LMFDB - Embedded newform 2300.2.c.j.1749.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1749,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 29x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 29*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.6
Root			$0.631352i$ of defining polynomial
Character	$\chi$	$=$	2300.1749
Dual form			2300.2.c.j.1749.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.631352i q^{3} -3.16780i q^{7} +2.60139 q^{9} +O(q^{10})$	$q+0.631352i q^{3} -3.16780i q^{7} +2.60139 q^{9} -1.90510 q^{11} +6.76920i q^{13} +0.737296i q^{17} -3.90510 q^{19} +2.00000 q^{21} +1.00000i q^{23} +3.53645i q^{27} +0.935058 q^{29} +1.02996 q^{31} -1.20279i q^{33} +5.94008i q^{37} -4.27375 q^{39} +11.0019 q^{41} +3.90510i q^{43} +5.47151i q^{47} -3.03498 q^{49} -0.465493 q^{51} -3.07290i q^{53} -2.46549i q^{57} -4.90510 q^{59} +11.0729 q^{61} -8.24071i q^{63} -3.20279i q^{67} -0.631352 q^{69} +8.86410 q^{71} +9.40055i q^{73} +6.03498i q^{77} -1.56949 q^{79} +5.57144 q^{81} +2.64240i q^{83} +0.590351i q^{87} +18.6113 q^{89} +21.4435 q^{91} +0.650266i q^{93} -15.3486i q^{97} -4.95592 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^9	$ 8 q - 4 q^{9} + 2 q^{11} - 14 q^{19} + 16 q^{21} + 10 q^{29} + 28 q^{31} - 22 q^{39} + 6 q^{41} - 2 q^{49} + 56 q^{51} - 22 q^{59} + 44 q^{61} + 36 q^{71} - 50 q^{79} + 50 q^{91} + 18 q^{99}+O(q^{100}) $ 8 * q - 4 * q^9 + 2 * q^11 - 14 * q^19 + 16 * q^21 + 10 * q^29 + 28 * q^31 - 22 * q^39 + 6 * q^41 - 2 * q^49 + 56 * q^51 - 22 * q^59 + 44 * q^61 + 36 * q^71 - 50 * q^79 + 50 * q^91 + 18 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0.631352i	0.364511i	0.983251	+	0.182256i	$0.0583399\pi$
$3$	0.631352i	0.364511i	−0.983251	+	0.182256i	$0.941660\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.16780i	− 1.19732i	−0.801004	−	0.598659i	$-0.795701\pi$
$7$	− 3.16780i	− 1.19732i	0.801004	−	0.598659i	$-0.204299\pi$
$8$	0	0
$9$	2.60139	0.867132
$10$	0	0
$11$	−1.90510	−0.574409	−0.287205	−	0.957869i	$-0.592726\pi$
$11$	−1.90510	−0.574409	−0.287205	+	0.957869i	$0.592726\pi$
$12$	0	0
$13$	6.76920i	1.87744i	0.344683	+	0.938719i	$0.387986\pi$
$13$	6.76920i	1.87744i	−0.344683	+	0.938719i	$0.612014\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.737296i	0.178820i	0.995995	+	0.0894102i	$0.0284982\pi$
$17$	0.737296i	0.178820i	−0.995995	+	0.0894102i	$0.971502\pi$
$18$	0	0
$19$	−3.90510	−0.895891	−0.447946	−	0.894061i	$-0.647844\pi$
$19$	−3.90510	−0.895891	−0.447946	+	0.894061i	$0.647844\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.53645i	0.680591i
$28$	0	0
$29$	0.935058	0.173636	0.0868179	−	0.996224i	$-0.472330\pi$
$29$	0.935058	0.173636	0.0868179	+	0.996224i	$0.472330\pi$
$30$	0	0
$31$	1.02996	0.184986	0.0924929	−	0.995713i	$-0.470516\pi$
$31$	1.02996	0.184986	0.0924929	+	0.995713i	$0.470516\pi$
$32$	0	0
$33$	− 1.20279i	− 0.209379i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.94008i	0.976544i	0.872692	+	0.488272i	$0.162373\pi$
$37$	5.94008i	0.976544i	−0.872692	+	0.488272i	$0.837627\pi$
$38$	0	0
$39$	−4.27375	−0.684347
$40$	0	0
$41$	11.0019	1.71822	0.859108	−	0.511795i	$-0.171019\pi$
$41$	11.0019	1.71822	0.859108	+	0.511795i	$0.171019\pi$
$42$	0	0
$43$	3.90510i	0.595522i	0.954640	+	0.297761i	$0.0962399\pi$
$43$	3.90510i	0.595522i	−0.954640	+	0.297761i	$0.903760\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.47151i	0.798102i	0.916929	+	0.399051i	$0.130660\pi$
$47$	5.47151i	0.798102i	−0.916929	+	0.399051i	$0.869340\pi$
$48$	0	0
$49$	−3.03498	−0.433569
$50$	0	0
$51$	−0.465493	−0.0651821
$52$	0	0
$53$	− 3.07290i	− 0.422096i	−0.977476	−	0.211048i	$-0.932312\pi$
$53$	− 3.07290i	− 0.422096i	0.977476	−	0.211048i	$-0.0676876\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.46549i	− 0.326563i
$58$	0	0
$59$	−4.90510	−0.638590	−0.319295	−	0.947655i	$-0.603446\pi$
$59$	−4.90510	−0.638590	−0.319295	+	0.947655i	$0.603446\pi$
$60$	0	0
$61$	11.0729	1.41774	0.708870	−	0.705339i	$-0.249205\pi$
$61$	11.0729	1.41774	0.708870	+	0.705339i	$0.249205\pi$
$62$	0	0
$63$	− 8.24071i	− 1.03823i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.20279i	− 0.391283i	−0.980675	−	0.195641i	$-0.937321\pi$
$67$	− 3.20279i	− 0.391283i	0.980675	−	0.195641i	$-0.0626789\pi$
$68$	0	0
$69$	−0.631352	−0.0760059
$70$	0	0
$71$	8.86410	1.05197	0.525987	−	0.850492i	$-0.323696\pi$
$71$	8.86410	1.05197	0.525987	+	0.850492i	$0.323696\pi$
$72$	0	0
$73$	9.40055i	1.10025i	0.835082	+	0.550126i	$0.185420\pi$
$73$	9.40055i	1.10025i	−0.835082	+	0.550126i	$0.814580\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.03498i	0.687750i
$78$	0	0
$79$	−1.56949	−0.176582	−0.0882908	−	0.996095i	$-0.528140\pi$
$79$	−1.56949	−0.176582	−0.0882908	+	0.996095i	$0.528140\pi$
$80$	0	0
$81$	5.57144	0.619049
$82$	0	0
$83$	2.64240i	0.290041i	0.989429	+	0.145020i	$0.0463248\pi$
$83$	2.64240i	0.290041i	−0.989429	+	0.145020i	$0.953675\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.590351i	0.0632922i
$88$	0	0
$89$	18.6113	1.97279	0.986397	−	0.164380i	$-0.0525624\pi$
$89$	18.6113	1.97279	0.986397	+	0.164380i	$0.0525624\pi$
$90$	0	0
$91$	21.4435	2.24789
$92$	0	0
$93$	0.650266i	0.0674294i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 15.3486i	− 1.55841i	−0.626767	−	0.779207i	$-0.715622\pi$
$97$	− 15.3486i	− 1.55841i	0.626767	−	0.779207i	$-0.284378\pi$
$98$	0	0
$99$	−4.95592	−0.498088
$100$	0	0
$101$	−9.18079	−0.913523	−0.456762	−	0.889589i	$-0.650991\pi$
$101$	−9.18079	−0.913523	−0.456762	+	0.889589i	$0.650991\pi$
$102$	0	0
$103$	− 7.50341i	− 0.739333i	−0.929164	−	0.369667i	$-0.879472\pi$
$103$	− 7.50341i	− 0.739333i	0.929164	−	0.369667i	$-0.120528\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.5384i	1.50215i	0.660215	+	0.751077i	$0.270465\pi$
$107$	15.5384i	1.50215i	−0.660215	+	0.751077i	$0.729535\pi$
$108$	0	0
$109$	−3.13282	−0.300070	−0.150035	−	0.988681i	$-0.547939\pi$
$109$	−3.13282	−0.300070	−0.150035	+	0.988681i	$0.547939\pi$
$110$	0	0
$111$	−3.75029	−0.355961
$112$	0	0
$113$	17.0130i	1.60045i	0.599702	+	0.800224i	$0.295286\pi$
$113$	17.0130i	1.60045i	−0.599702	+	0.800224i	$0.704714\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	17.6094i	1.62799i
$118$	0	0
$119$	2.33561	0.214105
$120$	0	0
$121$	−7.37059	−0.670054
$122$	0	0
$123$	6.94610i	0.626309i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.97004i	− 0.263549i	−0.991280	−	0.131774i	$-0.957933\pi$
$127$	− 2.97004i	− 0.263549i	0.991280	−	0.131774i	$-0.0420674\pi$
$128$	0	0
$129$	−2.46549	−0.217075
$130$	0	0
$131$	−4.22967	−0.369548	−0.184774	−	0.982781i	$-0.559155\pi$
$131$	−4.22967	−0.369548	−0.184774	+	0.982781i	$0.559155\pi$
$132$	0	0
$133$	12.3706i	1.07267i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.52541i	0.386632i	0.981137	+	0.193316i	$0.0619242\pi$
$137$	4.52541i	0.386632i	−0.981137	+	0.193316i	$0.938076\pi$
$138$	0	0
$139$	7.00991	0.594573	0.297286	−	0.954788i	$-0.403918\pi$
$139$	7.00991	0.594573	0.297286	+	0.954788i	$0.403918\pi$
$140$	0	0
$141$	−3.45445	−0.290917
$142$	0	0
$143$	− 12.8960i	− 1.07842i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.91614i	− 0.158041i
$148$	0	0
$149$	−8.73113	−0.715282	−0.357641	−	0.933859i	$-0.616419\pi$
$149$	−8.73113	−0.715282	−0.357641	+	0.933859i	$0.616419\pi$
$150$	0	0
$151$	19.2697	1.56814	0.784072	−	0.620670i	$-0.213139\pi$
$151$	19.2697	1.56814	0.784072	+	0.620670i	$0.213139\pi$
$152$	0	0
$153$	1.91800i	0.155061i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.65529i	0.531150i	0.964090	+	0.265575i	$0.0855618\pi$
$157$	6.65529i	0.531150i	−0.964090	+	0.265575i	$0.914438\pi$
$158$	0	0
$159$	1.94008	0.153859
$160$	0	0
$161$	3.16780	0.249658
$162$	0	0
$163$	9.11585i	0.714009i	0.934103	+	0.357004i	$0.116202\pi$
$163$	9.11585i	0.714009i	−0.934103	+	0.357004i	$0.883798\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.6463i	1.67504i	0.546407	+	0.837520i	$0.315995\pi$
$167$	21.6463i	1.67504i	−0.546407	+	0.837520i	$0.684005\pi$
$168$	0	0
$169$	−32.8221	−2.52477
$170$	0	0
$171$	−10.1587	−0.776856
$172$	0	0
$173$	12.8102i	0.973941i	0.873418	+	0.486971i	$0.161898\pi$
$173$	12.8102i	0.973941i	−0.873418	+	0.486971i	$0.838102\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3.09685i	− 0.232773i
$178$	0	0
$179$	−25.9769	−1.94160	−0.970801	−	0.239886i	$-0.922890\pi$
$179$	−25.9769	−1.94160	−0.970801	+	0.239886i	$0.922890\pi$
$180$	0	0
$181$	−13.1966	−0.980897	−0.490449	−	0.871470i	$-0.663167\pi$
$181$	−13.1966	−0.980897	−0.490449	+	0.871470i	$0.663167\pi$
$182$	0	0
$183$	6.99090i	0.516782i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.40462i	− 0.102716i
$188$	0	0
$189$	11.2028	0.814883
$190$	0	0
$191$	19.3136	1.39748	0.698742	−	0.715374i	$-0.253744\pi$
$191$	19.3136	1.39748	0.698742	+	0.715374i	$0.253744\pi$
$192$	0	0
$193$	− 6.70132i	− 0.482372i	−0.970479	−	0.241186i	$-0.922464\pi$
$193$	− 6.70132i	− 0.482372i	0.970479	−	0.241186i	$-0.0775363\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.9320i	0.778871i	0.921054	+	0.389436i	$0.127330\pi$
$197$	10.9320i	0.778871i	−0.921054	+	0.389436i	$0.872670\pi$
$198$	0	0
$199$	−8.77228	−0.621850	−0.310925	−	0.950434i	$-0.600639\pi$
$199$	−8.77228	−0.621850	−0.310925	+	0.950434i	$0.600639\pi$
$200$	0	0
$201$	2.02209	0.142627
$202$	0	0
$203$	− 2.96208i	− 0.207897i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.60139i	0.180809i
$208$	0	0
$209$	7.43961	0.514608
$210$	0	0
$211$	−6.71530	−0.462300	−0.231150	−	0.972918i	$-0.574249\pi$
$211$	−6.71530	−0.462300	−0.231150	+	0.972918i	$0.574249\pi$
$212$	0	0
$213$	5.59637i	0.383457i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.26270i	− 0.221487i
$218$	0	0
$219$	−5.93506	−0.401054
$220$	0	0
$221$	−4.99090	−0.335724
$222$	0	0
$223$	− 21.7661i	− 1.45757i	−0.684744	−	0.728784i	$-0.740086\pi$
$223$	− 21.7661i	− 1.45757i	0.684744	−	0.728784i	$-0.259914\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 23.5485i	− 1.56297i	−0.623927	−	0.781483i	$-0.714464\pi$
$227$	− 23.5485i	− 1.56297i	0.623927	−	0.781483i	$-0.285536\pi$
$228$	0	0
$229$	11.6204	0.767898	0.383949	−	0.923354i	$-0.374564\pi$
$229$	11.6204	0.767898	0.383949	+	0.923354i	$0.374564\pi$
$230$	0	0
$231$	−3.81020	−0.250693
$232$	0	0
$233$	− 12.1178i	− 0.793863i	−0.917848	−	0.396932i	$-0.870075\pi$
$233$	− 12.1178i	− 0.793863i	0.917848	−	0.396932i	$-0.129925\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 0.990902i	− 0.0643660i
$238$	0	0
$239$	5.63429	0.364452	0.182226	−	0.983257i	$-0.441670\pi$
$239$	5.63429	0.364452	0.182226	+	0.983257i	$0.441670\pi$
$240$	0	0
$241$	−5.07907	−0.327172	−0.163586	−	0.986529i	$-0.552306\pi$
$241$	−5.07907	−0.327172	−0.163586	+	0.986529i	$0.552306\pi$
$242$	0	0
$243$	14.1269i	0.906241i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 26.4344i	− 1.68198i
$248$	0	0
$249$	−1.66828	−0.105723
$250$	0	0
$251$	19.4186	1.22569	0.612845	−	0.790203i	$-0.290025\pi$
$251$	19.4186	1.22569	0.612845	+	0.790203i	$0.290025\pi$
$252$	0	0
$253$	− 1.90510i	− 0.119773i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.75507i	0.546126i	0.961996	+	0.273063i	$0.0880368\pi$
$257$	8.75507i	0.546126i	−0.961996	+	0.273063i	$0.911963\pi$
$258$	0	0
$259$	18.8170	1.16923
$260$	0	0
$261$	2.43245	0.150565
$262$	0	0
$263$	16.3457i	1.00792i	0.863728	+	0.503958i	$0.168124\pi$
$263$	16.3457i	1.00792i	−0.863728	+	0.503958i	$0.831876\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.7503i	0.719106i
$268$	0	0
$269$	3.14880	0.191986	0.0959928	−	0.995382i	$-0.469397\pi$
$269$	3.14880	0.191986	0.0959928	+	0.995382i	$0.469397\pi$
$270$	0	0
$271$	25.1525	1.52791	0.763954	−	0.645271i	$-0.223255\pi$
$271$	25.1525	1.52791	0.763954	+	0.645271i	$0.223255\pi$
$272$	0	0
$273$	13.5384i	0.819381i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.8630i	0.953113i	0.879144	+	0.476557i	$0.158115\pi$
$277$	15.8630i	0.953113i	−0.879144	+	0.476557i	$0.841885\pi$
$278$	0	0
$279$	2.67933	0.160407
$280$	0	0
$281$	−16.2057	−0.966752	−0.483376	−	0.875413i	$-0.660590\pi$
$281$	−16.2057	−0.966752	−0.483376	+	0.875413i	$0.660590\pi$
$282$	0	0
$283$	− 30.4814i	− 1.81193i	−0.423350	−	0.905966i	$-0.639146\pi$
$283$	− 30.4814i	− 1.81193i	0.423350	−	0.905966i	$-0.360854\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 34.8520i	− 2.05725i
$288$	0	0
$289$	16.4564	0.968023
$290$	0	0
$291$	9.69037	0.568060
$292$	0	0
$293$	− 15.9339i	− 0.930870i	−0.885082	−	0.465435i	$-0.845898\pi$
$293$	− 15.9339i	− 0.930870i	0.885082	−	0.465435i	$-0.154102\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 6.73730i	− 0.390938i
$298$	0	0
$299$	−6.76920	−0.391473
$300$	0	0
$301$	12.3706	0.713029
$302$	0	0
$303$	− 5.79631i	− 0.332990i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7.93003i	− 0.452591i	−0.974059	−	0.226295i	$-0.927339\pi$
$307$	− 7.93003i	− 0.452591i	0.974059	−	0.226295i	$-0.0726615\pi$
$308$	0	0
$309$	4.73730	0.269495
$310$	0	0
$311$	−28.9960	−1.64421	−0.822107	−	0.569333i	$-0.807201\pi$
$311$	−28.9960	−1.64421	−0.822107	+	0.569333i	$0.807201\pi$
$312$	0	0
$313$	− 21.4564i	− 1.21279i	−0.795165	−	0.606394i	$-0.792616\pi$
$313$	− 21.4564i	− 1.21279i	0.795165	−	0.606394i	$-0.207384\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.70620i	− 0.264327i	−0.991228	−	0.132163i	$-0.957808\pi$
$317$	− 4.70620i	− 0.264327i	0.991228	−	0.132163i	$-0.0421923\pi$
$318$	0	0
$319$	−1.78138	−0.0997381
$320$	0	0
$321$	−9.81020	−0.547552
$322$	0	0
$323$	− 2.87921i	− 0.160204i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.97791i	− 0.109379i
$328$	0	0
$329$	17.3327	0.955581
$330$	0	0
$331$	−5.98611	−0.329027	−0.164513	−	0.986375i	$-0.552605\pi$
$331$	−5.98611	−0.329027	−0.164513	+	0.986375i	$0.552605\pi$
$332$	0	0
$333$	15.4525i	0.846792i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.82319i	0.371683i	0.982580	+	0.185841i	$0.0595011\pi$
$337$	6.82319i	0.371683i	−0.982580	+	0.185841i	$0.940499\pi$
$338$	0	0
$339$	−10.7412	−0.583381
$340$	0	0
$341$	−1.96217	−0.106258
$342$	0	0
$343$	− 12.5604i	− 0.678197i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.50048i	0.187915i	0.995576	+	0.0939577i	$0.0299519\pi$
$347$	3.50048i	0.187915i	−0.995576	+	0.0939577i	$0.970048\pi$
$348$	0	0
$349$	7.44042	0.398276	0.199138	−	0.979971i	$-0.436186\pi$
$349$	7.44042	0.398276	0.199138	+	0.979971i	$0.436186\pi$
$350$	0	0
$351$	−23.9389	−1.27777
$352$	0	0
$353$	4.87822i	0.259642i	0.991537	+	0.129821i	$0.0414402\pi$
$353$	4.87822i	0.259642i	−0.991537	+	0.129821i	$0.958560\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.47459i	0.0780437i
$358$	0	0
$359$	−0.0570722	−0.00301216	−0.00150608	−	0.999999i	$-0.500479\pi$
$359$	−0.0570722	−0.00301216	−0.00150608	+	0.999999i	$0.500479\pi$
$360$	0	0
$361$	−3.75019	−0.197379
$362$	0	0
$363$	− 4.65344i	− 0.244242i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.11082i	0.214583i	0.994228	+	0.107292i	$0.0342179\pi$
$367$	4.11082i	0.214583i	−0.994228	+	0.107292i	$0.965782\pi$
$368$	0	0
$369$	28.6204	1.48992
$370$	0	0
$371$	−9.73436	−0.505383
$372$	0	0
$373$	− 26.2057i	− 1.35688i	−0.734655	−	0.678440i	$-0.762656\pi$
$373$	− 26.2057i	− 1.35688i	0.734655	−	0.678440i	$-0.237344\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.32959i	0.325991i
$378$	0	0
$379$	−27.9938	−1.43795	−0.718973	−	0.695038i	$-0.755388\pi$
$379$	−27.9938	−1.43795	−0.718973	+	0.695038i	$0.755388\pi$
$380$	0	0
$381$	1.87514	0.0960665
$382$	0	0
$383$	− 4.16487i	− 0.212815i	−0.994323	−	0.106407i	$-0.966065\pi$
$383$	− 4.16487i	− 0.212815i	0.994323	−	0.106407i	$-0.0339348\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.1587i	0.516396i
$388$	0	0
$389$	26.4593	1.34154	0.670771	−	0.741665i	$-0.265963\pi$
$389$	26.4593	1.34154	0.670771	+	0.741665i	$0.265963\pi$
$390$	0	0
$391$	−0.737296	−0.0372866
$392$	0	0
$393$	− 2.67041i	− 0.134704i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.65036i	0.233395i	0.993168	+	0.116697i	$0.0372308\pi$
$397$	4.65036i	0.233395i	−0.993168	+	0.116697i	$0.962769\pi$
$398$	0	0
$399$	−7.81020	−0.390999
$400$	0	0
$401$	−25.1328	−1.25507	−0.627537	−	0.778587i	$-0.715937\pi$
$401$	−25.1328	−1.25507	−0.627537	+	0.778587i	$0.715937\pi$
$402$	0	0
$403$	6.97199i	0.347299i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 11.3165i	− 0.560936i
$408$	0	0
$409$	−16.8659	−0.833965	−0.416982	−	0.908915i	$-0.636912\pi$
$409$	−16.8659	−0.833965	−0.416982	+	0.908915i	$0.636912\pi$
$410$	0	0
$411$	−2.85713	−0.140932
$412$	0	0
$413$	15.5384i	0.764595i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.42572i	0.216728i
$418$	0	0
$419$	−19.4594	−0.950655	−0.475328	−	0.879809i	$-0.657670\pi$
$419$	−19.4594	−0.950655	−0.475328	+	0.879809i	$0.657670\pi$
$420$	0	0
$421$	−4.41761	−0.215301	−0.107651	−	0.994189i	$-0.534333\pi$
$421$	−4.41761	−0.215301	−0.107651	+	0.994189i	$0.534333\pi$
$422$	0	0
$423$	14.2336i	0.692059i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 35.0768i	− 1.69749i
$428$	0	0
$429$	8.14192	0.393096
$430$	0	0
$431$	1.89220	0.0911442	0.0455721	−	0.998961i	$-0.485489\pi$
$431$	1.89220	0.0911442	0.0455721	+	0.998961i	$0.485489\pi$
$432$	0	0
$433$	− 25.4147i	− 1.22135i	−0.791881	−	0.610676i	$-0.790898\pi$
$433$	− 25.4147i	− 1.22135i	0.791881	−	0.610676i	$-0.209102\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.90510i	− 0.186806i
$438$	0	0
$439$	15.8979	0.758768	0.379384	−	0.925239i	$-0.376136\pi$
$439$	15.8979	0.758768	0.379384	+	0.925239i	$0.376136\pi$
$440$	0	0
$441$	−7.89519	−0.375962
$442$	0	0
$443$	− 2.34485i	− 0.111407i	−0.998447	−	0.0557037i	$-0.982260\pi$
$443$	− 2.34485i	− 0.111407i	0.998447	−	0.0557037i	$-0.0177402\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5.51242i	− 0.260728i
$448$	0	0
$449$	36.3774	1.71676	0.858378	−	0.513017i	$-0.171472\pi$
$449$	36.3774	1.71676	0.858378	+	0.513017i	$0.171472\pi$
$450$	0	0
$451$	−20.9598	−0.986959
$452$	0	0
$453$	12.1660i	0.571606i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.5035i	0.865557i	0.901500	+	0.432779i	$0.142467\pi$
$457$	18.5035i	0.865557i	−0.901500	+	0.432779i	$0.857533\pi$
$458$	0	0
$459$	−2.60741	−0.121704
$460$	0	0
$461$	34.0296	1.58492	0.792459	−	0.609925i	$-0.208801\pi$
$461$	34.0296	1.58492	0.792459	+	0.609925i	$0.208801\pi$
$462$	0	0
$463$	16.7153i	0.776826i	0.921485	+	0.388413i	$0.126977\pi$
$463$	16.7153i	0.776826i	−0.921485	+	0.388413i	$0.873023\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 11.0480i	− 0.511239i	−0.966777	−	0.255620i	$-0.917721\pi$
$467$	− 11.0480i	− 0.511239i	0.966777	−	0.255620i	$-0.0822795\pi$
$468$	0	0
$469$	−10.1458	−0.468490
$470$	0	0
$471$	−4.20183	−0.193610
$472$	0	0
$473$	− 7.43961i	− 0.342073i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 7.99384i	− 0.366013i
$478$	0	0
$479$	30.2508	1.38219	0.691096	−	0.722763i	$-0.257128\pi$
$479$	30.2508	1.38219	0.691096	+	0.722763i	$0.257128\pi$
$480$	0	0
$481$	−40.2096	−1.83340
$482$	0	0
$483$	2.00000i	0.0910032i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 17.9211i	− 0.812082i	−0.913855	−	0.406041i	$-0.866909\pi$
$487$	− 17.9211i	− 0.812082i	0.913855	−	0.406041i	$-0.133091\pi$
$488$	0	0
$489$	−5.75531	−0.260264
$490$	0	0
$491$	−34.5861	−1.56085	−0.780425	−	0.625249i	$-0.784997\pi$
$491$	−34.5861	−1.56085	−0.780425	+	0.625249i	$0.784997\pi$
$492$	0	0
$493$	0.689414i	0.0310497i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 28.0797i	− 1.25955i
$498$	0	0
$499$	0.653440	0.0292520	0.0146260	−	0.999893i	$-0.495344\pi$
$499$	0.653440	0.0292520	0.0146260	+	0.999893i	$0.495344\pi$
$500$	0	0
$501$	−13.6664	−0.610571
$502$	0	0
$503$	− 6.50048i	− 0.289842i	−0.989443	−	0.144921i	$-0.953707\pi$
$503$	− 6.50048i	− 0.289842i	0.989443	−	0.144921i	$-0.0462928\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 20.7223i	− 0.920308i
$508$	0	0
$509$	−34.6612	−1.53633	−0.768166	−	0.640251i	$-0.778830\pi$
$509$	−34.6612	−1.53633	−0.768166	+	0.640251i	$0.778830\pi$
$510$	0	0
$511$	29.7791	1.31735
$512$	0	0
$513$	− 13.8102i	− 0.609735i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 10.4238i	− 0.458437i
$518$	0	0
$519$	−8.08775	−0.355013
$520$	0	0
$521$	−12.8073	−0.561096	−0.280548	−	0.959840i	$-0.590516\pi$
$521$	−12.8073	−0.561096	−0.280548	+	0.959840i	$0.590516\pi$
$522$	0	0
$523$	− 37.9689i	− 1.66026i	−0.557567	−	0.830132i	$-0.688265\pi$
$523$	− 37.9689i	− 1.66026i	0.557567	−	0.830132i	$-0.311735\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.759383i	0.0330793i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−12.7601	−0.553741
$532$	0	0
$533$	74.4744i	3.22584i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 16.4006i	− 0.707736i
$538$	0	0
$539$	5.78195	0.249046
$540$	0	0
$541$	12.3088	0.529198	0.264599	−	0.964359i	$-0.414760\pi$
$541$	12.3088	0.529198	0.264599	+	0.964359i	$0.414760\pi$
$542$	0	0
$543$	− 8.33172i	− 0.357548i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.3009i	− 0.868002i	−0.900912	−	0.434001i	$-0.857101\pi$
$547$	− 20.3009i	− 0.868002i	0.900912	−	0.434001i	$-0.142899\pi$
$548$	0	0
$549$	28.8050	1.22937
$550$	0	0
$551$	−3.65149	−0.155559
$552$	0	0
$553$	4.97184i	0.211424i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.59149i	0.279290i	0.990202	+	0.139645i	$0.0445962\pi$
$557$	6.59149i	0.279290i	−0.990202	+	0.139645i	$0.955404\pi$
$558$	0	0
$559$	−26.4344	−1.11806
$560$	0	0
$561$	0.886811	0.0374412
$562$	0	0
$563$	5.08191i	0.214177i	0.994249	+	0.107088i	$0.0341528\pi$
$563$	5.08191i	0.214177i	−0.994249	+	0.107088i	$0.965847\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 17.6492i	− 0.741198i
$568$	0	0
$569$	16.8672	0.707109	0.353554	−	0.935414i	$-0.384973\pi$
$569$	16.8672	0.707109	0.353554	+	0.935414i	$0.384973\pi$
$570$	0	0
$571$	38.3174	1.60353	0.801767	−	0.597637i	$-0.203894\pi$
$571$	38.3174	1.60353	0.801767	+	0.597637i	$0.203894\pi$
$572$	0	0
$573$	12.1937i	0.509399i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 19.0240i	− 0.791981i	−0.918255	−	0.395990i	$-0.870401\pi$
$577$	− 19.0240i	− 0.791981i	0.918255	−	0.395990i	$-0.129599\pi$
$578$	0	0
$579$	4.23089	0.175830
$580$	0	0
$581$	8.37059	0.347271
$582$	0	0
$583$	5.85419i	0.242456i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.66747i	− 0.233922i	−0.993137	−	0.116961i	$-0.962685\pi$
$587$	− 5.66747i	− 0.233922i	0.993137	−	0.116961i	$-0.0373152\pi$
$588$	0	0
$589$	−4.02209	−0.165727
$590$	0	0
$591$	−6.90193	−0.283907
$592$	0	0
$593$	− 35.7130i	− 1.46656i	−0.679928	−	0.733279i	$-0.737989\pi$
$593$	− 35.7130i	− 1.46656i	0.679928	−	0.733279i	$-0.262011\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5.53840i	− 0.226672i
$598$	0	0
$599$	−20.1717	−0.824193	−0.412097	−	0.911140i	$-0.635203\pi$
$599$	−20.1717	−0.824193	−0.412097	+	0.911140i	$0.635203\pi$
$600$	0	0
$601$	−41.9069	−1.70942	−0.854709	−	0.519107i	$-0.826264\pi$
$601$	−41.9069	−1.70942	−0.854709	+	0.519107i	$0.826264\pi$
$602$	0	0
$603$	− 8.33172i	− 0.339294i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	35.8678i	1.45583i	0.685667	+	0.727915i	$0.259511\pi$
$607$	35.8678i	1.45583i	−0.685667	+	0.727915i	$0.740489\pi$
$608$	0	0
$609$	1.87012	0.0757809
$610$	0	0
$611$	−37.0377	−1.49839
$612$	0	0
$613$	− 21.7819i	− 0.879765i	−0.898055	−	0.439882i	$-0.855020\pi$
$613$	− 21.7819i	− 0.879765i	0.898055	−	0.439882i	$-0.144980\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.8480i	0.557501i	0.960364	+	0.278750i	$0.0899202\pi$
$617$	13.8480i	0.557501i	−0.960364	+	0.278750i	$0.910080\pi$
$618$	0	0
$619$	−38.6113	−1.55192	−0.775960	−	0.630783i	$-0.782734\pi$
$619$	−38.6113	−1.55192	−0.775960	+	0.630783i	$0.782734\pi$
$620$	0	0
$621$	−3.53645	−0.141913
$622$	0	0
$623$	− 58.9570i	− 2.36206i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.69701i	0.187581i
$628$	0	0
$629$	−4.37960	−0.174626
$630$	0	0
$631$	−0.598221	−0.0238148	−0.0119074	−	0.999929i	$-0.503790\pi$
$631$	−0.598221	−0.0238148	−0.0119074	+	0.999929i	$0.503790\pi$
$632$	0	0
$633$	− 4.23972i	− 0.168514i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 20.5444i	− 0.813999i
$638$	0	0
$639$	23.0590	0.912201
$640$	0	0
$641$	−22.5254	−0.889700	−0.444850	−	0.895605i	$-0.646743\pi$
$641$	−22.5254	−0.889700	−0.444850	+	0.895605i	$0.646743\pi$
$642$	0	0
$643$	− 36.1908i	− 1.42723i	−0.700539	−	0.713614i	$-0.747057\pi$
$643$	− 36.1908i	− 1.42723i	0.700539	−	0.713614i	$-0.252943\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.33067i	0.248884i	0.992227	+	0.124442i	$0.0397141\pi$
$647$	6.33067i	0.248884i	−0.992227	+	0.124442i	$0.960286\pi$
$648$	0	0
$649$	9.34471	0.366812
$650$	0	0
$651$	2.05992	0.0807344
$652$	0	0
$653$	− 14.3057i	− 0.559827i	−0.960025	−	0.279914i	$-0.909694\pi$
$653$	− 14.3057i	− 0.559827i	0.960025	−	0.279914i	$-0.0903058\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	24.4545i	0.954063i
$658$	0	0
$659$	38.6563	1.50584	0.752919	−	0.658114i	$-0.228645\pi$
$659$	38.6563	1.50584	0.752919	+	0.658114i	$0.228645\pi$
$660$	0	0
$661$	12.5954	0.489903	0.244952	−	0.969535i	$-0.421228\pi$
$661$	12.5954	0.489903	0.244952	+	0.969535i	$0.421228\pi$
$662$	0	0
$663$	− 3.15102i	− 0.122375i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.935058i	0.0362056i
$668$	0	0
$669$	13.7421	0.531300
$670$	0	0
$671$	−21.0950	−0.814363
$672$	0	0
$673$	0.356707i	0.0137500i	0.999976	+	0.00687501i	$0.00218840\pi$
$673$	0.356707i	0.0137500i	−0.999976	+	0.00687501i	$0.997812\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.8831i	0.725737i	0.931840	+	0.362868i	$0.118203\pi$
$677$	18.8831i	0.725737i	−0.931840	+	0.362868i	$0.881797\pi$
$678$	0	0
$679$	−48.6214	−1.86592
$680$	0	0
$681$	14.8674	0.569719
$682$	0	0
$683$	− 18.7520i	− 0.717525i	−0.933429	−	0.358763i	$-0.883199\pi$
$683$	− 18.7520i	− 0.717525i	0.933429	−	0.358763i	$-0.116801\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.33656i	0.279907i
$688$	0	0
$689$	20.8011	0.792459
$690$	0	0
$691$	−18.3175	−0.696831	−0.348415	−	0.937340i	$-0.613280\pi$
$691$	−18.3175	−0.696831	−0.348415	+	0.937340i	$0.613280\pi$
$692$	0	0
$693$	15.6994i	0.596370i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.11169i	0.307252i
$698$	0	0
$699$	7.65060	0.289372
$700$	0	0
$701$	27.0007	1.01980	0.509900	−	0.860233i	$-0.329682\pi$
$701$	27.0007	1.01980	0.509900	+	0.860233i	$0.329682\pi$
$702$	0	0
$703$	− 23.1966i	− 0.874877i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.0830i	1.09378i
$708$	0	0
$709$	−16.1875	−0.607935	−0.303968	−	0.952682i	$-0.598311\pi$
$709$	−16.1875	−0.607935	−0.303968	+	0.952682i	$0.598311\pi$
$710$	0	0
$711$	−4.08287	−0.153119
$712$	0	0
$713$	1.02996i	0.0385722i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.55722i	0.132847i
$718$	0	0
$719$	49.2926	1.83830	0.919151	−	0.393904i	$-0.128876\pi$
$719$	49.2926	1.83830	0.919151	+	0.393904i	$0.128876\pi$
$720$	0	0
$721$	−23.7693	−0.885217
$722$	0	0
$723$	− 3.20668i	− 0.119258i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.2447i	0.862098i	0.902329	+	0.431049i	$0.141856\pi$
$727$	23.2447i	0.862098i	−0.902329	+	0.431049i	$0.858144\pi$
$728$	0	0
$729$	7.79527	0.288714
$730$	0	0
$731$	−2.87921	−0.106492
$732$	0	0
$733$	25.8203i	0.953693i	0.878987	+	0.476846i	$0.158220\pi$
$733$	25.8203i	0.953693i	−0.878987	+	0.476846i	$0.841780\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.10163i	0.224757i
$738$	0	0
$739$	−21.6853	−0.797705	−0.398852	−	0.917015i	$-0.630591\pi$
$739$	−21.6853	−0.797705	−0.398852	+	0.917015i	$0.630591\pi$
$740$	0	0
$741$	16.6894	0.613101
$742$	0	0
$743$	27.5932i	1.01230i	0.862447	+	0.506148i	$0.168931\pi$
$743$	27.5932i	1.01230i	−0.862447	+	0.506148i	$0.831069\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.87391i	0.251503i
$748$	0	0
$749$	49.2226	1.79855
$750$	0	0
$751$	−48.2644	−1.76119	−0.880597	−	0.473866i	$-0.842858\pi$
$751$	−48.2644	−1.76119	−0.880597	+	0.473866i	$0.842858\pi$
$752$	0	0
$753$	12.2600i	0.446778i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.87012i	0.213353i	0.994294	+	0.106676i	$0.0340209\pi$
$757$	5.87012i	0.213353i	−0.994294	+	0.106676i	$0.965979\pi$
$758$	0	0
$759$	1.20279	0.0436585
$760$	0	0
$761$	−5.67624	−0.205764	−0.102882	−	0.994694i	$-0.532806\pi$
$761$	−5.67624	−0.205764	−0.102882	+	0.994694i	$0.532806\pi$
$762$	0	0
$763$	9.92416i	0.359279i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 33.2036i	− 1.19891i
$768$	0	0
$769$	−17.3226	−0.624670	−0.312335	−	0.949972i	$-0.601111\pi$
$769$	−17.3226	−0.624670	−0.312335	+	0.949972i	$0.601111\pi$
$770$	0	0
$771$	−5.52753	−0.199069
$772$	0	0
$773$	− 28.3773i	− 1.02066i	−0.859978	−	0.510331i	$-0.829523\pi$
$773$	− 28.3773i	− 1.02066i	0.859978	−	0.510331i	$-0.170477\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	11.8802i	0.426199i
$778$	0	0
$779$	−42.9637	−1.53933
$780$	0	0
$781$	−16.8870	−0.604264
$782$	0	0
$783$	3.30679i	0.118175i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31.3995i	1.11927i	0.828739	+	0.559636i	$0.189059\pi$
$787$	31.3995i	1.11927i	−0.828739	+	0.559636i	$0.810941\pi$
$788$	0	0
$789$	−10.3199	−0.367397
$790$	0	0
$791$	53.8938	1.91624
$792$	0	0
$793$	74.9547i	2.66172i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 21.0169i	− 0.744456i	−0.928141	−	0.372228i	$-0.878594\pi$
$797$	− 21.0169i	− 0.744456i	0.928141	−	0.372228i	$-0.121406\pi$
$798$	0	0
$799$	−4.03412	−0.142717
$800$	0	0
$801$	48.4153	1.71067
$802$	0	0
$803$	− 17.9090i	− 0.631995i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.98800i	0.0699809i
$808$	0	0
$809$	−25.6712	−0.902552	−0.451276	−	0.892384i	$-0.649031\pi$
$809$	−25.6712	−0.902552	−0.451276	+	0.892384i	$0.649031\pi$
$810$	0	0
$811$	40.3647	1.41740	0.708698	−	0.705512i	$-0.249283\pi$
$811$	40.3647	1.41740	0.708698	+	0.705512i	$0.249283\pi$
$812$	0	0
$813$	15.8801i	0.556940i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 15.2498i	− 0.533523i
$818$	0	0
$819$	55.7830	1.94922
$820$	0	0
$821$	25.6396	0.894827	0.447413	−	0.894327i	$-0.352345\pi$
$821$	25.6396	0.894827	0.447413	+	0.894327i	$0.352345\pi$
$822$	0	0
$823$	− 15.6726i	− 0.546312i	−0.961970	−	0.273156i	$-0.911932\pi$
$823$	− 15.6726i	− 0.546312i	0.961970	−	0.273156i	$-0.0880676\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.8429i	1.59411i	0.603904	+	0.797057i	$0.293611\pi$
$827$	45.8429i	1.59411i	−0.603904	+	0.797057i	$0.706389\pi$
$828$	0	0
$829$	15.6214	0.542552	0.271276	−	0.962502i	$-0.412554\pi$
$829$	15.6214	0.542552	0.271276	+	0.962502i	$0.412554\pi$
$830$	0	0
$831$	−10.0151	−0.347420
$832$	0	0
$833$	− 2.23768i	− 0.0775311i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.64240i	0.125900i
$838$	0	0
$839$	−35.7853	−1.23544	−0.617722	−	0.786396i	$-0.711944\pi$
$839$	−35.7853	−1.23544	−0.617722	+	0.786396i	$0.711944\pi$
$840$	0	0
$841$	−28.1257	−0.969851
$842$	0	0
$843$	− 10.2315i	− 0.352392i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	23.3486i	0.802267i
$848$	0	0
$849$	19.2445	0.660470
$850$	0	0
$851$	−5.94008	−0.203623
$852$	0	0
$853$	− 28.0350i	− 0.959900i	−0.877296	−	0.479950i	$-0.840655\pi$
$853$	− 28.0350i	− 0.959900i	0.877296	−	0.479950i	$-0.159345\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.37562i	− 0.320265i	−0.987096	−	0.160133i	$-0.948808\pi$
$857$	− 9.37562i	− 0.320265i	0.987096	−	0.160133i	$-0.0511922\pi$
$858$	0	0
$859$	9.85704	0.336318	0.168159	−	0.985760i	$-0.446218\pi$
$859$	9.85704	0.336318	0.168159	+	0.985760i	$0.446218\pi$
$860$	0	0
$861$	22.0039	0.749891
$862$	0	0
$863$	53.1607i	1.80961i	0.425827	+	0.904805i	$0.359983\pi$
$863$	53.1607i	1.80961i	−0.425827	+	0.904805i	$0.640017\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.3898i	0.352855i
$868$	0	0
$869$	2.99004	0.101430
$870$	0	0
$871$	21.6803	0.734609
$872$	0	0
$873$	− 39.9278i	− 1.35135i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 55.8780i	− 1.88687i	−0.331564	−	0.943433i	$-0.607576\pi$
$877$	− 55.8780i	− 1.88687i	0.331564	−	0.943433i	$-0.392424\pi$
$878$	0	0
$879$	10.0599	0.339313
$880$	0	0
$881$	35.8961	1.20937	0.604685	−	0.796465i	$-0.293299\pi$
$881$	35.8961	1.20937	0.604685	+	0.796465i	$0.293299\pi$
$882$	0	0
$883$	31.2225i	1.05072i	0.850880	+	0.525361i	$0.176070\pi$
$883$	31.2225i	1.05072i	−0.850880	+	0.525361i	$0.823930\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 44.3094i	− 1.48776i	−0.668311	−	0.743882i	$-0.732982\pi$
$887$	− 44.3094i	− 1.48776i	0.668311	−	0.743882i	$-0.267018\pi$
$888$	0	0
$889$	−9.40851	−0.315551
$890$	0	0
$891$	−10.6141	−0.355587
$892$	0	0
$893$	− 21.3668i	− 0.715013i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.27375i	− 0.142696i
$898$	0	0
$899$	0.963070	0.0321202
$900$	0	0
$901$	2.26564	0.0754794
$902$	0	0
$903$	7.81020i	0.259907i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 50.5424i	− 1.67823i	−0.543952	−	0.839116i	$-0.683073\pi$
$907$	− 50.5424i	− 1.67823i	0.543952	−	0.839116i	$-0.316927\pi$
$908$	0	0
$909$	−23.8829	−0.792145
$910$	0	0
$911$	0.446726	0.0148007	0.00740034	−	0.999973i	$-0.497644\pi$
$911$	0.446726	0.0148007	0.00740034	+	0.999973i	$0.497644\pi$
$912$	0	0
$913$	− 5.03403i	− 0.166602i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.3988i	0.442466i
$918$	0	0
$919$	−36.6373	−1.20855	−0.604276	−	0.796775i	$-0.706538\pi$
$919$	−36.6373	−1.20855	−0.604276	+	0.796775i	$0.706538\pi$
$920$	0	0
$921$	5.00664	0.164974
$922$	0	0
$923$	60.0028i	1.97502i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 19.5193i	− 0.641099i
$928$	0	0
$929$	17.8243	0.584797	0.292399	−	0.956297i	$-0.405547\pi$
$929$	17.8243	0.584797	0.292399	+	0.956297i	$0.405547\pi$
$930$	0	0
$931$	11.8519	0.388431
$932$	0	0
$933$	− 18.3067i	− 0.599334i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.503505i	0.0164488i	0.999966	+	0.00822440i	$0.00261794\pi$
$937$	0.503505i	0.0164488i	−0.999966	+	0.00822440i	$0.997382\pi$
$938$	0	0
$939$	13.5465	0.442075
$940$	0	0
$941$	−42.3593	−1.38087	−0.690437	−	0.723392i	$-0.742582\pi$
$941$	−42.3593	−1.38087	−0.690437	+	0.723392i	$0.742582\pi$
$942$	0	0
$943$	11.0019i	0.358273i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.89002i	− 0.158904i	−0.996839	−	0.0794521i	$-0.974683\pi$
$947$	− 4.89002i	− 0.158904i	0.996839	−	0.0794521i	$-0.0253171\pi$
$948$	0	0
$949$	−63.6342	−2.06565
$950$	0	0
$951$	2.97127	0.0963501
$952$	0	0
$953$	8.04399i	0.260570i	0.991477	+	0.130285i	$0.0415893\pi$
$953$	8.04399i	0.260570i	−0.991477	+	0.130285i	$0.958411\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.12468i	− 0.0363557i
$958$	0	0
$959$	14.3356	0.462921
$960$	0	0
$961$	−29.9392	−0.965780
$962$	0	0
$963$	40.4215i	1.30256i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21.1938i	0.681548i	0.940145	+	0.340774i	$0.110689\pi$
$967$	21.1938i	0.681548i	−0.940145	+	0.340774i	$0.889311\pi$
$968$	0	0
$969$	1.81780	0.0583961
$970$	0	0
$971$	61.5773	1.97611	0.988054	−	0.154106i	$-0.0492496\pi$
$971$	61.5773	1.97611	0.988054	+	0.154106i	$0.0492496\pi$
$972$	0	0
$973$	− 22.2060i	− 0.711892i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.8339i	1.40237i	0.712979	+	0.701185i	$0.247346\pi$
$977$	43.8339i	1.40237i	−0.712979	+	0.701185i	$0.752654\pi$
$978$	0	0
$979$	−35.4564	−1.13319
$980$	0	0
$981$	−8.14970	−0.260200
$982$	0	0
$983$	37.6752i	1.20165i	0.799380	+	0.600826i	$0.205162\pi$
$983$	37.6752i	1.20165i	−0.799380	+	0.600826i	$0.794838\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	10.9430i	0.348320i
$988$	0	0
$989$	−3.90510	−0.124175
$990$	0	0
$991$	17.1831	0.545838	0.272919	−	0.962037i	$-0.412011\pi$
$991$	17.1831	0.545838	0.272919	+	0.962037i	$0.412011\pi$
$992$	0	0
$993$	− 3.77935i	− 0.119934i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 13.3297i	− 0.422157i	−0.977469	−	0.211079i	$-0.932302\pi$
$997$	− 13.3297i	− 0.422157i	0.977469	−	0.211079i	$-0.0676976\pi$
$998$	0	0
$999$	−21.0068	−0.664627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.c.j.1749.6		8
5.2	odd	4		2300.2.a.l.1.3	✓	4
5.3	odd	4		2300.2.a.m.1.2	yes	4
5.4	even	2	inner	2300.2.c.j.1749.3		8
20.3	even	4		9200.2.a.cm.1.3		4
20.7	even	4		9200.2.a.co.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.a.l.1.3	✓	4	5.2	odd	4
2300.2.a.m.1.2	yes	4	5.3	odd	4
2300.2.c.j.1749.3		8	5.4	even	2	inner
2300.2.c.j.1749.6		8	1.1	even	1	trivial
9200.2.a.cm.1.3		4	20.3	even	4
9200.2.a.co.1.2		4	20.7	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.631352i q^{3} -3.16780i q^{7} +2.60139 q^{9} +O(q^{10})\)	\(q+0.631352i q^{3} -3.16780i q^{7} +2.60139 q^{9} -1.90510 q^{11} +6.76920i q^{13} +0.737296i q^{17} -3.90510 q^{19} +2.00000 q^{21} +1.00000i q^{23} +3.53645i q^{27} +0.935058 q^{29} +1.02996 q^{31} -1.20279i q^{33} +5.94008i q^{37} -4.27375 q^{39} +11.0019 q^{41} +3.90510i q^{43} +5.47151i q^{47} -3.03498 q^{49} -0.465493 q^{51} -3.07290i q^{53} -2.46549i q^{57} -4.90510 q^{59} +11.0729 q^{61} -8.24071i q^{63} -3.20279i q^{67} -0.631352 q^{69} +8.86410 q^{71} +9.40055i q^{73} +6.03498i q^{77} -1.56949 q^{79} +5.57144 q^{81} +2.64240i q^{83} +0.590351i q^{87} +18.6113 q^{89} +21.4435 q^{91} +0.650266i q^{93} -15.3486i q^{97} -4.95592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^9	\( 8 q - 4 q^{9} + 2 q^{11} - 14 q^{19} + 16 q^{21} + 10 q^{29} + 28 q^{31} - 22 q^{39} + 6 q^{41} - 2 q^{49} + 56 q^{51} - 22 q^{59} + 44 q^{61} + 36 q^{71} - 50 q^{79} + 50 q^{91} + 18 q^{99}+O(q^{100}) \) 8 * q - 4 * q^9 + 2 * q^11 - 14 * q^19 + 16 * q^21 + 10 * q^29 + 28 * q^31 - 22 * q^39 + 6 * q^41 - 2 * q^49 + 56 * q^51 - 22 * q^59 + 44 * q^61 + 36 * q^71 - 50 * q^79 + 50 * q^91 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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