LMFDB - Embedded newform 3744.2.c.o.3457.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3744,2,Mod(3457,3744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3744.3457"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.134560000.4
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{6} + 13x^{4} + 7x^{2} + 1 $ x^8 + 7x^6 + 13x^4 + 7*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 1248)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3457.4
Root			$-0.477260i$ of defining polynomial
Character	$\chi$	$=$	3744.3457
Dual form			3744.2.c.o.3457.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.954520i q^{5} -4.78051i q^{7} -2.64614i q^{11} +(-1.58993 + 3.23607i) q^{13} -4.47214 q^{17} -2.30838i q^{19} -8.38118 q^{23} +4.08889 q^{25} +3.90904 q^{29} -2.87147i q^{31} -4.56310 q^{35} -8.38118i q^{37} +4.95452i q^{41} +7.65199 q^{43} +10.2981i q^{47} -15.8533 q^{49} -10.3812 q^{53} -2.52580 q^{55} -4.55518i q^{59} -14.0332 q^{61} +(3.08889 + 1.51762i) q^{65} -0.399338i q^{67} +3.44482i q^{71} +12.2687i q^{73} -12.6499 q^{77} +12.7624 q^{79} -0.737102i q^{83} +4.26874i q^{85} -11.8078i q^{89} +(15.4701 + 7.60066i) q^{91} -2.20339 q^{95} +6.09096i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{23} - 16 q^{25} + 8 q^{29} - 24 q^{35} - 32 q^{49} - 24 q^{53} + 16 q^{55} + 8 q^{61} - 24 q^{65} + 32 q^{77} - 16 q^{79} + 16 q^{91} - 56 q^{95}+O(q^{100}) $ 8 * q - 8 * q^23 - 16 * q^25 + 8 * q^29 - 24 * q^35 - 32 * q^49 - 24 * q^53 + 16 * q^55 + 8 * q^61 - 24 * q^65 + 32 * q^77 - 16 * q^79 + 16 * q^91 - 56 * q^95

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	0.954520i		−	0.426874i	−0.976957	−	0.213437i	$-0.931534\pi$
$5$		−	0.954520i		−	0.426874i	0.976957	−	0.213437i	$-0.0684659\pi$
$6$		0			0
$7$		−	4.78051i		−	1.80686i	−0.428731	−	0.903432i	$-0.641039\pi$
$7$		−	4.78051i		−	1.80686i	0.428731	−	0.903432i	$-0.358961\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.64614i		−	0.797842i	−0.916985	−	0.398921i	$-0.869385\pi$
$11$		−	2.64614i		−	0.797842i	0.916985	−	0.398921i	$-0.130615\pi$
$12$		0			0
$13$	−1.58993	+	3.23607i	−0.440966	+	0.897524i
$13$	−1.58993	+	3.23607i	−0.440966	+	0.897524i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.47214			−1.08465			−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214			−1.08465			−0.542326	+	0.840168i	$0.682456\pi$
$18$		0			0
$19$		−	2.30838i		−	0.529578i	−0.964306	−	0.264789i	$-0.914698\pi$
$19$		−	2.30838i		−	0.529578i	0.964306	−	0.264789i	$-0.0853023\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.38118			−1.74760			−0.873798	−	0.486289i	$-0.838350\pi$
$23$	−8.38118			−1.74760			−0.873798	+	0.486289i	$0.838350\pi$
$24$		0			0
$25$	4.08889			0.817778
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.90904			0.725891			0.362945	−	0.931810i	$-0.381771\pi$
$29$	3.90904			0.725891			0.362945	+	0.931810i	$0.381771\pi$
$30$		0			0
$31$		−	2.87147i		−	0.515732i	−0.966181	−	0.257866i	$-0.916981\pi$
$31$		−	2.87147i		−	0.515732i	0.966181	−	0.257866i	$-0.0830193\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.56310			−0.771304
$36$		0			0
$37$		−	8.38118i		−	1.37786i	−0.724829	−	0.688928i	$-0.758081\pi$
$37$		−	8.38118i		−	1.37786i	0.724829	−	0.688928i	$-0.241919\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.95452i			0.773766i	0.922129	+	0.386883i	$0.126448\pi$
$41$			4.95452i			0.773766i	−0.922129	+	0.386883i	$0.873552\pi$
$42$		0			0
$43$	7.65199			1.16692			0.583459	−	0.812143i	$-0.301699\pi$
$43$	7.65199			1.16692			0.583459	+	0.812143i	$0.301699\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.2981i			1.50214i	0.660225	+	0.751068i	$0.270461\pi$
$47$			10.2981i			1.50214i	−0.660225	+	0.751068i	$0.729539\pi$
$48$		0			0
$49$	−15.8533			−2.26476
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.3812			−1.42596			−0.712982	−	0.701182i	$-0.752656\pi$
$53$	−10.3812			−1.42596			−0.712982	+	0.701182i	$0.752656\pi$
$54$		0			0
$55$	−2.52580			−0.340578
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	4.55518i		−	0.593034i	−0.955028	−	0.296517i	$-0.904175\pi$
$59$		−	4.55518i		−	0.593034i	0.955028	−	0.296517i	$-0.0958252\pi$
$60$		0			0
$61$	−14.0332			−1.79676			−0.898381	−	0.439217i	$-0.855256\pi$
$61$	−14.0332			−1.79676			−0.898381	+	0.439217i	$0.855256\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.08889	+	1.51762i	0.383130	+	0.188237i
$66$		0			0
$67$		−	0.399338i		−	0.0487869i	−0.999702	−	0.0243934i	$-0.992235\pi$
$67$		−	0.399338i		−	0.0487869i	0.999702	−	0.0243934i	$-0.00776544\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.44482i			0.408825i	0.978885	+	0.204412i	$0.0655283\pi$
$71$			3.44482i			0.408825i	−0.978885	+	0.204412i	$0.934472\pi$
$72$		0			0
$73$			12.2687i			1.43595i	0.696070	+	0.717974i	$0.254930\pi$
$73$			12.2687i			1.43595i	−0.696070	+	0.717974i	$0.745070\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.6499			−1.44159
$78$		0			0
$79$	12.7624			1.43588			0.717938	−	0.696107i	$-0.245086\pi$
$79$	12.7624			1.43588			0.717938	+	0.696107i	$0.245086\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	0.737102i		−	0.0809075i	−0.999181	−	0.0404537i	$-0.987120\pi$
$83$		−	0.737102i		−	0.0809075i	0.999181	−	0.0404537i	$-0.0128803\pi$
$84$		0			0
$85$			4.26874i			0.463010i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	11.8078i		−	1.25163i	−0.779972	−	0.625814i	$-0.784767\pi$
$89$		−	11.8078i		−	1.25163i	0.779972	−	0.625814i	$-0.215233\pi$
$90$		0			0
$91$	15.4701	+	7.60066i	1.62170	+	0.796766i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.20339			−0.226063
$96$		0			0
$97$			6.09096i			0.618443i	0.950990	+	0.309222i	$0.100068\pi$
$97$			6.09096i			0.618443i	−0.950990	+	0.309222i	$0.899932\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.38118			0.634951			0.317475	−	0.948267i	$-0.397165\pi$
$101$	6.38118			0.634951			0.317475	+	0.948267i	$0.397165\pi$
$102$		0			0
$103$	0.348012			0.0342907			0.0171453	−	0.999853i	$-0.494542\pi$
$103$	0.348012			0.0342907			0.0171453	+	0.999853i	$0.494542\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.02147			0.195423			0.0977116	−	0.995215i	$-0.468848\pi$
$107$	2.02147			0.195423			0.0977116	+	0.995215i	$0.468848\pi$
$108$		0			0
$109$			8.83184i			0.845937i	0.906144	+	0.422968i	$0.139012\pi$
$109$			8.83184i			0.845937i	−0.906144	+	0.422968i	$0.860988\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.8318			1.01897			0.509487	−	0.860478i	$-0.329835\pi$
$113$	10.8318			1.01897			0.509487	+	0.860478i	$0.329835\pi$
$114$		0			0
$115$			8.00000i			0.746004i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			21.3791i			1.95982i
$120$		0			0
$121$	3.99793			0.363448
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	8.67553i		−	0.775963i
$126$		0			0
$127$	−19.3040			−1.71295			−0.856475	−	0.516188i	$-0.827351\pi$
$127$	−19.3040			−1.71295			−0.856475	+	0.516188i	$0.827351\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.563096			0.0491979			0.0245990	−	0.999697i	$-0.492169\pi$
$131$	0.563096			0.0491979			0.0245990	+	0.999697i	$0.492169\pi$
$132$		0			0
$133$	−11.0352			−0.956876
$134$		0			0
$135$		0			0
$136$		0			0
$137$			14.1675i			1.21041i	0.796068	+	0.605207i	$0.206910\pi$
$137$			14.1675i			1.21041i	−0.796068	+	0.605207i	$0.793090\pi$
$138$		0			0
$139$	−1.11036			−0.0941799			−0.0470899	−	0.998891i	$-0.514995\pi$
$139$	−1.11036			−0.0941799			−0.0470899	+	0.998891i	$0.514995\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.56310	+	4.20717i	0.716082	+	0.351821i
$144$		0			0
$145$		−	3.73126i		−	0.309864i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			0.954520i			0.0781973i	0.999235	+	0.0390987i	$0.0124487\pi$
$149$			0.954520i			0.0781973i	−0.999235	+	0.0390987i	$0.987551\pi$
$150$		0			0
$151$			10.0728i			0.819713i	0.912150	+	0.409856i	$0.134421\pi$
$151$			10.0728i			0.819713i	−0.912150	+	0.409856i	$0.865579\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.74088			−0.220153
$156$		0			0
$157$	−7.90904			−0.631210			−0.315605	−	0.948891i	$-0.602207\pi$
$157$	−7.90904			−0.631210			−0.315605	+	0.948891i	$0.602207\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			40.0663i			3.15767i
$162$		0			0
$163$			3.86941i			0.303075i	0.988451	+	0.151538i	$0.0484225\pi$
$163$			3.86941i			0.303075i	−0.988451	+	0.151538i	$0.951578\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	24.8239i		−	1.92093i	−0.278396	−	0.960467i	$-0.589803\pi$
$167$		−	24.8239i		−	1.92093i	0.278396	−	0.960467i	$-0.410197\pi$
$168$		0			0
$169$	−7.94427	−	10.2902i	−0.611098	−	0.791555i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16.9228			−1.28662			−0.643308	−	0.765607i	$-0.722439\pi$
$173$	−16.9228			−1.28662			−0.643308	+	0.765607i	$0.722439\pi$
$174$		0			0
$175$		−	19.5470i		−	1.47761i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.1993			−1.21079			−0.605395	−	0.795925i	$-0.706985\pi$
$179$	−16.1993			−1.21079			−0.605395	+	0.795925i	$0.706985\pi$
$180$		0			0
$181$	−7.90904			−0.587874			−0.293937	−	0.955825i	$-0.594966\pi$
$181$	−7.90904			−0.587874			−0.293937	+	0.955825i	$0.594966\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.00000			−0.588172
$186$		0			0
$187$			11.8339i			0.865381i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.5590			1.63231			0.816154	−	0.577834i	$-0.196102\pi$
$191$	22.5590			1.63231			0.816154	+	0.577834i	$0.196102\pi$
$192$		0			0
$193$			19.1221i			1.37644i	0.725504	+	0.688218i	$0.241607\pi$
$193$			19.1221i			1.37644i	−0.725504	+	0.688218i	$0.758393\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3.53909i		−	0.252150i	−0.992021	−	0.126075i	$-0.959762\pi$
$197$		−	3.53909i		−	0.252150i	0.992021	−	0.126075i	$-0.0402379\pi$
$198$		0			0
$199$	−6.58457			−0.466768			−0.233384	−	0.972385i	$-0.574980\pi$
$199$	−6.58457			−0.466768			−0.233384	+	0.972385i	$0.574980\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	18.6872i		−	1.31159i
$204$		0			0
$205$	4.72919			0.330301
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.10830			−0.422520
$210$		0			0
$211$	−10.1778			−0.700667			−0.350334	−	0.936625i	$-0.613932\pi$
$211$	−10.1778			−0.700667			−0.350334	+	0.936625i	$0.613932\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	7.30398i		−	0.498127i
$216$		0			0
$217$	−13.7271			−0.931858
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.11036	−	14.4721i	0.478295	−	0.973501i
$222$		0			0
$223$		−	8.51177i		−	0.569990i	−0.958529	−	0.284995i	$-0.908008\pi$
$223$		−	8.51177i		−	0.569990i	0.958529	−	0.284995i	$-0.0919920\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			19.7565i			1.31129i	0.755071	+	0.655643i	$0.227602\pi$
$227$			19.7565i			1.31129i	−0.755071	+	0.655643i	$0.772398\pi$
$228$		0			0
$229$		−	3.41641i		−	0.225763i	−0.993608	−	0.112881i	$-0.963992\pi$
$229$		−	3.41641i		−	0.225763i	0.993608	−	0.112881i	$-0.0360080\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.5846			−0.824443			−0.412221	−	0.911084i	$-0.635247\pi$
$233$	−12.5846			−0.824443			−0.412221	+	0.911084i	$0.635247\pi$
$234$		0			0
$235$	9.82977			0.641224
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	13.7887i		−	0.891916i	−0.895054	−	0.445958i	$-0.852863\pi$
$239$		−	13.7887i		−	0.891916i	0.895054	−	0.445958i	$-0.147137\pi$
$240$		0			0
$241$			3.03523i			0.195516i	0.995210	+	0.0977582i	$0.0311672\pi$
$241$			3.03523i			0.195516i	−0.995210	+	0.0977582i	$0.968833\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			15.1323i			0.966767i
$246$		0			0
$247$	7.47007	+	3.67015i	0.475309	+	0.233526i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.1778			0.642416			0.321208	−	0.947009i	$-0.395911\pi$
$251$	10.1778			0.642416			0.321208	+	0.947009i	$0.395911\pi$
$252$		0			0
$253$			22.1778i			1.39431i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.58457			−0.285978			−0.142989	−	0.989724i	$-0.545671\pi$
$257$	−4.58457			−0.285978			−0.142989	+	0.989724i	$0.545671\pi$
$258$		0			0
$259$	−40.0663			−2.48960
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.5846			0.652672			0.326336	−	0.945254i	$-0.394186\pi$
$263$	10.5846			0.652672			0.326336	+	0.945254i	$0.394186\pi$
$264$		0			0
$265$			9.90904i			0.608707i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−27.0311			−1.64811			−0.824057	−	0.566506i	$-0.808295\pi$
$269$	−27.0311			−1.64811			−0.824057	+	0.566506i	$0.808295\pi$
$270$		0			0
$271$		−	5.54287i		−	0.336705i	−0.985727	−	0.168353i	$-0.946155\pi$
$271$		−	5.54287i		−	0.336705i	0.985727	−	0.168353i	$-0.0538447\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	10.8198i		−	0.652458i
$276$		0			0
$277$	−3.12619			−0.187835			−0.0939173	−	0.995580i	$-0.529939\pi$
$277$	−3.12619			−0.187835			−0.0939173	+	0.995580i	$0.529939\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	11.1323i		−	0.664098i	−0.943262	−	0.332049i	$-0.892260\pi$
$281$		−	11.1323i		−	0.664098i	0.943262	−	0.332049i	$-0.107740\pi$
$282$		0			0
$283$	−28.0663			−1.66837			−0.834185	−	0.551485i	$-0.814061\pi$
$283$	−28.0663			−1.66837			−0.834185	+	0.551485i	$0.814061\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	23.6852			1.39809
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	28.0766i		−	1.64025i	−0.572184	−	0.820126i	$-0.693904\pi$
$293$		−	28.0766i		−	1.64025i	0.572184	−	0.820126i	$-0.306096\pi$
$294$		0			0
$295$	−4.34801			−0.253151
$296$		0			0
$297$		0			0
$298$		0			0
$299$	13.3254	−	27.1221i	0.770631	−	1.56851i
$300$		0			0
$301$		−	36.5804i		−	2.10846i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			13.3949i			0.766992i
$306$		0			0
$307$		−	19.7463i		−	1.12698i	−0.826123	−	0.563489i	$-0.809459\pi$
$307$		−	19.7463i		−	1.12698i	0.826123	−	0.563489i	$-0.190541\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.7966			−0.782334			−0.391167	−	0.920320i	$-0.627928\pi$
$311$	−13.7966			−0.782334			−0.391167	+	0.920320i	$0.627928\pi$
$312$		0			0
$313$	−18.5268			−1.04720			−0.523598	−	0.851966i	$-0.675410\pi$
$313$	−18.5268			−1.04720			−0.523598	+	0.851966i	$0.675410\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	17.0208i		−	0.955986i	−0.878364	−	0.477993i	$-0.841364\pi$
$317$		−	17.0208i		−	0.955986i	0.878364	−	0.477993i	$-0.158636\pi$
$318$		0			0
$319$		−	10.3439i		−	0.579146i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			10.3234i			0.574408i
$324$		0			0
$325$	−6.50103	+	13.2319i	−0.360612	+	0.733975i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	49.2304			2.71416
$330$		0			0
$331$		−	7.27315i		−	0.399768i	−0.979820	−	0.199884i	$-0.935943\pi$
$331$		−	7.27315i		−	0.399768i	0.979820	−	0.199884i	$-0.0640566\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.381176			−0.0208259
$336$		0			0
$337$	−7.94220			−0.432639			−0.216320	−	0.976323i	$-0.569405\pi$
$337$	−7.94220			−0.432639			−0.216320	+	0.976323i	$0.569405\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.59833			−0.411473
$342$		0			0
$343$			42.3234i			2.28525i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.8738			−0.583737			−0.291868	−	0.956459i	$-0.594277\pi$
$347$	−10.8738			−0.583737			−0.291868	+	0.956459i	$0.594277\pi$
$348$		0			0
$349$		−	21.3254i		−	1.14153i	−0.821115	−	0.570763i	$-0.806648\pi$
$349$		−	21.3254i		−	1.14153i	0.821115	−	0.570763i	$-0.193352\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			3.72101i			0.198049i	0.995085	+	0.0990247i	$0.0315723\pi$
$353$			3.72101i			0.198049i	−0.995085	+	0.0990247i	$0.968428\pi$
$354$		0			0
$355$	3.28815			0.174517
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.0058i		−	0.686422i	−0.939258	−	0.343211i	$-0.888485\pi$
$359$		−	13.0058i		−	0.686422i	0.939258	−	0.343211i	$-0.111515\pi$
$360$		0			0
$361$	13.6714			0.719547
$362$		0			0
$363$		0			0
$364$		0			0
$365$	11.7108			0.612969
$366$		0			0
$367$	−25.5405			−1.33320			−0.666602	−	0.745413i	$-0.732252\pi$
$367$	−25.5405			−1.33320			−0.666602	+	0.745413i	$0.732252\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			49.6274i			2.57652i
$372$		0			0
$373$	10.9979			0.569451			0.284726	−	0.958609i	$-0.408097\pi$
$373$	10.9979			0.569451			0.284726	+	0.958609i	$0.408097\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.21508	+	12.6499i	−0.320093	+	0.651504i
$378$		0			0
$379$		−	35.0707i		−	1.80146i	−0.434377	−	0.900731i	$-0.643031\pi$
$379$		−	35.0707i		−	1.80146i	0.434377	−	0.900731i	$-0.356969\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			8.40959i			0.429710i	0.976646	+	0.214855i	$0.0689279\pi$
$383$			8.40959i			0.429710i	−0.976646	+	0.214855i	$0.931072\pi$
$384$		0			0
$385$			12.0746i			0.615379i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.2130			1.37975			0.689877	−	0.723926i	$-0.257664\pi$
$389$	27.2130			1.37975			0.689877	+	0.723926i	$0.257664\pi$
$390$		0			0
$391$	37.4818			1.89553
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	12.1819i		−	0.612939i
$396$		0			0
$397$		−	26.7409i		−	1.34209i	−0.741418	−	0.671043i	$-0.765847\pi$
$397$		−	26.7409i		−	1.34209i	0.741418	−	0.671043i	$-0.234153\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			24.3014i			1.21356i	0.794871	+	0.606778i	$0.207538\pi$
$401$			24.3014i			1.21356i	−0.794871	+	0.606778i	$0.792462\pi$
$402$		0			0
$403$	9.29228	+	4.56543i	0.462882	+	0.227420i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−22.1778			−1.09931
$408$		0			0
$409$			21.3949i			1.05791i	0.848649	+	0.528956i	$0.177416\pi$
$409$			21.3949i			1.05791i	−0.848649	+	0.528956i	$0.822584\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−21.7761			−1.07153
$414$		0			0
$415$	−0.703579			−0.0345373
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.97853			0.292070			0.146035	−	0.989279i	$-0.453349\pi$
$419$	5.97853			0.292070			0.146035	+	0.989279i	$0.453349\pi$
$420$		0			0
$421$			3.35008i			0.163273i	0.996662	+	0.0816365i	$0.0260146\pi$
$421$			3.35008i			0.163273i	−0.996662	+	0.0816365i	$0.973985\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−18.2861			−0.887005
$426$		0			0
$427$			67.0857i			3.24651i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			23.7136i			1.14224i	0.820866	+	0.571121i	$0.193491\pi$
$431$			23.7136i			1.14224i	−0.820866	+	0.571121i	$0.806509\pi$
$432$		0			0
$433$	−13.0889			−0.629012			−0.314506	−	0.949255i	$-0.601839\pi$
$433$	−13.0889			−0.629012			−0.314506	+	0.949255i	$0.601839\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			19.3469i			0.925489i
$438$		0			0
$439$	−25.8727			−1.23484			−0.617419	−	0.786635i	$-0.711822\pi$
$439$	−25.8727			−1.23484			−0.617419	+	0.786635i	$0.711822\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.6147			1.02694			0.513472	−	0.858106i	$-0.328359\pi$
$443$	21.6147			1.02694			0.513472	+	0.858106i	$0.328359\pi$
$444$		0			0
$445$	−11.2708			−0.534288
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	13.0413i		−	0.615459i	−0.951474	−	0.307730i	$-0.900431\pi$
$449$		−	13.0413i		−	0.615459i	0.951474	−	0.307730i	$-0.0995692\pi$
$450$		0			0
$451$	13.1104			0.617343
$452$		0			0
$453$		0			0
$454$		0			0
$455$	7.25498	−	14.7665i	0.340119	−	0.692264i
$456$		0			0
$457$			12.1298i			0.567407i	0.958912	+	0.283703i	$0.0915631\pi$
$457$			12.1298i			0.567407i	−0.958912	+	0.283703i	$0.908437\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	26.8431i		−	1.25021i	−0.780542	−	0.625103i	$-0.785057\pi$
$461$		−	26.8431i		−	1.25021i	0.780542	−	0.625103i	$-0.214943\pi$
$462$		0			0
$463$			23.7999i			1.10608i	0.833156	+	0.553038i	$0.186532\pi$
$463$			23.7999i			1.10608i	−0.833156	+	0.553038i	$0.813468\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	27.3040			1.26348			0.631739	−	0.775181i	$-0.282342\pi$
$467$	27.3040			1.26348			0.631739	+	0.775181i	$0.282342\pi$
$468$		0			0
$469$	−1.90904			−0.0881513
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	20.2482i		−	0.931015i
$474$		0			0
$475$		−	9.43871i		−	0.433078i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	32.0411i		−	1.46399i	−0.681308	−	0.731997i	$-0.738588\pi$
$479$		−	32.0411i		−	1.46399i	0.681308	−	0.731997i	$-0.261412\pi$
$480$		0			0
$481$	27.1221	+	13.3254i	1.23666	+	0.607588i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.81394			0.263998
$486$		0			0
$487$		−	13.9272i		−	0.631102i	−0.948909	−	0.315551i	$-0.897811\pi$
$487$		−	13.9272i		−	0.631102i	0.948909	−	0.315551i	$-0.102189\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−28.4302			−1.28304			−0.641518	−	0.767108i	$-0.721695\pi$
$491$	−28.4302			−1.28304			−0.641518	+	0.767108i	$0.721695\pi$
$492$		0			0
$493$	−17.4818			−0.787339
$494$		0			0
$495$		0			0
$496$		0			0
$497$	16.4680			0.738691
$498$		0			0
$499$		−	31.9945i		−	1.43227i	−0.697961	−	0.716135i	$-0.745909\pi$
$499$		−	31.9945i		−	1.43227i	0.697961	−	0.716135i	$-0.254091\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−22.2441			−0.991816			−0.495908	−	0.868375i	$-0.665165\pi$
$503$	−22.2441			−0.991816			−0.495908	+	0.868375i	$0.665165\pi$
$504$		0			0
$505$		−	6.09096i		−	0.271044i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			1.63005i			0.0722506i	0.999347	+	0.0361253i	$0.0115015\pi$
$509$			1.63005i			0.0722506i	−0.999347	+	0.0361253i	$0.988498\pi$
$510$		0			0
$511$	58.6509			2.59456
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	0.332185i		−	0.0146378i
$516$		0			0
$517$	27.2503			1.19847
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.93051			−0.259820			−0.129910	−	0.991526i	$-0.541469\pi$
$521$	−5.93051			−0.259820			−0.129910	+	0.991526i	$0.541469\pi$
$522$		0			0
$523$	31.2881			1.36814			0.684068	−	0.729419i	$-0.260209\pi$
$523$	31.2881			1.36814			0.684068	+	0.729419i	$0.260209\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			12.8416i			0.559390i
$528$		0			0
$529$	47.2441			2.05409
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−16.0332	−	7.87732i	−0.694473	−	0.341205i
$534$		0			0
$535$		−	1.92954i		−	0.0834211i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			41.9501i			1.80692i
$540$		0			0
$541$			28.5109i			1.22578i	0.790168	+	0.612891i	$0.209993\pi$
$541$			28.5109i			1.22578i	−0.790168	+	0.612891i	$0.790007\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.43017			0.361109
$546$		0			0
$547$	18.2366			0.779739			0.389869	−	0.920870i	$-0.372520\pi$
$547$	18.2366			0.779739			0.389869	+	0.920870i	$0.372520\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	9.02354i		−	0.384416i
$552$		0			0
$553$		−	61.0106i		−	2.59443i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	29.5350i		−	1.25144i	−0.780049	−	0.625718i	$-0.784806\pi$
$557$		−	29.5350i		−	1.25144i	0.780049	−	0.625718i	$-0.215194\pi$
$558$		0			0
$559$	−12.1661	+	24.7624i	−0.514571	+	1.04734i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	38.7409			1.63273			0.816367	−	0.577534i	$-0.195985\pi$
$563$	38.7409			1.63273			0.816367	+	0.577534i	$0.195985\pi$
$564$		0			0
$565$		−	10.3392i		−	0.434974i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.584569			0.0245064			0.0122532	−	0.999925i	$-0.496100\pi$
$569$	0.584569			0.0245064			0.0122532	+	0.999925i	$0.496100\pi$
$570$		0			0
$571$	−8.28924			−0.346894			−0.173447	−	0.984843i	$-0.555491\pi$
$571$	−8.28924			−0.346894			−0.173447	+	0.984843i	$0.555491\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−34.2697			−1.42915
$576$		0			0
$577$			0.696025i			0.0289759i	0.999895	+	0.0144879i	$0.00461182\pi$
$577$			0.696025i			0.0289759i	−0.999895	+	0.0144879i	$0.995388\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.52373			−0.146189
$582$		0			0
$583$			27.4701i			1.13769i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	2.62565i		−	0.108372i	−0.998531	−	0.0541860i	$-0.982744\pi$
$587$		−	2.62565i		−	0.108372i	0.998531	−	0.0541860i	$-0.0172564\pi$
$588$		0			0
$589$	−6.62845			−0.273120
$590$		0			0
$591$		0			0
$592$		0			0
$593$			18.8001i			0.772028i	0.922493	+	0.386014i	$0.126148\pi$
$593$			18.8001i			0.772028i	−0.922493	+	0.386014i	$0.873852\pi$
$594$		0			0
$595$	20.4068			0.836597
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−22.5416			−0.921026			−0.460513	−	0.887653i	$-0.652334\pi$
$599$	−22.5416			−0.921026			−0.460513	+	0.887653i	$0.652334\pi$
$600$		0			0
$601$	40.8865			1.66779			0.833897	−	0.551920i	$-0.186105\pi$
$601$	40.8865			1.66779			0.833897	+	0.551920i	$0.186105\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	3.81611i		−	0.155147i
$606$		0			0
$607$	−6.88964			−0.279642			−0.139821	−	0.990177i	$-0.544653\pi$
$607$	−6.88964			−0.279642			−0.139821	+	0.990177i	$0.544653\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−33.3254	−	16.3733i	−1.34820	−	0.662391i
$612$		0			0
$613$		−	2.31071i		−	0.0933288i	−0.998911	−	0.0466644i	$-0.985141\pi$
$613$		−	2.31071i		−	0.0933288i	0.998911	−	0.0466644i	$-0.0148591\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			4.41703i			0.177823i	0.996040	+	0.0889115i	$0.0283388\pi$
$617$			4.41703i			0.177823i	−0.996040	+	0.0889115i	$0.971661\pi$
$618$		0			0
$619$		−	12.9798i		−	0.521701i	−0.965379	−	0.260850i	$-0.915997\pi$
$619$		−	12.9798i		−	0.521701i	0.965379	−	0.260850i	$-0.0840030\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−56.4475			−2.26152
$624$		0			0
$625$	12.1635			0.486540
$626$		0			0
$627$		0			0
$628$		0			0
$629$			37.4818i			1.49450i
$630$		0			0
$631$			1.17147i			0.0466356i	0.999728	+	0.0233178i	$0.00742295\pi$
$631$			1.17147i			0.0466356i	−0.999728	+	0.0233178i	$0.992577\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			18.4260i			0.731215i
$636$		0			0
$637$	25.2056	−	51.3024i	0.998682	−	2.03267i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.8318			0.743813			0.371906	−	0.928270i	$-0.378704\pi$
$641$	18.8318			0.743813			0.371906	+	0.928270i	$0.378704\pi$
$642$		0			0
$643$		−	26.7590i		−	1.05527i	−0.849470	−	0.527637i	$-0.823078\pi$
$643$		−	26.7590i		−	1.05527i	0.849470	−	0.527637i	$-0.176922\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.45838			0.371847			0.185924	−	0.982564i	$-0.440472\pi$
$647$	9.45838			0.371847			0.185924	+	0.982564i	$0.440472\pi$
$648$		0			0
$649$	−12.0537			−0.473148
$650$		0			0
$651$		0			0
$652$		0			0
$653$	33.0782			1.29445			0.647225	−	0.762299i	$-0.275930\pi$
$653$	33.0782			1.29445			0.647225	+	0.762299i	$0.275930\pi$
$654$		0			0
$655$		−	0.537486i		−	0.0210013i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.719406			0.0280241			0.0140120	−	0.999902i	$-0.495540\pi$
$659$	0.719406			0.0280241			0.0140120	+	0.999902i	$0.495540\pi$
$660$		0			0
$661$			28.8543i			1.12230i	0.827714	+	0.561151i	$0.189641\pi$
$661$			28.8543i			1.12230i	−0.827714	+	0.561151i	$0.810359\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			10.5333i			0.408466i
$666$		0			0
$667$	−32.7624			−1.26856
$668$		0			0
$669$		0			0
$670$		0			0
$671$			37.1337i			1.43353i
$672$		0			0
$673$	−20.1339			−0.776105			−0.388053	−	0.921637i	$-0.626852\pi$
$673$	−20.1339			−0.776105			−0.388053	+	0.921637i	$0.626852\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−27.8396			−1.06996			−0.534980	−	0.844864i	$-0.679681\pi$
$677$	−27.8396			−1.06996			−0.534980	+	0.844864i	$0.679681\pi$
$678$		0			0
$679$	29.1179			1.11744
$680$		0			0
$681$		0			0
$682$		0			0
$683$			20.6578i			0.790450i	0.918584	+	0.395225i	$0.129333\pi$
$683$			20.6578i			0.790450i	−0.918584	+	0.395225i	$0.870667\pi$
$684$		0			0
$685$	13.5232			0.516695
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.5053	−	33.5942i	0.628802	−	1.27984i
$690$		0			0
$691$		−	43.1892i		−	1.64299i	−0.570212	−	0.821497i	$-0.693139\pi$
$691$		−	43.1892i		−	1.64299i	0.570212	−	0.821497i	$-0.306861\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			1.05986i			0.0402030i
$696$		0			0
$697$		−	22.1573i		−	0.839267i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.4936			0.849572			0.424786	−	0.905294i	$-0.360349\pi$
$701$	22.4936			0.849572			0.424786	+	0.905294i	$0.360349\pi$
$702$		0			0
$703$	−19.3469			−0.729683
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	30.5053i		−	1.14727i
$708$		0			0
$709$		−	14.9433i		−	0.561207i	−0.959824	−	0.280604i	$-0.909465\pi$
$709$		−	14.9433i		−	0.561207i	0.959824	−	0.280604i	$-0.0905346\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			24.0663i			0.901291i
$714$		0			0
$715$	4.01583	−	8.17365i	0.150183	−	0.305677i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10.2697			0.382996			0.191498	−	0.981493i	$-0.438665\pi$
$719$	10.2697			0.382996			0.191498	+	0.981493i	$0.438665\pi$
$720$		0			0
$721$		−	1.66368i		−	0.0619586i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	15.9836			0.593618
$726$		0			0
$727$	15.5932			0.578320			0.289160	−	0.957281i	$-0.406624\pi$
$727$	15.5932			0.578320			0.289160	+	0.957281i	$0.406624\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−34.2207			−1.26570
$732$		0			0
$733$			8.76551i			0.323762i	0.986810	+	0.161881i	$0.0517560\pi$
$733$			8.76551i			0.323762i	−0.986810	+	0.161881i	$0.948244\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.05670			−0.0389242
$738$		0			0
$739$		−	21.7421i		−	0.799797i	−0.916559	−	0.399898i	$-0.869045\pi$
$739$		−	21.7421i		−	0.799797i	0.916559	−	0.399898i	$-0.130955\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			45.5070i			1.66949i	0.550637	+	0.834745i	$0.314385\pi$
$743$			45.5070i			1.66949i	−0.550637	+	0.834745i	$0.685615\pi$
$744$		0			0
$745$	0.911108			0.0333804
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	9.66368i		−	0.353103i
$750$		0			0
$751$	3.36275			0.122708			0.0613542	−	0.998116i	$-0.480458\pi$
$751$	3.36275			0.122708			0.0613542	+	0.998116i	$0.480458\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.61469			0.349914
$756$		0			0
$757$	35.1594			1.27789			0.638944	−	0.769253i	$-0.279371\pi$
$757$	35.1594			1.27789			0.638944	+	0.769253i	$0.279371\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	26.8001i		−	0.971504i	−0.874097	−	0.485752i	$-0.838546\pi$
$761$		−	26.8001i		−	0.971504i	0.874097	−	0.485752i	$-0.161454\pi$
$762$		0			0
$763$	42.2207			1.52849
$764$		0			0
$765$		0			0
$766$		0			0
$767$	14.7409	+	7.24240i	0.532262	+	0.261508i
$768$		0			0
$769$			2.22073i			0.0800815i	0.999198	+	0.0400408i	$0.0127488\pi$
$769$			2.22073i			0.0800815i	−0.999198	+	0.0400408i	$0.987251\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			47.8947i			1.72265i	0.508054	+	0.861326i	$0.330365\pi$
$773$			47.8947i			1.72265i	−0.508054	+	0.861326i	$0.669635\pi$
$774$		0			0
$775$		−	11.7411i		−	0.421754i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11.4369			0.409770
$780$		0			0
$781$	9.11548			0.326177
$782$		0			0
$783$		0			0
$784$		0			0
$785$			7.54934i			0.269447i
$786$		0			0
$787$		−	9.49387i		−	0.338420i	−0.985580	−	0.169210i	$-0.945878\pi$
$787$		−	9.49387i		−	0.338420i	0.985580	−	0.169210i	$-0.0541216\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	51.7818i		−	1.84115i
$792$		0			0
$793$	22.3117	−	45.4123i	0.792311	−	1.61264i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.38020			−0.296842			−0.148421	−	0.988924i	$-0.547419\pi$
$797$	−8.38020			−0.296842			−0.148421	+	0.988924i	$0.547419\pi$
$798$		0			0
$799$		−	46.0546i		−	1.62930i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	32.4648			1.14566
$804$		0			0
$805$	38.2441			1.34793
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11.7098			−0.411694			−0.205847	−	0.978584i	$-0.565995\pi$
$809$	−11.7098			−0.411694			−0.205847	+	0.978584i	$0.565995\pi$
$810$		0			0
$811$			41.8009i			1.46783i	0.679242	+	0.733914i	$0.262308\pi$
$811$			41.8009i			1.46783i	−0.679242	+	0.733914i	$0.737692\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.69342			0.129375
$816$		0			0
$817$		−	17.6637i		−	0.617974i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	49.5788i		−	1.73031i	−0.501502	−	0.865157i	$-0.667219\pi$
$821$		−	49.5788i		−	1.73031i	0.501502	−	0.865157i	$-0.332781\pi$
$822$		0			0
$823$	17.4742			0.609112			0.304556	−	0.952494i	$-0.401492\pi$
$823$	17.4742			0.609112			0.304556	+	0.952494i	$0.401492\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	29.0722i		−	1.01094i	−0.862845	−	0.505469i	$-0.831319\pi$
$827$		−	29.0722i		−	1.01094i	0.862845	−	0.505469i	$-0.168681\pi$
$828$		0			0
$829$	−0.981572			−0.0340914			−0.0170457	−	0.999855i	$-0.505426\pi$
$829$	−0.981572			−0.0340914			−0.0170457	+	0.999855i	$0.505426\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	70.8982			2.45648
$834$		0			0
$835$	−23.6949			−0.819997
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	49.2019i		−	1.69864i	−0.527879	−	0.849320i	$-0.677012\pi$
$839$		−	49.2019i		−	1.69864i	0.527879	−	0.849320i	$-0.322988\pi$
$840$		0			0
$841$	−13.7194			−0.473083
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.82222	+	7.58297i	−0.337895	+	0.260862i
$846$		0			0
$847$		−	19.1122i		−	0.656702i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			70.2441i			2.40794i
$852$		0			0
$853$			7.68515i			0.263135i	0.991307	+	0.131567i	$0.0420009\pi$
$853$			7.68515i			0.263135i	−0.991307	+	0.131567i	$0.957999\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.16914			0.108256			0.0541278	−	0.998534i	$-0.482762\pi$
$857$	3.16914			0.108256			0.0541278	+	0.998534i	$0.482762\pi$
$858$		0			0
$859$	−35.3040			−1.20456			−0.602278	−	0.798286i	$-0.705740\pi$
$859$	−35.3040			−1.20456			−0.602278	+	0.798286i	$0.705740\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	13.0058i		−	0.442724i	−0.975192	−	0.221362i	$-0.928950\pi$
$863$		−	13.0058i		−	0.442724i	0.975192	−	0.221362i	$-0.0710503\pi$
$864$		0			0
$865$			16.1532i			0.549224i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	33.7710i		−	1.14560i
$870$		0			0
$871$	1.29228	+	0.634918i	0.0437874	+	0.0215134i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−41.4735			−1.40206
$876$		0			0
$877$			42.4946i			1.43494i	0.696589	+	0.717470i	$0.254700\pi$
$877$			42.4946i			1.43494i	−0.696589	+	0.717470i	$0.745300\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.82222			−0.128774			−0.0643869	−	0.997925i	$-0.520509\pi$
$881$	−3.82222			−0.128774			−0.0643869	+	0.997925i	$0.520509\pi$
$882$		0			0
$883$	−3.23765			−0.108956			−0.0544778	−	0.998515i	$-0.517349\pi$
$883$	−3.23765			−0.108956			−0.0544778	+	0.998515i	$0.517349\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.3382			0.682891			0.341445	−	0.939902i	$-0.389084\pi$
$887$	20.3382			0.682891			0.341445	+	0.939902i	$0.389084\pi$
$888$		0			0
$889$			92.2829i			3.09507i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	23.7720			0.795499
$894$		0			0
$895$			15.4625i			0.516855i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	11.2247i		−	0.374365i
$900$		0			0
$901$	46.4260			1.54668
$902$		0			0
$903$		0			0
$904$		0			0
$905$			7.54934i			0.250948i
$906$		0			0
$907$	6.13484			0.203704			0.101852	−	0.994800i	$-0.467523\pi$
$907$	6.13484			0.203704			0.101852	+	0.994800i	$0.467523\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−44.1583			−1.46303			−0.731514	−	0.681826i	$-0.761186\pi$
$911$	−44.1583			−1.46303			−0.731514	+	0.681826i	$0.761186\pi$
$912$		0			0
$913$	−1.95048			−0.0645514
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2.69189i		−	0.0888940i
$918$		0			0
$919$	−53.8885			−1.77762			−0.888810	−	0.458277i	$-0.848467\pi$
$919$	−53.8885			−1.77762			−0.888810	+	0.458277i	$0.848467\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−11.1477	−	5.47700i	−0.366930	−	0.180278i
$924$		0			0
$925$		−	34.2697i		−	1.12678i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			33.5350i			1.10025i	0.835084	+	0.550123i	$0.185419\pi$
$929$			33.5350i			1.10025i	−0.835084	+	0.550123i	$0.814581\pi$
$930$		0			0
$931$			36.5954i			1.19937i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.2957			0.369409
$936$		0			0
$937$	52.0720			1.70112			0.850559	−	0.525880i	$-0.176264\pi$
$937$	52.0720			1.70112			0.850559	+	0.525880i	$0.176264\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	23.5830i		−	0.768783i	−0.923170	−	0.384391i	$-0.874411\pi$
$941$		−	23.5830i		−	0.768783i	0.923170	−	0.384391i	$-0.125589\pi$
$942$		0			0
$943$		−	41.5247i		−	1.35223i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	49.2746i		−	1.60121i	−0.599193	−	0.800604i	$-0.704512\pi$
$947$		−	49.2746i		−	1.60121i	0.599193	−	0.800604i	$-0.295488\pi$
$948$		0			0
$949$	−39.7025	−	19.5064i	−1.28880	−	0.633204i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−50.1327			−1.62396			−0.811978	−	0.583688i	$-0.801609\pi$
$953$	−50.1327			−1.62396			−0.811978	+	0.583688i	$0.801609\pi$
$954$		0			0
$955$		−	21.5330i		−	0.696791i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	67.7281			2.18705
$960$		0			0
$961$	22.7546			0.734021
$962$		0			0
$963$		0			0
$964$		0			0
$965$	18.2524			0.587565
$966$		0			0
$967$		−	4.55472i		−	0.146470i	−0.997315	−	0.0732349i	$-0.976668\pi$
$967$		−	4.55472i		−	0.146470i	0.997315	−	0.0732349i	$-0.0233323\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−48.0234			−1.54114			−0.770572	−	0.637353i	$-0.780029\pi$
$971$	−48.0234			−1.54114			−0.770572	+	0.637353i	$0.780029\pi$
$972$		0			0
$973$			5.30811i			0.170170i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	22.9820i		−	0.735261i	−0.929972	−	0.367630i	$-0.880169\pi$
$977$		−	22.9820i		−	0.735261i	0.929972	−	0.367630i	$-0.119831\pi$
$978$		0			0
$979$	−31.2452			−0.998601
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	9.63843i		−	0.307418i	−0.988116	−	0.153709i	$-0.950878\pi$
$983$		−	9.63843i		−	0.307418i	0.988116	−	0.153709i	$-0.0491219\pi$
$984$		0			0
$985$	−3.37813			−0.107636
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−64.1327			−2.03930
$990$		0			0
$991$	10.2366			0.325175			0.162587	−	0.986694i	$-0.448016\pi$
$991$	10.2366			0.325175			0.162587	+	0.986694i	$0.448016\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			6.28510i			0.199251i
$996$		0			0
$997$	−30.3556			−0.961370			−0.480685	−	0.876893i	$-0.659612\pi$
$997$	−30.3556			−0.961370			−0.480685	+	0.876893i	$0.659612\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.c.o.3457.4		8
3.2	odd	2		1248.2.c.d.961.5	yes	8
4.3	odd	2		3744.2.c.p.3457.4		8
12.11	even	2		1248.2.c.c.961.5	yes	8
13.12	even	2	inner	3744.2.c.o.3457.5		8
24.5	odd	2		2496.2.c.q.961.4		8
24.11	even	2		2496.2.c.r.961.4		8
39.38	odd	2		1248.2.c.d.961.4	yes	8
52.51	odd	2		3744.2.c.p.3457.5		8
156.155	even	2		1248.2.c.c.961.4	✓	8
312.77	odd	2		2496.2.c.q.961.5		8
312.155	even	2		2496.2.c.r.961.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.c.c.961.4	✓	8	156.155	even	2
1248.2.c.c.961.5	yes	8	12.11	even	2
1248.2.c.d.961.4	yes	8	39.38	odd	2
1248.2.c.d.961.5	yes	8	3.2	odd	2
2496.2.c.q.961.4		8	24.5	odd	2
2496.2.c.q.961.5		8	312.77	odd	2
2496.2.c.r.961.4		8	24.11	even	2
2496.2.c.r.961.5		8	312.155	even	2
3744.2.c.o.3457.4		8	1.1	even	1	trivial
3744.2.c.o.3457.5		8	13.12	even	2	inner
3744.2.c.p.3457.4		8	4.3	odd	2
3744.2.c.p.3457.5		8	52.51	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.954520i q^{5} -4.78051i q^{7} -2.64614i q^{11} +(-1.58993 + 3.23607i) q^{13} -4.47214 q^{17} -2.30838i q^{19} -8.38118 q^{23} +4.08889 q^{25} +3.90904 q^{29} -2.87147i q^{31} -4.56310 q^{35} -8.38118i q^{37} +4.95452i q^{41} +7.65199 q^{43} +10.2981i q^{47} -15.8533 q^{49} -10.3812 q^{53} -2.52580 q^{55} -4.55518i q^{59} -14.0332 q^{61} +(3.08889 + 1.51762i) q^{65} -0.399338i q^{67} +3.44482i q^{71} +12.2687i q^{73} -12.6499 q^{77} +12.7624 q^{79} -0.737102i q^{83} +4.26874i q^{85} -11.8078i q^{89} +(15.4701 + 7.60066i) q^{91} -2.20339 q^{95} +6.09096i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{23} - 16 q^{25} + 8 q^{29} - 24 q^{35} - 32 q^{49} - 24 q^{53} + 16 q^{55} + 8 q^{61} - 24 q^{65} + 32 q^{77} - 16 q^{79} + 16 q^{91} - 56 q^{95}+O(q^{100}) \) 8 * q - 8 * q^23 - 16 * q^25 + 8 * q^29 - 24 * q^35 - 32 * q^49 - 24 * q^53 + 16 * q^55 + 8 * q^61 - 24 * q^65 + 32 * q^77 - 16 * q^79 + 16 * q^91 - 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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