LMFDB - Embedded newform 1024.4.b.j.513.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,4,Mod(513,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1024.513");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1024.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$60.4179558459$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 36x^{8} + 405x^{6} + 1380x^{4} + 420x^{2} + 32 $ x^10 + 36x^8 + 405x^6 + 1380x^4 + 420x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			513.8
Root			$0.446984i$ of defining polynomial
Character	$\chi$	$=$	1024.513
Dual form			1024.4.b.j.513.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.62644i q^{3} +17.8826i q^{5} -13.8754 q^{7} +5.59607 q^{9} +O(q^{10})$	$q+4.62644i q^{3} +17.8826i q^{5} -13.8754 q^{7} +5.59607 q^{9} +2.18771i q^{11} +46.3685i q^{13} -82.7326 q^{15} -18.6531 q^{17} -122.194i q^{19} -64.1936i q^{21} -134.006 q^{23} -194.786 q^{25} +150.804i q^{27} +84.5628i q^{29} -31.5391 q^{31} -10.1213 q^{33} -248.128i q^{35} -126.129i q^{37} -214.521 q^{39} -210.504 q^{41} +168.860i q^{43} +100.072i q^{45} +182.902 q^{47} -150.474 q^{49} -86.2972i q^{51} -37.0020i q^{53} -39.1219 q^{55} +565.323 q^{57} -624.494i q^{59} +246.759i q^{61} -77.6476 q^{63} -829.188 q^{65} -129.763i q^{67} -619.970i q^{69} -348.360 q^{71} -299.436 q^{73} -901.168i q^{75} -30.3553i q^{77} +943.487 q^{79} -546.590 q^{81} +443.034i q^{83} -333.565i q^{85} -391.224 q^{87} -1412.35 q^{89} -643.381i q^{91} -145.914i q^{93} +2185.14 q^{95} +1515.29 q^{97} +12.2426i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 28 q^{7} - 54 q^{9}+O(q^{10}) $ 10 * q - 28 * q^7 - 54 * q^9	$ 10 q - 28 q^{7} - 54 q^{9} + 124 q^{15} + 4 q^{17} - 276 q^{23} - 50 q^{25} + 368 q^{31} - 4 q^{33} - 732 q^{39} + 944 q^{47} - 94 q^{49} - 1380 q^{55} - 108 q^{57} + 2628 q^{63} - 492 q^{65} - 3468 q^{71} + 296 q^{73} + 4416 q^{79} - 482 q^{81} - 6036 q^{87} - 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q - 28 * q^7 - 54 * q^9 + 124 * q^15 + 4 * q^17 - 276 * q^23 - 50 * q^25 + 368 * q^31 - 4 * q^33 - 732 * q^39 + 944 * q^47 - 94 * q^49 - 1380 * q^55 - 108 * q^57 + 2628 * q^63 - 492 * q^65 - 3468 * q^71 + 296 * q^73 + 4416 * q^79 - 482 * q^81 - 6036 * q^87 - 88 * q^89 + 6900 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$-1$	$1$

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
$3$	4.62644i	0.890358i	0.895441	+	0.445179i	$0.146860\pi$
$3$	4.62644i	0.890358i	−0.895441	+	0.445179i	$0.853140\pi$
$4$	0	0
$5$	17.8826i	1.59947i	0.600356	+	0.799733i	$0.295026\pi$
$5$	17.8826i	1.59947i	−0.600356	+	0.799733i	$0.704974\pi$
$6$	0	0
$7$	−13.8754	−0.749200	−0.374600	−	0.927187i	$-0.622220\pi$
$7$	−13.8754	−0.749200	−0.374600	+	0.927187i	$0.622220\pi$
$8$	0	0
$9$	5.59607	0.207262
$10$	0	0
$11$	2.18771i	0.0599653i	0.999550	+	0.0299827i	$0.00954521\pi$
$11$	2.18771i	0.0599653i	−0.999550	+	0.0299827i	$0.990455\pi$
$12$	0	0
$13$	46.3685i	0.989255i	0.869105	+	0.494627i	$0.164695\pi$
$13$	46.3685i	0.989255i	−0.869105	+	0.494627i	$0.835305\pi$
$14$	0	0
$15$	−82.7326	−1.42410
$16$	0	0
$17$	−18.6531	−0.266119	−0.133060	−	0.991108i	$-0.542480\pi$
$17$	−18.6531	−0.266119	−0.133060	+	0.991108i	$0.542480\pi$
$18$	0	0
$19$	− 122.194i	− 1.47543i	−0.675111	−	0.737716i	$-0.735904\pi$
$19$	− 122.194i	− 1.47543i	0.675111	−	0.737716i	$-0.264096\pi$
$20$	0	0
$21$	− 64.1936i	− 0.667057i
$22$	0	0
$23$	−134.006	−1.21488	−0.607438	−	0.794367i	$-0.707803\pi$
$23$	−134.006	−1.21488	−0.607438	+	0.794367i	$0.707803\pi$
$24$	0	0
$25$	−194.786	−1.55829
$26$	0	0
$27$	150.804i	1.07490i
$28$	0	0
$29$	84.5628i	0.541480i	0.962653	+	0.270740i	$0.0872683\pi$
$29$	84.5628i	0.541480i	−0.962653	+	0.270740i	$0.912732\pi$
$30$	0	0
$31$	−31.5391	−0.182729	−0.0913645	−	0.995818i	$-0.529123\pi$
$31$	−31.5391	−0.182729	−0.0913645	+	0.995818i	$0.529123\pi$
$32$	0	0
$33$	−10.1213	−0.0533906
$34$	0	0
$35$	− 248.128i	− 1.19832i
$36$	0	0
$37$	− 126.129i	− 0.560418i	−0.959939	−	0.280209i	$-0.909596\pi$
$37$	− 126.129i	− 0.560418i	0.959939	−	0.280209i	$-0.0904038\pi$
$38$	0	0
$39$	−214.521	−0.880791
$40$	0	0
$41$	−210.504	−0.801834	−0.400917	−	0.916114i	$-0.631308\pi$
$41$	−210.504	−0.801834	−0.400917	+	0.916114i	$0.631308\pi$
$42$	0	0
$43$	168.860i	0.598857i	0.954119	+	0.299429i	$0.0967961\pi$
$43$	168.860i	0.598857i	−0.954119	+	0.299429i	$0.903204\pi$
$44$	0	0
$45$	100.072i	0.331508i
$46$	0	0
$47$	182.902	0.567638	0.283819	−	0.958878i	$-0.408398\pi$
$47$	182.902	0.567638	0.283819	+	0.958878i	$0.408398\pi$
$48$	0	0
$49$	−150.474	−0.438699
$50$	0	0
$51$	− 86.2972i	− 0.236942i
$52$	0	0
$53$	− 37.0020i	− 0.0958984i	−0.998850	−	0.0479492i	$-0.984731\pi$
$53$	− 37.0020i	− 0.0958984i	0.998850	−	0.0479492i	$-0.0152686\pi$
$54$	0	0
$55$	−39.1219	−0.0959125
$56$	0	0
$57$	565.323	1.31366
$58$	0	0
$59$	− 624.494i	− 1.37800i	−0.724760	−	0.689001i	$-0.758050\pi$
$59$	− 624.494i	− 1.37800i	0.724760	−	0.689001i	$-0.241950\pi$
$60$	0	0
$61$	246.759i	0.517938i	0.965886	+	0.258969i	$0.0833828\pi$
$61$	246.759i	0.517938i	−0.965886	+	0.258969i	$0.916617\pi$
$62$	0	0
$63$	−77.6476	−0.155281
$64$	0	0
$65$	−829.188	−1.58228
$66$	0	0
$67$	− 129.763i	− 0.236613i	−0.992977	−	0.118306i	$-0.962253\pi$
$67$	− 129.763i	− 0.236613i	0.992977	−	0.118306i	$-0.0377465\pi$
$68$	0	0
$69$	− 619.970i	− 1.08168i
$70$	0	0
$71$	−348.360	−0.582291	−0.291146	−	0.956679i	$-0.594036\pi$
$71$	−348.360	−0.582291	−0.291146	+	0.956679i	$0.594036\pi$
$72$	0	0
$73$	−299.436	−0.480087	−0.240043	−	0.970762i	$-0.577162\pi$
$73$	−299.436	−0.480087	−0.240043	+	0.970762i	$0.577162\pi$
$74$	0	0
$75$	− 901.168i	− 1.38744i
$76$	0	0
$77$	− 30.3553i	− 0.0449260i
$78$	0	0
$79$	943.487	1.34368	0.671839	−	0.740697i	$-0.265505\pi$
$79$	943.487	1.34368	0.671839	+	0.740697i	$0.265505\pi$
$80$	0	0
$81$	−546.590	−0.749781
$82$	0	0
$83$	443.034i	0.585895i	0.956129	+	0.292947i	$0.0946361\pi$
$83$	443.034i	0.585895i	−0.956129	+	0.292947i	$0.905364\pi$
$84$	0	0
$85$	− 333.565i	− 0.425649i
$86$	0	0
$87$	−391.224	−0.482111
$88$	0	0
$89$	−1412.35	−1.68212	−0.841060	−	0.540942i	$-0.818068\pi$
$89$	−1412.35	−1.68212	−0.841060	+	0.540942i	$0.818068\pi$
$90$	0	0
$91$	− 643.381i	− 0.741150i
$92$	0	0
$93$	− 145.914i	− 0.162694i
$94$	0	0
$95$	2185.14	2.35990
$96$	0	0
$97$	1515.29	1.58613	0.793063	−	0.609140i	$-0.208485\pi$
$97$	1515.29	1.58613	0.793063	+	0.609140i	$0.208485\pi$
$98$	0	0
$99$	12.2426i	0.0124285i
$100$	0	0
$101$	810.631i	0.798621i	0.916816	+	0.399311i	$0.130751\pi$
$101$	810.631i	0.798621i	−0.916816	+	0.399311i	$0.869249\pi$
$102$	0	0
$103$	1021.00	0.976717	0.488359	−	0.872643i	$-0.337596\pi$
$103$	1021.00	0.976717	0.488359	+	0.872643i	$0.337596\pi$
$104$	0	0
$105$	1147.95	1.06693
$106$	0	0
$107$	− 1754.75i	− 1.58540i	−0.609612	−	0.792700i	$-0.708675\pi$
$107$	− 1754.75i	− 1.58540i	0.609612	−	0.792700i	$-0.291325\pi$
$108$	0	0
$109$	− 153.624i	− 0.134996i	−0.997719	−	0.0674980i	$-0.978498\pi$
$109$	− 153.624i	− 0.134996i	0.997719	−	0.0674980i	$-0.0215016\pi$
$110$	0	0
$111$	583.528	0.498973
$112$	0	0
$113$	1722.22	1.43374	0.716870	−	0.697207i	$-0.245574\pi$
$113$	1722.22	1.43374	0.716870	+	0.697207i	$0.245574\pi$
$114$	0	0
$115$	− 2396.37i	− 1.94315i
$116$	0	0
$117$	259.481i	0.205035i
$118$	0	0
$119$	258.818	0.199377
$120$	0	0
$121$	1326.21	0.996404
$122$	0	0
$123$	− 973.883i	− 0.713919i
$124$	0	0
$125$	− 1247.96i	− 0.892969i
$126$	0	0
$127$	699.127	0.488484	0.244242	−	0.969714i	$-0.421461\pi$
$127$	699.127	0.488484	0.244242	+	0.969714i	$0.421461\pi$
$128$	0	0
$129$	−781.219	−0.533198
$130$	0	0
$131$	− 279.972i	− 0.186727i	−0.995632	−	0.0933636i	$-0.970238\pi$
$131$	− 279.972i	− 0.186727i	0.995632	−	0.0933636i	$-0.0297619\pi$
$132$	0	0
$133$	1695.49i	1.10539i
$134$	0	0
$135$	−2696.76	−1.71926
$136$	0	0
$137$	−271.386	−0.169242	−0.0846209	−	0.996413i	$-0.526968\pi$
$137$	−271.386	−0.169242	−0.0846209	+	0.996413i	$0.526968\pi$
$138$	0	0
$139$	650.449i	0.396909i	0.980110	+	0.198455i	$0.0635922\pi$
$139$	650.449i	0.396909i	−0.980110	+	0.198455i	$0.936408\pi$
$140$	0	0
$141$	846.185i	0.505402i
$142$	0	0
$143$	−101.441	−0.0593210
$144$	0	0
$145$	−1512.20	−0.866079
$146$	0	0
$147$	− 696.158i	− 0.390600i
$148$	0	0
$149$	856.692i	0.471026i	0.971871	+	0.235513i	$0.0756771\pi$
$149$	856.692i	0.471026i	−0.971871	+	0.235513i	$0.924323\pi$
$150$	0	0
$151$	−3534.47	−1.90484	−0.952421	−	0.304785i	$-0.901415\pi$
$151$	−3534.47	−1.90484	−0.952421	+	0.304785i	$0.901415\pi$
$152$	0	0
$153$	−104.384	−0.0551564
$154$	0	0
$155$	− 564.001i	− 0.292269i
$156$	0	0
$157$	1744.48i	0.886784i	0.896328	+	0.443392i	$0.146225\pi$
$157$	1744.48i	0.886784i	−0.896328	+	0.443392i	$0.853775\pi$
$158$	0	0
$159$	171.187	0.0853839
$160$	0	0
$161$	1859.38	0.910186
$162$	0	0
$163$	− 3633.63i	− 1.74606i	−0.487666	−	0.873030i	$-0.662152\pi$
$163$	− 3633.63i	− 1.74606i	0.487666	−	0.873030i	$-0.337848\pi$
$164$	0	0
$165$	− 180.995i	− 0.0853965i
$166$	0	0
$167$	−3048.39	−1.41252	−0.706262	−	0.707950i	$-0.749620\pi$
$167$	−3048.39	−1.41252	−0.706262	+	0.707950i	$0.749620\pi$
$168$	0	0
$169$	46.9618	0.0213754
$170$	0	0
$171$	− 683.806i	− 0.305801i
$172$	0	0
$173$	− 2152.51i	− 0.945965i	−0.881072	−	0.472983i	$-0.843177\pi$
$173$	− 2152.51i	− 0.945965i	0.881072	−	0.472983i	$-0.156823\pi$
$174$	0	0
$175$	2702.74	1.16747
$176$	0	0
$177$	2889.18	1.22692
$178$	0	0
$179$	427.457i	0.178489i	0.996010	+	0.0892447i	$0.0284453\pi$
$179$	427.457i	0.178489i	−0.996010	+	0.0892447i	$0.971555\pi$
$180$	0	0
$181$	2399.83i	0.985515i	0.870167	+	0.492758i	$0.164011\pi$
$181$	2399.83i	0.985515i	−0.870167	+	0.492758i	$0.835989\pi$
$182$	0	0
$183$	−1141.62	−0.461151
$184$	0	0
$185$	2255.51	0.896370
$186$	0	0
$187$	− 40.8074i	− 0.0159579i
$188$	0	0
$189$	− 2092.46i	− 0.805312i
$190$	0	0
$191$	−4035.31	−1.52872	−0.764358	−	0.644792i	$-0.776944\pi$
$191$	−4035.31	−1.52872	−0.764358	+	0.644792i	$0.776944\pi$
$192$	0	0
$193$	−886.172	−0.330508	−0.165254	−	0.986251i	$-0.552844\pi$
$193$	−886.172	−0.330508	−0.165254	+	0.986251i	$0.552844\pi$
$194$	0	0
$195$	− 3836.19i	− 1.40880i
$196$	0	0
$197$	− 4624.65i	− 1.67255i	−0.548308	−	0.836276i	$-0.684728\pi$
$197$	− 4624.65i	− 1.67255i	0.548308	−	0.836276i	$-0.315272\pi$
$198$	0	0
$199$	−222.513	−0.0792639	−0.0396319	−	0.999214i	$-0.512619\pi$
$199$	−222.513	−0.0792639	−0.0396319	+	0.999214i	$0.512619\pi$
$200$	0	0
$201$	600.340	0.210670
$202$	0	0
$203$	− 1173.34i	− 0.405677i
$204$	0	0
$205$	− 3764.35i	− 1.28251i
$206$	0	0
$207$	−749.907	−0.251798
$208$	0	0
$209$	267.325	0.0884748
$210$	0	0
$211$	− 4988.58i	− 1.62762i	−0.581131	−	0.813810i	$-0.697390\pi$
$211$	− 4988.58i	− 1.62762i	0.581131	−	0.813810i	$-0.302610\pi$
$212$	0	0
$213$	− 1611.66i	− 0.518448i
$214$	0	0
$215$	−3019.65	−0.957852
$216$	0	0
$217$	437.618	0.136901
$218$	0	0
$219$	− 1385.32i	− 0.427449i
$220$	0	0
$221$	− 864.914i	− 0.263260i
$222$	0	0
$223$	5841.90	1.75427	0.877136	−	0.480242i	$-0.159451\pi$
$223$	5841.90	1.75427	0.877136	+	0.480242i	$0.159451\pi$
$224$	0	0
$225$	−1090.04	−0.322975
$226$	0	0
$227$	1597.41i	0.467065i	0.972349	+	0.233533i	$0.0750286\pi$
$227$	1597.41i	0.467065i	−0.972349	+	0.233533i	$0.924971\pi$
$228$	0	0
$229$	4108.86i	1.18568i	0.805320	+	0.592840i	$0.201994\pi$
$229$	4108.86i	1.18568i	−0.805320	+	0.592840i	$0.798006\pi$
$230$	0	0
$231$	140.437	0.0400003
$232$	0	0
$233$	−734.054	−0.206393	−0.103196	−	0.994661i	$-0.532907\pi$
$233$	−734.054	−0.206393	−0.103196	+	0.994661i	$0.532907\pi$
$234$	0	0
$235$	3270.76i	0.907918i
$236$	0	0
$237$	4364.98i	1.19635i
$238$	0	0
$239$	511.807	0.138519	0.0692595	−	0.997599i	$-0.477936\pi$
$239$	511.807	0.138519	0.0692595	+	0.997599i	$0.477936\pi$
$240$	0	0
$241$	−5920.31	−1.58241	−0.791204	−	0.611552i	$-0.790545\pi$
$241$	−5920.31	−1.58241	−0.791204	+	0.611552i	$0.790545\pi$
$242$	0	0
$243$	1542.93i	0.407322i
$244$	0	0
$245$	− 2690.86i	− 0.701685i
$246$	0	0
$247$	5665.95	1.45958
$248$	0	0
$249$	−2049.67	−0.521656
$250$	0	0
$251$	437.462i	0.110009i	0.998486	+	0.0550046i	$0.0175174\pi$
$251$	437.462i	0.110009i	−0.998486	+	0.0550046i	$0.982483\pi$
$252$	0	0
$253$	− 293.166i	− 0.0728505i
$254$	0	0
$255$	1543.22	0.378980
$256$	0	0
$257$	323.723	0.0785730	0.0392865	−	0.999228i	$-0.487491\pi$
$257$	323.723	0.0785730	0.0392865	+	0.999228i	$0.487491\pi$
$258$	0	0
$259$	1750.09i	0.419865i
$260$	0	0
$261$	473.219i	0.112228i
$262$	0	0
$263$	−2689.15	−0.630495	−0.315248	−	0.949009i	$-0.602088\pi$
$263$	−2689.15	−0.630495	−0.315248	+	0.949009i	$0.602088\pi$
$264$	0	0
$265$	661.691	0.153386
$266$	0	0
$267$	− 6534.14i	− 1.49769i
$268$	0	0
$269$	6652.15i	1.50776i	0.657009	+	0.753882i	$0.271821\pi$
$269$	6652.15i	1.50776i	−0.657009	+	0.753882i	$0.728179\pi$
$270$	0	0
$271$	2018.97	0.452561	0.226280	−	0.974062i	$-0.427343\pi$
$271$	2018.97	0.452561	0.226280	+	0.974062i	$0.427343\pi$
$272$	0	0
$273$	2976.56	0.659889
$274$	0	0
$275$	− 426.136i	− 0.0934435i
$276$	0	0
$277$	− 4356.63i	− 0.944999i	−0.881331	−	0.472499i	$-0.843352\pi$
$277$	− 4356.63i	− 0.944999i	0.881331	−	0.472499i	$-0.156648\pi$
$278$	0	0
$279$	−176.495	−0.0378728
$280$	0	0
$281$	−3893.51	−0.826575	−0.413287	−	0.910601i	$-0.635619\pi$
$281$	−3893.51	−0.826575	−0.413287	+	0.910601i	$0.635619\pi$
$282$	0	0
$283$	− 2865.74i	− 0.601945i	−0.953633	−	0.300972i	$-0.902689\pi$
$283$	− 2865.74i	− 0.601945i	0.953633	−	0.300972i	$-0.0973112\pi$
$284$	0	0
$285$	10109.4i	2.10116i
$286$	0	0
$287$	2920.82	0.600734
$288$	0	0
$289$	−4565.06	−0.929180
$290$	0	0
$291$	7010.39i	1.41222i
$292$	0	0
$293$	1416.60i	0.282452i	0.989977	+	0.141226i	$0.0451044\pi$
$293$	1416.60i	0.282452i	−0.989977	+	0.141226i	$0.954896\pi$
$294$	0	0
$295$	11167.6	2.20407
$296$	0	0
$297$	−329.914	−0.0644565
$298$	0	0
$299$	− 6213.65i	− 1.20182i
$300$	0	0
$301$	− 2342.99i	− 0.448664i
$302$	0	0
$303$	−3750.33	−0.711059
$304$	0	0
$305$	−4412.69	−0.828425
$306$	0	0
$307$	4195.32i	0.779934i	0.920829	+	0.389967i	$0.127514\pi$
$307$	4195.32i	0.779934i	−0.920829	+	0.389967i	$0.872486\pi$
$308$	0	0
$309$	4723.58i	0.869628i
$310$	0	0
$311$	−2911.18	−0.530797	−0.265399	−	0.964139i	$-0.585504\pi$
$311$	−2911.18	−0.530797	−0.265399	+	0.964139i	$0.585504\pi$
$312$	0	0
$313$	−8287.74	−1.49665	−0.748324	−	0.663333i	$-0.769141\pi$
$313$	−8287.74	−1.49665	−0.748324	+	0.663333i	$0.769141\pi$
$314$	0	0
$315$	− 1388.54i	− 0.248366i
$316$	0	0
$317$	− 8120.67i	− 1.43881i	−0.694592	−	0.719404i	$-0.744415\pi$
$317$	− 8120.67i	− 1.43881i	0.694592	−	0.719404i	$-0.255585\pi$
$318$	0	0
$319$	−184.999	−0.0324700
$320$	0	0
$321$	8118.23	1.41157
$322$	0	0
$323$	2279.29i	0.392641i
$324$	0	0
$325$	− 9031.96i	− 1.54155i
$326$	0	0
$327$	710.734	0.120195
$328$	0	0
$329$	−2537.84	−0.425275
$330$	0	0
$331$	6362.92i	1.05661i	0.849055	+	0.528305i	$0.177172\pi$
$331$	6362.92i	1.05661i	−0.849055	+	0.528305i	$0.822828\pi$
$332$	0	0
$333$	− 705.827i	− 0.116153i
$334$	0	0
$335$	2320.50	0.378454
$336$	0	0
$337$	5860.06	0.947234	0.473617	−	0.880731i	$-0.342948\pi$
$337$	5860.06	0.947234	0.473617	+	0.880731i	$0.342948\pi$
$338$	0	0
$339$	7967.73i	1.27654i
$340$	0	0
$341$	− 68.9984i	− 0.0109574i
$342$	0	0
$343$	6847.14	1.07787
$344$	0	0
$345$	11086.7	1.73010
$346$	0	0
$347$	2869.60i	0.443942i	0.975053	+	0.221971i	$0.0712491\pi$
$347$	2869.60i	0.443942i	−0.975053	+	0.221971i	$0.928751\pi$
$348$	0	0
$349$	2748.19i	0.421511i	0.977539	+	0.210755i	$0.0675923\pi$
$349$	2748.19i	0.421511i	−0.977539	+	0.210755i	$0.932408\pi$
$350$	0	0
$351$	−6992.54	−1.06335
$352$	0	0
$353$	−7548.63	−1.13817	−0.569084	−	0.822280i	$-0.692702\pi$
$353$	−7548.63	−1.13817	−0.569084	+	0.822280i	$0.692702\pi$
$354$	0	0
$355$	− 6229.57i	− 0.931355i
$356$	0	0
$357$	1197.41i	0.177517i
$358$	0	0
$359$	−5554.15	−0.816537	−0.408269	−	0.912862i	$-0.633867\pi$
$359$	−5554.15	−0.816537	−0.408269	+	0.912862i	$0.633867\pi$
$360$	0	0
$361$	−8072.37	−1.17690
$362$	0	0
$363$	6135.65i	0.887157i
$364$	0	0
$365$	− 5354.69i	− 0.767883i
$366$	0	0
$367$	3610.98	0.513601	0.256800	−	0.966464i	$-0.417332\pi$
$367$	3610.98	0.513601	0.256800	+	0.966464i	$0.417332\pi$
$368$	0	0
$369$	−1177.99	−0.166190
$370$	0	0
$371$	513.417i	0.0718471i
$372$	0	0
$373$	1718.96i	0.238618i	0.992857	+	0.119309i	$0.0380679\pi$
$373$	1718.96i	0.238618i	−0.992857	+	0.119309i	$0.961932\pi$
$374$	0	0
$375$	5773.62	0.795062
$376$	0	0
$377$	−3921.05	−0.535661
$378$	0	0
$379$	10391.4i	1.40836i	0.710021	+	0.704180i	$0.248685\pi$
$379$	10391.4i	1.40836i	−0.710021	+	0.704180i	$0.751315\pi$
$380$	0	0
$381$	3234.47i	0.434926i
$382$	0	0
$383$	−7668.98	−1.02315	−0.511575	−	0.859238i	$-0.670938\pi$
$383$	−7668.98	−1.02315	−0.511575	+	0.859238i	$0.670938\pi$
$384$	0	0
$385$	542.831	0.0718577
$386$	0	0
$387$	944.951i	0.124120i
$388$	0	0
$389$	284.150i	0.0370359i	0.999829	+	0.0185180i	$0.00589479\pi$
$389$	284.150i	0.0370359i	−0.999829	+	0.0185180i	$0.994105\pi$
$390$	0	0
$391$	2499.62	0.323302
$392$	0	0
$393$	1295.27	0.166254
$394$	0	0
$395$	16872.0i	2.14917i
$396$	0	0
$397$	9209.66i	1.16428i	0.813088	+	0.582141i	$0.197785\pi$
$397$	9209.66i	1.16428i	−0.813088	+	0.582141i	$0.802215\pi$
$398$	0	0
$399$	−7844.07	−0.984197
$400$	0	0
$401$	−5565.10	−0.693036	−0.346518	−	0.938043i	$-0.612636\pi$
$401$	−5565.10	−0.693036	−0.346518	+	0.938043i	$0.612636\pi$
$402$	0	0
$403$	− 1462.42i	− 0.180765i
$404$	0	0
$405$	− 9774.44i	− 1.19925i
$406$	0	0
$407$	275.933	0.0336057
$408$	0	0
$409$	12077.6	1.46014	0.730072	−	0.683370i	$-0.239486\pi$
$409$	12077.6	1.46014	0.730072	+	0.683370i	$0.239486\pi$
$410$	0	0
$411$	− 1255.55i	− 0.150686i
$412$	0	0
$413$	8665.08i	1.03240i
$414$	0	0
$415$	−7922.58	−0.937119
$416$	0	0
$417$	−3009.26	−0.353391
$418$	0	0
$419$	2054.89i	0.239590i	0.992799	+	0.119795i	$0.0382237\pi$
$419$	2054.89i	0.239590i	−0.992799	+	0.119795i	$0.961776\pi$
$420$	0	0
$421$	6819.69i	0.789480i	0.918793	+	0.394740i	$0.129165\pi$
$421$	6819.69i	0.789480i	−0.918793	+	0.394740i	$0.870835\pi$
$422$	0	0
$423$	1023.53	0.117650
$424$	0	0
$425$	3633.36	0.414692
$426$	0	0
$427$	− 3423.87i	− 0.388040i
$428$	0	0
$429$	− 469.309i	− 0.0528169i
$430$	0	0
$431$	−12519.2	−1.39914	−0.699571	−	0.714563i	$-0.746626\pi$
$431$	−12519.2	−1.39914	−0.699571	+	0.714563i	$0.746626\pi$
$432$	0	0
$433$	2921.40	0.324235	0.162117	−	0.986771i	$-0.448168\pi$
$433$	2921.40	0.324235	0.162117	+	0.986771i	$0.448168\pi$
$434$	0	0
$435$	− 6996.10i	− 0.771120i
$436$	0	0
$437$	16374.7i	1.79247i
$438$	0	0
$439$	1140.50	0.123993	0.0619967	−	0.998076i	$-0.480253\pi$
$439$	1140.50	0.123993	0.0619967	+	0.998076i	$0.480253\pi$
$440$	0	0
$441$	−842.062	−0.0909256
$442$	0	0
$443$	2606.46i	0.279541i	0.990184	+	0.139771i	$0.0446365\pi$
$443$	2606.46i	0.279541i	−0.990184	+	0.139771i	$0.955363\pi$
$444$	0	0
$445$	− 25256.4i	− 2.69049i
$446$	0	0
$447$	−3963.43	−0.419382
$448$	0	0
$449$	1752.13	0.184161	0.0920805	−	0.995752i	$-0.470648\pi$
$449$	1752.13	0.184161	0.0920805	+	0.995752i	$0.470648\pi$
$450$	0	0
$451$	− 460.521i	− 0.0480822i
$452$	0	0
$453$	− 16352.0i	− 1.69599i
$454$	0	0
$455$	11505.3	1.18544
$456$	0	0
$457$	12875.6	1.31794	0.658968	−	0.752171i	$-0.270993\pi$
$457$	12875.6	1.31794	0.658968	+	0.752171i	$0.270993\pi$
$458$	0	0
$459$	− 2812.95i	− 0.286051i
$460$	0	0
$461$	− 19346.0i	− 1.95452i	−0.212050	−	0.977259i	$-0.568014\pi$
$461$	− 19346.0i	− 1.95452i	0.212050	−	0.977259i	$-0.431986\pi$
$462$	0	0
$463$	−15002.4	−1.50588	−0.752938	−	0.658091i	$-0.771364\pi$
$463$	−15002.4	−1.50588	−0.752938	+	0.658091i	$0.771364\pi$
$464$	0	0
$465$	2609.32	0.260224
$466$	0	0
$467$	13674.6i	1.35500i	0.735522	+	0.677501i	$0.236937\pi$
$467$	13674.6i	1.35500i	−0.735522	+	0.677501i	$0.763063\pi$
$468$	0	0
$469$	1800.51i	0.177270i
$470$	0	0
$471$	−8070.75	−0.789555
$472$	0	0
$473$	−369.416	−0.0359107
$474$	0	0
$475$	23801.7i	2.29915i
$476$	0	0
$477$	− 207.066i	− 0.0198761i
$478$	0	0
$479$	−3072.68	−0.293099	−0.146550	−	0.989203i	$-0.546817\pi$
$479$	−3072.68	−0.293099	−0.146550	+	0.989203i	$0.546817\pi$
$480$	0	0
$481$	5848.41	0.554396
$482$	0	0
$483$	8602.32i	0.810392i
$484$	0	0
$485$	27097.3i	2.53695i
$486$	0	0
$487$	−8689.64	−0.808553	−0.404276	−	0.914637i	$-0.632477\pi$
$487$	−8689.64	−0.808553	−0.404276	+	0.914637i	$0.632477\pi$
$488$	0	0
$489$	16810.8	1.55462
$490$	0	0
$491$	15831.2i	1.45509i	0.686057	+	0.727547i	$0.259340\pi$
$491$	15831.2i	1.45509i	−0.686057	+	0.727547i	$0.740660\pi$
$492$	0	0
$493$	− 1577.35i	− 0.144098i
$494$	0	0
$495$	−218.929	−0.0198790
$496$	0	0
$497$	4833.62	0.436253
$498$	0	0
$499$	− 2309.01i	− 0.207146i	−0.994622	−	0.103573i	$-0.966973\pi$
$499$	− 2309.01i	− 0.207146i	0.994622	−	0.103573i	$-0.0330275\pi$
$500$	0	0
$501$	− 14103.2i	− 1.25765i
$502$	0	0
$503$	−6901.81	−0.611802	−0.305901	−	0.952063i	$-0.598958\pi$
$503$	−6901.81	−0.611802	−0.305901	+	0.952063i	$0.598958\pi$
$504$	0	0
$505$	−14496.2	−1.27737
$506$	0	0
$507$	217.266i	0.0190318i
$508$	0	0
$509$	− 131.481i	− 0.0114495i	−0.999984	−	0.00572475i	$-0.998178\pi$
$509$	− 131.481i	− 0.0114495i	0.999984	−	0.00572475i	$-0.00182226\pi$
$510$	0	0
$511$	4154.79	0.359681
$512$	0	0
$513$	18427.3	1.58594
$514$	0	0
$515$	18258.1i	1.56223i
$516$	0	0
$517$	400.136i	0.0340386i
$518$	0	0
$519$	9958.43	0.842248
$520$	0	0
$521$	−11931.3	−1.00330	−0.501649	−	0.865071i	$-0.667273\pi$
$521$	−11931.3	−1.00330	−0.501649	+	0.865071i	$0.667273\pi$
$522$	0	0
$523$	− 13721.4i	− 1.14721i	−0.819131	−	0.573607i	$-0.805544\pi$
$523$	− 13721.4i	− 1.14721i	0.819131	−	0.573607i	$-0.194456\pi$
$524$	0	0
$525$	12504.0i	1.03947i
$526$	0	0
$527$	588.301	0.0486277
$528$	0	0
$529$	5790.59	0.475926
$530$	0	0
$531$	− 3494.71i	− 0.285607i
$532$	0	0
$533$	− 9760.75i	− 0.793218i
$534$	0	0
$535$	31379.4	2.53579
$536$	0	0
$537$	−1977.60	−0.158920
$538$	0	0
$539$	− 329.193i	− 0.0263067i
$540$	0	0
$541$	12101.0i	0.961665i	0.876812	+	0.480833i	$0.159666\pi$
$541$	12101.0i	0.961665i	−0.876812	+	0.480833i	$0.840334\pi$
$542$	0	0
$543$	−11102.7	−0.877462
$544$	0	0
$545$	2747.20	0.215921
$546$	0	0
$547$	63.9157i	0.00499605i	0.999997	+	0.00249803i	$0.000795147\pi$
$547$	63.9157i	0.00499605i	−0.999997	+	0.00249803i	$0.999205\pi$
$548$	0	0
$549$	1380.88i	0.107349i
$550$	0	0
$551$	10333.1	0.798917
$552$	0	0
$553$	−13091.2	−1.00668
$554$	0	0
$555$	10435.0i	0.798090i
$556$	0	0
$557$	6052.26i	0.460400i	0.973143	+	0.230200i	$0.0739380\pi$
$557$	6052.26i	0.460400i	−0.973143	+	0.230200i	$0.926062\pi$
$558$	0	0
$559$	−7829.77	−0.592422
$560$	0	0
$561$	188.793	0.0142083
$562$	0	0
$563$	20638.9i	1.54498i	0.635026	+	0.772491i	$0.280989\pi$
$563$	20638.9i	1.54498i	−0.635026	+	0.772491i	$0.719011\pi$
$564$	0	0
$565$	30797.7i	2.29322i
$566$	0	0
$567$	7584.14	0.561736
$568$	0	0
$569$	21728.1	1.60086	0.800431	−	0.599425i	$-0.204604\pi$
$569$	21728.1	1.60086	0.800431	+	0.599425i	$0.204604\pi$
$570$	0	0
$571$	− 22737.8i	− 1.66646i	−0.552929	−	0.833229i	$-0.686490\pi$
$571$	− 22737.8i	− 1.66646i	0.552929	−	0.833229i	$-0.313510\pi$
$572$	0	0
$573$	− 18669.1i	− 1.36110i
$574$	0	0
$575$	26102.5	1.89313
$576$	0	0
$577$	−26648.2	−1.92267	−0.961335	−	0.275383i	$-0.911195\pi$
$577$	−26648.2	−1.92267	−0.961335	+	0.275383i	$0.911195\pi$
$578$	0	0
$579$	− 4099.82i	− 0.294270i
$580$	0	0
$581$	− 6147.26i	− 0.438952i
$582$	0	0
$583$	80.9495	0.00575058
$584$	0	0
$585$	−4640.20	−0.327946
$586$	0	0
$587$	1898.75i	0.133509i	0.997769	+	0.0667544i	$0.0212644\pi$
$587$	1898.75i	0.133509i	−0.997769	+	0.0667544i	$0.978736\pi$
$588$	0	0
$589$	3853.89i	0.269604i
$590$	0	0
$591$	21395.7	1.48917
$592$	0	0
$593$	−4474.79	−0.309878	−0.154939	−	0.987924i	$-0.549518\pi$
$593$	−4474.79	−0.309878	−0.154939	+	0.987924i	$0.549518\pi$
$594$	0	0
$595$	4628.34i	0.318896i
$596$	0	0
$597$	− 1029.44i	− 0.0705733i
$598$	0	0
$599$	12603.8	0.859725	0.429863	−	0.902894i	$-0.358562\pi$
$599$	12603.8	0.859725	0.429863	+	0.902894i	$0.358562\pi$
$600$	0	0
$601$	−7220.64	−0.490077	−0.245038	−	0.969513i	$-0.578801\pi$
$601$	−7220.64	−0.490077	−0.245038	+	0.969513i	$0.578801\pi$
$602$	0	0
$603$	− 726.163i	− 0.0490408i
$604$	0	0
$605$	23716.1i	1.59371i
$606$	0	0
$607$	−13695.6	−0.915796	−0.457898	−	0.889005i	$-0.651397\pi$
$607$	−13695.6	−0.915796	−0.457898	+	0.889005i	$0.651397\pi$
$608$	0	0
$609$	5428.39	0.361198
$610$	0	0
$611$	8480.89i	0.561539i
$612$	0	0
$613$	3335.20i	0.219751i	0.993945	+	0.109876i	$0.0350452\pi$
$613$	3335.20i	0.219751i	−0.993945	+	0.109876i	$0.964955\pi$
$614$	0	0
$615$	17415.5	1.14189
$616$	0	0
$617$	4186.39	0.273157	0.136579	−	0.990629i	$-0.456389\pi$
$617$	4186.39	0.273157	0.136579	+	0.990629i	$0.456389\pi$
$618$	0	0
$619$	− 6788.94i	− 0.440825i	−0.975407	−	0.220412i	$-0.929260\pi$
$619$	− 6788.94i	− 0.440825i	0.975407	−	0.220412i	$-0.0707403\pi$
$620$	0	0
$621$	− 20208.6i	− 1.30587i
$622$	0	0
$623$	19596.9	1.26024
$624$	0	0
$625$	−2031.54	−0.130018
$626$	0	0
$627$	1236.76i	0.0787743i
$628$	0	0
$629$	2352.69i	0.149138i
$630$	0	0
$631$	−16106.3	−1.01614	−0.508069	−	0.861316i	$-0.669640\pi$
$631$	−16106.3	−1.01614	−0.508069	+	0.861316i	$0.669640\pi$
$632$	0	0
$633$	23079.3	1.44917
$634$	0	0
$635$	12502.2i	0.781314i
$636$	0	0
$637$	− 6977.25i	− 0.433985i
$638$	0	0
$639$	−1949.45	−0.120687
$640$	0	0
$641$	−6682.21	−0.411749	−0.205875	−	0.978578i	$-0.566004\pi$
$641$	−6682.21	−0.411749	−0.205875	+	0.978578i	$0.566004\pi$
$642$	0	0
$643$	7047.69i	0.432245i	0.976366	+	0.216123i	$0.0693411\pi$
$643$	7047.69i	0.432245i	−0.976366	+	0.216123i	$0.930659\pi$
$644$	0	0
$645$	− 13970.2i	− 0.852832i
$646$	0	0
$647$	−5078.45	−0.308585	−0.154292	−	0.988025i	$-0.549310\pi$
$647$	−5078.45	−0.308585	−0.154292	+	0.988025i	$0.549310\pi$
$648$	0	0
$649$	1366.21	0.0826324
$650$	0	0
$651$	2024.61i	0.121891i
$652$	0	0
$653$	8753.86i	0.524602i	0.964986	+	0.262301i	$0.0844813\pi$
$653$	8753.86i	0.524602i	−0.964986	+	0.262301i	$0.915519\pi$
$654$	0	0
$655$	5006.62	0.298664
$656$	0	0
$657$	−1675.67	−0.0995037
$658$	0	0
$659$	8133.42i	0.480778i	0.970677	+	0.240389i	$0.0772750\pi$
$659$	8133.42i	0.480778i	−0.970677	+	0.240389i	$0.922725\pi$
$660$	0	0
$661$	− 8917.15i	− 0.524716i	−0.964971	−	0.262358i	$-0.915500\pi$
$661$	− 8917.15i	− 0.524716i	0.964971	−	0.262358i	$-0.0845001\pi$
$662$	0	0
$663$	4001.47	0.234396
$664$	0	0
$665$	−30319.7	−1.76804
$666$	0	0
$667$	− 11331.9i	− 0.657831i
$668$	0	0
$669$	27027.2i	1.56193i
$670$	0	0
$671$	−539.837	−0.0310584
$672$	0	0
$673$	14664.4	0.839925	0.419963	−	0.907541i	$-0.362043\pi$
$673$	14664.4	0.839925	0.419963	+	0.907541i	$0.362043\pi$
$674$	0	0
$675$	− 29374.5i	− 1.67500i
$676$	0	0
$677$	− 8195.60i	− 0.465262i	−0.972565	−	0.232631i	$-0.925267\pi$
$677$	− 8195.60i	− 0.465262i	0.972565	−	0.232631i	$-0.0747334\pi$
$678$	0	0
$679$	−21025.2	−1.18833
$680$	0	0
$681$	−7390.32	−0.415855
$682$	0	0
$683$	− 18574.4i	− 1.04060i	−0.853983	−	0.520300i	$-0.825820\pi$
$683$	− 18574.4i	− 1.04060i	0.853983	−	0.520300i	$-0.174180\pi$
$684$	0	0
$685$	− 4853.09i	− 0.270696i
$686$	0	0
$687$	−19009.4	−1.05568
$688$	0	0
$689$	1715.73	0.0948679
$690$	0	0
$691$	20257.2i	1.11523i	0.830101	+	0.557613i	$0.188282\pi$
$691$	20257.2i	1.11523i	−0.830101	+	0.557613i	$0.811718\pi$
$692$	0	0
$693$	− 169.870i	− 0.00931146i
$694$	0	0
$695$	−11631.7	−0.634843
$696$	0	0
$697$	3926.54	0.213383
$698$	0	0
$699$	− 3396.06i	− 0.183763i
$700$	0	0
$701$	− 22609.3i	− 1.21818i	−0.793102	−	0.609089i	$-0.791535\pi$
$701$	− 22609.3i	− 1.21818i	0.793102	−	0.609089i	$-0.208465\pi$
$702$	0	0
$703$	−15412.2	−0.826859
$704$	0	0
$705$	−15132.0	−0.808373
$706$	0	0
$707$	− 11247.8i	− 0.598327i
$708$	0	0
$709$	− 27690.9i	− 1.46679i	−0.679802	−	0.733395i	$-0.737934\pi$
$709$	− 27690.9i	− 1.46679i	0.679802	−	0.733395i	$-0.262066\pi$
$710$	0	0
$711$	5279.82	0.278493
$712$	0	0
$713$	4226.43	0.221993
$714$	0	0
$715$	− 1814.02i	− 0.0948819i
$716$	0	0
$717$	2367.84i	0.123331i
$718$	0	0
$719$	2111.24	0.109507	0.0547537	−	0.998500i	$-0.482563\pi$
$719$	2111.24	0.109507	0.0547537	+	0.998500i	$0.482563\pi$
$720$	0	0
$721$	−14166.7	−0.731757
$722$	0	0
$723$	− 27389.9i	− 1.40891i
$724$	0	0
$725$	− 16471.7i	− 0.843784i
$726$	0	0
$727$	−14763.6	−0.753164	−0.376582	−	0.926383i	$-0.622901\pi$
$727$	−14763.6	−0.753164	−0.376582	+	0.926383i	$0.622901\pi$
$728$	0	0
$729$	−21896.2	−1.11244
$730$	0	0
$731$	− 3149.75i	− 0.159368i
$732$	0	0
$733$	− 4836.29i	− 0.243700i	−0.992549	−	0.121850i	$-0.961117\pi$
$733$	− 4836.29i	− 0.243700i	0.992549	−	0.121850i	$-0.0388827\pi$
$734$	0	0
$735$	12449.1	0.624751
$736$	0	0
$737$	283.883	0.0141886
$738$	0	0
$739$	16015.7i	0.797224i	0.917120	+	0.398612i	$0.130508\pi$
$739$	16015.7i	0.797224i	−0.917120	+	0.398612i	$0.869492\pi$
$740$	0	0
$741$	26213.2i	1.29955i
$742$	0	0
$743$	−20313.3	−1.00299	−0.501495	−	0.865160i	$-0.667217\pi$
$743$	−20313.3	−1.00299	−0.501495	+	0.865160i	$0.667217\pi$
$744$	0	0
$745$	−15319.9	−0.753391
$746$	0	0
$747$	2479.25i	0.121434i
$748$	0	0
$749$	24347.8i	1.18778i
$750$	0	0
$751$	−28755.3	−1.39720	−0.698598	−	0.715514i	$-0.746192\pi$
$751$	−28755.3	−1.39720	−0.698598	+	0.715514i	$0.746192\pi$
$752$	0	0
$753$	−2023.89	−0.0979477
$754$	0	0
$755$	− 63205.4i	− 3.04673i
$756$	0	0
$757$	− 32535.4i	− 1.56211i	−0.624460	−	0.781057i	$-0.714681\pi$
$757$	− 32535.4i	− 1.56211i	0.624460	−	0.781057i	$-0.285319\pi$
$758$	0	0
$759$	1356.31	0.0648631
$760$	0	0
$761$	9298.53	0.442932	0.221466	−	0.975168i	$-0.428916\pi$
$761$	9298.53	0.442932	0.221466	+	0.975168i	$0.428916\pi$
$762$	0	0
$763$	2131.60i	0.101139i
$764$	0	0
$765$	− 1866.65i	− 0.0882208i
$766$	0	0
$767$	28956.8	1.36319
$768$	0	0
$769$	−20402.0	−0.956717	−0.478358	−	0.878165i	$-0.658768\pi$
$769$	−20402.0	−0.956717	−0.478358	+	0.878165i	$0.658768\pi$
$770$	0	0
$771$	1497.68i	0.0699582i
$772$	0	0
$773$	− 10376.1i	− 0.482798i	−0.970426	−	0.241399i	$-0.922394\pi$
$773$	− 10376.1i	− 0.482798i	0.970426	−	0.241399i	$-0.0776063\pi$
$774$	0	0
$775$	6143.40	0.284745
$776$	0	0
$777$	−8096.67	−0.373831
$778$	0	0
$779$	25722.3i	1.18305i
$780$	0	0
$781$	− 762.109i	− 0.0349173i
$782$	0	0
$783$	−12752.4	−0.582034
$784$	0	0
$785$	−31195.9	−1.41838
$786$	0	0
$787$	− 16869.6i	− 0.764088i	−0.924144	−	0.382044i	$-0.875220\pi$
$787$	− 16869.6i	− 0.764088i	0.924144	−	0.382044i	$-0.124780\pi$
$788$	0	0
$789$	− 12441.2i	− 0.561367i
$790$	0	0
$791$	−23896.4	−1.07416
$792$	0	0
$793$	−11441.8	−0.512373
$794$	0	0
$795$	3061.27i	0.136569i
$796$	0	0
$797$	9300.12i	0.413334i	0.978411	+	0.206667i	$0.0662617\pi$
$797$	9300.12i	0.413334i	−0.978411	+	0.206667i	$0.933738\pi$
$798$	0	0
$799$	−3411.68	−0.151060
$800$	0	0
$801$	−7903.61	−0.348639
$802$	0	0
$803$	− 655.079i	− 0.0287886i
$804$	0	0
$805$	33250.6i	1.45581i
$806$	0	0
$807$	−30775.8	−1.34245
$808$	0	0
$809$	29320.9	1.27425	0.637126	−	0.770760i	$-0.280123\pi$
$809$	29320.9	1.27425	0.637126	+	0.770760i	$0.280123\pi$
$810$	0	0
$811$	− 20488.9i	− 0.887132i	−0.896242	−	0.443566i	$-0.853713\pi$
$811$	− 20488.9i	− 0.887132i	0.896242	−	0.443566i	$-0.146287\pi$
$812$	0	0
$813$	9340.66i	0.402941i
$814$	0	0
$815$	64978.7	2.79276
$816$	0	0
$817$	20633.6	0.883574
$818$	0	0
$819$	− 3600.40i	− 0.153612i
$820$	0	0
$821$	28650.9i	1.21793i	0.793196	+	0.608966i	$0.208415\pi$
$821$	28650.9i	1.21793i	−0.793196	+	0.608966i	$0.791585\pi$
$822$	0	0
$823$	−24605.9	−1.04217	−0.521086	−	0.853504i	$-0.674473\pi$
$823$	−24605.9	−1.04217	−0.521086	+	0.853504i	$0.674473\pi$
$824$	0	0
$825$	1971.49	0.0831982
$826$	0	0
$827$	34076.6i	1.43284i	0.697668	+	0.716421i	$0.254221\pi$
$827$	34076.6i	1.43284i	−0.697668	+	0.716421i	$0.745779\pi$
$828$	0	0
$829$	1293.46i	0.0541904i	0.999633	+	0.0270952i	$0.00862572\pi$
$829$	1293.46i	0.0541904i	−0.999633	+	0.0270952i	$0.991374\pi$
$830$	0	0
$831$	20155.7	0.841388
$832$	0	0
$833$	2806.80	0.116746
$834$	0	0
$835$	− 54513.1i	− 2.25929i
$836$	0	0
$837$	− 4756.22i	− 0.196415i
$838$	0	0
$839$	−28847.5	−1.18704	−0.593521	−	0.804819i	$-0.702263\pi$
$839$	−28847.5	−1.18704	−0.593521	+	0.804819i	$0.702263\pi$
$840$	0	0
$841$	17238.1	0.706800
$842$	0	0
$843$	− 18013.1i	− 0.735948i
$844$	0	0
$845$	839.798i	0.0341893i
$846$	0	0
$847$	−18401.7	−0.746506
$848$	0	0
$849$	13258.2	0.535947
$850$	0	0
$851$	16902.0i	0.680839i
$852$	0	0
$853$	58.7693i	0.00235900i	0.999999	+	0.00117950i	$0.000375446\pi$
$853$	58.7693i	0.00235900i	−0.999999	+	0.00117950i	$0.999625\pi$
$854$	0	0
$855$	12228.2	0.489118
$856$	0	0
$857$	20953.6	0.835194	0.417597	−	0.908632i	$-0.362872\pi$
$857$	20953.6	0.835194	0.417597	+	0.908632i	$0.362872\pi$
$858$	0	0
$859$	− 41459.5i	− 1.64677i	−0.567481	−	0.823387i	$-0.692082\pi$
$859$	− 41459.5i	− 1.64677i	0.567481	−	0.823387i	$-0.307918\pi$
$860$	0	0
$861$	13513.0i	0.534868i
$862$	0	0
$863$	−3389.59	−0.133700	−0.0668499	−	0.997763i	$-0.521295\pi$
$863$	−3389.59	−0.133700	−0.0668499	+	0.997763i	$0.521295\pi$
$864$	0	0
$865$	38492.3	1.51304
$866$	0	0
$867$	− 21120.0i	− 0.827304i
$868$	0	0
$869$	2064.07i	0.0805741i
$870$	0	0
$871$	6016.91	0.234070
$872$	0	0
$873$	8479.66	0.328743
$874$	0	0
$875$	17315.9i	0.669012i
$876$	0	0
$877$	− 19675.3i	− 0.757568i	−0.925485	−	0.378784i	$-0.876342\pi$
$877$	− 19675.3i	− 0.757568i	0.925485	−	0.378784i	$-0.123658\pi$
$878$	0	0
$879$	−6553.79	−0.251484
$880$	0	0
$881$	1497.48	0.0572660	0.0286330	−	0.999590i	$-0.490885\pi$
$881$	1497.48	0.0572660	0.0286330	+	0.999590i	$0.490885\pi$
$882$	0	0
$883$	− 5860.23i	− 0.223344i	−0.993745	−	0.111672i	$-0.964379\pi$
$883$	− 5860.23i	− 0.223344i	0.993745	−	0.111672i	$-0.0356206\pi$
$884$	0	0
$885$	51666.0i	1.96241i
$886$	0	0
$887$	−18058.0	−0.683573	−0.341787	−	0.939778i	$-0.611032\pi$
$887$	−18058.0	−0.683573	−0.341787	+	0.939778i	$0.611032\pi$
$888$	0	0
$889$	−9700.66	−0.365973
$890$	0	0
$891$	− 1195.78i	− 0.0449609i
$892$	0	0
$893$	− 22349.5i	− 0.837512i
$894$	0	0
$895$	−7644.03	−0.285488
$896$	0	0
$897$	28747.1	1.07005
$898$	0	0
$899$	− 2667.04i	− 0.0989441i
$900$	0	0
$901$	690.200i	0.0255204i
$902$	0	0
$903$	10839.7	0.399472
$904$	0	0
$905$	−42915.2	−1.57630
$906$	0	0
$907$	− 30297.4i	− 1.10916i	−0.832130	−	0.554580i	$-0.812879\pi$
$907$	− 30297.4i	− 1.10916i	0.832130	−	0.554580i	$-0.187121\pi$
$908$	0	0
$909$	4536.35i	0.165524i
$910$	0	0
$911$	31977.7	1.16297	0.581487	−	0.813556i	$-0.302471\pi$
$911$	31977.7	1.16297	0.581487	+	0.813556i	$0.302471\pi$
$912$	0	0
$913$	−969.228	−0.0351334
$914$	0	0
$915$	− 20415.0i	− 0.737595i
$916$	0	0
$917$	3884.72i	0.139896i
$918$	0	0
$919$	−40696.7	−1.46078	−0.730391	−	0.683029i	$-0.760662\pi$
$919$	−40696.7	−1.46078	−0.730391	+	0.683029i	$0.760662\pi$
$920$	0	0
$921$	−19409.4	−0.694421
$922$	0	0
$923$	− 16152.9i	− 0.576034i
$924$	0	0
$925$	24568.2i	0.873295i
$926$	0	0
$927$	5713.57	0.202436
$928$	0	0
$929$	−11467.5	−0.404989	−0.202495	−	0.979283i	$-0.564905\pi$
$929$	−11467.5	−0.404989	−0.202495	+	0.979283i	$0.564905\pi$
$930$	0	0
$931$	18387.0i	0.647271i
$932$	0	0
$933$	− 13468.4i	− 0.472600i
$934$	0	0
$935$	729.742	0.0255242
$936$	0	0
$937$	−14100.2	−0.491603	−0.245802	−	0.969320i	$-0.579051\pi$
$937$	−14100.2	−0.491603	−0.245802	+	0.969320i	$0.579051\pi$
$938$	0	0
$939$	− 38342.7i	− 1.33255i
$940$	0	0
$941$	− 29781.6i	− 1.03172i	−0.856672	−	0.515862i	$-0.827472\pi$
$941$	− 29781.6i	− 1.03172i	0.856672	−	0.515862i	$-0.172528\pi$
$942$	0	0
$943$	28208.8	0.974129
$944$	0	0
$945$	37418.5	1.28807
$946$	0	0
$947$	35353.8i	1.21314i	0.795030	+	0.606571i	$0.207455\pi$
$947$	35353.8i	1.21314i	−0.795030	+	0.606571i	$0.792545\pi$
$948$	0	0
$949$	− 13884.4i	− 0.474928i
$950$	0	0
$951$	37569.8	1.28106
$952$	0	0
$953$	−6456.01	−0.219445	−0.109722	−	0.993962i	$-0.534996\pi$
$953$	−6456.01	−0.219445	−0.109722	+	0.993962i	$0.534996\pi$
$954$	0	0
$955$	− 72161.7i	− 2.44513i
$956$	0	0
$957$	− 855.885i	− 0.0289100i
$958$	0	0
$959$	3765.59	0.126796
$960$	0	0
$961$	−28796.3	−0.966610
$962$	0	0
$963$	− 9819.69i	− 0.328593i
$964$	0	0
$965$	− 15847.0i	− 0.528636i
$966$	0	0
$967$	−15099.9	−0.502153	−0.251076	−	0.967967i	$-0.580785\pi$
$967$	−15099.9	−0.502153	−0.251076	+	0.967967i	$0.580785\pi$
$968$	0	0
$969$	−10545.0	−0.349591
$970$	0	0
$971$	− 8800.82i	− 0.290867i	−0.989368	−	0.145433i	$-0.953542\pi$
$971$	− 8800.82i	− 0.290867i	0.989368	−	0.145433i	$-0.0464577\pi$
$972$	0	0
$973$	− 9025.23i	− 0.297364i
$974$	0	0
$975$	41785.8	1.37253
$976$	0	0
$977$	34900.0	1.14284	0.571418	−	0.820659i	$-0.306393\pi$
$977$	34900.0	1.14284	0.571418	+	0.820659i	$0.306393\pi$
$978$	0	0
$979$	− 3089.81i	− 0.100869i
$980$	0	0
$981$	− 859.693i	− 0.0279795i
$982$	0	0
$983$	21221.5	0.688567	0.344283	−	0.938866i	$-0.388122\pi$
$983$	21221.5	0.688567	0.344283	+	0.938866i	$0.388122\pi$
$984$	0	0
$985$	82700.7	2.67519
$986$	0	0
$987$	− 11741.1i	− 0.378647i
$988$	0	0
$989$	− 22628.2i	− 0.727538i
$990$	0	0
$991$	23985.3	0.768838	0.384419	−	0.923159i	$-0.374402\pi$
$991$	23985.3	0.768838	0.384419	+	0.923159i	$0.374402\pi$
$992$	0	0
$993$	−29437.7	−0.940761
$994$	0	0
$995$	− 3979.10i	− 0.126780i
$996$	0	0
$997$	− 15222.3i	− 0.483547i	−0.970333	−	0.241773i	$-0.922271\pi$
$997$	− 15222.3i	− 0.483547i	0.970333	−	0.241773i	$-0.0777290\pi$
$998$	0	0
$999$	19020.7	0.602391

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.b.j.513.8		10
4.3	odd	2		1024.4.b.k.513.3		10
8.3	odd	2		1024.4.b.k.513.8		10
8.5	even	2	inner	1024.4.b.j.513.3		10
16.3	odd	4		1024.4.a.m.1.8		10
16.5	even	4		1024.4.a.n.1.8		10
16.11	odd	4		1024.4.a.m.1.3		10
16.13	even	4		1024.4.a.n.1.3		10
32.3	odd	8		64.4.e.a.49.4		10
32.5	even	8		16.4.e.a.13.5	yes	10
32.11	odd	8		128.4.e.a.33.2		10
32.13	even	8		128.4.e.b.97.4		10
32.19	odd	8		128.4.e.a.97.2		10
32.21	even	8		128.4.e.b.33.4		10
32.27	odd	8		64.4.e.a.17.4		10
32.29	even	8		16.4.e.a.5.5	✓	10
96.5	odd	8		144.4.k.a.109.1		10
96.29	odd	8		144.4.k.a.37.1		10
96.35	even	8		576.4.k.a.433.5		10
96.59	even	8		576.4.k.a.145.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.5	✓	10	32.29	even	8
16.4.e.a.13.5	yes	10	32.5	even	8
64.4.e.a.17.4		10	32.27	odd	8
64.4.e.a.49.4		10	32.3	odd	8
128.4.e.a.33.2		10	32.11	odd	8
128.4.e.a.97.2		10	32.19	odd	8
128.4.e.b.33.4		10	32.21	even	8
128.4.e.b.97.4		10	32.13	even	8
144.4.k.a.37.1		10	96.29	odd	8
144.4.k.a.109.1		10	96.5	odd	8
576.4.k.a.145.5		10	96.59	even	8
576.4.k.a.433.5		10	96.35	even	8
1024.4.a.m.1.3		10	16.11	odd	4
1024.4.a.m.1.8		10	16.3	odd	4
1024.4.a.n.1.3		10	16.13	even	4
1024.4.a.n.1.8		10	16.5	even	4
1024.4.b.j.513.3		10	8.5	even	2	inner
1024.4.b.j.513.8		10	1.1	even	1	trivial
1024.4.b.k.513.3		10	4.3	odd	2
1024.4.b.k.513.8		10	8.3	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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.62644i q^{3} +17.8826i q^{5} -13.8754 q^{7} +5.59607 q^{9} +O(q^{10})\)	\(q+4.62644i q^{3} +17.8826i q^{5} -13.8754 q^{7} +5.59607 q^{9} +2.18771i q^{11} +46.3685i q^{13} -82.7326 q^{15} -18.6531 q^{17} -122.194i q^{19} -64.1936i q^{21} -134.006 q^{23} -194.786 q^{25} +150.804i q^{27} +84.5628i q^{29} -31.5391 q^{31} -10.1213 q^{33} -248.128i q^{35} -126.129i q^{37} -214.521 q^{39} -210.504 q^{41} +168.860i q^{43} +100.072i q^{45} +182.902 q^{47} -150.474 q^{49} -86.2972i q^{51} -37.0020i q^{53} -39.1219 q^{55} +565.323 q^{57} -624.494i q^{59} +246.759i q^{61} -77.6476 q^{63} -829.188 q^{65} -129.763i q^{67} -619.970i q^{69} -348.360 q^{71} -299.436 q^{73} -901.168i q^{75} -30.3553i q^{77} +943.487 q^{79} -546.590 q^{81} +443.034i q^{83} -333.565i q^{85} -391.224 q^{87} -1412.35 q^{89} -643.381i q^{91} -145.914i q^{93} +2185.14 q^{95} +1515.29 q^{97} +12.2426i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{7} - 54 q^{9}+O(q^{10}) \) 10 * q - 28 * q^7 - 54 * q^9	\( 10 q - 28 q^{7} - 54 q^{9} + 124 q^{15} + 4 q^{17} - 276 q^{23} - 50 q^{25} + 368 q^{31} - 4 q^{33} - 732 q^{39} + 944 q^{47} - 94 q^{49} - 1380 q^{55} - 108 q^{57} + 2628 q^{63} - 492 q^{65} - 3468 q^{71} + 296 q^{73} + 4416 q^{79} - 482 q^{81} - 6036 q^{87} - 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q - 28 * q^7 - 54 * q^9 + 124 * q^15 + 4 * q^17 - 276 * q^23 - 50 * q^25 + 368 * q^31 - 4 * q^33 - 732 * q^39 + 944 * q^47 - 94 * q^49 - 1380 * q^55 - 108 * q^57 + 2628 * q^63 - 492 * q^65 - 3468 * q^71 + 296 * q^73 + 4416 * q^79 - 482 * q^81 - 6036 * q^87 - 88 * q^89 + 6900 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more