LMFDB - Embedded newform 4004.2.m.c.2157.26

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			2157.26
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.19267 q^{3} +3.34069i q^{5} +1.00000i q^{7} -1.57755 q^{9} +O(q^{10})$	$q+1.19267 q^{3} +3.34069i q^{5} +1.00000i q^{7} -1.57755 q^{9} -1.00000i q^{11} +(1.17513 - 3.40867i) q^{13} +3.98433i q^{15} -6.04428 q^{17} +2.04334i q^{19} +1.19267i q^{21} -2.12473 q^{23} -6.16024 q^{25} -5.45948 q^{27} -6.58526 q^{29} -11.0672i q^{31} -1.19267i q^{33} -3.34069 q^{35} -8.33828i q^{37} +(1.40154 - 4.06541i) q^{39} -9.38380i q^{41} +5.02523 q^{43} -5.27011i q^{45} +4.19380i q^{47} -1.00000 q^{49} -7.20880 q^{51} +5.62088 q^{53} +3.34069 q^{55} +2.43702i q^{57} -0.541261i q^{59} -2.94398 q^{61} -1.57755i q^{63} +(11.3873 + 3.92576i) q^{65} +3.32892i q^{67} -2.53409 q^{69} +1.30325i q^{71} -10.3206i q^{73} -7.34710 q^{75} +1.00000 q^{77} +5.90676 q^{79} -1.77869 q^{81} +2.01243i q^{83} -20.1921i q^{85} -7.85401 q^{87} +14.4302i q^{89} +(3.40867 + 1.17513i) q^{91} -13.1995i q^{93} -6.82617 q^{95} +12.3726i q^{97} +1.57755i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.19267			0.688586			0.344293	−	0.938862i	$-0.388119\pi$
$3$	1.19267			0.688586			0.344293	+	0.938862i	$0.388119\pi$
$4$		0			0
$5$			3.34069i			1.49400i	0.664822	+	0.747002i	$0.268507\pi$
$5$			3.34069i			1.49400i	−0.664822	+	0.747002i	$0.731493\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.57755			−0.525850
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	1.17513	−	3.40867i	0.325923	−	0.945396i
$13$	1.17513	−	3.40867i	0.325923	−	0.945396i
$14$		0			0
$15$			3.98433i			1.02875i
$16$		0			0
$17$	−6.04428			−1.46595			−0.732976	−	0.680254i	$-0.761869\pi$
$17$	−6.04428			−1.46595			−0.732976	+	0.680254i	$0.761869\pi$
$18$		0			0
$19$			2.04334i			0.468774i	0.972143	+	0.234387i	$0.0753083\pi$
$19$			2.04334i			0.468774i	−0.972143	+	0.234387i	$0.924692\pi$
$20$		0			0
$21$			1.19267i			0.260261i
$22$		0			0
$23$	−2.12473			−0.443036			−0.221518	−	0.975156i	$-0.571101\pi$
$23$	−2.12473			−0.443036			−0.221518	+	0.975156i	$0.571101\pi$
$24$		0			0
$25$	−6.16024			−1.23205
$26$		0			0
$27$	−5.45948			−1.05068
$28$		0			0
$29$	−6.58526			−1.22285			−0.611426	−	0.791302i	$-0.709404\pi$
$29$	−6.58526			−1.22285			−0.611426	+	0.791302i	$0.709404\pi$
$30$		0			0
$31$		−	11.0672i		−	1.98773i	−0.110589	−	0.993866i	$-0.535274\pi$
$31$		−	11.0672i		−	1.98773i	0.110589	−	0.993866i	$-0.464726\pi$
$32$		0			0
$33$		−	1.19267i		−	0.207616i
$34$		0			0
$35$	−3.34069			−0.564680
$36$		0			0
$37$		−	8.33828i		−	1.37080i	−0.728165	−	0.685402i	$-0.759627\pi$
$37$		−	8.33828i		−	1.37080i	0.728165	−	0.685402i	$-0.240373\pi$
$38$		0			0
$39$	1.40154	−	4.06541i	0.224426	−	0.650986i
$40$		0			0
$41$		−	9.38380i		−	1.46550i	−0.680496	−	0.732752i	$-0.738236\pi$
$41$		−	9.38380i		−	1.46550i	0.680496	−	0.732752i	$-0.261764\pi$
$42$		0			0
$43$	5.02523			0.766341			0.383171	−	0.923678i	$-0.374832\pi$
$43$	5.02523			0.766341			0.383171	+	0.923678i	$0.374832\pi$
$44$		0			0
$45$		−	5.27011i		−	0.785622i
$46$		0			0
$47$			4.19380i			0.611729i	0.952075	+	0.305864i	$0.0989454\pi$
$47$			4.19380i			0.611729i	−0.952075	+	0.305864i	$0.901055\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.20880			−1.00943
$52$		0			0
$53$	5.62088			0.772087			0.386044	−	0.922480i	$-0.373841\pi$
$53$	5.62088			0.772087			0.386044	+	0.922480i	$0.373841\pi$
$54$		0			0
$55$	3.34069			0.450459
$56$		0			0
$57$			2.43702i			0.322791i
$58$		0			0
$59$		−	0.541261i		−	0.0704661i	−0.999379	−	0.0352331i	$-0.988783\pi$
$59$		−	0.541261i		−	0.0704661i	0.999379	−	0.0352331i	$-0.0112174\pi$
$60$		0			0
$61$	−2.94398			−0.376938			−0.188469	−	0.982079i	$-0.560353\pi$
$61$	−2.94398			−0.376938			−0.188469	+	0.982079i	$0.560353\pi$
$62$		0			0
$63$		−	1.57755i		−	0.198753i
$64$		0			0
$65$	11.3873	+	3.92576i	1.41243	+	0.486931i
$66$		0			0
$67$			3.32892i			0.406692i	0.979107	+	0.203346i	$0.0651817\pi$
$67$			3.32892i			0.406692i	−0.979107	+	0.203346i	$0.934818\pi$
$68$		0			0
$69$	−2.53409			−0.305068
$70$		0			0
$71$			1.30325i			0.154667i	0.997005	+	0.0773336i	$0.0246406\pi$
$71$			1.30325i			0.154667i	−0.997005	+	0.0773336i	$0.975359\pi$
$72$		0			0
$73$		−	10.3206i		−	1.20794i	−0.797007	−	0.603970i	$-0.793585\pi$
$73$		−	10.3206i		−	1.20794i	0.797007	−	0.603970i	$-0.206415\pi$
$74$		0			0
$75$	−7.34710			−0.848370
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	5.90676			0.664562			0.332281	−	0.943180i	$-0.392182\pi$
$79$	5.90676			0.664562			0.332281	+	0.943180i	$0.392182\pi$
$80$		0			0
$81$	−1.77869			−0.197632
$82$		0			0
$83$			2.01243i			0.220893i	0.993882	+	0.110447i	$0.0352281\pi$
$83$			2.01243i			0.220893i	−0.993882	+	0.110447i	$0.964772\pi$
$84$		0			0
$85$		−	20.1921i		−	2.19014i
$86$		0			0
$87$	−7.85401			−0.842038
$88$		0			0
$89$			14.4302i			1.52960i	0.644266	+	0.764801i	$0.277163\pi$
$89$			14.4302i			1.52960i	−0.644266	+	0.764801i	$0.722837\pi$
$90$		0			0
$91$	3.40867	+	1.17513i	0.357326	+	0.123187i
$92$		0			0
$93$		−	13.1995i		−	1.36872i
$94$		0			0
$95$	−6.82617			−0.700350
$96$		0			0
$97$			12.3726i			1.25624i	0.778115	+	0.628122i	$0.216176\pi$
$97$			12.3726i			1.25624i	−0.778115	+	0.628122i	$0.783824\pi$
$98$		0			0
$99$			1.57755i			0.158550i
$100$		0			0
$101$	−7.69707			−0.765887			−0.382943	−	0.923772i	$-0.625090\pi$
$101$	−7.69707			−0.765887			−0.382943	+	0.923772i	$0.625090\pi$
$102$		0			0
$103$	−5.34884			−0.527036			−0.263518	−	0.964654i	$-0.584883\pi$
$103$	−5.34884			−0.527036			−0.263518	+	0.964654i	$0.584883\pi$
$104$		0			0
$105$	−3.98433			−0.388831
$106$		0			0
$107$	−10.8645			−1.05032			−0.525158	−	0.851005i	$-0.675994\pi$
$107$	−10.8645			−1.05032			−0.525158	+	0.851005i	$0.675994\pi$
$108$		0			0
$109$			14.4600i			1.38502i	0.721408	+	0.692510i	$0.243495\pi$
$109$			14.4600i			1.38502i	−0.721408	+	0.692510i	$0.756505\pi$
$110$		0			0
$111$		−	9.94477i		−	0.943916i
$112$		0			0
$113$	0.750074			0.0705610			0.0352805	−	0.999377i	$-0.488768\pi$
$113$	0.750074			0.0705610			0.0352805	+	0.999377i	$0.488768\pi$
$114$		0			0
$115$		−	7.09806i		−	0.661898i
$116$		0			0
$117$	−1.85383	+	5.37735i	−0.171387	+	0.497136i
$118$		0			0
$119$		−	6.04428i		−	0.554078i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	11.1917i		−	1.00913i
$124$		0			0
$125$		−	3.87599i		−	0.346679i
$126$		0			0
$127$	−4.02434			−0.357102			−0.178551	−	0.983931i	$-0.557141\pi$
$127$	−4.02434			−0.357102			−0.178551	+	0.983931i	$0.557141\pi$
$128$		0			0
$129$	5.99342			0.527691
$130$		0			0
$131$	3.72921			0.325823			0.162911	−	0.986641i	$-0.447912\pi$
$131$	3.72921			0.325823			0.162911	+	0.986641i	$0.447912\pi$
$132$		0			0
$133$	−2.04334			−0.177180
$134$		0			0
$135$		−	18.2385i		−	1.56972i
$136$		0			0
$137$		−	12.6364i		−	1.07960i	−0.841793	−	0.539801i	$-0.818499\pi$
$137$		−	12.6364i		−	1.07960i	0.841793	−	0.539801i	$-0.181501\pi$
$138$		0			0
$139$	13.0511			1.10698			0.553488	−	0.832857i	$-0.313296\pi$
$139$	13.0511			1.10698			0.553488	+	0.832857i	$0.313296\pi$
$140$		0			0
$141$			5.00180i			0.421228i
$142$		0			0
$143$	−3.40867	−	1.17513i	−0.285048	−	0.0982696i
$144$		0			0
$145$		−	21.9993i		−	1.82695i
$146$		0			0
$147$	−1.19267			−0.0983694
$148$		0			0
$149$		−	5.20098i		−	0.426081i	−0.977043	−	0.213041i	$-0.931663\pi$
$149$		−	5.20098i		−	0.426081i	0.977043	−	0.213041i	$-0.0683367\pi$
$150$		0			0
$151$		−	11.8865i		−	0.967310i	−0.875259	−	0.483655i	$-0.839309\pi$
$151$		−	11.8865i		−	0.967310i	0.875259	−	0.483655i	$-0.160691\pi$
$152$		0			0
$153$	9.53514			0.770871
$154$		0			0
$155$	36.9722			2.96968
$156$		0			0
$157$	12.1908			0.972934			0.486467	−	0.873699i	$-0.338285\pi$
$157$	12.1908			0.972934			0.486467	+	0.873699i	$0.338285\pi$
$158$		0			0
$159$	6.70383			0.531648
$160$		0			0
$161$		−	2.12473i		−	0.167452i
$162$		0			0
$163$			4.95715i			0.388274i	0.980974	+	0.194137i	$0.0621906\pi$
$163$			4.95715i			0.388274i	−0.980974	+	0.194137i	$0.937809\pi$
$164$		0			0
$165$	3.98433			0.310180
$166$		0			0
$167$			2.90201i			0.224564i	0.993676	+	0.112282i	$0.0358160\pi$
$167$			2.90201i			0.224564i	−0.993676	+	0.112282i	$0.964184\pi$
$168$		0			0
$169$	−10.2381	−	8.01130i	−0.787548	−	0.616253i
$170$		0			0
$171$		−	3.22347i		−	0.246505i
$172$		0			0
$173$	14.5331			1.10493			0.552464	−	0.833537i	$-0.313688\pi$
$173$	14.5331			1.10493			0.552464	+	0.833537i	$0.313688\pi$
$174$		0			0
$175$		−	6.16024i		−	0.465670i
$176$		0			0
$177$		−	0.645543i		−	0.0485220i
$178$		0			0
$179$	−13.4214			−1.00316			−0.501581	−	0.865111i	$-0.667248\pi$
$179$	−13.4214			−1.00316			−0.501581	+	0.865111i	$0.667248\pi$
$180$		0			0
$181$	−7.03853			−0.523170			−0.261585	−	0.965180i	$-0.584245\pi$
$181$	−7.03853			−0.523170			−0.261585	+	0.965180i	$0.584245\pi$
$182$		0			0
$183$	−3.51118			−0.259554
$184$		0			0
$185$	27.8556			2.04799
$186$		0			0
$187$			6.04428i			0.442001i
$188$		0			0
$189$		−	5.45948i		−	0.397119i
$190$		0			0
$191$	−21.7668			−1.57499			−0.787494	−	0.616323i	$-0.788622\pi$
$191$	−21.7668			−1.57499			−0.787494	+	0.616323i	$0.788622\pi$
$192$		0			0
$193$		−	24.1276i		−	1.73674i	−0.495915	−	0.868371i	$-0.665167\pi$
$193$		−	24.1276i		−	1.73674i	0.495915	−	0.868371i	$-0.334833\pi$
$194$		0			0
$195$	13.5813	+	4.68212i	0.972576	+	0.335294i
$196$		0			0
$197$			10.2937i			0.733397i	0.930340	+	0.366699i	$0.119512\pi$
$197$			10.2937i			0.733397i	−0.930340	+	0.366699i	$0.880488\pi$
$198$		0			0
$199$	−27.0241			−1.91568			−0.957842	−	0.287295i	$-0.907244\pi$
$199$	−27.0241			−1.91568			−0.957842	+	0.287295i	$0.907244\pi$
$200$		0			0
$201$			3.97029i			0.280042i
$202$		0			0
$203$		−	6.58526i		−	0.462195i
$204$		0			0
$205$	31.3484			2.18947
$206$		0			0
$207$	3.35186			0.232971
$208$		0			0
$209$	2.04334			0.141341
$210$		0			0
$211$	5.07083			0.349090			0.174545	−	0.984649i	$-0.444154\pi$
$211$	5.07083			0.349090			0.174545	+	0.984649i	$0.444154\pi$
$212$		0			0
$213$			1.55434i			0.106502i
$214$		0			0
$215$			16.7878i			1.14492i
$216$		0			0
$217$	11.0672			0.751292
$218$		0			0
$219$		−	12.3091i		−	0.831771i
$220$		0			0
$221$	−7.10283	+	20.6030i	−0.477788	+	1.38591i
$222$		0			0
$223$			8.26209i			0.553270i	0.960975	+	0.276635i	$0.0892193\pi$
$223$			8.26209i			0.553270i	−0.960975	+	0.276635i	$0.910781\pi$
$224$		0			0
$225$	9.71807			0.647872
$226$		0			0
$227$		−	0.817533i		−	0.0542615i	−0.999632	−	0.0271308i	$-0.991363\pi$
$227$		−	0.817533i		−	0.0542615i	0.999632	−	0.0271308i	$-0.00863705\pi$
$228$		0			0
$229$			2.37277i			0.156797i	0.996922	+	0.0783985i	$0.0249806\pi$
$229$			2.37277i			0.156797i	−0.996922	+	0.0783985i	$0.975019\pi$
$230$		0			0
$231$	1.19267			0.0784716
$232$		0			0
$233$	−22.2549			−1.45796			−0.728982	−	0.684532i	$-0.760006\pi$
$233$	−22.2549			−1.45796			−0.728982	+	0.684532i	$0.760006\pi$
$234$		0			0
$235$	−14.0102			−0.913925
$236$		0			0
$237$	7.04478			0.457608
$238$		0			0
$239$		−	5.58292i		−	0.361129i	−0.983563	−	0.180565i	$-0.942208\pi$
$239$		−	5.58292i		−	0.361129i	0.983563	−	0.180565i	$-0.0577925\pi$
$240$		0			0
$241$			6.05005i			0.389718i	0.980831	+	0.194859i	$0.0624250\pi$
$241$			6.05005i			0.389718i	−0.980831	+	0.194859i	$0.937575\pi$
$242$		0			0
$243$	14.2571			0.914592
$244$		0			0
$245$		−	3.34069i		−	0.213429i
$246$		0			0
$247$	6.96507	+	2.40119i	0.443177	+	0.152784i
$248$		0			0
$249$			2.40016i			0.152104i
$250$		0			0
$251$	−14.6616			−0.925434			−0.462717	−	0.886506i	$-0.653125\pi$
$251$	−14.6616			−0.925434			−0.462717	+	0.886506i	$0.653125\pi$
$252$		0			0
$253$			2.12473i			0.133580i
$254$		0			0
$255$		−	24.0824i		−	1.50810i
$256$		0			0
$257$	1.00509			0.0626961			0.0313480	−	0.999509i	$-0.490020\pi$
$257$	1.00509			0.0626961			0.0313480	+	0.999509i	$0.490020\pi$
$258$		0			0
$259$	8.33828			0.518115
$260$		0			0
$261$	10.3886			0.643036
$262$		0			0
$263$	−26.1393			−1.61182			−0.805909	−	0.592040i	$-0.798323\pi$
$263$	−26.1393			−1.61182			−0.805909	+	0.592040i	$0.798323\pi$
$264$		0			0
$265$			18.7776i			1.15350i
$266$		0			0
$267$			17.2105i			1.05326i
$268$		0			0
$269$	−13.5569			−0.826576			−0.413288	−	0.910600i	$-0.635620\pi$
$269$	−13.5569			−0.826576			−0.413288	+	0.910600i	$0.635620\pi$
$270$		0			0
$271$		−	22.2781i		−	1.35330i	−0.736306	−	0.676649i	$-0.763431\pi$
$271$		−	22.2781i		−	1.35330i	0.736306	−	0.676649i	$-0.236569\pi$
$272$		0			0
$273$	4.06541	+	1.40154i	0.246050	+	0.0848251i
$274$		0			0
$275$			6.16024i			0.371476i
$276$		0			0
$277$	−21.0835			−1.26679			−0.633393	−	0.773830i	$-0.718338\pi$
$277$	−21.0835			−1.26679			−0.633393	+	0.773830i	$0.718338\pi$
$278$		0			0
$279$			17.4591i			1.04525i
$280$		0			0
$281$		−	27.5818i		−	1.64539i	−0.568483	−	0.822695i	$-0.692469\pi$
$281$		−	27.5818i		−	1.64539i	0.568483	−	0.822695i	$-0.307531\pi$
$282$		0			0
$283$	−16.1815			−0.961892			−0.480946	−	0.876750i	$-0.659707\pi$
$283$	−16.1815			−0.961892			−0.480946	+	0.876750i	$0.659707\pi$
$284$		0			0
$285$	−8.14133			−0.482251
$286$		0			0
$287$	9.38380			0.553908
$288$		0			0
$289$	19.5333			1.14902
$290$		0			0
$291$			14.7563i			0.865031i
$292$		0			0
$293$		−	11.7705i		−	0.687642i	−0.939035	−	0.343821i	$-0.888279\pi$
$293$		−	11.7705i		−	0.687642i	0.939035	−	0.343821i	$-0.111721\pi$
$294$		0			0
$295$	1.80819			0.105277
$296$		0			0
$297$			5.45948i			0.316791i
$298$		0			0
$299$	−2.49684	+	7.24250i	−0.144396	+	0.418845i
$300$		0			0
$301$			5.02523i			0.289650i
$302$		0			0
$303$	−9.18003			−0.527379
$304$		0			0
$305$		−	9.83494i		−	0.563147i
$306$		0			0
$307$		−	18.1759i		−	1.03735i	−0.854971	−	0.518676i	$-0.826425\pi$
$307$		−	18.1759i		−	1.03735i	0.854971	−	0.518676i	$-0.173575\pi$
$308$		0			0
$309$	−6.37937			−0.362910
$310$		0			0
$311$	4.66713			0.264649			0.132324	−	0.991206i	$-0.457756\pi$
$311$	4.66713			0.264649			0.132324	+	0.991206i	$0.457756\pi$
$312$		0			0
$313$	−3.61157			−0.204138			−0.102069	−	0.994777i	$-0.532546\pi$
$313$	−3.61157			−0.204138			−0.102069	+	0.994777i	$0.532546\pi$
$314$		0			0
$315$	5.27011			0.296937
$316$		0			0
$317$			7.07744i			0.397509i	0.980049	+	0.198754i	$0.0636896\pi$
$317$			7.07744i			0.397509i	−0.980049	+	0.198754i	$0.936310\pi$
$318$		0			0
$319$			6.58526i			0.368704i
$320$		0			0
$321$	−12.9578			−0.723232
$322$		0			0
$323$		−	12.3505i		−	0.687200i
$324$		0			0
$325$	−7.23910	+	20.9982i	−0.401553	+	1.16477i
$326$		0			0
$327$			17.2460i			0.953705i
$328$		0			0
$329$	−4.19380			−0.231212
$330$		0			0
$331$		−	15.1141i		−	0.830747i	−0.909651	−	0.415374i	$-0.863651\pi$
$331$		−	15.1141i		−	0.830747i	0.909651	−	0.415374i	$-0.136349\pi$
$332$		0			0
$333$			13.1540i			0.720837i
$334$		0			0
$335$	−11.1209			−0.607600
$336$		0			0
$337$	−3.57706			−0.194855			−0.0974274	−	0.995243i	$-0.531061\pi$
$337$	−3.57706			−0.194855			−0.0974274	+	0.995243i	$0.531061\pi$
$338$		0			0
$339$	0.894587			0.0485873
$340$		0			0
$341$	−11.0672			−0.599324
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		−	8.46561i		−	0.455773i
$346$		0			0
$347$	−34.8605			−1.87141			−0.935705	−	0.352783i	$-0.885235\pi$
$347$	−34.8605			−1.87141			−0.935705	+	0.352783i	$0.885235\pi$
$348$		0			0
$349$		−	15.5191i		−	0.830721i	−0.909657	−	0.415360i	$-0.863655\pi$
$349$		−	15.5191i		−	0.830721i	0.909657	−	0.415360i	$-0.136345\pi$
$350$		0			0
$351$	−6.41562	+	18.6096i	−0.342441	+	0.993307i
$352$		0			0
$353$		−	31.6119i		−	1.68253i	−0.540622	−	0.841266i	$-0.681811\pi$
$353$		−	31.6119i		−	1.68253i	0.540622	−	0.841266i	$-0.318189\pi$
$354$		0			0
$355$	−4.35375			−0.231073
$356$		0			0
$357$		−	7.20880i		−	0.381530i
$358$		0			0
$359$		−	25.7429i		−	1.35866i	−0.733834	−	0.679328i	$-0.762271\pi$
$359$		−	25.7429i		−	1.35866i	0.733834	−	0.679328i	$-0.237729\pi$
$360$		0			0
$361$	14.8248			0.780251
$362$		0			0
$363$	−1.19267			−0.0625987
$364$		0			0
$365$	34.4781			1.80467
$366$		0			0
$367$	5.29064			0.276169			0.138085	−	0.990420i	$-0.455905\pi$
$367$	5.29064			0.276169			0.138085	+	0.990420i	$0.455905\pi$
$368$		0			0
$369$			14.8034i			0.770635i
$370$		0			0
$371$			5.62088i			0.291822i
$372$		0			0
$373$	10.4084			0.538925			0.269463	−	0.963011i	$-0.413154\pi$
$373$	10.4084			0.538925			0.269463	+	0.963011i	$0.413154\pi$
$374$		0			0
$375$		−	4.62276i		−	0.238718i
$376$		0			0
$377$	−7.73856	+	22.4470i	−0.398556	+	1.15608i
$378$		0			0
$379$			20.8842i			1.07275i	0.843979	+	0.536376i	$0.180207\pi$
$379$			20.8842i			1.07275i	−0.843979	+	0.536376i	$0.819793\pi$
$380$		0			0
$381$	−4.79969			−0.245896
$382$		0			0
$383$			27.8780i			1.42450i	0.701926	+	0.712250i	$0.252324\pi$
$383$			27.8780i			1.42450i	−0.701926	+	0.712250i	$0.747676\pi$
$384$		0			0
$385$			3.34069i			0.170258i
$386$		0			0
$387$	−7.92756			−0.402980
$388$		0			0
$389$	−20.4129			−1.03498			−0.517488	−	0.855690i	$-0.673133\pi$
$389$	−20.4129			−1.03498			−0.517488	+	0.855690i	$0.673133\pi$
$390$		0			0
$391$	12.8424			0.649470
$392$		0			0
$393$	4.44770			0.224357
$394$		0			0
$395$			19.7327i			0.992858i
$396$		0			0
$397$			28.0782i			1.40921i	0.709602	+	0.704603i	$0.248875\pi$
$397$			28.0782i			1.40921i	−0.709602	+	0.704603i	$0.751125\pi$
$398$		0			0
$399$	−2.43702			−0.122003
$400$		0			0
$401$			1.70290i			0.0850386i	0.999096	+	0.0425193i	$0.0135384\pi$
$401$			1.70290i			0.0850386i	−0.999096	+	0.0425193i	$0.986462\pi$
$402$		0			0
$403$	−37.7246	−	13.0055i	−1.87919	−	0.647849i
$404$		0			0
$405$		−	5.94206i		−	0.295263i
$406$		0			0
$407$	−8.33828			−0.413313
$408$		0			0
$409$			24.5419i			1.21352i	0.794886	+	0.606758i	$0.207530\pi$
$409$			24.5419i			1.21352i	−0.794886	+	0.606758i	$0.792470\pi$
$410$		0			0
$411$		−	15.0710i		−	0.743399i
$412$		0			0
$413$	0.541261			0.0266337
$414$		0			0
$415$	−6.72292			−0.330015
$416$		0			0
$417$	15.5656			0.762248
$418$		0			0
$419$	−10.8729			−0.531174			−0.265587	−	0.964087i	$-0.585566\pi$
$419$	−10.8729			−0.531174			−0.265587	+	0.964087i	$0.585566\pi$
$420$		0			0
$421$			34.4254i			1.67779i	0.544294	+	0.838895i	$0.316798\pi$
$421$			34.4254i			1.67779i	−0.544294	+	0.838895i	$0.683202\pi$
$422$		0			0
$423$		−	6.61593i		−	0.321677i
$424$		0			0
$425$	37.2342			1.80612
$426$		0			0
$427$		−	2.94398i		−	0.142469i
$428$		0			0
$429$	−4.06541	−	1.40154i	−0.196280	−	0.0676670i
$430$		0			0
$431$		−	17.1389i		−	0.825553i	−0.910832	−	0.412777i	$-0.864559\pi$
$431$		−	17.1389i		−	0.825553i	0.910832	−	0.412777i	$-0.135441\pi$
$432$		0			0
$433$	−39.3387			−1.89050			−0.945248	−	0.326353i	$-0.894180\pi$
$433$	−39.3387			−1.89050			−0.945248	+	0.326353i	$0.894180\pi$
$434$		0			0
$435$		−	26.2378i		−	1.25801i
$436$		0			0
$437$		−	4.34154i		−	0.207684i
$438$		0			0
$439$	10.3528			0.494111			0.247055	−	0.969001i	$-0.420537\pi$
$439$	10.3528			0.494111			0.247055	+	0.969001i	$0.420537\pi$
$440$		0			0
$441$	1.57755			0.0751214
$442$		0			0
$443$	24.7349			1.17519			0.587596	−	0.809155i	$-0.300075\pi$
$443$	24.7349			1.17519			0.587596	+	0.809155i	$0.300075\pi$
$444$		0			0
$445$	−48.2070			−2.28523
$446$		0			0
$447$		−	6.20303i		−	0.293393i
$448$		0			0
$449$			16.5195i			0.779605i	0.920899	+	0.389802i	$0.127457\pi$
$449$			16.5195i			0.779605i	−0.920899	+	0.389802i	$0.872543\pi$
$450$		0			0
$451$	−9.38380			−0.441866
$452$		0			0
$453$		−	14.1766i		−	0.666076i
$454$		0			0
$455$	−3.92576	+	11.3873i	−0.184043	+	0.533847i
$456$		0			0
$457$		−	26.6674i		−	1.24745i	−0.781645	−	0.623723i	$-0.785619\pi$
$457$		−	26.6674i		−	1.24745i	0.781645	−	0.623723i	$-0.214381\pi$
$458$		0			0
$459$	32.9986			1.54024
$460$		0			0
$461$			23.3543i			1.08772i	0.839177	+	0.543859i	$0.183037\pi$
$461$			23.3543i			1.08772i	−0.839177	+	0.543859i	$0.816963\pi$
$462$		0			0
$463$		−	8.08830i		−	0.375895i	−0.982179	−	0.187948i	$-0.939816\pi$
$463$		−	8.08830i		−	0.375895i	0.982179	−	0.187948i	$-0.0601835\pi$
$464$		0			0
$465$	44.0955			2.04488
$466$		0			0
$467$	−24.4186			−1.12996			−0.564979	−	0.825105i	$-0.691116\pi$
$467$	−24.4186			−1.12996			−0.564979	+	0.825105i	$0.691116\pi$
$468$		0			0
$469$	−3.32892			−0.153715
$470$		0			0
$471$	14.5396			0.669949
$472$		0			0
$473$		−	5.02523i		−	0.231061i
$474$		0			0
$475$		−	12.5874i		−	0.577551i
$476$		0			0
$477$	−8.86722			−0.406002
$478$		0			0
$479$			36.1194i			1.65034i	0.564888	+	0.825168i	$0.308919\pi$
$479$			36.1194i			1.65034i	−0.564888	+	0.825168i	$0.691081\pi$
$480$		0			0
$481$	−28.4225	−	9.79859i	−1.29595	−	0.446777i
$482$		0			0
$483$		−	2.53409i		−	0.115305i
$484$		0			0
$485$	−41.3329			−1.87683
$486$		0			0
$487$		−	41.5504i		−	1.88283i	−0.337250	−	0.941415i	$-0.609497\pi$
$487$		−	41.5504i		−	1.88283i	0.337250	−	0.941415i	$-0.390503\pi$
$488$		0			0
$489$			5.91222i			0.267360i
$490$		0			0
$491$	37.2990			1.68328			0.841641	−	0.540037i	$-0.181590\pi$
$491$	37.2990			1.68328			0.841641	+	0.540037i	$0.181590\pi$
$492$		0			0
$493$	39.8031			1.79264
$494$		0			0
$495$	−5.27011			−0.236874
$496$		0			0
$497$	−1.30325			−0.0584587
$498$		0			0
$499$			12.0012i			0.537246i	0.963245	+	0.268623i	$0.0865686\pi$
$499$			12.0012i			0.537246i	−0.963245	+	0.268623i	$0.913431\pi$
$500$		0			0
$501$			3.46113i			0.154632i
$502$		0			0
$503$	−2.40604			−0.107280			−0.0536399	−	0.998560i	$-0.517082\pi$
$503$	−2.40604			−0.107280			−0.0536399	+	0.998560i	$0.517082\pi$
$504$		0			0
$505$		−	25.7135i		−	1.14424i
$506$		0			0
$507$	−12.2107	−	9.55479i	−0.542294	−	0.424343i
$508$		0			0
$509$			8.52715i			0.377959i	0.981981	+	0.188980i	$0.0605180\pi$
$509$			8.52715i			0.377959i	−0.981981	+	0.188980i	$0.939482\pi$
$510$		0			0
$511$	10.3206			0.456559
$512$		0			0
$513$		−	11.1556i		−	0.492530i
$514$		0			0
$515$		−	17.8688i		−	0.787394i
$516$		0			0
$517$	4.19380			0.184443
$518$		0			0
$519$	17.3331			0.760838
$520$		0			0
$521$	41.7634			1.82969			0.914844	−	0.403807i	$-0.132313\pi$
$521$	41.7634			1.82969			0.914844	+	0.403807i	$0.132313\pi$
$522$		0			0
$523$	33.5930			1.46892			0.734460	−	0.678652i	$-0.237435\pi$
$523$	33.5930			1.46892			0.734460	+	0.678652i	$0.237435\pi$
$524$		0			0
$525$		−	7.34710i		−	0.320654i
$526$		0			0
$527$			66.8934i			2.91392i
$528$		0			0
$529$	−18.4855			−0.803719
$530$		0			0
$531$			0.853865i			0.0370546i
$532$		0			0
$533$	−31.9863	−	11.0272i	−1.38548	−	0.477642i
$534$		0			0
$535$		−	36.2951i		−	1.56917i
$536$		0			0
$537$	−16.0072			−0.690763
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$			11.8845i			0.510955i	0.966815	+	0.255477i	$0.0822327\pi$
$541$			11.8845i			0.510955i	−0.966815	+	0.255477i	$0.917767\pi$
$542$		0			0
$543$	−8.39461			−0.360247
$544$		0			0
$545$	−48.3066			−2.06923
$546$		0			0
$547$	12.7500			0.545149			0.272575	−	0.962135i	$-0.412125\pi$
$547$	12.7500			0.545149			0.272575	+	0.962135i	$0.412125\pi$
$548$		0			0
$549$	4.64428			0.198213
$550$		0			0
$551$		−	13.4559i		−	0.573241i
$552$		0			0
$553$			5.90676i			0.251181i
$554$		0			0
$555$	33.2224			1.41021
$556$		0			0
$557$			16.6628i			0.706026i	0.935618	+	0.353013i	$0.114843\pi$
$557$			16.6628i			0.706026i	−0.935618	+	0.353013i	$0.885157\pi$
$558$		0			0
$559$	5.90532	−	17.1294i	0.249769	−	0.724496i
$560$		0			0
$561$			7.20880i			0.304356i
$562$		0			0
$563$	23.8635			1.00573			0.502864	−	0.864366i	$-0.332280\pi$
$563$	23.8635			1.00573			0.502864	+	0.864366i	$0.332280\pi$
$564$		0			0
$565$			2.50577i			0.105418i
$566$		0			0
$567$		−	1.77869i		−	0.0746980i
$568$		0			0
$569$	−10.6444			−0.446238			−0.223119	−	0.974791i	$-0.571624\pi$
$569$	−10.6444			−0.446238			−0.223119	+	0.974791i	$0.571624\pi$
$570$		0			0
$571$	11.5810			0.484648			0.242324	−	0.970195i	$-0.422090\pi$
$571$	11.5810			0.484648			0.242324	+	0.970195i	$0.422090\pi$
$572$		0			0
$573$	−25.9605			−1.08451
$574$		0			0
$575$	13.0888			0.545842
$576$		0			0
$577$			45.2654i			1.88442i	0.335020	+	0.942211i	$0.391257\pi$
$577$			45.2654i			1.88442i	−0.335020	+	0.942211i	$0.608743\pi$
$578$		0			0
$579$		−	28.7761i		−	1.19590i
$580$		0			0
$581$	−2.01243			−0.0834898
$582$		0			0
$583$		−	5.62088i		−	0.232793i
$584$		0			0
$585$	−17.9641	−	6.19308i	−0.742724	−	0.256052i
$586$		0			0
$587$		−	41.6169i		−	1.71771i	−0.512216	−	0.858856i	$-0.671175\pi$
$587$		−	41.6169i		−	1.71771i	0.512216	−	0.858856i	$-0.328825\pi$
$588$		0			0
$589$	22.6141			0.931797
$590$		0			0
$591$			12.2770i			0.505007i
$592$		0			0
$593$			23.2591i			0.955135i	0.878595	+	0.477568i	$0.158481\pi$
$593$			23.2591i			0.955135i	−0.878595	+	0.477568i	$0.841519\pi$
$594$		0			0
$595$	20.1921			0.827794
$596$		0			0
$597$	−32.2306			−1.31911
$598$		0			0
$599$	−22.8211			−0.932445			−0.466222	−	0.884668i	$-0.654385\pi$
$599$	−22.8211			−0.932445			−0.466222	+	0.884668i	$0.654385\pi$
$600$		0			0
$601$	12.6166			0.514640			0.257320	−	0.966326i	$-0.417160\pi$
$601$	12.6166			0.514640			0.257320	+	0.966326i	$0.417160\pi$
$602$		0			0
$603$		−	5.25154i		−	0.213859i
$604$		0			0
$605$		−	3.34069i		−	0.135819i
$606$		0			0
$607$	7.31241			0.296801			0.148401	−	0.988927i	$-0.452587\pi$
$607$	7.31241			0.296801			0.148401	+	0.988927i	$0.452587\pi$
$608$		0			0
$609$		−	7.85401i		−	0.318261i
$610$		0			0
$611$	14.2953	+	4.92828i	0.578326	+	0.199377i
$612$		0			0
$613$			22.3135i			0.901236i	0.892717	+	0.450618i	$0.148796\pi$
$613$			22.3135i			0.901236i	−0.892717	+	0.450618i	$0.851204\pi$
$614$		0			0
$615$	37.3882			1.50764
$616$		0			0
$617$		−	23.4608i		−	0.944496i	−0.881466	−	0.472248i	$-0.843443\pi$
$617$		−	23.4608i		−	0.944496i	0.881466	−	0.472248i	$-0.156557\pi$
$618$		0			0
$619$		−	7.04552i		−	0.283183i	−0.989925	−	0.141592i	$-0.954778\pi$
$619$		−	7.04552i		−	0.283183i	0.989925	−	0.141592i	$-0.0452220\pi$
$620$		0			0
$621$	11.5999			0.465489
$622$		0			0
$623$	−14.4302			−0.578135
$624$		0			0
$625$	−17.8527			−0.714107
$626$		0			0
$627$	2.43702			0.0973251
$628$		0			0
$629$			50.3988i			2.00953i
$630$		0			0
$631$			12.1224i			0.482585i	0.970452	+	0.241292i	$0.0775713\pi$
$631$			12.1224i			0.482585i	−0.970452	+	0.241292i	$0.922429\pi$
$632$		0			0
$633$	6.04780			0.240379
$634$		0			0
$635$		−	13.4441i		−	0.533512i
$636$		0			0
$637$	−1.17513	+	3.40867i	−0.0465605	+	0.135057i
$638$		0			0
$639$		−	2.05594i		−	0.0813317i
$640$		0			0
$641$	31.3897			1.23982			0.619910	−	0.784673i	$-0.287169\pi$
$641$	31.3897			1.23982			0.619910	+	0.784673i	$0.287169\pi$
$642$		0			0
$643$		−	21.7448i		−	0.857530i	−0.903416	−	0.428765i	$-0.858949\pi$
$643$		−	21.7448i		−	0.857530i	0.903416	−	0.428765i	$-0.141051\pi$
$644$		0			0
$645$			20.0222i			0.788373i
$646$		0			0
$647$	−29.2740			−1.15088			−0.575440	−	0.817844i	$-0.695169\pi$
$647$	−29.2740			−1.15088			−0.575440	+	0.817844i	$0.695169\pi$
$648$		0			0
$649$	−0.541261			−0.0212463
$650$		0			0
$651$	13.1995			0.517329
$652$		0			0
$653$	−10.6898			−0.418326			−0.209163	−	0.977881i	$-0.567074\pi$
$653$	−10.6898			−0.418326			−0.209163	+	0.977881i	$0.567074\pi$
$654$		0			0
$655$			12.4582i			0.486780i
$656$		0			0
$657$			16.2813i			0.635195i
$658$		0			0
$659$	−11.5838			−0.451241			−0.225621	−	0.974215i	$-0.572441\pi$
$659$	−11.5838			−0.451241			−0.225621	+	0.974215i	$0.572441\pi$
$660$		0			0
$661$		−	21.2312i		−	0.825796i	−0.910777	−	0.412898i	$-0.864517\pi$
$661$		−	21.2312i		−	0.825796i	0.910777	−	0.412898i	$-0.135483\pi$
$662$		0			0
$663$	−8.47130	+	24.5724i	−0.328998	+	0.954315i
$664$		0			0
$665$		−	6.82617i		−	0.264707i
$666$		0			0
$667$	13.9919			0.541768
$668$		0			0
$669$			9.85390i			0.380974i
$670$		0			0
$671$			2.94398i			0.113651i
$672$		0			0
$673$	−8.66402			−0.333973			−0.166987	−	0.985959i	$-0.553404\pi$
$673$	−8.66402			−0.333973			−0.166987	+	0.985959i	$0.553404\pi$
$674$		0			0
$675$	33.6317			1.29449
$676$		0			0
$677$	38.7548			1.48947			0.744735	−	0.667361i	$-0.232576\pi$
$677$	38.7548			1.48947			0.744735	+	0.667361i	$0.232576\pi$
$678$		0			0
$679$	−12.3726			−0.474815
$680$		0			0
$681$		−	0.975043i		−	0.0373637i
$682$		0			0
$683$			18.1796i			0.695624i	0.937564	+	0.347812i	$0.113075\pi$
$683$			18.1796i			0.695624i	−0.937564	+	0.347812i	$0.886925\pi$
$684$		0			0
$685$	42.2144			1.61293
$686$		0			0
$687$			2.82992i			0.107968i
$688$		0			0
$689$	6.60529	−	19.1598i	0.251641	−	0.729928i
$690$		0			0
$691$		−	30.2307i		−	1.15003i	−0.818143	−	0.575015i	$-0.804996\pi$
$691$		−	30.2307i		−	1.15003i	0.818143	−	0.575015i	$-0.195004\pi$
$692$		0			0
$693$	−1.57755			−0.0599261
$694$		0			0
$695$			43.5996i			1.65383i
$696$		0			0
$697$			56.7183i			2.14836i
$698$		0			0
$699$	−26.5426			−1.00393
$700$		0			0
$701$	−13.7681			−0.520012			−0.260006	−	0.965607i	$-0.583725\pi$
$701$	−13.7681			−0.520012			−0.260006	+	0.965607i	$0.583725\pi$
$702$		0			0
$703$	17.0379			0.642597
$704$		0			0
$705$	−16.7095			−0.629316
$706$		0			0
$707$		−	7.69707i		−	0.289478i
$708$		0			0
$709$			5.53856i			0.208005i	0.994577	+	0.104003i	$0.0331650\pi$
$709$			5.53856i			0.208005i	−0.994577	+	0.104003i	$0.966835\pi$
$710$		0			0
$711$	−9.31820			−0.349460
$712$		0			0
$713$			23.5148i			0.880638i
$714$		0			0
$715$	3.92576	−	11.3873i	0.146815	−	0.425862i
$716$		0			0
$717$		−	6.65856i		−	0.248668i
$718$		0			0
$719$	−22.3234			−0.832523			−0.416262	−	0.909245i	$-0.636660\pi$
$719$	−22.3234			−0.832523			−0.416262	+	0.909245i	$0.636660\pi$
$720$		0			0
$721$		−	5.34884i		−	0.199201i
$722$		0			0
$723$			7.21569i			0.268354i
$724$		0			0
$725$	40.5667			1.50661
$726$		0			0
$727$	−15.8861			−0.589184			−0.294592	−	0.955623i	$-0.595184\pi$
$727$	−15.8861			−0.589184			−0.294592	+	0.955623i	$0.595184\pi$
$728$		0			0
$729$	22.3400			0.827407
$730$		0			0
$731$	−30.3739			−1.12342
$732$		0			0
$733$		−	27.4219i		−	1.01285i	−0.862283	−	0.506426i	$-0.830966\pi$
$733$		−	27.4219i		−	1.01285i	0.862283	−	0.506426i	$-0.169034\pi$
$734$		0			0
$735$		−	3.98433i		−	0.146964i
$736$		0			0
$737$	3.32892			0.122622
$738$		0			0
$739$		−	30.9472i		−	1.13841i	−0.822196	−	0.569205i	$-0.807251\pi$
$739$		−	30.9472i		−	1.13841i	0.822196	−	0.569205i	$-0.192749\pi$
$740$		0			0
$741$	8.30700	+	2.86382i	0.305165	+	0.105205i
$742$		0			0
$743$			28.9080i			1.06053i	0.847832	+	0.530266i	$0.177908\pi$
$743$			28.9080i			1.06053i	−0.847832	+	0.530266i	$0.822092\pi$
$744$		0			0
$745$	17.3749			0.636567
$746$		0			0
$747$		−	3.17471i		−	0.116157i
$748$		0			0
$749$		−	10.8645i		−	0.396982i
$750$		0			0
$751$	30.0280			1.09574			0.547869	−	0.836564i	$-0.315439\pi$
$751$	30.0280			1.09574			0.547869	+	0.836564i	$0.315439\pi$
$752$		0			0
$753$	−17.4864			−0.637240
$754$		0			0
$755$	39.7092			1.44517
$756$		0			0
$757$	29.9081			1.08703			0.543515	−	0.839400i	$-0.317093\pi$
$757$	29.9081			1.08703			0.543515	+	0.839400i	$0.317093\pi$
$758$		0			0
$759$			2.53409i			0.0919816i
$760$		0			0
$761$			9.91967i			0.359588i	0.983704	+	0.179794i	$0.0575431\pi$
$761$			9.91967i			0.359588i	−0.983704	+	0.179794i	$0.942457\pi$
$762$		0			0
$763$	−14.4600			−0.523489
$764$		0			0
$765$			31.8540i			1.15168i
$766$		0			0
$767$	−1.84498	−	0.636054i	−0.0666184	−	0.0229666i
$768$		0			0
$769$		−	6.41521i		−	0.231339i	−0.993288	−	0.115669i	$-0.963099\pi$
$769$		−	6.41521i		−	0.231339i	0.993288	−	0.115669i	$-0.0369013\pi$
$770$		0			0
$771$	1.19874			0.0431716
$772$		0			0
$773$			37.5827i			1.35176i	0.737014	+	0.675878i	$0.236235\pi$
$773$			37.5827i			1.35176i	−0.737014	+	0.675878i	$0.763765\pi$
$774$		0			0
$775$			68.1767i			2.44898i
$776$		0			0
$777$	9.94477			0.356767
$778$		0			0
$779$	19.1743			0.686990
$780$		0			0
$781$	1.30325			0.0466339
$782$		0			0
$783$	35.9521			1.28482
$784$		0			0
$785$			40.7258i			1.45357i
$786$		0			0
$787$			20.3144i			0.724131i	0.932153	+	0.362065i	$0.117928\pi$
$787$			20.3144i			0.724131i	−0.932153	+	0.362065i	$0.882072\pi$
$788$		0			0
$789$	−31.1754			−1.10987
$790$		0			0
$791$			0.750074i			0.0266696i
$792$		0			0
$793$	−3.45957	+	10.0351i	−0.122853	+	0.356356i
$794$		0			0
$795$			22.3954i			0.794285i
$796$		0			0
$797$	34.6540			1.22751			0.613753	−	0.789498i	$-0.289659\pi$
$797$	34.6540			1.22751			0.613753	+	0.789498i	$0.289659\pi$
$798$		0			0
$799$		−	25.3485i		−	0.896765i
$800$		0			0
$801$		−	22.7644i		−	0.804341i
$802$		0			0
$803$	−10.3206			−0.364208
$804$		0			0
$805$	7.09806			0.250174
$806$		0			0
$807$	−16.1688			−0.569169
$808$		0			0
$809$	−49.5135			−1.74080			−0.870400	−	0.492345i	$-0.836140\pi$
$809$	−49.5135			−1.74080			−0.870400	+	0.492345i	$0.836140\pi$
$810$		0			0
$811$		−	45.8185i		−	1.60890i	−0.594018	−	0.804452i	$-0.702459\pi$
$811$		−	45.8185i		−	1.60890i	0.594018	−	0.804452i	$-0.297541\pi$
$812$		0			0
$813$		−	26.5703i		−	0.931862i
$814$		0			0
$815$	−16.5603			−0.580083
$816$		0			0
$817$			10.2683i			0.359241i
$818$		0			0
$819$	−5.37735	−	1.85383i	−0.187900	−	0.0647781i
$820$		0			0
$821$		−	18.3126i		−	0.639115i	−0.947567	−	0.319557i	$-0.896466\pi$
$821$		−	18.3126i		−	0.639115i	0.947567	−	0.319557i	$-0.103534\pi$
$822$		0			0
$823$	−22.4864			−0.783827			−0.391914	−	0.920002i	$-0.628187\pi$
$823$	−22.4864			−0.783827			−0.391914	+	0.920002i	$0.628187\pi$
$824$		0			0
$825$			7.34710i			0.255793i
$826$		0			0
$827$		−	30.7815i		−	1.07038i	−0.844732	−	0.535189i	$-0.820240\pi$
$827$		−	30.7815i		−	1.07038i	0.844732	−	0.535189i	$-0.179760\pi$
$828$		0			0
$829$	−12.9274			−0.448989			−0.224494	−	0.974475i	$-0.572073\pi$
$829$	−12.9274			−0.448989			−0.224494	+	0.974475i	$0.572073\pi$
$830$		0			0
$831$	−25.1456			−0.872290
$832$		0			0
$833$	6.04428			0.209422
$834$		0			0
$835$	−9.69472			−0.335500
$836$		0			0
$837$			60.4213i			2.08847i
$838$		0			0
$839$		−	9.22637i		−	0.318529i	−0.987236	−	0.159265i	$-0.949088\pi$
$839$		−	9.22637i		−	0.318529i	0.987236	−	0.159265i	$-0.0509123\pi$
$840$		0			0
$841$	14.3656			0.495367
$842$		0			0
$843$		−	32.8958i		−	1.13299i
$844$		0			0
$845$	26.7633	−	34.2024i	0.920685	−	1.17660i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	−19.2991			−0.662345
$850$		0			0
$851$			17.7166i			0.607316i
$852$		0			0
$853$			44.4458i			1.52179i	0.648872	+	0.760897i	$0.275241\pi$
$853$			44.4458i			1.52179i	−0.648872	+	0.760897i	$0.724759\pi$
$854$		0			0
$855$	10.7686			0.368279
$856$		0			0
$857$	51.2030			1.74906			0.874530	−	0.484972i	$-0.161170\pi$
$857$	51.2030			1.74906			0.874530	+	0.484972i	$0.161170\pi$
$858$		0			0
$859$	−41.9986			−1.43297			−0.716486	−	0.697601i	$-0.754251\pi$
$859$	−41.9986			−1.43297			−0.716486	+	0.697601i	$0.754251\pi$
$860$		0			0
$861$	11.1917			0.381413
$862$		0			0
$863$		−	32.1825i		−	1.09551i	−0.836640	−	0.547753i	$-0.815483\pi$
$863$		−	32.1825i		−	1.09551i	0.836640	−	0.547753i	$-0.184517\pi$
$864$		0			0
$865$			48.5505i			1.65077i
$866$		0			0
$867$	23.2967			0.791196
$868$		0			0
$869$		−	5.90676i		−	0.200373i
$870$		0			0
$871$	11.3472	+	3.91193i	0.384485	+	0.132551i
$872$		0			0
$873$		−	19.5183i		−	0.660595i
$874$		0			0
$875$	3.87599			0.131032
$876$		0			0
$877$		−	13.3496i		−	0.450785i	−0.974268	−	0.225392i	$-0.927634\pi$
$877$		−	13.3496i		−	0.450785i	0.974268	−	0.225392i	$-0.0723664\pi$
$878$		0			0
$879$		−	14.0383i		−	0.473500i
$880$		0			0
$881$	26.5742			0.895306			0.447653	−	0.894207i	$-0.352260\pi$
$881$	26.5742			0.895306			0.447653	+	0.894207i	$0.352260\pi$
$882$		0			0
$883$	−33.8057			−1.13765			−0.568826	−	0.822458i	$-0.692602\pi$
$883$	−33.8057			−1.13765			−0.568826	+	0.822458i	$0.692602\pi$
$884$		0			0
$885$	2.15656			0.0724920
$886$		0			0
$887$	−20.4314			−0.686019			−0.343009	−	0.939332i	$-0.611446\pi$
$887$	−20.4314			−0.686019			−0.343009	+	0.939332i	$0.611446\pi$
$888$		0			0
$889$		−	4.02434i		−	0.134972i
$890$		0			0
$891$			1.77869i			0.0595884i
$892$		0			0
$893$	−8.56935			−0.286762
$894$		0			0
$895$		−	44.8368i		−	1.49873i
$896$		0			0
$897$	−2.97789	+	8.63788i	−0.0994289	+	0.288411i
$898$		0			0
$899$			72.8805i			2.43070i
$900$		0			0
$901$	−33.9742			−1.13184
$902$		0			0
$903$			5.99342i			0.199449i
$904$		0			0
$905$		−	23.5136i		−	0.781618i
$906$		0			0
$907$	10.3350			0.343169			0.171584	−	0.985169i	$-0.445111\pi$
$907$	10.3350			0.343169			0.171584	+	0.985169i	$0.445111\pi$
$908$		0			0
$909$	12.1425			0.402741
$910$		0			0
$911$	−1.83892			−0.0609263			−0.0304631	−	0.999536i	$-0.509698\pi$
$911$	−1.83892			−0.0609263			−0.0304631	+	0.999536i	$0.509698\pi$
$912$		0			0
$913$	2.01243			0.0666018
$914$		0			0
$915$		−	11.7298i		−	0.387775i
$916$		0			0
$917$			3.72921i			0.123149i
$918$		0			0
$919$	28.2384			0.931499			0.465750	−	0.884917i	$-0.345785\pi$
$919$	28.2384			0.931499			0.465750	+	0.884917i	$0.345785\pi$
$920$		0			0
$921$		−	21.6777i		−	0.714305i
$922$		0			0
$923$	4.44235	+	1.53149i	0.146222	+	0.0504097i
$924$		0			0
$925$			51.3657i			1.68889i
$926$		0			0
$927$	8.43805			0.277142
$928$		0			0
$929$		−	16.1784i		−	0.530795i	−0.964139	−	0.265397i	$-0.914497\pi$
$929$		−	16.1784i		−	0.530795i	0.964139	−	0.265397i	$-0.0855031\pi$
$930$		0			0
$931$		−	2.04334i		−	0.0669677i
$932$		0			0
$933$	5.56632			0.182233
$934$		0			0
$935$	−20.1921			−0.660351
$936$		0			0
$937$	39.8163			1.30074			0.650371	−	0.759617i	$-0.274613\pi$
$937$	39.8163			1.30074			0.650371	+	0.759617i	$0.274613\pi$
$938$		0			0
$939$	−4.30740			−0.140567
$940$		0			0
$941$		−	25.2779i		−	0.824036i	−0.911176	−	0.412018i	$-0.864824\pi$
$941$		−	25.2779i		−	0.824036i	0.911176	−	0.412018i	$-0.135176\pi$
$942$		0			0
$943$			19.9380i			0.649271i
$944$		0			0
$945$	18.2385			0.593297
$946$		0			0
$947$		−	18.0734i		−	0.587307i	−0.955912	−	0.293654i	$-0.905129\pi$
$947$		−	18.0734i		−	0.587307i	0.955912	−	0.293654i	$-0.0948712\pi$
$948$		0			0
$949$	−35.1797	−	12.1281i	−1.14198	−	0.393696i
$950$		0			0
$951$			8.44102i			0.273719i
$952$		0			0
$953$	22.5608			0.730815			0.365407	−	0.930848i	$-0.380930\pi$
$953$	22.5608			0.730815			0.365407	+	0.930848i	$0.380930\pi$
$954$		0			0
$955$		−	72.7161i		−	2.35304i
$956$		0			0
$957$			7.85401i			0.253884i
$958$		0			0
$959$	12.6364			0.408051
$960$		0			0
$961$	−91.4835			−2.95108
$962$		0			0
$963$	17.1394			0.552308
$964$		0			0
$965$	80.6029			2.59470
$966$		0			0
$967$		−	14.4274i		−	0.463954i	−0.972721	−	0.231977i	$-0.925481\pi$
$967$		−	14.4274i		−	0.463954i	0.972721	−	0.231977i	$-0.0745195\pi$
$968$		0			0
$969$		−	14.7300i		−	0.473196i
$970$		0			0
$971$	3.14109			0.100802			0.0504012	−	0.998729i	$-0.483950\pi$
$971$	3.14109			0.100802			0.0504012	+	0.998729i	$0.483950\pi$
$972$		0			0
$973$			13.0511i			0.418398i
$974$		0			0
$975$	−8.63382	+	25.0439i	−0.276504	+	0.802046i
$976$		0			0
$977$			22.6663i			0.725161i	0.931953	+	0.362580i	$0.118104\pi$
$977$			22.6663i			0.725161i	−0.931953	+	0.362580i	$0.881896\pi$
$978$		0			0
$979$	14.4302			0.461193
$980$		0			0
$981$		−	22.8114i		−	0.728313i
$982$		0			0
$983$		−	4.09966i		−	0.130759i	−0.997860	−	0.0653794i	$-0.979174\pi$
$983$		−	4.09966i		−	0.130759i	0.997860	−	0.0653794i	$-0.0208258\pi$
$984$		0			0
$985$	−34.3882			−1.09570
$986$		0			0
$987$	−5.00180			−0.159209
$988$		0			0
$989$	−10.6773			−0.339517
$990$		0			0
$991$	39.7946			1.26412			0.632058	−	0.774921i	$-0.282210\pi$
$991$	39.7946			1.26412			0.632058	+	0.774921i	$0.282210\pi$
$992$		0			0
$993$		−	18.0261i		−	0.572041i
$994$		0			0
$995$		−	90.2791i		−	2.86204i
$996$		0			0
$997$	−36.1265			−1.14414			−0.572069	−	0.820206i	$-0.693859\pi$
$997$	−36.1265			−1.14414			−0.572069	+	0.820206i	$0.693859\pi$
$998$		0			0
$999$			45.5227i			1.44027i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.26	yes	36
13.12	even	2	inner	4004.2.m.c.2157.25	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.25	✓	36	13.12	even	2	inner
4004.2.m.c.2157.26	yes	36	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.19267 q^{3} +3.34069i q^{5} +1.00000i q^{7} -1.57755 q^{9} +O(q^{10})\)	\(q+1.19267 q^{3} +3.34069i q^{5} +1.00000i q^{7} -1.57755 q^{9} -1.00000i q^{11} +(1.17513 - 3.40867i) q^{13} +3.98433i q^{15} -6.04428 q^{17} +2.04334i q^{19} +1.19267i q^{21} -2.12473 q^{23} -6.16024 q^{25} -5.45948 q^{27} -6.58526 q^{29} -11.0672i q^{31} -1.19267i q^{33} -3.34069 q^{35} -8.33828i q^{37} +(1.40154 - 4.06541i) q^{39} -9.38380i q^{41} +5.02523 q^{43} -5.27011i q^{45} +4.19380i q^{47} -1.00000 q^{49} -7.20880 q^{51} +5.62088 q^{53} +3.34069 q^{55} +2.43702i q^{57} -0.541261i q^{59} -2.94398 q^{61} -1.57755i q^{63} +(11.3873 + 3.92576i) q^{65} +3.32892i q^{67} -2.53409 q^{69} +1.30325i q^{71} -10.3206i q^{73} -7.34710 q^{75} +1.00000 q^{77} +5.90676 q^{79} -1.77869 q^{81} +2.01243i q^{83} -20.1921i q^{85} -7.85401 q^{87} +14.4302i q^{89} +(3.40867 + 1.17513i) q^{91} -13.1995i q^{93} -6.82617 q^{95} +12.3726i q^{97} +1.57755i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 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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