LMFDB - Embedded newform 4620.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.a (trivial)

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:	yes
Analytic conductor:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -0.354249 q^{13} +1.00000 q^{15} -1.00000 q^{17} -6.29150 q^{19} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.645751 q^{29} +10.9373 q^{31} +1.00000 q^{33} +1.00000 q^{35} -7.64575 q^{37} +0.354249 q^{39} -4.93725 q^{41} -6.64575 q^{43} -1.00000 q^{45} +6.93725 q^{47} +1.00000 q^{49} +1.00000 q^{51} -3.00000 q^{53} +1.00000 q^{55} +6.29150 q^{57} +8.64575 q^{59} -6.29150 q^{61} -1.00000 q^{63} +0.354249 q^{65} +8.58301 q^{67} +1.00000 q^{69} -1.64575 q^{71} -3.29150 q^{73} -1.00000 q^{75} +1.00000 q^{77} +1.64575 q^{79} +1.00000 q^{81} -7.58301 q^{83} +1.00000 q^{85} -0.645751 q^{87} +11.9373 q^{89} +0.354249 q^{91} -10.9373 q^{93} +6.29150 q^{95} +5.35425 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 6 q^{13} + 2 q^{15} - 2 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 2 q^{35} - 10 q^{37} + 6 q^{39} + 6 q^{41} - 8 q^{43} - 2 q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} - 6 q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} - 2 q^{61} - 2 q^{63} + 6 q^{65} - 4 q^{67} + 2 q^{69} + 2 q^{71} + 4 q^{73} - 2 q^{75} + 2 q^{77} - 2 q^{79} + 2 q^{81} + 6 q^{83} + 2 q^{85} + 4 q^{87} + 8 q^{89} + 6 q^{91} - 6 q^{93} + 2 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - 6 * q^13 + 2 * q^15 - 2 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 + 6 * q^31 + 2 * q^33 + 2 * q^35 - 10 * q^37 + 6 * q^39 + 6 * q^41 - 8 * q^43 - 2 * q^45 - 2 * q^47 + 2 * q^49 + 2 * q^51 - 6 * q^53 + 2 * q^55 + 2 * q^57 + 12 * q^59 - 2 * q^61 - 2 * q^63 + 6 * q^65 - 4 * q^67 + 2 * q^69 + 2 * q^71 + 4 * q^73 - 2 * q^75 + 2 * q^77 - 2 * q^79 + 2 * q^81 + 6 * q^83 + 2 * q^85 + 4 * q^87 + 8 * q^89 + 6 * q^91 - 6 * q^93 + 2 * q^95 + 16 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.354249	−0.0982509	−0.0491255	−	0.998793i	$-0.515643\pi$
$13$	−0.354249	−0.0982509	−0.0491255	+	0.998793i	$0.515643\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−6.29150	−1.44337	−0.721685	−	0.692222i	$-0.756632\pi$
$19$	−6.29150	−1.44337	−0.721685	+	0.692222i	$0.756632\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.645751	0.119913	0.0599565	−	0.998201i	$-0.480904\pi$
$29$	0.645751	0.119913	0.0599565	+	0.998201i	$0.480904\pi$
$30$	0	0
$31$	10.9373	1.96439	0.982194	−	0.187867i	$-0.0601575\pi$
$31$	10.9373	1.96439	0.982194	+	0.187867i	$0.0601575\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−7.64575	−1.25695	−0.628477	−	0.777828i	$-0.716321\pi$
$37$	−7.64575	−1.25695	−0.628477	+	0.777828i	$0.716321\pi$
$38$	0	0
$39$	0.354249	0.0567252
$40$	0	0
$41$	−4.93725	−0.771070	−0.385535	−	0.922693i	$-0.625983\pi$
$41$	−4.93725	−0.771070	−0.385535	+	0.922693i	$0.625983\pi$
$42$	0	0
$43$	−6.64575	−1.01347	−0.506734	−	0.862103i	$-0.669147\pi$
$43$	−6.64575	−1.01347	−0.506734	+	0.862103i	$0.669147\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.93725	1.01190	0.505951	−	0.862562i	$-0.331142\pi$
$47$	6.93725	1.01190	0.505951	+	0.862562i	$0.331142\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	6.29150	0.833330
$58$	0	0
$59$	8.64575	1.12558	0.562790	−	0.826600i	$-0.309728\pi$
$59$	8.64575	1.12558	0.562790	+	0.826600i	$0.309728\pi$
$60$	0	0
$61$	−6.29150	−0.805544	−0.402772	−	0.915300i	$-0.631953\pi$
$61$	−6.29150	−0.805544	−0.402772	+	0.915300i	$0.631953\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0.354249	0.0439391
$66$	0	0
$67$	8.58301	1.04858	0.524290	−	0.851539i	$-0.324331\pi$
$67$	8.58301	1.04858	0.524290	+	0.851539i	$0.324331\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−1.64575	−0.195315	−0.0976574	−	0.995220i	$-0.531135\pi$
$71$	−1.64575	−0.195315	−0.0976574	+	0.995220i	$0.531135\pi$
$72$	0	0
$73$	−3.29150	−0.385241	−0.192621	−	0.981273i	$-0.561699\pi$
$73$	−3.29150	−0.385241	−0.192621	+	0.981273i	$0.561699\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	1.64575	0.185161	0.0925807	−	0.995705i	$-0.470488\pi$
$79$	1.64575	0.185161	0.0925807	+	0.995705i	$0.470488\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.58301	−0.832343	−0.416171	−	0.909286i	$-0.636628\pi$
$83$	−7.58301	−0.832343	−0.416171	+	0.909286i	$0.636628\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−0.645751	−0.0692318
$88$	0	0
$89$	11.9373	1.26535	0.632673	−	0.774419i	$-0.281958\pi$
$89$	11.9373	1.26535	0.632673	+	0.774419i	$0.281958\pi$
$90$	0	0
$91$	0.354249	0.0371354
$92$	0	0
$93$	−10.9373	−1.13414
$94$	0	0
$95$	6.29150	0.645495
$96$	0	0
$97$	5.35425	0.543642	0.271821	−	0.962348i	$-0.412374\pi$
$97$	5.35425	0.543642	0.271821	+	0.962348i	$0.412374\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	14.6458	1.44309	0.721544	−	0.692368i	$-0.243433\pi$
$103$	14.6458	1.44309	0.721544	+	0.692368i	$0.243433\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	3.64575	0.352448	0.176224	−	0.984350i	$-0.443612\pi$
$107$	3.64575	0.352448	0.176224	+	0.984350i	$0.443612\pi$
$108$	0	0
$109$	−19.6458	−1.88172	−0.940861	−	0.338793i	$-0.889981\pi$
$109$	−19.6458	−1.88172	−0.940861	+	0.338793i	$0.889981\pi$
$110$	0	0
$111$	7.64575	0.725703
$112$	0	0
$113$	−1.70850	−0.160722	−0.0803610	−	0.996766i	$-0.525607\pi$
$113$	−1.70850	−0.160722	−0.0803610	+	0.996766i	$0.525607\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−0.354249	−0.0327503
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.93725	0.445177
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.35425	0.297641	0.148821	−	0.988864i	$-0.452452\pi$
$127$	3.35425	0.297641	0.148821	+	0.988864i	$0.452452\pi$
$128$	0	0
$129$	6.64575	0.585126
$130$	0	0
$131$	8.35425	0.729914	0.364957	−	0.931024i	$-0.381084\pi$
$131$	8.35425	0.729914	0.364957	+	0.931024i	$0.381084\pi$
$132$	0	0
$133$	6.29150	0.545542
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−13.8745	−1.18538	−0.592690	−	0.805431i	$-0.701934\pi$
$137$	−13.8745	−1.18538	−0.592690	+	0.805431i	$0.701934\pi$
$138$	0	0
$139$	−5.29150	−0.448819	−0.224410	−	0.974495i	$-0.572045\pi$
$139$	−5.29150	−0.448819	−0.224410	+	0.974495i	$0.572045\pi$
$140$	0	0
$141$	−6.93725	−0.584222
$142$	0	0
$143$	0.354249	0.0296238
$144$	0	0
$145$	−0.645751	−0.0536267
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	13.2915	1.08888	0.544441	−	0.838799i	$-0.316742\pi$
$149$	13.2915	1.08888	0.544441	+	0.838799i	$0.316742\pi$
$150$	0	0
$151$	8.58301	0.698475	0.349238	−	0.937034i	$-0.386441\pi$
$151$	8.58301	0.698475	0.349238	+	0.937034i	$0.386441\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−10.9373	−0.878501
$156$	0	0
$157$	−4.64575	−0.370771	−0.185386	−	0.982666i	$-0.559353\pi$
$157$	−4.64575	−0.370771	−0.185386	+	0.982666i	$0.559353\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−2.22876	−0.174570	−0.0872848	−	0.996183i	$-0.527819\pi$
$163$	−2.22876	−0.174570	−0.0872848	+	0.996183i	$0.527819\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−9.29150	−0.718998	−0.359499	−	0.933145i	$-0.617052\pi$
$167$	−9.29150	−0.718998	−0.359499	+	0.933145i	$0.617052\pi$
$168$	0	0
$169$	−12.8745	−0.990347
$170$	0	0
$171$	−6.29150	−0.481123
$172$	0	0
$173$	21.8745	1.66309	0.831544	−	0.555459i	$-0.187457\pi$
$173$	21.8745	1.66309	0.831544	+	0.555459i	$0.187457\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−8.64575	−0.649854
$178$	0	0
$179$	6.35425	0.474939	0.237469	−	0.971395i	$-0.423682\pi$
$179$	6.35425	0.474939	0.237469	+	0.971395i	$0.423682\pi$
$180$	0	0
$181$	4.58301	0.340652	0.170326	−	0.985388i	$-0.445518\pi$
$181$	4.58301	0.340652	0.170326	+	0.985388i	$0.445518\pi$
$182$	0	0
$183$	6.29150	0.465081
$184$	0	0
$185$	7.64575	0.562127
$186$	0	0
$187$	1.00000	0.0731272
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	11.8745	0.854746	0.427373	−	0.904075i	$-0.359439\pi$
$193$	11.8745	0.854746	0.427373	+	0.904075i	$0.359439\pi$
$194$	0	0
$195$	−0.354249	−0.0253683
$196$	0	0
$197$	9.87451	0.703530	0.351765	−	0.936088i	$-0.385582\pi$
$197$	9.87451	0.703530	0.351765	+	0.936088i	$0.385582\pi$
$198$	0	0
$199$	−2.58301	−0.183104	−0.0915522	−	0.995800i	$-0.529183\pi$
$199$	−2.58301	−0.183104	−0.0915522	+	0.995800i	$0.529183\pi$
$200$	0	0
$201$	−8.58301	−0.605399
$202$	0	0
$203$	−0.645751	−0.0453229
$204$	0	0
$205$	4.93725	0.344833
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	6.29150	0.435192
$210$	0	0
$211$	1.29150	0.0889107	0.0444554	−	0.999011i	$-0.485845\pi$
$211$	1.29150	0.0889107	0.0444554	+	0.999011i	$0.485845\pi$
$212$	0	0
$213$	1.64575	0.112765
$214$	0	0
$215$	6.64575	0.453236
$216$	0	0
$217$	−10.9373	−0.742469
$218$	0	0
$219$	3.29150	0.222419
$220$	0	0
$221$	0.354249	0.0238293
$222$	0	0
$223$	15.2288	1.01979	0.509896	−	0.860236i	$-0.329684\pi$
$223$	15.2288	1.01979	0.509896	+	0.860236i	$0.329684\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	13.0000	0.862840	0.431420	−	0.902151i	$-0.358013\pi$
$227$	13.0000	0.862840	0.431420	+	0.902151i	$0.358013\pi$
$228$	0	0
$229$	29.5203	1.95075	0.975377	−	0.220545i	$-0.0707836\pi$
$229$	29.5203	1.95075	0.975377	+	0.220545i	$0.0707836\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	5.64575	0.369865	0.184933	−	0.982751i	$-0.440793\pi$
$233$	5.64575	0.369865	0.184933	+	0.982751i	$0.440793\pi$
$234$	0	0
$235$	−6.93725	−0.452537
$236$	0	0
$237$	−1.64575	−0.106903
$238$	0	0
$239$	−10.5203	−0.680499	−0.340249	−	0.940335i	$-0.610512\pi$
$239$	−10.5203	−0.680499	−0.340249	+	0.940335i	$0.610512\pi$
$240$	0	0
$241$	26.5830	1.71236	0.856181	−	0.516676i	$-0.172831\pi$
$241$	26.5830	1.71236	0.856181	+	0.516676i	$0.172831\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	2.22876	0.141812
$248$	0	0
$249$	7.58301	0.480553
$250$	0	0
$251$	26.5830	1.67790	0.838952	−	0.544205i	$-0.183169\pi$
$251$	26.5830	1.67790	0.838952	+	0.544205i	$0.183169\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	28.9373	1.80506	0.902528	−	0.430631i	$-0.141709\pi$
$257$	28.9373	1.80506	0.902528	+	0.430631i	$0.141709\pi$
$258$	0	0
$259$	7.64575	0.475084
$260$	0	0
$261$	0.645751	0.0399710
$262$	0	0
$263$	7.29150	0.449613	0.224807	−	0.974403i	$-0.427825\pi$
$263$	7.29150	0.449613	0.224807	+	0.974403i	$0.427825\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	0	0
$267$	−11.9373	−0.730548
$268$	0	0
$269$	25.9373	1.58142	0.790711	−	0.612189i	$-0.209711\pi$
$269$	25.9373	1.58142	0.790711	+	0.612189i	$0.209711\pi$
$270$	0	0
$271$	10.4170	0.632787	0.316394	−	0.948628i	$-0.397528\pi$
$271$	10.4170	0.632787	0.316394	+	0.948628i	$0.397528\pi$
$272$	0	0
$273$	−0.354249	−0.0214401
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−29.1660	−1.75242	−0.876208	−	0.481933i	$-0.839935\pi$
$277$	−29.1660	−1.75242	−0.876208	+	0.481933i	$0.839935\pi$
$278$	0	0
$279$	10.9373	0.654796
$280$	0	0
$281$	−6.58301	−0.392709	−0.196355	−	0.980533i	$-0.562910\pi$
$281$	−6.58301	−0.392709	−0.196355	+	0.980533i	$0.562910\pi$
$282$	0	0
$283$	−30.2288	−1.79691	−0.898457	−	0.439062i	$-0.855311\pi$
$283$	−30.2288	−1.79691	−0.898457	+	0.439062i	$0.855311\pi$
$284$	0	0
$285$	−6.29150	−0.372676
$286$	0	0
$287$	4.93725	0.291437
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−5.35425	−0.313872
$292$	0	0
$293$	0.291503	0.0170298	0.00851488	−	0.999964i	$-0.497290\pi$
$293$	0.291503	0.0170298	0.00851488	+	0.999964i	$0.497290\pi$
$294$	0	0
$295$	−8.64575	−0.503375
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	0.354249	0.0204867
$300$	0	0
$301$	6.64575	0.383055
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	6.29150	0.360250
$306$	0	0
$307$	6.70850	0.382874	0.191437	−	0.981505i	$-0.438685\pi$
$307$	6.70850	0.382874	0.191437	+	0.981505i	$0.438685\pi$
$308$	0	0
$309$	−14.6458	−0.833168
$310$	0	0
$311$	4.70850	0.266994	0.133497	−	0.991049i	$-0.457379\pi$
$311$	4.70850	0.266994	0.133497	+	0.991049i	$0.457379\pi$
$312$	0	0
$313$	−23.8118	−1.34592	−0.672960	−	0.739679i	$-0.734977\pi$
$313$	−23.8118	−1.34592	−0.672960	+	0.739679i	$0.734977\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	5.41699	0.304249	0.152124	−	0.988361i	$-0.451389\pi$
$317$	5.41699	0.304249	0.152124	+	0.988361i	$0.451389\pi$
$318$	0	0
$319$	−0.645751	−0.0361551
$320$	0	0
$321$	−3.64575	−0.203486
$322$	0	0
$323$	6.29150	0.350069
$324$	0	0
$325$	−0.354249	−0.0196502
$326$	0	0
$327$	19.6458	1.08641
$328$	0	0
$329$	−6.93725	−0.382463
$330$	0	0
$331$	15.7085	0.863417	0.431709	−	0.902013i	$-0.357911\pi$
$331$	15.7085	0.863417	0.431709	+	0.902013i	$0.357911\pi$
$332$	0	0
$333$	−7.64575	−0.418985
$334$	0	0
$335$	−8.58301	−0.468940
$336$	0	0
$337$	12.5203	0.682022	0.341011	−	0.940059i	$-0.389231\pi$
$337$	12.5203	0.682022	0.341011	+	0.940059i	$0.389231\pi$
$338$	0	0
$339$	1.70850	0.0927928
$340$	0	0
$341$	−10.9373	−0.592286
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−5.87451	−0.315360	−0.157680	−	0.987490i	$-0.550401\pi$
$347$	−5.87451	−0.315360	−0.157680	+	0.987490i	$0.550401\pi$
$348$	0	0
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	0	0
$351$	0.354249	0.0189084
$352$	0	0
$353$	−30.4575	−1.62109	−0.810545	−	0.585676i	$-0.800829\pi$
$353$	−30.4575	−1.62109	−0.810545	+	0.585676i	$0.800829\pi$
$354$	0	0
$355$	1.64575	0.0873474
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	18.0627	0.953315	0.476658	−	0.879089i	$-0.341848\pi$
$359$	18.0627	0.953315	0.476658	+	0.879089i	$0.341848\pi$
$360$	0	0
$361$	20.5830	1.08332
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	3.29150	0.172285
$366$	0	0
$367$	−0.520259	−0.0271573	−0.0135787	−	0.999908i	$-0.504322\pi$
$367$	−0.520259	−0.0271573	−0.0135787	+	0.999908i	$0.504322\pi$
$368$	0	0
$369$	−4.93725	−0.257023
$370$	0	0
$371$	3.00000	0.155752
$372$	0	0
$373$	−6.64575	−0.344104	−0.172052	−	0.985088i	$-0.555040\pi$
$373$	−6.64575	−0.344104	−0.172052	+	0.985088i	$0.555040\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−0.228757	−0.0117816
$378$	0	0
$379$	36.7490	1.88767	0.943835	−	0.330417i	$-0.107189\pi$
$379$	36.7490	1.88767	0.943835	+	0.330417i	$0.107189\pi$
$380$	0	0
$381$	−3.35425	−0.171843
$382$	0	0
$383$	−20.9373	−1.06984	−0.534922	−	0.844902i	$-0.679659\pi$
$383$	−20.9373	−1.06984	−0.534922	+	0.844902i	$0.679659\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−6.64575	−0.337823
$388$	0	0
$389$	−12.8118	−0.649582	−0.324791	−	0.945786i	$-0.605294\pi$
$389$	−12.8118	−0.649582	−0.324791	+	0.945786i	$0.605294\pi$
$390$	0	0
$391$	1.00000	0.0505722
$392$	0	0
$393$	−8.35425	−0.421416
$394$	0	0
$395$	−1.64575	−0.0828067
$396$	0	0
$397$	−10.5830	−0.531146	−0.265573	−	0.964091i	$-0.585561\pi$
$397$	−10.5830	−0.531146	−0.265573	+	0.964091i	$0.585561\pi$
$398$	0	0
$399$	−6.29150	−0.314969
$400$	0	0
$401$	34.8118	1.73842	0.869208	−	0.494446i	$-0.164629\pi$
$401$	34.8118	1.73842	0.869208	+	0.494446i	$0.164629\pi$
$402$	0	0
$403$	−3.87451	−0.193003
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	7.64575	0.378986
$408$	0	0
$409$	13.1660	0.651017	0.325509	−	0.945539i	$-0.394464\pi$
$409$	13.1660	0.651017	0.325509	+	0.945539i	$0.394464\pi$
$410$	0	0
$411$	13.8745	0.684379
$412$	0	0
$413$	−8.64575	−0.425430
$414$	0	0
$415$	7.58301	0.372235
$416$	0	0
$417$	5.29150	0.259126
$418$	0	0
$419$	−3.35425	−0.163866	−0.0819329	−	0.996638i	$-0.526109\pi$
$419$	−3.35425	−0.163866	−0.0819329	+	0.996638i	$0.526109\pi$
$420$	0	0
$421$	−8.41699	−0.410219	−0.205110	−	0.978739i	$-0.565755\pi$
$421$	−8.41699	−0.410219	−0.205110	+	0.978739i	$0.565755\pi$
$422$	0	0
$423$	6.93725	0.337301
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	6.29150	0.304467
$428$	0	0
$429$	−0.354249	−0.0171033
$430$	0	0
$431$	−7.29150	−0.351219	−0.175610	−	0.984460i	$-0.556190\pi$
$431$	−7.29150	−0.351219	−0.175610	+	0.984460i	$0.556190\pi$
$432$	0	0
$433$	−6.70850	−0.322390	−0.161195	−	0.986923i	$-0.551535\pi$
$433$	−6.70850	−0.322390	−0.161195	+	0.986923i	$0.551535\pi$
$434$	0	0
$435$	0.645751	0.0309614
$436$	0	0
$437$	6.29150	0.300963
$438$	0	0
$439$	26.1660	1.24884	0.624418	−	0.781091i	$-0.285336\pi$
$439$	26.1660	1.24884	0.624418	+	0.781091i	$0.285336\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	15.8745	0.754221	0.377110	−	0.926168i	$-0.376918\pi$
$443$	15.8745	0.754221	0.377110	+	0.926168i	$0.376918\pi$
$444$	0	0
$445$	−11.9373	−0.565880
$446$	0	0
$447$	−13.2915	−0.628667
$448$	0	0
$449$	2.35425	0.111104	0.0555519	−	0.998456i	$-0.482308\pi$
$449$	2.35425	0.111104	0.0555519	+	0.998456i	$0.482308\pi$
$450$	0	0
$451$	4.93725	0.232486
$452$	0	0
$453$	−8.58301	−0.403265
$454$	0	0
$455$	−0.354249	−0.0166074
$456$	0	0
$457$	20.5203	0.959897	0.479949	−	0.877297i	$-0.340655\pi$
$457$	20.5203	0.959897	0.479949	+	0.877297i	$0.340655\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	27.8745	1.29825	0.649123	−	0.760684i	$-0.275136\pi$
$461$	27.8745	1.29825	0.649123	+	0.760684i	$0.275136\pi$
$462$	0	0
$463$	−42.4575	−1.97317	−0.986584	−	0.163255i	$-0.947801\pi$
$463$	−42.4575	−1.97317	−0.986584	+	0.163255i	$0.947801\pi$
$464$	0	0
$465$	10.9373	0.507203
$466$	0	0
$467$	−17.8745	−0.827134	−0.413567	−	0.910474i	$-0.635717\pi$
$467$	−17.8745	−0.827134	−0.413567	+	0.910474i	$0.635717\pi$
$468$	0	0
$469$	−8.58301	−0.396326
$470$	0	0
$471$	4.64575	0.214065
$472$	0	0
$473$	6.64575	0.305572
$474$	0	0
$475$	−6.29150	−0.288674
$476$	0	0
$477$	−3.00000	−0.137361
$478$	0	0
$479$	28.1033	1.28407	0.642035	−	0.766675i	$-0.278090\pi$
$479$	28.1033	1.28407	0.642035	+	0.766675i	$0.278090\pi$
$480$	0	0
$481$	2.70850	0.123497
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−5.35425	−0.243124
$486$	0	0
$487$	17.2915	0.783553	0.391776	−	0.920060i	$-0.371861\pi$
$487$	17.2915	0.783553	0.391776	+	0.920060i	$0.371861\pi$
$488$	0	0
$489$	2.22876	0.100788
$490$	0	0
$491$	29.8118	1.34539	0.672693	−	0.739922i	$-0.265137\pi$
$491$	29.8118	1.34539	0.672693	+	0.739922i	$0.265137\pi$
$492$	0	0
$493$	−0.645751	−0.0290832
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	1.64575	0.0738220
$498$	0	0
$499$	10.2915	0.460711	0.230355	−	0.973107i	$-0.426011\pi$
$499$	10.2915	0.460711	0.230355	+	0.973107i	$0.426011\pi$
$500$	0	0
$501$	9.29150	0.415114
$502$	0	0
$503$	2.29150	0.102173	0.0510865	−	0.998694i	$-0.483732\pi$
$503$	2.29150	0.102173	0.0510865	+	0.998694i	$0.483732\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	12.8745	0.571777
$508$	0	0
$509$	−37.8118	−1.67598	−0.837988	−	0.545688i	$-0.816268\pi$
$509$	−37.8118	−1.67598	−0.837988	+	0.545688i	$0.816268\pi$
$510$	0	0
$511$	3.29150	0.145608
$512$	0	0
$513$	6.29150	0.277777
$514$	0	0
$515$	−14.6458	−0.645369
$516$	0	0
$517$	−6.93725	−0.305100
$518$	0	0
$519$	−21.8745	−0.960184
$520$	0	0
$521$	−23.8118	−1.04321	−0.521606	−	0.853186i	$-0.674667\pi$
$521$	−23.8118	−1.04321	−0.521606	+	0.853186i	$0.674667\pi$
$522$	0	0
$523$	5.87451	0.256874	0.128437	−	0.991718i	$-0.459004\pi$
$523$	5.87451	0.256874	0.128437	+	0.991718i	$0.459004\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−10.9373	−0.476434
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	8.64575	0.375194
$532$	0	0
$533$	1.74902	0.0757583
$534$	0	0
$535$	−3.64575	−0.157620
$536$	0	0
$537$	−6.35425	−0.274206
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−12.8118	−0.550821	−0.275410	−	0.961327i	$-0.588814\pi$
$541$	−12.8118	−0.550821	−0.275410	+	0.961327i	$0.588814\pi$
$542$	0	0
$543$	−4.58301	−0.196676
$544$	0	0
$545$	19.6458	0.841532
$546$	0	0
$547$	11.9373	0.510400	0.255200	−	0.966888i	$-0.417859\pi$
$547$	11.9373	0.510400	0.255200	+	0.966888i	$0.417859\pi$
$548$	0	0
$549$	−6.29150	−0.268515
$550$	0	0
$551$	−4.06275	−0.173079
$552$	0	0
$553$	−1.64575	−0.0699845
$554$	0	0
$555$	−7.64575	−0.324544
$556$	0	0
$557$	27.8745	1.18108	0.590540	−	0.807008i	$-0.298915\pi$
$557$	27.8745	1.18108	0.590540	+	0.807008i	$0.298915\pi$
$558$	0	0
$559$	2.35425	0.0995741
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	0	0
$563$	7.16601	0.302011	0.151006	−	0.988533i	$-0.451749\pi$
$563$	7.16601	0.302011	0.151006	+	0.988533i	$0.451749\pi$
$564$	0	0
$565$	1.70850	0.0718770
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−4.64575	−0.194760	−0.0973800	−	0.995247i	$-0.531046\pi$
$569$	−4.64575	−0.194760	−0.0973800	+	0.995247i	$0.531046\pi$
$570$	0	0
$571$	−4.22876	−0.176968	−0.0884840	−	0.996078i	$-0.528202\pi$
$571$	−4.22876	−0.176968	−0.0884840	+	0.996078i	$0.528202\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−4.58301	−0.190793	−0.0953965	−	0.995439i	$-0.530412\pi$
$577$	−4.58301	−0.190793	−0.0953965	+	0.995439i	$0.530412\pi$
$578$	0	0
$579$	−11.8745	−0.493488
$580$	0	0
$581$	7.58301	0.314596
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	0.354249	0.0146464
$586$	0	0
$587$	34.1033	1.40759	0.703796	−	0.710402i	$-0.251487\pi$
$587$	34.1033	1.40759	0.703796	+	0.710402i	$0.251487\pi$
$588$	0	0
$589$	−68.8118	−2.83534
$590$	0	0
$591$	−9.87451	−0.406183
$592$	0	0
$593$	−23.7490	−0.975255	−0.487628	−	0.873052i	$-0.662138\pi$
$593$	−23.7490	−0.975255	−0.487628	+	0.873052i	$0.662138\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	0	0
$597$	2.58301	0.105715
$598$	0	0
$599$	−25.2915	−1.03338	−0.516691	−	0.856172i	$-0.672837\pi$
$599$	−25.2915	−1.03338	−0.516691	+	0.856172i	$0.672837\pi$
$600$	0	0
$601$	−36.7490	−1.49902	−0.749512	−	0.661991i	$-0.769712\pi$
$601$	−36.7490	−1.49902	−0.749512	+	0.661991i	$0.769712\pi$
$602$	0	0
$603$	8.58301	0.349527
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−2.70850	−0.109935	−0.0549673	−	0.998488i	$-0.517505\pi$
$607$	−2.70850	−0.109935	−0.0549673	+	0.998488i	$0.517505\pi$
$608$	0	0
$609$	0.645751	0.0261672
$610$	0	0
$611$	−2.45751	−0.0994203
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	−4.93725	−0.199089
$616$	0	0
$617$	−21.4170	−0.862216	−0.431108	−	0.902300i	$-0.641877\pi$
$617$	−21.4170	−0.862216	−0.431108	+	0.902300i	$0.641877\pi$
$618$	0	0
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−11.9373	−0.478256
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.29150	−0.251258
$628$	0	0
$629$	7.64575	0.304856
$630$	0	0
$631$	−5.58301	−0.222256	−0.111128	−	0.993806i	$-0.535446\pi$
$631$	−5.58301	−0.222256	−0.111128	+	0.993806i	$0.535446\pi$
$632$	0	0
$633$	−1.29150	−0.0513326
$634$	0	0
$635$	−3.35425	−0.133109
$636$	0	0
$637$	−0.354249	−0.0140358
$638$	0	0
$639$	−1.64575	−0.0651049
$640$	0	0
$641$	40.5830	1.60293	0.801466	−	0.598040i	$-0.204054\pi$
$641$	40.5830	1.60293	0.801466	+	0.598040i	$0.204054\pi$
$642$	0	0
$643$	−8.06275	−0.317964	−0.158982	−	0.987282i	$-0.550821\pi$
$643$	−8.06275	−0.317964	−0.158982	+	0.987282i	$0.550821\pi$
$644$	0	0
$645$	−6.64575	−0.261676
$646$	0	0
$647$	28.7085	1.12865	0.564324	−	0.825554i	$-0.309137\pi$
$647$	28.7085	1.12865	0.564324	+	0.825554i	$0.309137\pi$
$648$	0	0
$649$	−8.64575	−0.339375
$650$	0	0
$651$	10.9373	0.428665
$652$	0	0
$653$	−15.5830	−0.609810	−0.304905	−	0.952383i	$-0.598625\pi$
$653$	−15.5830	−0.609810	−0.304905	+	0.952383i	$0.598625\pi$
$654$	0	0
$655$	−8.35425	−0.326428
$656$	0	0
$657$	−3.29150	−0.128414
$658$	0	0
$659$	21.3542	0.831843	0.415922	−	0.909400i	$-0.363459\pi$
$659$	21.3542	0.831843	0.415922	+	0.909400i	$0.363459\pi$
$660$	0	0
$661$	21.5203	0.837041	0.418521	−	0.908207i	$-0.362549\pi$
$661$	21.5203	0.837041	0.418521	+	0.908207i	$0.362549\pi$
$662$	0	0
$663$	−0.354249	−0.0137579
$664$	0	0
$665$	−6.29150	−0.243974
$666$	0	0
$667$	−0.645751	−0.0250036
$668$	0	0
$669$	−15.2288	−0.588778
$670$	0	0
$671$	6.29150	0.242881
$672$	0	0
$673$	−2.06275	−0.0795130	−0.0397565	−	0.999209i	$-0.512658\pi$
$673$	−2.06275	−0.0795130	−0.0397565	+	0.999209i	$0.512658\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	46.7490	1.79671	0.898355	−	0.439270i	$-0.144763\pi$
$677$	46.7490	1.79671	0.898355	+	0.439270i	$0.144763\pi$
$678$	0	0
$679$	−5.35425	−0.205477
$680$	0	0
$681$	−13.0000	−0.498161
$682$	0	0
$683$	1.16601	0.0446162	0.0223081	−	0.999751i	$-0.492899\pi$
$683$	1.16601	0.0446162	0.0223081	+	0.999751i	$0.492899\pi$
$684$	0	0
$685$	13.8745	0.530118
$686$	0	0
$687$	−29.5203	−1.12627
$688$	0	0
$689$	1.06275	0.0404874
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	5.29150	0.200718
$696$	0	0
$697$	4.93725	0.187012
$698$	0	0
$699$	−5.64575	−0.213542
$700$	0	0
$701$	−7.47974	−0.282506	−0.141253	−	0.989974i	$-0.545113\pi$
$701$	−7.47974	−0.282506	−0.141253	+	0.989974i	$0.545113\pi$
$702$	0	0
$703$	48.1033	1.81425
$704$	0	0
$705$	6.93725	0.261272
$706$	0	0
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−37.4575	−1.40675	−0.703373	−	0.710821i	$-0.748324\pi$
$709$	−37.4575	−1.40675	−0.703373	+	0.710821i	$0.748324\pi$
$710$	0	0
$711$	1.64575	0.0617205
$712$	0	0
$713$	−10.9373	−0.409603
$714$	0	0
$715$	−0.354249	−0.0132481
$716$	0	0
$717$	10.5203	0.392886
$718$	0	0
$719$	1.81176	0.0675673	0.0337837	−	0.999429i	$-0.489244\pi$
$719$	1.81176	0.0675673	0.0337837	+	0.999429i	$0.489244\pi$
$720$	0	0
$721$	−14.6458	−0.545436
$722$	0	0
$723$	−26.5830	−0.988633
$724$	0	0
$725$	0.645751	0.0239826
$726$	0	0
$727$	−20.6458	−0.765709	−0.382854	−	0.923809i	$-0.625059\pi$
$727$	−20.6458	−0.765709	−0.382854	+	0.923809i	$0.625059\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.64575	0.245802
$732$	0	0
$733$	−6.70850	−0.247784	−0.123892	−	0.992296i	$-0.539538\pi$
$733$	−6.70850	−0.247784	−0.123892	+	0.992296i	$0.539538\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−8.58301	−0.316159
$738$	0	0
$739$	−34.7085	−1.27677	−0.638386	−	0.769716i	$-0.720398\pi$
$739$	−34.7085	−1.27677	−0.638386	+	0.769716i	$0.720398\pi$
$740$	0	0
$741$	−2.22876	−0.0818754
$742$	0	0
$743$	2.70850	0.0993651	0.0496826	−	0.998765i	$-0.484179\pi$
$743$	2.70850	0.0993651	0.0496826	+	0.998765i	$0.484179\pi$
$744$	0	0
$745$	−13.2915	−0.486963
$746$	0	0
$747$	−7.58301	−0.277448
$748$	0	0
$749$	−3.64575	−0.133213
$750$	0	0
$751$	−17.4575	−0.637034	−0.318517	−	0.947917i	$-0.603185\pi$
$751$	−17.4575	−0.637034	−0.318517	+	0.947917i	$0.603185\pi$
$752$	0	0
$753$	−26.5830	−0.968739
$754$	0	0
$755$	−8.58301	−0.312368
$756$	0	0
$757$	−2.45751	−0.0893198	−0.0446599	−	0.999002i	$-0.514220\pi$
$757$	−2.45751	−0.0893198	−0.0446599	+	0.999002i	$0.514220\pi$
$758$	0	0
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	48.5830	1.76113	0.880566	−	0.473923i	$-0.157163\pi$
$761$	48.5830	1.76113	0.880566	+	0.473923i	$0.157163\pi$
$762$	0	0
$763$	19.6458	0.711224
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	−3.06275	−0.110589
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	0	0
$771$	−28.9373	−1.04215
$772$	0	0
$773$	−2.45751	−0.0883906	−0.0441953	−	0.999023i	$-0.514072\pi$
$773$	−2.45751	−0.0883906	−0.0441953	+	0.999023i	$0.514072\pi$
$774$	0	0
$775$	10.9373	0.392878
$776$	0	0
$777$	−7.64575	−0.274290
$778$	0	0
$779$	31.0627	1.11294
$780$	0	0
$781$	1.64575	0.0588896
$782$	0	0
$783$	−0.645751	−0.0230773
$784$	0	0
$785$	4.64575	0.165814
$786$	0	0
$787$	17.1660	0.611902	0.305951	−	0.952047i	$-0.401026\pi$
$787$	17.1660	0.611902	0.305951	+	0.952047i	$0.401026\pi$
$788$	0	0
$789$	−7.29150	−0.259584
$790$	0	0
$791$	1.70850	0.0607472
$792$	0	0
$793$	2.22876	0.0791455
$794$	0	0
$795$	−3.00000	−0.106399
$796$	0	0
$797$	−27.8745	−0.987366	−0.493683	−	0.869642i	$-0.664350\pi$
$797$	−27.8745	−0.987366	−0.493683	+	0.869642i	$0.664350\pi$
$798$	0	0
$799$	−6.93725	−0.245422
$800$	0	0
$801$	11.9373	0.421782
$802$	0	0
$803$	3.29150	0.116155
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	−25.9373	−0.913035
$808$	0	0
$809$	53.0405	1.86481	0.932403	−	0.361421i	$-0.117708\pi$
$809$	53.0405	1.86481	0.932403	+	0.361421i	$0.117708\pi$
$810$	0	0
$811$	−10.5830	−0.371620	−0.185810	−	0.982586i	$-0.559491\pi$
$811$	−10.5830	−0.371620	−0.185810	+	0.982586i	$0.559491\pi$
$812$	0	0
$813$	−10.4170	−0.365340
$814$	0	0
$815$	2.22876	0.0780699
$816$	0	0
$817$	41.8118	1.46281
$818$	0	0
$819$	0.354249	0.0123785
$820$	0	0
$821$	2.06275	0.0719903	0.0359952	−	0.999352i	$-0.488540\pi$
$821$	2.06275	0.0719903	0.0359952	+	0.999352i	$0.488540\pi$
$822$	0	0
$823$	−14.9373	−0.520680	−0.260340	−	0.965517i	$-0.583835\pi$
$823$	−14.9373	−0.520680	−0.260340	+	0.965517i	$0.583835\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	−53.6235	−1.86467	−0.932336	−	0.361592i	$-0.882233\pi$
$827$	−53.6235	−1.86467	−0.932336	+	0.361592i	$0.882233\pi$
$828$	0	0
$829$	37.0405	1.28647	0.643235	−	0.765669i	$-0.277592\pi$
$829$	37.0405	1.28647	0.643235	+	0.765669i	$0.277592\pi$
$830$	0	0
$831$	29.1660	1.01176
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	9.29150	0.321546
$836$	0	0
$837$	−10.9373	−0.378047
$838$	0	0
$839$	−39.2288	−1.35433	−0.677164	−	0.735833i	$-0.736791\pi$
$839$	−39.2288	−1.35433	−0.677164	+	0.735833i	$0.736791\pi$
$840$	0	0
$841$	−28.5830	−0.985621
$842$	0	0
$843$	6.58301	0.226731
$844$	0	0
$845$	12.8745	0.442897
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	30.2288	1.03745
$850$	0	0
$851$	7.64575	0.262093
$852$	0	0
$853$	−52.9373	−1.81254	−0.906269	−	0.422702i	$-0.861082\pi$
$853$	−52.9373	−1.81254	−0.906269	+	0.422702i	$0.861082\pi$
$854$	0	0
$855$	6.29150	0.215165
$856$	0	0
$857$	−31.1660	−1.06461	−0.532305	−	0.846552i	$-0.678674\pi$
$857$	−31.1660	−1.06461	−0.532305	+	0.846552i	$0.678674\pi$
$858$	0	0
$859$	−45.5203	−1.55313	−0.776566	−	0.630036i	$-0.783040\pi$
$859$	−45.5203	−1.55313	−0.776566	+	0.630036i	$0.783040\pi$
$860$	0	0
$861$	−4.93725	−0.168261
$862$	0	0
$863$	9.00000	0.306364	0.153182	−	0.988198i	$-0.451048\pi$
$863$	9.00000	0.306364	0.153182	+	0.988198i	$0.451048\pi$
$864$	0	0
$865$	−21.8745	−0.743756
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	−1.64575	−0.0558283
$870$	0	0
$871$	−3.04052	−0.103024
$872$	0	0
$873$	5.35425	0.181214
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−16.7712	−0.566325	−0.283162	−	0.959072i	$-0.591384\pi$
$877$	−16.7712	−0.566325	−0.283162	+	0.959072i	$0.591384\pi$
$878$	0	0
$879$	−0.291503	−0.00983214
$880$	0	0
$881$	10.0627	0.339023	0.169511	−	0.985528i	$-0.445781\pi$
$881$	10.0627	0.339023	0.169511	+	0.985528i	$0.445781\pi$
$882$	0	0
$883$	−37.8745	−1.27458	−0.637289	−	0.770625i	$-0.719944\pi$
$883$	−37.8745	−1.27458	−0.637289	+	0.770625i	$0.719944\pi$
$884$	0	0
$885$	8.64575	0.290624
$886$	0	0
$887$	−20.1660	−0.677108	−0.338554	−	0.940947i	$-0.609938\pi$
$887$	−20.1660	−0.677108	−0.338554	+	0.940947i	$0.609938\pi$
$888$	0	0
$889$	−3.35425	−0.112498
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−43.6458	−1.46055
$894$	0	0
$895$	−6.35425	−0.212399
$896$	0	0
$897$	−0.354249	−0.0118280
$898$	0	0
$899$	7.06275	0.235556
$900$	0	0
$901$	3.00000	0.0999445
$902$	0	0
$903$	−6.64575	−0.221157
$904$	0	0
$905$	−4.58301	−0.152344
$906$	0	0
$907$	−2.47974	−0.0823384	−0.0411692	−	0.999152i	$-0.513108\pi$
$907$	−2.47974	−0.0823384	−0.0411692	+	0.999152i	$0.513108\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−21.8745	−0.724735	−0.362367	−	0.932035i	$-0.618031\pi$
$911$	−21.8745	−0.724735	−0.362367	+	0.932035i	$0.618031\pi$
$912$	0	0
$913$	7.58301	0.250961
$914$	0	0
$915$	−6.29150	−0.207991
$916$	0	0
$917$	−8.35425	−0.275882
$918$	0	0
$919$	−13.7490	−0.453538	−0.226769	−	0.973949i	$-0.572816\pi$
$919$	−13.7490	−0.453538	−0.226769	+	0.973949i	$0.572816\pi$
$920$	0	0
$921$	−6.70850	−0.221053
$922$	0	0
$923$	0.583005	0.0191899
$924$	0	0
$925$	−7.64575	−0.251391
$926$	0	0
$927$	14.6458	0.481030
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−6.29150	−0.206196
$932$	0	0
$933$	−4.70850	−0.154149
$934$	0	0
$935$	−1.00000	−0.0327035
$936$	0	0
$937$	44.6863	1.45984	0.729918	−	0.683534i	$-0.239558\pi$
$937$	44.6863	1.45984	0.729918	+	0.683534i	$0.239558\pi$
$938$	0	0
$939$	23.8118	0.777067
$940$	0	0
$941$	−60.1033	−1.95931	−0.979655	−	0.200688i	$-0.935682\pi$
$941$	−60.1033	−1.95931	−0.979655	+	0.200688i	$0.935682\pi$
$942$	0	0
$943$	4.93725	0.160779
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	6.29150	0.204446	0.102223	−	0.994761i	$-0.467404\pi$
$947$	6.29150	0.204446	0.102223	+	0.994761i	$0.467404\pi$
$948$	0	0
$949$	1.16601	0.0378503
$950$	0	0
$951$	−5.41699	−0.175658
$952$	0	0
$953$	−11.1660	−0.361703	−0.180851	−	0.983510i	$-0.557885\pi$
$953$	−11.1660	−0.361703	−0.180851	+	0.983510i	$0.557885\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	0	0
$957$	0.645751	0.0208742
$958$	0	0
$959$	13.8745	0.448031
$960$	0	0
$961$	88.6235	2.85882
$962$	0	0
$963$	3.64575	0.117483
$964$	0	0
$965$	−11.8745	−0.382254
$966$	0	0
$967$	13.1033	0.421373	0.210686	−	0.977554i	$-0.432430\pi$
$967$	13.1033	0.421373	0.210686	+	0.977554i	$0.432430\pi$
$968$	0	0
$969$	−6.29150	−0.202112
$970$	0	0
$971$	17.2288	0.552897	0.276449	−	0.961029i	$-0.410842\pi$
$971$	17.2288	0.552897	0.276449	+	0.961029i	$0.410842\pi$
$972$	0	0
$973$	5.29150	0.169638
$974$	0	0
$975$	0.354249	0.0113450
$976$	0	0
$977$	3.58301	0.114630	0.0573152	−	0.998356i	$-0.481746\pi$
$977$	3.58301	0.114630	0.0573152	+	0.998356i	$0.481746\pi$
$978$	0	0
$979$	−11.9373	−0.381516
$980$	0	0
$981$	−19.6458	−0.627241
$982$	0	0
$983$	7.87451	0.251158	0.125579	−	0.992084i	$-0.459921\pi$
$983$	7.87451	0.251158	0.125579	+	0.992084i	$0.459921\pi$
$984$	0	0
$985$	−9.87451	−0.314628
$986$	0	0
$987$	6.93725	0.220815
$988$	0	0
$989$	6.64575	0.211323
$990$	0	0
$991$	−22.7490	−0.722646	−0.361323	−	0.932441i	$-0.617675\pi$
$991$	−22.7490	−0.722646	−0.361323	+	0.932441i	$0.617675\pi$
$992$	0	0
$993$	−15.7085	−0.498494
$994$	0	0
$995$	2.58301	0.0818868
$996$	0	0
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	0	0
$999$	7.64575	0.241901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.o.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.o.1.2	✓	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -0.354249 q^{13} +1.00000 q^{15} -1.00000 q^{17} -6.29150 q^{19} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.645751 q^{29} +10.9373 q^{31} +1.00000 q^{33} +1.00000 q^{35} -7.64575 q^{37} +0.354249 q^{39} -4.93725 q^{41} -6.64575 q^{43} -1.00000 q^{45} +6.93725 q^{47} +1.00000 q^{49} +1.00000 q^{51} -3.00000 q^{53} +1.00000 q^{55} +6.29150 q^{57} +8.64575 q^{59} -6.29150 q^{61} -1.00000 q^{63} +0.354249 q^{65} +8.58301 q^{67} +1.00000 q^{69} -1.64575 q^{71} -3.29150 q^{73} -1.00000 q^{75} +1.00000 q^{77} +1.64575 q^{79} +1.00000 q^{81} -7.58301 q^{83} +1.00000 q^{85} -0.645751 q^{87} +11.9373 q^{89} +0.354249 q^{91} -10.9373 q^{93} +6.29150 q^{95} +5.35425 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 6 q^{13} + 2 q^{15} - 2 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 2 q^{35} - 10 q^{37} + 6 q^{39} + 6 q^{41} - 8 q^{43} - 2 q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} - 6 q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} - 2 q^{61} - 2 q^{63} + 6 q^{65} - 4 q^{67} + 2 q^{69} + 2 q^{71} + 4 q^{73} - 2 q^{75} + 2 q^{77} - 2 q^{79} + 2 q^{81} + 6 q^{83} + 2 q^{85} + 4 q^{87} + 8 q^{89} + 6 q^{91} - 6 q^{93} + 2 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - 6 * q^13 + 2 * q^15 - 2 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 + 6 * q^31 + 2 * q^33 + 2 * q^35 - 10 * q^37 + 6 * q^39 + 6 * q^41 - 8 * q^43 - 2 * q^45 - 2 * q^47 + 2 * q^49 + 2 * q^51 - 6 * q^53 + 2 * q^55 + 2 * q^57 + 12 * q^59 - 2 * q^61 - 2 * q^63 + 6 * q^65 - 4 * q^67 + 2 * q^69 + 2 * q^71 + 4 * q^73 - 2 * q^75 + 2 * q^77 - 2 * q^79 + 2 * q^81 + 6 * q^83 + 2 * q^85 + 4 * q^87 + 8 * q^89 + 6 * q^91 - 6 * q^93 + 2 * q^95 + 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more