LMFDB - Embedded newform 1176.3.f.d.97.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,3,Mod(97,1176)] 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("1176.97"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1176.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$32.0436790888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 66 x^{14} + 136 x^{13} + 3441 x^{12} - 4512 x^{11} - 100322 x^{10} + \cdots + 13841287201 $ x^16 - 4x^15 - 66x^14 + 136x^13 + 3441x^12 - 4512x^11 - 100322x^10 + 24868x^9 + 2325060x^8 + 260012x^7 - 34273754x^6 - 26559312x^5 + 402103713x^4 + 286444088x^3 - 2633783418x^2 - 3060285788*x + 13841287201
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.5
Root			$4.67475 + 2.12162i$ of defining polynomial
Character	$\chi$	$=$	1176.97
Dual form			1176.3.f.d.97.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +5.23547i q^{5} -3.00000 q^{9} +2.22851 q^{11} +5.78000i q^{13} +9.06809 q^{15} -13.2146i q^{17} +16.6237i q^{19} +0.256737 q^{23} -2.41011 q^{25} +5.19615i q^{27} -5.16218 q^{29} -5.46230i q^{31} -3.85990i q^{33} -60.2999 q^{37} +10.0113 q^{39} +36.4900i q^{41} +38.4669 q^{43} -15.7064i q^{45} +93.4861i q^{47} -22.8883 q^{51} -91.8568 q^{53} +11.6673i q^{55} +28.7932 q^{57} +49.6804i q^{59} -91.2350i q^{61} -30.2610 q^{65} -118.777 q^{67} -0.444682i q^{69} +3.59274 q^{71} +69.1547i q^{73} +4.17444i q^{75} -6.29178 q^{79} +9.00000 q^{81} +116.377i q^{83} +69.1844 q^{85} +8.94116i q^{87} -7.56417i q^{89} -9.46099 q^{93} -87.0330 q^{95} +37.8133i q^{97} -6.68554 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} - 112 q^{23} - 256 q^{25} + 112 q^{29} - 128 q^{37} - 112 q^{43} + 112 q^{53} + 192 q^{57} - 112 q^{65} - 224 q^{67} + 160 q^{71} + 304 q^{79} + 144 q^{81} - 416 q^{85} - 192 q^{93} + 912 q^{95}+O(q^{100}) $ 16 * q - 48 * q^9 - 112 * q^23 - 256 * q^25 + 112 * q^29 - 128 * q^37 - 112 * q^43 + 112 * q^53 + 192 * q^57 - 112 * q^65 - 224 * q^67 + 160 * q^71 + 304 * q^79 + 144 * q^81 - 416 * q^85 - 192 * q^93 + 912 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	5.23547i	1.04709i	0.851997	+	0.523547i	$0.175391\pi$
$5$	5.23547i	1.04709i	−0.851997	+	0.523547i	$0.824609\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	2.22851	0.202592	0.101296	−	0.994856i	$-0.467701\pi$
$11$	2.22851	0.202592	0.101296	+	0.994856i	$0.467701\pi$
$12$	0	0
$13$	5.78000i	0.444615i	0.974977	+	0.222308i	$0.0713589\pi$
$13$	5.78000i	0.444615i	−0.974977	+	0.222308i	$0.928641\pi$
$14$	0	0
$15$	9.06809	0.604540
$16$	0	0
$17$	− 13.2146i	− 0.777327i	−0.921380	−	0.388663i	$-0.872937\pi$
$17$	− 13.2146i	− 0.777327i	0.921380	−	0.388663i	$-0.127063\pi$
$18$	0	0
$19$	16.6237i	0.874933i	0.899235	+	0.437467i	$0.144124\pi$
$19$	16.6237i	0.874933i	−0.899235	+	0.437467i	$0.855876\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.256737	0.0111625	0.00558124	−	0.999984i	$-0.498223\pi$
$23$	0.256737	0.0111625	0.00558124	+	0.999984i	$0.498223\pi$
$24$	0	0
$25$	−2.41011	−0.0964046
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−5.16218	−0.178006	−0.0890031	−	0.996031i	$-0.528368\pi$
$29$	−5.16218	−0.178006	−0.0890031	+	0.996031i	$0.528368\pi$
$30$	0	0
$31$	− 5.46230i	− 0.176203i	−0.996111	−	0.0881017i	$-0.971920\pi$
$31$	− 5.46230i	− 0.176203i	0.996111	−	0.0881017i	$-0.0280800\pi$
$32$	0	0
$33$	− 3.85990i	− 0.116967i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−60.2999	−1.62973	−0.814864	−	0.579653i	$-0.803188\pi$
$37$	−60.2999	−1.62973	−0.814864	+	0.579653i	$0.803188\pi$
$38$	0	0
$39$	10.0113	0.256699
$40$	0	0
$41$	36.4900i	0.890001i	0.895530	+	0.445001i	$0.146797\pi$
$41$	36.4900i	0.890001i	−0.895530	+	0.445001i	$0.853203\pi$
$42$	0	0
$43$	38.4669	0.894578	0.447289	−	0.894389i	$-0.352389\pi$
$43$	38.4669	0.894578	0.447289	+	0.894389i	$0.352389\pi$
$44$	0	0
$45$	− 15.7064i	− 0.349031i
$46$	0	0
$47$	93.4861i	1.98907i	0.104421	+	0.994533i	$0.466701\pi$
$47$	93.4861i	1.98907i	−0.104421	+	0.994533i	$0.533299\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−22.8883	−0.448790
$52$	0	0
$53$	−91.8568	−1.73315	−0.866573	−	0.499050i	$-0.833682\pi$
$53$	−91.8568	−1.73315	−0.866573	+	0.499050i	$0.833682\pi$
$54$	0	0
$55$	11.6673i	0.212133i
$56$	0	0
$57$	28.7932	0.505143
$58$	0	0
$59$	49.6804i	0.842040i	0.907051	+	0.421020i	$0.138328\pi$
$59$	49.6804i	0.842040i	−0.907051	+	0.421020i	$0.861672\pi$
$60$	0	0
$61$	− 91.2350i	− 1.49566i	−0.663892	−	0.747828i	$-0.731097\pi$
$61$	− 91.2350i	− 1.49566i	0.663892	−	0.747828i	$-0.268903\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−30.2610	−0.465554
$66$	0	0
$67$	−118.777	−1.77279	−0.886397	−	0.462926i	$-0.846800\pi$
$67$	−118.777	−1.77279	−0.886397	+	0.462926i	$0.846800\pi$
$68$	0	0
$69$	− 0.444682i	− 0.00644466i
$70$	0	0
$71$	3.59274	0.0506020	0.0253010	−	0.999680i	$-0.491946\pi$
$71$	3.59274	0.0506020	0.0253010	+	0.999680i	$0.491946\pi$
$72$	0	0
$73$	69.1547i	0.947325i	0.880706	+	0.473662i	$0.157068\pi$
$73$	69.1547i	0.947325i	−0.880706	+	0.473662i	$0.842932\pi$
$74$	0	0
$75$	4.17444i	0.0556592i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.29178	−0.0796427	−0.0398214	−	0.999207i	$-0.512679\pi$
$79$	−6.29178	−0.0796427	−0.0398214	+	0.999207i	$0.512679\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	116.377i	1.40213i	0.713096	+	0.701067i	$0.247292\pi$
$83$	116.377i	1.40213i	−0.713096	+	0.701067i	$0.752708\pi$
$84$	0	0
$85$	69.1844	0.813934
$86$	0	0
$87$	8.94116i	0.102772i
$88$	0	0
$89$	− 7.56417i	− 0.0849906i	−0.999097	−	0.0424953i	$-0.986469\pi$
$89$	− 7.56417i	− 0.0849906i	0.999097	−	0.0424953i	$-0.0135308\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.46099	−0.101731
$94$	0	0
$95$	−87.0330	−0.916137
$96$	0	0
$97$	37.8133i	0.389828i	0.980820	+	0.194914i	$0.0624428\pi$
$97$	37.8133i	0.389828i	−0.980820	+	0.194914i	$0.937557\pi$
$98$	0	0
$99$	−6.68554	−0.0675307
$100$	0	0
$101$	− 0.146269i	− 0.00144821i	−1.00000		0.000724105i	$-0.999770\pi$
$101$	− 0.146269i	− 0.00144821i	1.00000		0.000724105i	$-0.000230490\pi$
$102$	0	0
$103$	− 18.5291i	− 0.179895i	−0.995947	−	0.0899473i	$-0.971330\pi$
$103$	− 18.5291i	− 0.179895i	0.995947	−	0.0899473i	$-0.0286699\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	54.6749	0.510981	0.255490	−	0.966812i	$-0.417763\pi$
$107$	54.6749	0.510981	0.255490	+	0.966812i	$0.417763\pi$
$108$	0	0
$109$	−133.619	−1.22587	−0.612933	−	0.790135i	$-0.710010\pi$
$109$	−133.619	−1.22587	−0.612933	+	0.790135i	$0.710010\pi$
$110$	0	0
$111$	104.443i	0.940923i
$112$	0	0
$113$	133.049	1.17742	0.588711	−	0.808343i	$-0.299635\pi$
$113$	133.049	1.17742	0.588711	+	0.808343i	$0.299635\pi$
$114$	0	0
$115$	1.34414i	0.0116882i
$116$	0	0
$117$	− 17.3400i	− 0.148205i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−116.034	−0.958956
$122$	0	0
$123$	63.2026	0.513842
$124$	0	0
$125$	118.269i	0.946149i
$126$	0	0
$127$	−32.5687	−0.256446	−0.128223	−	0.991745i	$-0.540927\pi$
$127$	−32.5687	−0.256446	−0.128223	+	0.991745i	$0.540927\pi$
$128$	0	0
$129$	− 66.6266i	− 0.516485i
$130$	0	0
$131$	146.297i	1.11677i	0.829581	+	0.558386i	$0.188579\pi$
$131$	146.297i	1.11677i	−0.829581	+	0.558386i	$0.811421\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−27.2043	−0.201513
$136$	0	0
$137$	169.811	1.23950	0.619748	−	0.784801i	$-0.287235\pi$
$137$	169.811	1.23950	0.619748	+	0.784801i	$0.287235\pi$
$138$	0	0
$139$	170.309i	1.22525i	0.790375	+	0.612623i	$0.209886\pi$
$139$	170.309i	1.22525i	−0.790375	+	0.612623i	$0.790114\pi$
$140$	0	0
$141$	161.923	1.14839
$142$	0	0
$143$	12.8808i	0.0900755i
$144$	0	0
$145$	− 27.0264i	− 0.186389i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−86.7765	−0.582392	−0.291196	−	0.956663i	$-0.594053\pi$
$149$	−86.7765	−0.582392	−0.291196	+	0.956663i	$0.594053\pi$
$150$	0	0
$151$	−233.764	−1.54811	−0.774053	−	0.633121i	$-0.781774\pi$
$151$	−233.764	−1.54811	−0.774053	+	0.633121i	$0.781774\pi$
$152$	0	0
$153$	39.6437i	0.259109i
$154$	0	0
$155$	28.5977	0.184501
$156$	0	0
$157$	274.301i	1.74714i	0.486696	+	0.873571i	$0.338202\pi$
$157$	274.301i	1.74714i	−0.486696	+	0.873571i	$0.661798\pi$
$158$	0	0
$159$	159.101i	1.00063i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	161.450	0.990489	0.495245	−	0.868754i	$-0.335078\pi$
$163$	161.450	0.990489	0.495245	+	0.868754i	$0.335078\pi$
$164$	0	0
$165$	20.2084	0.122475
$166$	0	0
$167$	115.083i	0.689122i	0.938764	+	0.344561i	$0.111972\pi$
$167$	115.083i	0.689122i	−0.938764	+	0.344561i	$0.888028\pi$
$168$	0	0
$169$	135.592	0.802317
$170$	0	0
$171$	− 49.8712i	− 0.291644i
$172$	0	0
$173$	− 142.849i	− 0.825719i	−0.910795	−	0.412860i	$-0.864530\pi$
$173$	− 142.849i	− 0.825719i	0.910795	−	0.412860i	$-0.135470\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	86.0489	0.486152
$178$	0	0
$179$	118.449	0.661724	0.330862	−	0.943679i	$-0.392661\pi$
$179$	118.449	0.661724	0.330862	+	0.943679i	$0.392661\pi$
$180$	0	0
$181$	− 34.3439i	− 0.189745i	−0.995489	−	0.0948727i	$-0.969756\pi$
$181$	− 34.3439i	− 0.189745i	0.995489	−	0.0948727i	$-0.0302444\pi$
$182$	0	0
$183$	−158.024	−0.863517
$184$	0	0
$185$	− 315.698i	− 1.70648i
$186$	0	0
$187$	− 29.4488i	− 0.157480i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.2598	−0.0851298	−0.0425649	−	0.999094i	$-0.513553\pi$
$191$	−16.2598	−0.0851298	−0.0425649	+	0.999094i	$0.513553\pi$
$192$	0	0
$193$	126.120	0.653470	0.326735	−	0.945116i	$-0.394052\pi$
$193$	126.120	0.653470	0.326735	+	0.945116i	$0.394052\pi$
$194$	0	0
$195$	52.4136i	0.268788i
$196$	0	0
$197$	232.932	1.18239	0.591197	−	0.806527i	$-0.298656\pi$
$197$	232.932	1.18239	0.591197	+	0.806527i	$0.298656\pi$
$198$	0	0
$199$	31.3046i	0.157310i	0.996902	+	0.0786548i	$0.0250625\pi$
$199$	31.3046i	0.157310i	−0.996902	+	0.0786548i	$0.974938\pi$
$200$	0	0
$201$	205.728i	1.02352i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−191.042	−0.931914
$206$	0	0
$207$	−0.770211	−0.00372083
$208$	0	0
$209$	37.0462i	0.177255i
$210$	0	0
$211$	−384.129	−1.82052	−0.910259	−	0.414040i	$-0.864117\pi$
$211$	−384.129	−1.82052	−0.910259	+	0.414040i	$0.864117\pi$
$212$	0	0
$213$	− 6.22281i	− 0.0292151i
$214$	0	0
$215$	201.392i	0.936707i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	119.779	0.546938
$220$	0	0
$221$	76.3801	0.345611
$222$	0	0
$223$	− 305.758i	− 1.37111i	−0.728021	−	0.685555i	$-0.759560\pi$
$223$	− 305.758i	− 1.37111i	0.728021	−	0.685555i	$-0.240440\pi$
$224$	0	0
$225$	7.23034	0.0321349
$226$	0	0
$227$	69.1219i	0.304502i	0.988342	+	0.152251i	$0.0486522\pi$
$227$	69.1219i	0.304502i	−0.988342	+	0.152251i	$0.951348\pi$
$228$	0	0
$229$	− 227.306i	− 0.992602i	−0.868151	−	0.496301i	$-0.834691\pi$
$229$	− 227.306i	− 0.992602i	0.868151	−	0.496301i	$-0.165309\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−123.501	−0.530048	−0.265024	−	0.964242i	$-0.585380\pi$
$233$	−123.501	−0.530048	−0.265024	+	0.964242i	$0.585380\pi$
$234$	0	0
$235$	−489.443	−2.08274
$236$	0	0
$237$	10.8977i	0.0459818i
$238$	0	0
$239$	61.0715	0.255529	0.127765	−	0.991805i	$-0.459220\pi$
$239$	61.0715	0.255529	0.127765	+	0.991805i	$0.459220\pi$
$240$	0	0
$241$	− 20.0040i	− 0.0830042i	−0.999138	−	0.0415021i	$-0.986786\pi$
$241$	− 20.0040i	− 0.0830042i	0.999138	−	0.0415021i	$-0.0132143\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−96.0852	−0.389009
$248$	0	0
$249$	201.571	0.809522
$250$	0	0
$251$	− 214.599i	− 0.854975i	−0.904021	−	0.427488i	$-0.859399\pi$
$251$	− 214.599i	− 0.854975i	0.904021	−	0.427488i	$-0.140601\pi$
$252$	0	0
$253$	0.572142	0.00226143
$254$	0	0
$255$	− 119.831i	− 0.469925i
$256$	0	0
$257$	− 248.522i	− 0.967010i	−0.875342	−	0.483505i	$-0.839364\pi$
$257$	− 248.522i	− 0.967010i	0.875342	−	0.483505i	$-0.160636\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	15.4865	0.0593354
$262$	0	0
$263$	−35.4619	−0.134836	−0.0674181	−	0.997725i	$-0.521476\pi$
$263$	−35.4619	−0.134836	−0.0674181	+	0.997725i	$0.521476\pi$
$264$	0	0
$265$	− 480.913i	− 1.81477i
$266$	0	0
$267$	−13.1015	−0.0490694
$268$	0	0
$269$	− 288.057i	− 1.07084i	−0.844585	−	0.535422i	$-0.820153\pi$
$269$	− 288.057i	− 1.07084i	0.844585	−	0.535422i	$-0.179847\pi$
$270$	0	0
$271$	− 138.450i	− 0.510885i	−0.966824	−	0.255442i	$-0.917779\pi$
$271$	− 138.450i	− 0.510885i	0.966824	−	0.255442i	$-0.0822211\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.37097	−0.0195308
$276$	0	0
$277$	−332.780	−1.20137	−0.600686	−	0.799485i	$-0.705106\pi$
$277$	−332.780	−1.20137	−0.600686	+	0.799485i	$0.705106\pi$
$278$	0	0
$279$	16.3869i	0.0587344i
$280$	0	0
$281$	−309.983	−1.10314	−0.551570	−	0.834128i	$-0.685971\pi$
$281$	−309.983	−1.10314	−0.551570	+	0.834128i	$0.685971\pi$
$282$	0	0
$283$	532.631i	1.88209i	0.338286	+	0.941043i	$0.390153\pi$
$283$	532.631i	1.88209i	−0.338286	+	0.941043i	$0.609847\pi$
$284$	0	0
$285$	150.746i	0.528932i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	114.375	0.395763
$290$	0	0
$291$	65.4946	0.225067
$292$	0	0
$293$	64.6662i	0.220704i	0.993893	+	0.110352i	$0.0351978\pi$
$293$	64.6662i	0.220704i	−0.993893	+	0.110352i	$0.964802\pi$
$294$	0	0
$295$	−260.100	−0.881694
$296$	0	0
$297$	11.5797i	0.0389888i
$298$	0	0
$299$	1.48394i	0.00496301i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.253346	−0.000836125 0
$304$	0	0
$305$	477.658	1.56609
$306$	0	0
$307$	− 285.057i	− 0.928524i	−0.885698	−	0.464262i	$-0.846320\pi$
$307$	− 285.057i	− 0.928524i	0.885698	−	0.464262i	$-0.153680\pi$
$308$	0	0
$309$	−32.0934	−0.103862
$310$	0	0
$311$	243.498i	0.782952i	0.920188	+	0.391476i	$0.128035\pi$
$311$	243.498i	0.782952i	−0.920188	+	0.391476i	$0.871965\pi$
$312$	0	0
$313$	− 408.999i	− 1.30671i	−0.757054	−	0.653353i	$-0.773362\pi$
$313$	− 408.999i	− 1.30671i	0.757054	−	0.653353i	$-0.226638\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−255.288	−0.805325	−0.402663	−	0.915348i	$-0.631915\pi$
$317$	−255.288	−0.805325	−0.402663	+	0.915348i	$0.631915\pi$
$318$	0	0
$319$	−11.5040	−0.0360626
$320$	0	0
$321$	− 94.6998i	− 0.295015i
$322$	0	0
$323$	219.675	0.680109
$324$	0	0
$325$	− 13.9305i	− 0.0428630i
$326$	0	0
$327$	231.436i	0.707754i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	447.342	1.35149	0.675743	−	0.737137i	$-0.263823\pi$
$331$	447.342	1.35149	0.675743	+	0.737137i	$0.263823\pi$
$332$	0	0
$333$	180.900	0.543242
$334$	0	0
$335$	− 621.854i	− 1.85628i
$336$	0	0
$337$	−327.784	−0.972652	−0.486326	−	0.873777i	$-0.661663\pi$
$337$	−327.784	−0.972652	−0.486326	+	0.873777i	$0.661663\pi$
$338$	0	0
$339$	− 230.447i	− 0.679785i
$340$	0	0
$341$	− 12.1728i	− 0.0356974i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.32812	0.00674816
$346$	0	0
$347$	486.064	1.40076	0.700380	−	0.713770i	$-0.253014\pi$
$347$	486.064	1.40076	0.700380	+	0.713770i	$0.253014\pi$
$348$	0	0
$349$	− 202.844i	− 0.581216i	−0.956842	−	0.290608i	$-0.906143\pi$
$349$	− 202.844i	− 0.581216i	0.956842	−	0.290608i	$-0.0938575\pi$
$350$	0	0
$351$	−30.0338	−0.0855663
$352$	0	0
$353$	− 425.994i	− 1.20678i	−0.797446	−	0.603391i	$-0.793816\pi$
$353$	− 425.994i	− 1.20678i	0.797446	−	0.603391i	$-0.206184\pi$
$354$	0	0
$355$	18.8097i	0.0529850i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	568.523	1.58363	0.791814	−	0.610762i	$-0.209137\pi$
$359$	568.523	1.58363	0.791814	+	0.610762i	$0.209137\pi$
$360$	0	0
$361$	84.6514	0.234491
$362$	0	0
$363$	200.976i	0.553654i
$364$	0	0
$365$	−362.057	−0.991938
$366$	0	0
$367$	− 143.432i	− 0.390822i	−0.980721	−	0.195411i	$-0.937396\pi$
$367$	− 143.432i	− 0.390822i	0.980721	−	0.195411i	$-0.0626041\pi$
$368$	0	0
$369$	− 109.470i	− 0.296667i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−47.7053	−0.127896	−0.0639481	−	0.997953i	$-0.520369\pi$
$373$	−47.7053	−0.127896	−0.0639481	+	0.997953i	$0.520369\pi$
$374$	0	0
$375$	204.847	0.546259
$376$	0	0
$377$	− 29.8374i	− 0.0791443i
$378$	0	0
$379$	−458.796	−1.21054	−0.605272	−	0.796019i	$-0.706936\pi$
$379$	−458.796	−1.21054	−0.605272	+	0.796019i	$0.706936\pi$
$380$	0	0
$381$	56.4106i	0.148059i
$382$	0	0
$383$	484.770i	1.26572i	0.774267	+	0.632859i	$0.218119\pi$
$383$	484.770i	1.26572i	−0.774267	+	0.632859i	$0.781881\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−115.401	−0.298193
$388$	0	0
$389$	247.347	0.635855	0.317927	−	0.948115i	$-0.397013\pi$
$389$	247.347	0.635855	0.317927	+	0.948115i	$0.397013\pi$
$390$	0	0
$391$	− 3.39267i	− 0.00867690i
$392$	0	0
$393$	253.394	0.644769
$394$	0	0
$395$	− 32.9404i	− 0.0833934i
$396$	0	0
$397$	− 191.231i	− 0.481690i	−0.970564	−	0.240845i	$-0.922575\pi$
$397$	− 191.231i	− 0.481690i	0.970564	−	0.240845i	$-0.0774245\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	426.177	1.06279	0.531393	−	0.847125i	$-0.321669\pi$
$401$	426.177	1.06279	0.531393	+	0.847125i	$0.321669\pi$
$402$	0	0
$403$	31.5721	0.0783427
$404$	0	0
$405$	47.1192i	0.116344i
$406$	0	0
$407$	−134.379	−0.330170
$408$	0	0
$409$	183.266i	0.448083i	0.974580	+	0.224042i	$0.0719252\pi$
$409$	183.266i	0.448083i	−0.974580	+	0.224042i	$0.928075\pi$
$410$	0	0
$411$	− 294.121i	− 0.715624i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−609.288	−1.46816
$416$	0	0
$417$	294.984	0.707396
$418$	0	0
$419$	31.5067i	0.0751950i	0.999293	+	0.0375975i	$0.0119705\pi$
$419$	31.5067i	0.0751950i	−0.999293	+	0.0375975i	$0.988030\pi$
$420$	0	0
$421$	488.569	1.16050	0.580248	−	0.814440i	$-0.302956\pi$
$421$	488.569	1.16050	0.580248	+	0.814440i	$0.302956\pi$
$422$	0	0
$423$	− 280.458i	− 0.663022i
$424$	0	0
$425$	31.8486i	0.0749379i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	22.3102	0.0520051
$430$	0	0
$431$	240.669	0.558397	0.279199	−	0.960233i	$-0.409931\pi$
$431$	240.669	0.558397	0.279199	+	0.960233i	$0.409931\pi$
$432$	0	0
$433$	− 698.068i	− 1.61217i	−0.591803	−	0.806083i	$-0.701584\pi$
$433$	− 698.068i	− 1.61217i	0.591803	−	0.806083i	$-0.298416\pi$
$434$	0	0
$435$	−46.8111	−0.107612
$436$	0	0
$437$	4.26793i	0.00976643i
$438$	0	0
$439$	− 677.096i	− 1.54236i	−0.636617	−	0.771180i	$-0.719667\pi$
$439$	− 677.096i	− 1.54236i	0.636617	−	0.771180i	$-0.280333\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	778.917	1.75828	0.879139	−	0.476565i	$-0.158118\pi$
$443$	778.917	1.75828	0.879139	+	0.476565i	$0.158118\pi$
$444$	0	0
$445$	39.6019	0.0889931
$446$	0	0
$447$	150.301i	0.336244i
$448$	0	0
$449$	−719.694	−1.60288	−0.801441	−	0.598073i	$-0.795933\pi$
$449$	−719.694	−1.60288	−0.801441	+	0.598073i	$0.795933\pi$
$450$	0	0
$451$	81.3185i	0.180307i
$452$	0	0
$453$	404.891i	0.893800i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	876.825	1.91865	0.959327	−	0.282298i	$-0.0910968\pi$
$457$	876.825	1.91865	0.959327	+	0.282298i	$0.0910968\pi$
$458$	0	0
$459$	68.6649	0.149597
$460$	0	0
$461$	807.236i	1.75105i	0.483169	+	0.875527i	$0.339486\pi$
$461$	807.236i	1.75105i	−0.483169	+	0.875527i	$0.660514\pi$
$462$	0	0
$463$	−6.94553	−0.0150011	−0.00750057	−	0.999972i	$-0.502388\pi$
$463$	−6.94553	−0.0150011	−0.00750057	+	0.999972i	$0.502388\pi$
$464$	0	0
$465$	− 49.5327i	− 0.106522i
$466$	0	0
$467$	391.956i	0.839306i	0.907685	+	0.419653i	$0.137848\pi$
$467$	391.956i	0.839306i	−0.907685	+	0.419653i	$0.862152\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	475.104	1.00871
$472$	0	0
$473$	85.7239	0.181234
$474$	0	0
$475$	− 40.0651i	− 0.0843476i
$476$	0	0
$477$	275.570	0.577716
$478$	0	0
$479$	− 309.384i	− 0.645895i	−0.946417	−	0.322948i	$-0.895326\pi$
$479$	− 309.384i	− 0.645895i	0.946417	−	0.322948i	$-0.104674\pi$
$480$	0	0
$481$	− 348.533i	− 0.724602i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−197.970	−0.408186
$486$	0	0
$487$	720.847	1.48018	0.740089	−	0.672509i	$-0.234783\pi$
$487$	720.847	1.48018	0.740089	+	0.672509i	$0.234783\pi$
$488$	0	0
$489$	− 279.639i	− 0.571859i
$490$	0	0
$491$	573.346	1.16771	0.583855	−	0.811858i	$-0.301543\pi$
$491$	573.346	1.16771	0.583855	+	0.811858i	$0.301543\pi$
$492$	0	0
$493$	68.2159i	0.138369i
$494$	0	0
$495$	− 35.0019i	− 0.0707109i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−420.382	−0.842448	−0.421224	−	0.906957i	$-0.638399\pi$
$499$	−420.382	−0.842448	−0.421224	+	0.906957i	$0.638399\pi$
$500$	0	0
$501$	199.330	0.397865
$502$	0	0
$503$	− 342.620i	− 0.681153i	−0.940217	−	0.340576i	$-0.889378\pi$
$503$	− 342.620i	− 0.681153i	0.940217	−	0.340576i	$-0.110622\pi$
$504$	0	0
$505$	0.765788	0.00151641
$506$	0	0
$507$	− 234.852i	− 0.463218i
$508$	0	0
$509$	174.621i	0.343067i	0.985178	+	0.171534i	$0.0548722\pi$
$509$	174.621i	0.343067i	−0.985178	+	0.171534i	$0.945128\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−86.3795	−0.168381
$514$	0	0
$515$	97.0087	0.188366
$516$	0	0
$517$	208.335i	0.402969i
$518$	0	0
$519$	−247.422	−0.476729
$520$	0	0
$521$	599.697i	1.15105i	0.817784	+	0.575525i	$0.195202\pi$
$521$	599.697i	1.15105i	−0.817784	+	0.575525i	$0.804798\pi$
$522$	0	0
$523$	− 613.538i	− 1.17311i	−0.809908	−	0.586556i	$-0.800483\pi$
$523$	− 613.538i	− 1.17311i	0.809908	−	0.586556i	$-0.199517\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−72.1819	−0.136968
$528$	0	0
$529$	−528.934	−0.999875
$530$	0	0
$531$	− 149.041i	− 0.280680i
$532$	0	0
$533$	−210.912	−0.395708
$534$	0	0
$535$	286.249i	0.535045i
$536$	0	0
$537$	− 205.159i	− 0.382046i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	230.177	0.425465	0.212733	−	0.977110i	$-0.431764\pi$
$541$	230.177	0.425465	0.212733	+	0.977110i	$0.431764\pi$
$542$	0	0
$543$	−59.4854	−0.109550
$544$	0	0
$545$	− 699.560i	− 1.28360i
$546$	0	0
$547$	−552.736	−1.01049	−0.505243	−	0.862977i	$-0.668597\pi$
$547$	−552.736	−1.01049	−0.505243	+	0.862977i	$0.668597\pi$
$548$	0	0
$549$	273.705i	0.498552i
$550$	0	0
$551$	− 85.8147i	− 0.155744i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−546.805	−0.985235
$556$	0	0
$557$	−760.252	−1.36491	−0.682453	−	0.730930i	$-0.739087\pi$
$557$	−760.252	−1.36491	−0.682453	+	0.730930i	$0.739087\pi$
$558$	0	0
$559$	222.338i	0.397743i
$560$	0	0
$561$	−51.0068	−0.0909212
$562$	0	0
$563$	53.1579i	0.0944189i	0.998885	+	0.0472095i	$0.0150328\pi$
$563$	53.1579i	0.0944189i	−0.998885	+	0.0472095i	$0.984967\pi$
$564$	0	0
$565$	696.572i	1.23287i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	308.772	0.542658	0.271329	−	0.962487i	$-0.412537\pi$
$569$	308.772	0.542658	0.271329	+	0.962487i	$0.412537\pi$
$570$	0	0
$571$	118.434	0.207416	0.103708	−	0.994608i	$-0.466929\pi$
$571$	118.434	0.207416	0.103708	+	0.994608i	$0.466929\pi$
$572$	0	0
$573$	28.1628i	0.0491497i
$574$	0	0
$575$	−0.618766	−0.00107611
$576$	0	0
$577$	584.347i	1.01273i	0.862319	+	0.506366i	$0.169012\pi$
$577$	584.347i	1.01273i	−0.862319	+	0.506366i	$0.830988\pi$
$578$	0	0
$579$	− 218.446i	− 0.377281i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−204.704	−0.351122
$584$	0	0
$585$	90.7830	0.155185
$586$	0	0
$587$	502.710i	0.856405i	0.903683	+	0.428203i	$0.140853\pi$
$587$	502.710i	0.856405i	−0.903683	+	0.428203i	$0.859147\pi$
$588$	0	0
$589$	90.8039	0.154166
$590$	0	0
$591$	− 403.450i	− 0.682656i
$592$	0	0
$593$	− 83.1023i	− 0.140139i	−0.997542	−	0.0700694i	$-0.977678\pi$
$593$	− 83.1023i	− 0.140139i	0.997542	−	0.0700694i	$-0.0223221\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	54.2211	0.0908227
$598$	0	0
$599$	−503.438	−0.840463	−0.420232	−	0.907417i	$-0.638051\pi$
$599$	−503.438	−0.840463	−0.420232	+	0.907417i	$0.638051\pi$
$600$	0	0
$601$	− 951.776i	− 1.58365i	−0.610745	−	0.791827i	$-0.709130\pi$
$601$	− 951.776i	− 1.58365i	0.610745	−	0.791827i	$-0.290870\pi$
$602$	0	0
$603$	356.332	0.590931
$604$	0	0
$605$	− 607.491i	− 1.00412i
$606$	0	0
$607$	− 323.836i	− 0.533502i	−0.963765	−	0.266751i	$-0.914050\pi$
$607$	− 323.836i	− 0.533502i	0.963765	−	0.266751i	$-0.0859502\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−540.350	−0.884369
$612$	0	0
$613$	−950.707	−1.55091	−0.775455	−	0.631403i	$-0.782479\pi$
$613$	−950.707	−1.55091	−0.775455	+	0.631403i	$0.782479\pi$
$614$	0	0
$615$	330.895i	0.538041i
$616$	0	0
$617$	−984.822	−1.59615	−0.798073	−	0.602561i	$-0.794147\pi$
$617$	−984.822	−1.59615	−0.798073	+	0.602561i	$0.794147\pi$
$618$	0	0
$619$	− 711.849i	− 1.15000i	−0.818154	−	0.574999i	$-0.805002\pi$
$619$	− 711.849i	− 1.15000i	0.818154	−	0.574999i	$-0.194998\pi$
$620$	0	0
$621$	1.33404i	0.00214822i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−679.444	−1.08711
$626$	0	0
$627$	64.1659	0.102338
$628$	0	0
$629$	796.837i	1.26683i
$630$	0	0
$631$	663.992	1.05229	0.526143	−	0.850396i	$-0.323638\pi$
$631$	663.992	1.05229	0.526143	+	0.850396i	$0.323638\pi$
$632$	0	0
$633$	665.331i	1.05108i
$634$	0	0
$635$	− 170.512i	− 0.268523i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.7782	−0.0168673
$640$	0	0
$641$	−332.864	−0.519289	−0.259644	−	0.965704i	$-0.583605\pi$
$641$	−332.864	−0.519289	−0.259644	+	0.965704i	$0.583605\pi$
$642$	0	0
$643$	260.954i	0.405839i	0.979195	+	0.202919i	$0.0650429\pi$
$643$	260.954i	0.405839i	−0.979195	+	0.202919i	$0.934957\pi$
$644$	0	0
$645$	348.821	0.540808
$646$	0	0
$647$	1121.14i	1.73283i	0.499321	+	0.866417i	$0.333583\pi$
$647$	1121.14i	1.73283i	−0.499321	+	0.866417i	$0.666417\pi$
$648$	0	0
$649$	110.713i	0.170591i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−76.7439	−0.117525	−0.0587625	−	0.998272i	$-0.518715\pi$
$653$	−76.7439	−0.117525	−0.0587625	+	0.998272i	$0.518715\pi$
$654$	0	0
$655$	−765.934	−1.16936
$656$	0	0
$657$	− 207.464i	− 0.315775i
$658$	0	0
$659$	898.262	1.36307	0.681534	−	0.731786i	$-0.261313\pi$
$659$	898.262	1.36307	0.681534	+	0.731786i	$0.261313\pi$
$660$	0	0
$661$	554.687i	0.839164i	0.907718	+	0.419582i	$0.137823\pi$
$661$	554.687i	0.839164i	−0.907718	+	0.419582i	$0.862177\pi$
$662$	0	0
$663$	− 132.294i	− 0.199539i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.32532	−0.00198699
$668$	0	0
$669$	−529.588	−0.791611
$670$	0	0
$671$	− 203.318i	− 0.303008i
$672$	0	0
$673$	−160.773	−0.238891	−0.119445	−	0.992841i	$-0.538112\pi$
$673$	−160.773	−0.238891	−0.119445	+	0.992841i	$0.538112\pi$
$674$	0	0
$675$	− 12.5233i	− 0.0185531i
$676$	0	0
$677$	214.781i	0.317254i	0.987339	+	0.158627i	$0.0507068\pi$
$677$	214.781i	0.317254i	−0.987339	+	0.158627i	$0.949293\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	119.723	0.175804
$682$	0	0
$683$	726.677	1.06395	0.531975	−	0.846760i	$-0.321450\pi$
$683$	726.677	1.06395	0.531975	+	0.846760i	$0.321450\pi$
$684$	0	0
$685$	889.040i	1.29787i
$686$	0	0
$687$	−393.705	−0.573079
$688$	0	0
$689$	− 530.932i	− 0.770584i
$690$	0	0
$691$	686.413i	0.993362i	0.867933	+	0.496681i	$0.165448\pi$
$691$	686.413i	0.993362i	−0.867933	+	0.496681i	$0.834552\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−891.648	−1.28295
$696$	0	0
$697$	482.200	0.691822
$698$	0	0
$699$	213.910i	0.306023i
$700$	0	0
$701$	749.879	1.06973	0.534864	−	0.844938i	$-0.320363\pi$
$701$	749.879	1.06973	0.534864	+	0.844938i	$0.320363\pi$
$702$	0	0
$703$	− 1002.41i	− 1.42590i
$704$	0	0
$705$	847.741i	1.20247i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	273.543	0.385816	0.192908	−	0.981217i	$-0.438208\pi$
$709$	273.543	0.385816	0.192908	+	0.981217i	$0.438208\pi$
$710$	0	0
$711$	18.8753	0.0265476
$712$	0	0
$713$	− 1.40238i	− 0.00196687i
$714$	0	0
$715$	−67.4370	−0.0943175
$716$	0	0
$717$	− 105.779i	− 0.147530i
$718$	0	0
$719$	− 262.653i	− 0.365303i	−0.983178	−	0.182652i	$-0.941532\pi$
$719$	− 262.653i	− 0.365303i	0.983178	−	0.182652i	$-0.0584680\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−34.6479	−0.0479225
$724$	0	0
$725$	12.4414	0.0171606
$726$	0	0
$727$	672.986i	0.925703i	0.886436	+	0.462851i	$0.153174\pi$
$727$	672.986i	0.925703i	−0.886436	+	0.462851i	$0.846826\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 508.323i	− 0.695380i
$732$	0	0
$733$	1356.89i	1.85115i	0.378567	+	0.925574i	$0.376417\pi$
$733$	1356.89i	1.85115i	−0.378567	+	0.925574i	$0.623583\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−264.696	−0.359154
$738$	0	0
$739$	−403.926	−0.546584	−0.273292	−	0.961931i	$-0.588113\pi$
$739$	−403.926	−0.546584	−0.273292	+	0.961931i	$0.588113\pi$
$740$	0	0
$741$	166.424i	0.224594i
$742$	0	0
$743$	1352.63	1.82050	0.910251	−	0.414056i	$-0.135888\pi$
$743$	1352.63	1.82050	0.910251	+	0.414056i	$0.135888\pi$
$744$	0	0
$745$	− 454.315i	− 0.609819i
$746$	0	0
$747$	− 349.131i	− 0.467378i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−393.655	−0.524174	−0.262087	−	0.965044i	$-0.584411\pi$
$751$	−393.655	−0.524174	−0.262087	+	0.965044i	$0.584411\pi$
$752$	0	0
$753$	−371.696	−0.493620
$754$	0	0
$755$	− 1223.86i	− 1.62101i
$756$	0	0
$757$	895.196	1.18256	0.591279	−	0.806467i	$-0.298623\pi$
$757$	895.196	1.18256	0.591279	+	0.806467i	$0.298623\pi$
$758$	0	0
$759$	− 0.990978i	− 0.00130564i
$760$	0	0
$761$	− 220.636i	− 0.289929i	−0.989437	−	0.144965i	$-0.953693\pi$
$761$	− 220.636i	− 0.289929i	0.989437	−	0.144965i	$-0.0463068\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−207.553	−0.271311
$766$	0	0
$767$	−287.152	−0.374384
$768$	0	0
$769$	562.340i	0.731261i	0.930760	+	0.365630i	$0.119147\pi$
$769$	562.340i	0.731261i	−0.930760	+	0.365630i	$0.880853\pi$
$770$	0	0
$771$	−430.452	−0.558303
$772$	0	0
$773$	− 150.801i	− 0.195085i	−0.995231	−	0.0975427i	$-0.968902\pi$
$773$	− 150.801i	− 0.195085i	0.995231	−	0.0975427i	$-0.0310983\pi$
$774$	0	0
$775$	13.1648i	0.0169868i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−606.601	−0.778692
$780$	0	0
$781$	8.00647	0.0102516
$782$	0	0
$783$	− 26.8235i	− 0.0342573i
$784$	0	0
$785$	−1436.10	−1.82942
$786$	0	0
$787$	578.147i	0.734622i	0.930098	+	0.367311i	$0.119721\pi$
$787$	578.147i	0.734622i	−0.930098	+	0.367311i	$0.880279\pi$
$788$	0	0
$789$	61.4218i	0.0778477i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	527.338	0.664992
$794$	0	0
$795$	−832.966	−1.04776
$796$	0	0
$797$	1300.96i	1.63232i	0.577828	+	0.816159i	$0.303900\pi$
$797$	1300.96i	1.63232i	−0.577828	+	0.816159i	$0.696100\pi$
$798$	0	0
$799$	1235.38	1.54615
$800$	0	0
$801$	22.6925i	0.0283302i
$802$	0	0
$803$	154.112i	0.191920i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−498.929	−0.618252
$808$	0	0
$809$	−751.353	−0.928743	−0.464371	−	0.885641i	$-0.653720\pi$
$809$	−751.353	−0.928743	−0.464371	+	0.885641i	$0.653720\pi$
$810$	0	0
$811$	233.292i	0.287660i	0.989602	+	0.143830i	$0.0459418\pi$
$811$	233.292i	0.287660i	−0.989602	+	0.143830i	$0.954058\pi$
$812$	0	0
$813$	−239.802	−0.294959
$814$	0	0
$815$	845.265i	1.03713i
$816$	0	0
$817$	639.463i	0.782697i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−614.893	−0.748956	−0.374478	−	0.927236i	$-0.622178\pi$
$821$	−614.893	−0.748956	−0.374478	+	0.927236i	$0.622178\pi$
$822$	0	0
$823$	1161.81	1.41168	0.705838	−	0.708373i	$-0.250571\pi$
$823$	1161.81	1.41168	0.705838	+	0.708373i	$0.250571\pi$
$824$	0	0
$825$	9.30279i	0.0112761i
$826$	0	0
$827$	1636.11	1.97836	0.989181	−	0.146700i	$-0.0468653\pi$
$827$	1636.11	1.97836	0.989181	+	0.146700i	$0.0468653\pi$
$828$	0	0
$829$	− 553.882i	− 0.668132i	−0.942550	−	0.334066i	$-0.891579\pi$
$829$	− 553.882i	− 0.668132i	0.942550	−	0.334066i	$-0.108421\pi$
$830$	0	0
$831$	576.392i	0.693612i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−602.516	−0.721575
$836$	0	0
$837$	28.3830	0.0339103
$838$	0	0
$839$	863.198i	1.02884i	0.857538	+	0.514421i	$0.171993\pi$
$839$	863.198i	1.02884i	−0.857538	+	0.514421i	$0.828007\pi$
$840$	0	0
$841$	−814.352	−0.968314
$842$	0	0
$843$	536.905i	0.636899i
$844$	0	0
$845$	709.885i	0.840101i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	922.543	1.08662
$850$	0	0
$851$	−15.4812	−0.0181918
$852$	0	0
$853$	1522.15i	1.78447i	0.451572	+	0.892234i	$0.350863\pi$
$853$	1522.15i	1.78447i	−0.451572	+	0.892234i	$0.649137\pi$
$854$	0	0
$855$	261.099	0.305379
$856$	0	0
$857$	− 368.770i	− 0.430304i	−0.976581	−	0.215152i	$-0.930975\pi$
$857$	− 368.770i	− 0.430304i	0.976581	−	0.215152i	$-0.0690246\pi$
$858$	0	0
$859$	− 121.566i	− 0.141520i	−0.997493	−	0.0707600i	$-0.977458\pi$
$859$	− 121.566i	− 0.141520i	0.997493	−	0.0707600i	$-0.0225425\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	524.468	0.607727	0.303863	−	0.952716i	$-0.401723\pi$
$863$	524.468	0.607727	0.303863	+	0.952716i	$0.401723\pi$
$864$	0	0
$865$	747.883	0.864605
$866$	0	0
$867$	− 198.104i	− 0.228494i
$868$	0	0
$869$	−14.0213	−0.0161350
$870$	0	0
$871$	− 686.532i	− 0.788212i
$872$	0	0
$873$	− 113.440i	− 0.129943i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1071.45	−1.22172	−0.610858	−	0.791740i	$-0.709175\pi$
$877$	−1071.45	−1.22172	−0.610858	+	0.791740i	$0.709175\pi$
$878$	0	0
$879$	112.005	0.127423
$880$	0	0
$881$	829.184i	0.941185i	0.882351	+	0.470593i	$0.155960\pi$
$881$	829.184i	0.941185i	−0.882351	+	0.470593i	$0.844040\pi$
$882$	0	0
$883$	447.952	0.507307	0.253653	−	0.967295i	$-0.418368\pi$
$883$	447.952	0.507307	0.253653	+	0.967295i	$0.418368\pi$
$884$	0	0
$885$	450.506i	0.509047i
$886$	0	0
$887$	− 842.622i	− 0.949969i	−0.879994	−	0.474984i	$-0.842454\pi$
$887$	− 842.622i	− 0.949969i	0.879994	−	0.474984i	$-0.157546\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	20.0566	0.0225102
$892$	0	0
$893$	−1554.09	−1.74030
$894$	0	0
$895$	620.133i	0.692887i
$896$	0	0
$897$	2.57026	0.00286540
$898$	0	0
$899$	28.1974i	0.0313653i
$900$	0	0
$901$	1213.85i	1.34722i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	179.807	0.198681
$906$	0	0
$907$	−437.221	−0.482052	−0.241026	−	0.970519i	$-0.577484\pi$
$907$	−437.221	−0.482052	−0.241026	+	0.970519i	$0.577484\pi$
$908$	0	0
$909$	0.438808i	0 0.000482737i
$910$	0	0
$911$	263.771	0.289540	0.144770	−	0.989465i	$-0.453756\pi$
$911$	263.771	0.289540	0.144770	+	0.989465i	$0.453756\pi$
$912$	0	0
$913$	259.348i	0.284061i
$914$	0	0
$915$	− 827.328i	− 0.904183i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	667.035	0.725827	0.362914	−	0.931823i	$-0.381782\pi$
$919$	667.035	0.725827	0.362914	+	0.931823i	$0.381782\pi$
$920$	0	0
$921$	−493.733	−0.536084
$922$	0	0
$923$	20.7660i	0.0224984i
$924$	0	0
$925$	145.330	0.157113
$926$	0	0
$927$	55.5874i	0.0599649i
$928$	0	0
$929$	− 1552.88i	− 1.67156i	−0.549064	−	0.835780i	$-0.685016\pi$
$929$	− 1552.88i	− 1.67156i	0.549064	−	0.835780i	$-0.314984\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	421.751	0.452038
$934$	0	0
$935$	154.178	0.164896
$936$	0	0
$937$	103.127i	0.110061i	0.998485	+	0.0550303i	$0.0175255\pi$
$937$	103.127i	0.110061i	−0.998485	+	0.0550303i	$0.982474\pi$
$938$	0	0
$939$	−708.407	−0.754427
$940$	0	0
$941$	357.382i	0.379789i	0.981804	+	0.189895i	$0.0608147\pi$
$941$	357.382i	0.379789i	−0.981804	+	0.189895i	$0.939185\pi$
$942$	0	0
$943$	9.36835i	0.00993462i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1418.02	−1.49739	−0.748693	−	0.662917i	$-0.769318\pi$
$947$	−1418.02	−1.49739	−0.748693	+	0.662917i	$0.769318\pi$
$948$	0	0
$949$	−399.714	−0.421195
$950$	0	0
$951$	442.172i	0.464955i
$952$	0	0
$953$	1473.18	1.54583	0.772917	−	0.634508i	$-0.218797\pi$
$953$	1473.18	1.54583	0.772917	+	0.634508i	$0.218797\pi$
$954$	0	0
$955$	− 85.1276i	− 0.0891389i
$956$	0	0
$957$	19.9255i	0.0208208i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	931.163	0.968952
$962$	0	0
$963$	−164.025	−0.170327
$964$	0	0
$965$	660.296i	0.684244i
$966$	0	0
$967$	−356.112	−0.368265	−0.184132	−	0.982901i	$-0.558947\pi$
$967$	−356.112	−0.368265	−0.184132	+	0.982901i	$0.558947\pi$
$968$	0	0
$969$	− 380.489i	− 0.392661i
$970$	0	0
$971$	− 121.790i	− 0.125427i	−0.998032	−	0.0627136i	$-0.980025\pi$
$971$	− 121.790i	− 0.125427i	0.998032	−	0.0627136i	$-0.0199755\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−24.1283	−0.0247469
$976$	0	0
$977$	1092.00	1.11770	0.558852	−	0.829267i	$-0.311242\pi$
$977$	1092.00	1.11770	0.558852	+	0.829267i	$0.311242\pi$
$978$	0	0
$979$	− 16.8568i	− 0.0172184i
$980$	0	0
$981$	400.858	0.408622
$982$	0	0
$983$	1638.86i	1.66720i	0.552367	+	0.833601i	$0.313725\pi$
$983$	1638.86i	1.66720i	−0.552367	+	0.833601i	$0.686275\pi$
$984$	0	0
$985$	1219.51i	1.23808i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.87587	0.00998571
$990$	0	0
$991$	213.967	0.215910	0.107955	−	0.994156i	$-0.465570\pi$
$991$	213.967	0.215910	0.107955	+	0.994156i	$0.465570\pi$
$992$	0	0
$993$	− 774.819i	− 0.780281i
$994$	0	0
$995$	−163.894	−0.164718
$996$	0	0
$997$	1137.45i	1.14087i	0.821342	+	0.570436i	$0.193226\pi$
$997$	1137.45i	1.14087i	−0.821342	+	0.570436i	$0.806774\pi$
$998$	0	0
$999$	− 313.328i	− 0.313641i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.3.f.d.97.5	✓	16
3.2	odd	2		3528.3.f.j.2449.7		16
4.3	odd	2		2352.3.f.m.97.13		16
7.2	even	3		1176.3.z.h.913.5		16
7.3	odd	6		1176.3.z.h.313.5		16
7.4	even	3		1176.3.z.g.313.4		16
7.5	odd	6		1176.3.z.g.913.4		16
7.6	odd	2	inner	1176.3.f.d.97.12	yes	16
21.20	even	2		3528.3.f.j.2449.10		16
28.27	even	2		2352.3.f.m.97.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.3.f.d.97.5	✓	16	1.1	even	1	trivial
1176.3.f.d.97.12	yes	16	7.6	odd	2	inner
1176.3.z.g.313.4		16	7.4	even	3
1176.3.z.g.913.4		16	7.5	odd	6
1176.3.z.h.313.5		16	7.3	odd	6
1176.3.z.h.913.5		16	7.2	even	3
2352.3.f.m.97.4		16	28.27	even	2
2352.3.f.m.97.13		16	4.3	odd	2
3528.3.f.j.2449.7		16	3.2	odd	2
3528.3.f.j.2449.10		16	21.20	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1176.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +5.23547i q^{5} -3.00000 q^{9} +2.22851 q^{11} +5.78000i q^{13} +9.06809 q^{15} -13.2146i q^{17} +16.6237i q^{19} +0.256737 q^{23} -2.41011 q^{25} +5.19615i q^{27} -5.16218 q^{29} -5.46230i q^{31} -3.85990i q^{33} -60.2999 q^{37} +10.0113 q^{39} +36.4900i q^{41} +38.4669 q^{43} -15.7064i q^{45} +93.4861i q^{47} -22.8883 q^{51} -91.8568 q^{53} +11.6673i q^{55} +28.7932 q^{57} +49.6804i q^{59} -91.2350i q^{61} -30.2610 q^{65} -118.777 q^{67} -0.444682i q^{69} +3.59274 q^{71} +69.1547i q^{73} +4.17444i q^{75} -6.29178 q^{79} +9.00000 q^{81} +116.377i q^{83} +69.1844 q^{85} +8.94116i q^{87} -7.56417i q^{89} -9.46099 q^{93} -87.0330 q^{95} +37.8133i q^{97} -6.68554 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 112 q^{23} - 256 q^{25} + 112 q^{29} - 128 q^{37} - 112 q^{43} + 112 q^{53} + 192 q^{57} - 112 q^{65} - 224 q^{67} + 160 q^{71} + 304 q^{79} + 144 q^{81} - 416 q^{85} - 192 q^{93} + 912 q^{95}+O(q^{100}) \) 16 * q - 48 * q^9 - 112 * q^23 - 256 * q^25 + 112 * q^29 - 128 * q^37 - 112 * q^43 + 112 * q^53 + 192 * q^57 - 112 * q^65 - 224 * q^67 + 160 * q^71 + 304 * q^79 + 144 * q^81 - 416 * q^85 - 192 * q^93 + 912 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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