LMFDB - Embedded newform 2116.3.d.c.1057.34

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2116,3,Mod(1057,2116)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2116.1057"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2116, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2116 = 2^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2116.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$57.6568239386$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 92)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1057.34
Character	$\chi$	$=$	2116.1057
Dual form			2116.3.d.c.1057.33

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.15436 q^{3} +0.508267i q^{5} +5.31530i q^{7} +8.25870 q^{9} -12.0624i q^{11} +0.283237 q^{13} +2.11153i q^{15} +9.14306i q^{17} -20.9092i q^{19} +22.0817i q^{21} +24.7417 q^{25} -3.07962 q^{27} +12.1261 q^{29} +34.8352 q^{31} -50.1115i q^{33} -2.70159 q^{35} -21.6541i q^{37} +1.17667 q^{39} +69.0300 q^{41} +75.5813i q^{43} +4.19763i q^{45} +14.5226 q^{47} +20.7476 q^{49} +37.9836i q^{51} -41.1074i q^{53} +6.13091 q^{55} -86.8642i q^{57} +104.578 q^{59} -56.5574i q^{61} +43.8975i q^{63} +0.143960i q^{65} +68.9338i q^{67} -86.0067 q^{71} +32.7488 q^{73} +102.786 q^{75} +64.1152 q^{77} -137.649i q^{79} -87.1222 q^{81} -21.5253i q^{83} -4.64712 q^{85} +50.3761 q^{87} +83.9505i q^{89} +1.50549i q^{91} +144.718 q^{93} +10.6274 q^{95} +29.0611i q^{97} -99.6196i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 2 q^{3} + 126 q^{9} - 2 q^{13} - 208 q^{25} - 26 q^{27} + 56 q^{29} - 20 q^{31} + 174 q^{35} - 42 q^{39} + 248 q^{41} - 58 q^{47} - 478 q^{49} + 56 q^{55} + 234 q^{59} - 776 q^{71} + 62 q^{73} + 246 q^{75}+ \cdots - 1232 q^{95}+O(q^{100}) $ 40 * q - 2 * q^3 + 126 * q^9 - 2 * q^13 - 208 * q^25 - 26 * q^27 + 56 * q^29 - 20 * q^31 + 174 * q^35 - 42 * q^39 + 248 * q^41 - 58 * q^47 - 478 * q^49 + 56 * q^55 + 234 * q^59 - 776 * q^71 + 62 * q^73 + 246 * q^75 - 8 * q^77 + 1164 * q^81 + 746 * q^85 + 942 * q^87 - 346 * q^93 - 1232 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2116\mathbb{Z}\right)^\times$.

$n$	$5$	$1059$
$\chi(n)$	$-1$	$1$

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
$3$	4.15436	1.38479	0.692393	−	0.721520i	$-0.256556\pi$
$3$	4.15436	1.38479	0.692393	+	0.721520i	$0.256556\pi$
$4$	0	0
$5$	0.508267i	0.101653i	0.998707	+	0.0508267i	$0.0161856\pi$
$5$	0.508267i	0.101653i	−0.998707	+	0.0508267i	$0.983814\pi$
$6$	0	0
$7$	5.31530i	0.759329i	0.925124	+	0.379664i	$0.123961\pi$
$7$	5.31530i	0.759329i	−0.925124	+	0.379664i	$0.876039\pi$
$8$	0	0
$9$	8.25870	0.917634
$10$	0	0
$11$	− 12.0624i	− 1.09658i	−0.836288	−	0.548290i	$-0.815279\pi$
$11$	− 12.0624i	− 1.09658i	0.836288	−	0.548290i	$-0.184721\pi$
$12$	0	0
$13$	0.283237	0.0217875	0.0108937	−	0.999941i	$-0.496532\pi$
$13$	0.283237	0.0217875	0.0108937	+	0.999941i	$0.496532\pi$
$14$	0	0
$15$	2.11153i	0.140768i
$16$	0	0
$17$	9.14306i	0.537827i	0.963164	+	0.268913i	$0.0866646\pi$
$17$	9.14306i	0.537827i	−0.963164	+	0.268913i	$0.913335\pi$
$18$	0	0
$19$	− 20.9092i	− 1.10048i	−0.835006	−	0.550241i	$-0.814536\pi$
$19$	− 20.9092i	− 1.10048i	0.835006	−	0.550241i	$-0.185464\pi$
$20$	0	0
$21$	22.0817i	1.05151i
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	24.7417	0.989667
$26$	0	0
$27$	−3.07962	−0.114060
$28$	0	0
$29$	12.1261	0.418141	0.209070	−	0.977901i	$-0.432956\pi$
$29$	12.1261	0.418141	0.209070	+	0.977901i	$0.432956\pi$
$30$	0	0
$31$	34.8352	1.12372	0.561858	−	0.827233i	$-0.310087\pi$
$31$	34.8352	1.12372	0.561858	+	0.827233i	$0.310087\pi$
$32$	0	0
$33$	− 50.1115i	− 1.51853i
$34$	0	0
$35$	−2.70159	−0.0771884
$36$	0	0
$37$	− 21.6541i	− 0.585245i	−0.956228	−	0.292622i	$-0.905472\pi$
$37$	− 21.6541i	− 0.585245i	0.956228	−	0.292622i	$-0.0945279\pi$
$38$	0	0
$39$	1.17667	0.0301710
$40$	0	0
$41$	69.0300	1.68366	0.841829	−	0.539744i	$-0.181479\pi$
$41$	69.0300	1.68366	0.841829	+	0.539744i	$0.181479\pi$
$42$	0	0
$43$	75.5813i	1.75770i	0.477094	+	0.878852i	$0.341690\pi$
$43$	75.5813i	1.75770i	−0.477094	+	0.878852i	$0.658310\pi$
$44$	0	0
$45$	4.19763i	0.0932806i
$46$	0	0
$47$	14.5226	0.308991	0.154496	−	0.987993i	$-0.450625\pi$
$47$	14.5226	0.308991	0.154496	+	0.987993i	$0.450625\pi$
$48$	0	0
$49$	20.7476	0.423420
$50$	0	0
$51$	37.9836i	0.744776i
$52$	0	0
$53$	− 41.1074i	− 0.775611i	−0.921741	−	0.387805i	$-0.873233\pi$
$53$	− 41.1074i	− 0.775611i	0.921741	−	0.387805i	$-0.126767\pi$
$54$	0	0
$55$	6.13091	0.111471
$56$	0	0
$57$	− 86.8642i	− 1.52393i
$58$	0	0
$59$	104.578	1.77251	0.886253	−	0.463202i	$-0.153300\pi$
$59$	104.578	1.77251	0.886253	+	0.463202i	$0.153300\pi$
$60$	0	0
$61$	− 56.5574i	− 0.927171i	−0.886052	−	0.463586i	$-0.846563\pi$
$61$	− 56.5574i	− 0.927171i	0.886052	−	0.463586i	$-0.153437\pi$
$62$	0	0
$63$	43.8975i	0.696786i
$64$	0	0
$65$	0.143960i	0.00221477i
$66$	0	0
$67$	68.9338i	1.02886i	0.857532	+	0.514431i	$0.171997\pi$
$67$	68.9338i	1.02886i	−0.857532	+	0.514431i	$0.828003\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−86.0067	−1.21136	−0.605681	−	0.795707i	$-0.707099\pi$
$71$	−86.0067	−1.21136	−0.605681	+	0.795707i	$0.707099\pi$
$72$	0	0
$73$	32.7488	0.448614	0.224307	−	0.974519i	$-0.427988\pi$
$73$	32.7488	0.448614	0.224307	+	0.974519i	$0.427988\pi$
$74$	0	0
$75$	102.786	1.37048
$76$	0	0
$77$	64.1152	0.832665
$78$	0	0
$79$	− 137.649i	− 1.74239i	−0.490940	−	0.871193i	$-0.663347\pi$
$79$	− 137.649i	− 1.74239i	0.490940	−	0.871193i	$-0.336653\pi$
$80$	0	0
$81$	−87.1222	−1.07558
$82$	0	0
$83$	− 21.5253i	− 0.259341i	−0.991557	−	0.129671i	$-0.958608\pi$
$83$	− 21.5253i	− 0.259341i	0.991557	−	0.129671i	$-0.0413920\pi$
$84$	0	0
$85$	−4.64712	−0.0546720
$86$	0	0
$87$	50.3761	0.579036
$88$	0	0
$89$	83.9505i	0.943264i	0.881795	+	0.471632i	$0.156335\pi$
$89$	83.9505i	0.943264i	−0.881795	+	0.471632i	$0.843665\pi$
$90$	0	0
$91$	1.50549i	0.0165438i
$92$	0	0
$93$	144.718	1.55611
$94$	0	0
$95$	10.6274	0.111868
$96$	0	0
$97$	29.0611i	0.299599i	0.988716	+	0.149799i	$0.0478628\pi$
$97$	29.0611i	0.299599i	−0.988716	+	0.149799i	$0.952137\pi$
$98$	0	0
$99$	− 99.6196i	− 1.00626i
$100$	0	0
$101$	−4.06916	−0.0402887	−0.0201444	−	0.999797i	$-0.506413\pi$
$101$	−4.06916	−0.0402887	−0.0201444	+	0.999797i	$0.506413\pi$
$102$	0	0
$103$	94.5172i	0.917642i	0.888529	+	0.458821i	$0.151728\pi$
$103$	94.5172i	0.917642i	−0.888529	+	0.458821i	$0.848272\pi$
$104$	0	0
$105$	−11.2234	−0.106889
$106$	0	0
$107$	41.5836i	0.388631i	0.980939	+	0.194316i	$0.0622486\pi$
$107$	41.5836i	0.388631i	−0.980939	+	0.194316i	$0.937751\pi$
$108$	0	0
$109$	178.849i	1.64081i	0.571781	+	0.820406i	$0.306253\pi$
$109$	178.849i	1.64081i	−0.571781	+	0.820406i	$0.693747\pi$
$110$	0	0
$111$	− 89.9588i	− 0.810439i
$112$	0	0
$113$	89.5484i	0.792463i	0.918151	+	0.396232i	$0.129682\pi$
$113$	89.5484i	0.792463i	−0.918151	+	0.396232i	$0.870318\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.33917	0.0199929
$118$	0	0
$119$	−48.5981	−0.408388
$120$	0	0
$121$	−24.5011	−0.202488
$122$	0	0
$123$	286.775	2.33151
$124$	0	0
$125$	25.2821i	0.202256i
$126$	0	0
$127$	0.908587	0.00715423	0.00357712	−	0.999994i	$-0.498861\pi$
$127$	0.908587	0.00715423	0.00357712	+	0.999994i	$0.498861\pi$
$128$	0	0
$129$	313.992i	2.43405i
$130$	0	0
$131$	192.805	1.47180	0.735899	−	0.677092i	$-0.236760\pi$
$131$	192.805	1.47180	0.735899	+	0.677092i	$0.236760\pi$
$132$	0	0
$133$	111.139	0.835628
$134$	0	0
$135$	− 1.56527i	− 0.0115946i
$136$	0	0
$137$	− 34.0085i	− 0.248237i	−0.992267	−	0.124119i	$-0.960390\pi$
$137$	− 34.0085i	− 0.248237i	0.992267	−	0.124119i	$-0.0396103\pi$
$138$	0	0
$139$	3.78150	0.0272050	0.0136025	−	0.999907i	$-0.495670\pi$
$139$	3.78150	0.0272050	0.0136025	+	0.999907i	$0.495670\pi$
$140$	0	0
$141$	60.3320	0.427887
$142$	0	0
$143$	− 3.41651i	− 0.0238917i
$144$	0	0
$145$	6.16329i	0.0425054i
$146$	0	0
$147$	86.1929	0.586346
$148$	0	0
$149$	− 85.1711i	− 0.571618i	−0.958287	−	0.285809i	$-0.907738\pi$
$149$	− 85.1711i	− 0.571618i	0.958287	−	0.285809i	$-0.0922623\pi$
$150$	0	0
$151$	−193.069	−1.27860	−0.639302	−	0.768956i	$-0.720777\pi$
$151$	−193.069	−1.27860	−0.639302	+	0.768956i	$0.720777\pi$
$152$	0	0
$153$	75.5098i	0.493528i
$154$	0	0
$155$	17.7056i	0.114230i
$156$	0	0
$157$	− 260.833i	− 1.66136i	−0.556751	−	0.830679i	$-0.687952\pi$
$157$	− 260.833i	− 1.66136i	0.556751	−	0.830679i	$-0.312048\pi$
$158$	0	0
$159$	− 170.775i	− 1.07406i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−180.996	−1.11040	−0.555201	−	0.831716i	$-0.687359\pi$
$163$	−180.996	−1.11040	−0.555201	+	0.831716i	$0.687359\pi$
$164$	0	0
$165$	25.4700	0.154364
$166$	0	0
$167$	279.988	1.67658	0.838288	−	0.545227i	$-0.183557\pi$
$167$	279.988	1.67658	0.838288	+	0.545227i	$0.183557\pi$
$168$	0	0
$169$	−168.920	−0.999525
$170$	0	0
$171$	− 172.683i	− 1.00984i
$172$	0	0
$173$	139.144	0.804298	0.402149	−	0.915574i	$-0.368263\pi$
$173$	139.144	0.804298	0.402149	+	0.915574i	$0.368263\pi$
$174$	0	0
$175$	131.509i	0.751482i
$176$	0	0
$177$	434.454	2.45454
$178$	0	0
$179$	−207.474	−1.15907	−0.579536	−	0.814946i	$-0.696766\pi$
$179$	−207.474	−1.15907	−0.579536	+	0.814946i	$0.696766\pi$
$180$	0	0
$181$	− 70.5976i	− 0.390042i	−0.980799	−	0.195021i	$-0.937522\pi$
$181$	− 70.5976i	− 0.390042i	0.980799	−	0.195021i	$-0.0624775\pi$
$182$	0	0
$183$	− 234.960i	− 1.28393i
$184$	0	0
$185$	11.0061	0.0594922
$186$	0	0
$187$	110.287	0.589770
$188$	0	0
$189$	− 16.3691i	− 0.0866089i
$190$	0	0
$191$	− 164.020i	− 0.858742i	−0.903128	−	0.429371i	$-0.858735\pi$
$191$	− 164.020i	− 0.858742i	0.903128	−	0.429371i	$-0.141265\pi$
$192$	0	0
$193$	95.1296	0.492899	0.246450	−	0.969156i	$-0.420736\pi$
$193$	95.1296	0.492899	0.246450	+	0.969156i	$0.420736\pi$
$194$	0	0
$195$	0.598062i	0.00306699i
$196$	0	0
$197$	−356.259	−1.80842	−0.904211	−	0.427086i	$-0.859540\pi$
$197$	−356.259	−1.80842	−0.904211	+	0.427086i	$0.859540\pi$
$198$	0	0
$199$	283.950i	1.42688i	0.700715	+	0.713441i	$0.252864\pi$
$199$	283.950i	1.42688i	−0.700715	+	0.713441i	$0.747136\pi$
$200$	0	0
$201$	286.376i	1.42475i
$202$	0	0
$203$	64.4538i	0.317506i
$204$	0	0
$205$	35.0857i	0.171150i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−252.214	−1.20677
$210$	0	0
$211$	76.0174	0.360272	0.180136	−	0.983642i	$-0.442346\pi$
$211$	76.0174	0.360272	0.180136	+	0.983642i	$0.442346\pi$
$212$	0	0
$213$	−357.303	−1.67748
$214$	0	0
$215$	−38.4155	−0.178677
$216$	0	0
$217$	185.160i	0.853270i
$218$	0	0
$219$	136.050	0.621234
$220$	0	0
$221$	2.58965i	0.0117179i
$222$	0	0
$223$	261.972	1.17476	0.587381	−	0.809311i	$-0.300159\pi$
$223$	261.972	1.17476	0.587381	+	0.809311i	$0.300159\pi$
$224$	0	0
$225$	204.334	0.908151
$226$	0	0
$227$	91.8956i	0.404827i	0.979300	+	0.202413i	$0.0648784\pi$
$227$	91.8956i	0.404827i	−0.979300	+	0.202413i	$0.935122\pi$
$228$	0	0
$229$	− 260.193i	− 1.13621i	−0.822954	−	0.568107i	$-0.807676\pi$
$229$	− 260.193i	− 1.13621i	0.822954	−	0.568107i	$-0.192324\pi$
$230$	0	0
$231$	266.358	1.15306
$232$	0	0
$233$	−390.311	−1.67515	−0.837577	−	0.546320i	$-0.816028\pi$
$233$	−390.311	−1.67515	−0.837577	+	0.546320i	$0.816028\pi$
$234$	0	0
$235$	7.38135i	0.0314100i
$236$	0	0
$237$	− 571.842i	− 2.41283i
$238$	0	0
$239$	−96.7843	−0.404955	−0.202478	−	0.979287i	$-0.564899\pi$
$239$	−96.7843	−0.404955	−0.202478	+	0.979287i	$0.564899\pi$
$240$	0	0
$241$	154.614i	0.641554i	0.947155	+	0.320777i	$0.103944\pi$
$241$	154.614i	0.641554i	−0.947155	+	0.320777i	$0.896056\pi$
$242$	0	0
$243$	−334.220	−1.37539
$244$	0	0
$245$	10.5453i	0.0430421i
$246$	0	0
$247$	− 5.92225i	− 0.0239767i
$248$	0	0
$249$	− 89.4239i	− 0.359132i
$250$	0	0
$251$	− 466.210i	− 1.85741i	−0.370820	−	0.928705i	$-0.620923\pi$
$251$	− 466.210i	− 1.85741i	0.370820	−	0.928705i	$-0.379077\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−19.3058	−0.0757090
$256$	0	0
$257$	87.1695	0.339181	0.169590	−	0.985515i	$-0.445756\pi$
$257$	87.1695	0.339181	0.169590	+	0.985515i	$0.445756\pi$
$258$	0	0
$259$	115.098	0.444393
$260$	0	0
$261$	100.146	0.383700
$262$	0	0
$263$	− 373.550i	− 1.42034i	−0.704029	−	0.710171i	$-0.748618\pi$
$263$	− 373.550i	− 1.42034i	0.704029	−	0.710171i	$-0.251382\pi$
$264$	0	0
$265$	20.8935	0.0788435
$266$	0	0
$267$	348.761i	1.30622i
$268$	0	0
$269$	237.024	0.881129	0.440565	−	0.897721i	$-0.354778\pi$
$269$	237.024	0.881129	0.440565	+	0.897721i	$0.354778\pi$
$270$	0	0
$271$	−346.690	−1.27930	−0.639649	−	0.768667i	$-0.720920\pi$
$271$	−346.690	−1.27930	−0.639649	+	0.768667i	$0.720920\pi$
$272$	0	0
$273$	6.25435i	0.0229097i
$274$	0	0
$275$	− 298.443i	− 1.08525i
$276$	0	0
$277$	−400.755	−1.44677	−0.723385	−	0.690445i	$-0.757415\pi$
$277$	−400.755	−1.44677	−0.723385	+	0.690445i	$0.757415\pi$
$278$	0	0
$279$	287.694	1.03116
$280$	0	0
$281$	− 264.318i	− 0.940632i	−0.882498	−	0.470316i	$-0.844140\pi$
$281$	− 264.318i	− 0.940632i	0.882498	−	0.470316i	$-0.155860\pi$
$282$	0	0
$283$	148.688i	0.525401i	0.964877	+	0.262700i	$0.0846131\pi$
$283$	148.688i	0.525401i	−0.964877	+	0.262700i	$0.915387\pi$
$284$	0	0
$285$	44.1502	0.154913
$286$	0	0
$287$	366.915i	1.27845i
$288$	0	0
$289$	205.404	0.710742
$290$	0	0
$291$	120.730i	0.414880i
$292$	0	0
$293$	− 194.311i	− 0.663179i	−0.943424	−	0.331589i	$-0.892415\pi$
$293$	− 194.311i	− 0.663179i	0.943424	−	0.331589i	$-0.107585\pi$
$294$	0	0
$295$	53.1535i	0.180181i
$296$	0	0
$297$	37.1475i	0.125076i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−401.737	−1.33468
$302$	0	0
$303$	−16.9048	−0.0557913
$304$	0	0
$305$	28.7463	0.0942501
$306$	0	0
$307$	255.410	0.831956	0.415978	−	0.909375i	$-0.363439\pi$
$307$	255.410	0.831956	0.415978	+	0.909375i	$0.363439\pi$
$308$	0	0
$309$	392.658i	1.27074i
$310$	0	0
$311$	−219.934	−0.707182	−0.353591	−	0.935400i	$-0.615040\pi$
$311$	−219.934	−0.707182	−0.353591	+	0.935400i	$0.615040\pi$
$312$	0	0
$313$	513.093i	1.63927i	0.572884	+	0.819637i	$0.305825\pi$
$313$	513.093i	1.63927i	−0.572884	+	0.819637i	$0.694175\pi$
$314$	0	0
$315$	−22.3117	−0.0708307
$316$	0	0
$317$	170.387	0.537498	0.268749	−	0.963210i	$-0.413390\pi$
$317$	170.387	0.537498	0.268749	+	0.963210i	$0.413390\pi$
$318$	0	0
$319$	− 146.269i	− 0.458525i
$320$	0	0
$321$	172.753i	0.538171i
$322$	0	0
$323$	191.174	0.591869
$324$	0	0
$325$	7.00776	0.0215623
$326$	0	0
$327$	743.001i	2.27218i
$328$	0	0
$329$	77.1919i	0.234626i
$330$	0	0
$331$	−13.6244	−0.0411612	−0.0205806	−	0.999788i	$-0.506551\pi$
$331$	−13.6244	−0.0411612	−0.0205806	+	0.999788i	$0.506551\pi$
$332$	0	0
$333$	− 178.834i	− 0.537040i
$334$	0	0
$335$	−35.0368	−0.104587
$336$	0	0
$337$	− 364.928i	− 1.08287i	−0.840742	−	0.541436i	$-0.817881\pi$
$337$	− 364.928i	− 1.08287i	0.840742	−	0.541436i	$-0.182119\pi$
$338$	0	0
$339$	372.016i	1.09739i
$340$	0	0
$341$	− 420.196i	− 1.23225i
$342$	0	0
$343$	370.729i	1.08084i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−148.585	−0.428199	−0.214099	−	0.976812i	$-0.568682\pi$
$347$	−148.585	−0.428199	−0.214099	+	0.976812i	$0.568682\pi$
$348$	0	0
$349$	−165.974	−0.475571	−0.237785	−	0.971318i	$-0.576421\pi$
$349$	−165.974	−0.475571	−0.237785	+	0.971318i	$0.576421\pi$
$350$	0	0
$351$	−0.872261	−0.00248508
$352$	0	0
$353$	−161.960	−0.458812	−0.229406	−	0.973331i	$-0.573678\pi$
$353$	−161.960	−0.458812	−0.229406	+	0.973331i	$0.573678\pi$
$354$	0	0
$355$	− 43.7144i	− 0.123139i
$356$	0	0
$357$	−201.894	−0.565529
$358$	0	0
$359$	− 433.184i	− 1.20664i	−0.797499	−	0.603320i	$-0.793844\pi$
$359$	− 433.184i	− 1.20664i	0.797499	−	0.603320i	$-0.206156\pi$
$360$	0	0
$361$	−76.1936	−0.211063
$362$	0	0
$363$	−101.786	−0.280403
$364$	0	0
$365$	16.6451i	0.0456031i
$366$	0	0
$367$	293.723i	0.800334i	0.916442	+	0.400167i	$0.131048\pi$
$367$	293.723i	0.800334i	−0.916442	+	0.400167i	$0.868952\pi$
$368$	0	0
$369$	570.098	1.54498
$370$	0	0
$371$	218.498	0.588944
$372$	0	0
$373$	− 369.382i	− 0.990301i	−0.868807	−	0.495151i	$-0.835113\pi$
$373$	− 369.382i	− 0.990301i	0.868807	−	0.495151i	$-0.164887\pi$
$374$	0	0
$375$	105.031i	0.280082i
$376$	0	0
$377$	3.43455	0.00911022
$378$	0	0
$379$	− 234.319i	− 0.618256i	−0.951020	−	0.309128i	$-0.899963\pi$
$379$	− 234.319i	− 0.618256i	0.951020	−	0.309128i	$-0.100037\pi$
$380$	0	0
$381$	3.77460	0.00990708
$382$	0	0
$383$	200.282i	0.522930i	0.965213	+	0.261465i	$0.0842055\pi$
$383$	200.282i	0.522930i	−0.965213	+	0.261465i	$0.915794\pi$
$384$	0	0
$385$	32.5877i	0.0846433i
$386$	0	0
$387$	624.203i	1.61293i
$388$	0	0
$389$	357.888i	0.920022i	0.887913	+	0.460011i	$0.152154\pi$
$389$	357.888i	0.920022i	−0.887913	+	0.460011i	$0.847846\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	800.983	2.03813
$394$	0	0
$395$	69.9623	0.177120
$396$	0	0
$397$	115.738	0.291531	0.145766	−	0.989319i	$-0.453435\pi$
$397$	115.738	0.291531	0.145766	+	0.989319i	$0.453435\pi$
$398$	0	0
$399$	461.710	1.15717
$400$	0	0
$401$	− 179.718i	− 0.448174i	−0.974569	−	0.224087i	$-0.928060\pi$
$401$	− 179.718i	− 0.448174i	0.974569	−	0.224087i	$-0.0719400\pi$
$402$	0	0
$403$	9.86662	0.0244829
$404$	0	0
$405$	− 44.2813i	− 0.109337i
$406$	0	0
$407$	−261.200	−0.641768
$408$	0	0
$409$	378.502	0.925434	0.462717	−	0.886506i	$-0.346875\pi$
$409$	378.502	0.925434	0.462717	+	0.886506i	$0.346875\pi$
$410$	0	0
$411$	− 141.284i	− 0.343756i
$412$	0	0
$413$	555.863i	1.34591i
$414$	0	0
$415$	10.9406	0.0263629
$416$	0	0
$417$	15.7097	0.0376732
$418$	0	0
$419$	− 293.323i	− 0.700054i	−0.936740	−	0.350027i	$-0.886172\pi$
$419$	− 293.323i	− 0.700054i	0.936740	−	0.350027i	$-0.113828\pi$
$420$	0	0
$421$	145.020i	0.344465i	0.985056	+	0.172232i	$0.0550980\pi$
$421$	145.020i	0.344465i	−0.985056	+	0.172232i	$0.944902\pi$
$422$	0	0
$423$	119.938	0.283541
$424$	0	0
$425$	226.214i	0.532269i
$426$	0	0
$427$	300.620	0.704028
$428$	0	0
$429$	− 14.1934i	− 0.0330849i
$430$	0	0
$431$	− 33.0532i	− 0.0766896i	−0.999265	−	0.0383448i	$-0.987791\pi$
$431$	− 33.0532i	− 0.0766896i	0.999265	−	0.0383448i	$-0.0122085\pi$
$432$	0	0
$433$	− 547.450i	− 1.26432i	−0.774838	−	0.632159i	$-0.782169\pi$
$433$	− 547.450i	− 1.26432i	0.774838	−	0.632159i	$-0.217831\pi$
$434$	0	0
$435$	25.6045i	0.0588610i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	61.5770	0.140267	0.0701333	−	0.997538i	$-0.477658\pi$
$439$	61.5770	0.140267	0.0701333	+	0.997538i	$0.477658\pi$
$440$	0	0
$441$	171.348	0.388544
$442$	0	0
$443$	774.392	1.74806	0.874032	−	0.485868i	$-0.161496\pi$
$443$	774.392	1.74806	0.874032	+	0.485868i	$0.161496\pi$
$444$	0	0
$445$	−42.6693	−0.0958861
$446$	0	0
$447$	− 353.832i	− 0.791569i
$448$	0	0
$449$	529.477	1.17924	0.589618	−	0.807682i	$-0.299278\pi$
$449$	529.477	1.17924	0.589618	+	0.807682i	$0.299278\pi$
$450$	0	0
$451$	− 832.666i	− 1.84627i
$452$	0	0
$453$	−802.079	−1.77059
$454$	0	0
$455$	−0.765191	−0.00168174
$456$	0	0
$457$	521.935i	1.14209i	0.820919	+	0.571045i	$0.193462\pi$
$457$	521.935i	1.14209i	−0.820919	+	0.571045i	$0.806538\pi$
$458$	0	0
$459$	− 28.1571i	− 0.0613445i
$460$	0	0
$461$	−527.521	−1.14430	−0.572149	−	0.820150i	$-0.693890\pi$
$461$	−527.521	−1.14430	−0.572149	+	0.820150i	$0.693890\pi$
$462$	0	0
$463$	−867.919	−1.87456	−0.937278	−	0.348583i	$-0.886663\pi$
$463$	−867.919	−1.87456	−0.937278	+	0.348583i	$0.886663\pi$
$464$	0	0
$465$	73.5554i	0.158184i
$466$	0	0
$467$	411.687i	0.881557i	0.897616	+	0.440779i	$0.145298\pi$
$467$	411.687i	0.881557i	−0.897616	+	0.440779i	$0.854702\pi$
$468$	0	0
$469$	−366.404	−0.781245
$470$	0	0
$471$	− 1083.60i	− 2.30063i
$472$	0	0
$473$	911.690	1.92746
$474$	0	0
$475$	− 517.328i	− 1.08911i
$476$	0	0
$477$	− 339.494i	− 0.711727i
$478$	0	0
$479$	221.817i	0.463083i	0.972825	+	0.231541i	$0.0743769\pi$
$479$	221.817i	0.463083i	−0.972825	+	0.231541i	$0.925623\pi$
$480$	0	0
$481$	− 6.13323i	− 0.0127510i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.7708	−0.0304552
$486$	0	0
$487$	−566.561	−1.16337	−0.581685	−	0.813414i	$-0.697606\pi$
$487$	−566.561	−1.16337	−0.581685	+	0.813414i	$0.697606\pi$
$488$	0	0
$489$	−751.920	−1.53767
$490$	0	0
$491$	−44.3921	−0.0904115	−0.0452058	−	0.998978i	$-0.514394\pi$
$491$	−44.3921	−0.0904115	−0.0452058	+	0.998978i	$0.514394\pi$
$492$	0	0
$493$	110.869i	0.224887i
$494$	0	0
$495$	50.6334	0.102290
$496$	0	0
$497$	− 457.152i	− 0.919822i
$498$	0	0
$499$	−62.6920	−0.125635	−0.0628176	−	0.998025i	$-0.520009\pi$
$499$	−62.6920	−0.125635	−0.0628176	+	0.998025i	$0.520009\pi$
$500$	0	0
$501$	1163.17	2.32170
$502$	0	0
$503$	697.131i	1.38595i	0.720963	+	0.692974i	$0.243700\pi$
$503$	697.131i	1.38595i	−0.720963	+	0.692974i	$0.756300\pi$
$504$	0	0
$505$	− 2.06822i	− 0.00409549i
$506$	0	0
$507$	−701.753	−1.38413
$508$	0	0
$509$	936.009	1.83892	0.919458	−	0.393187i	$-0.128628\pi$
$509$	936.009	1.83892	0.919458	+	0.393187i	$0.128628\pi$
$510$	0	0
$511$	174.070i	0.340645i
$512$	0	0
$513$	64.3922i	0.125521i
$514$	0	0
$515$	−48.0400	−0.0932815
$516$	0	0
$517$	− 175.177i	− 0.338834i
$518$	0	0
$519$	578.052	1.11378
$520$	0	0
$521$	− 37.0135i	− 0.0710432i	−0.999369	−	0.0355216i	$-0.988691\pi$
$521$	− 37.0135i	− 0.0710432i	0.999369	−	0.0355216i	$-0.0113093\pi$
$522$	0	0
$523$	− 147.800i	− 0.282600i	−0.989967	−	0.141300i	$-0.954872\pi$
$523$	− 147.800i	− 0.282600i	0.989967	−	0.141300i	$-0.0451282\pi$
$524$	0	0
$525$	546.337i	1.04064i
$526$	0	0
$527$	318.500i	0.604365i
$528$	0	0
$529$	0	0
$530$	0	0
$531$	863.677	1.62651
$532$	0	0
$533$	19.5519	0.0366827
$534$	0	0
$535$	−21.1356	−0.0395057
$536$	0	0
$537$	−861.921	−1.60507
$538$	0	0
$539$	− 250.265i	− 0.464314i
$540$	0	0
$541$	−954.893	−1.76505	−0.882526	−	0.470263i	$-0.844159\pi$
$541$	−954.893	−1.76505	−0.882526	+	0.470263i	$0.844159\pi$
$542$	0	0
$543$	− 293.288i	− 0.540125i
$544$	0	0
$545$	−90.9029	−0.166794
$546$	0	0
$547$	−593.730	−1.08543	−0.542715	−	0.839917i	$-0.682603\pi$
$547$	−593.730	−1.08543	−0.542715	+	0.839917i	$0.682603\pi$
$548$	0	0
$549$	− 467.091i	− 0.850803i
$550$	0	0
$551$	− 253.546i	− 0.460157i
$552$	0	0
$553$	731.644	1.32304
$554$	0	0
$555$	45.7231	0.0823840
$556$	0	0
$557$	205.791i	0.369464i	0.982789	+	0.184732i	$0.0591416\pi$
$557$	205.791i	0.369464i	−0.982789	+	0.184732i	$0.940858\pi$
$558$	0	0
$559$	21.4074i	0.0382959i
$560$	0	0
$561$	458.172	0.816706
$562$	0	0
$563$	220.646i	0.391911i	0.980613	+	0.195955i	$0.0627808\pi$
$563$	220.646i	0.391911i	−0.980613	+	0.195955i	$0.937219\pi$
$564$	0	0
$565$	−45.5145	−0.0805566
$566$	0	0
$567$	− 463.081i	− 0.816720i
$568$	0	0
$569$	− 772.399i	− 1.35747i	−0.734384	−	0.678734i	$-0.762529\pi$
$569$	− 772.399i	− 1.35747i	0.734384	−	0.678734i	$-0.237471\pi$
$570$	0	0
$571$	38.5863i	0.0675767i	0.999429	+	0.0337883i	$0.0107572\pi$
$571$	38.5863i	0.0675767i	−0.999429	+	0.0337883i	$0.989243\pi$
$572$	0	0
$573$	− 681.397i	− 1.18917i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	519.583	0.900491	0.450245	−	0.892905i	$-0.351337\pi$
$577$	519.583	0.900491	0.450245	+	0.892905i	$0.351337\pi$
$578$	0	0
$579$	395.202	0.682560
$580$	0	0
$581$	114.414	0.196925
$582$	0	0
$583$	−495.853	−0.850519
$584$	0	0
$585$	1.18892i	0.00203235i
$586$	0	0
$587$	−195.145	−0.332444	−0.166222	−	0.986088i	$-0.553157\pi$
$587$	−195.145	−0.332444	−0.166222	+	0.986088i	$0.553157\pi$
$588$	0	0
$589$	− 728.376i	− 1.23663i
$590$	0	0
$591$	−1480.03	−2.50428
$592$	0	0
$593$	−757.862	−1.27801	−0.639007	−	0.769201i	$-0.720655\pi$
$593$	−757.862	−1.27801	−0.639007	+	0.769201i	$0.720655\pi$
$594$	0	0
$595$	− 24.7008i	− 0.0415140i
$596$	0	0
$597$	1179.63i	1.97593i
$598$	0	0
$599$	−734.619	−1.22641	−0.613204	−	0.789924i	$-0.710120\pi$
$599$	−734.619	−1.22641	−0.613204	+	0.789924i	$0.710120\pi$
$600$	0	0
$601$	−448.835	−0.746814	−0.373407	−	0.927668i	$-0.621810\pi$
$601$	−448.835	−0.746814	−0.373407	+	0.927668i	$0.621810\pi$
$602$	0	0
$603$	569.304i	0.944119i
$604$	0	0
$605$	− 12.4531i	− 0.0205836i
$606$	0	0
$607$	−323.961	−0.533709	−0.266854	−	0.963737i	$-0.585984\pi$
$607$	−323.961	−0.533709	−0.266854	+	0.963737i	$0.585984\pi$
$608$	0	0
$609$	267.764i	0.439678i
$610$	0	0
$611$	4.11333	0.00673213
$612$	0	0
$613$	− 104.712i	− 0.170818i	−0.996346	−	0.0854092i	$-0.972780\pi$
$613$	− 104.712i	− 0.170818i	0.996346	−	0.0854092i	$-0.0272198\pi$
$614$	0	0
$615$	145.759i	0.237006i
$616$	0	0
$617$	614.459i	0.995882i	0.867211	+	0.497941i	$0.165910\pi$
$617$	614.459i	0.995882i	−0.867211	+	0.497941i	$0.834090\pi$
$618$	0	0
$619$	1083.29i	1.75007i	0.484058	+	0.875036i	$0.339162\pi$
$619$	1083.29i	1.75007i	−0.484058	+	0.875036i	$0.660838\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−446.222	−0.716248
$624$	0	0
$625$	605.692	0.969107
$626$	0	0
$627$	−1047.79	−1.67112
$628$	0	0
$629$	197.984	0.314761
$630$	0	0
$631$	767.024i	1.21557i	0.794102	+	0.607784i	$0.207941\pi$
$631$	767.024i	1.21557i	−0.794102	+	0.607784i	$0.792059\pi$
$632$	0	0
$633$	315.804	0.498900
$634$	0	0
$635$	0.461805i	0 0.000727252i
$636$	0	0
$637$	5.87648	0.00922524
$638$	0	0
$639$	−710.304	−1.11159
$640$	0	0
$641$	91.6885i	0.143040i	0.997439	+	0.0715199i	$0.0227850\pi$
$641$	91.6885i	0.143040i	−0.997439	+	0.0715199i	$0.977215\pi$
$642$	0	0
$643$	− 596.760i	− 0.928088i	−0.885812	−	0.464044i	$-0.846398\pi$
$643$	− 596.760i	− 0.928088i	0.885812	−	0.464044i	$-0.153602\pi$
$644$	0	0
$645$	−159.592	−0.247429
$646$	0	0
$647$	−487.575	−0.753594	−0.376797	−	0.926296i	$-0.622975\pi$
$647$	−487.575	−0.753594	−0.376797	+	0.926296i	$0.622975\pi$
$648$	0	0
$649$	− 1261.46i	− 1.94369i
$650$	0	0
$651$	769.220i	1.18160i
$652$	0	0
$653$	1023.55	1.56746	0.783728	−	0.621104i	$-0.213316\pi$
$653$	1023.55	1.56746	0.783728	+	0.621104i	$0.213316\pi$
$654$	0	0
$655$	97.9967i	0.149613i
$656$	0	0
$657$	270.462	0.411663
$658$	0	0
$659$	264.022i	0.400640i	0.979731	+	0.200320i	$0.0641982\pi$
$659$	264.022i	0.400640i	−0.979731	+	0.200320i	$0.935802\pi$
$660$	0	0
$661$	− 394.818i	− 0.597304i	−0.954362	−	0.298652i	$-0.903463\pi$
$661$	− 394.818i	− 0.597304i	0.954362	−	0.298652i	$-0.0965370\pi$
$662$	0	0
$663$	10.7583i	0.0162268i
$664$	0	0
$665$	56.4881i	0.0849445i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1088.32	1.62679
$670$	0	0
$671$	−682.217	−1.01672
$672$	0	0
$673$	−421.203	−0.625859	−0.312929	−	0.949776i	$-0.601310\pi$
$673$	−421.203	−0.625859	−0.312929	+	0.949776i	$0.601310\pi$
$674$	0	0
$675$	−76.1948	−0.112881
$676$	0	0
$677$	999.372i	1.47618i	0.674704	+	0.738089i	$0.264271\pi$
$677$	999.372i	1.47618i	−0.674704	+	0.738089i	$0.735729\pi$
$678$	0	0
$679$	−154.468	−0.227494
$680$	0	0
$681$	381.767i	0.560598i
$682$	0	0
$683$	−541.266	−0.792484	−0.396242	−	0.918146i	$-0.629686\pi$
$683$	−541.266	−0.792484	−0.396242	+	0.918146i	$0.629686\pi$
$684$	0	0
$685$	17.2854	0.0252342
$686$	0	0
$687$	− 1080.94i	− 1.57342i
$688$	0	0
$689$	− 11.6431i	− 0.0168986i
$690$	0	0
$691$	−819.906	−1.18655	−0.593275	−	0.805000i	$-0.702165\pi$
$691$	−819.906	−1.18655	−0.593275	+	0.805000i	$0.702165\pi$
$692$	0	0
$693$	529.508	0.764081
$694$	0	0
$695$	1.92201i	0.00276549i
$696$	0	0
$697$	631.145i	0.905517i
$698$	0	0
$699$	−1621.49	−2.31973
$700$	0	0
$701$	1113.91i	1.58903i	0.607243	+	0.794516i	$0.292276\pi$
$701$	1113.91i	1.58903i	−0.607243	+	0.794516i	$0.707724\pi$
$702$	0	0
$703$	−452.769	−0.644052
$704$	0	0
$705$	30.6648i	0.0434962i
$706$	0	0
$707$	− 21.6288i	− 0.0305924i
$708$	0	0
$709$	489.110i	0.689860i	0.938628	+	0.344930i	$0.112097\pi$
$709$	489.110i	0.689860i	−0.938628	+	0.344930i	$0.887903\pi$
$710$	0	0
$711$	− 1136.80i	− 1.59887i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.73650	0.00242867
$716$	0	0
$717$	−402.077	−0.560777
$718$	0	0
$719$	144.297	0.200692	0.100346	−	0.994953i	$-0.468005\pi$
$719$	144.297	0.200692	0.100346	+	0.994953i	$0.468005\pi$
$720$	0	0
$721$	−502.387	−0.696792
$722$	0	0
$723$	642.324i	0.888415i
$724$	0	0
$725$	300.019	0.413820
$726$	0	0
$727$	238.511i	0.328076i	0.986454	+	0.164038i	$0.0524520\pi$
$727$	238.511i	0.328076i	−0.986454	+	0.164038i	$0.947548\pi$
$728$	0	0
$729$	−604.371	−0.829042
$730$	0	0
$731$	−691.044	−0.945341
$732$	0	0
$733$	− 339.321i	− 0.462921i	−0.972844	−	0.231461i	$-0.925650\pi$
$733$	− 339.321i	− 0.462921i	0.972844	−	0.231461i	$-0.0743504\pi$
$734$	0	0
$735$	43.8090i	0.0596041i
$736$	0	0
$737$	831.506	1.12823
$738$	0	0
$739$	−105.802	−0.143169	−0.0715844	−	0.997435i	$-0.522806\pi$
$739$	−105.802	−0.143169	−0.0715844	+	0.997435i	$0.522806\pi$
$740$	0	0
$741$	− 24.6032i	− 0.0332027i
$742$	0	0
$743$	1472.92i	1.98239i	0.132413	+	0.991195i	$0.457728\pi$
$743$	1472.92i	1.98239i	−0.132413	+	0.991195i	$0.542272\pi$
$744$	0	0
$745$	43.2897	0.0581070
$746$	0	0
$747$	− 177.771i	− 0.237980i
$748$	0	0
$749$	−221.029	−0.295099
$750$	0	0
$751$	− 405.512i	− 0.539963i	−0.962866	−	0.269981i	$-0.912982\pi$
$751$	− 405.512i	− 0.539963i	0.962866	−	0.269981i	$-0.0870176\pi$
$752$	0	0
$753$	− 1936.80i	− 2.57212i
$754$	0	0
$755$	− 98.1307i	− 0.129974i
$756$	0	0
$757$	− 1185.83i	− 1.56649i	−0.621714	−	0.783244i	$-0.713564\pi$
$757$	− 1185.83i	− 1.56649i	0.621714	−	0.783244i	$-0.286436\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1350.67	−1.77486	−0.887430	−	0.460942i	$-0.847512\pi$
$761$	−1350.67	−1.77486	−0.887430	+	0.460942i	$0.847512\pi$
$762$	0	0
$763$	−950.634	−1.24592
$764$	0	0
$765$	−38.3792	−0.0501688
$766$	0	0
$767$	29.6203	0.0386184
$768$	0	0
$769$	646.846i	0.841152i	0.907257	+	0.420576i	$0.138172\pi$
$769$	646.846i	0.841152i	−0.907257	+	0.420576i	$0.861828\pi$
$770$	0	0
$771$	362.133	0.469693
$772$	0	0
$773$	− 492.183i	− 0.636718i	−0.947970	−	0.318359i	$-0.896868\pi$
$773$	− 492.183i	− 0.636718i	0.947970	−	0.318359i	$-0.103132\pi$
$774$	0	0
$775$	861.881	1.11210
$776$	0	0
$777$	478.158	0.615390
$778$	0	0
$779$	− 1443.36i	− 1.85284i
$780$	0	0
$781$	1037.45i	1.32836i
$782$	0	0
$783$	−37.3437	−0.0476931
$784$	0	0
$785$	132.573	0.168883
$786$	0	0
$787$	− 230.960i	− 0.293469i	−0.989176	−	0.146734i	$-0.953124\pi$
$787$	− 230.960i	− 0.293469i	0.989176	−	0.146734i	$-0.0468762\pi$
$788$	0	0
$789$	− 1551.86i	− 1.96687i
$790$	0	0
$791$	−475.977	−0.601740
$792$	0	0
$793$	− 16.0192i	− 0.0202007i
$794$	0	0
$795$	86.7992	0.109181
$796$	0	0
$797$	1398.28i	1.75443i	0.480100	+	0.877214i	$0.340600\pi$
$797$	1398.28i	1.75443i	−0.480100	+	0.877214i	$0.659400\pi$
$798$	0	0
$799$	132.781i	0.166184i
$800$	0	0
$801$	693.322i	0.865571i
$802$	0	0
$803$	− 395.028i	− 0.491941i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	984.682	1.22018
$808$	0	0
$809$	1221.99	1.51049	0.755246	−	0.655441i	$-0.227517\pi$
$809$	1221.99	1.51049	0.755246	+	0.655441i	$0.227517\pi$
$810$	0	0
$811$	−415.745	−0.512632	−0.256316	−	0.966593i	$-0.582509\pi$
$811$	−415.745	−0.512632	−0.256316	+	0.966593i	$0.582509\pi$
$812$	0	0
$813$	−1440.27	−1.77156
$814$	0	0
$815$	− 91.9941i	− 0.112876i
$816$	0	0
$817$	1580.34	1.93432
$818$	0	0
$819$	12.4334i	0.0151812i
$820$	0	0
$821$	−387.891	−0.472462	−0.236231	−	0.971697i	$-0.575912\pi$
$821$	−387.891	−0.472462	−0.236231	+	0.971697i	$0.575912\pi$
$822$	0	0
$823$	−34.6628	−0.0421176	−0.0210588	−	0.999778i	$-0.506704\pi$
$823$	−34.6628	−0.0421176	−0.0210588	+	0.999778i	$0.506704\pi$
$824$	0	0
$825$	− 1239.84i	− 1.50284i
$826$	0	0
$827$	203.231i	0.245744i	0.992423	+	0.122872i	$0.0392105\pi$
$827$	203.231i	0.245744i	−0.992423	+	0.122872i	$0.960789\pi$
$828$	0	0
$829$	−1341.34	−1.61802	−0.809011	−	0.587794i	$-0.799997\pi$
$829$	−1341.34	−1.61802	−0.809011	+	0.587794i	$0.799997\pi$
$830$	0	0
$831$	−1664.88	−2.00347
$832$	0	0
$833$	189.696i	0.227727i
$834$	0	0
$835$	142.309i	0.170430i
$836$	0	0
$837$	−107.279	−0.128171
$838$	0	0
$839$	669.716i	0.798231i	0.916901	+	0.399115i	$0.130683\pi$
$839$	669.716i	0.798231i	−0.916901	+	0.399115i	$0.869317\pi$
$840$	0	0
$841$	−693.958	−0.825158
$842$	0	0
$843$	− 1098.07i	− 1.30257i
$844$	0	0
$845$	− 85.8564i	− 0.101605i
$846$	0	0
$847$	− 130.230i	− 0.153755i
$848$	0	0
$849$	617.705i	0.727568i
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−943.444	−1.10603	−0.553015	−	0.833171i	$-0.686523\pi$
$853$	−943.444	−1.10603	−0.553015	+	0.833171i	$0.686523\pi$
$854$	0	0
$855$	87.7689	0.102654
$856$	0	0
$857$	1157.75	1.35093	0.675465	−	0.737392i	$-0.263943\pi$
$857$	1157.75	1.35093	0.675465	+	0.737392i	$0.263943\pi$
$858$	0	0
$859$	−1146.35	−1.33452	−0.667261	−	0.744824i	$-0.732533\pi$
$859$	−1146.35	−1.33452	−0.667261	+	0.744824i	$0.732533\pi$
$860$	0	0
$861$	1524.30i	1.77038i
$862$	0	0
$863$	−484.304	−0.561186	−0.280593	−	0.959827i	$-0.590531\pi$
$863$	−484.304	−0.561186	−0.280593	+	0.959827i	$0.590531\pi$
$864$	0	0
$865$	70.7221i	0.0817597i
$866$	0	0
$867$	853.324	0.984226
$868$	0	0
$869$	−1660.37	−1.91067
$870$	0	0
$871$	19.5246i	0.0224163i
$872$	0	0
$873$	240.007i	0.274922i
$874$	0	0
$875$	−134.382	−0.153579
$876$	0	0
$877$	−5.88765	−0.00671340	−0.00335670	−	0.999994i	$-0.501068\pi$
$877$	−5.88765	−0.00671340	−0.00335670	+	0.999994i	$0.501068\pi$
$878$	0	0
$879$	− 807.240i	− 0.918361i
$880$	0	0
$881$	151.544i	0.172014i	0.996295	+	0.0860071i	$0.0274108\pi$
$881$	151.544i	0.172014i	−0.996295	+	0.0860071i	$0.972589\pi$
$882$	0	0
$883$	−955.362	−1.08195	−0.540975	−	0.841039i	$-0.681945\pi$
$883$	−955.362	−1.08195	−0.540975	+	0.841039i	$0.681945\pi$
$884$	0	0
$885$	220.819i	0.249513i
$886$	0	0
$887$	−1385.80	−1.56235	−0.781173	−	0.624315i	$-0.785378\pi$
$887$	−1385.80	−1.56235	−0.781173	+	0.624315i	$0.785378\pi$
$888$	0	0
$889$	4.82942i	0.00543241i
$890$	0	0
$891$	1050.90i	1.17946i
$892$	0	0
$893$	− 303.655i	− 0.340039i
$894$	0	0
$895$	− 105.452i	− 0.117824i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	422.415	0.469872
$900$	0	0
$901$	375.847	0.417144
$902$	0	0
$903$	−1668.96	−1.84824
$904$	0	0
$905$	35.8825	0.0396491
$906$	0	0
$907$	15.8519i	0.0174773i	0.999962	+	0.00873864i	$0.00278163\pi$
$907$	15.8519i	0.0174773i	−0.999962	+	0.00873864i	$0.997218\pi$
$908$	0	0
$909$	−33.6060	−0.0369703
$910$	0	0
$911$	− 402.861i	− 0.442219i	−0.975249	−	0.221109i	$-0.929032\pi$
$911$	− 402.861i	− 0.442219i	0.975249	−	0.221109i	$-0.0709678\pi$
$912$	0	0
$913$	−259.647	−0.284388
$914$	0	0
$915$	119.422	0.130516
$916$	0	0
$917$	1024.82i	1.11758i
$918$	0	0
$919$	− 1163.05i	− 1.26557i	−0.774329	−	0.632783i	$-0.781913\pi$
$919$	− 1163.05i	− 1.26557i	0.774329	−	0.632783i	$-0.218087\pi$
$920$	0	0
$921$	1061.07	1.15208
$922$	0	0
$923$	−24.3603	−0.0263925
$924$	0	0
$925$	− 535.758i	− 0.579197i
$926$	0	0
$927$	780.589i	0.842060i
$928$	0	0
$929$	1180.21	1.27041	0.635206	−	0.772342i	$-0.280915\pi$
$929$	1180.21	1.27041	0.635206	+	0.772342i	$0.280915\pi$
$930$	0	0
$931$	− 433.815i	− 0.465966i
$932$	0	0
$933$	−913.683	−0.979296
$934$	0	0
$935$	56.0553i	0.0599522i
$936$	0	0
$937$	− 913.355i	− 0.974765i	−0.873189	−	0.487382i	$-0.837952\pi$
$937$	− 913.355i	− 0.974765i	0.873189	−	0.487382i	$-0.162048\pi$
$938$	0	0
$939$	2131.57i	2.27004i
$940$	0	0
$941$	− 1557.71i	− 1.65537i	−0.561191	−	0.827686i	$-0.689657\pi$
$941$	− 1557.71i	− 1.65537i	0.561191	−	0.827686i	$-0.310343\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	8.31987	0.00880410
$946$	0	0
$947$	−493.466	−0.521084	−0.260542	−	0.965463i	$-0.583901\pi$
$947$	−493.466	−0.521084	−0.260542	+	0.965463i	$0.583901\pi$
$948$	0	0
$949$	9.27567	0.00977415
$950$	0	0
$951$	707.849	0.744320
$952$	0	0
$953$	− 394.363i	− 0.413812i	−0.978361	−	0.206906i	$-0.933661\pi$
$953$	− 394.363i	− 0.413812i	0.978361	−	0.206906i	$-0.0663395\pi$
$954$	0	0
$955$	83.3659	0.0872941
$956$	0	0
$957$	− 607.656i	− 0.634959i
$958$	0	0
$959$	180.765	0.188494
$960$	0	0
$961$	252.492	0.262739
$962$	0	0
$963$	343.426i	0.356621i
$964$	0	0
$965$	48.3513i	0.0501049i
$966$	0	0
$967$	−951.697	−0.984175	−0.492087	−	0.870546i	$-0.663766\pi$
$967$	−951.697	−0.984175	−0.492087	+	0.870546i	$0.663766\pi$
$968$	0	0
$969$	794.205	0.819613
$970$	0	0
$971$	156.973i	0.161662i	0.996728	+	0.0808308i	$0.0257573\pi$
$971$	156.973i	0.161662i	−0.996728	+	0.0808308i	$0.974243\pi$
$972$	0	0
$973$	20.0998i	0.0206576i
$974$	0	0
$975$	29.1127	0.0298592
$976$	0	0
$977$	1568.21i	1.60512i	0.596569	+	0.802562i	$0.296530\pi$
$977$	1568.21i	1.60512i	−0.596569	+	0.802562i	$0.703470\pi$
$978$	0	0
$979$	1012.64	1.03436
$980$	0	0
$981$	1477.06i	1.50567i
$982$	0	0
$983$	− 1629.30i	− 1.65748i	−0.559636	−	0.828739i	$-0.689059\pi$
$983$	− 1629.30i	− 1.65748i	0.559636	−	0.828739i	$-0.310941\pi$
$984$	0	0
$985$	− 181.075i	− 0.183832i
$986$	0	0
$987$	320.683i	0.324907i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	360.223	0.363495	0.181747	−	0.983345i	$-0.441825\pi$
$991$	360.223	0.363495	0.181747	+	0.983345i	$0.441825\pi$
$992$	0	0
$993$	−56.6005	−0.0569995
$994$	0	0
$995$	−144.322	−0.145048
$996$	0	0
$997$	−162.968	−0.163459	−0.0817294	−	0.996655i	$-0.526044\pi$
$997$	−162.968	−0.163459	−0.0817294	+	0.996655i	$0.526044\pi$
$998$	0	0
$999$	66.6862i	0.0667530i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2116.3.d.c.1057.34		40
23.14	odd	22		92.3.f.a.57.1	yes	40
23.18	even	11		92.3.f.a.21.1	✓	40
23.22	odd	2	inner	2116.3.d.c.1057.33		40
92.83	even	22		368.3.p.c.241.4		40
92.87	odd	22		368.3.p.c.113.4		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.3.f.a.21.1	✓	40	23.18	even	11
92.3.f.a.57.1	yes	40	23.14	odd	22
368.3.p.c.113.4		40	92.87	odd	22
368.3.p.c.241.4		40	92.83	even	22
2116.3.d.c.1057.33		40	23.22	odd	2	inner
2116.3.d.c.1057.34		40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2116 = 2^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2116.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.15436 q^{3} +0.508267i q^{5} +5.31530i q^{7} +8.25870 q^{9} -12.0624i q^{11} +0.283237 q^{13} +2.11153i q^{15} +9.14306i q^{17} -20.9092i q^{19} +22.0817i q^{21} +24.7417 q^{25} -3.07962 q^{27} +12.1261 q^{29} +34.8352 q^{31} -50.1115i q^{33} -2.70159 q^{35} -21.6541i q^{37} +1.17667 q^{39} +69.0300 q^{41} +75.5813i q^{43} +4.19763i q^{45} +14.5226 q^{47} +20.7476 q^{49} +37.9836i q^{51} -41.1074i q^{53} +6.13091 q^{55} -86.8642i q^{57} +104.578 q^{59} -56.5574i q^{61} +43.8975i q^{63} +0.143960i q^{65} +68.9338i q^{67} -86.0067 q^{71} +32.7488 q^{73} +102.786 q^{75} +64.1152 q^{77} -137.649i q^{79} -87.1222 q^{81} -21.5253i q^{83} -4.64712 q^{85} +50.3761 q^{87} +83.9505i q^{89} +1.50549i q^{91} +144.718 q^{93} +10.6274 q^{95} +29.0611i q^{97} -99.6196i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{3} + 126 q^{9} - 2 q^{13} - 208 q^{25} - 26 q^{27} + 56 q^{29} - 20 q^{31} + 174 q^{35} - 42 q^{39} + 248 q^{41} - 58 q^{47} - 478 q^{49} + 56 q^{55} + 234 q^{59} - 776 q^{71} + 62 q^{73} + 246 q^{75}+ \cdots - 1232 q^{95}+O(q^{100}) \) 40 * q - 2 * q^3 + 126 * q^9 - 2 * q^13 - 208 * q^25 - 26 * q^27 + 56 * q^29 - 20 * q^31 + 174 * q^35 - 42 * q^39 + 248 * q^41 - 58 * q^47 - 478 * q^49 + 56 * q^55 + 234 * q^59 - 776 * q^71 + 62 * q^73 + 246 * q^75 - 8 * q^77 + 1164 * q^81 + 746 * q^85 + 942 * q^87 - 346 * q^93 - 1232 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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