LMFDB - Embedded newform 3168.2.h.i.2287.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3168,2,Mod(2287,3168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3168, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3168.2287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.h (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:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.3342602057661458415616.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 $ x^16 - 20x^14 + 164x^12 - 666x^10 + 1300x^8 - 924x^6 + 273x^4 + 404*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2287.3
Root			$0.946412 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3168.2287
Dual form			3168.2.h.i.2287.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{5} -0.936426 q^{7} +(-3.09218 + 1.19935i) q^{11} +4.27156 q^{13} +3.33513i q^{17} -2.89560i q^{19} +5.12311i q^{23} +1.00000 q^{25} +6.60421 q^{29} -1.73642i q^{31} +1.87285i q^{35} +6.18435i q^{37} +1.46228i q^{41} -10.3128i q^{43} +6.00000i q^{47} -6.12311 q^{49} -8.24621i q^{53} +(2.39871 + 6.18435i) q^{55} +9.65719 q^{59} +9.06897 q^{61} -8.54312i q^{65} +10.2462 q^{67} +4.24621i q^{71} -13.2084i q^{73} +(2.89560 - 1.12311i) q^{77} +3.86098 q^{79} -2.39871i q^{83} +6.67026 q^{85} -3.47284 q^{89} -4.00000 q^{91} -5.79119 q^{95} +11.3693 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{25} - 32 q^{49} + 32 q^{67} - 64 q^{91} - 16 q^{97}+O(q^{100}) $ 16 * q + 16 * q^25 - 32 * q^49 + 32 * q^67 - 64 * q^91 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$	−0.936426			−0.353936			−0.176968	−	0.984217i	$-0.556629\pi$
$7$	−0.936426			−0.353936			−0.176968	+	0.984217i	$0.556629\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.09218	+	1.19935i	−0.932326	+	0.361618i
$11$	−3.09218	+	1.19935i	−0.932326	+	0.361618i
$12$		0			0
$13$	4.27156			1.18472			0.592359	−	0.805674i	$-0.298197\pi$
$13$	4.27156			1.18472			0.592359	+	0.805674i	$0.298197\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.33513i			0.808888i	0.914563	+	0.404444i	$0.132535\pi$
$17$			3.33513i			0.808888i	−0.914563	+	0.404444i	$0.867465\pi$
$18$		0			0
$19$		−	2.89560i		−	0.664295i	−0.943227	−	0.332148i	$-0.892227\pi$
$19$		−	2.89560i		−	0.664295i	0.943227	−	0.332148i	$-0.107773\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.12311i			1.06824i	0.845408	+	0.534121i	$0.179357\pi$
$23$			5.12311i			1.06824i	−0.845408	+	0.534121i	$0.820643\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.60421			1.22637			0.613185	−	0.789939i	$-0.289888\pi$
$29$	6.60421			1.22637			0.613185	+	0.789939i	$0.289888\pi$
$30$		0			0
$31$		−	1.73642i		−	0.311870i	−0.987767	−	0.155935i	$-0.950161\pi$
$31$		−	1.73642i		−	0.311870i	0.987767	−	0.155935i	$-0.0498391\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			1.87285i			0.316570i
$36$		0			0
$37$			6.18435i			1.01670i	0.861150	+	0.508351i	$0.169745\pi$
$37$			6.18435i			1.01670i	−0.861150	+	0.508351i	$0.830255\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.46228i			0.228370i	0.993460	+	0.114185i	$0.0364256\pi$
$41$			1.46228i			0.228370i	−0.993460	+	0.114185i	$0.963574\pi$
$42$		0			0
$43$		−	10.3128i		−	1.57269i	−0.617788	−	0.786345i	$-0.711971\pi$
$43$		−	10.3128i		−	1.57269i	0.617788	−	0.786345i	$-0.288029\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	−6.12311			−0.874729
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.24621i		−	1.13270i	−0.824163	−	0.566352i	$-0.808354\pi$
$53$		−	8.24621i		−	1.13270i	0.824163	−	0.566352i	$-0.191646\pi$
$54$		0			0
$55$	2.39871	+	6.18435i	0.323441	+	0.833898i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.65719			1.25726			0.628630	−	0.777705i	$-0.283616\pi$
$59$	9.65719			1.25726			0.628630	+	0.777705i	$0.283616\pi$
$60$		0			0
$61$	9.06897			1.16116			0.580581	−	0.814202i	$-0.302825\pi$
$61$	9.06897			1.16116			0.580581	+	0.814202i	$0.302825\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	8.54312i		−	1.05964i
$66$		0			0
$67$	10.2462			1.25177			0.625887	−	0.779914i	$-0.284737\pi$
$67$	10.2462			1.25177			0.625887	+	0.779914i	$0.284737\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.24621i			0.503933i	0.967736	+	0.251966i	$0.0810772\pi$
$71$			4.24621i			0.503933i	−0.967736	+	0.251966i	$0.918923\pi$
$72$		0			0
$73$		−	13.2084i		−	1.54593i	−0.634450	−	0.772964i	$-0.718773\pi$
$73$		−	13.2084i		−	1.54593i	0.634450	−	0.772964i	$-0.281227\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.89560	−	1.12311i	0.329984	−	0.127990i
$78$		0			0
$79$	3.86098			0.434395			0.217197	−	0.976128i	$-0.430308\pi$
$79$	3.86098			0.434395			0.217197	+	0.976128i	$0.430308\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	2.39871i		−	0.263292i	−0.991297	−	0.131646i	$-0.957974\pi$
$83$		−	2.39871i		−	0.263292i	0.991297	−	0.131646i	$-0.0420262\pi$
$84$		0			0
$85$	6.67026			0.723492
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.47284			−0.368120			−0.184060	−	0.982915i	$-0.558924\pi$
$89$	−3.47284			−0.368120			−0.184060	+	0.982915i	$0.558924\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.79119			−0.594164
$96$		0			0
$97$	11.3693			1.15438			0.577190	−	0.816610i	$-0.304149\pi$
$97$	11.3693			1.15438			0.577190	+	0.816610i	$0.304149\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.60421			−0.657143			−0.328571	−	0.944479i	$-0.606567\pi$
$101$	−6.60421			−0.657143			−0.328571	+	0.944479i	$0.606567\pi$
$102$		0			0
$103$		−	1.73642i		−	0.171095i	−0.996334	−	0.0855473i	$-0.972736\pi$
$103$		−	1.73642i		−	0.171095i	0.996334	−	0.0855473i	$-0.0272639\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9.06897i		−	0.876730i	−0.898797	−	0.438365i	$-0.855558\pi$
$107$		−	9.06897i		−	0.876730i	0.898797	−	0.438365i	$-0.144442\pi$
$108$		0			0
$109$	−1.34700			−0.129019			−0.0645096	−	0.997917i	$-0.520548\pi$
$109$	−1.34700			−0.129019			−0.0645096	+	0.997917i	$0.520548\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−15.8415			−1.49025			−0.745124	−	0.666926i	$-0.767610\pi$
$113$	−15.8415			−1.49025			−0.745124	+	0.666926i	$0.767610\pi$
$114$		0			0
$115$	10.2462			0.955464
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	3.12311i		−	0.286295i
$120$		0			0
$121$	8.12311	−	7.41722i	0.738464	−	0.674293i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	16.1498			1.43306			0.716532	−	0.697554i	$-0.245728\pi$
$127$	16.1498			1.43306			0.716532	+	0.697554i	$0.245728\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			12.8147i			1.11962i	0.828620	+	0.559812i	$0.189127\pi$
$131$			12.8147i			1.11962i	−0.828620	+	0.559812i	$0.810873\pi$
$132$		0			0
$133$			2.71151i			0.235118i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.8415			1.35343			0.676717	−	0.736243i	$-0.263402\pi$
$137$	15.8415			1.35343			0.676717	+	0.736243i	$0.263402\pi$
$138$		0			0
$139$		−	10.3128i		−	0.874722i	−0.899286	−	0.437361i	$-0.855913\pi$
$139$		−	10.3128i		−	0.874722i	0.899286	−	0.437361i	$-0.144087\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−13.2084	+	5.12311i	−1.10454	+	0.428416i
$144$		0			0
$145$		−	13.2084i		−	1.09690i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.813015			−0.0666048			−0.0333024	−	0.999445i	$-0.510602\pi$
$149$	−0.813015			−0.0666048			−0.0333024	+	0.999445i	$0.510602\pi$
$150$		0			0
$151$	20.9472			1.70466			0.852330	−	0.523004i	$-0.175189\pi$
$151$	20.9472			1.70466			0.852330	+	0.523004i	$0.175189\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.47284			−0.278945
$156$		0			0
$157$		−	15.8415i		−	1.26429i	−0.774849	−	0.632146i	$-0.782174\pi$
$157$		−	15.8415i		−	1.26429i	0.774849	−	0.632146i	$-0.217826\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.79741i		−	0.378089i
$162$		0			0
$163$	−12.4924			−0.978482			−0.489241	−	0.872149i	$-0.662726\pi$
$163$	−12.4924			−0.978482			−0.489241	+	0.872149i	$0.662726\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.79119			0.448136			0.224068	−	0.974574i	$-0.428066\pi$
$167$	5.79119			0.448136			0.224068	+	0.974574i	$0.428066\pi$
$168$		0			0
$169$	5.24621			0.403555
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.813015			0.0618124			0.0309062	−	0.999522i	$-0.490161\pi$
$173$	0.813015			0.0618124			0.0309062	+	0.999522i	$0.490161\pi$
$174$		0			0
$175$	−0.936426			−0.0707872
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$			8.89586i			0.661224i	0.943767	+	0.330612i	$0.107255\pi$
$181$			8.89586i			0.661224i	−0.943767	+	0.330612i	$0.892745\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.3687			0.909365
$186$		0			0
$187$	−4.00000	−	10.3128i	−0.292509	−	0.754148i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.2462i		−	0.886105i	−0.896496	−	0.443052i	$-0.853896\pi$
$191$		−	12.2462i		−	0.886105i	0.896496	−	0.443052i	$-0.146104\pi$
$192$		0			0
$193$			1.62603i			0.117044i	0.998286	+	0.0585221i	$0.0186388\pi$
$193$			1.62603i			0.117044i	−0.998286	+	0.0585221i	$0.981361\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.3954			−0.883135			−0.441568	−	0.897228i	$-0.645577\pi$
$197$	−12.3954			−0.883135			−0.441568	+	0.897228i	$0.645577\pi$
$198$		0			0
$199$			14.1051i			0.999886i	0.866058	+	0.499943i	$0.166646\pi$
$199$			14.1051i			0.999886i	−0.866058	+	0.499943i	$0.833354\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.18435			−0.434056
$204$		0			0
$205$	2.92456			0.204260
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.47284	+	8.95369i	0.240221	+	0.619340i
$210$		0			0
$211$		−	2.89560i		−	0.199341i	−0.995020	−	0.0996705i	$-0.968221\pi$
$211$		−	2.89560i		−	0.199341i	0.995020	−	0.0996705i	$-0.0317789\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−20.6256			−1.40666
$216$		0			0
$217$			1.62603i			0.110382i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			14.2462i			0.958304i
$222$		0			0
$223$			26.4738i			1.77282i	0.462902	+	0.886409i	$0.346808\pi$
$223$			26.4738i			1.77282i	−0.462902	+	0.886409i	$0.653192\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	26.9764i		−	1.79048i	−0.445581	−	0.895242i	$-0.647003\pi$
$227$		−	26.9764i		−	1.79048i	0.445581	−	0.895242i	$-0.352997\pi$
$228$		0			0
$229$			25.4987i			1.68500i	0.538693	+	0.842502i	$0.318918\pi$
$229$			25.4987i			1.68500i	−0.538693	+	0.842502i	$0.681082\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.28343i		−	0.149592i	−0.997199	−	0.0747961i	$-0.976169\pi$
$233$		−	2.28343i		−	0.149592i	0.997199	−	0.0747961i	$-0.0238306\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$		0			0
$237$		0			0
$238$		0			0
$239$	20.6256			1.33416			0.667081	−	0.744986i	$-0.267544\pi$
$239$	20.6256			1.33416			0.667081	+	0.744986i	$0.267544\pi$
$240$		0			0
$241$		−	28.0429i		−	1.80640i	−0.429221	−	0.903199i	$-0.641212\pi$
$241$		−	28.0429i		−	1.80640i	0.429221	−	0.903199i	$-0.358788\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			12.2462i			0.782382i
$246$		0			0
$247$		−	12.3687i		−	0.787002i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.47284			−0.219204			−0.109602	−	0.993976i	$-0.534958\pi$
$251$	−3.47284			−0.219204			−0.109602	+	0.993976i	$0.534958\pi$
$252$		0			0
$253$	−6.14441	−	15.8415i	−0.386296	−	0.995949i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.89586			−0.554909			−0.277454	−	0.960739i	$-0.589491\pi$
$257$	−8.89586			−0.554909			−0.277454	+	0.960739i	$0.589491\pi$
$258$		0			0
$259$		−	5.79119i		−	0.359847i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	32.2080			1.98603			0.993016	−	0.117984i	$-0.0376430\pi$
$263$	32.2080			1.98603			0.993016	+	0.117984i	$0.0376430\pi$
$264$		0			0
$265$	−16.4924			−1.01312
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	0.246211i		−	0.0150118i	−0.999972	−	0.00750588i	$-0.997611\pi$
$269$		−	0.246211i		−	0.0150118i	0.999972	−	0.00750588i	$-0.00238922\pi$
$270$		0			0
$271$	5.73384			0.348306			0.174153	−	0.984719i	$-0.444281\pi$
$271$	5.73384			0.348306			0.174153	+	0.984719i	$0.444281\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.09218	+	1.19935i	−0.186465	+	0.0723237i
$276$		0			0
$277$	2.39871			0.144124			0.0720621	−	0.997400i	$-0.477042\pi$
$277$	2.39871			0.144124			0.0720621	+	0.997400i	$0.477042\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			24.1671i			1.44169i	0.693098	+	0.720843i	$0.256245\pi$
$281$			24.1671i			1.44169i	−0.693098	+	0.720843i	$0.743755\pi$
$282$		0			0
$283$			17.7300i			1.05394i	0.849884	+	0.526971i	$0.176672\pi$
$283$			17.7300i			1.05394i	−0.849884	+	0.526971i	$0.823328\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.36932i		−	0.0808282i
$288$		0			0
$289$	5.87689			0.345700
$290$		0			0
$291$		0			0
$292$		0			0
$293$	27.2298			1.59078			0.795392	−	0.606095i	$-0.207265\pi$
$293$	27.2298			1.59078			0.795392	+	0.606095i	$0.207265\pi$
$294$		0			0
$295$		−	19.3144i		−	1.12453i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			21.8836i			1.26556i
$300$		0			0
$301$			9.65719i			0.556631i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	18.1379i		−	1.03858i
$306$		0			0
$307$		−	4.52162i		−	0.258063i	−0.991641	−	0.129031i	$-0.958813\pi$
$307$		−	4.52162i		−	0.258063i	0.991641	−	0.129031i	$-0.0411868\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	13.1231i		−	0.744143i	−0.928204	−	0.372072i	$-0.878648\pi$
$311$		−	13.1231i		−	0.744143i	0.928204	−	0.372072i	$-0.121352\pi$
$312$		0			0
$313$	−5.12311			−0.289575			−0.144788	−	0.989463i	$-0.546250\pi$
$313$	−5.12311			−0.289575			−0.144788	+	0.989463i	$0.546250\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			22.4924i			1.26330i	0.775254	+	0.631650i	$0.217622\pi$
$317$			22.4924i			1.26330i	−0.775254	+	0.631650i	$0.782378\pi$
$318$		0			0
$319$	−20.4214	+	7.92077i	−1.14338	+	0.443478i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9.65719			0.537341
$324$		0			0
$325$	4.27156			0.236943
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	5.61856i		−	0.309761i
$330$		0			0
$331$	−1.75379			−0.0963969			−0.0481985	−	0.998838i	$-0.515348\pi$
$331$	−1.75379			−0.0963969			−0.0481985	+	0.998838i	$0.515348\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	20.4924i		−	1.11962i
$336$		0			0
$337$			24.7908i			1.35044i	0.737616	+	0.675220i	$0.235951\pi$
$337$			24.7908i			1.35044i	−0.737616	+	0.675220i	$0.764049\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.08258	+	5.36932i	0.112778	+	0.290765i
$342$		0			0
$343$	12.2888			0.663534
$344$		0			0
$345$		0			0
$346$		0			0
$347$			20.5366i			1.10246i	0.834352	+	0.551232i	$0.185842\pi$
$347$			20.5366i			1.10246i	−0.834352	+	0.551232i	$0.814158\pi$
$348$		0			0
$349$	31.7738			1.70081			0.850405	−	0.526128i	$-0.176357\pi$
$349$	31.7738			1.70081			0.850405	+	0.526128i	$0.176357\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.3687			−0.658320			−0.329160	−	0.944274i	$-0.606765\pi$
$353$	−12.3687			−0.658320			−0.329160	+	0.944274i	$0.606765\pi$
$354$		0			0
$355$	8.49242			0.450731
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.04325			0.477284			0.238642	−	0.971108i	$-0.423298\pi$
$359$	9.04325			0.477284			0.238642	+	0.971108i	$0.423298\pi$
$360$		0			0
$361$	10.6155			0.558712
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−26.4168			−1.38272
$366$		0			0
$367$		−	29.9467i		−	1.56320i	−0.623778	−	0.781602i	$-0.714403\pi$
$367$		−	29.9467i		−	1.56320i	0.623778	−	0.781602i	$-0.285597\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			7.72197i			0.400905i
$372$		0			0
$373$	−11.9935			−0.621001			−0.310501	−	0.950573i	$-0.600497\pi$
$373$	−11.9935			−0.621001			−0.310501	+	0.950573i	$0.600497\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	28.2102			1.45290
$378$		0			0
$379$	14.2462			0.731779			0.365889	−	0.930658i	$-0.380765\pi$
$379$	14.2462			0.731779			0.365889	+	0.930658i	$0.380765\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	0.630683i		−	0.0322264i	−0.999870	−	0.0161132i	$-0.994871\pi$
$383$		−	0.630683i		−	0.0322264i	0.999870	−	0.0161132i	$-0.00512921\pi$
$384$		0			0
$385$	−2.24621	−	5.79119i	−0.114478	−	0.295146i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			30.4924i			1.54603i	0.634389	+	0.773014i	$0.281252\pi$
$389$			30.4924i			1.54603i	−0.634389	+	0.773014i	$0.718748\pi$
$390$		0			0
$391$	−17.0862			−0.864088
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	7.72197i		−	0.388534i
$396$		0			0
$397$		−	18.5531i		−	0.931151i	−0.885008	−	0.465576i	$-0.845847\pi$
$397$		−	18.5531i		−	0.931151i	0.885008	−	0.465576i	$-0.154153\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−24.7374			−1.23533			−0.617664	−	0.786442i	$-0.711921\pi$
$401$	−24.7374			−1.23533			−0.617664	+	0.786442i	$0.711921\pi$
$402$		0			0
$403$		−	7.41722i		−	0.369478i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.41722	−	19.1231i	−0.367658	−	0.947897i
$408$		0			0
$409$			26.4168i			1.30623i	0.757260	+	0.653114i	$0.226538\pi$
$409$			26.4168i			1.30623i	−0.757260	+	0.653114i	$0.773462\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.04325			−0.444989
$414$		0			0
$415$	−4.79741			−0.235496
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.47284			−0.169659			−0.0848297	−	0.996395i	$-0.527035\pi$
$419$	−3.47284			−0.169659			−0.0848297	+	0.996395i	$0.527035\pi$
$420$		0			0
$421$		−	15.8415i		−	0.772070i	−0.922484	−	0.386035i	$-0.873844\pi$
$421$		−	15.8415i		−	0.772070i	0.922484	−	0.386035i	$-0.126156\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3.33513i			0.161778i
$426$		0			0
$427$	−8.49242			−0.410977
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11.5824			0.557904			0.278952	−	0.960305i	$-0.410013\pi$
$431$	11.5824			0.557904			0.278952	+	0.960305i	$0.410013\pi$
$432$		0			0
$433$	−6.87689			−0.330482			−0.165241	−	0.986253i	$-0.552840\pi$
$433$	−6.87689			−0.330482			−0.165241	+	0.986253i	$0.552840\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.8344			0.709628
$438$		0			0
$439$	−30.3115			−1.44669			−0.723344	−	0.690488i	$-0.757396\pi$
$439$	−30.3115			−1.44669			−0.723344	+	0.690488i	$0.757396\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.18435			0.293827			0.146914	−	0.989149i	$-0.453066\pi$
$443$	6.18435			0.293827			0.146914	+	0.989149i	$0.453066\pi$
$444$		0			0
$445$			6.94568i			0.329257i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−28.2102			−1.33132			−0.665662	−	0.746253i	$-0.731851\pi$
$449$	−28.2102			−1.33132			−0.665662	+	0.746253i	$0.731851\pi$
$450$		0			0
$451$	−1.75379	−	4.52162i	−0.0825827	−	0.212915i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			8.00000i			0.375046i
$456$		0			0
$457$			11.5824i			0.541801i	0.962607	+	0.270900i	$0.0873214\pi$
$457$			11.5824i			0.541801i	−0.962607	+	0.270900i	$0.912679\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−31.3950			−1.46221			−0.731105	−	0.682265i	$-0.760995\pi$
$461$	−31.3950			−1.46221			−0.731105	+	0.682265i	$0.760995\pi$
$462$		0			0
$463$		−	7.15944i		−	0.332728i	−0.986064	−	0.166364i	$-0.946797\pi$
$463$		−	7.15944i		−	0.332728i	0.986064	−	0.166364i	$-0.0532026\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.18435			−0.286178			−0.143089	−	0.989710i	$-0.545703\pi$
$467$	−6.18435			−0.286178			−0.143089	+	0.989710i	$0.545703\pi$
$468$		0			0
$469$	−9.59482			−0.443048
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.3687	+	31.8890i	0.568714	+	1.46626i
$474$		0			0
$475$		−	2.89560i		−	0.132859i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.8344			−0.677803			−0.338901	−	0.940822i	$-0.610055\pi$
$479$	−14.8344			−0.677803			−0.338901	+	0.940822i	$0.610055\pi$
$480$		0			0
$481$			26.4168i			1.20450i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	22.7386i		−	1.03251i
$486$		0			0
$487$			21.0508i			0.953903i	0.878930	+	0.476952i	$0.158258\pi$
$487$			21.0508i			0.953903i	−0.878930	+	0.476952i	$0.841742\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	29.9009i		−	1.34941i	−0.738088	−	0.674705i	$-0.764271\pi$
$491$		−	29.9009i		−	1.34941i	0.738088	−	0.674705i	$-0.235729\pi$
$492$		0			0
$493$			22.0259i			0.991996i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	3.97626i		−	0.178360i
$498$		0			0
$499$	−34.7386			−1.55511			−0.777557	−	0.628812i	$-0.783542\pi$
$499$	−34.7386			−1.55511			−0.777557	+	0.628812i	$0.783542\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−11.5824			−0.516433			−0.258216	−	0.966087i	$-0.583135\pi$
$503$	−11.5824			−0.516433			−0.258216	+	0.966087i	$0.583135\pi$
$504$		0			0
$505$			13.2084i			0.587767i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	10.0000i		−	0.443242i	−0.975133	−	0.221621i	$-0.928865\pi$
$509$		−	10.0000i		−	0.443242i	0.975133	−	0.221621i	$-0.0711348\pi$
$510$		0			0
$511$			12.3687i			0.547159i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.47284			−0.153032
$516$		0			0
$517$	−7.19612	−	18.5531i	−0.316485	−	0.815962i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.7374			1.08377			0.541883	−	0.840454i	$-0.317712\pi$
$521$	24.7374			1.08377			0.541883	+	0.840454i	$0.317712\pi$
$522$		0			0
$523$			25.1473i			1.09961i	0.835292	+	0.549806i	$0.185298\pi$
$523$			25.1473i			1.09961i	−0.835292	+	0.549806i	$0.814702\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.79119			0.252268
$528$		0			0
$529$	−3.24621			−0.141140
$530$		0			0
$531$		0			0
$532$		0			0
$533$			6.24621i			0.270553i
$534$		0			0
$535$	−18.1379			−0.784172
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.9337	−	7.34376i	0.815533	−	0.316318i
$540$		0			0
$541$	−14.6875			−0.631466			−0.315733	−	0.948848i	$-0.602250\pi$
$541$	−14.6875			−0.631466			−0.315733	+	0.948848i	$0.602250\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.69400i			0.115398i
$546$		0			0
$547$			23.5212i			1.00570i	0.864375	+	0.502848i	$0.167714\pi$
$547$			23.5212i			1.00570i	−0.864375	+	0.502848i	$0.832286\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	19.1231i		−	0.814672i
$552$		0			0
$553$	−3.61553			−0.153748
$554$		0			0
$555$		0			0
$556$		0			0
$557$	42.0643			1.78232			0.891160	−	0.453689i	$-0.149892\pi$
$557$	42.0643			1.78232			0.891160	+	0.453689i	$0.149892\pi$
$558$		0			0
$559$		−	44.0518i		−	1.86319i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			0.295294i			0.0124452i	0.999981	+	0.00622258i	$0.00198072\pi$
$563$			0.295294i			0.0124452i	−0.999981	+	0.00622258i	$0.998019\pi$
$564$		0			0
$565$			31.6831i			1.33292i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	34.5830i		−	1.44980i	−0.688856	−	0.724898i	$-0.741887\pi$
$569$		−	34.5830i		−	1.44980i	0.688856	−	0.724898i	$-0.258113\pi$
$570$		0			0
$571$			16.1040i			0.673932i	0.941517	+	0.336966i	$0.109401\pi$
$571$			16.1040i			0.673932i	−0.941517	+	0.336966i	$0.890599\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			5.12311i			0.213648i
$576$		0			0
$577$	8.24621			0.343294			0.171647	−	0.985158i	$-0.445091\pi$
$577$	8.24621			0.343294			0.171647	+	0.985158i	$0.445091\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			2.24621i			0.0931885i
$582$		0			0
$583$	9.89012	+	25.4987i	0.409607	+	1.05605i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−41.3403			−1.70630			−0.853148	−	0.521669i	$-0.825310\pi$
$587$	−41.3403			−1.70630			−0.853148	+	0.521669i	$0.825310\pi$
$588$		0			0
$589$	−5.02797			−0.207174
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	17.4968i		−	0.718508i	−0.933240	−	0.359254i	$-0.883031\pi$
$593$		−	17.4968i		−	0.718508i	0.933240	−	0.359254i	$-0.116969\pi$
$594$		0			0
$595$	−6.24621			−0.256070
$596$		0			0
$597$		0			0
$598$		0			0
$599$			8.63068i			0.352640i	0.984333	+	0.176320i	$0.0564194\pi$
$599$			8.63068i			0.352640i	−0.984333	+	0.176320i	$0.943581\pi$
$600$		0			0
$601$		−	24.7908i		−	1.01124i	−0.862757	−	0.505619i	$-0.831264\pi$
$601$		−	24.7908i		−	1.01124i	0.862757	−	0.505619i	$-0.168736\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.8344	−	16.2462i	−0.603106	−	0.660502i
$606$		0			0
$607$	−43.8826			−1.78114			−0.890569	−	0.454848i	$-0.849694\pi$
$607$	−43.8826			−1.78114			−0.890569	+	0.454848i	$0.849694\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			25.6294i			1.03685i
$612$		0			0
$613$	−25.1035			−1.01392			−0.506960	−	0.861969i	$-0.669231\pi$
$613$	−25.1035			−1.01392			−0.506960	+	0.861969i	$0.669231\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	40.5790			1.63365			0.816824	−	0.576888i	$-0.195733\pi$
$617$	40.5790			1.63365			0.816824	+	0.576888i	$0.195733\pi$
$618$		0			0
$619$	8.49242			0.341339			0.170670	−	0.985328i	$-0.445407\pi$
$619$	8.49242			0.341339			0.170670	+	0.985328i	$0.445407\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.25206			0.130291
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.6256			−0.822398
$630$		0			0
$631$		−	26.4738i		−	1.05391i	−0.849894	−	0.526953i	$-0.823334\pi$
$631$		−	26.4738i		−	1.05391i	0.849894	−	0.526953i	$-0.176666\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	32.2996i		−	1.28177i
$636$		0			0
$637$	−26.1552			−1.03631
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.42302			−0.214197			−0.107098	−	0.994248i	$-0.534156\pi$
$641$	−5.42302			−0.214197			−0.107098	+	0.994248i	$0.534156\pi$
$642$		0			0
$643$	−36.9848			−1.45854			−0.729270	−	0.684226i	$-0.760140\pi$
$643$	−36.9848			−1.45854			−0.729270	+	0.684226i	$0.760140\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			45.1231i			1.77397i	0.461796	+	0.886986i	$0.347205\pi$
$647$			45.1231i			1.77397i	−0.461796	+	0.886986i	$0.652795\pi$
$648$		0			0
$649$	−29.8617	+	11.5824i	−1.17218	+	0.454648i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	22.9848i		−	0.899466i	−0.893163	−	0.449733i	$-0.851519\pi$
$653$		−	22.9848i		−	0.899466i	0.893163	−	0.449733i	$-0.148481\pi$
$654$		0			0
$655$	25.6294			1.00142
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	0.525853i		−	0.0204843i	−0.999948	−	0.0102422i	$-0.996740\pi$
$659$		−	0.525853i		−	0.0204843i	0.999948	−	0.0102422i	$-0.00326024\pi$
$660$		0			0
$661$		−	40.5790i		−	1.57834i	−0.614176	−	0.789169i	$-0.710511\pi$
$661$		−	40.5790i		−	1.57834i	0.614176	−	0.789169i	$-0.289489\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.42302			0.210296
$666$		0			0
$667$			33.8340i			1.31006i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−28.0429	+	10.8769i	−1.08258	+	0.419898i
$672$		0			0
$673$			14.8344i			0.571826i	0.958256	+	0.285913i	$0.0922968\pi$
$673$			14.8344i			0.571826i	−0.958256	+	0.285913i	$0.907703\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.6475			−0.601381			−0.300690	−	0.953722i	$-0.597217\pi$
$677$	−15.6475			−0.601381			−0.300690	+	0.953722i	$0.597217\pi$
$678$		0			0
$679$	−10.6465			−0.408576
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.4185			0.398654			0.199327	−	0.979933i	$-0.436124\pi$
$683$	10.4185			0.398654			0.199327	+	0.979933i	$0.436124\pi$
$684$		0			0
$685$		−	31.6831i		−	1.21055i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	35.2242i		−	1.34193i
$690$		0			0
$691$	28.0000			1.06517			0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000			1.06517			0.532585	+	0.846376i	$0.321221\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.6256			−0.782375
$696$		0			0
$697$	−4.87689			−0.184726
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.2298			1.02846			0.514228	−	0.857653i	$-0.328078\pi$
$701$	27.2298			1.02846			0.514228	+	0.857653i	$0.328078\pi$
$702$		0			0
$703$	17.9074			0.675390
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.18435			0.232586
$708$		0			0
$709$		−	18.5531i		−	0.696775i	−0.937351	−	0.348387i	$-0.886729\pi$
$709$		−	18.5531i		−	0.696775i	0.937351	−	0.348387i	$-0.113271\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8.89586			0.333153
$714$		0			0
$715$	10.2462	+	26.4168i	0.383187	+	0.987933i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			14.9848i			0.558840i	0.960169	+	0.279420i	$0.0901422\pi$
$719$			14.9848i			0.558840i	−0.960169	+	0.279420i	$0.909858\pi$
$720$		0			0
$721$			1.62603i			0.0605565i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.60421			0.245274
$726$		0			0
$727$			14.1051i			0.523130i	0.965186	+	0.261565i	$0.0842386\pi$
$727$			14.1051i			0.523130i	−0.965186	+	0.261565i	$0.915761\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	34.3946			1.27213
$732$		0			0
$733$	−37.6229			−1.38963			−0.694816	−	0.719187i	$-0.744514\pi$
$733$	−37.6229			−1.38963			−0.694816	+	0.719187i	$0.744514\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−31.6831	+	12.2888i	−1.16706	+	0.452665i
$738$		0			0
$739$			14.4780i			0.532581i	0.963893	+	0.266290i	$0.0857980\pi$
$739$			14.4780i			0.532581i	−0.963893	+	0.266290i	$0.914202\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−41.2513			−1.51336			−0.756681	−	0.653784i	$-0.773180\pi$
$743$	−41.2513			−1.51336			−0.756681	+	0.653784i	$0.773180\pi$
$744$		0			0
$745$			1.62603i			0.0595731i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			8.49242i			0.310306i
$750$		0			0
$751$			24.5236i			0.894881i	0.894314	+	0.447440i	$0.147664\pi$
$751$			24.5236i			0.894881i	−0.894314	+	0.447440i	$0.852336\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	41.8944i		−	1.52469i
$756$		0			0
$757$			22.7872i			0.828216i	0.910228	+	0.414108i	$0.135906\pi$
$757$			22.7872i			0.828216i	−0.910228	+	0.414108i	$0.864094\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			29.1950i			1.05832i	0.848522	+	0.529160i	$0.177493\pi$
$761$			29.1950i			1.05832i	−0.848522	+	0.529160i	$0.822507\pi$
$762$		0			0
$763$	1.26137			0.0456645
$764$		0			0
$765$		0			0
$766$		0			0
$767$	41.2513			1.48950
$768$		0			0
$769$			39.6252i			1.42892i	0.699675	+	0.714461i	$0.253328\pi$
$769$			39.6252i			1.42892i	−0.699675	+	0.714461i	$0.746672\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	49.2311i		−	1.77072i	−0.464908	−	0.885359i	$-0.653913\pi$
$773$		−	49.2311i		−	1.77072i	0.464908	−	0.885359i	$-0.346087\pi$
$774$		0			0
$775$		−	1.73642i		−	0.0623741i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.23417			0.151705
$780$		0			0
$781$	−5.09271	−	13.1300i	−0.182231	−	0.469830i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−31.6831			−1.13082
$786$		0			0
$787$			48.3120i			1.72214i	0.508489	+	0.861069i	$0.330204\pi$
$787$			48.3120i			1.72214i	−0.508489	+	0.861069i	$0.669796\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	14.8344			0.527452
$792$		0			0
$793$	38.7386			1.37565
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	11.7538i		−	0.416341i	−0.978093	−	0.208170i	$-0.933249\pi$
$797$		−	11.7538i		−	0.416341i	0.978093	−	0.208170i	$-0.0667508\pi$
$798$		0			0
$799$	−20.0108			−0.707931
$800$		0			0
$801$		0			0
$802$		0			0
$803$	15.8415	+	40.8427i	0.559036	+	1.44131i
$804$		0			0
$805$	−9.59482			−0.338173
$806$		0			0
$807$		0			0
$808$		0			0
$809$			48.1541i			1.69301i	0.532382	+	0.846504i	$0.321297\pi$
$809$			48.1541i			1.69301i	−0.532382	+	0.846504i	$0.678703\pi$
$810$		0			0
$811$		−	30.9384i		−	1.08640i	−0.839605	−	0.543198i	$-0.817213\pi$
$811$		−	30.9384i		−	1.08640i	0.839605	−	0.543198i	$-0.182787\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			24.9848i			0.875181i
$816$		0			0
$817$	−29.8617			−1.04473
$818$		0			0
$819$		0			0
$820$		0			0
$821$	28.8559			1.00708			0.503538	−	0.863973i	$-0.332031\pi$
$821$	28.8559			1.00708			0.503538	+	0.863973i	$0.332031\pi$
$822$		0			0
$823$		−	47.7384i		−	1.66406i	−0.554734	−	0.832028i	$-0.687180\pi$
$823$		−	47.7384i		−	1.66406i	0.554734	−	0.832028i	$-0.312820\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			51.7846i			1.80073i	0.435140	+	0.900363i	$0.356699\pi$
$827$			51.7846i			1.80073i	−0.435140	+	0.900363i	$0.643301\pi$
$828$		0			0
$829$			25.4987i			0.885608i	0.896618	+	0.442804i	$0.146016\pi$
$829$			25.4987i			0.885608i	−0.896618	+	0.442804i	$0.853984\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	20.4214i		−	0.707558i
$834$		0			0
$835$		−	11.5824i		−	0.400825i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	18.8769i		−	0.651703i	−0.945421	−	0.325851i	$-0.894349\pi$
$839$		−	18.8769i		−	0.651703i	0.945421	−	0.325851i	$-0.105651\pi$
$840$		0			0
$841$	14.6155			0.503984
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	10.4924i		−	0.360950i
$846$		0			0
$847$	−7.60669	+	6.94568i	−0.261369	+	0.238656i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−31.6831			−1.08608
$852$		0			0
$853$	−17.3815			−0.595132			−0.297566	−	0.954701i	$-0.596175\pi$
$853$	−17.3815			−0.595132			−0.297566	+	0.954701i	$0.596175\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			11.0571i			0.377703i	0.982006	+	0.188852i	$0.0604765\pi$
$857$			11.0571i			0.377703i	−0.982006	+	0.188852i	$0.939523\pi$
$858$		0			0
$859$	−24.0000			−0.818869			−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000			−0.818869			−0.409435	+	0.912339i	$0.634274\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			48.3542i			1.64599i	0.568045	+	0.822997i	$0.307700\pi$
$863$			48.3542i			1.64599i	−0.568045	+	0.822997i	$0.692300\pi$
$864$		0			0
$865$		−	1.62603i		−	0.0552867i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−11.9388	+	4.63068i	−0.404998	+	0.157085i
$870$		0			0
$871$	43.7673			1.48300
$872$		0			0
$873$		0			0
$874$		0			0
$875$			11.2371i			0.379884i
$876$		0			0
$877$	−50.1423			−1.69318			−0.846592	−	0.532243i	$-0.821349\pi$
$877$	−50.1423			−1.69318			−0.846592	+	0.532243i	$0.821349\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−37.1061			−1.25014			−0.625068	−	0.780570i	$-0.714929\pi$
$881$	−37.1061			−1.25014			−0.625068	+	0.780570i	$0.714929\pi$
$882$		0			0
$883$	−5.75379			−0.193630			−0.0968152	−	0.995302i	$-0.530866\pi$
$883$	−5.75379			−0.193630			−0.0968152	+	0.995302i	$0.530866\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	29.6689			0.996184			0.498092	−	0.867124i	$-0.334034\pi$
$887$	29.6689			0.996184			0.498092	+	0.867124i	$0.334034\pi$
$888$		0			0
$889$	−15.1231			−0.507213
$890$		0			0
$891$		0			0
$892$		0			0
$893$	17.3736			0.581384
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	11.4677i		−	0.382468i
$900$		0			0
$901$	27.5022			0.916231
$902$		0			0
$903$		0			0
$904$		0			0
$905$	17.7917			0.591417
$906$		0			0
$907$	−30.7386			−1.02066			−0.510330	−	0.859979i	$-0.670477\pi$
$907$	−30.7386			−1.02066			−0.510330	+	0.859979i	$0.670477\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	48.7386i		−	1.61478i	−0.590016	−	0.807391i	$-0.700879\pi$
$911$		−	48.7386i		−	1.61478i	0.590016	−	0.807391i	$-0.299121\pi$
$912$		0			0
$913$	2.87689	+	7.41722i	0.0952113	+	0.245474i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$	−33.2360			−1.09636			−0.548178	−	0.836362i	$-0.684678\pi$
$919$	−33.2360			−1.09636			−0.548178	+	0.836362i	$0.684678\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			18.1379i			0.597018i
$924$		0			0
$925$			6.18435i			0.203340i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−28.2102			−0.925548			−0.462774	−	0.886476i	$-0.653146\pi$
$929$	−28.2102			−0.925548			−0.462774	+	0.886476i	$0.653146\pi$
$930$		0			0
$931$			17.7300i			0.581078i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−20.6256	+	8.00000i	−0.674530	+	0.261628i
$936$		0			0
$937$		−	14.8344i		−	0.484620i	−0.970199	−	0.242310i	$-0.922095\pi$
$937$		−	14.8344i		−	0.484620i	0.970199	−	0.242310i	$-0.0779051\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.6475			−0.510092			−0.255046	−	0.966929i	$-0.582091\pi$
$941$	−15.6475			−0.510092			−0.255046	+	0.966929i	$0.582091\pi$
$942$		0			0
$943$	−7.49141			−0.243954
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.94568			0.225704			0.112852	−	0.993612i	$-0.464001\pi$
$947$	6.94568			0.225704			0.112852	+	0.993612i	$0.464001\pi$
$948$		0			0
$949$		−	56.4205i		−	1.83149i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	43.9473i		−	1.42359i	−0.702386	−	0.711796i	$-0.747882\pi$
$953$		−	43.9473i		−	1.42359i	0.702386	−	0.711796i	$-0.252118\pi$
$954$		0			0
$955$	−24.4924			−0.792556
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.8344			−0.479029
$960$		0			0
$961$	27.9848			0.902737
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.25206			0.104687
$966$		0			0
$967$	−26.7963			−0.861712			−0.430856	−	0.902421i	$-0.641788\pi$
$967$	−26.7963			−0.861712			−0.430856	+	0.902421i	$0.641788\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−30.9218			−0.992327			−0.496163	−	0.868229i	$-0.665258\pi$
$971$	−30.9218			−0.992327			−0.496163	+	0.868229i	$0.665258\pi$
$972$		0			0
$973$			9.65719i			0.309595i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−30.1604			−0.964918			−0.482459	−	0.875919i	$-0.660256\pi$
$977$	−30.1604			−0.964918			−0.482459	+	0.875919i	$0.660256\pi$
$978$		0			0
$979$	10.7386	−	4.16516i	0.343208	−	0.133119i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			47.3693i			1.51085i	0.655237	+	0.755423i	$0.272569\pi$
$983$			47.3693i			1.51085i	−0.655237	+	0.755423i	$0.727431\pi$
$984$		0			0
$985$			24.7908i			0.789900i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	52.8336			1.68001
$990$		0			0
$991$			47.7384i			1.51646i	0.651987	+	0.758230i	$0.273936\pi$
$991$			47.7384i			1.51646i	−0.651987	+	0.758230i	$0.726064\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	28.2102			0.894325
$996$		0			0
$997$	28.0281			0.887657			0.443829	−	0.896112i	$-0.353620\pi$
$997$	28.0281			0.887657			0.443829	+	0.896112i	$0.353620\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.h.i.2287.3		16
3.2	odd	2	inner	3168.2.h.i.2287.12		16
4.3	odd	2		792.2.h.i.307.11	yes	16
8.3	odd	2	inner	3168.2.h.i.2287.13		16
8.5	even	2		792.2.h.i.307.8	yes	16
11.10	odd	2	inner	3168.2.h.i.2287.5		16
12.11	even	2		792.2.h.i.307.6	yes	16
24.5	odd	2		792.2.h.i.307.9	yes	16
24.11	even	2	inner	3168.2.h.i.2287.6		16
33.32	even	2	inner	3168.2.h.i.2287.14		16
44.43	even	2		792.2.h.i.307.5	✓	16
88.21	odd	2		792.2.h.i.307.10	yes	16
88.43	even	2	inner	3168.2.h.i.2287.11		16
132.131	odd	2		792.2.h.i.307.12	yes	16
264.131	odd	2	inner	3168.2.h.i.2287.4		16
264.197	even	2		792.2.h.i.307.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.h.i.307.5	✓	16	44.43	even	2
792.2.h.i.307.6	yes	16	12.11	even	2
792.2.h.i.307.7	yes	16	264.197	even	2
792.2.h.i.307.8	yes	16	8.5	even	2
792.2.h.i.307.9	yes	16	24.5	odd	2
792.2.h.i.307.10	yes	16	88.21	odd	2
792.2.h.i.307.11	yes	16	4.3	odd	2
792.2.h.i.307.12	yes	16	132.131	odd	2
3168.2.h.i.2287.3		16	1.1	even	1	trivial
3168.2.h.i.2287.4		16	264.131	odd	2	inner
3168.2.h.i.2287.5		16	11.10	odd	2	inner
3168.2.h.i.2287.6		16	24.11	even	2	inner
3168.2.h.i.2287.11		16	88.43	even	2	inner
3168.2.h.i.2287.12		16	3.2	odd	2	inner
3168.2.h.i.2287.13		16	8.3	odd	2	inner
3168.2.h.i.2287.14		16	33.32	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{5} -0.936426 q^{7} +(-3.09218 + 1.19935i) q^{11} +4.27156 q^{13} +3.33513i q^{17} -2.89560i q^{19} +5.12311i q^{23} +1.00000 q^{25} +6.60421 q^{29} -1.73642i q^{31} +1.87285i q^{35} +6.18435i q^{37} +1.46228i q^{41} -10.3128i q^{43} +6.00000i q^{47} -6.12311 q^{49} -8.24621i q^{53} +(2.39871 + 6.18435i) q^{55} +9.65719 q^{59} +9.06897 q^{61} -8.54312i q^{65} +10.2462 q^{67} +4.24621i q^{71} -13.2084i q^{73} +(2.89560 - 1.12311i) q^{77} +3.86098 q^{79} -2.39871i q^{83} +6.67026 q^{85} -3.47284 q^{89} -4.00000 q^{91} -5.79119 q^{95} +11.3693 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{25} - 32 q^{49} + 32 q^{67} - 64 q^{91} - 16 q^{97}+O(q^{100}) \) 16 * q + 16 * q^25 - 32 * q^49 + 32 * q^67 - 64 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more