LMFDB - Embedded newform 2300.2.i.b.1057.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1057,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("2300.1057");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.i (of order $4$, degree $2$, 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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1057.3
Root			$1.28897 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	2300.1057
Dual form			2300.2.i.b.1793.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.87083 - 1.87083i) q^{3} +(-1.41421 + 1.41421i) q^{7} -4.00000i q^{9} +O(q^{10})$	$q+(1.87083 - 1.87083i) q^{3} +(-1.41421 + 1.41421i) q^{7} -4.00000i q^{9} +5.29150i q^{11} +(1.87083 - 1.87083i) q^{13} +5.29150 q^{19} +5.29150i q^{21} +(0.957598 + 4.69926i) q^{23} +(-1.87083 - 1.87083i) q^{27} +1.00000i q^{29} +3.00000 q^{31} +(9.89949 + 9.89949i) q^{33} +(7.07107 - 7.07107i) q^{37} -7.00000i q^{39} +9.00000 q^{41} +(-4.24264 - 4.24264i) q^{43} +(-1.87083 - 1.87083i) q^{47} +3.00000i q^{49} +(-2.82843 - 2.82843i) q^{53} +(9.89949 - 9.89949i) q^{57} +5.29150i q^{61} +(5.65685 + 5.65685i) q^{63} +(1.41421 - 1.41421i) q^{67} +(10.5830 + 7.00000i) q^{69} -9.00000 q^{71} +(9.35414 - 9.35414i) q^{73} +(-7.48331 - 7.48331i) q^{77} +5.29150 q^{79} +5.00000 q^{81} +(8.48528 + 8.48528i) q^{83} +(1.87083 + 1.87083i) q^{87} -15.8745 q^{89} +5.29150i q^{91} +(5.61249 - 5.61249i) q^{93} +(1.41421 - 1.41421i) q^{97} +21.1660 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 24 q^{31} + 72 q^{41} - 72 q^{71} + 40 q^{81}+O(q^{100}) $ 8 * q + 24 * q^31 + 72 * q^41 - 72 * q^71 + 40 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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
$3$	1.87083	−	1.87083i	1.08012	−	1.08012i	0.0836263	−	0.996497i	$-0.473350\pi$
$3$	1.87083	−	1.87083i	1.08012	−	1.08012i	0.996497	−	0.0836263i	$-0.0266502\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.41421	+	1.41421i	−0.534522	+	0.534522i	−0.921915	−	0.387392i	$-0.873376\pi$
$7$	−1.41421	+	1.41421i	−0.534522	+	0.534522i	0.387392	+	0.921915i	$0.373376\pi$
$8$		0			0
$9$		−	4.00000i		−	1.33333i
$10$		0			0
$11$			5.29150i			1.59545i	0.603023	+	0.797724i	$0.293963\pi$
$11$			5.29150i			1.59545i	−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	1.87083	−	1.87083i	0.518875	−	0.518875i	−0.398356	−	0.917231i	$-0.630419\pi$
$13$	1.87083	−	1.87083i	0.518875	−	0.518875i	0.917231	+	0.398356i	$0.130419\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$	5.29150			1.21395			0.606977	−	0.794719i	$-0.292382\pi$
$19$	5.29150			1.21395			0.606977	+	0.794719i	$0.292382\pi$
$20$		0			0
$21$			5.29150i			1.15470i
$22$		0			0
$23$	0.957598	+	4.69926i	0.199673	+	0.979863i
$23$	0.957598	+	4.69926i	0.199673	+	0.979863i
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.87083	−	1.87083i	−0.360041	−	0.360041i
$28$		0			0
$29$			1.00000i			0.185695i	0.995680	+	0.0928477i	$0.0295970\pi$
$29$			1.00000i			0.185695i	−0.995680	+	0.0928477i	$0.970403\pi$
$30$		0			0
$31$	3.00000			0.538816			0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000			0.538816			0.269408	+	0.963026i	$0.413172\pi$
$32$		0			0
$33$	9.89949	+	9.89949i	1.72328	+	1.72328i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.07107	−	7.07107i	1.16248	−	1.16248i	0.178545	−	0.983932i	$-0.442861\pi$
$37$	7.07107	−	7.07107i	1.16248	−	1.16248i	0.983932	−	0.178545i	$-0.0571389\pi$
$38$		0			0
$39$		−	7.00000i		−	1.12090i
$40$		0			0
$41$	9.00000			1.40556			0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000			1.40556			0.702782	+	0.711405i	$0.251941\pi$
$42$		0			0
$43$	−4.24264	−	4.24264i	−0.646997	−	0.646997i	0.305269	−	0.952266i	$-0.401253\pi$
$43$	−4.24264	−	4.24264i	−0.646997	−	0.646997i	−0.952266	+	0.305269i	$0.901253\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.87083	−	1.87083i	−0.272888	−	0.272888i	0.557373	−	0.830262i	$-0.311809\pi$
$47$	−1.87083	−	1.87083i	−0.272888	−	0.272888i	−0.830262	+	0.557373i	$0.811809\pi$
$48$		0			0
$49$			3.00000i			0.428571i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.82843	−	2.82843i	−0.388514	−	0.388514i	0.485643	−	0.874157i	$-0.338586\pi$
$53$	−2.82843	−	2.82843i	−0.388514	−	0.388514i	−0.874157	+	0.485643i	$0.838586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	9.89949	−	9.89949i	1.31122	−	1.31122i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$			5.29150i			0.677507i	0.940875	+	0.338754i	$0.110005\pi$
$61$			5.29150i			0.677507i	−0.940875	+	0.338754i	$0.889995\pi$
$62$		0			0
$63$	5.65685	+	5.65685i	0.712697	+	0.712697i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.41421	−	1.41421i	0.172774	−	0.172774i	−0.615423	−	0.788197i	$-0.711015\pi$
$67$	1.41421	−	1.41421i	0.172774	−	0.172774i	0.788197	+	0.615423i	$0.211015\pi$
$68$		0			0
$69$	10.5830	+	7.00000i	1.27404	+	0.842701i
$70$		0			0
$71$	−9.00000			−1.06810			−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000			−1.06810			−0.534052	+	0.845452i	$0.679331\pi$
$72$		0			0
$73$	9.35414	−	9.35414i	1.09482	−	1.09482i	0.0998135	−	0.995006i	$-0.468175\pi$
$73$	9.35414	−	9.35414i	1.09482	−	1.09482i	0.995006	−	0.0998135i	$-0.0318246\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.48331	−	7.48331i	−0.852803	−	0.852803i
$78$		0			0
$79$	5.29150			0.595341			0.297670	−	0.954669i	$-0.403790\pi$
$79$	5.29150			0.595341			0.297670	+	0.954669i	$0.403790\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	8.48528	+	8.48528i	0.931381	+	0.931381i	0.997792	−	0.0664117i	$-0.0211551\pi$
$83$	8.48528	+	8.48528i	0.931381	+	0.931381i	−0.0664117	+	0.997792i	$0.521155\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.87083	+	1.87083i	0.200574	+	0.200574i
$88$		0			0
$89$	−15.8745			−1.68269			−0.841347	−	0.540495i	$-0.818237\pi$
$89$	−15.8745			−1.68269			−0.841347	+	0.540495i	$0.818237\pi$
$90$		0			0
$91$			5.29150i			0.554700i
$92$		0			0
$93$	5.61249	−	5.61249i	0.581988	−	0.581988i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.41421	−	1.41421i	0.143592	−	0.143592i	−0.631657	−	0.775248i	$-0.717625\pi$
$97$	1.41421	−	1.41421i	0.143592	−	0.143592i	0.775248	+	0.631657i	$0.217625\pi$
$98$		0			0
$99$	21.1660			2.12726
$100$		0			0
$101$	14.0000			1.39305			0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000			1.39305			0.696526	+	0.717532i	$0.254728\pi$
$102$		0			0
$103$	5.65685	+	5.65685i	0.557386	+	0.557386i	0.928562	−	0.371176i	$-0.121045\pi$
$103$	5.65685	+	5.65685i	0.557386	+	0.557386i	−0.371176	+	0.928562i	$0.621045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.24264	+	4.24264i	−0.410152	+	0.410152i	−0.881791	−	0.471640i	$-0.843662\pi$
$107$	−4.24264	+	4.24264i	−0.410152	+	0.410152i	0.471640	+	0.881791i	$0.343662\pi$
$108$		0			0
$109$	−15.8745			−1.52050			−0.760251	−	0.649629i	$-0.774924\pi$
$109$	−15.8745			−1.52050			−0.760251	+	0.649629i	$0.774924\pi$
$110$		0			0
$111$		−	26.4575i		−	2.51124i
$112$		0			0
$113$	4.24264	+	4.24264i	0.399114	+	0.399114i	0.877920	−	0.478806i	$-0.158930\pi$
$113$	4.24264	+	4.24264i	0.399114	+	0.399114i	−0.478806	+	0.877920i	$0.658930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.48331	−	7.48331i	−0.691833	−	0.691833i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−17.0000			−1.54545
$122$		0			0
$123$	16.8375	−	16.8375i	1.51818	−	1.51818i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−5.61249	−	5.61249i	−0.498028	−	0.498028i	0.412796	−	0.910824i	$-0.364552\pi$
$127$	−5.61249	−	5.61249i	−0.498028	−	0.498028i	−0.910824	+	0.412796i	$0.864552\pi$
$128$		0			0
$129$	−15.8745			−1.39767
$130$		0			0
$131$	−1.00000			−0.0873704			−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000			−0.0873704			−0.0436852	+	0.999045i	$0.513910\pi$
$132$		0			0
$133$	−7.48331	+	7.48331i	−0.648886	+	0.648886i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.48528	+	8.48528i	−0.724947	+	0.724947i	−0.969608	−	0.244662i	$-0.921323\pi$
$137$	−8.48528	+	8.48528i	−0.724947	+	0.724947i	0.244662	+	0.969608i	$0.421323\pi$
$138$		0			0
$139$			5.00000i			0.424094i	0.977259	+	0.212047i	$0.0680131\pi$
$139$			5.00000i			0.424094i	−0.977259	+	0.212047i	$0.931987\pi$
$140$		0			0
$141$	−7.00000			−0.589506
$142$		0			0
$143$	9.89949	+	9.89949i	0.827837	+	0.827837i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.61249	+	5.61249i	0.462910	+	0.462910i
$148$		0			0
$149$	−10.5830			−0.866994			−0.433497	−	0.901155i	$-0.642720\pi$
$149$	−10.5830			−0.866994			−0.433497	+	0.901155i	$0.642720\pi$
$150$		0			0
$151$	−23.0000			−1.87171			−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−23.0000			−1.87171			−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	12.7279	−	12.7279i	1.01580	−	1.01580i	0.0159256	−	0.999873i	$-0.494931\pi$
$157$	12.7279	−	12.7279i	1.01580	−	1.01580i	0.999873	−	0.0159256i	$-0.00506949\pi$
$158$		0			0
$159$	−10.5830			−0.839287
$160$		0			0
$161$	−8.00000	−	5.29150i	−0.630488	−	0.417029i
$162$		0			0
$163$	9.35414	−	9.35414i	0.732673	−	0.732673i	−0.238475	−	0.971149i	$-0.576648\pi$
$163$	9.35414	−	9.35414i	0.732673	−	0.732673i	0.971149	+	0.238475i	$0.0766477\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.2250	+	11.2250i	0.868614	+	0.868614i	0.992319	−	0.123705i	$-0.0394775\pi$
$167$	11.2250	+	11.2250i	0.868614	+	0.868614i	−0.123705	+	0.992319i	$0.539478\pi$
$168$		0			0
$169$			6.00000i			0.461538i
$170$		0			0
$171$		−	21.1660i		−	1.61861i
$172$		0			0
$173$	7.48331	−	7.48331i	0.568946	−	0.568946i	−0.362887	−	0.931833i	$-0.618209\pi$
$173$	7.48331	−	7.48331i	0.568946	−	0.568946i	0.931833	+	0.362887i	$0.118209\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	5.00000i		−	0.373718i	−0.982387	−	0.186859i	$-0.940169\pi$
$179$		−	5.00000i		−	0.373718i	0.982387	−	0.186859i	$-0.0598307\pi$
$180$		0			0
$181$		−	5.29150i		−	0.393314i	−0.980472	−	0.196657i	$-0.936991\pi$
$181$		−	5.29150i		−	0.393314i	0.980472	−	0.196657i	$-0.0630086\pi$
$182$		0			0
$183$	9.89949	+	9.89949i	0.731792	+	0.731792i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	5.29150			0.384900
$190$		0			0
$191$		−	10.5830i		−	0.765759i	−0.923798	−	0.382880i	$-0.874932\pi$
$191$		−	10.5830i		−	0.765759i	0.923798	−	0.382880i	$-0.125068\pi$
$192$		0			0
$193$	−9.35414	+	9.35414i	−0.673326	+	0.673326i	−0.958481	−	0.285155i	$-0.907955\pi$
$193$	−9.35414	+	9.35414i	−0.673326	+	0.673326i	0.285155	+	0.958481i	$0.407955\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.35414	−	9.35414i	−0.666455	−	0.666455i	0.290439	−	0.956894i	$-0.406199\pi$
$197$	−9.35414	−	9.35414i	−0.666455	−	0.666455i	−0.956894	+	0.290439i	$0.906199\pi$
$198$		0			0
$199$	−5.29150			−0.375105			−0.187552	−	0.982255i	$-0.560055\pi$
$199$	−5.29150			−0.375105			−0.187552	+	0.982255i	$0.560055\pi$
$200$		0			0
$201$		−	5.29150i		−	0.373234i
$202$		0			0
$203$	−1.41421	−	1.41421i	−0.0992583	−	0.0992583i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	18.7970	−	3.83039i	1.30648	−	0.266231i
$208$		0			0
$209$			28.0000i			1.93680i
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$	−16.8375	+	16.8375i	−1.15368	+	1.15368i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−4.24264	+	4.24264i	−0.288009	+	0.288009i
$218$		0			0
$219$		−	35.0000i		−	2.36508i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−11.2250	+	11.2250i	−0.751680	+	0.751680i	−0.974793	−	0.223113i	$-0.928378\pi$
$223$	−11.2250	+	11.2250i	−0.751680	+	0.751680i	0.223113	+	0.974793i	$0.428378\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.7279	−	12.7279i	0.844782	−	0.844782i	−0.144695	−	0.989476i	$-0.546220\pi$
$227$	12.7279	−	12.7279i	0.844782	−	0.844782i	0.989476	+	0.144695i	$0.0462199\pi$
$228$		0			0
$229$	−26.4575			−1.74836			−0.874181	−	0.485601i	$-0.838601\pi$
$229$	−26.4575			−1.74836			−0.874181	+	0.485601i	$0.838601\pi$
$230$		0			0
$231$	−28.0000			−1.84226
$232$		0			0
$233$	−1.87083	+	1.87083i	−0.122562	+	0.122562i	−0.765727	−	0.643165i	$-0.777621\pi$
$233$	−1.87083	+	1.87083i	−0.122562	+	0.122562i	0.643165	+	0.765727i	$0.277621\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	9.89949	−	9.89949i	0.643041	−	0.643041i
$238$		0			0
$239$		−	1.00000i		−	0.0646846i	−0.999477	−	0.0323423i	$-0.989703\pi$
$239$		−	1.00000i		−	0.0646846i	0.999477	−	0.0323423i	$-0.0102967\pi$
$240$		0			0
$241$			21.1660i			1.36342i	0.731621	+	0.681711i	$0.238764\pi$
$241$			21.1660i			1.36342i	−0.731621	+	0.681711i	$0.761236\pi$
$242$		0			0
$243$	14.9666	−	14.9666i	0.960110	−	0.960110i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	9.89949	−	9.89949i	0.629890	−	0.629890i
$248$		0			0
$249$	31.7490			2.01201
$250$		0			0
$251$			21.1660i			1.33599i	0.744167	+	0.667993i	$0.232847\pi$
$251$			21.1660i			1.33599i	−0.744167	+	0.667993i	$0.767153\pi$
$252$		0			0
$253$	−24.8661	+	5.06713i	−1.56332	+	0.318568i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.8375	−	16.8375i	−1.05029	−	1.05029i	−0.998667	−	0.0516253i	$-0.983560\pi$
$257$	−16.8375	−	16.8375i	−1.05029	−	1.05029i	−0.0516253	−	0.998667i	$-0.516440\pi$
$258$		0			0
$259$			20.0000i			1.24274i
$260$		0			0
$261$	4.00000			0.247594
$262$		0			0
$263$	−18.3848	−	18.3848i	−1.13365	−	1.13365i	−0.989565	−	0.144089i	$-0.953975\pi$
$263$	−18.3848	−	18.3848i	−1.13365	−	1.13365i	−0.144089	−	0.989565i	$-0.546025\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−29.6985	+	29.6985i	−1.81752	+	1.81752i
$268$		0			0
$269$		−	3.00000i		−	0.182913i	−0.995809	−	0.0914566i	$-0.970848\pi$
$269$		−	3.00000i		−	0.182913i	0.995809	−	0.0914566i	$-0.0291523\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$	9.89949	+	9.89949i	0.599145	+	0.599145i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.61249	−	5.61249i	−0.337222	−	0.337222i	0.518099	−	0.855321i	$-0.326640\pi$
$277$	−5.61249	−	5.61249i	−0.337222	−	0.337222i	−0.855321	+	0.518099i	$0.826640\pi$
$278$		0			0
$279$		−	12.0000i		−	0.718421i
$280$		0			0
$281$		−	15.8745i		−	0.946994i	−0.880795	−	0.473497i	$-0.842992\pi$
$281$		−	15.8745i		−	0.946994i	0.880795	−	0.473497i	$-0.157008\pi$
$282$		0			0
$283$	16.9706	+	16.9706i	1.00880	+	1.00880i	0.999961	+	0.00883427i	$0.00281207\pi$
$283$	16.9706	+	16.9706i	1.00880	+	1.00880i	0.00883427	+	0.999961i	$0.497188\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.7279	+	12.7279i	−0.751305	+	0.751305i
$288$		0			0
$289$			17.0000i			1.00000i
$290$		0			0
$291$		−	5.29150i		−	0.310193i
$292$		0			0
$293$	−11.3137	−	11.3137i	−0.660954	−	0.660954i	0.294651	−	0.955605i	$-0.404797\pi$
$293$	−11.3137	−	11.3137i	−0.660954	−	0.660954i	−0.955605	+	0.294651i	$0.904797\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	9.89949	−	9.89949i	0.574427	−	0.574427i
$298$		0			0
$299$	10.5830	+	7.00000i	0.612031	+	0.404820i
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$	26.1916	−	26.1916i	1.50467	−	1.50467i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−11.2250	−	11.2250i	−0.640643	−	0.640643i	0.310071	−	0.950714i	$-0.399647\pi$
$307$	−11.2250	−	11.2250i	−0.640643	−	0.640643i	−0.950714	+	0.310071i	$0.899647\pi$
$308$		0			0
$309$	21.1660			1.20409
$310$		0			0
$311$	21.0000			1.19080			0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000			1.19080			0.595400	+	0.803429i	$0.296993\pi$
$312$		0			0
$313$	−14.1421	−	14.1421i	−0.799361	−	0.799361i	0.183634	−	0.982995i	$-0.441214\pi$
$313$	−14.1421	−	14.1421i	−0.799361	−	0.799361i	−0.982995	+	0.183634i	$0.941214\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.4499	+	22.4499i	1.26091	+	1.26091i	0.950650	+	0.310264i	$0.100417\pi$
$317$	22.4499	+	22.4499i	1.26091	+	1.26091i	0.310264	+	0.950650i	$0.399583\pi$
$318$		0			0
$319$	−5.29150			−0.296267
$320$		0			0
$321$			15.8745i			0.886029i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−29.6985	+	29.6985i	−1.64233	+	1.64233i
$328$		0			0
$329$	5.29150			0.291730
$330$		0			0
$331$	29.0000			1.59398			0.796992	−	0.603990i	$-0.206423\pi$
$331$	29.0000			1.59398			0.796992	+	0.603990i	$0.206423\pi$
$332$		0			0
$333$	−28.2843	−	28.2843i	−1.54997	−	1.54997i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.41421	−	1.41421i	0.0770371	−	0.0770371i	−0.667538	−	0.744575i	$-0.732652\pi$
$337$	1.41421	−	1.41421i	0.0770371	−	0.0770371i	0.744575	+	0.667538i	$0.232652\pi$
$338$		0			0
$339$	15.8745			0.862185
$340$		0			0
$341$			15.8745i			0.859653i
$342$		0			0
$343$	−14.1421	−	14.1421i	−0.763604	−	0.763604i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.2250	+	11.2250i	0.602588	+	0.602588i	0.940999	−	0.338411i	$-0.109889\pi$
$347$	11.2250	+	11.2250i	0.602588	+	0.602588i	−0.338411	+	0.940999i	$0.609889\pi$
$348$		0			0
$349$		−	23.0000i		−	1.23116i	−0.788074	−	0.615581i	$-0.788921\pi$
$349$		−	23.0000i		−	1.23116i	0.788074	−	0.615581i	$-0.211079\pi$
$350$		0			0
$351$	−7.00000			−0.373632
$352$		0			0
$353$	16.8375	−	16.8375i	0.896167	−	0.896167i	−0.0989272	−	0.995095i	$-0.531541\pi$
$353$	16.8375	−	16.8375i	0.896167	−	0.896167i	0.995095	+	0.0989272i	$0.0315411\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.8745			−0.837824			−0.418912	−	0.908027i	$-0.637589\pi$
$359$	−15.8745			−0.837824			−0.418912	+	0.908027i	$0.637589\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$		0			0
$363$	−31.8041	+	31.8041i	−1.66928	+	1.66928i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	2.82843	−	2.82843i	0.147643	−	0.147643i	−0.629421	−	0.777064i	$-0.716708\pi$
$367$	2.82843	−	2.82843i	0.147643	−	0.147643i	0.777064	+	0.629421i	$0.216708\pi$
$368$		0			0
$369$		−	36.0000i		−	1.87409i
$370$		0			0
$371$	8.00000			0.415339
$372$		0			0
$373$	−2.82843	−	2.82843i	−0.146450	−	0.146450i	0.630080	−	0.776530i	$-0.283022\pi$
$373$	−2.82843	−	2.82843i	−0.146450	−	0.146450i	−0.776530	+	0.630080i	$0.783022\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.87083	+	1.87083i	0.0963526	+	0.0963526i
$378$		0			0
$379$	10.5830			0.543612			0.271806	−	0.962352i	$-0.412379\pi$
$379$	10.5830			0.543612			0.271806	+	0.962352i	$0.412379\pi$
$380$		0			0
$381$	−21.0000			−1.07586
$382$		0			0
$383$	24.0416	+	24.0416i	1.22847	+	1.22847i	0.964543	+	0.263927i	$0.0850179\pi$
$383$	24.0416	+	24.0416i	1.22847	+	1.22847i	0.263927	+	0.964543i	$0.414982\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−16.9706	+	16.9706i	−0.862662	+	0.862662i
$388$		0			0
$389$	−10.5830			−0.536580			−0.268290	−	0.963338i	$-0.586458\pi$
$389$	−10.5830			−0.536580			−0.268290	+	0.963338i	$0.586458\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−1.87083	+	1.87083i	−0.0943708	+	0.0943708i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−5.61249	−	5.61249i	−0.281683	−	0.281683i	0.552097	−	0.833780i	$-0.313828\pi$
$397$	−5.61249	−	5.61249i	−0.281683	−	0.281683i	−0.833780	+	0.552097i	$0.813828\pi$
$398$		0			0
$399$			28.0000i			1.40175i
$400$		0			0
$401$			31.7490i			1.58547i	0.609566	+	0.792735i	$0.291344\pi$
$401$			31.7490i			1.58547i	−0.609566	+	0.792735i	$0.708656\pi$
$402$		0			0
$403$	5.61249	−	5.61249i	0.279578	−	0.279578i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	37.4166	+	37.4166i	1.85467	+	1.85467i
$408$		0			0
$409$			7.00000i			0.346128i	0.984911	+	0.173064i	$0.0553667\pi$
$409$			7.00000i			0.346128i	−0.984911	+	0.173064i	$0.944633\pi$
$410$		0			0
$411$			31.7490i			1.56606i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	9.35414	+	9.35414i	0.458074	+	0.458074i
$418$		0			0
$419$	15.8745			0.775520			0.387760	−	0.921760i	$-0.373249\pi$
$419$	15.8745			0.775520			0.387760	+	0.921760i	$0.373249\pi$
$420$		0			0
$421$		−	5.29150i		−	0.257892i	−0.991652	−	0.128946i	$-0.958841\pi$
$421$		−	5.29150i		−	0.257892i	0.991652	−	0.128946i	$-0.0411594\pi$
$422$		0			0
$423$	−7.48331	+	7.48331i	−0.363851	+	0.363851i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.48331	−	7.48331i	−0.362143	−	0.362143i
$428$		0			0
$429$	37.0405			1.78833
$430$		0			0
$431$			15.8745i			0.764648i	0.924028	+	0.382324i	$0.124876\pi$
$431$			15.8745i			0.764648i	−0.924028	+	0.382324i	$0.875124\pi$
$432$		0			0
$433$	−8.48528	−	8.48528i	−0.407777	−	0.407777i	0.473186	−	0.880963i	$-0.343104\pi$
$433$	−8.48528	−	8.48528i	−0.407777	−	0.407777i	−0.880963	+	0.473186i	$0.843104\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.06713	+	24.8661i	0.242394	+	1.18951i
$438$		0			0
$439$			15.0000i			0.715911i	0.933739	+	0.357955i	$0.116526\pi$
$439$			15.0000i			0.715911i	−0.933739	+	0.357955i	$0.883474\pi$
$440$		0			0
$441$	12.0000			0.571429
$442$		0			0
$443$	−20.5791	+	20.5791i	−0.977743	+	0.977743i	−0.999758	−	0.0220144i	$-0.992992\pi$
$443$	−20.5791	+	20.5791i	−0.977743	+	0.977743i	0.0220144	+	0.999758i	$0.492992\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−19.7990	+	19.7990i	−0.936460	+	0.936460i
$448$		0			0
$449$		−	26.0000i		−	1.22702i	−0.789689	−	0.613508i	$-0.789758\pi$
$449$		−	26.0000i		−	1.22702i	0.789689	−	0.613508i	$-0.210242\pi$
$450$		0			0
$451$			47.6235i			2.24250i
$452$		0			0
$453$	−43.0291	+	43.0291i	−2.02168	+	2.02168i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	25.4558	−	25.4558i	1.19077	−	1.19077i	0.213924	−	0.976850i	$-0.431376\pi$
$457$	25.4558	−	25.4558i	1.19077	−	1.19077i	0.976850	−	0.213924i	$-0.0686244\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.0000			−0.884918			−0.442459	−	0.896789i	$-0.645894\pi$
$461$	−19.0000			−0.884918			−0.442459	+	0.896789i	$0.645894\pi$
$462$		0			0
$463$	−26.1916	+	26.1916i	−1.21723	+	1.21723i	−0.248628	+	0.968599i	$0.579980\pi$
$463$	−26.1916	+	26.1916i	−1.21723	+	1.21723i	−0.968599	+	0.248628i	$0.920020\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.7990	+	19.7990i	−0.916188	+	0.916188i	−0.996750	−	0.0805616i	$-0.974329\pi$
$467$	−19.7990	+	19.7990i	−0.916188	+	0.916188i	0.0805616	+	0.996750i	$0.474329\pi$
$468$		0			0
$469$			4.00000i			0.184703i
$470$		0			0
$471$		−	47.6235i		−	2.19438i
$472$		0			0
$473$	22.4499	−	22.4499i	1.03225	−	1.03225i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−11.3137	+	11.3137i	−0.518019	+	0.518019i
$478$		0			0
$479$	−31.7490			−1.45065			−0.725325	−	0.688407i	$-0.758310\pi$
$479$	−31.7490			−1.45065			−0.725325	+	0.688407i	$0.758310\pi$
$480$		0			0
$481$		−	26.4575i		−	1.20636i
$482$		0			0
$483$	−24.8661	+	5.06713i	−1.13145	+	0.230563i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−20.5791	−	20.5791i	−0.932529	−	0.932529i	0.0653347	−	0.997863i	$-0.479189\pi$
$487$	−20.5791	−	20.5791i	−0.932529	−	0.932529i	−0.997863	+	0.0653347i	$0.979189\pi$
$488$		0			0
$489$		−	35.0000i		−	1.58275i
$490$		0			0
$491$	23.0000			1.03798			0.518988	−	0.854782i	$-0.326309\pi$
$491$	23.0000			1.03798			0.518988	+	0.854782i	$0.326309\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.7279	−	12.7279i	0.570925	−	0.570925i
$498$		0			0
$499$		−	39.0000i		−	1.74588i	−0.487828	−	0.872940i	$-0.662211\pi$
$499$		−	39.0000i		−	1.74588i	0.487828	−	0.872940i	$-0.337789\pi$
$500$		0			0
$501$	42.0000			1.87642
$502$		0			0
$503$	−19.7990	−	19.7990i	−0.882793	−	0.882793i	0.111024	−	0.993818i	$-0.464587\pi$
$503$	−19.7990	−	19.7990i	−0.882793	−	0.882793i	−0.993818	+	0.111024i	$0.964587\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	11.2250	+	11.2250i	0.498519	+	0.498519i
$508$		0			0
$509$			21.0000i			0.930809i	0.885098	+	0.465404i	$0.154091\pi$
$509$			21.0000i			0.930809i	−0.885098	+	0.465404i	$0.845909\pi$
$510$		0			0
$511$			26.4575i			1.17041i
$512$		0			0
$513$	−9.89949	−	9.89949i	−0.437073	−	0.437073i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	9.89949	−	9.89949i	0.435379	−	0.435379i
$518$		0			0
$519$		−	28.0000i		−	1.22906i
$520$		0			0
$521$		−	10.5830i		−	0.463650i	−0.972758	−	0.231825i	$-0.925530\pi$
$521$		−	10.5830i		−	0.463650i	0.972758	−	0.231825i	$-0.0744697\pi$
$522$		0			0
$523$	−24.0416	−	24.0416i	−1.05127	−	1.05127i	−0.998613	−	0.0526543i	$-0.983232\pi$
$523$	−24.0416	−	24.0416i	−1.05127	−	1.05127i	−0.0526543	−	0.998613i	$-0.516768\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−21.1660	+	9.00000i	−0.920261	+	0.391304i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	16.8375	−	16.8375i	0.729311	−	0.729311i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−9.35414	−	9.35414i	−0.403661	−	0.403661i
$538$		0			0
$539$	−15.8745			−0.683763
$540$		0			0
$541$	−25.0000			−1.07483			−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000			−1.07483			−0.537417	+	0.843317i	$0.680600\pi$
$542$		0			0
$543$	−9.89949	−	9.89949i	−0.424828	−	0.424828i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.87083	+	1.87083i	0.0799909	+	0.0799909i	0.745970	−	0.665979i	$-0.231986\pi$
$547$	1.87083	+	1.87083i	0.0799909	+	0.0799909i	−0.665979	+	0.745970i	$0.731986\pi$
$548$		0			0
$549$	21.1660			0.903343
$550$		0			0
$551$			5.29150i			0.225426i
$552$		0			0
$553$	−7.48331	+	7.48331i	−0.318223	+	0.318223i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.41421	−	1.41421i	0.0599222	−	0.0599222i	−0.676511	−	0.736433i	$-0.736509\pi$
$557$	1.41421	−	1.41421i	0.0599222	−	0.0599222i	0.736433	+	0.676511i	$0.236509\pi$
$558$		0			0
$559$	−15.8745			−0.671420
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.82843	+	2.82843i	0.119204	+	0.119204i	0.764192	−	0.644988i	$-0.223138\pi$
$563$	2.82843	+	2.82843i	0.119204	+	0.119204i	−0.644988	+	0.764192i	$0.723138\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−7.07107	+	7.07107i	−0.296957	+	0.296957i
$568$		0			0
$569$	31.7490			1.33099			0.665494	−	0.746403i	$-0.268221\pi$
$569$	31.7490			1.33099			0.665494	+	0.746403i	$0.268221\pi$
$570$		0			0
$571$			10.5830i			0.442885i	0.975173	+	0.221442i	$0.0710765\pi$
$571$			10.5830i			0.442885i	−0.975173	+	0.221442i	$0.928923\pi$
$572$		0			0
$573$	−19.7990	−	19.7990i	−0.827115	−	0.827115i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−20.5791	−	20.5791i	−0.856720	−	0.856720i	0.134230	−	0.990950i	$-0.457144\pi$
$577$	−20.5791	−	20.5791i	−0.856720	−	0.856720i	−0.990950	+	0.134230i	$0.957144\pi$
$578$		0			0
$579$			35.0000i			1.45455i
$580$		0			0
$581$	−24.0000			−0.995688
$582$		0			0
$583$	14.9666	−	14.9666i	0.619854	−	0.619854i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.35414	−	9.35414i	−0.386087	−	0.386087i	0.487202	−	0.873289i	$-0.338017\pi$
$587$	−9.35414	−	9.35414i	−0.386087	−	0.386087i	−0.873289	+	0.487202i	$0.838017\pi$
$588$		0			0
$589$	15.8745			0.654098
$590$		0			0
$591$	−35.0000			−1.43971
$592$		0			0
$593$	−22.4499	+	22.4499i	−0.921909	+	0.921909i	−0.997164	−	0.0752556i	$-0.976023\pi$
$593$	−22.4499	+	22.4499i	−0.921909	+	0.921909i	0.0752556	+	0.997164i	$0.476023\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−9.89949	+	9.89949i	−0.405159	+	0.405159i
$598$		0			0
$599$			12.0000i			0.490307i	0.969484	+	0.245153i	$0.0788383\pi$
$599$			12.0000i			0.490307i	−0.969484	+	0.245153i	$0.921162\pi$
$600$		0			0
$601$	−9.00000			−0.367118			−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000			−0.367118			−0.183559	+	0.983009i	$0.558762\pi$
$602$		0			0
$603$	−5.65685	−	5.65685i	−0.230365	−	0.230365i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	3.74166	+	3.74166i	0.151869	+	0.151869i	0.778952	−	0.627083i	$-0.215751\pi$
$607$	3.74166	+	3.74166i	0.151869	+	0.151869i	−0.627083	+	0.778952i	$0.715751\pi$
$608$		0			0
$609$	−5.29150			−0.214423
$610$		0			0
$611$	−7.00000			−0.283190
$612$		0			0
$613$	−22.6274	−	22.6274i	−0.913913	−	0.913913i	0.0826647	−	0.996577i	$-0.473657\pi$
$613$	−22.6274	−	22.6274i	−0.913913	−	0.913913i	−0.996577	+	0.0826647i	$0.973657\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.5563	−	15.5563i	0.626275	−	0.626275i	−0.320854	−	0.947129i	$-0.603970\pi$
$617$	15.5563	−	15.5563i	0.626275	−	0.626275i	0.947129	+	0.320854i	$0.103970\pi$
$618$		0			0
$619$	−21.1660			−0.850734			−0.425367	−	0.905021i	$-0.639855\pi$
$619$	−21.1660			−0.850734			−0.425367	+	0.905021i	$0.639855\pi$
$620$		0			0
$621$	7.00000	−	10.5830i	0.280900	−	0.424681i
$622$		0			0
$623$	22.4499	−	22.4499i	0.899438	−	0.899438i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	52.3832	+	52.3832i	2.09198	+	2.09198i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	37.0405i		−	1.47456i	−0.675587	−	0.737280i	$-0.736110\pi$
$631$		−	37.0405i		−	1.47456i	0.675587	−	0.737280i	$-0.263890\pi$
$632$		0			0
$633$	37.4166	−	37.4166i	1.48718	−	1.48718i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	5.61249	+	5.61249i	0.222375	+	0.222375i
$638$		0			0
$639$			36.0000i			1.42414i
$640$		0			0
$641$		−	15.8745i		−	0.627005i	−0.949587	−	0.313503i	$-0.898498\pi$
$641$		−	15.8745i		−	0.627005i	0.949587	−	0.313503i	$-0.101502\pi$
$642$		0			0
$643$	−8.48528	−	8.48528i	−0.334627	−	0.334627i	0.519714	−	0.854341i	$-0.326039\pi$
$643$	−8.48528	−	8.48528i	−0.334627	−	0.334627i	−0.854341	+	0.519714i	$0.826039\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.35414	+	9.35414i	0.367749	+	0.367749i	0.866656	−	0.498906i	$-0.166265\pi$
$647$	9.35414	+	9.35414i	0.367749	+	0.367749i	−0.498906	+	0.866656i	$0.666265\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$			15.8745i			0.622171i
$652$		0			0
$653$	−9.35414	+	9.35414i	−0.366056	+	0.366056i	−0.866037	−	0.499981i	$-0.833340\pi$
$653$	−9.35414	+	9.35414i	−0.366056	+	0.366056i	0.499981	+	0.866037i	$0.333340\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−37.4166	−	37.4166i	−1.45976	−	1.45976i
$658$		0			0
$659$	−42.3320			−1.64902			−0.824511	−	0.565846i	$-0.808550\pi$
$659$	−42.3320			−1.64902			−0.824511	+	0.565846i	$0.808550\pi$
$660$		0			0
$661$		−	21.1660i		−	0.823262i	−0.911351	−	0.411631i	$-0.864959\pi$
$661$		−	21.1660i		−	0.823262i	0.911351	−	0.411631i	$-0.135041\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4.69926	+	0.957598i	−0.181956	+	0.0370784i
$668$		0			0
$669$			42.0000i			1.62381i
$670$		0			0
$671$	−28.0000			−1.08093
$672$		0			0
$673$	−35.5457	+	35.5457i	−1.37019	+	1.37019i	−0.510033	+	0.860155i	$0.670367\pi$
$673$	−35.5457	+	35.5457i	−1.37019	+	1.37019i	−0.860155	+	0.510033i	$0.829633\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.65685	+	5.65685i	−0.217411	+	0.217411i	−0.807406	−	0.589996i	$-0.799129\pi$
$677$	−5.65685	+	5.65685i	−0.217411	+	0.217411i	0.589996	+	0.807406i	$0.299129\pi$
$678$		0			0
$679$			4.00000i			0.153506i
$680$		0			0
$681$		−	47.6235i		−	1.82494i
$682$		0			0
$683$	−24.3208	+	24.3208i	−0.930609	+	0.930609i	−0.997744	−	0.0671354i	$-0.978614\pi$
$683$	−24.3208	+	24.3208i	−0.930609	+	0.930609i	0.0671354	+	0.997744i	$0.478614\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−49.4975	+	49.4975i	−1.88845	+	1.88845i
$688$		0			0
$689$	−10.5830			−0.403180
$690$		0			0
$691$	36.0000			1.36950			0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000			1.36950			0.684752	+	0.728776i	$0.259910\pi$
$692$		0			0
$693$	−29.9333	+	29.9333i	−1.13707	+	1.13707i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$			7.00000i			0.264764i
$700$		0			0
$701$		−	15.8745i		−	0.599572i	−0.954006	−	0.299786i	$-0.903085\pi$
$701$		−	15.8745i		−	0.599572i	0.954006	−	0.299786i	$-0.0969153\pi$
$702$		0			0
$703$	37.4166	−	37.4166i	1.41119	−	1.41119i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.7990	+	19.7990i	−0.744618	+	0.744618i
$708$		0			0
$709$	5.29150			0.198727			0.0993633	−	0.995051i	$-0.468319\pi$
$709$	5.29150			0.198727			0.0993633	+	0.995051i	$0.468319\pi$
$710$		0			0
$711$		−	21.1660i		−	0.793787i
$712$		0			0
$713$	2.87280	+	14.0978i	0.107587	+	0.527966i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1.87083	−	1.87083i	−0.0698674	−	0.0698674i
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$	39.5980	+	39.5980i	1.47266	+	1.47266i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	19.7990	−	19.7990i	0.734304	−	0.734304i	−0.237165	−	0.971469i	$-0.576218\pi$
$727$	19.7990	−	19.7990i	0.734304	−	0.734304i	0.971469	+	0.237165i	$0.0762183\pi$
$728$		0			0
$729$		−	41.0000i		−	1.51852i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−25.4558	−	25.4558i	−0.940233	−	0.940233i	0.0580789	−	0.998312i	$-0.481502\pi$
$733$	−25.4558	−	25.4558i	−0.940233	−	0.940233i	−0.998312	+	0.0580789i	$0.981502\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.48331	+	7.48331i	0.275651	+	0.275651i
$738$		0			0
$739$		−	3.00000i		−	0.110357i	−0.998477	−	0.0551784i	$-0.982427\pi$
$739$		−	3.00000i		−	0.110357i	0.998477	−	0.0551784i	$-0.0175728\pi$
$740$		0			0
$741$		−	37.0405i		−	1.36072i
$742$		0			0
$743$	−33.9411	−	33.9411i	−1.24518	−	1.24518i	−0.957824	−	0.287355i	$-0.907224\pi$
$743$	−33.9411	−	33.9411i	−1.24518	−	1.24518i	−0.287355	−	0.957824i	$-0.592776\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	33.9411	−	33.9411i	1.24184	−	1.24184i
$748$		0			0
$749$		−	12.0000i		−	0.438470i
$750$		0			0
$751$		−	37.0405i		−	1.35163i	−0.737072	−	0.675814i	$-0.763792\pi$
$751$		−	37.0405i		−	1.35163i	0.737072	−	0.675814i	$-0.236208\pi$
$752$		0			0
$753$	39.5980	+	39.5980i	1.44303	+	1.44303i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−24.0416	+	24.0416i	−0.873808	+	0.873808i	−0.992885	−	0.119077i	$-0.962006\pi$
$757$	−24.0416	+	24.0416i	−0.873808	+	0.873808i	0.119077	+	0.992885i	$0.462006\pi$
$758$		0			0
$759$	−37.0405	+	56.0000i	−1.34449	+	2.03267i
$760$		0			0
$761$	27.0000			0.978749			0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000			0.978749			0.489375	+	0.872074i	$0.337225\pi$
$762$		0			0
$763$	22.4499	−	22.4499i	0.812743	−	0.812743i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−10.5830			−0.381633			−0.190816	−	0.981626i	$-0.561114\pi$
$769$	−10.5830			−0.381633			−0.190816	+	0.981626i	$0.561114\pi$
$770$		0			0
$771$	−63.0000			−2.26889
$772$		0			0
$773$	26.8701	+	26.8701i	0.966449	+	0.966449i	0.999455	−	0.0330063i	$-0.0105082\pi$
$773$	26.8701	+	26.8701i	0.966449	+	0.966449i	−0.0330063	+	0.999455i	$0.510508\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	37.4166	+	37.4166i	1.34231	+	1.34231i
$778$		0			0
$779$	47.6235			1.70629
$780$		0			0
$781$		−	47.6235i		−	1.70410i
$782$		0			0
$783$	1.87083	−	1.87083i	0.0668580	−	0.0668580i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	19.7990	−	19.7990i	0.705758	−	0.705758i	−0.259882	−	0.965640i	$-0.583684\pi$
$787$	19.7990	−	19.7990i	0.705758	−	0.705758i	0.965640	+	0.259882i	$0.0836838\pi$
$788$		0			0
$789$	−68.7895			−2.44897
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$	9.89949	+	9.89949i	0.351541	+	0.351541i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.3848	+	18.3848i	−0.651222	+	0.651222i	−0.953287	−	0.302065i	$-0.902324\pi$
$797$	−18.3848	+	18.3848i	−0.651222	+	0.651222i	0.302065	+	0.953287i	$0.402324\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			63.4980i			2.24359i
$802$		0			0
$803$	49.4975	+	49.4975i	1.74673	+	1.74673i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−5.61249	−	5.61249i	−0.197569	−	0.197569i
$808$		0			0
$809$			10.0000i			0.351581i	0.984428	+	0.175791i	$0.0562482\pi$
$809$			10.0000i			0.351581i	−0.984428	+	0.175791i	$0.943752\pi$
$810$		0			0
$811$	9.00000			0.316033			0.158016	−	0.987436i	$-0.449490\pi$
$811$	9.00000			0.316033			0.158016	+	0.987436i	$0.449490\pi$
$812$		0			0
$813$	14.9666	−	14.9666i	0.524903	−	0.524903i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−22.4499	−	22.4499i	−0.785424	−	0.785424i
$818$		0			0
$819$	21.1660			0.739600
$820$		0			0
$821$	−38.0000			−1.32621			−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000			−1.32621			−0.663105	+	0.748527i	$0.730762\pi$
$822$		0			0
$823$	−35.5457	+	35.5457i	−1.23905	+	1.23905i	−0.278656	+	0.960391i	$0.589889\pi$
$823$	−35.5457	+	35.5457i	−1.23905	+	1.23905i	−0.960391	+	0.278656i	$0.910111\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−33.9411	+	33.9411i	−1.18025	+	1.18025i	−0.200569	+	0.979680i	$0.564279\pi$
$827$	−33.9411	+	33.9411i	−1.18025	+	1.18025i	−0.979680	+	0.200569i	$0.935721\pi$
$828$		0			0
$829$		−	38.0000i		−	1.31979i	−0.751356	−	0.659897i	$-0.770600\pi$
$829$		−	38.0000i		−	1.31979i	0.751356	−	0.659897i	$-0.229400\pi$
$830$		0			0
$831$	−21.0000			−0.728482
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−5.61249	−	5.61249i	−0.193996	−	0.193996i
$838$		0			0
$839$	−5.29150			−0.182683			−0.0913415	−	0.995820i	$-0.529115\pi$
$839$	−5.29150			−0.182683			−0.0913415	+	0.995820i	$0.529115\pi$
$840$		0			0
$841$	28.0000			0.965517
$842$		0			0
$843$	−29.6985	−	29.6985i	−1.02287	−	1.02287i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	24.0416	−	24.0416i	0.826080	−	0.826080i
$848$		0			0
$849$	63.4980			2.17925
$850$		0			0
$851$	40.0000	+	26.4575i	1.37118	+	0.906952i
$852$		0			0
$853$	−7.48331	+	7.48331i	−0.256224	+	0.256224i	−0.823516	−	0.567293i	$-0.807991\pi$
$853$	−7.48331	+	7.48331i	−0.256224	+	0.256224i	0.567293	+	0.823516i	$0.307991\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−31.8041	−	31.8041i	−1.08641	−	1.08641i	−0.995895	−	0.0905115i	$-0.971150\pi$
$857$	−31.8041	−	31.8041i	−1.08641	−	1.08641i	−0.0905115	−	0.995895i	$-0.528850\pi$
$858$		0			0
$859$			43.0000i			1.46714i	0.679613	+	0.733571i	$0.262148\pi$
$859$			43.0000i			1.46714i	−0.679613	+	0.733571i	$0.737852\pi$
$860$		0			0
$861$			47.6235i			1.62301i
$862$		0			0
$863$	20.5791	−	20.5791i	0.700521	−	0.700521i	−0.264001	−	0.964522i	$-0.585042\pi$
$863$	20.5791	−	20.5791i	0.700521	−	0.700521i	0.964522	+	0.264001i	$0.0850423\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	31.8041	+	31.8041i	1.08012	+	1.08012i
$868$		0			0
$869$			28.0000i			0.949835i
$870$		0			0
$871$		−	5.29150i		−	0.179296i
$872$		0			0
$873$	−5.65685	−	5.65685i	−0.191456	−	0.191456i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	37.4166	+	37.4166i	1.26347	+	1.26347i	0.949401	+	0.314067i	$0.101692\pi$
$877$	37.4166	+	37.4166i	1.26347	+	1.26347i	0.314067	+	0.949401i	$0.398308\pi$
$878$		0			0
$879$	−42.3320			−1.42782
$880$		0			0
$881$		−	31.7490i		−	1.06965i	−0.844962	−	0.534826i	$-0.820377\pi$
$881$		−	31.7490i		−	1.06965i	0.844962	−	0.534826i	$-0.179623\pi$
$882$		0			0
$883$	18.7083	−	18.7083i	0.629584	−	0.629584i	−0.318379	−	0.947963i	$-0.603139\pi$
$883$	18.7083	−	18.7083i	0.629584	−	0.629584i	0.947963	+	0.318379i	$0.103139\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.61249	−	5.61249i	−0.188449	−	0.188449i	0.606576	−	0.795025i	$-0.292542\pi$
$887$	−5.61249	−	5.61249i	−0.188449	−	0.188449i	−0.795025	+	0.606576i	$0.792542\pi$
$888$		0			0
$889$	15.8745			0.532414
$890$		0			0
$891$			26.4575i			0.886360i
$892$		0			0
$893$	−9.89949	−	9.89949i	−0.331274	−	0.331274i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	32.8948	−	6.70319i	1.09833	−	0.223813i
$898$		0			0
$899$			3.00000i			0.100056i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	22.4499	−	22.4499i	0.747087	−	0.747087i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	32.5269	−	32.5269i	1.08004	−	1.08004i	0.0835334	−	0.996505i	$-0.473379\pi$
$907$	32.5269	−	32.5269i	1.08004	−	1.08004i	0.996505	−	0.0835334i	$-0.0266205\pi$
$908$		0			0
$909$		−	56.0000i		−	1.85740i
$910$		0			0
$911$		−	15.8745i		−	0.525946i	−0.964803	−	0.262973i	$-0.915297\pi$
$911$		−	15.8745i		−	0.525946i	0.964803	−	0.262973i	$-0.0847030\pi$
$912$		0			0
$913$	−44.8999	+	44.8999i	−1.48597	+	1.48597i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.41421	−	1.41421i	0.0467014	−	0.0467014i
$918$		0			0
$919$	47.6235			1.57096			0.785478	−	0.618890i	$-0.212417\pi$
$919$	47.6235			1.57096			0.785478	+	0.618890i	$0.212417\pi$
$920$		0			0
$921$	−42.0000			−1.38395
$922$		0			0
$923$	−16.8375	+	16.8375i	−0.554212	+	0.554212i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	22.6274	−	22.6274i	0.743182	−	0.743182i
$928$		0			0
$929$			49.0000i			1.60764i	0.594874	+	0.803819i	$0.297202\pi$
$929$			49.0000i			1.60764i	−0.594874	+	0.803819i	$0.702798\pi$
$930$		0			0
$931$			15.8745i			0.520266i
$932$		0			0
$933$	39.2874	−	39.2874i	1.28621	−	1.28621i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	41.0122	−	41.0122i	1.33981	−	1.33981i	0.443570	−	0.896240i	$-0.353712\pi$
$937$	41.0122	−	41.0122i	1.33981	−	1.33981i	0.896240	−	0.443570i	$-0.146288\pi$
$938$		0			0
$939$	−52.9150			−1.72682
$940$		0			0
$941$		−	10.5830i		−	0.344996i	−0.985010	−	0.172498i	$-0.944816\pi$
$941$		−	10.5830i		−	0.344996i	0.985010	−	0.172498i	$-0.0551839\pi$
$942$		0			0
$943$	8.61839	+	42.2933i	0.280653	+	1.37726i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.3208	−	24.3208i	−0.790319	−	0.790319i	0.191227	−	0.981546i	$-0.438753\pi$
$947$	−24.3208	−	24.3208i	−0.790319	−	0.790319i	−0.981546	+	0.191227i	$0.938753\pi$
$948$		0			0
$949$		−	35.0000i		−	1.13615i
$950$		0			0
$951$	84.0000			2.72389
$952$		0			0
$953$	−5.65685	−	5.65685i	−0.183243	−	0.183243i	0.609524	−	0.792768i	$-0.291361\pi$
$953$	−5.65685	−	5.65685i	−0.183243	−	0.183243i	−0.792768	+	0.609524i	$0.791361\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−9.89949	+	9.89949i	−0.320005	+	0.320005i
$958$		0			0
$959$		−	24.0000i		−	0.775000i
$960$		0			0
$961$	−22.0000			−0.709677
$962$		0			0
$963$	16.9706	+	16.9706i	0.546869	+	0.546869i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−31.8041	−	31.8041i	−1.02275	−	1.02275i	−0.999735	−	0.0230154i	$-0.992673\pi$
$967$	−31.8041	−	31.8041i	−1.02275	−	1.02275i	−0.0230154	−	0.999735i	$-0.507327\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			21.1660i			0.679250i	0.940561	+	0.339625i	$0.110300\pi$
$971$			21.1660i			0.679250i	−0.940561	+	0.339625i	$0.889700\pi$
$972$		0			0
$973$	−7.07107	−	7.07107i	−0.226688	−	0.226688i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−26.8701	+	26.8701i	−0.859649	+	0.859649i	−0.991297	−	0.131647i	$-0.957973\pi$
$977$	−26.8701	+	26.8701i	−0.859649	+	0.859649i	0.131647	+	0.991297i	$0.457973\pi$
$978$		0			0
$979$		−	84.0000i		−	2.68465i
$980$		0			0
$981$			63.4980i			2.02734i
$982$		0			0
$983$	16.9706	+	16.9706i	0.541277	+	0.541277i	0.923903	−	0.382626i	$-0.124980\pi$
$983$	16.9706	+	16.9706i	0.541277	+	0.541277i	−0.382626	+	0.923903i	$0.624980\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	9.89949	−	9.89949i	0.315104	−	0.315104i
$988$		0			0
$989$	15.8745	−	24.0000i	0.504780	−	0.763156i
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$	54.2540	−	54.2540i	1.72170	−	1.72170i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−37.4166	−	37.4166i	−1.18499	−	1.18499i	−0.978434	−	0.206561i	$-0.933773\pi$
$997$	−37.4166	−	37.4166i	−1.18499	−	1.18499i	−0.206561	−	0.978434i	$-0.566227\pi$
$998$		0			0
$999$	−26.4575			−0.837079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.i.b.1057.3	yes	8
5.2	odd	4	inner	2300.2.i.b.1793.1	yes	8
5.3	odd	4	inner	2300.2.i.b.1793.4	yes	8
5.4	even	2	inner	2300.2.i.b.1057.2	yes	8
23.22	odd	2	inner	2300.2.i.b.1057.4	yes	8
115.22	even	4	inner	2300.2.i.b.1793.2	yes	8
115.68	even	4	inner	2300.2.i.b.1793.3	yes	8
115.114	odd	2	inner	2300.2.i.b.1057.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.i.b.1057.1	✓	8	115.114	odd	2	inner
2300.2.i.b.1057.2	yes	8	5.4	even	2	inner
2300.2.i.b.1057.3	yes	8	1.1	even	1	trivial
2300.2.i.b.1057.4	yes	8	23.22	odd	2	inner
2300.2.i.b.1793.1	yes	8	5.2	odd	4	inner
2300.2.i.b.1793.2	yes	8	115.22	even	4	inner
2300.2.i.b.1793.3	yes	8	115.68	even	4	inner
2300.2.i.b.1793.4	yes	8	5.3	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.87083 - 1.87083i) q^{3} +(-1.41421 + 1.41421i) q^{7} -4.00000i q^{9} +O(q^{10})\)	\(q+(1.87083 - 1.87083i) q^{3} +(-1.41421 + 1.41421i) q^{7} -4.00000i q^{9} +5.29150i q^{11} +(1.87083 - 1.87083i) q^{13} +5.29150 q^{19} +5.29150i q^{21} +(0.957598 + 4.69926i) q^{23} +(-1.87083 - 1.87083i) q^{27} +1.00000i q^{29} +3.00000 q^{31} +(9.89949 + 9.89949i) q^{33} +(7.07107 - 7.07107i) q^{37} -7.00000i q^{39} +9.00000 q^{41} +(-4.24264 - 4.24264i) q^{43} +(-1.87083 - 1.87083i) q^{47} +3.00000i q^{49} +(-2.82843 - 2.82843i) q^{53} +(9.89949 - 9.89949i) q^{57} +5.29150i q^{61} +(5.65685 + 5.65685i) q^{63} +(1.41421 - 1.41421i) q^{67} +(10.5830 + 7.00000i) q^{69} -9.00000 q^{71} +(9.35414 - 9.35414i) q^{73} +(-7.48331 - 7.48331i) q^{77} +5.29150 q^{79} +5.00000 q^{81} +(8.48528 + 8.48528i) q^{83} +(1.87083 + 1.87083i) q^{87} -15.8745 q^{89} +5.29150i q^{91} +(5.61249 - 5.61249i) q^{93} +(1.41421 - 1.41421i) q^{97} +21.1660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 24 q^{31} + 72 q^{41} - 72 q^{71} + 40 q^{81}+O(q^{100}) \) 8 * q + 24 * q^31 + 72 * q^41 - 72 * q^71 + 40 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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