LMFDB - Embedded newform 6024.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 43 x^{12} + 119 x^{11} + 679 x^{10} - 1667 x^{9} - 4890 x^{8} + 9662 x^{7} + \cdots - 416 $ x^14 - 3x^13 - 43x^12 + 119x^11 + 679x^10 - 1667x^9 - 4890x^8 + 9662x^7 + 16575x^6 - 20277x^5 - 25196x^4 + 8040x^3 + 10776x^2 + 912*x - 416
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.38301$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.38301 q^{5} +0.798768 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.38301 q^{5} +0.798768 q^{7} +1.00000 q^{9} -0.995645 q^{11} +0.108967 q^{13} +4.38301 q^{15} -4.71468 q^{17} +2.17507 q^{19} -0.798768 q^{21} -0.209665 q^{23} +14.2108 q^{25} -1.00000 q^{27} +1.12306 q^{29} +6.28610 q^{31} +0.995645 q^{33} -3.50101 q^{35} -11.2066 q^{37} -0.108967 q^{39} -4.79289 q^{41} +4.40594 q^{43} -4.38301 q^{45} +0.227577 q^{47} -6.36197 q^{49} +4.71468 q^{51} +10.8381 q^{53} +4.36392 q^{55} -2.17507 q^{57} -0.194921 q^{59} +10.8382 q^{61} +0.798768 q^{63} -0.477604 q^{65} +9.06537 q^{67} +0.209665 q^{69} -0.886650 q^{71} +16.0380 q^{73} -14.2108 q^{75} -0.795290 q^{77} +13.7624 q^{79} +1.00000 q^{81} -3.64179 q^{83} +20.6645 q^{85} -1.12306 q^{87} +8.58888 q^{89} +0.0870395 q^{91} -6.28610 q^{93} -9.53336 q^{95} -17.5455 q^{97} -0.995645 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9} - 22 q^{11} + 9 q^{13} + 3 q^{15} - 7 q^{21} - 17 q^{23} + 25 q^{25} - 14 q^{27} - 18 q^{29} - 7 q^{31} + 22 q^{33} - 27 q^{35} + 9 q^{37} - 9 q^{39} - 6 q^{41} - 14 q^{43} - 3 q^{45} - 15 q^{47} + 15 q^{49} - 11 q^{53} + 2 q^{55} - 36 q^{59} + 2 q^{61} + 7 q^{63} + 8 q^{65} + 3 q^{67} + 17 q^{69} - 29 q^{71} + 2 q^{73} - 25 q^{75} + 8 q^{77} + 23 q^{79} + 14 q^{81} - 55 q^{83} + 7 q^{85} + 18 q^{87} + 9 q^{89} - 22 q^{91} + 7 q^{93} - 27 q^{95} + 17 q^{97} - 22 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9 - 22 * q^11 + 9 * q^13 + 3 * q^15 - 7 * q^21 - 17 * q^23 + 25 * q^25 - 14 * q^27 - 18 * q^29 - 7 * q^31 + 22 * q^33 - 27 * q^35 + 9 * q^37 - 9 * q^39 - 6 * q^41 - 14 * q^43 - 3 * q^45 - 15 * q^47 + 15 * q^49 - 11 * q^53 + 2 * q^55 - 36 * q^59 + 2 * q^61 + 7 * q^63 + 8 * q^65 + 3 * q^67 + 17 * q^69 - 29 * q^71 + 2 * q^73 - 25 * q^75 + 8 * q^77 + 23 * q^79 + 14 * q^81 - 55 * q^83 + 7 * q^85 + 18 * q^87 + 9 * q^89 - 22 * q^91 + 7 * q^93 - 27 * q^95 + 17 * q^97 - 22 * 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
$5$	−4.38301	−1.96014	−0.980070	−	0.198650i	$-0.936344\pi$
$5$	−4.38301	−1.96014	−0.980070	+	0.198650i	$0.936344\pi$
$6$	0	0
$7$	0.798768	0.301906	0.150953	−	0.988541i	$-0.451766\pi$
$7$	0.798768	0.301906	0.150953	+	0.988541i	$0.451766\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.995645	−0.300198	−0.150099	−	0.988671i	$-0.547959\pi$
$11$	−0.995645	−0.300198	−0.150099	+	0.988671i	$0.547959\pi$
$12$	0	0
$13$	0.108967	0.0302220	0.0151110	−	0.999886i	$-0.495190\pi$
$13$	0.108967	0.0302220	0.0151110	+	0.999886i	$0.495190\pi$
$14$	0	0
$15$	4.38301	1.13169
$16$	0	0
$17$	−4.71468	−1.14348	−0.571739	−	0.820436i	$-0.693731\pi$
$17$	−4.71468	−1.14348	−0.571739	+	0.820436i	$0.693731\pi$
$18$	0	0
$19$	2.17507	0.498996	0.249498	−	0.968375i	$-0.419734\pi$
$19$	2.17507	0.498996	0.249498	+	0.968375i	$0.419734\pi$
$20$	0	0
$21$	−0.798768	−0.174306
$22$	0	0
$23$	−0.209665	−0.0437182	−0.0218591	−	0.999761i	$-0.506959\pi$
$23$	−0.209665	−0.0437182	−0.0218591	+	0.999761i	$0.506959\pi$
$24$	0	0
$25$	14.2108	2.84215
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.12306	0.208548	0.104274	−	0.994549i	$-0.466748\pi$
$29$	1.12306	0.208548	0.104274	+	0.994549i	$0.466748\pi$
$30$	0	0
$31$	6.28610	1.12902	0.564508	−	0.825427i	$-0.309066\pi$
$31$	6.28610	1.12902	0.564508	+	0.825427i	$0.309066\pi$
$32$	0	0
$33$	0.995645	0.173320
$34$	0	0
$35$	−3.50101	−0.591778
$36$	0	0
$37$	−11.2066	−1.84236	−0.921179	−	0.389138i	$-0.872773\pi$
$37$	−11.2066	−1.84236	−0.921179	+	0.389138i	$0.872773\pi$
$38$	0	0
$39$	−0.108967	−0.0174487
$40$	0	0
$41$	−4.79289	−0.748524	−0.374262	−	0.927323i	$-0.622104\pi$
$41$	−4.79289	−0.748524	−0.374262	+	0.927323i	$0.622104\pi$
$42$	0	0
$43$	4.40594	0.671900	0.335950	−	0.941880i	$-0.390943\pi$
$43$	4.40594	0.671900	0.335950	+	0.941880i	$0.390943\pi$
$44$	0	0
$45$	−4.38301	−0.653380
$46$	0	0
$47$	0.227577	0.0331955	0.0165978	−	0.999862i	$-0.494717\pi$
$47$	0.227577	0.0331955	0.0165978	+	0.999862i	$0.494717\pi$
$48$	0	0
$49$	−6.36197	−0.908853
$50$	0	0
$51$	4.71468	0.660187
$52$	0	0
$53$	10.8381	1.48873	0.744366	−	0.667772i	$-0.232752\pi$
$53$	10.8381	1.48873	0.744366	+	0.667772i	$0.232752\pi$
$54$	0	0
$55$	4.36392	0.588431
$56$	0	0
$57$	−2.17507	−0.288095
$58$	0	0
$59$	−0.194921	−0.0253766	−0.0126883	−	0.999920i	$-0.504039\pi$
$59$	−0.194921	−0.0253766	−0.0126883	+	0.999920i	$0.504039\pi$
$60$	0	0
$61$	10.8382	1.38769	0.693844	−	0.720125i	$-0.255916\pi$
$61$	10.8382	1.38769	0.693844	+	0.720125i	$0.255916\pi$
$62$	0	0
$63$	0.798768	0.100635
$64$	0	0
$65$	−0.477604	−0.0592395
$66$	0	0
$67$	9.06537	1.10751	0.553756	−	0.832679i	$-0.313194\pi$
$67$	9.06537	1.10751	0.553756	+	0.832679i	$0.313194\pi$
$68$	0	0
$69$	0.209665	0.0252407
$70$	0	0
$71$	−0.886650	−0.105226	−0.0526130	−	0.998615i	$-0.516755\pi$
$71$	−0.886650	−0.105226	−0.0526130	+	0.998615i	$0.516755\pi$
$72$	0	0
$73$	16.0380	1.87710	0.938552	−	0.345139i	$-0.112168\pi$
$73$	16.0380	1.87710	0.938552	+	0.345139i	$0.112168\pi$
$74$	0	0
$75$	−14.2108	−1.64092
$76$	0	0
$77$	−0.795290	−0.0906317
$78$	0	0
$79$	13.7624	1.54839	0.774194	−	0.632949i	$-0.218156\pi$
$79$	13.7624	1.54839	0.774194	+	0.632949i	$0.218156\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.64179	−0.399738	−0.199869	−	0.979823i	$-0.564052\pi$
$83$	−3.64179	−0.399738	−0.199869	+	0.979823i	$0.564052\pi$
$84$	0	0
$85$	20.6645	2.24138
$86$	0	0
$87$	−1.12306	−0.120405
$88$	0	0
$89$	8.58888	0.910420	0.455210	−	0.890384i	$-0.349564\pi$
$89$	8.58888	0.910420	0.455210	+	0.890384i	$0.349564\pi$
$90$	0	0
$91$	0.0870395	0.00912422
$92$	0	0
$93$	−6.28610	−0.651838
$94$	0	0
$95$	−9.53336	−0.978102
$96$	0	0
$97$	−17.5455	−1.78148	−0.890739	−	0.454516i	$-0.849812\pi$
$97$	−17.5455	−1.78148	−0.890739	+	0.454516i	$0.849812\pi$
$98$	0	0
$99$	−0.995645	−0.100066
$100$	0	0
$101$	−0.0727733	−0.00724121	−0.00362061	−	0.999993i	$-0.501152\pi$
$101$	−0.0727733	−0.00724121	−0.00362061	+	0.999993i	$0.501152\pi$
$102$	0	0
$103$	−9.33792	−0.920092	−0.460046	−	0.887895i	$-0.652167\pi$
$103$	−9.33792	−0.920092	−0.460046	+	0.887895i	$0.652167\pi$
$104$	0	0
$105$	3.50101	0.341663
$106$	0	0
$107$	1.38323	0.133722	0.0668609	−	0.997762i	$-0.478702\pi$
$107$	1.38323	0.133722	0.0668609	+	0.997762i	$0.478702\pi$
$108$	0	0
$109$	−15.2580	−1.46146	−0.730728	−	0.682669i	$-0.760819\pi$
$109$	−15.2580	−1.46146	−0.730728	+	0.682669i	$0.760819\pi$
$110$	0	0
$111$	11.2066	1.06369
$112$	0	0
$113$	12.0973	1.13802	0.569008	−	0.822332i	$-0.307327\pi$
$113$	12.0973	1.13802	0.569008	+	0.822332i	$0.307327\pi$
$114$	0	0
$115$	0.918963	0.0856938
$116$	0	0
$117$	0.108967	0.0100740
$118$	0	0
$119$	−3.76594	−0.345223
$120$	0	0
$121$	−10.0087	−0.909881
$122$	0	0
$123$	4.79289	0.432160
$124$	0	0
$125$	−40.3708	−3.61088
$126$	0	0
$127$	16.6443	1.47694	0.738469	−	0.674287i	$-0.235549\pi$
$127$	16.6443	1.47694	0.738469	+	0.674287i	$0.235549\pi$
$128$	0	0
$129$	−4.40594	−0.387922
$130$	0	0
$131$	−17.0119	−1.48633	−0.743167	−	0.669106i	$-0.766677\pi$
$131$	−17.0119	−1.48633	−0.743167	+	0.669106i	$0.766677\pi$
$132$	0	0
$133$	1.73738	0.150650
$134$	0	0
$135$	4.38301	0.377229
$136$	0	0
$137$	1.66007	0.141829	0.0709145	−	0.997482i	$-0.477408\pi$
$137$	1.66007	0.141829	0.0709145	+	0.997482i	$0.477408\pi$
$138$	0	0
$139$	−19.0158	−1.61290	−0.806448	−	0.591306i	$-0.798613\pi$
$139$	−19.0158	−1.61290	−0.806448	+	0.591306i	$0.798613\pi$
$140$	0	0
$141$	−0.227577	−0.0191654
$142$	0	0
$143$	−0.108493	−0.00907260
$144$	0	0
$145$	−4.92240	−0.408783
$146$	0	0
$147$	6.36197	0.524726
$148$	0	0
$149$	−3.64658	−0.298739	−0.149370	−	0.988781i	$-0.547724\pi$
$149$	−3.64658	−0.298739	−0.149370	+	0.988781i	$0.547724\pi$
$150$	0	0
$151$	14.7690	1.20188	0.600942	−	0.799292i	$-0.294792\pi$
$151$	14.7690	1.20188	0.600942	+	0.799292i	$0.294792\pi$
$152$	0	0
$153$	−4.71468	−0.381159
$154$	0	0
$155$	−27.5520	−2.21303
$156$	0	0
$157$	−1.74500	−0.139266	−0.0696330	−	0.997573i	$-0.522183\pi$
$157$	−1.74500	−0.139266	−0.0696330	+	0.997573i	$0.522183\pi$
$158$	0	0
$159$	−10.8381	−0.859520
$160$	0	0
$161$	−0.167474	−0.0131988
$162$	0	0
$163$	−2.35561	−0.184505	−0.0922526	−	0.995736i	$-0.529407\pi$
$163$	−2.35561	−0.184505	−0.0922526	+	0.995736i	$0.529407\pi$
$164$	0	0
$165$	−4.36392	−0.339731
$166$	0	0
$167$	13.0719	1.01153	0.505766	−	0.862671i	$-0.331210\pi$
$167$	13.0719	1.01153	0.505766	+	0.862671i	$0.331210\pi$
$168$	0	0
$169$	−12.9881	−0.999087
$170$	0	0
$171$	2.17507	0.166332
$172$	0	0
$173$	−5.27078	−0.400730	−0.200365	−	0.979721i	$-0.564213\pi$
$173$	−5.27078	−0.400730	−0.200365	+	0.979721i	$0.564213\pi$
$174$	0	0
$175$	11.3511	0.858063
$176$	0	0
$177$	0.194921	0.0146512
$178$	0	0
$179$	−26.4293	−1.97542	−0.987708	−	0.156313i	$-0.950039\pi$
$179$	−26.4293	−1.97542	−0.987708	+	0.156313i	$0.950039\pi$
$180$	0	0
$181$	−8.14733	−0.605586	−0.302793	−	0.953056i	$-0.597919\pi$
$181$	−8.14733	−0.605586	−0.302793	+	0.953056i	$0.597919\pi$
$182$	0	0
$183$	−10.8382	−0.801182
$184$	0	0
$185$	49.1188	3.61128
$186$	0	0
$187$	4.69415	0.343270
$188$	0	0
$189$	−0.798768	−0.0581019
$190$	0	0
$191$	22.1131	1.60005	0.800024	−	0.599968i	$-0.204820\pi$
$191$	22.1131	1.60005	0.800024	+	0.599968i	$0.204820\pi$
$192$	0	0
$193$	11.4091	0.821246	0.410623	−	0.911805i	$-0.365311\pi$
$193$	11.4091	0.821246	0.410623	+	0.911805i	$0.365311\pi$
$194$	0	0
$195$	0.477604	0.0342019
$196$	0	0
$197$	−5.66936	−0.403925	−0.201963	−	0.979393i	$-0.564732\pi$
$197$	−5.66936	−0.403925	−0.201963	+	0.979393i	$0.564732\pi$
$198$	0	0
$199$	−17.2548	−1.22316	−0.611578	−	0.791184i	$-0.709465\pi$
$199$	−17.2548	−1.22316	−0.611578	+	0.791184i	$0.709465\pi$
$200$	0	0
$201$	−9.06537	−0.639422
$202$	0	0
$203$	0.897067	0.0629618
$204$	0	0
$205$	21.0073	1.46721
$206$	0	0
$207$	−0.209665	−0.0145727
$208$	0	0
$209$	−2.16560	−0.149798
$210$	0	0
$211$	−16.4330	−1.13129	−0.565647	−	0.824648i	$-0.691373\pi$
$211$	−16.4330	−1.13129	−0.565647	+	0.824648i	$0.691373\pi$
$212$	0	0
$213$	0.886650	0.0607523
$214$	0	0
$215$	−19.3113	−1.31702
$216$	0	0
$217$	5.02114	0.340857
$218$	0	0
$219$	−16.0380	−1.08375
$220$	0	0
$221$	−0.513745	−0.0345582
$222$	0	0
$223$	7.81174	0.523113	0.261556	−	0.965188i	$-0.415764\pi$
$223$	7.81174	0.523113	0.261556	+	0.965188i	$0.415764\pi$
$224$	0	0
$225$	14.2108	0.947384
$226$	0	0
$227$	−13.8883	−0.921800	−0.460900	−	0.887452i	$-0.652473\pi$
$227$	−13.8883	−0.921800	−0.460900	+	0.887452i	$0.652473\pi$
$228$	0	0
$229$	−0.383732	−0.0253577	−0.0126789	−	0.999920i	$-0.504036\pi$
$229$	−0.383732	−0.0253577	−0.0126789	+	0.999920i	$0.504036\pi$
$230$	0	0
$231$	0.795290	0.0523262
$232$	0	0
$233$	10.4946	0.687527	0.343763	−	0.939056i	$-0.388298\pi$
$233$	10.4946	0.687527	0.343763	+	0.939056i	$0.388298\pi$
$234$	0	0
$235$	−0.997471	−0.0650679
$236$	0	0
$237$	−13.7624	−0.893962
$238$	0	0
$239$	−8.45576	−0.546958	−0.273479	−	0.961878i	$-0.588174\pi$
$239$	−8.45576	−0.546958	−0.273479	+	0.961878i	$0.588174\pi$
$240$	0	0
$241$	−4.65783	−0.300037	−0.150019	−	0.988683i	$-0.547933\pi$
$241$	−4.65783	−0.300037	−0.150019	+	0.988683i	$0.547933\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	27.8846	1.78148
$246$	0	0
$247$	0.237011	0.0150807
$248$	0	0
$249$	3.64179	0.230789
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	0.208752	0.0131241
$254$	0	0
$255$	−20.6645	−1.29406
$256$	0	0
$257$	16.1227	1.00570	0.502852	−	0.864372i	$-0.332284\pi$
$257$	16.1227	1.00570	0.502852	+	0.864372i	$0.332284\pi$
$258$	0	0
$259$	−8.95150	−0.556219
$260$	0	0
$261$	1.12306	0.0695159
$262$	0	0
$263$	−15.4603	−0.953323	−0.476661	−	0.879087i	$-0.658153\pi$
$263$	−15.4603	−0.953323	−0.476661	+	0.879087i	$0.658153\pi$
$264$	0	0
$265$	−47.5036	−2.91812
$266$	0	0
$267$	−8.58888	−0.525631
$268$	0	0
$269$	−5.32049	−0.324396	−0.162198	−	0.986758i	$-0.551858\pi$
$269$	−5.32049	−0.324396	−0.162198	+	0.986758i	$0.551858\pi$
$270$	0	0
$271$	−18.4567	−1.12117	−0.560583	−	0.828098i	$-0.689423\pi$
$271$	−18.4567	−1.12117	−0.560583	+	0.828098i	$0.689423\pi$
$272$	0	0
$273$	−0.0870395	−0.00526787
$274$	0	0
$275$	−14.1489	−0.853209
$276$	0	0
$277$	13.3252	0.800634	0.400317	−	0.916377i	$-0.368900\pi$
$277$	13.3252	0.800634	0.400317	+	0.916377i	$0.368900\pi$
$278$	0	0
$279$	6.28610	0.376339
$280$	0	0
$281$	−3.45059	−0.205845	−0.102922	−	0.994689i	$-0.532819\pi$
$281$	−3.45059	−0.205845	−0.102922	+	0.994689i	$0.532819\pi$
$282$	0	0
$283$	2.97544	0.176872	0.0884359	−	0.996082i	$-0.471813\pi$
$283$	2.97544	0.176872	0.0884359	+	0.996082i	$0.471813\pi$
$284$	0	0
$285$	9.53336	0.564708
$286$	0	0
$287$	−3.82841	−0.225984
$288$	0	0
$289$	5.22820	0.307541
$290$	0	0
$291$	17.5455	1.02854
$292$	0	0
$293$	−13.6692	−0.798565	−0.399282	−	0.916828i	$-0.630741\pi$
$293$	−13.6692	−0.798565	−0.399282	+	0.916828i	$0.630741\pi$
$294$	0	0
$295$	0.854341	0.0497417
$296$	0	0
$297$	0.995645	0.0577732
$298$	0	0
$299$	−0.0228466	−0.00132125
$300$	0	0
$301$	3.51933	0.202851
$302$	0	0
$303$	0.0727733	0.00418072
$304$	0	0
$305$	−47.5039	−2.72006
$306$	0	0
$307$	−23.8513	−1.36126	−0.680631	−	0.732626i	$-0.738294\pi$
$307$	−23.8513	−1.36126	−0.680631	+	0.732626i	$0.738294\pi$
$308$	0	0
$309$	9.33792	0.531216
$310$	0	0
$311$	−29.0572	−1.64768	−0.823841	−	0.566821i	$-0.808173\pi$
$311$	−29.0572	−1.64768	−0.823841	+	0.566821i	$0.808173\pi$
$312$	0	0
$313$	8.05336	0.455203	0.227601	−	0.973754i	$-0.426912\pi$
$313$	8.05336	0.455203	0.227601	+	0.973754i	$0.426912\pi$
$314$	0	0
$315$	−3.50101	−0.197259
$316$	0	0
$317$	−15.1824	−0.852726	−0.426363	−	0.904552i	$-0.640205\pi$
$317$	−15.1824	−0.852726	−0.426363	+	0.904552i	$0.640205\pi$
$318$	0	0
$319$	−1.11817	−0.0626056
$320$	0	0
$321$	−1.38323	−0.0772043
$322$	0	0
$323$	−10.2548	−0.570591
$324$	0	0
$325$	1.54851	0.0858957
$326$	0	0
$327$	15.2580	0.843772
$328$	0	0
$329$	0.181781	0.0100219
$330$	0	0
$331$	6.57908	0.361619	0.180809	−	0.983518i	$-0.442128\pi$
$331$	6.57908	0.361619	0.180809	+	0.983518i	$0.442128\pi$
$332$	0	0
$333$	−11.2066	−0.614120
$334$	0	0
$335$	−39.7336	−2.17088
$336$	0	0
$337$	−15.8822	−0.865160	−0.432580	−	0.901595i	$-0.642397\pi$
$337$	−15.8822	−0.865160	−0.432580	+	0.901595i	$0.642397\pi$
$338$	0	0
$339$	−12.0973	−0.657033
$340$	0	0
$341$	−6.25872	−0.338929
$342$	0	0
$343$	−10.6731	−0.576294
$344$	0	0
$345$	−0.918963	−0.0494753
$346$	0	0
$347$	−25.5444	−1.37130	−0.685649	−	0.727932i	$-0.740481\pi$
$347$	−25.5444	−1.37130	−0.685649	+	0.727932i	$0.740481\pi$
$348$	0	0
$349$	−11.7374	−0.628286	−0.314143	−	0.949376i	$-0.601717\pi$
$349$	−11.7374	−0.628286	−0.314143	+	0.949376i	$0.601717\pi$
$350$	0	0
$351$	−0.108967	−0.00581623
$352$	0	0
$353$	6.46582	0.344141	0.172070	−	0.985085i	$-0.444954\pi$
$353$	6.46582	0.344141	0.172070	+	0.985085i	$0.444954\pi$
$354$	0	0
$355$	3.88620	0.206258
$356$	0	0
$357$	3.76594	0.199314
$358$	0	0
$359$	6.01710	0.317570	0.158785	−	0.987313i	$-0.449242\pi$
$359$	6.01710	0.317570	0.158785	+	0.987313i	$0.449242\pi$
$360$	0	0
$361$	−14.2691	−0.751003
$362$	0	0
$363$	10.0087	0.525320
$364$	0	0
$365$	−70.2946	−3.67939
$366$	0	0
$367$	16.5713	0.865017	0.432508	−	0.901630i	$-0.357629\pi$
$367$	16.5713	0.865017	0.432508	+	0.901630i	$0.357629\pi$
$368$	0	0
$369$	−4.79289	−0.249508
$370$	0	0
$371$	8.65716	0.449457
$372$	0	0
$373$	17.3738	0.899579	0.449790	−	0.893135i	$-0.351499\pi$
$373$	17.3738	0.899579	0.449790	+	0.893135i	$0.351499\pi$
$374$	0	0
$375$	40.3708	2.08474
$376$	0	0
$377$	0.122377	0.00630273
$378$	0	0
$379$	−25.0985	−1.28922	−0.644612	−	0.764510i	$-0.722981\pi$
$379$	−25.0985	−1.28922	−0.644612	+	0.764510i	$0.722981\pi$
$380$	0	0
$381$	−16.6443	−0.852711
$382$	0	0
$383$	−34.6802	−1.77208	−0.886038	−	0.463613i	$-0.846553\pi$
$383$	−34.6802	−1.77208	−0.886038	+	0.463613i	$0.846553\pi$
$384$	0	0
$385$	3.48576	0.177651
$386$	0	0
$387$	4.40594	0.223967
$388$	0	0
$389$	10.6785	0.541421	0.270710	−	0.962661i	$-0.412741\pi$
$389$	10.6785	0.541421	0.270710	+	0.962661i	$0.412741\pi$
$390$	0	0
$391$	0.988503	0.0499907
$392$	0	0
$393$	17.0119	0.858135
$394$	0	0
$395$	−60.3206	−3.03506
$396$	0	0
$397$	6.11219	0.306762	0.153381	−	0.988167i	$-0.450984\pi$
$397$	6.11219	0.306762	0.153381	+	0.988167i	$0.450984\pi$
$398$	0	0
$399$	−1.73738	−0.0869778
$400$	0	0
$401$	20.0054	0.999023	0.499512	−	0.866307i	$-0.333513\pi$
$401$	20.0054	0.999023	0.499512	+	0.866307i	$0.333513\pi$
$402$	0	0
$403$	0.684978	0.0341212
$404$	0	0
$405$	−4.38301	−0.217793
$406$	0	0
$407$	11.1578	0.553073
$408$	0	0
$409$	30.6910	1.51757	0.758787	−	0.651338i	$-0.225792\pi$
$409$	30.6910	1.51757	0.758787	+	0.651338i	$0.225792\pi$
$410$	0	0
$411$	−1.66007	−0.0818850
$412$	0	0
$413$	−0.155697	−0.00766135
$414$	0	0
$415$	15.9620	0.783544
$416$	0	0
$417$	19.0158	0.931205
$418$	0	0
$419$	10.9668	0.535764	0.267882	−	0.963452i	$-0.413676\pi$
$419$	10.9668	0.535764	0.267882	+	0.963452i	$0.413676\pi$
$420$	0	0
$421$	39.1677	1.90892	0.954459	−	0.298343i	$-0.0964340\pi$
$421$	39.1677	1.90892	0.954459	+	0.298343i	$0.0964340\pi$
$422$	0	0
$423$	0.227577	0.0110652
$424$	0	0
$425$	−66.9992	−3.24994
$426$	0	0
$427$	8.65720	0.418951
$428$	0	0
$429$	0.108493	0.00523807
$430$	0	0
$431$	−24.7697	−1.19312	−0.596558	−	0.802570i	$-0.703466\pi$
$431$	−24.7697	−1.19312	−0.596558	+	0.802570i	$0.703466\pi$
$432$	0	0
$433$	32.4735	1.56058	0.780288	−	0.625420i	$-0.215072\pi$
$433$	32.4735	1.56058	0.780288	+	0.625420i	$0.215072\pi$
$434$	0	0
$435$	4.92240	0.236011
$436$	0	0
$437$	−0.456037	−0.0218152
$438$	0	0
$439$	−15.1198	−0.721628	−0.360814	−	0.932638i	$-0.617501\pi$
$439$	−15.1198	−0.721628	−0.360814	+	0.932638i	$0.617501\pi$
$440$	0	0
$441$	−6.36197	−0.302951
$442$	0	0
$443$	24.5377	1.16582	0.582911	−	0.812536i	$-0.301914\pi$
$443$	24.5377	1.16582	0.582911	+	0.812536i	$0.301914\pi$
$444$	0	0
$445$	−37.6451	−1.78455
$446$	0	0
$447$	3.64658	0.172477
$448$	0	0
$449$	22.0947	1.04271	0.521357	−	0.853339i	$-0.325426\pi$
$449$	22.0947	1.04271	0.521357	+	0.853339i	$0.325426\pi$
$450$	0	0
$451$	4.77202	0.224705
$452$	0	0
$453$	−14.7690	−0.693908
$454$	0	0
$455$	−0.381495	−0.0178848
$456$	0	0
$457$	−20.6882	−0.967755	−0.483878	−	0.875136i	$-0.660772\pi$
$457$	−20.6882	−0.967755	−0.483878	+	0.875136i	$0.660772\pi$
$458$	0	0
$459$	4.71468	0.220062
$460$	0	0
$461$	−12.3863	−0.576887	−0.288444	−	0.957497i	$-0.593138\pi$
$461$	−12.3863	−0.576887	−0.288444	+	0.957497i	$0.593138\pi$
$462$	0	0
$463$	4.73676	0.220136	0.110068	−	0.993924i	$-0.464893\pi$
$463$	4.73676	0.220136	0.110068	+	0.993924i	$0.464893\pi$
$464$	0	0
$465$	27.5520	1.27769
$466$	0	0
$467$	21.8211	1.00976	0.504880	−	0.863190i	$-0.331537\pi$
$467$	21.8211	1.00976	0.504880	+	0.863190i	$0.331537\pi$
$468$	0	0
$469$	7.24113	0.334364
$470$	0	0
$471$	1.74500	0.0804053
$472$	0	0
$473$	−4.38675	−0.201703
$474$	0	0
$475$	30.9094	1.41822
$476$	0	0
$477$	10.8381	0.496244
$478$	0	0
$479$	41.2802	1.88614	0.943071	−	0.332592i	$-0.107923\pi$
$479$	41.2802	1.88614	0.943071	+	0.332592i	$0.107923\pi$
$480$	0	0
$481$	−1.22115	−0.0556798
$482$	0	0
$483$	0.167474	0.00762032
$484$	0	0
$485$	76.9022	3.49195
$486$	0	0
$487$	−34.3016	−1.55435	−0.777177	−	0.629283i	$-0.783349\pi$
$487$	−34.3016	−1.55435	−0.777177	+	0.629283i	$0.783349\pi$
$488$	0	0
$489$	2.35561	0.106524
$490$	0	0
$491$	−24.1463	−1.08971	−0.544853	−	0.838532i	$-0.683414\pi$
$491$	−24.1463	−1.08971	−0.544853	+	0.838532i	$0.683414\pi$
$492$	0	0
$493$	−5.29488	−0.238470
$494$	0	0
$495$	4.36392	0.196144
$496$	0	0
$497$	−0.708228	−0.0317684
$498$	0	0
$499$	−19.9739	−0.894157	−0.447078	−	0.894495i	$-0.647535\pi$
$499$	−19.9739	−0.894157	−0.447078	+	0.894495i	$0.647535\pi$
$500$	0	0
$501$	−13.0719	−0.584008
$502$	0	0
$503$	7.67677	0.342290	0.171145	−	0.985246i	$-0.445253\pi$
$503$	7.67677	0.342290	0.171145	+	0.985246i	$0.445253\pi$
$504$	0	0
$505$	0.318966	0.0141938
$506$	0	0
$507$	12.9881	0.576823
$508$	0	0
$509$	32.2333	1.42872	0.714359	−	0.699779i	$-0.246718\pi$
$509$	32.2333	1.42872	0.714359	+	0.699779i	$0.246718\pi$
$510$	0	0
$511$	12.8106	0.566709
$512$	0	0
$513$	−2.17507	−0.0960318
$514$	0	0
$515$	40.9282	1.80351
$516$	0	0
$517$	−0.226586	−0.00996523
$518$	0	0
$519$	5.27078	0.231362
$520$	0	0
$521$	−10.8326	−0.474584	−0.237292	−	0.971438i	$-0.576260\pi$
$521$	−10.8326	−0.474584	−0.237292	+	0.971438i	$0.576260\pi$
$522$	0	0
$523$	−29.3420	−1.28304	−0.641518	−	0.767108i	$-0.721695\pi$
$523$	−29.3420	−1.28304	−0.641518	+	0.767108i	$0.721695\pi$
$524$	0	0
$525$	−11.3511	−0.495403
$526$	0	0
$527$	−29.6369	−1.29101
$528$	0	0
$529$	−22.9560	−0.998089
$530$	0	0
$531$	−0.194921	−0.00845886
$532$	0	0
$533$	−0.522267	−0.0226219
$534$	0	0
$535$	−6.06270	−0.262113
$536$	0	0
$537$	26.4293	1.14051
$538$	0	0
$539$	6.33426	0.272836
$540$	0	0
$541$	19.3257	0.830877	0.415439	−	0.909621i	$-0.363628\pi$
$541$	19.3257	0.830877	0.415439	+	0.909621i	$0.363628\pi$
$542$	0	0
$543$	8.14733	0.349635
$544$	0	0
$545$	66.8762	2.86466
$546$	0	0
$547$	−24.5799	−1.05096	−0.525480	−	0.850806i	$-0.676114\pi$
$547$	−24.5799	−1.05096	−0.525480	+	0.850806i	$0.676114\pi$
$548$	0	0
$549$	10.8382	0.462563
$550$	0	0
$551$	2.44274	0.104064
$552$	0	0
$553$	10.9929	0.467467
$554$	0	0
$555$	−49.1188	−2.08497
$556$	0	0
$557$	4.80617	0.203644	0.101822	−	0.994803i	$-0.467533\pi$
$557$	4.80617	0.203644	0.101822	+	0.994803i	$0.467533\pi$
$558$	0	0
$559$	0.480103	0.0203062
$560$	0	0
$561$	−4.69415	−0.198187
$562$	0	0
$563$	−27.4837	−1.15830	−0.579151	−	0.815220i	$-0.696616\pi$
$563$	−27.4837	−1.15830	−0.579151	+	0.815220i	$0.696616\pi$
$564$	0	0
$565$	−53.0224	−2.23067
$566$	0	0
$567$	0.798768	0.0335451
$568$	0	0
$569$	−31.5350	−1.32202	−0.661008	−	0.750379i	$-0.729871\pi$
$569$	−31.5350	−1.32202	−0.661008	+	0.750379i	$0.729871\pi$
$570$	0	0
$571$	−26.8243	−1.12256	−0.561281	−	0.827626i	$-0.689691\pi$
$571$	−26.8243	−1.12256	−0.561281	+	0.827626i	$0.689691\pi$
$572$	0	0
$573$	−22.1131	−0.923788
$574$	0	0
$575$	−2.97950	−0.124254
$576$	0	0
$577$	−21.9418	−0.913449	−0.456724	−	0.889608i	$-0.650977\pi$
$577$	−21.9418	−0.913449	−0.456724	+	0.889608i	$0.650977\pi$
$578$	0	0
$579$	−11.4091	−0.474146
$580$	0	0
$581$	−2.90895	−0.120683
$582$	0	0
$583$	−10.7909	−0.446915
$584$	0	0
$585$	−0.477604	−0.0197465
$586$	0	0
$587$	−20.9392	−0.864255	−0.432128	−	0.901812i	$-0.642237\pi$
$587$	−20.9392	−0.864255	−0.432128	+	0.901812i	$0.642237\pi$
$588$	0	0
$589$	13.6727	0.563375
$590$	0	0
$591$	5.66936	0.233206
$592$	0	0
$593$	−21.8246	−0.896229	−0.448114	−	0.893976i	$-0.647904\pi$
$593$	−21.8246	−0.896229	−0.448114	+	0.893976i	$0.647904\pi$
$594$	0	0
$595$	16.5061	0.676685
$596$	0	0
$597$	17.2548	0.706190
$598$	0	0
$599$	24.8758	1.01640	0.508198	−	0.861240i	$-0.330312\pi$
$599$	24.8758	1.01640	0.508198	+	0.861240i	$0.330312\pi$
$600$	0	0
$601$	8.73366	0.356253	0.178127	−	0.984008i	$-0.442996\pi$
$601$	8.73366	0.356253	0.178127	+	0.984008i	$0.442996\pi$
$602$	0	0
$603$	9.06537	0.369170
$604$	0	0
$605$	43.8682	1.78350
$606$	0	0
$607$	38.0448	1.54419	0.772095	−	0.635508i	$-0.219209\pi$
$607$	38.0448	1.54419	0.772095	+	0.635508i	$0.219209\pi$
$608$	0	0
$609$	−0.897067	−0.0363510
$610$	0	0
$611$	0.0247984	0.00100324
$612$	0	0
$613$	21.7819	0.879763	0.439881	−	0.898056i	$-0.355020\pi$
$613$	21.7819	0.879763	0.439881	+	0.898056i	$0.355020\pi$
$614$	0	0
$615$	−21.0073	−0.847095
$616$	0	0
$617$	12.6947	0.511069	0.255535	−	0.966800i	$-0.417748\pi$
$617$	12.6947	0.511069	0.255535	+	0.966800i	$0.417748\pi$
$618$	0	0
$619$	21.0759	0.847114	0.423557	−	0.905870i	$-0.360781\pi$
$619$	21.0759	0.847114	0.423557	+	0.905870i	$0.360781\pi$
$620$	0	0
$621$	0.209665	0.00841357
$622$	0	0
$623$	6.86053	0.274861
$624$	0	0
$625$	105.892	4.23568
$626$	0	0
$627$	2.16560	0.0864857
$628$	0	0
$629$	52.8357	2.10670
$630$	0	0
$631$	−25.6739	−1.02206	−0.511030	−	0.859563i	$-0.670736\pi$
$631$	−25.6739	−1.02206	−0.511030	+	0.859563i	$0.670736\pi$
$632$	0	0
$633$	16.4330	0.653152
$634$	0	0
$635$	−72.9519	−2.89501
$636$	0	0
$637$	−0.693245	−0.0274674
$638$	0	0
$639$	−0.886650	−0.0350753
$640$	0	0
$641$	−0.841768	−0.0332478	−0.0166239	−	0.999862i	$-0.505292\pi$
$641$	−0.841768	−0.0332478	−0.0166239	+	0.999862i	$0.505292\pi$
$642$	0	0
$643$	−25.5765	−1.00864	−0.504320	−	0.863517i	$-0.668257\pi$
$643$	−25.5765	−1.00864	−0.504320	+	0.863517i	$0.668257\pi$
$644$	0	0
$645$	19.3113	0.760381
$646$	0	0
$647$	−42.5334	−1.67216	−0.836080	−	0.548607i	$-0.815158\pi$
$647$	−42.5334	−1.67216	−0.836080	+	0.548607i	$0.815158\pi$
$648$	0	0
$649$	0.194072	0.00761801
$650$	0	0
$651$	−5.02114	−0.196794
$652$	0	0
$653$	16.4557	0.643959	0.321980	−	0.946747i	$-0.395652\pi$
$653$	16.4557	0.643959	0.321980	+	0.946747i	$0.395652\pi$
$654$	0	0
$655$	74.5632	2.91342
$656$	0	0
$657$	16.0380	0.625701
$658$	0	0
$659$	−17.3241	−0.674853	−0.337426	−	0.941352i	$-0.609556\pi$
$659$	−17.3241	−0.674853	−0.337426	+	0.941352i	$0.609556\pi$
$660$	0	0
$661$	−4.40934	−0.171503	−0.0857517	−	0.996317i	$-0.527329\pi$
$661$	−4.40934	−0.171503	−0.0857517	+	0.996317i	$0.527329\pi$
$662$	0	0
$663$	0.513745	0.0199522
$664$	0	0
$665$	−7.61495	−0.295295
$666$	0	0
$667$	−0.235467	−0.00911732
$668$	0	0
$669$	−7.81174	−0.302019
$670$	0	0
$671$	−10.7910	−0.416581
$672$	0	0
$673$	−22.9339	−0.884036	−0.442018	−	0.897006i	$-0.645737\pi$
$673$	−22.9339	−0.884036	−0.442018	+	0.897006i	$0.645737\pi$
$674$	0	0
$675$	−14.2108	−0.546972
$676$	0	0
$677$	32.8553	1.26273	0.631366	−	0.775485i	$-0.282495\pi$
$677$	32.8553	1.26273	0.631366	+	0.775485i	$0.282495\pi$
$678$	0	0
$679$	−14.0148	−0.537839
$680$	0	0
$681$	13.8883	0.532201
$682$	0	0
$683$	−19.1327	−0.732091	−0.366046	−	0.930597i	$-0.619289\pi$
$683$	−19.1327	−0.732091	−0.366046	+	0.930597i	$0.619289\pi$
$684$	0	0
$685$	−7.27608	−0.278005
$686$	0	0
$687$	0.383732	0.0146403
$688$	0	0
$689$	1.18100	0.0449925
$690$	0	0
$691$	−51.2203	−1.94851	−0.974255	−	0.225447i	$-0.927616\pi$
$691$	−51.2203	−1.94851	−0.974255	+	0.225447i	$0.927616\pi$
$692$	0	0
$693$	−0.795290	−0.0302106
$694$	0	0
$695$	83.3462	3.16150
$696$	0	0
$697$	22.5969	0.855920
$698$	0	0
$699$	−10.4946	−0.396944
$700$	0	0
$701$	−5.79805	−0.218989	−0.109495	−	0.993987i	$-0.534923\pi$
$701$	−5.79805	−0.218989	−0.109495	+	0.993987i	$0.534923\pi$
$702$	0	0
$703$	−24.3752	−0.919330
$704$	0	0
$705$	0.997471	0.0375669
$706$	0	0
$707$	−0.0581290	−0.00218617
$708$	0	0
$709$	−34.0405	−1.27842	−0.639208	−	0.769034i	$-0.720738\pi$
$709$	−34.0405	−1.27842	−0.639208	+	0.769034i	$0.720738\pi$
$710$	0	0
$711$	13.7624	0.516129
$712$	0	0
$713$	−1.31797	−0.0493586
$714$	0	0
$715$	0.475524	0.0177836
$716$	0	0
$717$	8.45576	0.315786
$718$	0	0
$719$	−5.11011	−0.190575	−0.0952875	−	0.995450i	$-0.530377\pi$
$719$	−5.11011	−0.190575	−0.0952875	+	0.995450i	$0.530377\pi$
$720$	0	0
$721$	−7.45883	−0.277781
$722$	0	0
$723$	4.65783	0.173227
$724$	0	0
$725$	15.9596	0.592724
$726$	0	0
$727$	24.1076	0.894101	0.447050	−	0.894509i	$-0.352474\pi$
$727$	24.1076	0.894101	0.447050	+	0.894509i	$0.352474\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−20.7726	−0.768303
$732$	0	0
$733$	4.79356	0.177054	0.0885270	−	0.996074i	$-0.471784\pi$
$733$	4.79356	0.177054	0.0885270	+	0.996074i	$0.471784\pi$
$734$	0	0
$735$	−27.8846	−1.02854
$736$	0	0
$737$	−9.02589	−0.332473
$738$	0	0
$739$	10.7520	0.395517	0.197759	−	0.980251i	$-0.436634\pi$
$739$	10.7520	0.395517	0.197759	+	0.980251i	$0.436634\pi$
$740$	0	0
$741$	−0.237011	−0.00870683
$742$	0	0
$743$	41.2906	1.51481	0.757403	−	0.652948i	$-0.226468\pi$
$743$	41.2906	1.51481	0.757403	+	0.652948i	$0.226468\pi$
$744$	0	0
$745$	15.9830	0.585571
$746$	0	0
$747$	−3.64179	−0.133246
$748$	0	0
$749$	1.10488	0.0403714
$750$	0	0
$751$	−30.0669	−1.09716	−0.548578	−	0.836100i	$-0.684830\pi$
$751$	−30.0669	−1.09716	−0.548578	+	0.836100i	$0.684830\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−64.7327	−2.35586
$756$	0	0
$757$	−8.09635	−0.294267	−0.147133	−	0.989117i	$-0.547005\pi$
$757$	−8.09635	−0.294267	−0.147133	+	0.989117i	$0.547005\pi$
$758$	0	0
$759$	−0.208752	−0.00757721
$760$	0	0
$761$	16.9705	0.615181	0.307591	−	0.951519i	$-0.400477\pi$
$761$	16.9705	0.615181	0.307591	+	0.951519i	$0.400477\pi$
$762$	0	0
$763$	−12.1876	−0.441222
$764$	0	0
$765$	20.6645	0.747126
$766$	0	0
$767$	−0.0212400	−0.000766932 0
$768$	0	0
$769$	−48.8482	−1.76151	−0.880756	−	0.473570i	$-0.842965\pi$
$769$	−48.8482	−1.76151	−0.880756	+	0.473570i	$0.842965\pi$
$770$	0	0
$771$	−16.1227	−0.580644
$772$	0	0
$773$	−34.2067	−1.23033	−0.615165	−	0.788398i	$-0.710911\pi$
$773$	−34.2067	−1.23033	−0.615165	+	0.788398i	$0.710911\pi$
$774$	0	0
$775$	89.3303	3.20884
$776$	0	0
$777$	8.95150	0.321133
$778$	0	0
$779$	−10.4249	−0.373510
$780$	0	0
$781$	0.882789	0.0315887
$782$	0	0
$783$	−1.12306	−0.0401350
$784$	0	0
$785$	7.64834	0.272981
$786$	0	0
$787$	41.1247	1.46594	0.732969	−	0.680262i	$-0.238134\pi$
$787$	41.1247	1.46594	0.732969	+	0.680262i	$0.238134\pi$
$788$	0	0
$789$	15.4603	0.550401
$790$	0	0
$791$	9.66292	0.343574
$792$	0	0
$793$	1.18101	0.0419388
$794$	0	0
$795$	47.5036	1.68478
$796$	0	0
$797$	46.0390	1.63078	0.815392	−	0.578909i	$-0.196521\pi$
$797$	46.0390	1.63078	0.815392	+	0.578909i	$0.196521\pi$
$798$	0	0
$799$	−1.07295	−0.0379583
$800$	0	0
$801$	8.58888	0.303473
$802$	0	0
$803$	−15.9681	−0.563503
$804$	0	0
$805$	0.734039	0.0258715
$806$	0	0
$807$	5.32049	0.187290
$808$	0	0
$809$	−22.2832	−0.783436	−0.391718	−	0.920085i	$-0.628119\pi$
$809$	−22.2832	−0.783436	−0.391718	+	0.920085i	$0.628119\pi$
$810$	0	0
$811$	43.2294	1.51799	0.758994	−	0.651097i	$-0.225691\pi$
$811$	43.2294	1.51799	0.758994	+	0.651097i	$0.225691\pi$
$812$	0	0
$813$	18.4567	0.647305
$814$	0	0
$815$	10.3246	0.361656
$816$	0	0
$817$	9.58325	0.335275
$818$	0	0
$819$	0.0870395	0.00304141
$820$	0	0
$821$	33.6590	1.17471	0.587354	−	0.809330i	$-0.300169\pi$
$821$	33.6590	1.17471	0.587354	+	0.809330i	$0.300169\pi$
$822$	0	0
$823$	−19.5437	−0.681252	−0.340626	−	0.940199i	$-0.610639\pi$
$823$	−19.5437	−0.681252	−0.340626	+	0.940199i	$0.610639\pi$
$824$	0	0
$825$	14.1489	0.492600
$826$	0	0
$827$	20.3693	0.708309	0.354154	−	0.935187i	$-0.384769\pi$
$827$	20.3693	0.708309	0.354154	+	0.935187i	$0.384769\pi$
$828$	0	0
$829$	−6.41724	−0.222880	−0.111440	−	0.993771i	$-0.535546\pi$
$829$	−6.41724	−0.222880	−0.111440	+	0.993771i	$0.535546\pi$
$830$	0	0
$831$	−13.3252	−0.462246
$832$	0	0
$833$	29.9946	1.03925
$834$	0	0
$835$	−57.2941	−1.98274
$836$	0	0
$837$	−6.28610	−0.217279
$838$	0	0
$839$	−43.0661	−1.48681	−0.743404	−	0.668843i	$-0.766790\pi$
$839$	−43.0661	−1.48681	−0.743404	+	0.668843i	$0.766790\pi$
$840$	0	0
$841$	−27.7387	−0.956508
$842$	0	0
$843$	3.45059	0.118845
$844$	0	0
$845$	56.9271	1.95835
$846$	0	0
$847$	−7.99463	−0.274699
$848$	0	0
$849$	−2.97544	−0.102117
$850$	0	0
$851$	2.34964	0.0805445
$852$	0	0
$853$	6.77213	0.231873	0.115937	−	0.993257i	$-0.463013\pi$
$853$	6.77213	0.231873	0.115937	+	0.993257i	$0.463013\pi$
$854$	0	0
$855$	−9.53336	−0.326034
$856$	0	0
$857$	47.0964	1.60878	0.804391	−	0.594100i	$-0.202492\pi$
$857$	47.0964	1.60878	0.804391	+	0.594100i	$0.202492\pi$
$858$	0	0
$859$	57.5233	1.96267	0.981335	−	0.192308i	$-0.0615974\pi$
$859$	57.5233	1.96267	0.981335	+	0.192308i	$0.0615974\pi$
$860$	0	0
$861$	3.82841	0.130472
$862$	0	0
$863$	32.1750	1.09525	0.547625	−	0.836724i	$-0.315532\pi$
$863$	32.1750	1.09525	0.547625	+	0.836724i	$0.315532\pi$
$864$	0	0
$865$	23.1019	0.785487
$866$	0	0
$867$	−5.22820	−0.177559
$868$	0	0
$869$	−13.7024	−0.464823
$870$	0	0
$871$	0.987827	0.0334712
$872$	0	0
$873$	−17.5455	−0.593826
$874$	0	0
$875$	−32.2470	−1.09015
$876$	0	0
$877$	−22.7766	−0.769113	−0.384556	−	0.923102i	$-0.625646\pi$
$877$	−22.7766	−0.769113	−0.384556	+	0.923102i	$0.625646\pi$
$878$	0	0
$879$	13.6692	0.461052
$880$	0	0
$881$	−8.68786	−0.292702	−0.146351	−	0.989233i	$-0.546753\pi$
$881$	−8.68786	−0.292702	−0.146351	+	0.989233i	$0.546753\pi$
$882$	0	0
$883$	16.7668	0.564249	0.282124	−	0.959378i	$-0.408961\pi$
$883$	16.7668	0.564249	0.282124	+	0.959378i	$0.408961\pi$
$884$	0	0
$885$	−0.854341	−0.0287184
$886$	0	0
$887$	46.9696	1.57708	0.788542	−	0.614981i	$-0.210836\pi$
$887$	46.9696	1.57708	0.788542	+	0.614981i	$0.210836\pi$
$888$	0	0
$889$	13.2949	0.445897
$890$	0	0
$891$	−0.995645	−0.0333554
$892$	0	0
$893$	0.494996	0.0165644
$894$	0	0
$895$	115.840	3.87209
$896$	0	0
$897$	0.0228466	0.000762825 0
$898$	0	0
$899$	7.05969	0.235454
$900$	0	0
$901$	−51.0983	−1.70233
$902$	0	0
$903$	−3.51933	−0.117116
$904$	0	0
$905$	35.7098	1.18703
$906$	0	0
$907$	12.3528	0.410167	0.205084	−	0.978744i	$-0.434253\pi$
$907$	12.3528	0.410167	0.205084	+	0.978744i	$0.434253\pi$
$908$	0	0
$909$	−0.0727733	−0.00241374
$910$	0	0
$911$	43.6650	1.44668	0.723342	−	0.690490i	$-0.242605\pi$
$911$	43.6650	1.44668	0.723342	+	0.690490i	$0.242605\pi$
$912$	0	0
$913$	3.62593	0.120001
$914$	0	0
$915$	47.5039	1.57043
$916$	0	0
$917$	−13.5885	−0.448733
$918$	0	0
$919$	43.9873	1.45101	0.725504	−	0.688218i	$-0.241607\pi$
$919$	43.9873	1.45101	0.725504	+	0.688218i	$0.241607\pi$
$920$	0	0
$921$	23.8513	0.785926
$922$	0	0
$923$	−0.0966157	−0.00318015
$924$	0	0
$925$	−159.255	−5.23626
$926$	0	0
$927$	−9.33792	−0.306697
$928$	0	0
$929$	−42.6507	−1.39933	−0.699663	−	0.714473i	$-0.746666\pi$
$929$	−42.6507	−1.39933	−0.699663	+	0.714473i	$0.746666\pi$
$930$	0	0
$931$	−13.8377	−0.453514
$932$	0	0
$933$	29.0572	0.951290
$934$	0	0
$935$	−20.5745	−0.672857
$936$	0	0
$937$	13.8874	0.453681	0.226841	−	0.973932i	$-0.427160\pi$
$937$	13.8874	0.453681	0.226841	+	0.973932i	$0.427160\pi$
$938$	0	0
$939$	−8.05336	−0.262811
$940$	0	0
$941$	−7.01869	−0.228803	−0.114401	−	0.993435i	$-0.536495\pi$
$941$	−7.01869	−0.228803	−0.114401	+	0.993435i	$0.536495\pi$
$942$	0	0
$943$	1.00490	0.0327241
$944$	0	0
$945$	3.50101	0.113888
$946$	0	0
$947$	−23.0069	−0.747624	−0.373812	−	0.927504i	$-0.621950\pi$
$947$	−23.0069	−0.747624	−0.373812	+	0.927504i	$0.621950\pi$
$948$	0	0
$949$	1.74761	0.0567299
$950$	0	0
$951$	15.1824	0.492322
$952$	0	0
$953$	−0.908729	−0.0294366	−0.0147183	−	0.999892i	$-0.504685\pi$
$953$	−0.908729	−0.0294366	−0.0147183	+	0.999892i	$0.504685\pi$
$954$	0	0
$955$	−96.9220	−3.13632
$956$	0	0
$957$	1.11817	0.0361454
$958$	0	0
$959$	1.32601	0.0428190
$960$	0	0
$961$	8.51505	0.274679
$962$	0	0
$963$	1.38323	0.0445739
$964$	0	0
$965$	−50.0062	−1.60976
$966$	0	0
$967$	16.5722	0.532927	0.266464	−	0.963845i	$-0.414145\pi$
$967$	16.5722	0.532927	0.266464	+	0.963845i	$0.414145\pi$
$968$	0	0
$969$	10.2548	0.329431
$970$	0	0
$971$	−13.2218	−0.424309	−0.212154	−	0.977236i	$-0.568048\pi$
$971$	−13.2218	−0.424309	−0.212154	+	0.977236i	$0.568048\pi$
$972$	0	0
$973$	−15.1892	−0.486943
$974$	0	0
$975$	−1.54851	−0.0495919
$976$	0	0
$977$	−9.21742	−0.294891	−0.147446	−	0.989070i	$-0.547105\pi$
$977$	−9.21742	−0.294891	−0.147446	+	0.989070i	$0.547105\pi$
$978$	0	0
$979$	−8.55148	−0.273306
$980$	0	0
$981$	−15.2580	−0.487152
$982$	0	0
$983$	−60.6755	−1.93525	−0.967624	−	0.252397i	$-0.918781\pi$
$983$	−60.6755	−1.93525	−0.967624	+	0.252397i	$0.918781\pi$
$984$	0	0
$985$	24.8489	0.791750
$986$	0	0
$987$	−0.181781	−0.00578616
$988$	0	0
$989$	−0.923772	−0.0293742
$990$	0	0
$991$	−0.521848	−0.0165770	−0.00828852	−	0.999966i	$-0.502638\pi$
$991$	−0.521848	−0.0165770	−0.00828852	+	0.999966i	$0.502638\pi$
$992$	0	0
$993$	−6.57908	−0.208781
$994$	0	0
$995$	75.6277	2.39756
$996$	0	0
$997$	−18.3034	−0.579675	−0.289837	−	0.957076i	$-0.593601\pi$
$997$	−18.3034	−0.579675	−0.289837	+	0.957076i	$0.593601\pi$
$998$	0	0
$999$	11.2066	0.354562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.o.1.1	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.o.1.1	✓	14	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.38301 q^{5} +0.798768 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.38301 q^{5} +0.798768 q^{7} +1.00000 q^{9} -0.995645 q^{11} +0.108967 q^{13} +4.38301 q^{15} -4.71468 q^{17} +2.17507 q^{19} -0.798768 q^{21} -0.209665 q^{23} +14.2108 q^{25} -1.00000 q^{27} +1.12306 q^{29} +6.28610 q^{31} +0.995645 q^{33} -3.50101 q^{35} -11.2066 q^{37} -0.108967 q^{39} -4.79289 q^{41} +4.40594 q^{43} -4.38301 q^{45} +0.227577 q^{47} -6.36197 q^{49} +4.71468 q^{51} +10.8381 q^{53} +4.36392 q^{55} -2.17507 q^{57} -0.194921 q^{59} +10.8382 q^{61} +0.798768 q^{63} -0.477604 q^{65} +9.06537 q^{67} +0.209665 q^{69} -0.886650 q^{71} +16.0380 q^{73} -14.2108 q^{75} -0.795290 q^{77} +13.7624 q^{79} +1.00000 q^{81} -3.64179 q^{83} +20.6645 q^{85} -1.12306 q^{87} +8.58888 q^{89} +0.0870395 q^{91} -6.28610 q^{93} -9.53336 q^{95} -17.5455 q^{97} -0.995645 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9} - 22 q^{11} + 9 q^{13} + 3 q^{15} - 7 q^{21} - 17 q^{23} + 25 q^{25} - 14 q^{27} - 18 q^{29} - 7 q^{31} + 22 q^{33} - 27 q^{35} + 9 q^{37} - 9 q^{39} - 6 q^{41} - 14 q^{43} - 3 q^{45} - 15 q^{47} + 15 q^{49} - 11 q^{53} + 2 q^{55} - 36 q^{59} + 2 q^{61} + 7 q^{63} + 8 q^{65} + 3 q^{67} + 17 q^{69} - 29 q^{71} + 2 q^{73} - 25 q^{75} + 8 q^{77} + 23 q^{79} + 14 q^{81} - 55 q^{83} + 7 q^{85} + 18 q^{87} + 9 q^{89} - 22 q^{91} + 7 q^{93} - 27 q^{95} + 17 q^{97} - 22 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9 - 22 * q^11 + 9 * q^13 + 3 * q^15 - 7 * q^21 - 17 * q^23 + 25 * q^25 - 14 * q^27 - 18 * q^29 - 7 * q^31 + 22 * q^33 - 27 * q^35 + 9 * q^37 - 9 * q^39 - 6 * q^41 - 14 * q^43 - 3 * q^45 - 15 * q^47 + 15 * q^49 - 11 * q^53 + 2 * q^55 - 36 * q^59 + 2 * q^61 + 7 * q^63 + 8 * q^65 + 3 * q^67 + 17 * q^69 - 29 * q^71 + 2 * q^73 - 25 * q^75 + 8 * q^77 + 23 * q^79 + 14 * q^81 - 55 * q^83 + 7 * q^85 + 18 * q^87 + 9 * q^89 - 22 * q^91 + 7 * q^93 - 27 * q^95 + 17 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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