LMFDB - Embedded newform 4024.2.a.d.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0730681 q^{3} -4.10981 q^{5} -1.52145 q^{7} -2.99466 q^{9} +O(q^{10})$	$q-0.0730681 q^{3} -4.10981 q^{5} -1.52145 q^{7} -2.99466 q^{9} +0.141296 q^{11} +5.15872 q^{13} +0.300296 q^{15} -0.728702 q^{17} +4.77534 q^{19} +0.111169 q^{21} -3.29945 q^{23} +11.8905 q^{25} +0.438018 q^{27} +5.05296 q^{29} +6.69716 q^{31} -0.0103242 q^{33} +6.25287 q^{35} +0.889403 q^{37} -0.376938 q^{39} -3.88933 q^{41} +10.3047 q^{43} +12.3075 q^{45} -10.1280 q^{47} -4.68519 q^{49} +0.0532449 q^{51} -5.37737 q^{53} -0.580698 q^{55} -0.348925 q^{57} -10.8751 q^{59} +2.06054 q^{61} +4.55623 q^{63} -21.2014 q^{65} +0.840216 q^{67} +0.241084 q^{69} -3.16133 q^{71} +1.65463 q^{73} -0.868819 q^{75} -0.214974 q^{77} -15.3526 q^{79} +8.95198 q^{81} -4.48134 q^{83} +2.99483 q^{85} -0.369210 q^{87} +9.40735 q^{89} -7.84874 q^{91} -0.489349 q^{93} -19.6258 q^{95} +3.59608 q^{97} -0.423132 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9} - 14 q^{11} - 31 q^{13} - 2 q^{15} - 9 q^{17} - 8 q^{19} - 28 q^{21} - 4 q^{23} + 22 q^{25} - 4 q^{27} - 47 q^{29} - 5 q^{31} - 26 q^{33} - 13 q^{35} - 67 q^{37} - 9 q^{39} - 28 q^{41} + 15 q^{43} - 57 q^{45} - 10 q^{47} + 20 q^{49} - 11 q^{51} - 58 q^{53} + 15 q^{55} - 31 q^{57} - 32 q^{59} - 55 q^{61} - 16 q^{63} - 44 q^{65} + 22 q^{67} - 44 q^{69} - 47 q^{71} - 5 q^{73} - 25 q^{75} - 50 q^{77} - 14 q^{79} - 28 q^{81} - 16 q^{83} - 78 q^{85} - 11 q^{87} - 20 q^{89} - 15 q^{91} - 83 q^{93} - 27 q^{95} - 8 q^{97} - 70 q^{99}+O(q^{100}) $ 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9 - 14 * q^11 - 31 * q^13 - 2 * q^15 - 9 * q^17 - 8 * q^19 - 28 * q^21 - 4 * q^23 + 22 * q^25 - 4 * q^27 - 47 * q^29 - 5 * q^31 - 26 * q^33 - 13 * q^35 - 67 * q^37 - 9 * q^39 - 28 * q^41 + 15 * q^43 - 57 * q^45 - 10 * q^47 + 20 * q^49 - 11 * q^51 - 58 * q^53 + 15 * q^55 - 31 * q^57 - 32 * q^59 - 55 * q^61 - 16 * q^63 - 44 * q^65 + 22 * q^67 - 44 * q^69 - 47 * q^71 - 5 * q^73 - 25 * q^75 - 50 * q^77 - 14 * q^79 - 28 * q^81 - 16 * q^83 - 78 * q^85 - 11 * q^87 - 20 * q^89 - 15 * q^91 - 83 * q^93 - 27 * q^95 - 8 * q^97 - 70 * 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
$3$	−0.0730681	−0.0421859	−0.0210929	−	0.999778i	$-0.506715\pi$
$3$	−0.0730681	−0.0421859	−0.0210929	+	0.999778i	$0.506715\pi$
$4$	0	0
$5$	−4.10981	−1.83796	−0.918981	−	0.394301i	$-0.870987\pi$
$5$	−4.10981	−1.83796	−0.918981	+	0.394301i	$0.870987\pi$
$6$	0	0
$7$	−1.52145	−0.575054	−0.287527	−	0.957773i	$-0.592833\pi$
$7$	−1.52145	−0.575054	−0.287527	+	0.957773i	$0.592833\pi$
$8$	0	0
$9$	−2.99466	−0.998220
$10$	0	0
$11$	0.141296	0.0426022	0.0213011	−	0.999773i	$-0.493219\pi$
$11$	0.141296	0.0426022	0.0213011	+	0.999773i	$0.493219\pi$
$12$	0	0
$13$	5.15872	1.43077	0.715386	−	0.698729i	$-0.246251\pi$
$13$	5.15872	1.43077	0.715386	+	0.698729i	$0.246251\pi$
$14$	0	0
$15$	0.300296	0.0775361
$16$	0	0
$17$	−0.728702	−0.176736	−0.0883681	−	0.996088i	$-0.528165\pi$
$17$	−0.728702	−0.176736	−0.0883681	+	0.996088i	$0.528165\pi$
$18$	0	0
$19$	4.77534	1.09554	0.547770	−	0.836629i	$-0.315477\pi$
$19$	4.77534	1.09554	0.547770	+	0.836629i	$0.315477\pi$
$20$	0	0
$21$	0.111169	0.0242592
$22$	0	0
$23$	−3.29945	−0.687982	−0.343991	−	0.938973i	$-0.611779\pi$
$23$	−3.29945	−0.687982	−0.343991	+	0.938973i	$0.611779\pi$
$24$	0	0
$25$	11.8905	2.37811
$26$	0	0
$27$	0.438018	0.0842967
$28$	0	0
$29$	5.05296	0.938311	0.469156	−	0.883116i	$-0.344558\pi$
$29$	5.05296	0.938311	0.469156	+	0.883116i	$0.344558\pi$
$30$	0	0
$31$	6.69716	1.20285	0.601423	−	0.798931i	$-0.294601\pi$
$31$	6.69716	1.20285	0.601423	+	0.798931i	$0.294601\pi$
$32$	0	0
$33$	−0.0103242	−0.00179721
$34$	0	0
$35$	6.25287	1.05693
$36$	0	0
$37$	0.889403	0.146217	0.0731084	−	0.997324i	$-0.476708\pi$
$37$	0.889403	0.146217	0.0731084	+	0.997324i	$0.476708\pi$
$38$	0	0
$39$	−0.376938	−0.0603584
$40$	0	0
$41$	−3.88933	−0.607411	−0.303705	−	0.952766i	$-0.598224\pi$
$41$	−3.88933	−0.607411	−0.303705	+	0.952766i	$0.598224\pi$
$42$	0	0
$43$	10.3047	1.57146	0.785729	−	0.618571i	$-0.212288\pi$
$43$	10.3047	1.57146	0.785729	+	0.618571i	$0.212288\pi$
$44$	0	0
$45$	12.3075	1.83469
$46$	0	0
$47$	−10.1280	−1.47732	−0.738661	−	0.674077i	$-0.764542\pi$
$47$	−10.1280	−1.47732	−0.738661	+	0.674077i	$0.764542\pi$
$48$	0	0
$49$	−4.68519	−0.669313
$50$	0	0
$51$	0.0532449	0.00745577
$52$	0	0
$53$	−5.37737	−0.738638	−0.369319	−	0.929303i	$-0.620409\pi$
$53$	−5.37737	−0.738638	−0.369319	+	0.929303i	$0.620409\pi$
$54$	0	0
$55$	−0.580698	−0.0783013
$56$	0	0
$57$	−0.348925	−0.0462163
$58$	0	0
$59$	−10.8751	−1.41581	−0.707906	−	0.706307i	$-0.750360\pi$
$59$	−10.8751	−1.41581	−0.707906	+	0.706307i	$0.750360\pi$
$60$	0	0
$61$	2.06054	0.263825	0.131912	−	0.991261i	$-0.457888\pi$
$61$	2.06054	0.263825	0.131912	+	0.991261i	$0.457888\pi$
$62$	0	0
$63$	4.55623	0.574031
$64$	0	0
$65$	−21.2014	−2.62971
$66$	0	0
$67$	0.840216	0.102649	0.0513244	−	0.998682i	$-0.483656\pi$
$67$	0.840216	0.102649	0.0513244	+	0.998682i	$0.483656\pi$
$68$	0	0
$69$	0.241084	0.0290231
$70$	0	0
$71$	−3.16133	−0.375180	−0.187590	−	0.982247i	$-0.560068\pi$
$71$	−3.16133	−0.375180	−0.187590	+	0.982247i	$0.560068\pi$
$72$	0	0
$73$	1.65463	0.193660	0.0968299	−	0.995301i	$-0.469130\pi$
$73$	1.65463	0.193660	0.0968299	+	0.995301i	$0.469130\pi$
$74$	0	0
$75$	−0.868819	−0.100323
$76$	0	0
$77$	−0.214974	−0.0244986
$78$	0	0
$79$	−15.3526	−1.72731	−0.863653	−	0.504086i	$-0.831829\pi$
$79$	−15.3526	−1.72731	−0.863653	+	0.504086i	$0.831829\pi$
$80$	0	0
$81$	8.95198	0.994664
$82$	0	0
$83$	−4.48134	−0.491891	−0.245945	−	0.969284i	$-0.579098\pi$
$83$	−4.48134	−0.491891	−0.245945	+	0.969284i	$0.579098\pi$
$84$	0	0
$85$	2.99483	0.324835
$86$	0	0
$87$	−0.369210	−0.0395835
$88$	0	0
$89$	9.40735	0.997177	0.498589	−	0.866839i	$-0.333852\pi$
$89$	9.40735	0.997177	0.498589	+	0.866839i	$0.333852\pi$
$90$	0	0
$91$	−7.84874	−0.822771
$92$	0	0
$93$	−0.489349	−0.0507431
$94$	0	0
$95$	−19.6258	−2.01356
$96$	0	0
$97$	3.59608	0.365127	0.182563	−	0.983194i	$-0.441561\pi$
$97$	3.59608	0.365127	0.182563	+	0.983194i	$0.441561\pi$
$98$	0	0
$99$	−0.423132	−0.0425264
$100$	0	0
$101$	2.53397	0.252140	0.126070	−	0.992021i	$-0.459764\pi$
$101$	2.53397	0.252140	0.126070	+	0.992021i	$0.459764\pi$
$102$	0	0
$103$	4.92495	0.485270	0.242635	−	0.970118i	$-0.421988\pi$
$103$	4.92495	0.485270	0.242635	+	0.970118i	$0.421988\pi$
$104$	0	0
$105$	−0.456885	−0.0445874
$106$	0	0
$107$	−14.2378	−1.37642	−0.688210	−	0.725511i	$-0.741603\pi$
$107$	−14.2378	−1.37642	−0.688210	+	0.725511i	$0.741603\pi$
$108$	0	0
$109$	−17.8156	−1.70643	−0.853215	−	0.521560i	$-0.825350\pi$
$109$	−17.8156	−1.70643	−0.853215	+	0.521560i	$0.825350\pi$
$110$	0	0
$111$	−0.0649869	−0.00616829
$112$	0	0
$113$	20.4837	1.92694	0.963472	−	0.267808i	$-0.0862994\pi$
$113$	20.4837	1.92694	0.963472	+	0.267808i	$0.0862994\pi$
$114$	0	0
$115$	13.5601	1.26449
$116$	0	0
$117$	−15.4486	−1.42823
$118$	0	0
$119$	1.10868	0.101633
$120$	0	0
$121$	−10.9800	−0.998185
$122$	0	0
$123$	0.284186	0.0256242
$124$	0	0
$125$	−28.3188	−2.53291
$126$	0	0
$127$	5.92105	0.525408	0.262704	−	0.964876i	$-0.415386\pi$
$127$	5.92105	0.525408	0.262704	+	0.964876i	$0.415386\pi$
$128$	0	0
$129$	−0.752948	−0.0662933
$130$	0	0
$131$	9.91998	0.866713	0.433356	−	0.901223i	$-0.357329\pi$
$131$	9.91998	0.866713	0.433356	+	0.901223i	$0.357329\pi$
$132$	0	0
$133$	−7.26545	−0.629994
$134$	0	0
$135$	−1.80017	−0.154934
$136$	0	0
$137$	−7.38820	−0.631216	−0.315608	−	0.948890i	$-0.602209\pi$
$137$	−7.38820	−0.631216	−0.315608	+	0.948890i	$0.602209\pi$
$138$	0	0
$139$	−8.65267	−0.733910	−0.366955	−	0.930239i	$-0.619600\pi$
$139$	−8.65267	−0.733910	−0.366955	+	0.930239i	$0.619600\pi$
$140$	0	0
$141$	0.740034	0.0623221
$142$	0	0
$143$	0.728905	0.0609541
$144$	0	0
$145$	−20.7667	−1.72458
$146$	0	0
$147$	0.342338	0.0282355
$148$	0	0
$149$	−6.04411	−0.495152	−0.247576	−	0.968868i	$-0.579634\pi$
$149$	−6.04411	−0.495152	−0.247576	+	0.968868i	$0.579634\pi$
$150$	0	0
$151$	17.6839	1.43909	0.719546	−	0.694445i	$-0.244350\pi$
$151$	17.6839	1.43909	0.719546	+	0.694445i	$0.244350\pi$
$152$	0	0
$153$	2.18222	0.176422
$154$	0	0
$155$	−27.5241	−2.21079
$156$	0	0
$157$	−17.6441	−1.40815	−0.704076	−	0.710125i	$-0.748638\pi$
$157$	−17.6441	−1.40815	−0.704076	+	0.710125i	$0.748638\pi$
$158$	0	0
$159$	0.392914	0.0311601
$160$	0	0
$161$	5.01994	0.395627
$162$	0	0
$163$	−8.29214	−0.649491	−0.324745	−	0.945801i	$-0.605279\pi$
$163$	−8.29214	−0.649491	−0.324745	+	0.945801i	$0.605279\pi$
$164$	0	0
$165$	0.0424305	0.00330321
$166$	0	0
$167$	−5.51612	−0.426850	−0.213425	−	0.976959i	$-0.568462\pi$
$167$	−5.51612	−0.426850	−0.213425	+	0.976959i	$0.568462\pi$
$168$	0	0
$169$	13.6124	1.04711
$170$	0	0
$171$	−14.3005	−1.09359
$172$	0	0
$173$	−17.2594	−1.31221	−0.656103	−	0.754672i	$-0.727796\pi$
$173$	−17.2594	−1.31221	−0.656103	+	0.754672i	$0.727796\pi$
$174$	0	0
$175$	−18.0909	−1.36754
$176$	0	0
$177$	0.794619	0.0597273
$178$	0	0
$179$	19.4972	1.45729	0.728643	−	0.684893i	$-0.240151\pi$
$179$	19.4972	1.45729	0.728643	+	0.684893i	$0.240151\pi$
$180$	0	0
$181$	−15.6670	−1.16452	−0.582261	−	0.813002i	$-0.697832\pi$
$181$	−15.6670	−1.16452	−0.582261	+	0.813002i	$0.697832\pi$
$182$	0	0
$183$	−0.150560	−0.0111297
$184$	0	0
$185$	−3.65528	−0.268741
$186$	0	0
$187$	−0.102962	−0.00752936
$188$	0	0
$189$	−0.666423	−0.0484751
$190$	0	0
$191$	2.96938	0.214857	0.107428	−	0.994213i	$-0.465738\pi$
$191$	2.96938	0.214857	0.107428	+	0.994213i	$0.465738\pi$
$192$	0	0
$193$	0.575182	0.0414025	0.0207012	−	0.999786i	$-0.493410\pi$
$193$	0.575182	0.0414025	0.0207012	+	0.999786i	$0.493410\pi$
$194$	0	0
$195$	1.54914	0.110936
$196$	0	0
$197$	10.5996	0.755190	0.377595	−	0.925971i	$-0.376751\pi$
$197$	10.5996	0.755190	0.377595	+	0.925971i	$0.376751\pi$
$198$	0	0
$199$	8.21873	0.582611	0.291305	−	0.956630i	$-0.405910\pi$
$199$	8.21873	0.582611	0.291305	+	0.956630i	$0.405910\pi$
$200$	0	0
$201$	−0.0613930	−0.00433033
$202$	0	0
$203$	−7.68783	−0.539580
$204$	0	0
$205$	15.9844	1.11640
$206$	0	0
$207$	9.88073	0.686758
$208$	0	0
$209$	0.674735	0.0466724
$210$	0	0
$211$	5.30392	0.365137	0.182568	−	0.983193i	$-0.441559\pi$
$211$	5.30392	0.365137	0.182568	+	0.983193i	$0.441559\pi$
$212$	0	0
$213$	0.230992	0.0158273
$214$	0	0
$215$	−42.3505	−2.88828
$216$	0	0
$217$	−10.1894	−0.691702
$218$	0	0
$219$	−0.120901	−0.00816971
$220$	0	0
$221$	−3.75917	−0.252869
$222$	0	0
$223$	−15.0828	−1.01002	−0.505009	−	0.863114i	$-0.668511\pi$
$223$	−15.0828	−1.01002	−0.505009	+	0.863114i	$0.668511\pi$
$224$	0	0
$225$	−35.6081	−2.37388
$226$	0	0
$227$	19.3241	1.28259	0.641293	−	0.767296i	$-0.278398\pi$
$227$	19.3241	1.28259	0.641293	+	0.767296i	$0.278398\pi$
$228$	0	0
$229$	−14.6934	−0.970965	−0.485482	−	0.874246i	$-0.661356\pi$
$229$	−14.6934	−0.970965	−0.485482	+	0.874246i	$0.661356\pi$
$230$	0	0
$231$	0.0157078	0.00103349
$232$	0	0
$233$	−24.7379	−1.62064	−0.810318	−	0.585990i	$-0.800706\pi$
$233$	−24.7379	−1.62064	−0.810318	+	0.585990i	$0.800706\pi$
$234$	0	0
$235$	41.6242	2.71526
$236$	0	0
$237$	1.12179	0.0728679
$238$	0	0
$239$	13.5630	0.877320	0.438660	−	0.898653i	$-0.355453\pi$
$239$	13.5630	0.877320	0.438660	+	0.898653i	$0.355453\pi$
$240$	0	0
$241$	−12.5435	−0.808000	−0.404000	−	0.914759i	$-0.632380\pi$
$241$	−12.5435	−0.808000	−0.404000	+	0.914759i	$0.632380\pi$
$242$	0	0
$243$	−1.96816	−0.126257
$244$	0	0
$245$	19.2552	1.23017
$246$	0	0
$247$	24.6347	1.56747
$248$	0	0
$249$	0.327443	0.0207508
$250$	0	0
$251$	−1.66001	−0.104779	−0.0523895	−	0.998627i	$-0.516684\pi$
$251$	−1.66001	−0.104779	−0.0523895	+	0.998627i	$0.516684\pi$
$252$	0	0
$253$	−0.466197	−0.0293096
$254$	0	0
$255$	−0.218826	−0.0137034
$256$	0	0
$257$	23.4693	1.46397	0.731986	−	0.681320i	$-0.238594\pi$
$257$	23.4693	1.46397	0.731986	+	0.681320i	$0.238594\pi$
$258$	0	0
$259$	−1.35318	−0.0840826
$260$	0	0
$261$	−15.1319	−0.936641
$262$	0	0
$263$	23.7494	1.46445	0.732225	−	0.681063i	$-0.238482\pi$
$263$	23.7494	1.46445	0.732225	+	0.681063i	$0.238482\pi$
$264$	0	0
$265$	22.1000	1.35759
$266$	0	0
$267$	−0.687377	−0.0420668
$268$	0	0
$269$	−26.1291	−1.59312	−0.796560	−	0.604559i	$-0.793349\pi$
$269$	−26.1291	−1.59312	−0.796560	+	0.604559i	$0.793349\pi$
$270$	0	0
$271$	−2.72457	−0.165506	−0.0827530	−	0.996570i	$-0.526371\pi$
$271$	−2.72457	−0.165506	−0.0827530	+	0.996570i	$0.526371\pi$
$272$	0	0
$273$	0.573492	0.0347093
$274$	0	0
$275$	1.68008	0.101313
$276$	0	0
$277$	−20.6569	−1.24115	−0.620577	−	0.784146i	$-0.713101\pi$
$277$	−20.6569	−1.24115	−0.620577	+	0.784146i	$0.713101\pi$
$278$	0	0
$279$	−20.0557	−1.20071
$280$	0	0
$281$	−14.5258	−0.866539	−0.433270	−	0.901264i	$-0.642640\pi$
$281$	−14.5258	−0.866539	−0.433270	+	0.901264i	$0.642640\pi$
$282$	0	0
$283$	−12.4114	−0.737783	−0.368892	−	0.929472i	$-0.620263\pi$
$283$	−12.4114	−0.737783	−0.368892	+	0.929472i	$0.620263\pi$
$284$	0	0
$285$	1.43402	0.0849438
$286$	0	0
$287$	5.91742	0.349294
$288$	0	0
$289$	−16.4690	−0.968764
$290$	0	0
$291$	−0.262759	−0.0154032
$292$	0	0
$293$	27.6191	1.61352	0.806762	−	0.590876i	$-0.201218\pi$
$293$	27.6191	1.61352	0.806762	+	0.590876i	$0.201218\pi$
$294$	0	0
$295$	44.6944	2.60221
$296$	0	0
$297$	0.0618901	0.00359123
$298$	0	0
$299$	−17.0209	−0.984346
$300$	0	0
$301$	−15.6781	−0.903674
$302$	0	0
$303$	−0.185153	−0.0106367
$304$	0	0
$305$	−8.46842	−0.484900
$306$	0	0
$307$	27.7198	1.58205	0.791025	−	0.611783i	$-0.209548\pi$
$307$	27.7198	1.58205	0.791025	+	0.611783i	$0.209548\pi$
$308$	0	0
$309$	−0.359857	−0.0204715
$310$	0	0
$311$	26.7891	1.51907	0.759536	−	0.650466i	$-0.225426\pi$
$311$	26.7891	1.51907	0.759536	+	0.650466i	$0.225426\pi$
$312$	0	0
$313$	32.2082	1.82051	0.910257	−	0.414043i	$-0.135884\pi$
$313$	32.2082	1.82051	0.910257	+	0.414043i	$0.135884\pi$
$314$	0	0
$315$	−18.7252	−1.05505
$316$	0	0
$317$	5.60285	0.314687	0.157344	−	0.987544i	$-0.449707\pi$
$317$	5.60285	0.314687	0.157344	+	0.987544i	$0.449707\pi$
$318$	0	0
$319$	0.713961	0.0399741
$320$	0	0
$321$	1.04033	0.0580655
$322$	0	0
$323$	−3.47980	−0.193622
$324$	0	0
$325$	61.3400	3.40253
$326$	0	0
$327$	1.30175	0.0719872
$328$	0	0
$329$	15.4093	0.849541
$330$	0	0
$331$	−6.98909	−0.384155	−0.192078	−	0.981380i	$-0.561523\pi$
$331$	−6.98909	−0.384155	−0.192078	+	0.981380i	$0.561523\pi$
$332$	0	0
$333$	−2.66346	−0.145957
$334$	0	0
$335$	−3.45313	−0.188665
$336$	0	0
$337$	−6.65743	−0.362653	−0.181327	−	0.983423i	$-0.558039\pi$
$337$	−6.65743	−0.362653	−0.181327	+	0.983423i	$0.558039\pi$
$338$	0	0
$339$	−1.49670	−0.0812898
$340$	0	0
$341$	0.946280	0.0512439
$342$	0	0
$343$	17.7784	0.959945
$344$	0	0
$345$	−0.990810	−0.0533434
$346$	0	0
$347$	−22.7192	−1.21963	−0.609814	−	0.792545i	$-0.708756\pi$
$347$	−22.7192	−1.21963	−0.609814	+	0.792545i	$0.708756\pi$
$348$	0	0
$349$	−26.8265	−1.43599	−0.717995	−	0.696049i	$-0.754940\pi$
$349$	−26.8265	−1.43599	−0.717995	+	0.696049i	$0.754940\pi$
$350$	0	0
$351$	2.25961	0.120609
$352$	0	0
$353$	−22.8463	−1.21598	−0.607992	−	0.793943i	$-0.708025\pi$
$353$	−22.8463	−1.21598	−0.607992	+	0.793943i	$0.708025\pi$
$354$	0	0
$355$	12.9925	0.689568
$356$	0	0
$357$	−0.0810094	−0.00428747
$358$	0	0
$359$	−0.532684	−0.0281140	−0.0140570	−	0.999901i	$-0.504475\pi$
$359$	−0.532684	−0.0281140	−0.0140570	+	0.999901i	$0.504475\pi$
$360$	0	0
$361$	3.80392	0.200206
$362$	0	0
$363$	0.802290	0.0421093
$364$	0	0
$365$	−6.80022	−0.355940
$366$	0	0
$367$	−18.7563	−0.979072	−0.489536	−	0.871983i	$-0.662834\pi$
$367$	−18.7563	−0.979072	−0.489536	+	0.871983i	$0.662834\pi$
$368$	0	0
$369$	11.6472	0.606330
$370$	0	0
$371$	8.18139	0.424757
$372$	0	0
$373$	−0.686733	−0.0355577	−0.0177788	−	0.999842i	$-0.505659\pi$
$373$	−0.686733	−0.0355577	−0.0177788	+	0.999842i	$0.505659\pi$
$374$	0	0
$375$	2.06920	0.106853
$376$	0	0
$377$	26.0668	1.34251
$378$	0	0
$379$	17.5577	0.901879	0.450939	−	0.892555i	$-0.351089\pi$
$379$	17.5577	0.901879	0.450939	+	0.892555i	$0.351089\pi$
$380$	0	0
$381$	−0.432639	−0.0221648
$382$	0	0
$383$	−28.1269	−1.43722	−0.718609	−	0.695414i	$-0.755221\pi$
$383$	−28.1269	−1.43722	−0.718609	+	0.695414i	$0.755221\pi$
$384$	0	0
$385$	0.883503	0.0450275
$386$	0	0
$387$	−30.8592	−1.56866
$388$	0	0
$389$	−28.6249	−1.45134	−0.725669	−	0.688043i	$-0.758470\pi$
$389$	−28.6249	−1.45134	−0.725669	+	0.688043i	$0.758470\pi$
$390$	0	0
$391$	2.40431	0.121591
$392$	0	0
$393$	−0.724834	−0.0365630
$394$	0	0
$395$	63.0964	3.17473
$396$	0	0
$397$	−26.0198	−1.30590	−0.652949	−	0.757402i	$-0.726468\pi$
$397$	−26.0198	−1.30590	−0.652949	+	0.757402i	$0.726468\pi$
$398$	0	0
$399$	0.530872	0.0265769
$400$	0	0
$401$	27.3587	1.36623	0.683115	−	0.730311i	$-0.260625\pi$
$401$	27.3587	1.36623	0.683115	+	0.730311i	$0.260625\pi$
$402$	0	0
$403$	34.5488	1.72100
$404$	0	0
$405$	−36.7909	−1.82816
$406$	0	0
$407$	0.125669	0.00622917
$408$	0	0
$409$	−4.25359	−0.210326	−0.105163	−	0.994455i	$-0.533537\pi$
$409$	−4.25359	−0.210326	−0.105163	+	0.994455i	$0.533537\pi$
$410$	0	0
$411$	0.539841	0.0266284
$412$	0	0
$413$	16.5459	0.814168
$414$	0	0
$415$	18.4174	0.904077
$416$	0	0
$417$	0.632234	0.0309606
$418$	0	0
$419$	22.9112	1.11929	0.559643	−	0.828734i	$-0.310938\pi$
$419$	22.9112	1.11929	0.559643	+	0.828734i	$0.310938\pi$
$420$	0	0
$421$	−20.2561	−0.987224	−0.493612	−	0.869682i	$-0.664324\pi$
$421$	−20.2561	−0.987224	−0.493612	+	0.869682i	$0.664324\pi$
$422$	0	0
$423$	30.3300	1.47469
$424$	0	0
$425$	−8.66466	−0.420298
$426$	0	0
$427$	−3.13501	−0.151714
$428$	0	0
$429$	−0.0532597	−0.00257140
$430$	0	0
$431$	−32.2661	−1.55421	−0.777103	−	0.629374i	$-0.783311\pi$
$431$	−32.2661	−1.55421	−0.777103	+	0.629374i	$0.783311\pi$
$432$	0	0
$433$	−12.3984	−0.595831	−0.297916	−	0.954592i	$-0.596291\pi$
$433$	−12.3984	−0.595831	−0.297916	+	0.954592i	$0.596291\pi$
$434$	0	0
$435$	1.51738	0.0727530
$436$	0	0
$437$	−15.7560	−0.753712
$438$	0	0
$439$	−23.4324	−1.11837	−0.559183	−	0.829044i	$-0.688885\pi$
$439$	−23.4324	−1.11837	−0.559183	+	0.829044i	$0.688885\pi$
$440$	0	0
$441$	14.0306	0.668122
$442$	0	0
$443$	−16.6583	−0.791459	−0.395729	−	0.918367i	$-0.629508\pi$
$443$	−16.6583	−0.791459	−0.395729	+	0.918367i	$0.629508\pi$
$444$	0	0
$445$	−38.6624	−1.83277
$446$	0	0
$447$	0.441631	0.0208884
$448$	0	0
$449$	−38.2161	−1.80353	−0.901765	−	0.432227i	$-0.857728\pi$
$449$	−38.2161	−1.80353	−0.901765	+	0.432227i	$0.857728\pi$
$450$	0	0
$451$	−0.549545	−0.0258771
$452$	0	0
$453$	−1.29213	−0.0607093
$454$	0	0
$455$	32.2568	1.51222
$456$	0	0
$457$	12.3565	0.578013	0.289006	−	0.957327i	$-0.406675\pi$
$457$	12.3565	0.578013	0.289006	+	0.957327i	$0.406675\pi$
$458$	0	0
$459$	−0.319185	−0.0148983
$460$	0	0
$461$	2.87272	0.133796	0.0668979	−	0.997760i	$-0.478690\pi$
$461$	2.87272	0.133796	0.0668979	+	0.997760i	$0.478690\pi$
$462$	0	0
$463$	17.2179	0.800184	0.400092	−	0.916475i	$-0.368978\pi$
$463$	17.2179	0.800184	0.400092	+	0.916475i	$0.368978\pi$
$464$	0	0
$465$	2.01113	0.0932640
$466$	0	0
$467$	−24.2286	−1.12117	−0.560583	−	0.828098i	$-0.689423\pi$
$467$	−24.2286	−1.12117	−0.560583	+	0.828098i	$0.689423\pi$
$468$	0	0
$469$	−1.27835	−0.0590286
$470$	0	0
$471$	1.28922	0.0594041
$472$	0	0
$473$	1.45601	0.0669476
$474$	0	0
$475$	56.7814	2.60531
$476$	0	0
$477$	16.1034	0.737323
$478$	0	0
$479$	−6.90956	−0.315706	−0.157853	−	0.987463i	$-0.550457\pi$
$479$	−6.90956	−0.315706	−0.157853	+	0.987463i	$0.550457\pi$
$480$	0	0
$481$	4.58818	0.209203
$482$	0	0
$483$	−0.366798	−0.0166899
$484$	0	0
$485$	−14.7792	−0.671090
$486$	0	0
$487$	−33.2651	−1.50738	−0.753692	−	0.657228i	$-0.771729\pi$
$487$	−33.2651	−1.50738	−0.753692	+	0.657228i	$0.771729\pi$
$488$	0	0
$489$	0.605891	0.0273993
$490$	0	0
$491$	20.7578	0.936785	0.468393	−	0.883520i	$-0.344833\pi$
$491$	20.7578	0.936785	0.468393	+	0.883520i	$0.344833\pi$
$492$	0	0
$493$	−3.68210	−0.165834
$494$	0	0
$495$	1.73899	0.0781620
$496$	0	0
$497$	4.80980	0.215749
$498$	0	0
$499$	15.4688	0.692481	0.346240	−	0.938146i	$-0.387458\pi$
$499$	15.4688	0.692481	0.346240	+	0.938146i	$0.387458\pi$
$500$	0	0
$501$	0.403053	0.0180071
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−10.4141	−0.463424
$506$	0	0
$507$	−0.994633	−0.0441732
$508$	0	0
$509$	−13.7838	−0.610956	−0.305478	−	0.952199i	$-0.598816\pi$
$509$	−13.7838	−0.610956	−0.305478	+	0.952199i	$0.598816\pi$
$510$	0	0
$511$	−2.51744	−0.111365
$512$	0	0
$513$	2.09169	0.0923503
$514$	0	0
$515$	−20.2406	−0.891908
$516$	0	0
$517$	−1.43104	−0.0629372
$518$	0	0
$519$	1.26111	0.0553565
$520$	0	0
$521$	7.46908	0.327226	0.163613	−	0.986525i	$-0.447685\pi$
$521$	7.46908	0.327226	0.163613	+	0.986525i	$0.447685\pi$
$522$	0	0
$523$	−9.98612	−0.436663	−0.218331	−	0.975875i	$-0.570061\pi$
$523$	−9.98612	−0.436663	−0.218331	+	0.975875i	$0.570061\pi$
$524$	0	0
$525$	1.32186	0.0576909
$526$	0	0
$527$	−4.88024	−0.212587
$528$	0	0
$529$	−12.1136	−0.526680
$530$	0	0
$531$	32.5671	1.41329
$532$	0	0
$533$	−20.0640	−0.869066
$534$	0	0
$535$	58.5147	2.52981
$536$	0	0
$537$	−1.42462	−0.0614769
$538$	0	0
$539$	−0.661997	−0.0285142
$540$	0	0
$541$	−32.9816	−1.41799	−0.708996	−	0.705213i	$-0.750852\pi$
$541$	−32.9816	−1.41799	−0.708996	+	0.705213i	$0.750852\pi$
$542$	0	0
$543$	1.14476	0.0491264
$544$	0	0
$545$	73.2189	3.13635
$546$	0	0
$547$	−29.2017	−1.24857	−0.624287	−	0.781195i	$-0.714610\pi$
$547$	−29.2017	−1.24857	−0.624287	+	0.781195i	$0.714610\pi$
$548$	0	0
$549$	−6.17061	−0.263355
$550$	0	0
$551$	24.1296	1.02796
$552$	0	0
$553$	23.3583	0.993295
$554$	0	0
$555$	0.267084	0.0113371
$556$	0	0
$557$	−3.99891	−0.169439	−0.0847197	−	0.996405i	$-0.526999\pi$
$557$	−3.99891	−0.169439	−0.0847197	+	0.996405i	$0.526999\pi$
$558$	0	0
$559$	53.1593	2.24840
$560$	0	0
$561$	0.00752327	0.000317633 0
$562$	0	0
$563$	−18.3964	−0.775315	−0.387657	−	0.921804i	$-0.626716\pi$
$563$	−18.3964	−0.775315	−0.387657	+	0.921804i	$0.626716\pi$
$564$	0	0
$565$	−84.1841	−3.54165
$566$	0	0
$567$	−13.6200	−0.571986
$568$	0	0
$569$	34.6703	1.45346	0.726728	−	0.686926i	$-0.241040\pi$
$569$	34.6703	1.45346	0.726728	+	0.686926i	$0.241040\pi$
$570$	0	0
$571$	8.27969	0.346494	0.173247	−	0.984878i	$-0.444574\pi$
$571$	8.27969	0.346494	0.173247	+	0.984878i	$0.444574\pi$
$572$	0	0
$573$	−0.216967	−0.00906392
$574$	0	0
$575$	−39.2322	−1.63610
$576$	0	0
$577$	−4.92315	−0.204953	−0.102477	−	0.994735i	$-0.532677\pi$
$577$	−4.92315	−0.204953	−0.102477	+	0.994735i	$0.532677\pi$
$578$	0	0
$579$	−0.0420274	−0.00174660
$580$	0	0
$581$	6.81813	0.282864
$582$	0	0
$583$	−0.759798	−0.0314676
$584$	0	0
$585$	63.4909	2.62503
$586$	0	0
$587$	−9.75924	−0.402807	−0.201403	−	0.979508i	$-0.564550\pi$
$587$	−9.75924	−0.402807	−0.201403	+	0.979508i	$0.564550\pi$
$588$	0	0
$589$	31.9813	1.31777
$590$	0	0
$591$	−0.774492	−0.0318583
$592$	0	0
$593$	28.1136	1.15449	0.577244	−	0.816572i	$-0.304128\pi$
$593$	28.1136	1.15449	0.577244	+	0.816572i	$0.304128\pi$
$594$	0	0
$595$	−4.55648	−0.186798
$596$	0	0
$597$	−0.600527	−0.0245779
$598$	0	0
$599$	2.03151	0.0830054	0.0415027	−	0.999138i	$-0.486785\pi$
$599$	2.03151	0.0830054	0.0415027	+	0.999138i	$0.486785\pi$
$600$	0	0
$601$	26.8019	1.09327	0.546637	−	0.837370i	$-0.315908\pi$
$601$	26.8019	1.09327	0.546637	+	0.837370i	$0.315908\pi$
$602$	0	0
$603$	−2.51616	−0.102466
$604$	0	0
$605$	45.1259	1.83463
$606$	0	0
$607$	23.7433	0.963711	0.481856	−	0.876251i	$-0.339963\pi$
$607$	23.7433	0.963711	0.481856	+	0.876251i	$0.339963\pi$
$608$	0	0
$609$	0.561735	0.0227626
$610$	0	0
$611$	−52.2476	−2.11371
$612$	0	0
$613$	12.3499	0.498809	0.249404	−	0.968399i	$-0.419765\pi$
$613$	12.3499	0.498809	0.249404	+	0.968399i	$0.419765\pi$
$614$	0	0
$615$	−1.16795	−0.0470962
$616$	0	0
$617$	−40.9551	−1.64879	−0.824395	−	0.566015i	$-0.808484\pi$
$617$	−40.9551	−1.64879	−0.824395	+	0.566015i	$0.808484\pi$
$618$	0	0
$619$	33.2176	1.33513	0.667564	−	0.744553i	$-0.267337\pi$
$619$	33.2176	1.33513	0.667564	+	0.744553i	$0.267337\pi$
$620$	0	0
$621$	−1.44522	−0.0579946
$622$	0	0
$623$	−14.3128	−0.573431
$624$	0	0
$625$	56.9322	2.27729
$626$	0	0
$627$	−0.0493016	−0.00196892
$628$	0	0
$629$	−0.648110	−0.0258418
$630$	0	0
$631$	−32.5839	−1.29715	−0.648573	−	0.761153i	$-0.724634\pi$
$631$	−32.5839	−1.29715	−0.648573	+	0.761153i	$0.724634\pi$
$632$	0	0
$633$	−0.387547	−0.0154036
$634$	0	0
$635$	−24.3344	−0.965680
$636$	0	0
$637$	−24.1696	−0.957634
$638$	0	0
$639$	9.46710	0.374513
$640$	0	0
$641$	24.5080	0.968007	0.484003	−	0.875066i	$-0.339182\pi$
$641$	24.5080	0.968007	0.484003	+	0.875066i	$0.339182\pi$
$642$	0	0
$643$	−42.0710	−1.65912	−0.829559	−	0.558419i	$-0.811408\pi$
$643$	−42.0710	−1.65912	−0.829559	+	0.558419i	$0.811408\pi$
$644$	0	0
$645$	3.09447	0.121845
$646$	0	0
$647$	−16.7124	−0.657033	−0.328517	−	0.944498i	$-0.606549\pi$
$647$	−16.7124	−0.657033	−0.328517	+	0.944498i	$0.606549\pi$
$648$	0	0
$649$	−1.53660	−0.0603167
$650$	0	0
$651$	0.744520	0.0291800
$652$	0	0
$653$	−22.4510	−0.878577	−0.439288	−	0.898346i	$-0.644769\pi$
$653$	−22.4510	−0.878577	−0.439288	+	0.898346i	$0.644769\pi$
$654$	0	0
$655$	−40.7692	−1.59299
$656$	0	0
$657$	−4.95506	−0.193315
$658$	0	0
$659$	49.5169	1.92890	0.964452	−	0.264260i	$-0.0851276\pi$
$659$	49.5169	1.92890	0.964452	+	0.264260i	$0.0851276\pi$
$660$	0	0
$661$	−19.5166	−0.759109	−0.379554	−	0.925169i	$-0.623923\pi$
$661$	−19.5166	−0.759109	−0.379554	+	0.925169i	$0.623923\pi$
$662$	0	0
$663$	0.274676	0.0106675
$664$	0	0
$665$	29.8596	1.15791
$666$	0	0
$667$	−16.6720	−0.645541
$668$	0	0
$669$	1.10207	0.0426084
$670$	0	0
$671$	0.291145	0.0112395
$672$	0	0
$673$	−2.03282	−0.0783595	−0.0391797	−	0.999232i	$-0.512474\pi$
$673$	−2.03282	−0.0783595	−0.0391797	+	0.999232i	$0.512474\pi$
$674$	0	0
$675$	5.20827	0.200467
$676$	0	0
$677$	26.7787	1.02919	0.514595	−	0.857433i	$-0.327942\pi$
$677$	26.7787	1.02919	0.514595	+	0.857433i	$0.327942\pi$
$678$	0	0
$679$	−5.47126	−0.209968
$680$	0	0
$681$	−1.41197	−0.0541070
$682$	0	0
$683$	13.8120	0.528500	0.264250	−	0.964454i	$-0.414876\pi$
$683$	13.8120	0.528500	0.264250	+	0.964454i	$0.414876\pi$
$684$	0	0
$685$	30.3641	1.16015
$686$	0	0
$687$	1.07362	0.0409610
$688$	0	0
$689$	−27.7403	−1.05682
$690$	0	0
$691$	−12.6835	−0.482504	−0.241252	−	0.970462i	$-0.577558\pi$
$691$	−12.6835	−0.482504	−0.241252	+	0.970462i	$0.577558\pi$
$692$	0	0
$693$	0.643775	0.0244550
$694$	0	0
$695$	35.5608	1.34890
$696$	0	0
$697$	2.83416	0.107352
$698$	0	0
$699$	1.80755	0.0683679
$700$	0	0
$701$	−15.4620	−0.583992	−0.291996	−	0.956419i	$-0.594319\pi$
$701$	−15.4620	−0.583992	−0.291996	+	0.956419i	$0.594319\pi$
$702$	0	0
$703$	4.24720	0.160186
$704$	0	0
$705$	−3.04140	−0.114546
$706$	0	0
$707$	−3.85531	−0.144994
$708$	0	0
$709$	−25.8423	−0.970527	−0.485263	−	0.874368i	$-0.661276\pi$
$709$	−25.8423	−0.970527	−0.485263	+	0.874368i	$0.661276\pi$
$710$	0	0
$711$	45.9759	1.72423
$712$	0	0
$713$	−22.0969	−0.827537
$714$	0	0
$715$	−2.99566	−0.112031
$716$	0	0
$717$	−0.991025	−0.0370105
$718$	0	0
$719$	−0.741955	−0.0276703	−0.0138351	−	0.999904i	$-0.504404\pi$
$719$	−0.741955	−0.0276703	−0.0138351	+	0.999904i	$0.504404\pi$
$720$	0	0
$721$	−7.49307	−0.279057
$722$	0	0
$723$	0.916532	0.0340862
$724$	0	0
$725$	60.0824	2.23140
$726$	0	0
$727$	11.5068	0.426765	0.213382	−	0.976969i	$-0.431552\pi$
$727$	11.5068	0.426765	0.213382	+	0.976969i	$0.431552\pi$
$728$	0	0
$729$	−26.7121	−0.989338
$730$	0	0
$731$	−7.50909	−0.277734
$732$	0	0
$733$	28.6560	1.05843	0.529217	−	0.848486i	$-0.322486\pi$
$733$	28.6560	1.05843	0.529217	+	0.848486i	$0.322486\pi$
$734$	0	0
$735$	−1.40694	−0.0518959
$736$	0	0
$737$	0.118719	0.00437307
$738$	0	0
$739$	−12.4543	−0.458140	−0.229070	−	0.973410i	$-0.573569\pi$
$739$	−12.4543	−0.458140	−0.229070	+	0.973410i	$0.573569\pi$
$740$	0	0
$741$	−1.80001	−0.0661250
$742$	0	0
$743$	−29.9540	−1.09891	−0.549453	−	0.835525i	$-0.685164\pi$
$743$	−29.9540	−1.09891	−0.549453	+	0.835525i	$0.685164\pi$
$744$	0	0
$745$	24.8401	0.910072
$746$	0	0
$747$	13.4201	0.491015
$748$	0	0
$749$	21.6621	0.791516
$750$	0	0
$751$	−41.4823	−1.51371	−0.756856	−	0.653582i	$-0.773266\pi$
$751$	−41.4823	−1.51371	−0.756856	+	0.653582i	$0.773266\pi$
$752$	0	0
$753$	0.121294	0.00442019
$754$	0	0
$755$	−72.6773	−2.64500
$756$	0	0
$757$	27.8676	1.01287	0.506433	−	0.862279i	$-0.330964\pi$
$757$	27.8676	1.01287	0.506433	+	0.862279i	$0.330964\pi$
$758$	0	0
$759$	0.0340641	0.00123645
$760$	0	0
$761$	26.3231	0.954210	0.477105	−	0.878846i	$-0.341686\pi$
$761$	26.3231	0.954210	0.477105	+	0.878846i	$0.341686\pi$
$762$	0	0
$763$	27.1056	0.981289
$764$	0	0
$765$	−8.96849	−0.324257
$766$	0	0
$767$	−56.1014	−2.02570
$768$	0	0
$769$	−11.6606	−0.420494	−0.210247	−	0.977648i	$-0.567427\pi$
$769$	−11.6606	−0.420494	−0.210247	+	0.977648i	$0.567427\pi$
$770$	0	0
$771$	−1.71485	−0.0617589
$772$	0	0
$773$	−5.19436	−0.186828	−0.0934141	−	0.995627i	$-0.529778\pi$
$773$	−5.19436	−0.186828	−0.0934141	+	0.995627i	$0.529778\pi$
$774$	0	0
$775$	79.6329	2.86050
$776$	0	0
$777$	0.0988744	0.00354710
$778$	0	0
$779$	−18.5729	−0.665442
$780$	0	0
$781$	−0.446682	−0.0159835
$782$	0	0
$783$	2.21329	0.0790965
$784$	0	0
$785$	72.5138	2.58813
$786$	0	0
$787$	30.1729	1.07555	0.537774	−	0.843089i	$-0.319265\pi$
$787$	30.1729	1.07555	0.537774	+	0.843089i	$0.319265\pi$
$788$	0	0
$789$	−1.73532	−0.0617791
$790$	0	0
$791$	−31.1649	−1.10810
$792$	0	0
$793$	10.6297	0.377473
$794$	0	0
$795$	−1.61480	−0.0572711
$796$	0	0
$797$	−37.8145	−1.33946	−0.669729	−	0.742605i	$-0.733590\pi$
$797$	−37.8145	−1.33946	−0.669729	+	0.742605i	$0.733590\pi$
$798$	0	0
$799$	7.38031	0.261096
$800$	0	0
$801$	−28.1718	−0.995403
$802$	0	0
$803$	0.233792	0.00825034
$804$	0	0
$805$	−20.6310	−0.727148
$806$	0	0
$807$	1.90920	0.0672072
$808$	0	0
$809$	−22.6375	−0.795893	−0.397947	−	0.917409i	$-0.630277\pi$
$809$	−22.6375	−0.795893	−0.397947	+	0.917409i	$0.630277\pi$
$810$	0	0
$811$	17.4411	0.612440	0.306220	−	0.951961i	$-0.400936\pi$
$811$	17.4411	0.612440	0.306220	+	0.951961i	$0.400936\pi$
$812$	0	0
$813$	0.199079	0.00698202
$814$	0	0
$815$	34.0791	1.19374
$816$	0	0
$817$	49.2087	1.72159
$818$	0	0
$819$	23.5043	0.821307
$820$	0	0
$821$	−52.3011	−1.82532	−0.912660	−	0.408720i	$-0.865975\pi$
$821$	−52.3011	−1.82532	−0.912660	+	0.408720i	$0.865975\pi$
$822$	0	0
$823$	18.3400	0.639293	0.319647	−	0.947537i	$-0.396436\pi$
$823$	18.3400	0.639293	0.319647	+	0.947537i	$0.396436\pi$
$824$	0	0
$825$	−0.122760	−0.00427396
$826$	0	0
$827$	−20.4391	−0.710738	−0.355369	−	0.934726i	$-0.615645\pi$
$827$	−20.4391	−0.710738	−0.355369	+	0.934726i	$0.615645\pi$
$828$	0	0
$829$	−31.9572	−1.10992	−0.554959	−	0.831878i	$-0.687266\pi$
$829$	−31.9572	−1.10992	−0.554959	+	0.831878i	$0.687266\pi$
$830$	0	0
$831$	1.50936	0.0523591
$832$	0	0
$833$	3.41411	0.118292
$834$	0	0
$835$	22.6702	0.784535
$836$	0	0
$837$	2.93348	0.101396
$838$	0	0
$839$	5.32138	0.183715	0.0918573	−	0.995772i	$-0.470720\pi$
$839$	5.32138	0.183715	0.0918573	+	0.995772i	$0.470720\pi$
$840$	0	0
$841$	−3.46760	−0.119572
$842$	0	0
$843$	1.06138	0.0365557
$844$	0	0
$845$	−55.9444	−1.92455
$846$	0	0
$847$	16.7056	0.574010
$848$	0	0
$849$	0.906880	0.0311240
$850$	0	0
$851$	−2.93454	−0.100595
$852$	0	0
$853$	20.3101	0.695405	0.347702	−	0.937605i	$-0.386962\pi$
$853$	20.3101	0.695405	0.347702	+	0.937605i	$0.386962\pi$
$854$	0	0
$855$	58.7725	2.00998
$856$	0	0
$857$	−25.7179	−0.878506	−0.439253	−	0.898363i	$-0.644757\pi$
$857$	−25.7179	−0.878506	−0.439253	+	0.898363i	$0.644757\pi$
$858$	0	0
$859$	−32.0127	−1.09226	−0.546130	−	0.837700i	$-0.683900\pi$
$859$	−32.0127	−1.09226	−0.546130	+	0.837700i	$0.683900\pi$
$860$	0	0
$861$	−0.432374	−0.0147353
$862$	0	0
$863$	−4.70980	−0.160324	−0.0801618	−	0.996782i	$-0.525544\pi$
$863$	−4.70980	−0.160324	−0.0801618	+	0.996782i	$0.525544\pi$
$864$	0	0
$865$	70.9327	2.41178
$866$	0	0
$867$	1.20336	0.0408682
$868$	0	0
$869$	−2.16926	−0.0735871
$870$	0	0
$871$	4.33444	0.146867
$872$	0	0
$873$	−10.7690	−0.364477
$874$	0	0
$875$	43.0856	1.45656
$876$	0	0
$877$	34.8968	1.17838	0.589191	−	0.807994i	$-0.299447\pi$
$877$	34.8968	1.17838	0.589191	+	0.807994i	$0.299447\pi$
$878$	0	0
$879$	−2.01807	−0.0680679
$880$	0	0
$881$	15.0888	0.508356	0.254178	−	0.967157i	$-0.418195\pi$
$881$	15.0888	0.508356	0.254178	+	0.967157i	$0.418195\pi$
$882$	0	0
$883$	20.0939	0.676215	0.338108	−	0.941107i	$-0.390213\pi$
$883$	20.0939	0.676215	0.338108	+	0.941107i	$0.390213\pi$
$884$	0	0
$885$	−3.26573	−0.109776
$886$	0	0
$887$	−13.3512	−0.448289	−0.224144	−	0.974556i	$-0.571959\pi$
$887$	−13.3512	−0.448289	−0.224144	+	0.974556i	$0.571959\pi$
$888$	0	0
$889$	−9.00858	−0.302138
$890$	0	0
$891$	1.26488	0.0423749
$892$	0	0
$893$	−48.3648	−1.61846
$894$	0	0
$895$	−80.1297	−2.67844
$896$	0	0
$897$	1.24369	0.0415255
$898$	0	0
$899$	33.8405	1.12864
$900$	0	0
$901$	3.91850	0.130544
$902$	0	0
$903$	1.14557	0.0381223
$904$	0	0
$905$	64.3885	2.14035
$906$	0	0
$907$	−28.2979	−0.939617	−0.469809	−	0.882768i	$-0.655677\pi$
$907$	−28.2979	−0.939617	−0.469809	+	0.882768i	$0.655677\pi$
$908$	0	0
$909$	−7.58839	−0.251691
$910$	0	0
$911$	−11.4782	−0.380289	−0.190145	−	0.981756i	$-0.560896\pi$
$911$	−11.4782	−0.380289	−0.190145	+	0.981756i	$0.560896\pi$
$912$	0	0
$913$	−0.633193	−0.0209556
$914$	0	0
$915$	0.618771	0.0204559
$916$	0	0
$917$	−15.0928	−0.498407
$918$	0	0
$919$	10.2410	0.337821	0.168910	−	0.985631i	$-0.445975\pi$
$919$	10.2410	0.337821	0.168910	+	0.985631i	$0.445975\pi$
$920$	0	0
$921$	−2.02543	−0.0667402
$922$	0	0
$923$	−16.3084	−0.536798
$924$	0	0
$925$	10.5755	0.347720
$926$	0	0
$927$	−14.7486	−0.484406
$928$	0	0
$929$	−39.8486	−1.30739	−0.653696	−	0.756758i	$-0.726782\pi$
$929$	−39.8486	−1.30739	−0.653696	+	0.756758i	$0.726782\pi$
$930$	0	0
$931$	−22.3734	−0.733258
$932$	0	0
$933$	−1.95743	−0.0640833
$934$	0	0
$935$	0.423156	0.0138387
$936$	0	0
$937$	−30.7233	−1.00368	−0.501842	−	0.864959i	$-0.667344\pi$
$937$	−30.7233	−1.00368	−0.501842	+	0.864959i	$0.667344\pi$
$938$	0	0
$939$	−2.35339	−0.0768000
$940$	0	0
$941$	−33.5974	−1.09524	−0.547622	−	0.836726i	$-0.684467\pi$
$941$	−33.5974	−1.09524	−0.547622	+	0.836726i	$0.684467\pi$
$942$	0	0
$943$	12.8326	0.417888
$944$	0	0
$945$	2.73887	0.0890955
$946$	0	0
$947$	44.7905	1.45549	0.727747	−	0.685845i	$-0.240567\pi$
$947$	44.7905	1.45549	0.727747	+	0.685845i	$0.240567\pi$
$948$	0	0
$949$	8.53578	0.277083
$950$	0	0
$951$	−0.409389	−0.0132754
$952$	0	0
$953$	42.3911	1.37318	0.686591	−	0.727044i	$-0.259106\pi$
$953$	42.3911	1.37318	0.686591	+	0.727044i	$0.259106\pi$
$954$	0	0
$955$	−12.2036	−0.394899
$956$	0	0
$957$	−0.0521678	−0.00168634
$958$	0	0
$959$	11.2408	0.362984
$960$	0	0
$961$	13.8520	0.446839
$962$	0	0
$963$	42.6374	1.37397
$964$	0	0
$965$	−2.36389	−0.0760963
$966$	0	0
$967$	−34.5556	−1.11123	−0.555617	−	0.831438i	$-0.687518\pi$
$967$	−34.5556	−1.11123	−0.555617	+	0.831438i	$0.687518\pi$
$968$	0	0
$969$	0.254263	0.00816809
$970$	0	0
$971$	−25.3833	−0.814590	−0.407295	−	0.913297i	$-0.633528\pi$
$971$	−25.3833	−0.814590	−0.407295	+	0.913297i	$0.633528\pi$
$972$	0	0
$973$	13.1646	0.422038
$974$	0	0
$975$	−4.48199	−0.143539
$976$	0	0
$977$	−9.73604	−0.311484	−0.155742	−	0.987798i	$-0.549777\pi$
$977$	−9.73604	−0.311484	−0.155742	+	0.987798i	$0.549777\pi$
$978$	0	0
$979$	1.32922	0.0424820
$980$	0	0
$981$	53.3518	1.70339
$982$	0	0
$983$	25.0925	0.800325	0.400162	−	0.916444i	$-0.368954\pi$
$983$	25.0925	0.800325	0.400162	+	0.916444i	$0.368954\pi$
$984$	0	0
$985$	−43.5623	−1.38801
$986$	0	0
$987$	−1.12593	−0.0358386
$988$	0	0
$989$	−33.9999	−1.08114
$990$	0	0
$991$	12.5639	0.399104	0.199552	−	0.979887i	$-0.436051\pi$
$991$	12.5639	0.399104	0.199552	+	0.979887i	$0.436051\pi$
$992$	0	0
$993$	0.510680	0.0162059
$994$	0	0
$995$	−33.7774	−1.07082
$996$	0	0
$997$	33.0261	1.04595	0.522974	−	0.852349i	$-0.324823\pi$
$997$	33.0261	1.04595	0.522974	+	0.852349i	$0.324823\pi$
$998$	0	0
$999$	0.389575	0.0123256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.d.1.14	✓	28
4.3	odd	2		8048.2.a.v.1.15		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.14	✓	28	1.1	even	1	trivial
8048.2.a.v.1.15		28	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0730681 q^{3} -4.10981 q^{5} -1.52145 q^{7} -2.99466 q^{9} +O(q^{10})\)	\(q-0.0730681 q^{3} -4.10981 q^{5} -1.52145 q^{7} -2.99466 q^{9} +0.141296 q^{11} +5.15872 q^{13} +0.300296 q^{15} -0.728702 q^{17} +4.77534 q^{19} +0.111169 q^{21} -3.29945 q^{23} +11.8905 q^{25} +0.438018 q^{27} +5.05296 q^{29} +6.69716 q^{31} -0.0103242 q^{33} +6.25287 q^{35} +0.889403 q^{37} -0.376938 q^{39} -3.88933 q^{41} +10.3047 q^{43} +12.3075 q^{45} -10.1280 q^{47} -4.68519 q^{49} +0.0532449 q^{51} -5.37737 q^{53} -0.580698 q^{55} -0.348925 q^{57} -10.8751 q^{59} +2.06054 q^{61} +4.55623 q^{63} -21.2014 q^{65} +0.840216 q^{67} +0.241084 q^{69} -3.16133 q^{71} +1.65463 q^{73} -0.868819 q^{75} -0.214974 q^{77} -15.3526 q^{79} +8.95198 q^{81} -4.48134 q^{83} +2.99483 q^{85} -0.369210 q^{87} +9.40735 q^{89} -7.84874 q^{91} -0.489349 q^{93} -19.6258 q^{95} +3.59608 q^{97} -0.423132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9} - 14 q^{11} - 31 q^{13} - 2 q^{15} - 9 q^{17} - 8 q^{19} - 28 q^{21} - 4 q^{23} + 22 q^{25} - 4 q^{27} - 47 q^{29} - 5 q^{31} - 26 q^{33} - 13 q^{35} - 67 q^{37} - 9 q^{39} - 28 q^{41} + 15 q^{43} - 57 q^{45} - 10 q^{47} + 20 q^{49} - 11 q^{51} - 58 q^{53} + 15 q^{55} - 31 q^{57} - 32 q^{59} - 55 q^{61} - 16 q^{63} - 44 q^{65} + 22 q^{67} - 44 q^{69} - 47 q^{71} - 5 q^{73} - 25 q^{75} - 50 q^{77} - 14 q^{79} - 28 q^{81} - 16 q^{83} - 78 q^{85} - 11 q^{87} - 20 q^{89} - 15 q^{91} - 83 q^{93} - 27 q^{95} - 8 q^{97} - 70 q^{99}+O(q^{100}) \) 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9 - 14 * q^11 - 31 * q^13 - 2 * q^15 - 9 * q^17 - 8 * q^19 - 28 * q^21 - 4 * q^23 + 22 * q^25 - 4 * q^27 - 47 * q^29 - 5 * q^31 - 26 * q^33 - 13 * q^35 - 67 * q^37 - 9 * q^39 - 28 * q^41 + 15 * q^43 - 57 * q^45 - 10 * q^47 + 20 * q^49 - 11 * q^51 - 58 * q^53 + 15 * q^55 - 31 * q^57 - 32 * q^59 - 55 * q^61 - 16 * q^63 - 44 * q^65 + 22 * q^67 - 44 * q^69 - 47 * q^71 - 5 * q^73 - 25 * q^75 - 50 * q^77 - 14 * q^79 - 28 * q^81 - 16 * q^83 - 78 * q^85 - 11 * q^87 - 20 * q^89 - 15 * q^91 - 83 * q^93 - 27 * q^95 - 8 * q^97 - 70 * q^99

Introduction

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