LMFDB - Embedded newform 1960.2.g.g.1569.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1569,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1960.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.g (of order $2$, degree $1$, 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:	$15.6506787962$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 122x^{16} + 2481x^{12} + 15576x^{8} + 27792x^{4} + 16 $ x^20 + 122x^16 + 2481x^12 + 15576x^8 + 27792x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{25} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.7
Root			$0.934141 - 0.934141i$ of defining polynomial
Character	$\chi$	$=$	1960.1569
Dual form			1960.2.g.g.1569.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.12212i q^{3} +(-0.981033 - 2.00937i) q^{5} +1.74084 q^{9} +O(q^{10})$	$q-1.12212i q^{3} +(-0.981033 - 2.00937i) q^{5} +1.74084 q^{9} -4.94252 q^{11} +4.64101i q^{13} +(-2.25476 + 1.10084i) q^{15} +5.72505i q^{17} +7.53763 q^{19} +6.87122i q^{23} +(-3.07515 + 3.94252i) q^{25} -5.31980i q^{27} -8.11636 q^{29} +3.87614 q^{31} +5.54611i q^{33} +5.18786i q^{37} +5.20778 q^{39} +9.45170 q^{41} +0.706904i q^{43} +(-1.70782 - 3.49800i) q^{45} -2.20616i q^{47} +6.42420 q^{51} +1.32166i q^{53} +(4.84877 + 9.93136i) q^{55} -8.45814i q^{57} -3.94064 q^{59} +6.07542 q^{61} +(9.32551 - 4.55298i) q^{65} -8.24674i q^{67} +7.71035 q^{69} -4.50952 q^{71} +2.93470i q^{73} +(4.42399 + 3.45069i) q^{75} +11.0249 q^{79} -0.746943 q^{81} -5.27356i q^{83} +(11.5037 - 5.61646i) q^{85} +9.10755i q^{87} -9.49970 q^{89} -4.34950i q^{93} +(-7.39466 - 15.1459i) q^{95} +1.74808i q^{97} -8.60415 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 28 q^{9}+O(q^{10}) $ 20 * q - 28 * q^9	$ 20 q - 28 q^{9} - 24 q^{11} - 8 q^{15} + 8 q^{25} - 24 q^{29} + 40 q^{39} - 72 q^{51} - 20 q^{65} - 16 q^{71} + 8 q^{79} + 100 q^{81} + 60 q^{85} - 48 q^{95} + 144 q^{99}+O(q^{100}) $ 20 * q - 28 * q^9 - 24 * q^11 - 8 * q^15 + 8 * q^25 - 24 * q^29 + 40 * q^39 - 72 * q^51 - 20 * q^65 - 16 * q^71 + 8 * q^79 + 100 * q^81 + 60 * q^85 - 48 * q^95 + 144 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$.

$n$	$981$	$1081$	$1177$	$1471$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		−	1.12212i		−	0.647857i	−0.946081	−	0.323929i	$-0.894996\pi$
$3$		−	1.12212i		−	0.647857i	0.946081	−	0.323929i	$-0.105004\pi$
$4$		0			0
$5$	−0.981033	−	2.00937i	−0.438731	−	0.898618i
$5$	−0.981033	−	2.00937i	−0.438731	−	0.898618i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.74084			0.580281
$10$		0			0
$11$	−4.94252			−1.49023			−0.745113	−	0.666938i	$-0.767604\pi$
$11$	−4.94252			−1.49023			−0.745113	+	0.666938i	$0.767604\pi$
$12$		0			0
$13$			4.64101i			1.28718i	0.765369	+	0.643592i	$0.222557\pi$
$13$			4.64101i			1.28718i	−0.765369	+	0.643592i	$0.777443\pi$
$14$		0			0
$15$	−2.25476	+	1.10084i	−0.582177	+	0.284235i
$16$		0			0
$17$			5.72505i			1.38853i	0.719720	+	0.694264i	$0.244270\pi$
$17$			5.72505i			1.38853i	−0.719720	+	0.694264i	$0.755730\pi$
$18$		0			0
$19$	7.53763			1.72925			0.864625	−	0.502417i	$-0.167556\pi$
$19$	7.53763			1.72925			0.864625	+	0.502417i	$0.167556\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.87122i			1.43275i	0.697716	+	0.716374i	$0.254200\pi$
$23$			6.87122i			1.43275i	−0.697716	+	0.716374i	$0.745800\pi$
$24$		0			0
$25$	−3.07515	+	3.94252i	−0.615030	+	0.788504i
$26$		0			0
$27$		−	5.31980i		−	1.02380i
$28$		0			0
$29$	−8.11636			−1.50717			−0.753585	−	0.657350i	$-0.771677\pi$
$29$	−8.11636			−1.50717			−0.753585	+	0.657350i	$0.771677\pi$
$30$		0			0
$31$	3.87614			0.696175			0.348087	−	0.937462i	$-0.386831\pi$
$31$	3.87614			0.696175			0.348087	+	0.937462i	$0.386831\pi$
$32$		0			0
$33$			5.54611i			0.965454i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			5.18786i			0.852879i	0.904516	+	0.426439i	$0.140232\pi$
$37$			5.18786i			0.852879i	−0.904516	+	0.426439i	$0.859768\pi$
$38$		0			0
$39$	5.20778			0.833912
$40$		0			0
$41$	9.45170			1.47611			0.738054	−	0.674742i	$-0.235745\pi$
$41$	9.45170			1.47611			0.738054	+	0.674742i	$0.235745\pi$
$42$		0			0
$43$			0.706904i			0.107802i	0.998546	+	0.0539009i	$0.0171655\pi$
$43$			0.706904i			0.107802i	−0.998546	+	0.0539009i	$0.982834\pi$
$44$		0			0
$45$	−1.70782	−	3.49800i	−0.254587	−	0.521451i
$46$		0			0
$47$		−	2.20616i		−	0.321802i	−0.986971	−	0.160901i	$-0.948560\pi$
$47$		−	2.20616i		−	0.321802i	0.986971	−	0.160901i	$-0.0514399\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	6.42420			0.899568
$52$		0			0
$53$			1.32166i			0.181544i	0.995872	+	0.0907721i	$0.0289335\pi$
$53$			1.32166i			0.181544i	−0.995872	+	0.0907721i	$0.971067\pi$
$54$		0			0
$55$	4.84877	+	9.93136i	0.653809	+	1.33914i
$56$		0			0
$57$		−	8.45814i		−	1.12031i
$58$		0			0
$59$	−3.94064			−0.513027			−0.256514	−	0.966541i	$-0.582574\pi$
$59$	−3.94064			−0.513027			−0.256514	+	0.966541i	$0.582574\pi$
$60$		0			0
$61$	6.07542			0.777878			0.388939	−	0.921264i	$-0.372842\pi$
$61$	6.07542			0.777878			0.388939	+	0.921264i	$0.372842\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.32551	−	4.55298i	1.15669	−	0.564728i
$66$		0			0
$67$		−	8.24674i		−	1.00750i	−0.863850	−	0.503750i	$-0.831953\pi$
$67$		−	8.24674i		−	1.00750i	0.863850	−	0.503750i	$-0.168047\pi$
$68$		0			0
$69$	7.71035			0.928217
$70$		0			0
$71$	−4.50952			−0.535182			−0.267591	−	0.963533i	$-0.586228\pi$
$71$	−4.50952			−0.535182			−0.267591	+	0.963533i	$0.586228\pi$
$72$		0			0
$73$			2.93470i			0.343481i	0.985142	+	0.171741i	$0.0549391\pi$
$73$			2.93470i			0.343481i	−0.985142	+	0.171741i	$0.945061\pi$
$74$		0			0
$75$	4.42399	+	3.45069i	0.510838	+	0.398452i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	11.0249			1.24040			0.620201	−	0.784443i	$-0.287051\pi$
$79$	11.0249			1.24040			0.620201	+	0.784443i	$0.287051\pi$
$80$		0			0
$81$	−0.746943			−0.0829937
$82$		0			0
$83$		−	5.27356i		−	0.578848i	−0.957201	−	0.289424i	$-0.906536\pi$
$83$		−	5.27356i		−	0.578848i	0.957201	−	0.289424i	$-0.0934638\pi$
$84$		0			0
$85$	11.5037	−	5.61646i	1.24776	−	0.609191i
$86$		0			0
$87$			9.10755i			0.976432i
$88$		0			0
$89$	−9.49970			−1.00697			−0.503483	−	0.864005i	$-0.667948\pi$
$89$	−9.49970			−1.00697			−0.503483	+	0.864005i	$0.667948\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	4.34950i		−	0.451022i
$94$		0			0
$95$	−7.39466	−	15.1459i	−0.758676	−	1.55394i
$96$		0			0
$97$			1.74808i			0.177491i	0.996054	+	0.0887454i	$0.0282857\pi$
$97$			1.74808i			0.177491i	−0.996054	+	0.0887454i	$0.971714\pi$
$98$		0			0
$99$	−8.60415			−0.864749
$100$		0			0
$101$	−5.22585			−0.519991			−0.259996	−	0.965610i	$-0.583721\pi$
$101$	−5.22585			−0.519991			−0.259996	+	0.965610i	$0.583721\pi$
$102$		0			0
$103$		−	4.15144i		−	0.409053i	−0.978861	−	0.204527i	$-0.934434\pi$
$103$		−	4.15144i		−	0.409053i	0.978861	−	0.204527i	$-0.0655655\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			16.7234i			1.61671i	0.588694	+	0.808356i	$0.299642\pi$
$107$			16.7234i			1.61671i	−0.588694	+	0.808356i	$0.700358\pi$
$108$		0			0
$109$	−11.3182			−1.08409			−0.542045	−	0.840349i	$-0.682350\pi$
$109$	−11.3182			−1.08409			−0.542045	+	0.840349i	$0.682350\pi$
$110$		0			0
$111$	5.82141			0.552544
$112$		0			0
$113$			0.614757i			0.0578315i	0.999582	+	0.0289157i	$0.00920545\pi$
$113$			0.614757i			0.0578315i	−0.999582	+	0.0289157i	$0.990795\pi$
$114$		0			0
$115$	13.8068	−	6.74089i	1.28749	−	0.628592i
$116$		0			0
$117$			8.07926i			0.746928i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	13.4285			1.22077
$122$		0			0
$123$		−	10.6060i		−	0.956307i
$124$		0			0
$125$	10.9388	+	2.31138i	0.978397	+	0.206736i
$126$		0			0
$127$			5.32166i			0.472221i	0.971726	+	0.236111i	$0.0758727\pi$
$127$			5.32166i			0.472221i	−0.971726	+	0.236111i	$0.924127\pi$
$128$		0			0
$129$	0.793233			0.0698402
$130$		0			0
$131$	7.30041			0.637840			0.318920	−	0.947782i	$-0.396680\pi$
$131$	7.30041			0.637840			0.318920	+	0.947782i	$0.396680\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−10.6895	+	5.21890i	−0.920002	+	0.449172i
$136$		0			0
$137$			21.9675i			1.87681i	0.345542	+	0.938403i	$0.387695\pi$
$137$			21.9675i			1.87681i	−0.345542	+	0.938403i	$0.612305\pi$
$138$		0			0
$139$	9.46849			0.803107			0.401553	−	0.915836i	$-0.368470\pi$
$139$	9.46849			0.803107			0.401553	+	0.915836i	$0.368470\pi$
$140$		0			0
$141$	−2.47558			−0.208482
$142$		0			0
$143$		−	22.9383i		−	1.91820i
$144$		0			0
$145$	7.96242	+	16.3088i	0.661243	+	1.35437i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.7823			1.29293			0.646467	−	0.762942i	$-0.276246\pi$
$149$	15.7823			1.29293			0.646467	+	0.762942i	$0.276246\pi$
$150$		0			0
$151$	11.2669			0.916884			0.458442	−	0.888724i	$-0.348408\pi$
$151$	11.2669			0.916884			0.458442	+	0.888724i	$0.348408\pi$
$152$		0			0
$153$			9.96640i			0.805736i
$154$		0			0
$155$	−3.80262	−	7.78860i	−0.305434	−	0.625595i
$156$		0			0
$157$			8.66981i			0.691926i	0.938248	+	0.345963i	$0.112448\pi$
$157$			8.66981i			0.691926i	−0.938248	+	0.345963i	$0.887552\pi$
$158$		0			0
$159$	1.48307			0.117615
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	24.1275i		−	1.88981i	−0.327344	−	0.944905i	$-0.606154\pi$
$163$		−	24.1275i		−	1.88981i	0.327344	−	0.944905i	$-0.393846\pi$
$164$		0			0
$165$	11.1442	−	5.44092i	0.867575	−	0.423575i
$166$		0			0
$167$			16.9825i			1.31414i	0.753829	+	0.657071i	$0.228205\pi$
$167$			16.9825i			1.31414i	−0.753829	+	0.657071i	$0.771795\pi$
$168$		0			0
$169$	−8.53896			−0.656843
$170$		0			0
$171$	13.1218			1.00345
$172$		0			0
$173$			19.7691i			1.50302i	0.659723	+	0.751509i	$0.270673\pi$
$173$			19.7691i			1.50302i	−0.659723	+	0.751509i	$0.729327\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			4.42188i			0.332368i
$178$		0			0
$179$	2.11496			0.158080			0.0790398	−	0.996871i	$-0.474815\pi$
$179$	2.11496			0.158080			0.0790398	+	0.996871i	$0.474815\pi$
$180$		0			0
$181$	−9.23706			−0.686585			−0.343293	−	0.939229i	$-0.611542\pi$
$181$	−9.23706			−0.686585			−0.343293	+	0.939229i	$0.611542\pi$
$182$		0			0
$183$		−	6.81736i		−	0.503954i
$184$		0			0
$185$	10.4243	−	5.08946i	0.766413	−	0.374185i
$186$		0			0
$187$		−	28.2962i		−	2.06922i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.75734			−0.488944			−0.244472	−	0.969656i	$-0.578615\pi$
$191$	−6.75734			−0.488944			−0.244472	+	0.969656i	$0.578615\pi$
$192$		0			0
$193$		−	2.71570i		−	0.195480i	−0.995212	−	0.0977402i	$-0.968839\pi$
$193$		−	2.71570i		−	0.195480i	0.995212	−	0.0977402i	$-0.0311614\pi$
$194$		0			0
$195$	−5.10900	−	10.4644i	−0.365863	−	0.749369i
$196$		0			0
$197$			17.2189i			1.22680i	0.789774	+	0.613398i	$0.210198\pi$
$197$			17.2189i			1.22680i	−0.789774	+	0.613398i	$0.789802\pi$
$198$		0			0
$199$	−14.0215			−0.993956			−0.496978	−	0.867763i	$-0.665557\pi$
$199$	−14.0215			−0.993956			−0.496978	+	0.867763i	$0.665557\pi$
$200$		0			0
$201$	−9.25385			−0.652716
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−9.27243	−	18.9920i	−0.647615	−	1.32646i
$206$		0			0
$207$			11.9617i			0.831396i
$208$		0			0
$209$	−37.2549			−2.57697
$210$		0			0
$211$	−9.78518			−0.673639			−0.336820	−	0.941569i	$-0.609351\pi$
$211$	−9.78518			−0.673639			−0.336820	+	0.941569i	$0.609351\pi$
$212$		0			0
$213$			5.06023i			0.346721i
$214$		0			0
$215$	1.42043	−	0.693496i	0.0968727	−	0.0472960i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	3.29310			0.222527
$220$		0			0
$221$	−26.5700			−1.78729
$222$		0			0
$223$			7.13608i			0.477867i	0.971036	+	0.238934i	$0.0767978\pi$
$223$			7.13608i			0.477867i	−0.971036	+	0.238934i	$0.923202\pi$
$224$		0			0
$225$	−5.35335	+	6.86330i	−0.356890	+	0.457554i
$226$		0			0
$227$			0.773180i			0.0513178i	0.999671	+	0.0256589i	$0.00816838\pi$
$227$			0.773180i			0.0513178i	−0.999671	+	0.0256589i	$0.991832\pi$
$228$		0			0
$229$	14.2336			0.940580			0.470290	−	0.882512i	$-0.344149\pi$
$229$	14.2336			0.940580			0.470290	+	0.882512i	$0.344149\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			11.7275i			0.768294i	0.923272	+	0.384147i	$0.125504\pi$
$233$			11.7275i			0.768294i	−0.923272	+	0.384147i	$0.874496\pi$
$234$		0			0
$235$	−4.43300	+	2.16432i	−0.289177	+	0.141184i
$236$		0			0
$237$		−	12.3713i		−	0.803604i
$238$		0			0
$239$	14.3330			0.927124			0.463562	−	0.886065i	$-0.346571\pi$
$239$	14.3330			0.927124			0.463562	+	0.886065i	$0.346571\pi$
$240$		0			0
$241$	−2.07456			−0.133634			−0.0668171	−	0.997765i	$-0.521284\pi$
$241$	−2.07456			−0.133634			−0.0668171	+	0.997765i	$0.521284\pi$
$242$		0			0
$243$		−	15.1212i		−	0.970029i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			34.9822i			2.22586i
$248$		0			0
$249$	−5.91758			−0.375011
$250$		0			0
$251$	2.81192			0.177487			0.0887435	−	0.996055i	$-0.471715\pi$
$251$	2.81192			0.177487			0.0887435	+	0.996055i	$0.471715\pi$
$252$		0			0
$253$		−	33.9611i		−	2.13512i
$254$		0			0
$255$	−6.30235	−	12.9086i	−0.394669	−	0.808369i
$256$		0			0
$257$		−	21.5852i		−	1.34644i	−0.739440	−	0.673222i	$-0.764910\pi$
$257$		−	21.5852i		−	1.34644i	0.739440	−	0.673222i	$-0.235090\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−14.1293			−0.874582
$262$		0			0
$263$		−	16.9163i		−	1.04310i	−0.853220	−	0.521551i	$-0.825354\pi$
$263$		−	16.9163i		−	1.04310i	0.853220	−	0.521551i	$-0.174646\pi$
$264$		0			0
$265$	2.65571	−	1.29659i	0.163139	−	0.0796491i
$266$		0			0
$267$			10.6598i			0.652370i
$268$		0			0
$269$	−11.3325			−0.690953			−0.345477	−	0.938427i	$-0.612283\pi$
$269$	−11.3325			−0.690953			−0.345477	+	0.938427i	$0.612283\pi$
$270$		0			0
$271$	3.11364			0.189140			0.0945701	−	0.995518i	$-0.469852\pi$
$271$	3.11364			0.189140			0.0945701	+	0.995518i	$0.469852\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.1990	−	19.4860i	0.916533	−	1.17505i
$276$		0			0
$277$			26.6607i			1.60189i	0.598739	+	0.800945i	$0.295669\pi$
$277$			26.6607i			1.60189i	−0.598739	+	0.800945i	$0.704331\pi$
$278$		0			0
$279$	6.74774			0.403977
$280$		0			0
$281$	28.2579			1.68572			0.842861	−	0.538131i	$-0.180869\pi$
$281$	28.2579			1.68572			0.842861	+	0.538131i	$0.180869\pi$
$282$		0			0
$283$			13.9531i			0.829428i	0.909952	+	0.414714i	$0.136118\pi$
$283$			13.9531i			0.829428i	−0.909952	+	0.414714i	$0.863882\pi$
$284$		0			0
$285$	−16.9955	+	8.29771i	−1.00673	+	0.491514i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.7762			−0.928011
$290$		0			0
$291$	1.96156			0.114989
$292$		0			0
$293$			6.50405i			0.379970i	0.981787	+	0.189985i	$0.0608440\pi$
$293$			6.50405i			0.379970i	−0.981787	+	0.189985i	$0.939156\pi$
$294$		0			0
$295$	3.86589	+	7.91820i	0.225081	+	0.461016i
$296$		0			0
$297$			26.2932i			1.52569i
$298$		0			0
$299$	−31.8894			−1.84421
$300$		0			0
$301$		0			0
$302$		0			0
$303$			5.86404i			0.336880i
$304$		0			0
$305$	−5.96019	−	12.2078i	−0.341279	−	0.699015i
$306$		0			0
$307$		−	28.4580i		−	1.62418i	−0.583531	−	0.812091i	$-0.698329\pi$
$307$		−	28.4580i		−	1.62418i	0.583531	−	0.812091i	$-0.301671\pi$
$308$		0			0
$309$	−4.65842			−0.265008
$310$		0			0
$311$	6.10465			0.346163			0.173081	−	0.984908i	$-0.444628\pi$
$311$	6.10465			0.346163			0.173081	+	0.984908i	$0.444628\pi$
$312$		0			0
$313$			4.42029i			0.249850i	0.992166	+	0.124925i	$0.0398690\pi$
$313$			4.42029i			0.249850i	−0.992166	+	0.124925i	$0.960131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	5.67611i		−	0.318802i	−0.987214	−	0.159401i	$-0.949044\pi$
$317$		−	5.67611i		−	0.318802i	0.987214	−	0.159401i	$-0.0509563\pi$
$318$		0			0
$319$	40.1153			2.24602
$320$		0			0
$321$	18.7657			1.04740
$322$		0			0
$323$			43.1533i			2.40111i
$324$		0			0
$325$	−18.2973	−	14.2718i	−1.01495	−	0.791657i
$326$		0			0
$327$			12.7004i			0.702336i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−24.4091			−1.34165			−0.670823	−	0.741618i	$-0.734059\pi$
$331$	−24.4091			−1.34165			−0.670823	+	0.741618i	$0.734059\pi$
$332$		0			0
$333$			9.03124i			0.494909i
$334$		0			0
$335$	−16.5708	+	8.09032i	−0.905358	+	0.442022i
$336$		0			0
$337$			2.84018i			0.154714i	0.997003	+	0.0773572i	$0.0246482\pi$
$337$			2.84018i			0.154714i	−0.997003	+	0.0773572i	$0.975352\pi$
$338$		0			0
$339$	0.689832			0.0374666
$340$		0			0
$341$	−19.1579			−1.03746
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−7.56411	−	15.4930i	−0.407238	−	0.834113i
$346$		0			0
$347$			5.47921i			0.294139i	0.989126	+	0.147070i	$0.0469841\pi$
$347$			5.47921i			0.294139i	−0.989126	+	0.147070i	$0.953016\pi$
$348$		0			0
$349$	−11.5197			−0.616637			−0.308318	−	0.951283i	$-0.599766\pi$
$349$	−11.5197			−0.616637			−0.308318	+	0.951283i	$0.599766\pi$
$350$		0			0
$351$	24.6893			1.31781
$352$		0			0
$353$		−	11.1927i		−	0.595726i	−0.954609	−	0.297863i	$-0.903726\pi$
$353$		−	11.1927i		−	0.595726i	0.954609	−	0.297863i	$-0.0962739\pi$
$354$		0			0
$355$	4.42399	+	9.06130i	0.234801	+	0.480924i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−35.2726			−1.86161			−0.930807	−	0.365511i	$-0.880894\pi$
$359$	−35.2726			−1.86161			−0.930807	+	0.365511i	$0.880894\pi$
$360$		0			0
$361$	37.8159			1.99031
$362$		0			0
$363$		−	15.0684i		−	0.790887i
$364$		0			0
$365$	5.89691	−	2.87904i	0.308658	−	0.150696i
$366$		0			0
$367$			31.1383i			1.62541i	0.582677	+	0.812704i	$0.302005\pi$
$367$			31.1383i			1.62541i	−0.582677	+	0.812704i	$0.697995\pi$
$368$		0			0
$369$	16.4539			0.856557
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	10.9531i		−	0.567131i	−0.958953	−	0.283565i	$-0.908483\pi$
$373$		−	10.9531i		−	0.567131i	0.958953	−	0.283565i	$-0.0915173\pi$
$374$		0			0
$375$	2.59365	−	12.2747i	0.133935	−	0.633862i
$376$		0			0
$377$		−	37.6681i		−	1.94001i
$378$		0			0
$379$	−7.34338			−0.377204			−0.188602	−	0.982054i	$-0.560396\pi$
$379$	−7.34338			−0.377204			−0.188602	+	0.982054i	$0.560396\pi$
$380$		0			0
$381$	5.97155			0.305932
$382$		0			0
$383$		−	3.73491i		−	0.190845i	−0.995437	−	0.0954224i	$-0.969580\pi$
$383$		−	3.73491i		−	0.190845i	0.995437	−	0.0954224i	$-0.0304202\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			1.23061i			0.0625553i
$388$		0			0
$389$	−9.29745			−0.471399			−0.235700	−	0.971826i	$-0.575738\pi$
$389$	−9.29745			−0.471399			−0.235700	+	0.971826i	$0.575738\pi$
$390$		0			0
$391$	−39.3381			−1.98941
$392$		0			0
$393$		−	8.19195i		−	0.413229i
$394$		0			0
$395$	−10.8158	−	22.1532i	−0.544203	−	1.11465i
$396$		0			0
$397$		−	13.2764i		−	0.666325i	−0.942869	−	0.333163i	$-0.891884\pi$
$397$		−	13.2764i		−	0.666325i	0.942869	−	0.333163i	$-0.108116\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.01019			−0.200260			−0.100130	−	0.994974i	$-0.531926\pi$
$401$	−4.01019			−0.200260			−0.100130	+	0.994974i	$0.531926\pi$
$402$		0			0
$403$			17.9892i			0.896105i
$404$		0			0
$405$	0.732776	+	1.50089i	0.0364119	+	0.0745797i
$406$		0			0
$407$		−	25.6411i		−	1.27098i
$408$		0			0
$409$	29.2260			1.44513			0.722567	−	0.691301i	$-0.242962\pi$
$409$	29.2260			1.44513			0.722567	+	0.691301i	$0.242962\pi$
$410$		0			0
$411$	24.6502			1.21590
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−10.5965	+	5.17354i	−0.520164	+	0.253959i
$416$		0			0
$417$		−	10.6248i		−	0.520299i
$418$		0			0
$419$	29.8552			1.45852			0.729260	−	0.684237i	$-0.239864\pi$
$419$	29.8552			1.45852			0.729260	+	0.684237i	$0.239864\pi$
$420$		0			0
$421$	−9.81281			−0.478247			−0.239124	−	0.970989i	$-0.576860\pi$
$421$	−9.81281			−0.478247			−0.239124	+	0.970989i	$0.576860\pi$
$422$		0			0
$423$		−	3.84058i		−	0.186735i
$424$		0			0
$425$	−22.5711	−	17.6054i	−1.09486	−	0.853986i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−25.7395			−1.24272
$430$		0			0
$431$	−17.0840			−0.822908			−0.411454	−	0.911430i	$-0.634979\pi$
$431$	−17.0840			−0.822908			−0.411454	+	0.911430i	$0.634979\pi$
$432$		0			0
$433$		−	9.70709i		−	0.466493i	−0.972418	−	0.233246i	$-0.925065\pi$
$433$		−	9.70709i		−	0.466493i	0.972418	−	0.233246i	$-0.0749349\pi$
$434$		0			0
$435$	18.3004	−	8.93480i	0.877439	−	0.428391i
$436$		0			0
$437$			51.7927i			2.47758i
$438$		0			0
$439$	−10.4222			−0.497425			−0.248713	−	0.968577i	$-0.580007\pi$
$439$	−10.4222			−0.497425			−0.248713	+	0.968577i	$0.580007\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	34.8412i		−	1.65535i	−0.561205	−	0.827677i	$-0.689662\pi$
$443$		−	34.8412i		−	1.65535i	0.561205	−	0.827677i	$-0.310338\pi$
$444$		0			0
$445$	9.31951	+	19.0884i	0.441787	+	0.904878i
$446$		0			0
$447$		−	17.7096i		−	0.837637i
$448$		0			0
$449$	8.64043			0.407767			0.203883	−	0.978995i	$-0.434644\pi$
$449$	8.64043			0.407767			0.203883	+	0.978995i	$0.434644\pi$
$450$		0			0
$451$	−46.7152			−2.19973
$452$		0			0
$453$		−	12.6428i		−	0.594010i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			25.6058i			1.19779i	0.800829	+	0.598893i	$0.204393\pi$
$457$			25.6058i			1.19779i	−0.800829	+	0.598893i	$0.795607\pi$
$458$		0			0
$459$	30.4561			1.42157
$460$		0			0
$461$	−23.8105			−1.10896			−0.554482	−	0.832196i	$-0.687083\pi$
$461$	−23.8105			−1.10896			−0.554482	+	0.832196i	$0.687083\pi$
$462$		0			0
$463$		−	3.27660i		−	0.152276i	−0.997097	−	0.0761382i	$-0.975741\pi$
$463$		−	3.27660i		−	0.152276i	0.997097	−	0.0761382i	$-0.0242590\pi$
$464$		0			0
$465$	−8.73976	+	4.26700i	−0.405297	+	0.197877i
$466$		0			0
$467$		−	29.1321i		−	1.34807i	−0.738699	−	0.674036i	$-0.764559\pi$
$467$		−	29.1321i		−	1.34807i	0.738699	−	0.674036i	$-0.235441\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	9.72859			0.448270
$472$		0			0
$473$		−	3.49389i		−	0.160649i
$474$		0			0
$475$	−23.1793	+	29.7173i	−1.06354	+	1.36352i
$476$		0			0
$477$			2.30080i			0.105347i
$478$		0			0
$479$	−7.58771			−0.346692			−0.173346	−	0.984861i	$-0.555458\pi$
$479$	−7.58771			−0.346692			−0.173346	+	0.984861i	$0.555458\pi$
$480$		0			0
$481$	−24.0769			−1.09781
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.51255	−	1.71493i	0.159496	−	0.0778707i
$486$		0			0
$487$			4.86118i			0.220281i	0.993916	+	0.110140i	$0.0351301\pi$
$487$			4.86118i			0.220281i	−0.993916	+	0.110140i	$0.964870\pi$
$488$		0			0
$489$	−27.0740			−1.22433
$490$		0			0
$491$	5.95002			0.268521			0.134260	−	0.990946i	$-0.457134\pi$
$491$	5.95002			0.268521			0.134260	+	0.990946i	$0.457134\pi$
$492$		0			0
$493$		−	46.4666i		−	2.09275i
$494$		0			0
$495$	8.44095	+	17.2889i	0.379392	+	0.777079i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.08297			0.0932467			0.0466234	−	0.998913i	$-0.485154\pi$
$499$	2.08297			0.0932467			0.0466234	+	0.998913i	$0.485154\pi$
$500$		0			0
$501$	19.0564			0.851377
$502$		0			0
$503$		−	2.22949i		−	0.0994081i	−0.998764	−	0.0497041i	$-0.984172\pi$
$503$		−	2.22949i		−	0.0994081i	0.998764	−	0.0497041i	$-0.0158278\pi$
$504$		0			0
$505$	5.12673	+	10.5007i	0.228136	+	0.467274i
$506$		0			0
$507$			9.58176i			0.425541i
$508$		0			0
$509$	−1.75149			−0.0776334			−0.0388167	−	0.999246i	$-0.512359\pi$
$509$	−1.75149			−0.0776334			−0.0388167	+	0.999246i	$0.512359\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	40.0987i		−	1.77040i
$514$		0			0
$515$	−8.34178	+	4.07270i	−0.367583	+	0.179464i
$516$		0			0
$517$			10.9040i			0.479557i
$518$		0			0
$519$	22.1833			0.973741
$520$		0			0
$521$	−16.7933			−0.735727			−0.367864	−	0.929880i	$-0.619911\pi$
$521$	−16.7933			−0.735727			−0.367864	+	0.929880i	$0.619911\pi$
$522$		0			0
$523$			20.3763i			0.890995i	0.895283	+	0.445497i	$0.146973\pi$
$523$			20.3763i			0.890995i	−0.895283	+	0.445497i	$0.853027\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			22.1911i			0.966658i
$528$		0			0
$529$	−24.2137			−1.05277
$530$		0			0
$531$	−6.86002			−0.297700
$532$		0			0
$533$			43.8654i			1.90002i
$534$		0			0
$535$	33.6035	−	16.4062i	1.45281	−	0.709302i
$536$		0			0
$537$		−	2.37324i		−	0.102413i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−12.4407			−0.534867			−0.267434	−	0.963576i	$-0.586176\pi$
$541$	−12.4407			−0.534867			−0.267434	+	0.963576i	$0.586176\pi$
$542$		0			0
$543$			10.3651i			0.444809i
$544$		0			0
$545$	11.1036	+	22.7425i	0.475624	+	0.974184i
$546$		0			0
$547$			33.2004i			1.41955i	0.704431	+	0.709773i	$0.251202\pi$
$547$			33.2004i			1.41955i	−0.704431	+	0.709773i	$0.748798\pi$
$548$		0			0
$549$	10.5763			0.451388
$550$		0			0
$551$	−61.1781			−2.60628
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−5.71100	−	11.6974i	−0.242418	−	0.496526i
$556$		0			0
$557$		−	26.0813i		−	1.10510i	−0.833479	−	0.552551i	$-0.813655\pi$
$557$		−	26.0813i		−	1.10510i	0.833479	−	0.552551i	$-0.186345\pi$
$558$		0			0
$559$	−3.28075			−0.138761
$560$		0			0
$561$	−31.7517			−1.34056
$562$		0			0
$563$		−	39.4346i		−	1.66197i	−0.556294	−	0.830986i	$-0.687777\pi$
$563$		−	39.4346i		−	1.66197i	0.556294	−	0.830986i	$-0.312223\pi$
$564$		0			0
$565$	1.23528	−	0.603097i	0.0519684	−	0.0253725i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.09101			0.255348			0.127674	−	0.991816i	$-0.459249\pi$
$569$	6.09101			0.255348			0.127674	+	0.991816i	$0.459249\pi$
$570$		0			0
$571$	−24.9957			−1.04604			−0.523019	−	0.852321i	$-0.675194\pi$
$571$	−24.9957			−1.04604			−0.523019	+	0.852321i	$0.675194\pi$
$572$		0			0
$573$			7.58256i			0.316766i
$574$		0			0
$575$	−27.0899	−	21.1300i	−1.12973	−	0.881183i
$576$		0			0
$577$		−	23.4463i		−	0.976083i	−0.872820	−	0.488042i	$-0.837711\pi$
$577$		−	23.4463i		−	0.976083i	0.872820	−	0.488042i	$-0.162289\pi$
$578$		0			0
$579$	−3.04735			−0.126643
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	6.53234i		−	0.270542i
$584$		0			0
$585$	16.2342	−	7.92602i	0.671203	−	0.327701i
$586$		0			0
$587$		−	27.6033i		−	1.13931i	−0.821884	−	0.569656i	$-0.807077\pi$
$587$		−	27.6033i		−	1.13931i	0.821884	−	0.569656i	$-0.192923\pi$
$588$		0			0
$589$	29.2169			1.20386
$590$		0			0
$591$	19.3217			0.794789
$592$		0			0
$593$			5.40275i			0.221864i	0.993828	+	0.110932i	$0.0353836\pi$
$593$			5.40275i			0.221864i	−0.993828	+	0.110932i	$0.964616\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			15.7338i			0.643942i
$598$		0			0
$599$	−42.1269			−1.72126			−0.860628	−	0.509234i	$-0.829929\pi$
$599$	−42.1269			−1.72126			−0.860628	+	0.509234i	$0.829929\pi$
$600$		0			0
$601$	−1.70777			−0.0696615			−0.0348307	−	0.999393i	$-0.511089\pi$
$601$	−1.70777			−0.0696615			−0.0348307	+	0.999393i	$0.511089\pi$
$602$		0			0
$603$		−	14.3563i		−	0.584632i
$604$		0			0
$605$	−13.1738	−	26.9828i	−0.535591	−	1.09701i
$606$		0			0
$607$			13.1419i			0.533412i	0.963778	+	0.266706i	$0.0859353\pi$
$607$			13.1419i			0.533412i	−0.963778	+	0.266706i	$0.914065\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.2388			0.414218
$612$		0			0
$613$		−	38.3179i		−	1.54764i	−0.633403	−	0.773822i	$-0.718342\pi$
$613$		−	38.3179i		−	1.54764i	0.633403	−	0.773822i	$-0.281658\pi$
$614$		0			0
$615$	−21.3113	+	10.4048i	−0.859355	+	0.419562i
$616$		0			0
$617$		−	9.55459i		−	0.384653i	−0.981331	−	0.192327i	$-0.938397\pi$
$617$		−	9.55459i		−	0.384653i	0.981331	−	0.192327i	$-0.0616033\pi$
$618$		0			0
$619$	30.6354			1.23134			0.615670	−	0.788004i	$-0.288885\pi$
$619$	30.6354			1.23134			0.615670	+	0.788004i	$0.288885\pi$
$620$		0			0
$621$	36.5535			1.46684
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−6.08692	−	24.2477i	−0.243477	−	0.969907i
$626$		0			0
$627$			41.8045i			1.66951i
$628$		0			0
$629$	−29.7007			−1.18425
$630$		0			0
$631$	−31.4786			−1.25314			−0.626571	−	0.779364i	$-0.715542\pi$
$631$	−31.4786			−1.25314			−0.626571	+	0.779364i	$0.715542\pi$
$632$		0			0
$633$			10.9802i			0.436422i
$634$		0			0
$635$	10.6932	−	5.22072i	0.424346	−	0.207178i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−7.85036			−0.310556
$640$		0			0
$641$	21.4201			0.846041			0.423021	−	0.906120i	$-0.360970\pi$
$641$	21.4201			0.846041			0.423021	+	0.906120i	$0.360970\pi$
$642$		0			0
$643$		−	5.83210i		−	0.229996i	−0.993366	−	0.114998i	$-0.963314\pi$
$643$		−	5.83210i		−	0.229996i	0.993366	−	0.114998i	$-0.0366861\pi$
$644$		0			0
$645$	−0.778187	−	1.59390i	−0.0306411	−	0.0627597i
$646$		0			0
$647$			26.1428i			1.02778i	0.857856	+	0.513890i	$0.171796\pi$
$647$			26.1428i			1.02778i	−0.857856	+	0.513890i	$0.828204\pi$
$648$		0			0
$649$	19.4767			0.764526
$650$		0			0
$651$		0			0
$652$		0			0
$653$			37.1019i			1.45191i	0.687742	+	0.725955i	$0.258602\pi$
$653$			37.1019i			1.45191i	−0.687742	+	0.725955i	$0.741398\pi$
$654$		0			0
$655$	−7.16194	−	14.6692i	−0.279840	−	0.573175i
$656$		0			0
$657$			5.10886i			0.199315i
$658$		0			0
$659$	49.4758			1.92730			0.963652	−	0.267159i	$-0.0860849\pi$
$659$	49.4758			1.92730			0.963652	+	0.267159i	$0.0860849\pi$
$660$		0			0
$661$	28.1885			1.09640			0.548202	−	0.836346i	$-0.315313\pi$
$661$	28.1885			1.09640			0.548202	+	0.836346i	$0.315313\pi$
$662$		0			0
$663$			29.8148i			1.15791i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	55.7693i		−	2.15940i
$668$		0			0
$669$	8.00755			0.309590
$670$		0			0
$671$	−30.0279			−1.15921
$672$		0			0
$673$			14.0752i			0.542558i	0.962501	+	0.271279i	$0.0874466\pi$
$673$			14.0752i			0.542558i	−0.962501	+	0.271279i	$0.912553\pi$
$674$		0			0
$675$	20.9734	+	16.3592i	0.807268	+	0.629665i
$676$		0			0
$677$		−	27.9999i		−	1.07612i	−0.842906	−	0.538062i	$-0.819157\pi$
$677$		−	27.9999i		−	1.07612i	0.842906	−	0.538062i	$-0.180843\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	0.867603			0.0332466
$682$		0			0
$683$			25.1558i			0.962562i	0.876566	+	0.481281i	$0.159828\pi$
$683$			25.1558i			0.962562i	−0.876566	+	0.481281i	$0.840172\pi$
$684$		0			0
$685$	44.1408	−	21.5508i	1.68653	−	0.823414i
$686$		0			0
$687$		−	15.9718i		−	0.609362i
$688$		0			0
$689$	−6.13384			−0.233681
$690$		0			0
$691$	18.6990			0.711344			0.355672	−	0.934611i	$-0.384252\pi$
$691$	18.6990			0.711344			0.355672	+	0.934611i	$0.384252\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.28890	−	19.0257i	−0.352348	−	0.721687i
$696$		0			0
$697$			54.1114i			2.04962i
$698$		0			0
$699$	13.1597			0.497745
$700$		0			0
$701$	25.1354			0.949351			0.474676	−	0.880161i	$-0.342565\pi$
$701$	25.1354			0.949351			0.474676	+	0.880161i	$0.342565\pi$
$702$		0			0
$703$			39.1042i			1.47484i
$704$		0			0
$705$	2.42863	+	4.97437i	0.0914674	+	0.187345i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−17.6659			−0.663458			−0.331729	−	0.943375i	$-0.607632\pi$
$709$	−17.6659			−0.663458			−0.331729	+	0.943375i	$0.607632\pi$
$710$		0			0
$711$	19.1927			0.719782
$712$		0			0
$713$			26.6338i			0.997443i
$714$		0			0
$715$	−46.0915	+	22.5032i	−1.72373	+	0.841572i
$716$		0			0
$717$		−	16.0834i		−	0.600644i
$718$		0			0
$719$	−4.06812			−0.151715			−0.0758576	−	0.997119i	$-0.524169\pi$
$719$	−4.06812			−0.151715			−0.0758576	+	0.997119i	$0.524169\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			2.32791i			0.0865760i
$724$		0			0
$725$	24.9590	−	31.9989i	0.926955	−	1.18841i
$726$		0			0
$727$			35.5626i			1.31894i	0.751730	+	0.659471i	$0.229220\pi$
$727$			35.5626i			1.31894i	−0.751730	+	0.659471i	$0.770780\pi$
$728$		0			0
$729$	−19.2087			−0.711434
$730$		0			0
$731$	−4.04706			−0.149686
$732$		0			0
$733$		−	1.90581i		−	0.0703929i	−0.999380	−	0.0351964i	$-0.988794\pi$
$733$		−	1.90581i		−	0.0703929i	0.999380	−	0.0351964i	$-0.0112057\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			40.7597i			1.50140i
$738$		0			0
$739$	5.44004			0.200115			0.100058	−	0.994982i	$-0.468097\pi$
$739$	5.44004			0.200115			0.100058	+	0.994982i	$0.468097\pi$
$740$		0			0
$741$	39.2543			1.44204
$742$		0			0
$743$		−	42.7120i		−	1.56695i	−0.621423	−	0.783475i	$-0.713445\pi$
$743$		−	42.7120i		−	1.56695i	0.621423	−	0.783475i	$-0.286555\pi$
$744$		0			0
$745$	−15.4829	−	31.7125i	−0.567251	−	1.16185i
$746$		0			0
$747$		−	9.18043i		−	0.335895i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−15.9029			−0.580306			−0.290153	−	0.956980i	$-0.593706\pi$
$751$	−15.9029			−0.580306			−0.290153	+	0.956980i	$0.593706\pi$
$752$		0			0
$753$		−	3.15532i		−	0.114986i
$754$		0			0
$755$	−11.0532	−	22.6393i	−0.402266	−	0.823929i
$756$		0			0
$757$			1.28840i			0.0468277i	0.999726	+	0.0234138i	$0.00745353\pi$
$757$			1.28840i			0.0468277i	−0.999726	+	0.0234138i	$0.992546\pi$
$758$		0			0
$759$	−38.1085			−1.38325
$760$		0			0
$761$	18.7660			0.680266			0.340133	−	0.940377i	$-0.389528\pi$
$761$	18.7660			0.680266			0.340133	+	0.940377i	$0.389528\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	20.0262	−	9.77737i	0.724049	−	0.353502i
$766$		0			0
$767$		−	18.2885i		−	0.660360i
$768$		0			0
$769$	39.3190			1.41788			0.708939	−	0.705269i	$-0.249174\pi$
$769$	39.3190			1.41788			0.708939	+	0.705269i	$0.249174\pi$
$770$		0			0
$771$	−24.2212			−0.872304
$772$		0			0
$773$			11.9516i			0.429868i	0.976629	+	0.214934i	$0.0689536\pi$
$773$			11.9516i			0.429868i	−0.976629	+	0.214934i	$0.931046\pi$
$774$		0			0
$775$	−11.9197	+	15.2817i	−0.428168	+	0.548936i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	71.2434			2.55256
$780$		0			0
$781$	22.2884			0.797541
$782$		0			0
$783$			43.1774i			1.54304i
$784$		0			0
$785$	17.4209	−	8.50537i	0.621778	−	0.303570i
$786$		0			0
$787$			36.6318i			1.30578i	0.757451	+	0.652892i	$0.226445\pi$
$787$			36.6318i			1.30578i	−0.757451	+	0.652892i	$0.773555\pi$
$788$		0			0
$789$	−18.9821			−0.675782
$790$		0			0
$791$		0			0
$792$		0			0
$793$			28.1961i			1.00127i
$794$		0			0
$795$	−1.45494	−	2.98003i	−0.0516012	−	0.105691i
$796$		0			0
$797$			1.35871i			0.0481281i	0.999710	+	0.0240641i	$0.00766057\pi$
$797$			1.35871i			0.0481281i	−0.999710	+	0.0240641i	$0.992339\pi$
$798$		0			0
$799$	12.6304			0.446831
$800$		0			0
$801$	−16.5375			−0.584323
$802$		0			0
$803$		−	14.5048i		−	0.511864i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			12.7164i			0.447639i
$808$		0			0
$809$	−55.6391			−1.95617			−0.978084	−	0.208211i	$-0.933236\pi$
$809$	−55.6391			−1.95617			−0.978084	+	0.208211i	$0.933236\pi$
$810$		0			0
$811$	54.1690			1.90213			0.951065	−	0.308990i	$-0.0999909\pi$
$811$	54.1690			1.90213			0.951065	+	0.308990i	$0.0999909\pi$
$812$		0			0
$813$		−	3.49389i		−	0.122536i
$814$		0			0
$815$	−48.4811	+	23.6699i	−1.69822	+	0.829119i
$816$		0			0
$817$			5.32838i			0.186416i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	46.2489			1.61410			0.807048	−	0.590486i	$-0.201064\pi$
$821$	46.2489			1.61410			0.807048	+	0.590486i	$0.201064\pi$
$822$		0			0
$823$		−	54.1540i		−	1.88769i	−0.330388	−	0.943845i	$-0.607179\pi$
$823$		−	54.1540i		−	1.88769i	0.330388	−	0.943845i	$-0.392821\pi$
$824$		0			0
$825$	−21.8656	−	17.0551i	−0.761264	−	0.593783i
$826$		0			0
$827$			10.0834i			0.350632i	0.984512	+	0.175316i	$0.0560948\pi$
$827$			10.0834i			0.350632i	−0.984512	+	0.175316i	$0.943905\pi$
$828$		0			0
$829$	39.0572			1.35651			0.678256	−	0.734825i	$-0.262736\pi$
$829$	39.0572			1.35651			0.678256	+	0.734825i	$0.262736\pi$
$830$		0			0
$831$	29.9166			1.03780
$832$		0			0
$833$		0			0
$834$		0			0
$835$	34.1241	−	16.6603i	1.18091	−	0.576555i
$836$		0			0
$837$		−	20.6203i		−	0.712741i
$838$		0			0
$839$	44.4278			1.53382			0.766909	−	0.641756i	$-0.221794\pi$
$839$	44.4278			1.53382			0.766909	+	0.641756i	$0.221794\pi$
$840$		0			0
$841$	36.8753			1.27156
$842$		0			0
$843$		−	31.7088i		−	1.09211i
$844$		0			0
$845$	8.37700	+	17.1580i	0.288178	+	0.590251i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	15.6571			0.537351
$850$		0			0
$851$	−35.6469			−1.22196
$852$		0			0
$853$		−	49.0774i		−	1.68038i	−0.542293	−	0.840189i	$-0.682444\pi$
$853$		−	49.0774i		−	1.68038i	0.542293	−	0.840189i	$-0.317556\pi$
$854$		0			0
$855$	−12.8729	−	26.3666i	−0.440245	−	0.901719i
$856$		0			0
$857$		−	33.7257i		−	1.15205i	−0.817432	−	0.576025i	$-0.804603\pi$
$857$		−	33.7257i		−	1.15205i	0.817432	−	0.576025i	$-0.195397\pi$
$858$		0			0
$859$	−27.8680			−0.950843			−0.475421	−	0.879758i	$-0.657704\pi$
$859$	−27.8680			−0.950843			−0.475421	+	0.879758i	$0.657704\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			23.3459i			0.794702i	0.917667	+	0.397351i	$0.130070\pi$
$863$			23.3459i			0.794702i	−0.917667	+	0.397351i	$0.869930\pi$
$864$		0			0
$865$	39.7235	−	19.3941i	1.35064	−	0.659421i
$866$		0			0
$867$			17.7028i			0.601219i
$868$		0			0
$869$	−54.4910			−1.84848
$870$		0			0
$871$	38.2732			1.29684
$872$		0			0
$873$			3.04313i			0.102994i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			22.4536i			0.758206i	0.925355	+	0.379103i	$0.123767\pi$
$877$			22.4536i			0.758206i	−0.925355	+	0.379103i	$0.876233\pi$
$878$		0			0
$879$	7.29833			0.246167
$880$		0			0
$881$	2.44467			0.0823629			0.0411814	−	0.999152i	$-0.486888\pi$
$881$	2.44467			0.0823629			0.0411814	+	0.999152i	$0.486888\pi$
$882$		0			0
$883$		−	42.2730i		−	1.42260i	−0.702889	−	0.711299i	$-0.748107\pi$
$883$		−	42.2730i		−	1.42260i	0.702889	−	0.711299i	$-0.251893\pi$
$884$		0			0
$885$	8.88519	−	4.33800i	0.298672	−	0.145820i
$886$		0			0
$887$			2.29839i			0.0771725i	0.999255	+	0.0385863i	$0.0122854\pi$
$887$			2.29839i			0.0771725i	−0.999255	+	0.0385863i	$0.987715\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	3.69178			0.123679
$892$		0			0
$893$		−	16.6292i		−	0.556476i
$894$		0			0
$895$	−2.07485	−	4.24974i	−0.0693545	−	0.142053i
$896$		0			0
$897$			35.7838i			1.19479i
$898$		0			0
$899$	−31.4601			−1.04925
$900$		0			0
$901$	−7.56657			−0.252079
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.06186	+	18.5607i	0.301226	+	0.616978i
$906$		0			0
$907$			54.1626i			1.79844i	0.437497	+	0.899220i	$0.355865\pi$
$907$			54.1626i			1.79844i	−0.437497	+	0.899220i	$0.644135\pi$
$908$		0			0
$909$	−9.09738			−0.301741
$910$		0			0
$911$	7.71481			0.255603			0.127801	−	0.991800i	$-0.459208\pi$
$911$	7.71481			0.255603			0.127801	+	0.991800i	$0.459208\pi$
$912$		0			0
$913$			26.0647i			0.862615i
$914$		0			0
$915$	−13.6986	+	6.68806i	−0.452862	+	0.221100i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−6.86010			−0.226294			−0.113147	−	0.993578i	$-0.536093\pi$
$919$	−6.86010			−0.226294			−0.113147	+	0.993578i	$0.536093\pi$
$920$		0			0
$921$	−31.9333			−1.05224
$922$		0			0
$923$		−	20.9287i		−	0.688877i
$924$		0			0
$925$	−20.4532	−	15.9534i	−0.672498	−	0.524546i
$926$		0			0
$927$		−	7.22700i		−	0.237366i
$928$		0			0
$929$	19.2013			0.629975			0.314988	−	0.949096i	$-0.398000\pi$
$929$	19.2013			0.629975			0.314988	+	0.949096i	$0.398000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	6.85016i		−	0.224264i
$934$		0			0
$935$	−56.8575	+	27.7595i	−1.85944	+	0.907832i
$936$		0			0
$937$		−	20.3868i		−	0.666009i	−0.942925	−	0.333004i	$-0.891938\pi$
$937$		−	20.3868i		−	0.666009i	0.942925	−	0.333004i	$-0.108062\pi$
$938$		0			0
$939$	4.96011			0.161867
$940$		0			0
$941$	−44.2362			−1.44206			−0.721030	−	0.692904i	$-0.756331\pi$
$941$	−44.2362			−1.44206			−0.721030	+	0.692904i	$0.756331\pi$
$942$		0			0
$943$			64.9447i			2.11489i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			36.4842i			1.18558i	0.805359	+	0.592788i	$0.201973\pi$
$947$			36.4842i			1.18558i	−0.805359	+	0.592788i	$0.798027\pi$
$948$		0			0
$949$	−13.6200			−0.442123
$950$		0			0
$951$	−6.36929			−0.206538
$952$		0			0
$953$		−	2.51474i		−	0.0814606i	−0.999170	−	0.0407303i	$-0.987032\pi$
$953$		−	2.51474i		−	0.0814606i	0.999170	−	0.0407303i	$-0.0129684\pi$
$954$		0			0
$955$	6.62917	+	13.5780i	0.214515	+	0.439374i
$956$		0			0
$957$		−	45.0142i		−	1.45510i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.9756			−0.515341
$962$		0			0
$963$			29.1128i			0.938147i
$964$		0			0
$965$	−5.45685	+	2.66419i	−0.175662	+	0.0857633i
$966$		0			0
$967$			2.49369i			0.0801915i	0.999196	+	0.0400958i	$0.0127663\pi$
$967$			2.49369i			0.0801915i	−0.999196	+	0.0400958i	$0.987234\pi$
$968$		0			0
$969$	48.4233			1.55558
$970$		0			0
$971$	−15.3021			−0.491069			−0.245534	−	0.969388i	$-0.578963\pi$
$971$	−15.3021			−0.491069			−0.245534	+	0.969388i	$0.578963\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−16.0147	+	20.5318i	−0.512881	+	0.657543i
$976$		0			0
$977$		−	38.5170i		−	1.23227i	−0.787641	−	0.616134i	$-0.788698\pi$
$977$		−	38.5170i		−	1.23227i	0.787641	−	0.616134i	$-0.211302\pi$
$978$		0			0
$979$	46.9524			1.50061
$980$		0			0
$981$	−19.7033			−0.629077
$982$		0			0
$983$			19.9438i			0.636107i	0.948073	+	0.318054i	$0.103029\pi$
$983$			19.9438i			0.636107i	−0.948073	+	0.318054i	$0.896971\pi$
$984$		0			0
$985$	34.5992	−	16.8923i	1.10242	−	0.538234i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.85729			−0.154453
$990$		0			0
$991$	−1.09531			−0.0347937			−0.0173968	−	0.999849i	$-0.505538\pi$
$991$	−1.09531			−0.0347937			−0.0173968	+	0.999849i	$0.505538\pi$
$992$		0			0
$993$			27.3900i			0.869195i
$994$		0			0
$995$	13.7555	+	28.1744i	0.436080	+	0.893187i
$996$		0			0
$997$			43.8274i			1.38803i	0.719961	+	0.694014i	$0.244159\pi$
$997$			43.8274i			1.38803i	−0.719961	+	0.694014i	$0.755841\pi$
$998$		0			0
$999$	27.5984			0.873175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.g.g.1569.7	✓	20
5.2	odd	4		9800.2.a.db.1.4		10
5.3	odd	4		9800.2.a.dc.1.7		10
5.4	even	2	inner	1960.2.g.g.1569.13	yes	20
7.6	odd	2	inner	1960.2.g.g.1569.14	yes	20
35.13	even	4		9800.2.a.dc.1.4		10
35.27	even	4		9800.2.a.db.1.7		10
35.34	odd	2	inner	1960.2.g.g.1569.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.g.g.1569.7	✓	20	1.1	even	1	trivial
1960.2.g.g.1569.8	yes	20	35.34	odd	2	inner
1960.2.g.g.1569.13	yes	20	5.4	even	2	inner
1960.2.g.g.1569.14	yes	20	7.6	odd	2	inner
9800.2.a.db.1.4		10	5.2	odd	4
9800.2.a.db.1.7		10	35.27	even	4
9800.2.a.dc.1.4		10	35.13	even	4
9800.2.a.dc.1.7		10	5.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.12212i q^{3} +(-0.981033 - 2.00937i) q^{5} +1.74084 q^{9} +O(q^{10})\)	\(q-1.12212i q^{3} +(-0.981033 - 2.00937i) q^{5} +1.74084 q^{9} -4.94252 q^{11} +4.64101i q^{13} +(-2.25476 + 1.10084i) q^{15} +5.72505i q^{17} +7.53763 q^{19} +6.87122i q^{23} +(-3.07515 + 3.94252i) q^{25} -5.31980i q^{27} -8.11636 q^{29} +3.87614 q^{31} +5.54611i q^{33} +5.18786i q^{37} +5.20778 q^{39} +9.45170 q^{41} +0.706904i q^{43} +(-1.70782 - 3.49800i) q^{45} -2.20616i q^{47} +6.42420 q^{51} +1.32166i q^{53} +(4.84877 + 9.93136i) q^{55} -8.45814i q^{57} -3.94064 q^{59} +6.07542 q^{61} +(9.32551 - 4.55298i) q^{65} -8.24674i q^{67} +7.71035 q^{69} -4.50952 q^{71} +2.93470i q^{73} +(4.42399 + 3.45069i) q^{75} +11.0249 q^{79} -0.746943 q^{81} -5.27356i q^{83} +(11.5037 - 5.61646i) q^{85} +9.10755i q^{87} -9.49970 q^{89} -4.34950i q^{93} +(-7.39466 - 15.1459i) q^{95} +1.74808i q^{97} -8.60415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 28 q^{9}+O(q^{10}) \) 20 * q - 28 * q^9	\( 20 q - 28 q^{9} - 24 q^{11} - 8 q^{15} + 8 q^{25} - 24 q^{29} + 40 q^{39} - 72 q^{51} - 20 q^{65} - 16 q^{71} + 8 q^{79} + 100 q^{81} + 60 q^{85} - 48 q^{95} + 144 q^{99}+O(q^{100}) \) 20 * q - 28 * q^9 - 24 * q^11 - 8 * q^15 + 8 * q^25 - 24 * q^29 + 40 * q^39 - 72 * q^51 - 20 * q^65 - 16 * q^71 + 8 * q^79 + 100 * q^81 + 60 * q^85 - 48 * q^95 + 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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