LMFDB - Embedded newform 4020.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, 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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1257629.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 9x^{2} + 17x - 18 $ x^5 - x^4 - 9x^3 + 9x^2 + 17*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.48010$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +0.397192 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +0.397192 q^{7} +1.00000 q^{9} +2.34874 q^{11} -3.80931 q^{13} -1.00000 q^{15} -1.91709 q^{17} +0.0633716 q^{19} -0.397192 q^{21} -8.11049 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.46861 q^{29} +5.21367 q^{31} -2.34874 q^{33} +0.397192 q^{35} -10.3732 q^{37} +3.80931 q^{39} +6.96825 q^{41} +6.01046 q^{43} +1.00000 q^{45} -12.2975 q^{47} -6.84224 q^{49} +1.91709 q^{51} -11.1378 q^{53} +2.34874 q^{55} -0.0633716 q^{57} -11.9161 q^{59} -1.97271 q^{61} +0.397192 q^{63} -3.80931 q^{65} -1.00000 q^{67} +8.11049 q^{69} +1.95155 q^{71} -4.57551 q^{73} -1.00000 q^{75} +0.932903 q^{77} -1.20175 q^{79} +1.00000 q^{81} +10.9135 q^{83} -1.91709 q^{85} -6.46861 q^{87} -8.12359 q^{89} -1.51303 q^{91} -5.21367 q^{93} +0.0633716 q^{95} +10.3686 q^{97} +2.34874 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 5 * q^5 - 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9} + 3 q^{11} - 11 q^{13} - 5 q^{15} - 11 q^{17} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 5 q^{25} - 5 q^{27} - 2 q^{29} + q^{31} - 3 q^{33} - 3 q^{35} - 13 q^{37} + 11 q^{39} - 13 q^{41} - 9 q^{43} + 5 q^{45} - 12 q^{47} - 2 q^{49} + 11 q^{51} - 19 q^{53} + 3 q^{55} - 6 q^{57} + 6 q^{59} - 3 q^{61} - 3 q^{63} - 11 q^{65} - 5 q^{67} - 3 q^{69} + 6 q^{71} - 21 q^{73} - 5 q^{75} - 15 q^{77} - q^{79} + 5 q^{81} - 8 q^{83} - 11 q^{85} + 2 q^{87} - 29 q^{89} + 23 q^{91} - q^{93} + 6 q^{95} - 13 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 5 * q^5 - 3 * q^7 + 5 * q^9 + 3 * q^11 - 11 * q^13 - 5 * q^15 - 11 * q^17 + 6 * q^19 + 3 * q^21 + 3 * q^23 + 5 * q^25 - 5 * q^27 - 2 * q^29 + q^31 - 3 * q^33 - 3 * q^35 - 13 * q^37 + 11 * q^39 - 13 * q^41 - 9 * q^43 + 5 * q^45 - 12 * q^47 - 2 * q^49 + 11 * q^51 - 19 * q^53 + 3 * q^55 - 6 * q^57 + 6 * q^59 - 3 * q^61 - 3 * q^63 - 11 * q^65 - 5 * q^67 - 3 * q^69 + 6 * q^71 - 21 * q^73 - 5 * q^75 - 15 * q^77 - q^79 + 5 * q^81 - 8 * q^83 - 11 * q^85 + 2 * q^87 - 29 * q^89 + 23 * q^91 - q^93 + 6 * q^95 - 13 * q^97 + 3 * 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
$7$	0.397192	0.150125	0.0750623	−	0.997179i	$-0.476084\pi$
$7$	0.397192	0.150125	0.0750623	+	0.997179i	$0.476084\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.34874	0.708173	0.354087	−	0.935213i	$-0.384792\pi$
$11$	2.34874	0.708173	0.354087	+	0.935213i	$0.384792\pi$
$12$	0	0
$13$	−3.80931	−1.05651	−0.528256	−	0.849085i	$-0.677154\pi$
$13$	−3.80931	−1.05651	−0.528256	+	0.849085i	$0.677154\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.91709	−0.464964	−0.232482	−	0.972601i	$-0.574685\pi$
$17$	−1.91709	−0.464964	−0.232482	+	0.972601i	$0.574685\pi$
$18$	0	0
$19$	0.0633716	0.0145384	0.00726922	−	0.999974i	$-0.497686\pi$
$19$	0.0633716	0.0145384	0.00726922	+	0.999974i	$0.497686\pi$
$20$	0	0
$21$	−0.397192	−0.0866745
$22$	0	0
$23$	−8.11049	−1.69115	−0.845577	−	0.533853i	$-0.820743\pi$
$23$	−8.11049	−1.69115	−0.845577	+	0.533853i	$0.820743\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.46861	1.20119	0.600596	−	0.799553i	$-0.294930\pi$
$29$	6.46861	1.20119	0.600596	+	0.799553i	$0.294930\pi$
$30$	0	0
$31$	5.21367	0.936402	0.468201	−	0.883622i	$-0.344902\pi$
$31$	5.21367	0.936402	0.468201	+	0.883622i	$0.344902\pi$
$32$	0	0
$33$	−2.34874	−0.408864
$34$	0	0
$35$	0.397192	0.0671378
$36$	0	0
$37$	−10.3732	−1.70534	−0.852672	−	0.522447i	$-0.825019\pi$
$37$	−10.3732	−1.70534	−0.852672	+	0.522447i	$0.825019\pi$
$38$	0	0
$39$	3.80931	0.609978
$40$	0	0
$41$	6.96825	1.08826	0.544129	−	0.839002i	$-0.316860\pi$
$41$	6.96825	1.08826	0.544129	+	0.839002i	$0.316860\pi$
$42$	0	0
$43$	6.01046	0.916587	0.458294	−	0.888801i	$-0.348461\pi$
$43$	6.01046	0.916587	0.458294	+	0.888801i	$0.348461\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−12.2975	−1.79377	−0.896884	−	0.442265i	$-0.854175\pi$
$47$	−12.2975	−1.79377	−0.896884	+	0.442265i	$0.854175\pi$
$48$	0	0
$49$	−6.84224	−0.977463
$50$	0	0
$51$	1.91709	0.268447
$52$	0	0
$53$	−11.1378	−1.52989	−0.764946	−	0.644094i	$-0.777234\pi$
$53$	−11.1378	−1.52989	−0.764946	+	0.644094i	$0.777234\pi$
$54$	0	0
$55$	2.34874	0.316705
$56$	0	0
$57$	−0.0633716	−0.00839377
$58$	0	0
$59$	−11.9161	−1.55134	−0.775670	−	0.631138i	$-0.782588\pi$
$59$	−11.9161	−1.55134	−0.775670	+	0.631138i	$0.782588\pi$
$60$	0	0
$61$	−1.97271	−0.252579	−0.126290	−	0.991993i	$-0.540307\pi$
$61$	−1.97271	−0.252579	−0.126290	+	0.991993i	$0.540307\pi$
$62$	0	0
$63$	0.397192	0.0500415
$64$	0	0
$65$	−3.80931	−0.472487
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	8.11049	0.976388
$70$	0	0
$71$	1.95155	0.231607	0.115803	−	0.993272i	$-0.463056\pi$
$71$	1.95155	0.231607	0.115803	+	0.993272i	$0.463056\pi$
$72$	0	0
$73$	−4.57551	−0.535523	−0.267762	−	0.963485i	$-0.586284\pi$
$73$	−4.57551	−0.535523	−0.267762	+	0.963485i	$0.586284\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0.932903	0.106314
$78$	0	0
$79$	−1.20175	−0.135207	−0.0676037	−	0.997712i	$-0.521535\pi$
$79$	−1.20175	−0.135207	−0.0676037	+	0.997712i	$0.521535\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.9135	1.19791	0.598957	−	0.800781i	$-0.295582\pi$
$83$	10.9135	1.19791	0.598957	+	0.800781i	$0.295582\pi$
$84$	0	0
$85$	−1.91709	−0.207938
$86$	0	0
$87$	−6.46861	−0.693508
$88$	0	0
$89$	−8.12359	−0.861099	−0.430550	−	0.902567i	$-0.641680\pi$
$89$	−8.12359	−0.861099	−0.430550	+	0.902567i	$0.641680\pi$
$90$	0	0
$91$	−1.51303	−0.158608
$92$	0	0
$93$	−5.21367	−0.540632
$94$	0	0
$95$	0.0633716	0.00650179
$96$	0	0
$97$	10.3686	1.05277	0.526384	−	0.850247i	$-0.323547\pi$
$97$	10.3686	1.05277	0.526384	+	0.850247i	$0.323547\pi$
$98$	0	0
$99$	2.34874	0.236058
$100$	0	0
$101$	9.62411	0.957635	0.478818	−	0.877914i	$-0.341066\pi$
$101$	9.62411	0.957635	0.478818	+	0.877914i	$0.341066\pi$
$102$	0	0
$103$	9.44775	0.930915	0.465457	−	0.885070i	$-0.345890\pi$
$103$	9.44775	0.930915	0.465457	+	0.885070i	$0.345890\pi$
$104$	0	0
$105$	−0.397192	−0.0387620
$106$	0	0
$107$	10.7492	1.03917	0.519584	−	0.854419i	$-0.326087\pi$
$107$	10.7492	1.03917	0.519584	+	0.854419i	$0.326087\pi$
$108$	0	0
$109$	−9.87195	−0.945561	−0.472780	−	0.881180i	$-0.656750\pi$
$109$	−9.87195	−0.945561	−0.472780	+	0.881180i	$0.656750\pi$
$110$	0	0
$111$	10.3732	0.984580
$112$	0	0
$113$	14.0319	1.32001	0.660006	−	0.751261i	$-0.270554\pi$
$113$	14.0319	1.32001	0.660006	+	0.751261i	$0.270554\pi$
$114$	0	0
$115$	−8.11049	−0.756307
$116$	0	0
$117$	−3.80931	−0.352171
$118$	0	0
$119$	−0.761455	−0.0698025
$120$	0	0
$121$	−5.48340	−0.498491
$122$	0	0
$123$	−6.96825	−0.628306
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.71640	0.684720	0.342360	−	0.939569i	$-0.388774\pi$
$127$	7.71640	0.684720	0.342360	+	0.939569i	$0.388774\pi$
$128$	0	0
$129$	−6.01046	−0.529192
$130$	0	0
$131$	−1.11825	−0.0977020	−0.0488510	−	0.998806i	$-0.515556\pi$
$131$	−1.11825	−0.0977020	−0.0488510	+	0.998806i	$0.515556\pi$
$132$	0	0
$133$	0.0251707	0.00218258
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−0.147588	−0.0126093	−0.00630466	−	0.999980i	$-0.502007\pi$
$137$	−0.147588	−0.0126093	−0.00630466	+	0.999980i	$0.502007\pi$
$138$	0	0
$139$	−1.92960	−0.163667	−0.0818334	−	0.996646i	$-0.526078\pi$
$139$	−1.92960	−0.163667	−0.0818334	+	0.996646i	$0.526078\pi$
$140$	0	0
$141$	12.2975	1.03563
$142$	0	0
$143$	−8.94709	−0.748194
$144$	0	0
$145$	6.46861	0.537189
$146$	0	0
$147$	6.84224	0.564338
$148$	0	0
$149$	−13.3665	−1.09502	−0.547512	−	0.836798i	$-0.684425\pi$
$149$	−13.3665	−1.09502	−0.547512	+	0.836798i	$0.684425\pi$
$150$	0	0
$151$	−9.76204	−0.794423	−0.397212	−	0.917727i	$-0.630022\pi$
$151$	−9.76204	−0.794423	−0.397212	+	0.917727i	$0.630022\pi$
$152$	0	0
$153$	−1.91709	−0.154988
$154$	0	0
$155$	5.21367	0.418772
$156$	0	0
$157$	−17.1661	−1.37000	−0.685002	−	0.728541i	$-0.740199\pi$
$157$	−17.1661	−1.37000	−0.685002	+	0.728541i	$0.740199\pi$
$158$	0	0
$159$	11.1378	0.883284
$160$	0	0
$161$	−3.22142	−0.253884
$162$	0	0
$163$	−2.14941	−0.168355	−0.0841773	−	0.996451i	$-0.526826\pi$
$163$	−2.14941	−0.168355	−0.0841773	+	0.996451i	$0.526826\pi$
$164$	0	0
$165$	−2.34874	−0.182850
$166$	0	0
$167$	−14.2473	−1.10249	−0.551246	−	0.834343i	$-0.685847\pi$
$167$	−14.2473	−1.10249	−0.551246	+	0.834343i	$0.685847\pi$
$168$	0	0
$169$	1.51083	0.116218
$170$	0	0
$171$	0.0633716	0.00484614
$172$	0	0
$173$	−13.6510	−1.03786	−0.518931	−	0.854816i	$-0.673670\pi$
$173$	−13.6510	−1.03786	−0.518931	+	0.854816i	$0.673670\pi$
$174$	0	0
$175$	0.397192	0.0300249
$176$	0	0
$177$	11.9161	0.895667
$178$	0	0
$179$	−3.59118	−0.268418	−0.134209	−	0.990953i	$-0.542849\pi$
$179$	−3.59118	−0.268418	−0.134209	+	0.990953i	$0.542849\pi$
$180$	0	0
$181$	−17.0301	−1.26584	−0.632920	−	0.774217i	$-0.718144\pi$
$181$	−17.0301	−1.26584	−0.632920	+	0.774217i	$0.718144\pi$
$182$	0	0
$183$	1.97271	0.145827
$184$	0	0
$185$	−10.3732	−0.762653
$186$	0	0
$187$	−4.50276	−0.329275
$188$	0	0
$189$	−0.397192	−0.0288915
$190$	0	0
$191$	−10.9841	−0.794778	−0.397389	−	0.917650i	$-0.630084\pi$
$191$	−10.9841	−0.794778	−0.397389	+	0.917650i	$0.630084\pi$
$192$	0	0
$193$	−16.6357	−1.19747	−0.598733	−	0.800948i	$-0.704329\pi$
$193$	−16.6357	−1.19747	−0.598733	+	0.800948i	$0.704329\pi$
$194$	0	0
$195$	3.80931	0.272790
$196$	0	0
$197$	−4.36426	−0.310941	−0.155470	−	0.987841i	$-0.549689\pi$
$197$	−4.36426	−0.310941	−0.155470	+	0.987841i	$0.549689\pi$
$198$	0	0
$199$	−12.5836	−0.892025	−0.446013	−	0.895027i	$-0.647156\pi$
$199$	−12.5836	−0.892025	−0.446013	+	0.895027i	$0.647156\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	2.56928	0.180328
$204$	0	0
$205$	6.96825	0.486684
$206$	0	0
$207$	−8.11049	−0.563718
$208$	0	0
$209$	0.148844	0.0102957
$210$	0	0
$211$	5.28568	0.363881	0.181941	−	0.983310i	$-0.441762\pi$
$211$	5.28568	0.363881	0.181941	+	0.983310i	$0.441762\pi$
$212$	0	0
$213$	−1.95155	−0.133718
$214$	0	0
$215$	6.01046	0.409910
$216$	0	0
$217$	2.07083	0.140577
$218$	0	0
$219$	4.57551	0.309185
$220$	0	0
$221$	7.30280	0.491240
$222$	0	0
$223$	−17.3216	−1.15994	−0.579971	−	0.814637i	$-0.696936\pi$
$223$	−17.3216	−1.15994	−0.579971	+	0.814637i	$0.696936\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	27.1785	1.80390	0.901950	−	0.431841i	$-0.142136\pi$
$227$	27.1785	1.80390	0.901950	+	0.431841i	$0.142136\pi$
$228$	0	0
$229$	−2.03636	−0.134567	−0.0672833	−	0.997734i	$-0.521433\pi$
$229$	−2.03636	−0.134567	−0.0672833	+	0.997734i	$0.521433\pi$
$230$	0	0
$231$	−0.932903	−0.0613805
$232$	0	0
$233$	−18.1887	−1.19158	−0.595792	−	0.803139i	$-0.703162\pi$
$233$	−18.1887	−1.19158	−0.595792	+	0.803139i	$0.703162\pi$
$234$	0	0
$235$	−12.2975	−0.802198
$236$	0	0
$237$	1.20175	0.0780621
$238$	0	0
$239$	13.5209	0.874597	0.437298	−	0.899316i	$-0.355935\pi$
$239$	13.5209	0.874597	0.437298	+	0.899316i	$0.355935\pi$
$240$	0	0
$241$	11.2339	0.723642	0.361821	−	0.932248i	$-0.382155\pi$
$241$	11.2339	0.723642	0.361821	+	0.932248i	$0.382155\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.84224	−0.437135
$246$	0	0
$247$	−0.241402	−0.0153600
$248$	0	0
$249$	−10.9135	−0.691616
$250$	0	0
$251$	−0.499056	−0.0315002	−0.0157501	−	0.999876i	$-0.505014\pi$
$251$	−0.499056	−0.0315002	−0.0157501	+	0.999876i	$0.505014\pi$
$252$	0	0
$253$	−19.0495	−1.19763
$254$	0	0
$255$	1.91709	0.120053
$256$	0	0
$257$	−12.7173	−0.793282	−0.396641	−	0.917974i	$-0.629824\pi$
$257$	−12.7173	−0.793282	−0.396641	+	0.917974i	$0.629824\pi$
$258$	0	0
$259$	−4.12015	−0.256014
$260$	0	0
$261$	6.46861	0.400397
$262$	0	0
$263$	12.4458	0.767440	0.383720	−	0.923449i	$-0.374643\pi$
$263$	12.4458	0.767440	0.383720	+	0.923449i	$0.374643\pi$
$264$	0	0
$265$	−11.1378	−0.684189
$266$	0	0
$267$	8.12359	0.497156
$268$	0	0
$269$	−4.08561	−0.249104	−0.124552	−	0.992213i	$-0.539749\pi$
$269$	−4.08561	−0.249104	−0.124552	+	0.992213i	$0.539749\pi$
$270$	0	0
$271$	−12.3515	−0.750297	−0.375149	−	0.926965i	$-0.622408\pi$
$271$	−12.3515	−0.750297	−0.375149	+	0.926965i	$0.622408\pi$
$272$	0	0
$273$	1.51303	0.0915726
$274$	0	0
$275$	2.34874	0.141635
$276$	0	0
$277$	−3.34320	−0.200873	−0.100437	−	0.994943i	$-0.532024\pi$
$277$	−3.34320	−0.200873	−0.100437	+	0.994943i	$0.532024\pi$
$278$	0	0
$279$	5.21367	0.312134
$280$	0	0
$281$	−20.2159	−1.20598	−0.602991	−	0.797748i	$-0.706025\pi$
$281$	−20.2159	−1.20598	−0.602991	+	0.797748i	$0.706025\pi$
$282$	0	0
$283$	−33.5100	−1.99196	−0.995980	−	0.0895725i	$-0.971450\pi$
$283$	−33.5100	−1.99196	−0.995980	+	0.0895725i	$0.971450\pi$
$284$	0	0
$285$	−0.0633716	−0.00375381
$286$	0	0
$287$	2.76773	0.163374
$288$	0	0
$289$	−13.3248	−0.783809
$290$	0	0
$291$	−10.3686	−0.607816
$292$	0	0
$293$	16.7795	0.980267	0.490133	−	0.871647i	$-0.336948\pi$
$293$	16.7795	0.980267	0.490133	+	0.871647i	$0.336948\pi$
$294$	0	0
$295$	−11.9161	−0.693781
$296$	0	0
$297$	−2.34874	−0.136288
$298$	0	0
$299$	30.8954	1.78673
$300$	0	0
$301$	2.38731	0.137602
$302$	0	0
$303$	−9.62411	−0.552891
$304$	0	0
$305$	−1.97271	−0.112957
$306$	0	0
$307$	−5.11109	−0.291705	−0.145853	−	0.989306i	$-0.546592\pi$
$307$	−5.11109	−0.291705	−0.145853	+	0.989306i	$0.546592\pi$
$308$	0	0
$309$	−9.44775	−0.537464
$310$	0	0
$311$	19.8265	1.12426	0.562129	−	0.827049i	$-0.309982\pi$
$311$	19.8265	1.12426	0.562129	+	0.827049i	$0.309982\pi$
$312$	0	0
$313$	−5.24537	−0.296486	−0.148243	−	0.988951i	$-0.547362\pi$
$313$	−5.24537	−0.296486	−0.148243	+	0.988951i	$0.547362\pi$
$314$	0	0
$315$	0.397192	0.0223793
$316$	0	0
$317$	−19.4054	−1.08992	−0.544959	−	0.838463i	$-0.683455\pi$
$317$	−19.4054	−1.08992	−0.544959	+	0.838463i	$0.683455\pi$
$318$	0	0
$319$	15.1931	0.850652
$320$	0	0
$321$	−10.7492	−0.599964
$322$	0	0
$323$	−0.121489	−0.00675984
$324$	0	0
$325$	−3.80931	−0.211302
$326$	0	0
$327$	9.87195	0.545920
$328$	0	0
$329$	−4.88446	−0.269289
$330$	0	0
$331$	21.1446	1.16221	0.581107	−	0.813827i	$-0.302620\pi$
$331$	21.1446	1.16221	0.581107	+	0.813827i	$0.302620\pi$
$332$	0	0
$333$	−10.3732	−0.568448
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	0.241705	0.0131665	0.00658325	−	0.999978i	$-0.497904\pi$
$337$	0.241705	0.0131665	0.00658325	+	0.999978i	$0.497904\pi$
$338$	0	0
$339$	−14.0319	−0.762109
$340$	0	0
$341$	12.2456	0.663135
$342$	0	0
$343$	−5.49803	−0.296866
$344$	0	0
$345$	8.11049	0.436654
$346$	0	0
$347$	26.2981	1.41175	0.705877	−	0.708335i	$-0.250553\pi$
$347$	26.2981	1.41175	0.705877	+	0.708335i	$0.250553\pi$
$348$	0	0
$349$	−6.12481	−0.327854	−0.163927	−	0.986472i	$-0.552416\pi$
$349$	−6.12481	−0.327854	−0.163927	+	0.986472i	$0.552416\pi$
$350$	0	0
$351$	3.80931	0.203326
$352$	0	0
$353$	18.7841	0.999778	0.499889	−	0.866089i	$-0.333374\pi$
$353$	18.7841	0.999778	0.499889	+	0.866089i	$0.333374\pi$
$354$	0	0
$355$	1.95155	0.103578
$356$	0	0
$357$	0.761455	0.0403005
$358$	0	0
$359$	33.5626	1.77136	0.885682	−	0.464292i	$-0.153691\pi$
$359$	33.5626	1.77136	0.885682	+	0.464292i	$0.153691\pi$
$360$	0	0
$361$	−18.9960	−0.999789
$362$	0	0
$363$	5.48340	0.287804
$364$	0	0
$365$	−4.57551	−0.239493
$366$	0	0
$367$	−31.9340	−1.66694	−0.833471	−	0.552564i	$-0.813650\pi$
$367$	−31.9340	−1.66694	−0.833471	+	0.552564i	$0.813650\pi$
$368$	0	0
$369$	6.96825	0.362752
$370$	0	0
$371$	−4.42384	−0.229674
$372$	0	0
$373$	−6.71463	−0.347670	−0.173835	−	0.984775i	$-0.555616\pi$
$373$	−6.71463	−0.347670	−0.173835	+	0.984775i	$0.555616\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−24.6409	−1.26907
$378$	0	0
$379$	13.9624	0.717200	0.358600	−	0.933491i	$-0.383254\pi$
$379$	13.9624	0.717200	0.358600	+	0.933491i	$0.383254\pi$
$380$	0	0
$381$	−7.71640	−0.395323
$382$	0	0
$383$	27.6756	1.41416	0.707078	−	0.707136i	$-0.250013\pi$
$383$	27.6756	1.41416	0.707078	+	0.707136i	$0.250013\pi$
$384$	0	0
$385$	0.932903	0.0475452
$386$	0	0
$387$	6.01046	0.305529
$388$	0	0
$389$	−14.0852	−0.714147	−0.357074	−	0.934076i	$-0.616225\pi$
$389$	−14.0852	−0.714147	−0.357074	+	0.934076i	$0.616225\pi$
$390$	0	0
$391$	15.5486	0.786325
$392$	0	0
$393$	1.11825	0.0564083
$394$	0	0
$395$	−1.20175	−0.0604666
$396$	0	0
$397$	−18.3423	−0.920574	−0.460287	−	0.887770i	$-0.652253\pi$
$397$	−18.3423	−0.920574	−0.460287	+	0.887770i	$0.652253\pi$
$398$	0	0
$399$	−0.0251707	−0.00126011
$400$	0	0
$401$	−7.10544	−0.354829	−0.177414	−	0.984136i	$-0.556773\pi$
$401$	−7.10544	−0.354829	−0.177414	+	0.984136i	$0.556773\pi$
$402$	0	0
$403$	−19.8605	−0.989320
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−24.3640	−1.20768
$408$	0	0
$409$	−24.2408	−1.19863	−0.599315	−	0.800513i	$-0.704560\pi$
$409$	−24.2408	−1.19863	−0.599315	+	0.800513i	$0.704560\pi$
$410$	0	0
$411$	0.147588	0.00728000
$412$	0	0
$413$	−4.73297	−0.232894
$414$	0	0
$415$	10.9135	0.535724
$416$	0	0
$417$	1.92960	0.0944931
$418$	0	0
$419$	−1.21976	−0.0595890	−0.0297945	−	0.999556i	$-0.509485\pi$
$419$	−1.21976	−0.0595890	−0.0297945	+	0.999556i	$0.509485\pi$
$420$	0	0
$421$	35.5703	1.73359	0.866796	−	0.498663i	$-0.166176\pi$
$421$	35.5703	1.73359	0.866796	+	0.498663i	$0.166176\pi$
$422$	0	0
$423$	−12.2975	−0.597923
$424$	0	0
$425$	−1.91709	−0.0929927
$426$	0	0
$427$	−0.783544	−0.0379183
$428$	0	0
$429$	8.94709	0.431970
$430$	0	0
$431$	9.73820	0.469072	0.234536	−	0.972107i	$-0.424643\pi$
$431$	9.73820	0.469072	0.234536	+	0.972107i	$0.424643\pi$
$432$	0	0
$433$	−18.4073	−0.884601	−0.442300	−	0.896867i	$-0.645837\pi$
$433$	−18.4073	−0.884601	−0.442300	+	0.896867i	$0.645837\pi$
$434$	0	0
$435$	−6.46861	−0.310146
$436$	0	0
$437$	−0.513975	−0.0245867
$438$	0	0
$439$	−2.08292	−0.0994125	−0.0497063	−	0.998764i	$-0.515829\pi$
$439$	−2.08292	−0.0994125	−0.0497063	+	0.998764i	$0.515829\pi$
$440$	0	0
$441$	−6.84224	−0.325821
$442$	0	0
$443$	−6.88447	−0.327091	−0.163546	−	0.986536i	$-0.552293\pi$
$443$	−6.88447	−0.327091	−0.163546	+	0.986536i	$0.552293\pi$
$444$	0	0
$445$	−8.12359	−0.385095
$446$	0	0
$447$	13.3665	0.632212
$448$	0	0
$449$	−28.8460	−1.36133	−0.680665	−	0.732595i	$-0.738309\pi$
$449$	−28.8460	−1.36133	−0.680665	+	0.732595i	$0.738309\pi$
$450$	0	0
$451$	16.3666	0.770675
$452$	0	0
$453$	9.76204	0.458661
$454$	0	0
$455$	−1.51303	−0.0709319
$456$	0	0
$457$	22.8928	1.07088	0.535441	−	0.844573i	$-0.320145\pi$
$457$	22.8928	1.07088	0.535441	+	0.844573i	$0.320145\pi$
$458$	0	0
$459$	1.91709	0.0894823
$460$	0	0
$461$	13.2827	0.618637	0.309319	−	0.950958i	$-0.399899\pi$
$461$	13.2827	0.618637	0.309319	+	0.950958i	$0.399899\pi$
$462$	0	0
$463$	32.5196	1.51132	0.755658	−	0.654967i	$-0.227317\pi$
$463$	32.5196	1.51132	0.755658	+	0.654967i	$0.227317\pi$
$464$	0	0
$465$	−5.21367	−0.241778
$466$	0	0
$467$	11.9911	0.554881	0.277440	−	0.960743i	$-0.410514\pi$
$467$	11.9911	0.554881	0.277440	+	0.960743i	$0.410514\pi$
$468$	0	0
$469$	−0.397192	−0.0183406
$470$	0	0
$471$	17.1661	0.790972
$472$	0	0
$473$	14.1170	0.649103
$474$	0	0
$475$	0.0633716	0.00290769
$476$	0	0
$477$	−11.1378	−0.509964
$478$	0	0
$479$	−41.7697	−1.90851	−0.954253	−	0.299000i	$-0.903347\pi$
$479$	−41.7697	−1.90851	−0.954253	+	0.299000i	$0.903347\pi$
$480$	0	0
$481$	39.5147	1.80172
$482$	0	0
$483$	3.22142	0.146580
$484$	0	0
$485$	10.3686	0.470813
$486$	0	0
$487$	16.8583	0.763921	0.381960	−	0.924179i	$-0.375249\pi$
$487$	16.8583	0.763921	0.381960	+	0.924179i	$0.375249\pi$
$488$	0	0
$489$	2.14941	0.0971996
$490$	0	0
$491$	−13.0424	−0.588594	−0.294297	−	0.955714i	$-0.595086\pi$
$491$	−13.0424	−0.588594	−0.294297	+	0.955714i	$0.595086\pi$
$492$	0	0
$493$	−12.4009	−0.558510
$494$	0	0
$495$	2.34874	0.105568
$496$	0	0
$497$	0.775142	0.0347699
$498$	0	0
$499$	−40.1555	−1.79761	−0.898804	−	0.438351i	$-0.855563\pi$
$499$	−40.1555	−1.79761	−0.898804	+	0.438351i	$0.855563\pi$
$500$	0	0
$501$	14.2473	0.636524
$502$	0	0
$503$	−25.0599	−1.11737	−0.558684	−	0.829381i	$-0.688693\pi$
$503$	−25.0599	−1.11737	−0.558684	+	0.829381i	$0.688693\pi$
$504$	0	0
$505$	9.62411	0.428267
$506$	0	0
$507$	−1.51083	−0.0670984
$508$	0	0
$509$	32.3161	1.43239	0.716193	−	0.697902i	$-0.245883\pi$
$509$	32.3161	1.43239	0.716193	+	0.697902i	$0.245883\pi$
$510$	0	0
$511$	−1.81736	−0.0803952
$512$	0	0
$513$	−0.0633716	−0.00279792
$514$	0	0
$515$	9.44775	0.416318
$516$	0	0
$517$	−28.8836	−1.27030
$518$	0	0
$519$	13.6510	0.599210
$520$	0	0
$521$	−4.84699	−0.212350	−0.106175	−	0.994347i	$-0.533860\pi$
$521$	−4.84699	−0.212350	−0.106175	+	0.994347i	$0.533860\pi$
$522$	0	0
$523$	28.2430	1.23498	0.617489	−	0.786579i	$-0.288150\pi$
$523$	28.2430	1.23498	0.617489	+	0.786579i	$0.288150\pi$
$524$	0	0
$525$	−0.397192	−0.0173349
$526$	0	0
$527$	−9.99509	−0.435393
$528$	0	0
$529$	42.7801	1.86000
$530$	0	0
$531$	−11.9161	−0.517114
$532$	0	0
$533$	−26.5442	−1.14976
$534$	0	0
$535$	10.7492	0.464730
$536$	0	0
$537$	3.59118	0.154971
$538$	0	0
$539$	−16.0707	−0.692213
$540$	0	0
$541$	−29.4313	−1.26535	−0.632675	−	0.774417i	$-0.718043\pi$
$541$	−29.4313	−1.26535	−0.632675	+	0.774417i	$0.718043\pi$
$542$	0	0
$543$	17.0301	0.730833
$544$	0	0
$545$	−9.87195	−0.422868
$546$	0	0
$547$	21.4392	0.916675	0.458338	−	0.888778i	$-0.348445\pi$
$547$	21.4392	0.916675	0.458338	+	0.888778i	$0.348445\pi$
$548$	0	0
$549$	−1.97271	−0.0841931
$550$	0	0
$551$	0.409926	0.0174634
$552$	0	0
$553$	−0.477326	−0.0202980
$554$	0	0
$555$	10.3732	0.440318
$556$	0	0
$557$	39.6295	1.67916	0.839578	−	0.543238i	$-0.182802\pi$
$557$	39.6295	1.67916	0.839578	+	0.543238i	$0.182802\pi$
$558$	0	0
$559$	−22.8957	−0.968386
$560$	0	0
$561$	4.50276	0.190107
$562$	0	0
$563$	23.0023	0.969430	0.484715	−	0.874672i	$-0.338923\pi$
$563$	23.0023	0.969430	0.484715	+	0.874672i	$0.338923\pi$
$564$	0	0
$565$	14.0319	0.590327
$566$	0	0
$567$	0.397192	0.0166805
$568$	0	0
$569$	15.3534	0.643647	0.321823	−	0.946800i	$-0.395704\pi$
$569$	15.3534	0.643647	0.321823	+	0.946800i	$0.395704\pi$
$570$	0	0
$571$	33.2216	1.39028	0.695140	−	0.718874i	$-0.255342\pi$
$571$	33.2216	1.39028	0.695140	+	0.718874i	$0.255342\pi$
$572$	0	0
$573$	10.9841	0.458866
$574$	0	0
$575$	−8.11049	−0.338231
$576$	0	0
$577$	28.3789	1.18143	0.590714	−	0.806881i	$-0.298846\pi$
$577$	28.3789	1.18143	0.590714	+	0.806881i	$0.298846\pi$
$578$	0	0
$579$	16.6357	0.691358
$580$	0	0
$581$	4.33477	0.179836
$582$	0	0
$583$	−26.1598	−1.08343
$584$	0	0
$585$	−3.80931	−0.157496
$586$	0	0
$587$	17.0182	0.702415	0.351207	−	0.936298i	$-0.385771\pi$
$587$	17.0182	0.702415	0.351207	+	0.936298i	$0.385771\pi$
$588$	0	0
$589$	0.330398	0.0136138
$590$	0	0
$591$	4.36426	0.179522
$592$	0	0
$593$	−32.3947	−1.33029	−0.665146	−	0.746713i	$-0.731631\pi$
$593$	−32.3947	−1.33029	−0.665146	+	0.746713i	$0.731631\pi$
$594$	0	0
$595$	−0.761455	−0.0312166
$596$	0	0
$597$	12.5836	0.515011
$598$	0	0
$599$	−22.4838	−0.918664	−0.459332	−	0.888265i	$-0.651911\pi$
$599$	−22.4838	−0.918664	−0.459332	+	0.888265i	$0.651911\pi$
$600$	0	0
$601$	−8.44869	−0.344629	−0.172315	−	0.985042i	$-0.555125\pi$
$601$	−8.44869	−0.344629	−0.172315	+	0.985042i	$0.555125\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−5.48340	−0.222932
$606$	0	0
$607$	−10.2378	−0.415541	−0.207770	−	0.978178i	$-0.566621\pi$
$607$	−10.2378	−0.415541	−0.207770	+	0.978178i	$0.566621\pi$
$608$	0	0
$609$	−2.56928	−0.104113
$610$	0	0
$611$	46.8448	1.89514
$612$	0	0
$613$	0.224064	0.00904984	0.00452492	−	0.999990i	$-0.498560\pi$
$613$	0.224064	0.00904984	0.00452492	+	0.999990i	$0.498560\pi$
$614$	0	0
$615$	−6.96825	−0.280987
$616$	0	0
$617$	9.33433	0.375786	0.187893	−	0.982190i	$-0.439834\pi$
$617$	9.33433	0.375786	0.187893	+	0.982190i	$0.439834\pi$
$618$	0	0
$619$	−6.41585	−0.257875	−0.128937	−	0.991653i	$-0.541157\pi$
$619$	−6.41585	−0.257875	−0.128937	+	0.991653i	$0.541157\pi$
$620$	0	0
$621$	8.11049	0.325463
$622$	0	0
$623$	−3.22663	−0.129272
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.148844	−0.00594424
$628$	0	0
$629$	19.8864	0.792923
$630$	0	0
$631$	6.80599	0.270942	0.135471	−	0.990781i	$-0.456745\pi$
$631$	6.80599	0.270942	0.135471	+	0.990781i	$0.456745\pi$
$632$	0	0
$633$	−5.28568	−0.210087
$634$	0	0
$635$	7.71640	0.306216
$636$	0	0
$637$	26.0642	1.03270
$638$	0	0
$639$	1.95155	0.0772022
$640$	0	0
$641$	32.7375	1.29305	0.646527	−	0.762891i	$-0.276221\pi$
$641$	32.7375	1.29305	0.646527	+	0.762891i	$0.276221\pi$
$642$	0	0
$643$	−41.6599	−1.64291	−0.821453	−	0.570276i	$-0.806836\pi$
$643$	−41.6599	−1.64291	−0.821453	+	0.570276i	$0.806836\pi$
$644$	0	0
$645$	−6.01046	−0.236662
$646$	0	0
$647$	2.84237	0.111745	0.0558726	−	0.998438i	$-0.482206\pi$
$647$	2.84237	0.111745	0.0558726	+	0.998438i	$0.482206\pi$
$648$	0	0
$649$	−27.9878	−1.09862
$650$	0	0
$651$	−2.07083	−0.0811622
$652$	0	0
$653$	25.4967	0.997762	0.498881	−	0.866670i	$-0.333744\pi$
$653$	25.4967	0.997762	0.498881	+	0.866670i	$0.333744\pi$
$654$	0	0
$655$	−1.11825	−0.0436936
$656$	0	0
$657$	−4.57551	−0.178508
$658$	0	0
$659$	4.02639	0.156846	0.0784230	−	0.996920i	$-0.475012\pi$
$659$	4.02639	0.156846	0.0784230	+	0.996920i	$0.475012\pi$
$660$	0	0
$661$	−33.1266	−1.28848	−0.644238	−	0.764825i	$-0.722825\pi$
$661$	−33.1266	−1.28848	−0.644238	+	0.764825i	$0.722825\pi$
$662$	0	0
$663$	−7.30280	−0.283617
$664$	0	0
$665$	0.0251707	0.000976078 0
$666$	0	0
$667$	−52.4636	−2.03140
$668$	0	0
$669$	17.3216	0.669693
$670$	0	0
$671$	−4.63338	−0.178870
$672$	0	0
$673$	−48.8999	−1.88495	−0.942476	−	0.334275i	$-0.891508\pi$
$673$	−48.8999	−1.88495	−0.942476	+	0.334275i	$0.891508\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	28.3461	1.08943	0.544714	−	0.838622i	$-0.316638\pi$
$677$	28.3461	1.08943	0.544714	+	0.838622i	$0.316638\pi$
$678$	0	0
$679$	4.11832	0.158046
$680$	0	0
$681$	−27.1785	−1.04148
$682$	0	0
$683$	−29.0107	−1.11007	−0.555033	−	0.831829i	$-0.687294\pi$
$683$	−29.0107	−1.11007	−0.555033	+	0.831829i	$0.687294\pi$
$684$	0	0
$685$	−0.147588	−0.00563906
$686$	0	0
$687$	2.03636	0.0776921
$688$	0	0
$689$	42.4273	1.61635
$690$	0	0
$691$	5.59712	0.212925	0.106462	−	0.994317i	$-0.466048\pi$
$691$	5.59712	0.212925	0.106462	+	0.994317i	$0.466048\pi$
$692$	0	0
$693$	0.932903	0.0354381
$694$	0	0
$695$	−1.92960	−0.0731940
$696$	0	0
$697$	−13.3588	−0.506000
$698$	0	0
$699$	18.1887	0.687961
$700$	0	0
$701$	38.7983	1.46539	0.732696	−	0.680556i	$-0.238262\pi$
$701$	38.7983	1.46539	0.732696	+	0.680556i	$0.238262\pi$
$702$	0	0
$703$	−0.657366	−0.0247930
$704$	0	0
$705$	12.2975	0.463149
$706$	0	0
$707$	3.82262	0.143765
$708$	0	0
$709$	9.13556	0.343093	0.171547	−	0.985176i	$-0.445124\pi$
$709$	9.13556	0.343093	0.171547	+	0.985176i	$0.445124\pi$
$710$	0	0
$711$	−1.20175	−0.0450692
$712$	0	0
$713$	−42.2854	−1.58360
$714$	0	0
$715$	−8.94709	−0.334602
$716$	0	0
$717$	−13.5209	−0.504949
$718$	0	0
$719$	−45.2578	−1.68783	−0.843916	−	0.536475i	$-0.819756\pi$
$719$	−45.2578	−1.68783	−0.843916	+	0.536475i	$0.819756\pi$
$720$	0	0
$721$	3.75257	0.139753
$722$	0	0
$723$	−11.2339	−0.417795
$724$	0	0
$725$	6.46861	0.240238
$726$	0	0
$727$	7.80696	0.289544	0.144772	−	0.989465i	$-0.453755\pi$
$727$	7.80696	0.289544	0.144772	+	0.989465i	$0.453755\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.5226	−0.426180
$732$	0	0
$733$	−3.41700	−0.126210	−0.0631049	−	0.998007i	$-0.520100\pi$
$733$	−3.41700	−0.126210	−0.0631049	+	0.998007i	$0.520100\pi$
$734$	0	0
$735$	6.84224	0.252380
$736$	0	0
$737$	−2.34874	−0.0865171
$738$	0	0
$739$	39.7723	1.46305	0.731524	−	0.681816i	$-0.238810\pi$
$739$	39.7723	1.46305	0.731524	+	0.681816i	$0.238810\pi$
$740$	0	0
$741$	0.241402	0.00886812
$742$	0	0
$743$	47.3647	1.73764	0.868821	−	0.495126i	$-0.164878\pi$
$743$	47.3647	1.73764	0.868821	+	0.495126i	$0.164878\pi$
$744$	0	0
$745$	−13.3665	−0.489709
$746$	0	0
$747$	10.9135	0.399305
$748$	0	0
$749$	4.26951	0.156005
$750$	0	0
$751$	−43.0115	−1.56951	−0.784756	−	0.619805i	$-0.787212\pi$
$751$	−43.0115	−1.56951	−0.784756	+	0.619805i	$0.787212\pi$
$752$	0	0
$753$	0.499056	0.0181866
$754$	0	0
$755$	−9.76204	−0.355277
$756$	0	0
$757$	48.0162	1.74518	0.872589	−	0.488455i	$-0.162439\pi$
$757$	48.0162	1.74518	0.872589	+	0.488455i	$0.162439\pi$
$758$	0	0
$759$	19.0495	0.691452
$760$	0	0
$761$	20.3559	0.737901	0.368951	−	0.929449i	$-0.379717\pi$
$761$	20.3559	0.737901	0.368951	+	0.929449i	$0.379717\pi$
$762$	0	0
$763$	−3.92106	−0.141952
$764$	0	0
$765$	−1.91709	−0.0693127
$766$	0	0
$767$	45.3920	1.63901
$768$	0	0
$769$	32.9541	1.18836	0.594178	−	0.804334i	$-0.297478\pi$
$769$	32.9541	1.18836	0.594178	+	0.804334i	$0.297478\pi$
$770$	0	0
$771$	12.7173	0.458002
$772$	0	0
$773$	13.2766	0.477526	0.238763	−	0.971078i	$-0.423258\pi$
$773$	13.2766	0.477526	0.238763	+	0.971078i	$0.423258\pi$
$774$	0	0
$775$	5.21367	0.187280
$776$	0	0
$777$	4.12015	0.147810
$778$	0	0
$779$	0.441589	0.0158216
$780$	0	0
$781$	4.58370	0.164018
$782$	0	0
$783$	−6.46861	−0.231169
$784$	0	0
$785$	−17.1661	−0.612685
$786$	0	0
$787$	20.5520	0.732598	0.366299	−	0.930497i	$-0.380625\pi$
$787$	20.5520	0.732598	0.366299	+	0.930497i	$0.380625\pi$
$788$	0	0
$789$	−12.4458	−0.443082
$790$	0	0
$791$	5.57337	0.198166
$792$	0	0
$793$	7.51465	0.266853
$794$	0	0
$795$	11.1378	0.395017
$796$	0	0
$797$	−34.2795	−1.21424	−0.607120	−	0.794610i	$-0.707675\pi$
$797$	−34.2795	−1.21424	−0.607120	+	0.794610i	$0.707675\pi$
$798$	0	0
$799$	23.5754	0.834037
$800$	0	0
$801$	−8.12359	−0.287033
$802$	0	0
$803$	−10.7467	−0.379243
$804$	0	0
$805$	−3.22142	−0.113540
$806$	0	0
$807$	4.08561	0.143820
$808$	0	0
$809$	39.0408	1.37260	0.686300	−	0.727319i	$-0.259234\pi$
$809$	39.0408	1.37260	0.686300	+	0.727319i	$0.259234\pi$
$810$	0	0
$811$	18.3503	0.644368	0.322184	−	0.946677i	$-0.395583\pi$
$811$	18.3503	0.644368	0.322184	+	0.946677i	$0.395583\pi$
$812$	0	0
$813$	12.3515	0.433184
$814$	0	0
$815$	−2.14941	−0.0752905
$816$	0	0
$817$	0.380893	0.0133257
$818$	0	0
$819$	−1.51303	−0.0528695
$820$	0	0
$821$	50.5111	1.76285	0.881425	−	0.472323i	$-0.156585\pi$
$821$	50.5111	1.76285	0.881425	+	0.472323i	$0.156585\pi$
$822$	0	0
$823$	20.1580	0.702664	0.351332	−	0.936251i	$-0.385729\pi$
$823$	20.1580	0.702664	0.351332	+	0.936251i	$0.385729\pi$
$824$	0	0
$825$	−2.34874	−0.0817728
$826$	0	0
$827$	52.4790	1.82487	0.912437	−	0.409217i	$-0.134198\pi$
$827$	52.4790	1.82487	0.912437	+	0.409217i	$0.134198\pi$
$828$	0	0
$829$	8.54257	0.296696	0.148348	−	0.988935i	$-0.452604\pi$
$829$	8.54257	0.296696	0.148348	+	0.988935i	$0.452604\pi$
$830$	0	0
$831$	3.34320	0.115974
$832$	0	0
$833$	13.1172	0.454484
$834$	0	0
$835$	−14.2473	−0.493050
$836$	0	0
$837$	−5.21367	−0.180211
$838$	0	0
$839$	12.5475	0.433188	0.216594	−	0.976262i	$-0.430505\pi$
$839$	12.5475	0.433188	0.216594	+	0.976262i	$0.430505\pi$
$840$	0	0
$841$	12.8430	0.442861
$842$	0	0
$843$	20.2159	0.696274
$844$	0	0
$845$	1.51083	0.0519742
$846$	0	0
$847$	−2.17796	−0.0748357
$848$	0	0
$849$	33.5100	1.15006
$850$	0	0
$851$	84.1317	2.88400
$852$	0	0
$853$	16.2514	0.556437	0.278218	−	0.960518i	$-0.410256\pi$
$853$	16.2514	0.556437	0.278218	+	0.960518i	$0.410256\pi$
$854$	0	0
$855$	0.0633716	0.00216726
$856$	0	0
$857$	9.60596	0.328133	0.164067	−	0.986449i	$-0.447539\pi$
$857$	9.60596	0.328133	0.164067	+	0.986449i	$0.447539\pi$
$858$	0	0
$859$	32.7529	1.11752	0.558758	−	0.829331i	$-0.311278\pi$
$859$	32.7529	1.11752	0.558758	+	0.829331i	$0.311278\pi$
$860$	0	0
$861$	−2.76773	−0.0943241
$862$	0	0
$863$	−24.6073	−0.837642	−0.418821	−	0.908069i	$-0.637557\pi$
$863$	−24.6073	−0.837642	−0.418821	+	0.908069i	$0.637557\pi$
$864$	0	0
$865$	−13.6510	−0.464146
$866$	0	0
$867$	13.3248	0.452532
$868$	0	0
$869$	−2.82260	−0.0957503
$870$	0	0
$871$	3.80931	0.129073
$872$	0	0
$873$	10.3686	0.350923
$874$	0	0
$875$	0.397192	0.0134276
$876$	0	0
$877$	−28.6398	−0.967096	−0.483548	−	0.875318i	$-0.660652\pi$
$877$	−28.6398	−0.967096	−0.483548	+	0.875318i	$0.660652\pi$
$878$	0	0
$879$	−16.7795	−0.565957
$880$	0	0
$881$	−19.5612	−0.659034	−0.329517	−	0.944150i	$-0.606886\pi$
$881$	−19.5612	−0.659034	−0.329517	+	0.944150i	$0.606886\pi$
$882$	0	0
$883$	−39.8339	−1.34052	−0.670259	−	0.742127i	$-0.733817\pi$
$883$	−39.8339	−1.34052	−0.670259	+	0.742127i	$0.733817\pi$
$884$	0	0
$885$	11.9161	0.400554
$886$	0	0
$887$	25.6705	0.861931	0.430965	−	0.902368i	$-0.358173\pi$
$887$	25.6705	0.861931	0.430965	+	0.902368i	$0.358173\pi$
$888$	0	0
$889$	3.06489	0.102793
$890$	0	0
$891$	2.34874	0.0786859
$892$	0	0
$893$	−0.779309	−0.0260786
$894$	0	0
$895$	−3.59118	−0.120040
$896$	0	0
$897$	−30.8954	−1.03157
$898$	0	0
$899$	33.7252	1.12480
$900$	0	0
$901$	21.3522	0.711344
$902$	0	0
$903$	−2.38731	−0.0794447
$904$	0	0
$905$	−17.0301	−0.566101
$906$	0	0
$907$	−0.840872	−0.0279207	−0.0139603	−	0.999903i	$-0.504444\pi$
$907$	−0.840872	−0.0279207	−0.0139603	+	0.999903i	$0.504444\pi$
$908$	0	0
$909$	9.62411	0.319212
$910$	0	0
$911$	10.3180	0.341852	0.170926	−	0.985284i	$-0.445324\pi$
$911$	10.3180	0.341852	0.170926	+	0.985284i	$0.445324\pi$
$912$	0	0
$913$	25.6331	0.848331
$914$	0	0
$915$	1.97271	0.0652157
$916$	0	0
$917$	−0.444160	−0.0146675
$918$	0	0
$919$	36.7899	1.21359	0.606794	−	0.794859i	$-0.292455\pi$
$919$	36.7899	1.21359	0.606794	+	0.794859i	$0.292455\pi$
$920$	0	0
$921$	5.11109	0.168416
$922$	0	0
$923$	−7.43407	−0.244695
$924$	0	0
$925$	−10.3732	−0.341069
$926$	0	0
$927$	9.44775	0.310305
$928$	0	0
$929$	−45.8396	−1.50395	−0.751974	−	0.659193i	$-0.770898\pi$
$929$	−45.8396	−1.50395	−0.751974	+	0.659193i	$0.770898\pi$
$930$	0	0
$931$	−0.433603	−0.0142108
$932$	0	0
$933$	−19.8265	−0.649091
$934$	0	0
$935$	−4.50276	−0.147256
$936$	0	0
$937$	44.1457	1.44218	0.721089	−	0.692843i	$-0.243642\pi$
$937$	44.1457	1.44218	0.721089	+	0.692843i	$0.243642\pi$
$938$	0	0
$939$	5.24537	0.171176
$940$	0	0
$941$	50.1282	1.63413	0.817067	−	0.576543i	$-0.195599\pi$
$941$	50.1282	1.63413	0.817067	+	0.576543i	$0.195599\pi$
$942$	0	0
$943$	−56.5159	−1.84041
$944$	0	0
$945$	−0.397192	−0.0129207
$946$	0	0
$947$	39.8718	1.29566	0.647830	−	0.761785i	$-0.275677\pi$
$947$	39.8718	1.29566	0.647830	+	0.761785i	$0.275677\pi$
$948$	0	0
$949$	17.4295	0.565787
$950$	0	0
$951$	19.4054	0.629265
$952$	0	0
$953$	35.8597	1.16161	0.580805	−	0.814043i	$-0.302738\pi$
$953$	35.8597	1.16161	0.580805	+	0.814043i	$0.302738\pi$
$954$	0	0
$955$	−10.9841	−0.355436
$956$	0	0
$957$	−15.1931	−0.491124
$958$	0	0
$959$	−0.0586210	−0.00189297
$960$	0	0
$961$	−3.81769	−0.123151
$962$	0	0
$963$	10.7492	0.346389
$964$	0	0
$965$	−16.6357	−0.535523
$966$	0	0
$967$	−23.3274	−0.750158	−0.375079	−	0.926993i	$-0.622384\pi$
$967$	−23.3274	−0.750158	−0.375079	+	0.926993i	$0.622384\pi$
$968$	0	0
$969$	0.121489	0.00390280
$970$	0	0
$971$	−0.746337	−0.0239511	−0.0119755	−	0.999928i	$-0.503812\pi$
$971$	−0.746337	−0.0239511	−0.0119755	+	0.999928i	$0.503812\pi$
$972$	0	0
$973$	−0.766424	−0.0245704
$974$	0	0
$975$	3.80931	0.121996
$976$	0	0
$977$	−49.3709	−1.57951	−0.789757	−	0.613419i	$-0.789794\pi$
$977$	−49.3709	−1.57951	−0.789757	+	0.613419i	$0.789794\pi$
$978$	0	0
$979$	−19.0803	−0.609807
$980$	0	0
$981$	−9.87195	−0.315187
$982$	0	0
$983$	−7.70615	−0.245788	−0.122894	−	0.992420i	$-0.539218\pi$
$983$	−7.70615	−0.245788	−0.122894	+	0.992420i	$0.539218\pi$
$984$	0	0
$985$	−4.36426	−0.139057
$986$	0	0
$987$	4.88446	0.155474
$988$	0	0
$989$	−48.7478	−1.55009
$990$	0	0
$991$	−25.4495	−0.808431	−0.404215	−	0.914664i	$-0.632455\pi$
$991$	−25.4495	−0.808431	−0.404215	+	0.914664i	$0.632455\pi$
$992$	0	0
$993$	−21.1446	−0.671005
$994$	0	0
$995$	−12.5836	−0.398926
$996$	0	0
$997$	−22.5648	−0.714633	−0.357317	−	0.933983i	$-0.616308\pi$
$997$	−22.5648	−0.714633	−0.357317	+	0.933983i	$0.616308\pi$
$998$	0	0
$999$	10.3732	0.328193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.e.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.e.1.4	✓	5	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +0.397192 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +0.397192 q^{7} +1.00000 q^{9} +2.34874 q^{11} -3.80931 q^{13} -1.00000 q^{15} -1.91709 q^{17} +0.0633716 q^{19} -0.397192 q^{21} -8.11049 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.46861 q^{29} +5.21367 q^{31} -2.34874 q^{33} +0.397192 q^{35} -10.3732 q^{37} +3.80931 q^{39} +6.96825 q^{41} +6.01046 q^{43} +1.00000 q^{45} -12.2975 q^{47} -6.84224 q^{49} +1.91709 q^{51} -11.1378 q^{53} +2.34874 q^{55} -0.0633716 q^{57} -11.9161 q^{59} -1.97271 q^{61} +0.397192 q^{63} -3.80931 q^{65} -1.00000 q^{67} +8.11049 q^{69} +1.95155 q^{71} -4.57551 q^{73} -1.00000 q^{75} +0.932903 q^{77} -1.20175 q^{79} +1.00000 q^{81} +10.9135 q^{83} -1.91709 q^{85} -6.46861 q^{87} -8.12359 q^{89} -1.51303 q^{91} -5.21367 q^{93} +0.0633716 q^{95} +10.3686 q^{97} +2.34874 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 5 * q^5 - 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9} + 3 q^{11} - 11 q^{13} - 5 q^{15} - 11 q^{17} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 5 q^{25} - 5 q^{27} - 2 q^{29} + q^{31} - 3 q^{33} - 3 q^{35} - 13 q^{37} + 11 q^{39} - 13 q^{41} - 9 q^{43} + 5 q^{45} - 12 q^{47} - 2 q^{49} + 11 q^{51} - 19 q^{53} + 3 q^{55} - 6 q^{57} + 6 q^{59} - 3 q^{61} - 3 q^{63} - 11 q^{65} - 5 q^{67} - 3 q^{69} + 6 q^{71} - 21 q^{73} - 5 q^{75} - 15 q^{77} - q^{79} + 5 q^{81} - 8 q^{83} - 11 q^{85} + 2 q^{87} - 29 q^{89} + 23 q^{91} - q^{93} + 6 q^{95} - 13 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 5 * q^5 - 3 * q^7 + 5 * q^9 + 3 * q^11 - 11 * q^13 - 5 * q^15 - 11 * q^17 + 6 * q^19 + 3 * q^21 + 3 * q^23 + 5 * q^25 - 5 * q^27 - 2 * q^29 + q^31 - 3 * q^33 - 3 * q^35 - 13 * q^37 + 11 * q^39 - 13 * q^41 - 9 * q^43 + 5 * q^45 - 12 * q^47 - 2 * q^49 + 11 * q^51 - 19 * q^53 + 3 * q^55 - 6 * q^57 + 6 * q^59 - 3 * q^61 - 3 * q^63 - 11 * q^65 - 5 * q^67 - 3 * q^69 + 6 * q^71 - 21 * q^73 - 5 * q^75 - 15 * q^77 - q^79 + 5 * q^81 - 8 * q^83 - 11 * q^85 + 2 * q^87 - 29 * q^89 + 23 * q^91 - q^93 + 6 * q^95 - 13 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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