LMFDB - Embedded newform 1200.4.f.x.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,4,Mod(49,1200)] 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("1200.49"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-36,0,32,0,0,0,0,0,0,0,-420] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{109})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 55x^{2} + 729 $ x^4 + 55*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-4.72015i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.x.49.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +19.8806i q^{7} -9.00000 q^{9} +70.6418 q^{11} +0.761226i q^{13} +108.403i q^{17} -125.881 q^{19} +59.6418 q^{21} +40.8806i q^{23} +27.0000i q^{27} +140.881 q^{29} -296.926 q^{31} -211.926i q^{33} -26.8061i q^{37} +2.28368 q^{39} -20.6418 q^{41} +32.5969i q^{43} -11.4326i q^{47} -52.2388 q^{49} +325.209 q^{51} -159.358i q^{53} +377.642i q^{57} -374.955 q^{59} +303.478 q^{61} -178.926i q^{63} -877.583i q^{67} +122.642 q^{69} -1159.54 q^{71} -120.716i q^{73} +1404.40i q^{77} -1248.66 q^{79} +81.0000 q^{81} +654.180i q^{83} -422.642i q^{87} -795.522 q^{89} -15.1336 q^{91} +890.777i q^{93} +850.910i q^{97} -635.777 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9} + 32 q^{11} - 420 q^{19} - 12 q^{21} + 480 q^{29} - 436 q^{31} - 492 q^{39} + 168 q^{41} - 376 q^{49} + 48 q^{51} - 2168 q^{59} + 1548 q^{61} + 240 q^{69} - 2216 q^{71} - 2656 q^{79} + 324 q^{81}+ \cdots - 288 q^{99}+O(q^{100}) $ 4 * q - 36 * q^9 + 32 * q^11 - 420 * q^19 - 12 * q^21 + 480 * q^29 - 436 * q^31 - 492 * q^39 + 168 * q^41 - 376 * q^49 + 48 * q^51 - 2168 * q^59 + 1548 * q^61 + 240 * q^69 - 2216 * q^71 - 2656 * q^79 + 324 * q^81 - 2848 * q^89 - 3652 * q^91 - 288 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	19.8806i	1.07345i	0.843757	+	0.536726i	$0.180339\pi$
$7$	19.8806i	1.07345i	−0.843757	+	0.536726i	$0.819661\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	70.6418	1.93630	0.968151	−	0.250368i	$-0.0805516\pi$
$11$	70.6418	1.93630	0.968151	+	0.250368i	$0.0805516\pi$
$12$	0	0
$13$	0.761226i	0.0162405i	0.999967	+	0.00812024i	$0.00258478\pi$
$13$	0.761226i	0.0162405i	−0.999967	+	0.00812024i	$0.997415\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	108.403i	1.54657i	0.634062	+	0.773283i	$0.281387\pi$
$17$	108.403i	1.54657i	−0.634062	+	0.773283i	$0.718613\pi$
$18$	0	0
$19$	−125.881	−1.51995	−0.759974	−	0.649954i	$-0.774788\pi$
$19$	−125.881	−1.51995	−0.759974	+	0.649954i	$0.774788\pi$
$20$	0	0
$21$	59.6418	0.619758
$22$	0	0
$23$	40.8806i	0.370617i	0.982680	+	0.185309i	$0.0593285\pi$
$23$	40.8806i	0.370617i	−0.982680	+	0.185309i	$0.940672\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	140.881	0.902099	0.451050	−	0.892499i	$-0.351050\pi$
$29$	140.881	0.902099	0.451050	+	0.892499i	$0.351050\pi$
$30$	0	0
$31$	−296.926	−1.72030	−0.860152	−	0.510039i	$-0.829631\pi$
$31$	−296.926	−1.72030	−0.860152	+	0.510039i	$0.829631\pi$
$32$	0	0
$33$	− 211.926i	− 1.11792i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 26.8061i	− 0.119105i	−0.998225	−	0.0595527i	$-0.981033\pi$
$37$	− 26.8061i	− 0.119105i	0.998225	−	0.0595527i	$-0.0189674\pi$
$38$	0	0
$39$	2.28368	0.00937644
$40$	0	0
$41$	−20.6418	−0.0786272	−0.0393136	−	0.999227i	$-0.512517\pi$
$41$	−20.6418	−0.0786272	−0.0393136	+	0.999227i	$0.512517\pi$
$42$	0	0
$43$	32.5969i	0.115604i	0.998328	+	0.0578022i	$0.0184093\pi$
$43$	32.5969i	0.115604i	−0.998328	+	0.0578022i	$0.981591\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 11.4326i	− 0.0354813i	−0.999843	−	0.0177407i	$-0.994353\pi$
$47$	− 11.4326i	− 0.0354813i	0.999843	−	0.0177407i	$-0.00564733\pi$
$48$	0	0
$49$	−52.2388	−0.152300
$50$	0	0
$51$	325.209	0.892910
$52$	0	0
$53$	− 159.358i	− 0.413010i	−0.978446	−	0.206505i	$-0.933791\pi$
$53$	− 159.358i	− 0.413010i	0.978446	−	0.206505i	$-0.0662090\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	377.642i	0.877542i
$58$	0	0
$59$	−374.955	−0.827373	−0.413686	−	0.910419i	$-0.635759\pi$
$59$	−374.955	−0.827373	−0.413686	+	0.910419i	$0.635759\pi$
$60$	0	0
$61$	303.478	0.636989	0.318494	−	0.947925i	$-0.396823\pi$
$61$	303.478	0.636989	0.318494	+	0.947925i	$0.396823\pi$
$62$	0	0
$63$	− 178.926i	− 0.357817i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 877.583i	− 1.60021i	−0.599863	−	0.800103i	$-0.704778\pi$
$67$	− 877.583i	− 1.60021i	0.599863	−	0.800103i	$-0.295222\pi$
$68$	0	0
$69$	122.642	0.213976
$70$	0	0
$71$	−1159.54	−1.93819	−0.969097	−	0.246679i	$-0.920661\pi$
$71$	−1159.54	−1.93819	−0.969097	+	0.246679i	$0.920661\pi$
$72$	0	0
$73$	− 120.716i	− 0.193545i	−0.995307	−	0.0967724i	$-0.969148\pi$
$73$	− 120.716i	− 0.193545i	0.995307	−	0.0967724i	$-0.0308519\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1404.40i	2.07853i
$78$	0	0
$79$	−1248.66	−1.77829	−0.889145	−	0.457626i	$-0.848700\pi$
$79$	−1248.66	−1.77829	−0.889145	+	0.457626i	$0.848700\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	654.180i	0.865127i	0.901603	+	0.432564i	$0.142391\pi$
$83$	654.180i	0.865127i	−0.901603	+	0.432564i	$0.857609\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 422.642i	− 0.520827i
$88$	0	0
$89$	−795.522	−0.947474	−0.473737	−	0.880666i	$-0.657095\pi$
$89$	−795.522	−0.947474	−0.473737	+	0.880666i	$0.657095\pi$
$90$	0	0
$91$	−15.1336	−0.0174334
$92$	0	0
$93$	890.777i	0.993217i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	850.910i	0.890689i	0.895359	+	0.445345i	$0.146919\pi$
$97$	850.910i	0.890689i	−0.895359	+	0.445345i	$0.853081\pi$
$98$	0	0
$99$	−635.777	−0.645434
$100$	0	0
$101$	−149.777	−0.147558	−0.0737788	−	0.997275i	$-0.523506\pi$
$101$	−149.777	−0.147558	−0.0737788	+	0.997275i	$0.523506\pi$
$102$	0	0
$103$	1176.81i	1.12577i	0.826536	+	0.562884i	$0.190308\pi$
$103$	1176.81i	1.12577i	−0.826536	+	0.562884i	$0.809692\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 402.552i	− 0.363703i	−0.983326	−	0.181851i	$-0.941791\pi$
$107$	− 402.552i	− 0.363703i	0.983326	−	0.181851i	$-0.0582089\pi$
$108$	0	0
$109$	−1650.31	−1.45020	−0.725098	−	0.688645i	$-0.758206\pi$
$109$	−1650.31	−1.45020	−0.725098	+	0.688645i	$0.758206\pi$
$110$	0	0
$111$	−80.4184	−0.0687655
$112$	0	0
$113$	1353.13i	1.12648i	0.826293	+	0.563240i	$0.190445\pi$
$113$	1353.13i	1.12648i	−0.826293	+	0.563240i	$0.809555\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.85103i	− 0.00541349i
$118$	0	0
$119$	−2155.12	−1.66016
$120$	0	0
$121$	3659.27	2.74926
$122$	0	0
$123$	61.9255i	0.0453954i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1836.21i	1.28297i	0.767135	+	0.641485i	$0.221682\pi$
$127$	1836.21i	1.28297i	−0.767135	+	0.641485i	$0.778318\pi$
$128$	0	0
$129$	97.7908	0.0667442
$130$	0	0
$131$	1111.98	0.741638	0.370819	−	0.928705i	$-0.379077\pi$
$131$	1111.98	0.741638	0.370819	+	0.928705i	$0.379077\pi$
$132$	0	0
$133$	− 2502.58i	− 1.63159i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	567.284i	0.353769i	0.984232	+	0.176884i	$0.0566019\pi$
$137$	567.284i	0.353769i	−0.984232	+	0.176884i	$0.943398\pi$
$138$	0	0
$139$	1968.87	1.20142	0.600709	−	0.799468i	$-0.294885\pi$
$139$	1968.87	1.20142	0.600709	+	0.799468i	$0.294885\pi$
$140$	0	0
$141$	−34.2979	−0.0204852
$142$	0	0
$143$	53.7744i	0.0314464i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	156.716i	0.0879302i
$148$	0	0
$149$	1810.87	0.995651	0.497826	−	0.867277i	$-0.334132\pi$
$149$	1810.87	0.995651	0.497826	+	0.867277i	$0.334132\pi$
$150$	0	0
$151$	−266.746	−0.143758	−0.0718791	−	0.997413i	$-0.522900\pi$
$151$	−266.746	−0.143758	−0.0718791	+	0.997413i	$0.522900\pi$
$152$	0	0
$153$	− 975.628i	− 0.515522i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2993.33i	1.52162i	0.648977	+	0.760808i	$0.275197\pi$
$157$	2993.33i	1.52162i	−0.648977	+	0.760808i	$0.724803\pi$
$158$	0	0
$159$	−478.074	−0.238451
$160$	0	0
$161$	−812.732	−0.397840
$162$	0	0
$163$	2356.36i	1.13230i	0.824304	+	0.566148i	$0.191567\pi$
$163$	2356.36i	1.13230i	−0.824304	+	0.566148i	$0.808433\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3148.55i	− 1.45894i	−0.684015	−	0.729468i	$-0.739768\pi$
$167$	− 3148.55i	− 1.45894i	0.684015	−	0.729468i	$-0.260232\pi$
$168$	0	0
$169$	2196.42	0.999736
$170$	0	0
$171$	1132.93	0.506649
$172$	0	0
$173$	3747.12i	1.64675i	0.567496	+	0.823376i	$0.307912\pi$
$173$	3747.12i	1.64675i	−0.567496	+	0.823376i	$0.692088\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1124.87i	0.477684i
$178$	0	0
$179$	−2309.37	−0.964305	−0.482153	−	0.876087i	$-0.660145\pi$
$179$	−2309.37	−0.964305	−0.482153	+	0.876087i	$0.660145\pi$
$180$	0	0
$181$	257.447	0.105723	0.0528615	−	0.998602i	$-0.483166\pi$
$181$	257.447	0.105723	0.0528615	+	0.998602i	$0.483166\pi$
$182$	0	0
$183$	− 910.433i	− 0.367766i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7657.79i	2.99462i
$188$	0	0
$189$	−536.777	−0.206586
$190$	0	0
$191$	4184.82	1.58536	0.792679	−	0.609640i	$-0.208686\pi$
$191$	4184.82	1.58536	0.792679	+	0.609640i	$0.208686\pi$
$192$	0	0
$193$	3636.79i	1.35638i	0.734885	+	0.678192i	$0.237236\pi$
$193$	3636.79i	1.35638i	−0.734885	+	0.678192i	$0.762764\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3519.52i	− 1.27287i	−0.771330	−	0.636436i	$-0.780408\pi$
$197$	− 3519.52i	− 1.27287i	0.771330	−	0.636436i	$-0.219592\pi$
$198$	0	0
$199$	−1937.28	−0.690103	−0.345051	−	0.938584i	$-0.612139\pi$
$199$	−1937.28	−0.690103	−0.345051	+	0.938584i	$0.612139\pi$
$200$	0	0
$201$	−2632.75	−0.923879
$202$	0	0
$203$	2800.79i	0.968360i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 367.926i	− 0.123539i
$208$	0	0
$209$	−8892.44	−2.94308
$210$	0	0
$211$	−2928.63	−0.955522	−0.477761	−	0.878490i	$-0.658552\pi$
$211$	−2928.63	−0.955522	−0.477761	+	0.878490i	$0.658552\pi$
$212$	0	0
$213$	3478.61i	1.11902i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5903.06i	− 1.84666i
$218$	0	0
$219$	−362.149	−0.111743
$220$	0	0
$221$	−82.5192	−0.0251169
$222$	0	0
$223$	2637.07i	0.791890i	0.918274	+	0.395945i	$0.129583\pi$
$223$	2637.07i	0.791890i	−0.918274	+	0.395945i	$0.870417\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3287.27i	0.961162i	0.876950	+	0.480581i	$0.159574\pi$
$227$	3287.27i	0.961162i	−0.876950	+	0.480581i	$0.840426\pi$
$228$	0	0
$229$	−6382.02	−1.84164	−0.920820	−	0.389988i	$-0.872479\pi$
$229$	−6382.02	−1.84164	−0.920820	+	0.389988i	$0.872479\pi$
$230$	0	0
$231$	4213.21	1.20004
$232$	0	0
$233$	− 2952.91i	− 0.830265i	−0.909761	−	0.415133i	$-0.863735\pi$
$233$	− 2952.91i	− 0.830265i	0.909761	−	0.415133i	$-0.136265\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3745.97i	1.02670i
$238$	0	0
$239$	−1384.34	−0.374668	−0.187334	−	0.982296i	$-0.559985\pi$
$239$	−1384.34	−0.374668	−0.187334	+	0.982296i	$0.559985\pi$
$240$	0	0
$241$	4236.82	1.13244	0.566219	−	0.824255i	$-0.308406\pi$
$241$	4236.82	1.13244	0.566219	+	0.824255i	$0.308406\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 95.8236i	− 0.0246847i
$248$	0	0
$249$	1962.54	0.499481
$250$	0	0
$251$	1861.22	0.468045	0.234023	−	0.972231i	$-0.424811\pi$
$251$	1861.22	0.468045	0.234023	+	0.972231i	$0.424811\pi$
$252$	0	0
$253$	2887.88i	0.717627i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 347.251i	− 0.0842837i	−0.999112	−	0.0421419i	$-0.986582\pi$
$257$	− 347.251i	− 0.0842837i	0.999112	−	0.0421419i	$-0.0134181\pi$
$258$	0	0
$259$	532.922	0.127854
$260$	0	0
$261$	−1267.93	−0.300700
$262$	0	0
$263$	− 1489.39i	− 0.349200i	−0.984639	−	0.174600i	$-0.944137\pi$
$263$	− 1489.39i	− 0.349200i	0.984639	−	0.174600i	$-0.0558632\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2386.57i	0.547025i
$268$	0	0
$269$	−1570.88	−0.356053	−0.178026	−	0.984026i	$-0.556971\pi$
$269$	−1570.88	−0.356053	−0.178026	+	0.984026i	$0.556971\pi$
$270$	0	0
$271$	−3014.42	−0.675693	−0.337847	−	0.941201i	$-0.609699\pi$
$271$	−3014.42	−0.675693	−0.337847	+	0.941201i	$0.609699\pi$
$272$	0	0
$273$	45.4009i	0.0100652i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2792.41i	− 0.605702i	−0.953038	−	0.302851i	$-0.902062\pi$
$277$	− 2792.41i	− 0.605702i	0.953038	−	0.302851i	$-0.0979384\pi$
$278$	0	0
$279$	2672.33	0.573434
$280$	0	0
$281$	−1745.81	−0.370626	−0.185313	−	0.982680i	$-0.559330\pi$
$281$	−1745.81	−0.370626	−0.185313	+	0.982680i	$0.559330\pi$
$282$	0	0
$283$	− 2664.33i	− 0.559639i	−0.960053	−	0.279820i	$-0.909725\pi$
$283$	− 2664.33i	− 0.559639i	0.960053	−	0.279820i	$-0.0902747\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 410.372i	− 0.0844025i
$288$	0	0
$289$	−6838.22	−1.39186
$290$	0	0
$291$	2552.73	0.514240
$292$	0	0
$293$	5927.79i	1.18193i	0.806697	+	0.590965i	$0.201253\pi$
$293$	5927.79i	1.18193i	−0.806697	+	0.590965i	$0.798747\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1907.33i	0.372641i
$298$	0	0
$299$	−31.1194	−0.00601900
$300$	0	0
$301$	−648.047	−0.124096
$302$	0	0
$303$	449.330i	0.0851925i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4634.48i	0.861575i	0.902453	+	0.430788i	$0.141764\pi$
$307$	4634.48i	0.861575i	−0.902453	+	0.430788i	$0.858236\pi$
$308$	0	0
$309$	3530.42	0.649963
$310$	0	0
$311$	−761.969	−0.138930	−0.0694651	−	0.997584i	$-0.522129\pi$
$311$	−761.969	−0.138930	−0.0694651	+	0.997584i	$0.522129\pi$
$312$	0	0
$313$	− 8734.32i	− 1.57729i	−0.614847	−	0.788647i	$-0.710782\pi$
$313$	− 8734.32i	− 1.57729i	0.614847	−	0.788647i	$-0.289218\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4428.21i	− 0.784584i	−0.919841	−	0.392292i	$-0.871682\pi$
$317$	− 4428.21i	− 0.784584i	0.919841	−	0.392292i	$-0.128318\pi$
$318$	0	0
$319$	9952.07	1.74674
$320$	0	0
$321$	−1207.66	−0.209984
$322$	0	0
$323$	− 13645.8i	− 2.35070i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4950.94i	0.837271i
$328$	0	0
$329$	227.288	0.0380875
$330$	0	0
$331$	2239.68	0.371915	0.185957	−	0.982558i	$-0.440461\pi$
$331$	2239.68	0.371915	0.185957	+	0.982558i	$0.440461\pi$
$332$	0	0
$333$	241.255i	0.0397018i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8459.21i	− 1.36737i	−0.729779	−	0.683683i	$-0.760377\pi$
$337$	− 8459.21i	− 1.36737i	0.729779	−	0.683683i	$-0.239623\pi$
$338$	0	0
$339$	4059.40	0.650373
$340$	0	0
$341$	−20975.4	−3.33103
$342$	0	0
$343$	5780.51i	0.909966i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9253.53i	− 1.43157i	−0.698320	−	0.715786i	$-0.746069\pi$
$347$	− 9253.53i	− 1.43157i	0.698320	−	0.715786i	$-0.253931\pi$
$348$	0	0
$349$	1630.53	0.250087	0.125044	−	0.992151i	$-0.460093\pi$
$349$	1630.53	0.250087	0.125044	+	0.992151i	$0.460093\pi$
$350$	0	0
$351$	−20.5531	−0.00312548
$352$	0	0
$353$	8148.98i	1.22869i	0.789039	+	0.614343i	$0.210579\pi$
$353$	8148.98i	1.22869i	−0.789039	+	0.614343i	$0.789421\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6465.36i	0.958496i
$358$	0	0
$359$	−5850.67	−0.860130	−0.430065	−	0.902798i	$-0.641509\pi$
$359$	−5850.67	−0.860130	−0.430065	+	0.902798i	$0.641509\pi$
$360$	0	0
$361$	8986.93	1.31024
$362$	0	0
$363$	− 10977.8i	− 1.58729i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7205.40i	1.02485i	0.858733	+	0.512424i	$0.171252\pi$
$367$	7205.40i	1.02485i	−0.858733	+	0.512424i	$0.828748\pi$
$368$	0	0
$369$	185.777	0.0262091
$370$	0	0
$371$	3168.14	0.443346
$372$	0	0
$373$	− 8708.52i	− 1.20887i	−0.796653	−	0.604437i	$-0.793398\pi$
$373$	− 8708.52i	− 1.20887i	0.796653	−	0.604437i	$-0.206602\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	107.242i	0.0146505i
$378$	0	0
$379$	−4153.83	−0.562975	−0.281488	−	0.959565i	$-0.590828\pi$
$379$	−4153.83	−0.562975	−0.281488	+	0.959565i	$0.590828\pi$
$380$	0	0
$381$	5508.63	0.740724
$382$	0	0
$383$	10362.0i	1.38244i	0.722645	+	0.691219i	$0.242926\pi$
$383$	10362.0i	1.38244i	−0.722645	+	0.691219i	$0.757074\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 293.372i	− 0.0385348i
$388$	0	0
$389$	3056.19	0.398343	0.199171	−	0.979965i	$-0.436175\pi$
$389$	3056.19	0.398343	0.199171	+	0.979965i	$0.436175\pi$
$390$	0	0
$391$	−4431.58	−0.573184
$392$	0	0
$393$	− 3335.95i	− 0.428185i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14305.4i	1.80849i	0.427019	+	0.904243i	$0.359564\pi$
$397$	14305.4i	1.80849i	−0.427019	+	0.904243i	$0.640436\pi$
$398$	0	0
$399$	−7507.75	−0.941999
$400$	0	0
$401$	11391.2	1.41858	0.709289	−	0.704917i	$-0.249016\pi$
$401$	11391.2	1.41858	0.709289	+	0.704917i	$0.249016\pi$
$402$	0	0
$403$	− 226.027i	− 0.0279385i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1893.63i	− 0.230624i
$408$	0	0
$409$	5689.18	0.687804	0.343902	−	0.939006i	$-0.388251\pi$
$409$	5689.18	0.687804	0.343902	+	0.939006i	$0.388251\pi$
$410$	0	0
$411$	1701.85	0.204248
$412$	0	0
$413$	− 7454.34i	− 0.888145i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5906.60i	− 0.693639i
$418$	0	0
$419$	2086.17	0.243236	0.121618	−	0.992577i	$-0.461192\pi$
$419$	2086.17	0.243236	0.121618	+	0.992577i	$0.461192\pi$
$420$	0	0
$421$	661.557	0.0765851	0.0382926	−	0.999267i	$-0.487808\pi$
$421$	661.557	0.0765851	0.0382926	+	0.999267i	$0.487808\pi$
$422$	0	0
$423$	102.894i	0.0118271i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6033.32i	0.683777i
$428$	0	0
$429$	161.323	0.0181556
$430$	0	0
$431$	−14508.1	−1.62142	−0.810709	−	0.585449i	$-0.800918\pi$
$431$	−14508.1	−1.62142	−0.810709	+	0.585449i	$0.800918\pi$
$432$	0	0
$433$	− 11584.2i	− 1.28568i	−0.766000	−	0.642840i	$-0.777756\pi$
$433$	− 11584.2i	− 1.28568i	0.766000	−	0.642840i	$-0.222244\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5146.08i	− 0.563319i
$438$	0	0
$439$	10554.6	1.14748	0.573740	−	0.819038i	$-0.305492\pi$
$439$	10554.6	1.14748	0.573740	+	0.819038i	$0.305492\pi$
$440$	0	0
$441$	470.149	0.0507665
$442$	0	0
$443$	14791.1i	1.58634i	0.609001	+	0.793169i	$0.291571\pi$
$443$	14791.1i	1.58634i	−0.609001	+	0.793169i	$0.708429\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5432.60i	− 0.574840i
$448$	0	0
$449$	−540.276	−0.0567866	−0.0283933	−	0.999597i	$-0.509039\pi$
$449$	−540.276	−0.0567866	−0.0283933	+	0.999597i	$0.509039\pi$
$450$	0	0
$451$	−1458.18	−0.152246
$452$	0	0
$453$	800.238i	0.0829988i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8022.12i	0.821136i	0.911830	+	0.410568i	$0.134670\pi$
$457$	8022.12i	0.821136i	−0.911830	+	0.410568i	$0.865330\pi$
$458$	0	0
$459$	−2926.88	−0.297637
$460$	0	0
$461$	−11830.8	−1.19526	−0.597632	−	0.801771i	$-0.703891\pi$
$461$	−11830.8	−1.19526	−0.597632	+	0.801771i	$0.703891\pi$
$462$	0	0
$463$	− 12268.5i	− 1.23145i	−0.787959	−	0.615727i	$-0.788862\pi$
$463$	− 12268.5i	− 1.23145i	0.787959	−	0.615727i	$-0.211138\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12567.2i	− 1.24526i	−0.782515	−	0.622632i	$-0.786063\pi$
$467$	− 12567.2i	− 1.24526i	0.782515	−	0.622632i	$-0.213937\pi$
$468$	0	0
$469$	17446.9	1.71774
$470$	0	0
$471$	8979.99	0.878506
$472$	0	0
$473$	2302.71i	0.223845i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1434.22i	0.137670i
$478$	0	0
$479$	−5686.36	−0.542414	−0.271207	−	0.962521i	$-0.587423\pi$
$479$	−5686.36	−0.542414	−0.271207	+	0.962521i	$0.587423\pi$
$480$	0	0
$481$	20.4055	0.00193433
$482$	0	0
$483$	2438.19i	0.229693i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 9429.19i	− 0.877367i	−0.898642	−	0.438683i	$-0.855445\pi$
$487$	− 9429.19i	− 0.877367i	0.898642	−	0.438683i	$-0.144555\pi$
$488$	0	0
$489$	7069.07	0.653731
$490$	0	0
$491$	−19869.3	−1.82625	−0.913126	−	0.407677i	$-0.866339\pi$
$491$	−19869.3	−1.82625	−0.913126	+	0.407677i	$0.866339\pi$
$492$	0	0
$493$	15271.9i	1.39515i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 23052.3i	− 2.08056i
$498$	0	0
$499$	4160.66	0.373260	0.186630	−	0.982430i	$-0.440243\pi$
$499$	4160.66	0.373260	0.186630	+	0.982430i	$0.440243\pi$
$500$	0	0
$501$	−9445.66	−0.842318
$502$	0	0
$503$	− 9817.15i	− 0.870229i	−0.900375	−	0.435114i	$-0.856708\pi$
$503$	− 9817.15i	− 0.870229i	0.900375	−	0.435114i	$-0.143292\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6589.26i	− 0.577198i
$508$	0	0
$509$	8709.02	0.758390	0.379195	−	0.925317i	$-0.376201\pi$
$509$	8709.02	0.758390	0.379195	+	0.925317i	$0.376201\pi$
$510$	0	0
$511$	2399.91	0.207761
$512$	0	0
$513$	− 3398.78i	− 0.292514i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 807.623i	− 0.0687025i
$518$	0	0
$519$	11241.4	0.950753
$520$	0	0
$521$	−1711.84	−0.143948	−0.0719742	−	0.997406i	$-0.522930\pi$
$521$	−1711.84	−0.143948	−0.0719742	+	0.997406i	$0.522930\pi$
$522$	0	0
$523$	− 1901.43i	− 0.158975i	−0.996836	−	0.0794875i	$-0.974672\pi$
$523$	− 1901.43i	− 0.158975i	0.996836	−	0.0794875i	$-0.0253284\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 32187.6i	− 2.66056i
$528$	0	0
$529$	10495.8	0.862643
$530$	0	0
$531$	3374.60	0.275791
$532$	0	0
$533$	− 15.7131i	− 0.00127694i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6928.12i	0.556742i
$538$	0	0
$539$	−3690.24	−0.294898
$540$	0	0
$541$	10469.9	0.832043	0.416021	−	0.909355i	$-0.363424\pi$
$541$	10469.9	0.832043	0.416021	+	0.909355i	$0.363424\pi$
$542$	0	0
$543$	− 772.341i	− 0.0610392i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 23357.6i	− 1.82578i	−0.408207	−	0.912889i	$-0.633846\pi$
$547$	− 23357.6i	− 1.82578i	0.408207	−	0.912889i	$-0.366154\pi$
$548$	0	0
$549$	−2731.30	−0.212330
$550$	0	0
$551$	−17734.1	−1.37114
$552$	0	0
$553$	− 24824.1i	− 1.90891i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7925.47i	0.602896i	0.953483	+	0.301448i	$0.0974699\pi$
$557$	7925.47i	0.602896i	−0.953483	+	0.301448i	$0.902530\pi$
$558$	0	0
$559$	−24.8136	−0.00187747
$560$	0	0
$561$	22973.4	1.72894
$562$	0	0
$563$	11846.1i	0.886770i	0.896331	+	0.443385i	$0.146223\pi$
$563$	11846.1i	0.886770i	−0.896331	+	0.443385i	$0.853777\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1610.33i	0.119272i
$568$	0	0
$569$	−5807.33	−0.427866	−0.213933	−	0.976848i	$-0.568627\pi$
$569$	−5807.33	−0.427866	−0.213933	+	0.976848i	$0.568627\pi$
$570$	0	0
$571$	2520.62	0.184737	0.0923685	−	0.995725i	$-0.470556\pi$
$571$	2520.62	0.184737	0.0923685	+	0.995725i	$0.470556\pi$
$572$	0	0
$573$	− 12554.5i	− 0.915306i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8634.70i	0.622994i	0.950247	+	0.311497i	$0.100830\pi$
$577$	8634.70i	0.622994i	−0.950247	+	0.311497i	$0.899170\pi$
$578$	0	0
$579$	10910.4	0.783109
$580$	0	0
$581$	−13005.5	−0.928672
$582$	0	0
$583$	− 11257.4i	− 0.799712i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 22165.7i	− 1.55856i	−0.626674	−	0.779282i	$-0.715584\pi$
$587$	− 22165.7i	− 1.55856i	0.626674	−	0.779282i	$-0.284416\pi$
$588$	0	0
$589$	37377.2	2.61477
$590$	0	0
$591$	−10558.6	−0.734893
$592$	0	0
$593$	− 17874.0i	− 1.23777i	−0.785483	−	0.618883i	$-0.787585\pi$
$593$	− 17874.0i	− 1.23777i	0.785483	−	0.618883i	$-0.212415\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5811.85i	0.398431i
$598$	0	0
$599$	11051.4	0.753839	0.376919	−	0.926246i	$-0.376983\pi$
$599$	11051.4	0.753839	0.376919	+	0.926246i	$0.376983\pi$
$600$	0	0
$601$	17452.3	1.18452	0.592258	−	0.805749i	$-0.298237\pi$
$601$	17452.3	1.18452	0.592258	+	0.805749i	$0.298237\pi$
$602$	0	0
$603$	7898.24i	0.533402i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4557.98i	0.304782i	0.988320	+	0.152391i	$0.0486973\pi$
$607$	4557.98i	0.304782i	−0.988320	+	0.152391i	$0.951303\pi$
$608$	0	0
$609$	8402.38	0.559083
$610$	0	0
$611$	8.70283	0.000576233 0
$612$	0	0
$613$	26741.1i	1.76193i	0.473185	+	0.880963i	$0.343104\pi$
$613$	26741.1i	1.76193i	−0.473185	+	0.880963i	$0.656896\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6569.62i	− 0.428659i	−0.976761	−	0.214330i	$-0.931243\pi$
$617$	− 6569.62i	− 0.428659i	0.976761	−	0.214330i	$-0.0687567\pi$
$618$	0	0
$619$	12947.7	0.840729	0.420365	−	0.907355i	$-0.361902\pi$
$619$	12947.7	0.840729	0.420365	+	0.907355i	$0.361902\pi$
$620$	0	0
$621$	−1103.78	−0.0713253
$622$	0	0
$623$	− 15815.5i	− 1.01707i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	26677.3i	1.69919i
$628$	0	0
$629$	2905.87	0.184204
$630$	0	0
$631$	15799.6	0.996785	0.498393	−	0.866951i	$-0.333924\pi$
$631$	15799.6	0.996785	0.498393	+	0.866951i	$0.333924\pi$
$632$	0	0
$633$	8785.89i	0.551671i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 39.7655i	− 0.00247342i
$638$	0	0
$639$	10435.8	0.646065
$640$	0	0
$641$	7372.39	0.454278	0.227139	−	0.973862i	$-0.427063\pi$
$641$	7372.39	0.454278	0.227139	+	0.973862i	$0.427063\pi$
$642$	0	0
$643$	3238.37i	0.198614i	0.995057	+	0.0993070i	$0.0316626\pi$
$643$	3238.37i	0.198614i	−0.995057	+	0.0993070i	$0.968337\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23260.8i	1.41341i	0.707507	+	0.706707i	$0.249820\pi$
$647$	23260.8i	1.41341i	−0.707507	+	0.706707i	$0.750180\pi$
$648$	0	0
$649$	−26487.5	−1.60204
$650$	0	0
$651$	−17709.2	−1.06617
$652$	0	0
$653$	20129.6i	1.20633i	0.797618	+	0.603163i	$0.206093\pi$
$653$	20129.6i	1.20633i	−0.797618	+	0.603163i	$0.793907\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1086.45i	0.0645150i
$658$	0	0
$659$	12381.4	0.731882	0.365941	−	0.930638i	$-0.380747\pi$
$659$	12381.4	0.731882	0.365941	+	0.930638i	$0.380747\pi$
$660$	0	0
$661$	−24627.1	−1.44914	−0.724572	−	0.689199i	$-0.757963\pi$
$661$	−24627.1	−1.44914	−0.724572	+	0.689199i	$0.757963\pi$
$662$	0	0
$663$	247.558i	0.0145013i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5759.29i	0.334333i
$668$	0	0
$669$	7911.22	0.457198
$670$	0	0
$671$	21438.2	1.23340
$672$	0	0
$673$	− 5211.65i	− 0.298506i	−0.988799	−	0.149253i	$-0.952313\pi$
$673$	− 5211.65i	− 0.298506i	0.988799	−	0.149253i	$-0.0476868\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18402.6i	1.04471i	0.852728	+	0.522354i	$0.174946\pi$
$677$	18402.6i	1.04471i	−0.852728	+	0.522354i	$0.825054\pi$
$678$	0	0
$679$	−16916.6	−0.956112
$680$	0	0
$681$	9861.81	0.554927
$682$	0	0
$683$	30347.2i	1.70015i	0.526659	+	0.850076i	$0.323444\pi$
$683$	30347.2i	1.70015i	−0.526659	+	0.850076i	$0.676556\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	19146.0i	1.06327i
$688$	0	0
$689$	121.308	0.00670748
$690$	0	0
$691$	−18853.1	−1.03792	−0.518961	−	0.854798i	$-0.673681\pi$
$691$	−18853.1	−1.03792	−0.518961	+	0.854798i	$0.673681\pi$
$692$	0	0
$693$	− 12639.6i	− 0.692842i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2237.64i	− 0.121602i
$698$	0	0
$699$	−8858.74	−0.479354
$700$	0	0
$701$	11008.1	0.593109	0.296555	−	0.955016i	$-0.404162\pi$
$701$	11008.1	0.593109	0.296555	+	0.955016i	$0.404162\pi$
$702$	0	0
$703$	3374.37i	0.181034i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2977.65i	− 0.158396i
$708$	0	0
$709$	−18670.6	−0.988981	−0.494491	−	0.869183i	$-0.664645\pi$
$709$	−18670.6	−0.988981	−0.494491	+	0.869183i	$0.664645\pi$
$710$	0	0
$711$	11237.9	0.592763
$712$	0	0
$713$	− 12138.5i	− 0.637574i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4153.03i	0.216315i
$718$	0	0
$719$	−26208.7	−1.35941	−0.679707	−	0.733484i	$-0.737893\pi$
$719$	−26208.7	−1.35941	−0.679707	+	0.733484i	$0.737893\pi$
$720$	0	0
$721$	−23395.6	−1.20846
$722$	0	0
$723$	− 12710.5i	− 0.653813i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 24977.6i	− 1.27424i	−0.770767	−	0.637118i	$-0.780127\pi$
$727$	− 24977.6i	− 1.27424i	0.770767	−	0.637118i	$-0.219873\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−3533.61	−0.178790
$732$	0	0
$733$	1491.96i	0.0751800i	0.999293	+	0.0375900i	$0.0119681\pi$
$733$	1491.96i	0.0751800i	−0.999293	+	0.0375900i	$0.988032\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 61994.1i	− 3.09848i
$738$	0	0
$739$	5276.36	0.262644	0.131322	−	0.991340i	$-0.458078\pi$
$739$	5276.36	0.262644	0.131322	+	0.991340i	$0.458078\pi$
$740$	0	0
$741$	−287.471	−0.0142517
$742$	0	0
$743$	− 23611.8i	− 1.16586i	−0.812522	−	0.582930i	$-0.801906\pi$
$743$	− 23611.8i	− 1.16586i	0.812522	−	0.582930i	$-0.198094\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5887.62i	− 0.288376i
$748$	0	0
$749$	8002.98	0.390417
$750$	0	0
$751$	31848.5	1.54749	0.773747	−	0.633494i	$-0.218380\pi$
$751$	31848.5	1.54749	0.773747	+	0.633494i	$0.218380\pi$
$752$	0	0
$753$	− 5583.67i	− 0.270226i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	19974.7i	0.959041i	0.877531	+	0.479521i	$0.159189\pi$
$757$	19974.7i	0.959041i	−0.877531	+	0.479521i	$0.840811\pi$
$758$	0	0
$759$	8663.65	0.414322
$760$	0	0
$761$	27224.0	1.29680	0.648402	−	0.761298i	$-0.275438\pi$
$761$	27224.0	1.29680	0.648402	+	0.761298i	$0.275438\pi$
$762$	0	0
$763$	− 32809.3i	− 1.55672i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 285.426i	− 0.0134369i
$768$	0	0
$769$	29987.8	1.40623	0.703113	−	0.711078i	$-0.251793\pi$
$769$	29987.8	1.40623	0.703113	+	0.711078i	$0.251793\pi$
$770$	0	0
$771$	−1041.75	−0.0486612
$772$	0	0
$773$	− 2856.40i	− 0.132908i	−0.997790	−	0.0664538i	$-0.978831\pi$
$773$	− 2856.40i	− 0.132908i	0.997790	−	0.0664538i	$-0.0211685\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1598.77i	− 0.0738165i
$778$	0	0
$779$	2598.41	0.119509
$780$	0	0
$781$	−81911.9	−3.75293
$782$	0	0
$783$	3803.78i	0.173609i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 16704.1i	− 0.756590i	−0.925685	−	0.378295i	$-0.876511\pi$
$787$	− 16704.1i	− 0.756590i	0.925685	−	0.378295i	$-0.123489\pi$
$788$	0	0
$789$	−4468.16	−0.201611
$790$	0	0
$791$	−26901.1	−1.20922
$792$	0	0
$793$	231.015i	0.0103450i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 33709.4i	− 1.49818i	−0.662468	−	0.749090i	$-0.730491\pi$
$797$	− 33709.4i	− 1.49818i	0.662468	−	0.749090i	$-0.269509\pi$
$798$	0	0
$799$	1239.33	0.0548742
$800$	0	0
$801$	7159.70	0.315825
$802$	0	0
$803$	− 8527.62i	− 0.374761i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4712.64i	0.205567i
$808$	0	0
$809$	−21582.4	−0.937944	−0.468972	−	0.883213i	$-0.655375\pi$
$809$	−21582.4	−0.937944	−0.468972	+	0.883213i	$0.655375\pi$
$810$	0	0
$811$	40451.0	1.75145	0.875726	−	0.482809i	$-0.160384\pi$
$811$	40451.0	1.75145	0.875726	+	0.482809i	$0.160384\pi$
$812$	0	0
$813$	9043.26i	0.390112i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4103.32i	− 0.175712i
$818$	0	0
$819$	136.203	0.00581112
$820$	0	0
$821$	−2857.45	−0.121468	−0.0607342	−	0.998154i	$-0.519344\pi$
$821$	−2857.45	−0.121468	−0.0607342	+	0.998154i	$0.519344\pi$
$822$	0	0
$823$	15150.8i	0.641707i	0.947129	+	0.320853i	$0.103970\pi$
$823$	15150.8i	0.641707i	−0.947129	+	0.320853i	$0.896030\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12171.5i	0.511784i	0.966705	+	0.255892i	$0.0823692\pi$
$827$	12171.5i	0.511784i	−0.966705	+	0.255892i	$0.917631\pi$
$828$	0	0
$829$	13543.0	0.567392	0.283696	−	0.958914i	$-0.408439\pi$
$829$	13543.0	0.567392	0.283696	+	0.958914i	$0.408439\pi$
$830$	0	0
$831$	−8377.22	−0.349702
$832$	0	0
$833$	− 5662.84i	− 0.235541i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8016.99i	− 0.331072i
$838$	0	0
$839$	25376.2	1.04420	0.522101	−	0.852884i	$-0.325149\pi$
$839$	25376.2	1.04420	0.522101	+	0.852884i	$0.325149\pi$
$840$	0	0
$841$	−4541.65	−0.186217
$842$	0	0
$843$	5237.42i	0.213981i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	72748.5i	2.95120i
$848$	0	0
$849$	−7992.99	−0.323108
$850$	0	0
$851$	1095.85	0.0441425
$852$	0	0
$853$	10225.0i	0.410431i	0.978717	+	0.205215i	$0.0657895\pi$
$853$	10225.0i	0.410431i	−0.978717	+	0.205215i	$0.934211\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 35735.9i	− 1.42440i	−0.701975	−	0.712202i	$-0.747698\pi$
$857$	− 35735.9i	− 1.42440i	0.701975	−	0.712202i	$-0.252302\pi$
$858$	0	0
$859$	−3972.20	−0.157776	−0.0788881	−	0.996883i	$-0.525137\pi$
$859$	−3972.20	−0.157776	−0.0788881	+	0.996883i	$0.525137\pi$
$860$	0	0
$861$	−1231.12	−0.0487298
$862$	0	0
$863$	22682.0i	0.894673i	0.894366	+	0.447337i	$0.147627\pi$
$863$	22682.0i	0.894673i	−0.894366	+	0.447337i	$0.852373\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	20514.7i	0.803593i
$868$	0	0
$869$	−88207.4	−3.44331
$870$	0	0
$871$	668.039	0.0259881
$872$	0	0
$873$	− 7658.19i	− 0.296896i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12622.0i	0.485993i	0.970027	+	0.242997i	$0.0781304\pi$
$877$	12622.0i	0.485993i	−0.970027	+	0.242997i	$0.921870\pi$
$878$	0	0
$879$	17783.4	0.682387
$880$	0	0
$881$	−5343.08	−0.204328	−0.102164	−	0.994768i	$-0.532577\pi$
$881$	−5343.08	−0.204328	−0.102164	+	0.994768i	$0.532577\pi$
$882$	0	0
$883$	− 21649.7i	− 0.825110i	−0.910933	−	0.412555i	$-0.864637\pi$
$883$	− 21649.7i	− 0.825110i	0.910933	−	0.412555i	$-0.135363\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30808.6i	1.16624i	0.812388	+	0.583118i	$0.198167\pi$
$887$	30808.6i	1.16624i	−0.812388	+	0.583118i	$0.801833\pi$
$888$	0	0
$889$	−36505.0	−1.37721
$890$	0	0
$891$	5721.99	0.215145
$892$	0	0
$893$	1439.15i	0.0539297i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	93.3582i	0.00347507i
$898$	0	0
$899$	−41831.0	−1.55188
$900$	0	0
$901$	17274.9	0.638747
$902$	0	0
$903$	1944.14i	0.0716467i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26499.7i	0.970131i	0.874478	+	0.485065i	$0.161204\pi$
$907$	26499.7i	0.970131i	−0.874478	+	0.485065i	$0.838796\pi$
$908$	0	0
$909$	1347.99	0.0491859
$910$	0	0
$911$	−25861.3	−0.940532	−0.470266	−	0.882525i	$-0.655842\pi$
$911$	−25861.3	−0.940532	−0.470266	+	0.882525i	$0.655842\pi$
$912$	0	0
$913$	46212.5i	1.67515i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	22106.9i	0.796113i
$918$	0	0
$919$	10296.7	0.369595	0.184798	−	0.982777i	$-0.440837\pi$
$919$	10296.7	0.369595	0.184798	+	0.982777i	$0.440837\pi$
$920$	0	0
$921$	13903.4	0.497431
$922$	0	0
$923$	− 882.670i	− 0.0314772i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 10591.3i	− 0.375256i
$928$	0	0
$929$	−271.470	−0.00958734	−0.00479367	−	0.999989i	$-0.501526\pi$
$929$	−271.470	−0.00958734	−0.00479367	+	0.999989i	$0.501526\pi$
$930$	0	0
$931$	6575.85	0.231487
$932$	0	0
$933$	2285.91i	0.0802114i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8739.10i	0.304689i	0.988327	+	0.152345i	$0.0486824\pi$
$937$	8739.10i	0.304689i	−0.988327	+	0.152345i	$0.951318\pi$
$938$	0	0
$939$	−26202.9	−0.910651
$940$	0	0
$941$	−55104.2	−1.90897	−0.954487	−	0.298252i	$-0.903597\pi$
$941$	−55104.2	−1.90897	−0.954487	+	0.298252i	$0.903597\pi$
$942$	0	0
$943$	− 843.851i	− 0.0291406i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20552.8i	− 0.705256i	−0.935764	−	0.352628i	$-0.885288\pi$
$947$	− 20552.8i	− 0.705256i	0.935764	−	0.352628i	$-0.114712\pi$
$948$	0	0
$949$	91.8924	0.00314326
$950$	0	0
$951$	−13284.6	−0.452980
$952$	0	0
$953$	− 5642.08i	− 0.191778i	−0.995392	−	0.0958892i	$-0.969431\pi$
$953$	− 5642.08i	− 0.191778i	0.995392	−	0.0958892i	$-0.0305694\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 29856.2i	− 1.00848i
$958$	0	0
$959$	−11277.9	−0.379754
$960$	0	0
$961$	58373.8	1.95944
$962$	0	0
$963$	3622.97i	0.121234i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 21996.2i	− 0.731489i	−0.930715	−	0.365745i	$-0.880814\pi$
$967$	− 21996.2i	− 0.731489i	0.930715	−	0.365745i	$-0.119186\pi$
$968$	0	0
$969$	−40937.5	−1.35718
$970$	0	0
$971$	43179.9	1.42710	0.713548	−	0.700606i	$-0.247087\pi$
$971$	43179.9	1.42710	0.713548	+	0.700606i	$0.247087\pi$
$972$	0	0
$973$	39142.3i	1.28967i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50887.5i	1.66636i	0.553001	+	0.833181i	$0.313483\pi$
$977$	50887.5i	1.66636i	−0.553001	+	0.833181i	$0.686517\pi$
$978$	0	0
$979$	−56197.2	−1.83460
$980$	0	0
$981$	14852.8	0.483399
$982$	0	0
$983$	13150.8i	0.426700i	0.976976	+	0.213350i	$0.0684375\pi$
$983$	13150.8i	0.426700i	−0.976976	+	0.213350i	$0.931562\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 681.864i	− 0.0219898i
$988$	0	0
$989$	−1332.58	−0.0428450
$990$	0	0
$991$	49196.4	1.57697	0.788483	−	0.615056i	$-0.210867\pi$
$991$	49196.4	1.57697	0.788483	+	0.615056i	$0.210867\pi$
$992$	0	0
$993$	− 6719.03i	− 0.214725i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39419.0i	1.25217i	0.779755	+	0.626085i	$0.215344\pi$
$997$	39419.0i	1.25217i	−0.779755	+	0.626085i	$0.784656\pi$
$998$	0	0
$999$	723.766	0.0229218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.x.49.2		4
4.3	odd	2		600.4.f.j.49.3		4
5.2	odd	4		1200.4.a.bp.1.1		2
5.3	odd	4		1200.4.a.br.1.2		2
5.4	even	2	inner	1200.4.f.x.49.3		4
12.11	even	2		1800.4.f.z.649.2		4
20.3	even	4		600.4.a.s.1.1	✓	2
20.7	even	4		600.4.a.u.1.2	yes	2
20.19	odd	2		600.4.f.j.49.2		4
60.23	odd	4		1800.4.a.bo.1.1		2
60.47	odd	4		1800.4.a.bm.1.2		2
60.59	even	2		1800.4.f.z.649.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.s.1.1	✓	2	20.3	even	4
600.4.a.u.1.2	yes	2	20.7	even	4
600.4.f.j.49.2		4	20.19	odd	2
600.4.f.j.49.3		4	4.3	odd	2
1200.4.a.bp.1.1		2	5.2	odd	4
1200.4.a.br.1.2		2	5.3	odd	4
1200.4.f.x.49.2		4	1.1	even	1	trivial
1200.4.f.x.49.3		4	5.4	even	2	inner
1800.4.a.bm.1.2		2	60.47	odd	4
1800.4.a.bo.1.1		2	60.23	odd	4
1800.4.f.z.649.2		4	12.11	even	2
1800.4.f.z.649.3		4	60.59	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +19.8806i q^{7} -9.00000 q^{9} +70.6418 q^{11} +0.761226i q^{13} +108.403i q^{17} -125.881 q^{19} +59.6418 q^{21} +40.8806i q^{23} +27.0000i q^{27} +140.881 q^{29} -296.926 q^{31} -211.926i q^{33} -26.8061i q^{37} +2.28368 q^{39} -20.6418 q^{41} +32.5969i q^{43} -11.4326i q^{47} -52.2388 q^{49} +325.209 q^{51} -159.358i q^{53} +377.642i q^{57} -374.955 q^{59} +303.478 q^{61} -178.926i q^{63} -877.583i q^{67} +122.642 q^{69} -1159.54 q^{71} -120.716i q^{73} +1404.40i q^{77} -1248.66 q^{79} +81.0000 q^{81} +654.180i q^{83} -422.642i q^{87} -795.522 q^{89} -15.1336 q^{91} +890.777i q^{93} +850.910i q^{97} -635.777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9} + 32 q^{11} - 420 q^{19} - 12 q^{21} + 480 q^{29} - 436 q^{31} - 492 q^{39} + 168 q^{41} - 376 q^{49} + 48 q^{51} - 2168 q^{59} + 1548 q^{61} + 240 q^{69} - 2216 q^{71} - 2656 q^{79} + 324 q^{81}+ \cdots - 288 q^{99}+O(q^{100}) \) 4 * q - 36 * q^9 + 32 * q^11 - 420 * q^19 - 12 * q^21 + 480 * q^29 - 436 * q^31 - 492 * q^39 + 168 * q^41 - 376 * q^49 + 48 * q^51 - 2168 * q^59 + 1548 * q^61 + 240 * q^69 - 2216 * q^71 - 2656 * q^79 + 324 * q^81 - 2848 * q^89 - 3652 * q^91 - 288 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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