LMFDB - Embedded newform 3380.2.f.h.3041.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.4
Root			$0.675970 + 0.675970i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.h.3041.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35194 q^{3} +1.00000i q^{5} -4.17226i q^{7} -1.17226 q^{9} +O(q^{10})$	$q+1.35194 q^{3} +1.00000i q^{5} -4.17226i q^{7} -1.17226 q^{9} +5.52420i q^{11} +1.35194i q^{15} +0.703878 q^{17} +6.82032i q^{19} -5.64064i q^{21} +2.64806 q^{23} -1.00000 q^{25} -5.64064 q^{27} +8.17226 q^{29} -9.52420i q^{31} +7.46838i q^{33} +4.17226 q^{35} +6.87614i q^{37} +0.703878i q^{41} +1.35194 q^{43} -1.17226i q^{45} +8.17226i q^{47} -10.4078 q^{49} +0.951601 q^{51} +5.04840 q^{53} -5.52420 q^{55} +9.22066i q^{57} +12.2281i q^{59} -0.172260 q^{61} +4.89098i q^{63} +10.8761i q^{67} +3.58002 q^{69} -5.52420i q^{71} +11.2207i q^{73} -1.35194 q^{75} +23.0484 q^{77} +1.29612 q^{79} -4.10902 q^{81} -9.58002i q^{83} +0.703878i q^{85} +11.0484 q^{87} -11.7523i q^{89} -12.8761i q^{93} -6.82032 q^{95} +18.3445i q^{97} -6.47580i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 22 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 + 22 * q^9	$ 6 q + 4 q^{3} + 22 q^{9} - 4 q^{17} + 20 q^{23} - 6 q^{25} + 16 q^{27} + 20 q^{29} - 4 q^{35} + 4 q^{43} - 46 q^{49} + 72 q^{51} - 36 q^{53} + 28 q^{61} - 24 q^{69} - 4 q^{75} + 72 q^{77} + 16 q^{79} + 46 q^{81} - 16 q^{95}+O(q^{100}) $ 6 * q + 4 * q^3 + 22 * q^9 - 4 * q^17 + 20 * q^23 - 6 * q^25 + 16 * q^27 + 20 * q^29 - 4 * q^35 + 4 * q^43 - 46 * q^49 + 72 * q^51 - 36 * q^53 + 28 * q^61 - 24 * q^69 - 4 * q^75 + 72 * q^77 + 16 * q^79 + 46 * q^81 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.35194	0.780542	0.390271	−	0.920700i	$-0.372381\pi$
$3$	1.35194	0.780542	0.390271	+	0.920700i	$0.372381\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	− 4.17226i	− 1.57697i	−0.615056	−	0.788483i	$-0.710867\pi$
$7$	− 4.17226i	− 1.57697i	0.615056	−	0.788483i	$-0.289133\pi$
$8$	0	0
$9$	−1.17226	−0.390753
$10$	0	0
$11$	5.52420i	1.66561i	0.553567	+	0.832804i	$0.313266\pi$
$11$	5.52420i	1.66561i	−0.553567	+	0.832804i	$0.686734\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.35194i	0.349069i
$16$	0	0
$17$	0.703878	0.170716	0.0853578	−	0.996350i	$-0.472797\pi$
$17$	0.703878	0.170716	0.0853578	+	0.996350i	$0.472797\pi$
$18$	0	0
$19$	6.82032i	1.56469i	0.622846	+	0.782344i	$0.285976\pi$
$19$	6.82032i	1.56469i	−0.622846	+	0.782344i	$0.714024\pi$
$20$	0	0
$21$	− 5.64064i	− 1.23089i
$22$	0	0
$23$	2.64806	0.552159	0.276079	−	0.961135i	$-0.410965\pi$
$23$	2.64806	0.552159	0.276079	+	0.961135i	$0.410965\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.64064	−1.08554
$28$	0	0
$29$	8.17226	1.51755	0.758775	−	0.651352i	$-0.225798\pi$
$29$	8.17226	1.51755	0.758775	+	0.651352i	$0.225798\pi$
$30$	0	0
$31$	− 9.52420i	− 1.71060i	−0.518136	−	0.855298i	$-0.673374\pi$
$31$	− 9.52420i	− 1.71060i	0.518136	−	0.855298i	$-0.326626\pi$
$32$	0	0
$33$	7.46838i	1.30008i
$34$	0	0
$35$	4.17226	0.705241
$36$	0	0
$37$	6.87614i	1.13043i	0.824944	+	0.565215i	$0.191207\pi$
$37$	6.87614i	1.13043i	−0.824944	+	0.565215i	$0.808793\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.703878i	0.109927i	0.998488	+	0.0549637i	$0.0175043\pi$
$41$	0.703878i	0.109927i	−0.998488	+	0.0549637i	$0.982496\pi$
$42$	0	0
$43$	1.35194	0.206169	0.103084	−	0.994673i	$-0.467129\pi$
$43$	1.35194	0.206169	0.103084	+	0.994673i	$0.467129\pi$
$44$	0	0
$45$	− 1.17226i	− 0.174750i
$46$	0	0
$47$	8.17226i	1.19205i	0.802967	+	0.596023i	$0.203253\pi$
$47$	8.17226i	1.19205i	−0.802967	+	0.596023i	$0.796747\pi$
$48$	0	0
$49$	−10.4078	−1.48682
$50$	0	0
$51$	0.951601	0.133251
$52$	0	0
$53$	5.04840	0.693451	0.346725	−	0.937967i	$-0.387294\pi$
$53$	5.04840	0.693451	0.346725	+	0.937967i	$0.387294\pi$
$54$	0	0
$55$	−5.52420	−0.744883
$56$	0	0
$57$	9.22066i	1.22131i
$58$	0	0
$59$	12.2281i	1.59196i	0.605323	+	0.795980i	$0.293044\pi$
$59$	12.2281i	1.59196i	−0.605323	+	0.795980i	$0.706956\pi$
$60$	0	0
$61$	−0.172260	−0.0220557	−0.0110278	−	0.999939i	$-0.503510\pi$
$61$	−0.172260	−0.0220557	−0.0110278	+	0.999939i	$0.503510\pi$
$62$	0	0
$63$	4.89098i	0.616205i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.8761i	1.32873i	0.747407	+	0.664366i	$0.231298\pi$
$67$	10.8761i	1.32873i	−0.747407	+	0.664366i	$0.768702\pi$
$68$	0	0
$69$	3.58002	0.430983
$70$	0	0
$71$	− 5.52420i	− 0.655602i	−0.944747	−	0.327801i	$-0.893692\pi$
$71$	− 5.52420i	− 0.655602i	0.944747	−	0.327801i	$-0.106308\pi$
$72$	0	0
$73$	11.2207i	1.31328i	0.754205	+	0.656639i	$0.228023\pi$
$73$	11.2207i	1.31328i	−0.754205	+	0.656639i	$0.771977\pi$
$74$	0	0
$75$	−1.35194	−0.156108
$76$	0	0
$77$	23.0484	2.62661
$78$	0	0
$79$	1.29612	0.145825	0.0729125	−	0.997338i	$-0.476771\pi$
$79$	1.29612	0.145825	0.0729125	+	0.997338i	$0.476771\pi$
$80$	0	0
$81$	−4.10902	−0.456558
$82$	0	0
$83$	− 9.58002i	− 1.05154i	−0.850626	−	0.525772i	$-0.823777\pi$
$83$	− 9.58002i	− 1.05154i	0.850626	−	0.525772i	$-0.176223\pi$
$84$	0	0
$85$	0.703878i	0.0763463i
$86$	0	0
$87$	11.0484	1.18451
$88$	0	0
$89$	− 11.7523i	− 1.24574i	−0.782326	−	0.622869i	$-0.785967\pi$
$89$	− 11.7523i	− 1.24574i	0.782326	−	0.622869i	$-0.214033\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 12.8761i	− 1.33519i
$94$	0	0
$95$	−6.82032	−0.699750
$96$	0	0
$97$	18.3445i	1.86260i	0.364248	+	0.931302i	$0.381326\pi$
$97$	18.3445i	1.86260i	−0.364248	+	0.931302i	$0.618674\pi$
$98$	0	0
$99$	− 6.47580i	− 0.650842i
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	10.9926	1.08313	0.541566	−	0.840658i	$-0.317832\pi$
$103$	10.9926	1.08313	0.541566	+	0.840658i	$0.317832\pi$
$104$	0	0
$105$	5.64064	0.550470
$106$	0	0
$107$	6.99258	0.675998	0.337999	−	0.941146i	$-0.390250\pi$
$107$	6.99258	0.675998	0.337999	+	0.941146i	$0.390250\pi$
$108$	0	0
$109$	3.40776i	0.326404i	0.986593	+	0.163202i	$0.0521822\pi$
$109$	3.40776i	0.326404i	−0.986593	+	0.163202i	$0.947818\pi$
$110$	0	0
$111$	9.29612i	0.882349i
$112$	0	0
$113$	3.64064	0.342483	0.171241	−	0.985229i	$-0.445222\pi$
$113$	3.64064	0.342483	0.171241	+	0.985229i	$0.445222\pi$
$114$	0	0
$115$	2.64806i	0.246933i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 2.93676i	− 0.269213i
$120$	0	0
$121$	−19.5168	−1.77425
$122$	0	0
$123$	0.951601i	0.0858030i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	5.69646	0.505479	0.252740	−	0.967534i	$-0.418668\pi$
$127$	5.69646	0.505479	0.252740	+	0.967534i	$0.418668\pi$
$128$	0	0
$129$	1.82774	0.160923
$130$	0	0
$131$	17.7523	1.55102	0.775512	−	0.631333i	$-0.217492\pi$
$131$	17.7523	1.55102	0.775512	+	0.631333i	$0.217492\pi$
$132$	0	0
$133$	28.4562	2.46746
$134$	0	0
$135$	− 5.64064i	− 0.485469i
$136$	0	0
$137$	8.93676i	0.763519i	0.924262	+	0.381760i	$0.124682\pi$
$137$	8.93676i	0.763519i	−0.924262	+	0.381760i	$0.875318\pi$
$138$	0	0
$139$	13.2961	1.12776	0.563881	−	0.825856i	$-0.309308\pi$
$139$	13.2961	1.12776	0.563881	+	0.825856i	$0.309308\pi$
$140$	0	0
$141$	11.0484i	0.930443i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.17226i	0.678669i
$146$	0	0
$147$	−14.0707	−1.16053
$148$	0	0
$149$	− 8.93676i	− 0.732128i	−0.930590	−	0.366064i	$-0.880705\pi$
$149$	− 8.93676i	− 0.732128i	0.930590	−	0.366064i	$-0.119295\pi$
$150$	0	0
$151$	− 2.93196i	− 0.238599i	−0.992858	−	0.119300i	$-0.961935\pi$
$151$	− 2.93196i	− 0.238599i	0.992858	−	0.119300i	$-0.0380649\pi$
$152$	0	0
$153$	−0.825129	−0.0667077
$154$	0	0
$155$	9.52420	0.765002
$156$	0	0
$157$	0.592243	0.0472662	0.0236331	−	0.999721i	$-0.492477\pi$
$157$	0.592243	0.0472662	0.0236331	+	0.999721i	$0.492477\pi$
$158$	0	0
$159$	6.82513	0.541268
$160$	0	0
$161$	− 11.0484i	− 0.870736i
$162$	0	0
$163$	− 16.6284i	− 1.30244i	−0.758890	−	0.651219i	$-0.774258\pi$
$163$	− 16.6284i	− 1.30244i	0.758890	−	0.651219i	$-0.225742\pi$
$164$	0	0
$165$	−7.46838	−0.581413
$166$	0	0
$167$	3.82774i	0.296199i	0.988972	+	0.148100i	$0.0473156\pi$
$167$	3.82774i	0.296199i	−0.988972	+	0.148100i	$0.952684\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 7.99519i	− 0.611408i
$172$	0	0
$173$	−14.1116	−1.07289	−0.536444	−	0.843936i	$-0.680233\pi$
$173$	−14.1116	−1.07289	−0.536444	+	0.843936i	$0.680233\pi$
$174$	0	0
$175$	4.17226i	0.315393i
$176$	0	0
$177$	16.5316i	1.24259i
$178$	0	0
$179$	−5.75228	−0.429945	−0.214973	−	0.976620i	$-0.568966\pi$
$179$	−5.75228	−0.429945	−0.214973	+	0.976620i	$0.568966\pi$
$180$	0	0
$181$	−2.76450	−0.205484	−0.102742	−	0.994708i	$-0.532762\pi$
$181$	−2.76450	−0.205484	−0.102742	+	0.994708i	$0.532762\pi$
$182$	0	0
$183$	−0.232886	−0.0172154
$184$	0	0
$185$	−6.87614	−0.505544
$186$	0	0
$187$	3.88836i	0.284345i
$188$	0	0
$189$	23.5342i	1.71186i
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	14.3445i	1.03254i	0.856426	+	0.516271i	$0.172680\pi$
$193$	14.3445i	1.03254i	−0.856426	+	0.516271i	$0.827320\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.40776i	− 0.527781i	−0.964553	−	0.263890i	$-0.914994\pi$
$197$	− 7.40776i	− 0.527781i	0.964553	−	0.263890i	$-0.0850057\pi$
$198$	0	0
$199$	−1.75228	−0.124216	−0.0621078	−	0.998069i	$-0.519782\pi$
$199$	−1.75228	−0.124216	−0.0621078	+	0.998069i	$0.519782\pi$
$200$	0	0
$201$	14.7039i	1.03713i
$202$	0	0
$203$	− 34.0968i	− 2.39313i
$204$	0	0
$205$	−0.703878	−0.0491610
$206$	0	0
$207$	−3.10422	−0.215758
$208$	0	0
$209$	−37.6768	−2.60616
$210$	0	0
$211$	−9.75228	−0.671374	−0.335687	−	0.941974i	$-0.608969\pi$
$211$	−9.75228	−0.671374	−0.335687	+	0.941974i	$0.608969\pi$
$212$	0	0
$213$	− 7.46838i	− 0.511725i
$214$	0	0
$215$	1.35194i	0.0922015i
$216$	0	0
$217$	−39.7374	−2.69755
$218$	0	0
$219$	15.1696i	1.02507i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.76450i	0.185125i	0.995707	+	0.0925624i	$0.0295058\pi$
$223$	2.76450i	0.185125i	−0.995707	+	0.0925624i	$0.970494\pi$
$224$	0	0
$225$	1.17226	0.0781507
$226$	0	0
$227$	17.8129i	1.18228i	0.806568	+	0.591142i	$0.201323\pi$
$227$	17.8129i	1.18228i	−0.806568	+	0.591142i	$0.798677\pi$
$228$	0	0
$229$	− 10.1116i	− 0.668196i	−0.942538	−	0.334098i	$-0.891568\pi$
$229$	− 10.1116i	− 0.668196i	0.942538	−	0.334098i	$-0.108432\pi$
$230$	0	0
$231$	31.1600	2.05018
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−8.17226	−0.533099
$236$	0	0
$237$	1.75228	0.113823
$238$	0	0
$239$	9.86872i	0.638354i	0.947695	+	0.319177i	$0.103407\pi$
$239$	9.86872i	0.638354i	−0.947695	+	0.319177i	$0.896593\pi$
$240$	0	0
$241$	1.88836i	0.121640i	0.998149	+	0.0608201i	$0.0193716\pi$
$241$	1.88836i	0.121640i	−0.998149	+	0.0608201i	$0.980628\pi$
$242$	0	0
$243$	11.3668	0.729179
$244$	0	0
$245$	− 10.4078i	− 0.664927i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 12.9516i	− 0.820774i
$250$	0	0
$251$	−2.35936	−0.148921	−0.0744607	−	0.997224i	$-0.523724\pi$
$251$	−2.35936	−0.148921	−0.0744607	+	0.997224i	$0.523724\pi$
$252$	0	0
$253$	14.6284i	0.919681i
$254$	0	0
$255$	0.951601i	0.0595916i
$256$	0	0
$257$	1.65548	0.103266	0.0516330	−	0.998666i	$-0.483557\pi$
$257$	1.65548	0.103266	0.0516330	+	0.998666i	$0.483557\pi$
$258$	0	0
$259$	28.6890	1.78265
$260$	0	0
$261$	−9.58002	−0.592988
$262$	0	0
$263$	−16.6332	−1.02565	−0.512824	−	0.858494i	$-0.671401\pi$
$263$	−16.6332	−1.02565	−0.512824	+	0.858494i	$0.671401\pi$
$264$	0	0
$265$	5.04840i	0.310121i
$266$	0	0
$267$	− 15.8884i	− 0.972352i
$268$	0	0
$269$	16.0968	0.981439	0.490720	−	0.871318i	$-0.336734\pi$
$269$	16.0968	0.981439	0.490720	+	0.871318i	$0.336734\pi$
$270$	0	0
$271$	10.4758i	0.636360i	0.948030	+	0.318180i	$0.103072\pi$
$271$	10.4758i	0.636360i	−0.948030	+	0.318180i	$0.896928\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 5.52420i	− 0.333122i
$276$	0	0
$277$	−17.3929	−1.04504	−0.522520	−	0.852627i	$-0.675008\pi$
$277$	−17.3929	−1.04504	−0.522520	+	0.852627i	$0.675008\pi$
$278$	0	0
$279$	11.1648i	0.668422i
$280$	0	0
$281$	− 11.2961i	− 0.673870i	−0.941528	−	0.336935i	$-0.890610\pi$
$281$	− 11.2961i	− 0.673870i	0.941528	−	0.336935i	$-0.109390\pi$
$282$	0	0
$283$	−9.24030	−0.549279	−0.274640	−	0.961547i	$-0.588559\pi$
$283$	−9.24030	−0.549279	−0.274640	+	0.961547i	$0.588559\pi$
$284$	0	0
$285$	−9.22066	−0.546185
$286$	0	0
$287$	2.93676	0.173352
$288$	0	0
$289$	−16.5046	−0.970856
$290$	0	0
$291$	24.8007i	1.45384i
$292$	0	0
$293$	16.4051i	0.958399i	0.877706	+	0.479199i	$0.159073\pi$
$293$	16.4051i	0.958399i	−0.877706	+	0.479199i	$0.840927\pi$
$294$	0	0
$295$	−12.2281	−0.711946
$296$	0	0
$297$	− 31.1600i	− 1.80809i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 5.64064i	− 0.325121i
$302$	0	0
$303$	−8.11164	−0.466001
$304$	0	0
$305$	− 0.172260i	− 0.00986360i
$306$	0	0
$307$	15.1090i	0.862318i	0.902276	+	0.431159i	$0.141895\pi$
$307$	15.1090i	0.862318i	−0.902276	+	0.431159i	$0.858105\pi$
$308$	0	0
$309$	14.8613	0.845430
$310$	0	0
$311$	8.11164	0.459969	0.229984	−	0.973194i	$-0.426133\pi$
$311$	8.11164	0.459969	0.229984	+	0.973194i	$0.426133\pi$
$312$	0	0
$313$	−1.39292	−0.0787325	−0.0393662	−	0.999225i	$-0.512534\pi$
$313$	−1.39292	−0.0787325	−0.0393662	+	0.999225i	$0.512534\pi$
$314$	0	0
$315$	−4.89098	−0.275575
$316$	0	0
$317$	− 29.8129i	− 1.67446i	−0.546851	−	0.837230i	$-0.684174\pi$
$317$	− 29.8129i	− 1.67446i	0.546851	−	0.837230i	$-0.315826\pi$
$318$	0	0
$319$	45.1452i	2.52765i
$320$	0	0
$321$	9.45355	0.527645
$322$	0	0
$323$	4.80068i	0.267117i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.60708i	0.254772i
$328$	0	0
$329$	34.0968	1.87982
$330$	0	0
$331$	− 19.1648i	− 1.05339i	−0.850053	−	0.526697i	$-0.823430\pi$
$331$	− 19.1648i	− 1.05339i	0.850053	−	0.526697i	$-0.176570\pi$
$332$	0	0
$333$	− 8.06063i	− 0.441720i
$334$	0	0
$335$	−10.8761	−0.594227
$336$	0	0
$337$	−11.6406	−0.634106	−0.317053	−	0.948408i	$-0.602693\pi$
$337$	−11.6406	−0.634106	−0.317053	+	0.948408i	$0.602693\pi$
$338$	0	0
$339$	4.92193	0.267322
$340$	0	0
$341$	52.6136	2.84919
$342$	0	0
$343$	14.2180i	0.767702i
$344$	0	0
$345$	3.58002i	0.192742i
$346$	0	0
$347$	−22.7597	−1.22180	−0.610902	−	0.791706i	$-0.709193\pi$
$347$	−22.7597	−1.22180	−0.610902	+	0.791706i	$0.709193\pi$
$348$	0	0
$349$	− 1.04840i	− 0.0561195i	−0.999606	−	0.0280598i	$-0.991067\pi$
$349$	− 1.04840i	− 0.0561195i	0.999606	−	0.0280598i	$-0.00893287\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.40515i	0.234462i	0.993105	+	0.117231i	$0.0374018\pi$
$353$	4.40515i	0.234462i	−0.993105	+	0.117231i	$0.962598\pi$
$354$	0	0
$355$	5.52420	0.293194
$356$	0	0
$357$	− 3.97033i	− 0.210132i
$358$	0	0
$359$	− 24.2281i	− 1.27871i	−0.768912	−	0.639355i	$-0.779202\pi$
$359$	− 24.2281i	− 1.27871i	0.768912	−	0.639355i	$-0.220798\pi$
$360$	0	0
$361$	−27.5168	−1.44825
$362$	0	0
$363$	−26.3855	−1.38488
$364$	0	0
$365$	−11.2207	−0.587316
$366$	0	0
$367$	−27.3371	−1.42699	−0.713493	−	0.700663i	$-0.752888\pi$
$367$	−27.3371	−1.42699	−0.713493	+	0.700663i	$0.752888\pi$
$368$	0	0
$369$	− 0.825129i	− 0.0429545i
$370$	0	0
$371$	− 21.0632i	− 1.09355i
$372$	0	0
$373$	22.6890	1.17479	0.587397	−	0.809299i	$-0.300153\pi$
$373$	22.6890	1.17479	0.587397	+	0.809299i	$0.300153\pi$
$374$	0	0
$375$	− 1.35194i	− 0.0698138i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.3249i	1.55768i	0.627220	+	0.778842i	$0.284193\pi$
$379$	30.3249i	1.55768i	−0.627220	+	0.778842i	$0.715807\pi$
$380$	0	0
$381$	7.70127	0.394548
$382$	0	0
$383$	− 5.23550i	− 0.267521i	−0.991014	−	0.133761i	$-0.957295\pi$
$383$	− 5.23550i	− 0.267521i	0.991014	−	0.133761i	$-0.0427053\pi$
$384$	0	0
$385$	23.0484i	1.17466i
$386$	0	0
$387$	−1.58482	−0.0805612
$388$	0	0
$389$	−23.8735	−1.21044	−0.605218	−	0.796060i	$-0.706914\pi$
$389$	−23.8735	−1.21044	−0.605218	+	0.796060i	$0.706914\pi$
$390$	0	0
$391$	1.86391	0.0942621
$392$	0	0
$393$	24.0000	1.21064
$394$	0	0
$395$	1.29612i	0.0652150i
$396$	0	0
$397$	− 39.2207i	− 1.96843i	−0.176981	−	0.984214i	$-0.556633\pi$
$397$	− 39.2207i	− 1.96843i	0.176981	−	0.984214i	$-0.443367\pi$
$398$	0	0
$399$	38.4710	1.92596
$400$	0	0
$401$	− 31.0336i	− 1.54974i	−0.632119	−	0.774871i	$-0.717815\pi$
$401$	− 31.0336i	− 1.54974i	0.632119	−	0.774871i	$-0.282185\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 4.10902i	− 0.204179i
$406$	0	0
$407$	−37.9852	−1.88285
$408$	0	0
$409$	− 18.1116i	− 0.895563i	−0.894143	−	0.447781i	$-0.852214\pi$
$409$	− 18.1116i	− 0.895563i	0.894143	−	0.447781i	$-0.147786\pi$
$410$	0	0
$411$	12.0820i	0.595959i
$412$	0	0
$413$	51.0187	2.51047
$414$	0	0
$415$	9.58002	0.470265
$416$	0	0
$417$	17.9755	0.880266
$418$	0	0
$419$	−23.0484	−1.12599	−0.562994	−	0.826461i	$-0.690351\pi$
$419$	−23.0484	−1.12599	−0.562994	+	0.826461i	$0.690351\pi$
$420$	0	0
$421$	− 34.9123i	− 1.70152i	−0.525553	−	0.850761i	$-0.676141\pi$
$421$	− 34.9123i	− 1.70152i	0.525553	−	0.850761i	$-0.323859\pi$
$422$	0	0
$423$	− 9.58002i	− 0.465796i
$424$	0	0
$425$	−0.703878	−0.0341431
$426$	0	0
$427$	0.718715i	0.0347811i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.47580i	0.311928i	0.987763	+	0.155964i	$0.0498484\pi$
$431$	6.47580i	0.311928i	−0.987763	+	0.155964i	$0.950152\pi$
$432$	0	0
$433$	24.8155	1.19256	0.596279	−	0.802777i	$-0.296645\pi$
$433$	24.8155	1.19256	0.596279	+	0.802777i	$0.296645\pi$
$434$	0	0
$435$	11.0484i	0.529730i
$436$	0	0
$437$	18.0606i	0.863957i
$438$	0	0
$439$	−10.8155	−0.516196	−0.258098	−	0.966119i	$-0.583096\pi$
$439$	−10.8155	−0.516196	−0.258098	+	0.966119i	$0.583096\pi$
$440$	0	0
$441$	12.2006	0.580981
$442$	0	0
$443$	18.0410	0.857153	0.428576	−	0.903506i	$-0.359015\pi$
$443$	18.0410	0.857153	0.428576	+	0.903506i	$0.359015\pi$
$444$	0	0
$445$	11.7523	0.557111
$446$	0	0
$447$	− 12.0820i	− 0.571457i
$448$	0	0
$449$	− 27.1452i	− 1.28106i	−0.767933	−	0.640531i	$-0.778714\pi$
$449$	− 27.1452i	− 1.28106i	0.767933	−	0.640531i	$-0.221286\pi$
$450$	0	0
$451$	−3.88836	−0.183096
$452$	0	0
$453$	− 3.96383i	− 0.186237i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.75228i	− 0.175524i	−0.996141	−	0.0877621i	$-0.972028\pi$
$457$	− 3.75228i	− 0.175524i	0.996141	−	0.0877621i	$-0.0279715\pi$
$458$	0	0
$459$	−3.97033	−0.185319
$460$	0	0
$461$	36.3297i	1.69204i	0.533150	+	0.846021i	$0.321008\pi$
$461$	36.3297i	1.69204i	−0.533150	+	0.846021i	$0.678992\pi$
$462$	0	0
$463$	29.4684i	1.36951i	0.728772	+	0.684756i	$0.240091\pi$
$463$	29.4684i	1.36951i	−0.728772	+	0.684756i	$0.759909\pi$
$464$	0	0
$465$	12.8761	0.597117
$466$	0	0
$467$	17.5848	0.813729	0.406864	−	0.913489i	$-0.366622\pi$
$467$	17.5848	0.813729	0.406864	+	0.913489i	$0.366622\pi$
$468$	0	0
$469$	45.3781	2.09537
$470$	0	0
$471$	0.800677	0.0368932
$472$	0	0
$473$	7.46838i	0.343397i
$474$	0	0
$475$	− 6.82032i	− 0.312938i
$476$	0	0
$477$	−5.91804	−0.270968
$478$	0	0
$479$	11.2765i	0.515235i	0.966247	+	0.257618i	$0.0829375\pi$
$479$	11.2765i	0.515235i	−0.966247	+	0.257618i	$0.917062\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 14.9368i	− 0.679646i
$484$	0	0
$485$	−18.3445	−0.832982
$486$	0	0
$487$	− 26.1574i	− 1.18531i	−0.805458	−	0.592653i	$-0.798081\pi$
$487$	− 26.1574i	− 1.18531i	0.805458	−	0.592653i	$-0.201919\pi$
$488$	0	0
$489$	− 22.4806i	− 1.01661i
$490$	0	0
$491$	−0.456156	−0.0205860	−0.0102930	−	0.999947i	$-0.503276\pi$
$491$	−0.456156	−0.0205860	−0.0102930	+	0.999947i	$0.503276\pi$
$492$	0	0
$493$	5.75228	0.259070
$494$	0	0
$495$	6.47580	0.291066
$496$	0	0
$497$	−23.0484	−1.03386
$498$	0	0
$499$	− 12.4610i	− 0.557829i	−0.960316	−	0.278915i	$-0.910025\pi$
$499$	− 12.4610i	− 0.557829i	0.960316	−	0.278915i	$-0.0899747\pi$
$500$	0	0
$501$	5.17487i	0.231196i
$502$	0	0
$503$	−11.7933	−0.525835	−0.262918	−	0.964818i	$-0.584685\pi$
$503$	−11.7933	−0.525835	−0.262918	+	0.964818i	$0.584685\pi$
$504$	0	0
$505$	− 6.00000i	− 0.266996i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 29.0484i	− 1.28755i	−0.765216	−	0.643774i	$-0.777368\pi$
$509$	− 29.0484i	− 1.28755i	0.765216	−	0.643774i	$-0.222632\pi$
$510$	0	0
$511$	46.8155	2.07100
$512$	0	0
$513$	− 38.4710i	− 1.69854i
$514$	0	0
$515$	10.9926i	0.484391i
$516$	0	0
$517$	−45.1452	−1.98548
$518$	0	0
$519$	−19.0781	−0.837434
$520$	0	0
$521$	−15.3323	−0.671720	−0.335860	−	0.941912i	$-0.609027\pi$
$521$	−15.3323	−0.671720	−0.335860	+	0.941912i	$0.609027\pi$
$522$	0	0
$523$	15.4487	0.675526	0.337763	−	0.941231i	$-0.390330\pi$
$523$	15.4487	0.675526	0.337763	+	0.941231i	$0.390330\pi$
$524$	0	0
$525$	5.64064i	0.246178i
$526$	0	0
$527$	− 6.70388i	− 0.292026i
$528$	0	0
$529$	−15.9878	−0.695121
$530$	0	0
$531$	− 14.3345i	− 0.622064i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.99258i	0.302316i
$536$	0	0
$537$	−7.77673	−0.335591
$538$	0	0
$539$	− 57.4945i	− 2.47646i
$540$	0	0
$541$	38.8007i	1.66817i	0.551635	+	0.834086i	$0.314004\pi$
$541$	38.8007i	1.66817i	−0.551635	+	0.834086i	$0.685996\pi$
$542$	0	0
$543$	−3.73744	−0.160389
$544$	0	0
$545$	−3.40776	−0.145972
$546$	0	0
$547$	14.9926	0.641036	0.320518	−	0.947242i	$-0.396143\pi$
$547$	14.9926	0.641036	0.320518	+	0.947242i	$0.396143\pi$
$548$	0	0
$549$	0.201934	0.00861833
$550$	0	0
$551$	55.7374i	2.37449i
$552$	0	0
$553$	− 5.40776i	− 0.229961i
$554$	0	0
$555$	−9.29612	−0.394598
$556$	0	0
$557$	26.3807i	1.11779i	0.829240	+	0.558893i	$0.188774\pi$
$557$	26.3807i	1.11779i	−0.829240	+	0.558893i	$0.811226\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	5.25683i	0.221944i
$562$	0	0
$563$	42.0410	1.77182	0.885908	−	0.463861i	$-0.153536\pi$
$563$	42.0410	1.77182	0.885908	+	0.463861i	$0.153536\pi$
$564$	0	0
$565$	3.64064i	0.153163i
$566$	0	0
$567$	17.1439i	0.719977i
$568$	0	0
$569$	−30.2691	−1.26894	−0.634472	−	0.772945i	$-0.718783\pi$
$569$	−30.2691	−1.26894	−0.634472	+	0.772945i	$0.718783\pi$
$570$	0	0
$571$	12.1116	0.506856	0.253428	−	0.967354i	$-0.418442\pi$
$571$	12.1116	0.506856	0.253428	+	0.967354i	$0.418442\pi$
$572$	0	0
$573$	16.2233	0.677737
$574$	0	0
$575$	−2.64806	−0.110432
$576$	0	0
$577$	− 8.40515i	− 0.349911i	−0.984576	−	0.174955i	$-0.944022\pi$
$577$	− 8.40515i	− 0.349911i	0.984576	−	0.174955i	$-0.0559781\pi$
$578$	0	0
$579$	19.3929i	0.805942i
$580$	0	0
$581$	−39.9703	−1.65825
$582$	0	0
$583$	27.8884i	1.15502i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	47.4439i	1.95822i	0.203330	+	0.979110i	$0.434824\pi$
$587$	47.4439i	1.95822i	−0.203330	+	0.979110i	$0.565176\pi$
$588$	0	0
$589$	64.9581	2.67655
$590$	0	0
$591$	− 10.0148i	− 0.411955i
$592$	0	0
$593$	− 32.4413i	− 1.33221i	−0.745860	−	0.666103i	$-0.767961\pi$
$593$	− 32.4413i	− 1.33221i	0.745860	−	0.666103i	$-0.232039\pi$
$594$	0	0
$595$	2.93676	0.120396
$596$	0	0
$597$	−2.36897	−0.0969556
$598$	0	0
$599$	−34.6742	−1.41675	−0.708375	−	0.705836i	$-0.750571\pi$
$599$	−34.6742	−1.41675	−0.708375	+	0.705836i	$0.750571\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	− 12.7497i	− 0.519207i
$604$	0	0
$605$	− 19.5168i	− 0.793470i
$606$	0	0
$607$	−2.30354	−0.0934978	−0.0467489	−	0.998907i	$-0.514886\pi$
$607$	−2.30354	−0.0934978	−0.0467489	+	0.998907i	$0.514886\pi$
$608$	0	0
$609$	− 46.0968i	− 1.86794i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 15.8735i	− 0.641126i	−0.947227	−	0.320563i	$-0.896128\pi$
$613$	− 15.8735i	− 0.641126i	0.947227	−	0.320563i	$-0.103872\pi$
$614$	0	0
$615$	−0.951601	−0.0383722
$616$	0	0
$617$	10.3445i	0.416455i	0.978080	+	0.208227i	$0.0667694\pi$
$617$	10.3445i	0.416455i	−0.978080	+	0.208227i	$0.933231\pi$
$618$	0	0
$619$	− 37.1500i	− 1.49318i	−0.665282	−	0.746592i	$-0.731689\pi$
$619$	− 37.1500i	− 1.49318i	0.665282	−	0.746592i	$-0.268311\pi$
$620$	0	0
$621$	−14.9368	−0.599392
$622$	0	0
$623$	−49.0336	−1.96449
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−50.9368	−2.03422
$628$	0	0
$629$	4.83997i	0.192982i
$630$	0	0
$631$	− 10.2132i	− 0.406583i	−0.979118	−	0.203291i	$-0.934836\pi$
$631$	− 10.2132i	− 0.406583i	0.979118	−	0.203291i	$-0.0651639\pi$
$632$	0	0
$633$	−13.1845	−0.524036
$634$	0	0
$635$	5.69646i	0.226057i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.47580i	0.256179i
$640$	0	0
$641$	16.0968	0.635785	0.317893	−	0.948127i	$-0.397025\pi$
$641$	16.0968	0.635785	0.317893	+	0.948127i	$0.397025\pi$
$642$	0	0
$643$	− 18.0458i	− 0.711656i	−0.934551	−	0.355828i	$-0.884199\pi$
$643$	− 18.0458i	− 0.711656i	0.934551	−	0.355828i	$-0.115801\pi$
$644$	0	0
$645$	1.82774i	0.0719672i
$646$	0	0
$647$	43.5700	1.71291	0.856456	−	0.516219i	$-0.172661\pi$
$647$	43.5700	1.71291	0.856456	+	0.516219i	$0.172661\pi$
$648$	0	0
$649$	−67.5503	−2.65158
$650$	0	0
$651$	−53.7226	−2.10555
$652$	0	0
$653$	−22.3445	−0.874409	−0.437204	−	0.899362i	$-0.644031\pi$
$653$	−22.3445	−0.874409	−0.437204	+	0.899362i	$0.644031\pi$
$654$	0	0
$655$	17.7523i	0.693639i
$656$	0	0
$657$	− 13.1535i	− 0.513168i
$658$	0	0
$659$	28.9219	1.12664	0.563319	−	0.826239i	$-0.309524\pi$
$659$	28.9219	1.12664	0.563319	+	0.826239i	$0.309524\pi$
$660$	0	0
$661$	18.6890i	0.726919i	0.931610	+	0.363460i	$0.118405\pi$
$661$	18.6890i	0.726919i	−0.931610	+	0.363460i	$0.881595\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	28.4562i	1.10348i
$666$	0	0
$667$	21.6406	0.837929
$668$	0	0
$669$	3.73744i	0.144498i
$670$	0	0
$671$	− 0.951601i	− 0.0367361i
$672$	0	0
$673$	−23.5194	−0.906606	−0.453303	−	0.891357i	$-0.649754\pi$
$673$	−23.5194	−0.906606	−0.453303	+	0.891357i	$0.649754\pi$
$674$	0	0
$675$	5.64064	0.217108
$676$	0	0
$677$	−8.35936	−0.321276	−0.160638	−	0.987013i	$-0.551355\pi$
$677$	−8.35936	−0.321276	−0.160638	+	0.987013i	$0.551355\pi$
$678$	0	0
$679$	76.5381	2.93726
$680$	0	0
$681$	24.0820i	0.922823i
$682$	0	0
$683$	11.9394i	0.456847i	0.973562	+	0.228424i	$0.0733572\pi$
$683$	11.9394i	0.456847i	−0.973562	+	0.228424i	$0.926643\pi$
$684$	0	0
$685$	−8.93676	−0.341456
$686$	0	0
$687$	− 13.6703i	− 0.521555i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	15.4274i	0.586886i	0.955977	+	0.293443i	$0.0948011\pi$
$691$	15.4274i	0.586886i	−0.955977	+	0.293443i	$0.905199\pi$
$692$	0	0
$693$	−27.0187	−1.02636
$694$	0	0
$695$	13.2961i	0.504351i
$696$	0	0
$697$	0.495445i	0.0187663i
$698$	0	0
$699$	8.11164	0.306810
$700$	0	0
$701$	−37.2813	−1.40809	−0.704047	−	0.710153i	$-0.748626\pi$
$701$	−37.2813	−1.40809	−0.704047	+	0.710153i	$0.748626\pi$
$702$	0	0
$703$	−46.8975	−1.76877
$704$	0	0
$705$	−11.0484	−0.416107
$706$	0	0
$707$	25.0336i	0.941484i
$708$	0	0
$709$	11.9852i	0.450112i	0.974346	+	0.225056i	$0.0722565\pi$
$709$	11.9852i	0.450112i	−0.974346	+	0.225056i	$0.927743\pi$
$710$	0	0
$711$	−1.51939	−0.0569817
$712$	0	0
$713$	− 25.2207i	− 0.944521i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.3419i	0.498263i
$718$	0	0
$719$	−23.6258	−0.881094	−0.440547	−	0.897730i	$-0.645215\pi$
$719$	−23.6258	−0.881094	−0.440547	+	0.897730i	$0.645215\pi$
$720$	0	0
$721$	− 45.8639i	− 1.70806i
$722$	0	0
$723$	2.55295i	0.0949454i
$724$	0	0
$725$	−8.17226	−0.303510
$726$	0	0
$727$	−22.1526	−0.821595	−0.410798	−	0.911727i	$-0.634750\pi$
$727$	−22.1526	−0.821595	−0.410798	+	0.911727i	$0.634750\pi$
$728$	0	0
$729$	27.6943	1.02571
$730$	0	0
$731$	0.951601	0.0351962
$732$	0	0
$733$	− 10.0000i	− 0.369358i	−0.982799	−	0.184679i	$-0.940875\pi$
$733$	− 10.0000i	− 0.369358i	0.982799	−	0.184679i	$-0.0591246\pi$
$734$	0	0
$735$	− 14.0707i	− 0.519004i
$736$	0	0
$737$	−60.0820	−2.21315
$738$	0	0
$739$	19.0436i	0.700530i	0.936651	+	0.350265i	$0.113908\pi$
$739$	19.0436i	0.700530i	−0.936651	+	0.350265i	$0.886092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.26906i	0.229989i	0.993366	+	0.114995i	$0.0366851\pi$
$743$	6.26906i	0.229989i	−0.993366	+	0.114995i	$0.963315\pi$
$744$	0	0
$745$	8.93676	0.327418
$746$	0	0
$747$	11.2303i	0.410894i
$748$	0	0
$749$	− 29.1749i	− 1.06603i
$750$	0	0
$751$	20.9219	0.763452	0.381726	−	0.924276i	$-0.375330\pi$
$751$	20.9219	0.763452	0.381726	+	0.924276i	$0.375330\pi$
$752$	0	0
$753$	−3.18971	−0.116239
$754$	0	0
$755$	2.93196	0.106705
$756$	0	0
$757$	−48.8975	−1.77721	−0.888604	−	0.458674i	$-0.848324\pi$
$757$	−48.8975	−1.77721	−0.888604	+	0.458674i	$0.848324\pi$
$758$	0	0
$759$	19.7767i	0.717850i
$760$	0	0
$761$	4.09680i	0.148509i	0.997239	+	0.0742544i	$0.0236577\pi$
$761$	4.09680i	0.148509i	−0.997239	+	0.0742544i	$0.976342\pi$
$762$	0	0
$763$	14.2180	0.514728
$764$	0	0
$765$	− 0.825129i	− 0.0298326i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 28.7858i	− 1.03804i	−0.854761	−	0.519022i	$-0.826296\pi$
$769$	− 28.7858i	− 1.03804i	0.854761	−	0.519022i	$-0.173704\pi$
$770$	0	0
$771$	2.23811	0.0806035
$772$	0	0
$773$	7.22066i	0.259709i	0.991533	+	0.129855i	$0.0414510\pi$
$773$	7.22066i	0.259709i	−0.991533	+	0.129855i	$0.958549\pi$
$774$	0	0
$775$	9.52420i	0.342119i
$776$	0	0
$777$	38.7858	1.39143
$778$	0	0
$779$	−4.80068	−0.172002
$780$	0	0
$781$	30.5168	1.09198
$782$	0	0
$783$	−46.0968	−1.64737
$784$	0	0
$785$	0.592243i	0.0211381i
$786$	0	0
$787$	− 41.5800i	− 1.48217i	−0.671413	−	0.741084i	$-0.734312\pi$
$787$	− 41.5800i	− 1.48217i	0.671413	−	0.741084i	$-0.265688\pi$
$788$	0	0
$789$	−22.4871	−0.800562
$790$	0	0
$791$	− 15.1897i	− 0.540084i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	6.82513i	0.242062i
$796$	0	0
$797$	−19.4078	−0.687458	−0.343729	−	0.939069i	$-0.611690\pi$
$797$	−19.4078	−0.687458	−0.343729	+	0.939069i	$0.611690\pi$
$798$	0	0
$799$	5.75228i	0.203501i
$800$	0	0
$801$	13.7767i	0.486777i
$802$	0	0
$803$	−61.9852	−2.18741
$804$	0	0
$805$	11.0484	0.389405
$806$	0	0
$807$	21.7619	0.766055
$808$	0	0
$809$	43.7981	1.53986	0.769929	−	0.638130i	$-0.220292\pi$
$809$	43.7981	1.53986	0.769929	+	0.638130i	$0.220292\pi$
$810$	0	0
$811$	− 5.75709i	− 0.202159i	−0.994878	−	0.101079i	$-0.967770\pi$
$811$	− 5.75709i	− 0.202159i	0.994878	−	0.101079i	$-0.0322296\pi$
$812$	0	0
$813$	14.1626i	0.496706i
$814$	0	0
$815$	16.6284	0.582468
$816$	0	0
$817$	9.22066i	0.322590i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 32.9368i	− 1.14950i	−0.818329	−	0.574750i	$-0.805099\pi$
$821$	− 32.9368i	− 1.14950i	0.818329	−	0.574750i	$-0.194901\pi$
$822$	0	0
$823$	57.0894	1.99001	0.995005	−	0.0998217i	$-0.0318272\pi$
$823$	57.0894	1.99001	0.995005	+	0.0998217i	$0.0318272\pi$
$824$	0	0
$825$	− 7.46838i	− 0.260016i
$826$	0	0
$827$	27.4535i	0.954653i	0.878726	+	0.477327i	$0.158394\pi$
$827$	27.4535i	0.954653i	−0.878726	+	0.477327i	$0.841606\pi$
$828$	0	0
$829$	−10.4200	−0.361901	−0.180950	−	0.983492i	$-0.557917\pi$
$829$	−10.4200	−0.361901	−0.180950	+	0.983492i	$0.557917\pi$
$830$	0	0
$831$	−23.5142	−0.815698
$832$	0	0
$833$	−7.32580	−0.253824
$834$	0	0
$835$	−3.82774	−0.132464
$836$	0	0
$837$	53.7226i	1.85692i
$838$	0	0
$839$	− 40.1984i	− 1.38780i	−0.720070	−	0.693902i	$-0.755890\pi$
$839$	− 40.1984i	− 1.38780i	0.720070	−	0.693902i	$-0.244110\pi$
$840$	0	0
$841$	37.7858	1.30296
$842$	0	0
$843$	− 15.2717i	− 0.525984i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	81.4291i	2.79794i
$848$	0	0
$849$	−12.4923	−0.428736
$850$	0	0
$851$	18.2084i	0.624177i
$852$	0	0
$853$	− 34.8761i	− 1.19414i	−0.802191	−	0.597068i	$-0.796332\pi$
$853$	− 34.8761i	− 1.19414i	0.802191	−	0.597068i	$-0.203668\pi$
$854$	0	0
$855$	7.99519	0.273430
$856$	0	0
$857$	−25.1600	−0.859450	−0.429725	−	0.902960i	$-0.641390\pi$
$857$	−25.1600	−0.859450	−0.429725	+	0.902960i	$0.641390\pi$
$858$	0	0
$859$	−15.0484	−0.513445	−0.256722	−	0.966485i	$-0.582643\pi$
$859$	−15.0484	−0.513445	−0.256722	+	0.966485i	$0.582643\pi$
$860$	0	0
$861$	3.97033	0.135308
$862$	0	0
$863$	2.29873i	0.0782498i	0.999234	+	0.0391249i	$0.0124570\pi$
$863$	2.29873i	0.0782498i	−0.999234	+	0.0391249i	$0.987543\pi$
$864$	0	0
$865$	− 14.1116i	− 0.479810i
$866$	0	0
$867$	−22.3132	−0.757794
$868$	0	0
$869$	7.16003i	0.242888i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 21.5046i	− 0.727819i
$874$	0	0
$875$	−4.17226	−0.141048
$876$	0	0
$877$	− 10.0000i	− 0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$	− 10.0000i	− 0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$	0	0
$879$	22.1788i	0.748071i
$880$	0	0
$881$	48.1426	1.62196	0.810982	−	0.585070i	$-0.198933\pi$
$881$	48.1426	1.62196	0.810982	+	0.585070i	$0.198933\pi$
$882$	0	0
$883$	12.8565	0.432655	0.216328	−	0.976321i	$-0.430592\pi$
$883$	12.8565	0.432655	0.216328	+	0.976321i	$0.430592\pi$
$884$	0	0
$885$	−16.5316	−0.555704
$886$	0	0
$887$	−32.5216	−1.09197	−0.545984	−	0.837796i	$-0.683844\pi$
$887$	−32.5216	−1.09197	−0.545984	+	0.837796i	$0.683844\pi$
$888$	0	0
$889$	− 23.7671i	− 0.797123i
$890$	0	0
$891$	− 22.6991i	− 0.760447i
$892$	0	0
$893$	−55.7374	−1.86518
$894$	0	0
$895$	− 5.75228i	− 0.192277i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 77.8342i	− 2.59592i
$900$	0	0
$901$	3.55346	0.118383
$902$	0	0
$903$	− 7.62581i	− 0.253771i
$904$	0	0
$905$	− 2.76450i	− 0.0918952i
$906$	0	0
$907$	21.0894	0.700261	0.350131	−	0.936701i	$-0.386137\pi$
$907$	21.0894	0.700261	0.350131	+	0.936701i	$0.386137\pi$
$908$	0	0
$909$	7.03356	0.233289
$910$	0	0
$911$	43.1600	1.42996	0.714978	−	0.699147i	$-0.246437\pi$
$911$	43.1600	1.42996	0.714978	+	0.699147i	$0.246437\pi$
$912$	0	0
$913$	52.9219	1.75146
$914$	0	0
$915$	− 0.232886i	− 0.00769896i
$916$	0	0
$917$	− 74.0671i	− 2.44591i
$918$	0	0
$919$	11.8884	0.392161	0.196080	−	0.980588i	$-0.437179\pi$
$919$	11.8884	0.392161	0.196080	+	0.980588i	$0.437179\pi$
$920$	0	0
$921$	20.4265i	0.673075i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 6.87614i	− 0.226086i
$926$	0	0
$927$	−12.8862	−0.423237
$928$	0	0
$929$	− 34.4265i	− 1.12950i	−0.825263	−	0.564748i	$-0.808973\pi$
$929$	− 34.4265i	− 1.12950i	0.825263	−	0.564748i	$-0.191027\pi$
$930$	0	0
$931$	− 70.9842i	− 2.32641i
$932$	0	0
$933$	10.9664	0.359025
$934$	0	0
$935$	−3.88836	−0.127163
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−1.88314	−0.0614541
$940$	0	0
$941$	− 38.1116i	− 1.24240i	−0.783651	−	0.621202i	$-0.786645\pi$
$941$	− 38.1116i	− 1.24240i	0.783651	−	0.621202i	$-0.213355\pi$
$942$	0	0
$943$	1.86391i	0.0606974i
$944$	0	0
$945$	−23.5342	−0.765569
$946$	0	0
$947$	− 13.9245i	− 0.452487i	−0.974071	−	0.226243i	$-0.927356\pi$
$947$	− 13.9245i	− 0.452487i	0.974071	−	0.226243i	$-0.0726445\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 40.3052i	− 1.30699i
$952$	0	0
$953$	23.7523	0.769412	0.384706	−	0.923039i	$-0.374303\pi$
$953$	23.7523	0.769412	0.384706	+	0.923039i	$0.374303\pi$
$954$	0	0
$955$	12.0000i	0.388311i
$956$	0	0
$957$	61.0336i	1.97293i
$958$	0	0
$959$	37.2865	1.20404
$960$	0	0
$961$	−59.7104	−1.92614
$962$	0	0
$963$	−8.19713	−0.264149
$964$	0	0
$965$	−14.3445	−0.461766
$966$	0	0
$967$	− 6.87614i	− 0.221122i	−0.993869	−	0.110561i	$-0.964735\pi$
$967$	− 6.87614i	− 0.221122i	0.993869	−	0.110561i	$-0.0352647\pi$
$968$	0	0
$969$	6.49022i	0.208496i
$970$	0	0
$971$	6.24772	0.200499	0.100249	−	0.994962i	$-0.468036\pi$
$971$	6.24772	0.200499	0.100249	+	0.994962i	$0.468036\pi$
$972$	0	0
$973$	− 55.4749i	− 1.77844i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34.5316i	− 1.10476i	−0.833591	−	0.552382i	$-0.813719\pi$
$977$	− 34.5316i	− 1.10476i	0.833591	−	0.552382i	$-0.186281\pi$
$978$	0	0
$979$	64.9219	2.07491
$980$	0	0
$981$	− 3.99478i	− 0.127543i
$982$	0	0
$983$	11.5652i	0.368872i	0.982845	+	0.184436i	$0.0590458\pi$
$983$	11.5652i	0.368872i	−0.982845	+	0.184436i	$0.940954\pi$
$984$	0	0
$985$	7.40776	0.236031
$986$	0	0
$987$	46.0968	1.46728
$988$	0	0
$989$	3.58002	0.113838
$990$	0	0
$991$	3.77673	0.119972	0.0599859	−	0.998199i	$-0.480894\pi$
$991$	3.77673	0.119972	0.0599859	+	0.998199i	$0.480894\pi$
$992$	0	0
$993$	− 25.9097i	− 0.822220i
$994$	0	0
$995$	− 1.75228i	− 0.0555509i
$996$	0	0
$997$	54.4265	1.72370	0.861852	−	0.507160i	$-0.169305\pi$
$997$	54.4265	1.72370	0.861852	+	0.507160i	$0.169305\pi$
$998$	0	0
$999$	− 38.7858i	− 1.22713i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.h.3041.4		6
13.5	odd	4		260.2.a.b.1.2	✓	3
13.8	odd	4		3380.2.a.o.1.2		3
13.12	even	2	inner	3380.2.f.h.3041.3		6
39.5	even	4		2340.2.a.n.1.3		3
52.31	even	4		1040.2.a.o.1.2		3
65.18	even	4		1300.2.c.f.1249.4		6
65.44	odd	4		1300.2.a.i.1.2		3
65.57	even	4		1300.2.c.f.1249.3		6
104.5	odd	4		4160.2.a.bo.1.2		3
104.83	even	4		4160.2.a.br.1.2		3
156.83	odd	4		9360.2.a.da.1.1		3
260.239	even	4		5200.2.a.ci.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.a.b.1.2	✓	3	13.5	odd	4
1040.2.a.o.1.2		3	52.31	even	4
1300.2.a.i.1.2		3	65.44	odd	4
1300.2.c.f.1249.3		6	65.57	even	4
1300.2.c.f.1249.4		6	65.18	even	4
2340.2.a.n.1.3		3	39.5	even	4
3380.2.a.o.1.2		3	13.8	odd	4
3380.2.f.h.3041.3		6	13.12	even	2	inner
3380.2.f.h.3041.4		6	1.1	even	1	trivial
4160.2.a.bo.1.2		3	104.5	odd	4
4160.2.a.br.1.2		3	104.83	even	4
5200.2.a.ci.1.2		3	260.239	even	4
9360.2.a.da.1.1		3	156.83	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.35194 q^{3} +1.00000i q^{5} -4.17226i q^{7} -1.17226 q^{9} +O(q^{10})\)	\(q+1.35194 q^{3} +1.00000i q^{5} -4.17226i q^{7} -1.17226 q^{9} +5.52420i q^{11} +1.35194i q^{15} +0.703878 q^{17} +6.82032i q^{19} -5.64064i q^{21} +2.64806 q^{23} -1.00000 q^{25} -5.64064 q^{27} +8.17226 q^{29} -9.52420i q^{31} +7.46838i q^{33} +4.17226 q^{35} +6.87614i q^{37} +0.703878i q^{41} +1.35194 q^{43} -1.17226i q^{45} +8.17226i q^{47} -10.4078 q^{49} +0.951601 q^{51} +5.04840 q^{53} -5.52420 q^{55} +9.22066i q^{57} +12.2281i q^{59} -0.172260 q^{61} +4.89098i q^{63} +10.8761i q^{67} +3.58002 q^{69} -5.52420i q^{71} +11.2207i q^{73} -1.35194 q^{75} +23.0484 q^{77} +1.29612 q^{79} -4.10902 q^{81} -9.58002i q^{83} +0.703878i q^{85} +11.0484 q^{87} -11.7523i q^{89} -12.8761i q^{93} -6.82032 q^{95} +18.3445i q^{97} -6.47580i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 22 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 + 22 * q^9	\( 6 q + 4 q^{3} + 22 q^{9} - 4 q^{17} + 20 q^{23} - 6 q^{25} + 16 q^{27} + 20 q^{29} - 4 q^{35} + 4 q^{43} - 46 q^{49} + 72 q^{51} - 36 q^{53} + 28 q^{61} - 24 q^{69} - 4 q^{75} + 72 q^{77} + 16 q^{79} + 46 q^{81} - 16 q^{95}+O(q^{100}) \) 6 * q + 4 * q^3 + 22 * q^9 - 4 * q^17 + 20 * q^23 - 6 * q^25 + 16 * q^27 + 20 * q^29 - 4 * q^35 + 4 * q^43 - 46 * q^49 + 72 * q^51 - 36 * q^53 + 28 * q^61 - 24 * q^69 - 4 * q^75 + 72 * q^77 + 16 * q^79 + 46 * q^81 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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