LMFDB - Embedded newform 2880.2.t.c.2161.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2880,2,Mod(721,2880)] 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("2880.721"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2880, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.t (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 $ x^16 - 4x^15 + 4x^14 + 7x^12 - 8x^11 - 28x^10 + 28x^9 + 17x^8 + 56x^7 - 112x^6 - 64x^5 + 112x^4 + 256x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2161.6
Root			$1.32070 - 0.505727i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.c.721.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} -2.89402i q^{7} +(1.84462 + 1.84462i) q^{11} +(-3.08011 + 3.08011i) q^{13} -7.29875 q^{17} +(1.23593 - 1.23593i) q^{19} -4.60490i q^{23} +1.00000i q^{25} +(-4.24680 + 4.24680i) q^{29} -2.06299 q^{31} +(2.04638 - 2.04638i) q^{35} +(-1.17899 - 1.17899i) q^{37} -4.61484i q^{41} +(-3.03019 - 3.03019i) q^{43} -11.7111 q^{47} -1.37537 q^{49} +(-2.73048 - 2.73048i) q^{53} +2.60869i q^{55} +(3.11306 + 3.11306i) q^{59} +(2.34962 - 2.34962i) q^{61} -4.35593 q^{65} +(-8.24311 + 8.24311i) q^{67} -3.25937i q^{71} +12.6877i q^{73} +(5.33839 - 5.33839i) q^{77} +0.113885 q^{79} +(9.76813 - 9.76813i) q^{83} +(-5.16100 - 5.16100i) q^{85} +3.74593i q^{89} +(8.91390 + 8.91390i) q^{91} +1.74787 q^{95} -13.9853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) $ 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$		−	2.89402i		−	1.09384i	−0.837186	−	0.546919i	$-0.815801\pi$
$7$		−	2.89402i		−	1.09384i	0.837186	−	0.546919i	$-0.184199\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.84462	+	1.84462i	0.556175	+	0.556175i	0.928216	−	0.372041i	$-0.121342\pi$
$11$	1.84462	+	1.84462i	0.556175	+	0.556175i	−0.372041	+	0.928216i	$0.621342\pi$
$12$		0			0
$13$	−3.08011	+	3.08011i	−0.854268	+	0.854268i	−0.990656	−	0.136388i	$-0.956451\pi$
$13$	−3.08011	+	3.08011i	−0.854268	+	0.854268i	0.136388	+	0.990656i	$0.456451\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.29875			−1.77021			−0.885104	−	0.465393i	$-0.845913\pi$
$17$	−7.29875			−1.77021			−0.885104	+	0.465393i	$0.845913\pi$
$18$		0			0
$19$	1.23593	−	1.23593i	0.283542	−	0.283542i	−0.550978	−	0.834520i	$-0.685745\pi$
$19$	1.23593	−	1.23593i	0.283542	−	0.283542i	0.834520	+	0.550978i	$0.185745\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	4.60490i		−	0.960189i	−0.877217	−	0.480094i	$-0.840602\pi$
$23$		−	4.60490i		−	0.960189i	0.877217	−	0.480094i	$-0.159398\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.24680	+	4.24680i	−0.788611	+	0.788611i	−0.981266	−	0.192656i	$-0.938290\pi$
$29$	−4.24680	+	4.24680i	−0.788611	+	0.788611i	0.192656	+	0.981266i	$0.438290\pi$
$30$		0			0
$31$	−2.06299			−0.370524			−0.185262	−	0.982689i	$-0.559313\pi$
$31$	−2.06299			−0.370524			−0.185262	+	0.982689i	$0.559313\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.04638	−	2.04638i	0.345902	−	0.345902i
$36$		0			0
$37$	−1.17899	−	1.17899i	−0.193825	−	0.193825i	0.603522	−	0.797346i	$-0.293764\pi$
$37$	−1.17899	−	1.17899i	−0.193825	−	0.193825i	−0.797346	+	0.603522i	$0.793764\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.61484i		−	0.720717i	−0.932814	−	0.360359i	$-0.882654\pi$
$41$		−	4.61484i		−	0.720717i	0.932814	−	0.360359i	$-0.117346\pi$
$42$		0			0
$43$	−3.03019	−	3.03019i	−0.462099	−	0.462099i	0.437244	−	0.899343i	$-0.355955\pi$
$43$	−3.03019	−	3.03019i	−0.462099	−	0.462099i	−0.899343	+	0.437244i	$0.855955\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.7111			−1.70823			−0.854117	−	0.520081i	$-0.825902\pi$
$47$	−11.7111			−1.70823			−0.854117	+	0.520081i	$0.825902\pi$
$48$		0			0
$49$	−1.37537			−0.196481
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.73048	−	2.73048i	−0.375061	−	0.375061i	0.494256	−	0.869316i	$-0.335441\pi$
$53$	−2.73048	−	2.73048i	−0.375061	−	0.375061i	−0.869316	+	0.494256i	$0.835441\pi$
$54$		0			0
$55$			2.60869i			0.351756i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.11306	+	3.11306i	0.405285	+	0.405285i	0.880091	−	0.474805i	$-0.157482\pi$
$59$	3.11306	+	3.11306i	0.405285	+	0.405285i	−0.474805	+	0.880091i	$0.657482\pi$
$60$		0			0
$61$	2.34962	−	2.34962i	0.300838	−	0.300838i	−0.540503	−	0.841342i	$-0.681766\pi$
$61$	2.34962	−	2.34962i	0.300838	−	0.300838i	0.841342	+	0.540503i	$0.181766\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.35593			−0.540286
$66$		0			0
$67$	−8.24311	+	8.24311i	−1.00706	+	1.00706i	−0.00708173	+	0.999975i	$0.502254\pi$
$67$	−8.24311	+	8.24311i	−1.00706	+	1.00706i	−0.999975	+	0.00708173i	$0.997746\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.25937i		−	0.386816i	−0.981118	−	0.193408i	$-0.938046\pi$
$71$		−	3.25937i		−	0.386816i	0.981118	−	0.193408i	$-0.0619541\pi$
$72$		0			0
$73$			12.6877i			1.48499i	0.669853	+	0.742494i	$0.266357\pi$
$73$			12.6877i			1.48499i	−0.669853	+	0.742494i	$0.733643\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.33839	−	5.33839i	0.608365	−	0.608365i
$78$		0			0
$79$	0.113885			0.0128130			0.00640652	−	0.999979i	$-0.497961\pi$
$79$	0.113885			0.0128130			0.00640652	+	0.999979i	$0.497961\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.76813	−	9.76813i	1.07219	−	1.07219i	0.0750089	−	0.997183i	$-0.476101\pi$
$83$	9.76813	−	9.76813i	1.07219	−	1.07219i	0.997183	−	0.0750089i	$-0.0238985\pi$
$84$		0			0
$85$	−5.16100	−	5.16100i	−0.559789	−	0.559789i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3.74593i			0.397068i	0.980094	+	0.198534i	$0.0636180\pi$
$89$			3.74593i			0.397068i	−0.980094	+	0.198534i	$0.936382\pi$
$90$		0			0
$91$	8.91390	+	8.91390i	0.934430	+	0.934430i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.74787			0.179328
$96$		0			0
$97$	−13.9853			−1.41999			−0.709995	−	0.704206i	$-0.751303\pi$
$97$	−13.9853			−1.41999			−0.709995	+	0.704206i	$0.751303\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.52228	−	3.52228i	−0.350480	−	0.350480i	0.509808	−	0.860288i	$-0.329716\pi$
$101$	−3.52228	−	3.52228i	−0.350480	−	0.350480i	−0.860288	+	0.509808i	$0.829716\pi$
$102$		0			0
$103$		−	0.150216i		−	0.0148013i	−0.999973	−	0.00740063i	$-0.997644\pi$
$103$		−	0.150216i		−	0.0148013i	0.999973	−	0.00740063i	$-0.00235572\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.75062	−	2.75062i	−0.265912	−	0.265912i	0.561539	−	0.827451i	$-0.310210\pi$
$107$	−2.75062	−	2.75062i	−0.265912	−	0.265912i	−0.827451	+	0.561539i	$0.810210\pi$
$108$		0			0
$109$	−6.90778	+	6.90778i	−0.661646	+	0.661646i	−0.955768	−	0.294122i	$-0.904973\pi$
$109$	−6.90778	+	6.90778i	−0.661646	+	0.661646i	0.294122	+	0.955768i	$0.404973\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.49507			0.328788			0.164394	−	0.986395i	$-0.447433\pi$
$113$	3.49507			0.328788			0.164394	+	0.986395i	$0.447433\pi$
$114$		0			0
$115$	3.25616	−	3.25616i	0.303638	−	0.303638i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			21.1228i			1.93632i
$120$		0			0
$121$		−	4.19472i		−	0.381338i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	−6.25357			−0.554915			−0.277458	−	0.960738i	$-0.589492\pi$
$127$	−6.25357			−0.554915			−0.277458	+	0.960738i	$0.589492\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.16490	+	5.16490i	−0.451259	+	0.451259i	−0.895772	−	0.444513i	$-0.853377\pi$
$131$	−5.16490	+	5.16490i	−0.451259	+	0.451259i	0.444513	+	0.895772i	$0.353377\pi$
$132$		0			0
$133$	−3.57681	−	3.57681i	−0.310149	−	0.310149i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			18.9408i			1.61823i	0.587654	+	0.809113i	$0.300052\pi$
$137$			18.9408i			1.61823i	−0.587654	+	0.809113i	$0.699948\pi$
$138$		0			0
$139$	−2.79057	−	2.79057i	−0.236693	−	0.236693i	0.578786	−	0.815479i	$-0.303527\pi$
$139$	−2.79057	−	2.79057i	−0.236693	−	0.236693i	−0.815479	+	0.578786i	$0.803527\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.3633			−0.950245
$144$		0			0
$145$	−6.00588			−0.498761
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.60372	+	1.60372i	0.131382	+	0.131382i	0.769740	−	0.638358i	$-0.220386\pi$
$149$	1.60372	+	1.60372i	0.131382	+	0.131382i	−0.638358	+	0.769740i	$0.720386\pi$
$150$		0			0
$151$		−	2.53754i		−	0.206502i	−0.994655	−	0.103251i	$-0.967076\pi$
$151$		−	2.53754i		−	0.206502i	0.994655	−	0.103251i	$-0.0329245\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.45875	−	1.45875i	−0.117170	−	0.117170i
$156$		0			0
$157$	10.2405	−	10.2405i	0.817278	−	0.817278i	−0.168435	−	0.985713i	$-0.553871\pi$
$157$	10.2405	−	10.2405i	0.817278	−	0.817278i	0.985713	+	0.168435i	$0.0538712\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13.3267			−1.05029
$162$		0			0
$163$	−8.02607	+	8.02607i	−0.628650	+	0.628650i	−0.947728	−	0.319078i	$-0.896627\pi$
$163$	−8.02607	+	8.02607i	−0.628650	+	0.628650i	0.319078	+	0.947728i	$0.396627\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	6.82611i		−	0.528221i	−0.964492	−	0.264110i	$-0.914922\pi$
$167$		−	6.82611i		−	0.528221i	0.964492	−	0.264110i	$-0.0850783\pi$
$168$		0			0
$169$		−	5.97411i		−	0.459547i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.08901	+	5.08901i	−0.386910	+	0.386910i	−0.873584	−	0.486674i	$-0.838210\pi$
$173$	−5.08901	+	5.08901i	−0.386910	+	0.386910i	0.486674	+	0.873584i	$0.338210\pi$
$174$		0			0
$175$	2.89402			0.218768
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.63797	−	1.63797i	0.122428	−	0.122428i	−0.643238	−	0.765666i	$-0.722410\pi$
$179$	1.63797	−	1.63797i	0.122428	−	0.122428i	0.765666	+	0.643238i	$0.222410\pi$
$180$		0			0
$181$	−16.7757	−	16.7757i	−1.24693	−	1.24693i	−0.957071	−	0.289855i	$-0.906393\pi$
$181$	−16.7757	−	16.7757i	−1.24693	−	1.24693i	−0.289855	−	0.957071i	$-0.593607\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1.66734i		−	0.122585i
$186$		0			0
$187$	−13.4635	−	13.4635i	−0.984546	−	0.984546i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.85815			0.423881			0.211940	−	0.977283i	$-0.432022\pi$
$191$	5.85815			0.423881			0.211940	+	0.977283i	$0.432022\pi$
$192$		0			0
$193$	−0.0241155			−0.00173587			−0.000867935	−	1.00000i	$-0.500276\pi$
$193$	−0.0241155			−0.00173587			−0.000867935		1.00000i	$0.500276\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.9086	−	14.9086i	−1.06219	−	1.06219i	−0.997933	−	0.0642576i	$-0.979532\pi$
$197$	−14.9086	−	14.9086i	−1.06219	−	1.06219i	−0.0642576	−	0.997933i	$-0.520468\pi$
$198$		0			0
$199$		−	13.6525i		−	0.967801i	−0.875123	−	0.483900i	$-0.839220\pi$
$199$		−	13.6525i		−	0.967801i	0.875123	−	0.483900i	$-0.160780\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.2903	+	12.2903i	0.862612	+	0.862612i
$204$		0			0
$205$	3.26319	−	3.26319i	0.227911	−	0.227911i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.55966			0.315398
$210$		0			0
$211$	2.45103	−	2.45103i	0.168736	−	0.168736i	−0.617688	−	0.786424i	$-0.711930\pi$
$211$	2.45103	−	2.45103i	0.168736	−	0.168736i	0.786424	+	0.617688i	$0.211930\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4.28533i		−	0.292257i
$216$		0			0
$217$			5.97033i			0.405293i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	22.4809	−	22.4809i	1.51223	−	1.51223i
$222$		0			0
$223$	−13.9483			−0.934045			−0.467023	−	0.884245i	$-0.654673\pi$
$223$	−13.9483			−0.934045			−0.467023	+	0.884245i	$0.654673\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.43883	−	4.43883i	0.294616	−	0.294616i	−0.544285	−	0.838901i	$-0.683199\pi$
$227$	4.43883	−	4.43883i	0.294616	−	0.294616i	0.838901	+	0.544285i	$0.183199\pi$
$228$		0			0
$229$	−5.35068	−	5.35068i	−0.353583	−	0.353583i	0.507858	−	0.861441i	$-0.330438\pi$
$229$	−5.35068	−	5.35068i	−0.353583	−	0.353583i	−0.861441	+	0.507858i	$0.830438\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			11.9370i			0.782019i	0.920387	+	0.391010i	$0.127874\pi$
$233$			11.9370i			0.782019i	−0.920387	+	0.391010i	$0.872126\pi$
$234$		0			0
$235$	−8.28097	−	8.28097i	−0.540191	−	0.540191i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.7720			−1.08489			−0.542445	−	0.840091i	$-0.682501\pi$
$239$	−16.7720			−1.08489			−0.542445	+	0.840091i	$0.682501\pi$
$240$		0			0
$241$	−22.0294			−1.41904			−0.709519	−	0.704686i	$-0.751088\pi$
$241$	−22.0294			−1.41904			−0.709519	+	0.704686i	$0.751088\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.972532	−	0.972532i	−0.0621328	−	0.0621328i
$246$		0			0
$247$			7.61360i			0.484442i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.63925	+	6.63925i	0.419066	+	0.419066i	0.884882	−	0.465816i	$-0.154239\pi$
$251$	6.63925	+	6.63925i	0.419066	+	0.419066i	−0.465816	+	0.884882i	$0.654239\pi$
$252$		0			0
$253$	8.49432	−	8.49432i	0.534033	−	0.534033i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.25821			0.452755			0.226377	−	0.974040i	$-0.427312\pi$
$257$	7.25821			0.452755			0.226377	+	0.974040i	$0.427312\pi$
$258$		0			0
$259$	−3.41202	+	3.41202i	−0.212013	+	0.212013i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	9.27431i		−	0.571878i	−0.958248	−	0.285939i	$-0.907694\pi$
$263$		−	9.27431i		−	0.571878i	0.958248	−	0.285939i	$-0.0923055\pi$
$264$		0			0
$265$		−	3.86149i		−	0.237209i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.4195	−	13.4195i	0.818199	−	0.818199i	−0.167648	−	0.985847i	$-0.553617\pi$
$269$	13.4195	−	13.4195i	0.818199	−	0.818199i	0.985847	+	0.167648i	$0.0536173\pi$
$270$		0			0
$271$	−22.5999			−1.37285			−0.686423	−	0.727202i	$-0.740820\pi$
$271$	−22.5999			−1.37285			−0.686423	+	0.727202i	$0.740820\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.84462	+	1.84462i	−0.111235	+	0.111235i
$276$		0			0
$277$	16.2015	+	16.2015i	0.973451	+	0.973451i	0.999657	−	0.0262056i	$-0.00834245\pi$
$277$	16.2015	+	16.2015i	0.973451	+	0.973451i	−0.0262056	+	0.999657i	$0.508342\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			8.84793i			0.527824i	0.964547	+	0.263912i	$0.0850128\pi$
$281$			8.84793i			0.527824i	−0.964547	+	0.263912i	$0.914987\pi$
$282$		0			0
$283$	20.3062	+	20.3062i	1.20708	+	1.20708i	0.971969	+	0.235109i	$0.0755447\pi$
$283$	20.3062	+	20.3062i	1.20708	+	1.20708i	0.235109	+	0.971969i	$0.424455\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−13.3555			−0.788348
$288$		0			0
$289$	36.2718			2.13364
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.16936	+	7.16936i	0.418839	+	0.418839i	0.884803	−	0.465965i	$-0.154293\pi$
$293$	7.16936	+	7.16936i	0.418839	+	0.418839i	−0.465965	+	0.884803i	$0.654293\pi$
$294$		0			0
$295$			4.40253i			0.256325i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	14.1836	+	14.1836i	0.820258	+	0.820258i
$300$		0			0
$301$	−8.76943	+	8.76943i	−0.505461	+	0.505461i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.32287			0.190267
$306$		0			0
$307$	18.4308	−	18.4308i	1.05190	−	1.05190i	0.0533241	−	0.998577i	$-0.483018\pi$
$307$	18.4308	−	18.4308i	1.05190	−	1.05190i	0.998577	−	0.0533241i	$-0.0169816\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			7.08961i			0.402015i	0.979590	+	0.201007i	$0.0644215\pi$
$311$			7.08961i			0.402015i	−0.979590	+	0.201007i	$0.935578\pi$
$312$		0			0
$313$			22.0477i			1.24621i	0.782139	+	0.623104i	$0.214129\pi$
$313$			22.0477i			1.24621i	−0.782139	+	0.623104i	$0.785871\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.19670	−	6.19670i	0.348042	−	0.348042i	−0.511338	−	0.859380i	$-0.670850\pi$
$317$	6.19670	−	6.19670i	0.348042	−	0.348042i	0.859380	+	0.511338i	$0.170850\pi$
$318$		0			0
$319$	−15.6675			−0.877211
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9.02076	+	9.02076i	−0.501929	+	0.501929i
$324$		0			0
$325$	−3.08011	−	3.08011i	−0.170854	−	0.170854i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			33.8921i			1.86853i
$330$		0			0
$331$	18.6174	+	18.6174i	1.02330	+	1.02330i	0.999722	+	0.0235823i	$0.00750716\pi$
$331$	18.6174	+	18.6174i	1.02330	+	1.02330i	0.0235823	+	0.999722i	$0.492493\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.6575			−0.636919
$336$		0			0
$337$	14.2577			0.776666			0.388333	−	0.921519i	$-0.373051\pi$
$337$	14.2577			0.776666			0.388333	+	0.921519i	$0.373051\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.80544	−	3.80544i	−0.206076	−	0.206076i
$342$		0			0
$343$		−	16.2778i		−	0.878919i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−23.5395	−	23.5395i	−1.26367	−	1.26367i	−0.949303	−	0.314363i	$-0.898209\pi$
$347$	−23.5395	−	23.5395i	−1.26367	−	1.26367i	−0.314363	−	0.949303i	$-0.601791\pi$
$348$		0			0
$349$	−1.56682	+	1.56682i	−0.0838701	+	0.0838701i	−0.747797	−	0.663927i	$-0.768888\pi$
$349$	−1.56682	+	1.56682i	−0.0838701	+	0.0838701i	0.663927	+	0.747797i	$0.268888\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.44678			−0.502801			−0.251401	−	0.967883i	$-0.580891\pi$
$353$	−9.44678			−0.502801			−0.251401	+	0.967883i	$0.580891\pi$
$354$		0			0
$355$	2.30472	−	2.30472i	0.122322	−	0.122322i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	18.0452i		−	0.952392i	−0.879339	−	0.476196i	$-0.842015\pi$
$359$		−	18.0452i		−	0.952392i	0.879339	−	0.476196i	$-0.157985\pi$
$360$		0			0
$361$			15.9449i			0.839208i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.97159	+	8.97159i	−0.469594	+	0.469594i
$366$		0			0
$367$	−29.1329			−1.52073			−0.760363	−	0.649498i	$-0.774979\pi$
$367$	−29.1329			−1.52073			−0.760363	+	0.649498i	$0.774979\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.90208	+	7.90208i	−0.410255	+	0.410255i
$372$		0			0
$373$	3.35598	+	3.35598i	0.173766	+	0.173766i	0.788632	−	0.614866i	$-0.210790\pi$
$373$	3.35598	+	3.35598i	0.173766	+	0.173766i	−0.614866	+	0.788632i	$0.710790\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	26.1612i		−	1.34737i
$378$		0			0
$379$	−11.6507	−	11.6507i	−0.598457	−	0.598457i	0.341445	−	0.939902i	$-0.389084\pi$
$379$	−11.6507	−	11.6507i	−0.598457	−	0.598457i	−0.939902	+	0.341445i	$0.889084\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−21.8044			−1.11415			−0.557077	−	0.830461i	$-0.688077\pi$
$383$	−21.8044			−1.11415			−0.557077	+	0.830461i	$0.688077\pi$
$384$		0			0
$385$	7.54962			0.384764
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.8899	+	11.8899i	0.602842	+	0.602842i	0.941066	−	0.338224i	$-0.109826\pi$
$389$	11.8899	+	11.8899i	0.602842	+	0.602842i	−0.338224	+	0.941066i	$0.609826\pi$
$390$		0			0
$391$			33.6101i			1.69973i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.0805287	+	0.0805287i	0.00405184	+	0.00405184i
$396$		0			0
$397$	−9.23905	+	9.23905i	−0.463694	+	0.463694i	−0.899864	−	0.436170i	$-0.856335\pi$
$397$	−9.23905	+	9.23905i	−0.463694	+	0.463694i	0.436170	+	0.899864i	$0.356335\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.4744			0.722818			0.361409	−	0.932407i	$-0.382296\pi$
$401$	14.4744			0.722818			0.361409	+	0.932407i	$0.382296\pi$
$402$		0			0
$403$	6.35422	−	6.35422i	0.316526	−	0.316526i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4.34958i		−	0.215601i
$408$		0			0
$409$		−	9.54117i		−	0.471781i	−0.971780	−	0.235890i	$-0.924199\pi$
$409$		−	9.54117i		−	0.471781i	0.971780	−	0.235890i	$-0.0758006\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.00925	−	9.00925i	0.443316	−	0.443316i
$414$		0			0
$415$	13.8142			0.678114
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.837667	−	0.837667i	0.0409227	−	0.0409227i	−0.686349	−	0.727272i	$-0.740788\pi$
$419$	0.837667	−	0.837667i	0.0409227	−	0.0409227i	0.727272	+	0.686349i	$0.240788\pi$
$420$		0			0
$421$	17.9679	+	17.9679i	0.875702	+	0.875702i	0.993087	−	0.117385i	$-0.0374511\pi$
$421$	17.9679	+	17.9679i	0.875702	+	0.875702i	−0.117385	+	0.993087i	$0.537451\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	7.29875i		−	0.354042i
$426$		0			0
$427$	−6.79986	−	6.79986i	−0.329068	−	0.329068i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.85473			−0.185676			−0.0928380	−	0.995681i	$-0.529594\pi$
$431$	−3.85473			−0.185676			−0.0928380	+	0.995681i	$0.529594\pi$
$432$		0			0
$433$	−25.5651			−1.22858			−0.614289	−	0.789081i	$-0.710557\pi$
$433$	−25.5651			−1.22858			−0.614289	+	0.789081i	$0.710557\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.69135	−	5.69135i	−0.272254	−	0.272254i
$438$		0			0
$439$			30.1311i			1.43808i	0.694970	+	0.719039i	$0.255418\pi$
$439$			30.1311i			1.43808i	−0.694970	+	0.719039i	$0.744582\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.1625	+	20.1625i	0.957948	+	0.957948i	0.999151	−	0.0412027i	$-0.0131189\pi$
$443$	20.1625	+	20.1625i	0.957948	+	0.957948i	−0.0412027	+	0.999151i	$0.513119\pi$
$444$		0			0
$445$	−2.64877	+	2.64877i	−0.125564	+	0.125564i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	36.5827			1.72644			0.863221	−	0.504826i	$-0.168443\pi$
$449$	36.5827			1.72644			0.863221	+	0.504826i	$0.168443\pi$
$450$		0			0
$451$	8.51265	−	8.51265i	0.400845	−	0.400845i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			12.6062i			0.590986i
$456$		0			0
$457$		−	16.7340i		−	0.782785i	−0.920224	−	0.391392i	$-0.871994\pi$
$457$		−	16.7340i		−	0.782785i	0.920224	−	0.391392i	$-0.128006\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.8377	+	11.8377i	−0.551335	+	0.551335i	−0.926826	−	0.375491i	$-0.877474\pi$
$461$	−11.8377	+	11.8377i	−0.551335	+	0.551335i	0.375491	+	0.926826i	$0.377474\pi$
$462$		0			0
$463$	32.2711			1.49976			0.749882	−	0.661572i	$-0.230110\pi$
$463$	32.2711			1.49976			0.749882	+	0.661572i	$0.230110\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.22565	−	1.22565i	0.0567163	−	0.0567163i	−0.678180	−	0.734896i	$-0.737231\pi$
$467$	1.22565	−	1.22565i	0.0567163	−	0.0567163i	0.734896	+	0.678180i	$0.237231\pi$
$468$		0			0
$469$	23.8558	+	23.8558i	1.10156	+	1.10156i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	11.1791i		−	0.514016i
$474$		0			0
$475$	1.23593	+	1.23593i	0.0567084	+	0.0567084i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	28.8399			1.31773			0.658865	−	0.752261i	$-0.271037\pi$
$479$	28.8399			1.31773			0.658865	+	0.752261i	$0.271037\pi$
$480$		0			0
$481$	7.26282			0.331156
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.88909	−	9.88909i	−0.449041	−	0.449041i
$486$		0			0
$487$		−	32.1668i		−	1.45762i	−0.684718	−	0.728808i	$-0.740075\pi$
$487$		−	32.1668i		−	1.45762i	0.684718	−	0.728808i	$-0.259925\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.43607	−	5.43607i	−0.245326	−	0.245326i	0.573723	−	0.819049i	$-0.305499\pi$
$491$	−5.43607	−	5.43607i	−0.245326	−	0.245326i	−0.819049	+	0.573723i	$0.805499\pi$
$492$		0			0
$493$	30.9963	−	30.9963i	1.39600	−	1.39600i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.43268			−0.423114
$498$		0			0
$499$	17.1282	−	17.1282i	0.766762	−	0.766762i	−0.210773	−	0.977535i	$-0.567598\pi$
$499$	17.1282	−	17.1282i	0.766762	−	0.766762i	0.977535	+	0.210773i	$0.0675981\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	23.5180i		−	1.04862i	−0.851529	−	0.524308i	$-0.824324\pi$
$503$		−	23.5180i		−	1.04862i	0.851529	−	0.524308i	$-0.175676\pi$
$504$		0			0
$505$		−	4.98126i		−	0.221663i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−20.3147	+	20.3147i	−0.900434	+	0.900434i	−0.995474	−	0.0950391i	$-0.969702\pi$
$509$	−20.3147	+	20.3147i	−0.900434	+	0.900434i	0.0950391	+	0.995474i	$0.469702\pi$
$510$		0			0
$511$	36.7186			1.62434
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.106219	−	0.106219i	0.00468057	−	0.00468057i
$516$		0			0
$517$	−21.6025	−	21.6025i	−0.950077	−	0.950077i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			35.5082i			1.55564i	0.628487	+	0.777820i	$0.283675\pi$
$521$			35.5082i			1.55564i	−0.628487	+	0.777820i	$0.716325\pi$
$522$		0			0
$523$	−0.677766	−	0.677766i	−0.0296366	−	0.0296366i	0.692133	−	0.721770i	$-0.256671\pi$
$523$	−0.677766	−	0.677766i	−0.0296366	−	0.0296366i	−0.721770	+	0.692133i	$0.756671\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	15.0572			0.655904
$528$		0			0
$529$	1.79485			0.0780371
$530$		0			0
$531$		0			0
$532$		0			0
$533$	14.2142	+	14.2142i	0.615686	+	0.615686i
$534$		0			0
$535$		−	3.88996i		−	0.168178i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.53704	−	2.53704i	−0.109278	−	0.109278i
$540$		0			0
$541$	5.37099	−	5.37099i	0.230917	−	0.230917i	−0.582158	−	0.813075i	$-0.697792\pi$
$541$	5.37099	−	5.37099i	0.230917	−	0.230917i	0.813075	+	0.582158i	$0.197792\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.76908			−0.418461
$546$		0			0
$547$	−8.86782	+	8.86782i	−0.379161	+	0.379161i	−0.870799	−	0.491639i	$-0.836398\pi$
$547$	−8.86782	+	8.86782i	−0.379161	+	0.379161i	0.491639	+	0.870799i	$0.336398\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			10.4975i			0.447209i
$552$		0			0
$553$		−	0.329585i		−	0.0140154i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22.8089	−	22.8089i	0.966446	−	0.966446i	−0.0330091	−	0.999455i	$-0.510509\pi$
$557$	22.8089	−	22.8089i	0.966446	−	0.966446i	0.999455	+	0.0330091i	$0.0105090\pi$
$558$		0			0
$559$	18.6666			0.789513
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.9711	−	20.9711i	0.883826	−	0.883826i	−0.110095	−	0.993921i	$-0.535116\pi$
$563$	20.9711	−	20.9711i	0.883826	−	0.883826i	0.993921	+	0.110095i	$0.0351156\pi$
$564$		0			0
$565$	2.47139	+	2.47139i	0.103972	+	0.103972i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			8.05295i			0.337597i	0.985651	+	0.168799i	$0.0539888\pi$
$569$			8.05295i			0.337597i	−0.985651	+	0.168799i	$0.946011\pi$
$570$		0			0
$571$	22.5040	+	22.5040i	0.941762	+	0.941762i	0.998395	−	0.0566333i	$-0.0180366\pi$
$571$	22.5040	+	22.5040i	0.941762	+	0.941762i	−0.0566333	+	0.998395i	$0.518037\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.60490			0.192038
$576$		0			0
$577$	15.9819			0.665334			0.332667	−	0.943044i	$-0.392051\pi$
$577$	15.9819			0.665334			0.332667	+	0.943044i	$0.392051\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−28.2692	−	28.2692i	−1.17280	−	1.17280i
$582$		0			0
$583$		−	10.0734i		−	0.417199i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.25752	−	5.25752i	−0.217001	−	0.217001i	0.590232	−	0.807233i	$-0.299036\pi$
$587$	−5.25752	−	5.25752i	−0.217001	−	0.217001i	−0.807233	+	0.590232i	$0.799036\pi$
$588$		0			0
$589$	−2.54971	+	2.54971i	−0.105059	+	0.105059i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.96571			−0.162852			−0.0814260	−	0.996679i	$-0.525947\pi$
$593$	−3.96571			−0.162852			−0.0814260	+	0.996679i	$0.525947\pi$
$594$		0			0
$595$	−14.9360	+	14.9360i	−0.612318	+	0.612318i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			8.31600i			0.339783i	0.985463	+	0.169891i	$0.0543417\pi$
$599$			8.31600i			0.339783i	−0.985463	+	0.169891i	$0.945658\pi$
$600$		0			0
$601$		−	46.0550i		−	1.87862i	−0.343068	−	0.939310i	$-0.611466\pi$
$601$		−	46.0550i		−	1.87862i	0.343068	−	0.939310i	$-0.388534\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.96612	−	2.96612i	0.120590	−	0.120590i
$606$		0			0
$607$	5.05760			0.205282			0.102641	−	0.994718i	$-0.467271\pi$
$607$	5.05760			0.205282			0.102641	+	0.994718i	$0.467271\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	36.0713	−	36.0713i	1.45929	−	1.45929i
$612$		0			0
$613$	31.2000	+	31.2000i	1.26016	+	1.26016i	0.951016	+	0.309141i	$0.100042\pi$
$613$	31.2000	+	31.2000i	1.26016	+	1.26016i	0.309141	+	0.951016i	$0.399958\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	30.7412i		−	1.23759i	−0.785551	−	0.618796i	$-0.787621\pi$
$617$		−	30.7412i		−	1.23759i	0.785551	−	0.618796i	$-0.212379\pi$
$618$		0			0
$619$	16.8766	+	16.8766i	0.678329	+	0.678329i	0.959622	−	0.281293i	$-0.0907632\pi$
$619$	16.8766	+	16.8766i	0.678329	+	0.678329i	−0.281293	+	0.959622i	$0.590763\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.8408			0.434328
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.60515	+	8.60515i	0.343110	+	0.343110i
$630$		0			0
$631$		−	30.7318i		−	1.22342i	−0.791084	−	0.611708i	$-0.790483\pi$
$631$		−	30.7318i		−	1.22342i	0.791084	−	0.611708i	$-0.209517\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.42195	−	4.42195i	−0.175480	−	0.175480i
$636$		0			0
$637$	4.23628	−	4.23628i	0.167848	−	0.167848i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−22.1658			−0.875496			−0.437748	−	0.899098i	$-0.644224\pi$
$641$	−22.1658			−0.875496			−0.437748	+	0.899098i	$0.644224\pi$
$642$		0			0
$643$	−0.975773	+	0.975773i	−0.0384807	+	0.0384807i	−0.726085	−	0.687605i	$-0.758662\pi$
$643$	−0.975773	+	0.975773i	−0.0384807	+	0.0384807i	0.687605	+	0.726085i	$0.258662\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	23.2610i		−	0.914484i	−0.889342	−	0.457242i	$-0.848837\pi$
$647$		−	23.2610i		−	0.914484i	0.889342	−	0.457242i	$-0.151163\pi$
$648$		0			0
$649$			11.4848i			0.450819i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.9372	−	23.9372i	0.936735	−	0.936735i	−0.0613792	−	0.998115i	$-0.519550\pi$
$653$	23.9372	−	23.9372i	0.936735	−	0.936735i	0.998115	+	0.0613792i	$0.0195499\pi$
$654$		0			0
$655$	−7.30427			−0.285401
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−14.1064	+	14.1064i	−0.549508	+	0.549508i	−0.926299	−	0.376790i	$-0.877028\pi$
$659$	−14.1064	+	14.1064i	−0.549508	+	0.549508i	0.376790	+	0.926299i	$0.377028\pi$
$660$		0			0
$661$	−3.04121	−	3.04121i	−0.118289	−	0.118289i	0.645484	−	0.763774i	$-0.276656\pi$
$661$	−3.04121	−	3.04121i	−0.118289	−	0.118289i	−0.763774	+	0.645484i	$0.776656\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	5.05838i		−	0.196156i
$666$		0			0
$667$	19.5561	+	19.5561i	0.757215	+	0.757215i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8.66835			0.334638
$672$		0			0
$673$	−25.3628			−0.977662			−0.488831	−	0.872378i	$-0.662577\pi$
$673$	−25.3628			−0.977662			−0.488831	+	0.872378i	$0.662577\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9.36526	−	9.36526i	−0.359936	−	0.359936i	0.503853	−	0.863789i	$-0.331915\pi$
$677$	−9.36526	−	9.36526i	−0.359936	−	0.359936i	−0.863789	+	0.503853i	$0.831915\pi$
$678$		0			0
$679$			40.4737i			1.55324i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.20530	+	4.20530i	0.160911	+	0.160911i	0.782970	−	0.622059i	$-0.213704\pi$
$683$	4.20530	+	4.20530i	0.160911	+	0.160911i	−0.622059	+	0.782970i	$0.713704\pi$
$684$		0			0
$685$	−13.3932	+	13.3932i	−0.511728	+	0.511728i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.8204			0.640804
$690$		0			0
$691$	−5.79295	+	5.79295i	−0.220374	+	0.220374i	−0.808656	−	0.588282i	$-0.799805\pi$
$691$	−5.79295	+	5.79295i	−0.220374	+	0.220374i	0.588282	+	0.808656i	$0.299805\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	3.94646i		−	0.149698i
$696$		0			0
$697$			33.6826i			1.27582i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	0.258991	−	0.258991i	0.00978196	−	0.00978196i	−0.702199	−	0.711981i	$-0.747798\pi$
$701$	0.258991	−	0.258991i	0.00978196	−	0.00978196i	0.711981	+	0.702199i	$0.247798\pi$
$702$		0			0
$703$	−2.91430			−0.109915
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.1936	+	10.1936i	−0.383368	+	0.383368i
$708$		0			0
$709$	−0.751674	−	0.751674i	−0.0282297	−	0.0282297i	0.692851	−	0.721081i	$-0.256354\pi$
$709$	−0.751674	−	0.751674i	−0.0282297	−	0.0282297i	−0.721081	+	0.692851i	$0.756354\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			9.49986i			0.355773i
$714$		0			0
$715$	−8.03505	−	8.03505i	−0.300494	−	0.300494i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	39.6557			1.47891			0.739455	−	0.673206i	$-0.235083\pi$
$719$	39.6557			1.47891			0.739455	+	0.673206i	$0.235083\pi$
$720$		0			0
$721$	−0.434730			−0.0161902
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.24680	−	4.24680i	−0.157722	−	0.157722i
$726$		0			0
$727$		−	22.2952i		−	0.826881i	−0.910531	−	0.413441i	$-0.864327\pi$
$727$		−	22.2952i		−	0.826881i	0.910531	−	0.413441i	$-0.135673\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	22.1166	+	22.1166i	0.818011	+	0.818011i
$732$		0			0
$733$	28.2309	−	28.2309i	1.04273	−	1.04273i	0.0436851	−	0.999045i	$-0.486090\pi$
$733$	28.2309	−	28.2309i	1.04273	−	1.04273i	0.999045	−	0.0436851i	$-0.0139098\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−30.4109			−1.12020
$738$		0			0
$739$	5.45140	−	5.45140i	0.200533	−	0.200533i	−0.599695	−	0.800228i	$-0.704712\pi$
$739$	5.45140	−	5.45140i	0.200533	−	0.200533i	0.800228	+	0.599695i	$0.204712\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			52.5667i			1.92849i	0.265020	+	0.964243i	$0.414621\pi$
$743$			52.5667i			1.92849i	−0.265020	+	0.964243i	$0.585379\pi$
$744$		0			0
$745$			2.26800i			0.0830931i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.96035	+	7.96035i	−0.290865	+	0.290865i
$750$		0			0
$751$	−31.0189			−1.13190			−0.565948	−	0.824441i	$-0.691490\pi$
$751$	−31.0189			−1.13190			−0.565948	+	0.824441i	$0.691490\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.79431	−	1.79431i	0.0653016	−	0.0653016i
$756$		0			0
$757$	−2.47389	−	2.47389i	−0.0899152	−	0.0899152i	0.660719	−	0.750634i	$-0.270252\pi$
$757$	−2.47389	−	2.47389i	−0.0899152	−	0.0899152i	−0.750634	+	0.660719i	$0.770252\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			2.48375i			0.0900358i	0.998986	+	0.0450179i	$0.0143345\pi$
$761$			2.48375i			0.0900358i	−0.998986	+	0.0450179i	$0.985666\pi$
$762$		0			0
$763$	19.9913	+	19.9913i	0.723733	+	0.723733i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−19.1771			−0.692444
$768$		0			0
$769$	43.4690			1.56753			0.783767	−	0.621055i	$-0.213296\pi$
$769$	43.4690			1.56753			0.783767	+	0.621055i	$0.213296\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.297026	−	0.297026i	−0.0106833	−	0.0106833i	0.701745	−	0.712428i	$-0.252405\pi$
$773$	−0.297026	−	0.297026i	−0.0106833	−	0.0106833i	−0.712428	+	0.701745i	$0.752405\pi$
$774$		0			0
$775$		−	2.06299i		−	0.0741047i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.70363	−	5.70363i	−0.204354	−	0.204354i
$780$		0			0
$781$	6.01231	−	6.01231i	0.215137	−	0.215137i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	14.4822			0.516892
$786$		0			0
$787$	23.6931	−	23.6931i	0.844567	−	0.844567i	−0.144882	−	0.989449i	$-0.546280\pi$
$787$	23.6931	−	23.6931i	0.844567	−	0.844567i	0.989449	+	0.144882i	$0.0462802\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	10.1148i		−	0.359641i
$792$		0			0
$793$			14.4742i			0.513993i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−38.2292	+	38.2292i	−1.35415	+	1.35415i	−0.473186	+	0.880963i	$0.656896\pi$
$797$	−38.2292	+	38.2292i	−1.35415	+	1.35415i	−0.880963	+	0.473186i	$0.843104\pi$
$798$		0			0
$799$	85.4762			3.02393
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−23.4041	+	23.4041i	−0.825913	+	0.825913i
$804$		0			0
$805$	−9.42340	−	9.42340i	−0.332131	−	0.332131i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	53.8310i		−	1.89260i	−0.323296	−	0.946298i	$-0.604791\pi$
$809$		−	53.8310i		−	1.89260i	0.323296	−	0.946298i	$-0.395209\pi$
$810$		0			0
$811$	−27.0549	−	27.0549i	−0.950025	−	0.950025i	0.0487847	−	0.998809i	$-0.484465\pi$
$811$	−27.0549	−	27.0549i	−0.950025	−	0.950025i	−0.998809	+	0.0487847i	$0.984465\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−11.3506			−0.397593
$816$		0			0
$817$	−7.49020			−0.262049
$818$		0			0
$819$		0			0
$820$		0			0
$821$	24.2170	+	24.2170i	0.845180	+	0.845180i	0.989527	−	0.144347i	$-0.0461082\pi$
$821$	24.2170	+	24.2170i	0.845180	+	0.845180i	−0.144347	+	0.989527i	$0.546108\pi$
$822$		0			0
$823$			41.3013i			1.43967i	0.694144	+	0.719836i	$0.255783\pi$
$823$			41.3013i			1.43967i	−0.694144	+	0.719836i	$0.744217\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15.7264	+	15.7264i	0.546862	+	0.546862i	0.925532	−	0.378670i	$-0.123618\pi$
$827$	15.7264	+	15.7264i	0.546862	+	0.546862i	−0.378670	+	0.925532i	$0.623618\pi$
$828$		0			0
$829$	−20.7323	+	20.7323i	−0.720061	+	0.720061i	−0.968618	−	0.248556i	$-0.920044\pi$
$829$	−20.7323	+	20.7323i	−0.720061	+	0.720061i	0.248556	+	0.968618i	$0.420044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.0385			0.347813
$834$		0			0
$835$	4.82679	−	4.82679i	0.167038	−	0.167038i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			43.6919i			1.50841i	0.656638	+	0.754206i	$0.271978\pi$
$839$			43.6919i			1.50841i	−0.656638	+	0.754206i	$0.728022\pi$
$840$		0			0
$841$		−	7.07060i		−	0.243814i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.22433	−	4.22433i	0.145321	−	0.145321i
$846$		0			0
$847$	−12.1396			−0.417122
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.42913	+	5.42913i	−0.186108	+	0.186108i
$852$		0			0
$853$	−35.0610	−	35.0610i	−1.20046	−	1.20046i	−0.974025	−	0.226439i	$-0.927292\pi$
$853$	−35.0610	−	35.0610i	−1.20046	−	1.20046i	−0.226439	−	0.974025i	$-0.572708\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			45.3397i			1.54878i	0.632711	+	0.774388i	$0.281942\pi$
$857$			45.3397i			1.54878i	−0.632711	+	0.774388i	$0.718058\pi$
$858$		0			0
$859$	−32.1229	−	32.1229i	−1.09602	−	1.09602i	−0.994871	−	0.101147i	$-0.967749\pi$
$859$	−32.1229	−	32.1229i	−1.09602	−	1.09602i	−0.101147	−	0.994871i	$-0.532251\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	36.9142			1.25657			0.628287	−	0.777981i	$-0.283756\pi$
$863$	36.9142			1.25657			0.628287	+	0.777981i	$0.283756\pi$
$864$		0			0
$865$	−7.19695			−0.244704
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.210075	+	0.210075i	0.00712630	+	0.00712630i
$870$		0			0
$871$		−	50.7793i		−	1.72059i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.04638	+	2.04638i	0.0691804	+	0.0691804i
$876$		0			0
$877$	−15.7178	+	15.7178i	−0.530753	+	0.530753i	−0.920796	−	0.390044i	$-0.872460\pi$
$877$	−15.7178	+	15.7178i	−0.530753	+	0.530753i	0.390044	+	0.920796i	$0.372460\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.16748			−0.0393335			−0.0196667	−	0.999807i	$-0.506261\pi$
$881$	−1.16748			−0.0393335			−0.0196667	+	0.999807i	$0.506261\pi$
$882$		0			0
$883$	−32.2410	+	32.2410i	−1.08500	+	1.08500i	−0.0889621	+	0.996035i	$0.528355\pi$
$883$	−32.2410	+	32.2410i	−1.08500	+	1.08500i	−0.996035	+	0.0889621i	$0.971645\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	42.7282i		−	1.43467i	−0.696728	−	0.717336i	$-0.745361\pi$
$887$		−	42.7282i		−	1.43467i	0.696728	−	0.717336i	$-0.254639\pi$
$888$		0			0
$889$			18.0980i			0.606987i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−14.4741	+	14.4741i	−0.484356	+	0.484356i
$894$		0			0
$895$	2.31644			0.0774302
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.76109	−	8.76109i	0.292199	−	0.292199i
$900$		0			0
$901$	19.9291	+	19.9291i	0.663935	+	0.663935i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	23.7244i		−	0.788625i
$906$		0			0
$907$	−1.23335	−	1.23335i	−0.0409528	−	0.0409528i	0.686334	−	0.727287i	$-0.259219\pi$
$907$	−1.23335	−	1.23335i	−0.0409528	−	0.0409528i	−0.727287	+	0.686334i	$0.759219\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	23.9284			0.792785			0.396392	−	0.918081i	$-0.370262\pi$
$911$	23.9284			0.792785			0.396392	+	0.918081i	$0.370262\pi$
$912$		0			0
$913$	36.0371			1.19265
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.9473	+	14.9473i	0.493605	+	0.493605i
$918$		0			0
$919$			45.3844i			1.49709i	0.663082	+	0.748546i	$0.269248\pi$
$919$			45.3844i			1.49709i	−0.663082	+	0.748546i	$0.730752\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.0392	+	10.0392i	0.330444	+	0.330444i
$924$		0			0
$925$	1.17899	−	1.17899i	0.0387649	−	0.0387649i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6.51036			0.213598			0.106799	−	0.994281i	$-0.465940\pi$
$929$	6.51036			0.213598			0.106799	+	0.994281i	$0.465940\pi$
$930$		0			0
$931$	−1.69986	+	1.69986i	−0.0557107	+	0.0557107i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	19.0402i		−	0.622681i
$936$		0			0
$937$		−	40.2986i		−	1.31650i	−0.752801	−	0.658248i	$-0.771298\pi$
$937$		−	40.2986i		−	1.31650i	0.752801	−	0.658248i	$-0.228702\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.10649	+	1.10649i	−0.0360705	+	0.0360705i	−0.724912	−	0.688841i	$-0.758120\pi$
$941$	−1.10649	+	1.10649i	−0.0360705	+	0.0360705i	0.688841	+	0.724912i	$0.258120\pi$
$942$		0			0
$943$	−21.2509			−0.692025
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.83833	+	8.83833i	−0.287207	+	0.287207i	−0.835975	−	0.548768i	$-0.815097\pi$
$947$	−8.83833	+	8.83833i	−0.287207	+	0.287207i	0.548768	+	0.835975i	$0.315097\pi$
$948$		0			0
$949$	−39.0796	−	39.0796i	−1.26858	−	1.26858i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	14.9610i		−	0.484636i	−0.970197	−	0.242318i	$-0.922092\pi$
$953$		−	14.9610i		−	0.484636i	0.970197	−	0.242318i	$-0.0779077\pi$
$954$		0			0
$955$	4.14234	+	4.14234i	0.134043	+	0.134043i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	54.8152			1.77008
$960$		0			0
$961$	−26.7441			−0.862712
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.0170522	−	0.0170522i	−0.000548930	−	0.000548930i
$966$		0			0
$967$			3.95287i			0.127116i	0.997978	+	0.0635578i	$0.0202447\pi$
$967$			3.95287i			0.127116i	−0.997978	+	0.0635578i	$0.979755\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−29.0538	−	29.0538i	−0.932380	−	0.932380i	0.0654740	−	0.997854i	$-0.479144\pi$
$971$	−29.0538	−	29.0538i	−0.932380	−	0.932380i	−0.997854	+	0.0654740i	$0.979144\pi$
$972$		0			0
$973$	−8.07597	+	8.07597i	−0.258904	+	0.258904i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.8962			−0.828494			−0.414247	−	0.910164i	$-0.635955\pi$
$977$	−25.8962			−0.828494			−0.414247	+	0.910164i	$0.635955\pi$
$978$		0			0
$979$	−6.90984	+	6.90984i	−0.220839	+	0.220839i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			22.0151i			0.702173i	0.936343	+	0.351087i	$0.114188\pi$
$983$			22.0151i			0.702173i	−0.936343	+	0.351087i	$0.885812\pi$
$984$		0			0
$985$		−	21.0839i		−	0.671789i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13.9537	+	13.9537i	−0.443702	+	0.443702i
$990$		0			0
$991$	54.3207			1.72556			0.862778	−	0.505583i	$-0.168723\pi$
$991$	54.3207			1.72556			0.862778	+	0.505583i	$0.168723\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9.65378	−	9.65378i	0.306045	−	0.306045i
$996$		0			0
$997$	8.14405	+	8.14405i	0.257925	+	0.257925i	0.824210	−	0.566285i	$-0.191620\pi$
$997$	8.14405	+	8.14405i	0.257925	+	0.257925i	−0.566285	+	0.824210i	$0.691620\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.c.2161.6		16
3.2	odd	2		320.2.l.a.241.7		16
4.3	odd	2		720.2.t.c.181.5		16
12.11	even	2		80.2.l.a.21.4	✓	16
15.2	even	4		1600.2.q.h.49.2		16
15.8	even	4		1600.2.q.g.49.7		16
15.14	odd	2		1600.2.l.i.1201.2		16
16.3	odd	4		720.2.t.c.541.5		16
16.13	even	4	inner	2880.2.t.c.721.7		16
24.5	odd	2		640.2.l.a.481.2		16
24.11	even	2		640.2.l.b.481.7		16
48.5	odd	4		640.2.l.a.161.2		16
48.11	even	4		640.2.l.b.161.7		16
48.29	odd	4		320.2.l.a.81.7		16
48.35	even	4		80.2.l.a.61.4	yes	16
60.23	odd	4		400.2.q.h.149.7		16
60.47	odd	4		400.2.q.g.149.2		16
60.59	even	2		400.2.l.h.101.5		16
96.29	odd	8		5120.2.a.t.1.8		8
96.35	even	8		5120.2.a.v.1.1		8
96.77	odd	8		5120.2.a.u.1.1		8
96.83	even	8		5120.2.a.s.1.8		8
240.29	odd	4		1600.2.l.i.401.2		16
240.77	even	4		1600.2.q.g.849.7		16
240.83	odd	4		400.2.q.g.349.2		16
240.173	even	4		1600.2.q.h.849.2		16
240.179	even	4		400.2.l.h.301.5		16
240.227	odd	4		400.2.q.h.349.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.4	✓	16	12.11	even	2
80.2.l.a.61.4	yes	16	48.35	even	4
320.2.l.a.81.7		16	48.29	odd	4
320.2.l.a.241.7		16	3.2	odd	2
400.2.l.h.101.5		16	60.59	even	2
400.2.l.h.301.5		16	240.179	even	4
400.2.q.g.149.2		16	60.47	odd	4
400.2.q.g.349.2		16	240.83	odd	4
400.2.q.h.149.7		16	60.23	odd	4
400.2.q.h.349.7		16	240.227	odd	4
640.2.l.a.161.2		16	48.5	odd	4
640.2.l.a.481.2		16	24.5	odd	2
640.2.l.b.161.7		16	48.11	even	4
640.2.l.b.481.7		16	24.11	even	2
720.2.t.c.181.5		16	4.3	odd	2
720.2.t.c.541.5		16	16.3	odd	4
1600.2.l.i.401.2		16	240.29	odd	4
1600.2.l.i.1201.2		16	15.14	odd	2
1600.2.q.g.49.7		16	15.8	even	4
1600.2.q.g.849.7		16	240.77	even	4
1600.2.q.h.49.2		16	15.2	even	4
1600.2.q.h.849.2		16	240.173	even	4
2880.2.t.c.721.7		16	16.13	even	4	inner
2880.2.t.c.2161.6		16	1.1	even	1	trivial
5120.2.a.s.1.8		8	96.83	even	8
5120.2.a.t.1.8		8	96.29	odd	8
5120.2.a.u.1.1		8	96.77	odd	8
5120.2.a.v.1.1		8	96.35	even	8

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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} -2.89402i q^{7} +(1.84462 + 1.84462i) q^{11} +(-3.08011 + 3.08011i) q^{13} -7.29875 q^{17} +(1.23593 - 1.23593i) q^{19} -4.60490i q^{23} +1.00000i q^{25} +(-4.24680 + 4.24680i) q^{29} -2.06299 q^{31} +(2.04638 - 2.04638i) q^{35} +(-1.17899 - 1.17899i) q^{37} -4.61484i q^{41} +(-3.03019 - 3.03019i) q^{43} -11.7111 q^{47} -1.37537 q^{49} +(-2.73048 - 2.73048i) q^{53} +2.60869i q^{55} +(3.11306 + 3.11306i) q^{59} +(2.34962 - 2.34962i) q^{61} -4.35593 q^{65} +(-8.24311 + 8.24311i) q^{67} -3.25937i q^{71} +12.6877i q^{73} +(5.33839 - 5.33839i) q^{77} +0.113885 q^{79} +(9.76813 - 9.76813i) q^{83} +(-5.16100 - 5.16100i) q^{85} +3.74593i q^{89} +(8.91390 + 8.91390i) q^{91} +1.74787 q^{95} -13.9853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) \) 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Introduction

L-functions

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