LMFDB - Embedded newform 324.4.e.a.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$324 = 2^{2} \cdot 3^{4}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	324.e (of order $3$ , degree $2$ , 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:	$19.1166188419$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.4.e.a.217.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.00000 + 15.5885i) q^{5} +(-4.00000 - 6.92820i) q^{7} +(18.0000 + 31.1769i) q^{11} +(5.00000 - 8.66025i) q^{13} -18.0000 q^{17} -100.000 q^{19} +(36.0000 - 62.3538i) q^{23} +(-99.5000 - 172.339i) q^{25} +(-117.000 - 202.650i) q^{29} +(8.00000 - 13.8564i) q^{31} +144.000 q^{35} -226.000 q^{37} +(45.0000 - 77.9423i) q^{41} +(-226.000 - 391.443i) q^{43} +(216.000 + 374.123i) q^{47} +(139.500 - 241.621i) q^{49} -414.000 q^{53} -648.000 q^{55} +(-342.000 + 592.361i) q^{59} +(-211.000 - 365.463i) q^{61} +(90.0000 + 155.885i) q^{65} +(-166.000 + 287.520i) q^{67} +360.000 q^{71} +26.0000 q^{73} +(144.000 - 249.415i) q^{77} +(-256.000 - 443.405i) q^{79} +(-594.000 - 1028.84i) q^{83} +(162.000 - 280.592i) q^{85} +630.000 q^{89} -80.0000 q^{91} +(900.000 - 1558.85i) q^{95} +(527.000 + 912.791i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 18 q^{5} - 8 q^{7} + 36 q^{11} + 10 q^{13} - 36 q^{17} - 200 q^{19} + 72 q^{23} - 199 q^{25} - 234 q^{29} + 16 q^{31} + 288 q^{35} - 452 q^{37} + 90 q^{41} - 452 q^{43} + 432 q^{47} + 279 q^{49} - 828 q^{53}+ \cdots + 1054 q^{97}+O(q^{100})$ 2 * q - 18 * q^5 - 8 * q^7 + 36 * q^11 + 10 * q^13 - 36 * q^17 - 200 * q^19 + 72 * q^23 - 199 * q^25 - 234 * q^29 + 16 * q^31 + 288 * q^35 - 452 * q^37 + 90 * q^41 - 452 * q^43 + 432 * q^47 + 279 * q^49 - 828 * q^53 - 1296 * q^55 - 684 * q^59 - 422 * q^61 + 180 * q^65 - 332 * q^67 + 720 * q^71 + 52 * q^73 + 288 * q^77 - 512 * q^79 - 1188 * q^83 + 324 * q^85 + 1260 * q^89 - 160 * q^91 + 1800 * q^95 + 1054 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$ .

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−9.00000	+	15.5885i	−0.804984	+	1.39427i	0.111317	+	0.993785i	$0.464493\pi$
$5$	−9.00000	+	15.5885i	−0.804984	+	1.39427i	−0.916302	+	0.400489i	$0.868840\pi$
$6$		0			0
$7$	−4.00000	−	6.92820i	−0.215980	−	0.374088i	0.737595	−	0.675243i	$-0.235961\pi$
$7$	−4.00000	−	6.92820i	−0.215980	−	0.374088i	−0.953575	+	0.301155i	$0.902628\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18.0000	+	31.1769i	0.493382	+	0.854563i	0.999971	−	0.00762479i	$-0.00242707\pi$
$11$	18.0000	+	31.1769i	0.493382	+	0.854563i	−0.506589	+	0.862188i	$0.669094\pi$
$12$		0			0
$13$	5.00000	−	8.66025i	0.106673	−	0.184763i	−0.807747	−	0.589529i	$-0.799313\pi$
$13$	5.00000	−	8.66025i	0.106673	−	0.184763i	0.914421	+	0.404765i	$0.132647\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−18.0000			−0.256802			−0.128401	−	0.991722i	$-0.540985\pi$
$17$	−18.0000			−0.256802			−0.128401	+	0.991722i	$0.540985\pi$
$18$		0			0
$19$	−100.000			−1.20745			−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000			−1.20745			−0.603726	+	0.797192i	$0.706318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	36.0000	−	62.3538i	0.326370	−	0.565290i	−0.655418	−	0.755266i	$-0.727508\pi$
$23$	36.0000	−	62.3538i	0.326370	−	0.565290i	0.981789	+	0.189976i	$0.0608410\pi$
$24$		0			0
$25$	−99.5000	−	172.339i	−0.796000	−	1.37871i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−117.000	−	202.650i	−0.749185	−	1.29763i	−0.948214	−	0.317632i	$-0.897112\pi$
$29$	−117.000	−	202.650i	−0.749185	−	1.29763i	0.199029	−	0.979994i	$-0.436221\pi$
$30$		0			0
$31$	8.00000	−	13.8564i	0.0463498	−	0.0802801i	−0.841920	−	0.539603i	$-0.818574\pi$
$31$	8.00000	−	13.8564i	0.0463498	−	0.0802801i	0.888270	+	0.459323i	$0.151908\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	144.000			0.695441
$36$		0			0
$37$	−226.000			−1.00417			−0.502083	−	0.864819i	$-0.667433\pi$
$37$	−226.000			−1.00417			−0.502083	+	0.864819i	$0.667433\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	45.0000	−	77.9423i	0.171410	−	0.296891i	−0.767503	−	0.641045i	$-0.778501\pi$
$41$	45.0000	−	77.9423i	0.171410	−	0.296891i	0.938913	+	0.344154i	$0.111834\pi$
$42$		0			0
$43$	−226.000	−	391.443i	−0.801504	−	1.38825i	−0.918626	−	0.395128i	$-0.870700\pi$
$43$	−226.000	−	391.443i	−0.801504	−	1.38825i	0.117122	−	0.993118i	$-0.462633\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	216.000	+	374.123i	0.670358	+	1.16109i	0.977803	+	0.209528i	$0.0671929\pi$
$47$	216.000	+	374.123i	0.670358	+	1.16109i	−0.307444	+	0.951566i	$0.599474\pi$
$48$		0			0
$49$	139.500	−	241.621i	0.406706	−	0.704435i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−414.000			−1.07297			−0.536484	−	0.843911i	$-0.680248\pi$
$53$	−414.000			−1.07297			−0.536484	+	0.843911i	$0.680248\pi$
$54$		0			0
$55$	−648.000			−1.58866
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−342.000	+	592.361i	−0.754654	+	1.30710i	0.190892	+	0.981611i	$0.438862\pi$
$59$	−342.000	+	592.361i	−0.754654	+	1.30710i	−0.945546	+	0.325489i	$0.894471\pi$
$60$		0			0
$61$	−211.000	−	365.463i	−0.442882	−	0.767094i	0.555020	−	0.831837i	$-0.312710\pi$
$61$	−211.000	−	365.463i	−0.442882	−	0.767094i	−0.997902	+	0.0647433i	$0.979377\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	90.0000	+	155.885i	0.171740	+	0.297463i
$66$		0			0
$67$	−166.000	+	287.520i	−0.302688	+	0.524272i	−0.976744	−	0.214409i	$-0.931218\pi$
$67$	−166.000	+	287.520i	−0.302688	+	0.524272i	0.674056	+	0.738681i	$0.264551\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	360.000			0.601748			0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000			0.601748			0.300874	+	0.953664i	$0.402722\pi$
$72$		0			0
$73$	26.0000			0.0416859			0.0208429	−	0.999783i	$-0.493365\pi$
$73$	26.0000			0.0416859			0.0208429	+	0.999783i	$0.493365\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	144.000	−	249.415i	0.213121	−	0.369137i
$78$		0			0
$79$	−256.000	−	443.405i	−0.364585	−	0.631481i	0.624124	−	0.781325i	$-0.285456\pi$
$79$	−256.000	−	443.405i	−0.364585	−	0.631481i	−0.988710	+	0.149845i	$0.952123\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−594.000	−	1028.84i	−0.785542	−	1.36060i	−0.928675	−	0.370895i	$-0.879051\pi$
$83$	−594.000	−	1028.84i	−0.785542	−	1.36060i	0.143133	−	0.989703i	$-0.454282\pi$
$84$		0			0
$85$	162.000	−	280.592i	0.206722	−	0.358053i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	630.000			0.750336			0.375168	−	0.926957i	$-0.377585\pi$
$89$	630.000			0.750336			0.375168	+	0.926957i	$0.377585\pi$
$90$		0			0
$91$	−80.0000			−0.0921569
$92$		0			0
$93$		0			0
$94$		0			0
$95$	900.000	−	1558.85i	0.971979	−	1.68352i
$96$		0			0
$97$	527.000	+	912.791i	0.551637	+	0.955462i	0.998157	+	0.0606889i	$0.0193298\pi$
$97$	527.000	+	912.791i	0.551637	+	0.955462i	−0.446520	+	0.894774i	$0.647337\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	279.000	+	483.242i	0.274867	+	0.476083i	0.970101	−	0.242700i	$-0.0780329\pi$
$101$	279.000	+	483.242i	0.274867	+	0.476083i	−0.695235	+	0.718783i	$0.744700\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.00382652	+	0.00662773i	−0.867932	−	0.496682i	$-0.834551\pi$
$103$	−4.00000	+	6.92820i	−0.00382652	+	0.00662773i	0.864106	+	0.503310i	$0.167885\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1764.00			−1.59376			−0.796880	−	0.604138i	$-0.793518\pi$
$107$	−1764.00			−1.59376			−0.796880	+	0.604138i	$0.793518\pi$
$108$		0			0
$109$	1622.00			1.42532			0.712658	−	0.701512i	$-0.247491\pi$
$109$	1622.00			1.42532			0.712658	+	0.701512i	$0.247491\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−567.000	+	982.073i	−0.472025	+	0.817572i	−0.999488	−	0.0320065i	$-0.989810\pi$
$113$	−567.000	+	982.073i	−0.472025	+	0.817572i	0.527462	+	0.849578i	$0.323144\pi$
$114$		0			0
$115$	648.000	+	1122.37i	0.525446	+	0.910099i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	72.0000	+	124.708i	0.0554641	+	0.0960667i
$120$		0			0
$121$	17.5000	−	30.3109i	0.0131480	−	0.0227730i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1332.00			0.953102
$126$		0			0
$127$	−592.000			−0.413634			−0.206817	−	0.978380i	$-0.566310\pi$
$127$	−592.000			−0.413634			−0.206817	+	0.978380i	$0.566310\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−954.000	+	1652.38i	−0.636270	+	1.10205i	0.349975	+	0.936759i	$0.386190\pi$
$131$	−954.000	+	1652.38i	−0.636270	+	1.10205i	−0.986245	+	0.165293i	$0.947143\pi$
$132$		0			0
$133$	400.000	+	692.820i	0.260785	+	0.451693i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	477.000	+	826.188i	0.297466	+	0.515226i	0.975556	−	0.219753i	$-0.0705251\pi$
$137$	477.000	+	826.188i	0.297466	+	0.515226i	−0.678089	+	0.734979i	$0.737192\pi$
$138$		0			0
$139$	−1282.00	+	2220.49i	−0.782286	+	1.35496i	0.148320	+	0.988939i	$0.452613\pi$
$139$	−1282.00	+	2220.49i	−0.782286	+	1.35496i	−0.930607	+	0.366021i	$0.880720\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	360.000			0.210522
$144$		0			0
$145$	4212.00			2.41233
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−369.000	+	639.127i	−0.202884	+	0.351405i	−0.949456	−	0.313899i	$-0.898365\pi$
$149$	−369.000	+	639.127i	−0.202884	+	0.351405i	0.746573	+	0.665304i	$0.231698\pi$
$150$		0			0
$151$	1220.00	+	2113.10i	0.657498	+	1.13882i	0.981261	+	0.192682i	$0.0617185\pi$
$151$	1220.00	+	2113.10i	0.657498	+	1.13882i	−0.323763	+	0.946138i	$0.604948\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	144.000	+	249.415i	0.0746217	+	0.129249i
$156$		0			0
$157$	1277.00	−	2211.83i	0.649145	−	1.12435i	−0.334183	−	0.942508i	$-0.608460\pi$
$157$	1277.00	−	2211.83i	0.649145	−	1.12435i	0.983328	−	0.181843i	$-0.0582063\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−576.000			−0.281958
$162$		0			0
$163$	−820.000			−0.394033			−0.197016	−	0.980400i	$-0.563125\pi$
$163$	−820.000			−0.394033			−0.197016	+	0.980400i	$0.563125\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	972.000	−	1683.55i	0.450393	−	0.780103i	−0.548017	−	0.836467i	$-0.684617\pi$
$167$	972.000	−	1683.55i	0.450393	−	0.780103i	0.998410	+	0.0563635i	$0.0179506\pi$
$168$		0			0
$169$	1048.50	+	1816.06i	0.477242	+	0.826607i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−621.000	−	1075.60i	−0.272912	−	0.472697i	0.696694	−	0.717368i	$-0.254653\pi$
$173$	−621.000	−	1075.60i	−0.272912	−	0.472697i	−0.969606	+	0.244671i	$0.921320\pi$
$174$		0			0
$175$	−796.000	+	1378.71i	−0.343840	+	0.595548i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1116.00			−0.465999			−0.232999	−	0.972477i	$-0.574854\pi$
$179$	−1116.00			−0.465999			−0.232999	+	0.972477i	$0.574854\pi$
$180$		0			0
$181$	1070.00			0.439406			0.219703	−	0.975567i	$-0.429491\pi$
$181$	1070.00			0.439406			0.219703	+	0.975567i	$0.429491\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2034.00	−	3522.99i	0.808339	−	1.40008i
$186$		0			0
$187$	−324.000	−	561.184i	−0.126702	−	0.219454i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−288.000	−	498.831i	−0.109104	−	0.188974i	0.806303	−	0.591502i	$-0.201465\pi$
$191$	−288.000	−	498.831i	−0.109104	−	0.188974i	−0.915408	+	0.402528i	$0.868132\pi$
$192$		0			0
$193$	671.000	−	1162.21i	0.250257	−	0.433458i	−0.713339	−	0.700819i	$-0.752818\pi$
$193$	671.000	−	1162.21i	0.250257	−	0.433458i	0.963597	+	0.267361i	$0.0861515\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1422.00			−0.514281			−0.257140	−	0.966374i	$-0.582780\pi$
$197$	−1422.00			−0.514281			−0.257140	+	0.966374i	$0.582780\pi$
$198$		0			0
$199$	872.000			0.310625			0.155313	−	0.987865i	$-0.450361\pi$
$199$	872.000			0.310625			0.155313	+	0.987865i	$0.450361\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−936.000	+	1621.20i	−0.323617	+	0.560522i
$204$		0			0
$205$	810.000	+	1402.96i	0.275965	+	0.477986i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1800.00	−	3117.69i	−0.595735	−	1.03184i
$210$		0			0
$211$	−670.000	+	1160.47i	−0.218600	+	0.378627i	−0.954380	−	0.298594i	$-0.903482\pi$
$211$	−670.000	+	1160.47i	−0.218600	+	0.378627i	0.735780	+	0.677221i	$0.236816\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8136.00			2.58079
$216$		0			0
$217$	−128.000			−0.0400424
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−90.0000	+	155.885i	−0.0273939	+	0.0474477i
$222$		0			0
$223$	−2440.00	−	4226.20i	−0.732711	−	1.26909i	−0.955720	−	0.294276i	$-0.904922\pi$
$223$	−2440.00	−	4226.20i	−0.732711	−	1.26909i	0.223010	−	0.974816i	$-0.428412\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1350.00	+	2338.27i	0.394725	+	0.683684i	0.993066	−	0.117557i	$-0.0375064\pi$
$227$	1350.00	+	2338.27i	0.394725	+	0.683684i	−0.598341	+	0.801242i	$0.704173\pi$
$228$		0			0
$229$	−127.000	+	219.970i	−0.0366480	+	0.0634762i	−0.883768	−	0.467926i	$-0.845001\pi$
$229$	−127.000	+	219.970i	−0.0366480	+	0.0634762i	0.847120	+	0.531402i	$0.178335\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4410.00			−1.23995			−0.619976	−	0.784621i	$-0.712858\pi$
$233$	−4410.00			−1.23995			−0.619976	+	0.784621i	$0.712858\pi$
$234$		0			0
$235$	−7776.00			−2.15851
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1944.00	+	3367.11i	−0.526138	+	0.911297i	0.473399	+	0.880848i	$0.343027\pi$
$239$	−1944.00	+	3367.11i	−0.526138	+	0.911297i	−0.999536	+	0.0304489i	$0.990306\pi$
$240$		0			0
$241$	−2569.00	−	4449.64i	−0.686655	−	1.18932i	−0.972914	−	0.231169i	$-0.925745\pi$
$241$	−2569.00	−	4449.64i	−0.686655	−	1.18932i	0.286259	−	0.958152i	$-0.407588\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2511.00	+	4349.18i	0.654783	+	1.13412i
$246$		0			0
$247$	−500.000	+	866.025i	−0.128803	+	0.223093i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4788.00			−1.20405			−0.602024	−	0.798478i	$-0.705639\pi$
$251$	−4788.00			−1.20405			−0.602024	+	0.798478i	$0.705639\pi$
$252$		0			0
$253$	2592.00			0.644101
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2943.00	+	5097.43i	−0.714316	+	1.23723i	0.248906	+	0.968528i	$0.419929\pi$
$257$	−2943.00	+	5097.43i	−0.714316	+	1.23723i	−0.963223	+	0.268705i	$0.913404\pi$
$258$		0			0
$259$	904.000	+	1565.77i	0.216880	+	0.375647i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1116.00	+	1932.97i	0.261656	+	0.453201i	0.966682	−	0.255980i	$-0.0823981\pi$
$263$	1116.00	+	1932.97i	0.261656	+	0.453201i	−0.705026	+	0.709181i	$0.749065\pi$
$264$		0			0
$265$	3726.00	−	6453.62i	0.863722	−	1.49601i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	666.000			0.150954			0.0754772	−	0.997148i	$-0.475952\pi$
$269$	666.000			0.150954			0.0754772	+	0.997148i	$0.475952\pi$
$270$		0			0
$271$	−5536.00			−1.24092			−0.620458	−	0.784240i	$-0.713053\pi$
$271$	−5536.00			−1.24092			−0.620458	+	0.784240i	$0.713053\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3582.00	−	6204.21i	0.785464	−	1.36046i
$276$		0			0
$277$	−1063.00	−	1841.17i	−0.230576	−	0.399369i	0.727402	−	0.686212i	$-0.240728\pi$
$277$	−1063.00	−	1841.17i	−0.230576	−	0.399369i	−0.957978	+	0.286843i	$0.907394\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1467.00	−	2540.92i	−0.311437	−	0.539425i	0.667236	−	0.744846i	$-0.267477\pi$
$281$	−1467.00	−	2540.92i	−0.311437	−	0.539425i	−0.978674	+	0.205421i	$0.934144\pi$
$282$		0			0
$283$	−1018.00	+	1763.23i	−0.213830	+	0.370364i	−0.952910	−	0.303253i	$-0.901927\pi$
$283$	−1018.00	+	1763.23i	−0.213830	+	0.370364i	0.739080	+	0.673617i	$0.235260\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−720.000			−0.148085
$288$		0			0
$289$	−4589.00			−0.934053
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1143.00	−	1979.73i	0.227900	−	0.394735i	−0.729285	−	0.684210i	$-0.760147\pi$
$293$	1143.00	−	1979.73i	0.227900	−	0.394735i	0.957186	+	0.289475i	$0.0934807\pi$
$294$		0			0
$295$	−6156.00	−	10662.5i	−1.21497	−	2.10439i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−360.000	−	623.538i	−0.0696299	−	0.120603i
$300$		0			0
$301$	−1808.00	+	3131.55i	−0.346217	+	0.599666i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7596.00			1.42605
$306$		0			0
$307$	1244.00			0.231267			0.115633	−	0.993292i	$-0.463110\pi$
$307$	1244.00			0.231267			0.115633	+	0.993292i	$0.463110\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	612.000	−	1060.02i	0.111586	−	0.193273i	−0.804824	−	0.593514i	$-0.797740\pi$
$311$	612.000	−	1060.02i	0.111586	−	0.193273i	0.916410	+	0.400241i	$0.131074\pi$
$312$		0			0
$313$	−949.000	−	1643.72i	−0.171376	−	0.296832i	0.767525	−	0.641019i	$-0.221488\pi$
$313$	−949.000	−	1643.72i	−0.171376	−	0.296832i	−0.938901	+	0.344187i	$0.888155\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4581.00	−	7934.52i	−0.811655	−	1.40583i	−0.911705	−	0.410845i	$-0.865234\pi$
$317$	−4581.00	−	7934.52i	−0.811655	−	1.40583i	0.100050	−	0.994982i	$-0.468100\pi$
$318$		0			0
$319$	4212.00	−	7295.40i	0.739269	−	1.28045i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1800.00			0.310076
$324$		0			0
$325$	−1990.00			−0.339647
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1728.00	−	2992.98i	0.289568	−	0.501546i
$330$		0			0
$331$	2174.00	+	3765.48i	0.361009	+	0.625285i	0.988127	−	0.153639i	$-0.0490992\pi$
$331$	2174.00	+	3765.48i	0.361009	+	0.625285i	−0.627119	+	0.778924i	$0.715766\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2988.00	−	5175.37i	−0.487319	−	0.844061i
$336$		0			0
$337$	−3577.00	+	6195.55i	−0.578195	+	1.00146i	0.417492	+	0.908681i	$0.362909\pi$
$337$	−3577.00	+	6195.55i	−0.578195	+	1.00146i	−0.995686	+	0.0927819i	$0.970424\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	576.000			0.0914726
$342$		0			0
$343$	−4976.00			−0.783320
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−918.000	+	1590.02i	−0.142020	+	0.245985i	−0.928257	−	0.371939i	$-0.878693\pi$
$347$	−918.000	+	1590.02i	−0.142020	+	0.245985i	0.786237	+	0.617925i	$0.212026\pi$
$348$		0			0
$349$	−2947.00	−	5104.35i	−0.452004	−	0.782893i	0.546507	−	0.837455i	$-0.315957\pi$
$349$	−2947.00	−	5104.35i	−0.452004	−	0.782893i	−0.998510	+	0.0545613i	$0.982624\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5553.00	+	9618.08i	0.837270	+	1.45019i	0.892169	+	0.451703i	$0.149183\pi$
$353$	5553.00	+	9618.08i	0.837270	+	1.45019i	−0.0548984	+	0.998492i	$0.517484\pi$
$354$		0			0
$355$	−3240.00	+	5611.84i	−0.484398	+	0.839002i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−13176.0			−1.93705			−0.968527	−	0.248907i	$-0.919929\pi$
$359$	−13176.0			−1.93705			−0.968527	+	0.248907i	$0.919929\pi$
$360$		0			0
$361$	3141.00			0.457938
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−234.000	+	405.300i	−0.0335565	+	0.0581216i
$366$		0			0
$367$	3056.00	+	5293.15i	0.434665	+	0.752861i	0.997268	−	0.0738656i	$-0.0235336\pi$
$367$	3056.00	+	5293.15i	0.434665	+	0.752861i	−0.562604	+	0.826727i	$0.690200\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1656.00	+	2868.28i	0.231739	+	0.401384i
$372$		0			0
$373$	6809.00	−	11793.5i	0.945192	−	1.63712i	0.189826	−	0.981818i	$-0.439208\pi$
$373$	6809.00	−	11793.5i	0.945192	−	1.63712i	0.755366	−	0.655303i	$-0.227459\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2340.00			−0.319671
$378$		0			0
$379$	692.000			0.0937880			0.0468940	−	0.998900i	$-0.485068\pi$
$379$	692.000			0.0937880			0.0468940	+	0.998900i	$0.485068\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4032.00	+	6983.63i	−0.537926	+	0.931715i	0.461090	+	0.887353i	$0.347459\pi$
$383$	−4032.00	+	6983.63i	−0.537926	+	0.931715i	−0.999016	+	0.0443613i	$0.985875\pi$
$384$		0			0
$385$	2592.00	+	4489.48i	0.343118	+	0.594298i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6327.00	+	10958.7i	0.824657	+	1.42835i	0.902181	+	0.431358i	$0.141965\pi$
$389$	6327.00	+	10958.7i	0.824657	+	1.42835i	−0.0775239	+	0.996990i	$0.524701\pi$
$390$		0			0
$391$	−648.000	+	1122.37i	−0.0838127	+	0.145168i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9216.00			1.17394
$396$		0			0
$397$	−106.000			−0.0134005			−0.00670024	−	0.999978i	$-0.502133\pi$
$397$	−106.000			−0.0134005			−0.00670024	+	0.999978i	$0.502133\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2007.00	+	3476.23i	−0.249937	+	0.432904i	−0.963508	−	0.267679i	$-0.913743\pi$
$401$	−2007.00	+	3476.23i	−0.249937	+	0.432904i	0.713571	+	0.700583i	$0.247077\pi$
$402$		0			0
$403$	−80.0000	−	138.564i	−0.00988855	−	0.0171275i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4068.00	−	7045.98i	−0.495438	−	0.858124i
$408$		0			0
$409$	−1957.00	+	3389.62i	−0.236595	+	0.409795i	−0.959735	−	0.280907i	$-0.909365\pi$
$409$	−1957.00	+	3389.62i	−0.236595	+	0.409795i	0.723140	+	0.690702i	$0.242698\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5472.00			0.651960
$414$		0			0
$415$	21384.0			2.52940
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2214.00	−	3834.76i	0.258141	−	0.447113i	−0.707603	−	0.706610i	$-0.750224\pi$
$419$	2214.00	−	3834.76i	0.258141	−	0.447113i	0.965744	+	0.259497i	$0.0835568\pi$
$420$		0			0
$421$	7745.00	+	13414.7i	0.896599	+	1.55296i	0.831813	+	0.555056i	$0.187303\pi$
$421$	7745.00	+	13414.7i	0.896599	+	1.55296i	0.0647860	+	0.997899i	$0.479364\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1791.00	+	3102.10i	0.204415	+	0.354057i
$426$		0			0
$427$	−1688.00	+	2923.70i	−0.191307	+	0.331353i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6768.00			−0.756388			−0.378194	−	0.925726i	$-0.623455\pi$
$431$	−6768.00			−0.756388			−0.378194	+	0.925726i	$0.623455\pi$
$432$		0			0
$433$	1298.00			0.144060			0.0720299	−	0.997402i	$-0.477052\pi$
$433$	1298.00			0.144060			0.0720299	+	0.997402i	$0.477052\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3600.00	+	6235.38i	−0.394076	+	0.682560i
$438$		0			0
$439$	1124.00	+	1946.83i	0.122200	+	0.211656i	0.920635	−	0.390425i	$-0.127672\pi$
$439$	1124.00	+	1946.83i	0.122200	+	0.211656i	−0.798435	+	0.602081i	$0.794339\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4806.00	−	8324.24i	−0.515440	−	0.892768i	−0.999839	−	0.0179214i	$-0.994295\pi$
$443$	−4806.00	−	8324.24i	−0.515440	−	0.892768i	0.484399	−	0.874847i	$-0.339038\pi$
$444$		0			0
$445$	−5670.00	+	9820.73i	−0.604008	+	1.04617i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−162.000			−0.0170273			−0.00851364	−	0.999964i	$-0.502710\pi$
$449$	−162.000			−0.0170273			−0.00851364	+	0.999964i	$0.502710\pi$
$450$		0			0
$451$	3240.00			0.338283
$452$		0			0
$453$		0			0
$454$		0			0
$455$	720.000	−	1247.08i	0.0741849	−	0.128492i
$456$		0			0
$457$	−685.000	−	1186.45i	−0.0701159	−	0.121444i	0.828836	−	0.559492i	$-0.189004\pi$
$457$	−685.000	−	1186.45i	−0.0701159	−	0.121444i	−0.898952	+	0.438047i	$0.855670\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7677.00	−	13297.0i	−0.775604	−	1.34339i	−0.934454	−	0.356083i	$-0.884112\pi$
$461$	−7677.00	−	13297.0i	−0.775604	−	1.34339i	0.158850	−	0.987303i	$-0.449221\pi$
$462$		0			0
$463$	6512.00	−	11279.1i	0.653646	−	1.13215i	−0.328585	−	0.944474i	$-0.606572\pi$
$463$	6512.00	−	11279.1i	0.653646	−	1.13215i	0.982231	−	0.187674i	$-0.0600950\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14436.0			1.43045			0.715223	−	0.698896i	$-0.246325\pi$
$467$	14436.0			1.43045			0.715223	+	0.698896i	$0.246325\pi$
$468$		0			0
$469$	2656.00			0.261498
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8136.00	−	14092.0i	0.790896	−	1.36987i
$474$		0			0
$475$	9950.00	+	17233.9i	0.961131	+	1.66473i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6048.00	+	10475.4i	0.576911	+	0.999238i	0.995831	+	0.0912156i	$0.0290752\pi$
$479$	6048.00	+	10475.4i	0.576911	+	0.999238i	−0.418921	+	0.908023i	$0.637591\pi$
$480$		0			0
$481$	−1130.00	+	1957.22i	−0.107118	+	0.185533i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18972.0			−1.77624
$486$		0			0
$487$	6056.00			0.563498			0.281749	−	0.959488i	$-0.409085\pi$
$487$	6056.00			0.563498			0.281749	+	0.959488i	$0.409085\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3762.00	−	6515.98i	0.345777	−	0.598904i	−0.639717	−	0.768610i	$-0.720949\pi$
$491$	3762.00	−	6515.98i	0.345777	−	0.598904i	0.985495	+	0.169706i	$0.0542819\pi$
$492$		0			0
$493$	2106.00	+	3647.70i	0.192392	+	0.333233i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1440.00	−	2494.15i	−0.129965	−	0.225107i
$498$		0			0
$499$	−2638.00	+	4569.15i	−0.236660	+	0.409906i	−0.959754	−	0.280843i	$-0.909386\pi$
$499$	−2638.00	+	4569.15i	−0.236660	+	0.409906i	0.723094	+	0.690749i	$0.242719\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4968.00			−0.440382			−0.220191	−	0.975457i	$-0.570668\pi$
$503$	−4968.00			−0.440382			−0.220191	+	0.975457i	$0.570668\pi$
$504$		0			0
$505$	−10044.0			−0.885054
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5499.00	−	9524.55i	0.478858	−	0.829407i	−0.520848	−	0.853650i	$-0.674384\pi$
$509$	5499.00	−	9524.55i	0.478858	−	0.829407i	0.999706	+	0.0242426i	$0.00771742\pi$
$510$		0			0
$511$	−104.000	−	180.133i	−0.00900331	−	0.0155942i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−72.0000	−	124.708i	−0.00616058	−	0.0106704i
$516$		0			0
$517$	−7776.00	+	13468.4i	−0.661486	+	1.14573i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8838.00			0.743186			0.371593	−	0.928396i	$-0.378812\pi$
$521$	8838.00			0.743186			0.371593	+	0.928396i	$0.378812\pi$
$522$		0			0
$523$	22436.0			1.87583			0.937914	−	0.346869i	$-0.112755\pi$
$523$	22436.0			1.87583			0.937914	+	0.346869i	$0.112755\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−144.000	+	249.415i	−0.0119027	+	0.0206161i
$528$		0			0
$529$	3491.50	+	6047.46i	0.286965	+	0.497038i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−450.000	−	779.423i	−0.0365697	−	0.0633406i
$534$		0			0
$535$	15876.0	−	27498.0i	1.28295	−	2.22214i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10044.0			0.802645
$540$		0			0
$541$	−4762.00			−0.378437			−0.189218	−	0.981935i	$-0.560595\pi$
$541$	−4762.00			−0.378437			−0.189218	+	0.981935i	$0.560595\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14598.0	+	25284.5i	−1.14736	+	1.98728i
$546$		0			0
$547$	3002.00	+	5199.62i	0.234655	+	0.406434i	0.959172	−	0.282822i	$-0.0912706\pi$
$547$	3002.00	+	5199.62i	0.234655	+	0.406434i	−0.724517	+	0.689257i	$0.757937\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	11700.0	+	20265.0i	0.904604	+	1.56682i
$552$		0			0
$553$	−2048.00	+	3547.24i	−0.157486	+	0.272774i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5274.00			0.401197			0.200598	−	0.979674i	$-0.435711\pi$
$557$	5274.00			0.401197			0.200598	+	0.979674i	$0.435711\pi$
$558$		0			0
$559$	−4520.00			−0.341996
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6210.00	+	10756.0i	−0.464867	+	0.805174i	−0.999196	−	0.0401033i	$-0.987231\pi$
$563$	−6210.00	+	10756.0i	−0.464867	+	0.805174i	0.534328	+	0.845277i	$0.320565\pi$
$564$		0			0
$565$	−10206.0	−	17677.3i	−0.759946	−	1.31627i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10683.0	−	18503.5i	−0.787091	−	1.36328i	−0.927742	−	0.373222i	$-0.878253\pi$
$569$	−10683.0	−	18503.5i	−0.787091	−	1.36328i	0.140651	−	0.990059i	$-0.455080\pi$
$570$		0			0
$571$	−10570.0	+	18307.8i	−0.774677	+	1.34178i	0.160298	+	0.987069i	$0.448754\pi$
$571$	−10570.0	+	18307.8i	−0.774677	+	1.34178i	−0.934976	+	0.354712i	$0.884579\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−14328.0			−1.03916
$576$		0			0
$577$	3266.00			0.235642			0.117821	−	0.993035i	$-0.462409\pi$
$577$	3266.00			0.235642			0.117821	+	0.993035i	$0.462409\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4752.00	+	8230.71i	−0.339322	+	0.587723i
$582$		0			0
$583$	−7452.00	−	12907.2i	−0.529383	−	0.916918i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8514.00	+	14746.7i	0.598655	+	1.03690i	0.993020	+	0.117947i	$0.0376312\pi$
$587$	8514.00	+	14746.7i	0.598655	+	1.03690i	−0.394365	+	0.918954i	$0.629035\pi$
$588$		0			0
$589$	−800.000	+	1385.64i	−0.0559651	+	0.0969343i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9522.00			−0.659396			−0.329698	−	0.944086i	$-0.606947\pi$
$593$	−9522.00			−0.659396			−0.329698	+	0.944086i	$0.606947\pi$
$594$		0			0
$595$	−2592.00			−0.178591
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5148.00	+	8916.60i	−0.351155	+	0.608218i	−0.986452	−	0.164050i	$-0.947544\pi$
$599$	−5148.00	+	8916.60i	−0.351155	+	0.608218i	0.635297	+	0.772268i	$0.280877\pi$
$600$		0			0
$601$	1691.00	+	2928.90i	0.114771	+	0.198789i	0.917688	−	0.397301i	$-0.130053\pi$
$601$	1691.00	+	2928.90i	0.114771	+	0.198789i	−0.802917	+	0.596091i	$0.796720\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	315.000	+	545.596i	0.0211679	+	0.0366639i
$606$		0			0
$607$	10328.0	−	17888.6i	0.690611	−	1.19617i	−0.281028	−	0.959700i	$-0.590675\pi$
$607$	10328.0	−	17888.6i	0.690611	−	1.19617i	0.971638	−	0.236473i	$-0.0759914\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4320.00			0.286037
$612$		0			0
$613$	−22114.0			−1.45706			−0.728529	−	0.685015i	$-0.759795\pi$
$613$	−22114.0			−1.45706			−0.728529	+	0.685015i	$0.759795\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9981.00	−	17287.6i	0.651248	−	1.12799i	−0.331573	−	0.943430i	$-0.607579\pi$
$617$	9981.00	−	17287.6i	0.651248	−	1.12799i	0.982820	−	0.184565i	$-0.0590875\pi$
$618$		0			0
$619$	302.000	+	523.079i	0.0196097	+	0.0339650i	0.875664	−	0.482921i	$-0.160424\pi$
$619$	302.000	+	523.079i	0.0196097	+	0.0339650i	−0.856054	+	0.516886i	$0.827091\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2520.00	−	4364.77i	−0.162057	−	0.280691i
$624$		0			0
$625$	449.500	−	778.557i	0.0287680	−	0.0498276i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4068.00			0.257872
$630$		0			0
$631$	152.000			0.00958958			0.00479479	−	0.999989i	$-0.498474\pi$
$631$	152.000			0.00958958			0.00479479	+	0.999989i	$0.498474\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5328.00	−	9228.37i	0.332969	−	0.576719i
$636$		0			0
$637$	−1395.00	−	2416.21i	−0.0867691	−	0.150288i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2097.00	+	3632.11i	0.129215	+	0.223806i	0.923372	−	0.383905i	$-0.125421\pi$
$641$	2097.00	+	3632.11i	0.129215	+	0.223806i	−0.794158	+	0.607711i	$0.792088\pi$
$642$		0			0
$643$	3626.00	−	6280.42i	0.222388	−	0.385187i	−0.733145	−	0.680073i	$-0.761948\pi$
$643$	3626.00	−	6280.42i	0.222388	−	0.385187i	0.955533	+	0.294885i	$0.0952815\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6696.00			0.406873			0.203437	−	0.979088i	$-0.434789\pi$
$647$	6696.00			0.406873			0.203437	+	0.979088i	$0.434789\pi$
$648$		0			0
$649$	−24624.0			−1.48933
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14211.0	−	24614.2i	0.851638	−	1.47508i	−0.0280919	−	0.999605i	$-0.508943\pi$
$653$	14211.0	−	24614.2i	0.851638	−	1.47508i	0.879730	−	0.475474i	$-0.157724\pi$
$654$		0			0
$655$	−17172.0	−	29742.8i	−1.02437	−	1.77427i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9954.00	−	17240.8i	−0.588396	−	1.01913i	−0.994443	−	0.105279i	$-0.966426\pi$
$659$	−9954.00	−	17240.8i	−0.588396	−	1.01913i	0.406047	−	0.913852i	$-0.366907\pi$
$660$		0			0
$661$	−7159.00	+	12399.8i	−0.421260	+	0.729644i	−0.996063	−	0.0886482i	$-0.971745\pi$
$661$	−7159.00	+	12399.8i	−0.421260	+	0.729644i	0.574803	+	0.818292i	$0.305079\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−14400.0			−0.839711
$666$		0			0
$667$	−16848.0			−0.978047
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7596.00	−	13156.7i	0.437020	−	0.756941i
$672$		0			0
$673$	−15025.0	−	26024.1i	−0.860581	−	1.49057i	−0.871369	−	0.490629i	$-0.836767\pi$
$673$	−15025.0	−	26024.1i	−0.860581	−	1.49057i	0.0107875	−	0.999942i	$-0.496566\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11079.0	+	19189.4i	0.628952	+	1.08938i	0.987762	+	0.155966i	$0.0498492\pi$
$677$	11079.0	+	19189.4i	0.628952	+	1.08938i	−0.358810	+	0.933411i	$0.616817\pi$
$678$		0			0
$679$	4216.00	−	7302.33i	0.238285	−	0.412721i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3132.00			0.175465			0.0877325	−	0.996144i	$-0.472038\pi$
$683$	3132.00			0.175465			0.0877325	+	0.996144i	$0.472038\pi$
$684$		0			0
$685$	−17172.0			−0.957822
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2070.00	+	3585.35i	−0.114457	+	0.198245i
$690$		0			0
$691$	10466.0	+	18127.6i	0.576187	+	0.997986i	0.995912	+	0.0903340i	$0.0287934\pi$
$691$	10466.0	+	18127.6i	0.576187	+	0.997986i	−0.419724	+	0.907652i	$0.637873\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−23076.0	−	39968.8i	−1.25946	−	2.18144i
$696$		0			0
$697$	−810.000	+	1402.96i	−0.0440186	+	0.0762424i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21834.0			1.17640			0.588202	−	0.808714i	$-0.299836\pi$
$701$	21834.0			1.17640			0.588202	+	0.808714i	$0.299836\pi$
$702$		0			0
$703$	22600.0			1.21248
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2232.00	−	3865.94i	0.118731	−	0.205649i
$708$		0			0
$709$	−6223.00	−	10778.6i	−0.329633	−	0.570941i	0.652806	−	0.757525i	$-0.273592\pi$
$709$	−6223.00	−	10778.6i	−0.329633	−	0.570941i	−0.982439	+	0.186584i	$0.940258\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−576.000	−	997.661i	−0.0302544	−	0.0524021i
$714$		0			0
$715$	−3240.00	+	5611.84i	−0.169467	+	0.293526i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12528.0			0.649813			0.324907	−	0.945746i	$-0.394667\pi$
$719$	12528.0			0.649813			0.324907	+	0.945746i	$0.394667\pi$
$720$		0			0
$721$	64.0000			0.00330580
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−23283.0	+	40327.3i	−1.19270	+	2.06582i
$726$		0			0
$727$	−5788.00	−	10025.1i	−0.295275	−	0.511432i	0.679774	−	0.733422i	$-0.262078\pi$
$727$	−5788.00	−	10025.1i	−0.295275	−	0.511432i	−0.975049	+	0.221990i	$0.928745\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4068.00	+	7045.98i	0.205828	+	0.356505i
$732$		0			0
$733$	14669.0	−	25407.5i	0.739170	−	1.28028i	−0.213699	−	0.976900i	$-0.568551\pi$
$733$	14669.0	−	25407.5i	0.739170	−	1.28028i	0.952869	−	0.303381i	$-0.0981155\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11952.0			−0.597364
$738$		0			0
$739$	2540.00			0.126435			0.0632175	−	0.998000i	$-0.479864\pi$
$739$	2540.00			0.126435			0.0632175	+	0.998000i	$0.479864\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9396.00	+	16274.3i	−0.463938	+	0.803564i	−0.999153	−	0.0411519i	$-0.986897\pi$
$743$	−9396.00	+	16274.3i	−0.463938	+	0.803564i	0.535215	+	0.844716i	$0.320231\pi$
$744$		0			0
$745$	−6642.00	−	11504.3i	−0.326636	−	0.565751i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7056.00	+	12221.4i	0.344220	+	0.596206i
$750$		0			0
$751$	−2416.00	+	4184.63i	−0.117392	+	0.203328i	−0.918733	−	0.394879i	$-0.870787\pi$
$751$	−2416.00	+	4184.63i	−0.117392	+	0.203328i	0.801342	+	0.598207i	$0.204120\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−43920.0			−2.11710
$756$		0			0
$757$	−20818.0			−0.999529			−0.499764	−	0.866161i	$-0.666580\pi$
$757$	−20818.0			−0.999529			−0.499764	+	0.866161i	$0.666580\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6021.00	−	10428.7i	0.286808	−	0.496766i	−0.686238	−	0.727377i	$-0.740739\pi$
$761$	6021.00	−	10428.7i	0.286808	−	0.496766i	0.973046	+	0.230611i	$0.0740724\pi$
$762$		0			0
$763$	−6488.00	−	11237.5i	−0.307839	−	0.533193i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3420.00	+	5923.61i	0.161003	+	0.278865i
$768$		0			0
$769$	−6529.00	+	11308.6i	−0.306166	+	0.530295i	−0.977520	−	0.210841i	$-0.932380\pi$
$769$	−6529.00	+	11308.6i	−0.306166	+	0.530295i	0.671354	+	0.741137i	$0.265713\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11826.0			0.550261			0.275130	−	0.961407i	$-0.411279\pi$
$773$	11826.0			0.550261			0.275130	+	0.961407i	$0.411279\pi$
$774$		0			0
$775$	−3184.00			−0.147578
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4500.00	+	7794.23i	−0.206969	+	0.358482i
$780$		0			0
$781$	6480.00	+	11223.7i	0.296892	+	0.514232i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22986.0	+	39812.9i	1.04510	+	1.81017i
$786$		0			0
$787$	−5998.00	+	10388.8i	−0.271672	+	0.470549i	−0.969290	−	0.245920i	$-0.920910\pi$
$787$	−5998.00	+	10388.8i	−0.271672	+	0.470549i	0.697618	+	0.716470i	$0.254243\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9072.00			0.407792
$792$		0			0
$793$	−4220.00			−0.188974
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3483.00	−	6032.73i	0.154798	−	0.268118i	−0.778187	−	0.628032i	$-0.783861\pi$
$797$	3483.00	−	6032.73i	0.154798	−	0.268118i	0.932986	+	0.359914i	$0.117194\pi$
$798$		0			0
$799$	−3888.00	−	6734.21i	−0.172150	−	0.298172i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	468.000	+	810.600i	0.0205671	+	0.0356232i
$804$		0			0
$805$	5184.00	−	8978.95i	0.226971	−	0.393126i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	40806.0			1.77338			0.886689	−	0.462367i	$-0.153000\pi$
$809$	40806.0			1.77338			0.886689	+	0.462367i	$0.153000\pi$
$810$		0			0
$811$	−17980.0			−0.778500			−0.389250	−	0.921132i	$-0.627266\pi$
$811$	−17980.0			−0.778500			−0.389250	+	0.921132i	$0.627266\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	7380.00	−	12782.5i	0.317190	−	0.549390i
$816$		0			0
$817$	22600.0	+	39144.3i	0.967777	+	1.67624i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6417.00	−	11114.6i	−0.272783	−	0.472474i	0.696790	−	0.717275i	$-0.254611\pi$
$821$	−6417.00	−	11114.6i	−0.272783	−	0.472474i	−0.969573	+	0.244801i	$0.921277\pi$
$822$		0			0
$823$	18932.0	−	32791.2i	0.801857	−	1.38886i	−0.116536	−	0.993186i	$-0.537179\pi$
$823$	18932.0	−	32791.2i	0.801857	−	1.38886i	0.918393	−	0.395670i	$-0.129488\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−42516.0			−1.78770			−0.893849	−	0.448368i	$-0.852005\pi$
$827$	−42516.0			−1.78770			−0.893849	+	0.448368i	$0.852005\pi$
$828$		0			0
$829$	45638.0			1.91203			0.956015	−	0.293317i	$-0.0947591\pi$
$829$	45638.0			1.91203			0.956015	+	0.293317i	$0.0947591\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2511.00	+	4349.18i	−0.104443	+	0.180901i
$834$		0			0
$835$	17496.0	+	30304.0i	0.725119	+	1.25594i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8748.00	+	15152.0i	0.359970	+	0.623486i	0.987955	−	0.154739i	$-0.0494537\pi$
$839$	8748.00	+	15152.0i	0.359970	+	0.623486i	−0.627986	+	0.778225i	$0.716120\pi$
$840$		0			0
$841$	−15183.5	+	26298.6i	−0.622555	+	1.07830i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−37746.0			−1.53669
$846$		0			0
$847$	−280.000			−0.0113588
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8136.00	+	14092.0i	−0.327730	+	0.567646i
$852$		0			0
$853$	−16087.0	−	27863.5i	−0.645731	−	1.11844i	−0.984132	−	0.177437i	$-0.943219\pi$
$853$	−16087.0	−	27863.5i	−0.645731	−	1.11844i	0.338401	−	0.941002i	$-0.390114\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19467.0	−	33717.8i	−0.775939	−	1.34397i	−0.934265	−	0.356579i	$-0.883943\pi$
$857$	−19467.0	−	33717.8i	−0.775939	−	1.34397i	0.158326	−	0.987387i	$-0.449390\pi$
$858$		0			0
$859$	−14890.0	+	25790.2i	−0.591432	+	1.02439i	0.402608	+	0.915373i	$0.368104\pi$
$859$	−14890.0	+	25790.2i	−0.591432	+	1.02439i	−0.994040	+	0.109018i	$0.965229\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	48096.0			1.89711			0.948556	−	0.316611i	$-0.102545\pi$
$863$	48096.0			1.89711			0.948556	+	0.316611i	$0.102545\pi$
$864$		0			0
$865$	22356.0			0.878759
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9216.00	−	15962.6i	0.359760	−	0.623122i
$870$		0			0
$871$	1660.00	+	2875.20i	0.0645774	+	0.111851i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5328.00	−	9228.37i	−0.205851	−	0.356544i
$876$		0			0
$877$	−10651.0	+	18448.1i	−0.410101	+	0.710316i	−0.994900	−	0.100862i	$-0.967840\pi$
$877$	−10651.0	+	18448.1i	−0.410101	+	0.710316i	0.584799	+	0.811178i	$0.301173\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	7470.00			0.285665			0.142832	−	0.989747i	$-0.454379\pi$
$881$	7470.00			0.285665			0.142832	+	0.989747i	$0.454379\pi$
$882$		0			0
$883$	764.000			0.0291174			0.0145587	−	0.999894i	$-0.495366\pi$
$883$	764.000			0.0291174			0.0145587	+	0.999894i	$0.495366\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16164.0	−	27996.9i	0.611876	−	1.05980i	−0.379048	−	0.925377i	$-0.623748\pi$
$887$	16164.0	−	27996.9i	0.611876	−	1.05980i	0.990924	−	0.134423i	$-0.0429182\pi$
$888$		0			0
$889$	2368.00	+	4101.50i	0.0893365	+	0.154735i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−21600.0	−	37412.3i	−0.809425	−	1.40197i
$894$		0			0
$895$	10044.0	−	17396.7i	0.375122	−	0.649730i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3744.00			−0.138898
$900$		0			0
$901$	7452.00			0.275541
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9630.00	+	16679.6i	−0.353715	+	0.612652i
$906$		0			0
$907$	18158.0	+	31450.6i	0.664748	+	1.15138i	0.979354	+	0.202155i	$0.0647944\pi$
$907$	18158.0	+	31450.6i	0.664748	+	1.15138i	−0.314605	+	0.949223i	$0.601872\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6696.00	−	11597.8i	−0.243522	−	0.421792i	0.718193	−	0.695844i	$-0.244969\pi$
$911$	−6696.00	−	11597.8i	−0.243522	−	0.421792i	−0.961715	+	0.274052i	$0.911636\pi$
$912$		0			0
$913$	21384.0	−	37038.2i	0.775145	−	1.34259i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15264.0			0.549686
$918$		0			0
$919$	38072.0			1.36657			0.683286	−	0.730151i	$-0.260550\pi$
$919$	38072.0			1.36657			0.683286	+	0.730151i	$0.260550\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1800.00	−	3117.69i	0.0641904	−	0.111181i
$924$		0			0
$925$	22487.0	+	38948.6i	0.799317	+	1.38446i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6399.00	−	11083.4i	−0.225990	−	0.391426i	0.730626	−	0.682778i	$-0.239228\pi$
$929$	−6399.00	−	11083.4i	−0.225990	−	0.391426i	−0.956616	+	0.291352i	$0.905895\pi$
$930$		0			0
$931$	−13950.0	+	24162.1i	−0.491077	+	0.850570i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11664.0			0.407972
$936$		0			0
$937$	34874.0			1.21588			0.607942	−	0.793981i	$-0.291995\pi$
$937$	34874.0			1.21588			0.607942	+	0.793981i	$0.291995\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8595.00	−	14887.0i	0.297757	−	0.515730i	−0.677866	−	0.735186i	$-0.737095\pi$
$941$	8595.00	−	14887.0i	0.297757	−	0.515730i	0.975622	+	0.219456i	$0.0704283\pi$
$942$		0			0
$943$	−3240.00	−	5611.84i	−0.111886	−	0.193793i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20142.0	+	34887.0i	0.691158	+	1.19712i	0.971459	+	0.237209i	$0.0762327\pi$
$947$	20142.0	+	34887.0i	0.691158	+	1.19712i	−0.280300	+	0.959912i	$0.590434\pi$
$948$		0			0
$949$	130.000	−	225.167i	0.00444676	−	0.00770202i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15498.0			−0.526789			−0.263394	−	0.964688i	$-0.584842\pi$
$953$	−15498.0			−0.526789			−0.263394	+	0.964688i	$0.584842\pi$
$954$		0			0
$955$	10368.0			0.351310
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3816.00	−	6609.51i	0.128493	−	0.222557i
$960$		0			0
$961$	14767.5	+	25578.1i	0.495703	+	0.858583i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12078.0	+	20919.7i	0.402906	+	0.697854i
$966$		0			0
$967$	−18580.0	+	32181.5i	−0.617883	+	1.07020i	0.371989	+	0.928237i	$0.378676\pi$
$967$	−18580.0	+	32181.5i	−0.617883	+	1.07020i	−0.989871	+	0.141967i	$0.954657\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18468.0			−0.610367			−0.305183	−	0.952294i	$-0.598718\pi$
$971$	−18468.0			−0.610367			−0.305183	+	0.952294i	$0.598718\pi$
$972$		0			0
$973$	20512.0			0.675832
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5193.00	−	8994.54i	0.170050	−	0.294535i	−0.768387	−	0.639985i	$-0.778940\pi$
$977$	5193.00	−	8994.54i	0.170050	−	0.294535i	0.938437	+	0.345450i	$0.112274\pi$
$978$		0			0
$979$	11340.0	+	19641.5i	0.370202	+	0.641209i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22068.0	+	38222.9i	0.716032	+	1.24020i	0.962560	+	0.271069i	$0.0873771\pi$
$983$	22068.0	+	38222.9i	0.716032	+	1.24020i	−0.246528	+	0.969136i	$0.579290\pi$
$984$		0			0
$985$	12798.0	−	22166.8i	0.413988	−	0.717048i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32544.0			−1.04635
$990$		0			0
$991$	−28432.0			−0.911375			−0.455687	−	0.890140i	$-0.650606\pi$
$991$	−28432.0			−0.911375			−0.455687	+	0.890140i	$0.650606\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7848.00	+	13593.1i	−0.250049	+	0.433097i
$996$		0			0
$997$	19889.0	+	34448.8i	0.631786	+	1.09429i	0.987186	+	0.159572i	$0.0510113\pi$
$997$	19889.0	+	34448.8i	0.631786	+	1.09429i	−0.355400	+	0.934714i	$0.615655\pi$
$998$		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	324.4.e.a.109.1		2
3.2	odd	2		324.4.e.h.109.1		2
9.2	odd	6		324.4.e.h.217.1		2
9.4	even	3		36.4.a.a.1.1		1
9.5	odd	6		12.4.a.a.1.1	✓	1
9.7	even	3	inner	324.4.e.a.217.1		2
36.23	even	6		48.4.a.a.1.1		1
36.31	odd	6		144.4.a.g.1.1		1
45.4	even	6		900.4.a.g.1.1		1
45.13	odd	12		900.4.d.c.649.1		2
45.14	odd	6		300.4.a.b.1.1		1
45.22	odd	12		900.4.d.c.649.2		2
45.23	even	12		300.4.d.e.49.2		2
45.32	even	12		300.4.d.e.49.1		2
63.4	even	3		1764.4.k.b.1549.1		2
63.5	even	6		588.4.i.e.361.1		2
63.13	odd	6		1764.4.a.b.1.1		1
63.23	odd	6		588.4.i.d.361.1		2
63.31	odd	6		1764.4.k.o.1549.1		2
63.32	odd	6		588.4.i.d.373.1		2
63.40	odd	6		1764.4.k.o.361.1		2
63.41	even	6		588.4.a.c.1.1		1
63.58	even	3		1764.4.k.b.361.1		2
63.59	even	6		588.4.i.e.373.1		2
72.5	odd	6		192.4.a.f.1.1		1
72.13	even	6		576.4.a.b.1.1		1
72.59	even	6		192.4.a.l.1.1		1
72.67	odd	6		576.4.a.a.1.1		1
99.32	even	6		1452.4.a.d.1.1		1
117.5	even	12		2028.4.b.c.337.2		2
117.77	odd	6		2028.4.a.c.1.1		1
117.86	even	12		2028.4.b.c.337.1		2
144.5	odd	12		768.4.d.g.385.1		2
144.59	even	12		768.4.d.j.385.2		2
144.77	odd	12		768.4.d.g.385.2		2
144.131	even	12		768.4.d.j.385.1		2
180.23	odd	12		1200.4.f.d.49.1		2
180.59	even	6		1200.4.a.be.1.1		1
180.167	odd	12		1200.4.f.d.49.2		2
252.167	odd	6		2352.4.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.4.a.a.1.1	✓	1	9.5	odd	6
36.4.a.a.1.1		1	9.4	even	3
48.4.a.a.1.1		1	36.23	even	6
144.4.a.g.1.1		1	36.31	odd	6
192.4.a.f.1.1		1	72.5	odd	6
192.4.a.l.1.1		1	72.59	even	6
300.4.a.b.1.1		1	45.14	odd	6
300.4.d.e.49.1		2	45.32	even	12
300.4.d.e.49.2		2	45.23	even	12
324.4.e.a.109.1		2	1.1	even	1	trivial
324.4.e.a.217.1		2	9.7	even	3	inner
324.4.e.h.109.1		2	3.2	odd	2
324.4.e.h.217.1		2	9.2	odd	6
576.4.a.a.1.1		1	72.67	odd	6
576.4.a.b.1.1		1	72.13	even	6
588.4.a.c.1.1		1	63.41	even	6
588.4.i.d.361.1		2	63.23	odd	6
588.4.i.d.373.1		2	63.32	odd	6
588.4.i.e.361.1		2	63.5	even	6
588.4.i.e.373.1		2	63.59	even	6
768.4.d.g.385.1		2	144.5	odd	12
768.4.d.g.385.2		2	144.77	odd	12
768.4.d.j.385.1		2	144.131	even	12
768.4.d.j.385.2		2	144.59	even	12
900.4.a.g.1.1		1	45.4	even	6
900.4.d.c.649.1		2	45.13	odd	12
900.4.d.c.649.2		2	45.22	odd	12
1200.4.a.be.1.1		1	180.59	even	6
1200.4.f.d.49.1		2	180.23	odd	12
1200.4.f.d.49.2		2	180.167	odd	12
1452.4.a.d.1.1		1	99.32	even	6
1764.4.a.b.1.1		1	63.13	odd	6
1764.4.k.b.361.1		2	63.58	even	3
1764.4.k.b.1549.1		2	63.4	even	3
1764.4.k.o.361.1		2	63.40	odd	6
1764.4.k.o.1549.1		2	63.31	odd	6
2028.4.a.c.1.1		1	117.77	odd	6
2028.4.b.c.337.1		2	117.86	even	12
2028.4.b.c.337.2		2	117.5	even	12
2352.4.a.bk.1.1		1	252.167	odd	6

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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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	324.4.e.a.109.1

Level	$324$
Weight	$4$
Character	324.109
Analytic conductor	$19.117$
Analytic rank	$1$
Dimension	$2$
Inner twists	$2$