LMFDB - Embedded newform 4028.2.a.f.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.a (trivial)

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:	yes
Analytic conductor:	$32.1637419342$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 30 x^{17} + 124 x^{16} + 364 x^{15} - 1554 x^{14} - 2310 x^{13} + 10113 x^{12} + \cdots + 139 $ x^19 - 4x^18 - 30x^17 + 124x^16 + 364x^15 - 1554x^14 - 2310x^13 + 10113x^12 + 8368x^11 - 36567x^10 - 18074x^9 + 72868x^8 + 23819x^7 - 73816x^6 - 17877x^5 + 30973x^4 + 5074x^3 - 4299x^2 - 454x + 139
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$0.667966$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.667966 q^{3} -3.08795 q^{5} -2.50903 q^{7} -2.55382 q^{9} +O(q^{10})$	$q+0.667966 q^{3} -3.08795 q^{5} -2.50903 q^{7} -2.55382 q^{9} -1.87869 q^{11} -0.792221 q^{13} -2.06264 q^{15} -3.96066 q^{17} -1.00000 q^{19} -1.67595 q^{21} +0.00275991 q^{23} +4.53544 q^{25} -3.70976 q^{27} +4.17060 q^{29} -0.345514 q^{31} -1.25490 q^{33} +7.74777 q^{35} -5.04885 q^{37} -0.529177 q^{39} -3.80178 q^{41} +0.706552 q^{43} +7.88608 q^{45} -7.21615 q^{47} -0.704749 q^{49} -2.64559 q^{51} -1.00000 q^{53} +5.80129 q^{55} -0.667966 q^{57} +6.10049 q^{59} +10.9378 q^{61} +6.40763 q^{63} +2.44634 q^{65} -6.76803 q^{67} +0.00184353 q^{69} -1.86865 q^{71} +8.42669 q^{73} +3.02952 q^{75} +4.71369 q^{77} +5.68450 q^{79} +5.18347 q^{81} +6.54771 q^{83} +12.2303 q^{85} +2.78582 q^{87} +1.84138 q^{89} +1.98771 q^{91} -0.230791 q^{93} +3.08795 q^{95} +8.44483 q^{97} +4.79783 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * q^99

Display 100 coefficients 10 coefficients

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
$3$	0.667966	0.385650	0.192825	−	0.981233i	$-0.438235\pi$
$3$	0.667966	0.385650	0.192825	+	0.981233i	$0.438235\pi$
$4$	0	0
$5$	−3.08795	−1.38097	−0.690487	−	0.723345i	$-0.742604\pi$
$5$	−3.08795	−1.38097	−0.690487	+	0.723345i	$0.742604\pi$
$6$	0	0
$7$	−2.50903	−0.948326	−0.474163	−	0.880437i	$-0.657249\pi$
$7$	−2.50903	−0.948326	−0.474163	+	0.880437i	$0.657249\pi$
$8$	0	0
$9$	−2.55382	−0.851274
$10$	0	0
$11$	−1.87869	−0.566446	−0.283223	−	0.959054i	$-0.591404\pi$
$11$	−1.87869	−0.566446	−0.283223	+	0.959054i	$0.591404\pi$
$12$	0	0
$13$	−0.792221	−0.219723	−0.109861	−	0.993947i	$-0.535041\pi$
$13$	−0.792221	−0.219723	−0.109861	+	0.993947i	$0.535041\pi$
$14$	0	0
$15$	−2.06264	−0.532573
$16$	0	0
$17$	−3.96066	−0.960602	−0.480301	−	0.877104i	$-0.659473\pi$
$17$	−3.96066	−0.960602	−0.480301	+	0.877104i	$0.659473\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.67595	−0.365722
$22$	0	0
$23$	0.00275991	0.000575481 0	0.000287741	−	1.00000i	$-0.499908\pi$
$23$	0.00275991	0.000575481 0	0.000287741		1.00000i	$0.499908\pi$
$24$	0	0
$25$	4.53544	0.907088
$26$	0	0
$27$	−3.70976	−0.713944
$28$	0	0
$29$	4.17060	0.774461	0.387230	−	0.921983i	$-0.373432\pi$
$29$	4.17060	0.774461	0.387230	+	0.921983i	$0.373432\pi$
$30$	0	0
$31$	−0.345514	−0.0620561	−0.0310281	−	0.999519i	$-0.509878\pi$
$31$	−0.345514	−0.0620561	−0.0310281	+	0.999519i	$0.509878\pi$
$32$	0	0
$33$	−1.25490	−0.218450
$34$	0	0
$35$	7.74777	1.30961
$36$	0	0
$37$	−5.04885	−0.830026	−0.415013	−	0.909816i	$-0.636223\pi$
$37$	−5.04885	−0.830026	−0.415013	+	0.909816i	$0.636223\pi$
$38$	0	0
$39$	−0.529177	−0.0847361
$40$	0	0
$41$	−3.80178	−0.593738	−0.296869	−	0.954918i	$-0.595942\pi$
$41$	−3.80178	−0.593738	−0.296869	+	0.954918i	$0.595942\pi$
$42$	0	0
$43$	0.706552	0.107748	0.0538741	−	0.998548i	$-0.482843\pi$
$43$	0.706552	0.107748	0.0538741	+	0.998548i	$0.482843\pi$
$44$	0	0
$45$	7.88608	1.17559
$46$	0	0
$47$	−7.21615	−1.05258	−0.526292	−	0.850304i	$-0.676418\pi$
$47$	−7.21615	−1.05258	−0.526292	+	0.850304i	$0.676418\pi$
$48$	0	0
$49$	−0.704749	−0.100678
$50$	0	0
$51$	−2.64559	−0.370456
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	5.80129	0.782246
$56$	0	0
$57$	−0.667966	−0.0884742
$58$	0	0
$59$	6.10049	0.794216	0.397108	−	0.917772i	$-0.370014\pi$
$59$	6.10049	0.794216	0.397108	+	0.917772i	$0.370014\pi$
$60$	0	0
$61$	10.9378	1.40044	0.700219	−	0.713928i	$-0.253086\pi$
$61$	10.9378	1.40044	0.700219	+	0.713928i	$0.253086\pi$
$62$	0	0
$63$	6.40763	0.807285
$64$	0	0
$65$	2.44634	0.303431
$66$	0	0
$67$	−6.76803	−0.826846	−0.413423	−	0.910539i	$-0.635667\pi$
$67$	−6.76803	−0.826846	−0.413423	+	0.910539i	$0.635667\pi$
$68$	0	0
$69$	0.00184353	0.000221934 0
$70$	0	0
$71$	−1.86865	−0.221767	−0.110884	−	0.993833i	$-0.535368\pi$
$71$	−1.86865	−0.221767	−0.110884	+	0.993833i	$0.535368\pi$
$72$	0	0
$73$	8.42669	0.986269	0.493135	−	0.869953i	$-0.335851\pi$
$73$	8.42669	0.986269	0.493135	+	0.869953i	$0.335851\pi$
$74$	0	0
$75$	3.02952	0.349819
$76$	0	0
$77$	4.71369	0.537175
$78$	0	0
$79$	5.68450	0.639557	0.319778	−	0.947492i	$-0.396392\pi$
$79$	5.68450	0.639557	0.319778	+	0.947492i	$0.396392\pi$
$80$	0	0
$81$	5.18347	0.575941
$82$	0	0
$83$	6.54771	0.718705	0.359352	−	0.933202i	$-0.382998\pi$
$83$	6.54771	0.718705	0.359352	+	0.933202i	$0.382998\pi$
$84$	0	0
$85$	12.2303	1.32657
$86$	0	0
$87$	2.78582	0.298671
$88$	0	0
$89$	1.84138	0.195186	0.0975928	−	0.995226i	$-0.468886\pi$
$89$	1.84138	0.195186	0.0975928	+	0.995226i	$0.468886\pi$
$90$	0	0
$91$	1.98771	0.208369
$92$	0	0
$93$	−0.230791	−0.0239319
$94$	0	0
$95$	3.08795	0.316817
$96$	0	0
$97$	8.44483	0.857442	0.428721	−	0.903437i	$-0.358964\pi$
$97$	8.44483	0.857442	0.428721	+	0.903437i	$0.358964\pi$
$98$	0	0
$99$	4.79783	0.482200
$100$	0	0
$101$	−12.4057	−1.23441	−0.617206	−	0.786801i	$-0.711736\pi$
$101$	−12.4057	−1.23441	−0.617206	+	0.786801i	$0.711736\pi$
$102$	0	0
$103$	9.23534	0.909985	0.454992	−	0.890495i	$-0.349642\pi$
$103$	9.23534	0.909985	0.454992	+	0.890495i	$0.349642\pi$
$104$	0	0
$105$	5.17525	0.505052
$106$	0	0
$107$	−13.3226	−1.28794	−0.643972	−	0.765049i	$-0.722715\pi$
$107$	−13.3226	−1.28794	−0.643972	+	0.765049i	$0.722715\pi$
$108$	0	0
$109$	−17.0813	−1.63609	−0.818047	−	0.575151i	$-0.804943\pi$
$109$	−17.0813	−1.63609	−0.818047	+	0.575151i	$0.804943\pi$
$110$	0	0
$111$	−3.37246	−0.320099
$112$	0	0
$113$	19.7224	1.85533	0.927663	−	0.373417i	$-0.121814\pi$
$113$	19.7224	1.85533	0.927663	+	0.373417i	$0.121814\pi$
$114$	0	0
$115$	−0.00852247	−0.000794725 0
$116$	0	0
$117$	2.02319	0.187044
$118$	0	0
$119$	9.93744	0.910963
$120$	0	0
$121$	−7.47053	−0.679139
$122$	0	0
$123$	−2.53946	−0.228975
$124$	0	0
$125$	1.43454	0.128309
$126$	0	0
$127$	16.6801	1.48012	0.740062	−	0.672539i	$-0.234796\pi$
$127$	16.6801	1.48012	0.740062	+	0.672539i	$0.234796\pi$
$128$	0	0
$129$	0.471952	0.0415531
$130$	0	0
$131$	−6.05310	−0.528861	−0.264431	−	0.964405i	$-0.585184\pi$
$131$	−6.05310	−0.528861	−0.264431	+	0.964405i	$0.585184\pi$
$132$	0	0
$133$	2.50903	0.217561
$134$	0	0
$135$	11.4556	0.985938
$136$	0	0
$137$	8.24481	0.704401	0.352201	−	0.935924i	$-0.385433\pi$
$137$	8.24481	0.704401	0.352201	+	0.935924i	$0.385433\pi$
$138$	0	0
$139$	−11.3482	−0.962541	−0.481271	−	0.876572i	$-0.659825\pi$
$139$	−11.3482	−0.962541	−0.481271	+	0.876572i	$0.659825\pi$
$140$	0	0
$141$	−4.82014	−0.405929
$142$	0	0
$143$	1.48834	0.124461
$144$	0	0
$145$	−12.8786	−1.06951
$146$	0	0
$147$	−0.470748	−0.0388267
$148$	0	0
$149$	10.9372	0.896007	0.448004	−	0.894032i	$-0.352135\pi$
$149$	10.9372	0.896007	0.448004	+	0.894032i	$0.352135\pi$
$150$	0	0
$151$	16.0767	1.30830	0.654151	−	0.756364i	$-0.273026\pi$
$151$	16.0767	1.30830	0.654151	+	0.756364i	$0.273026\pi$
$152$	0	0
$153$	10.1148	0.817735
$154$	0	0
$155$	1.06693	0.0856978
$156$	0	0
$157$	0.837071	0.0668055	0.0334028	−	0.999442i	$-0.489366\pi$
$157$	0.837071	0.0668055	0.0334028	+	0.999442i	$0.489366\pi$
$158$	0	0
$159$	−0.667966	−0.0529731
$160$	0	0
$161$	−0.00692471	−0.000545744 0
$162$	0	0
$163$	12.7684	1.00010	0.500049	−	0.865997i	$-0.333315\pi$
$163$	12.7684	1.00010	0.500049	+	0.865997i	$0.333315\pi$
$164$	0	0
$165$	3.87507	0.301673
$166$	0	0
$167$	−0.293372	−0.0227018	−0.0113509	−	0.999936i	$-0.503613\pi$
$167$	−0.293372	−0.0227018	−0.0113509	+	0.999936i	$0.503613\pi$
$168$	0	0
$169$	−12.3724	−0.951722
$170$	0	0
$171$	2.55382	0.195296
$172$	0	0
$173$	11.7549	0.893711	0.446855	−	0.894606i	$-0.352544\pi$
$173$	11.7549	0.893711	0.446855	+	0.894606i	$0.352544\pi$
$174$	0	0
$175$	−11.3796	−0.860215
$176$	0	0
$177$	4.07492	0.306290
$178$	0	0
$179$	−18.3248	−1.36966	−0.684829	−	0.728704i	$-0.740123\pi$
$179$	−18.3248	−1.36966	−0.684829	+	0.728704i	$0.740123\pi$
$180$	0	0
$181$	−4.56331	−0.339188	−0.169594	−	0.985514i	$-0.554246\pi$
$181$	−4.56331	−0.339188	−0.169594	+	0.985514i	$0.554246\pi$
$182$	0	0
$183$	7.30606	0.540079
$184$	0	0
$185$	15.5906	1.14624
$186$	0	0
$187$	7.44085	0.544129
$188$	0	0
$189$	9.30792	0.677051
$190$	0	0
$191$	−18.9417	−1.37057	−0.685287	−	0.728273i	$-0.740323\pi$
$191$	−18.9417	−1.37057	−0.685287	+	0.728273i	$0.740323\pi$
$192$	0	0
$193$	10.8478	0.780841	0.390421	−	0.920637i	$-0.372330\pi$
$193$	10.8478	0.780841	0.390421	+	0.920637i	$0.372330\pi$
$194$	0	0
$195$	1.63407	0.117018
$196$	0	0
$197$	13.5644	0.966420	0.483210	−	0.875504i	$-0.339471\pi$
$197$	13.5644	0.966420	0.483210	+	0.875504i	$0.339471\pi$
$198$	0	0
$199$	4.07822	0.289098	0.144549	−	0.989498i	$-0.453827\pi$
$199$	4.07822	0.289098	0.144549	+	0.989498i	$0.453827\pi$
$200$	0	0
$201$	−4.52081	−0.318873
$202$	0	0
$203$	−10.4642	−0.734441
$204$	0	0
$205$	11.7397	0.819936
$206$	0	0
$207$	−0.00704832	−0.000489892 0
$208$	0	0
$209$	1.87869	0.129952
$210$	0	0
$211$	27.3697	1.88421	0.942104	−	0.335320i	$-0.108844\pi$
$211$	27.3697	1.88421	0.942104	+	0.335320i	$0.108844\pi$
$212$	0	0
$213$	−1.24819	−0.0855246
$214$	0	0
$215$	−2.18180	−0.148797
$216$	0	0
$217$	0.866906	0.0588494
$218$	0	0
$219$	5.62874	0.380355
$220$	0	0
$221$	3.13772	0.211066
$222$	0	0
$223$	3.15495	0.211271	0.105635	−	0.994405i	$-0.466312\pi$
$223$	3.15495	0.211271	0.105635	+	0.994405i	$0.466312\pi$
$224$	0	0
$225$	−11.5827	−0.772180
$226$	0	0
$227$	−13.4915	−0.895461	−0.447731	−	0.894169i	$-0.647768\pi$
$227$	−13.4915	−0.895461	−0.447731	+	0.894169i	$0.647768\pi$
$228$	0	0
$229$	13.3562	0.882600	0.441300	−	0.897360i	$-0.354517\pi$
$229$	13.3562	0.882600	0.441300	+	0.897360i	$0.354517\pi$
$230$	0	0
$231$	3.14858	0.207162
$232$	0	0
$233$	−25.1979	−1.65077	−0.825383	−	0.564573i	$-0.809041\pi$
$233$	−25.1979	−1.65077	−0.825383	+	0.564573i	$0.809041\pi$
$234$	0	0
$235$	22.2831	1.45359
$236$	0	0
$237$	3.79705	0.246645
$238$	0	0
$239$	−9.41098	−0.608745	−0.304373	−	0.952553i	$-0.598447\pi$
$239$	−9.41098	−0.608745	−0.304373	+	0.952553i	$0.598447\pi$
$240$	0	0
$241$	4.68431	0.301743	0.150871	−	0.988553i	$-0.451792\pi$
$241$	4.68431	0.301743	0.150871	+	0.988553i	$0.451792\pi$
$242$	0	0
$243$	14.5917	0.936056
$244$	0	0
$245$	2.17623	0.139034
$246$	0	0
$247$	0.792221	0.0504078
$248$	0	0
$249$	4.37365	0.277169
$250$	0	0
$251$	−9.68417	−0.611260	−0.305630	−	0.952150i	$-0.598867\pi$
$251$	−9.68417	−0.611260	−0.305630	+	0.952150i	$0.598867\pi$
$252$	0	0
$253$	−0.00518501	−0.000325979 0
$254$	0	0
$255$	8.16944	0.511590
$256$	0	0
$257$	−8.92953	−0.557009	−0.278504	−	0.960435i	$-0.589839\pi$
$257$	−8.92953	−0.557009	−0.278504	+	0.960435i	$0.589839\pi$
$258$	0	0
$259$	12.6677	0.787135
$260$	0	0
$261$	−10.6510	−0.659278
$262$	0	0
$263$	−11.9788	−0.738642	−0.369321	−	0.929302i	$-0.620410\pi$
$263$	−11.9788	−0.738642	−0.369321	+	0.929302i	$0.620410\pi$
$264$	0	0
$265$	3.08795	0.189691
$266$	0	0
$267$	1.22998	0.0752733
$268$	0	0
$269$	−15.4113	−0.939643	−0.469822	−	0.882761i	$-0.655682\pi$
$269$	−15.4113	−0.939643	−0.469822	+	0.882761i	$0.655682\pi$
$270$	0	0
$271$	−10.6603	−0.647567	−0.323783	−	0.946131i	$-0.604955\pi$
$271$	−10.6603	−0.647567	−0.323783	+	0.946131i	$0.604955\pi$
$272$	0	0
$273$	1.32772	0.0803574
$274$	0	0
$275$	−8.52067	−0.513816
$276$	0	0
$277$	−17.1614	−1.03113	−0.515565	−	0.856850i	$-0.672418\pi$
$277$	−17.1614	−1.03113	−0.515565	+	0.856850i	$0.672418\pi$
$278$	0	0
$279$	0.882381	0.0528267
$280$	0	0
$281$	−2.12149	−0.126558	−0.0632789	−	0.997996i	$-0.520156\pi$
$281$	−2.12149	−0.126558	−0.0632789	+	0.997996i	$0.520156\pi$
$282$	0	0
$283$	−1.83829	−0.109275	−0.0546376	−	0.998506i	$-0.517400\pi$
$283$	−1.83829	−0.109275	−0.0546376	+	0.998506i	$0.517400\pi$
$284$	0	0
$285$	2.06264	0.122181
$286$	0	0
$287$	9.53878	0.563057
$288$	0	0
$289$	−1.31315	−0.0772443
$290$	0	0
$291$	5.64085	0.330673
$292$	0	0
$293$	6.70050	0.391447	0.195724	−	0.980659i	$-0.437294\pi$
$293$	6.70050	0.391447	0.195724	+	0.980659i	$0.437294\pi$
$294$	0	0
$295$	−18.8380	−1.09679
$296$	0	0
$297$	6.96948	0.404411
$298$	0	0
$299$	−0.00218646	−0.000126446 0
$300$	0	0
$301$	−1.77276	−0.102180
$302$	0	0
$303$	−8.28658	−0.476051
$304$	0	0
$305$	−33.7753	−1.93397
$306$	0	0
$307$	19.1145	1.09092	0.545461	−	0.838136i	$-0.316354\pi$
$307$	19.1145	1.09092	0.545461	+	0.838136i	$0.316354\pi$
$308$	0	0
$309$	6.16889	0.350936
$310$	0	0
$311$	−0.946295	−0.0536595	−0.0268297	−	0.999640i	$-0.508541\pi$
$311$	−0.946295	−0.0536595	−0.0268297	+	0.999640i	$0.508541\pi$
$312$	0	0
$313$	−25.3574	−1.43328	−0.716642	−	0.697441i	$-0.754322\pi$
$313$	−25.3574	−1.43328	−0.716642	+	0.697441i	$0.754322\pi$
$314$	0	0
$315$	−19.7864	−1.11484
$316$	0	0
$317$	25.9585	1.45798	0.728989	−	0.684526i	$-0.239991\pi$
$317$	25.9585	1.45798	0.728989	+	0.684526i	$0.239991\pi$
$318$	0	0
$319$	−7.83525	−0.438690
$320$	0	0
$321$	−8.89904	−0.496696
$322$	0	0
$323$	3.96066	0.220377
$324$	0	0
$325$	−3.59307	−0.199308
$326$	0	0
$327$	−11.4097	−0.630960
$328$	0	0
$329$	18.1056	0.998192
$330$	0	0
$331$	−6.78684	−0.373039	−0.186519	−	0.982451i	$-0.559721\pi$
$331$	−6.78684	−0.373039	−0.186519	+	0.982451i	$0.559721\pi$
$332$	0	0
$333$	12.8939	0.706579
$334$	0	0
$335$	20.8993	1.14185
$336$	0	0
$337$	−19.9624	−1.08742	−0.543712	−	0.839272i	$-0.682982\pi$
$337$	−19.9624	−1.08742	−0.543712	+	0.839272i	$0.682982\pi$
$338$	0	0
$339$	13.1739	0.715507
$340$	0	0
$341$	0.649112	0.0351514
$342$	0	0
$343$	19.3315	1.04380
$344$	0	0
$345$	−0.00569272	−0.000306486 0
$346$	0	0
$347$	−13.2909	−0.713490	−0.356745	−	0.934202i	$-0.616114\pi$
$347$	−13.2909	−0.713490	−0.356745	+	0.934202i	$0.616114\pi$
$348$	0	0
$349$	19.3591	1.03627	0.518133	−	0.855300i	$-0.326627\pi$
$349$	19.3591	1.03627	0.518133	+	0.855300i	$0.326627\pi$
$350$	0	0
$351$	2.93895	0.156870
$352$	0	0
$353$	−30.9467	−1.64712	−0.823562	−	0.567226i	$-0.808017\pi$
$353$	−30.9467	−1.64712	−0.823562	+	0.567226i	$0.808017\pi$
$354$	0	0
$355$	5.77028	0.306255
$356$	0	0
$357$	6.63787	0.351313
$358$	0	0
$359$	27.3653	1.44429	0.722144	−	0.691743i	$-0.243157\pi$
$359$	27.3653	1.44429	0.722144	+	0.691743i	$0.243157\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.99006	−0.261910
$364$	0	0
$365$	−26.0212	−1.36201
$366$	0	0
$367$	−26.8014	−1.39902	−0.699510	−	0.714623i	$-0.746598\pi$
$367$	−26.8014	−1.39902	−0.699510	+	0.714623i	$0.746598\pi$
$368$	0	0
$369$	9.70906	0.505433
$370$	0	0
$371$	2.50903	0.130263
$372$	0	0
$373$	14.0104	0.725429	0.362714	−	0.931900i	$-0.381850\pi$
$373$	14.0104	0.725429	0.362714	+	0.931900i	$0.381850\pi$
$374$	0	0
$375$	0.958224	0.0494825
$376$	0	0
$377$	−3.30404	−0.170167
$378$	0	0
$379$	−1.04299	−0.0535747	−0.0267874	−	0.999641i	$-0.508528\pi$
$379$	−1.04299	−0.0535747	−0.0267874	+	0.999641i	$0.508528\pi$
$380$	0	0
$381$	11.1418	0.570810
$382$	0	0
$383$	5.31610	0.271640	0.135820	−	0.990734i	$-0.456633\pi$
$383$	5.31610	0.271640	0.135820	+	0.990734i	$0.456633\pi$
$384$	0	0
$385$	−14.5556	−0.741824
$386$	0	0
$387$	−1.80441	−0.0917232
$388$	0	0
$389$	−8.58927	−0.435493	−0.217747	−	0.976005i	$-0.569871\pi$
$389$	−8.58927	−0.435493	−0.217747	+	0.976005i	$0.569871\pi$
$390$	0	0
$391$	−0.0109311	−0.000552808 0
$392$	0	0
$393$	−4.04326	−0.203955
$394$	0	0
$395$	−17.5535	−0.883211
$396$	0	0
$397$	−11.0515	−0.554658	−0.277329	−	0.960775i	$-0.589449\pi$
$397$	−11.0515	−0.554658	−0.277329	+	0.960775i	$0.589449\pi$
$398$	0	0
$399$	1.67595	0.0839024
$400$	0	0
$401$	−8.21297	−0.410136	−0.205068	−	0.978748i	$-0.565742\pi$
$401$	−8.21297	−0.410136	−0.205068	+	0.978748i	$0.565742\pi$
$402$	0	0
$403$	0.273723	0.0136351
$404$	0	0
$405$	−16.0063	−0.795360
$406$	0	0
$407$	9.48521	0.470164
$408$	0	0
$409$	−23.1237	−1.14340	−0.571698	−	0.820464i	$-0.693715\pi$
$409$	−23.1237	−1.14340	−0.571698	+	0.820464i	$0.693715\pi$
$410$	0	0
$411$	5.50725	0.271653
$412$	0	0
$413$	−15.3063	−0.753176
$414$	0	0
$415$	−20.2190	−0.992512
$416$	0	0
$417$	−7.58020	−0.371204
$418$	0	0
$419$	−6.12374	−0.299164	−0.149582	−	0.988749i	$-0.547793\pi$
$419$	−6.12374	−0.299164	−0.149582	+	0.988749i	$0.547793\pi$
$420$	0	0
$421$	18.6459	0.908747	0.454373	−	0.890811i	$-0.349863\pi$
$421$	18.6459	0.908747	0.454373	+	0.890811i	$0.349863\pi$
$422$	0	0
$423$	18.4288	0.896037
$424$	0	0
$425$	−17.9633	−0.871350
$426$	0	0
$427$	−27.4432	−1.32807
$428$	0	0
$429$	0.994157	0.0479984
$430$	0	0
$431$	37.9651	1.82871	0.914357	−	0.404908i	$-0.132697\pi$
$431$	37.9651	1.82871	0.914357	+	0.404908i	$0.132697\pi$
$432$	0	0
$433$	−4.31528	−0.207379	−0.103689	−	0.994610i	$-0.533065\pi$
$433$	−4.31528	−0.207379	−0.103689	+	0.994610i	$0.533065\pi$
$434$	0	0
$435$	−8.60247	−0.412457
$436$	0	0
$437$	−0.00275991	−0.000132024 0
$438$	0	0
$439$	9.19764	0.438979	0.219490	−	0.975615i	$-0.429561\pi$
$439$	9.19764	0.438979	0.219490	+	0.975615i	$0.429561\pi$
$440$	0	0
$441$	1.79980	0.0857050
$442$	0	0
$443$	−13.0897	−0.621909	−0.310954	−	0.950425i	$-0.600649\pi$
$443$	−13.0897	−0.621909	−0.310954	+	0.950425i	$0.600649\pi$
$444$	0	0
$445$	−5.68608	−0.269546
$446$	0	0
$447$	7.30565	0.345545
$448$	0	0
$449$	7.12843	0.336411	0.168206	−	0.985752i	$-0.446203\pi$
$449$	7.12843	0.336411	0.168206	+	0.985752i	$0.446203\pi$
$450$	0	0
$451$	7.14235	0.336320
$452$	0	0
$453$	10.7387	0.504547
$454$	0	0
$455$	−6.13795	−0.287751
$456$	0	0
$457$	12.1188	0.566893	0.283446	−	0.958988i	$-0.408522\pi$
$457$	12.1188	0.566893	0.283446	+	0.958988i	$0.408522\pi$
$458$	0	0
$459$	14.6931	0.685816
$460$	0	0
$461$	−20.2062	−0.941096	−0.470548	−	0.882374i	$-0.655944\pi$
$461$	−20.2062	−0.941096	−0.470548	+	0.882374i	$0.655944\pi$
$462$	0	0
$463$	35.6793	1.65816	0.829079	−	0.559131i	$-0.188865\pi$
$463$	35.6793	1.65816	0.829079	+	0.559131i	$0.188865\pi$
$464$	0	0
$465$	0.712672	0.0330494
$466$	0	0
$467$	34.5611	1.59930	0.799649	−	0.600468i	$-0.205019\pi$
$467$	34.5611	1.59930	0.799649	+	0.600468i	$0.205019\pi$
$468$	0	0
$469$	16.9812	0.784119
$470$	0	0
$471$	0.559134	0.0257636
$472$	0	0
$473$	−1.32739	−0.0610334
$474$	0	0
$475$	−4.53544	−0.208100
$476$	0	0
$477$	2.55382	0.116931
$478$	0	0
$479$	−24.5570	−1.12204	−0.561018	−	0.827804i	$-0.689590\pi$
$479$	−24.5570	−1.12204	−0.561018	+	0.827804i	$0.689590\pi$
$480$	0	0
$481$	3.99980	0.182375
$482$	0	0
$483$	−0.00462547	−0.000210466 0
$484$	0	0
$485$	−26.0772	−1.18410
$486$	0	0
$487$	−12.1408	−0.550153	−0.275076	−	0.961422i	$-0.588703\pi$
$487$	−12.1408	−0.550153	−0.275076	+	0.961422i	$0.588703\pi$
$488$	0	0
$489$	8.52885	0.385688
$490$	0	0
$491$	−6.89427	−0.311134	−0.155567	−	0.987825i	$-0.549720\pi$
$491$	−6.89427	−0.311134	−0.155567	+	0.987825i	$0.549720\pi$
$492$	0	0
$493$	−16.5183	−0.743949
$494$	0	0
$495$	−14.8155	−0.665906
$496$	0	0
$497$	4.68849	0.210308
$498$	0	0
$499$	19.1431	0.856965	0.428482	−	0.903550i	$-0.359048\pi$
$499$	19.1431	0.856965	0.428482	+	0.903550i	$0.359048\pi$
$500$	0	0
$501$	−0.195962	−0.00875495
$502$	0	0
$503$	3.89638	0.173731	0.0868656	−	0.996220i	$-0.472315\pi$
$503$	3.89638	0.173731	0.0868656	+	0.996220i	$0.472315\pi$
$504$	0	0
$505$	38.3082	1.70469
$506$	0	0
$507$	−8.26433	−0.367032
$508$	0	0
$509$	22.2425	0.985882	0.492941	−	0.870063i	$-0.335922\pi$
$509$	22.2425	0.985882	0.492941	+	0.870063i	$0.335922\pi$
$510$	0	0
$511$	−21.1428	−0.935304
$512$	0	0
$513$	3.70976	0.163790
$514$	0	0
$515$	−28.5183	−1.25666
$516$	0	0
$517$	13.5569	0.596231
$518$	0	0
$519$	7.85189	0.344660
$520$	0	0
$521$	−8.91621	−0.390626	−0.195313	−	0.980741i	$-0.562572\pi$
$521$	−8.91621	−0.390626	−0.195313	+	0.980741i	$0.562572\pi$
$522$	0	0
$523$	−1.90421	−0.0832652	−0.0416326	−	0.999133i	$-0.513256\pi$
$523$	−1.90421	−0.0832652	−0.0416326	+	0.999133i	$0.513256\pi$
$524$	0	0
$525$	−7.60116	−0.331742
$526$	0	0
$527$	1.36846	0.0596112
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−15.5796	−0.676096
$532$	0	0
$533$	3.01185	0.130458
$534$	0	0
$535$	41.1395	1.77862
$536$	0	0
$537$	−12.2403	−0.528209
$538$	0	0
$539$	1.32400	0.0570289
$540$	0	0
$541$	14.4964	0.623247	0.311623	−	0.950206i	$-0.399127\pi$
$541$	14.4964	0.623247	0.311623	+	0.950206i	$0.399127\pi$
$542$	0	0
$543$	−3.04813	−0.130808
$544$	0	0
$545$	52.7463	2.25940
$546$	0	0
$547$	20.5310	0.877841	0.438920	−	0.898526i	$-0.355361\pi$
$547$	20.5310	0.877841	0.438920	+	0.898526i	$0.355361\pi$
$548$	0	0
$549$	−27.9331	−1.19216
$550$	0	0
$551$	−4.17060	−0.177674
$552$	0	0
$553$	−14.2626	−0.606508
$554$	0	0
$555$	10.4140	0.442049
$556$	0	0
$557$	12.7396	0.539795	0.269897	−	0.962889i	$-0.413010\pi$
$557$	12.7396	0.539795	0.269897	+	0.962889i	$0.413010\pi$
$558$	0	0
$559$	−0.559745	−0.0236747
$560$	0	0
$561$	4.97023	0.209843
$562$	0	0
$563$	−10.5933	−0.446453	−0.223226	−	0.974767i	$-0.571659\pi$
$563$	−10.5933	−0.446453	−0.223226	+	0.974767i	$0.571659\pi$
$564$	0	0
$565$	−60.9018	−2.56216
$566$	0	0
$567$	−13.0055	−0.546180
$568$	0	0
$569$	24.8920	1.04353	0.521764	−	0.853090i	$-0.325274\pi$
$569$	24.8920	1.04353	0.521764	+	0.853090i	$0.325274\pi$
$570$	0	0
$571$	27.7949	1.16318	0.581590	−	0.813482i	$-0.302431\pi$
$571$	27.7949	1.16318	0.581590	+	0.813482i	$0.302431\pi$
$572$	0	0
$573$	−12.6524	−0.528562
$574$	0	0
$575$	0.0125174	0.000522012 0
$576$	0	0
$577$	−7.81048	−0.325155	−0.162577	−	0.986696i	$-0.551981\pi$
$577$	−7.81048	−0.325155	−0.162577	+	0.986696i	$0.551981\pi$
$578$	0	0
$579$	7.24595	0.301131
$580$	0	0
$581$	−16.4284	−0.681566
$582$	0	0
$583$	1.87869	0.0778073
$584$	0	0
$585$	−6.24752	−0.258303
$586$	0	0
$587$	−14.6013	−0.602662	−0.301331	−	0.953520i	$-0.597431\pi$
$587$	−14.6013	−0.602662	−0.301331	+	0.953520i	$0.597431\pi$
$588$	0	0
$589$	0.345514	0.0142366
$590$	0	0
$591$	9.06052	0.372700
$592$	0	0
$593$	−35.6738	−1.46495	−0.732474	−	0.680795i	$-0.761634\pi$
$593$	−35.6738	−1.46495	−0.732474	+	0.680795i	$0.761634\pi$
$594$	0	0
$595$	−30.6863	−1.25802
$596$	0	0
$597$	2.72411	0.111491
$598$	0	0
$599$	14.2179	0.580927	0.290464	−	0.956886i	$-0.406191\pi$
$599$	14.2179	0.580927	0.290464	+	0.956886i	$0.406191\pi$
$600$	0	0
$601$	1.35419	0.0552386	0.0276193	−	0.999619i	$-0.491207\pi$
$601$	1.35419	0.0552386	0.0276193	+	0.999619i	$0.491207\pi$
$602$	0	0
$603$	17.2843	0.703873
$604$	0	0
$605$	23.0686	0.937873
$606$	0	0
$607$	8.37984	0.340127	0.170064	−	0.985433i	$-0.445603\pi$
$607$	8.37984	0.340127	0.170064	+	0.985433i	$0.445603\pi$
$608$	0	0
$609$	−6.98971	−0.283237
$610$	0	0
$611$	5.71678	0.231276
$612$	0	0
$613$	0.834445	0.0337029	0.0168515	−	0.999858i	$-0.494636\pi$
$613$	0.834445	0.0337029	0.0168515	+	0.999858i	$0.494636\pi$
$614$	0	0
$615$	7.84171	0.316208
$616$	0	0
$617$	−5.99861	−0.241495	−0.120747	−	0.992683i	$-0.538529\pi$
$617$	−5.99861	−0.241495	−0.120747	+	0.992683i	$0.538529\pi$
$618$	0	0
$619$	−23.0770	−0.927544	−0.463772	−	0.885954i	$-0.653504\pi$
$619$	−23.0770	−0.927544	−0.463772	+	0.885954i	$0.653504\pi$
$620$	0	0
$621$	−0.0102386	−0.000410862 0
$622$	0	0
$623$	−4.62008	−0.185099
$624$	0	0
$625$	−27.1070	−1.08428
$626$	0	0
$627$	1.25490	0.0501158
$628$	0	0
$629$	19.9968	0.797324
$630$	0	0
$631$	1.82417	0.0726192	0.0363096	−	0.999341i	$-0.488440\pi$
$631$	1.82417	0.0726192	0.0363096	+	0.999341i	$0.488440\pi$
$632$	0	0
$633$	18.2820	0.726645
$634$	0	0
$635$	−51.5075	−2.04401
$636$	0	0
$637$	0.558317	0.0221213
$638$	0	0
$639$	4.77219	0.188785
$640$	0	0
$641$	−11.8783	−0.469164	−0.234582	−	0.972096i	$-0.575372\pi$
$641$	−11.8783	−0.469164	−0.234582	+	0.972096i	$0.575372\pi$
$642$	0	0
$643$	19.8125	0.781327	0.390663	−	0.920534i	$-0.372246\pi$
$643$	19.8125	0.781327	0.390663	+	0.920534i	$0.372246\pi$
$644$	0	0
$645$	−1.45737	−0.0573837
$646$	0	0
$647$	14.0060	0.550634	0.275317	−	0.961354i	$-0.411217\pi$
$647$	14.0060	0.550634	0.275317	+	0.961354i	$0.411217\pi$
$648$	0	0
$649$	−11.4609	−0.449880
$650$	0	0
$651$	0.579063	0.0226953
$652$	0	0
$653$	−12.1367	−0.474947	−0.237473	−	0.971394i	$-0.576319\pi$
$653$	−12.1367	−0.474947	−0.237473	+	0.971394i	$0.576319\pi$
$654$	0	0
$655$	18.6917	0.730344
$656$	0	0
$657$	−21.5203	−0.839585
$658$	0	0
$659$	6.96698	0.271395	0.135698	−	0.990750i	$-0.456672\pi$
$659$	6.96698	0.271395	0.135698	+	0.990750i	$0.456672\pi$
$660$	0	0
$661$	18.4072	0.715957	0.357979	−	0.933730i	$-0.383466\pi$
$661$	18.4072	0.715957	0.357979	+	0.933730i	$0.383466\pi$
$662$	0	0
$663$	2.09589	0.0813976
$664$	0	0
$665$	−7.74777	−0.300446
$666$	0	0
$667$	0.0115105	0.000445688 0
$668$	0	0
$669$	2.10740	0.0814766
$670$	0	0
$671$	−20.5487	−0.793272
$672$	0	0
$673$	5.43220	0.209396	0.104698	−	0.994504i	$-0.466612\pi$
$673$	5.43220	0.209396	0.104698	+	0.994504i	$0.466612\pi$
$674$	0	0
$675$	−16.8254	−0.647610
$676$	0	0
$677$	−12.5758	−0.483327	−0.241664	−	0.970360i	$-0.577693\pi$
$677$	−12.5758	−0.483327	−0.241664	+	0.970360i	$0.577693\pi$
$678$	0	0
$679$	−21.1884	−0.813134
$680$	0	0
$681$	−9.01185	−0.345335
$682$	0	0
$683$	26.4815	1.01329	0.506643	−	0.862156i	$-0.330886\pi$
$683$	26.4815	1.01329	0.506643	+	0.862156i	$0.330886\pi$
$684$	0	0
$685$	−25.4596	−0.972760
$686$	0	0
$687$	8.92146	0.340375
$688$	0	0
$689$	0.792221	0.0301812
$690$	0	0
$691$	30.5680	1.16286	0.581430	−	0.813596i	$-0.302493\pi$
$691$	30.5680	1.16286	0.581430	+	0.813596i	$0.302493\pi$
$692$	0	0
$693$	−12.0379	−0.457283
$694$	0	0
$695$	35.0427	1.32924
$696$	0	0
$697$	15.0575	0.570345
$698$	0	0
$699$	−16.8313	−0.636618
$700$	0	0
$701$	−17.7368	−0.669908	−0.334954	−	0.942234i	$-0.608721\pi$
$701$	−17.7368	−0.669908	−0.334954	+	0.942234i	$0.608721\pi$
$702$	0	0
$703$	5.04885	0.190421
$704$	0	0
$705$	14.8843	0.560577
$706$	0	0
$707$	31.1263	1.17062
$708$	0	0
$709$	37.2843	1.40024	0.700120	−	0.714025i	$-0.253130\pi$
$709$	37.2843	1.40024	0.700120	+	0.714025i	$0.253130\pi$
$710$	0	0
$711$	−14.5172	−0.544438
$712$	0	0
$713$	−0.000953587 0	−3.57121e−5 0
$714$	0	0
$715$	−4.59591	−0.171877
$716$	0	0
$717$	−6.28621	−0.234763
$718$	0	0
$719$	13.8196	0.515384	0.257692	−	0.966227i	$-0.417038\pi$
$719$	13.8196	0.515384	0.257692	+	0.966227i	$0.417038\pi$
$720$	0	0
$721$	−23.1718	−0.862962
$722$	0	0
$723$	3.12896	0.116367
$724$	0	0
$725$	18.9155	0.702504
$726$	0	0
$727$	1.59740	0.0592441	0.0296221	−	0.999561i	$-0.490570\pi$
$727$	1.59740	0.0592441	0.0296221	+	0.999561i	$0.490570\pi$
$728$	0	0
$729$	−5.80368	−0.214951
$730$	0	0
$731$	−2.79841	−0.103503
$732$	0	0
$733$	36.8542	1.36124	0.680620	−	0.732637i	$-0.261710\pi$
$733$	36.8542	1.36124	0.680620	+	0.732637i	$0.261710\pi$
$734$	0	0
$735$	1.45365	0.0536186
$736$	0	0
$737$	12.7150	0.468363
$738$	0	0
$739$	21.4158	0.787794	0.393897	−	0.919155i	$-0.371127\pi$
$739$	21.4158	0.787794	0.393897	+	0.919155i	$0.371127\pi$
$740$	0	0
$741$	0.529177	0.0194398
$742$	0	0
$743$	−0.619440	−0.0227250	−0.0113625	−	0.999935i	$-0.503617\pi$
$743$	−0.619440	−0.0227250	−0.0113625	+	0.999935i	$0.503617\pi$
$744$	0	0
$745$	−33.7734	−1.23736
$746$	0	0
$747$	−16.7217	−0.611815
$748$	0	0
$749$	33.4268	1.22139
$750$	0	0
$751$	−29.5440	−1.07808	−0.539039	−	0.842281i	$-0.681212\pi$
$751$	−29.5440	−1.07808	−0.539039	+	0.842281i	$0.681212\pi$
$752$	0	0
$753$	−6.46870	−0.235732
$754$	0	0
$755$	−49.6440	−1.80673
$756$	0	0
$757$	−12.4944	−0.454118	−0.227059	−	0.973881i	$-0.572911\pi$
$757$	−12.4944	−0.454118	−0.227059	+	0.973881i	$0.572911\pi$
$758$	0	0
$759$	−0.00346341	−0.000125714 0
$760$	0	0
$761$	−45.6574	−1.65508	−0.827540	−	0.561406i	$-0.810260\pi$
$761$	−45.6574	−1.65508	−0.827540	+	0.561406i	$0.810260\pi$
$762$	0	0
$763$	42.8576	1.55155
$764$	0	0
$765$	−31.2341	−1.12927
$766$	0	0
$767$	−4.83294	−0.174507
$768$	0	0
$769$	15.3873	0.554882	0.277441	−	0.960743i	$-0.410514\pi$
$769$	15.3873	0.554882	0.277441	+	0.960743i	$0.410514\pi$
$770$	0	0
$771$	−5.96462	−0.214810
$772$	0	0
$773$	−5.75084	−0.206843	−0.103422	−	0.994638i	$-0.532979\pi$
$773$	−5.75084	−0.206843	−0.103422	+	0.994638i	$0.532979\pi$
$774$	0	0
$775$	−1.56706	−0.0562903
$776$	0	0
$777$	8.46161	0.303559
$778$	0	0
$779$	3.80178	0.136213
$780$	0	0
$781$	3.51060	0.125619
$782$	0	0
$783$	−15.4719	−0.552922
$784$	0	0
$785$	−2.58483	−0.0922566
$786$	0	0
$787$	30.5812	1.09010	0.545051	−	0.838403i	$-0.316510\pi$
$787$	30.5812	1.09010	0.545051	+	0.838403i	$0.316510\pi$
$788$	0	0
$789$	−8.00140	−0.284857
$790$	0	0
$791$	−49.4842	−1.75945
$792$	0	0
$793$	−8.66513	−0.307708
$794$	0	0
$795$	2.06264	0.0731545
$796$	0	0
$797$	−27.7543	−0.983109	−0.491555	−	0.870847i	$-0.663571\pi$
$797$	−27.7543	−0.983109	−0.491555	+	0.870847i	$0.663571\pi$
$798$	0	0
$799$	28.5807	1.01111
$800$	0	0
$801$	−4.70255	−0.166156
$802$	0	0
$803$	−15.8311	−0.558668
$804$	0	0
$805$	0.0213832	0.000753658 0
$806$	0	0
$807$	−10.2942	−0.362374
$808$	0	0
$809$	−39.9697	−1.40526	−0.702630	−	0.711555i	$-0.747991\pi$
$809$	−39.9697	−1.40526	−0.702630	+	0.711555i	$0.747991\pi$
$810$	0	0
$811$	−20.1310	−0.706894	−0.353447	−	0.935454i	$-0.614991\pi$
$811$	−20.1310	−0.706894	−0.353447	+	0.935454i	$0.614991\pi$
$812$	0	0
$813$	−7.12071	−0.249734
$814$	0	0
$815$	−39.4282	−1.38111
$816$	0	0
$817$	−0.706552	−0.0247191
$818$	0	0
$819$	−5.07626	−0.177379
$820$	0	0
$821$	8.30885	0.289981	0.144990	−	0.989433i	$-0.453685\pi$
$821$	8.30885	0.289981	0.144990	+	0.989433i	$0.453685\pi$
$822$	0	0
$823$	34.6906	1.20924	0.604618	−	0.796515i	$-0.293326\pi$
$823$	34.6906	1.20924	0.604618	+	0.796515i	$0.293326\pi$
$824$	0	0
$825$	−5.69152	−0.198153
$826$	0	0
$827$	43.9204	1.52726	0.763630	−	0.645654i	$-0.223415\pi$
$827$	43.9204	1.52726	0.763630	+	0.645654i	$0.223415\pi$
$828$	0	0
$829$	45.1851	1.56934	0.784671	−	0.619912i	$-0.212832\pi$
$829$	45.1851	1.56934	0.784671	+	0.619912i	$0.212832\pi$
$830$	0	0
$831$	−11.4632	−0.397655
$832$	0	0
$833$	2.79127	0.0967119
$834$	0	0
$835$	0.905917	0.0313506
$836$	0	0
$837$	1.28177	0.0443046
$838$	0	0
$839$	28.5343	0.985113	0.492557	−	0.870280i	$-0.336062\pi$
$839$	28.5343	0.985113	0.492557	+	0.870280i	$0.336062\pi$
$840$	0	0
$841$	−11.6061	−0.400210
$842$	0	0
$843$	−1.41708	−0.0488070
$844$	0	0
$845$	38.2053	1.31430
$846$	0	0
$847$	18.7438	0.644045
$848$	0	0
$849$	−1.22792	−0.0421420
$850$	0	0
$851$	−0.0139344	−0.000477664 0
$852$	0	0
$853$	−29.2778	−1.00245	−0.501226	−	0.865316i	$-0.667118\pi$
$853$	−29.2778	−1.00245	−0.501226	+	0.865316i	$0.667118\pi$
$854$	0	0
$855$	−7.88608	−0.269698
$856$	0	0
$857$	−25.6269	−0.875400	−0.437700	−	0.899121i	$-0.644207\pi$
$857$	−25.6269	−0.875400	−0.437700	+	0.899121i	$0.644207\pi$
$858$	0	0
$859$	2.68954	0.0917659	0.0458830	−	0.998947i	$-0.485390\pi$
$859$	2.68954	0.0917659	0.0458830	+	0.998947i	$0.485390\pi$
$860$	0	0
$861$	6.37158	0.217143
$862$	0	0
$863$	29.0023	0.987248	0.493624	−	0.869675i	$-0.335672\pi$
$863$	29.0023	0.987248	0.493624	+	0.869675i	$0.335672\pi$
$864$	0	0
$865$	−36.2986	−1.23419
$866$	0	0
$867$	−0.877141	−0.0297893
$868$	0	0
$869$	−10.6794	−0.362274
$870$	0	0
$871$	5.36177	0.181677
$872$	0	0
$873$	−21.5666	−0.729918
$874$	0	0
$875$	−3.59931	−0.121679
$876$	0	0
$877$	−32.9427	−1.11240	−0.556199	−	0.831049i	$-0.687741\pi$
$877$	−32.9427	−1.11240	−0.556199	+	0.831049i	$0.687741\pi$
$878$	0	0
$879$	4.47570	0.150962
$880$	0	0
$881$	6.63334	0.223483	0.111742	−	0.993737i	$-0.464357\pi$
$881$	6.63334	0.223483	0.111742	+	0.993737i	$0.464357\pi$
$882$	0	0
$883$	−28.1977	−0.948927	−0.474463	−	0.880275i	$-0.657358\pi$
$883$	−28.1977	−0.948927	−0.474463	+	0.880275i	$0.657358\pi$
$884$	0	0
$885$	−12.5831	−0.422978
$886$	0	0
$887$	51.0836	1.71522	0.857610	−	0.514301i	$-0.171949\pi$
$887$	51.0836	1.71522	0.857610	+	0.514301i	$0.171949\pi$
$888$	0	0
$889$	−41.8511	−1.40364
$890$	0	0
$891$	−9.73812	−0.326239
$892$	0	0
$893$	7.21615	0.241479
$894$	0	0
$895$	56.5860	1.89146
$896$	0	0
$897$	−0.00146048	−4.87640e−5 0
$898$	0	0
$899$	−1.44100	−0.0480600
$900$	0	0
$901$	3.96066	0.131949
$902$	0	0
$903$	−1.18414	−0.0394058
$904$	0	0
$905$	14.0913	0.468409
$906$	0	0
$907$	32.8396	1.09042	0.545210	−	0.838299i	$-0.316450\pi$
$907$	32.8396	1.09042	0.545210	+	0.838299i	$0.316450\pi$
$908$	0	0
$909$	31.6819	1.05082
$910$	0	0
$911$	44.1485	1.46270	0.731352	−	0.682000i	$-0.238890\pi$
$911$	44.1485	1.46270	0.731352	+	0.682000i	$0.238890\pi$
$912$	0	0
$913$	−12.3011	−0.407107
$914$	0	0
$915$	−22.5607	−0.745835
$916$	0	0
$917$	15.1874	0.501533
$918$	0	0
$919$	25.5020	0.841232	0.420616	−	0.907239i	$-0.361814\pi$
$919$	25.5020	0.841232	0.420616	+	0.907239i	$0.361814\pi$
$920$	0	0
$921$	12.7678	0.420715
$922$	0	0
$923$	1.48038	0.0487273
$924$	0	0
$925$	−22.8987	−0.752906
$926$	0	0
$927$	−23.5854	−0.774646
$928$	0	0
$929$	−41.9920	−1.37771	−0.688856	−	0.724898i	$-0.741887\pi$
$929$	−41.9920	−1.37771	−0.688856	+	0.724898i	$0.741887\pi$
$930$	0	0
$931$	0.704749	0.0230972
$932$	0	0
$933$	−0.632093	−0.0206938
$934$	0	0
$935$	−22.9770	−0.751427
$936$	0	0
$937$	43.1442	1.40946	0.704730	−	0.709476i	$-0.251068\pi$
$937$	43.1442	1.40946	0.704730	+	0.709476i	$0.251068\pi$
$938$	0	0
$939$	−16.9379	−0.552746
$940$	0	0
$941$	−7.69322	−0.250792	−0.125396	−	0.992107i	$-0.540020\pi$
$941$	−7.69322	−0.250792	−0.125396	+	0.992107i	$0.540020\pi$
$942$	0	0
$943$	−0.0104926	−0.000341685 0
$944$	0	0
$945$	−28.7424	−0.934990
$946$	0	0
$947$	−32.5881	−1.05897	−0.529486	−	0.848319i	$-0.677615\pi$
$947$	−32.5881	−1.05897	−0.529486	+	0.848319i	$0.677615\pi$
$948$	0	0
$949$	−6.67580	−0.216706
$950$	0	0
$951$	17.3394	0.562269
$952$	0	0
$953$	46.3902	1.50273	0.751363	−	0.659889i	$-0.229397\pi$
$953$	46.3902	1.50273	0.751363	+	0.659889i	$0.229397\pi$
$954$	0	0
$955$	58.4911	1.89273
$956$	0	0
$957$	−5.23368	−0.169181
$958$	0	0
$959$	−20.6865	−0.668002
$960$	0	0
$961$	−30.8806	−0.996149
$962$	0	0
$963$	34.0235	1.09639
$964$	0	0
$965$	−33.4974	−1.07832
$966$	0	0
$967$	−12.6735	−0.407551	−0.203775	−	0.979018i	$-0.565321\pi$
$967$	−12.6735	−0.407551	−0.203775	+	0.979018i	$0.565321\pi$
$968$	0	0
$969$	2.64559	0.0849885
$970$	0	0
$971$	−14.5987	−0.468493	−0.234247	−	0.972177i	$-0.575262\pi$
$971$	−14.5987	−0.468493	−0.234247	+	0.972177i	$0.575262\pi$
$972$	0	0
$973$	28.4730	0.912802
$974$	0	0
$975$	−2.40005	−0.0768630
$976$	0	0
$977$	−10.7914	−0.345247	−0.172623	−	0.984988i	$-0.555224\pi$
$977$	−10.7914	−0.345247	−0.172623	+	0.984988i	$0.555224\pi$
$978$	0	0
$979$	−3.45937	−0.110562
$980$	0	0
$981$	43.6227	1.39276
$982$	0	0
$983$	−18.0395	−0.575372	−0.287686	−	0.957725i	$-0.592886\pi$
$983$	−18.0395	−0.575372	−0.287686	+	0.957725i	$0.592886\pi$
$984$	0	0
$985$	−41.8861	−1.33460
$986$	0	0
$987$	12.0939	0.384953
$988$	0	0
$989$	0.00195002	6.20070e−5 0
$990$	0	0
$991$	55.5673	1.76515	0.882577	−	0.470168i	$-0.155807\pi$
$991$	55.5673	1.76515	0.882577	+	0.470168i	$0.155807\pi$
$992$	0	0
$993$	−4.53338	−0.143862
$994$	0	0
$995$	−12.5933	−0.399236
$996$	0	0
$997$	−17.0384	−0.539613	−0.269806	−	0.962915i	$-0.586960\pi$
$997$	−17.0384	−0.539613	−0.269806	+	0.962915i	$0.586960\pi$
$998$	0	0
$999$	18.7300	0.592592

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.f.1.12	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.f.1.12	✓	19	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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q+0.667966 q^{3} -3.08795 q^{5} -2.50903 q^{7} -2.55382 q^{9} +O(q^{10})\)	\(q+0.667966 q^{3} -3.08795 q^{5} -2.50903 q^{7} -2.55382 q^{9} -1.87869 q^{11} -0.792221 q^{13} -2.06264 q^{15} -3.96066 q^{17} -1.00000 q^{19} -1.67595 q^{21} +0.00275991 q^{23} +4.53544 q^{25} -3.70976 q^{27} +4.17060 q^{29} -0.345514 q^{31} -1.25490 q^{33} +7.74777 q^{35} -5.04885 q^{37} -0.529177 q^{39} -3.80178 q^{41} +0.706552 q^{43} +7.88608 q^{45} -7.21615 q^{47} -0.704749 q^{49} -2.64559 q^{51} -1.00000 q^{53} +5.80129 q^{55} -0.667966 q^{57} +6.10049 q^{59} +10.9378 q^{61} +6.40763 q^{63} +2.44634 q^{65} -6.76803 q^{67} +0.00184353 q^{69} -1.86865 q^{71} +8.42669 q^{73} +3.02952 q^{75} +4.71369 q^{77} +5.68450 q^{79} +5.18347 q^{81} +6.54771 q^{83} +12.2303 q^{85} +2.78582 q^{87} +1.84138 q^{89} +1.98771 q^{91} -0.230791 q^{93} +3.08795 q^{95} +8.44483 q^{97} +4.79783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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