LMFDB - Embedded newform 6040.2.a.o.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 $ x^15 - 5x^14 - 19x^13 + 119x^12 + 106x^11 - 1063x^10 - 48x^9 + 4510x^8 - 1130x^7 - 9512x^6 + 2839x^5 + 9383x^4 - 1781x^3 - 3692x^2 - 92x + 272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$2.38022$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.38022 q^{3} +1.00000 q^{5} -1.25626 q^{7} +2.66546 q^{9} +O(q^{10})$	$q+2.38022 q^{3} +1.00000 q^{5} -1.25626 q^{7} +2.66546 q^{9} -0.522516 q^{11} +5.53122 q^{13} +2.38022 q^{15} +7.81458 q^{17} +2.25452 q^{19} -2.99018 q^{21} -0.0687708 q^{23} +1.00000 q^{25} -0.796285 q^{27} -4.41337 q^{29} +4.67884 q^{31} -1.24370 q^{33} -1.25626 q^{35} -2.62736 q^{37} +13.1655 q^{39} -6.67284 q^{41} +1.95670 q^{43} +2.66546 q^{45} +7.37690 q^{47} -5.42180 q^{49} +18.6004 q^{51} +0.411478 q^{53} -0.522516 q^{55} +5.36625 q^{57} +1.50728 q^{59} +1.38076 q^{61} -3.34851 q^{63} +5.53122 q^{65} +3.55045 q^{67} -0.163690 q^{69} +7.24182 q^{71} -15.0784 q^{73} +2.38022 q^{75} +0.656417 q^{77} -2.28572 q^{79} -9.89171 q^{81} +4.27243 q^{83} +7.81458 q^{85} -10.5048 q^{87} -6.50052 q^{89} -6.94867 q^{91} +11.1367 q^{93} +2.25452 q^{95} +7.46349 q^{97} -1.39274 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * 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
$3$	2.38022	1.37422	0.687111	−	0.726552i	$-0.258879\pi$
$3$	2.38022	1.37422	0.687111	+	0.726552i	$0.258879\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.25626	−0.474823	−0.237411	−	0.971409i	$-0.576299\pi$
$7$	−1.25626	−0.474823	−0.237411	+	0.971409i	$0.576299\pi$
$8$	0	0
$9$	2.66546	0.888486
$10$	0	0
$11$	−0.522516	−0.157545	−0.0787723	−	0.996893i	$-0.525100\pi$
$11$	−0.522516	−0.157545	−0.0787723	+	0.996893i	$0.525100\pi$
$12$	0	0
$13$	5.53122	1.53408	0.767042	−	0.641596i	$-0.221728\pi$
$13$	5.53122	1.53408	0.767042	+	0.641596i	$0.221728\pi$
$14$	0	0
$15$	2.38022	0.614571
$16$	0	0
$17$	7.81458	1.89531	0.947657	−	0.319290i	$-0.103444\pi$
$17$	7.81458	1.89531	0.947657	+	0.319290i	$0.103444\pi$
$18$	0	0
$19$	2.25452	0.517222	0.258611	−	0.965982i	$-0.416735\pi$
$19$	2.25452	0.517222	0.258611	+	0.965982i	$0.416735\pi$
$20$	0	0
$21$	−2.99018	−0.652512
$22$	0	0
$23$	−0.0687708	−0.0143397	−0.00716985	−	0.999974i	$-0.502282\pi$
$23$	−0.0687708	−0.0143397	−0.00716985	+	0.999974i	$0.502282\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.796285	−0.153245
$28$	0	0
$29$	−4.41337	−0.819542	−0.409771	−	0.912188i	$-0.634391\pi$
$29$	−4.41337	−0.819542	−0.409771	+	0.912188i	$0.634391\pi$
$30$	0	0
$31$	4.67884	0.840344	0.420172	−	0.907444i	$-0.361970\pi$
$31$	4.67884	0.840344	0.420172	+	0.907444i	$0.361970\pi$
$32$	0	0
$33$	−1.24370	−0.216501
$34$	0	0
$35$	−1.25626	−0.212347
$36$	0	0
$37$	−2.62736	−0.431935	−0.215968	−	0.976401i	$-0.569291\pi$
$37$	−2.62736	−0.431935	−0.215968	+	0.976401i	$0.569291\pi$
$38$	0	0
$39$	13.1655	2.10817
$40$	0	0
$41$	−6.67284	−1.04212	−0.521061	−	0.853519i	$-0.674464\pi$
$41$	−6.67284	−1.04212	−0.521061	+	0.853519i	$0.674464\pi$
$42$	0	0
$43$	1.95670	0.298395	0.149197	−	0.988807i	$-0.452331\pi$
$43$	1.95670	0.298395	0.149197	+	0.988807i	$0.452331\pi$
$44$	0	0
$45$	2.66546	0.397343
$46$	0	0
$47$	7.37690	1.07603	0.538016	−	0.842935i	$-0.319174\pi$
$47$	7.37690	1.07603	0.538016	+	0.842935i	$0.319174\pi$
$48$	0	0
$49$	−5.42180	−0.774543
$50$	0	0
$51$	18.6004	2.60458
$52$	0	0
$53$	0.411478	0.0565208	0.0282604	−	0.999601i	$-0.491003\pi$
$53$	0.411478	0.0565208	0.0282604	+	0.999601i	$0.491003\pi$
$54$	0	0
$55$	−0.522516	−0.0704561
$56$	0	0
$57$	5.36625	0.710778
$58$	0	0
$59$	1.50728	0.196231	0.0981153	−	0.995175i	$-0.468719\pi$
$59$	1.50728	0.196231	0.0981153	+	0.995175i	$0.468719\pi$
$60$	0	0
$61$	1.38076	0.176788	0.0883940	−	0.996086i	$-0.471827\pi$
$61$	1.38076	0.176788	0.0883940	+	0.996086i	$0.471827\pi$
$62$	0	0
$63$	−3.34851	−0.421873
$64$	0	0
$65$	5.53122	0.686064
$66$	0	0
$67$	3.55045	0.433757	0.216878	−	0.976199i	$-0.430412\pi$
$67$	3.55045	0.433757	0.216878	+	0.976199i	$0.430412\pi$
$68$	0	0
$69$	−0.163690	−0.0197059
$70$	0	0
$71$	7.24182	0.859446	0.429723	−	0.902961i	$-0.358611\pi$
$71$	7.24182	0.859446	0.429723	+	0.902961i	$0.358611\pi$
$72$	0	0
$73$	−15.0784	−1.76479	−0.882394	−	0.470511i	$-0.844070\pi$
$73$	−15.0784	−1.76479	−0.882394	+	0.470511i	$0.844070\pi$
$74$	0	0
$75$	2.38022	0.274844
$76$	0	0
$77$	0.656417	0.0748057
$78$	0	0
$79$	−2.28572	−0.257164	−0.128582	−	0.991699i	$-0.541043\pi$
$79$	−2.28572	−0.257164	−0.128582	+	0.991699i	$0.541043\pi$
$80$	0	0
$81$	−9.89171	−1.09908
$82$	0	0
$83$	4.27243	0.468961	0.234480	−	0.972121i	$-0.424661\pi$
$83$	4.27243	0.468961	0.234480	+	0.972121i	$0.424661\pi$
$84$	0	0
$85$	7.81458	0.847610
$86$	0	0
$87$	−10.5048	−1.12623
$88$	0	0
$89$	−6.50052	−0.689054	−0.344527	−	0.938776i	$-0.611961\pi$
$89$	−6.50052	−0.689054	−0.344527	+	0.938776i	$0.611961\pi$
$90$	0	0
$91$	−6.94867	−0.728418
$92$	0	0
$93$	11.1367	1.15482
$94$	0	0
$95$	2.25452	0.231309
$96$	0	0
$97$	7.46349	0.757802	0.378901	−	0.925437i	$-0.376302\pi$
$97$	7.46349	0.757802	0.378901	+	0.925437i	$0.376302\pi$
$98$	0	0
$99$	−1.39274	−0.139976
$100$	0	0
$101$	−7.23226	−0.719636	−0.359818	−	0.933022i	$-0.617161\pi$
$101$	−7.23226	−0.719636	−0.359818	+	0.933022i	$0.617161\pi$
$102$	0	0
$103$	3.45353	0.340287	0.170143	−	0.985419i	$-0.445577\pi$
$103$	3.45353	0.340287	0.170143	+	0.985419i	$0.445577\pi$
$104$	0	0
$105$	−2.99018	−0.291812
$106$	0	0
$107$	12.9147	1.24851	0.624256	−	0.781220i	$-0.285402\pi$
$107$	12.9147	1.24851	0.624256	+	0.781220i	$0.285402\pi$
$108$	0	0
$109$	−1.79149	−0.171594	−0.0857970	−	0.996313i	$-0.527344\pi$
$109$	−1.79149	−0.171594	−0.0857970	+	0.996313i	$0.527344\pi$
$110$	0	0
$111$	−6.25370	−0.593575
$112$	0	0
$113$	−5.59252	−0.526100	−0.263050	−	0.964782i	$-0.584728\pi$
$113$	−5.59252	−0.526100	−0.263050	+	0.964782i	$0.584728\pi$
$114$	0	0
$115$	−0.0687708	−0.00641291
$116$	0	0
$117$	14.7432	1.36301
$118$	0	0
$119$	−9.81717	−0.899938
$120$	0	0
$121$	−10.7270	−0.975180
$122$	0	0
$123$	−15.8828	−1.43211
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.8368	1.22782	0.613911	−	0.789375i	$-0.289595\pi$
$127$	13.8368	1.22782	0.613911	+	0.789375i	$0.289595\pi$
$128$	0	0
$129$	4.65739	0.410061
$130$	0	0
$131$	−11.9743	−1.04620	−0.523101	−	0.852271i	$-0.675225\pi$
$131$	−11.9743	−1.04620	−0.523101	+	0.852271i	$0.675225\pi$
$132$	0	0
$133$	−2.83227	−0.245589
$134$	0	0
$135$	−0.796285	−0.0685333
$136$	0	0
$137$	11.6180	0.992592	0.496296	−	0.868153i	$-0.334693\pi$
$137$	11.6180	0.992592	0.496296	+	0.868153i	$0.334693\pi$
$138$	0	0
$139$	10.0588	0.853174	0.426587	−	0.904446i	$-0.359716\pi$
$139$	10.0588	0.853174	0.426587	+	0.904446i	$0.359716\pi$
$140$	0	0
$141$	17.5587	1.47871
$142$	0	0
$143$	−2.89015	−0.241687
$144$	0	0
$145$	−4.41337	−0.366510
$146$	0	0
$147$	−12.9051	−1.06439
$148$	0	0
$149$	6.75997	0.553798	0.276899	−	0.960899i	$-0.410693\pi$
$149$	6.75997	0.553798	0.276899	+	0.960899i	$0.410693\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	20.8294	1.68396
$154$	0	0
$155$	4.67884	0.375813
$156$	0	0
$157$	6.73063	0.537163	0.268581	−	0.963257i	$-0.413445\pi$
$157$	6.73063	0.537163	0.268581	+	0.963257i	$0.413445\pi$
$158$	0	0
$159$	0.979409	0.0776722
$160$	0	0
$161$	0.0863942	0.00680882
$162$	0	0
$163$	1.53003	0.119841	0.0599207	−	0.998203i	$-0.480915\pi$
$163$	1.53003	0.119841	0.0599207	+	0.998203i	$0.480915\pi$
$164$	0	0
$165$	−1.24370	−0.0968223
$166$	0	0
$167$	15.5085	1.20009	0.600044	−	0.799967i	$-0.295150\pi$
$167$	15.5085	1.20009	0.600044	+	0.799967i	$0.295150\pi$
$168$	0	0
$169$	17.5944	1.35342
$170$	0	0
$171$	6.00932	0.459544
$172$	0	0
$173$	7.11380	0.540852	0.270426	−	0.962741i	$-0.412835\pi$
$173$	7.11380	0.540852	0.270426	+	0.962741i	$0.412835\pi$
$174$	0	0
$175$	−1.25626	−0.0949645
$176$	0	0
$177$	3.58765	0.269664
$178$	0	0
$179$	2.50505	0.187236	0.0936182	−	0.995608i	$-0.470157\pi$
$179$	2.50505	0.187236	0.0936182	+	0.995608i	$0.470157\pi$
$180$	0	0
$181$	6.37499	0.473849	0.236925	−	0.971528i	$-0.423861\pi$
$181$	6.37499	0.473849	0.236925	+	0.971528i	$0.423861\pi$
$182$	0	0
$183$	3.28651	0.242946
$184$	0	0
$185$	−2.62736	−0.193167
$186$	0	0
$187$	−4.08324	−0.298596
$188$	0	0
$189$	1.00034	0.0727643
$190$	0	0
$191$	8.97690	0.649546	0.324773	−	0.945792i	$-0.394712\pi$
$191$	8.97690	0.649546	0.324773	+	0.945792i	$0.394712\pi$
$192$	0	0
$193$	3.24456	0.233548	0.116774	−	0.993159i	$-0.462745\pi$
$193$	3.24456	0.233548	0.116774	+	0.993159i	$0.462745\pi$
$194$	0	0
$195$	13.1655	0.942804
$196$	0	0
$197$	21.7073	1.54658	0.773289	−	0.634053i	$-0.218610\pi$
$197$	21.7073	1.54658	0.773289	+	0.634053i	$0.218610\pi$
$198$	0	0
$199$	−5.56000	−0.394138	−0.197069	−	0.980390i	$-0.563142\pi$
$199$	−5.56000	−0.394138	−0.197069	+	0.980390i	$0.563142\pi$
$200$	0	0
$201$	8.45087	0.596078
$202$	0	0
$203$	5.54435	0.389137
$204$	0	0
$205$	−6.67284	−0.466051
$206$	0	0
$207$	−0.183306	−0.0127406
$208$	0	0
$209$	−1.17802	−0.0814855
$210$	0	0
$211$	12.9994	0.894915	0.447457	−	0.894305i	$-0.352330\pi$
$211$	12.9994	0.894915	0.447457	+	0.894305i	$0.352330\pi$
$212$	0	0
$213$	17.2372	1.18107
$214$	0	0
$215$	1.95670	0.133446
$216$	0	0
$217$	−5.87785	−0.399014
$218$	0	0
$219$	−35.8898	−2.42521
$220$	0	0
$221$	43.2242	2.90757
$222$	0	0
$223$	−6.96459	−0.466384	−0.233192	−	0.972431i	$-0.574917\pi$
$223$	−6.96459	−0.466384	−0.233192	+	0.972431i	$0.574917\pi$
$224$	0	0
$225$	2.66546	0.177697
$226$	0	0
$227$	15.7795	1.04732	0.523660	−	0.851927i	$-0.324566\pi$
$227$	15.7795	1.04732	0.523660	+	0.851927i	$0.324566\pi$
$228$	0	0
$229$	−19.4775	−1.28711	−0.643555	−	0.765399i	$-0.722541\pi$
$229$	−19.4775	−1.28711	−0.643555	+	0.765399i	$0.722541\pi$
$230$	0	0
$231$	1.56242	0.102800
$232$	0	0
$233$	−14.0744	−0.922046	−0.461023	−	0.887388i	$-0.652517\pi$
$233$	−14.0744	−0.922046	−0.461023	+	0.887388i	$0.652517\pi$
$234$	0	0
$235$	7.37690	0.481216
$236$	0	0
$237$	−5.44053	−0.353400
$238$	0	0
$239$	−18.5448	−1.19956	−0.599782	−	0.800163i	$-0.704746\pi$
$239$	−18.5448	−1.19956	−0.599782	+	0.800163i	$0.704746\pi$
$240$	0	0
$241$	5.15986	0.332376	0.166188	−	0.986094i	$-0.446854\pi$
$241$	5.15986	0.332376	0.166188	+	0.986094i	$0.446854\pi$
$242$	0	0
$243$	−21.1556	−1.35713
$244$	0	0
$245$	−5.42180	−0.346386
$246$	0	0
$247$	12.4702	0.793462
$248$	0	0
$249$	10.1693	0.644456
$250$	0	0
$251$	6.12822	0.386810	0.193405	−	0.981119i	$-0.438047\pi$
$251$	6.12822	0.386810	0.193405	+	0.981119i	$0.438047\pi$
$252$	0	0
$253$	0.0359339	0.00225914
$254$	0	0
$255$	18.6004	1.16480
$256$	0	0
$257$	−0.783119	−0.0488497	−0.0244248	−	0.999702i	$-0.507775\pi$
$257$	−0.783119	−0.0488497	−0.0244248	+	0.999702i	$0.507775\pi$
$258$	0	0
$259$	3.30066	0.205093
$260$	0	0
$261$	−11.7636	−0.728151
$262$	0	0
$263$	17.1647	1.05842	0.529210	−	0.848491i	$-0.322489\pi$
$263$	17.1647	1.05842	0.529210	+	0.848491i	$0.322489\pi$
$264$	0	0
$265$	0.411478	0.0252769
$266$	0	0
$267$	−15.4727	−0.946913
$268$	0	0
$269$	2.69495	0.164314	0.0821569	−	0.996619i	$-0.473819\pi$
$269$	2.69495	0.164314	0.0821569	+	0.996619i	$0.473819\pi$
$270$	0	0
$271$	15.2490	0.926308	0.463154	−	0.886278i	$-0.346718\pi$
$271$	15.2490	0.926308	0.463154	+	0.886278i	$0.346718\pi$
$272$	0	0
$273$	−16.5394	−1.00101
$274$	0	0
$275$	−0.522516	−0.0315089
$276$	0	0
$277$	−21.1816	−1.27268	−0.636339	−	0.771410i	$-0.719552\pi$
$277$	−21.1816	−1.27268	−0.636339	+	0.771410i	$0.719552\pi$
$278$	0	0
$279$	12.4712	0.746634
$280$	0	0
$281$	−21.2183	−1.26578	−0.632890	−	0.774242i	$-0.718131\pi$
$281$	−21.2183	−1.26578	−0.632890	+	0.774242i	$0.718131\pi$
$282$	0	0
$283$	−9.26169	−0.550550	−0.275275	−	0.961365i	$-0.588769\pi$
$283$	−9.26169	−0.550550	−0.275275	+	0.961365i	$0.588769\pi$
$284$	0	0
$285$	5.36625	0.317869
$286$	0	0
$287$	8.38284	0.494823
$288$	0	0
$289$	44.0677	2.59222
$290$	0	0
$291$	17.7648	1.04139
$292$	0	0
$293$	−32.5402	−1.90102	−0.950508	−	0.310700i	$-0.899437\pi$
$293$	−32.5402	−1.90102	−0.950508	+	0.310700i	$0.899437\pi$
$294$	0	0
$295$	1.50728	0.0877570
$296$	0	0
$297$	0.416072	0.0241429
$298$	0	0
$299$	−0.380386	−0.0219983
$300$	0	0
$301$	−2.45814	−0.141685
$302$	0	0
$303$	−17.2144	−0.988940
$304$	0	0
$305$	1.38076	0.0790620
$306$	0	0
$307$	−27.4098	−1.56436	−0.782179	−	0.623054i	$-0.785892\pi$
$307$	−27.4098	−1.56436	−0.782179	+	0.623054i	$0.785892\pi$
$308$	0	0
$309$	8.22018	0.467630
$310$	0	0
$311$	10.8488	0.615180	0.307590	−	0.951519i	$-0.400477\pi$
$311$	10.8488	0.615180	0.307590	+	0.951519i	$0.400477\pi$
$312$	0	0
$313$	−9.43444	−0.533266	−0.266633	−	0.963798i	$-0.585911\pi$
$313$	−9.43444	−0.533266	−0.266633	+	0.963798i	$0.585911\pi$
$314$	0	0
$315$	−3.34851	−0.188667
$316$	0	0
$317$	8.15919	0.458266	0.229133	−	0.973395i	$-0.426411\pi$
$317$	8.15919	0.458266	0.229133	+	0.973395i	$0.426411\pi$
$318$	0	0
$319$	2.30606	0.129114
$320$	0	0
$321$	30.7399	1.71573
$322$	0	0
$323$	17.6181	0.980298
$324$	0	0
$325$	5.53122	0.306817
$326$	0	0
$327$	−4.26415	−0.235808
$328$	0	0
$329$	−9.26733	−0.510924
$330$	0	0
$331$	−31.1761	−1.71359	−0.856797	−	0.515654i	$-0.827549\pi$
$331$	−31.1761	−1.71359	−0.856797	+	0.515654i	$0.827549\pi$
$332$	0	0
$333$	−7.00312	−0.383769
$334$	0	0
$335$	3.55045	0.193982
$336$	0	0
$337$	23.0151	1.25371	0.626857	−	0.779134i	$-0.284341\pi$
$337$	23.0151	1.25371	0.626857	+	0.779134i	$0.284341\pi$
$338$	0	0
$339$	−13.3114	−0.722978
$340$	0	0
$341$	−2.44477	−0.132392
$342$	0	0
$343$	15.6050	0.842593
$344$	0	0
$345$	−0.163690	−0.00881276
$346$	0	0
$347$	−31.1913	−1.67444	−0.837220	−	0.546867i	$-0.815820\pi$
$347$	−31.1913	−1.67444	−0.837220	+	0.546867i	$0.815820\pi$
$348$	0	0
$349$	−14.8317	−0.793922	−0.396961	−	0.917835i	$-0.629935\pi$
$349$	−14.8317	−0.793922	−0.396961	+	0.917835i	$0.629935\pi$
$350$	0	0
$351$	−4.40443	−0.235091
$352$	0	0
$353$	−6.08359	−0.323797	−0.161899	−	0.986807i	$-0.551762\pi$
$353$	−6.08359	−0.323797	−0.161899	+	0.986807i	$0.551762\pi$
$354$	0	0
$355$	7.24182	0.384356
$356$	0	0
$357$	−23.3670	−1.23671
$358$	0	0
$359$	−3.83328	−0.202313	−0.101156	−	0.994871i	$-0.532254\pi$
$359$	−3.83328	−0.202313	−0.101156	+	0.994871i	$0.532254\pi$
$360$	0	0
$361$	−13.9171	−0.732481
$362$	0	0
$363$	−25.5326	−1.34011
$364$	0	0
$365$	−15.0784	−0.789237
$366$	0	0
$367$	35.5046	1.85332	0.926662	−	0.375895i	$-0.122665\pi$
$367$	35.5046	1.85332	0.926662	+	0.375895i	$0.122665\pi$
$368$	0	0
$369$	−17.7862	−0.925911
$370$	0	0
$371$	−0.516924	−0.0268374
$372$	0	0
$373$	12.0095	0.621827	0.310913	−	0.950438i	$-0.399365\pi$
$373$	12.0095	0.621827	0.310913	+	0.950438i	$0.399365\pi$
$374$	0	0
$375$	2.38022	0.122914
$376$	0	0
$377$	−24.4113	−1.25725
$378$	0	0
$379$	−20.9004	−1.07358	−0.536791	−	0.843715i	$-0.680364\pi$
$379$	−20.9004	−1.07358	−0.536791	+	0.843715i	$0.680364\pi$
$380$	0	0
$381$	32.9348	1.68730
$382$	0	0
$383$	6.06705	0.310012	0.155006	−	0.987914i	$-0.450460\pi$
$383$	6.06705	0.310012	0.155006	+	0.987914i	$0.450460\pi$
$384$	0	0
$385$	0.656417	0.0334541
$386$	0	0
$387$	5.21551	0.265119
$388$	0	0
$389$	−15.4878	−0.785262	−0.392631	−	0.919696i	$-0.628435\pi$
$389$	−15.4878	−0.785262	−0.392631	+	0.919696i	$0.628435\pi$
$390$	0	0
$391$	−0.537415	−0.0271782
$392$	0	0
$393$	−28.5016	−1.43771
$394$	0	0
$395$	−2.28572	−0.115007
$396$	0	0
$397$	−27.7401	−1.39224	−0.696118	−	0.717927i	$-0.745091\pi$
$397$	−27.7401	−1.39224	−0.696118	+	0.717927i	$0.745091\pi$
$398$	0	0
$399$	−6.74142	−0.337493
$400$	0	0
$401$	−12.2523	−0.611850	−0.305925	−	0.952056i	$-0.598966\pi$
$401$	−12.2523	−0.611850	−0.305925	+	0.952056i	$0.598966\pi$
$402$	0	0
$403$	25.8797	1.28916
$404$	0	0
$405$	−9.89171	−0.491523
$406$	0	0
$407$	1.37284	0.0680491
$408$	0	0
$409$	30.7179	1.51890	0.759450	−	0.650565i	$-0.225468\pi$
$409$	30.7179	1.51890	0.759450	+	0.650565i	$0.225468\pi$
$410$	0	0
$411$	27.6534	1.36404
$412$	0	0
$413$	−1.89353	−0.0931748
$414$	0	0
$415$	4.27243	0.209726
$416$	0	0
$417$	23.9421	1.17245
$418$	0	0
$419$	−31.4527	−1.53657	−0.768283	−	0.640111i	$-0.778889\pi$
$419$	−31.4527	−1.53657	−0.768283	+	0.640111i	$0.778889\pi$
$420$	0	0
$421$	5.91170	0.288119	0.144059	−	0.989569i	$-0.453984\pi$
$421$	5.91170	0.288119	0.144059	+	0.989569i	$0.453984\pi$
$422$	0	0
$423$	19.6628	0.956039
$424$	0	0
$425$	7.81458	0.379063
$426$	0	0
$427$	−1.73460	−0.0839430
$428$	0	0
$429$	−6.87920	−0.332131
$430$	0	0
$431$	−12.1771	−0.586551	−0.293276	−	0.956028i	$-0.594745\pi$
$431$	−12.1771	−0.586551	−0.293276	+	0.956028i	$0.594745\pi$
$432$	0	0
$433$	−17.4346	−0.837854	−0.418927	−	0.908020i	$-0.637594\pi$
$433$	−17.4346	−0.837854	−0.418927	+	0.908020i	$0.637594\pi$
$434$	0	0
$435$	−10.5048	−0.503666
$436$	0	0
$437$	−0.155045	−0.00741681
$438$	0	0
$439$	−16.3422	−0.779970	−0.389985	−	0.920821i	$-0.627520\pi$
$439$	−16.3422	−0.779970	−0.389985	+	0.920821i	$0.627520\pi$
$440$	0	0
$441$	−14.4516	−0.688171
$442$	0	0
$443$	7.41071	0.352094	0.176047	−	0.984382i	$-0.443669\pi$
$443$	7.41071	0.352094	0.176047	+	0.984382i	$0.443669\pi$
$444$	0	0
$445$	−6.50052	−0.308154
$446$	0	0
$447$	16.0902	0.761042
$448$	0	0
$449$	−15.2910	−0.721625	−0.360812	−	0.932638i	$-0.617500\pi$
$449$	−15.2910	−0.721625	−0.360812	+	0.932638i	$0.617500\pi$
$450$	0	0
$451$	3.48667	0.164181
$452$	0	0
$453$	−2.38022	−0.111833
$454$	0	0
$455$	−6.94867	−0.325759
$456$	0	0
$457$	15.3789	0.719397	0.359698	−	0.933069i	$-0.382880\pi$
$457$	15.3789	0.719397	0.359698	+	0.933069i	$0.382880\pi$
$458$	0	0
$459$	−6.22264	−0.290448
$460$	0	0
$461$	17.8134	0.829651	0.414826	−	0.909901i	$-0.363843\pi$
$461$	17.8134	0.829651	0.414826	+	0.909901i	$0.363843\pi$
$462$	0	0
$463$	−2.89137	−0.134373	−0.0671867	−	0.997740i	$-0.521402\pi$
$463$	−2.89137	−0.134373	−0.0671867	+	0.997740i	$0.521402\pi$
$464$	0	0
$465$	11.1367	0.516451
$466$	0	0
$467$	−0.107744	−0.00498580	−0.00249290	−	0.999997i	$-0.500794\pi$
$467$	−0.107744	−0.00498580	−0.00249290	+	0.999997i	$0.500794\pi$
$468$	0	0
$469$	−4.46030	−0.205958
$470$	0	0
$471$	16.0204	0.738181
$472$	0	0
$473$	−1.02241	−0.0470105
$474$	0	0
$475$	2.25452	0.103444
$476$	0	0
$477$	1.09678	0.0502180
$478$	0	0
$479$	−6.16795	−0.281821	−0.140910	−	0.990022i	$-0.545003\pi$
$479$	−6.16795	−0.281821	−0.140910	+	0.990022i	$0.545003\pi$
$480$	0	0
$481$	−14.5325	−0.662626
$482$	0	0
$483$	0.205637	0.00935682
$484$	0	0
$485$	7.46349	0.338899
$486$	0	0
$487$	−12.8774	−0.583529	−0.291765	−	0.956490i	$-0.594242\pi$
$487$	−12.8774	−0.583529	−0.291765	+	0.956490i	$0.594242\pi$
$488$	0	0
$489$	3.64182	0.164689
$490$	0	0
$491$	−27.6707	−1.24876	−0.624381	−	0.781120i	$-0.714649\pi$
$491$	−27.6707	−1.24876	−0.624381	+	0.781120i	$0.714649\pi$
$492$	0	0
$493$	−34.4886	−1.55329
$494$	0	0
$495$	−1.39274	−0.0625992
$496$	0	0
$497$	−9.09763	−0.408085
$498$	0	0
$499$	6.50009	0.290984	0.145492	−	0.989359i	$-0.453523\pi$
$499$	6.50009	0.290984	0.145492	+	0.989359i	$0.453523\pi$
$500$	0	0
$501$	36.9138	1.64919
$502$	0	0
$503$	−40.0461	−1.78557	−0.892784	−	0.450484i	$-0.851251\pi$
$503$	−40.0461	−1.78557	−0.892784	+	0.450484i	$0.851251\pi$
$504$	0	0
$505$	−7.23226	−0.321831
$506$	0	0
$507$	41.8786	1.85989
$508$	0	0
$509$	38.4135	1.70265	0.851324	−	0.524640i	$-0.175800\pi$
$509$	38.4135	1.70265	0.851324	+	0.524640i	$0.175800\pi$
$510$	0	0
$511$	18.9424	0.837962
$512$	0	0
$513$	−1.79524	−0.0792618
$514$	0	0
$515$	3.45353	0.152181
$516$	0	0
$517$	−3.85455	−0.169523
$518$	0	0
$519$	16.9324	0.743251
$520$	0	0
$521$	−4.43835	−0.194448	−0.0972239	−	0.995263i	$-0.530996\pi$
$521$	−4.43835	−0.194448	−0.0972239	+	0.995263i	$0.530996\pi$
$522$	0	0
$523$	−0.0143339	−0.000626778 0	−0.000313389	−	1.00000i	$-0.500100\pi$
$523$	−0.0143339	−0.000626778 0	−0.000313389		1.00000i	$0.500100\pi$
$524$	0	0
$525$	−2.99018	−0.130502
$526$	0	0
$527$	36.5632	1.59272
$528$	0	0
$529$	−22.9953	−0.999794
$530$	0	0
$531$	4.01758	0.174348
$532$	0	0
$533$	−36.9089	−1.59870
$534$	0	0
$535$	12.9147	0.558351
$536$	0	0
$537$	5.96258	0.257304
$538$	0	0
$539$	2.83298	0.122025
$540$	0	0
$541$	−12.1463	−0.522211	−0.261105	−	0.965310i	$-0.584087\pi$
$541$	−12.1463	−0.522211	−0.261105	+	0.965310i	$0.584087\pi$
$542$	0	0
$543$	15.1739	0.651174
$544$	0	0
$545$	−1.79149	−0.0767392
$546$	0	0
$547$	−39.3253	−1.68143	−0.840713	−	0.541480i	$-0.817864\pi$
$547$	−39.3253	−1.68143	−0.840713	+	0.541480i	$0.817864\pi$
$548$	0	0
$549$	3.68035	0.157074
$550$	0	0
$551$	−9.95002	−0.423885
$552$	0	0
$553$	2.87147	0.122107
$554$	0	0
$555$	−6.25370	−0.265455
$556$	0	0
$557$	5.31912	0.225378	0.112689	−	0.993630i	$-0.464054\pi$
$557$	5.31912	0.225378	0.112689	+	0.993630i	$0.464054\pi$
$558$	0	0
$559$	10.8230	0.457763
$560$	0	0
$561$	−9.71903	−0.410338
$562$	0	0
$563$	−12.0898	−0.509523	−0.254762	−	0.967004i	$-0.581997\pi$
$563$	−12.0898	−0.509523	−0.254762	+	0.967004i	$0.581997\pi$
$564$	0	0
$565$	−5.59252	−0.235279
$566$	0	0
$567$	12.4266	0.521867
$568$	0	0
$569$	−26.8121	−1.12402	−0.562012	−	0.827129i	$-0.689972\pi$
$569$	−26.8121	−1.12402	−0.562012	+	0.827129i	$0.689972\pi$
$570$	0	0
$571$	35.5512	1.48777	0.743885	−	0.668308i	$-0.232981\pi$
$571$	35.5512	1.48777	0.743885	+	0.668308i	$0.232981\pi$
$572$	0	0
$573$	21.3670	0.892620
$574$	0	0
$575$	−0.0687708	−0.00286794
$576$	0	0
$577$	−31.4571	−1.30958	−0.654788	−	0.755813i	$-0.727242\pi$
$577$	−31.4571	−1.30958	−0.654788	+	0.755813i	$0.727242\pi$
$578$	0	0
$579$	7.72276	0.320947
$580$	0	0
$581$	−5.36730	−0.222673
$582$	0	0
$583$	−0.215004	−0.00890455
$584$	0	0
$585$	14.7432	0.609558
$586$	0	0
$587$	25.8239	1.06587	0.532933	−	0.846158i	$-0.321090\pi$
$587$	25.8239	1.06587	0.532933	+	0.846158i	$0.321090\pi$
$588$	0	0
$589$	10.5485	0.434645
$590$	0	0
$591$	51.6681	2.12534
$592$	0	0
$593$	−19.6666	−0.807610	−0.403805	−	0.914845i	$-0.632313\pi$
$593$	−19.6666	−0.807610	−0.403805	+	0.914845i	$0.632313\pi$
$594$	0	0
$595$	−9.81717	−0.402465
$596$	0	0
$597$	−13.2340	−0.541633
$598$	0	0
$599$	−41.1999	−1.68338	−0.841692	−	0.539959i	$-0.818440\pi$
$599$	−41.1999	−1.68338	−0.841692	+	0.539959i	$0.818440\pi$
$600$	0	0
$601$	10.6303	0.433619	0.216809	−	0.976214i	$-0.430435\pi$
$601$	10.6303	0.433619	0.216809	+	0.976214i	$0.430435\pi$
$602$	0	0
$603$	9.46358	0.385387
$604$	0	0
$605$	−10.7270	−0.436114
$606$	0	0
$607$	−10.1062	−0.410198	−0.205099	−	0.978741i	$-0.565752\pi$
$607$	−10.1062	−0.410198	−0.205099	+	0.978741i	$0.565752\pi$
$608$	0	0
$609$	13.1968	0.534761
$610$	0	0
$611$	40.8033	1.65072
$612$	0	0
$613$	−8.66585	−0.350010	−0.175005	−	0.984568i	$-0.555994\pi$
$613$	−8.66585	−0.350010	−0.175005	+	0.984568i	$0.555994\pi$
$614$	0	0
$615$	−15.8828	−0.640458
$616$	0	0
$617$	22.2708	0.896587	0.448294	−	0.893886i	$-0.352032\pi$
$617$	22.2708	0.896587	0.448294	+	0.893886i	$0.352032\pi$
$618$	0	0
$619$	21.9293	0.881415	0.440707	−	0.897651i	$-0.354728\pi$
$619$	21.9293	0.881415	0.440707	+	0.897651i	$0.354728\pi$
$620$	0	0
$621$	0.0547612	0.00219749
$622$	0	0
$623$	8.16637	0.327179
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.80395	−0.111979
$628$	0	0
$629$	−20.5317	−0.818653
$630$	0	0
$631$	−15.8985	−0.632911	−0.316456	−	0.948607i	$-0.602493\pi$
$631$	−15.8985	−0.632911	−0.316456	+	0.948607i	$0.602493\pi$
$632$	0	0
$633$	30.9414	1.22981
$634$	0	0
$635$	13.8368	0.549099
$636$	0	0
$637$	−29.9892	−1.18822
$638$	0	0
$639$	19.3028	0.763606
$640$	0	0
$641$	24.6323	0.972917	0.486458	−	0.873704i	$-0.338289\pi$
$641$	24.6323	0.972917	0.486458	+	0.873704i	$0.338289\pi$
$642$	0	0
$643$	26.5130	1.04557	0.522785	−	0.852464i	$-0.324893\pi$
$643$	26.5130	1.04557	0.522785	+	0.852464i	$0.324893\pi$
$644$	0	0
$645$	4.65739	0.183385
$646$	0	0
$647$	29.1643	1.14657	0.573283	−	0.819358i	$-0.305670\pi$
$647$	29.1643	1.14657	0.573283	+	0.819358i	$0.305670\pi$
$648$	0	0
$649$	−0.787576	−0.0309151
$650$	0	0
$651$	−13.9906	−0.548334
$652$	0	0
$653$	−19.2823	−0.754574	−0.377287	−	0.926096i	$-0.623143\pi$
$653$	−19.2823	−0.754574	−0.377287	+	0.926096i	$0.623143\pi$
$654$	0	0
$655$	−11.9743	−0.467876
$656$	0	0
$657$	−40.1907	−1.56799
$658$	0	0
$659$	−23.3381	−0.909125	−0.454563	−	0.890715i	$-0.650204\pi$
$659$	−23.3381	−0.909125	−0.454563	+	0.890715i	$0.650204\pi$
$660$	0	0
$661$	−44.6902	−1.73825	−0.869124	−	0.494594i	$-0.835317\pi$
$661$	−44.6902	−1.73825	−0.869124	+	0.494594i	$0.835317\pi$
$662$	0	0
$663$	102.883	3.99565
$664$	0	0
$665$	−2.83227	−0.109831
$666$	0	0
$667$	0.303511	0.0117520
$668$	0	0
$669$	−16.5773	−0.640915
$670$	0	0
$671$	−0.721469	−0.0278520
$672$	0	0
$673$	−17.1154	−0.659750	−0.329875	−	0.944025i	$-0.607007\pi$
$673$	−17.1154	−0.659750	−0.329875	+	0.944025i	$0.607007\pi$
$674$	0	0
$675$	−0.796285	−0.0306490
$676$	0	0
$677$	15.7478	0.605235	0.302618	−	0.953112i	$-0.402139\pi$
$677$	15.7478	0.605235	0.302618	+	0.953112i	$0.402139\pi$
$678$	0	0
$679$	−9.37610	−0.359822
$680$	0	0
$681$	37.5586	1.43925
$682$	0	0
$683$	12.4301	0.475625	0.237813	−	0.971311i	$-0.423570\pi$
$683$	12.4301	0.475625	0.237813	+	0.971311i	$0.423570\pi$
$684$	0	0
$685$	11.6180	0.443901
$686$	0	0
$687$	−46.3608	−1.76878
$688$	0	0
$689$	2.27598	0.0867078
$690$	0	0
$691$	1.14110	0.0434095	0.0217048	−	0.999764i	$-0.493091\pi$
$691$	1.14110	0.0434095	0.0217048	+	0.999764i	$0.493091\pi$
$692$	0	0
$693$	1.74965	0.0664638
$694$	0	0
$695$	10.0588	0.381551
$696$	0	0
$697$	−52.1454	−1.97515
$698$	0	0
$699$	−33.5002	−1.26710
$700$	0	0
$701$	21.8869	0.826655	0.413328	−	0.910582i	$-0.364366\pi$
$701$	21.8869	0.826655	0.413328	+	0.910582i	$0.364366\pi$
$702$	0	0
$703$	−5.92343	−0.223407
$704$	0	0
$705$	17.5587	0.661298
$706$	0	0
$707$	9.08561	0.341700
$708$	0	0
$709$	−10.3487	−0.388655	−0.194328	−	0.980937i	$-0.562252\pi$
$709$	−10.3487	−0.388655	−0.194328	+	0.980937i	$0.562252\pi$
$710$	0	0
$711$	−6.09249	−0.228486
$712$	0	0
$713$	−0.321767	−0.0120503
$714$	0	0
$715$	−2.89015	−0.108086
$716$	0	0
$717$	−44.1408	−1.64847
$718$	0	0
$719$	13.1516	0.490470	0.245235	−	0.969464i	$-0.421135\pi$
$719$	13.1516	0.490470	0.245235	+	0.969464i	$0.421135\pi$
$720$	0	0
$721$	−4.33855	−0.161576
$722$	0	0
$723$	12.2816	0.456758
$724$	0	0
$725$	−4.41337	−0.163908
$726$	0	0
$727$	0.497784	0.0184618	0.00923090	−	0.999957i	$-0.497062\pi$
$727$	0.497784	0.0184618	0.00923090	+	0.999957i	$0.497062\pi$
$728$	0	0
$729$	−20.6799	−0.765923
$730$	0	0
$731$	15.2908	0.565552
$732$	0	0
$733$	29.8488	1.10249	0.551245	−	0.834344i	$-0.314153\pi$
$733$	29.8488	1.10249	0.551245	+	0.834344i	$0.314153\pi$
$734$	0	0
$735$	−12.9051	−0.476012
$736$	0	0
$737$	−1.85517	−0.0683360
$738$	0	0
$739$	−29.3829	−1.08087	−0.540435	−	0.841386i	$-0.681740\pi$
$739$	−29.3829	−1.08087	−0.540435	+	0.841386i	$0.681740\pi$
$740$	0	0
$741$	29.6819	1.09039
$742$	0	0
$743$	−18.7307	−0.687163	−0.343582	−	0.939123i	$-0.611640\pi$
$743$	−18.7307	−0.687163	−0.343582	+	0.939123i	$0.611640\pi$
$744$	0	0
$745$	6.75997	0.247666
$746$	0	0
$747$	11.3880	0.416665
$748$	0	0
$749$	−16.2243	−0.592822
$750$	0	0
$751$	−27.4206	−1.00059	−0.500297	−	0.865854i	$-0.666776\pi$
$751$	−27.4206	−1.00059	−0.500297	+	0.865854i	$0.666776\pi$
$752$	0	0
$753$	14.5865	0.531562
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−34.9422	−1.26999	−0.634997	−	0.772514i	$-0.718999\pi$
$757$	−34.9422	−1.26999	−0.634997	+	0.772514i	$0.718999\pi$
$758$	0	0
$759$	0.0855305	0.00310456
$760$	0	0
$761$	−15.7306	−0.570233	−0.285117	−	0.958493i	$-0.592032\pi$
$761$	−15.7306	−0.570233	−0.285117	+	0.958493i	$0.592032\pi$
$762$	0	0
$763$	2.25059	0.0814767
$764$	0	0
$765$	20.8294	0.753090
$766$	0	0
$767$	8.33708	0.301034
$768$	0	0
$769$	−20.2871	−0.731573	−0.365786	−	0.930699i	$-0.619200\pi$
$769$	−20.2871	−0.731573	−0.365786	+	0.930699i	$0.619200\pi$
$770$	0	0
$771$	−1.86400	−0.0671303
$772$	0	0
$773$	23.5242	0.846106	0.423053	−	0.906105i	$-0.360958\pi$
$773$	23.5242	0.846106	0.423053	+	0.906105i	$0.360958\pi$
$774$	0	0
$775$	4.67884	0.168069
$776$	0	0
$777$	7.85629	0.281843
$778$	0	0
$779$	−15.0440	−0.539009
$780$	0	0
$781$	−3.78397	−0.135401
$782$	0	0
$783$	3.51430	0.125591
$784$	0	0
$785$	6.73063	0.240227
$786$	0	0
$787$	−6.41798	−0.228776	−0.114388	−	0.993436i	$-0.536491\pi$
$787$	−6.41798	−0.228776	−0.114388	+	0.993436i	$0.536491\pi$
$788$	0	0
$789$	40.8557	1.45450
$790$	0	0
$791$	7.02567	0.249804
$792$	0	0
$793$	7.63728	0.271208
$794$	0	0
$795$	0.979409	0.0347361
$796$	0	0
$797$	−22.6763	−0.803234	−0.401617	−	0.915808i	$-0.631552\pi$
$797$	−22.6763	−0.803234	−0.401617	+	0.915808i	$0.631552\pi$
$798$	0	0
$799$	57.6474	2.03942
$800$	0	0
$801$	−17.3269	−0.612215
$802$	0	0
$803$	7.87869	0.278033
$804$	0	0
$805$	0.0863942	0.00304499
$806$	0	0
$807$	6.41458	0.225804
$808$	0	0
$809$	−1.16288	−0.0408848	−0.0204424	−	0.999791i	$-0.506507\pi$
$809$	−1.16288	−0.0408848	−0.0204424	+	0.999791i	$0.506507\pi$
$810$	0	0
$811$	−6.36953	−0.223664	−0.111832	−	0.993727i	$-0.535672\pi$
$811$	−6.36953	−0.223664	−0.111832	+	0.993727i	$0.535672\pi$
$812$	0	0
$813$	36.2959	1.27295
$814$	0	0
$815$	1.53003	0.0535947
$816$	0	0
$817$	4.41143	0.154336
$818$	0	0
$819$	−18.5214	−0.647189
$820$	0	0
$821$	−24.7664	−0.864354	−0.432177	−	0.901789i	$-0.642254\pi$
$821$	−24.7664	−0.864354	−0.432177	+	0.901789i	$0.642254\pi$
$822$	0	0
$823$	18.8917	0.658523	0.329262	−	0.944239i	$-0.393200\pi$
$823$	18.8917	0.658523	0.329262	+	0.944239i	$0.393200\pi$
$824$	0	0
$825$	−1.24370	−0.0433002
$826$	0	0
$827$	−41.2425	−1.43414	−0.717072	−	0.696999i	$-0.754518\pi$
$827$	−41.2425	−1.43414	−0.717072	+	0.696999i	$0.754518\pi$
$828$	0	0
$829$	42.7166	1.48361	0.741805	−	0.670616i	$-0.233970\pi$
$829$	42.7166	1.48361	0.741805	+	0.670616i	$0.233970\pi$
$830$	0	0
$831$	−50.4169	−1.74894
$832$	0	0
$833$	−42.3691	−1.46800
$834$	0	0
$835$	15.5085	0.536695
$836$	0	0
$837$	−3.72569	−0.128779
$838$	0	0
$839$	−28.4270	−0.981408	−0.490704	−	0.871326i	$-0.663260\pi$
$839$	−28.4270	−0.981408	−0.490704	+	0.871326i	$0.663260\pi$
$840$	0	0
$841$	−9.52219	−0.328351
$842$	0	0
$843$	−50.5043	−1.73946
$844$	0	0
$845$	17.5944	0.605266
$846$	0	0
$847$	13.4759	0.463037
$848$	0	0
$849$	−22.0449	−0.756578
$850$	0	0
$851$	0.180686	0.00619383
$852$	0	0
$853$	−10.8148	−0.370292	−0.185146	−	0.982711i	$-0.559276\pi$
$853$	−10.8148	−0.370292	−0.185146	+	0.982711i	$0.559276\pi$
$854$	0	0
$855$	6.00932	0.205515
$856$	0	0
$857$	−2.04777	−0.0699505	−0.0349753	−	0.999388i	$-0.511135\pi$
$857$	−2.04777	−0.0699505	−0.0349753	+	0.999388i	$0.511135\pi$
$858$	0	0
$859$	13.5866	0.463567	0.231784	−	0.972767i	$-0.425544\pi$
$859$	13.5866	0.463567	0.231784	+	0.972767i	$0.425544\pi$
$860$	0	0
$861$	19.9530	0.679997
$862$	0	0
$863$	45.3226	1.54280	0.771400	−	0.636351i	$-0.219557\pi$
$863$	45.3226	1.54280	0.771400	+	0.636351i	$0.219557\pi$
$864$	0	0
$865$	7.11380	0.241876
$866$	0	0
$867$	104.891	3.56228
$868$	0	0
$869$	1.19433	0.0405147
$870$	0	0
$871$	19.6383	0.665420
$872$	0	0
$873$	19.8936	0.673297
$874$	0	0
$875$	−1.25626	−0.0424694
$876$	0	0
$877$	−26.0476	−0.879564	−0.439782	−	0.898105i	$-0.644944\pi$
$877$	−26.0476	−0.879564	−0.439782	+	0.898105i	$0.644944\pi$
$878$	0	0
$879$	−77.4528	−2.61242
$880$	0	0
$881$	−10.0135	−0.337365	−0.168682	−	0.985670i	$-0.553951\pi$
$881$	−10.0135	−0.337365	−0.168682	+	0.985670i	$0.553951\pi$
$882$	0	0
$883$	−26.8431	−0.903344	−0.451672	−	0.892184i	$-0.649172\pi$
$883$	−26.8431	−0.903344	−0.451672	+	0.892184i	$0.649172\pi$
$884$	0	0
$885$	3.58765	0.120598
$886$	0	0
$887$	3.85822	0.129546	0.0647732	−	0.997900i	$-0.479368\pi$
$887$	3.85822	0.129546	0.0647732	+	0.997900i	$0.479368\pi$
$888$	0	0
$889$	−17.3827	−0.582998
$890$	0	0
$891$	5.16858	0.173154
$892$	0	0
$893$	16.6314	0.556547
$894$	0	0
$895$	2.50505	0.0837347
$896$	0	0
$897$	−0.905404	−0.0302306
$898$	0	0
$899$	−20.6494	−0.688697
$900$	0	0
$901$	3.21553	0.107125
$902$	0	0
$903$	−5.85091	−0.194706
$904$	0	0
$905$	6.37499	0.211912
$906$	0	0
$907$	−1.97932	−0.0657224	−0.0328612	−	0.999460i	$-0.510462\pi$
$907$	−1.97932	−0.0657224	−0.0328612	+	0.999460i	$0.510462\pi$
$908$	0	0
$909$	−19.2773	−0.639387
$910$	0	0
$911$	38.8735	1.28794	0.643968	−	0.765052i	$-0.277287\pi$
$911$	38.8735	1.28794	0.643968	+	0.765052i	$0.277287\pi$
$912$	0	0
$913$	−2.23242	−0.0738822
$914$	0	0
$915$	3.28651	0.108649
$916$	0	0
$917$	15.0429	0.496760
$918$	0	0
$919$	−17.8160	−0.587695	−0.293847	−	0.955852i	$-0.594936\pi$
$919$	−17.8160	−0.587695	−0.293847	+	0.955852i	$0.594936\pi$
$920$	0	0
$921$	−65.2413	−2.14977
$922$	0	0
$923$	40.0561	1.31846
$924$	0	0
$925$	−2.62736	−0.0863871
$926$	0	0
$927$	9.20525	0.302340
$928$	0	0
$929$	44.1799	1.44949	0.724747	−	0.689015i	$-0.241956\pi$
$929$	44.1799	1.44949	0.724747	+	0.689015i	$0.241956\pi$
$930$	0	0
$931$	−12.2236	−0.400611
$932$	0	0
$933$	25.8226	0.845394
$934$	0	0
$935$	−4.08324	−0.133536
$936$	0	0
$937$	−60.5239	−1.97723	−0.988616	−	0.150463i	$-0.951923\pi$
$937$	−60.5239	−1.97723	−0.988616	+	0.150463i	$0.951923\pi$
$938$	0	0
$939$	−22.4561	−0.732826
$940$	0	0
$941$	15.3637	0.500842	0.250421	−	0.968137i	$-0.419431\pi$
$941$	15.3637	0.500842	0.250421	+	0.968137i	$0.419431\pi$
$942$	0	0
$943$	0.458896	0.0149437
$944$	0	0
$945$	1.00034	0.0325412
$946$	0	0
$947$	37.3391	1.21336	0.606679	−	0.794947i	$-0.292501\pi$
$947$	37.3391	1.21336	0.606679	+	0.794947i	$0.292501\pi$
$948$	0	0
$949$	−83.4017	−2.70734
$950$	0	0
$951$	19.4207	0.629759
$952$	0	0
$953$	0.671011	0.0217362	0.0108681	−	0.999941i	$-0.496541\pi$
$953$	0.671011	0.0217362	0.0108681	+	0.999941i	$0.496541\pi$
$954$	0	0
$955$	8.97690	0.290486
$956$	0	0
$957$	5.48892	0.177432
$958$	0	0
$959$	−14.5952	−0.471305
$960$	0	0
$961$	−9.10847	−0.293821
$962$	0	0
$963$	34.4236	1.10928
$964$	0	0
$965$	3.24456	0.104446
$966$	0	0
$967$	33.6619	1.08249	0.541247	−	0.840864i	$-0.317953\pi$
$967$	33.6619	1.08249	0.541247	+	0.840864i	$0.317953\pi$
$968$	0	0
$969$	41.9350	1.34715
$970$	0	0
$971$	22.7376	0.729684	0.364842	−	0.931069i	$-0.381123\pi$
$971$	22.7376	0.729684	0.364842	+	0.931069i	$0.381123\pi$
$972$	0	0
$973$	−12.6365	−0.405106
$974$	0	0
$975$	13.1655	0.421635
$976$	0	0
$977$	31.1648	0.997050	0.498525	−	0.866875i	$-0.333875\pi$
$977$	31.1648	0.997050	0.498525	+	0.866875i	$0.333875\pi$
$978$	0	0
$979$	3.39663	0.108557
$980$	0	0
$981$	−4.77515	−0.152459
$982$	0	0
$983$	10.3456	0.329974	0.164987	−	0.986296i	$-0.447242\pi$
$983$	10.3456	0.329974	0.164987	+	0.986296i	$0.447242\pi$
$984$	0	0
$985$	21.7073	0.691651
$986$	0	0
$987$	−22.0583	−0.702123
$988$	0	0
$989$	−0.134564	−0.00427889
$990$	0	0
$991$	9.96716	0.316617	0.158309	−	0.987390i	$-0.449396\pi$
$991$	9.96716	0.316617	0.158309	+	0.987390i	$0.449396\pi$
$992$	0	0
$993$	−74.2061	−2.35486
$994$	0	0
$995$	−5.56000	−0.176264
$996$	0	0
$997$	37.6018	1.19086	0.595431	−	0.803407i	$-0.296981\pi$
$997$	37.6018	1.19086	0.595431	+	0.803407i	$0.296981\pi$
$998$	0	0
$999$	2.09213	0.0661920

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.12	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.12	✓	15	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+2.38022 q^{3} +1.00000 q^{5} -1.25626 q^{7} +2.66546 q^{9} +O(q^{10})\)	\(q+2.38022 q^{3} +1.00000 q^{5} -1.25626 q^{7} +2.66546 q^{9} -0.522516 q^{11} +5.53122 q^{13} +2.38022 q^{15} +7.81458 q^{17} +2.25452 q^{19} -2.99018 q^{21} -0.0687708 q^{23} +1.00000 q^{25} -0.796285 q^{27} -4.41337 q^{29} +4.67884 q^{31} -1.24370 q^{33} -1.25626 q^{35} -2.62736 q^{37} +13.1655 q^{39} -6.67284 q^{41} +1.95670 q^{43} +2.66546 q^{45} +7.37690 q^{47} -5.42180 q^{49} +18.6004 q^{51} +0.411478 q^{53} -0.522516 q^{55} +5.36625 q^{57} +1.50728 q^{59} +1.38076 q^{61} -3.34851 q^{63} +5.53122 q^{65} +3.55045 q^{67} -0.163690 q^{69} +7.24182 q^{71} -15.0784 q^{73} +2.38022 q^{75} +0.656417 q^{77} -2.28572 q^{79} -9.89171 q^{81} +4.27243 q^{83} +7.81458 q^{85} -10.5048 q^{87} -6.50052 q^{89} -6.94867 q^{91} +11.1367 q^{93} +2.25452 q^{95} +7.46349 q^{97} -1.39274 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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