LMFDB - Embedded newform 9200.2.a.cv.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.521397.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 3x^{2} + 18x + 12 $ x^5 - 9x^3 - 3x^2 + 18*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4600)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.61696$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61696 q^{3} +3.83744 q^{7} +3.84849 q^{9} +O(q^{10})$	$q+2.61696 q^{3} +3.83744 q^{7} +3.84849 q^{9} +0.508005 q^{11} -1.01106 q^{13} -1.44705 q^{17} +0.508005 q^{19} +10.0424 q^{21} -1.00000 q^{23} +2.22047 q^{27} +7.51040 q^{29} +0.439038 q^{31} +1.32943 q^{33} +7.02642 q^{37} -2.64590 q^{39} +5.47041 q^{41} +6.72592 q^{43} -2.64098 q^{47} +7.72592 q^{49} -3.78688 q^{51} -4.77648 q^{53} +1.32943 q^{57} -3.85345 q^{59} -9.05844 q^{61} +14.7683 q^{63} +3.45696 q^{67} -2.61696 q^{69} +2.73649 q^{71} +9.21300 q^{73} +1.94944 q^{77} -10.5504 q^{79} -5.73458 q^{81} -1.40211 q^{83} +19.6544 q^{87} +6.77086 q^{89} -3.87986 q^{91} +1.14895 q^{93} +0.313420 q^{97} +1.95506 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 5 * q + 4 * q^7 + 3 * q^9	$ 5 q + 4 q^{7} + 3 q^{9} - 4 q^{13} - 6 q^{17} - 5 q^{23} + 9 q^{27} + 12 q^{29} + 18 q^{31} - 6 q^{33} - 10 q^{37} - 9 q^{39} - 6 q^{41} + 10 q^{43} + 22 q^{47} + 15 q^{49} + 6 q^{51} - 10 q^{53} - 6 q^{57} + q^{59} + 10 q^{61} + 8 q^{67} - 8 q^{71} - 6 q^{73} - 27 q^{81} - 2 q^{83} + 39 q^{87} + 14 q^{89} + 46 q^{91} + 3 q^{93} - 6 q^{97} + 6 q^{99}+O(q^{100}) $ 5 * q + 4 * q^7 + 3 * q^9 - 4 * q^13 - 6 * q^17 - 5 * q^23 + 9 * q^27 + 12 * q^29 + 18 * q^31 - 6 * q^33 - 10 * q^37 - 9 * q^39 - 6 * q^41 + 10 * q^43 + 22 * q^47 + 15 * q^49 + 6 * q^51 - 10 * q^53 - 6 * q^57 + q^59 + 10 * q^61 + 8 * q^67 - 8 * q^71 - 6 * q^73 - 27 * q^81 - 2 * q^83 + 39 * q^87 + 14 * q^89 + 46 * q^91 + 3 * q^93 - 6 * 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
$3$	2.61696	1.51090	0.755452	−	0.655204i	$-0.227417\pi$
$3$	2.61696	1.51090	0.755452	+	0.655204i	$0.227417\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.83744	1.45041	0.725207	−	0.688531i	$-0.241744\pi$
$7$	3.83744	1.45041	0.725207	+	0.688531i	$0.241744\pi$
$8$	0	0
$9$	3.84849	1.28283
$10$	0	0
$11$	0.508005	0.153169	0.0765847	−	0.997063i	$-0.475598\pi$
$11$	0.508005	0.153169	0.0765847	+	0.997063i	$0.475598\pi$
$12$	0	0
$13$	−1.01106	−0.280417	−0.140208	−	0.990122i	$-0.544777\pi$
$13$	−1.01106	−0.280417	−0.140208	+	0.990122i	$0.544777\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.44705	−0.350961	−0.175481	−	0.984483i	$-0.556148\pi$
$17$	−1.44705	−0.350961	−0.175481	+	0.984483i	$0.556148\pi$
$18$	0	0
$19$	0.508005	0.116544	0.0582722	−	0.998301i	$-0.481441\pi$
$19$	0.508005	0.116544	0.0582722	+	0.998301i	$0.481441\pi$
$20$	0	0
$21$	10.0424	2.19144
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.22047	0.427330
$28$	0	0
$29$	7.51040	1.39465	0.697323	−	0.716757i	$-0.254374\pi$
$29$	7.51040	1.39465	0.697323	+	0.716757i	$0.254374\pi$
$30$	0	0
$31$	0.439038	0.0788536	0.0394268	−	0.999222i	$-0.487447\pi$
$31$	0.439038	0.0788536	0.0394268	+	0.999222i	$0.487447\pi$
$32$	0	0
$33$	1.32943	0.231424
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.02642	1.15514	0.577568	−	0.816343i	$-0.304002\pi$
$37$	7.02642	1.15514	0.577568	+	0.816343i	$0.304002\pi$
$38$	0	0
$39$	−2.64590	−0.423683
$40$	0	0
$41$	5.47041	0.854335	0.427167	−	0.904173i	$-0.359512\pi$
$41$	5.47041	0.854335	0.427167	+	0.904173i	$0.359512\pi$
$42$	0	0
$43$	6.72592	1.02569	0.512847	−	0.858480i	$-0.328591\pi$
$43$	6.72592	1.02569	0.512847	+	0.858480i	$0.328591\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.64098	−0.385227	−0.192614	−	0.981275i	$-0.561696\pi$
$47$	−2.64098	−0.385227	−0.192614	+	0.981275i	$0.561696\pi$
$48$	0	0
$49$	7.72592	1.10370
$50$	0	0
$51$	−3.78688	−0.530269
$52$	0	0
$53$	−4.77648	−0.656100	−0.328050	−	0.944660i	$-0.606391\pi$
$53$	−4.77648	−0.656100	−0.328050	+	0.944660i	$0.606391\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.32943	0.176087
$58$	0	0
$59$	−3.85345	−0.501676	−0.250838	−	0.968029i	$-0.580706\pi$
$59$	−3.85345	−0.501676	−0.250838	+	0.968029i	$0.580706\pi$
$60$	0	0
$61$	−9.05844	−1.15981	−0.579907	−	0.814683i	$-0.696911\pi$
$61$	−9.05844	−1.15981	−0.579907	+	0.814683i	$0.696911\pi$
$62$	0	0
$63$	14.7683	1.86064
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.45696	0.422335	0.211167	−	0.977450i	$-0.432273\pi$
$67$	3.45696	0.422335	0.211167	+	0.977450i	$0.432273\pi$
$68$	0	0
$69$	−2.61696	−0.315045
$70$	0	0
$71$	2.73649	0.324762	0.162381	−	0.986728i	$-0.448083\pi$
$71$	2.73649	0.324762	0.162381	+	0.986728i	$0.448083\pi$
$72$	0	0
$73$	9.21300	1.07830	0.539150	−	0.842210i	$-0.318746\pi$
$73$	9.21300	1.07830	0.539150	+	0.842210i	$0.318746\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.94944	0.222159
$78$	0	0
$79$	−10.5504	−1.18702	−0.593508	−	0.804828i	$-0.702258\pi$
$79$	−10.5504	−1.18702	−0.593508	+	0.804828i	$0.702258\pi$
$80$	0	0
$81$	−5.73458	−0.637176
$82$	0	0
$83$	−1.40211	−0.153901	−0.0769505	−	0.997035i	$-0.524518\pi$
$83$	−1.40211	−0.153901	−0.0769505	+	0.997035i	$0.524518\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19.6544	2.10718
$88$	0	0
$89$	6.77086	0.717710	0.358855	−	0.933393i	$-0.383167\pi$
$89$	6.77086	0.717710	0.358855	+	0.933393i	$0.383167\pi$
$90$	0	0
$91$	−3.87986	−0.406720
$92$	0	0
$93$	1.14895	0.119140
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.313420	0.0318230	0.0159115	−	0.999873i	$-0.494935\pi$
$97$	0.313420	0.0318230	0.0159115	+	0.999873i	$0.494935\pi$
$98$	0	0
$99$	1.95506	0.196490
$100$	0	0
$101$	−4.83182	−0.480784	−0.240392	−	0.970676i	$-0.577276\pi$
$101$	−4.83182	−0.480784	−0.240392	+	0.970676i	$0.577276\pi$
$102$	0	0
$103$	8.07746	0.795896	0.397948	−	0.917408i	$-0.369722\pi$
$103$	8.07746	0.795896	0.397948	+	0.917408i	$0.369722\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.1587	−1.07876	−0.539378	−	0.842064i	$-0.681341\pi$
$107$	−11.1587	−1.07876	−0.539378	+	0.842064i	$0.681341\pi$
$108$	0	0
$109$	17.6015	1.68592	0.842958	−	0.537979i	$-0.180812\pi$
$109$	17.6015	1.68592	0.842958	+	0.537979i	$0.180812\pi$
$110$	0	0
$111$	18.3879	1.74530
$112$	0	0
$113$	4.06454	0.382360	0.191180	−	0.981555i	$-0.438769\pi$
$113$	4.06454	0.382360	0.191180	+	0.981555i	$0.438769\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.89104	−0.359727
$118$	0	0
$119$	−5.55296	−0.509039
$120$	0	0
$121$	−10.7419	−0.976539
$122$	0	0
$123$	14.3159	1.29082
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.75246	0.421713	0.210856	−	0.977517i	$-0.432375\pi$
$127$	4.75246	0.421713	0.210856	+	0.977517i	$0.432375\pi$
$128$	0	0
$129$	17.6015	1.54972
$130$	0	0
$131$	−1.96851	−0.171989	−0.0859946	−	0.996296i	$-0.527407\pi$
$131$	−1.96851	−0.171989	−0.0859946	+	0.996296i	$0.527407\pi$
$132$	0	0
$133$	1.94944	0.169038
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.64285	0.482101	0.241051	−	0.970513i	$-0.422508\pi$
$137$	5.64285	0.482101	0.241051	+	0.970513i	$0.422508\pi$
$138$	0	0
$139$	7.59724	0.644390	0.322195	−	0.946673i	$-0.395579\pi$
$139$	7.59724	0.644390	0.322195	+	0.946673i	$0.395579\pi$
$140$	0	0
$141$	−6.91136	−0.582041
$142$	0	0
$143$	−0.513622	−0.0429512
$144$	0	0
$145$	0	0
$146$	0	0
$147$	20.2184	1.66759
$148$	0	0
$149$	10.7709	0.882384	0.441192	−	0.897413i	$-0.354556\pi$
$149$	10.7709	0.882384	0.441192	+	0.897413i	$0.354556\pi$
$150$	0	0
$151$	16.5333	1.34546	0.672729	−	0.739889i	$-0.265122\pi$
$151$	16.5333	1.34546	0.672729	+	0.739889i	$0.265122\pi$
$152$	0	0
$153$	−5.56896	−0.450224
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.5436	0.921281	0.460640	−	0.887587i	$-0.347620\pi$
$157$	11.5436	0.921281	0.460640	+	0.887587i	$0.347620\pi$
$158$	0	0
$159$	−12.4999	−0.991304
$160$	0	0
$161$	−3.83744	−0.302432
$162$	0	0
$163$	3.32143	0.260155	0.130077	−	0.991504i	$-0.458477\pi$
$163$	3.32143	0.260155	0.130077	+	0.991504i	$0.458477\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.6742	0.980755	0.490378	−	0.871510i	$-0.336859\pi$
$167$	12.6742	0.980755	0.490378	+	0.871510i	$0.336859\pi$
$168$	0	0
$169$	−11.9778	−0.921367
$170$	0	0
$171$	1.95506	0.149507
$172$	0	0
$173$	−17.5733	−1.33607	−0.668035	−	0.744130i	$-0.732864\pi$
$173$	−17.5733	−1.33607	−0.668035	+	0.744130i	$0.732864\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.0843	−0.757984
$178$	0	0
$179$	14.8952	1.11332	0.556659	−	0.830741i	$-0.312083\pi$
$179$	14.8952	1.11332	0.556659	+	0.830741i	$0.312083\pi$
$180$	0	0
$181$	−23.2868	−1.73089	−0.865446	−	0.501003i	$-0.832965\pi$
$181$	−23.2868	−1.73089	−0.865446	+	0.501003i	$0.832965\pi$
$182$	0	0
$183$	−23.7056	−1.75237
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.735109	−0.0537565
$188$	0	0
$189$	8.52093	0.619806
$190$	0	0
$191$	−6.92360	−0.500974	−0.250487	−	0.968120i	$-0.580591\pi$
$191$	−6.92360	−0.500974	−0.250487	+	0.968120i	$0.580591\pi$
$192$	0	0
$193$	7.74318	0.557366	0.278683	−	0.960383i	$-0.410102\pi$
$193$	7.74318	0.557366	0.278683	+	0.960383i	$0.410102\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.8908	1.70215	0.851074	−	0.525045i	$-0.175952\pi$
$197$	23.8908	1.70215	0.851074	+	0.525045i	$0.175952\pi$
$198$	0	0
$199$	23.9444	1.69737	0.848687	−	0.528896i	$-0.177394\pi$
$199$	23.9444	1.69737	0.848687	+	0.528896i	$0.177394\pi$
$200$	0	0
$201$	9.04673	0.638107
$202$	0	0
$203$	28.8207	2.02282
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.84849	−0.267489
$208$	0	0
$209$	0.258070	0.0178510
$210$	0	0
$211$	−14.4709	−0.996220	−0.498110	−	0.867114i	$-0.665973\pi$
$211$	−14.4709	−0.996220	−0.498110	+	0.867114i	$0.665973\pi$
$212$	0	0
$213$	7.16129	0.490684
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.68478	0.114370
$218$	0	0
$219$	24.1101	1.62921
$220$	0	0
$221$	1.46305	0.0984153
$222$	0	0
$223$	−4.74755	−0.317919	−0.158960	−	0.987285i	$-0.550814\pi$
$223$	−4.74755	−0.317919	−0.158960	+	0.987285i	$0.550814\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.98399	0.596288	0.298144	−	0.954521i	$-0.403632\pi$
$227$	8.98399	0.596288	0.298144	+	0.954521i	$0.403632\pi$
$228$	0	0
$229$	20.7290	1.36981	0.684906	−	0.728631i	$-0.259843\pi$
$229$	20.7290	1.36981	0.684906	+	0.728631i	$0.259843\pi$
$230$	0	0
$231$	5.10161	0.335661
$232$	0	0
$233$	−26.0634	−1.70747	−0.853735	−	0.520707i	$-0.825668\pi$
$233$	−26.0634	−1.70747	−0.853735	+	0.520707i	$0.825668\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−27.6101	−1.79347
$238$	0	0
$239$	2.94209	0.190308	0.0951540	−	0.995463i	$-0.469666\pi$
$239$	2.94209	0.190308	0.0951540	+	0.995463i	$0.469666\pi$
$240$	0	0
$241$	−24.9413	−1.60661	−0.803306	−	0.595567i	$-0.796927\pi$
$241$	−24.9413	−1.60661	−0.803306	+	0.595567i	$0.796927\pi$
$242$	0	0
$243$	−21.6686	−1.39004
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.513622	−0.0326810
$248$	0	0
$249$	−3.66926	−0.232530
$250$	0	0
$251$	−21.1908	−1.33755	−0.668774	−	0.743465i	$-0.733181\pi$
$251$	−21.1908	−1.33755	−0.668774	+	0.743465i	$0.733181\pi$
$252$	0	0
$253$	−0.508005	−0.0319380
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.2110	−1.63500	−0.817500	−	0.575929i	$-0.804640\pi$
$257$	−26.2110	−1.63500	−0.817500	+	0.575929i	$0.804640\pi$
$258$	0	0
$259$	26.9634	1.67543
$260$	0	0
$261$	28.9037	1.78910
$262$	0	0
$263$	−5.65276	−0.348564	−0.174282	−	0.984696i	$-0.555760\pi$
$263$	−5.65276	−0.348564	−0.174282	+	0.984696i	$0.555760\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	17.7191	1.08439
$268$	0	0
$269$	9.65772	0.588841	0.294421	−	0.955676i	$-0.404873\pi$
$269$	9.65772	0.588841	0.294421	+	0.955676i	$0.404873\pi$
$270$	0	0
$271$	−8.39040	−0.509680	−0.254840	−	0.966983i	$-0.582023\pi$
$271$	−8.39040	−0.509680	−0.254840	+	0.966983i	$0.582023\pi$
$272$	0	0
$273$	−10.1535	−0.614515
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−30.3317	−1.82246	−0.911229	−	0.411900i	$-0.864865\pi$
$277$	−30.3317	−1.82246	−0.911229	+	0.411900i	$0.864865\pi$
$278$	0	0
$279$	1.68964	0.101156
$280$	0	0
$281$	−25.3132	−1.51006	−0.755028	−	0.655692i	$-0.772377\pi$
$281$	−25.3132	−1.51006	−0.755028	+	0.655692i	$0.772377\pi$
$282$	0	0
$283$	−12.9629	−0.770566	−0.385283	−	0.922798i	$-0.625896\pi$
$283$	−12.9629	−0.770566	−0.385283	+	0.922798i	$0.625896\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.9924	1.23914
$288$	0	0
$289$	−14.9060	−0.876826
$290$	0	0
$291$	0.820209	0.0480815
$292$	0	0
$293$	−25.9469	−1.51584	−0.757918	−	0.652350i	$-0.773783\pi$
$293$	−25.9469	−1.51584	−0.757918	+	0.652350i	$0.773783\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.12801	0.0654539
$298$	0	0
$299$	1.01106	0.0584709
$300$	0	0
$301$	25.8103	1.48768
$302$	0	0
$303$	−12.6447	−0.726419
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0640	0.688531	0.344266	−	0.938872i	$-0.388128\pi$
$307$	12.0640	0.688531	0.344266	+	0.938872i	$0.388128\pi$
$308$	0	0
$309$	21.1384	1.20252
$310$	0	0
$311$	31.8125	1.80392	0.901959	−	0.431821i	$-0.142129\pi$
$311$	31.8125	1.80392	0.901959	+	0.431821i	$0.142129\pi$
$312$	0	0
$313$	4.30122	0.243119	0.121560	−	0.992584i	$-0.461210\pi$
$313$	4.30122	0.243119	0.121560	+	0.992584i	$0.461210\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.28317	0.352898	0.176449	−	0.984310i	$-0.443539\pi$
$317$	6.28317	0.352898	0.176449	+	0.984310i	$0.443539\pi$
$318$	0	0
$319$	3.81532	0.213617
$320$	0	0
$321$	−29.2020	−1.62990
$322$	0	0
$323$	−0.735109	−0.0409026
$324$	0	0
$325$	0	0
$326$	0	0
$327$	46.0624	2.54726
$328$	0	0
$329$	−10.1346	−0.558739
$330$	0	0
$331$	8.00053	0.439749	0.219874	−	0.975528i	$-0.429435\pi$
$331$	8.00053	0.439749	0.219874	+	0.975528i	$0.429435\pi$
$332$	0	0
$333$	27.0411	1.48184
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10.1436	−0.552554	−0.276277	−	0.961078i	$-0.589101\pi$
$337$	−10.1436	−0.552554	−0.276277	+	0.961078i	$0.589101\pi$
$338$	0	0
$339$	10.6367	0.577709
$340$	0	0
$341$	0.223034	0.0120780
$342$	0	0
$343$	2.78567	0.150412
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.8747	−1.01325	−0.506625	−	0.862167i	$-0.669107\pi$
$347$	−18.8747	−1.01325	−0.506625	+	0.862167i	$0.669107\pi$
$348$	0	0
$349$	9.26611	0.496004	0.248002	−	0.968760i	$-0.420226\pi$
$349$	9.26611	0.496004	0.248002	+	0.968760i	$0.420226\pi$
$350$	0	0
$351$	−2.24502	−0.119831
$352$	0	0
$353$	−8.04442	−0.428161	−0.214081	−	0.976816i	$-0.568676\pi$
$353$	−8.04442	−0.428161	−0.214081	+	0.976816i	$0.568676\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−14.5319	−0.769109
$358$	0	0
$359$	−20.5443	−1.08429	−0.542144	−	0.840285i	$-0.682387\pi$
$359$	−20.5443	−1.08429	−0.542144	+	0.840285i	$0.682387\pi$
$360$	0	0
$361$	−18.7419	−0.986417
$362$	0	0
$363$	−28.1112	−1.47546
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.64032	−0.451021	−0.225511	−	0.974241i	$-0.572405\pi$
$367$	−8.64032	−0.451021	−0.225511	+	0.974241i	$0.572405\pi$
$368$	0	0
$369$	21.0528	1.09597
$370$	0	0
$371$	−18.3294	−0.951617
$372$	0	0
$373$	19.0442	0.986073	0.493037	−	0.870009i	$-0.335887\pi$
$373$	19.0442	0.986073	0.493037	+	0.870009i	$0.335887\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.59344	−0.391082
$378$	0	0
$379$	−0.926121	−0.0475717	−0.0237858	−	0.999717i	$-0.507572\pi$
$379$	−0.926121	−0.0475717	−0.0237858	+	0.999717i	$0.507572\pi$
$380$	0	0
$381$	12.4370	0.637167
$382$	0	0
$383$	−9.38526	−0.479564	−0.239782	−	0.970827i	$-0.577076\pi$
$383$	−9.38526	−0.479564	−0.239782	+	0.970827i	$0.577076\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	25.8847	1.31579
$388$	0	0
$389$	2.22865	0.112997	0.0564985	−	0.998403i	$-0.482006\pi$
$389$	2.22865	0.112997	0.0564985	+	0.998403i	$0.482006\pi$
$390$	0	0
$391$	1.44705	0.0731805
$392$	0	0
$393$	−5.15151	−0.259859
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.0448	−0.654701	−0.327350	−	0.944903i	$-0.606156\pi$
$397$	−13.0448	−0.654701	−0.327350	+	0.944903i	$0.606156\pi$
$398$	0	0
$399$	5.10161	0.255400
$400$	0	0
$401$	−2.22221	−0.110972	−0.0554859	−	0.998459i	$-0.517671\pi$
$401$	−2.22221	−0.110972	−0.0554859	+	0.998459i	$0.517671\pi$
$402$	0	0
$403$	−0.443893	−0.0221119
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.56946	0.176931
$408$	0	0
$409$	−19.5232	−0.965362	−0.482681	−	0.875796i	$-0.660337\pi$
$409$	−19.5232	−0.965362	−0.482681	+	0.875796i	$0.660337\pi$
$410$	0	0
$411$	14.7671	0.728409
$412$	0	0
$413$	−14.7874	−0.727638
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.8817	0.973611
$418$	0	0
$419$	24.9180	1.21732	0.608662	−	0.793430i	$-0.291707\pi$
$419$	24.9180	1.21732	0.608662	+	0.793430i	$0.291707\pi$
$420$	0	0
$421$	6.10721	0.297647	0.148824	−	0.988864i	$-0.452451\pi$
$421$	6.10721	0.297647	0.148824	+	0.988864i	$0.452451\pi$
$422$	0	0
$423$	−10.1638	−0.494181
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−34.7612	−1.68221
$428$	0	0
$429$	−1.34413	−0.0648952
$430$	0	0
$431$	−4.44095	−0.213913	−0.106956	−	0.994264i	$-0.534111\pi$
$431$	−4.44095	−0.213913	−0.106956	+	0.994264i	$0.534111\pi$
$432$	0	0
$433$	−3.25554	−0.156451	−0.0782257	−	0.996936i	$-0.524925\pi$
$433$	−3.25554	−0.156451	−0.0782257	+	0.996936i	$0.524925\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.508005	−0.0243012
$438$	0	0
$439$	26.9611	1.28678	0.643392	−	0.765537i	$-0.277527\pi$
$439$	26.9611	1.28678	0.643392	+	0.765537i	$0.277527\pi$
$440$	0	0
$441$	29.7331	1.41586
$442$	0	0
$443$	−16.0072	−0.760526	−0.380263	−	0.924878i	$-0.624167\pi$
$443$	−16.0072	−0.760526	−0.380263	+	0.924878i	$0.624167\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	28.1869	1.33320
$448$	0	0
$449$	−37.4278	−1.76633	−0.883163	−	0.469066i	$-0.844591\pi$
$449$	−37.4278	−1.76633	−0.883163	+	0.469066i	$0.844591\pi$
$450$	0	0
$451$	2.77900	0.130858
$452$	0	0
$453$	43.2670	2.03286
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.75949	0.409752	0.204876	−	0.978788i	$-0.434321\pi$
$457$	8.75949	0.409752	0.204876	+	0.978788i	$0.434321\pi$
$458$	0	0
$459$	−3.21314	−0.149976
$460$	0	0
$461$	11.4459	0.533089	0.266545	−	0.963823i	$-0.414118\pi$
$461$	11.4459	0.533089	0.266545	+	0.963823i	$0.414118\pi$
$462$	0	0
$463$	8.23570	0.382745	0.191373	−	0.981517i	$-0.438706\pi$
$463$	8.23570	0.382745	0.191373	+	0.981517i	$0.438706\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.9436	−1.33935	−0.669675	−	0.742654i	$-0.733567\pi$
$467$	−28.9436	−1.33935	−0.669675	+	0.742654i	$0.733567\pi$
$468$	0	0
$469$	13.2659	0.612561
$470$	0	0
$471$	30.2092	1.39197
$472$	0	0
$473$	3.41680	0.157105
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−18.3823	−0.841666
$478$	0	0
$479$	−31.1031	−1.42114	−0.710569	−	0.703627i	$-0.751563\pi$
$479$	−31.1031	−1.42114	−0.710569	+	0.703627i	$0.751563\pi$
$480$	0	0
$481$	−7.10410	−0.323919
$482$	0	0
$483$	−10.0424	−0.456946
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.90588	0.0863635	0.0431817	−	0.999067i	$-0.486251\pi$
$487$	1.90588	0.0863635	0.0431817	+	0.999067i	$0.486251\pi$
$488$	0	0
$489$	8.69206	0.393069
$490$	0	0
$491$	27.2222	1.22852	0.614259	−	0.789104i	$-0.289455\pi$
$491$	27.2222	1.22852	0.614259	+	0.789104i	$0.289455\pi$
$492$	0	0
$493$	−10.8679	−0.489467
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.5011	0.471039
$498$	0	0
$499$	9.38913	0.420315	0.210158	−	0.977668i	$-0.432602\pi$
$499$	9.38913	0.420315	0.210158	+	0.977668i	$0.432602\pi$
$500$	0	0
$501$	33.1678	1.48183
$502$	0	0
$503$	29.5613	1.31807	0.659037	−	0.752110i	$-0.270964\pi$
$503$	29.5613	1.31807	0.659037	+	0.752110i	$0.270964\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−31.3454	−1.39210
$508$	0	0
$509$	0.448890	0.0198967	0.00994835	−	0.999951i	$-0.496833\pi$
$509$	0.448890	0.0198967	0.00994835	+	0.999951i	$0.496833\pi$
$510$	0	0
$511$	35.3543	1.56398
$512$	0	0
$513$	1.12801	0.0498030
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.34163	−0.0590050
$518$	0	0
$519$	−45.9886	−2.01867
$520$	0	0
$521$	26.8712	1.17725	0.588623	−	0.808407i	$-0.299670\pi$
$521$	26.8712	1.17725	0.588623	+	0.808407i	$0.299670\pi$
$522$	0	0
$523$	2.90473	0.127015	0.0635075	−	0.997981i	$-0.479771\pi$
$523$	2.90473	0.127015	0.0635075	+	0.997981i	$0.479771\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.635311	−0.0276746
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−14.8300	−0.643566
$532$	0	0
$533$	−5.53089	−0.239570
$534$	0	0
$535$	0	0
$536$	0	0
$537$	38.9801	1.68212
$538$	0	0
$539$	3.92481	0.169053
$540$	0	0
$541$	34.0386	1.46343	0.731716	−	0.681610i	$-0.238720\pi$
$541$	34.0386	1.46343	0.731716	+	0.681610i	$0.238720\pi$
$542$	0	0
$543$	−60.9406	−2.61521
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−23.8364	−1.01917	−0.509586	−	0.860420i	$-0.670202\pi$
$547$	−23.8364	−1.01917	−0.509586	+	0.860420i	$0.670202\pi$
$548$	0	0
$549$	−34.8613	−1.48785
$550$	0	0
$551$	3.81532	0.162538
$552$	0	0
$553$	−40.4866	−1.72167
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−40.4201	−1.71265	−0.856326	−	0.516435i	$-0.827259\pi$
$557$	−40.4201	−1.71265	−0.856326	+	0.516435i	$0.827259\pi$
$558$	0	0
$559$	−6.80028	−0.287621
$560$	0	0
$561$	−1.92375	−0.0812209
$562$	0	0
$563$	−6.54758	−0.275948	−0.137974	−	0.990436i	$-0.544059\pi$
$563$	−6.54758	−0.275948	−0.137974	+	0.990436i	$0.544059\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−22.0061	−0.924169
$568$	0	0
$569$	20.4590	0.857686	0.428843	−	0.903379i	$-0.358921\pi$
$569$	20.4590	0.857686	0.428843	+	0.903379i	$0.358921\pi$
$570$	0	0
$571$	−4.71300	−0.197233	−0.0986164	−	0.995126i	$-0.531442\pi$
$571$	−4.71300	−0.197233	−0.0986164	+	0.995126i	$0.531442\pi$
$572$	0	0
$573$	−18.1188	−0.756924
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.9331	1.07961	0.539805	−	0.841790i	$-0.318498\pi$
$577$	25.9331	1.07961	0.539805	+	0.841790i	$0.318498\pi$
$578$	0	0
$579$	20.2636	0.842127
$580$	0	0
$581$	−5.38049	−0.223220
$582$	0	0
$583$	−2.42648	−0.100494
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.4711	0.432189	0.216095	−	0.976372i	$-0.430668\pi$
$587$	10.4711	0.432189	0.216095	+	0.976372i	$0.430668\pi$
$588$	0	0
$589$	0.223034	0.00918995
$590$	0	0
$591$	62.5213	2.57178
$592$	0	0
$593$	−20.8158	−0.854803	−0.427401	−	0.904062i	$-0.640571\pi$
$593$	−20.8158	−0.854803	−0.427401	+	0.904062i	$0.640571\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	62.6616	2.56457
$598$	0	0
$599$	14.7951	0.604511	0.302256	−	0.953227i	$-0.402260\pi$
$599$	14.7951	0.604511	0.302256	+	0.953227i	$0.402260\pi$
$600$	0	0
$601$	−33.8995	−1.38279	−0.691394	−	0.722477i	$-0.743003\pi$
$601$	−33.8995	−1.38279	−0.691394	+	0.722477i	$0.743003\pi$
$602$	0	0
$603$	13.3041	0.541784
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.9350	1.33679	0.668394	−	0.743807i	$-0.266982\pi$
$607$	32.9350	1.33679	0.668394	+	0.743807i	$0.266982\pi$
$608$	0	0
$609$	75.4227	3.05628
$610$	0	0
$611$	2.67018	0.108024
$612$	0	0
$613$	24.8664	1.00434	0.502172	−	0.864768i	$-0.332535\pi$
$613$	24.8664	1.00434	0.502172	+	0.864768i	$0.332535\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.1280	0.810324	0.405162	−	0.914245i	$-0.367215\pi$
$617$	20.1280	0.810324	0.405162	+	0.914245i	$0.367215\pi$
$618$	0	0
$619$	15.3719	0.617847	0.308924	−	0.951087i	$-0.400031\pi$
$619$	15.3719	0.617847	0.308924	+	0.951087i	$0.400031\pi$
$620$	0	0
$621$	−2.22047	−0.0891046
$622$	0	0
$623$	25.9828	1.04098
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.675358	0.0269712
$628$	0	0
$629$	−10.1676	−0.405408
$630$	0	0
$631$	25.7115	1.02356	0.511779	−	0.859117i	$-0.328987\pi$
$631$	25.7115	1.02356	0.511779	+	0.859117i	$0.328987\pi$
$632$	0	0
$633$	−37.8699	−1.50519
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.81134	−0.309497
$638$	0	0
$639$	10.5314	0.416614
$640$	0	0
$641$	2.95684	0.116788	0.0583941	−	0.998294i	$-0.481402\pi$
$641$	2.95684	0.116788	0.0583941	+	0.998294i	$0.481402\pi$
$642$	0	0
$643$	−16.7764	−0.661596	−0.330798	−	0.943702i	$-0.607318\pi$
$643$	−16.7764	−0.661596	−0.330798	+	0.943702i	$0.607318\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.99247	−0.0783322	−0.0391661	−	0.999233i	$-0.512470\pi$
$647$	−1.99247	−0.0783322	−0.0391661	+	0.999233i	$0.512470\pi$
$648$	0	0
$649$	−1.95757	−0.0768414
$650$	0	0
$651$	4.40901	0.172803
$652$	0	0
$653$	−46.4989	−1.81964	−0.909822	−	0.414999i	$-0.863782\pi$
$653$	−46.4989	−1.81964	−0.909822	+	0.414999i	$0.863782\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	35.4562	1.38328
$658$	0	0
$659$	37.1468	1.44703	0.723517	−	0.690307i	$-0.242524\pi$
$659$	37.1468	1.44703	0.723517	+	0.690307i	$0.242524\pi$
$660$	0	0
$661$	−7.01756	−0.272952	−0.136476	−	0.990643i	$-0.543578\pi$
$661$	−7.01756	−0.272952	−0.136476	+	0.990643i	$0.543578\pi$
$662$	0	0
$663$	3.82874	0.148696
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.51040	−0.290804
$668$	0	0
$669$	−12.4242	−0.480346
$670$	0	0
$671$	−4.60174	−0.177648
$672$	0	0
$673$	6.81155	0.262566	0.131283	−	0.991345i	$-0.458090\pi$
$673$	6.81155	0.262566	0.131283	+	0.991345i	$0.458090\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.522812	−0.0200933	−0.0100466	−	0.999950i	$-0.503198\pi$
$677$	−0.522812	−0.0200933	−0.0100466	+	0.999950i	$0.503198\pi$
$678$	0	0
$679$	1.20273	0.0461566
$680$	0	0
$681$	23.5108	0.900934
$682$	0	0
$683$	2.93412	0.112271	0.0561355	−	0.998423i	$-0.482122\pi$
$683$	2.93412	0.112271	0.0561355	+	0.998423i	$0.482122\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	54.2471	2.06965
$688$	0	0
$689$	4.82929	0.183981
$690$	0	0
$691$	31.8456	1.21146	0.605731	−	0.795669i	$-0.292881\pi$
$691$	31.8456	1.21146	0.605731	+	0.795669i	$0.292881\pi$
$692$	0	0
$693$	7.50240	0.284993
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−7.91596	−0.299838
$698$	0	0
$699$	−68.2070	−2.57982
$700$	0	0
$701$	11.2070	0.423283	0.211642	−	0.977347i	$-0.432119\pi$
$701$	11.2070	0.423283	0.211642	+	0.977347i	$0.432119\pi$
$702$	0	0
$703$	3.56946	0.134625
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.5418	−0.697336
$708$	0	0
$709$	23.7300	0.891198	0.445599	−	0.895233i	$-0.352991\pi$
$709$	23.7300	0.891198	0.445599	+	0.895233i	$0.352991\pi$
$710$	0	0
$711$	−40.6033	−1.52274
$712$	0	0
$713$	−0.439038	−0.0164421
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.69934	0.287537
$718$	0	0
$719$	27.8687	1.03933	0.519664	−	0.854371i	$-0.326057\pi$
$719$	27.8687	1.03933	0.519664	+	0.854371i	$0.326057\pi$
$720$	0	0
$721$	30.9968	1.15438
$722$	0	0
$723$	−65.2705	−2.42744
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−47.5773	−1.76455	−0.882273	−	0.470739i	$-0.843987\pi$
$727$	−47.5773	−1.76455	−0.882273	+	0.470739i	$0.843987\pi$
$728$	0	0
$729$	−39.5022	−1.46304
$730$	0	0
$731$	−9.73274	−0.359978
$732$	0	0
$733$	−15.4086	−0.569129	−0.284565	−	0.958657i	$-0.591849\pi$
$733$	−15.4086	−0.569129	−0.284565	+	0.958657i	$0.591849\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.75615	0.0646888
$738$	0	0
$739$	−53.0212	−1.95042	−0.975208	−	0.221292i	$-0.928973\pi$
$739$	−53.0212	−1.95042	−0.975208	+	0.221292i	$0.928973\pi$
$740$	0	0
$741$	−1.34413	−0.0493778
$742$	0	0
$743$	15.6611	0.574551	0.287275	−	0.957848i	$-0.407250\pi$
$743$	15.6611	0.574551	0.287275	+	0.957848i	$0.407250\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.39599	−0.197429
$748$	0	0
$749$	−42.8209	−1.56464
$750$	0	0
$751$	−34.6216	−1.26336	−0.631679	−	0.775230i	$-0.717634\pi$
$751$	−34.6216	−1.26336	−0.631679	+	0.775230i	$0.717634\pi$
$752$	0	0
$753$	−55.4554	−2.02091
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.4389	1.39709	0.698543	−	0.715568i	$-0.253832\pi$
$757$	38.4389	1.39709	0.698543	+	0.715568i	$0.253832\pi$
$758$	0	0
$759$	−1.32943	−0.0482553
$760$	0	0
$761$	42.0932	1.52588	0.762939	−	0.646470i	$-0.223755\pi$
$761$	42.0932	1.52588	0.762939	+	0.646470i	$0.223755\pi$
$762$	0	0
$763$	67.5446	2.44528
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.89605	0.140678
$768$	0	0
$769$	34.2859	1.23638	0.618191	−	0.786028i	$-0.287866\pi$
$769$	34.2859	1.23638	0.618191	+	0.786028i	$0.287866\pi$
$770$	0	0
$771$	−68.5933	−2.47033
$772$	0	0
$773$	12.7422	0.458304	0.229152	−	0.973391i	$-0.426405\pi$
$773$	12.7422	0.458304	0.229152	+	0.973391i	$0.426405\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	70.5623	2.53141
$778$	0	0
$779$	2.77900	0.0995679
$780$	0	0
$781$	1.39015	0.0497436
$782$	0	0
$783$	16.6766	0.595975
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.5534	1.01782	0.508910	−	0.860820i	$-0.330049\pi$
$787$	28.5534	1.01782	0.508910	+	0.860820i	$0.330049\pi$
$788$	0	0
$789$	−14.7931	−0.526647
$790$	0	0
$791$	15.5974	0.554580
$792$	0	0
$793$	9.15859	0.325231
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.39925	0.297517	0.148758	−	0.988874i	$-0.452472\pi$
$797$	8.39925	0.297517	0.148758	+	0.988874i	$0.452472\pi$
$798$	0	0
$799$	3.82164	0.135200
$800$	0	0
$801$	26.0576	0.920701
$802$	0	0
$803$	4.68026	0.165163
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25.2739	0.889683
$808$	0	0
$809$	−28.8816	−1.01542	−0.507712	−	0.861527i	$-0.669509\pi$
$809$	−28.8816	−1.01542	−0.507712	+	0.861527i	$0.669509\pi$
$810$	0	0
$811$	46.5292	1.63386	0.816931	−	0.576736i	$-0.195674\pi$
$811$	46.5292	1.63386	0.816931	+	0.576736i	$0.195674\pi$
$812$	0	0
$813$	−21.9574	−0.770078
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.41680	0.119539
$818$	0	0
$819$	−14.9316	−0.521753
$820$	0	0
$821$	−2.99065	−0.104374	−0.0521872	−	0.998637i	$-0.516619\pi$
$821$	−2.99065	−0.104374	−0.0521872	+	0.998637i	$0.516619\pi$
$822$	0	0
$823$	5.14357	0.179294	0.0896468	−	0.995974i	$-0.471426\pi$
$823$	5.14357	0.179294	0.0896468	+	0.995974i	$0.471426\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−53.4288	−1.85790	−0.928950	−	0.370205i	$-0.879287\pi$
$827$	−53.4288	−1.85790	−0.928950	+	0.370205i	$0.879287\pi$
$828$	0	0
$829$	11.5290	0.400420	0.200210	−	0.979753i	$-0.435838\pi$
$829$	11.5290	0.400420	0.200210	+	0.979753i	$0.435838\pi$
$830$	0	0
$831$	−79.3770	−2.75356
$832$	0	0
$833$	−11.1798	−0.387357
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.974874	0.0336966
$838$	0	0
$839$	−12.0983	−0.417678	−0.208839	−	0.977950i	$-0.566968\pi$
$839$	−12.0983	−0.417678	−0.208839	+	0.977950i	$0.566968\pi$
$840$	0	0
$841$	27.4061	0.945038
$842$	0	0
$843$	−66.2436	−2.28155
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−41.2215	−1.41639
$848$	0	0
$849$	−33.9235	−1.16425
$850$	0	0
$851$	−7.02642	−0.240862
$852$	0	0
$853$	−3.10103	−0.106177	−0.0530886	−	0.998590i	$-0.516907\pi$
$853$	−3.10103	−0.106177	−0.0530886	+	0.998590i	$0.516907\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.39457	−0.150116	−0.0750578	−	0.997179i	$-0.523914\pi$
$857$	−4.39457	−0.150116	−0.0750578	+	0.997179i	$0.523914\pi$
$858$	0	0
$859$	29.0453	0.991014	0.495507	−	0.868604i	$-0.334982\pi$
$859$	29.0453	0.991014	0.495507	+	0.868604i	$0.334982\pi$
$860$	0	0
$861$	54.9362	1.87222
$862$	0	0
$863$	−40.1020	−1.36509	−0.682543	−	0.730845i	$-0.739126\pi$
$863$	−40.1020	−1.36509	−0.682543	+	0.730845i	$0.739126\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−39.0086	−1.32480
$868$	0	0
$869$	−5.35968	−0.181815
$870$	0	0
$871$	−3.49518	−0.118430
$872$	0	0
$873$	1.20620	0.0408235
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.7503	−1.27474	−0.637368	−	0.770559i	$-0.719977\pi$
$877$	−37.7503	−1.27474	−0.637368	+	0.770559i	$0.719977\pi$
$878$	0	0
$879$	−67.9021	−2.29028
$880$	0	0
$881$	−29.1541	−0.982227	−0.491113	−	0.871096i	$-0.663410\pi$
$881$	−29.1541	−0.982227	−0.491113	+	0.871096i	$0.663410\pi$
$882$	0	0
$883$	−43.3971	−1.46043	−0.730214	−	0.683218i	$-0.760580\pi$
$883$	−43.3971	−1.46043	−0.730214	+	0.683218i	$0.760580\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.4091	−0.853155	−0.426578	−	0.904451i	$-0.640281\pi$
$887$	−25.4091	−0.853155	−0.426578	+	0.904451i	$0.640281\pi$
$888$	0	0
$889$	18.2373	0.611658
$890$	0	0
$891$	−2.91320	−0.0975958
$892$	0	0
$893$	−1.34163	−0.0448961
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.64590	0.0883439
$898$	0	0
$899$	3.29735	0.109973
$900$	0	0
$901$	6.91181	0.230266
$902$	0	0
$903$	67.5446	2.24774
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.00778	0.0666672	0.0333336	−	0.999444i	$-0.489388\pi$
$907$	2.00778	0.0666672	0.0333336	+	0.999444i	$0.489388\pi$
$908$	0	0
$909$	−18.5952	−0.616765
$910$	0	0
$911$	8.45792	0.280223	0.140112	−	0.990136i	$-0.455254\pi$
$911$	8.45792	0.280223	0.140112	+	0.990136i	$0.455254\pi$
$912$	0	0
$913$	−0.712277	−0.0235729
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.55402	−0.249456
$918$	0	0
$919$	−0.840028	−0.0277100	−0.0138550	−	0.999904i	$-0.504410\pi$
$919$	−0.840028	−0.0277100	−0.0138550	+	0.999904i	$0.504410\pi$
$920$	0	0
$921$	31.5711	1.04030
$922$	0	0
$923$	−2.76675	−0.0910686
$924$	0	0
$925$	0	0
$926$	0	0
$927$	31.0861	1.02100
$928$	0	0
$929$	15.7877	0.517979	0.258989	−	0.965880i	$-0.416611\pi$
$929$	15.7877	0.517979	0.258989	+	0.965880i	$0.416611\pi$
$930$	0	0
$931$	3.92481	0.128630
$932$	0	0
$933$	83.2520	2.72555
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−31.6568	−1.03418	−0.517091	−	0.855930i	$-0.672985\pi$
$937$	−31.6568	−1.03418	−0.517091	+	0.855930i	$0.672985\pi$
$938$	0	0
$939$	11.2561	0.367330
$940$	0	0
$941$	−38.3932	−1.25158	−0.625792	−	0.779990i	$-0.715224\pi$
$941$	−38.3932	−1.25158	−0.625792	+	0.779990i	$0.715224\pi$
$942$	0	0
$943$	−5.47041	−0.178141
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.3783	−1.11714	−0.558572	−	0.829456i	$-0.688651\pi$
$947$	−34.3783	−1.11714	−0.558572	+	0.829456i	$0.688651\pi$
$948$	0	0
$949$	−9.31486	−0.302373
$950$	0	0
$951$	16.4428	0.533195
$952$	0	0
$953$	−48.3782	−1.56712	−0.783562	−	0.621314i	$-0.786599\pi$
$953$	−48.3782	−1.56712	−0.783562	+	0.621314i	$0.786599\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.98456	0.322755
$958$	0	0
$959$	21.6541	0.699247
$960$	0	0
$961$	−30.8072	−0.993782
$962$	0	0
$963$	−42.9443	−1.38386
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−51.3705	−1.65196	−0.825982	−	0.563697i	$-0.809379\pi$
$967$	−51.3705	−1.65196	−0.825982	+	0.563697i	$0.809379\pi$
$968$	0	0
$969$	−1.92375	−0.0617999
$970$	0	0
$971$	−50.4587	−1.61930	−0.809649	−	0.586914i	$-0.800343\pi$
$971$	−50.4587	−1.61930	−0.809649	+	0.586914i	$0.800343\pi$
$972$	0	0
$973$	29.1539	0.934633
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.1406	−1.02827	−0.514134	−	0.857710i	$-0.671887\pi$
$977$	−32.1406	−1.02827	−0.514134	+	0.857710i	$0.671887\pi$
$978$	0	0
$979$	3.43964	0.109931
$980$	0	0
$981$	67.7392	2.16275
$982$	0	0
$983$	24.1306	0.769647	0.384824	−	0.922990i	$-0.374262\pi$
$983$	24.1306	0.769647	0.384824	+	0.922990i	$0.374262\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−26.5219	−0.844201
$988$	0	0
$989$	−6.72592	−0.213872
$990$	0	0
$991$	−40.1267	−1.27467	−0.637333	−	0.770588i	$-0.719962\pi$
$991$	−40.1267	−1.27467	−0.637333	+	0.770588i	$0.719962\pi$
$992$	0	0
$993$	20.9371	0.664418
$994$	0	0
$995$	0	0
$996$	0	0
$997$	48.9890	1.55150	0.775749	−	0.631042i	$-0.217372\pi$
$997$	48.9890	1.55150	0.775749	+	0.631042i	$0.217372\pi$
$998$	0	0
$999$	15.6020	0.493625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cv.1.5		5
4.3	odd	2		4600.2.a.bd.1.1	✓	5
5.4	even	2		9200.2.a.ct.1.1		5
20.3	even	4		4600.2.e.w.4049.1		10
20.7	even	4		4600.2.e.w.4049.10		10
20.19	odd	2		4600.2.a.bf.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bd.1.1	✓	5	4.3	odd	2
4600.2.a.bf.1.5	yes	5	20.19	odd	2
4600.2.e.w.4049.1		10	20.3	even	4
4600.2.e.w.4049.10		10	20.7	even	4
9200.2.a.ct.1.1		5	5.4	even	2
9200.2.a.cv.1.5		5	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61696 q^{3} +3.83744 q^{7} +3.84849 q^{9} +O(q^{10})\)	\(q+2.61696 q^{3} +3.83744 q^{7} +3.84849 q^{9} +0.508005 q^{11} -1.01106 q^{13} -1.44705 q^{17} +0.508005 q^{19} +10.0424 q^{21} -1.00000 q^{23} +2.22047 q^{27} +7.51040 q^{29} +0.439038 q^{31} +1.32943 q^{33} +7.02642 q^{37} -2.64590 q^{39} +5.47041 q^{41} +6.72592 q^{43} -2.64098 q^{47} +7.72592 q^{49} -3.78688 q^{51} -4.77648 q^{53} +1.32943 q^{57} -3.85345 q^{59} -9.05844 q^{61} +14.7683 q^{63} +3.45696 q^{67} -2.61696 q^{69} +2.73649 q^{71} +9.21300 q^{73} +1.94944 q^{77} -10.5504 q^{79} -5.73458 q^{81} -1.40211 q^{83} +19.6544 q^{87} +6.77086 q^{89} -3.87986 q^{91} +1.14895 q^{93} +0.313420 q^{97} +1.95506 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 5 * q + 4 * q^7 + 3 * q^9	\( 5 q + 4 q^{7} + 3 q^{9} - 4 q^{13} - 6 q^{17} - 5 q^{23} + 9 q^{27} + 12 q^{29} + 18 q^{31} - 6 q^{33} - 10 q^{37} - 9 q^{39} - 6 q^{41} + 10 q^{43} + 22 q^{47} + 15 q^{49} + 6 q^{51} - 10 q^{53} - 6 q^{57} + q^{59} + 10 q^{61} + 8 q^{67} - 8 q^{71} - 6 q^{73} - 27 q^{81} - 2 q^{83} + 39 q^{87} + 14 q^{89} + 46 q^{91} + 3 q^{93} - 6 q^{97} + 6 q^{99}+O(q^{100}) \) 5 * q + 4 * q^7 + 3 * q^9 - 4 * q^13 - 6 * q^17 - 5 * q^23 + 9 * q^27 + 12 * q^29 + 18 * q^31 - 6 * q^33 - 10 * q^37 - 9 * q^39 - 6 * q^41 + 10 * q^43 + 22 * q^47 + 15 * q^49 + 6 * q^51 - 10 * q^53 - 6 * q^57 + q^59 + 10 * q^61 + 8 * q^67 - 8 * q^71 - 6 * q^73 - 27 * q^81 - 2 * q^83 + 39 * q^87 + 14 * q^89 + 46 * q^91 + 3 * q^93 - 6 * q^97 + 6 * q^99

Introduction

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