LMFDB - Embedded newform 3744.2.g.e.1873.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1873,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("3744.1873");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.g (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:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 2x^14 - 4x^13 + 9x^12 - 10x^11 + 2x^10 - 8x^9 + 28x^8 - 16x^7 + 8x^6 - 80x^5 + 144x^4 - 128x^3 + 128x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1873.7
Root			$-0.414573 - 1.35208i$ of defining polynomial
Character	$\chi$	$=$	3744.1873
Dual form			3744.2.g.e.1873.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.550135i q^{5} -4.37841 q^{7} +O(q^{10})$	$q-0.550135i q^{5} -4.37841 q^{7} -2.79676i q^{11} -1.00000i q^{13} -4.67772 q^{17} -4.05320i q^{19} -1.34132 q^{23} +4.69735 q^{25} -7.77294i q^{29} +8.12845 q^{31} +2.40872i q^{35} +9.06663i q^{37} -6.32307 q^{41} +6.38891i q^{43} -2.92854 q^{47} +12.1705 q^{49} +8.02256i q^{53} -1.53860 q^{55} +2.05355i q^{59} -3.20410i q^{61} -0.550135 q^{65} +12.5601i q^{67} -5.81639 q^{71} -10.4792 q^{73} +12.2454i q^{77} +0.975314 q^{79} -1.58722i q^{83} +2.57338i q^{85} +7.89497 q^{89} +4.37841i q^{91} -2.22981 q^{95} -13.8234 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7}+O(q^{10}) $ 16 * q - 4 * q^7	$ 16 q - 4 q^{7} - 16 q^{17} - 8 q^{23} - 32 q^{25} + 4 q^{31} + 36 q^{41} + 24 q^{47} + 48 q^{49} - 24 q^{55} - 4 q^{65} - 32 q^{73} + 60 q^{89} - 24 q^{95} + 40 q^{97}+O(q^{100}) $ 16 * q - 4 * q^7 - 16 * q^17 - 8 * q^23 - 32 * q^25 + 4 * q^31 + 36 * q^41 + 24 * q^47 + 48 * q^49 - 24 * q^55 - 4 * q^65 - 32 * q^73 + 60 * q^89 - 24 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	− 0.550135i	− 0.246028i	−0.992405	−	0.123014i	$-0.960744\pi$
$5$	− 0.550135i	− 0.246028i	0.992405	−	0.123014i	$-0.0392560\pi$
$6$	0	0
$7$	−4.37841	−1.65488	−0.827441	−	0.561552i	$-0.810204\pi$
$7$	−4.37841	−1.65488	−0.827441	+	0.561552i	$0.810204\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.79676i	− 0.843255i	−0.906769	−	0.421627i	$-0.861459\pi$
$11$	− 2.79676i	− 0.843255i	0.906769	−	0.421627i	$-0.138541\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.67772	−1.13451	−0.567257	−	0.823541i	$-0.691995\pi$
$17$	−4.67772	−1.13451	−0.567257	+	0.823541i	$0.691995\pi$
$18$	0	0
$19$	− 4.05320i	− 0.929869i	−0.885345	−	0.464934i	$-0.846078\pi$
$19$	− 4.05320i	− 0.929869i	0.885345	−	0.464934i	$-0.153922\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.34132	−0.279685	−0.139843	−	0.990174i	$-0.544660\pi$
$23$	−1.34132	−0.279685	−0.139843	+	0.990174i	$0.544660\pi$
$24$	0	0
$25$	4.69735	0.939470
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.77294i	− 1.44340i	−0.692207	−	0.721699i	$-0.743362\pi$
$29$	− 7.77294i	− 1.44340i	0.692207	−	0.721699i	$-0.256638\pi$
$30$	0	0
$31$	8.12845	1.45991	0.729956	−	0.683494i	$-0.239540\pi$
$31$	8.12845	1.45991	0.729956	+	0.683494i	$0.239540\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.40872i	0.407147i
$36$	0	0
$37$	9.06663i	1.49054i	0.666760	+	0.745272i	$0.267680\pi$
$37$	9.06663i	1.49054i	−0.666760	+	0.745272i	$0.732320\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.32307	−0.987498	−0.493749	−	0.869605i	$-0.664374\pi$
$41$	−6.32307	−0.987498	−0.493749	+	0.869605i	$0.664374\pi$
$42$	0	0
$43$	6.38891i	0.974299i	0.873319	+	0.487149i	$0.161963\pi$
$43$	6.38891i	0.974299i	−0.873319	+	0.487149i	$0.838037\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.92854	−0.427172	−0.213586	−	0.976924i	$-0.568514\pi$
$47$	−2.92854	−0.427172	−0.213586	+	0.976924i	$0.568514\pi$
$48$	0	0
$49$	12.1705	1.73864
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.02256i	1.10198i	0.834511	+	0.550991i	$0.185750\pi$
$53$	8.02256i	1.10198i	−0.834511	+	0.550991i	$0.814250\pi$
$54$	0	0
$55$	−1.53860	−0.207464
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.05355i	0.267350i	0.991025	+	0.133675i	$0.0426778\pi$
$59$	2.05355i	0.267350i	−0.991025	+	0.133675i	$0.957322\pi$
$60$	0	0
$61$	− 3.20410i	− 0.410243i	−0.978736	−	0.205122i	$-0.934241\pi$
$61$	− 3.20410i	− 0.410243i	0.978736	−	0.205122i	$-0.0657590\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.550135	−0.0682359
$66$	0	0
$67$	12.5601i	1.53446i	0.641375	+	0.767228i	$0.278364\pi$
$67$	12.5601i	1.53446i	−0.641375	+	0.767228i	$0.721636\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.81639	−0.690278	−0.345139	−	0.938551i	$-0.612168\pi$
$71$	−5.81639	−0.690278	−0.345139	+	0.938551i	$0.612168\pi$
$72$	0	0
$73$	−10.4792	−1.22650	−0.613251	−	0.789888i	$-0.710139\pi$
$73$	−10.4792	−1.22650	−0.613251	+	0.789888i	$0.710139\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.2454i	1.39549i
$78$	0	0
$79$	0.975314	0.109731	0.0548657	−	0.998494i	$-0.482527\pi$
$79$	0.975314	0.109731	0.0548657	+	0.998494i	$0.482527\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1.58722i	− 0.174220i	−0.996199	−	0.0871100i	$-0.972237\pi$
$83$	− 1.58722i	− 0.174220i	0.996199	−	0.0871100i	$-0.0277632\pi$
$84$	0	0
$85$	2.57338i	0.279122i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.89497	0.836865	0.418433	−	0.908248i	$-0.362580\pi$
$89$	7.89497	0.836865	0.418433	+	0.908248i	$0.362580\pi$
$90$	0	0
$91$	4.37841i	0.458982i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.22981	−0.228774
$96$	0	0
$97$	−13.8234	−1.40356	−0.701779	−	0.712395i	$-0.747611\pi$
$97$	−13.8234	−1.40356	−0.701779	+	0.712395i	$0.747611\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.8909i	1.58120i	0.612331	+	0.790601i	$0.290232\pi$
$101$	15.8909i	1.58120i	−0.612331	+	0.790601i	$0.709768\pi$
$102$	0	0
$103$	−3.33134	−0.328247	−0.164123	−	0.986440i	$-0.552480\pi$
$103$	−3.33134	−0.328247	−0.164123	+	0.986440i	$0.552480\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.9485i	− 1.44512i	−0.691308	−	0.722561i	$-0.742965\pi$
$107$	− 14.9485i	− 1.44512i	0.691308	−	0.722561i	$-0.257035\pi$
$108$	0	0
$109$	14.8678i	1.42408i	0.702140	+	0.712039i	$0.252228\pi$
$109$	14.8678i	1.42408i	−0.702140	+	0.712039i	$0.747772\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.3553	−1.53857	−0.769287	−	0.638904i	$-0.779388\pi$
$113$	−16.3553	−1.53857	−0.769287	+	0.638904i	$0.779388\pi$
$114$	0	0
$115$	0.737909i	0.0688104i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.4810	1.87749
$120$	0	0
$121$	3.17814	0.288922
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 5.33485i	− 0.477164i
$126$	0	0
$127$	10.2860	0.912734	0.456367	−	0.889792i	$-0.349150\pi$
$127$	10.2860	0.912734	0.456367	+	0.889792i	$0.349150\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.9485i	0.956571i	0.878205	+	0.478285i	$0.158742\pi$
$131$	10.9485i	0.956571i	−0.878205	+	0.478285i	$0.841258\pi$
$132$	0	0
$133$	17.7466i	1.53882i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.86357	−0.757266	−0.378633	−	0.925547i	$-0.623606\pi$
$137$	−8.86357	−0.757266	−0.378633	+	0.925547i	$0.623606\pi$
$138$	0	0
$139$	− 8.94518i	− 0.758720i	−0.925249	−	0.379360i	$-0.876144\pi$
$139$	− 8.94518i	− 0.758720i	0.925249	−	0.379360i	$-0.123856\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.79676	−0.233877
$144$	0	0
$145$	−4.27616	−0.355116
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.2283i	1.32947i	0.747077	+	0.664737i	$0.231456\pi$
$149$	16.2283i	1.32947i	−0.747077	+	0.664737i	$0.768544\pi$
$150$	0	0
$151$	9.58509	0.780023	0.390012	−	0.920810i	$-0.372471\pi$
$151$	9.58509	0.780023	0.390012	+	0.920810i	$0.372471\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4.47174i	− 0.359179i
$156$	0	0
$157$	0.940535i	0.0750628i	0.999295	+	0.0375314i	$0.0119494\pi$
$157$	0.940535i	0.0750628i	−0.999295	+	0.0375314i	$0.988051\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.87286	0.462846
$162$	0	0
$163$	10.0472i	0.786955i	0.919334	+	0.393478i	$0.128728\pi$
$163$	10.0472i	0.786955i	−0.919334	+	0.393478i	$0.871272\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.90093	−0.534010	−0.267005	−	0.963695i	$-0.586034\pi$
$167$	−6.90093	−0.534010	−0.267005	+	0.963695i	$0.586034\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.859257i	0.0653281i	0.999466	+	0.0326641i	$0.0103991\pi$
$173$	0.859257i	0.0653281i	−0.999466	+	0.0326641i	$0.989601\pi$
$174$	0	0
$175$	−20.5669	−1.55471
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 2.40872i	− 0.180036i	−0.995940	−	0.0900180i	$-0.971308\pi$
$179$	− 2.40872i	− 0.180036i	0.995940	−	0.0900180i	$-0.0286924\pi$
$180$	0	0
$181$	5.09873i	0.378985i	0.981882	+	0.189493i	$0.0606843\pi$
$181$	5.09873i	0.378985i	−0.981882	+	0.189493i	$0.939316\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.98787	0.366715
$186$	0	0
$187$	13.0825i	0.956684i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.9034	−0.861303	−0.430651	−	0.902518i	$-0.641716\pi$
$191$	−11.9034	−0.861303	−0.430651	+	0.902518i	$0.641716\pi$
$192$	0	0
$193$	−10.0615	−0.724244	−0.362122	−	0.932131i	$-0.617948\pi$
$193$	−10.0615	−0.724244	−0.362122	+	0.932131i	$0.617948\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.94994i	− 0.495163i	−0.968867	−	0.247582i	$-0.920364\pi$
$197$	− 6.94994i	− 0.495163i	0.968867	−	0.247582i	$-0.0796358\pi$
$198$	0	0
$199$	17.9719	1.27400	0.636998	−	0.770865i	$-0.280176\pi$
$199$	17.9719	1.27400	0.636998	+	0.770865i	$0.280176\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	34.0331i	2.38865i
$204$	0	0
$205$	3.47854i	0.242952i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−11.3358	−0.784116
$210$	0	0
$211$	12.2779i	0.845245i	0.906306	+	0.422622i	$0.138890\pi$
$211$	12.2779i	0.845245i	−0.906306	+	0.422622i	$0.861110\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.51476	0.239705
$216$	0	0
$217$	−35.5897	−2.41598
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.67772i	0.314658i
$222$	0	0
$223$	−22.2617	−1.49075	−0.745377	−	0.666643i	$-0.767730\pi$
$223$	−22.2617	−1.49075	−0.745377	+	0.666643i	$0.767730\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 17.6758i	− 1.17318i	−0.809882	−	0.586592i	$-0.800469\pi$
$227$	− 17.6758i	− 1.17318i	0.809882	−	0.586592i	$-0.199531\pi$
$228$	0	0
$229$	5.97901i	0.395104i	0.980292	+	0.197552i	$0.0632991\pi$
$229$	5.97901i	0.395104i	−0.980292	+	0.197552i	$0.936701\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.4636	1.79920	0.899599	−	0.436716i	$-0.143859\pi$
$233$	27.4636	1.79920	0.899599	+	0.436716i	$0.143859\pi$
$234$	0	0
$235$	1.61109i	0.105096i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.3680	0.929389	0.464695	−	0.885471i	$-0.346164\pi$
$239$	14.3680	0.929389	0.464695	+	0.885471i	$0.346164\pi$
$240$	0	0
$241$	9.76192	0.628821	0.314410	−	0.949287i	$-0.398193\pi$
$241$	9.76192	0.628821	0.314410	+	0.949287i	$0.398193\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 6.69540i	− 0.427753i
$246$	0	0
$247$	−4.05320	−0.257899
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0017i	0.757542i	0.925490	+	0.378771i	$0.123653\pi$
$251$	12.0017i	0.757542i	−0.925490	+	0.378771i	$0.876347\pi$
$252$	0	0
$253$	3.75136i	0.235846i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.86392	0.428160	0.214080	−	0.976816i	$-0.431325\pi$
$257$	6.86392	0.428160	0.214080	+	0.976816i	$0.431325\pi$
$258$	0	0
$259$	− 39.6974i	− 2.46668i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.3713	0.639524	0.319762	−	0.947498i	$-0.396397\pi$
$263$	10.3713	0.639524	0.319762	+	0.947498i	$0.396397\pi$
$264$	0	0
$265$	4.41349	0.271119
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 31.2235i	− 1.90373i	−0.306513	−	0.951866i	$-0.599162\pi$
$269$	− 31.2235i	− 1.90373i	0.306513	−	0.951866i	$-0.400838\pi$
$270$	0	0
$271$	−6.43367	−0.390818	−0.195409	−	0.980722i	$-0.562603\pi$
$271$	−6.43367	−0.390818	−0.195409	+	0.980722i	$0.562603\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 13.1374i	− 0.792213i
$276$	0	0
$277$	− 15.4506i	− 0.928337i	−0.885747	−	0.464168i	$-0.846353\pi$
$277$	− 15.4506i	− 0.928337i	0.885747	−	0.464168i	$-0.153647\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	29.4587	1.75736	0.878679	−	0.477412i	$-0.158425\pi$
$281$	29.4587	1.75736	0.878679	+	0.477412i	$0.158425\pi$
$282$	0	0
$283$	19.3341i	1.14929i	0.818402	+	0.574646i	$0.194860\pi$
$283$	19.3341i	1.14929i	−0.818402	+	0.574646i	$0.805140\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	27.6850	1.63419
$288$	0	0
$289$	4.88107	0.287122
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.6955i	1.03378i	0.856051	+	0.516892i	$0.172911\pi$
$293$	17.6955i	1.03378i	−0.856051	+	0.516892i	$0.827089\pi$
$294$	0	0
$295$	1.12973	0.0657755
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.34132i	0.0775707i
$300$	0	0
$301$	− 27.9732i	− 1.61235i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.76269	−0.100931
$306$	0	0
$307$	− 27.7919i	− 1.58617i	−0.609114	−	0.793083i	$-0.708475\pi$
$307$	− 27.7919i	− 1.58617i	0.609114	−	0.793083i	$-0.291525\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.4546	−1.10317	−0.551585	−	0.834119i	$-0.685977\pi$
$311$	−19.4546	−1.10317	−0.551585	+	0.834119i	$0.685977\pi$
$312$	0	0
$313$	−1.42732	−0.0806767	−0.0403384	−	0.999186i	$-0.512844\pi$
$313$	−1.42732	−0.0806767	−0.0403384	+	0.999186i	$0.512844\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.6858i	0.656341i	0.944618	+	0.328171i	$0.106432\pi$
$317$	11.6858i	0.656341i	−0.944618	+	0.328171i	$0.893568\pi$
$318$	0	0
$319$	−21.7390	−1.21715
$320$	0	0
$321$	0	0
$322$	0	0
$323$	18.9598i	1.05495i
$324$	0	0
$325$	− 4.69735i	− 0.260562i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.8224	0.706920
$330$	0	0
$331$	27.5055i	1.51184i	0.654664	+	0.755920i	$0.272810\pi$
$331$	27.5055i	1.51184i	−0.654664	+	0.755920i	$0.727190\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.90973	0.377519
$336$	0	0
$337$	22.2411	1.21155	0.605774	−	0.795637i	$-0.292864\pi$
$337$	22.2411	1.21155	0.605774	+	0.795637i	$0.292864\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 22.7333i	− 1.23108i
$342$	0	0
$343$	−22.6384	−1.22236
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.8691i	1.65714i	0.559886	+	0.828569i	$0.310845\pi$
$347$	30.8691i	1.65714i	−0.559886	+	0.828569i	$0.689155\pi$
$348$	0	0
$349$	2.50165i	0.133910i	0.997756	+	0.0669550i	$0.0213284\pi$
$349$	2.50165i	0.133910i	−0.997756	+	0.0669550i	$0.978672\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11.7826	−0.627123	−0.313561	−	0.949568i	$-0.601522\pi$
$353$	−11.7826	−0.627123	−0.313561	+	0.949568i	$0.601522\pi$
$354$	0	0
$355$	3.19980i	0.169828i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.9569	−1.68662	−0.843311	−	0.537425i	$-0.819397\pi$
$359$	−31.9569	−1.68662	−0.843311	+	0.537425i	$0.819397\pi$
$360$	0	0
$361$	2.57154	0.135344
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.76500i	0.301754i
$366$	0	0
$367$	15.0432	0.785249	0.392624	−	0.919699i	$-0.371567\pi$
$367$	15.0432	0.785249	0.392624	+	0.919699i	$0.371567\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 35.1260i	− 1.82365i
$372$	0	0
$373$	− 1.18257i	− 0.0612310i	−0.999531	−	0.0306155i	$-0.990253\pi$
$373$	− 1.18257i	− 0.0612310i	0.999531	−	0.0306155i	$-0.00974674\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.77294	−0.400327
$378$	0	0
$379$	− 15.8003i	− 0.811605i	−0.913961	−	0.405802i	$-0.866992\pi$
$379$	− 15.8003i	− 0.811605i	0.913961	−	0.405802i	$-0.133008\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.30964	0.322407	0.161204	−	0.986921i	$-0.448462\pi$
$383$	6.30964	0.322407	0.161204	+	0.986921i	$0.448462\pi$
$384$	0	0
$385$	6.73660	0.343329
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.62899i	0.0825932i	0.999147	+	0.0412966i	$0.0131489\pi$
$389$	1.62899i	0.0825932i	−0.999147	+	0.0412966i	$0.986851\pi$
$390$	0	0
$391$	6.27434	0.317307
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 0.536554i	− 0.0269970i
$396$	0	0
$397$	2.56226i	0.128596i	0.997931	+	0.0642981i	$0.0204809\pi$
$397$	2.56226i	0.128596i	−0.997931	+	0.0642981i	$0.979519\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−28.4098	−1.41872	−0.709360	−	0.704847i	$-0.751016\pi$
$401$	−28.4098	−1.41872	−0.709360	+	0.704847i	$0.751016\pi$
$402$	0	0
$403$	− 8.12845i	− 0.404907i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.3572	1.25691
$408$	0	0
$409$	−12.2044	−0.603470	−0.301735	−	0.953392i	$-0.597566\pi$
$409$	−12.2044	−0.603470	−0.301735	+	0.953392i	$0.597566\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 8.99129i	− 0.442432i
$414$	0	0
$415$	−0.873185	−0.0428630
$416$	0	0
$417$	0	0
$418$	0	0
$419$	38.4519i	1.87850i	0.343238	+	0.939248i	$0.388476\pi$
$419$	38.4519i	1.87850i	−0.343238	+	0.939248i	$0.611524\pi$
$420$	0	0
$421$	33.4328i	1.62941i	0.579872	+	0.814707i	$0.303102\pi$
$421$	33.4328i	1.62941i	−0.579872	+	0.814707i	$0.696898\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−21.9729	−1.06584
$426$	0	0
$427$	14.0289i	0.678905i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.3849	−1.55992	−0.779962	−	0.625827i	$-0.784762\pi$
$431$	−32.3849	−1.55992	−0.779962	+	0.625827i	$0.784762\pi$
$432$	0	0
$433$	24.5534	1.17996	0.589982	−	0.807417i	$-0.299135\pi$
$433$	24.5534	1.17996	0.589982	+	0.807417i	$0.299135\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.43666i	0.260071i
$438$	0	0
$439$	0.478098	0.0228184	0.0114092	−	0.999935i	$-0.496368\pi$
$439$	0.478098	0.0228184	0.0114092	+	0.999935i	$0.496368\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.2409i	1.24674i	0.781927	+	0.623371i	$0.214237\pi$
$443$	26.2409i	1.24674i	−0.781927	+	0.623371i	$0.785763\pi$
$444$	0	0
$445$	− 4.34330i	− 0.205892i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.58963	−0.122212	−0.0611061	−	0.998131i	$-0.519463\pi$
$449$	−2.58963	−0.122212	−0.0611061	+	0.998131i	$0.519463\pi$
$450$	0	0
$451$	17.6841i	0.832712i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.40872	0.112922
$456$	0	0
$457$	−18.3773	−0.859654	−0.429827	−	0.902911i	$-0.641425\pi$
$457$	−18.3773	−0.859654	−0.429827	+	0.902911i	$0.641425\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 32.6379i	− 1.52010i	−0.649867	−	0.760048i	$-0.725175\pi$
$461$	− 32.6379i	− 1.52010i	0.649867	−	0.760048i	$-0.274825\pi$
$462$	0	0
$463$	−21.7101	−1.00895	−0.504477	−	0.863425i	$-0.668315\pi$
$463$	−21.7101	−1.00895	−0.504477	+	0.863425i	$0.668315\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 19.5025i	− 0.902467i	−0.892406	−	0.451234i	$-0.850984\pi$
$467$	− 19.5025i	− 0.902467i	0.892406	−	0.451234i	$-0.149016\pi$
$468$	0	0
$469$	− 54.9931i	− 2.53934i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17.8682	0.821582
$474$	0	0
$475$	− 19.0393i	− 0.873584i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.9078	1.13807	0.569034	−	0.822314i	$-0.307317\pi$
$479$	24.9078	1.13807	0.569034	+	0.822314i	$0.307317\pi$
$480$	0	0
$481$	9.06663	0.413403
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.60476i	0.345314i
$486$	0	0
$487$	−12.4029	−0.562029	−0.281015	−	0.959704i	$-0.590671\pi$
$487$	−12.4029	−0.562029	−0.281015	+	0.959704i	$0.590671\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.14291i	0.0967083i	0.998830	+	0.0483542i	$0.0153976\pi$
$491$	2.14291i	0.0967083i	−0.998830	+	0.0483542i	$0.984602\pi$
$492$	0	0
$493$	36.3596i	1.63755i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.4665	1.14233
$498$	0	0
$499$	17.7972i	0.796713i	0.917231	+	0.398356i	$0.130419\pi$
$499$	17.7972i	0.796713i	−0.917231	+	0.398356i	$0.869581\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.40291	−0.107140	−0.0535702	−	0.998564i	$-0.517060\pi$
$503$	−2.40291	−0.107140	−0.0535702	+	0.998564i	$0.517060\pi$
$504$	0	0
$505$	8.74213	0.389020
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.0822i	0.668508i	0.942483	+	0.334254i	$0.108484\pi$
$509$	15.0822i	0.668508i	−0.942483	+	0.334254i	$0.891516\pi$
$510$	0	0
$511$	45.8824	2.02972
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.83269i	0.0807579i
$516$	0	0
$517$	8.19043i	0.360215i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13.8911	0.608579	0.304289	−	0.952580i	$-0.401581\pi$
$521$	13.8911	0.608579	0.304289	+	0.952580i	$0.401581\pi$
$522$	0	0
$523$	33.5425i	1.46671i	0.679844	+	0.733356i	$0.262047\pi$
$523$	33.5425i	1.46671i	−0.679844	+	0.733356i	$0.737953\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−38.0226	−1.65629
$528$	0	0
$529$	−21.2009	−0.921776
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.32307i	0.273883i
$534$	0	0
$535$	−8.22367	−0.355540
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 34.0378i	− 1.46611i
$540$	0	0
$541$	− 16.5069i	− 0.709688i	−0.934926	−	0.354844i	$-0.884534\pi$
$541$	− 16.5069i	− 0.709688i	0.934926	−	0.354844i	$-0.115466\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.17931	0.350363
$546$	0	0
$547$	− 3.37437i	− 0.144278i	−0.997395	−	0.0721388i	$-0.977018\pi$
$547$	− 3.37437i	− 0.144278i	0.997395	−	0.0721388i	$-0.0229825\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−31.5053	−1.34217
$552$	0	0
$553$	−4.27032	−0.181593
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 18.3736i	− 0.778516i	−0.921129	−	0.389258i	$-0.872731\pi$
$557$	− 18.3736i	− 0.778516i	0.921129	−	0.389258i	$-0.127269\pi$
$558$	0	0
$559$	6.38891	0.270222
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 8.76400i	− 0.369358i	−0.982799	−	0.184679i	$-0.940875\pi$
$563$	− 8.76400i	− 0.369358i	0.982799	−	0.184679i	$-0.0591246\pi$
$564$	0	0
$565$	8.99760i	0.378532i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.1069	−0.926768	−0.463384	−	0.886158i	$-0.653365\pi$
$569$	−22.1069	−0.926768	−0.463384	+	0.886158i	$0.653365\pi$
$570$	0	0
$571$	8.29924i	0.347312i	0.984806	+	0.173656i	$0.0555581\pi$
$571$	8.29924i	0.347312i	−0.984806	+	0.173656i	$0.944442\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.30067	−0.262756
$576$	0	0
$577$	29.8701	1.24351	0.621753	−	0.783213i	$-0.286421\pi$
$577$	29.8701	1.24351	0.621753	+	0.783213i	$0.286421\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.94950i	0.288314i
$582$	0	0
$583$	22.4372	0.929252
$584$	0	0
$585$	0	0
$586$	0	0
$587$	46.6027i	1.92350i	0.273930	+	0.961750i	$0.411676\pi$
$587$	46.6027i	1.92350i	−0.273930	+	0.961750i	$0.588324\pi$
$588$	0	0
$589$	− 32.9463i	− 1.35753i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.1676	−1.15671	−0.578353	−	0.815786i	$-0.696304\pi$
$593$	−28.1676	−1.15671	−0.578353	+	0.815786i	$0.696304\pi$
$594$	0	0
$595$	− 11.2673i	− 0.461914i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.02174	−0.286900	−0.143450	−	0.989658i	$-0.545820\pi$
$599$	−7.02174	−0.286900	−0.143450	+	0.989658i	$0.545820\pi$
$600$	0	0
$601$	−20.1536	−0.822083	−0.411041	−	0.911617i	$-0.634835\pi$
$601$	−20.1536	−0.822083	−0.411041	+	0.911617i	$0.634835\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1.74841i	− 0.0710828i
$606$	0	0
$607$	−19.7136	−0.800149	−0.400075	−	0.916483i	$-0.631016\pi$
$607$	−19.7136	−0.800149	−0.400075	+	0.916483i	$0.631016\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.92854i	0.118476i
$612$	0	0
$613$	− 7.16179i	− 0.289262i	−0.989486	−	0.144631i	$-0.953800\pi$
$613$	− 7.16179i	− 0.289262i	0.989486	−	0.144631i	$-0.0461995\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.9287	1.76850	0.884251	−	0.467011i	$-0.154669\pi$
$617$	43.9287	1.76850	0.884251	+	0.467011i	$0.154669\pi$
$618$	0	0
$619$	− 19.1198i	− 0.768489i	−0.923231	−	0.384244i	$-0.874462\pi$
$619$	− 19.1198i	− 0.768489i	0.923231	−	0.384244i	$-0.125538\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−34.5674	−1.38491
$624$	0	0
$625$	20.5519	0.822075
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 42.4111i	− 1.69104i
$630$	0	0
$631$	−28.8405	−1.14812	−0.574061	−	0.818812i	$-0.694633\pi$
$631$	−28.8405	−1.14812	−0.574061	+	0.818812i	$0.694633\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 5.65869i	− 0.224558i
$636$	0	0
$637$	− 12.1705i	− 0.482211i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−22.0321	−0.870217	−0.435109	−	0.900378i	$-0.643290\pi$
$641$	−22.0321	−0.870217	−0.435109	+	0.900378i	$0.643290\pi$
$642$	0	0
$643$	26.2180i	1.03394i	0.856005	+	0.516968i	$0.172939\pi$
$643$	26.2180i	1.03394i	−0.856005	+	0.516968i	$0.827061\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.1790	0.714692	0.357346	−	0.933972i	$-0.383682\pi$
$647$	18.1790	0.714692	0.357346	+	0.933972i	$0.383682\pi$
$648$	0	0
$649$	5.74329	0.225444
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 47.1721i	− 1.84599i	−0.384815	−	0.922994i	$-0.625735\pi$
$653$	− 47.1721i	− 1.84599i	0.384815	−	0.922994i	$-0.374265\pi$
$654$	0	0
$655$	6.02313	0.235343
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 39.2629i	− 1.52947i	−0.644347	−	0.764733i	$-0.722871\pi$
$659$	− 39.2629i	− 1.52947i	0.644347	−	0.764733i	$-0.277129\pi$
$660$	0	0
$661$	− 18.7516i	− 0.729354i	−0.931134	−	0.364677i	$-0.881179\pi$
$661$	− 18.7516i	− 0.729354i	0.931134	−	0.364677i	$-0.118821\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.76302	0.378594
$666$	0	0
$667$	10.4260i	0.403697i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.96111	−0.345940
$672$	0	0
$673$	12.5340	0.483152	0.241576	−	0.970382i	$-0.422336\pi$
$673$	12.5340	0.483152	0.241576	+	0.970382i	$0.422336\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4.37154i	− 0.168012i	−0.996465	−	0.0840059i	$-0.973229\pi$
$677$	− 4.37154i	− 0.168012i	0.996465	−	0.0840059i	$-0.0267715\pi$
$678$	0	0
$679$	60.5247	2.32272
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 18.2155i	− 0.696999i	−0.937309	−	0.348499i	$-0.886691\pi$
$683$	− 18.2155i	− 0.696999i	0.937309	−	0.348499i	$-0.113309\pi$
$684$	0	0
$685$	4.87616i	0.186309i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.02256	0.305635
$690$	0	0
$691$	− 13.8806i	− 0.528043i	−0.964517	−	0.264021i	$-0.914951\pi$
$691$	− 13.8806i	− 0.528043i	0.964517	−	0.264021i	$-0.0850489\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.92106	−0.186666
$696$	0	0
$697$	29.5776	1.12033
$698$	0	0
$699$	0	0
$700$	0	0
$701$	41.1168i	1.55296i	0.630142	+	0.776480i	$0.282997\pi$
$701$	41.1168i	1.55296i	−0.630142	+	0.776480i	$0.717003\pi$
$702$	0	0
$703$	36.7489	1.38601
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 69.5768i	− 2.61670i
$708$	0	0
$709$	− 31.8699i	− 1.19690i	−0.801160	−	0.598450i	$-0.795784\pi$
$709$	− 31.8699i	− 1.19690i	0.801160	−	0.598450i	$-0.204216\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10.9029	−0.408316
$714$	0	0
$715$	1.53860i	0.0575402i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.8550	0.404822	0.202411	−	0.979301i	$-0.435122\pi$
$719$	10.8550	0.404822	0.202411	+	0.979301i	$0.435122\pi$
$720$	0	0
$721$	14.5860	0.543210
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 36.5122i	− 1.35603i
$726$	0	0
$727$	−5.04295	−0.187033	−0.0935164	−	0.995618i	$-0.529811\pi$
$727$	−5.04295	−0.187033	−0.0935164	+	0.995618i	$0.529811\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 29.8855i	− 1.10536i
$732$	0	0
$733$	− 24.9094i	− 0.920049i	−0.887906	−	0.460024i	$-0.847841\pi$
$733$	− 24.9094i	− 0.920049i	0.887906	−	0.460024i	$-0.152159\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35.1275	1.29394
$738$	0	0
$739$	− 4.44667i	− 0.163573i	−0.996650	−	0.0817867i	$-0.973937\pi$
$739$	− 4.44667i	− 0.163573i	0.996650	−	0.0817867i	$-0.0260626\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.4660	−0.824198	−0.412099	−	0.911139i	$-0.635204\pi$
$743$	−22.4660	−0.824198	−0.412099	+	0.911139i	$0.635204\pi$
$744$	0	0
$745$	8.92776	0.327088
$746$	0	0
$747$	0	0
$748$	0	0
$749$	65.4504i	2.39151i
$750$	0	0
$751$	50.2976	1.83539	0.917693	−	0.397290i	$-0.130049\pi$
$751$	50.2976	1.83539	0.917693	+	0.397290i	$0.130049\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 5.27309i	− 0.191907i
$756$	0	0
$757$	25.6163i	0.931039i	0.885038	+	0.465519i	$0.154132\pi$
$757$	25.6163i	0.931039i	−0.885038	+	0.465519i	$0.845868\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10.9247	−0.396020	−0.198010	−	0.980200i	$-0.563448\pi$
$761$	−10.9247	−0.396020	−0.198010	+	0.980200i	$0.563448\pi$
$762$	0	0
$763$	− 65.0973i	− 2.35668i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.05355	0.0741494
$768$	0	0
$769$	1.99231	0.0718447	0.0359223	−	0.999355i	$-0.488563\pi$
$769$	1.99231	0.0718447	0.0359223	+	0.999355i	$0.488563\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 16.2934i	− 0.586032i	−0.956108	−	0.293016i	$-0.905341\pi$
$773$	− 16.2934i	− 0.586032i	0.956108	−	0.293016i	$-0.0946589\pi$
$774$	0	0
$775$	38.1822	1.37154
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.6287i	0.918243i
$780$	0	0
$781$	16.2670i	0.582080i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.517421	0.0184676
$786$	0	0
$787$	− 30.0717i	− 1.07194i	−0.844237	−	0.535970i	$-0.819946\pi$
$787$	− 30.0717i	− 1.07194i	0.844237	−	0.535970i	$-0.180054\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	71.6100	2.54616
$792$	0	0
$793$	−3.20410	−0.113781
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.8074i	1.41005i	0.709183	+	0.705024i	$0.249064\pi$
$797$	39.8074i	1.41005i	−0.709183	+	0.705024i	$0.750936\pi$
$798$	0	0
$799$	13.6989	0.484633
$800$	0	0
$801$	0	0
$802$	0	0
$803$	29.3079i	1.03425i
$804$	0	0
$805$	− 3.23087i	− 0.113873i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.97006	−0.0692635	−0.0346318	−	0.999400i	$-0.511026\pi$
$809$	−1.97006	−0.0692635	−0.0346318	+	0.999400i	$0.511026\pi$
$810$	0	0
$811$	− 33.7621i	− 1.18555i	−0.805369	−	0.592774i	$-0.798033\pi$
$811$	− 33.7621i	− 1.18555i	0.805369	−	0.592774i	$-0.201967\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.52730	0.193613
$816$	0	0
$817$	25.8955	0.905970
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.01692i	0.244892i	0.992475	+	0.122446i	$0.0390739\pi$
$821$	7.01692i	0.244892i	−0.992475	+	0.122446i	$0.960926\pi$
$822$	0	0
$823$	−54.2820	−1.89215	−0.946076	−	0.323944i	$-0.894991\pi$
$823$	−54.2820	−1.89215	−0.946076	+	0.323944i	$0.894991\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5.78711i	− 0.201238i	−0.994925	−	0.100619i	$-0.967918\pi$
$827$	− 5.78711i	− 0.201238i	0.994925	−	0.100619i	$-0.0320823\pi$
$828$	0	0
$829$	19.7851i	0.687164i	0.939123	+	0.343582i	$0.111640\pi$
$829$	19.7851i	0.687164i	−0.939123	+	0.343582i	$0.888360\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−56.9300	−1.97251
$834$	0	0
$835$	3.79645i	0.131381i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.3886	0.703893	0.351947	−	0.936020i	$-0.385520\pi$
$839$	20.3886	0.703893	0.351947	+	0.936020i	$0.385520\pi$
$840$	0	0
$841$	−31.4185	−1.08340
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.550135i	0.0189252i
$846$	0	0
$847$	−13.9152	−0.478132
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 12.1613i	− 0.416883i
$852$	0	0
$853$	2.15969i	0.0739465i	0.999316	+	0.0369732i	$0.0117716\pi$
$853$	2.15969i	0.0739465i	−0.999316	+	0.0369732i	$0.988228\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.9199	−0.748768	−0.374384	−	0.927274i	$-0.622146\pi$
$857$	−21.9199	−0.748768	−0.374384	+	0.927274i	$0.622146\pi$
$858$	0	0
$859$	− 58.0347i	− 1.98012i	−0.140648	−	0.990060i	$-0.544919\pi$
$859$	− 58.0347i	− 1.98012i	0.140648	−	0.990060i	$-0.455081\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.5554	−0.495471	−0.247736	−	0.968828i	$-0.579686\pi$
$863$	−14.5554	−0.495471	−0.247736	+	0.968828i	$0.579686\pi$
$864$	0	0
$865$	0.472708	0.0160725
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.72772i	− 0.0925315i
$870$	0	0
$871$	12.5601	0.425581
$872$	0	0
$873$	0	0
$874$	0	0
$875$	23.3582i	0.789650i
$876$	0	0
$877$	− 40.7360i	− 1.37556i	−0.725920	−	0.687779i	$-0.758586\pi$
$877$	− 40.7360i	− 1.37556i	0.725920	−	0.687779i	$-0.241414\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	43.8603	1.47769	0.738845	−	0.673875i	$-0.235371\pi$
$881$	43.8603	1.47769	0.738845	+	0.673875i	$0.235371\pi$
$882$	0	0
$883$	− 34.7347i	− 1.16891i	−0.811425	−	0.584457i	$-0.801307\pi$
$883$	− 34.7347i	− 1.16891i	0.811425	−	0.584457i	$-0.198693\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.2309	1.58586	0.792928	−	0.609315i	$-0.208555\pi$
$887$	47.2309	1.58586	0.792928	+	0.609315i	$0.208555\pi$
$888$	0	0
$889$	−45.0363	−1.51047
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.8700i	0.397214i
$894$	0	0
$895$	−1.32512	−0.0442939
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 63.1819i	− 2.10723i
$900$	0	0
$901$	− 37.5273i	− 1.25021i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.80499	0.0932410
$906$	0	0
$907$	− 4.14840i	− 0.137745i	−0.997625	−	0.0688726i	$-0.978060\pi$
$907$	− 4.14840i	− 0.137745i	0.997625	−	0.0688726i	$-0.0219402\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.54285	−0.0842485	−0.0421243	−	0.999112i	$-0.513413\pi$
$911$	−2.54285	−0.0842485	−0.0421243	+	0.999112i	$0.513413\pi$
$912$	0	0
$913$	−4.43907	−0.146912
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 47.9368i	− 1.58301i
$918$	0	0
$919$	−12.3864	−0.408590	−0.204295	−	0.978909i	$-0.565490\pi$
$919$	−12.3864	−0.408590	−0.204295	+	0.978909i	$0.565490\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.81639i	0.191449i
$924$	0	0
$925$	42.5891i	1.40032i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−59.0139	−1.93618	−0.968092	−	0.250595i	$-0.919374\pi$
$929$	−59.0139	−1.93618	−0.968092	+	0.250595i	$0.919374\pi$
$930$	0	0
$931$	− 49.3294i	− 1.61670i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.19712	0.235371
$936$	0	0
$937$	−22.6251	−0.739128	−0.369564	−	0.929205i	$-0.620493\pi$
$937$	−22.6251	−0.739128	−0.369564	+	0.929205i	$0.620493\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 14.0164i	− 0.456923i	−0.973553	−	0.228461i	$-0.926630\pi$
$941$	− 14.0164i	− 0.456923i	0.973553	−	0.228461i	$-0.0733695\pi$
$942$	0	0
$943$	8.48128	0.276189
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.06570i	0.262100i	0.991376	+	0.131050i	$0.0418348\pi$
$947$	8.06570i	0.262100i	−0.991376	+	0.131050i	$0.958165\pi$
$948$	0	0
$949$	10.4792i	0.340171i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−31.5021	−1.02045	−0.510226	−	0.860040i	$-0.670438\pi$
$953$	−31.5021	−1.02045	−0.510226	+	0.860040i	$0.670438\pi$
$954$	0	0
$955$	6.54850i	0.211905i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	38.8083	1.25319
$960$	0	0
$961$	35.0717	1.13134
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.53520i	0.178184i
$966$	0	0
$967$	−55.9561	−1.79943	−0.899714	−	0.436480i	$-0.856225\pi$
$967$	−55.9561	−1.79943	−0.899714	+	0.436480i	$0.856225\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	43.2515i	1.38801i	0.719972	+	0.694003i	$0.244154\pi$
$971$	43.2515i	1.38801i	−0.719972	+	0.694003i	$0.755846\pi$
$972$	0	0
$973$	39.1657i	1.25559i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.7198	−0.502920	−0.251460	−	0.967868i	$-0.580911\pi$
$977$	−15.7198	−0.502920	−0.251460	+	0.967868i	$0.580911\pi$
$978$	0	0
$979$	− 22.0803i	− 0.705690i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.47680	0.110893	0.0554464	−	0.998462i	$-0.482342\pi$
$983$	3.47680	0.110893	0.0554464	+	0.998462i	$0.482342\pi$
$984$	0	0
$985$	−3.82341	−0.121824
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 8.56959i	− 0.272497i
$990$	0	0
$991$	−24.7650	−0.786685	−0.393342	−	0.919392i	$-0.628681\pi$
$991$	−24.7650	−0.786685	−0.393342	+	0.919392i	$0.628681\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 9.88699i	− 0.313439i
$996$	0	0
$997$	− 47.1632i	− 1.49367i	−0.665007	−	0.746837i	$-0.731571\pi$
$997$	− 47.1632i	− 1.49367i	0.665007	−	0.746837i	$-0.268429\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.g.e.1873.7		16
3.2	odd	2		1248.2.g.b.625.5		16
4.3	odd	2		936.2.g.e.469.15		16
8.3	odd	2		936.2.g.e.469.16		16
8.5	even	2	inner	3744.2.g.e.1873.10		16
12.11	even	2		312.2.g.b.157.2	yes	16
24.5	odd	2		1248.2.g.b.625.12		16
24.11	even	2		312.2.g.b.157.1	✓	16
48.5	odd	4		9984.2.a.bs.1.4		8
48.11	even	4		9984.2.a.bu.1.4		8
48.29	odd	4		9984.2.a.bv.1.5		8
48.35	even	4		9984.2.a.bt.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.b.157.1	✓	16	24.11	even	2
312.2.g.b.157.2	yes	16	12.11	even	2
936.2.g.e.469.15		16	4.3	odd	2
936.2.g.e.469.16		16	8.3	odd	2
1248.2.g.b.625.5		16	3.2	odd	2
1248.2.g.b.625.12		16	24.5	odd	2
3744.2.g.e.1873.7		16	1.1	even	1	trivial
3744.2.g.e.1873.10		16	8.5	even	2	inner
9984.2.a.bs.1.4		8	48.5	odd	4
9984.2.a.bt.1.5		8	48.35	even	4
9984.2.a.bu.1.4		8	48.11	even	4
9984.2.a.bv.1.5		8	48.29	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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.550135i q^{5} -4.37841 q^{7} +O(q^{10})\)	\(q-0.550135i q^{5} -4.37841 q^{7} -2.79676i q^{11} -1.00000i q^{13} -4.67772 q^{17} -4.05320i q^{19} -1.34132 q^{23} +4.69735 q^{25} -7.77294i q^{29} +8.12845 q^{31} +2.40872i q^{35} +9.06663i q^{37} -6.32307 q^{41} +6.38891i q^{43} -2.92854 q^{47} +12.1705 q^{49} +8.02256i q^{53} -1.53860 q^{55} +2.05355i q^{59} -3.20410i q^{61} -0.550135 q^{65} +12.5601i q^{67} -5.81639 q^{71} -10.4792 q^{73} +12.2454i q^{77} +0.975314 q^{79} -1.58722i q^{83} +2.57338i q^{85} +7.89497 q^{89} +4.37841i q^{91} -2.22981 q^{95} -13.8234 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7}+O(q^{10}) \) 16 * q - 4 * q^7	\( 16 q - 4 q^{7} - 16 q^{17} - 8 q^{23} - 32 q^{25} + 4 q^{31} + 36 q^{41} + 24 q^{47} + 48 q^{49} - 24 q^{55} - 4 q^{65} - 32 q^{73} + 60 q^{89} - 24 q^{95} + 40 q^{97}+O(q^{100}) \) 16 * q - 4 * q^7 - 16 * q^17 - 8 * q^23 - 32 * q^25 + 4 * q^31 + 36 * q^41 + 24 * q^47 + 48 * q^49 - 24 * q^55 - 4 * q^65 - 32 * q^73 + 60 * q^89 - 24 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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