LMFDB - Embedded newform 1134.3.b.c.323.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1134,3,Mod(323,1134)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1134, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1134.323"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1134.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$30.8992619785$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.8
Character	$\chi$	$=$	1134.323
Dual form			1134.3.b.c.323.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} +2.95738i q^{5} -2.64575 q^{7} +2.82843i q^{8} +4.18237 q^{10} -2.24372i q^{11} -4.48140 q^{13} +3.74166i q^{14} +4.00000 q^{16} +3.17017i q^{17} +15.1070 q^{19} -5.91476i q^{20} -3.17310 q^{22} -20.9186i q^{23} +16.2539 q^{25} +6.33766i q^{26} +5.29150 q^{28} +33.3094i q^{29} -51.5504 q^{31} -5.65685i q^{32} +4.48330 q^{34} -7.82449i q^{35} -8.74834 q^{37} -21.3645i q^{38} -8.36473 q^{40} -70.6247i q^{41} -44.9345 q^{43} +4.48744i q^{44} -29.5834 q^{46} +18.6949i q^{47} +7.00000 q^{49} -22.9865i q^{50} +8.96281 q^{52} -38.6329i q^{53} +6.63553 q^{55} -7.48331i q^{56} +47.1066 q^{58} -38.4541i q^{59} -73.3167 q^{61} +72.9033i q^{62} -8.00000 q^{64} -13.2532i q^{65} -108.915 q^{67} -6.34035i q^{68} -11.0655 q^{70} -23.2803i q^{71} +47.8485 q^{73} +12.3720i q^{74} -30.2140 q^{76} +5.93632i q^{77} -78.4605 q^{79} +11.8295i q^{80} -99.8784 q^{82} -113.968i q^{83} -9.37541 q^{85} +63.5470i q^{86} +6.34619 q^{88} -164.207i q^{89} +11.8567 q^{91} +41.8372i q^{92} +26.4385 q^{94} +44.6771i q^{95} -87.7018 q^{97} -9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) $ 24 * q - 48 * q^4 + 96 * q^16 + 24 * q^19 - 48 * q^22 - 144 * q^25 + 120 * q^31 + 96 * q^34 - 168 * q^37 - 120 * q^43 + 168 * q^49 + 264 * q^55 - 192 * q^64 - 144 * q^67 + 24 * q^73 - 48 * q^76 - 24 * q^79 - 288 * q^82 + 312 * q^85 + 96 * q^88 - 168 * q^91 - 96 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	2.95738i	0.591476i	0.955269	+	0.295738i	$0.0955655\pi$
$5$	2.95738i	0.591476i	−0.955269	+	0.295738i	$0.904434\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	2.82843i	0.353553i
$9$	0	0
$10$	4.18237	0.418237
$11$	− 2.24372i	− 0.203974i	−0.994786	−	0.101987i	$-0.967480\pi$
$11$	− 2.24372i	− 0.203974i	0.994786	−	0.101987i	$-0.0325201\pi$
$12$	0	0
$13$	−4.48140	−0.344723	−0.172362	−	0.985034i	$-0.555140\pi$
$13$	−4.48140	−0.344723	−0.172362	+	0.985034i	$0.555140\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	3.17017i	0.186481i	0.995644	+	0.0932404i	$0.0297225\pi$
$17$	3.17017i	0.186481i	−0.995644	+	0.0932404i	$0.970277\pi$
$18$	0	0
$19$	15.1070	0.795105	0.397553	−	0.917579i	$-0.369860\pi$
$19$	15.1070	0.795105	0.397553	+	0.917579i	$0.369860\pi$
$20$	− 5.91476i	− 0.295738i
$21$	0	0
$22$	−3.17310	−0.144232
$23$	− 20.9186i	− 0.909505i	−0.890618	−	0.454753i	$-0.849728\pi$
$23$	− 20.9186i	− 0.909505i	0.890618	−	0.454753i	$-0.150272\pi$
$24$	0	0
$25$	16.2539	0.650156
$26$	6.33766i	0.243756i
$27$	0	0
$28$	5.29150	0.188982
$29$	33.3094i	1.14860i	0.818645	+	0.574300i	$0.194726\pi$
$29$	33.3094i	1.14860i	−0.818645	+	0.574300i	$0.805274\pi$
$30$	0	0
$31$	−51.5504	−1.66292	−0.831458	−	0.555588i	$-0.812493\pi$
$31$	−51.5504	−1.66292	−0.831458	+	0.555588i	$0.812493\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	4.48330	0.131862
$35$	− 7.82449i	− 0.223557i
$36$	0	0
$37$	−8.74834	−0.236442	−0.118221	−	0.992987i	$-0.537719\pi$
$37$	−8.74834	−0.236442	−0.118221	+	0.992987i	$0.537719\pi$
$38$	− 21.3645i	− 0.562224i
$39$	0	0
$40$	−8.36473	−0.209118
$41$	− 70.6247i	− 1.72255i	−0.508137	−	0.861276i	$-0.669666\pi$
$41$	− 70.6247i	− 1.72255i	0.508137	−	0.861276i	$-0.330334\pi$
$42$	0	0
$43$	−44.9345	−1.04499	−0.522494	−	0.852643i	$-0.674998\pi$
$43$	−44.9345	−1.04499	−0.522494	+	0.852643i	$0.674998\pi$
$44$	4.48744i	0.101987i
$45$	0	0
$46$	−29.5834	−0.643117
$47$	18.6949i	0.397763i	0.980024	+	0.198881i	$0.0637309\pi$
$47$	18.6949i	0.397763i	−0.980024	+	0.198881i	$0.936269\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	− 22.9865i	− 0.459730i
$51$	0	0
$52$	8.96281	0.172362
$53$	− 38.6329i	− 0.728923i	−0.931218	−	0.364462i	$-0.881253\pi$
$53$	− 38.6329i	− 0.728923i	0.931218	−	0.364462i	$-0.118747\pi$
$54$	0	0
$55$	6.63553	0.120646
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	47.1066	0.812183
$59$	− 38.4541i	− 0.651765i	−0.945410	−	0.325882i	$-0.894339\pi$
$59$	− 38.4541i	− 0.651765i	0.945410	−	0.325882i	$-0.105661\pi$
$60$	0	0
$61$	−73.3167	−1.20191	−0.600957	−	0.799282i	$-0.705214\pi$
$61$	−73.3167	−1.20191	−0.600957	+	0.799282i	$0.705214\pi$
$62$	72.9033i	1.17586i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 13.2532i	− 0.203896i
$66$	0	0
$67$	−108.915	−1.62559	−0.812795	−	0.582550i	$-0.802055\pi$
$67$	−108.915	−1.62559	−0.812795	+	0.582550i	$0.802055\pi$
$68$	− 6.34035i	− 0.0932404i
$69$	0	0
$70$	−11.0655	−0.158079
$71$	− 23.2803i	− 0.327892i	−0.986469	−	0.163946i	$-0.947578\pi$
$71$	− 23.2803i	− 0.327892i	0.986469	−	0.163946i	$-0.0524223\pi$
$72$	0	0
$73$	47.8485	0.655458	0.327729	−	0.944772i	$-0.393717\pi$
$73$	47.8485	0.655458	0.327729	+	0.944772i	$0.393717\pi$
$74$	12.3720i	0.167189i
$75$	0	0
$76$	−30.2140	−0.397553
$77$	5.93632i	0.0770951i
$78$	0	0
$79$	−78.4605	−0.993170	−0.496585	−	0.867988i	$-0.665413\pi$
$79$	−78.4605	−0.993170	−0.496585	+	0.867988i	$0.665413\pi$
$80$	11.8295i	0.147869i
$81$	0	0
$82$	−99.8784	−1.21803
$83$	− 113.968i	− 1.37310i	−0.727081	−	0.686551i	$-0.759124\pi$
$83$	− 113.968i	− 1.37310i	0.727081	−	0.686551i	$-0.240876\pi$
$84$	0	0
$85$	−9.37541	−0.110299
$86$	63.5470i	0.738918i
$87$	0	0
$88$	6.34619	0.0721158
$89$	− 164.207i	− 1.84502i	−0.385973	−	0.922510i	$-0.626134\pi$
$89$	− 164.207i	− 1.84502i	0.385973	−	0.922510i	$-0.373866\pi$
$90$	0	0
$91$	11.8567	0.130293
$92$	41.8372i	0.454753i
$93$	0	0
$94$	26.4385	0.281261
$95$	44.6771i	0.470286i
$96$	0	0
$97$	−87.7018	−0.904142	−0.452071	−	0.891982i	$-0.649315\pi$
$97$	−87.7018	−0.904142	−0.452071	+	0.891982i	$0.649315\pi$
$98$	− 9.89949i	− 0.101015i
$99$	0	0
$100$	−32.5078	−0.325078
$101$	− 30.5249i	− 0.302226i	−0.988516	−	0.151113i	$-0.951714\pi$
$101$	− 30.5249i	− 0.302226i	0.988516	−	0.151113i	$-0.0482858\pi$
$102$	0	0
$103$	−41.7413	−0.405255	−0.202628	−	0.979256i	$-0.564948\pi$
$103$	−41.7413	−0.405255	−0.202628	+	0.979256i	$0.564948\pi$
$104$	− 12.6753i	− 0.121878i
$105$	0	0
$106$	−54.6352	−0.515426
$107$	− 52.6534i	− 0.492088i	−0.969259	−	0.246044i	$-0.920869\pi$
$107$	− 52.6534i	− 0.492088i	0.969259	−	0.246044i	$-0.0791307\pi$
$108$	0	0
$109$	−193.765	−1.77766	−0.888832	−	0.458232i	$-0.848483\pi$
$109$	−193.765	−1.77766	−0.888832	+	0.458232i	$0.848483\pi$
$110$	− 9.38405i	− 0.0853096i
$111$	0	0
$112$	−10.5830	−0.0944911
$113$	125.172i	1.10772i	0.832610	+	0.553860i	$0.186846\pi$
$113$	125.172i	1.10772i	−0.832610	+	0.553860i	$0.813154\pi$
$114$	0	0
$115$	61.8643	0.537950
$116$	− 66.6188i	− 0.574300i
$117$	0	0
$118$	−54.3823	−0.460867
$119$	− 8.38749i	− 0.0704831i
$120$	0	0
$121$	115.966	0.958394
$122$	103.685i	0.849881i
$123$	0	0
$124$	103.101	0.831458
$125$	122.003i	0.976028i
$126$	0	0
$127$	−76.1997	−0.599997	−0.299999	−	0.953940i	$-0.596986\pi$
$127$	−76.1997	−0.599997	−0.299999	+	0.953940i	$0.596986\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−18.7429	−0.144176
$131$	175.145i	1.33698i	0.743719	+	0.668492i	$0.233060\pi$
$131$	175.145i	1.33698i	−0.743719	+	0.668492i	$0.766940\pi$
$132$	0	0
$133$	−39.9694	−0.300521
$134$	154.028i	1.14947i
$135$	0	0
$136$	−8.96661	−0.0659309
$137$	65.9238i	0.481196i	0.970625	+	0.240598i	$0.0773435\pi$
$137$	65.9238i	0.481196i	−0.970625	+	0.240598i	$0.922657\pi$
$138$	0	0
$139$	−37.0302	−0.266404	−0.133202	−	0.991089i	$-0.542526\pi$
$139$	−37.0302	−0.266404	−0.133202	+	0.991089i	$0.542526\pi$
$140$	15.6490i	0.111778i
$141$	0	0
$142$	−32.9233	−0.231855
$143$	10.0550i	0.0703147i
$144$	0	0
$145$	−98.5085	−0.679369
$146$	− 67.6680i	− 0.463479i
$147$	0	0
$148$	17.4967	0.118221
$149$	− 140.856i	− 0.945341i	−0.881239	−	0.472670i	$-0.843290\pi$
$149$	− 140.856i	− 0.945341i	0.881239	−	0.472670i	$-0.156710\pi$
$150$	0	0
$151$	287.779	1.90582	0.952911	−	0.303252i	$-0.0980722\pi$
$151$	287.779	1.90582	0.952911	+	0.303252i	$0.0980722\pi$
$152$	42.7290i	0.281112i
$153$	0	0
$154$	8.39522	0.0545144
$155$	− 152.454i	− 0.983575i
$156$	0	0
$157$	88.5085	0.563748	0.281874	−	0.959451i	$-0.409044\pi$
$157$	88.5085	0.563748	0.281874	+	0.959451i	$0.409044\pi$
$158$	110.960i	0.702277i
$159$	0	0
$160$	16.7295	0.104559
$161$	55.3455i	0.343761i
$162$	0	0
$163$	−190.795	−1.17052	−0.585261	−	0.810845i	$-0.699008\pi$
$163$	−190.795	−1.17052	−0.585261	+	0.810845i	$0.699008\pi$
$164$	141.249i	0.861276i
$165$	0	0
$166$	−161.174	−0.970930
$167$	− 148.863i	− 0.891394i	−0.895184	−	0.445697i	$-0.852956\pi$
$167$	− 148.863i	− 0.891394i	0.895184	−	0.445697i	$-0.147044\pi$
$168$	0	0
$169$	−148.917	−0.881166
$170$	13.2588i	0.0779931i
$171$	0	0
$172$	89.8690	0.522494
$173$	69.0028i	0.398860i	0.979912	+	0.199430i	$0.0639091\pi$
$173$	69.0028i	0.398860i	−0.979912	+	0.199430i	$0.936091\pi$
$174$	0	0
$175$	−43.0038	−0.245736
$176$	− 8.97487i	− 0.0509936i
$177$	0	0
$178$	−232.224	−1.30463
$179$	− 297.138i	− 1.65999i	−0.557772	−	0.829994i	$-0.688344\pi$
$179$	− 297.138i	− 1.65999i	0.557772	−	0.829994i	$-0.311656\pi$
$180$	0	0
$181$	8.68202	0.0479669	0.0239835	−	0.999712i	$-0.492365\pi$
$181$	8.68202	0.0479669	0.0239835	+	0.999712i	$0.492365\pi$
$182$	− 16.7679i	− 0.0921312i
$183$	0	0
$184$	59.1668	0.321559
$185$	− 25.8722i	− 0.139850i
$186$	0	0
$187$	7.11298	0.0380373
$188$	− 37.3897i	− 0.198881i
$189$	0	0
$190$	63.1830	0.332542
$191$	215.791i	1.12980i	0.825160	+	0.564898i	$0.191085\pi$
$191$	215.791i	1.12980i	−0.825160	+	0.564898i	$0.808915\pi$
$192$	0	0
$193$	193.075	1.00039	0.500195	−	0.865913i	$-0.333262\pi$
$193$	193.075	1.00039	0.500195	+	0.865913i	$0.333262\pi$
$194$	124.029i	0.639325i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	328.601i	1.66802i	0.551746	+	0.834012i	$0.313962\pi$
$197$	328.601i	1.66802i	−0.551746	+	0.834012i	$0.686038\pi$
$198$	0	0
$199$	301.035	1.51274	0.756371	−	0.654143i	$-0.226971\pi$
$199$	301.035	1.51274	0.756371	+	0.654143i	$0.226971\pi$
$200$	45.9730i	0.229865i
$201$	0	0
$202$	−43.1687	−0.213706
$203$	− 88.1284i	− 0.434130i
$204$	0	0
$205$	208.864	1.01885
$206$	59.0311i	0.286559i
$207$	0	0
$208$	−17.9256	−0.0861809
$209$	− 33.8958i	− 0.162181i
$210$	0	0
$211$	−187.617	−0.889182	−0.444591	−	0.895734i	$-0.646651\pi$
$211$	−187.617	−0.889182	−0.444591	+	0.895734i	$0.646651\pi$
$212$	77.2658i	0.364462i
$213$	0	0
$214$	−74.4631	−0.347959
$215$	− 132.888i	− 0.618085i
$216$	0	0
$217$	136.390	0.628523
$218$	274.026i	1.25700i
$219$	0	0
$220$	−13.2711	−0.0603230
$221$	− 14.2068i	− 0.0642843i
$222$	0	0
$223$	−415.288	−1.86228	−0.931139	−	0.364665i	$-0.881184\pi$
$223$	−415.288	−1.86228	−0.931139	+	0.364665i	$0.881184\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	177.020	0.783276
$227$	− 343.850i	− 1.51476i	−0.652975	−	0.757379i	$-0.726480\pi$
$227$	− 343.850i	− 1.51476i	0.652975	−	0.757379i	$-0.273520\pi$
$228$	0	0
$229$	−72.2331	−0.315429	−0.157714	−	0.987485i	$-0.550412\pi$
$229$	−72.2331	−0.315429	−0.157714	+	0.987485i	$0.550412\pi$
$230$	− 87.4893i	− 0.380388i
$231$	0	0
$232$	−94.2132	−0.406091
$233$	351.362i	1.50799i	0.656880	+	0.753995i	$0.271876\pi$
$233$	351.362i	1.50799i	−0.656880	+	0.753995i	$0.728124\pi$
$234$	0	0
$235$	−55.2878	−0.235267
$236$	76.9082i	0.325882i
$237$	0	0
$238$	−11.8617	−0.0498391
$239$	101.620i	0.425190i	0.977140	+	0.212595i	$0.0681915\pi$
$239$	101.620i	0.425190i	−0.977140	+	0.212595i	$0.931809\pi$
$240$	0	0
$241$	−17.2373	−0.0715239	−0.0357619	−	0.999360i	$-0.511386\pi$
$241$	−17.2373	−0.0715239	−0.0357619	+	0.999360i	$0.511386\pi$
$242$	− 164.000i	− 0.677687i
$243$	0	0
$244$	146.633	0.600957
$245$	20.7017i	0.0844966i
$246$	0	0
$247$	−67.7006	−0.274091
$248$	− 145.807i	− 0.587930i
$249$	0	0
$250$	172.539	0.690156
$251$	− 248.646i	− 0.990621i	−0.868716	−	0.495311i	$-0.835054\pi$
$251$	− 248.646i	− 0.990621i	0.868716	−	0.495311i	$-0.164946\pi$
$252$	0	0
$253$	−46.9355	−0.185516
$254$	107.763i	0.424262i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 239.914i	− 0.933516i	−0.884385	−	0.466758i	$-0.845422\pi$
$257$	− 239.914i	− 0.933516i	0.884385	−	0.466758i	$-0.154578\pi$
$258$	0	0
$259$	23.1459	0.0893665
$260$	26.5064i	0.101948i
$261$	0	0
$262$	247.692	0.945391
$263$	90.5805i	0.344412i	0.985061	+	0.172206i	$0.0550896\pi$
$263$	90.5805i	0.344412i	−0.985061	+	0.172206i	$0.944910\pi$
$264$	0	0
$265$	114.252	0.431140
$266$	56.5252i	0.212501i
$267$	0	0
$268$	217.829	0.812795
$269$	241.010i	0.895949i	0.894046	+	0.447975i	$0.147855\pi$
$269$	241.010i	0.895949i	−0.894046	+	0.447975i	$0.852145\pi$
$270$	0	0
$271$	−118.017	−0.435487	−0.217744	−	0.976006i	$-0.569870\pi$
$271$	−118.017	−0.435487	−0.217744	+	0.976006i	$0.569870\pi$
$272$	12.6807i	0.0466202i
$273$	0	0
$274$	93.2303	0.340257
$275$	− 36.4692i	− 0.132615i
$276$	0	0
$277$	427.318	1.54267	0.771333	−	0.636432i	$-0.219590\pi$
$277$	427.318	1.54267	0.771333	+	0.636432i	$0.219590\pi$
$278$	52.3685i	0.188376i
$279$	0	0
$280$	22.1310	0.0790393
$281$	− 189.711i	− 0.675127i	−0.941303	−	0.337563i	$-0.890397\pi$
$281$	− 189.711i	− 0.675127i	0.941303	−	0.337563i	$-0.109603\pi$
$282$	0	0
$283$	65.8693	0.232754	0.116377	−	0.993205i	$-0.462872\pi$
$283$	65.8693	0.232754	0.116377	+	0.993205i	$0.462872\pi$
$284$	46.5606i	0.163946i
$285$	0	0
$286$	14.2199	0.0497200
$287$	186.855i	0.651064i
$288$	0	0
$289$	278.950	0.965225
$290$	139.312i	0.480387i
$291$	0	0
$292$	−95.6969	−0.327729
$293$	− 377.260i	− 1.28758i	−0.765204	−	0.643788i	$-0.777362\pi$
$293$	− 377.260i	− 1.28758i	0.765204	−	0.643788i	$-0.222638\pi$
$294$	0	0
$295$	113.723	0.385503
$296$	− 24.7440i	− 0.0835947i
$297$	0	0
$298$	−199.200	−0.668457
$299$	93.7448i	0.313528i
$300$	0	0
$301$	118.885	0.394968
$302$	− 406.981i	− 1.34762i
$303$	0	0
$304$	60.4280	0.198776
$305$	− 216.825i	− 0.710903i
$306$	0	0
$307$	397.032	1.29326	0.646632	−	0.762802i	$-0.276177\pi$
$307$	397.032	1.29326	0.646632	+	0.762802i	$0.276177\pi$
$308$	− 11.8726i	− 0.0385475i
$309$	0	0
$310$	−215.603	−0.695492
$311$	− 72.5394i	− 0.233246i	−0.993176	−	0.116623i	$-0.962793\pi$
$311$	− 72.5394i	− 0.233246i	0.993176	−	0.116623i	$-0.0372069\pi$
$312$	0	0
$313$	−9.22611	−0.0294764	−0.0147382	−	0.999891i	$-0.504691\pi$
$313$	−9.22611	−0.0294764	−0.0147382	+	0.999891i	$0.504691\pi$
$314$	− 125.170i	− 0.398630i
$315$	0	0
$316$	156.921	0.496585
$317$	− 225.068i	− 0.709993i	−0.934868	−	0.354997i	$-0.884482\pi$
$317$	− 225.068i	− 0.709993i	0.934868	−	0.354997i	$-0.115518\pi$
$318$	0	0
$319$	74.7369	0.234285
$320$	− 23.6590i	− 0.0739345i
$321$	0	0
$322$	78.2703	0.243075
$323$	47.8918i	0.148272i
$324$	0	0
$325$	−72.8403	−0.224124
$326$	269.825i	0.827684i
$327$	0	0
$328$	199.757	0.609014
$329$	− 49.4619i	− 0.150340i
$330$	0	0
$331$	391.558	1.18295	0.591477	−	0.806322i	$-0.298545\pi$
$331$	391.558	1.18295	0.591477	+	0.806322i	$0.298545\pi$
$332$	227.935i	0.686551i
$333$	0	0
$334$	−210.524	−0.630311
$335$	− 322.102i	− 0.961498i
$336$	0	0
$337$	−315.229	−0.935397	−0.467698	−	0.883888i	$-0.654917\pi$
$337$	−315.229	−0.935397	−0.467698	+	0.883888i	$0.654917\pi$
$338$	210.600i	0.623078i
$339$	0	0
$340$	18.7508	0.0551495
$341$	115.665i	0.339192i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	− 127.094i	− 0.369459i
$345$	0	0
$346$	97.5847	0.282037
$347$	241.178i	0.695038i	0.937673	+	0.347519i	$0.112976\pi$
$347$	241.178i	0.695038i	−0.937673	+	0.347519i	$0.887024\pi$
$348$	0	0
$349$	−389.051	−1.11476	−0.557379	−	0.830258i	$-0.688193\pi$
$349$	−389.051	−1.11476	−0.557379	+	0.830258i	$0.688193\pi$
$350$	60.8165i	0.173762i
$351$	0	0
$352$	−12.6924	−0.0360579
$353$	− 169.653i	− 0.480603i	−0.970698	−	0.240302i	$-0.922754\pi$
$353$	− 169.653i	− 0.480603i	0.970698	−	0.240302i	$-0.0772463\pi$
$354$	0	0
$355$	68.8487	0.193940
$356$	328.414i	0.922510i
$357$	0	0
$358$	−420.216	−1.17379
$359$	− 480.422i	− 1.33822i	−0.743162	−	0.669111i	$-0.766675\pi$
$359$	− 480.422i	− 1.33822i	0.743162	−	0.669111i	$-0.233325\pi$
$360$	0	0
$361$	−132.779	−0.367808
$362$	− 12.2782i	− 0.0339177i
$363$	0	0
$364$	−23.7134	−0.0651466
$365$	141.506i	0.387688i
$366$	0	0
$367$	334.462	0.911342	0.455671	−	0.890148i	$-0.349399\pi$
$367$	334.462	0.911342	0.455671	+	0.890148i	$0.349399\pi$
$368$	− 83.6745i	− 0.227376i
$369$	0	0
$370$	−36.5888	−0.0988885
$371$	102.213i	0.275507i
$372$	0	0
$373$	−19.8698	−0.0532702	−0.0266351	−	0.999645i	$-0.508479\pi$
$373$	−19.8698	−0.0532702	−0.0266351	+	0.999645i	$0.508479\pi$
$374$	− 10.0593i	− 0.0268964i
$375$	0	0
$376$	−52.8770	−0.140630
$377$	− 149.273i	− 0.395949i
$378$	0	0
$379$	−103.780	−0.273825	−0.136913	−	0.990583i	$-0.543718\pi$
$379$	−103.780	−0.273825	−0.136913	+	0.990583i	$0.543718\pi$
$380$	− 89.3543i	− 0.235143i
$381$	0	0
$382$	305.175	0.798887
$383$	− 365.733i	− 0.954917i	−0.878654	−	0.477458i	$-0.841558\pi$
$383$	− 365.733i	− 0.954917i	0.878654	−	0.477458i	$-0.158442\pi$
$384$	0	0
$385$	−17.5560	−0.0455999
$386$	− 273.050i	− 0.707383i
$387$	0	0
$388$	175.404	0.452071
$389$	− 147.384i	− 0.378880i	−0.981892	−	0.189440i	$-0.939333\pi$
$389$	− 147.384i	− 0.378880i	0.981892	−	0.189440i	$-0.0606673\pi$
$390$	0	0
$391$	66.3157	0.169605
$392$	19.7990i	0.0505076i
$393$	0	0
$394$	464.712	1.17947
$395$	− 232.037i	− 0.587436i
$396$	0	0
$397$	−241.232	−0.607638	−0.303819	−	0.952730i	$-0.598262\pi$
$397$	−241.232	−0.607638	−0.303819	+	0.952730i	$0.598262\pi$
$398$	− 425.728i	− 1.06967i
$399$	0	0
$400$	65.0156	0.162539
$401$	658.660i	1.64254i	0.570537	+	0.821272i	$0.306735\pi$
$401$	658.660i	1.64254i	−0.570537	+	0.821272i	$0.693265\pi$
$402$	0	0
$403$	231.018	0.573246
$404$	61.0497i	0.151113i
$405$	0	0
$406$	−124.632	−0.306976
$407$	19.6288i	0.0482280i
$408$	0	0
$409$	467.667	1.14344	0.571720	−	0.820449i	$-0.306277\pi$
$409$	467.667	1.14344	0.571720	+	0.820449i	$0.306277\pi$
$410$	− 295.378i	− 0.720435i
$411$	0	0
$412$	83.4826	0.202628
$413$	101.740i	0.246344i
$414$	0	0
$415$	337.045	0.812157
$416$	25.3507i	0.0609391i
$417$	0	0
$418$	−47.9360	−0.114679
$419$	225.059i	0.537133i	0.963261	+	0.268567i	$0.0865500\pi$
$419$	225.059i	0.537133i	−0.963261	+	0.268567i	$0.913450\pi$
$420$	0	0
$421$	500.783	1.18951	0.594754	−	0.803908i	$-0.297249\pi$
$421$	500.783	1.18951	0.594754	+	0.803908i	$0.297249\pi$
$422$	265.331i	0.628747i
$423$	0	0
$424$	109.270	0.257713
$425$	51.5277i	0.121242i
$426$	0	0
$427$	193.978	0.454280
$428$	105.307i	0.246044i
$429$	0	0
$430$	−187.932	−0.437052
$431$	− 465.507i	− 1.08006i	−0.841645	−	0.540031i	$-0.818413\pi$
$431$	− 465.507i	− 1.08006i	0.841645	−	0.540031i	$-0.181587\pi$
$432$	0	0
$433$	−229.008	−0.528887	−0.264443	−	0.964401i	$-0.585188\pi$
$433$	−229.008	−0.528887	−0.264443	+	0.964401i	$0.585188\pi$
$434$	− 192.884i	− 0.444433i
$435$	0	0
$436$	387.531	0.888832
$437$	− 316.017i	− 0.723152i
$438$	0	0
$439$	−769.472	−1.75278	−0.876392	−	0.481599i	$-0.840056\pi$
$439$	−769.472	−1.75278	−0.876392	+	0.481599i	$0.840056\pi$
$440$	18.7681i	0.0426548i
$441$	0	0
$442$	−20.0915	−0.0454559
$443$	257.695i	0.581705i	0.956768	+	0.290853i	$0.0939389\pi$
$443$	257.695i	0.581705i	−0.956768	+	0.290853i	$0.906061\pi$
$444$	0	0
$445$	485.622	1.09129
$446$	587.306i	1.31683i
$447$	0	0
$448$	21.1660	0.0472456
$449$	90.2076i	0.200908i	0.994942	+	0.100454i	$0.0320295\pi$
$449$	90.2076i	0.200908i	−0.994942	+	0.100454i	$0.967971\pi$
$450$	0	0
$451$	−158.462	−0.351357
$452$	− 250.345i	− 0.553860i
$453$	0	0
$454$	−486.277	−1.07110
$455$	35.0647i	0.0770653i
$456$	0	0
$457$	−328.002	−0.717728	−0.358864	−	0.933390i	$-0.616836\pi$
$457$	−328.002	−0.717728	−0.358864	+	0.933390i	$0.616836\pi$
$458$	102.153i	0.223042i
$459$	0	0
$460$	−123.729	−0.268975
$461$	729.439i	1.58230i	0.611624	+	0.791149i	$0.290517\pi$
$461$	729.439i	1.58230i	−0.611624	+	0.791149i	$0.709483\pi$
$462$	0	0
$463$	100.082	0.216159	0.108080	−	0.994142i	$-0.465530\pi$
$463$	100.082	0.216159	0.108080	+	0.994142i	$0.465530\pi$
$464$	133.238i	0.287150i
$465$	0	0
$466$	496.901	1.06631
$467$	− 57.7466i	− 0.123654i	−0.998087	−	0.0618272i	$-0.980307\pi$
$467$	− 57.7466i	− 0.123654i	0.998087	−	0.0618272i	$-0.0196928\pi$
$468$	0	0
$469$	288.161	0.614415
$470$	78.1887i	0.166359i
$471$	0	0
$472$	108.765	0.230434
$473$	100.820i	0.213151i
$474$	0	0
$475$	245.548	0.516942
$476$	16.7750i	0.0352416i
$477$	0	0
$478$	143.713	0.300655
$479$	− 413.163i	− 0.862554i	−0.902220	−	0.431277i	$-0.858063\pi$
$479$	− 413.163i	− 0.862554i	0.902220	−	0.431277i	$-0.141937\pi$
$480$	0	0
$481$	39.2048	0.0815070
$482$	24.3772i	0.0505750i
$483$	0	0
$484$	−231.931	−0.479197
$485$	− 259.368i	− 0.534778i
$486$	0	0
$487$	851.825	1.74913	0.874564	−	0.484910i	$-0.161148\pi$
$487$	851.825	1.74913	0.874564	+	0.484910i	$0.161148\pi$
$488$	− 207.371i	− 0.424940i
$489$	0	0
$490$	29.2766	0.0597481
$491$	108.134i	0.220232i	0.993919	+	0.110116i	$0.0351223\pi$
$491$	108.134i	0.220232i	−0.993919	+	0.110116i	$0.964878\pi$
$492$	0	0
$493$	−105.597	−0.214192
$494$	95.7431i	0.193812i
$495$	0	0
$496$	−206.202	−0.415729
$497$	61.5939i	0.123931i
$498$	0	0
$499$	436.336	0.874421	0.437211	−	0.899359i	$-0.355966\pi$
$499$	436.336	0.874421	0.437211	+	0.899359i	$0.355966\pi$
$500$	− 244.007i	− 0.488014i
$501$	0	0
$502$	−351.639	−0.700475
$503$	309.461i	0.615230i	0.951511	+	0.307615i	$0.0995309\pi$
$503$	309.461i	0.615230i	−0.951511	+	0.307615i	$0.900469\pi$
$504$	0	0
$505$	90.2736	0.178760
$506$	66.3768i	0.131179i
$507$	0	0
$508$	152.399	0.299999
$509$	245.031i	0.481397i	0.970600	+	0.240698i	$0.0773764\pi$
$509$	245.031i	0.481397i	−0.970600	+	0.240698i	$0.922624\pi$
$510$	0	0
$511$	−126.595	−0.247740
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−339.289	−0.660095
$515$	− 123.445i	− 0.239699i
$516$	0	0
$517$	41.9460	0.0811334
$518$	− 32.7333i	− 0.0631917i
$519$	0	0
$520$	37.4858	0.0720880
$521$	− 727.090i	− 1.39557i	−0.716309	−	0.697783i	$-0.754170\pi$
$521$	− 727.090i	− 1.39557i	0.716309	−	0.697783i	$-0.245830\pi$
$522$	0	0
$523$	−434.448	−0.830684	−0.415342	−	0.909665i	$-0.636338\pi$
$523$	−434.448	−0.830684	−0.415342	+	0.909665i	$0.636338\pi$
$524$	− 350.290i	− 0.668492i
$525$	0	0
$526$	128.100	0.243536
$527$	− 163.424i	− 0.310102i
$528$	0	0
$529$	91.4114	0.172800
$530$	− 161.577i	− 0.304862i
$531$	0	0
$532$	79.9387	0.150261
$533$	316.498i	0.593804i
$534$	0	0
$535$	155.716	0.291058
$536$	− 308.057i	− 0.574733i
$537$	0	0
$538$	340.840	0.633532
$539$	− 15.7060i	− 0.0291392i
$540$	0	0
$541$	126.002	0.232906	0.116453	−	0.993196i	$-0.462848\pi$
$541$	126.002	0.232906	0.116453	+	0.993196i	$0.462848\pi$
$542$	166.901i	0.307936i
$543$	0	0
$544$	17.9332	0.0329655
$545$	− 573.038i	− 1.05145i
$546$	0	0
$547$	533.893	0.976039	0.488019	−	0.872833i	$-0.337720\pi$
$547$	533.893	0.976039	0.488019	+	0.872833i	$0.337720\pi$
$548$	− 131.848i	− 0.240598i
$549$	0	0
$550$	−51.5752	−0.0937731
$551$	503.205i	0.913258i
$552$	0	0
$553$	207.587	0.375383
$554$	− 604.319i	− 1.09083i
$555$	0	0
$556$	74.0603	0.133202
$557$	1055.89i	1.89567i	0.318766	+	0.947833i	$0.396731\pi$
$557$	1055.89i	1.89567i	−0.318766	+	0.947833i	$0.603269\pi$
$558$	0	0
$559$	201.370	0.360232
$560$	− 31.2980i	− 0.0558892i
$561$	0	0
$562$	−268.291	−0.477387
$563$	187.605i	0.333224i	0.986022	+	0.166612i	$0.0532828\pi$
$563$	187.605i	0.333224i	−0.986022	+	0.166612i	$0.946717\pi$
$564$	0	0
$565$	−370.182	−0.655189
$566$	− 93.1532i	− 0.164582i
$567$	0	0
$568$	65.8467	0.115927
$569$	593.618i	1.04327i	0.853170	+	0.521633i	$0.174677\pi$
$569$	593.618i	1.04327i	−0.853170	+	0.521633i	$0.825323\pi$
$570$	0	0
$571$	−306.586	−0.536927	−0.268464	−	0.963290i	$-0.586516\pi$
$571$	−306.586	−0.536927	−0.268464	+	0.963290i	$0.586516\pi$
$572$	− 20.1100i	− 0.0351574i
$573$	0	0
$574$	264.253	0.460372
$575$	− 340.009i	− 0.591320i
$576$	0	0
$577$	452.891	0.784907	0.392454	−	0.919772i	$-0.371626\pi$
$577$	452.891	0.784907	0.392454	+	0.919772i	$0.371626\pi$
$578$	− 394.495i	− 0.682517i
$579$	0	0
$580$	197.017	0.339685
$581$	301.530i	0.518984i
$582$	0	0
$583$	−86.6814	−0.148682
$584$	135.336i	0.231740i
$585$	0	0
$586$	−533.526	−0.910454
$587$	− 665.661i	− 1.13400i	−0.823716	−	0.567002i	$-0.808103\pi$
$587$	− 665.661i	− 1.13400i	0.823716	−	0.567002i	$-0.191897\pi$
$588$	0	0
$589$	−778.772	−1.32219
$590$	− 160.829i	− 0.272592i
$591$	0	0
$592$	−34.9934	−0.0591104
$593$	960.335i	1.61945i	0.586809	+	0.809726i	$0.300384\pi$
$593$	960.335i	1.61945i	−0.586809	+	0.809726i	$0.699616\pi$
$594$	0	0
$595$	24.8050	0.0416891
$596$	281.712i	0.472670i
$597$	0	0
$598$	132.575	0.221698
$599$	− 552.783i	− 0.922844i	−0.887181	−	0.461422i	$-0.847339\pi$
$599$	− 552.783i	− 0.922844i	0.887181	−	0.461422i	$-0.152661\pi$
$600$	0	0
$601$	663.737	1.10439	0.552194	−	0.833716i	$-0.313791\pi$
$601$	663.737	1.10439	0.552194	+	0.833716i	$0.313791\pi$
$602$	− 168.129i	− 0.279285i
$603$	0	0
$604$	−575.558	−0.952911
$605$	342.955i	0.566867i
$606$	0	0
$607$	−772.801	−1.27315	−0.636574	−	0.771216i	$-0.719649\pi$
$607$	−772.801	−1.27315	−0.636574	+	0.771216i	$0.719649\pi$
$608$	− 85.4581i	− 0.140556i
$609$	0	0
$610$	−306.637	−0.502684
$611$	− 83.7792i	− 0.137118i
$612$	0	0
$613$	4.25996	0.00694936	0.00347468	−	0.999994i	$-0.498894\pi$
$613$	4.25996	0.00694936	0.00347468	+	0.999994i	$0.498894\pi$
$614$	− 561.488i	− 0.914475i
$615$	0	0
$616$	−16.7904	−0.0272572
$617$	1195.23i	1.93717i	0.248687	+	0.968584i	$0.420001\pi$
$617$	1195.23i	1.93717i	−0.248687	+	0.968584i	$0.579999\pi$
$618$	0	0
$619$	214.058	0.345813	0.172906	−	0.984938i	$-0.444684\pi$
$619$	214.058	0.345813	0.172906	+	0.984938i	$0.444684\pi$
$620$	304.908i	0.491787i
$621$	0	0
$622$	−102.586	−0.164930
$623$	434.450i	0.697352i
$624$	0	0
$625$	45.5370	0.0728593
$626$	13.0477i	0.0208430i
$627$	0	0
$628$	−177.017	−0.281874
$629$	− 27.7338i	− 0.0440918i
$630$	0	0
$631$	−646.049	−1.02385	−0.511925	−	0.859030i	$-0.671067\pi$
$631$	−646.049	−1.02385	−0.511925	+	0.859030i	$0.671067\pi$
$632$	− 221.920i	− 0.351139i
$633$	0	0
$634$	−318.294	−0.502041
$635$	− 225.351i	− 0.354884i
$636$	0	0
$637$	−31.3698	−0.0492462
$638$	− 105.694i	− 0.165664i
$639$	0	0
$640$	−33.4589	−0.0522796
$641$	− 107.335i	− 0.167449i	−0.996489	−	0.0837246i	$-0.973318\pi$
$641$	− 107.335i	− 0.167449i	0.996489	−	0.0837246i	$-0.0266816\pi$
$642$	0	0
$643$	−1165.81	−1.81308	−0.906538	−	0.422124i	$-0.861285\pi$
$643$	−1165.81	−1.81308	−0.906538	+	0.422124i	$0.861285\pi$
$644$	− 110.691i	− 0.171880i
$645$	0	0
$646$	67.7293	0.104844
$647$	246.501i	0.380991i	0.981688	+	0.190495i	$0.0610094\pi$
$647$	246.501i	0.380991i	−0.981688	+	0.190495i	$0.938991\pi$
$648$	0	0
$649$	−86.2802	−0.132943
$650$	103.012i	0.158480i
$651$	0	0
$652$	381.590	0.585261
$653$	113.119i	0.173230i	0.996242	+	0.0866151i	$0.0276050\pi$
$653$	113.119i	0.173230i	−0.996242	+	0.0866151i	$0.972395\pi$
$654$	0	0
$655$	−517.970	−0.790794
$656$	− 282.499i	− 0.430638i
$657$	0	0
$658$	−69.9497	−0.106307
$659$	− 643.090i	− 0.975858i	−0.872883	−	0.487929i	$-0.837753\pi$
$659$	− 643.090i	− 0.975858i	0.872883	−	0.487929i	$-0.162247\pi$
$660$	0	0
$661$	−975.829	−1.47629	−0.738146	−	0.674641i	$-0.764298\pi$
$661$	−975.829	−1.47629	−0.738146	+	0.674641i	$0.764298\pi$
$662$	− 553.747i	− 0.836475i
$663$	0	0
$664$	322.349	0.485465
$665$	− 118.205i	− 0.177751i
$666$	0	0
$667$	696.787	1.04466
$668$	297.726i	0.445697i
$669$	0	0
$670$	−455.521	−0.679882
$671$	164.502i	0.245159i
$672$	0	0
$673$	−803.825	−1.19439	−0.597195	−	0.802096i	$-0.703718\pi$
$673$	−803.825	−1.19439	−0.597195	+	0.802096i	$0.703718\pi$
$674$	445.801i	0.661425i
$675$	0	0
$676$	297.834	0.440583
$677$	508.559i	0.751195i	0.926783	+	0.375598i	$0.122562\pi$
$677$	508.559i	0.751195i	−0.926783	+	0.375598i	$0.877438\pi$
$678$	0	0
$679$	232.037	0.341734
$680$	− 26.5177i	− 0.0389966i
$681$	0	0
$682$	163.574	0.239845
$683$	184.107i	0.269557i	0.990876	+	0.134778i	$0.0430323\pi$
$683$	184.107i	0.269557i	−0.990876	+	0.134778i	$0.956968\pi$
$684$	0	0
$685$	−194.962	−0.284616
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	−179.738	−0.261247
$689$	173.130i	0.251277i
$690$	0	0
$691$	−532.248	−0.770258	−0.385129	−	0.922863i	$-0.625843\pi$
$691$	−532.248	−0.770258	−0.385129	+	0.922863i	$0.625843\pi$
$692$	− 138.006i	− 0.199430i
$693$	0	0
$694$	341.077	0.491466
$695$	− 109.512i	− 0.157572i
$696$	0	0
$697$	223.893	0.321223
$698$	550.201i	0.788253i
$699$	0	0
$700$	86.0076	0.122868
$701$	− 316.163i	− 0.451017i	−0.974241	−	0.225509i	$-0.927596\pi$
$701$	− 316.163i	− 0.451017i	0.974241	−	0.225509i	$-0.0724044\pi$
$702$	0	0
$703$	−132.161	−0.187996
$704$	17.9497i	0.0254968i
$705$	0	0
$706$	−239.925	−0.339838
$707$	80.7612i	0.114231i
$708$	0	0
$709$	−995.000	−1.40339	−0.701693	−	0.712480i	$-0.747572\pi$
$709$	−995.000	−1.40339	−0.701693	+	0.712480i	$0.747572\pi$
$710$	− 97.3668i	− 0.137136i
$711$	0	0
$712$	464.447	0.652313
$713$	1078.36i	1.51243i
$714$	0	0
$715$	−29.7365	−0.0415895
$716$	594.276i	0.829994i
$717$	0	0
$718$	−679.419	−0.946266
$719$	486.059i	0.676020i	0.941142	+	0.338010i	$0.109754\pi$
$719$	486.059i	0.676020i	−0.941142	+	0.338010i	$0.890246\pi$
$720$	0	0
$721$	110.437	0.153172
$722$	187.777i	0.260079i
$723$	0	0
$724$	−17.3640	−0.0239835
$725$	541.408i	0.746769i
$726$	0	0
$727$	−417.265	−0.573954	−0.286977	−	0.957937i	$-0.592650\pi$
$727$	−417.265	−0.573954	−0.286977	+	0.957937i	$0.592650\pi$
$728$	33.5358i	0.0460656i
$729$	0	0
$730$	200.120	0.274137
$731$	− 142.450i	− 0.194870i
$732$	0	0
$733$	−977.052	−1.33295	−0.666475	−	0.745527i	$-0.732198\pi$
$733$	−977.052	−1.33295	−0.666475	+	0.745527i	$0.732198\pi$
$734$	− 473.001i	− 0.644416i
$735$	0	0
$736$	−118.334	−0.160779
$737$	244.374i	0.331579i
$738$	0	0
$739$	1165.38	1.57697	0.788484	−	0.615056i	$-0.210866\pi$
$739$	1165.38	1.57697	0.788484	+	0.615056i	$0.210866\pi$
$740$	51.7443i	0.0699248i
$741$	0	0
$742$	144.551	0.194813
$743$	− 88.5990i	− 0.119245i	−0.998221	−	0.0596224i	$-0.981010\pi$
$743$	− 88.5990i	− 0.119245i	0.998221	−	0.0596224i	$-0.0189897\pi$
$744$	0	0
$745$	416.564	0.559146
$746$	28.1001i	0.0376677i
$747$	0	0
$748$	−14.2260	−0.0190187
$749$	139.308i	0.185992i
$750$	0	0
$751$	226.828	0.302035	0.151017	−	0.988531i	$-0.451745\pi$
$751$	226.828	0.302035	0.151017	+	0.988531i	$0.451745\pi$
$752$	74.7794i	0.0994407i
$753$	0	0
$754$	−211.104	−0.279978
$755$	851.072i	1.12725i
$756$	0	0
$757$	−870.736	−1.15025	−0.575123	−	0.818067i	$-0.695046\pi$
$757$	−870.736	−1.15025	−0.575123	+	0.818067i	$0.695046\pi$
$758$	146.767i	0.193624i
$759$	0	0
$760$	−126.366	−0.166271
$761$	− 305.625i	− 0.401610i	−0.979631	−	0.200805i	$-0.935644\pi$
$761$	− 305.625i	− 0.401610i	0.979631	−	0.200805i	$-0.0643558\pi$
$762$	0	0
$763$	512.655	0.671894
$764$	− 431.582i	− 0.564898i
$765$	0	0
$766$	−517.225	−0.675228
$767$	172.328i	0.224679i
$768$	0	0
$769$	58.8251	0.0764956	0.0382478	−	0.999268i	$-0.487822\pi$
$769$	58.8251	0.0764956	0.0382478	+	0.999268i	$0.487822\pi$
$770$	24.8279i	0.0322440i
$771$	0	0
$772$	−386.151	−0.500195
$773$	180.353i	0.233315i	0.993172	+	0.116658i	$0.0372180\pi$
$773$	180.353i	0.233315i	−0.993172	+	0.116658i	$0.962782\pi$
$774$	0	0
$775$	−837.895	−1.08115
$776$	− 248.058i	− 0.319663i
$777$	0	0
$778$	−208.433	−0.267909
$779$	− 1066.93i	− 1.36961i
$780$	0	0
$781$	−52.2345	−0.0668815
$782$	− 93.7845i	− 0.119929i
$783$	0	0
$784$	28.0000	0.0357143
$785$	261.753i	0.333444i
$786$	0	0
$787$	−269.961	−0.343025	−0.171513	−	0.985182i	$-0.554865\pi$
$787$	−269.961	−0.343025	−0.171513	+	0.985182i	$0.554865\pi$
$788$	− 657.201i	− 0.834012i
$789$	0	0
$790$	−328.150	−0.415380
$791$	− 331.175i	− 0.418678i
$792$	0	0
$793$	328.562	0.414328
$794$	341.154i	0.429665i
$795$	0	0
$796$	−602.071	−0.756371
$797$	934.360i	1.17235i	0.810186	+	0.586173i	$0.199366\pi$
$797$	934.360i	1.17235i	−0.810186	+	0.586173i	$0.800634\pi$
$798$	0	0
$799$	−59.2660	−0.0741752
$800$	− 91.9460i	− 0.114932i
$801$	0	0
$802$	931.486	1.16145
$803$	− 107.358i	− 0.133697i
$804$	0	0
$805$	−163.678	−0.203326
$806$	− 326.709i	− 0.405346i
$807$	0	0
$808$	86.3373	0.106853
$809$	1442.65i	1.78326i	0.452768	+	0.891628i	$0.350436\pi$
$809$	1442.65i	1.78326i	−0.452768	+	0.891628i	$0.649564\pi$
$810$	0	0
$811$	−1560.38	−1.92402	−0.962009	−	0.273017i	$-0.911979\pi$
$811$	−1560.38	−1.92402	−0.962009	+	0.273017i	$0.911979\pi$
$812$	176.257i	0.217065i
$813$	0	0
$814$	27.7593	0.0341024
$815$	− 564.253i	− 0.692335i
$816$	0	0
$817$	−678.825	−0.830875
$818$	− 661.380i	− 0.808533i
$819$	0	0
$820$	−417.728	−0.509424
$821$	− 295.257i	− 0.359631i	−0.983700	−	0.179815i	$-0.942450\pi$
$821$	− 295.257i	− 0.359631i	0.983700	−	0.179815i	$-0.0575500\pi$
$822$	0	0
$823$	1085.01	1.31836	0.659178	−	0.751987i	$-0.270904\pi$
$823$	1085.01	1.31836	0.659178	+	0.751987i	$0.270904\pi$
$824$	− 118.062i	− 0.143279i
$825$	0	0
$826$	143.882	0.174191
$827$	1304.29i	1.57714i	0.614948	+	0.788568i	$0.289177\pi$
$827$	1304.29i	1.57714i	−0.614948	+	0.788568i	$0.710823\pi$
$828$	0	0
$829$	−912.447	−1.10066	−0.550330	−	0.834947i	$-0.685498\pi$
$829$	−912.447	−1.10066	−0.550330	+	0.834947i	$0.685498\pi$
$830$	− 476.654i	− 0.574282i
$831$	0	0
$832$	35.8512	0.0430904
$833$	22.1912i	0.0266401i
$834$	0	0
$835$	440.244	0.527238
$836$	67.7917i	0.0810905i
$837$	0	0
$838$	318.281	0.379811
$839$	709.512i	0.845664i	0.906208	+	0.422832i	$0.138964\pi$
$839$	709.512i	0.845664i	−0.906208	+	0.422832i	$0.861036\pi$
$840$	0	0
$841$	−268.516	−0.319282
$842$	− 708.214i	− 0.841109i
$843$	0	0
$844$	375.235	0.444591
$845$	− 440.404i	− 0.521188i
$846$	0	0
$847$	−306.816	−0.362239
$848$	− 154.532i	− 0.182231i
$849$	0	0
$850$	72.8712	0.0857308
$851$	183.003i	0.215045i
$852$	0	0
$853$	37.7697	0.0442787	0.0221393	−	0.999755i	$-0.492952\pi$
$853$	37.7697	0.0442787	0.0221393	+	0.999755i	$0.492952\pi$
$854$	− 274.326i	− 0.321225i
$855$	0	0
$856$	148.926	0.173979
$857$	− 369.235i	− 0.430846i	−0.976521	−	0.215423i	$-0.930887\pi$
$857$	− 369.235i	− 0.430846i	0.976521	−	0.215423i	$-0.0691130\pi$
$858$	0	0
$859$	−308.004	−0.358561	−0.179280	−	0.983798i	$-0.557377\pi$
$859$	−308.004	−0.358561	−0.179280	+	0.983798i	$0.557377\pi$
$860$	265.777i	0.309043i
$861$	0	0
$862$	−658.326	−0.763719
$863$	915.511i	1.06085i	0.847733	+	0.530423i	$0.177967\pi$
$863$	915.511i	1.06085i	−0.847733	+	0.530423i	$0.822033\pi$
$864$	0	0
$865$	−204.067	−0.235916
$866$	323.866i	0.373979i
$867$	0	0
$868$	−272.779	−0.314262
$869$	176.043i	0.202581i
$870$	0	0
$871$	488.090	0.560379
$872$	− 548.051i	− 0.628499i
$873$	0	0
$874$	−446.916	−0.511346
$875$	− 322.791i	− 0.368904i
$876$	0	0
$877$	1072.04	1.22240	0.611198	−	0.791478i	$-0.290688\pi$
$877$	1072.04	1.22240	0.611198	+	0.791478i	$0.290688\pi$
$878$	1088.20i	1.23941i
$879$	0	0
$880$	26.5421	0.0301615
$881$	− 1203.17i	− 1.36569i	−0.730562	−	0.682846i	$-0.760742\pi$
$881$	− 1203.17i	− 1.36569i	0.730562	−	0.682846i	$-0.239258\pi$
$882$	0	0
$883$	−1529.93	−1.73265	−0.866325	−	0.499481i	$-0.833524\pi$
$883$	−1529.93	−1.73265	−0.866325	+	0.499481i	$0.833524\pi$
$884$	28.4137i	0.0321422i
$885$	0	0
$886$	364.436	0.411328
$887$	− 679.061i	− 0.765571i	−0.923837	−	0.382785i	$-0.874965\pi$
$887$	− 679.061i	− 0.765571i	0.923837	−	0.382785i	$-0.125035\pi$
$888$	0	0
$889$	201.605	0.226778
$890$	− 686.773i	− 0.771655i
$891$	0	0
$892$	830.576	0.931139
$893$	282.423i	0.316263i
$894$	0	0
$895$	878.750	0.981843
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	127.573	0.142063
$899$	− 1717.11i	− 1.91002i
$900$	0	0
$901$	122.473	0.135930
$902$	224.099i	0.248447i
$903$	0	0
$904$	−354.041	−0.391638
$905$	25.6760i	0.0283713i
$906$	0	0
$907$	−575.405	−0.634405	−0.317203	−	0.948358i	$-0.602743\pi$
$907$	−575.405	−0.634405	−0.317203	+	0.948358i	$0.602743\pi$
$908$	687.700i	0.757379i
$909$	0	0
$910$	49.5890	0.0544934
$911$	817.676i	0.897559i	0.893643	+	0.448779i	$0.148141\pi$
$911$	817.676i	0.897559i	−0.893643	+	0.448779i	$0.851859\pi$
$912$	0	0
$913$	−255.711	−0.280078
$914$	463.864i	0.507510i
$915$	0	0
$916$	144.466	0.157714
$917$	− 463.390i	− 0.505333i
$918$	0	0
$919$	−883.992	−0.961906	−0.480953	−	0.876746i	$-0.659709\pi$
$919$	−883.992	−0.961906	−0.480953	+	0.876746i	$0.659709\pi$
$920$	174.979i	0.190194i
$921$	0	0
$922$	1031.58	1.11885
$923$	104.329i	0.113032i
$924$	0	0
$925$	−142.195	−0.153724
$926$	− 141.537i	− 0.152848i
$927$	0	0
$928$	188.426	0.203046
$929$	− 1321.59i	− 1.42260i	−0.702890	−	0.711299i	$-0.748107\pi$
$929$	− 1321.59i	− 1.42260i	0.702890	−	0.711299i	$-0.251893\pi$
$930$	0	0
$931$	105.749	0.113586
$932$	− 702.724i	− 0.753995i
$933$	0	0
$934$	−81.6661	−0.0874369
$935$	21.0358i	0.0224982i
$936$	0	0
$937$	−526.249	−0.561632	−0.280816	−	0.959762i	$-0.590605\pi$
$937$	−526.249	−0.561632	−0.280816	+	0.959762i	$0.590605\pi$
$938$	− 407.521i	− 0.434457i
$939$	0	0
$940$	110.576	0.117634
$941$	1363.58i	1.44908i	0.689234	+	0.724539i	$0.257947\pi$
$941$	1363.58i	1.44908i	−0.689234	+	0.724539i	$0.742053\pi$
$942$	0	0
$943$	−1477.37	−1.56667
$944$	− 153.816i	− 0.162941i
$945$	0	0
$946$	142.581	0.150720
$947$	− 1510.90i	− 1.59546i	−0.603016	−	0.797729i	$-0.706035\pi$
$947$	− 1510.90i	− 1.59546i	0.603016	−	0.797729i	$-0.293965\pi$
$948$	0	0
$949$	−214.428	−0.225952
$950$	− 347.257i	− 0.365534i
$951$	0	0
$952$	23.7234	0.0249196
$953$	− 473.540i	− 0.496895i	−0.968645	−	0.248447i	$-0.920080\pi$
$953$	− 473.540i	− 0.496895i	0.968645	−	0.248447i	$-0.0799203\pi$
$954$	0	0
$955$	−638.177	−0.668248
$956$	− 203.241i	− 0.212595i
$957$	0	0
$958$	−584.301	−0.609918
$959$	− 174.418i	− 0.181875i
$960$	0	0
$961$	1696.44	1.76529
$962$	− 55.4440i	− 0.0576341i
$963$	0	0
$964$	34.4745	0.0357619
$965$	570.997i	0.591707i
$966$	0	0
$967$	915.249	0.946483	0.473242	−	0.880933i	$-0.343084\pi$
$967$	915.249	0.946483	0.473242	+	0.880933i	$0.343084\pi$
$968$	328.001i	0.338844i
$969$	0	0
$970$	−366.801	−0.378145
$971$	− 603.984i	− 0.622023i	−0.950406	−	0.311011i	$-0.899332\pi$
$971$	− 603.984i	− 0.622023i	0.950406	−	0.311011i	$-0.100668\pi$
$972$	0	0
$973$	97.9726	0.100691
$974$	− 1204.66i	− 1.23682i
$975$	0	0
$976$	−293.267	−0.300478
$977$	725.053i	0.742122i	0.928608	+	0.371061i	$0.121006\pi$
$977$	725.053i	0.742122i	−0.928608	+	0.371061i	$0.878994\pi$
$978$	0	0
$979$	−368.434	−0.376337
$980$	− 41.4033i	− 0.0422483i
$981$	0	0
$982$	152.925	0.155728
$983$	− 1023.10i	− 1.04079i	−0.853926	−	0.520395i	$-0.825785\pi$
$983$	− 1023.10i	− 1.04079i	0.853926	−	0.520395i	$-0.174215\pi$
$984$	0	0
$985$	−971.797	−0.986596
$986$	149.336i	0.151457i
$987$	0	0
$988$	135.401	0.137046
$989$	939.967i	0.950422i
$990$	0	0
$991$	1489.57	1.50310	0.751551	−	0.659675i	$-0.229306\pi$
$991$	1489.57	1.50310	0.751551	+	0.659675i	$0.229306\pi$
$992$	291.613i	0.293965i
$993$	0	0
$994$	87.1070	0.0876328
$995$	890.276i	0.894750i
$996$	0	0
$997$	476.635	0.478069	0.239035	−	0.971011i	$-0.423169\pi$
$997$	476.635	0.478069	0.239035	+	0.971011i	$0.423169\pi$
$998$	− 617.073i	− 0.618309i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.3.b.c.323.8		24
3.2	odd	2	inner	1134.3.b.c.323.17		24
9.2	odd	6		126.3.q.a.113.10	yes	24
9.4	even	3		126.3.q.a.29.10	✓	24
9.5	odd	6		378.3.q.a.197.3		24
9.7	even	3		378.3.q.a.71.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.q.a.29.10	✓	24	9.4	even	3
126.3.q.a.113.10	yes	24	9.2	odd	6
378.3.q.a.71.3		24	9.7	even	3
378.3.q.a.197.3		24	9.5	odd	6
1134.3.b.c.323.8		24	1.1	even	1	trivial
1134.3.b.c.323.17		24	3.2	odd	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:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1134.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.95738i q^{5} -2.64575 q^{7} +2.82843i q^{8} +4.18237 q^{10} -2.24372i q^{11} -4.48140 q^{13} +3.74166i q^{14} +4.00000 q^{16} +3.17017i q^{17} +15.1070 q^{19} -5.91476i q^{20} -3.17310 q^{22} -20.9186i q^{23} +16.2539 q^{25} +6.33766i q^{26} +5.29150 q^{28} +33.3094i q^{29} -51.5504 q^{31} -5.65685i q^{32} +4.48330 q^{34} -7.82449i q^{35} -8.74834 q^{37} -21.3645i q^{38} -8.36473 q^{40} -70.6247i q^{41} -44.9345 q^{43} +4.48744i q^{44} -29.5834 q^{46} +18.6949i q^{47} +7.00000 q^{49} -22.9865i q^{50} +8.96281 q^{52} -38.6329i q^{53} +6.63553 q^{55} -7.48331i q^{56} +47.1066 q^{58} -38.4541i q^{59} -73.3167 q^{61} +72.9033i q^{62} -8.00000 q^{64} -13.2532i q^{65} -108.915 q^{67} -6.34035i q^{68} -11.0655 q^{70} -23.2803i q^{71} +47.8485 q^{73} +12.3720i q^{74} -30.2140 q^{76} +5.93632i q^{77} -78.4605 q^{79} +11.8295i q^{80} -99.8784 q^{82} -113.968i q^{83} -9.37541 q^{85} +63.5470i q^{86} +6.34619 q^{88} -164.207i q^{89} +11.8567 q^{91} +41.8372i q^{92} +26.4385 q^{94} +44.6771i q^{95} -87.7018 q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) \) 24 * q - 48 * q^4 + 96 * q^16 + 24 * q^19 - 48 * q^22 - 144 * q^25 + 120 * q^31 + 96 * q^34 - 168 * q^37 - 120 * q^43 + 168 * q^49 + 264 * q^55 - 192 * q^64 - 144 * q^67 + 24 * q^73 - 48 * q^76 - 24 * q^79 - 288 * q^82 + 312 * q^85 + 96 * q^88 - 168 * q^91 - 96 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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