LMFDB - Embedded newform 4012.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.89300 q^{3} -2.37849 q^{5} +4.40428 q^{7} +0.583446 q^{9} +O(q^{10})$	$q-1.89300 q^{3} -2.37849 q^{5} +4.40428 q^{7} +0.583446 q^{9} -5.07247 q^{11} -4.39295 q^{13} +4.50247 q^{15} +1.00000 q^{17} -2.53212 q^{19} -8.33729 q^{21} -0.707852 q^{23} +0.657202 q^{25} +4.57454 q^{27} +2.39269 q^{29} -9.02514 q^{31} +9.60219 q^{33} -10.4755 q^{35} -1.30855 q^{37} +8.31586 q^{39} +11.5883 q^{41} -5.33553 q^{43} -1.38772 q^{45} +4.98460 q^{47} +12.3976 q^{49} -1.89300 q^{51} -5.84816 q^{53} +12.0648 q^{55} +4.79331 q^{57} +1.00000 q^{59} -15.2741 q^{61} +2.56966 q^{63} +10.4486 q^{65} +3.60953 q^{67} +1.33996 q^{69} -2.63835 q^{71} -14.6962 q^{73} -1.24408 q^{75} -22.3406 q^{77} -0.361230 q^{79} -10.4099 q^{81} +8.85220 q^{83} -2.37849 q^{85} -4.52937 q^{87} +8.27378 q^{89} -19.3478 q^{91} +17.0846 q^{93} +6.02263 q^{95} +6.59391 q^{97} -2.95951 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * 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
$3$	−1.89300	−1.09292	−0.546462	−	0.837484i	$-0.684026\pi$
$3$	−1.89300	−1.09292	−0.546462	+	0.837484i	$0.684026\pi$
$4$	0	0
$5$	−2.37849	−1.06369	−0.531846	−	0.846841i	$-0.678501\pi$
$5$	−2.37849	−1.06369	−0.531846	+	0.846841i	$0.678501\pi$
$6$	0	0
$7$	4.40428	1.66466	0.832330	−	0.554281i	$-0.187007\pi$
$7$	4.40428	1.66466	0.832330	+	0.554281i	$0.187007\pi$
$8$	0	0
$9$	0.583446	0.194482
$10$	0	0
$11$	−5.07247	−1.52941	−0.764704	−	0.644381i	$-0.777115\pi$
$11$	−5.07247	−1.52941	−0.764704	+	0.644381i	$0.777115\pi$
$12$	0	0
$13$	−4.39295	−1.21839	−0.609193	−	0.793022i	$-0.708507\pi$
$13$	−4.39295	−1.21839	−0.609193	+	0.793022i	$0.708507\pi$
$14$	0	0
$15$	4.50247	1.16253
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.53212	−0.580909	−0.290455	−	0.956889i	$-0.593807\pi$
$19$	−2.53212	−0.580909	−0.290455	+	0.956889i	$0.593807\pi$
$20$	0	0
$21$	−8.33729	−1.81935
$22$	0	0
$23$	−0.707852	−0.147597	−0.0737987	−	0.997273i	$-0.523512\pi$
$23$	−0.707852	−0.147597	−0.0737987	+	0.997273i	$0.523512\pi$
$24$	0	0
$25$	0.657202	0.131440
$26$	0	0
$27$	4.57454	0.880370
$28$	0	0
$29$	2.39269	0.444312	0.222156	−	0.975011i	$-0.428691\pi$
$29$	2.39269	0.444312	0.222156	+	0.975011i	$0.428691\pi$
$30$	0	0
$31$	−9.02514	−1.62096	−0.810481	−	0.585765i	$-0.800794\pi$
$31$	−9.02514	−1.62096	−0.810481	+	0.585765i	$0.800794\pi$
$32$	0	0
$33$	9.60219	1.67153
$34$	0	0
$35$	−10.4755	−1.77068
$36$	0	0
$37$	−1.30855	−0.215124	−0.107562	−	0.994198i	$-0.534304\pi$
$37$	−1.30855	−0.215124	−0.107562	+	0.994198i	$0.534304\pi$
$38$	0	0
$39$	8.31586	1.33160
$40$	0	0
$41$	11.5883	1.80978	0.904890	−	0.425644i	$-0.139953\pi$
$41$	11.5883	1.80978	0.904890	+	0.425644i	$0.139953\pi$
$42$	0	0
$43$	−5.33553	−0.813661	−0.406830	−	0.913504i	$-0.633366\pi$
$43$	−5.33553	−0.813661	−0.406830	+	0.913504i	$0.633366\pi$
$44$	0	0
$45$	−1.38772	−0.206869
$46$	0	0
$47$	4.98460	0.727079	0.363539	−	0.931579i	$-0.381568\pi$
$47$	4.98460	0.727079	0.363539	+	0.931579i	$0.381568\pi$
$48$	0	0
$49$	12.3976	1.77109
$50$	0	0
$51$	−1.89300	−0.265073
$52$	0	0
$53$	−5.84816	−0.803307	−0.401653	−	0.915792i	$-0.631564\pi$
$53$	−5.84816	−0.803307	−0.401653	+	0.915792i	$0.631564\pi$
$54$	0	0
$55$	12.0648	1.62682
$56$	0	0
$57$	4.79331	0.634889
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−15.2741	−1.95565	−0.977823	−	0.209434i	$-0.932838\pi$
$61$	−15.2741	−1.95565	−0.977823	+	0.209434i	$0.932838\pi$
$62$	0	0
$63$	2.56966	0.323746
$64$	0	0
$65$	10.4486	1.29599
$66$	0	0
$67$	3.60953	0.440975	0.220487	−	0.975390i	$-0.429235\pi$
$67$	3.60953	0.440975	0.220487	+	0.975390i	$0.429235\pi$
$68$	0	0
$69$	1.33996	0.161313
$70$	0	0
$71$	−2.63835	−0.313114	−0.156557	−	0.987669i	$-0.550040\pi$
$71$	−2.63835	−0.313114	−0.156557	+	0.987669i	$0.550040\pi$
$72$	0	0
$73$	−14.6962	−1.72006	−0.860031	−	0.510241i	$-0.829556\pi$
$73$	−14.6962	−1.72006	−0.860031	+	0.510241i	$0.829556\pi$
$74$	0	0
$75$	−1.24408	−0.143654
$76$	0	0
$77$	−22.3406	−2.54595
$78$	0	0
$79$	−0.361230	−0.0406416	−0.0203208	−	0.999794i	$-0.506469\pi$
$79$	−0.361230	−0.0406416	−0.0203208	+	0.999794i	$0.506469\pi$
$80$	0	0
$81$	−10.4099	−1.15666
$82$	0	0
$83$	8.85220	0.971655	0.485828	−	0.874055i	$-0.338518\pi$
$83$	8.85220	0.971655	0.485828	+	0.874055i	$0.338518\pi$
$84$	0	0
$85$	−2.37849	−0.257983
$86$	0	0
$87$	−4.52937	−0.485599
$88$	0	0
$89$	8.27378	0.877019	0.438510	−	0.898726i	$-0.355507\pi$
$89$	8.27378	0.877019	0.438510	+	0.898726i	$0.355507\pi$
$90$	0	0
$91$	−19.3478	−2.02820
$92$	0	0
$93$	17.0846	1.77159
$94$	0	0
$95$	6.02263	0.617908
$96$	0	0
$97$	6.59391	0.669510	0.334755	−	0.942305i	$-0.391346\pi$
$97$	6.59391	0.669510	0.334755	+	0.942305i	$0.391346\pi$
$98$	0	0
$99$	−2.95951	−0.297442
$100$	0	0
$101$	3.23724	0.322117	0.161058	−	0.986945i	$-0.448509\pi$
$101$	3.23724	0.322117	0.161058	+	0.986945i	$0.448509\pi$
$102$	0	0
$103$	−7.80929	−0.769472	−0.384736	−	0.923027i	$-0.625708\pi$
$103$	−7.80929	−0.769472	−0.384736	+	0.923027i	$0.625708\pi$
$104$	0	0
$105$	19.8301	1.93522
$106$	0	0
$107$	−10.1126	−0.977619	−0.488809	−	0.872391i	$-0.662569\pi$
$107$	−10.1126	−0.977619	−0.488809	+	0.872391i	$0.662569\pi$
$108$	0	0
$109$	13.4693	1.29012	0.645061	−	0.764131i	$-0.276832\pi$
$109$	13.4693	1.29012	0.645061	+	0.764131i	$0.276832\pi$
$110$	0	0
$111$	2.47708	0.235114
$112$	0	0
$113$	−0.917007	−0.0862648	−0.0431324	−	0.999069i	$-0.513734\pi$
$113$	−0.917007	−0.0862648	−0.0431324	+	0.999069i	$0.513734\pi$
$114$	0	0
$115$	1.68362	0.156998
$116$	0	0
$117$	−2.56305	−0.236954
$118$	0	0
$119$	4.40428	0.403739
$120$	0	0
$121$	14.7300	1.33909
$122$	0	0
$123$	−21.9366	−1.97795
$124$	0	0
$125$	10.3293	0.923880
$126$	0	0
$127$	0.757130	0.0671845	0.0335922	−	0.999436i	$-0.489305\pi$
$127$	0.757130	0.0671845	0.0335922	+	0.999436i	$0.489305\pi$
$128$	0	0
$129$	10.1002	0.889269
$130$	0	0
$131$	−2.89315	−0.252776	−0.126388	−	0.991981i	$-0.540338\pi$
$131$	−2.89315	−0.252776	−0.126388	+	0.991981i	$0.540338\pi$
$132$	0	0
$133$	−11.1522	−0.967016
$134$	0	0
$135$	−10.8805	−0.936442
$136$	0	0
$137$	12.3651	1.05642	0.528210	−	0.849114i	$-0.322863\pi$
$137$	12.3651	1.05642	0.528210	+	0.849114i	$0.322863\pi$
$138$	0	0
$139$	11.0101	0.933862	0.466931	−	0.884294i	$-0.345360\pi$
$139$	11.0101	0.933862	0.466931	+	0.884294i	$0.345360\pi$
$140$	0	0
$141$	−9.43584	−0.794641
$142$	0	0
$143$	22.2831	1.86341
$144$	0	0
$145$	−5.69099	−0.472611
$146$	0	0
$147$	−23.4687	−1.93567
$148$	0	0
$149$	6.12164	0.501504	0.250752	−	0.968051i	$-0.419322\pi$
$149$	6.12164	0.501504	0.250752	+	0.968051i	$0.419322\pi$
$150$	0	0
$151$	0.415148	0.0337843	0.0168921	−	0.999857i	$-0.494623\pi$
$151$	0.415148	0.0337843	0.0168921	+	0.999857i	$0.494623\pi$
$152$	0	0
$153$	0.583446	0.0471688
$154$	0	0
$155$	21.4662	1.72420
$156$	0	0
$157$	−18.1526	−1.44874	−0.724368	−	0.689414i	$-0.757868\pi$
$157$	−18.1526	−1.44874	−0.724368	+	0.689414i	$0.757868\pi$
$158$	0	0
$159$	11.0706	0.877953
$160$	0	0
$161$	−3.11758	−0.245699
$162$	0	0
$163$	6.54783	0.512865	0.256433	−	0.966562i	$-0.417453\pi$
$163$	6.54783	0.512865	0.256433	+	0.966562i	$0.417453\pi$
$164$	0	0
$165$	−22.8387	−1.77799
$166$	0	0
$167$	−16.5201	−1.27836	−0.639182	−	0.769056i	$-0.720727\pi$
$167$	−16.5201	−1.27836	−0.639182	+	0.769056i	$0.720727\pi$
$168$	0	0
$169$	6.29804	0.484465
$170$	0	0
$171$	−1.47736	−0.112976
$172$	0	0
$173$	−2.29041	−0.174137	−0.0870685	−	0.996202i	$-0.527750\pi$
$173$	−2.29041	−0.174137	−0.0870685	+	0.996202i	$0.527750\pi$
$174$	0	0
$175$	2.89450	0.218803
$176$	0	0
$177$	−1.89300	−0.142287
$178$	0	0
$179$	4.00207	0.299129	0.149564	−	0.988752i	$-0.452213\pi$
$179$	4.00207	0.299129	0.149564	+	0.988752i	$0.452213\pi$
$180$	0	0
$181$	15.8275	1.17645	0.588224	−	0.808698i	$-0.299827\pi$
$181$	15.8275	1.17645	0.588224	+	0.808698i	$0.299827\pi$
$182$	0	0
$183$	28.9138	2.13737
$184$	0	0
$185$	3.11236	0.228826
$186$	0	0
$187$	−5.07247	−0.370936
$188$	0	0
$189$	20.1475	1.46552
$190$	0	0
$191$	20.5778	1.48896	0.744479	−	0.667646i	$-0.232698\pi$
$191$	20.5778	1.48896	0.744479	+	0.667646i	$0.232698\pi$
$192$	0	0
$193$	−22.9047	−1.64872	−0.824359	−	0.566067i	$-0.808464\pi$
$193$	−22.9047	−1.64872	−0.824359	+	0.566067i	$0.808464\pi$
$194$	0	0
$195$	−19.7792	−1.41642
$196$	0	0
$197$	21.4975	1.53163	0.765816	−	0.643060i	$-0.222335\pi$
$197$	21.4975	1.53163	0.765816	+	0.643060i	$0.222335\pi$
$198$	0	0
$199$	19.0664	1.35158	0.675790	−	0.737095i	$-0.263803\pi$
$199$	19.0664	1.35158	0.675790	+	0.737095i	$0.263803\pi$
$200$	0	0
$201$	−6.83285	−0.481952
$202$	0	0
$203$	10.5381	0.739629
$204$	0	0
$205$	−27.5625	−1.92505
$206$	0	0
$207$	−0.412993	−0.0287050
$208$	0	0
$209$	12.8441	0.888448
$210$	0	0
$211$	15.4617	1.06442	0.532212	−	0.846611i	$-0.321361\pi$
$211$	15.4617	1.06442	0.532212	+	0.846611i	$0.321361\pi$
$212$	0	0
$213$	4.99439	0.342210
$214$	0	0
$215$	12.6905	0.865484
$216$	0	0
$217$	−39.7492	−2.69835
$218$	0	0
$219$	27.8199	1.87990
$220$	0	0
$221$	−4.39295	−0.295502
$222$	0	0
$223$	18.0937	1.21165	0.605823	−	0.795600i	$-0.292844\pi$
$223$	18.0937	1.21165	0.605823	+	0.795600i	$0.292844\pi$
$224$	0	0
$225$	0.383441	0.0255628
$226$	0	0
$227$	28.1788	1.87029	0.935145	−	0.354265i	$-0.115269\pi$
$227$	28.1788	1.87029	0.935145	+	0.354265i	$0.115269\pi$
$228$	0	0
$229$	−5.73616	−0.379056	−0.189528	−	0.981875i	$-0.560696\pi$
$229$	−5.73616	−0.379056	−0.189528	+	0.981875i	$0.560696\pi$
$230$	0	0
$231$	42.2907	2.78252
$232$	0	0
$233$	2.91341	0.190864	0.0954318	−	0.995436i	$-0.469577\pi$
$233$	2.91341	0.190864	0.0954318	+	0.995436i	$0.469577\pi$
$234$	0	0
$235$	−11.8558	−0.773388
$236$	0	0
$237$	0.683808	0.0444181
$238$	0	0
$239$	−18.2010	−1.17732	−0.588662	−	0.808380i	$-0.700345\pi$
$239$	−18.2010	−1.17732	−0.588662	+	0.808380i	$0.700345\pi$
$240$	0	0
$241$	2.23828	0.144180	0.0720901	−	0.997398i	$-0.477033\pi$
$241$	2.23828	0.144180	0.0720901	+	0.997398i	$0.477033\pi$
$242$	0	0
$243$	5.98238	0.383770
$244$	0	0
$245$	−29.4876	−1.88390
$246$	0	0
$247$	11.1235	0.707772
$248$	0	0
$249$	−16.7572	−1.06194
$250$	0	0
$251$	6.51159	0.411008	0.205504	−	0.978656i	$-0.434117\pi$
$251$	6.51159	0.411008	0.205504	+	0.978656i	$0.434117\pi$
$252$	0	0
$253$	3.59056	0.225737
$254$	0	0
$255$	4.50247	0.281956
$256$	0	0
$257$	0.933766	0.0582467	0.0291234	−	0.999576i	$-0.490728\pi$
$257$	0.933766	0.0582467	0.0291234	+	0.999576i	$0.490728\pi$
$258$	0	0
$259$	−5.76320	−0.358108
$260$	0	0
$261$	1.39601	0.0864107
$262$	0	0
$263$	7.31630	0.451142	0.225571	−	0.974227i	$-0.427575\pi$
$263$	7.31630	0.451142	0.225571	+	0.974227i	$0.427575\pi$
$264$	0	0
$265$	13.9098	0.854471
$266$	0	0
$267$	−15.6623	−0.958515
$268$	0	0
$269$	6.81567	0.415558	0.207779	−	0.978176i	$-0.433376\pi$
$269$	6.81567	0.415558	0.207779	+	0.978176i	$0.433376\pi$
$270$	0	0
$271$	9.32471	0.566436	0.283218	−	0.959056i	$-0.408598\pi$
$271$	9.32471	0.566436	0.283218	+	0.959056i	$0.408598\pi$
$272$	0	0
$273$	36.6253	2.21667
$274$	0	0
$275$	−3.33364	−0.201026
$276$	0	0
$277$	2.81439	0.169100	0.0845501	−	0.996419i	$-0.473055\pi$
$277$	2.81439	0.169100	0.0845501	+	0.996419i	$0.473055\pi$
$278$	0	0
$279$	−5.26568	−0.315248
$280$	0	0
$281$	−12.7445	−0.760276	−0.380138	−	0.924930i	$-0.624123\pi$
$281$	−12.7445	−0.760276	−0.380138	+	0.924930i	$0.624123\pi$
$282$	0	0
$283$	22.9940	1.36685	0.683426	−	0.730020i	$-0.260489\pi$
$283$	22.9940	1.36685	0.683426	+	0.730020i	$0.260489\pi$
$284$	0	0
$285$	−11.4008	−0.675327
$286$	0	0
$287$	51.0379	3.01267
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−12.4823	−0.731723
$292$	0	0
$293$	4.08090	0.238408	0.119204	−	0.992870i	$-0.461966\pi$
$293$	4.08090	0.238408	0.119204	+	0.992870i	$0.461966\pi$
$294$	0	0
$295$	−2.37849	−0.138481
$296$	0	0
$297$	−23.2042	−1.34645
$298$	0	0
$299$	3.10956	0.179831
$300$	0	0
$301$	−23.4991	−1.35447
$302$	0	0
$303$	−6.12808	−0.352049
$304$	0	0
$305$	36.3292	2.08020
$306$	0	0
$307$	22.4623	1.28199	0.640997	−	0.767544i	$-0.278521\pi$
$307$	22.4623	1.28199	0.640997	+	0.767544i	$0.278521\pi$
$308$	0	0
$309$	14.7830	0.840974
$310$	0	0
$311$	11.9009	0.674839	0.337420	−	0.941354i	$-0.390446\pi$
$311$	11.9009	0.674839	0.337420	+	0.941354i	$0.390446\pi$
$312$	0	0
$313$	9.06760	0.512531	0.256265	−	0.966606i	$-0.417508\pi$
$313$	9.06760	0.512531	0.256265	+	0.966606i	$0.417508\pi$
$314$	0	0
$315$	−6.11189	−0.344366
$316$	0	0
$317$	33.8407	1.90068	0.950342	−	0.311209i	$-0.100734\pi$
$317$	33.8407	1.90068	0.950342	+	0.311209i	$0.100734\pi$
$318$	0	0
$319$	−12.1369	−0.679535
$320$	0	0
$321$	19.1431	1.06846
$322$	0	0
$323$	−2.53212	−0.140891
$324$	0	0
$325$	−2.88706	−0.160145
$326$	0	0
$327$	−25.4973	−1.41000
$328$	0	0
$329$	21.9536	1.21034
$330$	0	0
$331$	−17.7123	−0.973557	−0.486779	−	0.873525i	$-0.661828\pi$
$331$	−17.7123	−0.973557	−0.486779	+	0.873525i	$0.661828\pi$
$332$	0	0
$333$	−0.763466	−0.0418377
$334$	0	0
$335$	−8.58523	−0.469061
$336$	0	0
$337$	9.65368	0.525869	0.262935	−	0.964814i	$-0.415310\pi$
$337$	9.65368	0.525869	0.262935	+	0.964814i	$0.415310\pi$
$338$	0	0
$339$	1.73589	0.0942808
$340$	0	0
$341$	45.7798	2.47911
$342$	0	0
$343$	23.7727	1.28361
$344$	0	0
$345$	−3.18709	−0.171587
$346$	0	0
$347$	0.0908954	0.00487952	0.00243976	−	0.999997i	$-0.499223\pi$
$347$	0.0908954	0.00487952	0.00243976	+	0.999997i	$0.499223\pi$
$348$	0	0
$349$	−28.9061	−1.54731	−0.773654	−	0.633608i	$-0.781573\pi$
$349$	−28.9061	−1.54731	−0.773654	+	0.633608i	$0.781573\pi$
$350$	0	0
$351$	−20.0957	−1.07263
$352$	0	0
$353$	−33.3764	−1.77645	−0.888224	−	0.459412i	$-0.848060\pi$
$353$	−33.3764	−1.77645	−0.888224	+	0.459412i	$0.848060\pi$
$354$	0	0
$355$	6.27527	0.333057
$356$	0	0
$357$	−8.33729	−0.441256
$358$	0	0
$359$	16.4970	0.870679	0.435339	−	0.900266i	$-0.356628\pi$
$359$	16.4970	0.870679	0.435339	+	0.900266i	$0.356628\pi$
$360$	0	0
$361$	−12.5883	−0.662544
$362$	0	0
$363$	−27.8839	−1.46352
$364$	0	0
$365$	34.9548	1.82962
$366$	0	0
$367$	22.2727	1.16262	0.581312	−	0.813681i	$-0.302540\pi$
$367$	22.2727	1.16262	0.581312	+	0.813681i	$0.302540\pi$
$368$	0	0
$369$	6.76112	0.351970
$370$	0	0
$371$	−25.7569	−1.33723
$372$	0	0
$373$	−33.7632	−1.74819	−0.874095	−	0.485755i	$-0.838545\pi$
$373$	−33.7632	−1.74819	−0.874095	+	0.485755i	$0.838545\pi$
$374$	0	0
$375$	−19.5533	−1.00973
$376$	0	0
$377$	−10.5110	−0.541344
$378$	0	0
$379$	24.3138	1.24892	0.624459	−	0.781058i	$-0.285320\pi$
$379$	24.3138	1.24892	0.624459	+	0.781058i	$0.285320\pi$
$380$	0	0
$381$	−1.43325	−0.0734275
$382$	0	0
$383$	−1.00767	−0.0514894	−0.0257447	−	0.999669i	$-0.508196\pi$
$383$	−1.00767	−0.0514894	−0.0257447	+	0.999669i	$0.508196\pi$
$384$	0	0
$385$	53.1368	2.70810
$386$	0	0
$387$	−3.11299	−0.158242
$388$	0	0
$389$	22.5472	1.14319	0.571594	−	0.820537i	$-0.306325\pi$
$389$	22.5472	1.14319	0.571594	+	0.820537i	$0.306325\pi$
$390$	0	0
$391$	−0.707852	−0.0357976
$392$	0	0
$393$	5.47673	0.276264
$394$	0	0
$395$	0.859181	0.0432301
$396$	0	0
$397$	−16.1011	−0.808093	−0.404047	−	0.914738i	$-0.632397\pi$
$397$	−16.1011	−0.808093	−0.404047	+	0.914738i	$0.632397\pi$
$398$	0	0
$399$	21.1111	1.05687
$400$	0	0
$401$	−15.7437	−0.786202	−0.393101	−	0.919495i	$-0.628598\pi$
$401$	−15.7437	−0.786202	−0.393101	+	0.919495i	$0.628598\pi$
$402$	0	0
$403$	39.6470	1.97496
$404$	0	0
$405$	24.7599	1.23033
$406$	0	0
$407$	6.63757	0.329012
$408$	0	0
$409$	−19.6150	−0.969898	−0.484949	−	0.874542i	$-0.661162\pi$
$409$	−19.6150	−0.969898	−0.484949	+	0.874542i	$0.661162\pi$
$410$	0	0
$411$	−23.4071	−1.15459
$412$	0	0
$413$	4.40428	0.216720
$414$	0	0
$415$	−21.0548	−1.03354
$416$	0	0
$417$	−20.8420	−1.02064
$418$	0	0
$419$	35.9286	1.75523	0.877613	−	0.479369i	$-0.159134\pi$
$419$	35.9286	1.75523	0.877613	+	0.479369i	$0.159134\pi$
$420$	0	0
$421$	7.40371	0.360835	0.180417	−	0.983590i	$-0.442255\pi$
$421$	7.40371	0.360835	0.180417	+	0.983590i	$0.442255\pi$
$422$	0	0
$423$	2.90824	0.141404
$424$	0	0
$425$	0.657202	0.0318790
$426$	0	0
$427$	−67.2713	−3.25548
$428$	0	0
$429$	−42.1820	−2.03657
$430$	0	0
$431$	16.9900	0.818378	0.409189	−	0.912450i	$-0.365812\pi$
$431$	16.9900	0.818378	0.409189	+	0.912450i	$0.365812\pi$
$432$	0	0
$433$	−3.32106	−0.159600	−0.0797999	−	0.996811i	$-0.525428\pi$
$433$	−3.32106	−0.159600	−0.0797999	+	0.996811i	$0.525428\pi$
$434$	0	0
$435$	10.7730	0.516528
$436$	0	0
$437$	1.79237	0.0857407
$438$	0	0
$439$	31.9932	1.52695	0.763475	−	0.645838i	$-0.223492\pi$
$439$	31.9932	1.52695	0.763475	+	0.645838i	$0.223492\pi$
$440$	0	0
$441$	7.23335	0.344445
$442$	0	0
$443$	12.5497	0.596254	0.298127	−	0.954526i	$-0.403638\pi$
$443$	12.5497	0.596254	0.298127	+	0.954526i	$0.403638\pi$
$444$	0	0
$445$	−19.6791	−0.932878
$446$	0	0
$447$	−11.5883	−0.548106
$448$	0	0
$449$	−29.1467	−1.37552	−0.687759	−	0.725939i	$-0.741405\pi$
$449$	−29.1467	−1.37552	−0.687759	+	0.725939i	$0.741405\pi$
$450$	0	0
$451$	−58.7811	−2.76789
$452$	0	0
$453$	−0.785875	−0.0369236
$454$	0	0
$455$	46.0184	2.15738
$456$	0	0
$457$	−22.2668	−1.04160	−0.520798	−	0.853680i	$-0.674366\pi$
$457$	−22.2668	−1.04160	−0.520798	+	0.853680i	$0.674366\pi$
$458$	0	0
$459$	4.57454	0.213521
$460$	0	0
$461$	−2.57166	−0.119774	−0.0598872	−	0.998205i	$-0.519074\pi$
$461$	−2.57166	−0.119774	−0.0598872	+	0.998205i	$0.519074\pi$
$462$	0	0
$463$	38.7171	1.79934	0.899668	−	0.436575i	$-0.143809\pi$
$463$	38.7171	1.79934	0.899668	+	0.436575i	$0.143809\pi$
$464$	0	0
$465$	−40.6354	−1.88442
$466$	0	0
$467$	32.3214	1.49566	0.747829	−	0.663892i	$-0.231097\pi$
$467$	32.3214	1.49566	0.747829	+	0.663892i	$0.231097\pi$
$468$	0	0
$469$	15.8974	0.734073
$470$	0	0
$471$	34.3629	1.58336
$472$	0	0
$473$	27.0643	1.24442
$474$	0	0
$475$	−1.66412	−0.0763549
$476$	0	0
$477$	−3.41208	−0.156229
$478$	0	0
$479$	−28.6426	−1.30871	−0.654357	−	0.756186i	$-0.727061\pi$
$479$	−28.6426	−1.30871	−0.654357	+	0.756186i	$0.727061\pi$
$480$	0	0
$481$	5.74839	0.262104
$482$	0	0
$483$	5.90157	0.268531
$484$	0	0
$485$	−15.6835	−0.712152
$486$	0	0
$487$	−25.9100	−1.17409	−0.587046	−	0.809553i	$-0.699709\pi$
$487$	−25.9100	−1.17409	−0.587046	+	0.809553i	$0.699709\pi$
$488$	0	0
$489$	−12.3950	−0.560523
$490$	0	0
$491$	−33.1727	−1.49706	−0.748531	−	0.663100i	$-0.769241\pi$
$491$	−33.1727	−1.49706	−0.748531	+	0.663100i	$0.769241\pi$
$492$	0	0
$493$	2.39269	0.107762
$494$	0	0
$495$	7.03916	0.316387
$496$	0	0
$497$	−11.6200	−0.521228
$498$	0	0
$499$	−7.45259	−0.333624	−0.166812	−	0.985989i	$-0.553347\pi$
$499$	−7.45259	−0.333624	−0.166812	+	0.985989i	$0.553347\pi$
$500$	0	0
$501$	31.2725	1.39715
$502$	0	0
$503$	−12.4655	−0.555810	−0.277905	−	0.960609i	$-0.589640\pi$
$503$	−12.4655	−0.555810	−0.277905	+	0.960609i	$0.589640\pi$
$504$	0	0
$505$	−7.69972	−0.342633
$506$	0	0
$507$	−11.9222	−0.529483
$508$	0	0
$509$	−18.8029	−0.833422	−0.416711	−	0.909039i	$-0.636817\pi$
$509$	−18.8029	−0.833422	−0.416711	+	0.909039i	$0.636817\pi$
$510$	0	0
$511$	−64.7262	−2.86332
$512$	0	0
$513$	−11.5833	−0.511415
$514$	0	0
$515$	18.5743	0.818481
$516$	0	0
$517$	−25.2843	−1.11200
$518$	0	0
$519$	4.33575	0.190318
$520$	0	0
$521$	18.4193	0.806965	0.403482	−	0.914987i	$-0.367800\pi$
$521$	18.4193	0.806965	0.403482	+	0.914987i	$0.367800\pi$
$522$	0	0
$523$	−20.3156	−0.888340	−0.444170	−	0.895942i	$-0.646501\pi$
$523$	−20.3156	−0.888340	−0.444170	+	0.895942i	$0.646501\pi$
$524$	0	0
$525$	−5.47928	−0.239135
$526$	0	0
$527$	−9.02514	−0.393141
$528$	0	0
$529$	−22.4989	−0.978215
$530$	0	0
$531$	0.583446	0.0253194
$532$	0	0
$533$	−50.9067	−2.20501
$534$	0	0
$535$	24.0526	1.03989
$536$	0	0
$537$	−7.57591	−0.326925
$538$	0	0
$539$	−62.8867	−2.70872
$540$	0	0
$541$	−29.8813	−1.28470	−0.642349	−	0.766412i	$-0.722040\pi$
$541$	−29.8813	−1.28470	−0.642349	+	0.766412i	$0.722040\pi$
$542$	0	0
$543$	−29.9614	−1.28577
$544$	0	0
$545$	−32.0365	−1.37229
$546$	0	0
$547$	−4.62407	−0.197711	−0.0988554	−	0.995102i	$-0.531518\pi$
$547$	−4.62407	−0.197711	−0.0988554	+	0.995102i	$0.531518\pi$
$548$	0	0
$549$	−8.91159	−0.380338
$550$	0	0
$551$	−6.05860	−0.258105
$552$	0	0
$553$	−1.59096	−0.0676544
$554$	0	0
$555$	−5.89170	−0.250089
$556$	0	0
$557$	−45.6524	−1.93435	−0.967176	−	0.254106i	$-0.918219\pi$
$557$	−45.6524	−1.93435	−0.967176	+	0.254106i	$0.918219\pi$
$558$	0	0
$559$	23.4387	0.991353
$560$	0	0
$561$	9.60219	0.405405
$562$	0	0
$563$	−7.52438	−0.317115	−0.158557	−	0.987350i	$-0.550684\pi$
$563$	−7.52438	−0.317115	−0.158557	+	0.987350i	$0.550684\pi$
$564$	0	0
$565$	2.18109	0.0917591
$566$	0	0
$567$	−45.8482	−1.92544
$568$	0	0
$569$	−5.91161	−0.247828	−0.123914	−	0.992293i	$-0.539545\pi$
$569$	−5.91161	−0.247828	−0.123914	+	0.992293i	$0.539545\pi$
$570$	0	0
$571$	1.57021	0.0657112	0.0328556	−	0.999460i	$-0.489540\pi$
$571$	1.57021	0.0657112	0.0328556	+	0.999460i	$0.489540\pi$
$572$	0	0
$573$	−38.9538	−1.62732
$574$	0	0
$575$	−0.465202	−0.0194003
$576$	0	0
$577$	13.9644	0.581345	0.290673	−	0.956823i	$-0.406121\pi$
$577$	13.9644	0.581345	0.290673	+	0.956823i	$0.406121\pi$
$578$	0	0
$579$	43.3586	1.80192
$580$	0	0
$581$	38.9875	1.61748
$582$	0	0
$583$	29.6646	1.22858
$584$	0	0
$585$	6.09618	0.252046
$586$	0	0
$587$	−20.2793	−0.837014	−0.418507	−	0.908213i	$-0.637447\pi$
$587$	−20.2793	−0.837014	−0.418507	+	0.908213i	$0.637447\pi$
$588$	0	0
$589$	22.8528	0.941632
$590$	0	0
$591$	−40.6947	−1.67396
$592$	0	0
$593$	5.91036	0.242709	0.121355	−	0.992609i	$-0.461276\pi$
$593$	5.91036	0.242709	0.121355	+	0.992609i	$0.461276\pi$
$594$	0	0
$595$	−10.4755	−0.429454
$596$	0	0
$597$	−36.0926	−1.47717
$598$	0	0
$599$	−33.3390	−1.36220	−0.681098	−	0.732192i	$-0.738497\pi$
$599$	−33.3390	−1.36220	−0.681098	+	0.732192i	$0.738497\pi$
$600$	0	0
$601$	5.20052	0.212133	0.106067	−	0.994359i	$-0.466174\pi$
$601$	5.20052	0.212133	0.106067	+	0.994359i	$0.466174\pi$
$602$	0	0
$603$	2.10597	0.0857616
$604$	0	0
$605$	−35.0351	−1.42438
$606$	0	0
$607$	17.7508	0.720483	0.360242	−	0.932859i	$-0.382694\pi$
$607$	17.7508	0.720483	0.360242	+	0.932859i	$0.382694\pi$
$608$	0	0
$609$	−19.9486	−0.808358
$610$	0	0
$611$	−21.8971	−0.885863
$612$	0	0
$613$	−8.05335	−0.325272	−0.162636	−	0.986686i	$-0.552000\pi$
$613$	−8.05335	−0.325272	−0.162636	+	0.986686i	$0.552000\pi$
$614$	0	0
$615$	52.1758	2.10393
$616$	0	0
$617$	25.1093	1.01086	0.505432	−	0.862866i	$-0.331333\pi$
$617$	25.1093	1.01086	0.505432	+	0.862866i	$0.331333\pi$
$618$	0	0
$619$	27.4401	1.10291	0.551456	−	0.834204i	$-0.314073\pi$
$619$	27.4401	1.10291	0.551456	+	0.834204i	$0.314073\pi$
$620$	0	0
$621$	−3.23810	−0.129940
$622$	0	0
$623$	36.4400	1.45994
$624$	0	0
$625$	−27.8541	−1.11416
$626$	0	0
$627$	−24.3139	−0.971005
$628$	0	0
$629$	−1.30855	−0.0521752
$630$	0	0
$631$	−18.1477	−0.722450	−0.361225	−	0.932479i	$-0.617641\pi$
$631$	−18.1477	−0.722450	−0.361225	+	0.932479i	$0.617641\pi$
$632$	0	0
$633$	−29.2689	−1.16333
$634$	0	0
$635$	−1.80083	−0.0714636
$636$	0	0
$637$	−54.4623	−2.15787
$638$	0	0
$639$	−1.53933	−0.0608950
$640$	0	0
$641$	37.5071	1.48144	0.740721	−	0.671813i	$-0.234484\pi$
$641$	37.5071	1.48144	0.740721	+	0.671813i	$0.234484\pi$
$642$	0	0
$643$	−25.4430	−1.00338	−0.501688	−	0.865049i	$-0.667287\pi$
$643$	−25.4430	−1.00338	−0.501688	+	0.865049i	$0.667287\pi$
$644$	0	0
$645$	−24.0231	−0.945908
$646$	0	0
$647$	−0.469511	−0.0184584	−0.00922920	−	0.999957i	$-0.502938\pi$
$647$	−0.469511	−0.0184584	−0.00922920	+	0.999957i	$0.502938\pi$
$648$	0	0
$649$	−5.07247	−0.199112
$650$	0	0
$651$	75.2452	2.94909
$652$	0	0
$653$	−40.8797	−1.59975	−0.799874	−	0.600168i	$-0.795100\pi$
$653$	−40.8797	−1.59975	−0.799874	+	0.600168i	$0.795100\pi$
$654$	0	0
$655$	6.88132	0.268875
$656$	0	0
$657$	−8.57445	−0.334521
$658$	0	0
$659$	−42.2549	−1.64602	−0.823008	−	0.568030i	$-0.807706\pi$
$659$	−42.2549	−1.64602	−0.823008	+	0.568030i	$0.807706\pi$
$660$	0	0
$661$	−7.66118	−0.297985	−0.148993	−	0.988838i	$-0.547603\pi$
$661$	−7.66118	−0.297985	−0.148993	+	0.988838i	$0.547603\pi$
$662$	0	0
$663$	8.31586	0.322961
$664$	0	0
$665$	26.5253	1.02861
$666$	0	0
$667$	−1.69367	−0.0655793
$668$	0	0
$669$	−34.2514	−1.32424
$670$	0	0
$671$	77.4774	2.99098
$672$	0	0
$673$	45.3388	1.74768	0.873840	−	0.486213i	$-0.161622\pi$
$673$	45.3388	1.74768	0.873840	+	0.486213i	$0.161622\pi$
$674$	0	0
$675$	3.00639	0.115716
$676$	0	0
$677$	38.0313	1.46166	0.730830	−	0.682559i	$-0.239133\pi$
$677$	38.0313	1.46166	0.730830	+	0.682559i	$0.239133\pi$
$678$	0	0
$679$	29.0414	1.11451
$680$	0	0
$681$	−53.3424	−2.04408
$682$	0	0
$683$	23.9172	0.915166	0.457583	−	0.889167i	$-0.348715\pi$
$683$	23.9172	0.915166	0.457583	+	0.889167i	$0.348715\pi$
$684$	0	0
$685$	−29.4102	−1.12371
$686$	0	0
$687$	10.8585	0.414279
$688$	0	0
$689$	25.6907	0.978738
$690$	0	0
$691$	10.0999	0.384220	0.192110	−	0.981373i	$-0.438467\pi$
$691$	10.0999	0.384220	0.192110	+	0.981373i	$0.438467\pi$
$692$	0	0
$693$	−13.0345	−0.495140
$694$	0	0
$695$	−26.1873	−0.993341
$696$	0	0
$697$	11.5883	0.438936
$698$	0	0
$699$	−5.51507	−0.208599
$700$	0	0
$701$	−21.8589	−0.825600	−0.412800	−	0.910822i	$-0.635449\pi$
$701$	−21.8589	−0.825600	−0.412800	+	0.910822i	$0.635449\pi$
$702$	0	0
$703$	3.31341	0.124967
$704$	0	0
$705$	22.4430	0.845254
$706$	0	0
$707$	14.2577	0.536215
$708$	0	0
$709$	19.1455	0.719025	0.359513	−	0.933140i	$-0.382943\pi$
$709$	19.1455	0.719025	0.359513	+	0.933140i	$0.382943\pi$
$710$	0	0
$711$	−0.210758	−0.00790405
$712$	0	0
$713$	6.38847	0.239250
$714$	0	0
$715$	−53.0002	−1.98209
$716$	0	0
$717$	34.4544	1.28672
$718$	0	0
$719$	33.5187	1.25004	0.625018	−	0.780610i	$-0.285091\pi$
$719$	33.5187	1.25004	0.625018	+	0.780610i	$0.285091\pi$
$720$	0	0
$721$	−34.3943	−1.28091
$722$	0	0
$723$	−4.23706	−0.157578
$724$	0	0
$725$	1.57248	0.0584005
$726$	0	0
$727$	23.4045	0.868026	0.434013	−	0.900907i	$-0.357097\pi$
$727$	23.4045	0.868026	0.434013	+	0.900907i	$0.357097\pi$
$728$	0	0
$729$	19.9051	0.737228
$730$	0	0
$731$	−5.33553	−0.197342
$732$	0	0
$733$	38.8987	1.43676	0.718378	−	0.695653i	$-0.244885\pi$
$733$	38.8987	1.43676	0.718378	+	0.695653i	$0.244885\pi$
$734$	0	0
$735$	55.8201	2.05895
$736$	0	0
$737$	−18.3093	−0.674431
$738$	0	0
$739$	−1.18203	−0.0434817	−0.0217408	−	0.999764i	$-0.506921\pi$
$739$	−1.18203	−0.0434817	−0.0217408	+	0.999764i	$0.506921\pi$
$740$	0	0
$741$	−21.0568	−0.773540
$742$	0	0
$743$	15.4151	0.565526	0.282763	−	0.959190i	$-0.408749\pi$
$743$	15.4151	0.565526	0.282763	+	0.959190i	$0.408749\pi$
$744$	0	0
$745$	−14.5602	−0.533446
$746$	0	0
$747$	5.16478	0.188969
$748$	0	0
$749$	−44.5385	−1.62740
$750$	0	0
$751$	15.7537	0.574862	0.287431	−	0.957801i	$-0.407199\pi$
$751$	15.7537	0.574862	0.287431	+	0.957801i	$0.407199\pi$
$752$	0	0
$753$	−12.3264	−0.449200
$754$	0	0
$755$	−0.987424	−0.0359360
$756$	0	0
$757$	44.2197	1.60719	0.803597	−	0.595174i	$-0.202917\pi$
$757$	44.2197	1.60719	0.803597	+	0.595174i	$0.202917\pi$
$758$	0	0
$759$	−6.79693	−0.246713
$760$	0	0
$761$	0.325670	0.0118055	0.00590277	−	0.999983i	$-0.498121\pi$
$761$	0.325670	0.0118055	0.00590277	+	0.999983i	$0.498121\pi$
$762$	0	0
$763$	59.3224	2.14761
$764$	0	0
$765$	−1.38772	−0.0501730
$766$	0	0
$767$	−4.39295	−0.158620
$768$	0	0
$769$	3.05371	0.110120	0.0550598	−	0.998483i	$-0.482465\pi$
$769$	3.05371	0.110120	0.0550598	+	0.998483i	$0.482465\pi$
$770$	0	0
$771$	−1.76762	−0.0636592
$772$	0	0
$773$	−21.7358	−0.781781	−0.390890	−	0.920437i	$-0.627833\pi$
$773$	−21.7358	−0.781781	−0.390890	+	0.920437i	$0.627833\pi$
$774$	0	0
$775$	−5.93133	−0.213060
$776$	0	0
$777$	10.9097	0.391385
$778$	0	0
$779$	−29.3429	−1.05132
$780$	0	0
$781$	13.3829	0.478879
$782$	0	0
$783$	10.9455	0.391159
$784$	0	0
$785$	43.1757	1.54101
$786$	0	0
$787$	1.12421	0.0400739	0.0200370	−	0.999799i	$-0.493622\pi$
$787$	1.12421	0.0400739	0.0200370	+	0.999799i	$0.493622\pi$
$788$	0	0
$789$	−13.8497	−0.493064
$790$	0	0
$791$	−4.03875	−0.143601
$792$	0	0
$793$	67.0983	2.38273
$794$	0	0
$795$	−26.3312	−0.933871
$796$	0	0
$797$	29.2686	1.03675	0.518374	−	0.855154i	$-0.326538\pi$
$797$	29.2686	1.03675	0.518374	+	0.855154i	$0.326538\pi$
$798$	0	0
$799$	4.98460	0.176342
$800$	0	0
$801$	4.82730	0.170564
$802$	0	0
$803$	74.5462	2.63068
$804$	0	0
$805$	7.41512	0.261349
$806$	0	0
$807$	−12.9020	−0.454174
$808$	0	0
$809$	27.8881	0.980492	0.490246	−	0.871584i	$-0.336907\pi$
$809$	27.8881	0.980492	0.490246	+	0.871584i	$0.336907\pi$
$810$	0	0
$811$	−29.7359	−1.04417	−0.522084	−	0.852894i	$-0.674845\pi$
$811$	−29.7359	−1.04417	−0.522084	+	0.852894i	$0.674845\pi$
$812$	0	0
$813$	−17.6517	−0.619071
$814$	0	0
$815$	−15.5739	−0.545531
$816$	0	0
$817$	13.5102	0.472663
$818$	0	0
$819$	−11.2884	−0.394448
$820$	0	0
$821$	−31.4948	−1.09918	−0.549589	−	0.835435i	$-0.685216\pi$
$821$	−31.4948	−1.09918	−0.549589	+	0.835435i	$0.685216\pi$
$822$	0	0
$823$	8.33120	0.290407	0.145204	−	0.989402i	$-0.453616\pi$
$823$	8.33120	0.290407	0.145204	+	0.989402i	$0.453616\pi$
$824$	0	0
$825$	6.31057	0.219706
$826$	0	0
$827$	0.0862540	0.00299935	0.00149967	−	0.999999i	$-0.499523\pi$
$827$	0.0862540	0.00299935	0.00149967	+	0.999999i	$0.499523\pi$
$828$	0	0
$829$	43.9474	1.52636	0.763178	−	0.646189i	$-0.223638\pi$
$829$	43.9474	1.52636	0.763178	+	0.646189i	$0.223638\pi$
$830$	0	0
$831$	−5.32764	−0.184814
$832$	0	0
$833$	12.3976	0.429553
$834$	0	0
$835$	39.2928	1.35979
$836$	0	0
$837$	−41.2858	−1.42705
$838$	0	0
$839$	2.90851	0.100413	0.0502064	−	0.998739i	$-0.484012\pi$
$839$	2.90851	0.100413	0.0502064	+	0.998739i	$0.484012\pi$
$840$	0	0
$841$	−23.2750	−0.802587
$842$	0	0
$843$	24.1254	0.830923
$844$	0	0
$845$	−14.9798	−0.515321
$846$	0	0
$847$	64.8750	2.22913
$848$	0	0
$849$	−43.5277	−1.49387
$850$	0	0
$851$	0.926258	0.0317517
$852$	0	0
$853$	−21.3879	−0.732307	−0.366154	−	0.930554i	$-0.619326\pi$
$853$	−21.3879	−0.732307	−0.366154	+	0.930554i	$0.619326\pi$
$854$	0	0
$855$	3.51387	0.120172
$856$	0	0
$857$	−25.4202	−0.868338	−0.434169	−	0.900831i	$-0.642958\pi$
$857$	−25.4202	−0.868338	−0.434169	+	0.900831i	$0.642958\pi$
$858$	0	0
$859$	0.0961723	0.00328136	0.00164068	−	0.999999i	$-0.499478\pi$
$859$	0.0961723	0.00328136	0.00164068	+	0.999999i	$0.499478\pi$
$860$	0	0
$861$	−96.6146	−3.29262
$862$	0	0
$863$	11.9041	0.405220	0.202610	−	0.979259i	$-0.435058\pi$
$863$	11.9041	0.405220	0.202610	+	0.979259i	$0.435058\pi$
$864$	0	0
$865$	5.44772	0.185228
$866$	0	0
$867$	−1.89300	−0.0642896
$868$	0	0
$869$	1.83233	0.0621576
$870$	0	0
$871$	−15.8565	−0.537278
$872$	0	0
$873$	3.84719	0.130208
$874$	0	0
$875$	45.4930	1.53795
$876$	0	0
$877$	34.0109	1.14847	0.574234	−	0.818691i	$-0.305300\pi$
$877$	34.0109	1.14847	0.574234	+	0.818691i	$0.305300\pi$
$878$	0	0
$879$	−7.72513	−0.260562
$880$	0	0
$881$	−16.8338	−0.567145	−0.283572	−	0.958951i	$-0.591520\pi$
$881$	−16.8338	−0.567145	−0.283572	+	0.958951i	$0.591520\pi$
$882$	0	0
$883$	13.7658	0.463258	0.231629	−	0.972804i	$-0.425595\pi$
$883$	13.7658	0.463258	0.231629	+	0.972804i	$0.425595\pi$
$884$	0	0
$885$	4.50247	0.151349
$886$	0	0
$887$	−31.1028	−1.04433	−0.522164	−	0.852845i	$-0.674875\pi$
$887$	−31.1028	−1.04433	−0.522164	+	0.852845i	$0.674875\pi$
$888$	0	0
$889$	3.33461	0.111839
$890$	0	0
$891$	52.8041	1.76900
$892$	0	0
$893$	−12.6216	−0.422367
$894$	0	0
$895$	−9.51887	−0.318181
$896$	0	0
$897$	−5.88640	−0.196541
$898$	0	0
$899$	−21.5944	−0.720213
$900$	0	0
$901$	−5.84816	−0.194830
$902$	0	0
$903$	44.4839	1.48033
$904$	0	0
$905$	−37.6455	−1.25138
$906$	0	0
$907$	−21.1141	−0.701082	−0.350541	−	0.936547i	$-0.614002\pi$
$907$	−21.1141	−0.701082	−0.350541	+	0.936547i	$0.614002\pi$
$908$	0	0
$909$	1.88875	0.0626459
$910$	0	0
$911$	−11.5231	−0.381778	−0.190889	−	0.981612i	$-0.561137\pi$
$911$	−11.5231	−0.381778	−0.190889	+	0.981612i	$0.561137\pi$
$912$	0	0
$913$	−44.9026	−1.48606
$914$	0	0
$915$	−68.7711	−2.27350
$916$	0	0
$917$	−12.7422	−0.420785
$918$	0	0
$919$	−54.8980	−1.81092	−0.905459	−	0.424435i	$-0.860473\pi$
$919$	−54.8980	−1.81092	−0.905459	+	0.424435i	$0.860473\pi$
$920$	0	0
$921$	−42.5212	−1.40112
$922$	0	0
$923$	11.5901	0.381494
$924$	0	0
$925$	−0.859980	−0.0282760
$926$	0	0
$927$	−4.55630	−0.149648
$928$	0	0
$929$	−24.6266	−0.807973	−0.403987	−	0.914765i	$-0.632376\pi$
$929$	−24.6266	−0.807973	−0.403987	+	0.914765i	$0.632376\pi$
$930$	0	0
$931$	−31.3924	−1.02884
$932$	0	0
$933$	−22.5284	−0.737547
$934$	0	0
$935$	12.0648	0.394562
$936$	0	0
$937$	−50.4089	−1.64679	−0.823394	−	0.567470i	$-0.807922\pi$
$937$	−50.4089	−1.64679	−0.823394	+	0.567470i	$0.807922\pi$
$938$	0	0
$939$	−17.1650	−0.560157
$940$	0	0
$941$	49.9360	1.62787	0.813934	−	0.580957i	$-0.197322\pi$
$941$	49.9360	1.62787	0.813934	+	0.580957i	$0.197322\pi$
$942$	0	0
$943$	−8.20277	−0.267119
$944$	0	0
$945$	−47.9206	−1.55886
$946$	0	0
$947$	−37.0636	−1.20440	−0.602202	−	0.798344i	$-0.705710\pi$
$947$	−37.0636	−1.20440	−0.602202	+	0.798344i	$0.705710\pi$
$948$	0	0
$949$	64.5598	2.09570
$950$	0	0
$951$	−64.0604	−2.07730
$952$	0	0
$953$	−29.1426	−0.944020	−0.472010	−	0.881593i	$-0.656471\pi$
$953$	−29.1426	−0.944020	−0.472010	+	0.881593i	$0.656471\pi$
$954$	0	0
$955$	−48.9441	−1.58379
$956$	0	0
$957$	22.9751	0.742680
$958$	0	0
$959$	54.4592	1.75858
$960$	0	0
$961$	50.4531	1.62752
$962$	0	0
$963$	−5.90013	−0.190129
$964$	0	0
$965$	54.4786	1.75373
$966$	0	0
$967$	−47.9428	−1.54174	−0.770869	−	0.636994i	$-0.780178\pi$
$967$	−47.9428	−1.54174	−0.770869	+	0.636994i	$0.780178\pi$
$968$	0	0
$969$	4.79331	0.153983
$970$	0	0
$971$	31.6304	1.01507	0.507534	−	0.861632i	$-0.330557\pi$
$971$	31.6304	1.01507	0.507534	+	0.861632i	$0.330557\pi$
$972$	0	0
$973$	48.4914	1.55456
$974$	0	0
$975$	5.46519	0.175026
$976$	0	0
$977$	−24.4253	−0.781436	−0.390718	−	0.920510i	$-0.627773\pi$
$977$	−24.4253	−0.781436	−0.390718	+	0.920510i	$0.627773\pi$
$978$	0	0
$979$	−41.9686	−1.34132
$980$	0	0
$981$	7.85859	0.250905
$982$	0	0
$983$	−23.2555	−0.741734	−0.370867	−	0.928686i	$-0.620939\pi$
$983$	−23.2555	−0.741734	−0.370867	+	0.928686i	$0.620939\pi$
$984$	0	0
$985$	−51.1315	−1.62918
$986$	0	0
$987$	−41.5581	−1.32281
$988$	0	0
$989$	3.77677	0.120094
$990$	0	0
$991$	−25.7219	−0.817083	−0.408542	−	0.912740i	$-0.633963\pi$
$991$	−25.7219	−0.817083	−0.408542	+	0.912740i	$0.633963\pi$
$992$	0	0
$993$	33.5294	1.06402
$994$	0	0
$995$	−45.3491	−1.43766
$996$	0	0
$997$	16.4621	0.521359	0.260680	−	0.965425i	$-0.416053\pi$
$997$	16.4621	0.521359	0.260680	+	0.965425i	$0.416053\pi$
$998$	0	0
$999$	−5.98600	−0.189389

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.5	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.5	✓	21	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-1.89300 q^{3} -2.37849 q^{5} +4.40428 q^{7} +0.583446 q^{9} +O(q^{10})\)	\(q-1.89300 q^{3} -2.37849 q^{5} +4.40428 q^{7} +0.583446 q^{9} -5.07247 q^{11} -4.39295 q^{13} +4.50247 q^{15} +1.00000 q^{17} -2.53212 q^{19} -8.33729 q^{21} -0.707852 q^{23} +0.657202 q^{25} +4.57454 q^{27} +2.39269 q^{29} -9.02514 q^{31} +9.60219 q^{33} -10.4755 q^{35} -1.30855 q^{37} +8.31586 q^{39} +11.5883 q^{41} -5.33553 q^{43} -1.38772 q^{45} +4.98460 q^{47} +12.3976 q^{49} -1.89300 q^{51} -5.84816 q^{53} +12.0648 q^{55} +4.79331 q^{57} +1.00000 q^{59} -15.2741 q^{61} +2.56966 q^{63} +10.4486 q^{65} +3.60953 q^{67} +1.33996 q^{69} -2.63835 q^{71} -14.6962 q^{73} -1.24408 q^{75} -22.3406 q^{77} -0.361230 q^{79} -10.4099 q^{81} +8.85220 q^{83} -2.37849 q^{85} -4.52937 q^{87} +8.27378 q^{89} -19.3478 q^{91} +17.0846 q^{93} +6.02263 q^{95} +6.59391 q^{97} -2.95951 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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