LMFDB - Embedded newform 1968.2.j.f.1393.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1968,2,Mod(1393,1968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1968, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1968.1393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1968 = 2^{4} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1968.j (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:	$15.7145591178$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.265727878144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 984)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1393.5
Root			$-2.60520i$ of defining polynomial
Character	$\chi$	$=$	1968.1393
Dual form			1968.2.j.f.1393.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.60520 q^{5} +0.181860i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.60520 q^{5} +0.181860i q^{7} -1.00000 q^{9} +2.05044i q^{11} -0.949556i q^{13} -1.60520i q^{15} -3.83750i q^{17} -6.15995i q^{19} -0.181860 q^{21} +4.73661 q^{23} -2.42334 q^{25} -1.00000i q^{27} -3.26084i q^{29} +7.12887 q^{31} -2.05044 q^{33} -0.291922i q^{35} -4.42334 q^{37} +0.949556 q^{39} +(-0.372894 + 6.39226i) q^{41} +7.54359 q^{43} +1.60520 q^{45} +10.2802i q^{47} +6.96693 q^{49} +3.83750 q^{51} -0.554754i q^{53} -3.29137i q^{55} +6.15995 q^{57} +11.6531 q^{59} -9.80642 q^{61} -0.181860i q^{63} +1.52423i q^{65} -7.91593i q^{67} +4.73661i q^{69} +3.73661i q^{71} +12.1095 q^{73} -2.42334i q^{75} -0.372894 q^{77} -8.84668i q^{79} +1.00000 q^{81} +1.70609 q^{83} +6.15995i q^{85} +3.26084 q^{87} -4.76770i q^{89} +0.172687 q^{91} +7.12887i q^{93} +9.88795i q^{95} -9.39226i q^{97} -2.05044i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 10 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q + 10 * q^5 - 8 * q^9	$ 8 q + 10 q^{5} - 8 q^{9} + 20 q^{23} + 2 q^{25} - 8 q^{31} - 10 q^{33} - 14 q^{37} + 14 q^{39} + 12 q^{41} - 6 q^{43} - 10 q^{45} - 32 q^{49} + 10 q^{57} - 6 q^{59} - 22 q^{61} + 64 q^{73} + 12 q^{77} + 8 q^{81} - 22 q^{83} - 26 q^{87} + 12 q^{91}+O(q^{100}) $ 8 * q + 10 * q^5 - 8 * q^9 + 20 * q^23 + 2 * q^25 - 8 * q^31 - 10 * q^33 - 14 * q^37 + 14 * q^39 + 12 * q^41 - 6 * q^43 - 10 * q^45 - 32 * q^49 + 10 * q^57 - 6 * q^59 - 22 * q^61 + 64 * q^73 + 12 * q^77 + 8 * q^81 - 22 * q^83 - 26 * q^87 + 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1968\mathbb{Z}\right)^\times$.

$n$	$1231$	$1313$	$1441$	$1477$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−1.60520			−0.717867			−0.358933	−	0.933363i	$-0.616859\pi$
$5$	−1.60520			−0.717867			−0.358933	+	0.933363i	$0.616859\pi$
$6$		0			0
$7$			0.181860i			0.0687367i	0.999409	+	0.0343684i	$0.0109419\pi$
$7$			0.181860i			0.0687367i	−0.999409	+	0.0343684i	$0.989058\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			2.05044i			0.618232i	0.951024	+	0.309116i	$0.100033\pi$
$11$			2.05044i			0.618232i	−0.951024	+	0.309116i	$0.899967\pi$
$12$		0			0
$13$		−	0.949556i		−	0.263359i	−0.991292	−	0.131680i	$-0.957963\pi$
$13$		−	0.949556i		−	0.263359i	0.991292	−	0.131680i	$-0.0420370\pi$
$14$		0			0
$15$		−	1.60520i		−	0.414460i
$16$		0			0
$17$		−	3.83750i		−	0.930731i	−0.885119	−	0.465366i	$-0.845923\pi$
$17$		−	3.83750i		−	0.930731i	0.885119	−	0.465366i	$-0.154077\pi$
$18$		0			0
$19$		−	6.15995i		−	1.41319i	−0.707618	−	0.706595i	$-0.750230\pi$
$19$		−	6.15995i		−	1.41319i	0.707618	−	0.706595i	$-0.249770\pi$
$20$		0			0
$21$	−0.181860			−0.0396852
$22$		0			0
$23$	4.73661			0.987652			0.493826	−	0.869561i	$-0.335598\pi$
$23$	4.73661			0.987652			0.493826	+	0.869561i	$0.335598\pi$
$24$		0			0
$25$	−2.42334			−0.484668
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$		−	3.26084i		−	0.605523i	−0.953066	−	0.302761i	$-0.902091\pi$
$29$		−	3.26084i		−	0.605523i	0.953066	−	0.302761i	$-0.0979085\pi$
$30$		0			0
$31$	7.12887			1.28038			0.640192	−	0.768215i	$-0.278855\pi$
$31$	7.12887			1.28038			0.640192	+	0.768215i	$0.278855\pi$
$32$		0			0
$33$	−2.05044			−0.356936
$34$		0			0
$35$		−	0.291922i		−	0.0493438i
$36$		0			0
$37$	−4.42334			−0.727192			−0.363596	−	0.931557i	$-0.618451\pi$
$37$	−4.42334			−0.727192			−0.363596	+	0.931557i	$0.618451\pi$
$38$		0			0
$39$	0.949556			0.152051
$40$		0			0
$41$	−0.372894	+	6.39226i	−0.0582363	+	0.998303i
$41$	−0.372894	+	6.39226i	−0.0582363	+	0.998303i
$42$		0			0
$43$	7.54359			1.15039			0.575193	−	0.818018i	$-0.304927\pi$
$43$	7.54359			1.15039			0.575193	+	0.818018i	$0.304927\pi$
$44$		0			0
$45$	1.60520			0.239289
$46$		0			0
$47$			10.2802i			1.49952i	0.661709	+	0.749761i	$0.269832\pi$
$47$			10.2802i			1.49952i	−0.661709	+	0.749761i	$0.730168\pi$
$48$		0			0
$49$	6.96693			0.995275
$50$		0			0
$51$	3.83750			0.537358
$52$		0			0
$53$		−	0.554754i		−	0.0762014i	−0.999274	−	0.0381007i	$-0.987869\pi$
$53$		−	0.554754i		−	0.0762014i	0.999274	−	0.0381007i	$-0.0121308\pi$
$54$		0			0
$55$		−	3.29137i		−	0.443808i
$56$		0			0
$57$	6.15995			0.815906
$58$		0			0
$59$	11.6531			1.51710			0.758552	−	0.651612i	$-0.225907\pi$
$59$	11.6531			1.51710			0.758552	+	0.651612i	$0.225907\pi$
$60$		0			0
$61$	−9.80642			−1.25558			−0.627792	−	0.778381i	$-0.716041\pi$
$61$	−9.80642			−1.25558			−0.627792	+	0.778381i	$0.716041\pi$
$62$		0			0
$63$		−	0.181860i		−	0.0229122i
$64$		0			0
$65$			1.52423i			0.189057i
$66$		0			0
$67$		−	7.91593i		−	0.967085i	−0.875321	−	0.483542i	$-0.839350\pi$
$67$		−	7.91593i		−	0.967085i	0.875321	−	0.483542i	$-0.160650\pi$
$68$		0			0
$69$			4.73661i			0.570221i
$70$		0			0
$71$			3.73661i			0.443455i	0.975109	+	0.221727i	$0.0711695\pi$
$71$			3.73661i			0.443455i	−0.975109	+	0.221727i	$0.928831\pi$
$72$		0			0
$73$	12.1095			1.41731			0.708655	−	0.705555i	$-0.249302\pi$
$73$	12.1095			1.41731			0.708655	+	0.705555i	$0.249302\pi$
$74$		0			0
$75$		−	2.42334i		−	0.279823i
$76$		0			0
$77$	−0.372894			−0.0424952
$78$		0			0
$79$		−	8.84668i		−	0.995329i	−0.867370	−	0.497664i	$-0.834191\pi$
$79$		−	8.84668i		−	0.995329i	0.867370	−	0.497664i	$-0.165809\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	1.70609			0.187267			0.0936336	−	0.995607i	$-0.470152\pi$
$83$	1.70609			0.187267			0.0936336	+	0.995607i	$0.470152\pi$
$84$		0			0
$85$			6.15995i			0.668141i
$86$		0			0
$87$	3.26084			0.349599
$88$		0			0
$89$		−	4.76770i		−	0.505375i	−0.967548	−	0.252687i	$-0.918686\pi$
$89$		−	4.76770i		−	0.505375i	0.967548	−	0.252687i	$-0.0813144\pi$
$90$		0			0
$91$	0.172687			0.0181025
$92$		0			0
$93$			7.12887i			0.739230i
$94$		0			0
$95$			9.88795i			1.01448i
$96$		0			0
$97$		−	9.39226i		−	0.953639i	−0.879001	−	0.476820i	$-0.841789\pi$
$97$		−	9.39226i		−	0.953639i	0.879001	−	0.476820i	$-0.158211\pi$
$98$		0			0
$99$		−	2.05044i		−	0.206077i
$100$		0			0
$101$			0.241478i			0.0240279i	0.999928	+	0.0120140i	$0.00382426\pi$
$101$			0.241478i			0.0240279i	−0.999928	+	0.0120140i	$0.996176\pi$
$102$		0			0
$103$	−0.836950			−0.0824671			−0.0412336	−	0.999150i	$-0.513129\pi$
$103$	−0.836950			−0.0824671			−0.0412336	+	0.999150i	$0.513129\pi$
$104$		0			0
$105$	0.291922			0.0284886
$106$		0			0
$107$	13.0876			1.26523			0.632613	−	0.774468i	$-0.281982\pi$
$107$	13.0876			1.26523			0.632613	+	0.774468i	$0.281982\pi$
$108$		0			0
$109$		−	10.6027i		−	1.01555i	−0.861490	−	0.507775i	$-0.830468\pi$
$109$		−	10.6027i		−	1.01555i	0.861490	−	0.507775i	$-0.169532\pi$
$110$		0			0
$111$		−	4.42334i		−	0.419845i
$112$		0			0
$113$	14.1488			1.33101			0.665503	−	0.746395i	$-0.268217\pi$
$113$	14.1488			1.33101			0.665503	+	0.746395i	$0.268217\pi$
$114$		0			0
$115$	−7.60321			−0.709003
$116$		0			0
$117$			0.949556i			0.0877865i
$118$		0			0
$119$	0.697889			0.0639754
$120$		0			0
$121$	6.79568			0.617789
$122$		0			0
$123$	−6.39226	−	0.372894i	−0.576370	−	0.0336227i
$124$		0			0
$125$	11.9159			1.06579
$126$		0			0
$127$	9.36118			0.830670			0.415335	−	0.909669i	$-0.363664\pi$
$127$	9.36118			0.830670			0.415335	+	0.909669i	$0.363664\pi$
$128$		0			0
$129$			7.54359i			0.664176i
$130$		0			0
$131$	−8.09171			−0.706976			−0.353488	−	0.935439i	$-0.615005\pi$
$131$	−8.09171			−0.706976			−0.353488	+	0.935439i	$0.615005\pi$
$132$		0			0
$133$	1.12025			0.0971380
$134$		0			0
$135$			1.60520i			0.138153i
$136$		0			0
$137$		−	9.90574i		−	0.846305i	−0.906059	−	0.423152i	$-0.860924\pi$
$137$		−	9.90574i		−	0.846305i	0.906059	−	0.423152i	$-0.139076\pi$
$138$		0			0
$139$	3.44525			0.292222			0.146111	−	0.989268i	$-0.453324\pi$
$139$	3.44525			0.292222			0.146111	+	0.989268i	$0.453324\pi$
$140$		0			0
$141$	−10.2802			−0.865749
$142$		0			0
$143$	1.94701			0.162817
$144$		0			0
$145$			5.23430i			0.434685i
$146$		0			0
$147$			6.96693i			0.574622i
$148$		0			0
$149$		−	1.57412i		−	0.128957i	−0.997919	−	0.0644784i	$-0.979462\pi$
$149$		−	1.57412i		−	0.128957i	0.997919	−	0.0644784i	$-0.0205383\pi$
$150$		0			0
$151$			9.53030i			0.775565i	0.921751	+	0.387783i	$0.126759\pi$
$151$			9.53030i			0.775565i	−0.921751	+	0.387783i	$0.873241\pi$
$152$		0			0
$153$			3.83750i			0.310244i
$154$		0			0
$155$	−11.4433			−0.919144
$156$		0			0
$157$		−	0.688715i		−	0.0549655i	−0.999622	−	0.0274827i	$-0.991251\pi$
$157$		−	0.688715i		−	0.0549655i	0.999622	−	0.0274827i	$-0.00874913\pi$
$158$		0			0
$159$	0.554754			0.0439949
$160$		0			0
$161$			0.861402i			0.0678880i
$162$		0			0
$163$	−11.2980			−0.884928			−0.442464	−	0.896786i	$-0.645895\pi$
$163$	−11.2980			−0.884928			−0.442464	+	0.896786i	$0.645895\pi$
$164$		0			0
$165$	3.29137			0.256233
$166$		0			0
$167$			4.89657i			0.378908i	0.981890	+	0.189454i	$0.0606718\pi$
$167$			4.89657i			0.378908i	−0.981890	+	0.189454i	$0.939328\pi$
$168$		0			0
$169$	12.0983			0.930642
$170$		0			0
$171$			6.15995i			0.471063i
$172$		0			0
$173$	−8.14879			−0.619541			−0.309770	−	0.950811i	$-0.600252\pi$
$173$	−8.14879			−0.619541			−0.309770	+	0.950811i	$0.600252\pi$
$174$		0			0
$175$		−	0.440709i		−	0.0333145i
$176$		0			0
$177$			11.6531i			0.875900i
$178$		0			0
$179$			17.8105i			1.33122i	0.746300	+	0.665610i	$0.231828\pi$
$179$			17.8105i			1.33122i	−0.746300	+	0.665610i	$0.768172\pi$
$180$		0			0
$181$			15.7540i			1.17098i	0.810678	+	0.585492i	$0.199099\pi$
$181$			15.7540i			1.17098i	−0.810678	+	0.585492i	$0.800901\pi$
$182$		0			0
$183$		−	9.80642i		−	0.724912i
$184$		0			0
$185$	7.10034			0.522027
$186$		0			0
$187$	7.86858			0.575408
$188$		0			0
$189$	0.181860			0.0132284
$190$		0			0
$191$			13.3113i			0.963171i	0.876399	+	0.481585i	$0.159939\pi$
$191$			13.3113i			0.963171i	−0.876399	+	0.481585i	$0.840061\pi$
$192$		0			0
$193$		−	8.88383i		−	0.639472i	−0.947507	−	0.319736i	$-0.896406\pi$
$193$		−	8.88383i		−	0.639472i	0.947507	−	0.319736i	$-0.103594\pi$
$194$		0			0
$195$	−1.52423			−0.109152
$196$		0			0
$197$	−9.24148			−0.658428			−0.329214	−	0.944255i	$-0.606784\pi$
$197$	−9.24148			−0.658428			−0.329214	+	0.944255i	$0.606784\pi$
$198$		0			0
$199$		−	5.65509i		−	0.400879i	−0.979706	−	0.200439i	$-0.935763\pi$
$199$		−	5.65509i		−	0.400879i	0.979706	−	0.200439i	$-0.0642370\pi$
$200$		0			0
$201$	7.91593			0.558347
$202$		0			0
$203$	0.593017			0.0416217
$204$		0			0
$205$	0.598569	−	10.2608i	0.0418059	−	0.716648i
$206$		0			0
$207$	−4.73661			−0.329217
$208$		0			0
$209$	12.6306			0.873679
$210$		0			0
$211$		−	21.6918i		−	1.49333i	−0.665202	−	0.746663i	$-0.731655\pi$
$211$		−	21.6918i		−	1.49333i	0.665202	−	0.746663i	$-0.268345\pi$
$212$		0			0
$213$	−3.73661			−0.256029
$214$		0			0
$215$	−12.1090			−0.825824
$216$		0			0
$217$			1.29646i			0.0880093i
$218$		0			0
$219$			12.1095i			0.818285i
$220$		0			0
$221$	−3.64392			−0.245117
$222$		0			0
$223$	−17.9159			−1.19974			−0.599870	−	0.800098i	$-0.704781\pi$
$223$	−17.9159			−1.19974			−0.599870	+	0.800098i	$0.704781\pi$
$224$		0			0
$225$	2.42334			0.161556
$226$		0			0
$227$			11.0978i			0.736586i	0.929710	+	0.368293i	$0.120058\pi$
$227$			11.0978i			0.736586i	−0.929710	+	0.368293i	$0.879942\pi$
$228$		0			0
$229$		−	23.5502i		−	1.55624i	−0.628114	−	0.778121i	$-0.716173\pi$
$229$		−	23.5502i		−	1.55624i	0.628114	−	0.778121i	$-0.283827\pi$
$230$		0			0
$231$		−	0.372894i		−	0.0245346i
$232$		0			0
$233$		−	0.863494i		−	0.0565694i	−0.999600	−	0.0282847i	$-0.990995\pi$
$233$		−	0.863494i		−	0.0565694i	0.999600	−	0.0282847i	$-0.00900450\pi$
$234$		0			0
$235$		−	16.5018i		−	1.07646i
$236$		0			0
$237$	8.84668			0.574653
$238$		0			0
$239$		−	22.6617i		−	1.46586i	−0.680302	−	0.732932i	$-0.738151\pi$
$239$		−	22.6617i		−	1.46586i	0.680302	−	0.732932i	$-0.261849\pi$
$240$		0			0
$241$	−2.82731			−0.182123			−0.0910616	−	0.995845i	$-0.529026\pi$
$241$	−2.82731			−0.182123			−0.0910616	+	0.995845i	$0.529026\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−11.1833			−0.714475
$246$		0			0
$247$	−5.84922			−0.372177
$248$		0			0
$249$			1.70609i			0.108119i
$250$		0			0
$251$	−10.5217			−0.664123			−0.332061	−	0.943258i	$-0.607744\pi$
$251$	−10.5217			−0.664123			−0.332061	+	0.943258i	$0.607744\pi$
$252$		0			0
$253$			9.71216i			0.610598i
$254$		0			0
$255$	−6.15995			−0.385751
$256$		0			0
$257$			2.44325i			0.152406i	0.997092	+	0.0762030i	$0.0242797\pi$
$257$			2.44325i			0.152406i	−0.997092	+	0.0762030i	$0.975720\pi$
$258$		0			0
$259$		−	0.804429i		−	0.0499848i
$260$		0			0
$261$			3.26084i			0.201841i
$262$		0			0
$263$			6.34844i			0.391462i	0.980658	+	0.195731i	$0.0627079\pi$
$263$			6.34844i			0.391462i	−0.980658	+	0.195731i	$0.937292\pi$
$264$		0			0
$265$			0.890491i			0.0547024i
$266$		0			0
$267$	4.76770			0.291778
$268$		0			0
$269$	1.03163			0.0628998			0.0314499	−	0.999505i	$-0.489988\pi$
$269$	1.03163			0.0628998			0.0314499	+	0.999505i	$0.489988\pi$
$270$		0			0
$271$	4.36627			0.265232			0.132616	−	0.991168i	$-0.457662\pi$
$271$	4.36627			0.265232			0.132616	+	0.991168i	$0.457662\pi$
$272$		0			0
$273$			0.172687i			0.0104515i
$274$		0			0
$275$		−	4.96892i		−	0.299637i
$276$		0			0
$277$	4.10343			0.246551			0.123276	−	0.992372i	$-0.460660\pi$
$277$	4.10343			0.246551			0.123276	+	0.992372i	$0.460660\pi$
$278$		0			0
$279$	−7.12887			−0.426794
$280$		0			0
$281$		−	8.03363i		−	0.479246i	−0.970866	−	0.239623i	$-0.922976\pi$
$281$		−	8.03363i		−	0.479246i	0.970866	−	0.239623i	$-0.0770238\pi$
$282$		0			0
$283$	−9.53285			−0.566669			−0.283334	−	0.959021i	$-0.591441\pi$
$283$	−9.53285			−0.566669			−0.283334	+	0.959021i	$0.591441\pi$
$284$		0			0
$285$	−9.88795			−0.585711
$286$		0			0
$287$	−1.16250	−	0.0678146i	−0.0686201	−	0.00400297i
$288$		0			0
$289$	2.27357			0.133740
$290$		0			0
$291$	9.39226			0.550584
$292$		0			0
$293$		−	30.8019i		−	1.79947i	−0.436441	−	0.899733i	$-0.643761\pi$
$293$		−	30.8019i		−	1.79947i	0.436441	−	0.899733i	$-0.356239\pi$
$294$		0			0
$295$	−18.7055			−1.08908
$296$		0			0
$297$	2.05044			0.118979
$298$		0			0
$299$		−	4.49768i		−	0.260108i
$300$		0			0
$301$			1.37188i			0.0790738i
$302$		0			0
$303$	−0.241478			−0.0138725
$304$		0			0
$305$	15.7413			0.901341
$306$		0			0
$307$	−13.9536			−0.796376			−0.398188	−	0.917304i	$-0.630361\pi$
$307$	−13.9536			−0.796376			−0.398188	+	0.917304i	$0.630361\pi$
$308$		0			0
$309$		−	0.836950i		−	0.0476124i
$310$		0			0
$311$			0.986714i			0.0559514i	0.999609	+	0.0279757i	$0.00890611\pi$
$311$			0.986714i			0.0559514i	−0.999609	+	0.0279757i	$0.991094\pi$
$312$		0			0
$313$		−	5.19048i		−	0.293383i	−0.989182	−	0.146692i	$-0.953137\pi$
$313$		−	5.19048i		−	0.293383i	0.989182	−	0.146692i	$-0.0468625\pi$
$314$		0			0
$315$			0.291922i			0.0164479i
$316$		0			0
$317$			33.5757i			1.88580i	0.333080	+	0.942898i	$0.391912\pi$
$317$			33.5757i			1.88580i	−0.333080	+	0.942898i	$0.608088\pi$
$318$		0			0
$319$	6.68617			0.374354
$320$		0			0
$321$			13.0876i			0.730479i
$322$		0			0
$323$	−23.6388			−1.31530
$324$		0			0
$325$			2.30110i			0.127642i
$326$		0			0
$327$	10.6027			0.586328
$328$		0			0
$329$	−1.86956			−0.103072
$330$		0			0
$331$		−	20.7646i		−	1.14133i	−0.821185	−	0.570663i	$-0.806686\pi$
$331$		−	20.7646i		−	1.14133i	0.821185	−	0.570663i	$-0.193314\pi$
$332$		0			0
$333$	4.42334			0.242397
$334$		0			0
$335$			12.7066i			0.694238i
$336$		0			0
$337$	−4.01936			−0.218949			−0.109474	−	0.993990i	$-0.534917\pi$
$337$	−4.01936			−0.218949			−0.109474	+	0.993990i	$0.534917\pi$
$338$		0			0
$339$			14.1488i			0.768457i
$340$		0			0
$341$			14.6174i			0.791574i
$342$		0			0
$343$			2.54003i			0.137149i
$344$		0			0
$345$		−	7.60321i		−	0.409343i
$346$		0			0
$347$		−	24.7010i		−	1.32602i	−0.748611	−	0.663009i	$-0.769279\pi$
$347$		−	24.7010i		−	1.32602i	0.748611	−	0.663009i	$-0.230721\pi$
$348$		0			0
$349$	−12.5522			−0.671904			−0.335952	−	0.941879i	$-0.609058\pi$
$349$	−12.5522			−0.671904			−0.335952	+	0.941879i	$0.609058\pi$
$350$		0			0
$351$	−0.949556			−0.0506836
$352$		0			0
$353$	14.3148			0.761901			0.380950	−	0.924595i	$-0.375597\pi$
$353$	14.3148			0.761901			0.380950	+	0.924595i	$0.375597\pi$
$354$		0			0
$355$		−	5.99801i		−	0.318341i
$356$		0			0
$357$			0.697889i			0.0369362i
$358$		0			0
$359$	−0.754962			−0.0398454			−0.0199227	−	0.999802i	$-0.506342\pi$
$359$	−0.754962			−0.0398454			−0.0199227	+	0.999802i	$0.506342\pi$
$360$		0			0
$361$	−18.9450			−0.997106
$362$		0			0
$363$			6.79568i			0.356681i
$364$		0			0
$365$	−19.4382			−1.01744
$366$		0			0
$367$	−4.79780			−0.250443			−0.125222	−	0.992129i	$-0.539964\pi$
$367$	−4.79780			−0.250443			−0.125222	+	0.992129i	$0.539964\pi$
$368$		0			0
$369$	0.372894	−	6.39226i	0.0194121	−	0.332768i
$370$		0			0
$371$	0.100888			0.00523783
$372$		0			0
$373$	−6.97809			−0.361312			−0.180656	−	0.983546i	$-0.557822\pi$
$373$	−6.97809			−0.361312			−0.180656	+	0.983546i	$0.557822\pi$
$374$		0			0
$375$			11.9159i			0.615336i
$376$		0			0
$377$	−3.09635			−0.159470
$378$		0			0
$379$	37.5798			1.93034			0.965172	−	0.261618i	$-0.0842560\pi$
$379$	37.5798			1.93034			0.965172	+	0.261618i	$0.0842560\pi$
$380$		0			0
$381$			9.36118i			0.479588i
$382$		0			0
$383$		−	1.24967i		−	0.0638554i	−0.999490	−	0.0319277i	$-0.989835\pi$
$383$		−	1.24967i		−	0.0638554i	0.999490	−	0.0319277i	$-0.0101646\pi$
$384$		0			0
$385$	0.598569			0.0305059
$386$		0			0
$387$	−7.54359			−0.383462
$388$		0			0
$389$	−23.8019			−1.20680			−0.603402	−	0.797437i	$-0.706188\pi$
$389$	−23.8019			−1.20680			−0.603402	+	0.797437i	$0.706188\pi$
$390$		0			0
$391$		−	18.1768i		−	0.919239i
$392$		0			0
$393$		−	8.09171i		−	0.408173i
$394$		0			0
$395$			14.2007i			0.714513i
$396$		0			0
$397$			6.52478i			0.327469i	0.986504	+	0.163735i	$0.0523541\pi$
$397$			6.52478i			0.327469i	−0.986504	+	0.163735i	$0.947646\pi$
$398$		0			0
$399$			1.12025i			0.0560827i
$400$		0			0
$401$	−18.9241			−0.945026			−0.472513	−	0.881324i	$-0.656653\pi$
$401$	−18.9241			−0.945026			−0.472513	+	0.881324i	$0.656653\pi$
$402$		0			0
$403$		−	6.76926i		−	0.337201i
$404$		0			0
$405$	−1.60520			−0.0797629
$406$		0			0
$407$		−	9.06981i		−	0.449574i
$408$		0			0
$409$	−16.4681			−0.814297			−0.407149	−	0.913362i	$-0.633477\pi$
$409$	−16.4681			−0.814297			−0.407149	+	0.913362i	$0.633477\pi$
$410$		0			0
$411$	9.90574			0.488614
$412$		0			0
$413$			2.11924i			0.104281i
$414$		0			0
$415$	−2.73861			−0.134433
$416$		0			0
$417$			3.44525i			0.168714i
$418$		0			0
$419$	−39.3373			−1.92175			−0.960876	−	0.276979i	$-0.910667\pi$
$419$	−39.3373			−1.92175			−0.960876	+	0.276979i	$0.910667\pi$
$420$		0			0
$421$		−	36.6832i		−	1.78783i	−0.448237	−	0.893915i	$-0.647948\pi$
$421$		−	36.6832i		−	1.78783i	0.448237	−	0.893915i	$-0.352052\pi$
$422$		0			0
$423$		−	10.2802i		−	0.499841i
$424$		0			0
$425$			9.29957i			0.451095i
$426$		0			0
$427$		−	1.78340i		−	0.0863047i
$428$		0			0
$429$			1.94701i			0.0940026i
$430$		0			0
$431$	31.2196			1.50379			0.751897	−	0.659281i	$-0.229139\pi$
$431$	31.2196			1.50379			0.751897	+	0.659281i	$0.229139\pi$
$432$		0			0
$433$	17.8136			0.856067			0.428034	−	0.903763i	$-0.359207\pi$
$433$	17.8136			0.856067			0.428034	+	0.903763i	$0.359207\pi$
$434$		0			0
$435$	−5.23430			−0.250965
$436$		0			0
$437$		−	29.1773i		−	1.39574i
$438$		0			0
$439$			37.7270i			1.80061i	0.435259	+	0.900305i	$0.356657\pi$
$439$			37.7270i			1.80061i	−0.435259	+	0.900305i	$0.643343\pi$
$440$		0			0
$441$	−6.96693			−0.331758
$442$		0			0
$443$	31.0204			1.47382			0.736911	−	0.675990i	$-0.236284\pi$
$443$	31.0204			1.47382			0.736911	+	0.675990i	$0.236284\pi$
$444$		0			0
$445$			7.65310i			0.362792i
$446$		0			0
$447$	1.57412			0.0744532
$448$		0			0
$449$	−25.6078			−1.20850			−0.604252	−	0.796793i	$-0.706528\pi$
$449$	−25.6078			−1.20850			−0.604252	+	0.796793i	$0.706528\pi$
$450$		0			0
$451$	−13.1070	−	0.764599i	−0.617183	−	0.0360035i
$452$		0			0
$453$	−9.53030			−0.447773
$454$		0			0
$455$	−0.277196			−0.0129952
$456$		0			0
$457$		−	31.4692i		−	1.47207i	−0.676944	−	0.736035i	$-0.736696\pi$
$457$		−	31.4692i		−	1.47207i	0.676944	−	0.736035i	$-0.263304\pi$
$458$		0			0
$459$	−3.83750			−0.179119
$460$		0			0
$461$	−7.73606			−0.360304			−0.180152	−	0.983639i	$-0.557659\pi$
$461$	−7.73606			−0.360304			−0.180152	+	0.983639i	$0.557659\pi$
$462$		0			0
$463$			6.54049i			0.303962i	0.988383	+	0.151981i	$0.0485653\pi$
$463$			6.54049i			0.303962i	−0.988383	+	0.151981i	$0.951435\pi$
$464$		0			0
$465$		−	11.4433i		−	0.530668i
$466$		0			0
$467$	19.4075			0.898072			0.449036	−	0.893514i	$-0.351767\pi$
$467$	19.4075			0.898072			0.449036	+	0.893514i	$0.351767\pi$
$468$		0			0
$469$	1.43959			0.0664742
$470$		0			0
$471$	0.688715			0.0317343
$472$		0			0
$473$			15.4677i			0.711206i
$474$		0			0
$475$			14.9276i			0.684927i
$476$		0			0
$477$			0.554754i			0.0254005i
$478$		0			0
$479$		−	6.25264i		−	0.285691i	−0.989745	−	0.142845i	$-0.954375\pi$
$479$		−	6.25264i		−	0.285691i	0.989745	−	0.142845i	$-0.0456251\pi$
$480$		0			0
$481$			4.20021i			0.191513i
$482$		0			0
$483$	−0.861402			−0.0391951
$484$		0			0
$485$			15.0764i			0.684586i
$486$		0			0
$487$	24.7213			1.12023			0.560115	−	0.828415i	$-0.310757\pi$
$487$	24.7213			1.12023			0.560115	+	0.828415i	$0.310757\pi$
$488$		0			0
$489$		−	11.2980i		−	0.510913i
$490$		0			0
$491$	15.9079			0.717912			0.358956	−	0.933355i	$-0.383133\pi$
$491$	15.9079			0.717912			0.358956	+	0.933355i	$0.383133\pi$
$492$		0			0
$493$	−12.5135			−0.563579
$494$		0			0
$495$			3.29137i			0.147936i
$496$		0			0
$497$	−0.679542			−0.0304816
$498$		0			0
$499$			1.29336i			0.0578988i	0.999581	+	0.0289494i	$0.00921617\pi$
$499$			1.29336i			0.0578988i	−0.999581	+	0.0289494i	$0.990784\pi$
$500$		0			0
$501$	−4.89657			−0.218763
$502$		0			0
$503$		−	24.2241i		−	1.08010i	−0.841633	−	0.540050i	$-0.818405\pi$
$503$		−	24.2241i		−	1.08010i	0.841633	−	0.540050i	$-0.181595\pi$
$504$		0			0
$505$		−	0.387620i		−	0.0172489i
$506$		0			0
$507$			12.0983i			0.537306i
$508$		0			0
$509$			39.5369i			1.75244i	0.481908	+	0.876222i	$0.339944\pi$
$509$			39.5369i			1.75244i	−0.481908	+	0.876222i	$0.660056\pi$
$510$		0			0
$511$			2.20224i			0.0974213i
$512$		0			0
$513$	−6.15995			−0.271969
$514$		0			0
$515$	1.34347			0.0592004
$516$		0			0
$517$	−21.0790			−0.927052
$518$		0			0
$519$		−	8.14879i		−	0.357692i
$520$		0			0
$521$			9.09932i			0.398648i	0.979934	+	0.199324i	$0.0638747\pi$
$521$			9.09932i			0.398648i	−0.979934	+	0.199324i	$0.936125\pi$
$522$		0			0
$523$	−21.8844			−0.956937			−0.478469	−	0.878105i	$-0.658808\pi$
$523$	−21.8844			−0.956937			−0.478469	+	0.878105i	$0.658808\pi$
$524$		0			0
$525$	0.440709			0.0192341
$526$		0			0
$527$		−	27.3571i		−	1.19169i
$528$		0			0
$529$	−0.564481			−0.0245426
$530$		0			0
$531$	−11.6531			−0.505701
$532$		0			0
$533$	6.06981	+	0.354084i	0.262913	+	0.0153371i
$534$		0			0
$535$	−21.0082			−0.908264
$536$		0			0
$537$	−17.8105			−0.768580
$538$		0			0
$539$			14.2853i			0.615311i
$540$		0			0
$541$	−33.2878			−1.43116			−0.715578	−	0.698533i	$-0.753837\pi$
$541$	−33.2878			−1.43116			−0.715578	+	0.698533i	$0.753837\pi$
$542$		0			0
$543$	−15.7540			−0.676068
$544$		0			0
$545$			17.0194i			0.729029i
$546$		0			0
$547$			37.8936i			1.62021i	0.586283	+	0.810107i	$0.300591\pi$
$547$			37.8936i			1.62021i	−0.586283	+	0.810107i	$0.699409\pi$
$548$		0			0
$549$	9.80642			0.418528
$550$		0			0
$551$	−20.0866			−0.855719
$552$		0			0
$553$	1.60886			0.0684156
$554$		0			0
$555$			7.10034i			0.301392i
$556$		0			0
$557$			4.57102i			0.193680i	0.995300	+	0.0968402i	$0.0308736\pi$
$557$			4.57102i			0.193680i	−0.995300	+	0.0968402i	$0.969126\pi$
$558$		0			0
$559$		−	7.16306i		−	0.302965i
$560$		0			0
$561$			7.86858i			0.332212i
$562$		0			0
$563$			10.6235i			0.447729i	0.974620	+	0.223865i	$0.0718673\pi$
$563$			10.6235i			0.447729i	−0.974620	+	0.223865i	$0.928133\pi$
$564$		0			0
$565$	−22.7116			−0.955485
$566$		0			0
$567$			0.181860i			0.00763741i
$568$		0			0
$569$	7.44368			0.312055			0.156028	−	0.987753i	$-0.450131\pi$
$569$	7.44368			0.312055			0.156028	+	0.987753i	$0.450131\pi$
$570$		0			0
$571$		−	35.9949i		−	1.50634i	−0.657826	−	0.753170i	$-0.728524\pi$
$571$		−	35.9949i		−	1.50634i	0.657826	−	0.753170i	$-0.271476\pi$
$572$		0			0
$573$	−13.3113			−0.556087
$574$		0			0
$575$	−11.4784			−0.478683
$576$		0			0
$577$			14.4584i			0.601912i	0.953638	+	0.300956i	$0.0973057\pi$
$577$			14.4584i			0.601912i	−0.953638	+	0.300956i	$0.902694\pi$
$578$		0			0
$579$	8.88383			0.369200
$580$		0			0
$581$			0.310269i			0.0128721i
$582$		0			0
$583$	1.13749			0.0471101
$584$		0			0
$585$		−	1.52423i		−	0.0630190i
$586$		0			0
$587$			38.1172i			1.57326i	0.617423	+	0.786632i	$0.288177\pi$
$587$			38.1172i			1.57326i	−0.617423	+	0.786632i	$0.711823\pi$
$588$		0			0
$589$		−	43.9135i		−	1.80942i
$590$		0			0
$591$		−	9.24148i		−	0.380144i
$592$		0			0
$593$			5.15277i			0.211599i	0.994387	+	0.105799i	$0.0337402\pi$
$593$			5.15277i			0.211599i	−0.994387	+	0.105799i	$0.966260\pi$
$594$		0			0
$595$	−1.12025			−0.0459258
$596$		0			0
$597$	5.65509			0.231447
$598$		0			0
$599$	−16.1273			−0.658944			−0.329472	−	0.944165i	$-0.606871\pi$
$599$	−16.1273			−0.658944			−0.329472	+	0.944165i	$0.606871\pi$
$600$		0			0
$601$		−	47.7259i		−	1.94678i	−0.229156	−	0.973390i	$-0.573597\pi$
$601$		−	47.7259i		−	1.94678i	0.229156	−	0.973390i	$-0.426403\pi$
$602$		0			0
$603$			7.91593i			0.322362i
$604$		0			0
$605$	−10.9084			−0.443490
$606$		0			0
$607$	10.4937			0.425926			0.212963	−	0.977060i	$-0.431689\pi$
$607$	10.4937			0.425926			0.212963	+	0.977060i	$0.431689\pi$
$608$		0			0
$609$			0.593017i			0.0240303i
$610$		0			0
$611$	9.76163			0.394913
$612$		0			0
$613$	18.1249			0.732057			0.366029	−	0.930604i	$-0.380717\pi$
$613$	18.1249			0.732057			0.366029	+	0.930604i	$0.380717\pi$
$614$		0			0
$615$	10.2608	+	0.598569i	0.413757	+	0.0241366i
$616$		0			0
$617$	17.8860			0.720061			0.360031	−	0.932940i	$-0.382766\pi$
$617$	17.8860			0.720061			0.360031	+	0.932940i	$0.382766\pi$
$618$		0			0
$619$	16.7432			0.672968			0.336484	−	0.941689i	$-0.390762\pi$
$619$	16.7432			0.672968			0.336484	+	0.941689i	$0.390762\pi$
$620$		0			0
$621$		−	4.73661i		−	0.190074i
$622$		0			0
$623$	0.867054			0.0347378
$624$		0			0
$625$	−7.01074			−0.280430
$626$		0			0
$627$			12.6306i			0.504419i
$628$		0			0
$629$			16.9746i			0.676820i
$630$		0			0
$631$	−4.35762			−0.173474			−0.0867370	−	0.996231i	$-0.527644\pi$
$631$	−4.35762			−0.173474			−0.0867370	+	0.996231i	$0.527644\pi$
$632$		0			0
$633$	21.6918			0.862173
$634$		0			0
$635$	−15.0265			−0.596310
$636$		0			0
$637$		−	6.61549i		−	0.262115i
$638$		0			0
$639$		−	3.73661i		−	0.147818i
$640$		0			0
$641$			34.4263i			1.35976i	0.733324	+	0.679879i	$0.237968\pi$
$641$			34.4263i			1.35976i	−0.733324	+	0.679879i	$0.762032\pi$
$642$		0			0
$643$			5.23230i			0.206342i	0.994664	+	0.103171i	$0.0328989\pi$
$643$			5.23230i			0.206342i	−0.994664	+	0.103171i	$0.967101\pi$
$644$		0			0
$645$		−	12.1090i		−	0.476790i
$646$		0			0
$647$	11.8421			0.465563			0.232781	−	0.972529i	$-0.425217\pi$
$647$	11.8421			0.465563			0.232781	+	0.972529i	$0.425217\pi$
$648$		0			0
$649$			23.8940i			0.937922i
$650$		0			0
$651$	−1.29646			−0.0508122
$652$		0			0
$653$			8.73407i			0.341791i	0.985289	+	0.170895i	$0.0546660\pi$
$653$			8.73407i			0.341791i	−0.985289	+	0.170895i	$0.945334\pi$
$654$		0			0
$655$	12.9888			0.507515
$656$		0			0
$657$	−12.1095			−0.472437
$658$		0			0
$659$		−	15.9378i		−	0.620850i	−0.950598	−	0.310425i	$-0.899529\pi$
$659$		−	15.9378i		−	0.620850i	0.950598	−	0.310425i	$-0.100471\pi$
$660$		0			0
$661$	−28.8472			−1.12203			−0.561014	−	0.827806i	$-0.689589\pi$
$661$	−28.8472			−1.12203			−0.561014	+	0.827806i	$0.689589\pi$
$662$		0			0
$663$		−	3.64392i		−	0.141518i
$664$		0			0
$665$	−1.79822			−0.0697322
$666$		0			0
$667$		−	15.4453i		−	0.598046i
$668$		0			0
$669$		−	17.9159i		−	0.692670i
$670$		0			0
$671$		−	20.1075i		−	0.776242i
$672$		0			0
$673$			16.2976i			0.628225i	0.949386	+	0.314113i	$0.101707\pi$
$673$			16.2976i			0.628225i	−0.949386	+	0.314113i	$0.898293\pi$
$674$		0			0
$675$			2.42334i			0.0932743i
$676$		0			0
$677$	−7.90676			−0.303881			−0.151941	−	0.988390i	$-0.548552\pi$
$677$	−7.90676			−0.303881			−0.151941	+	0.988390i	$0.548552\pi$
$678$		0			0
$679$	1.70808			0.0655500
$680$		0			0
$681$	−11.0978			−0.425268
$682$		0			0
$683$			5.66537i			0.216779i	0.994108	+	0.108390i	$0.0345694\pi$
$683$			5.66537i			0.216779i	−0.994108	+	0.108390i	$0.965431\pi$
$684$		0			0
$685$			15.9007i			0.607534i
$686$		0			0
$687$	23.5502			0.898497
$688$		0			0
$689$	−0.526771			−0.0200684
$690$		0			0
$691$			12.1380i			0.461753i	0.972983	+	0.230877i	$0.0741593\pi$
$691$			12.1380i			0.461753i	−0.972983	+	0.230877i	$0.925841\pi$
$692$		0			0
$693$	0.372894			0.0141651
$694$		0			0
$695$	−5.53030			−0.209776
$696$		0			0
$697$	24.5303	+	1.43098i	0.929151	+	0.0542023i
$698$		0			0
$699$	0.863494			0.0326604
$700$		0			0
$701$	−25.9949			−0.981814			−0.490907	−	0.871212i	$-0.663335\pi$
$701$	−25.9949			−0.981814			−0.490907	+	0.871212i	$0.663335\pi$
$702$		0			0
$703$			27.2476i			1.02766i
$704$		0			0
$705$	16.5018			0.621492
$706$		0			0
$707$	−0.0439152			−0.00165160
$708$		0			0
$709$			27.3515i			1.02721i	0.858027	+	0.513604i	$0.171690\pi$
$709$			27.3515i			1.02721i	−0.858027	+	0.513604i	$0.828310\pi$
$710$		0			0
$711$			8.84668i			0.331776i
$712$		0			0
$713$	33.7667			1.26457
$714$		0			0
$715$	−3.12534			−0.116881
$716$		0			0
$717$	22.6617			0.846317
$718$		0			0
$719$		−	41.1940i		−	1.53628i	−0.640283	−	0.768139i	$-0.721183\pi$
$719$		−	41.1940i		−	1.53628i	0.640283	−	0.768139i	$-0.278817\pi$
$720$		0			0
$721$		−	0.152208i		−	0.00566852i
$722$		0			0
$723$		−	2.82731i		−	0.105149i
$724$		0			0
$725$			7.90212i			0.293477i
$726$		0			0
$727$			11.0845i			0.411102i	0.978646	+	0.205551i	$0.0658986\pi$
$727$			11.0845i			0.411102i	−0.978646	+	0.205551i	$0.934101\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	28.9485i		−	1.07070i
$732$		0			0
$733$	0.675429			0.0249475			0.0124738	−	0.999922i	$-0.496029\pi$
$733$	0.675429			0.0249475			0.0124738	+	0.999922i	$0.496029\pi$
$734$		0			0
$735$		−	11.1833i		−	0.412502i
$736$		0			0
$737$	16.2312			0.597883
$738$		0			0
$739$	−15.1788			−0.558360			−0.279180	−	0.960239i	$-0.590063\pi$
$739$	−15.1788			−0.558360			−0.279180	+	0.960239i	$0.590063\pi$
$740$		0			0
$741$		−	5.84922i		−	0.214876i
$742$		0			0
$743$	−22.8405			−0.837936			−0.418968	−	0.908001i	$-0.637608\pi$
$743$	−22.8405			−0.837936			−0.418968	+	0.908001i	$0.637608\pi$
$744$		0			0
$745$			2.52677i			0.0925737i
$746$		0			0
$747$	−1.70609			−0.0624224
$748$		0			0
$749$			2.38011i			0.0869675i
$750$		0			0
$751$			48.0918i			1.75490i	0.479672	+	0.877448i	$0.340755\pi$
$751$			48.0918i			1.75490i	−0.479672	+	0.877448i	$0.659245\pi$
$752$		0			0
$753$		−	10.5217i		−	0.383431i
$754$		0			0
$755$		−	15.2980i		−	0.556752i
$756$		0			0
$757$			40.6597i			1.47780i	0.673814	+	0.738901i	$0.264655\pi$
$757$			40.6597i			1.47780i	−0.673814	+	0.738901i	$0.735345\pi$
$758$		0			0
$759$	−9.71216			−0.352529
$760$		0			0
$761$	−37.3062			−1.35235			−0.676174	−	0.736742i	$-0.736363\pi$
$761$	−37.3062			−1.35235			−0.676174	+	0.736742i	$0.736363\pi$
$762$		0			0
$763$	1.92820			0.0698056
$764$		0			0
$765$		−	6.15995i		−	0.222714i
$766$		0			0
$767$		−	11.0653i		−	0.399544i
$768$		0			0
$769$	40.3051			1.45344			0.726719	−	0.686934i	$-0.241044\pi$
$769$	40.3051			1.45344			0.726719	+	0.686934i	$0.241044\pi$
$770$		0			0
$771$	−2.44325			−0.0879916
$772$		0			0
$773$		−	43.8956i		−	1.57881i	−0.613870	−	0.789407i	$-0.710388\pi$
$773$		−	43.8956i		−	1.57881i	0.613870	−	0.789407i	$-0.289612\pi$
$774$		0			0
$775$	−17.2757			−0.620560
$776$		0			0
$777$	0.804429			0.0288587
$778$		0			0
$779$	39.3760	+	2.29701i	1.41079	+	0.0822990i
$780$		0			0
$781$	−7.66172			−0.274158
$782$		0			0
$783$	−3.26084			−0.116533
$784$		0			0
$785$			1.10552i			0.0394579i
$786$		0			0
$787$	13.8024			0.492004			0.246002	−	0.969269i	$-0.420883\pi$
$787$	13.8024			0.492004			0.246002	+	0.969269i	$0.420883\pi$
$788$		0			0
$789$	−6.34844			−0.226011
$790$		0			0
$791$			2.57310i			0.0914890i
$792$		0			0
$793$			9.31175i			0.330670i
$794$		0			0
$795$	−0.890491			−0.0315825
$796$		0			0
$797$	25.5084			0.903554			0.451777	−	0.892131i	$-0.350790\pi$
$797$	25.5084			0.903554			0.451777	+	0.892131i	$0.350790\pi$
$798$		0			0
$799$	39.4503			1.39565
$800$		0			0
$801$			4.76770i			0.168458i
$802$		0			0
$803$			24.8299i			0.876227i
$804$		0			0
$805$		−	1.38272i		−	0.0487345i
$806$		0			0
$807$			1.03163i			0.0363152i
$808$		0			0
$809$			36.6617i			1.28896i	0.764622	+	0.644479i	$0.222926\pi$
$809$			36.6617i			1.28896i	−0.764622	+	0.644479i	$0.777074\pi$
$810$		0			0
$811$	−9.89148			−0.347337			−0.173668	−	0.984804i	$-0.555562\pi$
$811$	−9.89148			−0.347337			−0.173668	+	0.984804i	$0.555562\pi$
$812$		0			0
$813$			4.36627i			0.153132i
$814$		0			0
$815$	18.1355			0.635260
$816$		0			0
$817$		−	46.4682i		−	1.62571i
$818$		0			0
$819$	−0.172687			−0.00603416
$820$		0			0
$821$	−31.6216			−1.10360			−0.551801	−	0.833976i	$-0.686059\pi$
$821$	−31.6216			−1.10360			−0.551801	+	0.833976i	$0.686059\pi$
$822$		0			0
$823$			10.1956i			0.355396i	0.984085	+	0.177698i	$0.0568650\pi$
$823$			10.1956i			0.355396i	−0.984085	+	0.177698i	$0.943135\pi$
$824$		0			0
$825$	4.96892			0.172996
$826$		0			0
$827$			38.0744i			1.32398i	0.749513	+	0.661989i	$0.230287\pi$
$827$			38.0744i			1.32398i	−0.749513	+	0.661989i	$0.769713\pi$
$828$		0			0
$829$	48.8874			1.69793			0.848965	−	0.528449i	$-0.177226\pi$
$829$	48.8874			1.69793			0.848965	+	0.528449i	$0.177226\pi$
$830$		0			0
$831$			4.10343i			0.142346i
$832$		0			0
$833$		−	26.7356i		−	0.926334i
$834$		0			0
$835$		−	7.85996i		−	0.272005i
$836$		0			0
$837$		−	7.12887i		−	0.246410i
$838$		0			0
$839$		−	36.2186i		−	1.25040i	−0.780463	−	0.625202i	$-0.785016\pi$
$839$		−	36.2186i		−	1.25040i	0.780463	−	0.625202i	$-0.214984\pi$
$840$		0			0
$841$	18.3669			0.633342
$842$		0			0
$843$	8.03363			0.276693
$844$		0			0
$845$	−19.4202			−0.668077
$846$		0			0
$847$			1.23586i			0.0424648i
$848$		0			0
$849$		−	9.53285i		−	0.327166i
$850$		0			0
$851$	−20.9516			−0.718213
$852$		0			0
$853$	58.2185			1.99336			0.996681	−	0.0814023i	$-0.0259398\pi$
$853$	58.2185			1.99336			0.996681	+	0.0814023i	$0.0259398\pi$
$854$		0			0
$855$		−	9.88795i		−	0.338161i
$856$		0			0
$857$	48.0297			1.64066			0.820331	−	0.571889i	$-0.193789\pi$
$857$	48.0297			1.64066			0.820331	+	0.571889i	$0.193789\pi$
$858$		0			0
$859$	−20.7203			−0.706969			−0.353485	−	0.935440i	$-0.615003\pi$
$859$	−20.7203			−0.706969			−0.353485	+	0.935440i	$0.615003\pi$
$860$		0			0
$861$	0.0678146	−	1.16250i	0.00231112	−	0.0396178i
$862$		0			0
$863$	−30.3067			−1.03165			−0.515827	−	0.856693i	$-0.672515\pi$
$863$	−30.3067			−1.03165			−0.515827	+	0.856693i	$0.672515\pi$
$864$		0			0
$865$	13.0804			0.444748
$866$		0			0
$867$			2.27357i			0.0772146i
$868$		0			0
$869$	18.1396			0.615344
$870$		0			0
$871$	−7.51662			−0.254691
$872$		0			0
$873$			9.39226i			0.317880i
$874$		0			0
$875$			2.16703i			0.0732591i
$876$		0			0
$877$	6.12743			0.206909			0.103454	−	0.994634i	$-0.467010\pi$
$877$	6.12743			0.206909			0.103454	+	0.994634i	$0.467010\pi$
$878$		0			0
$879$	30.8019			1.03892
$880$		0			0
$881$	−4.70706			−0.158585			−0.0792925	−	0.996851i	$-0.525266\pi$
$881$	−4.70706			−0.158585			−0.0792925	+	0.996851i	$0.525266\pi$
$882$		0			0
$883$		−	34.2812i		−	1.15365i	−0.816866	−	0.576827i	$-0.804291\pi$
$883$		−	34.2812i		−	1.15365i	0.816866	−	0.576827i	$-0.195709\pi$
$884$		0			0
$885$		−	18.7055i		−	0.628780i
$886$		0			0
$887$		−	7.53086i		−	0.252861i	−0.991975	−	0.126431i	$-0.959648\pi$
$887$		−	7.53086i		−	0.252861i	0.991975	−	0.126431i	$-0.0403521\pi$
$888$		0			0
$889$			1.70243i			0.0570975i
$890$		0			0
$891$			2.05044i			0.0686925i
$892$		0			0
$893$	63.3256			2.11911
$894$		0			0
$895$		−	28.5894i		−	0.955638i
$896$		0			0
$897$	4.49768			0.150173
$898$		0			0
$899$		−	23.2461i		−	0.775301i
$900$		0			0
$901$	−2.12887			−0.0709230
$902$		0			0
$903$	−1.37188			−0.0456533
$904$		0			0
$905$		−	25.2883i		−	0.840611i
$906$		0			0
$907$	36.1410			1.20004			0.600022	−	0.799984i	$-0.295159\pi$
$907$	36.1410			1.20004			0.600022	+	0.799984i	$0.295159\pi$
$908$		0			0
$909$		−	0.241478i		−	0.00800932i
$910$		0			0
$911$	54.5543			1.80747			0.903733	−	0.428098i	$-0.140816\pi$
$911$	54.5543			1.80747			0.903733	+	0.428098i	$0.140816\pi$
$912$		0			0
$913$			3.49823i			0.115775i
$914$		0			0
$915$			15.7413i			0.520390i
$916$		0			0
$917$		−	1.47156i		−	0.0485952i
$918$		0			0
$919$		−	9.85797i		−	0.325184i	−0.986693	−	0.162592i	$-0.948014\pi$
$919$		−	9.85797i		−	0.325184i	0.986693	−	0.162592i	$-0.0519855\pi$
$920$		0			0
$921$		−	13.9536i		−	0.459788i
$922$		0			0
$923$	3.54813			0.116788
$924$		0			0
$925$	10.7192			0.352447
$926$		0			0
$927$	0.836950			0.0274890
$928$		0			0
$929$		−	18.0684i		−	0.592804i	−0.955063	−	0.296402i	$-0.904213\pi$
$929$		−	18.0684i		−	0.592804i	0.955063	−	0.296402i	$-0.0957868\pi$
$930$		0			0
$931$		−	42.9159i		−	1.40651i
$932$		0			0
$933$	−0.986714			−0.0323036
$934$		0			0
$935$	−12.6306			−0.413066
$936$		0			0
$937$			3.14823i			0.102848i	0.998677	+	0.0514242i	$0.0163760\pi$
$937$			3.14823i			0.102848i	−0.998677	+	0.0514242i	$0.983624\pi$
$938$		0			0
$939$	5.19048			0.169385
$940$		0			0
$941$	0.0433906			0.00141449			0.000707246	−	1.00000i	$-0.499775\pi$
$941$	0.0433906			0.00141449			0.000707246		1.00000i	$0.499775\pi$
$942$		0			0
$943$	−1.76626	+	30.2777i	−0.0575172	+	0.985976i
$944$		0			0
$945$	−0.291922			−0.00949622
$946$		0			0
$947$	−8.73917			−0.283985			−0.141992	−	0.989868i	$-0.545351\pi$
$947$	−8.73917			−0.283985			−0.141992	+	0.989868i	$0.545351\pi$
$948$		0			0
$949$		−	11.4987i		−	0.373262i
$950$		0			0
$951$	−33.5757			−1.08877
$952$		0			0
$953$	26.4727			0.857534			0.428767	−	0.903415i	$-0.358948\pi$
$953$	26.4727			0.857534			0.428767	+	0.903415i	$0.358948\pi$
$954$		0			0
$955$		−	21.3673i		−	0.691428i
$956$		0			0
$957$			6.68617i			0.216133i
$958$		0			0
$959$	1.80146			0.0581722
$960$		0			0
$961$	19.8208			0.639381
$962$		0			0
$963$	−13.0876			−0.421742
$964$		0			0
$965$			14.2603i			0.459056i
$966$		0			0
$967$			57.9064i			1.86215i	0.364834	+	0.931073i	$0.381126\pi$
$967$			57.9064i			1.86215i	−0.364834	+	0.931073i	$0.618874\pi$
$968$		0			0
$969$		−	23.6388i		−	0.759389i
$970$		0			0
$971$			1.25408i			0.0402454i	0.999798	+	0.0201227i	$0.00640569\pi$
$971$			1.25408i			0.0402454i	−0.999798	+	0.0201227i	$0.993594\pi$
$972$		0			0
$973$			0.626553i			0.0200864i
$974$		0			0
$975$	−2.30110			−0.0736940
$976$		0			0
$977$		−	6.54346i		−	0.209344i	−0.994507	−	0.104672i	$-0.966621\pi$
$977$		−	6.54346i		−	0.209344i	0.994507	−	0.104672i	$-0.0333792\pi$
$978$		0			0
$979$	9.77589			0.312439
$980$		0			0
$981$			10.6027i			0.338517i
$982$		0			0
$983$	−48.5860			−1.54965			−0.774826	−	0.632175i	$-0.782163\pi$
$983$	−48.5860			−1.54965			−0.774826	+	0.632175i	$0.782163\pi$
$984$		0			0
$985$	14.8344			0.472663
$986$		0			0
$987$		−	1.86956i		−	0.0595088i
$988$		0			0
$989$	35.7311			1.13618
$990$		0			0
$991$			58.6164i			1.86201i	0.365006	+	0.931005i	$0.381067\pi$
$991$			58.6164i			1.86201i	−0.365006	+	0.931005i	$0.618933\pi$
$992$		0			0
$993$	20.7646			0.658945
$994$		0			0
$995$			9.07754i			0.287777i
$996$		0			0
$997$			21.2870i			0.674165i	0.941475	+	0.337082i	$0.109440\pi$
$997$			21.2870i			0.674165i	−0.941475	+	0.337082i	$0.890560\pi$
$998$		0			0
$999$			4.42334i			0.139948i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1968.2.j.f.1393.5		8
4.3	odd	2		984.2.j.b.409.1	✓	8
12.11	even	2		2952.2.j.d.2377.7		8
41.40	even	2	inner	1968.2.j.f.1393.1		8
164.163	odd	2		984.2.j.b.409.5	yes	8
492.491	even	2		2952.2.j.d.2377.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.2.j.b.409.1	✓	8	4.3	odd	2
984.2.j.b.409.5	yes	8	164.163	odd	2
1968.2.j.f.1393.1		8	41.40	even	2	inner
1968.2.j.f.1393.5		8	1.1	even	1	trivial
2952.2.j.d.2377.7		8	12.11	even	2
2952.2.j.d.2377.8		8	492.491	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1968 = 2^{4} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1968.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.60520 q^{5} +0.181860i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.60520 q^{5} +0.181860i q^{7} -1.00000 q^{9} +2.05044i q^{11} -0.949556i q^{13} -1.60520i q^{15} -3.83750i q^{17} -6.15995i q^{19} -0.181860 q^{21} +4.73661 q^{23} -2.42334 q^{25} -1.00000i q^{27} -3.26084i q^{29} +7.12887 q^{31} -2.05044 q^{33} -0.291922i q^{35} -4.42334 q^{37} +0.949556 q^{39} +(-0.372894 + 6.39226i) q^{41} +7.54359 q^{43} +1.60520 q^{45} +10.2802i q^{47} +6.96693 q^{49} +3.83750 q^{51} -0.554754i q^{53} -3.29137i q^{55} +6.15995 q^{57} +11.6531 q^{59} -9.80642 q^{61} -0.181860i q^{63} +1.52423i q^{65} -7.91593i q^{67} +4.73661i q^{69} +3.73661i q^{71} +12.1095 q^{73} -2.42334i q^{75} -0.372894 q^{77} -8.84668i q^{79} +1.00000 q^{81} +1.70609 q^{83} +6.15995i q^{85} +3.26084 q^{87} -4.76770i q^{89} +0.172687 q^{91} +7.12887i q^{93} +9.88795i q^{95} -9.39226i q^{97} -2.05044i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q + 10 * q^5 - 8 * q^9	\( 8 q + 10 q^{5} - 8 q^{9} + 20 q^{23} + 2 q^{25} - 8 q^{31} - 10 q^{33} - 14 q^{37} + 14 q^{39} + 12 q^{41} - 6 q^{43} - 10 q^{45} - 32 q^{49} + 10 q^{57} - 6 q^{59} - 22 q^{61} + 64 q^{73} + 12 q^{77} + 8 q^{81} - 22 q^{83} - 26 q^{87} + 12 q^{91}+O(q^{100}) \) 8 * q + 10 * q^5 - 8 * q^9 + 20 * q^23 + 2 * q^25 - 8 * q^31 - 10 * q^33 - 14 * q^37 + 14 * q^39 + 12 * q^41 - 6 * q^43 - 10 * q^45 - 32 * q^49 + 10 * q^57 - 6 * q^59 - 22 * q^61 + 64 * q^73 + 12 * q^77 + 8 * q^81 - 22 * q^83 - 26 * q^87 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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