LMFDB - Embedded newform 2880.2.o.e.2879.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(2879,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2880.2879");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.o (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2879.8
Root			$-0.394157 + 1.35818i$ of defining polynomial
Character	$\chi$	$=$	2880.2879
Dual form			2880.2.o.e.2879.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.256912 + 2.22126i) q^{5} +3.50466 q^{7} +O(q^{10})$	$q+(0.256912 + 2.22126i) q^{5} +3.50466 q^{7} -1.92804 q^{11} -5.50466i q^{13} +4.44252 q^{17} -7.00933i q^{19} -1.10027i q^{23} +(-4.86799 + 1.14134i) q^{25} -5.47017i q^{29} -8.28267i q^{31} +(0.900390 + 7.78477i) q^{35} -0.778008i q^{37} +2.44186i q^{41} -9.55602 q^{43} -11.7135i q^{47} +5.28267 q^{49} -11.5268 q^{53} +(-0.495336 - 4.28267i) q^{55} +9.78543 q^{59} +3.45331 q^{61} +(12.2273 - 1.41421i) q^{65} +5.45331 q^{67} +4.25583 q^{71} +7.27334i q^{73} -6.75712 q^{77} -2.82936i q^{79} +4.25583i q^{83} +(1.14134 + 9.86799i) q^{85} -0.386566i q^{89} -19.2920i q^{91} +(15.5695 - 1.80078i) q^{95} +9.29200i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 8 q^{25} - 64 q^{43} - 4 q^{49} - 48 q^{55} + 8 q^{61} + 32 q^{67} - 20 q^{85}+O(q^{100}) $ 12 * q - 8 * q^25 - 64 * q^43 - 4 * q^49 - 48 * q^55 + 8 * q^61 + 32 * q^67 - 20 * q^85

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$5$	0.256912	+	2.22126i	0.114894	+	0.993378i
$5$	0.256912	+	2.22126i	0.114894	+	0.993378i
$6$		0			0
$7$	3.50466			1.32464			0.662319	−	0.749222i	$-0.269572\pi$
$7$	3.50466			1.32464			0.662319	+	0.749222i	$0.269572\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.92804			−0.581325			−0.290663	−	0.956826i	$-0.593876\pi$
$11$	−1.92804			−0.581325			−0.290663	+	0.956826i	$0.593876\pi$
$12$		0			0
$13$		−	5.50466i		−	1.52672i	−0.645974	−	0.763360i	$-0.723548\pi$
$13$		−	5.50466i		−	1.52672i	0.645974	−	0.763360i	$-0.276452\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.44252			1.07747			0.538735	−	0.842475i	$-0.318903\pi$
$17$	4.44252			1.07747			0.538735	+	0.842475i	$0.318903\pi$
$18$		0			0
$19$		−	7.00933i		−	1.60805i	−0.594595	−	0.804025i	$-0.702688\pi$
$19$		−	7.00933i		−	1.60805i	0.594595	−	0.804025i	$-0.297312\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.10027i		−	0.229422i	−0.993399	−	0.114711i	$-0.963406\pi$
$23$		−	1.10027i		−	0.229422i	0.993399	−	0.114711i	$-0.0365942\pi$
$24$		0			0
$25$	−4.86799	+	1.14134i	−0.973599	+	0.228267i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.47017i		−	1.01578i	−0.861421	−	0.507892i	$-0.830425\pi$
$29$		−	5.47017i		−	1.01578i	0.861421	−	0.507892i	$-0.169575\pi$
$30$		0			0
$31$		−	8.28267i		−	1.48761i	−0.668396	−	0.743806i	$-0.733019\pi$
$31$		−	8.28267i		−	1.48761i	0.668396	−	0.743806i	$-0.266981\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.900390	+	7.78477i	0.152194	+	1.31587i
$36$		0			0
$37$		−	0.778008i		−	0.127904i	−0.997953	−	0.0639519i	$-0.979630\pi$
$37$		−	0.778008i		−	0.127904i	0.997953	−	0.0639519i	$-0.0203704\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.44186i			0.381355i	0.981653	+	0.190677i	$0.0610684\pi$
$41$			2.44186i			0.381355i	−0.981653	+	0.190677i	$0.938932\pi$
$42$		0			0
$43$	−9.55602			−1.45728			−0.728639	−	0.684898i	$-0.759847\pi$
$43$	−9.55602			−1.45728			−0.728639	+	0.684898i	$0.759847\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	11.7135i		−	1.70858i	−0.519793	−	0.854292i	$-0.673991\pi$
$47$		−	11.7135i		−	1.70858i	0.519793	−	0.854292i	$-0.326009\pi$
$48$		0			0
$49$	5.28267			0.754667
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−11.5268			−1.58333			−0.791663	−	0.610959i	$-0.790784\pi$
$53$	−11.5268			−1.58333			−0.791663	+	0.610959i	$0.790784\pi$
$54$		0			0
$55$	−0.495336	−	4.28267i	−0.0667910	−	0.577475i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.78543			1.27395			0.636977	−	0.770883i	$-0.280185\pi$
$59$	9.78543			1.27395			0.636977	+	0.770883i	$0.280185\pi$
$60$		0			0
$61$	3.45331			0.442151			0.221076	−	0.975257i	$-0.429043\pi$
$61$	3.45331			0.442151			0.221076	+	0.975257i	$0.429043\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12.2273	−	1.41421i	1.51661	−	0.175412i
$66$		0			0
$67$	5.45331			0.666228			0.333114	−	0.942887i	$-0.391901\pi$
$67$	5.45331			0.666228			0.333114	+	0.942887i	$0.391901\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.25583			0.505075			0.252537	−	0.967587i	$-0.418735\pi$
$71$	4.25583			0.505075			0.252537	+	0.967587i	$0.418735\pi$
$72$		0			0
$73$			7.27334i			0.851280i	0.904892	+	0.425640i	$0.139951\pi$
$73$			7.27334i			0.851280i	−0.904892	+	0.425640i	$0.860049\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.75712			−0.770046
$78$		0			0
$79$		−	2.82936i		−	0.318328i	−0.987252	−	0.159164i	$-0.949120\pi$
$79$		−	2.82936i		−	0.318328i	0.987252	−	0.159164i	$-0.0508798\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			4.25583i			0.467138i	0.972340	+	0.233569i	$0.0750405\pi$
$83$			4.25583i			0.467138i	−0.972340	+	0.233569i	$0.924959\pi$
$84$		0			0
$85$	1.14134	+	9.86799i	0.123795	+	1.07033i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	0.386566i		−	0.0409759i	−0.999790	−	0.0204880i	$-0.993478\pi$
$89$		−	0.386566i		−	0.0409759i	0.999790	−	0.0204880i	$-0.00652198\pi$
$90$		0			0
$91$		−	19.2920i		−	2.02235i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	15.5695	−	1.80078i	1.59740	−	0.184756i
$96$		0			0
$97$			9.29200i			0.943460i	0.881743	+	0.471730i	$0.156370\pi$
$97$			9.29200i			0.943460i	−0.881743	+	0.471730i	$0.843630\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			7.27095i			0.723486i	0.932278	+	0.361743i	$0.117818\pi$
$101$			7.27095i			0.723486i	−0.932278	+	0.361743i	$0.882182\pi$
$102$		0			0
$103$	16.9580			1.67092			0.835460	−	0.549552i	$-0.185202\pi$
$103$	16.9580			1.67092			0.835460	+	0.549552i	$0.185202\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			9.91269i			0.958296i	0.877734	+	0.479148i	$0.159054\pi$
$107$			9.91269i			0.958296i	−0.877734	+	0.479148i	$0.840946\pi$
$108$		0			0
$109$	−8.28267			−0.793336			−0.396668	−	0.917962i	$-0.629834\pi$
$109$	−8.28267			−0.793336			−0.396668	+	0.917962i	$0.629834\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.8115			−1.67557			−0.837784	−	0.546002i	$-0.816149\pi$
$113$	−17.8115			−1.67557			−0.837784	+	0.546002i	$0.816149\pi$
$114$		0			0
$115$	2.44398	−	0.282672i	0.227903	−	0.0263593i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	15.5695			1.42726
$120$		0			0
$121$	−7.28267			−0.662061
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.78585	−	10.5199i	−0.338617	−	0.940924i
$126$		0			0
$127$	0.957977			0.0850067			0.0425034	−	0.999096i	$-0.486467\pi$
$127$	0.957977			0.0850067			0.0425034	+	0.999096i	$0.486467\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.38441			0.732549			0.366275	−	0.930507i	$-0.380633\pi$
$131$	8.38441			0.732549			0.366275	+	0.930507i	$0.380633\pi$
$132$		0			0
$133$		−	24.5653i		−	2.13009i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.1270			0.950646			0.475323	−	0.879811i	$-0.342331\pi$
$137$	11.1270			0.950646			0.475323	+	0.879811i	$0.342331\pi$
$138$		0			0
$139$			1.17064i			0.0992924i	0.998767	+	0.0496462i	$0.0158094\pi$
$139$			1.17064i			0.0992924i	−0.998767	+	0.0496462i	$0.984191\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			10.6132i			0.887520i
$144$		0			0
$145$	12.1507	−	1.40535i	1.00906	−	0.116708i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			11.5268i			0.944311i	0.881515	+	0.472155i	$0.156524\pi$
$149$			11.5268i			0.944311i	−0.881515	+	0.472155i	$0.843476\pi$
$150$		0			0
$151$		−	10.8294i		−	0.881281i	−0.897684	−	0.440640i	$-0.854751\pi$
$151$		−	10.8294i		−	0.881281i	0.897684	−	0.440640i	$-0.145249\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	18.3980	−	2.12792i	1.47776	−	0.170918i
$156$		0			0
$157$			10.3340i			0.824745i	0.911015	+	0.412372i	$0.135300\pi$
$157$			10.3340i			0.824745i	−0.911015	+	0.412372i	$0.864700\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	3.85607i		−	0.303901i
$162$		0			0
$163$	−8.99067			−0.704204			−0.352102	−	0.935962i	$-0.614533\pi$
$163$	−8.99067			−0.704204			−0.352102	+	0.935962i	$0.614533\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			13.0683i			1.01125i	0.862753	+	0.505626i	$0.168738\pi$
$167$			13.0683i			1.01125i	−0.862753	+	0.505626i	$0.831262\pi$
$168$		0			0
$169$	−17.3013			−1.33087
$170$		0			0
$171$		0			0
$172$		0			0
$173$	19.6123			1.49110			0.745548	−	0.666452i	$-0.232188\pi$
$173$	19.6123			1.49110			0.745548	+	0.666452i	$0.232188\pi$
$174$		0			0
$175$	−17.0607	+	4.00000i	−1.28967	+	0.302372i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.58489			0.566921			0.283461	−	0.958984i	$-0.408517\pi$
$179$	7.58489			0.566921			0.283461	+	0.958984i	$0.408517\pi$
$180$		0			0
$181$	2.82936			0.210305			0.105152	−	0.994456i	$-0.466467\pi$
$181$	2.82936			0.210305			0.105152	+	0.994456i	$0.466467\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.72816	−	0.199879i	0.127057	−	0.0146954i
$186$		0			0
$187$	−8.56534			−0.626360
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.1698			−1.09765			−0.548823	−	0.835938i	$-0.684924\pi$
$191$	−15.1698			−1.09765			−0.548823	+	0.835938i	$0.684924\pi$
$192$		0			0
$193$		−	0.565344i		−	0.0406944i	−0.999793	−	0.0203472i	$-0.993523\pi$
$193$		−	0.565344i		−	0.0406944i	0.999793	−	0.0203472i	$-0.00647716\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.37122			0.596424			0.298212	−	0.954500i	$-0.403610\pi$
$197$	8.37122			0.596424			0.298212	+	0.954500i	$0.403610\pi$
$198$		0			0
$199$		−	19.7546i		−	1.40037i	−0.713962	−	0.700185i	$-0.753101\pi$
$199$		−	19.7546i		−	1.40037i	0.713962	−	0.700185i	$-0.246899\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	19.1711i		−	1.34555i
$204$		0			0
$205$	−5.42401	+	0.627343i	−0.378829	+	0.0438155i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			13.5142i			0.934800i
$210$		0			0
$211$		−	14.8294i		−	1.02090i	−0.859909	−	0.510448i	$-0.829480\pi$
$211$		−	14.8294i		−	1.02090i	0.859909	−	0.510448i	$-0.170520\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.45505	−	21.2264i	−0.167433	−	1.44763i
$216$		0			0
$217$		−	29.0280i		−	1.97055i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	24.4546i		−	1.64499i
$222$		0			0
$223$	19.5047			1.30613			0.653064	−	0.757302i	$-0.273483\pi$
$223$	19.5047			1.30613			0.653064	+	0.757302i	$0.273483\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	6.45637i		−	0.428524i	−0.976776	−	0.214262i	$-0.931265\pi$
$227$		−	6.45637i		−	0.428524i	0.976776	−	0.214262i	$-0.0687347\pi$
$228$		0			0
$229$	24.8480			1.64200			0.821002	−	0.570926i	$-0.193416\pi$
$229$	24.8480			1.64200			0.821002	+	0.570926i	$0.193416\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.04408			−0.526985			−0.263493	−	0.964661i	$-0.584874\pi$
$233$	−8.04408			−0.526985			−0.263493	+	0.964661i	$0.584874\pi$
$234$		0			0
$235$	26.0187	−	3.00933i	1.69727	−	0.196307i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	19.5709			1.26593			0.632967	−	0.774179i	$-0.281837\pi$
$239$	19.5709			1.26593			0.632967	+	0.774179i	$0.281837\pi$
$240$		0			0
$241$	25.1307			1.61881			0.809405	−	0.587251i	$-0.199790\pi$
$241$	25.1307			1.61881			0.809405	+	0.587251i	$0.199790\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.35718	+	11.7342i	0.0867071	+	0.749670i
$246$		0			0
$247$	−38.5840			−2.45504
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−26.7560			−1.68882			−0.844412	−	0.535695i	$-0.820050\pi$
$251$	−26.7560			−1.68882			−0.844412	+	0.535695i	$0.820050\pi$
$252$		0			0
$253$			2.12136i			0.133369i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.69835			−0.542588			−0.271294	−	0.962496i	$-0.587452\pi$
$257$	−8.69835			−0.542588			−0.271294	+	0.962496i	$0.587452\pi$
$258$		0			0
$259$		−	2.72666i		−	0.169426i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			18.0708i			1.11430i	0.830413	+	0.557148i	$0.188104\pi$
$263$			18.0708i			1.11430i	−0.830413	+	0.557148i	$0.811896\pi$
$264$		0			0
$265$	−2.96137	−	25.6040i	−0.181915	−	1.57284i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			9.95413i			0.606914i	0.952845	+	0.303457i	$0.0981409\pi$
$269$			9.95413i			0.606914i	−0.952845	+	0.303457i	$0.901859\pi$
$270$		0			0
$271$		−	27.1893i		−	1.65163i	−0.563939	−	0.825816i	$-0.690715\pi$
$271$		−	27.1893i		−	1.65163i	0.563939	−	0.825816i	$-0.309285\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.38567	−	2.20054i	0.565977	−	0.132697i
$276$		0			0
$277$			9.40196i			0.564909i	0.959281	+	0.282455i	$0.0911486\pi$
$277$			9.40196i			0.564909i	−0.959281	+	0.282455i	$0.908851\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	9.52612i		−	0.568281i	−0.958783	−	0.284140i	$-0.908292\pi$
$281$		−	9.52612i		−	0.568281i	0.958783	−	0.284140i	$-0.0917082\pi$
$282$		0			0
$283$	14.5840			0.866929			0.433464	−	0.901171i	$-0.357291\pi$
$283$	14.5840			0.866929			0.433464	+	0.901171i	$0.357291\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			8.55790i			0.505157i
$288$		0			0
$289$	2.73599			0.160940
$290$		0			0
$291$		0			0
$292$		0			0
$293$	21.8855			1.27856			0.639281	−	0.768973i	$-0.279232\pi$
$293$	21.8855			1.27856			0.639281	+	0.768973i	$0.279232\pi$
$294$		0			0
$295$	2.51399	+	21.7360i	0.146370	+	1.26552i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.05661			−0.350263
$300$		0			0
$301$	−33.4906			−1.93037
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.887197	+	7.67071i	0.0508008	+	0.439223i
$306$		0			0
$307$	25.1307			1.43428			0.717142	−	0.696927i	$-0.245450\pi$
$307$	25.1307			1.43428			0.717142	+	0.696927i	$0.245450\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−22.8819			−1.29752			−0.648758	−	0.760995i	$-0.724711\pi$
$311$	−22.8819			−1.29752			−0.648758	+	0.760995i	$0.724711\pi$
$312$		0			0
$313$			32.5653i			1.84070i	0.391093	+	0.920351i	$0.372097\pi$
$313$			32.5653i			1.84070i	−0.391093	+	0.920351i	$0.627903\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.91484			−0.107548			−0.0537742	−	0.998553i	$-0.517125\pi$
$317$	−1.91484			−0.107548			−0.0537742	+	0.998553i	$0.517125\pi$
$318$		0			0
$319$			10.5467i			0.590501i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	31.1391i		−	1.73262i
$324$		0			0
$325$	6.28267	+	26.7967i	0.348500	+	1.48641i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	41.0518i		−	2.26326i
$330$		0			0
$331$		−	6.44398i		−	0.354193i	−0.984193	−	0.177097i	$-0.943329\pi$
$331$		−	6.44398i		−	0.354193i	0.984193	−	0.177097i	$-0.0566705\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.40102	+	12.1132i	0.0765459	+	0.661816i
$336$		0			0
$337$		−	8.93206i		−	0.486560i	−0.969956	−	0.243280i	$-0.921777\pi$
$337$		−	8.93206i		−	0.486560i	0.969956	−	0.243280i	$-0.0782235\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			15.9693i			0.864786i
$342$		0			0
$343$	−6.01866			−0.324977
$344$		0			0
$345$		0			0
$346$		0			0
$347$			23.2817i			1.24983i	0.780694	+	0.624913i	$0.214866\pi$
$347$			23.2817i			1.24983i	−0.780694	+	0.624913i	$0.785134\pi$
$348$		0			0
$349$	−17.1120			−0.915986			−0.457993	−	0.888956i	$-0.651432\pi$
$349$	−17.1120			−0.915986			−0.457993	+	0.888956i	$0.651432\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.7286			−0.783923			−0.391962	−	0.919982i	$-0.628203\pi$
$353$	−14.7286			−0.783923			−0.391962	+	0.919982i	$0.628203\pi$
$354$		0			0
$355$	1.09337	+	9.45331i	0.0580303	+	0.501730i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.2251			−1.06744			−0.533721	−	0.845661i	$-0.679207\pi$
$359$	−20.2251			−1.06744			−0.533721	+	0.845661i	$0.679207\pi$
$360$		0			0
$361$	−30.1307			−1.58583
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.1560	+	1.86861i	−0.845643	+	0.0978074i
$366$		0			0
$367$	12.0700			0.630049			0.315025	−	0.949083i	$-0.397987\pi$
$367$	12.0700			0.630049			0.315025	+	0.949083i	$0.397987\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−40.3975			−2.09733
$372$		0			0
$373$			12.8153i			0.663552i	0.943358	+	0.331776i	$0.107648\pi$
$373$			12.8153i			0.663552i	−0.943358	+	0.331776i	$0.892352\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−30.1114			−1.55082
$378$		0			0
$379$			17.7360i			0.911036i	0.890226	+	0.455518i	$0.150546\pi$
$379$			17.7360i			0.911036i	−0.890226	+	0.455518i	$0.849454\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			25.0825i			1.28165i	0.767685	+	0.640827i	$0.221408\pi$
$383$			25.0825i			1.28165i	−0.767685	+	0.640827i	$0.778592\pi$
$384$		0			0
$385$	−1.73599	−	15.0093i	−0.0884740	−	0.764946i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	23.4684i		−	1.18989i	−0.803765	−	0.594947i	$-0.797173\pi$
$389$		−	23.4684i		−	1.18989i	0.803765	−	0.594947i	$-0.202827\pi$
$390$		0			0
$391$		−	4.88797i		−	0.247195i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.28474	−	0.726896i	0.316220	−	0.0365741i
$396$		0			0
$397$		−	2.69396i		−	0.135206i	−0.997712	−	0.0676031i	$-0.978465\pi$
$397$		−	2.69396i		−	0.135206i	0.997712	−	0.0676031i	$-0.0215351\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			32.3252i			1.61424i	0.590386	+	0.807121i	$0.298975\pi$
$401$			32.3252i			1.61424i	−0.590386	+	0.807121i	$0.701025\pi$
$402$		0			0
$403$	−45.5933			−2.27117
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.50003i			0.0743536i
$408$		0			0
$409$	20.6426			1.02071			0.510356	−	0.859963i	$-0.329514\pi$
$409$	20.6426			1.02071			0.510356	+	0.859963i	$0.329514\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	34.2946			1.68753
$414$		0			0
$415$	−9.45331	+	1.09337i	−0.464045	+	0.0536716i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	23.1544			1.13117			0.565584	−	0.824691i	$-0.308651\pi$
$419$	23.1544			1.13117			0.565584	+	0.824691i	$0.308651\pi$
$420$		0			0
$421$	28.3200			1.38023			0.690116	−	0.723699i	$-0.257560\pi$
$421$	28.3200			1.38023			0.690116	+	0.723699i	$0.257560\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−21.6262	+	5.07041i	−1.04902	+	0.245951i
$426$		0			0
$427$	12.1027			0.585691
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−26.6287			−1.28266			−0.641331	−	0.767265i	$-0.721617\pi$
$431$	−26.6287			−1.28266			−0.641331	+	0.767265i	$0.721617\pi$
$432$		0			0
$433$			23.3693i			1.12306i	0.827458	+	0.561528i	$0.189786\pi$
$433$			23.3693i			1.12306i	−0.827458	+	0.561528i	$0.810214\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.71215			−0.368922
$438$		0			0
$439$			28.3200i			1.35164i	0.737067	+	0.675820i	$0.236210\pi$
$439$			28.3200i			1.35164i	−0.737067	+	0.675820i	$0.763790\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	23.2817i		−	1.10615i	−0.833133	−	0.553073i	$-0.813455\pi$
$443$		−	23.2817i		−	1.10615i	0.833133	−	0.553073i	$-0.186545\pi$
$444$		0			0
$445$	0.858664	−	0.0993134i	0.0407046	−	0.00470791i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			29.3251i			1.38394i	0.721927	+	0.691969i	$0.243256\pi$
$449$			29.3251i			1.38394i	−0.721927	+	0.691969i	$0.756744\pi$
$450$		0			0
$451$		−	4.70800i		−	0.221691i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	42.8526	−	4.95634i	2.00896	−	0.232357i
$456$		0			0
$457$			18.9507i			0.886477i	0.896404	+	0.443239i	$0.146171\pi$
$457$			18.9507i			0.886477i	−0.896404	+	0.443239i	$0.853829\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			10.3539i			0.482229i	0.970497	+	0.241114i	$0.0775129\pi$
$461$			10.3539i			0.482229i	−0.970497	+	0.241114i	$0.922487\pi$
$462$		0			0
$463$	−3.07934			−0.143109			−0.0715545	−	0.997437i	$-0.522796\pi$
$463$	−3.07934			−0.143109			−0.0715545	+	0.997437i	$0.522796\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	40.9065i		−	1.89293i	−0.322809	−	0.946464i	$-0.604627\pi$
$467$		−	40.9065i		−	1.89293i	0.322809	−	0.946464i	$-0.395373\pi$
$468$		0			0
$469$	19.1120			0.882512
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18.4244			0.847153
$474$		0			0
$475$	8.00000	+	34.1214i	0.367065	+	1.56560i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.3691			−0.747921			−0.373961	−	0.927445i	$-0.622001\pi$
$479$	−16.3691			−0.747921			−0.373961	+	0.927445i	$0.622001\pi$
$480$		0			0
$481$	−4.28267			−0.195273
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−20.6400	+	2.38723i	−0.937212	+	0.108398i
$486$		0			0
$487$	−0.957977			−0.0434101			−0.0217050	−	0.999764i	$-0.506909\pi$
$487$	−0.957977			−0.0434101			−0.0217050	+	0.999764i	$0.506909\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.8395			0.489178			0.244589	−	0.969627i	$-0.421347\pi$
$491$	10.8395			0.489178			0.244589	+	0.969627i	$0.421347\pi$
$492$		0			0
$493$		−	24.3013i		−	1.09448i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14.9153			0.669041
$498$		0			0
$499$			31.5747i			1.41348i	0.707475	+	0.706738i	$0.249834\pi$
$499$			31.5747i			1.41348i	−0.707475	+	0.706738i	$0.750166\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	15.8241i		−	0.705560i	−0.935706	−	0.352780i	$-0.885236\pi$
$503$		−	15.8241i		−	0.705560i	0.935706	−	0.352780i	$-0.114764\pi$
$504$		0			0
$505$	−16.1507	+	1.86799i	−0.718695	+	0.0831246i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	10.1258i		−	0.448816i	−0.974495	−	0.224408i	$-0.927955\pi$
$509$		−	10.1258i		−	0.448816i	0.974495	−	0.224408i	$-0.0720449\pi$
$510$		0			0
$511$			25.4906i			1.12764i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.35671	+	37.6681i	0.191979	+	1.65985i
$516$		0			0
$517$			22.5840i			0.993243i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	33.3528i		−	1.46121i	−0.682799	−	0.730607i	$-0.739237\pi$
$521$		−	33.3528i		−	1.46121i	0.682799	−	0.730607i	$-0.260763\pi$
$522$		0			0
$523$	−25.9160			−1.13323			−0.566613	−	0.823984i	$-0.691746\pi$
$523$	−25.9160			−1.13323			−0.566613	+	0.823984i	$0.691746\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	36.7959i		−	1.60286i
$528$		0			0
$529$	21.7894			0.947366
$530$		0			0
$531$		0			0
$532$		0			0
$533$	13.4416			0.582221
$534$		0			0
$535$	−22.0187	+	2.54669i	−0.951950	+	0.110103i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−10.1852			−0.438707
$540$		0			0
$541$	−3.39470			−0.145950			−0.0729749	−	0.997334i	$-0.523249\pi$
$541$	−3.39470			−0.145950			−0.0729749	+	0.997334i	$0.523249\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.12792	−	18.3980i	−0.0911499	−	0.788082i
$546$		0			0
$547$	−7.57467			−0.323870			−0.161935	−	0.986801i	$-0.551773\pi$
$547$	−7.57467			−0.323870			−0.161935	+	0.986801i	$0.551773\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−38.3422			−1.63343
$552$		0			0
$553$		−	9.91595i		−	0.421669i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.38596			−0.0587252			−0.0293626	−	0.999569i	$-0.509348\pi$
$557$	−1.38596			−0.0587252			−0.0293626	+	0.999569i	$0.509348\pi$
$558$		0			0
$559$			52.6027i			2.22486i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			31.7934i			1.33993i	0.742393	+	0.669965i	$0.233691\pi$
$563$			31.7934i			1.33993i	−0.742393	+	0.669965i	$0.766309\pi$
$564$		0			0
$565$	−4.57599	−	39.5640i	−0.192513	−	1.66447i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	11.3005i		−	0.473742i	−0.971541	−	0.236871i	$-0.923878\pi$
$569$		−	11.3005i		−	0.473742i	0.971541	−	0.236871i	$-0.0761219\pi$
$570$		0			0
$571$			13.5933i			0.568863i	0.958696	+	0.284432i	$0.0918049\pi$
$571$			13.5933i			0.568863i	−0.958696	+	0.284432i	$0.908195\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.25578	+	5.35610i	0.0523695	+	0.223365i
$576$		0			0
$577$		−	43.0466i		−	1.79206i	−0.443998	−	0.896028i	$-0.646440\pi$
$577$		−	43.0466i		−	1.79206i	0.443998	−	0.896028i	$-0.353560\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			14.9153i			0.618790i
$582$		0			0
$583$	22.2241			0.920427
$584$		0			0
$585$		0			0
$586$		0			0
$587$			5.80210i			0.239478i	0.992805	+	0.119739i	$0.0382058\pi$
$587$			5.80210i			0.239478i	−0.992805	+	0.119739i	$0.961794\pi$
$588$		0			0
$589$	−58.0560			−2.39215
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.9015			0.652995			0.326498	−	0.945198i	$-0.394131\pi$
$593$	15.9015			0.652995			0.326498	+	0.945198i	$0.394131\pi$
$594$		0			0
$595$	4.00000	+	34.5840i	0.163984	+	1.41781i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	39.2510			1.60375			0.801876	−	0.597490i	$-0.203835\pi$
$599$	39.2510			1.60375			0.801876	+	0.597490i	$0.203835\pi$
$600$		0			0
$601$	−7.73599			−0.315557			−0.157779	−	0.987474i	$-0.550433\pi$
$601$	−7.73599			−0.315557			−0.157779	+	0.987474i	$0.550433\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.87100	−	16.1767i	−0.0760672	−	0.657677i
$606$		0			0
$607$	7.60737			0.308774			0.154387	−	0.988010i	$-0.450660\pi$
$607$	7.60737			0.308774			0.154387	+	0.988010i	$0.450660\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−64.4787			−2.60853
$612$		0			0
$613$		−	0.418069i		−	0.0168856i	−0.999964	−	0.00844282i	$-0.997313\pi$
$613$		−	0.418069i		−	0.0168856i	0.999964	−	0.00844282i	$-0.00268747\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	44.3214			1.78431			0.892156	−	0.451727i	$-0.149192\pi$
$617$	44.3214			1.78431			0.892156	+	0.451727i	$0.149192\pi$
$618$		0			0
$619$		−	4.07727i		−	0.163879i	−0.996637	−	0.0819396i	$-0.973889\pi$
$619$		−	4.07727i		−	0.163879i	0.996637	−	0.0819396i	$-0.0261115\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1.35478i		−	0.0542783i
$624$		0			0
$625$	22.3947	−	11.1120i	0.895788	−	0.444481i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	3.45632i		−	0.137812i
$630$		0			0
$631$		−	7.71733i		−	0.307222i	−0.988131	−	0.153611i	$-0.950910\pi$
$631$		−	7.71733i		−	0.307222i	0.988131	−	0.153611i	$-0.0490903\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.246116	+	2.12792i	0.00976680	+	0.0844438i
$636$		0			0
$637$		−	29.0793i		−	1.15217i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	6.18866i		−	0.244438i	−0.992503	−	0.122219i	$-0.960999\pi$
$641$		−	6.18866i		−	0.244438i	0.992503	−	0.122219i	$-0.0390010\pi$
$642$		0			0
$643$	26.1214			1.03013			0.515063	−	0.857152i	$-0.327769\pi$
$643$	26.1214			1.03013			0.515063	+	0.857152i	$0.327769\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			3.20180i			0.125876i	0.998017	+	0.0629379i	$0.0200470\pi$
$647$			3.20180i			0.125876i	−0.998017	+	0.0629379i	$0.979953\pi$
$648$		0			0
$649$	−18.8667			−0.740582
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.4568			−0.761403			−0.380702	−	0.924698i	$-0.624317\pi$
$653$	−19.4568			−0.761403			−0.380702	+	0.924698i	$0.624317\pi$
$654$		0			0
$655$	2.15405	+	18.6240i	0.0841659	+	0.727698i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.32780			−0.0906781			−0.0453390	−	0.998972i	$-0.514437\pi$
$659$	−2.32780			−0.0906781			−0.0453390	+	0.998972i	$0.514437\pi$
$660$		0			0
$661$	38.0373			1.47948			0.739740	−	0.672893i	$-0.234948\pi$
$661$	38.0373			1.47948			0.739740	+	0.672893i	$0.234948\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	54.5660	−	6.31113i	2.11598	−	0.244735i
$666$		0			0
$667$	−6.01866			−0.233043
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.65812			−0.257034
$672$		0			0
$673$			34.6867i			1.33707i	0.743679	+	0.668537i	$0.233079\pi$
$673$			34.6867i			1.33707i	−0.743679	+	0.668537i	$0.766921\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−45.1672			−1.73591			−0.867957	−	0.496639i	$-0.834567\pi$
$677$	−45.1672			−1.73591			−0.867957	+	0.496639i	$0.834567\pi$
$678$		0			0
$679$			32.5653i			1.24974i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			44.7986i			1.71417i	0.515175	+	0.857085i	$0.327727\pi$
$683$			44.7986i			1.71417i	−0.515175	+	0.857085i	$0.672273\pi$
$684$		0			0
$685$	2.85866	+	24.7160i	0.109224	+	0.944350i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			63.4511i			2.41729i
$690$		0			0
$691$		−	6.80392i		−	0.258833i	−0.991590	−	0.129417i	$-0.958690\pi$
$691$		−	6.80392i		−	0.258833i	0.991590	−	0.129417i	$-0.0413105\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.60030	+	0.300751i	−0.0986349	+	0.0114082i
$696$		0			0
$697$			10.8480i			0.410898i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	29.5250i		−	1.11514i	−0.830129	−	0.557572i	$-0.811733\pi$
$701$		−	29.5250i		−	1.11514i	0.830129	−	0.557572i	$-0.188267\pi$
$702$		0			0
$703$	−5.45331			−0.205676
$704$		0			0
$705$		0			0
$706$		0			0
$707$			25.4822i			0.958358i
$708$		0			0
$709$	27.1893			1.02112			0.510558	−	0.859843i	$-0.329439\pi$
$709$	27.1893			1.02112			0.510558	+	0.859843i	$0.329439\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.11317			−0.341291
$714$		0			0
$715$	−23.5747	+	2.72666i	−0.881643	+	0.101971i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.1145			−0.600971			−0.300486	−	0.953786i	$-0.597149\pi$
$719$	−16.1145			−0.600971			−0.300486	+	0.953786i	$0.597149\pi$
$720$		0			0
$721$	59.4320			2.21336
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.24330	+	26.6287i	0.231870	+	0.988966i
$726$		0			0
$727$	31.1820			1.15648			0.578239	−	0.815867i	$-0.303740\pi$
$727$	31.1820			1.15648			0.578239	+	0.815867i	$0.303740\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−42.4528			−1.57017
$732$		0			0
$733$		−	4.41129i		−	0.162935i	−0.996676	−	0.0814674i	$-0.974039\pi$
$733$		−	4.41129i		−	0.162935i	0.996676	−	0.0814674i	$-0.0259606\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10.5142			−0.387295
$738$		0			0
$739$			17.5560i			0.645808i	0.946432	+	0.322904i	$0.104659\pi$
$739$			17.5560i			0.645808i	−0.946432	+	0.322904i	$0.895341\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	9.65817i		−	0.354324i	−0.984182	−	0.177162i	$-0.943308\pi$
$743$		−	9.65817i		−	0.354324i	0.984182	−	0.177162i	$-0.0566917\pi$
$744$		0			0
$745$	−25.6040	+	2.96137i	−0.938057	+	0.108496i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			34.7406i			1.26940i
$750$		0			0
$751$		−	12.6053i		−	0.459974i	−0.973194	−	0.229987i	$-0.926132\pi$
$751$		−	12.6053i		−	0.459974i	0.973194	−	0.229987i	$-0.0738683\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24.0548	−	2.78219i	0.875445	−	0.101254i
$756$		0			0
$757$		−	36.0887i		−	1.31166i	−0.754906	−	0.655832i	$-0.772318\pi$
$757$		−	36.0887i		−	1.31166i	0.754906	−	0.655832i	$-0.227682\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			16.8385i			0.610396i	0.952289	+	0.305198i	$0.0987226\pi$
$761$			16.8385i			0.610396i	−0.952289	+	0.305198i	$0.901277\pi$
$762$		0			0
$763$	−29.0280			−1.05088
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	53.8655i		−	1.94497i
$768$		0			0
$769$	−54.3200			−1.95883			−0.979414	−	0.201860i	$-0.935301\pi$
$769$	−54.3200			−1.95883			−0.979414	+	0.201860i	$0.935301\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.17068			−0.221944			−0.110972	−	0.993824i	$-0.535396\pi$
$773$	−6.17068			−0.221944			−0.110972	+	0.993824i	$0.535396\pi$
$774$		0			0
$775$	9.45331	+	40.3200i	0.339573	+	1.44834i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17.1158			0.613237
$780$		0			0
$781$	−8.20541			−0.293612
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−22.9546	+	2.65493i	−0.819283	+	0.0947586i
$786$		0			0
$787$	2.12136			0.0756183			0.0378092	−	0.999285i	$-0.487962\pi$
$787$	2.12136			0.0756183			0.0378092	+	0.999285i	$0.487962\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−62.4234			−2.21952
$792$		0			0
$793$		−	19.0093i		−	0.675041i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.92522			0.280726			0.140363	−	0.990100i	$-0.455173\pi$
$797$	7.92522			0.280726			0.140363	+	0.990100i	$0.455173\pi$
$798$		0			0
$799$		−	52.0373i		−	1.84095i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	14.0233i		−	0.494871i
$804$		0			0
$805$	8.56534	−	0.990671i	0.301889	−	0.0349166i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	29.2158i		−	1.02717i	−0.858037	−	0.513587i	$-0.828316\pi$
$809$		−	29.2158i		−	1.02717i	0.858037	−	0.513587i	$-0.171684\pi$
$810$		0			0
$811$			10.3013i			0.361729i	0.983508	+	0.180864i	$0.0578895\pi$
$811$			10.3013i			0.361729i	−0.983508	+	0.180864i	$0.942111\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.30981	−	19.9706i	−0.0809091	−	0.699540i
$816$		0			0
$817$			66.9813i			2.34338i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	11.1270i		−	0.388336i	−0.980968	−	0.194168i	$-0.937799\pi$
$821$		−	11.1270i		−	0.388336i	0.980968	−	0.194168i	$-0.0622006\pi$
$822$		0			0
$823$	35.6447			1.24250			0.621248	−	0.783614i	$-0.286626\pi$
$823$	35.6447			1.24250			0.621248	+	0.783614i	$0.286626\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	9.11317i		−	0.316896i	−0.987367	−	0.158448i	$-0.949351\pi$
$827$		−	9.11317i		−	0.316896i	0.987367	−	0.158448i	$-0.0506490\pi$
$828$		0			0
$829$	−10.2640			−0.356484			−0.178242	−	0.983987i	$-0.557041\pi$
$829$	−10.2640			−0.356484			−0.178242	+	0.983987i	$0.557041\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	23.4684			0.813131
$834$		0			0
$835$	−29.0280	+	3.35739i	−1.00455	+	0.116187i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−31.5388			−1.08884			−0.544421	−	0.838812i	$-0.683251\pi$
$839$	−31.5388			−1.08884			−0.544421	+	0.838812i	$0.683251\pi$
$840$		0			0
$841$	−0.922733			−0.0318184
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.44492	−	38.4308i	−0.152910	−	1.32206i
$846$		0			0
$847$	−25.5233			−0.876992
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.856018			−0.0293439
$852$		0			0
$853$		−	15.3620i		−	0.525985i	−0.964798	−	0.262993i	$-0.915291\pi$
$853$		−	15.3620i		−	0.525985i	0.964798	−	0.262993i	$-0.0847095\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−16.4105			−0.560572			−0.280286	−	0.959917i	$-0.590429\pi$
$857$	−16.4105			−0.560572			−0.280286	+	0.959917i	$0.590429\pi$
$858$		0			0
$859$			14.8294i			0.505971i	0.967470	+	0.252986i	$0.0814125\pi$
$859$			14.8294i			0.505971i	−0.967470	+	0.252986i	$0.918587\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	24.3820i		−	0.829972i	−0.909828	−	0.414986i	$-0.863787\pi$
$863$		−	24.3820i		−	0.829972i	0.909828	−	0.414986i	$-0.136213\pi$
$864$		0			0
$865$	5.03863	+	43.5640i	0.171319	+	1.48122i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			5.45511i			0.185052i
$870$		0			0
$871$		−	30.0187i		−	1.01714i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−13.2681	−	36.8686i	−0.448545	−	1.24638i
$876$		0			0
$877$			19.6846i			0.664703i	0.943156	+	0.332351i	$0.107842\pi$
$877$			19.6846i			0.664703i	−0.943156	+	0.332351i	$0.892158\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			15.7280i			0.529889i	0.964264	+	0.264945i	$0.0853536\pi$
$881$			15.7280i			0.529889i	−0.964264	+	0.264945i	$0.914646\pi$
$882$		0			0
$883$	8.99067			0.302560			0.151280	−	0.988491i	$-0.451660\pi$
$883$	8.99067			0.302560			0.151280	+	0.988491i	$0.451660\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			8.66717i			0.291015i	0.989357	+	0.145508i	$0.0464815\pi$
$887$			8.66717i			0.291015i	−0.989357	+	0.145508i	$0.953518\pi$
$888$		0			0
$889$	3.35739			0.112603
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−82.1035			−2.74749
$894$		0			0
$895$	1.94865	+	16.8480i	0.0651361	+	0.563167i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−45.3076			−1.51109
$900$		0			0
$901$	−51.2080			−1.70598
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.726896	+	6.28474i	0.0241628	+	0.208912i
$906$		0			0
$907$	−39.5747			−1.31406			−0.657028	−	0.753866i	$-0.728187\pi$
$907$	−39.5747			−1.31406			−0.657028	+	0.753866i	$0.728187\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.60156			−0.119325			−0.0596625	−	0.998219i	$-0.519002\pi$
$911$	−3.60156			−0.119325			−0.0596625	+	0.998219i	$0.519002\pi$
$912$		0			0
$913$		−	8.20541i		−	0.271559i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	29.3845			0.970363
$918$		0			0
$919$			15.7173i			0.518467i	0.965815	+	0.259233i	$0.0834699\pi$
$919$			15.7173i			0.518467i	−0.965815	+	0.259233i	$0.916530\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	23.4269i		−	0.771107i
$924$		0			0
$925$	0.887968	+	3.78734i	0.0291962	+	0.124527i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	44.6929i		−	1.46633i	−0.680053	−	0.733163i	$-0.738043\pi$
$929$		−	44.6929i		−	1.46633i	0.680053	−	0.733163i	$-0.261957\pi$
$930$		0			0
$931$		−	37.0280i		−	1.21354i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.20054	−	19.0259i	−0.0719653	−	0.622212i
$936$		0			0
$937$			17.9813i			0.587425i	0.955894	+	0.293712i	$0.0948908\pi$
$937$			17.9813i			0.587425i	−0.955894	+	0.293712i	$0.905109\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	13.3276i		−	0.434466i	−0.976120	−	0.217233i	$-0.930297\pi$
$941$		−	13.3276i		−	0.434466i	0.976120	−	0.217233i	$-0.0697032\pi$
$942$		0			0
$943$	2.68670			0.0874911
$944$		0			0
$945$		0			0
$946$		0			0
$947$			43.8538i			1.42506i	0.701643	+	0.712529i	$0.252450\pi$
$947$			43.8538i			1.42506i	−0.701643	+	0.712529i	$0.747550\pi$
$948$		0			0
$949$	40.0373			1.29967
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.4382			−0.662058			−0.331029	−	0.943621i	$-0.607396\pi$
$953$	−20.4382			−0.662058			−0.331029	+	0.943621i	$0.607396\pi$
$954$		0			0
$955$	−3.89730	−	33.6960i	−0.126114	−	1.09038i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	38.9965			1.25926
$960$		0			0
$961$	−37.6027			−1.21299
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.25578	−	0.145244i	0.0404249	−	0.00467556i
$966$		0			0
$967$	−14.6167			−0.470041			−0.235021	−	0.971990i	$-0.575516\pi$
$967$	−14.6167			−0.470041			−0.235021	+	0.971990i	$0.575516\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−50.4374			−1.61861			−0.809307	−	0.587385i	$-0.800157\pi$
$971$	−50.4374			−1.61861			−0.809307	+	0.587385i	$0.800157\pi$
$972$		0			0
$973$			4.10270i			0.131527i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	17.0744			0.546257			0.273129	−	0.961978i	$-0.411942\pi$
$977$	17.0744			0.546257			0.273129	+	0.961978i	$0.411942\pi$
$978$		0			0
$979$			0.745314i			0.0238203i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			13.7688i			0.439155i	0.975595	+	0.219578i	$0.0704679\pi$
$983$			13.7688i			0.439155i	−0.975595	+	0.219578i	$0.929532\pi$
$984$		0			0
$985$	2.15066	+	18.5946i	0.0685259	+	0.592475i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			10.5142i			0.334332i
$990$		0			0
$991$			14.0959i			0.447772i	0.974615	+	0.223886i	$0.0718743\pi$
$991$			14.0959i			0.447772i	−0.974615	+	0.223886i	$0.928126\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	43.8802	−	5.07520i	1.39110	−	0.160895i
$996$		0			0
$997$			32.7126i			1.03602i	0.855375	+	0.518010i	$0.173327\pi$
$997$			32.7126i			1.03602i	−0.855375	+	0.518010i	$0.826673\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.o.e.2879.8		12
3.2	odd	2	inner	2880.2.o.e.2879.5		12
4.3	odd	2		2880.2.o.f.2879.8		12
5.4	even	2		2880.2.o.f.2879.6		12
8.3	odd	2		1440.2.o.a.1439.5	✓	12
8.5	even	2		1440.2.o.b.1439.5	yes	12
12.11	even	2		2880.2.o.f.2879.5		12
15.14	odd	2		2880.2.o.f.2879.7		12
20.19	odd	2	inner	2880.2.o.e.2879.6		12
24.5	odd	2		1440.2.o.b.1439.8	yes	12
24.11	even	2		1440.2.o.a.1439.8	yes	12
40.3	even	4		7200.2.h.l.1151.11		12
40.13	odd	4		7200.2.h.l.1151.2		12
40.19	odd	2		1440.2.o.b.1439.7	yes	12
40.27	even	4		7200.2.h.m.1151.1		12
40.29	even	2		1440.2.o.a.1439.7	yes	12
40.37	odd	4		7200.2.h.m.1151.12		12
60.59	even	2	inner	2880.2.o.e.2879.7		12
120.29	odd	2		1440.2.o.a.1439.6	yes	12
120.53	even	4		7200.2.h.l.1151.1		12
120.59	even	2		1440.2.o.b.1439.6	yes	12
120.77	even	4		7200.2.h.m.1151.11		12
120.83	odd	4		7200.2.h.l.1151.12		12
120.107	odd	4		7200.2.h.m.1151.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.o.a.1439.5	✓	12	8.3	odd	2
1440.2.o.a.1439.6	yes	12	120.29	odd	2
1440.2.o.a.1439.7	yes	12	40.29	even	2
1440.2.o.a.1439.8	yes	12	24.11	even	2
1440.2.o.b.1439.5	yes	12	8.5	even	2
1440.2.o.b.1439.6	yes	12	120.59	even	2
1440.2.o.b.1439.7	yes	12	40.19	odd	2
1440.2.o.b.1439.8	yes	12	24.5	odd	2
2880.2.o.e.2879.5		12	3.2	odd	2	inner
2880.2.o.e.2879.6		12	20.19	odd	2	inner
2880.2.o.e.2879.7		12	60.59	even	2	inner
2880.2.o.e.2879.8		12	1.1	even	1	trivial
2880.2.o.f.2879.5		12	12.11	even	2
2880.2.o.f.2879.6		12	5.4	even	2
2880.2.o.f.2879.7		12	15.14	odd	2
2880.2.o.f.2879.8		12	4.3	odd	2
7200.2.h.l.1151.1		12	120.53	even	4
7200.2.h.l.1151.2		12	40.13	odd	4
7200.2.h.l.1151.11		12	40.3	even	4
7200.2.h.l.1151.12		12	120.83	odd	4
7200.2.h.m.1151.1		12	40.27	even	4
7200.2.h.m.1151.2		12	120.107	odd	4
7200.2.h.m.1151.11		12	120.77	even	4
7200.2.h.m.1151.12		12	40.37	odd	4

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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.256912 + 2.22126i) q^{5} +3.50466 q^{7} +O(q^{10})\)	\(q+(0.256912 + 2.22126i) q^{5} +3.50466 q^{7} -1.92804 q^{11} -5.50466i q^{13} +4.44252 q^{17} -7.00933i q^{19} -1.10027i q^{23} +(-4.86799 + 1.14134i) q^{25} -5.47017i q^{29} -8.28267i q^{31} +(0.900390 + 7.78477i) q^{35} -0.778008i q^{37} +2.44186i q^{41} -9.55602 q^{43} -11.7135i q^{47} +5.28267 q^{49} -11.5268 q^{53} +(-0.495336 - 4.28267i) q^{55} +9.78543 q^{59} +3.45331 q^{61} +(12.2273 - 1.41421i) q^{65} +5.45331 q^{67} +4.25583 q^{71} +7.27334i q^{73} -6.75712 q^{77} -2.82936i q^{79} +4.25583i q^{83} +(1.14134 + 9.86799i) q^{85} -0.386566i q^{89} -19.2920i q^{91} +(15.5695 - 1.80078i) q^{95} +9.29200i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 8 q^{25} - 64 q^{43} - 4 q^{49} - 48 q^{55} + 8 q^{61} + 32 q^{67} - 20 q^{85}+O(q^{100}) \) 12 * q - 8 * q^25 - 64 * q^43 - 4 * q^49 - 48 * q^55 + 8 * q^61 + 32 * q^67 - 20 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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