LMFDB - Embedded newform 108.6.e.a.73.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(37,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("108.37");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.e (of order $3$, degree $2$, 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:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 175x^{8} + 8800x^{6} + 124623x^{4} + 498609x^{2} + 442368 $ x^10 + 175x^8 + 8800x^6 + 124623x^4 + 498609x^2 + 442368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{16} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			73.4
Root			$2.13639i$ of defining polynomial
Character	$\chi$	$=$	108.73
Dual form			108.6.e.a.37.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(14.0718 + 24.3731i) q^{5} +(75.7039 - 131.123i) q^{7} +O(q^{10})$	$q+(14.0718 + 24.3731i) q^{5} +(75.7039 - 131.123i) q^{7} +(-138.873 + 240.536i) q^{11} +(-291.929 - 505.636i) q^{13} +1612.01 q^{17} +1368.76 q^{19} +(428.014 + 741.342i) q^{23} +(1166.47 - 2020.38i) q^{25} +(4267.49 - 7391.50i) q^{29} +(-1469.19 - 2544.71i) q^{31} +4261.17 q^{35} +4036.80 q^{37} +(9449.81 + 16367.5i) q^{41} +(10158.6 - 17595.1i) q^{43} +(-147.890 + 256.152i) q^{47} +(-3058.67 - 5297.78i) q^{49} -3039.13 q^{53} -7816.81 q^{55} +(-8618.31 - 14927.4i) q^{59} +(-12826.2 + 22215.7i) q^{61} +(8215.95 - 14230.4i) q^{65} +(13140.1 + 22759.4i) q^{67} -76665.7 q^{71} +1496.33 q^{73} +(21026.5 + 36419.0i) q^{77} +(-49637.1 + 85974.0i) q^{79} +(25025.7 - 43345.7i) q^{83} +(22683.8 + 39289.6i) q^{85} -136635. q^{89} -88400.8 q^{91} +(19261.0 + 33361.0i) q^{95} +(33325.0 - 57720.5i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 21 q^{5} + 29 q^{7}+O(q^{10}) $ 10 * q + 21 * q^5 + 29 * q^7	$ 10 q + 21 q^{5} + 29 q^{7} - 177 q^{11} - 181 q^{13} - 2280 q^{17} - 832 q^{19} - 399 q^{23} - 4778 q^{25} + 6033 q^{29} + 2759 q^{31} - 37146 q^{35} - 15172 q^{37} + 18435 q^{41} + 1469 q^{43} + 25155 q^{47} - 4056 q^{49} - 116844 q^{53} + 14778 q^{55} + 90537 q^{59} + 1403 q^{61} + 148407 q^{65} + 13907 q^{67} - 229368 q^{71} + 15200 q^{73} + 211983 q^{77} + 29993 q^{79} + 228951 q^{83} - 49662 q^{85} - 598332 q^{89} + 124930 q^{91} + 394764 q^{95} + 40541 q^{97}+O(q^{100}) $ 10 * q + 21 * q^5 + 29 * q^7 - 177 * q^11 - 181 * q^13 - 2280 * q^17 - 832 * q^19 - 399 * q^23 - 4778 * q^25 + 6033 * q^29 + 2759 * q^31 - 37146 * q^35 - 15172 * q^37 + 18435 * q^41 + 1469 * q^43 + 25155 * q^47 - 4056 * q^49 - 116844 * q^53 + 14778 * q^55 + 90537 * q^59 + 1403 * q^61 + 148407 * q^65 + 13907 * q^67 - 229368 * q^71 + 15200 * q^73 + 211983 * q^77 + 29993 * q^79 + 228951 * q^83 - 49662 * q^85 - 598332 * q^89 + 124930 * q^91 + 394764 * q^95 + 40541 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	14.0718	+	24.3731i	0.251724	+	0.435999i	0.964001	−	0.265900i	$-0.0856690\pi$
$5$	14.0718	+	24.3731i	0.251724	+	0.435999i	−0.712276	+	0.701899i	$0.752336\pi$
$6$		0			0
$7$	75.7039	−	131.123i	0.583947	−	1.01143i	−0.411059	−	0.911609i	$-0.634841\pi$
$7$	75.7039	−	131.123i	0.583947	−	1.01143i	0.995006	−	0.0998170i	$-0.0318257\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−138.873	+	240.536i	−0.346049	+	0.599374i	−0.985544	−	0.169421i	$-0.945810\pi$
$11$	−138.873	+	240.536i	−0.346049	+	0.599374i	0.639495	+	0.768795i	$0.279144\pi$
$12$		0			0
$13$	−291.929	−	505.636i	−0.479092	−	0.829812i	0.520620	−	0.853788i	$-0.325701\pi$
$13$	−291.929	−	505.636i	−0.479092	−	0.829812i	−0.999713	+	0.0239762i	$0.992367\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1612.01			1.35283			0.676417	−	0.736519i	$-0.263532\pi$
$17$	1612.01			1.35283			0.676417	+	0.736519i	$0.263532\pi$
$18$		0			0
$19$	1368.76			0.869851			0.434925	−	0.900467i	$-0.356775\pi$
$19$	1368.76			0.869851			0.434925	+	0.900467i	$0.356775\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	428.014	+	741.342i	0.168709	+	0.292213i	0.937966	−	0.346727i	$-0.112707\pi$
$23$	428.014	+	741.342i	0.168709	+	0.292213i	−0.769257	+	0.638939i	$0.779373\pi$
$24$		0			0
$25$	1166.47	−	2020.38i	0.373270	−	0.646522i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4267.49	−	7391.50i	0.942274	−	1.63207i	0.181154	−	0.983455i	$-0.442017\pi$
$29$	4267.49	−	7391.50i	0.942274	−	1.63207i	0.761120	−	0.648611i	$-0.224650\pi$
$30$		0			0
$31$	−1469.19	−	2544.71i	−0.274583	−	0.475591i	0.695447	−	0.718577i	$-0.255206\pi$
$31$	−1469.19	−	2544.71i	−0.274583	−	0.475591i	−0.970030	+	0.242986i	$0.921873\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4261.17			0.587975
$36$		0			0
$37$	4036.80			0.484767			0.242383	−	0.970181i	$-0.422071\pi$
$37$	4036.80			0.484767			0.242383	+	0.970181i	$0.422071\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	9449.81	+	16367.5i	0.877937	+	1.52063i	0.853601	+	0.520928i	$0.174414\pi$
$41$	9449.81	+	16367.5i	0.877937	+	1.52063i	0.0243361	+	0.999704i	$0.492253\pi$
$42$		0			0
$43$	10158.6	−	17595.1i	0.837840	−	1.45118i	−0.0538576	−	0.998549i	$-0.517152\pi$
$43$	10158.6	−	17595.1i	0.837840	−	1.45118i	0.891697	−	0.452632i	$-0.149515\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−147.890	+	256.152i	−0.00976546	+	0.0169143i	−0.870867	−	0.491519i	$-0.836442\pi$
$47$	−147.890	+	256.152i	−0.00976546	+	0.0169143i	0.861101	+	0.508433i	$0.169775\pi$
$48$		0			0
$49$	−3058.67	−	5297.78i	−0.181988	−	0.315213i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3039.13			−0.148614			−0.0743069	−	0.997235i	$-0.523674\pi$
$53$	−3039.13			−0.148614			−0.0743069	+	0.997235i	$0.523674\pi$
$54$		0			0
$55$	−7816.81			−0.348436
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8618.31	−	14927.4i	−0.322324	−	0.558281i	0.658643	−	0.752455i	$-0.271131\pi$
$59$	−8618.31	−	14927.4i	−0.322324	−	0.558281i	−0.980967	+	0.194174i	$0.937797\pi$
$60$		0			0
$61$	−12826.2	+	22215.7i	−0.441342	+	0.764426i	−0.997789	−	0.0664565i	$-0.978831\pi$
$61$	−12826.2	+	22215.7i	−0.441342	+	0.764426i	0.556448	+	0.830883i	$0.312164\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8215.95	−	14230.4i	0.241198	−	0.417768i
$66$		0			0
$67$	13140.1	+	22759.4i	0.357613	+	0.619403i	0.987561	−	0.157233i	$-0.0502575\pi$
$67$	13140.1	+	22759.4i	0.357613	+	0.619403i	−0.629949	+	0.776637i	$0.716924\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−76665.7			−1.80491			−0.902454	−	0.430786i	$-0.858236\pi$
$71$	−76665.7			−1.80491			−0.902454	+	0.430786i	$0.858236\pi$
$72$		0			0
$73$	1496.33			0.0328640			0.0164320	−	0.999865i	$-0.494769\pi$
$73$	1496.33			0.0328640			0.0164320	+	0.999865i	$0.494769\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	21026.5	+	36419.0i	0.404148	+	0.700006i
$78$		0			0
$79$	−49637.1	+	85974.0i	−0.894826	+	1.54988i	−0.0608070	+	0.998150i	$0.519367\pi$
$79$	−49637.1	+	85974.0i	−0.894826	+	1.54988i	−0.834019	+	0.551735i	$0.813966\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	25025.7	−	43345.7i	0.398740	−	0.690639i	−0.594830	−	0.803851i	$-0.702781\pi$
$83$	25025.7	−	43345.7i	0.398740	−	0.690639i	0.993571	+	0.113213i	$0.0361141\pi$
$84$		0			0
$85$	22683.8	+	39289.6i	0.340541	+	0.589834i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−136635.			−1.82847			−0.914235	−	0.405185i	$-0.867207\pi$
$89$	−136635.			−1.82847			−0.914235	+	0.405185i	$0.867207\pi$
$90$		0			0
$91$	−88400.8			−1.11906
$92$		0			0
$93$		0			0
$94$		0			0
$95$	19261.0	+	33361.0i	0.218963	+	0.379254i
$96$		0			0
$97$	33325.0	−	57720.5i	0.359617	−	0.622875i	−0.628280	−	0.777987i	$-0.716241\pi$
$97$	33325.0	−	57720.5i	0.359617	−	0.622875i	0.987897	+	0.155113i	$0.0495740\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12338.7	+	21371.2i	−0.120355	+	0.208462i	−0.919908	−	0.392135i	$-0.871737\pi$
$101$	−12338.7	+	21371.2i	−0.120355	+	0.208462i	0.799552	+	0.600596i	$0.205070\pi$
$102$		0			0
$103$	−57883.5	−	100257.i	−0.537603	−	0.931155i	−0.999032	−	0.0439785i	$-0.985997\pi$
$103$	−57883.5	−	100257.i	−0.537603	−	0.931155i	0.461430	−	0.887177i	$-0.347337\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−84364.3			−0.712360			−0.356180	−	0.934417i	$-0.615921\pi$
$107$	−84364.3			−0.712360			−0.356180	+	0.934417i	$0.615921\pi$
$108$		0			0
$109$	198400.			1.59947			0.799735	−	0.600354i	$-0.204973\pi$
$109$	198400.			1.59947			0.799735	+	0.600354i	$0.204973\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	114453.	+	198238.i	0.843199	+	1.46046i	0.887176	+	0.461430i	$0.152664\pi$
$113$	114453.	+	198238.i	0.843199	+	1.46046i	−0.0439777	+	0.999033i	$0.514003\pi$
$114$		0			0
$115$	−12045.9	+	20864.1i	−0.0849364	+	0.147114i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	122035.	−	211371.i	0.789983	−	1.36829i
$120$		0			0
$121$	41953.8	+	72666.2i	0.260500	+	0.451200i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	153606.			0.879293
$126$		0			0
$127$	−246629.			−1.35686			−0.678430	−	0.734665i	$-0.737339\pi$
$127$	−246629.			−1.35686			−0.678430	+	0.734665i	$0.737339\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−84612.6	−	146553.i	−0.430781	−	0.746135i	0.566159	−	0.824296i	$-0.308429\pi$
$131$	−84612.6	−	146553.i	−0.430781	−	0.746135i	−0.996941	+	0.0781606i	$0.975095\pi$
$132$		0			0
$133$	103621.	−	179477.i	0.507947	−	0.879789i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	49985.3	−	86577.0i	0.227531	−	0.394095i	−0.729545	−	0.683933i	$-0.760268\pi$
$137$	49985.3	−	86577.0i	0.227531	−	0.394095i	0.957076	+	0.289838i	$0.0936013\pi$
$138$		0			0
$139$	18699.4	+	32388.2i	0.0820899	+	0.142184i	0.904147	−	0.427221i	$-0.140507\pi$
$139$	18699.4	+	32388.2i	0.0820899	+	0.142184i	−0.822058	+	0.569404i	$0.807174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	162165.			0.663158
$144$		0			0
$145$	240205.			0.948773
$146$		0			0
$147$		0			0
$148$		0			0
$149$	127267.	+	220433.i	0.469623	+	0.813411i	0.999397	−	0.0347281i	$-0.0110565\pi$
$149$	127267.	+	220433.i	0.469623	+	0.813411i	−0.529774	+	0.848139i	$0.677723\pi$
$150$		0			0
$151$	118363.	−	205010.i	0.422448	−	0.731701i	−0.573731	−	0.819044i	$-0.694504\pi$
$151$	118363.	−	205010.i	0.422448	−	0.731701i	0.996178	+	0.0873434i	$0.0278377\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	41348.3	−	71617.4i	0.138238	−	0.239436i
$156$		0			0
$157$	−127406.	−	220673.i	−0.412515	−	0.714497i	0.582649	−	0.812724i	$-0.302016\pi$
$157$	−127406.	−	220673.i	−0.412515	−	0.714497i	−0.995164	+	0.0982266i	$0.968683\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	129609.			0.394069
$162$		0			0
$163$	215050.			0.633973			0.316987	−	0.948430i	$-0.397329\pi$
$163$	215050.			0.633973			0.316987	+	0.948430i	$0.397329\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	96168.5	+	166569.i	0.266834	+	0.462171i	0.968043	−	0.250786i	$-0.0806890\pi$
$167$	96168.5	+	166569.i	0.266834	+	0.462171i	−0.701208	+	0.712956i	$0.747356\pi$
$168$		0			0
$169$	15201.2	−	26329.2i	0.0409411	−	0.0709121i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4150.57	+	7189.00i	−0.0105437	+	0.0182622i	−0.871249	−	0.490841i	$-0.836690\pi$
$173$	−4150.57	+	7189.00i	−0.0105437	+	0.0182622i	0.860705	+	0.509103i	$0.170023\pi$
$174$		0			0
$175$	−176612.	−	305902.i	−0.435939	−	0.755069i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−574496.			−1.34015			−0.670077	−	0.742292i	$-0.733739\pi$
$179$	−574496.			−1.34015			−0.670077	+	0.742292i	$0.733739\pi$
$180$		0			0
$181$	−224707.			−0.509823			−0.254912	−	0.966964i	$-0.582046\pi$
$181$	−224707.			−0.509823			−0.254912	+	0.966964i	$0.582046\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	56805.1	+	98389.3i	0.122028	+	0.211358i
$186$		0			0
$187$	−223865.	+	387745.i	−0.468147	+	0.810854i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−239041.	+	414031.i	−0.474121	+	0.821201i	−0.999561	−	0.0296294i	$-0.990567\pi$
$191$	−239041.	+	414031.i	−0.474121	+	0.821201i	0.525440	+	0.850830i	$0.323901\pi$
$192$		0			0
$193$	263025.	+	455572.i	0.508281	+	0.880368i	0.999954	+	0.00958824i	$0.00305208\pi$
$193$	263025.	+	455572.i	0.508281	+	0.880368i	−0.491673	+	0.870780i	$0.663615\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	222278.			0.408067			0.204034	−	0.978964i	$-0.434595\pi$
$197$	222278.			0.408067			0.204034	+	0.978964i	$0.434595\pi$
$198$		0			0
$199$	−109696.			−0.196363			−0.0981813	−	0.995169i	$-0.531303\pi$
$199$	−109696.			−0.196363			−0.0981813	+	0.995169i	$0.531303\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−646131.	−	1.11913e6i	−1.10048	−	1.90608i
$204$		0			0
$205$	−265952.	+	460642.i	−0.441996	+	0.765560i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−190085.	+	329237.i	−0.301011	+	0.521366i
$210$		0			0
$211$	309436.	+	535959.i	0.478482	+	0.828754i	0.999696	−	0.0246717i	$-0.00785404\pi$
$211$	309436.	+	535959.i	0.478482	+	0.828754i	−0.521214	+	0.853426i	$0.674521\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	571797.			0.843618
$216$		0			0
$217$	−444893.			−0.641367
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−470592.	−	815089.i	−0.648132	−	1.12260i
$222$		0			0
$223$	−231653.	+	401234.i	−0.311943	+	0.540301i	−0.978783	−	0.204900i	$-0.934313\pi$
$223$	−231653.	+	401234.i	−0.311943	+	0.540301i	0.666840	+	0.745201i	$0.267646\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−573326.	+	993029.i	−0.738477	+	1.27908i	0.214704	+	0.976679i	$0.431121\pi$
$227$	−573326.	+	993029.i	−0.738477	+	1.27908i	−0.953181	+	0.302400i	$0.902212\pi$
$228$		0			0
$229$	175527.	+	304022.i	0.221185	+	0.383104i	0.955168	−	0.296064i	$-0.0956741\pi$
$229$	175527.	+	304022.i	0.221185	+	0.383104i	−0.733983	+	0.679168i	$0.762341\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−814275.			−0.982610			−0.491305	−	0.870988i	$-0.663480\pi$
$233$	−814275.			−0.982610			−0.491305	+	0.870988i	$0.663480\pi$
$234$		0			0
$235$	−8324.30			−0.00983281
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−357452.	−	619124.i	−0.404783	−	0.701105i	0.589513	−	0.807759i	$-0.299320\pi$
$239$	−357452.	−	619124.i	−0.404783	−	0.701105i	−0.994296	+	0.106654i	$0.965986\pi$
$240$		0			0
$241$	−22648.5	+	39228.3i	−0.0251186	+	0.0435068i	−0.878311	−	0.478089i	$-0.841330\pi$
$241$	−22648.5	+	39228.3i	−0.0251186	+	0.0435068i	0.853193	+	0.521596i	$0.174663\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	86082.2	−	149099.i	0.0916216	−	0.158693i
$246$		0			0
$247$	−399582.	−	692097.i	−0.416739	−	0.721813i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.35110e6			1.35364			0.676822	−	0.736147i	$-0.263357\pi$
$251$	1.35110e6			1.35364			0.676822	+	0.736147i	$0.263357\pi$
$252$		0			0
$253$	−237759.			−0.233526
$254$		0			0
$255$		0			0
$256$		0			0
$257$	600097.	+	1.03940e6i	0.566747	+	0.981634i	0.996885	+	0.0788708i	$0.0251315\pi$
$257$	600097.	+	1.03940e6i	0.566747	+	0.981634i	−0.430138	+	0.902763i	$0.641535\pi$
$258$		0			0
$259$	305602.	−	529318.i	0.283078	−	0.490306i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−656802.	+	1.13761e6i	−0.585525	+	1.01416i	0.409285	+	0.912407i	$0.365778\pi$
$263$	−656802.	+	1.13761e6i	−0.585525	+	1.01416i	−0.994810	+	0.101752i	$0.967555\pi$
$264$		0			0
$265$	−42766.1	−	74073.0i	−0.0374097	−	0.0647956i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−321363.			−0.270779			−0.135389	−	0.990792i	$-0.543229\pi$
$269$	−321363.			−0.270779			−0.135389	+	0.990792i	$0.543229\pi$
$270$		0			0
$271$	384928.			0.318388			0.159194	−	0.987247i	$-0.449110\pi$
$271$	384928.			0.318388			0.159194	+	0.987247i	$0.449110\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	323983.	+	561155.i	0.258339	+	0.447457i
$276$		0			0
$277$	847964.	−	1.46872e6i	0.664015	−	1.15011i	−0.315536	−	0.948913i	$-0.602184\pi$
$277$	847964.	−	1.46872e6i	0.664015	−	1.15011i	0.979551	−	0.201194i	$-0.0644822\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	530669.	−	919146.i	0.400920	−	0.694415i	−0.592917	−	0.805264i	$-0.702024\pi$
$281$	530669.	−	919146.i	0.400920	−	0.694415i	0.993837	+	0.110849i	$0.0353570\pi$
$282$		0			0
$283$	192313.	+	333096.i	0.142739	+	0.247231i	0.928527	−	0.371265i	$-0.121076\pi$
$283$	192313.	+	333096.i	0.142739	+	0.247231i	−0.785788	+	0.618496i	$0.787742\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.86155e6			2.05067
$288$		0			0
$289$	1.17871e6			0.830158
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−667643.	−	1.15639e6i	−0.454334	−	0.786929i	0.544316	−	0.838880i	$-0.316789\pi$
$293$	−667643.	−	1.15639e6i	−0.454334	−	0.786929i	−0.998650	+	0.0519513i	$0.983456\pi$
$294$		0			0
$295$	242551.	−	420110.i	0.162273	−	0.281066i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	249900.	−	432839.i	0.161655	−	0.279994i
$300$		0			0
$301$	−1.53809e6	−	2.66404e6i	−0.978508	−	1.69483i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−721954.			−0.444386
$306$		0			0
$307$	636269.			0.385296			0.192648	−	0.981268i	$-0.438292\pi$
$307$	636269.			0.385296			0.192648	+	0.981268i	$0.438292\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−269890.	−	467464.i	−0.158229	−	0.274061i	0.776001	−	0.630732i	$-0.217245\pi$
$311$	−269890.	−	467464.i	−0.158229	−	0.274061i	−0.934230	+	0.356671i	$0.883912\pi$
$312$		0			0
$313$	−976605.	+	1.69153e6i	−0.563454	+	0.975931i	0.433738	+	0.901039i	$0.357194\pi$
$313$	−976605.	+	1.69153e6i	−0.563454	+	0.975931i	−0.997192	+	0.0748915i	$0.976139\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	781983.	−	1.35443e6i	0.437068	−	0.757024i	−0.560394	−	0.828226i	$-0.689350\pi$
$317$	781983.	−	1.35443e6i	0.437068	−	0.757024i	0.997462	+	0.0712021i	$0.0226835\pi$
$318$		0			0
$319$	1.18528e6	+	2.05297e6i	0.652146	+	1.12955i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.20646e6			1.17676
$324$		0			0
$325$	−1.36210e6			−0.715323
$326$		0			0
$327$		0			0
$328$		0			0
$329$	22391.6	+	38783.5i	0.0114050	+	0.0197541i
$330$		0			0
$331$	1.28000e6	−	2.21702e6i	0.642154	−	1.11224i	−0.342796	−	0.939410i	$-0.611374\pi$
$331$	1.28000e6	−	2.21702e6i	0.642154	−	1.11224i	0.984951	−	0.172834i	$-0.0552925\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−369811.	+	640532.i	−0.180040	+	0.311838i
$336$		0			0
$337$	299646.	+	519002.i	0.143725	+	0.248940i	0.928897	−	0.370339i	$-0.120758\pi$
$337$	299646.	+	519002.i	0.143725	+	0.248940i	−0.785171	+	0.619279i	$0.787425\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	816125.			0.380076
$342$		0			0
$343$	1.61850e6			0.742808
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.34386e6	+	2.32764e6i	0.599144	+	1.03775i	0.992948	+	0.118553i	$0.0378256\pi$
$347$	1.34386e6	+	2.32764e6i	0.599144	+	1.03775i	−0.393804	+	0.919195i	$0.628841\pi$
$348$		0			0
$349$	−335807.	+	581635.i	−0.147580	+	0.255615i	−0.930332	−	0.366717i	$-0.880482\pi$
$349$	−335807.	+	581635.i	−0.147580	+	0.255615i	0.782753	+	0.622333i	$0.213815\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−764881.	+	1.32481e6i	−0.326706	+	0.565871i	−0.981856	−	0.189627i	$-0.939272\pi$
$353$	−764881.	+	1.32481e6i	−0.326706	+	0.565871i	0.655150	+	0.755499i	$0.272605\pi$
$354$		0			0
$355$	−1.07883e6	−	1.86858e6i	−0.454339	−	0.786939i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.27210e6			−1.33996			−0.669978	−	0.742381i	$-0.733697\pi$
$359$	−3.27210e6			−1.33996			−0.669978	+	0.742381i	$0.733697\pi$
$360$		0			0
$361$	−602583.			−0.243360
$362$		0			0
$363$		0			0
$364$		0			0
$365$	21056.1	+	36470.2i	0.00827267	+	0.0143287i
$366$		0			0
$367$	−313571.	+	543121.i	−0.121526	+	0.210490i	−0.920370	−	0.391049i	$-0.872112\pi$
$367$	−313571.	+	543121.i	−0.121526	+	0.210490i	0.798843	+	0.601539i	$0.205446\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−230074.	+	398500.i	−0.0867826	+	0.150312i
$372$		0			0
$373$	−66186.8	−	114639.i	−0.0246320	−	0.0426638i	0.853447	−	0.521180i	$-0.174508\pi$
$373$	−66186.8	−	114639.i	−0.0246320	−	0.0426638i	−0.878079	+	0.478516i	$0.841175\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.98322e6			−1.80574
$378$		0			0
$379$	−163225.			−0.0583700			−0.0291850	−	0.999574i	$-0.509291\pi$
$379$	−163225.			−0.0583700			−0.0291850	+	0.999574i	$0.509291\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−398301.	−	689877.i	−0.138744	−	0.240312i	0.788277	−	0.615320i	$-0.210973\pi$
$383$	−398301.	−	689877.i	−0.138744	−	0.240312i	−0.927021	+	0.375008i	$0.877640\pi$
$384$		0			0
$385$	−591763.	+	1.02496e6i	−0.203468	+	0.352417i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.04745e6	−	3.54629e6i	0.686024	−	1.18823i	−0.287089	−	0.957904i	$-0.592688\pi$
$389$	2.04745e6	−	3.54629e6i	0.686024	−	1.18823i	0.973114	−	0.230325i	$-0.0739790\pi$
$390$		0			0
$391$	689961.	+	1.19505e6i	0.228235	+	0.395315i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.79394e6			−0.900998
$396$		0			0
$397$	−5.58867e6			−1.77964			−0.889820	−	0.456312i	$-0.849170\pi$
$397$	−5.58867e6			−1.77964			−0.889820	+	0.456312i	$0.849170\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.11676e6	+	1.93429e6i	0.346817	+	0.600705i	0.985682	−	0.168614i	$-0.0539291\pi$
$401$	1.11676e6	+	1.93429e6i	0.346817	+	0.600705i	−0.638865	+	0.769319i	$0.720596\pi$
$402$		0			0
$403$	−857798.	+	1.48575e6i	−0.263101	+	0.455704i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−560604.	+	970995.i	−0.167753	+	0.290557i
$408$		0			0
$409$	2.30256e6	+	3.98815e6i	0.680617	+	1.17886i	0.974793	+	0.223111i	$0.0716214\pi$
$409$	2.30256e6	+	3.98815e6i	0.680617	+	1.17886i	−0.294176	+	0.955751i	$0.595045\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.60976e6			−0.752880
$414$		0			0
$415$	1.40863e6			0.401491
$416$		0			0
$417$		0			0
$418$		0			0
$419$	571064.	+	989112.i	0.158909	+	0.275239i	0.934476	−	0.356027i	$-0.115869\pi$
$419$	571064.	+	989112.i	0.158909	+	0.275239i	−0.775566	+	0.631266i	$0.782536\pi$
$420$		0			0
$421$	−962383.	+	1.66690e6i	−0.264632	+	0.458356i	−0.967467	−	0.252996i	$-0.918584\pi$
$421$	−962383.	+	1.66690e6i	−0.264632	+	0.458356i	0.702835	+	0.711353i	$0.251917\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.88035e6	−	3.25687e6i	0.504972	−	0.874637i
$426$		0			0
$427$	1.94199e6	+	3.36363e6i	0.515440	+	0.892769i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.21303e6			−0.573843			−0.286922	−	0.957954i	$-0.592632\pi$
$431$	−2.21303e6			−0.573843			−0.286922	+	0.957954i	$0.592632\pi$
$432$		0			0
$433$	3.00235e6			0.769558			0.384779	−	0.923009i	$-0.374278\pi$
$433$	3.00235e6			0.769558			0.384779	+	0.923009i	$0.374278\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	585851.	+	1.01472e6i	0.146752	+	0.254181i
$438$		0			0
$439$	−960857.	+	1.66425e6i	−0.237956	+	0.412153i	−0.960128	−	0.279562i	$-0.909811\pi$
$439$	−960857.	+	1.66425e6i	−0.237956	+	0.412153i	0.722171	+	0.691714i	$0.243144\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−924258.	+	1.60086e6i	−0.223761	+	0.387565i	−0.955947	−	0.293540i	$-0.905167\pi$
$443$	−924258.	+	1.60086e6i	−0.223761	+	0.387565i	0.732186	+	0.681105i	$0.238500\pi$
$444$		0			0
$445$	−1.92271e6	−	3.33022e6i	−0.460270	−	0.797211i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3.45640e6			−0.809110			−0.404555	−	0.914514i	$-0.632574\pi$
$449$	−3.45640e6			−0.809110			−0.404555	+	0.914514i	$0.632574\pi$
$450$		0			0
$451$	−5.24931e6			−1.21524
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.24396e6	−	2.15460e6i	−0.281694	−	0.487908i
$456$		0			0
$457$	−1.79319e6	+	3.10589e6i	−0.401638	+	0.695658i	−0.993924	−	0.110070i	$-0.964892\pi$
$457$	−1.79319e6	+	3.10589e6i	−0.401638	+	0.695658i	0.592286	+	0.805728i	$0.298226\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.21775e6	−	7.30536e6i	0.924333	−	1.60099i	0.131703	−	0.991289i	$-0.457955\pi$
$461$	4.21775e6	−	7.30536e6i	0.924333	−	1.60099i	0.792630	−	0.609703i	$-0.208711\pi$
$462$		0			0
$463$	−2.47237e6	−	4.28227e6i	−0.535995	−	0.928370i	−0.999114	−	0.0420745i	$-0.986603\pi$
$463$	−2.47237e6	−	4.28227e6i	−0.535995	−	0.928370i	0.463120	−	0.886296i	$-0.346730\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.12589e6			0.875438			0.437719	−	0.899112i	$-0.355787\pi$
$467$	4.12589e6			0.875438			0.437719	+	0.899112i	$0.355787\pi$
$468$		0			0
$469$	3.97904e6			0.835307
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.82151e6	+	4.88700e6i	0.579867	+	1.00436i
$474$		0			0
$475$	1.59662e6	−	2.76543e6i	0.324689	−	0.562378i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−561301.	+	972203.i	−0.111778	+	0.193606i	−0.916487	−	0.400064i	$-0.868988\pi$
$479$	−561301.	+	972203.i	−0.111778	+	0.193606i	0.804709	+	0.593669i	$0.202321\pi$
$480$		0			0
$481$	−1.17846e6	−	2.04115e6i	−0.232248	−	0.402265i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.87577e6			0.362097
$486$		0			0
$487$	8.11380e6			1.55025			0.775126	−	0.631807i	$-0.217687\pi$
$487$	8.11380e6			1.55025			0.775126	+	0.631807i	$0.217687\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.66481e6	−	2.88353e6i	−0.311645	−	0.539786i	0.667073	−	0.744992i	$-0.267547\pi$
$491$	−1.66481e6	−	2.88353e6i	−0.311645	−	0.539786i	−0.978719	+	0.205207i	$0.934213\pi$
$492$		0			0
$493$	6.87921e6	−	1.19151e7i	1.27474	−	2.20791i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.80389e6	+	1.00526e7i	−1.05397	+	1.82553i
$498$		0			0
$499$	−940630.	−	1.62922e6i	−0.169109	−	0.292906i	0.768998	−	0.639252i	$-0.220756\pi$
$499$	−940630.	−	1.62922e6i	−0.169109	−	0.292906i	−0.938107	+	0.346346i	$0.887422\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.96749e6			1.40411			0.702056	−	0.712122i	$-0.252266\pi$
$503$	7.96749e6			1.40411			0.702056	+	0.712122i	$0.252266\pi$
$504$		0			0
$505$	−694511.			−0.121186
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.40561e6	+	9.36279e6i	0.924805	+	1.60181i	0.791874	+	0.610684i	$0.209106\pi$
$509$	5.40561e6	+	9.36279e6i	0.924805	+	1.60181i	0.132931	+	0.991125i	$0.457561\pi$
$510$		0			0
$511$	113278.	−	196204.i	0.0191908	−	0.0332395i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.62905e6	−	2.82160e6i	0.270655	−	0.468789i
$516$		0			0
$517$	−41075.9	−	71145.5i	−0.00675865	−	0.0117063i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.19670e6			0.193148			0.0965740	−	0.995326i	$-0.469212\pi$
$521$	1.19670e6			0.193148			0.0965740	+	0.995326i	$0.469212\pi$
$522$		0			0
$523$	−6.43371e6			−1.02851			−0.514254	−	0.857638i	$-0.671931\pi$
$523$	−6.43371e6			−1.02851			−0.514254	+	0.857638i	$0.671931\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.36834e6	−	4.10209e6i	−0.371465	−	0.643396i
$528$		0			0
$529$	2.85178e6	−	4.93943e6i	0.443074	−	0.767427i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5.51735e6	−	9.55633e6i	0.841226	−	1.45705i
$534$		0			0
$535$	−1.18716e6	−	2.05622e6i	−0.179318	−	0.310588i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.69907e6			0.251907
$540$		0			0
$541$	5.85989e6			0.860788			0.430394	−	0.902641i	$-0.358375\pi$
$541$	5.85989e6			0.860788			0.430394	+	0.902641i	$0.358375\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.79185e6	+	4.83563e6i	0.402625	+	0.697368i
$546$		0			0
$547$	−2.59436e6	+	4.49357e6i	−0.370734	+	0.642130i	−0.989679	−	0.143305i	$-0.954227\pi$
$547$	−2.59436e6	+	4.49357e6i	−0.370734	+	0.642130i	0.618945	+	0.785434i	$0.287560\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.84118e6	−	1.01172e7i	0.819637	−	1.41965i
$552$		0			0
$553$	7.51545e6	+	1.30171e7i	1.04506	+	1.81010i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.07405e6			−0.692973			−0.346487	−	0.938055i	$-0.612625\pi$
$557$	−5.07405e6			−0.692973			−0.346487	+	0.938055i	$0.612625\pi$
$558$		0			0
$559$	−1.18623e7			−1.60561
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.05439e6	−	5.29036e6i	−0.406119	−	0.703419i	0.588332	−	0.808620i	$-0.299785\pi$
$563$	−3.05439e6	−	5.29036e6i	−0.406119	−	0.703419i	−0.994451	+	0.105200i	$0.966452\pi$
$564$		0			0
$565$	−3.22111e6	+	5.57913e6i	−0.424507	+	0.735268i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.40483e6	+	9.36143e6i	−0.699844	+	1.21216i	0.268677	+	0.963230i	$0.413414\pi$
$569$	−5.40483e6	+	9.36143e6i	−0.699844	+	1.21216i	−0.968520	+	0.248934i	$0.919920\pi$
$570$		0			0
$571$	−6.02796e6	−	1.04407e7i	−0.773714	−	1.34011i	−0.935515	−	0.353288i	$-0.885063\pi$
$571$	−6.02796e6	−	1.04407e7i	−0.773714	−	1.34011i	0.161801	−	0.986823i	$-0.448270\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.99706e6			0.251896
$576$		0			0
$577$	1.22323e7			1.52957			0.764786	−	0.644285i	$-0.222845\pi$
$577$	1.22323e7			1.52957			0.764786	+	0.644285i	$0.222845\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.78908e6	−	6.56288e6i	−0.465686	−	0.806593i
$582$		0			0
$583$	422054.	−	731019.i	0.0514277	−	0.0890754i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.29817e6	+	7.44465e6i	−0.514859	+	0.891762i	0.484992	+	0.874519i	$0.338823\pi$
$587$	−4.29817e6	+	7.44465e6i	−0.514859	+	0.891762i	−0.999851	+	0.0172439i	$0.994511\pi$
$588$		0			0
$589$	−2.01097e6	−	3.48311e6i	−0.238846	−	0.413693i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.78455e6			0.675511			0.337756	−	0.941234i	$-0.390332\pi$
$593$	5.78455e6			0.675511			0.337756	+	0.941234i	$0.390332\pi$
$594$		0			0
$595$	6.86903e6			0.795432
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.62557e6	−	4.54761e6i	−0.298989	−	0.517865i	0.676916	−	0.736061i	$-0.263316\pi$
$599$	−2.62557e6	−	4.54761e6i	−0.298989	−	0.517865i	−0.975905	+	0.218196i	$0.929983\pi$
$600$		0			0
$601$	3.11169e6	−	5.38960e6i	0.351407	−	0.608654i	−0.635090	−	0.772439i	$-0.719037\pi$
$601$	3.11169e6	−	5.38960e6i	0.351407	−	0.608654i	0.986496	+	0.163784i	$0.0523701\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.18073e6	+	2.04509e6i	−0.131148	+	0.227156i
$606$		0			0
$607$	7.94049e6	+	1.37533e7i	0.874733	+	1.51508i	0.857046	+	0.515239i	$0.172297\pi$
$607$	7.94049e6	+	1.37533e7i	0.874733	+	1.51508i	0.0176871	+	0.999844i	$0.494370\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	172693.			0.0187142
$612$		0			0
$613$	−1.28661e7			−1.38292			−0.691458	−	0.722417i	$-0.743031\pi$
$613$	−1.28661e7			−1.38292			−0.691458	+	0.722417i	$0.743031\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.43797e6	−	5.95473e6i	−0.363571	−	0.629723i	0.624975	−	0.780645i	$-0.285109\pi$
$617$	−3.43797e6	−	5.95473e6i	−0.363571	−	0.629723i	−0.988546	+	0.150922i	$0.951776\pi$
$618$		0			0
$619$	−6.52408e6	+	1.13000e7i	−0.684373	+	1.18537i	0.289261	+	0.957250i	$0.406591\pi$
$619$	−6.52408e6	+	1.13000e7i	−0.684373	+	1.18537i	−0.973633	+	0.228118i	$0.926743\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.03438e7	+	1.79160e7i	−1.06773	+	1.84936i
$624$		0			0
$625$	−1.48369e6	−	2.56983e6i	−0.151930	−	0.263151i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.50734e6			0.655809
$630$		0			0
$631$	−1.52784e7			−1.52758			−0.763790	−	0.645464i	$-0.776664\pi$
$631$	−1.52784e7			−1.52758			−0.763790	+	0.645464i	$0.776664\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.47052e6	−	6.01111e6i	−0.341554	−	0.591590i
$636$		0			0
$637$	−1.78583e6	+	3.09315e6i	−0.174378	+	0.302032i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.96157e6	+	1.03257e7i	−0.573080	+	0.992604i	0.423167	+	0.906052i	$0.360918\pi$
$641$	−5.96157e6	+	1.03257e7i	−0.573080	+	0.992604i	−0.996247	+	0.0865524i	$0.972415\pi$
$642$		0			0
$643$	3.00302e6	+	5.20137e6i	0.286438	+	0.496125i	0.972957	−	0.230987i	$-0.0741955\pi$
$643$	3.00302e6	+	5.20137e6i	0.286438	+	0.496125i	−0.686519	+	0.727112i	$0.740862\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.19440e7			−1.12173			−0.560864	−	0.827908i	$-0.689531\pi$
$647$	−1.19440e7			−1.12173			−0.560864	+	0.827908i	$0.689531\pi$
$648$		0			0
$649$	4.78742e6			0.446159
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.53297e6	−	2.65518e6i	−0.140686	−	0.243675i	0.787069	−	0.616864i	$-0.211597\pi$
$653$	−1.53297e6	−	2.65518e6i	−0.140686	−	0.243675i	−0.927755	+	0.373190i	$0.878264\pi$
$654$		0			0
$655$	2.38131e6	−	4.12454e6i	0.216876	−	0.375641i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.55065e6	+	2.68581e6i	−0.139092	+	0.240914i	−0.927153	−	0.374683i	$-0.877752\pi$
$659$	−1.55065e6	+	2.68581e6i	−0.139092	+	0.240914i	0.788061	+	0.615597i	$0.211085\pi$
$660$		0			0
$661$	−6.48853e6	−	1.12385e7i	−0.577621	−	1.00047i	−0.995751	−	0.0920819i	$-0.970648\pi$
$661$	−6.48853e6	−	1.12385e7i	−0.577621	−	1.00047i	0.418130	−	0.908387i	$-0.362685\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.83253e6			0.511450
$666$		0			0
$667$	7.30618e6			0.635881
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.56245e6	−	6.17034e6i	−0.305452	−	0.529058i
$672$		0			0
$673$	−1.10703e7	+	1.91743e7i	−0.942154	+	1.63186i	−0.180801	+	0.983520i	$0.557869\pi$
$673$	−1.10703e7	+	1.91743e7i	−0.942154	+	1.63186i	−0.761352	+	0.648338i	$0.775464\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	799850.	−	1.38538e6i	0.0670713	−	0.116171i	−0.830540	−	0.556960i	$-0.811968\pi$
$677$	799850.	−	1.38538e6i	0.0670713	−	0.116171i	0.897611	+	0.440789i	$0.145301\pi$
$678$		0			0
$679$	−5.04566e6	−	8.73934e6i	−0.419994	−	0.727452i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.36986e7			−1.12363			−0.561816	−	0.827262i	$-0.689897\pi$
$683$	−1.36986e7			−1.12363			−0.561816	+	0.827262i	$0.689897\pi$
$684$		0			0
$685$	2.81353e6			0.229100
$686$		0			0
$687$		0			0
$688$		0			0
$689$	887210.	+	1.53669e6i	0.0711998	+	0.123322i
$690$		0			0
$691$	2.09107e6	−	3.62183e6i	0.166599	−	0.288558i	−0.770623	−	0.637291i	$-0.780055\pi$
$691$	2.09107e6	−	3.62183e6i	0.166599	−	0.288558i	0.937222	+	0.348733i	$0.113388\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−526268.	+	911522.i	−0.0413280	+	0.0715822i
$696$		0			0
$697$	1.52331e7	+	2.63846e7i	1.18770	+	2.05716i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.47134e7			1.13089			0.565443	−	0.824788i	$-0.308705\pi$
$701$	1.47134e7			1.13089			0.565443	+	0.824788i	$0.308705\pi$
$702$		0			0
$703$	5.52543e6			0.421675
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.86818e6	+	3.23577e6i	0.140562	+	0.243461i
$708$		0			0
$709$	4.25908e6	−	7.37694e6i	0.318200	−	0.551139i	−0.661913	−	0.749581i	$-0.730255\pi$
$709$	4.25908e6	−	7.37694e6i	0.318200	−	0.551139i	0.980113	+	0.198442i	$0.0635883\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.25767e6	−	2.17834e6i	0.0926493	−	0.160473i
$714$		0			0
$715$	2.28195e6	+	3.95246e6i	0.166933	+	0.289136i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.20183e7			0.867003			0.433502	−	0.901153i	$-0.357278\pi$
$719$	1.20183e7			0.867003			0.433502	+	0.901153i	$0.357278\pi$
$720$		0			0
$721$	−1.75280e7			−1.25573
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.95577e6	−	1.72439e7i	−0.703445	−	1.21840i
$726$		0			0
$727$	−1.49257e6	+	2.58521e6i	−0.104737	+	0.181409i	−0.913631	−	0.406545i	$-0.866733\pi$
$727$	−1.49257e6	+	2.58521e6i	−0.104737	+	0.181409i	0.808894	+	0.587955i	$0.200067\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.63757e7	−	2.83635e7i	1.13346	−	1.96321i
$732$		0			0
$733$	−1.05675e7	−	1.83035e7i	−0.726464	−	1.25827i	−0.958369	−	0.285534i	$-0.907829\pi$
$733$	−1.05675e7	−	1.83035e7i	−0.726464	−	1.25827i	0.231905	−	0.972739i	$-0.425504\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.29926e6			−0.495006
$738$		0			0
$739$	1.33961e7			0.902337			0.451168	−	0.892439i	$-0.351007\pi$
$739$	1.33961e7			0.902337			0.451168	+	0.892439i	$0.351007\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.95418e6	+	1.03129e7i	0.395685	+	0.685347i	0.993188	−	0.116519i	$-0.0371737\pi$
$743$	5.95418e6	+	1.03129e7i	0.395685	+	0.685347i	−0.597503	+	0.801867i	$0.703840\pi$
$744$		0			0
$745$	−3.58175e6	+	6.20377e6i	−0.236431	+	0.409511i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.38671e6	+	1.10621e7i	−0.415980	+	0.720499i
$750$		0			0
$751$	−2.33184e6	−	4.03886e6i	−0.150868	−	0.261312i	0.780679	−	0.624933i	$-0.214874\pi$
$751$	−2.33184e6	−	4.03886e6i	−0.150868	−	0.261312i	−0.931547	+	0.363621i	$0.881540\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.66232e6			0.425361
$756$		0			0
$757$	138599.			0.00879065			0.00439532	−	0.999990i	$-0.498601\pi$
$757$	138599.			0.00879065			0.00439532	+	0.999990i	$0.498601\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.19856e6	+	2.07597e6i	0.0750237	+	0.129945i	0.901097	−	0.433618i	$-0.142763\pi$
$761$	1.19856e6	+	2.07597e6i	0.0750237	+	0.129945i	−0.826073	+	0.563563i	$0.809430\pi$
$762$		0			0
$763$	1.50197e7	−	2.60148e7i	0.934005	−	1.61774i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.03188e6	+	8.71546e6i	−0.308846	+	0.534936i
$768$		0			0
$769$	−399739.	−	692369.i	−0.0243759	−	0.0422203i	0.853580	−	0.520962i	$-0.174427\pi$
$769$	−399739.	−	692369.i	−0.0243759	−	0.0422203i	−0.877956	+	0.478741i	$0.841093\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.07743e6			−0.0648546			−0.0324273	−	0.999474i	$-0.510324\pi$
$773$	−1.07743e6			−0.0648546			−0.0324273	+	0.999474i	$0.510324\pi$
$774$		0			0
$775$	−6.85505e6			−0.409974
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.29346e7	+	2.24033e7i	0.763674	+	1.32272i
$780$		0			0
$781$	1.06468e7	−	1.84408e7i	0.624586	−	1.08182i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.58566e6	−	6.21055e6i	0.207680	−	0.359713i
$786$		0			0
$787$	−1.53546e6	−	2.65949e6i	−0.0883692	−	0.153060i	0.818453	−	0.574574i	$-0.194832\pi$
$787$	−1.53546e6	−	2.65949e6i	−0.0883692	−	0.153060i	−0.906822	+	0.421514i	$0.861499\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.46581e7			1.96953
$792$		0			0
$793$	1.49774e7			0.845774
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.49242e6	+	1.29772e7i	0.417808	+	0.723664i	0.995719	−	0.0924359i	$-0.0294653\pi$
$797$	7.49242e6	+	1.29772e7i	0.417808	+	0.723664i	−0.577911	+	0.816100i	$0.696132\pi$
$798$		0			0
$799$	−238399.	+	412919.i	−0.0132110	+	0.0228822i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−207801.	+	359921.i	−0.0113726	+	0.0196978i
$804$		0			0
$805$	1.82384e6	+	3.15898e6i	0.0991967	+	0.171814i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.11069e6			0.167103			0.0835516	−	0.996503i	$-0.473374\pi$
$809$	3.11069e6			0.167103			0.0835516	+	0.996503i	$0.473374\pi$
$810$		0			0
$811$	1.12694e7			0.601659			0.300830	−	0.953678i	$-0.402736\pi$
$811$	1.12694e7			0.601659			0.300830	+	0.953678i	$0.402736\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.02615e6	+	5.24144e6i	0.159586	+	0.276412i
$816$		0			0
$817$	1.39047e7	−	2.40836e7i	0.728795	−	1.26231i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.04769e7	−	1.81465e7i	0.542467	−	0.939580i	−0.456295	−	0.889829i	$-0.650824\pi$
$821$	1.04769e7	−	1.81465e7i	0.542467	−	0.939580i	0.998762	−	0.0497515i	$-0.0158430\pi$
$822$		0			0
$823$	−974079.	−	1.68715e6i	−0.0501296	−	0.0868271i	0.839872	−	0.542785i	$-0.182630\pi$
$823$	−974079.	−	1.68715e6i	−0.0501296	−	0.0868271i	−0.890001	+	0.455958i	$0.849297\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.11807e7			−1.58534			−0.792671	−	0.609650i	$-0.791310\pi$
$827$	−3.11807e7			−1.58534			−0.792671	+	0.609650i	$0.791310\pi$
$828$		0			0
$829$	−1.67293e7			−0.845455			−0.422727	−	0.906257i	$-0.638927\pi$
$829$	−1.67293e7			−0.845455			−0.422727	+	0.906257i	$0.638927\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.93060e6	−	8.54005e6i	−0.246199	−	0.426430i
$834$		0			0
$835$	−2.70653e6	+	4.68785e6i	−0.134337	+	0.232679i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4.13660e6	+	7.16481e6i	−0.202880	+	0.351398i	−0.949455	−	0.313903i	$-0.898363\pi$
$839$	−4.13660e6	+	7.16481e6i	−0.202880	+	0.351398i	0.746575	+	0.665301i	$0.231697\pi$
$840$		0			0
$841$	−2.61673e7	−	4.53231e7i	−1.27576	−	2.20968i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	855632.			0.0412235
$846$		0			0
$847$	1.27043e7			0.608473
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.72781e6	+	2.99265e6i	0.0817846	+	0.141655i
$852$		0			0
$853$	−1.73469e7	+	3.00458e7i	−0.816300	+	1.41387i	0.0920905	+	0.995751i	$0.470645\pi$
$853$	−1.73469e7	+	3.00458e7i	−0.816300	+	1.41387i	−0.908391	+	0.418123i	$0.862688\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.51863e6	−	4.36239e6i	0.117142	−	0.202895i	−0.801492	−	0.598005i	$-0.795960\pi$
$857$	2.51863e6	−	4.36239e6i	0.117142	−	0.202895i	0.918634	+	0.395110i	$0.129293\pi$
$858$		0			0
$859$	2.13910e7	+	3.70502e7i	0.989116	+	1.71320i	0.621984	+	0.783030i	$0.286327\pi$
$859$	2.13910e7	+	3.70502e7i	0.989116	+	1.71320i	0.367132	+	0.930169i	$0.380340\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.50904e6			−0.0689721			−0.0344861	−	0.999405i	$-0.510979\pi$
$863$	−1.50904e6			−0.0689721			−0.0344861	+	0.999405i	$0.510979\pi$
$864$		0			0
$865$	−233624.			−0.0106164
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.37866e7	−	2.38790e7i	−0.619307	−	1.07267i
$870$		0			0
$871$	7.67198e6	−	1.32883e7i	0.342659	−	0.593503i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.16286e7	−	2.01413e7i	0.513460	−	0.889339i
$876$		0			0
$877$	1.64700e7	+	2.85268e7i	0.723092	+	1.25243i	0.959755	+	0.280840i	$0.0906131\pi$
$877$	1.64700e7	+	2.85268e7i	0.723092	+	1.25243i	−0.236663	+	0.971592i	$0.576054\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.62885e7			−1.14110			−0.570552	−	0.821261i	$-0.693271\pi$
$881$	−2.62885e7			−1.14110			−0.570552	+	0.821261i	$0.693271\pi$
$882$		0			0
$883$	−2.07499e7			−0.895602			−0.447801	−	0.894133i	$-0.647793\pi$
$883$	−2.07499e7			−0.895602			−0.447801	+	0.894133i	$0.647793\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.64291e7	+	2.84561e7i	0.701140	+	1.21441i	0.968066	+	0.250694i	$0.0806587\pi$
$887$	1.64291e7	+	2.84561e7i	0.701140	+	1.21441i	−0.266926	+	0.963717i	$0.586008\pi$
$888$		0			0
$889$	−1.86708e7	+	3.23387e7i	−0.792334	+	1.37236i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−202426.	+	350612.i	−0.00849449	+	0.0147129i
$894$		0			0
$895$	−8.08420e6	−	1.40022e7i	−0.337349	−	0.584306i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.50790e7			−1.03493
$900$		0			0
$901$	−4.89909e6			−0.201050
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.16203e6	−	5.47680e6i	−0.128335	−	0.222283i
$906$		0			0
$907$	−2.35836e7	+	4.08480e7i	−0.951900	+	1.64874i	−0.210593	+	0.977574i	$0.567540\pi$
$907$	−2.35836e7	+	4.08480e7i	−0.951900	+	1.64874i	−0.741307	+	0.671166i	$0.765794\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.90246e7	−	3.29516e7i	0.759486	−	1.31547i	−0.183627	−	0.982996i	$-0.558784\pi$
$911$	1.90246e7	−	3.29516e7i	0.759486	−	1.31547i	0.943113	−	0.332472i	$-0.107883\pi$
$912$		0			0
$913$	6.95080e6	+	1.20391e7i	0.275967	+	0.477990i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.56220e7			−1.00621
$918$		0			0
$919$	2.54292e7			0.993215			0.496607	−	0.867975i	$-0.334579\pi$
$919$	2.54292e7			0.993215			0.496607	+	0.867975i	$0.334579\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.23809e7	+	3.87649e7i	0.864717	+	1.49773i
$924$		0			0
$925$	4.70880e6	−	8.15588e6i	0.180949	−	0.313412i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.00472e6	+	1.21325e7i	−0.266288	+	0.461224i	−0.967900	−	0.251335i	$-0.919131\pi$
$929$	−7.00472e6	+	1.21325e7i	−0.266288	+	0.461224i	0.701612	+	0.712559i	$0.252464\pi$
$930$		0			0
$931$	−4.18660e6	−	7.25141e6i	−0.158302	−	0.274188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.26007e7			−0.471375
$936$		0			0
$937$	−1.29313e7			−0.481164			−0.240582	−	0.970629i	$-0.577338\pi$
$937$	−1.29313e7			−0.481164			−0.240582	+	0.970629i	$0.577338\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.79973e6	−	8.31337e6i	−0.176702	−	0.306058i	0.764047	−	0.645161i	$-0.223210\pi$
$941$	−4.79973e6	−	8.31337e6i	−0.176702	−	0.306058i	−0.940749	+	0.339103i	$0.889876\pi$
$942$		0			0
$943$	−8.08930e6	+	1.40111e7i	−0.296232	+	0.513089i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.16687e6	+	1.24134e7i	−0.259690	+	0.449796i	−0.966159	−	0.257948i	$-0.916954\pi$
$947$	−7.16687e6	+	1.24134e7i	−0.259690	+	0.449796i	0.706469	+	0.707744i	$0.250287\pi$
$948$		0			0
$949$	−436823.	−	756599.i	−0.0157449	−	0.0272710i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.14184e7			−1.12060			−0.560302	−	0.828289i	$-0.689315\pi$
$953$	−3.14184e7			−1.12060			−0.560302	+	0.828289i	$0.689315\pi$
$954$		0			0
$955$	−1.34550e7			−0.477391
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.56817e6	−	1.31084e7i	−0.265732	−	0.460262i
$960$		0			0
$961$	9.99755e6	−	1.73163e7i	0.349209	−	0.604847i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.40248e6	+	1.28215e7i	−0.255893	+	0.443220i
$966$		0			0
$967$	6.76319e6	+	1.17142e7i	0.232587	+	0.402852i	0.958569	−	0.284862i	$-0.0919477\pi$
$967$	6.76319e6	+	1.17142e7i	0.232587	+	0.402852i	−0.725982	+	0.687714i	$0.758614\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.90534e7			−1.32926			−0.664632	−	0.747171i	$-0.731412\pi$
$971$	−3.90534e7			−1.32926			−0.664632	+	0.747171i	$0.731412\pi$
$972$		0			0
$973$	5.66246e6			0.191744
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.45244e6	+	1.63721e7i	0.316816	+	0.548742i	0.979822	−	0.199873i	$-0.0640528\pi$
$977$	9.45244e6	+	1.63721e7i	0.316816	+	0.548742i	−0.663006	+	0.748614i	$0.730720\pi$
$978$		0			0
$979$	1.89750e7	−	3.28657e7i	0.632740	−	1.09594i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.95013e6	+	5.10977e6i	−0.0973771	+	0.168662i	−0.910598	−	0.413293i	$-0.864379\pi$
$983$	−2.95013e6	+	5.10977e6i	−0.0973771	+	0.168662i	0.813221	+	0.581955i	$0.197712\pi$
$984$		0			0
$985$	3.12786e6	+	5.41762e6i	0.102720	+	0.177917i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.73920e7			0.565405
$990$		0			0
$991$	−3.67515e7			−1.18875			−0.594375	−	0.804188i	$-0.702601\pi$
$991$	−3.67515e7			−1.18875			−0.594375	+	0.804188i	$0.702601\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.54362e6	−	2.67364e6i	−0.0494293	−	0.0856140i
$996$		0			0
$997$	−1.46074e7	+	2.53008e7i	−0.465410	+	0.806114i	−0.999220	−	0.0394905i	$-0.987427\pi$
$997$	−1.46074e7	+	2.53008e7i	−0.465410	+	0.806114i	0.533810	+	0.845605i	$0.320760\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.e.a.73.4		10
3.2	odd	2		36.6.e.a.25.1	yes	10
4.3	odd	2		432.6.i.d.289.4		10
9.2	odd	6		324.6.a.e.1.4		5
9.4	even	3	inner	108.6.e.a.37.4		10
9.5	odd	6		36.6.e.a.13.1	✓	10
9.7	even	3		324.6.a.d.1.2		5
12.11	even	2		144.6.i.d.97.5		10
36.23	even	6		144.6.i.d.49.5		10
36.31	odd	6		432.6.i.d.145.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.e.a.13.1	✓	10	9.5	odd	6
36.6.e.a.25.1	yes	10	3.2	odd	2
108.6.e.a.37.4		10	9.4	even	3	inner
108.6.e.a.73.4		10	1.1	even	1	trivial
144.6.i.d.49.5		10	36.23	even	6
144.6.i.d.97.5		10	12.11	even	2
324.6.a.d.1.2		5	9.7	even	3
324.6.a.e.1.4		5	9.2	odd	6
432.6.i.d.145.4		10	36.31	odd	6
432.6.i.d.289.4		10	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(14.0718 + 24.3731i) q^{5} +(75.7039 - 131.123i) q^{7} +O(q^{10})\)	\(q+(14.0718 + 24.3731i) q^{5} +(75.7039 - 131.123i) q^{7} +(-138.873 + 240.536i) q^{11} +(-291.929 - 505.636i) q^{13} +1612.01 q^{17} +1368.76 q^{19} +(428.014 + 741.342i) q^{23} +(1166.47 - 2020.38i) q^{25} +(4267.49 - 7391.50i) q^{29} +(-1469.19 - 2544.71i) q^{31} +4261.17 q^{35} +4036.80 q^{37} +(9449.81 + 16367.5i) q^{41} +(10158.6 - 17595.1i) q^{43} +(-147.890 + 256.152i) q^{47} +(-3058.67 - 5297.78i) q^{49} -3039.13 q^{53} -7816.81 q^{55} +(-8618.31 - 14927.4i) q^{59} +(-12826.2 + 22215.7i) q^{61} +(8215.95 - 14230.4i) q^{65} +(13140.1 + 22759.4i) q^{67} -76665.7 q^{71} +1496.33 q^{73} +(21026.5 + 36419.0i) q^{77} +(-49637.1 + 85974.0i) q^{79} +(25025.7 - 43345.7i) q^{83} +(22683.8 + 39289.6i) q^{85} -136635. q^{89} -88400.8 q^{91} +(19261.0 + 33361.0i) q^{95} +(33325.0 - 57720.5i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 21 q^{5} + 29 q^{7}+O(q^{10}) \) 10 * q + 21 * q^5 + 29 * q^7	\( 10 q + 21 q^{5} + 29 q^{7} - 177 q^{11} - 181 q^{13} - 2280 q^{17} - 832 q^{19} - 399 q^{23} - 4778 q^{25} + 6033 q^{29} + 2759 q^{31} - 37146 q^{35} - 15172 q^{37} + 18435 q^{41} + 1469 q^{43} + 25155 q^{47} - 4056 q^{49} - 116844 q^{53} + 14778 q^{55} + 90537 q^{59} + 1403 q^{61} + 148407 q^{65} + 13907 q^{67} - 229368 q^{71} + 15200 q^{73} + 211983 q^{77} + 29993 q^{79} + 228951 q^{83} - 49662 q^{85} - 598332 q^{89} + 124930 q^{91} + 394764 q^{95} + 40541 q^{97}+O(q^{100}) \) 10 * q + 21 * q^5 + 29 * q^7 - 177 * q^11 - 181 * q^13 - 2280 * q^17 - 832 * q^19 - 399 * q^23 - 4778 * q^25 + 6033 * q^29 + 2759 * q^31 - 37146 * q^35 - 15172 * q^37 + 18435 * q^41 + 1469 * q^43 + 25155 * q^47 - 4056 * q^49 - 116844 * q^53 + 14778 * q^55 + 90537 * q^59 + 1403 * q^61 + 148407 * q^65 + 13907 * q^67 - 229368 * q^71 + 15200 * q^73 + 211983 * q^77 + 29993 * q^79 + 228951 * q^83 - 49662 * q^85 - 598332 * q^89 + 124930 * q^91 + 394764 * q^95 + 40541 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more