LMFDB - Embedded newform 4140.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.14345904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12 $ x^5 - 2x^4 - 13x^3 + 34x^2 - 11x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.84503$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -3.57723 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.57723 q^{7} +4.75505 q^{11} -6.44510 q^{13} +3.46021 q^{17} -2.75505 q^{19} +1.00000 q^{23} +1.00000 q^{25} +2.86787 q^{29} -0.395216 q^{31} -3.57723 q^{35} +0.112824 q^{37} -2.32389 q^{41} +12.6105 q^{43} -6.80074 q^{47} +5.79655 q^{49} -1.41452 q^{53} +4.75505 q^{55} +4.51223 q^{59} -0.165379 q^{61} -6.44510 q^{65} +5.53153 q^{67} +13.4783 q^{71} +10.4451 q^{73} -17.0099 q^{77} +0.498285 q^{79} -12.8487 q^{83} +3.46021 q^{85} +5.15445 q^{89} +23.0556 q^{91} -2.75505 q^{95} -6.96612 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} + 4 q^{7}+O(q^{10}) $ 5 * q + 5 * q^5 + 4 * q^7	$ 5 q + 5 q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{23} + 5 q^{25} + 2 q^{29} + 8 q^{31} + 4 q^{35} + 8 q^{37} + 2 q^{41} + 18 q^{43} - 4 q^{47} + 13 q^{49} - 2 q^{53} + 4 q^{55} + 6 q^{59} + 10 q^{61} + 2 q^{65} + 16 q^{67} + 10 q^{71} + 18 q^{73} - 16 q^{77} + 14 q^{79} + 8 q^{83} + 2 q^{85} - 18 q^{89} + 36 q^{91} + 6 q^{95} + 6 q^{97}+O(q^{100}) $ 5 * q + 5 * q^5 + 4 * q^7 + 4 * q^11 + 2 * q^13 + 2 * q^17 + 6 * q^19 + 5 * q^23 + 5 * q^25 + 2 * q^29 + 8 * q^31 + 4 * q^35 + 8 * q^37 + 2 * q^41 + 18 * q^43 - 4 * q^47 + 13 * q^49 - 2 * q^53 + 4 * q^55 + 6 * q^59 + 10 * q^61 + 2 * q^65 + 16 * q^67 + 10 * q^71 + 18 * q^73 - 16 * q^77 + 14 * q^79 + 8 * q^83 + 2 * q^85 - 18 * q^89 + 36 * q^91 + 6 * q^95 + 6 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.57723	−1.35206	−0.676032	−	0.736872i	$-0.736302\pi$
$7$	−3.57723	−1.35206	−0.676032	+	0.736872i	$0.736302\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.75505	1.43370	0.716850	−	0.697227i	$-0.245583\pi$
$11$	4.75505	1.43370	0.716850	+	0.697227i	$0.245583\pi$
$12$	0	0
$13$	−6.44510	−1.78755	−0.893774	−	0.448517i	$-0.851952\pi$
$13$	−6.44510	−1.78755	−0.893774	+	0.448517i	$0.851952\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46021	0.839225	0.419612	−	0.907703i	$-0.362166\pi$
$17$	3.46021	0.839225	0.419612	+	0.907703i	$0.362166\pi$
$18$	0	0
$19$	−2.75505	−0.632051	−0.316026	−	0.948751i	$-0.602349\pi$
$19$	−2.75505	−0.632051	−0.316026	+	0.948751i	$0.602349\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.86787	0.532550	0.266275	−	0.963897i	$-0.414207\pi$
$29$	2.86787	0.532550	0.266275	+	0.963897i	$0.414207\pi$
$30$	0	0
$31$	−0.395216	−0.0709828	−0.0354914	−	0.999370i	$-0.511300\pi$
$31$	−0.395216	−0.0709828	−0.0354914	+	0.999370i	$0.511300\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.57723	−0.604662
$36$	0	0
$37$	0.112824	0.0185481	0.00927404	−	0.999957i	$-0.497048\pi$
$37$	0.112824	0.0185481	0.00927404	+	0.999957i	$0.497048\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.32389	−0.362931	−0.181466	−	0.983397i	$-0.558084\pi$
$41$	−2.32389	−0.362931	−0.181466	+	0.983397i	$0.558084\pi$
$42$	0	0
$43$	12.6105	1.92308	0.961540	−	0.274666i	$-0.0885672\pi$
$43$	12.6105	1.92308	0.961540	+	0.274666i	$0.0885672\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.80074	−0.991990	−0.495995	−	0.868325i	$-0.665197\pi$
$47$	−6.80074	−0.991990	−0.495995	+	0.868325i	$0.665197\pi$
$48$	0	0
$49$	5.79655	0.828079
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.41452	−0.194299	−0.0971497	−	0.995270i	$-0.530973\pi$
$53$	−1.41452	−0.194299	−0.0971497	+	0.995270i	$0.530973\pi$
$54$	0	0
$55$	4.75505	0.641170
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.51223	0.587442	0.293721	−	0.955891i	$-0.405106\pi$
$59$	4.51223	0.587442	0.293721	+	0.955891i	$0.405106\pi$
$60$	0	0
$61$	−0.165379	−0.0211746	−0.0105873	−	0.999944i	$-0.503370\pi$
$61$	−0.165379	−0.0211746	−0.0105873	+	0.999944i	$0.503370\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.44510	−0.799416
$66$	0	0
$67$	5.53153	0.675785	0.337892	−	0.941185i	$-0.390286\pi$
$67$	5.53153	0.675785	0.337892	+	0.941185i	$0.390286\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.4783	1.59959	0.799793	−	0.600276i	$-0.204943\pi$
$71$	13.4783	1.59959	0.799793	+	0.600276i	$0.204943\pi$
$72$	0	0
$73$	10.4451	1.22251	0.611253	−	0.791435i	$-0.290666\pi$
$73$	10.4451	1.22251	0.611253	+	0.791435i	$0.290666\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−17.0099	−1.93846
$78$	0	0
$79$	0.498285	0.0560614	0.0280307	−	0.999607i	$-0.491076\pi$
$79$	0.498285	0.0560614	0.0280307	+	0.999607i	$0.491076\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.8487	−1.41033	−0.705164	−	0.709045i	$-0.749126\pi$
$83$	−12.8487	−1.41033	−0.705164	+	0.709045i	$0.749126\pi$
$84$	0	0
$85$	3.46021	0.375313
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.15445	0.546371	0.273185	−	0.961961i	$-0.411923\pi$
$89$	5.15445	0.546371	0.273185	+	0.961961i	$0.411923\pi$
$90$	0	0
$91$	23.0556	2.41688
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.75505	−0.282662
$96$	0	0
$97$	−6.96612	−0.707302	−0.353651	−	0.935377i	$-0.615060\pi$
$97$	−6.96612	−0.707302	−0.353651	+	0.935377i	$0.615060\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.1227	1.50477	0.752383	−	0.658726i	$-0.228904\pi$
$101$	15.1227	1.50477	0.752383	+	0.658726i	$0.228904\pi$
$102$	0	0
$103$	0.899619	0.0886421	0.0443210	−	0.999017i	$-0.485888\pi$
$103$	0.899619	0.0886421	0.0443210	+	0.999017i	$0.485888\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.99581	−0.192942	−0.0964711	−	0.995336i	$-0.530756\pi$
$107$	−1.99581	−0.192942	−0.0964711	+	0.995336i	$0.530756\pi$
$108$	0	0
$109$	−2.75505	−0.263886	−0.131943	−	0.991257i	$-0.542122\pi$
$109$	−2.75505	−0.263886	−0.131943	+	0.991257i	$0.542122\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.8064	1.86323	0.931616	−	0.363444i	$-0.118399\pi$
$113$	19.8064	1.86323	0.931616	+	0.363444i	$0.118399\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.3780	−1.13469
$120$	0	0
$121$	11.6105	1.05550
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.2111	1.08356	0.541779	−	0.840521i	$-0.317751\pi$
$127$	12.2111	1.08356	0.541779	+	0.840521i	$0.317751\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.1545	0.974569	0.487285	−	0.873243i	$-0.337987\pi$
$131$	11.1545	0.974569	0.487285	+	0.873243i	$0.337987\pi$
$132$	0	0
$133$	9.85543	0.854574
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.4258	−1.31791	−0.658957	−	0.752180i	$-0.729002\pi$
$137$	−15.4258	−1.31791	−0.658957	+	0.752180i	$0.729002\pi$
$138$	0	0
$139$	−0.796550	−0.0675624	−0.0337812	−	0.999429i	$-0.510755\pi$
$139$	−0.796550	−0.0675624	−0.0337812	+	0.999429i	$0.510755\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−30.6467	−2.56281
$144$	0	0
$145$	2.86787	0.238164
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.26514	0.677107	0.338553	−	0.940947i	$-0.390062\pi$
$149$	8.26514	0.677107	0.338553	+	0.940947i	$0.390062\pi$
$150$	0	0
$151$	18.6105	1.51450	0.757250	−	0.653126i	$-0.226543\pi$
$151$	18.6105	1.51450	0.757250	+	0.653126i	$0.226543\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.395216	−0.0317445
$156$	0	0
$157$	−18.8231	−1.50224	−0.751122	−	0.660163i	$-0.770487\pi$
$157$	−18.8231	−1.50224	−0.751122	+	0.660163i	$0.770487\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.57723	−0.281925
$162$	0	0
$163$	1.07957	0.0845588	0.0422794	−	0.999106i	$-0.486538\pi$
$163$	1.07957	0.0845588	0.0422794	+	0.999106i	$0.486538\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.94681	0.769707	0.384854	−	0.922978i	$-0.374252\pi$
$167$	9.94681	0.769707	0.384854	+	0.922978i	$0.374252\pi$
$168$	0	0
$169$	28.5393	2.19533
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.945311	−0.0718706	−0.0359353	−	0.999354i	$-0.511441\pi$
$173$	−0.945311	−0.0718706	−0.0359353	+	0.999354i	$0.511441\pi$
$174$	0	0
$175$	−3.57723	−0.270413
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.4762	1.38098	0.690488	−	0.723343i	$-0.257396\pi$
$179$	18.4762	1.38098	0.690488	+	0.723343i	$0.257396\pi$
$180$	0	0
$181$	−19.6806	−1.46285	−0.731425	−	0.681922i	$-0.761144\pi$
$181$	−19.6806	−1.46285	−0.731425	+	0.681922i	$0.761144\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.112824	0.00829495
$186$	0	0
$187$	16.4535	1.20320
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.7248	1.06545	0.532725	−	0.846288i	$-0.321168\pi$
$191$	14.7248	1.06545	0.532725	+	0.846288i	$0.321168\pi$
$192$	0	0
$193$	−21.8106	−1.56996	−0.784981	−	0.619519i	$-0.787328\pi$
$193$	−21.8106	−1.56996	−0.784981	+	0.619519i	$0.787328\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.0446	1.14313	0.571567	−	0.820555i	$-0.306336\pi$
$197$	16.0446	1.14313	0.571567	+	0.820555i	$0.306336\pi$
$198$	0	0
$199$	−0.928806	−0.0658413	−0.0329207	−	0.999458i	$-0.510481\pi$
$199$	−0.928806	−0.0658413	−0.0329207	+	0.999458i	$0.510481\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.2590	−0.720042
$204$	0	0
$205$	−2.32389	−0.162308
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−13.1004	−0.906172
$210$	0	0
$211$	5.88450	0.405106	0.202553	−	0.979271i	$-0.435076\pi$
$211$	5.88450	0.405106	0.202553	+	0.979271i	$0.435076\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.6105	0.860027
$216$	0	0
$217$	1.41378	0.0959734
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−22.3014	−1.50016
$222$	0	0
$223$	14.9204	0.999146	0.499573	−	0.866272i	$-0.333490\pi$
$223$	14.9204	0.999146	0.499573	+	0.866272i	$0.333490\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.6890	1.17406	0.587030	−	0.809565i	$-0.300297\pi$
$227$	17.6890	1.17406	0.587030	+	0.809565i	$0.300297\pi$
$228$	0	0
$229$	−3.87839	−0.256291	−0.128145	−	0.991755i	$-0.540902\pi$
$229$	−3.87839	−0.256291	−0.128145	+	0.991755i	$0.540902\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.41871	−0.354992	−0.177496	−	0.984122i	$-0.556800\pi$
$233$	−5.41871	−0.354992	−0.177496	+	0.984122i	$0.556800\pi$
$234$	0	0
$235$	−6.80074	−0.443631
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.1613	−0.851335	−0.425667	−	0.904880i	$-0.639961\pi$
$239$	−13.1613	−0.851335	−0.425667	+	0.904880i	$0.639961\pi$
$240$	0	0
$241$	27.9760	1.80209	0.901046	−	0.433723i	$-0.142800\pi$
$241$	27.9760	1.80209	0.901046	+	0.433723i	$0.142800\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.79655	0.370328
$246$	0	0
$247$	17.7565	1.12982
$248$	0	0
$249$	0	0
$250$	0	0
$251$	30.3173	1.91361	0.956805	−	0.290730i	$-0.0938982\pi$
$251$	30.3173	1.91361	0.956805	+	0.290730i	$0.0938982\pi$
$252$	0	0
$253$	4.75505	0.298947
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.8336	−1.48670	−0.743349	−	0.668904i	$-0.766764\pi$
$257$	−23.8336	−1.48670	−0.743349	+	0.668904i	$0.766764\pi$
$258$	0	0
$259$	−0.403595	−0.0250782
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.5261	−1.14237	−0.571184	−	0.820822i	$-0.693516\pi$
$263$	−18.5261	−1.14237	−0.571184	+	0.820822i	$0.693516\pi$
$264$	0	0
$265$	−1.41452	−0.0868933
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.9125	−1.88477	−0.942385	−	0.334531i	$-0.891422\pi$
$269$	−30.9125	−1.88477	−0.942385	+	0.334531i	$0.891422\pi$
$270$	0	0
$271$	−8.51579	−0.517297	−0.258649	−	0.965971i	$-0.583277\pi$
$271$	−8.51579	−0.517297	−0.258649	+	0.965971i	$0.583277\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.75505	0.286740
$276$	0	0
$277$	21.8936	1.31546	0.657730	−	0.753254i	$-0.271517\pi$
$277$	21.8936	1.31546	0.657730	+	0.753254i	$0.271517\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.8215	1.24211	0.621055	−	0.783767i	$-0.286704\pi$
$281$	20.8215	1.24211	0.621055	+	0.783767i	$0.286704\pi$
$282$	0	0
$283$	7.09159	0.421551	0.210776	−	0.977534i	$-0.432401\pi$
$283$	7.09159	0.421551	0.210776	+	0.977534i	$0.432401\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.31310	0.490707
$288$	0	0
$289$	−5.02693	−0.295702
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.04645	0.177976	0.0889878	−	0.996033i	$-0.471637\pi$
$293$	3.04645	0.177976	0.0889878	+	0.996033i	$0.471637\pi$
$294$	0	0
$295$	4.51223	0.262712
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.44510	−0.372730
$300$	0	0
$301$	−45.1105	−2.60013
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.165379	−0.00946956
$306$	0	0
$307$	28.6416	1.63466	0.817331	−	0.576168i	$-0.195453\pi$
$307$	28.6416	1.63466	0.817331	+	0.576168i	$0.195453\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.6259	−0.602542	−0.301271	−	0.953539i	$-0.597411\pi$
$311$	−10.6259	−0.602542	−0.301271	+	0.953539i	$0.597411\pi$
$312$	0	0
$313$	−8.30932	−0.469670	−0.234835	−	0.972035i	$-0.575455\pi$
$313$	−8.30932	−0.469670	−0.234835	+	0.972035i	$0.575455\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.67074	0.262335	0.131168	−	0.991360i	$-0.458127\pi$
$317$	4.67074	0.262335	0.131168	+	0.991360i	$0.458127\pi$
$318$	0	0
$319$	13.6369	0.763518
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−9.53305	−0.530433
$324$	0	0
$325$	−6.44510	−0.357510
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.3278	1.34123
$330$	0	0
$331$	11.7753	0.647230	0.323615	−	0.946189i	$-0.395102\pi$
$331$	11.7753	0.647230	0.323615	+	0.946189i	$0.395102\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.53153	0.302220
$336$	0	0
$337$	−9.01181	−0.490905	−0.245452	−	0.969409i	$-0.578936\pi$
$337$	−9.01181	−0.490905	−0.245452	+	0.969409i	$0.578936\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.87927	−0.101768
$342$	0	0
$343$	4.30501	0.232449
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.61990	−0.248009	−0.124005	−	0.992282i	$-0.539574\pi$
$347$	−4.61990	−0.248009	−0.124005	+	0.992282i	$0.539574\pi$
$348$	0	0
$349$	8.93492	0.478276	0.239138	−	0.970986i	$-0.423135\pi$
$349$	8.93492	0.478276	0.239138	+	0.970986i	$0.423135\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.12807	−0.432613	−0.216307	−	0.976325i	$-0.569401\pi$
$353$	−8.12807	−0.432613	−0.216307	+	0.976325i	$0.569401\pi$
$354$	0	0
$355$	13.4783	0.715356
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.5985	−1.77326	−0.886631	−	0.462478i	$-0.846960\pi$
$359$	−33.5985	−1.77326	−0.886631	+	0.462478i	$0.846960\pi$
$360$	0	0
$361$	−11.4097	−0.600511
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.4451	0.546721
$366$	0	0
$367$	−35.8254	−1.87007	−0.935037	−	0.354551i	$-0.884634\pi$
$367$	−35.8254	−1.87007	−0.935037	+	0.354551i	$0.884634\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.06006	0.262705
$372$	0	0
$373$	−25.4728	−1.31893	−0.659465	−	0.751735i	$-0.729217\pi$
$373$	−25.4728	−1.31893	−0.659465	+	0.751735i	$0.729217\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.4837	−0.951959
$378$	0	0
$379$	8.68105	0.445916	0.222958	−	0.974828i	$-0.428429\pi$
$379$	8.68105	0.445916	0.222958	+	0.974828i	$0.428429\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.7725	1.16362	0.581811	−	0.813324i	$-0.302344\pi$
$383$	22.7725	1.16362	0.581811	+	0.813324i	$0.302344\pi$
$384$	0	0
$385$	−17.0099	−0.866904
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.4221	−0.731232	−0.365616	−	0.930766i	$-0.619142\pi$
$389$	−14.4221	−0.731232	−0.365616	+	0.930766i	$0.619142\pi$
$390$	0	0
$391$	3.46021	0.174990
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.498285	0.0250714
$396$	0	0
$397$	−0.806936	−0.0404990	−0.0202495	−	0.999795i	$-0.506446\pi$
$397$	−0.806936	−0.0404990	−0.0202495	+	0.999795i	$0.506446\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.6562	1.53090	0.765448	−	0.643498i	$-0.222517\pi$
$401$	30.6562	1.53090	0.765448	+	0.643498i	$0.222517\pi$
$402$	0	0
$403$	2.54720	0.126885
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.536481	0.0265924
$408$	0	0
$409$	6.80493	0.336482	0.168241	−	0.985746i	$-0.446191\pi$
$409$	6.80493	0.336482	0.168241	+	0.985746i	$0.446191\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−16.1413	−0.794260
$414$	0	0
$415$	−12.8487	−0.630718
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.38568	−0.116548	−0.0582740	−	0.998301i	$-0.518560\pi$
$419$	−2.38568	−0.116548	−0.0582740	+	0.998301i	$0.518560\pi$
$420$	0	0
$421$	−1.01093	−0.0492695	−0.0246348	−	0.999697i	$-0.507842\pi$
$421$	−1.01093	−0.0492695	−0.0246348	+	0.999697i	$0.507842\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.46021	0.167845
$426$	0	0
$427$	0.591597	0.0286294
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.59737	0.365952	0.182976	−	0.983117i	$-0.441427\pi$
$431$	7.59737	0.365952	0.182976	+	0.983117i	$0.441427\pi$
$432$	0	0
$433$	−10.4009	−0.499836	−0.249918	−	0.968267i	$-0.580404\pi$
$433$	−10.4009	−0.499836	−0.249918	+	0.968267i	$0.580404\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.75505	−0.131792
$438$	0	0
$439$	16.4805	0.786570	0.393285	−	0.919417i	$-0.371338\pi$
$439$	16.4805	0.786570	0.393285	+	0.919417i	$0.371338\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.18084	−0.103615	−0.0518074	−	0.998657i	$-0.516498\pi$
$443$	−2.18084	−0.103615	−0.0518074	+	0.998657i	$0.516498\pi$
$444$	0	0
$445$	5.15445	0.244345
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−23.2900	−1.09912	−0.549562	−	0.835453i	$-0.685205\pi$
$449$	−23.2900	−1.09912	−0.549562	+	0.835453i	$0.685205\pi$
$450$	0	0
$451$	−11.0502	−0.520335
$452$	0	0
$453$	0	0
$454$	0	0
$455$	23.0556	1.08086
$456$	0	0
$457$	40.5964	1.89902	0.949509	−	0.313739i	$-0.101582\pi$
$457$	40.5964	1.89902	0.949509	+	0.313739i	$0.101582\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.5980	1.05250	0.526248	−	0.850331i	$-0.323598\pi$
$461$	22.5980	1.05250	0.526248	+	0.850331i	$0.323598\pi$
$462$	0	0
$463$	26.2943	1.22200	0.611001	−	0.791630i	$-0.290767\pi$
$463$	26.2943	1.22200	0.611001	+	0.791630i	$0.290767\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.81481	0.454175	0.227088	−	0.973874i	$-0.427080\pi$
$467$	9.81481	0.454175	0.227088	+	0.973874i	$0.427080\pi$
$468$	0	0
$469$	−19.7876	−0.913704
$470$	0	0
$471$	0	0
$472$	0	0
$473$	59.9634	2.75712
$474$	0	0
$475$	−2.75505	−0.126410
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.47060	0.158576	0.0792879	−	0.996852i	$-0.474735\pi$
$479$	3.47060	0.158576	0.0792879	+	0.996852i	$0.474735\pi$
$480$	0	0
$481$	−0.727159	−0.0331556
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.96612	−0.316315
$486$	0	0
$487$	−9.95519	−0.451113	−0.225556	−	0.974230i	$-0.572420\pi$
$487$	−9.95519	−0.451113	−0.225556	+	0.974230i	$0.572420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−27.8786	−1.25815	−0.629073	−	0.777347i	$-0.716565\pi$
$491$	−27.8786	−1.25815	−0.629073	+	0.777347i	$0.716565\pi$
$492$	0	0
$493$	9.92344	0.446929
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−48.2151	−2.16274
$498$	0	0
$499$	2.53359	0.113419	0.0567095	−	0.998391i	$-0.481939\pi$
$499$	2.53359	0.113419	0.0567095	+	0.998391i	$0.481939\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	43.2198	1.92708	0.963538	−	0.267570i	$-0.0862206\pi$
$503$	43.2198	1.92708	0.963538	+	0.267570i	$0.0862206\pi$
$504$	0	0
$505$	15.1227	0.672952
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.2085	−0.940051	−0.470026	−	0.882653i	$-0.655755\pi$
$509$	−21.2085	−0.940051	−0.470026	+	0.882653i	$0.655755\pi$
$510$	0	0
$511$	−37.3645	−1.65291
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.899619	0.0396419
$516$	0	0
$517$	−32.3378	−1.42222
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−37.2053	−1.63000	−0.814998	−	0.579464i	$-0.803262\pi$
$521$	−37.2053	−1.63000	−0.814998	+	0.579464i	$0.803262\pi$
$522$	0	0
$523$	1.78273	0.0779535	0.0389767	−	0.999240i	$-0.487590\pi$
$523$	1.78273	0.0779535	0.0389767	+	0.999240i	$0.487590\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.36753	−0.0595706
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14.9777	0.648757
$534$	0	0
$535$	−1.99581	−0.0862864
$536$	0	0
$537$	0	0
$538$	0	0
$539$	27.5629	1.18722
$540$	0	0
$541$	13.9687	0.600562	0.300281	−	0.953851i	$-0.402919\pi$
$541$	13.9687	0.600562	0.300281	+	0.953851i	$0.402919\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.75505	−0.118013
$546$	0	0
$547$	38.6343	1.65188	0.825942	−	0.563755i	$-0.190644\pi$
$547$	38.6343	1.65188	0.825942	+	0.563755i	$0.190644\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.90112	−0.336599
$552$	0	0
$553$	−1.78248	−0.0757987
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.53271	−0.192057	−0.0960285	−	0.995379i	$-0.530614\pi$
$557$	−4.53271	−0.192057	−0.0960285	+	0.995379i	$0.530614\pi$
$558$	0	0
$559$	−81.2757	−3.43760
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.2474	−0.431877	−0.215939	−	0.976407i	$-0.569281\pi$
$563$	−10.2474	−0.431877	−0.215939	+	0.976407i	$0.569281\pi$
$564$	0	0
$565$	19.8064	0.833263
$566$	0	0
$567$	0	0
$568$	0	0
$569$	41.4786	1.73887	0.869437	−	0.494044i	$-0.164482\pi$
$569$	41.4786	1.73887	0.869437	+	0.494044i	$0.164482\pi$
$570$	0	0
$571$	−13.3501	−0.558683	−0.279341	−	0.960192i	$-0.590116\pi$
$571$	−13.3501	−0.558683	−0.279341	+	0.960192i	$0.590116\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	37.1356	1.54598	0.772988	−	0.634421i	$-0.218761\pi$
$577$	37.1356	1.54598	0.772988	+	0.634421i	$0.218761\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	45.9627	1.90685
$582$	0	0
$583$	−6.72612	−0.278567
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.6990	−0.524146	−0.262073	−	0.965048i	$-0.584406\pi$
$587$	−12.6990	−0.524146	−0.262073	+	0.965048i	$0.584406\pi$
$588$	0	0
$589$	1.08884	0.0448648
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.69671	−0.192871	−0.0964353	−	0.995339i	$-0.530744\pi$
$593$	−4.69671	−0.192871	−0.0964353	+	0.995339i	$0.530744\pi$
$594$	0	0
$595$	−12.3780	−0.507447
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.9567	−1.10142	−0.550710	−	0.834696i	$-0.685643\pi$
$599$	−26.9567	−1.10142	−0.550710	+	0.834696i	$0.685643\pi$
$600$	0	0
$601$	−15.2770	−0.623163	−0.311582	−	0.950219i	$-0.600859\pi$
$601$	−15.2770	−0.623163	−0.311582	+	0.950219i	$0.600859\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11.6105	0.472033
$606$	0	0
$607$	−16.2623	−0.660068	−0.330034	−	0.943969i	$-0.607060\pi$
$607$	−16.2623	−0.660068	−0.330034	+	0.943969i	$0.607060\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	43.8314	1.77323
$612$	0	0
$613$	11.8492	0.478586	0.239293	−	0.970947i	$-0.423084\pi$
$613$	11.8492	0.478586	0.239293	+	0.970947i	$0.423084\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−42.8298	−1.72426	−0.862132	−	0.506683i	$-0.830871\pi$
$617$	−42.8298	−1.72426	−0.862132	+	0.506683i	$0.830871\pi$
$618$	0	0
$619$	36.5054	1.46728	0.733638	−	0.679541i	$-0.237821\pi$
$619$	36.5054	1.46728	0.733638	+	0.679541i	$0.237821\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−18.4386	−0.738729
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.390394	0.0155660
$630$	0	0
$631$	31.5110	1.25443	0.627216	−	0.778845i	$-0.284194\pi$
$631$	31.5110	1.25443	0.627216	+	0.778845i	$0.284194\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.2111	0.484582
$636$	0	0
$637$	−37.3593	−1.48023
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.8728	0.666436	0.333218	−	0.942850i	$-0.391866\pi$
$641$	16.8728	0.666436	0.333218	+	0.942850i	$0.391866\pi$
$642$	0	0
$643$	10.9781	0.432935	0.216468	−	0.976290i	$-0.430546\pi$
$643$	10.9781	0.432935	0.216468	+	0.976290i	$0.430546\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.2604	−0.757205	−0.378602	−	0.925559i	$-0.623595\pi$
$647$	−19.2604	−0.757205	−0.378602	+	0.925559i	$0.623595\pi$
$648$	0	0
$649$	21.4559	0.842216
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.9754	0.977362	0.488681	−	0.872462i	$-0.337478\pi$
$653$	24.9754	0.977362	0.488681	+	0.872462i	$0.337478\pi$
$654$	0	0
$655$	11.1545	0.435841
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−19.9011	−0.775238	−0.387619	−	0.921820i	$-0.626702\pi$
$659$	−19.9011	−0.775238	−0.387619	+	0.921820i	$0.626702\pi$
$660$	0	0
$661$	−32.5521	−1.26613	−0.633066	−	0.774098i	$-0.718204\pi$
$661$	−32.5521	−1.26613	−0.633066	+	0.774098i	$0.718204\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.85543	0.382177
$666$	0	0
$667$	2.86787	0.111044
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.786383	−0.0303580
$672$	0	0
$673$	6.67074	0.257138	0.128569	−	0.991701i	$-0.458962\pi$
$673$	6.67074	0.257138	0.128569	+	0.991701i	$0.458962\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.87420	−0.0720312	−0.0360156	−	0.999351i	$-0.511467\pi$
$677$	−1.87420	−0.0720312	−0.0360156	+	0.999351i	$0.511467\pi$
$678$	0	0
$679$	24.9194	0.956318
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.2195	−0.926732	−0.463366	−	0.886167i	$-0.653358\pi$
$683$	−24.2195	−0.926732	−0.463366	+	0.886167i	$0.653358\pi$
$684$	0	0
$685$	−15.4258	−0.589389
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.11673	0.347320
$690$	0	0
$691$	−40.4823	−1.54002	−0.770009	−	0.638033i	$-0.779748\pi$
$691$	−40.4823	−1.54002	−0.770009	+	0.638033i	$0.779748\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.796550	−0.0302148
$696$	0	0
$697$	−8.04117	−0.304581
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.2778	0.690343	0.345171	−	0.938540i	$-0.387821\pi$
$701$	18.2778	0.690343	0.345171	+	0.938540i	$0.387821\pi$
$702$	0	0
$703$	−0.310834	−0.0117233
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−54.0973	−2.03454
$708$	0	0
$709$	18.1798	0.682757	0.341378	−	0.939926i	$-0.389106\pi$
$709$	18.1798	0.682757	0.341378	+	0.939926i	$0.389106\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.395216	−0.0148009
$714$	0	0
$715$	−30.6467	−1.14612
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.9743	−1.56538	−0.782689	−	0.622413i	$-0.786152\pi$
$719$	−41.9743	−1.56538	−0.782689	+	0.622413i	$0.786152\pi$
$720$	0	0
$721$	−3.21814	−0.119850
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.86787	0.106510
$726$	0	0
$727$	−10.0380	−0.372288	−0.186144	−	0.982522i	$-0.559599\pi$
$727$	−10.0380	−0.372288	−0.186144	+	0.982522i	$0.559599\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	43.6349	1.61390
$732$	0	0
$733$	−24.7327	−0.913524	−0.456762	−	0.889589i	$-0.650991\pi$
$733$	−24.7327	−0.913524	−0.456762	+	0.889589i	$0.650991\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.3027	0.968873
$738$	0	0
$739$	−24.8341	−0.913538	−0.456769	−	0.889585i	$-0.650993\pi$
$739$	−24.8341	−0.913538	−0.456769	+	0.889585i	$0.650993\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.78039	0.358808	0.179404	−	0.983776i	$-0.442583\pi$
$743$	9.78039	0.358808	0.179404	+	0.983776i	$0.442583\pi$
$744$	0	0
$745$	8.26514	0.302811
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.13946	0.260870
$750$	0	0
$751$	45.6650	1.66634	0.833170	−	0.553017i	$-0.186524\pi$
$751$	45.6650	1.66634	0.833170	+	0.553017i	$0.186524\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.6105	0.677305
$756$	0	0
$757$	−34.5974	−1.25746	−0.628732	−	0.777622i	$-0.716426\pi$
$757$	−34.5974	−1.25746	−0.628732	+	0.777622i	$0.716426\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.1810	−0.550312	−0.275156	−	0.961400i	$-0.588730\pi$
$761$	−15.1810	−0.550312	−0.275156	+	0.961400i	$0.588730\pi$
$762$	0	0
$763$	9.85543	0.356790
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−29.0818	−1.05008
$768$	0	0
$769$	−37.3810	−1.34799	−0.673996	−	0.738735i	$-0.735424\pi$
$769$	−37.3810	−1.34799	−0.673996	+	0.738735i	$0.735424\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.5135	0.665885	0.332943	−	0.942947i	$-0.391958\pi$
$773$	18.5135	0.665885	0.332943	+	0.942947i	$0.391958\pi$
$774$	0	0
$775$	−0.395216	−0.0141966
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.40244	0.229391
$780$	0	0
$781$	64.0902	2.29333
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−18.8231	−0.671824
$786$	0	0
$787$	−8.04206	−0.286668	−0.143334	−	0.989674i	$-0.545782\pi$
$787$	−8.04206	−0.286668	−0.143334	+	0.989674i	$0.545782\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−70.8521	−2.51921
$792$	0	0
$793$	1.06588	0.0378506
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.0328	−1.09924	−0.549618	−	0.835416i	$-0.685227\pi$
$797$	−31.0328	−1.09924	−0.549618	+	0.835416i	$0.685227\pi$
$798$	0	0
$799$	−23.5320	−0.832503
$800$	0	0
$801$	0	0
$802$	0	0
$803$	49.6669	1.75271
$804$	0	0
$805$	−3.57723	−0.126081
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−37.6748	−1.32458	−0.662288	−	0.749249i	$-0.730415\pi$
$809$	−37.6748	−1.32458	−0.662288	+	0.749249i	$0.730415\pi$
$810$	0	0
$811$	8.18503	0.287415	0.143708	−	0.989620i	$-0.454098\pi$
$811$	8.18503	0.287415	0.143708	+	0.989620i	$0.454098\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.07957	0.0378158
$816$	0	0
$817$	−34.7425	−1.21548
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.51952	0.122832	0.0614160	−	0.998112i	$-0.480438\pi$
$821$	3.51952	0.122832	0.0614160	+	0.998112i	$0.480438\pi$
$822$	0	0
$823$	−46.5138	−1.62137	−0.810684	−	0.585484i	$-0.800905\pi$
$823$	−46.5138	−1.62137	−0.810684	+	0.585484i	$0.800905\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.1550	−0.422669	−0.211335	−	0.977414i	$-0.567781\pi$
$827$	−12.1550	−0.422669	−0.211335	+	0.977414i	$0.567781\pi$
$828$	0	0
$829$	−27.2024	−0.944779	−0.472389	−	0.881390i	$-0.656608\pi$
$829$	−27.2024	−0.944779	−0.472389	+	0.881390i	$0.656608\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	20.0573	0.694944
$834$	0	0
$835$	9.94681	0.344224
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27.8522	0.961566	0.480783	−	0.876840i	$-0.340352\pi$
$839$	27.8522	0.961566	0.480783	+	0.876840i	$0.340352\pi$
$840$	0	0
$841$	−20.7753	−0.716390
$842$	0	0
$843$	0	0
$844$	0	0
$845$	28.5393	0.981781
$846$	0	0
$847$	−41.5333	−1.42710
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.112824	0.00386754
$852$	0	0
$853$	−26.7437	−0.915687	−0.457843	−	0.889033i	$-0.651378\pi$
$853$	−26.7437	−0.915687	−0.457843	+	0.889033i	$0.651378\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30.6843	−1.04816	−0.524078	−	0.851670i	$-0.675590\pi$
$857$	−30.6843	−1.04816	−0.524078	+	0.851670i	$0.675590\pi$
$858$	0	0
$859$	0.561699	0.0191649	0.00958247	−	0.999954i	$-0.496950\pi$
$859$	0.561699	0.0191649	0.00958247	+	0.999954i	$0.496950\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.6645	−1.11191	−0.555957	−	0.831211i	$-0.687648\pi$
$863$	−32.6645	−1.11191	−0.555957	+	0.831211i	$0.687648\pi$
$864$	0	0
$865$	−0.945311	−0.0321415
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.36937	0.0803753
$870$	0	0
$871$	−35.6513	−1.20800
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.57723	−0.120932
$876$	0	0
$877$	17.7126	0.598113	0.299056	−	0.954235i	$-0.403328\pi$
$877$	17.7126	0.598113	0.299056	+	0.954235i	$0.403328\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−0.955810	−0.0322021	−0.0161010	−	0.999870i	$-0.505125\pi$
$881$	−0.955810	−0.0322021	−0.0161010	+	0.999870i	$0.505125\pi$
$882$	0	0
$883$	0.357570	0.0120332	0.00601660	−	0.999982i	$-0.498085\pi$
$883$	0.357570	0.0120332	0.00601660	+	0.999982i	$0.498085\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.1907	−1.04728	−0.523641	−	0.851939i	$-0.675427\pi$
$887$	−31.1907	−1.04728	−0.523641	+	0.851939i	$0.675427\pi$
$888$	0	0
$889$	−43.6818	−1.46504
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.7364	0.626988
$894$	0	0
$895$	18.4762	0.617592
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.13343	−0.0378019
$900$	0	0
$901$	−4.89454	−0.163061
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.6806	−0.654206
$906$	0	0
$907$	−8.67226	−0.287958	−0.143979	−	0.989581i	$-0.545990\pi$
$907$	−8.67226	−0.287958	−0.143979	+	0.989581i	$0.545990\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.72762	−0.123501	−0.0617507	−	0.998092i	$-0.519668\pi$
$911$	−3.72762	−0.123501	−0.0617507	+	0.998092i	$0.519668\pi$
$912$	0	0
$913$	−61.0961	−2.02199
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−39.9020	−1.31768
$918$	0	0
$919$	−46.9765	−1.54961	−0.774805	−	0.632200i	$-0.782152\pi$
$919$	−46.9765	−1.54961	−0.774805	+	0.632200i	$0.782152\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−86.8693	−2.85934
$924$	0	0
$925$	0.112824	0.00370962
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.7134	−0.909248	−0.454624	−	0.890684i	$-0.650226\pi$
$929$	−27.7134	−0.909248	−0.454624	+	0.890684i	$0.650226\pi$
$930$	0	0
$931$	−15.9698	−0.523388
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16.4535	0.538086
$936$	0	0
$937$	−30.4376	−0.994353	−0.497177	−	0.867649i	$-0.665630\pi$
$937$	−30.4376	−0.994353	−0.497177	+	0.867649i	$0.665630\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.8447	0.353526	0.176763	−	0.984253i	$-0.443437\pi$
$941$	10.8447	0.353526	0.176763	+	0.984253i	$0.443437\pi$
$942$	0	0
$943$	−2.32389	−0.0756764
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.412938	0.0134187	0.00670934	−	0.999977i	$-0.497864\pi$
$947$	0.412938	0.0134187	0.00670934	+	0.999977i	$0.497864\pi$
$948$	0	0
$949$	−67.3197	−2.18529
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−9.56287	−0.309772	−0.154886	−	0.987932i	$-0.549501\pi$
$953$	−9.56287	−0.309772	−0.154886	+	0.987932i	$0.549501\pi$
$954$	0	0
$955$	14.7248	0.476484
$956$	0	0
$957$	0	0
$958$	0	0
$959$	55.1816	1.78191
$960$	0	0
$961$	−30.8438	−0.994961
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−21.8106	−0.702109
$966$	0	0
$967$	8.43589	0.271280	0.135640	−	0.990758i	$-0.456691\pi$
$967$	8.43589	0.271280	0.135640	+	0.990758i	$0.456691\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−39.9520	−1.28212	−0.641060	−	0.767490i	$-0.721505\pi$
$971$	−39.9520	−1.28212	−0.641060	+	0.767490i	$0.721505\pi$
$972$	0	0
$973$	2.84944	0.0913488
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.4770	−0.623124	−0.311562	−	0.950226i	$-0.600852\pi$
$977$	−19.4770	−0.623124	−0.311562	+	0.950226i	$0.600852\pi$
$978$	0	0
$979$	24.5097	0.783332
$980$	0	0
$981$	0	0
$982$	0	0
$983$	43.6295	1.39157	0.695783	−	0.718252i	$-0.255058\pi$
$983$	43.6295	1.39157	0.695783	+	0.718252i	$0.255058\pi$
$984$	0	0
$985$	16.0446	0.511225
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.6105	0.400990
$990$	0	0
$991$	−5.34375	−0.169750	−0.0848749	−	0.996392i	$-0.527049\pi$
$991$	−5.34375	−0.169750	−0.0848749	+	0.996392i	$0.527049\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.928806	−0.0294451
$996$	0	0
$997$	−19.7774	−0.626355	−0.313178	−	0.949695i	$-0.601394\pi$
$997$	−19.7774	−0.626355	−0.313178	+	0.949695i	$0.601394\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.u.1.1	yes	5
3.2	odd	2		4140.2.a.t.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.a.t.1.1	✓	5	3.2	odd	2
4140.2.a.u.1.1	yes	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.57723 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.57723 q^{7} +4.75505 q^{11} -6.44510 q^{13} +3.46021 q^{17} -2.75505 q^{19} +1.00000 q^{23} +1.00000 q^{25} +2.86787 q^{29} -0.395216 q^{31} -3.57723 q^{35} +0.112824 q^{37} -2.32389 q^{41} +12.6105 q^{43} -6.80074 q^{47} +5.79655 q^{49} -1.41452 q^{53} +4.75505 q^{55} +4.51223 q^{59} -0.165379 q^{61} -6.44510 q^{65} +5.53153 q^{67} +13.4783 q^{71} +10.4451 q^{73} -17.0099 q^{77} +0.498285 q^{79} -12.8487 q^{83} +3.46021 q^{85} +5.15445 q^{89} +23.0556 q^{91} -2.75505 q^{95} -6.96612 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} + 4 q^{7}+O(q^{10}) \) 5 * q + 5 * q^5 + 4 * q^7	\( 5 q + 5 q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{23} + 5 q^{25} + 2 q^{29} + 8 q^{31} + 4 q^{35} + 8 q^{37} + 2 q^{41} + 18 q^{43} - 4 q^{47} + 13 q^{49} - 2 q^{53} + 4 q^{55} + 6 q^{59} + 10 q^{61} + 2 q^{65} + 16 q^{67} + 10 q^{71} + 18 q^{73} - 16 q^{77} + 14 q^{79} + 8 q^{83} + 2 q^{85} - 18 q^{89} + 36 q^{91} + 6 q^{95} + 6 q^{97}+O(q^{100}) \) 5 * q + 5 * q^5 + 4 * q^7 + 4 * q^11 + 2 * q^13 + 2 * q^17 + 6 * q^19 + 5 * q^23 + 5 * q^25 + 2 * q^29 + 8 * q^31 + 4 * q^35 + 8 * q^37 + 2 * q^41 + 18 * q^43 - 4 * q^47 + 13 * q^49 - 2 * q^53 + 4 * q^55 + 6 * q^59 + 10 * q^61 + 2 * q^65 + 16 * q^67 + 10 * q^71 + 18 * q^73 - 16 * q^77 + 14 * q^79 + 8 * q^83 + 2 * q^85 - 18 * q^89 + 36 * q^91 + 6 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more