LMFDB - Embedded newform 1872.2.t.d.1153.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(289,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1872.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.t (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:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1153.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1872.1153
Dual form			1872.2.t.d.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +O(q^{10})$	$q-2.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{17} +(3.50000 + 6.06218i) q^{19} +(-0.500000 + 0.866025i) q^{23} -1.00000 q^{25} +(1.50000 - 2.59808i) q^{29} -8.00000 q^{31} +(1.00000 + 1.73205i) q^{35} +(0.500000 - 0.866025i) q^{37} +(5.50000 - 9.52628i) q^{41} +(5.50000 + 9.52628i) q^{43} +12.0000 q^{47} +(3.00000 - 5.19615i) q^{49} +6.00000 q^{53} +(1.00000 - 1.73205i) q^{55} +(4.50000 + 7.79423i) q^{59} +(4.50000 + 7.79423i) q^{61} +(2.00000 + 6.92820i) q^{65} +(-1.50000 + 2.59808i) q^{67} +(2.50000 + 4.33013i) q^{71} -2.00000 q^{73} +1.00000 q^{77} +12.0000 q^{79} -4.00000 q^{83} +(-3.00000 - 5.19615i) q^{85} +(-0.500000 + 0.866025i) q^{89} +(-2.50000 + 2.59808i) q^{91} +(-7.00000 - 12.1244i) q^{95} +(0.500000 + 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 - q^7	$ 2 q - 4 q^{5} - q^{7} - q^{11} - 2 q^{13} + 3 q^{17} + 7 q^{19} - q^{23} - 2 q^{25} + 3 q^{29} - 16 q^{31} + 2 q^{35} + q^{37} + 11 q^{41} + 11 q^{43} + 24 q^{47} + 6 q^{49} + 12 q^{53} + 2 q^{55} + 9 q^{59} + 9 q^{61} + 4 q^{65} - 3 q^{67} + 5 q^{71} - 4 q^{73} + 2 q^{77} + 24 q^{79} - 8 q^{83} - 6 q^{85} - q^{89} - 5 q^{91} - 14 q^{95} + q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - q^7 - q^11 - 2 * q^13 + 3 * q^17 + 7 * q^19 - q^23 - 2 * q^25 + 3 * q^29 - 16 * q^31 + 2 * q^35 + q^37 + 11 * q^41 + 11 * q^43 + 24 * q^47 + 6 * q^49 + 12 * q^53 + 2 * q^55 + 9 * q^59 + 9 * q^61 + 4 * q^65 - 3 * q^67 + 5 * q^71 - 4 * q^73 + 2 * q^77 + 24 * q^79 - 8 * q^83 - 6 * q^85 - q^89 - 5 * q^91 - 14 * q^95 + q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.00000			−0.894427			−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000			−0.894427			−0.447214	+	0.894427i	$0.647584\pi$
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i	0.755929	−	0.654654i	$-0.227186\pi$
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i	−0.931505	−	0.363727i	$-0.881504\pi$
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i	0.780750	+	0.624844i	$0.214837\pi$
$12$		0			0
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.500000	+	0.866025i	−0.104257	+	0.180579i	−0.913434	−	0.406986i	$-0.866580\pi$
$23$	−0.500000	+	0.866025i	−0.104257	+	0.180579i	0.809177	+	0.587565i	$0.199913\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	−0.692480	−	0.721437i	$-0.743482\pi$
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	0.971023	+	0.238987i	$0.0768152\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	+	1.73205i	0.169031	+	0.292770i
$36$		0			0
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	0.904194	+	0.427121i	$0.140472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.50000	−	9.52628i	0.858956	−	1.48775i	−0.0139704	−	0.999902i	$-0.504447\pi$
$41$	5.50000	−	9.52628i	0.858956	−	1.48775i	0.872926	−	0.487852i	$-0.162220\pi$
$42$		0			0
$43$	5.50000	+	9.52628i	0.838742	+	1.45274i	0.890947	+	0.454108i	$0.150042\pi$
$43$	5.50000	+	9.52628i	0.838742	+	1.45274i	−0.0522047	+	0.998636i	$0.516625\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.0000			1.75038			0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000			1.75038			0.875190	+	0.483779i	$0.160736\pi$
$48$		0			0
$49$	3.00000	−	5.19615i	0.428571	−	0.742307i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	1.00000	−	1.73205i	0.134840	−	0.233550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.50000	+	7.79423i	0.585850	+	1.01472i	0.994769	+	0.102151i	$0.0325726\pi$
$59$	4.50000	+	7.79423i	0.585850	+	1.01472i	−0.408919	+	0.912571i	$0.634094\pi$
$60$		0			0
$61$	4.50000	+	7.79423i	0.576166	+	0.997949i	0.995914	+	0.0903080i	$0.0287851\pi$
$61$	4.50000	+	7.79423i	0.576166	+	0.997949i	−0.419748	+	0.907641i	$0.637882\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000	+	6.92820i	0.248069	+	0.859338i
$66$		0			0
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	−0.942987	−	0.332830i	$-0.891996\pi$
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	0.759733	+	0.650236i	$0.225330\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.50000	+	4.33013i	0.296695	+	0.513892i	0.975378	−	0.220540i	$-0.0707821\pi$
$71$	2.50000	+	4.33013i	0.296695	+	0.513892i	−0.678682	+	0.734432i	$0.737449\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	12.0000			1.35011			0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000			1.35011			0.675053	+	0.737769i	$0.264121\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$	−3.00000	−	5.19615i	−0.325396	−	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.500000	+	0.866025i	−0.0529999	+	0.0917985i	−0.891308	−	0.453398i	$-0.850212\pi$
$89$	−0.500000	+	0.866025i	−0.0529999	+	0.0917985i	0.838308	+	0.545197i	$0.183545\pi$
$90$		0			0
$91$	−2.50000	+	2.59808i	−0.262071	+	0.272352i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.00000	−	12.1244i	−0.718185	−	1.24393i
$96$		0			0
$97$	0.500000	+	0.866025i	0.0507673	+	0.0879316i	0.890292	−	0.455389i	$-0.150500\pi$
$97$	0.500000	+	0.866025i	0.0507673	+	0.0879316i	−0.839525	+	0.543321i	$0.817167\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.50000	+	14.7224i	−0.845782	+	1.46494i	0.0391591	+	0.999233i	$0.487532\pi$
$101$	−8.50000	+	14.7224i	−0.845782	+	1.46494i	−0.884941	+	0.465704i	$0.845801\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.50000	+	14.7224i	−0.821726	+	1.42327i	0.0826699	+	0.996577i	$0.473655\pi$
$107$	−8.50000	+	14.7224i	−0.821726	+	1.42327i	−0.904396	+	0.426694i	$0.859678\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.50000	+	9.52628i	0.517396	+	0.896157i	0.999796	+	0.0202056i	$0.00643208\pi$
$113$	5.50000	+	9.52628i	0.517396	+	0.896157i	−0.482399	+	0.875951i	$0.660235\pi$
$114$		0			0
$115$	1.00000	−	1.73205i	0.0932505	−	0.161515i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.50000	−	2.59808i	0.137505	−	0.238165i
$120$		0			0
$121$	5.00000	+	8.66025i	0.454545	+	0.787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	6.50000	−	11.2583i	0.576782	−	0.999015i	−0.419064	−	0.907957i	$-0.637642\pi$
$127$	6.50000	−	11.2583i	0.576782	−	0.999015i	0.995846	−	0.0910585i	$-0.0290250\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	3.50000	−	6.06218i	0.303488	−	0.525657i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	0.607233	−	0.794524i	$-0.292279\pi$
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	−0.991694	+	0.128618i	$0.958946\pi$
$138$		0			0
$139$	−4.50000	−	7.79423i	−0.381685	−	0.661098i	0.609618	−	0.792695i	$-0.291323\pi$
$139$	−4.50000	−	7.79423i	−0.381685	−	0.661098i	−0.991303	+	0.131597i	$0.957989\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.50000	+	0.866025i	0.292685	+	0.0724207i
$144$		0			0
$145$	−3.00000	+	5.19615i	−0.249136	+	0.431517i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.500000	−	0.866025i	−0.0409616	−	0.0709476i	0.844818	−	0.535054i	$-0.179709\pi$
$149$	−0.500000	−	0.866025i	−0.0409616	−	0.0709476i	−0.885779	+	0.464107i	$0.846375\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.0000			1.28515
$156$		0			0
$157$	10.0000			0.798087			0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000			0.798087			0.399043	+	0.916932i	$0.369342\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.00000			0.0788110
$162$		0			0
$163$	−2.50000	−	4.33013i	−0.195815	−	0.339162i	0.751352	−	0.659901i	$-0.229402\pi$
$163$	−2.50000	−	4.33013i	−0.195815	−	0.339162i	−0.947167	+	0.320740i	$0.896069\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.50000	−	6.06218i	0.270838	−	0.469105i	−0.698239	−	0.715865i	$-0.746033\pi$
$167$	3.50000	−	6.06218i	0.270838	−	0.469105i	0.969077	+	0.246760i	$0.0793659\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.50000	−	7.79423i	−0.342129	−	0.592584i	0.642699	−	0.766119i	$-0.277815\pi$
$173$	−4.50000	−	7.79423i	−0.342129	−	0.592584i	−0.984828	+	0.173534i	$0.944481\pi$
$174$		0			0
$175$	0.500000	+	0.866025i	0.0377964	+	0.0654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.50000	+	7.79423i	−0.336346	+	0.582568i	−0.983742	−	0.179585i	$-0.942524\pi$
$179$	−4.50000	+	7.79423i	−0.336346	+	0.582568i	0.647397	+	0.762153i	$0.275858\pi$
$180$		0			0
$181$	−14.0000			−1.04061			−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000			−1.04061			−0.520306	+	0.853980i	$0.674182\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.00000	+	1.73205i	−0.0735215	+	0.127343i
$186$		0			0
$187$	−3.00000			−0.219382
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.50000	−	6.06218i	−0.253251	−	0.438644i	0.711168	−	0.703022i	$-0.248167\pi$
$191$	−3.50000	−	6.06218i	−0.253251	−	0.438644i	−0.964419	+	0.264378i	$0.914833\pi$
$192$		0			0
$193$	2.50000	−	4.33013i	0.179954	−	0.311689i	−0.761911	−	0.647682i	$-0.775738\pi$
$193$	2.50000	−	4.33013i	0.179954	−	0.311689i	0.941865	+	0.335993i	$0.109072\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.50000	+	11.2583i	−0.463106	+	0.802123i	−0.999114	−	0.0420901i	$-0.986598\pi$
$197$	−6.50000	+	11.2583i	−0.463106	+	0.802123i	0.536008	+	0.844213i	$0.319932\pi$
$198$		0			0
$199$	1.50000	+	2.59808i	0.106332	+	0.184173i	0.914282	−	0.405079i	$-0.132756\pi$
$199$	1.50000	+	2.59808i	0.106332	+	0.184173i	−0.807950	+	0.589252i	$0.799423\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.00000			−0.210559
$204$		0			0
$205$	−11.0000	+	19.0526i	−0.768273	+	1.33069i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.00000			−0.484200
$210$		0			0
$211$	0.500000	−	0.866025i	0.0344214	−	0.0596196i	−0.848301	−	0.529514i	$-0.822374\pi$
$211$	0.500000	−	0.866025i	0.0344214	−	0.0596196i	0.882723	+	0.469894i	$0.155708\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−11.0000	−	19.0526i	−0.750194	−	1.29937i
$216$		0			0
$217$	4.00000	+	6.92820i	0.271538	+	0.470317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.50000	−	7.79423i	0.504505	−	0.524297i
$222$		0			0
$223$	0.500000	−	0.866025i	0.0334825	−	0.0579934i	−0.848799	−	0.528716i	$-0.822674\pi$
$223$	0.500000	−	0.866025i	0.0334825	−	0.0579934i	0.882281	+	0.470723i	$0.156007\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.500000	+	0.866025i	0.0331862	+	0.0574801i	0.882141	−	0.470985i	$-0.156101\pi$
$227$	0.500000	+	0.866025i	0.0331862	+	0.0574801i	−0.848955	+	0.528465i	$0.822768\pi$
$228$		0			0
$229$	−6.00000			−0.396491			−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000			−0.396491			−0.198246	+	0.980152i	$0.563524\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	26.0000			1.70332			0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000			1.70332			0.851658	+	0.524097i	$0.175597\pi$
$234$		0			0
$235$	−24.0000			−1.56559
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	−5.50000	−	9.52628i	−0.354286	−	0.613642i	0.632709	−	0.774389i	$-0.281943\pi$
$241$	−5.50000	−	9.52628i	−0.354286	−	0.613642i	−0.986996	+	0.160748i	$0.948609\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.00000	+	10.3923i	−0.383326	+	0.663940i
$246$		0			0
$247$	17.5000	−	18.1865i	1.11350	−	1.15718i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.50000	+	11.2583i	0.410276	+	0.710620i	0.994920	−	0.100671i	$-0.0320989\pi$
$251$	6.50000	+	11.2583i	0.410276	+	0.710620i	−0.584643	+	0.811290i	$0.698766\pi$
$252$		0			0
$253$	−0.500000	−	0.866025i	−0.0314347	−	0.0544466i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.50000	+	11.2583i	−0.405459	+	0.702275i	−0.994375	−	0.105919i	$-0.966222\pi$
$257$	−6.50000	+	11.2583i	−0.405459	+	0.702275i	0.588916	+	0.808194i	$0.299555\pi$
$258$		0			0
$259$	−1.00000			−0.0621370
$260$		0			0
$261$		0			0
$262$		0			0
$263$	15.5000	−	26.8468i	0.955771	−	1.65544i	0.223177	−	0.974778i	$-0.428357\pi$
$263$	15.5000	−	26.8468i	0.955771	−	1.65544i	0.732594	−	0.680666i	$-0.238309\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.50000	−	4.33013i	−0.152428	−	0.264013i	0.779692	−	0.626164i	$-0.215376\pi$
$269$	−2.50000	−	4.33013i	−0.152428	−	0.264013i	−0.932119	+	0.362151i	$0.882042\pi$
$270$		0			0
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	−0.952531	−	0.304443i	$-0.901530\pi$
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	0.739921	+	0.672694i	$0.234863\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.500000	−	0.866025i	0.0301511	−	0.0522233i
$276$		0			0
$277$	10.5000	+	18.1865i	0.630884	+	1.09272i	0.987371	+	0.158423i	$0.0506409\pi$
$277$	10.5000	+	18.1865i	0.630884	+	1.09272i	−0.356488	+	0.934300i	$0.616026\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−14.0000			−0.835170			−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000			−0.835170			−0.417585	+	0.908638i	$0.637123\pi$
$282$		0			0
$283$	12.5000	−	21.6506i	0.743048	−	1.28700i	−0.208053	−	0.978117i	$-0.566713\pi$
$283$	12.5000	−	21.6506i	0.743048	−	1.28700i	0.951101	−	0.308879i	$-0.0999539\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−11.0000			−0.649309
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.50000	−	7.79423i	−0.262893	−	0.455344i	0.704117	−	0.710084i	$-0.251343\pi$
$293$	−4.50000	−	7.79423i	−0.262893	−	0.455344i	−0.967009	+	0.254741i	$0.918010\pi$
$294$		0			0
$295$	−9.00000	−	15.5885i	−0.524000	−	0.907595i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.50000	+	0.866025i	0.202410	+	0.0500835i
$300$		0			0
$301$	5.50000	−	9.52628i	0.317015	−	0.549086i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.00000	−	15.5885i	−0.515339	−	0.892592i
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$	30.0000			1.69570			0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000			1.69570			0.847850	+	0.530236i	$0.177897\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.00000			0.112331			0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000			0.112331			0.0561656	+	0.998421i	$0.482113\pi$
$318$		0			0
$319$	1.50000	+	2.59808i	0.0839839	+	0.145464i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.5000	+	18.1865i	−0.584236	+	1.01193i
$324$		0			0
$325$	1.00000	+	3.46410i	0.0554700	+	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.00000	−	10.3923i	−0.330791	−	0.572946i
$330$		0			0
$331$	13.5000	+	23.3827i	0.742027	+	1.28523i	0.951571	+	0.307429i	$0.0994688\pi$
$331$	13.5000	+	23.3827i	0.742027	+	1.28523i	−0.209544	+	0.977799i	$0.567198\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.00000	−	5.19615i	0.163908	−	0.283896i
$336$		0			0
$337$	6.00000			0.326841			0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000			0.326841			0.163420	+	0.986557i	$0.447747\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.00000	−	6.92820i	0.216612	−	0.375183i
$342$		0			0
$343$	−13.0000			−0.701934
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.5000	−	23.3827i	−0.724718	−	1.25525i	−0.959090	−	0.283101i	$-0.908637\pi$
$347$	−13.5000	−	23.3827i	−0.724718	−	1.25525i	0.234372	−	0.972147i	$-0.424697\pi$
$348$		0			0
$349$	4.50000	−	7.79423i	0.240879	−	0.417215i	−0.720086	−	0.693885i	$-0.755897\pi$
$349$	4.50000	−	7.79423i	0.240879	−	0.417215i	0.960965	+	0.276670i	$0.0892308\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.5000	+	21.6506i	−0.665308	+	1.15235i	0.313894	+	0.949458i	$0.398366\pi$
$353$	−12.5000	+	21.6506i	−0.665308	+	1.15235i	−0.979202	+	0.202889i	$0.934967\pi$
$354$		0			0
$355$	−5.00000	−	8.66025i	−0.265372	−	0.459639i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−15.0000	+	25.9808i	−0.789474	+	1.36741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.00000			0.209370
$366$		0			0
$367$	−7.50000	+	12.9904i	−0.391497	+	0.678092i	−0.992647	−	0.121044i	$-0.961376\pi$
$367$	−7.50000	+	12.9904i	−0.391497	+	0.678092i	0.601150	+	0.799136i	$0.294709\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.00000	−	5.19615i	−0.155752	−	0.269771i
$372$		0			0
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	0.687776	−	0.725923i	$-0.258587\pi$
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	−0.972556	+	0.232671i	$0.925254\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10.5000	−	2.59808i	−0.540778	−	0.133808i
$378$		0			0
$379$	−7.50000	+	12.9904i	−0.385249	+	0.667271i	−0.991804	−	0.127771i	$-0.959218\pi$
$379$	−7.50000	+	12.9904i	−0.385249	+	0.667271i	0.606555	+	0.795042i	$0.292551\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.5000	+	28.5788i	0.843111	+	1.46031i	0.887252	+	0.461285i	$0.152611\pi$
$383$	16.5000	+	28.5788i	0.843111	+	1.46031i	−0.0441413	+	0.999025i	$0.514055\pi$
$384$		0			0
$385$	−2.00000			−0.101929
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	−3.00000			−0.151717
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−24.0000			−1.20757
$396$		0			0
$397$	−9.50000	−	16.4545i	−0.476791	−	0.825827i	0.522855	−	0.852422i	$-0.324867\pi$
$397$	−9.50000	−	16.4545i	−0.476791	−	0.825827i	−0.999646	+	0.0265948i	$0.991534\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.5000	−	23.3827i	0.674158	−	1.16768i	−0.302556	−	0.953131i	$-0.597840\pi$
$401$	13.5000	−	23.3827i	0.674158	−	1.16768i	0.976714	−	0.214544i	$-0.0688266\pi$
$402$		0			0
$403$	8.00000	+	27.7128i	0.398508	+	1.38047i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.500000	+	0.866025i	0.0247841	+	0.0429273i
$408$		0			0
$409$	−17.5000	−	30.3109i	−0.865319	−	1.49878i	−0.866730	−	0.498778i	$-0.833782\pi$
$409$	−17.5000	−	30.3109i	−0.865319	−	1.49878i	0.00141047	−	0.999999i	$-0.499551\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.50000	−	7.79423i	0.221431	−	0.383529i
$414$		0			0
$415$	8.00000			0.392705
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.5000	+	18.1865i	−0.512959	+	0.888470i	0.486928	+	0.873442i	$0.338117\pi$
$419$	−10.5000	+	18.1865i	−0.512959	+	0.888470i	−0.999887	+	0.0150285i	$0.995216\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.50000	−	2.59808i	−0.0727607	−	0.126025i
$426$		0			0
$427$	4.50000	−	7.79423i	0.217770	−	0.377189i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.50000	−	9.52628i	0.264926	−	0.458865i	−0.702618	−	0.711567i	$-0.747986\pi$
$431$	5.50000	−	9.52628i	0.264926	−	0.458865i	0.967544	+	0.252702i	$0.0813192\pi$
$432$		0			0
$433$	−9.50000	−	16.4545i	−0.456541	−	0.790752i	0.542234	−	0.840227i	$-0.317578\pi$
$433$	−9.50000	−	16.4545i	−0.456541	−	0.790752i	−0.998775	+	0.0494752i	$0.984245\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.00000			−0.334855
$438$		0			0
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	0.0586141	+	0.998281i	$0.481332\pi$
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	−0.893843	+	0.448379i	$0.852001\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.0000			0.570137			0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000			0.570137			0.285069	+	0.958507i	$0.407984\pi$
$444$		0			0
$445$	1.00000	−	1.73205i	0.0474045	−	0.0821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10.5000	−	18.1865i	−0.495526	−	0.858276i	0.504461	−	0.863434i	$-0.331691\pi$
$449$	−10.5000	−	18.1865i	−0.495526	−	0.858276i	−0.999987	+	0.00515887i	$0.998358\pi$
$450$		0			0
$451$	5.50000	+	9.52628i	0.258985	+	0.448575i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.00000	−	5.19615i	0.234404	−	0.243599i
$456$		0			0
$457$	−3.50000	+	6.06218i	−0.163723	+	0.283577i	−0.936201	−	0.351465i	$-0.885684\pi$
$457$	−3.50000	+	6.06218i	−0.163723	+	0.283577i	0.772478	+	0.635042i	$0.219017\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.50000	+	2.59808i	0.0698620	+	0.121004i	0.898840	−	0.438276i	$-0.144411\pi$
$461$	1.50000	+	2.59808i	0.0698620	+	0.121004i	−0.828978	+	0.559281i	$0.811077\pi$
$462$		0			0
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$		0			0
$469$	3.00000			0.138527
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.0000			−0.505781
$474$		0			0
$475$	−3.50000	−	6.06218i	−0.160591	−	0.278152i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.5000	−	23.3827i	0.616831	−	1.06838i	−0.373230	−	0.927739i	$-0.621750\pi$
$479$	13.5000	−	23.3827i	0.616831	−	1.06838i	0.990060	−	0.140643i	$-0.0449170\pi$
$480$		0			0
$481$	−3.50000	−	0.866025i	−0.159586	−	0.0394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.00000	−	1.73205i	−0.0454077	−	0.0786484i
$486$		0			0
$487$	−6.50000	−	11.2583i	−0.294543	−	0.510164i	0.680335	−	0.732901i	$-0.261834\pi$
$487$	−6.50000	−	11.2583i	−0.294543	−	0.510164i	−0.974879	+	0.222737i	$0.928501\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.50000	−	2.59808i	0.0676941	−	0.117250i	−0.830192	−	0.557478i	$-0.811769\pi$
$491$	1.50000	−	2.59808i	0.0676941	−	0.117250i	0.897886	+	0.440228i	$0.145102\pi$
$492$		0			0
$493$	9.00000			0.405340
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.50000	−	4.33013i	0.112140	−	0.194233i
$498$		0			0
$499$	−36.0000			−1.61158			−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000			−1.61158			−0.805791	+	0.592200i	$0.798259\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.5000	+	21.6506i	0.557347	+	0.965354i	0.997717	+	0.0675374i	$0.0215142\pi$
$503$	12.5000	+	21.6506i	0.557347	+	0.965354i	−0.440369	+	0.897817i	$0.645152\pi$
$504$		0			0
$505$	17.0000	−	29.4449i	0.756490	−	1.31028i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.500000	+	0.866025i	−0.0221621	+	0.0383859i	−0.876894	−	0.480684i	$-0.840388\pi$
$509$	−0.500000	+	0.866025i	−0.0221621	+	0.0383859i	0.854732	+	0.519070i	$0.173722\pi$
$510$		0			0
$511$	1.00000	+	1.73205i	0.0442374	+	0.0766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.0000			−0.705044
$516$		0			0
$517$	−6.00000	+	10.3923i	−0.263880	+	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$		0			0
$523$	−9.50000	+	16.4545i	−0.415406	+	0.719504i	−0.995471	−	0.0950659i	$-0.969694\pi$
$523$	−9.50000	+	16.4545i	−0.415406	+	0.719504i	0.580065	+	0.814570i	$0.303027\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000	−	20.7846i	−0.522728	−	0.905392i
$528$		0			0
$529$	11.0000	+	19.0526i	0.478261	+	0.828372i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−38.5000	−	9.52628i	−1.66762	−	0.412629i
$534$		0			0
$535$	17.0000	−	29.4449i	0.734974	−	1.27301i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.00000	+	5.19615i	0.129219	+	0.223814i
$540$		0			0
$541$	30.0000			1.28980			0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000			1.28980			0.644900	+	0.764267i	$0.276899\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.0000			−0.856706
$546$		0			0
$547$	−12.0000			−0.513083			−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000			−0.513083			−0.256541	+	0.966533i	$0.582583\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	21.0000			0.894630
$552$		0			0
$553$	−6.00000	−	10.3923i	−0.255146	−	0.441926i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.50000	−	16.4545i	0.402528	−	0.697199i	−0.591502	−	0.806303i	$-0.701465\pi$
$557$	9.50000	−	16.4545i	0.402528	−	0.697199i	0.994030	+	0.109104i	$0.0347983\pi$
$558$		0			0
$559$	27.5000	−	28.5788i	1.16313	−	1.20876i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.50000	+	7.79423i	0.189652	+	0.328488i	0.945134	−	0.326682i	$-0.105931\pi$
$563$	4.50000	+	7.79423i	0.189652	+	0.328488i	−0.755482	+	0.655169i	$0.772597\pi$
$564$		0			0
$565$	−11.0000	−	19.0526i	−0.462773	−	0.801547i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.50000	+	14.7224i	−0.356339	+	0.617196i	−0.987346	−	0.158580i	$-0.949308\pi$
$569$	−8.50000	+	14.7224i	−0.356339	+	0.617196i	0.631008	+	0.775777i	$0.282642\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.500000	−	0.866025i	0.0208514	−	0.0361158i
$576$		0			0
$577$	22.0000			0.915872			0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000			0.915872			0.457936	+	0.888985i	$0.348589\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.00000	+	3.46410i	0.0829740	+	0.143715i
$582$		0			0
$583$	−3.00000	+	5.19615i	−0.124247	+	0.215203i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.5000	+	18.1865i	−0.433381	+	0.750639i	−0.997162	−	0.0752860i	$-0.976013\pi$
$587$	−10.5000	+	18.1865i	−0.433381	+	0.750639i	0.563781	+	0.825925i	$0.309346\pi$
$588$		0			0
$589$	−28.0000	−	48.4974i	−1.15372	−	1.99830i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−14.0000			−0.574911			−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000			−0.574911			−0.287456	+	0.957794i	$0.592809\pi$
$594$		0			0
$595$	−3.00000	+	5.19615i	−0.122988	+	0.213021i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−44.0000			−1.79779			−0.898896	−	0.438163i	$-0.855629\pi$
$599$	−44.0000			−1.79779			−0.898896	+	0.438163i	$0.855629\pi$
$600$		0			0
$601$	10.5000	−	18.1865i	0.428304	−	0.741844i	−0.568419	−	0.822739i	$-0.692445\pi$
$601$	10.5000	−	18.1865i	0.428304	−	0.741844i	0.996723	+	0.0808953i	$0.0257779\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0000	−	17.3205i	−0.406558	−	0.704179i
$606$		0			0
$607$	−6.50000	−	11.2583i	−0.263827	−	0.456962i	0.703429	−	0.710766i	$-0.251651\pi$
$607$	−6.50000	−	11.2583i	−0.263827	−	0.456962i	−0.967256	+	0.253804i	$0.918318\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	−	41.5692i	−0.485468	−	1.68171i
$612$		0			0
$613$	−19.5000	+	33.7750i	−0.787598	+	1.36416i	0.139837	+	0.990174i	$0.455342\pi$
$613$	−19.5000	+	33.7750i	−0.787598	+	1.36416i	−0.927435	+	0.373985i	$0.877991\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	23.5000	+	40.7032i	0.946074	+	1.63865i	0.753586	+	0.657350i	$0.228323\pi$
$617$	23.5000	+	40.7032i	0.946074	+	1.63865i	0.192489	+	0.981299i	$0.438344\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.00000			0.0400642
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.00000			0.119618
$630$		0			0
$631$	13.5000	+	23.3827i	0.537427	+	0.930850i	0.999042	+	0.0437697i	$0.0139368\pi$
$631$	13.5000	+	23.3827i	0.537427	+	0.930850i	−0.461615	+	0.887080i	$0.652730\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.0000	+	22.5167i	−0.515889	+	0.893546i
$636$		0			0
$637$	−21.0000	−	5.19615i	−0.832050	−	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.50000	+	9.52628i	0.217237	+	0.376265i	0.953962	−	0.299927i	$-0.0969622\pi$
$641$	5.50000	+	9.52628i	0.217237	+	0.376265i	−0.736725	+	0.676192i	$0.763629\pi$
$642$		0			0
$643$	9.50000	+	16.4545i	0.374643	+	0.648901i	0.990274	−	0.139134i	$-0.0444318\pi$
$643$	9.50000	+	16.4545i	0.374643	+	0.648901i	−0.615630	+	0.788035i	$0.711098\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.50000	−	2.59808i	0.0589711	−	0.102141i	−0.835033	−	0.550200i	$-0.814551\pi$
$647$	1.50000	−	2.59808i	0.0589711	−	0.102141i	0.894004	+	0.448059i	$0.147885\pi$
$648$		0			0
$649$	−9.00000			−0.353281
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.50000	+	7.79423i	−0.176099	+	0.305012i	−0.940541	−	0.339680i	$-0.889681\pi$
$653$	−4.50000	+	7.79423i	−0.176099	+	0.305012i	0.764442	+	0.644692i	$0.223014\pi$
$654$		0			0
$655$	−24.0000			−0.937758
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.5000	+	25.1147i	0.564840	+	0.978331i	0.997065	+	0.0765653i	$0.0243954\pi$
$659$	14.5000	+	25.1147i	0.564840	+	0.978331i	−0.432225	+	0.901766i	$0.642271\pi$
$660$		0			0
$661$	12.5000	−	21.6506i	0.486194	−	0.842112i	−0.513680	−	0.857982i	$-0.671718\pi$
$661$	12.5000	−	21.6506i	0.486194	−	0.842112i	0.999874	+	0.0158695i	$0.00505163\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.00000	+	12.1244i	−0.271448	+	0.470162i
$666$		0			0
$667$	1.50000	+	2.59808i	0.0580802	+	0.100598i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.00000			−0.347441
$672$		0			0
$673$	−19.5000	+	33.7750i	−0.751670	+	1.30193i	0.195343	+	0.980735i	$0.437418\pi$
$673$	−19.5000	+	33.7750i	−0.751670	+	1.30193i	−0.947013	+	0.321195i	$0.895915\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$	0.500000	−	0.866025i	0.0191882	−	0.0332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.50000	+	7.79423i	0.172188	+	0.298238i	0.939184	−	0.343413i	$-0.111583\pi$
$683$	4.50000	+	7.79423i	0.172188	+	0.298238i	−0.766997	+	0.641651i	$0.778250\pi$
$684$		0			0
$685$	9.00000	+	15.5885i	0.343872	+	0.595604i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.00000	−	20.7846i	−0.228582	−	0.791831i
$690$		0			0
$691$	12.5000	−	21.6506i	0.475522	−	0.823629i	−0.524084	−	0.851666i	$-0.675592\pi$
$691$	12.5000	−	21.6506i	0.475522	−	0.823629i	0.999607	+	0.0280373i	$0.00892572\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.00000	+	15.5885i	0.341389	+	0.591304i
$696$		0			0
$697$	33.0000			1.24996
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		0			0
$703$	7.00000			0.264010
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.0000			0.639351
$708$		0			0
$709$	14.5000	+	25.1147i	0.544559	+	0.943204i	0.998635	+	0.0522406i	$0.0166363\pi$
$709$	14.5000	+	25.1147i	0.544559	+	0.943204i	−0.454076	+	0.890963i	$0.650030\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.00000	−	6.92820i	0.149801	−	0.259463i
$714$		0			0
$715$	−7.00000	−	1.73205i	−0.261785	−	0.0647750i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−19.5000	−	33.7750i	−0.727227	−	1.25959i	−0.958051	−	0.286599i	$-0.907475\pi$
$719$	−19.5000	−	33.7750i	−0.727227	−	1.25959i	0.230823	−	0.972996i	$-0.425858\pi$
$720$		0			0
$721$	−4.00000	−	6.92820i	−0.148968	−	0.258020i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.50000	+	2.59808i	−0.0557086	+	0.0964901i
$726$		0			0
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−16.5000	+	28.5788i	−0.610275	+	1.05703i
$732$		0			0
$733$	46.0000			1.69905			0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000			1.69905			0.849524	+	0.527549i	$0.176889\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.50000	−	2.59808i	−0.0552532	−	0.0957014i
$738$		0			0
$739$	−5.50000	+	9.52628i	−0.202321	+	0.350430i	−0.949276	−	0.314445i	$-0.898182\pi$
$739$	−5.50000	+	9.52628i	−0.202321	+	0.350430i	0.746955	+	0.664875i	$0.231515\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.5000	+	25.1147i	−0.531953	+	0.921370i	0.467351	+	0.884072i	$0.345209\pi$
$743$	−14.5000	+	25.1147i	−0.531953	+	0.921370i	−0.999304	+	0.0372984i	$0.988125\pi$
$744$		0			0
$745$	1.00000	+	1.73205i	0.0366372	+	0.0634574i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	17.0000			0.621166
$750$		0			0
$751$	8.50000	−	14.7224i	0.310169	−	0.537229i	−0.668229	−	0.743955i	$-0.732948\pi$
$751$	8.50000	−	14.7224i	0.310169	−	0.537229i	0.978399	+	0.206726i	$0.0662809\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	32.0000			1.16460
$756$		0			0
$757$	−23.5000	+	40.7032i	−0.854122	+	1.47938i	0.0233351	+	0.999728i	$0.492572\pi$
$757$	−23.5000	+	40.7032i	−0.854122	+	1.47938i	−0.877457	+	0.479655i	$0.840762\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.50000	+	2.59808i	0.0543750	+	0.0941802i	0.891932	−	0.452170i	$-0.149350\pi$
$761$	1.50000	+	2.59808i	0.0543750	+	0.0941802i	−0.837557	+	0.546350i	$0.816017\pi$
$762$		0			0
$763$	−5.00000	−	8.66025i	−0.181012	−	0.313522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22.5000	−	23.3827i	0.812428	−	0.844300i
$768$		0			0
$769$	2.50000	−	4.33013i	0.0901523	−	0.156148i	−0.817423	−	0.576038i	$-0.804598\pi$
$769$	2.50000	−	4.33013i	0.0901523	−	0.156148i	0.907575	+	0.419890i	$0.137931\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.50000	+	2.59808i	0.0539513	+	0.0934463i	0.891740	−	0.452549i	$-0.149485\pi$
$773$	1.50000	+	2.59808i	0.0539513	+	0.0934463i	−0.837788	+	0.545995i	$0.816152\pi$
$774$		0			0
$775$	8.00000			0.287368
$776$		0			0
$777$		0			0
$778$		0			0
$779$	77.0000			2.75881
$780$		0			0
$781$	−5.00000			−0.178914
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−20.0000			−0.713831
$786$		0			0
$787$	−10.5000	−	18.1865i	−0.374285	−	0.648280i	0.615935	−	0.787797i	$-0.288778\pi$
$787$	−10.5000	−	18.1865i	−0.374285	−	0.648280i	−0.990220	+	0.139517i	$0.955445\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.50000	−	9.52628i	0.195557	−	0.338716i
$792$		0			0
$793$	22.5000	−	23.3827i	0.798998	−	0.830344i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.5000	+	19.9186i	0.407351	+	0.705552i	0.994592	−	0.103860i	$-0.0331193\pi$
$797$	11.5000	+	19.9186i	0.407351	+	0.705552i	−0.587241	+	0.809412i	$0.699786\pi$
$798$		0			0
$799$	18.0000	+	31.1769i	0.636794	+	1.10296i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.00000	−	1.73205i	0.0352892	−	0.0611227i
$804$		0			0
$805$	−2.00000			−0.0704907
$806$		0			0
$807$		0			0
$808$		0			0
$809$	13.5000	−	23.3827i	0.474635	−	0.822091i	−0.524943	−	0.851137i	$-0.675914\pi$
$809$	13.5000	−	23.3827i	0.474635	−	0.822091i	0.999578	+	0.0290457i	$0.00924684\pi$
$810$		0			0
$811$	−32.0000			−1.12367			−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000			−1.12367			−0.561836	+	0.827249i	$0.689905\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.00000	+	8.66025i	0.175142	+	0.303355i
$816$		0			0
$817$	−38.5000	+	66.6840i	−1.34694	+	2.33298i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.50000	+	4.33013i	−0.0872506	+	0.151122i	−0.906348	−	0.422532i	$-0.861141\pi$
$821$	−2.50000	+	4.33013i	−0.0872506	+	0.151122i	0.819097	+	0.573654i	$0.194475\pi$
$822$		0			0
$823$	−14.5000	−	25.1147i	−0.505438	−	0.875445i	−0.999980	−	0.00629095i	$-0.997998\pi$
$823$	−14.5000	−	25.1147i	−0.505438	−	0.875445i	0.494542	−	0.869154i	$-0.335336\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.0000			0.695468			0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000			0.695468			0.347734	+	0.937593i	$0.386951\pi$
$828$		0			0
$829$	−1.50000	+	2.59808i	−0.0520972	+	0.0902349i	−0.890898	−	0.454204i	$-0.849924\pi$
$829$	−1.50000	+	2.59808i	−0.0520972	+	0.0902349i	0.838801	+	0.544438i	$0.183257\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	18.0000			0.623663
$834$		0			0
$835$	−7.00000	+	12.1244i	−0.242245	+	0.419581i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.5000	−	23.3827i	−0.466072	−	0.807260i	0.533177	−	0.846003i	$-0.320998\pi$
$839$	−13.5000	−	23.3827i	−0.466072	−	0.807260i	−0.999249	+	0.0387435i	$0.987664\pi$
$840$		0			0
$841$	10.0000	+	17.3205i	0.344828	+	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	22.0000	−	13.8564i	0.756823	−	0.476675i
$846$		0			0
$847$	5.00000	−	8.66025i	0.171802	−	0.297570i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.500000	+	0.866025i	0.0171398	+	0.0296870i
$852$		0			0
$853$	10.0000			0.342393			0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000			0.342393			0.171197	+	0.985237i	$0.445237\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.0000			0.614868			0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000			0.614868			0.307434	+	0.951569i	$0.400530\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16.0000			−0.544646			−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000			−0.544646			−0.272323	+	0.962206i	$0.587792\pi$
$864$		0			0
$865$	9.00000	+	15.5885i	0.306009	+	0.530023i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.00000	+	10.3923i	−0.203536	+	0.352535i
$870$		0			0
$871$	10.5000	+	2.59808i	0.355779	+	0.0880325i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.00000	−	10.3923i	−0.202837	−	0.351324i
$876$		0			0
$877$	0.500000	+	0.866025i	0.0168838	+	0.0292436i	0.874344	−	0.485307i	$-0.161292\pi$
$877$	0.500000	+	0.866025i	0.0168838	+	0.0292436i	−0.857460	+	0.514551i	$0.827959\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−22.5000	+	38.9711i	−0.758044	+	1.31297i	0.185802	+	0.982587i	$0.440512\pi$
$881$	−22.5000	+	38.9711i	−0.758044	+	1.31297i	−0.943847	+	0.330384i	$0.892822\pi$
$882$		0			0
$883$	12.0000			0.403832			0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000			0.403832			0.201916	+	0.979403i	$0.435283\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.5000	−	23.3827i	0.453286	−	0.785114i	−0.545302	−	0.838240i	$-0.683585\pi$
$887$	13.5000	−	23.3827i	0.453286	−	0.785114i	0.998588	+	0.0531258i	$0.0169184\pi$
$888$		0			0
$889$	−13.0000			−0.436006
$890$		0			0
$891$		0			0
$892$		0			0
$893$	42.0000	+	72.7461i	1.40548	+	2.43436i
$894$		0			0
$895$	9.00000	−	15.5885i	0.300837	−	0.521065i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.0000	+	20.7846i	−0.400222	+	0.693206i
$900$		0			0
$901$	9.00000	+	15.5885i	0.299833	+	0.519327i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	28.0000			0.930751
$906$		0			0
$907$	12.5000	−	21.6506i	0.415056	−	0.718898i	−0.580379	−	0.814347i	$-0.697095\pi$
$907$	12.5000	−	21.6506i	0.415056	−	0.718898i	0.995434	+	0.0954492i	$0.0304288\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$	2.00000	−	3.46410i	0.0661903	−	0.114645i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.00000	−	10.3923i	−0.198137	−	0.343184i
$918$		0			0
$919$	7.50000	+	12.9904i	0.247402	+	0.428513i	0.962804	−	0.270200i	$-0.0870898\pi$
$919$	7.50000	+	12.9904i	0.247402	+	0.428513i	−0.715402	+	0.698713i	$0.753756\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.5000	−	12.9904i	0.411443	−	0.427584i
$924$		0			0
$925$	−0.500000	+	0.866025i	−0.0164399	+	0.0284747i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.50000	−	4.33013i	−0.0820223	−	0.142067i	0.822096	−	0.569349i	$-0.192805\pi$
$929$	−2.50000	−	4.33013i	−0.0820223	−	0.142067i	−0.904118	+	0.427282i	$0.859471\pi$
$930$		0			0
$931$	42.0000			1.37649
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.00000			0.196221
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.0000			−0.456387			−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000			−0.456387			−0.228193	+	0.973616i	$0.573282\pi$
$942$		0			0
$943$	5.50000	+	9.52628i	0.179105	+	0.310218i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.5000	−	33.7750i	0.633665	−	1.09754i	−0.353131	−	0.935574i	$-0.614883\pi$
$947$	19.5000	−	33.7750i	0.633665	−	1.09754i	0.986796	−	0.161966i	$-0.0517835\pi$
$948$		0			0
$949$	2.00000	+	6.92820i	0.0649227	+	0.224899i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4.50000	−	7.79423i	−0.145769	−	0.252480i	0.783890	−	0.620899i	$-0.213232\pi$
$953$	−4.50000	−	7.79423i	−0.145769	−	0.252480i	−0.929660	+	0.368419i	$0.879899\pi$
$954$		0			0
$955$	7.00000	+	12.1244i	0.226515	+	0.392335i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.50000	+	7.79423i	−0.145313	+	0.251689i
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.00000	+	8.66025i	−0.160956	+	0.278783i
$966$		0			0
$967$	16.0000			0.514525			0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000			0.514525			0.257263	+	0.966342i	$0.417179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.5000	−	23.3827i	−0.433236	−	0.750386i	0.563914	−	0.825833i	$-0.309295\pi$
$971$	−13.5000	−	23.3827i	−0.433236	−	0.750386i	−0.997150	+	0.0754473i	$0.975962\pi$
$972$		0			0
$973$	−4.50000	+	7.79423i	−0.144263	+	0.249871i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.50000	+	7.79423i	−0.143968	+	0.249359i	−0.928987	−	0.370111i	$-0.879319\pi$
$977$	−4.50000	+	7.79423i	−0.143968	+	0.249359i	0.785020	+	0.619471i	$0.212653\pi$
$978$		0			0
$979$	−0.500000	−	0.866025i	−0.0159801	−	0.0276783i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	32.0000			1.02064			0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000			1.02064			0.510321	+	0.859984i	$0.329527\pi$
$984$		0			0
$985$	13.0000	−	22.5167i	0.414214	−	0.717440i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.0000			−0.349780
$990$		0			0
$991$	6.50000	−	11.2583i	0.206479	−	0.357633i	−0.744124	−	0.668042i	$-0.767133\pi$
$991$	6.50000	−	11.2583i	0.206479	−	0.357633i	0.950603	+	0.310409i	$0.100466\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.00000	−	5.19615i	−0.0951064	−	0.164729i
$996$		0			0
$997$	12.5000	+	21.6506i	0.395879	+	0.685682i	0.993213	−	0.116310i	$-0.0371066\pi$
$997$	12.5000	+	21.6506i	0.395879	+	0.685682i	−0.597334	+	0.801993i	$0.703773\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.t.d.1153.1		2
3.2	odd	2		208.2.i.c.113.1		2
4.3	odd	2		936.2.t.c.217.1		2
12.11	even	2		104.2.i.a.9.1	✓	2
13.3	even	3	inner	1872.2.t.d.289.1		2
24.5	odd	2		832.2.i.d.321.1		2
24.11	even	2		832.2.i.g.321.1		2
39.17	odd	6		2704.2.a.c.1.1		1
39.20	even	12		2704.2.f.c.337.2		2
39.29	odd	6		208.2.i.c.81.1		2
39.32	even	12		2704.2.f.c.337.1		2
39.35	odd	6		2704.2.a.e.1.1		1
52.3	odd	6		936.2.t.c.289.1		2
156.11	odd	12		1352.2.o.b.361.2		4
156.23	even	6		1352.2.i.a.1329.1		2
156.35	even	6		1352.2.a.c.1.1		1
156.47	odd	4		1352.2.o.b.1161.2		4
156.59	odd	12		1352.2.f.a.337.2		2
156.71	odd	12		1352.2.f.a.337.1		2
156.83	odd	4		1352.2.o.b.1161.1		4
156.95	even	6		1352.2.a.a.1.1		1
156.107	even	6		104.2.i.a.81.1	yes	2
156.119	odd	12		1352.2.o.b.361.1		4
156.155	even	2		1352.2.i.a.529.1		2
312.29	odd	6		832.2.i.d.705.1		2
312.107	even	6		832.2.i.g.705.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.i.a.9.1	✓	2	12.11	even	2
104.2.i.a.81.1	yes	2	156.107	even	6
208.2.i.c.81.1		2	39.29	odd	6
208.2.i.c.113.1		2	3.2	odd	2
832.2.i.d.321.1		2	24.5	odd	2
832.2.i.d.705.1		2	312.29	odd	6
832.2.i.g.321.1		2	24.11	even	2
832.2.i.g.705.1		2	312.107	even	6
936.2.t.c.217.1		2	4.3	odd	2
936.2.t.c.289.1		2	52.3	odd	6
1352.2.a.a.1.1		1	156.95	even	6
1352.2.a.c.1.1		1	156.35	even	6
1352.2.f.a.337.1		2	156.71	odd	12
1352.2.f.a.337.2		2	156.59	odd	12
1352.2.i.a.529.1		2	156.155	even	2
1352.2.i.a.1329.1		2	156.23	even	6
1352.2.o.b.361.1		4	156.119	odd	12
1352.2.o.b.361.2		4	156.11	odd	12
1352.2.o.b.1161.1		4	156.83	odd	4
1352.2.o.b.1161.2		4	156.47	odd	4
1872.2.t.d.289.1		2	13.3	even	3	inner
1872.2.t.d.1153.1		2	1.1	even	1	trivial
2704.2.a.c.1.1		1	39.17	odd	6
2704.2.a.e.1.1		1	39.35	odd	6
2704.2.f.c.337.1		2	39.32	even	12
2704.2.f.c.337.2		2	39.20	even	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.t (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +O(q^{10})\)	\(q-2.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{17} +(3.50000 + 6.06218i) q^{19} +(-0.500000 + 0.866025i) q^{23} -1.00000 q^{25} +(1.50000 - 2.59808i) q^{29} -8.00000 q^{31} +(1.00000 + 1.73205i) q^{35} +(0.500000 - 0.866025i) q^{37} +(5.50000 - 9.52628i) q^{41} +(5.50000 + 9.52628i) q^{43} +12.0000 q^{47} +(3.00000 - 5.19615i) q^{49} +6.00000 q^{53} +(1.00000 - 1.73205i) q^{55} +(4.50000 + 7.79423i) q^{59} +(4.50000 + 7.79423i) q^{61} +(2.00000 + 6.92820i) q^{65} +(-1.50000 + 2.59808i) q^{67} +(2.50000 + 4.33013i) q^{71} -2.00000 q^{73} +1.00000 q^{77} +12.0000 q^{79} -4.00000 q^{83} +(-3.00000 - 5.19615i) q^{85} +(-0.500000 + 0.866025i) q^{89} +(-2.50000 + 2.59808i) q^{91} +(-7.00000 - 12.1244i) q^{95} +(0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 - q^7	\( 2 q - 4 q^{5} - q^{7} - q^{11} - 2 q^{13} + 3 q^{17} + 7 q^{19} - q^{23} - 2 q^{25} + 3 q^{29} - 16 q^{31} + 2 q^{35} + q^{37} + 11 q^{41} + 11 q^{43} + 24 q^{47} + 6 q^{49} + 12 q^{53} + 2 q^{55} + 9 q^{59} + 9 q^{61} + 4 q^{65} - 3 q^{67} + 5 q^{71} - 4 q^{73} + 2 q^{77} + 24 q^{79} - 8 q^{83} - 6 q^{85} - q^{89} - 5 q^{91} - 14 q^{95} + q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - q^7 - q^11 - 2 * q^13 + 3 * q^17 + 7 * q^19 - q^23 - 2 * q^25 + 3 * q^29 - 16 * q^31 + 2 * q^35 + q^37 + 11 * q^41 + 11 * q^43 + 24 * q^47 + 6 * q^49 + 12 * q^53 + 2 * q^55 + 9 * q^59 + 9 * q^61 + 4 * q^65 - 3 * q^67 + 5 * q^71 - 4 * q^73 + 2 * q^77 + 24 * q^79 - 8 * q^83 - 6 * q^85 - q^89 - 5 * q^91 - 14 * q^95 + q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more