LMFDB - Embedded newform 4600.2.e.q.4049.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4600,2,Mod(4049,4600)] 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("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,0,-14] 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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3356224.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$0.254102i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.q.4049.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93543i q^{3} -4.93543i q^{7} -0.745898 q^{9} +0.745898 q^{11} +1.74590i q^{13} +6.10856i q^{17} -5.44364 q^{19} -9.55220 q^{21} -1.00000i q^{23} -4.36266i q^{27} +1.66492 q^{29} -1.61676 q^{31} -1.44364i q^{33} -4.34625i q^{37} +3.37907 q^{39} -6.95184 q^{41} +5.01641i q^{43} -2.68133i q^{47} -17.3585 q^{49} +11.8227 q^{51} +13.7417i q^{53} +10.5358i q^{57} -12.2171 q^{59} -13.9794 q^{61} +3.68133i q^{63} -13.1044i q^{67} -1.93543 q^{69} -9.67716 q^{71} +5.69774i q^{73} -3.68133i q^{77} -10.3791 q^{79} -10.6813 q^{81} +0.637339i q^{83} -3.22235i q^{87} -2.72532 q^{89} +8.61676 q^{91} +3.12914i q^{93} -7.12497i q^{97} -0.556364 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9} + 6 q^{11} - 14 q^{19} - 12 q^{21} + 2 q^{29} + 20 q^{31} - 14 q^{39} - 20 q^{41} - 12 q^{49} + 18 q^{51} - 20 q^{59} - 26 q^{61} + 4 q^{69} + 20 q^{71} - 28 q^{79} - 50 q^{81} + 40 q^{89}+ \cdots - 22 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 6 * q^11 - 14 * q^19 - 12 * q^21 + 2 * q^29 + 20 * q^31 - 14 * q^39 - 20 * q^41 - 12 * q^49 + 18 * q^51 - 20 * q^59 - 26 * q^61 + 4 * q^69 + 20 * q^71 - 28 * q^79 - 50 * q^81 + 40 * q^89 + 22 * q^91 - 22 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.93543i	− 1.11742i	−0.829362	−	0.558711i	$-0.811296\pi$
$3$	− 1.93543i	− 1.11742i	0.829362	−	0.558711i	$-0.188704\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.93543i	− 1.86542i	−0.360630	−	0.932709i	$-0.617438\pi$
$7$	− 4.93543i	− 1.86542i	0.360630	−	0.932709i	$-0.382562\pi$
$8$	0	0
$9$	−0.745898	−0.248633
$10$	0	0
$11$	0.745898	0.224897	0.112448	−	0.993658i	$-0.464131\pi$
$11$	0.745898	0.224897	0.112448	+	0.993658i	$0.464131\pi$
$12$	0	0
$13$	1.74590i	0.484225i	0.970248	+	0.242113i	$0.0778403\pi$
$13$	1.74590i	0.484225i	−0.970248	+	0.242113i	$0.922160\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.10856i	1.48154i	0.671757	+	0.740772i	$0.265540\pi$
$17$	6.10856i	1.48154i	−0.671757	+	0.740772i	$0.734460\pi$
$18$	0	0
$19$	−5.44364	−1.24886	−0.624428	−	0.781083i	$-0.714668\pi$
$19$	−5.44364	−1.24886	−0.624428	+	0.781083i	$0.714668\pi$
$20$	0	0
$21$	−9.55220	−2.08446
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.36266i	− 0.839595i
$28$	0	0
$29$	1.66492	0.309169	0.154584	−	0.987980i	$-0.450596\pi$
$29$	1.66492	0.309169	0.154584	+	0.987980i	$0.450596\pi$
$30$	0	0
$31$	−1.61676	−0.290379	−0.145190	−	0.989404i	$-0.546379\pi$
$31$	−1.61676	−0.290379	−0.145190	+	0.989404i	$0.546379\pi$
$32$	0	0
$33$	− 1.44364i	− 0.251305i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.34625i	− 0.714520i	−0.934005	−	0.357260i	$-0.883711\pi$
$37$	− 4.34625i	− 0.714520i	0.934005	−	0.357260i	$-0.116289\pi$
$38$	0	0
$39$	3.37907	0.541084
$40$	0	0
$41$	−6.95184	−1.08569	−0.542847	−	0.839831i	$-0.682654\pi$
$41$	−6.95184	−1.08569	−0.542847	+	0.839831i	$0.682654\pi$
$42$	0	0
$43$	5.01641i	0.764995i	0.923957	+	0.382497i	$0.124936\pi$
$43$	5.01641i	0.764995i	−0.923957	+	0.382497i	$0.875064\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.68133i	− 0.391112i	−0.980693	−	0.195556i	$-0.937349\pi$
$47$	− 2.68133i	− 0.391112i	0.980693	−	0.195556i	$-0.0626512\pi$
$48$	0	0
$49$	−17.3585	−2.47978
$50$	0	0
$51$	11.8227	1.65551
$52$	0	0
$53$	13.7417i	1.88757i	0.330558	+	0.943786i	$0.392763\pi$
$53$	13.7417i	1.88757i	−0.330558	+	0.943786i	$0.607237\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.5358i	1.39550i
$58$	0	0
$59$	−12.2171	−1.59053	−0.795267	−	0.606260i	$-0.792669\pi$
$59$	−12.2171	−1.59053	−0.795267	+	0.606260i	$0.792669\pi$
$60$	0	0
$61$	−13.9794	−1.78988	−0.894941	−	0.446185i	$-0.852782\pi$
$61$	−13.9794	−1.78988	−0.894941	+	0.446185i	$0.852782\pi$
$62$	0	0
$63$	3.68133i	0.463804i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.1044i	− 1.60096i	−0.599362	−	0.800478i	$-0.704579\pi$
$67$	− 13.1044i	− 1.60096i	0.599362	−	0.800478i	$-0.295421\pi$
$68$	0	0
$69$	−1.93543	−0.232999
$70$	0	0
$71$	−9.67716	−1.14847	−0.574234	−	0.818691i	$-0.694700\pi$
$71$	−9.67716	−1.14847	−0.574234	+	0.818691i	$0.694700\pi$
$72$	0	0
$73$	5.69774i	0.666870i	0.942773	+	0.333435i	$0.108208\pi$
$73$	5.69774i	0.666870i	−0.942773	+	0.333435i	$0.891792\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.68133i	− 0.419527i
$78$	0	0
$79$	−10.3791	−1.16774	−0.583868	−	0.811848i	$-0.698462\pi$
$79$	−10.3791	−1.16774	−0.583868	+	0.811848i	$0.698462\pi$
$80$	0	0
$81$	−10.6813	−1.18681
$82$	0	0
$83$	0.637339i	0.0699570i	0.999388	+	0.0349785i	$0.0111363\pi$
$83$	0.637339i	0.0699570i	−0.999388	+	0.0349785i	$0.988864\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.22235i	− 0.345472i
$88$	0	0
$89$	−2.72532	−0.288884	−0.144442	−	0.989513i	$-0.546139\pi$
$89$	−2.72532	−0.288884	−0.144442	+	0.989513i	$0.546139\pi$
$90$	0	0
$91$	8.61676	0.903282
$92$	0	0
$93$	3.12914i	0.324476i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 7.12497i	− 0.723431i	−0.932289	−	0.361715i	$-0.882191\pi$
$97$	− 7.12497i	− 0.723431i	0.932289	−	0.361715i	$-0.117809\pi$
$98$	0	0
$99$	−0.556364	−0.0559167
$100$	0	0
$101$	14.9753	1.49009	0.745047	−	0.667012i	$-0.232427\pi$
$101$	14.9753	1.49009	0.745047	+	0.667012i	$0.232427\pi$
$102$	0	0
$103$	− 10.8667i	− 1.07073i	−0.844622	−	0.535364i	$-0.820175\pi$
$103$	− 10.8667i	− 1.07073i	0.844622	−	0.535364i	$-0.179825\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.49180i	0.724259i	0.932128	+	0.362130i	$0.117950\pi$
$107$	7.49180i	0.724259i	−0.932128	+	0.362130i	$0.882050\pi$
$108$	0	0
$109$	18.3309	1.75578	0.877891	−	0.478860i	$-0.158950\pi$
$109$	18.3309	1.75578	0.877891	+	0.478860i	$0.158950\pi$
$110$	0	0
$111$	−8.41188	−0.798420
$112$	0	0
$113$	0.379068i	0.0356597i	0.999841	+	0.0178299i	$0.00567572\pi$
$113$	0.379068i	0.0356597i	−0.999841	+	0.0178299i	$0.994324\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.30226i	− 0.120394i
$118$	0	0
$119$	30.1484	2.76370
$120$	0	0
$121$	−10.4436	−0.949421
$122$	0	0
$123$	13.4548i	1.21318i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.64852i	− 0.235018i	−0.993072	−	0.117509i	$-0.962509\pi$
$127$	− 2.64852i	− 0.235018i	0.993072	−	0.117509i	$-0.0374909\pi$
$128$	0	0
$129$	9.70892	0.854822
$130$	0	0
$131$	−5.53579	−0.483664	−0.241832	−	0.970318i	$-0.577748\pi$
$131$	−5.53579	−0.483664	−0.241832	+	0.970318i	$0.577748\pi$
$132$	0	0
$133$	26.8667i	2.32964i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.16896i	− 0.527050i	−0.964653	−	0.263525i	$-0.915115\pi$
$137$	− 6.16896i	− 0.527050i	0.964653	−	0.263525i	$-0.0848851\pi$
$138$	0	0
$139$	20.7693	1.76163	0.880815	−	0.473460i	$-0.156995\pi$
$139$	20.7693	1.76163	0.880815	+	0.473460i	$0.156995\pi$
$140$	0	0
$141$	−5.18953	−0.437038
$142$	0	0
$143$	1.30226i	0.108901i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	33.5962i	2.77097i
$148$	0	0
$149$	1.06457	0.0872128	0.0436064	−	0.999049i	$-0.486115\pi$
$149$	1.06457	0.0872128	0.0436064	+	0.999049i	$0.486115\pi$
$150$	0	0
$151$	−12.6608	−1.03032	−0.515159	−	0.857095i	$-0.672267\pi$
$151$	−12.6608	−1.03032	−0.515159	+	0.857095i	$0.672267\pi$
$152$	0	0
$153$	− 4.55636i	− 0.368360i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.8873i	− 1.18813i	−0.804416	−	0.594067i	$-0.797521\pi$
$157$	− 14.8873i	− 1.18813i	0.804416	−	0.594067i	$-0.202479\pi$
$158$	0	0
$159$	26.5962	2.10921
$160$	0	0
$161$	−4.93543	−0.388967
$162$	0	0
$163$	6.72949i	0.527094i	0.964646	+	0.263547i	$0.0848925\pi$
$163$	6.72949i	0.527094i	−0.964646	+	0.263547i	$0.915108\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.98359i	0.230877i	0.993315	+	0.115439i	$0.0368274\pi$
$167$	2.98359i	0.230877i	−0.993315	+	0.115439i	$0.963173\pi$
$168$	0	0
$169$	9.95184	0.765526
$170$	0	0
$171$	4.06040	0.310506
$172$	0	0
$173$	− 15.8503i	− 1.20508i	−0.798091	−	0.602538i	$-0.794156\pi$
$173$	− 15.8503i	− 1.20508i	0.798091	−	0.602538i	$-0.205844\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	23.6454i	1.77730i
$178$	0	0
$179$	−3.02759	−0.226292	−0.113146	−	0.993578i	$-0.536093\pi$
$179$	−3.02759	−0.226292	−0.113146	+	0.993578i	$0.536093\pi$
$180$	0	0
$181$	1.82270	0.135481	0.0677403	−	0.997703i	$-0.478421\pi$
$181$	1.82270	0.135481	0.0677403	+	0.997703i	$0.478421\pi$
$182$	0	0
$183$	27.0562i	2.00005i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.55636i	0.333194i
$188$	0	0
$189$	−21.5316	−1.56619
$190$	0	0
$191$	19.6126	1.41912	0.709559	−	0.704646i	$-0.248894\pi$
$191$	19.6126	1.41912	0.709559	+	0.704646i	$0.248894\pi$
$192$	0	0
$193$	− 0.335076i	− 0.0241193i	−0.999927	−	0.0120597i	$-0.996161\pi$
$193$	− 0.335076i	− 0.0241193i	0.999927	−	0.0120597i	$-0.00383880\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.5316i	1.46282i	0.681939	+	0.731409i	$0.261137\pi$
$197$	20.5316i	1.46282i	−0.681939	+	0.731409i	$0.738863\pi$
$198$	0	0
$199$	−16.7253	−1.18563	−0.592813	−	0.805340i	$-0.701983\pi$
$199$	−16.7253	−1.18563	−0.592813	+	0.805340i	$0.701983\pi$
$200$	0	0
$201$	−25.3627	−1.78894
$202$	0	0
$203$	− 8.21712i	− 0.576729i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.745898i	0.0518435i
$208$	0	0
$209$	−4.06040	−0.280864
$210$	0	0
$211$	−13.3955	−0.922183	−0.461091	−	0.887353i	$-0.652542\pi$
$211$	−13.3955	−0.922183	−0.461091	+	0.887353i	$0.652542\pi$
$212$	0	0
$213$	18.7295i	1.28332i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.97942i	0.541679i
$218$	0	0
$219$	11.0276	0.745175
$220$	0	0
$221$	−10.6649	−0.717400
$222$	0	0
$223$	1.27468i	0.0853587i	0.999089	+	0.0426794i	$0.0135894\pi$
$223$	1.27468i	0.0853587i	−0.999089	+	0.0426794i	$0.986411\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.57978i	− 0.237598i	−0.992918	−	0.118799i	$-0.962096\pi$
$227$	− 3.57978i	− 0.237598i	0.992918	−	0.118799i	$-0.0379045\pi$
$228$	0	0
$229$	−10.4119	−0.688036	−0.344018	−	0.938963i	$-0.611788\pi$
$229$	−10.4119	−0.688036	−0.344018	+	0.938963i	$0.611788\pi$
$230$	0	0
$231$	−7.12497	−0.468788
$232$	0	0
$233$	3.06040i	0.200493i	0.994963	+	0.100247i	$0.0319632\pi$
$233$	3.06040i	0.200493i	−0.994963	+	0.100247i	$0.968037\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	20.0880i	1.30485i
$238$	0	0
$239$	19.1895	1.24127	0.620634	−	0.784100i	$-0.286875\pi$
$239$	19.1895	1.24127	0.620634	+	0.784100i	$0.286875\pi$
$240$	0	0
$241$	0.346255	0.0223042	0.0111521	−	0.999938i	$-0.496450\pi$
$241$	0.346255	0.0223042	0.0111521	+	0.999938i	$0.496450\pi$
$242$	0	0
$243$	7.58501i	0.486579i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 9.50403i	− 0.604727i
$248$	0	0
$249$	1.23353	0.0781715
$250$	0	0
$251$	14.3861	0.908041	0.454021	−	0.890991i	$-0.349989\pi$
$251$	14.3861	0.908041	0.454021	+	0.890991i	$0.349989\pi$
$252$	0	0
$253$	− 0.745898i	− 0.0468942i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.28586i	− 0.0802095i	−0.999195	−	0.0401047i	$-0.987231\pi$
$257$	− 1.28586i	− 0.0802095i	0.999195	−	0.0401047i	$-0.0127692\pi$
$258$	0	0
$259$	−21.4506	−1.33288
$260$	0	0
$261$	−1.24186	−0.0768694
$262$	0	0
$263$	28.2294i	1.74070i	0.492437	+	0.870348i	$0.336106\pi$
$263$	28.2294i	1.74070i	−0.492437	+	0.870348i	$0.663894\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.27468i	0.322805i
$268$	0	0
$269$	19.4395	1.18525	0.592623	−	0.805480i	$-0.298093\pi$
$269$	19.4395	1.18525	0.592623	+	0.805480i	$0.298093\pi$
$270$	0	0
$271$	−6.42723	−0.390426	−0.195213	−	0.980761i	$-0.562540\pi$
$271$	−6.42723	−0.390426	−0.195213	+	0.980761i	$0.562540\pi$
$272$	0	0
$273$	− 16.6772i	− 1.00935i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4.08514i	− 0.245452i	−0.992441	−	0.122726i	$-0.960836\pi$
$277$	− 4.08514i	− 0.245452i	0.992441	−	0.122726i	$-0.0391637\pi$
$278$	0	0
$279$	1.20594	0.0721978
$280$	0	0
$281$	−6.38741	−0.381041	−0.190520	−	0.981683i	$-0.561018\pi$
$281$	−6.38741	−0.381041	−0.190520	+	0.981683i	$0.561018\pi$
$282$	0	0
$283$	− 21.4835i	− 1.27706i	−0.769597	−	0.638530i	$-0.779543\pi$
$283$	− 21.4835i	− 1.27706i	0.769597	−	0.638530i	$-0.220457\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	34.3103i	2.02527i
$288$	0	0
$289$	−20.3145	−1.19497
$290$	0	0
$291$	−13.7899	−0.808378
$292$	0	0
$293$	− 16.1208i	− 0.941787i	−0.882190	−	0.470894i	$-0.843932\pi$
$293$	− 16.1208i	− 0.941787i	0.882190	−	0.470894i	$-0.156068\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.25410i	− 0.188822i
$298$	0	0
$299$	1.74590	0.100968
$300$	0	0
$301$	24.7581	1.42704
$302$	0	0
$303$	− 28.9836i	− 1.66506i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.96302i	0.397400i	0.980060	+	0.198700i	$0.0636720\pi$
$307$	6.96302i	0.397400i	−0.980060	+	0.198700i	$0.936328\pi$
$308$	0	0
$309$	−21.0318	−1.19645
$310$	0	0
$311$	−4.14031	−0.234776	−0.117388	−	0.993086i	$-0.537452\pi$
$311$	−4.14031	−0.234776	−0.117388	+	0.993086i	$0.537452\pi$
$312$	0	0
$313$	6.26528i	0.354135i	0.984199	+	0.177067i	$0.0566610\pi$
$313$	6.26528i	0.354135i	−0.984199	+	0.177067i	$0.943339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.53996i	− 0.535817i	−0.963444	−	0.267909i	$-0.913667\pi$
$317$	− 9.53996i	− 0.535817i	0.963444	−	0.267909i	$-0.0863326\pi$
$318$	0	0
$319$	1.24186	0.0695310
$320$	0	0
$321$	14.4999	0.809304
$322$	0	0
$323$	− 33.2528i	− 1.85023i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 35.4782i	− 1.96195i
$328$	0	0
$329$	−13.2335	−0.729588
$330$	0	0
$331$	−14.3023	−0.786123	−0.393062	−	0.919512i	$-0.628584\pi$
$331$	−14.3023	−0.786123	−0.393062	+	0.919512i	$0.628584\pi$
$332$	0	0
$333$	3.24186i	0.177653i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.34731i	0.182340i	0.995835	+	0.0911699i	$0.0290606\pi$
$337$	3.34731i	0.182340i	−0.995835	+	0.0911699i	$0.970939\pi$
$338$	0	0
$339$	0.733661	0.0398470
$340$	0	0
$341$	−1.20594	−0.0653054
$342$	0	0
$343$	51.1236i	2.76042i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.9383i	1.28507i	0.766255	+	0.642537i	$0.222118\pi$
$347$	23.9383i	1.28507i	−0.766255	+	0.642537i	$0.777882\pi$
$348$	0	0
$349$	11.9149	0.637788	0.318894	−	0.947790i	$-0.396689\pi$
$349$	11.9149	0.637788	0.318894	+	0.947790i	$0.396689\pi$
$350$	0	0
$351$	7.61676	0.406553
$352$	0	0
$353$	− 25.2447i	− 1.34364i	−0.740714	−	0.671820i	$-0.765513\pi$
$353$	− 25.2447i	− 1.34364i	0.740714	−	0.671820i	$-0.234487\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 58.3502i	− 3.08822i
$358$	0	0
$359$	−20.4671	−1.08021	−0.540105	−	0.841598i	$-0.681615\pi$
$359$	−20.4671	−1.08021	−0.540105	+	0.841598i	$0.681615\pi$
$360$	0	0
$361$	10.6332	0.559641
$362$	0	0
$363$	20.2130i	1.06090i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	36.4999i	1.90528i	0.304104	+	0.952639i	$0.401643\pi$
$367$	36.4999i	1.90528i	−0.304104	+	0.952639i	$0.598357\pi$
$368$	0	0
$369$	5.18537	0.269939
$370$	0	0
$371$	67.8214	3.52111
$372$	0	0
$373$	− 5.57978i	− 0.288910i	−0.989511	−	0.144455i	$-0.953857\pi$
$373$	− 5.57978i	− 0.288910i	0.989511	−	0.144455i	$-0.0461429\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.90679i	0.149707i
$378$	0	0
$379$	−36.6168	−1.88088	−0.940438	−	0.339964i	$-0.889585\pi$
$379$	−36.6168	−1.88088	−0.940438	+	0.339964i	$0.889585\pi$
$380$	0	0
$381$	−5.12603	−0.262614
$382$	0	0
$383$	− 28.7581i	− 1.46947i	−0.678353	−	0.734736i	$-0.737306\pi$
$383$	− 28.7581i	− 1.46947i	0.678353	−	0.734736i	$-0.262694\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3.74173i	− 0.190203i
$388$	0	0
$389$	−26.3913	−1.33809	−0.669046	−	0.743221i	$-0.733297\pi$
$389$	−26.3913	−1.33809	−0.669046	+	0.743221i	$0.733297\pi$
$390$	0	0
$391$	6.10856	0.308923
$392$	0	0
$393$	10.7141i	0.540457i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.08931i	− 0.0546710i	−0.999626	−	0.0273355i	$-0.991298\pi$
$397$	− 1.08931i	− 0.0546710i	0.999626	−	0.0273355i	$-0.00870225\pi$
$398$	0	0
$399$	51.9987	2.60319
$400$	0	0
$401$	−22.5962	−1.12840	−0.564200	−	0.825638i	$-0.690815\pi$
$401$	−22.5962	−1.12840	−0.564200	+	0.825638i	$0.690815\pi$
$402$	0	0
$403$	− 2.82270i	− 0.140609i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.24186i	− 0.160693i
$408$	0	0
$409$	16.0510	0.793671	0.396835	−	0.917890i	$-0.370108\pi$
$409$	16.0510	0.793671	0.396835	+	0.917890i	$0.370108\pi$
$410$	0	0
$411$	−11.9396	−0.588937
$412$	0	0
$413$	60.2968i	2.96701i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 40.1976i	− 1.96849i
$418$	0	0
$419$	24.4342	1.19369	0.596845	−	0.802356i	$-0.296421\pi$
$419$	24.4342	1.19369	0.596845	+	0.802356i	$0.296421\pi$
$420$	0	0
$421$	−11.3473	−0.553034	−0.276517	−	0.961009i	$-0.589180\pi$
$421$	−11.3473	−0.553034	−0.276517	+	0.961009i	$0.589180\pi$
$422$	0	0
$423$	2.00000i	0.0972433i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	68.9945i	3.33888i
$428$	0	0
$429$	2.52044	0.121688
$430$	0	0
$431$	7.83805	0.377546	0.188773	−	0.982021i	$-0.439549\pi$
$431$	7.83805	0.377546	0.188773	+	0.982021i	$0.439549\pi$
$432$	0	0
$433$	− 37.9023i	− 1.82147i	−0.412990	−	0.910735i	$-0.635516\pi$
$433$	− 37.9023i	− 1.82147i	0.412990	−	0.910735i	$-0.364484\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.44364i	0.260404i
$438$	0	0
$439$	17.7704	0.848134	0.424067	−	0.905631i	$-0.360602\pi$
$439$	17.7704	0.848134	0.424067	+	0.905631i	$0.360602\pi$
$440$	0	0
$441$	12.9477	0.616556
$442$	0	0
$443$	16.9466i	0.805158i	0.915385	+	0.402579i	$0.131886\pi$
$443$	16.9466i	0.805158i	−0.915385	+	0.402579i	$0.868114\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2.06040i	− 0.0974535i
$448$	0	0
$449$	21.0890	0.995253	0.497627	−	0.867391i	$-0.334205\pi$
$449$	21.0890	0.995253	0.497627	+	0.867391i	$0.334205\pi$
$450$	0	0
$451$	−5.18537	−0.244169
$452$	0	0
$453$	24.5040i	1.15130i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 13.1372i	− 0.614532i	−0.951624	−	0.307266i	$-0.900586\pi$
$457$	− 13.1372i	− 0.614532i	0.951624	−	0.307266i	$-0.0994142\pi$
$458$	0	0
$459$	26.6496	1.24390
$460$	0	0
$461$	−14.5439	−0.677375	−0.338687	−	0.940899i	$-0.609983\pi$
$461$	−14.5439	−0.677375	−0.338687	+	0.940899i	$0.609983\pi$
$462$	0	0
$463$	19.4283i	0.902909i	0.892294	+	0.451455i	$0.149095\pi$
$463$	19.4283i	0.902909i	−0.892294	+	0.451455i	$0.850905\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.61259i	0.167171i	0.996501	+	0.0835855i	$0.0266372\pi$
$467$	3.61259i	0.167171i	−0.996501	+	0.0835855i	$0.973363\pi$
$468$	0	0
$469$	−64.6758	−2.98645
$470$	0	0
$471$	−28.8133	−1.32765
$472$	0	0
$473$	3.74173i	0.172045i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.2499i	− 0.469312i
$478$	0	0
$479$	−5.13720	−0.234725	−0.117362	−	0.993089i	$-0.537444\pi$
$479$	−5.13720	−0.234725	−0.117362	+	0.993089i	$0.537444\pi$
$480$	0	0
$481$	7.58812	0.345988
$482$	0	0
$483$	9.55220i	0.434640i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.7693i	1.12240i	0.827679	+	0.561202i	$0.189661\pi$
$487$	24.7693i	1.12240i	−0.827679	+	0.561202i	$0.810339\pi$
$488$	0	0
$489$	13.0245	0.588987
$490$	0	0
$491$	3.54413	0.159944	0.0799721	−	0.996797i	$-0.474517\pi$
$491$	3.54413	0.159944	0.0799721	+	0.996797i	$0.474517\pi$
$492$	0	0
$493$	10.1703i	0.458047i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	47.7610i	2.14237i
$498$	0	0
$499$	−31.5358	−1.41174	−0.705868	−	0.708344i	$-0.749443\pi$
$499$	−31.5358	−1.41174	−0.705868	+	0.708344i	$0.749443\pi$
$500$	0	0
$501$	5.77454	0.257988
$502$	0	0
$503$	− 32.2346i	− 1.43727i	−0.695388	−	0.718635i	$-0.744767\pi$
$503$	− 32.2346i	− 1.43727i	0.695388	−	0.718635i	$-0.255233\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 19.2611i	− 0.855416i
$508$	0	0
$509$	−0.302263	−0.0133976	−0.00669878	−	0.999978i	$-0.502132\pi$
$509$	−0.302263	−0.0133976	−0.00669878	+	0.999978i	$0.502132\pi$
$510$	0	0
$511$	28.1208	1.24399
$512$	0	0
$513$	23.7487i	1.04853i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.00000i	− 0.0879599i
$518$	0	0
$519$	−30.6772	−1.34658
$520$	0	0
$521$	−30.3463	−1.32949	−0.664747	−	0.747069i	$-0.731461\pi$
$521$	−30.3463	−1.32949	−0.664747	+	0.747069i	$0.731461\pi$
$522$	0	0
$523$	− 42.5878i	− 1.86224i	−0.364717	−	0.931118i	$-0.618834\pi$
$523$	− 42.5878i	− 1.86224i	0.364717	−	0.931118i	$-0.381166\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9.87609i	− 0.430209i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	9.11273	0.395459
$532$	0	0
$533$	− 12.1372i	− 0.525721i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.85969i	0.252864i
$538$	0	0
$539$	−12.9477	−0.557696
$540$	0	0
$541$	38.9313	1.67379	0.836893	−	0.547367i	$-0.184370\pi$
$541$	38.9313	1.67379	0.836893	+	0.547367i	$0.184370\pi$
$542$	0	0
$543$	− 3.52772i	− 0.151389i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16.6004i	− 0.709780i	−0.934908	−	0.354890i	$-0.884518\pi$
$547$	− 16.6004i	− 0.709780i	0.934908	−	0.354890i	$-0.115482\pi$
$548$	0	0
$549$	10.4272	0.445023
$550$	0	0
$551$	−9.06324	−0.386107
$552$	0	0
$553$	51.2252i	2.17832i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 17.4506i	− 0.739408i	−0.929150	−	0.369704i	$-0.879459\pi$
$557$	− 17.4506i	− 0.739408i	0.929150	−	0.369704i	$-0.120541\pi$
$558$	0	0
$559$	−8.75814	−0.370430
$560$	0	0
$561$	8.81853	0.372319
$562$	0	0
$563$	8.25827i	0.348045i	0.984742	+	0.174022i	$0.0556765\pi$
$563$	8.25827i	0.348045i	−0.984742	+	0.174022i	$0.944324\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	52.7170i	2.21391i
$568$	0	0
$569$	41.4178	1.73633	0.868163	−	0.496279i	$-0.165301\pi$
$569$	41.4178	1.73633	0.868163	+	0.496279i	$0.165301\pi$
$570$	0	0
$571$	7.19059	0.300917	0.150458	−	0.988616i	$-0.451925\pi$
$571$	7.19059	0.300917	0.150458	+	0.988616i	$0.451925\pi$
$572$	0	0
$573$	− 37.9588i	− 1.58575i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 39.3103i	− 1.63651i	−0.574855	−	0.818255i	$-0.694942\pi$
$577$	− 39.3103i	− 1.63651i	0.574855	−	0.818255i	$-0.305058\pi$
$578$	0	0
$579$	−0.648517	−0.0269515
$580$	0	0
$581$	3.14554	0.130499
$582$	0	0
$583$	10.2499i	0.424509i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.7899i	− 1.18829i	−0.804359	−	0.594143i	$-0.797491\pi$
$587$	− 28.7899i	− 1.18829i	0.804359	−	0.594143i	$-0.202509\pi$
$588$	0	0
$589$	8.80107	0.362642
$590$	0	0
$591$	39.7376	1.63458
$592$	0	0
$593$	− 40.8873i	− 1.67904i	−0.543330	−	0.839519i	$-0.682837\pi$
$593$	− 40.8873i	− 1.67904i	0.543330	−	0.839519i	$-0.317163\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	32.3707i	1.32485i
$598$	0	0
$599$	−19.1801	−0.783679	−0.391840	−	0.920034i	$-0.628161\pi$
$599$	−19.1801	−0.783679	−0.391840	+	0.920034i	$0.628161\pi$
$600$	0	0
$601$	3.36683	0.137336	0.0686679	−	0.997640i	$-0.478125\pi$
$601$	3.36683	0.137336	0.0686679	+	0.997640i	$0.478125\pi$
$602$	0	0
$603$	9.77454i	0.398050i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 13.1924i	− 0.535462i	−0.963494	−	0.267731i	$-0.913726\pi$
$607$	− 13.1924i	− 0.535462i	0.963494	−	0.267731i	$-0.0862738\pi$
$608$	0	0
$609$	−15.9037	−0.644450
$610$	0	0
$611$	4.68133	0.189386
$612$	0	0
$613$	32.6207i	1.31754i	0.752346	+	0.658768i	$0.228922\pi$
$613$	32.6207i	1.31754i	−0.752346	+	0.658768i	$0.771078\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 41.7212i	− 1.67963i	−0.542872	−	0.839815i	$-0.682663\pi$
$617$	− 41.7212i	− 1.67963i	0.542872	−	0.839815i	$-0.317337\pi$
$618$	0	0
$619$	0.339245	0.0136354	0.00681771	−	0.999977i	$-0.497830\pi$
$619$	0.339245	0.0136354	0.00681771	+	0.999977i	$0.497830\pi$
$620$	0	0
$621$	−4.36266	−0.175068
$622$	0	0
$623$	13.4506i	0.538889i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.85863i	0.313843i
$628$	0	0
$629$	26.5494	1.05859
$630$	0	0
$631$	3.71725	0.147982	0.0739908	−	0.997259i	$-0.476426\pi$
$631$	3.71725	0.147982	0.0739908	+	0.997259i	$0.476426\pi$
$632$	0	0
$633$	25.9260i	1.03047i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 30.3062i	− 1.20077i
$638$	0	0
$639$	7.21818	0.285547
$640$	0	0
$641$	−22.0328	−0.870244	−0.435122	−	0.900372i	$-0.643295\pi$
$641$	−22.0328	−0.870244	−0.435122	+	0.900372i	$0.643295\pi$
$642$	0	0
$643$	− 24.6842i	− 0.973449i	−0.873556	−	0.486724i	$-0.838192\pi$
$643$	− 24.6842i	− 0.973449i	0.873556	−	0.486724i	$-0.161808\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 32.1731i	− 1.26486i	−0.774619	−	0.632428i	$-0.782058\pi$
$647$	− 32.1731i	− 1.26486i	0.774619	−	0.632428i	$-0.217942\pi$
$648$	0	0
$649$	−9.11273	−0.357706
$650$	0	0
$651$	15.4436	0.605284
$652$	0	0
$653$	− 26.9466i	− 1.05450i	−0.849709	−	0.527251i	$-0.823223\pi$
$653$	− 26.9466i	− 1.05450i	0.849709	−	0.527251i	$-0.176777\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.24993i	− 0.165806i
$658$	0	0
$659$	19.7417	0.769029	0.384514	−	0.923119i	$-0.374369\pi$
$659$	19.7417	0.769029	0.384514	+	0.923119i	$0.374369\pi$
$660$	0	0
$661$	−1.28692	−0.0500552	−0.0250276	−	0.999687i	$-0.507967\pi$
$661$	−1.28692	−0.0500552	−0.0250276	+	0.999687i	$0.507967\pi$
$662$	0	0
$663$	20.6412i	0.801639i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.66492i	− 0.0644661i
$668$	0	0
$669$	2.46705	0.0953817
$670$	0	0
$671$	−10.4272	−0.402539
$672$	0	0
$673$	12.4970i	0.481725i	0.970559	+	0.240862i	$0.0774303\pi$
$673$	12.4970i	0.481725i	−0.970559	+	0.240862i	$0.922570\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.69251i	0.103482i	0.998661	+	0.0517408i	$0.0164770\pi$
$677$	2.69251i	0.103482i	−0.998661	+	0.0517408i	$0.983523\pi$
$678$	0	0
$679$	−35.1648	−1.34950
$680$	0	0
$681$	−6.92842	−0.265498
$682$	0	0
$683$	− 2.72115i	− 0.104122i	−0.998644	−	0.0520610i	$-0.983421\pi$
$683$	− 2.72115i	− 0.104122i	0.998644	−	0.0520610i	$-0.0165790\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	20.1515i	0.768827i
$688$	0	0
$689$	−23.9917	−0.914010
$690$	0	0
$691$	38.3051	1.45719	0.728597	−	0.684942i	$-0.240173\pi$
$691$	38.3051	1.45719	0.728597	+	0.684942i	$0.240173\pi$
$692$	0	0
$693$	2.74590i	0.104308i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 42.4657i	− 1.60850i
$698$	0	0
$699$	5.92319	0.224036
$700$	0	0
$701$	39.8472	1.50501	0.752504	−	0.658588i	$-0.228846\pi$
$701$	39.8472	1.50501	0.752504	+	0.658588i	$0.228846\pi$
$702$	0	0
$703$	23.6594i	0.892332i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 73.9094i	− 2.77965i
$708$	0	0
$709$	44.8995	1.68624	0.843118	−	0.537728i	$-0.180717\pi$
$709$	44.8995	1.68624	0.843118	+	0.537728i	$0.180717\pi$
$710$	0	0
$711$	7.74173	0.290338
$712$	0	0
$713$	1.61676i	0.0605482i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 37.1400i	− 1.38702i
$718$	0	0
$719$	31.5400	1.17624	0.588121	−	0.808773i	$-0.299868\pi$
$719$	31.5400	1.17624	0.588121	+	0.808773i	$0.299868\pi$
$720$	0	0
$721$	−53.6318	−1.99735
$722$	0	0
$723$	− 0.670152i	− 0.0249232i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	48.9219i	1.81441i	0.420687	+	0.907206i	$0.361789\pi$
$727$	48.9219i	1.81441i	−0.420687	+	0.907206i	$0.638211\pi$
$728$	0	0
$729$	−17.3637	−0.643101
$730$	0	0
$731$	−30.6430	−1.13337
$732$	0	0
$733$	− 14.1536i	− 0.522776i	−0.965234	−	0.261388i	$-0.915820\pi$
$733$	− 14.1536i	− 0.522776i	0.965234	−	0.261388i	$-0.0841801\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.77454i	− 0.360050i
$738$	0	0
$739$	−4.46421	−0.164219	−0.0821093	−	0.996623i	$-0.526166\pi$
$739$	−4.46421	−0.164219	−0.0821093	+	0.996623i	$0.526166\pi$
$740$	0	0
$741$	−18.3944	−0.675736
$742$	0	0
$743$	− 34.2898i	− 1.25797i	−0.777418	−	0.628985i	$-0.783471\pi$
$743$	− 34.2898i	− 1.25797i	0.777418	−	0.628985i	$-0.216529\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 0.475390i	− 0.0173936i
$748$	0	0
$749$	36.9753	1.35105
$750$	0	0
$751$	−8.28275	−0.302242	−0.151121	−	0.988515i	$-0.548288\pi$
$751$	−8.28275	−0.302242	−0.151121	+	0.988515i	$0.548288\pi$
$752$	0	0
$753$	− 27.8433i	− 1.01467i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.32985i	− 0.121025i	−0.998167	−	0.0605127i	$-0.980726\pi$
$757$	− 3.32985i	− 0.121025i	0.998167	−	0.0605127i	$-0.0192736\pi$
$758$	0	0
$759$	−1.44364	−0.0524007
$760$	0	0
$761$	−18.4077	−0.667279	−0.333640	−	0.942701i	$-0.608277\pi$
$761$	−18.4077	−0.667279	−0.333640	+	0.942701i	$0.608277\pi$
$762$	0	0
$763$	− 90.4710i	− 3.27527i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 21.3298i	− 0.770176i
$768$	0	0
$769$	−51.4506	−1.85536	−0.927679	−	0.373379i	$-0.878199\pi$
$769$	−51.4506	−1.85536	−0.927679	+	0.373379i	$0.878199\pi$
$770$	0	0
$771$	−2.48869	−0.0896279
$772$	0	0
$773$	17.4506i	0.627656i	0.949480	+	0.313828i	$0.101612\pi$
$773$	17.4506i	0.627656i	−0.949480	+	0.313828i	$0.898388\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	41.5163i	1.48939i
$778$	0	0
$779$	37.8433	1.35588
$780$	0	0
$781$	−7.21818	−0.258287
$782$	0	0
$783$	− 7.26350i	− 0.259576i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	21.2335i	0.756893i	0.925623	+	0.378447i	$0.123542\pi$
$787$	21.2335i	0.756893i	−0.925623	+	0.378447i	$0.876458\pi$
$788$	0	0
$789$	54.6360	1.94509
$790$	0	0
$791$	1.87086	0.0665203
$792$	0	0
$793$	− 24.4067i	− 0.866706i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.3215i	0.896934i	0.893799	+	0.448467i	$0.148030\pi$
$797$	25.3215i	0.896934i	−0.893799	+	0.448467i	$0.851970\pi$
$798$	0	0
$799$	16.3791	0.579450
$800$	0	0
$801$	2.03281	0.0718259
$802$	0	0
$803$	4.24993i	0.149977i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 37.6238i	− 1.32442i
$808$	0	0
$809$	−6.14137	−0.215919	−0.107960	−	0.994155i	$-0.534432\pi$
$809$	−6.14137	−0.215919	−0.107960	+	0.994155i	$0.534432\pi$
$810$	0	0
$811$	15.5051	0.544457	0.272229	−	0.962233i	$-0.412239\pi$
$811$	15.5051	0.544457	0.272229	+	0.962233i	$0.412239\pi$
$812$	0	0
$813$	12.4395i	0.436271i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 27.3075i	− 0.955368i
$818$	0	0
$819$	−6.42723	−0.224586
$820$	0	0
$821$	−21.4835	−0.749778	−0.374889	−	0.927070i	$-0.622319\pi$
$821$	−21.4835	−0.749778	−0.374889	+	0.927070i	$0.622319\pi$
$822$	0	0
$823$	40.1565i	1.39977i	0.714258	+	0.699883i	$0.246765\pi$
$823$	40.1565i	1.39977i	−0.714258	+	0.699883i	$0.753235\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 38.0573i	− 1.32338i	−0.749777	−	0.661691i	$-0.769839\pi$
$827$	− 38.0573i	− 1.32338i	0.749777	−	0.661691i	$-0.230161\pi$
$828$	0	0
$829$	51.2580	1.78026	0.890132	−	0.455703i	$-0.150612\pi$
$829$	51.2580	1.78026	0.890132	+	0.455703i	$0.150612\pi$
$830$	0	0
$831$	−7.90652	−0.274274
$832$	0	0
$833$	− 106.035i	− 3.67391i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.05339i	0.243801i
$838$	0	0
$839$	−45.4200	−1.56807	−0.784035	−	0.620716i	$-0.786842\pi$
$839$	−45.4200	−1.56807	−0.784035	+	0.620716i	$0.786842\pi$
$840$	0	0
$841$	−26.2280	−0.904415
$842$	0	0
$843$	12.3624i	0.425783i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	51.5439i	1.77107i
$848$	0	0
$849$	−41.5798	−1.42701
$850$	0	0
$851$	−4.34625	−0.148988
$852$	0	0
$853$	40.0867i	1.37254i	0.727346	+	0.686270i	$0.240753\pi$
$853$	40.0867i	1.37254i	−0.727346	+	0.686270i	$0.759247\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.63423i	0.124143i	0.998072	+	0.0620715i	$0.0197707\pi$
$857$	3.63423i	0.124143i	−0.998072	+	0.0620715i	$0.980229\pi$
$858$	0	0
$859$	37.7529	1.28811	0.644056	−	0.764978i	$-0.277250\pi$
$859$	37.7529	1.28811	0.644056	+	0.764978i	$0.277250\pi$
$860$	0	0
$861$	66.4053	2.26309
$862$	0	0
$863$	− 3.94555i	− 0.134308i	−0.997743	−	0.0671541i	$-0.978608\pi$
$863$	− 3.94555i	− 0.134308i	0.997743	−	0.0671541i	$-0.0213919\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	39.3173i	1.33529i
$868$	0	0
$869$	−7.74173	−0.262620
$870$	0	0
$871$	22.8789	0.775223
$872$	0	0
$873$	5.31450i	0.179869i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.64958i	0.292075i	0.989279	+	0.146038i	$0.0466521\pi$
$877$	8.64958i	0.292075i	−0.989279	+	0.146038i	$0.953348\pi$
$878$	0	0
$879$	−31.2007	−1.05237
$880$	0	0
$881$	−56.1760	−1.89262	−0.946308	−	0.323266i	$-0.895219\pi$
$881$	−56.1760	−1.89262	−0.946308	+	0.323266i	$0.895219\pi$
$882$	0	0
$883$	− 12.1054i	− 0.407381i	−0.979035	−	0.203690i	$-0.934706\pi$
$883$	− 12.1054i	− 0.407381i	0.979035	−	0.203690i	$-0.0652936\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.7058i	0.829540i	0.909926	+	0.414770i	$0.136138\pi$
$887$	24.7058i	0.829540i	−0.909926	+	0.414770i	$0.863862\pi$
$888$	0	0
$889$	−13.0716	−0.438407
$890$	0	0
$891$	−7.96719	−0.266911
$892$	0	0
$893$	14.5962i	0.488443i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.37907i	− 0.112824i
$898$	0	0
$899$	−2.69179	−0.0897761
$900$	0	0
$901$	−83.9422	−2.79652
$902$	0	0
$903$	− 47.9177i	− 1.59460i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 22.7170i	− 0.754305i	−0.926151	−	0.377153i	$-0.876903\pi$
$907$	− 22.7170i	− 0.754305i	0.926151	−	0.377153i	$-0.123097\pi$
$908$	0	0
$909$	−11.1700	−0.370486
$910$	0	0
$911$	−12.3156	−0.408033	−0.204016	−	0.978967i	$-0.565400\pi$
$911$	−12.3156	−0.408033	−0.204016	+	0.978967i	$0.565400\pi$
$912$	0	0
$913$	0.475390i	0.0157331i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.3215i	0.902236i
$918$	0	0
$919$	−46.2088	−1.52429	−0.762144	−	0.647408i	$-0.775853\pi$
$919$	−46.2088	−1.52429	−0.762144	+	0.647408i	$0.775853\pi$
$920$	0	0
$921$	13.4764	0.444064
$922$	0	0
$923$	− 16.8953i	− 0.556117i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.10545i	0.266218i
$928$	0	0
$929$	−9.12391	−0.299346	−0.149673	−	0.988736i	$-0.547822\pi$
$929$	−9.12391	−0.299346	−0.149673	+	0.988736i	$0.547822\pi$
$930$	0	0
$931$	94.4933	3.09689
$932$	0	0
$933$	8.01330i	0.262344i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33.9190i	1.10809i	0.832488	+	0.554043i	$0.186916\pi$
$937$	33.9190i	1.10809i	−0.832488	+	0.554043i	$0.813084\pi$
$938$	0	0
$939$	12.1260	0.395718
$940$	0	0
$941$	−37.2926	−1.21570	−0.607852	−	0.794050i	$-0.707969\pi$
$941$	−37.2926	−1.21570	−0.607852	+	0.794050i	$0.707969\pi$
$942$	0	0
$943$	6.95184i	0.226383i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 33.7047i	− 1.09526i	−0.836722	−	0.547629i	$-0.815531\pi$
$947$	− 33.7047i	− 1.09526i	0.836722	−	0.547629i	$-0.184469\pi$
$948$	0	0
$949$	−9.94767	−0.322915
$950$	0	0
$951$	−18.4639	−0.598734
$952$	0	0
$953$	− 48.4465i	− 1.56934i	−0.619917	−	0.784668i	$-0.712834\pi$
$953$	− 48.4465i	− 1.56934i	0.619917	−	0.784668i	$-0.287166\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.40354i	− 0.0776955i
$958$	0	0
$959$	−30.4465	−0.983168
$960$	0	0
$961$	−28.3861	−0.915680
$962$	0	0
$963$	− 5.58812i	− 0.180075i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.2528i	0.908548i	0.890862	+	0.454274i	$0.150101\pi$
$967$	28.2528i	0.908548i	−0.890862	+	0.454274i	$0.849899\pi$
$968$	0	0
$969$	−64.3585	−2.06749
$970$	0	0
$971$	1.65553	0.0531284	0.0265642	−	0.999647i	$-0.491543\pi$
$971$	1.65553	0.0531284	0.0265642	+	0.999647i	$0.491543\pi$
$972$	0	0
$973$	− 102.506i	− 3.28618i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.1630i	0.357136i	0.983928	+	0.178568i	$0.0571465\pi$
$977$	11.1630i	0.357136i	−0.983928	+	0.178568i	$0.942854\pi$
$978$	0	0
$979$	−2.03281	−0.0649690
$980$	0	0
$981$	−13.6730	−0.436545
$982$	0	0
$983$	15.0615i	0.480386i	0.970725	+	0.240193i	$0.0772107\pi$
$983$	15.0615i	0.480386i	−0.970725	+	0.240193i	$0.922789\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	25.6126i	0.815258i
$988$	0	0
$989$	5.01641	0.159512
$990$	0	0
$991$	−43.8971	−1.39444	−0.697219	−	0.716858i	$-0.745579\pi$
$991$	−43.8971	−1.39444	−0.697219	+	0.716858i	$0.745579\pi$
$992$	0	0
$993$	27.6811i	0.878432i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.5655i	− 1.22138i	−0.791869	−	0.610691i	$-0.790892\pi$
$997$	− 38.5655i	− 1.22138i	0.791869	−	0.610691i	$-0.209108\pi$
$998$	0	0
$999$	−18.9612	−0.599907

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.q.4049.2		6
5.2	odd	4		920.2.a.i.1.1	✓	3
5.3	odd	4		4600.2.a.v.1.3		3
5.4	even	2	inner	4600.2.e.q.4049.5		6
15.2	even	4		8280.2.a.bl.1.3		3
20.3	even	4		9200.2.a.ci.1.1		3
20.7	even	4		1840.2.a.q.1.3		3
40.27	even	4		7360.2.a.cf.1.1		3
40.37	odd	4		7360.2.a.bw.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.i.1.1	✓	3	5.2	odd	4
1840.2.a.q.1.3		3	20.7	even	4
4600.2.a.v.1.3		3	5.3	odd	4
4600.2.e.q.4049.2		6	1.1	even	1	trivial
4600.2.e.q.4049.5		6	5.4	even	2	inner
7360.2.a.bw.1.3		3	40.37	odd	4
7360.2.a.cf.1.1		3	40.27	even	4
8280.2.a.bl.1.3		3	15.2	even	4
9200.2.a.ci.1.1		3	20.3	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.93543i q^{3} -4.93543i q^{7} -0.745898 q^{9} +0.745898 q^{11} +1.74590i q^{13} +6.10856i q^{17} -5.44364 q^{19} -9.55220 q^{21} -1.00000i q^{23} -4.36266i q^{27} +1.66492 q^{29} -1.61676 q^{31} -1.44364i q^{33} -4.34625i q^{37} +3.37907 q^{39} -6.95184 q^{41} +5.01641i q^{43} -2.68133i q^{47} -17.3585 q^{49} +11.8227 q^{51} +13.7417i q^{53} +10.5358i q^{57} -12.2171 q^{59} -13.9794 q^{61} +3.68133i q^{63} -13.1044i q^{67} -1.93543 q^{69} -9.67716 q^{71} +5.69774i q^{73} -3.68133i q^{77} -10.3791 q^{79} -10.6813 q^{81} +0.637339i q^{83} -3.22235i q^{87} -2.72532 q^{89} +8.61676 q^{91} +3.12914i q^{93} -7.12497i q^{97} -0.556364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9} + 6 q^{11} - 14 q^{19} - 12 q^{21} + 2 q^{29} + 20 q^{31} - 14 q^{39} - 20 q^{41} - 12 q^{49} + 18 q^{51} - 20 q^{59} - 26 q^{61} + 4 q^{69} + 20 q^{71} - 28 q^{79} - 50 q^{81} + 40 q^{89}+ \cdots - 22 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 6 * q^11 - 14 * q^19 - 12 * q^21 + 2 * q^29 + 20 * q^31 - 14 * q^39 - 20 * q^41 - 12 * q^49 + 18 * q^51 - 20 * q^59 - 26 * q^61 + 4 * q^69 + 20 * q^71 - 28 * q^79 - 50 * q^81 + 40 * q^89 + 22 * q^91 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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