LMFDB - Embedded newform 2300.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1,2300)]

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, 0, 0]))

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

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

chi := DirichletCharacter("2300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.a (trivial)

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:	yes
Analytic conductor:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} +1.56155 q^{7} +3.56155 q^{9} +O(q^{10})$	$q-2.56155 q^{3} +1.56155 q^{7} +3.56155 q^{9} +2.00000 q^{11} -0.561553 q^{13} +1.56155 q^{17} +6.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} -1.43845 q^{27} -2.12311 q^{29} -9.24621 q^{31} -5.12311 q^{33} +0.438447 q^{37} +1.43845 q^{39} -4.12311 q^{41} +7.68466 q^{47} -4.56155 q^{49} -4.00000 q^{51} +0.438447 q^{53} -15.3693 q^{57} +8.68466 q^{59} +1.12311 q^{61} +5.56155 q^{63} +4.43845 q^{67} +2.56155 q^{69} +1.87689 q^{71} +8.56155 q^{73} +3.12311 q^{77} +13.1231 q^{79} -7.00000 q^{81} -14.9309 q^{83} +5.43845 q^{87} -2.24621 q^{89} -0.876894 q^{91} +23.6847 q^{93} +4.87689 q^{97} +7.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - q^7 + 3 * q^9	$ 2 q - q^{3} - q^{7} + 3 q^{9} + 4 q^{11} + 3 q^{13} - q^{17} + 12 q^{19} - 8 q^{21} - 2 q^{23} - 7 q^{27} + 4 q^{29} - 2 q^{31} - 2 q^{33} + 5 q^{37} + 7 q^{39} + 3 q^{47} - 5 q^{49} - 8 q^{51} + 5 q^{53} - 6 q^{57} + 5 q^{59} - 6 q^{61} + 7 q^{63} + 13 q^{67} + q^{69} + 12 q^{71} + 13 q^{73} - 2 q^{77} + 18 q^{79} - 14 q^{81} - q^{83} + 15 q^{87} + 12 q^{89} - 10 q^{91} + 35 q^{93} + 18 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^7 + 3 * q^9 + 4 * q^11 + 3 * q^13 - q^17 + 12 * q^19 - 8 * q^21 - 2 * q^23 - 7 * q^27 + 4 * q^29 - 2 * q^31 - 2 * q^33 + 5 * q^37 + 7 * q^39 + 3 * q^47 - 5 * q^49 - 8 * q^51 + 5 * q^53 - 6 * q^57 + 5 * q^59 - 6 * q^61 + 7 * q^63 + 13 * q^67 + q^69 + 12 * q^71 + 13 * q^73 - 2 * q^77 + 18 * q^79 - 14 * q^81 - q^83 + 15 * q^87 + 12 * q^89 - 10 * q^91 + 35 * q^93 + 18 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.56155	0.590211	0.295106	−	0.955465i	$-0.404645\pi$
$7$	1.56155	0.590211	0.295106	+	0.955465i	$0.404645\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−0.561553	−0.155747	−0.0778734	−	0.996963i	$-0.524813\pi$
$13$	−0.561553	−0.155747	−0.0778734	+	0.996963i	$0.524813\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.56155	0.378732	0.189366	−	0.981907i	$-0.439357\pi$
$17$	1.56155	0.378732	0.189366	+	0.981907i	$0.439357\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	−2.12311	−0.394251	−0.197125	−	0.980378i	$-0.563161\pi$
$29$	−2.12311	−0.394251	−0.197125	+	0.980378i	$0.563161\pi$
$30$	0	0
$31$	−9.24621	−1.66067	−0.830334	−	0.557266i	$-0.811851\pi$
$31$	−9.24621	−1.66067	−0.830334	+	0.557266i	$0.811851\pi$
$32$	0	0
$33$	−5.12311	−0.891818
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.438447	0.0720803	0.0360401	−	0.999350i	$-0.488526\pi$
$37$	0.438447	0.0720803	0.0360401	+	0.999350i	$0.488526\pi$
$38$	0	0
$39$	1.43845	0.230336
$40$	0	0
$41$	−4.12311	−0.643921	−0.321960	−	0.946753i	$-0.604342\pi$
$41$	−4.12311	−0.643921	−0.321960	+	0.946753i	$0.604342\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.68466	1.12092	0.560461	−	0.828181i	$-0.310624\pi$
$47$	7.68466	1.12092	0.560461	+	0.828181i	$0.310624\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	0.438447	0.0602254	0.0301127	−	0.999547i	$-0.490413\pi$
$53$	0.438447	0.0602254	0.0301127	+	0.999547i	$0.490413\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−15.3693	−2.03572
$58$	0	0
$59$	8.68466	1.13065	0.565323	−	0.824870i	$-0.308751\pi$
$59$	8.68466	1.13065	0.565323	+	0.824870i	$0.308751\pi$
$60$	0	0
$61$	1.12311	0.143799	0.0718995	−	0.997412i	$-0.477094\pi$
$61$	1.12311	0.143799	0.0718995	+	0.997412i	$0.477094\pi$
$62$	0	0
$63$	5.56155	0.700690
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.43845	0.542243	0.271121	−	0.962545i	$-0.412606\pi$
$67$	4.43845	0.542243	0.271121	+	0.962545i	$0.412606\pi$
$68$	0	0
$69$	2.56155	0.308375
$70$	0	0
$71$	1.87689	0.222746	0.111373	−	0.993779i	$-0.464475\pi$
$71$	1.87689	0.222746	0.111373	+	0.993779i	$0.464475\pi$
$72$	0	0
$73$	8.56155	1.00205	0.501027	−	0.865432i	$-0.332956\pi$
$73$	8.56155	1.00205	0.501027	+	0.865432i	$0.332956\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.12311	0.355911
$78$	0	0
$79$	13.1231	1.47646	0.738232	−	0.674546i	$-0.235661\pi$
$79$	13.1231	1.47646	0.738232	+	0.674546i	$0.235661\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−14.9309	−1.63888	−0.819438	−	0.573168i	$-0.805714\pi$
$83$	−14.9309	−1.63888	−0.819438	+	0.573168i	$0.805714\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.43845	0.583063
$88$	0	0
$89$	−2.24621	−0.238098	−0.119049	−	0.992888i	$-0.537985\pi$
$89$	−2.24621	−0.238098	−0.119049	+	0.992888i	$0.537985\pi$
$90$	0	0
$91$	−0.876894	−0.0919235
$92$	0	0
$93$	23.6847	2.45598
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.87689	0.495174	0.247587	−	0.968866i	$-0.420362\pi$
$97$	4.87689	0.495174	0.247587	+	0.968866i	$0.420362\pi$
$98$	0	0
$99$	7.12311	0.715899
$100$	0	0
$101$	−5.31534	−0.528896	−0.264448	−	0.964400i	$-0.585190\pi$
$101$	−5.31534	−0.528896	−0.264448	+	0.964400i	$0.585190\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.56155	−0.731003	−0.365501	−	0.930811i	$-0.619102\pi$
$107$	−7.56155	−0.731003	−0.365501	+	0.930811i	$0.619102\pi$
$108$	0	0
$109$	−9.36932	−0.897418	−0.448709	−	0.893678i	$-0.648116\pi$
$109$	−9.36932	−0.897418	−0.448709	+	0.893678i	$0.648116\pi$
$110$	0	0
$111$	−1.12311	−0.106600
$112$	0	0
$113$	7.80776	0.734493	0.367246	−	0.930124i	$-0.380301\pi$
$113$	7.80776	0.734493	0.367246	+	0.930124i	$0.380301\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	2.43845	0.223532
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	10.5616	0.952303
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.8078	1.13651	0.568253	−	0.822854i	$-0.307620\pi$
$127$	12.8078	1.13651	0.568253	+	0.822854i	$0.307620\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.8078	1.81798	0.908991	−	0.416815i	$-0.136854\pi$
$131$	20.8078	1.81798	0.908991	+	0.416815i	$0.136854\pi$
$132$	0	0
$133$	9.36932	0.812423
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.1231	−1.29205	−0.646027	−	0.763315i	$-0.723571\pi$
$137$	−15.1231	−1.29205	−0.646027	+	0.763315i	$0.723571\pi$
$138$	0	0
$139$	5.24621	0.444978	0.222489	−	0.974935i	$-0.428582\pi$
$139$	5.24621	0.444978	0.222489	+	0.974935i	$0.428582\pi$
$140$	0	0
$141$	−19.6847	−1.65775
$142$	0	0
$143$	−1.12311	−0.0939188
$144$	0	0
$145$	0	0
$146$	0	0
$147$	11.6847	0.963734
$148$	0	0
$149$	15.3693	1.25910	0.629552	−	0.776959i	$-0.283239\pi$
$149$	15.3693	1.25910	0.629552	+	0.776959i	$0.283239\pi$
$150$	0	0
$151$	23.0540	1.87611	0.938053	−	0.346492i	$-0.112627\pi$
$151$	23.0540	1.87611	0.938053	+	0.346492i	$0.112627\pi$
$152$	0	0
$153$	5.56155	0.449625
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.6847	1.33158	0.665790	−	0.746139i	$-0.268095\pi$
$157$	16.6847	1.33158	0.665790	+	0.746139i	$0.268095\pi$
$158$	0	0
$159$	−1.12311	−0.0890681
$160$	0	0
$161$	−1.56155	−0.123068
$162$	0	0
$163$	11.9309	0.934498	0.467249	−	0.884126i	$-0.345245\pi$
$163$	11.9309	0.934498	0.467249	+	0.884126i	$0.345245\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	21.3693	1.63415
$172$	0	0
$173$	19.3693	1.47262	0.736311	−	0.676643i	$-0.236566\pi$
$173$	19.3693	1.47262	0.736311	+	0.676643i	$0.236566\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−22.2462	−1.67213
$178$	0	0
$179$	21.9309	1.63919	0.819595	−	0.572943i	$-0.194198\pi$
$179$	21.9309	1.63919	0.819595	+	0.572943i	$0.194198\pi$
$180$	0	0
$181$	19.3693	1.43971	0.719855	−	0.694124i	$-0.244208\pi$
$181$	19.3693	1.43971	0.719855	+	0.694124i	$0.244208\pi$
$182$	0	0
$183$	−2.87689	−0.212666
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.12311	0.228384
$188$	0	0
$189$	−2.24621	−0.163388
$190$	0	0
$191$	−7.36932	−0.533225	−0.266613	−	0.963804i	$-0.585904\pi$
$191$	−7.36932	−0.533225	−0.266613	+	0.963804i	$0.585904\pi$
$192$	0	0
$193$	−6.56155	−0.472311	−0.236155	−	0.971715i	$-0.575887\pi$
$193$	−6.56155	−0.472311	−0.236155	+	0.971715i	$0.575887\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.31534	0.449949	0.224975	−	0.974365i	$-0.427770\pi$
$197$	6.31534	0.449949	0.224975	+	0.974365i	$0.427770\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	−11.3693	−0.801930
$202$	0	0
$203$	−3.31534	−0.232691
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.56155	−0.247545
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	12.6847	0.873248	0.436624	−	0.899644i	$-0.356174\pi$
$211$	12.6847	0.873248	0.436624	+	0.899644i	$0.356174\pi$
$212$	0	0
$213$	−4.80776	−0.329423
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−14.4384	−0.980146
$218$	0	0
$219$	−21.9309	−1.48195
$220$	0	0
$221$	−0.876894	−0.0589863
$222$	0	0
$223$	−23.6155	−1.58141	−0.790706	−	0.612196i	$-0.790287\pi$
$223$	−23.6155	−1.58141	−0.790706	+	0.612196i	$0.790287\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.75379	−0.647382	−0.323691	−	0.946163i	$-0.604924\pi$
$227$	−9.75379	−0.647382	−0.323691	+	0.946163i	$0.604924\pi$
$228$	0	0
$229$	22.7386	1.50261	0.751306	−	0.659954i	$-0.229424\pi$
$229$	22.7386	1.50261	0.751306	+	0.659954i	$0.229424\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−11.6847	−0.765487	−0.382744	−	0.923855i	$-0.625021\pi$
$233$	−11.6847	−0.765487	−0.382744	+	0.923855i	$0.625021\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−33.6155	−2.18356
$238$	0	0
$239$	19.2462	1.24493	0.622467	−	0.782646i	$-0.286131\pi$
$239$	19.2462	1.24493	0.622467	+	0.782646i	$0.286131\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.36932	−0.214384
$248$	0	0
$249$	38.2462	2.42376
$250$	0	0
$251$	17.3693	1.09634	0.548171	−	0.836366i	$-0.315324\pi$
$251$	17.3693	1.09634	0.548171	+	0.836366i	$0.315324\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.1922	−0.698152	−0.349076	−	0.937094i	$-0.613505\pi$
$257$	−11.1922	−0.698152	−0.349076	+	0.937094i	$0.613505\pi$
$258$	0	0
$259$	0.684658	0.0425426
$260$	0	0
$261$	−7.56155	−0.468048
$262$	0	0
$263$	−28.9309	−1.78395	−0.891977	−	0.452081i	$-0.850682\pi$
$263$	−28.9309	−1.78395	−0.891977	+	0.452081i	$0.850682\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.75379	0.352126
$268$	0	0
$269$	6.75379	0.411786	0.205893	−	0.978575i	$-0.433990\pi$
$269$	6.75379	0.411786	0.205893	+	0.978575i	$0.433990\pi$
$270$	0	0
$271$	−6.93087	−0.421020	−0.210510	−	0.977592i	$-0.567513\pi$
$271$	−6.93087	−0.421020	−0.210510	+	0.977592i	$0.567513\pi$
$272$	0	0
$273$	2.24621	0.135947
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.68466	−0.341558	−0.170779	−	0.985309i	$-0.554628\pi$
$277$	−5.68466	−0.341558	−0.170779	+	0.985309i	$0.554628\pi$
$278$	0	0
$279$	−32.9309	−1.97152
$280$	0	0
$281$	−15.1231	−0.902169	−0.451084	−	0.892481i	$-0.648963\pi$
$281$	−15.1231	−0.902169	−0.451084	+	0.892481i	$0.648963\pi$
$282$	0	0
$283$	5.80776	0.345236	0.172618	−	0.984989i	$-0.444777\pi$
$283$	5.80776	0.345236	0.172618	+	0.984989i	$0.444777\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.43845	−0.380050
$288$	0	0
$289$	−14.5616	−0.856562
$290$	0	0
$291$	−12.4924	−0.732319
$292$	0	0
$293$	−3.80776	−0.222452	−0.111226	−	0.993795i	$-0.535478\pi$
$293$	−3.80776	−0.222452	−0.111226	+	0.993795i	$0.535478\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.87689	−0.166934
$298$	0	0
$299$	0.561553	0.0324754
$300$	0	0
$301$	0	0
$302$	0	0
$303$	13.6155	0.782192
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.8769	0.734923	0.367462	−	0.930039i	$-0.380227\pi$
$307$	12.8769	0.734923	0.367462	+	0.930039i	$0.380227\pi$
$308$	0	0
$309$	−40.9848	−2.33155
$310$	0	0
$311$	−3.68466	−0.208938	−0.104469	−	0.994528i	$-0.533314\pi$
$311$	−3.68466	−0.208938	−0.104469	+	0.994528i	$0.533314\pi$
$312$	0	0
$313$	−19.8078	−1.11960	−0.559801	−	0.828627i	$-0.689122\pi$
$313$	−19.8078	−1.11960	−0.559801	+	0.828627i	$0.689122\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.87689	−0.386245	−0.193122	−	0.981175i	$-0.561861\pi$
$317$	−6.87689	−0.386245	−0.193122	+	0.981175i	$0.561861\pi$
$318$	0	0
$319$	−4.24621	−0.237742
$320$	0	0
$321$	19.3693	1.08109
$322$	0	0
$323$	9.36932	0.521323
$324$	0	0
$325$	0	0
$326$	0	0
$327$	24.0000	1.32720
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	−12.6155	−0.693412	−0.346706	−	0.937974i	$-0.612700\pi$
$331$	−12.6155	−0.693412	−0.346706	+	0.937974i	$0.612700\pi$
$332$	0	0
$333$	1.56155	0.0855726
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.6155	1.50431	0.752157	−	0.658984i	$-0.229014\pi$
$337$	27.6155	1.50431	0.752157	+	0.658984i	$0.229014\pi$
$338$	0	0
$339$	−20.0000	−1.08625
$340$	0	0
$341$	−18.4924	−1.00142
$342$	0	0
$343$	−18.0540	−0.974823
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.4924	0.885360	0.442680	−	0.896680i	$-0.354028\pi$
$347$	16.4924	0.885360	0.442680	+	0.896680i	$0.354028\pi$
$348$	0	0
$349$	−1.63068	−0.0872885	−0.0436442	−	0.999047i	$-0.513897\pi$
$349$	−1.63068	−0.0872885	−0.0436442	+	0.999047i	$0.513897\pi$
$350$	0	0
$351$	0.807764	0.0431153
$352$	0	0
$353$	−35.0540	−1.86573	−0.932867	−	0.360220i	$-0.882702\pi$
$353$	−35.0540	−1.86573	−0.932867	+	0.360220i	$0.882702\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.24621	−0.330585
$358$	0	0
$359$	−25.6155	−1.35194	−0.675968	−	0.736931i	$-0.736274\pi$
$359$	−25.6155	−1.35194	−0.675968	+	0.736931i	$0.736274\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	17.9309	0.941127
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.4384	0.962479	0.481240	−	0.876589i	$-0.340187\pi$
$367$	18.4384	0.962479	0.481240	+	0.876589i	$0.340187\pi$
$368$	0	0
$369$	−14.6847	−0.764453
$370$	0	0
$371$	0.684658	0.0355457
$372$	0	0
$373$	3.75379	0.194364	0.0971819	−	0.995267i	$-0.469017\pi$
$373$	3.75379	0.194364	0.0971819	+	0.995267i	$0.469017\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.19224	0.0614033
$378$	0	0
$379$	−3.50758	−0.180172	−0.0900861	−	0.995934i	$-0.528714\pi$
$379$	−3.50758	−0.180172	−0.0900861	+	0.995934i	$0.528714\pi$
$380$	0	0
$381$	−32.8078	−1.68079
$382$	0	0
$383$	−23.8078	−1.21652	−0.608260	−	0.793738i	$-0.708132\pi$
$383$	−23.8078	−1.21652	−0.608260	+	0.793738i	$0.708132\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−19.1231	−0.969580	−0.484790	−	0.874631i	$-0.661104\pi$
$389$	−19.1231	−0.969580	−0.484790	+	0.874631i	$0.661104\pi$
$390$	0	0
$391$	−1.56155	−0.0789711
$392$	0	0
$393$	−53.3002	−2.68864
$394$	0	0
$395$	0	0
$396$	0	0
$397$	35.5464	1.78402	0.892011	−	0.452013i	$-0.149294\pi$
$397$	35.5464	1.78402	0.892011	+	0.452013i	$0.149294\pi$
$398$	0	0
$399$	−24.0000	−1.20150
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	5.19224	0.258644
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.876894	0.0434660
$408$	0	0
$409$	29.9848	1.48266	0.741328	−	0.671143i	$-0.234197\pi$
$409$	29.9848	1.48266	0.741328	+	0.671143i	$0.234197\pi$
$410$	0	0
$411$	38.7386	1.91084
$412$	0	0
$413$	13.5616	0.667320
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−13.4384	−0.658084
$418$	0	0
$419$	−27.1231	−1.32505	−0.662525	−	0.749040i	$-0.730515\pi$
$419$	−27.1231	−1.32505	−0.662525	+	0.749040i	$0.730515\pi$
$420$	0	0
$421$	16.8769	0.822530	0.411265	−	0.911516i	$-0.365087\pi$
$421$	16.8769	0.822530	0.411265	+	0.911516i	$0.365087\pi$
$422$	0	0
$423$	27.3693	1.33074
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.75379	0.0848718
$428$	0	0
$429$	2.87689	0.138898
$430$	0	0
$431$	6.24621	0.300869	0.150435	−	0.988620i	$-0.451933\pi$
$431$	6.24621	0.300869	0.150435	+	0.988620i	$0.451933\pi$
$432$	0	0
$433$	−11.0691	−0.531948	−0.265974	−	0.963980i	$-0.585694\pi$
$433$	−11.0691	−0.531948	−0.265974	+	0.963980i	$0.585694\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−0.807764	−0.0385525	−0.0192762	−	0.999814i	$-0.506136\pi$
$439$	−0.807764	−0.0385525	−0.0192762	+	0.999814i	$0.506136\pi$
$440$	0	0
$441$	−16.2462	−0.773629
$442$	0	0
$443$	−8.80776	−0.418469	−0.209235	−	0.977865i	$-0.567097\pi$
$443$	−8.80776	−0.418469	−0.209235	+	0.977865i	$0.567097\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−39.3693	−1.86210
$448$	0	0
$449$	−9.31534	−0.439618	−0.219809	−	0.975543i	$-0.570543\pi$
$449$	−9.31534	−0.439618	−0.219809	+	0.975543i	$0.570543\pi$
$450$	0	0
$451$	−8.24621	−0.388299
$452$	0	0
$453$	−59.0540	−2.77460
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13.5616	−0.634383	−0.317191	−	0.948362i	$-0.602740\pi$
$457$	−13.5616	−0.634383	−0.317191	+	0.948362i	$0.602740\pi$
$458$	0	0
$459$	−2.24621	−0.104844
$460$	0	0
$461$	−12.0691	−0.562115	−0.281058	−	0.959691i	$-0.590685\pi$
$461$	−12.0691	−0.562115	−0.281058	+	0.959691i	$0.590685\pi$
$462$	0	0
$463$	6.63068	0.308154	0.154077	−	0.988059i	$-0.450760\pi$
$463$	6.63068	0.308154	0.154077	+	0.988059i	$0.450760\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.6847	1.60501	0.802507	−	0.596642i	$-0.203499\pi$
$467$	34.6847	1.60501	0.802507	+	0.596642i	$0.203499\pi$
$468$	0	0
$469$	6.93087	0.320038
$470$	0	0
$471$	−42.7386	−1.96929
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.56155	0.0714986
$478$	0	0
$479$	39.2311	1.79251	0.896256	−	0.443536i	$-0.146276\pi$
$479$	39.2311	1.79251	0.896256	+	0.443536i	$0.146276\pi$
$480$	0	0
$481$	−0.246211	−0.0112263
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.8078	0.580375	0.290188	−	0.956970i	$-0.406282\pi$
$487$	12.8078	0.580375	0.290188	+	0.956970i	$0.406282\pi$
$488$	0	0
$489$	−30.5616	−1.38204
$490$	0	0
$491$	−21.4924	−0.969939	−0.484970	−	0.874531i	$-0.661169\pi$
$491$	−21.4924	−0.969939	−0.484970	+	0.874531i	$0.661169\pi$
$492$	0	0
$493$	−3.31534	−0.149315
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.93087	0.131467
$498$	0	0
$499$	−18.6155	−0.833345	−0.416673	−	0.909057i	$-0.636804\pi$
$499$	−18.6155	−0.833345	−0.416673	+	0.909057i	$0.636804\pi$
$500$	0	0
$501$	−20.4924	−0.915534
$502$	0	0
$503$	−24.9309	−1.11161	−0.555806	−	0.831312i	$-0.687590\pi$
$503$	−24.9309	−1.11161	−0.555806	+	0.831312i	$0.687590\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	32.4924	1.44304
$508$	0	0
$509$	18.1771	0.805685	0.402842	−	0.915269i	$-0.368022\pi$
$509$	18.1771	0.805685	0.402842	+	0.915269i	$0.368022\pi$
$510$	0	0
$511$	13.3693	0.591424
$512$	0	0
$513$	−8.63068	−0.381054
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15.3693	0.675942
$518$	0	0
$519$	−49.6155	−2.17788
$520$	0	0
$521$	27.6155	1.20986	0.604929	−	0.796279i	$-0.293201\pi$
$521$	27.6155	1.20986	0.604929	+	0.796279i	$0.293201\pi$
$522$	0	0
$523$	−34.7386	−1.51901	−0.759507	−	0.650499i	$-0.774560\pi$
$523$	−34.7386	−1.51901	−0.759507	+	0.650499i	$0.774560\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.4384	−0.628949
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	30.9309	1.34229
$532$	0	0
$533$	2.31534	0.100289
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−56.1771	−2.42422
$538$	0	0
$539$	−9.12311	−0.392960
$540$	0	0
$541$	9.68466	0.416376	0.208188	−	0.978089i	$-0.433243\pi$
$541$	9.68466	0.416376	0.208188	+	0.978089i	$0.433243\pi$
$542$	0	0
$543$	−49.6155	−2.12921
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−24.1771	−1.03374	−0.516869	−	0.856065i	$-0.672902\pi$
$547$	−24.1771	−1.03374	−0.516869	+	0.856065i	$0.672902\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	−12.7386	−0.542684
$552$	0	0
$553$	20.4924	0.871426
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.31534	0.140476	0.0702378	−	0.997530i	$-0.477624\pi$
$557$	3.31534	0.140476	0.0702378	+	0.997530i	$0.477624\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	45.6695	1.92474	0.962370	−	0.271742i	$-0.0875998\pi$
$563$	45.6695	1.92474	0.962370	+	0.271742i	$0.0875998\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−10.9309	−0.459053
$568$	0	0
$569$	−32.7386	−1.37247	−0.686237	−	0.727378i	$-0.740739\pi$
$569$	−32.7386	−1.37247	−0.686237	+	0.727378i	$0.740739\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	18.8769	0.788594
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.4384	1.05902	0.529508	−	0.848305i	$-0.322376\pi$
$577$	25.4384	1.05902	0.529508	+	0.848305i	$0.322376\pi$
$578$	0	0
$579$	16.8078	0.698507
$580$	0	0
$581$	−23.3153	−0.967283
$582$	0	0
$583$	0.876894	0.0363173
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.9309	−1.48303	−0.741513	−	0.670939i	$-0.765891\pi$
$587$	−35.9309	−1.48303	−0.741513	+	0.670939i	$0.765891\pi$
$588$	0	0
$589$	−55.4773	−2.28590
$590$	0	0
$591$	−16.1771	−0.665436
$592$	0	0
$593$	25.6155	1.05190	0.525952	−	0.850514i	$-0.323709\pi$
$593$	25.6155	1.05190	0.525952	+	0.850514i	$0.323709\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	25.6155	1.04837
$598$	0	0
$599$	32.9848	1.34772	0.673862	−	0.738857i	$-0.264634\pi$
$599$	32.9848	1.34772	0.673862	+	0.738857i	$0.264634\pi$
$600$	0	0
$601$	11.6307	0.474425	0.237213	−	0.971458i	$-0.423766\pi$
$601$	11.6307	0.474425	0.237213	+	0.971458i	$0.423766\pi$
$602$	0	0
$603$	15.8078	0.643742
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24.4924	0.994117	0.497058	−	0.867717i	$-0.334413\pi$
$607$	24.4924	0.994117	0.497058	+	0.867717i	$0.334413\pi$
$608$	0	0
$609$	8.49242	0.344130
$610$	0	0
$611$	−4.31534	−0.174580
$612$	0	0
$613$	−35.6155	−1.43850	−0.719249	−	0.694753i	$-0.755514\pi$
$613$	−35.6155	−1.43850	−0.719249	+	0.694753i	$0.755514\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.56155	0.143383	0.0716914	−	0.997427i	$-0.477160\pi$
$617$	3.56155	0.143383	0.0716914	+	0.997427i	$0.477160\pi$
$618$	0	0
$619$	−20.4924	−0.823660	−0.411830	−	0.911261i	$-0.635110\pi$
$619$	−20.4924	−0.823660	−0.411830	+	0.911261i	$0.635110\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	−3.50758	−0.140528
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−30.7386	−1.22758
$628$	0	0
$629$	0.684658	0.0272991
$630$	0	0
$631$	13.7538	0.547530	0.273765	−	0.961797i	$-0.411731\pi$
$631$	13.7538	0.547530	0.273765	+	0.961797i	$0.411731\pi$
$632$	0	0
$633$	−32.4924	−1.29146
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.56155	0.101492
$638$	0	0
$639$	6.68466	0.264441
$640$	0	0
$641$	15.1231	0.597327	0.298663	−	0.954359i	$-0.403459\pi$
$641$	15.1231	0.597327	0.298663	+	0.954359i	$0.403459\pi$
$642$	0	0
$643$	41.4233	1.63358	0.816788	−	0.576939i	$-0.195753\pi$
$643$	41.4233	1.63358	0.816788	+	0.576939i	$0.195753\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.6847	−0.695256	−0.347628	−	0.937633i	$-0.613013\pi$
$647$	−17.6847	−0.695256	−0.347628	+	0.937633i	$0.613013\pi$
$648$	0	0
$649$	17.3693	0.681805
$650$	0	0
$651$	36.9848	1.44955
$652$	0	0
$653$	38.6695	1.51325	0.756627	−	0.653846i	$-0.226846\pi$
$653$	38.6695	1.51325	0.756627	+	0.653846i	$0.226846\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.4924	1.18962
$658$	0	0
$659$	−9.36932	−0.364977	−0.182488	−	0.983208i	$-0.558415\pi$
$659$	−9.36932	−0.364977	−0.182488	+	0.983208i	$0.558415\pi$
$660$	0	0
$661$	26.4924	1.03044	0.515218	−	0.857059i	$-0.327711\pi$
$661$	26.4924	1.03044	0.515218	+	0.857059i	$0.327711\pi$
$662$	0	0
$663$	2.24621	0.0872356
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.12311	0.0822070
$668$	0	0
$669$	60.4924	2.33877
$670$	0	0
$671$	2.24621	0.0867140
$672$	0	0
$673$	35.5464	1.37021	0.685106	−	0.728443i	$-0.259756\pi$
$673$	35.5464	1.37021	0.685106	+	0.728443i	$0.259756\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.19224	−0.237987	−0.118993	−	0.992895i	$-0.537967\pi$
$677$	−6.19224	−0.237987	−0.118993	+	0.992895i	$0.537967\pi$
$678$	0	0
$679$	7.61553	0.292257
$680$	0	0
$681$	24.9848	0.957421
$682$	0	0
$683$	−4.94602	−0.189254	−0.0946272	−	0.995513i	$-0.530166\pi$
$683$	−4.94602	−0.189254	−0.0946272	+	0.995513i	$0.530166\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−58.2462	−2.22223
$688$	0	0
$689$	−0.246211	−0.00937990
$690$	0	0
$691$	−16.4924	−0.627401	−0.313701	−	0.949522i	$-0.601569\pi$
$691$	−16.4924	−0.627401	−0.313701	+	0.949522i	$0.601569\pi$
$692$	0	0
$693$	11.1231	0.422532
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.43845	−0.243874
$698$	0	0
$699$	29.9309	1.13209
$700$	0	0
$701$	−18.2462	−0.689150	−0.344575	−	0.938759i	$-0.611977\pi$
$701$	−18.2462	−0.689150	−0.344575	+	0.938759i	$0.611977\pi$
$702$	0	0
$703$	2.63068	0.0992181
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.30019	−0.312161
$708$	0	0
$709$	46.2462	1.73681	0.868406	−	0.495853i	$-0.165145\pi$
$709$	46.2462	1.73681	0.868406	+	0.495853i	$0.165145\pi$
$710$	0	0
$711$	46.7386	1.75284
$712$	0	0
$713$	9.24621	0.346273
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−49.3002	−1.84115
$718$	0	0
$719$	−18.0540	−0.673300	−0.336650	−	0.941630i	$-0.609294\pi$
$719$	−18.0540	−0.673300	−0.336650	+	0.941630i	$0.609294\pi$
$720$	0	0
$721$	24.9848	0.930484
$722$	0	0
$723$	−15.3693	−0.571591
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.8078	1.55056	0.775282	−	0.631615i	$-0.217608\pi$
$727$	41.8078	1.55056	0.775282	+	0.631615i	$0.217608\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−12.6847	−0.468519	−0.234259	−	0.972174i	$-0.575266\pi$
$733$	−12.6847	−0.468519	−0.234259	+	0.972174i	$0.575266\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.87689	0.326985
$738$	0	0
$739$	−40.6155	−1.49407	−0.747033	−	0.664787i	$-0.768522\pi$
$739$	−40.6155	−1.49407	−0.747033	+	0.664787i	$0.768522\pi$
$740$	0	0
$741$	8.63068	0.317056
$742$	0	0
$743$	−36.4924	−1.33878	−0.669389	−	0.742912i	$-0.733444\pi$
$743$	−36.4924	−1.33878	−0.669389	+	0.742912i	$0.733444\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−53.1771	−1.94565
$748$	0	0
$749$	−11.8078	−0.431446
$750$	0	0
$751$	−4.87689	−0.177960	−0.0889802	−	0.996033i	$-0.528361\pi$
$751$	−4.87689	−0.177960	−0.0889802	+	0.996033i	$0.528361\pi$
$752$	0	0
$753$	−44.4924	−1.62139
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.43845	−0.306701	−0.153350	−	0.988172i	$-0.549006\pi$
$757$	−8.43845	−0.306701	−0.153350	+	0.988172i	$0.549006\pi$
$758$	0	0
$759$	5.12311	0.185957
$760$	0	0
$761$	21.9848	0.796950	0.398475	−	0.917179i	$-0.369540\pi$
$761$	21.9848	0.796950	0.398475	+	0.917179i	$0.369540\pi$
$762$	0	0
$763$	−14.6307	−0.529666
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.87689	−0.176094
$768$	0	0
$769$	−35.3693	−1.27545	−0.637725	−	0.770264i	$-0.720124\pi$
$769$	−35.3693	−1.27545	−0.637725	+	0.770264i	$0.720124\pi$
$770$	0	0
$771$	28.6695	1.03251
$772$	0	0
$773$	32.8769	1.18250	0.591250	−	0.806488i	$-0.298635\pi$
$773$	32.8769	1.18250	0.591250	+	0.806488i	$0.298635\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.75379	−0.0629168
$778$	0	0
$779$	−24.7386	−0.886354
$780$	0	0
$781$	3.75379	0.134321
$782$	0	0
$783$	3.05398	0.109140
$784$	0	0
$785$	0	0
$786$	0	0
$787$	27.3153	0.973687	0.486843	−	0.873489i	$-0.338148\pi$
$787$	27.3153	0.973687	0.486843	+	0.873489i	$0.338148\pi$
$788$	0	0
$789$	74.1080	2.63831
$790$	0	0
$791$	12.1922	0.433506
$792$	0	0
$793$	−0.630683	−0.0223962
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.42329	−0.333790	−0.166895	−	0.985975i	$-0.553374\pi$
$797$	−9.42329	−0.333790	−0.166895	+	0.985975i	$0.553374\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	17.1231	0.604261
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17.3002	−0.608995
$808$	0	0
$809$	−8.82292	−0.310197	−0.155099	−	0.987899i	$-0.549570\pi$
$809$	−8.82292	−0.310197	−0.155099	+	0.987899i	$0.549570\pi$
$810$	0	0
$811$	30.7538	1.07991	0.539956	−	0.841693i	$-0.318441\pi$
$811$	30.7538	1.07991	0.539956	+	0.841693i	$0.318441\pi$
$812$	0	0
$813$	17.7538	0.622653
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−3.12311	−0.109130
$820$	0	0
$821$	−1.50758	−0.0526148	−0.0263074	−	0.999654i	$-0.508375\pi$
$821$	−1.50758	−0.0526148	−0.0263074	+	0.999654i	$0.508375\pi$
$822$	0	0
$823$	0.946025	0.0329763	0.0164882	−	0.999864i	$-0.494751\pi$
$823$	0.946025	0.0329763	0.0164882	+	0.999864i	$0.494751\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.31534	−0.254379	−0.127190	−	0.991878i	$-0.540596\pi$
$827$	−7.31534	−0.254379	−0.127190	+	0.991878i	$0.540596\pi$
$828$	0	0
$829$	40.5464	1.40823	0.704117	−	0.710084i	$-0.251343\pi$
$829$	40.5464	1.40823	0.704117	+	0.710084i	$0.251343\pi$
$830$	0	0
$831$	14.5616	0.505135
$832$	0	0
$833$	−7.12311	−0.246801
$834$	0	0
$835$	0	0
$836$	0	0
$837$	13.3002	0.459722
$838$	0	0
$839$	−51.1231	−1.76497	−0.882483	−	0.470345i	$-0.844130\pi$
$839$	−51.1231	−1.76497	−0.882483	+	0.470345i	$0.844130\pi$
$840$	0	0
$841$	−24.4924	−0.844566
$842$	0	0
$843$	38.7386	1.33423
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.9309	−0.375589
$848$	0	0
$849$	−14.8769	−0.510574
$850$	0	0
$851$	−0.438447	−0.0150298
$852$	0	0
$853$	3.75379	0.128527	0.0642636	−	0.997933i	$-0.479530\pi$
$853$	3.75379	0.128527	0.0642636	+	0.997933i	$0.479530\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.6847	1.56056	0.780279	−	0.625431i	$-0.215077\pi$
$857$	45.6847	1.56056	0.780279	+	0.625431i	$0.215077\pi$
$858$	0	0
$859$	−47.4924	−1.62042	−0.810210	−	0.586139i	$-0.800647\pi$
$859$	−47.4924	−1.62042	−0.810210	+	0.586139i	$0.800647\pi$
$860$	0	0
$861$	16.4924	0.562060
$862$	0	0
$863$	3.43845	0.117046	0.0585231	−	0.998286i	$-0.481361\pi$
$863$	3.43845	0.117046	0.0585231	+	0.998286i	$0.481361\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	37.3002	1.26678
$868$	0	0
$869$	26.2462	0.890342
$870$	0	0
$871$	−2.49242	−0.0844525
$872$	0	0
$873$	17.3693	0.587862
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−46.9848	−1.58657	−0.793283	−	0.608853i	$-0.791630\pi$
$877$	−46.9848	−1.58657	−0.793283	+	0.608853i	$0.791630\pi$
$878$	0	0
$879$	9.75379	0.328987
$880$	0	0
$881$	−57.4773	−1.93646	−0.968229	−	0.250065i	$-0.919548\pi$
$881$	−57.4773	−1.93646	−0.968229	+	0.250065i	$0.919548\pi$
$882$	0	0
$883$	38.7386	1.30366	0.651829	−	0.758366i	$-0.274002\pi$
$883$	38.7386	1.30366	0.651829	+	0.758366i	$0.274002\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.7926	1.47041	0.735206	−	0.677844i	$-0.237085\pi$
$887$	43.7926	1.47041	0.735206	+	0.677844i	$0.237085\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	−14.0000	−0.469018
$892$	0	0
$893$	46.1080	1.54294
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.43845	−0.0480284
$898$	0	0
$899$	19.6307	0.654720
$900$	0	0
$901$	0.684658	0.0228093
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−9.31534	−0.309311	−0.154655	−	0.987968i	$-0.549427\pi$
$907$	−9.31534	−0.309311	−0.154655	+	0.987968i	$0.549427\pi$
$908$	0	0
$909$	−18.9309	−0.627897
$910$	0	0
$911$	5.12311	0.169736	0.0848680	−	0.996392i	$-0.472953\pi$
$911$	5.12311	0.169736	0.0848680	+	0.996392i	$0.472953\pi$
$912$	0	0
$913$	−29.8617	−0.988279
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32.4924	1.07299
$918$	0	0
$919$	−2.73863	−0.0903392	−0.0451696	−	0.998979i	$-0.514383\pi$
$919$	−2.73863	−0.0903392	−0.0451696	+	0.998979i	$0.514383\pi$
$920$	0	0
$921$	−32.9848	−1.08689
$922$	0	0
$923$	−1.05398	−0.0346920
$924$	0	0
$925$	0	0
$926$	0	0
$927$	56.9848	1.87163
$928$	0	0
$929$	58.7235	1.92665	0.963327	−	0.268329i	$-0.0864713\pi$
$929$	58.7235	1.92665	0.963327	+	0.268329i	$0.0864713\pi$
$930$	0	0
$931$	−27.3693	−0.896993
$932$	0	0
$933$	9.43845	0.309001
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.246211	−0.00804337	−0.00402169	−	0.999992i	$-0.501280\pi$
$937$	−0.246211	−0.00804337	−0.00402169	+	0.999992i	$0.501280\pi$
$938$	0	0
$939$	50.7386	1.65579
$940$	0	0
$941$	−54.2462	−1.76838	−0.884188	−	0.467131i	$-0.845288\pi$
$941$	−54.2462	−1.76838	−0.884188	+	0.467131i	$0.845288\pi$
$942$	0	0
$943$	4.12311	0.134267
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44.3153	−1.44006	−0.720028	−	0.693945i	$-0.755871\pi$
$947$	−44.3153	−1.44006	−0.720028	+	0.693945i	$0.755871\pi$
$948$	0	0
$949$	−4.80776	−0.156067
$950$	0	0
$951$	17.6155	0.571223
$952$	0	0
$953$	5.50758	0.178408	0.0892040	−	0.996013i	$-0.471568\pi$
$953$	5.50758	0.178408	0.0892040	+	0.996013i	$0.471568\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10.8769	0.351600
$958$	0	0
$959$	−23.6155	−0.762585
$960$	0	0
$961$	54.4924	1.75782
$962$	0	0
$963$	−26.9309	−0.867835
$964$	0	0
$965$	0	0
$966$	0	0
$967$	44.3153	1.42509	0.712543	−	0.701629i	$-0.247543\pi$
$967$	44.3153	1.42509	0.712543	+	0.701629i	$0.247543\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−47.3693	−1.52015	−0.760077	−	0.649833i	$-0.774839\pi$
$971$	−47.3693	−1.52015	−0.760077	+	0.649833i	$0.774839\pi$
$972$	0	0
$973$	8.19224	0.262631
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.7926	0.857172	0.428586	−	0.903501i	$-0.359012\pi$
$977$	26.7926	0.857172	0.428586	+	0.903501i	$0.359012\pi$
$978$	0	0
$979$	−4.49242	−0.143578
$980$	0	0
$981$	−33.3693	−1.06540
$982$	0	0
$983$	−0.0539753	−0.00172155	−0.000860773	−	1.00000i	$-0.500274\pi$
$983$	−0.0539753	−0.00172155	−0.000860773		1.00000i	$0.500274\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−30.7386	−0.978421
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−0.300187	−0.00953574	−0.00476787	−	0.999989i	$-0.501518\pi$
$991$	−0.300187	−0.00953574	−0.00476787	+	0.999989i	$0.501518\pi$
$992$	0	0
$993$	32.3153	1.02550
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.3845	−0.455561	−0.227780	−	0.973713i	$-0.573147\pi$
$997$	−14.3845	−0.455561	−0.227780	+	0.973713i	$0.573147\pi$
$998$	0	0
$999$	−0.630683	−0.0199539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.a.i.1.1		2
4.3	odd	2		9200.2.a.bv.1.2		2
5.2	odd	4		2300.2.c.h.1749.4		4
5.3	odd	4		2300.2.c.h.1749.1		4
5.4	even	2		460.2.a.e.1.2	✓	2
15.14	odd	2		4140.2.a.m.1.1		2
20.19	odd	2		1840.2.a.m.1.1		2
40.19	odd	2		7360.2.a.bo.1.2		2
40.29	even	2		7360.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.a.e.1.2	✓	2	5.4	even	2
1840.2.a.m.1.1		2	20.19	odd	2
2300.2.a.i.1.1		2	1.1	even	1	trivial
2300.2.c.h.1749.1		4	5.3	odd	4
2300.2.c.h.1749.4		4	5.2	odd	4
4140.2.a.m.1.1		2	15.14	odd	2
7360.2.a.bi.1.1		2	40.29	even	2
7360.2.a.bo.1.2		2	40.19	odd	2
9200.2.a.bv.1.2		2	4.3	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} +1.56155 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q-2.56155 q^{3} +1.56155 q^{7} +3.56155 q^{9} +2.00000 q^{11} -0.561553 q^{13} +1.56155 q^{17} +6.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} -1.43845 q^{27} -2.12311 q^{29} -9.24621 q^{31} -5.12311 q^{33} +0.438447 q^{37} +1.43845 q^{39} -4.12311 q^{41} +7.68466 q^{47} -4.56155 q^{49} -4.00000 q^{51} +0.438447 q^{53} -15.3693 q^{57} +8.68466 q^{59} +1.12311 q^{61} +5.56155 q^{63} +4.43845 q^{67} +2.56155 q^{69} +1.87689 q^{71} +8.56155 q^{73} +3.12311 q^{77} +13.1231 q^{79} -7.00000 q^{81} -14.9309 q^{83} +5.43845 q^{87} -2.24621 q^{89} -0.876894 q^{91} +23.6847 q^{93} +4.87689 q^{97} +7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - q^7 + 3 * q^9	\( 2 q - q^{3} - q^{7} + 3 q^{9} + 4 q^{11} + 3 q^{13} - q^{17} + 12 q^{19} - 8 q^{21} - 2 q^{23} - 7 q^{27} + 4 q^{29} - 2 q^{31} - 2 q^{33} + 5 q^{37} + 7 q^{39} + 3 q^{47} - 5 q^{49} - 8 q^{51} + 5 q^{53} - 6 q^{57} + 5 q^{59} - 6 q^{61} + 7 q^{63} + 13 q^{67} + q^{69} + 12 q^{71} + 13 q^{73} - 2 q^{77} + 18 q^{79} - 14 q^{81} - q^{83} + 15 q^{87} + 12 q^{89} - 10 q^{91} + 35 q^{93} + 18 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^7 + 3 * q^9 + 4 * q^11 + 3 * q^13 - q^17 + 12 * q^19 - 8 * q^21 - 2 * q^23 - 7 * q^27 + 4 * q^29 - 2 * q^31 - 2 * q^33 + 5 * q^37 + 7 * q^39 + 3 * q^47 - 5 * q^49 - 8 * q^51 + 5 * q^53 - 6 * q^57 + 5 * q^59 - 6 * q^61 + 7 * q^63 + 13 * q^67 + q^69 + 12 * q^71 + 13 * q^73 - 2 * q^77 + 18 * q^79 - 14 * q^81 - q^83 + 15 * q^87 + 12 * q^89 - 10 * q^91 + 35 * q^93 + 18 * q^97 + 6 * q^99

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