LMFDB - Embedded newform 1792.2.b.p.897.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(897,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1792.897");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			897.6
Root			$1.40680 + 0.144584i$ of defining polynomial
Character	$\chi$	$=$	1792.897
Dual form			1792.2.b.p.897.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.10278i q^{3} -2.52444i q^{5} +1.00000 q^{7} -6.62721 q^{9} +O(q^{10})$	$q+3.10278i q^{3} -2.52444i q^{5} +1.00000 q^{7} -6.62721 q^{9} +3.62721i q^{11} -4.72999i q^{13} +7.83276 q^{15} +4.20555 q^{17} +7.10278i q^{19} +3.10278i q^{21} +0.578337 q^{23} -1.37279 q^{25} -11.2544i q^{27} +8.20555i q^{29} -5.04888 q^{31} -11.2544 q^{33} -2.52444i q^{35} +3.04888i q^{37} +14.6761 q^{39} +0.205550 q^{41} +4.78389i q^{43} +16.7300i q^{45} -6.20555 q^{47} +1.00000 q^{49} +13.0489i q^{51} +2.00000i q^{53} +9.15667 q^{55} -22.0383 q^{57} +12.5628i q^{59} +10.5244i q^{61} -6.62721 q^{63} -11.9406 q^{65} -6.57834i q^{67} +1.79445i q^{69} +9.25443 q^{73} -4.25945i q^{75} +3.62721i q^{77} -5.15667 q^{79} +15.0383 q^{81} -5.94610i q^{83} -10.6167i q^{85} -25.4600 q^{87} +10.4111 q^{89} -4.72999i q^{91} -15.6655i q^{93} +17.9305 q^{95} -9.36222 q^{97} -24.0383i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{7} - 14 q^{9}+O(q^{10}) $ 6 * q + 6 * q^7 - 14 * q^9	$ 6 q + 6 q^{7} - 14 q^{9} - 8 q^{15} - 4 q^{17} - 34 q^{25} - 8 q^{31} - 16 q^{33} + 40 q^{39} - 28 q^{41} - 8 q^{47} + 6 q^{49} + 48 q^{55} - 48 q^{57} - 14 q^{63} - 32 q^{65} + 4 q^{73} - 24 q^{79} + 6 q^{81} - 72 q^{87} + 4 q^{89} + 8 q^{95} - 20 q^{97}+O(q^{100}) $ 6 * q + 6 * q^7 - 14 * q^9 - 8 * q^15 - 4 * q^17 - 34 * q^25 - 8 * q^31 - 16 * q^33 + 40 * q^39 - 28 * q^41 - 8 * q^47 + 6 * q^49 + 48 * q^55 - 48 * q^57 - 14 * q^63 - 32 * q^65 + 4 * q^73 - 24 * q^79 + 6 * q^81 - 72 * q^87 + 4 * q^89 + 8 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	3.10278i	1.79139i	0.444671	+	0.895694i	$0.353321\pi$
$3$	3.10278i	1.79139i	−0.444671	+	0.895694i	$0.646679\pi$
$4$	0	0
$5$	− 2.52444i	− 1.12896i	−0.825446	−	0.564481i	$-0.809076\pi$
$5$	− 2.52444i	− 1.12896i	0.825446	−	0.564481i	$-0.190924\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−6.62721	−2.20907
$10$	0	0
$11$	3.62721i	1.09365i	0.837248	+	0.546823i	$0.184163\pi$
$11$	3.62721i	1.09365i	−0.837248	+	0.546823i	$0.815837\pi$
$12$	0	0
$13$	− 4.72999i	− 1.31186i	−0.754821	−	0.655931i	$-0.772276\pi$
$13$	− 4.72999i	− 1.31186i	0.754821	−	0.655931i	$-0.227724\pi$
$14$	0	0
$15$	7.83276	2.02241
$16$	0	0
$17$	4.20555	1.02000	0.509998	−	0.860176i	$-0.329646\pi$
$17$	4.20555	1.02000	0.509998	+	0.860176i	$0.329646\pi$
$18$	0	0
$19$	7.10278i	1.62949i	0.579821	+	0.814744i	$0.303123\pi$
$19$	7.10278i	1.62949i	−0.579821	+	0.814744i	$0.696877\pi$
$20$	0	0
$21$	3.10278i	0.677081i
$22$	0	0
$23$	0.578337	0.120592	0.0602958	−	0.998181i	$-0.480796\pi$
$23$	0.578337	0.120592	0.0602958	+	0.998181i	$0.480796\pi$
$24$	0	0
$25$	−1.37279	−0.274557
$26$	0	0
$27$	− 11.2544i	− 2.16592i
$28$	0	0
$29$	8.20555i	1.52373i	0.647734	+	0.761866i	$0.275717\pi$
$29$	8.20555i	1.52373i	−0.647734	+	0.761866i	$0.724283\pi$
$30$	0	0
$31$	−5.04888	−0.906805	−0.453402	−	0.891306i	$-0.649790\pi$
$31$	−5.04888	−0.906805	−0.453402	+	0.891306i	$0.649790\pi$
$32$	0	0
$33$	−11.2544	−1.95914
$34$	0	0
$35$	− 2.52444i	− 0.426708i
$36$	0	0
$37$	3.04888i	0.501232i	0.968087	+	0.250616i	$0.0806332\pi$
$37$	3.04888i	0.501232i	−0.968087	+	0.250616i	$0.919367\pi$
$38$	0	0
$39$	14.6761	2.35006
$40$	0	0
$41$	0.205550	0.0321015	0.0160508	−	0.999871i	$-0.494891\pi$
$41$	0.205550	0.0321015	0.0160508	+	0.999871i	$0.494891\pi$
$42$	0	0
$43$	4.78389i	0.729536i	0.931098	+	0.364768i	$0.118852\pi$
$43$	4.78389i	0.729536i	−0.931098	+	0.364768i	$0.881148\pi$
$44$	0	0
$45$	16.7300i	2.49396i
$46$	0	0
$47$	−6.20555	−0.905173	−0.452586	−	0.891721i	$-0.649499\pi$
$47$	−6.20555	−0.905173	−0.452586	+	0.891721i	$0.649499\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	13.0489i	1.82721i
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	9.15667	1.23469
$56$	0	0
$57$	−22.0383	−2.91905
$58$	0	0
$59$	12.5628i	1.63553i	0.575552	+	0.817765i	$0.304787\pi$
$59$	12.5628i	1.63553i	−0.575552	+	0.817765i	$0.695213\pi$
$60$	0	0
$61$	10.5244i	1.34752i	0.738952	+	0.673758i	$0.235321\pi$
$61$	10.5244i	1.34752i	−0.738952	+	0.673758i	$0.764679\pi$
$62$	0	0
$63$	−6.62721	−0.834950
$64$	0	0
$65$	−11.9406	−1.48104
$66$	0	0
$67$	− 6.57834i	− 0.803672i	−0.915712	−	0.401836i	$-0.868372\pi$
$67$	− 6.57834i	− 0.803672i	0.915712	−	0.401836i	$-0.131628\pi$
$68$	0	0
$69$	1.79445i	0.216026i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	9.25443	1.08315	0.541574	−	0.840653i	$-0.317828\pi$
$73$	9.25443	1.08315	0.541574	+	0.840653i	$0.317828\pi$
$74$	0	0
$75$	− 4.25945i	− 0.491839i
$76$	0	0
$77$	3.62721i	0.413359i
$78$	0	0
$79$	−5.15667	−0.580171	−0.290086	−	0.957001i	$-0.593684\pi$
$79$	−5.15667	−0.580171	−0.290086	+	0.957001i	$0.593684\pi$
$80$	0	0
$81$	15.0383	1.67092
$82$	0	0
$83$	− 5.94610i	− 0.652669i	−0.945254	−	0.326335i	$-0.894186\pi$
$83$	− 5.94610i	− 0.652669i	0.945254	−	0.326335i	$-0.105814\pi$
$84$	0	0
$85$	− 10.6167i	− 1.15154i
$86$	0	0
$87$	−25.4600	−2.72960
$88$	0	0
$89$	10.4111	1.10357	0.551787	−	0.833985i	$-0.313946\pi$
$89$	10.4111	1.10357	0.551787	+	0.833985i	$0.313946\pi$
$90$	0	0
$91$	− 4.72999i	− 0.495837i
$92$	0	0
$93$	− 15.6655i	− 1.62444i
$94$	0	0
$95$	17.9305	1.83963
$96$	0	0
$97$	−9.36222	−0.950590	−0.475295	−	0.879827i	$-0.657659\pi$
$97$	−9.36222	−0.950590	−0.475295	+	0.879827i	$0.657659\pi$
$98$	0	0
$99$	− 24.0383i	− 2.41594i
$100$	0	0
$101$	− 4.31889i	− 0.429745i	−0.976642	−	0.214873i	$-0.931066\pi$
$101$	− 4.31889i	− 0.429745i	0.976642	−	0.214873i	$-0.0689337\pi$
$102$	0	0
$103$	5.04888	0.497481	0.248740	−	0.968570i	$-0.419983\pi$
$103$	5.04888	0.497481	0.248740	+	0.968570i	$0.419983\pi$
$104$	0	0
$105$	7.83276	0.764399
$106$	0	0
$107$	− 9.83276i	− 0.950569i	−0.879832	−	0.475285i	$-0.842345\pi$
$107$	− 9.83276i	− 0.950569i	0.879832	−	0.475285i	$-0.157655\pi$
$108$	0	0
$109$	19.4600i	1.86393i	0.362551	+	0.931964i	$0.381906\pi$
$109$	19.4600i	1.86393i	−0.362551	+	0.931964i	$0.618094\pi$
$110$	0	0
$111$	−9.45998	−0.897901
$112$	0	0
$113$	8.78389	0.826319	0.413159	−	0.910659i	$-0.364425\pi$
$113$	8.78389	0.826319	0.413159	+	0.910659i	$0.364425\pi$
$114$	0	0
$115$	− 1.45998i	− 0.136143i
$116$	0	0
$117$	31.3466i	2.89800i
$118$	0	0
$119$	4.20555	0.385522
$120$	0	0
$121$	−2.15667	−0.196061
$122$	0	0
$123$	0.637776i	0.0575063i
$124$	0	0
$125$	− 9.15667i	− 0.818998i
$126$	0	0
$127$	−14.6761	−1.30229	−0.651146	−	0.758952i	$-0.725712\pi$
$127$	−14.6761	−1.30229	−0.651146	+	0.758952i	$0.725712\pi$
$128$	0	0
$129$	−14.8433	−1.30688
$130$	0	0
$131$	4.15165i	0.362731i	0.983416	+	0.181366i	$0.0580518\pi$
$131$	4.15165i	0.362731i	−0.983416	+	0.181366i	$0.941948\pi$
$132$	0	0
$133$	7.10278i	0.615889i
$134$	0	0
$135$	−28.4111	−2.44524
$136$	0	0
$137$	−5.25443	−0.448916	−0.224458	−	0.974484i	$-0.572061\pi$
$137$	−5.25443	−0.448916	−0.224458	+	0.974484i	$0.572061\pi$
$138$	0	0
$139$	− 8.89722i	− 0.754653i	−0.926080	−	0.377326i	$-0.876843\pi$
$139$	− 8.89722i	− 0.754653i	0.926080	−	0.377326i	$-0.123157\pi$
$140$	0	0
$141$	− 19.2544i	− 1.62152i
$142$	0	0
$143$	17.1567	1.43471
$144$	0	0
$145$	20.7144	1.72024
$146$	0	0
$147$	3.10278i	0.255913i
$148$	0	0
$149$	− 21.6655i	− 1.77491i	−0.460895	−	0.887455i	$-0.652472\pi$
$149$	− 21.6655i	− 1.77491i	0.460895	−	0.887455i	$-0.347528\pi$
$150$	0	0
$151$	−4.98944	−0.406035	−0.203017	−	0.979175i	$-0.565075\pi$
$151$	−4.98944	−0.406035	−0.203017	+	0.979175i	$0.565075\pi$
$152$	0	0
$153$	−27.8711	−2.25324
$154$	0	0
$155$	12.7456i	1.02375i
$156$	0	0
$157$	20.0922i	1.60353i	0.597637	+	0.801767i	$0.296106\pi$
$157$	20.0922i	1.60353i	−0.597637	+	0.801767i	$0.703894\pi$
$158$	0	0
$159$	−6.20555	−0.492132
$160$	0	0
$161$	0.578337	0.0455793
$162$	0	0
$163$	16.0383i	1.25622i	0.778126	+	0.628109i	$0.216171\pi$
$163$	16.0383i	1.25622i	−0.778126	+	0.628109i	$0.783829\pi$
$164$	0	0
$165$	28.4111i	2.21180i
$166$	0	0
$167$	1.79445	0.138859	0.0694294	−	0.997587i	$-0.477882\pi$
$167$	1.79445	0.138859	0.0694294	+	0.997587i	$0.477882\pi$
$168$	0	0
$169$	−9.37279	−0.720984
$170$	0	0
$171$	− 47.0716i	− 3.59966i
$172$	0	0
$173$	− 22.8277i	− 1.73556i	−0.496948	−	0.867780i	$-0.665546\pi$
$173$	− 22.8277i	− 1.73556i	0.496948	−	0.867780i	$-0.334454\pi$
$174$	0	0
$175$	−1.37279	−0.103773
$176$	0	0
$177$	−38.9794	−2.92987
$178$	0	0
$179$	3.51941i	0.263053i	0.991313	+	0.131527i	$0.0419879\pi$
$179$	3.51941i	0.263053i	−0.991313	+	0.131527i	$0.958012\pi$
$180$	0	0
$181$	10.9355i	0.812832i	0.913688	+	0.406416i	$0.133222\pi$
$181$	10.9355i	0.812832i	−0.913688	+	0.406416i	$0.866778\pi$
$182$	0	0
$183$	−32.6550	−2.41392
$184$	0	0
$185$	7.69670	0.565872
$186$	0	0
$187$	15.2544i	1.11551i
$188$	0	0
$189$	− 11.2544i	− 0.818639i
$190$	0	0
$191$	0.745574	0.0539478	0.0269739	−	0.999636i	$-0.491413\pi$
$191$	0.745574	0.0539478	0.0269739	+	0.999636i	$0.491413\pi$
$192$	0	0
$193$	−14.1361	−1.01754	−0.508768	−	0.860904i	$-0.669899\pi$
$193$	−14.1361	−1.01754	−0.508768	+	0.860904i	$0.669899\pi$
$194$	0	0
$195$	− 37.0489i	− 2.65313i
$196$	0	0
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	0	0
$199$	−0.637776	−0.0452107	−0.0226054	−	0.999744i	$-0.507196\pi$
$199$	−0.637776	−0.0452107	−0.0226054	+	0.999744i	$0.507196\pi$
$200$	0	0
$201$	20.4111	1.43969
$202$	0	0
$203$	8.20555i	0.575917i
$204$	0	0
$205$	− 0.518898i	− 0.0362414i
$206$	0	0
$207$	−3.83276	−0.266395
$208$	0	0
$209$	−25.7633	−1.78208
$210$	0	0
$211$	− 0.676089i	− 0.0465439i	−0.999729	−	0.0232719i	$-0.992592\pi$
$211$	− 0.676089i	− 0.0465439i	0.999729	−	0.0232719i	$-0.00740836\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.0766	0.823619
$216$	0	0
$217$	−5.04888	−0.342740
$218$	0	0
$219$	28.7144i	1.94034i
$220$	0	0
$221$	− 19.8922i	− 1.33809i
$222$	0	0
$223$	23.6655	1.58476	0.792380	−	0.610027i	$-0.208842\pi$
$223$	23.6655	1.58476	0.792380	+	0.610027i	$0.208842\pi$
$224$	0	0
$225$	9.09775	0.606517
$226$	0	0
$227$	− 8.89722i	− 0.590530i	−0.955415	−	0.295265i	$-0.904592\pi$
$227$	− 8.89722i	− 0.590530i	0.955415	−	0.295265i	$-0.0954079\pi$
$228$	0	0
$229$	− 7.68111i	− 0.507582i	−0.967259	−	0.253791i	$-0.918322\pi$
$229$	− 7.68111i	− 0.507582i	0.967259	−	0.253791i	$-0.0816776\pi$
$230$	0	0
$231$	−11.2544	−0.740487
$232$	0	0
$233$	−5.66553	−0.371161	−0.185580	−	0.982629i	$-0.559417\pi$
$233$	−5.66553	−0.371161	−0.185580	+	0.982629i	$0.559417\pi$
$234$	0	0
$235$	15.6655i	1.02191i
$236$	0	0
$237$	− 16.0000i	− 1.03931i
$238$	0	0
$239$	−11.4217	−0.738806	−0.369403	−	0.929269i	$-0.620438\pi$
$239$	−11.4217	−0.738806	−0.369403	+	0.929269i	$0.620438\pi$
$240$	0	0
$241$	17.5577	1.13099	0.565496	−	0.824751i	$-0.308685\pi$
$241$	17.5577	1.13099	0.565496	+	0.824751i	$0.308685\pi$
$242$	0	0
$243$	12.8972i	0.827357i
$244$	0	0
$245$	− 2.52444i	− 0.161280i
$246$	0	0
$247$	33.5960	2.13766
$248$	0	0
$249$	18.4494	1.16918
$250$	0	0
$251$	12.1517i	0.767005i	0.923540	+	0.383503i	$0.125282\pi$
$251$	12.1517i	0.767005i	−0.923540	+	0.383503i	$0.874718\pi$
$252$	0	0
$253$	2.09775i	0.131885i
$254$	0	0
$255$	32.9411	2.06285
$256$	0	0
$257$	6.41110	0.399913	0.199957	−	0.979805i	$-0.435920\pi$
$257$	6.41110	0.399913	0.199957	+	0.979805i	$0.435920\pi$
$258$	0	0
$259$	3.04888i	0.189448i
$260$	0	0
$261$	− 54.3799i	− 3.36603i
$262$	0	0
$263$	10.3133	0.635948	0.317974	−	0.948099i	$-0.396997\pi$
$263$	10.3133	0.635948	0.317974	+	0.948099i	$0.396997\pi$
$264$	0	0
$265$	5.04888	0.310150
$266$	0	0
$267$	32.3033i	1.97693i
$268$	0	0
$269$	15.6811i	0.956094i	0.878334	+	0.478047i	$0.158655\pi$
$269$	15.6811i	0.956094i	−0.878334	+	0.478047i	$0.841345\pi$
$270$	0	0
$271$	25.7633	1.56501	0.782504	−	0.622646i	$-0.213942\pi$
$271$	25.7633	1.56501	0.782504	+	0.622646i	$0.213942\pi$
$272$	0	0
$273$	14.6761	0.888237
$274$	0	0
$275$	− 4.97939i	− 0.300269i
$276$	0	0
$277$	− 17.2544i	− 1.03672i	−0.855163	−	0.518359i	$-0.826543\pi$
$277$	− 17.2544i	− 1.03672i	0.855163	−	0.518359i	$-0.173457\pi$
$278$	0	0
$279$	33.4600	2.00320
$280$	0	0
$281$	32.5089	1.93932	0.969658	−	0.244466i	$-0.0786128\pi$
$281$	32.5089	1.93932	0.969658	+	0.244466i	$0.0786128\pi$
$282$	0	0
$283$	− 22.9950i	− 1.36691i	−0.729993	−	0.683455i	$-0.760477\pi$
$283$	− 22.9950i	− 1.36691i	0.729993	−	0.683455i	$-0.239523\pi$
$284$	0	0
$285$	55.6344i	3.29549i
$286$	0	0
$287$	0.205550	0.0121332
$288$	0	0
$289$	0.686652	0.0403913
$290$	0	0
$291$	− 29.0489i	− 1.70288i
$292$	0	0
$293$	− 8.72999i	− 0.510011i	−0.966940	−	0.255006i	$-0.917923\pi$
$293$	− 8.72999i	− 0.510011i	0.966940	−	0.255006i	$-0.0820773\pi$
$294$	0	0
$295$	31.7139	1.84645
$296$	0	0
$297$	40.8222	2.36874
$298$	0	0
$299$	− 2.73553i	− 0.158200i
$300$	0	0
$301$	4.78389i	0.275739i
$302$	0	0
$303$	13.4005	0.769841
$304$	0	0
$305$	26.5683	1.52130
$306$	0	0
$307$	26.6605i	1.52160i	0.648989	+	0.760798i	$0.275192\pi$
$307$	26.6605i	1.52160i	−0.648989	+	0.760798i	$0.724808\pi$
$308$	0	0
$309$	15.6655i	0.891181i
$310$	0	0
$311$	10.0978	0.572591	0.286295	−	0.958141i	$-0.407576\pi$
$311$	10.0978	0.572591	0.286295	+	0.958141i	$0.407576\pi$
$312$	0	0
$313$	0.205550	0.0116184	0.00580919	−	0.999983i	$-0.498151\pi$
$313$	0.205550	0.0116184	0.00580919	+	0.999983i	$0.498151\pi$
$314$	0	0
$315$	16.7300i	0.942628i
$316$	0	0
$317$	− 30.4111i	− 1.70806i	−0.520226	−	0.854029i	$-0.674152\pi$
$317$	− 30.4111i	− 1.70806i	0.520226	−	0.854029i	$-0.325848\pi$
$318$	0	0
$319$	−29.7633	−1.66642
$320$	0	0
$321$	30.5089	1.70284
$322$	0	0
$323$	29.8711i	1.66207i
$324$	0	0
$325$	6.49327i	0.360182i
$326$	0	0
$327$	−60.3799	−3.33902
$328$	0	0
$329$	−6.20555	−0.342123
$330$	0	0
$331$	2.47054i	0.135793i	0.997692	+	0.0678965i	$0.0216288\pi$
$331$	2.47054i	0.135793i	−0.997692	+	0.0678965i	$0.978371\pi$
$332$	0	0
$333$	− 20.2056i	− 1.10726i
$334$	0	0
$335$	−16.6066	−0.907316
$336$	0	0
$337$	−11.2927	−0.615155	−0.307577	−	0.951523i	$-0.599518\pi$
$337$	−11.2927	−0.615155	−0.307577	+	0.951523i	$0.599518\pi$
$338$	0	0
$339$	27.2544i	1.48026i
$340$	0	0
$341$	− 18.3133i	− 0.991723i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	4.52998	0.243886
$346$	0	0
$347$	2.68614i	0.144199i	0.997397	+	0.0720997i	$0.0229700\pi$
$347$	2.68614i	0.144199i	−0.997397	+	0.0720997i	$0.977030\pi$
$348$	0	0
$349$	17.4756i	0.935445i	0.883875	+	0.467723i	$0.154925\pi$
$349$	17.4756i	0.935445i	−0.883875	+	0.467723i	$0.845075\pi$
$350$	0	0
$351$	−53.2333	−2.84138
$352$	0	0
$353$	1.05892	0.0563608	0.0281804	−	0.999603i	$-0.491029\pi$
$353$	1.05892	0.0563608	0.0281804	+	0.999603i	$0.491029\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13.0489i	0.690620i
$358$	0	0
$359$	−11.8328	−0.624509	−0.312255	−	0.949998i	$-0.601084\pi$
$359$	−11.8328	−0.624509	−0.312255	+	0.949998i	$0.601084\pi$
$360$	0	0
$361$	−31.4494	−1.65523
$362$	0	0
$363$	− 6.69167i	− 0.351222i
$364$	0	0
$365$	− 23.3622i	− 1.22283i
$366$	0	0
$367$	19.2544	1.00507	0.502536	−	0.864556i	$-0.332400\pi$
$367$	19.2544	1.00507	0.502536	+	0.864556i	$0.332400\pi$
$368$	0	0
$369$	−1.36222	−0.0709146
$370$	0	0
$371$	2.00000i	0.103835i
$372$	0	0
$373$	9.66553i	0.500462i	0.968186	+	0.250231i	$0.0805066\pi$
$373$	9.66553i	0.500462i	−0.968186	+	0.250231i	$0.919493\pi$
$374$	0	0
$375$	28.4111	1.46714
$376$	0	0
$377$	38.8122	1.99893
$378$	0	0
$379$	− 12.0383i	− 0.618367i	−0.951002	−	0.309183i	$-0.899944\pi$
$379$	− 12.0383i	− 0.618367i	0.951002	−	0.309183i	$-0.100056\pi$
$380$	0	0
$381$	− 45.5366i	− 2.33291i
$382$	0	0
$383$	31.0278	1.58544	0.792722	−	0.609583i	$-0.208663\pi$
$383$	31.0278	1.58544	0.792722	+	0.609583i	$0.208663\pi$
$384$	0	0
$385$	9.15667	0.466667
$386$	0	0
$387$	− 31.7038i	− 1.61160i
$388$	0	0
$389$	13.1466i	0.666560i	0.942828	+	0.333280i	$0.108156\pi$
$389$	13.1466i	0.666560i	−0.942828	+	0.333280i	$0.891844\pi$
$390$	0	0
$391$	2.43223	0.123003
$392$	0	0
$393$	−12.8816	−0.649793
$394$	0	0
$395$	13.0177i	0.654992i
$396$	0	0
$397$	1.47556i	0.0740563i	0.999314	+	0.0370282i	$0.0117891\pi$
$397$	1.47556i	0.0740563i	−0.999314	+	0.0370282i	$0.988211\pi$
$398$	0	0
$399$	−22.0383	−1.10330
$400$	0	0
$401$	−12.9794	−0.648160	−0.324080	−	0.946030i	$-0.605055\pi$
$401$	−12.9794	−0.648160	−0.324080	+	0.946030i	$0.605055\pi$
$402$	0	0
$403$	23.8811i	1.18960i
$404$	0	0
$405$	− 37.9633i	− 1.88641i
$406$	0	0
$407$	−11.0589	−0.548170
$408$	0	0
$409$	1.36222	0.0673577	0.0336788	−	0.999433i	$-0.489278\pi$
$409$	1.36222	0.0673577	0.0336788	+	0.999433i	$0.489278\pi$
$410$	0	0
$411$	− 16.3033i	− 0.804183i
$412$	0	0
$413$	12.5628i	0.618173i
$414$	0	0
$415$	−15.0106	−0.736840
$416$	0	0
$417$	27.6061	1.35188
$418$	0	0
$419$	− 25.3083i	− 1.23639i	−0.786024	−	0.618196i	$-0.787864\pi$
$419$	− 25.3083i	− 1.23639i	0.786024	−	0.618196i	$-0.212136\pi$
$420$	0	0
$421$	1.05892i	0.0516087i	0.999667	+	0.0258044i	$0.00821469\pi$
$421$	1.05892i	0.0516087i	−0.999667	+	0.0258044i	$0.991785\pi$
$422$	0	0
$423$	41.1255	1.99959
$424$	0	0
$425$	−5.77332	−0.280047
$426$	0	0
$427$	10.5244i	0.509313i
$428$	0	0
$429$	53.2333i	2.57013i
$430$	0	0
$431$	12.2439	0.589766	0.294883	−	0.955533i	$-0.404719\pi$
$431$	12.2439	0.589766	0.294883	+	0.955533i	$0.404719\pi$
$432$	0	0
$433$	37.0278	1.77944	0.889720	−	0.456507i	$-0.150899\pi$
$433$	37.0278	1.77944	0.889720	+	0.456507i	$0.150899\pi$
$434$	0	0
$435$	64.2721i	3.08161i
$436$	0	0
$437$	4.10780i	0.196503i
$438$	0	0
$439$	8.33447	0.397783	0.198891	−	0.980022i	$-0.436266\pi$
$439$	8.33447	0.397783	0.198891	+	0.980022i	$0.436266\pi$
$440$	0	0
$441$	−6.62721	−0.315582
$442$	0	0
$443$	− 27.5960i	− 1.31113i	−0.755140	−	0.655564i	$-0.772431\pi$
$443$	− 27.5960i	− 1.31113i	0.755140	−	0.655564i	$-0.227569\pi$
$444$	0	0
$445$	− 26.2822i	− 1.24589i
$446$	0	0
$447$	67.2233	3.17955
$448$	0	0
$449$	−1.66553	−0.0786010	−0.0393005	−	0.999227i	$-0.512513\pi$
$449$	−1.66553	−0.0786010	−0.0393005	+	0.999227i	$0.512513\pi$
$450$	0	0
$451$	0.745574i	0.0351077i
$452$	0	0
$453$	− 15.4811i	− 0.727366i
$454$	0	0
$455$	−11.9406	−0.559782
$456$	0	0
$457$	−25.7250	−1.20336	−0.601682	−	0.798736i	$-0.705502\pi$
$457$	−25.7250	−1.20336	−0.601682	+	0.798736i	$0.705502\pi$
$458$	0	0
$459$	− 47.3311i	− 2.20922i
$460$	0	0
$461$	− 12.4267i	− 0.578768i	−0.957213	−	0.289384i	$-0.906549\pi$
$461$	− 12.4267i	− 0.578768i	0.957213	−	0.289384i	$-0.0934505\pi$
$462$	0	0
$463$	−4.33447	−0.201440	−0.100720	−	0.994915i	$-0.532115\pi$
$463$	−4.33447	−0.201440	−0.100720	+	0.994915i	$0.532115\pi$
$464$	0	0
$465$	−39.5466	−1.83393
$466$	0	0
$467$	10.6917i	0.494752i	0.968920	+	0.247376i	$0.0795682\pi$
$467$	10.6917i	0.494752i	−0.968920	+	0.247376i	$0.920432\pi$
$468$	0	0
$469$	− 6.57834i	− 0.303759i
$470$	0	0
$471$	−62.3416	−2.87255
$472$	0	0
$473$	−17.3522	−0.797854
$474$	0	0
$475$	− 9.75060i	− 0.447388i
$476$	0	0
$477$	− 13.2544i	− 0.606878i
$478$	0	0
$479$	−2.95112	−0.134840	−0.0674202	−	0.997725i	$-0.521477\pi$
$479$	−2.95112	−0.134840	−0.0674202	+	0.997725i	$0.521477\pi$
$480$	0	0
$481$	14.4211	0.657548
$482$	0	0
$483$	1.79445i	0.0816503i
$484$	0	0
$485$	23.6344i	1.07318i
$486$	0	0
$487$	−5.93051	−0.268737	−0.134369	−	0.990931i	$-0.542901\pi$
$487$	−5.93051	−0.268737	−0.134369	+	0.990931i	$0.542901\pi$
$488$	0	0
$489$	−49.7633	−2.25037
$490$	0	0
$491$	− 21.3028i	− 0.961381i	−0.876890	−	0.480691i	$-0.840386\pi$
$491$	− 21.3028i	− 0.961381i	0.876890	−	0.480691i	$-0.159614\pi$
$492$	0	0
$493$	34.5089i	1.55420i
$494$	0	0
$495$	−60.6832	−2.72751
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.6761i	1.64185i	0.571038	+	0.820924i	$0.306541\pi$
$499$	36.6761i	1.64185i	−0.571038	+	0.820924i	$0.693459\pi$
$500$	0	0
$501$	5.56777i	0.248750i
$502$	0	0
$503$	20.7456	0.924999	0.462500	−	0.886619i	$-0.346953\pi$
$503$	20.7456	0.924999	0.462500	+	0.886619i	$0.346953\pi$
$504$	0	0
$505$	−10.9028	−0.485167
$506$	0	0
$507$	− 29.0816i	− 1.29156i
$508$	0	0
$509$	− 21.0333i	− 0.932284i	−0.884710	−	0.466142i	$-0.845644\pi$
$509$	− 21.0333i	− 0.932284i	0.884710	−	0.466142i	$-0.154356\pi$
$510$	0	0
$511$	9.25443	0.409392
$512$	0	0
$513$	79.9377	3.52933
$514$	0	0
$515$	− 12.7456i	− 0.561637i
$516$	0	0
$517$	− 22.5089i	− 0.989938i
$518$	0	0
$519$	70.8293	3.10906
$520$	0	0
$521$	12.7355	0.557954	0.278977	−	0.960298i	$-0.410005\pi$
$521$	12.7355	0.557954	0.278977	+	0.960298i	$0.410005\pi$
$522$	0	0
$523$	0.151651i	0.00663123i	0.999995	+	0.00331562i	$0.00105540\pi$
$523$	0.151651i	0.00663123i	−0.999995	+	0.00331562i	$0.998945\pi$
$524$	0	0
$525$	− 4.25945i	− 0.185898i
$526$	0	0
$527$	−21.2333	−0.924937
$528$	0	0
$529$	−22.6655	−0.985458
$530$	0	0
$531$	− 83.2560i	− 3.61300i
$532$	0	0
$533$	− 0.972250i	− 0.0421128i
$534$	0	0
$535$	−24.8222	−1.07316
$536$	0	0
$537$	−10.9200	−0.471231
$538$	0	0
$539$	3.62721i	0.156235i
$540$	0	0
$541$	11.5678i	0.497337i	0.968589	+	0.248669i	$0.0799930\pi$
$541$	11.5678i	0.497337i	−0.968589	+	0.248669i	$0.920007\pi$
$542$	0	0
$543$	−33.9305	−1.45610
$544$	0	0
$545$	49.1255	2.10431
$546$	0	0
$547$	− 10.0594i	− 0.430111i	−0.976602	−	0.215055i	$-0.931007\pi$
$547$	− 10.0594i	− 0.430111i	0.976602	−	0.215055i	$-0.0689932\pi$
$548$	0	0
$549$	− 69.7477i	− 2.97676i
$550$	0	0
$551$	−58.2822	−2.48290
$552$	0	0
$553$	−5.15667	−0.219284
$554$	0	0
$555$	23.8811i	1.01370i
$556$	0	0
$557$	4.50885i	0.191046i	0.995427	+	0.0955231i	$0.0304524\pi$
$557$	4.50885i	0.191046i	−0.995427	+	0.0955231i	$0.969548\pi$
$558$	0	0
$559$	22.6277	0.957051
$560$	0	0
$561$	−47.3311	−1.99832
$562$	0	0
$563$	24.9739i	1.05252i	0.850323	+	0.526261i	$0.176407\pi$
$563$	24.9739i	1.05252i	−0.850323	+	0.526261i	$0.823593\pi$
$564$	0	0
$565$	− 22.1744i	− 0.932883i
$566$	0	0
$567$	15.0383	0.631550
$568$	0	0
$569$	21.9406	0.919796	0.459898	−	0.887972i	$-0.347886\pi$
$569$	21.9406	0.919796	0.459898	+	0.887972i	$0.347886\pi$
$570$	0	0
$571$	− 19.2161i	− 0.804169i	−0.915602	−	0.402085i	$-0.868286\pi$
$571$	− 19.2161i	− 0.804169i	0.915602	−	0.402085i	$-0.131714\pi$
$572$	0	0
$573$	2.31335i	0.0966415i
$574$	0	0
$575$	−0.793934	−0.0331093
$576$	0	0
$577$	−0.432226	−0.0179938	−0.00899689	−	0.999960i	$-0.502864\pi$
$577$	−0.432226	−0.0179938	−0.00899689	+	0.999960i	$0.502864\pi$
$578$	0	0
$579$	− 43.8610i	− 1.82280i
$580$	0	0
$581$	− 5.94610i	− 0.246686i
$582$	0	0
$583$	−7.25443	−0.300448
$584$	0	0
$585$	79.1326	3.27173
$586$	0	0
$587$	− 13.3083i	− 0.549293i	−0.961545	−	0.274647i	$-0.911439\pi$
$587$	− 13.3083i	− 0.549293i	0.961545	−	0.274647i	$-0.0885609\pi$
$588$	0	0
$589$	− 35.8610i	− 1.47763i
$590$	0	0
$591$	−31.0278	−1.27631
$592$	0	0
$593$	7.68665	0.315653	0.157826	−	0.987467i	$-0.449551\pi$
$593$	7.68665	0.315653	0.157826	+	0.987467i	$0.449551\pi$
$594$	0	0
$595$	− 10.6167i	− 0.435240i
$596$	0	0
$597$	− 1.97887i	− 0.0809899i
$598$	0	0
$599$	−37.0177	−1.51250	−0.756251	−	0.654281i	$-0.772971\pi$
$599$	−37.0177	−1.51250	−0.756251	+	0.654281i	$0.772971\pi$
$600$	0	0
$601$	−9.78440	−0.399114	−0.199557	−	0.979886i	$-0.563950\pi$
$601$	−9.78440	−0.399114	−0.199557	+	0.979886i	$0.563950\pi$
$602$	0	0
$603$	43.5960i	1.77537i
$604$	0	0
$605$	5.44439i	0.221346i
$606$	0	0
$607$	−40.6066	−1.64817	−0.824086	−	0.566465i	$-0.808311\pi$
$607$	−40.6066	−1.64817	−0.824086	+	0.566465i	$0.808311\pi$
$608$	0	0
$609$	−25.4600	−1.03169
$610$	0	0
$611$	29.3522i	1.18746i
$612$	0	0
$613$	− 24.8122i	− 1.00215i	−0.865403	−	0.501077i	$-0.832937\pi$
$613$	− 24.8122i	− 1.00215i	0.865403	−	0.501077i	$-0.167063\pi$
$614$	0	0
$615$	1.61003	0.0649225
$616$	0	0
$617$	40.4494	1.62843	0.814216	−	0.580562i	$-0.197167\pi$
$617$	40.4494	1.62843	0.814216	+	0.580562i	$0.197167\pi$
$618$	0	0
$619$	5.20053i	0.209027i	0.994523	+	0.104513i	$0.0333285\pi$
$619$	5.20053i	0.209027i	−0.994523	+	0.104513i	$0.966671\pi$
$620$	0	0
$621$	− 6.50885i	− 0.261191i
$622$	0	0
$623$	10.4111	0.417112
$624$	0	0
$625$	−29.9794	−1.19918
$626$	0	0
$627$	− 79.9377i	− 3.19240i
$628$	0	0
$629$	12.8222i	0.511255i
$630$	0	0
$631$	−43.7422	−1.74135	−0.870674	−	0.491861i	$-0.836317\pi$
$631$	−43.7422	−1.74135	−0.870674	+	0.491861i	$0.836317\pi$
$632$	0	0
$633$	2.09775	0.0833781
$634$	0	0
$635$	37.0489i	1.47024i
$636$	0	0
$637$	− 4.72999i	− 0.187409i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.21611	0.285019	0.142510	−	0.989793i	$-0.454483\pi$
$641$	7.21611	0.285019	0.142510	+	0.989793i	$0.454483\pi$
$642$	0	0
$643$	20.3472i	0.802413i	0.915988	+	0.401207i	$0.131409\pi$
$643$	20.3472i	0.802413i	−0.915988	+	0.401207i	$0.868591\pi$
$644$	0	0
$645$	37.4711i	1.47542i
$646$	0	0
$647$	11.5577	0.454381	0.227191	−	0.973850i	$-0.427046\pi$
$647$	11.5577	0.454381	0.227191	+	0.973850i	$0.427046\pi$
$648$	0	0
$649$	−45.5678	−1.78869
$650$	0	0
$651$	− 15.6655i	− 0.613980i
$652$	0	0
$653$	− 14.3033i	− 0.559731i	−0.960039	−	0.279866i	$-0.909710\pi$
$653$	− 14.3033i	− 0.559731i	0.960039	−	0.279866i	$-0.0902900\pi$
$654$	0	0
$655$	10.4806	0.409510
$656$	0	0
$657$	−61.3311	−2.39275
$658$	0	0
$659$	− 19.7038i	− 0.767553i	−0.923426	−	0.383776i	$-0.874623\pi$
$659$	− 19.7038i	− 0.767553i	0.923426	−	0.383776i	$-0.125377\pi$
$660$	0	0
$661$	29.7789i	1.15826i	0.815234	+	0.579132i	$0.196608\pi$
$661$	29.7789i	1.15826i	−0.815234	+	0.579132i	$0.803392\pi$
$662$	0	0
$663$	61.7210	2.39705
$664$	0	0
$665$	17.9305	0.695316
$666$	0	0
$667$	4.74557i	0.183749i
$668$	0	0
$669$	73.4288i	2.83892i
$670$	0	0
$671$	−38.1744	−1.47371
$672$	0	0
$673$	32.9200	1.26897	0.634485	−	0.772935i	$-0.281212\pi$
$673$	32.9200	1.26897	0.634485	+	0.772935i	$0.281212\pi$
$674$	0	0
$675$	15.4499i	0.594668i
$676$	0	0
$677$	− 22.7089i	− 0.872772i	−0.899759	−	0.436386i	$-0.856258\pi$
$677$	− 22.7089i	− 0.872772i	0.899759	−	0.436386i	$-0.143742\pi$
$678$	0	0
$679$	−9.36222	−0.359289
$680$	0	0
$681$	27.6061	1.05787
$682$	0	0
$683$	− 23.5194i	− 0.899945i	−0.893042	−	0.449973i	$-0.851434\pi$
$683$	− 23.5194i	− 0.899945i	0.893042	−	0.449973i	$-0.148566\pi$
$684$	0	0
$685$	13.2645i	0.506809i
$686$	0	0
$687$	23.8328	0.909277
$688$	0	0
$689$	9.45998	0.360396
$690$	0	0
$691$	− 35.1794i	− 1.33829i	−0.743133	−	0.669144i	$-0.766661\pi$
$691$	− 35.1794i	− 1.33829i	0.743133	−	0.669144i	$-0.233339\pi$
$692$	0	0
$693$	− 24.0383i	− 0.913140i
$694$	0	0
$695$	−22.4605	−0.851975
$696$	0	0
$697$	0.864451	0.0327434
$698$	0	0
$699$	− 17.5789i	− 0.664893i
$700$	0	0
$701$	1.48110i	0.0559404i	0.999609	+	0.0279702i	$0.00890436\pi$
$701$	1.48110i	0.0559404i	−0.999609	+	0.0279702i	$0.991096\pi$
$702$	0	0
$703$	−21.6555	−0.816752
$704$	0	0
$705$	−48.6066	−1.83063
$706$	0	0
$707$	− 4.31889i	− 0.162428i
$708$	0	0
$709$	29.3622i	1.10272i	0.834267	+	0.551361i	$0.185891\pi$
$709$	29.3622i	1.10272i	−0.834267	+	0.551361i	$0.814109\pi$
$710$	0	0
$711$	34.1744	1.28164
$712$	0	0
$713$	−2.91995	−0.109353
$714$	0	0
$715$	− 43.3110i	− 1.61974i
$716$	0	0
$717$	− 35.4389i	− 1.32349i
$718$	0	0
$719$	−13.3833	−0.499115	−0.249557	−	0.968360i	$-0.580285\pi$
$719$	−13.3833	−0.499115	−0.249557	+	0.968360i	$0.580285\pi$
$720$	0	0
$721$	5.04888	0.188030
$722$	0	0
$723$	54.4777i	2.02605i
$724$	0	0
$725$	− 11.2645i	− 0.418352i
$726$	0	0
$727$	21.5366	0.798748	0.399374	−	0.916788i	$-0.369227\pi$
$727$	21.5366	0.798748	0.399374	+	0.916788i	$0.369227\pi$
$728$	0	0
$729$	5.09775	0.188806
$730$	0	0
$731$	20.1189i	0.744124i
$732$	0	0
$733$	− 30.2978i	− 1.11907i	−0.828806	−	0.559537i	$-0.810979\pi$
$733$	− 30.2978i	− 1.11907i	0.828806	−	0.559537i	$-0.189021\pi$
$734$	0	0
$735$	7.83276	0.288916
$736$	0	0
$737$	23.8610	0.878932
$738$	0	0
$739$	− 12.9794i	− 0.477455i	−0.971087	−	0.238727i	$-0.923270\pi$
$739$	− 12.9794i	− 0.477455i	0.971087	−	0.238727i	$-0.0767302\pi$
$740$	0	0
$741$	104.241i	3.82939i
$742$	0	0
$743$	50.0071	1.83458	0.917292	−	0.398215i	$-0.130370\pi$
$743$	50.0071	1.83458	0.917292	+	0.398215i	$0.130370\pi$
$744$	0	0
$745$	−54.6933	−2.00381
$746$	0	0
$747$	39.4061i	1.44179i
$748$	0	0
$749$	− 9.83276i	− 0.359281i
$750$	0	0
$751$	−22.8917	−0.835329	−0.417665	−	0.908601i	$-0.637151\pi$
$751$	−22.8917	−0.835329	−0.417665	+	0.908601i	$0.637151\pi$
$752$	0	0
$753$	−37.7038	−1.37400
$754$	0	0
$755$	12.5955i	0.458398i
$756$	0	0
$757$	29.0278i	1.05503i	0.849545	+	0.527516i	$0.176876\pi$
$757$	29.0278i	1.05503i	−0.849545	+	0.527516i	$0.823124\pi$
$758$	0	0
$759$	−6.50885	−0.236256
$760$	0	0
$761$	−36.8122	−1.33444	−0.667220	−	0.744861i	$-0.732516\pi$
$761$	−36.8122	−1.33444	−0.667220	+	0.744861i	$0.732516\pi$
$762$	0	0
$763$	19.4600i	0.704498i
$764$	0	0
$765$	70.3588i	2.54383i
$766$	0	0
$767$	59.4217	2.14559
$768$	0	0
$769$	−47.8711	−1.72628	−0.863138	−	0.504969i	$-0.831504\pi$
$769$	−47.8711	−1.72628	−0.863138	+	0.504969i	$0.831504\pi$
$770$	0	0
$771$	19.8922i	0.716400i
$772$	0	0
$773$	− 41.2489i	− 1.48362i	−0.670610	−	0.741810i	$-0.733968\pi$
$773$	− 41.2489i	− 1.48362i	0.670610	−	0.741810i	$-0.266032\pi$
$774$	0	0
$775$	6.93103	0.248970
$776$	0	0
$777$	−9.45998	−0.339375
$778$	0	0
$779$	1.45998i	0.0523091i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	92.3488	3.30028
$784$	0	0
$785$	50.7215	1.81033
$786$	0	0
$787$	− 6.79947i	− 0.242375i	−0.992630	−	0.121188i	$-0.961330\pi$
$787$	− 6.79947i	− 0.242375i	0.992630	−	0.121188i	$-0.0386702\pi$
$788$	0	0
$789$	32.0000i	1.13923i
$790$	0	0
$791$	8.78389	0.312319
$792$	0	0
$793$	49.7805	1.76776
$794$	0	0
$795$	15.6655i	0.555599i
$796$	0	0
$797$	6.18996i	0.219260i	0.993972	+	0.109630i	$0.0349666\pi$
$797$	6.18996i	0.219260i	−0.993972	+	0.109630i	$0.965033\pi$
$798$	0	0
$799$	−26.0978	−0.923272
$800$	0	0
$801$	−68.9966	−2.43787
$802$	0	0
$803$	33.5678i	1.18458i
$804$	0	0
$805$	− 1.45998i	− 0.0514574i
$806$	0	0
$807$	−48.6550	−1.71274
$808$	0	0
$809$	−30.3517	−1.06711	−0.533554	−	0.845766i	$-0.679144\pi$
$809$	−30.3517	−1.06711	−0.533554	+	0.845766i	$0.679144\pi$
$810$	0	0
$811$	− 40.0328i	− 1.40574i	−0.711318	−	0.702870i	$-0.751901\pi$
$811$	− 40.0328i	− 1.40574i	0.711318	−	0.702870i	$-0.248099\pi$
$812$	0	0
$813$	79.9377i	2.80354i
$814$	0	0
$815$	40.4877	1.41822
$816$	0	0
$817$	−33.9789	−1.18877
$818$	0	0
$819$	31.3466i	1.09534i
$820$	0	0
$821$	51.9789i	1.81408i	0.421050	+	0.907038i	$0.361662\pi$
$821$	51.9789i	1.81408i	−0.421050	+	0.907038i	$0.638338\pi$
$822$	0	0
$823$	26.9200	0.938371	0.469185	−	0.883100i	$-0.344548\pi$
$823$	26.9200	0.938371	0.469185	+	0.883100i	$0.344548\pi$
$824$	0	0
$825$	15.4499	0.537898
$826$	0	0
$827$	47.5960i	1.65508i	0.561409	+	0.827538i	$0.310259\pi$
$827$	47.5960i	1.65508i	−0.561409	+	0.827538i	$0.689741\pi$
$828$	0	0
$829$	8.21109i	0.285183i	0.989782	+	0.142591i	$0.0455435\pi$
$829$	8.21109i	0.285183i	−0.989782	+	0.142591i	$0.954456\pi$
$830$	0	0
$831$	53.5366	1.85716
$832$	0	0
$833$	4.20555	0.145714
$834$	0	0
$835$	− 4.52998i	− 0.156766i
$836$	0	0
$837$	56.8222i	1.96406i
$838$	0	0
$839$	−5.04888	−0.174307	−0.0871533	−	0.996195i	$-0.527777\pi$
$839$	−5.04888	−0.174307	−0.0871533	+	0.996195i	$0.527777\pi$
$840$	0	0
$841$	−38.3311	−1.32176
$842$	0	0
$843$	100.868i	3.47407i
$844$	0	0
$845$	23.6610i	0.813964i
$846$	0	0
$847$	−2.15667	−0.0741042
$848$	0	0
$849$	71.3482	2.44867
$850$	0	0
$851$	1.76328i	0.0604444i
$852$	0	0
$853$	1.14109i	0.0390701i	0.999809	+	0.0195351i	$0.00621860\pi$
$853$	1.14109i	0.0390701i	−0.999809	+	0.0195351i	$0.993781\pi$
$854$	0	0
$855$	−118.829	−4.06388
$856$	0	0
$857$	3.45998	0.118191	0.0590953	−	0.998252i	$-0.481178\pi$
$857$	3.45998	0.118191	0.0590953	+	0.998252i	$0.481178\pi$
$858$	0	0
$859$	4.15165i	0.141653i	0.997489	+	0.0708263i	$0.0225636\pi$
$859$	4.15165i	0.141653i	−0.997489	+	0.0708263i	$0.977436\pi$
$860$	0	0
$861$	0.637776i	0.0217353i
$862$	0	0
$863$	13.7633	0.468507	0.234254	−	0.972175i	$-0.424735\pi$
$863$	13.7633	0.468507	0.234254	+	0.972175i	$0.424735\pi$
$864$	0	0
$865$	−57.6272	−1.95938
$866$	0	0
$867$	2.13053i	0.0723564i
$868$	0	0
$869$	− 18.7044i	− 0.634502i
$870$	0	0
$871$	−31.1155	−1.05431
$872$	0	0
$873$	62.0455	2.09992
$874$	0	0
$875$	− 9.15667i	− 0.309552i
$876$	0	0
$877$	− 53.9688i	− 1.82240i	−0.411967	−	0.911199i	$-0.635158\pi$
$877$	− 53.9688i	− 1.82240i	0.411967	−	0.911199i	$-0.364842\pi$
$878$	0	0
$879$	27.0872	0.913628
$880$	0	0
$881$	56.1744	1.89256	0.946281	−	0.323344i	$-0.104807\pi$
$881$	56.1744	1.89256	0.946281	+	0.323344i	$0.104807\pi$
$882$	0	0
$883$	− 52.5371i	− 1.76801i	−0.467473	−	0.884007i	$-0.654835\pi$
$883$	− 52.5371i	− 1.76801i	0.467473	−	0.884007i	$-0.345165\pi$
$884$	0	0
$885$	98.4011i	3.30772i
$886$	0	0
$887$	−44.4988	−1.49412	−0.747062	−	0.664755i	$-0.768536\pi$
$887$	−44.4988	−1.49412	−0.747062	+	0.664755i	$0.768536\pi$
$888$	0	0
$889$	−14.6761	−0.492220
$890$	0	0
$891$	54.5472i	1.82740i
$892$	0	0
$893$	− 44.0766i	− 1.47497i
$894$	0	0
$895$	8.88454	0.296978
$896$	0	0
$897$	8.48773	0.283397
$898$	0	0
$899$	− 41.4288i	− 1.38173i
$900$	0	0
$901$	8.41110i	0.280214i
$902$	0	0
$903$	−14.8433	−0.493955
$904$	0	0
$905$	27.6061	0.917657
$906$	0	0
$907$	− 44.4182i	− 1.47488i	−0.675411	−	0.737442i	$-0.736034\pi$
$907$	− 44.4182i	− 1.47488i	0.675411	−	0.737442i	$-0.263966\pi$
$908$	0	0
$909$	28.6222i	0.949338i
$910$	0	0
$911$	48.9894	1.62309	0.811546	−	0.584288i	$-0.198626\pi$
$911$	48.9894	1.62309	0.811546	+	0.584288i	$0.198626\pi$
$912$	0	0
$913$	21.5678	0.713789
$914$	0	0
$915$	82.4354i	2.72523i
$916$	0	0
$917$	4.15165i	0.137100i
$918$	0	0
$919$	34.9200	1.15190	0.575951	−	0.817484i	$-0.304632\pi$
$919$	34.9200	1.15190	0.575951	+	0.817484i	$0.304632\pi$
$920$	0	0
$921$	−82.7215	−2.72577
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 4.18546i	− 0.137617i
$926$	0	0
$927$	−33.4600	−1.09897
$928$	0	0
$929$	3.98995	0.130906	0.0654531	−	0.997856i	$-0.479151\pi$
$929$	3.98995	0.130906	0.0654531	+	0.997856i	$0.479151\pi$
$930$	0	0
$931$	7.10278i	0.232784i
$932$	0	0
$933$	31.3311i	1.02573i
$934$	0	0
$935$	38.5089	1.25937
$936$	0	0
$937$	47.9789	1.56740	0.783701	−	0.621139i	$-0.213330\pi$
$937$	47.9789	1.56740	0.783701	+	0.621139i	$0.213330\pi$
$938$	0	0
$939$	0.637776i	0.0208130i
$940$	0	0
$941$	− 26.1900i	− 0.853768i	−0.904306	−	0.426884i	$-0.859611\pi$
$941$	− 26.1900i	− 0.853768i	0.904306	−	0.426884i	$-0.140389\pi$
$942$	0	0
$943$	0.118877	0.00387118
$944$	0	0
$945$	−28.4111	−0.924213
$946$	0	0
$947$	− 57.3905i	− 1.86494i	−0.361247	−	0.932470i	$-0.617649\pi$
$947$	− 57.3905i	− 1.86494i	0.361247	−	0.932470i	$-0.382351\pi$
$948$	0	0
$949$	− 43.7733i	− 1.42094i
$950$	0	0
$951$	94.3588	3.05979
$952$	0	0
$953$	3.68665	0.119422	0.0597112	−	0.998216i	$-0.480982\pi$
$953$	3.68665	0.119422	0.0597112	+	0.998216i	$0.480982\pi$
$954$	0	0
$955$	− 1.88216i	− 0.0609051i
$956$	0	0
$957$	− 92.3488i	− 2.98521i
$958$	0	0
$959$	−5.25443	−0.169674
$960$	0	0
$961$	−5.50885	−0.177705
$962$	0	0
$963$	65.1638i	2.09987i
$964$	0	0
$965$	35.6856i	1.14876i
$966$	0	0
$967$	−27.6172	−0.888108	−0.444054	−	0.896000i	$-0.646460\pi$
$967$	−27.6172	−0.888108	−0.444054	+	0.896000i	$0.646460\pi$
$968$	0	0
$969$	−92.6832	−2.97741
$970$	0	0
$971$	− 9.94610i	− 0.319186i	−0.987183	−	0.159593i	$-0.948982\pi$
$971$	− 9.94610i	− 0.319186i	0.987183	−	0.159593i	$-0.0510181\pi$
$972$	0	0
$973$	− 8.89722i	− 0.285232i
$974$	0	0
$975$	−20.1471	−0.645225
$976$	0	0
$977$	−21.2544	−0.679989	−0.339995	−	0.940427i	$-0.610425\pi$
$977$	−21.2544	−0.679989	−0.339995	+	0.940427i	$0.610425\pi$
$978$	0	0
$979$	37.7633i	1.20692i
$980$	0	0
$981$	− 128.965i	− 4.11755i
$982$	0	0
$983$	−16.1844	−0.516203	−0.258101	−	0.966118i	$-0.583097\pi$
$983$	−16.1844	−0.516203	−0.258101	+	0.966118i	$0.583097\pi$
$984$	0	0
$985$	25.2444	0.804353
$986$	0	0
$987$	− 19.2544i	− 0.612875i
$988$	0	0
$989$	2.76670i	0.0879759i
$990$	0	0
$991$	16.0766	0.510691	0.255345	−	0.966850i	$-0.417811\pi$
$991$	16.0766	0.510691	0.255345	+	0.966850i	$0.417811\pi$
$992$	0	0
$993$	−7.66553	−0.243258
$994$	0	0
$995$	1.61003i	0.0510412i
$996$	0	0
$997$	− 29.4444i	− 0.932513i	−0.884650	−	0.466257i	$-0.845602\pi$
$997$	− 29.4444i	− 0.932513i	0.884650	−	0.466257i	$-0.154398\pi$
$998$	0	0
$999$	34.3133	1.08563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.b.p.897.6		6
4.3	odd	2		1792.2.b.o.897.1		6
8.3	odd	2		1792.2.b.o.897.6		6
8.5	even	2	inner	1792.2.b.p.897.1		6
16.3	odd	4		896.2.a.k.1.3	yes	3
16.5	even	4		896.2.a.l.1.3	yes	3
16.11	odd	4		896.2.a.j.1.1	yes	3
16.13	even	4		896.2.a.i.1.1	✓	3
48.5	odd	4		8064.2.a.bu.1.2		3
48.11	even	4		8064.2.a.cb.1.2		3
48.29	odd	4		8064.2.a.ce.1.2		3
48.35	even	4		8064.2.a.ch.1.2		3
112.13	odd	4		6272.2.a.x.1.3		3
112.27	even	4		6272.2.a.w.1.3		3
112.69	odd	4		6272.2.a.u.1.1		3
112.83	even	4		6272.2.a.v.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.a.i.1.1	✓	3	16.13	even	4
896.2.a.j.1.1	yes	3	16.11	odd	4
896.2.a.k.1.3	yes	3	16.3	odd	4
896.2.a.l.1.3	yes	3	16.5	even	4
1792.2.b.o.897.1		6	4.3	odd	2
1792.2.b.o.897.6		6	8.3	odd	2
1792.2.b.p.897.1		6	8.5	even	2	inner
1792.2.b.p.897.6		6	1.1	even	1	trivial
6272.2.a.u.1.1		3	112.69	odd	4
6272.2.a.v.1.1		3	112.83	even	4
6272.2.a.w.1.3		3	112.27	even	4
6272.2.a.x.1.3		3	112.13	odd	4
8064.2.a.bu.1.2		3	48.5	odd	4
8064.2.a.cb.1.2		3	48.11	even	4
8064.2.a.ce.1.2		3	48.29	odd	4
8064.2.a.ch.1.2		3	48.35	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.10278i q^{3} -2.52444i q^{5} +1.00000 q^{7} -6.62721 q^{9} +O(q^{10})\)	\(q+3.10278i q^{3} -2.52444i q^{5} +1.00000 q^{7} -6.62721 q^{9} +3.62721i q^{11} -4.72999i q^{13} +7.83276 q^{15} +4.20555 q^{17} +7.10278i q^{19} +3.10278i q^{21} +0.578337 q^{23} -1.37279 q^{25} -11.2544i q^{27} +8.20555i q^{29} -5.04888 q^{31} -11.2544 q^{33} -2.52444i q^{35} +3.04888i q^{37} +14.6761 q^{39} +0.205550 q^{41} +4.78389i q^{43} +16.7300i q^{45} -6.20555 q^{47} +1.00000 q^{49} +13.0489i q^{51} +2.00000i q^{53} +9.15667 q^{55} -22.0383 q^{57} +12.5628i q^{59} +10.5244i q^{61} -6.62721 q^{63} -11.9406 q^{65} -6.57834i q^{67} +1.79445i q^{69} +9.25443 q^{73} -4.25945i q^{75} +3.62721i q^{77} -5.15667 q^{79} +15.0383 q^{81} -5.94610i q^{83} -10.6167i q^{85} -25.4600 q^{87} +10.4111 q^{89} -4.72999i q^{91} -15.6655i q^{93} +17.9305 q^{95} -9.36222 q^{97} -24.0383i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{7} - 14 q^{9}+O(q^{10}) \) 6 * q + 6 * q^7 - 14 * q^9	\( 6 q + 6 q^{7} - 14 q^{9} - 8 q^{15} - 4 q^{17} - 34 q^{25} - 8 q^{31} - 16 q^{33} + 40 q^{39} - 28 q^{41} - 8 q^{47} + 6 q^{49} + 48 q^{55} - 48 q^{57} - 14 q^{63} - 32 q^{65} + 4 q^{73} - 24 q^{79} + 6 q^{81} - 72 q^{87} + 4 q^{89} + 8 q^{95} - 20 q^{97}+O(q^{100}) \) 6 * q + 6 * q^7 - 14 * q^9 - 8 * q^15 - 4 * q^17 - 34 * q^25 - 8 * q^31 - 16 * q^33 + 40 * q^39 - 28 * q^41 - 8 * q^47 + 6 * q^49 + 48 * q^55 - 48 * q^57 - 14 * q^63 - 32 * q^65 + 4 * q^73 - 24 * q^79 + 6 * q^81 - 72 * q^87 + 4 * q^89 + 8 * q^95 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more