LMFDB - Embedded newform 832.2.b.c.417.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [832,2,Mod(417,832)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("832.417"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-12,0,0,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.64355344817$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.195105024.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 $ x^8 - 4x^7 + 5x^6 + 4x^5 - 20x^4 + 12x^3 + 45x^2 - 108*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.3
Root			$-1.58726 + 0.693255i$ of defining polynomial
Character	$\chi$	$=$	832.417
Dual form			832.2.b.c.417.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.38651i q^{3} -1.38651i q^{5} +2.44247 q^{7} +1.07759 q^{9} +1.05596i q^{11} -1.00000i q^{13} -1.92241 q^{15} +6.96254 q^{17} -0.408139i q^{19} -3.38651i q^{21} -3.57603 q^{23} +3.07759 q^{25} -5.65362i q^{27} +4.34904i q^{29} -5.74705 q^{31} +1.46410 q^{33} -3.38651i q^{35} -7.49843i q^{37} -1.38651 q^{39} -1.57603 q^{41} +2.27145i q^{43} -1.49409i q^{45} -4.33055 q^{47} -1.03433 q^{49} -9.65362i q^{51} +0.647823i q^{53} +1.46410 q^{55} -0.565889 q^{57} -3.82898i q^{59} -4.80301i q^{61} +2.63199 q^{63} -1.38651 q^{65} -11.4050i q^{67} +4.95819i q^{69} +7.66945 q^{71} +7.68795 q^{73} -4.26711i q^{75} +2.57916i q^{77} -15.8131 q^{79} -4.60602 q^{81} +15.4050i q^{83} -9.65362i q^{85} +6.02999 q^{87} +2.88494 q^{89} -2.44247i q^{91} +7.96833i q^{93} -0.565889 q^{95} -3.30892 q^{97} +1.13790i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{9} - 36 q^{15} - 4 q^{17} + 24 q^{23} + 4 q^{25} - 20 q^{31} - 16 q^{33} - 4 q^{39} + 40 q^{41} - 40 q^{47} - 4 q^{49} - 16 q^{55} - 8 q^{57} - 44 q^{63} - 4 q^{65} + 56 q^{71} - 16 q^{73} - 32 q^{79}+ \cdots - 40 q^{97}+O(q^{100}) $ 8 * q - 12 * q^9 - 36 * q^15 - 4 * q^17 + 24 * q^23 + 4 * q^25 - 20 * q^31 - 16 * q^33 - 4 * q^39 + 40 * q^41 - 40 * q^47 - 4 * q^49 - 16 * q^55 - 8 * q^57 - 44 * q^63 - 4 * q^65 + 56 * q^71 - 16 * q^73 - 32 * q^79 + 40 * q^81 + 24 * q^87 - 16 * q^89 - 8 * q^95 - 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.38651i	− 0.800501i	−0.916406	−	0.400251i	$-0.868923\pi$
$3$	− 1.38651i	− 0.800501i	0.916406	−	0.400251i	$-0.131077\pi$
$4$	0	0
$5$	− 1.38651i	− 0.620066i	−0.950726	−	0.310033i	$-0.899660\pi$
$5$	− 1.38651i	− 0.620066i	0.950726	−	0.310033i	$-0.100340\pi$
$6$	0	0
$7$	2.44247	0.923167	0.461584	−	0.887097i	$-0.347281\pi$
$7$	2.44247	0.923167	0.461584	+	0.887097i	$0.347281\pi$
$8$	0	0
$9$	1.07759	0.359198
$10$	0	0
$11$	1.05596i	0.318385i	0.987248	+	0.159192i	$0.0508890\pi$
$11$	1.05596i	0.318385i	−0.987248	+	0.159192i	$0.949111\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−1.92241	−0.496363
$16$	0	0
$17$	6.96254	1.68866	0.844331	−	0.535821i	$-0.179998\pi$
$17$	6.96254	1.68866	0.844331	+	0.535821i	$0.179998\pi$
$18$	0	0
$19$	− 0.408139i	− 0.0936335i	−0.998903	−	0.0468168i	$-0.985092\pi$
$19$	− 0.408139i	− 0.0936335i	0.998903	−	0.0468168i	$-0.0149077\pi$
$20$	0	0
$21$	− 3.38651i	− 0.738997i
$22$	0	0
$23$	−3.57603	−0.745653	−0.372827	−	0.927901i	$-0.621611\pi$
$23$	−3.57603	−0.745653	−0.372827	+	0.927901i	$0.621611\pi$
$24$	0	0
$25$	3.07759	0.615519
$26$	0	0
$27$	− 5.65362i	− 1.08804i
$28$	0	0
$29$	4.34904i	0.807597i	0.914848	+	0.403799i	$0.132310\pi$
$29$	4.34904i	0.807597i	−0.914848	+	0.403799i	$0.867690\pi$
$30$	0	0
$31$	−5.74705	−1.03220	−0.516100	−	0.856528i	$-0.672617\pi$
$31$	−5.74705	−1.03220	−0.516100	+	0.856528i	$0.672617\pi$
$32$	0	0
$33$	1.46410	0.254867
$34$	0	0
$35$	− 3.38651i	− 0.572425i
$36$	0	0
$37$	− 7.49843i	− 1.23273i	−0.787459	−	0.616367i	$-0.788604\pi$
$37$	− 7.49843i	− 1.23273i	0.787459	−	0.616367i	$-0.211396\pi$
$38$	0	0
$39$	−1.38651	−0.222019
$40$	0	0
$41$	−1.57603	−0.246134	−0.123067	−	0.992398i	$-0.539273\pi$
$41$	−1.57603	−0.246134	−0.123067	+	0.992398i	$0.539273\pi$
$42$	0	0
$43$	2.27145i	0.346393i	0.984887	+	0.173197i	$0.0554096\pi$
$43$	2.27145i	0.346393i	−0.984887	+	0.173197i	$0.944590\pi$
$44$	0	0
$45$	− 1.49409i	− 0.222726i
$46$	0	0
$47$	−4.33055	−0.631675	−0.315838	−	0.948813i	$-0.602285\pi$
$47$	−4.33055	−0.631675	−0.315838	+	0.948813i	$0.602285\pi$
$48$	0	0
$49$	−1.03433	−0.147762
$50$	0	0
$51$	− 9.65362i	− 1.35178i
$52$	0	0
$53$	0.647823i	0.0889854i	0.999010	+	0.0444927i	$0.0141671\pi$
$53$	0.647823i	0.0889854i	−0.999010	+	0.0444927i	$0.985833\pi$
$54$	0	0
$55$	1.46410	0.197419
$56$	0	0
$57$	−0.565889	−0.0749538
$58$	0	0
$59$	− 3.82898i	− 0.498491i	−0.968440	−	0.249245i	$-0.919817\pi$
$59$	− 3.82898i	− 0.498491i	0.968440	−	0.249245i	$-0.0801826\pi$
$60$	0	0
$61$	− 4.80301i	− 0.614962i	−0.951554	−	0.307481i	$-0.900514\pi$
$61$	− 4.80301i	− 0.614962i	0.951554	−	0.307481i	$-0.0994861\pi$
$62$	0	0
$63$	2.63199	0.331599
$64$	0	0
$65$	−1.38651	−0.171975
$66$	0	0
$67$	− 11.4050i	− 1.39334i	−0.717390	−	0.696672i	$-0.754663\pi$
$67$	− 11.4050i	− 1.39334i	0.717390	−	0.696672i	$-0.245337\pi$
$68$	0	0
$69$	4.95819i	0.596896i
$70$	0	0
$71$	7.66945	0.910197	0.455098	−	0.890441i	$-0.349604\pi$
$71$	7.66945	0.910197	0.455098	+	0.890441i	$0.349604\pi$
$72$	0	0
$73$	7.68795	0.899807	0.449903	−	0.893077i	$-0.351458\pi$
$73$	7.68795	0.899807	0.449903	+	0.893077i	$0.351458\pi$
$74$	0	0
$75$	− 4.26711i	− 0.492723i
$76$	0	0
$77$	2.57916i	0.293922i
$78$	0	0
$79$	−15.8131	−1.77912	−0.889559	−	0.456820i	$-0.848988\pi$
$79$	−15.8131	−1.77912	−0.889559	+	0.456820i	$0.848988\pi$
$80$	0	0
$81$	−4.60602	−0.511780
$82$	0	0
$83$	15.4050i	1.69092i	0.534039	+	0.845460i	$0.320673\pi$
$83$	15.4050i	1.69092i	−0.534039	+	0.845460i	$0.679327\pi$
$84$	0	0
$85$	− 9.65362i	− 1.04708i
$86$	0	0
$87$	6.02999	0.646483
$88$	0	0
$89$	2.88494	0.305803	0.152902	−	0.988241i	$-0.451138\pi$
$89$	2.88494	0.305803	0.152902	+	0.988241i	$0.451138\pi$
$90$	0	0
$91$	− 2.44247i	− 0.256041i
$92$	0	0
$93$	7.96833i	0.826277i
$94$	0	0
$95$	−0.565889	−0.0580589
$96$	0	0
$97$	−3.30892	−0.335970	−0.167985	−	0.985790i	$-0.553726\pi$
$97$	−3.30892	−0.335970	−0.167985	+	0.985790i	$0.553726\pi$
$98$	0	0
$99$	1.13790i	0.114363i
$100$	0	0
$101$	12.2238i	1.21632i	0.793815	+	0.608159i	$0.208092\pi$
$101$	12.2238i	1.21632i	−0.793815	+	0.608159i	$0.791908\pi$
$102$	0	0
$103$	13.7012	1.35002	0.675011	−	0.737808i	$-0.264139\pi$
$103$	13.7012	1.35002	0.675011	+	0.737808i	$0.264139\pi$
$104$	0	0
$105$	−4.69543	−0.458227
$106$	0	0
$107$	8.26711i	0.799212i	0.916687	+	0.399606i	$0.130853\pi$
$107$	8.26711i	0.799212i	−0.916687	+	0.399606i	$0.869147\pi$
$108$	0	0
$109$	11.7223i	1.12279i	0.827548	+	0.561396i	$0.189735\pi$
$109$	11.7223i	1.12279i	−0.827548	+	0.561396i	$0.810265\pi$
$110$	0	0
$111$	−10.3966	−0.986806
$112$	0	0
$113$	−3.58930	−0.337653	−0.168826	−	0.985646i	$-0.553998\pi$
$113$	−3.58930	−0.337653	−0.168826	+	0.985646i	$0.553998\pi$
$114$	0	0
$115$	4.95819i	0.462354i
$116$	0	0
$117$	− 1.07759i	− 0.0996235i
$118$	0	0
$119$	17.0058	1.55892
$120$	0	0
$121$	9.88494	0.898631
$122$	0	0
$123$	2.18518i	0.197031i
$124$	0	0
$125$	− 11.1997i	− 1.00173i
$126$	0	0
$127$	−6.02999	−0.535075	−0.267538	−	0.963547i	$-0.586210\pi$
$127$	−6.02999	−0.535075	−0.267538	+	0.963547i	$0.586210\pi$
$128$	0	0
$129$	3.14939	0.277288
$130$	0	0
$131$	15.2429i	1.33178i	0.746050	+	0.665890i	$0.231948\pi$
$131$	15.2429i	1.33178i	−0.746050	+	0.665890i	$0.768052\pi$
$132$	0	0
$133$	− 0.996868i	− 0.0864394i
$134$	0	0
$135$	−7.83879	−0.674656
$136$	0	0
$137$	−20.8233	−1.77905	−0.889527	−	0.456883i	$-0.848966\pi$
$137$	−20.8233	−1.77905	−0.889527	+	0.456883i	$0.848966\pi$
$138$	0	0
$139$	− 0.314712i	− 0.0266936i	−0.999911	−	0.0133468i	$-0.995751\pi$
$139$	− 0.314712i	− 0.0266936i	0.999911	−	0.0133468i	$-0.00424854\pi$
$140$	0	0
$141$	6.00434i	0.505657i
$142$	0	0
$143$	1.05596	0.0883040
$144$	0	0
$145$	6.02999	0.500763
$146$	0	0
$147$	1.43411i	0.118284i
$148$	0	0
$149$	− 10.2238i	− 0.837570i	−0.908085	−	0.418785i	$-0.862456\pi$
$149$	− 10.2238i	− 0.837570i	0.908085	−	0.418785i	$-0.137544\pi$
$150$	0	0
$151$	0.741250	0.0603221	0.0301610	−	0.999545i	$-0.490398\pi$
$151$	0.741250	0.0603221	0.0301610	+	0.999545i	$0.490398\pi$
$152$	0	0
$153$	7.50278	0.606564
$154$	0	0
$155$	7.96833i	0.640032i
$156$	0	0
$157$	− 20.6681i	− 1.64949i	−0.565502	−	0.824747i	$-0.691318\pi$
$157$	− 20.6681i	− 1.64949i	0.565502	−	0.824747i	$-0.308682\pi$
$158$	0	0
$159$	0.898213	0.0712329
$160$	0	0
$161$	−8.73434	−0.688363
$162$	0	0
$163$	15.1679i	1.18804i	0.804450	+	0.594020i	$0.202460\pi$
$163$	15.1679i	1.18804i	−0.804450	+	0.594020i	$0.797540\pi$
$164$	0	0
$165$	− 2.02999i	− 0.158035i
$166$	0	0
$167$	21.4050	1.65637	0.828185	−	0.560455i	$-0.189374\pi$
$167$	21.4050	1.65637	0.828185	+	0.560455i	$0.189374\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 0.439808i	− 0.0336329i
$172$	0	0
$173$	14.6091i	1.11071i	0.831612	+	0.555357i	$0.187418\pi$
$173$	14.6091i	1.11071i	−0.831612	+	0.555357i	$0.812582\pi$
$174$	0	0
$175$	7.51693	0.568227
$176$	0	0
$177$	−5.30892	−0.399043
$178$	0	0
$179$	6.20279i	0.463618i	0.972761	+	0.231809i	$0.0744645\pi$
$179$	6.20279i	0.463618i	−0.972761	+	0.231809i	$0.925536\pi$
$180$	0	0
$181$	21.6263i	1.60747i	0.594988	+	0.803735i	$0.297157\pi$
$181$	21.6263i	1.60747i	−0.594988	+	0.803735i	$0.702843\pi$
$182$	0	0
$183$	−6.65941	−0.492278
$184$	0	0
$185$	−10.3966	−0.764377
$186$	0	0
$187$	7.35218i	0.537644i
$188$	0	0
$189$	− 13.8088i	− 1.00444i
$190$	0	0
$191$	12.6162	0.912873	0.456436	−	0.889756i	$-0.349126\pi$
$191$	12.6162	0.912873	0.456436	+	0.889756i	$0.349126\pi$
$192$	0	0
$193$	−15.2040	−1.09441	−0.547204	−	0.836999i	$-0.684308\pi$
$193$	−15.2040	−1.09441	−0.547204	+	0.836999i	$0.684308\pi$
$194$	0	0
$195$	1.92241i	0.137666i
$196$	0	0
$197$	23.7488i	1.69203i	0.533156	+	0.846017i	$0.321006\pi$
$197$	23.7488i	1.69203i	−0.533156	+	0.846017i	$0.678994\pi$
$198$	0	0
$199$	−0.0686650	−0.00486753	−0.00243377	−	0.999997i	$-0.500775\pi$
$199$	−0.0686650	−0.00486753	−0.00243377	+	0.999997i	$0.500775\pi$
$200$	0	0
$201$	−15.8131	−1.11537
$202$	0	0
$203$	10.6224i	0.745548i
$204$	0	0
$205$	2.18518i	0.152619i
$206$	0	0
$207$	−3.85350	−0.267837
$208$	0	0
$209$	0.430980	0.0298115
$210$	0	0
$211$	− 6.61349i	− 0.455291i	−0.973744	−	0.227646i	$-0.926897\pi$
$211$	− 6.61349i	− 0.455291i	0.973744	−	0.227646i	$-0.0731028\pi$
$212$	0	0
$213$	− 10.6338i	− 0.728614i
$214$	0	0
$215$	3.14939	0.214787
$216$	0	0
$217$	−14.0370	−0.952893
$218$	0	0
$219$	− 10.6594i	− 0.720297i
$220$	0	0
$221$	− 6.96254i	− 0.468351i
$222$	0	0
$223$	20.2989	1.35931	0.679657	−	0.733530i	$-0.262129\pi$
$223$	20.2989	1.35931	0.679657	+	0.733530i	$0.262129\pi$
$224$	0	0
$225$	3.31639	0.221093
$226$	0	0
$227$	7.09922i	0.471192i	0.971851	+	0.235596i	$0.0757042\pi$
$227$	7.09922i	0.471192i	−0.971851	+	0.235596i	$0.924296\pi$
$228$	0	0
$229$	4.22819i	0.279407i	0.990193	+	0.139703i	$0.0446149\pi$
$229$	4.22819i	0.279407i	−0.990193	+	0.139703i	$0.955385\pi$
$230$	0	0
$231$	3.57603	0.235285
$232$	0	0
$233$	8.14626	0.533679	0.266840	−	0.963741i	$-0.414021\pi$
$233$	8.14626	0.533679	0.266840	+	0.963741i	$0.414021\pi$
$234$	0	0
$235$	6.00434i	0.391680i
$236$	0	0
$237$	21.9251i	1.42419i
$238$	0	0
$239$	−29.3390	−1.89778	−0.948891	−	0.315603i	$-0.897793\pi$
$239$	−29.3390	−1.89778	−0.948891	+	0.315603i	$0.897793\pi$
$240$	0	0
$241$	3.04881	0.196391	0.0981956	−	0.995167i	$-0.468693\pi$
$241$	3.04881	0.196391	0.0981956	+	0.995167i	$0.468693\pi$
$242$	0	0
$243$	− 10.5746i	− 0.678359i
$244$	0	0
$245$	1.43411i	0.0916220i
$246$	0	0
$247$	−0.408139	−0.0259693
$248$	0	0
$249$	21.3592	1.35358
$250$	0	0
$251$	− 11.4508i	− 0.722770i	−0.932417	−	0.361385i	$-0.882304\pi$
$251$	− 11.4508i	− 0.722770i	0.932417	−	0.361385i	$-0.117696\pi$
$252$	0	0
$253$	− 3.77615i	− 0.237405i
$254$	0	0
$255$	−13.3848	−0.838191
$256$	0	0
$257$	−13.7788	−0.859499	−0.429749	−	0.902948i	$-0.641398\pi$
$257$	−13.7788	−0.859499	−0.429749	+	0.902948i	$0.641398\pi$
$258$	0	0
$259$	− 18.3147i	− 1.13802i
$260$	0	0
$261$	4.68650i	0.290087i
$262$	0	0
$263$	15.7629	0.971981	0.485990	−	0.873964i	$-0.338459\pi$
$263$	15.7629	0.971981	0.485990	+	0.873964i	$0.338459\pi$
$264$	0	0
$265$	0.898213	0.0551768
$266$	0	0
$267$	− 4.00000i	− 0.244796i
$268$	0	0
$269$	− 0.193860i	− 0.0118198i	−0.999983	−	0.00590992i	$-0.998119\pi$
$269$	− 0.193860i	− 0.0118198i	0.999983	−	0.00590992i	$-0.00188120\pi$
$270$	0	0
$271$	−24.3446	−1.47883	−0.739413	−	0.673252i	$-0.764897\pi$
$271$	−24.3446	−1.47883	−0.739413	+	0.673252i	$0.764897\pi$
$272$	0	0
$273$	−3.38651	−0.204961
$274$	0	0
$275$	3.24982i	0.195972i
$276$	0	0
$277$	29.9753i	1.80104i	0.434811	+	0.900522i	$0.356815\pi$
$277$	29.9753i	1.80104i	−0.434811	+	0.900522i	$0.643185\pi$
$278$	0	0
$279$	−6.19297	−0.370764
$280$	0	0
$281$	−33.1890	−1.97989	−0.989946	−	0.141443i	$-0.954826\pi$
$281$	−33.1890	−1.97989	−0.989946	+	0.141443i	$0.954826\pi$
$282$	0	0
$283$	23.0771i	1.37179i	0.727699	+	0.685896i	$0.240590\pi$
$283$	23.0771i	1.37179i	−0.727699	+	0.685896i	$0.759410\pi$
$284$	0	0
$285$	0.784610i	0.0464763i
$286$	0	0
$287$	−3.84940	−0.227223
$288$	0	0
$289$	31.4769	1.85158
$290$	0	0
$291$	4.58784i	0.268944i
$292$	0	0
$293$	3.34325i	0.195315i	0.995220	+	0.0976573i	$0.0311349\pi$
$293$	3.34325i	0.195315i	−0.995220	+	0.0976573i	$0.968865\pi$
$294$	0	0
$295$	−5.30892	−0.309097
$296$	0	0
$297$	5.97001	0.346415
$298$	0	0
$299$	3.57603i	0.206807i
$300$	0	0
$301$	5.54796i	0.319779i
$302$	0	0
$303$	16.9485	0.973665
$304$	0	0
$305$	−6.65941	−0.381317
$306$	0	0
$307$	− 17.6854i	− 1.00936i	−0.863307	−	0.504679i	$-0.831611\pi$
$307$	− 17.6854i	− 1.00936i	0.863307	−	0.504679i	$-0.168389\pi$
$308$	0	0
$309$	− 18.9969i	− 1.08069i
$310$	0	0
$311$	28.9485	1.64152	0.820759	−	0.571275i	$-0.193551\pi$
$311$	28.9485	1.64152	0.820759	+	0.571275i	$0.193551\pi$
$312$	0	0
$313$	−9.51170	−0.537633	−0.268817	−	0.963191i	$-0.586633\pi$
$313$	−9.51170	−0.537633	−0.268817	+	0.963191i	$0.586633\pi$
$314$	0	0
$315$	− 3.64928i	− 0.205613i
$316$	0	0
$317$	− 20.6727i	− 1.16109i	−0.814227	−	0.580547i	$-0.802839\pi$
$317$	− 20.6727i	− 1.16109i	0.814227	−	0.580547i	$-0.197161\pi$
$318$	0	0
$319$	−4.59243	−0.257127
$320$	0	0
$321$	11.4624	0.639770
$322$	0	0
$323$	− 2.84168i	− 0.158115i
$324$	0	0
$325$	− 3.07759i	− 0.170714i
$326$	0	0
$327$	16.2531	0.898796
$328$	0	0
$329$	−10.5772	−0.583142
$330$	0	0
$331$	− 21.0110i	− 1.15487i	−0.816437	−	0.577435i	$-0.804054\pi$
$331$	− 21.0110i	− 1.15487i	0.816437	−	0.577435i	$-0.195946\pi$
$332$	0	0
$333$	− 8.08026i	− 0.442795i
$334$	0	0
$335$	−15.8131	−0.863964
$336$	0	0
$337$	13.5117	0.736029	0.368015	−	0.929820i	$-0.380038\pi$
$337$	13.5117	0.736029	0.368015	+	0.929820i	$0.380038\pi$
$338$	0	0
$339$	4.97659i	0.270291i
$340$	0	0
$341$	− 6.06866i	− 0.328637i
$342$	0	0
$343$	−19.6236	−1.05958
$344$	0	0
$345$	6.87458	0.370115
$346$	0	0
$347$	8.15953i	0.438026i	0.975722	+	0.219013i	$0.0702838\pi$
$347$	8.15953i	0.438026i	−0.975722	+	0.219013i	$0.929716\pi$
$348$	0	0
$349$	− 17.5417i	− 0.938985i	−0.882936	−	0.469493i	$-0.844437\pi$
$349$	− 17.5417i	− 0.938985i	0.882936	−	0.469493i	$-0.155563\pi$
$350$	0	0
$351$	−5.65362	−0.301768
$352$	0	0
$353$	1.52576	0.0812080	0.0406040	−	0.999175i	$-0.487072\pi$
$353$	1.52576	0.0812080	0.0406040	+	0.999175i	$0.487072\pi$
$354$	0	0
$355$	− 10.6338i	− 0.564382i
$356$	0	0
$357$	− 23.5787i	− 1.24792i
$358$	0	0
$359$	−17.4050	−0.918601	−0.459301	−	0.888281i	$-0.651900\pi$
$359$	−17.4050	−0.918601	−0.459301	+	0.888281i	$0.651900\pi$
$360$	0	0
$361$	18.8334	0.991233
$362$	0	0
$363$	− 13.7056i	− 0.719356i
$364$	0	0
$365$	− 10.6594i	− 0.557939i
$366$	0	0
$367$	25.5830	1.33542	0.667712	−	0.744420i	$-0.267274\pi$
$367$	25.5830	1.33542	0.667712	+	0.744420i	$0.267274\pi$
$368$	0	0
$369$	−1.69831	−0.0884107
$370$	0	0
$371$	1.58229i	0.0821484i
$372$	0	0
$373$	− 8.77302i	− 0.454250i	−0.973866	−	0.227125i	$-0.927067\pi$
$373$	− 8.77302i	− 0.454250i	0.973866	−	0.227125i	$-0.0729325\pi$
$374$	0	0
$375$	−15.5284	−0.801884
$376$	0	0
$377$	4.34904	0.223987
$378$	0	0
$379$	− 32.2945i	− 1.65886i	−0.558611	−	0.829429i	$-0.688666\pi$
$379$	− 32.2945i	− 1.65886i	0.558611	−	0.829429i	$-0.311334\pi$
$380$	0	0
$381$	8.36064i	0.428328i
$382$	0	0
$383$	−10.0052	−0.511243	−0.255622	−	0.966777i	$-0.582280\pi$
$383$	−10.0052	−0.511243	−0.255622	+	0.966777i	$0.582280\pi$
$384$	0	0
$385$	3.57603	0.182251
$386$	0	0
$387$	2.44770i	0.124424i
$388$	0	0
$389$	17.6625i	0.895527i	0.894152	+	0.447763i	$0.147779\pi$
$389$	17.6625i	0.895527i	−0.894152	+	0.447763i	$0.852221\pi$
$390$	0	0
$391$	−24.8982	−1.25916
$392$	0	0
$393$	21.1344	1.06609
$394$	0	0
$395$	21.9251i	1.10317i
$396$	0	0
$397$	− 6.81628i	− 0.342099i	−0.985262	−	0.171050i	$-0.945284\pi$
$397$	− 6.81628i	− 0.342099i	0.985262	−	0.171050i	$-0.0547158\pi$
$398$	0	0
$399$	−1.38217	−0.0691949
$400$	0	0
$401$	−17.5530	−0.876557	−0.438279	−	0.898839i	$-0.644412\pi$
$401$	−17.5530	−0.876557	−0.438279	+	0.898839i	$0.644412\pi$
$402$	0	0
$403$	5.74705i	0.286281i
$404$	0	0
$405$	6.38628i	0.317337i
$406$	0	0
$407$	7.91807	0.392484
$408$	0	0
$409$	28.8833	1.42819	0.714093	−	0.700051i	$-0.246839\pi$
$409$	28.8833	1.42819	0.714093	+	0.700051i	$0.246839\pi$
$410$	0	0
$411$	28.8717i	1.42413i
$412$	0	0
$413$	− 9.35218i	− 0.460191i
$414$	0	0
$415$	21.3592	1.04848
$416$	0	0
$417$	−0.436352	−0.0213682
$418$	0	0
$419$	− 22.7687i	− 1.11232i	−0.831074	−	0.556161i	$-0.812274\pi$
$419$	− 22.7687i	− 1.11232i	0.831074	−	0.556161i	$-0.187726\pi$
$420$	0	0
$421$	− 23.4668i	− 1.14370i	−0.820358	−	0.571850i	$-0.806226\pi$
$421$	− 23.4668i	− 1.14370i	0.820358	−	0.571850i	$-0.193774\pi$
$422$	0	0
$423$	−4.66656	−0.226896
$424$	0	0
$425$	21.4278	1.03940
$426$	0	0
$427$	− 11.7312i	− 0.567713i
$428$	0	0
$429$	− 1.46410i	− 0.0706875i
$430$	0	0
$431$	22.3421	1.07618	0.538091	−	0.842886i	$-0.319145\pi$
$431$	22.3421	1.07618	0.538091	+	0.842886i	$0.319145\pi$
$432$	0	0
$433$	9.55496	0.459182	0.229591	−	0.973287i	$-0.426261\pi$
$433$	9.55496	0.459182	0.229591	+	0.973287i	$0.426261\pi$
$434$	0	0
$435$	− 8.36064i	− 0.400862i
$436$	0	0
$437$	1.45952i	0.0698181i
$438$	0	0
$439$	0.966878	0.0461466	0.0230733	−	0.999734i	$-0.492655\pi$
$439$	0.966878	0.0461466	0.0230733	+	0.999734i	$0.492655\pi$
$440$	0	0
$441$	−1.11459	−0.0530757
$442$	0	0
$443$	27.2164i	1.29309i	0.762876	+	0.646545i	$0.223787\pi$
$443$	27.2164i	1.29309i	−0.762876	+	0.646545i	$0.776213\pi$
$444$	0	0
$445$	− 4.00000i	− 0.189618i
$446$	0	0
$447$	−14.1755	−0.670476
$448$	0	0
$449$	7.55931	0.356746	0.178373	−	0.983963i	$-0.442917\pi$
$449$	7.55931	0.356746	0.178373	+	0.983963i	$0.442917\pi$
$450$	0	0
$451$	− 1.66423i	− 0.0783653i
$452$	0	0
$453$	− 1.02775i	− 0.0482879i
$454$	0	0
$455$	−3.38651	−0.158762
$456$	0	0
$457$	27.7012	1.29581	0.647904	−	0.761722i	$-0.275646\pi$
$457$	27.7012	1.29581	0.647904	+	0.761722i	$0.275646\pi$
$458$	0	0
$459$	− 39.3635i	− 1.83733i
$460$	0	0
$461$	33.8025i	1.57434i	0.616735	+	0.787171i	$0.288455\pi$
$461$	33.8025i	1.57434i	−0.616735	+	0.787171i	$0.711545\pi$
$462$	0	0
$463$	1.97090	0.0915953	0.0457977	−	0.998951i	$-0.485417\pi$
$463$	1.97090	0.0915953	0.0457977	+	0.998951i	$0.485417\pi$
$464$	0	0
$465$	11.0482	0.512346
$466$	0	0
$467$	15.2410i	0.705269i	0.935761	+	0.352635i	$0.114714\pi$
$467$	15.2410i	0.705269i	−0.935761	+	0.352635i	$0.885286\pi$
$468$	0	0
$469$	− 27.8564i	− 1.28629i
$470$	0	0
$471$	−28.6565	−1.32042
$472$	0	0
$473$	−2.39857	−0.110286
$474$	0	0
$475$	− 1.25609i	− 0.0576332i
$476$	0	0
$477$	0.698090i	0.0319633i
$478$	0	0
$479$	10.4795	0.478819	0.239410	−	0.970919i	$-0.423046\pi$
$479$	10.4795	0.478819	0.239410	+	0.970919i	$0.423046\pi$
$480$	0	0
$481$	−7.49843	−0.341899
$482$	0	0
$483$	12.1102i	0.551035i
$484$	0	0
$485$	4.58784i	0.208323i
$486$	0	0
$487$	−42.0714	−1.90644	−0.953219	−	0.302280i	$-0.902252\pi$
$487$	−42.0714	−1.90644	−0.953219	+	0.302280i	$0.902252\pi$
$488$	0	0
$489$	21.0304	0.951028
$490$	0	0
$491$	− 17.8025i	− 0.803417i	−0.915768	−	0.401709i	$-0.868416\pi$
$491$	− 17.8025i	− 0.803417i	0.915768	−	0.401709i	$-0.131584\pi$
$492$	0	0
$493$	30.2804i	1.36376i
$494$	0	0
$495$	1.57770	0.0709126
$496$	0	0
$497$	18.7324	0.840264
$498$	0	0
$499$	− 40.1534i	− 1.79751i	−0.438450	−	0.898756i	$-0.644472\pi$
$499$	− 40.1534i	− 1.79751i	0.438450	−	0.898756i	$-0.355528\pi$
$500$	0	0
$501$	− 29.6782i	− 1.32593i
$502$	0	0
$503$	5.88808	0.262536	0.131268	−	0.991347i	$-0.458095\pi$
$503$	5.88808	0.262536	0.131268	+	0.991347i	$0.458095\pi$
$504$	0	0
$505$	16.9485	0.754197
$506$	0	0
$507$	1.38651i	0.0615770i
$508$	0	0
$509$	26.6064i	1.17931i	0.807655	+	0.589655i	$0.200736\pi$
$509$	26.6064i	1.17931i	−0.807655	+	0.589655i	$0.799264\pi$
$510$	0	0
$511$	18.7776	0.830672
$512$	0	0
$513$	−2.30746	−0.101877
$514$	0	0
$515$	− 18.9969i	− 0.837102i
$516$	0	0
$517$	− 4.57289i	− 0.201116i
$518$	0	0
$519$	20.2557	0.889127
$520$	0	0
$521$	−32.6638	−1.43103	−0.715513	−	0.698600i	$-0.753807\pi$
$521$	−32.6638	−1.43103	−0.715513	+	0.698600i	$0.753807\pi$
$522$	0	0
$523$	− 2.88494i	− 0.126150i	−0.998009	−	0.0630749i	$-0.979909\pi$
$523$	− 2.88494i	− 0.126150i	0.998009	−	0.0630749i	$-0.0200907\pi$
$524$	0	0
$525$	− 10.4223i	− 0.454866i
$526$	0	0
$527$	−40.0140	−1.74304
$528$	0	0
$529$	−10.2120	−0.444001
$530$	0	0
$531$	− 4.12608i	− 0.179057i
$532$	0	0
$533$	1.57603i	0.0682653i
$534$	0	0
$535$	11.4624	0.495564
$536$	0	0
$537$	8.60022	0.371127
$538$	0	0
$539$	− 1.09222i	− 0.0470451i
$540$	0	0
$541$	− 31.4984i	− 1.35422i	−0.735880	−	0.677112i	$-0.763231\pi$
$541$	− 31.4984i	− 1.35422i	0.735880	−	0.677112i	$-0.236769\pi$
$542$	0	0
$543$	29.9851	1.28678
$544$	0	0
$545$	16.2531	0.696204
$546$	0	0
$547$	− 4.28817i	− 0.183349i	−0.995789	−	0.0916745i	$-0.970778\pi$
$547$	− 4.28817i	− 0.183349i	0.995789	−	0.0916745i	$-0.0292219\pi$
$548$	0	0
$549$	− 5.17569i	− 0.220893i
$550$	0	0
$551$	1.77502	0.0756182
$552$	0	0
$553$	−38.6232	−1.64242
$554$	0	0
$555$	14.4150i	0.611885i
$556$	0	0
$557$	23.9640i	1.01539i	0.861538	+	0.507693i	$0.169502\pi$
$557$	23.9640i	1.01539i	−0.861538	+	0.507693i	$0.830498\pi$
$558$	0	0
$559$	2.27145	0.0960722
$560$	0	0
$561$	10.1939	0.430385
$562$	0	0
$563$	39.1247i	1.64891i	0.565927	+	0.824455i	$0.308518\pi$
$563$	39.1247i	1.64891i	−0.565927	+	0.824455i	$0.691482\pi$
$564$	0	0
$565$	4.97659i	0.209367i
$566$	0	0
$567$	−11.2501	−0.472458
$568$	0	0
$569$	1.17593	0.0492975	0.0246488	−	0.999696i	$-0.492153\pi$
$569$	1.17593	0.0492975	0.0246488	+	0.999696i	$0.492153\pi$
$570$	0	0
$571$	− 19.0295i	− 0.796361i	−0.917307	−	0.398180i	$-0.869642\pi$
$571$	− 19.0295i	− 0.796361i	0.917307	−	0.398180i	$-0.130358\pi$
$572$	0	0
$573$	− 17.4924i	− 0.730756i
$574$	0	0
$575$	−11.0056	−0.458963
$576$	0	0
$577$	−26.5245	−1.10423	−0.552115	−	0.833768i	$-0.686179\pi$
$577$	−26.5245	−1.10423	−0.552115	+	0.833768i	$0.686179\pi$
$578$	0	0
$579$	21.0805i	0.876075i
$580$	0	0
$581$	37.6263i	1.56100i
$582$	0	0
$583$	−0.684077	−0.0283316
$584$	0	0
$585$	−1.49409	−0.0617731
$586$	0	0
$587$	− 46.1418i	− 1.90447i	−0.305359	−	0.952237i	$-0.598776\pi$
$587$	− 46.1418i	− 1.90447i	0.305359	−	0.952237i	$-0.401224\pi$
$588$	0	0
$589$	2.34559i	0.0966485i
$590$	0	0
$591$	32.9280	1.35448
$592$	0	0
$593$	−37.3575	−1.53409	−0.767044	−	0.641594i	$-0.778273\pi$
$593$	−37.3575	−1.53409	−0.767044	+	0.641594i	$0.778273\pi$
$594$	0	0
$595$	− 23.5787i	− 0.966632i
$596$	0	0
$597$	0.0952046i	0.00389647i
$598$	0	0
$599$	−15.4757	−0.632320	−0.316160	−	0.948706i	$-0.602394\pi$
$599$	−15.4757	−0.632320	−0.316160	+	0.948706i	$0.602394\pi$
$600$	0	0
$601$	−44.5226	−1.81611	−0.908057	−	0.418846i	$-0.862435\pi$
$601$	−44.5226	−1.81611	−0.908057	+	0.418846i	$0.862435\pi$
$602$	0	0
$603$	− 12.2900i	− 0.500485i
$604$	0	0
$605$	− 13.7056i	− 0.557210i
$606$	0	0
$607$	−14.2908	−0.580047	−0.290024	−	0.957019i	$-0.593663\pi$
$607$	−14.2908	−0.580047	−0.290024	+	0.957019i	$0.593663\pi$
$608$	0	0
$609$	14.7281	0.596812
$610$	0	0
$611$	4.33055i	0.175195i
$612$	0	0
$613$	27.4827i	1.11002i	0.831845	+	0.555008i	$0.187285\pi$
$613$	27.4827i	1.11002i	−0.831845	+	0.555008i	$0.812715\pi$
$614$	0	0
$615$	3.02977	0.122172
$616$	0	0
$617$	−7.79287	−0.313729	−0.156865	−	0.987620i	$-0.550139\pi$
$617$	−7.79287	−0.313729	−0.156865	+	0.987620i	$0.550139\pi$
$618$	0	0
$619$	41.9092i	1.68447i	0.539107	+	0.842237i	$0.318762\pi$
$619$	41.9092i	1.68447i	−0.539107	+	0.842237i	$0.681238\pi$
$620$	0	0
$621$	20.2175i	0.811300i
$622$	0	0
$623$	7.04639	0.282308
$624$	0	0
$625$	−0.140462	−0.00561847
$626$	0	0
$627$	− 0.597557i	− 0.0238641i
$628$	0	0
$629$	− 52.2081i	− 2.08167i
$630$	0	0
$631$	−24.3905	−0.970972	−0.485486	−	0.874245i	$-0.661357\pi$
$631$	−24.3905	−0.970972	−0.485486	+	0.874245i	$0.661357\pi$
$632$	0	0
$633$	−9.16967	−0.364461
$634$	0	0
$635$	8.36064i	0.331782i
$636$	0	0
$637$	1.03433i	0.0409817i
$638$	0	0
$639$	8.26455	0.326940
$640$	0	0
$641$	−2.12978	−0.0841213	−0.0420606	−	0.999115i	$-0.513392\pi$
$641$	−2.12978	−0.0841213	−0.0420606	+	0.999115i	$0.513392\pi$
$642$	0	0
$643$	30.7475i	1.21256i	0.795251	+	0.606281i	$0.207339\pi$
$643$	30.7475i	1.21256i	−0.795251	+	0.606281i	$0.792661\pi$
$644$	0	0
$645$	− 4.36666i	− 0.171937i
$646$	0	0
$647$	−11.0471	−0.434308	−0.217154	−	0.976137i	$-0.569677\pi$
$647$	−11.0471	−0.434308	−0.217154	+	0.976137i	$0.569677\pi$
$648$	0	0
$649$	4.04326	0.158712
$650$	0	0
$651$	19.4624i	0.762793i
$652$	0	0
$653$	3.16700i	0.123934i	0.998078	+	0.0619672i	$0.0197374\pi$
$653$	3.16700i	0.123934i	−0.998078	+	0.0619672i	$0.980263\pi$
$654$	0	0
$655$	21.1344	0.825791
$656$	0	0
$657$	8.28448	0.323208
$658$	0	0
$659$	− 18.5572i	− 0.722886i	−0.932394	−	0.361443i	$-0.882284\pi$
$659$	− 18.5572i	− 0.722886i	0.932394	−	0.361443i	$-0.117716\pi$
$660$	0	0
$661$	12.2758i	0.477473i	0.971084	+	0.238737i	$0.0767332\pi$
$661$	12.2758i	0.477473i	−0.971084	+	0.238737i	$0.923267\pi$
$662$	0	0
$663$	−9.65362	−0.374916
$664$	0	0
$665$	−1.38217	−0.0535981
$666$	0	0
$667$	− 15.5523i	− 0.602187i
$668$	0	0
$669$	− 28.1446i	− 1.08813i
$670$	0	0
$671$	5.07180	0.195795
$672$	0	0
$673$	−33.4484	−1.28934	−0.644670	−	0.764461i	$-0.723005\pi$
$673$	−33.4484	−1.28934	−0.644670	+	0.764461i	$0.723005\pi$
$674$	0	0
$675$	− 17.3995i	− 0.669708i
$676$	0	0
$677$	− 20.5429i	− 0.789528i	−0.918783	−	0.394764i	$-0.870826\pi$
$677$	− 20.5429i	− 0.789528i	0.918783	−	0.394764i	$-0.129174\pi$
$678$	0	0
$679$	−8.08193	−0.310156
$680$	0	0
$681$	9.84314	0.377190
$682$	0	0
$683$	− 0.878823i	− 0.0336272i	−0.999859	−	0.0168136i	$-0.994648\pi$
$683$	− 0.878823i	− 0.0336272i	0.999859	−	0.0168136i	$-0.00535219\pi$
$684$	0	0
$685$	28.8717i	1.10313i
$686$	0	0
$687$	5.86243	0.223666
$688$	0	0
$689$	0.647823	0.0246801
$690$	0	0
$691$	30.0864i	1.14454i	0.820065	+	0.572270i	$0.193937\pi$
$691$	30.0864i	1.14454i	−0.820065	+	0.572270i	$0.806063\pi$
$692$	0	0
$693$	2.77928i	0.105576i
$694$	0	0
$695$	−0.436352	−0.0165518
$696$	0	0
$697$	−10.9731	−0.415637
$698$	0	0
$699$	− 11.2949i	− 0.427211i
$700$	0	0
$701$	21.7312i	0.820777i	0.911911	+	0.410388i	$0.134607\pi$
$701$	21.7312i	0.820777i	−0.911911	+	0.410388i	$0.865393\pi$
$702$	0	0
$703$	−3.06040	−0.115425
$704$	0	0
$705$	8.32508	0.313540
$706$	0	0
$707$	29.8564i	1.12287i
$708$	0	0
$709$	− 4.42230i	− 0.166083i	−0.996546	−	0.0830414i	$-0.973537\pi$
$709$	− 4.42230i	− 0.166083i	0.996546	−	0.0830414i	$-0.0264634\pi$
$710$	0	0
$711$	−17.0401	−0.639055
$712$	0	0
$713$	20.5516	0.769663
$714$	0	0
$715$	− 1.46410i	− 0.0547543i
$716$	0	0
$717$	40.6788i	1.51918i
$718$	0	0
$719$	44.1427	1.64624	0.823122	−	0.567865i	$-0.192230\pi$
$719$	44.1427	1.64624	0.823122	+	0.567865i	$0.192230\pi$
$720$	0	0
$721$	33.4648	1.24630
$722$	0	0
$723$	− 4.22721i	− 0.157212i
$724$	0	0
$725$	13.3846i	0.497091i
$726$	0	0
$727$	−14.9519	−0.554536	−0.277268	−	0.960793i	$-0.589429\pi$
$727$	−14.9519	−0.554536	−0.277268	+	0.960793i	$0.589429\pi$
$728$	0	0
$729$	−28.4798	−1.05481
$730$	0	0
$731$	15.8151i	0.584941i
$732$	0	0
$733$	− 17.4465i	− 0.644401i	−0.946671	−	0.322200i	$-0.895578\pi$
$733$	− 17.4465i	− 0.644401i	0.946671	−	0.322200i	$-0.104422\pi$
$734$	0	0
$735$	1.98841	0.0733436
$736$	0	0
$737$	12.0433	0.443619
$738$	0	0
$739$	24.4406i	0.899060i	0.893265	+	0.449530i	$0.148409\pi$
$739$	24.4406i	0.899060i	−0.893265	+	0.449530i	$0.851591\pi$
$740$	0	0
$741$	0.565889i	0.0207884i
$742$	0	0
$743$	10.5290	0.386271	0.193136	−	0.981172i	$-0.438134\pi$
$743$	10.5290	0.386271	0.193136	+	0.981172i	$0.438134\pi$
$744$	0	0
$745$	−14.1755	−0.519349
$746$	0	0
$747$	16.6003i	0.607374i
$748$	0	0
$749$	20.1922i	0.737806i
$750$	0	0
$751$	37.4411	1.36625	0.683123	−	0.730303i	$-0.260621\pi$
$751$	37.4411	1.36625	0.683123	+	0.730303i	$0.260621\pi$
$752$	0	0
$753$	−15.8767	−0.578578
$754$	0	0
$755$	− 1.02775i	− 0.0374037i
$756$	0	0
$757$	15.1423i	0.550358i	0.961393	+	0.275179i	$0.0887371\pi$
$757$	15.1423i	0.550358i	−0.961393	+	0.275179i	$0.911263\pi$
$758$	0	0
$759$	−5.23567	−0.190043
$760$	0	0
$761$	−15.6828	−0.568502	−0.284251	−	0.958750i	$-0.591745\pi$
$761$	−15.6828	−0.568502	−0.284251	+	0.958750i	$0.591745\pi$
$762$	0	0
$763$	28.6313i	1.03652i
$764$	0	0
$765$	− 10.4027i	− 0.376109i
$766$	0	0
$767$	−3.82898	−0.138256
$768$	0	0
$769$	−8.81870	−0.318010	−0.159005	−	0.987278i	$-0.550829\pi$
$769$	−8.81870	−0.318010	−0.159005	+	0.987278i	$0.550829\pi$
$770$	0	0
$771$	19.1045i	0.688030i
$772$	0	0
$773$	47.6819i	1.71500i	0.514484	+	0.857500i	$0.327983\pi$
$773$	47.6819i	1.71500i	−0.514484	+	0.857500i	$0.672017\pi$
$774$	0	0
$775$	−17.6871	−0.635338
$776$	0	0
$777$	−25.3935	−0.910987
$778$	0	0
$779$	0.643238i	0.0230464i
$780$	0	0
$781$	8.09866i	0.289793i
$782$	0	0
$783$	24.5878	0.878698
$784$	0	0
$785$	−28.6565	−1.02279
$786$	0	0
$787$	38.0874i	1.35767i	0.734291	+	0.678835i	$0.237515\pi$
$787$	38.0874i	1.35767i	−0.734291	+	0.678835i	$0.762485\pi$
$788$	0	0
$789$	− 21.8554i	− 0.778072i
$790$	0	0
$791$	−8.76675	−0.311710
$792$	0	0
$793$	−4.80301	−0.170560
$794$	0	0
$795$	− 1.24538i	− 0.0441691i
$796$	0	0
$797$	− 41.6950i	− 1.47691i	−0.674302	−	0.738456i	$-0.735555\pi$
$797$	− 41.6950i	− 1.47691i	0.674302	−	0.738456i	$-0.264445\pi$
$798$	0	0
$799$	−30.1516	−1.06669
$800$	0	0
$801$	3.10879	0.109844
$802$	0	0
$803$	8.11819i	0.286485i
$804$	0	0
$805$	12.1102i	0.426830i
$806$	0	0
$807$	−0.268788	−0.00946180
$808$	0	0
$809$	−0.353387	−0.0124244	−0.00621221	−	0.999981i	$-0.501977\pi$
$809$	−0.353387	−0.0124244	−0.00621221	+	0.999981i	$0.501977\pi$
$810$	0	0
$811$	22.6257i	0.794497i	0.917711	+	0.397248i	$0.130035\pi$
$811$	22.6257i	0.794497i	−0.917711	+	0.397248i	$0.869965\pi$
$812$	0	0
$813$	33.7540i	1.18380i
$814$	0	0
$815$	21.0304	0.736663
$816$	0	0
$817$	0.927069	0.0324340
$818$	0	0
$819$	− 2.63199i	− 0.0919691i
$820$	0	0
$821$	− 12.9263i	− 0.451130i	−0.974228	−	0.225565i	$-0.927577\pi$
$821$	− 12.9263i	− 0.451130i	0.974228	−	0.225565i	$-0.0724229\pi$
$822$	0	0
$823$	46.3631	1.61611	0.808057	−	0.589104i	$-0.200519\pi$
$823$	46.3631	1.61611	0.808057	+	0.589104i	$0.200519\pi$
$824$	0	0
$825$	4.50591	0.156876
$826$	0	0
$827$	26.0458i	0.905702i	0.891586	+	0.452851i	$0.149593\pi$
$827$	26.0458i	0.905702i	−0.891586	+	0.452851i	$0.850407\pi$
$828$	0	0
$829$	− 10.9969i	− 0.381937i	−0.981596	−	0.190969i	$-0.938837\pi$
$829$	− 10.9969i	− 0.381937i	0.981596	−	0.190969i	$-0.0611628\pi$
$830$	0	0
$831$	41.5611	1.44174
$832$	0	0
$833$	−7.20158	−0.249520
$834$	0	0
$835$	− 29.6782i	− 1.02706i
$836$	0	0
$837$	32.4916i	1.12307i
$838$	0	0
$839$	7.82730	0.270228	0.135114	−	0.990830i	$-0.456860\pi$
$839$	7.82730	0.270228	0.135114	+	0.990830i	$0.456860\pi$
$840$	0	0
$841$	10.0858	0.347787
$842$	0	0
$843$	46.0169i	1.58491i
$844$	0	0
$845$	1.38651i	0.0476974i
$846$	0	0
$847$	24.1437	0.829587
$848$	0	0
$849$	31.9966	1.09812
$850$	0	0
$851$	26.8146i	0.919193i
$852$	0	0
$853$	− 25.6157i	− 0.877064i	−0.898715	−	0.438532i	$-0.855499\pi$
$853$	− 25.6157i	− 0.877064i	0.898715	−	0.438532i	$-0.144501\pi$
$854$	0	0
$855$	−0.609797	−0.0208546
$856$	0	0
$857$	22.4477	0.766799	0.383399	−	0.923583i	$-0.374753\pi$
$857$	22.4477	0.766799	0.383399	+	0.923583i	$0.374753\pi$
$858$	0	0
$859$	− 33.1025i	− 1.12944i	−0.825281	−	0.564722i	$-0.808983\pi$
$859$	− 33.1025i	− 1.12944i	0.825281	−	0.564722i	$-0.191017\pi$
$860$	0	0
$861$	5.33723i	0.181892i
$862$	0	0
$863$	2.22908	0.0758787	0.0379394	−	0.999280i	$-0.487921\pi$
$863$	2.22908	0.0758787	0.0379394	+	0.999280i	$0.487921\pi$
$864$	0	0
$865$	20.2557	0.688715
$866$	0	0
$867$	− 43.6430i	− 1.48219i
$868$	0	0
$869$	− 16.6981i	− 0.566444i
$870$	0	0
$871$	−11.4050	−0.386444
$872$	0	0
$873$	−3.56566	−0.120679
$874$	0	0
$875$	− 27.3548i	− 0.924762i
$876$	0	0
$877$	− 17.3405i	− 0.585548i	−0.956182	−	0.292774i	$-0.905422\pi$
$877$	− 17.3405i	− 0.585548i	0.956182	−	0.292774i	$-0.0945784\pi$
$878$	0	0
$879$	4.63545	0.156350
$880$	0	0
$881$	−41.7063	−1.40512	−0.702561	−	0.711624i	$-0.747960\pi$
$881$	−41.7063	−1.40512	−0.702561	+	0.711624i	$0.747960\pi$
$882$	0	0
$883$	4.06788i	0.136895i	0.997655	+	0.0684475i	$0.0218046\pi$
$883$	4.06788i	0.136895i	−0.997655	+	0.0684475i	$0.978195\pi$
$884$	0	0
$885$	7.36086i	0.247433i
$886$	0	0
$887$	45.8571	1.53973	0.769866	−	0.638205i	$-0.220323\pi$
$887$	45.8571	1.53973	0.769866	+	0.638205i	$0.220323\pi$
$888$	0	0
$889$	−14.7281	−0.493964
$890$	0	0
$891$	− 4.86378i	− 0.162943i
$892$	0	0
$893$	1.76747i	0.0591460i
$894$	0	0
$895$	8.60022	0.287474
$896$	0	0
$897$	4.95819	0.165549
$898$	0	0
$899$	− 24.9942i	− 0.833602i
$900$	0	0
$901$	4.51049i	0.150266i
$902$	0	0
$903$	7.69229	0.255984
$904$	0	0
$905$	29.9851	0.996737
$906$	0	0
$907$	28.6621i	0.951709i	0.879524	+	0.475854i	$0.157861\pi$
$907$	28.6621i	0.951709i	−0.879524	+	0.475854i	$0.842139\pi$
$908$	0	0
$909$	13.1723i	0.436899i
$910$	0	0
$911$	−13.4041	−0.444098	−0.222049	−	0.975035i	$-0.571275\pi$
$911$	−13.4041	−0.444098	−0.222049	+	0.975035i	$0.571275\pi$
$912$	0	0
$913$	−16.2671	−0.538363
$914$	0	0
$915$	9.23334i	0.305245i
$916$	0	0
$917$	37.2304i	1.22946i
$918$	0	0
$919$	−32.7124	−1.07908	−0.539541	−	0.841959i	$-0.681402\pi$
$919$	−32.7124	−1.07908	−0.539541	+	0.841959i	$0.681402\pi$
$920$	0	0
$921$	−24.5210	−0.807993
$922$	0	0
$923$	− 7.66945i	− 0.252443i
$924$	0	0
$925$	− 23.0771i	− 0.758771i
$926$	0	0
$927$	14.7643	0.484924
$928$	0	0
$929$	−0.0167204	−0.000548578 0	−0.000274289	−	1.00000i	$-0.500087\pi$
$929$	−0.0167204	−0.000548578 0	−0.000274289		1.00000i	$0.500087\pi$
$930$	0	0
$931$	0.422152i	0.0138355i
$932$	0	0
$933$	− 40.1373i	− 1.31404i
$934$	0	0
$935$	10.1939	0.333375
$936$	0	0
$937$	12.5139	0.408813	0.204406	−	0.978886i	$-0.434474\pi$
$937$	12.5139	0.408813	0.204406	+	0.978886i	$0.434474\pi$
$938$	0	0
$939$	13.1881i	0.430376i
$940$	0	0
$941$	− 48.0150i	− 1.56524i	−0.622497	−	0.782622i	$-0.713882\pi$
$941$	− 48.0150i	− 1.56524i	0.622497	−	0.782622i	$-0.286118\pi$
$942$	0	0
$943$	5.63591	0.183531
$944$	0	0
$945$	−19.1460	−0.622820
$946$	0	0
$947$	3.81113i	0.123845i	0.998081	+	0.0619225i	$0.0197232\pi$
$947$	3.81113i	0.123845i	−0.998081	+	0.0619225i	$0.980277\pi$
$948$	0	0
$949$	− 7.68795i	− 0.249561i
$950$	0	0
$951$	−28.6629	−0.929457
$952$	0	0
$953$	−0.770354	−0.0249542	−0.0124771	−	0.999922i	$-0.503972\pi$
$953$	−0.770354	−0.0249542	−0.0124771	+	0.999922i	$0.503972\pi$
$954$	0	0
$955$	− 17.4924i	− 0.566041i
$956$	0	0
$957$	6.36744i	0.205830i
$958$	0	0
$959$	−50.8603	−1.64236
$960$	0	0
$961$	2.02854	0.0654367
$962$	0	0
$963$	8.90858i	0.287075i
$964$	0	0
$965$	21.0805i	0.678605i
$966$	0	0
$967$	−2.10043	−0.0675454	−0.0337727	−	0.999430i	$-0.510752\pi$
$967$	−2.10043	−0.0675454	−0.0337727	+	0.999430i	$0.510752\pi$
$968$	0	0
$969$	−3.94002	−0.126572
$970$	0	0
$971$	57.1733i	1.83478i	0.397991	+	0.917389i	$0.369708\pi$
$971$	57.1733i	1.83478i	−0.397991	+	0.917389i	$0.630292\pi$
$972$	0	0
$973$	− 0.768676i	− 0.0246426i
$974$	0	0
$975$	−4.26711	−0.136657
$976$	0	0
$977$	−33.4216	−1.06925	−0.534626	−	0.845089i	$-0.679547\pi$
$977$	−33.4216	−1.06925	−0.534626	+	0.845089i	$0.679547\pi$
$978$	0	0
$979$	3.04639i	0.0973631i
$980$	0	0
$981$	12.6318i	0.403304i
$982$	0	0
$983$	20.3905	0.650357	0.325178	−	0.945653i	$-0.394576\pi$
$983$	20.3905	0.650357	0.325178	+	0.945653i	$0.394576\pi$
$984$	0	0
$985$	32.9280	1.04917
$986$	0	0
$987$	14.6654i	0.466806i
$988$	0	0
$989$	− 8.12277i	− 0.258289i
$990$	0	0
$991$	11.2702	0.358011	0.179006	−	0.983848i	$-0.442712\pi$
$991$	11.2702	0.358011	0.179006	+	0.983848i	$0.442712\pi$
$992$	0	0
$993$	−29.1320	−0.924475
$994$	0	0
$995$	0.0952046i	0.00301819i
$996$	0	0
$997$	27.3662i	0.866696i	0.901227	+	0.433348i	$0.142668\pi$
$997$	27.3662i	0.866696i	−0.901227	+	0.433348i	$0.857332\pi$
$998$	0	0
$999$	−42.3933	−1.34126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.b.c.417.3	✓	8
4.3	odd	2		832.2.b.d.417.6	yes	8
8.3	odd	2		832.2.b.d.417.3	yes	8
8.5	even	2	inner	832.2.b.c.417.6	yes	8
16.3	odd	4		3328.2.a.bi.1.2		4
16.5	even	4		3328.2.a.bj.1.2		4
16.11	odd	4		3328.2.a.bn.1.3		4
16.13	even	4		3328.2.a.bm.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.c.417.3	✓	8	1.1	even	1	trivial
832.2.b.c.417.6	yes	8	8.5	even	2	inner
832.2.b.d.417.3	yes	8	8.3	odd	2
832.2.b.d.417.6	yes	8	4.3	odd	2
3328.2.a.bi.1.2		4	16.3	odd	4
3328.2.a.bj.1.2		4	16.5	even	4
3328.2.a.bm.1.3		4	16.13	even	4
3328.2.a.bn.1.3		4	16.11	odd	4

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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.38651i q^{3} -1.38651i q^{5} +2.44247 q^{7} +1.07759 q^{9} +1.05596i q^{11} -1.00000i q^{13} -1.92241 q^{15} +6.96254 q^{17} -0.408139i q^{19} -3.38651i q^{21} -3.57603 q^{23} +3.07759 q^{25} -5.65362i q^{27} +4.34904i q^{29} -5.74705 q^{31} +1.46410 q^{33} -3.38651i q^{35} -7.49843i q^{37} -1.38651 q^{39} -1.57603 q^{41} +2.27145i q^{43} -1.49409i q^{45} -4.33055 q^{47} -1.03433 q^{49} -9.65362i q^{51} +0.647823i q^{53} +1.46410 q^{55} -0.565889 q^{57} -3.82898i q^{59} -4.80301i q^{61} +2.63199 q^{63} -1.38651 q^{65} -11.4050i q^{67} +4.95819i q^{69} +7.66945 q^{71} +7.68795 q^{73} -4.26711i q^{75} +2.57916i q^{77} -15.8131 q^{79} -4.60602 q^{81} +15.4050i q^{83} -9.65362i q^{85} +6.02999 q^{87} +2.88494 q^{89} -2.44247i q^{91} +7.96833i q^{93} -0.565889 q^{95} -3.30892 q^{97} +1.13790i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{9} - 36 q^{15} - 4 q^{17} + 24 q^{23} + 4 q^{25} - 20 q^{31} - 16 q^{33} - 4 q^{39} + 40 q^{41} - 40 q^{47} - 4 q^{49} - 16 q^{55} - 8 q^{57} - 44 q^{63} - 4 q^{65} + 56 q^{71} - 16 q^{73} - 32 q^{79}+ \cdots - 40 q^{97}+O(q^{100}) \) 8 * q - 12 * q^9 - 36 * q^15 - 4 * q^17 + 24 * q^23 + 4 * q^25 - 20 * q^31 - 16 * q^33 - 4 * q^39 + 40 * q^41 - 40 * q^47 - 4 * q^49 - 16 * q^55 - 8 * q^57 - 44 * q^63 - 4 * q^65 + 56 * q^71 - 16 * q^73 - 32 * q^79 + 40 * q^81 + 24 * q^87 - 16 * q^89 - 8 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more