LMFDB - Embedded newform 4600.2.e.l.4049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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, 1, 0]))

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

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.l.4049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607i q^{3} -0.236068i q^{7} +1.47214 q^{9} +1.00000 q^{11} +2.23607i q^{13} +2.47214i q^{17} -1.00000 q^{19} -0.291796 q^{21} -1.00000i q^{23} -5.52786i q^{27} +6.23607 q^{29} -8.47214 q^{31} -1.23607i q^{33} -6.76393i q^{37} +2.76393 q^{39} +11.9443 q^{41} +11.4721i q^{43} +1.70820i q^{47} +6.94427 q^{49} +3.05573 q^{51} -3.23607i q^{53} +1.23607i q^{57} +1.23607 q^{59} -2.76393 q^{61} -0.347524i q^{63} +4.94427i q^{67} -1.23607 q^{69} +10.0000 q^{71} -0.527864i q^{73} -0.236068i q^{77} +7.18034 q^{79} -2.41641 q^{81} +9.00000i q^{83} -7.70820i q^{87} -2.00000 q^{89} +0.527864 q^{91} +10.4721i q^{93} +16.1803i q^{97} +1.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9} + 4 q^{11} - 4 q^{19} - 28 q^{21} + 16 q^{29} - 16 q^{31} + 20 q^{39} + 12 q^{41} - 8 q^{49} + 48 q^{51} - 4 q^{59} - 20 q^{61} + 4 q^{69} + 40 q^{71} - 16 q^{79} + 44 q^{81} - 8 q^{89} + 20 q^{91}+ \cdots - 12 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 + 4 * q^11 - 4 * q^19 - 28 * q^21 + 16 * q^29 - 16 * q^31 + 20 * q^39 + 12 * q^41 - 8 * q^49 + 48 * q^51 - 4 * q^59 - 20 * q^61 + 4 * q^69 + 40 * q^71 - 16 * q^79 + 44 * q^81 - 8 * q^89 + 20 * q^91 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.23607i	− 0.713644i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	− 1.23607i	− 0.713644i	0.934172	−	0.356822i	$-0.116140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.236068i	− 0.0892253i	−0.999004	−	0.0446127i	$-0.985795\pi$
$7$	− 0.236068i	− 0.0892253i	0.999004	−	0.0446127i	$-0.0142054\pi$
$8$	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
$13$	2.23607i	0.620174i	0.950708	+	0.310087i	$0.100358\pi$
$13$	2.23607i	0.620174i	−0.950708	+	0.310087i	$0.899642\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.47214i	0.599581i	0.954005	+	0.299791i	$0.0969168\pi$
$17$	2.47214i	0.599581i	−0.954005	+	0.299791i	$0.903083\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−0.291796	−0.0636751
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.52786i	− 1.06384i
$28$	0	0
$29$	6.23607	1.15801	0.579004	−	0.815324i	$-0.303441\pi$
$29$	6.23607	1.15801	0.579004	+	0.815324i	$0.303441\pi$
$30$	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
$33$	− 1.23607i	− 0.215172i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.76393i	− 1.11198i	−0.831188	−	0.555992i	$-0.812339\pi$
$37$	− 6.76393i	− 1.11198i	0.831188	−	0.555992i	$-0.187661\pi$
$38$	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
$43$	11.4721i	1.74948i	0.484589	+	0.874742i	$0.338969\pi$
$43$	11.4721i	1.74948i	−0.484589	+	0.874742i	$0.661031\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.70820i	0.249167i	0.992209	+	0.124584i	$0.0397595\pi$
$47$	1.70820i	0.249167i	−0.992209	+	0.124584i	$0.960241\pi$
$48$	0	0
$49$	6.94427	0.992039
$50$	0	0
$51$	3.05573	0.427888
$52$	0	0
$53$	− 3.23607i	− 0.444508i	−0.974989	−	0.222254i	$-0.928659\pi$
$53$	− 3.23607i	− 0.444508i	0.974989	−	0.222254i	$-0.0713414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.23607i	0.163721i
$58$	0	0
$59$	1.23607	0.160922	0.0804612	−	0.996758i	$-0.474361\pi$
$59$	1.23607	0.160922	0.0804612	+	0.996758i	$0.474361\pi$
$60$	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
$63$	− 0.347524i	− 0.0437839i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.94427i	0.604039i	0.953302	+	0.302019i	$0.0976608\pi$
$67$	4.94427i	0.604039i	−0.953302	+	0.302019i	$0.902339\pi$
$68$	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
$73$	− 0.527864i	− 0.0617818i	−0.999523	−	0.0308909i	$-0.990166\pi$
$73$	− 0.527864i	− 0.0617818i	0.999523	−	0.0308909i	$-0.00983445\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.236068i	− 0.0269024i
$78$	0	0
$79$	7.18034	0.807851	0.403926	−	0.914792i	$-0.367645\pi$
$79$	7.18034	0.807851	0.403926	+	0.914792i	$0.367645\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7.70820i	− 0.826406i
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0.527864	0.0553352
$92$	0	0
$93$	10.4721i	1.08591i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.1803i	1.64286i	0.570306	+	0.821432i	$0.306825\pi$
$97$	16.1803i	1.64286i	−0.570306	+	0.821432i	$0.693175\pi$
$98$	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
$103$	2.23607i	0.220326i	0.993914	+	0.110163i	$0.0351373\pi$
$103$	2.23607i	0.220326i	−0.993914	+	0.110163i	$0.964863\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.41641i	− 0.716971i	−0.933535	−	0.358486i	$-0.883293\pi$
$107$	− 7.41641i	− 0.716971i	0.933535	−	0.358486i	$-0.116707\pi$
$108$	0	0
$109$	17.2361	1.65092	0.825458	−	0.564464i	$-0.190917\pi$
$109$	17.2361	1.65092	0.825458	+	0.564464i	$0.190917\pi$
$110$	0	0
$111$	−8.36068	−0.793561
$112$	0	0
$113$	11.2361i	1.05700i	0.848933	+	0.528500i	$0.177245\pi$
$113$	11.2361i	1.05700i	−0.848933	+	0.528500i	$0.822755\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.29180i	0.304327i
$118$	0	0
$119$	0.583592	0.0534978
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	− 14.7639i	− 1.33122i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 18.1803i	− 1.61324i	−0.591067	−	0.806622i	$-0.701293\pi$
$127$	− 18.1803i	− 1.61324i	0.591067	−	0.806622i	$-0.298707\pi$
$128$	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
$133$	0.236068i	0.0204697i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 20.4721i	− 1.74905i	−0.484978	−	0.874526i	$-0.661172\pi$
$137$	− 20.4721i	− 1.74905i	0.484978	−	0.874526i	$-0.338828\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	2.11146	0.177817
$142$	0	0
$143$	2.23607i	0.186989i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 8.58359i	− 0.707963i
$148$	0	0
$149$	6.94427	0.568897	0.284448	−	0.958691i	$-0.408190\pi$
$149$	6.94427	0.568897	0.284448	+	0.958691i	$0.408190\pi$
$150$	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
$153$	3.63932i	0.294222i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 13.7082i	− 1.09403i	−0.837122	−	0.547017i	$-0.815763\pi$
$157$	− 13.7082i	− 1.09403i	0.837122	−	0.547017i	$-0.184237\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−0.236068	−0.0186048
$162$	0	0
$163$	1.70820i	0.133797i	0.997760	+	0.0668984i	$0.0213103\pi$
$163$	1.70820i	0.133797i	−0.997760	+	0.0668984i	$0.978690\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 21.4164i	− 1.65725i	−0.559803	−	0.828626i	$-0.689123\pi$
$167$	− 21.4164i	− 1.65725i	0.559803	−	0.828626i	$-0.310877\pi$
$168$	0	0
$169$	8.00000	0.615385
$170$	0	0
$171$	−1.47214	−0.112577
$172$	0	0
$173$	9.76393i	0.742338i	0.928565	+	0.371169i	$0.121043\pi$
$173$	9.76393i	0.742338i	−0.928565	+	0.371169i	$0.878957\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.52786i	− 0.114841i
$178$	0	0
$179$	2.29180	0.171297	0.0856484	−	0.996325i	$-0.472704\pi$
$179$	2.29180	0.171297	0.0856484	+	0.996325i	$0.472704\pi$
$180$	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
$183$	3.41641i	0.252548i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.47214i	0.180780i
$188$	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
$193$	− 14.9443i	− 1.07571i	−0.843037	−	0.537856i	$-0.819234\pi$
$193$	− 14.9443i	− 1.07571i	0.843037	−	0.537856i	$-0.180766\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.76393i	0.268169i	0.990970	+	0.134085i	$0.0428094\pi$
$197$	3.76393i	0.268169i	−0.990970	+	0.134085i	$0.957191\pi$
$198$	0	0
$199$	0.708204	0.0502032	0.0251016	−	0.999685i	$-0.492009\pi$
$199$	0.708204	0.0502032	0.0251016	+	0.999685i	$0.492009\pi$
$200$	0	0
$201$	6.11146	0.431069
$202$	0	0
$203$	− 1.47214i	− 0.103324i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.47214i	− 0.102321i
$208$	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
$213$	− 12.3607i	− 0.846940i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.00000i	0.135769i
$218$	0	0
$219$	−0.652476	−0.0440902
$220$	0	0
$221$	−5.52786	−0.371844
$222$	0	0
$223$	8.76393i	0.586876i	0.955978	+	0.293438i	$0.0947995\pi$
$223$	8.76393i	0.586876i	−0.955978	+	0.293438i	$0.905201\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.94427i	0.328163i	0.986447	+	0.164081i	$0.0524660\pi$
$227$	4.94427i	0.328163i	−0.986447	+	0.164081i	$0.947534\pi$
$228$	0	0
$229$	10.9443	0.723218	0.361609	−	0.932330i	$-0.382228\pi$
$229$	10.9443	0.723218	0.361609	+	0.932330i	$0.382228\pi$
$230$	0	0
$231$	−0.291796	−0.0191988
$232$	0	0
$233$	13.4721i	0.882589i	0.897362	+	0.441294i	$0.145481\pi$
$233$	13.4721i	0.882589i	−0.897362	+	0.441294i	$0.854519\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.87539i	− 0.576518i
$238$	0	0
$239$	−11.2361	−0.726801	−0.363400	−	0.931633i	$-0.618384\pi$
$239$	−11.2361	−0.726801	−0.363400	+	0.931633i	$0.618384\pi$
$240$	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
$243$	− 13.5967i	− 0.872232i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.23607i	− 0.142278i
$248$	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
$253$	− 1.00000i	− 0.0628695i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	0	0
$259$	−1.59675	−0.0992171
$260$	0	0
$261$	9.18034	0.568249
$262$	0	0
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.47214i	0.151292i
$268$	0	0
$269$	2.81966	0.171918	0.0859589	−	0.996299i	$-0.472605\pi$
$269$	2.81966	0.171918	0.0859589	+	0.996299i	$0.472605\pi$
$270$	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
$273$	− 0.652476i	− 0.0394896i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 9.76393i	− 0.586658i	−0.956012	−	0.293329i	$-0.905237\pi$
$277$	− 9.76393i	− 0.586658i	0.956012	−	0.293329i	$-0.0947631\pi$
$278$	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
$283$	− 16.3607i	− 0.972541i	−0.873808	−	0.486271i	$-0.838357\pi$
$283$	− 16.3607i	− 0.972541i	0.873808	−	0.486271i	$-0.161643\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.81966i	− 0.166439i
$288$	0	0
$289$	10.8885	0.640503
$290$	0	0
$291$	20.0000	1.17242
$292$	0	0
$293$	− 5.52786i	− 0.322941i	−0.986878	−	0.161471i	$-0.948376\pi$
$293$	− 5.52786i	− 0.322941i	0.986878	−	0.161471i	$-0.0516237\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.52786i	− 0.320759i
$298$	0	0
$299$	2.23607	0.129315
$300$	0	0
$301$	2.70820	0.156098
$302$	0	0
$303$	1.30495i	0.0749675i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 24.0000i	− 1.36975i	−0.728659	−	0.684876i	$-0.759856\pi$
$307$	− 24.0000i	− 1.36975i	0.728659	−	0.684876i	$-0.240144\pi$
$308$	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
$313$	20.6525i	1.16735i	0.811988	+	0.583673i	$0.198385\pi$
$313$	20.6525i	1.16735i	−0.811988	+	0.583673i	$0.801615\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 18.1246i	− 1.01798i	−0.860773	−	0.508990i	$-0.830019\pi$
$317$	− 18.1246i	− 1.01798i	0.860773	−	0.508990i	$-0.169981\pi$
$318$	0	0
$319$	6.23607	0.349153
$320$	0	0
$321$	−9.16718	−0.511662
$322$	0	0
$323$	− 2.47214i	− 0.137553i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 21.3050i	− 1.17817i
$328$	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
$333$	− 9.95743i	− 0.545664i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 6.65248i	− 0.362383i	−0.983448	−	0.181192i	$-0.942005\pi$
$337$	− 6.65248i	− 0.362383i	0.983448	−	0.181192i	$-0.0579955\pi$
$338$	0	0
$339$	13.8885	0.754322
$340$	0	0
$341$	−8.47214	−0.458792
$342$	0	0
$343$	− 3.29180i	− 0.177740i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.9443i	− 1.12435i	−0.827019	−	0.562174i	$-0.809965\pi$
$347$	− 20.9443i	− 1.12435i	0.827019	−	0.562174i	$-0.190035\pi$
$348$	0	0
$349$	−17.1803	−0.919643	−0.459821	−	0.888011i	$-0.652087\pi$
$349$	−17.1803	−0.919643	−0.459821	+	0.888011i	$0.652087\pi$
$350$	0	0
$351$	12.3607	0.659764
$352$	0	0
$353$	30.8885i	1.64403i	0.569465	+	0.822016i	$0.307150\pi$
$353$	30.8885i	1.64403i	−0.569465	+	0.822016i	$0.692850\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 0.721360i	− 0.0381784i
$358$	0	0
$359$	−24.2361	−1.27913	−0.639565	−	0.768737i	$-0.720886\pi$
$359$	−24.2361	−1.27913	−0.639565	+	0.768737i	$0.720886\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	12.3607i	0.648767i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7.65248i	− 0.399456i	−0.979851	−	0.199728i	$-0.935994\pi$
$367$	− 7.65248i	− 0.399456i	0.979851	−	0.199728i	$-0.0640059\pi$
$368$	0	0
$369$	17.5836	0.915365
$370$	0	0
$371$	−0.763932	−0.0396614
$372$	0	0
$373$	− 12.0000i	− 0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$	− 12.0000i	− 0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.9443i	0.718167i
$378$	0	0
$379$	−20.9443	−1.07583	−0.537917	−	0.842997i	$-0.680789\pi$
$379$	−20.9443	−1.07583	−0.537917	+	0.842997i	$0.680789\pi$
$380$	0	0
$381$	−22.4721	−1.15128
$382$	0	0
$383$	4.23607i	0.216453i	0.994126	+	0.108226i	$0.0345172\pi$
$383$	4.23607i	0.216453i	−0.994126	+	0.108226i	$0.965483\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.8885i	0.858493i
$388$	0	0
$389$	16.1803	0.820376	0.410188	−	0.912001i	$-0.365463\pi$
$389$	16.1803	0.820376	0.410188	+	0.912001i	$0.365463\pi$
$390$	0	0
$391$	2.47214	0.125021
$392$	0	0
$393$	− 3.63932i	− 0.183579i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.3607i	1.12225i	0.827731	+	0.561125i	$0.189631\pi$
$397$	22.3607i	1.12225i	−0.827731	+	0.561125i	$0.810369\pi$
$398$	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
$403$	− 18.9443i	− 0.943681i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.76393i	− 0.335276i
$408$	0	0
$409$	−0.416408	−0.0205900	−0.0102950	−	0.999947i	$-0.503277\pi$
$409$	−0.416408	−0.0205900	−0.0102950	+	0.999947i	$0.503277\pi$
$410$	0	0
$411$	−25.3050	−1.24820
$412$	0	0
$413$	− 0.291796i	− 0.0143583i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	24.7214i	1.21061i
$418$	0	0
$419$	−34.7771	−1.69897	−0.849486	−	0.527611i	$-0.823088\pi$
$419$	−34.7771	−1.69897	−0.849486	+	0.527611i	$0.823088\pi$
$420$	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
$423$	2.51471i	0.122269i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.652476i	0.0315755i
$428$	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
$433$	11.1246i	0.534615i	0.963611	+	0.267307i	$0.0861339\pi$
$433$	11.1246i	0.534615i	−0.963611	+	0.267307i	$0.913866\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.00000i	0.0478365i
$438$	0	0
$439$	32.3607	1.54449	0.772245	−	0.635324i	$-0.219134\pi$
$439$	32.3607	1.54449	0.772245	+	0.635324i	$0.219134\pi$
$440$	0	0
$441$	10.2229	0.486805
$442$	0	0
$443$	− 10.0000i	− 0.475114i	−0.971374	−	0.237557i	$-0.923653\pi$
$443$	− 10.0000i	− 0.475114i	0.971374	−	0.237557i	$-0.0763467\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.58359i	− 0.405990i
$448$	0	0
$449$	−16.8328	−0.794390	−0.397195	−	0.917734i	$-0.630016\pi$
$449$	−16.8328	−0.794390	−0.397195	+	0.917734i	$0.630016\pi$
$450$	0	0
$451$	11.9443	0.562434
$452$	0	0
$453$	− 25.3050i	− 1.18893i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.6525i	− 0.498302i	−0.968465	−	0.249151i	$-0.919848\pi$
$457$	− 10.6525i	− 0.498302i	0.968465	−	0.249151i	$-0.0801515\pi$
$458$	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
$463$	− 17.5279i	− 0.814589i	−0.913297	−	0.407294i	$-0.866472\pi$
$463$	− 17.5279i	− 0.814589i	0.913297	−	0.407294i	$-0.133528\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.0000i	0.786666i	0.919396	+	0.393333i	$0.128678\pi$
$467$	17.0000i	0.786666i	−0.919396	+	0.393333i	$0.871322\pi$
$468$	0	0
$469$	1.16718	0.0538956
$470$	0	0
$471$	−16.9443	−0.780751
$472$	0	0
$473$	11.4721i	0.527489i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.76393i	− 0.218125i
$478$	0	0
$479$	−28.1246	−1.28505	−0.642523	−	0.766266i	$-0.722112\pi$
$479$	−28.1246	−1.28505	−0.642523	+	0.766266i	$0.722112\pi$
$480$	0	0
$481$	15.1246	0.689623
$482$	0	0
$483$	0.291796i	0.0132772i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.12461i	− 0.0509610i	−0.999675	−	0.0254805i	$-0.991888\pi$
$487$	− 1.12461i	− 0.0509610i	0.999675	−	0.0254805i	$-0.00811158\pi$
$488$	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
$493$	15.4164i	0.694320i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.36068i	− 0.105891i
$498$	0	0
$499$	−24.5410	−1.09861	−0.549303	−	0.835623i	$-0.685107\pi$
$499$	−24.5410	−1.09861	−0.549303	+	0.835623i	$0.685107\pi$
$500$	0	0
$501$	−26.4721	−1.18269
$502$	0	0
$503$	− 32.5967i	− 1.45342i	−0.686946	−	0.726709i	$-0.741049\pi$
$503$	− 32.5967i	− 1.45342i	0.686946	−	0.726709i	$-0.258951\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.88854i	− 0.439166i
$508$	0	0
$509$	14.3607	0.636526	0.318263	−	0.948002i	$-0.396901\pi$
$509$	14.3607	0.636526	0.318263	+	0.948002i	$0.396901\pi$
$510$	0	0
$511$	−0.124612	−0.00551250
$512$	0	0
$513$	5.52786i	0.244061i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.70820i	0.0751267i
$518$	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
$523$	16.8885i	0.738484i	0.929333	+	0.369242i	$0.120383\pi$
$523$	16.8885i	0.738484i	−0.929333	+	0.369242i	$0.879617\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 20.9443i	− 0.912347i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	1.81966	0.0789665
$532$	0	0
$533$	26.7082i	1.15686i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.83282i	− 0.122245i
$538$	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
$543$	− 20.9443i	− 0.898805i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.29180i	0.0979901i	0.998799	+	0.0489951i	$0.0156019\pi$
$547$	2.29180i	0.0979901i	−0.998799	+	0.0489951i	$0.984398\pi$
$548$	0	0
$549$	−4.06888	−0.173656
$550$	0	0
$551$	−6.23607	−0.265665
$552$	0	0
$553$	− 1.69505i	− 0.0720808i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 13.8197i	− 0.585558i	−0.956180	−	0.292779i	$-0.905420\pi$
$557$	− 13.8197i	− 0.585558i	0.956180	−	0.292779i	$-0.0945800\pi$
$558$	0	0
$559$	−25.6525	−1.08498
$560$	0	0
$561$	3.05573	0.129013
$562$	0	0
$563$	5.36068i	0.225926i	0.993599	+	0.112963i	$0.0360341\pi$
$563$	5.36068i	0.225926i	−0.993599	+	0.112963i	$0.963966\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.570437i	0.0239561i
$568$	0	0
$569$	−39.4164	−1.65242	−0.826211	−	0.563361i	$-0.809508\pi$
$569$	−39.4164	−1.65242	−0.826211	+	0.563361i	$0.809508\pi$
$570$	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
$573$	− 3.93112i	− 0.164225i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 9.00000i	− 0.374675i	−0.982296	−	0.187337i	$-0.940014\pi$
$577$	− 9.00000i	− 0.374675i	0.982296	−	0.187337i	$-0.0599858\pi$
$578$	0	0
$579$	−18.4721	−0.767676
$580$	0	0
$581$	2.12461	0.0881437
$582$	0	0
$583$	− 3.23607i	− 0.134024i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.5967i	0.891393i	0.895184	+	0.445697i	$0.147044\pi$
$587$	21.5967i	0.891393i	−0.895184	+	0.445697i	$0.852956\pi$
$588$	0	0
$589$	8.47214	0.349088
$590$	0	0
$591$	4.65248	0.191377
$592$	0	0
$593$	9.11146i	0.374163i	0.982344	+	0.187081i	$0.0599028\pi$
$593$	9.11146i	0.374163i	−0.982344	+	0.187081i	$0.940097\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 0.875388i	− 0.0358273i
$598$	0	0
$599$	36.8328	1.50495	0.752474	−	0.658622i	$-0.228860\pi$
$599$	36.8328	1.50495	0.752474	+	0.658622i	$0.228860\pi$
$600$	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
$603$	7.27864i	0.296409i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.1246i	0.695067i	0.937668	+	0.347533i	$0.112981\pi$
$607$	17.1246i	0.695067i	−0.937668	+	0.347533i	$0.887019\pi$
$608$	0	0
$609$	−1.81966	−0.0737363
$610$	0	0
$611$	−3.81966	−0.154527
$612$	0	0
$613$	40.0689i	1.61837i	0.587556	+	0.809183i	$0.300090\pi$
$613$	40.0689i	1.61837i	−0.587556	+	0.809183i	$0.699910\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.3050i	1.50184i	0.660393	+	0.750920i	$0.270390\pi$
$617$	37.3050i	1.50184i	−0.660393	+	0.750920i	$0.729610\pi$
$618$	0	0
$619$	18.4721	0.742458	0.371229	−	0.928541i	$-0.378937\pi$
$619$	18.4721	0.742458	0.371229	+	0.928541i	$0.378937\pi$
$620$	0	0
$621$	−5.52786	−0.221826
$622$	0	0
$623$	0.472136i	0.0189157i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.23607i	0.0493638i
$628$	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
$633$	− 3.63932i	− 0.144650i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15.5279i	0.615236i
$638$	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
$643$	− 15.9443i	− 0.628781i	−0.949294	−	0.314390i	$-0.898200\pi$
$643$	− 15.9443i	− 0.628781i	0.949294	−	0.314390i	$-0.101800\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.1803i	0.478859i	0.970914	+	0.239429i	$0.0769603\pi$
$647$	12.1803i	0.478859i	−0.970914	+	0.239429i	$0.923040\pi$
$648$	0	0
$649$	1.23607	0.0485199
$650$	0	0
$651$	2.47214	0.0968906
$652$	0	0
$653$	27.5410i	1.07776i	0.842381	+	0.538882i	$0.181153\pi$
$653$	27.5410i	1.07776i	−0.842381	+	0.538882i	$0.818847\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 0.777088i	− 0.0303171i
$658$	0	0
$659$	−4.52786	−0.176381	−0.0881903	−	0.996104i	$-0.528108\pi$
$659$	−4.52786	−0.176381	−0.0881903	+	0.996104i	$0.528108\pi$
$660$	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
$663$	6.83282i	0.265365i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6.23607i	− 0.241462i
$668$	0	0
$669$	10.8328	0.418821
$670$	0	0
$671$	−2.76393	−0.106700
$672$	0	0
$673$	− 14.3050i	− 0.551415i	−0.961242	−	0.275708i	$-0.911088\pi$
$673$	− 14.3050i	− 0.551415i	0.961242	−	0.275708i	$-0.0889122\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.29180i	0.164947i	0.996593	+	0.0824736i	$0.0262820\pi$
$677$	4.29180i	0.164947i	−0.996593	+	0.0824736i	$0.973718\pi$
$678$	0	0
$679$	3.81966	0.146585
$680$	0	0
$681$	6.11146	0.234192
$682$	0	0
$683$	− 29.7771i	− 1.13939i	−0.821857	−	0.569694i	$-0.807062\pi$
$683$	− 29.7771i	− 1.13939i	0.821857	−	0.569694i	$-0.192938\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 13.5279i	− 0.516120i
$688$	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
$693$	− 0.347524i	− 0.0132014i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	29.5279i	1.11845i
$698$	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
$703$	6.76393i	0.255107i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.249224i	0.00937302i
$708$	0	0
$709$	17.4164	0.654087	0.327043	−	0.945009i	$-0.393948\pi$
$709$	17.4164	0.654087	0.327043	+	0.945009i	$0.393948\pi$
$710$	0	0
$711$	10.5704	0.396422
$712$	0	0
$713$	8.47214i	0.317284i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.8885i	0.518677i
$718$	0	0
$719$	−24.2918	−0.905931	−0.452966	−	0.891528i	$-0.649634\pi$
$719$	−24.2918	−0.905931	−0.452966	+	0.891528i	$0.649634\pi$
$720$	0	0
$721$	0.527864	0.0196587
$722$	0	0
$723$	− 20.7214i	− 0.770636i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	44.7214i	1.65862i	0.558786	+	0.829312i	$0.311267\pi$
$727$	44.7214i	1.65862i	−0.558786	+	0.829312i	$0.688733\pi$
$728$	0	0
$729$	−24.0557	−0.890953
$730$	0	0
$731$	−28.3607	−1.04896
$732$	0	0
$733$	− 48.4721i	− 1.79036i	−0.445706	−	0.895180i	$-0.647047\pi$
$733$	− 48.4721i	− 1.79036i	0.445706	−	0.895180i	$-0.352953\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.94427i	0.182125i
$738$	0	0
$739$	−26.9443	−0.991161	−0.495581	−	0.868562i	$-0.665045\pi$
$739$	−26.9443	−0.991161	−0.495581	+	0.868562i	$0.665045\pi$
$740$	0	0
$741$	−2.76393	−0.101536
$742$	0	0
$743$	16.8197i	0.617053i	0.951216	+	0.308527i	$0.0998359\pi$
$743$	16.8197i	0.617053i	−0.951216	+	0.308527i	$0.900164\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.2492i	0.484764i
$748$	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
$753$	17.8885i	0.651895i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21.2361i	0.771838i	0.922533	+	0.385919i	$0.126116\pi$
$757$	21.2361i	0.771838i	−0.922533	+	0.385919i	$0.873884\pi$
$758$	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
$763$	− 4.06888i	− 0.147303i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.76393i	0.0997998i
$768$	0	0
$769$	−12.0689	−0.435215	−0.217608	−	0.976036i	$-0.569825\pi$
$769$	−12.0689	−0.435215	−0.217608	+	0.976036i	$0.569825\pi$
$770$	0	0
$771$	17.3050	0.623223
$772$	0	0
$773$	− 9.23607i	− 0.332198i	−0.986109	−	0.166099i	$-0.946883\pi$
$773$	− 9.23607i	− 0.332198i	0.986109	−	0.166099i	$-0.0531172\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.97369i	0.0708057i
$778$	0	0
$779$	−11.9443	−0.427948
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	− 34.4721i	− 1.23193i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	49.7214i	1.77238i	0.463327	+	0.886188i	$0.346656\pi$
$787$	49.7214i	1.77238i	−0.463327	+	0.886188i	$0.653344\pi$
$788$	0	0
$789$	9.88854	0.352041
$790$	0	0
$791$	2.65248	0.0943112
$792$	0	0
$793$	− 6.18034i	− 0.219470i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 10.7639i	− 0.381278i	−0.981660	−	0.190639i	$-0.938944\pi$
$797$	− 10.7639i	− 0.381278i	0.981660	−	0.190639i	$-0.0610560\pi$
$798$	0	0
$799$	−4.22291	−0.149396
$800$	0	0
$801$	−2.94427	−0.104031
$802$	0	0
$803$	− 0.527864i	− 0.0186279i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3.48529i	− 0.122688i
$808$	0	0
$809$	44.8885	1.57820	0.789099	−	0.614267i	$-0.210548\pi$
$809$	44.8885	1.57820	0.789099	+	0.614267i	$0.210548\pi$
$810$	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
$813$	− 32.2229i	− 1.13011i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11.4721i	− 0.401359i
$818$	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
$823$	− 8.94427i	− 0.311778i	−0.987775	−	0.155889i	$-0.950176\pi$
$823$	− 8.94427i	− 0.311778i	0.987775	−	0.155889i	$-0.0498242\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.00000i	− 0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$	− 3.00000i	− 0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$	0	0
$829$	−0.236068	−0.00819898	−0.00409949	−	0.999992i	$-0.501305\pi$
$829$	−0.236068	−0.00819898	−0.00409949	+	0.999992i	$0.501305\pi$
$830$	0	0
$831$	−12.0689	−0.418665
$832$	0	0
$833$	17.1672i	0.594808i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	46.8328i	1.61878i
$838$	0	0
$839$	40.5967	1.40156	0.700778	−	0.713380i	$-0.252836\pi$
$839$	40.5967	1.40156	0.700778	+	0.713380i	$0.252836\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	0	0
$843$	− 0.222912i	− 0.00767751i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.36068i	0.0811139i
$848$	0	0
$849$	−20.2229	−0.694049
$850$	0	0
$851$	−6.76393	−0.231865
$852$	0	0
$853$	23.7639i	0.813662i	0.913504	+	0.406831i	$0.133366\pi$
$853$	23.7639i	0.813662i	−0.913504	+	0.406831i	$0.866634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 27.3050i	− 0.932719i	−0.884595	−	0.466360i	$-0.845565\pi$
$857$	− 27.3050i	− 0.932719i	0.884595	−	0.466360i	$-0.154435\pi$
$858$	0	0
$859$	8.29180	0.282912	0.141456	−	0.989945i	$-0.454822\pi$
$859$	8.29180	0.282912	0.141456	+	0.989945i	$0.454822\pi$
$860$	0	0
$861$	−3.48529	−0.118778
$862$	0	0
$863$	32.5410i	1.10771i	0.832613	+	0.553855i	$0.186844\pi$
$863$	32.5410i	1.10771i	−0.832613	+	0.553855i	$0.813156\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 13.4590i	− 0.457091i
$868$	0	0
$869$	7.18034	0.243576
$870$	0	0
$871$	−11.0557	−0.374609
$872$	0	0
$873$	23.8197i	0.806173i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.4164i	0.993322i	0.867945	+	0.496661i	$0.165441\pi$
$877$	29.4164i	0.993322i	−0.867945	+	0.496661i	$0.834559\pi$
$878$	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
$883$	− 20.9443i	− 0.704831i	−0.935844	−	0.352415i	$-0.885360\pi$
$883$	− 20.9443i	− 0.704831i	0.935844	−	0.352415i	$-0.114640\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 32.3607i	− 1.08657i	−0.839550	−	0.543283i	$-0.817181\pi$
$887$	− 32.3607i	− 1.08657i	0.839550	−	0.543283i	$-0.182819\pi$
$888$	0	0
$889$	−4.29180	−0.143942
$890$	0	0
$891$	−2.41641	−0.0809527
$892$	0	0
$893$	− 1.70820i	− 0.0571629i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.76393i	− 0.0922850i
$898$	0	0
$899$	−52.8328	−1.76207
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	− 3.34752i	− 0.111399i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.4164i	1.07637i	0.842827	+	0.538185i	$0.180890\pi$
$907$	32.4164i	1.07637i	−0.842827	+	0.538185i	$0.819110\pi$
$908$	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
$913$	9.00000i	0.297857i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.695048i	− 0.0229525i
$918$	0	0
$919$	27.7771	0.916282	0.458141	−	0.888880i	$-0.348516\pi$
$919$	27.7771	0.916282	0.458141	+	0.888880i	$0.348516\pi$
$920$	0	0
$921$	−29.6656	−0.977516
$922$	0	0
$923$	22.3607i	0.736011i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.29180i	0.108117i
$928$	0	0
$929$	−31.0000	−1.01708	−0.508539	−	0.861039i	$-0.669814\pi$
$929$	−31.0000	−1.01708	−0.508539	+	0.861039i	$0.669814\pi$
$930$	0	0
$931$	−6.94427	−0.227589
$932$	0	0
$933$	− 1.52786i	− 0.0500200i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33.1246i	1.08213i	0.840980	+	0.541067i	$0.181979\pi$
$937$	33.1246i	1.08213i	−0.840980	+	0.541067i	$0.818021\pi$
$938$	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
$943$	− 11.9443i	− 0.388959i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	53.1935i	1.72856i	0.503014	+	0.864278i	$0.332224\pi$
$947$	53.1935i	1.72856i	−0.503014	+	0.864278i	$0.667776\pi$
$948$	0	0
$949$	1.18034	0.0383155
$950$	0	0
$951$	−22.4033	−0.726475
$952$	0	0
$953$	36.6525i	1.18729i	0.804727	+	0.593645i	$0.202312\pi$
$953$	36.6525i	1.18729i	−0.804727	+	0.593645i	$0.797688\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 7.70820i	− 0.249171i
$958$	0	0
$959$	−4.83282	−0.156060
$960$	0	0
$961$	40.7771	1.31539
$962$	0	0
$963$	− 10.9180i	− 0.351826i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.8885i	0.961151i	0.876953	+	0.480575i	$0.159572\pi$
$967$	29.8885i	0.961151i	−0.876953	+	0.480575i	$0.840428\pi$
$968$	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
$973$	4.72136i	0.151360i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44.5410i	1.42499i	0.701675	+	0.712497i	$0.252436\pi$
$977$	44.5410i	1.42499i	−0.701675	+	0.712497i	$0.747564\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	25.3738	0.810124
$982$	0	0
$983$	− 3.65248i	− 0.116496i	−0.998302	−	0.0582479i	$-0.981449\pi$
$983$	− 3.65248i	− 0.116496i	0.998302	−	0.0582479i	$-0.0185514\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.498447i	− 0.0158657i
$988$	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
$993$	− 16.1378i	− 0.512117i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.1246i	− 1.20742i	−0.797205	−	0.603709i	$-0.793689\pi$
$997$	− 38.1246i	− 1.20742i	0.797205	−	0.603709i	$-0.206311\pi$
$998$	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.e.l.4049.2		4
5.2	odd	4		4600.2.a.u.1.1	yes	2
5.3	odd	4		4600.2.a.q.1.2	✓	2
5.4	even	2	inner	4600.2.e.l.4049.3		4
20.3	even	4		9200.2.a.bz.1.1		2
20.7	even	4		9200.2.a.bn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.q.1.2	✓	2	5.3	odd	4
4600.2.a.u.1.1	yes	2	5.2	odd	4
4600.2.e.l.4049.2		4	1.1	even	1	trivial
4600.2.e.l.4049.3		4	5.4	even	2	inner
9200.2.a.bn.1.2		2	20.7	even	4
9200.2.a.bz.1.1		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.23607i q^{3} -0.236068i q^{7} +1.47214 q^{9} +1.00000 q^{11} +2.23607i q^{13} +2.47214i q^{17} -1.00000 q^{19} -0.291796 q^{21} -1.00000i q^{23} -5.52786i q^{27} +6.23607 q^{29} -8.47214 q^{31} -1.23607i q^{33} -6.76393i q^{37} +2.76393 q^{39} +11.9443 q^{41} +11.4721i q^{43} +1.70820i q^{47} +6.94427 q^{49} +3.05573 q^{51} -3.23607i q^{53} +1.23607i q^{57} +1.23607 q^{59} -2.76393 q^{61} -0.347524i q^{63} +4.94427i q^{67} -1.23607 q^{69} +10.0000 q^{71} -0.527864i q^{73} -0.236068i q^{77} +7.18034 q^{79} -2.41641 q^{81} +9.00000i q^{83} -7.70820i q^{87} -2.00000 q^{89} +0.527864 q^{91} +10.4721i q^{93} +16.1803i q^{97} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9} + 4 q^{11} - 4 q^{19} - 28 q^{21} + 16 q^{29} - 16 q^{31} + 20 q^{39} + 12 q^{41} - 8 q^{49} + 48 q^{51} - 4 q^{59} - 20 q^{61} + 4 q^{69} + 40 q^{71} - 16 q^{79} + 44 q^{81} - 8 q^{89} + 20 q^{91}+ \cdots - 12 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 + 4 * q^11 - 4 * q^19 - 28 * q^21 + 16 * q^29 - 16 * q^31 + 20 * q^39 + 12 * q^41 - 8 * q^49 + 48 * q^51 - 4 * q^59 - 20 * q^61 + 4 * q^69 + 40 * q^71 - 16 * q^79 + 44 * q^81 - 8 * q^89 + 20 * q^91 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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