LMFDB - Embedded newform 9904.2.a.n.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9904,2,Mod(1,9904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9904, 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("9904.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9904 = 2^{4} \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9904.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:	$79.0838381619$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 619)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	9904.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.456554 q^{3} +0.324059 q^{5} -2.73608 q^{7} -2.79156 q^{9} +O(q^{10})$	$q-0.456554 q^{3} +0.324059 q^{5} -2.73608 q^{7} -2.79156 q^{9} -5.58448 q^{11} +6.14956 q^{13} -0.147950 q^{15} -1.80252 q^{17} -1.89984 q^{19} +1.24917 q^{21} -0.651915 q^{23} -4.89499 q^{25} +2.64416 q^{27} +3.25305 q^{29} -7.40763 q^{31} +2.54962 q^{33} -0.886652 q^{35} -11.3565 q^{37} -2.80761 q^{39} +4.27888 q^{41} +8.46794 q^{43} -0.904630 q^{45} +6.21248 q^{47} +0.486153 q^{49} +0.822950 q^{51} -8.37397 q^{53} -1.80970 q^{55} +0.867380 q^{57} -8.13212 q^{59} -15.0533 q^{61} +7.63794 q^{63} +1.99282 q^{65} -0.0636738 q^{67} +0.297634 q^{69} -5.20384 q^{71} +11.9435 q^{73} +2.23482 q^{75} +15.2796 q^{77} +10.6355 q^{79} +7.16747 q^{81} +0.958841 q^{83} -0.584124 q^{85} -1.48519 q^{87} -15.8425 q^{89} -16.8257 q^{91} +3.38198 q^{93} -0.615661 q^{95} -4.18127 q^{97} +15.5894 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9}+O(q^{10}) $ 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9	$ 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9} - 23 q^{11} + 9 q^{13} + 2 q^{15} + 4 q^{17} + q^{19} + 30 q^{21} - 4 q^{23} + 35 q^{25} + 5 q^{27} + 90 q^{29} - 2 q^{31} - 6 q^{33} - 9 q^{35} + 19 q^{37} - 32 q^{39} + 59 q^{41} + 4 q^{43} + 30 q^{45} - 4 q^{47} + 30 q^{49} + 34 q^{53} + 17 q^{55} - 8 q^{57} - 13 q^{59} + 16 q^{61} + 40 q^{63} + 31 q^{65} + 11 q^{67} + 6 q^{69} - 42 q^{71} - 4 q^{73} + 52 q^{75} + 29 q^{77} - 3 q^{79} + 30 q^{81} + 11 q^{83} + 19 q^{85} + 20 q^{87} + 58 q^{89} + 39 q^{91} - 15 q^{93} - 23 q^{95} - 9 q^{97} + 8 q^{99}+O(q^{100}) $ 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9 - 23 * q^11 + 9 * q^13 + 2 * q^15 + 4 * q^17 + q^19 + 30 * q^21 - 4 * q^23 + 35 * q^25 + 5 * q^27 + 90 * q^29 - 2 * q^31 - 6 * q^33 - 9 * q^35 + 19 * q^37 - 32 * q^39 + 59 * q^41 + 4 * q^43 + 30 * q^45 - 4 * q^47 + 30 * q^49 + 34 * q^53 + 17 * q^55 - 8 * q^57 - 13 * q^59 + 16 * q^61 + 40 * q^63 + 31 * q^65 + 11 * q^67 + 6 * q^69 - 42 * q^71 - 4 * q^73 + 52 * q^75 + 29 * q^77 - 3 * q^79 + 30 * q^81 + 11 * q^83 + 19 * q^85 + 20 * q^87 + 58 * q^89 + 39 * q^91 - 15 * q^93 - 23 * q^95 - 9 * q^97 + 8 * 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
$3$	−0.456554	−0.263592	−0.131796	−	0.991277i	$-0.542074\pi$
$3$	−0.456554	−0.263592	−0.131796	+	0.991277i	$0.542074\pi$
$4$	0	0
$5$	0.324059	0.144924	0.0724618	−	0.997371i	$-0.476914\pi$
$5$	0.324059	0.144924	0.0724618	+	0.997371i	$0.476914\pi$
$6$	0	0
$7$	−2.73608	−1.03414	−0.517071	−	0.855942i	$-0.672978\pi$
$7$	−2.73608	−1.03414	−0.517071	+	0.855942i	$0.672978\pi$
$8$	0	0
$9$	−2.79156	−0.930520
$10$	0	0
$11$	−5.58448	−1.68378	−0.841892	−	0.539646i	$-0.818558\pi$
$11$	−5.58448	−1.68378	−0.841892	+	0.539646i	$0.818558\pi$
$12$	0	0
$13$	6.14956	1.70558	0.852790	−	0.522253i	$-0.174908\pi$
$13$	6.14956	1.70558	0.852790	+	0.522253i	$0.174908\pi$
$14$	0	0
$15$	−0.147950	−0.0382006
$16$	0	0
$17$	−1.80252	−0.437176	−0.218588	−	0.975817i	$-0.570145\pi$
$17$	−1.80252	−0.437176	−0.218588	+	0.975817i	$0.570145\pi$
$18$	0	0
$19$	−1.89984	−0.435853	−0.217927	−	0.975965i	$-0.569929\pi$
$19$	−1.89984	−0.435853	−0.217927	+	0.975965i	$0.569929\pi$
$20$	0	0
$21$	1.24917	0.272591
$22$	0	0
$23$	−0.651915	−0.135934	−0.0679668	−	0.997688i	$-0.521651\pi$
$23$	−0.651915	−0.135934	−0.0679668	+	0.997688i	$0.521651\pi$
$24$	0	0
$25$	−4.89499	−0.978997
$26$	0	0
$27$	2.64416	0.508869
$28$	0	0
$29$	3.25305	0.604076	0.302038	−	0.953296i	$-0.402333\pi$
$29$	3.25305	0.604076	0.302038	+	0.953296i	$0.402333\pi$
$30$	0	0
$31$	−7.40763	−1.33045	−0.665225	−	0.746643i	$-0.731664\pi$
$31$	−7.40763	−1.33045	−0.665225	+	0.746643i	$0.731664\pi$
$32$	0	0
$33$	2.54962	0.443831
$34$	0	0
$35$	−0.886652	−0.149872
$36$	0	0
$37$	−11.3565	−1.86699	−0.933495	−	0.358591i	$-0.883257\pi$
$37$	−11.3565	−1.86699	−0.933495	+	0.358591i	$0.883257\pi$
$38$	0	0
$39$	−2.80761	−0.449577
$40$	0	0
$41$	4.27888	0.668248	0.334124	−	0.942529i	$-0.391559\pi$
$41$	4.27888	0.668248	0.334124	+	0.942529i	$0.391559\pi$
$42$	0	0
$43$	8.46794	1.29135	0.645675	−	0.763613i	$-0.276576\pi$
$43$	8.46794	1.29135	0.645675	+	0.763613i	$0.276576\pi$
$44$	0	0
$45$	−0.904630	−0.134854
$46$	0	0
$47$	6.21248	0.906183	0.453092	−	0.891464i	$-0.350321\pi$
$47$	6.21248	0.906183	0.453092	+	0.891464i	$0.350321\pi$
$48$	0	0
$49$	0.486153	0.0694505
$50$	0	0
$51$	0.822950	0.115236
$52$	0	0
$53$	−8.37397	−1.15025	−0.575126	−	0.818065i	$-0.695047\pi$
$53$	−8.37397	−1.15025	−0.575126	+	0.818065i	$0.695047\pi$
$54$	0	0
$55$	−1.80970	−0.244020
$56$	0	0
$57$	0.867380	0.114887
$58$	0	0
$59$	−8.13212	−1.05871	−0.529356	−	0.848400i	$-0.677566\pi$
$59$	−8.13212	−1.05871	−0.529356	+	0.848400i	$0.677566\pi$
$60$	0	0
$61$	−15.0533	−1.92738	−0.963692	−	0.267017i	$-0.913962\pi$
$61$	−15.0533	−1.92738	−0.963692	+	0.267017i	$0.913962\pi$
$62$	0	0
$63$	7.63794	0.962290
$64$	0	0
$65$	1.99282	0.247179
$66$	0	0
$67$	−0.0636738	−0.00777899	−0.00388950	−	0.999992i	$-0.501238\pi$
$67$	−0.0636738	−0.00777899	−0.00388950	+	0.999992i	$0.501238\pi$
$68$	0	0
$69$	0.297634	0.0358310
$70$	0	0
$71$	−5.20384	−0.617582	−0.308791	−	0.951130i	$-0.599924\pi$
$71$	−5.20384	−0.617582	−0.308791	+	0.951130i	$0.599924\pi$
$72$	0	0
$73$	11.9435	1.39788	0.698939	−	0.715181i	$-0.253656\pi$
$73$	11.9435	1.39788	0.698939	+	0.715181i	$0.253656\pi$
$74$	0	0
$75$	2.23482	0.258055
$76$	0	0
$77$	15.2796	1.74127
$78$	0	0
$79$	10.6355	1.19658	0.598291	−	0.801279i	$-0.295847\pi$
$79$	10.6355	1.19658	0.598291	+	0.801279i	$0.295847\pi$
$80$	0	0
$81$	7.16747	0.796386
$82$	0	0
$83$	0.958841	0.105246	0.0526232	−	0.998614i	$-0.483242\pi$
$83$	0.958841	0.105246	0.0526232	+	0.998614i	$0.483242\pi$
$84$	0	0
$85$	−0.584124	−0.0633572
$86$	0	0
$87$	−1.48519	−0.159229
$88$	0	0
$89$	−15.8425	−1.67930	−0.839651	−	0.543127i	$-0.817240\pi$
$89$	−15.8425	−1.67930	−0.839651	+	0.543127i	$0.817240\pi$
$90$	0	0
$91$	−16.8257	−1.76381
$92$	0	0
$93$	3.38198	0.350695
$94$	0	0
$95$	−0.615661	−0.0631654
$96$	0	0
$97$	−4.18127	−0.424544	−0.212272	−	0.977211i	$-0.568086\pi$
$97$	−4.18127	−0.424544	−0.212272	+	0.977211i	$0.568086\pi$
$98$	0	0
$99$	15.5894	1.56679
$100$	0	0
$101$	−0.989339	−0.0984429	−0.0492215	−	0.998788i	$-0.515674\pi$
$101$	−0.989339	−0.0984429	−0.0492215	+	0.998788i	$0.515674\pi$
$102$	0	0
$103$	1.05602	0.104053	0.0520263	−	0.998646i	$-0.483432\pi$
$103$	1.05602	0.104053	0.0520263	+	0.998646i	$0.483432\pi$
$104$	0	0
$105$	0.404805	0.0395049
$106$	0	0
$107$	−11.4285	−1.10483	−0.552417	−	0.833568i	$-0.686294\pi$
$107$	−11.4285	−1.10483	−0.552417	+	0.833568i	$0.686294\pi$
$108$	0	0
$109$	−5.02427	−0.481238	−0.240619	−	0.970620i	$-0.577350\pi$
$109$	−5.02427	−0.481238	−0.240619	+	0.970620i	$0.577350\pi$
$110$	0	0
$111$	5.18483	0.492123
$112$	0	0
$113$	1.58608	0.149205	0.0746027	−	0.997213i	$-0.476231\pi$
$113$	1.58608	0.149205	0.0746027	+	0.997213i	$0.476231\pi$
$114$	0	0
$115$	−0.211259	−0.0197000
$116$	0	0
$117$	−17.1669	−1.58708
$118$	0	0
$119$	4.93186	0.452103
$120$	0	0
$121$	20.1864	1.83513
$122$	0	0
$123$	−1.95354	−0.176145
$124$	0	0
$125$	−3.20656	−0.286803
$126$	0	0
$127$	9.80901	0.870409	0.435205	−	0.900332i	$-0.356676\pi$
$127$	9.80901	0.870409	0.435205	+	0.900332i	$0.356676\pi$
$128$	0	0
$129$	−3.86607	−0.340389
$130$	0	0
$131$	−6.66849	−0.582629	−0.291314	−	0.956627i	$-0.594093\pi$
$131$	−6.66849	−0.582629	−0.291314	+	0.956627i	$0.594093\pi$
$132$	0	0
$133$	5.19812	0.450735
$134$	0	0
$135$	0.856863	0.0737470
$136$	0	0
$137$	14.5814	1.24577	0.622886	−	0.782313i	$-0.285960\pi$
$137$	14.5814	1.24577	0.622886	+	0.782313i	$0.285960\pi$
$138$	0	0
$139$	8.43054	0.715069	0.357534	−	0.933900i	$-0.383618\pi$
$139$	8.43054	0.715069	0.357534	+	0.933900i	$0.383618\pi$
$140$	0	0
$141$	−2.83633	−0.238862
$142$	0	0
$143$	−34.3421	−2.87183
$144$	0	0
$145$	1.05418	0.0875449
$146$	0	0
$147$	−0.221955	−0.0183065
$148$	0	0
$149$	−3.52837	−0.289055	−0.144527	−	0.989501i	$-0.546166\pi$
$149$	−3.52837	−0.289055	−0.144527	+	0.989501i	$0.546166\pi$
$150$	0	0
$151$	−14.7327	−1.19893	−0.599464	−	0.800401i	$-0.704620\pi$
$151$	−14.7327	−1.19893	−0.599464	+	0.800401i	$0.704620\pi$
$152$	0	0
$153$	5.03185	0.406801
$154$	0	0
$155$	−2.40051	−0.192813
$156$	0	0
$157$	−5.46385	−0.436063	−0.218031	−	0.975942i	$-0.569964\pi$
$157$	−5.46385	−0.436063	−0.218031	+	0.975942i	$0.569964\pi$
$158$	0	0
$159$	3.82317	0.303197
$160$	0	0
$161$	1.78369	0.140575
$162$	0	0
$163$	15.9649	1.25047	0.625235	−	0.780437i	$-0.285003\pi$
$163$	15.9649	1.25047	0.625235	+	0.780437i	$0.285003\pi$
$164$	0	0
$165$	0.826226	0.0643216
$166$	0	0
$167$	10.1368	0.784413	0.392206	−	0.919877i	$-0.371712\pi$
$167$	10.1368	0.784413	0.392206	+	0.919877i	$0.371712\pi$
$168$	0	0
$169$	24.8171	1.90901
$170$	0	0
$171$	5.30352	0.405570
$172$	0	0
$173$	9.93076	0.755022	0.377511	−	0.926005i	$-0.376780\pi$
$173$	9.93076	0.755022	0.377511	+	0.926005i	$0.376780\pi$
$174$	0	0
$175$	13.3931	1.01242
$176$	0	0
$177$	3.71275	0.279067
$178$	0	0
$179$	19.3305	1.44483	0.722416	−	0.691459i	$-0.243032\pi$
$179$	19.3305	1.44483	0.722416	+	0.691459i	$0.243032\pi$
$180$	0	0
$181$	−2.97160	−0.220877	−0.110439	−	0.993883i	$-0.535226\pi$
$181$	−2.97160	−0.220877	−0.110439	+	0.993883i	$0.535226\pi$
$182$	0	0
$183$	6.87266	0.508042
$184$	0	0
$185$	−3.68016	−0.270571
$186$	0	0
$187$	10.0662	0.736111
$188$	0	0
$189$	−7.23464	−0.526243
$190$	0	0
$191$	6.72781	0.486807	0.243404	−	0.969925i	$-0.421736\pi$
$191$	6.72781	0.486807	0.243404	+	0.969925i	$0.421736\pi$
$192$	0	0
$193$	−6.50657	−0.468353	−0.234177	−	0.972194i	$-0.575239\pi$
$193$	−6.50657	−0.468353	−0.234177	+	0.972194i	$0.575239\pi$
$194$	0	0
$195$	−0.909830	−0.0651542
$196$	0	0
$197$	0.618271	0.0440500	0.0220250	−	0.999757i	$-0.492989\pi$
$197$	0.618271	0.0440500	0.0220250	+	0.999757i	$0.492989\pi$
$198$	0	0
$199$	−14.2336	−1.00899	−0.504497	−	0.863414i	$-0.668322\pi$
$199$	−14.2336	−1.00899	−0.504497	+	0.863414i	$0.668322\pi$
$200$	0	0
$201$	0.0290705	0.00205048
$202$	0	0
$203$	−8.90062	−0.624701
$204$	0	0
$205$	1.38661	0.0968449
$206$	0	0
$207$	1.81986	0.126489
$208$	0	0
$209$	10.6096	0.733883
$210$	0	0
$211$	−14.5538	−1.00192	−0.500962	−	0.865469i	$-0.667020\pi$
$211$	−14.5538	−1.00192	−0.500962	+	0.865469i	$0.667020\pi$
$212$	0	0
$213$	2.37583	0.162789
$214$	0	0
$215$	2.74411	0.187147
$216$	0	0
$217$	20.2679	1.37587
$218$	0	0
$219$	−5.45284	−0.368469
$220$	0	0
$221$	−11.0847	−0.745640
$222$	0	0
$223$	−19.8528	−1.32944	−0.664722	−	0.747091i	$-0.731450\pi$
$223$	−19.8528	−1.32944	−0.664722	+	0.747091i	$0.731450\pi$
$224$	0	0
$225$	13.6646	0.910976
$226$	0	0
$227$	29.6461	1.96768	0.983841	−	0.179042i	$-0.0572998\pi$
$227$	29.6461	1.96768	0.983841	+	0.179042i	$0.0572998\pi$
$228$	0	0
$229$	18.4358	1.21827	0.609136	−	0.793066i	$-0.291517\pi$
$229$	18.4358	1.21827	0.609136	+	0.793066i	$0.291517\pi$
$230$	0	0
$231$	−6.97596	−0.458984
$232$	0	0
$233$	4.29821	0.281585	0.140793	−	0.990039i	$-0.455035\pi$
$233$	4.29821	0.281585	0.140793	+	0.990039i	$0.455035\pi$
$234$	0	0
$235$	2.01321	0.131327
$236$	0	0
$237$	−4.85566	−0.315409
$238$	0	0
$239$	6.86439	0.444020	0.222010	−	0.975044i	$-0.428738\pi$
$239$	6.86439	0.444020	0.222010	+	0.975044i	$0.428738\pi$
$240$	0	0
$241$	−4.99351	−0.321660	−0.160830	−	0.986982i	$-0.551417\pi$
$241$	−4.99351	−0.321660	−0.160830	+	0.986982i	$0.551417\pi$
$242$	0	0
$243$	−11.2048	−0.718789
$244$	0	0
$245$	0.157542	0.0100650
$246$	0	0
$247$	−11.6832	−0.743383
$248$	0	0
$249$	−0.437763	−0.0277421
$250$	0	0
$251$	−23.6247	−1.49118	−0.745588	−	0.666407i	$-0.767831\pi$
$251$	−23.6247	−1.49118	−0.745588	+	0.666407i	$0.767831\pi$
$252$	0	0
$253$	3.64061	0.228883
$254$	0	0
$255$	0.266684	0.0167004
$256$	0	0
$257$	31.3858	1.95779	0.978896	−	0.204357i	$-0.0655104\pi$
$257$	31.3858	1.95779	0.978896	+	0.204357i	$0.0655104\pi$
$258$	0	0
$259$	31.0722	1.93073
$260$	0	0
$261$	−9.08108	−0.562105
$262$	0	0
$263$	4.05291	0.249913	0.124956	−	0.992162i	$-0.460121\pi$
$263$	4.05291	0.249913	0.124956	+	0.992162i	$0.460121\pi$
$264$	0	0
$265$	−2.71366	−0.166699
$266$	0	0
$267$	7.23296	0.442650
$268$	0	0
$269$	13.4531	0.820253	0.410126	−	0.912029i	$-0.365485\pi$
$269$	13.4531	0.820253	0.410126	+	0.912029i	$0.365485\pi$
$270$	0	0
$271$	−8.19178	−0.497615	−0.248808	−	0.968553i	$-0.580039\pi$
$271$	−8.19178	−0.497615	−0.248808	+	0.968553i	$0.580039\pi$
$272$	0	0
$273$	7.68184	0.464926
$274$	0	0
$275$	27.3359	1.64842
$276$	0	0
$277$	8.98587	0.539908	0.269954	−	0.962873i	$-0.412991\pi$
$277$	8.98587	0.539908	0.269954	+	0.962873i	$0.412991\pi$
$278$	0	0
$279$	20.6788	1.23801
$280$	0	0
$281$	−6.69336	−0.399292	−0.199646	−	0.979868i	$-0.563979\pi$
$281$	−6.69336	−0.399292	−0.199646	+	0.979868i	$0.563979\pi$
$282$	0	0
$283$	1.11110	0.0660479	0.0330240	−	0.999455i	$-0.489486\pi$
$283$	1.11110	0.0660479	0.0330240	+	0.999455i	$0.489486\pi$
$284$	0	0
$285$	0.281082	0.0166499
$286$	0	0
$287$	−11.7074	−0.691064
$288$	0	0
$289$	−13.7509	−0.808877
$290$	0	0
$291$	1.90898	0.111906
$292$	0	0
$293$	13.9330	0.813975	0.406988	−	0.913434i	$-0.366579\pi$
$293$	13.9330	0.813975	0.406988	+	0.913434i	$0.366579\pi$
$294$	0	0
$295$	−2.63529	−0.153432
$296$	0	0
$297$	−14.7662	−0.856825
$298$	0	0
$299$	−4.00899	−0.231846
$300$	0	0
$301$	−23.1690	−1.33544
$302$	0	0
$303$	0.451687	0.0259487
$304$	0	0
$305$	−4.87817	−0.279323
$306$	0	0
$307$	22.6694	1.29381	0.646906	−	0.762570i	$-0.276063\pi$
$307$	22.6694	1.29381	0.646906	+	0.762570i	$0.276063\pi$
$308$	0	0
$309$	−0.482129	−0.0274274
$310$	0	0
$311$	7.07052	0.400932	0.200466	−	0.979701i	$-0.435754\pi$
$311$	7.07052	0.400932	0.200466	+	0.979701i	$0.435754\pi$
$312$	0	0
$313$	19.6863	1.11274	0.556369	−	0.830935i	$-0.312194\pi$
$313$	19.6863	1.11274	0.556369	+	0.830935i	$0.312194\pi$
$314$	0	0
$315$	2.47514	0.139458
$316$	0	0
$317$	34.4847	1.93686	0.968428	−	0.249294i	$-0.0801985\pi$
$317$	34.4847	1.93686	0.968428	+	0.249294i	$0.0801985\pi$
$318$	0	0
$319$	−18.1666	−1.01713
$320$	0	0
$321$	5.21772	0.291225
$322$	0	0
$323$	3.42451	0.190545
$324$	0	0
$325$	−30.1020	−1.66976
$326$	0	0
$327$	2.29385	0.126850
$328$	0	0
$329$	−16.9979	−0.937123
$330$	0	0
$331$	16.5345	0.908817	0.454409	−	0.890793i	$-0.349851\pi$
$331$	16.5345	0.908817	0.454409	+	0.890793i	$0.349851\pi$
$332$	0	0
$333$	31.7022	1.73727
$334$	0	0
$335$	−0.0206341	−0.00112736
$336$	0	0
$337$	−12.7375	−0.693857	−0.346929	−	0.937892i	$-0.612775\pi$
$337$	−12.7375	−0.693857	−0.346929	+	0.937892i	$0.612775\pi$
$338$	0	0
$339$	−0.724129	−0.0393293
$340$	0	0
$341$	41.3677	2.24019
$342$	0	0
$343$	17.8224	0.962321
$344$	0	0
$345$	0.0964511	0.00519275
$346$	0	0
$347$	−24.3982	−1.30977	−0.654883	−	0.755731i	$-0.727282\pi$
$347$	−24.3982	−1.30977	−0.654883	+	0.755731i	$0.727282\pi$
$348$	0	0
$349$	−0.345203	−0.0184783	−0.00923915	−	0.999957i	$-0.502941\pi$
$349$	−0.345203	−0.0184783	−0.00923915	+	0.999957i	$0.502941\pi$
$350$	0	0
$351$	16.2604	0.867916
$352$	0	0
$353$	7.53129	0.400850	0.200425	−	0.979709i	$-0.435768\pi$
$353$	7.53129	0.400850	0.200425	+	0.979709i	$0.435768\pi$
$354$	0	0
$355$	−1.68635	−0.0895022
$356$	0	0
$357$	−2.25166	−0.119170
$358$	0	0
$359$	−37.0601	−1.95596	−0.977978	−	0.208709i	$-0.933074\pi$
$359$	−37.0601	−1.95596	−0.977978	+	0.208709i	$0.933074\pi$
$360$	0	0
$361$	−15.3906	−0.810032
$362$	0	0
$363$	−9.21618	−0.483724
$364$	0	0
$365$	3.87039	0.202586
$366$	0	0
$367$	34.0706	1.77847	0.889237	−	0.457448i	$-0.151236\pi$
$367$	34.0706	1.77847	0.889237	+	0.457448i	$0.151236\pi$
$368$	0	0
$369$	−11.9447	−0.621818
$370$	0	0
$371$	22.9119	1.18953
$372$	0	0
$373$	11.0987	0.574668	0.287334	−	0.957831i	$-0.407231\pi$
$373$	11.0987	0.574668	0.287334	+	0.957831i	$0.407231\pi$
$374$	0	0
$375$	1.46397	0.0755989
$376$	0	0
$377$	20.0048	1.03030
$378$	0	0
$379$	4.13884	0.212598	0.106299	−	0.994334i	$-0.466100\pi$
$379$	4.13884	0.212598	0.106299	+	0.994334i	$0.466100\pi$
$380$	0	0
$381$	−4.47834	−0.229432
$382$	0	0
$383$	22.6404	1.15687	0.578435	−	0.815729i	$-0.303664\pi$
$383$	22.6404	1.15687	0.578435	+	0.815729i	$0.303664\pi$
$384$	0	0
$385$	4.95149	0.252351
$386$	0	0
$387$	−23.6388	−1.20163
$388$	0	0
$389$	11.5595	0.586089	0.293045	−	0.956099i	$-0.405332\pi$
$389$	11.5595	0.586089	0.293045	+	0.956099i	$0.405332\pi$
$390$	0	0
$391$	1.17509	0.0594270
$392$	0	0
$393$	3.04452	0.153576
$394$	0	0
$395$	3.44652	0.173413
$396$	0	0
$397$	9.58919	0.481268	0.240634	−	0.970616i	$-0.422645\pi$
$397$	9.58919	0.481268	0.240634	+	0.970616i	$0.422645\pi$
$398$	0	0
$399$	−2.37322	−0.118810
$400$	0	0
$401$	7.08014	0.353565	0.176783	−	0.984250i	$-0.443431\pi$
$401$	7.08014	0.353565	0.176783	+	0.984250i	$0.443431\pi$
$402$	0	0
$403$	−45.5537	−2.26919
$404$	0	0
$405$	2.32268	0.115415
$406$	0	0
$407$	63.4199	3.14361
$408$	0	0
$409$	−4.82226	−0.238445	−0.119223	−	0.992868i	$-0.538040\pi$
$409$	−4.82226	−0.238445	−0.119223	+	0.992868i	$0.538040\pi$
$410$	0	0
$411$	−6.65718	−0.328375
$412$	0	0
$413$	22.2502	1.09486
$414$	0	0
$415$	0.310721	0.0152527
$416$	0	0
$417$	−3.84899	−0.188486
$418$	0	0
$419$	30.6610	1.49789	0.748945	−	0.662632i	$-0.230561\pi$
$419$	30.6610	1.49789	0.748945	+	0.662632i	$0.230561\pi$
$420$	0	0
$421$	38.8553	1.89369	0.946847	−	0.321685i	$-0.104249\pi$
$421$	38.8553	1.89369	0.946847	+	0.321685i	$0.104249\pi$
$422$	0	0
$423$	−17.3425	−0.843221
$424$	0	0
$425$	8.82333	0.427995
$426$	0	0
$427$	41.1872	1.99319
$428$	0	0
$429$	15.6790	0.756990
$430$	0	0
$431$	4.42582	0.213184	0.106592	−	0.994303i	$-0.466006\pi$
$431$	4.42582	0.213184	0.106592	+	0.994303i	$0.466006\pi$
$432$	0	0
$433$	−32.5424	−1.56389	−0.781945	−	0.623348i	$-0.785772\pi$
$433$	−32.5424	−1.56389	−0.781945	+	0.623348i	$0.785772\pi$
$434$	0	0
$435$	−0.481290	−0.0230761
$436$	0	0
$437$	1.23854	0.0592472
$438$	0	0
$439$	9.32797	0.445200	0.222600	−	0.974910i	$-0.428546\pi$
$439$	9.32797	0.445200	0.222600	+	0.974910i	$0.428546\pi$
$440$	0	0
$441$	−1.35713	−0.0646250
$442$	0	0
$443$	−6.77507	−0.321893	−0.160947	−	0.986963i	$-0.551455\pi$
$443$	−6.77507	−0.321893	−0.160947	+	0.986963i	$0.551455\pi$
$444$	0	0
$445$	−5.13390	−0.243370
$446$	0	0
$447$	1.61089	0.0761924
$448$	0	0
$449$	4.35830	0.205681	0.102840	−	0.994698i	$-0.467207\pi$
$449$	4.35830	0.205681	0.102840	+	0.994698i	$0.467207\pi$
$450$	0	0
$451$	−23.8953	−1.12519
$452$	0	0
$453$	6.72626	0.316027
$454$	0	0
$455$	−5.45252	−0.255618
$456$	0	0
$457$	−29.2326	−1.36744	−0.683722	−	0.729742i	$-0.739640\pi$
$457$	−29.2326	−1.36744	−0.683722	+	0.729742i	$0.739640\pi$
$458$	0	0
$459$	−4.76616	−0.222465
$460$	0	0
$461$	1.96271	0.0914127	0.0457063	−	0.998955i	$-0.485446\pi$
$461$	1.96271	0.0914127	0.0457063	+	0.998955i	$0.485446\pi$
$462$	0	0
$463$	−32.1118	−1.49236	−0.746181	−	0.665743i	$-0.768115\pi$
$463$	−32.1118	−1.49236	−0.746181	+	0.665743i	$0.768115\pi$
$464$	0	0
$465$	1.09596	0.0508240
$466$	0	0
$467$	−29.0701	−1.34520	−0.672602	−	0.740004i	$-0.734823\pi$
$467$	−29.0701	−1.34520	−0.672602	+	0.740004i	$0.734823\pi$
$468$	0	0
$469$	0.174217	0.00804459
$470$	0	0
$471$	2.49454	0.114942
$472$	0	0
$473$	−47.2891	−2.17435
$474$	0	0
$475$	9.29970	0.426699
$476$	0	0
$477$	23.3764	1.07033
$478$	0	0
$479$	−12.8787	−0.588441	−0.294220	−	0.955738i	$-0.595060\pi$
$479$	−12.8787	−0.588441	−0.294220	+	0.955738i	$0.595060\pi$
$480$	0	0
$481$	−69.8372	−3.18430
$482$	0	0
$483$	−0.814353	−0.0370543
$484$	0	0
$485$	−1.35498	−0.0615264
$486$	0	0
$487$	−3.18335	−0.144251	−0.0721256	−	0.997396i	$-0.522978\pi$
$487$	−3.18335	−0.144251	−0.0721256	+	0.997396i	$0.522978\pi$
$488$	0	0
$489$	−7.28885	−0.329613
$490$	0	0
$491$	−37.2465	−1.68091	−0.840455	−	0.541881i	$-0.817712\pi$
$491$	−37.2465	−1.68091	−0.840455	+	0.541881i	$0.817712\pi$
$492$	0	0
$493$	−5.86370	−0.264088
$494$	0	0
$495$	5.05188	0.227065
$496$	0	0
$497$	14.2381	0.638668
$498$	0	0
$499$	16.8511	0.754360	0.377180	−	0.926140i	$-0.376894\pi$
$499$	16.8511	0.754360	0.377180	+	0.926140i	$0.376894\pi$
$500$	0	0
$501$	−4.62802	−0.206764
$502$	0	0
$503$	33.2138	1.48093	0.740466	−	0.672094i	$-0.234605\pi$
$503$	33.2138	1.48093	0.740466	+	0.672094i	$0.234605\pi$
$504$	0	0
$505$	−0.320604	−0.0142667
$506$	0	0
$507$	−11.3303	−0.503198
$508$	0	0
$509$	−9.90862	−0.439192	−0.219596	−	0.975591i	$-0.570474\pi$
$509$	−9.90862	−0.439192	−0.219596	+	0.975591i	$0.570474\pi$
$510$	0	0
$511$	−32.6784	−1.44561
$512$	0	0
$513$	−5.02348	−0.221792
$514$	0	0
$515$	0.342212	0.0150797
$516$	0	0
$517$	−34.6935	−1.52582
$518$	0	0
$519$	−4.53393	−0.199017
$520$	0	0
$521$	−19.1173	−0.837545	−0.418773	−	0.908091i	$-0.637540\pi$
$521$	−19.1173	−0.837545	−0.418773	+	0.908091i	$0.637540\pi$
$522$	0	0
$523$	28.2256	1.23422	0.617109	−	0.786878i	$-0.288304\pi$
$523$	28.2256	1.23422	0.617109	+	0.786878i	$0.288304\pi$
$524$	0	0
$525$	−6.11467	−0.266866
$526$	0	0
$527$	13.3524	0.581641
$528$	0	0
$529$	−22.5750	−0.981522
$530$	0	0
$531$	22.7013	0.985152
$532$	0	0
$533$	26.3132	1.13975
$534$	0	0
$535$	−3.70350	−0.160116
$536$	0	0
$537$	−8.82543	−0.380845
$538$	0	0
$539$	−2.71491	−0.116940
$540$	0	0
$541$	−18.5735	−0.798537	−0.399269	−	0.916834i	$-0.630736\pi$
$541$	−18.5735	−0.798537	−0.399269	+	0.916834i	$0.630736\pi$
$542$	0	0
$543$	1.35670	0.0582214
$544$	0	0
$545$	−1.62816	−0.0697427
$546$	0	0
$547$	10.4076	0.444995	0.222497	−	0.974933i	$-0.428579\pi$
$547$	10.4076	0.444995	0.222497	+	0.974933i	$0.428579\pi$
$548$	0	0
$549$	42.0223	1.79347
$550$	0	0
$551$	−6.18028	−0.263289
$552$	0	0
$553$	−29.0995	−1.23744
$554$	0	0
$555$	1.68019	0.0713202
$556$	0	0
$557$	−42.0014	−1.77966	−0.889829	−	0.456294i	$-0.849176\pi$
$557$	−42.0014	−1.77966	−0.889829	+	0.456294i	$0.849176\pi$
$558$	0	0
$559$	52.0741	2.20250
$560$	0	0
$561$	−4.59574	−0.194032
$562$	0	0
$563$	−12.3917	−0.522249	−0.261124	−	0.965305i	$-0.584093\pi$
$563$	−12.3917	−0.522249	−0.261124	+	0.965305i	$0.584093\pi$
$564$	0	0
$565$	0.513982	0.0216234
$566$	0	0
$567$	−19.6108	−0.823577
$568$	0	0
$569$	−14.7083	−0.616604	−0.308302	−	0.951289i	$-0.599761\pi$
$569$	−14.7083	−0.616604	−0.308302	+	0.951289i	$0.599761\pi$
$570$	0	0
$571$	−18.3299	−0.767082	−0.383541	−	0.923524i	$-0.625295\pi$
$571$	−18.3299	−0.767082	−0.383541	+	0.923524i	$0.625295\pi$
$572$	0	0
$573$	−3.07161	−0.128318
$574$	0	0
$575$	3.19112	0.133079
$576$	0	0
$577$	31.8509	1.32597	0.662984	−	0.748633i	$-0.269290\pi$
$577$	31.8509	1.32597	0.662984	+	0.748633i	$0.269290\pi$
$578$	0	0
$579$	2.97060	0.123454
$580$	0	0
$581$	−2.62347	−0.108840
$582$	0	0
$583$	46.7642	1.93678
$584$	0	0
$585$	−5.56307	−0.230005
$586$	0	0
$587$	−8.04388	−0.332006	−0.166003	−	0.986125i	$-0.553086\pi$
$587$	−8.04388	−0.332006	−0.166003	+	0.986125i	$0.553086\pi$
$588$	0	0
$589$	14.0733	0.579881
$590$	0	0
$591$	−0.282274	−0.0116112
$592$	0	0
$593$	15.2793	0.627447	0.313724	−	0.949514i	$-0.398423\pi$
$593$	15.2793	0.627447	0.313724	+	0.949514i	$0.398423\pi$
$594$	0	0
$595$	1.59821	0.0655203
$596$	0	0
$597$	6.49840	0.265962
$598$	0	0
$599$	−26.2320	−1.07181	−0.535905	−	0.844278i	$-0.680029\pi$
$599$	−26.2320	−1.07181	−0.535905	+	0.844278i	$0.680029\pi$
$600$	0	0
$601$	−8.49717	−0.346607	−0.173303	−	0.984868i	$-0.555444\pi$
$601$	−8.49717	−0.346607	−0.173303	+	0.984868i	$0.555444\pi$
$602$	0	0
$603$	0.177749	0.00723851
$604$	0	0
$605$	6.54158	0.265953
$606$	0	0
$607$	−4.08646	−0.165864	−0.0829321	−	0.996555i	$-0.526428\pi$
$607$	−4.08646	−0.165864	−0.0829321	+	0.996555i	$0.526428\pi$
$608$	0	0
$609$	4.06361	0.164666
$610$	0	0
$611$	38.2040	1.54557
$612$	0	0
$613$	25.9544	1.04829	0.524145	−	0.851629i	$-0.324385\pi$
$613$	25.9544	1.04829	0.524145	+	0.851629i	$0.324385\pi$
$614$	0	0
$615$	−0.633062	−0.0255275
$616$	0	0
$617$	39.7318	1.59954	0.799771	−	0.600305i	$-0.204954\pi$
$617$	39.7318	1.59954	0.799771	+	0.600305i	$0.204954\pi$
$618$	0	0
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	0	0
$621$	−1.72377	−0.0691724
$622$	0	0
$623$	43.3464	1.73664
$624$	0	0
$625$	23.4358	0.937433
$626$	0	0
$627$	−4.84386	−0.193445
$628$	0	0
$629$	20.4703	0.816204
$630$	0	0
$631$	−12.9854	−0.516942	−0.258471	−	0.966019i	$-0.583219\pi$
$631$	−12.9854	−0.516942	−0.258471	+	0.966019i	$0.583219\pi$
$632$	0	0
$633$	6.64459	0.264099
$634$	0	0
$635$	3.17870	0.126143
$636$	0	0
$637$	2.98963	0.118453
$638$	0	0
$639$	14.5268	0.574672
$640$	0	0
$641$	29.0223	1.14631	0.573157	−	0.819446i	$-0.305719\pi$
$641$	29.0223	1.14631	0.573157	+	0.819446i	$0.305719\pi$
$642$	0	0
$643$	26.7834	1.05623	0.528116	−	0.849172i	$-0.322898\pi$
$643$	26.7834	1.05623	0.528116	+	0.849172i	$0.322898\pi$
$644$	0	0
$645$	−1.25284	−0.0493304
$646$	0	0
$647$	−6.87426	−0.270255	−0.135128	−	0.990828i	$-0.543144\pi$
$647$	−6.87426	−0.270255	−0.135128	+	0.990828i	$0.543144\pi$
$648$	0	0
$649$	45.4136	1.78264
$650$	0	0
$651$	−9.25339	−0.362669
$652$	0	0
$653$	4.82987	0.189007	0.0945037	−	0.995525i	$-0.469874\pi$
$653$	4.82987	0.189007	0.0945037	+	0.995525i	$0.469874\pi$
$654$	0	0
$655$	−2.16098	−0.0844366
$656$	0	0
$657$	−33.3409	−1.30075
$658$	0	0
$659$	−44.6713	−1.74015	−0.870074	−	0.492921i	$-0.835929\pi$
$659$	−44.6713	−1.74015	−0.870074	+	0.492921i	$0.835929\pi$
$660$	0	0
$661$	−37.1820	−1.44621	−0.723107	−	0.690736i	$-0.757287\pi$
$661$	−37.1820	−1.44621	−0.723107	+	0.690736i	$0.757287\pi$
$662$	0	0
$663$	5.06078	0.196544
$664$	0	0
$665$	1.68450	0.0653221
$666$	0	0
$667$	−2.12071	−0.0821143
$668$	0	0
$669$	9.06389	0.350430
$670$	0	0
$671$	84.0651	3.24530
$672$	0	0
$673$	−38.2119	−1.47296	−0.736481	−	0.676459i	$-0.763514\pi$
$673$	−38.2119	−1.47296	−0.736481	+	0.676459i	$0.763514\pi$
$674$	0	0
$675$	−12.9431	−0.498181
$676$	0	0
$677$	−14.4022	−0.553521	−0.276761	−	0.960939i	$-0.589261\pi$
$677$	−14.4022	−0.553521	−0.276761	+	0.960939i	$0.589261\pi$
$678$	0	0
$679$	11.4403	0.439039
$680$	0	0
$681$	−13.5351	−0.518664
$682$	0	0
$683$	13.3703	0.511600	0.255800	−	0.966730i	$-0.417661\pi$
$683$	13.3703	0.511600	0.255800	+	0.966730i	$0.417661\pi$
$684$	0	0
$685$	4.72523	0.180542
$686$	0	0
$687$	−8.41693	−0.321126
$688$	0	0
$689$	−51.4962	−1.96185
$690$	0	0
$691$	37.2174	1.41581	0.707907	−	0.706305i	$-0.249639\pi$
$691$	37.2174	1.41581	0.707907	+	0.706305i	$0.249639\pi$
$692$	0	0
$693$	−42.6539	−1.62029
$694$	0	0
$695$	2.73199	0.103630
$696$	0	0
$697$	−7.71278	−0.292142
$698$	0	0
$699$	−1.96237	−0.0742235
$700$	0	0
$701$	−11.3492	−0.428654	−0.214327	−	0.976762i	$-0.568756\pi$
$701$	−11.3492	−0.428654	−0.214327	+	0.976762i	$0.568756\pi$
$702$	0	0
$703$	21.5755	0.813734
$704$	0	0
$705$	−0.919139	−0.0346168
$706$	0	0
$707$	2.70691	0.101804
$708$	0	0
$709$	−33.9228	−1.27400	−0.636998	−	0.770865i	$-0.719824\pi$
$709$	−33.9228	−1.27400	−0.636998	+	0.770865i	$0.719824\pi$
$710$	0	0
$711$	−29.6895	−1.11344
$712$	0	0
$713$	4.82915	0.180853
$714$	0	0
$715$	−11.1289	−0.416196
$716$	0	0
$717$	−3.13396	−0.117040
$718$	0	0
$719$	13.4168	0.500362	0.250181	−	0.968199i	$-0.419510\pi$
$719$	13.4168	0.500362	0.250181	+	0.968199i	$0.419510\pi$
$720$	0	0
$721$	−2.88935	−0.107605
$722$	0	0
$723$	2.27981	0.0847869
$724$	0	0
$725$	−15.9236	−0.591389
$726$	0	0
$727$	17.6240	0.653639	0.326820	−	0.945087i	$-0.394023\pi$
$727$	17.6240	0.653639	0.326820	+	0.945087i	$0.394023\pi$
$728$	0	0
$729$	−16.3868	−0.606919
$730$	0	0
$731$	−15.2637	−0.564548
$732$	0	0
$733$	2.71813	0.100397	0.0501983	−	0.998739i	$-0.484015\pi$
$733$	2.71813	0.100397	0.0501983	+	0.998739i	$0.484015\pi$
$734$	0	0
$735$	−0.0719265	−0.00265305
$736$	0	0
$737$	0.355585	0.0130981
$738$	0	0
$739$	−30.5087	−1.12228	−0.561139	−	0.827721i	$-0.689637\pi$
$739$	−30.5087	−1.12228	−0.561139	+	0.827721i	$0.689637\pi$
$740$	0	0
$741$	5.33400	0.195950
$742$	0	0
$743$	49.5983	1.81959	0.909793	−	0.415062i	$-0.136240\pi$
$743$	49.5983	1.81959	0.909793	+	0.415062i	$0.136240\pi$
$744$	0	0
$745$	−1.14340	−0.0418909
$746$	0	0
$747$	−2.67666	−0.0979339
$748$	0	0
$749$	31.2693	1.14256
$750$	0	0
$751$	−42.9033	−1.56556	−0.782781	−	0.622297i	$-0.786200\pi$
$751$	−42.9033	−1.56556	−0.782781	+	0.622297i	$0.786200\pi$
$752$	0	0
$753$	10.7859	0.393061
$754$	0	0
$755$	−4.77426	−0.173753
$756$	0	0
$757$	−26.9768	−0.980489	−0.490244	−	0.871585i	$-0.663092\pi$
$757$	−26.9768	−0.980489	−0.490244	+	0.871585i	$0.663092\pi$
$758$	0	0
$759$	−1.66213	−0.0603316
$760$	0	0
$761$	−13.0038	−0.471389	−0.235694	−	0.971827i	$-0.575736\pi$
$761$	−13.0038	−0.471389	−0.235694	+	0.971827i	$0.575736\pi$
$762$	0	0
$763$	13.7468	0.497669
$764$	0	0
$765$	1.63062	0.0589551
$766$	0	0
$767$	−50.0089	−1.80572
$768$	0	0
$769$	14.1357	0.509746	0.254873	−	0.966975i	$-0.417966\pi$
$769$	14.1357	0.509746	0.254873	+	0.966975i	$0.417966\pi$
$770$	0	0
$771$	−14.3293	−0.516058
$772$	0	0
$773$	2.81969	0.101417	0.0507085	−	0.998713i	$-0.483852\pi$
$773$	2.81969	0.101417	0.0507085	+	0.998713i	$0.483852\pi$
$774$	0	0
$775$	36.2602	1.30251
$776$	0	0
$777$	−14.1861	−0.508925
$778$	0	0
$779$	−8.12919	−0.291258
$780$	0	0
$781$	29.0607	1.03987
$782$	0	0
$783$	8.60158	0.307395
$784$	0	0
$785$	−1.77061	−0.0631958
$786$	0	0
$787$	33.3985	1.19053	0.595264	−	0.803530i	$-0.297047\pi$
$787$	33.3985	1.19053	0.595264	+	0.803530i	$0.297047\pi$
$788$	0	0
$789$	−1.85037	−0.0658749
$790$	0	0
$791$	−4.33964	−0.154300
$792$	0	0
$793$	−92.5714	−3.28731
$794$	0	0
$795$	1.23893	0.0439404
$796$	0	0
$797$	34.5741	1.22468	0.612338	−	0.790596i	$-0.290229\pi$
$797$	34.5741	1.22468	0.612338	+	0.790596i	$0.290229\pi$
$798$	0	0
$799$	−11.1982	−0.396162
$800$	0	0
$801$	44.2253	1.56262
$802$	0	0
$803$	−66.6981	−2.35373
$804$	0	0
$805$	0.578022	0.0203726
$806$	0	0
$807$	−6.14208	−0.216212
$808$	0	0
$809$	35.2664	1.23990	0.619951	−	0.784641i	$-0.287153\pi$
$809$	35.2664	1.23990	0.619951	+	0.784641i	$0.287153\pi$
$810$	0	0
$811$	32.2493	1.13243	0.566213	−	0.824259i	$-0.308408\pi$
$811$	32.2493	1.13243	0.566213	+	0.824259i	$0.308408\pi$
$812$	0	0
$813$	3.73999	0.131167
$814$	0	0
$815$	5.17358	0.181223
$816$	0	0
$817$	−16.0878	−0.562839
$818$	0	0
$819$	46.9699	1.64126
$820$	0	0
$821$	33.6734	1.17521	0.587605	−	0.809148i	$-0.300071\pi$
$821$	33.6734	1.17521	0.587605	+	0.809148i	$0.300071\pi$
$822$	0	0
$823$	−21.8129	−0.760351	−0.380175	−	0.924914i	$-0.624136\pi$
$823$	−21.8129	−0.760351	−0.380175	+	0.924914i	$0.624136\pi$
$824$	0	0
$825$	−12.4803	−0.434509
$826$	0	0
$827$	−13.1546	−0.457430	−0.228715	−	0.973493i	$-0.573452\pi$
$827$	−13.1546	−0.457430	−0.228715	+	0.973493i	$0.573452\pi$
$828$	0	0
$829$	29.1249	1.01155	0.505774	−	0.862666i	$-0.331207\pi$
$829$	29.1249	1.01155	0.505774	+	0.862666i	$0.331207\pi$
$830$	0	0
$831$	−4.10253	−0.142315
$832$	0	0
$833$	−0.876303	−0.0303621
$834$	0	0
$835$	3.28493	0.113680
$836$	0	0
$837$	−19.5869	−0.677024
$838$	0	0
$839$	52.4610	1.81115	0.905577	−	0.424182i	$-0.139438\pi$
$839$	52.4610	1.81115	0.905577	+	0.424182i	$0.139438\pi$
$840$	0	0
$841$	−18.4177	−0.635092
$842$	0	0
$843$	3.05588	0.105250
$844$	0	0
$845$	8.04220	0.276660
$846$	0	0
$847$	−55.2317	−1.89778
$848$	0	0
$849$	−0.507276	−0.0174097
$850$	0	0
$851$	7.40344	0.253787
$852$	0	0
$853$	−54.1042	−1.85249	−0.926247	−	0.376918i	$-0.876984\pi$
$853$	−54.1042	−1.85249	−0.926247	+	0.376918i	$0.876984\pi$
$854$	0	0
$855$	1.71865	0.0587767
$856$	0	0
$857$	−25.7757	−0.880481	−0.440241	−	0.897880i	$-0.645107\pi$
$857$	−25.7757	−0.880481	−0.440241	+	0.897880i	$0.645107\pi$
$858$	0	0
$859$	−29.5571	−1.00847	−0.504237	−	0.863565i	$-0.668226\pi$
$859$	−29.5571	−1.00847	−0.504237	+	0.863565i	$0.668226\pi$
$860$	0	0
$861$	5.34504	0.182159
$862$	0	0
$863$	−21.8902	−0.745151	−0.372576	−	0.928002i	$-0.621525\pi$
$863$	−21.8902	−0.745151	−0.372576	+	0.928002i	$0.621525\pi$
$864$	0	0
$865$	3.21815	0.109420
$866$	0	0
$867$	6.27803	0.213213
$868$	0	0
$869$	−59.3935	−2.01479
$870$	0	0
$871$	−0.391566	−0.0132677
$872$	0	0
$873$	11.6723	0.395046
$874$	0	0
$875$	8.77341	0.296595
$876$	0	0
$877$	41.9789	1.41753	0.708764	−	0.705446i	$-0.249253\pi$
$877$	41.9789	1.41753	0.708764	+	0.705446i	$0.249253\pi$
$878$	0	0
$879$	−6.36117	−0.214557
$880$	0	0
$881$	41.4755	1.39735	0.698673	−	0.715441i	$-0.253774\pi$
$881$	41.4755	1.39735	0.698673	+	0.715441i	$0.253774\pi$
$882$	0	0
$883$	−5.31405	−0.178832	−0.0894161	−	0.995994i	$-0.528500\pi$
$883$	−5.31405	−0.178832	−0.0894161	+	0.995994i	$0.528500\pi$
$884$	0	0
$885$	1.20315	0.0404435
$886$	0	0
$887$	16.9302	0.568459	0.284230	−	0.958756i	$-0.408262\pi$
$887$	16.9302	0.568459	0.284230	+	0.958756i	$0.408262\pi$
$888$	0	0
$889$	−26.8383	−0.900127
$890$	0	0
$891$	−40.0266	−1.34094
$892$	0	0
$893$	−11.8027	−0.394963
$894$	0	0
$895$	6.26423	0.209390
$896$	0	0
$897$	1.83032	0.0611126
$898$	0	0
$899$	−24.0974	−0.803693
$900$	0	0
$901$	15.0943	0.502863
$902$	0	0
$903$	10.5779	0.352010
$904$	0	0
$905$	−0.962973	−0.0320103
$906$	0	0
$907$	−51.8587	−1.72194	−0.860969	−	0.508657i	$-0.830142\pi$
$907$	−51.8587	−1.72194	−0.860969	+	0.508657i	$0.830142\pi$
$908$	0	0
$909$	2.76180	0.0916031
$910$	0	0
$911$	−18.1143	−0.600154	−0.300077	−	0.953915i	$-0.597012\pi$
$911$	−18.1143	−0.600154	−0.300077	+	0.953915i	$0.597012\pi$
$912$	0	0
$913$	−5.35463	−0.177212
$914$	0	0
$915$	2.22715	0.0736273
$916$	0	0
$917$	18.2455	0.602521
$918$	0	0
$919$	−7.78573	−0.256828	−0.128414	−	0.991721i	$-0.540989\pi$
$919$	−7.78573	−0.256828	−0.128414	+	0.991721i	$0.540989\pi$
$920$	0	0
$921$	−10.3498	−0.341038
$922$	0	0
$923$	−32.0013	−1.05334
$924$	0	0
$925$	55.5897	1.82778
$926$	0	0
$927$	−2.94794	−0.0968229
$928$	0	0
$929$	18.7248	0.614341	0.307171	−	0.951654i	$-0.400618\pi$
$929$	18.7248	0.614341	0.307171	+	0.951654i	$0.400618\pi$
$930$	0	0
$931$	−0.923614	−0.0302702
$932$	0	0
$933$	−3.22807	−0.105682
$934$	0	0
$935$	3.26203	0.106680
$936$	0	0
$937$	10.6166	0.346828	0.173414	−	0.984849i	$-0.444520\pi$
$937$	10.6166	0.346828	0.173414	+	0.984849i	$0.444520\pi$
$938$	0	0
$939$	−8.98788	−0.293308
$940$	0	0
$941$	18.6697	0.608615	0.304307	−	0.952574i	$-0.401575\pi$
$941$	18.6697	0.608615	0.304307	+	0.952574i	$0.401575\pi$
$942$	0	0
$943$	−2.78946	−0.0908375
$944$	0	0
$945$	−2.34445	−0.0762649
$946$	0	0
$947$	32.9368	1.07030	0.535150	−	0.844757i	$-0.320255\pi$
$947$	32.9368	1.07030	0.535150	+	0.844757i	$0.320255\pi$
$948$	0	0
$949$	73.4472	2.38420
$950$	0	0
$951$	−15.7441	−0.510539
$952$	0	0
$953$	−41.0145	−1.32859	−0.664295	−	0.747471i	$-0.731268\pi$
$953$	−41.0145	−1.32859	−0.664295	+	0.747471i	$0.731268\pi$
$954$	0	0
$955$	2.18021	0.0705499
$956$	0	0
$957$	8.29403	0.268108
$958$	0	0
$959$	−39.8959	−1.28830
$960$	0	0
$961$	23.8730	0.770096
$962$	0	0
$963$	31.9033	1.02807
$964$	0	0
$965$	−2.10851	−0.0678754
$966$	0	0
$967$	10.1307	0.325780	0.162890	−	0.986644i	$-0.447918\pi$
$967$	10.1307	0.325780	0.162890	+	0.986644i	$0.447918\pi$
$968$	0	0
$969$	−1.56347	−0.0502260
$970$	0	0
$971$	39.8541	1.27898	0.639490	−	0.768800i	$-0.279146\pi$
$971$	39.8541	1.27898	0.639490	+	0.768800i	$0.279146\pi$
$972$	0	0
$973$	−23.0667	−0.739483
$974$	0	0
$975$	13.7432	0.440134
$976$	0	0
$977$	20.5984	0.659003	0.329501	−	0.944155i	$-0.393119\pi$
$977$	20.5984	0.659003	0.329501	+	0.944155i	$0.393119\pi$
$978$	0	0
$979$	88.4721	2.82758
$980$	0	0
$981$	14.0255	0.447801
$982$	0	0
$983$	9.91205	0.316145	0.158073	−	0.987427i	$-0.449472\pi$
$983$	9.91205	0.316145	0.158073	+	0.987427i	$0.449472\pi$
$984$	0	0
$985$	0.200356	0.00638388
$986$	0	0
$987$	7.76044	0.247018
$988$	0	0
$989$	−5.52038	−0.175538
$990$	0	0
$991$	−35.1371	−1.11617	−0.558084	−	0.829785i	$-0.688463\pi$
$991$	−35.1371	−1.11617	−0.558084	+	0.829785i	$0.688463\pi$
$992$	0	0
$993$	−7.54888	−0.239557
$994$	0	0
$995$	−4.61253	−0.146227
$996$	0	0
$997$	41.1380	1.30285	0.651426	−	0.758712i	$-0.274171\pi$
$997$	41.1380	1.30285	0.651426	+	0.758712i	$0.274171\pi$
$998$	0	0
$999$	−30.0283	−0.950052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9904.2.a.n.1.14		30
4.3	odd	2		619.2.a.b.1.21	✓	30
12.11	even	2		5571.2.a.g.1.10		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.b.1.21	✓	30	4.3	odd	2
5571.2.a.g.1.10		30	12.11	even	2
9904.2.a.n.1.14		30	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:	\( N \)	\(=\)	\( 9904 = 2^{4} \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9904.a (trivial)

\(f(q)\)	\(=\)	\(q-0.456554 q^{3} +0.324059 q^{5} -2.73608 q^{7} -2.79156 q^{9} +O(q^{10})\)	\(q-0.456554 q^{3} +0.324059 q^{5} -2.73608 q^{7} -2.79156 q^{9} -5.58448 q^{11} +6.14956 q^{13} -0.147950 q^{15} -1.80252 q^{17} -1.89984 q^{19} +1.24917 q^{21} -0.651915 q^{23} -4.89499 q^{25} +2.64416 q^{27} +3.25305 q^{29} -7.40763 q^{31} +2.54962 q^{33} -0.886652 q^{35} -11.3565 q^{37} -2.80761 q^{39} +4.27888 q^{41} +8.46794 q^{43} -0.904630 q^{45} +6.21248 q^{47} +0.486153 q^{49} +0.822950 q^{51} -8.37397 q^{53} -1.80970 q^{55} +0.867380 q^{57} -8.13212 q^{59} -15.0533 q^{61} +7.63794 q^{63} +1.99282 q^{65} -0.0636738 q^{67} +0.297634 q^{69} -5.20384 q^{71} +11.9435 q^{73} +2.23482 q^{75} +15.2796 q^{77} +10.6355 q^{79} +7.16747 q^{81} +0.958841 q^{83} -0.584124 q^{85} -1.48519 q^{87} -15.8425 q^{89} -16.8257 q^{91} +3.38198 q^{93} -0.615661 q^{95} -4.18127 q^{97} +15.5894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9}+O(q^{10}) \) 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9	\( 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9} - 23 q^{11} + 9 q^{13} + 2 q^{15} + 4 q^{17} + q^{19} + 30 q^{21} - 4 q^{23} + 35 q^{25} + 5 q^{27} + 90 q^{29} - 2 q^{31} - 6 q^{33} - 9 q^{35} + 19 q^{37} - 32 q^{39} + 59 q^{41} + 4 q^{43} + 30 q^{45} - 4 q^{47} + 30 q^{49} + 34 q^{53} + 17 q^{55} - 8 q^{57} - 13 q^{59} + 16 q^{61} + 40 q^{63} + 31 q^{65} + 11 q^{67} + 6 q^{69} - 42 q^{71} - 4 q^{73} + 52 q^{75} + 29 q^{77} - 3 q^{79} + 30 q^{81} + 11 q^{83} + 19 q^{85} + 20 q^{87} + 58 q^{89} + 39 q^{91} - 15 q^{93} - 23 q^{95} - 9 q^{97} + 8 q^{99}+O(q^{100}) \) 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9 - 23 * q^11 + 9 * q^13 + 2 * q^15 + 4 * q^17 + q^19 + 30 * q^21 - 4 * q^23 + 35 * q^25 + 5 * q^27 + 90 * q^29 - 2 * q^31 - 6 * q^33 - 9 * q^35 + 19 * q^37 - 32 * q^39 + 59 * q^41 + 4 * q^43 + 30 * q^45 - 4 * q^47 + 30 * q^49 + 34 * q^53 + 17 * q^55 - 8 * q^57 - 13 * q^59 + 16 * q^61 + 40 * q^63 + 31 * q^65 + 11 * q^67 + 6 * q^69 - 42 * q^71 - 4 * q^73 + 52 * q^75 + 29 * q^77 - 3 * q^79 + 30 * q^81 + 11 * q^83 + 19 * q^85 + 20 * q^87 + 58 * q^89 + 39 * q^91 - 15 * q^93 - 23 * q^95 - 9 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more