LMFDB - Embedded newform 4140.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.a (trivial)

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:	yes
Analytic conductor:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +1.00000 q^{7} +4.44949 q^{11} +2.44949 q^{13} +3.44949 q^{17} +7.34847 q^{19} -1.00000 q^{23} +1.00000 q^{25} +9.44949 q^{29} -1.89898 q^{31} +1.00000 q^{35} -9.89898 q^{37} +0.550510 q^{41} -7.79796 q^{43} -7.34847 q^{47} -6.00000 q^{49} -4.34847 q^{53} +4.44949 q^{55} +6.55051 q^{59} +0.449490 q^{61} +2.44949 q^{65} +8.79796 q^{67} -2.34847 q^{71} -7.34847 q^{73} +4.44949 q^{77} -13.7980 q^{79} +15.4495 q^{83} +3.44949 q^{85} +3.10102 q^{89} +2.44949 q^{91} +7.34847 q^{95} +4.89898 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 2 * q^7	$ 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} + 2 q^{17} - 2 q^{23} + 2 q^{25} + 14 q^{29} + 6 q^{31} + 2 q^{35} - 10 q^{37} + 6 q^{41} + 4 q^{43} - 12 q^{49} + 6 q^{53} + 4 q^{55} + 18 q^{59} - 4 q^{61} - 2 q^{67} + 10 q^{71} + 4 q^{77} - 8 q^{79} + 26 q^{83} + 2 q^{85} + 16 q^{89}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 + 2 * q^17 - 2 * q^23 + 2 * q^25 + 14 * q^29 + 6 * q^31 + 2 * q^35 - 10 * q^37 + 6 * q^41 + 4 * q^43 - 12 * q^49 + 6 * q^53 + 4 * q^55 + 18 * q^59 - 4 * q^61 - 2 * q^67 + 10 * q^71 + 4 * q^77 - 8 * q^79 + 26 * q^83 + 2 * q^85 + 16 * q^89

Display 100 coefficients 10 coefficients

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.44949	1.34157	0.670786	−	0.741651i	$-0.265957\pi$
$11$	4.44949	1.34157	0.670786	+	0.741651i	$0.265957\pi$
$12$	0	0
$13$	2.44949	0.679366	0.339683	−	0.940540i	$-0.389680\pi$
$13$	2.44949	0.679366	0.339683	+	0.940540i	$0.389680\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.44949	0.836624	0.418312	−	0.908303i	$-0.362622\pi$
$17$	3.44949	0.836624	0.418312	+	0.908303i	$0.362622\pi$
$18$	0	0
$19$	7.34847	1.68585	0.842927	−	0.538028i	$-0.180830\pi$
$19$	7.34847	1.68585	0.842927	+	0.538028i	$0.180830\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.44949	1.75473	0.877363	−	0.479827i	$-0.159301\pi$
$29$	9.44949	1.75473	0.877363	+	0.479827i	$0.159301\pi$
$30$	0	0
$31$	−1.89898	−0.341067	−0.170533	−	0.985352i	$-0.554549\pi$
$31$	−1.89898	−0.341067	−0.170533	+	0.985352i	$0.554549\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−9.89898	−1.62738	−0.813691	−	0.581298i	$-0.802545\pi$
$37$	−9.89898	−1.62738	−0.813691	+	0.581298i	$0.802545\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.550510	0.0859753	0.0429876	−	0.999076i	$-0.486312\pi$
$41$	0.550510	0.0859753	0.0429876	+	0.999076i	$0.486312\pi$
$42$	0	0
$43$	−7.79796	−1.18918	−0.594589	−	0.804030i	$-0.702685\pi$
$43$	−7.79796	−1.18918	−0.594589	+	0.804030i	$0.702685\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.34847	−1.07188	−0.535942	−	0.844255i	$-0.680044\pi$
$47$	−7.34847	−1.07188	−0.535942	+	0.844255i	$0.680044\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.34847	−0.597308	−0.298654	−	0.954361i	$-0.596538\pi$
$53$	−4.34847	−0.597308	−0.298654	+	0.954361i	$0.596538\pi$
$54$	0	0
$55$	4.44949	0.599969
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.55051	0.852804	0.426402	−	0.904534i	$-0.359781\pi$
$59$	6.55051	0.852804	0.426402	+	0.904534i	$0.359781\pi$
$60$	0	0
$61$	0.449490	0.0575513	0.0287756	−	0.999586i	$-0.490839\pi$
$61$	0.449490	0.0575513	0.0287756	+	0.999586i	$0.490839\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.44949	0.303822
$66$	0	0
$67$	8.79796	1.07484	0.537421	−	0.843314i	$-0.319399\pi$
$67$	8.79796	1.07484	0.537421	+	0.843314i	$0.319399\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.34847	−0.278712	−0.139356	−	0.990242i	$-0.544503\pi$
$71$	−2.34847	−0.278712	−0.139356	+	0.990242i	$0.544503\pi$
$72$	0	0
$73$	−7.34847	−0.860073	−0.430037	−	0.902811i	$-0.641499\pi$
$73$	−7.34847	−0.860073	−0.430037	+	0.902811i	$0.641499\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.44949	0.507066
$78$	0	0
$79$	−13.7980	−1.55239	−0.776196	−	0.630492i	$-0.782853\pi$
$79$	−13.7980	−1.55239	−0.776196	+	0.630492i	$0.782853\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.4495	1.69580	0.847901	−	0.530155i	$-0.177866\pi$
$83$	15.4495	1.69580	0.847901	+	0.530155i	$0.177866\pi$
$84$	0	0
$85$	3.44949	0.374150
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.10102	0.328708	0.164354	−	0.986401i	$-0.447446\pi$
$89$	3.10102	0.328708	0.164354	+	0.986401i	$0.447446\pi$
$90$	0	0
$91$	2.44949	0.256776
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.34847	0.753937
$96$	0	0
$97$	4.89898	0.497416	0.248708	−	0.968579i	$-0.419994\pi$
$97$	4.89898	0.497416	0.248708	+	0.968579i	$0.419994\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.3485	1.62673	0.813367	−	0.581751i	$-0.197632\pi$
$101$	16.3485	1.62673	0.813367	+	0.581751i	$0.197632\pi$
$102$	0	0
$103$	2.89898	0.285645	0.142822	−	0.989748i	$-0.454382\pi$
$103$	2.89898	0.285645	0.142822	+	0.989748i	$0.454382\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.65153	−0.159660	−0.0798298	−	0.996809i	$-0.525438\pi$
$107$	−1.65153	−0.159660	−0.0798298	+	0.996809i	$0.525438\pi$
$108$	0	0
$109$	−18.4495	−1.76714	−0.883570	−	0.468299i	$-0.844867\pi$
$109$	−18.4495	−1.76714	−0.883570	+	0.468299i	$0.844867\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.2474	1.43436	0.717180	−	0.696888i	$-0.245433\pi$
$113$	15.2474	1.43436	0.717180	+	0.696888i	$0.245433\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.44949	0.316214
$120$	0	0
$121$	8.79796	0.799814
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.24745	−0.554371	−0.277186	−	0.960816i	$-0.589402\pi$
$127$	−6.24745	−0.554371	−0.277186	+	0.960816i	$0.589402\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	7.34847	0.637193
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.6969	−1.59739	−0.798694	−	0.601738i	$-0.794475\pi$
$137$	−18.6969	−1.59739	−0.798694	+	0.601738i	$0.794475\pi$
$138$	0	0
$139$	−10.7980	−0.915871	−0.457935	−	0.888985i	$-0.651411\pi$
$139$	−10.7980	−0.915871	−0.457935	+	0.888985i	$0.651411\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.8990	0.911418
$144$	0	0
$145$	9.44949	0.784737
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.55051	0.454716	0.227358	−	0.973811i	$-0.426991\pi$
$149$	5.55051	0.454716	0.227358	+	0.973811i	$0.426991\pi$
$150$	0	0
$151$	−3.79796	−0.309074	−0.154537	−	0.987987i	$-0.549389\pi$
$151$	−3.79796	−0.309074	−0.154537	+	0.987987i	$0.549389\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.89898	−0.152530
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	21.5959	1.69152	0.845761	−	0.533561i	$-0.179147\pi$
$163$	21.5959	1.69152	0.845761	+	0.533561i	$0.179147\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.4495	−1.11814	−0.559068	−	0.829122i	$-0.688841\pi$
$167$	−14.4495	−1.11814	−0.559068	+	0.829122i	$0.688841\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.79796	−0.744925	−0.372463	−	0.928047i	$-0.621486\pi$
$173$	−9.79796	−0.744925	−0.372463	+	0.928047i	$0.621486\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−4.89898	−0.364138	−0.182069	−	0.983286i	$-0.558279\pi$
$181$	−4.89898	−0.364138	−0.182069	+	0.983286i	$0.558279\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.89898	−0.727787
$186$	0	0
$187$	15.3485	1.12239
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.1464	1.24067	0.620336	−	0.784336i	$-0.286996\pi$
$191$	17.1464	1.24067	0.620336	+	0.784336i	$0.286996\pi$
$192$	0	0
$193$	−0.898979	−0.0647100	−0.0323550	−	0.999476i	$-0.510301\pi$
$193$	−0.898979	−0.0647100	−0.0323550	+	0.999476i	$0.510301\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.8990	0.776520	0.388260	−	0.921550i	$-0.373076\pi$
$197$	10.8990	0.776520	0.388260	+	0.921550i	$0.373076\pi$
$198$	0	0
$199$	24.6969	1.75072	0.875360	−	0.483472i	$-0.160625\pi$
$199$	24.6969	1.75072	0.875360	+	0.483472i	$0.160625\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.44949	0.663224
$204$	0	0
$205$	0.550510	0.0384493
$206$	0	0
$207$	0	0
$208$	0	0
$209$	32.6969	2.26169
$210$	0	0
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.79796	−0.531816
$216$	0	0
$217$	−1.89898	−0.128911
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.44949	0.568374
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.20204	−0.146155	−0.0730773	−	0.997326i	$-0.523282\pi$
$227$	−2.20204	−0.146155	−0.0730773	+	0.997326i	$0.523282\pi$
$228$	0	0
$229$	−22.6969	−1.49986	−0.749928	−	0.661520i	$-0.769912\pi$
$229$	−22.6969	−1.49986	−0.749928	+	0.661520i	$0.769912\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−23.5959	−1.54582	−0.772910	−	0.634516i	$-0.781200\pi$
$233$	−23.5959	−1.54582	−0.772910	+	0.634516i	$0.781200\pi$
$234$	0	0
$235$	−7.34847	−0.479361
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.0454	−1.49068	−0.745342	−	0.666683i	$-0.767714\pi$
$239$	−23.0454	−1.49068	−0.745342	+	0.666683i	$0.767714\pi$
$240$	0	0
$241$	14.0454	0.904744	0.452372	−	0.891829i	$-0.350578\pi$
$241$	14.0454	0.904744	0.452372	+	0.891829i	$0.350578\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.00000	−0.383326
$246$	0	0
$247$	18.0000	1.14531
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−4.44949	−0.279737
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.2474	−1.26300	−0.631501	−	0.775375i	$-0.717561\pi$
$257$	−20.2474	−1.26300	−0.631501	+	0.775375i	$0.717561\pi$
$258$	0	0
$259$	−9.89898	−0.615093
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.3485	−0.884765	−0.442382	−	0.896827i	$-0.645867\pi$
$263$	−14.3485	−0.884765	−0.442382	+	0.896827i	$0.645867\pi$
$264$	0	0
$265$	−4.34847	−0.267124
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.24745	0.563827	0.281913	−	0.959440i	$-0.409031\pi$
$269$	9.24745	0.563827	0.281913	+	0.959440i	$0.409031\pi$
$270$	0	0
$271$	15.8990	0.965794	0.482897	−	0.875677i	$-0.339585\pi$
$271$	15.8990	0.965794	0.482897	+	0.875677i	$0.339585\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.44949	0.268314
$276$	0	0
$277$	−19.7980	−1.18954	−0.594772	−	0.803894i	$-0.702758\pi$
$277$	−19.7980	−1.18954	−0.594772	+	0.803894i	$0.702758\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.3485	1.03492	0.517461	−	0.855707i	$-0.326877\pi$
$281$	17.3485	1.03492	0.517461	+	0.855707i	$0.326877\pi$
$282$	0	0
$283$	−5.89898	−0.350658	−0.175329	−	0.984510i	$-0.556099\pi$
$283$	−5.89898	−0.350658	−0.175329	+	0.984510i	$0.556099\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.550510	0.0324956
$288$	0	0
$289$	−5.10102	−0.300060
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.0454	0.995803	0.497902	−	0.867233i	$-0.334104\pi$
$293$	17.0454	0.995803	0.497902	+	0.867233i	$0.334104\pi$
$294$	0	0
$295$	6.55051	0.381385
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.44949	−0.141658
$300$	0	0
$301$	−7.79796	−0.449467
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.449490	0.0257377
$306$	0	0
$307$	4.24745	0.242415	0.121207	−	0.992627i	$-0.461323\pi$
$307$	4.24745	0.242415	0.121207	+	0.992627i	$0.461323\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−23.5959	−1.33800	−0.669001	−	0.743262i	$-0.733278\pi$
$311$	−23.5959	−1.33800	−0.669001	+	0.743262i	$0.733278\pi$
$312$	0	0
$313$	−33.6969	−1.90466	−0.952332	−	0.305064i	$-0.901322\pi$
$313$	−33.6969	−1.90466	−0.952332	+	0.305064i	$0.901322\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.6515	−0.710581	−0.355290	−	0.934756i	$-0.615618\pi$
$317$	−12.6515	−0.710581	−0.355290	+	0.934756i	$0.615618\pi$
$318$	0	0
$319$	42.0454	2.35409
$320$	0	0
$321$	0	0
$322$	0	0
$323$	25.3485	1.41043
$324$	0	0
$325$	2.44949	0.135873
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.34847	−0.405134
$330$	0	0
$331$	1.20204	0.0660702	0.0330351	−	0.999454i	$-0.489483\pi$
$331$	1.20204	0.0660702	0.0330351	+	0.999454i	$0.489483\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.79796	0.480684
$336$	0	0
$337$	4.20204	0.228900	0.114450	−	0.993429i	$-0.463489\pi$
$337$	4.20204	0.228900	0.114450	+	0.993429i	$0.463489\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8.44949	−0.457566
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.4949	1.10022	0.550112	−	0.835091i	$-0.314585\pi$
$347$	20.4949	1.10022	0.550112	+	0.835091i	$0.314585\pi$
$348$	0	0
$349$	15.4949	0.829423	0.414711	−	0.909953i	$-0.363883\pi$
$349$	15.4949	0.829423	0.414711	+	0.909953i	$0.363883\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.4495	−0.556170	−0.278085	−	0.960556i	$-0.589700\pi$
$353$	−10.4495	−0.556170	−0.278085	+	0.960556i	$0.589700\pi$
$354$	0	0
$355$	−2.34847	−0.124644
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.4495	0.762615	0.381307	−	0.924448i	$-0.375474\pi$
$359$	14.4495	0.762615	0.381307	+	0.924448i	$0.375474\pi$
$360$	0	0
$361$	35.0000	1.84211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.34847	−0.384636
$366$	0	0
$367$	20.1010	1.04926	0.524632	−	0.851329i	$-0.324203\pi$
$367$	20.1010	1.04926	0.524632	+	0.851329i	$0.324203\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.34847	−0.225761
$372$	0	0
$373$	29.7980	1.54288	0.771440	−	0.636302i	$-0.219537\pi$
$373$	29.7980	1.54288	0.771440	+	0.636302i	$0.219537\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.1464	1.19210
$378$	0	0
$379$	−5.79796	−0.297821	−0.148911	−	0.988851i	$-0.547577\pi$
$379$	−5.79796	−0.297821	−0.148911	+	0.988851i	$0.547577\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.24745	0.165937	0.0829684	−	0.996552i	$-0.473560\pi$
$383$	3.24745	0.165937	0.0829684	+	0.996552i	$0.473560\pi$
$384$	0	0
$385$	4.44949	0.226767
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−34.4949	−1.74896	−0.874480	−	0.485061i	$-0.838797\pi$
$389$	−34.4949	−1.74896	−0.874480	+	0.485061i	$0.838797\pi$
$390$	0	0
$391$	−3.44949	−0.174448
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.7980	−0.694251
$396$	0	0
$397$	−18.2020	−0.913534	−0.456767	−	0.889586i	$-0.650993\pi$
$397$	−18.2020	−0.913534	−0.456767	+	0.889586i	$0.650993\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.8990	−1.14352	−0.571760	−	0.820421i	$-0.693739\pi$
$401$	−22.8990	−1.14352	−0.571760	+	0.820421i	$0.693739\pi$
$402$	0	0
$403$	−4.65153	−0.231709
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−44.0454	−2.18325
$408$	0	0
$409$	−9.69694	−0.479483	−0.239741	−	0.970837i	$-0.577063\pi$
$409$	−9.69694	−0.479483	−0.239741	+	0.970837i	$0.577063\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.55051	0.322330
$414$	0	0
$415$	15.4495	0.758386
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.3485	1.62918	0.814590	−	0.580038i	$-0.196962\pi$
$419$	33.3485	1.62918	0.814590	+	0.580038i	$0.196962\pi$
$420$	0	0
$421$	7.34847	0.358142	0.179071	−	0.983836i	$-0.442691\pi$
$421$	7.34847	0.358142	0.179071	+	0.983836i	$0.442691\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.44949	0.167325
$426$	0	0
$427$	0.449490	0.0217523
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.4949	−0.601858	−0.300929	−	0.953647i	$-0.597297\pi$
$431$	−12.4949	−0.601858	−0.300929	+	0.953647i	$0.597297\pi$
$432$	0	0
$433$	−24.7980	−1.19171	−0.595857	−	0.803091i	$-0.703187\pi$
$433$	−24.7980	−1.19171	−0.595857	+	0.803091i	$0.703187\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.34847	−0.351525
$438$	0	0
$439$	−24.6969	−1.17872	−0.589360	−	0.807870i	$-0.700620\pi$
$439$	−24.6969	−1.17872	−0.589360	+	0.807870i	$0.700620\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.5505	0.833850	0.416925	−	0.908941i	$-0.363108\pi$
$443$	17.5505	0.833850	0.416925	+	0.908941i	$0.363108\pi$
$444$	0	0
$445$	3.10102	0.147002
$446$	0	0
$447$	0	0
$448$	0	0
$449$	38.8434	1.83313	0.916566	−	0.399884i	$-0.130949\pi$
$449$	38.8434	1.83313	0.916566	+	0.399884i	$0.130949\pi$
$450$	0	0
$451$	2.44949	0.115342
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.44949	0.114834
$456$	0	0
$457$	21.4949	1.00549	0.502744	−	0.864435i	$-0.332324\pi$
$457$	21.4949	1.00549	0.502744	+	0.864435i	$0.332324\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16.6969	−0.777654	−0.388827	−	0.921311i	$-0.627120\pi$
$461$	−16.6969	−0.777654	−0.388827	+	0.921311i	$0.627120\pi$
$462$	0	0
$463$	7.34847	0.341512	0.170756	−	0.985313i	$-0.445379\pi$
$463$	7.34847	0.341512	0.170756	+	0.985313i	$0.445379\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.1464	1.48756	0.743780	−	0.668425i	$-0.233031\pi$
$467$	32.1464	1.48756	0.743780	+	0.668425i	$0.233031\pi$
$468$	0	0
$469$	8.79796	0.406252
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−34.6969	−1.59537
$474$	0	0
$475$	7.34847	0.337171
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.9444	1.41389	0.706943	−	0.707271i	$-0.250074\pi$
$479$	30.9444	1.41389	0.706943	+	0.707271i	$0.250074\pi$
$480$	0	0
$481$	−24.2474	−1.10559
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.89898	0.222451
$486$	0	0
$487$	−37.8434	−1.71485	−0.857423	−	0.514612i	$-0.827936\pi$
$487$	−37.8434	−1.71485	−0.857423	+	0.514612i	$0.827936\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.24745	−0.146555	−0.0732777	−	0.997312i	$-0.523346\pi$
$491$	−3.24745	−0.146555	−0.0732777	+	0.997312i	$0.523346\pi$
$492$	0	0
$493$	32.5959	1.46805
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.34847	−0.105343
$498$	0	0
$499$	7.20204	0.322408	0.161204	−	0.986921i	$-0.448462\pi$
$499$	7.20204	0.322408	0.161204	+	0.986921i	$0.448462\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.1464	−0.809109	−0.404555	−	0.914514i	$-0.632573\pi$
$503$	−18.1464	−0.809109	−0.404555	+	0.914514i	$0.632573\pi$
$504$	0	0
$505$	16.3485	0.727497
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.30306	−0.0577572	−0.0288786	−	0.999583i	$-0.509194\pi$
$509$	−1.30306	−0.0577572	−0.0288786	+	0.999583i	$0.509194\pi$
$510$	0	0
$511$	−7.34847	−0.325077
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.89898	0.127744
$516$	0	0
$517$	−32.6969	−1.43801
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.2474	0.448949	0.224474	−	0.974480i	$-0.427933\pi$
$521$	10.2474	0.448949	0.224474	+	0.974480i	$0.427933\pi$
$522$	0	0
$523$	−29.7980	−1.30297	−0.651487	−	0.758660i	$-0.725854\pi$
$523$	−29.7980	−1.30297	−0.651487	+	0.758660i	$0.725854\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.55051	−0.285345
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.34847	0.0584087
$534$	0	0
$535$	−1.65153	−0.0714019
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−26.6969	−1.14992
$540$	0	0
$541$	−16.4949	−0.709171	−0.354586	−	0.935024i	$-0.615378\pi$
$541$	−16.4949	−0.709171	−0.354586	+	0.935024i	$0.615378\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−18.4495	−0.790289
$546$	0	0
$547$	−2.20204	−0.0941525	−0.0470762	−	0.998891i	$-0.514990\pi$
$547$	−2.20204	−0.0941525	−0.0470762	+	0.998891i	$0.514990\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	69.4393	2.95821
$552$	0	0
$553$	−13.7980	−0.586749
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.3485	−1.54013	−0.770067	−	0.637963i	$-0.779777\pi$
$557$	−36.3485	−1.54013	−0.770067	+	0.637963i	$0.779777\pi$
$558$	0	0
$559$	−19.1010	−0.807887
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.6515	0.743923	0.371962	−	0.928248i	$-0.378685\pi$
$563$	17.6515	0.743923	0.371962	+	0.928248i	$0.378685\pi$
$564$	0	0
$565$	15.2474	0.641465
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.69694	−0.364595	−0.182297	−	0.983243i	$-0.558353\pi$
$569$	−8.69694	−0.364595	−0.182297	+	0.983243i	$0.558353\pi$
$570$	0	0
$571$	−3.55051	−0.148584	−0.0742921	−	0.997237i	$-0.523670\pi$
$571$	−3.55051	−0.148584	−0.0742921	+	0.997237i	$0.523670\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	5.79796	0.241372	0.120686	−	0.992691i	$-0.461491\pi$
$577$	5.79796	0.241372	0.120686	+	0.992691i	$0.461491\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.4495	0.640953
$582$	0	0
$583$	−19.3485	−0.801332
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.3939	−0.800471	−0.400235	−	0.916412i	$-0.631072\pi$
$587$	−19.3939	−0.800471	−0.400235	+	0.916412i	$0.631072\pi$
$588$	0	0
$589$	−13.9546	−0.574989
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.2474	0.995723	0.497862	−	0.867256i	$-0.334119\pi$
$593$	24.2474	0.995723	0.497862	+	0.867256i	$0.334119\pi$
$594$	0	0
$595$	3.44949	0.141415
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.4949	1.00083	0.500417	−	0.865784i	$-0.333180\pi$
$599$	24.4949	1.00083	0.500417	+	0.865784i	$0.333180\pi$
$600$	0	0
$601$	−36.3939	−1.48454	−0.742269	−	0.670102i	$-0.766250\pi$
$601$	−36.3939	−1.48454	−0.742269	+	0.670102i	$0.766250\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.79796	0.357688
$606$	0	0
$607$	32.0454	1.30068	0.650341	−	0.759642i	$-0.274626\pi$
$607$	32.0454	1.30068	0.650341	+	0.759642i	$0.274626\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.0000	−0.728202
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−42.1464	−1.69675	−0.848376	−	0.529395i	$-0.822419\pi$
$617$	−42.1464	−1.69675	−0.848376	+	0.529395i	$0.822419\pi$
$618$	0	0
$619$	−26.2020	−1.05315	−0.526574	−	0.850129i	$-0.676524\pi$
$619$	−26.2020	−1.05315	−0.526574	+	0.850129i	$0.676524\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.10102	0.124240
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−34.1464	−1.36151
$630$	0	0
$631$	36.0454	1.43495	0.717473	−	0.696587i	$-0.245299\pi$
$631$	36.0454	1.43495	0.717473	+	0.696587i	$0.245299\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.24745	−0.247922
$636$	0	0
$637$	−14.6969	−0.582314
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.44949	0.175744	0.0878721	−	0.996132i	$-0.471993\pi$
$641$	4.44949	0.175744	0.0878721	+	0.996132i	$0.471993\pi$
$642$	0	0
$643$	23.4949	0.926548	0.463274	−	0.886215i	$-0.346675\pi$
$643$	23.4949	0.926548	0.463274	+	0.886215i	$0.346675\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.6515	−0.418755	−0.209377	−	0.977835i	$-0.567144\pi$
$647$	−10.6515	−0.418755	−0.209377	+	0.977835i	$0.567144\pi$
$648$	0	0
$649$	29.1464	1.14410
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.34847	0.365834	0.182917	−	0.983128i	$-0.441446\pi$
$653$	9.34847	0.365834	0.182917	+	0.983128i	$0.441446\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.9444	−0.815877	−0.407939	−	0.913009i	$-0.633752\pi$
$659$	−20.9444	−0.815877	−0.407939	+	0.913009i	$0.633752\pi$
$660$	0	0
$661$	44.6969	1.73851	0.869255	−	0.494364i	$-0.164599\pi$
$661$	44.6969	1.73851	0.869255	+	0.494364i	$0.164599\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.34847	0.284961
$666$	0	0
$667$	−9.44949	−0.365886
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	−27.5505	−1.06199	−0.530997	−	0.847374i	$-0.678182\pi$
$673$	−27.5505	−1.06199	−0.530997	+	0.847374i	$0.678182\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.34847	0.320858	0.160429	−	0.987047i	$-0.448712\pi$
$677$	8.34847	0.320858	0.160429	+	0.987047i	$0.448712\pi$
$678$	0	0
$679$	4.89898	0.188006
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.95459	−0.0747904	−0.0373952	−	0.999301i	$-0.511906\pi$
$683$	−1.95459	−0.0747904	−0.0373952	+	0.999301i	$0.511906\pi$
$684$	0	0
$685$	−18.6969	−0.714373
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.6515	−0.405791
$690$	0	0
$691$	29.5959	1.12588	0.562941	−	0.826497i	$-0.309670\pi$
$691$	29.5959	1.12588	0.562941	+	0.826497i	$0.309670\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.7980	−0.409590
$696$	0	0
$697$	1.89898	0.0719290
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16.6515	0.628920	0.314460	−	0.949271i	$-0.398177\pi$
$701$	16.6515	0.628920	0.314460	+	0.949271i	$0.398177\pi$
$702$	0	0
$703$	−72.7423	−2.74353
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.3485	0.614847
$708$	0	0
$709$	−33.3485	−1.25243	−0.626214	−	0.779651i	$-0.715396\pi$
$709$	−33.3485	−1.25243	−0.626214	+	0.779651i	$0.715396\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.89898	0.0711173
$714$	0	0
$715$	10.8990	0.407599
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.3485	−0.385933	−0.192966	−	0.981205i	$-0.561811\pi$
$719$	−10.3485	−0.385933	−0.192966	+	0.981205i	$0.561811\pi$
$720$	0	0
$721$	2.89898	0.107964
$722$	0	0
$723$	0	0
$724$	0	0
$725$	9.44949	0.350945
$726$	0	0
$727$	−8.10102	−0.300450	−0.150225	−	0.988652i	$-0.548000\pi$
$727$	−8.10102	−0.300450	−0.150225	+	0.988652i	$0.548000\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−26.8990	−0.994895
$732$	0	0
$733$	−26.3939	−0.974880	−0.487440	−	0.873156i	$-0.662069\pi$
$733$	−26.3939	−0.974880	−0.487440	+	0.873156i	$0.662069\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.1464	1.44198
$738$	0	0
$739$	7.40408	0.272364	0.136182	−	0.990684i	$-0.456517\pi$
$739$	7.40408	0.272364	0.136182	+	0.990684i	$0.456517\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.2020	1.25475	0.627376	−	0.778717i	$-0.284129\pi$
$743$	34.2020	1.25475	0.627376	+	0.778717i	$0.284129\pi$
$744$	0	0
$745$	5.55051	0.203355
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.65153	−0.0603456
$750$	0	0
$751$	45.1464	1.64742	0.823708	−	0.567014i	$-0.191901\pi$
$751$	45.1464	1.64742	0.823708	+	0.567014i	$0.191901\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.79796	−0.138222
$756$	0	0
$757$	−19.6969	−0.715897	−0.357949	−	0.933741i	$-0.616524\pi$
$757$	−19.6969	−0.715897	−0.357949	+	0.933741i	$0.616524\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.85357	0.284692	0.142346	−	0.989817i	$-0.454535\pi$
$761$	7.85357	0.284692	0.142346	+	0.989817i	$0.454535\pi$
$762$	0	0
$763$	−18.4495	−0.667916
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.0454	0.579366
$768$	0	0
$769$	−35.1464	−1.26741	−0.633706	−	0.773574i	$-0.718467\pi$
$769$	−35.1464	−1.26741	−0.633706	+	0.773574i	$0.718467\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10.8990	−0.392009	−0.196005	−	0.980603i	$-0.562797\pi$
$773$	−10.8990	−0.392009	−0.196005	+	0.980603i	$0.562797\pi$
$774$	0	0
$775$	−1.89898	−0.0682134
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.04541	0.144942
$780$	0	0
$781$	−10.4495	−0.373912
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.0000	0.392607
$786$	0	0
$787$	−47.2929	−1.68581	−0.842904	−	0.538064i	$-0.819156\pi$
$787$	−47.2929	−1.68581	−0.842904	+	0.538064i	$0.819156\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.2474	0.542137
$792$	0	0
$793$	1.10102	0.0390984
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.4495	−1.32653	−0.663264	−	0.748385i	$-0.730830\pi$
$797$	−37.4495	−1.32653	−0.663264	+	0.748385i	$0.730830\pi$
$798$	0	0
$799$	−25.3485	−0.896764
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−32.6969	−1.15385
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36.5505	−1.28505	−0.642524	−	0.766266i	$-0.722113\pi$
$809$	−36.5505	−1.28505	−0.642524	+	0.766266i	$0.722113\pi$
$810$	0	0
$811$	−32.7980	−1.15169	−0.575846	−	0.817558i	$-0.695327\pi$
$811$	−32.7980	−1.15169	−0.575846	+	0.817558i	$0.695327\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21.5959	0.756472
$816$	0	0
$817$	−57.3031	−2.00478
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44.4949	−1.55288	−0.776441	−	0.630190i	$-0.782977\pi$
$821$	−44.4949	−1.55288	−0.776441	+	0.630190i	$0.782977\pi$
$822$	0	0
$823$	17.7980	0.620398	0.310199	−	0.950672i	$-0.399604\pi$
$823$	17.7980	0.620398	0.310199	+	0.950672i	$0.399604\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.4393	−1.75395	−0.876973	−	0.480540i	$-0.840441\pi$
$827$	−50.4393	−1.75395	−0.876973	+	0.480540i	$0.840441\pi$
$828$	0	0
$829$	−2.30306	−0.0799886	−0.0399943	−	0.999200i	$-0.512734\pi$
$829$	−2.30306	−0.0799886	−0.0399943	+	0.999200i	$0.512734\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20.6969	−0.717106
$834$	0	0
$835$	−14.4495	−0.500045
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.00000	0.0690477	0.0345238	−	0.999404i	$-0.489009\pi$
$839$	2.00000	0.0690477	0.0345238	+	0.999404i	$0.489009\pi$
$840$	0	0
$841$	60.2929	2.07906
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.00000	−0.240807
$846$	0	0
$847$	8.79796	0.302301
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.89898	0.339333
$852$	0	0
$853$	−5.10102	−0.174656	−0.0873278	−	0.996180i	$-0.527833\pi$
$853$	−5.10102	−0.174656	−0.0873278	+	0.996180i	$0.527833\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.5959	0.464428	0.232214	−	0.972665i	$-0.425403\pi$
$857$	13.5959	0.464428	0.232214	+	0.972665i	$0.425403\pi$
$858$	0	0
$859$	−22.5959	−0.770963	−0.385481	−	0.922716i	$-0.625965\pi$
$859$	−22.5959	−0.770963	−0.385481	+	0.922716i	$0.625965\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.2929	0.963100	0.481550	−	0.876419i	$-0.340074\pi$
$863$	28.2929	0.963100	0.481550	+	0.876419i	$0.340074\pi$
$864$	0	0
$865$	−9.79796	−0.333141
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−61.3939	−2.08264
$870$	0	0
$871$	21.5505	0.730211
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	24.6969	0.833956	0.416978	−	0.908916i	$-0.363089\pi$
$877$	24.6969	0.833956	0.416978	+	0.908916i	$0.363089\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19.7526	0.665480	0.332740	−	0.943019i	$-0.392027\pi$
$881$	19.7526	0.665480	0.332740	+	0.943019i	$0.392027\pi$
$882$	0	0
$883$	−42.2474	−1.42174	−0.710870	−	0.703324i	$-0.751699\pi$
$883$	−42.2474	−1.42174	−0.710870	+	0.703324i	$0.751699\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.6969	−1.09786	−0.548928	−	0.835870i	$-0.684964\pi$
$887$	−32.6969	−1.09786	−0.548928	+	0.835870i	$0.684964\pi$
$888$	0	0
$889$	−6.24745	−0.209533
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−54.0000	−1.80704
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17.9444	−0.598479
$900$	0	0
$901$	−15.0000	−0.499722
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.89898	−0.162848
$906$	0	0
$907$	−45.0000	−1.49420	−0.747100	−	0.664711i	$-0.768555\pi$
$907$	−45.0000	−1.49420	−0.747100	+	0.664711i	$0.768555\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.40408	−0.145914	−0.0729569	−	0.997335i	$-0.523244\pi$
$911$	−4.40408	−0.145914	−0.0729569	+	0.997335i	$0.523244\pi$
$912$	0	0
$913$	68.7423	2.27504
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	25.7980	0.850996	0.425498	−	0.904959i	$-0.360099\pi$
$919$	25.7980	0.850996	0.425498	+	0.904959i	$0.360099\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.75255	−0.189348
$924$	0	0
$925$	−9.89898	−0.325476
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.8434	−0.946320	−0.473160	−	0.880976i	$-0.656887\pi$
$929$	−28.8434	−0.946320	−0.473160	+	0.880976i	$0.656887\pi$
$930$	0	0
$931$	−44.0908	−1.44502
$932$	0	0
$933$	0	0
$934$	0	0
$935$	15.3485	0.501949
$936$	0	0
$937$	12.4949	0.408191	0.204095	−	0.978951i	$-0.434575\pi$
$937$	12.4949	0.408191	0.204095	+	0.978951i	$0.434575\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.4495	0.992625	0.496312	−	0.868144i	$-0.334687\pi$
$941$	30.4495	0.992625	0.496312	+	0.868144i	$0.334687\pi$
$942$	0	0
$943$	−0.550510	−0.0179271
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.6969	1.38746	0.693732	−	0.720233i	$-0.255965\pi$
$947$	42.6969	1.38746	0.693732	+	0.720233i	$0.255965\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.59592	0.181270	0.0906348	−	0.995884i	$-0.471110\pi$
$953$	5.59592	0.181270	0.0906348	+	0.995884i	$0.471110\pi$
$954$	0	0
$955$	17.1464	0.554845
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.6969	−0.603756
$960$	0	0
$961$	−27.3939	−0.883673
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.898979	−0.0289392
$966$	0	0
$967$	−6.65153	−0.213899	−0.106949	−	0.994264i	$-0.534108\pi$
$967$	−6.65153	−0.213899	−0.106949	+	0.994264i	$0.534108\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.8990	1.44088	0.720438	−	0.693519i	$-0.243941\pi$
$971$	44.8990	1.44088	0.720438	+	0.693519i	$0.243941\pi$
$972$	0	0
$973$	−10.7980	−0.346167
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.3485	0.714991	0.357495	−	0.933915i	$-0.383631\pi$
$977$	22.3485	0.714991	0.357495	+	0.933915i	$0.383631\pi$
$978$	0	0
$979$	13.7980	0.440985
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.14643	−0.259831	−0.129915	−	0.991525i	$-0.541471\pi$
$983$	−8.14643	−0.259831	−0.129915	+	0.991525i	$0.541471\pi$
$984$	0	0
$985$	10.8990	0.347270
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.79796	0.247961
$990$	0	0
$991$	−18.5959	−0.590719	−0.295359	−	0.955386i	$-0.595439\pi$
$991$	−18.5959	−0.590719	−0.295359	+	0.955386i	$0.595439\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	24.6969	0.782946
$996$	0	0
$997$	−36.8990	−1.16860	−0.584301	−	0.811537i	$-0.698631\pi$
$997$	−36.8990	−1.16860	−0.584301	+	0.811537i	$0.698631\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.r.1.2		2
3.2	odd	2		1380.2.a.f.1.1	✓	2
12.11	even	2		5520.2.a.bl.1.2		2
15.2	even	4		6900.2.f.h.6349.3		4
15.8	even	4		6900.2.f.h.6349.1		4
15.14	odd	2		6900.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.f.1.1	✓	2	3.2	odd	2
4140.2.a.r.1.2		2	1.1	even	1	trivial
5520.2.a.bl.1.2		2	12.11	even	2
6900.2.a.r.1.1		2	15.14	odd	2
6900.2.f.h.6349.1		4	15.8	even	4
6900.2.f.h.6349.3		4	15.2	even	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +1.00000 q^{7} +4.44949 q^{11} +2.44949 q^{13} +3.44949 q^{17} +7.34847 q^{19} -1.00000 q^{23} +1.00000 q^{25} +9.44949 q^{29} -1.89898 q^{31} +1.00000 q^{35} -9.89898 q^{37} +0.550510 q^{41} -7.79796 q^{43} -7.34847 q^{47} -6.00000 q^{49} -4.34847 q^{53} +4.44949 q^{55} +6.55051 q^{59} +0.449490 q^{61} +2.44949 q^{65} +8.79796 q^{67} -2.34847 q^{71} -7.34847 q^{73} +4.44949 q^{77} -13.7980 q^{79} +15.4495 q^{83} +3.44949 q^{85} +3.10102 q^{89} +2.44949 q^{91} +7.34847 q^{95} +4.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} + 2 q^{17} - 2 q^{23} + 2 q^{25} + 14 q^{29} + 6 q^{31} + 2 q^{35} - 10 q^{37} + 6 q^{41} + 4 q^{43} - 12 q^{49} + 6 q^{53} + 4 q^{55} + 18 q^{59} - 4 q^{61} - 2 q^{67} + 10 q^{71} + 4 q^{77} - 8 q^{79} + 26 q^{83} + 2 q^{85} + 16 q^{89}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 + 2 * q^17 - 2 * q^23 + 2 * q^25 + 14 * q^29 + 6 * q^31 + 2 * q^35 - 10 * q^37 + 6 * q^41 + 4 * q^43 - 12 * q^49 + 6 * q^53 + 4 * q^55 + 18 * q^59 - 4 * q^61 - 2 * q^67 + 10 * q^71 + 4 * q^77 - 8 * q^79 + 26 * q^83 + 2 * q^85 + 16 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more