LMFDB - Embedded newform 2400.3.e.h.1951.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,3,Mod(1951,2400)] 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("2400.1951"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2400.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-36,0,0,0,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$65.3952634465$
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} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 $ x^12 - 6x^11 + 49x^10 - 190x^9 + 792x^8 - 2094x^7 + 5517x^6 - 9954x^5 + 17189x^4 - 19884x^3 + 20996x^2 - 12416*x + 5584
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1951.12
Root			$0.500000 + 1.80359i$ of defining polynomial
Character	$\chi$	$=$	2400.1951
Dual form			2400.3.e.h.1951.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +9.43574i q^{7} -3.00000 q^{9} -3.73048i q^{11} -12.5929 q^{13} -30.0484 q^{17} +10.7454i q^{19} -16.3432 q^{21} -38.0158i q^{23} -5.19615i q^{27} +27.0083 q^{29} +16.9678i q^{31} +6.46138 q^{33} -45.2400 q^{37} -21.8116i q^{39} +68.6923 q^{41} -81.0133i q^{43} -17.1188i q^{47} -40.0331 q^{49} -52.0454i q^{51} +50.3769 q^{53} -18.6115 q^{57} -63.3848i q^{59} -12.1092 q^{61} -28.3072i q^{63} -10.4760i q^{67} +65.8452 q^{69} +1.92327i q^{71} +31.3735 q^{73} +35.1998 q^{77} +69.1535i q^{79} +9.00000 q^{81} +67.3300i q^{83} +46.7798i q^{87} -63.1761 q^{89} -118.823i q^{91} -29.3890 q^{93} -51.3786 q^{97} +11.1914i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 36 q^{9} - 80 q^{17} + 40 q^{29} - 320 q^{37} + 136 q^{41} - 212 q^{49} + 176 q^{53} - 48 q^{57} + 40 q^{61} - 448 q^{73} - 448 q^{77} + 108 q^{81} + 8 q^{89} + 144 q^{93} - 224 q^{97}+O(q^{100}) $ 12 * q - 36 * q^9 - 80 * q^17 + 40 * q^29 - 320 * q^37 + 136 * q^41 - 212 * q^49 + 176 * q^53 - 48 * q^57 + 40 * q^61 - 448 * q^73 - 448 * q^77 + 108 * q^81 + 8 * q^89 + 144 * q^93 - 224 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.43574i	1.34796i	0.738748	+	0.673981i	$0.235417\pi$
$7$	9.43574i	1.34796i	−0.738748	+	0.673981i	$0.764583\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 3.73048i	− 0.339134i	−0.985519	−	0.169567i	$-0.945763\pi$
$11$	− 3.73048i	− 0.339134i	0.985519	−	0.169567i	$-0.0542370\pi$
$12$	0	0
$13$	−12.5929	−0.968686	−0.484343	−	0.874878i	$-0.660941\pi$
$13$	−12.5929	−0.968686	−0.484343	+	0.874878i	$0.660941\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−30.0484	−1.76755	−0.883777	−	0.467907i	$-0.845008\pi$
$17$	−30.0484	−1.76755	−0.883777	+	0.467907i	$0.845008\pi$
$18$	0	0
$19$	10.7454i	0.565546i	0.959187	+	0.282773i	$0.0912543\pi$
$19$	10.7454i	0.565546i	−0.959187	+	0.282773i	$0.908746\pi$
$20$	0	0
$21$	−16.3432	−0.778246
$22$	0	0
$23$	− 38.0158i	− 1.65286i	−0.563040	−	0.826430i	$-0.690368\pi$
$23$	− 38.0158i	− 1.65286i	0.563040	−	0.826430i	$-0.309632\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	27.0083	0.931321	0.465661	−	0.884963i	$-0.345817\pi$
$29$	27.0083	0.931321	0.465661	+	0.884963i	$0.345817\pi$
$30$	0	0
$31$	16.9678i	0.547348i	0.961823	+	0.273674i	$0.0882389\pi$
$31$	16.9678i	0.547348i	−0.961823	+	0.273674i	$0.911761\pi$
$32$	0	0
$33$	6.46138	0.195799
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−45.2400	−1.22270	−0.611352	−	0.791359i	$-0.709374\pi$
$37$	−45.2400	−1.22270	−0.611352	+	0.791359i	$0.709374\pi$
$38$	0	0
$39$	− 21.8116i	− 0.559271i
$40$	0	0
$41$	68.6923	1.67542	0.837711	−	0.546113i	$-0.183893\pi$
$41$	68.6923	1.67542	0.837711	+	0.546113i	$0.183893\pi$
$42$	0	0
$43$	− 81.0133i	− 1.88403i	−0.335569	−	0.942016i	$-0.608929\pi$
$43$	− 81.0133i	− 1.88403i	0.335569	−	0.942016i	$-0.391071\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 17.1188i	− 0.364230i	−0.983277	−	0.182115i	$-0.941706\pi$
$47$	− 17.1188i	− 0.364230i	0.983277	−	0.182115i	$-0.0582943\pi$
$48$	0	0
$49$	−40.0331	−0.817003
$50$	0	0
$51$	− 52.0454i	− 1.02050i
$52$	0	0
$53$	50.3769	0.950507	0.475254	−	0.879849i	$-0.342356\pi$
$53$	50.3769	0.950507	0.475254	+	0.879849i	$0.342356\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−18.6115	−0.326518
$58$	0	0
$59$	− 63.3848i	− 1.07432i	−0.843481	−	0.537160i	$-0.819497\pi$
$59$	− 63.3848i	− 1.07432i	0.843481	−	0.537160i	$-0.180503\pi$
$60$	0	0
$61$	−12.1092	−0.198512	−0.0992560	−	0.995062i	$-0.531646\pi$
$61$	−12.1092	−0.198512	−0.0992560	+	0.995062i	$0.531646\pi$
$62$	0	0
$63$	− 28.3072i	− 0.449321i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.4760i	− 0.156358i	−0.996939	−	0.0781790i	$-0.975089\pi$
$67$	− 10.4760i	− 0.156358i	0.996939	−	0.0781790i	$-0.0249106\pi$
$68$	0	0
$69$	65.8452	0.954279
$70$	0	0
$71$	1.92327i	0.0270883i	0.999908	+	0.0135442i	$0.00431137\pi$
$71$	1.92327i	0.0270883i	−0.999908	+	0.0135442i	$0.995689\pi$
$72$	0	0
$73$	31.3735	0.429775	0.214887	−	0.976639i	$-0.431062\pi$
$73$	31.3735	0.429775	0.214887	+	0.976639i	$0.431062\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	35.1998	0.457140
$78$	0	0
$79$	69.1535i	0.875361i	0.899131	+	0.437681i	$0.144200\pi$
$79$	69.1535i	0.875361i	−0.899131	+	0.437681i	$0.855800\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	67.3300i	0.811205i	0.914050	+	0.405603i	$0.132938\pi$
$83$	67.3300i	0.811205i	−0.914050	+	0.405603i	$0.867062\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	46.7798i	0.537699i
$88$	0	0
$89$	−63.1761	−0.709844	−0.354922	−	0.934896i	$-0.615493\pi$
$89$	−63.1761	−0.709844	−0.354922	+	0.934896i	$0.615493\pi$
$90$	0	0
$91$	− 118.823i	− 1.30575i
$92$	0	0
$93$	−29.3890	−0.316011
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−51.3786	−0.529676	−0.264838	−	0.964293i	$-0.585319\pi$
$97$	−51.3786	−0.529676	−0.264838	+	0.964293i	$0.585319\pi$
$98$	0	0
$99$	11.1914i	0.113045i
$100$	0	0
$101$	68.0951	0.674209	0.337105	−	0.941467i	$-0.390552\pi$
$101$	68.0951	0.674209	0.337105	+	0.941467i	$0.390552\pi$
$102$	0	0
$103$	155.320i	1.50796i	0.656895	+	0.753982i	$0.271869\pi$
$103$	155.320i	1.50796i	−0.656895	+	0.753982i	$0.728131\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	21.7681i	0.203440i	0.994813	+	0.101720i	$0.0324346\pi$
$107$	21.7681i	0.203440i	−0.994813	+	0.101720i	$0.967565\pi$
$108$	0	0
$109$	126.784	1.16315	0.581577	−	0.813492i	$-0.302436\pi$
$109$	126.784	1.16315	0.581577	+	0.813492i	$0.302436\pi$
$110$	0	0
$111$	− 78.3580i	− 0.705928i
$112$	0	0
$113$	−80.0613	−0.708507	−0.354253	−	0.935149i	$-0.615265\pi$
$113$	−80.0613	−0.708507	−0.354253	+	0.935149i	$0.615265\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	37.7788	0.322895
$118$	0	0
$119$	− 283.529i	− 2.38260i
$120$	0	0
$121$	107.084	0.884988
$122$	0	0
$123$	118.979i	0.967306i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 156.543i	− 1.23262i	−0.787502	−	0.616312i	$-0.788626\pi$
$127$	− 156.543i	− 1.23262i	0.787502	−	0.616312i	$-0.211374\pi$
$128$	0	0
$129$	140.319	1.08775
$130$	0	0
$131$	35.5220i	0.271160i	0.990766	+	0.135580i	$0.0432898\pi$
$131$	35.5220i	0.271160i	−0.990766	+	0.135580i	$0.956710\pi$
$132$	0	0
$133$	−101.391	−0.762335
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−105.233	−0.768122	−0.384061	−	0.923308i	$-0.625475\pi$
$137$	−105.233	−0.768122	−0.384061	+	0.923308i	$0.625475\pi$
$138$	0	0
$139$	181.835i	1.30817i	0.756423	+	0.654083i	$0.226945\pi$
$139$	181.835i	1.30817i	−0.756423	+	0.654083i	$0.773055\pi$
$140$	0	0
$141$	29.6506	0.210288
$142$	0	0
$143$	46.9776i	0.328515i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 69.3394i	− 0.471697i
$148$	0	0
$149$	−132.241	−0.887527	−0.443763	−	0.896144i	$-0.646357\pi$
$149$	−132.241	−0.887527	−0.443763	+	0.896144i	$0.646357\pi$
$150$	0	0
$151$	− 250.226i	− 1.65713i	−0.559895	−	0.828564i	$-0.689158\pi$
$151$	− 250.226i	− 1.65713i	0.559895	−	0.828564i	$-0.310842\pi$
$152$	0	0
$153$	90.1453	0.589185
$154$	0	0
$155$	0	0
$156$	0	0
$157$	283.974	1.80875	0.904377	−	0.426735i	$-0.140336\pi$
$157$	283.974	1.80875	0.904377	+	0.426735i	$0.140336\pi$
$158$	0	0
$159$	87.2553i	0.548775i
$160$	0	0
$161$	358.707	2.22799
$162$	0	0
$163$	− 142.259i	− 0.872753i	−0.899764	−	0.436376i	$-0.856262\pi$
$163$	− 142.259i	− 0.872753i	0.899764	−	0.436376i	$-0.143738\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 59.4298i	− 0.355867i	−0.984042	−	0.177934i	$-0.943059\pi$
$167$	− 59.4298i	− 0.355867i	0.984042	−	0.177934i	$-0.0569412\pi$
$168$	0	0
$169$	−10.4183	−0.0616470
$170$	0	0
$171$	− 32.2361i	− 0.188515i
$172$	0	0
$173$	235.203	1.35956	0.679778	−	0.733418i	$-0.262076\pi$
$173$	235.203	1.35956	0.679778	+	0.733418i	$0.262076\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	109.786	0.620258
$178$	0	0
$179$	− 52.4818i	− 0.293194i	−0.989196	−	0.146597i	$-0.953168\pi$
$179$	− 52.4818i	− 0.293194i	0.989196	−	0.146597i	$-0.0468321\pi$
$180$	0	0
$181$	−14.9662	−0.0826864	−0.0413432	−	0.999145i	$-0.513164\pi$
$181$	−14.9662	−0.0826864	−0.0413432	+	0.999145i	$0.513164\pi$
$182$	0	0
$183$	− 20.9738i	− 0.114611i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	112.095i	0.599439i
$188$	0	0
$189$	49.0295	0.259415
$190$	0	0
$191$	− 208.113i	− 1.08960i	−0.838567	−	0.544798i	$-0.816606\pi$
$191$	− 208.113i	− 1.08960i	0.838567	−	0.544798i	$-0.183394\pi$
$192$	0	0
$193$	−303.975	−1.57500	−0.787499	−	0.616316i	$-0.788625\pi$
$193$	−303.975	−1.57500	−0.787499	+	0.616316i	$0.788625\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	318.751	1.61803	0.809014	−	0.587790i	$-0.200002\pi$
$197$	318.751	1.61803	0.809014	+	0.587790i	$0.200002\pi$
$198$	0	0
$199$	− 180.444i	− 0.906756i	−0.891318	−	0.453378i	$-0.850219\pi$
$199$	− 180.444i	− 0.906756i	0.891318	−	0.453378i	$-0.149781\pi$
$200$	0	0
$201$	18.1449	0.0902733
$202$	0	0
$203$	254.843i	1.25539i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	114.047i	0.550953i
$208$	0	0
$209$	40.0854	0.191796
$210$	0	0
$211$	47.1478i	0.223449i	0.993739	+	0.111725i	$0.0356375\pi$
$211$	47.1478i	0.223449i	−0.993739	+	0.111725i	$0.964363\pi$
$212$	0	0
$213$	−3.33120	−0.0156395
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−160.103	−0.737804
$218$	0	0
$219$	54.3406i	0.248130i
$220$	0	0
$221$	378.398	1.71221
$222$	0	0
$223$	− 281.448i	− 1.26210i	−0.775742	−	0.631050i	$-0.782624\pi$
$223$	− 281.448i	− 1.26210i	0.775742	−	0.631050i	$-0.217376\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 241.837i	− 1.06536i	−0.846316	−	0.532682i	$-0.821184\pi$
$227$	− 241.837i	− 1.06536i	0.846316	−	0.532682i	$-0.178816\pi$
$228$	0	0
$229$	−103.354	−0.451328	−0.225664	−	0.974205i	$-0.572455\pi$
$229$	−103.354	−0.451328	−0.225664	+	0.974205i	$0.572455\pi$
$230$	0	0
$231$	60.9679i	0.263930i
$232$	0	0
$233$	−356.103	−1.52834	−0.764170	−	0.645015i	$-0.776851\pi$
$233$	−356.103	−1.52834	−0.764170	+	0.645015i	$0.776851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−119.777	−0.505390
$238$	0	0
$239$	− 61.3511i	− 0.256699i	−0.991729	−	0.128350i	$-0.959032\pi$
$239$	− 61.3511i	− 0.256699i	0.991729	−	0.128350i	$-0.0409680\pi$
$240$	0	0
$241$	−101.184	−0.419851	−0.209925	−	0.977717i	$-0.567322\pi$
$241$	−101.184	−0.419851	−0.209925	+	0.977717i	$0.567322\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 135.316i	− 0.547837i
$248$	0	0
$249$	−116.619	−0.468349
$250$	0	0
$251$	− 115.712i	− 0.461005i	−0.973072	−	0.230502i	$-0.925963\pi$
$251$	− 115.712i	− 0.461005i	0.973072	−	0.230502i	$-0.0740370\pi$
$252$	0	0
$253$	−141.817	−0.560542
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−159.654	−0.621222	−0.310611	−	0.950537i	$-0.600534\pi$
$257$	−159.654	−0.621222	−0.310611	+	0.950537i	$0.600534\pi$
$258$	0	0
$259$	− 426.873i	− 1.64816i
$260$	0	0
$261$	−81.0249	−0.310440
$262$	0	0
$263$	6.73896i	0.0256234i	0.999918	+	0.0128117i	$0.00407821\pi$
$263$	6.73896i	0.0256234i	−0.999918	+	0.0128117i	$0.995922\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 109.424i	− 0.409829i
$268$	0	0
$269$	333.834	1.24102	0.620509	−	0.784199i	$-0.286926\pi$
$269$	333.834	1.24102	0.620509	+	0.784199i	$0.286926\pi$
$270$	0	0
$271$	− 485.690i	− 1.79221i	−0.443839	−	0.896107i	$-0.646384\pi$
$271$	− 485.690i	− 1.79221i	0.443839	−	0.896107i	$-0.353616\pi$
$272$	0	0
$273$	205.808	0.753877
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−132.443	−0.478134	−0.239067	−	0.971003i	$-0.576842\pi$
$277$	−132.443	−0.478134	−0.239067	+	0.971003i	$0.576842\pi$
$278$	0	0
$279$	− 50.9033i	− 0.182449i
$280$	0	0
$281$	−265.883	−0.946202	−0.473101	−	0.881008i	$-0.656865\pi$
$281$	−265.883	−0.946202	−0.473101	+	0.881008i	$0.656865\pi$
$282$	0	0
$283$	− 352.065i	− 1.24405i	−0.782998	−	0.622024i	$-0.786311\pi$
$283$	− 352.065i	− 1.24405i	0.782998	−	0.622024i	$-0.213689\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	648.163i	2.25841i
$288$	0	0
$289$	613.908	2.12425
$290$	0	0
$291$	− 88.9904i	− 0.305809i
$292$	0	0
$293$	246.313	0.840657	0.420329	−	0.907372i	$-0.361915\pi$
$293$	246.313	0.840657	0.420329	+	0.907372i	$0.361915\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−19.3841	−0.0652665
$298$	0	0
$299$	478.730i	1.60110i
$300$	0	0
$301$	764.421	2.53960
$302$	0	0
$303$	117.944i	0.389255i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	479.633i	1.56232i	0.624329	+	0.781162i	$0.285373\pi$
$307$	479.633i	1.56232i	−0.624329	+	0.781162i	$0.714627\pi$
$308$	0	0
$309$	−269.023	−0.870623
$310$	0	0
$311$	− 238.106i	− 0.765613i	−0.923828	−	0.382807i	$-0.874958\pi$
$311$	− 238.106i	− 0.765613i	0.923828	−	0.382807i	$-0.125042\pi$
$312$	0	0
$313$	158.297	0.505741	0.252870	−	0.967500i	$-0.418625\pi$
$313$	158.297	0.505741	0.252870	+	0.967500i	$0.418625\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	172.575	0.544399	0.272200	−	0.962241i	$-0.412249\pi$
$317$	172.575	0.544399	0.272200	+	0.962241i	$0.412249\pi$
$318$	0	0
$319$	− 100.754i	− 0.315843i
$320$	0	0
$321$	−37.7035	−0.117456
$322$	0	0
$323$	− 322.882i	− 0.999634i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	219.596i	0.671547i
$328$	0	0
$329$	161.528	0.490968
$330$	0	0
$331$	− 394.897i	− 1.19304i	−0.802598	−	0.596521i	$-0.796549\pi$
$331$	− 394.897i	− 1.19304i	0.802598	−	0.596521i	$-0.203451\pi$
$332$	0	0
$333$	135.720	0.407568
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−552.712	−1.64010	−0.820048	−	0.572295i	$-0.806053\pi$
$337$	−552.712	−1.64010	−0.820048	+	0.572295i	$0.806053\pi$
$338$	0	0
$339$	− 138.670i	− 0.409057i
$340$	0	0
$341$	63.2979	0.185624
$342$	0	0
$343$	84.6090i	0.246674i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	342.973i	0.988395i	0.869350	+	0.494197i	$0.164538\pi$
$347$	342.973i	0.988395i	−0.869350	+	0.494197i	$0.835462\pi$
$348$	0	0
$349$	21.4938	0.0615869	0.0307935	−	0.999526i	$-0.490197\pi$
$349$	21.4938	0.0615869	0.0307935	+	0.999526i	$0.490197\pi$
$350$	0	0
$351$	65.4347i	0.186424i
$352$	0	0
$353$	623.505	1.76630	0.883152	−	0.469086i	$-0.155417\pi$
$353$	623.505	1.76630	0.883152	+	0.469086i	$0.155417\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	491.087	1.37559
$358$	0	0
$359$	283.468i	0.789605i	0.918766	+	0.394802i	$0.129187\pi$
$359$	283.468i	0.789605i	−0.918766	+	0.394802i	$0.870813\pi$
$360$	0	0
$361$	245.537	0.680158
$362$	0	0
$363$	185.474i	0.510948i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 542.611i	− 1.47850i	−0.673429	−	0.739252i	$-0.735179\pi$
$367$	− 542.611i	− 1.47850i	0.673429	−	0.739252i	$-0.264821\pi$
$368$	0	0
$369$	−206.077	−0.558474
$370$	0	0
$371$	475.343i	1.28125i
$372$	0	0
$373$	248.299	0.665682	0.332841	−	0.942983i	$-0.391993\pi$
$373$	248.299	0.665682	0.332841	+	0.942983i	$0.391993\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−340.114	−0.902158
$378$	0	0
$379$	− 304.847i	− 0.804345i	−0.915564	−	0.402173i	$-0.868255\pi$
$379$	− 304.847i	− 0.804345i	0.915564	−	0.402173i	$-0.131745\pi$
$380$	0	0
$381$	271.141	0.711656
$382$	0	0
$383$	3.78168i	0.00987383i	0.999988	+	0.00493692i	$0.00157148\pi$
$383$	3.78168i	0.00987383i	−0.999988	+	0.00493692i	$0.998429\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	243.040i	0.628010i
$388$	0	0
$389$	−394.970	−1.01535	−0.507674	−	0.861549i	$-0.669495\pi$
$389$	−394.970	−1.01535	−0.507674	+	0.861549i	$0.669495\pi$
$390$	0	0
$391$	1142.31i	2.92152i
$392$	0	0
$393$	−61.5259	−0.156554
$394$	0	0
$395$	0	0
$396$	0	0
$397$	287.721	0.724739	0.362369	−	0.932035i	$-0.381968\pi$
$397$	287.721	0.724739	0.362369	+	0.932035i	$0.381968\pi$
$398$	0	0
$399$	− 175.614i	− 0.440134i
$400$	0	0
$401$	−640.735	−1.59784	−0.798922	−	0.601435i	$-0.794596\pi$
$401$	−640.735	−1.59784	−0.798922	+	0.601435i	$0.794596\pi$
$402$	0	0
$403$	− 213.674i	− 0.530208i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	168.767i	0.414661i
$408$	0	0
$409$	−8.13388	−0.0198872	−0.00994361	−	0.999951i	$-0.503165\pi$
$409$	−8.13388	−0.0198872	−0.00994361	+	0.999951i	$0.503165\pi$
$410$	0	0
$411$	− 182.268i	− 0.443476i
$412$	0	0
$413$	598.083	1.44814
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−314.947	−0.755269
$418$	0	0
$419$	− 335.898i	− 0.801665i	−0.916151	−	0.400832i	$-0.868721\pi$
$419$	− 335.898i	− 0.801665i	0.916151	−	0.400832i	$-0.131279\pi$
$420$	0	0
$421$	−101.764	−0.241720	−0.120860	−	0.992670i	$-0.538565\pi$
$421$	−101.764	−0.241720	−0.120860	+	0.992670i	$0.538565\pi$
$422$	0	0
$423$	51.3564i	0.121410i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 114.260i	− 0.267587i
$428$	0	0
$429$	−81.3676	−0.189668
$430$	0	0
$431$	− 709.379i	− 1.64589i	−0.568120	−	0.822946i	$-0.692329\pi$
$431$	− 709.379i	− 1.64589i	0.568120	−	0.822946i	$-0.307671\pi$
$432$	0	0
$433$	−775.129	−1.79014	−0.895068	−	0.445929i	$-0.852873\pi$
$433$	−775.129	−1.79014	−0.895068	+	0.445929i	$0.852873\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	408.494	0.934768
$438$	0	0
$439$	− 49.7622i	− 0.113354i	−0.998393	−	0.0566768i	$-0.981950\pi$
$439$	− 49.7622i	− 0.113354i	0.998393	−	0.0566768i	$-0.0180505\pi$
$440$	0	0
$441$	120.099	0.272334
$442$	0	0
$443$	625.947i	1.41297i	0.707726	+	0.706487i	$0.249721\pi$
$443$	625.947i	1.41297i	−0.707726	+	0.706487i	$0.750279\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 229.049i	− 0.512414i
$448$	0	0
$449$	444.155	0.989209	0.494605	−	0.869118i	$-0.335313\pi$
$449$	444.155	0.989209	0.494605	+	0.869118i	$0.335313\pi$
$450$	0	0
$451$	− 256.255i	− 0.568194i
$452$	0	0
$453$	433.405	0.956743
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.9532	−0.0633550	−0.0316775	−	0.999498i	$-0.510085\pi$
$457$	−28.9532	−0.0633550	−0.0316775	+	0.999498i	$0.510085\pi$
$458$	0	0
$459$	156.136i	0.340166i
$460$	0	0
$461$	−651.815	−1.41392	−0.706958	−	0.707256i	$-0.749933\pi$
$461$	−651.815	−1.41392	−0.706958	+	0.707256i	$0.749933\pi$
$462$	0	0
$463$	− 329.121i	− 0.710845i	−0.934706	−	0.355422i	$-0.884337\pi$
$463$	− 329.121i	− 0.710845i	0.934706	−	0.355422i	$-0.115663\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 760.999i	− 1.62955i	−0.579778	−	0.814775i	$-0.696861\pi$
$467$	− 760.999i	− 1.62955i	0.579778	−	0.814775i	$-0.303139\pi$
$468$	0	0
$469$	98.8486	0.210765
$470$	0	0
$471$	491.858i	1.04428i
$472$	0	0
$473$	−302.219	−0.638940
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−151.131	−0.316836
$478$	0	0
$479$	536.008i	1.11902i	0.828825	+	0.559508i	$0.189010\pi$
$479$	536.008i	1.11902i	−0.828825	+	0.559508i	$0.810990\pi$
$480$	0	0
$481$	569.704	1.18442
$482$	0	0
$483$	621.298i	1.28633i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.2364i	0.0456600i	0.999739	+	0.0228300i	$0.00726764\pi$
$487$	22.2364i	0.0456600i	−0.999739	+	0.0228300i	$0.992732\pi$
$488$	0	0
$489$	246.399	0.503884
$490$	0	0
$491$	− 762.724i	− 1.55341i	−0.629865	−	0.776705i	$-0.716890\pi$
$491$	− 762.724i	− 1.55341i	0.629865	−	0.776705i	$-0.283110\pi$
$492$	0	0
$493$	−811.558	−1.64616
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.1475	−0.0365141
$498$	0	0
$499$	− 162.461i	− 0.325572i	−0.986661	−	0.162786i	$-0.947952\pi$
$499$	− 162.461i	− 0.325572i	0.986661	−	0.162786i	$-0.0520481\pi$
$500$	0	0
$501$	102.935	0.205460
$502$	0	0
$503$	517.965i	1.02975i	0.857265	+	0.514876i	$0.172162\pi$
$503$	517.965i	1.02975i	−0.857265	+	0.514876i	$0.827838\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 18.0451i	− 0.0355919i
$508$	0	0
$509$	−143.469	−0.281865	−0.140932	−	0.990019i	$-0.545010\pi$
$509$	−143.469	−0.281865	−0.140932	+	0.990019i	$0.545010\pi$
$510$	0	0
$511$	296.033i	0.579320i
$512$	0	0
$513$	55.8346	0.108839
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−63.8613	−0.123523
$518$	0	0
$519$	407.384i	0.784940i
$520$	0	0
$521$	542.459	1.04119	0.520594	−	0.853804i	$-0.325710\pi$
$521$	542.459	1.04119	0.520594	+	0.853804i	$0.325710\pi$
$522$	0	0
$523$	− 572.228i	− 1.09413i	−0.837091	−	0.547063i	$-0.815746\pi$
$523$	− 572.228i	− 1.09413i	0.837091	−	0.547063i	$-0.184254\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 509.855i	− 0.967467i
$528$	0	0
$529$	−916.199	−1.73194
$530$	0	0
$531$	190.154i	0.358106i
$532$	0	0
$533$	−865.037	−1.62296
$534$	0	0
$535$	0	0
$536$	0	0
$537$	90.9011	0.169276
$538$	0	0
$539$	149.343i	0.277074i
$540$	0	0
$541$	−875.914	−1.61906	−0.809532	−	0.587075i	$-0.800279\pi$
$541$	−875.914	−1.61906	−0.809532	+	0.587075i	$0.800279\pi$
$542$	0	0
$543$	− 25.9223i	− 0.0477390i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 776.471i	− 1.41951i	−0.704449	−	0.709754i	$-0.748806\pi$
$547$	− 776.471i	− 1.41951i	0.704449	−	0.709754i	$-0.251194\pi$
$548$	0	0
$549$	36.3277	0.0661707
$550$	0	0
$551$	290.215i	0.526705i
$552$	0	0
$553$	−652.514	−1.17995
$554$	0	0
$555$	0	0
$556$	0	0
$557$	417.952	0.750363	0.375182	−	0.926951i	$-0.377580\pi$
$557$	417.952	0.750363	0.375182	+	0.926951i	$0.377580\pi$
$558$	0	0
$559$	1020.19i	1.82504i
$560$	0	0
$561$	−194.154	−0.346086
$562$	0	0
$563$	131.170i	0.232984i	0.993192	+	0.116492i	$0.0371649\pi$
$563$	131.170i	0.232984i	−0.993192	+	0.116492i	$0.962835\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	84.9216i	0.149774i
$568$	0	0
$569$	−241.675	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$569$	−241.675	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$570$	0	0
$571$	311.504i	0.545542i	0.962079	+	0.272771i	$0.0879401\pi$
$571$	311.504i	0.545542i	−0.962079	+	0.272771i	$0.912060\pi$
$572$	0	0
$573$	360.462	0.629079
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−293.075	−0.507929	−0.253964	−	0.967214i	$-0.581735\pi$
$577$	−293.075	−0.507929	−0.253964	+	0.967214i	$0.581735\pi$
$578$	0	0
$579$	− 526.500i	− 0.909326i
$580$	0	0
$581$	−635.308	−1.09347
$582$	0	0
$583$	− 187.930i	− 0.322350i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	528.326i	0.900044i	0.893018	+	0.450022i	$0.148584\pi$
$587$	528.326i	0.900044i	−0.893018	+	0.450022i	$0.851416\pi$
$588$	0	0
$589$	−182.325	−0.309550
$590$	0	0
$591$	552.094i	0.934169i
$592$	0	0
$593$	216.255	0.364680	0.182340	−	0.983236i	$-0.441633\pi$
$593$	216.255	0.364680	0.182340	+	0.983236i	$0.441633\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	312.539	0.523516
$598$	0	0
$599$	− 300.236i	− 0.501229i	−0.968087	−	0.250615i	$-0.919367\pi$
$599$	− 300.236i	− 0.501229i	0.968087	−	0.250615i	$-0.0806327\pi$
$600$	0	0
$601$	−387.274	−0.644383	−0.322191	−	0.946675i	$-0.604419\pi$
$601$	−387.274	−0.644383	−0.322191	+	0.946675i	$0.604419\pi$
$602$	0	0
$603$	31.4279i	0.0521193i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 160.885i	− 0.265049i	−0.991180	−	0.132524i	$-0.957692\pi$
$607$	− 160.885i	− 0.265049i	0.991180	−	0.132524i	$-0.0423083\pi$
$608$	0	0
$609$	−441.402	−0.724797
$610$	0	0
$611$	215.576i	0.352824i
$612$	0	0
$613$	−150.755	−0.245930	−0.122965	−	0.992411i	$-0.539240\pi$
$613$	−150.755	−0.245930	−0.122965	+	0.992411i	$0.539240\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−252.185	−0.408728	−0.204364	−	0.978895i	$-0.565513\pi$
$617$	−252.185	−0.408728	−0.204364	+	0.978895i	$0.565513\pi$
$618$	0	0
$619$	22.1097i	0.0357184i	0.999841	+	0.0178592i	$0.00568506\pi$
$619$	22.1097i	0.0357184i	−0.999841	+	0.0178592i	$0.994315\pi$
$620$	0	0
$621$	−197.536	−0.318093
$622$	0	0
$623$	− 596.113i	− 0.956843i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	69.4300i	0.110734i
$628$	0	0
$629$	1359.39	2.16120
$630$	0	0
$631$	− 33.7222i	− 0.0534424i	−0.999643	−	0.0267212i	$-0.991493\pi$
$631$	− 33.7222i	− 0.0534424i	0.999643	−	0.0267212i	$-0.00850664\pi$
$632$	0	0
$633$	−81.6624	−0.129009
$634$	0	0
$635$	0	0
$636$	0	0
$637$	504.134	0.791419
$638$	0	0
$639$	− 5.76982i	− 0.00902944i
$640$	0	0
$641$	−492.458	−0.768265	−0.384133	−	0.923278i	$-0.625499\pi$
$641$	−492.458	−0.768265	−0.384133	+	0.923278i	$0.625499\pi$
$642$	0	0
$643$	− 981.203i	− 1.52598i	−0.646413	−	0.762988i	$-0.723732\pi$
$643$	− 981.203i	− 1.52598i	0.646413	−	0.762988i	$-0.276268\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 635.264i	− 0.981861i	−0.871199	−	0.490930i	$-0.836657\pi$
$647$	− 635.264i	− 0.981861i	0.871199	−	0.490930i	$-0.163343\pi$
$648$	0	0
$649$	−236.456	−0.364339
$650$	0	0
$651$	− 277.307i	− 0.425971i
$652$	0	0
$653$	12.2032	0.0186879	0.00934396	−	0.999956i	$-0.497026\pi$
$653$	12.2032	0.0186879	0.00934396	+	0.999956i	$0.497026\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−94.1206	−0.143258
$658$	0	0
$659$	420.257i	0.637720i	0.947802	+	0.318860i	$0.103300\pi$
$659$	420.257i	0.637720i	−0.947802	+	0.318860i	$0.896700\pi$
$660$	0	0
$661$	−344.639	−0.521390	−0.260695	−	0.965421i	$-0.583952\pi$
$661$	−344.639	−0.521390	−0.260695	+	0.965421i	$0.583952\pi$
$662$	0	0
$663$	655.404i	0.988543i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1026.74i	− 1.53934i
$668$	0	0
$669$	487.483	0.728674
$670$	0	0
$671$	45.1733i	0.0673223i
$672$	0	0
$673$	−1307.73	−1.94314	−0.971571	−	0.236747i	$-0.923919\pi$
$673$	−1307.73	−1.94314	−0.971571	+	0.236747i	$0.923919\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	247.644	0.365796	0.182898	−	0.983132i	$-0.441452\pi$
$677$	247.644	0.365796	0.182898	+	0.983132i	$0.441452\pi$
$678$	0	0
$679$	− 484.795i	− 0.713984i
$680$	0	0
$681$	418.875	0.615088
$682$	0	0
$683$	242.635i	0.355249i	0.984098	+	0.177624i	$0.0568412\pi$
$683$	242.635i	0.355249i	−0.984098	+	0.177624i	$0.943159\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 179.015i	− 0.260574i
$688$	0	0
$689$	−634.392	−0.920743
$690$	0	0
$691$	753.499i	1.09045i	0.838291	+	0.545224i	$0.183555\pi$
$691$	753.499i	1.09045i	−0.838291	+	0.545224i	$0.816445\pi$
$692$	0	0
$693$	−105.599	−0.152380
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2064.10	−2.96140
$698$	0	0
$699$	− 616.789i	− 0.882387i
$700$	0	0
$701$	367.412	0.524126	0.262063	−	0.965051i	$-0.415597\pi$
$701$	367.412	0.524126	0.262063	+	0.965051i	$0.415597\pi$
$702$	0	0
$703$	− 486.121i	− 0.691495i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	642.528i	0.908809i
$708$	0	0
$709$	−394.799	−0.556839	−0.278419	−	0.960460i	$-0.589810\pi$
$709$	−394.799	−0.556839	−0.278419	+	0.960460i	$0.589810\pi$
$710$	0	0
$711$	− 207.461i	− 0.291787i
$712$	0	0
$713$	645.043	0.904689
$714$	0	0
$715$	0	0
$716$	0	0
$717$	106.263	0.148205
$718$	0	0
$719$	− 475.466i	− 0.661288i	−0.943756	−	0.330644i	$-0.892734\pi$
$719$	− 475.466i	− 0.661288i	0.943756	−	0.330644i	$-0.107266\pi$
$720$	0	0
$721$	−1465.56	−2.03268
$722$	0	0
$723$	− 175.256i	− 0.242401i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	452.918i	0.622996i	0.950247	+	0.311498i	$0.100831\pi$
$727$	452.918i	0.622996i	−0.950247	+	0.311498i	$0.899169\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	2434.32i	3.33013i
$732$	0	0
$733$	−92.6649	−0.126419	−0.0632094	−	0.998000i	$-0.520134\pi$
$733$	−92.6649	−0.126419	−0.0632094	+	0.998000i	$0.520134\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−39.0804	−0.0530264
$738$	0	0
$739$	1373.49i	1.85857i	0.369360	+	0.929286i	$0.379577\pi$
$739$	1373.49i	1.85857i	−0.369360	+	0.929286i	$0.620423\pi$
$740$	0	0
$741$	234.374	0.316294
$742$	0	0
$743$	423.832i	0.570434i	0.958463	+	0.285217i	$0.0920656\pi$
$743$	423.832i	0.570434i	−0.958463	+	0.285217i	$0.907934\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 201.990i	− 0.270402i
$748$	0	0
$749$	−205.398	−0.274230
$750$	0	0
$751$	781.600i	1.04075i	0.853939	+	0.520373i	$0.174207\pi$
$751$	781.600i	1.04075i	−0.853939	+	0.520373i	$0.825793\pi$
$752$	0	0
$753$	200.419	0.266161
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−122.603	−0.161959	−0.0809793	−	0.996716i	$-0.525805\pi$
$757$	−122.603	−0.161959	−0.0809793	+	0.996716i	$0.525805\pi$
$758$	0	0
$759$	− 245.634i	− 0.323629i
$760$	0	0
$761$	481.169	0.632285	0.316142	−	0.948712i	$-0.397612\pi$
$761$	481.169	0.632285	0.316142	+	0.948712i	$0.397612\pi$
$762$	0	0
$763$	1196.30i	1.56789i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	798.200i	1.04068i
$768$	0	0
$769$	−1369.26	−1.78057	−0.890287	−	0.455400i	$-0.849496\pi$
$769$	−1369.26	−1.78057	−0.890287	+	0.455400i	$0.849496\pi$
$770$	0	0
$771$	− 276.529i	− 0.358663i
$772$	0	0
$773$	248.772	0.321826	0.160913	−	0.986969i	$-0.448556\pi$
$773$	248.772	0.321826	0.160913	+	0.986969i	$0.448556\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	739.366	0.951564
$778$	0	0
$779$	738.125i	0.947529i
$780$	0	0
$781$	7.17472	0.00918659
$782$	0	0
$783$	− 140.339i	− 0.179233i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 73.4073i	− 0.0932749i	−0.998912	−	0.0466374i	$-0.985149\pi$
$787$	− 73.4073i	− 0.0932749i	0.998912	−	0.0466374i	$-0.0148505\pi$
$788$	0	0
$789$	−11.6722	−0.0147937
$790$	0	0
$791$	− 755.437i	− 0.955040i
$792$	0	0
$793$	152.491	0.192296
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−845.614	−1.06100	−0.530498	−	0.847686i	$-0.677995\pi$
$797$	−845.614	−1.06100	−0.530498	+	0.847686i	$0.677995\pi$
$798$	0	0
$799$	514.393i	0.643796i
$800$	0	0
$801$	189.528	0.236615
$802$	0	0
$803$	− 117.038i	− 0.145751i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	578.218i	0.716503i
$808$	0	0
$809$	−46.2307	−0.0571455	−0.0285728	−	0.999592i	$-0.509096\pi$
$809$	−46.2307	−0.0571455	−0.0285728	+	0.999592i	$0.509096\pi$
$810$	0	0
$811$	− 614.528i	− 0.757742i	−0.925450	−	0.378871i	$-0.876312\pi$
$811$	− 614.528i	− 0.757742i	0.925450	−	0.378871i	$-0.123688\pi$
$812$	0	0
$813$	841.239	1.03473
$814$	0	0
$815$	0	0
$816$	0	0
$817$	870.519	1.06551
$818$	0	0
$819$	356.470i	0.435251i
$820$	0	0
$821$	−1006.35	−1.22576	−0.612880	−	0.790176i	$-0.709989\pi$
$821$	−1006.35	−1.22576	−0.612880	+	0.790176i	$0.709989\pi$
$822$	0	0
$823$	− 742.969i	− 0.902757i	−0.892332	−	0.451379i	$-0.850932\pi$
$823$	− 742.969i	− 0.902757i	0.892332	−	0.451379i	$-0.149068\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1080.66i	1.30672i	0.757047	+	0.653361i	$0.226642\pi$
$827$	1080.66i	1.30672i	−0.757047	+	0.653361i	$0.773358\pi$
$828$	0	0
$829$	−33.5681	−0.0404922	−0.0202461	−	0.999795i	$-0.506445\pi$
$829$	−33.5681	−0.0404922	−0.0202461	+	0.999795i	$0.506445\pi$
$830$	0	0
$831$	− 229.398i	− 0.276051i
$832$	0	0
$833$	1202.93	1.44410
$834$	0	0
$835$	0	0
$836$	0	0
$837$	88.1671	0.105337
$838$	0	0
$839$	− 228.003i	− 0.271756i	−0.990726	−	0.135878i	$-0.956614\pi$
$839$	− 228.003i	− 0.271756i	0.990726	−	0.135878i	$-0.0433855\pi$
$840$	0	0
$841$	−111.551	−0.132641
$842$	0	0
$843$	− 460.522i	− 0.546290i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1010.41i	1.19293i
$848$	0	0
$849$	609.795	0.718251
$850$	0	0
$851$	1719.83i	2.02096i
$852$	0	0
$853$	1499.35	1.75774	0.878868	−	0.477065i	$-0.158299\pi$
$853$	1499.35	1.75774	0.878868	+	0.477065i	$0.158299\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1462.41	1.70643	0.853216	−	0.521558i	$-0.174649\pi$
$857$	1462.41	1.70643	0.853216	+	0.521558i	$0.174649\pi$
$858$	0	0
$859$	− 1082.73i	− 1.26045i	−0.776412	−	0.630225i	$-0.782963\pi$
$859$	− 1082.73i	− 1.26045i	0.776412	−	0.630225i	$-0.217037\pi$
$860$	0	0
$861$	−1122.65	−1.30389
$862$	0	0
$863$	1359.98i	1.57587i	0.615758	+	0.787935i	$0.288850\pi$
$863$	1359.98i	1.57587i	−0.615758	+	0.787935i	$0.711150\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1063.32i	1.22644i
$868$	0	0
$869$	257.976	0.296865
$870$	0	0
$871$	131.923i	0.151462i
$872$	0	0
$873$	154.136	0.176559
$874$	0	0
$875$	0	0
$876$	0	0
$877$	814.748	0.929018	0.464509	−	0.885569i	$-0.346231\pi$
$877$	814.748	0.929018	0.464509	+	0.885569i	$0.346231\pi$
$878$	0	0
$879$	426.626i	0.485354i
$880$	0	0
$881$	−477.767	−0.542301	−0.271150	−	0.962537i	$-0.587404\pi$
$881$	−477.767	−0.542301	−0.271150	+	0.962537i	$0.587404\pi$
$882$	0	0
$883$	− 1044.41i	− 1.18280i	−0.806378	−	0.591401i	$-0.798575\pi$
$883$	− 1044.41i	− 1.18280i	0.806378	−	0.591401i	$-0.201425\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1042.60i	− 1.17543i	−0.809069	−	0.587713i	$-0.800028\pi$
$887$	− 1042.60i	− 1.17543i	0.809069	−	0.587713i	$-0.199972\pi$
$888$	0	0
$889$	1477.10	1.66153
$890$	0	0
$891$	− 33.5743i	− 0.0376816i
$892$	0	0
$893$	183.948	0.205989
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−829.184	−0.924397
$898$	0	0
$899$	458.271i	0.509756i
$900$	0	0
$901$	−1513.75	−1.68007
$902$	0	0
$903$	1324.02i	1.46624i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	615.703i	0.678835i	0.940636	+	0.339417i	$0.110230\pi$
$907$	615.703i	0.678835i	−0.940636	+	0.339417i	$0.889770\pi$
$908$	0	0
$909$	−204.285	−0.224736
$910$	0	0
$911$	400.257i	0.439360i	0.975572	+	0.219680i	$0.0705013\pi$
$911$	400.257i	0.439360i	−0.975572	+	0.219680i	$0.929499\pi$
$912$	0	0
$913$	251.173	0.275108
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−335.176	−0.365514
$918$	0	0
$919$	1614.36i	1.75665i	0.478066	+	0.878324i	$0.341338\pi$
$919$	1614.36i	1.75665i	−0.478066	+	0.878324i	$0.658662\pi$
$920$	0	0
$921$	−830.749	−0.902008
$922$	0	0
$923$	− 24.2196i	− 0.0262401i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 465.961i	− 0.502654i
$928$	0	0
$929$	−153.105	−0.164807	−0.0824033	−	0.996599i	$-0.526260\pi$
$929$	−153.105	−0.164807	−0.0824033	+	0.996599i	$0.526260\pi$
$930$	0	0
$931$	− 430.171i	− 0.462053i
$932$	0	0
$933$	412.411	0.442027
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−296.081	−0.315989	−0.157994	−	0.987440i	$-0.550503\pi$
$937$	−296.081	−0.315989	−0.157994	+	0.987440i	$0.550503\pi$
$938$	0	0
$939$	274.178i	0.291989i
$940$	0	0
$941$	129.275	0.137380	0.0686902	−	0.997638i	$-0.478118\pi$
$941$	129.275	0.137380	0.0686902	+	0.997638i	$0.478118\pi$
$942$	0	0
$943$	− 2611.39i	− 2.76924i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 141.017i	− 0.148910i	−0.997224	−	0.0744548i	$-0.976278\pi$
$947$	− 141.017i	− 0.148910i	0.997224	−	0.0744548i	$-0.0237216\pi$
$948$	0	0
$949$	−395.085	−0.416317
$950$	0	0
$951$	298.908i	0.314309i
$952$	0	0
$953$	806.942	0.846738	0.423369	−	0.905957i	$-0.360847\pi$
$953$	806.942	0.846738	0.423369	+	0.905957i	$0.360847\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	174.511	0.182352
$958$	0	0
$959$	− 992.948i	− 1.03540i
$960$	0	0
$961$	673.095	0.700411
$962$	0	0
$963$	− 65.3043i	− 0.0678134i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	889.210i	0.919555i	0.888034	+	0.459778i	$0.152071\pi$
$967$	889.210i	0.919555i	−0.888034	+	0.459778i	$0.847929\pi$
$968$	0	0
$969$	559.248	0.577139
$970$	0	0
$971$	1331.48i	1.37125i	0.727955	+	0.685625i	$0.240471\pi$
$971$	1331.48i	1.37125i	−0.727955	+	0.685625i	$0.759529\pi$
$972$	0	0
$973$	−1715.75	−1.76336
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1024.81	−1.04894	−0.524468	−	0.851430i	$-0.675736\pi$
$977$	−1024.81	−1.04894	−0.524468	+	0.851430i	$0.675736\pi$
$978$	0	0
$979$	235.677i	0.240733i
$980$	0	0
$981$	−380.351	−0.387718
$982$	0	0
$983$	566.908i	0.576712i	0.957523	+	0.288356i	$0.0931087\pi$
$983$	566.908i	0.576712i	−0.957523	+	0.288356i	$0.906891\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	279.775i	0.283460i
$988$	0	0
$989$	−3079.78	−3.11404
$990$	0	0
$991$	1522.10i	1.53592i	0.640497	+	0.767961i	$0.278729\pi$
$991$	1522.10i	1.53592i	−0.640497	+	0.767961i	$0.721271\pi$
$992$	0	0
$993$	683.981	0.688803
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1574.81	−1.57954	−0.789772	−	0.613400i	$-0.789801\pi$
$997$	−1574.81	−1.57954	−0.789772	+	0.613400i	$0.789801\pi$
$998$	0	0
$999$	235.074i	0.235309i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.3.e.h.1951.12		12
4.3	odd	2	inner	2400.3.e.h.1951.1		12
5.2	odd	4		480.3.j.a.319.9	✓	12
5.3	odd	4		480.3.j.b.319.4	yes	12
5.4	even	2		2400.3.e.i.1951.1		12
15.2	even	4		1440.3.j.d.1279.8		12
15.8	even	4		1440.3.j.c.1279.7		12
20.3	even	4		480.3.j.a.319.10	yes	12
20.7	even	4		480.3.j.b.319.3	yes	12
20.19	odd	2		2400.3.e.i.1951.12		12
40.3	even	4		960.3.j.f.319.3		12
40.13	odd	4		960.3.j.g.319.9		12
40.27	even	4		960.3.j.g.319.10		12
40.37	odd	4		960.3.j.f.319.4		12
60.23	odd	4		1440.3.j.d.1279.7		12
60.47	odd	4		1440.3.j.c.1279.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.j.a.319.9	✓	12	5.2	odd	4
480.3.j.a.319.10	yes	12	20.3	even	4
480.3.j.b.319.3	yes	12	20.7	even	4
480.3.j.b.319.4	yes	12	5.3	odd	4
960.3.j.f.319.3		12	40.3	even	4
960.3.j.f.319.4		12	40.37	odd	4
960.3.j.g.319.9		12	40.13	odd	4
960.3.j.g.319.10		12	40.27	even	4
1440.3.j.c.1279.7		12	15.8	even	4
1440.3.j.c.1279.8		12	60.47	odd	4
1440.3.j.d.1279.7		12	60.23	odd	4
1440.3.j.d.1279.8		12	15.2	even	4
2400.3.e.h.1951.1		12	4.3	odd	2	inner
2400.3.e.h.1951.12		12	1.1	even	1	trivial
2400.3.e.i.1951.1		12	5.4	even	2
2400.3.e.i.1951.12		12	20.19	odd	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2400.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +9.43574i q^{7} -3.00000 q^{9} -3.73048i q^{11} -12.5929 q^{13} -30.0484 q^{17} +10.7454i q^{19} -16.3432 q^{21} -38.0158i q^{23} -5.19615i q^{27} +27.0083 q^{29} +16.9678i q^{31} +6.46138 q^{33} -45.2400 q^{37} -21.8116i q^{39} +68.6923 q^{41} -81.0133i q^{43} -17.1188i q^{47} -40.0331 q^{49} -52.0454i q^{51} +50.3769 q^{53} -18.6115 q^{57} -63.3848i q^{59} -12.1092 q^{61} -28.3072i q^{63} -10.4760i q^{67} +65.8452 q^{69} +1.92327i q^{71} +31.3735 q^{73} +35.1998 q^{77} +69.1535i q^{79} +9.00000 q^{81} +67.3300i q^{83} +46.7798i q^{87} -63.1761 q^{89} -118.823i q^{91} -29.3890 q^{93} -51.3786 q^{97} +11.1914i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{9} - 80 q^{17} + 40 q^{29} - 320 q^{37} + 136 q^{41} - 212 q^{49} + 176 q^{53} - 48 q^{57} + 40 q^{61} - 448 q^{73} - 448 q^{77} + 108 q^{81} + 8 q^{89} + 144 q^{93} - 224 q^{97}+O(q^{100}) \) 12 * q - 36 * q^9 - 80 * q^17 + 40 * q^29 - 320 * q^37 + 136 * q^41 - 212 * q^49 + 176 * q^53 - 48 * q^57 + 40 * q^61 - 448 * q^73 - 448 * q^77 + 108 * q^81 + 8 * q^89 + 144 * q^93 - 224 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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