LMFDB - Embedded newform 4004.2.m.c.2157.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(2157,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("4004.2157");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.m (of order $2$, degree $1$, 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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.23
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.810205 q^{3} +3.82551i q^{5} -1.00000i q^{7} -2.34357 q^{9} +O(q^{10})$	$q+0.810205 q^{3} +3.82551i q^{5} -1.00000i q^{7} -2.34357 q^{9} +1.00000i q^{11} +(3.12587 - 1.79692i) q^{13} +3.09945i q^{15} +1.91054 q^{17} -2.75048i q^{19} -0.810205i q^{21} -7.88826 q^{23} -9.63452 q^{25} -4.32939 q^{27} -8.14739 q^{29} +6.00616i q^{31} +0.810205i q^{33} +3.82551 q^{35} +5.83504i q^{37} +(2.53260 - 1.45587i) q^{39} -5.43959i q^{41} -5.59610 q^{43} -8.96534i q^{45} -12.1911i q^{47} -1.00000 q^{49} +1.54793 q^{51} -7.25361 q^{53} -3.82551 q^{55} -2.22845i q^{57} +10.5590i q^{59} +10.1797 q^{61} +2.34357i q^{63} +(6.87414 + 11.9581i) q^{65} +1.53230i q^{67} -6.39111 q^{69} +10.9857i q^{71} -9.21307i q^{73} -7.80594 q^{75} +1.00000 q^{77} -6.37576 q^{79} +3.52301 q^{81} +4.00995i q^{83} +7.30879i q^{85} -6.60106 q^{87} -8.61636i q^{89} +(-1.79692 - 3.12587i) q^{91} +4.86622i q^{93} +10.5220 q^{95} +1.03847i q^{97} -2.34357i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) $ 36 * q - 4 * q^3 + 40 * q^9	$ 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) $ 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	0.810205			0.467772			0.233886	−	0.972264i	$-0.424856\pi$
$3$	0.810205			0.467772			0.233886	+	0.972264i	$0.424856\pi$
$4$		0			0
$5$			3.82551i			1.71082i	0.517952	+	0.855410i	$0.326695\pi$
$5$			3.82551i			1.71082i	−0.517952	+	0.855410i	$0.673305\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−2.34357			−0.781189
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	3.12587	−	1.79692i	0.866961	−	0.498376i
$13$	3.12587	−	1.79692i	0.866961	−	0.498376i
$14$		0			0
$15$			3.09945i			0.800274i
$16$		0			0
$17$	1.91054			0.463374			0.231687	−	0.972790i	$-0.425575\pi$
$17$	1.91054			0.463374			0.231687	+	0.972790i	$0.425575\pi$
$18$		0			0
$19$		−	2.75048i		−	0.631003i	−0.948925	−	0.315502i	$-0.897827\pi$
$19$		−	2.75048i		−	0.631003i	0.948925	−	0.315502i	$-0.102173\pi$
$20$		0			0
$21$		−	0.810205i		−	0.176801i
$22$		0			0
$23$	−7.88826			−1.64482			−0.822408	−	0.568898i	$-0.807370\pi$
$23$	−7.88826			−1.64482			−0.822408	+	0.568898i	$0.807370\pi$
$24$		0			0
$25$	−9.63452			−1.92690
$26$		0			0
$27$	−4.32939			−0.833191
$28$		0			0
$29$	−8.14739			−1.51293			−0.756467	−	0.654032i	$-0.773076\pi$
$29$	−8.14739			−1.51293			−0.756467	+	0.654032i	$0.773076\pi$
$30$		0			0
$31$			6.00616i			1.07874i	0.842070	+	0.539369i	$0.181337\pi$
$31$			6.00616i			1.07874i	−0.842070	+	0.539369i	$0.818663\pi$
$32$		0			0
$33$			0.810205i			0.141039i
$34$		0			0
$35$	3.82551			0.646629
$36$		0			0
$37$			5.83504i			0.959274i	0.877467	+	0.479637i	$0.159232\pi$
$37$			5.83504i			0.959274i	−0.877467	+	0.479637i	$0.840768\pi$
$38$		0			0
$39$	2.53260	−	1.45587i	0.405540	−	0.233126i
$40$		0			0
$41$		−	5.43959i		−	0.849521i	−0.905306	−	0.424761i	$-0.860358\pi$
$41$		−	5.43959i		−	0.849521i	0.905306	−	0.424761i	$-0.139642\pi$
$42$		0			0
$43$	−5.59610			−0.853397			−0.426699	−	0.904394i	$-0.640324\pi$
$43$	−5.59610			−0.853397			−0.426699	+	0.904394i	$0.640324\pi$
$44$		0			0
$45$		−	8.96534i		−	1.33647i
$46$		0			0
$47$		−	12.1911i		−	1.77825i	−0.457661	−	0.889127i	$-0.651313\pi$
$47$		−	12.1911i		−	1.77825i	0.457661	−	0.889127i	$-0.348687\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.54793			0.216754
$52$		0			0
$53$	−7.25361			−0.996361			−0.498180	−	0.867073i	$-0.665998\pi$
$53$	−7.25361			−0.996361			−0.498180	+	0.867073i	$0.665998\pi$
$54$		0			0
$55$	−3.82551			−0.515832
$56$		0			0
$57$		−	2.22845i		−	0.295166i
$58$		0			0
$59$			10.5590i			1.37466i	0.726345	+	0.687330i	$0.241217\pi$
$59$			10.5590i			1.37466i	−0.726345	+	0.687330i	$0.758783\pi$
$60$		0			0
$61$	10.1797			1.30338			0.651691	−	0.758484i	$-0.274060\pi$
$61$	10.1797			1.30338			0.651691	+	0.758484i	$0.274060\pi$
$62$		0			0
$63$			2.34357i			0.295262i
$64$		0			0
$65$	6.87414	+	11.9581i	0.852632	+	1.48321i
$66$		0			0
$67$			1.53230i			0.187200i	0.995610	+	0.0936000i	$0.0298375\pi$
$67$			1.53230i			0.187200i	−0.995610	+	0.0936000i	$0.970163\pi$
$68$		0			0
$69$	−6.39111			−0.769399
$70$		0			0
$71$			10.9857i			1.30377i	0.758320	+	0.651883i	$0.226021\pi$
$71$			10.9857i			1.30377i	−0.758320	+	0.651883i	$0.773979\pi$
$72$		0			0
$73$		−	9.21307i		−	1.07831i	−0.842207	−	0.539154i	$-0.818744\pi$
$73$		−	9.21307i		−	1.07831i	0.842207	−	0.539154i	$-0.181256\pi$
$74$		0			0
$75$	−7.80594			−0.901352
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	−6.37576			−0.717329			−0.358664	−	0.933467i	$-0.616768\pi$
$79$	−6.37576			−0.717329			−0.358664	+	0.933467i	$0.616768\pi$
$80$		0			0
$81$	3.52301			0.391446
$82$		0			0
$83$			4.00995i			0.440149i	0.975483	+	0.220075i	$0.0706301\pi$
$83$			4.00995i			0.440149i	−0.975483	+	0.220075i	$0.929370\pi$
$84$		0			0
$85$			7.30879i			0.792750i
$86$		0			0
$87$	−6.60106			−0.707708
$88$		0			0
$89$		−	8.61636i		−	0.913332i	−0.889638	−	0.456666i	$-0.849043\pi$
$89$		−	8.61636i		−	0.913332i	0.889638	−	0.456666i	$-0.150957\pi$
$90$		0			0
$91$	−1.79692	−	3.12587i	−0.188368	−	0.327680i
$92$		0			0
$93$			4.86622i			0.504604i
$94$		0			0
$95$	10.5220			1.07953
$96$		0			0
$97$			1.03847i			0.105441i	0.998609	+	0.0527203i	$0.0167892\pi$
$97$			1.03847i			0.105441i	−0.998609	+	0.0527203i	$0.983211\pi$
$98$		0			0
$99$		−	2.34357i		−	0.235537i
$100$		0			0
$101$	16.3268			1.62458			0.812290	−	0.583254i	$-0.198221\pi$
$101$	16.3268			1.62458			0.812290	+	0.583254i	$0.198221\pi$
$102$		0			0
$103$	−17.4748			−1.72184			−0.860921	−	0.508738i	$-0.830112\pi$
$103$	−17.4748			−1.72184			−0.860921	+	0.508738i	$0.830112\pi$
$104$		0			0
$105$	3.09945			0.302475
$106$		0			0
$107$	8.44728			0.816629			0.408314	−	0.912841i	$-0.366117\pi$
$107$	8.44728			0.816629			0.408314	+	0.912841i	$0.366117\pi$
$108$		0			0
$109$		−	12.7130i		−	1.21768i	−0.793291	−	0.608842i	$-0.791634\pi$
$109$		−	12.7130i		−	1.21768i	0.793291	−	0.608842i	$-0.208366\pi$
$110$		0			0
$111$			4.72758i			0.448722i
$112$		0			0
$113$	−17.9640			−1.68991			−0.844955	−	0.534837i	$-0.820373\pi$
$113$	−17.9640			−1.68991			−0.844955	+	0.534837i	$0.820373\pi$
$114$		0			0
$115$		−	30.1766i		−	2.81398i
$116$		0			0
$117$	−7.32569	+	4.21120i	−0.677261	+	0.389326i
$118$		0			0
$119$		−	1.91054i		−	0.175139i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	4.40718i		−	0.397382i
$124$		0			0
$125$		−	17.7294i		−	1.58577i
$126$		0			0
$127$	−10.2539			−0.909887			−0.454944	−	0.890520i	$-0.650341\pi$
$127$	−10.2539			−0.909887			−0.454944	+	0.890520i	$0.650341\pi$
$128$		0			0
$129$	−4.53399			−0.399196
$130$		0			0
$131$	−14.2031			−1.24093			−0.620467	−	0.784232i	$-0.713057\pi$
$131$	−14.2031			−1.24093			−0.620467	+	0.784232i	$0.713057\pi$
$132$		0			0
$133$	−2.75048			−0.238497
$134$		0			0
$135$		−	16.5621i		−	1.42544i
$136$		0			0
$137$			3.55443i			0.303676i	0.988405	+	0.151838i	$0.0485192\pi$
$137$			3.55443i			0.303676i	−0.988405	+	0.151838i	$0.951481\pi$
$138$		0			0
$139$	6.82549			0.578930			0.289465	−	0.957189i	$-0.406523\pi$
$139$	6.82549			0.578930			0.289465	+	0.957189i	$0.406523\pi$
$140$		0			0
$141$		−	9.87728i		−	0.831817i
$142$		0			0
$143$	1.79692	+	3.12587i	0.150266	+	0.261399i
$144$		0			0
$145$		−	31.1679i		−	2.58836i
$146$		0			0
$147$	−0.810205			−0.0668246
$148$		0			0
$149$		−	0.609809i		−	0.0499575i	−0.999688	−	0.0249788i	$-0.992048\pi$
$149$		−	0.609809i		−	0.0499575i	0.999688	−	0.0249788i	$-0.00795181\pi$
$150$		0			0
$151$		−	16.5888i		−	1.34998i	−0.737826	−	0.674990i	$-0.764148\pi$
$151$		−	16.5888i		−	1.34998i	0.737826	−	0.674990i	$-0.235852\pi$
$152$		0			0
$153$	−4.47748			−0.361983
$154$		0			0
$155$	−22.9766			−1.84553
$156$		0			0
$157$	20.0683			1.60162			0.800811	−	0.598918i	$-0.204402\pi$
$157$	20.0683			1.60162			0.800811	+	0.598918i	$0.204402\pi$
$158$		0			0
$159$	−5.87692			−0.466070
$160$		0			0
$161$			7.88826i			0.621682i
$162$		0			0
$163$			3.36355i			0.263454i	0.991286	+	0.131727i	$0.0420522\pi$
$163$			3.36355i			0.263454i	−0.991286	+	0.131727i	$0.957948\pi$
$164$		0			0
$165$	−3.09945			−0.241292
$166$		0			0
$167$			18.6361i			1.44210i	0.692882	+	0.721051i	$0.256341\pi$
$167$			18.6361i			1.44210i	−0.692882	+	0.721051i	$0.743659\pi$
$168$		0			0
$169$	6.54215	−	11.2339i	0.503243	−	0.864145i
$170$		0			0
$171$			6.44593i			0.492933i
$172$		0			0
$173$	−17.8784			−1.35927			−0.679636	−	0.733549i	$-0.737862\pi$
$173$	−17.8784			−1.35927			−0.679636	+	0.733549i	$0.737862\pi$
$174$		0			0
$175$			9.63452i			0.728301i
$176$		0			0
$177$			8.55493i			0.643028i
$178$		0			0
$179$	−1.99525			−0.149132			−0.0745661	−	0.997216i	$-0.523757\pi$
$179$	−1.99525			−0.149132			−0.0745661	+	0.997216i	$0.523757\pi$
$180$		0			0
$181$	−22.9006			−1.70219			−0.851093	−	0.525015i	$-0.824060\pi$
$181$	−22.9006			−1.70219			−0.851093	+	0.525015i	$0.824060\pi$
$182$		0			0
$183$	8.24768			0.609686
$184$		0			0
$185$	−22.3220			−1.64114
$186$		0			0
$187$			1.91054i			0.139713i
$188$		0			0
$189$			4.32939i			0.314917i
$190$		0			0
$191$	−20.5446			−1.48656			−0.743279	−	0.668982i	$-0.766730\pi$
$191$	−20.5446			−1.48656			−0.743279	+	0.668982i	$0.766730\pi$
$192$		0			0
$193$		−	19.1022i		−	1.37501i	−0.726181	−	0.687503i	$-0.758707\pi$
$193$		−	19.1022i		−	1.37501i	0.726181	−	0.687503i	$-0.241293\pi$
$194$		0			0
$195$	5.56946	+	9.68848i	0.398837	+	0.693806i
$196$		0			0
$197$		−	12.1514i		−	0.865748i	−0.901454	−	0.432874i	$-0.857499\pi$
$197$		−	12.1514i		−	0.865748i	0.901454	−	0.432874i	$-0.142501\pi$
$198$		0			0
$199$	−7.12897			−0.505359			−0.252680	−	0.967550i	$-0.581312\pi$
$199$	−7.12897			−0.505359			−0.252680	+	0.967550i	$0.581312\pi$
$200$		0			0
$201$			1.24148i			0.0875669i
$202$		0			0
$203$			8.14739i			0.571835i
$204$		0			0
$205$	20.8092			1.45338
$206$		0			0
$207$	18.4867			1.28491
$208$		0			0
$209$	2.75048			0.190255
$210$		0			0
$211$	−2.27714			−0.156765			−0.0783824	−	0.996923i	$-0.524976\pi$
$211$	−2.27714			−0.156765			−0.0783824	+	0.996923i	$0.524976\pi$
$212$		0			0
$213$			8.90069i			0.609865i
$214$		0			0
$215$		−	21.4079i		−	1.46001i
$216$		0			0
$217$	6.00616			0.407725
$218$		0			0
$219$		−	7.46448i		−	0.504403i
$220$		0			0
$221$	5.97211	−	3.43309i	0.401727	−	0.230935i
$222$		0			0
$223$			14.9117i			0.998563i	0.866440	+	0.499282i	$0.166403\pi$
$223$			14.9117i			0.998563i	−0.866440	+	0.499282i	$0.833597\pi$
$224$		0			0
$225$	22.5791			1.50528
$226$		0			0
$227$			22.1082i			1.46737i	0.679488	+	0.733687i	$0.262202\pi$
$227$			22.1082i			1.46737i	−0.679488	+	0.733687i	$0.737798\pi$
$228$		0			0
$229$		−	0.280435i		−	0.0185316i	−0.999957	−	0.00926582i	$-0.997051\pi$
$229$		−	0.280435i		−	0.0185316i	0.999957	−	0.00926582i	$-0.00294944\pi$
$230$		0			0
$231$	0.810205			0.0533076
$232$		0			0
$233$	16.6953			1.09374			0.546872	−	0.837216i	$-0.315818\pi$
$233$	16.6953			1.09374			0.546872	+	0.837216i	$0.315818\pi$
$234$		0			0
$235$	46.6371			3.04227
$236$		0			0
$237$	−5.16567			−0.335546
$238$		0			0
$239$			24.2526i			1.56877i	0.620275	+	0.784384i	$0.287021\pi$
$239$			24.2526i			1.56877i	−0.620275	+	0.784384i	$0.712979\pi$
$240$		0			0
$241$		−	9.63690i		−	0.620767i	−0.950611	−	0.310384i	$-0.899542\pi$
$241$		−	9.63690i		−	0.620767i	0.950611	−	0.310384i	$-0.100458\pi$
$242$		0			0
$243$	15.8425			1.01630
$244$		0			0
$245$		−	3.82551i		−	0.244403i
$246$		0			0
$247$	−4.94239	−	8.59765i	−0.314477	−	0.547055i
$248$		0			0
$249$			3.24888i			0.205890i
$250$		0			0
$251$	3.41128			0.215318			0.107659	−	0.994188i	$-0.465664\pi$
$251$	3.41128			0.215318			0.107659	+	0.994188i	$0.465664\pi$
$252$		0			0
$253$		−	7.88826i		−	0.495931i
$254$		0			0
$255$			5.92162i			0.370826i
$256$		0			0
$257$	18.0645			1.12684			0.563418	−	0.826172i	$-0.309486\pi$
$257$	18.0645			1.12684			0.563418	+	0.826172i	$0.309486\pi$
$258$		0			0
$259$	5.83504			0.362571
$260$		0			0
$261$	19.0940			1.18189
$262$		0			0
$263$	1.54515			0.0952779			0.0476389	−	0.998865i	$-0.484830\pi$
$263$	1.54515			0.0952779			0.0476389	+	0.998865i	$0.484830\pi$
$264$		0			0
$265$		−	27.7488i		−	1.70459i
$266$		0			0
$267$		−	6.98102i		−	0.427231i
$268$		0			0
$269$	15.5203			0.946287			0.473143	−	0.880985i	$-0.343119\pi$
$269$	15.5203			0.946287			0.473143	+	0.880985i	$0.343119\pi$
$270$		0			0
$271$		−	5.86316i		−	0.356162i	−0.984016	−	0.178081i	$-0.943011\pi$
$271$		−	5.86316i		−	0.356162i	0.984016	−	0.178081i	$-0.0569889\pi$
$272$		0			0
$273$	−1.45587	−	2.53260i	−0.0881135	−	0.153280i
$274$		0			0
$275$		−	9.63452i		−	0.580983i
$276$		0			0
$277$	−18.3570			−1.10296			−0.551481	−	0.834187i	$-0.685937\pi$
$277$	−18.3570			−1.10296			−0.551481	+	0.834187i	$0.685937\pi$
$278$		0			0
$279$		−	14.0758i		−	0.842698i
$280$		0			0
$281$		−	15.8478i		−	0.945398i	−0.881224	−	0.472699i	$-0.843280\pi$
$281$		−	15.8478i		−	0.945398i	0.881224	−	0.472699i	$-0.156720\pi$
$282$		0			0
$283$	27.2037			1.61709			0.808545	−	0.588434i	$-0.200255\pi$
$283$	27.2037			1.61709			0.808545	+	0.588434i	$0.200255\pi$
$284$		0			0
$285$	8.52496			0.504975
$286$		0			0
$287$	−5.43959			−0.321089
$288$		0			0
$289$	−13.3498			−0.785284
$290$		0			0
$291$			0.841373i			0.0493222i
$292$		0			0
$293$			10.9766i			0.641262i	0.947204	+	0.320631i	$0.103895\pi$
$293$			10.9766i			0.641262i	−0.947204	+	0.320631i	$0.896105\pi$
$294$		0			0
$295$	−40.3934			−2.35179
$296$		0			0
$297$		−	4.32939i		−	0.251216i
$298$		0			0
$299$	−24.6577	+	14.1746i	−1.42599	+	0.819737i
$300$		0			0
$301$			5.59610i			0.322554i
$302$		0			0
$303$	13.2281			0.759933
$304$		0			0
$305$			38.9427i			2.22985i
$306$		0			0
$307$		−	29.6137i		−	1.69015i	−0.534652	−	0.845073i	$-0.679557\pi$
$307$		−	29.6137i		−	1.69015i	0.534652	−	0.845073i	$-0.320443\pi$
$308$		0			0
$309$	−14.1582			−0.805430
$310$		0			0
$311$	−19.4926			−1.10532			−0.552662	−	0.833405i	$-0.686388\pi$
$311$	−19.4926			−1.10532			−0.552662	+	0.833405i	$0.686388\pi$
$312$		0			0
$313$	−15.1618			−0.856994			−0.428497	−	0.903543i	$-0.640957\pi$
$313$	−15.1618			−0.856994			−0.428497	+	0.903543i	$0.640957\pi$
$314$		0			0
$315$	−8.96534			−0.505140
$316$		0			0
$317$			26.2137i			1.47231i	0.676814	+	0.736154i	$0.263360\pi$
$317$			26.2137i			1.47231i	−0.676814	+	0.736154i	$0.736640\pi$
$318$		0			0
$319$		−	8.14739i		−	0.456166i
$320$		0			0
$321$	6.84403			0.381996
$322$		0			0
$323$		−	5.25490i		−	0.292391i
$324$		0			0
$325$	−30.1163	+	17.3125i	−1.67055	+	0.960323i
$326$		0			0
$327$		−	10.3001i		−	0.569599i
$328$		0			0
$329$	−12.1911			−0.672117
$330$		0			0
$331$			8.31485i			0.457026i	0.973541	+	0.228513i	$0.0733863\pi$
$331$			8.31485i			0.457026i	−0.973541	+	0.228513i	$0.926614\pi$
$332$		0			0
$333$		−	13.6748i		−	0.749374i
$334$		0			0
$335$	−5.86182			−0.320265
$336$		0			0
$337$	1.87403			0.102085			0.0510426	−	0.998696i	$-0.483746\pi$
$337$	1.87403			0.102085			0.0510426	+	0.998696i	$0.483746\pi$
$338$		0			0
$339$	−14.5545			−0.790493
$340$		0			0
$341$	−6.00616			−0.325252
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	24.4493i		−	1.31630i
$346$		0			0
$347$	−16.3694			−0.878755			−0.439378	−	0.898302i	$-0.644801\pi$
$347$	−16.3694			−0.878755			−0.439378	+	0.898302i	$0.644801\pi$
$348$		0			0
$349$			14.7403i			0.789031i	0.918889	+	0.394515i	$0.129087\pi$
$349$			14.7403i			0.789031i	−0.918889	+	0.394515i	$0.870913\pi$
$350$		0			0
$351$	−13.5331	+	7.77956i	−0.722344	+	0.415242i
$352$		0			0
$353$			6.90003i			0.367252i	0.982996	+	0.183626i	$0.0587835\pi$
$353$			6.90003i			0.367252i	−0.982996	+	0.183626i	$0.941217\pi$
$354$		0			0
$355$	−42.0260			−2.23051
$356$		0			0
$357$		−	1.54793i		−	0.0819251i
$358$		0			0
$359$			27.7588i			1.46505i	0.680740	+	0.732526i	$0.261659\pi$
$359$			27.7588i			1.46505i	−0.680740	+	0.732526i	$0.738341\pi$
$360$		0			0
$361$	11.4349			0.601835
$362$		0			0
$363$	−0.810205			−0.0425247
$364$		0			0
$365$	35.2447			1.84479
$366$		0			0
$367$	32.5352			1.69833			0.849163	−	0.528131i	$-0.177107\pi$
$367$	32.5352			1.69833			0.849163	+	0.528131i	$0.177107\pi$
$368$		0			0
$369$			12.7480i			0.663637i
$370$		0			0
$371$			7.25361i			0.376589i
$372$		0			0
$373$	−13.4309			−0.695424			−0.347712	−	0.937601i	$-0.613041\pi$
$373$	−13.4309			−0.695424			−0.347712	+	0.937601i	$0.613041\pi$
$374$		0			0
$375$		−	14.3645i		−	0.741777i
$376$		0			0
$377$	−25.4677	+	14.6402i	−1.31165	+	0.754010i
$378$		0			0
$379$		−	20.4132i		−	1.04856i	−0.851547	−	0.524279i	$-0.824335\pi$
$379$		−	20.4132i		−	1.04856i	0.851547	−	0.524279i	$-0.175665\pi$
$380$		0			0
$381$	−8.30777			−0.425620
$382$		0			0
$383$			17.8138i			0.910242i	0.890430	+	0.455121i	$0.150404\pi$
$383$			17.8138i			0.910242i	−0.890430	+	0.455121i	$0.849596\pi$
$384$		0			0
$385$			3.82551i			0.194966i
$386$		0			0
$387$	13.1148			0.666665
$388$		0			0
$389$	33.1676			1.68166			0.840831	−	0.541298i	$-0.182067\pi$
$389$	33.1676			1.68166			0.840831	+	0.541298i	$0.182067\pi$
$390$		0			0
$391$	−15.0708			−0.762165
$392$		0			0
$393$	−11.5075			−0.580475
$394$		0			0
$395$		−	24.3905i		−	1.22722i
$396$		0			0
$397$		−	7.90138i		−	0.396559i	−0.980146	−	0.198279i	$-0.936465\pi$
$397$		−	7.90138i		−	0.396559i	0.980146	−	0.198279i	$-0.0635353\pi$
$398$		0			0
$399$	−2.22845			−0.111562
$400$		0			0
$401$		−	37.1332i		−	1.85435i	−0.374634	−	0.927173i	$-0.622232\pi$
$401$		−	37.1332i		−	1.85435i	0.374634	−	0.927173i	$-0.377768\pi$
$402$		0			0
$403$	10.7926	+	18.7745i	0.537617	+	0.935224i
$404$		0			0
$405$			13.4773i			0.669693i
$406$		0			0
$407$	−5.83504			−0.289232
$408$		0			0
$409$			7.93769i			0.392493i	0.980555	+	0.196247i	$0.0628754\pi$
$409$			7.93769i			0.392493i	−0.980555	+	0.196247i	$0.937125\pi$
$410$		0			0
$411$			2.87982i			0.142051i
$412$		0			0
$413$	10.5590			0.519573
$414$		0			0
$415$	−15.3401			−0.753016
$416$		0			0
$417$	5.53005			0.270808
$418$		0			0
$419$	2.23501			0.109187			0.0545937	−	0.998509i	$-0.482614\pi$
$419$	2.23501			0.109187			0.0545937	+	0.998509i	$0.482614\pi$
$420$		0			0
$421$			33.9151i			1.65292i	0.562994	+	0.826461i	$0.309649\pi$
$421$			33.9151i			1.65292i	−0.562994	+	0.826461i	$0.690351\pi$
$422$		0			0
$423$			28.5706i			1.38915i
$424$		0			0
$425$	−18.4071			−0.892877
$426$		0			0
$427$		−	10.1797i		−	0.492632i
$428$		0			0
$429$	1.45587	+	2.53260i	0.0702903	+	0.122275i
$430$		0			0
$431$			15.9604i			0.768784i	0.923170	+	0.384392i	$0.125589\pi$
$431$			15.9604i			0.768784i	−0.923170	+	0.384392i	$0.874411\pi$
$432$		0			0
$433$	−3.87431			−0.186188			−0.0930938	−	0.995657i	$-0.529676\pi$
$433$	−3.87431			−0.186188			−0.0930938	+	0.995657i	$0.529676\pi$
$434$		0			0
$435$		−	25.2524i		−	1.21076i
$436$		0			0
$437$			21.6965i			1.03788i
$438$		0			0
$439$	−21.4656			−1.02450			−0.512248	−	0.858838i	$-0.671187\pi$
$439$	−21.4656			−1.02450			−0.512248	+	0.858838i	$0.671187\pi$
$440$		0			0
$441$	2.34357			0.111598
$442$		0			0
$443$	−8.03686			−0.381843			−0.190921	−	0.981605i	$-0.561148\pi$
$443$	−8.03686			−0.381843			−0.190921	+	0.981605i	$0.561148\pi$
$444$		0			0
$445$	32.9620			1.56255
$446$		0			0
$447$		−	0.494070i		−	0.0233687i
$448$		0			0
$449$			33.5666i			1.58411i	0.610452	+	0.792054i	$0.290988\pi$
$449$			33.5666i			1.58411i	−0.610452	+	0.792054i	$0.709012\pi$
$450$		0			0
$451$	5.43959			0.256140
$452$		0			0
$453$		−	13.4404i		−	0.631484i
$454$		0			0
$455$	11.9581	−	6.87414i	0.560602	−	0.322264i
$456$		0			0
$457$			22.5771i			1.05611i	0.849209	+	0.528057i	$0.177079\pi$
$457$			22.5771i			1.05611i	−0.849209	+	0.528057i	$0.822921\pi$
$458$		0			0
$459$	−8.27147			−0.386079
$460$		0			0
$461$		−	38.8120i		−	1.80765i	−0.427897	−	0.903827i	$-0.640745\pi$
$461$		−	38.8120i		−	1.80765i	0.427897	−	0.903827i	$-0.359255\pi$
$462$		0			0
$463$		−	22.0060i		−	1.02270i	−0.859371	−	0.511352i	$-0.829145\pi$
$463$		−	22.0060i		−	1.02270i	0.859371	−	0.511352i	$-0.170855\pi$
$464$		0			0
$465$	−18.6158			−0.863286
$466$		0			0
$467$	25.5139			1.18064			0.590322	−	0.807168i	$-0.299001\pi$
$467$	25.5139			1.18064			0.590322	+	0.807168i	$0.299001\pi$
$468$		0			0
$469$	1.53230			0.0707549
$470$		0			0
$471$	16.2594			0.749194
$472$		0			0
$473$		−	5.59610i		−	0.257309i
$474$		0			0
$475$			26.4995i			1.21588i
$476$		0			0
$477$	16.9993			0.778346
$478$		0			0
$479$		−	12.6343i		−	0.577274i	−0.957439	−	0.288637i	$-0.906798\pi$
$479$		−	12.6343i		−	0.577274i	0.957439	−	0.288637i	$-0.0932021\pi$
$480$		0			0
$481$	10.4851	+	18.2396i	0.478079	+	0.831653i
$482$		0			0
$483$			6.39111i			0.290806i
$484$		0			0
$485$	−3.97267			−0.180390
$486$		0			0
$487$		−	11.8074i		−	0.535043i	−0.963552	−	0.267522i	$-0.913795\pi$
$487$		−	11.8074i		−	0.535043i	0.963552	−	0.267522i	$-0.0862047\pi$
$488$		0			0
$489$			2.72517i			0.123236i
$490$		0			0
$491$	−14.1537			−0.638747			−0.319374	−	0.947629i	$-0.603472\pi$
$491$	−14.1537			−0.638747			−0.319374	+	0.947629i	$0.603472\pi$
$492$		0			0
$493$	−15.5659			−0.701054
$494$		0			0
$495$	8.96534			0.402962
$496$		0			0
$497$	10.9857			0.492777
$498$		0			0
$499$			17.3630i			0.777276i	0.921390	+	0.388638i	$0.127054\pi$
$499$			17.3630i			0.777276i	−0.921390	+	0.388638i	$0.872946\pi$
$500$		0			0
$501$			15.0990i			0.674576i
$502$		0			0
$503$	−26.0761			−1.16267			−0.581337	−	0.813663i	$-0.697470\pi$
$503$	−26.0761			−1.16267			−0.581337	+	0.813663i	$0.697470\pi$
$504$		0			0
$505$			62.4584i			2.77936i
$506$		0			0
$507$	5.30049	−	9.10175i	0.235403	−	0.404223i
$508$		0			0
$509$		−	5.29781i		−	0.234821i	−0.993083	−	0.117411i	$-0.962541\pi$
$509$		−	5.29781i		−	0.234821i	0.993083	−	0.117411i	$-0.0374593\pi$
$510$		0			0
$511$	−9.21307			−0.407562
$512$		0			0
$513$			11.9079i			0.525746i
$514$		0			0
$515$		−	66.8500i		−	2.94576i
$516$		0			0
$517$	12.1911			0.536164
$518$		0			0
$519$	−14.4852			−0.635830
$520$		0			0
$521$	6.39819			0.280310			0.140155	−	0.990130i	$-0.455240\pi$
$521$	6.39819			0.280310			0.140155	+	0.990130i	$0.455240\pi$
$522$		0			0
$523$	−6.20893			−0.271498			−0.135749	−	0.990743i	$-0.543344\pi$
$523$	−6.20893			−0.271498			−0.135749	+	0.990743i	$0.543344\pi$
$524$		0			0
$525$			7.80594i			0.340679i
$526$		0			0
$527$			11.4750i			0.499859i
$528$		0			0
$529$	39.2247			1.70542
$530$		0			0
$531$		−	24.7456i		−	1.07387i
$532$		0			0
$533$	−9.77451	−	17.0035i	−0.423381	−	0.736502i
$534$		0			0
$535$			32.3151i			1.39710i
$536$		0			0
$537$	−1.61656			−0.0697599
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$			26.3385i			1.13238i	0.824275	+	0.566190i	$0.191583\pi$
$541$			26.3385i			1.13238i	−0.824275	+	0.566190i	$0.808417\pi$
$542$		0			0
$543$	−18.5542			−0.796235
$544$		0			0
$545$	48.6337			2.08324
$546$		0			0
$547$	−20.0106			−0.855593			−0.427796	−	0.903875i	$-0.640710\pi$
$547$	−20.0106			−0.855593			−0.427796	+	0.903875i	$0.640710\pi$
$548$		0			0
$549$	−23.8569			−1.01819
$550$		0			0
$551$			22.4092i			0.954665i
$552$		0			0
$553$			6.37576i			0.271125i
$554$		0			0
$555$	−18.0854			−0.767682
$556$		0			0
$557$			29.5912i			1.25382i	0.779093	+	0.626909i	$0.215680\pi$
$557$			29.5912i			1.25382i	−0.779093	+	0.626909i	$0.784320\pi$
$558$		0			0
$559$	−17.4927	+	10.0557i	−0.739862	+	0.425313i
$560$		0			0
$561$			1.54793i			0.0653537i
$562$		0			0
$563$	−17.3869			−0.732768			−0.366384	−	0.930464i	$-0.619404\pi$
$563$	−17.3869			−0.732768			−0.366384	+	0.930464i	$0.619404\pi$
$564$		0			0
$565$		−	68.7214i		−	2.89113i
$566$		0			0
$567$		−	3.52301i		−	0.147953i
$568$		0			0
$569$	19.9401			0.835931			0.417965	−	0.908463i	$-0.362743\pi$
$569$	19.9401			0.835931			0.417965	+	0.908463i	$0.362743\pi$
$570$		0			0
$571$	−14.9167			−0.624243			−0.312122	−	0.950042i	$-0.601040\pi$
$571$	−14.9167			−0.624243			−0.312122	+	0.950042i	$0.601040\pi$
$572$		0			0
$573$	−16.6454			−0.695370
$574$		0			0
$575$	75.9996			3.16940
$576$		0			0
$577$			34.6442i			1.44226i	0.692801	+	0.721128i	$0.256376\pi$
$577$			34.6442i			1.44226i	−0.692801	+	0.721128i	$0.743624\pi$
$578$		0			0
$579$		−	15.4767i		−	0.643190i
$580$		0			0
$581$	4.00995			0.166361
$582$		0			0
$583$		−	7.25361i		−	0.300414i
$584$		0			0
$585$	−16.1100	−	28.0245i	−0.666066	−	1.15867i
$586$		0			0
$587$		−	25.7114i		−	1.06122i	−0.847616	−	0.530611i	$-0.821963\pi$
$587$		−	25.7114i		−	1.06122i	0.847616	−	0.530611i	$-0.178037\pi$
$588$		0			0
$589$	16.5198			0.680687
$590$		0			0
$591$		−	9.84509i		−	0.404973i
$592$		0			0
$593$			30.6866i			1.26015i	0.776535	+	0.630074i	$0.216975\pi$
$593$			30.6866i			1.26015i	−0.776535	+	0.630074i	$0.783025\pi$
$594$		0			0
$595$	7.30879			0.299631
$596$		0			0
$597$	−5.77593			−0.236393
$598$		0			0
$599$	35.4819			1.44975			0.724875	−	0.688880i	$-0.241897\pi$
$599$	35.4819			1.44975			0.724875	+	0.688880i	$0.241897\pi$
$600$		0			0
$601$	−8.62650			−0.351882			−0.175941	−	0.984401i	$-0.556297\pi$
$601$	−8.62650			−0.351882			−0.175941	+	0.984401i	$0.556297\pi$
$602$		0			0
$603$		−	3.59104i		−	0.146239i
$604$		0			0
$605$		−	3.82551i		−	0.155529i
$606$		0			0
$607$	15.6481			0.635136			0.317568	−	0.948235i	$-0.397134\pi$
$607$	15.6481			0.635136			0.317568	+	0.948235i	$0.397134\pi$
$608$		0			0
$609$			6.60106i			0.267488i
$610$		0			0
$611$	−21.9064	−	38.1078i	−0.886239	−	1.54168i
$612$		0			0
$613$		−	24.6208i		−	0.994426i	−0.867628	−	0.497213i	$-0.834357\pi$
$613$		−	24.6208i		−	0.994426i	0.867628	−	0.497213i	$-0.165643\pi$
$614$		0			0
$615$	16.8597			0.679850
$616$		0			0
$617$			44.8982i			1.80754i	0.428024	+	0.903768i	$0.359210\pi$
$617$			44.8982i			1.80754i	−0.428024	+	0.903768i	$0.640790\pi$
$618$		0			0
$619$			10.9600i			0.440520i	0.975441	+	0.220260i	$0.0706906\pi$
$619$			10.9600i			0.440520i	−0.975441	+	0.220260i	$0.929309\pi$
$620$		0			0
$621$	34.1513			1.37045
$622$		0			0
$623$	−8.61636			−0.345207
$624$		0			0
$625$	19.6514			0.786055
$626$		0			0
$627$	2.22845			0.0889958
$628$		0			0
$629$			11.1481i			0.444503i
$630$		0			0
$631$			4.96234i			0.197548i	0.995110	+	0.0987738i	$0.0314920\pi$
$631$			4.96234i			0.197548i	−0.995110	+	0.0987738i	$0.968508\pi$
$632$		0			0
$633$	−1.84495			−0.0733302
$634$		0			0
$635$		−	39.2264i		−	1.55665i
$636$		0			0
$637$	−3.12587	+	1.79692i	−0.123852	+	0.0711966i
$638$		0			0
$639$		−	25.7458i		−	1.01849i
$640$		0			0
$641$	−48.0510			−1.89790			−0.948951	−	0.315423i	$-0.897854\pi$
$641$	−48.0510			−1.89790			−0.948951	+	0.315423i	$0.897854\pi$
$642$		0			0
$643$			13.3346i			0.525864i	0.964814	+	0.262932i	$0.0846895\pi$
$643$			13.3346i			0.525864i	−0.964814	+	0.262932i	$0.915310\pi$
$644$		0			0
$645$		−	17.3448i		−	0.682952i
$646$		0			0
$647$	−24.5437			−0.964912			−0.482456	−	0.875920i	$-0.660255\pi$
$647$	−24.5437			−0.964912			−0.482456	+	0.875920i	$0.660255\pi$
$648$		0			0
$649$	−10.5590			−0.414475
$650$		0			0
$651$	4.86622			0.190722
$652$		0			0
$653$	5.18967			0.203088			0.101544	−	0.994831i	$-0.467622\pi$
$653$	5.18967			0.203088			0.101544	+	0.994831i	$0.467622\pi$
$654$		0			0
$655$		−	54.3343i		−	2.12302i
$656$		0			0
$657$			21.5915i			0.842363i
$658$		0			0
$659$	−25.2362			−0.983063			−0.491532	−	0.870860i	$-0.663563\pi$
$659$	−25.2362			−0.983063			−0.491532	+	0.870860i	$0.663563\pi$
$660$		0			0
$661$		−	4.79114i		−	0.186354i	−0.995650	−	0.0931770i	$-0.970298\pi$
$661$		−	4.79114i		−	0.186354i	0.995650	−	0.0931770i	$-0.0297022\pi$
$662$		0			0
$663$	4.83863	−	2.78151i	0.187917	−	0.108025i
$664$		0			0
$665$		−	10.5220i		−	0.408025i
$666$		0			0
$667$	64.2688			2.48850
$668$		0			0
$669$			12.0816i			0.467100i
$670$		0			0
$671$			10.1797i			0.392985i
$672$		0			0
$673$	−33.1840			−1.27915			−0.639575	−	0.768729i	$-0.720890\pi$
$673$	−33.1840			−1.27915			−0.639575	+	0.768729i	$0.720890\pi$
$674$		0			0
$675$	41.7116			1.60548
$676$		0			0
$677$	13.5797			0.521911			0.260955	−	0.965351i	$-0.415962\pi$
$677$	13.5797			0.521911			0.260955	+	0.965351i	$0.415962\pi$
$678$		0			0
$679$	1.03847			0.0398528
$680$		0			0
$681$			17.9122i			0.686396i
$682$		0			0
$683$			44.7327i			1.71165i	0.517267	+	0.855824i	$0.326949\pi$
$683$			44.7327i			1.71165i	−0.517267	+	0.855824i	$0.673051\pi$
$684$		0			0
$685$	−13.5975			−0.519534
$686$		0			0
$687$		−	0.227210i		−	0.00866859i
$688$		0			0
$689$	−22.6739	+	13.0342i	−0.863806	+	0.496562i
$690$		0			0
$691$		−	10.3633i		−	0.394238i	−0.980380	−	0.197119i	$-0.936841\pi$
$691$		−	10.3633i		−	0.394238i	0.980380	−	0.197119i	$-0.0631585\pi$
$692$		0			0
$693$	−2.34357			−0.0890248
$694$		0			0
$695$			26.1110i			0.990445i
$696$		0			0
$697$		−	10.3926i		−	0.393646i
$698$		0			0
$699$	13.5266			0.511623
$700$		0			0
$701$	35.6871			1.34788			0.673941	−	0.738785i	$-0.264600\pi$
$701$	35.6871			1.34788			0.673941	+	0.738785i	$0.264600\pi$
$702$		0			0
$703$	16.0491			0.605305
$704$		0			0
$705$	37.7856			1.42309
$706$		0			0
$707$		−	16.3268i		−	0.614033i
$708$		0			0
$709$		−	24.8188i		−	0.932089i	−0.884761	−	0.466045i	$-0.845679\pi$
$709$		−	24.8188i		−	0.932089i	0.884761	−	0.466045i	$-0.154321\pi$
$710$		0			0
$711$	14.9420			0.560369
$712$		0			0
$713$		−	47.3782i		−	1.77433i
$714$		0			0
$715$	−11.9581	+	6.87414i	−0.447206	+	0.257078i
$716$		0			0
$717$			19.6496i			0.733826i
$718$		0			0
$719$	5.03720			0.187856			0.0939279	−	0.995579i	$-0.470058\pi$
$719$	5.03720			0.187856			0.0939279	+	0.995579i	$0.470058\pi$
$720$		0			0
$721$			17.4748i			0.650795i
$722$		0			0
$723$		−	7.80787i		−	0.290378i
$724$		0			0
$725$	78.4962			2.91528
$726$		0			0
$727$	7.85553			0.291345			0.145673	−	0.989333i	$-0.453465\pi$
$727$	7.85553			0.291345			0.145673	+	0.989333i	$0.453465\pi$
$728$		0			0
$729$	2.26666			0.0839504
$730$		0			0
$731$	−10.6916			−0.395442
$732$		0			0
$733$			50.9180i			1.88070i	0.340208	+	0.940350i	$0.389503\pi$
$733$			50.9180i			1.88070i	−0.340208	+	0.940350i	$0.610497\pi$
$734$		0			0
$735$		−	3.09945i		−	0.114325i
$736$		0			0
$737$	−1.53230			−0.0564429
$738$		0			0
$739$		−	39.2431i		−	1.44358i	−0.692111	−	0.721791i	$-0.743319\pi$
$739$		−	39.2431i		−	1.44358i	0.692111	−	0.721791i	$-0.256681\pi$
$740$		0			0
$741$	−4.00435	−	6.96586i	−0.147104	−	0.255897i
$742$		0			0
$743$			11.6893i			0.428839i	0.976742	+	0.214419i	$0.0687859\pi$
$743$			11.6893i			0.428839i	−0.976742	+	0.214419i	$0.931214\pi$
$744$		0			0
$745$	2.33283			0.0854683
$746$		0			0
$747$		−	9.39759i		−	0.343840i
$748$		0			0
$749$		−	8.44728i		−	0.308657i
$750$		0			0
$751$	−30.9287			−1.12860			−0.564302	−	0.825568i	$-0.690855\pi$
$751$	−30.9287			−1.12860			−0.564302	+	0.825568i	$0.690855\pi$
$752$		0			0
$753$	2.76384			0.100720
$754$		0			0
$755$	63.4608			2.30957
$756$		0			0
$757$	−8.11518			−0.294951			−0.147476	−	0.989066i	$-0.547115\pi$
$757$	−8.11518			−0.294951			−0.147476	+	0.989066i	$0.547115\pi$
$758$		0			0
$759$		−	6.39111i		−	0.231983i
$760$		0			0
$761$		−	8.76604i		−	0.317769i	−0.987297	−	0.158884i	$-0.949210\pi$
$761$		−	8.76604i		−	0.317769i	0.987297	−	0.158884i	$-0.0507897\pi$
$762$		0			0
$763$	−12.7130			−0.460241
$764$		0			0
$765$		−	17.1286i		−	0.619287i
$766$		0			0
$767$	18.9736	+	33.0060i	0.685097	+	1.19178i
$768$		0			0
$769$		−	1.81454i		−	0.0654342i	−0.999465	−	0.0327171i	$-0.989584\pi$
$769$		−	1.81454i		−	0.0654342i	0.999465	−	0.0327171i	$-0.0104160\pi$
$770$		0			0
$771$	14.6360			0.527102
$772$		0			0
$773$			18.8071i			0.676445i	0.941066	+	0.338223i	$0.109826\pi$
$773$			18.8071i			0.676445i	−0.941066	+	0.338223i	$0.890174\pi$
$774$		0			0
$775$		−	57.8665i		−	2.07862i
$776$		0			0
$777$	4.72758			0.169601
$778$		0			0
$779$	−14.9615			−0.536050
$780$		0			0
$781$	−10.9857			−0.393100
$782$		0			0
$783$	35.2732			1.26056
$784$		0			0
$785$			76.7713i			2.74008i
$786$		0			0
$787$			9.49371i			0.338414i	0.985581	+	0.169207i	$0.0541207\pi$
$787$			9.49371i			0.338414i	−0.985581	+	0.169207i	$0.945879\pi$
$788$		0			0
$789$	1.25189			0.0445683
$790$		0			0
$791$			17.9640i			0.638726i
$792$		0			0
$793$	31.8206	−	18.2922i	1.12998	−	0.649575i
$794$		0			0
$795$		−	22.4822i		−	0.797361i
$796$		0			0
$797$	2.52015			0.0892682			0.0446341	−	0.999003i	$-0.485788\pi$
$797$	2.52015			0.0892682			0.0446341	+	0.999003i	$0.485788\pi$
$798$		0			0
$799$		−	23.2916i		−	0.823997i
$800$		0			0
$801$			20.1930i			0.713485i
$802$		0			0
$803$	9.21307			0.325122
$804$		0			0
$805$	−30.1766			−1.06359
$806$		0			0
$807$	12.5746			0.442647
$808$		0			0
$809$	18.9859			0.667508			0.333754	−	0.942660i	$-0.391684\pi$
$809$	18.9859			0.667508			0.333754	+	0.942660i	$0.391684\pi$
$810$		0			0
$811$		−	17.8344i		−	0.626251i	−0.949712	−	0.313125i	$-0.898624\pi$
$811$		−	17.8344i		−	0.626251i	0.949712	−	0.313125i	$-0.101376\pi$
$812$		0			0
$813$		−	4.75036i		−	0.166603i
$814$		0			0
$815$	−12.8673			−0.450722
$816$		0			0
$817$			15.3920i			0.538496i
$818$		0			0
$819$	4.21120	+	7.32569i	0.147151	+	0.255980i
$820$		0			0
$821$			17.7404i			0.619144i	0.950876	+	0.309572i	$0.100186\pi$
$821$			17.7404i			0.619144i	−0.950876	+	0.309572i	$0.899814\pi$
$822$		0			0
$823$	9.94418			0.346632			0.173316	−	0.984866i	$-0.444552\pi$
$823$	9.94418			0.346632			0.173316	+	0.984866i	$0.444552\pi$
$824$		0			0
$825$		−	7.80594i		−	0.271768i
$826$		0			0
$827$			38.2279i			1.32931i	0.747148	+	0.664657i	$0.231422\pi$
$827$			38.2279i			1.32931i	−0.747148	+	0.664657i	$0.768578\pi$
$828$		0			0
$829$	41.3467			1.43603			0.718015	−	0.696028i	$-0.245051\pi$
$829$	41.3467			1.43603			0.718015	+	0.696028i	$0.245051\pi$
$830$		0			0
$831$	−14.8729			−0.515935
$832$		0			0
$833$	−1.91054			−0.0661963
$834$		0			0
$835$	−71.2925			−2.46718
$836$		0			0
$837$		−	26.0030i		−	0.898795i
$838$		0			0
$839$			54.4252i			1.87897i	0.342597	+	0.939483i	$0.388694\pi$
$839$			54.4252i			1.87897i	−0.342597	+	0.939483i	$0.611306\pi$
$840$		0			0
$841$	37.3800			1.28897
$842$		0			0
$843$		−	12.8399i		−	0.442231i
$844$		0			0
$845$	42.9753	+	25.0271i	1.47840	+	0.860957i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	22.0406			0.756430
$850$		0			0
$851$		−	46.0283i		−	1.57783i
$852$		0			0
$853$			35.9279i			1.23015i	0.788470	+	0.615073i	$0.210874\pi$
$853$			35.9279i			1.23015i	−0.788470	+	0.615073i	$0.789126\pi$
$854$		0			0
$855$	−24.6590			−0.843319
$856$		0			0
$857$	−51.0644			−1.74433			−0.872164	−	0.489214i	$-0.837284\pi$
$857$	−51.0644			−1.74433			−0.872164	+	0.489214i	$0.837284\pi$
$858$		0			0
$859$	48.7308			1.66267			0.831337	−	0.555769i	$-0.187576\pi$
$859$	48.7308			1.66267			0.831337	+	0.555769i	$0.187576\pi$
$860$		0			0
$861$	−4.40718			−0.150196
$862$		0			0
$863$		−	45.5246i		−	1.54968i	−0.632159	−	0.774838i	$-0.717831\pi$
$863$		−	45.5246i		−	1.54968i	0.632159	−	0.774838i	$-0.282169\pi$
$864$		0			0
$865$		−	68.3941i		−	2.32547i
$866$		0			0
$867$	−10.8161			−0.367334
$868$		0			0
$869$		−	6.37576i		−	0.216283i
$870$		0			0
$871$	2.75342	+	4.78977i	0.0932960	+	0.162295i
$872$		0			0
$873$		−	2.43372i		−	0.0823690i
$874$		0			0
$875$	−17.7294			−0.599363
$876$		0			0
$877$			29.0033i			0.979371i	0.871899	+	0.489685i	$0.162888\pi$
$877$			29.0033i			0.979371i	−0.871899	+	0.489685i	$0.837112\pi$
$878$		0			0
$879$			8.89333i			0.299965i
$880$		0			0
$881$	−33.4304			−1.12630			−0.563150	−	0.826355i	$-0.690411\pi$
$881$	−33.4304			−1.12630			−0.563150	+	0.826355i	$0.690411\pi$
$882$		0			0
$883$	32.3234			1.08777			0.543885	−	0.839160i	$-0.316953\pi$
$883$	32.3234			1.08777			0.543885	+	0.839160i	$0.316953\pi$
$884$		0			0
$885$	−32.7269			−1.10010
$886$		0			0
$887$	41.0092			1.37696			0.688478	−	0.725257i	$-0.258279\pi$
$887$	41.0092			1.37696			0.688478	+	0.725257i	$0.258279\pi$
$888$		0			0
$889$			10.2539i			0.343905i
$890$		0			0
$891$			3.52301i			0.118025i
$892$		0			0
$893$	−33.5313			−1.12208
$894$		0			0
$895$		−	7.63286i		−	0.255138i
$896$		0			0
$897$	−19.9778	+	11.4843i	−0.667039	+	0.383450i
$898$		0			0
$899$		−	48.9345i		−	1.63206i
$900$		0			0
$901$	−13.8583			−0.461688
$902$		0			0
$903$			4.53399i			0.150882i
$904$		0			0
$905$		−	87.6063i		−	2.91213i
$906$		0			0
$907$	−53.4637			−1.77523			−0.887617	−	0.460582i	$-0.847641\pi$
$907$	−53.4637			−1.77523			−0.887617	+	0.460582i	$0.847641\pi$
$908$		0			0
$909$	−38.2630			−1.26910
$910$		0			0
$911$	47.5349			1.57490			0.787450	−	0.616379i	$-0.211401\pi$
$911$	47.5349			1.57490			0.787450	+	0.616379i	$0.211401\pi$
$912$		0			0
$913$	−4.00995			−0.132710
$914$		0			0
$915$			31.5516i			1.04306i
$916$		0			0
$917$			14.2031i			0.469029i
$918$		0			0
$919$	1.20359			0.0397027			0.0198514	−	0.999803i	$-0.493681\pi$
$919$	1.20359			0.0397027			0.0198514	+	0.999803i	$0.493681\pi$
$920$		0			0
$921$		−	23.9932i		−	0.790603i
$922$		0			0
$923$	19.7405	+	34.3400i	0.649766	+	1.13031i
$924$		0			0
$925$		−	56.2178i		−	1.84843i
$926$		0			0
$927$	40.9534			1.34508
$928$		0			0
$929$			33.2327i			1.09033i	0.838329	+	0.545164i	$0.183533\pi$
$929$			33.2327i			1.09033i	−0.838329	+	0.545164i	$0.816467\pi$
$930$		0			0
$931$			2.75048i			0.0901433i
$932$		0			0
$933$	−15.7930			−0.517040
$934$		0			0
$935$	−7.30879			−0.239023
$936$		0			0
$937$	15.7970			0.516066			0.258033	−	0.966136i	$-0.416926\pi$
$937$	15.7970			0.516066			0.258033	+	0.966136i	$0.416926\pi$
$938$		0			0
$939$	−12.2842			−0.400878
$940$		0			0
$941$		−	21.8276i		−	0.711560i	−0.934570	−	0.355780i	$-0.884215\pi$
$941$		−	21.8276i		−	0.711560i	0.934570	−	0.355780i	$-0.115785\pi$
$942$		0			0
$943$			42.9089i			1.39731i
$944$		0			0
$945$	−16.5621			−0.538765
$946$		0			0
$947$		−	25.0365i		−	0.813579i	−0.913522	−	0.406789i	$-0.866648\pi$
$947$		−	25.0365i		−	0.813579i	0.913522	−	0.406789i	$-0.133352\pi$
$948$		0			0
$949$	−16.5552	−	28.7989i	−0.537403	−	0.934851i
$950$		0			0
$951$			21.2385i			0.688705i
$952$		0			0
$953$	19.0867			0.618278			0.309139	−	0.951017i	$-0.399959\pi$
$953$	19.0867			0.618278			0.309139	+	0.951017i	$0.399959\pi$
$954$		0			0
$955$		−	78.5937i		−	2.54323i
$956$		0			0
$957$		−	6.60106i		−	0.213382i
$958$		0			0
$959$	3.55443			0.114779
$960$		0			0
$961$	−5.07394			−0.163675
$962$		0			0
$963$	−19.7968			−0.637942
$964$		0			0
$965$	73.0757			2.35239
$966$		0			0
$967$			29.9788i			0.964054i	0.876156	+	0.482027i	$0.160099\pi$
$967$			29.9788i			0.964054i	−0.876156	+	0.482027i	$0.839901\pi$
$968$		0			0
$969$		−	4.25755i		−	0.136772i
$970$		0			0
$971$	46.5837			1.49494			0.747471	−	0.664295i	$-0.231268\pi$
$971$	46.5837			1.49494			0.747471	+	0.664295i	$0.231268\pi$
$972$		0			0
$973$		−	6.82549i		−	0.218815i
$974$		0			0
$975$	−24.4004	+	14.0267i	−0.781437	+	0.449212i
$976$		0			0
$977$			26.7240i			0.854975i	0.904021	+	0.427488i	$0.140601\pi$
$977$			26.7240i			0.854975i	−0.904021	+	0.427488i	$0.859399\pi$
$978$		0			0
$979$	8.61636			0.275380
$980$		0			0
$981$			29.7938i			0.951242i
$982$		0			0
$983$			6.97158i			0.222359i	0.993800	+	0.111179i	$0.0354628\pi$
$983$			6.97158i			0.222359i	−0.993800	+	0.111179i	$0.964537\pi$
$984$		0			0
$985$	46.4851			1.48114
$986$		0			0
$987$	−9.87728			−0.314397
$988$		0			0
$989$	44.1435			1.40368
$990$		0			0
$991$	11.1471			0.354098			0.177049	−	0.984202i	$-0.443345\pi$
$991$	11.1471			0.354098			0.177049	+	0.984202i	$0.443345\pi$
$992$		0			0
$993$			6.73674i			0.213784i
$994$		0			0
$995$		−	27.2719i		−	0.864578i
$996$		0			0
$997$	48.7855			1.54505			0.772525	−	0.634984i	$-0.218993\pi$
$997$	48.7855			1.54505			0.772525	+	0.634984i	$0.218993\pi$
$998$		0			0
$999$		−	25.2621i		−	0.799258i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.23	✓	36
13.12	even	2	inner	4004.2.m.c.2157.24	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.23	✓	36	1.1	even	1	trivial
4004.2.m.c.2157.24	yes	36	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.810205 q^{3} +3.82551i q^{5} -1.00000i q^{7} -2.34357 q^{9} +O(q^{10})\)	\(q+0.810205 q^{3} +3.82551i q^{5} -1.00000i q^{7} -2.34357 q^{9} +1.00000i q^{11} +(3.12587 - 1.79692i) q^{13} +3.09945i q^{15} +1.91054 q^{17} -2.75048i q^{19} -0.810205i q^{21} -7.88826 q^{23} -9.63452 q^{25} -4.32939 q^{27} -8.14739 q^{29} +6.00616i q^{31} +0.810205i q^{33} +3.82551 q^{35} +5.83504i q^{37} +(2.53260 - 1.45587i) q^{39} -5.43959i q^{41} -5.59610 q^{43} -8.96534i q^{45} -12.1911i q^{47} -1.00000 q^{49} +1.54793 q^{51} -7.25361 q^{53} -3.82551 q^{55} -2.22845i q^{57} +10.5590i q^{59} +10.1797 q^{61} +2.34357i q^{63} +(6.87414 + 11.9581i) q^{65} +1.53230i q^{67} -6.39111 q^{69} +10.9857i q^{71} -9.21307i q^{73} -7.80594 q^{75} +1.00000 q^{77} -6.37576 q^{79} +3.52301 q^{81} +4.00995i q^{83} +7.30879i q^{85} -6.60106 q^{87} -8.61636i q^{89} +(-1.79692 - 3.12587i) q^{91} +4.86622i q^{93} +10.5220 q^{95} +1.03847i q^{97} -2.34357i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 36 * q - 4 * q^3 + 40 * q^9	\( 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) \) 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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