LMFDB - Embedded newform 5472.2.f.c.1025.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5472,2,Mod(1025,5472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5472.1025"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,28,0,0,0,-16,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(59)] == traces)

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

Self dual:	no
Analytic conductor:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 38 x^{18} + 528 x^{16} + 3442 x^{14} + 11480 x^{12} + 20550 x^{10} + 20369 x^{8} + 11136 x^{6} + \cdots + 16 $ x^20 + 38x^18 + 528x^16 + 3442x^14 + 11480x^12 + 20550x^10 + 20369x^8 + 11136x^6 + 3192x^4 + 416*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.11
Root			$-2.39458i$ of defining polynomial
Character	$\chi$	$=$	5472.1025
Dual form			5472.2.f.c.1025.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.375576i q^{5} -4.49473 q^{7} +5.61106i q^{11} +5.52430i q^{13} +1.09153i q^{17} +(3.38510 + 2.74610i) q^{19} -0.447559i q^{23} +4.85894 q^{25} -7.25896 q^{29} -4.33970i q^{31} -1.68811i q^{35} +9.75290i q^{37} +8.27147 q^{41} -6.29271 q^{43} -2.13567i q^{47} +13.2026 q^{49} +9.33376 q^{53} -2.10738 q^{55} -10.9555 q^{59} -7.93374 q^{61} -2.07480 q^{65} -3.84460i q^{67} -6.18083 q^{71} +1.42246 q^{73} -25.2202i q^{77} +4.52873i q^{79} +10.2795i q^{83} -0.409951 q^{85} -0.458924 q^{89} -24.8303i q^{91} +(-1.03137 + 1.27136i) q^{95} -13.8453i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{25} - 16 q^{41} + 28 q^{49} - 16 q^{53} + 16 q^{65} + 16 q^{73} - 32 q^{85} + 48 q^{89}+O(q^{100}) $ 20 * q - 4 * q^25 - 16 * q^41 + 28 * q^49 - 16 * q^53 + 16 * q^65 + 16 * q^73 - 32 * q^85 + 48 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			0.375576i			0.167963i	0.996467	+	0.0839813i	$0.0267636\pi$
$5$			0.375576i			0.167963i	−0.996467	+	0.0839813i	$0.973236\pi$
$6$		0			0
$7$	−4.49473			−1.69885			−0.849425	−	0.527710i	$-0.823051\pi$
$7$	−4.49473			−1.69885			−0.849425	+	0.527710i	$0.823051\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.61106i			1.69180i	0.533342	+	0.845900i	$0.320936\pi$
$11$			5.61106i			1.69180i	−0.533342	+	0.845900i	$0.679064\pi$
$12$		0			0
$13$			5.52430i			1.53217i	0.642742	+	0.766083i	$0.277797\pi$
$13$			5.52430i			1.53217i	−0.642742	+	0.766083i	$0.722203\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.09153i			0.264734i	0.991201	+	0.132367i	$0.0422577\pi$
$17$			1.09153i			0.264734i	−0.991201	+	0.132367i	$0.957742\pi$
$18$		0			0
$19$	3.38510	+	2.74610i	0.776596	+	0.629999i
$19$	3.38510	+	2.74610i	0.776596	+	0.629999i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.447559i		−	0.0933225i	−0.998911	−	0.0466612i	$-0.985142\pi$
$23$		−	0.447559i		−	0.0933225i	0.998911	−	0.0466612i	$-0.0148581\pi$
$24$		0			0
$25$	4.85894			0.971789
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.25896			−1.34796			−0.673978	−	0.738752i	$-0.735416\pi$
$29$	−7.25896			−1.34796			−0.673978	+	0.738752i	$0.735416\pi$
$30$		0			0
$31$		−	4.33970i		−	0.779433i	−0.920935	−	0.389717i	$-0.872573\pi$
$31$		−	4.33970i		−	0.779433i	0.920935	−	0.389717i	$-0.127427\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.68811i		−	0.285343i
$36$		0			0
$37$			9.75290i			1.60337i	0.597749	+	0.801683i	$0.296062\pi$
$37$			9.75290i			1.60337i	−0.597749	+	0.801683i	$0.703938\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.27147			1.29179			0.645893	−	0.763428i	$-0.276485\pi$
$41$	8.27147			1.29179			0.645893	+	0.763428i	$0.276485\pi$
$42$		0			0
$43$	−6.29271			−0.959630			−0.479815	−	0.877370i	$-0.659296\pi$
$43$	−6.29271			−0.959630			−0.479815	+	0.877370i	$0.659296\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.13567i		−	0.311520i	−0.987795	−	0.155760i	$-0.950217\pi$
$47$		−	2.13567i		−	0.311520i	0.987795	−	0.155760i	$-0.0497826\pi$
$48$		0			0
$49$	13.2026			1.88609
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.33376			1.28209			0.641045	−	0.767503i	$-0.278501\pi$
$53$	9.33376			1.28209			0.641045	+	0.767503i	$0.278501\pi$
$54$		0			0
$55$	−2.10738			−0.284159
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.9555			−1.42629			−0.713145	−	0.701017i	$-0.752730\pi$
$59$	−10.9555			−1.42629			−0.713145	+	0.701017i	$0.752730\pi$
$60$		0			0
$61$	−7.93374			−1.01581			−0.507906	−	0.861413i	$-0.669580\pi$
$61$	−7.93374			−1.01581			−0.507906	+	0.861413i	$0.669580\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.07480			−0.257347
$66$		0			0
$67$		−	3.84460i		−	0.469692i	−0.972033	−	0.234846i	$-0.924541\pi$
$67$		−	3.84460i		−	0.469692i	0.972033	−	0.234846i	$-0.0754586\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.18083			−0.733530			−0.366765	−	0.930314i	$-0.619535\pi$
$71$	−6.18083			−0.733530			−0.366765	+	0.930314i	$0.619535\pi$
$72$		0			0
$73$	1.42246			0.166486			0.0832430	−	0.996529i	$-0.473472\pi$
$73$	1.42246			0.166486			0.0832430	+	0.996529i	$0.473472\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	25.2202i		−	2.87411i
$78$		0			0
$79$			4.52873i			0.509522i	0.967004	+	0.254761i	$0.0819968\pi$
$79$			4.52873i			0.509522i	−0.967004	+	0.254761i	$0.918003\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			10.2795i			1.12832i	0.825666	+	0.564159i	$0.190799\pi$
$83$			10.2795i			1.12832i	−0.825666	+	0.564159i	$0.809201\pi$
$84$		0			0
$85$	−0.409951			−0.0444654
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.458924			−0.0486458			−0.0243229	−	0.999704i	$-0.507743\pi$
$89$	−0.458924			−0.0486458			−0.0243229	+	0.999704i	$0.507743\pi$
$90$		0			0
$91$		−	24.8303i		−	2.60292i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.03137	+	1.27136i	−0.105816	+	0.130439i
$96$		0			0
$97$		−	13.8453i		−	1.40578i	−0.711300	−	0.702889i	$-0.751893\pi$
$97$		−	13.8453i		−	1.40578i	0.711300	−	0.702889i	$-0.248107\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			13.6247i			1.35570i	0.735198	+	0.677852i	$0.237089\pi$
$101$			13.6247i			1.35570i	−0.735198	+	0.677852i	$0.762911\pi$
$102$		0			0
$103$		−	11.5605i		−	1.13909i	−0.821959	−	0.569546i	$-0.807119\pi$
$103$		−	11.5605i		−	1.13909i	0.821959	−	0.569546i	$-0.192881\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.46733			0.818568			0.409284	−	0.912407i	$-0.365779\pi$
$107$	8.46733			0.818568			0.409284	+	0.912407i	$0.365779\pi$
$108$		0			0
$109$		−	8.45851i		−	0.810178i	−0.914277	−	0.405089i	$-0.867241\pi$
$109$		−	8.45851i		−	0.810178i	0.914277	−	0.405089i	$-0.132759\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.44642			0.324212			0.162106	−	0.986773i	$-0.448171\pi$
$113$	3.44642			0.324212			0.162106	+	0.986773i	$0.448171\pi$
$114$		0			0
$115$	0.168092			0.0156747
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	4.90612i		−	0.449743i
$120$		0			0
$121$	−20.4840			−1.86218
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3.70278i			0.331187i
$126$		0			0
$127$		−	3.74392i		−	0.332219i	−0.986107	−	0.166109i	$-0.946879\pi$
$127$		−	3.74392i		−	0.332219i	0.986107	−	0.166109i	$-0.0531205\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			11.9676i			1.04561i	0.852452	+	0.522806i	$0.175115\pi$
$131$			11.9676i			1.04561i	−0.852452	+	0.522806i	$0.824885\pi$
$132$		0			0
$133$	−15.2151	−	12.3430i	−1.31932	−	1.07027i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			7.19427i			0.614648i	0.951605	+	0.307324i	$0.0994335\pi$
$137$			7.19427i			0.614648i	−0.951605	+	0.307324i	$0.900566\pi$
$138$		0			0
$139$	2.86485			0.242993			0.121497	−	0.992592i	$-0.461231\pi$
$139$	2.86485			0.242993			0.121497	+	0.992592i	$0.461231\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−30.9972			−2.59212
$144$		0			0
$145$		−	2.72629i		−	0.226406i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	0.955334i		−	0.0782640i	−0.999234	−	0.0391320i	$-0.987541\pi$
$149$		−	0.955334i		−	0.0782640i	0.999234	−	0.0391320i	$-0.0124593\pi$
$150$		0			0
$151$		−	10.9647i		−	0.892298i	−0.894959	−	0.446149i	$-0.852795\pi$
$151$		−	10.9647i		−	0.892298i	0.894959	−	0.446149i	$-0.147205\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.62989			0.130916
$156$		0			0
$157$	8.87483			0.708289			0.354144	−	0.935191i	$-0.384772\pi$
$157$	8.87483			0.708289			0.354144	+	0.935191i	$0.384772\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2.01166i			0.158541i
$162$		0			0
$163$	−13.2715			−1.03950			−0.519751	−	0.854318i	$-0.673975\pi$
$163$	−13.2715			−1.03950			−0.519751	+	0.854318i	$0.673975\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.02340			−0.543487			−0.271743	−	0.962370i	$-0.587600\pi$
$167$	−7.02340			−0.543487			−0.271743	+	0.962370i	$0.587600\pi$
$168$		0			0
$169$	−17.5179			−1.34753
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.2965			−1.08694			−0.543471	−	0.839428i	$-0.682890\pi$
$173$	−14.2965			−1.08694			−0.543471	+	0.839428i	$0.682890\pi$
$174$		0			0
$175$	−21.8397			−1.65092
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−22.5299			−1.68396			−0.841981	−	0.539507i	$-0.818611\pi$
$179$	−22.5299			−1.68396			−0.841981	+	0.539507i	$0.818611\pi$
$180$		0			0
$181$		−	15.7937i		−	1.17393i	−0.809611	−	0.586967i	$-0.800322\pi$
$181$		−	15.7937i		−	1.17393i	0.809611	−	0.586967i	$-0.199678\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.66295			−0.269306
$186$		0			0
$187$	−6.12462			−0.447877
$188$		0			0
$189$		0			0
$190$		0			0
$191$			13.1201i			0.949336i	0.880165	+	0.474668i	$0.157432\pi$
$191$			13.1201i			0.949336i	−0.880165	+	0.474668i	$0.842568\pi$
$192$		0			0
$193$			7.40556i			0.533064i	0.963826	+	0.266532i	$0.0858778\pi$
$193$			7.40556i			0.533064i	−0.963826	+	0.266532i	$0.914122\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.47255i			0.603644i	0.953364	+	0.301822i	$0.0975949\pi$
$197$			8.47255i			0.603644i	−0.953364	+	0.301822i	$0.902405\pi$
$198$		0			0
$199$	−26.3344			−1.86680			−0.933398	−	0.358844i	$-0.883171\pi$
$199$	−26.3344			−1.86680			−0.933398	+	0.358844i	$0.883171\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	32.6271			2.28997
$204$		0			0
$205$			3.10656i			0.216972i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−15.4086	+	18.9940i	−1.06583	+	1.31384i
$210$		0			0
$211$		−	3.84460i		−	0.264673i	−0.991205	−	0.132336i	$-0.957752\pi$
$211$		−	3.84460i		−	0.264673i	0.991205	−	0.132336i	$-0.0422479\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2.36339i		−	0.161182i
$216$		0			0
$217$			19.5058i			1.32414i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.02992			−0.405616
$222$		0			0
$223$			13.8655i			0.928505i	0.885703	+	0.464252i	$0.153677\pi$
$223$			13.8655i			0.928505i	−0.885703	+	0.464252i	$0.846323\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.5854			0.835324			0.417662	−	0.908602i	$-0.362850\pi$
$227$	12.5854			0.835324			0.417662	+	0.908602i	$0.362850\pi$
$228$		0			0
$229$	9.04090			0.597440			0.298720	−	0.954341i	$-0.403440\pi$
$229$	9.04090			0.597440			0.298720	+	0.954341i	$0.403440\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	12.1401i		−	0.795326i	−0.917531	−	0.397663i	$-0.869821\pi$
$233$		−	12.1401i		−	0.795326i	0.917531	−	0.397663i	$-0.130179\pi$
$234$		0			0
$235$	0.802107			0.0523237
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	0.644772i		−	0.0417069i	−0.999783	−	0.0208534i	$-0.993362\pi$
$239$		−	0.644772i		−	0.0417069i	0.999783	−	0.0208534i	$-0.00663833\pi$
$240$		0			0
$241$		−	15.6575i		−	1.00859i	−0.863532	−	0.504293i	$-0.831753\pi$
$241$		−	15.6575i		−	1.00859i	0.863532	−	0.504293i	$-0.168247\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			4.95859i			0.316793i
$246$		0			0
$247$	−15.1703	+	18.7003i	−0.965263	+	1.18987i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			15.8430i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$251$			15.8430i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$	2.51128			0.157883
$254$		0			0
$255$		0			0
$256$		0			0
$257$	23.1713			1.44539			0.722693	−	0.691169i	$-0.242904\pi$
$257$	23.1713			1.44539			0.722693	+	0.691169i	$0.242904\pi$
$258$		0			0
$259$		−	43.8367i		−	2.72388i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			13.2783i			0.818775i	0.912361	+	0.409388i	$0.134258\pi$
$263$			13.2783i			0.818775i	−0.912361	+	0.409388i	$0.865742\pi$
$264$		0			0
$265$			3.50553i			0.215343i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−27.9961			−1.70695			−0.853477	−	0.521131i	$-0.825510\pi$
$269$	−27.9961			−1.70695			−0.853477	+	0.521131i	$0.825510\pi$
$270$		0			0
$271$	21.9111			1.33100			0.665502	−	0.746396i	$-0.268218\pi$
$271$	21.9111			1.33100			0.665502	+	0.746396i	$0.268218\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			27.2638i			1.64407i
$276$		0			0
$277$	24.2343			1.45610			0.728049	−	0.685526i	$-0.240428\pi$
$277$	24.2343			1.45610			0.728049	+	0.685526i	$0.240428\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−28.6648			−1.71000			−0.855000	−	0.518629i	$-0.826443\pi$
$281$	−28.6648			−1.71000			−0.855000	+	0.518629i	$0.826443\pi$
$282$		0			0
$283$	−8.57921			−0.509981			−0.254991	−	0.966944i	$-0.582072\pi$
$283$	−8.57921			−0.509981			−0.254991	+	0.966944i	$0.582072\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−37.1780			−2.19455
$288$		0			0
$289$	15.8086			0.929916
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.1020			0.823848			0.411924	−	0.911218i	$-0.364857\pi$
$293$	14.1020			0.823848			0.411924	+	0.911218i	$0.364857\pi$
$294$		0			0
$295$		−	4.11464i		−	0.239563i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.47245			0.142986
$300$		0			0
$301$	28.2841			1.63027
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	2.97972i		−	0.170618i
$306$		0			0
$307$			0.595783i			0.0340031i	0.999855	+	0.0170016i	$0.00541202\pi$
$307$			0.595783i			0.0340031i	−0.999855	+	0.0170016i	$0.994588\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	32.2691i		−	1.82981i	−0.403665	−	0.914907i	$-0.632264\pi$
$311$		−	32.2691i		−	1.82981i	0.403665	−	0.914907i	$-0.367736\pi$
$312$		0			0
$313$	−17.6501			−0.997643			−0.498821	−	0.866705i	$-0.666234\pi$
$313$	−17.6501			−0.997643			−0.498821	+	0.866705i	$0.666234\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.4888			0.589112			0.294556	−	0.955634i	$-0.404828\pi$
$317$	10.4888			0.589112			0.294556	+	0.955634i	$0.404828\pi$
$318$		0			0
$319$		−	40.7305i		−	2.28047i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.99744	+	3.69493i	−0.166782	+	0.205591i
$324$		0			0
$325$			26.8423i			1.48894i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			9.59928i			0.529225i
$330$		0			0
$331$		−	34.7629i		−	1.91074i	−0.295413	−	0.955370i	$-0.595457\pi$
$331$		−	34.7629i		−	1.91074i	0.295413	−	0.955370i	$-0.404543\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.44394			0.0788907
$336$		0			0
$337$			24.8235i			1.35222i	0.736800	+	0.676111i	$0.236336\pi$
$337$			24.8235i			1.35222i	−0.736800	+	0.676111i	$0.763664\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	24.3503			1.31864
$342$		0			0
$343$	−27.8792			−1.50533
$344$		0			0
$345$		0			0
$346$		0			0
$347$			20.9463i			1.12446i	0.826982	+	0.562228i	$0.190056\pi$
$347$			20.9463i			1.12446i	−0.826982	+	0.562228i	$0.809944\pi$
$348$		0			0
$349$	−25.8277			−1.38253			−0.691263	−	0.722603i	$-0.742945\pi$
$349$	−25.8277			−1.38253			−0.691263	+	0.722603i	$0.742945\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	19.3569i		−	1.03026i	−0.857111	−	0.515131i	$-0.827743\pi$
$353$		−	19.3569i		−	1.03026i	0.857111	−	0.515131i	$-0.172257\pi$
$354$		0			0
$355$		−	2.32137i		−	0.123206i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	23.7658i		−	1.25431i	−0.778894	−	0.627156i	$-0.784219\pi$
$359$		−	23.7658i		−	1.25431i	0.778894	−	0.627156i	$-0.215781\pi$
$360$		0			0
$361$	3.91785	+	18.5917i	0.206202	+	0.978509i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			0.534241i			0.0279634i
$366$		0			0
$367$	−17.9789			−0.938493			−0.469246	−	0.883067i	$-0.655474\pi$
$367$	−17.9789			−0.938493			−0.469246	+	0.883067i	$0.655474\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−41.9528			−2.17808
$372$		0			0
$373$		−	14.9331i		−	0.773206i	−0.922246	−	0.386603i	$-0.873648\pi$
$373$		−	14.9331i		−	0.773206i	0.922246	−	0.386603i	$-0.126352\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	40.1007i		−	2.06529i
$378$		0			0
$379$		−	8.74102i		−	0.448996i	−0.974475	−	0.224498i	$-0.927926\pi$
$379$		−	8.74102i		−	0.448996i	0.974475	−	0.224498i	$-0.0720742\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.3490			0.835397			0.417699	−	0.908586i	$-0.362837\pi$
$383$	16.3490			0.835397			0.417699	+	0.908586i	$0.362837\pi$
$384$		0			0
$385$	9.47211			0.482744
$386$		0			0
$387$		0			0
$388$		0			0
$389$			18.2781i			0.926735i	0.886166	+	0.463368i	$0.153359\pi$
$389$			18.2781i			0.926735i	−0.886166	+	0.463368i	$0.846641\pi$
$390$		0			0
$391$	0.488522			0.0247056
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.70088			−0.0855807
$396$		0			0
$397$	−1.57205			−0.0788988			−0.0394494	−	0.999222i	$-0.512560\pi$
$397$	−1.57205			−0.0788988			−0.0394494	+	0.999222i	$0.512560\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.2584			0.911780			0.455890	−	0.890036i	$-0.349321\pi$
$401$	18.2584			0.911780			0.455890	+	0.890036i	$0.349321\pi$
$402$		0			0
$403$	23.9738			1.19422
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−54.7241			−2.71257
$408$		0			0
$409$			20.1475i			0.996231i	0.867111	+	0.498116i	$0.165975\pi$
$409$			20.1475i			0.996231i	−0.867111	+	0.498116i	$0.834025\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	49.2422			2.42305
$414$		0			0
$415$	−3.86072			−0.189515
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	28.9114i		−	1.41242i	−0.708004	−	0.706208i	$-0.750404\pi$
$419$		−	28.9114i		−	1.41242i	0.708004	−	0.706208i	$-0.249596\pi$
$420$		0			0
$421$		−	0.707523i		−	0.0344826i	−0.999851	−	0.0172413i	$-0.994512\pi$
$421$		−	0.707523i		−	0.0344826i	0.999851	−	0.0172413i	$-0.00548834\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			5.30366i			0.257265i
$426$		0			0
$427$	35.6600			1.72571
$428$		0			0
$429$		0			0
$430$		0			0
$431$	37.4018			1.80158			0.900791	−	0.434253i	$-0.142988\pi$
$431$	37.4018			1.80158			0.900791	+	0.434253i	$0.142988\pi$
$432$		0			0
$433$		−	39.8612i		−	1.91561i	−0.287421	−	0.957804i	$-0.592798\pi$
$433$		−	39.8612i		−	1.91561i	0.287421	−	0.957804i	$-0.407202\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.22904	−	1.51503i	0.0587931	−	0.0724739i
$438$		0			0
$439$			25.2444i			1.20485i	0.798176	+	0.602424i	$0.205798\pi$
$439$			25.2444i			1.20485i	−0.798176	+	0.602424i	$0.794202\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			28.3362i			1.34629i	0.739510	+	0.673146i	$0.235057\pi$
$443$			28.3362i			1.34629i	−0.739510	+	0.673146i	$0.764943\pi$
$444$		0			0
$445$		−	0.172361i		−	0.00817068i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17.2907			−0.816001			−0.408000	−	0.912982i	$-0.633774\pi$
$449$	−17.2907			−0.816001			−0.408000	+	0.912982i	$0.633774\pi$
$450$		0			0
$451$			46.4117i			2.18544i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.32565			0.437193
$456$		0			0
$457$	32.1961			1.50607			0.753034	−	0.657982i	$-0.228590\pi$
$457$	32.1961			1.50607			0.753034	+	0.657982i	$0.228590\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	10.9148i		−	0.508354i	−0.967158	−	0.254177i	$-0.918195\pi$
$461$		−	10.9148i		−	0.508354i	0.967158	−	0.254177i	$-0.0818046\pi$
$462$		0			0
$463$	−9.62348			−0.447241			−0.223621	−	0.974676i	$-0.571788\pi$
$463$	−9.62348			−0.447241			−0.223621	+	0.974676i	$0.571788\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	0.151182i		−	0.00699588i	−0.999994	−	0.00349794i	$-0.998887\pi$
$467$		−	0.151182i		−	0.00699588i	0.999994	−	0.00349794i	$-0.00111343\pi$
$468$		0			0
$469$			17.2804i			0.797936i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	35.3088i		−	1.62350i
$474$		0			0
$475$	16.4480	+	13.3432i	0.754687	+	0.612226i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			12.4222i			0.567584i	0.958886	+	0.283792i	$0.0915925\pi$
$479$			12.4222i			0.567584i	−0.958886	+	0.283792i	$0.908408\pi$
$480$		0			0
$481$	−53.8780			−2.45662
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.19996			0.236118
$486$		0			0
$487$		−	35.9840i		−	1.63059i	−0.579044	−	0.815296i	$-0.696574\pi$
$487$		−	35.9840i		−	1.63059i	0.579044	−	0.815296i	$-0.303426\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	16.5565i		−	0.747184i	−0.927593	−	0.373592i	$-0.878126\pi$
$491$		−	16.5565i		−	0.747184i	0.927593	−	0.373592i	$-0.121874\pi$
$492$		0			0
$493$		−	7.92334i		−	0.356849i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	27.7812			1.24616
$498$		0			0
$499$	−39.0879			−1.74981			−0.874907	−	0.484290i	$-0.839078\pi$
$499$	−39.0879			−1.74981			−0.874907	+	0.484290i	$0.839078\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	22.5784i		−	1.00672i	−0.864077	−	0.503360i	$-0.832097\pi$
$503$		−	22.5784i		−	1.00672i	0.864077	−	0.503360i	$-0.167903\pi$
$504$		0			0
$505$	−5.11709			−0.227708
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.55417			−0.0688874			−0.0344437	−	0.999407i	$-0.510966\pi$
$509$	−1.55417			−0.0688874			−0.0344437	+	0.999407i	$0.510966\pi$
$510$		0			0
$511$	−6.39357			−0.282835
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.34185			0.191325
$516$		0			0
$517$	11.9834			0.527029
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.15425			0.138190			0.0690950	−	0.997610i	$-0.477989\pi$
$521$	3.15425			0.138190			0.0690950	+	0.997610i	$0.477989\pi$
$522$		0			0
$523$		−	22.1772i		−	0.969743i	−0.874585	−	0.484872i	$-0.838866\pi$
$523$		−	22.1772i		−	0.969743i	0.874585	−	0.484872i	$-0.161134\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.73689			0.206342
$528$		0			0
$529$	22.7997			0.991291
$530$		0			0
$531$		0			0
$532$		0			0
$533$			45.6941i			1.97923i
$534$		0			0
$535$			3.18013i			0.137489i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			74.0808i			3.19089i
$540$		0			0
$541$	−34.8451			−1.49811			−0.749055	−	0.662508i	$-0.769492\pi$
$541$	−34.8451			−1.49811			−0.749055	+	0.662508i	$0.769492\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.17681			0.136080
$546$		0			0
$547$		−	28.9390i		−	1.23734i	−0.785650	−	0.618671i	$-0.787671\pi$
$547$		−	28.9390i		−	1.23734i	0.785650	−	0.618671i	$-0.212329\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−24.5723	−	19.9338i	−1.04682	−	0.849211i
$552$		0			0
$553$		−	20.3554i		−	0.865601i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	23.2239i		−	0.984030i	−0.870587	−	0.492015i	$-0.836261\pi$
$557$		−	23.2239i		−	0.984030i	0.870587	−	0.492015i	$-0.163739\pi$
$558$		0			0
$559$		−	34.7629i		−	1.47031i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	30.7735			1.29695			0.648473	−	0.761237i	$-0.275408\pi$
$563$	30.7735			1.29695			0.648473	+	0.761237i	$0.275408\pi$
$564$		0			0
$565$			1.29439i			0.0544555i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−26.5091			−1.11132			−0.555660	−	0.831409i	$-0.687535\pi$
$569$	−26.5091			−1.11132			−0.555660	+	0.831409i	$0.687535\pi$
$570$		0			0
$571$	36.3355			1.52059			0.760296	−	0.649576i	$-0.225054\pi$
$571$	36.3355			1.52059			0.760296	+	0.649576i	$0.225054\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	2.17466i		−	0.0906897i
$576$		0			0
$577$	21.4443			0.892738			0.446369	−	0.894849i	$-0.352717\pi$
$577$	21.4443			0.892738			0.446369	+	0.894849i	$0.352717\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	46.2034i		−	1.91684i
$582$		0			0
$583$			52.3723i			2.16904i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			22.9699i			0.948068i	0.880507	+	0.474034i	$0.157203\pi$
$587$			22.9699i			0.948068i	−0.880507	+	0.474034i	$0.842797\pi$
$588$		0			0
$589$	11.9173	−	14.6903i	0.491042	−	0.605305i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	5.71685i		−	0.234763i	−0.993087	−	0.117382i	$-0.962550\pi$
$593$		−	5.71685i		−	0.234763i	0.993087	−	0.117382i	$-0.0374500\pi$
$594$		0			0
$595$	1.84262			0.0755400
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−34.2570			−1.39970			−0.699851	−	0.714289i	$-0.746750\pi$
$599$	−34.2570			−1.39970			−0.699851	+	0.714289i	$0.746750\pi$
$600$		0			0
$601$			7.40556i			0.302079i	0.988528	+	0.151040i	$0.0482621\pi$
$601$			7.40556i			0.302079i	−0.988528	+	0.151040i	$0.951738\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	7.69331i		−	0.312777i
$606$		0			0
$607$			20.4906i			0.831687i	0.909436	+	0.415843i	$0.136514\pi$
$607$			20.4906i			0.831687i	−0.909436	+	0.415843i	$0.863486\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.7981			0.477300
$612$		0			0
$613$	−15.5020			−0.626121			−0.313061	−	0.949733i	$-0.601354\pi$
$613$	−15.5020			−0.626121			−0.313061	+	0.949733i	$0.601354\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			31.0662i			1.25068i	0.780354	+	0.625338i	$0.215039\pi$
$617$			31.0662i			1.25068i	−0.780354	+	0.625338i	$0.784961\pi$
$618$		0			0
$619$	−19.5794			−0.786962			−0.393481	−	0.919333i	$-0.628729\pi$
$619$	−19.5794			−0.786962			−0.393481	+	0.919333i	$0.628729\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.06274			0.0826419
$624$		0			0
$625$	22.9040			0.916162
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−10.6455			−0.424465
$630$		0			0
$631$	−8.42688			−0.335469			−0.167734	−	0.985832i	$-0.553645\pi$
$631$	−8.42688			−0.335469			−0.167734	+	0.985832i	$0.553645\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.40613			0.0558004
$636$		0			0
$637$			72.9353i			2.88980i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−39.4719			−1.55904			−0.779522	−	0.626374i	$-0.784538\pi$
$641$	−39.4719			−1.55904			−0.779522	+	0.626374i	$0.784538\pi$
$642$		0			0
$643$	2.96194			0.116808			0.0584038	−	0.998293i	$-0.481399\pi$
$643$	2.96194			0.116808			0.0584038	+	0.998293i	$0.481399\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			33.1798i			1.30443i	0.758032	+	0.652217i	$0.226161\pi$
$647$			33.1798i			1.30443i	−0.758032	+	0.652217i	$0.773839\pi$
$648$		0			0
$649$		−	61.4722i		−	2.41300i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			25.8127i			1.01013i	0.863081	+	0.505065i	$0.168531\pi$
$653$			25.8127i			1.01013i	−0.863081	+	0.505065i	$0.831469\pi$
$654$		0			0
$655$	−4.49473			−0.175624
$656$		0			0
$657$		0			0
$658$		0			0
$659$	42.7953			1.66707			0.833534	−	0.552468i	$-0.186314\pi$
$659$	42.7953			1.66707			0.833534	+	0.552468i	$0.186314\pi$
$660$		0			0
$661$		−	42.2086i		−	1.64172i	−0.571127	−	0.820862i	$-0.693494\pi$
$661$		−	42.2086i		−	1.64172i	0.571127	−	0.820862i	$-0.306506\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.63573	−	5.71444i	0.179766	−	0.221596i
$666$		0			0
$667$			3.24881i			0.125795i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	44.5167i		−	1.71855i
$672$		0			0
$673$			0.849647i			0.0327515i	0.999866	+	0.0163757i	$0.00521279\pi$
$673$			0.849647i			0.0327515i	−0.999866	+	0.0163757i	$0.994787\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.653149			−0.0251025			−0.0125513	−	0.999921i	$-0.503995\pi$
$677$	−0.653149			−0.0251025			−0.0125513	+	0.999921i	$0.503995\pi$
$678$		0			0
$679$			62.2309i			2.38820i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.7160			0.410037			0.205019	−	0.978758i	$-0.434275\pi$
$683$	10.7160			0.410037			0.205019	+	0.978758i	$0.434275\pi$
$684$		0			0
$685$	−2.70199			−0.103238
$686$		0			0
$687$		0			0
$688$		0			0
$689$			51.5625i			1.96437i
$690$		0			0
$691$	−10.2249			−0.388972			−0.194486	−	0.980905i	$-0.562304\pi$
$691$	−10.2249			−0.388972			−0.194486	+	0.980905i	$0.562304\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			1.07597i			0.0408138i
$696$		0			0
$697$			9.02852i			0.341980i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			43.1180i			1.62855i	0.580482	+	0.814273i	$0.302864\pi$
$701$			43.1180i			1.62855i	−0.580482	+	0.814273i	$0.697136\pi$
$702$		0			0
$703$	−26.7825	+	33.0146i	−1.01012	+	1.24517i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	61.2392i		−	2.30314i
$708$		0			0
$709$	−20.2688			−0.761210			−0.380605	−	0.924738i	$-0.624284\pi$
$709$	−20.2688			−0.761210			−0.380605	+	0.924738i	$0.624284\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.94227			−0.0727386
$714$		0			0
$715$		−	11.6418i		−	0.435379i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			40.2532i			1.50119i	0.660761	+	0.750596i	$0.270234\pi$
$719$			40.2532i			1.50119i	−0.660761	+	0.750596i	$0.729766\pi$
$720$		0			0
$721$			51.9615i			1.93515i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−35.2709			−1.30993
$726$		0			0
$727$	−0.970200			−0.0359827			−0.0179914	−	0.999838i	$-0.505727\pi$
$727$	−0.970200			−0.0359827			−0.0179914	+	0.999838i	$0.505727\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	6.86866i		−	0.254047i
$732$		0			0
$733$	−24.6175			−0.909268			−0.454634	−	0.890678i	$-0.650230\pi$
$733$	−24.6175			−0.909268			−0.454634	+	0.890678i	$0.650230\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	21.5723			0.794625
$738$		0			0
$739$	15.6740			0.576579			0.288289	−	0.957543i	$-0.406914\pi$
$739$	15.6740			0.576579			0.288289	+	0.957543i	$0.406914\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.54627			−0.203473			−0.101737	−	0.994811i	$-0.532440\pi$
$743$	−5.54627			−0.203473			−0.101737	+	0.994811i	$0.532440\pi$
$744$		0			0
$745$	0.358800			0.0131454
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−38.0584			−1.39062
$750$		0			0
$751$			1.62794i			0.0594044i	0.999559	+	0.0297022i	$0.00945590\pi$
$751$			1.62794i			0.0594044i	−0.999559	+	0.0297022i	$0.990544\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.11809			0.149873
$756$		0			0
$757$	−0.941337			−0.0342135			−0.0171067	−	0.999854i	$-0.505446\pi$
$757$	−0.941337			−0.0342135			−0.0171067	+	0.999854i	$0.505446\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			8.80696i			0.319252i	0.987178	+	0.159626i	$0.0510288\pi$
$761$			8.80696i			0.319252i	−0.987178	+	0.159626i	$0.948971\pi$
$762$		0			0
$763$			38.0187i			1.37637i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	60.5217i		−	2.18531i
$768$		0			0
$769$	−24.2773			−0.875462			−0.437731	−	0.899106i	$-0.644218\pi$
$769$	−24.2773			−0.875462			−0.437731	+	0.899106i	$0.644218\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−53.0322			−1.90744			−0.953718	−	0.300702i	$-0.902779\pi$
$773$	−53.0322			−1.90744			−0.953718	+	0.300702i	$0.902779\pi$
$774$		0			0
$775$		−	21.0864i		−	0.757444i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	27.9998	+	22.7143i	1.00320	+	0.813824i
$780$		0			0
$781$		−	34.6810i		−	1.24099i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			3.33317i			0.118966i
$786$		0			0
$787$		−	28.9913i		−	1.03343i	−0.856158	−	0.516714i	$-0.827155\pi$
$787$		−	28.9913i		−	1.03343i	0.856158	−	0.516714i	$-0.172845\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−15.4907			−0.550787
$792$		0			0
$793$		−	43.8284i		−	1.55639i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.714596			−0.0253123			−0.0126561	−	0.999920i	$-0.504029\pi$
$797$	−0.714596			−0.0253123			−0.0126561	+	0.999920i	$0.504029\pi$
$798$		0			0
$799$	2.33114			0.0824699
$800$		0			0
$801$		0			0
$802$		0			0
$803$			7.98150i			0.281661i
$804$		0			0
$805$	−0.755530			−0.0266289
$806$		0			0
$807$		0			0
$808$		0			0
$809$			43.0231i			1.51261i	0.654218	+	0.756306i	$0.272998\pi$
$809$			43.0231i			1.51261i	−0.654218	+	0.756306i	$0.727002\pi$
$810$		0			0
$811$			16.9644i			0.595700i	0.954613	+	0.297850i	$0.0962695\pi$
$811$			16.9644i			0.595700i	−0.954613	+	0.297850i	$0.903731\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	4.98444i		−	0.174597i
$816$		0			0
$817$	−21.3015	−	17.2804i	−0.745245	−	0.604566i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	40.4763i		−	1.41263i	−0.707897	−	0.706316i	$-0.750356\pi$
$821$		−	40.4763i		−	1.41263i	0.707897	−	0.706316i	$-0.249644\pi$
$822$		0			0
$823$	−13.5966			−0.473949			−0.236974	−	0.971516i	$-0.576156\pi$
$823$	−13.5966			−0.473949			−0.236974	+	0.971516i	$0.576156\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.48821			−0.0865234			−0.0432617	−	0.999064i	$-0.513775\pi$
$827$	−2.48821			−0.0865234			−0.0432617	+	0.999064i	$0.513775\pi$
$828$		0			0
$829$		−	9.09694i		−	0.315950i	−0.987443	−	0.157975i	$-0.949503\pi$
$829$		−	9.09694i		−	0.315950i	0.987443	−	0.157975i	$-0.0504965\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			14.4110i			0.499312i
$834$		0			0
$835$		−	2.63782i		−	0.0912855i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.2042			−0.455861			−0.227930	−	0.973677i	$-0.573196\pi$
$839$	−13.2042			−0.455861			−0.227930	+	0.973677i	$0.573196\pi$
$840$		0			0
$841$	23.6925			0.816984
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	6.57931i		−	0.226335i
$846$		0			0
$847$	92.0703			3.16357
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.36500			0.149630
$852$		0			0
$853$	14.1496			0.484473			0.242236	−	0.970217i	$-0.422119\pi$
$853$	14.1496			0.484473			0.242236	+	0.970217i	$0.422119\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11.1781			0.381836			0.190918	−	0.981606i	$-0.438854\pi$
$857$	11.1781			0.381836			0.190918	+	0.981606i	$0.438854\pi$
$858$		0			0
$859$	31.8960			1.08828			0.544139	−	0.838995i	$-0.316857\pi$
$859$	31.8960			1.08828			0.544139	+	0.838995i	$0.316857\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.90530			−0.0988977			−0.0494488	−	0.998777i	$-0.515746\pi$
$863$	−2.90530			−0.0988977			−0.0494488	+	0.998777i	$0.515746\pi$
$864$		0			0
$865$		−	5.36941i		−	0.182566i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−25.4110			−0.862009
$870$		0			0
$871$	21.2387			0.719646
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	16.6430i		−	0.562637i
$876$		0			0
$877$		−	16.7613i		−	0.565990i	−0.959121	−	0.282995i	$-0.908672\pi$
$877$		−	16.7613i		−	0.565990i	0.959121	−	0.282995i	$-0.0913280\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			27.4606i			0.925172i	0.886575	+	0.462586i	$0.153078\pi$
$881$			27.4606i			0.925172i	−0.886575	+	0.462586i	$0.846922\pi$
$882$		0			0
$883$	22.2342			0.748239			0.374119	−	0.927381i	$-0.377945\pi$
$883$	22.2342			0.748239			0.374119	+	0.927381i	$0.377945\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	35.9736			1.20788			0.603938	−	0.797031i	$-0.293597\pi$
$887$	35.9736			1.20788			0.603938	+	0.797031i	$0.293597\pi$
$888$		0			0
$889$			16.8279i			0.564390i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.86478	−	7.22947i	0.196257	−	0.241925i
$894$		0			0
$895$		−	8.46168i		−	0.282843i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			31.5017i			1.05064i
$900$		0			0
$901$			10.1880i			0.339413i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.93172			0.197177
$906$		0			0
$907$		−	6.60565i		−	0.219337i	−0.993968	−	0.109668i	$-0.965021\pi$
$907$		−	6.60565i		−	0.219337i	0.993968	−	0.109668i	$-0.0349789\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	21.6716			0.718011			0.359005	−	0.933335i	$-0.383116\pi$
$911$	21.6716			0.718011			0.359005	+	0.933335i	$0.383116\pi$
$912$		0			0
$913$	−57.6787			−1.90889
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	53.7911i		−	1.77634i
$918$		0			0
$919$	40.7446			1.34404			0.672020	−	0.740533i	$-0.265427\pi$
$919$	40.7446			1.34404			0.672020	+	0.740533i	$0.265427\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	34.1448i		−	1.12389i
$924$		0			0
$925$			47.3888i			1.55813i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			24.4809i			0.803192i	0.915817	+	0.401596i	$0.131544\pi$
$929$			24.4809i			0.803192i	−0.915817	+	0.401596i	$0.868456\pi$
$930$		0			0
$931$	44.6923	+	36.2558i	1.46473	+	1.18823i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	2.30026i		−	0.0752265i
$936$		0			0
$937$	21.7162			0.709439			0.354719	−	0.934973i	$-0.384576\pi$
$937$	21.7162			0.709439			0.354719	+	0.934973i	$0.384576\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.7451			0.774068			0.387034	−	0.922065i	$-0.373500\pi$
$941$	23.7451			0.774068			0.387034	+	0.922065i	$0.373500\pi$
$942$		0			0
$943$		−	3.70197i		−	0.120553i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	53.0643i		−	1.72436i	−0.506603	−	0.862179i	$-0.669099\pi$
$947$		−	53.0643i		−	1.72436i	0.506603	−	0.862179i	$-0.330901\pi$
$948$		0			0
$949$			7.85808i			0.255084i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−52.7366			−1.70831			−0.854153	−	0.520023i	$-0.825924\pi$
$953$	−52.7366			−1.70831			−0.854153	+	0.520023i	$0.825924\pi$
$954$		0			0
$955$	−4.92759			−0.159453
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	32.3363i		−	1.04419i
$960$		0			0
$961$	12.1670			0.392484
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.78135			−0.0895349
$966$		0			0
$967$	17.6547			0.567738			0.283869	−	0.958863i	$-0.408382\pi$
$967$	17.6547			0.567738			0.283869	+	0.958863i	$0.408382\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.71197			−0.0870312			−0.0435156	−	0.999053i	$-0.513856\pi$
$971$	−2.71197			−0.0870312			−0.0435156	+	0.999053i	$0.513856\pi$
$972$		0			0
$973$	−12.8767			−0.412809
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.6515			0.532730			0.266365	−	0.963872i	$-0.414177\pi$
$977$	16.6515			0.532730			0.266365	+	0.963872i	$0.414177\pi$
$978$		0			0
$979$		−	2.57505i		−	0.0822989i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	29.7928			0.950243			0.475121	−	0.879920i	$-0.342404\pi$
$983$	29.7928			0.950243			0.475121	+	0.879920i	$0.342404\pi$
$984$		0			0
$985$	−3.18209			−0.101390
$986$		0			0
$987$		0			0
$988$		0			0
$989$			2.81636i			0.0895550i
$990$		0			0
$991$			34.2748i			1.08878i	0.838834	+	0.544388i	$0.183238\pi$
$991$			34.2748i			1.08878i	−0.838834	+	0.544388i	$0.816762\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	9.89056i		−	0.313552i
$996$		0			0
$997$	46.7522			1.48066			0.740329	−	0.672245i	$-0.234670\pi$
$997$	46.7522			1.48066			0.740329	+	0.672245i	$0.234670\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.f.c.1025.11	yes	20
3.2	odd	2		5472.2.f.d.1025.9	yes	20
4.3	odd	2	inner	5472.2.f.c.1025.12	yes	20
12.11	even	2		5472.2.f.d.1025.10	yes	20
19.18	odd	2		5472.2.f.d.1025.11	yes	20
57.56	even	2	inner	5472.2.f.c.1025.9	✓	20
76.75	even	2		5472.2.f.d.1025.12	yes	20
228.227	odd	2	inner	5472.2.f.c.1025.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.f.c.1025.9	✓	20	57.56	even	2	inner
5472.2.f.c.1025.10	yes	20	228.227	odd	2	inner
5472.2.f.c.1025.11	yes	20	1.1	even	1	trivial
5472.2.f.c.1025.12	yes	20	4.3	odd	2	inner
5472.2.f.d.1025.9	yes	20	3.2	odd	2
5472.2.f.d.1025.10	yes	20	12.11	even	2
5472.2.f.d.1025.11	yes	20	19.18	odd	2
5472.2.f.d.1025.12	yes	20	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.375576i q^{5} -4.49473 q^{7} +5.61106i q^{11} +5.52430i q^{13} +1.09153i q^{17} +(3.38510 + 2.74610i) q^{19} -0.447559i q^{23} +4.85894 q^{25} -7.25896 q^{29} -4.33970i q^{31} -1.68811i q^{35} +9.75290i q^{37} +8.27147 q^{41} -6.29271 q^{43} -2.13567i q^{47} +13.2026 q^{49} +9.33376 q^{53} -2.10738 q^{55} -10.9555 q^{59} -7.93374 q^{61} -2.07480 q^{65} -3.84460i q^{67} -6.18083 q^{71} +1.42246 q^{73} -25.2202i q^{77} +4.52873i q^{79} +10.2795i q^{83} -0.409951 q^{85} -0.458924 q^{89} -24.8303i q^{91} +(-1.03137 + 1.27136i) q^{95} -13.8453i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{25} - 16 q^{41} + 28 q^{49} - 16 q^{53} + 16 q^{65} + 16 q^{73} - 32 q^{85} + 48 q^{89}+O(q^{100}) \) 20 * q - 4 * q^25 - 16 * q^41 + 28 * q^49 - 16 * q^53 + 16 * q^65 + 16 * q^73 - 32 * q^85 + 48 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more