LMFDB - Embedded newform 252.6.k.e.109.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,6,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$252 = 2^{2} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	252.k (of order $3$ , degree $2$ , 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:	$40.4167225929$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7081})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900$ x^4 - x^3 + 1771x^2 + 1770x + 3132900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.2
Root			$-20.7872 - 36.0044i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.6.k.e.37.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(32.7872 + 56.7890i) q^{5} +(40.6487 + 123.104i) q^{7} +(-122.787 + 212.674i) q^{11} +434.872 q^{13} +(-551.149 + 954.618i) q^{17} +(1438.65 + 2491.81i) q^{19} +(-2114.72 - 3662.80i) q^{23} +(-587.497 + 1017.57i) q^{25} -4969.60 q^{29} +(4391.32 - 7606.00i) q^{31} +(-5658.22 + 6344.64i) q^{35} +(1220.03 + 2113.15i) q^{37} +3668.55 q^{41} -7198.06 q^{43} +(1636.66 + 2834.78i) q^{47} +(-13502.4 + 10008.1i) q^{49} +(-1510.84 + 2616.84i) q^{53} -16103.4 q^{55} +(-25743.7 + 44589.5i) q^{59} +(6656.65 + 11529.6i) q^{61} +(14258.2 + 24695.9i) q^{65} +(-15447.9 + 26756.5i) q^{67} +41882.8 q^{71} +(-17308.7 + 29979.5i) q^{73} +(-31172.2 - 6470.74i) q^{77} +(-38771.7 - 67154.6i) q^{79} -100908. q^{83} -72282.4 q^{85} +(20391.8 + 35319.7i) q^{89} +(17677.0 + 53534.6i) q^{91} +(-94338.2 + 163399. i) q^{95} +140147. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 47 q^{5} - 174 q^{7} - 407 q^{11} + 898 q^{13} - 1868 q^{17} + 1463 q^{19} - 44 q^{23} + 1605 q^{25} - 1534 q^{29} + 11170 q^{31} - 9674 q^{35} + 3113 q^{37} + 15684 q^{41} - 25258 q^{43} + 9576 q^{47}+ \cdots + 369570 q^{97}+O(q^{100})$ 4 * q + 47 * q^5 - 174 * q^7 - 407 * q^11 + 898 * q^13 - 1868 * q^17 + 1463 * q^19 - 44 * q^23 + 1605 * q^25 - 1534 * q^29 + 11170 * q^31 - 9674 * q^35 + 3113 * q^37 + 15684 * q^41 - 25258 * q^43 + 9576 * q^47 + 4558 * q^49 + 13395 * q^53 - 26210 * q^55 - 47521 * q^59 + 63652 * q^61 + 28254 * q^65 - 44541 * q^67 + 251680 * q^71 - 6039 * q^73 - 35407 * q^77 - 17588 * q^79 - 78650 * q^83 - 116120 * q^85 + 83082 * q^89 + 31747 * q^91 - 214946 * q^95 + 369570 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$ .

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	32.7872	+	56.7890i	0.586515	+	1.01587i	0.994685	+	0.102967i	$0.0328337\pi$
$5$	32.7872	+	56.7890i	0.586515	+	1.01587i	−0.408170	+	0.912906i	$0.633833\pi$
$6$		0			0
$7$	40.6487	+	123.104i	0.313546	+	0.949573i
$7$	40.6487	+	123.104i	0.313546	+	0.949573i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−122.787	+	212.674i	−0.305965	+	0.529946i	−0.977476	−	0.211048i	$-0.932312\pi$
$11$	−122.787	+	212.674i	−0.305965	+	0.529946i	0.671511	+	0.740995i	$0.265646\pi$
$12$		0			0
$13$	434.872			0.713679			0.356839	−	0.934166i	$-0.383854\pi$
$13$	434.872			0.713679			0.356839	+	0.934166i	$0.383854\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−551.149	+	954.618i	−0.462537	+	0.801138i	−0.999087	−	0.0427312i	$-0.986394\pi$
$17$	−551.149	+	954.618i	−0.462537	+	0.801138i	0.536550	+	0.843869i	$0.319727\pi$
$18$		0			0
$19$	1438.65	+	2491.81i	0.914260	+	1.58355i	0.807981	+	0.589209i	$0.200560\pi$
$19$	1438.65	+	2491.81i	0.914260	+	1.58355i	0.106279	+	0.994336i	$0.466106\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2114.72	−	3662.80i	−0.833552	−	1.44375i	−0.895204	−	0.445657i	$-0.852970\pi$
$23$	−2114.72	−	3662.80i	−0.833552	−	1.44375i	0.0616521	−	0.998098i	$-0.480363\pi$
$24$		0			0
$25$	−587.497	+	1017.57i	−0.187999	+	0.325624i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4969.60			−1.09730			−0.548652	−	0.836051i	$-0.684859\pi$
$29$	−4969.60			−1.09730			−0.548652	+	0.836051i	$0.684859\pi$
$30$		0			0
$31$	4391.32	−	7606.00i	0.820713	−	1.42152i	−0.0844390	−	0.996429i	$-0.526910\pi$
$31$	4391.32	−	7606.00i	0.820713	−	1.42152i	0.905152	−	0.425088i	$-0.139757\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5658.22	+	6344.64i	−0.780746	+	0.875462i
$36$		0			0
$37$	1220.03	+	2113.15i	0.146510	+	0.253762i	0.929935	−	0.367723i	$-0.119863\pi$
$37$	1220.03	+	2113.15i	0.146510	+	0.253762i	−0.783425	+	0.621486i	$0.786529\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3668.55			0.340828			0.170414	−	0.985373i	$-0.445489\pi$
$41$	3668.55			0.340828			0.170414	+	0.985373i	$0.445489\pi$
$42$		0			0
$43$	−7198.06			−0.593669			−0.296835	−	0.954929i	$-0.595931\pi$
$43$	−7198.06			−0.593669			−0.296835	+	0.954929i	$0.595931\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1636.66	+	2834.78i	0.108072	+	0.187187i	0.914989	−	0.403478i	$-0.132199\pi$
$47$	1636.66	+	2834.78i	0.108072	+	0.187187i	−0.806917	+	0.590665i	$0.798866\pi$
$48$		0			0
$49$	−13502.4	+	10008.1i	−0.803378	+	0.595470i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1510.84	+	2616.84i	−0.0738801	+	0.127964i	−0.900599	−	0.434651i	$-0.856872\pi$
$53$	−1510.84	+	2616.84i	−0.0738801	+	0.127964i	0.826719	+	0.562616i	$0.190205\pi$
$54$		0			0
$55$	−16103.4			−0.717811
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−25743.7	+	44589.5i	−0.962812	+	1.66764i	−0.247432	+	0.968905i	$0.579587\pi$
$59$	−25743.7	+	44589.5i	−0.962812	+	1.66764i	−0.715381	+	0.698735i	$0.753747\pi$
$60$		0			0
$61$	6656.65	+	11529.6i	0.229050	+	0.396727i	0.957527	−	0.288344i	$-0.0931046\pi$
$61$	6656.65	+	11529.6i	0.229050	+	0.396727i	−0.728477	+	0.685071i	$0.759771\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	14258.2	+	24695.9i	0.418583	+	0.725007i
$66$		0			0
$67$	−15447.9	+	26756.5i	−0.420418	+	0.728186i	−0.995980	−	0.0895723i	$-0.971450\pi$
$67$	−15447.9	+	26756.5i	−0.420418	+	0.728186i	0.575562	+	0.817758i	$0.304783\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	41882.8			0.986030			0.493015	−	0.870021i	$-0.335895\pi$
$71$	41882.8			0.986030			0.493015	+	0.870021i	$0.335895\pi$
$72$		0			0
$73$	−17308.7	+	29979.5i	−0.380151	+	0.658441i	−0.991084	−	0.133242i	$-0.957461\pi$
$73$	−17308.7	+	29979.5i	−0.380151	+	0.658441i	0.610932	+	0.791683i	$0.290795\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−31172.2	−	6470.74i	−0.599157	−	0.124373i
$78$		0			0
$79$	−38771.7	−	67154.6i	−0.698952	−	1.21062i	−0.968830	−	0.247726i	$-0.920317\pi$
$79$	−38771.7	−	67154.6i	−0.698952	−	1.21062i	0.269878	−	0.962895i	$-0.413017\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−100908.			−1.60779			−0.803897	−	0.594768i	$-0.797244\pi$
$83$	−100908.			−1.60779			−0.803897	+	0.594768i	$0.797244\pi$
$84$		0			0
$85$	−72282.4			−1.08514
$86$		0			0
$87$		0			0
$88$		0			0
$89$	20391.8	+	35319.7i	0.272886	+	0.472652i	0.969600	−	0.244697i	$-0.0786885\pi$
$89$	20391.8	+	35319.7i	0.272886	+	0.472652i	−0.696714	+	0.717349i	$0.745355\pi$
$90$		0			0
$91$	17677.0	+	53534.6i	0.223771	+	0.677690i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−94338.2	+	163399.i	−1.07245	+	1.85755i
$96$		0			0
$97$	140147.			1.51236			0.756178	−	0.654366i	$-0.227064\pi$
$97$	140147.			1.51236			0.756178	+	0.654366i	$0.227064\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4288.23	−	7427.43i	0.0418287	−	0.0724494i	−0.844353	−	0.535787i	$-0.820015\pi$
$101$	4288.23	−	7427.43i	0.0418287	−	0.0724494i	0.886182	+	0.463338i	$0.153348\pi$
$102$		0			0
$103$	17630.1	+	30536.2i	0.163742	+	0.283610i	0.936208	−	0.351447i	$-0.114310\pi$
$103$	17630.1	+	30536.2i	0.163742	+	0.283610i	−0.772466	+	0.635057i	$0.780977\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−96314.3	−	166821.i	−0.813264	−	1.40861i	−0.910568	−	0.413360i	$-0.864355\pi$
$107$	−96314.3	−	166821.i	−0.813264	−	1.40861i	0.0973041	−	0.995255i	$-0.468978\pi$
$108$		0			0
$109$	−61646.1	+	106774.i	−0.496980	+	0.860795i	−0.999994	−	0.00348322i	$-0.998891\pi$
$109$	−61646.1	+	106774.i	−0.496980	+	0.860795i	0.503014	+	0.864279i	$0.332225\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−28400.9			−0.209236			−0.104618	−	0.994512i	$-0.533362\pi$
$113$	−28400.9			−0.209236			−0.104618	+	0.994512i	$0.533362\pi$
$114$		0			0
$115$	138671.	−	240186.i	0.977781	−	1.69357i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−139921.	−	29044.9i	−0.905765	−	0.188019i
$120$		0			0
$121$	50372.1	+	87247.1i	0.312771	+	0.541736i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	127870.			0.731973
$126$		0			0
$127$	−47198.8			−0.259670			−0.129835	−	0.991536i	$-0.541445\pi$
$127$	−47198.8			−0.259670			−0.129835	+	0.991536i	$0.541445\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	39904.7	+	69116.9i	0.203163	+	0.351889i	0.949546	−	0.313628i	$-0.101544\pi$
$131$	39904.7	+	69116.9i	0.203163	+	0.351889i	−0.746383	+	0.665517i	$0.768211\pi$
$132$		0			0
$133$	−248273.	+	278392.i	−1.21703	+	1.36467i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−43032.1	+	74533.9i	−0.195881	+	0.339275i	−0.947189	−	0.320676i	$-0.896090\pi$
$137$	−43032.1	+	74533.9i	−0.195881	+	0.339275i	0.751308	+	0.659952i	$0.229423\pi$
$138$		0			0
$139$	270587.			1.18787			0.593935	−	0.804513i	$-0.297573\pi$
$139$	270587.			1.18787			0.593935	+	0.804513i	$0.297573\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−53396.7	+	92485.7i	−0.218360	+	0.378211i
$144$		0			0
$145$	−162939.	−	282219.i	−0.643585	−	1.11472i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−86273.9	−	149431.i	−0.318356	−	0.551410i	0.661789	−	0.749690i	$-0.269798\pi$
$149$	−86273.9	−	149431.i	−0.318356	−	0.551410i	−0.980145	+	0.198281i	$0.936464\pi$
$150$		0			0
$151$	8949.75	−	15501.4i	0.0319425	−	0.0553260i	−0.849612	−	0.527408i	$-0.823164\pi$
$151$	8949.75	−	15501.4i	0.0319425	−	0.0553260i	0.881555	+	0.472082i	$0.156497\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	575916.			1.92544
$156$		0			0
$157$	89509.3	−	155035.i	0.289814	−	0.501972i	−0.683951	−	0.729528i	$-0.739740\pi$
$157$	89509.3	−	155035.i	0.289814	−	0.501972i	0.973765	+	0.227555i	$0.0730733\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	364946.	−	409219.i	1.10959	−	1.24420i
$162$		0			0
$163$	118645.	+	205498.i	0.349767	+	0.605814i	0.986208	−	0.165511i	$-0.0529275\pi$
$163$	118645.	+	205498.i	0.349767	+	0.605814i	−0.636441	+	0.771325i	$0.719594\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	94040.0			0.260928			0.130464	−	0.991453i	$-0.458353\pi$
$167$	94040.0			0.260928			0.130464	+	0.991453i	$0.458353\pi$
$168$		0			0
$169$	−182180.			−0.490663
$170$		0			0
$171$		0			0
$172$		0			0
$173$	244357.	+	423238.i	0.620739	+	1.07515i	0.989348	+	0.145567i	$0.0465006\pi$
$173$	244357.	+	423238.i	0.620739	+	1.07515i	−0.368609	+	0.929584i	$0.620166\pi$
$174$		0			0
$175$	−149149.	−	30960.4i	−0.368150	−	0.0764207i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	406498.	−	704075.i	0.948257	−	1.64243i	0.199161	−	0.979967i	$-0.436178\pi$
$179$	406498.	−	704075.i	0.948257	−	1.64243i	0.749095	−	0.662462i	$-0.230488\pi$
$180$		0			0
$181$	−332961.			−0.755434			−0.377717	−	0.925921i	$-0.623291\pi$
$181$	−332961.			−0.755434			−0.377717	+	0.925921i	$0.623291\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−80002.7	+	138569.i	−0.171860	+	0.297671i
$186$		0			0
$187$	−135348.	−	234430.i	−0.283040	−	0.490240i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−72143.9	−	124957.i	−0.143092	−	0.247843i	0.785567	−	0.618776i	$-0.212371\pi$
$191$	−72143.9	−	124957.i	−0.143092	−	0.247843i	−0.928660	+	0.370933i	$0.879038\pi$
$192$		0			0
$193$	447310.	−	774763.i	0.864400	−	1.49719i	−0.00324119	−	0.999995i	$-0.501032\pi$
$193$	447310.	−	774763.i	0.864400	−	1.49719i	0.867641	−	0.497190i	$-0.165635\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	599462.			1.10052			0.550258	−	0.834995i	$-0.314530\pi$
$197$	599462.			1.10052			0.550258	+	0.834995i	$0.314530\pi$
$198$		0			0
$199$	389129.	−	673992.i	0.696564	−	1.20648i	−0.273086	−	0.961990i	$-0.588044\pi$
$199$	389129.	−	673992.i	0.696564	−	1.20648i	0.969651	−	0.244495i	$-0.0786222\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−202008.	−	611780.i	−0.344055	−	1.04197i
$204$		0			0
$205$	120282.	+	208334.i	0.199901	+	0.346238i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−706589.			−1.11893
$210$		0			0
$211$	−810532.			−1.25333			−0.626663	−	0.779291i	$-0.715580\pi$
$211$	−810532.			−1.25333			−0.626663	+	0.779291i	$0.715580\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−236004.	−	408771.i	−0.348196	−	0.603093i
$216$		0			0
$217$	1.11483e6	+	231418.i	1.60717	+	0.333616i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−239679.	+	415136.i	−0.330103	+	0.571755i
$222$		0			0
$223$	220486.			0.296905			0.148453	−	0.988920i	$-0.452571\pi$
$223$	220486.			0.296905			0.148453	+	0.988920i	$0.452571\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	671890.	−	1.16375e6i	0.865433	−	1.49897i	−0.00118391	−	0.999999i	$-0.500377\pi$
$227$	671890.	−	1.16375e6i	0.865433	−	1.49897i	0.866617	−	0.498974i	$-0.166290\pi$
$228$		0			0
$229$	521292.	+	902904.i	0.656889	+	1.13777i	0.981417	+	0.191889i	$0.0614614\pi$
$229$	521292.	+	902904.i	0.656889	+	1.13777i	−0.324528	+	0.945876i	$0.605205\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	316256.	+	547771.i	0.381635	+	0.661011i	0.991296	−	0.131651i	$-0.0420278\pi$
$233$	316256.	+	547771.i	0.381635	+	0.661011i	−0.609661	+	0.792662i	$0.708694\pi$
$234$		0			0
$235$	−107323.	+	185889.i	−0.126772	+	0.219575i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	684919.			0.775612			0.387806	−	0.921741i	$-0.373233\pi$
$239$	684919.			0.775612			0.387806	+	0.921741i	$0.373233\pi$
$240$		0			0
$241$	3866.44	−	6696.87i	0.00428814	−	0.00742728i	−0.863873	−	0.503709i	$-0.831968\pi$
$241$	3866.44	−	6696.87i	0.00428814	−	0.00742728i	0.868162	+	0.496282i	$0.165302\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.01105e6	−	438651.i	−1.07611	−	0.466878i
$246$		0			0
$247$	625626.	+	1.08362e6i	0.652488	+	1.13014i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.09614e6			−1.09820			−0.549098	−	0.835758i	$-0.685029\pi$
$251$	−1.09614e6			−1.09820			−0.549098	+	0.835758i	$0.685029\pi$
$252$		0			0
$253$	1.03864e6			1.02015
$254$		0			0
$255$		0			0
$256$		0			0
$257$	357460.	+	619139.i	0.337594	+	0.584730i	0.983980	−	0.178281i	$-0.0570536\pi$
$257$	357460.	+	619139.i	0.337594	+	0.584730i	−0.646386	+	0.763011i	$0.723720\pi$
$258$		0			0
$259$	−210546.	+	236088.i	−0.195028	+	0.218688i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−298256.	+	516595.i	−0.265889	+	0.460533i	−0.967796	−	0.251735i	$-0.918999\pi$
$263$	−298256.	+	516595.i	−0.265889	+	0.460533i	0.701907	+	0.712268i	$0.252332\pi$
$264$		0			0
$265$	−198144.			−0.173327
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.01654e6	+	1.76070e6i	−0.856531	+	1.48356i	0.0186864	+	0.999825i	$0.494052\pi$
$269$	−1.01654e6	+	1.76070e6i	−0.856531	+	1.48356i	−0.875217	+	0.483730i	$0.839282\pi$
$270$		0			0
$271$	−897277.	−	1.55413e6i	−0.742170	−	1.28548i	−0.951505	−	0.307632i	$-0.900463\pi$
$271$	−897277.	−	1.55413e6i	−0.742170	−	1.28548i	0.209336	−	0.977844i	$-0.432870\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−144274.	−	249890.i	−0.115042	−	0.199259i
$276$		0			0
$277$	−211790.	+	366831.i	−0.165846	+	0.287254i	−0.936956	−	0.349449i	$-0.886369\pi$
$277$	−211790.	+	366831.i	−0.165846	+	0.287254i	0.771109	+	0.636703i	$0.219702\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.63799e6			1.23750			0.618749	−	0.785589i	$-0.287640\pi$
$281$	1.63799e6			1.23750			0.618749	+	0.785589i	$0.287640\pi$
$282$		0			0
$283$	147018.	−	254642.i	0.109120	−	0.189001i	−0.806294	−	0.591515i	$-0.798530\pi$
$283$	147018.	−	254642.i	0.109120	−	0.189001i	0.915414	+	0.402514i	$0.131863\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	149122.	+	451615.i	0.106865	+	0.323641i
$288$		0			0
$289$	102399.	+	177360.i	0.0721191	+	0.124914i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	356961.			0.242913			0.121457	−	0.992597i	$-0.461243\pi$
$293$	356961.			0.242913			0.121457	+	0.992597i	$0.461243\pi$
$294$		0			0
$295$	−3.37626e6			−2.25881
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−919631.	−	1.59285e6i	−0.594888	−	1.03038i
$300$		0			0
$301$	−292592.	−	886113.i	−0.186143	−	0.563732i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−436505.	+	756049.i	−0.268683	+	0.465372i
$306$		0			0
$307$	−2.04097e6			−1.23592			−0.617960	−	0.786210i	$-0.712041\pi$
$307$	−2.04097e6			−1.23592			−0.617960	+	0.786210i	$0.712041\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.02337e6	−	1.77253e6i	0.599972	−	1.03918i	−0.392852	−	0.919602i	$-0.628512\pi$
$311$	1.02337e6	−	1.77253e6i	0.599972	−	1.03918i	0.992824	−	0.119581i	$-0.0381552\pi$
$312$		0			0
$313$	−319272.	−	552996.i	−0.184205	−	0.319052i	0.759104	−	0.650970i	$-0.225638\pi$
$313$	−319272.	−	552996.i	−0.184205	−	0.319052i	−0.943308	+	0.331918i	$0.892304\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.74954e6	+	3.03030e6i	0.977860	+	1.69370i	0.670155	+	0.742221i	$0.266228\pi$
$317$	1.74954e6	+	3.03030e6i	0.977860	+	1.69370i	0.307705	+	0.951482i	$0.400439\pi$
$318$		0			0
$319$	610203.	−	1.05690e6i	0.335736	−	0.581512i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.17163e6			−1.69152
$324$		0			0
$325$	−255486.	+	442514.i	−0.134171	+	0.232391i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−282446.	+	316710.i	−0.143862	+	0.161314i
$330$		0			0
$331$	1.47089e6	+	2.54766e6i	0.737923	+	1.27812i	0.953429	+	0.301618i	$0.0975266\pi$
$331$	1.47089e6	+	2.54766e6i	0.737923	+	1.27812i	−0.215506	+	0.976503i	$0.569140\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.02597e6			−0.986326
$336$		0			0
$337$	2.77854e6			1.33273			0.666364	−	0.745627i	$-0.267850\pi$
$337$	2.77854e6			1.33273			0.666364	+	0.745627i	$0.267850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.07840e6	+	1.86784e6i	0.502218	+	0.869868i
$342$		0			0
$343$	−1.78089e6	−	1.25539e6i	−0.817338	−	0.576159i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−86001.9	+	148960.i	−0.0383429	+	0.0664118i	−0.884560	−	0.466426i	$-0.845541\pi$
$347$	−86001.9	+	148960.i	−0.0383429	+	0.0664118i	0.846217	+	0.532838i	$0.178875\pi$
$348$		0			0
$349$	−3.88321e6			−1.70658			−0.853290	−	0.521436i	$-0.825397\pi$
$349$	−3.88321e6			−1.70658			−0.853290	+	0.521436i	$0.825397\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−385467.	+	667649.i	−0.164646	+	0.285175i	−0.936529	−	0.350589i	$-0.885981\pi$
$353$	−385467.	+	667649.i	−0.164646	+	0.285175i	0.771884	+	0.635764i	$0.219315\pi$
$354$		0			0
$355$	1.37322e6	+	2.37849e6i	0.578321	+	1.00168i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.57017e6	+	2.71961e6i	0.642998	+	1.11370i	0.984760	+	0.173918i	$0.0556428\pi$
$359$	1.57017e6	+	2.71961e6i	0.642998	+	1.11370i	−0.341763	+	0.939786i	$0.611024\pi$
$360$		0			0
$361$	−2.90135e6	+	5.02529e6i	−1.17174	+	2.02952i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.27001e6			−0.891857
$366$		0			0
$367$	182842.	−	316691.i	0.0708615	−	0.122736i	−0.828418	−	0.560111i	$-0.810758\pi$
$367$	182842.	−	316691.i	0.0708615	−	0.122736i	0.899279	+	0.437375i	$0.144092\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−383559.	−	79619.2i	−0.144676	−	0.0300319i
$372$		0			0
$373$	77129.9	+	133593.i	0.0287045	+	0.0497177i	0.880021	−	0.474935i	$-0.157528\pi$
$373$	77129.9	+	133593.i	0.0287045	+	0.0497177i	−0.851316	+	0.524653i	$0.824195\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.16114e6			−0.783122
$378$		0			0
$379$	4.06013e6			1.45192			0.725959	−	0.687738i	$-0.241396\pi$
$379$	4.06013e6			1.45192			0.725959	+	0.687738i	$0.241396\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.02152e6	+	1.76932e6i	0.355834	+	0.616323i	0.987260	−	0.159113i	$-0.0508633\pi$
$383$	1.02152e6	+	1.76932e6i	0.355834	+	0.616323i	−0.631426	+	0.775436i	$0.717530\pi$
$384$		0			0
$385$	−654581.	−	1.98240e6i	−0.225067	−	0.681614i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.06183e6	+	1.83915e6i	−0.355780	+	0.616229i	−0.987251	−	0.159170i	$-0.949118\pi$
$389$	−1.06183e6	+	1.83915e6i	−0.355780	+	0.616229i	0.631471	+	0.775399i	$0.282451\pi$
$390$		0			0
$391$	4.66209e6			1.54219
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.54243e6	−	4.40362e6i	0.819892	−	1.42009i
$396$		0			0
$397$	1.83182e6	+	3.17280e6i	0.583319	+	1.01034i	0.995083	+	0.0990473i	$0.0315795\pi$
$397$	1.83182e6	+	3.17280e6i	0.583319	+	1.01034i	−0.411764	+	0.911291i	$0.635087\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−32044.9	−	55503.3i	−0.00995170	−	0.0172369i	0.861007	−	0.508594i	$-0.169834\pi$
$401$	−32044.9	−	55503.3i	−0.00995170	−	0.0172369i	−0.870958	+	0.491357i	$0.836501\pi$
$402$		0			0
$403$	1.90966e6	−	3.30763e6i	0.585725	−	1.01451i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−599216.			−0.179307
$408$		0			0
$409$	−1.01287e6	+	1.75434e6i	−0.299395	+	0.518568i	−0.975998	−	0.217781i	$-0.930118\pi$
$409$	−1.01287e6	+	1.75434e6i	−0.299395	+	0.518568i	0.676603	+	0.736348i	$0.263452\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.53561e6	−	1.35666e6i	−1.88543	−	0.391379i
$414$		0			0
$415$	−3.30849e6	−	5.73047e6i	−0.942995	−	1.63332i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.38986e6			−0.665025			−0.332513	−	0.943099i	$-0.607896\pi$
$419$	−2.38986e6			−0.665025			−0.332513	+	0.943099i	$0.607896\pi$
$420$		0			0
$421$	3.46875e6			0.953822			0.476911	−	0.878952i	$-0.341756\pi$
$421$	3.46875e6			0.953822			0.476911	+	0.878952i	$0.341756\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−647596.	−	1.12167e6i	−0.173913	−	0.301226i
$426$		0			0
$427$	−1.14877e6	+	1.28813e6i	−0.304903	+	0.341892i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−802201.	+	1.38945e6i	−0.208013	+	0.360289i	−0.951088	−	0.308919i	$-0.900033\pi$
$431$	−802201.	+	1.38945e6i	−0.208013	+	0.360289i	0.743076	+	0.669207i	$0.233366\pi$
$432$		0			0
$433$	741661.			0.190102			0.0950508	−	0.995472i	$-0.469699\pi$
$433$	741661.			0.190102			0.0950508	+	0.995472i	$0.469699\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.08466e6	−	1.05389e7i	1.52417	−	2.63993i
$438$		0			0
$439$	−1.43394e6	−	2.48365e6i	−0.355115	−	0.615077i	0.632023	−	0.774950i	$-0.282225\pi$
$439$	−1.43394e6	−	2.48365e6i	−0.355115	−	0.615077i	−0.987138	+	0.159873i	$0.948892\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−89148.9	−	154410.i	−0.0215827	−	0.0373824i	0.855032	−	0.518575i	$-0.173537\pi$
$443$	−89148.9	−	154410.i	−0.0215827	−	0.0373824i	−0.876615	+	0.481192i	$0.840204\pi$
$444$		0			0
$445$	−1.33718e6	+	2.31607e6i	−0.320103	+	0.554435i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.77890e6			0.650515			0.325258	−	0.945625i	$-0.394549\pi$
$449$	2.77890e6			0.650515			0.325258	+	0.945625i	$0.394549\pi$
$450$		0			0
$451$	−450451.	+	780205.i	−0.104281	+	0.180621i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.46060e6	+	2.75911e6i	−0.557202	+	0.624798i
$456$		0			0
$457$	−3.20669e6	−	5.55416e6i	−0.718236	−	1.24402i	−0.961698	−	0.274111i	$-0.911617\pi$
$457$	−3.20669e6	−	5.55416e6i	−0.718236	−	1.24402i	0.243462	−	0.969910i	$-0.421717\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.88393e6			1.50864			0.754318	−	0.656510i	$-0.227968\pi$
$461$	6.88393e6			1.50864			0.754318	+	0.656510i	$0.227968\pi$
$462$		0			0
$463$	6.70530e6			1.45367			0.726835	−	0.686812i	$-0.240991\pi$
$463$	6.70530e6			1.45367			0.726835	+	0.686812i	$0.240991\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.62836e6	−	2.82041e6i	−0.345509	−	0.598439i	0.639937	−	0.768427i	$-0.278960\pi$
$467$	−1.62836e6	−	2.82041e6i	−0.345509	−	0.598439i	−0.985446	+	0.169988i	$0.945627\pi$
$468$		0			0
$469$	−3.92178e6	−	814084.i	−0.823286	−	0.170898i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	883830.	−	1.53084e6i	0.181642	−	0.314613i
$474$		0			0
$475$	−3.38080e6			−0.687520
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.67916e6	−	8.10454e6i	0.931814	−	1.61395i	0.151595	−	0.988443i	$-0.451559\pi$
$479$	4.67916e6	−	8.10454e6i	0.931814	−	1.61395i	0.780219	−	0.625506i	$-0.215108\pi$
$480$		0			0
$481$	530557.	+	918951.i	0.104561	+	0.181105i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.59502e6	+	7.95881e6i	0.887019	+	1.53636i
$486$		0			0
$487$	1.31470e6	−	2.27712e6i	0.251191	−	0.435075i	−0.712663	−	0.701506i	$-0.752511\pi$
$487$	1.31470e6	−	2.27712e6i	0.251191	−	0.435075i	0.963854	+	0.266431i	$0.0858445\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.81856e6			0.902015			0.451008	−	0.892520i	$-0.351065\pi$
$491$	4.81856e6			0.902015			0.451008	+	0.892520i	$0.351065\pi$
$492$		0			0
$493$	2.73899e6	−	4.74407e6i	0.507543	−	0.879091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.70248e6	+	5.15596e6i	0.309166	+	0.936308i
$498$		0			0
$499$	2.09549e6	+	3.62949e6i	0.376733	+	0.652521i	0.990585	−	0.136900i	$-0.0437139\pi$
$499$	2.09549e6	+	3.62949e6i	0.376733	+	0.652521i	−0.613851	+	0.789422i	$0.710381\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.75338e6			0.308999			0.154500	−	0.987993i	$-0.450623\pi$
$503$	1.75338e6			0.308999			0.154500	+	0.987993i	$0.450623\pi$
$504$		0			0
$505$	562395.			0.0981326
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.80923e6	+	4.86573e6i	0.480610	+	0.832441i	0.999753	−	0.0222464i	$-0.00708185\pi$
$509$	2.80923e6	+	4.86573e6i	0.480610	+	0.832441i	−0.519142	+	0.854688i	$0.673749\pi$
$510$		0			0
$511$	−4.39418e6	−	912146.i	−0.744433	−	0.154530i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.15608e6	+	2.00239e6i	−0.192075	+	0.332683i
$516$		0			0
$517$	−803844.			−0.132265
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.53589e6	−	9.58845e6i	0.893498	−	1.54758i	0.0578446	−	0.998326i	$-0.481577\pi$
$521$	5.53589e6	−	9.58845e6i	0.893498	−	1.54758i	0.835653	−	0.549258i	$-0.185089\pi$
$522$		0			0
$523$	3.79894e6	+	6.57996e6i	0.607307	+	1.05189i	0.991682	+	0.128710i	$0.0410836\pi$
$523$	3.79894e6	+	6.57996e6i	0.607307	+	1.05189i	−0.384375	+	0.923177i	$0.625583\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.84055e6	+	8.38407e6i	0.759220	+	1.31501i
$528$		0			0
$529$	−5.72588e6	+	9.91752e6i	−0.889618	+	1.54086i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.59535e6			0.243242
$534$		0			0
$535$	6.31575e6	−	1.09392e7i	0.953982	−	1.65235i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−470532.	−	4.10046e6i	−0.0697618	−	0.607940i
$540$		0			0
$541$	1.67539e6	+	2.90187e6i	0.246107	+	0.426270i	0.962442	−	0.271486i	$-0.0875152\pi$
$541$	1.67539e6	+	2.90187e6i	0.246107	+	0.426270i	−0.716335	+	0.697756i	$0.754182\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.08480e6			−1.16595
$546$		0			0
$547$	−1.00856e7			−1.44123			−0.720615	−	0.693335i	$-0.756140\pi$
$547$	−1.00856e7			−1.44123			−0.720615	+	0.693335i	$0.756140\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.14950e6	−	1.23833e7i	−1.00322	−	1.73763i
$552$		0			0
$553$	6.69101e6	−	7.50272e6i	0.930419	−	1.04329i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.78337e6	−	1.17491e7i	0.926419	−	1.60460i	0.137155	−	0.990550i	$-0.456204\pi$
$557$	6.78337e6	−	1.17491e7i	0.926419	−	1.60460i	0.789263	−	0.614055i	$-0.210463\pi$
$558$		0			0
$559$	−3.13023e6			−0.423689
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.98494e6	+	3.43802e6i	−0.263922	+	0.457127i	−0.967281	−	0.253708i	$-0.918350\pi$
$563$	−1.98494e6	+	3.43802e6i	−0.263922	+	0.457127i	0.703358	+	0.710835i	$0.251683\pi$
$564$		0			0
$565$	−931185.	−	1.61286e6i	−0.122720	−	0.212557i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.08724e6	−	3.61521e6i	−0.270267	−	0.468116i	0.698663	−	0.715451i	$-0.253779\pi$
$569$	−2.08724e6	−	3.61521e6i	−0.270267	−	0.468116i	−0.968930	+	0.247335i	$0.920445\pi$
$570$		0			0
$571$	−1.92833e6	+	3.33996e6i	−0.247509	+	0.428698i	−0.962834	−	0.270094i	$-0.912945\pi$
$571$	−1.92833e6	+	3.33996e6i	−0.247509	+	0.428698i	0.715325	+	0.698792i	$0.246279\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.96956e6			0.626828
$576$		0			0
$577$	1.12288e6	−	1.94488e6i	0.140408	−	0.243194i	−0.787242	−	0.616644i	$-0.788492\pi$
$577$	1.12288e6	−	1.94488e6i	0.140408	−	0.243194i	0.927650	+	0.373450i	$0.121825\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.10178e6	−	1.24222e7i	−0.504118	−	1.52672i
$582$		0			0
$583$	−371023.	−	642630.i	−0.0452094	−	0.0783050i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.40082e6			−0.766726			−0.383363	−	0.923598i	$-0.625234\pi$
$587$	−6.40082e6			−0.766726			−0.383363	+	0.923598i	$0.625234\pi$
$588$		0			0
$589$	2.52702e7			3.00138
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.56031e6	+	9.63074e6i	0.649325	+	1.12466i	0.983284	+	0.182077i	$0.0582820\pi$
$593$	5.56031e6	+	9.63074e6i	0.649325	+	1.12466i	−0.333959	+	0.942588i	$0.608385\pi$
$594$		0			0
$595$	−2.93818e6	−	8.89828e6i	−0.340241	−	1.03042i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.56707e6	+	1.13745e7i	−0.747833	+	1.29528i	0.201027	+	0.979586i	$0.435572\pi$
$599$	−6.56707e6	+	1.13745e7i	−0.747833	+	1.29528i	−0.948860	+	0.315698i	$0.897761\pi$
$600$		0			0
$601$	−7.88546e6			−0.890514			−0.445257	−	0.895403i	$-0.646888\pi$
$601$	−7.88546e6			−0.890514			−0.445257	+	0.895403i	$0.646888\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.30312e6	+	5.72117e6i	−0.366890	+	0.635472i
$606$		0			0
$607$	3.82967e6	+	6.63319e6i	0.421881	+	0.730719i	0.996123	−	0.0879660i	$-0.0280367\pi$
$607$	3.82967e6	+	6.63319e6i	0.421881	+	0.730719i	−0.574243	+	0.818685i	$0.694703\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	711738.	+	1.23277e6i	0.0771289	+	0.133591i
$612$		0			0
$613$	7.62066e6	−	1.31994e7i	0.819108	−	1.41874i	−0.0872322	−	0.996188i	$-0.527802\pi$
$613$	7.62066e6	−	1.31994e7i	0.819108	−	1.41874i	0.906340	−	0.422549i	$-0.138864\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.35844e7			−1.43657			−0.718285	−	0.695749i	$-0.755073\pi$
$617$	−1.35844e7			−1.43657			−0.718285	+	0.695749i	$0.755073\pi$
$618$		0			0
$619$	−3.18709e6	+	5.52019e6i	−0.334324	+	0.579066i	−0.983355	−	0.181696i	$-0.941841\pi$
$619$	−3.18709e6	+	5.52019e6i	−0.334324	+	0.579066i	0.649031	+	0.760762i	$0.275175\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.51911e6	+	3.94602e6i	−0.363256	+	0.407323i
$624$		0			0
$625$	6.02844e6	+	1.04416e7i	0.617312	+	1.06922i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.68967e6			−0.271065
$630$		0			0
$631$	−1.42736e7			−1.42712			−0.713561	−	0.700593i	$-0.752919\pi$
$631$	−1.42736e7			−1.42712			−0.713561	+	0.700593i	$0.752919\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.54751e6	−	2.68037e6i	−0.152300	−	0.263792i
$636$		0			0
$637$	−5.87180e6	+	4.35222e6i	−0.573354	+	0.424974i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.63480e6	+	8.02771e6i	−0.445539	+	0.771697i	−0.998090	−	0.0617828i	$-0.980321\pi$
$641$	−4.63480e6	+	8.02771e6i	−0.445539	+	0.771697i	0.552550	+	0.833480i	$0.313655\pi$
$642$		0			0
$643$	1.17375e7			1.11956			0.559780	−	0.828641i	$-0.310885\pi$
$643$	1.17375e7			1.11956			0.559780	+	0.828641i	$0.310885\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.12724e6	−	8.88063e6i	0.481529	−	0.834033i	−0.518246	−	0.855232i	$-0.673415\pi$
$647$	5.12724e6	−	8.88063e6i	0.481529	−	0.834033i	0.999775	+	0.0211984i	$0.00674818\pi$
$648$		0			0
$649$	−6.32200e6	−	1.09500e7i	−0.589173	−	1.02048i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.14489e6	+	7.17915e6i	0.380391	+	0.658856i	0.991118	−	0.132985i	$-0.0424562\pi$
$653$	4.14489e6	+	7.17915e6i	0.380391	+	0.658856i	−0.610727	+	0.791841i	$0.709123\pi$
$654$		0			0
$655$	−2.61672e6	+	4.53230e6i	−0.238317	+	0.412776i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.06731e7			1.85435			0.927174	−	0.374631i	$-0.122230\pi$
$659$	2.06731e7			1.85435			0.927174	+	0.374631i	$0.122230\pi$
$660$		0			0
$661$	−97171.2	+	168305.i	−0.00865035	+	0.0149829i	−0.870318	−	0.492490i	$-0.836087\pi$
$661$	−97171.2	+	168305.i	−0.00865035	+	0.0149829i	0.861668	+	0.507473i	$0.169420\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.39498e7	−	4.97151e6i	−2.10014	−	0.435948i
$666$		0			0
$667$	1.05093e7	+	1.82026e7i	0.914659	+	1.58424i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.26940e6			−0.280325
$672$		0			0
$673$	−1.14437e7			−0.973929			−0.486965	−	0.873422i	$-0.661896\pi$
$673$	−1.14437e7			−0.973929			−0.486965	+	0.873422i	$0.661896\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.51869e6	+	7.82660e6i	0.378914	+	0.656299i	0.990905	−	0.134567i	$-0.0429642\pi$
$677$	4.51869e6	+	7.82660e6i	0.378914	+	0.656299i	−0.611990	+	0.790865i	$0.709631\pi$
$678$		0			0
$679$	5.69679e6	+	1.72527e7i	0.474193	+	1.43609i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5.59212e6	+	9.68584e6i	−0.458696	+	0.794485i	−0.998892	−	0.0470541i	$-0.985017\pi$
$683$	−5.59212e6	+	9.68584e6i	−0.458696	+	0.794485i	0.540196	+	0.841539i	$0.318350\pi$
$684$		0			0
$685$	−5.64361e6			−0.459548
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−657020.	+	1.13799e6i	−0.0527267	+	0.0913253i
$690$		0			0
$691$	−2.83465e6	−	4.90976e6i	−0.225842	−	0.391170i	0.730730	−	0.682667i	$-0.239180\pi$
$691$	−2.83465e6	−	4.90976e6i	−0.225842	−	0.391170i	−0.956572	+	0.291497i	$0.905847\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.87177e6	+	1.53664e7i	0.696704	+	1.20673i
$696$		0			0
$697$	−2.02192e6	+	3.50207e6i	−0.157646	+	0.273050i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.31822e6			0.101320			0.0506599	−	0.998716i	$-0.483868\pi$
$701$	1.31822e6			0.101320			0.0506599	+	0.998716i	$0.483868\pi$
$702$		0			0
$703$	−3.51038e6	+	6.08016e6i	−0.267896	+	0.464009i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.08866e6	+	225984.i	0.0819112	+	0.0170032i
$708$		0			0
$709$	−4.13018e6	−	7.15368e6i	−0.308570	−	0.534458i	0.669480	−	0.742830i	$-0.266517\pi$
$709$	−4.13018e6	−	7.15368e6i	−0.308570	−	0.534458i	−0.978050	+	0.208372i	$0.933184\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.71456e7			−2.73643
$714$		0			0
$715$	−7.00290e6			−0.512287
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.13372e7	+	1.96365e7i	0.817865	+	1.41658i	0.907252	+	0.420587i	$0.138176\pi$
$719$	1.13372e7	+	1.96365e7i	0.817865	+	1.41658i	−0.0893870	+	0.995997i	$0.528491\pi$
$720$		0			0
$721$	−3.04250e6	+	3.41159e6i	−0.217968	+	0.244410i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.91963e6	−	5.05694e6i	0.206292	−	0.357308i
$726$		0			0
$727$	−1.93477e7			−1.35767			−0.678833	−	0.734293i	$-0.737514\pi$
$727$	−1.93477e7			−1.35767			−0.678833	+	0.734293i	$0.737514\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.96720e6	−	6.87140e6i	0.274594	−	0.475611i
$732$		0			0
$733$	−7.42922e6	−	1.28678e7i	−0.510720	−	0.884593i	−0.999923	−	0.0124232i	$-0.996045\pi$
$733$	−7.42922e6	−	1.28678e7i	−0.510720	−	0.884593i	0.489203	−	0.872170i	$-0.337288\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.79360e6	−	6.57071e6i	−0.257266	−	0.445598i
$738$		0			0
$739$	8.84089e6	−	1.53129e7i	0.595504	−	1.03144i	−0.397971	−	0.917398i	$-0.630286\pi$
$739$	8.84089e6	−	1.53129e7i	0.595504	−	1.03144i	0.993476	−	0.114045i	$-0.0363810\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.28756e6			0.550750			0.275375	−	0.961337i	$-0.411198\pi$
$743$	8.28756e6			0.550750			0.275375	+	0.961337i	$0.411198\pi$
$744$		0			0
$745$	5.65735e6	−	9.79882e6i	0.373442	−	0.646820i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.66214e7	−	1.86378e7i	1.08259	−	1.21392i
$750$		0			0
$751$	1.15682e7	+	2.00368e7i	0.748457	+	1.29637i	0.948562	+	0.316592i	$0.102539\pi$
$751$	1.15682e7	+	2.00368e7i	0.748457	+	1.29637i	−0.200104	+	0.979775i	$0.564128\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.17375e6			0.0749389
$756$		0			0
$757$	−1.95475e7			−1.23980			−0.619900	−	0.784681i	$-0.712827\pi$
$757$	−1.95475e7			−1.23980			−0.619900	+	0.784681i	$0.712827\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.91233e6	+	6.77635e6i	0.244891	+	0.424164i	0.962101	−	0.272693i	$-0.0879144\pi$
$761$	3.91233e6	+	6.77635e6i	0.244891	+	0.424164i	−0.717210	+	0.696857i	$0.754581\pi$
$762$		0			0
$763$	−1.56502e7	−	3.24867e6i	−0.973214	−	0.202020i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.11952e7	+	1.93907e7i	−0.687139	+	1.19016i
$768$		0			0
$769$	−8.27325e6			−0.504499			−0.252250	−	0.967662i	$-0.581170\pi$
$769$	−8.27325e6			−0.504499			−0.252250	+	0.967662i	$0.581170\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.41844e6	+	1.63132e7i	−0.566931	+	0.981953i	0.429936	+	0.902859i	$0.358536\pi$
$773$	−9.41844e6	+	1.63132e7i	−0.566931	+	0.981953i	−0.996867	+	0.0790941i	$0.974797\pi$
$774$		0			0
$775$	5.15978e6	+	8.93700e6i	0.308587	+	0.534488i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.27775e6	+	9.14133e6i	0.311605	+	0.539717i
$780$		0			0
$781$	−5.14267e6	+	8.90737e6i	−0.301690	+	0.522543i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.17390e7			0.679920
$786$		0			0
$787$	9.19734e6	−	1.59303e7i	0.529329	−	0.916824i	−0.470086	−	0.882621i	$-0.655777\pi$
$787$	9.19734e6	−	1.59303e7i	0.529329	−	0.916824i	0.999415	−	0.0342037i	$-0.0108895\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.15446e6	−	3.49628e6i	−0.0656051	−	0.198685i
$792$		0			0
$793$	2.89479e6	+	5.01392e6i	0.163468	+	0.283135i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.70208e6			−0.206443			−0.103222	−	0.994658i	$-0.532915\pi$
$797$	−3.70208e6			−0.206443			−0.103222	+	0.994658i	$0.532915\pi$
$798$		0			0
$799$	−3.60818e6			−0.199950
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.25056e6	−	7.36219e6i	−0.232626	−	0.402920i
$804$		0			0
$805$	3.52047e7	+	7.30780e6i	1.91474	+	0.397463i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.09563e6	−	1.05579e7i	0.327451	−	0.567163i	−0.654554	−	0.756015i	$-0.727144\pi$
$809$	6.09563e6	−	1.05579e7i	0.327451	−	0.567163i	0.982005	+	0.188853i	$0.0604769\pi$
$810$		0			0
$811$	−2.52433e6			−0.134770			−0.0673851	−	0.997727i	$-0.521466\pi$
$811$	−2.52433e6			−0.134770			−0.0673851	+	0.997727i	$0.521466\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.78004e6	+	1.34754e7i	−0.410287	+	0.710638i
$816$		0			0
$817$	−1.03555e7	−	1.79362e7i	−0.542768	−	0.940102i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.75026e6	−	8.22769e6i	−0.245957	−	0.426010i	0.716443	−	0.697645i	$-0.245769\pi$
$821$	−4.75026e6	−	8.22769e6i	−0.245957	−	0.426010i	−0.962400	+	0.271635i	$0.912436\pi$
$822$		0			0
$823$	1.13168e7	−	1.96013e7i	0.582403	−	1.00875i	−0.412791	−	0.910826i	$-0.635446\pi$
$823$	1.13168e7	−	1.96013e7i	0.582403	−	1.00875i	0.995194	−	0.0979259i	$-0.0312208\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.09007e7			−0.554230			−0.277115	−	0.960837i	$-0.589378\pi$
$827$	−1.09007e7			−0.554230			−0.277115	+	0.960837i	$0.589378\pi$
$828$		0			0
$829$	1.18060e7	−	2.04486e7i	0.596647	−	1.03342i	−0.396665	−	0.917963i	$-0.629833\pi$
$829$	1.18060e7	−	2.04486e7i	0.596647	−	1.03342i	0.993312	−	0.115459i	$-0.0368340\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.11206e6	−	1.84055e7i	−0.105461	−	0.919043i
$834$		0			0
$835$	3.08331e6	+	5.34044e6i	0.153038	+	0.265070i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.67438e7			−1.31165			−0.655826	−	0.754912i	$-0.727679\pi$
$839$	−2.67438e7			−1.31165			−0.655826	+	0.754912i	$0.727679\pi$
$840$		0			0
$841$	4.18580e6			0.204075
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.97315e6	−	1.03458e7i	−0.287781	−	0.498451i
$846$		0			0
$847$	−8.69294e6	+	9.74751e6i	−0.416349	+	0.466858i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.16004e6	−	8.93745e6i	0.244247	−	0.423048i
$852$		0			0
$853$	5.00667e6			0.235600			0.117800	−	0.993037i	$-0.462416\pi$
$853$	5.00667e6			0.235600			0.117800	+	0.993037i	$0.462416\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.21812e7	−	2.10984e7i	0.566549	−	0.981292i	−0.430355	−	0.902660i	$-0.641612\pi$
$857$	1.21812e7	−	2.10984e7i	0.566549	−	0.981292i	0.996904	−	0.0786317i	$-0.0250551\pi$
$858$		0			0
$859$	1.49577e7	+	2.59074e7i	0.691641	+	1.19796i	0.971300	+	0.237858i	$0.0764454\pi$
$859$	1.49577e7	+	2.59074e7i	0.691641	+	1.19796i	−0.279659	+	0.960100i	$0.590221\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.30052e7	−	2.25256e7i	−0.594414	−	1.02955i	−0.993629	−	0.112697i	$-0.964051\pi$
$863$	−1.30052e7	−	2.25256e7i	−0.594414	−	1.02955i	0.399216	−	0.916857i	$-0.369282\pi$
$864$		0			0
$865$	−1.60235e7	+	2.77536e7i	−0.728145	+	1.26118i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.90427e7			0.855419
$870$		0			0
$871$	−6.71784e6	+	1.16356e7i	−0.300044	+	0.519691i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.19776e6	+	1.57414e7i	0.229507	+	0.695061i
$876$		0			0
$877$	1.00687e7	+	1.74396e7i	0.442055	+	0.765661i	0.997842	−	0.0656634i	$-0.0209164\pi$
$877$	1.00687e7	+	1.74396e7i	0.442055	+	0.765661i	−0.555787	+	0.831325i	$0.687583\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3.69393e7			1.60343			0.801713	−	0.597710i	$-0.203922\pi$
$881$	3.69393e7			1.60343			0.801713	+	0.597710i	$0.203922\pi$
$882$		0			0
$883$	2.27031e7			0.979904			0.489952	−	0.871749i	$-0.337014\pi$
$883$	2.27031e7			0.979904			0.489952	+	0.871749i	$0.337014\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.79894e6	−	1.35082e7i	−0.332833	−	0.576484i	0.650233	−	0.759735i	$-0.274671\pi$
$887$	−7.79894e6	−	1.35082e7i	−0.332833	−	0.576484i	−0.983066	+	0.183251i	$0.941338\pi$
$888$		0			0
$889$	−1.91857e6	−	5.81038e6i	−0.0814185	−	0.246575i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.70915e6	+	8.15649e6i	−0.197612	+	0.342275i
$894$		0			0
$895$	5.33117e7			2.22467
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.18231e7	+	3.77988e7i	−0.900571	+	1.55983i
$900$		0			0
$901$	−1.66539e6	−	2.88454e6i	−0.0683446	−	0.118376i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.09168e7	−	1.89085e7i	−0.443073	−	0.767425i
$906$		0			0
$907$	9.67425e6	−	1.67563e7i	0.390480	−	0.676332i	−0.602032	−	0.798472i	$-0.705642\pi$
$907$	9.67425e6	−	1.67563e7i	0.390480	−	0.676332i	0.992513	+	0.122140i	$0.0389756\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.84661e7			1.53561			0.767807	−	0.640682i	$-0.221348\pi$
$911$	3.84661e7			1.53561			0.767807	+	0.640682i	$0.221348\pi$
$912$		0			0
$913$	1.23902e7	−	2.14605e7i	0.491928	−	0.852045i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.88652e6	+	7.72195e6i	−0.270443	+	0.303252i
$918$		0			0
$919$	−9.79189e6	−	1.69600e7i	−0.382453	−	0.662427i	0.608960	−	0.793201i	$-0.291587\pi$
$919$	−9.79189e6	−	1.69600e7i	−0.382453	−	0.662427i	−0.991412	+	0.130774i	$0.958254\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.82137e7			0.703709
$924$		0			0
$925$	−2.86706e6			−0.110175
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.41281e6	−	1.11073e7i	−0.243786	−	0.422250i	0.718003	−	0.696040i	$-0.245056\pi$
$929$	−6.41281e6	−	1.11073e7i	−0.243786	−	0.422250i	−0.961790	+	0.273790i	$0.911723\pi$
$930$		0			0
$931$	−4.43633e7	−	1.92473e7i	−1.67745	−	0.727771i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.87535e6	−	1.53726e7i	0.332014	−	0.575065i
$936$		0			0
$937$	5.60430e6			0.208532			0.104266	−	0.994549i	$-0.466751\pi$
$937$	5.60430e6			0.208532			0.104266	+	0.994549i	$0.466751\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.72174e7	−	2.98214e7i	0.633861	−	1.09788i	−0.352895	−	0.935663i	$-0.614803\pi$
$941$	1.72174e7	−	2.98214e7i	0.633861	−	1.09788i	0.986755	−	0.162216i	$-0.0518640\pi$
$942$		0			0
$943$	−7.75795e6	−	1.34372e7i	−0.284098	−	0.492072i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.55313e6	−	6.15420e6i	−0.128747	−	0.222996i	0.794445	−	0.607337i	$-0.207762\pi$
$947$	−3.55313e6	−	6.15420e6i	−0.128747	−	0.222996i	−0.923191	+	0.384341i	$0.874429\pi$
$948$		0			0
$949$	−7.52705e6	+	1.30372e7i	−0.271306	+	0.469916i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.78058e7			−1.34842			−0.674212	−	0.738538i	$-0.735517\pi$
$953$	−3.78058e7			−1.34842			−0.674212	+	0.738538i	$0.735517\pi$
$954$		0			0
$955$	4.73079e6	−	8.19396e6i	0.167851	−	0.290727i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.09246e7	−	2.26774e6i	−0.383584	−	0.0796246i
$960$		0			0
$961$	−2.42529e7	−	4.20072e7i	−0.847140	−	1.46729i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.86641e7			2.02793
$966$		0			0
$967$	1.33643e7			0.459601			0.229800	−	0.973238i	$-0.426193\pi$
$967$	1.33643e7			0.459601			0.229800	+	0.973238i	$0.426193\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.34334e7	+	4.05878e7i	0.797604	+	1.38149i	0.921172	+	0.389155i	$0.127233\pi$
$971$	2.34334e7	+	4.05878e7i	0.797604	+	1.38149i	−0.123568	+	0.992336i	$0.539434\pi$
$972$		0			0
$973$	1.09990e7	+	3.33104e7i	0.372452	+	1.12797i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.12969e7	−	3.68874e7i	0.713807	−	1.23635i	−0.249611	−	0.968346i	$-0.580303\pi$
$977$	2.12969e7	−	3.68874e7i	0.713807	−	1.23635i	0.963418	−	0.268004i	$-0.0863639\pi$
$978$		0			0
$979$	−1.00154e7			−0.333974
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.60288e7	−	4.50833e7i	0.859154	−	1.48810i	−0.0135836	−	0.999908i	$-0.504324\pi$
$983$	2.60288e7	−	4.50833e7i	0.859154	−	1.48810i	0.872737	−	0.488190i	$-0.162343\pi$
$984$		0			0
$985$	1.96547e7	+	3.40429e7i	0.645468	+	1.11798i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.52219e7	+	2.63650e7i	0.494854	+	0.857112i
$990$		0			0
$991$	−1.45768e7	+	2.52477e7i	−0.471495	+	0.816652i	−0.999468	−	0.0326083i	$-0.989619\pi$
$991$	−1.45768e7	+	2.52477e7i	−0.471495	+	0.816652i	0.527974	+	0.849261i	$0.322952\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.10338e7			1.63418
$996$		0			0
$997$	−4.49652e6	+	7.78820e6i	−0.143264	+	0.248141i	−0.928724	−	0.370772i	$-0.879093\pi$
$997$	−4.49652e6	+	7.78820e6i	−0.143264	+	0.248141i	0.785460	+	0.618913i	$0.212427\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.6.k.e.109.2		4
3.2	odd	2		84.6.i.b.25.1	✓	4
7.2	even	3	inner	252.6.k.e.37.2		4
12.11	even	2		336.6.q.g.193.1		4
21.2	odd	6		84.6.i.b.37.1	yes	4
21.5	even	6		588.6.i.m.373.2		4
21.11	odd	6		588.6.a.l.1.2		2
21.17	even	6		588.6.a.h.1.1		2
21.20	even	2		588.6.i.m.361.2		4
84.23	even	6		336.6.q.g.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.b.25.1	✓	4	3.2	odd	2
84.6.i.b.37.1	yes	4	21.2	odd	6
252.6.k.e.37.2		4	7.2	even	3	inner
252.6.k.e.109.2		4	1.1	even	1	trivial
336.6.q.g.193.1		4	12.11	even	2
336.6.q.g.289.1		4	84.23	even	6
588.6.a.h.1.1		2	21.17	even	6
588.6.a.l.1.2		2	21.11	odd	6
588.6.i.m.361.2		4	21.20	even	2
588.6.i.m.373.2		4	21.5	even	6

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	252.6.k.e.109.2

Level	$252$
Weight	$6$
Character	252.109
Analytic conductor	$40.417$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$