LMFDB - Embedded newform 4024.2.a.d.1.28

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			1.28
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)

$f(q)$	$=$	$q+3.18633 q^{3} -1.57145 q^{5} -3.06067 q^{7} +7.15268 q^{9} +O(q^{10})$	$q+3.18633 q^{3} -1.57145 q^{5} -3.06067 q^{7} +7.15268 q^{9} +0.545150 q^{11} -1.64668 q^{13} -5.00716 q^{15} -3.65160 q^{17} -2.61130 q^{19} -9.75229 q^{21} -3.92971 q^{23} -2.53054 q^{25} +13.2318 q^{27} +7.11113 q^{29} -7.75620 q^{31} +1.73703 q^{33} +4.80969 q^{35} -9.57716 q^{37} -5.24686 q^{39} -1.04104 q^{41} +10.0422 q^{43} -11.2401 q^{45} -2.76111 q^{47} +2.36768 q^{49} -11.6352 q^{51} -9.38236 q^{53} -0.856678 q^{55} -8.32047 q^{57} -2.80183 q^{59} +5.26624 q^{61} -21.8920 q^{63} +2.58768 q^{65} +7.28320 q^{67} -12.5214 q^{69} -3.46671 q^{71} -15.1421 q^{73} -8.06313 q^{75} -1.66852 q^{77} -9.12623 q^{79} +20.7028 q^{81} +5.33358 q^{83} +5.73832 q^{85} +22.6584 q^{87} -12.2732 q^{89} +5.03994 q^{91} -24.7138 q^{93} +4.10354 q^{95} +12.0744 q^{97} +3.89929 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.18633	1.83963	0.919814	−	0.392356i	$-0.128340\pi$
$3$	3.18633	1.83963	0.919814	+	0.392356i	$0.128340\pi$
$4$	0	0
$5$	−1.57145	−0.702775	−0.351387	−	0.936230i	$-0.614290\pi$
$5$	−1.57145	−0.702775	−0.351387	+	0.936230i	$0.614290\pi$
$6$	0	0
$7$	−3.06067	−1.15682	−0.578412	−	0.815745i	$-0.696327\pi$
$7$	−3.06067	−1.15682	−0.578412	+	0.815745i	$0.696327\pi$
$8$	0	0
$9$	7.15268	2.38423
$10$	0	0
$11$	0.545150	0.164369	0.0821845	−	0.996617i	$-0.473810\pi$
$11$	0.545150	0.164369	0.0821845	+	0.996617i	$0.473810\pi$
$12$	0	0
$13$	−1.64668	−0.456707	−0.228353	−	0.973578i	$-0.573334\pi$
$13$	−1.64668	−0.456707	−0.228353	+	0.973578i	$0.573334\pi$
$14$	0	0
$15$	−5.00716	−1.29284
$16$	0	0
$17$	−3.65160	−0.885644	−0.442822	−	0.896610i	$-0.646023\pi$
$17$	−3.65160	−0.885644	−0.442822	+	0.896610i	$0.646023\pi$
$18$	0	0
$19$	−2.61130	−0.599074	−0.299537	−	0.954085i	$-0.596832\pi$
$19$	−2.61130	−0.599074	−0.299537	+	0.954085i	$0.596832\pi$
$20$	0	0
$21$	−9.75229	−2.12812
$22$	0	0
$23$	−3.92971	−0.819402	−0.409701	−	0.912220i	$-0.634367\pi$
$23$	−3.92971	−0.819402	−0.409701	+	0.912220i	$0.634367\pi$
$24$	0	0
$25$	−2.53054	−0.506108
$26$	0	0
$27$	13.2318	2.54646
$28$	0	0
$29$	7.11113	1.32050	0.660252	−	0.751044i	$-0.270450\pi$
$29$	7.11113	1.32050	0.660252	+	0.751044i	$0.270450\pi$
$30$	0	0
$31$	−7.75620	−1.39305	−0.696527	−	0.717531i	$-0.745272\pi$
$31$	−7.75620	−1.39305	−0.696527	+	0.717531i	$0.745272\pi$
$32$	0	0
$33$	1.73703	0.302378
$34$	0	0
$35$	4.80969	0.812986
$36$	0	0
$37$	−9.57716	−1.57448	−0.787238	−	0.616650i	$-0.788490\pi$
$37$	−9.57716	−1.57448	−0.787238	+	0.616650i	$0.788490\pi$
$38$	0	0
$39$	−5.24686	−0.840170
$40$	0	0
$41$	−1.04104	−0.162583	−0.0812917	−	0.996690i	$-0.525905\pi$
$41$	−1.04104	−0.162583	−0.0812917	+	0.996690i	$0.525905\pi$
$42$	0	0
$43$	10.0422	1.53141	0.765707	−	0.643190i	$-0.222389\pi$
$43$	10.0422	1.53141	0.765707	+	0.643190i	$0.222389\pi$
$44$	0	0
$45$	−11.2401	−1.67557
$46$	0	0
$47$	−2.76111	−0.402749	−0.201375	−	0.979514i	$-0.564541\pi$
$47$	−2.76111	−0.402749	−0.201375	+	0.979514i	$0.564541\pi$
$48$	0	0
$49$	2.36768	0.338240
$50$	0	0
$51$	−11.6352	−1.62925
$52$	0	0
$53$	−9.38236	−1.28877	−0.644383	−	0.764703i	$-0.722886\pi$
$53$	−9.38236	−1.28877	−0.644383	+	0.764703i	$0.722886\pi$
$54$	0	0
$55$	−0.856678	−0.115514
$56$	0	0
$57$	−8.32047	−1.10207
$58$	0	0
$59$	−2.80183	−0.364767	−0.182383	−	0.983228i	$-0.558381\pi$
$59$	−2.80183	−0.364767	−0.182383	+	0.983228i	$0.558381\pi$
$60$	0	0
$61$	5.26624	0.674273	0.337136	−	0.941456i	$-0.390542\pi$
$61$	5.26624	0.674273	0.337136	+	0.941456i	$0.390542\pi$
$62$	0	0
$63$	−21.8920	−2.75813
$64$	0	0
$65$	2.58768	0.320962
$66$	0	0
$67$	7.28320	0.889784	0.444892	−	0.895584i	$-0.353242\pi$
$67$	7.28320	0.889784	0.444892	+	0.895584i	$0.353242\pi$
$68$	0	0
$69$	−12.5214	−1.50739
$70$	0	0
$71$	−3.46671	−0.411422	−0.205711	−	0.978613i	$-0.565951\pi$
$71$	−3.46671	−0.411422	−0.205711	+	0.978613i	$0.565951\pi$
$72$	0	0
$73$	−15.1421	−1.77225	−0.886123	−	0.463450i	$-0.846612\pi$
$73$	−15.1421	−1.77225	−0.886123	+	0.463450i	$0.846612\pi$
$74$	0	0
$75$	−8.06313	−0.931050
$76$	0	0
$77$	−1.66852	−0.190146
$78$	0	0
$79$	−9.12623	−1.02678	−0.513391	−	0.858155i	$-0.671611\pi$
$79$	−9.12623	−1.02678	−0.513391	+	0.858155i	$0.671611\pi$
$80$	0	0
$81$	20.7028	2.30031
$82$	0	0
$83$	5.33358	0.585436	0.292718	−	0.956199i	$-0.405440\pi$
$83$	5.33358	0.585436	0.292718	+	0.956199i	$0.405440\pi$
$84$	0	0
$85$	5.73832	0.622408
$86$	0	0
$87$	22.6584	2.42923
$88$	0	0
$89$	−12.2732	−1.30096	−0.650481	−	0.759523i	$-0.725432\pi$
$89$	−12.2732	−1.30096	−0.650481	+	0.759523i	$0.725432\pi$
$90$	0	0
$91$	5.03994	0.528329
$92$	0	0
$93$	−24.7138	−2.56270
$94$	0	0
$95$	4.10354	0.421014
$96$	0	0
$97$	12.0744	1.22597	0.612983	−	0.790096i	$-0.289970\pi$
$97$	12.0744	1.22597	0.612983	+	0.790096i	$0.289970\pi$
$98$	0	0
$99$	3.89929	0.391893
$100$	0	0
$101$	18.6218	1.85294	0.926468	−	0.376374i	$-0.122829\pi$
$101$	18.6218	1.85294	0.926468	+	0.376374i	$0.122829\pi$
$102$	0	0
$103$	−1.34086	−0.132119	−0.0660596	−	0.997816i	$-0.521043\pi$
$103$	−1.34086	−0.132119	−0.0660596	+	0.997816i	$0.521043\pi$
$104$	0	0
$105$	15.3252	1.49559
$106$	0	0
$107$	0.0262517	0.00253785	0.00126892	−	0.999999i	$-0.499596\pi$
$107$	0.0262517	0.00253785	0.00126892	+	0.999999i	$0.499596\pi$
$108$	0	0
$109$	0.350614	0.0335828	0.0167914	−	0.999859i	$-0.494655\pi$
$109$	0.350614	0.0335828	0.0167914	+	0.999859i	$0.494655\pi$
$110$	0	0
$111$	−30.5160	−2.89645
$112$	0	0
$113$	2.16637	0.203795	0.101897	−	0.994795i	$-0.467509\pi$
$113$	2.16637	0.203795	0.101897	+	0.994795i	$0.467509\pi$
$114$	0	0
$115$	6.17535	0.575855
$116$	0	0
$117$	−11.7782	−1.08889
$118$	0	0
$119$	11.1763	1.02453
$120$	0	0
$121$	−10.7028	−0.972983
$122$	0	0
$123$	−3.31710	−0.299093
$124$	0	0
$125$	11.8339	1.05845
$126$	0	0
$127$	−3.24877	−0.288282	−0.144141	−	0.989557i	$-0.546042\pi$
$127$	−3.24877	−0.288282	−0.144141	+	0.989557i	$0.546042\pi$
$128$	0	0
$129$	31.9976	2.81723
$130$	0	0
$131$	−5.29059	−0.462241	−0.231120	−	0.972925i	$-0.574239\pi$
$131$	−5.29059	−0.462241	−0.231120	+	0.972925i	$0.574239\pi$
$132$	0	0
$133$	7.99233	0.693023
$134$	0	0
$135$	−20.7932	−1.78959
$136$	0	0
$137$	16.5680	1.41550	0.707751	−	0.706462i	$-0.249710\pi$
$137$	16.5680	1.41550	0.707751	+	0.706462i	$0.249710\pi$
$138$	0	0
$139$	−14.2940	−1.21240	−0.606202	−	0.795311i	$-0.707308\pi$
$139$	−14.2940	−1.21240	−0.606202	+	0.795311i	$0.707308\pi$
$140$	0	0
$141$	−8.79780	−0.740909
$142$	0	0
$143$	−0.897688	−0.0750685
$144$	0	0
$145$	−11.1748	−0.928016
$146$	0	0
$147$	7.54421	0.622236
$148$	0	0
$149$	2.98494	0.244536	0.122268	−	0.992497i	$-0.460983\pi$
$149$	2.98494	0.244536	0.122268	+	0.992497i	$0.460983\pi$
$150$	0	0
$151$	−11.2406	−0.914744	−0.457372	−	0.889276i	$-0.651209\pi$
$151$	−11.2406	−0.914744	−0.457372	+	0.889276i	$0.651209\pi$
$152$	0	0
$153$	−26.1188	−2.11158
$154$	0	0
$155$	12.1885	0.979003
$156$	0	0
$157$	11.9185	0.951201	0.475601	−	0.879661i	$-0.342231\pi$
$157$	11.9185	0.951201	0.475601	+	0.879661i	$0.342231\pi$
$158$	0	0
$159$	−29.8953	−2.37085
$160$	0	0
$161$	12.0275	0.947903
$162$	0	0
$163$	11.8331	0.926840	0.463420	−	0.886139i	$-0.346622\pi$
$163$	11.8331	0.926840	0.463420	+	0.886139i	$0.346622\pi$
$164$	0	0
$165$	−2.72966	−0.212503
$166$	0	0
$167$	−5.94973	−0.460404	−0.230202	−	0.973143i	$-0.573939\pi$
$167$	−5.94973	−0.460404	−0.230202	+	0.973143i	$0.573939\pi$
$168$	0	0
$169$	−10.2884	−0.791419
$170$	0	0
$171$	−18.6778	−1.42833
$172$	0	0
$173$	−8.47330	−0.644213	−0.322107	−	0.946703i	$-0.604391\pi$
$173$	−8.47330	−0.644213	−0.322107	+	0.946703i	$0.604391\pi$
$174$	0	0
$175$	7.74514	0.585477
$176$	0	0
$177$	−8.92754	−0.671035
$178$	0	0
$179$	−18.8938	−1.41219	−0.706094	−	0.708118i	$-0.749545\pi$
$179$	−18.8938	−1.41219	−0.706094	+	0.708118i	$0.749545\pi$
$180$	0	0
$181$	8.01854	0.596013	0.298007	−	0.954564i	$-0.403678\pi$
$181$	8.01854	0.596013	0.298007	+	0.954564i	$0.403678\pi$
$182$	0	0
$183$	16.7800	1.24041
$184$	0	0
$185$	15.0500	1.10650
$186$	0	0
$187$	−1.99067	−0.145572
$188$	0	0
$189$	−40.4982	−2.94581
$190$	0	0
$191$	24.1165	1.74501	0.872503	−	0.488609i	$-0.162496\pi$
$191$	24.1165	1.74501	0.872503	+	0.488609i	$0.162496\pi$
$192$	0	0
$193$	−8.50608	−0.612281	−0.306140	−	0.951986i	$-0.599038\pi$
$193$	−8.50608	−0.612281	−0.306140	+	0.951986i	$0.599038\pi$
$194$	0	0
$195$	8.24519	0.590450
$196$	0	0
$197$	−25.3232	−1.80420	−0.902101	−	0.431524i	$-0.857976\pi$
$197$	−25.3232	−1.80420	−0.902101	+	0.431524i	$0.857976\pi$
$198$	0	0
$199$	3.62162	0.256730	0.128365	−	0.991727i	$-0.459027\pi$
$199$	3.62162	0.256730	0.128365	+	0.991727i	$0.459027\pi$
$200$	0	0
$201$	23.2066	1.63687
$202$	0	0
$203$	−21.7648	−1.52759
$204$	0	0
$205$	1.63595	0.114259
$206$	0	0
$207$	−28.1080	−1.95364
$208$	0	0
$209$	−1.42355	−0.0984692
$210$	0	0
$211$	3.15954	0.217512	0.108756	−	0.994068i	$-0.465313\pi$
$211$	3.15954	0.217512	0.108756	+	0.994068i	$0.465313\pi$
$212$	0	0
$213$	−11.0461	−0.756864
$214$	0	0
$215$	−15.7808	−1.07624
$216$	0	0
$217$	23.7391	1.61152
$218$	0	0
$219$	−48.2476	−3.26027
$220$	0	0
$221$	6.01302	0.404480
$222$	0	0
$223$	1.82132	0.121965	0.0609824	−	0.998139i	$-0.480577\pi$
$223$	1.82132	0.121965	0.0609824	+	0.998139i	$0.480577\pi$
$224$	0	0
$225$	−18.1001	−1.20668
$226$	0	0
$227$	−19.8468	−1.31728	−0.658638	−	0.752460i	$-0.728867\pi$
$227$	−19.8468	−1.31728	−0.658638	+	0.752460i	$0.728867\pi$
$228$	0	0
$229$	−15.0063	−0.991645	−0.495822	−	0.868424i	$-0.665133\pi$
$229$	−15.0063	−0.991645	−0.495822	+	0.868424i	$0.665133\pi$
$230$	0	0
$231$	−5.31646	−0.349798
$232$	0	0
$233$	−7.99972	−0.524079	−0.262039	−	0.965057i	$-0.584395\pi$
$233$	−7.99972	−0.524079	−0.262039	+	0.965057i	$0.584395\pi$
$234$	0	0
$235$	4.33895	0.283042
$236$	0	0
$237$	−29.0792	−1.88889
$238$	0	0
$239$	−10.0846	−0.652318	−0.326159	−	0.945315i	$-0.605754\pi$
$239$	−10.0846	−0.652318	−0.326159	+	0.945315i	$0.605754\pi$
$240$	0	0
$241$	8.19257	0.527730	0.263865	−	0.964560i	$-0.415003\pi$
$241$	8.19257	0.527730	0.263865	+	0.964560i	$0.415003\pi$
$242$	0	0
$243$	26.2706	1.68526
$244$	0	0
$245$	−3.72070	−0.237707
$246$	0	0
$247$	4.29998	0.273601
$248$	0	0
$249$	16.9945	1.07698
$250$	0	0
$251$	14.1224	0.891396	0.445698	−	0.895183i	$-0.352956\pi$
$251$	14.1224	0.891396	0.445698	+	0.895183i	$0.352956\pi$
$252$	0	0
$253$	−2.14228	−0.134684
$254$	0	0
$255$	18.2842	1.14500
$256$	0	0
$257$	1.57153	0.0980294	0.0490147	−	0.998798i	$-0.484392\pi$
$257$	1.57153	0.0980294	0.0490147	+	0.998798i	$0.484392\pi$
$258$	0	0
$259$	29.3125	1.82139
$260$	0	0
$261$	50.8637	3.14838
$262$	0	0
$263$	5.79538	0.357358	0.178679	−	0.983907i	$-0.442818\pi$
$263$	5.79538	0.357358	0.178679	+	0.983907i	$0.442818\pi$
$264$	0	0
$265$	14.7439	0.905713
$266$	0	0
$267$	−39.1066	−2.39328
$268$	0	0
$269$	1.77372	0.108145	0.0540727	−	0.998537i	$-0.482780\pi$
$269$	1.77372	0.108145	0.0540727	+	0.998537i	$0.482780\pi$
$270$	0	0
$271$	16.7882	1.01981	0.509905	−	0.860231i	$-0.329681\pi$
$271$	16.7882	1.01981	0.509905	+	0.860231i	$0.329681\pi$
$272$	0	0
$273$	16.0589	0.971928
$274$	0	0
$275$	−1.37952	−0.0831885
$276$	0	0
$277$	15.1740	0.911716	0.455858	−	0.890053i	$-0.349333\pi$
$277$	15.1740	0.911716	0.455858	+	0.890053i	$0.349333\pi$
$278$	0	0
$279$	−55.4776	−3.32136
$280$	0	0
$281$	−31.1977	−1.86110	−0.930548	−	0.366170i	$-0.880669\pi$
$281$	−31.1977	−1.86110	−0.930548	+	0.366170i	$0.880669\pi$
$282$	0	0
$283$	−14.6509	−0.870908	−0.435454	−	0.900211i	$-0.643412\pi$
$283$	−14.6509	−0.870908	−0.435454	+	0.900211i	$0.643412\pi$
$284$	0	0
$285$	13.0752	0.774509
$286$	0	0
$287$	3.18628	0.188080
$288$	0	0
$289$	−3.66579	−0.215635
$290$	0	0
$291$	38.4729	2.25532
$292$	0	0
$293$	6.59868	0.385499	0.192749	−	0.981248i	$-0.438260\pi$
$293$	6.59868	0.385499	0.192749	+	0.981248i	$0.438260\pi$
$294$	0	0
$295$	4.40293	0.256349
$296$	0	0
$297$	7.21333	0.418560
$298$	0	0
$299$	6.47098	0.374226
$300$	0	0
$301$	−30.7357	−1.77158
$302$	0	0
$303$	59.3351	3.40871
$304$	0	0
$305$	−8.27564	−0.473862
$306$	0	0
$307$	27.4053	1.56410	0.782052	−	0.623213i	$-0.214173\pi$
$307$	27.4053	1.56410	0.782052	+	0.623213i	$0.214173\pi$
$308$	0	0
$309$	−4.27243	−0.243050
$310$	0	0
$311$	2.28546	0.129596	0.0647982	−	0.997898i	$-0.479360\pi$
$311$	2.28546	0.129596	0.0647982	+	0.997898i	$0.479360\pi$
$312$	0	0
$313$	21.2333	1.20018	0.600090	−	0.799933i	$-0.295132\pi$
$313$	21.2333	1.20018	0.600090	+	0.799933i	$0.295132\pi$
$314$	0	0
$315$	34.4022	1.93834
$316$	0	0
$317$	27.7858	1.56060	0.780302	−	0.625403i	$-0.215065\pi$
$317$	27.7858	1.56060	0.780302	+	0.625403i	$0.215065\pi$
$318$	0	0
$319$	3.87664	0.217050
$320$	0	0
$321$	0.0836465	0.00466869
$322$	0	0
$323$	9.53545	0.530566
$324$	0	0
$325$	4.16699	0.231143
$326$	0	0
$327$	1.11717	0.0617798
$328$	0	0
$329$	8.45084	0.465910
$330$	0	0
$331$	28.7972	1.58284	0.791418	−	0.611275i	$-0.209343\pi$
$331$	28.7972	1.58284	0.791418	+	0.611275i	$0.209343\pi$
$332$	0	0
$333$	−68.5024	−3.75391
$334$	0	0
$335$	−11.4452	−0.625318
$336$	0	0
$337$	14.5387	0.791974	0.395987	−	0.918256i	$-0.370402\pi$
$337$	14.5387	0.791974	0.395987	+	0.918256i	$0.370402\pi$
$338$	0	0
$339$	6.90275	0.374906
$340$	0	0
$341$	−4.22829	−0.228975
$342$	0	0
$343$	14.1780	0.765539
$344$	0	0
$345$	19.6767	1.05936
$346$	0	0
$347$	4.66898	0.250644	0.125322	−	0.992116i	$-0.460004\pi$
$347$	4.66898	0.250644	0.125322	+	0.992116i	$0.460004\pi$
$348$	0	0
$349$	−7.22552	−0.386773	−0.193387	−	0.981123i	$-0.561947\pi$
$349$	−7.22552	−0.386773	−0.193387	+	0.981123i	$0.561947\pi$
$350$	0	0
$351$	−21.7886	−1.16299
$352$	0	0
$353$	4.51906	0.240525	0.120263	−	0.992742i	$-0.461626\pi$
$353$	4.51906	0.240525	0.120263	+	0.992742i	$0.461626\pi$
$354$	0	0
$355$	5.44776	0.289137
$356$	0	0
$357$	35.6115	1.88476
$358$	0	0
$359$	−12.1764	−0.642644	−0.321322	−	0.946970i	$-0.604127\pi$
$359$	−12.1764	−0.642644	−0.321322	+	0.946970i	$0.604127\pi$
$360$	0	0
$361$	−12.1811	−0.641110
$362$	0	0
$363$	−34.1027	−1.78993
$364$	0	0
$365$	23.7951	1.24549
$366$	0	0
$367$	30.4937	1.59176	0.795878	−	0.605457i	$-0.207010\pi$
$367$	30.4937	1.59176	0.795878	+	0.605457i	$0.207010\pi$
$368$	0	0
$369$	−7.44624	−0.387636
$370$	0	0
$371$	28.7163	1.49088
$372$	0	0
$373$	−26.7772	−1.38647	−0.693235	−	0.720711i	$-0.743815\pi$
$373$	−26.7772	−1.38647	−0.693235	+	0.720711i	$0.743815\pi$
$374$	0	0
$375$	37.7066	1.94716
$376$	0	0
$377$	−11.7098	−0.603083
$378$	0	0
$379$	18.4204	0.946193	0.473096	−	0.881011i	$-0.343136\pi$
$379$	18.4204	0.946193	0.473096	+	0.881011i	$0.343136\pi$
$380$	0	0
$381$	−10.3516	−0.530331
$382$	0	0
$383$	−12.8232	−0.655236	−0.327618	−	0.944810i	$-0.606246\pi$
$383$	−12.8232	−0.655236	−0.327618	+	0.944810i	$0.606246\pi$
$384$	0	0
$385$	2.62200	0.133630
$386$	0	0
$387$	71.8283	3.65124
$388$	0	0
$389$	5.97406	0.302897	0.151448	−	0.988465i	$-0.451606\pi$
$389$	5.97406	0.302897	0.151448	+	0.988465i	$0.451606\pi$
$390$	0	0
$391$	14.3498	0.725698
$392$	0	0
$393$	−16.8575	−0.850351
$394$	0	0
$395$	14.3414	0.721596
$396$	0	0
$397$	4.22139	0.211866	0.105933	−	0.994373i	$-0.466217\pi$
$397$	4.22139	0.211866	0.105933	+	0.994373i	$0.466217\pi$
$398$	0	0
$399$	25.4662	1.27490
$400$	0	0
$401$	−24.8341	−1.24015	−0.620077	−	0.784541i	$-0.712899\pi$
$401$	−24.8341	−1.24015	−0.620077	+	0.784541i	$0.712899\pi$
$402$	0	0
$403$	12.7720	0.636217
$404$	0	0
$405$	−32.5335	−1.61660
$406$	0	0
$407$	−5.22099	−0.258795
$408$	0	0
$409$	−7.51909	−0.371795	−0.185897	−	0.982569i	$-0.559519\pi$
$409$	−7.51909	−0.371795	−0.185897	+	0.982569i	$0.559519\pi$
$410$	0	0
$411$	52.7912	2.60400
$412$	0	0
$413$	8.57546	0.421971
$414$	0	0
$415$	−8.38146	−0.411430
$416$	0	0
$417$	−45.5454	−2.23037
$418$	0	0
$419$	30.7864	1.50402	0.752008	−	0.659154i	$-0.229085\pi$
$419$	30.7864	1.50402	0.752008	+	0.659154i	$0.229085\pi$
$420$	0	0
$421$	−36.6540	−1.78641	−0.893203	−	0.449654i	$-0.851547\pi$
$421$	−36.6540	−1.78641	−0.893203	+	0.449654i	$0.851547\pi$
$422$	0	0
$423$	−19.7493	−0.960246
$424$	0	0
$425$	9.24053	0.448231
$426$	0	0
$427$	−16.1182	−0.780014
$428$	0	0
$429$	−2.86033	−0.138098
$430$	0	0
$431$	−6.47534	−0.311906	−0.155953	−	0.987764i	$-0.549845\pi$
$431$	−6.47534	−0.311906	−0.155953	+	0.987764i	$0.549845\pi$
$432$	0	0
$433$	−5.97327	−0.287057	−0.143529	−	0.989646i	$-0.545845\pi$
$433$	−5.97327	−0.287057	−0.143529	+	0.989646i	$0.545845\pi$
$434$	0	0
$435$	−35.6066	−1.70720
$436$	0	0
$437$	10.2617	0.490882
$438$	0	0
$439$	−12.5237	−0.597724	−0.298862	−	0.954296i	$-0.596607\pi$
$439$	−12.5237	−0.597724	−0.298862	+	0.954296i	$0.596607\pi$
$440$	0	0
$441$	16.9353	0.806442
$442$	0	0
$443$	−28.1272	−1.33636	−0.668182	−	0.743997i	$-0.732927\pi$
$443$	−28.1272	−1.33636	−0.668182	+	0.743997i	$0.732927\pi$
$444$	0	0
$445$	19.2868	0.914282
$446$	0	0
$447$	9.51099	0.449855
$448$	0	0
$449$	3.84589	0.181499	0.0907493	−	0.995874i	$-0.471074\pi$
$449$	3.84589	0.181499	0.0907493	+	0.995874i	$0.471074\pi$
$450$	0	0
$451$	−0.567524	−0.0267237
$452$	0	0
$453$	−35.8161	−1.68279
$454$	0	0
$455$	−7.92002	−0.371296
$456$	0	0
$457$	29.3780	1.37424	0.687122	−	0.726542i	$-0.258874\pi$
$457$	29.3780	1.37424	0.687122	+	0.726542i	$0.258874\pi$
$458$	0	0
$459$	−48.3173	−2.25526
$460$	0	0
$461$	11.2331	0.523179	0.261590	−	0.965179i	$-0.415753\pi$
$461$	11.2331	0.523179	0.261590	+	0.965179i	$0.415753\pi$
$462$	0	0
$463$	6.77063	0.314658	0.157329	−	0.987546i	$-0.449712\pi$
$463$	6.77063	0.314658	0.157329	+	0.987546i	$0.449712\pi$
$464$	0	0
$465$	38.8365	1.80100
$466$	0	0
$467$	37.2784	1.72504	0.862518	−	0.506026i	$-0.168886\pi$
$467$	37.2784	1.72504	0.862518	+	0.506026i	$0.168886\pi$
$468$	0	0
$469$	−22.2914	−1.02932
$470$	0	0
$471$	37.9763	1.74986
$472$	0	0
$473$	5.47448	0.251717
$474$	0	0
$475$	6.60801	0.303196
$476$	0	0
$477$	−67.1091	−3.07271
$478$	0	0
$479$	−32.4426	−1.48234	−0.741171	−	0.671317i	$-0.765729\pi$
$479$	−32.4426	−1.48234	−0.741171	+	0.671317i	$0.765729\pi$
$480$	0	0
$481$	15.7705	0.719074
$482$	0	0
$483$	38.3237	1.74379
$484$	0	0
$485$	−18.9743	−0.861577
$486$	0	0
$487$	19.8614	0.900004	0.450002	−	0.893028i	$-0.351423\pi$
$487$	19.8614	0.900004	0.450002	+	0.893028i	$0.351423\pi$
$488$	0	0
$489$	37.7041	1.70504
$490$	0	0
$491$	1.44852	0.0653710	0.0326855	−	0.999466i	$-0.489594\pi$
$491$	1.44852	0.0653710	0.0326855	+	0.999466i	$0.489594\pi$
$492$	0	0
$493$	−25.9670	−1.16950
$494$	0	0
$495$	−6.12754	−0.275413
$496$	0	0
$497$	10.6104	0.475943
$498$	0	0
$499$	7.13499	0.319406	0.159703	−	0.987165i	$-0.448946\pi$
$499$	7.13499	0.319406	0.159703	+	0.987165i	$0.448946\pi$
$500$	0	0
$501$	−18.9578	−0.846972
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−29.2632	−1.30220
$506$	0	0
$507$	−32.7824	−1.45592
$508$	0	0
$509$	−13.4239	−0.595005	−0.297502	−	0.954721i	$-0.596154\pi$
$509$	−13.4239	−0.595005	−0.297502	+	0.954721i	$0.596154\pi$
$510$	0	0
$511$	46.3449	2.05018
$512$	0	0
$513$	−34.5523	−1.52552
$514$	0	0
$515$	2.10710	0.0928501
$516$	0	0
$517$	−1.50522	−0.0661995
$518$	0	0
$519$	−26.9987	−1.18511
$520$	0	0
$521$	−15.8555	−0.694640	−0.347320	−	0.937747i	$-0.612908\pi$
$521$	−15.8555	−0.694640	−0.347320	+	0.937747i	$0.612908\pi$
$522$	0	0
$523$	37.0927	1.62195	0.810974	−	0.585082i	$-0.198938\pi$
$523$	37.0927	1.62195	0.810974	+	0.585082i	$0.198938\pi$
$524$	0	0
$525$	24.6785	1.07706
$526$	0	0
$527$	28.3226	1.23375
$528$	0	0
$529$	−7.55736	−0.328581
$530$	0	0
$531$	−20.0406	−0.869687
$532$	0	0
$533$	1.71426	0.0742529
$534$	0	0
$535$	−0.0412533	−0.00178353
$536$	0	0
$537$	−60.2018	−2.59790
$538$	0	0
$539$	1.29074	0.0555962
$540$	0	0
$541$	−33.4083	−1.43633	−0.718167	−	0.695871i	$-0.755019\pi$
$541$	−33.4083	−1.43633	−0.718167	+	0.695871i	$0.755019\pi$
$542$	0	0
$543$	25.5497	1.09644
$544$	0	0
$545$	−0.550973	−0.0236011
$546$	0	0
$547$	15.3912	0.658081	0.329041	−	0.944316i	$-0.393275\pi$
$547$	15.3912	0.658081	0.329041	+	0.944316i	$0.393275\pi$
$548$	0	0
$549$	37.6677	1.60762
$550$	0	0
$551$	−18.5693	−0.791079
$552$	0	0
$553$	27.9323	1.18780
$554$	0	0
$555$	47.9544	2.03555
$556$	0	0
$557$	36.8442	1.56114	0.780570	−	0.625068i	$-0.214929\pi$
$557$	36.8442	1.56114	0.780570	+	0.625068i	$0.214929\pi$
$558$	0	0
$559$	−16.5362	−0.699407
$560$	0	0
$561$	−6.34294	−0.267799
$562$	0	0
$563$	−22.6230	−0.953445	−0.476723	−	0.879054i	$-0.658175\pi$
$563$	−22.6230	−0.953445	−0.476723	+	0.879054i	$0.658175\pi$
$564$	0	0
$565$	−3.40434	−0.143222
$566$	0	0
$567$	−63.3645	−2.66106
$568$	0	0
$569$	−29.5554	−1.23903	−0.619514	−	0.784986i	$-0.712670\pi$
$569$	−29.5554	−1.23903	−0.619514	+	0.784986i	$0.712670\pi$
$570$	0	0
$571$	29.1394	1.21945	0.609723	−	0.792614i	$-0.291281\pi$
$571$	29.1394	1.21945	0.609723	+	0.792614i	$0.291281\pi$
$572$	0	0
$573$	76.8429	3.21016
$574$	0	0
$575$	9.94429	0.414706
$576$	0	0
$577$	2.88102	0.119939	0.0599693	−	0.998200i	$-0.480900\pi$
$577$	2.88102	0.119939	0.0599693	+	0.998200i	$0.480900\pi$
$578$	0	0
$579$	−27.1031	−1.12637
$580$	0	0
$581$	−16.3243	−0.677246
$582$	0	0
$583$	−5.11480	−0.211833
$584$	0	0
$585$	18.5088	0.765246
$586$	0	0
$587$	21.1371	0.872422	0.436211	−	0.899844i	$-0.356320\pi$
$587$	21.1371	0.872422	0.436211	+	0.899844i	$0.356320\pi$
$588$	0	0
$589$	20.2538	0.834542
$590$	0	0
$591$	−80.6880	−3.31906
$592$	0	0
$593$	−20.4767	−0.840879	−0.420439	−	0.907321i	$-0.638124\pi$
$593$	−20.4767	−0.840879	−0.420439	+	0.907321i	$0.638124\pi$
$594$	0	0
$595$	−17.5631	−0.720016
$596$	0	0
$597$	11.5397	0.472287
$598$	0	0
$599$	24.4801	1.00023	0.500115	−	0.865959i	$-0.333291\pi$
$599$	24.4801	1.00023	0.500115	+	0.865959i	$0.333291\pi$
$600$	0	0
$601$	−34.3885	−1.40274	−0.701368	−	0.712799i	$-0.747427\pi$
$601$	−34.3885	−1.40274	−0.701368	+	0.712799i	$0.747427\pi$
$602$	0	0
$603$	52.0944	2.12145
$604$	0	0
$605$	16.8190	0.683788
$606$	0	0
$607$	20.6276	0.837247	0.418624	−	0.908160i	$-0.362513\pi$
$607$	20.6276	0.837247	0.418624	+	0.908160i	$0.362513\pi$
$608$	0	0
$609$	−69.3498	−2.81019
$610$	0	0
$611$	4.54666	0.183938
$612$	0	0
$613$	1.90684	0.0770164	0.0385082	−	0.999258i	$-0.487739\pi$
$613$	1.90684	0.0770164	0.0385082	+	0.999258i	$0.487739\pi$
$614$	0	0
$615$	5.21266	0.210195
$616$	0	0
$617$	19.1072	0.769227	0.384613	−	0.923078i	$-0.374335\pi$
$617$	19.1072	0.769227	0.384613	+	0.923078i	$0.374335\pi$
$618$	0	0
$619$	−19.2943	−0.775504	−0.387752	−	0.921764i	$-0.626748\pi$
$619$	−19.2943	−0.775504	−0.387752	+	0.921764i	$0.626748\pi$
$620$	0	0
$621$	−51.9972	−2.08658
$622$	0	0
$623$	37.5643	1.50498
$624$	0	0
$625$	−5.94368	−0.237747
$626$	0	0
$627$	−4.53591	−0.181147
$628$	0	0
$629$	34.9720	1.39442
$630$	0	0
$631$	−44.1125	−1.75609	−0.878045	−	0.478577i	$-0.841153\pi$
$631$	−44.1125	−1.75609	−0.878045	+	0.478577i	$0.841153\pi$
$632$	0	0
$633$	10.0673	0.400140
$634$	0	0
$635$	5.10529	0.202597
$636$	0	0
$637$	−3.89881	−0.154477
$638$	0	0
$639$	−24.7962	−0.980924
$640$	0	0
$641$	13.0671	0.516120	0.258060	−	0.966129i	$-0.416917\pi$
$641$	13.0671	0.516120	0.258060	+	0.966129i	$0.416917\pi$
$642$	0	0
$643$	23.6402	0.932279	0.466139	−	0.884711i	$-0.345644\pi$
$643$	23.6402	0.932279	0.466139	+	0.884711i	$0.345644\pi$
$644$	0	0
$645$	−50.2827	−1.97988
$646$	0	0
$647$	−17.2476	−0.678073	−0.339036	−	0.940773i	$-0.610101\pi$
$647$	−17.2476	−0.678073	−0.339036	+	0.940773i	$0.610101\pi$
$648$	0	0
$649$	−1.52742	−0.0599563
$650$	0	0
$651$	75.6407	2.96459
$652$	0	0
$653$	−4.73847	−0.185431	−0.0927154	−	0.995693i	$-0.529555\pi$
$653$	−4.73847	−0.185431	−0.0927154	+	0.995693i	$0.529555\pi$
$654$	0	0
$655$	8.31390	0.324851
$656$	0	0
$657$	−108.307	−4.22544
$658$	0	0
$659$	−27.0030	−1.05189	−0.525943	−	0.850520i	$-0.676288\pi$
$659$	−27.0030	−1.05189	−0.525943	+	0.850520i	$0.676288\pi$
$660$	0	0
$661$	40.0848	1.55912	0.779560	−	0.626328i	$-0.215443\pi$
$661$	40.0848	1.55912	0.779560	+	0.626328i	$0.215443\pi$
$662$	0	0
$663$	19.1595	0.744092
$664$	0	0
$665$	−12.5596	−0.487039
$666$	0	0
$667$	−27.9447	−1.08202
$668$	0	0
$669$	5.80333	0.224370
$670$	0	0
$671$	2.87089	0.110830
$672$	0	0
$673$	−17.5878	−0.677959	−0.338980	−	0.940794i	$-0.610082\pi$
$673$	−17.5878	−0.677959	−0.338980	+	0.940794i	$0.610082\pi$
$674$	0	0
$675$	−33.4836	−1.28878
$676$	0	0
$677$	−27.6314	−1.06196	−0.530981	−	0.847384i	$-0.678177\pi$
$677$	−27.6314	−1.06196	−0.530981	+	0.847384i	$0.678177\pi$
$678$	0	0
$679$	−36.9556	−1.41823
$680$	0	0
$681$	−63.2383	−2.42330
$682$	0	0
$683$	−15.4621	−0.591639	−0.295820	−	0.955244i	$-0.595593\pi$
$683$	−15.4621	−0.591639	−0.295820	+	0.955244i	$0.595593\pi$
$684$	0	0
$685$	−26.0359	−0.994780
$686$	0	0
$687$	−47.8150	−1.82426
$688$	0	0
$689$	15.4498	0.588589
$690$	0	0
$691$	9.92880	0.377709	0.188855	−	0.982005i	$-0.439522\pi$
$691$	9.92880	0.377709	0.188855	+	0.982005i	$0.439522\pi$
$692$	0	0
$693$	−11.9344	−0.453351
$694$	0	0
$695$	22.4624	0.852046
$696$	0	0
$697$	3.80147	0.143991
$698$	0	0
$699$	−25.4897	−0.964110
$700$	0	0
$701$	−50.4195	−1.90432	−0.952158	−	0.305605i	$-0.901141\pi$
$701$	−50.4195	−1.90432	−0.952158	+	0.305605i	$0.901141\pi$
$702$	0	0
$703$	25.0089	0.943227
$704$	0	0
$705$	13.8253	0.520692
$706$	0	0
$707$	−56.9950	−2.14352
$708$	0	0
$709$	−49.2180	−1.84842	−0.924210	−	0.381885i	$-0.875275\pi$
$709$	−49.2180	−1.84842	−0.924210	+	0.381885i	$0.875275\pi$
$710$	0	0
$711$	−65.2770	−2.44808
$712$	0	0
$713$	30.4796	1.14147
$714$	0	0
$715$	1.41067	0.0527562
$716$	0	0
$717$	−32.1328	−1.20002
$718$	0	0
$719$	−10.8606	−0.405033	−0.202516	−	0.979279i	$-0.564912\pi$
$719$	−10.8606	−0.405033	−0.202516	+	0.979279i	$0.564912\pi$
$720$	0	0
$721$	4.10394	0.152839
$722$	0	0
$723$	26.1042	0.970826
$724$	0	0
$725$	−17.9950	−0.668317
$726$	0	0
$727$	14.7206	0.545958	0.272979	−	0.962020i	$-0.411991\pi$
$727$	14.7206	0.545958	0.272979	+	0.962020i	$0.411991\pi$
$728$	0	0
$729$	21.5981	0.799931
$730$	0	0
$731$	−36.6700	−1.35629
$732$	0	0
$733$	35.9113	1.32642	0.663208	−	0.748435i	$-0.269195\pi$
$733$	35.9113	1.32642	0.663208	+	0.748435i	$0.269195\pi$
$734$	0	0
$735$	−11.8554	−0.437291
$736$	0	0
$737$	3.97044	0.146253
$738$	0	0
$739$	31.0887	1.14362	0.571808	−	0.820387i	$-0.306242\pi$
$739$	31.0887	1.14362	0.571808	+	0.820387i	$0.306242\pi$
$740$	0	0
$741$	13.7011	0.503324
$742$	0	0
$743$	22.9725	0.842779	0.421389	−	0.906880i	$-0.361543\pi$
$743$	22.9725	0.842779	0.421389	+	0.906880i	$0.361543\pi$
$744$	0	0
$745$	−4.69069	−0.171853
$746$	0	0
$747$	38.1494	1.39581
$748$	0	0
$749$	−0.0803477	−0.00293584
$750$	0	0
$751$	−39.7768	−1.45148	−0.725739	−	0.687971i	$-0.758502\pi$
$751$	−39.7768	−1.45148	−0.725739	+	0.687971i	$0.758502\pi$
$752$	0	0
$753$	44.9985	1.63984
$754$	0	0
$755$	17.6640	0.642859
$756$	0	0
$757$	−9.07248	−0.329745	−0.164872	−	0.986315i	$-0.552721\pi$
$757$	−9.07248	−0.329745	−0.164872	+	0.986315i	$0.552721\pi$
$758$	0	0
$759$	−6.82602	−0.247769
$760$	0	0
$761$	−3.44813	−0.124995	−0.0624973	−	0.998045i	$-0.519906\pi$
$761$	−3.44813	−0.124995	−0.0624973	+	0.998045i	$0.519906\pi$
$762$	0	0
$763$	−1.07311	−0.0388493
$764$	0	0
$765$	41.0444	1.48396
$766$	0	0
$767$	4.61371	0.166591
$768$	0	0
$769$	−16.8510	−0.607664	−0.303832	−	0.952726i	$-0.598266\pi$
$769$	−16.8510	−0.607664	−0.303832	+	0.952726i	$0.598266\pi$
$770$	0	0
$771$	5.00741	0.180337
$772$	0	0
$773$	−2.58677	−0.0930396	−0.0465198	−	0.998917i	$-0.514813\pi$
$773$	−2.58677	−0.0930396	−0.0465198	+	0.998917i	$0.514813\pi$
$774$	0	0
$775$	19.6274	0.705036
$776$	0	0
$777$	93.3992	3.35068
$778$	0	0
$779$	2.71848	0.0973995
$780$	0	0
$781$	−1.88988	−0.0676251
$782$	0	0
$783$	94.0931	3.36261
$784$	0	0
$785$	−18.7294	−0.668480
$786$	0	0
$787$	−33.6266	−1.19866	−0.599329	−	0.800503i	$-0.704566\pi$
$787$	−33.6266	−1.19866	−0.599329	+	0.800503i	$0.704566\pi$
$788$	0	0
$789$	18.4660	0.657406
$790$	0	0
$791$	−6.63053	−0.235754
$792$	0	0
$793$	−8.67181	−0.307945
$794$	0	0
$795$	46.9790	1.66617
$796$	0	0
$797$	42.9826	1.52252	0.761261	−	0.648445i	$-0.224580\pi$
$797$	42.9826	1.52252	0.761261	+	0.648445i	$0.224580\pi$
$798$	0	0
$799$	10.0825	0.356693
$800$	0	0
$801$	−87.7866	−3.10179
$802$	0	0
$803$	−8.25471	−0.291302
$804$	0	0
$805$	−18.9007	−0.666162
$806$	0	0
$807$	5.65164	0.198947
$808$	0	0
$809$	−12.7197	−0.447202	−0.223601	−	0.974681i	$-0.571781\pi$
$809$	−12.7197	−0.447202	−0.223601	+	0.974681i	$0.571781\pi$
$810$	0	0
$811$	23.3898	0.821326	0.410663	−	0.911787i	$-0.365297\pi$
$811$	23.3898	0.821326	0.410663	+	0.911787i	$0.365297\pi$
$812$	0	0
$813$	53.4927	1.87607
$814$	0	0
$815$	−18.5952	−0.651360
$816$	0	0
$817$	−26.2231	−0.917430
$818$	0	0
$819$	36.0491	1.25966
$820$	0	0
$821$	−20.6853	−0.721923	−0.360961	−	0.932581i	$-0.617551\pi$
$821$	−20.6853	−0.721923	−0.360961	+	0.932581i	$0.617551\pi$
$822$	0	0
$823$	16.0268	0.558658	0.279329	−	0.960195i	$-0.409888\pi$
$823$	16.0268	0.558658	0.279329	+	0.960195i	$0.409888\pi$
$824$	0	0
$825$	−4.39562	−0.153036
$826$	0	0
$827$	38.5677	1.34113	0.670565	−	0.741851i	$-0.266052\pi$
$827$	38.5677	1.34113	0.670565	+	0.741851i	$0.266052\pi$
$828$	0	0
$829$	38.3558	1.33215	0.666075	−	0.745885i	$-0.267973\pi$
$829$	38.3558	1.33215	0.666075	+	0.745885i	$0.267973\pi$
$830$	0	0
$831$	48.3493	1.67722
$832$	0	0
$833$	−8.64583	−0.299560
$834$	0	0
$835$	9.34972	0.323560
$836$	0	0
$837$	−102.629	−3.54736
$838$	0	0
$839$	16.6698	0.575505	0.287752	−	0.957705i	$-0.407092\pi$
$839$	16.6698	0.575505	0.287752	+	0.957705i	$0.407092\pi$
$840$	0	0
$841$	21.5682	0.743730
$842$	0	0
$843$	−99.4059	−3.42372
$844$	0	0
$845$	16.1678	0.556189
$846$	0	0
$847$	32.7577	1.12557
$848$	0	0
$849$	−46.6827	−1.60215
$850$	0	0
$851$	37.6355	1.29013
$852$	0	0
$853$	15.3583	0.525859	0.262929	−	0.964815i	$-0.415311\pi$
$853$	15.3583	0.525859	0.262929	+	0.964815i	$0.415311\pi$
$854$	0	0
$855$	29.3513	1.00379
$856$	0	0
$857$	6.90657	0.235924	0.117962	−	0.993018i	$-0.462364\pi$
$857$	6.90657	0.235924	0.117962	+	0.993018i	$0.462364\pi$
$858$	0	0
$859$	−42.8992	−1.46370	−0.731851	−	0.681465i	$-0.761343\pi$
$859$	−42.8992	−1.46370	−0.731851	+	0.681465i	$0.761343\pi$
$860$	0	0
$861$	10.1525	0.345997
$862$	0	0
$863$	−42.2767	−1.43912	−0.719558	−	0.694432i	$-0.755656\pi$
$863$	−42.2767	−1.43912	−0.719558	+	0.694432i	$0.755656\pi$
$864$	0	0
$865$	13.3154	0.452737
$866$	0	0
$867$	−11.6804	−0.396687
$868$	0	0
$869$	−4.97517	−0.168771
$870$	0	0
$871$	−11.9931	−0.406370
$872$	0	0
$873$	86.3641	2.92298
$874$	0	0
$875$	−36.2196	−1.22444
$876$	0	0
$877$	10.5272	0.355478	0.177739	−	0.984078i	$-0.443122\pi$
$877$	10.5272	0.355478	0.177739	+	0.984078i	$0.443122\pi$
$878$	0	0
$879$	21.0255	0.709174
$880$	0	0
$881$	26.8995	0.906266	0.453133	−	0.891443i	$-0.350306\pi$
$881$	26.8995	0.906266	0.453133	+	0.891443i	$0.350306\pi$
$882$	0	0
$883$	27.1797	0.914669	0.457334	−	0.889295i	$-0.348804\pi$
$883$	27.1797	0.914669	0.457334	+	0.889295i	$0.348804\pi$
$884$	0	0
$885$	14.0292	0.471586
$886$	0	0
$887$	−13.2469	−0.444787	−0.222393	−	0.974957i	$-0.571387\pi$
$887$	−13.2469	−0.444787	−0.222393	+	0.974957i	$0.571387\pi$
$888$	0	0
$889$	9.94340	0.333491
$890$	0	0
$891$	11.2862	0.378100
$892$	0	0
$893$	7.21010	0.241277
$894$	0	0
$895$	29.6907	0.992450
$896$	0	0
$897$	20.6187	0.688437
$898$	0	0
$899$	−55.1553	−1.83953
$900$	0	0
$901$	34.2607	1.14139
$902$	0	0
$903$	−97.9339	−3.25904
$904$	0	0
$905$	−12.6007	−0.418863
$906$	0	0
$907$	−28.9672	−0.961840	−0.480920	−	0.876764i	$-0.659697\pi$
$907$	−28.9672	−0.961840	−0.480920	+	0.876764i	$0.659697\pi$
$908$	0	0
$909$	133.196	4.41782
$910$	0	0
$911$	−9.28341	−0.307573	−0.153787	−	0.988104i	$-0.549147\pi$
$911$	−9.28341	−0.307573	−0.153787	+	0.988104i	$0.549147\pi$
$912$	0	0
$913$	2.90760	0.0962275
$914$	0	0
$915$	−26.3689	−0.871729
$916$	0	0
$917$	16.1927	0.534731
$918$	0	0
$919$	19.2457	0.634859	0.317429	−	0.948282i	$-0.397180\pi$
$919$	19.2457	0.634859	0.317429	+	0.948282i	$0.397180\pi$
$920$	0	0
$921$	87.3223	2.87737
$922$	0	0
$923$	5.70855	0.187899
$924$	0	0
$925$	24.2354	0.796854
$926$	0	0
$927$	−9.59078	−0.315002
$928$	0	0
$929$	38.1294	1.25098	0.625492	−	0.780230i	$-0.284898\pi$
$929$	38.1294	1.25098	0.625492	+	0.780230i	$0.284898\pi$
$930$	0	0
$931$	−6.18273	−0.202631
$932$	0	0
$933$	7.28222	0.238409
$934$	0	0
$935$	3.12825	0.102305
$936$	0	0
$937$	−0.539014	−0.0176088	−0.00880440	−	0.999961i	$-0.502803\pi$
$937$	−0.539014	−0.0176088	−0.00880440	+	0.999961i	$0.502803\pi$
$938$	0	0
$939$	67.6564	2.20788
$940$	0	0
$941$	11.9849	0.390695	0.195348	−	0.980734i	$-0.437416\pi$
$941$	11.9849	0.390695	0.195348	+	0.980734i	$0.437416\pi$
$942$	0	0
$943$	4.09099	0.133221
$944$	0	0
$945$	63.6409	2.07024
$946$	0	0
$947$	−59.1562	−1.92232	−0.961158	−	0.275997i	$-0.910992\pi$
$947$	−59.1562	−1.92232	−0.961158	+	0.275997i	$0.910992\pi$
$948$	0	0
$949$	24.9342	0.809397
$950$	0	0
$951$	88.5345	2.87093
$952$	0	0
$953$	−45.7815	−1.48301	−0.741505	−	0.670948i	$-0.765887\pi$
$953$	−45.7815	−1.48301	−0.741505	+	0.670948i	$0.765887\pi$
$954$	0	0
$955$	−37.8979	−1.22635
$956$	0	0
$957$	12.3522	0.399291
$958$	0	0
$959$	−50.7092	−1.63749
$960$	0	0
$961$	29.1586	0.940599
$962$	0	0
$963$	0.187770	0.00605080
$964$	0	0
$965$	13.3669	0.430295
$966$	0	0
$967$	44.8467	1.44217	0.721086	−	0.692846i	$-0.243643\pi$
$967$	44.8467	1.44217	0.721086	+	0.692846i	$0.243643\pi$
$968$	0	0
$969$	30.3831	0.976044
$970$	0	0
$971$	−10.2798	−0.329895	−0.164948	−	0.986302i	$-0.552746\pi$
$971$	−10.2798	−0.329895	−0.164948	+	0.986302i	$0.552746\pi$
$972$	0	0
$973$	43.7492	1.40254
$974$	0	0
$975$	13.2774	0.425217
$976$	0	0
$977$	9.74691	0.311831	0.155916	−	0.987770i	$-0.450167\pi$
$977$	9.74691	0.311831	0.155916	+	0.987770i	$0.450167\pi$
$978$	0	0
$979$	−6.69076	−0.213838
$980$	0	0
$981$	2.50783	0.0800689
$982$	0	0
$983$	39.1336	1.24817	0.624084	−	0.781357i	$-0.285472\pi$
$983$	39.1336	1.24817	0.624084	+	0.781357i	$0.285472\pi$
$984$	0	0
$985$	39.7942	1.26795
$986$	0	0
$987$	26.9271	0.857100
$988$	0	0
$989$	−39.4628	−1.25484
$990$	0	0
$991$	37.3199	1.18551	0.592754	−	0.805384i	$-0.298041\pi$
$991$	37.3199	1.18551	0.592754	+	0.805384i	$0.298041\pi$
$992$	0	0
$993$	91.7573	2.91183
$994$	0	0
$995$	−5.69120	−0.180423
$996$	0	0
$997$	−55.6608	−1.76279	−0.881397	−	0.472376i	$-0.843396\pi$
$997$	−55.6608	−1.76279	−0.881397	+	0.472376i	$0.843396\pi$
$998$	0	0
$999$	−126.723	−4.00934

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.d.1.28	✓	28
4.3	odd	2		8048.2.a.v.1.1		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.28	✓	28	1.1	even	1	trivial
8048.2.a.v.1.1		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)

\(f(q)\)	\(=\)	\(q+3.18633 q^{3} -1.57145 q^{5} -3.06067 q^{7} +7.15268 q^{9} +O(q^{10})\)	\(q+3.18633 q^{3} -1.57145 q^{5} -3.06067 q^{7} +7.15268 q^{9} +0.545150 q^{11} -1.64668 q^{13} -5.00716 q^{15} -3.65160 q^{17} -2.61130 q^{19} -9.75229 q^{21} -3.92971 q^{23} -2.53054 q^{25} +13.2318 q^{27} +7.11113 q^{29} -7.75620 q^{31} +1.73703 q^{33} +4.80969 q^{35} -9.57716 q^{37} -5.24686 q^{39} -1.04104 q^{41} +10.0422 q^{43} -11.2401 q^{45} -2.76111 q^{47} +2.36768 q^{49} -11.6352 q^{51} -9.38236 q^{53} -0.856678 q^{55} -8.32047 q^{57} -2.80183 q^{59} +5.26624 q^{61} -21.8920 q^{63} +2.58768 q^{65} +7.28320 q^{67} -12.5214 q^{69} -3.46671 q^{71} -15.1421 q^{73} -8.06313 q^{75} -1.66852 q^{77} -9.12623 q^{79} +20.7028 q^{81} +5.33358 q^{83} +5.73832 q^{85} +22.6584 q^{87} -12.2732 q^{89} +5.03994 q^{91} -24.7138 q^{93} +4.10354 q^{95} +12.0744 q^{97} +3.89929 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more