LMFDB - Embedded newform 4600.2.e.u.4049.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256 $ x^10 + 28x^8 + 260x^6 + 897x^4 + 1056x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.8
Root			$1.93283i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.u.4049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93283i q^{3} -2.38236i q^{7} -0.735829 q^{9} +O(q^{10})$	$q+1.93283i q^{3} -2.38236i q^{7} -0.735829 q^{9} +5.33368 q^{11} -4.53752i q^{13} -1.81464i q^{17} -7.00233 q^{19} +4.60469 q^{21} -1.00000i q^{23} +4.37626i q^{27} +0.118188 q^{29} -0.884147 q^{31} +10.3091i q^{33} -7.51903i q^{37} +8.77026 q^{39} -1.45186 q^{41} -10.4389i q^{47} +1.32437 q^{49} +3.50739 q^{51} -9.42167i q^{53} -13.5343i q^{57} -7.79239 q^{59} -2.80533 q^{61} +1.75301i q^{63} +3.11134i q^{67} +1.93283 q^{69} -13.5909 q^{71} +12.4389i q^{73} -12.7067i q^{77} -6.80169 q^{79} -10.6660 q^{81} -13.5190i q^{83} +0.228437i q^{87} -2.89906 q^{89} -10.8100 q^{91} -1.70890i q^{93} +1.97774i q^{97} -3.92468 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 26 q^{9}+O(q^{10}) $ 10 * q - 26 * q^9	$ 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) $ 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.93283i	1.11592i	0.829868	+	0.557960i	$0.188416\pi$
$3$	1.93283i	1.11592i	−0.829868	+	0.557960i	$0.811584\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.38236i	− 0.900447i	−0.892916	−	0.450223i	$-0.851344\pi$
$7$	− 2.38236i	− 0.900447i	0.892916	−	0.450223i	$-0.148656\pi$
$8$	0	0
$9$	−0.735829	−0.245276
$10$	0	0
$11$	5.33368	1.60816	0.804082	−	0.594519i	$-0.202657\pi$
$11$	5.33368	1.60816	0.804082	+	0.594519i	$0.202657\pi$
$12$	0	0
$13$	− 4.53752i	− 1.25848i	−0.777210	−	0.629241i	$-0.783366\pi$
$13$	− 4.53752i	− 1.25848i	0.777210	−	0.629241i	$-0.216634\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.81464i	− 0.440115i	−0.975487	−	0.220058i	$-0.929375\pi$
$17$	− 1.81464i	− 0.440115i	0.975487	−	0.220058i	$-0.0706245\pi$
$18$	0	0
$19$	−7.00233	−1.60645	−0.803223	−	0.595679i	$-0.796883\pi$
$19$	−7.00233	−1.60645	−0.803223	+	0.595679i	$0.796883\pi$
$20$	0	0
$21$	4.60469	1.00483
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.37626i	0.842211i
$28$	0	0
$29$	0.118188	0.0219469	0.0109735	−	0.999940i	$-0.496507\pi$
$29$	0.118188	0.0219469	0.0109735	+	0.999940i	$0.496507\pi$
$30$	0	0
$31$	−0.884147	−0.158797	−0.0793987	−	0.996843i	$-0.525300\pi$
$31$	−0.884147	−0.158797	−0.0793987	+	0.996843i	$0.525300\pi$
$32$	0	0
$33$	10.3091i	1.79458i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.51903i	− 1.23612i	−0.786130	−	0.618061i	$-0.787919\pi$
$37$	− 7.51903i	− 1.23612i	0.786130	−	0.618061i	$-0.212081\pi$
$38$	0	0
$39$	8.77026	1.40436
$40$	0	0
$41$	−1.45186	−0.226743	−0.113371	−	0.993553i	$-0.536165\pi$
$41$	−1.45186	−0.226743	−0.113371	+	0.993553i	$0.536165\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.4389i	− 1.52267i	−0.648357	−	0.761336i	$-0.724544\pi$
$47$	− 10.4389i	− 1.52267i	0.648357	−	0.761336i	$-0.275456\pi$
$48$	0	0
$49$	1.32437	0.189195
$50$	0	0
$51$	3.50739	0.491133
$52$	0	0
$53$	− 9.42167i	− 1.29417i	−0.762420	−	0.647083i	$-0.775989\pi$
$53$	− 9.42167i	− 1.29417i	0.762420	−	0.647083i	$-0.224011\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 13.5343i	− 1.79266i
$58$	0	0
$59$	−7.79239	−1.01448	−0.507241	−	0.861804i	$-0.669335\pi$
$59$	−7.79239	−1.01448	−0.507241	+	0.861804i	$0.669335\pi$
$60$	0	0
$61$	−2.80533	−0.359186	−0.179593	−	0.983741i	$-0.557478\pi$
$61$	−2.80533	−0.359186	−0.179593	+	0.983741i	$0.557478\pi$
$62$	0	0
$63$	1.75301i	0.220858i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.11134i	0.380111i	0.981773	+	0.190055i	$0.0608668\pi$
$67$	3.11134i	0.380111i	−0.981773	+	0.190055i	$0.939133\pi$
$68$	0	0
$69$	1.93283	0.232685
$70$	0	0
$71$	−13.5909	−1.61294	−0.806470	−	0.591275i	$-0.798625\pi$
$71$	−13.5909	−1.61294	−0.806470	+	0.591275i	$0.798625\pi$
$72$	0	0
$73$	12.4389i	1.45586i	0.685649	+	0.727932i	$0.259519\pi$
$73$	12.4389i	1.45586i	−0.685649	+	0.727932i	$0.740481\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 12.7067i	− 1.44807i
$78$	0	0
$79$	−6.80169	−0.765250	−0.382625	−	0.923904i	$-0.624980\pi$
$79$	−6.80169	−0.765250	−0.382625	+	0.923904i	$0.624980\pi$
$80$	0	0
$81$	−10.6660	−1.18512
$82$	0	0
$83$	− 13.5190i	− 1.48391i	−0.670451	−	0.741953i	$-0.733900\pi$
$83$	− 13.5190i	− 1.48391i	0.670451	−	0.741953i	$-0.266100\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.228437i	0.0244910i
$88$	0	0
$89$	−2.89906	−0.307300	−0.153650	−	0.988125i	$-0.549103\pi$
$89$	−2.89906	−0.307300	−0.153650	+	0.988125i	$0.549103\pi$
$90$	0	0
$91$	−10.8100	−1.13320
$92$	0	0
$93$	− 1.70890i	− 0.177205i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.97774i	0.200809i	0.994947	+	0.100405i	$0.0320138\pi$
$97$	1.97774i	0.200809i	−0.994947	+	0.100405i	$0.967986\pi$
$98$	0	0
$99$	−3.92468	−0.394445
$100$	0	0
$101$	−12.2838	−1.22228	−0.611139	−	0.791523i	$-0.709289\pi$
$101$	−12.2838	−1.22228	−0.611139	+	0.791523i	$0.709289\pi$
$102$	0	0
$103$	− 12.5061i	− 1.23226i	−0.787644	−	0.616131i	$-0.788699\pi$
$103$	− 12.5061i	− 1.23226i	0.787644	−	0.616131i	$-0.211301\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.3847i	1.68064i	0.542089	+	0.840321i	$0.317633\pi$
$107$	17.3847i	1.68064i	−0.542089	+	0.840321i	$0.682367\pi$
$108$	0	0
$109$	−4.30908	−0.412735	−0.206368	−	0.978475i	$-0.566164\pi$
$109$	−4.30908	−0.412735	−0.206368	+	0.978475i	$0.566164\pi$
$110$	0	0
$111$	14.5330	1.37941
$112$	0	0
$113$	− 12.1494i	− 1.14292i	−0.820630	−	0.571460i	$-0.806377\pi$
$113$	− 12.1494i	− 1.14292i	0.820630	−	0.571460i	$-0.193623\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.33884i	0.308676i
$118$	0	0
$119$	−4.32313	−0.396300
$120$	0	0
$121$	17.4481	1.58619
$122$	0	0
$123$	− 2.80620i	− 0.253027i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.22953i	0.109103i	0.998511	+	0.0545515i	$0.0173729\pi$
$127$	1.22953i	0.109103i	−0.998511	+	0.0545515i	$0.982627\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.97345	0.172421	0.0862104	−	0.996277i	$-0.472524\pi$
$131$	1.97345	0.172421	0.0862104	+	0.996277i	$0.472524\pi$
$132$	0	0
$133$	16.6821i	1.44652i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13.7671i	− 1.17620i	−0.808789	−	0.588099i	$-0.799876\pi$
$137$	− 13.7671i	− 1.17620i	0.808789	−	0.588099i	$-0.200124\pi$
$138$	0	0
$139$	−5.30280	−0.449778	−0.224889	−	0.974384i	$-0.572202\pi$
$139$	−5.30280	−0.449778	−0.224889	+	0.974384i	$0.572202\pi$
$140$	0	0
$141$	20.1766	1.69918
$142$	0	0
$143$	− 24.2017i	− 2.02385i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.55978i	0.211127i
$148$	0	0
$149$	−7.73256	−0.633476	−0.316738	−	0.948513i	$-0.602588\pi$
$149$	−7.73256	−0.633476	−0.316738	+	0.948513i	$0.602588\pi$
$150$	0	0
$151$	21.4156	1.74277	0.871387	−	0.490596i	$-0.163221\pi$
$151$	21.4156	1.74277	0.871387	+	0.490596i	$0.163221\pi$
$152$	0	0
$153$	1.33527i	0.107950i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.04738i	0.323016i	0.986871	+	0.161508i	$0.0516357\pi$
$157$	4.04738i	0.323016i	−0.986871	+	0.161508i	$0.948364\pi$
$158$	0	0
$159$	18.2105	1.44418
$160$	0	0
$161$	−2.38236	−0.187756
$162$	0	0
$163$	9.04725i	0.708635i	0.935125	+	0.354318i	$0.115287\pi$
$163$	9.04725i	0.708635i	−0.935125	+	0.354318i	$0.884713\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.5467i	0.970893i	0.874266	+	0.485446i	$0.161343\pi$
$167$	12.5467i	0.970893i	−0.874266	+	0.485446i	$0.838657\pi$
$168$	0	0
$169$	−7.58910	−0.583777
$170$	0	0
$171$	5.15252	0.394023
$172$	0	0
$173$	− 14.1354i	− 1.07469i	−0.843362	−	0.537346i	$-0.819427\pi$
$173$	− 14.1354i	− 1.07469i	0.843362	−	0.537346i	$-0.180573\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 15.0614i	− 1.13208i
$178$	0	0
$179$	26.2442	1.96158	0.980791	−	0.195061i	$-0.0624904\pi$
$179$	26.2442	1.96158	0.980791	+	0.195061i	$0.0624904\pi$
$180$	0	0
$181$	−18.3461	−1.36365	−0.681826	−	0.731514i	$-0.738814\pi$
$181$	−18.3461	−1.36365	−0.681826	+	0.731514i	$0.738814\pi$
$182$	0	0
$183$	− 5.42223i	− 0.400823i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 9.67871i	− 0.707777i
$188$	0	0
$189$	10.4258	0.758366
$190$	0	0
$191$	−3.86566	−0.279709	−0.139855	−	0.990172i	$-0.544664\pi$
$191$	−3.86566	−0.279709	−0.139855	+	0.990172i	$0.544664\pi$
$192$	0	0
$193$	− 23.8599i	− 1.71747i	−0.512417	−	0.858737i	$-0.671250\pi$
$193$	− 23.8599i	− 1.71747i	0.512417	−	0.858737i	$-0.328750\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.7356i	− 0.836128i	−0.908418	−	0.418064i	$-0.862709\pi$
$197$	− 11.7356i	− 0.836128i	0.908418	−	0.418064i	$-0.137291\pi$
$198$	0	0
$199$	8.89906	0.630838	0.315419	−	0.948953i	$-0.397855\pi$
$199$	8.89906	0.630838	0.315419	+	0.948953i	$0.397855\pi$
$200$	0	0
$201$	−6.01369	−0.424173
$202$	0	0
$203$	− 0.281566i	− 0.0197620i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.735829i	0.0511437i
$208$	0	0
$209$	−37.3482	−2.58343
$210$	0	0
$211$	23.1224	1.59181	0.795906	−	0.605420i	$-0.206995\pi$
$211$	23.1224	1.59181	0.795906	+	0.605420i	$0.206995\pi$
$212$	0	0
$213$	− 26.2688i	− 1.79991i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.10635i	0.142989i
$218$	0	0
$219$	−24.0423	−1.62463
$220$	0	0
$221$	−8.23398	−0.553877
$222$	0	0
$223$	1.50863i	0.101026i	0.998723	+	0.0505128i	$0.0160856\pi$
$223$	1.50863i	0.101026i	−0.998723	+	0.0505128i	$0.983914\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.0047i	1.46050i	0.683179	+	0.730251i	$0.260597\pi$
$227$	22.0047i	1.46050i	−0.683179	+	0.730251i	$0.739403\pi$
$228$	0	0
$229$	11.2931	0.746266	0.373133	−	0.927778i	$-0.378283\pi$
$229$	11.2931	0.746266	0.373133	+	0.927778i	$0.378283\pi$
$230$	0	0
$231$	24.5599	1.61593
$232$	0	0
$233$	− 17.6789i	− 1.15818i	−0.815263	−	0.579091i	$-0.803408\pi$
$233$	− 17.6789i	− 1.15818i	0.815263	−	0.579091i	$-0.196592\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 13.1465i	− 0.853958i
$238$	0	0
$239$	25.2583	1.63382	0.816911	−	0.576763i	$-0.195684\pi$
$239$	25.2583	1.63382	0.816911	+	0.576763i	$0.195684\pi$
$240$	0	0
$241$	0.790615	0.0509280	0.0254640	−	0.999676i	$-0.491894\pi$
$241$	0.790615	0.0509280	0.0254640	+	0.999676i	$0.491894\pi$
$242$	0	0
$243$	− 7.48688i	− 0.480283i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	31.7732i	2.02168i
$248$	0	0
$249$	26.1300	1.65592
$250$	0	0
$251$	12.3461	0.779276	0.389638	−	0.920968i	$-0.372600\pi$
$251$	12.3461	0.779276	0.389638	+	0.920968i	$0.372600\pi$
$252$	0	0
$253$	− 5.33368i	− 0.335325i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.40442i	− 0.399497i	−0.979847	−	0.199748i	$-0.935988\pi$
$257$	− 6.40442i	− 0.399497i	0.979847	−	0.199748i	$-0.0640125\pi$
$258$	0	0
$259$	−17.9130	−1.11306
$260$	0	0
$261$	−0.0869661	−0.00538307
$262$	0	0
$263$	19.6914i	1.21422i	0.794617	+	0.607111i	$0.207672\pi$
$263$	19.6914i	1.21422i	−0.794617	+	0.607111i	$0.792328\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5.60338i	− 0.342922i
$268$	0	0
$269$	3.42494	0.208822	0.104411	−	0.994534i	$-0.466704\pi$
$269$	3.42494	0.208822	0.104411	+	0.994534i	$0.466704\pi$
$270$	0	0
$271$	13.7197	0.833411	0.416705	−	0.909042i	$-0.363185\pi$
$271$	13.7197	0.833411	0.416705	+	0.909042i	$0.363185\pi$
$272$	0	0
$273$	− 20.8939i	− 1.26456i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 0.636129i	− 0.0382213i	−0.999817	−	0.0191107i	$-0.993917\pi$
$277$	− 0.636129i	− 0.0382213i	0.999817	−	0.0191107i	$-0.00608348\pi$
$278$	0	0
$279$	0.650581	0.0389493
$280$	0	0
$281$	18.4124	1.09839	0.549195	−	0.835694i	$-0.314935\pi$
$281$	18.4124	1.09839	0.549195	+	0.835694i	$0.314935\pi$
$282$	0	0
$283$	− 7.95729i	− 0.473012i	−0.971630	−	0.236506i	$-0.923998\pi$
$283$	− 7.95729i	− 0.473012i	0.971630	−	0.236506i	$-0.0760023\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.45886i	0.204170i
$288$	0	0
$289$	13.7071	0.806299
$290$	0	0
$291$	−3.82264	−0.224087
$292$	0	0
$293$	− 20.9166i	− 1.22196i	−0.791645	−	0.610981i	$-0.790775\pi$
$293$	− 20.9166i	− 1.22196i	0.791645	−	0.610981i	$-0.209225\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	23.3415i	1.35441i
$298$	0	0
$299$	−4.53752	−0.262412
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 23.7424i	− 1.36396i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.78807i	0.330342i	0.986265	+	0.165171i	$0.0528177\pi$
$307$	5.78807i	0.330342i	−0.986265	+	0.165171i	$0.947182\pi$
$308$	0	0
$309$	24.1721	1.37510
$310$	0	0
$311$	18.3674	1.04152	0.520761	−	0.853702i	$-0.325648\pi$
$311$	18.3674	1.04152	0.520761	+	0.853702i	$0.325648\pi$
$312$	0	0
$313$	9.85869i	0.557246i	0.960401	+	0.278623i	$0.0898780\pi$
$313$	9.85869i	0.557246i	−0.960401	+	0.278623i	$0.910122\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.9488i	1.85059i	0.379250	+	0.925294i	$0.376182\pi$
$317$	32.9488i	1.85059i	−0.379250	+	0.925294i	$0.623818\pi$
$318$	0	0
$319$	0.630376	0.0352943
$320$	0	0
$321$	−33.6016	−1.87546
$322$	0	0
$323$	12.7067i	0.707021i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8.32873i	− 0.460580i
$328$	0	0
$329$	−24.8692	−1.37109
$330$	0	0
$331$	−6.85308	−0.376679	−0.188340	−	0.982104i	$-0.560311\pi$
$331$	−6.85308	−0.376679	−0.188340	+	0.982104i	$0.560311\pi$
$332$	0	0
$333$	5.53273i	0.303192i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 35.8432i	− 1.95250i	−0.216641	−	0.976251i	$-0.569510\pi$
$337$	− 35.8432i	− 1.95250i	0.216641	−	0.976251i	$-0.430490\pi$
$338$	0	0
$339$	23.4827	1.27541
$340$	0	0
$341$	−4.71575	−0.255372
$342$	0	0
$343$	− 19.8316i	− 1.07081i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.09198i	0.219669i	0.993950	+	0.109835i	$0.0350321\pi$
$347$	4.09198i	0.219669i	−0.993950	+	0.109835i	$0.964968\pi$
$348$	0	0
$349$	11.3229	0.606101	0.303051	−	0.952974i	$-0.401995\pi$
$349$	11.3229	0.606101	0.303051	+	0.952974i	$0.401995\pi$
$350$	0	0
$351$	19.8574	1.05991
$352$	0	0
$353$	26.1443i	1.39152i	0.718273	+	0.695761i	$0.244933\pi$
$353$	26.1443i	1.39152i	−0.718273	+	0.695761i	$0.755067\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 8.35587i	− 0.442239i
$358$	0	0
$359$	12.2058	0.644198	0.322099	−	0.946706i	$-0.395612\pi$
$359$	12.2058	0.644198	0.322099	+	0.946706i	$0.395612\pi$
$360$	0	0
$361$	30.0327	1.58067
$362$	0	0
$363$	33.7242i	1.77006i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.15299i	0.164585i	0.996608	+	0.0822923i	$0.0262241\pi$
$367$	3.15299i	0.164585i	−0.996608	+	0.0822923i	$0.973776\pi$
$368$	0	0
$369$	1.06832	0.0556147
$370$	0	0
$371$	−22.4458	−1.16533
$372$	0	0
$373$	9.59872i	0.497003i	0.968632	+	0.248501i	$0.0799380\pi$
$373$	9.59872i	0.497003i	−0.968632	+	0.248501i	$0.920062\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 0.536280i	− 0.0276198i
$378$	0	0
$379$	−32.7841	−1.68401	−0.842003	−	0.539473i	$-0.818624\pi$
$379$	−32.7841	−1.68401	−0.842003	+	0.539473i	$0.818624\pi$
$380$	0	0
$381$	−2.37647	−0.121750
$382$	0	0
$383$	34.7378i	1.77502i	0.460792	+	0.887508i	$0.347565\pi$
$383$	34.7378i	1.77502i	−0.460792	+	0.887508i	$0.652435\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.8965	−0.704581	−0.352290	−	0.935891i	$-0.614597\pi$
$389$	−13.8965	−0.704581	−0.352290	+	0.935891i	$0.614597\pi$
$390$	0	0
$391$	−1.81464	−0.0917704
$392$	0	0
$393$	3.81434i	0.192408i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.1135i	1.31060i	0.755370	+	0.655299i	$0.227457\pi$
$397$	26.1135i	1.31060i	−0.755370	+	0.655299i	$0.772543\pi$
$398$	0	0
$399$	−32.2436	−1.61420
$400$	0	0
$401$	18.0715	0.902446	0.451223	−	0.892411i	$-0.350988\pi$
$401$	18.0715	0.902446	0.451223	+	0.892411i	$0.350988\pi$
$402$	0	0
$403$	4.01183i	0.199844i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 40.1041i	− 1.98789i
$408$	0	0
$409$	0.865532	0.0427978	0.0213989	−	0.999771i	$-0.493188\pi$
$409$	0.865532	0.0427978	0.0213989	+	0.999771i	$0.493188\pi$
$410$	0	0
$411$	26.6094	1.31254
$412$	0	0
$413$	18.5643i	0.913487i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 10.2494i	− 0.501916i
$418$	0	0
$419$	20.5675	1.00479	0.502394	−	0.864639i	$-0.332453\pi$
$419$	20.5675	1.00479	0.502394	+	0.864639i	$0.332453\pi$
$420$	0	0
$421$	26.2423	1.27897	0.639485	−	0.768803i	$-0.279147\pi$
$421$	26.2423	1.27897	0.639485	+	0.768803i	$0.279147\pi$
$422$	0	0
$423$	7.68126i	0.373476i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.68331i	0.323428i
$428$	0	0
$429$	46.7777	2.25845
$430$	0	0
$431$	23.2654	1.12066	0.560328	−	0.828271i	$-0.310675\pi$
$431$	23.2654	1.12066	0.560328	+	0.828271i	$0.310675\pi$
$432$	0	0
$433$	− 34.7365i	− 1.66933i	−0.550758	−	0.834665i	$-0.685661\pi$
$433$	− 34.7365i	− 1.66933i	0.550758	−	0.834665i	$-0.314339\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.00233i	0.334967i
$438$	0	0
$439$	−28.5238	−1.36137	−0.680684	−	0.732577i	$-0.738317\pi$
$439$	−28.5238	−1.36137	−0.680684	+	0.732577i	$0.738317\pi$
$440$	0	0
$441$	−0.974509	−0.0464052
$442$	0	0
$443$	26.3428i	1.25158i	0.779990	+	0.625792i	$0.215224\pi$
$443$	26.3428i	1.25158i	−0.779990	+	0.625792i	$0.784776\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 14.9457i	− 0.706908i
$448$	0	0
$449$	−29.5411	−1.39413	−0.697065	−	0.717008i	$-0.745511\pi$
$449$	−29.5411	−1.39413	−0.697065	+	0.717008i	$0.745511\pi$
$450$	0	0
$451$	−7.74377	−0.364640
$452$	0	0
$453$	41.3926i	1.94480i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.16668i	− 0.148131i	−0.997253	−	0.0740655i	$-0.976403\pi$
$457$	− 3.16668i	− 0.148131i	0.997253	−	0.0740655i	$-0.0235974\pi$
$458$	0	0
$459$	7.94133	0.370670
$460$	0	0
$461$	8.43891	0.393039	0.196520	−	0.980500i	$-0.437036\pi$
$461$	8.43891	0.393039	0.196520	+	0.980500i	$0.437036\pi$
$462$	0	0
$463$	9.62461i	0.447294i	0.974670	+	0.223647i	$0.0717962\pi$
$463$	9.62461i	0.447294i	−0.974670	+	0.223647i	$0.928204\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.1041i	1.06913i	0.845128	+	0.534564i	$0.179524\pi$
$467$	23.1041i	1.06913i	−0.845128	+	0.534564i	$0.820476\pi$
$468$	0	0
$469$	7.41233	0.342270
$470$	0	0
$471$	−7.82289	−0.360460
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.93274i	0.317428i
$478$	0	0
$479$	−20.5981	−0.941150	−0.470575	−	0.882360i	$-0.655953\pi$
$479$	−20.5981	−0.941150	−0.470575	+	0.882360i	$0.655953\pi$
$480$	0	0
$481$	−34.1178	−1.55564
$482$	0	0
$483$	− 4.60469i	− 0.209521i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 40.1145i	− 1.81776i	−0.417056	−	0.908881i	$-0.636938\pi$
$487$	− 40.1145i	− 1.81776i	0.417056	−	0.908881i	$-0.363062\pi$
$488$	0	0
$489$	−17.4868	−0.790780
$490$	0	0
$491$	−18.4799	−0.833985	−0.416993	−	0.908910i	$-0.636916\pi$
$491$	−18.4799	−0.833985	−0.416993	+	0.908910i	$0.636916\pi$
$492$	0	0
$493$	− 0.214469i	− 0.00965918i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.3783i	1.45237i
$498$	0	0
$499$	−0.0506452	−0.00226719	−0.00113360	−	0.999999i	$-0.500361\pi$
$499$	−0.0506452	−0.00226719	−0.00113360	+	0.999999i	$0.500361\pi$
$500$	0	0
$501$	−24.2506	−1.08344
$502$	0	0
$503$	− 31.1887i	− 1.39064i	−0.718702	−	0.695318i	$-0.755263\pi$
$503$	− 31.1887i	− 1.39064i	0.718702	−	0.695318i	$-0.244737\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 14.6684i	− 0.651448i
$508$	0	0
$509$	−3.42426	−0.151778	−0.0758888	−	0.997116i	$-0.524179\pi$
$509$	−3.42426	−0.151778	−0.0758888	+	0.997116i	$0.524179\pi$
$510$	0	0
$511$	29.6340	1.31093
$512$	0	0
$513$	− 30.6440i	− 1.35297i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 55.6778i	− 2.44871i
$518$	0	0
$519$	27.3213	1.19927
$520$	0	0
$521$	−9.78442	−0.428663	−0.214332	−	0.976761i	$-0.568757\pi$
$521$	−9.78442	−0.428663	−0.214332	+	0.976761i	$0.568757\pi$
$522$	0	0
$523$	6.22884i	0.272368i	0.990684	+	0.136184i	$0.0434838\pi$
$523$	6.22884i	0.272368i	−0.990684	+	0.136184i	$0.956516\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.60441i	0.0698892i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	5.73387	0.248829
$532$	0	0
$533$	6.58786i	0.285352i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	50.7255i	2.18897i
$538$	0	0
$539$	7.06375	0.304257
$540$	0	0
$541$	19.8477	0.853319	0.426660	−	0.904412i	$-0.359690\pi$
$541$	19.8477	0.853319	0.426660	+	0.904412i	$0.359690\pi$
$542$	0	0
$543$	− 35.4598i	− 1.52173i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.68024i	0.328383i	0.986428	+	0.164192i	$0.0525016\pi$
$547$	7.68024i	0.328383i	−0.986428	+	0.164192i	$0.947498\pi$
$548$	0	0
$549$	2.06425	0.0880999
$550$	0	0
$551$	−0.827591	−0.0352566
$552$	0	0
$553$	16.2041i	0.689067i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 18.1260i	− 0.768023i	−0.923328	−	0.384012i	$-0.874542\pi$
$557$	− 18.1260i	− 0.768023i	0.923328	−	0.384012i	$-0.125458\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	18.7073	0.789823
$562$	0	0
$563$	− 16.8772i	− 0.711287i	−0.934622	−	0.355644i	$-0.884262\pi$
$563$	− 16.8772i	− 0.711287i	0.934622	−	0.355644i	$-0.115738\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	25.4103i	1.06713i
$568$	0	0
$569$	−36.0022	−1.50929	−0.754645	−	0.656133i	$-0.772191\pi$
$569$	−36.0022	−1.50929	−0.754645	+	0.656133i	$0.772191\pi$
$570$	0	0
$571$	26.6267	1.11429	0.557147	−	0.830414i	$-0.311896\pi$
$571$	26.6267	1.11429	0.557147	+	0.830414i	$0.311896\pi$
$572$	0	0
$573$	− 7.47166i	− 0.312133i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 30.0190i	− 1.24971i	−0.780742	−	0.624854i	$-0.785158\pi$
$577$	− 30.0190i	− 1.24971i	0.780742	−	0.624854i	$-0.214842\pi$
$578$	0	0
$579$	46.1171	1.91656
$580$	0	0
$581$	−32.2072	−1.33618
$582$	0	0
$583$	− 50.2521i	− 2.08123i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.1814i	1.28699i	0.765449	+	0.643497i	$0.222517\pi$
$587$	31.1814i	1.28699i	−0.765449	+	0.643497i	$0.777483\pi$
$588$	0	0
$589$	6.19109	0.255099
$590$	0	0
$591$	22.6829	0.933051
$592$	0	0
$593$	15.0334i	0.617348i	0.951168	+	0.308674i	$0.0998852\pi$
$593$	15.0334i	0.617348i	−0.951168	+	0.308674i	$0.900115\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.2004i	0.703964i
$598$	0	0
$599$	−20.3534	−0.831617	−0.415808	−	0.909452i	$-0.636501\pi$
$599$	−20.3534	−0.831617	−0.415808	+	0.909452i	$0.636501\pi$
$600$	0	0
$601$	46.5338	1.89816	0.949078	−	0.315043i	$-0.102019\pi$
$601$	46.5338	1.89816	0.949078	+	0.315043i	$0.102019\pi$
$602$	0	0
$603$	− 2.28942i	− 0.0932323i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.52725i	0.305522i	0.988263	+	0.152761i	$0.0488164\pi$
$607$	7.52725i	0.305522i	−0.988263	+	0.152761i	$0.951184\pi$
$608$	0	0
$609$	0.544219	0.0220529
$610$	0	0
$611$	−47.3668	−1.91626
$612$	0	0
$613$	32.2643i	1.30314i	0.758587	+	0.651572i	$0.225890\pi$
$613$	32.2643i	1.30314i	−0.758587	+	0.651572i	$0.774110\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.52158i	− 0.182032i	−0.995849	−	0.0910161i	$-0.970989\pi$
$617$	− 4.52158i	− 0.182032i	0.995849	−	0.0910161i	$-0.0290115\pi$
$618$	0	0
$619$	−17.7807	−0.714668	−0.357334	−	0.933977i	$-0.616314\pi$
$619$	−17.7807	−0.714668	−0.357334	+	0.933977i	$0.616314\pi$
$620$	0	0
$621$	4.37626	0.175613
$622$	0	0
$623$	6.90660i	0.276707i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 72.1877i	− 2.88290i
$628$	0	0
$629$	−13.6444	−0.544036
$630$	0	0
$631$	27.4554	1.09298	0.546490	−	0.837466i	$-0.315964\pi$
$631$	27.4554	1.09298	0.546490	+	0.837466i	$0.315964\pi$
$632$	0	0
$633$	44.6917i	1.77634i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.00935i	− 0.238099i
$638$	0	0
$639$	10.0006	0.395616
$640$	0	0
$641$	−48.9270	−1.93250	−0.966250	−	0.257605i	$-0.917067\pi$
$641$	−48.9270	−1.93250	−0.966250	+	0.257605i	$0.917067\pi$
$642$	0	0
$643$	2.89333i	0.114102i	0.998371	+	0.0570508i	$0.0181697\pi$
$643$	2.89333i	0.114102i	−0.998371	+	0.0570508i	$0.981830\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 15.0877i	− 0.593157i	−0.955009	−	0.296578i	$-0.904154\pi$
$647$	− 15.0877i	− 0.593157i	0.955009	−	0.296578i	$-0.0958456\pi$
$648$	0	0
$649$	−41.5621	−1.63145
$650$	0	0
$651$	−4.07122	−0.159564
$652$	0	0
$653$	− 13.3255i	− 0.521468i	−0.965411	−	0.260734i	$-0.916035\pi$
$653$	− 13.3255i	− 0.521468i	0.965411	−	0.260734i	$-0.0839645\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 9.15292i	− 0.357089i
$658$	0	0
$659$	39.9000	1.55428	0.777142	−	0.629325i	$-0.216669\pi$
$659$	39.9000	1.55428	0.777142	+	0.629325i	$0.216669\pi$
$660$	0	0
$661$	−37.3060	−1.45104	−0.725518	−	0.688203i	$-0.758400\pi$
$661$	−37.3060	−1.45104	−0.725518	+	0.688203i	$0.758400\pi$
$662$	0	0
$663$	− 15.9149i	− 0.618082i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 0.118188i	− 0.00457625i
$668$	0	0
$669$	−2.91593	−0.112736
$670$	0	0
$671$	−14.9627	−0.577630
$672$	0	0
$673$	− 18.6641i	− 0.719447i	−0.933059	−	0.359724i	$-0.882871\pi$
$673$	− 18.6641i	− 0.719447i	0.933059	−	0.359724i	$-0.117129\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 19.4978i	− 0.749361i	−0.927154	−	0.374681i	$-0.877752\pi$
$677$	− 19.4978i	− 0.749361i	0.927154	−	0.374681i	$-0.122248\pi$
$678$	0	0
$679$	4.71169	0.180818
$680$	0	0
$681$	−42.5313	−1.62980
$682$	0	0
$683$	− 41.6331i	− 1.59305i	−0.604608	−	0.796523i	$-0.706670\pi$
$683$	− 41.6331i	− 1.59305i	0.604608	−	0.796523i	$-0.293330\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.8276i	0.832773i
$688$	0	0
$689$	−42.7510	−1.62868
$690$	0	0
$691$	−9.53192	−0.362611	−0.181306	−	0.983427i	$-0.558032\pi$
$691$	−9.53192	−0.362611	−0.181306	+	0.983427i	$0.558032\pi$
$692$	0	0
$693$	9.34998i	0.355177i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.63461i	0.0997930i
$698$	0	0
$699$	34.1702	1.29244
$700$	0	0
$701$	27.9897	1.05716	0.528579	−	0.848884i	$-0.322725\pi$
$701$	27.9897	1.05716	0.528579	+	0.848884i	$0.322725\pi$
$702$	0	0
$703$	52.6508i	1.98576i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.2643i	1.10060i
$708$	0	0
$709$	−27.6832	−1.03966	−0.519831	−	0.854269i	$-0.674005\pi$
$709$	−27.6832	−1.03966	−0.519831	+	0.854269i	$0.674005\pi$
$710$	0	0
$711$	5.00489	0.187698
$712$	0	0
$713$	0.884147i	0.0331115i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.8200i	1.82322i
$718$	0	0
$719$	−50.8364	−1.89588	−0.947938	−	0.318454i	$-0.896836\pi$
$719$	−50.8364	−1.89588	−0.947938	+	0.318454i	$0.896836\pi$
$720$	0	0
$721$	−29.7940	−1.10959
$722$	0	0
$723$	1.52812i	0.0568316i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 37.5061i	− 1.39102i	−0.718514	−	0.695512i	$-0.755177\pi$
$727$	− 37.5061i	− 1.39102i	0.718514	−	0.695512i	$-0.244823\pi$
$728$	0	0
$729$	−17.5273	−0.649158
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 36.9374i	− 1.36431i	−0.731206	−	0.682157i	$-0.761042\pi$
$733$	− 36.9374i	− 1.36431i	0.731206	−	0.682157i	$-0.238958\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.5949i	0.611281i
$738$	0	0
$739$	12.3287	0.453517	0.226759	−	0.973951i	$-0.427187\pi$
$739$	12.3287	0.453517	0.226759	+	0.973951i	$0.427187\pi$
$740$	0	0
$741$	−61.4123	−2.25604
$742$	0	0
$743$	− 29.7534i	− 1.09154i	−0.837933	−	0.545772i	$-0.816236\pi$
$743$	− 29.7534i	− 1.09154i	0.837933	−	0.545772i	$-0.183764\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.94770i	0.363967i
$748$	0	0
$749$	41.4166	1.51333
$750$	0	0
$751$	−41.9834	−1.53200	−0.765999	−	0.642842i	$-0.777755\pi$
$751$	−41.9834	−1.53200	−0.765999	+	0.642842i	$0.777755\pi$
$752$	0	0
$753$	23.8628i	0.869610i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 11.4708i	− 0.416915i	−0.978031	−	0.208458i	$-0.933156\pi$
$757$	− 11.4708i	− 0.416915i	0.978031	−	0.208458i	$-0.0668443\pi$
$758$	0	0
$759$	10.3091	0.374196
$760$	0	0
$761$	−13.4122	−0.486193	−0.243097	−	0.970002i	$-0.578163\pi$
$761$	−13.4122	−0.486193	−0.243097	+	0.970002i	$0.578163\pi$
$762$	0	0
$763$	10.2658i	0.371646i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35.3581i	1.27671i
$768$	0	0
$769$	−2.18327	−0.0787307	−0.0393654	−	0.999225i	$-0.512534\pi$
$769$	−2.18327	−0.0787307	−0.0393654	+	0.999225i	$0.512534\pi$
$770$	0	0
$771$	12.3787	0.445806
$772$	0	0
$773$	5.73306i	0.206204i	0.994671	+	0.103102i	$0.0328768\pi$
$773$	5.73306i	0.206204i	−0.994671	+	0.103102i	$0.967123\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 34.6228i	− 1.24209i
$778$	0	0
$779$	10.1664	0.364250
$780$	0	0
$781$	−72.4893	−2.59387
$782$	0	0
$783$	0.517220i	0.0184839i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.13502i	0.289982i	0.989433	+	0.144991i	$0.0463153\pi$
$787$	8.13502i	0.289982i	−0.989433	+	0.144991i	$0.953685\pi$
$788$	0	0
$789$	−38.0601	−1.35497
$790$	0	0
$791$	−28.9442	−1.02914
$792$	0	0
$793$	12.7293i	0.452029i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.5139i	1.47050i	0.677797	+	0.735250i	$0.262935\pi$
$797$	41.5139i	1.47050i	−0.677797	+	0.735250i	$0.737065\pi$
$798$	0	0
$799$	−18.9429	−0.670151
$800$	0	0
$801$	2.13321	0.0753733
$802$	0	0
$803$	66.3451i	2.34127i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.61982i	0.233029i
$808$	0	0
$809$	22.8566	0.803596	0.401798	−	0.915728i	$-0.368385\pi$
$809$	22.8566	0.803596	0.401798	+	0.915728i	$0.368385\pi$
$810$	0	0
$811$	−42.1917	−1.48155	−0.740776	−	0.671753i	$-0.765542\pi$
$811$	−42.1917	−1.48155	−0.740776	+	0.671753i	$0.765542\pi$
$812$	0	0
$813$	26.5178i	0.930020i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	7.95432	0.277946
$820$	0	0
$821$	27.0195	0.942985	0.471493	−	0.881870i	$-0.343715\pi$
$821$	27.0195	0.942985	0.471493	+	0.881870i	$0.343715\pi$
$822$	0	0
$823$	27.2435i	0.949650i	0.880080	+	0.474825i	$0.157489\pi$
$823$	27.2435i	0.949650i	−0.880080	+	0.474825i	$0.842511\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 10.5732i	− 0.367667i	−0.982957	−	0.183834i	$-0.941149\pi$
$827$	− 10.5732i	− 0.367667i	0.982957	−	0.183834i	$-0.0588508\pi$
$828$	0	0
$829$	1.16445	0.0404429	0.0202215	−	0.999796i	$-0.493563\pi$
$829$	1.16445	0.0404429	0.0202215	+	0.999796i	$0.493563\pi$
$830$	0	0
$831$	1.22953	0.0426519
$832$	0	0
$833$	− 2.40325i	− 0.0832678i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.86925i	− 0.133741i
$838$	0	0
$839$	20.2920	0.700556	0.350278	−	0.936646i	$-0.386087\pi$
$839$	20.2920	0.700556	0.350278	+	0.936646i	$0.386087\pi$
$840$	0	0
$841$	−28.9860	−0.999518
$842$	0	0
$843$	35.5880i	1.22571i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 41.5676i	− 1.42828i
$848$	0	0
$849$	15.3801	0.527843
$850$	0	0
$851$	−7.51903	−0.257749
$852$	0	0
$853$	26.6154i	0.911295i	0.890160	+	0.455648i	$0.150592\pi$
$853$	26.6154i	0.911295i	−0.890160	+	0.455648i	$0.849408\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.4342i	0.356427i	0.983992	+	0.178214i	$0.0570318\pi$
$857$	10.4342i	0.356427i	−0.983992	+	0.178214i	$0.942968\pi$
$858$	0	0
$859$	24.6623	0.841466	0.420733	−	0.907185i	$-0.361773\pi$
$859$	24.6623	0.841466	0.420733	+	0.907185i	$0.361773\pi$
$860$	0	0
$861$	−6.68538	−0.227837
$862$	0	0
$863$	8.48317i	0.288771i	0.989522	+	0.144385i	$0.0461205\pi$
$863$	8.48317i	0.288771i	−0.989522	+	0.144385i	$0.953880\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.4934i	0.899764i
$868$	0	0
$869$	−36.2780	−1.23065
$870$	0	0
$871$	14.1178	0.478363
$872$	0	0
$873$	− 1.45528i	− 0.0492538i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 3.08796i	− 0.104273i	−0.998640	−	0.0521365i	$-0.983397\pi$
$877$	− 3.08796i	− 0.104273i	0.998640	−	0.0521365i	$-0.0166031\pi$
$878$	0	0
$879$	40.4282	1.36361
$880$	0	0
$881$	42.5879	1.43482	0.717412	−	0.696649i	$-0.245327\pi$
$881$	42.5879	1.43482	0.717412	+	0.696649i	$0.245327\pi$
$882$	0	0
$883$	− 48.3342i	− 1.62657i	−0.581863	−	0.813287i	$-0.697676\pi$
$883$	− 48.3342i	− 1.62657i	0.581863	−	0.813287i	$-0.302324\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.12173i	0.0712408i	0.999365	+	0.0356204i	$0.0113407\pi$
$887$	2.12173i	0.0712408i	−0.999365	+	0.0356204i	$0.988659\pi$
$888$	0	0
$889$	2.92918	0.0982415
$890$	0	0
$891$	−56.8892	−1.90586
$892$	0	0
$893$	73.0968i	2.44609i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8.77026i	− 0.292830i
$898$	0	0
$899$	−0.104495	−0.00348512
$900$	0	0
$901$	−17.0970	−0.569582
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 28.6914i	− 0.952684i	−0.879260	−	0.476342i	$-0.841962\pi$
$907$	− 28.6914i	− 0.952684i	0.879260	−	0.476342i	$-0.158038\pi$
$908$	0	0
$909$	9.03875	0.299796
$910$	0	0
$911$	−1.14899	−0.0380679	−0.0190339	−	0.999819i	$-0.506059\pi$
$911$	−1.14899	−0.0380679	−0.0190339	+	0.999819i	$0.506059\pi$
$912$	0	0
$913$	− 72.1061i	− 2.38636i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.70146i	− 0.155256i
$918$	0	0
$919$	55.4798	1.83011	0.915055	−	0.403329i	$-0.132147\pi$
$919$	55.4798	1.83011	0.915055	+	0.403329i	$0.132147\pi$
$920$	0	0
$921$	−11.1873	−0.368636
$922$	0	0
$923$	61.6689i	2.02986i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.20235i	0.302245i
$928$	0	0
$929$	4.05780	0.133132	0.0665660	−	0.997782i	$-0.478796\pi$
$929$	4.05780	0.133132	0.0665660	+	0.997782i	$0.478796\pi$
$930$	0	0
$931$	−9.27367	−0.303932
$932$	0	0
$933$	35.5011i	1.16226i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 16.0588i	− 0.524617i	−0.964984	−	0.262308i	$-0.915516\pi$
$937$	− 16.0588i	− 0.524617i	0.964984	−	0.262308i	$-0.0844838\pi$
$938$	0	0
$939$	−19.0552	−0.621842
$940$	0	0
$941$	4.73779	0.154448	0.0772238	−	0.997014i	$-0.475394\pi$
$941$	4.73779	0.154448	0.0772238	+	0.997014i	$0.475394\pi$
$942$	0	0
$943$	1.45186i	0.0472792i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.38634i	0.110041i	0.998485	+	0.0550207i	$0.0175225\pi$
$947$	3.38634i	0.110041i	−0.998485	+	0.0550207i	$0.982478\pi$
$948$	0	0
$949$	56.4418	1.83218
$950$	0	0
$951$	−63.6844	−2.06511
$952$	0	0
$953$	29.2967i	0.949013i	0.880252	+	0.474507i	$0.157373\pi$
$953$	29.2967i	0.949013i	−0.880252	+	0.474507i	$0.842627\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.21841i	0.0393856i
$958$	0	0
$959$	−32.7981	−1.05910
$960$	0	0
$961$	−30.2183	−0.974783
$962$	0	0
$963$	− 12.7922i	− 0.412222i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	15.5743i	0.500837i	0.968138	+	0.250419i	$0.0805683\pi$
$967$	15.5743i	0.500837i	−0.968138	+	0.250419i	$0.919432\pi$
$968$	0	0
$969$	−24.5599	−0.788979
$970$	0	0
$971$	51.5951	1.65577	0.827883	−	0.560900i	$-0.189545\pi$
$971$	51.5951	1.65577	0.827883	+	0.560900i	$0.189545\pi$
$972$	0	0
$973$	12.6332i	0.405001i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 9.04984i	− 0.289530i	−0.989466	−	0.144765i	$-0.953757\pi$
$977$	− 9.04984i	− 0.289530i	0.989466	−	0.144765i	$-0.0462426\pi$
$978$	0	0
$979$	−15.4626	−0.494188
$980$	0	0
$981$	3.17075	0.101234
$982$	0	0
$983$	− 39.3855i	− 1.25620i	−0.778132	−	0.628101i	$-0.783833\pi$
$983$	− 39.3855i	− 1.25620i	0.778132	−	0.628101i	$-0.216167\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 48.0680i	− 1.53002i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−18.9152	−0.600860	−0.300430	−	0.953804i	$-0.597130\pi$
$991$	−18.9152	−0.600860	−0.300430	+	0.953804i	$0.597130\pi$
$992$	0	0
$993$	− 13.2458i	− 0.420344i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.2966i	0.389438i	0.980859	+	0.194719i	$0.0623795\pi$
$997$	12.2966i	0.389438i	−0.980859	+	0.194719i	$0.937620\pi$
$998$	0	0
$999$	32.9052	1.04107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.u.4049.8		10
5.2	odd	4		920.2.a.j.1.4	✓	5
5.3	odd	4		4600.2.a.be.1.2		5
5.4	even	2	inner	4600.2.e.u.4049.3		10
15.2	even	4		8280.2.a.bs.1.4		5
20.3	even	4		9200.2.a.cu.1.4		5
20.7	even	4		1840.2.a.v.1.2		5
40.27	even	4		7360.2.a.cp.1.4		5
40.37	odd	4		7360.2.a.co.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.4	✓	5	5.2	odd	4
1840.2.a.v.1.2		5	20.7	even	4
4600.2.a.be.1.2		5	5.3	odd	4
4600.2.e.u.4049.3		10	5.4	even	2	inner
4600.2.e.u.4049.8		10	1.1	even	1	trivial
7360.2.a.co.1.2		5	40.37	odd	4
7360.2.a.cp.1.4		5	40.27	even	4
8280.2.a.bs.1.4		5	15.2	even	4
9200.2.a.cu.1.4		5	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+1.93283i q^{3} -2.38236i q^{7} -0.735829 q^{9} +O(q^{10})\)	\(q+1.93283i q^{3} -2.38236i q^{7} -0.735829 q^{9} +5.33368 q^{11} -4.53752i q^{13} -1.81464i q^{17} -7.00233 q^{19} +4.60469 q^{21} -1.00000i q^{23} +4.37626i q^{27} +0.118188 q^{29} -0.884147 q^{31} +10.3091i q^{33} -7.51903i q^{37} +8.77026 q^{39} -1.45186 q^{41} -10.4389i q^{47} +1.32437 q^{49} +3.50739 q^{51} -9.42167i q^{53} -13.5343i q^{57} -7.79239 q^{59} -2.80533 q^{61} +1.75301i q^{63} +3.11134i q^{67} +1.93283 q^{69} -13.5909 q^{71} +12.4389i q^{73} -12.7067i q^{77} -6.80169 q^{79} -10.6660 q^{81} -13.5190i q^{83} +0.228437i q^{87} -2.89906 q^{89} -10.8100 q^{91} -1.70890i q^{93} +1.97774i q^{97} -3.92468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 26 q^{9}+O(q^{10}) \) 10 * q - 26 * q^9	\( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) \) 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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