LMFDB - Embedded newform 2900.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2900,2,Mod(1,2900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2900, 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("2900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.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:	$23.1566165862$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2370465.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - x^{2} + 9x - 3 $ x^5 - 9x^3 - x^2 + 9x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.672645$ of defining polynomial
Character	$\chi$	$=$	2900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.672645 q^{3} +3.37701 q^{7} -2.54755 q^{9} +2.11471 q^{11} -1.91246 q^{13} -5.53382 q^{17} -7.99382 q^{19} +2.27153 q^{21} -3.27602 q^{23} -3.73153 q^{27} -1.00000 q^{29} -10.2833 q^{31} +1.42245 q^{33} +1.37246 q^{37} -1.28640 q^{39} -1.81453 q^{41} -1.60338 q^{43} -0.327355 q^{47} +4.40418 q^{49} -3.72230 q^{51} -4.18848 q^{53} -5.37701 q^{57} -3.30908 q^{59} +7.00306 q^{61} -8.60309 q^{63} +6.57443 q^{67} -2.20360 q^{69} +15.5005 q^{71} -5.43283 q^{73} +7.14140 q^{77} +2.72819 q^{79} +5.13265 q^{81} -14.9969 q^{83} -0.672645 q^{87} +12.6048 q^{89} -6.45838 q^{91} -6.91700 q^{93} +0.851941 q^{97} -5.38734 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{7} + 3 q^{9} + 4 q^{11} - 12 q^{13} - 5 q^{17} - 4 q^{19} + 3 q^{21} - 9 q^{23} + 3 q^{27} - 5 q^{29} + 3 q^{31} - 21 q^{33} - 4 q^{37} - 18 q^{39} + 5 q^{41} - 4 q^{43} - 5 q^{47} - 3 q^{49}+ \cdots - 36 q^{99}+O(q^{100}) $ 5 * q - 4 * q^7 + 3 * q^9 + 4 * q^11 - 12 * q^13 - 5 * q^17 - 4 * q^19 + 3 * q^21 - 9 * q^23 + 3 * q^27 - 5 * q^29 + 3 * q^31 - 21 * q^33 - 4 * q^37 - 18 * q^39 + 5 * q^41 - 4 * q^43 - 5 * q^47 - 3 * q^49 + 9 * q^51 - 16 * q^53 - 6 * q^57 - 23 * q^59 + 5 * q^61 - 21 * q^63 - 5 * q^67 - 30 * q^69 + 23 * q^71 - 18 * q^73 - 8 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 6 * q^89 + 27 * q^91 - 27 * q^93 - 21 * q^97 - 36 * q^99

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
$3$	0.672645	0.388352	0.194176	−	0.980967i	$-0.437797\pi$
$3$	0.672645	0.388352	0.194176	+	0.980967i	$0.437797\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.37701	1.27639	0.638194	−	0.769875i	$-0.279682\pi$
$7$	3.37701	1.27639	0.638194	+	0.769875i	$0.279682\pi$
$8$	0	0
$9$	−2.54755	−0.849183
$10$	0	0
$11$	2.11471	0.637610	0.318805	−	0.947820i	$-0.396718\pi$
$11$	2.11471	0.637610	0.318805	+	0.947820i	$0.396718\pi$
$12$	0	0
$13$	−1.91246	−0.530420	−0.265210	−	0.964191i	$-0.585441\pi$
$13$	−1.91246	−0.530420	−0.265210	+	0.964191i	$0.585441\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.53382	−1.34215	−0.671074	−	0.741390i	$-0.734167\pi$
$17$	−5.53382	−1.34215	−0.671074	+	0.741390i	$0.734167\pi$
$18$	0	0
$19$	−7.99382	−1.83391	−0.916955	−	0.398992i	$-0.869360\pi$
$19$	−7.99382	−1.83391	−0.916955	+	0.398992i	$0.869360\pi$
$20$	0	0
$21$	2.27153	0.495688
$22$	0	0
$23$	−3.27602	−0.683098	−0.341549	−	0.939864i	$-0.610951\pi$
$23$	−3.27602	−0.683098	−0.341549	+	0.939864i	$0.610951\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.73153	−0.718134
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−10.2833	−1.84693	−0.923466	−	0.383679i	$-0.874657\pi$
$31$	−10.2833	−1.84693	−0.923466	+	0.383679i	$0.874657\pi$
$32$	0	0
$33$	1.42245	0.247617
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.37246	0.225631	0.112815	−	0.993616i	$-0.464013\pi$
$37$	1.37246	0.225631	0.112815	+	0.993616i	$0.464013\pi$
$38$	0	0
$39$	−1.28640	−0.205989
$40$	0	0
$41$	−1.81453	−0.283382	−0.141691	−	0.989911i	$-0.545254\pi$
$41$	−1.81453	−0.283382	−0.141691	+	0.989911i	$0.545254\pi$
$42$	0	0
$43$	−1.60338	−0.244513	−0.122256	−	0.992499i	$-0.539013\pi$
$43$	−1.60338	−0.244513	−0.122256	+	0.992499i	$0.539013\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.327355	−0.0477496	−0.0238748	−	0.999715i	$-0.507600\pi$
$47$	−0.327355	−0.0477496	−0.0238748	+	0.999715i	$0.507600\pi$
$48$	0	0
$49$	4.40418	0.629168
$50$	0	0
$51$	−3.72230	−0.521226
$52$	0	0
$53$	−4.18848	−0.575332	−0.287666	−	0.957731i	$-0.592879\pi$
$53$	−4.18848	−0.575332	−0.287666	+	0.957731i	$0.592879\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.37701	−0.712202
$58$	0	0
$59$	−3.30908	−0.430805	−0.215403	−	0.976525i	$-0.569106\pi$
$59$	−3.30908	−0.430805	−0.215403	+	0.976525i	$0.569106\pi$
$60$	0	0
$61$	7.00306	0.896650	0.448325	−	0.893871i	$-0.352021\pi$
$61$	7.00306	0.896650	0.448325	+	0.893871i	$0.352021\pi$
$62$	0	0
$63$	−8.60309	−1.08389
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.57443	0.803195	0.401597	−	0.915816i	$-0.368455\pi$
$67$	6.57443	0.803195	0.401597	+	0.915816i	$0.368455\pi$
$68$	0	0
$69$	−2.20360	−0.265282
$70$	0	0
$71$	15.5005	1.83957	0.919784	−	0.392425i	$-0.128364\pi$
$71$	15.5005	1.83957	0.919784	+	0.392425i	$0.128364\pi$
$72$	0	0
$73$	−5.43283	−0.635865	−0.317933	−	0.948113i	$-0.602989\pi$
$73$	−5.43283	−0.635865	−0.317933	+	0.948113i	$0.602989\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.14140	0.813838
$78$	0	0
$79$	2.72819	0.306945	0.153472	−	0.988153i	$-0.450954\pi$
$79$	2.72819	0.306945	0.153472	+	0.988153i	$0.450954\pi$
$80$	0	0
$81$	5.13265	0.570294
$82$	0	0
$83$	−14.9969	−1.64612	−0.823061	−	0.567953i	$-0.807735\pi$
$83$	−14.9969	−1.64612	−0.823061	+	0.567953i	$0.807735\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.672645	−0.0721151
$88$	0	0
$89$	12.6048	1.33610	0.668051	−	0.744116i	$-0.267129\pi$
$89$	12.6048	1.33610	0.668051	+	0.744116i	$0.267129\pi$
$90$	0	0
$91$	−6.45838	−0.677022
$92$	0	0
$93$	−6.91700	−0.717260
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.851941	0.0865015	0.0432508	−	0.999064i	$-0.486229\pi$
$97$	0.851941	0.0865015	0.0432508	+	0.999064i	$0.486229\pi$
$98$	0	0
$99$	−5.38734	−0.541448
$100$	0	0
$101$	17.4522	1.73656	0.868279	−	0.496075i	$-0.165226\pi$
$101$	17.4522	1.73656	0.868279	+	0.496075i	$0.165226\pi$
$102$	0	0
$103$	−10.6951	−1.05382	−0.526911	−	0.849920i	$-0.676650\pi$
$103$	−10.6951	−1.05382	−0.526911	+	0.849920i	$0.676650\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−20.5895	−1.99047	−0.995233	−	0.0975276i	$-0.968907\pi$
$107$	−20.5895	−1.99047	−0.995233	+	0.0975276i	$0.968907\pi$
$108$	0	0
$109$	0.972490	0.0931477	0.0465738	−	0.998915i	$-0.485170\pi$
$109$	0.972490	0.0931477	0.0465738	+	0.998915i	$0.485170\pi$
$110$	0	0
$111$	0.923178	0.0876242
$112$	0	0
$113$	12.7192	1.19652	0.598260	−	0.801302i	$-0.295859\pi$
$113$	12.7192	1.19652	0.598260	+	0.801302i	$0.295859\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.87207	0.450423
$118$	0	0
$119$	−18.6878	−1.71310
$120$	0	0
$121$	−6.52799	−0.593453
$122$	0	0
$123$	−1.22053	−0.110052
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.2036	1.34910	0.674552	−	0.738227i	$-0.264337\pi$
$127$	15.2036	1.34910	0.674552	+	0.738227i	$0.264337\pi$
$128$	0	0
$129$	−1.07850	−0.0949569
$130$	0	0
$131$	6.03755	0.527503	0.263752	−	0.964591i	$-0.415040\pi$
$131$	6.03755	0.527503	0.263752	+	0.964591i	$0.415040\pi$
$132$	0	0
$133$	−26.9952	−2.34078
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.4901	−1.32341	−0.661704	−	0.749765i	$-0.730166\pi$
$137$	−15.4901	−1.32341	−0.661704	+	0.749765i	$0.730166\pi$
$138$	0	0
$139$	2.09639	0.177813	0.0889065	−	0.996040i	$-0.471663\pi$
$139$	2.09639	0.177813	0.0889065	+	0.996040i	$0.471663\pi$
$140$	0	0
$141$	−0.220194	−0.0185437
$142$	0	0
$143$	−4.04430	−0.338201
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.96245	0.244339
$148$	0	0
$149$	5.07653	0.415886	0.207943	−	0.978141i	$-0.433323\pi$
$149$	5.07653	0.415886	0.207943	+	0.978141i	$0.433323\pi$
$150$	0	0
$151$	12.6412	1.02872	0.514361	−	0.857574i	$-0.328029\pi$
$151$	12.6412	1.02872	0.514361	+	0.857574i	$0.328029\pi$
$152$	0	0
$153$	14.0977	1.13973
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−19.7495	−1.57618	−0.788090	−	0.615561i	$-0.788930\pi$
$157$	−19.7495	−1.57618	−0.788090	+	0.615561i	$0.788930\pi$
$158$	0	0
$159$	−2.81736	−0.223431
$160$	0	0
$161$	−11.0631	−0.871898
$162$	0	0
$163$	19.9484	1.56248	0.781240	−	0.624230i	$-0.214587\pi$
$163$	19.9484	1.56248	0.781240	+	0.624230i	$0.214587\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.34529	−0.181484	−0.0907420	−	0.995874i	$-0.528924\pi$
$167$	−2.34529	−0.181484	−0.0907420	+	0.995874i	$0.528924\pi$
$168$	0	0
$169$	−9.34251	−0.718655
$170$	0	0
$171$	20.3647	1.55732
$172$	0	0
$173$	−14.8187	−1.12665	−0.563324	−	0.826236i	$-0.690478\pi$
$173$	−14.8187	−1.12665	−0.563324	+	0.826236i	$0.690478\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.22584	−0.167304
$178$	0	0
$179$	−6.12381	−0.457715	−0.228857	−	0.973460i	$-0.573499\pi$
$179$	−6.12381	−0.457715	−0.228857	+	0.973460i	$0.573499\pi$
$180$	0	0
$181$	−18.4853	−1.37400	−0.686999	−	0.726658i	$-0.741072\pi$
$181$	−18.4853	−1.37400	−0.686999	+	0.726658i	$0.741072\pi$
$182$	0	0
$183$	4.71057	0.348216
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−11.7024	−0.855768
$188$	0	0
$189$	−12.6014	−0.916618
$190$	0	0
$191$	12.1585	0.879757	0.439878	−	0.898057i	$-0.355022\pi$
$191$	12.1585	0.879757	0.439878	+	0.898057i	$0.355022\pi$
$192$	0	0
$193$	−19.6332	−1.41323	−0.706614	−	0.707600i	$-0.749778\pi$
$193$	−19.6332	−1.41323	−0.706614	+	0.707600i	$0.749778\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.0656	1.21588	0.607938	−	0.793985i	$-0.291997\pi$
$197$	17.0656	1.21588	0.607938	+	0.793985i	$0.291997\pi$
$198$	0	0
$199$	−22.8668	−1.62098	−0.810491	−	0.585751i	$-0.800800\pi$
$199$	−22.8668	−1.62098	−0.810491	+	0.585751i	$0.800800\pi$
$200$	0	0
$201$	4.42226	0.311922
$202$	0	0
$203$	−3.37701	−0.237019
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.34582	0.580075
$208$	0	0
$209$	−16.9047	−1.16932
$210$	0	0
$211$	18.2332	1.25522	0.627611	−	0.778527i	$-0.284033\pi$
$211$	18.2332	1.25522	0.627611	+	0.778527i	$0.284033\pi$
$212$	0	0
$213$	10.4263	0.714400
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−34.7267	−2.35740
$218$	0	0
$219$	−3.65437	−0.246939
$220$	0	0
$221$	10.5832	0.711902
$222$	0	0
$223$	−3.50379	−0.234631	−0.117315	−	0.993095i	$-0.537429\pi$
$223$	−3.50379	−0.234631	−0.117315	+	0.993095i	$0.537429\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.0723	0.867641	0.433820	−	0.900999i	$-0.357165\pi$
$227$	13.0723	0.867641	0.433820	+	0.900999i	$0.357165\pi$
$228$	0	0
$229$	15.7072	1.03796	0.518981	−	0.854786i	$-0.326311\pi$
$229$	15.7072	1.03796	0.518981	+	0.854786i	$0.326311\pi$
$230$	0	0
$231$	4.80363	0.316056
$232$	0	0
$233$	5.96096	0.390515	0.195258	−	0.980752i	$-0.437446\pi$
$233$	5.96096	0.390515	0.195258	+	0.980752i	$0.437446\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.83510	0.119203
$238$	0	0
$239$	−5.20661	−0.336787	−0.168394	−	0.985720i	$-0.553858\pi$
$239$	−5.20661	−0.336787	−0.168394	+	0.985720i	$0.553858\pi$
$240$	0	0
$241$	−9.12261	−0.587639	−0.293819	−	0.955861i	$-0.594926\pi$
$241$	−9.12261	−0.587639	−0.293819	+	0.955861i	$0.594926\pi$
$242$	0	0
$243$	14.6470	0.939608
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.2878	0.972742
$248$	0	0
$249$	−10.0876	−0.639274
$250$	0	0
$251$	−23.8338	−1.50437	−0.752187	−	0.658950i	$-0.771001\pi$
$251$	−23.8338	−1.50437	−0.752187	+	0.658950i	$0.771001\pi$
$252$	0	0
$253$	−6.92785	−0.435550
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.79497	0.548615	0.274307	−	0.961642i	$-0.411551\pi$
$257$	8.79497	0.548615	0.274307	+	0.961642i	$0.411551\pi$
$258$	0	0
$259$	4.63481	0.287993
$260$	0	0
$261$	2.54755	0.157689
$262$	0	0
$263$	17.0631	1.05216	0.526079	−	0.850436i	$-0.323662\pi$
$263$	17.0631	1.05216	0.526079	+	0.850436i	$0.323662\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.47853	0.518877
$268$	0	0
$269$	18.1534	1.10683	0.553415	−	0.832906i	$-0.313324\pi$
$269$	18.1534	1.10683	0.553415	+	0.832906i	$0.313324\pi$
$270$	0	0
$271$	−13.2003	−0.801861	−0.400930	−	0.916109i	$-0.631313\pi$
$271$	−13.2003	−0.801861	−0.400930	+	0.916109i	$0.631313\pi$
$272$	0	0
$273$	−4.34420	−0.262923
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−15.7679	−0.947403	−0.473702	−	0.880685i	$-0.657083\pi$
$277$	−15.7679	−0.947403	−0.473702	+	0.880685i	$0.657083\pi$
$278$	0	0
$279$	26.1972	1.56838
$280$	0	0
$281$	−1.11500	−0.0665154	−0.0332577	−	0.999447i	$-0.510588\pi$
$281$	−1.11500	−0.0665154	−0.0332577	+	0.999447i	$0.510588\pi$
$282$	0	0
$283$	−17.7462	−1.05490	−0.527450	−	0.849586i	$-0.676852\pi$
$283$	−17.7462	−1.05490	−0.527450	+	0.849586i	$0.676852\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.12768	−0.361705
$288$	0	0
$289$	13.6232	0.801363
$290$	0	0
$291$	0.573054	0.0335930
$292$	0	0
$293$	1.44336	0.0843218	0.0421609	−	0.999111i	$-0.486576\pi$
$293$	1.44336	0.0843218	0.0421609	+	0.999111i	$0.486576\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−7.89112	−0.457889
$298$	0	0
$299$	6.26525	0.362328
$300$	0	0
$301$	−5.41461	−0.312093
$302$	0	0
$303$	11.7391	0.674396
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5.34395	−0.304995	−0.152498	−	0.988304i	$-0.548732\pi$
$307$	−5.34395	−0.304995	−0.152498	+	0.988304i	$0.548732\pi$
$308$	0	0
$309$	−7.19403	−0.409254
$310$	0	0
$311$	24.6634	1.39854	0.699268	−	0.714860i	$-0.253509\pi$
$311$	24.6634	1.39854	0.699268	+	0.714860i	$0.253509\pi$
$312$	0	0
$313$	−33.9786	−1.92058	−0.960292	−	0.278998i	$-0.909998\pi$
$313$	−33.9786	−1.92058	−0.960292	+	0.278998i	$0.909998\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.0876	−0.566575	−0.283287	−	0.959035i	$-0.591425\pi$
$317$	−10.0876	−0.566575	−0.283287	+	0.959035i	$0.591425\pi$
$318$	0	0
$319$	−2.11471	−0.118401
$320$	0	0
$321$	−13.8495	−0.773001
$322$	0	0
$323$	44.2364	2.46138
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.654141	0.0361741
$328$	0	0
$329$	−1.10548	−0.0609471
$330$	0	0
$331$	−16.3891	−0.900824	−0.450412	−	0.892821i	$-0.648723\pi$
$331$	−16.3891	−0.900824	−0.450412	+	0.892821i	$0.648723\pi$
$332$	0	0
$333$	−3.49641	−0.191602
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.3782	−1.00112	−0.500562	−	0.865700i	$-0.666873\pi$
$337$	−18.3782	−1.00112	−0.500562	+	0.865700i	$0.666873\pi$
$338$	0	0
$339$	8.55549	0.464671
$340$	0	0
$341$	−21.7462	−1.17762
$342$	0	0
$343$	−8.76611	−0.473326
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.6029	0.783925	0.391962	−	0.919981i	$-0.371796\pi$
$347$	14.6029	0.783925	0.391962	+	0.919981i	$0.371796\pi$
$348$	0	0
$349$	21.4716	1.14935	0.574675	−	0.818382i	$-0.305129\pi$
$349$	21.4716	1.14935	0.574675	+	0.818382i	$0.305129\pi$
$350$	0	0
$351$	7.13639	0.380912
$352$	0	0
$353$	−22.9674	−1.22243	−0.611214	−	0.791465i	$-0.709319\pi$
$353$	−22.9674	−1.22243	−0.611214	+	0.791465i	$0.709319\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−12.5702	−0.665287
$358$	0	0
$359$	1.62824	0.0859351	0.0429675	−	0.999076i	$-0.486319\pi$
$359$	1.62824	0.0859351	0.0429675	+	0.999076i	$0.486319\pi$
$360$	0	0
$361$	44.9012	2.36322
$362$	0	0
$363$	−4.39102	−0.230469
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.19451	−0.114552	−0.0572761	−	0.998358i	$-0.518242\pi$
$367$	−2.19451	−0.114552	−0.0572761	+	0.998358i	$0.518242\pi$
$368$	0	0
$369$	4.62260	0.240643
$370$	0	0
$371$	−14.1445	−0.734347
$372$	0	0
$373$	6.00186	0.310764	0.155382	−	0.987854i	$-0.450339\pi$
$373$	6.00186	0.310764	0.155382	+	0.987854i	$0.450339\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.91246	0.0984965
$378$	0	0
$379$	−0.966315	−0.0496363	−0.0248181	−	0.999692i	$-0.507901\pi$
$379$	−0.966315	−0.0496363	−0.0248181	+	0.999692i	$0.507901\pi$
$380$	0	0
$381$	10.2267	0.523927
$382$	0	0
$383$	−11.6799	−0.596816	−0.298408	−	0.954438i	$-0.596456\pi$
$383$	−11.6799	−0.596816	−0.298408	+	0.954438i	$0.596456\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.08468	0.207636
$388$	0	0
$389$	−2.55610	−0.129600	−0.0647998	−	0.997898i	$-0.520641\pi$
$389$	−2.55610	−0.129600	−0.0647998	+	0.997898i	$0.520641\pi$
$390$	0	0
$391$	18.1289	0.916819
$392$	0	0
$393$	4.06113	0.204857
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.08996	0.456212	0.228106	−	0.973636i	$-0.426747\pi$
$397$	9.08996	0.456212	0.228106	+	0.973636i	$0.426747\pi$
$398$	0	0
$399$	−18.1582	−0.909047
$400$	0	0
$401$	4.58696	0.229062	0.114531	−	0.993420i	$-0.463464\pi$
$401$	4.58696	0.229062	0.114531	+	0.993420i	$0.463464\pi$
$402$	0	0
$403$	19.6663	0.979650
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.90236	0.143865
$408$	0	0
$409$	−10.5214	−0.520248	−0.260124	−	0.965575i	$-0.583763\pi$
$409$	−10.5214	−0.520248	−0.260124	+	0.965575i	$0.583763\pi$
$410$	0	0
$411$	−10.4193	−0.513948
$412$	0	0
$413$	−11.1748	−0.549875
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.41012	0.0690540
$418$	0	0
$419$	5.75688	0.281242	0.140621	−	0.990064i	$-0.455090\pi$
$419$	5.75688	0.281242	0.140621	+	0.990064i	$0.455090\pi$
$420$	0	0
$421$	−29.0307	−1.41487	−0.707436	−	0.706778i	$-0.750148\pi$
$421$	−29.0307	−1.41487	−0.707436	+	0.706778i	$0.750148\pi$
$422$	0	0
$423$	0.833952	0.0405482
$424$	0	0
$425$	0	0
$426$	0	0
$427$	23.6494	1.14447
$428$	0	0
$429$	−2.72038	−0.131341
$430$	0	0
$431$	−6.64217	−0.319942	−0.159971	−	0.987122i	$-0.551140\pi$
$431$	−6.64217	−0.319942	−0.159971	+	0.987122i	$0.551140\pi$
$432$	0	0
$433$	23.9576	1.15133	0.575663	−	0.817687i	$-0.304744\pi$
$433$	23.9576	1.15133	0.575663	+	0.817687i	$0.304744\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.1879	1.25274
$438$	0	0
$439$	−14.7719	−0.705022	−0.352511	−	0.935808i	$-0.614672\pi$
$439$	−14.7719	−0.705022	−0.352511	+	0.935808i	$0.614672\pi$
$440$	0	0
$441$	−11.2199	−0.534279
$442$	0	0
$443$	−6.01027	−0.285557	−0.142778	−	0.989755i	$-0.545604\pi$
$443$	−6.01027	−0.285557	−0.142778	+	0.989755i	$0.545604\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.41471	0.161510
$448$	0	0
$449$	13.0960	0.618037	0.309018	−	0.951056i	$-0.400000\pi$
$449$	13.0960	0.618037	0.309018	+	0.951056i	$0.400000\pi$
$450$	0	0
$451$	−3.83721	−0.180687
$452$	0	0
$453$	8.50301	0.399506
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.1969	1.41255	0.706275	−	0.707938i	$-0.250374\pi$
$457$	30.1969	1.41255	0.706275	+	0.707938i	$0.250374\pi$
$458$	0	0
$459$	20.6496	0.963842
$460$	0	0
$461$	−8.78242	−0.409038	−0.204519	−	0.978863i	$-0.565563\pi$
$461$	−8.78242	−0.409038	−0.204519	+	0.978863i	$0.565563\pi$
$462$	0	0
$463$	33.7175	1.56699	0.783493	−	0.621401i	$-0.213436\pi$
$463$	33.7175	1.56699	0.783493	+	0.621401i	$0.213436\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.9838	0.924739	0.462370	−	0.886687i	$-0.346999\pi$
$467$	19.9838	0.924739	0.462370	+	0.886687i	$0.346999\pi$
$468$	0	0
$469$	22.2019	1.02519
$470$	0	0
$471$	−13.2844	−0.612112
$472$	0	0
$473$	−3.39068	−0.155904
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.6703	0.488562
$478$	0	0
$479$	−15.2093	−0.694930	−0.347465	−	0.937693i	$-0.612957\pi$
$479$	−15.2093	−0.694930	−0.347465	+	0.937693i	$0.612957\pi$
$480$	0	0
$481$	−2.62477	−0.119679
$482$	0	0
$483$	−7.44157	−0.338603
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−25.9309	−1.17504	−0.587520	−	0.809210i	$-0.699896\pi$
$487$	−25.9309	−1.17504	−0.587520	+	0.809210i	$0.699896\pi$
$488$	0	0
$489$	13.4182	0.606792
$490$	0	0
$491$	43.0917	1.94470	0.972351	−	0.233523i	$-0.0750254\pi$
$491$	43.0917	1.94470	0.972351	+	0.233523i	$0.0750254\pi$
$492$	0	0
$493$	5.53382	0.249231
$494$	0	0
$495$	0	0
$496$	0	0
$497$	52.3452	2.34800
$498$	0	0
$499$	15.1581	0.678568	0.339284	−	0.940684i	$-0.389815\pi$
$499$	15.1581	0.678568	0.339284	+	0.940684i	$0.389815\pi$
$500$	0	0
$501$	−1.57755	−0.0704796
$502$	0	0
$503$	−17.7730	−0.792461	−0.396230	−	0.918151i	$-0.629682\pi$
$503$	−17.7730	−0.792461	−0.396230	+	0.918151i	$0.629682\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.28420	−0.279091
$508$	0	0
$509$	13.7177	0.608029	0.304014	−	0.952667i	$-0.401673\pi$
$509$	13.7177	0.608029	0.304014	+	0.952667i	$0.401673\pi$
$510$	0	0
$511$	−18.3467	−0.811611
$512$	0	0
$513$	29.8292	1.31699
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.692262	−0.0304456
$518$	0	0
$519$	−9.96775	−0.437536
$520$	0	0
$521$	−28.4327	−1.24566	−0.622830	−	0.782357i	$-0.714017\pi$
$521$	−28.4327	−1.24566	−0.622830	+	0.782357i	$0.714017\pi$
$522$	0	0
$523$	16.0837	0.703289	0.351645	−	0.936134i	$-0.385623\pi$
$523$	16.0837	0.703289	0.351645	+	0.936134i	$0.385623\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	56.9059	2.47886
$528$	0	0
$529$	−12.2677	−0.533378
$530$	0	0
$531$	8.43004	0.365833
$532$	0	0
$533$	3.47021	0.150311
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.11915	−0.177754
$538$	0	0
$539$	9.31357	0.401164
$540$	0	0
$541$	−26.5985	−1.14356	−0.571779	−	0.820408i	$-0.693747\pi$
$541$	−26.5985	−1.14356	−0.571779	+	0.820408i	$0.693747\pi$
$542$	0	0
$543$	−12.4340	−0.533595
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−10.6965	−0.457350	−0.228675	−	0.973503i	$-0.573439\pi$
$547$	−10.6965	−0.457350	−0.228675	+	0.973503i	$0.573439\pi$
$548$	0	0
$549$	−17.8406	−0.761420
$550$	0	0
$551$	7.99382	0.340548
$552$	0	0
$553$	9.21310	0.391781
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−40.1771	−1.70236	−0.851179	−	0.524876i	$-0.824112\pi$
$557$	−40.1771	−1.70236	−0.851179	+	0.524876i	$0.824112\pi$
$558$	0	0
$559$	3.06639	0.129694
$560$	0	0
$561$	−7.87159	−0.332339
$562$	0	0
$563$	22.8080	0.961243	0.480622	−	0.876928i	$-0.340411\pi$
$563$	22.8080	0.961243	0.480622	+	0.876928i	$0.340411\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	17.3330	0.727917
$568$	0	0
$569$	−0.554721	−0.0232551	−0.0116275	−	0.999932i	$-0.503701\pi$
$569$	−0.554721	−0.0232551	−0.0116275	+	0.999932i	$0.503701\pi$
$570$	0	0
$571$	−10.8977	−0.456055	−0.228028	−	0.973655i	$-0.573228\pi$
$571$	−10.8977	−0.456055	−0.228028	+	0.973655i	$0.573228\pi$
$572$	0	0
$573$	8.17834	0.341655
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−36.1882	−1.50653	−0.753267	−	0.657715i	$-0.771523\pi$
$577$	−36.1882	−1.50653	−0.753267	+	0.657715i	$0.771523\pi$
$578$	0	0
$579$	−13.2062	−0.548829
$580$	0	0
$581$	−50.6446	−2.10109
$582$	0	0
$583$	−8.85743	−0.366837
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.6509	0.439609	0.219805	−	0.975544i	$-0.429458\pi$
$587$	10.6509	0.439609	0.219805	+	0.975544i	$0.429458\pi$
$588$	0	0
$589$	82.2028	3.38711
$590$	0	0
$591$	11.4791	0.472187
$592$	0	0
$593$	43.0193	1.76659	0.883296	−	0.468815i	$-0.155319\pi$
$593$	43.0193	1.76659	0.883296	+	0.468815i	$0.155319\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.3812	−0.629511
$598$	0	0
$599$	−4.72552	−0.193080	−0.0965398	−	0.995329i	$-0.530777\pi$
$599$	−4.72552	−0.193080	−0.0965398	+	0.995329i	$0.530777\pi$
$600$	0	0
$601$	−24.7076	−1.00784	−0.503921	−	0.863750i	$-0.668110\pi$
$601$	−24.7076	−1.00784	−0.503921	+	0.863750i	$0.668110\pi$
$602$	0	0
$603$	−16.7487	−0.682059
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.0370	−0.610331	−0.305166	−	0.952299i	$-0.598712\pi$
$607$	−15.0370	−0.610331	−0.305166	+	0.952299i	$0.598712\pi$
$608$	0	0
$609$	−2.27153	−0.0920469
$610$	0	0
$611$	0.626052	0.0253273
$612$	0	0
$613$	−22.1098	−0.893007	−0.446504	−	0.894782i	$-0.647331\pi$
$613$	−22.1098	−0.893007	−0.446504	+	0.894782i	$0.647331\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.9264	−1.32557	−0.662783	−	0.748811i	$-0.730625\pi$
$617$	−32.9264	−1.32557	−0.662783	+	0.748811i	$0.730625\pi$
$618$	0	0
$619$	−2.81002	−0.112944	−0.0564721	−	0.998404i	$-0.517985\pi$
$619$	−2.81002	−0.112944	−0.0564721	+	0.998404i	$0.517985\pi$
$620$	0	0
$621$	12.2246	0.490555
$622$	0	0
$623$	42.5663	1.70538
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−11.3708	−0.454107
$628$	0	0
$629$	−7.59495	−0.302830
$630$	0	0
$631$	5.73420	0.228275	0.114137	−	0.993465i	$-0.463590\pi$
$631$	5.73420	0.228275	0.114137	+	0.993465i	$0.463590\pi$
$632$	0	0
$633$	12.2644	0.487468
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8.42279	−0.333723
$638$	0	0
$639$	−39.4882	−1.56213
$640$	0	0
$641$	−12.3873	−0.489270	−0.244635	−	0.969615i	$-0.578668\pi$
$641$	−12.3873	−0.489270	−0.244635	+	0.969615i	$0.578668\pi$
$642$	0	0
$643$	46.2926	1.82560	0.912801	−	0.408404i	$-0.133915\pi$
$643$	46.2926	1.82560	0.912801	+	0.408404i	$0.133915\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−47.7387	−1.87680	−0.938400	−	0.345551i	$-0.887692\pi$
$647$	−47.7387	−1.87680	−0.938400	+	0.345551i	$0.887692\pi$
$648$	0	0
$649$	−6.99776	−0.274686
$650$	0	0
$651$	−23.3588	−0.915502
$652$	0	0
$653$	−0.227460	−0.00890118	−0.00445059	−	0.999990i	$-0.501417\pi$
$653$	−0.227460	−0.00890118	−0.00445059	+	0.999990i	$0.501417\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.8404	0.539966
$658$	0	0
$659$	−19.8569	−0.773513	−0.386757	−	0.922182i	$-0.626405\pi$
$659$	−19.8569	−0.773513	−0.386757	+	0.922182i	$0.626405\pi$
$660$	0	0
$661$	−25.7959	−1.00334	−0.501671	−	0.865058i	$-0.667281\pi$
$661$	−25.7959	−1.00334	−0.501671	+	0.865058i	$0.667281\pi$
$662$	0	0
$663$	7.11873	0.276468
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.27602	0.126848
$668$	0	0
$669$	−2.35680	−0.0911193
$670$	0	0
$671$	14.8095	0.571713
$672$	0	0
$673$	39.4615	1.52113	0.760564	−	0.649263i	$-0.224923\pi$
$673$	39.4615	1.52113	0.760564	+	0.649263i	$0.224923\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.31980	0.0507241	0.0253621	−	0.999678i	$-0.491926\pi$
$677$	1.31980	0.0507241	0.0253621	+	0.999678i	$0.491926\pi$
$678$	0	0
$679$	2.87701	0.110410
$680$	0	0
$681$	8.79304	0.336950
$682$	0	0
$683$	6.48925	0.248304	0.124152	−	0.992263i	$-0.460379\pi$
$683$	6.48925	0.248304	0.124152	+	0.992263i	$0.460379\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.5654	0.403095
$688$	0	0
$689$	8.01028	0.305167
$690$	0	0
$691$	−18.6676	−0.710150	−0.355075	−	0.934838i	$-0.615545\pi$
$691$	−18.6676	−0.710150	−0.355075	+	0.934838i	$0.615545\pi$
$692$	0	0
$693$	−18.1931	−0.691098
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.0413	0.380340
$698$	0	0
$699$	4.00961	0.151657
$700$	0	0
$701$	7.90767	0.298669	0.149334	−	0.988787i	$-0.452287\pi$
$701$	7.90767	0.298669	0.149334	+	0.988787i	$0.452287\pi$
$702$	0	0
$703$	−10.9712	−0.413787
$704$	0	0
$705$	0	0
$706$	0	0
$707$	58.9362	2.21652
$708$	0	0
$709$	24.1817	0.908163	0.454082	−	0.890960i	$-0.349967\pi$
$709$	24.1817	0.908163	0.454082	+	0.890960i	$0.349967\pi$
$710$	0	0
$711$	−6.95019	−0.260652
$712$	0	0
$713$	33.6883	1.26164
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.50220	−0.130792
$718$	0	0
$719$	43.1372	1.60875	0.804374	−	0.594123i	$-0.202501\pi$
$719$	43.1372	1.60875	0.804374	+	0.594123i	$0.202501\pi$
$720$	0	0
$721$	−36.1175	−1.34509
$722$	0	0
$723$	−6.13628	−0.228211
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−40.6603	−1.50801	−0.754004	−	0.656870i	$-0.771880\pi$
$727$	−40.6603	−1.50801	−0.754004	+	0.656870i	$0.771880\pi$
$728$	0	0
$729$	−5.54568	−0.205396
$730$	0	0
$731$	8.87280	0.328172
$732$	0	0
$733$	24.1828	0.893211	0.446606	−	0.894731i	$-0.352633\pi$
$733$	24.1828	0.893211	0.446606	+	0.894731i	$0.352633\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.9030	0.512125
$738$	0	0
$739$	30.4007	1.11831	0.559153	−	0.829064i	$-0.311126\pi$
$739$	30.4007	1.11831	0.559153	+	0.829064i	$0.311126\pi$
$740$	0	0
$741$	10.2833	0.377766
$742$	0	0
$743$	−24.6273	−0.903488	−0.451744	−	0.892148i	$-0.649198\pi$
$743$	−24.6273	−0.903488	−0.451744	+	0.892148i	$0.649198\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	38.2053	1.39786
$748$	0	0
$749$	−69.5310	−2.54061
$750$	0	0
$751$	−4.60339	−0.167980	−0.0839901	−	0.996467i	$-0.526766\pi$
$751$	−4.60339	−0.167980	−0.0839901	+	0.996467i	$0.526766\pi$
$752$	0	0
$753$	−16.0317	−0.584226
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−36.2417	−1.31723	−0.658613	−	0.752482i	$-0.728856\pi$
$757$	−36.2417	−1.31723	−0.658613	+	0.752482i	$0.728856\pi$
$758$	0	0
$759$	−4.65998	−0.169147
$760$	0	0
$761$	31.3704	1.13718	0.568588	−	0.822623i	$-0.307490\pi$
$761$	31.3704	1.13718	0.568588	+	0.822623i	$0.307490\pi$
$762$	0	0
$763$	3.28411	0.118893
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.32847	0.228508
$768$	0	0
$769$	−1.33146	−0.0480135	−0.0240068	−	0.999712i	$-0.507642\pi$
$769$	−1.33146	−0.0480135	−0.0240068	+	0.999712i	$0.507642\pi$
$770$	0	0
$771$	5.91589	0.213056
$772$	0	0
$773$	32.9499	1.18513	0.592563	−	0.805524i	$-0.298116\pi$
$773$	32.9499	1.18513	0.592563	+	0.805524i	$0.298116\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.11758	0.111843
$778$	0	0
$779$	14.5050	0.519696
$780$	0	0
$781$	32.7791	1.17293
$782$	0	0
$783$	3.73153	0.133354
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9.27094	−0.330473	−0.165237	−	0.986254i	$-0.552839\pi$
$787$	−9.27094	−0.330473	−0.165237	+	0.986254i	$0.552839\pi$
$788$	0	0
$789$	11.4774	0.408607
$790$	0	0
$791$	42.9528	1.52722
$792$	0	0
$793$	−13.3930	−0.475601
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.1758	1.21057	0.605285	−	0.796009i	$-0.293059\pi$
$797$	34.1758	1.21057	0.605285	+	0.796009i	$0.293059\pi$
$798$	0	0
$799$	1.81152	0.0640871
$800$	0	0
$801$	−32.1112	−1.13459
$802$	0	0
$803$	−11.4889	−0.405434
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.2108	0.429839
$808$	0	0
$809$	20.0891	0.706294	0.353147	−	0.935568i	$-0.385111\pi$
$809$	20.0891	0.706294	0.353147	+	0.935568i	$0.385111\pi$
$810$	0	0
$811$	−8.35300	−0.293314	−0.146657	−	0.989187i	$-0.546851\pi$
$811$	−8.35300	−0.293314	−0.146657	+	0.989187i	$0.546851\pi$
$812$	0	0
$813$	−8.87911	−0.311404
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.8171	0.448414
$818$	0	0
$819$	16.4530	0.574915
$820$	0	0
$821$	−5.83477	−0.203635	−0.101817	−	0.994803i	$-0.532466\pi$
$821$	−5.83477	−0.203635	−0.101817	+	0.994803i	$0.532466\pi$
$822$	0	0
$823$	−0.662009	−0.0230762	−0.0115381	−	0.999933i	$-0.503673\pi$
$823$	−0.662009	−0.0230762	−0.0115381	+	0.999933i	$0.503673\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.7516	1.48662	0.743310	−	0.668947i	$-0.233255\pi$
$827$	42.7516	1.48662	0.743310	+	0.668947i	$0.233255\pi$
$828$	0	0
$829$	24.2593	0.842562	0.421281	−	0.906930i	$-0.361581\pi$
$829$	24.2593	0.842562	0.421281	+	0.906930i	$0.361581\pi$
$830$	0	0
$831$	−10.6062	−0.367926
$832$	0	0
$833$	−24.3719	−0.844437
$834$	0	0
$835$	0	0
$836$	0	0
$837$	38.3724	1.32634
$838$	0	0
$839$	−16.5618	−0.571777	−0.285889	−	0.958263i	$-0.592289\pi$
$839$	−16.5618	−0.571777	−0.285889	+	0.958263i	$0.592289\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−0.750000	−0.0258314
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−22.0451	−0.757477
$848$	0	0
$849$	−11.9369	−0.409672
$850$	0	0
$851$	−4.49621	−0.154128
$852$	0	0
$853$	−2.34257	−0.0802081	−0.0401041	−	0.999196i	$-0.512769\pi$
$853$	−2.34257	−0.0802081	−0.0401041	+	0.999196i	$0.512769\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−55.3020	−1.88908	−0.944540	−	0.328397i	$-0.893492\pi$
$857$	−55.3020	−1.88908	−0.944540	+	0.328397i	$0.893492\pi$
$858$	0	0
$859$	44.9751	1.53453	0.767266	−	0.641329i	$-0.221617\pi$
$859$	44.9751	1.53453	0.767266	+	0.641329i	$0.221617\pi$
$860$	0	0
$861$	−4.12175	−0.140469
$862$	0	0
$863$	−48.3195	−1.64481	−0.822407	−	0.568900i	$-0.807369\pi$
$863$	−48.3195	−1.64481	−0.822407	+	0.568900i	$0.807369\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.16356	0.311211
$868$	0	0
$869$	5.76933	0.195711
$870$	0	0
$871$	−12.5733	−0.426030
$872$	0	0
$873$	−2.17036	−0.0734556
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−56.1140	−1.89483	−0.947417	−	0.320001i	$-0.896317\pi$
$877$	−56.1140	−1.89483	−0.947417	+	0.320001i	$0.896317\pi$
$878$	0	0
$879$	0.970867	0.0327465
$880$	0	0
$881$	58.1133	1.95789	0.978944	−	0.204129i	$-0.0654361\pi$
$881$	58.1133	1.95789	0.978944	+	0.204129i	$0.0654361\pi$
$882$	0	0
$883$	21.9911	0.740061	0.370030	−	0.929020i	$-0.379347\pi$
$883$	21.9911	0.740061	0.370030	+	0.929020i	$0.379347\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.1035	0.439973	0.219986	−	0.975503i	$-0.429399\pi$
$887$	13.1035	0.439973	0.219986	+	0.975503i	$0.429399\pi$
$888$	0	0
$889$	51.3428	1.72198
$890$	0	0
$891$	10.8541	0.363626
$892$	0	0
$893$	2.61682	0.0875685
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.21429	0.140711
$898$	0	0
$899$	10.2833	0.342967
$900$	0	0
$901$	23.1783	0.772180
$902$	0	0
$903$	−3.64211	−0.121202
$904$	0	0
$905$	0	0
$906$	0	0
$907$	51.7491	1.71830	0.859150	−	0.511724i	$-0.170993\pi$
$907$	51.7491	1.71830	0.859150	+	0.511724i	$0.170993\pi$
$908$	0	0
$909$	−44.4603	−1.47466
$910$	0	0
$911$	−28.9974	−0.960726	−0.480363	−	0.877070i	$-0.659495\pi$
$911$	−28.9974	−0.960726	−0.480363	+	0.877070i	$0.659495\pi$
$912$	0	0
$913$	−31.7141	−1.04958
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20.3889	0.673299
$918$	0	0
$919$	25.0943	0.827784	0.413892	−	0.910326i	$-0.364169\pi$
$919$	25.0943	0.827784	0.413892	+	0.910326i	$0.364169\pi$
$920$	0	0
$921$	−3.59458	−0.118445
$922$	0	0
$923$	−29.6440	−0.975743
$924$	0	0
$925$	0	0
$926$	0	0
$927$	27.2464	0.894888
$928$	0	0
$929$	50.3675	1.65250	0.826251	−	0.563301i	$-0.190469\pi$
$929$	50.3675	1.65250	0.826251	+	0.563301i	$0.190469\pi$
$930$	0	0
$931$	−35.2062	−1.15384
$932$	0	0
$933$	16.5897	0.543124
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−39.2746	−1.28304	−0.641522	−	0.767104i	$-0.721697\pi$
$937$	−39.2746	−1.28304	−0.641522	+	0.767104i	$0.721697\pi$
$938$	0	0
$939$	−22.8555	−0.745862
$940$	0	0
$941$	4.60332	0.150064	0.0750320	−	0.997181i	$-0.476094\pi$
$941$	4.60332	0.150064	0.0750320	+	0.997181i	$0.476094\pi$
$942$	0	0
$943$	5.94443	0.193577
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.40704	0.0457226	0.0228613	−	0.999739i	$-0.492722\pi$
$947$	1.40704	0.0457226	0.0228613	+	0.999739i	$0.492722\pi$
$948$	0	0
$949$	10.3901	0.337275
$950$	0	0
$951$	−6.78536	−0.220030
$952$	0	0
$953$	45.4121	1.47104	0.735521	−	0.677502i	$-0.236938\pi$
$953$	45.4121	1.47104	0.735521	+	0.677502i	$0.236938\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.42245	−0.0459813
$958$	0	0
$959$	−52.3102	−1.68918
$960$	0	0
$961$	74.7460	2.41116
$962$	0	0
$963$	52.4528	1.69027
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.6658	0.407305	0.203652	−	0.979043i	$-0.434719\pi$
$967$	12.6658	0.407305	0.203652	+	0.979043i	$0.434719\pi$
$968$	0	0
$969$	29.7554	0.955881
$970$	0	0
$971$	−24.5447	−0.787676	−0.393838	−	0.919180i	$-0.628853\pi$
$971$	−24.5447	−0.787676	−0.393838	+	0.919180i	$0.628853\pi$
$972$	0	0
$973$	7.07951	0.226959
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.3008	−0.809444	−0.404722	−	0.914440i	$-0.632632\pi$
$977$	−25.3008	−0.809444	−0.404722	+	0.914440i	$0.632632\pi$
$978$	0	0
$979$	26.6554	0.851912
$980$	0	0
$981$	−2.47747	−0.0790994
$982$	0	0
$983$	25.1822	0.803188	0.401594	−	0.915818i	$-0.368456\pi$
$983$	25.1822	0.803188	0.401594	+	0.915818i	$0.368456\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.743596	−0.0236689
$988$	0	0
$989$	5.25269	0.167026
$990$	0	0
$991$	13.0936	0.415933	0.207967	−	0.978136i	$-0.433315\pi$
$991$	13.0936	0.415933	0.207967	+	0.978136i	$0.433315\pi$
$992$	0	0
$993$	−11.0240	−0.349837
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−31.0809	−0.984341	−0.492171	−	0.870499i	$-0.663796\pi$
$997$	−31.0809	−0.984341	−0.492171	+	0.870499i	$0.663796\pi$
$998$	0	0
$999$	−5.12138	−0.162033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.a.i.1.4	✓	5
5.2	odd	4		2900.2.c.i.349.4		10
5.3	odd	4		2900.2.c.i.349.7		10
5.4	even	2		2900.2.a.k.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2900.2.a.i.1.4	✓	5	1.1	even	1	trivial
2900.2.a.k.1.2	yes	5	5.4	even	2
2900.2.c.i.349.4		10	5.2	odd	4
2900.2.c.i.349.7		10	5.3	odd	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:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.a (trivial)

\(f(q)\)	\(=\)	\(q+0.672645 q^{3} +3.37701 q^{7} -2.54755 q^{9} +2.11471 q^{11} -1.91246 q^{13} -5.53382 q^{17} -7.99382 q^{19} +2.27153 q^{21} -3.27602 q^{23} -3.73153 q^{27} -1.00000 q^{29} -10.2833 q^{31} +1.42245 q^{33} +1.37246 q^{37} -1.28640 q^{39} -1.81453 q^{41} -1.60338 q^{43} -0.327355 q^{47} +4.40418 q^{49} -3.72230 q^{51} -4.18848 q^{53} -5.37701 q^{57} -3.30908 q^{59} +7.00306 q^{61} -8.60309 q^{63} +6.57443 q^{67} -2.20360 q^{69} +15.5005 q^{71} -5.43283 q^{73} +7.14140 q^{77} +2.72819 q^{79} +5.13265 q^{81} -14.9969 q^{83} -0.672645 q^{87} +12.6048 q^{89} -6.45838 q^{91} -6.91700 q^{93} +0.851941 q^{97} -5.38734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{7} + 3 q^{9} + 4 q^{11} - 12 q^{13} - 5 q^{17} - 4 q^{19} + 3 q^{21} - 9 q^{23} + 3 q^{27} - 5 q^{29} + 3 q^{31} - 21 q^{33} - 4 q^{37} - 18 q^{39} + 5 q^{41} - 4 q^{43} - 5 q^{47} - 3 q^{49}+ \cdots - 36 q^{99}+O(q^{100}) \) 5 * q - 4 * q^7 + 3 * q^9 + 4 * q^11 - 12 * q^13 - 5 * q^17 - 4 * q^19 + 3 * q^21 - 9 * q^23 + 3 * q^27 - 5 * q^29 + 3 * q^31 - 21 * q^33 - 4 * q^37 - 18 * q^39 + 5 * q^41 - 4 * q^43 - 5 * q^47 - 3 * q^49 + 9 * q^51 - 16 * q^53 - 6 * q^57 - 23 * q^59 + 5 * q^61 - 21 * q^63 - 5 * q^67 - 30 * q^69 + 23 * q^71 - 18 * q^73 - 8 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 6 * q^89 + 27 * q^91 - 27 * q^93 - 21 * q^97 - 36 * q^99

Introduction

L-functions

Modular forms

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