LMFDB - Embedded newform 4028.2.a.d.1.2

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:	$1$
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} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 $ x^19 - 4x^18 - 30x^17 + 132x^16 + 332x^15 - 1714x^14 - 1598x^13 + 11179x^12 + 2544x^11 - 38897x^10 + 3416x^9 + 71354x^8 - 10941x^7 - 64854x^6 + 219x^5 + 26515x^4 + 5796x^3 - 2375x^2 - 826x - 49
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			1.2
Root			$2.97522$ 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)

$f(q)$	$=$	$q-2.97522 q^{3} +0.840694 q^{5} -4.22061 q^{7} +5.85193 q^{9} +O(q^{10})$	$q-2.97522 q^{3} +0.840694 q^{5} -4.22061 q^{7} +5.85193 q^{9} +3.71552 q^{11} -1.07012 q^{13} -2.50125 q^{15} +1.37513 q^{17} -1.00000 q^{19} +12.5573 q^{21} -4.51456 q^{23} -4.29323 q^{25} -8.48511 q^{27} +0.356792 q^{29} -4.24901 q^{31} -11.0545 q^{33} -3.54825 q^{35} +4.55793 q^{37} +3.18386 q^{39} +10.9742 q^{41} +5.17832 q^{43} +4.91968 q^{45} -4.68548 q^{47} +10.8136 q^{49} -4.09133 q^{51} +1.00000 q^{53} +3.12361 q^{55} +2.97522 q^{57} +7.35327 q^{59} +6.24655 q^{61} -24.6987 q^{63} -0.899648 q^{65} +3.54664 q^{67} +13.4318 q^{69} -5.04982 q^{71} +5.95668 q^{73} +12.7733 q^{75} -15.6818 q^{77} +11.2077 q^{79} +7.68928 q^{81} -3.23503 q^{83} +1.15607 q^{85} -1.06153 q^{87} -0.365040 q^{89} +4.51658 q^{91} +12.6417 q^{93} -0.840694 q^{95} -4.23679 q^{97} +21.7429 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + 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
$3$	−2.97522	−1.71774	−0.858872	−	0.512191i	$-0.828834\pi$
$3$	−2.97522	−1.71774	−0.858872	+	0.512191i	$0.828834\pi$
$4$	0	0
$5$	0.840694	0.375970	0.187985	−	0.982172i	$-0.439804\pi$
$5$	0.840694	0.375970	0.187985	+	0.982172i	$0.439804\pi$
$6$	0	0
$7$	−4.22061	−1.59524	−0.797621	−	0.603159i	$-0.793908\pi$
$7$	−4.22061	−1.59524	−0.797621	+	0.603159i	$0.793908\pi$
$8$	0	0
$9$	5.85193	1.95064
$10$	0	0
$11$	3.71552	1.12027	0.560135	−	0.828401i	$-0.310749\pi$
$11$	3.71552	1.12027	0.560135	+	0.828401i	$0.310749\pi$
$12$	0	0
$13$	−1.07012	−0.296799	−0.148400	−	0.988927i	$-0.547412\pi$
$13$	−1.07012	−0.296799	−0.148400	+	0.988927i	$0.547412\pi$
$14$	0	0
$15$	−2.50125	−0.645820
$16$	0	0
$17$	1.37513	0.333519	0.166760	−	0.985998i	$-0.446670\pi$
$17$	1.37513	0.333519	0.166760	+	0.985998i	$0.446670\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	12.5573	2.74022
$22$	0	0
$23$	−4.51456	−0.941351	−0.470676	−	0.882306i	$-0.655990\pi$
$23$	−4.51456	−0.941351	−0.470676	+	0.882306i	$0.655990\pi$
$24$	0	0
$25$	−4.29323	−0.858647
$26$	0	0
$27$	−8.48511	−1.63296
$28$	0	0
$29$	0.356792	0.0662546	0.0331273	−	0.999451i	$-0.489453\pi$
$29$	0.356792	0.0662546	0.0331273	+	0.999451i	$0.489453\pi$
$30$	0	0
$31$	−4.24901	−0.763144	−0.381572	−	0.924339i	$-0.624617\pi$
$31$	−4.24901	−0.763144	−0.381572	+	0.924339i	$0.624617\pi$
$32$	0	0
$33$	−11.0545	−1.92434
$34$	0	0
$35$	−3.54825	−0.599763
$36$	0	0
$37$	4.55793	0.749319	0.374659	−	0.927163i	$-0.377760\pi$
$37$	4.55793	0.749319	0.374659	+	0.927163i	$0.377760\pi$
$38$	0	0
$39$	3.18386	0.509825
$40$	0	0
$41$	10.9742	1.71388	0.856942	−	0.515412i	$-0.172361\pi$
$41$	10.9742	1.71388	0.856942	+	0.515412i	$0.172361\pi$
$42$	0	0
$43$	5.17832	0.789686	0.394843	−	0.918749i	$-0.370799\pi$
$43$	5.17832	0.789686	0.394843	+	0.918749i	$0.370799\pi$
$44$	0	0
$45$	4.91968	0.733383
$46$	0	0
$47$	−4.68548	−0.683448	−0.341724	−	0.939800i	$-0.611011\pi$
$47$	−4.68548	−0.683448	−0.341724	+	0.939800i	$0.611011\pi$
$48$	0	0
$49$	10.8136	1.54480
$50$	0	0
$51$	−4.09133	−0.572900
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	3.12361	0.421188
$56$	0	0
$57$	2.97522	0.394077
$58$	0	0
$59$	7.35327	0.957315	0.478657	−	0.878002i	$-0.341124\pi$
$59$	7.35327	0.957315	0.478657	+	0.878002i	$0.341124\pi$
$60$	0	0
$61$	6.24655	0.799789	0.399895	−	0.916561i	$-0.369047\pi$
$61$	6.24655	0.799789	0.399895	+	0.916561i	$0.369047\pi$
$62$	0	0
$63$	−24.6987	−3.11175
$64$	0	0
$65$	−0.899648	−0.111588
$66$	0	0
$67$	3.54664	0.433291	0.216646	−	0.976250i	$-0.430488\pi$
$67$	3.54664	0.433291	0.216646	+	0.976250i	$0.430488\pi$
$68$	0	0
$69$	13.4318	1.61700
$70$	0	0
$71$	−5.04982	−0.599303	−0.299652	−	0.954049i	$-0.596870\pi$
$71$	−5.04982	−0.599303	−0.299652	+	0.954049i	$0.596870\pi$
$72$	0	0
$73$	5.95668	0.697177	0.348589	−	0.937276i	$-0.386661\pi$
$73$	5.95668	0.697177	0.348589	+	0.937276i	$0.386661\pi$
$74$	0	0
$75$	12.7733	1.47493
$76$	0	0
$77$	−15.6818	−1.78710
$78$	0	0
$79$	11.2077	1.26096	0.630482	−	0.776204i	$-0.282857\pi$
$79$	11.2077	1.26096	0.630482	+	0.776204i	$0.282857\pi$
$80$	0	0
$81$	7.68928	0.854364
$82$	0	0
$83$	−3.23503	−0.355091	−0.177545	−	0.984113i	$-0.556816\pi$
$83$	−3.23503	−0.355091	−0.177545	+	0.984113i	$0.556816\pi$
$84$	0	0
$85$	1.15607	0.125393
$86$	0	0
$87$	−1.06153	−0.113808
$88$	0	0
$89$	−0.365040	−0.0386942	−0.0193471	−	0.999813i	$-0.506159\pi$
$89$	−0.365040	−0.0386942	−0.0193471	+	0.999813i	$0.506159\pi$
$90$	0	0
$91$	4.51658	0.473467
$92$	0	0
$93$	12.6417	1.31089
$94$	0	0
$95$	−0.840694	−0.0862534
$96$	0	0
$97$	−4.23679	−0.430181	−0.215090	−	0.976594i	$-0.569005\pi$
$97$	−4.23679	−0.430181	−0.215090	+	0.976594i	$0.569005\pi$
$98$	0	0
$99$	21.7429	2.18525
$100$	0	0
$101$	−5.24757	−0.522153	−0.261076	−	0.965318i	$-0.584077\pi$
$101$	−5.24757	−0.522153	−0.261076	+	0.965318i	$0.584077\pi$
$102$	0	0
$103$	11.5467	1.13773	0.568867	−	0.822429i	$-0.307382\pi$
$103$	11.5467	1.13773	0.568867	+	0.822429i	$0.307382\pi$
$104$	0	0
$105$	10.5568	1.03024
$106$	0	0
$107$	−11.9642	−1.15662	−0.578310	−	0.815817i	$-0.696288\pi$
$107$	−11.9642	−1.15662	−0.578310	+	0.815817i	$0.696288\pi$
$108$	0	0
$109$	−9.53419	−0.913209	−0.456605	−	0.889670i	$-0.650935\pi$
$109$	−9.53419	−0.913209	−0.456605	+	0.889670i	$0.650935\pi$
$110$	0	0
$111$	−13.5608	−1.28714
$112$	0	0
$113$	−7.29833	−0.686569	−0.343285	−	0.939231i	$-0.611540\pi$
$113$	−7.29833	−0.686569	−0.343285	+	0.939231i	$0.611540\pi$
$114$	0	0
$115$	−3.79537	−0.353920
$116$	0	0
$117$	−6.26229	−0.578949
$118$	0	0
$119$	−5.80391	−0.532044
$120$	0	0
$121$	2.80506	0.255006
$122$	0	0
$123$	−32.6507	−2.94401
$124$	0	0
$125$	−7.81277	−0.698795
$126$	0	0
$127$	−17.5407	−1.55649	−0.778245	−	0.627961i	$-0.783890\pi$
$127$	−17.5407	−1.55649	−0.778245	+	0.627961i	$0.783890\pi$
$128$	0	0
$129$	−15.4066	−1.35648
$130$	0	0
$131$	−1.77183	−0.154806	−0.0774028	−	0.997000i	$-0.524663\pi$
$131$	−1.77183	−0.154806	−0.0774028	+	0.997000i	$0.524663\pi$
$132$	0	0
$133$	4.22061	0.365974
$134$	0	0
$135$	−7.13338	−0.613944
$136$	0	0
$137$	−8.58520	−0.733483	−0.366741	−	0.930323i	$-0.619527\pi$
$137$	−8.58520	−0.733483	−0.366741	+	0.930323i	$0.619527\pi$
$138$	0	0
$139$	−3.52199	−0.298731	−0.149366	−	0.988782i	$-0.547723\pi$
$139$	−3.52199	−0.298731	−0.149366	+	0.988782i	$0.547723\pi$
$140$	0	0
$141$	13.9403	1.17399
$142$	0	0
$143$	−3.97607	−0.332495
$144$	0	0
$145$	0.299953	0.0249097
$146$	0	0
$147$	−32.1728	−2.65357
$148$	0	0
$149$	5.33047	0.436689	0.218345	−	0.975872i	$-0.429934\pi$
$149$	5.33047	0.436689	0.218345	+	0.975872i	$0.429934\pi$
$150$	0	0
$151$	−0.0625813	−0.00509280	−0.00254640	−	0.999997i	$-0.500811\pi$
$151$	−0.0625813	−0.00509280	−0.00254640	+	0.999997i	$0.500811\pi$
$152$	0	0
$153$	8.04719	0.650577
$154$	0	0
$155$	−3.57211	−0.286919
$156$	0	0
$157$	14.1905	1.13253	0.566263	−	0.824225i	$-0.308389\pi$
$157$	14.1905	1.13253	0.566263	+	0.824225i	$0.308389\pi$
$158$	0	0
$159$	−2.97522	−0.235950
$160$	0	0
$161$	19.0542	1.50168
$162$	0	0
$163$	0.0663669	0.00519825	0.00259913	−	0.999997i	$-0.499173\pi$
$163$	0.0663669	0.00519825	0.00259913	+	0.999997i	$0.499173\pi$
$164$	0	0
$165$	−9.29343	−0.723493
$166$	0	0
$167$	5.05248	0.390973	0.195486	−	0.980706i	$-0.437371\pi$
$167$	5.05248	0.390973	0.195486	+	0.980706i	$0.437371\pi$
$168$	0	0
$169$	−11.8548	−0.911910
$170$	0	0
$171$	−5.85193	−0.447508
$172$	0	0
$173$	−13.7174	−1.04292	−0.521458	−	0.853277i	$-0.674611\pi$
$173$	−13.7174	−1.04292	−0.521458	+	0.853277i	$0.674611\pi$
$174$	0	0
$175$	18.1201	1.36975
$176$	0	0
$177$	−21.8776	−1.64442
$178$	0	0
$179$	1.86867	0.139671	0.0698355	−	0.997559i	$-0.477753\pi$
$179$	1.86867	0.139671	0.0698355	+	0.997559i	$0.477753\pi$
$180$	0	0
$181$	12.3639	0.919001	0.459500	−	0.888178i	$-0.348028\pi$
$181$	12.3639	0.919001	0.459500	+	0.888178i	$0.348028\pi$
$182$	0	0
$183$	−18.5849	−1.37383
$184$	0	0
$185$	3.83182	0.281721
$186$	0	0
$187$	5.10934	0.373632
$188$	0	0
$189$	35.8124	2.60497
$190$	0	0
$191$	−14.8145	−1.07194	−0.535971	−	0.844236i	$-0.680054\pi$
$191$	−14.8145	−1.07194	−0.535971	+	0.844236i	$0.680054\pi$
$192$	0	0
$193$	7.96136	0.573071	0.286536	−	0.958070i	$-0.407496\pi$
$193$	7.96136	0.573071	0.286536	+	0.958070i	$0.407496\pi$
$194$	0	0
$195$	2.67665	0.191679
$196$	0	0
$197$	−9.57542	−0.682220	−0.341110	−	0.940023i	$-0.610803\pi$
$197$	−9.57542	−0.682220	−0.341110	+	0.940023i	$0.610803\pi$
$198$	0	0
$199$	−0.699774	−0.0496056	−0.0248028	−	0.999692i	$-0.507896\pi$
$199$	−0.699774	−0.0496056	−0.0248028	+	0.999692i	$0.507896\pi$
$200$	0	0
$201$	−10.5520	−0.744284
$202$	0	0
$203$	−1.50588	−0.105692
$204$	0	0
$205$	9.22596	0.644369
$206$	0	0
$207$	−26.4189	−1.83624
$208$	0	0
$209$	−3.71552	−0.257008
$210$	0	0
$211$	−15.8027	−1.08790	−0.543950	−	0.839118i	$-0.683072\pi$
$211$	−15.8027	−1.08790	−0.543950	+	0.839118i	$0.683072\pi$
$212$	0	0
$213$	15.0243	1.02945
$214$	0	0
$215$	4.35338	0.296898
$216$	0	0
$217$	17.9334	1.21740
$218$	0	0
$219$	−17.7224	−1.19757
$220$	0	0
$221$	−1.47157	−0.0989882
$222$	0	0
$223$	−2.10067	−0.140671	−0.0703356	−	0.997523i	$-0.522407\pi$
$223$	−2.10067	−0.140671	−0.0703356	+	0.997523i	$0.522407\pi$
$224$	0	0
$225$	−25.1237	−1.67491
$226$	0	0
$227$	−2.78860	−0.185086	−0.0925431	−	0.995709i	$-0.529500\pi$
$227$	−2.78860	−0.185086	−0.0925431	+	0.995709i	$0.529500\pi$
$228$	0	0
$229$	−21.3824	−1.41299	−0.706495	−	0.707718i	$-0.749725\pi$
$229$	−21.3824	−1.41299	−0.706495	+	0.707718i	$0.749725\pi$
$230$	0	0
$231$	46.6567	3.06978
$232$	0	0
$233$	−15.8652	−1.03936	−0.519681	−	0.854361i	$-0.673949\pi$
$233$	−15.8652	−1.03936	−0.519681	+	0.854361i	$0.673949\pi$
$234$	0	0
$235$	−3.93906	−0.256956
$236$	0	0
$237$	−33.3453	−2.16601
$238$	0	0
$239$	3.04028	0.196659	0.0983295	−	0.995154i	$-0.468650\pi$
$239$	3.04028	0.196659	0.0983295	+	0.995154i	$0.468650\pi$
$240$	0	0
$241$	1.15242	0.0742338	0.0371169	−	0.999311i	$-0.488183\pi$
$241$	1.15242	0.0742338	0.0371169	+	0.999311i	$0.488183\pi$
$242$	0	0
$243$	2.57804	0.165382
$244$	0	0
$245$	9.09092	0.580797
$246$	0	0
$247$	1.07012	0.0680904
$248$	0	0
$249$	9.62493	0.609955
$250$	0	0
$251$	8.88283	0.560679	0.280340	−	0.959901i	$-0.409553\pi$
$251$	8.88283	0.560679	0.280340	+	0.959901i	$0.409553\pi$
$252$	0	0
$253$	−16.7739	−1.05457
$254$	0	0
$255$	−3.43955	−0.215393
$256$	0	0
$257$	2.86797	0.178899	0.0894496	−	0.995991i	$-0.471489\pi$
$257$	2.86797	0.178899	0.0894496	+	0.995991i	$0.471489\pi$
$258$	0	0
$259$	−19.2373	−1.19535
$260$	0	0
$261$	2.08792	0.129239
$262$	0	0
$263$	−11.0299	−0.680131	−0.340065	−	0.940402i	$-0.610449\pi$
$263$	−11.0299	−0.680131	−0.340065	+	0.940402i	$0.610449\pi$
$264$	0	0
$265$	0.840694	0.0516434
$266$	0	0
$267$	1.08607	0.0664666
$268$	0	0
$269$	11.3740	0.693488	0.346744	−	0.937960i	$-0.387287\pi$
$269$	11.3740	0.693488	0.346744	+	0.937960i	$0.387287\pi$
$270$	0	0
$271$	−29.6727	−1.80249	−0.901244	−	0.433311i	$-0.857345\pi$
$271$	−29.6727	−1.80249	−0.901244	+	0.433311i	$0.857345\pi$
$272$	0	0
$273$	−13.4378	−0.813294
$274$	0	0
$275$	−15.9516	−0.961916
$276$	0	0
$277$	−30.2485	−1.81745	−0.908727	−	0.417390i	$-0.862945\pi$
$277$	−30.2485	−1.81745	−0.908727	+	0.417390i	$0.862945\pi$
$278$	0	0
$279$	−24.8649	−1.48862
$280$	0	0
$281$	28.5911	1.70560	0.852802	−	0.522234i	$-0.174901\pi$
$281$	28.5911	1.70560	0.852802	+	0.522234i	$0.174901\pi$
$282$	0	0
$283$	−15.2731	−0.907894	−0.453947	−	0.891029i	$-0.649984\pi$
$283$	−15.2731	−0.907894	−0.453947	+	0.891029i	$0.649984\pi$
$284$	0	0
$285$	2.50125	0.148161
$286$	0	0
$287$	−46.3179	−2.73406
$288$	0	0
$289$	−15.1090	−0.888765
$290$	0	0
$291$	12.6054	0.738940
$292$	0	0
$293$	−25.0308	−1.46232	−0.731158	−	0.682208i	$-0.761020\pi$
$293$	−25.0308	−1.46232	−0.731158	+	0.682208i	$0.761020\pi$
$294$	0	0
$295$	6.18185	0.359921
$296$	0	0
$297$	−31.5266	−1.82936
$298$	0	0
$299$	4.83114	0.279392
$300$	0	0
$301$	−21.8557	−1.25974
$302$	0	0
$303$	15.6127	0.896925
$304$	0	0
$305$	5.25144	0.300697
$306$	0	0
$307$	22.0251	1.25704	0.628521	−	0.777793i	$-0.283661\pi$
$307$	22.0251	1.25704	0.628521	+	0.777793i	$0.283661\pi$
$308$	0	0
$309$	−34.3541	−1.95434
$310$	0	0
$311$	4.76915	0.270434	0.135217	−	0.990816i	$-0.456827\pi$
$311$	4.76915	0.270434	0.135217	+	0.990816i	$0.456827\pi$
$312$	0	0
$313$	24.7241	1.39749	0.698745	−	0.715371i	$-0.253742\pi$
$313$	24.7241	1.39749	0.698745	+	0.715371i	$0.253742\pi$
$314$	0	0
$315$	−20.7641	−1.16992
$316$	0	0
$317$	−7.67668	−0.431165	−0.215583	−	0.976486i	$-0.569165\pi$
$317$	−7.67668	−0.431165	−0.215583	+	0.976486i	$0.569165\pi$
$318$	0	0
$319$	1.32567	0.0742230
$320$	0	0
$321$	35.5960	1.98678
$322$	0	0
$323$	−1.37513	−0.0765145
$324$	0	0
$325$	4.59429	0.254846
$326$	0	0
$327$	28.3663	1.56866
$328$	0	0
$329$	19.7756	1.09027
$330$	0	0
$331$	−30.0635	−1.65244	−0.826219	−	0.563349i	$-0.809513\pi$
$331$	−30.0635	−1.65244	−0.826219	+	0.563349i	$0.809513\pi$
$332$	0	0
$333$	26.6727	1.46165
$334$	0	0
$335$	2.98164	0.162905
$336$	0	0
$337$	2.64397	0.144026	0.0720130	−	0.997404i	$-0.477058\pi$
$337$	2.64397	0.144026	0.0720130	+	0.997404i	$0.477058\pi$
$338$	0	0
$339$	21.7141	1.17935
$340$	0	0
$341$	−15.7872	−0.854928
$342$	0	0
$343$	−16.0957	−0.869084
$344$	0	0
$345$	11.2920	0.607943
$346$	0	0
$347$	−23.9516	−1.28579	−0.642896	−	0.765954i	$-0.722267\pi$
$347$	−23.9516	−1.28579	−0.642896	+	0.765954i	$0.722267\pi$
$348$	0	0
$349$	−3.59212	−0.192282	−0.0961409	−	0.995368i	$-0.530650\pi$
$349$	−3.59212	−0.192282	−0.0961409	+	0.995368i	$0.530650\pi$
$350$	0	0
$351$	9.08013	0.484661
$352$	0	0
$353$	10.7761	0.573554	0.286777	−	0.957997i	$-0.407416\pi$
$353$	10.7761	0.573554	0.286777	+	0.957997i	$0.407416\pi$
$354$	0	0
$355$	−4.24535	−0.225320
$356$	0	0
$357$	17.2679	0.913915
$358$	0	0
$359$	−23.7132	−1.25153	−0.625767	−	0.780010i	$-0.715214\pi$
$359$	−23.7132	−1.25153	−0.625767	+	0.780010i	$0.715214\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−8.34567	−0.438034
$364$	0	0
$365$	5.00775	0.262118
$366$	0	0
$367$	−24.5017	−1.27898	−0.639489	−	0.768800i	$-0.720854\pi$
$367$	−24.5017	−1.27898	−0.639489	+	0.768800i	$0.720854\pi$
$368$	0	0
$369$	64.2203	3.34318
$370$	0	0
$371$	−4.22061	−0.219123
$372$	0	0
$373$	17.0247	0.881505	0.440752	−	0.897629i	$-0.354712\pi$
$373$	17.0247	0.881505	0.440752	+	0.897629i	$0.354712\pi$
$374$	0	0
$375$	23.2447	1.20035
$376$	0	0
$377$	−0.381812	−0.0196643
$378$	0	0
$379$	22.2487	1.14284	0.571418	−	0.820659i	$-0.306393\pi$
$379$	22.2487	1.14284	0.571418	+	0.820659i	$0.306393\pi$
$380$	0	0
$381$	52.1876	2.67365
$382$	0	0
$383$	−15.2777	−0.780654	−0.390327	−	0.920676i	$-0.627638\pi$
$383$	−15.2777	−0.780654	−0.390327	+	0.920676i	$0.627638\pi$
$384$	0	0
$385$	−13.1836	−0.671897
$386$	0	0
$387$	30.3032	1.54040
$388$	0	0
$389$	−27.2173	−1.37997	−0.689987	−	0.723822i	$-0.742384\pi$
$389$	−27.2173	−1.37997	−0.689987	+	0.723822i	$0.742384\pi$
$390$	0	0
$391$	−6.20813	−0.313959
$392$	0	0
$393$	5.27159	0.265916
$394$	0	0
$395$	9.42224	0.474084
$396$	0	0
$397$	−14.1764	−0.711492	−0.355746	−	0.934583i	$-0.615773\pi$
$397$	−14.1764	−0.711492	−0.355746	+	0.934583i	$0.615773\pi$
$398$	0	0
$399$	−12.5573	−0.628649
$400$	0	0
$401$	−4.72260	−0.235835	−0.117918	−	0.993023i	$-0.537622\pi$
$401$	−4.72260	−0.235835	−0.117918	+	0.993023i	$0.537622\pi$
$402$	0	0
$403$	4.54697	0.226500
$404$	0	0
$405$	6.46433	0.321215
$406$	0	0
$407$	16.9351	0.839440
$408$	0	0
$409$	11.3145	0.559465	0.279733	−	0.960078i	$-0.409754\pi$
$409$	11.3145	0.559465	0.279733	+	0.960078i	$0.409754\pi$
$410$	0	0
$411$	25.5428	1.25994
$412$	0	0
$413$	−31.0353	−1.52715
$414$	0	0
$415$	−2.71967	−0.133503
$416$	0	0
$417$	10.4787	0.513144
$418$	0	0
$419$	−5.08754	−0.248543	−0.124271	−	0.992248i	$-0.539659\pi$
$419$	−5.08754	−0.248543	−0.124271	+	0.992248i	$0.539659\pi$
$420$	0	0
$421$	16.1078	0.785048	0.392524	−	0.919742i	$-0.371602\pi$
$421$	16.1078	0.785048	0.392524	+	0.919742i	$0.371602\pi$
$422$	0	0
$423$	−27.4191	−1.33316
$424$	0	0
$425$	−5.90377	−0.286375
$426$	0	0
$427$	−26.3643	−1.27586
$428$	0	0
$429$	11.8297	0.571142
$430$	0	0
$431$	7.87058	0.379113	0.189556	−	0.981870i	$-0.439295\pi$
$431$	7.87058	0.379113	0.189556	+	0.981870i	$0.439295\pi$
$432$	0	0
$433$	7.98769	0.383864	0.191932	−	0.981408i	$-0.438525\pi$
$433$	7.98769	0.383864	0.191932	+	0.981408i	$0.438525\pi$
$434$	0	0
$435$	−0.892425	−0.0427885
$436$	0	0
$437$	4.51456	0.215961
$438$	0	0
$439$	−0.796487	−0.0380143	−0.0190071	−	0.999819i	$-0.506051\pi$
$439$	−0.796487	−0.0380143	−0.0190071	+	0.999819i	$0.506051\pi$
$440$	0	0
$441$	63.2803	3.01335
$442$	0	0
$443$	27.3640	1.30010	0.650050	−	0.759891i	$-0.274748\pi$
$443$	27.3640	1.30010	0.650050	+	0.759891i	$0.274748\pi$
$444$	0	0
$445$	−0.306887	−0.0145478
$446$	0	0
$447$	−15.8593	−0.750120
$448$	0	0
$449$	−30.3688	−1.43319	−0.716597	−	0.697488i	$-0.754301\pi$
$449$	−30.3688	−1.43319	−0.716597	+	0.697488i	$0.754301\pi$
$450$	0	0
$451$	40.7749	1.92001
$452$	0	0
$453$	0.186193	0.00874812
$454$	0	0
$455$	3.79706	0.178009
$456$	0	0
$457$	17.1263	0.801137	0.400568	−	0.916267i	$-0.368813\pi$
$457$	17.1263	0.801137	0.400568	+	0.916267i	$0.368813\pi$
$458$	0	0
$459$	−11.6682	−0.544624
$460$	0	0
$461$	−18.1700	−0.846260	−0.423130	−	0.906069i	$-0.639069\pi$
$461$	−18.1700	−0.846260	−0.423130	+	0.906069i	$0.639069\pi$
$462$	0	0
$463$	−24.8465	−1.15471	−0.577357	−	0.816492i	$-0.695916\pi$
$463$	−24.8465	−1.15471	−0.577357	+	0.816492i	$0.695916\pi$
$464$	0	0
$465$	10.6278	0.492853
$466$	0	0
$467$	17.0875	0.790714	0.395357	−	0.918528i	$-0.370621\pi$
$467$	17.0875	0.790714	0.395357	+	0.918528i	$0.370621\pi$
$468$	0	0
$469$	−14.9690	−0.691205
$470$	0	0
$471$	−42.2199	−1.94539
$472$	0	0
$473$	19.2401	0.884662
$474$	0	0
$475$	4.29323	0.196987
$476$	0	0
$477$	5.85193	0.267941
$478$	0	0
$479$	35.4945	1.62179	0.810893	−	0.585194i	$-0.198982\pi$
$479$	35.4945	1.62179	0.810893	+	0.585194i	$0.198982\pi$
$480$	0	0
$481$	−4.87755	−0.222397
$482$	0	0
$483$	−56.6905	−2.57951
$484$	0	0
$485$	−3.56184	−0.161735
$486$	0	0
$487$	15.5860	0.706269	0.353134	−	0.935573i	$-0.385116\pi$
$487$	15.5860	0.706269	0.353134	+	0.935573i	$0.385116\pi$
$488$	0	0
$489$	−0.197456	−0.00892927
$490$	0	0
$491$	−4.33672	−0.195713	−0.0978567	−	0.995201i	$-0.531199\pi$
$491$	−4.33672	−0.195713	−0.0978567	+	0.995201i	$0.531199\pi$
$492$	0	0
$493$	0.490637	0.0220972
$494$	0	0
$495$	18.2792	0.821587
$496$	0	0
$497$	21.3133	0.956034
$498$	0	0
$499$	−25.6908	−1.15008	−0.575038	−	0.818127i	$-0.695013\pi$
$499$	−25.6908	−1.15008	−0.575038	+	0.818127i	$0.695013\pi$
$500$	0	0
$501$	−15.0322	−0.671591
$502$	0	0
$503$	12.0218	0.536024	0.268012	−	0.963416i	$-0.413633\pi$
$503$	12.0218	0.536024	0.268012	+	0.963416i	$0.413633\pi$
$504$	0	0
$505$	−4.41160	−0.196314
$506$	0	0
$507$	35.2707	1.56643
$508$	0	0
$509$	−30.8346	−1.36672	−0.683361	−	0.730081i	$-0.739482\pi$
$509$	−30.8346	−1.36672	−0.683361	+	0.730081i	$0.739482\pi$
$510$	0	0
$511$	−25.1409	−1.11217
$512$	0	0
$513$	8.48511	0.374627
$514$	0	0
$515$	9.70728	0.427754
$516$	0	0
$517$	−17.4090	−0.765647
$518$	0	0
$519$	40.8123	1.79146
$520$	0	0
$521$	−32.0129	−1.40251	−0.701255	−	0.712910i	$-0.747377\pi$
$521$	−32.0129	−1.40251	−0.701255	+	0.712910i	$0.747377\pi$
$522$	0	0
$523$	13.8228	0.604431	0.302215	−	0.953240i	$-0.402274\pi$
$523$	13.8228	0.604431	0.302215	+	0.953240i	$0.402274\pi$
$524$	0	0
$525$	−53.9112	−2.35288
$526$	0	0
$527$	−5.84296	−0.254523
$528$	0	0
$529$	−2.61874	−0.113858
$530$	0	0
$531$	43.0308	1.86738
$532$	0	0
$533$	−11.7438	−0.508680
$534$	0	0
$535$	−10.0582	−0.434854
$536$	0	0
$537$	−5.55971	−0.239919
$538$	0	0
$539$	40.1780	1.73059
$540$	0	0
$541$	8.13116	0.349586	0.174793	−	0.984605i	$-0.444074\pi$
$541$	8.13116	0.349586	0.174793	+	0.984605i	$0.444074\pi$
$542$	0	0
$543$	−36.7853	−1.57861
$544$	0	0
$545$	−8.01534	−0.343339
$546$	0	0
$547$	−35.8326	−1.53209	−0.766046	−	0.642786i	$-0.777779\pi$
$547$	−35.8326	−1.53209	−0.766046	+	0.642786i	$0.777779\pi$
$548$	0	0
$549$	36.5544	1.56010
$550$	0	0
$551$	−0.356792	−0.0151998
$552$	0	0
$553$	−47.3033	−2.01154
$554$	0	0
$555$	−11.4005	−0.483925
$556$	0	0
$557$	−20.1896	−0.855461	−0.427730	−	0.903906i	$-0.640687\pi$
$557$	−20.1896	−0.855461	−0.427730	+	0.903906i	$0.640687\pi$
$558$	0	0
$559$	−5.54145	−0.234378
$560$	0	0
$561$	−15.2014	−0.641803
$562$	0	0
$563$	−35.0089	−1.47545	−0.737725	−	0.675102i	$-0.764100\pi$
$563$	−35.0089	−1.47545	−0.737725	+	0.675102i	$0.764100\pi$
$564$	0	0
$565$	−6.13566	−0.258129
$566$	0	0
$567$	−32.4535	−1.36292
$568$	0	0
$569$	19.6655	0.824422	0.412211	−	0.911088i	$-0.364757\pi$
$569$	19.6655	0.824422	0.412211	+	0.911088i	$0.364757\pi$
$570$	0	0
$571$	0.832125	0.0348234	0.0174117	−	0.999848i	$-0.494457\pi$
$571$	0.832125	0.0348234	0.0174117	+	0.999848i	$0.494457\pi$
$572$	0	0
$573$	44.0765	1.84132
$574$	0	0
$575$	19.3821	0.808288
$576$	0	0
$577$	−14.0792	−0.586127	−0.293063	−	0.956093i	$-0.594675\pi$
$577$	−14.0792	−0.586127	−0.293063	+	0.956093i	$0.594675\pi$
$578$	0	0
$579$	−23.6868	−0.984389
$580$	0	0
$581$	13.6538	0.566456
$582$	0	0
$583$	3.71552	0.153881
$584$	0	0
$585$	−5.26467	−0.217667
$586$	0	0
$587$	−13.5544	−0.559451	−0.279725	−	0.960080i	$-0.590243\pi$
$587$	−13.5544	−0.559451	−0.279725	+	0.960080i	$0.590243\pi$
$588$	0	0
$589$	4.24901	0.175077
$590$	0	0
$591$	28.4890	1.17188
$592$	0	0
$593$	−9.31694	−0.382601	−0.191301	−	0.981532i	$-0.561271\pi$
$593$	−9.31694	−0.382601	−0.191301	+	0.981532i	$0.561271\pi$
$594$	0	0
$595$	−4.87932	−0.200032
$596$	0	0
$597$	2.08198	0.0852098
$598$	0	0
$599$	26.2330	1.07185	0.535926	−	0.844265i	$-0.319963\pi$
$599$	26.2330	1.07185	0.535926	+	0.844265i	$0.319963\pi$
$600$	0	0
$601$	−21.0201	−0.857427	−0.428713	−	0.903441i	$-0.641033\pi$
$601$	−21.0201	−0.857427	−0.428713	+	0.903441i	$0.641033\pi$
$602$	0	0
$603$	20.7547	0.845197
$604$	0	0
$605$	2.35820	0.0958744
$606$	0	0
$607$	5.61623	0.227956	0.113978	−	0.993483i	$-0.463641\pi$
$607$	5.61623	0.227956	0.113978	+	0.993483i	$0.463641\pi$
$608$	0	0
$609$	4.48032	0.181552
$610$	0	0
$611$	5.01405	0.202847
$612$	0	0
$613$	−48.9131	−1.97558	−0.987790	−	0.155790i	$-0.950208\pi$
$613$	−48.9131	−1.97558	−0.987790	+	0.155790i	$0.950208\pi$
$614$	0	0
$615$	−27.4493	−1.10686
$616$	0	0
$617$	30.2094	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$617$	30.2094	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$618$	0	0
$619$	−9.72482	−0.390874	−0.195437	−	0.980716i	$-0.562612\pi$
$619$	−9.72482	−0.390874	−0.195437	+	0.980716i	$0.562612\pi$
$620$	0	0
$621$	38.3066	1.53719
$622$	0	0
$623$	1.54069	0.0617266
$624$	0	0
$625$	14.8980	0.595921
$626$	0	0
$627$	11.0545	0.441473
$628$	0	0
$629$	6.26777	0.249912
$630$	0	0
$631$	41.1987	1.64009	0.820047	−	0.572296i	$-0.193947\pi$
$631$	41.1987	1.64009	0.820047	+	0.572296i	$0.193947\pi$
$632$	0	0
$633$	47.0164	1.86873
$634$	0	0
$635$	−14.7464	−0.585193
$636$	0	0
$637$	−11.5719	−0.458495
$638$	0	0
$639$	−29.5512	−1.16903
$640$	0	0
$641$	15.0491	0.594403	0.297201	−	0.954815i	$-0.403947\pi$
$641$	15.0491	0.594403	0.297201	+	0.954815i	$0.403947\pi$
$642$	0	0
$643$	20.7314	0.817567	0.408783	−	0.912631i	$-0.365953\pi$
$643$	20.7314	0.817567	0.408783	+	0.912631i	$0.365953\pi$
$644$	0	0
$645$	−12.9523	−0.509995
$646$	0	0
$647$	−4.60478	−0.181033	−0.0905163	−	0.995895i	$-0.528852\pi$
$647$	−4.60478	−0.181033	−0.0905163	+	0.995895i	$0.528852\pi$
$648$	0	0
$649$	27.3212	1.07245
$650$	0	0
$651$	−53.3558	−2.09118
$652$	0	0
$653$	21.4907	0.840994	0.420497	−	0.907294i	$-0.361856\pi$
$653$	21.4907	0.840994	0.420497	+	0.907294i	$0.361856\pi$
$654$	0	0
$655$	−1.48957	−0.0582023
$656$	0	0
$657$	34.8581	1.35994
$658$	0	0
$659$	42.4486	1.65356	0.826782	−	0.562523i	$-0.190169\pi$
$659$	42.4486	1.65356	0.826782	+	0.562523i	$0.190169\pi$
$660$	0	0
$661$	−23.9112	−0.930039	−0.465020	−	0.885300i	$-0.653953\pi$
$661$	−23.9112	−0.930039	−0.465020	+	0.885300i	$0.653953\pi$
$662$	0	0
$663$	4.37823	0.170036
$664$	0	0
$665$	3.54825	0.137595
$666$	0	0
$667$	−1.61076	−0.0623688
$668$	0	0
$669$	6.24995	0.241637
$670$	0	0
$671$	23.2092	0.895980
$672$	0	0
$673$	6.60761	0.254705	0.127352	−	0.991858i	$-0.459352\pi$
$673$	6.60761	0.254705	0.127352	+	0.991858i	$0.459352\pi$
$674$	0	0
$675$	36.4286	1.40214
$676$	0	0
$677$	−47.8559	−1.83925	−0.919626	−	0.392796i	$-0.871508\pi$
$677$	−47.8559	−1.83925	−0.919626	+	0.392796i	$0.871508\pi$
$678$	0	0
$679$	17.8818	0.686242
$680$	0	0
$681$	8.29671	0.317930
$682$	0	0
$683$	−1.39775	−0.0534835	−0.0267418	−	0.999642i	$-0.508513\pi$
$683$	−1.39775	−0.0534835	−0.0267418	+	0.999642i	$0.508513\pi$
$684$	0	0
$685$	−7.21752	−0.275767
$686$	0	0
$687$	63.6174	2.42716
$688$	0	0
$689$	−1.07012	−0.0407685
$690$	0	0
$691$	5.22951	0.198940	0.0994699	−	0.995041i	$-0.468285\pi$
$691$	5.22951	0.198940	0.0994699	+	0.995041i	$0.468285\pi$
$692$	0	0
$693$	−91.7685	−3.48600
$694$	0	0
$695$	−2.96092	−0.112314
$696$	0	0
$697$	15.0910	0.571613
$698$	0	0
$699$	47.2023	1.78536
$700$	0	0
$701$	−51.4662	−1.94385	−0.971925	−	0.235291i	$-0.924396\pi$
$701$	−51.4662	−1.94385	−0.971925	+	0.235291i	$0.924396\pi$
$702$	0	0
$703$	−4.55793	−0.171906
$704$	0	0
$705$	11.7196	0.441384
$706$	0	0
$707$	22.1480	0.832960
$708$	0	0
$709$	−17.9661	−0.674732	−0.337366	−	0.941374i	$-0.609536\pi$
$709$	−17.9661	−0.674732	−0.337366	+	0.941374i	$0.609536\pi$
$710$	0	0
$711$	65.5866	2.45969
$712$	0	0
$713$	19.1824	0.718386
$714$	0	0
$715$	−3.34265	−0.125008
$716$	0	0
$717$	−9.04548	−0.337810
$718$	0	0
$719$	−3.13026	−0.116739	−0.0583695	−	0.998295i	$-0.518590\pi$
$719$	−3.13026	−0.116739	−0.0583695	+	0.998295i	$0.518590\pi$
$720$	0	0
$721$	−48.7343	−1.81496
$722$	0	0
$723$	−3.42870	−0.127515
$724$	0	0
$725$	−1.53179	−0.0568893
$726$	0	0
$727$	6.94895	0.257722	0.128861	−	0.991663i	$-0.458868\pi$
$727$	6.94895	0.257722	0.128861	+	0.991663i	$0.458868\pi$
$728$	0	0
$729$	−30.7381	−1.13845
$730$	0	0
$731$	7.12089	0.263375
$732$	0	0
$733$	36.2491	1.33889	0.669446	−	0.742861i	$-0.266532\pi$
$733$	36.2491	1.33889	0.669446	+	0.742861i	$0.266532\pi$
$734$	0	0
$735$	−27.0475	−0.997661
$736$	0	0
$737$	13.1776	0.485404
$738$	0	0
$739$	0.968430	0.0356243	0.0178121	−	0.999841i	$-0.494330\pi$
$739$	0.968430	0.0356243	0.0178121	+	0.999841i	$0.494330\pi$
$740$	0	0
$741$	−3.18386	−0.116962
$742$	0	0
$743$	41.6326	1.52735	0.763677	−	0.645599i	$-0.223392\pi$
$743$	41.6326	1.52735	0.763677	+	0.645599i	$0.223392\pi$
$744$	0	0
$745$	4.48129	0.164182
$746$	0	0
$747$	−18.9312	−0.692655
$748$	0	0
$749$	50.4962	1.84509
$750$	0	0
$751$	27.9606	1.02030	0.510149	−	0.860086i	$-0.329590\pi$
$751$	27.9606	1.02030	0.510149	+	0.860086i	$0.329590\pi$
$752$	0	0
$753$	−26.4284	−0.963103
$754$	0	0
$755$	−0.0526118	−0.00191474
$756$	0	0
$757$	−8.32862	−0.302709	−0.151354	−	0.988480i	$-0.548363\pi$
$757$	−8.32862	−0.302709	−0.151354	+	0.988480i	$0.548363\pi$
$758$	0	0
$759$	49.9061	1.81148
$760$	0	0
$761$	1.42069	0.0514998	0.0257499	−	0.999668i	$-0.491803\pi$
$761$	1.42069	0.0514998	0.0257499	+	0.999668i	$0.491803\pi$
$762$	0	0
$763$	40.2401	1.45679
$764$	0	0
$765$	6.76523	0.244597
$766$	0	0
$767$	−7.86892	−0.284130
$768$	0	0
$769$	53.3760	1.92479	0.962394	−	0.271657i	$-0.0875715\pi$
$769$	53.3760	1.92479	0.962394	+	0.271657i	$0.0875715\pi$
$770$	0	0
$771$	−8.53285	−0.307303
$772$	0	0
$773$	24.5422	0.882722	0.441361	−	0.897330i	$-0.354496\pi$
$773$	24.5422	0.882722	0.441361	+	0.897330i	$0.354496\pi$
$774$	0	0
$775$	18.2420	0.655271
$776$	0	0
$777$	57.2351	2.05330
$778$	0	0
$779$	−10.9742	−0.393192
$780$	0	0
$781$	−18.7627	−0.671381
$782$	0	0
$783$	−3.02742	−0.108191
$784$	0	0
$785$	11.9299	0.425795
$786$	0	0
$787$	−53.6965	−1.91407	−0.957036	−	0.289969i	$-0.906355\pi$
$787$	−53.6965	−1.91407	−0.957036	+	0.289969i	$0.906355\pi$
$788$	0	0
$789$	32.8163	1.16829
$790$	0	0
$791$	30.8034	1.09524
$792$	0	0
$793$	−6.68459	−0.237377
$794$	0	0
$795$	−2.50125	−0.0887102
$796$	0	0
$797$	40.3147	1.42802	0.714010	−	0.700135i	$-0.246877\pi$
$797$	40.3147	1.42802	0.714010	+	0.700135i	$0.246877\pi$
$798$	0	0
$799$	−6.44317	−0.227943
$800$	0	0
$801$	−2.13619	−0.0754785
$802$	0	0
$803$	22.1322	0.781027
$804$	0	0
$805$	16.0188	0.564588
$806$	0	0
$807$	−33.8403	−1.19123
$808$	0	0
$809$	−11.8672	−0.417227	−0.208614	−	0.977998i	$-0.566895\pi$
$809$	−11.8672	−0.417227	−0.208614	+	0.977998i	$0.566895\pi$
$810$	0	0
$811$	−25.8087	−0.906266	−0.453133	−	0.891443i	$-0.649694\pi$
$811$	−25.8087	−0.906266	−0.453133	+	0.891443i	$0.649694\pi$
$812$	0	0
$813$	88.2828	3.09621
$814$	0	0
$815$	0.0557942	0.00195439
$816$	0	0
$817$	−5.17832	−0.181166
$818$	0	0
$819$	26.4307	0.923564
$820$	0	0
$821$	−36.8624	−1.28651	−0.643253	−	0.765654i	$-0.722416\pi$
$821$	−36.8624	−1.28651	−0.643253	+	0.765654i	$0.722416\pi$
$822$	0	0
$823$	−25.7764	−0.898509	−0.449255	−	0.893404i	$-0.648310\pi$
$823$	−25.7764	−0.898509	−0.449255	+	0.893404i	$0.648310\pi$
$824$	0	0
$825$	47.4594	1.65233
$826$	0	0
$827$	16.9229	0.588466	0.294233	−	0.955734i	$-0.404936\pi$
$827$	16.9229	0.588466	0.294233	+	0.955734i	$0.404936\pi$
$828$	0	0
$829$	27.5392	0.956475	0.478237	−	0.878231i	$-0.341276\pi$
$829$	27.5392	0.956475	0.478237	+	0.878231i	$0.341276\pi$
$830$	0	0
$831$	89.9958	3.12192
$832$	0	0
$833$	14.8701	0.515220
$834$	0	0
$835$	4.24759	0.146994
$836$	0	0
$837$	36.0533	1.24618
$838$	0	0
$839$	34.5599	1.19314	0.596570	−	0.802561i	$-0.296530\pi$
$839$	34.5599	1.19314	0.596570	+	0.802561i	$0.296530\pi$
$840$	0	0
$841$	−28.8727	−0.995610
$842$	0	0
$843$	−85.0649	−2.92979
$844$	0	0
$845$	−9.96629	−0.342851
$846$	0	0
$847$	−11.8391	−0.406796
$848$	0	0
$849$	45.4409	1.55953
$850$	0	0
$851$	−20.5770	−0.705372
$852$	0	0
$853$	29.3711	1.00565	0.502824	−	0.864389i	$-0.332294\pi$
$853$	29.3711	1.00565	0.502824	+	0.864389i	$0.332294\pi$
$854$	0	0
$855$	−4.91968	−0.168250
$856$	0	0
$857$	−37.4480	−1.27920	−0.639600	−	0.768708i	$-0.720900\pi$
$857$	−37.4480	−1.27920	−0.639600	+	0.768708i	$0.720900\pi$
$858$	0	0
$859$	10.4648	0.357054	0.178527	−	0.983935i	$-0.442867\pi$
$859$	10.4648	0.357054	0.178527	+	0.983935i	$0.442867\pi$
$860$	0	0
$861$	137.806	4.69642
$862$	0	0
$863$	−52.7941	−1.79713	−0.898567	−	0.438837i	$-0.855391\pi$
$863$	−52.7941	−1.79713	−0.898567	+	0.438837i	$0.855391\pi$
$864$	0	0
$865$	−11.5321	−0.392105
$866$	0	0
$867$	44.9526	1.52667
$868$	0	0
$869$	41.6424	1.41262
$870$	0	0
$871$	−3.79535	−0.128601
$872$	0	0
$873$	−24.7934	−0.839129
$874$	0	0
$875$	32.9747	1.11475
$876$	0	0
$877$	−50.6830	−1.71144	−0.855722	−	0.517437i	$-0.826886\pi$
$877$	−50.6830	−1.71144	−0.855722	+	0.517437i	$0.826886\pi$
$878$	0	0
$879$	74.4722	2.51189
$880$	0	0
$881$	34.8273	1.17336	0.586681	−	0.809818i	$-0.300434\pi$
$881$	34.8273	1.17336	0.586681	+	0.809818i	$0.300434\pi$
$882$	0	0
$883$	−6.49713	−0.218646	−0.109323	−	0.994006i	$-0.534868\pi$
$883$	−6.49713	−0.218646	−0.109323	+	0.994006i	$0.534868\pi$
$884$	0	0
$885$	−18.3924	−0.618253
$886$	0	0
$887$	59.3534	1.99289	0.996447	−	0.0842241i	$-0.0268412\pi$
$887$	59.3534	1.99289	0.996447	+	0.0842241i	$0.0268412\pi$
$888$	0	0
$889$	74.0327	2.48298
$890$	0	0
$891$	28.5696	0.957119
$892$	0	0
$893$	4.68548	0.156794
$894$	0	0
$895$	1.57098	0.0525121
$896$	0	0
$897$	−14.3737	−0.479924
$898$	0	0
$899$	−1.51601	−0.0505618
$900$	0	0
$901$	1.37513	0.0458124
$902$	0	0
$903$	65.0254	2.16391
$904$	0	0
$905$	10.3943	0.345517
$906$	0	0
$907$	7.55945	0.251007	0.125504	−	0.992093i	$-0.459945\pi$
$907$	7.55945	0.251007	0.125504	+	0.992093i	$0.459945\pi$
$908$	0	0
$909$	−30.7084	−1.01853
$910$	0	0
$911$	27.4756	0.910308	0.455154	−	0.890413i	$-0.349584\pi$
$911$	27.4756	0.910308	0.455154	+	0.890413i	$0.349584\pi$
$912$	0	0
$913$	−12.0198	−0.397798
$914$	0	0
$915$	−15.6242	−0.516520
$916$	0	0
$917$	7.47822	0.246952
$918$	0	0
$919$	−7.51453	−0.247882	−0.123941	−	0.992290i	$-0.539553\pi$
$919$	−7.51453	−0.247882	−0.123941	+	0.992290i	$0.539553\pi$
$920$	0	0
$921$	−65.5296	−2.15927
$922$	0	0
$923$	5.40393	0.177873
$924$	0	0
$925$	−19.5683	−0.643400
$926$	0	0
$927$	67.5707	2.21931
$928$	0	0
$929$	19.0716	0.625718	0.312859	−	0.949800i	$-0.398713\pi$
$929$	19.0716	0.625718	0.312859	+	0.949800i	$0.398713\pi$
$930$	0	0
$931$	−10.8136	−0.354401
$932$	0	0
$933$	−14.1893	−0.464536
$934$	0	0
$935$	4.29539	0.140474
$936$	0	0
$937$	−0.0609918	−0.00199252	−0.000996258	−	1.00000i	$-0.500317\pi$
$937$	−0.0609918	−0.00199252	−0.000996258		1.00000i	$0.500317\pi$
$938$	0	0
$939$	−73.5597	−2.40053
$940$	0	0
$941$	−53.9013	−1.75713	−0.878566	−	0.477621i	$-0.841499\pi$
$941$	−53.9013	−1.75713	−0.878566	+	0.477621i	$0.841499\pi$
$942$	0	0
$943$	−49.5438	−1.61337
$944$	0	0
$945$	30.1073	0.979389
$946$	0	0
$947$	35.1185	1.14120	0.570600	−	0.821228i	$-0.306711\pi$
$947$	35.1185	1.14120	0.570600	+	0.821228i	$0.306711\pi$
$948$	0	0
$949$	−6.37439	−0.206922
$950$	0	0
$951$	22.8398	0.740631
$952$	0	0
$953$	6.25933	0.202760	0.101380	−	0.994848i	$-0.467674\pi$
$953$	6.25933	0.202760	0.101380	+	0.994848i	$0.467674\pi$
$954$	0	0
$955$	−12.4545	−0.403018
$956$	0	0
$957$	−3.94415	−0.127496
$958$	0	0
$959$	36.2348	1.17008
$960$	0	0
$961$	−12.9460	−0.417611
$962$	0	0
$963$	−70.0135	−2.25615
$964$	0	0
$965$	6.69307	0.215457
$966$	0	0
$967$	−25.1928	−0.810147	−0.405073	−	0.914284i	$-0.632754\pi$
$967$	−25.1928	−0.810147	−0.405073	+	0.914284i	$0.632754\pi$
$968$	0	0
$969$	4.09133	0.131432
$970$	0	0
$971$	−12.4247	−0.398728	−0.199364	−	0.979926i	$-0.563888\pi$
$971$	−12.4247	−0.398728	−0.199364	+	0.979926i	$0.563888\pi$
$972$	0	0
$973$	14.8650	0.476549
$974$	0	0
$975$	−13.6690	−0.437759
$976$	0	0
$977$	−23.2577	−0.744079	−0.372040	−	0.928217i	$-0.621341\pi$
$977$	−23.2577	−0.744079	−0.372040	+	0.928217i	$0.621341\pi$
$978$	0	0
$979$	−1.35631	−0.0433479
$980$	0	0
$981$	−55.7934	−1.78135
$982$	0	0
$983$	40.5385	1.29298	0.646489	−	0.762923i	$-0.276236\pi$
$983$	40.5385	1.29298	0.646489	+	0.762923i	$0.276236\pi$
$984$	0	0
$985$	−8.05000	−0.256494
$986$	0	0
$987$	−58.8368	−1.87280
$988$	0	0
$989$	−23.3778	−0.743372
$990$	0	0
$991$	−31.8507	−1.01177	−0.505886	−	0.862600i	$-0.668834\pi$
$991$	−31.8507	−1.01177	−0.505886	+	0.862600i	$0.668834\pi$
$992$	0	0
$993$	89.4454	2.83846
$994$	0	0
$995$	−0.588296	−0.0186502
$996$	0	0
$997$	8.18740	0.259297	0.129649	−	0.991560i	$-0.458615\pi$
$997$	8.18740	0.259297	0.129649	+	0.991560i	$0.458615\pi$
$998$	0	0
$999$	−38.6745	−1.22361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.2	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.2	✓	19	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.97522 q^{3} +0.840694 q^{5} -4.22061 q^{7} +5.85193 q^{9} +O(q^{10})\)	\(q-2.97522 q^{3} +0.840694 q^{5} -4.22061 q^{7} +5.85193 q^{9} +3.71552 q^{11} -1.07012 q^{13} -2.50125 q^{15} +1.37513 q^{17} -1.00000 q^{19} +12.5573 q^{21} -4.51456 q^{23} -4.29323 q^{25} -8.48511 q^{27} +0.356792 q^{29} -4.24901 q^{31} -11.0545 q^{33} -3.54825 q^{35} +4.55793 q^{37} +3.18386 q^{39} +10.9742 q^{41} +5.17832 q^{43} +4.91968 q^{45} -4.68548 q^{47} +10.8136 q^{49} -4.09133 q^{51} +1.00000 q^{53} +3.12361 q^{55} +2.97522 q^{57} +7.35327 q^{59} +6.24655 q^{61} -24.6987 q^{63} -0.899648 q^{65} +3.54664 q^{67} +13.4318 q^{69} -5.04982 q^{71} +5.95668 q^{73} +12.7733 q^{75} -15.6818 q^{77} +11.2077 q^{79} +7.68928 q^{81} -3.23503 q^{83} +1.15607 q^{85} -1.06153 q^{87} -0.365040 q^{89} +4.51658 q^{91} +12.6417 q^{93} -0.840694 q^{95} -4.23679 q^{97} +21.7429 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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