LMFDB - Embedded newform 4028.2.a.e.1.1

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} - 3 x^{18} - 35 x^{17} + 103 x^{16} + 501 x^{15} - 1437 x^{14} - 3775 x^{13} + 10450 x^{12} + \cdots - 4 $ x^19 - 3x^18 - 35x^17 + 103x^16 + 501x^15 - 1437x^14 - 3775x^13 + 10450x^12 + 16076x^11 - 42255x^10 - 38701x^9 + 93907x^8 + 49522x^7 - 106284x^6 - 26703x^5 + 50522x^4 - 446x^3 - 6265x^2 + 719x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.88114$ 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)

$f(q)$	$=$	$q-2.88114 q^{3} -1.52019 q^{5} -1.29303 q^{7} +5.30096 q^{9} +O(q^{10})$	$q-2.88114 q^{3} -1.52019 q^{5} -1.29303 q^{7} +5.30096 q^{9} -0.667690 q^{11} -2.92971 q^{13} +4.37988 q^{15} -4.25584 q^{17} +1.00000 q^{19} +3.72538 q^{21} +7.86708 q^{23} -2.68902 q^{25} -6.62938 q^{27} -4.11936 q^{29} -9.26851 q^{31} +1.92371 q^{33} +1.96565 q^{35} +7.29374 q^{37} +8.44090 q^{39} -6.62916 q^{41} -7.54137 q^{43} -8.05847 q^{45} -12.0732 q^{47} -5.32809 q^{49} +12.2617 q^{51} +1.00000 q^{53} +1.01502 q^{55} -2.88114 q^{57} -4.86350 q^{59} +3.63228 q^{61} -6.85427 q^{63} +4.45372 q^{65} +13.1427 q^{67} -22.6661 q^{69} -2.79160 q^{71} -7.95239 q^{73} +7.74743 q^{75} +0.863340 q^{77} -2.75055 q^{79} +3.19728 q^{81} -1.26835 q^{83} +6.46969 q^{85} +11.8685 q^{87} +5.85129 q^{89} +3.78819 q^{91} +26.7039 q^{93} -1.52019 q^{95} -17.1415 q^{97} -3.53939 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * 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
$3$	−2.88114	−1.66343	−0.831713	−	0.555206i	$-0.812639\pi$
$3$	−2.88114	−1.66343	−0.831713	+	0.555206i	$0.812639\pi$
$4$	0	0
$5$	−1.52019	−0.679850	−0.339925	−	0.940452i	$-0.610402\pi$
$5$	−1.52019	−0.679850	−0.339925	+	0.940452i	$0.610402\pi$
$6$	0	0
$7$	−1.29303	−0.488718	−0.244359	−	0.969685i	$-0.578577\pi$
$7$	−1.29303	−0.488718	−0.244359	+	0.969685i	$0.578577\pi$
$8$	0	0
$9$	5.30096	1.76699
$10$	0	0
$11$	−0.667690	−0.201316	−0.100658	−	0.994921i	$-0.532095\pi$
$11$	−0.667690	−0.201316	−0.100658	+	0.994921i	$0.532095\pi$
$12$	0	0
$13$	−2.92971	−0.812555	−0.406278	−	0.913750i	$-0.633173\pi$
$13$	−2.92971	−0.812555	−0.406278	+	0.913750i	$0.633173\pi$
$14$	0	0
$15$	4.37988	1.13088
$16$	0	0
$17$	−4.25584	−1.03219	−0.516096	−	0.856531i	$-0.672615\pi$
$17$	−4.25584	−1.03219	−0.516096	+	0.856531i	$0.672615\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.72538	0.812946
$22$	0	0
$23$	7.86708	1.64040	0.820199	−	0.572078i	$-0.193862\pi$
$23$	7.86708	1.64040	0.820199	+	0.572078i	$0.193862\pi$
$24$	0	0
$25$	−2.68902	−0.537804
$26$	0	0
$27$	−6.62938	−1.27582
$28$	0	0
$29$	−4.11936	−0.764947	−0.382473	−	0.923966i	$-0.624928\pi$
$29$	−4.11936	−0.764947	−0.382473	+	0.923966i	$0.624928\pi$
$30$	0	0
$31$	−9.26851	−1.66467	−0.832337	−	0.554270i	$-0.812997\pi$
$31$	−9.26851	−1.66467	−0.832337	+	0.554270i	$0.812997\pi$
$32$	0	0
$33$	1.92371	0.334874
$34$	0	0
$35$	1.96565	0.332255
$36$	0	0
$37$	7.29374	1.19908	0.599542	−	0.800343i	$-0.295349\pi$
$37$	7.29374	1.19908	0.599542	+	0.800343i	$0.295349\pi$
$38$	0	0
$39$	8.44090	1.35163
$40$	0	0
$41$	−6.62916	−1.03530	−0.517651	−	0.855592i	$-0.673193\pi$
$41$	−6.62916	−1.03530	−0.517651	+	0.855592i	$0.673193\pi$
$42$	0	0
$43$	−7.54137	−1.15005	−0.575024	−	0.818137i	$-0.695007\pi$
$43$	−7.54137	−1.15005	−0.575024	+	0.818137i	$0.695007\pi$
$44$	0	0
$45$	−8.05847	−1.20129
$46$	0	0
$47$	−12.0732	−1.76105	−0.880525	−	0.473999i	$-0.842810\pi$
$47$	−12.0732	−1.76105	−0.880525	+	0.473999i	$0.842810\pi$
$48$	0	0
$49$	−5.32809	−0.761155
$50$	0	0
$51$	12.2617	1.71698
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	1.01502	0.136865
$56$	0	0
$57$	−2.88114	−0.381616
$58$	0	0
$59$	−4.86350	−0.633173	−0.316587	−	0.948564i	$-0.602537\pi$
$59$	−4.86350	−0.633173	−0.316587	+	0.948564i	$0.602537\pi$
$60$	0	0
$61$	3.63228	0.465065	0.232533	−	0.972589i	$-0.425299\pi$
$61$	3.63228	0.465065	0.232533	+	0.972589i	$0.425299\pi$
$62$	0	0
$63$	−6.85427	−0.863557
$64$	0	0
$65$	4.45372	0.552416
$66$	0	0
$67$	13.1427	1.60564	0.802820	−	0.596222i	$-0.203332\pi$
$67$	13.1427	1.60564	0.802820	+	0.596222i	$0.203332\pi$
$68$	0	0
$69$	−22.6661	−2.72868
$70$	0	0
$71$	−2.79160	−0.331302	−0.165651	−	0.986184i	$-0.552973\pi$
$71$	−2.79160	−0.331302	−0.165651	+	0.986184i	$0.552973\pi$
$72$	0	0
$73$	−7.95239	−0.930757	−0.465379	−	0.885112i	$-0.654082\pi$
$73$	−7.95239	−0.930757	−0.465379	+	0.885112i	$0.654082\pi$
$74$	0	0
$75$	7.74743	0.894596
$76$	0	0
$77$	0.863340	0.0983867
$78$	0	0
$79$	−2.75055	−0.309461	−0.154730	−	0.987957i	$-0.549451\pi$
$79$	−2.75055	−0.309461	−0.154730	+	0.987957i	$0.549451\pi$
$80$	0	0
$81$	3.19728	0.355253
$82$	0	0
$83$	−1.26835	−0.139220	−0.0696099	−	0.997574i	$-0.522175\pi$
$83$	−1.26835	−0.139220	−0.0696099	+	0.997574i	$0.522175\pi$
$84$	0	0
$85$	6.46969	0.701736
$86$	0	0
$87$	11.8685	1.27243
$88$	0	0
$89$	5.85129	0.620236	0.310118	−	0.950698i	$-0.399632\pi$
$89$	5.85129	0.620236	0.310118	+	0.950698i	$0.399632\pi$
$90$	0	0
$91$	3.78819	0.397110
$92$	0	0
$93$	26.7039	2.76906
$94$	0	0
$95$	−1.52019	−0.155968
$96$	0	0
$97$	−17.1415	−1.74046	−0.870229	−	0.492648i	$-0.836029\pi$
$97$	−17.1415	−1.74046	−0.870229	+	0.492648i	$0.836029\pi$
$98$	0	0
$99$	−3.53939	−0.355722
$100$	0	0
$101$	−14.8483	−1.47746	−0.738731	−	0.674000i	$-0.764575\pi$
$101$	−14.8483	−1.47746	−0.738731	+	0.674000i	$0.764575\pi$
$102$	0	0
$103$	10.3939	1.02414	0.512071	−	0.858943i	$-0.328879\pi$
$103$	10.3939	1.02414	0.512071	+	0.858943i	$0.328879\pi$
$104$	0	0
$105$	−5.66330	−0.552681
$106$	0	0
$107$	7.45335	0.720542	0.360271	−	0.932848i	$-0.382684\pi$
$107$	7.45335	0.720542	0.360271	+	0.932848i	$0.382684\pi$
$108$	0	0
$109$	−8.66813	−0.830256	−0.415128	−	0.909763i	$-0.636263\pi$
$109$	−8.66813	−0.830256	−0.415128	+	0.909763i	$0.636263\pi$
$110$	0	0
$111$	−21.0143	−1.99459
$112$	0	0
$113$	4.93572	0.464314	0.232157	−	0.972678i	$-0.425422\pi$
$113$	4.93572	0.464314	0.232157	+	0.972678i	$0.425422\pi$
$114$	0	0
$115$	−11.9595	−1.11523
$116$	0	0
$117$	−15.5303	−1.43577
$118$	0	0
$119$	5.50291	0.504451
$120$	0	0
$121$	−10.5542	−0.959472
$122$	0	0
$123$	19.0995	1.72215
$124$	0	0
$125$	11.6888	1.04548
$126$	0	0
$127$	−0.629681	−0.0558752	−0.0279376	−	0.999610i	$-0.508894\pi$
$127$	−0.629681	−0.0558752	−0.0279376	+	0.999610i	$0.508894\pi$
$128$	0	0
$129$	21.7277	1.91302
$130$	0	0
$131$	−2.67706	−0.233896	−0.116948	−	0.993138i	$-0.537311\pi$
$131$	−2.67706	−0.233896	−0.116948	+	0.993138i	$0.537311\pi$
$132$	0	0
$133$	−1.29303	−0.112120
$134$	0	0
$135$	10.0779	0.867369
$136$	0	0
$137$	10.5205	0.898826	0.449413	−	0.893324i	$-0.351633\pi$
$137$	10.5205	0.898826	0.449413	+	0.893324i	$0.351633\pi$
$138$	0	0
$139$	17.4647	1.48134	0.740670	−	0.671869i	$-0.234508\pi$
$139$	17.4647	1.48134	0.740670	+	0.671869i	$0.234508\pi$
$140$	0	0
$141$	34.7844	2.92938
$142$	0	0
$143$	1.95614	0.163580
$144$	0	0
$145$	6.26222	0.520049
$146$	0	0
$147$	15.3510	1.26613
$148$	0	0
$149$	−9.13385	−0.748274	−0.374137	−	0.927373i	$-0.622061\pi$
$149$	−9.13385	−0.748274	−0.374137	+	0.927373i	$0.622061\pi$
$150$	0	0
$151$	3.10117	0.252369	0.126185	−	0.992007i	$-0.459727\pi$
$151$	3.10117	0.252369	0.126185	+	0.992007i	$0.459727\pi$
$152$	0	0
$153$	−22.5600	−1.82387
$154$	0	0
$155$	14.0899	1.13173
$156$	0	0
$157$	−3.44177	−0.274683	−0.137341	−	0.990524i	$-0.543856\pi$
$157$	−3.44177	−0.274683	−0.137341	+	0.990524i	$0.543856\pi$
$158$	0	0
$159$	−2.88114	−0.228489
$160$	0	0
$161$	−10.1723	−0.801692
$162$	0	0
$163$	15.4333	1.20883	0.604415	−	0.796669i	$-0.293407\pi$
$163$	15.4333	1.20883	0.604415	+	0.796669i	$0.293407\pi$
$164$	0	0
$165$	−2.92440	−0.227664
$166$	0	0
$167$	−24.9590	−1.93139	−0.965693	−	0.259685i	$-0.916381\pi$
$167$	−24.9590	−1.93139	−0.965693	+	0.259685i	$0.916381\pi$
$168$	0	0
$169$	−4.41681	−0.339754
$170$	0	0
$171$	5.30096	0.405374
$172$	0	0
$173$	−4.74504	−0.360759	−0.180379	−	0.983597i	$-0.557733\pi$
$173$	−4.74504	−0.360759	−0.180379	+	0.983597i	$0.557733\pi$
$174$	0	0
$175$	3.47697	0.262834
$176$	0	0
$177$	14.0124	1.05324
$178$	0	0
$179$	16.1711	1.20868	0.604341	−	0.796726i	$-0.293437\pi$
$179$	16.1711	1.20868	0.604341	+	0.796726i	$0.293437\pi$
$180$	0	0
$181$	−23.5007	−1.74680	−0.873398	−	0.487006i	$-0.838089\pi$
$181$	−23.5007	−1.74680	−0.873398	+	0.487006i	$0.838089\pi$
$182$	0	0
$183$	−10.4651	−0.773602
$184$	0	0
$185$	−11.0879	−0.815198
$186$	0	0
$187$	2.84158	0.207797
$188$	0	0
$189$	8.57195	0.623518
$190$	0	0
$191$	25.5983	1.85223	0.926113	−	0.377247i	$-0.123129\pi$
$191$	25.5983	1.85223	0.926113	+	0.377247i	$0.123129\pi$
$192$	0	0
$193$	8.81746	0.634695	0.317347	−	0.948309i	$-0.397208\pi$
$193$	8.81746	0.634695	0.317347	+	0.948309i	$0.397208\pi$
$194$	0	0
$195$	−12.8318	−0.918903
$196$	0	0
$197$	15.2049	1.08330	0.541652	−	0.840603i	$-0.317799\pi$
$197$	15.2049	1.08330	0.541652	+	0.840603i	$0.317799\pi$
$198$	0	0
$199$	2.36348	0.167543	0.0837714	−	0.996485i	$-0.473303\pi$
$199$	2.36348	0.167543	0.0837714	+	0.996485i	$0.473303\pi$
$200$	0	0
$201$	−37.8660	−2.67086
$202$	0	0
$203$	5.32644	0.373843
$204$	0	0
$205$	10.0776	0.703850
$206$	0	0
$207$	41.7030	2.89856
$208$	0	0
$209$	−0.667690	−0.0461851
$210$	0	0
$211$	5.34144	0.367720	0.183860	−	0.982952i	$-0.441141\pi$
$211$	5.34144	0.367720	0.183860	+	0.982952i	$0.441141\pi$
$212$	0	0
$213$	8.04299	0.551097
$214$	0	0
$215$	11.4643	0.781860
$216$	0	0
$217$	11.9844	0.813555
$218$	0	0
$219$	22.9119	1.54825
$220$	0	0
$221$	12.4684	0.838713
$222$	0	0
$223$	28.7535	1.92548	0.962738	−	0.270436i	$-0.0871679\pi$
$223$	28.7535	1.92548	0.962738	+	0.270436i	$0.0871679\pi$
$224$	0	0
$225$	−14.2544	−0.950291
$226$	0	0
$227$	8.90059	0.590753	0.295376	−	0.955381i	$-0.404555\pi$
$227$	8.90059	0.590753	0.295376	+	0.955381i	$0.404555\pi$
$228$	0	0
$229$	−11.5067	−0.760382	−0.380191	−	0.924908i	$-0.624142\pi$
$229$	−11.5067	−0.760382	−0.380191	+	0.924908i	$0.624142\pi$
$230$	0	0
$231$	−2.48740	−0.163659
$232$	0	0
$233$	−9.72237	−0.636934	−0.318467	−	0.947934i	$-0.603168\pi$
$233$	−9.72237	−0.636934	−0.318467	+	0.947934i	$0.603168\pi$
$234$	0	0
$235$	18.3535	1.19725
$236$	0	0
$237$	7.92471	0.514765
$238$	0	0
$239$	−29.4808	−1.90696	−0.953478	−	0.301463i	$-0.902525\pi$
$239$	−29.4808	−1.90696	−0.953478	+	0.301463i	$0.902525\pi$
$240$	0	0
$241$	27.4454	1.76792	0.883958	−	0.467567i	$-0.154869\pi$
$241$	27.4454	1.76792	0.883958	+	0.467567i	$0.154869\pi$
$242$	0	0
$243$	10.6763	0.684887
$244$	0	0
$245$	8.09971	0.517472
$246$	0	0
$247$	−2.92971	−0.186413
$248$	0	0
$249$	3.65430	0.231582
$250$	0	0
$251$	16.1315	1.01821	0.509106	−	0.860704i	$-0.329976\pi$
$251$	16.1315	1.01821	0.509106	+	0.860704i	$0.329976\pi$
$252$	0	0
$253$	−5.25277	−0.330239
$254$	0	0
$255$	−18.6401	−1.16729
$256$	0	0
$257$	10.1751	0.634705	0.317352	−	0.948308i	$-0.397206\pi$
$257$	10.1751	0.634705	0.317352	+	0.948308i	$0.397206\pi$
$258$	0	0
$259$	−9.43100	−0.586014
$260$	0	0
$261$	−21.8366	−1.35165
$262$	0	0
$263$	−1.82124	−0.112302	−0.0561511	−	0.998422i	$-0.517883\pi$
$263$	−1.82124	−0.112302	−0.0561511	+	0.998422i	$0.517883\pi$
$264$	0	0
$265$	−1.52019	−0.0933846
$266$	0	0
$267$	−16.8584	−1.03172
$268$	0	0
$269$	15.5382	0.947378	0.473689	−	0.880692i	$-0.342922\pi$
$269$	15.5382	0.947378	0.473689	+	0.880692i	$0.342922\pi$
$270$	0	0
$271$	31.6526	1.92276	0.961378	−	0.275231i	$-0.0887544\pi$
$271$	31.6526	1.92276	0.961378	+	0.275231i	$0.0887544\pi$
$272$	0	0
$273$	−10.9143	−0.660563
$274$	0	0
$275$	1.79543	0.108268
$276$	0	0
$277$	0.831765	0.0499759	0.0249880	−	0.999688i	$-0.492045\pi$
$277$	0.831765	0.0499759	0.0249880	+	0.999688i	$0.492045\pi$
$278$	0	0
$279$	−49.1320	−2.94145
$280$	0	0
$281$	−30.8957	−1.84308	−0.921542	−	0.388280i	$-0.873069\pi$
$281$	−30.8957	−1.84308	−0.921542	+	0.388280i	$0.873069\pi$
$282$	0	0
$283$	20.9244	1.24383	0.621914	−	0.783086i	$-0.286356\pi$
$283$	20.9244	1.24383	0.621914	+	0.783086i	$0.286356\pi$
$284$	0	0
$285$	4.37988	0.259442
$286$	0	0
$287$	8.57167	0.505970
$288$	0	0
$289$	1.11216	0.0654209
$290$	0	0
$291$	49.3871	2.89512
$292$	0	0
$293$	31.9901	1.86888	0.934441	−	0.356119i	$-0.115900\pi$
$293$	31.9901	1.86888	0.934441	+	0.356119i	$0.115900\pi$
$294$	0	0
$295$	7.39345	0.430463
$296$	0	0
$297$	4.42637	0.256844
$298$	0	0
$299$	−23.0482	−1.33291
$300$	0	0
$301$	9.75118	0.562049
$302$	0	0
$303$	42.7800	2.45765
$304$	0	0
$305$	−5.52176	−0.316175
$306$	0	0
$307$	33.8496	1.93190	0.965951	−	0.258727i	$-0.0833029\pi$
$307$	33.8496	1.93190	0.965951	+	0.258727i	$0.0833029\pi$
$308$	0	0
$309$	−29.9463	−1.70358
$310$	0	0
$311$	10.8151	0.613270	0.306635	−	0.951827i	$-0.400797\pi$
$311$	10.8151	0.613270	0.306635	+	0.951827i	$0.400797\pi$
$312$	0	0
$313$	12.7644	0.721488	0.360744	−	0.932665i	$-0.382523\pi$
$313$	12.7644	0.721488	0.360744	+	0.932665i	$0.382523\pi$
$314$	0	0
$315$	10.4198	0.587090
$316$	0	0
$317$	18.3350	1.02980	0.514899	−	0.857251i	$-0.327829\pi$
$317$	18.3350	1.02980	0.514899	+	0.857251i	$0.327829\pi$
$318$	0	0
$319$	2.75046	0.153996
$320$	0	0
$321$	−21.4741	−1.19857
$322$	0	0
$323$	−4.25584	−0.236801
$324$	0	0
$325$	7.87804	0.436995
$326$	0	0
$327$	24.9741	1.38107
$328$	0	0
$329$	15.6109	0.860656
$330$	0	0
$331$	16.4509	0.904226	0.452113	−	0.891961i	$-0.350670\pi$
$331$	16.4509	0.904226	0.452113	+	0.891961i	$0.350670\pi$
$332$	0	0
$333$	38.6638	2.11876
$334$	0	0
$335$	−19.9795	−1.09159
$336$	0	0
$337$	−19.1657	−1.04402	−0.522012	−	0.852938i	$-0.674818\pi$
$337$	−19.1657	−1.04402	−0.522012	+	0.852938i	$0.674818\pi$
$338$	0	0
$339$	−14.2205	−0.772352
$340$	0	0
$341$	6.18849	0.335125
$342$	0	0
$343$	15.9405	0.860708
$344$	0	0
$345$	34.4569	1.85510
$346$	0	0
$347$	−23.4439	−1.25853	−0.629267	−	0.777189i	$-0.716645\pi$
$347$	−23.4439	−1.25853	−0.629267	+	0.777189i	$0.716645\pi$
$348$	0	0
$349$	13.9473	0.746579	0.373290	−	0.927715i	$-0.378230\pi$
$349$	13.9473	0.746579	0.373290	+	0.927715i	$0.378230\pi$
$350$	0	0
$351$	19.4221	1.03668
$352$	0	0
$353$	−29.9330	−1.59317	−0.796586	−	0.604525i	$-0.793363\pi$
$353$	−29.9330	−1.59317	−0.796586	+	0.604525i	$0.793363\pi$
$354$	0	0
$355$	4.24377	0.225236
$356$	0	0
$357$	−15.8546	−0.839116
$358$	0	0
$359$	−34.7192	−1.83241	−0.916203	−	0.400713i	$-0.868762\pi$
$359$	−34.7192	−1.83241	−0.916203	+	0.400713i	$0.868762\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	30.4081	1.59601
$364$	0	0
$365$	12.0892	0.632776
$366$	0	0
$367$	−9.52733	−0.497323	−0.248661	−	0.968590i	$-0.579991\pi$
$367$	−9.52733	−0.497323	−0.248661	+	0.968590i	$0.579991\pi$
$368$	0	0
$369$	−35.1409	−1.82936
$370$	0	0
$371$	−1.29303	−0.0671305
$372$	0	0
$373$	−8.04988	−0.416807	−0.208404	−	0.978043i	$-0.566827\pi$
$373$	−8.04988	−0.416807	−0.208404	+	0.978043i	$0.566827\pi$
$374$	0	0
$375$	−33.6770	−1.73907
$376$	0	0
$377$	12.0685	0.621561
$378$	0	0
$379$	−11.5053	−0.590987	−0.295494	−	0.955345i	$-0.595484\pi$
$379$	−11.5053	−0.590987	−0.295494	+	0.955345i	$0.595484\pi$
$380$	0	0
$381$	1.81420	0.0929442
$382$	0	0
$383$	13.5302	0.691360	0.345680	−	0.938352i	$-0.387648\pi$
$383$	13.5302	0.691360	0.345680	+	0.938352i	$0.387648\pi$
$384$	0	0
$385$	−1.31244	−0.0668882
$386$	0	0
$387$	−39.9765	−2.03212
$388$	0	0
$389$	−17.6471	−0.894742	−0.447371	−	0.894348i	$-0.647640\pi$
$389$	−17.6471	−0.894742	−0.447371	+	0.894348i	$0.647640\pi$
$390$	0	0
$391$	−33.4810	−1.69321
$392$	0	0
$393$	7.71299	0.389069
$394$	0	0
$395$	4.18136	0.210387
$396$	0	0
$397$	5.36745	0.269384	0.134692	−	0.990887i	$-0.456995\pi$
$397$	5.36745	0.269384	0.134692	+	0.990887i	$0.456995\pi$
$398$	0	0
$399$	3.72538	0.186503
$400$	0	0
$401$	−10.0209	−0.500421	−0.250210	−	0.968191i	$-0.580500\pi$
$401$	−10.0209	−0.500421	−0.250210	+	0.968191i	$0.580500\pi$
$402$	0	0
$403$	27.1540	1.35264
$404$	0	0
$405$	−4.86048	−0.241519
$406$	0	0
$407$	−4.86996	−0.241395
$408$	0	0
$409$	−0.213085	−0.0105364	−0.00526819	−	0.999986i	$-0.501677\pi$
$409$	−0.213085	−0.0105364	−0.00526819	+	0.999986i	$0.501677\pi$
$410$	0	0
$411$	−30.3110	−1.49513
$412$	0	0
$413$	6.28863	0.309443
$414$	0	0
$415$	1.92814	0.0946486
$416$	0	0
$417$	−50.3183	−2.46410
$418$	0	0
$419$	−17.2335	−0.841911	−0.420955	−	0.907081i	$-0.638305\pi$
$419$	−17.2335	−0.841911	−0.420955	+	0.907081i	$0.638305\pi$
$420$	0	0
$421$	−24.4399	−1.19113	−0.595563	−	0.803308i	$-0.703071\pi$
$421$	−24.4399	−1.19113	−0.595563	+	0.803308i	$0.703071\pi$
$422$	0	0
$423$	−63.9993	−3.11175
$424$	0	0
$425$	11.4440	0.555117
$426$	0	0
$427$	−4.69663	−0.227286
$428$	0	0
$429$	−5.63590	−0.272104
$430$	0	0
$431$	−10.9021	−0.525136	−0.262568	−	0.964913i	$-0.584569\pi$
$431$	−10.9021	−0.525136	−0.262568	+	0.964913i	$0.584569\pi$
$432$	0	0
$433$	30.0137	1.44237	0.721184	−	0.692744i	$-0.243598\pi$
$433$	30.0137	1.44237	0.721184	+	0.692744i	$0.243598\pi$
$434$	0	0
$435$	−18.0423	−0.865064
$436$	0	0
$437$	7.86708	0.376333
$438$	0	0
$439$	27.1054	1.29367	0.646835	−	0.762630i	$-0.276092\pi$
$439$	27.1054	1.29367	0.646835	+	0.762630i	$0.276092\pi$
$440$	0	0
$441$	−28.2440	−1.34495
$442$	0	0
$443$	−35.5681	−1.68989	−0.844945	−	0.534853i	$-0.820367\pi$
$443$	−35.5681	−1.68989	−0.844945	+	0.534853i	$0.820367\pi$
$444$	0	0
$445$	−8.89508	−0.421667
$446$	0	0
$447$	26.3159	1.24470
$448$	0	0
$449$	22.6260	1.06779	0.533894	−	0.845551i	$-0.320728\pi$
$449$	22.6260	1.06779	0.533894	+	0.845551i	$0.320728\pi$
$450$	0	0
$451$	4.42622	0.208423
$452$	0	0
$453$	−8.93489	−0.419798
$454$	0	0
$455$	−5.75877	−0.269975
$456$	0	0
$457$	−7.24072	−0.338707	−0.169353	−	0.985555i	$-0.554168\pi$
$457$	−7.24072	−0.338707	−0.169353	+	0.985555i	$0.554168\pi$
$458$	0	0
$459$	28.2136	1.31690
$460$	0	0
$461$	−14.4895	−0.674842	−0.337421	−	0.941354i	$-0.609555\pi$
$461$	−14.4895	−0.674842	−0.337421	+	0.941354i	$0.609555\pi$
$462$	0	0
$463$	−25.5237	−1.18619	−0.593094	−	0.805133i	$-0.702094\pi$
$463$	−25.5237	−1.18619	−0.593094	+	0.805133i	$0.702094\pi$
$464$	0	0
$465$	−40.5950	−1.88255
$466$	0	0
$467$	−13.8722	−0.641927	−0.320963	−	0.947092i	$-0.604007\pi$
$467$	−13.8722	−0.641927	−0.320963	+	0.947092i	$0.604007\pi$
$468$	0	0
$469$	−16.9939	−0.784704
$470$	0	0
$471$	9.91620	0.456915
$472$	0	0
$473$	5.03529	0.231523
$474$	0	0
$475$	−2.68902	−0.123381
$476$	0	0
$477$	5.30096	0.242714
$478$	0	0
$479$	−15.1350	−0.691537	−0.345768	−	0.938320i	$-0.612382\pi$
$479$	−15.1350	−0.691537	−0.345768	+	0.938320i	$0.612382\pi$
$480$	0	0
$481$	−21.3685	−0.974322
$482$	0	0
$483$	29.3079	1.33356
$484$	0	0
$485$	26.0584	1.18325
$486$	0	0
$487$	−27.8165	−1.26048	−0.630242	−	0.776399i	$-0.717044\pi$
$487$	−27.8165	−1.26048	−0.630242	+	0.776399i	$0.717044\pi$
$488$	0	0
$489$	−44.4655	−2.01080
$490$	0	0
$491$	21.5924	0.974449	0.487225	−	0.873277i	$-0.338009\pi$
$491$	21.5924	0.974449	0.487225	+	0.873277i	$0.338009\pi$
$492$	0	0
$493$	17.5313	0.789572
$494$	0	0
$495$	5.38056	0.241838
$496$	0	0
$497$	3.60961	0.161913
$498$	0	0
$499$	−2.60788	−0.116745	−0.0583723	−	0.998295i	$-0.518591\pi$
$499$	−2.60788	−0.116745	−0.0583723	+	0.998295i	$0.518591\pi$
$500$	0	0
$501$	71.9104	3.21272
$502$	0	0
$503$	−15.9770	−0.712378	−0.356189	−	0.934414i	$-0.615924\pi$
$503$	−15.9770	−0.712378	−0.356189	+	0.934414i	$0.615924\pi$
$504$	0	0
$505$	22.5723	1.00445
$506$	0	0
$507$	12.7254	0.565156
$508$	0	0
$509$	30.2367	1.34022	0.670110	−	0.742262i	$-0.266247\pi$
$509$	30.2367	1.34022	0.670110	+	0.742262i	$0.266247\pi$
$510$	0	0
$511$	10.2826	0.454877
$512$	0	0
$513$	−6.62938	−0.292694
$514$	0	0
$515$	−15.8007	−0.696263
$516$	0	0
$517$	8.06112	0.354528
$518$	0	0
$519$	13.6711	0.600096
$520$	0	0
$521$	−18.7679	−0.822237	−0.411118	−	0.911582i	$-0.634862\pi$
$521$	−18.7679	−0.822237	−0.411118	+	0.911582i	$0.634862\pi$
$522$	0	0
$523$	−29.2262	−1.27797	−0.638987	−	0.769217i	$-0.720646\pi$
$523$	−29.2262	−1.27797	−0.638987	+	0.769217i	$0.720646\pi$
$524$	0	0
$525$	−10.0176	−0.437205
$526$	0	0
$527$	39.4453	1.71826
$528$	0	0
$529$	38.8909	1.69091
$530$	0	0
$531$	−25.7812	−1.11881
$532$	0	0
$533$	19.4215	0.841239
$534$	0	0
$535$	−11.3305	−0.489861
$536$	0	0
$537$	−46.5910	−2.01055
$538$	0	0
$539$	3.55751	0.153233
$540$	0	0
$541$	12.4061	0.533379	0.266690	−	0.963782i	$-0.414070\pi$
$541$	12.4061	0.533379	0.266690	+	0.963782i	$0.414070\pi$
$542$	0	0
$543$	67.7089	2.90567
$544$	0	0
$545$	13.1772	0.564450
$546$	0	0
$547$	8.38221	0.358397	0.179199	−	0.983813i	$-0.442650\pi$
$547$	8.38221	0.358397	0.179199	+	0.983813i	$0.442650\pi$
$548$	0	0
$549$	19.2545	0.821764
$550$	0	0
$551$	−4.11936	−0.175491
$552$	0	0
$553$	3.55653	0.151239
$554$	0	0
$555$	31.9457	1.35602
$556$	0	0
$557$	32.6832	1.38483	0.692415	−	0.721500i	$-0.256547\pi$
$557$	32.6832	1.38483	0.692415	+	0.721500i	$0.256547\pi$
$558$	0	0
$559$	22.0940	0.934477
$560$	0	0
$561$	−8.18698	−0.345655
$562$	0	0
$563$	−43.2695	−1.82359	−0.911796	−	0.410644i	$-0.865304\pi$
$563$	−43.2695	−1.82359	−0.911796	+	0.410644i	$0.865304\pi$
$564$	0	0
$565$	−7.50324	−0.315664
$566$	0	0
$567$	−4.13416	−0.173618
$568$	0	0
$569$	−7.08846	−0.297164	−0.148582	−	0.988900i	$-0.547471\pi$
$569$	−7.08846	−0.297164	−0.148582	+	0.988900i	$0.547471\pi$
$570$	0	0
$571$	−20.9527	−0.876842	−0.438421	−	0.898770i	$-0.644462\pi$
$571$	−20.9527	−0.876842	−0.438421	+	0.898770i	$0.644462\pi$
$572$	0	0
$573$	−73.7521	−3.08104
$574$	0	0
$575$	−21.1547	−0.882212
$576$	0	0
$577$	4.21800	0.175598	0.0877989	−	0.996138i	$-0.472017\pi$
$577$	4.21800	0.175598	0.0877989	+	0.996138i	$0.472017\pi$
$578$	0	0
$579$	−25.4043	−1.05577
$580$	0	0
$581$	1.64001	0.0680391
$582$	0	0
$583$	−0.667690	−0.0276529
$584$	0	0
$585$	23.6090	0.976111
$586$	0	0
$587$	−0.0208307	−0.000859774 0	−0.000429887	−	1.00000i	$-0.500137\pi$
$587$	−0.0208307	−0.000859774 0	−0.000429887		1.00000i	$0.500137\pi$
$588$	0	0
$589$	−9.26851	−0.381902
$590$	0	0
$591$	−43.8074	−1.80200
$592$	0	0
$593$	−7.48240	−0.307265	−0.153633	−	0.988128i	$-0.549097\pi$
$593$	−7.48240	−0.307265	−0.153633	+	0.988128i	$0.549097\pi$
$594$	0	0
$595$	−8.36547	−0.342951
$596$	0	0
$597$	−6.80952	−0.278695
$598$	0	0
$599$	27.8291	1.13707	0.568534	−	0.822660i	$-0.307511\pi$
$599$	27.8291	1.13707	0.568534	+	0.822660i	$0.307511\pi$
$600$	0	0
$601$	16.8651	0.687941	0.343971	−	0.938980i	$-0.388228\pi$
$601$	16.8651	0.687941	0.343971	+	0.938980i	$0.388228\pi$
$602$	0	0
$603$	69.6690	2.83714
$604$	0	0
$605$	16.0444	0.652297
$606$	0	0
$607$	15.9361	0.646825	0.323412	−	0.946258i	$-0.395170\pi$
$607$	15.9361	0.646825	0.323412	+	0.946258i	$0.395170\pi$
$608$	0	0
$609$	−15.3462	−0.621860
$610$	0	0
$611$	35.3708	1.43095
$612$	0	0
$613$	31.0004	1.25210	0.626048	−	0.779784i	$-0.284671\pi$
$613$	31.0004	1.25210	0.626048	+	0.779784i	$0.284671\pi$
$614$	0	0
$615$	−29.0349	−1.17080
$616$	0	0
$617$	36.3085	1.46172	0.730862	−	0.682525i	$-0.239118\pi$
$617$	36.3085	1.46172	0.730862	+	0.682525i	$0.239118\pi$
$618$	0	0
$619$	−21.4843	−0.863527	−0.431763	−	0.901987i	$-0.642108\pi$
$619$	−21.4843	−0.863527	−0.431763	+	0.901987i	$0.642108\pi$
$620$	0	0
$621$	−52.1538	−2.09286
$622$	0	0
$623$	−7.56587	−0.303120
$624$	0	0
$625$	−4.32409	−0.172964
$626$	0	0
$627$	1.92371	0.0768254
$628$	0	0
$629$	−31.0410	−1.23769
$630$	0	0
$631$	40.4731	1.61121	0.805605	−	0.592453i	$-0.201840\pi$
$631$	40.4731	1.61121	0.805605	+	0.592453i	$0.201840\pi$
$632$	0	0
$633$	−15.3894	−0.611674
$634$	0	0
$635$	0.957236	0.0379867
$636$	0	0
$637$	15.6097	0.618480
$638$	0	0
$639$	−14.7982	−0.585406
$640$	0	0
$641$	−32.4592	−1.28206	−0.641030	−	0.767516i	$-0.721493\pi$
$641$	−32.4592	−1.28206	−0.641030	+	0.767516i	$0.721493\pi$
$642$	0	0
$643$	−18.7786	−0.740556	−0.370278	−	0.928921i	$-0.620738\pi$
$643$	−18.7786	−0.740556	−0.370278	+	0.928921i	$0.620738\pi$
$644$	0	0
$645$	−33.0303	−1.30057
$646$	0	0
$647$	3.18871	0.125361	0.0626805	−	0.998034i	$-0.480035\pi$
$647$	3.18871	0.125361	0.0626805	+	0.998034i	$0.480035\pi$
$648$	0	0
$649$	3.24731	0.127468
$650$	0	0
$651$	−34.5288	−1.35329
$652$	0	0
$653$	−20.2831	−0.793741	−0.396870	−	0.917875i	$-0.629904\pi$
$653$	−20.2831	−0.793741	−0.396870	+	0.917875i	$0.629904\pi$
$654$	0	0
$655$	4.06965	0.159014
$656$	0	0
$657$	−42.1553	−1.64463
$658$	0	0
$659$	−17.1174	−0.666799	−0.333400	−	0.942786i	$-0.608196\pi$
$659$	−17.1174	−0.666799	−0.333400	+	0.942786i	$0.608196\pi$
$660$	0	0
$661$	42.2253	1.64237	0.821187	−	0.570659i	$-0.193312\pi$
$661$	42.2253	1.64237	0.821187	+	0.570659i	$0.193312\pi$
$662$	0	0
$663$	−35.9231	−1.39514
$664$	0	0
$665$	1.96565	0.0762245
$666$	0	0
$667$	−32.4074	−1.25482
$668$	0	0
$669$	−82.8428	−3.20289
$670$	0	0
$671$	−2.42523	−0.0936251
$672$	0	0
$673$	−13.2915	−0.512349	−0.256174	−	0.966631i	$-0.582462\pi$
$673$	−13.2915	−0.512349	−0.256174	+	0.966631i	$0.582462\pi$
$674$	0	0
$675$	17.8265	0.686143
$676$	0	0
$677$	9.47654	0.364213	0.182106	−	0.983279i	$-0.441708\pi$
$677$	9.47654	0.364213	0.182106	+	0.983279i	$0.441708\pi$
$678$	0	0
$679$	22.1644	0.850592
$680$	0	0
$681$	−25.6438	−0.982674
$682$	0	0
$683$	−2.34886	−0.0898767	−0.0449383	−	0.998990i	$-0.514309\pi$
$683$	−2.34886	−0.0898767	−0.0449383	+	0.998990i	$0.514309\pi$
$684$	0	0
$685$	−15.9932	−0.611067
$686$	0	0
$687$	33.1523	1.26484
$688$	0	0
$689$	−2.92971	−0.111613
$690$	0	0
$691$	−45.3204	−1.72407	−0.862034	−	0.506851i	$-0.830810\pi$
$691$	−45.3204	−1.72407	−0.862034	+	0.506851i	$0.830810\pi$
$692$	0	0
$693$	4.57653	0.173848
$694$	0	0
$695$	−26.5497	−1.00709
$696$	0	0
$697$	28.2126	1.06863
$698$	0	0
$699$	28.0115	1.05949
$700$	0	0
$701$	22.5432	0.851443	0.425722	−	0.904854i	$-0.360020\pi$
$701$	22.5432	0.851443	0.425722	+	0.904854i	$0.360020\pi$
$702$	0	0
$703$	7.29374	0.275089
$704$	0	0
$705$	−52.8790	−1.99154
$706$	0	0
$707$	19.1992	0.722062
$708$	0	0
$709$	−40.9845	−1.53921	−0.769603	−	0.638523i	$-0.779546\pi$
$709$	−40.9845	−1.53921	−0.769603	+	0.638523i	$0.779546\pi$
$710$	0	0
$711$	−14.5805	−0.546813
$712$	0	0
$713$	−72.9161	−2.73073
$714$	0	0
$715$	−2.97370	−0.111210
$716$	0	0
$717$	84.9383	3.17208
$718$	0	0
$719$	−7.67167	−0.286105	−0.143053	−	0.989715i	$-0.545692\pi$
$719$	−7.67167	−0.286105	−0.143053	+	0.989715i	$0.545692\pi$
$720$	0	0
$721$	−13.4396	−0.500516
$722$	0	0
$723$	−79.0741	−2.94080
$724$	0	0
$725$	11.0770	0.411391
$726$	0	0
$727$	36.9635	1.37090	0.685451	−	0.728119i	$-0.259605\pi$
$727$	36.9635	1.37090	0.685451	+	0.728119i	$0.259605\pi$
$728$	0	0
$729$	−40.3518	−1.49451
$730$	0	0
$731$	32.0948	1.18707
$732$	0	0
$733$	14.6444	0.540904	0.270452	−	0.962733i	$-0.412827\pi$
$733$	14.6444	0.540904	0.270452	+	0.962733i	$0.412827\pi$
$734$	0	0
$735$	−23.3364	−0.860776
$736$	0	0
$737$	−8.77526	−0.323241
$738$	0	0
$739$	−18.0094	−0.662488	−0.331244	−	0.943545i	$-0.607468\pi$
$739$	−18.0094	−0.662488	−0.331244	+	0.943545i	$0.607468\pi$
$740$	0	0
$741$	8.44090	0.310084
$742$	0	0
$743$	−25.1105	−0.921214	−0.460607	−	0.887604i	$-0.652368\pi$
$743$	−25.1105	−0.921214	−0.460607	+	0.887604i	$0.652368\pi$
$744$	0	0
$745$	13.8852	0.508715
$746$	0	0
$747$	−6.72348	−0.245999
$748$	0	0
$749$	−9.63736	−0.352142
$750$	0	0
$751$	−13.9058	−0.507428	−0.253714	−	0.967279i	$-0.581652\pi$
$751$	−13.9058	−0.507428	−0.253714	+	0.967279i	$0.581652\pi$
$752$	0	0
$753$	−46.4771	−1.69372
$754$	0	0
$755$	−4.71437	−0.171573
$756$	0	0
$757$	16.7076	0.607248	0.303624	−	0.952792i	$-0.401803\pi$
$757$	16.7076	0.607248	0.303624	+	0.952792i	$0.401803\pi$
$758$	0	0
$759$	15.1339	0.549327
$760$	0	0
$761$	41.6997	1.51161	0.755806	−	0.654795i	$-0.227245\pi$
$761$	41.6997	1.51161	0.755806	+	0.654795i	$0.227245\pi$
$762$	0	0
$763$	11.2081	0.405761
$764$	0	0
$765$	34.2955	1.23996
$766$	0	0
$767$	14.2486	0.514488
$768$	0	0
$769$	10.2485	0.369572	0.184786	−	0.982779i	$-0.440841\pi$
$769$	10.2485	0.369572	0.184786	+	0.982779i	$0.440841\pi$
$770$	0	0
$771$	−29.3158	−1.05578
$772$	0	0
$773$	18.6833	0.671991	0.335996	−	0.941864i	$-0.390927\pi$
$773$	18.6833	0.671991	0.335996	+	0.941864i	$0.390927\pi$
$774$	0	0
$775$	24.9232	0.895267
$776$	0	0
$777$	27.1720	0.974790
$778$	0	0
$779$	−6.62916	−0.237514
$780$	0	0
$781$	1.86392	0.0666964
$782$	0	0
$783$	27.3088	0.975938
$784$	0	0
$785$	5.23214	0.186743
$786$	0	0
$787$	−29.2231	−1.04169	−0.520845	−	0.853651i	$-0.674383\pi$
$787$	−29.2231	−1.04169	−0.520845	+	0.853651i	$0.674383\pi$
$788$	0	0
$789$	5.24723	0.186806
$790$	0	0
$791$	−6.38202	−0.226918
$792$	0	0
$793$	−10.6415	−0.377891
$794$	0	0
$795$	4.37988	0.155338
$796$	0	0
$797$	−32.6336	−1.15594	−0.577971	−	0.816057i	$-0.696155\pi$
$797$	−32.6336	−1.15594	−0.577971	+	0.816057i	$0.696155\pi$
$798$	0	0
$799$	51.3814	1.81774
$800$	0	0
$801$	31.0174	1.09595
$802$	0	0
$803$	5.30973	0.187376
$804$	0	0
$805$	15.4639	0.545030
$806$	0	0
$807$	−44.7676	−1.57589
$808$	0	0
$809$	−6.66779	−0.234427	−0.117213	−	0.993107i	$-0.537396\pi$
$809$	−6.66779	−0.234427	−0.117213	+	0.993107i	$0.537396\pi$
$810$	0	0
$811$	−16.7347	−0.587634	−0.293817	−	0.955862i	$-0.594926\pi$
$811$	−16.7347	−0.587634	−0.293817	+	0.955862i	$0.594926\pi$
$812$	0	0
$813$	−91.1954	−3.19836
$814$	0	0
$815$	−23.4616	−0.821824
$816$	0	0
$817$	−7.54137	−0.263839
$818$	0	0
$819$	20.0810	0.701688
$820$	0	0
$821$	−15.5553	−0.542885	−0.271443	−	0.962455i	$-0.587501\pi$
$821$	−15.5553	−0.542885	−0.271443	+	0.962455i	$0.587501\pi$
$822$	0	0
$823$	48.0676	1.67553	0.837766	−	0.546030i	$-0.183861\pi$
$823$	48.0676	1.67553	0.837766	+	0.546030i	$0.183861\pi$
$824$	0	0
$825$	−5.17288	−0.180097
$826$	0	0
$827$	53.9186	1.87493	0.937467	−	0.348074i	$-0.113164\pi$
$827$	53.9186	1.87493	0.937467	+	0.348074i	$0.113164\pi$
$828$	0	0
$829$	−18.2870	−0.635134	−0.317567	−	0.948236i	$-0.602866\pi$
$829$	−18.2870	−0.635134	−0.317567	+	0.948236i	$0.602866\pi$
$830$	0	0
$831$	−2.39643	−0.0831312
$832$	0	0
$833$	22.6755	0.785658
$834$	0	0
$835$	37.9425	1.31305
$836$	0	0
$837$	61.4445	2.12383
$838$	0	0
$839$	29.7497	1.02707	0.513537	−	0.858067i	$-0.328335\pi$
$839$	29.7497	1.02707	0.513537	+	0.858067i	$0.328335\pi$
$840$	0	0
$841$	−12.0308	−0.414856
$842$	0	0
$843$	89.0148	3.06583
$844$	0	0
$845$	6.71439	0.230982
$846$	0	0
$847$	13.6468	0.468911
$848$	0	0
$849$	−60.2861	−2.06901
$850$	0	0
$851$	57.3804	1.96698
$852$	0	0
$853$	38.5838	1.32109	0.660543	−	0.750788i	$-0.270326\pi$
$853$	38.5838	1.32109	0.660543	+	0.750788i	$0.270326\pi$
$854$	0	0
$855$	−8.05847	−0.275594
$856$	0	0
$857$	−29.3651	−1.00309	−0.501547	−	0.865131i	$-0.667235\pi$
$857$	−29.3651	−1.00309	−0.501547	+	0.865131i	$0.667235\pi$
$858$	0	0
$859$	44.5300	1.51934	0.759672	−	0.650307i	$-0.225360\pi$
$859$	44.5300	1.51934	0.759672	+	0.650307i	$0.225360\pi$
$860$	0	0
$861$	−24.6962	−0.841643
$862$	0	0
$863$	−24.4538	−0.832415	−0.416208	−	0.909270i	$-0.636641\pi$
$863$	−24.4538	−0.832415	−0.416208	+	0.909270i	$0.636641\pi$
$864$	0	0
$865$	7.21338	0.245262
$866$	0	0
$867$	−3.20428	−0.108823
$868$	0	0
$869$	1.83651	0.0622994
$870$	0	0
$871$	−38.5043	−1.30467
$872$	0	0
$873$	−90.8664	−3.07536
$874$	0	0
$875$	−15.1139	−0.510943
$876$	0	0
$877$	−28.0241	−0.946307	−0.473154	−	0.880980i	$-0.656884\pi$
$877$	−28.0241	−0.946307	−0.473154	+	0.880980i	$0.656884\pi$
$878$	0	0
$879$	−92.1679	−3.10875
$880$	0	0
$881$	−36.3624	−1.22508	−0.612540	−	0.790439i	$-0.709852\pi$
$881$	−36.3624	−1.22508	−0.612540	+	0.790439i	$0.709852\pi$
$882$	0	0
$883$	0.962867	0.0324030	0.0162015	−	0.999869i	$-0.494843\pi$
$883$	0.962867	0.0324030	0.0162015	+	0.999869i	$0.494843\pi$
$884$	0	0
$885$	−21.3015	−0.716044
$886$	0	0
$887$	37.6769	1.26507	0.632534	−	0.774533i	$-0.282015\pi$
$887$	37.6769	1.26507	0.632534	+	0.774533i	$0.282015\pi$
$888$	0	0
$889$	0.814194	0.0273072
$890$	0	0
$891$	−2.13479	−0.0715181
$892$	0	0
$893$	−12.0732	−0.404013
$894$	0	0
$895$	−24.5831	−0.821722
$896$	0	0
$897$	66.4052	2.21720
$898$	0	0
$899$	38.1804	1.27339
$900$	0	0
$901$	−4.25584	−0.141783
$902$	0	0
$903$	−28.0945	−0.934926
$904$	0	0
$905$	35.7256	1.18756
$906$	0	0
$907$	26.0293	0.864288	0.432144	−	0.901805i	$-0.357757\pi$
$907$	26.0293	0.864288	0.432144	+	0.901805i	$0.357757\pi$
$908$	0	0
$909$	−78.7103	−2.61065
$910$	0	0
$911$	16.8021	0.556677	0.278338	−	0.960483i	$-0.410216\pi$
$911$	16.8021	0.556677	0.278338	+	0.960483i	$0.410216\pi$
$912$	0	0
$913$	0.846865	0.0280272
$914$	0	0
$915$	15.9089	0.525933
$916$	0	0
$917$	3.46151	0.114309
$918$	0	0
$919$	−17.4573	−0.575864	−0.287932	−	0.957651i	$-0.592968\pi$
$919$	−17.4573	−0.575864	−0.287932	+	0.957651i	$0.592968\pi$
$920$	0	0
$921$	−97.5255	−3.21357
$922$	0	0
$923$	8.17858	0.269201
$924$	0	0
$925$	−19.6130	−0.644872
$926$	0	0
$927$	55.0976	1.80964
$928$	0	0
$929$	11.4035	0.374138	0.187069	−	0.982347i	$-0.440101\pi$
$929$	11.4035	0.374138	0.187069	+	0.982347i	$0.440101\pi$
$930$	0	0
$931$	−5.32809	−0.174621
$932$	0	0
$933$	−31.1599	−1.02013
$934$	0	0
$935$	−4.31974	−0.141271
$936$	0	0
$937$	−0.230149	−0.00751865	−0.00375933	−	0.999993i	$-0.501197\pi$
$937$	−0.230149	−0.00751865	−0.00375933	+	0.999993i	$0.501197\pi$
$938$	0	0
$939$	−36.7761	−1.20014
$940$	0	0
$941$	−49.3730	−1.60951	−0.804756	−	0.593605i	$-0.797704\pi$
$941$	−49.3730	−1.60951	−0.804756	+	0.593605i	$0.797704\pi$
$942$	0	0
$943$	−52.1521	−1.69831
$944$	0	0
$945$	−13.0310	−0.423899
$946$	0	0
$947$	39.8578	1.29520	0.647602	−	0.761979i	$-0.275772\pi$
$947$	39.8578	1.29520	0.647602	+	0.761979i	$0.275772\pi$
$948$	0	0
$949$	23.2982	0.756292
$950$	0	0
$951$	−52.8258	−1.71299
$952$	0	0
$953$	2.83247	0.0917526	0.0458763	−	0.998947i	$-0.485392\pi$
$953$	2.83247	0.0917526	0.0458763	+	0.998947i	$0.485392\pi$
$954$	0	0
$955$	−38.9143	−1.25924
$956$	0	0
$957$	−7.92445	−0.256161
$958$	0	0
$959$	−13.6033	−0.439272
$960$	0	0
$961$	54.9053	1.77114
$962$	0	0
$963$	39.5099	1.27319
$964$	0	0
$965$	−13.4042	−0.431497
$966$	0	0
$967$	47.6102	1.53104	0.765521	−	0.643411i	$-0.222481\pi$
$967$	47.6102	1.53104	0.765521	+	0.643411i	$0.222481\pi$
$968$	0	0
$969$	12.2617	0.393901
$970$	0	0
$971$	−30.8876	−0.991229	−0.495615	−	0.868542i	$-0.665057\pi$
$971$	−30.8876	−0.991229	−0.495615	+	0.868542i	$0.665057\pi$
$972$	0	0
$973$	−22.5823	−0.723957
$974$	0	0
$975$	−22.6977	−0.726909
$976$	0	0
$977$	40.3458	1.29078	0.645389	−	0.763854i	$-0.276695\pi$
$977$	40.3458	1.29078	0.645389	+	0.763854i	$0.276695\pi$
$978$	0	0
$979$	−3.90685	−0.124863
$980$	0	0
$981$	−45.9494	−1.46705
$982$	0	0
$983$	−2.97852	−0.0950000	−0.0475000	−	0.998871i	$-0.515125\pi$
$983$	−2.97852	−0.0950000	−0.0475000	+	0.998871i	$0.515125\pi$
$984$	0	0
$985$	−23.1144	−0.736485
$986$	0	0
$987$	−44.9771	−1.43164
$988$	0	0
$989$	−59.3285	−1.88654
$990$	0	0
$991$	−39.4686	−1.25376	−0.626880	−	0.779116i	$-0.715669\pi$
$991$	−39.4686	−1.25376	−0.626880	+	0.779116i	$0.715669\pi$
$992$	0	0
$993$	−47.3974	−1.50411
$994$	0	0
$995$	−3.59294	−0.113904
$996$	0	0
$997$	−28.2316	−0.894105	−0.447052	−	0.894508i	$-0.647526\pi$
$997$	−28.2316	−0.894105	−0.447052	+	0.894508i	$0.647526\pi$
$998$	0	0
$999$	−48.3530	−1.52982

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.e.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-2.88114 q^{3} -1.52019 q^{5} -1.29303 q^{7} +5.30096 q^{9} +O(q^{10})\)	\(q-2.88114 q^{3} -1.52019 q^{5} -1.29303 q^{7} +5.30096 q^{9} -0.667690 q^{11} -2.92971 q^{13} +4.37988 q^{15} -4.25584 q^{17} +1.00000 q^{19} +3.72538 q^{21} +7.86708 q^{23} -2.68902 q^{25} -6.62938 q^{27} -4.11936 q^{29} -9.26851 q^{31} +1.92371 q^{33} +1.96565 q^{35} +7.29374 q^{37} +8.44090 q^{39} -6.62916 q^{41} -7.54137 q^{43} -8.05847 q^{45} -12.0732 q^{47} -5.32809 q^{49} +12.2617 q^{51} +1.00000 q^{53} +1.01502 q^{55} -2.88114 q^{57} -4.86350 q^{59} +3.63228 q^{61} -6.85427 q^{63} +4.45372 q^{65} +13.1427 q^{67} -22.6661 q^{69} -2.79160 q^{71} -7.95239 q^{73} +7.74743 q^{75} +0.863340 q^{77} -2.75055 q^{79} +3.19728 q^{81} -1.26835 q^{83} +6.46969 q^{85} +11.8685 q^{87} +5.85129 q^{89} +3.78819 q^{91} +26.7039 q^{93} -1.52019 q^{95} -17.1415 q^{97} -3.53939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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