−66806.1	+	79616.4i	−0.210765	+	0.251180i	0.650297	+	0.759680i	$0.274645\pi$
$564$		0			0
$565$	361338.	+	131516.i	1.13192	+	0.411986i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13515.7	+	37134.0i	−0.0417458	+	0.114696i	−0.958814	−	0.284035i	$-0.908327\pi$
$569$	−13515.7	+	37134.0i	−0.0417458	+	0.114696i	0.917068	+	0.398730i	$0.130549\pi$
$570$		0			0
$571$	78313.5	+	65712.8i	0.240195	+	0.201548i	0.754937	−	0.655798i	$-0.227668\pi$
$571$	78313.5	+	65712.8i	0.240195	+	0.201548i	−0.514741	+	0.857346i	$0.672112\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−101783.	+	58764.3i	−0.307849	+	0.177737i
$576$		0			0
$577$	−289987.	+	502273.i	−0.871019	+	1.50865i	−0.0100745	+	0.999949i	$0.503207\pi$
$577$	−289987.	+	502273.i	−0.871019	+	1.50865i	−0.860944	+	0.508699i	$0.830126\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−106092.	−	291487.i	−0.314291	−	0.863508i
$582$		0			0
$583$	−176035.	−	998345.i	−0.517920	−	2.93727i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	397931.	+	474236.i	1.15487	+	1.37632i	0.913978	+	0.405763i	$0.132994\pi$
$587$	397931.	+	474236.i	1.15487	+	1.37632i	0.240888	+	0.970553i	$0.422561\pi$
$588$		0			0
$589$	−71015.1	+	402747.i	−0.204701	+	1.16092i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	292678.i		−	0.832302i	−0.909296	−	0.416151i	$-0.863379\pi$
$593$		−	292678.i		−	0.832302i	0.909296	−	0.416151i	$-0.136621\pi$
$594$		0			0
$595$	214190.			0.605014
$596$		0			0
$597$		0			0
$598$		0			0
$599$	121157.	+	21363.2i	0.337671	+	0.0595406i	0.339913	−	0.940457i	$-0.389602\pi$
$599$	121157.	+	21363.2i	0.337671	+	0.0595406i	−0.00224158	+	0.999997i	$0.500714\pi$
$600$		0			0
$601$	239835.	−	201245.i	0.663992	−	0.557156i	−0.247288	−	0.968942i	$-0.579539\pi$
$601$	239835.	−	201245.i	0.663992	−	0.557156i	0.911280	+	0.411786i	$0.135095\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.32423e6	−	233497.i	3.61786	−	0.637927i
$606$		0			0
$607$	11528.8	−	4196.12i	0.0312900	−	0.0113886i	−0.326328	−	0.945257i	$-0.605811\pi$
$607$	11528.8	−	4196.12i	0.0312900	−	0.0113886i	0.357618	+	0.933868i	$0.383589\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	123864.	+	71512.7i	0.331789	+	0.191558i
$612$		0			0
$613$	−46812.8	−	81082.2i	−0.124579	−	0.215777i	0.796989	−	0.603993i	$-0.206425\pi$
$613$	−46812.8	−	81082.2i	−0.124579	−	0.215777i	−0.921568	+	0.388217i	$0.873091\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−221939.	+	264497.i	−0.582994	+	0.694785i	−0.974243	−	0.225500i	$-0.927599\pi$
$617$	−221939.	+	264497.i	−0.582994	+	0.694785i	0.391249	+	0.920285i	$0.372043\pi$
$618$		0			0
$619$	114983.	+	41850.2i	0.300090	+	0.109224i	0.487677	−	0.873024i	$-0.337844\pi$
$619$	114983.	+	41850.2i	0.300090	+	0.109224i	−0.187587	+	0.982248i	$0.560067\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−127734.	+	350945.i	−0.329101	+	0.904197i
$624$		0			0
$625$	−293606.	−	246365.i	−0.751631	−	0.630694i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−48508.7	+	28006.5i	−0.122608	+	0.0707876i
$630$		0			0
$631$	−92854.5	+	160829.i	−0.233208	+	0.403929i	−0.958750	−	0.284249i	$-0.908256\pi$
$631$	−92854.5	+	160829.i	−0.233208	+	0.403929i	0.725542	+	0.688178i	$0.241589\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	42909.7	+	117893.i	0.106416	+	0.292376i
$636$		0			0
$637$	−4337.55	−	24599.4i	−0.0106897	−	0.0606243i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−253987.	−	302690.i	−0.618153	−	0.736686i	0.362598	−	0.931945i	$-0.381890\pi$
$641$	−253987.	−	302690.i	−0.618153	−	0.736686i	−0.980752	+	0.195259i	$0.937445\pi$
$642$		0			0
$643$	−11892.1	+	67443.5i	−0.0287632	+	0.163124i	−0.995806	−	0.0914895i	$-0.970837\pi$
$643$	−11892.1	+	67443.5i	−0.0287632	+	0.163124i	0.967043	+	0.254614i	$0.0819483\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			83238.3i			0.198845i	0.995045	+	0.0994225i	$0.0316995\pi$
$647$			83238.3i			0.198845i	−0.995045	+	0.0994225i	$0.968300\pi$
$648$		0			0
$649$	393829.			0.935015
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−245223.	−	43239.4i	−0.575088	−	0.101404i	−0.121463	−	0.992596i	$-0.538759\pi$
$653$	−245223.	−	43239.4i	−0.575088	−	0.101404i	−0.453625	+	0.891192i	$0.649870\pi$
$654$		0			0
$655$	389781.	−	327065.i	0.908526	−	0.762344i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	611537.	−	107831.i	1.40816	−	0.248297i	0.582668	−	0.812710i	$-0.302009\pi$
$659$	611537.	−	107831.i	1.40816	−	0.248297i	0.825492	+	0.564413i	$0.190898\pi$
$660$		0			0
$661$	328192.	−	119452.i	0.751147	−	0.273395i	0.0620585	−	0.998073i	$-0.480233\pi$
$661$	328192.	−	119452.i	0.751147	−	0.273395i	0.689088	+	0.724677i	$0.258011\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−563940.	−	325591.i	−1.27523	−	0.736256i
$666$		0			0
$667$	22119.7	+	38312.5i	0.0497197	+	0.0861170i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	60063.5	−	71580.9i	0.133403	−	0.158984i
$672$		0			0
$673$	−835372.	−	304051.i	−1.84438	−	0.671299i	−0.987895	−	0.155122i	$-0.950423\pi$
$673$	−835372.	−	304051.i	−1.84438	−	0.671299i	−0.856482	−	0.516176i	$-0.827355\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−279851.	+	768884.i	−0.610590	+	1.67758i	0.118324	+	0.992975i	$0.462248\pi$
$677$	−279851.	+	768884.i	−0.610590	+	1.67758i	−0.728913	+	0.684606i	$0.759974\pi$
$678$		0			0
$679$	−139599.	−	117137.i	−0.302790	−	0.254071i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−319788.	+	184630.i	−0.685522	+	0.395786i	−0.801932	−	0.597415i	$-0.796195\pi$
$683$	−319788.	+	184630.i	−0.685522	+	0.395786i	0.116411	+	0.993201i	$0.462861\pi$
$684$		0			0
$685$	−333955.	+	578428.i	−0.711717	+	1.23273i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	67990.1	+	186801.i	0.143221	+	0.393497i
$690$		0			0
$691$	−8234.89	−	46702.4i	−0.0172465	−	0.0978100i	0.974969	−	0.222339i	$-0.0713691\pi$
$691$	−8234.89	−	46702.4i	−0.0172465	−	0.0978100i	−0.992216	+	0.124529i	$0.960258\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−597865.	−	712508.i	−1.23775	−	1.47510i
$696$		0			0
$697$	−35038.0	+	198710.i	−0.0721230	+	0.409030i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	528043.i		−	1.07457i	−0.843402	−	0.537283i	$-0.819451\pi$
$701$		−	528043.i		−	1.07457i	0.843402	−	0.537283i	$-0.180549\pi$
$702$		0			0
$703$	170291.			0.344573
$704$		0			0
$705$		0			0
$706$		0			0
$707$	183370.	+	32333.0i	0.366850	+	0.0646856i
$708$		0			0
$709$	−104633.	+	87797.7i	−0.208150	+	0.174659i	−0.740903	−	0.671612i	$-0.765602\pi$
$709$	−104633.	+	87797.7i	−0.208150	+	0.174659i	0.532753	+	0.846271i	$0.321158\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	174221.	−	30719.8i	0.342705	−	0.0604281i
$714$		0			0
$715$	−343602.	+	125061.i	−0.672114	+	0.244630i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	33941.4	+	19596.1i	0.0656557	+	0.0379063i	0.532468	−	0.846450i	$-0.321264\pi$
$719$	33941.4	+	19596.1i	0.0656557	+	0.0379063i	−0.466813	+	0.884356i	$0.654598\pi$
$720$		0			0
$721$	367650.	+	636788.i	0.707235	+	1.22497i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	98037.9	−	116837.i	0.186517	−	0.222282i
$726$		0			0
$727$	−514259.	−	187175.i	−0.973001	−	0.354144i	−0.193886	−	0.981024i	$-0.562109\pi$
$727$	−514259.	−	187175.i	−0.973001	−	0.354144i	−0.779115	+	0.626880i	$0.784331\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−150754.	+	414194.i	−0.282121	+	0.775121i
$732$		0			0
$733$	106906.	+	89705.0i	0.198974	+	0.166959i	0.736831	−	0.676077i	$-0.236322\pi$
$733$	106906.	+	89705.0i	0.198974	+	0.166959i	−0.537857	+	0.843036i	$0.680766\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.60263e6	−	925281.i	2.95052	−	1.70349i
$738$		0			0
$739$	420152.	−	727724.i	0.769338	−	1.33253i	−0.168584	−	0.985687i	$-0.553919\pi$
$739$	420152.	−	727724.i	0.769338	−	1.33253i	0.937922	−	0.346846i	$-0.112747\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−161078.	−	442557.i	−0.291781	−	0.801662i	−0.995806	−	0.0914872i	$-0.970838\pi$
$743$	−161078.	−	442557.i	−0.291781	−	0.801662i	0.704025	−	0.710175i	$-0.251384\pi$
$744$		0			0
$745$	98811.3	+	560387.i	0.178030	+	1.00966i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	329044.	+	392139.i	0.586530	+	0.698999i
$750$		0			0
$751$	−2767.89	+	15697.5i	−0.00490760	+	0.0278324i	−0.987163	−	0.159715i	$-0.948943\pi$
$751$	−2767.89	+	15697.5i	−0.00490760	+	0.0278324i	0.982256	+	0.187547i	$0.0600537\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			979807.i			1.71888i
$756$		0			0
$757$	100752.			0.175818			0.0879088	−	0.996129i	$-0.471982\pi$
$757$	100752.			0.175818			0.0879088	+	0.996129i	$0.471982\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.05323e6	+	185712.i	1.81866	+	0.320680i	0.976002	−	0.217763i	$-0.0698758\pi$
$761$	1.05323e6	+	185712.i	1.81866	+	0.320680i	0.842663	+	0.538442i	$0.180987\pi$
$762$		0			0
$763$	−668294.	+	560765.i	−1.14794	+	0.963234i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−76054.4	+	13410.4i	−0.129281	+	0.0227957i
$768$		0			0
$769$	783204.	−	285063.i	1.32441	−	0.482046i	0.419541	−	0.907736i	$-0.362191\pi$
$769$	783204.	−	285063.i	1.32441	−	0.482046i	0.904869	+	0.425691i	$0.139969\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−54654.0	−	31554.5i	−0.0914667	−	0.0528083i	0.453569	−	0.891221i	$-0.350151\pi$
$773$	−54654.0	−	31554.5i	−0.0914667	−	0.0528083i	−0.545036	+	0.838413i	$0.683484\pi$
$774$		0			0
$775$	−304954.	−	528196.i	−0.507728	−	0.879410i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	394312.	−	469922.i	0.649777	−	0.774374i
$780$		0			0
$781$	1.73699e6	+	632211.i	2.84770	+	1.03648i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	299973.	−	824169.i	0.486792	−	1.33745i
$786$		0			0
$787$	−121961.	−	102338.i	−0.196912	−	0.165229i	0.539000	−	0.842306i	$-0.318802\pi$
$787$	−121961.	−	102338.i	−0.196912	−	0.165229i	−0.735913	+	0.677077i	$0.763247\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	402727.	−	232515.i	0.643662	−	0.371619i
$792$		0			0
$793$	−9161.75	+	15868.6i	−0.0145691	+	0.0252344i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	150183.	+	412625.i	0.236431	+	0.649590i	0.999993	+	0.00385444i	$0.00122691\pi$
$797$	150183.	+	412625.i	0.236431	+	0.649590i	−0.763561	+	0.645736i	$0.776551\pi$
$798$		0			0
$799$	−77604.2	−	440115.i	−0.121560	−	0.689403i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−747344.	−	890650.i	−1.15902	−	1.38126i
$804$		0			0
$805$	−48914.8	+	277409.i	−0.0754828	+	0.428084i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	166385.i		−	0.254224i	−0.991888	−	0.127112i	$-0.959429\pi$
$809$		−	166385.i		−	0.254224i	0.991888	−	0.127112i	$-0.0405707\pi$
$810$		0			0
$811$	1.00277e6			1.52461			0.762303	−	0.647220i	$-0.224069\pi$
$811$	1.00277e6			1.52461			0.762303	+	0.647220i	$0.224069\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.52942e6	−	269679.i	−2.30257	−	0.406005i
$816$		0			0
$817$	1.02654e6	−	861368.i	1.53791	−	1.29046i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	531672.	−	93748.0i	0.788782	−	0.139084i	0.235276	−	0.971929i	$-0.424401\pi$
$821$	531672.	−	93748.0i	0.788782	−	0.139084i	0.553506	+	0.832845i	$0.313290\pi$
$822$		0			0
$823$	−705215.	+	256677.i	−1.04117	+	0.378955i	−0.805324	−	0.592836i	$-0.798008\pi$
$823$	−705215.	+	256677.i	−1.04117	+	0.378955i	−0.235846	+	0.971790i	$0.575786\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	617523.	+	356527.i	0.902905	+	0.521293i	0.878142	−	0.478401i	$-0.158783\pi$
$827$	617523.	+	356527.i	0.902905	+	0.521293i	0.0247636	+	0.999693i	$0.492117\pi$
$828$		0			0
$829$	305079.	+	528412.i	0.443919	+	0.768890i	0.997976	−	0.0635888i	$-0.0202546\pi$
$829$	305079.	+	528412.i	0.443919	+	0.768890i	−0.554058	+	0.832478i	$0.686921\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−50169.9	+	59790.1i	−0.0723025	+	0.0861667i
$834$		0			0
$835$	225813.	+	82189.1i	0.323874	+	0.117880i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	150107.	−	412415.i	0.213244	−	0.585883i	−0.786243	−	0.617918i	$-0.787977\pi$
$839$	150107.	−	412415.i	0.213244	−	0.585883i	0.999487	+	0.0320348i	$0.0101987\pi$
$840$		0			0
$841$	497830.	+	417729.i	0.703864	+	0.590612i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−816432.	+	471367.i	−1.14342	+	0.660155i
$846$		0			0
$847$	813082.	−	1.40830e6i	1.13336	−	1.96304i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−25194.8	−	69222.1i	−0.0347898	−	0.0955841i
$852$		0			0
$853$	89243.6	+	506126.i	0.122653	+	0.695601i	0.982674	+	0.185342i	$0.0593393\pi$
$853$	89243.6	+	506126.i	0.122653	+	0.695601i	−0.860021	+	0.510259i	$0.829550\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−237663.	−	283235.i	−0.323593	−	0.385643i	0.579583	−	0.814913i	$-0.303215\pi$
$857$	−237663.	−	283235.i	−0.323593	−	0.385643i	−0.903176	+	0.429270i	$0.858771\pi$
$858$		0			0
$859$	6305.57	−	35760.6i	0.00854551	−	0.0484640i	−0.980237	−	0.197828i	$-0.936611\pi$
$859$	6305.57	−	35760.6i	0.00854551	−	0.0484640i	0.988782	+	0.149364i	$0.0477225\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	525679.i		−	0.705828i	−0.935656	−	0.352914i	$-0.885191\pi$
$863$		−	525679.i		−	0.705828i	0.935656	−	0.352914i	$-0.114809\pi$
$864$		0			0
$865$	−1.48298e6			−1.98200
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.42549e6	+	251352.i	1.88766	+	0.332846i
$870$		0			0
$871$	−277986.	+	233258.i	−0.366426	+	0.307468i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	17347.7	−	3058.87i	0.0226583	−	0.00399526i
$876$		0			0
$877$	186764.	−	67976.7i	0.242826	−	0.0883814i	−0.217740	−	0.976007i	$-0.569869\pi$
$877$	186764.	−	67976.7i	0.242826	−	0.0883814i	0.460566	+	0.887625i	$0.347646\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−163652.	−	94484.7i	−0.210848	−	0.121733i	0.390857	−	0.920451i	$-0.372179\pi$
$881$	−163652.	−	94484.7i	−0.210848	−	0.121733i	−0.601706	+	0.798718i	$0.705512\pi$
$882$		0			0
$883$	−253391.	−	438887.i	−0.324990	−	0.562900i	0.656520	−	0.754309i	$-0.272028\pi$
$883$	−253391.	−	438887.i	−0.324990	−	0.562900i	−0.981510	+	0.191409i	$0.938694\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−329254.	+	392389.i	−0.418488	+	0.498735i	−0.933565	−	0.358409i	$-0.883319\pi$
$887$	−329254.	+	392389.i	−0.418488	+	0.498735i	0.515076	+	0.857144i	$0.327764\pi$
$888$		0			0
$889$	142575.	+	51893.0i	0.180401	+	0.0656606i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−464697.	+	1.27674e6i	−0.582729	+	1.60104i
$894$		0			0
$895$	618952.	+	519362.i	0.772700	+	0.648372i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−198821.	+	114789.i	−0.246004	+	0.142030i
$900$		0			0
$901$	310575.	−	537931.i	0.382575	−	0.662639i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−288311.	−	792128.i	−0.352017	−	0.967159i
$906$		0			0
$907$	274400.	+	1.55620e6i	0.333557	+	1.89170i	0.441038	+	0.897489i	$0.354610\pi$
$907$	274400.	+	1.55620e6i	0.333557	+	1.89170i	−0.107481	+	0.994207i	$0.534278\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−696406.	−	829945.i	−0.839124	−	1.00003i	−0.999915	−	0.0130404i	$-0.995849\pi$
$911$	−696406.	−	829945.i	−0.839124	−	1.00003i	0.160791	−	0.986988i	$-0.448595\pi$
$912$		0			0
$913$	287326.	−	1.62951e6i	0.344694	−	1.95485i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	615346.i		−	0.731780i
$918$		0			0
$919$	−775475.			−0.918199			−0.459099	−	0.888385i	$-0.651828\pi$
$919$	−775475.			−0.918199			−0.459099	+	0.888385i	$0.651828\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−356966.	−	62942.7i	−0.419009	−	0.0738826i
$924$		0			0
$925$	−194549.	+	163246.i	−0.227377	+	0.190792i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	10981.2	−	1936.28i	0.0127238	−	0.00224356i	−0.167283	−	0.985909i	$-0.553499\pi$
$929$	10981.2	−	1936.28i	0.0127238	−	0.00224356i	0.180006	+	0.983665i	$0.442388\pi$
$930$		0			0
$931$	222979.	−	81157.8i	0.257256	−	0.0936334i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	989468.	+	571270.i	1.13182	+	0.653458i
$936$		0			0
$937$	544968.	+	943913.i	0.620715	+	1.07511i	0.989353	+	0.145536i	$0.0464908\pi$
$937$	544968.	+	943913.i	0.620715	+	1.07511i	−0.368638	+	0.929573i	$0.620176\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	71038.7	−	84660.7i	0.0802261	−	0.0956098i	−0.724438	−	0.689340i	$-0.757901\pi$
$941$	71038.7	−	84660.7i	0.0802261	−	0.0956098i	0.804664	+	0.593730i	$0.202345\pi$
$942$		0			0
$943$	−249359.	−	90759.3i	−0.280415	−	0.102063i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−248.954	+	683.996i	−0.000277600	+	0.000762700i	−0.939831	−	0.341639i	$-0.889018\pi$
$947$	−248.954	+	683.996i	−0.000277600	+	0.000762700i	0.939554	+	0.342401i	$0.111240\pi$
$948$		0			0
$949$	174651.	+	146550.i	0.193928	+	0.162725i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−157177.	+	90746.3i	−0.173063	+	0.0999179i	−0.584029	−	0.811733i	$-0.698525\pi$
$953$	−157177.	+	90746.3i	−0.173063	+	0.0999179i	0.410966	+	0.911650i	$0.365191\pi$
$954$		0			0
$955$	−446871.	+	774004.i	−0.489977	+	0.848665i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	276263.	+	759027.i	0.300390	+	0.825315i
$960$		0			0
$961$	−949.163	−	5382.97i	−0.00102777	−	0.00582875i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−74483.9	−	88766.4i	−0.0799848	−	0.0953222i
$966$		0			0
$967$	170809.	−	968706.i	0.182666	−	1.03595i	−0.746251	−	0.665664i	$-0.768148\pi$
$967$	170809.	−	968706.i	0.182666	−	1.03595i	0.928918	−	0.370287i	$-0.120741\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.12658e6i			1.19488i	0.801915	+	0.597438i	$0.203814\pi$
$971$			1.12658e6i			1.19488i	−0.801915	+	0.597438i	$0.796186\pi$
$972$		0			0
$973$	−1.12483e6			−1.18813
$974$		0			0
$975$		0			0
$976$		0			0
$977$	106008.	+	18692.0i	0.111058	+	0.0195825i	0.228901	−	0.973450i	$-0.426487\pi$
$977$	106008.	+	18692.0i	0.111058	+	0.0195825i	−0.117843	+	0.993032i	$0.537598\pi$
$978$		0			0
$979$	−1.52609e6	+	1.28054e6i	−1.59226	+	1.33606i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.15740e6	+	204081.i	−1.19778	+	0.211200i	−0.736737	−	0.676179i	$-0.763634\pi$
$983$	−1.15740e6	+	204081.i	−1.19778	+	0.211200i	−0.461040	+	0.887380i	$0.652523\pi$
$984$		0			0
$985$	−1.06437e6	+	387401.i	−1.09704	+	0.399289i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−502018.	−	289840.i	−0.513247	−	0.296323i
$990$		0			0
$991$	−139241.	−	241172.i	−0.141781	−	0.245573i	0.786386	−	0.617735i	$-0.211950\pi$
$991$	−139241.	−	241172.i	−0.141781	−	0.245573i	−0.928168	+	0.372163i	$0.878616\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.25800e6	−	1.49923e6i	1.27067	−	1.51433i
$996$		0			0
$997$	−470590.	−	171281.i	−0.473426	−	0.172313i	0.0942775	−	0.995546i	$-0.469946\pi$
$997$	−470590.	−	171281.i	−0.473426	−	0.172313i	−0.567704	+	0.823233i	$0.692168\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.k.a.17.11		72
3.2	odd	2		108.5.k.a.77.10	✓	72
27.7	even	9		108.5.k.a.101.10	yes	72
27.20	odd	18	inner	324.5.k.a.305.11		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.77.10	✓	72	3.2	odd	2
108.5.k.a.101.10	yes	72	27.7	even	9
324.5.k.a.17.11		72	1.1	even	1	trivial
324.5.k.a.305.11		72	27.20	odd	18	inner

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.k (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(34.9786 + 6.16768i) q^{5} +(32.9047 - 27.6104i) q^{7} +O(q^{10})\)	\(q+(34.9786 + 6.16768i) q^{5} +(32.9047 - 27.6104i) q^{7} +(225.646 - 39.7875i) q^{11} +(-42.2209 + 15.3671i) q^{13} +(121.583 + 70.1961i) q^{17} +(-213.411 - 369.638i) q^{19} +(-118.681 + 141.438i) q^{23} +(598.158 + 217.712i) q^{25} +(81.9498 - 225.155i) q^{29} +(-733.987 - 615.888i) q^{31} +(1321.26 - 762.827i) q^{35} +(-199.487 + 345.522i) q^{37} +(491.561 + 1350.55i) q^{41} +(545.187 + 3091.91i) q^{43} +(-2046.16 - 2438.52i) q^{47} +(-96.5390 + 547.500i) q^{49} -4424.38i q^{53} +8138.19 q^{55} +(1692.71 + 298.471i) q^{59} +(312.407 - 262.141i) q^{61} +(-1571.61 + 277.117i) q^{65} +(7589.50 - 2762.35i) q^{67} +(6986.58 + 4033.70i) q^{71} +(-2537.15 - 4394.48i) q^{73} +(6326.28 - 7539.37i) q^{77} +(5936.39 + 2160.67i) q^{79} +(2469.90 - 6786.00i) q^{83} +(3819.87 + 3205.25i) q^{85} +(-7529.73 + 4347.29i) q^{89} +(-964.975 + 1671.39i) q^{91} +(-5185.01 - 14245.7i) q^{95} +(-736.704 - 4178.06i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{5}+O(q^{10}) \) 72 * q - 9 * q^5	\( 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) \) 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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