LMFDB - Embedded newform 4004.2.m.b.2157.9

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			2157.9
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.b.2157.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.49114 q^{3} -1.37882i q^{5} +1.00000i q^{7} -0.776492 q^{9} +O(q^{10})$	$q-1.49114 q^{3} -1.37882i q^{5} +1.00000i q^{7} -0.776492 q^{9} +1.00000i q^{11} +(-2.93568 + 2.09327i) q^{13} +2.05601i q^{15} +0.801929 q^{17} -0.0452303i q^{19} -1.49114i q^{21} +6.18836 q^{23} +3.09886 q^{25} +5.63129 q^{27} -7.84316 q^{29} +2.31339i q^{31} -1.49114i q^{33} +1.37882 q^{35} +3.74996i q^{37} +(4.37752 - 3.12136i) q^{39} -2.15820i q^{41} -4.55492 q^{43} +1.07064i q^{45} -1.91517i q^{47} -1.00000 q^{49} -1.19579 q^{51} +1.32588 q^{53} +1.37882 q^{55} +0.0674448i q^{57} -8.56985i q^{59} +7.82926 q^{61} -0.776492i q^{63} +(2.88623 + 4.04777i) q^{65} -13.0545i q^{67} -9.22773 q^{69} +1.36987i q^{71} +6.13718i q^{73} -4.62085 q^{75} -1.00000 q^{77} -8.88057 q^{79} -6.06758 q^{81} -4.20617i q^{83} -1.10571i q^{85} +11.6953 q^{87} -9.55584i q^{89} +(-2.09327 - 2.93568i) q^{91} -3.44960i q^{93} -0.0623643 q^{95} -5.59559i q^{97} -0.776492i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.49114			−0.860912			−0.430456	−	0.902612i	$-0.641647\pi$
$3$	−1.49114			−0.860912			−0.430456	+	0.902612i	$0.641647\pi$
$4$		0			0
$5$		−	1.37882i		−	0.616626i	−0.951285	−	0.308313i	$-0.900236\pi$
$5$		−	1.37882i		−	0.616626i	0.951285	−	0.308313i	$-0.0997643\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−0.776492			−0.258831
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−2.93568	+	2.09327i	−0.814212	+	0.580568i
$13$	−2.93568	+	2.09327i	−0.814212	+	0.580568i
$14$		0			0
$15$			2.05601i			0.530860i
$16$		0			0
$17$	0.801929			0.194496			0.0972482	−	0.995260i	$-0.468996\pi$
$17$	0.801929			0.194496			0.0972482	+	0.995260i	$0.468996\pi$
$18$		0			0
$19$		−	0.0452303i		−	0.0103765i	−0.999987	−	0.00518827i	$-0.998349\pi$
$19$		−	0.0452303i		−	0.0103765i	0.999987	−	0.00518827i	$-0.00165149\pi$
$20$		0			0
$21$		−	1.49114i		−	0.325394i
$22$		0			0
$23$	6.18836			1.29036			0.645181	−	0.764030i	$-0.276782\pi$
$23$	6.18836			1.29036			0.645181	+	0.764030i	$0.276782\pi$
$24$		0			0
$25$	3.09886			0.619773
$26$		0			0
$27$	5.63129			1.08374
$28$		0			0
$29$	−7.84316			−1.45644			−0.728219	−	0.685344i	$-0.759652\pi$
$29$	−7.84316			−1.45644			−0.728219	+	0.685344i	$0.759652\pi$
$30$		0			0
$31$			2.31339i			0.415497i	0.978182	+	0.207749i	$0.0666136\pi$
$31$			2.31339i			0.415497i	−0.978182	+	0.207749i	$0.933386\pi$
$32$		0			0
$33$		−	1.49114i		−	0.259575i
$34$		0			0
$35$	1.37882			0.233063
$36$		0			0
$37$			3.74996i			0.616489i	0.951307	+	0.308245i	$0.0997415\pi$
$37$			3.74996i			0.616489i	−0.951307	+	0.308245i	$0.900258\pi$
$38$		0			0
$39$	4.37752	−	3.12136i	0.700964	−	0.499818i
$40$		0			0
$41$		−	2.15820i		−	0.337054i	−0.985697	−	0.168527i	$-0.946099\pi$
$41$		−	2.15820i		−	0.337054i	0.985697	−	0.168527i	$-0.0539010\pi$
$42$		0			0
$43$	−4.55492			−0.694618			−0.347309	−	0.937751i	$-0.612905\pi$
$43$	−4.55492			−0.694618			−0.347309	+	0.937751i	$0.612905\pi$
$44$		0			0
$45$			1.07064i			0.159602i
$46$		0			0
$47$		−	1.91517i		−	0.279357i	−0.990197	−	0.139678i	$-0.955393\pi$
$47$		−	1.91517i		−	0.279357i	0.990197	−	0.139678i	$-0.0446069\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−1.19579			−0.167444
$52$		0			0
$53$	1.32588			0.182123			0.0910615	−	0.995845i	$-0.470974\pi$
$53$	1.32588			0.182123			0.0910615	+	0.995845i	$0.470974\pi$
$54$		0			0
$55$	1.37882			0.185920
$56$		0			0
$57$			0.0674448i			0.00893329i
$58$		0			0
$59$		−	8.56985i		−	1.11570i	−0.829942	−	0.557850i	$-0.811626\pi$
$59$		−	8.56985i		−	1.11570i	0.829942	−	0.557850i	$-0.188374\pi$
$60$		0			0
$61$	7.82926			1.00243			0.501217	−	0.865321i	$-0.332886\pi$
$61$	7.82926			1.00243			0.501217	+	0.865321i	$0.332886\pi$
$62$		0			0
$63$		−	0.776492i		−	0.0978288i
$64$		0			0
$65$	2.88623	+	4.04777i	0.357993	+	0.502064i
$66$		0			0
$67$		−	13.0545i		−	1.59487i	−0.603407	−	0.797433i	$-0.706191\pi$
$67$		−	13.0545i		−	1.59487i	0.603407	−	0.797433i	$-0.293809\pi$
$68$		0			0
$69$	−9.22773			−1.11089
$70$		0			0
$71$			1.36987i			0.162573i	0.996691	+	0.0812867i	$0.0259029\pi$
$71$			1.36987i			0.162573i	−0.996691	+	0.0812867i	$0.974097\pi$
$72$		0			0
$73$			6.13718i			0.718303i	0.933279	+	0.359151i	$0.116934\pi$
$73$			6.13718i			0.718303i	−0.933279	+	0.359151i	$0.883066\pi$
$74$		0			0
$75$	−4.62085			−0.533570
$76$		0			0
$77$	−1.00000			−0.113961
$78$		0			0
$79$	−8.88057			−0.999142			−0.499571	−	0.866273i	$-0.666509\pi$
$79$	−8.88057			−0.999142			−0.499571	+	0.866273i	$0.666509\pi$
$80$		0			0
$81$	−6.06758			−0.674176
$82$		0			0
$83$		−	4.20617i		−	0.461687i	−0.972991	−	0.230844i	$-0.925851\pi$
$83$		−	4.20617i		−	0.461687i	0.972991	−	0.230844i	$-0.0741486\pi$
$84$		0			0
$85$		−	1.10571i		−	0.119931i
$86$		0			0
$87$	11.6953			1.25387
$88$		0			0
$89$		−	9.55584i		−	1.01292i	−0.862264	−	0.506458i	$-0.830954\pi$
$89$		−	9.55584i		−	1.01292i	0.862264	−	0.506458i	$-0.169046\pi$
$90$		0			0
$91$	−2.09327	−	2.93568i	−0.219434	−	0.307743i
$92$		0			0
$93$		−	3.44960i		−	0.357707i
$94$		0			0
$95$	−0.0623643			−0.00639844
$96$		0			0
$97$		−	5.59559i		−	0.568146i	−0.958803	−	0.284073i	$-0.908314\pi$
$97$		−	5.59559i		−	0.568146i	0.958803	−	0.284073i	$-0.0916858\pi$
$98$		0			0
$99$		−	0.776492i		−	0.0780404i
$100$		0			0
$101$	−5.25485			−0.522877			−0.261439	−	0.965220i	$-0.584197\pi$
$101$	−5.25485			−0.522877			−0.261439	+	0.965220i	$0.584197\pi$
$102$		0			0
$103$	−13.0374			−1.28461			−0.642306	−	0.766448i	$-0.722022\pi$
$103$	−13.0374			−1.28461			−0.642306	+	0.766448i	$0.722022\pi$
$104$		0			0
$105$	−2.05601			−0.200646
$106$		0			0
$107$	−9.92739			−0.959717			−0.479859	−	0.877346i	$-0.659312\pi$
$107$	−9.92739			−0.959717			−0.479859	+	0.877346i	$0.659312\pi$
$108$		0			0
$109$			13.4848i			1.29161i	0.763503	+	0.645805i	$0.223478\pi$
$109$			13.4848i			1.29161i	−0.763503	+	0.645805i	$0.776522\pi$
$110$		0			0
$111$		−	5.59172i		−	0.530743i
$112$		0			0
$113$	13.0357			1.22629			0.613147	−	0.789969i	$-0.289903\pi$
$113$	13.0357			1.22629			0.613147	+	0.789969i	$0.289903\pi$
$114$		0			0
$115$		−	8.53261i		−	0.795670i
$116$		0			0
$117$	2.27953	−	1.62541i	0.210743	−	0.150269i
$118$		0			0
$119$			0.801929i			0.0735128i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			3.21818i			0.290173i
$124$		0			0
$125$		−	11.1668i		−	0.998793i
$126$		0			0
$127$	−10.0769			−0.894181			−0.447091	−	0.894489i	$-0.647540\pi$
$127$	−10.0769			−0.894181			−0.447091	+	0.894489i	$0.647540\pi$
$128$		0			0
$129$	6.79203			0.598005
$130$		0			0
$131$	−8.92342			−0.779643			−0.389821	−	0.920890i	$-0.627463\pi$
$131$	−8.92342			−0.779643			−0.389821	+	0.920890i	$0.627463\pi$
$132$		0			0
$133$	0.0452303			0.00392196
$134$		0			0
$135$		−	7.76452i		−	0.668263i
$136$		0			0
$137$			20.2513i			1.73018i	0.501615	+	0.865091i	$0.332740\pi$
$137$			20.2513i			1.73018i	−0.501615	+	0.865091i	$0.667260\pi$
$138$		0			0
$139$	−6.85480			−0.581416			−0.290708	−	0.956812i	$-0.593891\pi$
$139$	−6.85480			−0.581416			−0.290708	+	0.956812i	$0.593891\pi$
$140$		0			0
$141$			2.85580i			0.240502i
$142$		0			0
$143$	−2.09327	−	2.93568i	−0.175048	−	0.245494i
$144$		0			0
$145$			10.8143i			0.898077i
$146$		0			0
$147$	1.49114			0.122987
$148$		0			0
$149$		−	0.974168i		−	0.0798070i	−0.999204	−	0.0399035i	$-0.987295\pi$
$149$		−	0.974168i		−	0.0798070i	0.999204	−	0.0399035i	$-0.0127050\pi$
$150$		0			0
$151$			12.9920i			1.05728i	0.848847	+	0.528638i	$0.177297\pi$
$151$			12.9920i			1.05728i	−0.848847	+	0.528638i	$0.822703\pi$
$152$		0			0
$153$	−0.622692			−0.0503417
$154$		0			0
$155$	3.18974			0.256206
$156$		0			0
$157$	11.3467			0.905563			0.452782	−	0.891621i	$-0.350432\pi$
$157$	11.3467			0.905563			0.452782	+	0.891621i	$0.350432\pi$
$158$		0			0
$159$	−1.97707			−0.156792
$160$		0			0
$161$			6.18836i			0.487711i
$162$		0			0
$163$			4.04804i			0.317067i	0.987354	+	0.158533i	$0.0506765\pi$
$163$			4.04804i			0.317067i	−0.987354	+	0.158533i	$0.949323\pi$
$164$		0			0
$165$	−2.05601			−0.160060
$166$		0			0
$167$		−	7.22415i		−	0.559022i	−0.960143	−	0.279511i	$-0.909828\pi$
$167$		−	7.22415i		−	0.559022i	0.960143	−	0.279511i	$-0.0901723\pi$
$168$		0			0
$169$	4.23645	−	12.2903i	0.325881	−	0.945411i
$170$		0			0
$171$			0.0351210i			0.00268577i
$172$		0			0
$173$	−16.2479			−1.23530			−0.617651	−	0.786453i	$-0.711915\pi$
$173$	−16.2479			−1.23530			−0.617651	+	0.786453i	$0.711915\pi$
$174$		0			0
$175$			3.09886i			0.234252i
$176$		0			0
$177$			12.7789i			0.960519i
$178$		0			0
$179$	−23.9732			−1.79184			−0.895920	−	0.444216i	$-0.853482\pi$
$179$	−23.9732			−1.79184			−0.895920	+	0.444216i	$0.853482\pi$
$180$		0			0
$181$	7.32148			0.544201			0.272101	−	0.962269i	$-0.412282\pi$
$181$	7.32148			0.544201			0.272101	+	0.962269i	$0.412282\pi$
$182$		0			0
$183$	−11.6746			−0.863008
$184$		0			0
$185$	5.17050			0.380143
$186$		0			0
$187$			0.801929i			0.0586429i
$188$		0			0
$189$			5.63129i			0.409616i
$190$		0			0
$191$	15.4656			1.11905			0.559527	−	0.828812i	$-0.310983\pi$
$191$	15.4656			1.11905			0.559527	+	0.828812i	$0.310983\pi$
$192$		0			0
$193$		−	16.4032i		−	1.18073i	−0.807137	−	0.590364i	$-0.798984\pi$
$193$		−	16.4032i		−	1.18073i	0.807137	−	0.590364i	$-0.201016\pi$
$194$		0			0
$195$	−4.30379	−	6.03580i	−0.308201	−	0.432233i
$196$		0			0
$197$		−	4.53639i		−	0.323204i	−0.986856	−	0.161602i	$-0.948334\pi$
$197$		−	4.53639i		−	0.323204i	0.986856	−	0.161602i	$-0.0516661\pi$
$198$		0			0
$199$	−9.56527			−0.678064			−0.339032	−	0.940775i	$-0.610099\pi$
$199$	−9.56527			−0.678064			−0.339032	+	0.940775i	$0.610099\pi$
$200$		0			0
$201$			19.4662i			1.37304i
$202$		0			0
$203$		−	7.84316i		−	0.550482i
$204$		0			0
$205$	−2.97576			−0.207836
$206$		0			0
$207$	−4.80521			−0.333985
$208$		0			0
$209$	0.0452303			0.00312864
$210$		0			0
$211$	−24.5121			−1.68748			−0.843742	−	0.536748i	$-0.819652\pi$
$211$	−24.5121			−1.68748			−0.843742	+	0.536748i	$0.819652\pi$
$212$		0			0
$213$		−	2.04267i		−	0.139961i
$214$		0			0
$215$			6.28040i			0.428319i
$216$		0			0
$217$	−2.31339			−0.157043
$218$		0			0
$219$		−	9.15141i		−	0.618395i
$220$		0			0
$221$	−2.35421	+	1.67865i	−0.158361	+	0.112918i
$222$		0			0
$223$		−	4.87329i		−	0.326340i	−0.986598	−	0.163170i	$-0.947828\pi$
$223$		−	4.87329i		−	0.326340i	0.986598	−	0.163170i	$-0.0521719\pi$
$224$		0			0
$225$	−2.40624			−0.160416
$226$		0			0
$227$		−	27.4033i		−	1.81882i	−0.415897	−	0.909412i	$-0.636532\pi$
$227$		−	27.4033i		−	1.81882i	0.415897	−	0.909412i	$-0.363468\pi$
$228$		0			0
$229$			6.18332i			0.408606i	0.978908	+	0.204303i	$0.0654927\pi$
$229$			6.18332i			0.408606i	−0.978908	+	0.204303i	$0.934507\pi$
$230$		0			0
$231$	1.49114			0.0981100
$232$		0			0
$233$	−13.1947			−0.864411			−0.432205	−	0.901775i	$-0.642264\pi$
$233$	−13.1947			−0.864411			−0.432205	+	0.901775i	$0.642264\pi$
$234$		0			0
$235$	−2.64067			−0.172259
$236$		0			0
$237$	13.2422			0.860173
$238$		0			0
$239$		−	17.9098i		−	1.15849i	−0.815154	−	0.579244i	$-0.803348\pi$
$239$		−	17.9098i		−	1.15849i	0.815154	−	0.579244i	$-0.196652\pi$
$240$		0			0
$241$		−	19.4891i		−	1.25540i	−0.778454	−	0.627701i	$-0.783996\pi$
$241$		−	19.4891i		−	1.25540i	0.778454	−	0.627701i	$-0.216004\pi$
$242$		0			0
$243$	−7.84623			−0.503336
$244$		0			0
$245$			1.37882i			0.0880894i
$246$		0			0
$247$	0.0946792	+	0.132782i	0.00602429	+	0.00844870i
$248$		0			0
$249$			6.27200i			0.397472i
$250$		0			0
$251$	−4.04285			−0.255183			−0.127591	−	0.991827i	$-0.540725\pi$
$251$	−4.04285			−0.255183			−0.127591	+	0.991827i	$0.540725\pi$
$252$		0			0
$253$			6.18836i			0.389059i
$254$		0			0
$255$			1.64878i			0.103250i
$256$		0			0
$257$	−18.2244			−1.13681			−0.568405	−	0.822749i	$-0.692439\pi$
$257$	−18.2244			−1.13681			−0.568405	+	0.822749i	$0.692439\pi$
$258$		0			0
$259$	−3.74996			−0.233011
$260$		0			0
$261$	6.09015			0.376971
$262$		0			0
$263$	11.2306			0.692508			0.346254	−	0.938141i	$-0.387454\pi$
$263$	11.2306			0.692508			0.346254	+	0.938141i	$0.387454\pi$
$264$		0			0
$265$		−	1.82814i		−	0.112302i
$266$		0			0
$267$			14.2491i			0.872032i
$268$		0			0
$269$	−12.6499			−0.771281			−0.385640	−	0.922649i	$-0.626019\pi$
$269$	−12.6499			−0.771281			−0.385640	+	0.922649i	$0.626019\pi$
$270$		0			0
$271$		−	23.5896i		−	1.43297i	−0.697603	−	0.716484i	$-0.745750\pi$
$271$		−	23.5896i		−	1.43297i	0.697603	−	0.716484i	$-0.254250\pi$
$272$		0			0
$273$	3.12136	+	4.37752i	0.188914	+	0.264940i
$274$		0			0
$275$			3.09886i			0.186869i
$276$		0			0
$277$	−29.3386			−1.76279			−0.881394	−	0.472383i	$-0.843394\pi$
$277$	−29.3386			−1.76279			−0.881394	+	0.472383i	$0.843394\pi$
$278$		0			0
$279$		−	1.79633i		−	0.107543i
$280$		0			0
$281$		−	14.5026i		−	0.865153i	−0.901597	−	0.432577i	$-0.857604\pi$
$281$		−	14.5026i		−	0.865153i	0.901597	−	0.432577i	$-0.142396\pi$
$282$		0			0
$283$	10.2156			0.607254			0.303627	−	0.952791i	$-0.401802\pi$
$283$	10.2156			0.607254			0.303627	+	0.952791i	$0.401802\pi$
$284$		0			0
$285$	0.0929941			0.00550849
$286$		0			0
$287$	2.15820			0.127394
$288$		0			0
$289$	−16.3569			−0.962171
$290$		0			0
$291$			8.34382i			0.489123i
$292$		0			0
$293$		−	17.6529i		−	1.03129i	−0.856801	−	0.515647i	$-0.827552\pi$
$293$		−	17.6529i		−	1.03129i	0.856801	−	0.515647i	$-0.172448\pi$
$294$		0			0
$295$	−11.8163			−0.687969
$296$		0			0
$297$			5.63129i			0.326761i
$298$		0			0
$299$	−18.1670	+	12.9539i	−1.05063	+	0.749143i
$300$		0			0
$301$		−	4.55492i		−	0.262541i
$302$		0			0
$303$	7.83574			0.450151
$304$		0			0
$305$		−	10.7951i		−	0.618127i
$306$		0			0
$307$		−	29.9238i		−	1.70784i	−0.520405	−	0.853920i	$-0.674219\pi$
$307$		−	29.9238i		−	1.70784i	0.520405	−	0.853920i	$-0.325781\pi$
$308$		0			0
$309$	19.4406			1.10594
$310$		0			0
$311$	22.7093			1.28772			0.643862	−	0.765142i	$-0.277331\pi$
$311$	22.7093			1.28772			0.643862	+	0.765142i	$0.277331\pi$
$312$		0			0
$313$	−10.2929			−0.581791			−0.290896	−	0.956755i	$-0.593953\pi$
$313$	−10.2929			−0.581791			−0.290896	+	0.956755i	$0.593953\pi$
$314$		0			0
$315$	−1.07064			−0.0603237
$316$		0			0
$317$		−	2.55697i		−	0.143614i	−0.997419	−	0.0718068i	$-0.977124\pi$
$317$		−	2.55697i		−	0.143614i	0.997419	−	0.0718068i	$-0.0228765\pi$
$318$		0			0
$319$		−	7.84316i		−	0.439133i
$320$		0			0
$321$	14.8032			0.826232
$322$		0			0
$323$		−	0.0362715i		−	0.00201820i
$324$		0			0
$325$	−9.09728	+	6.48676i	−0.504626	+	0.359821i
$326$		0			0
$327$		−	20.1078i		−	1.11196i
$328$		0			0
$329$	1.91517			0.105587
$330$		0			0
$331$		−	20.4274i		−	1.12279i	−0.827547	−	0.561396i	$-0.810264\pi$
$331$		−	20.4274i		−	1.12279i	0.827547	−	0.561396i	$-0.189736\pi$
$332$		0			0
$333$		−	2.91181i		−	0.159566i
$334$		0			0
$335$	−17.9998			−0.983436
$336$		0			0
$337$	5.41864			0.295172			0.147586	−	0.989049i	$-0.452850\pi$
$337$	5.41864			0.295172			0.147586	+	0.989049i	$0.452850\pi$
$338$		0			0
$339$	−19.4381			−1.05573
$340$		0			0
$341$	−2.31339			−0.125277
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			12.7233i			0.685002i
$346$		0			0
$347$	−5.01725			−0.269340			−0.134670	−	0.990890i	$-0.542997\pi$
$347$	−5.01725			−0.269340			−0.134670	+	0.990890i	$0.542997\pi$
$348$		0			0
$349$		−	26.9128i		−	1.44061i	−0.693657	−	0.720306i	$-0.744002\pi$
$349$		−	26.9128i		−	1.44061i	0.693657	−	0.720306i	$-0.255998\pi$
$350$		0			0
$351$	−16.5317	+	11.7878i	−0.882395	+	0.629186i
$352$		0			0
$353$		−	6.73979i		−	0.358723i	−0.983783	−	0.179361i	$-0.942597\pi$
$353$		−	6.73979i		−	0.358723i	0.983783	−	0.179361i	$-0.0574031\pi$
$354$		0			0
$355$	1.88880			0.100247
$356$		0			0
$357$		−	1.19579i		−	0.0632880i
$358$		0			0
$359$			0.933367i			0.0492612i	0.999697	+	0.0246306i	$0.00784096\pi$
$359$			0.933367i			0.0492612i	−0.999697	+	0.0246306i	$0.992159\pi$
$360$		0			0
$361$	18.9980			0.999892
$362$		0			0
$363$	1.49114			0.0782647
$364$		0			0
$365$	8.46204			0.442924
$366$		0			0
$367$	−29.6499			−1.54771			−0.773856	−	0.633362i	$-0.781675\pi$
$367$	−29.6499			−1.54771			−0.773856	+	0.633362i	$0.781675\pi$
$368$		0			0
$369$			1.67582i			0.0872398i
$370$		0			0
$371$			1.32588i			0.0688360i
$372$		0			0
$373$	−20.5132			−1.06214			−0.531068	−	0.847329i	$-0.678209\pi$
$373$	−20.5132			−1.06214			−0.531068	+	0.847329i	$0.678209\pi$
$374$		0			0
$375$			16.6514i			0.859873i
$376$		0			0
$377$	23.0250	−	16.4178i	1.18585	−	0.845562i
$378$		0			0
$379$			27.0229i			1.38807i	0.719940	+	0.694037i	$0.244170\pi$
$379$			27.0229i			1.38807i	−0.719940	+	0.694037i	$0.755830\pi$
$380$		0			0
$381$	15.0261			0.769811
$382$		0			0
$383$			21.2421i			1.08542i	0.839921	+	0.542709i	$0.182601\pi$
$383$			21.2421i			1.08542i	−0.839921	+	0.542709i	$0.817399\pi$
$384$		0			0
$385$			1.37882i			0.0702710i
$386$		0			0
$387$	3.53686			0.179789
$388$		0			0
$389$	15.2249			0.771931			0.385966	−	0.922513i	$-0.373868\pi$
$389$	15.2249			0.771931			0.385966	+	0.922513i	$0.373868\pi$
$390$		0			0
$391$	4.96263			0.250971
$392$		0			0
$393$	13.3061			0.671204
$394$		0			0
$395$			12.2447i			0.616097i
$396$		0			0
$397$			25.7143i			1.29057i	0.763944	+	0.645283i	$0.223261\pi$
$397$			25.7143i			1.29057i	−0.763944	+	0.645283i	$0.776739\pi$
$398$		0			0
$399$	−0.0674448			−0.00337647
$400$		0			0
$401$			32.1744i			1.60671i	0.595497	+	0.803357i	$0.296955\pi$
$401$			32.1744i			1.60671i	−0.595497	+	0.803357i	$0.703045\pi$
$402$		0			0
$403$	−4.84255	−	6.79138i	−0.241225	−	0.338303i
$404$		0			0
$405$			8.36609i			0.415714i
$406$		0			0
$407$	−3.74996			−0.185879
$408$		0			0
$409$			22.2195i			1.09868i	0.835599	+	0.549341i	$0.185121\pi$
$409$			22.2195i			1.09868i	−0.835599	+	0.549341i	$0.814879\pi$
$410$		0			0
$411$		−	30.1975i		−	1.48953i
$412$		0			0
$413$	8.56985			0.421695
$414$		0			0
$415$	−5.79954			−0.284688
$416$		0			0
$417$	10.2215			0.500548
$418$		0			0
$419$	24.7205			1.20768			0.603839	−	0.797107i	$-0.293637\pi$
$419$	24.7205			1.20768			0.603839	+	0.797107i	$0.293637\pi$
$420$		0			0
$421$			15.5645i			0.758569i	0.925280	+	0.379285i	$0.123830\pi$
$421$			15.5645i			0.758569i	−0.925280	+	0.379285i	$0.876170\pi$
$422$		0			0
$423$			1.48712i			0.0723062i
$424$		0			0
$425$	2.48507			0.120544
$426$		0			0
$427$			7.82926i			0.378885i
$428$		0			0
$429$	3.12136	+	4.37752i	0.150701	+	0.211349i
$430$		0			0
$431$		−	5.08765i		−	0.245063i	−0.992465	−	0.122532i	$-0.960899\pi$
$431$		−	5.08765i		−	0.245063i	0.992465	−	0.122532i	$-0.0391013\pi$
$432$		0			0
$433$	22.6678			1.08934			0.544672	−	0.838649i	$-0.316654\pi$
$433$	22.6678			1.08934			0.544672	+	0.838649i	$0.316654\pi$
$434$		0			0
$435$		−	16.1256i		−	0.773165i
$436$		0			0
$437$		−	0.279901i		−	0.0133895i
$438$		0			0
$439$	−12.2165			−0.583061			−0.291531	−	0.956561i	$-0.594165\pi$
$439$	−12.2165			−0.583061			−0.291531	+	0.956561i	$0.594165\pi$
$440$		0			0
$441$	0.776492			0.0369758
$442$		0			0
$443$	−2.07771			−0.0987150			−0.0493575	−	0.998781i	$-0.515717\pi$
$443$	−2.07771			−0.0987150			−0.0493575	+	0.998781i	$0.515717\pi$
$444$		0			0
$445$	−13.1757			−0.624590
$446$		0			0
$447$			1.45262i			0.0687068i
$448$		0			0
$449$		−	35.9045i		−	1.69444i	−0.531244	−	0.847219i	$-0.678275\pi$
$449$		−	35.9045i		−	1.69444i	0.531244	−	0.847219i	$-0.321725\pi$
$450$		0			0
$451$	2.15820			0.101625
$452$		0			0
$453$		−	19.3730i		−	0.910222i
$454$		0			0
$455$	−4.04777	+	2.88623i	−0.189762	+	0.135309i
$456$		0			0
$457$			1.11925i			0.0523561i	0.999657	+	0.0261781i	$0.00833369\pi$
$457$			1.11925i			0.0523561i	−0.999657	+	0.0261781i	$0.991666\pi$
$458$		0			0
$459$	4.51590			0.210784
$460$		0			0
$461$			26.2572i			1.22292i	0.791276	+	0.611459i	$0.209417\pi$
$461$			26.2572i			1.22292i	−0.791276	+	0.611459i	$0.790583\pi$
$462$		0			0
$463$		−	37.1914i		−	1.72843i	−0.503121	−	0.864216i	$-0.667815\pi$
$463$		−	37.1914i		−	1.72843i	0.503121	−	0.864216i	$-0.332185\pi$
$464$		0			0
$465$	−4.75636			−0.220571
$466$		0			0
$467$	−8.96476			−0.414840			−0.207420	−	0.978252i	$-0.566507\pi$
$467$	−8.96476			−0.414840			−0.207420	+	0.978252i	$0.566507\pi$
$468$		0			0
$469$	13.0545			0.602803
$470$		0			0
$471$	−16.9195			−0.779610
$472$		0			0
$473$		−	4.55492i		−	0.209435i
$474$		0			0
$475$		−	0.140163i		−	0.00643110i
$476$		0			0
$477$	−1.02953			−0.0471390
$478$		0			0
$479$		−	40.2673i		−	1.83986i	−0.392081	−	0.919931i	$-0.628245\pi$
$479$		−	40.2673i		−	1.83986i	0.392081	−	0.919931i	$-0.371755\pi$
$480$		0			0
$481$	−7.84967	−	11.0087i	−0.357914	−	0.501953i
$482$		0			0
$483$		−	9.22773i		−	0.419876i
$484$		0			0
$485$	−7.71529			−0.350333
$486$		0			0
$487$		−	6.97904i		−	0.316250i	−0.987419	−	0.158125i	$-0.949455\pi$
$487$		−	6.97904i		−	0.316250i	0.987419	−	0.158125i	$-0.0505450\pi$
$488$		0			0
$489$		−	6.03620i		−	0.272966i
$490$		0			0
$491$	−18.3699			−0.829023			−0.414512	−	0.910044i	$-0.636048\pi$
$491$	−18.3699			−0.829023			−0.414512	+	0.910044i	$0.636048\pi$
$492$		0			0
$493$	−6.28966			−0.283272
$494$		0			0
$495$	−1.07064			−0.0481217
$496$		0			0
$497$	−1.36987			−0.0614469
$498$		0			0
$499$			18.3859i			0.823065i	0.911395	+	0.411533i	$0.135006\pi$
$499$			18.3859i			0.823065i	−0.911395	+	0.411533i	$0.864994\pi$
$500$		0			0
$501$			10.7722i			0.481268i
$502$		0			0
$503$	24.5148			1.09306			0.546531	−	0.837439i	$-0.315948\pi$
$503$	24.5148			1.09306			0.546531	+	0.837439i	$0.315948\pi$
$504$		0			0
$505$			7.24548i			0.322419i
$506$		0			0
$507$	−6.31715	+	18.3267i	−0.280555	+	0.813915i
$508$		0			0
$509$			5.95666i			0.264024i	0.991248	+	0.132012i	$0.0421438\pi$
$509$			5.95666i			0.264024i	−0.991248	+	0.132012i	$0.957856\pi$
$510$		0			0
$511$	−6.13718			−0.271493
$512$		0			0
$513$		−	0.254705i		−	0.0112455i
$514$		0			0
$515$			17.9762i			0.792125i
$516$		0			0
$517$	1.91517			0.0842293
$518$		0			0
$519$	24.2279			1.06349
$520$		0			0
$521$	18.6556			0.817315			0.408658	−	0.912688i	$-0.365997\pi$
$521$	18.6556			0.817315			0.408658	+	0.912688i	$0.365997\pi$
$522$		0			0
$523$	−21.6725			−0.947674			−0.473837	−	0.880612i	$-0.657131\pi$
$523$	−21.6725			−0.947674			−0.473837	+	0.880612i	$0.657131\pi$
$524$		0			0
$525$		−	4.62085i		−	0.201670i
$526$		0			0
$527$			1.85518i			0.0808128i
$528$		0			0
$529$	15.2958			0.665033
$530$		0			0
$531$			6.65442i			0.288777i
$532$		0			0
$533$	4.51769	+	6.33578i	0.195683	+	0.274433i
$534$		0			0
$535$			13.6881i			0.591786i
$536$		0			0
$537$	35.7474			1.54262
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$			23.6938i			1.01867i	0.860567	+	0.509337i	$0.170109\pi$
$541$			23.6938i			1.01867i	−0.860567	+	0.509337i	$0.829891\pi$
$542$		0			0
$543$	−10.9174			−0.468509
$544$		0			0
$545$	18.5931			0.796439
$546$		0			0
$547$	−18.1652			−0.776688			−0.388344	−	0.921514i	$-0.626953\pi$
$547$	−18.1652			−0.776688			−0.388344	+	0.921514i	$0.626953\pi$
$548$		0			0
$549$	−6.07936			−0.259461
$550$		0			0
$551$			0.354749i			0.0151128i
$552$		0			0
$553$		−	8.88057i		−	0.377640i
$554$		0			0
$555$	−7.70996			−0.327270
$556$		0			0
$557$			7.84539i			0.332420i	0.986090	+	0.166210i	$0.0531529\pi$
$557$			7.84539i			0.332420i	−0.986090	+	0.166210i	$0.946847\pi$
$558$		0			0
$559$	13.3718	−	9.53467i	0.565566	−	0.403273i
$560$		0			0
$561$		−	1.19579i		−	0.0504864i
$562$		0			0
$563$	23.3111			0.982446			0.491223	−	0.871034i	$-0.336550\pi$
$563$	23.3111			0.982446			0.491223	+	0.871034i	$0.336550\pi$
$564$		0			0
$565$		−	17.9738i		−	0.756164i
$566$		0			0
$567$		−	6.06758i		−	0.254815i
$568$		0			0
$569$	4.92117			0.206306			0.103153	−	0.994666i	$-0.467107\pi$
$569$	4.92117			0.206306			0.103153	+	0.994666i	$0.467107\pi$
$570$		0			0
$571$	23.9300			1.00144			0.500719	−	0.865610i	$-0.333069\pi$
$571$	23.9300			1.00144			0.500719	+	0.865610i	$0.333069\pi$
$572$		0			0
$573$	−23.0615			−0.963407
$574$		0			0
$575$	19.1769			0.799731
$576$		0			0
$577$		−	12.2779i		−	0.511134i	−0.966791	−	0.255567i	$-0.917738\pi$
$577$		−	12.2779i		−	0.511134i	0.966791	−	0.255567i	$-0.0822621\pi$
$578$		0			0
$579$			24.4595i			1.01650i
$580$		0			0
$581$	4.20617			0.174501
$582$		0			0
$583$			1.32588i			0.0549122i
$584$		0			0
$585$	−2.24114	−	3.14306i	−0.0926596	−	0.129949i
$586$		0			0
$587$		−	37.8757i		−	1.56330i	−0.623720	−	0.781648i	$-0.714379\pi$
$587$		−	37.8757i		−	1.56330i	0.623720	−	0.781648i	$-0.285621\pi$
$588$		0			0
$589$	0.104635			0.00431142
$590$		0			0
$591$			6.76441i			0.278251i
$592$		0			0
$593$			41.0910i			1.68740i	0.536811	+	0.843702i	$0.319629\pi$
$593$			41.0910i			1.68740i	−0.536811	+	0.843702i	$0.680371\pi$
$594$		0			0
$595$	1.10571			0.0453298
$596$		0			0
$597$	14.2632			0.583753
$598$		0			0
$599$	−15.4676			−0.631990			−0.315995	−	0.948761i	$-0.602338\pi$
$599$	−15.4676			−0.631990			−0.315995	+	0.948761i	$0.602338\pi$
$600$		0			0
$601$	−8.93017			−0.364269			−0.182135	−	0.983274i	$-0.558301\pi$
$601$	−8.93017			−0.364269			−0.182135	+	0.983274i	$0.558301\pi$
$602$		0			0
$603$			10.1368i			0.412800i
$604$		0			0
$605$			1.37882i			0.0560569i
$606$		0			0
$607$	4.19652			0.170331			0.0851657	−	0.996367i	$-0.472858\pi$
$607$	4.19652			0.170331			0.0851657	+	0.996367i	$0.472858\pi$
$608$		0			0
$609$			11.6953i			0.473917i
$610$		0			0
$611$	4.00898	+	5.62234i	0.162186	+	0.227456i
$612$		0			0
$613$		−	13.2786i		−	0.536316i	−0.963375	−	0.268158i	$-0.913585\pi$
$613$		−	13.2786i		−	0.536316i	0.963375	−	0.268158i	$-0.0864149\pi$
$614$		0			0
$615$	4.43728			0.178928
$616$		0			0
$617$			44.6391i			1.79710i	0.438870	+	0.898550i	$0.355379\pi$
$617$			44.6391i			1.79710i	−0.438870	+	0.898550i	$0.644621\pi$
$618$		0			0
$619$		−	9.48610i		−	0.381279i	−0.981660	−	0.190639i	$-0.938944\pi$
$619$		−	9.48610i		−	0.381279i	0.981660	−	0.190639i	$-0.0610561\pi$
$620$		0			0
$621$	34.8484			1.39842
$622$		0			0
$623$	9.55584			0.382846
$624$		0			0
$625$	0.0972860			0.00389144
$626$		0			0
$627$	−0.0674448			−0.00269349
$628$		0			0
$629$			3.00720i			0.119905i
$630$		0			0
$631$			10.8053i			0.430151i	0.976597	+	0.215075i	$0.0689997\pi$
$631$			10.8053i			0.430151i	−0.976597	+	0.215075i	$0.931000\pi$
$632$		0			0
$633$	36.5511			1.45278
$634$		0			0
$635$			13.8942i			0.551375i
$636$		0			0
$637$	2.93568	−	2.09327i	0.116316	−	0.0829383i
$638$		0			0
$639$		−	1.06369i		−	0.0420790i
$640$		0			0
$641$	11.5052			0.454427			0.227213	−	0.973845i	$-0.427039\pi$
$641$	11.5052			0.454427			0.227213	+	0.973845i	$0.427039\pi$
$642$		0			0
$643$			12.2080i			0.481438i	0.970595	+	0.240719i	$0.0773833\pi$
$643$			12.2080i			0.481438i	−0.970595	+	0.240719i	$0.922617\pi$
$644$		0			0
$645$		−	9.36497i		−	0.368745i
$646$		0			0
$647$	4.38289			0.172309			0.0861547	−	0.996282i	$-0.472542\pi$
$647$	4.38289			0.172309			0.0861547	+	0.996282i	$0.472542\pi$
$648$		0			0
$649$	8.56985			0.336396
$650$		0			0
$651$	3.44960			0.135200
$652$		0			0
$653$	12.6643			0.495592			0.247796	−	0.968812i	$-0.420294\pi$
$653$	12.6643			0.495592			0.247796	+	0.968812i	$0.420294\pi$
$654$		0			0
$655$			12.3038i			0.480748i
$656$		0			0
$657$		−	4.76547i		−	0.185919i
$658$		0			0
$659$	17.1021			0.666203			0.333101	−	0.942891i	$-0.391905\pi$
$659$	17.1021			0.666203			0.333101	+	0.942891i	$0.391905\pi$
$660$		0			0
$661$		−	49.0572i		−	1.90810i	−0.299643	−	0.954051i	$-0.596868\pi$
$661$		−	49.0572i		−	1.90810i	0.299643	−	0.954051i	$-0.403132\pi$
$662$		0			0
$663$	3.51046	−	2.50311i	0.136335	−	0.0972129i
$664$		0			0
$665$		−	0.0623643i		−	0.00241838i
$666$		0			0
$667$	−48.5363			−1.87933
$668$		0			0
$669$			7.26678i			0.280950i
$670$		0			0
$671$			7.82926i			0.302245i
$672$		0			0
$673$	−28.6787			−1.10548			−0.552742	−	0.833353i	$-0.686418\pi$
$673$	−28.6787			−1.10548			−0.552742	+	0.833353i	$0.686418\pi$
$674$		0			0
$675$	17.4506			0.671674
$676$		0			0
$677$	41.0793			1.57881			0.789403	−	0.613876i	$-0.210390\pi$
$677$	41.0793			1.57881			0.789403	+	0.613876i	$0.210390\pi$
$678$		0			0
$679$	5.59559			0.214739
$680$		0			0
$681$			40.8623i			1.56585i
$682$		0			0
$683$		−	2.10104i		−	0.0803940i	−0.999192	−	0.0401970i	$-0.987201\pi$
$683$		−	2.10104i		−	0.0803940i	0.999192	−	0.0401970i	$-0.0127986\pi$
$684$		0			0
$685$	27.9228			1.06687
$686$		0			0
$687$		−	9.22022i		−	0.351773i
$688$		0			0
$689$	−3.89235	+	2.77541i	−0.148287	+	0.105735i
$690$		0			0
$691$			48.7134i			1.85315i	0.376115	+	0.926573i	$0.377260\pi$
$691$			48.7134i			1.85315i	−0.376115	+	0.926573i	$0.622740\pi$
$692$		0			0
$693$	0.776492			0.0294965
$694$		0			0
$695$			9.45151i			0.358516i
$696$		0			0
$697$		−	1.73072i		−	0.0655557i
$698$		0			0
$699$	19.6751			0.744182
$700$		0			0
$701$	36.1876			1.36679			0.683393	−	0.730051i	$-0.260504\pi$
$701$	36.1876			1.36679			0.683393	+	0.730051i	$0.260504\pi$
$702$		0			0
$703$	0.169612			0.00639703
$704$		0			0
$705$	3.93762			0.148300
$706$		0			0
$707$		−	5.25485i		−	0.197629i
$708$		0			0
$709$		−	0.460582i		−	0.0172975i	−0.999963	−	0.00864876i	$-0.997247\pi$
$709$		−	0.460582i		−	0.0172975i	0.999963	−	0.00864876i	$-0.00275302\pi$
$710$		0			0
$711$	6.89569			0.258609
$712$		0			0
$713$			14.3161i			0.536142i
$714$		0			0
$715$	−4.04777	+	2.88623i	−0.151378	+	0.107939i
$716$		0			0
$717$			26.7061i			0.997357i
$718$		0			0
$719$	−41.0938			−1.53254			−0.766271	−	0.642518i	$-0.777890\pi$
$719$	−41.0938			−1.53254			−0.766271	+	0.642518i	$0.777890\pi$
$720$		0			0
$721$		−	13.0374i		−	0.485538i
$722$		0			0
$723$			29.0610i			1.08079i
$724$		0			0
$725$	−24.3049			−0.902661
$726$		0			0
$727$	17.5980			0.652675			0.326337	−	0.945253i	$-0.394185\pi$
$727$	17.5980			0.652675			0.326337	+	0.945253i	$0.394185\pi$
$728$		0			0
$729$	29.9026			1.10750
$730$		0			0
$731$	−3.65272			−0.135101
$732$		0			0
$733$		−	9.72639i		−	0.359252i	−0.983735	−	0.179626i	$-0.942511\pi$
$733$		−	9.72639i		−	0.359252i	0.983735	−	0.179626i	$-0.0574888\pi$
$734$		0			0
$735$		−	2.05601i		−	0.0758372i
$736$		0			0
$737$	13.0545			0.480870
$738$		0			0
$739$			39.8300i			1.46517i	0.680676	+	0.732585i	$0.261686\pi$
$739$			39.8300i			1.46517i	−0.680676	+	0.732585i	$0.738314\pi$
$740$		0			0
$741$	−0.141180	−	0.197997i	−0.00518638	−	0.00727359i
$742$		0			0
$743$			3.40607i			0.124957i	0.998046	+	0.0624784i	$0.0199004\pi$
$743$			3.40607i			0.124957i	−0.998046	+	0.0624784i	$0.980100\pi$
$744$		0			0
$745$	−1.34320			−0.0492110
$746$		0			0
$747$			3.26606i			0.119499i
$748$		0			0
$749$		−	9.92739i		−	0.362739i
$750$		0			0
$751$	13.0147			0.474912			0.237456	−	0.971398i	$-0.423686\pi$
$751$	13.0147			0.474912			0.237456	+	0.971398i	$0.423686\pi$
$752$		0			0
$753$	6.02848			0.219690
$754$		0			0
$755$	17.9136			0.651944
$756$		0			0
$757$	−44.8498			−1.63009			−0.815047	−	0.579395i	$-0.803289\pi$
$757$	−44.8498			−1.63009			−0.815047	+	0.579395i	$0.803289\pi$
$758$		0			0
$759$		−	9.22773i		−	0.334945i
$760$		0			0
$761$			34.7548i			1.25986i	0.776652	+	0.629930i	$0.216916\pi$
$761$			34.7548i			1.25986i	−0.776652	+	0.629930i	$0.783084\pi$
$762$		0			0
$763$	−13.4848			−0.488182
$764$		0			0
$765$			0.858578i			0.0310419i
$766$		0			0
$767$	17.9390	+	25.1584i	0.647740	+	0.908416i
$768$		0			0
$769$		−	37.7642i		−	1.36181i	−0.732372	−	0.680905i	$-0.761587\pi$
$769$		−	37.7642i		−	1.36181i	0.732372	−	0.680905i	$-0.238413\pi$
$770$		0			0
$771$	27.1753			0.978693
$772$		0			0
$773$		−	18.9080i		−	0.680074i	−0.940412	−	0.340037i	$-0.889560\pi$
$773$		−	18.9080i		−	0.680074i	0.940412	−	0.340037i	$-0.110440\pi$
$774$		0			0
$775$			7.16889i			0.257514i
$776$		0			0
$777$	5.59172			0.200602
$778$		0			0
$779$	−0.0976158			−0.00349745
$780$		0			0
$781$	−1.36987			−0.0490177
$782$		0			0
$783$	−44.1671			−1.57840
$784$		0			0
$785$		−	15.6450i		−	0.558393i
$786$		0			0
$787$		−	53.7701i		−	1.91670i	−0.285601	−	0.958349i	$-0.592193\pi$
$787$		−	53.7701i		−	1.91670i	0.285601	−	0.958349i	$-0.407807\pi$
$788$		0			0
$789$	−16.7464			−0.596188
$790$		0			0
$791$			13.0357i			0.463495i
$792$		0			0
$793$	−22.9842	+	16.3888i	−0.816194	+	0.581982i
$794$		0			0
$795$			2.72602i			0.0966819i
$796$		0			0
$797$	−13.1715			−0.466559			−0.233279	−	0.972410i	$-0.574946\pi$
$797$	−13.1715			−0.466559			−0.233279	+	0.972410i	$0.574946\pi$
$798$		0			0
$799$		−	1.53584i		−	0.0543339i
$800$		0			0
$801$			7.42003i			0.262174i
$802$		0			0
$803$	−6.13718			−0.216576
$804$		0			0
$805$	8.53261			0.300735
$806$		0			0
$807$	18.8629			0.664005
$808$		0			0
$809$	−43.3683			−1.52475			−0.762374	−	0.647137i	$-0.775966\pi$
$809$	−43.3683			−1.52475			−0.762374	+	0.647137i	$0.775966\pi$
$810$		0			0
$811$		−	38.4125i		−	1.34884i	−0.738346	−	0.674422i	$-0.764393\pi$
$811$		−	38.4125i		−	1.34884i	0.738346	−	0.674422i	$-0.235607\pi$
$812$		0			0
$813$			35.1755i			1.23366i
$814$		0			0
$815$	5.58150			0.195511
$816$		0			0
$817$			0.206020i			0.00720774i
$818$		0			0
$819$	1.62541	+	2.27953i	0.0567963	+	0.0796533i
$820$		0			0
$821$			34.5301i			1.20511i	0.798078	+	0.602554i	$0.205850\pi$
$821$			34.5301i			1.20511i	−0.798078	+	0.602554i	$0.794150\pi$
$822$		0			0
$823$	−39.8457			−1.38893			−0.694466	−	0.719525i	$-0.744359\pi$
$823$	−39.8457			−1.38893			−0.694466	+	0.719525i	$0.744359\pi$
$824$		0			0
$825$		−	4.62085i		−	0.160877i
$826$		0			0
$827$			3.90160i			0.135672i	0.997696	+	0.0678359i	$0.0216094\pi$
$827$			3.90160i			0.135672i	−0.997696	+	0.0678359i	$0.978391\pi$
$828$		0			0
$829$	37.1801			1.29132			0.645658	−	0.763626i	$-0.276583\pi$
$829$	37.1801			1.29132			0.645658	+	0.763626i	$0.276583\pi$
$830$		0			0
$831$	43.7481			1.51760
$832$		0			0
$833$	−0.801929			−0.0277852
$834$		0			0
$835$	−9.96078			−0.344707
$836$		0			0
$837$			13.0274i			0.450292i
$838$		0			0
$839$			10.4262i			0.359952i	0.983671	+	0.179976i	$0.0576020\pi$
$839$			10.4262i			0.359952i	−0.983671	+	0.179976i	$0.942398\pi$
$840$		0			0
$841$	32.5152			1.12121
$842$		0			0
$843$			21.6255i			0.744821i
$844$		0			0
$845$	−16.9461	−	5.84129i	−0.582964	−	0.200946i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	−15.2329			−0.522792
$850$		0			0
$851$			23.2061i			0.795494i
$852$		0			0
$853$		−	34.4123i		−	1.17826i	−0.808040	−	0.589128i	$-0.799471\pi$
$853$		−	34.4123i		−	1.17826i	0.808040	−	0.589128i	$-0.200529\pi$
$854$		0			0
$855$	0.0484254			0.00165611
$856$		0			0
$857$	−10.6969			−0.365401			−0.182700	−	0.983169i	$-0.558484\pi$
$857$	−10.6969			−0.365401			−0.182700	+	0.983169i	$0.558484\pi$
$858$		0			0
$859$	−12.8625			−0.438862			−0.219431	−	0.975628i	$-0.570420\pi$
$859$	−12.8625			−0.438862			−0.219431	+	0.975628i	$0.570420\pi$
$860$		0			0
$861$	−3.21818			−0.109675
$862$		0			0
$863$			1.29654i			0.0441348i	0.999756	+	0.0220674i	$0.00702484\pi$
$863$			1.29654i			0.0441348i	−0.999756	+	0.0220674i	$0.992975\pi$
$864$		0			0
$865$			22.4028i			0.761718i
$866$		0			0
$867$	24.3905			0.828345
$868$		0			0
$869$		−	8.88057i		−	0.301253i
$870$		0			0
$871$	27.3267	+	38.3240i	0.925929	+	1.29856i
$872$		0			0
$873$			4.34493i			0.147054i
$874$		0			0
$875$	11.1668			0.377508
$876$		0			0
$877$			4.67717i			0.157937i	0.996877	+	0.0789685i	$0.0251626\pi$
$877$			4.67717i			0.157937i	−0.996877	+	0.0789685i	$0.974837\pi$
$878$		0			0
$879$			26.3230i			0.887854i
$880$		0			0
$881$	15.8608			0.534364			0.267182	−	0.963646i	$-0.413907\pi$
$881$	15.8608			0.534364			0.267182	+	0.963646i	$0.413907\pi$
$882$		0			0
$883$	−18.7381			−0.630586			−0.315293	−	0.948994i	$-0.602103\pi$
$883$	−18.7381			−0.630586			−0.315293	+	0.948994i	$0.602103\pi$
$884$		0			0
$885$	17.6197			0.592281
$886$		0			0
$887$	−8.74195			−0.293526			−0.146763	−	0.989172i	$-0.546885\pi$
$887$	−8.74195			−0.293526			−0.146763	+	0.989172i	$0.546885\pi$
$888$		0			0
$889$		−	10.0769i		−	0.337969i
$890$		0			0
$891$		−	6.06758i		−	0.203272i
$892$		0			0
$893$	−0.0866239			−0.00289876
$894$		0			0
$895$			33.0546i			1.10489i
$896$		0			0
$897$	27.0897	−	19.3161i	0.904497	−	0.644946i
$898$		0			0
$899$		−	18.1443i		−	0.605146i
$900$		0			0
$901$	1.06326			0.0354223
$902$		0			0
$903$			6.79203i			0.226025i
$904$		0			0
$905$		−	10.0950i		−	0.335568i
$906$		0			0
$907$	−14.9691			−0.497040			−0.248520	−	0.968627i	$-0.579944\pi$
$907$	−14.9691			−0.497040			−0.248520	+	0.968627i	$0.579944\pi$
$908$		0			0
$909$	4.08035			0.135337
$910$		0			0
$911$	19.6555			0.651214			0.325607	−	0.945505i	$-0.394431\pi$
$911$	19.6555			0.651214			0.325607	+	0.945505i	$0.394431\pi$
$912$		0			0
$913$	4.20617			0.139204
$914$		0			0
$915$			16.0971i			0.532153i
$916$		0			0
$917$		−	8.92342i		−	0.294677i
$918$		0			0
$919$	59.1037			1.94965			0.974826	−	0.222966i	$-0.0715740\pi$
$919$	59.1037			1.94965			0.974826	+	0.222966i	$0.0715740\pi$
$920$		0			0
$921$			44.6206i			1.47030i
$922$		0			0
$923$	−2.86750	−	4.02149i	−0.0943849	−	0.132369i
$924$		0			0
$925$			11.6206i			0.382083i
$926$		0			0
$927$	10.1234			0.332497
$928$		0			0
$929$			34.1152i			1.11928i	0.828735	+	0.559642i	$0.189061\pi$
$929$			34.1152i			1.11928i	−0.828735	+	0.559642i	$0.810939\pi$
$930$		0			0
$931$			0.0452303i			0.00148236i
$932$		0			0
$933$	−33.8627			−1.10862
$934$		0			0
$935$	1.10571			0.0361607
$936$		0			0
$937$	−4.66720			−0.152471			−0.0762353	−	0.997090i	$-0.524290\pi$
$937$	−4.66720			−0.152471			−0.0762353	+	0.997090i	$0.524290\pi$
$938$		0			0
$939$	15.3482			0.500871
$940$		0			0
$941$			29.0694i			0.947634i	0.880623	+	0.473817i	$0.157124\pi$
$941$			29.0694i			0.947634i	−0.880623	+	0.473817i	$0.842876\pi$
$942$		0			0
$943$		−	13.3557i		−	0.434921i
$944$		0			0
$945$	7.76452			0.252580
$946$		0			0
$947$		−	60.3612i		−	1.96148i	−0.195329	−	0.980738i	$-0.562577\pi$
$947$		−	60.3612i		−	1.96148i	0.195329	−	0.980738i	$-0.437423\pi$
$948$		0			0
$949$	−12.8468	−	18.0168i	−0.417024	−	0.584850i
$950$		0			0
$951$			3.81280i			0.123639i
$952$		0			0
$953$	−4.49297			−0.145542			−0.0727709	−	0.997349i	$-0.523184\pi$
$953$	−4.49297			−0.145542			−0.0727709	+	0.997349i	$0.523184\pi$
$954$		0			0
$955$		−	21.3243i		−	0.690037i
$956$		0			0
$957$			11.6953i			0.378055i
$958$		0			0
$959$	−20.2513			−0.653947
$960$		0			0
$961$	25.6482			0.827362
$962$		0			0
$963$	7.70854			0.248404
$964$		0			0
$965$	−22.6170			−0.728067
$966$		0			0
$967$			2.07203i			0.0666321i	0.999445	+	0.0333160i	$0.0106068\pi$
$967$			2.07203i			0.0666321i	−0.999445	+	0.0333160i	$0.989393\pi$
$968$		0			0
$969$			0.0540860i			0.00173749i
$970$		0			0
$971$	35.2443			1.13104			0.565522	−	0.824733i	$-0.308675\pi$
$971$	35.2443			1.13104			0.565522	+	0.824733i	$0.308675\pi$
$972$		0			0
$973$		−	6.85480i		−	0.219755i
$974$		0			0
$975$	13.5653	−	9.67268i	0.434439	−	0.309774i
$976$		0			0
$977$			47.2830i			1.51272i	0.654157	+	0.756359i	$0.273024\pi$
$977$			47.2830i			1.51272i	−0.654157	+	0.756359i	$0.726976\pi$
$978$		0			0
$979$	9.55584			0.305406
$980$		0			0
$981$		−	10.4708i		−	0.334308i
$982$		0			0
$983$		−	30.1778i		−	0.962521i	−0.876578	−	0.481261i	$-0.840179\pi$
$983$		−	30.1778i		−	0.962521i	0.876578	−	0.481261i	$-0.159821\pi$
$984$		0			0
$985$	−6.25485			−0.199296
$986$		0			0
$987$	−2.85580			−0.0909011
$988$		0			0
$989$	−28.1874			−0.896309
$990$		0			0
$991$	−5.35538			−0.170119			−0.0850596	−	0.996376i	$-0.527108\pi$
$991$	−5.35538			−0.170119			−0.0850596	+	0.996376i	$0.527108\pi$
$992$		0			0
$993$			30.4602i			0.966625i
$994$		0			0
$995$			13.1888i			0.418112i
$996$		0			0
$997$	10.1781			0.322343			0.161171	−	0.986926i	$-0.448473\pi$
$997$	10.1781			0.322343			0.161171	+	0.986926i	$0.448473\pi$
$998$		0			0
$999$			21.1171i			0.668116i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.b.2157.9	✓	30
13.12	even	2	inner	4004.2.m.b.2157.10	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.b.2157.9	✓	30	1.1	even	1	trivial
4004.2.m.b.2157.10	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)

\(f(q)\)	\(=\)	\(q-1.49114 q^{3} -1.37882i q^{5} +1.00000i q^{7} -0.776492 q^{9} +O(q^{10})\)	\(q-1.49114 q^{3} -1.37882i q^{5} +1.00000i q^{7} -0.776492 q^{9} +1.00000i q^{11} +(-2.93568 + 2.09327i) q^{13} +2.05601i q^{15} +0.801929 q^{17} -0.0452303i q^{19} -1.49114i q^{21} +6.18836 q^{23} +3.09886 q^{25} +5.63129 q^{27} -7.84316 q^{29} +2.31339i q^{31} -1.49114i q^{33} +1.37882 q^{35} +3.74996i q^{37} +(4.37752 - 3.12136i) q^{39} -2.15820i q^{41} -4.55492 q^{43} +1.07064i q^{45} -1.91517i q^{47} -1.00000 q^{49} -1.19579 q^{51} +1.32588 q^{53} +1.37882 q^{55} +0.0674448i q^{57} -8.56985i q^{59} +7.82926 q^{61} -0.776492i q^{63} +(2.88623 + 4.04777i) q^{65} -13.0545i q^{67} -9.22773 q^{69} +1.36987i q^{71} +6.13718i q^{73} -4.62085 q^{75} -1.00000 q^{77} -8.88057 q^{79} -6.06758 q^{81} -4.20617i q^{83} -1.10571i q^{85} +11.6953 q^{87} -9.55584i q^{89} +(-2.09327 - 2.93568i) q^{91} -3.44960i q^{93} -0.0623643 q^{95} -5.59559i q^{97} -0.776492i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 18 q^{9}+O(q^{10}) \) 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

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