LMFDB - Embedded newform 684.3.e.a.305.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(305,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("684.305");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.e (of order $2$, degree $1$, 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:	$18.6376500822$
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} + 156x^{10} + 8721x^{8} + 208784x^{6} + 2024760x^{4} + 7117056x^{2} + 6533136 $ x^12 + 156x^10 + 8721x^8 + 208784x^6 + 2024760x^4 + 7117056*x^2 + 6533136
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			305.1
Root			$-7.53519i$ of defining polynomial
Character	$\chi$	$=$	684.305
Dual form			684.3.e.a.305.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.53519i q^{5} +5.18161 q^{7} +O(q^{10})$	$q-7.53519i q^{5} +5.18161 q^{7} -20.8851i q^{11} +5.35632 q^{13} +8.47306i q^{17} -4.35890 q^{19} +22.0052i q^{23} -31.7791 q^{25} -34.4800i q^{29} -35.1551 q^{31} -39.0444i q^{35} +15.3168 q^{37} +19.8541i q^{41} +24.9768 q^{43} -11.0976i q^{47} -22.1509 q^{49} -90.7505i q^{53} -157.373 q^{55} +51.2576i q^{59} -47.6464 q^{61} -40.3609i q^{65} +4.34056 q^{67} -30.5108i q^{71} +79.8288 q^{73} -108.219i q^{77} -152.066 q^{79} +22.6822i q^{83} +63.8461 q^{85} -134.723i q^{89} +27.7543 q^{91} +32.8451i q^{95} -32.4684 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{7}+O(q^{10}) $ 12 * q + 16 * q^7	$ 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97}+O(q^{100}) $ 12 * q + 16 * q^7 - 16 * q^13 - 12 * q^25 - 40 * q^31 - 32 * q^37 + 92 * q^43 - 84 * q^55 - 48 * q^61 - 88 * q^67 + 148 * q^73 - 56 * q^79 + 228 * q^85 - 8 * q^91 + 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	− 7.53519i	− 1.50704i	−0.657426	−	0.753519i	$-0.728355\pi$
$5$	− 7.53519i	− 1.50704i	0.657426	−	0.753519i	$-0.271645\pi$
$6$	0	0
$7$	5.18161	0.740230	0.370115	−	0.928986i	$-0.379318\pi$
$7$	5.18161	0.740230	0.370115	+	0.928986i	$0.379318\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 20.8851i	− 1.89865i	−0.314301	−	0.949323i	$-0.601770\pi$
$11$	− 20.8851i	− 1.89865i	0.314301	−	0.949323i	$-0.398230\pi$
$12$	0	0
$13$	5.35632	0.412024	0.206012	−	0.978549i	$-0.433951\pi$
$13$	5.35632	0.412024	0.206012	+	0.978549i	$0.433951\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	8.47306i	0.498415i	0.968450	+	0.249208i	$0.0801701\pi$
$17$	8.47306i	0.498415i	−0.968450	+	0.249208i	$0.919830\pi$
$18$	0	0
$19$	−4.35890	−0.229416
$19$	−4.35890	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	22.0052i	0.956746i	0.878157	+	0.478373i	$0.158773\pi$
$23$	22.0052i	0.956746i	−0.878157	+	0.478373i	$0.841227\pi$
$24$	0	0
$25$	−31.7791	−1.27116
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 34.4800i	− 1.18897i	−0.804108	−	0.594483i	$-0.797357\pi$
$29$	− 34.4800i	− 1.18897i	0.804108	−	0.594483i	$-0.202643\pi$
$30$	0	0
$31$	−35.1551	−1.13404	−0.567018	−	0.823705i	$-0.691903\pi$
$31$	−35.1551	−1.13404	−0.567018	+	0.823705i	$0.691903\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 39.0444i	− 1.11555i
$36$	0	0
$37$	15.3168	0.413967	0.206984	−	0.978344i	$-0.433635\pi$
$37$	15.3168	0.413967	0.206984	+	0.978344i	$0.433635\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	19.8541i	0.484247i	0.970245	+	0.242123i	$0.0778439\pi$
$41$	19.8541i	0.484247i	−0.970245	+	0.242123i	$0.922156\pi$
$42$	0	0
$43$	24.9768	0.580855	0.290427	−	0.956897i	$-0.406203\pi$
$43$	24.9768	0.580855	0.290427	+	0.956897i	$0.406203\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 11.0976i	− 0.236119i	−0.993007	−	0.118060i	$-0.962333\pi$
$47$	− 11.0976i	− 0.236119i	0.993007	−	0.118060i	$-0.0376674\pi$
$48$	0	0
$49$	−22.1509	−0.452060
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 90.7505i	− 1.71227i	−0.516749	−	0.856137i	$-0.672858\pi$
$53$	− 90.7505i	− 1.71227i	0.516749	−	0.856137i	$-0.327142\pi$
$54$	0	0
$55$	−157.373	−2.86133
$56$	0	0
$57$	0	0
$58$	0	0
$59$	51.2576i	0.868773i	0.900727	+	0.434387i	$0.143035\pi$
$59$	51.2576i	0.868773i	−0.900727	+	0.434387i	$0.856965\pi$
$60$	0	0
$61$	−47.6464	−0.781088	−0.390544	−	0.920584i	$-0.627713\pi$
$61$	−47.6464	−0.781088	−0.390544	+	0.920584i	$0.627713\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 40.3609i	− 0.620936i
$66$	0	0
$67$	4.34056	0.0647844	0.0323922	−	0.999475i	$-0.489687\pi$
$67$	4.34056	0.0647844	0.0323922	+	0.999475i	$0.489687\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 30.5108i	− 0.429730i	−0.976644	−	0.214865i	$-0.931069\pi$
$71$	− 30.5108i	− 0.429730i	0.976644	−	0.214865i	$-0.0689311\pi$
$72$	0	0
$73$	79.8288	1.09354	0.546772	−	0.837281i	$-0.315856\pi$
$73$	79.8288	1.09354	0.546772	+	0.837281i	$0.315856\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 108.219i	− 1.40544i
$78$	0	0
$79$	−152.066	−1.92488	−0.962442	−	0.271486i	$-0.912485\pi$
$79$	−152.066	−1.92488	−0.962442	+	0.271486i	$0.912485\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	22.6822i	0.273279i	0.990621	+	0.136639i	$0.0436302\pi$
$83$	22.6822i	0.273279i	−0.990621	+	0.136639i	$0.956370\pi$
$84$	0	0
$85$	63.8461	0.751130
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 134.723i	− 1.51375i	−0.653562	−	0.756873i	$-0.726726\pi$
$89$	− 134.723i	− 1.51375i	0.653562	−	0.756873i	$-0.273274\pi$
$90$	0	0
$91$	27.7543	0.304993
$92$	0	0
$93$	0	0
$94$	0	0
$95$	32.8451i	0.345738i
$96$	0	0
$97$	−32.4684	−0.334726	−0.167363	−	0.985895i	$-0.553525\pi$
$97$	−32.4684	−0.334726	−0.167363	+	0.985895i	$0.553525\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	20.3927i	0.201908i	0.994891	+	0.100954i	$0.0321895\pi$
$101$	20.3927i	0.201908i	−0.994891	+	0.100954i	$0.967811\pi$
$102$	0	0
$103$	146.777	1.42502	0.712510	−	0.701662i	$-0.247558\pi$
$103$	146.777	1.42502	0.712510	+	0.701662i	$0.247558\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 103.067i	− 0.963242i	−0.876380	−	0.481621i	$-0.840048\pi$
$107$	− 103.067i	− 0.963242i	0.876380	−	0.481621i	$-0.159952\pi$
$108$	0	0
$109$	141.461	1.29781	0.648906	−	0.760869i	$-0.275227\pi$
$109$	141.461	1.29781	0.648906	+	0.760869i	$0.275227\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 148.052i	− 1.31020i	−0.755544	−	0.655098i	$-0.772627\pi$
$113$	− 148.052i	− 1.31020i	0.755544	−	0.655098i	$-0.227373\pi$
$114$	0	0
$115$	165.813	1.44185
$116$	0	0
$117$	0	0
$118$	0	0
$119$	43.9041i	0.368942i
$120$	0	0
$121$	−315.188	−2.60486
$122$	0	0
$123$	0	0
$124$	0	0
$125$	51.0814i	0.408652i
$126$	0	0
$127$	222.419	1.75133	0.875666	−	0.482918i	$-0.160423\pi$
$127$	222.419	1.75133	0.875666	+	0.482918i	$0.160423\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	171.157i	1.30654i	0.757124	+	0.653271i	$0.226604\pi$
$131$	171.157i	1.30654i	−0.757124	+	0.653271i	$0.773396\pi$
$132$	0	0
$133$	−22.5861	−0.169820
$134$	0	0
$135$	0	0
$136$	0	0
$137$	222.459i	1.62379i	0.583805	+	0.811894i	$0.301563\pi$
$137$	222.459i	1.62379i	−0.583805	+	0.811894i	$0.698437\pi$
$138$	0	0
$139$	−4.22356	−0.0303853	−0.0151927	−	0.999885i	$-0.504836\pi$
$139$	−4.22356	−0.0303853	−0.0151927	+	0.999885i	$0.504836\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 111.867i	− 0.782289i
$144$	0	0
$145$	−259.814	−1.79182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 38.3311i	− 0.257255i	−0.991693	−	0.128628i	$-0.958943\pi$
$149$	− 38.3311i	− 0.257255i	0.991693	−	0.128628i	$-0.0410572\pi$
$150$	0	0
$151$	52.5136	0.347772	0.173886	−	0.984766i	$-0.444368\pi$
$151$	52.5136	0.347772	0.173886	+	0.984766i	$0.444368\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	264.901i	1.70904i
$156$	0	0
$157$	−308.922	−1.96765	−0.983827	−	0.179120i	$-0.942675\pi$
$157$	−308.922	−1.96765	−0.983827	+	0.179120i	$0.942675\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	114.022i	0.708212i
$162$	0	0
$163$	67.5587	0.414471	0.207235	−	0.978291i	$-0.433553\pi$
$163$	67.5587	0.414471	0.207235	+	0.978291i	$0.433553\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	279.364i	1.67284i	0.548092	+	0.836418i	$0.315354\pi$
$167$	279.364i	1.67284i	−0.548092	+	0.836418i	$0.684646\pi$
$168$	0	0
$169$	−140.310	−0.830236
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 75.4624i	− 0.436199i	−0.975927	−	0.218100i	$-0.930014\pi$
$173$	− 75.4624i	− 0.436199i	0.975927	−	0.218100i	$-0.0699857\pi$
$174$	0	0
$175$	−164.667	−0.940952
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 66.5626i	− 0.371858i	−0.982563	−	0.185929i	$-0.940471\pi$
$179$	− 66.5626i	− 0.371858i	0.982563	−	0.185929i	$-0.0595294\pi$
$180$	0	0
$181$	107.560	0.594253	0.297127	−	0.954838i	$-0.403972\pi$
$181$	107.560	0.594253	0.297127	+	0.954838i	$0.403972\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 115.415i	− 0.623864i
$186$	0	0
$187$	176.961	0.946314
$188$	0	0
$189$	0	0
$190$	0	0
$191$	81.2713i	0.425504i	0.977106	+	0.212752i	$0.0682427\pi$
$191$	81.2713i	0.425504i	−0.977106	+	0.212752i	$0.931757\pi$
$192$	0	0
$193$	181.379	0.939787	0.469893	−	0.882723i	$-0.344292\pi$
$193$	181.379	0.939787	0.469893	+	0.882723i	$0.344292\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	170.455i	0.865254i	0.901573	+	0.432627i	$0.142413\pi$
$197$	170.455i	0.865254i	−0.901573	+	0.432627i	$0.857587\pi$
$198$	0	0
$199$	174.952	0.879158	0.439579	−	0.898204i	$-0.355128\pi$
$199$	174.952	0.879158	0.439579	+	0.898204i	$0.355128\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 178.662i	− 0.880109i
$204$	0	0
$205$	149.605	0.729778
$206$	0	0
$207$	0	0
$208$	0	0
$209$	91.0361i	0.435579i
$210$	0	0
$211$	365.354	1.73154	0.865768	−	0.500445i	$-0.166830\pi$
$211$	365.354	1.73154	0.865768	+	0.500445i	$0.166830\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 188.205i	− 0.875370i
$216$	0	0
$217$	−182.160	−0.839448
$218$	0	0
$219$	0	0
$220$	0	0
$221$	45.3844i	0.205359i
$222$	0	0
$223$	303.853	1.36257	0.681284	−	0.732019i	$-0.261422\pi$
$223$	303.853	1.36257	0.681284	+	0.732019i	$0.261422\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	243.551i	1.07291i	0.843928	+	0.536457i	$0.180238\pi$
$227$	243.551i	1.07291i	−0.843928	+	0.536457i	$0.819762\pi$
$228$	0	0
$229$	226.668	0.989816	0.494908	−	0.868945i	$-0.335202\pi$
$229$	226.668	0.989816	0.494908	+	0.868945i	$0.335202\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	382.360i	1.64103i	0.571626	+	0.820514i	$0.306313\pi$
$233$	382.360i	1.64103i	−0.571626	+	0.820514i	$0.693687\pi$
$234$	0	0
$235$	−83.6225	−0.355841
$236$	0	0
$237$	0	0
$238$	0	0
$239$	77.8251i	0.325628i	0.986657	+	0.162814i	$0.0520570\pi$
$239$	77.8251i	0.325628i	−0.986657	+	0.162814i	$0.947943\pi$
$240$	0	0
$241$	248.887	1.03273	0.516363	−	0.856370i	$-0.327286\pi$
$241$	248.887	1.03273	0.516363	+	0.856370i	$0.327286\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	166.911i	0.681271i
$246$	0	0
$247$	−23.3476	−0.0945249
$248$	0	0
$249$	0	0
$250$	0	0
$251$	30.9963i	0.123491i	0.998092	+	0.0617456i	$0.0196667\pi$
$251$	30.9963i	0.123491i	−0.998092	+	0.0617456i	$0.980333\pi$
$252$	0	0
$253$	459.580	1.81652
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 297.917i	− 1.15921i	−0.814897	−	0.579606i	$-0.803207\pi$
$257$	− 297.917i	− 1.15921i	0.814897	−	0.579606i	$-0.196793\pi$
$258$	0	0
$259$	79.3657	0.306431
$260$	0	0
$261$	0	0
$262$	0	0
$263$	204.533i	0.777693i	0.921303	+	0.388846i	$0.127126\pi$
$263$	204.533i	0.777693i	−0.921303	+	0.388846i	$0.872874\pi$
$264$	0	0
$265$	−683.822	−2.58046
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 379.300i	− 1.41004i	−0.709189	−	0.705018i	$-0.750939\pi$
$269$	− 379.300i	− 1.41004i	0.709189	−	0.705018i	$-0.249061\pi$
$270$	0	0
$271$	349.060	1.28804	0.644022	−	0.765007i	$-0.277264\pi$
$271$	349.060	1.28804	0.644022	+	0.765007i	$0.277264\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	663.709i	2.41349i
$276$	0	0
$277$	−95.1319	−0.343436	−0.171718	−	0.985146i	$-0.554932\pi$
$277$	−95.1319	−0.343436	−0.171718	+	0.985146i	$0.554932\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 458.688i	− 1.63234i	−0.577811	−	0.816170i	$-0.696093\pi$
$281$	− 458.688i	− 1.63234i	0.577811	−	0.816170i	$-0.303907\pi$
$282$	0	0
$283$	−397.496	−1.40458	−0.702291	−	0.711890i	$-0.747839\pi$
$283$	−397.496	−1.40458	−0.702291	+	0.711890i	$0.747839\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	102.876i	0.358454i
$288$	0	0
$289$	217.207	0.751582
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 7.12469i	− 0.0243163i	−0.999926	−	0.0121582i	$-0.996130\pi$
$293$	− 7.12469i	− 0.0243163i	0.999926	−	0.0121582i	$-0.00387016\pi$
$294$	0	0
$295$	386.236	1.30927
$296$	0	0
$297$	0	0
$298$	0	0
$299$	117.867i	0.394203i
$300$	0	0
$301$	129.420	0.429966
$302$	0	0
$303$	0	0
$304$	0	0
$305$	359.024i	1.17713i
$306$	0	0
$307$	41.5633	0.135385	0.0676927	−	0.997706i	$-0.478436\pi$
$307$	41.5633	0.135385	0.0676927	+	0.997706i	$0.478436\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	91.9061i	0.295518i	0.989023	+	0.147759i	$0.0472060\pi$
$311$	91.9061i	0.295518i	−0.989023	+	0.147759i	$0.952794\pi$
$312$	0	0
$313$	−259.708	−0.829739	−0.414869	−	0.909881i	$-0.636173\pi$
$313$	−259.708	−0.829739	−0.414869	+	0.909881i	$0.636173\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 344.258i	− 1.08599i	−0.839737	−	0.542994i	$-0.817291\pi$
$317$	− 344.258i	− 1.08599i	0.839737	−	0.542994i	$-0.182709\pi$
$318$	0	0
$319$	−720.119	−2.25743
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 36.9332i	− 0.114344i
$324$	0	0
$325$	−170.219	−0.523750
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 57.5035i	− 0.174783i
$330$	0	0
$331$	−433.674	−1.31019	−0.655097	−	0.755545i	$-0.727372\pi$
$331$	−433.674	−1.31019	−0.655097	+	0.755545i	$0.727372\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 32.7069i	− 0.0976326i
$336$	0	0
$337$	502.810	1.49202	0.746009	−	0.665936i	$-0.231968\pi$
$337$	502.810	1.49202	0.746009	+	0.665936i	$0.231968\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	734.219i	2.15313i
$342$	0	0
$343$	−368.676	−1.07486
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 129.071i	− 0.371963i	−0.982553	−	0.185981i	$-0.940454\pi$
$347$	− 129.071i	− 0.371963i	0.982553	−	0.185981i	$-0.0595464\pi$
$348$	0	0
$349$	464.174	1.33001	0.665006	−	0.746838i	$-0.268429\pi$
$349$	464.174	1.33001	0.665006	+	0.746838i	$0.268429\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 669.313i	− 1.89607i	−0.318167	−	0.948035i	$-0.603067\pi$
$353$	− 669.313i	− 1.89607i	0.318167	−	0.948035i	$-0.396933\pi$
$354$	0	0
$355$	−229.905	−0.647619
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 36.1893i	− 0.100806i	−0.998729	−	0.0504029i	$-0.983949\pi$
$359$	− 36.1893i	− 0.100806i	0.998729	−	0.0504029i	$-0.0160505\pi$
$360$	0	0
$361$	19.0000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 601.525i	− 1.64801i
$366$	0	0
$367$	666.378	1.81574	0.907871	−	0.419249i	$-0.137706\pi$
$367$	666.378	1.81574	0.907871	+	0.419249i	$0.137706\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 470.234i	− 1.26748i
$372$	0	0
$373$	−452.256	−1.21248	−0.606241	−	0.795281i	$-0.707323\pi$
$373$	−452.256	−1.21248	−0.606241	+	0.795281i	$0.707323\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 184.686i	− 0.489883i
$378$	0	0
$379$	−186.492	−0.492062	−0.246031	−	0.969262i	$-0.579127\pi$
$379$	−186.492	−0.492062	−0.246031	+	0.969262i	$0.579127\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	344.258i	0.898846i	0.893319	+	0.449423i	$0.148370\pi$
$383$	344.258i	0.898846i	−0.893319	+	0.449423i	$0.851630\pi$
$384$	0	0
$385$	−815.447	−2.11804
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 236.543i	− 0.608079i	−0.952659	−	0.304039i	$-0.901665\pi$
$389$	− 236.543i	− 0.608079i	0.952659	−	0.304039i	$-0.0983355\pi$
$390$	0	0
$391$	−186.451	−0.476857
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1145.84i	2.90087i
$396$	0	0
$397$	334.506	0.842585	0.421292	−	0.906925i	$-0.361577\pi$
$397$	334.506	0.842585	0.421292	+	0.906925i	$0.361577\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	224.617i	0.560143i	0.959979	+	0.280071i	$0.0903581\pi$
$401$	224.617i	0.560143i	−0.959979	+	0.280071i	$0.909642\pi$
$402$	0	0
$403$	−188.302	−0.467251
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 319.893i	− 0.785978i
$408$	0	0
$409$	−277.610	−0.678753	−0.339377	−	0.940651i	$-0.610216\pi$
$409$	−277.610	−0.678753	−0.339377	+	0.940651i	$0.610216\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	265.597i	0.643092i
$414$	0	0
$415$	170.914	0.411842
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 36.3658i	− 0.0867919i	−0.999058	−	0.0433960i	$-0.986182\pi$
$419$	− 36.3658i	− 0.0867919i	0.999058	−	0.0433960i	$-0.0138177\pi$
$420$	0	0
$421$	349.661	0.830548	0.415274	−	0.909696i	$-0.363686\pi$
$421$	349.661	0.830548	0.415274	+	0.909696i	$0.363686\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 269.266i	− 0.633566i
$426$	0	0
$427$	−246.885	−0.578185
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 259.539i	− 0.602179i	−0.953596	−	0.301090i	$-0.902650\pi$
$431$	− 259.539i	− 0.602179i	0.953596	−	0.301090i	$-0.0973503\pi$
$432$	0	0
$433$	50.2963	0.116158	0.0580788	−	0.998312i	$-0.481503\pi$
$433$	50.2963	0.116158	0.0580788	+	0.998312i	$0.481503\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 95.9183i	− 0.219493i
$438$	0	0
$439$	277.042	0.631075	0.315538	−	0.948913i	$-0.397815\pi$
$439$	277.042	0.631075	0.315538	+	0.948913i	$0.397815\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 217.214i	− 0.490326i	−0.969482	−	0.245163i	$-0.921159\pi$
$443$	− 217.214i	− 0.490326i	0.969482	−	0.245163i	$-0.0788415\pi$
$444$	0	0
$445$	−1015.17	−2.28127
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 776.946i	− 1.73039i	−0.501434	−	0.865196i	$-0.667194\pi$
$449$	− 776.946i	− 1.73039i	0.501434	−	0.865196i	$-0.332806\pi$
$450$	0	0
$451$	414.656	0.919414
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 209.134i	− 0.459636i
$456$	0	0
$457$	−71.9468	−0.157433	−0.0787164	−	0.996897i	$-0.525082\pi$
$457$	−71.9468	−0.157433	−0.0787164	+	0.996897i	$0.525082\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 221.825i	− 0.481183i	−0.970626	−	0.240592i	$-0.922659\pi$
$461$	− 221.825i	− 0.481183i	0.970626	−	0.240592i	$-0.0773414\pi$
$462$	0	0
$463$	39.3588	0.0850081	0.0425041	−	0.999096i	$-0.486466\pi$
$463$	39.3588	0.0850081	0.0425041	+	0.999096i	$0.486466\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 639.280i	− 1.36891i	−0.729056	−	0.684454i	$-0.760041\pi$
$467$	− 639.280i	− 1.36891i	0.729056	−	0.684454i	$-0.239959\pi$
$468$	0	0
$469$	22.4911	0.0479554
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 521.642i	− 1.10284i
$474$	0	0
$475$	138.522	0.291625
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 338.214i	− 0.706084i	−0.935607	−	0.353042i	$-0.885147\pi$
$479$	− 338.214i	− 0.706084i	0.935607	−	0.353042i	$-0.114853\pi$
$480$	0	0
$481$	82.0416	0.170565
$482$	0	0
$483$	0	0
$484$	0	0
$485$	244.656i	0.504445i
$486$	0	0
$487$	24.0212	0.0493248	0.0246624	−	0.999696i	$-0.492149\pi$
$487$	24.0212	0.0493248	0.0246624	+	0.999696i	$0.492149\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 141.161i	− 0.287497i	−0.989614	−	0.143748i	$-0.954084\pi$
$491$	− 141.161i	− 0.287497i	0.989614	−	0.143748i	$-0.0459156\pi$
$492$	0	0
$493$	292.151	0.592599
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 158.095i	− 0.318099i
$498$	0	0
$499$	403.559	0.808736	0.404368	−	0.914596i	$-0.367492\pi$
$499$	403.559	0.808736	0.404368	+	0.914596i	$0.367492\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	535.176i	1.06397i	0.846754	+	0.531985i	$0.178554\pi$
$503$	535.176i	1.06397i	−0.846754	+	0.531985i	$0.821446\pi$
$504$	0	0
$505$	153.663	0.304283
$506$	0	0
$507$	0	0
$508$	0	0
$509$	91.6004i	0.179962i	0.995944	+	0.0899808i	$0.0286806\pi$
$509$	91.6004i	0.179962i	−0.995944	+	0.0899808i	$0.971319\pi$
$510$	0	0
$511$	413.641	0.809475
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1105.99i	− 2.14756i
$516$	0	0
$517$	−231.775	−0.448307
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 160.806i	− 0.308649i	−0.988020	−	0.154324i	$-0.950680\pi$
$521$	− 160.806i	− 0.308649i	0.988020	−	0.154324i	$-0.0493201\pi$
$522$	0	0
$523$	52.2686	0.0999399	0.0499700	−	0.998751i	$-0.484087\pi$
$523$	52.2686	0.0999399	0.0499700	+	0.998751i	$0.484087\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 297.871i	− 0.565221i
$528$	0	0
$529$	44.7728	0.0846367
$530$	0	0
$531$	0	0
$532$	0	0
$533$	106.345i	0.199522i
$534$	0	0
$535$	−776.628	−1.45164
$536$	0	0
$537$	0	0
$538$	0	0
$539$	462.624i	0.858301i
$540$	0	0
$541$	551.075	1.01862	0.509311	−	0.860582i	$-0.329900\pi$
$541$	551.075	1.01862	0.509311	+	0.860582i	$0.329900\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1065.94i	− 1.95585i
$546$	0	0
$547$	−135.636	−0.247963	−0.123981	−	0.992285i	$-0.539566\pi$
$547$	−135.636	−0.247963	−0.123981	+	0.992285i	$0.539566\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	150.295i	0.272768i
$552$	0	0
$553$	−787.946	−1.42486
$554$	0	0
$555$	0	0
$556$	0	0
$557$	652.252i	1.17101i	0.810669	+	0.585504i	$0.199104\pi$
$557$	652.252i	1.17101i	−0.810669	+	0.585504i	$0.800896\pi$
$558$	0	0
$559$	133.783	0.239326
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 322.081i	− 0.572080i	−0.958218	−	0.286040i	$-0.907661\pi$
$563$	− 322.081i	− 0.572080i	0.958218	−	0.286040i	$-0.0923390\pi$
$564$	0	0
$565$	−1115.60	−1.97451
$566$	0	0
$567$	0	0
$568$	0	0
$569$	528.919i	0.929559i	0.885426	+	0.464780i	$0.153866\pi$
$569$	528.919i	0.929559i	−0.885426	+	0.464780i	$0.846134\pi$
$570$	0	0
$571$	−536.810	−0.940123	−0.470062	−	0.882634i	$-0.655768\pi$
$571$	−536.810	−0.940123	−0.470062	+	0.882634i	$0.655768\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 699.303i	− 1.21618i
$576$	0	0
$577$	−560.381	−0.971197	−0.485599	−	0.874182i	$-0.661398\pi$
$577$	−560.381	−0.971197	−0.485599	+	0.874182i	$0.661398\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	117.530i	0.202289i
$582$	0	0
$583$	−1895.34	−3.25100
$584$	0	0
$585$	0	0
$586$	0	0
$587$	479.579i	0.817000i	0.912758	+	0.408500i	$0.133948\pi$
$587$	479.579i	0.817000i	−0.912758	+	0.408500i	$0.866052\pi$
$588$	0	0
$589$	153.238	0.260166
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 966.804i	− 1.63036i	−0.579207	−	0.815181i	$-0.696638\pi$
$593$	− 966.804i	− 1.63036i	0.579207	−	0.815181i	$-0.303362\pi$
$594$	0	0
$595$	330.825	0.556009
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1144.37i	1.91047i	0.295855	+	0.955233i	$0.404395\pi$
$599$	1144.37i	1.91047i	−0.295855	+	0.955233i	$0.595605\pi$
$600$	0	0
$601$	824.038	1.37111	0.685556	−	0.728020i	$-0.259559\pi$
$601$	824.038	1.37111	0.685556	+	0.728020i	$0.259559\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2375.00i	3.92562i
$606$	0	0
$607$	833.071	1.37244	0.686220	−	0.727394i	$-0.259269\pi$
$607$	833.071	1.37244	0.686220	+	0.727394i	$0.259269\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 59.4423i	− 0.0972869i
$612$	0	0
$613$	−527.455	−0.860449	−0.430225	−	0.902722i	$-0.641566\pi$
$613$	−527.455	−0.860449	−0.430225	+	0.902722i	$0.641566\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	421.181i	0.682628i	0.939949	+	0.341314i	$0.110872\pi$
$617$	421.181i	0.682628i	−0.939949	+	0.341314i	$0.889128\pi$
$618$	0	0
$619$	−855.134	−1.38148	−0.690738	−	0.723105i	$-0.742714\pi$
$619$	−855.134	−1.38148	−0.690738	+	0.723105i	$0.742714\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 698.084i	− 1.12052i
$624$	0	0
$625$	−409.568	−0.655309
$626$	0	0
$627$	0	0
$628$	0	0
$629$	129.780i	0.206328i
$630$	0	0
$631$	−828.566	−1.31310	−0.656550	−	0.754283i	$-0.727985\pi$
$631$	−828.566	−1.31310	−0.656550	+	0.754283i	$0.727985\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1675.97i	− 2.63932i
$636$	0	0
$637$	−118.647	−0.186260
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1279.90i	1.99672i	0.0572528	+	0.998360i	$0.481766\pi$
$641$	1279.90i	1.99672i	−0.0572528	+	0.998360i	$0.518234\pi$
$642$	0	0
$643$	−302.928	−0.471116	−0.235558	−	0.971860i	$-0.575692\pi$
$643$	−302.928	−0.471116	−0.235558	+	0.971860i	$0.575692\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	330.443i	0.510731i	0.966845	+	0.255366i	$0.0821959\pi$
$647$	330.443i	0.510731i	−0.966845	+	0.255366i	$0.917804\pi$
$648$	0	0
$649$	1070.52	1.64949
$650$	0	0
$651$	0	0
$652$	0	0
$653$	150.460i	0.230413i	0.993342	+	0.115207i	$0.0367530\pi$
$653$	150.460i	0.230413i	−0.993342	+	0.115207i	$0.963247\pi$
$654$	0	0
$655$	1289.70	1.96901
$656$	0	0
$657$	0	0
$658$	0	0
$659$	476.960i	0.723764i	0.932224	+	0.361882i	$0.117866\pi$
$659$	476.960i	0.723764i	−0.932224	+	0.361882i	$0.882134\pi$
$660$	0	0
$661$	1077.26	1.62975	0.814873	−	0.579639i	$-0.196806\pi$
$661$	1077.26	1.62975	0.814873	+	0.579639i	$0.196806\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	170.191i	0.255926i
$666$	0	0
$667$	758.739	1.13754
$668$	0	0
$669$	0	0
$670$	0	0
$671$	995.100i	1.48301i
$672$	0	0
$673$	258.876	0.384660	0.192330	−	0.981330i	$-0.438396\pi$
$673$	258.876	0.384660	0.192330	+	0.981330i	$0.438396\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	428.473i	0.632899i	0.948609	+	0.316450i	$0.102491\pi$
$677$	428.473i	0.632899i	−0.948609	+	0.316450i	$0.897509\pi$
$678$	0	0
$679$	−168.239	−0.247774
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 980.386i	− 1.43541i	−0.696347	−	0.717706i	$-0.745192\pi$
$683$	− 980.386i	− 1.43541i	0.696347	−	0.717706i	$-0.254808\pi$
$684$	0	0
$685$	1676.27	2.44711
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 486.089i	− 0.705499i
$690$	0	0
$691$	1193.72	1.72753	0.863766	−	0.503894i	$-0.168100\pi$
$691$	1193.72	1.72753	0.863766	+	0.503894i	$0.168100\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	31.8253i	0.0457918i
$696$	0	0
$697$	−168.225	−0.241356
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 885.039i	− 1.26254i	−0.775564	−	0.631269i	$-0.782534\pi$
$701$	− 885.039i	− 1.26254i	0.775564	−	0.631269i	$-0.217466\pi$
$702$	0	0
$703$	−66.7644	−0.0949706
$704$	0	0
$705$	0	0
$706$	0	0
$707$	105.667i	0.149458i
$708$	0	0
$709$	−1128.51	−1.59170	−0.795850	−	0.605494i	$-0.792975\pi$
$709$	−1128.51	−1.59170	−0.795850	+	0.605494i	$0.792975\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 773.594i	− 1.08499i
$714$	0	0
$715$	−842.941	−1.17894
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 383.648i	− 0.533586i	−0.963754	−	0.266793i	$-0.914036\pi$
$719$	− 383.648i	− 0.533586i	0.963754	−	0.266793i	$-0.0859640\pi$
$720$	0	0
$721$	760.541	1.05484
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1095.74i	1.51137i
$726$	0	0
$727$	827.848	1.13872	0.569359	−	0.822089i	$-0.307191\pi$
$727$	827.848	1.13872	0.569359	+	0.822089i	$0.307191\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	211.629i	0.289507i
$732$	0	0
$733$	−339.718	−0.463462	−0.231731	−	0.972780i	$-0.574439\pi$
$733$	−339.718	−0.463462	−0.231731	+	0.972780i	$0.574439\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 90.6530i	− 0.123003i
$738$	0	0
$739$	508.257	0.687763	0.343881	−	0.939013i	$-0.388258\pi$
$739$	508.257	0.687763	0.343881	+	0.939013i	$0.388258\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1102.51i	1.48386i	0.670479	+	0.741929i	$0.266089\pi$
$743$	1102.51i	1.48386i	−0.670479	+	0.741929i	$0.733911\pi$
$744$	0	0
$745$	−288.832	−0.387694
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 534.052i	− 0.713020i
$750$	0	0
$751$	−661.082	−0.880269	−0.440134	−	0.897932i	$-0.645069\pi$
$751$	−661.082	−0.880269	−0.440134	+	0.897932i	$0.645069\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 395.700i	− 0.524106i
$756$	0	0
$757$	882.880	1.16629	0.583144	−	0.812369i	$-0.301822\pi$
$757$	882.880	1.16629	0.583144	+	0.812369i	$0.301822\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 18.0466i	− 0.0237143i	−0.999930	−	0.0118571i	$-0.996226\pi$
$761$	− 18.0466i	− 0.0237143i	0.999930	−	0.0118571i	$-0.00377433\pi$
$762$	0	0
$763$	732.998	0.960679
$764$	0	0
$765$	0	0
$766$	0	0
$767$	274.552i	0.357956i
$768$	0	0
$769$	1302.05	1.69317	0.846584	−	0.532255i	$-0.178655\pi$
$769$	1302.05	1.69317	0.846584	+	0.532255i	$0.178655\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	924.196i	1.19560i	0.801647	+	0.597798i	$0.203958\pi$
$773$	924.196i	1.19560i	−0.801647	+	0.597798i	$0.796042\pi$
$774$	0	0
$775$	1117.20	1.44154
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 86.5421i	− 0.111094i
$780$	0	0
$781$	−637.222	−0.815905
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2327.78i	2.96533i
$786$	0	0
$787$	−281.015	−0.357071	−0.178535	−	0.983933i	$-0.557136\pi$
$787$	−281.015	−0.357071	−0.178535	+	0.983933i	$0.557136\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 767.148i	− 0.969846i
$792$	0	0
$793$	−255.209	−0.321827
$794$	0	0
$795$	0	0
$796$	0	0
$797$	614.820i	0.771418i	0.922620	+	0.385709i	$0.126043\pi$
$797$	614.820i	0.771418i	−0.922620	+	0.385709i	$0.873957\pi$
$798$	0	0
$799$	94.0306	0.117685
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1667.23i	− 2.07626i
$804$	0	0
$805$	859.178	1.06730
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1378.73i	− 1.70424i	−0.523348	−	0.852119i	$-0.675317\pi$
$809$	− 1378.73i	− 1.70424i	0.523348	−	0.852119i	$-0.324683\pi$
$810$	0	0
$811$	−1161.09	−1.43167	−0.715837	−	0.698268i	$-0.753954\pi$
$811$	−1161.09	−1.43167	−0.715837	+	0.698268i	$0.753954\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 509.067i	− 0.624623i
$816$	0	0
$817$	−108.871	−0.133257
$818$	0	0
$819$	0	0
$820$	0	0
$821$	548.846i	0.668510i	0.942483	+	0.334255i	$0.108485\pi$
$821$	548.846i	0.668510i	−0.942483	+	0.334255i	$0.891515\pi$
$822$	0	0
$823$	−1250.98	−1.52002	−0.760012	−	0.649909i	$-0.774807\pi$
$823$	−1250.98	−1.52002	−0.760012	+	0.649909i	$0.774807\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1327.93i	1.60572i	0.596165	+	0.802862i	$0.296691\pi$
$827$	1327.93i	1.60572i	−0.596165	+	0.802862i	$0.703309\pi$
$828$	0	0
$829$	−1015.57	−1.22506	−0.612528	−	0.790449i	$-0.709847\pi$
$829$	−1015.57	−1.22506	−0.612528	+	0.790449i	$0.709847\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 187.686i	− 0.225313i
$834$	0	0
$835$	2105.06	2.52103
$836$	0	0
$837$	0	0
$838$	0	0
$839$	950.704i	1.13314i	0.824014	+	0.566570i	$0.191730\pi$
$839$	950.704i	1.13314i	−0.824014	+	0.566570i	$0.808270\pi$
$840$	0	0
$841$	−347.873	−0.413642
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1057.26i	1.25120i
$846$	0	0
$847$	−1633.18	−1.92819
$848$	0	0
$849$	0	0
$850$	0	0
$851$	337.049i	0.396062i
$852$	0	0
$853$	1487.29	1.74359	0.871797	−	0.489868i	$-0.162955\pi$
$853$	1487.29	1.74359	0.871797	+	0.489868i	$0.162955\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.01645i	0.0105209i	0.999986	+	0.00526047i	$0.00167447\pi$
$857$	9.01645i	0.0105209i	−0.999986	+	0.00526047i	$0.998326\pi$
$858$	0	0
$859$	433.625	0.504802	0.252401	−	0.967623i	$-0.418780\pi$
$859$	433.625	0.504802	0.252401	+	0.967623i	$0.418780\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1301.22i	− 1.50778i	−0.656998	−	0.753892i	$-0.728174\pi$
$863$	− 1301.22i	− 1.50778i	0.656998	−	0.753892i	$-0.271826\pi$
$864$	0	0
$865$	−568.624	−0.657368
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3175.91i	3.65468i
$870$	0	0
$871$	23.2494	0.0266928
$872$	0	0
$873$	0	0
$874$	0	0
$875$	264.684i	0.302496i
$876$	0	0
$877$	−969.321	−1.10527	−0.552634	−	0.833424i	$-0.686377\pi$
$877$	−969.321	−1.10527	−0.552634	+	0.833424i	$0.686377\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	958.937i	1.08846i	0.838935	+	0.544232i	$0.183179\pi$
$881$	958.937i	1.08846i	−0.838935	+	0.544232i	$0.816821\pi$
$882$	0	0
$883$	−124.679	−0.141200	−0.0705999	−	0.997505i	$-0.522491\pi$
$883$	−124.679	−0.141200	−0.0705999	+	0.997505i	$0.522491\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1699.20i	− 1.91568i	−0.287309	−	0.957838i	$-0.592761\pi$
$887$	− 1699.20i	− 1.91568i	0.287309	−	0.957838i	$-0.407239\pi$
$888$	0	0
$889$	1152.49	1.29639
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48.3733i	0.0541695i
$894$	0	0
$895$	−501.561	−0.560404
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1212.15i	1.34833i
$900$	0	0
$901$	768.935	0.853423
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 810.484i	− 0.895562i
$906$	0	0
$907$	682.473	0.752450	0.376225	−	0.926528i	$-0.377222\pi$
$907$	682.473	0.752450	0.376225	+	0.926528i	$0.377222\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 162.119i	− 0.177957i	−0.996034	−	0.0889785i	$-0.971640\pi$
$911$	− 162.119i	− 0.177957i	0.996034	−	0.0889785i	$-0.0283602\pi$
$912$	0	0
$913$	473.719	0.518860
$914$	0	0
$915$	0	0
$916$	0	0
$917$	886.868i	0.967141i
$918$	0	0
$919$	1172.42	1.27576	0.637880	−	0.770135i	$-0.279811\pi$
$919$	1172.42	1.27576	0.637880	+	0.770135i	$0.279811\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 163.426i	− 0.177059i
$924$	0	0
$925$	−486.753	−0.526220
$926$	0	0
$927$	0	0
$928$	0	0
$929$	129.000i	0.138858i	0.997587	+	0.0694292i	$0.0221178\pi$
$929$	129.000i	0.138858i	−0.997587	+	0.0694292i	$0.977882\pi$
$930$	0	0
$931$	96.5536	0.103710
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1333.43i	− 1.42613i
$936$	0	0
$937$	−54.2758	−0.0579251	−0.0289625	−	0.999580i	$-0.509220\pi$
$937$	−54.2758	−0.0579251	−0.0289625	+	0.999580i	$0.509220\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	726.918i	0.772495i	0.922395	+	0.386247i	$0.126229\pi$
$941$	726.918i	0.772495i	−0.922395	+	0.386247i	$0.873771\pi$
$942$	0	0
$943$	−436.893	−0.463301
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1859.88i	1.96397i	0.188969	+	0.981983i	$0.439485\pi$
$947$	1859.88i	1.96397i	−0.188969	+	0.981983i	$0.560515\pi$
$948$	0	0
$949$	427.588	0.450567
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1412.84i	− 1.48252i	−0.671217	−	0.741261i	$-0.734228\pi$
$953$	− 1412.84i	− 1.48252i	0.671217	−	0.741261i	$-0.265772\pi$
$954$	0	0
$955$	612.395	0.641251
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1152.70i	1.20198i
$960$	0	0
$961$	274.883	0.286039
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1366.72i	− 1.41629i
$966$	0	0
$967$	−1127.14	−1.16561	−0.582804	−	0.812613i	$-0.698045\pi$
$967$	−1127.14	−1.16561	−0.582804	+	0.812613i	$0.698045\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1224.06i	1.26062i	0.776342	+	0.630311i	$0.217073\pi$
$971$	1224.06i	1.26062i	−0.776342	+	0.630311i	$0.782927\pi$
$972$	0	0
$973$	−21.8848	−0.0224921
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 873.223i	− 0.893780i	−0.894589	−	0.446890i	$-0.852532\pi$
$977$	− 873.223i	− 0.893780i	0.894589	−	0.446890i	$-0.147468\pi$
$978$	0	0
$979$	−2813.71	−2.87407
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1330.86i	1.35388i	0.736038	+	0.676940i	$0.236695\pi$
$983$	1330.86i	1.35388i	−0.736038	+	0.676940i	$0.763305\pi$
$984$	0	0
$985$	1284.41	1.30397
$986$	0	0
$987$	0	0
$988$	0	0
$989$	549.618i	0.555731i
$990$	0	0
$991$	−265.348	−0.267758	−0.133879	−	0.990998i	$-0.542743\pi$
$991$	−265.348	−0.267758	−0.133879	+	0.990998i	$0.542743\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1318.30i	− 1.32492i
$996$	0	0
$997$	1179.05	1.18260	0.591298	−	0.806453i	$-0.298616\pi$
$997$	1179.05	1.18260	0.591298	+	0.806453i	$0.298616\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.e.a.305.1	✓	12
3.2	odd	2	inner	684.3.e.a.305.12	yes	12
4.3	odd	2		2736.3.h.c.305.1		12
12.11	even	2		2736.3.h.c.305.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.e.a.305.1	✓	12	1.1	even	1	trivial
684.3.e.a.305.12	yes	12	3.2	odd	2	inner
2736.3.h.c.305.1		12	4.3	odd	2
2736.3.h.c.305.12		12	12.11	even	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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.53519i q^{5} +5.18161 q^{7} +O(q^{10})\)	\(q-7.53519i q^{5} +5.18161 q^{7} -20.8851i q^{11} +5.35632 q^{13} +8.47306i q^{17} -4.35890 q^{19} +22.0052i q^{23} -31.7791 q^{25} -34.4800i q^{29} -35.1551 q^{31} -39.0444i q^{35} +15.3168 q^{37} +19.8541i q^{41} +24.9768 q^{43} -11.0976i q^{47} -22.1509 q^{49} -90.7505i q^{53} -157.373 q^{55} +51.2576i q^{59} -47.6464 q^{61} -40.3609i q^{65} +4.34056 q^{67} -30.5108i q^{71} +79.8288 q^{73} -108.219i q^{77} -152.066 q^{79} +22.6822i q^{83} +63.8461 q^{85} -134.723i q^{89} +27.7543 q^{91} +32.8451i q^{95} -32.4684 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{7}+O(q^{10}) \) 12 * q + 16 * q^7	\( 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97}+O(q^{100}) \) 12 * q + 16 * q^7 - 16 * q^13 - 12 * q^25 - 40 * q^31 - 32 * q^37 + 92 * q^43 - 84 * q^55 - 48 * q^61 - 88 * q^67 + 148 * q^73 - 56 * q^79 + 228 * q^85 - 8 * q^91 + 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more