LMFDB - Embedded newform 4004.2.m.b.2157.11

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:	$30$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.11
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.b.2157.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.822838 q^{3} -0.417533i q^{5} -1.00000i q^{7} -2.32294 q^{9} +O(q^{10})$	$q-0.822838 q^{3} -0.417533i q^{5} -1.00000i q^{7} -2.32294 q^{9} -1.00000i q^{11} +(3.41087 - 1.16874i) q^{13} +0.343562i q^{15} -2.01565 q^{17} +6.08253i q^{19} +0.822838i q^{21} -3.72968 q^{23} +4.82567 q^{25} +4.37991 q^{27} -6.39229 q^{29} +5.62923i q^{31} +0.822838i q^{33} -0.417533 q^{35} -1.18609i q^{37} +(-2.80659 + 0.961687i) q^{39} +1.80439i q^{41} +4.65388 q^{43} +0.969904i q^{45} -4.48275i q^{47} -1.00000 q^{49} +1.65856 q^{51} -4.51746 q^{53} -0.417533 q^{55} -5.00493i q^{57} +10.8927i q^{59} +6.40671 q^{61} +2.32294i q^{63} +(-0.487990 - 1.42415i) q^{65} -6.82342i q^{67} +3.06892 q^{69} +1.67328i q^{71} +1.29221i q^{73} -3.97074 q^{75} -1.00000 q^{77} +13.0290 q^{79} +3.36485 q^{81} -14.6645i q^{83} +0.841603i q^{85} +5.25982 q^{87} +14.8484i q^{89} +(-1.16874 - 3.41087i) q^{91} -4.63194i q^{93} +2.53966 q^{95} -16.0714i q^{97} +2.32294i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 18 q^{9}+O(q^{10}) $ 30 * q + 18 * q^9	$ 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) $ 30 * q + 18 * q^9 + 4 * q^13 + 12 * q^17 + 6 * q^23 - 6 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 30 * q^49 + 60 * q^51 - 38 * q^53 - 2 * q^55 - 36 * q^61 + 10 * q^65 + 36 * q^69 - 20 * q^75 - 30 * q^77 - 10 * q^79 - 42 * q^81 + 36 * q^87 + 6 * q^91 - 2 * 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
$3$	−0.822838			−0.475066			−0.237533	−	0.971380i	$-0.576339\pi$
$3$	−0.822838			−0.475066			−0.237533	+	0.971380i	$0.576339\pi$
$4$		0			0
$5$		−	0.417533i		−	0.186727i	−0.995632	−	0.0933633i	$-0.970238\pi$
$5$		−	0.417533i		−	0.186727i	0.995632	−	0.0933633i	$-0.0297618\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−2.32294			−0.774313
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	3.41087	−	1.16874i	0.946005	−	0.324151i
$13$	3.41087	−	1.16874i	0.946005	−	0.324151i
$14$		0			0
$15$			0.343562i			0.0887074i
$16$		0			0
$17$	−2.01565			−0.488868			−0.244434	−	0.969666i	$-0.578602\pi$
$17$	−2.01565			−0.488868			−0.244434	+	0.969666i	$0.578602\pi$
$18$		0			0
$19$			6.08253i			1.39543i	0.716377	+	0.697714i	$0.245799\pi$
$19$			6.08253i			1.39543i	−0.716377	+	0.697714i	$0.754201\pi$
$20$		0			0
$21$			0.822838i			0.179558i
$22$		0			0
$23$	−3.72968			−0.777693			−0.388846	−	0.921303i	$-0.627126\pi$
$23$	−3.72968			−0.777693			−0.388846	+	0.921303i	$0.627126\pi$
$24$		0			0
$25$	4.82567			0.965133
$26$		0			0
$27$	4.37991			0.842915
$28$		0			0
$29$	−6.39229			−1.18702			−0.593509	−	0.804827i	$-0.702258\pi$
$29$	−6.39229			−1.18702			−0.593509	+	0.804827i	$0.702258\pi$
$30$		0			0
$31$			5.62923i			1.01104i	0.862815	+	0.505520i	$0.168699\pi$
$31$			5.62923i			1.01104i	−0.862815	+	0.505520i	$0.831301\pi$
$32$		0			0
$33$			0.822838i			0.143238i
$34$		0			0
$35$	−0.417533			−0.0705760
$36$		0			0
$37$		−	1.18609i		−	0.194992i	−0.995236	−	0.0974962i	$-0.968917\pi$
$37$		−	1.18609i		−	0.194992i	0.995236	−	0.0974962i	$-0.0310834\pi$
$38$		0			0
$39$	−2.80659	+	0.961687i	−0.449415	+	0.153993i
$40$		0			0
$41$			1.80439i			0.281798i	0.990024	+	0.140899i	$0.0449992\pi$
$41$			1.80439i			0.281798i	−0.990024	+	0.140899i	$0.955001\pi$
$42$		0			0
$43$	4.65388			0.709710			0.354855	−	0.934921i	$-0.384530\pi$
$43$	4.65388			0.709710			0.354855	+	0.934921i	$0.384530\pi$
$44$		0			0
$45$			0.969904i			0.144585i
$46$		0			0
$47$		−	4.48275i		−	0.653876i	−0.945046	−	0.326938i	$-0.893983\pi$
$47$		−	4.48275i		−	0.653876i	0.945046	−	0.326938i	$-0.106017\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.65856			0.232244
$52$		0			0
$53$	−4.51746			−0.620521			−0.310260	−	0.950652i	$-0.600416\pi$
$53$	−4.51746			−0.620521			−0.310260	+	0.950652i	$0.600416\pi$
$54$		0			0
$55$	−0.417533			−0.0563002
$56$		0			0
$57$		−	5.00493i		−	0.662920i
$58$		0			0
$59$			10.8927i			1.41811i	0.705154	+	0.709055i	$0.250878\pi$
$59$			10.8927i			1.41811i	−0.705154	+	0.709055i	$0.749122\pi$
$60$		0			0
$61$	6.40671			0.820296			0.410148	−	0.912019i	$-0.365477\pi$
$61$	6.40671			0.820296			0.410148	+	0.912019i	$0.365477\pi$
$62$		0			0
$63$			2.32294i			0.292663i
$64$		0			0
$65$	−0.487990	−	1.42415i	−0.0605277	−	0.176644i
$66$		0			0
$67$		−	6.82342i		−	0.833613i	−0.908995	−	0.416807i	$-0.863149\pi$
$67$		−	6.82342i		−	0.833613i	0.908995	−	0.416807i	$-0.136851\pi$
$68$		0			0
$69$	3.06892			0.369455
$70$		0			0
$71$			1.67328i			0.198582i	0.995058	+	0.0992910i	$0.0316575\pi$
$71$			1.67328i			0.198582i	−0.995058	+	0.0992910i	$0.968343\pi$
$72$		0			0
$73$			1.29221i			0.151242i	0.997137	+	0.0756208i	$0.0240939\pi$
$73$			1.29221i			0.151242i	−0.997137	+	0.0756208i	$0.975906\pi$
$74$		0			0
$75$	−3.97074			−0.458502
$76$		0			0
$77$	−1.00000			−0.113961
$78$		0			0
$79$	13.0290			1.46588			0.732939	−	0.680294i	$-0.238148\pi$
$79$	13.0290			1.46588			0.732939	+	0.680294i	$0.238148\pi$
$80$		0			0
$81$	3.36485			0.373873
$82$		0			0
$83$		−	14.6645i		−	1.60964i	−0.593519	−	0.804820i	$-0.702262\pi$
$83$		−	14.6645i		−	1.60964i	0.593519	−	0.804820i	$-0.297738\pi$
$84$		0			0
$85$			0.841603i			0.0912847i
$86$		0			0
$87$	5.25982			0.563912
$88$		0			0
$89$			14.8484i			1.57393i	0.616999	+	0.786964i	$0.288348\pi$
$89$			14.8484i			1.57393i	−0.616999	+	0.786964i	$0.711652\pi$
$90$		0			0
$91$	−1.16874	−	3.41087i	−0.122518	−	0.357556i
$92$		0			0
$93$		−	4.63194i		−	0.480310i
$94$		0			0
$95$	2.53966			0.260564
$96$		0			0
$97$		−	16.0714i		−	1.63181i	−0.578187	−	0.815904i	$-0.696240\pi$
$97$		−	16.0714i		−	1.63181i	0.578187	−	0.815904i	$-0.303760\pi$
$98$		0			0
$99$			2.32294i			0.233464i
$100$		0			0
$101$	−8.52647			−0.848415			−0.424208	−	0.905565i	$-0.639447\pi$
$101$	−8.52647			−0.848415			−0.424208	+	0.905565i	$0.639447\pi$
$102$		0			0
$103$	15.0709			1.48498			0.742488	−	0.669860i	$-0.233646\pi$
$103$	15.0709			1.48498			0.742488	+	0.669860i	$0.233646\pi$
$104$		0			0
$105$	0.343562			0.0335283
$106$		0			0
$107$	−0.378034			−0.0365459			−0.0182729	−	0.999833i	$-0.505817\pi$
$107$	−0.378034			−0.0365459			−0.0182729	+	0.999833i	$0.505817\pi$
$108$		0			0
$109$			7.14955i			0.684802i	0.939554	+	0.342401i	$0.111240\pi$
$109$			7.14955i			0.684802i	−0.939554	+	0.342401i	$0.888760\pi$
$110$		0			0
$111$			0.975961i			0.0926342i
$112$		0			0
$113$	20.2285			1.90294			0.951468	−	0.307747i	$-0.0995751\pi$
$113$	20.2285			1.90294			0.951468	+	0.307747i	$0.0995751\pi$
$114$		0			0
$115$			1.55727i			0.145216i
$116$		0			0
$117$	−7.92324	+	2.71492i	−0.732504	+	0.250995i
$118$		0			0
$119$			2.01565i			0.184775i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	1.48472i		−	0.133872i
$124$		0			0
$125$		−	4.10254i		−	0.366943i
$126$		0			0
$127$	−0.801712			−0.0711405			−0.0355702	−	0.999367i	$-0.511325\pi$
$127$	−0.801712			−0.0711405			−0.0355702	+	0.999367i	$0.511325\pi$
$128$		0			0
$129$	−3.82939			−0.337159
$130$		0			0
$131$	4.43254			0.387273			0.193636	−	0.981073i	$-0.437972\pi$
$131$	4.43254			0.387273			0.193636	+	0.981073i	$0.437972\pi$
$132$		0			0
$133$	6.08253			0.527422
$134$		0			0
$135$		−	1.82876i		−	0.157395i
$136$		0			0
$137$			2.68350i			0.229266i	0.993408	+	0.114633i	$0.0365693\pi$
$137$			2.68350i			0.229266i	−0.993408	+	0.114633i	$0.963431\pi$
$138$		0			0
$139$	18.1538			1.53978			0.769892	−	0.638174i	$-0.220310\pi$
$139$	18.1538			1.53978			0.769892	+	0.638174i	$0.220310\pi$
$140$		0			0
$141$			3.68857i			0.310634i
$142$		0			0
$143$	−1.16874	−	3.41087i	−0.0977353	−	0.285231i
$144$		0			0
$145$			2.66900i			0.221648i
$146$		0			0
$147$	0.822838			0.0678665
$148$		0			0
$149$			4.93859i			0.404585i	0.979325	+	0.202293i	$0.0648392\pi$
$149$			4.93859i			0.404585i	−0.979325	+	0.202293i	$0.935161\pi$
$150$		0			0
$151$			7.77779i			0.632948i	0.948601	+	0.316474i	$0.102499\pi$
$151$			7.77779i			0.632948i	−0.948601	+	0.316474i	$0.897501\pi$
$152$		0			0
$153$	4.68224			0.378537
$154$		0			0
$155$	2.35039			0.188788
$156$		0			0
$157$	−5.85647			−0.467397			−0.233698	−	0.972309i	$-0.575083\pi$
$157$	−5.85647			−0.467397			−0.233698	+	0.972309i	$0.575083\pi$
$158$		0			0
$159$	3.71714			0.294788
$160$		0			0
$161$			3.72968i			0.293940i
$162$		0			0
$163$			19.7238i			1.54489i	0.635083	+	0.772444i	$0.280966\pi$
$163$			19.7238i			1.54489i	−0.635083	+	0.772444i	$0.719034\pi$
$164$		0			0
$165$	0.343562			0.0267463
$166$		0			0
$167$			6.17917i			0.478158i	0.971000	+	0.239079i	$0.0768456\pi$
$167$			6.17917i			0.478158i	−0.971000	+	0.239079i	$0.923154\pi$
$168$		0			0
$169$	10.2681	−	7.97287i	0.789852	−	0.613298i
$170$		0			0
$171$		−	14.1293i		−	1.08050i
$172$		0			0
$173$	7.94731			0.604223			0.302111	−	0.953273i	$-0.402309\pi$
$173$	7.94731			0.604223			0.302111	+	0.953273i	$0.402309\pi$
$174$		0			0
$175$		−	4.82567i		−	0.364786i
$176$		0			0
$177$		−	8.96293i		−	0.673695i
$178$		0			0
$179$	12.0117			0.897796			0.448898	−	0.893583i	$-0.351817\pi$
$179$	12.0117			0.897796			0.448898	+	0.893583i	$0.351817\pi$
$180$		0			0
$181$	−9.61175			−0.714436			−0.357218	−	0.934021i	$-0.616275\pi$
$181$	−9.61175			−0.714436			−0.357218	+	0.934021i	$0.616275\pi$
$182$		0			0
$183$	−5.27169			−0.389694
$184$		0			0
$185$	−0.495233			−0.0364103
$186$		0			0
$187$			2.01565i			0.147399i
$188$		0			0
$189$		−	4.37991i		−	0.318592i
$190$		0			0
$191$	−4.89105			−0.353904			−0.176952	−	0.984219i	$-0.556624\pi$
$191$	−4.89105			−0.353904			−0.176952	+	0.984219i	$0.556624\pi$
$192$		0			0
$193$			11.7291i			0.844280i	0.906531	+	0.422140i	$0.138721\pi$
$193$			11.7291i			0.844280i	−0.906531	+	0.422140i	$0.861279\pi$
$194$		0			0
$195$	0.401537	+	1.17185i	0.0287546	+	0.0839177i
$196$		0			0
$197$			8.82421i			0.628699i	0.949307	+	0.314349i	$0.101786\pi$
$197$			8.82421i			0.628699i	−0.949307	+	0.314349i	$0.898214\pi$
$198$		0			0
$199$	19.5136			1.38329			0.691643	−	0.722240i	$-0.256887\pi$
$199$	19.5136			1.38329			0.691643	+	0.722240i	$0.256887\pi$
$200$		0			0
$201$			5.61457i			0.396021i
$202$		0			0
$203$			6.39229i			0.448651i
$204$		0			0
$205$	0.753391			0.0526191
$206$		0			0
$207$	8.66382			0.602177
$208$		0			0
$209$	6.08253			0.420737
$210$		0			0
$211$	−2.61715			−0.180172			−0.0900861	−	0.995934i	$-0.528714\pi$
$211$	−2.61715			−0.180172			−0.0900861	+	0.995934i	$0.528714\pi$
$212$		0			0
$213$		−	1.37684i		−	0.0943395i
$214$		0			0
$215$		−	1.94315i		−	0.132522i
$216$		0			0
$217$	5.62923			0.382137
$218$		0			0
$219$		−	1.06328i		−	0.0718497i
$220$		0			0
$221$	−6.87513	+	2.35578i	−0.462472	+	0.158467i
$222$		0			0
$223$			1.65428i			0.110779i	0.998465	+	0.0553893i	$0.0176400\pi$
$223$			1.65428i			0.110779i	−0.998465	+	0.0553893i	$0.982360\pi$
$224$		0			0
$225$	−11.2097			−0.747315
$226$		0			0
$227$		−	8.52161i		−	0.565599i	−0.959179	−	0.282800i	$-0.908737\pi$
$227$		−	8.52161i		−	0.565599i	0.959179	−	0.282800i	$-0.0912632\pi$
$228$		0			0
$229$			8.03101i			0.530704i	0.964152	+	0.265352i	$0.0854882\pi$
$229$			8.03101i			0.530704i	−0.964152	+	0.265352i	$0.914512\pi$
$230$		0			0
$231$	0.822838			0.0541388
$232$		0			0
$233$	19.3842			1.26990			0.634952	−	0.772552i	$-0.281020\pi$
$233$	19.3842			1.26990			0.634952	+	0.772552i	$0.281020\pi$
$234$		0			0
$235$	−1.87170			−0.122096
$236$		0			0
$237$	−10.7208			−0.696388
$238$		0			0
$239$			3.98275i			0.257623i	0.991669	+	0.128811i	$0.0411162\pi$
$239$			3.98275i			0.257623i	−0.991669	+	0.128811i	$0.958884\pi$
$240$		0			0
$241$		−	2.14464i		−	0.138149i	−0.997612	−	0.0690743i	$-0.977995\pi$
$241$		−	2.14464i		−	0.138149i	0.997612	−	0.0690743i	$-0.0220045\pi$
$242$		0			0
$243$	−15.9085			−1.02053
$244$		0			0
$245$			0.417533i			0.0266752i
$246$		0			0
$247$	7.10892	+	20.7467i	0.452330	+	1.32008i
$248$		0			0
$249$			12.0665i			0.764684i
$250$		0			0
$251$	19.1209			1.20690			0.603449	−	0.797401i	$-0.293793\pi$
$251$	19.1209			1.20690			0.603449	+	0.797401i	$0.293793\pi$
$252$		0			0
$253$			3.72968i			0.234483i
$254$		0			0
$255$		−	0.692503i		−	0.0433662i
$256$		0			0
$257$	−10.7935			−0.673283			−0.336641	−	0.941633i	$-0.609291\pi$
$257$	−10.7935			−0.673283			−0.336641	+	0.941633i	$0.609291\pi$
$258$		0			0
$259$	−1.18609			−0.0737002
$260$		0			0
$261$	14.8489			0.919124
$262$		0			0
$263$	−4.84360			−0.298669			−0.149335	−	0.988787i	$-0.547713\pi$
$263$	−4.84360			−0.298669			−0.149335	+	0.988787i	$0.547713\pi$
$264$		0			0
$265$			1.88619i			0.115868i
$266$		0			0
$267$		−	12.2178i		−	0.747719i
$268$		0			0
$269$	19.5558			1.19234			0.596169	−	0.802859i	$-0.296689\pi$
$269$	19.5558			1.19234			0.596169	+	0.802859i	$0.296689\pi$
$270$		0			0
$271$			4.81529i			0.292508i	0.989247	+	0.146254i	$0.0467217\pi$
$271$			4.81529i			0.292508i	−0.989247	+	0.146254i	$0.953278\pi$
$272$		0			0
$273$	0.961687	+	2.80659i	0.0582040	+	0.169863i
$274$		0			0
$275$		−	4.82567i		−	0.290999i
$276$		0			0
$277$	−4.26083			−0.256009			−0.128004	−	0.991774i	$-0.540857\pi$
$277$	−4.26083			−0.256009			−0.128004	+	0.991774i	$0.540857\pi$
$278$		0			0
$279$		−	13.0764i		−	0.782861i
$280$		0			0
$281$		−	28.1311i		−	1.67816i	−0.544007	−	0.839081i	$-0.683093\pi$
$281$		−	28.1311i		−	1.67816i	0.544007	−	0.839081i	$-0.316907\pi$
$282$		0			0
$283$	21.8713			1.30011			0.650056	−	0.759886i	$-0.274746\pi$
$283$	21.8713			1.30011			0.650056	+	0.759886i	$0.274746\pi$
$284$		0			0
$285$	−2.08973			−0.123785
$286$		0			0
$287$	1.80439			0.106510
$288$		0			0
$289$	−12.9371			−0.761008
$290$		0			0
$291$			13.2242i			0.775216i
$292$		0			0
$293$		−	8.14072i		−	0.475586i	−0.971316	−	0.237793i	$-0.923576\pi$
$293$		−	8.14072i		−	0.475586i	0.971316	−	0.237793i	$-0.0764239\pi$
$294$		0			0
$295$	4.54807			0.264799
$296$		0			0
$297$		−	4.37991i		−	0.254148i
$298$		0			0
$299$	−12.7215	+	4.35905i	−0.735701	+	0.252090i
$300$		0			0
$301$		−	4.65388i		−	0.268245i
$302$		0			0
$303$	7.01590			0.403053
$304$		0			0
$305$		−	2.67502i		−	0.153171i
$306$		0			0
$307$			14.0678i			0.802894i	0.915882	+	0.401447i	$0.131493\pi$
$307$			14.0678i			0.802894i	−0.915882	+	0.401447i	$0.868507\pi$
$308$		0			0
$309$	−12.4009			−0.705461
$310$		0			0
$311$	14.2929			0.810476			0.405238	−	0.914211i	$-0.367189\pi$
$311$	14.2929			0.810476			0.405238	+	0.914211i	$0.367189\pi$
$312$		0			0
$313$	−24.7501			−1.39896			−0.699478	−	0.714654i	$-0.746584\pi$
$313$	−24.7501			−1.39896			−0.699478	+	0.714654i	$0.746584\pi$
$314$		0			0
$315$	0.969904			0.0546479
$316$		0			0
$317$			31.8826i			1.79070i	0.445359	+	0.895352i	$0.353076\pi$
$317$			31.8826i			1.79070i	−0.445359	+	0.895352i	$0.646924\pi$
$318$		0			0
$319$			6.39229i			0.357900i
$320$		0			0
$321$	0.311060			0.0173617
$322$		0			0
$323$		−	12.2603i		−	0.682180i
$324$		0			0
$325$	16.4597	−	5.63997i	0.913021	−	0.312849i
$326$		0			0
$327$		−	5.88292i		−	0.325326i
$328$		0			0
$329$	−4.48275			−0.247142
$330$		0			0
$331$		−	1.87589i		−	0.103108i	−0.998670	−	0.0515542i	$-0.983583\pi$
$331$		−	1.87589i		−	0.103108i	0.998670	−	0.0515542i	$-0.0164175\pi$
$332$		0			0
$333$			2.75522i			0.150985i
$334$		0			0
$335$	−2.84901			−0.155658
$336$		0			0
$337$	−4.11116			−0.223949			−0.111974	−	0.993711i	$-0.535717\pi$
$337$	−4.11116			−0.223949			−0.111974	+	0.993711i	$0.535717\pi$
$338$		0			0
$339$	−16.6448			−0.904020
$340$		0			0
$341$	5.62923			0.304840
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	1.28138i		−	0.0689871i
$346$		0			0
$347$	27.6836			1.48613			0.743066	−	0.669218i	$-0.233371\pi$
$347$	27.6836			1.48613			0.743066	+	0.669218i	$0.233371\pi$
$348$		0			0
$349$		−	35.1301i		−	1.88047i	−0.340527	−	0.940235i	$-0.610605\pi$
$349$		−	35.1301i		−	1.88047i	0.340527	−	0.940235i	$-0.389395\pi$
$350$		0			0
$351$	14.9393	−	5.11900i	0.797402	−	0.273232i
$352$		0			0
$353$		−	26.2476i		−	1.39702i	−0.715602	−	0.698509i	$-0.753847\pi$
$353$		−	26.2476i		−	1.39702i	0.715602	−	0.698509i	$-0.246153\pi$
$354$		0			0
$355$	0.698651			0.0370806
$356$		0			0
$357$		−	1.65856i		−	0.0877801i
$358$		0			0
$359$		−	13.8645i		−	0.731740i	−0.930666	−	0.365870i	$-0.880772\pi$
$359$		−	13.8645i		−	0.731740i	0.930666	−	0.365870i	$-0.119228\pi$
$360$		0			0
$361$	−17.9972			−0.947219
$362$		0			0
$363$	0.822838			0.0431878
$364$		0			0
$365$	0.539540			0.0282408
$366$		0			0
$367$	16.9181			0.883118			0.441559	−	0.897232i	$-0.354426\pi$
$367$	16.9181			0.883118			0.441559	+	0.897232i	$0.354426\pi$
$368$		0			0
$369$		−	4.19148i		−	0.218200i
$370$		0			0
$371$			4.51746i			0.234535i
$372$		0			0
$373$	8.92535			0.462137			0.231068	−	0.972938i	$-0.425778\pi$
$373$	8.92535			0.462137			0.231068	+	0.972938i	$0.425778\pi$
$374$		0			0
$375$			3.37573i			0.174322i
$376$		0			0
$377$	−21.8033	+	7.47096i	−1.12293	+	0.384774i
$378$		0			0
$379$			17.9566i			0.922371i	0.887304	+	0.461185i	$0.152576\pi$
$379$			17.9566i			0.922371i	−0.887304	+	0.461185i	$0.847424\pi$
$380$		0			0
$381$	0.659679			0.0337964
$382$		0			0
$383$			32.0978i			1.64012i	0.572276	+	0.820061i	$0.306061\pi$
$383$			32.0978i			1.64012i	−0.572276	+	0.820061i	$0.693939\pi$
$384$		0			0
$385$			0.417533i			0.0212795i
$386$		0			0
$387$	−10.8107			−0.549538
$388$		0			0
$389$	19.2404			0.975526			0.487763	−	0.872976i	$-0.337813\pi$
$389$	19.2404			0.975526			0.487763	+	0.872976i	$0.337813\pi$
$390$		0			0
$391$	7.51775			0.380189
$392$		0			0
$393$	−3.64726			−0.183980
$394$		0			0
$395$		−	5.44005i		−	0.273718i
$396$		0			0
$397$		−	13.1486i		−	0.659911i	−0.943996	−	0.329955i	$-0.892966\pi$
$397$		−	13.1486i		−	0.659911i	0.943996	−	0.329955i	$-0.107034\pi$
$398$		0			0
$399$	−5.00493			−0.250560
$400$		0			0
$401$			10.6391i			0.531290i	0.964071	+	0.265645i	$0.0855848\pi$
$401$			10.6391i			0.531290i	−0.964071	+	0.265645i	$0.914415\pi$
$402$		0			0
$403$	6.57913	+	19.2006i	0.327730	+	0.956449i
$404$		0			0
$405$		−	1.40494i		−	0.0698120i
$406$		0			0
$407$	−1.18609			−0.0587924
$408$		0			0
$409$			4.31418i			0.213322i	0.994295	+	0.106661i	$0.0340160\pi$
$409$			4.31418i			0.213322i	−0.994295	+	0.106661i	$0.965984\pi$
$410$		0			0
$411$		−	2.20808i		−	0.108917i
$412$		0			0
$413$	10.8927			0.535995
$414$		0			0
$415$	−6.12293			−0.300563
$416$		0			0
$417$	−14.9376			−0.731499
$418$		0			0
$419$	23.7813			1.16179			0.580895	−	0.813978i	$-0.302703\pi$
$419$	23.7813			1.16179			0.580895	+	0.813978i	$0.302703\pi$
$420$		0			0
$421$			24.4301i			1.19065i	0.803486	+	0.595324i	$0.202976\pi$
$421$			24.4301i			1.19065i	−0.803486	+	0.595324i	$0.797024\pi$
$422$		0			0
$423$			10.4131i			0.506304i
$424$		0			0
$425$	−9.72687			−0.471823
$426$		0			0
$427$		−	6.40671i		−	0.310043i
$428$		0			0
$429$	0.961687	+	2.80659i	0.0464307	+	0.135504i
$430$		0			0
$431$			40.1860i			1.93569i	0.251549	+	0.967845i	$0.419060\pi$
$431$			40.1860i			1.93569i	−0.251549	+	0.967845i	$0.580940\pi$
$432$		0			0
$433$	−11.8191			−0.567990			−0.283995	−	0.958826i	$-0.591660\pi$
$433$	−11.8191			−0.567990			−0.283995	+	0.958826i	$0.591660\pi$
$434$		0			0
$435$		−	2.19615i		−	0.105297i
$436$		0			0
$437$		−	22.6859i		−	1.08521i
$438$		0			0
$439$	30.7214			1.46625			0.733126	−	0.680093i	$-0.238060\pi$
$439$	30.7214			1.46625			0.733126	+	0.680093i	$0.238060\pi$
$440$		0			0
$441$	2.32294			0.110616
$442$		0			0
$443$	−0.811487			−0.0385549			−0.0192775	−	0.999814i	$-0.506137\pi$
$443$	−0.811487			−0.0385549			−0.0192775	+	0.999814i	$0.506137\pi$
$444$		0			0
$445$	6.19970			0.293894
$446$		0			0
$447$		−	4.06366i		−	0.192205i
$448$		0			0
$449$			6.21639i			0.293369i	0.989183	+	0.146685i	$0.0468603\pi$
$449$			6.21639i			0.293369i	−0.989183	+	0.146685i	$0.953140\pi$
$450$		0			0
$451$	1.80439			0.0849652
$452$		0			0
$453$		−	6.39986i		−	0.300692i
$454$		0			0
$455$	−1.42415	+	0.487990i	−0.0667653	+	0.0228773i
$456$		0			0
$457$		−	23.1955i		−	1.08504i	−0.840043	−	0.542519i	$-0.817471\pi$
$457$		−	23.1955i		−	1.08504i	0.840043	−	0.542519i	$-0.182529\pi$
$458$		0			0
$459$	−8.82839			−0.412074
$460$		0			0
$461$			3.31043i			0.154182i	0.997024	+	0.0770910i	$0.0245632\pi$
$461$			3.31043i			0.154182i	−0.997024	+	0.0770910i	$0.975437\pi$
$462$		0			0
$463$			18.1650i			0.844197i	0.906550	+	0.422099i	$0.138706\pi$
$463$			18.1650i			0.844197i	−0.906550	+	0.422099i	$0.861294\pi$
$464$		0			0
$465$	−1.93399			−0.0896867
$466$		0			0
$467$	−5.01493			−0.232063			−0.116032	−	0.993246i	$-0.537017\pi$
$467$	−5.01493			−0.232063			−0.116032	+	0.993246i	$0.537017\pi$
$468$		0			0
$469$	−6.82342			−0.315076
$470$		0			0
$471$	4.81892			0.222044
$472$		0			0
$473$		−	4.65388i		−	0.213986i
$474$		0			0
$475$			29.3523i			1.34677i
$476$		0			0
$477$	10.4938			0.480477
$478$		0			0
$479$			13.2205i			0.604062i	0.953298	+	0.302031i	$0.0976646\pi$
$479$			13.2205i			0.604062i	−0.953298	+	0.302031i	$0.902335\pi$
$480$		0			0
$481$	−1.38624	−	4.04561i	−0.0632070	−	0.184464i
$482$		0			0
$483$		−	3.06892i		−	0.139641i
$484$		0			0
$485$	−6.71037			−0.304702
$486$		0			0
$487$		−	2.37117i		−	0.107448i	−0.998556	−	0.0537241i	$-0.982891\pi$
$487$		−	2.37117i		−	0.107448i	0.998556	−	0.0537241i	$-0.0171091\pi$
$488$		0			0
$489$		−	16.2295i		−	0.733923i
$490$		0			0
$491$	26.5678			1.19899			0.599493	−	0.800380i	$-0.295369\pi$
$491$	26.5678			1.19899			0.599493	+	0.800380i	$0.295369\pi$
$492$		0			0
$493$	12.8846			0.580295
$494$		0			0
$495$	0.969904			0.0435940
$496$		0			0
$497$	1.67328			0.0750570
$498$		0			0
$499$		−	10.7117i		−	0.479521i	−0.970832	−	0.239761i	$-0.922931\pi$
$499$		−	10.7117i		−	0.479521i	0.970832	−	0.239761i	$-0.0770689\pi$
$500$		0			0
$501$		−	5.08445i		−	0.227157i
$502$		0			0
$503$	5.56897			0.248308			0.124154	−	0.992263i	$-0.460378\pi$
$503$	5.56897			0.248308			0.124154	+	0.992263i	$0.460378\pi$
$504$		0			0
$505$			3.56009i			0.158422i
$506$		0			0
$507$	−8.44896	+	6.56038i	−0.375231	+	0.291357i
$508$		0			0
$509$			16.0979i			0.713529i	0.934194	+	0.356764i	$0.116120\pi$
$509$			16.0979i			0.713529i	−0.934194	+	0.356764i	$0.883880\pi$
$510$		0			0
$511$	1.29221			0.0571640
$512$		0			0
$513$			26.6410i			1.17623i
$514$		0			0
$515$		−	6.29258i		−	0.277284i
$516$		0			0
$517$	−4.48275			−0.197151
$518$		0			0
$519$	−6.53935			−0.287045
$520$		0			0
$521$	−4.72604			−0.207051			−0.103526	−	0.994627i	$-0.533012\pi$
$521$	−4.72604			−0.207051			−0.103526	+	0.994627i	$0.533012\pi$
$522$		0			0
$523$	16.4599			0.719739			0.359870	−	0.933003i	$-0.382821\pi$
$523$	16.4599			0.719739			0.359870	+	0.933003i	$0.382821\pi$
$524$		0			0
$525$			3.97074i			0.173297i
$526$		0			0
$527$		−	11.3466i		−	0.494265i
$528$		0			0
$529$	−9.08947			−0.395194
$530$		0			0
$531$		−	25.3031i		−	1.09806i
$532$		0			0
$533$	2.10887	+	6.15453i	0.0913451	+	0.266582i
$534$		0			0
$535$			0.157842i			0.00682409i
$536$		0			0
$537$	−9.88368			−0.426512
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$			40.0400i			1.72145i	0.509066	+	0.860727i	$0.329991\pi$
$541$			40.0400i			1.72145i	−0.509066	+	0.860727i	$0.670009\pi$
$542$		0			0
$543$	7.90891			0.339404
$544$		0			0
$545$	2.98517			0.127871
$546$		0			0
$547$	−13.7890			−0.589576			−0.294788	−	0.955563i	$-0.595249\pi$
$547$	−13.7890			−0.589576			−0.294788	+	0.955563i	$0.595249\pi$
$548$		0			0
$549$	−14.8824			−0.635165
$550$		0			0
$551$		−	38.8813i		−	1.65640i
$552$		0			0
$553$		−	13.0290i		−	0.554050i
$554$		0			0
$555$	0.407497			0.0172973
$556$		0			0
$557$		−	15.7122i		−	0.665746i	−0.942972	−	0.332873i	$-0.891982\pi$
$557$		−	15.7122i		−	0.665746i	0.942972	−	0.332873i	$-0.108018\pi$
$558$		0			0
$559$	15.8738	−	5.43920i	0.671389	−	0.230054i
$560$		0			0
$561$		−	1.65856i		−	0.0700243i
$562$		0			0
$563$	−40.4057			−1.70290			−0.851448	−	0.524438i	$-0.824275\pi$
$563$	−40.4057			−1.70290			−0.851448	+	0.524438i	$0.824275\pi$
$564$		0			0
$565$		−	8.44607i		−	0.355329i
$566$		0			0
$567$		−	3.36485i		−	0.141311i
$568$		0			0
$569$	−35.0705			−1.47023			−0.735116	−	0.677941i	$-0.762872\pi$
$569$	−35.0705			−1.47023			−0.735116	+	0.677941i	$0.762872\pi$
$570$		0			0
$571$	−26.3679			−1.10346			−0.551731	−	0.834022i	$-0.686033\pi$
$571$	−26.3679			−1.10346			−0.551731	+	0.834022i	$0.686033\pi$
$572$		0			0
$573$	4.02454			0.168128
$574$		0			0
$575$	−17.9982			−0.750577
$576$		0			0
$577$		−	27.6400i		−	1.15067i	−0.817919	−	0.575334i	$-0.804872\pi$
$577$		−	27.6400i		−	1.15067i	0.817919	−	0.575334i	$-0.195128\pi$
$578$		0			0
$579$		−	9.65115i		−	0.401088i
$580$		0			0
$581$	−14.6645			−0.608387
$582$		0			0
$583$			4.51746i			0.187094i
$584$		0			0
$585$	1.13357	+	3.30822i	0.0468674	+	0.136778i
$586$		0			0
$587$		−	8.58286i		−	0.354253i	−0.984188	−	0.177126i	$-0.943320\pi$
$587$		−	8.58286i		−	0.354253i	0.984188	−	0.177126i	$-0.0566801\pi$
$588$		0			0
$589$	−34.2399			−1.41083
$590$		0			0
$591$		−	7.26089i		−	0.298673i
$592$		0			0
$593$			20.5838i			0.845277i	0.906298	+	0.422638i	$0.138896\pi$
$593$			20.5838i			0.845277i	−0.906298	+	0.422638i	$0.861104\pi$
$594$		0			0
$595$	0.841603			0.0345024
$596$		0			0
$597$	−16.0566			−0.657151
$598$		0			0
$599$	−23.3021			−0.952096			−0.476048	−	0.879419i	$-0.657931\pi$
$599$	−23.3021			−0.952096			−0.476048	+	0.879419i	$0.657931\pi$
$600$		0			0
$601$	−25.2914			−1.03166			−0.515830	−	0.856691i	$-0.672516\pi$
$601$	−25.2914			−1.03166			−0.515830	+	0.856691i	$0.672516\pi$
$602$		0			0
$603$			15.8504i			0.645477i
$604$		0			0
$605$			0.417533i			0.0169751i
$606$		0			0
$607$	5.56433			0.225849			0.112925	−	0.993604i	$-0.463978\pi$
$607$	5.56433			0.225849			0.112925	+	0.993604i	$0.463978\pi$
$608$		0			0
$609$		−	5.25982i		−	0.213139i
$610$		0			0
$611$	−5.23919	−	15.2901i	−0.211955	−	0.618570i
$612$		0			0
$613$		−	10.9058i		−	0.440480i	−0.975446	−	0.220240i	$-0.929316\pi$
$613$		−	10.9058i		−	0.440480i	0.975446	−	0.220240i	$-0.0706840\pi$
$614$		0			0
$615$	−0.619919			−0.0249975
$616$		0			0
$617$		−	38.3778i		−	1.54503i	−0.634994	−	0.772517i	$-0.718998\pi$
$617$		−	38.3778i		−	1.54503i	0.634994	−	0.772517i	$-0.281002\pi$
$618$		0			0
$619$			8.44310i			0.339357i	0.985499	+	0.169678i	$0.0542729\pi$
$619$			8.44310i			0.339357i	−0.985499	+	0.169678i	$0.945727\pi$
$620$		0			0
$621$	−16.3357			−0.655529
$622$		0			0
$623$	14.8484			0.594889
$624$		0			0
$625$	22.4154			0.896615
$626$		0			0
$627$	−5.00493			−0.199878
$628$		0			0
$629$			2.39075i			0.0953255i
$630$		0			0
$631$		−	12.5679i		−	0.500320i	−0.968204	−	0.250160i	$-0.919517\pi$
$631$		−	12.5679i		−	0.500320i	0.968204	−	0.250160i	$-0.0804833\pi$
$632$		0			0
$633$	2.15349			0.0855936
$634$		0			0
$635$			0.334742i			0.0132838i
$636$		0			0
$637$	−3.41087	+	1.16874i	−0.135144	+	0.0463073i
$638$		0			0
$639$		−	3.88693i		−	0.153765i
$640$		0			0
$641$	3.67534			0.145167			0.0725837	−	0.997362i	$-0.476876\pi$
$641$	3.67534			0.145167			0.0725837	+	0.997362i	$0.476876\pi$
$642$		0			0
$643$		−	45.2814i		−	1.78573i	−0.450329	−	0.892863i	$-0.648693\pi$
$643$		−	45.2814i		−	1.78573i	0.450329	−	0.892863i	$-0.351307\pi$
$644$		0			0
$645$			1.59890i			0.0629565i
$646$		0			0
$647$	6.01059			0.236301			0.118150	−	0.992996i	$-0.462304\pi$
$647$	6.01059			0.236301			0.118150	+	0.992996i	$0.462304\pi$
$648$		0			0
$649$	10.8927			0.427576
$650$		0			0
$651$	−4.63194			−0.181540
$652$		0			0
$653$	−49.6476			−1.94286			−0.971430	−	0.237328i	$-0.923729\pi$
$653$	−49.6476			−1.94286			−0.971430	+	0.237328i	$0.923729\pi$
$654$		0			0
$655$		−	1.85073i		−	0.0723141i
$656$		0			0
$657$		−	3.00172i		−	0.117108i
$658$		0			0
$659$	9.73801			0.379339			0.189670	−	0.981848i	$-0.439258\pi$
$659$	9.73801			0.379339			0.189670	+	0.981848i	$0.439258\pi$
$660$		0			0
$661$		−	11.6255i		−	0.452181i	−0.974106	−	0.226091i	$-0.927405\pi$
$661$		−	11.6255i		−	0.452181i	0.974106	−	0.226091i	$-0.0725946\pi$
$662$		0			0
$663$	5.65712	−	1.93843i	0.219704	−	0.0752823i
$664$		0			0
$665$		−	2.53966i		−	0.0984838i
$666$		0			0
$667$	23.8412			0.923136
$668$		0			0
$669$		−	1.36120i		−	0.0526271i
$670$		0			0
$671$		−	6.40671i		−	0.247328i
$672$		0			0
$673$	−16.6280			−0.640962			−0.320481	−	0.947255i	$-0.603845\pi$
$673$	−16.6280			−0.640962			−0.320481	+	0.947255i	$0.603845\pi$
$674$		0			0
$675$	21.1360			0.813525
$676$		0			0
$677$	−10.1266			−0.389197			−0.194598	−	0.980883i	$-0.562340\pi$
$677$	−10.1266			−0.389197			−0.194598	+	0.980883i	$0.562340\pi$
$678$		0			0
$679$	−16.0714			−0.616765
$680$		0			0
$681$			7.01191i			0.268697i
$682$		0			0
$683$			4.90442i			0.187662i	0.995588	+	0.0938312i	$0.0299114\pi$
$683$			4.90442i			0.187662i	−0.995588	+	0.0938312i	$0.970089\pi$
$684$		0			0
$685$	1.12045			0.0428102
$686$		0			0
$687$		−	6.60822i		−	0.252119i
$688$		0			0
$689$	−15.4085	+	5.27976i	−0.587016	+	0.201143i
$690$		0			0
$691$			1.23021i			0.0467996i	0.999726	+	0.0233998i	$0.00744906\pi$
$691$			1.23021i			0.0467996i	−0.999726	+	0.0233998i	$0.992551\pi$
$692$		0			0
$693$	2.32294			0.0882411
$694$		0			0
$695$		−	7.57982i		−	0.287519i
$696$		0			0
$697$		−	3.63702i		−	0.137762i
$698$		0			0
$699$	−15.9501			−0.603287
$700$		0			0
$701$	−19.8899			−0.751230			−0.375615	−	0.926776i	$-0.622569\pi$
$701$	−19.8899			−0.751230			−0.375615	+	0.926776i	$0.622569\pi$
$702$		0			0
$703$	7.21444			0.272098
$704$		0			0
$705$	1.54010			0.0580036
$706$		0			0
$707$			8.52647i			0.320671i
$708$		0			0
$709$		−	31.3972i		−	1.17915i	−0.807715	−	0.589573i	$-0.799296\pi$
$709$		−	31.3972i		−	1.17915i	0.807715	−	0.589573i	$-0.200704\pi$
$710$		0			0
$711$	−30.2656			−1.13505
$712$		0			0
$713$		−	20.9952i		−	0.786278i
$714$		0			0
$715$	−1.42415	+	0.487990i	−0.0532603	+	0.0182498i
$716$		0			0
$717$		−	3.27716i		−	0.122388i
$718$		0			0
$719$	−49.2935			−1.83834			−0.919169	−	0.393863i	$-0.871139\pi$
$719$	−49.2935			−1.83834			−0.919169	+	0.393863i	$0.871139\pi$
$720$		0			0
$721$		−	15.0709i		−	0.561268i
$722$		0			0
$723$			1.76469i			0.0656296i
$724$		0			0
$725$	−30.8471			−1.14563
$726$		0			0
$727$	38.0880			1.41260			0.706302	−	0.707911i	$-0.250362\pi$
$727$	38.0880			1.41260			0.706302	+	0.707911i	$0.250362\pi$
$728$		0			0
$729$	2.99553			0.110946
$730$		0			0
$731$	−9.38061			−0.346955
$732$		0			0
$733$		−	12.7335i		−	0.470321i	−0.971957	−	0.235160i	$-0.924438\pi$
$733$		−	12.7335i		−	0.470321i	0.971957	−	0.235160i	$-0.0755616\pi$
$734$		0			0
$735$		−	0.343562i		−	0.0126725i
$736$		0			0
$737$	−6.82342			−0.251344
$738$		0			0
$739$			44.9803i			1.65463i	0.561740	+	0.827314i	$0.310132\pi$
$739$			44.9803i			1.65463i	−0.561740	+	0.827314i	$0.689868\pi$
$740$		0			0
$741$	−5.84949	−	17.0712i	−0.214886	−	0.627126i
$742$		0			0
$743$		−	11.9711i		−	0.439177i	−0.975593	−	0.219589i	$-0.929528\pi$
$743$		−	11.9711i		−	0.439177i	0.975593	−	0.219589i	$-0.0704715\pi$
$744$		0			0
$745$	2.06203			0.0755468
$746$		0			0
$747$			34.0648i			1.24636i
$748$		0			0
$749$			0.378034i			0.0138130i
$750$		0			0
$751$	−39.7724			−1.45131			−0.725657	−	0.688056i	$-0.758464\pi$
$751$	−39.7724			−1.45131			−0.725657	+	0.688056i	$0.758464\pi$
$752$		0			0
$753$	−15.7334			−0.573356
$754$		0			0
$755$	3.24749			0.118188
$756$		0			0
$757$	1.93864			0.0704609			0.0352304	−	0.999379i	$-0.488783\pi$
$757$	1.93864			0.0704609			0.0352304	+	0.999379i	$0.488783\pi$
$758$		0			0
$759$		−	3.06892i		−	0.111395i
$760$		0			0
$761$			24.7627i			0.897646i	0.893621	+	0.448823i	$0.148157\pi$
$761$			24.7627i			0.897646i	−0.893621	+	0.448823i	$0.851843\pi$
$762$		0			0
$763$	7.14955			0.258831
$764$		0			0
$765$		−	1.95499i		−	0.0706829i
$766$		0			0
$767$	12.7308	+	37.1536i	0.459682	+	1.34154i
$768$		0			0
$769$			28.6923i			1.03467i	0.855783	+	0.517335i	$0.173076\pi$
$769$			28.6923i			1.03467i	−0.855783	+	0.517335i	$0.826924\pi$
$770$		0			0
$771$	8.88134			0.319854
$772$		0			0
$773$		−	36.8140i		−	1.32411i	−0.749457	−	0.662053i	$-0.769685\pi$
$773$		−	36.8140i		−	1.32411i	0.749457	−	0.662053i	$-0.230315\pi$
$774$		0			0
$775$			27.1648i			0.975788i
$776$		0			0
$777$	0.975961			0.0350124
$778$		0			0
$779$	−10.9752			−0.393228
$780$		0			0
$781$	1.67328			0.0598747
$782$		0			0
$783$	−27.9977			−1.00056
$784$		0			0
$785$			2.44527i			0.0872754i
$786$		0			0
$787$		−	34.1976i		−	1.21901i	−0.792782	−	0.609506i	$-0.791368\pi$
$787$		−	34.1976i		−	1.21901i	0.792782	−	0.609506i	$-0.208632\pi$
$788$		0			0
$789$	3.98550			0.141888
$790$		0			0
$791$		−	20.2285i		−	0.719242i
$792$		0			0
$793$	21.8525	−	7.48781i	0.776004	−	0.265900i
$794$		0			0
$795$		−	1.55203i		−	0.0550448i
$796$		0			0
$797$	12.0522			0.426911			0.213456	−	0.976953i	$-0.431528\pi$
$797$	12.0522			0.426911			0.213456	+	0.976953i	$0.431528\pi$
$798$		0			0
$799$			9.03567i			0.319659i
$800$		0			0
$801$		−	34.4919i		−	1.21871i
$802$		0			0
$803$	1.29221			0.0456011
$804$		0			0
$805$	1.55727			0.0548865
$806$		0			0
$807$	−16.0913			−0.566439
$808$		0			0
$809$	−26.0602			−0.916228			−0.458114	−	0.888894i	$-0.651475\pi$
$809$	−26.0602			−0.916228			−0.458114	+	0.888894i	$0.651475\pi$
$810$		0			0
$811$		−	41.2789i		−	1.44950i	−0.689013	−	0.724749i	$-0.741956\pi$
$811$		−	41.2789i		−	1.44950i	0.689013	−	0.724749i	$-0.258044\pi$
$812$		0			0
$813$		−	3.96221i		−	0.138961i
$814$		0			0
$815$	8.23535			0.288472
$816$		0			0
$817$			28.3074i			0.990349i
$818$		0			0
$819$	2.71492	+	7.92324i	0.0948670	+	0.276860i
$820$		0			0
$821$			30.0236i			1.04783i	0.851771	+	0.523915i	$0.175529\pi$
$821$			30.0236i			1.04783i	−0.851771	+	0.523915i	$0.824471\pi$
$822$		0			0
$823$	−50.3999			−1.75683			−0.878414	−	0.477900i	$-0.841398\pi$
$823$	−50.3999			−1.75683			−0.878414	+	0.477900i	$0.841398\pi$
$824$		0			0
$825$			3.97074i			0.138243i
$826$		0			0
$827$			36.3728i			1.26481i	0.774639	+	0.632404i	$0.217932\pi$
$827$			36.3728i			1.26481i	−0.774639	+	0.632404i	$0.782068\pi$
$828$		0			0
$829$	4.38425			0.152271			0.0761357	−	0.997097i	$-0.475742\pi$
$829$	4.38425			0.152271			0.0761357	+	0.997097i	$0.475742\pi$
$830$		0			0
$831$	3.50598			0.121621
$832$		0			0
$833$	2.01565			0.0698383
$834$		0			0
$835$	2.58001			0.0892849
$836$		0			0
$837$			24.6555i			0.852220i
$838$		0			0
$839$		−	54.7376i		−	1.88975i	−0.327430	−	0.944875i	$-0.606183\pi$
$839$		−	54.7376i		−	1.88975i	0.327430	−	0.944875i	$-0.393817\pi$
$840$		0			0
$841$	11.8614			0.409014
$842$		0			0
$843$			23.1473i			0.797237i
$844$		0			0
$845$	−3.32894	−	4.28726i	−0.114519	−	0.147486i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	−17.9965			−0.617638
$850$		0			0
$851$			4.42375i			0.151644i
$852$		0			0
$853$			26.0219i			0.890973i	0.895289	+	0.445486i	$0.146969\pi$
$853$			26.0219i			0.890973i	−0.895289	+	0.445486i	$0.853031\pi$
$854$		0			0
$855$	−5.89947			−0.201758
$856$		0			0
$857$	1.50651			0.0514614			0.0257307	−	0.999669i	$-0.491809\pi$
$857$	1.50651			0.0514614			0.0257307	+	0.999669i	$0.491809\pi$
$858$		0			0
$859$	15.4793			0.528148			0.264074	−	0.964502i	$-0.414934\pi$
$859$	15.4793			0.528148			0.264074	+	0.964502i	$0.414934\pi$
$860$		0			0
$861$	−1.48472			−0.0505990
$862$		0			0
$863$			43.4092i			1.47767i	0.673888	+	0.738834i	$0.264623\pi$
$863$			43.4092i			1.47767i	−0.673888	+	0.738834i	$0.735377\pi$
$864$		0			0
$865$		−	3.31827i		−	0.112824i
$866$		0			0
$867$	10.6452			0.361529
$868$		0			0
$869$		−	13.0290i		−	0.441979i
$870$		0			0
$871$	−7.97484	−	23.2738i	−0.270217	−	0.788603i
$872$		0			0
$873$			37.3330i			1.26353i
$874$		0			0
$875$	−4.10254			−0.138691
$876$		0			0
$877$			46.0499i			1.55499i	0.628887	+	0.777497i	$0.283511\pi$
$877$			46.0499i			1.55499i	−0.628887	+	0.777497i	$0.716489\pi$
$878$		0			0
$879$			6.69849i			0.225934i
$880$		0			0
$881$	−42.2782			−1.42439			−0.712194	−	0.701982i	$-0.752298\pi$
$881$	−42.2782			−1.42439			−0.712194	+	0.701982i	$0.752298\pi$
$882$		0			0
$883$	52.0815			1.75268			0.876342	−	0.481690i	$-0.159977\pi$
$883$	52.0815			1.75268			0.876342	+	0.481690i	$0.159977\pi$
$884$		0			0
$885$	−3.74232			−0.125797
$886$		0			0
$887$	−52.1230			−1.75012			−0.875059	−	0.484016i	$-0.839178\pi$
$887$	−52.1230			−1.75012			−0.875059	+	0.484016i	$0.839178\pi$
$888$		0			0
$889$			0.801712i			0.0268886i
$890$		0			0
$891$		−	3.36485i		−	0.112727i
$892$		0			0
$893$	27.2664			0.912437
$894$		0			0
$895$		−	5.01529i		−	0.167643i
$896$		0			0
$897$	10.4677	−	3.58679i	0.349506	−	0.119759i
$898$		0			0
$899$		−	35.9837i		−	1.20012i
$900$		0			0
$901$	9.10563			0.303353
$902$		0			0
$903$			3.82939i			0.127434i
$904$		0			0
$905$			4.01323i			0.133404i
$906$		0			0
$907$	18.0897			0.600658			0.300329	−	0.953836i	$-0.402904\pi$
$907$	18.0897			0.600658			0.300329	+	0.953836i	$0.402904\pi$
$908$		0			0
$909$	19.8065			0.656939
$910$		0			0
$911$	−31.1178			−1.03098			−0.515489	−	0.856896i	$-0.672390\pi$
$911$	−31.1178			−1.03098			−0.515489	+	0.856896i	$0.672390\pi$
$912$		0			0
$913$	−14.6645			−0.485325
$914$		0			0
$915$			2.20111i			0.0727663i
$916$		0			0
$917$		−	4.43254i		−	0.146375i
$918$		0			0
$919$	44.8527			1.47956			0.739778	−	0.672851i	$-0.234931\pi$
$919$	44.8527			1.47956			0.739778	+	0.672851i	$0.234931\pi$
$920$		0			0
$921$		−	11.5755i		−	0.381427i
$922$		0			0
$923$	1.95564	+	5.70735i	0.0643707	+	0.187860i
$924$		0			0
$925$		−	5.72368i		−	0.188194i
$926$		0			0
$927$	−35.0087			−1.14983
$928$		0			0
$929$		−	8.14824i		−	0.267335i	−0.991026	−	0.133668i	$-0.957325\pi$
$929$		−	8.14824i		−	0.267335i	0.991026	−	0.133668i	$-0.0426754\pi$
$930$		0			0
$931$		−	6.08253i		−	0.199347i
$932$		0			0
$933$	−11.7607			−0.385029
$934$		0			0
$935$	0.841603			0.0275234
$936$		0			0
$937$	−18.4145			−0.601576			−0.300788	−	0.953691i	$-0.597250\pi$
$937$	−18.4145			−0.601576			−0.300788	+	0.953691i	$0.597250\pi$
$938$		0			0
$939$	20.3653			0.664596
$940$		0			0
$941$			4.39306i			0.143210i	0.997433	+	0.0716049i	$0.0228121\pi$
$941$			4.39306i			0.143210i	−0.997433	+	0.0716049i	$0.977188\pi$
$942$		0			0
$943$		−	6.72979i		−	0.219152i
$944$		0			0
$945$	−1.82876			−0.0594896
$946$		0			0
$947$			9.34268i			0.303596i	0.988412	+	0.151798i	$0.0485064\pi$
$947$			9.34268i			0.303596i	−0.988412	+	0.151798i	$0.951494\pi$
$948$		0			0
$949$	1.51026	+	4.40756i	0.0490252	+	0.143075i
$950$		0			0
$951$		−	26.2342i		−	0.850702i
$952$		0			0
$953$	36.7521			1.19052			0.595259	−	0.803534i	$-0.297049\pi$
$953$	36.7521			1.19052			0.595259	+	0.803534i	$0.297049\pi$
$954$		0			0
$955$			2.04218i			0.0660833i
$956$		0			0
$957$		−	5.25982i		−	0.170026i
$958$		0			0
$959$	2.68350			0.0866546
$960$		0			0
$961$	−0.688223			−0.0222007
$962$		0			0
$963$	0.878149			0.0282979
$964$		0			0
$965$	4.89729			0.157650
$966$		0			0
$967$		−	26.5345i		−	0.853292i	−0.904419	−	0.426646i	$-0.859695\pi$
$967$		−	26.5345i		−	0.853292i	0.904419	−	0.426646i	$-0.140305\pi$
$968$		0			0
$969$			10.0882i			0.324080i
$970$		0			0
$971$	15.9422			0.511608			0.255804	−	0.966729i	$-0.417660\pi$
$971$	15.9422			0.511608			0.255804	+	0.966729i	$0.417660\pi$
$972$		0			0
$973$		−	18.1538i		−	0.581984i
$974$		0			0
$975$	−13.5437	+	4.64078i	−0.433745	+	0.148624i
$976$		0			0
$977$		−	9.93517i		−	0.317854i	−0.987290	−	0.158927i	$-0.949196\pi$
$977$		−	9.93517i		−	0.317854i	0.987290	−	0.158927i	$-0.0508035\pi$
$978$		0			0
$979$	14.8484			0.474557
$980$		0			0
$981$		−	16.6080i		−	0.530251i
$982$		0			0
$983$		−	42.4523i		−	1.35402i	−0.735975	−	0.677009i	$-0.763276\pi$
$983$		−	42.4523i		−	1.35402i	0.735975	−	0.677009i	$-0.236724\pi$
$984$		0			0
$985$	3.68440			0.117395
$986$		0			0
$987$	3.68857			0.117409
$988$		0			0
$989$	−17.3575			−0.551936
$990$		0			0
$991$	−57.0180			−1.81124			−0.905619	−	0.424093i	$-0.860593\pi$
$991$	−57.0180			−1.81124			−0.905619	+	0.424093i	$0.860593\pi$
$992$		0			0
$993$			1.54355i			0.0489832i
$994$		0			0
$995$		−	8.14760i		−	0.258296i
$996$		0			0
$997$	−20.0842			−0.636072			−0.318036	−	0.948079i	$-0.603023\pi$
$997$	−20.0842			−0.636072			−0.318036	+	0.948079i	$0.603023\pi$
$998$		0			0
$999$		−	5.19498i		−	0.164362i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.b.2157.11	✓	30
13.12	even	2	inner	4004.2.m.b.2157.12	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.b.2157.11	✓	30	1.1	even	1	trivial
4004.2.m.b.2157.12	yes	30	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)

\(f(q)\)	\(=\)	\(q-0.822838 q^{3} -0.417533i q^{5} -1.00000i q^{7} -2.32294 q^{9} +O(q^{10})\)	\(q-0.822838 q^{3} -0.417533i q^{5} -1.00000i q^{7} -2.32294 q^{9} -1.00000i q^{11} +(3.41087 - 1.16874i) q^{13} +0.343562i q^{15} -2.01565 q^{17} +6.08253i q^{19} +0.822838i q^{21} -3.72968 q^{23} +4.82567 q^{25} +4.37991 q^{27} -6.39229 q^{29} +5.62923i q^{31} +0.822838i q^{33} -0.417533 q^{35} -1.18609i q^{37} +(-2.80659 + 0.961687i) q^{39} +1.80439i q^{41} +4.65388 q^{43} +0.969904i q^{45} -4.48275i q^{47} -1.00000 q^{49} +1.65856 q^{51} -4.51746 q^{53} -0.417533 q^{55} -5.00493i q^{57} +10.8927i q^{59} +6.40671 q^{61} +2.32294i q^{63} +(-0.487990 - 1.42415i) q^{65} -6.82342i q^{67} +3.06892 q^{69} +1.67328i q^{71} +1.29221i q^{73} -3.97074 q^{75} -1.00000 q^{77} +13.0290 q^{79} +3.36485 q^{81} -14.6645i q^{83} +0.841603i q^{85} +5.25982 q^{87} +14.8484i q^{89} +(-1.16874 - 3.41087i) q^{91} -4.63194i q^{93} +2.53966 q^{95} -16.0714i q^{97} +2.32294i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 18 q^{9}+O(q^{10}) \) 30 * q + 18 * q^9	\( 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) \) 30 * q + 18 * q^9 + 4 * q^13 + 12 * q^17 + 6 * q^23 - 6 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 30 * q^49 + 60 * q^51 - 38 * q^53 - 2 * q^55 - 36 * q^61 + 10 * q^65 + 36 * q^69 - 20 * q^75 - 30 * q^77 - 10 * q^79 - 42 * q^81 + 36 * q^87 + 6 * q^91 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more