LMFDB - Embedded newform 3808.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,2,Mod(1,3808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3808, 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("3808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.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:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4022000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 10x^{3} + 14x^{2} - 8x - 4 $ x^6 - 2x^5 - 7x^4 + 10x^3 + 14x^2 - 8*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.93381$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.07358 q^{3} -1.81319 q^{5} -1.00000 q^{7} -1.84742 q^{9} +4.90949 q^{11} -4.83339 q^{13} -1.94661 q^{15} +1.00000 q^{17} +5.27074 q^{19} -1.07358 q^{21} +3.94372 q^{23} -1.71233 q^{25} -5.20410 q^{27} -2.75728 q^{29} +1.28477 q^{31} +5.27074 q^{33} +1.81319 q^{35} +0.798599 q^{37} -5.18904 q^{39} +4.50888 q^{41} -5.71588 q^{43} +3.34973 q^{45} -3.76437 q^{47} +1.00000 q^{49} +1.07358 q^{51} -7.86929 q^{53} -8.90184 q^{55} +5.65857 q^{57} -0.722499 q^{59} -8.29276 q^{61} +1.84742 q^{63} +8.76386 q^{65} -0.598550 q^{67} +4.23390 q^{69} -15.1178 q^{71} -7.53075 q^{73} -1.83833 q^{75} -4.90949 q^{77} -0.156924 q^{79} -0.0447670 q^{81} +6.38107 q^{83} -1.81319 q^{85} -2.96017 q^{87} -0.304088 q^{89} +4.83339 q^{91} +1.37931 q^{93} -9.55686 q^{95} -16.5918 q^{97} -9.06989 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 2 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 10 * q^23 - 2 * q^27 + 2 * q^29 - 12 * q^31 + 2 * q^33 - 2 * q^35 + 2 * q^37 - 10 * q^39 - 8 * q^43 + 14 * q^45 - 22 * q^47 + 6 * q^49 - 2 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^57 - 8 * q^59 - 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 4 * q^69 - 18 * q^71 + 8 * q^73 - 30 * q^75 + 2 * q^77 - 14 * q^79 - 26 * q^81 + 2 * q^85 - 22 * q^87 + 2 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

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
$3$	1.07358	0.619833	0.309916	−	0.950764i	$-0.399699\pi$
$3$	1.07358	0.619833	0.309916	+	0.950764i	$0.399699\pi$
$4$	0	0
$5$	−1.81319	−0.810884	−0.405442	−	0.914121i	$-0.632882\pi$
$5$	−1.81319	−0.810884	−0.405442	+	0.914121i	$0.632882\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.84742	−0.615807
$10$	0	0
$11$	4.90949	1.48027	0.740133	−	0.672461i	$-0.234763\pi$
$11$	4.90949	1.48027	0.740133	+	0.672461i	$0.234763\pi$
$12$	0	0
$13$	−4.83339	−1.34054	−0.670270	−	0.742117i	$-0.733822\pi$
$13$	−4.83339	−1.34054	−0.670270	+	0.742117i	$0.733822\pi$
$14$	0	0
$15$	−1.94661	−0.502613
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.27074	1.20919	0.604595	−	0.796533i	$-0.293335\pi$
$19$	5.27074	1.20919	0.604595	+	0.796533i	$0.293335\pi$
$20$	0	0
$21$	−1.07358	−0.234275
$22$	0	0
$23$	3.94372	0.822321	0.411161	−	0.911563i	$-0.365123\pi$
$23$	3.94372	0.822321	0.411161	+	0.911563i	$0.365123\pi$
$24$	0	0
$25$	−1.71233	−0.342466
$26$	0	0
$27$	−5.20410	−1.00153
$28$	0	0
$29$	−2.75728	−0.512015	−0.256007	−	0.966675i	$-0.582407\pi$
$29$	−2.75728	−0.512015	−0.256007	+	0.966675i	$0.582407\pi$
$30$	0	0
$31$	1.28477	0.230752	0.115376	−	0.993322i	$-0.463193\pi$
$31$	1.28477	0.230752	0.115376	+	0.993322i	$0.463193\pi$
$32$	0	0
$33$	5.27074	0.917517
$34$	0	0
$35$	1.81319	0.306485
$36$	0	0
$37$	0.798599	0.131289	0.0656445	−	0.997843i	$-0.479090\pi$
$37$	0.798599	0.131289	0.0656445	+	0.997843i	$0.479090\pi$
$38$	0	0
$39$	−5.18904	−0.830911
$40$	0	0
$41$	4.50888	0.704169	0.352085	−	0.935968i	$-0.385473\pi$
$41$	4.50888	0.704169	0.352085	+	0.935968i	$0.385473\pi$
$42$	0	0
$43$	−5.71588	−0.871664	−0.435832	−	0.900028i	$-0.643546\pi$
$43$	−5.71588	−0.871664	−0.435832	+	0.900028i	$0.643546\pi$
$44$	0	0
$45$	3.34973	0.499349
$46$	0	0
$47$	−3.76437	−0.549090	−0.274545	−	0.961574i	$-0.588527\pi$
$47$	−3.76437	−0.549090	−0.274545	+	0.961574i	$0.588527\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.07358	0.150332
$52$	0	0
$53$	−7.86929	−1.08093	−0.540465	−	0.841367i	$-0.681752\pi$
$53$	−7.86929	−1.08093	−0.540465	+	0.841367i	$0.681752\pi$
$54$	0	0
$55$	−8.90184	−1.20032
$56$	0	0
$57$	5.65857	0.749495
$58$	0	0
$59$	−0.722499	−0.0940613	−0.0470307	−	0.998893i	$-0.514976\pi$
$59$	−0.722499	−0.0940613	−0.0470307	+	0.998893i	$0.514976\pi$
$60$	0	0
$61$	−8.29276	−1.06178	−0.530889	−	0.847441i	$-0.678142\pi$
$61$	−8.29276	−1.06178	−0.530889	+	0.847441i	$0.678142\pi$
$62$	0	0
$63$	1.84742	0.232753
$64$	0	0
$65$	8.76386	1.08702
$66$	0	0
$67$	−0.598550	−0.0731246	−0.0365623	−	0.999331i	$-0.511641\pi$
$67$	−0.598550	−0.0731246	−0.0365623	+	0.999331i	$0.511641\pi$
$68$	0	0
$69$	4.23390	0.509702
$70$	0	0
$71$	−15.1178	−1.79415	−0.897075	−	0.441878i	$-0.854312\pi$
$71$	−15.1178	−1.79415	−0.897075	+	0.441878i	$0.854312\pi$
$72$	0	0
$73$	−7.53075	−0.881407	−0.440704	−	0.897653i	$-0.645271\pi$
$73$	−7.53075	−0.881407	−0.440704	+	0.897653i	$0.645271\pi$
$74$	0	0
$75$	−1.83833	−0.212272
$76$	0	0
$77$	−4.90949	−0.559488
$78$	0	0
$79$	−0.156924	−0.0176554	−0.00882768	−	0.999961i	$-0.502810\pi$
$79$	−0.156924	−0.0176554	−0.00882768	+	0.999961i	$0.502810\pi$
$80$	0	0
$81$	−0.0447670	−0.00497411
$82$	0	0
$83$	6.38107	0.700413	0.350206	−	0.936673i	$-0.386111\pi$
$83$	6.38107	0.700413	0.350206	+	0.936673i	$0.386111\pi$
$84$	0	0
$85$	−1.81319	−0.196668
$86$	0	0
$87$	−2.96017	−0.317364
$88$	0	0
$89$	−0.304088	−0.0322333	−0.0161166	−	0.999870i	$-0.505130\pi$
$89$	−0.304088	−0.0322333	−0.0161166	+	0.999870i	$0.505130\pi$
$90$	0	0
$91$	4.83339	0.506676
$92$	0	0
$93$	1.37931	0.143028
$94$	0	0
$95$	−9.55686	−0.980513
$96$	0	0
$97$	−16.5918	−1.68464	−0.842321	−	0.538976i	$-0.818811\pi$
$97$	−16.5918	−1.68464	−0.842321	+	0.538976i	$0.818811\pi$
$98$	0	0
$99$	−9.06989	−0.911558
$100$	0	0
$101$	13.4633	1.33965	0.669825	−	0.742519i	$-0.266369\pi$
$101$	13.4633	1.33965	0.669825	+	0.742519i	$0.266369\pi$
$102$	0	0
$103$	−2.64584	−0.260702	−0.130351	−	0.991468i	$-0.541610\pi$
$103$	−2.64584	−0.260702	−0.130351	+	0.991468i	$0.541610\pi$
$104$	0	0
$105$	1.94661	0.189970
$106$	0	0
$107$	−12.0520	−1.16511	−0.582555	−	0.812791i	$-0.697947\pi$
$107$	−12.0520	−1.16511	−0.582555	+	0.812791i	$0.697947\pi$
$108$	0	0
$109$	−11.0750	−1.06079	−0.530394	−	0.847751i	$-0.677956\pi$
$109$	−11.0750	−1.06079	−0.530394	+	0.847751i	$0.677956\pi$
$110$	0	0
$111$	0.857362	0.0813772
$112$	0	0
$113$	−11.2646	−1.05968	−0.529842	−	0.848096i	$-0.677749\pi$
$113$	−11.2646	−1.05968	−0.529842	+	0.848096i	$0.677749\pi$
$114$	0	0
$115$	−7.15072	−0.666808
$116$	0	0
$117$	8.92930	0.825514
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	13.1031	1.19119
$122$	0	0
$123$	4.84066	0.436467
$124$	0	0
$125$	12.1708	1.08859
$126$	0	0
$127$	2.05335	0.182206	0.0911028	−	0.995841i	$-0.470961\pi$
$127$	2.05335	0.182206	0.0911028	+	0.995841i	$0.470961\pi$
$128$	0	0
$129$	−6.13647	−0.540286
$130$	0	0
$131$	−0.568667	−0.0496847	−0.0248423	−	0.999691i	$-0.507908\pi$
$131$	−0.568667	−0.0496847	−0.0248423	+	0.999691i	$0.507908\pi$
$132$	0	0
$133$	−5.27074	−0.457031
$134$	0	0
$135$	9.43604	0.812125
$136$	0	0
$137$	−10.9544	−0.935901	−0.467950	−	0.883755i	$-0.655007\pi$
$137$	−10.9544	−0.935901	−0.467950	+	0.883755i	$0.655007\pi$
$138$	0	0
$139$	−7.18567	−0.609480	−0.304740	−	0.952435i	$-0.598570\pi$
$139$	−7.18567	−0.609480	−0.304740	+	0.952435i	$0.598570\pi$
$140$	0	0
$141$	−4.04136	−0.340344
$142$	0	0
$143$	−23.7294	−1.98436
$144$	0	0
$145$	4.99949	0.415185
$146$	0	0
$147$	1.07358	0.0885475
$148$	0	0
$149$	−16.4980	−1.35157	−0.675785	−	0.737099i	$-0.736195\pi$
$149$	−16.4980	−1.35157	−0.675785	+	0.737099i	$0.736195\pi$
$150$	0	0
$151$	−17.7546	−1.44485	−0.722423	−	0.691451i	$-0.756972\pi$
$151$	−17.7546	−1.44485	−0.722423	+	0.691451i	$0.756972\pi$
$152$	0	0
$153$	−1.84742	−0.149355
$154$	0	0
$155$	−2.32954	−0.187113
$156$	0	0
$157$	14.2018	1.13342	0.566712	−	0.823916i	$-0.308215\pi$
$157$	14.2018	1.13342	0.566712	+	0.823916i	$0.308215\pi$
$158$	0	0
$159$	−8.44832	−0.669996
$160$	0	0
$161$	−3.94372	−0.310808
$162$	0	0
$163$	7.76653	0.608322	0.304161	−	0.952621i	$-0.401624\pi$
$163$	7.76653	0.608322	0.304161	+	0.952621i	$0.401624\pi$
$164$	0	0
$165$	−9.55686	−0.744000
$166$	0	0
$167$	−6.69909	−0.518391	−0.259196	−	0.965825i	$-0.583457\pi$
$167$	−6.69909	−0.518391	−0.259196	+	0.965825i	$0.583457\pi$
$168$	0	0
$169$	10.3616	0.797047
$170$	0	0
$171$	−9.73727	−0.744628
$172$	0	0
$173$	14.1551	1.07619	0.538096	−	0.842884i	$-0.319144\pi$
$173$	14.1551	1.07619	0.538096	+	0.842884i	$0.319144\pi$
$174$	0	0
$175$	1.71233	0.129440
$176$	0	0
$177$	−0.775662	−0.0583023
$178$	0	0
$179$	8.87466	0.663324	0.331662	−	0.943398i	$-0.392391\pi$
$179$	8.87466	0.663324	0.331662	+	0.943398i	$0.392391\pi$
$180$	0	0
$181$	12.4511	0.925480	0.462740	−	0.886494i	$-0.346866\pi$
$181$	12.4511	0.925480	0.462740	+	0.886494i	$0.346866\pi$
$182$	0	0
$183$	−8.90295	−0.658125
$184$	0	0
$185$	−1.44801	−0.106460
$186$	0	0
$187$	4.90949	0.359017
$188$	0	0
$189$	5.20410	0.378543
$190$	0	0
$191$	9.35155	0.676654	0.338327	−	0.941029i	$-0.390139\pi$
$191$	9.35155	0.676654	0.338327	+	0.941029i	$0.390139\pi$
$192$	0	0
$193$	−24.7662	−1.78271	−0.891356	−	0.453303i	$-0.850246\pi$
$193$	−24.7662	−1.78271	−0.891356	+	0.453303i	$0.850246\pi$
$194$	0	0
$195$	9.40872	0.673772
$196$	0	0
$197$	10.2222	0.728301	0.364151	−	0.931340i	$-0.381359\pi$
$197$	10.2222	0.728301	0.364151	+	0.931340i	$0.381359\pi$
$198$	0	0
$199$	−7.77047	−0.550834	−0.275417	−	0.961325i	$-0.588816\pi$
$199$	−7.77047	−0.550834	−0.275417	+	0.961325i	$0.588816\pi$
$200$	0	0
$201$	−0.642593	−0.0453250
$202$	0	0
$203$	2.75728	0.193523
$204$	0	0
$205$	−8.17547	−0.571000
$206$	0	0
$207$	−7.28571	−0.506392
$208$	0	0
$209$	25.8766	1.78992
$210$	0	0
$211$	−2.16867	−0.149298	−0.0746489	−	0.997210i	$-0.523784\pi$
$211$	−2.16867	−0.149298	−0.0746489	+	0.997210i	$0.523784\pi$
$212$	0	0
$213$	−16.2302	−1.11207
$214$	0	0
$215$	10.3640	0.706819
$216$	0	0
$217$	−1.28477	−0.0872160
$218$	0	0
$219$	−8.08487	−0.546325
$220$	0	0
$221$	−4.83339	−0.325129
$222$	0	0
$223$	−9.58979	−0.642180	−0.321090	−	0.947049i	$-0.604049\pi$
$223$	−9.58979	−0.642180	−0.321090	+	0.947049i	$0.604049\pi$
$224$	0	0
$225$	3.16340	0.210893
$226$	0	0
$227$	−7.36101	−0.488567	−0.244284	−	0.969704i	$-0.578553\pi$
$227$	−7.36101	−0.488567	−0.244284	+	0.969704i	$0.578553\pi$
$228$	0	0
$229$	−12.8695	−0.850443	−0.425221	−	0.905089i	$-0.639804\pi$
$229$	−12.8695	−0.850443	−0.425221	+	0.905089i	$0.639804\pi$
$230$	0	0
$231$	−5.27074	−0.346789
$232$	0	0
$233$	−17.0708	−1.11834	−0.559172	−	0.829052i	$-0.688881\pi$
$233$	−17.0708	−1.11834	−0.559172	+	0.829052i	$0.688881\pi$
$234$	0	0
$235$	6.82553	0.445248
$236$	0	0
$237$	−0.168471	−0.0109434
$238$	0	0
$239$	13.8787	0.897739	0.448869	−	0.893597i	$-0.351827\pi$
$239$	13.8787	0.897739	0.448869	+	0.893597i	$0.351827\pi$
$240$	0	0
$241$	15.9751	1.02905	0.514524	−	0.857476i	$-0.327968\pi$
$241$	15.9751	1.02905	0.514524	+	0.857476i	$0.327968\pi$
$242$	0	0
$243$	15.5643	0.998447
$244$	0	0
$245$	−1.81319	−0.115841
$246$	0	0
$247$	−25.4755	−1.62097
$248$	0	0
$249$	6.85060	0.434139
$250$	0	0
$251$	−10.1242	−0.639031	−0.319516	−	0.947581i	$-0.603520\pi$
$251$	−10.1242	−0.639031	−0.319516	+	0.947581i	$0.603520\pi$
$252$	0	0
$253$	19.3616	1.21725
$254$	0	0
$255$	−1.94661	−0.121901
$256$	0	0
$257$	15.8594	0.989282	0.494641	−	0.869098i	$-0.335300\pi$
$257$	15.8594	0.989282	0.494641	+	0.869098i	$0.335300\pi$
$258$	0	0
$259$	−0.798599	−0.0496226
$260$	0	0
$261$	5.09387	0.315303
$262$	0	0
$263$	−22.7178	−1.40084	−0.700420	−	0.713730i	$-0.747004\pi$
$263$	−22.7178	−1.40084	−0.700420	+	0.713730i	$0.747004\pi$
$264$	0	0
$265$	14.2685	0.876509
$266$	0	0
$267$	−0.326464	−0.0199792
$268$	0	0
$269$	5.44746	0.332138	0.166069	−	0.986114i	$-0.446893\pi$
$269$	5.44746	0.332138	0.166069	+	0.986114i	$0.446893\pi$
$270$	0	0
$271$	2.28550	0.138834	0.0694170	−	0.997588i	$-0.477886\pi$
$271$	2.28550	0.138834	0.0694170	+	0.997588i	$0.477886\pi$
$272$	0	0
$273$	5.18904	0.314055
$274$	0	0
$275$	−8.40667	−0.506941
$276$	0	0
$277$	−22.1473	−1.33070	−0.665350	−	0.746532i	$-0.731718\pi$
$277$	−22.1473	−1.33070	−0.665350	+	0.746532i	$0.731718\pi$
$278$	0	0
$279$	−2.37352	−0.142099
$280$	0	0
$281$	20.1996	1.20501	0.602503	−	0.798116i	$-0.294170\pi$
$281$	20.1996	1.20501	0.602503	+	0.798116i	$0.294170\pi$
$282$	0	0
$283$	31.4827	1.87145	0.935726	−	0.352728i	$-0.114746\pi$
$283$	31.4827	1.87145	0.935726	+	0.352728i	$0.114746\pi$
$284$	0	0
$285$	−10.2601	−0.607754
$286$	0	0
$287$	−4.50888	−0.266151
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−17.8127	−1.04420
$292$	0	0
$293$	8.12416	0.474619	0.237309	−	0.971434i	$-0.423735\pi$
$293$	8.12416	0.474619	0.237309	+	0.971434i	$0.423735\pi$
$294$	0	0
$295$	1.31003	0.0762729
$296$	0	0
$297$	−25.5495	−1.48253
$298$	0	0
$299$	−19.0615	−1.10235
$300$	0	0
$301$	5.71588	0.329458
$302$	0	0
$303$	14.4540	0.830359
$304$	0	0
$305$	15.0364	0.860980
$306$	0	0
$307$	−14.3805	−0.820738	−0.410369	−	0.911920i	$-0.634600\pi$
$307$	−14.3805	−0.820738	−0.410369	+	0.911920i	$0.634600\pi$
$308$	0	0
$309$	−2.84052	−0.161592
$310$	0	0
$311$	−23.4134	−1.32765	−0.663825	−	0.747888i	$-0.731068\pi$
$311$	−23.4134	−1.32765	−0.663825	+	0.747888i	$0.731068\pi$
$312$	0	0
$313$	−1.12926	−0.0638296	−0.0319148	−	0.999491i	$-0.510161\pi$
$313$	−1.12926	−0.0638296	−0.0319148	+	0.999491i	$0.510161\pi$
$314$	0	0
$315$	−3.34973	−0.188736
$316$	0	0
$317$	−9.60409	−0.539419	−0.269710	−	0.962942i	$-0.586928\pi$
$317$	−9.60409	−0.539419	−0.269710	+	0.962942i	$0.586928\pi$
$318$	0	0
$319$	−13.5369	−0.757918
$320$	0	0
$321$	−12.9388	−0.722174
$322$	0	0
$323$	5.27074	0.293272
$324$	0	0
$325$	8.27636	0.459090
$326$	0	0
$327$	−11.8899	−0.657512
$328$	0	0
$329$	3.76437	0.207536
$330$	0	0
$331$	29.2438	1.60739	0.803693	−	0.595044i	$-0.202865\pi$
$331$	29.2438	1.60739	0.803693	+	0.595044i	$0.202865\pi$
$332$	0	0
$333$	−1.47535	−0.0808487
$334$	0	0
$335$	1.08529	0.0592956
$336$	0	0
$337$	9.60718	0.523336	0.261668	−	0.965158i	$-0.415727\pi$
$337$	9.60718	0.523336	0.261668	+	0.965158i	$0.415727\pi$
$338$	0	0
$339$	−12.0935	−0.656827
$340$	0	0
$341$	6.30757	0.341574
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.67688	−0.413309
$346$	0	0
$347$	−11.7839	−0.632591	−0.316296	−	0.948661i	$-0.602439\pi$
$347$	−11.7839	−0.632591	−0.316296	+	0.948661i	$0.602439\pi$
$348$	0	0
$349$	8.74461	0.468088	0.234044	−	0.972226i	$-0.424804\pi$
$349$	8.74461	0.468088	0.234044	+	0.972226i	$0.424804\pi$
$350$	0	0
$351$	25.1534	1.34259
$352$	0	0
$353$	3.06697	0.163238	0.0816192	−	0.996664i	$-0.473991\pi$
$353$	3.06697	0.163238	0.0816192	+	0.996664i	$0.473991\pi$
$354$	0	0
$355$	27.4114	1.45485
$356$	0	0
$357$	−1.07358	−0.0568200
$358$	0	0
$359$	−22.9878	−1.21325	−0.606625	−	0.794988i	$-0.707477\pi$
$359$	−22.9878	−1.21325	−0.606625	+	0.794988i	$0.707477\pi$
$360$	0	0
$361$	8.78065	0.462140
$362$	0	0
$363$	14.0672	0.738337
$364$	0	0
$365$	13.6547	0.714719
$366$	0	0
$367$	16.4845	0.860482	0.430241	−	0.902714i	$-0.358429\pi$
$367$	16.4845	0.860482	0.430241	+	0.902714i	$0.358429\pi$
$368$	0	0
$369$	−8.32981	−0.433633
$370$	0	0
$371$	7.86929	0.408553
$372$	0	0
$373$	13.9705	0.723366	0.361683	−	0.932301i	$-0.382202\pi$
$373$	13.9705	0.723366	0.361683	+	0.932301i	$0.382202\pi$
$374$	0	0
$375$	13.0663	0.674741
$376$	0	0
$377$	13.3270	0.686376
$378$	0	0
$379$	30.7980	1.58199	0.790994	−	0.611824i	$-0.209564\pi$
$379$	30.7980	1.58199	0.790994	+	0.611824i	$0.209564\pi$
$380$	0	0
$381$	2.20444	0.112937
$382$	0	0
$383$	−21.7023	−1.10893	−0.554467	−	0.832206i	$-0.687078\pi$
$383$	−21.7023	−1.10893	−0.554467	+	0.832206i	$0.687078\pi$
$384$	0	0
$385$	8.90184	0.453680
$386$	0	0
$387$	10.5596	0.536777
$388$	0	0
$389$	13.0069	0.659476	0.329738	−	0.944072i	$-0.393040\pi$
$389$	13.0069	0.659476	0.329738	+	0.944072i	$0.393040\pi$
$390$	0	0
$391$	3.94372	0.199442
$392$	0	0
$393$	−0.610511	−0.0307962
$394$	0	0
$395$	0.284534	0.0143165
$396$	0	0
$397$	8.24209	0.413659	0.206829	−	0.978377i	$-0.433685\pi$
$397$	8.24209	0.413659	0.206829	+	0.978377i	$0.433685\pi$
$398$	0	0
$399$	−5.65857	−0.283283
$400$	0	0
$401$	23.5644	1.17675	0.588375	−	0.808588i	$-0.299768\pi$
$401$	23.5644	1.17675	0.588375	+	0.808588i	$0.299768\pi$
$402$	0	0
$403$	−6.20980	−0.309332
$404$	0	0
$405$	0.0811712	0.00403343
$406$	0	0
$407$	3.92071	0.194343
$408$	0	0
$409$	−7.40880	−0.366342	−0.183171	−	0.983081i	$-0.558636\pi$
$409$	−7.40880	−0.366342	−0.183171	+	0.983081i	$0.558636\pi$
$410$	0	0
$411$	−11.7605	−0.580102
$412$	0	0
$413$	0.722499	0.0355518
$414$	0	0
$415$	−11.5701	−0.567954
$416$	0	0
$417$	−7.71440	−0.377776
$418$	0	0
$419$	−27.7321	−1.35480	−0.677401	−	0.735614i	$-0.736894\pi$
$419$	−27.7321	−1.35480	−0.677401	+	0.735614i	$0.736894\pi$
$420$	0	0
$421$	9.54638	0.465262	0.232631	−	0.972565i	$-0.425267\pi$
$421$	9.54638	0.465262	0.232631	+	0.972565i	$0.425267\pi$
$422$	0	0
$423$	6.95438	0.338134
$424$	0	0
$425$	−1.71233	−0.0830603
$426$	0	0
$427$	8.29276	0.401315
$428$	0	0
$429$	−25.4755	−1.22997
$430$	0	0
$431$	36.2397	1.74560	0.872802	−	0.488075i	$-0.162301\pi$
$431$	36.2397	1.74560	0.872802	+	0.488075i	$0.162301\pi$
$432$	0	0
$433$	2.83494	0.136239	0.0681193	−	0.997677i	$-0.478300\pi$
$433$	2.83494	0.136239	0.0681193	+	0.997677i	$0.478300\pi$
$434$	0	0
$435$	5.36736	0.257345
$436$	0	0
$437$	20.7863	0.994343
$438$	0	0
$439$	25.6611	1.22474	0.612368	−	0.790573i	$-0.290217\pi$
$439$	25.6611	1.22474	0.612368	+	0.790573i	$0.290217\pi$
$440$	0	0
$441$	−1.84742	−0.0879725
$442$	0	0
$443$	−8.17868	−0.388581	−0.194291	−	0.980944i	$-0.562240\pi$
$443$	−8.17868	−0.388581	−0.194291	+	0.980944i	$0.562240\pi$
$444$	0	0
$445$	0.551370	0.0261375
$446$	0	0
$447$	−17.7120	−0.837748
$448$	0	0
$449$	−18.1445	−0.856294	−0.428147	−	0.903709i	$-0.640833\pi$
$449$	−18.1445	−0.856294	−0.428147	+	0.903709i	$0.640833\pi$
$450$	0	0
$451$	22.1363	1.04236
$452$	0	0
$453$	−19.0610	−0.895563
$454$	0	0
$455$	−8.76386	−0.410856
$456$	0	0
$457$	−19.3950	−0.907262	−0.453631	−	0.891190i	$-0.649872\pi$
$457$	−19.3950	−0.907262	−0.453631	+	0.891190i	$0.649872\pi$
$458$	0	0
$459$	−5.20410	−0.242907
$460$	0	0
$461$	0.965033	0.0449461	0.0224730	−	0.999747i	$-0.492846\pi$
$461$	0.965033	0.0449461	0.0224730	+	0.999747i	$0.492846\pi$
$462$	0	0
$463$	26.9383	1.25193	0.625965	−	0.779851i	$-0.284705\pi$
$463$	26.9383	1.25193	0.625965	+	0.779851i	$0.284705\pi$
$464$	0	0
$465$	−2.50095	−0.115979
$466$	0	0
$467$	10.0040	0.462928	0.231464	−	0.972843i	$-0.425648\pi$
$467$	10.0040	0.462928	0.231464	+	0.972843i	$0.425648\pi$
$468$	0	0
$469$	0.598550	0.0276385
$470$	0	0
$471$	15.2468	0.702534
$472$	0	0
$473$	−28.0621	−1.29029
$474$	0	0
$475$	−9.02525	−0.414107
$476$	0	0
$477$	14.5379	0.665644
$478$	0	0
$479$	−12.9941	−0.593714	−0.296857	−	0.954922i	$-0.595938\pi$
$479$	−12.9941	−0.593714	−0.296857	+	0.954922i	$0.595938\pi$
$480$	0	0
$481$	−3.85994	−0.175998
$482$	0	0
$483$	−4.23390	−0.192649
$484$	0	0
$485$	30.0841	1.36605
$486$	0	0
$487$	−7.54341	−0.341824	−0.170912	−	0.985286i	$-0.554671\pi$
$487$	−7.54341	−0.341824	−0.170912	+	0.985286i	$0.554671\pi$
$488$	0	0
$489$	8.33801	0.377058
$490$	0	0
$491$	29.9222	1.35037	0.675185	−	0.737648i	$-0.264064\pi$
$491$	29.9222	1.35037	0.675185	+	0.737648i	$0.264064\pi$
$492$	0	0
$493$	−2.75728	−0.124182
$494$	0	0
$495$	16.4455	0.739168
$496$	0	0
$497$	15.1178	0.678125
$498$	0	0
$499$	14.4721	0.647861	0.323931	−	0.946081i	$-0.394996\pi$
$499$	14.4721	0.647861	0.323931	+	0.946081i	$0.394996\pi$
$500$	0	0
$501$	−7.19202	−0.321316
$502$	0	0
$503$	−3.75068	−0.167235	−0.0836173	−	0.996498i	$-0.526647\pi$
$503$	−3.75068	−0.167235	−0.0836173	+	0.996498i	$0.526647\pi$
$504$	0	0
$505$	−24.4116	−1.08630
$506$	0	0
$507$	11.1240	0.494036
$508$	0	0
$509$	−43.2251	−1.91592	−0.957959	−	0.286905i	$-0.907374\pi$
$509$	−43.2251	−1.91592	−0.957959	+	0.286905i	$0.907374\pi$
$510$	0	0
$511$	7.53075	0.333141
$512$	0	0
$513$	−27.4295	−1.21104
$514$	0	0
$515$	4.79741	0.211399
$516$	0	0
$517$	−18.4811	−0.812799
$518$	0	0
$519$	15.1966	0.667059
$520$	0	0
$521$	−33.8581	−1.48335	−0.741675	−	0.670760i	$-0.765968\pi$
$521$	−33.8581	−1.48335	−0.741675	+	0.670760i	$0.765968\pi$
$522$	0	0
$523$	−30.1871	−1.31999	−0.659996	−	0.751269i	$-0.729442\pi$
$523$	−30.1871	−1.31999	−0.659996	+	0.751269i	$0.729442\pi$
$524$	0	0
$525$	1.83833	0.0802313
$526$	0	0
$527$	1.28477	0.0559655
$528$	0	0
$529$	−7.44711	−0.323787
$530$	0	0
$531$	1.33476	0.0579237
$532$	0	0
$533$	−21.7932	−0.943967
$534$	0	0
$535$	21.8526	0.944770
$536$	0	0
$537$	9.52768	0.411150
$538$	0	0
$539$	4.90949	0.211467
$540$	0	0
$541$	−21.9828	−0.945116	−0.472558	−	0.881300i	$-0.656669\pi$
$541$	−21.9828	−0.945116	−0.472558	+	0.881300i	$0.656669\pi$
$542$	0	0
$543$	13.3672	0.573643
$544$	0	0
$545$	20.0810	0.860177
$546$	0	0
$547$	−28.7173	−1.22786	−0.613932	−	0.789359i	$-0.710413\pi$
$547$	−28.7173	−1.22786	−0.613932	+	0.789359i	$0.710413\pi$
$548$	0	0
$549$	15.3202	0.653851
$550$	0	0
$551$	−14.5329	−0.619123
$552$	0	0
$553$	0.156924	0.00667310
$554$	0	0
$555$	−1.55456	−0.0659875
$556$	0	0
$557$	−20.0505	−0.849565	−0.424783	−	0.905295i	$-0.639649\pi$
$557$	−20.0505	−0.849565	−0.424783	+	0.905295i	$0.639649\pi$
$558$	0	0
$559$	27.6271	1.16850
$560$	0	0
$561$	5.27074	0.222531
$562$	0	0
$563$	24.3329	1.02551	0.512755	−	0.858535i	$-0.328625\pi$
$563$	24.3329	1.02551	0.512755	+	0.858535i	$0.328625\pi$
$564$	0	0
$565$	20.4249	0.859281
$566$	0	0
$567$	0.0447670	0.00188004
$568$	0	0
$569$	−44.6538	−1.87199	−0.935993	−	0.352018i	$-0.885496\pi$
$569$	−44.6538	−1.87199	−0.935993	+	0.352018i	$0.885496\pi$
$570$	0	0
$571$	9.07727	0.379872	0.189936	−	0.981796i	$-0.439172\pi$
$571$	9.07727	0.379872	0.189936	+	0.981796i	$0.439172\pi$
$572$	0	0
$573$	10.0397	0.419412
$574$	0	0
$575$	−6.75295	−0.281618
$576$	0	0
$577$	−6.57031	−0.273526	−0.136763	−	0.990604i	$-0.543670\pi$
$577$	−6.57031	−0.273526	−0.136763	+	0.990604i	$0.543670\pi$
$578$	0	0
$579$	−26.5886	−1.10498
$580$	0	0
$581$	−6.38107	−0.264731
$582$	0	0
$583$	−38.6341	−1.60006
$584$	0	0
$585$	−16.1905	−0.669397
$586$	0	0
$587$	−43.6330	−1.80092	−0.900462	−	0.434934i	$-0.856772\pi$
$587$	−43.6330	−1.80092	−0.900462	+	0.434934i	$0.856772\pi$
$588$	0	0
$589$	6.77169	0.279023
$590$	0	0
$591$	10.9744	0.451425
$592$	0	0
$593$	−16.8930	−0.693713	−0.346857	−	0.937918i	$-0.612751\pi$
$593$	−16.8930	−0.693713	−0.346857	+	0.937918i	$0.612751\pi$
$594$	0	0
$595$	1.81319	0.0743337
$596$	0	0
$597$	−8.34224	−0.341425
$598$	0	0
$599$	−24.6378	−1.00667	−0.503336	−	0.864091i	$-0.667894\pi$
$599$	−24.6378	−1.00667	−0.503336	+	0.864091i	$0.667894\pi$
$600$	0	0
$601$	7.59480	0.309798	0.154899	−	0.987930i	$-0.450495\pi$
$601$	7.59480	0.309798	0.154899	+	0.987930i	$0.450495\pi$
$602$	0	0
$603$	1.10577	0.0450306
$604$	0	0
$605$	−23.7584	−0.965915
$606$	0	0
$607$	28.5674	1.15952	0.579758	−	0.814789i	$-0.303147\pi$
$607$	28.5674	1.15952	0.579758	+	0.814789i	$0.303147\pi$
$608$	0	0
$609$	2.96017	0.119952
$610$	0	0
$611$	18.1947	0.736077
$612$	0	0
$613$	−1.91340	−0.0772814	−0.0386407	−	0.999253i	$-0.512303\pi$
$613$	−1.91340	−0.0772814	−0.0386407	+	0.999253i	$0.512303\pi$
$614$	0	0
$615$	−8.77704	−0.353924
$616$	0	0
$617$	0.668569	0.0269156	0.0134578	−	0.999909i	$-0.495716\pi$
$617$	0.668569	0.0269156	0.0134578	+	0.999909i	$0.495716\pi$
$618$	0	0
$619$	41.4618	1.66649	0.833246	−	0.552903i	$-0.186480\pi$
$619$	41.4618	1.66649	0.833246	+	0.552903i	$0.186480\pi$
$620$	0	0
$621$	−20.5235	−0.823580
$622$	0	0
$623$	0.304088	0.0121830
$624$	0	0
$625$	−13.5063	−0.540250
$626$	0	0
$627$	27.7807	1.10945
$628$	0	0
$629$	0.798599	0.0318422
$630$	0	0
$631$	−40.1471	−1.59823	−0.799115	−	0.601179i	$-0.794698\pi$
$631$	−40.1471	−1.59823	−0.799115	+	0.601179i	$0.794698\pi$
$632$	0	0
$633$	−2.32825	−0.0925396
$634$	0	0
$635$	−3.72313	−0.147748
$636$	0	0
$637$	−4.83339	−0.191506
$638$	0	0
$639$	27.9289	1.10485
$640$	0	0
$641$	17.7061	0.699349	0.349675	−	0.936871i	$-0.386292\pi$
$641$	17.7061	0.699349	0.349675	+	0.936871i	$0.386292\pi$
$642$	0	0
$643$	36.1351	1.42503	0.712514	−	0.701658i	$-0.247556\pi$
$643$	36.1351	1.42503	0.712514	+	0.701658i	$0.247556\pi$
$644$	0	0
$645$	11.1266	0.438110
$646$	0	0
$647$	−43.3926	−1.70594	−0.852970	−	0.521961i	$-0.825201\pi$
$647$	−43.3926	−1.70594	−0.852970	+	0.521961i	$0.825201\pi$
$648$	0	0
$649$	−3.54710	−0.139236
$650$	0	0
$651$	−1.37931	−0.0540593
$652$	0	0
$653$	−36.1857	−1.41606	−0.708028	−	0.706184i	$-0.750415\pi$
$653$	−36.1857	−1.41606	−0.708028	+	0.706184i	$0.750415\pi$
$654$	0	0
$655$	1.03110	0.0402885
$656$	0	0
$657$	13.9125	0.542777
$658$	0	0
$659$	9.60166	0.374028	0.187014	−	0.982357i	$-0.440119\pi$
$659$	9.60166	0.374028	0.187014	+	0.982357i	$0.440119\pi$
$660$	0	0
$661$	45.7424	1.77917	0.889587	−	0.456766i	$-0.150992\pi$
$661$	45.7424	1.77917	0.889587	+	0.456766i	$0.150992\pi$
$662$	0	0
$663$	−5.18904	−0.201525
$664$	0	0
$665$	9.55686	0.370599
$666$	0	0
$667$	−10.8739	−0.421041
$668$	0	0
$669$	−10.2954	−0.398044
$670$	0	0
$671$	−40.7132	−1.57171
$672$	0	0
$673$	21.9341	0.845498	0.422749	−	0.906247i	$-0.361065\pi$
$673$	21.9341	0.845498	0.422749	+	0.906247i	$0.361065\pi$
$674$	0	0
$675$	8.91116	0.342991
$676$	0	0
$677$	−7.53121	−0.289448	−0.144724	−	0.989472i	$-0.546229\pi$
$677$	−7.53121	−0.289448	−0.144724	+	0.989472i	$0.546229\pi$
$678$	0	0
$679$	16.5918	0.636735
$680$	0	0
$681$	−7.90265	−0.302830
$682$	0	0
$683$	−12.4800	−0.477533	−0.238766	−	0.971077i	$-0.576743\pi$
$683$	−12.4800	−0.477533	−0.238766	+	0.971077i	$0.576743\pi$
$684$	0	0
$685$	19.8625	0.758907
$686$	0	0
$687$	−13.8165	−0.527132
$688$	0	0
$689$	38.0353	1.44903
$690$	0	0
$691$	34.6379	1.31769	0.658843	−	0.752280i	$-0.271046\pi$
$691$	34.6379	1.31769	0.658843	+	0.752280i	$0.271046\pi$
$692$	0	0
$693$	9.06989	0.344537
$694$	0	0
$695$	13.0290	0.494218
$696$	0	0
$697$	4.50888	0.170786
$698$	0	0
$699$	−18.3269	−0.693186
$700$	0	0
$701$	18.8435	0.711711	0.355856	−	0.934541i	$-0.384190\pi$
$701$	18.8435	0.711711	0.355856	+	0.934541i	$0.384190\pi$
$702$	0	0
$703$	4.20921	0.158753
$704$	0	0
$705$	7.32776	0.275980
$706$	0	0
$707$	−13.4633	−0.506340
$708$	0	0
$709$	9.14557	0.343469	0.171735	−	0.985143i	$-0.445063\pi$
$709$	9.14557	0.343469	0.171735	+	0.985143i	$0.445063\pi$
$710$	0	0
$711$	0.289905	0.0108723
$712$	0	0
$713$	5.06677	0.189752
$714$	0	0
$715$	43.0260	1.60908
$716$	0	0
$717$	14.8999	0.556448
$718$	0	0
$719$	0.122120	0.00455429	0.00227715	−	0.999997i	$-0.499275\pi$
$719$	0.122120	0.00455429	0.00227715	+	0.999997i	$0.499275\pi$
$720$	0	0
$721$	2.64584	0.0985361
$722$	0	0
$723$	17.1506	0.637838
$724$	0	0
$725$	4.72139	0.175348
$726$	0	0
$727$	4.96621	0.184186	0.0920932	−	0.995750i	$-0.470644\pi$
$727$	4.96621	0.184186	0.0920932	+	0.995750i	$0.470644\pi$
$728$	0	0
$729$	16.8438	0.623844
$730$	0	0
$731$	−5.71588	−0.211410
$732$	0	0
$733$	41.7529	1.54218	0.771089	−	0.636727i	$-0.219712\pi$
$733$	41.7529	1.54218	0.771089	+	0.636727i	$0.219712\pi$
$734$	0	0
$735$	−1.94661	−0.0718018
$736$	0	0
$737$	−2.93857	−0.108244
$738$	0	0
$739$	14.2021	0.522432	0.261216	−	0.965280i	$-0.415877\pi$
$739$	14.2021	0.522432	0.261216	+	0.965280i	$0.415877\pi$
$740$	0	0
$741$	−27.3500	−1.00473
$742$	0	0
$743$	−6.65245	−0.244055	−0.122027	−	0.992527i	$-0.538940\pi$
$743$	−6.65245	−0.244055	−0.122027	+	0.992527i	$0.538940\pi$
$744$	0	0
$745$	29.9141	1.09597
$746$	0	0
$747$	−11.7885	−0.431319
$748$	0	0
$749$	12.0520	0.440370
$750$	0	0
$751$	1.99696	0.0728701	0.0364351	−	0.999336i	$-0.488400\pi$
$751$	1.99696	0.0728701	0.0364351	+	0.999336i	$0.488400\pi$
$752$	0	0
$753$	−10.8691	−0.396093
$754$	0	0
$755$	32.1925	1.17160
$756$	0	0
$757$	49.9076	1.81392	0.906962	−	0.421213i	$-0.138395\pi$
$757$	49.9076	1.81392	0.906962	+	0.421213i	$0.138395\pi$
$758$	0	0
$759$	20.7863	0.754494
$760$	0	0
$761$	0.778750	0.0282297	0.0141148	−	0.999900i	$-0.495507\pi$
$761$	0.778750	0.0282297	0.0141148	+	0.999900i	$0.495507\pi$
$762$	0	0
$763$	11.0750	0.400941
$764$	0	0
$765$	3.34973	0.121110
$766$	0	0
$767$	3.49212	0.126093
$768$	0	0
$769$	−19.9542	−0.719568	−0.359784	−	0.933036i	$-0.617150\pi$
$769$	−19.9542	−0.719568	−0.359784	+	0.933036i	$0.617150\pi$
$770$	0	0
$771$	17.0264	0.613189
$772$	0	0
$773$	42.9022	1.54309	0.771543	−	0.636177i	$-0.219485\pi$
$773$	42.9022	1.54309	0.771543	+	0.636177i	$0.219485\pi$
$774$	0	0
$775$	−2.19996	−0.0790248
$776$	0	0
$777$	−0.857362	−0.0307577
$778$	0	0
$779$	23.7651	0.851474
$780$	0	0
$781$	−74.2205	−2.65582
$782$	0	0
$783$	14.3492	0.512799
$784$	0	0
$785$	−25.7505	−0.919076
$786$	0	0
$787$	0.935933	0.0333624	0.0166812	−	0.999861i	$-0.494690\pi$
$787$	0.935933	0.0333624	0.0166812	+	0.999861i	$0.494690\pi$
$788$	0	0
$789$	−24.3894	−0.868287
$790$	0	0
$791$	11.2646	0.400523
$792$	0	0
$793$	40.0821	1.42336
$794$	0	0
$795$	15.3184	0.543289
$796$	0	0
$797$	1.16932	0.0414193	0.0207097	−	0.999786i	$-0.493407\pi$
$797$	1.16932	0.0414193	0.0207097	+	0.999786i	$0.493407\pi$
$798$	0	0
$799$	−3.76437	−0.133174
$800$	0	0
$801$	0.561779	0.0198495
$802$	0	0
$803$	−36.9721	−1.30472
$804$	0	0
$805$	7.15072	0.252030
$806$	0	0
$807$	5.84830	0.205870
$808$	0	0
$809$	−54.1573	−1.90407	−0.952035	−	0.305990i	$-0.901013\pi$
$809$	−54.1573	−1.90407	−0.952035	+	0.305990i	$0.901013\pi$
$810$	0	0
$811$	−37.0433	−1.30077	−0.650383	−	0.759607i	$-0.725391\pi$
$811$	−37.0433	−1.30077	−0.650383	+	0.759607i	$0.725391\pi$
$812$	0	0
$813$	2.45367	0.0860539
$814$	0	0
$815$	−14.0822	−0.493279
$816$	0	0
$817$	−30.1269	−1.05401
$818$	0	0
$819$	−8.92930	−0.312015
$820$	0	0
$821$	39.0894	1.36423	0.682115	−	0.731245i	$-0.261060\pi$
$821$	39.0894	1.36423	0.682115	+	0.731245i	$0.261060\pi$
$822$	0	0
$823$	−3.19521	−0.111378	−0.0556890	−	0.998448i	$-0.517736\pi$
$823$	−3.19521	−0.111378	−0.0556890	+	0.998448i	$0.517736\pi$
$824$	0	0
$825$	−9.02525	−0.314219
$826$	0	0
$827$	−11.9621	−0.415963	−0.207981	−	0.978133i	$-0.566689\pi$
$827$	−11.9621	−0.415963	−0.207981	+	0.978133i	$0.566689\pi$
$828$	0	0
$829$	28.6317	0.994421	0.497210	−	0.867630i	$-0.334358\pi$
$829$	28.6317	0.994421	0.497210	+	0.867630i	$0.334358\pi$
$830$	0	0
$831$	−23.7769	−0.824811
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	12.1467	0.420355
$836$	0	0
$837$	−6.68609	−0.231105
$838$	0	0
$839$	30.9393	1.06814	0.534071	−	0.845439i	$-0.320661\pi$
$839$	30.9393	1.06814	0.534071	+	0.845439i	$0.320661\pi$
$840$	0	0
$841$	−21.3974	−0.737841
$842$	0	0
$843$	21.6859	0.746903
$844$	0	0
$845$	−18.7876	−0.646313
$846$	0	0
$847$	−13.1031	−0.450226
$848$	0	0
$849$	33.7992	1.15999
$850$	0	0
$851$	3.14945	0.107962
$852$	0	0
$853$	−42.6934	−1.46179	−0.730897	−	0.682488i	$-0.760898\pi$
$853$	−42.6934	−1.46179	−0.730897	+	0.682488i	$0.760898\pi$
$854$	0	0
$855$	17.6555	0.603807
$856$	0	0
$857$	−39.3652	−1.34469	−0.672344	−	0.740238i	$-0.734713\pi$
$857$	−39.3652	−1.34469	−0.672344	+	0.740238i	$0.734713\pi$
$858$	0	0
$859$	17.7829	0.606744	0.303372	−	0.952872i	$-0.401888\pi$
$859$	17.7829	0.606744	0.303372	+	0.952872i	$0.401888\pi$
$860$	0	0
$861$	−4.84066	−0.164969
$862$	0	0
$863$	−11.3441	−0.386159	−0.193079	−	0.981183i	$-0.561847\pi$
$863$	−11.3441	−0.386159	−0.193079	+	0.981183i	$0.561847\pi$
$864$	0	0
$865$	−25.6659	−0.872667
$866$	0	0
$867$	1.07358	0.0364608
$868$	0	0
$869$	−0.770418	−0.0261346
$870$	0	0
$871$	2.89302	0.0980264
$872$	0	0
$873$	30.6520	1.03741
$874$	0	0
$875$	−12.1708	−0.411447
$876$	0	0
$877$	49.4992	1.67147	0.835735	−	0.549133i	$-0.185042\pi$
$877$	49.4992	1.67147	0.835735	+	0.549133i	$0.185042\pi$
$878$	0	0
$879$	8.72195	0.294184
$880$	0	0
$881$	59.0826	1.99054	0.995272	−	0.0971308i	$-0.0309665\pi$
$881$	59.0826	1.99054	0.995272	+	0.0971308i	$0.0309665\pi$
$882$	0	0
$883$	−17.5466	−0.590491	−0.295246	−	0.955421i	$-0.595402\pi$
$883$	−17.5466	−0.590491	−0.295246	+	0.955421i	$0.595402\pi$
$884$	0	0
$885$	1.40642	0.0472764
$886$	0	0
$887$	54.6195	1.83394	0.916972	−	0.398952i	$-0.130626\pi$
$887$	54.6195	1.83394	0.916972	+	0.398952i	$0.130626\pi$
$888$	0	0
$889$	−2.05335	−0.0688673
$890$	0	0
$891$	−0.219783	−0.00736301
$892$	0	0
$893$	−19.8410	−0.663954
$894$	0	0
$895$	−16.0915	−0.537879
$896$	0	0
$897$	−20.4641	−0.683276
$898$	0	0
$899$	−3.54248	−0.118148
$900$	0	0
$901$	−7.86929	−0.262164
$902$	0	0
$903$	6.13647	0.204209
$904$	0	0
$905$	−22.5762	−0.750457
$906$	0	0
$907$	59.0131	1.95950	0.979748	−	0.200233i	$-0.0641700\pi$
$907$	59.0131	1.95950	0.979748	+	0.200233i	$0.0641700\pi$
$908$	0	0
$909$	−24.8724	−0.824967
$910$	0	0
$911$	41.7407	1.38293	0.691465	−	0.722410i	$-0.256966\pi$
$911$	41.7407	1.38293	0.691465	+	0.722410i	$0.256966\pi$
$912$	0	0
$913$	31.3278	1.03680
$914$	0	0
$915$	16.1428	0.533663
$916$	0	0
$917$	0.568667	0.0187790
$918$	0	0
$919$	−58.3975	−1.92636	−0.963178	−	0.268865i	$-0.913351\pi$
$919$	−58.3975	−1.92636	−0.963178	+	0.268865i	$0.913351\pi$
$920$	0	0
$921$	−15.4386	−0.508720
$922$	0	0
$923$	73.0701	2.40513
$924$	0	0
$925$	−1.36747	−0.0449621
$926$	0	0
$927$	4.88798	0.160542
$928$	0	0
$929$	−17.6020	−0.577502	−0.288751	−	0.957404i	$-0.593240\pi$
$929$	−17.6020	−0.577502	−0.288751	+	0.957404i	$0.593240\pi$
$930$	0	0
$931$	5.27074	0.172741
$932$	0	0
$933$	−25.1362	−0.822921
$934$	0	0
$935$	−8.90184	−0.291121
$936$	0	0
$937$	37.3278	1.21945	0.609723	−	0.792614i	$-0.291281\pi$
$937$	37.3278	1.21945	0.609723	+	0.792614i	$0.291281\pi$
$938$	0	0
$939$	−1.21236	−0.0395637
$940$	0	0
$941$	−6.61281	−0.215572	−0.107786	−	0.994174i	$-0.534376\pi$
$941$	−6.61281	−0.215572	−0.107786	+	0.994174i	$0.534376\pi$
$942$	0	0
$943$	17.7818	0.579054
$944$	0	0
$945$	−9.43604	−0.306955
$946$	0	0
$947$	18.7834	0.610378	0.305189	−	0.952292i	$-0.401280\pi$
$947$	18.7834	0.610378	0.305189	+	0.952292i	$0.401280\pi$
$948$	0	0
$949$	36.3990	1.18156
$950$	0	0
$951$	−10.3108	−0.334350
$952$	0	0
$953$	−14.2542	−0.461737	−0.230869	−	0.972985i	$-0.574157\pi$
$953$	−14.2542	−0.461737	−0.230869	+	0.972985i	$0.574157\pi$
$954$	0	0
$955$	−16.9562	−0.548688
$956$	0	0
$957$	−14.5329	−0.469783
$958$	0	0
$959$	10.9544	0.353737
$960$	0	0
$961$	−29.3494	−0.946754
$962$	0	0
$963$	22.2651	0.717484
$964$	0	0
$965$	44.9060	1.44557
$966$	0	0
$967$	13.7293	0.441504	0.220752	−	0.975330i	$-0.429149\pi$
$967$	13.7293	0.441504	0.220752	+	0.975330i	$0.429149\pi$
$968$	0	0
$969$	5.65857	0.181779
$970$	0	0
$971$	−31.6982	−1.01724	−0.508622	−	0.860990i	$-0.669845\pi$
$971$	−31.6982	−1.01724	−0.508622	+	0.860990i	$0.669845\pi$
$972$	0	0
$973$	7.18567	0.230362
$974$	0	0
$975$	8.88535	0.284559
$976$	0	0
$977$	−20.1941	−0.646068	−0.323034	−	0.946387i	$-0.604703\pi$
$977$	−20.1941	−0.646068	−0.323034	+	0.946387i	$0.604703\pi$
$978$	0	0
$979$	−1.49292	−0.0477138
$980$	0	0
$981$	20.4601	0.653242
$982$	0	0
$983$	−59.3564	−1.89318	−0.946588	−	0.322447i	$-0.895495\pi$
$983$	−59.3564	−1.89318	−0.946588	+	0.322447i	$0.895495\pi$
$984$	0	0
$985$	−18.5348	−0.590568
$986$	0	0
$987$	4.04136	0.128638
$988$	0	0
$989$	−22.5418	−0.716788
$990$	0	0
$991$	40.2432	1.27837	0.639184	−	0.769054i	$-0.279272\pi$
$991$	40.2432	1.27837	0.639184	+	0.769054i	$0.279272\pi$
$992$	0	0
$993$	31.3956	0.996311
$994$	0	0
$995$	14.0894	0.446663
$996$	0	0
$997$	38.6001	1.22248	0.611239	−	0.791446i	$-0.290672\pi$
$997$	38.6001	1.22248	0.611239	+	0.791446i	$0.290672\pi$
$998$	0	0
$999$	−4.15600	−0.131490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.j.1.5	✓	6
4.3	odd	2		3808.2.a.n.1.2	yes	6
8.3	odd	2		7616.2.a.bw.1.5		6
8.5	even	2		7616.2.a.ca.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.j.1.5	✓	6	1.1	even	1	trivial
3808.2.a.n.1.2	yes	6	4.3	odd	2
7616.2.a.bw.1.5		6	8.3	odd	2
7616.2.a.ca.1.2		6	8.5	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q+1.07358 q^{3} -1.81319 q^{5} -1.00000 q^{7} -1.84742 q^{9} +4.90949 q^{11} -4.83339 q^{13} -1.94661 q^{15} +1.00000 q^{17} +5.27074 q^{19} -1.07358 q^{21} +3.94372 q^{23} -1.71233 q^{25} -5.20410 q^{27} -2.75728 q^{29} +1.28477 q^{31} +5.27074 q^{33} +1.81319 q^{35} +0.798599 q^{37} -5.18904 q^{39} +4.50888 q^{41} -5.71588 q^{43} +3.34973 q^{45} -3.76437 q^{47} +1.00000 q^{49} +1.07358 q^{51} -7.86929 q^{53} -8.90184 q^{55} +5.65857 q^{57} -0.722499 q^{59} -8.29276 q^{61} +1.84742 q^{63} +8.76386 q^{65} -0.598550 q^{67} +4.23390 q^{69} -15.1178 q^{71} -7.53075 q^{73} -1.83833 q^{75} -4.90949 q^{77} -0.156924 q^{79} -0.0447670 q^{81} +6.38107 q^{83} -1.81319 q^{85} -2.96017 q^{87} -0.304088 q^{89} +4.83339 q^{91} +1.37931 q^{93} -9.55686 q^{95} -16.5918 q^{97} -9.06989 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 2 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 10 * q^23 - 2 * q^27 + 2 * q^29 - 12 * q^31 + 2 * q^33 - 2 * q^35 + 2 * q^37 - 10 * q^39 - 8 * q^43 + 14 * q^45 - 22 * q^47 + 6 * q^49 - 2 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^57 - 8 * q^59 - 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 4 * q^69 - 18 * q^71 + 8 * q^73 - 30 * q^75 + 2 * q^77 - 14 * q^79 - 26 * q^81 + 2 * q^85 - 22 * q^87 + 2 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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