LMFDB - Embedded newform 4600.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} -0.236068 q^{7} -1.47214 q^{9} +1.00000 q^{11} -2.23607 q^{13} +2.47214 q^{17} +1.00000 q^{19} -0.291796 q^{21} +1.00000 q^{23} -5.52786 q^{27} -6.23607 q^{29} -8.47214 q^{31} +1.23607 q^{33} -6.76393 q^{37} -2.76393 q^{39} +11.9443 q^{41} -11.4721 q^{43} +1.70820 q^{47} -6.94427 q^{49} +3.05573 q^{51} +3.23607 q^{53} +1.23607 q^{57} -1.23607 q^{59} -2.76393 q^{61} +0.347524 q^{63} +4.94427 q^{67} +1.23607 q^{69} +10.0000 q^{71} +0.527864 q^{73} -0.236068 q^{77} -7.18034 q^{79} -2.41641 q^{81} -9.00000 q^{83} -7.70820 q^{87} +2.00000 q^{89} +0.527864 q^{91} -10.4721 q^{93} +16.1803 q^{97} -1.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{7} + 6 q^{9} + 2 q^{11} - 4 q^{17} + 2 q^{19} - 14 q^{21} + 2 q^{23} - 20 q^{27} - 8 q^{29} - 8 q^{31} - 2 q^{33} - 18 q^{37} - 10 q^{39} + 6 q^{41} - 14 q^{43} - 10 q^{47} + 4 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^7 + 6 * q^9 + 2 * q^11 - 4 * q^17 + 2 * q^19 - 14 * q^21 + 2 * q^23 - 20 * q^27 - 8 * q^29 - 8 * q^31 - 2 * q^33 - 18 * q^37 - 10 * q^39 + 6 * q^41 - 14 * q^43 - 10 * q^47 + 4 * q^49 + 24 * q^51 + 2 * q^53 - 2 * q^57 + 2 * q^59 - 10 * q^61 + 32 * q^63 - 8 * q^67 - 2 * q^69 + 20 * q^71 + 10 * q^73 + 4 * q^77 + 8 * q^79 + 22 * q^81 - 18 * q^83 - 2 * q^87 + 4 * q^89 + 10 * q^91 - 12 * q^93 + 10 * q^97 + 6 * 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
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.236068	−0.0892253	−0.0446127	−	0.999004i	$-0.514205\pi$
$7$	−0.236068	−0.0892253	−0.0446127	+	0.999004i	$0.514205\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−2.23607	−0.620174	−0.310087	−	0.950708i	$-0.600358\pi$
$13$	−2.23607	−0.620174	−0.310087	+	0.950708i	$0.600358\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−0.291796	−0.0636751
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	−6.23607	−1.15801	−0.579004	−	0.815324i	$-0.696559\pi$
$29$	−6.23607	−1.15801	−0.579004	+	0.815324i	$0.696559\pi$
$30$	0	0
$31$	−8.47214	−1.52164	−0.760820	−	0.648963i	$-0.775203\pi$
$31$	−8.47214	−1.52164	−0.760820	+	0.648963i	$0.775203\pi$
$32$	0	0
$33$	1.23607	0.215172
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.76393	−1.11198	−0.555992	−	0.831188i	$-0.687661\pi$
$37$	−6.76393	−1.11198	−0.555992	+	0.831188i	$0.687661\pi$
$38$	0	0
$39$	−2.76393	−0.442583
$40$	0	0
$41$	11.9443	1.86538	0.932691	−	0.360677i	$-0.117454\pi$
$41$	11.9443	1.86538	0.932691	+	0.360677i	$0.117454\pi$
$42$	0	0
$43$	−11.4721	−1.74948	−0.874742	−	0.484589i	$-0.838969\pi$
$43$	−11.4721	−1.74948	−0.874742	+	0.484589i	$0.838969\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.70820	0.249167	0.124584	−	0.992209i	$-0.460241\pi$
$47$	1.70820	0.249167	0.124584	+	0.992209i	$0.460241\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	3.05573	0.427888
$52$	0	0
$53$	3.23607	0.444508	0.222254	−	0.974989i	$-0.428659\pi$
$53$	3.23607	0.444508	0.222254	+	0.974989i	$0.428659\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.23607	0.163721
$58$	0	0
$59$	−1.23607	−0.160922	−0.0804612	−	0.996758i	$-0.525639\pi$
$59$	−1.23607	−0.160922	−0.0804612	+	0.996758i	$0.525639\pi$
$60$	0	0
$61$	−2.76393	−0.353885	−0.176943	−	0.984221i	$-0.556621\pi$
$61$	−2.76393	−0.353885	−0.176943	+	0.984221i	$0.556621\pi$
$62$	0	0
$63$	0.347524	0.0437839
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.94427	0.604039	0.302019	−	0.953302i	$-0.402339\pi$
$67$	4.94427	0.604039	0.302019	+	0.953302i	$0.402339\pi$
$68$	0	0
$69$	1.23607	0.148805
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	0.527864	0.0617818	0.0308909	−	0.999523i	$-0.490166\pi$
$73$	0.527864	0.0617818	0.0308909	+	0.999523i	$0.490166\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.236068	−0.0269024
$78$	0	0
$79$	−7.18034	−0.807851	−0.403926	−	0.914792i	$-0.632355\pi$
$79$	−7.18034	−0.807851	−0.403926	+	0.914792i	$0.632355\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.70820	−0.826406
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0.527864	0.0553352
$92$	0	0
$93$	−10.4721	−1.08591
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.1803	1.64286	0.821432	−	0.570306i	$-0.193175\pi$
$97$	16.1803	1.64286	0.821432	+	0.570306i	$0.193175\pi$
$98$	0	0
$99$	−1.47214	−0.147955
$100$	0	0
$101$	−1.05573	−0.105049	−0.0525244	−	0.998620i	$-0.516727\pi$
$101$	−1.05573	−0.105049	−0.0525244	+	0.998620i	$0.516727\pi$
$102$	0	0
$103$	−2.23607	−0.220326	−0.110163	−	0.993914i	$-0.535137\pi$
$103$	−2.23607	−0.220326	−0.110163	+	0.993914i	$0.535137\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.41641	−0.716971	−0.358486	−	0.933535i	$-0.616707\pi$
$107$	−7.41641	−0.716971	−0.358486	+	0.933535i	$0.616707\pi$
$108$	0	0
$109$	−17.2361	−1.65092	−0.825458	−	0.564464i	$-0.809083\pi$
$109$	−17.2361	−1.65092	−0.825458	+	0.564464i	$0.809083\pi$
$110$	0	0
$111$	−8.36068	−0.793561
$112$	0	0
$113$	−11.2361	−1.05700	−0.528500	−	0.848933i	$-0.677245\pi$
$113$	−11.2361	−1.05700	−0.528500	+	0.848933i	$0.677245\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.29180	0.304327
$118$	0	0
$119$	−0.583592	−0.0534978
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	14.7639	1.33122
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.1803	−1.61324	−0.806622	−	0.591067i	$-0.798707\pi$
$127$	−18.1803	−1.61324	−0.806622	+	0.591067i	$0.798707\pi$
$128$	0	0
$129$	−14.1803	−1.24851
$130$	0	0
$131$	2.94427	0.257242	0.128621	−	0.991694i	$-0.458945\pi$
$131$	2.94427	0.257242	0.128621	+	0.991694i	$0.458945\pi$
$132$	0	0
$133$	−0.236068	−0.0204697
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.4721	−1.74905	−0.874526	−	0.484978i	$-0.838828\pi$
$137$	−20.4721	−1.74905	−0.874526	+	0.484978i	$0.838828\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	2.11146	0.177817
$142$	0	0
$143$	−2.23607	−0.186989
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.58359	−0.707963
$148$	0	0
$149$	−6.94427	−0.568897	−0.284448	−	0.958691i	$-0.591810\pi$
$149$	−6.94427	−0.568897	−0.284448	+	0.958691i	$0.591810\pi$
$150$	0	0
$151$	20.4721	1.66600	0.832999	−	0.553274i	$-0.186622\pi$
$151$	20.4721	1.66600	0.832999	+	0.553274i	$0.186622\pi$
$152$	0	0
$153$	−3.63932	−0.294222
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.7082	−1.09403	−0.547017	−	0.837122i	$-0.684237\pi$
$157$	−13.7082	−1.09403	−0.547017	+	0.837122i	$0.684237\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	−0.236068	−0.0186048
$162$	0	0
$163$	−1.70820	−0.133797	−0.0668984	−	0.997760i	$-0.521310\pi$
$163$	−1.70820	−0.133797	−0.0668984	+	0.997760i	$0.521310\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.4164	−1.65725	−0.828626	−	0.559803i	$-0.810877\pi$
$167$	−21.4164	−1.65725	−0.828626	+	0.559803i	$0.810877\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	−1.47214	−0.112577
$172$	0	0
$173$	−9.76393	−0.742338	−0.371169	−	0.928565i	$-0.621043\pi$
$173$	−9.76393	−0.742338	−0.371169	+	0.928565i	$0.621043\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.52786	−0.114841
$178$	0	0
$179$	−2.29180	−0.171297	−0.0856484	−	0.996325i	$-0.527296\pi$
$179$	−2.29180	−0.171297	−0.0856484	+	0.996325i	$0.527296\pi$
$180$	0	0
$181$	16.9443	1.25946	0.629729	−	0.776815i	$-0.283166\pi$
$181$	16.9443	1.25946	0.629729	+	0.776815i	$0.283166\pi$
$182$	0	0
$183$	−3.41641	−0.252548
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.47214	0.180780
$188$	0	0
$189$	1.30495	0.0949213
$190$	0	0
$191$	3.18034	0.230121	0.115061	−	0.993358i	$-0.463294\pi$
$191$	3.18034	0.230121	0.115061	+	0.993358i	$0.463294\pi$
$192$	0	0
$193$	14.9443	1.07571	0.537856	−	0.843037i	$-0.319234\pi$
$193$	14.9443	1.07571	0.537856	+	0.843037i	$0.319234\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.76393	0.268169	0.134085	−	0.990970i	$-0.457191\pi$
$197$	3.76393	0.268169	0.134085	+	0.990970i	$0.457191\pi$
$198$	0	0
$199$	−0.708204	−0.0502032	−0.0251016	−	0.999685i	$-0.507991\pi$
$199$	−0.708204	−0.0502032	−0.0251016	+	0.999685i	$0.507991\pi$
$200$	0	0
$201$	6.11146	0.431069
$202$	0	0
$203$	1.47214	0.103324
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.47214	−0.102321
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	2.94427	0.202692	0.101346	−	0.994851i	$-0.467685\pi$
$211$	2.94427	0.202692	0.101346	+	0.994851i	$0.467685\pi$
$212$	0	0
$213$	12.3607	0.846940
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	0.652476	0.0440902
$220$	0	0
$221$	−5.52786	−0.371844
$222$	0	0
$223$	−8.76393	−0.586876	−0.293438	−	0.955978i	$-0.594799\pi$
$223$	−8.76393	−0.586876	−0.293438	+	0.955978i	$0.594799\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.94427	0.328163	0.164081	−	0.986447i	$-0.447534\pi$
$227$	4.94427	0.328163	0.164081	+	0.986447i	$0.447534\pi$
$228$	0	0
$229$	−10.9443	−0.723218	−0.361609	−	0.932330i	$-0.617772\pi$
$229$	−10.9443	−0.723218	−0.361609	+	0.932330i	$0.617772\pi$
$230$	0	0
$231$	−0.291796	−0.0191988
$232$	0	0
$233$	−13.4721	−0.882589	−0.441294	−	0.897362i	$-0.645481\pi$
$233$	−13.4721	−0.882589	−0.441294	+	0.897362i	$0.645481\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.87539	−0.576518
$238$	0	0
$239$	11.2361	0.726801	0.363400	−	0.931633i	$-0.381616\pi$
$239$	11.2361	0.726801	0.363400	+	0.931633i	$0.381616\pi$
$240$	0	0
$241$	16.7639	1.07986	0.539930	−	0.841710i	$-0.318451\pi$
$241$	16.7639	1.07986	0.539930	+	0.841710i	$0.318451\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.23607	−0.142278
$248$	0	0
$249$	−11.1246	−0.704994
$250$	0	0
$251$	−14.4721	−0.913473	−0.456737	−	0.889602i	$-0.650982\pi$
$251$	−14.4721	−0.913473	−0.456737	+	0.889602i	$0.650982\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	1.59675	0.0992171
$260$	0	0
$261$	9.18034	0.568249
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.47214	0.151292
$268$	0	0
$269$	−2.81966	−0.171918	−0.0859589	−	0.996299i	$-0.527395\pi$
$269$	−2.81966	−0.171918	−0.0859589	+	0.996299i	$0.527395\pi$
$270$	0	0
$271$	26.0689	1.58357	0.791786	−	0.610799i	$-0.209152\pi$
$271$	26.0689	1.58357	0.791786	+	0.610799i	$0.209152\pi$
$272$	0	0
$273$	0.652476	0.0394896
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9.76393	−0.586658	−0.293329	−	0.956012i	$-0.594763\pi$
$277$	−9.76393	−0.586658	−0.293329	+	0.956012i	$0.594763\pi$
$278$	0	0
$279$	12.4721	0.746687
$280$	0	0
$281$	0.180340	0.0107582	0.00537909	−	0.999986i	$-0.498288\pi$
$281$	0.180340	0.0107582	0.00537909	+	0.999986i	$0.498288\pi$
$282$	0	0
$283$	16.3607	0.972541	0.486271	−	0.873808i	$-0.338357\pi$
$283$	16.3607	0.972541	0.486271	+	0.873808i	$0.338357\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.81966	−0.166439
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	20.0000	1.17242
$292$	0	0
$293$	5.52786	0.322941	0.161471	−	0.986878i	$-0.448376\pi$
$293$	5.52786	0.322941	0.161471	+	0.986878i	$0.448376\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.52786	−0.320759
$298$	0	0
$299$	−2.23607	−0.129315
$300$	0	0
$301$	2.70820	0.156098
$302$	0	0
$303$	−1.30495	−0.0749675
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.0000	−1.36975	−0.684876	−	0.728659i	$-0.740144\pi$
$307$	−24.0000	−1.36975	−0.684876	+	0.728659i	$0.740144\pi$
$308$	0	0
$309$	−2.76393	−0.157235
$310$	0	0
$311$	1.23607	0.0700910	0.0350455	−	0.999386i	$-0.488842\pi$
$311$	1.23607	0.0700910	0.0350455	+	0.999386i	$0.488842\pi$
$312$	0	0
$313$	−20.6525	−1.16735	−0.583673	−	0.811988i	$-0.698385\pi$
$313$	−20.6525	−1.16735	−0.583673	+	0.811988i	$0.698385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.1246	−1.01798	−0.508990	−	0.860773i	$-0.669981\pi$
$317$	−18.1246	−1.01798	−0.508990	+	0.860773i	$0.669981\pi$
$318$	0	0
$319$	−6.23607	−0.349153
$320$	0	0
$321$	−9.16718	−0.511662
$322$	0	0
$323$	2.47214	0.137553
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−21.3050	−1.17817
$328$	0	0
$329$	−0.403252	−0.0222320
$330$	0	0
$331$	13.0557	0.717608	0.358804	−	0.933413i	$-0.383185\pi$
$331$	13.0557	0.717608	0.358804	+	0.933413i	$0.383185\pi$
$332$	0	0
$333$	9.95743	0.545664
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6.65248	−0.362383	−0.181192	−	0.983448i	$-0.557995\pi$
$337$	−6.65248	−0.362383	−0.181192	+	0.983448i	$0.557995\pi$
$338$	0	0
$339$	−13.8885	−0.754322
$340$	0	0
$341$	−8.47214	−0.458792
$342$	0	0
$343$	3.29180	0.177740
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.9443	−1.12435	−0.562174	−	0.827019i	$-0.690035\pi$
$347$	−20.9443	−1.12435	−0.562174	+	0.827019i	$0.690035\pi$
$348$	0	0
$349$	17.1803	0.919643	0.459821	−	0.888011i	$-0.347913\pi$
$349$	17.1803	0.919643	0.459821	+	0.888011i	$0.347913\pi$
$350$	0	0
$351$	12.3607	0.659764
$352$	0	0
$353$	−30.8885	−1.64403	−0.822016	−	0.569465i	$-0.807150\pi$
$353$	−30.8885	−1.64403	−0.822016	+	0.569465i	$0.807150\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.721360	−0.0381784
$358$	0	0
$359$	24.2361	1.27913	0.639565	−	0.768737i	$-0.279114\pi$
$359$	24.2361	1.27913	0.639565	+	0.768737i	$0.279114\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−12.3607	−0.648767
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7.65248	−0.399456	−0.199728	−	0.979851i	$-0.564006\pi$
$367$	−7.65248	−0.399456	−0.199728	+	0.979851i	$0.564006\pi$
$368$	0	0
$369$	−17.5836	−0.915365
$370$	0	0
$371$	−0.763932	−0.0396614
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.9443	0.718167
$378$	0	0
$379$	20.9443	1.07583	0.537917	−	0.842997i	$-0.319211\pi$
$379$	20.9443	1.07583	0.537917	+	0.842997i	$0.319211\pi$
$380$	0	0
$381$	−22.4721	−1.15128
$382$	0	0
$383$	−4.23607	−0.216453	−0.108226	−	0.994126i	$-0.534517\pi$
$383$	−4.23607	−0.216453	−0.108226	+	0.994126i	$0.534517\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.8885	0.858493
$388$	0	0
$389$	−16.1803	−0.820376	−0.410188	−	0.912001i	$-0.634537\pi$
$389$	−16.1803	−0.820376	−0.410188	+	0.912001i	$0.634537\pi$
$390$	0	0
$391$	2.47214	0.125021
$392$	0	0
$393$	3.63932	0.183579
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.3607	1.12225	0.561125	−	0.827731i	$-0.310369\pi$
$397$	22.3607	1.12225	0.561125	+	0.827731i	$0.310369\pi$
$398$	0	0
$399$	−0.291796	−0.0146081
$400$	0	0
$401$	27.1246	1.35454	0.677269	−	0.735735i	$-0.263163\pi$
$401$	27.1246	1.35454	0.677269	+	0.735735i	$0.263163\pi$
$402$	0	0
$403$	18.9443	0.943681
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.76393	−0.335276
$408$	0	0
$409$	0.416408	0.0205900	0.0102950	−	0.999947i	$-0.496723\pi$
$409$	0.416408	0.0205900	0.0102950	+	0.999947i	$0.496723\pi$
$410$	0	0
$411$	−25.3050	−1.24820
$412$	0	0
$413$	0.291796	0.0143583
$414$	0	0
$415$	0	0
$416$	0	0
$417$	24.7214	1.21061
$418$	0	0
$419$	34.7771	1.69897	0.849486	−	0.527611i	$-0.176912\pi$
$419$	34.7771	1.69897	0.849486	+	0.527611i	$0.176912\pi$
$420$	0	0
$421$	−19.8885	−0.969308	−0.484654	−	0.874706i	$-0.661055\pi$
$421$	−19.8885	−0.969308	−0.484654	+	0.874706i	$0.661055\pi$
$422$	0	0
$423$	−2.51471	−0.122269
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.652476	0.0315755
$428$	0	0
$429$	−2.76393	−0.133444
$430$	0	0
$431$	−4.94427	−0.238157	−0.119079	−	0.992885i	$-0.537994\pi$
$431$	−4.94427	−0.238157	−0.119079	+	0.992885i	$0.537994\pi$
$432$	0	0
$433$	−11.1246	−0.534615	−0.267307	−	0.963611i	$-0.586134\pi$
$433$	−11.1246	−0.534615	−0.267307	+	0.963611i	$0.586134\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.00000	0.0478365
$438$	0	0
$439$	−32.3607	−1.54449	−0.772245	−	0.635324i	$-0.780866\pi$
$439$	−32.3607	−1.54449	−0.772245	+	0.635324i	$0.780866\pi$
$440$	0	0
$441$	10.2229	0.486805
$442$	0	0
$443$	10.0000	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8.58359	−0.405990
$448$	0	0
$449$	16.8328	0.794390	0.397195	−	0.917734i	$-0.369984\pi$
$449$	16.8328	0.794390	0.397195	+	0.917734i	$0.369984\pi$
$450$	0	0
$451$	11.9443	0.562434
$452$	0	0
$453$	25.3050	1.18893
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.6525	−0.498302	−0.249151	−	0.968465i	$-0.580152\pi$
$457$	−10.6525	−0.498302	−0.249151	+	0.968465i	$0.580152\pi$
$458$	0	0
$459$	−13.6656	−0.637857
$460$	0	0
$461$	2.34752	0.109335	0.0546676	−	0.998505i	$-0.482590\pi$
$461$	2.34752	0.109335	0.0546676	+	0.998505i	$0.482590\pi$
$462$	0	0
$463$	17.5279	0.814589	0.407294	−	0.913297i	$-0.366472\pi$
$463$	17.5279	0.814589	0.407294	+	0.913297i	$0.366472\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.0000	0.786666	0.393333	−	0.919396i	$-0.371322\pi$
$467$	17.0000	0.786666	0.393333	+	0.919396i	$0.371322\pi$
$468$	0	0
$469$	−1.16718	−0.0538956
$470$	0	0
$471$	−16.9443	−0.780751
$472$	0	0
$473$	−11.4721	−0.527489
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.76393	−0.218125
$478$	0	0
$479$	28.1246	1.28505	0.642523	−	0.766266i	$-0.277888\pi$
$479$	28.1246	1.28505	0.642523	+	0.766266i	$0.277888\pi$
$480$	0	0
$481$	15.1246	0.689623
$482$	0	0
$483$	−0.291796	−0.0132772
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.12461	−0.0509610	−0.0254805	−	0.999675i	$-0.508112\pi$
$487$	−1.12461	−0.0509610	−0.0254805	+	0.999675i	$0.508112\pi$
$488$	0	0
$489$	−2.11146	−0.0954833
$490$	0	0
$491$	1.23607	0.0557830	0.0278915	−	0.999611i	$-0.491121\pi$
$491$	1.23607	0.0557830	0.0278915	+	0.999611i	$0.491121\pi$
$492$	0	0
$493$	−15.4164	−0.694320
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.36068	−0.105891
$498$	0	0
$499$	24.5410	1.09861	0.549303	−	0.835623i	$-0.314893\pi$
$499$	24.5410	1.09861	0.549303	+	0.835623i	$0.314893\pi$
$500$	0	0
$501$	−26.4721	−1.18269
$502$	0	0
$503$	32.5967	1.45342	0.726709	−	0.686946i	$-0.241049\pi$
$503$	32.5967	1.45342	0.726709	+	0.686946i	$0.241049\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.88854	−0.439166
$508$	0	0
$509$	−14.3607	−0.636526	−0.318263	−	0.948002i	$-0.603099\pi$
$509$	−14.3607	−0.636526	−0.318263	+	0.948002i	$0.603099\pi$
$510$	0	0
$511$	−0.124612	−0.00551250
$512$	0	0
$513$	−5.52786	−0.244061
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.70820	0.0751267
$518$	0	0
$519$	−12.0689	−0.529765
$520$	0	0
$521$	40.6525	1.78102	0.890509	−	0.454966i	$-0.150349\pi$
$521$	40.6525	1.78102	0.890509	+	0.454966i	$0.150349\pi$
$522$	0	0
$523$	−16.8885	−0.738484	−0.369242	−	0.929333i	$-0.620383\pi$
$523$	−16.8885	−0.738484	−0.369242	+	0.929333i	$0.620383\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.9443	−0.912347
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.81966	0.0789665
$532$	0	0
$533$	−26.7082	−1.15686
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.83282	−0.122245
$538$	0	0
$539$	−6.94427	−0.299111
$540$	0	0
$541$	−4.70820	−0.202421	−0.101211	−	0.994865i	$-0.532272\pi$
$541$	−4.70820	−0.202421	−0.101211	+	0.994865i	$0.532272\pi$
$542$	0	0
$543$	20.9443	0.898805
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.29180	0.0979901	0.0489951	−	0.998799i	$-0.484398\pi$
$547$	2.29180	0.0979901	0.0489951	+	0.998799i	$0.484398\pi$
$548$	0	0
$549$	4.06888	0.173656
$550$	0	0
$551$	−6.23607	−0.265665
$552$	0	0
$553$	1.69505	0.0720808
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13.8197	−0.585558	−0.292779	−	0.956180i	$-0.594580\pi$
$557$	−13.8197	−0.585558	−0.292779	+	0.956180i	$0.594580\pi$
$558$	0	0
$559$	25.6525	1.08498
$560$	0	0
$561$	3.05573	0.129013
$562$	0	0
$563$	−5.36068	−0.225926	−0.112963	−	0.993599i	$-0.536034\pi$
$563$	−5.36068	−0.225926	−0.112963	+	0.993599i	$0.536034\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.570437	0.0239561
$568$	0	0
$569$	39.4164	1.65242	0.826211	−	0.563361i	$-0.190492\pi$
$569$	39.4164	1.65242	0.826211	+	0.563361i	$0.190492\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	3.93112	0.164225
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−9.00000	−0.374675	−0.187337	−	0.982296i	$-0.559986\pi$
$577$	−9.00000	−0.374675	−0.187337	+	0.982296i	$0.559986\pi$
$578$	0	0
$579$	18.4721	0.767676
$580$	0	0
$581$	2.12461	0.0881437
$582$	0	0
$583$	3.23607	0.134024
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.5967	0.891393	0.445697	−	0.895184i	$-0.352956\pi$
$587$	21.5967	0.891393	0.445697	+	0.895184i	$0.352956\pi$
$588$	0	0
$589$	−8.47214	−0.349088
$590$	0	0
$591$	4.65248	0.191377
$592$	0	0
$593$	−9.11146	−0.374163	−0.187081	−	0.982344i	$-0.559903\pi$
$593$	−9.11146	−0.374163	−0.187081	+	0.982344i	$0.559903\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.875388	−0.0358273
$598$	0	0
$599$	−36.8328	−1.50495	−0.752474	−	0.658622i	$-0.771140\pi$
$599$	−36.8328	−1.50495	−0.752474	+	0.658622i	$0.771140\pi$
$600$	0	0
$601$	2.94427	0.120099	0.0600497	−	0.998195i	$-0.480874\pi$
$601$	2.94427	0.120099	0.0600497	+	0.998195i	$0.480874\pi$
$602$	0	0
$603$	−7.27864	−0.296409
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.1246	0.695067	0.347533	−	0.937668i	$-0.387019\pi$
$607$	17.1246	0.695067	0.347533	+	0.937668i	$0.387019\pi$
$608$	0	0
$609$	1.81966	0.0737363
$610$	0	0
$611$	−3.81966	−0.154527
$612$	0	0
$613$	−40.0689	−1.61837	−0.809183	−	0.587556i	$-0.800090\pi$
$613$	−40.0689	−1.61837	−0.809183	+	0.587556i	$0.800090\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.3050	1.50184	0.750920	−	0.660393i	$-0.229610\pi$
$617$	37.3050	1.50184	0.750920	+	0.660393i	$0.229610\pi$
$618$	0	0
$619$	−18.4721	−0.742458	−0.371229	−	0.928541i	$-0.621063\pi$
$619$	−18.4721	−0.742458	−0.371229	+	0.928541i	$0.621063\pi$
$620$	0	0
$621$	−5.52786	−0.221826
$622$	0	0
$623$	−0.472136	−0.0189157
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.23607	0.0493638
$628$	0	0
$629$	−16.7214	−0.666724
$630$	0	0
$631$	16.1246	0.641911	0.320955	−	0.947094i	$-0.395996\pi$
$631$	16.1246	0.641911	0.320955	+	0.947094i	$0.395996\pi$
$632$	0	0
$633$	3.63932	0.144650
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15.5279	0.615236
$638$	0	0
$639$	−14.7214	−0.582368
$640$	0	0
$641$	−43.7771	−1.72909	−0.864546	−	0.502555i	$-0.832394\pi$
$641$	−43.7771	−1.72909	−0.864546	+	0.502555i	$0.832394\pi$
$642$	0	0
$643$	15.9443	0.628781	0.314390	−	0.949294i	$-0.398200\pi$
$643$	15.9443	0.628781	0.314390	+	0.949294i	$0.398200\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.1803	0.478859	0.239429	−	0.970914i	$-0.423040\pi$
$647$	12.1803	0.478859	0.239429	+	0.970914i	$0.423040\pi$
$648$	0	0
$649$	−1.23607	−0.0485199
$650$	0	0
$651$	2.47214	0.0968906
$652$	0	0
$653$	−27.5410	−1.07776	−0.538882	−	0.842381i	$-0.681153\pi$
$653$	−27.5410	−1.07776	−0.538882	+	0.842381i	$0.681153\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.777088	−0.0303171
$658$	0	0
$659$	4.52786	0.176381	0.0881903	−	0.996104i	$-0.471892\pi$
$659$	4.52786	0.176381	0.0881903	+	0.996104i	$0.471892\pi$
$660$	0	0
$661$	−44.1803	−1.71842	−0.859208	−	0.511626i	$-0.829043\pi$
$661$	−44.1803	−1.71842	−0.859208	+	0.511626i	$0.829043\pi$
$662$	0	0
$663$	−6.83282	−0.265365
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.23607	−0.241462
$668$	0	0
$669$	−10.8328	−0.418821
$670$	0	0
$671$	−2.76393	−0.106700
$672$	0	0
$673$	14.3050	0.551415	0.275708	−	0.961242i	$-0.411088\pi$
$673$	14.3050	0.551415	0.275708	+	0.961242i	$0.411088\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.29180	0.164947	0.0824736	−	0.996593i	$-0.473718\pi$
$677$	4.29180	0.164947	0.0824736	+	0.996593i	$0.473718\pi$
$678$	0	0
$679$	−3.81966	−0.146585
$680$	0	0
$681$	6.11146	0.234192
$682$	0	0
$683$	29.7771	1.13939	0.569694	−	0.821857i	$-0.307062\pi$
$683$	29.7771	1.13939	0.569694	+	0.821857i	$0.307062\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−13.5279	−0.516120
$688$	0	0
$689$	−7.23607	−0.275672
$690$	0	0
$691$	−33.5279	−1.27546	−0.637730	−	0.770260i	$-0.720126\pi$
$691$	−33.5279	−1.27546	−0.637730	+	0.770260i	$0.720126\pi$
$692$	0	0
$693$	0.347524	0.0132014
$694$	0	0
$695$	0	0
$696$	0	0
$697$	29.5279	1.11845
$698$	0	0
$699$	−16.6525	−0.629854
$700$	0	0
$701$	−46.4721	−1.75523	−0.877614	−	0.479368i	$-0.840866\pi$
$701$	−46.4721	−1.75523	−0.877614	+	0.479368i	$0.840866\pi$
$702$	0	0
$703$	−6.76393	−0.255107
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.249224	0.00937302
$708$	0	0
$709$	−17.4164	−0.654087	−0.327043	−	0.945009i	$-0.606052\pi$
$709$	−17.4164	−0.654087	−0.327043	+	0.945009i	$0.606052\pi$
$710$	0	0
$711$	10.5704	0.396422
$712$	0	0
$713$	−8.47214	−0.317284
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.8885	0.518677
$718$	0	0
$719$	24.2918	0.905931	0.452966	−	0.891528i	$-0.350366\pi$
$719$	24.2918	0.905931	0.452966	+	0.891528i	$0.350366\pi$
$720$	0	0
$721$	0.527864	0.0196587
$722$	0	0
$723$	20.7214	0.770636
$724$	0	0
$725$	0	0
$726$	0	0
$727$	44.7214	1.65862	0.829312	−	0.558786i	$-0.188733\pi$
$727$	44.7214	1.65862	0.829312	+	0.558786i	$0.188733\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−28.3607	−1.04896
$732$	0	0
$733$	48.4721	1.79036	0.895180	−	0.445706i	$-0.147047\pi$
$733$	48.4721	1.79036	0.895180	+	0.445706i	$0.147047\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.94427	0.182125
$738$	0	0
$739$	26.9443	0.991161	0.495581	−	0.868562i	$-0.334955\pi$
$739$	26.9443	0.991161	0.495581	+	0.868562i	$0.334955\pi$
$740$	0	0
$741$	−2.76393	−0.101536
$742$	0	0
$743$	−16.8197	−0.617053	−0.308527	−	0.951216i	$-0.599836\pi$
$743$	−16.8197	−0.617053	−0.308527	+	0.951216i	$0.599836\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.2492	0.484764
$748$	0	0
$749$	1.75078	0.0639720
$750$	0	0
$751$	26.5967	0.970529	0.485265	−	0.874367i	$-0.338723\pi$
$751$	26.5967	0.970529	0.485265	+	0.874367i	$0.338723\pi$
$752$	0	0
$753$	−17.8885	−0.651895
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21.2361	0.771838	0.385919	−	0.922533i	$-0.373884\pi$
$757$	21.2361	0.771838	0.385919	+	0.922533i	$0.373884\pi$
$758$	0	0
$759$	1.23607	0.0448664
$760$	0	0
$761$	−17.1115	−0.620290	−0.310145	−	0.950689i	$-0.600378\pi$
$761$	−17.1115	−0.620290	−0.310145	+	0.950689i	$0.600378\pi$
$762$	0	0
$763$	4.06888	0.147303
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.76393	0.0997998
$768$	0	0
$769$	12.0689	0.435215	0.217608	−	0.976036i	$-0.430175\pi$
$769$	12.0689	0.435215	0.217608	+	0.976036i	$0.430175\pi$
$770$	0	0
$771$	17.3050	0.623223
$772$	0	0
$773$	9.23607	0.332198	0.166099	−	0.986109i	$-0.446883\pi$
$773$	9.23607	0.332198	0.166099	+	0.986109i	$0.446883\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.97369	0.0708057
$778$	0	0
$779$	11.9443	0.427948
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	34.4721	1.23193
$784$	0	0
$785$	0	0
$786$	0	0
$787$	49.7214	1.77238	0.886188	−	0.463327i	$-0.153344\pi$
$787$	49.7214	1.77238	0.886188	+	0.463327i	$0.153344\pi$
$788$	0	0
$789$	−9.88854	−0.352041
$790$	0	0
$791$	2.65248	0.0943112
$792$	0	0
$793$	6.18034	0.219470
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.7639	−0.381278	−0.190639	−	0.981660i	$-0.561056\pi$
$797$	−10.7639	−0.381278	−0.190639	+	0.981660i	$0.561056\pi$
$798$	0	0
$799$	4.22291	0.149396
$800$	0	0
$801$	−2.94427	−0.104031
$802$	0	0
$803$	0.527864	0.0186279
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.48529	−0.122688
$808$	0	0
$809$	−44.8885	−1.57820	−0.789099	−	0.614267i	$-0.789452\pi$
$809$	−44.8885	−1.57820	−0.789099	+	0.614267i	$0.789452\pi$
$810$	0	0
$811$	−11.4164	−0.400884	−0.200442	−	0.979706i	$-0.564238\pi$
$811$	−11.4164	−0.400884	−0.200442	+	0.979706i	$0.564238\pi$
$812$	0	0
$813$	32.2229	1.13011
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−11.4721	−0.401359
$818$	0	0
$819$	−0.777088	−0.0271536
$820$	0	0
$821$	−17.7639	−0.619966	−0.309983	−	0.950742i	$-0.600323\pi$
$821$	−17.7639	−0.619966	−0.309983	+	0.950742i	$0.600323\pi$
$822$	0	0
$823$	8.94427	0.311778	0.155889	−	0.987775i	$-0.450176\pi$
$823$	8.94427	0.311778	0.155889	+	0.987775i	$0.450176\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	0	0
$829$	0.236068	0.00819898	0.00409949	−	0.999992i	$-0.498695\pi$
$829$	0.236068	0.00819898	0.00409949	+	0.999992i	$0.498695\pi$
$830$	0	0
$831$	−12.0689	−0.418665
$832$	0	0
$833$	−17.1672	−0.594808
$834$	0	0
$835$	0	0
$836$	0	0
$837$	46.8328	1.61878
$838$	0	0
$839$	−40.5967	−1.40156	−0.700778	−	0.713380i	$-0.747164\pi$
$839$	−40.5967	−1.40156	−0.700778	+	0.713380i	$0.747164\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	0	0
$843$	0.222912	0.00767751
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.36068	0.0811139
$848$	0	0
$849$	20.2229	0.694049
$850$	0	0
$851$	−6.76393	−0.231865
$852$	0	0
$853$	−23.7639	−0.813662	−0.406831	−	0.913504i	$-0.633366\pi$
$853$	−23.7639	−0.813662	−0.406831	+	0.913504i	$0.633366\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.3050	−0.932719	−0.466360	−	0.884595i	$-0.654435\pi$
$857$	−27.3050	−0.932719	−0.466360	+	0.884595i	$0.654435\pi$
$858$	0	0
$859$	−8.29180	−0.282912	−0.141456	−	0.989945i	$-0.545178\pi$
$859$	−8.29180	−0.282912	−0.141456	+	0.989945i	$0.545178\pi$
$860$	0	0
$861$	−3.48529	−0.118778
$862$	0	0
$863$	−32.5410	−1.10771	−0.553855	−	0.832613i	$-0.686844\pi$
$863$	−32.5410	−1.10771	−0.553855	+	0.832613i	$0.686844\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−13.4590	−0.457091
$868$	0	0
$869$	−7.18034	−0.243576
$870$	0	0
$871$	−11.0557	−0.374609
$872$	0	0
$873$	−23.8197	−0.806173
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.4164	0.993322	0.496661	−	0.867945i	$-0.334559\pi$
$877$	29.4164	0.993322	0.496661	+	0.867945i	$0.334559\pi$
$878$	0	0
$879$	6.83282	0.230465
$880$	0	0
$881$	24.2918	0.818411	0.409206	−	0.912442i	$-0.365806\pi$
$881$	24.2918	0.818411	0.409206	+	0.912442i	$0.365806\pi$
$882$	0	0
$883$	20.9443	0.704831	0.352415	−	0.935844i	$-0.385360\pi$
$883$	20.9443	0.704831	0.352415	+	0.935844i	$0.385360\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.3607	−1.08657	−0.543283	−	0.839550i	$-0.682819\pi$
$887$	−32.3607	−1.08657	−0.543283	+	0.839550i	$0.682819\pi$
$888$	0	0
$889$	4.29180	0.143942
$890$	0	0
$891$	−2.41641	−0.0809527
$892$	0	0
$893$	1.70820	0.0571629
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.76393	−0.0922850
$898$	0	0
$899$	52.8328	1.76207
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	3.34752	0.111399
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.4164	1.07637	0.538185	−	0.842827i	$-0.319110\pi$
$907$	32.4164	1.07637	0.538185	+	0.842827i	$0.319110\pi$
$908$	0	0
$909$	1.55418	0.0515487
$910$	0	0
$911$	27.0689	0.896832	0.448416	−	0.893825i	$-0.351988\pi$
$911$	27.0689	0.896832	0.448416	+	0.893825i	$0.351988\pi$
$912$	0	0
$913$	−9.00000	−0.297857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.695048	−0.0229525
$918$	0	0
$919$	−27.7771	−0.916282	−0.458141	−	0.888880i	$-0.651484\pi$
$919$	−27.7771	−0.916282	−0.458141	+	0.888880i	$0.651484\pi$
$920$	0	0
$921$	−29.6656	−0.977516
$922$	0	0
$923$	−22.3607	−0.736011
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.29180	0.108117
$928$	0	0
$929$	31.0000	1.01708	0.508539	−	0.861039i	$-0.330186\pi$
$929$	31.0000	1.01708	0.508539	+	0.861039i	$0.330186\pi$
$930$	0	0
$931$	−6.94427	−0.227589
$932$	0	0
$933$	1.52786	0.0500200
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33.1246	1.08213	0.541067	−	0.840980i	$-0.318021\pi$
$937$	33.1246	1.08213	0.541067	+	0.840980i	$0.318021\pi$
$938$	0	0
$939$	−25.5279	−0.833070
$940$	0	0
$941$	−8.76393	−0.285696	−0.142848	−	0.989745i	$-0.545626\pi$
$941$	−8.76393	−0.285696	−0.142848	+	0.989745i	$0.545626\pi$
$942$	0	0
$943$	11.9443	0.388959
$944$	0	0
$945$	0	0
$946$	0	0
$947$	53.1935	1.72856	0.864278	−	0.503014i	$-0.167776\pi$
$947$	53.1935	1.72856	0.864278	+	0.503014i	$0.167776\pi$
$948$	0	0
$949$	−1.18034	−0.0383155
$950$	0	0
$951$	−22.4033	−0.726475
$952$	0	0
$953$	−36.6525	−1.18729	−0.593645	−	0.804727i	$-0.702312\pi$
$953$	−36.6525	−1.18729	−0.593645	+	0.804727i	$0.702312\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7.70820	−0.249171
$958$	0	0
$959$	4.83282	0.156060
$960$	0	0
$961$	40.7771	1.31539
$962$	0	0
$963$	10.9180	0.351826
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.8885	0.961151	0.480575	−	0.876953i	$-0.340428\pi$
$967$	29.8885	0.961151	0.480575	+	0.876953i	$0.340428\pi$
$968$	0	0
$969$	3.05573	0.0981641
$970$	0	0
$971$	−43.2492	−1.38793	−0.693967	−	0.720007i	$-0.744139\pi$
$971$	−43.2492	−1.38793	−0.693967	+	0.720007i	$0.744139\pi$
$972$	0	0
$973$	−4.72136	−0.151360
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44.5410	1.42499	0.712497	−	0.701675i	$-0.247564\pi$
$977$	44.5410	1.42499	0.712497	+	0.701675i	$0.247564\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	25.3738	0.810124
$982$	0	0
$983$	3.65248	0.116496	0.0582479	−	0.998302i	$-0.481449\pi$
$983$	3.65248	0.116496	0.0582479	+	0.998302i	$0.481449\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.498447	−0.0158657
$988$	0	0
$989$	−11.4721	−0.364793
$990$	0	0
$991$	7.59675	0.241319	0.120659	−	0.992694i	$-0.461499\pi$
$991$	7.59675	0.241319	0.120659	+	0.992694i	$0.461499\pi$
$992$	0	0
$993$	16.1378	0.512117
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−38.1246	−1.20742	−0.603709	−	0.797205i	$-0.706311\pi$
$997$	−38.1246	−1.20742	−0.603709	+	0.797205i	$0.706311\pi$
$998$	0	0
$999$	37.3901	1.18297

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.q.1.2	✓	2
4.3	odd	2		9200.2.a.bz.1.1		2
5.2	odd	4		4600.2.e.l.4049.2		4
5.3	odd	4		4600.2.e.l.4049.3		4
5.4	even	2		4600.2.a.u.1.1	yes	2
20.19	odd	2		9200.2.a.bn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.q.1.2	✓	2	1.1	even	1	trivial
4600.2.a.u.1.1	yes	2	5.4	even	2
4600.2.e.l.4049.2		4	5.2	odd	4
4600.2.e.l.4049.3		4	5.3	odd	4
9200.2.a.bn.1.2		2	20.19	odd	2
9200.2.a.bz.1.1		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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -0.236068 q^{7} -1.47214 q^{9} +1.00000 q^{11} -2.23607 q^{13} +2.47214 q^{17} +1.00000 q^{19} -0.291796 q^{21} +1.00000 q^{23} -5.52786 q^{27} -6.23607 q^{29} -8.47214 q^{31} +1.23607 q^{33} -6.76393 q^{37} -2.76393 q^{39} +11.9443 q^{41} -11.4721 q^{43} +1.70820 q^{47} -6.94427 q^{49} +3.05573 q^{51} +3.23607 q^{53} +1.23607 q^{57} -1.23607 q^{59} -2.76393 q^{61} +0.347524 q^{63} +4.94427 q^{67} +1.23607 q^{69} +10.0000 q^{71} +0.527864 q^{73} -0.236068 q^{77} -7.18034 q^{79} -2.41641 q^{81} -9.00000 q^{83} -7.70820 q^{87} +2.00000 q^{89} +0.527864 q^{91} -10.4721 q^{93} +16.1803 q^{97} -1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{7} + 6 q^{9} + 2 q^{11} - 4 q^{17} + 2 q^{19} - 14 q^{21} + 2 q^{23} - 20 q^{27} - 8 q^{29} - 8 q^{31} - 2 q^{33} - 18 q^{37} - 10 q^{39} + 6 q^{41} - 14 q^{43} - 10 q^{47} + 4 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^7 + 6 * q^9 + 2 * q^11 - 4 * q^17 + 2 * q^19 - 14 * q^21 + 2 * q^23 - 20 * q^27 - 8 * q^29 - 8 * q^31 - 2 * q^33 - 18 * q^37 - 10 * q^39 + 6 * q^41 - 14 * q^43 - 10 * q^47 + 4 * q^49 + 24 * q^51 + 2 * q^53 - 2 * q^57 + 2 * q^59 - 10 * q^61 + 32 * q^63 - 8 * q^67 - 2 * q^69 + 20 * q^71 + 10 * q^73 + 4 * q^77 + 8 * q^79 + 22 * q^81 - 18 * q^83 - 2 * q^87 + 4 * q^89 + 10 * q^91 - 12 * q^93 + 10 * q^97 + 6 * q^99

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