LMFDB - Embedded newform 3240.2.q.r.2161.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3240,2,Mod(1081,3240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3240.1081"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3240, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1,0,0,0,0,0,-1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$25.8715302549$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2161.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.2161
Dual form			3240.2.q.r.1081.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{11} -7.00000 q^{19} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(3.50000 - 6.06218i) q^{29} +(-0.500000 - 0.866025i) q^{31} -2.00000 q^{37} +(-4.50000 - 7.79423i) q^{41} +(3.00000 - 5.19615i) q^{43} +(1.00000 - 1.73205i) q^{47} +(3.50000 + 6.06218i) q^{49} -1.00000 q^{55} +(-1.50000 - 2.59808i) q^{59} +(5.00000 - 8.66025i) q^{61} +(1.00000 + 1.73205i) q^{67} +1.00000 q^{71} +(-2.00000 + 3.46410i) q^{79} +(3.00000 - 5.19615i) q^{83} -7.00000 q^{89} +(-3.50000 - 6.06218i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - q^{11} - 14 q^{19} - 6 q^{23} - q^{25} + 7 q^{29} - q^{31} - 4 q^{37} - 9 q^{41} + 6 q^{43} + 2 q^{47} + 7 q^{49} - 2 q^{55} - 3 q^{59} + 10 q^{61} + 2 q^{67} + 2 q^{71} - 4 q^{79} + 6 q^{83}+ \cdots - 2 q^{97}+O(q^{100}) $ 2 * q + q^5 - q^11 - 14 * q^19 - 6 * q^23 - q^25 + 7 * q^29 - q^31 - 4 * q^37 - 9 * q^41 + 6 * q^43 + 2 * q^47 + 7 * q^49 - 2 * q^55 - 3 * q^59 + 10 * q^61 + 2 * q^67 + 2 * q^71 - 4 * q^79 + 6 * q^83 - 14 * q^89 - 7 * q^95 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$		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
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−7.00000			−1.60591			−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000			−1.60591			−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.50000	−	6.06218i	0.649934	−	1.12572i	−0.333205	−	0.942855i	$-0.608130\pi$
$29$	3.50000	−	6.06218i	0.649934	−	1.12572i	0.983138	−	0.182864i	$-0.0585367\pi$
$30$		0			0
$31$	−0.500000	−	0.866025i	−0.0898027	−	0.155543i	0.817625	−	0.575751i	$-0.195290\pi$
$31$	−0.500000	−	0.866025i	−0.0898027	−	0.155543i	−0.907428	+	0.420208i	$0.861957\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.50000	−	7.79423i	−0.702782	−	1.21725i	−0.967486	−	0.252924i	$-0.918608\pi$
$41$	−4.50000	−	7.79423i	−0.702782	−	1.21725i	0.264704	−	0.964330i	$-0.414726\pi$
$42$		0			0
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	−0.541332	−	0.840809i	$-0.682080\pi$
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	0.998828	+	0.0484030i	$0.0154132\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	−0.783830	−	0.620975i	$-0.786737\pi$
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	0.929695	+	0.368329i	$0.120070\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−1.00000			−0.134840
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	$-0.229229\pi$
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	−0.946993	+	0.321253i	$0.895896\pi$
$60$		0			0
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	−0.345207	−	0.938527i	$-0.612191\pi$
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	0.985391	−	0.170305i	$-0.0544754\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.00000	+	1.73205i	0.122169	+	0.211604i	0.920623	−	0.390453i	$-0.127682\pi$
$67$	1.00000	+	1.73205i	0.122169	+	0.211604i	−0.798454	+	0.602056i	$0.794348\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.00000			0.118678			0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000			0.118678			0.0593391	+	0.998238i	$0.481101\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	0.731307	+	0.682048i	$0.238911\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	−0.653079	−	0.757290i	$-0.726523\pi$
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	0.982372	+	0.186938i	$0.0598564\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.00000			−0.741999			−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000			−0.741999			−0.370999	+	0.928633i	$0.620985\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.50000	−	6.06218i	−0.359092	−	0.621966i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.50000	−	7.79423i	0.447767	−	0.775555i	−0.550474	−	0.834853i	$-0.685553\pi$
$101$	4.50000	−	7.79423i	0.447767	−	0.775555i	0.998240	+	0.0592978i	$0.0188862\pi$
$102$		0			0
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	0.975124	−	0.221660i	$-0.0711475\pi$
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	−0.679525	+	0.733652i	$0.737814\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.00000			−0.193347			−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000			−0.193347			−0.0966736	+	0.995316i	$0.530820\pi$
$108$		0			0
$109$	3.00000			0.287348			0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000			0.287348			0.143674	+	0.989625i	$0.454108\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$	3.00000	−	5.19615i	0.279751	−	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000	+	8.66025i	0.454545	+	0.787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−20.0000			−1.77471			−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000			−1.77471			−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.50000	−	9.52628i	−0.480537	−	0.832315i	0.519213	−	0.854645i	$-0.326225\pi$
$131$	−5.50000	−	9.52628i	−0.480537	−	0.832315i	−0.999751	+	0.0223297i	$0.992892\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.0000	−	17.3205i	0.854358	−	1.47979i	−0.0228820	−	0.999738i	$-0.507284\pi$
$137$	10.0000	−	17.3205i	0.854358	−	1.47979i	0.877240	−	0.480053i	$-0.159382\pi$
$138$		0			0
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	0.678551	−	0.734553i	$-0.262608\pi$
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	7.00000			0.581318
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	0.585231	−	0.810867i	$-0.301004\pi$
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	−0.994847	+	0.101391i	$0.967671\pi$
$150$		0			0
$151$	5.50000	−	9.52628i	0.447584	−	0.775238i	−0.550645	−	0.834740i	$-0.685618\pi$
$151$	5.50000	−	9.52628i	0.447584	−	0.775238i	0.998228	+	0.0595022i	$0.0189513\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.500000	−	0.866025i	0.0401610	−	0.0695608i
$156$		0			0
$157$	9.00000	+	15.5885i	0.718278	+	1.24409i	0.961681	+	0.274169i	$0.0884028\pi$
$157$	9.00000	+	15.5885i	0.718278	+	1.24409i	−0.243403	+	0.969925i	$0.578264\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−20.0000			−1.56652			−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000			−1.56652			−0.783260	+	0.621694i	$0.786445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.0000	−	17.3205i	−0.773823	−	1.34030i	−0.935454	−	0.353450i	$-0.885009\pi$
$167$	−10.0000	−	17.3205i	−0.773823	−	1.34030i	0.161630	−	0.986851i	$-0.448325\pi$
$168$		0			0
$169$	6.50000	−	11.2583i	0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.00000	−	12.1244i	0.532200	−	0.921798i	−0.467093	−	0.884208i	$-0.654699\pi$
$173$	7.00000	−	12.1244i	0.532200	−	0.921798i	0.999293	−	0.0375896i	$-0.0119679\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.00000			0.224231			0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000			0.224231			0.112115	+	0.993695i	$0.464237\pi$
$180$		0			0
$181$	11.0000			0.817624			0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000			0.817624			0.408812	+	0.912619i	$0.365943\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.00000	−	1.73205i	−0.0735215	−	0.127343i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.50000	+	7.79423i	−0.325609	+	0.563971i	−0.981635	−	0.190767i	$-0.938902\pi$
$191$	−4.50000	+	7.79423i	−0.325609	+	0.563971i	0.656027	+	0.754738i	$0.272236\pi$
$192$		0			0
$193$	7.00000	+	12.1244i	0.503871	+	0.872730i	0.999990	+	0.00447566i	$0.00142465\pi$
$193$	7.00000	+	12.1244i	0.503871	+	0.872730i	−0.496119	+	0.868255i	$0.665242\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−20.0000			−1.42494			−0.712470	−	0.701702i	$-0.752424\pi$
$197$	−20.0000			−1.42494			−0.712470	+	0.701702i	$0.752424\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.50000	−	7.79423i	0.314294	−	0.544373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.50000	−	6.06218i	0.242100	−	0.419330i
$210$		0			0
$211$	−4.50000	−	7.79423i	−0.309793	−	0.536577i	0.668524	−	0.743690i	$-0.266926\pi$
$211$	−4.50000	−	7.79423i	−0.309793	−	0.536577i	−0.978317	+	0.207114i	$0.933593\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.00000			0.409197
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−10.0000	+	17.3205i	−0.669650	+	1.15987i	0.308353	+	0.951272i	$0.400222\pi$
$223$	−10.0000	+	17.3205i	−0.669650	+	1.15987i	−0.978002	+	0.208595i	$0.933111\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.0000	−	17.3205i	0.663723	−	1.14960i	−0.315906	−	0.948790i	$-0.602309\pi$
$227$	10.0000	−	17.3205i	0.663723	−	1.14960i	0.979630	−	0.200812i	$-0.0643581\pi$
$228$		0			0
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	0.982592	−	0.185776i	$-0.0594799\pi$
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	−0.652183	+	0.758062i	$0.726147\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.0000			−0.917170			−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$		0			0
$235$	2.00000			0.130466
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	−0.632709	−	0.774389i	$-0.718057\pi$
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	0.986996	+	0.160748i	$0.0513906\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.50000	+	6.06218i	−0.223607	+	0.387298i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$	6.00000			0.377217
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.0000	+	19.0526i	0.686161	+	1.18847i	0.973070	+	0.230508i	$0.0740389\pi$
$257$	11.0000	+	19.0526i	0.686161	+	1.18847i	−0.286909	+	0.957958i	$0.592628\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	−0.943572	−	0.331166i	$-0.892558\pi$
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	0.758585	+	0.651575i	$0.225891\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.00000			0.182913			0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000			0.182913			0.0914566	+	0.995809i	$0.470848\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.500000	−	0.866025i	−0.0301511	−	0.0522233i
$276$		0			0
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	0.319447	+	0.947604i	$0.396503\pi$
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	−0.980373	+	0.197153i	$0.936830\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.00000	+	8.66025i	−0.298275	+	0.516627i	−0.975741	−	0.218926i	$-0.929745\pi$
$281$	−5.00000	+	8.66025i	−0.298275	+	0.516627i	0.677466	+	0.735554i	$0.263078\pi$
$282$		0			0
$283$	−13.0000	−	22.5167i	−0.772770	−	1.33848i	−0.936039	−	0.351895i	$-0.885537\pi$
$283$	−13.0000	−	22.5167i	−0.772770	−	1.33848i	0.163270	−	0.986581i	$-0.447796\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.00000	−	12.1244i	−0.408944	−	0.708312i	0.585827	−	0.810436i	$-0.300770\pi$
$293$	−7.00000	−	12.1244i	−0.408944	−	0.708312i	−0.994772	+	0.102123i	$0.967436\pi$
$294$		0			0
$295$	1.50000	−	2.59808i	0.0873334	−	0.151266i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.0000			0.572598
$306$		0			0
$307$	−22.0000			−1.25561			−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000			−1.25561			−0.627803	+	0.778372i	$0.716046\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.5000	−	26.8468i	−0.878924	−	1.52234i	−0.852523	−	0.522690i	$-0.824928\pi$
$311$	−15.5000	−	26.8468i	−0.878924	−	1.52234i	−0.0264017	−	0.999651i	$-0.508405\pi$
$312$		0			0
$313$	10.0000	−	17.3205i	0.565233	−	0.979013i	−0.431795	−	0.901972i	$-0.642119\pi$
$313$	10.0000	−	17.3205i	0.565233	−	0.979013i	0.997028	−	0.0770410i	$-0.0245472\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.0000	+	19.0526i	−0.617822	+	1.07010i	0.372061	+	0.928208i	$0.378651\pi$
$317$	−11.0000	+	19.0526i	−0.617822	+	1.07010i	−0.989882	+	0.141890i	$0.954682\pi$
$318$		0			0
$319$	3.50000	+	6.06218i	0.195962	+	0.339417i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	15.5000	−	26.8468i	0.851957	−	1.47563i	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	15.5000	−	26.8468i	0.851957	−	1.47563i	0.879440	−	0.476011i	$-0.157918\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.00000	+	1.73205i	−0.0546358	+	0.0946320i
$336$		0			0
$337$	−10.0000	−	17.3205i	−0.544735	−	0.943508i	−0.998624	−	0.0524499i	$-0.983297\pi$
$337$	−10.0000	−	17.3205i	−0.544735	−	0.943508i	0.453889	−	0.891058i	$-0.350036\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.00000			0.0541530
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	−8.50000	+	14.7224i	−0.454995	+	0.788074i	−0.998688	−	0.0512103i	$-0.983692\pi$
$349$	−8.50000	+	14.7224i	−0.454995	+	0.788074i	0.543693	+	0.839284i	$0.317025\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	−0.617388	−	0.786659i	$-0.711809\pi$
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	0.989960	+	0.141344i	$0.0451425\pi$
$354$		0			0
$355$	0.500000	+	0.866025i	0.0265372	+	0.0459639i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.00000			−0.158334			−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000			−0.158334			−0.0791670	+	0.996861i	$0.525226\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−11.0000	+	19.0526i	−0.574195	+	0.994535i	0.421933	+	0.906627i	$0.361352\pi$
$367$	−11.0000	+	19.0526i	−0.574195	+	0.994535i	−0.996129	+	0.0879086i	$0.971982\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	10.0000	+	17.3205i	0.517780	+	0.896822i	0.999787	+	0.0206542i	$0.00657489\pi$
$373$	10.0000	+	17.3205i	0.517780	+	0.896822i	−0.482006	+	0.876168i	$0.660092\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	15.0000	−	25.9808i	0.760530	−	1.31728i	−0.182047	−	0.983290i	$-0.558272\pi$
$389$	15.0000	−	25.9808i	0.760530	−	1.31728i	0.942578	−	0.333987i	$-0.108394\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.00000			−0.201262
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.00000	−	1.73205i	−0.0499376	−	0.0864945i	0.839976	−	0.542623i	$-0.182569\pi$
$401$	−1.00000	−	1.73205i	−0.0499376	−	0.0864945i	−0.889914	+	0.456129i	$0.849236\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.00000	−	1.73205i	0.0495682	−	0.0858546i
$408$		0			0
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	0.962757	−	0.270367i	$-0.0871450\pi$
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	−0.715523	+	0.698589i	$0.753812\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	6.00000			0.294528
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000	+	20.7846i	0.586238	+	1.01539i	0.994720	+	0.102628i	$0.0327251\pi$
$419$	12.0000	+	20.7846i	0.586238	+	1.01539i	−0.408481	+	0.912767i	$0.633942\pi$
$420$		0			0
$421$	15.5000	−	26.8468i	0.755424	−	1.30843i	−0.189740	−	0.981834i	$-0.560764\pi$
$421$	15.5000	−	26.8468i	0.755424	−	1.30843i	0.945163	−	0.326598i	$-0.105902\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	19.0000			0.915198			0.457599	−	0.889159i	$-0.348710\pi$
$431$	19.0000			0.915198			0.457599	+	0.889159i	$0.348710\pi$
$432$		0			0
$433$	6.00000			0.288342			0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000			0.288342			0.144171	+	0.989553i	$0.453949\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	21.0000	+	36.3731i	1.00457	+	1.73996i
$438$		0			0
$439$	−18.5000	+	32.0429i	−0.882957	+	1.52933i	−0.0349192	+	0.999390i	$0.511117\pi$
$439$	−18.5000	+	32.0429i	−0.882957	+	1.52933i	−0.848038	+	0.529936i	$0.822216\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.0000	+	29.4449i	−0.807694	+	1.39897i	0.106763	+	0.994285i	$0.465952\pi$
$443$	−17.0000	+	29.4449i	−0.807694	+	1.39897i	−0.914457	+	0.404683i	$0.867382\pi$
$444$		0			0
$445$	−3.50000	−	6.06218i	−0.165916	−	0.287375i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.0000			−1.55737			−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000			−1.55737			−0.778683	+	0.627417i	$0.784112\pi$
$450$		0			0
$451$	9.00000			0.423793
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.0474665	+	0.998873i	$0.515115\pi$
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.841316	+	0.540544i	$0.818219\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.50000	+	16.4545i	−0.442459	+	0.766362i	−0.997871	−	0.0652135i	$-0.979227\pi$
$461$	−9.50000	+	16.4545i	−0.442459	+	0.766362i	0.555412	+	0.831575i	$0.312560\pi$
$462$		0			0
$463$	−7.00000	−	12.1244i	−0.325318	−	0.563467i	0.656259	−	0.754536i	$-0.272138\pi$
$463$	−7.00000	−	12.1244i	−0.325318	−	0.563467i	−0.981577	+	0.191069i	$0.938805\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.0000			−0.925490			−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000			−0.925490			−0.462745	+	0.886492i	$0.653135\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.00000	+	5.19615i	0.137940	+	0.238919i
$474$		0			0
$475$	3.50000	−	6.06218i	0.160591	−	0.278152i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.50000	+	14.7224i	−0.388375	+	0.672685i	−0.992231	−	0.124408i	$-0.960297\pi$
$479$	−8.50000	+	14.7224i	−0.388375	+	0.672685i	0.603856	+	0.797093i	$0.293630\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.00000			−0.0908153
$486$		0			0
$487$	−20.0000			−0.906287			−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000			−0.906287			−0.453143	+	0.891438i	$0.649697\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.50000	+	7.79423i	0.203082	+	0.351749i	0.949520	−	0.313707i	$-0.101571\pi$
$491$	4.50000	+	7.79423i	0.203082	+	0.351749i	−0.746438	+	0.665455i	$0.768237\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−18.5000	−	32.0429i	−0.828174	−	1.43444i	−0.899469	−	0.436984i	$-0.856047\pi$
$499$	−18.5000	−	32.0429i	−0.828174	−	1.43444i	0.0712957	−	0.997455i	$-0.477287\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	40.0000			1.78351			0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000			1.78351			0.891756	+	0.452517i	$0.149474\pi$
$504$		0			0
$505$	9.00000			0.400495
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.00000	+	8.66025i	0.221621	+	0.383859i	0.955300	−	0.295637i	$-0.0955319\pi$
$509$	5.00000	+	8.66025i	0.221621	+	0.383859i	−0.733679	+	0.679496i	$0.762199\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.00000	+	5.19615i	−0.132196	+	0.228970i
$516$		0			0
$517$	1.00000	+	1.73205i	0.0439799	+	0.0761755i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	30.0000			1.31432			0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000			1.31432			0.657162	+	0.753749i	$0.271757\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.00000	−	1.73205i	−0.0432338	−	0.0748831i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.00000			−0.301511
$540$		0			0
$541$	39.0000			1.67674			0.838370	−	0.545101i	$-0.183509\pi$
$541$	39.0000			1.67674			0.838370	+	0.545101i	$0.183509\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.50000	+	2.59808i	0.0642529	+	0.111289i
$546$		0			0
$547$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$547$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−24.5000	+	42.4352i	−1.04374	+	1.80780i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	17.0000	+	29.4449i	0.716465	+	1.24095i	0.962392	+	0.271665i	$0.0875742\pi$
$563$	17.0000	+	29.4449i	0.716465	+	1.24095i	−0.245927	+	0.969288i	$0.579092\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.50000	+	2.59808i	−0.0628833	+	0.108917i	−0.895753	−	0.444552i	$-0.853363\pi$
$569$	−1.50000	+	2.59808i	−0.0628833	+	0.108917i	0.832870	+	0.553469i	$0.186696\pi$
$570$		0			0
$571$	−19.5000	−	33.7750i	−0.816050	−	1.41344i	−0.908572	−	0.417729i	$-0.862826\pi$
$571$	−19.5000	−	33.7750i	−0.816050	−	1.41344i	0.0925222	−	0.995711i	$-0.470507\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.00000			0.250217
$576$		0			0
$577$	20.0000			0.832611			0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000			0.832611			0.416305	+	0.909225i	$0.363325\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.0000	−	32.9090i	0.784214	−	1.35830i	−0.145254	−	0.989394i	$-0.546400\pi$
$587$	19.0000	−	32.9090i	0.784214	−	1.35830i	0.929468	−	0.368904i	$-0.120267\pi$
$588$		0			0
$589$	3.50000	+	6.06218i	0.144215	+	0.249788i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.50000	+	14.7224i	0.347301	+	0.601542i	0.985769	−	0.168106i	$-0.0537650\pi$
$599$	8.50000	+	14.7224i	0.347301	+	0.601542i	−0.638468	+	0.769648i	$0.720432\pi$
$600$		0			0
$601$	9.50000	−	16.4545i	0.387513	−	0.671192i	−0.604601	−	0.796528i	$-0.706668\pi$
$601$	9.50000	−	16.4545i	0.387513	−	0.671192i	0.992114	+	0.125336i	$0.0400009\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.00000	+	8.66025i	−0.203279	+	0.352089i
$606$		0			0
$607$	19.0000	+	32.9090i	0.771186	+	1.33573i	0.936913	+	0.349562i	$0.113670\pi$
$607$	19.0000	+	32.9090i	0.771186	+	1.33573i	−0.165727	+	0.986172i	$0.552997\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−20.0000			−0.807792			−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000			−0.807792			−0.403896	+	0.914805i	$0.632344\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	10.0000	+	17.3205i	0.402585	+	0.697297i	0.994037	−	0.109043i	$-0.0347785\pi$
$617$	10.0000	+	17.3205i	0.402585	+	0.697297i	−0.591452	+	0.806340i	$0.701445\pi$
$618$		0			0
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	−0.993959	−	0.109749i	$-0.964995\pi$
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	0.592025	+	0.805919i	$0.298329\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	41.0000			1.63218			0.816092	−	0.577922i	$-0.196136\pi$
$631$	41.0000			1.63218			0.816092	+	0.577922i	$0.196136\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.0000	−	17.3205i	−0.396838	−	0.687343i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.5000	+	18.1865i	−0.414725	+	0.718325i	−0.995400	−	0.0958109i	$-0.969456\pi$
$641$	−10.5000	+	18.1865i	−0.414725	+	0.718325i	0.580674	+	0.814136i	$0.302789\pi$
$642$		0			0
$643$	7.00000	+	12.1244i	0.276053	+	0.478138i	0.970400	−	0.241502i	$-0.0776401\pi$
$643$	7.00000	+	12.1244i	0.276053	+	0.478138i	−0.694347	+	0.719640i	$0.744307\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.0000			−0.707653			−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000			−0.707653			−0.353827	+	0.935311i	$0.615120\pi$
$648$		0			0
$649$	3.00000			0.117760
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.0000	+	17.3205i	0.391330	+	0.677804i	0.992625	−	0.121223i	$-0.0386817\pi$
$653$	10.0000	+	17.3205i	0.391330	+	0.677804i	−0.601295	+	0.799027i	$0.705348\pi$
$654$		0			0
$655$	5.50000	−	9.52628i	0.214903	−	0.372223i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	−0.992388	−	0.123148i	$-0.960701\pi$
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	0.602844	+	0.797859i	$0.294034\pi$
$660$		0			0
$661$	−10.5000	−	18.1865i	−0.408403	−	0.707374i	0.586308	−	0.810088i	$-0.300581\pi$
$661$	−10.5000	−	18.1865i	−0.408403	−	0.707374i	−0.994711	+	0.102714i	$0.967247\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−42.0000			−1.62625
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.00000	+	8.66025i	0.193023	+	0.334325i
$672$		0			0
$673$	−23.0000	+	39.8372i	−0.886585	+	1.53561i	−0.0426985	+	0.999088i	$0.513595\pi$
$673$	−23.0000	+	39.8372i	−0.886585	+	1.53561i	−0.843886	+	0.536522i	$0.819738\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.00000	−	1.73205i	0.0384331	−	0.0665681i	−0.846169	−	0.532915i	$-0.821097\pi$
$677$	1.00000	−	1.73205i	0.0384331	−	0.0665681i	0.884602	+	0.466347i	$0.154430\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−40.0000			−1.53056			−0.765279	−	0.643699i	$-0.777399\pi$
$683$	−40.0000			−1.53056			−0.765279	+	0.643699i	$0.777399\pi$
$684$		0			0
$685$	20.0000			0.764161
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	−0.610704	−	0.791859i	$-0.709113\pi$
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	0.991122	+	0.132956i	$0.0424468\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.50000	−	6.06218i	0.132763	−	0.229952i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.00000			0.339925			0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000			0.339925			0.169963	+	0.985451i	$0.445635\pi$
$702$		0			0
$703$	14.0000			0.528020
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	−0.756730	−	0.653727i	$-0.773204\pi$
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	0.944509	+	0.328484i	$0.106538\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.00000	+	5.19615i	−0.112351	+	0.194597i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	27.0000			1.00693			0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000			1.00693			0.503465	+	0.864016i	$0.332058\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.50000	+	6.06218i	0.129987	+	0.225144i
$726$		0			0
$727$	−20.0000	+	34.6410i	−0.741759	+	1.28476i	0.209935	+	0.977715i	$0.432675\pi$
$727$	−20.0000	+	34.6410i	−0.741759	+	1.28476i	−0.951694	+	0.307049i	$0.900659\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−23.0000	−	39.8372i	−0.849524	−	1.47142i	−0.881633	−	0.471935i	$-0.843556\pi$
$733$	−23.0000	−	39.8372i	−0.849524	−	1.47142i	0.0321090	−	0.999484i	$-0.489778\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.00000			−0.0736709
$738$		0			0
$739$	47.0000			1.72892			0.864461	−	0.502699i	$-0.167660\pi$
$739$	47.0000			1.72892			0.864461	+	0.502699i	$0.167660\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	10.0000	+	17.3205i	0.366864	+	0.635428i	0.989073	−	0.147423i	$-0.0470980\pi$
$743$	10.0000	+	17.3205i	0.366864	+	0.635428i	−0.622209	+	0.782851i	$0.713765\pi$
$744$		0			0
$745$	5.00000	−	8.66025i	0.183186	−	0.317287i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	0.956963	+	0.290209i	$0.0937250\pi$
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	−0.227153	+	0.973859i	$0.572942\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.0000			0.400331
$756$		0			0
$757$	−20.0000			−0.726912			−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000			−0.726912			−0.363456	+	0.931611i	$0.618403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.50000	−	16.4545i	−0.344375	−	0.596475i	0.640865	−	0.767653i	$-0.278576\pi$
$761$	−9.50000	−	16.4545i	−0.344375	−	0.596475i	−0.985240	+	0.171179i	$0.945242\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.50000	+	2.59808i	0.0540914	+	0.0936890i	0.891803	−	0.452423i	$-0.149440\pi$
$769$	1.50000	+	2.59808i	0.0540914	+	0.0936890i	−0.837712	+	0.546113i	$0.816107\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$	1.00000			0.0359211
$776$		0			0
$777$		0			0
$778$		0			0
$779$	31.5000	+	54.5596i	1.12860	+	1.95480i
$780$		0			0
$781$	−0.500000	+	0.866025i	−0.0178914	+	0.0309888i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.00000	+	15.5885i	−0.321224	+	0.556376i
$786$		0			0
$787$	−20.0000	−	34.6410i	−0.712923	−	1.23482i	−0.963755	−	0.266788i	$-0.914038\pi$
$787$	−20.0000	−	34.6410i	−0.712923	−	1.23482i	0.250832	−	0.968031i	$-0.419296\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	20.0000	+	34.6410i	0.708436	+	1.22705i	0.965437	+	0.260637i	$0.0839324\pi$
$797$	20.0000	+	34.6410i	0.708436	+	1.22705i	−0.257001	+	0.966411i	$0.582734\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	33.0000			1.16022			0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000			1.16022			0.580109	+	0.814539i	$0.303010\pi$
$810$		0			0
$811$	41.0000			1.43970			0.719852	−	0.694127i	$-0.244209\pi$
$811$	41.0000			1.43970			0.719852	+	0.694127i	$0.244209\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.0000	−	17.3205i	−0.350285	−	0.606711i
$816$		0			0
$817$	−21.0000	+	36.3731i	−0.734697	+	1.27253i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.500000	+	0.866025i	−0.0174501	+	0.0302245i	−0.874619	−	0.484812i	$-0.838888\pi$
$821$	−0.500000	+	0.866025i	−0.0174501	+	0.0302245i	0.857168	+	0.515036i	$0.172221\pi$
$822$		0			0
$823$	3.00000	+	5.19615i	0.104573	+	0.181126i	0.913564	−	0.406695i	$-0.133319\pi$
$823$	3.00000	+	5.19615i	0.104573	+	0.181126i	−0.808990	+	0.587822i	$0.799986\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−17.0000			−0.590434			−0.295217	−	0.955430i	$-0.595392\pi$
$829$	−17.0000			−0.590434			−0.295217	+	0.955430i	$0.595392\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	10.0000	−	17.3205i	0.346064	−	0.599401i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−26.5000	+	45.8993i	−0.914882	+	1.58462i	−0.107807	+	0.994172i	$0.534383\pi$
$839$	−26.5000	+	45.8993i	−0.914882	+	1.58462i	−0.807075	+	0.590450i	$0.798950\pi$
$840$		0			0
$841$	−10.0000	−	17.3205i	−0.344828	−	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.0000			0.447214
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.00000	+	10.3923i	0.205677	+	0.356244i
$852$		0			0
$853$	−10.0000	+	17.3205i	−0.342393	+	0.593043i	−0.984877	−	0.173257i	$-0.944571\pi$
$853$	−10.0000	+	17.3205i	−0.342393	+	0.593043i	0.642483	+	0.766300i	$0.277904\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.0000	+	34.6410i	−0.683187	+	1.18331i	0.290816	+	0.956779i	$0.406073\pi$
$857$	−20.0000	+	34.6410i	−0.683187	+	1.18331i	−0.974003	+	0.226536i	$0.927260\pi$
$858$		0			0
$859$	−6.50000	−	11.2583i	−0.221777	−	0.384129i	0.733571	−	0.679613i	$-0.237852\pi$
$859$	−6.50000	−	11.2583i	−0.221777	−	0.384129i	−0.955348	+	0.295484i	$0.904519\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−20.0000			−0.680808			−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000			−0.680808			−0.340404	+	0.940279i	$0.610564\pi$
$864$		0			0
$865$	14.0000			0.476014
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.00000	−	3.46410i	−0.0678454	−	0.117512i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−20.0000	−	34.6410i	−0.675352	−	1.16974i	−0.976366	−	0.216124i	$-0.930658\pi$
$877$	−20.0000	−	34.6410i	−0.675352	−	1.16974i	0.301014	−	0.953620i	$-0.402675\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.00000			0.0336909			0.0168454	−	0.999858i	$-0.494638\pi$
$881$	1.00000			0.0336909			0.0168454	+	0.999858i	$0.494638\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.00000	+	15.5885i	0.302190	+	0.523409i	0.976632	−	0.214919i	$-0.0689488\pi$
$887$	9.00000	+	15.5885i	0.302190	+	0.523409i	−0.674441	+	0.738328i	$0.735615\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.00000	+	12.1244i	−0.234246	+	0.405726i
$894$		0			0
$895$	1.50000	+	2.59808i	0.0501395	+	0.0868441i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7.00000			−0.233463
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.50000	+	9.52628i	0.182826	+	0.316664i
$906$		0			0
$907$	1.00000	−	1.73205i	0.0332045	−	0.0575118i	−0.848946	−	0.528480i	$-0.822762\pi$
$907$	1.00000	−	1.73205i	0.0332045	−	0.0575118i	0.882150	+	0.470968i	$0.156095\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19.5000	+	33.7750i	−0.646064	+	1.11902i	0.337991	+	0.941149i	$0.390253\pi$
$911$	−19.5000	+	33.7750i	−0.646064	+	1.11902i	−0.984055	+	0.177866i	$0.943081\pi$
$912$		0			0
$913$	3.00000	+	5.19615i	0.0992855	+	0.171968i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−13.0000			−0.428830			−0.214415	−	0.976743i	$-0.568785\pi$
$919$	−13.0000			−0.428830			−0.214415	+	0.976743i	$0.568785\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.00000	−	1.73205i	0.0328798	−	0.0569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16.5000	−	28.5788i	0.541347	−	0.937641i	−0.457480	−	0.889220i	$-0.651248\pi$
$929$	16.5000	−	28.5788i	0.541347	−	0.937641i	0.998827	−	0.0484211i	$-0.0154190\pi$
$930$		0			0
$931$	−24.5000	−	42.4352i	−0.802955	−	1.39076i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−18.0000			−0.588034			−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000			−0.588034			−0.294017	+	0.955800i	$0.594992\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−25.0000	−	43.3013i	−0.814977	−	1.41158i	−0.909345	−	0.416044i	$-0.863416\pi$
$941$	−25.0000	−	43.3013i	−0.814977	−	1.41158i	0.0943679	−	0.995537i	$-0.469917\pi$
$942$		0			0
$943$	−27.0000	+	46.7654i	−0.879241	+	1.52289i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.00000	−	15.5885i	0.292461	−	0.506557i	−0.681930	−	0.731417i	$-0.738859\pi$
$947$	9.00000	−	15.5885i	0.292461	−	0.506557i	0.974391	+	0.224860i	$0.0721926\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20.0000			0.647864			0.323932	−	0.946080i	$-0.394995\pi$
$953$	20.0000			0.647864			0.323932	+	0.946080i	$0.394995\pi$
$954$		0			0
$955$	−9.00000			−0.291233
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.0000	−	25.9808i	0.483871	−	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.00000	+	12.1244i	−0.225338	+	0.390297i
$966$		0			0
$967$	30.0000	+	51.9615i	0.964735	+	1.67097i	0.710325	+	0.703873i	$0.248548\pi$
$967$	30.0000	+	51.9615i	0.964735	+	1.67097i	0.254410	+	0.967097i	$0.418119\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	9.00000			0.288824			0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000			0.288824			0.144412	+	0.989518i	$0.453871\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.0000	+	34.6410i	0.639857	+	1.10826i	0.985464	+	0.169885i	$0.0543396\pi$
$977$	20.0000	+	34.6410i	0.639857	+	1.10826i	−0.345607	+	0.938379i	$0.612327\pi$
$978$		0			0
$979$	3.50000	−	6.06218i	0.111860	−	0.193748i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10.0000	−	17.3205i	0.318950	−	0.552438i	−0.661319	−	0.750105i	$-0.730003\pi$
$983$	10.0000	−	17.3205i	0.318950	−	0.552438i	0.980269	+	0.197666i	$0.0633363\pi$
$984$		0			0
$985$	−10.0000	−	17.3205i	−0.318626	−	0.551877i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−36.0000			−1.14473
$990$		0			0
$991$	1.00000			0.0317660			0.0158830	−	0.999874i	$-0.494944\pi$
$991$	1.00000			0.0317660			0.0158830	+	0.999874i	$0.494944\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−30.0000	+	51.9615i	−0.950110	+	1.64564i	−0.204927	+	0.978777i	$0.565696\pi$
$997$	−30.0000	+	51.9615i	−0.950110	+	1.64564i	−0.745182	+	0.666861i	$0.767638\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.q.r.2161.1		2
3.2	odd	2		3240.2.q.f.2161.1		2
9.2	odd	6		3240.2.a.e.1.1	yes	1
9.4	even	3	inner	3240.2.q.r.1081.1		2
9.5	odd	6		3240.2.q.f.1081.1		2
9.7	even	3		3240.2.a.c.1.1	✓	1
36.7	odd	6		6480.2.a.d.1.1		1
36.11	even	6		6480.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.c.1.1	✓	1	9.7	even	3
3240.2.a.e.1.1	yes	1	9.2	odd	6
3240.2.q.f.1081.1		2	9.5	odd	6
3240.2.q.f.2161.1		2	3.2	odd	2
3240.2.q.r.1081.1		2	9.4	even	3	inner
3240.2.q.r.2161.1		2	1.1	even	1	trivial
6480.2.a.d.1.1		1	36.7	odd	6
6480.2.a.s.1.1		1	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{11} -7.00000 q^{19} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(3.50000 - 6.06218i) q^{29} +(-0.500000 - 0.866025i) q^{31} -2.00000 q^{37} +(-4.50000 - 7.79423i) q^{41} +(3.00000 - 5.19615i) q^{43} +(1.00000 - 1.73205i) q^{47} +(3.50000 + 6.06218i) q^{49} -1.00000 q^{55} +(-1.50000 - 2.59808i) q^{59} +(5.00000 - 8.66025i) q^{61} +(1.00000 + 1.73205i) q^{67} +1.00000 q^{71} +(-2.00000 + 3.46410i) q^{79} +(3.00000 - 5.19615i) q^{83} -7.00000 q^{89} +(-3.50000 - 6.06218i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - q^{11} - 14 q^{19} - 6 q^{23} - q^{25} + 7 q^{29} - q^{31} - 4 q^{37} - 9 q^{41} + 6 q^{43} + 2 q^{47} + 7 q^{49} - 2 q^{55} - 3 q^{59} + 10 q^{61} + 2 q^{67} + 2 q^{71} - 4 q^{79} + 6 q^{83}+ \cdots - 2 q^{97}+O(q^{100}) \) 2 * q + q^5 - q^11 - 14 * q^19 - 6 * q^23 - q^25 + 7 * q^29 - q^31 - 4 * q^37 - 9 * q^41 + 6 * q^43 + 2 * q^47 + 7 * q^49 - 2 * q^55 - 3 * q^59 + 10 * q^61 + 2 * q^67 + 2 * q^71 - 4 * q^79 + 6 * q^83 - 14 * q^89 - 7 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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