LMFDB - Embedded newform 2340.2.j.d.649.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(649,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.j (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12745506816.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9 $ x^8 + 16x^6 + 79x^4 + 120*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$2.29678i$ of defining polynomial
Character	$\chi$	$=$	2340.649
Dual form			2340.2.j.d.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 + 1.41421i) q^{5} +4.37780 q^{7} -2.53326i q^{11} +(-2.64575 + 2.44949i) q^{13} +1.29217i q^{17} +5.36169i q^{19} +1.80341i q^{23} +(1.00000 - 4.89898i) q^{25} +7.58258 q^{29} +3.12359i q^{31} +(-7.58258 + 6.19115i) q^{35} -2.55040 q^{37} -7.89495i q^{41} +4.38774i q^{43} -6.20520 q^{47} +12.1652 q^{49} +6.19115i q^{53} +(3.58258 + 4.38774i) q^{55} +10.4282i q^{59} +3.58258 q^{61} +(1.11847 - 7.98430i) q^{65} -2.55040 q^{67} -0.295164i q^{71} -2.55040 q^{73} -11.0901i q^{77} +11.1652 q^{79} -0.723000 q^{83} +(-1.82740 - 2.23810i) q^{85} +11.3137i q^{89} +(-11.5826 + 10.7234i) q^{91} +(-7.58258 - 9.28672i) q^{95} -12.2197 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{25} + 24 q^{29} - 24 q^{35} + 24 q^{49} - 8 q^{55} - 8 q^{61} + 16 q^{79} - 56 q^{91} - 24 q^{95}+O(q^{100}) $ 8 * q + 8 * q^25 + 24 * q^29 - 24 * q^35 + 24 * q^49 - 8 * q^55 - 8 * q^61 + 16 * q^79 - 56 * q^91 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$	−1.73205	+	1.41421i	−0.774597	+	0.632456i
$5$	−1.73205	+	1.41421i	−0.774597	+	0.632456i
$6$		0			0
$7$	4.37780			1.65465			0.827327	−	0.561721i	$-0.189860\pi$
$7$	4.37780			1.65465			0.827327	+	0.561721i	$0.189860\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.53326i		−	0.763808i	−0.924202	−	0.381904i	$-0.875269\pi$
$11$		−	2.53326i		−	0.763808i	0.924202	−	0.381904i	$-0.124731\pi$
$12$		0			0
$13$	−2.64575	+	2.44949i	−0.733799	+	0.679366i
$13$	−2.64575	+	2.44949i	−0.733799	+	0.679366i
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.29217i			0.313397i	0.987647	+	0.156698i	$0.0500850\pi$
$17$			1.29217i			0.313397i	−0.987647	+	0.156698i	$0.949915\pi$
$18$		0			0
$19$			5.36169i			1.23006i	0.788505	+	0.615028i	$0.210855\pi$
$19$			5.36169i			1.23006i	−0.788505	+	0.615028i	$0.789145\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.80341i			0.376036i	0.982166	+	0.188018i	$0.0602064\pi$
$23$			1.80341i			0.376036i	−0.982166	+	0.188018i	$0.939794\pi$
$24$		0			0
$25$	1.00000	−	4.89898i	0.200000	−	0.979796i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.58258			1.40805			0.704024	−	0.710176i	$-0.251385\pi$
$29$	7.58258			1.40805			0.704024	+	0.710176i	$0.251385\pi$
$30$		0			0
$31$			3.12359i			0.561013i	0.959852	+	0.280507i	$0.0905025\pi$
$31$			3.12359i			0.561013i	−0.959852	+	0.280507i	$0.909498\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.58258	+	6.19115i	−1.28169	+	1.04649i
$36$		0			0
$37$	−2.55040			−0.419283			−0.209642	−	0.977778i	$-0.567230\pi$
$37$	−2.55040			−0.419283			−0.209642	+	0.977778i	$0.567230\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	7.89495i		−	1.23298i	−0.787361	−	0.616492i	$-0.788553\pi$
$41$		−	7.89495i		−	1.23298i	0.787361	−	0.616492i	$-0.211447\pi$
$42$		0			0
$43$			4.38774i			0.669124i	0.942374	+	0.334562i	$0.108588\pi$
$43$			4.38774i			0.669124i	−0.942374	+	0.334562i	$0.891412\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.20520			−0.905122			−0.452561	−	0.891733i	$-0.649490\pi$
$47$	−6.20520			−0.905122			−0.452561	+	0.891733i	$0.649490\pi$
$48$		0			0
$49$	12.1652			1.73788
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.19115i			0.850419i	0.905095	+	0.425210i	$0.139800\pi$
$53$			6.19115i			0.850419i	−0.905095	+	0.425210i	$0.860200\pi$
$54$		0			0
$55$	3.58258	+	4.38774i	0.483074	+	0.591643i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			10.4282i			1.35764i	0.734305	+	0.678819i	$0.237508\pi$
$59$			10.4282i			1.35764i	−0.734305	+	0.678819i	$0.762492\pi$
$60$		0			0
$61$	3.58258			0.458702			0.229351	−	0.973344i	$-0.426340\pi$
$61$	3.58258			0.458702			0.229351	+	0.973344i	$0.426340\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.11847	−	7.98430i	0.138730	−	0.990330i
$66$		0			0
$67$	−2.55040			−0.311581			−0.155791	−	0.987790i	$-0.549792\pi$
$67$	−2.55040			−0.311581			−0.155791	+	0.987790i	$0.549792\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.295164i		−	0.0350295i	−0.999847	−	0.0175147i	$-0.994425\pi$
$71$		−	0.295164i		−	0.0350295i	0.999847	−	0.0175147i	$-0.00557540\pi$
$72$		0			0
$73$	−2.55040			−0.298502			−0.149251	−	0.988799i	$-0.547686\pi$
$73$	−2.55040			−0.298502			−0.149251	+	0.988799i	$0.547686\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	11.0901i		−	1.26384i
$78$		0			0
$79$	11.1652			1.25618			0.628089	−	0.778142i	$-0.283837\pi$
$79$	11.1652			1.25618			0.628089	+	0.778142i	$0.283837\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.723000			−0.0793596			−0.0396798	−	0.999212i	$-0.512634\pi$
$83$	−0.723000			−0.0793596			−0.0396798	+	0.999212i	$0.512634\pi$
$84$		0			0
$85$	−1.82740	−	2.23810i	−0.198209	−	0.242756i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			11.3137i			1.19925i	0.800281	+	0.599625i	$0.204684\pi$
$89$			11.3137i			1.19925i	−0.800281	+	0.599625i	$0.795316\pi$
$90$		0			0
$91$	−11.5826	+	10.7234i	−1.21418	+	1.12412i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.58258	−	9.28672i	−0.777956	−	0.952797i
$96$		0			0
$97$	−12.2197			−1.24072			−0.620362	−	0.784316i	$-0.713014\pi$
$97$	−12.2197			−1.24072			−0.620362	+	0.784316i	$0.713014\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.16515			0.911967			0.455983	−	0.889988i	$-0.349288\pi$
$101$	9.16515			0.911967			0.455983	+	0.889988i	$0.349288\pi$
$102$		0			0
$103$		−	9.28672i		−	0.915048i	−0.889197	−	0.457524i	$-0.848736\pi$
$103$		−	9.28672i		−	0.915048i	0.889197	−	0.457524i	$-0.151264\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			10.5789i			1.02270i	0.859373	+	0.511350i	$0.170854\pi$
$107$			10.5789i			1.02270i	−0.859373	+	0.511350i	$0.829146\pi$
$108$		0			0
$109$			8.48528i			0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$			8.48528i			0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			11.0901i			1.04327i	0.853168	+	0.521636i	$0.174678\pi$
$113$			11.0901i			1.04327i	−0.853168	+	0.521636i	$0.825322\pi$
$114$		0			0
$115$	−2.55040	−	3.12359i	−0.237826	−	0.291276i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.65685i			0.518563i
$120$		0			0
$121$	4.58258			0.416598
$122$		0			0
$123$		0			0
$124$		0			0
$125$	5.19615	+	9.89949i	0.464758	+	0.885438i
$126$		0			0
$127$			0.511238i			0.0453651i	0.999743	+	0.0226825i	$0.00722069\pi$
$127$			0.511238i			0.0453651i	−0.999743	+	0.0226825i	$0.992779\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.16515			0.276541			0.138270	−	0.990395i	$-0.455846\pi$
$131$	3.16515			0.276541			0.138270	+	0.990395i	$0.455846\pi$
$132$		0			0
$133$			23.4724i			2.03532i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.3923			−0.887875			−0.443937	−	0.896058i	$-0.646419\pi$
$137$	−10.3923			−0.887875			−0.443937	+	0.896058i	$0.646419\pi$
$138$		0			0
$139$	−7.16515			−0.607740			−0.303870	−	0.952713i	$-0.598279\pi$
$139$	−7.16515			−0.607740			−0.303870	+	0.952713i	$0.598279\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.20520	+	6.70239i	0.518905	+	0.560482i
$144$		0			0
$145$	−13.1334	+	10.7234i	−1.09067	+	0.890528i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			2.82843i			0.231714i	0.993266	+	0.115857i	$0.0369614\pi$
$149$			2.82843i			0.231714i	−0.993266	+	0.115857i	$0.963039\pi$
$150$		0			0
$151$			16.0851i			1.30898i	0.756069	+	0.654492i	$0.227118\pi$
$151$			16.0851i			1.30898i	−0.756069	+	0.654492i	$0.772882\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.41742	−	5.41022i	−0.354816	−	0.434559i
$156$		0			0
$157$			22.4499i			1.79170i	0.444356	+	0.895850i	$0.353433\pi$
$157$			22.4499i			1.79170i	−0.444356	+	0.895850i	$0.646567\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			7.89495i			0.622210i
$162$		0			0
$163$	11.3060			0.885555			0.442777	−	0.896632i	$-0.353993\pi$
$163$	11.3060			0.885555			0.442777	+	0.896632i	$0.353993\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.723000			0.0559474			0.0279737	−	0.999609i	$-0.491095\pi$
$167$	0.723000			0.0559474			0.0279737	+	0.999609i	$0.491095\pi$
$168$		0			0
$169$	1.00000	−	12.9615i	0.0769231	−	0.997037i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	3.60681i		−	0.274221i	−0.990556	−	0.137110i	$-0.956218\pi$
$173$		−	3.60681i		−	0.274221i	0.990556	−	0.137110i	$-0.0437815\pi$
$174$		0			0
$175$	4.37780	−	21.4468i	0.330931	−	1.62122i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−23.5826			−1.75288			−0.876440	−	0.481512i	$-0.840088\pi$
$181$	−23.5826			−1.75288			−0.876440	+	0.481512i	$0.840088\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.41742	−	3.60681i	0.324775	−	0.265178i
$186$		0			0
$187$	3.27340			0.239375
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$	0.190700			0.0137269			0.00686346	−	0.999976i	$-0.497815\pi$
$193$	0.190700			0.0137269			0.00686346	+	0.999976i	$0.497815\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	22.8027			1.62463			0.812313	−	0.583222i	$-0.198208\pi$
$197$	22.8027			1.62463			0.812313	+	0.583222i	$0.198208\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	33.1950			2.32983
$204$		0			0
$205$	11.1652	+	13.6745i	0.779808	+	0.955066i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	13.5826			0.939526
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.20520	−	7.59979i	−0.423191	−	0.518301i
$216$		0			0
$217$			13.6745i			0.928283i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.16515	−	3.41875i	−0.212911	−	0.229970i
$222$		0			0
$223$	−2.55040			−0.170787			−0.0853937	−	0.996347i	$-0.527215\pi$
$223$	−2.55040			−0.170787			−0.0853937	+	0.996347i	$0.527215\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−26.9898			−1.79138			−0.895688	−	0.444682i	$-0.853317\pi$
$227$	−26.9898			−1.79138			−0.895688	+	0.444682i	$0.853317\pi$
$228$		0			0
$229$			10.7234i			0.708621i	0.935128	+	0.354310i	$0.115284\pi$
$229$			10.7234i			0.708621i	−0.935128	+	0.354310i	$0.884716\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	12.3823i		−	0.811191i	−0.914053	−	0.405596i	$-0.867064\pi$
$233$		−	12.3823i		−	0.811191i	0.914053	−	0.405596i	$-0.132936\pi$
$234$		0			0
$235$	10.7477	−	8.77548i	0.701104	−	0.572449i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	30.2272i		−	1.95524i	−0.210389	−	0.977618i	$-0.567473\pi$
$239$		−	30.2272i		−	1.95524i	0.210389	−	0.977618i	$-0.432527\pi$
$240$		0			0
$241$			14.7325i			0.949001i	0.880255	+	0.474501i	$0.157371\pi$
$241$			14.7325i			0.949001i	−0.880255	+	0.474501i	$0.842629\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−21.0707	+	17.2041i	−1.34616	+	1.09913i
$246$		0			0
$247$	−13.1334	−	14.1857i	−0.835659	−	0.902614i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.1652			0.957216			0.478608	−	0.878029i	$-0.341142\pi$
$251$	15.1652			0.957216			0.478608	+	0.878029i	$0.341142\pi$
$252$		0			0
$253$	4.56850			0.287219
$254$		0			0
$255$		0			0
$256$		0			0
$257$			7.21362i			0.449973i	0.974362	+	0.224987i	$0.0722339\pi$
$257$			7.21362i			0.449973i	−0.974362	+	0.224987i	$0.927766\pi$
$258$		0			0
$259$	−11.1652			−0.693769
$260$		0			0
$261$		0			0
$262$		0			0
$263$			16.5003i			1.01745i	0.860928	+	0.508727i	$0.169884\pi$
$263$			16.5003i			1.01745i	−0.860928	+	0.508727i	$0.830116\pi$
$264$		0			0
$265$	−8.75560	−	10.7234i	−0.537852	−	0.658732i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.16515			−0.558809			−0.279405	−	0.960173i	$-0.590137\pi$
$269$	−9.16515			−0.558809			−0.279405	+	0.960173i	$0.590137\pi$
$270$		0			0
$271$		−	22.3323i		−	1.35659i	−0.734791	−	0.678294i	$-0.762720\pi$
$271$		−	22.3323i		−	1.35659i	0.734791	−	0.678294i	$-0.237280\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−12.4104	−	2.53326i	−0.748376	−	0.152762i
$276$		0			0
$277$		−	28.3714i		−	1.70467i	−0.522994	−	0.852336i	$-0.675185\pi$
$277$		−	28.3714i		−	1.70467i	0.522994	−	0.852336i	$-0.324815\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	12.3712i		−	0.738001i	−0.929429	−	0.369001i	$-0.879700\pi$
$281$		−	12.3712i		−	0.738001i	0.929429	−	0.369001i	$-0.120300\pi$
$282$		0			0
$283$			4.38774i			0.260824i	0.991460	+	0.130412i	$0.0416300\pi$
$283$			4.38774i			0.260824i	−0.991460	+	0.130412i	$0.958370\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	34.5625i		−	2.04016i
$288$		0			0
$289$	15.3303			0.901783
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.20520			0.362512			0.181256	−	0.983436i	$-0.441984\pi$
$293$	6.20520			0.362512			0.181256	+	0.983436i	$0.441984\pi$
$294$		0			0
$295$	−14.7477	−	18.0622i	−0.858646	−	1.05162i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.41742	−	4.77136i	−0.255466	−	0.275935i
$300$		0			0
$301$			19.2087i			1.10717i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.20520	+	5.06653i	−0.355309	+	0.290108i
$306$		0			0
$307$	−8.03260			−0.458445			−0.229222	−	0.973374i	$-0.573618\pi$
$307$	−8.03260			−0.458445			−0.229222	+	0.973374i	$0.573618\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.33030			−0.358959			−0.179479	−	0.983762i	$-0.557441\pi$
$311$	−6.33030			−0.358959			−0.179479	+	0.983762i	$0.557441\pi$
$312$		0			0
$313$		−	3.87650i		−	0.219113i	−0.993981	−	0.109556i	$-0.965057\pi$
$313$		−	3.87650i		−	0.219113i	0.993981	−	0.109556i	$-0.0349430\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.0616			1.12677			0.563386	−	0.826194i	$-0.309498\pi$
$317$	20.0616			1.12677			0.563386	+	0.826194i	$0.309498\pi$
$318$		0			0
$319$		−	19.2087i		−	1.07548i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.92820			−0.385496
$324$		0			0
$325$	9.35425	+	15.4110i	0.518880	+	0.854847i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−27.1652			−1.49766
$330$		0			0
$331$		−	24.5704i		−	1.35051i	−0.737585	−	0.675254i	$-0.764034\pi$
$331$		−	24.5704i		−	1.35051i	0.737585	−	0.675254i	$-0.235966\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.41742	−	3.60681i	0.241350	−	0.197061i
$336$		0			0
$337$		−	23.4724i		−	1.27862i	−0.768947	−	0.639312i	$-0.779219\pi$
$337$		−	23.4724i		−	1.27862i	0.768947	−	0.639312i	$-0.220781\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.91288			0.428506
$342$		0			0
$343$	22.6120			1.22093
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	2.07310i		−	0.111290i	−0.998451	−	0.0556448i	$-0.982279\pi$
$347$		−	2.07310i		−	0.111290i	0.998451	−	0.0556448i	$-0.0177215\pi$
$348$		0			0
$349$		−	32.1701i		−	1.72203i	−0.508581	−	0.861014i	$-0.669830\pi$
$349$		−	32.1701i		−	1.72203i	0.508581	−	0.861014i	$-0.330170\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.20520			0.330270			0.165135	−	0.986271i	$-0.447194\pi$
$353$	6.20520			0.330270			0.165135	+	0.986271i	$0.447194\pi$
$354$		0			0
$355$	0.417424	+	0.511238i	0.0221546	+	0.0271337i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	21.7419i		−	1.14749i	−0.819032	−	0.573747i	$-0.805489\pi$
$359$		−	21.7419i		−	1.14749i	0.819032	−	0.573747i	$-0.194511\pi$
$360$		0			0
$361$	−9.74773			−0.513038
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.41742	−	3.60681i	0.231219	−	0.188789i
$366$		0			0
$367$		−	9.28672i		−	0.484763i	−0.970181	−	0.242381i	$-0.922071\pi$
$367$		−	9.28672i		−	0.484763i	0.970181	−	0.242381i	$-0.0779285\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			27.1036i			1.40715i
$372$		0			0
$373$			14.6969i			0.760979i	0.924785	+	0.380489i	$0.124244\pi$
$373$			14.6969i			0.760979i	−0.924785	+	0.380489i	$0.875756\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.0616	+	18.5734i	−1.03323	+	0.956581i
$378$		0			0
$379$			1.35261i			0.0694789i	0.999396	+	0.0347394i	$0.0110601\pi$
$379$			1.35261i			0.0694789i	−0.999396	+	0.0347394i	$0.988940\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.6156			0.951213			0.475607	−	0.879658i	$-0.342229\pi$
$383$	18.6156			0.951213			0.475607	+	0.879658i	$0.342229\pi$
$384$		0			0
$385$	15.6838	+	19.2087i	0.799321	+	0.978964i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	−2.33030			−0.117848
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−19.3386	+	15.7899i	−0.973031	+	0.794477i
$396$		0			0
$397$	25.1624			1.26287			0.631433	−	0.775431i	$-0.282467\pi$
$397$	25.1624			1.26287			0.631433	+	0.775431i	$0.282467\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	22.6274i		−	1.12996i	−0.825105	−	0.564980i	$-0.808884\pi$
$401$		−	22.6274i		−	1.12996i	0.825105	−	0.564980i	$-0.191116\pi$
$402$		0			0
$403$	−7.65120	−	8.26424i	−0.381134	−	0.411671i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.46084i			0.320252i
$408$		0			0
$409$			6.71430i			0.332001i	0.986126	+	0.166000i	$0.0530853\pi$
$409$			6.71430i			0.332001i	−0.986126	+	0.166000i	$0.946915\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			45.6527i			2.24642i
$414$		0			0
$415$	1.25227	−	1.02248i	0.0614717	−	0.0501914i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−33.4955			−1.63636			−0.818180	−	0.574962i	$-0.805017\pi$
$419$	−33.4955			−1.63636			−0.818180	+	0.574962i	$0.805017\pi$
$420$		0			0
$421$			17.4377i			0.849861i	0.905226	+	0.424930i	$0.139701\pi$
$421$			17.4377i			0.849861i	−0.905226	+	0.424930i	$0.860299\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.33030	+	1.29217i	0.307065	+	0.0626793i
$426$		0			0
$427$	15.6838			0.758993
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	30.2272i		−	1.45599i	−0.685581	−	0.727997i	$-0.740452\pi$
$431$		−	30.2272i		−	1.45599i	0.685581	−	0.727997i	$-0.259548\pi$
$432$		0			0
$433$		−	9.79796i		−	0.470860i	−0.971891	−	0.235430i	$-0.924350\pi$
$433$		−	9.79796i		−	0.470860i	0.971891	−	0.235430i	$-0.0756498\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.66930			−0.462546
$438$		0			0
$439$	−37.4955			−1.78956			−0.894780	−	0.446507i	$-0.852668\pi$
$439$	−37.4955			−1.78956			−0.894780	+	0.446507i	$0.852668\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			25.2758i			1.20089i	0.799666	+	0.600445i	$0.205010\pi$
$443$			25.2758i			1.20089i	−0.799666	+	0.600445i	$0.794990\pi$
$444$		0			0
$445$	−16.0000	−	19.5959i	−0.758473	−	0.928936i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			34.9986i			1.65168i	0.563901	+	0.825842i	$0.309300\pi$
$449$			34.9986i			1.65168i	−0.563901	+	0.825842i	$0.690700\pi$
$450$		0			0
$451$	−20.0000			−0.941763
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.89646	−	34.9537i	0.229550	−	1.63865i
$456$		0			0
$457$	1.63670			0.0765616			0.0382808	−	0.999267i	$-0.487812\pi$
$457$	1.63670			0.0765616			0.0382808	+	0.999267i	$0.487812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			0.590327i			0.0274943i	0.999906	+	0.0137471i	$0.00437599\pi$
$461$			0.590327i			0.0274943i	−0.999906	+	0.0137471i	$0.995624\pi$
$462$		0			0
$463$	39.0188			1.81336			0.906679	−	0.421821i	$-0.138609\pi$
$463$	39.0188			1.81336			0.906679	+	0.421821i	$0.138609\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	31.4670i		−	1.45612i	−0.685515	−	0.728059i	$-0.740423\pi$
$467$		−	31.4670i		−	1.45612i	0.685515	−	0.728059i	$-0.259577\pi$
$468$		0			0
$469$	−11.1652			−0.515559
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11.1153			0.511082
$474$		0			0
$475$	26.2668	+	5.36169i	1.20520	+	0.246011i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	0.295164i		−	0.0134864i	−0.999977	−	0.00674318i	$-0.997854\pi$
$479$		−	0.295164i		−	0.0134864i	0.999977	−	0.00674318i	$-0.00214644\pi$
$480$		0			0
$481$	6.74773	−	6.24718i	0.307670	−	0.284847i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	21.1652	−	17.2813i	0.961060	−	0.784702i
$486$		0			0
$487$	−9.47860			−0.429517			−0.214758	−	0.976667i	$-0.568896\pi$
$487$	−9.47860			−0.429517			−0.214758	+	0.976667i	$0.568896\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.3303			−1.36879			−0.684394	−	0.729113i	$-0.739933\pi$
$491$	−30.3303			−1.36879			−0.684394	+	0.729113i	$0.739933\pi$
$492$		0			0
$493$			9.79796i			0.441278i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	1.29217i		−	0.0579616i
$498$		0			0
$499$			0.885491i			0.0396400i	0.999804	+	0.0198200i	$0.00630932\pi$
$499$			0.885491i			0.0396400i	−0.999804	+	0.0198200i	$0.993691\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	5.94960i		−	0.265280i	−0.991164	−	0.132640i	$-0.957655\pi$
$503$		−	5.94960i		−	0.265280i	0.991164	−	0.132640i	$-0.0423453\pi$
$504$		0			0
$505$	−15.8745	+	12.9615i	−0.706406	+	0.576778i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	11.9040i		−	0.527637i	−0.964572	−	0.263819i	$-0.915018\pi$
$509$		−	11.9040i		−	0.527637i	0.964572	−	0.263819i	$-0.0849820\pi$
$510$		0			0
$511$	−11.1652			−0.493917
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.1334	+	16.0851i	0.578727	+	0.708793i
$516$		0			0
$517$			15.7194i			0.691339i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−31.5826			−1.38366			−0.691829	−	0.722061i	$-0.743195\pi$
$521$	−31.5826			−1.38366			−0.691829	+	0.722061i	$0.743195\pi$
$522$		0			0
$523$			1.53371i			0.0670647i	0.999438	+	0.0335323i	$0.0106757\pi$
$523$			1.53371i			0.0670647i	−0.999438	+	0.0335323i	$0.989324\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.03620			−0.175820
$528$		0			0
$529$	19.7477			0.858597
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.3386	+	20.8881i	0.837648	+	0.904763i
$534$		0			0
$535$	−14.9608	−	18.3232i	−0.646812	−	0.792180i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	30.8175i		−	1.32741i
$540$		0			0
$541$		−	19.6758i		−	0.845928i	−0.906147	−	0.422964i	$-0.860990\pi$
$541$		−	19.6758i		−	0.845928i	0.906147	−	0.422964i	$-0.139010\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000	−	14.6969i	−0.514024	−	0.629548i
$546$		0			0
$547$		−	15.2082i		−	0.650255i	−0.945670	−	0.325127i	$-0.894593\pi$
$547$		−	15.2082i		−	0.650255i	0.945670	−	0.325127i	$-0.105407\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			40.6554i			1.73198i
$552$		0			0
$553$	48.8788			2.07854
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.65120			−0.324192			−0.162096	−	0.986775i	$-0.551825\pi$
$557$	−7.65120			−0.324192			−0.162096	+	0.986775i	$0.551825\pi$
$558$		0			0
$559$	−10.7477	−	11.6089i	−0.454580	−	0.491003i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	24.5230i		−	1.03352i	−0.856129	−	0.516761i	$-0.827137\pi$
$563$		−	24.5230i		−	1.03352i	0.856129	−	0.516761i	$-0.172863\pi$
$564$		0			0
$565$	−15.6838	−	19.2087i	−0.659823	−	0.808115i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.5826			0.820944			0.410472	−	0.911873i	$-0.365364\pi$
$569$	19.5826			0.820944			0.410472	+	0.911873i	$0.365364\pi$
$570$		0			0
$571$	17.4955			0.732162			0.366081	−	0.930583i	$-0.380699\pi$
$571$	17.4955			0.732162			0.366081	+	0.930583i	$0.380699\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.83485	+	1.80341i	0.368439	+	0.0752072i
$576$		0			0
$577$	22.2704			0.927129			0.463565	−	0.886063i	$-0.346570\pi$
$577$	22.2704			0.927129			0.463565	+	0.886063i	$0.346570\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.16515			−0.131313
$582$		0			0
$583$	15.6838			0.649557
$584$		0			0
$585$		0			0
$586$		0			0
$587$	26.9898			1.11399			0.556994	−	0.830516i	$-0.311954\pi$
$587$	26.9898			1.11399			0.556994	+	0.830516i	$0.311954\pi$
$588$		0			0
$589$	−16.7477			−0.690078
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.8745			0.651888			0.325944	−	0.945389i	$-0.394318\pi$
$593$	15.8745			0.651888			0.325944	+	0.945389i	$0.394318\pi$
$594$		0			0
$595$	−8.00000	−	9.79796i	−0.327968	−	0.401677i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.66970			0.231658			0.115829	−	0.993269i	$-0.463048\pi$
$599$	5.66970			0.231658			0.115829	+	0.993269i	$0.463048\pi$
$600$		0			0
$601$	7.66970			0.312853			0.156427	−	0.987690i	$-0.450002\pi$
$601$	7.66970			0.312853			0.156427	+	0.987690i	$0.450002\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.93725	+	6.48074i	−0.322695	+	0.263480i
$606$		0			0
$607$		−	21.9387i		−	0.890465i	−0.895415	−	0.445232i	$-0.853121\pi$
$607$		−	21.9387i		−	0.890465i	0.895415	−	0.445232i	$-0.146879\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16.4174	−	15.1996i	0.664178	−	0.614909i
$612$		0			0
$613$	−17.7019			−0.714973			−0.357487	−	0.933918i	$-0.616366\pi$
$613$	−17.7019			−0.714973			−0.357487	+	0.933918i	$0.616366\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−35.2131			−1.41763			−0.708813	−	0.705396i	$-0.750769\pi$
$617$	−35.2131			−1.41763			−0.708813	+	0.705396i	$0.750769\pi$
$618$		0			0
$619$			39.3028i			1.57971i	0.613291	+	0.789857i	$0.289845\pi$
$619$			39.3028i			1.57971i	−0.613291	+	0.789857i	$0.710155\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			49.5292i			1.98434i
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	3.29555i		−	0.131402i
$630$		0			0
$631$		−	30.8175i		−	1.22683i	−0.789762	−	0.613413i	$-0.789796\pi$
$631$		−	30.8175i		−	1.22683i	0.789762	−	0.613413i	$-0.210204\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.723000	−	0.885491i	−0.0286914	−	0.0351396i
$636$		0			0
$637$	−32.1860	+	29.7984i	−1.27525	+	1.18066i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.3303			−0.487018			−0.243509	−	0.969899i	$-0.578298\pi$
$641$	−12.3303			−0.487018			−0.243509	+	0.969899i	$0.578298\pi$
$642$		0			0
$643$	−35.7454			−1.40966			−0.704831	−	0.709375i	$-0.748977\pi$
$643$	−35.7454			−1.40966			−0.704831	+	0.709375i	$0.748977\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	32.4895i		−	1.27729i	−0.769501	−	0.638646i	$-0.779495\pi$
$647$		−	32.4895i		−	1.27729i	0.769501	−	0.638646i	$-0.220505\pi$
$648$		0			0
$649$	26.4174			1.03697
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	44.6302i		−	1.74651i	−0.487259	−	0.873257i	$-0.662003\pi$
$653$		−	44.6302i		−	1.74651i	0.487259	−	0.873257i	$-0.337997\pi$
$654$		0			0
$655$	−5.48220	+	4.47620i	−0.214207	+	0.174900i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	33.4955			1.30480			0.652399	−	0.757876i	$-0.273763\pi$
$659$	33.4955			1.30480			0.652399	+	0.757876i	$0.273763\pi$
$660$		0			0
$661$		−	8.48528i		−	0.330039i	−0.986290	−	0.165020i	$-0.947231\pi$
$661$		−	8.48528i		−	0.330039i	0.986290	−	0.165020i	$-0.0527687\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−33.1950	−	40.6554i	−1.28725	−	1.57655i
$666$		0			0
$667$			13.6745i			0.529477i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	9.07561i		−	0.350360i
$672$		0			0
$673$		−	23.4724i		−	0.904795i	−0.891816	−	0.452398i	$-0.850569\pi$
$673$		−	23.4724i		−	0.904795i	0.891816	−	0.452398i	$-0.149431\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			23.7421i			0.912483i	0.889856	+	0.456242i	$0.150805\pi$
$677$			23.7421i			0.912483i	−0.889856	+	0.456242i	$0.849195\pi$
$678$		0			0
$679$	−53.4955			−2.05297
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26.9898			1.03274			0.516368	−	0.856367i	$-0.327284\pi$
$683$	26.9898			1.03274			0.516368	+	0.856367i	$0.327284\pi$
$684$		0			0
$685$	18.0000	−	14.6969i	0.687745	−	0.561541i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.1652	−	16.3802i	−0.577746	−	0.624037i
$690$		0			0
$691$		−	20.0942i		−	0.764418i	−0.924076	−	0.382209i	$-0.875164\pi$
$691$		−	20.0942i		−	0.764418i	0.924076	−	0.382209i	$-0.124836\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12.4104	−	10.1331i	0.470754	−	0.384369i
$696$		0			0
$697$	10.2016			0.386413
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14.8348			0.560304			0.280152	−	0.959956i	$-0.409615\pi$
$701$	14.8348			0.560304			0.280152	+	0.959956i	$0.409615\pi$
$702$		0			0
$703$		−	13.6745i		−	0.515742i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	40.1232			1.50899
$708$		0			0
$709$		−	15.1996i		−	0.570832i	−0.958404	−	0.285416i	$-0.907868\pi$
$709$		−	15.1996i		−	0.570832i	0.958404	−	0.285416i	$-0.0921318\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.63310			−0.210961
$714$		0			0
$715$	−20.2263	−	2.83339i	−0.756422	−	0.105963i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	33.4955			1.24917			0.624585	−	0.780957i	$-0.285268\pi$
$719$	33.4955			1.24917			0.624585	+	0.780957i	$0.285268\pi$
$720$		0			0
$721$		−	40.6554i		−	1.51409i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.58258	−	37.1469i	0.281610	−	1.37960i
$726$		0			0
$727$			15.2082i			0.564040i	0.959409	+	0.282020i	$0.0910045\pi$
$727$			15.2082i			0.564040i	−0.959409	+	0.282020i	$0.908996\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.66970			−0.209701
$732$		0			0
$733$	−33.0043			−1.21904			−0.609521	−	0.792770i	$-0.708638\pi$
$733$	−33.0043			−1.21904			−0.609521	+	0.792770i	$0.708638\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			6.46084i			0.237988i
$738$		0			0
$739$		−	33.5228i		−	1.23315i	−0.787294	−	0.616577i	$-0.788519\pi$
$739$		−	33.5228i		−	1.23315i	0.787294	−	0.616577i	$-0.211481\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.6156			0.682940			0.341470	−	0.939893i	$-0.389075\pi$
$743$	18.6156			0.682940			0.341470	+	0.939893i	$0.389075\pi$
$744$		0			0
$745$	−4.00000	−	4.89898i	−0.146549	−	0.179485i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			46.3123i			1.69221i
$750$		0			0
$751$	4.83485			0.176426			0.0882131	−	0.996102i	$-0.471884\pi$
$751$	4.83485			0.176426			0.0882131	+	0.996102i	$0.471884\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−22.7477	−	27.8602i	−0.827875	−	1.01394i
$756$		0			0
$757$		−	26.3264i		−	0.956851i	−0.878128	−	0.478426i	$-0.841208\pi$
$757$		−	26.3264i		−	0.956851i	0.878128	−	0.478426i	$-0.158792\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			11.3137i			0.410122i	0.978749	+	0.205061i	$0.0657392\pi$
$761$			11.3137i			0.410122i	−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$			37.1469i			1.34481i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−25.5438	−	27.5905i	−0.922334	−	0.996234i
$768$		0			0
$769$			42.8935i			1.54678i	0.633930	+	0.773390i	$0.281441\pi$
$769$			42.8935i			1.54678i	−0.633930	+	0.773390i	$0.718559\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.20520			0.223186			0.111593	−	0.993754i	$-0.464405\pi$
$773$	6.20520			0.223186			0.111593	+	0.993754i	$0.464405\pi$
$774$		0			0
$775$	15.3024	+	3.12359i	0.549679	+	0.112203i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	42.3303			1.51664
$780$		0			0
$781$	−0.747727			−0.0267558
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−31.7490	−	38.8844i	−1.13317	−	1.38785i
$786$		0			0
$787$	−8.03260			−0.286331			−0.143166	−	0.989699i	$-0.545728\pi$
$787$	−8.03260			−0.286331			−0.143166	+	0.989699i	$0.545728\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			48.5504i			1.72625i
$792$		0			0
$793$	−9.47860	+	8.77548i	−0.336595	+	0.311627i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	34.8322i		−	1.23382i	−0.787033	−	0.616911i	$-0.788384\pi$
$797$		−	34.8322i		−	1.23382i	0.787033	−	0.616911i	$-0.211616\pi$
$798$		0			0
$799$		−	8.01816i		−	0.283662i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			6.46084i			0.227998i
$804$		0			0
$805$	−11.1652	−	13.6745i	−0.393520	−	0.481961i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22.7477			0.799767			0.399884	−	0.916566i	$-0.369050\pi$
$809$	22.7477			0.799767			0.399884	+	0.916566i	$0.369050\pi$
$810$		0			0
$811$		−	12.0760i		−	0.424045i	−0.977265	−	0.212023i	$-0.931995\pi$
$811$		−	12.0760i		−	0.424045i	0.977265	−	0.212023i	$-0.0680051\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.5826	+	15.9891i	−0.685948	+	0.560074i
$816$		0			0
$817$	−23.5257			−0.823060
$818$		0			0
$819$		0			0
$820$		0			0
$821$			19.7990i			0.690990i	0.938421	+	0.345495i	$0.112289\pi$
$821$			19.7990i			0.690990i	−0.938421	+	0.345495i	$0.887711\pi$
$822$		0			0
$823$			39.7031i			1.38396i	0.721916	+	0.691981i	$0.243262\pi$
$823$			39.7031i			1.38396i	−0.721916	+	0.691981i	$0.756738\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−32.4720			−1.12916			−0.564581	−	0.825377i	$-0.690962\pi$
$827$	−32.4720			−1.12916			−0.564581	+	0.825377i	$0.690962\pi$
$828$		0			0
$829$	40.2432			1.39770			0.698852	−	0.715267i	$-0.253695\pi$
$829$	40.2432			1.39770			0.698852	+	0.715267i	$0.253695\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			15.7194i			0.544645i
$834$		0			0
$835$	−1.25227	+	1.02248i	−0.0433367	+	0.0353843i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	0.762282i		−	0.0263169i	−0.999913	−	0.0131585i	$-0.995811\pi$
$839$		−	0.762282i		−	0.0263169i	0.999913	−	0.0131585i	$-0.00418859\pi$
$840$		0			0
$841$	28.4955			0.982602
$842$		0			0
$843$		0			0
$844$		0			0
$845$	16.5983	+	23.8642i	0.570997	+	0.820952i
$846$		0			0
$847$	20.0616			0.689325
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	4.59941i		−	0.157666i
$852$		0			0
$853$	39.0188			1.33598			0.667989	−	0.744171i	$-0.267155\pi$
$853$	39.0188			1.33598			0.667989	+	0.744171i	$0.267155\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			44.3605i			1.51533i	0.652646	+	0.757663i	$0.273659\pi$
$857$			44.3605i			1.51533i	−0.652646	+	0.757663i	$0.726341\pi$
$858$		0			0
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.20520			0.211228			0.105614	−	0.994407i	$-0.466319\pi$
$863$	6.20520			0.211228			0.105614	+	0.994407i	$0.466319\pi$
$864$		0			0
$865$	5.10080	+	6.24718i	0.173432	+	0.212411i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	28.2843i		−	0.959478i
$870$		0			0
$871$	6.74773	−	6.24718i	0.228638	−	0.211678i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	22.7477	+	43.3380i	0.769014	+	1.46509i
$876$		0			0
$877$	−20.5939			−0.695407			−0.347703	−	0.937605i	$-0.613038\pi$
$877$	−20.5939			−0.695407			−0.347703	+	0.937605i	$0.613038\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.41742			−0.148827			−0.0744134	−	0.997227i	$-0.523708\pi$
$881$	−4.41742			−0.148827			−0.0744134	+	0.997227i	$0.523708\pi$
$882$		0			0
$883$			31.7367i			1.06802i	0.845477	+	0.534012i	$0.179316\pi$
$883$			31.7367i			1.06802i	−0.845477	+	0.534012i	$0.820684\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	40.2425i		−	1.35121i	−0.737264	−	0.675605i	$-0.763883\pi$
$887$		−	40.2425i		−	1.35121i	0.737264	−	0.675605i	$-0.236117\pi$
$888$		0			0
$889$			2.23810i			0.0750635i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	33.2704i		−	1.11335i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			23.6849i			0.789934i
$900$		0			0
$901$	−8.00000			−0.266519
$902$		0			0
$903$		0			0
$904$		0			0
$905$	40.8462	−	33.3508i	1.35777	−	1.10862i
$906$		0			0
$907$		−	39.7031i		−	1.31832i	−0.752003	−	0.659159i	$-0.770912\pi$
$907$		−	39.7031i		−	1.31832i	0.752003	−	0.659159i	$-0.229088\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−17.6697			−0.585423			−0.292712	−	0.956201i	$-0.594558\pi$
$911$	−17.6697			−0.585423			−0.292712	+	0.956201i	$0.594558\pi$
$912$		0			0
$913$			1.83155i			0.0606155i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	13.8564			0.457579
$918$		0			0
$919$	13.6697			0.450922			0.225461	−	0.974252i	$-0.427611\pi$
$919$	13.6697			0.450922			0.225461	+	0.974252i	$0.427611\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.723000	+	0.780929i	0.0237978	+	0.0257046i
$924$		0			0
$925$	−2.55040	+	12.4944i	−0.0838567	+	0.410812i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			43.9510i			1.44198i	0.692943	+	0.720992i	$0.256314\pi$
$929$			43.9510i			1.44198i	−0.692943	+	0.720992i	$0.743686\pi$
$930$		0			0
$931$			65.2258i			2.13769i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.66970	+	4.62929i	−0.185419	+	0.151394i
$936$		0			0
$937$		−	7.75301i		−	0.253280i	−0.991949	−	0.126640i	$-0.959581\pi$
$937$		−	7.75301i		−	0.253280i	0.991949	−	0.126640i	$-0.0404192\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	20.8564i		−	0.679900i	−0.940443	−	0.339950i	$-0.889590\pi$
$941$		−	20.8564i		−	0.679900i	0.940443	−	0.339950i	$-0.110410\pi$
$942$		0			0
$943$	14.2378			0.463647
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−35.3640			−1.14918			−0.574588	−	0.818443i	$-0.694838\pi$
$947$	−35.3640			−1.14918			−0.574588	+	0.818443i	$0.694838\pi$
$948$		0			0
$949$	6.74773	−	6.24718i	0.219040	−	0.202792i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			52.1135i			1.68812i	0.536247	+	0.844061i	$0.319842\pi$
$953$			52.1135i			1.68812i	−0.536247	+	0.844061i	$0.680158\pi$
$954$		0			0
$955$	−20.7846	+	16.9706i	−0.672574	+	0.549155i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−45.4955			−1.46912
$960$		0			0
$961$	21.2432			0.685264
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.330303	+	0.269691i	−0.0106328	+	0.00868166i
$966$		0			0
$967$	44.5010			1.43106			0.715528	−	0.698584i	$-0.246186\pi$
$967$	44.5010			1.43106			0.715528	+	0.698584i	$0.246186\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−48.6606			−1.56159			−0.780797	−	0.624785i	$-0.785186\pi$
$971$	−48.6606			−1.56159			−0.780797	+	0.624785i	$0.785186\pi$
$972$		0			0
$973$	−31.3676			−1.00560
$974$		0			0
$975$		0			0
$976$		0			0
$977$	58.7388			1.87922			0.939611	−	0.342245i	$-0.111187\pi$
$977$	58.7388			1.87922			0.939611	+	0.342245i	$0.111187\pi$
$978$		0			0
$979$	28.6606			0.915997
$980$		0			0
$981$		0			0
$982$		0			0
$983$	58.7388			1.87348			0.936739	−	0.350029i	$-0.113828\pi$
$983$	58.7388			1.87348			0.936739	+	0.350029i	$0.113828\pi$
$984$		0			0
$985$	−39.4955	+	32.2479i	−1.25843	+	1.02750i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.91288			−0.251615
$990$		0			0
$991$	−25.4955			−0.809890			−0.404945	−	0.914341i	$-0.632709\pi$
$991$	−25.4955			−0.809890			−0.404945	+	0.914341i	$0.632709\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.92820	−	5.65685i	0.219639	−	0.179334i
$996$		0			0
$997$		−	30.4164i		−	0.963296i	−0.876365	−	0.481648i	$-0.840038\pi$
$997$		−	30.4164i		−	0.963296i	0.876365	−	0.481648i	$-0.159962\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.j.d.649.4		8
3.2	odd	2		260.2.d.a.129.2	yes	8
5.4	even	2	inner	2340.2.j.d.649.7		8
12.11	even	2		1040.2.f.e.129.8		8
13.12	even	2	inner	2340.2.j.d.649.5		8
15.2	even	4		1300.2.f.f.701.2		8
15.8	even	4		1300.2.f.f.701.7		8
15.14	odd	2		260.2.d.a.129.7	yes	8
39.5	even	4		3380.2.c.d.2029.1		8
39.8	even	4		3380.2.c.d.2029.2		8
39.38	odd	2		260.2.d.a.129.1	✓	8
60.59	even	2		1040.2.f.e.129.1		8
65.64	even	2	inner	2340.2.j.d.649.2		8
156.155	even	2		1040.2.f.e.129.7		8
195.38	even	4		1300.2.f.f.701.8		8
195.44	even	4		3380.2.c.d.2029.7		8
195.77	even	4		1300.2.f.f.701.1		8
195.164	even	4		3380.2.c.d.2029.8		8
195.194	odd	2		260.2.d.a.129.8	yes	8
780.779	even	2		1040.2.f.e.129.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.d.a.129.1	✓	8	39.38	odd	2
260.2.d.a.129.2	yes	8	3.2	odd	2
260.2.d.a.129.7	yes	8	15.14	odd	2
260.2.d.a.129.8	yes	8	195.194	odd	2
1040.2.f.e.129.1		8	60.59	even	2
1040.2.f.e.129.2		8	780.779	even	2
1040.2.f.e.129.7		8	156.155	even	2
1040.2.f.e.129.8		8	12.11	even	2
1300.2.f.f.701.1		8	195.77	even	4
1300.2.f.f.701.2		8	15.2	even	4
1300.2.f.f.701.7		8	15.8	even	4
1300.2.f.f.701.8		8	195.38	even	4
2340.2.j.d.649.2		8	65.64	even	2	inner
2340.2.j.d.649.4		8	1.1	even	1	trivial
2340.2.j.d.649.5		8	13.12	even	2	inner
2340.2.j.d.649.7		8	5.4	even	2	inner
3380.2.c.d.2029.1		8	39.5	even	4
3380.2.c.d.2029.2		8	39.8	even	4
3380.2.c.d.2029.7		8	195.44	even	4
3380.2.c.d.2029.8		8	195.164	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.41421i) q^{5} +4.37780 q^{7} -2.53326i q^{11} +(-2.64575 + 2.44949i) q^{13} +1.29217i q^{17} +5.36169i q^{19} +1.80341i q^{23} +(1.00000 - 4.89898i) q^{25} +7.58258 q^{29} +3.12359i q^{31} +(-7.58258 + 6.19115i) q^{35} -2.55040 q^{37} -7.89495i q^{41} +4.38774i q^{43} -6.20520 q^{47} +12.1652 q^{49} +6.19115i q^{53} +(3.58258 + 4.38774i) q^{55} +10.4282i q^{59} +3.58258 q^{61} +(1.11847 - 7.98430i) q^{65} -2.55040 q^{67} -0.295164i q^{71} -2.55040 q^{73} -11.0901i q^{77} +11.1652 q^{79} -0.723000 q^{83} +(-1.82740 - 2.23810i) q^{85} +11.3137i q^{89} +(-11.5826 + 10.7234i) q^{91} +(-7.58258 - 9.28672i) q^{95} -12.2197 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{25} + 24 q^{29} - 24 q^{35} + 24 q^{49} - 8 q^{55} - 8 q^{61} + 16 q^{79} - 56 q^{91} - 24 q^{95}+O(q^{100}) \) 8 * q + 8 * q^25 + 24 * q^29 - 24 * q^35 + 24 * q^49 - 8 * q^55 - 8 * q^61 + 16 * q^79 - 56 * q^91 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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