LMFDB - Embedded newform 1728.4.c.i.1727.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(1727,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.1727"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$101.955300490$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{10} + 112x^{8} - 368x^{6} + 928x^{4} - 256x^{2} + 64 $ x^12 - 12x^10 + 112x^8 - 368x^6 + 928x^4 - 256*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1727.5
Root			$-1.61829 - 0.934317i$ of defining polynomial
Character	$\chi$	$=$	1728.1727
Dual form			1728.4.c.i.1727.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.33155i q^{5} -16.9016i q^{7} +16.8325 q^{11} -25.0775 q^{13} -116.919i q^{17} +85.4801i q^{19} +158.740 q^{23} +113.901 q^{25} +269.955i q^{29} +36.0884i q^{31} -56.3085 q^{35} +353.780 q^{37} -144.415i q^{41} -368.186i q^{43} -397.333 q^{47} +57.3363 q^{49} -96.0857i q^{53} -56.0783i q^{55} -294.902 q^{59} -146.724 q^{61} +83.5471i q^{65} -301.433i q^{67} +687.794 q^{71} -312.138 q^{73} -284.496i q^{77} -602.063i q^{79} +1334.44 q^{83} -389.521 q^{85} +856.292i q^{89} +423.850i q^{91} +284.781 q^{95} +218.564 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 36 q^{13} - 132 q^{25} - 516 q^{37} - 720 q^{49} + 972 q^{61} + 660 q^{73} - 1056 q^{85} + 2532 q^{97}+O(q^{100}) $ 12 * q - 36 * q^13 - 132 * q^25 - 516 * q^37 - 720 * q^49 + 972 * q^61 + 660 * q^73 - 1056 * q^85 + 2532 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 3.33155i	− 0.297983i	−0.988838	−	0.148992i	$-0.952397\pi$
$5$	− 3.33155i	− 0.297983i	0.988838	−	0.148992i	$-0.0476027\pi$
$6$	0	0
$7$	− 16.9016i	− 0.912600i	−0.889826	−	0.456300i	$-0.849174\pi$
$7$	− 16.9016i	− 0.912600i	0.889826	−	0.456300i	$-0.150826\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.8325	0.461380	0.230690	−	0.973027i	$-0.425902\pi$
$11$	16.8325	0.461380	0.230690	+	0.973027i	$0.425902\pi$
$12$	0	0
$13$	−25.0775	−0.535020	−0.267510	−	0.963555i	$-0.586201\pi$
$13$	−25.0775	−0.535020	−0.267510	+	0.963555i	$0.586201\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 116.919i	− 1.66806i	−0.551722	−	0.834028i	$-0.686029\pi$
$17$	− 116.919i	− 1.66806i	0.551722	−	0.834028i	$-0.313971\pi$
$18$	0	0
$19$	85.4801i	1.03213i	0.856549	+	0.516065i	$0.172604\pi$
$19$	85.4801i	1.03213i	−0.856549	+	0.516065i	$0.827396\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	158.740	1.43911	0.719555	−	0.694436i	$-0.244346\pi$
$23$	158.740	1.43911	0.719555	+	0.694436i	$0.244346\pi$
$24$	0	0
$25$	113.901	0.911206
$26$	0	0
$27$	0	0
$28$	0	0
$29$	269.955i	1.72860i	0.502976	+	0.864300i	$0.332238\pi$
$29$	269.955i	1.72860i	−0.502976	+	0.864300i	$0.667762\pi$
$30$	0	0
$31$	36.0884i	0.209086i	0.994520	+	0.104543i	$0.0333380\pi$
$31$	36.0884i	0.209086i	−0.994520	+	0.104543i	$0.966662\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−56.3085	−0.271939
$36$	0	0
$37$	353.780	1.57192	0.785960	−	0.618277i	$-0.212169\pi$
$37$	353.780	1.57192	0.785960	+	0.618277i	$0.212169\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 144.415i	− 0.550093i	−0.961431	−	0.275046i	$-0.911307\pi$
$41$	− 144.415i	− 0.550093i	0.961431	−	0.275046i	$-0.0886932\pi$
$42$	0	0
$43$	− 368.186i	− 1.30576i	−0.757460	−	0.652881i	$-0.773560\pi$
$43$	− 368.186i	− 1.30576i	0.757460	−	0.652881i	$-0.226440\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−397.333	−1.23313	−0.616563	−	0.787305i	$-0.711476\pi$
$47$	−397.333	−1.23313	−0.616563	+	0.787305i	$0.711476\pi$
$48$	0	0
$49$	57.3363	0.167161
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 96.0857i	− 0.249026i	−0.992218	−	0.124513i	$-0.960263\pi$
$53$	− 96.0857i	− 0.249026i	0.992218	−	0.124513i	$-0.0397369\pi$
$54$	0	0
$55$	− 56.0783i	− 0.137484i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−294.902	−0.650727	−0.325364	−	0.945589i	$-0.605487\pi$
$59$	−294.902	−0.650727	−0.325364	+	0.945589i	$0.605487\pi$
$60$	0	0
$61$	−146.724	−0.307969	−0.153984	−	0.988073i	$-0.549211\pi$
$61$	−146.724	−0.307969	−0.153984	+	0.988073i	$0.549211\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	83.5471i	0.159427i
$66$	0	0
$67$	− 301.433i	− 0.549641i	−0.961496	−	0.274820i	$-0.911382\pi$
$67$	− 301.433i	− 0.549641i	0.961496	−	0.274820i	$-0.0886184\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	687.794	1.14966	0.574832	−	0.818272i	$-0.305067\pi$
$71$	687.794	1.14966	0.574832	+	0.818272i	$0.305067\pi$
$72$	0	0
$73$	−312.138	−0.500452	−0.250226	−	0.968187i	$-0.580505\pi$
$73$	−312.138	−0.500452	−0.250226	+	0.968187i	$0.580505\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 284.496i	− 0.421056i
$78$	0	0
$79$	− 602.063i	− 0.857436i	−0.903438	−	0.428718i	$-0.858965\pi$
$79$	− 602.063i	− 0.857436i	0.903438	−	0.428718i	$-0.141035\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1334.44	1.76475	0.882373	−	0.470551i	$-0.155945\pi$
$83$	1334.44	1.76475	0.882373	+	0.470551i	$0.155945\pi$
$84$	0	0
$85$	−389.521	−0.497053
$86$	0	0
$87$	0	0
$88$	0	0
$89$	856.292i	1.01985i	0.860218	+	0.509926i	$0.170327\pi$
$89$	856.292i	1.01985i	−0.860218	+	0.509926i	$0.829673\pi$
$90$	0	0
$91$	423.850i	0.488259i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	284.781	0.307557
$96$	0	0
$97$	218.564	0.228782	0.114391	−	0.993436i	$-0.463508\pi$
$97$	218.564	0.228782	0.114391	+	0.993436i	$0.463508\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 793.625i	− 0.781868i	−0.920419	−	0.390934i	$-0.872152\pi$
$101$	− 793.625i	− 0.781868i	0.920419	−	0.390934i	$-0.127848\pi$
$102$	0	0
$103$	− 1245.59i	− 1.19157i	−0.803145	−	0.595784i	$-0.796841\pi$
$103$	− 1245.59i	− 1.19157i	0.803145	−	0.595784i	$-0.203159\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	95.7189	0.0864813	0.0432406	−	0.999065i	$-0.486232\pi$
$107$	95.7189	0.0864813	0.0432406	+	0.999065i	$0.486232\pi$
$108$	0	0
$109$	−1349.09	−1.18550	−0.592749	−	0.805387i	$-0.701957\pi$
$109$	−1349.09	−1.18550	−0.592749	+	0.805387i	$0.701957\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1885.41i	1.56960i	0.619750	+	0.784799i	$0.287234\pi$
$113$	1885.41i	1.56960i	−0.619750	+	0.784799i	$0.712766\pi$
$114$	0	0
$115$	− 528.850i	− 0.428831i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1976.11	−1.52227
$120$	0	0
$121$	−1047.67	−0.787128
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 795.911i	− 0.569507i
$126$	0	0
$127$	714.330i	0.499106i	0.968361	+	0.249553i	$0.0802838\pi$
$127$	714.330i	0.499106i	−0.968361	+	0.249553i	$0.919716\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2153.89	−1.43654	−0.718268	−	0.695766i	$-0.755065\pi$
$131$	−2153.89	−1.43654	−0.718268	+	0.695766i	$0.755065\pi$
$132$	0	0
$133$	1444.75	0.941922
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2053.68i	− 1.28071i	−0.768079	−	0.640356i	$-0.778787\pi$
$137$	− 2053.68i	− 1.28071i	0.768079	−	0.640356i	$-0.221213\pi$
$138$	0	0
$139$	− 2000.40i	− 1.22066i	−0.792148	−	0.610329i	$-0.791037\pi$
$139$	− 2000.40i	− 1.22066i	0.792148	−	0.610329i	$-0.208963\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−422.117	−0.246847
$144$	0	0
$145$	899.370	0.515094
$146$	0	0
$147$	0	0
$148$	0	0
$149$	988.026i	0.543236i	0.962405	+	0.271618i	$0.0875588\pi$
$149$	988.026i	0.543236i	−0.962405	+	0.271618i	$0.912441\pi$
$150$	0	0
$151$	− 1630.56i	− 0.878764i	−0.898300	−	0.439382i	$-0.855197\pi$
$151$	− 1630.56i	− 0.878764i	0.898300	−	0.439382i	$-0.144803\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	120.231	0.0623042
$156$	0	0
$157$	690.002	0.350752	0.175376	−	0.984501i	$-0.443886\pi$
$157$	690.002	0.350752	0.175376	+	0.984501i	$0.443886\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2682.95i	− 1.31333i
$162$	0	0
$163$	− 3414.58i	− 1.64080i	−0.571790	−	0.820400i	$-0.693751\pi$
$163$	− 3414.58i	− 1.64080i	0.571790	−	0.820400i	$-0.306249\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1372.95	0.636180	0.318090	−	0.948061i	$-0.396959\pi$
$167$	1372.95	0.636180	0.318090	+	0.948061i	$0.396959\pi$
$168$	0	0
$169$	−1568.12	−0.713754
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1118.25i	0.491441i	0.969341	+	0.245721i	$0.0790246\pi$
$173$	1118.25i	0.491441i	−0.969341	+	0.245721i	$0.920975\pi$
$174$	0	0
$175$	− 1925.10i	− 0.831567i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1755.59	−0.733067	−0.366534	−	0.930405i	$-0.619456\pi$
$179$	−1755.59	−0.733067	−0.366534	+	0.930405i	$0.619456\pi$
$180$	0	0
$181$	−312.973	−0.128525	−0.0642626	−	0.997933i	$-0.520470\pi$
$181$	−312.973	−0.128525	−0.0642626	+	0.997933i	$0.520470\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1178.64i	− 0.468406i
$186$	0	0
$187$	− 1968.03i	− 0.769608i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2024.40	0.766915	0.383457	−	0.923559i	$-0.374733\pi$
$191$	2024.40	0.766915	0.383457	+	0.923559i	$0.374733\pi$
$192$	0	0
$193$	1138.00	0.424431	0.212216	−	0.977223i	$-0.431932\pi$
$193$	1138.00	0.424431	0.212216	+	0.977223i	$0.431932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2345.77i	0.848370i	0.905576	+	0.424185i	$0.139439\pi$
$197$	2345.77i	0.848370i	−0.905576	+	0.424185i	$0.860561\pi$
$198$	0	0
$199$	− 2839.16i	− 1.01137i	−0.862718	−	0.505686i	$-0.831240\pi$
$199$	− 2839.16i	− 1.01137i	0.862718	−	0.505686i	$-0.168760\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4562.67	1.57752
$204$	0	0
$205$	−481.126	−0.163918
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1438.84i	0.476204i
$210$	0	0
$211$	− 256.564i	− 0.0837091i	−0.999124	−	0.0418545i	$-0.986673\pi$
$211$	− 256.564i	− 0.0837091i	0.999124	−	0.0418545i	$-0.0133266\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1226.63	−0.389095
$216$	0	0
$217$	609.952	0.190812
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2932.03i	0.892443i
$222$	0	0
$223$	− 4020.58i	− 1.20734i	−0.797233	−	0.603672i	$-0.793704\pi$
$223$	− 4020.58i	− 1.20734i	0.797233	−	0.603672i	$-0.206296\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2958.44	−0.865016	−0.432508	−	0.901630i	$-0.642371\pi$
$227$	−2958.44	−0.865016	−0.432508	+	0.901630i	$0.642371\pi$
$228$	0	0
$229$	−226.069	−0.0652360	−0.0326180	−	0.999468i	$-0.510384\pi$
$229$	−226.069	−0.0652360	−0.0326180	+	0.999468i	$0.510384\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2356.56i	0.662591i	0.943527	+	0.331295i	$0.107486\pi$
$233$	2356.56i	0.662591i	−0.943527	+	0.331295i	$0.892514\pi$
$234$	0	0
$235$	1323.74i	0.367451i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2191.70	−0.593176	−0.296588	−	0.955006i	$-0.595849\pi$
$239$	−2191.70	−0.593176	−0.296588	+	0.955006i	$0.595849\pi$
$240$	0	0
$241$	6532.08	1.74593	0.872963	−	0.487787i	$-0.162196\pi$
$241$	6532.08	1.74593	0.872963	+	0.487787i	$0.162196\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 191.019i	− 0.0498113i
$246$	0	0
$247$	− 2143.63i	− 0.552210i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	34.6697	0.00871844	0.00435922	−	0.999990i	$-0.498612\pi$
$251$	34.6697	0.00871844	0.00435922	+	0.999990i	$0.498612\pi$
$252$	0	0
$253$	2671.98	0.663977
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3945.92i	− 0.957742i	−0.877885	−	0.478871i	$-0.841046\pi$
$257$	− 3945.92i	− 0.957742i	0.877885	−	0.478871i	$-0.158954\pi$
$258$	0	0
$259$	− 5979.44i	− 1.43453i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6558.79	1.53777	0.768883	−	0.639390i	$-0.220813\pi$
$263$	6558.79	1.53777	0.768883	+	0.639390i	$0.220813\pi$
$264$	0	0
$265$	−320.115	−0.0742056
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3741.27i	− 0.847991i	−0.905664	−	0.423995i	$-0.860627\pi$
$269$	− 3741.27i	− 0.847991i	0.905664	−	0.423995i	$-0.139373\pi$
$270$	0	0
$271$	− 6011.98i	− 1.34761i	−0.738910	−	0.673804i	$-0.764659\pi$
$271$	− 6011.98i	− 1.34761i	0.738910	−	0.673804i	$-0.235341\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1917.23	0.420412
$276$	0	0
$277$	4524.48	0.981407	0.490704	−	0.871327i	$-0.336740\pi$
$277$	4524.48	0.981407	0.490704	+	0.871327i	$0.336740\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4251.22i	0.902514i	0.892394	+	0.451257i	$0.149024\pi$
$281$	4251.22i	0.902514i	−0.892394	+	0.451257i	$0.850976\pi$
$282$	0	0
$283$	− 4433.04i	− 0.931154i	−0.885007	−	0.465577i	$-0.845847\pi$
$283$	− 4433.04i	− 0.931154i	0.885007	−	0.465577i	$-0.154153\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2440.84	−0.502015
$288$	0	0
$289$	−8756.99	−1.78241
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 7734.73i	− 1.54221i	−0.636708	−	0.771105i	$-0.719704\pi$
$293$	− 7734.73i	− 1.54221i	0.636708	−	0.771105i	$-0.280296\pi$
$294$	0	0
$295$	982.480i	0.193906i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3980.80	−0.769952
$300$	0	0
$301$	−6222.92	−1.19164
$302$	0	0
$303$	0	0
$304$	0	0
$305$	488.819i	0.0917694i
$306$	0	0
$307$	− 8510.18i	− 1.58209i	−0.611759	−	0.791045i	$-0.709538\pi$
$307$	− 8510.18i	− 1.58209i	0.611759	−	0.791045i	$-0.290462\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8133.65	−1.48301	−0.741507	−	0.670945i	$-0.765888\pi$
$311$	−8133.65	−1.48301	−0.741507	+	0.670945i	$0.765888\pi$
$312$	0	0
$313$	5066.73	0.914979	0.457489	−	0.889215i	$-0.348749\pi$
$313$	5066.73	0.914979	0.457489	+	0.889215i	$0.348749\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4122.54i	− 0.730425i	−0.930924	−	0.365212i	$-0.880996\pi$
$317$	− 4122.54i	− 0.730425i	0.930924	−	0.365212i	$-0.119004\pi$
$318$	0	0
$319$	4544.01i	0.797542i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9994.22	1.72165
$324$	0	0
$325$	−2856.35	−0.487513
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6715.55i	1.12535i
$330$	0	0
$331$	3819.96i	0.634332i	0.948370	+	0.317166i	$0.102731\pi$
$331$	3819.96i	0.634332i	−0.948370	+	0.317166i	$0.897269\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1004.24	−0.163784
$336$	0	0
$337$	−9712.35	−1.56993	−0.784963	−	0.619542i	$-0.787318\pi$
$337$	−9712.35	−1.56993	−0.784963	+	0.619542i	$0.787318\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	607.457i	0.0964682i
$342$	0	0
$343$	− 6766.32i	− 1.06515i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3315.20	−0.512880	−0.256440	−	0.966560i	$-0.582550\pi$
$347$	−3315.20	−0.512880	−0.256440	+	0.966560i	$0.582550\pi$
$348$	0	0
$349$	2864.88	0.439408	0.219704	−	0.975567i	$-0.429491\pi$
$349$	2864.88	0.439408	0.219704	+	0.975567i	$0.429491\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4904.72i	0.739523i	0.929127	+	0.369762i	$0.120561\pi$
$353$	4904.72i	0.739523i	−0.929127	+	0.369762i	$0.879439\pi$
$354$	0	0
$355$	− 2291.42i	− 0.342580i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4185.24	0.615288	0.307644	−	0.951501i	$-0.400459\pi$
$359$	4185.24	0.615288	0.307644	+	0.951501i	$0.400459\pi$
$360$	0	0
$361$	−447.841	−0.0652924
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1039.90i	0.149126i
$366$	0	0
$367$	4979.97i	0.708317i	0.935185	+	0.354158i	$0.115233\pi$
$367$	4979.97i	0.708317i	−0.935185	+	0.354158i	$0.884767\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1624.00	−0.227261
$372$	0	0
$373$	−6411.84	−0.890060	−0.445030	−	0.895516i	$-0.646807\pi$
$373$	−6411.84	−0.890060	−0.445030	+	0.895516i	$0.646807\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6769.81i	− 0.924835i
$378$	0	0
$379$	− 5294.51i	− 0.717574i	−0.933419	−	0.358787i	$-0.883190\pi$
$379$	− 5294.51i	− 0.717574i	0.933419	−	0.358787i	$-0.116810\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5923.23	0.790242	0.395121	−	0.918629i	$-0.370703\pi$
$383$	5923.23	0.790242	0.395121	+	0.918629i	$0.370703\pi$
$384$	0	0
$385$	−947.812	−0.125467
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4084.92i	0.532426i	0.963914	+	0.266213i	$0.0857724\pi$
$389$	4084.92i	0.532426i	−0.963914	+	0.266213i	$0.914228\pi$
$390$	0	0
$391$	− 18559.6i	− 2.40052i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2005.81	−0.255501
$396$	0	0
$397$	−2821.56	−0.356700	−0.178350	−	0.983967i	$-0.557076\pi$
$397$	−2821.56	−0.356700	−0.178350	+	0.983967i	$0.557076\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4215.14i	− 0.524923i	−0.964942	−	0.262461i	$-0.915466\pi$
$401$	− 4215.14i	− 0.524923i	0.964942	−	0.262461i	$-0.0845343\pi$
$402$	0	0
$403$	− 905.009i	− 0.111865i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5954.99	0.725253
$408$	0	0
$409$	605.285	0.0731771	0.0365885	−	0.999330i	$-0.488351\pi$
$409$	605.285	0.0731771	0.0365885	+	0.999330i	$0.488351\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4984.30i	0.593854i
$414$	0	0
$415$	− 4445.76i	− 0.525865i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6479.90	0.755523	0.377761	−	0.925903i	$-0.376694\pi$
$419$	6479.90	0.755523	0.377761	+	0.925903i	$0.376694\pi$
$420$	0	0
$421$	363.812	0.0421166	0.0210583	−	0.999778i	$-0.493296\pi$
$421$	363.812	0.0421166	0.0210583	+	0.999778i	$0.493296\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 13317.1i	− 1.51994i
$426$	0	0
$427$	2479.87i	0.281052i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1486.97	−0.166182	−0.0830912	−	0.996542i	$-0.526479\pi$
$431$	−1486.97	−0.166182	−0.0830912	+	0.996542i	$0.526479\pi$
$432$	0	0
$433$	12322.8	1.36765	0.683827	−	0.729644i	$-0.260314\pi$
$433$	12322.8	1.36765	0.683827	+	0.729644i	$0.260314\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13569.1i	1.48535i
$438$	0	0
$439$	− 10918.0i	− 1.18698i	−0.804840	−	0.593492i	$-0.797749\pi$
$439$	− 10918.0i	− 1.18698i	0.804840	−	0.593492i	$-0.202251\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3934.04	0.421923	0.210961	−	0.977494i	$-0.432341\pi$
$443$	3934.04	0.421923	0.210961	+	0.977494i	$0.432341\pi$
$444$	0	0
$445$	2852.78	0.303899
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9560.84i	1.00491i	0.864604	+	0.502454i	$0.167570\pi$
$449$	9560.84i	1.00491i	−0.864604	+	0.502454i	$0.832430\pi$
$450$	0	0
$451$	− 2430.86i	− 0.253802i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1412.08	0.145493
$456$	0	0
$457$	6479.90	0.663276	0.331638	−	0.943407i	$-0.392399\pi$
$457$	6479.90	0.663276	0.331638	+	0.943407i	$0.392399\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 3560.77i	− 0.359743i	−0.983690	−	0.179872i	$-0.942432\pi$
$461$	− 3560.77i	− 0.359743i	0.983690	−	0.179872i	$-0.0575682\pi$
$462$	0	0
$463$	15775.3i	1.58346i	0.610874	+	0.791728i	$0.290818\pi$
$463$	15775.3i	1.58346i	−0.610874	+	0.791728i	$0.709182\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14291.6	−1.41614	−0.708071	−	0.706141i	$-0.750434\pi$
$467$	−14291.6	−1.41614	−0.708071	+	0.706141i	$0.750434\pi$
$468$	0	0
$469$	−5094.70	−0.501602
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 6197.48i	− 0.602453i
$474$	0	0
$475$	9736.24i	0.940483i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14630.0	−1.39553	−0.697766	−	0.716326i	$-0.745822\pi$
$479$	−14630.0	−1.39553	−0.697766	+	0.716326i	$0.745822\pi$
$480$	0	0
$481$	−8871.92	−0.841008
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 728.159i	− 0.0681732i
$486$	0	0
$487$	16742.7i	1.55787i	0.627102	+	0.778937i	$0.284241\pi$
$487$	16742.7i	1.55787i	−0.627102	+	0.778937i	$0.715759\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7342.23	−0.674847	−0.337424	−	0.941353i	$-0.609555\pi$
$491$	−7342.23	−0.674847	−0.337424	+	0.941353i	$0.609555\pi$
$492$	0	0
$493$	31562.8	2.88340
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 11624.8i	− 1.04918i
$498$	0	0
$499$	− 1565.06i	− 0.140404i	−0.997533	−	0.0702020i	$-0.977636\pi$
$499$	− 1565.06i	− 0.140404i	0.997533	−	0.0702020i	$-0.0223644\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1257.87	−0.111503	−0.0557513	−	0.998445i	$-0.517755\pi$
$503$	−1257.87	−0.111503	−0.0557513	+	0.998445i	$0.517755\pi$
$504$	0	0
$505$	−2644.00	−0.232983
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3160.96i	− 0.275260i	−0.990484	−	0.137630i	$-0.956052\pi$
$509$	− 3160.96i	− 0.275260i	0.990484	−	0.137630i	$-0.0439484\pi$
$510$	0	0
$511$	5275.63i	0.456712i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4149.75	−0.355067
$516$	0	0
$517$	−6688.09	−0.568940
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2973.62i	0.250051i	0.992154	+	0.125026i	$0.0399013\pi$
$521$	2973.62i	0.250051i	−0.992154	+	0.125026i	$0.960099\pi$
$522$	0	0
$523$	8019.43i	0.670488i	0.942131	+	0.335244i	$0.108819\pi$
$523$	8019.43i	0.670488i	−0.942131	+	0.335244i	$0.891181\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4219.41	0.348768
$528$	0	0
$529$	13031.3	1.07104
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3621.57i	0.294311i
$534$	0	0
$535$	− 318.893i	− 0.0257700i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	965.112	0.0771249
$540$	0	0
$541$	−11815.3	−0.938961	−0.469480	−	0.882943i	$-0.655559\pi$
$541$	−11815.3	−0.938961	−0.469480	+	0.882943i	$0.655559\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4494.56i	0.353259i
$546$	0	0
$547$	− 15596.6i	− 1.21913i	−0.792737	−	0.609564i	$-0.791345\pi$
$547$	− 15596.6i	− 1.21913i	0.792737	−	0.609564i	$-0.208655\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−23075.8	−1.78414
$552$	0	0
$553$	−10175.8	−0.782496
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19104.3i	1.45327i	0.687022	+	0.726637i	$0.258918\pi$
$557$	19104.3i	1.45327i	−0.687022	+	0.726637i	$0.741082\pi$
$558$	0	0
$559$	9233.19i	0.698608i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13747.5	−1.02911	−0.514555	−	0.857458i	$-0.672043\pi$
$563$	−13747.5	−1.02911	−0.514555	+	0.857458i	$0.672043\pi$
$564$	0	0
$565$	6281.35	0.467714
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18303.4i	1.34854i	0.738486	+	0.674269i	$0.235541\pi$
$569$	18303.4i	1.34854i	−0.738486	+	0.674269i	$0.764459\pi$
$570$	0	0
$571$	8345.58i	0.611649i	0.952088	+	0.305824i	$0.0989320\pi$
$571$	8345.58i	0.611649i	−0.952088	+	0.305824i	$0.901068\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18080.6	1.31133
$576$	0	0
$577$	−21336.5	−1.53943	−0.769713	−	0.638390i	$-0.779601\pi$
$577$	−21336.5	−1.53943	−0.769713	+	0.638390i	$0.779601\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 22554.2i	− 1.61051i
$582$	0	0
$583$	− 1617.36i	− 0.114896i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26431.2	1.85849	0.929244	−	0.369467i	$-0.120460\pi$
$587$	26431.2	1.85849	0.929244	+	0.369467i	$0.120460\pi$
$588$	0	0
$589$	−3084.84	−0.215804
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 248.904i	− 0.0172366i	−0.999963	−	0.00861828i	$-0.997257\pi$
$593$	− 248.904i	− 0.0172366i	0.999963	−	0.00861828i	$-0.00274332\pi$
$594$	0	0
$595$	6583.52i	0.453610i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14995.9	1.02290	0.511449	−	0.859313i	$-0.329109\pi$
$599$	14995.9	1.02290	0.511449	+	0.859313i	$0.329109\pi$
$600$	0	0
$601$	23747.1	1.61175	0.805876	−	0.592085i	$-0.201695\pi$
$601$	23747.1	1.61175	0.805876	+	0.592085i	$0.201695\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3490.36i	0.234551i
$606$	0	0
$607$	6539.02i	0.437250i	0.975809	+	0.218625i	$0.0701571\pi$
$607$	6539.02i	0.437250i	−0.975809	+	0.218625i	$0.929843\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9964.12	0.659747
$612$	0	0
$613$	5985.95	0.394405	0.197202	−	0.980363i	$-0.436814\pi$
$613$	5985.95	0.394405	0.197202	+	0.980363i	$0.436814\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6168.16i	0.402465i	0.979544	+	0.201232i	$0.0644947\pi$
$617$	6168.16i	0.402465i	−0.979544	+	0.201232i	$0.935505\pi$
$618$	0	0
$619$	6667.31i	0.432927i	0.976291	+	0.216464i	$0.0694523\pi$
$619$	6667.31i	0.432927i	−0.976291	+	0.216464i	$0.930548\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14472.7	0.930717
$624$	0	0
$625$	11586.0	0.741502
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 41363.5i	− 2.62205i
$630$	0	0
$631$	− 8168.49i	− 0.515344i	−0.966232	−	0.257672i	$-0.917045\pi$
$631$	− 8168.49i	− 0.515344i	0.966232	−	0.257672i	$-0.0829554\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2379.83	0.148725
$636$	0	0
$637$	−1437.85	−0.0894346
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6332.36i	0.390192i	0.980784	+	0.195096i	$0.0625018\pi$
$641$	6332.36i	0.390192i	−0.980784	+	0.195096i	$0.937498\pi$
$642$	0	0
$643$	9433.98i	0.578600i	0.957239	+	0.289300i	$0.0934225\pi$
$643$	9433.98i	0.578600i	−0.957239	+	0.289300i	$0.906578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20479.8	−1.24442	−0.622212	−	0.782848i	$-0.713766\pi$
$647$	−20479.8	−1.24442	−0.622212	+	0.782848i	$0.713766\pi$
$648$	0	0
$649$	−4963.92	−0.300233
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 24435.7i	− 1.46438i	−0.681099	−	0.732192i	$-0.738498\pi$
$653$	− 24435.7i	− 1.46438i	0.681099	−	0.732192i	$-0.261502\pi$
$654$	0	0
$655$	7175.80i	0.428064i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	32183.0	1.90238	0.951192	−	0.308601i	$-0.0998607\pi$
$659$	32183.0	1.90238	0.951192	+	0.308601i	$0.0998607\pi$
$660$	0	0
$661$	−13990.4	−0.823245	−0.411622	−	0.911355i	$-0.635038\pi$
$661$	−13990.4	−0.823245	−0.411622	+	0.911355i	$0.635038\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 4813.26i	− 0.280677i
$666$	0	0
$667$	42852.6i	2.48765i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2469.73	−0.142091
$672$	0	0
$673$	12289.9	0.703923	0.351961	−	0.936014i	$-0.385515\pi$
$673$	12289.9	0.703923	0.351961	+	0.936014i	$0.385515\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16303.5i	0.925548i	0.886476	+	0.462774i	$0.153146\pi$
$677$	16303.5i	0.925548i	−0.886476	+	0.462774i	$0.846854\pi$
$678$	0	0
$679$	− 3694.09i	− 0.208786i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−9720.93	−0.544599	−0.272299	−	0.962213i	$-0.587784\pi$
$683$	−9720.93	−0.544599	−0.272299	+	0.962213i	$0.587784\pi$
$684$	0	0
$685$	−6841.93	−0.381630
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2409.59i	0.133234i
$690$	0	0
$691$	80.4083i	0.00442674i	0.999998	+	0.00221337i	$0.000704538\pi$
$691$	80.4083i	0.00442674i	−0.999998	+	0.00221337i	$0.999295\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6664.43	−0.363736
$696$	0	0
$697$	−16884.8	−0.917586
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4285.46i	− 0.230898i	−0.993313	−	0.115449i	$-0.963169\pi$
$701$	− 4285.46i	− 0.230898i	0.993313	−	0.115449i	$-0.0368307\pi$
$702$	0	0
$703$	30241.1i	1.62243i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13413.5	−0.713532
$708$	0	0
$709$	8587.44	0.454878	0.227439	−	0.973792i	$-0.426965\pi$
$709$	8587.44	0.454878	0.227439	+	0.973792i	$0.426965\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5728.67i	0.300898i
$714$	0	0
$715$	1406.30i	0.0735564i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−665.197	−0.0345030	−0.0172515	−	0.999851i	$-0.505492\pi$
$719$	−665.197	−0.0345030	−0.0172515	+	0.999851i	$0.505492\pi$
$720$	0	0
$721$	−21052.4	−1.08742
$722$	0	0
$723$	0	0
$724$	0	0
$725$	30748.1i	1.57511i
$726$	0	0
$727$	− 1269.94i	− 0.0647860i	−0.999475	−	0.0323930i	$-0.989687\pi$
$727$	− 1269.94i	− 0.0647860i	0.999475	−	0.0323930i	$-0.0103128\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−43047.8	−2.17809
$732$	0	0
$733$	−26425.0	−1.33155	−0.665777	−	0.746151i	$-0.731900\pi$
$733$	−26425.0	−1.33155	−0.665777	+	0.746151i	$0.731900\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5073.87i	− 0.253593i
$738$	0	0
$739$	− 4732.65i	− 0.235579i	−0.993039	−	0.117790i	$-0.962419\pi$
$739$	− 4732.65i	− 0.235579i	0.993039	−	0.117790i	$-0.0375809\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14771.2	−0.729346	−0.364673	−	0.931136i	$-0.618819\pi$
$743$	−14771.2	−0.729346	−0.364673	+	0.931136i	$0.618819\pi$
$744$	0	0
$745$	3291.66	0.161875
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1617.80i	− 0.0789228i
$750$	0	0
$751$	27071.9i	1.31540i	0.753280	+	0.657700i	$0.228471\pi$
$751$	27071.9i	1.31540i	−0.753280	+	0.657700i	$0.771529\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5432.31	−0.261857
$756$	0	0
$757$	12404.5	0.595576	0.297788	−	0.954632i	$-0.403751\pi$
$757$	12404.5	0.595576	0.297788	+	0.954632i	$0.403751\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 8564.67i	− 0.407975i	−0.978973	−	0.203988i	$-0.934610\pi$
$761$	− 8564.67i	− 0.407975i	0.978973	−	0.203988i	$-0.0653902\pi$
$762$	0	0
$763$	22801.8i	1.08189i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7395.40	0.348152
$768$	0	0
$769$	−19117.5	−0.896483	−0.448241	−	0.893913i	$-0.647949\pi$
$769$	−19117.5	−0.896483	−0.448241	+	0.893913i	$0.647949\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28145.2i	1.30959i	0.755807	+	0.654794i	$0.227245\pi$
$773$	28145.2i	1.30959i	−0.755807	+	0.654794i	$0.772755\pi$
$774$	0	0
$775$	4110.50i	0.190521i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12344.6	0.567767
$780$	0	0
$781$	11577.3	0.530432
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2298.78i	− 0.104518i
$786$	0	0
$787$	− 15895.3i	− 0.719959i	−0.932960	−	0.359979i	$-0.882784\pi$
$787$	− 15895.3i	− 0.719959i	0.932960	−	0.359979i	$-0.117216\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	31866.4	1.43242
$792$	0	0
$793$	3679.47	0.164769
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39051.2i	1.73559i	0.496924	+	0.867794i	$0.334463\pi$
$797$	39051.2i	1.73559i	−0.496924	+	0.867794i	$0.665537\pi$
$798$	0	0
$799$	46455.6i	2.05692i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5254.05	−0.230898
$804$	0	0
$805$	−8938.40	−0.391351
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 67.6844i	− 0.00294148i	−0.999999	−	0.00147074i	$-0.999532\pi$
$809$	− 67.6844i	− 0.00294148i	0.999999	−	0.00147074i	$-0.000468151\pi$
$810$	0	0
$811$	26390.7i	1.14267i	0.820718	+	0.571334i	$0.193574\pi$
$811$	26390.7i	1.14267i	−0.820718	+	0.571334i	$0.806426\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11375.9	−0.488931
$816$	0	0
$817$	31472.5	1.34772
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 15840.6i	− 0.673374i	−0.941617	−	0.336687i	$-0.890694\pi$
$821$	− 15840.6i	− 0.673374i	0.941617	−	0.336687i	$-0.109306\pi$
$822$	0	0
$823$	227.779i	0.00964747i	0.999988	+	0.00482373i	$0.00153545\pi$
$823$	227.779i	0.00964747i	−0.999988	+	0.00482373i	$0.998465\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11468.2	0.482209	0.241105	−	0.970499i	$-0.422490\pi$
$827$	11468.2	0.482209	0.241105	+	0.970499i	$0.422490\pi$
$828$	0	0
$829$	−15188.4	−0.636327	−0.318164	−	0.948036i	$-0.603066\pi$
$829$	−15188.4	−0.636327	−0.318164	+	0.948036i	$0.603066\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 6703.69i	− 0.278835i
$834$	0	0
$835$	− 4574.06i	− 0.189571i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28109.4	1.15667	0.578334	−	0.815800i	$-0.303703\pi$
$839$	28109.4	1.15667	0.578334	+	0.815800i	$0.303703\pi$
$840$	0	0
$841$	−48486.8	−1.98806
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5224.27i	0.212687i
$846$	0	0
$847$	17707.2i	0.718333i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	56158.9	2.26217
$852$	0	0
$853$	41817.8	1.67856	0.839281	−	0.543699i	$-0.182977\pi$
$853$	41817.8	1.67856	0.839281	+	0.543699i	$0.182977\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 38510.6i	− 1.53500i	−0.641048	−	0.767501i	$-0.721500\pi$
$857$	− 38510.6i	− 1.53500i	0.641048	−	0.767501i	$-0.278500\pi$
$858$	0	0
$859$	43912.4i	1.74421i	0.489322	+	0.872103i	$0.337244\pi$
$859$	43912.4i	1.74421i	−0.489322	+	0.872103i	$0.662756\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40298.8	−1.58956	−0.794779	−	0.606899i	$-0.792413\pi$
$863$	−40298.8	−1.58956	−0.794779	+	0.606899i	$0.792413\pi$
$864$	0	0
$865$	3725.52	0.146441
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 10134.2i	− 0.395604i
$870$	0	0
$871$	7559.20i	0.294069i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−13452.2	−0.519732
$876$	0	0
$877$	−17806.9	−0.685629	−0.342814	−	0.939403i	$-0.611380\pi$
$877$	−17806.9	−0.685629	−0.342814	+	0.939403i	$0.611380\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 4232.13i	− 0.161843i	−0.996720	−	0.0809217i	$-0.974214\pi$
$881$	− 4232.13i	− 0.161843i	0.996720	−	0.0809217i	$-0.0257864\pi$
$882$	0	0
$883$	− 36881.3i	− 1.40561i	−0.711382	−	0.702806i	$-0.751930\pi$
$883$	− 36881.3i	− 1.40561i	0.711382	−	0.702806i	$-0.248070\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31760.2	1.20226	0.601128	−	0.799152i	$-0.294718\pi$
$887$	31760.2	1.20226	0.601128	+	0.799152i	$0.294718\pi$
$888$	0	0
$889$	12073.3	0.455485
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 33964.0i	− 1.27275i
$894$	0	0
$895$	5848.84i	0.218442i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9742.26	−0.361427
$900$	0	0
$901$	−11234.2	−0.415390
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1042.68i	0.0382984i
$906$	0	0
$907$	50257.3i	1.83987i	0.392066	+	0.919937i	$0.371760\pi$
$907$	50257.3i	1.83987i	−0.392066	+	0.919937i	$0.628240\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22485.6	0.817763	0.408881	−	0.912588i	$-0.365919\pi$
$911$	22485.6	0.817763	0.408881	+	0.912588i	$0.365919\pi$
$912$	0	0
$913$	22461.9	0.814219
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36404.2i	1.31098i
$918$	0	0
$919$	− 26918.9i	− 0.966240i	−0.875554	−	0.483120i	$-0.839504\pi$
$919$	− 26918.9i	− 0.966240i	0.875554	−	0.483120i	$-0.160496\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−17248.2	−0.615093
$924$	0	0
$925$	40295.8	1.43234
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 11212.4i	− 0.395983i	−0.980204	−	0.197991i	$-0.936558\pi$
$929$	− 11212.4i	− 0.395983i	0.980204	−	0.197991i	$-0.0634418\pi$
$930$	0	0
$931$	4901.11i	0.172532i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6556.60	−0.229330
$936$	0	0
$937$	38358.1	1.33736	0.668679	−	0.743551i	$-0.266860\pi$
$937$	38358.1	1.33736	0.668679	+	0.743551i	$0.266860\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7558.00i	0.261832i	0.991393	+	0.130916i	$0.0417918\pi$
$941$	7558.00i	0.261832i	−0.991393	+	0.130916i	$0.958208\pi$
$942$	0	0
$943$	− 22924.4i	− 0.791644i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−50708.7	−1.74003	−0.870016	−	0.493023i	$-0.835892\pi$
$947$	−50708.7	−1.74003	−0.870016	+	0.493023i	$0.835892\pi$
$948$	0	0
$949$	7827.65	0.267751
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16440.9i	0.558837i	0.960169	+	0.279419i	$0.0901417\pi$
$953$	16440.9i	0.558837i	−0.960169	+	0.279419i	$0.909858\pi$
$954$	0	0
$955$	− 6744.41i	− 0.228528i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34710.4	−1.16878
$960$	0	0
$961$	28488.6	0.956283
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 3791.32i	− 0.126473i
$966$	0	0
$967$	− 488.917i	− 0.0162591i	−0.999967	−	0.00812953i	$-0.997412\pi$
$967$	− 488.917i	− 0.0162591i	0.999967	−	0.00812953i	$-0.00258774\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31950.1	1.05595	0.527975	−	0.849260i	$-0.322951\pi$
$971$	31950.1	1.05595	0.527975	+	0.849260i	$0.322951\pi$
$972$	0	0
$973$	−33809.9	−1.11397
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5271.73i	0.172628i	0.996268	+	0.0863140i	$0.0275088\pi$
$977$	5271.73i	0.172628i	−0.996268	+	0.0863140i	$0.972491\pi$
$978$	0	0
$979$	14413.5i	0.470539i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11052.1	−0.358603	−0.179302	−	0.983794i	$-0.557384\pi$
$983$	−11052.1	−0.358603	−0.179302	+	0.983794i	$0.557384\pi$
$984$	0	0
$985$	7815.04	0.252800
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 58445.7i	− 1.87914i
$990$	0	0
$991$	41203.4i	1.32076i	0.750933	+	0.660378i	$0.229604\pi$
$991$	41203.4i	1.32076i	−0.750933	+	0.660378i	$0.770396\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9458.82	−0.301372
$996$	0	0
$997$	14951.3	0.474938	0.237469	−	0.971395i	$-0.423682\pi$
$997$	14951.3	0.474938	0.237469	+	0.971395i	$0.423682\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.c.i.1727.5		12
3.2	odd	2	inner	1728.4.c.i.1727.7		12
4.3	odd	2	inner	1728.4.c.i.1727.6		12
8.3	odd	2		108.4.b.a.107.12	yes	12
8.5	even	2		108.4.b.a.107.2	yes	12
12.11	even	2	inner	1728.4.c.i.1727.8		12
24.5	odd	2		108.4.b.a.107.11	yes	12
24.11	even	2		108.4.b.a.107.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.b.a.107.1	✓	12	24.11	even	2
108.4.b.a.107.2	yes	12	8.5	even	2
108.4.b.a.107.11	yes	12	24.5	odd	2
108.4.b.a.107.12	yes	12	8.3	odd	2
1728.4.c.i.1727.5		12	1.1	even	1	trivial
1728.4.c.i.1727.6		12	4.3	odd	2	inner
1728.4.c.i.1727.7		12	3.2	odd	2	inner
1728.4.c.i.1727.8		12	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.33155i q^{5} -16.9016i q^{7} +16.8325 q^{11} -25.0775 q^{13} -116.919i q^{17} +85.4801i q^{19} +158.740 q^{23} +113.901 q^{25} +269.955i q^{29} +36.0884i q^{31} -56.3085 q^{35} +353.780 q^{37} -144.415i q^{41} -368.186i q^{43} -397.333 q^{47} +57.3363 q^{49} -96.0857i q^{53} -56.0783i q^{55} -294.902 q^{59} -146.724 q^{61} +83.5471i q^{65} -301.433i q^{67} +687.794 q^{71} -312.138 q^{73} -284.496i q^{77} -602.063i q^{79} +1334.44 q^{83} -389.521 q^{85} +856.292i q^{89} +423.850i q^{91} +284.781 q^{95} +218.564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{13} - 132 q^{25} - 516 q^{37} - 720 q^{49} + 972 q^{61} + 660 q^{73} - 1056 q^{85} + 2532 q^{97}+O(q^{100}) \) 12 * q - 36 * q^13 - 132 * q^25 - 516 * q^37 - 720 * q^49 + 972 * q^61 + 660 * q^73 - 1056 * q^85 + 2532 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more