LMFDB - Embedded newform 3120.2.g.o.961.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(961,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3120.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.g (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:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.3
Root			$-1.56155i$ of defining polynomial
Character	$\chi$	$=$	3120.961
Dual form			3120.2.g.o.961.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000i q^{5} -3.12311i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000i q^{5} -3.12311i q^{7} +1.00000 q^{9} -5.12311i q^{11} +(3.56155 - 0.561553i) q^{13} -1.00000i q^{15} +2.00000 q^{17} +6.00000i q^{19} +3.12311i q^{21} +5.12311 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -3.12311i q^{31} +5.12311i q^{33} +3.12311 q^{35} -5.12311i q^{37} +(-3.56155 + 0.561553i) q^{39} +0.876894i q^{41} -6.24621 q^{43} +1.00000i q^{45} +6.24621i q^{47} -2.75379 q^{49} -2.00000 q^{51} -13.3693 q^{53} +5.12311 q^{55} -6.00000i q^{57} +1.12311i q^{59} +10.0000 q^{61} -3.12311i q^{63} +(0.561553 + 3.56155i) q^{65} -4.87689i q^{67} -5.12311 q^{69} -10.2462i q^{71} -13.1231i q^{73} +1.00000 q^{75} -16.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.24621i q^{83} +2.00000i q^{85} -2.00000 q^{87} +3.12311i q^{89} +(-1.75379 - 11.1231i) q^{91} +3.12311i q^{93} -6.00000 q^{95} -13.1231i q^{97} -5.12311i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{9} + 6 q^{13} + 8 q^{17} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} - 4 q^{35} - 6 q^{39} + 8 q^{43} - 44 q^{49} - 8 q^{51} - 4 q^{53} + 4 q^{55} + 40 q^{61} - 6 q^{65} - 4 q^{69} + 4 q^{75} - 64 q^{77} - 32 q^{79} + 4 q^{81} - 8 q^{87} - 40 q^{91} - 24 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 + 6 * q^13 + 8 * q^17 + 4 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 - 4 * q^35 - 6 * q^39 + 8 * q^43 - 44 * q^49 - 8 * q^51 - 4 * q^53 + 4 * q^55 + 40 * q^61 - 6 * q^65 - 4 * q^69 + 4 * q^75 - 64 * q^77 - 32 * q^79 + 4 * q^81 - 8 * q^87 - 40 * q^91 - 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	3.12311i		−	1.18042i	−0.807249	−	0.590211i	$-0.799044\pi$
$7$		−	3.12311i		−	1.18042i	0.807249	−	0.590211i	$-0.200956\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	5.12311i		−	1.54467i	−0.635213	−	0.772337i	$-0.719088\pi$
$11$		−	5.12311i		−	1.54467i	0.635213	−	0.772337i	$-0.280912\pi$
$12$		0			0
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$			3.12311i			0.681518i
$22$		0			0
$23$	5.12311			1.06824			0.534121	−	0.845408i	$-0.320643\pi$
$23$	5.12311			1.06824			0.534121	+	0.845408i	$0.320643\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$		−	3.12311i		−	0.560926i	−0.959865	−	0.280463i	$-0.909512\pi$
$31$		−	3.12311i		−	0.560926i	0.959865	−	0.280463i	$-0.0904881\pi$
$32$		0			0
$33$			5.12311i			0.891818i
$34$		0			0
$35$	3.12311			0.527901
$36$		0			0
$37$		−	5.12311i		−	0.842233i	−0.907006	−	0.421117i	$-0.861638\pi$
$37$		−	5.12311i		−	0.842233i	0.907006	−	0.421117i	$-0.138362\pi$
$38$		0			0
$39$	−3.56155	+	0.561553i	−0.570305	+	0.0899204i
$40$		0			0
$41$			0.876894i			0.136948i	0.997653	+	0.0684739i	$0.0218130\pi$
$41$			0.876894i			0.136948i	−0.997653	+	0.0684739i	$0.978187\pi$
$42$		0			0
$43$	−6.24621			−0.952538			−0.476269	−	0.879300i	$-0.658011\pi$
$43$	−6.24621			−0.952538			−0.476269	+	0.879300i	$0.658011\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$			6.24621i			0.911104i	0.890209	+	0.455552i	$0.150558\pi$
$47$			6.24621i			0.911104i	−0.890209	+	0.455552i	$0.849442\pi$
$48$		0			0
$49$	−2.75379			−0.393398
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	−13.3693			−1.83642			−0.918208	−	0.396098i	$-0.870364\pi$
$53$	−13.3693			−1.83642			−0.918208	+	0.396098i	$0.870364\pi$
$54$		0			0
$55$	5.12311			0.690799
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$		0			0
$59$			1.12311i			0.146216i	0.997324	+	0.0731079i	$0.0232918\pi$
$59$			1.12311i			0.146216i	−0.997324	+	0.0731079i	$0.976708\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		−	3.12311i		−	0.393474i
$64$		0			0
$65$	0.561553	+	3.56155i	0.0696521	+	0.441756i
$66$		0			0
$67$		−	4.87689i		−	0.595807i	−0.954596	−	0.297904i	$-0.903713\pi$
$67$		−	4.87689i		−	0.595807i	0.954596	−	0.297904i	$-0.0962874\pi$
$68$		0			0
$69$	−5.12311			−0.616749
$70$		0			0
$71$		−	10.2462i		−	1.21600i	−0.793936	−	0.608001i	$-0.791972\pi$
$71$		−	10.2462i		−	1.21600i	0.793936	−	0.608001i	$-0.208028\pi$
$72$		0			0
$73$		−	13.1231i		−	1.53594i	−0.640484	−	0.767972i	$-0.721266\pi$
$73$		−	13.1231i		−	1.53594i	0.640484	−	0.767972i	$-0.278734\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	−16.0000			−1.82337
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			6.24621i			0.685611i	0.939406	+	0.342805i	$0.111377\pi$
$83$			6.24621i			0.685611i	−0.939406	+	0.342805i	$0.888623\pi$
$84$		0			0
$85$			2.00000i			0.216930i
$86$		0			0
$87$	−2.00000			−0.214423
$88$		0			0
$89$			3.12311i			0.331049i	0.986206	+	0.165524i	$0.0529316\pi$
$89$			3.12311i			0.331049i	−0.986206	+	0.165524i	$0.947068\pi$
$90$		0			0
$91$	−1.75379	−	11.1231i	−0.183847	−	1.16602i
$92$		0			0
$93$			3.12311i			0.323851i
$94$		0			0
$95$	−6.00000			−0.615587
$96$		0			0
$97$		−	13.1231i		−	1.33245i	−0.745751	−	0.666225i	$-0.767909\pi$
$97$		−	13.1231i		−	1.33245i	0.745751	−	0.666225i	$-0.232091\pi$
$98$		0			0
$99$		−	5.12311i		−	0.514891i
$100$		0			0
$101$	−12.2462			−1.21854			−0.609272	−	0.792961i	$-0.708538\pi$
$101$	−12.2462			−1.21854			−0.609272	+	0.792961i	$0.708538\pi$
$102$		0			0
$103$	13.1231			1.29306			0.646529	−	0.762889i	$-0.276220\pi$
$103$	13.1231			1.29306			0.646529	+	0.762889i	$0.276220\pi$
$104$		0			0
$105$	−3.12311			−0.304784
$106$		0			0
$107$	8.00000			0.773389			0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000			0.773389			0.386695	+	0.922208i	$0.373617\pi$
$108$		0			0
$109$			2.87689i			0.275557i	0.990463	+	0.137778i	$0.0439961\pi$
$109$			2.87689i			0.275557i	−0.990463	+	0.137778i	$0.956004\pi$
$110$		0			0
$111$			5.12311i			0.486264i
$112$		0			0
$113$	12.2462			1.15203			0.576013	−	0.817440i	$-0.304608\pi$
$113$	12.2462			1.15203			0.576013	+	0.817440i	$0.304608\pi$
$114$		0			0
$115$			5.12311i			0.477732i
$116$		0			0
$117$	3.56155	−	0.561553i	0.329266	−	0.0519156i
$118$		0			0
$119$		−	6.24621i		−	0.572589i
$120$		0			0
$121$	−15.2462			−1.38602
$122$		0			0
$123$		−	0.876894i		−	0.0790669i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	13.1231			1.16449			0.582244	−	0.813014i	$-0.302175\pi$
$127$	13.1231			1.16449			0.582244	+	0.813014i	$0.302175\pi$
$128$		0			0
$129$	6.24621			0.549948
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$	18.7386			1.62485
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$		−	10.4924i		−	0.896428i	−0.893926	−	0.448214i	$-0.852060\pi$
$137$		−	10.4924i		−	0.896428i	0.893926	−	0.448214i	$-0.147940\pi$
$138$		0			0
$139$	16.4924			1.39887			0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924			1.39887			0.699435	+	0.714697i	$0.253435\pi$
$140$		0			0
$141$		−	6.24621i		−	0.526026i
$142$		0			0
$143$	−2.87689	−	18.2462i	−0.240578	−	1.52582i
$144$		0			0
$145$			2.00000i			0.166091i
$146$		0			0
$147$	2.75379			0.227129
$148$		0			0
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$		−	13.3693i		−	1.08798i	−0.839092	−	0.543990i	$-0.816913\pi$
$151$		−	13.3693i		−	1.08798i	0.839092	−	0.543990i	$-0.183087\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$	3.12311			0.250854
$156$		0			0
$157$	21.3693			1.70546			0.852729	−	0.522354i	$-0.174946\pi$
$157$	21.3693			1.70546			0.852729	+	0.522354i	$0.174946\pi$
$158$		0			0
$159$	13.3693			1.06026
$160$		0			0
$161$		−	16.0000i		−	1.26098i
$162$		0			0
$163$			7.12311i			0.557925i	0.960302	+	0.278962i	$0.0899905\pi$
$163$			7.12311i			0.557925i	−0.960302	+	0.278962i	$0.910010\pi$
$164$		0			0
$165$	−5.12311			−0.398833
$166$		0			0
$167$			22.2462i			1.72146i	0.509059	+	0.860732i	$0.329994\pi$
$167$			22.2462i			1.72146i	−0.509059	+	0.860732i	$0.670006\pi$
$168$		0			0
$169$	12.3693	−	4.00000i	0.951486	−	0.307692i
$170$		0			0
$171$			6.00000i			0.458831i
$172$		0			0
$173$	23.1231			1.75802			0.879009	−	0.476806i	$-0.158206\pi$
$173$	23.1231			1.75802			0.879009	+	0.476806i	$0.158206\pi$
$174$		0			0
$175$			3.12311i			0.236085i
$176$		0			0
$177$		−	1.12311i		−	0.0844178i
$178$		0			0
$179$	−16.4924			−1.23270			−0.616351	−	0.787472i	$-0.711390\pi$
$179$	−16.4924			−1.23270			−0.616351	+	0.787472i	$0.711390\pi$
$180$		0			0
$181$	−20.2462			−1.50489			−0.752445	−	0.658656i	$-0.771125\pi$
$181$	−20.2462			−1.50489			−0.752445	+	0.658656i	$0.771125\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$	5.12311			0.376658
$186$		0			0
$187$		−	10.2462i		−	0.749277i
$188$		0			0
$189$			3.12311i			0.227173i
$190$		0			0
$191$	−16.4924			−1.19335			−0.596675	−	0.802483i	$-0.703512\pi$
$191$	−16.4924			−1.19335			−0.596675	+	0.802483i	$0.703512\pi$
$192$		0			0
$193$		−	25.1231i		−	1.80840i	−0.427109	−	0.904200i	$-0.640468\pi$
$193$		−	25.1231i		−	1.80840i	0.427109	−	0.904200i	$-0.359532\pi$
$194$		0			0
$195$	−0.561553	−	3.56155i	−0.0402136	−	0.255048i
$196$		0			0
$197$			16.2462i			1.15749i	0.815507	+	0.578747i	$0.196458\pi$
$197$			16.2462i			1.15749i	−0.815507	+	0.578747i	$0.803542\pi$
$198$		0			0
$199$	−18.2462			−1.29344			−0.646720	−	0.762728i	$-0.723860\pi$
$199$	−18.2462			−1.29344			−0.646720	+	0.762728i	$0.723860\pi$
$200$		0			0
$201$			4.87689i			0.343990i
$202$		0			0
$203$		−	6.24621i		−	0.438398i
$204$		0			0
$205$	−0.876894			−0.0612450
$206$		0			0
$207$	5.12311			0.356080
$208$		0			0
$209$	30.7386			2.12624
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$			10.2462i			0.702059i
$214$		0			0
$215$		−	6.24621i		−	0.425988i
$216$		0			0
$217$	−9.75379			−0.662130
$218$		0			0
$219$			13.1231i			0.886777i
$220$		0			0
$221$	7.12311	−	1.12311i	0.479152	−	0.0755483i
$222$		0			0
$223$		−	9.36932i		−	0.627416i	−0.949520	−	0.313708i	$-0.898429\pi$
$223$		−	9.36932i		−	0.627416i	0.949520	−	0.313708i	$-0.101571\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$		−	5.12311i		−	0.338544i	−0.985569	−	0.169272i	$-0.945858\pi$
$229$		−	5.12311i		−	0.338544i	0.985569	−	0.169272i	$-0.0541417\pi$
$230$		0			0
$231$	16.0000			1.05272
$232$		0			0
$233$	−7.75379			−0.507968			−0.253984	−	0.967208i	$-0.581741\pi$
$233$	−7.75379			−0.507968			−0.253984	+	0.967208i	$0.581741\pi$
$234$		0			0
$235$	−6.24621			−0.407458
$236$		0			0
$237$	8.00000			0.519656
$238$		0			0
$239$		−	4.49242i		−	0.290591i	−0.989388	−	0.145295i	$-0.953587\pi$
$239$		−	4.49242i		−	0.290591i	0.989388	−	0.145295i	$-0.0464132\pi$
$240$		0			0
$241$		−	14.2462i		−	0.917679i	−0.888519	−	0.458840i	$-0.848265\pi$
$241$		−	14.2462i		−	0.917679i	0.888519	−	0.458840i	$-0.151735\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	2.75379i		−	0.175933i
$246$		0			0
$247$	3.36932	+	21.3693i	0.214384	+	1.35970i
$248$		0			0
$249$		−	6.24621i		−	0.395838i
$250$		0			0
$251$	−26.2462			−1.65665			−0.828323	−	0.560251i	$-0.810705\pi$
$251$	−26.2462			−1.65665			−0.828323	+	0.560251i	$0.810705\pi$
$252$		0			0
$253$		−	26.2462i		−	1.65009i
$254$		0			0
$255$		−	2.00000i		−	0.125245i
$256$		0			0
$257$	−12.2462			−0.763898			−0.381949	−	0.924183i	$-0.624747\pi$
$257$	−12.2462			−0.763898			−0.381949	+	0.924183i	$0.624747\pi$
$258$		0			0
$259$	−16.0000			−0.994192
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$	27.3693			1.68766			0.843832	−	0.536607i	$-0.180294\pi$
$263$	27.3693			1.68766			0.843832	+	0.536607i	$0.180294\pi$
$264$		0			0
$265$		−	13.3693i		−	0.821271i
$266$		0			0
$267$		−	3.12311i		−	0.191131i
$268$		0			0
$269$	−16.2462			−0.990549			−0.495274	−	0.868737i	$-0.664933\pi$
$269$	−16.2462			−0.990549			−0.495274	+	0.868737i	$0.664933\pi$
$270$		0			0
$271$		−	23.1231i		−	1.40463i	−0.711867	−	0.702314i	$-0.752150\pi$
$271$		−	23.1231i		−	1.40463i	0.711867	−	0.702314i	$-0.247850\pi$
$272$		0			0
$273$	1.75379	+	11.1231i	0.106144	+	0.673201i
$274$		0			0
$275$			5.12311i			0.308935i
$276$		0			0
$277$	−29.8617			−1.79422			−0.897109	−	0.441809i	$-0.854337\pi$
$277$	−29.8617			−1.79422			−0.897109	+	0.441809i	$0.854337\pi$
$278$		0			0
$279$		−	3.12311i		−	0.186975i
$280$		0			0
$281$		−	3.12311i		−	0.186309i	−0.995652	−	0.0931544i	$-0.970305\pi$
$281$		−	3.12311i		−	0.186309i	0.995652	−	0.0931544i	$-0.0296950\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$	6.00000			0.355409
$286$		0			0
$287$	2.73863			0.161656
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$			13.1231i			0.769290i
$292$		0			0
$293$			28.7386i			1.67893i	0.543415	+	0.839464i	$0.317131\pi$
$293$			28.7386i			1.67893i	−0.543415	+	0.839464i	$0.682869\pi$
$294$		0			0
$295$	−1.12311			−0.0653897
$296$		0			0
$297$			5.12311i			0.297273i
$298$		0			0
$299$	18.2462	−	2.87689i	1.05521	−	0.166375i
$300$		0			0
$301$			19.5076i			1.12440i
$302$		0			0
$303$	12.2462			0.703526
$304$		0			0
$305$			10.0000i			0.572598i
$306$		0			0
$307$		−	31.1231i		−	1.77629i	−0.459564	−	0.888145i	$-0.651994\pi$
$307$		−	31.1231i		−	1.77629i	0.459564	−	0.888145i	$-0.348006\pi$
$308$		0			0
$309$	−13.1231			−0.746547
$310$		0			0
$311$	−8.49242			−0.481561			−0.240781	−	0.970580i	$-0.577403\pi$
$311$	−8.49242			−0.481561			−0.240781	+	0.970580i	$0.577403\pi$
$312$		0			0
$313$	16.2462			0.918290			0.459145	−	0.888361i	$-0.348156\pi$
$313$	16.2462			0.918290			0.459145	+	0.888361i	$0.348156\pi$
$314$		0			0
$315$	3.12311			0.175967
$316$		0			0
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$		−	10.2462i		−	0.573678i
$320$		0			0
$321$	−8.00000			−0.446516
$322$		0			0
$323$			12.0000i			0.667698i
$324$		0			0
$325$	−3.56155	+	0.561553i	−0.197559	+	0.0311493i
$326$		0			0
$327$		−	2.87689i		−	0.159093i
$328$		0			0
$329$	19.5076			1.07549
$330$		0			0
$331$			7.75379i			0.426187i	0.977032	+	0.213093i	$0.0683539\pi$
$331$			7.75379i			0.426187i	−0.977032	+	0.213093i	$0.931646\pi$
$332$		0			0
$333$		−	5.12311i		−	0.280744i
$334$		0			0
$335$	4.87689			0.266453
$336$		0			0
$337$	6.00000			0.326841			0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000			0.326841			0.163420	+	0.986557i	$0.447747\pi$
$338$		0			0
$339$	−12.2462			−0.665123
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$		−	13.2614i		−	0.716046i
$344$		0			0
$345$		−	5.12311i		−	0.275819i
$346$		0			0
$347$	14.2462			0.764777			0.382388	−	0.924002i	$-0.375102\pi$
$347$	14.2462			0.764777			0.382388	+	0.924002i	$0.375102\pi$
$348$		0			0
$349$			19.3693i			1.03682i	0.855133	+	0.518408i	$0.173475\pi$
$349$			19.3693i			1.03682i	−0.855133	+	0.518408i	$0.826525\pi$
$350$		0			0
$351$	−3.56155	+	0.561553i	−0.190102	+	0.0299735i
$352$		0			0
$353$		−	12.2462i		−	0.651800i	−0.945404	−	0.325900i	$-0.894333\pi$
$353$		−	12.2462i		−	0.651800i	0.945404	−	0.325900i	$-0.105667\pi$
$354$		0			0
$355$	10.2462			0.543812
$356$		0			0
$357$			6.24621i			0.330585i
$358$		0			0
$359$		−	17.7538i		−	0.937009i	−0.883461	−	0.468505i	$-0.844793\pi$
$359$		−	17.7538i		−	0.937009i	0.883461	−	0.468505i	$-0.155207\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	15.2462			0.800219
$364$		0			0
$365$	13.1231			0.686895
$366$		0			0
$367$	−8.63068			−0.450518			−0.225259	−	0.974299i	$-0.572323\pi$
$367$	−8.63068			−0.450518			−0.225259	+	0.974299i	$0.572323\pi$
$368$		0			0
$369$			0.876894i			0.0456493i
$370$		0			0
$371$			41.7538i			2.16775i
$372$		0			0
$373$	7.12311			0.368820			0.184410	−	0.982849i	$-0.440963\pi$
$373$	7.12311			0.368820			0.184410	+	0.982849i	$0.440963\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$	7.12311	−	1.12311i	0.366859	−	0.0578429i
$378$		0			0
$379$			6.00000i			0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$			6.00000i			0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$		0			0
$381$	−13.1231			−0.672317
$382$		0			0
$383$		−	1.75379i		−	0.0896144i	−0.998996	−	0.0448072i	$-0.985733\pi$
$383$		−	1.75379i		−	0.0896144i	0.998996	−	0.0448072i	$-0.0142674\pi$
$384$		0			0
$385$		−	16.0000i		−	0.815436i
$386$		0			0
$387$	−6.24621			−0.317513
$388$		0			0
$389$	−0.246211			−0.0124834			−0.00624170	−	0.999981i	$-0.501987\pi$
$389$	−0.246211			−0.0124834			−0.00624170	+	0.999981i	$0.501987\pi$
$390$		0			0
$391$	10.2462			0.518173
$392$		0			0
$393$	4.00000			0.201773
$394$		0			0
$395$		−	8.00000i		−	0.402524i
$396$		0			0
$397$		−	15.3693i		−	0.771364i	−0.922632	−	0.385682i	$-0.873966\pi$
$397$		−	15.3693i		−	0.771364i	0.922632	−	0.385682i	$-0.126034\pi$
$398$		0			0
$399$	−18.7386			−0.938105
$400$		0			0
$401$		−	1.36932i		−	0.0683804i	−0.999415	−	0.0341902i	$-0.989115\pi$
$401$		−	1.36932i		−	0.0683804i	0.999415	−	0.0341902i	$-0.0108852\pi$
$402$		0			0
$403$	−1.75379	−	11.1231i	−0.0873624	−	0.554081i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$	−26.2462			−1.30098
$408$		0			0
$409$			8.49242i			0.419923i	0.977710	+	0.209962i	$0.0673339\pi$
$409$			8.49242i			0.419923i	−0.977710	+	0.209962i	$0.932666\pi$
$410$		0			0
$411$			10.4924i			0.517553i
$412$		0			0
$413$	3.50758			0.172597
$414$		0			0
$415$	−6.24621			−0.306614
$416$		0			0
$417$	−16.4924			−0.807637
$418$		0			0
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$		−	0.630683i		−	0.0307376i	−0.999882	−	0.0153688i	$-0.995108\pi$
$421$		−	0.630683i		−	0.0307376i	0.999882	−	0.0153688i	$-0.00489224\pi$
$422$		0			0
$423$			6.24621i			0.303701i
$424$		0			0
$425$	−2.00000			−0.0970143
$426$		0			0
$427$		−	31.2311i		−	1.51138i
$428$		0			0
$429$	2.87689	+	18.2462i	0.138898	+	0.880935i
$430$		0			0
$431$			32.4924i			1.56510i	0.622585	+	0.782552i	$0.286083\pi$
$431$			32.4924i			1.56510i	−0.622585	+	0.782552i	$0.713917\pi$
$432$		0			0
$433$	−18.0000			−0.865025			−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000			−0.865025			−0.432512	+	0.901628i	$0.642373\pi$
$434$		0			0
$435$		−	2.00000i		−	0.0958927i
$436$		0			0
$437$			30.7386i			1.47043i
$438$		0			0
$439$	36.4924			1.74169			0.870844	−	0.491559i	$-0.163573\pi$
$439$	36.4924			1.74169			0.870844	+	0.491559i	$0.163573\pi$
$440$		0			0
$441$	−2.75379			−0.131133
$442$		0			0
$443$	3.50758			0.166650			0.0833250	−	0.996522i	$-0.473446\pi$
$443$	3.50758			0.166650			0.0833250	+	0.996522i	$0.473446\pi$
$444$		0			0
$445$	−3.12311			−0.148049
$446$		0			0
$447$			14.0000i			0.662177i
$448$		0			0
$449$		−	28.8769i		−	1.36278i	−0.731918	−	0.681392i	$-0.761375\pi$
$449$		−	28.8769i		−	1.36278i	0.731918	−	0.681392i	$-0.238625\pi$
$450$		0			0
$451$	4.49242			0.211540
$452$		0			0
$453$			13.3693i			0.628145i
$454$		0			0
$455$	11.1231	−	1.75379i	0.521459	−	0.0822189i
$456$		0			0
$457$			31.3693i			1.46739i	0.679476	+	0.733697i	$0.262207\pi$
$457$			31.3693i			1.46739i	−0.679476	+	0.733697i	$0.737793\pi$
$458$		0			0
$459$	−2.00000			−0.0933520
$460$		0			0
$461$			18.4924i			0.861278i	0.902524	+	0.430639i	$0.141712\pi$
$461$			18.4924i			0.861278i	−0.902524	+	0.430639i	$0.858288\pi$
$462$		0			0
$463$			21.8617i			1.01600i	0.861357	+	0.508001i	$0.169615\pi$
$463$			21.8617i			1.01600i	−0.861357	+	0.508001i	$0.830385\pi$
$464$		0			0
$465$	−3.12311			−0.144831
$466$		0			0
$467$	22.2462			1.02943			0.514716	−	0.857361i	$-0.327897\pi$
$467$	22.2462			1.02943			0.514716	+	0.857361i	$0.327897\pi$
$468$		0			0
$469$	−15.2311			−0.703305
$470$		0			0
$471$	−21.3693			−0.984646
$472$		0			0
$473$			32.0000i			1.47136i
$474$		0			0
$475$		−	6.00000i		−	0.275299i
$476$		0			0
$477$	−13.3693			−0.612139
$478$		0			0
$479$			12.4924i			0.570793i	0.958409	+	0.285397i	$0.0921253\pi$
$479$			12.4924i			0.570793i	−0.958409	+	0.285397i	$0.907875\pi$
$480$		0			0
$481$	−2.87689	−	18.2462i	−0.131175	−	0.831956i
$482$		0			0
$483$			16.0000i			0.728025i
$484$		0			0
$485$	13.1231			0.595890
$486$		0			0
$487$		−	17.3693i		−	0.787079i	−0.919308	−	0.393539i	$-0.871250\pi$
$487$		−	17.3693i		−	0.787079i	0.919308	−	0.393539i	$-0.128750\pi$
$488$		0			0
$489$		−	7.12311i		−	0.322118i
$490$		0			0
$491$	−38.7386			−1.74825			−0.874125	−	0.485701i	$-0.838564\pi$
$491$	−38.7386			−1.74825			−0.874125	+	0.485701i	$0.838564\pi$
$492$		0			0
$493$	4.00000			0.180151
$494$		0			0
$495$	5.12311			0.230266
$496$		0			0
$497$	−32.0000			−1.43540
$498$		0			0
$499$		−	34.4924i		−	1.54409i	−0.635566	−	0.772046i	$-0.719233\pi$
$499$		−	34.4924i		−	1.54409i	0.635566	−	0.772046i	$-0.280767\pi$
$500$		0			0
$501$		−	22.2462i		−	0.993887i
$502$		0			0
$503$	−35.3693			−1.57704			−0.788520	−	0.615009i	$-0.789152\pi$
$503$	−35.3693			−1.57704			−0.788520	+	0.615009i	$0.789152\pi$
$504$		0			0
$505$		−	12.2462i		−	0.544949i
$506$		0			0
$507$	−12.3693	+	4.00000i	−0.549341	+	0.177646i
$508$		0			0
$509$			32.2462i			1.42929i	0.699488	+	0.714644i	$0.253411\pi$
$509$			32.2462i			1.42929i	−0.699488	+	0.714644i	$0.746589\pi$
$510$		0			0
$511$	−40.9848			−1.81306
$512$		0			0
$513$		−	6.00000i		−	0.264906i
$514$		0			0
$515$			13.1231i			0.578273i
$516$		0			0
$517$	32.0000			1.40736
$518$		0			0
$519$	−23.1231			−1.01499
$520$		0			0
$521$	0.246211			0.0107867			0.00539336	−	0.999985i	$-0.498283\pi$
$521$	0.246211			0.0107867			0.00539336	+	0.999985i	$0.498283\pi$
$522$		0			0
$523$	26.7386			1.16920			0.584599	−	0.811322i	$-0.301252\pi$
$523$	26.7386			1.16920			0.584599	+	0.811322i	$0.301252\pi$
$524$		0			0
$525$		−	3.12311i		−	0.136304i
$526$		0			0
$527$		−	6.24621i		−	0.272089i
$528$		0			0
$529$	3.24621			0.141140
$530$		0			0
$531$			1.12311i			0.0487386i
$532$		0			0
$533$	0.492423	+	3.12311i	0.0213292	+	0.135277i
$534$		0			0
$535$			8.00000i			0.345870i
$536$		0			0
$537$	16.4924			0.711701
$538$		0			0
$539$			14.1080i			0.607672i
$540$		0			0
$541$			10.8769i			0.467634i	0.972281	+	0.233817i	$0.0751217\pi$
$541$			10.8769i			0.467634i	−0.972281	+	0.233817i	$0.924878\pi$
$542$		0			0
$543$	20.2462			0.868848
$544$		0			0
$545$	−2.87689			−0.123233
$546$		0			0
$547$	−20.9848			−0.897247			−0.448624	−	0.893721i	$-0.648086\pi$
$547$	−20.9848			−0.897247			−0.448624	+	0.893721i	$0.648086\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$			12.0000i			0.511217i
$552$		0			0
$553$			24.9848i			1.06246i
$554$		0			0
$555$	−5.12311			−0.217464
$556$		0			0
$557$			11.7538i			0.498024i	0.968500	+	0.249012i	$0.0801059\pi$
$557$			11.7538i			0.498024i	−0.968500	+	0.249012i	$0.919894\pi$
$558$		0			0
$559$	−22.2462	+	3.50758i	−0.940914	+	0.148355i
$560$		0			0
$561$			10.2462i			0.432595i
$562$		0			0
$563$	32.9848			1.39015			0.695073	−	0.718939i	$-0.255372\pi$
$563$	32.9848			1.39015			0.695073	+	0.718939i	$0.255372\pi$
$564$		0			0
$565$			12.2462i			0.515202i
$566$		0			0
$567$		−	3.12311i		−	0.131158i
$568$		0			0
$569$	−12.7386			−0.534031			−0.267016	−	0.963692i	$-0.586038\pi$
$569$	−12.7386			−0.534031			−0.267016	+	0.963692i	$0.586038\pi$
$570$		0			0
$571$	8.49242			0.355397			0.177698	−	0.984085i	$-0.443135\pi$
$571$	8.49242			0.355397			0.177698	+	0.984085i	$0.443135\pi$
$572$		0			0
$573$	16.4924			0.688981
$574$		0			0
$575$	−5.12311			−0.213648
$576$		0			0
$577$			27.3693i			1.13940i	0.821853	+	0.569700i	$0.192941\pi$
$577$			27.3693i			1.13940i	−0.821853	+	0.569700i	$0.807059\pi$
$578$		0			0
$579$			25.1231i			1.04408i
$580$		0			0
$581$	19.5076			0.809311
$582$		0			0
$583$			68.4924i			2.83667i
$584$		0			0
$585$	0.561553	+	3.56155i	0.0232174	+	0.147252i
$586$		0			0
$587$			16.4924i			0.680715i	0.940296	+	0.340358i	$0.110548\pi$
$587$			16.4924i			0.680715i	−0.940296	+	0.340358i	$0.889452\pi$
$588$		0			0
$589$	18.7386			0.772112
$590$		0			0
$591$		−	16.2462i		−	0.668280i
$592$		0			0
$593$		−	5.50758i		−	0.226169i	−0.993585	−	0.113085i	$-0.963927\pi$
$593$		−	5.50758i		−	0.226169i	0.993585	−	0.113085i	$-0.0360731\pi$
$594$		0			0
$595$	6.24621			0.256070
$596$		0			0
$597$	18.2462			0.746768
$598$		0			0
$599$	36.4924			1.49104			0.745520	−	0.666483i	$-0.232201\pi$
$599$	36.4924			1.49104			0.745520	+	0.666483i	$0.232201\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$		0			0
$603$		−	4.87689i		−	0.198602i
$604$		0			0
$605$		−	15.2462i		−	0.619847i
$606$		0			0
$607$	−15.3693			−0.623821			−0.311911	−	0.950111i	$-0.600969\pi$
$607$	−15.3693			−0.623821			−0.311911	+	0.950111i	$0.600969\pi$
$608$		0			0
$609$			6.24621i			0.253109i
$610$		0			0
$611$	3.50758	+	22.2462i	0.141901	+	0.899985i
$612$		0			0
$613$			39.3693i			1.59011i	0.606536	+	0.795056i	$0.292559\pi$
$613$			39.3693i			1.59011i	−0.606536	+	0.795056i	$0.707441\pi$
$614$		0			0
$615$	0.876894			0.0353598
$616$		0			0
$617$		−	40.7386i		−	1.64008i	−0.572309	−	0.820038i	$-0.693952\pi$
$617$		−	40.7386i		−	1.64008i	0.572309	−	0.820038i	$-0.306048\pi$
$618$		0			0
$619$		−	38.9848i		−	1.56693i	−0.621434	−	0.783467i	$-0.713450\pi$
$619$		−	38.9848i		−	1.56693i	0.621434	−	0.783467i	$-0.286550\pi$
$620$		0			0
$621$	−5.12311			−0.205583
$622$		0			0
$623$	9.75379			0.390777
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−30.7386			−1.22758
$628$		0			0
$629$		−	10.2462i		−	0.408543i
$630$		0			0
$631$			35.6155i			1.41783i	0.705293	+	0.708916i	$0.250815\pi$
$631$			35.6155i			1.41783i	−0.705293	+	0.708916i	$0.749185\pi$
$632$		0			0
$633$	4.00000			0.158986
$634$		0			0
$635$			13.1231i			0.520775i
$636$		0			0
$637$	−9.80776	+	1.54640i	−0.388598	+	0.0612705i
$638$		0			0
$639$		−	10.2462i		−	0.405334i
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$		−	17.3693i		−	0.684979i	−0.939522	−	0.342489i	$-0.888730\pi$
$643$		−	17.3693i		−	0.684979i	0.939522	−	0.342489i	$-0.111270\pi$
$644$		0			0
$645$			6.24621i			0.245944i
$646$		0			0
$647$	−29.6155			−1.16431			−0.582153	−	0.813079i	$-0.697790\pi$
$647$	−29.6155			−1.16431			−0.582153	+	0.813079i	$0.697790\pi$
$648$		0			0
$649$	5.75379			0.225856
$650$		0			0
$651$	9.75379			0.382281
$652$		0			0
$653$	−13.8617			−0.542452			−0.271226	−	0.962516i	$-0.587429\pi$
$653$	−13.8617			−0.542452			−0.271226	+	0.962516i	$0.587429\pi$
$654$		0			0
$655$		−	4.00000i		−	0.156293i
$656$		0			0
$657$		−	13.1231i		−	0.511981i
$658$		0			0
$659$	21.7538			0.847407			0.423704	−	0.905801i	$-0.360730\pi$
$659$	21.7538			0.847407			0.423704	+	0.905801i	$0.360730\pi$
$660$		0			0
$661$			9.12311i			0.354848i	0.984135	+	0.177424i	$0.0567763\pi$
$661$			9.12311i			0.354848i	−0.984135	+	0.177424i	$0.943224\pi$
$662$		0			0
$663$	−7.12311	+	1.12311i	−0.276638	+	0.0436178i
$664$		0			0
$665$			18.7386i			0.726653i
$666$		0			0
$667$	10.2462			0.396735
$668$		0			0
$669$			9.36932i			0.362239i
$670$		0			0
$671$		−	51.2311i		−	1.97775i
$672$		0			0
$673$	26.9848			1.04019			0.520095	−	0.854109i	$-0.325897\pi$
$673$	26.9848			1.04019			0.520095	+	0.854109i	$0.325897\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$	12.8769			0.494899			0.247450	−	0.968901i	$-0.420408\pi$
$677$	12.8769			0.494899			0.247450	+	0.968901i	$0.420408\pi$
$678$		0			0
$679$	−40.9848			−1.57285
$680$		0			0
$681$			8.00000i			0.306561i
$682$		0			0
$683$			16.9848i			0.649907i	0.945730	+	0.324954i	$0.105349\pi$
$683$			16.9848i			0.649907i	−0.945730	+	0.324954i	$0.894651\pi$
$684$		0			0
$685$	10.4924			0.400895
$686$		0			0
$687$			5.12311i			0.195459i
$688$		0			0
$689$	−47.6155	+	7.50758i	−1.81401	+	0.286016i
$690$		0			0
$691$		−	28.7386i		−	1.09327i	−0.837371	−	0.546635i	$-0.815909\pi$
$691$		−	28.7386i		−	1.09327i	0.837371	−	0.546635i	$-0.184091\pi$
$692$		0			0
$693$	−16.0000			−0.607790
$694$		0			0
$695$			16.4924i			0.625593i
$696$		0			0
$697$			1.75379i			0.0664295i
$698$		0			0
$699$	7.75379			0.293275
$700$		0			0
$701$	−14.0000			−0.528773			−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000			−0.528773			−0.264386	+	0.964417i	$0.585169\pi$
$702$		0			0
$703$	30.7386			1.15933
$704$		0			0
$705$	6.24621			0.235246
$706$		0			0
$707$			38.2462i			1.43840i
$708$		0			0
$709$		−	1.61553i		−	0.0606724i	−0.999540	−	0.0303362i	$-0.990342\pi$
$709$		−	1.61553i		−	0.0606724i	0.999540	−	0.0303362i	$-0.00965780\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$	18.2462	−	2.87689i	0.682370	−	0.107590i
$716$		0			0
$717$			4.49242i			0.167773i
$718$		0			0
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$		−	40.9848i		−	1.52636i
$722$		0			0
$723$			14.2462i			0.529822i
$724$		0			0
$725$	−2.00000			−0.0742781
$726$		0			0
$727$	−19.8617			−0.736631			−0.368316	−	0.929701i	$-0.620065\pi$
$727$	−19.8617			−0.736631			−0.368316	+	0.929701i	$0.620065\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−12.4924			−0.462049
$732$		0			0
$733$			35.3693i			1.30640i	0.757187	+	0.653198i	$0.226573\pi$
$733$			35.3693i			1.30640i	−0.757187	+	0.653198i	$0.773427\pi$
$734$		0			0
$735$			2.75379i			0.101575i
$736$		0			0
$737$	−24.9848			−0.920329
$738$		0			0
$739$			37.2311i			1.36957i	0.728747	+	0.684783i	$0.240103\pi$
$739$			37.2311i			1.36957i	−0.728747	+	0.684783i	$0.759897\pi$
$740$		0			0
$741$	−3.36932	−	21.3693i	−0.123775	−	0.785021i
$742$		0			0
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	14.0000			0.512920
$746$		0			0
$747$			6.24621i			0.228537i
$748$		0			0
$749$		−	24.9848i		−	0.912926i
$750$		0			0
$751$	26.2462			0.957738			0.478869	−	0.877886i	$-0.341047\pi$
$751$	26.2462			0.957738			0.478869	+	0.877886i	$0.341047\pi$
$752$		0			0
$753$	26.2462			0.956465
$754$		0			0
$755$	13.3693			0.486559
$756$		0			0
$757$	−3.12311			−0.113511			−0.0567556	−	0.998388i	$-0.518076\pi$
$757$	−3.12311			−0.113511			−0.0567556	+	0.998388i	$0.518076\pi$
$758$		0			0
$759$			26.2462i			0.952677i
$760$		0			0
$761$		−	3.12311i		−	0.113212i	−0.998397	−	0.0566062i	$-0.981972\pi$
$761$		−	3.12311i		−	0.113212i	0.998397	−	0.0566062i	$-0.0180280\pi$
$762$		0			0
$763$	8.98485			0.325273
$764$		0			0
$765$			2.00000i			0.0723102i
$766$		0			0
$767$	0.630683	+	4.00000i	0.0227726	+	0.144432i
$768$		0			0
$769$		−	32.9848i		−	1.18946i	−0.803924	−	0.594732i	$-0.797258\pi$
$769$		−	32.9848i		−	1.18946i	0.803924	−	0.594732i	$-0.202742\pi$
$770$		0			0
$771$	12.2462			0.441037
$772$		0			0
$773$		−	16.2462i		−	0.584336i	−0.956367	−	0.292168i	$-0.905623\pi$
$773$		−	16.2462i		−	0.584336i	0.956367	−	0.292168i	$-0.0943766\pi$
$774$		0			0
$775$			3.12311i			0.112185i
$776$		0			0
$777$	16.0000			0.573997
$778$		0			0
$779$	−5.26137			−0.188508
$780$		0			0
$781$	−52.4924			−1.87833
$782$		0			0
$783$	−2.00000			−0.0714742
$784$		0			0
$785$			21.3693i			0.762704i
$786$		0			0
$787$			12.3845i			0.441459i	0.975335	+	0.220729i	$0.0708438\pi$
$787$			12.3845i			0.441459i	−0.975335	+	0.220729i	$0.929156\pi$
$788$		0			0
$789$	−27.3693			−0.974373
$790$		0			0
$791$		−	38.2462i		−	1.35988i
$792$		0			0
$793$	35.6155	−	5.61553i	1.26474	−	0.199413i
$794$		0			0
$795$			13.3693i			0.474161i
$796$		0			0
$797$	25.8617			0.916070			0.458035	−	0.888934i	$-0.348553\pi$
$797$	25.8617			0.916070			0.458035	+	0.888934i	$0.348553\pi$
$798$		0			0
$799$			12.4924i			0.441950i
$800$		0			0
$801$			3.12311i			0.110350i
$802$		0			0
$803$	−67.2311			−2.37253
$804$		0			0
$805$	16.0000			0.563926
$806$		0			0
$807$	16.2462			0.571894
$808$		0			0
$809$	−13.5076			−0.474901			−0.237451	−	0.971400i	$-0.576312\pi$
$809$	−13.5076			−0.474901			−0.237451	+	0.971400i	$0.576312\pi$
$810$		0			0
$811$			27.7538i			0.974567i	0.873244	+	0.487284i	$0.162012\pi$
$811$			27.7538i			0.974567i	−0.873244	+	0.487284i	$0.837988\pi$
$812$		0			0
$813$			23.1231i			0.810963i
$814$		0			0
$815$	−7.12311			−0.249512
$816$		0			0
$817$		−	37.4773i		−	1.31116i
$818$		0			0
$819$	−1.75379	−	11.1231i	−0.0612823	−	0.388673i
$820$		0			0
$821$			44.2462i			1.54420i	0.635499	+	0.772102i	$0.280794\pi$
$821$			44.2462i			1.54420i	−0.635499	+	0.772102i	$0.719206\pi$
$822$		0			0
$823$	42.8769			1.49459			0.747297	−	0.664490i	$-0.231351\pi$
$823$	42.8769			1.49459			0.747297	+	0.664490i	$0.231351\pi$
$824$		0			0
$825$		−	5.12311i		−	0.178364i
$826$		0			0
$827$		−	1.26137i		−	0.0438620i	−0.999759	−	0.0219310i	$-0.993019\pi$
$827$		−	1.26137i		−	0.0438620i	0.999759	−	0.0219310i	$-0.00698141\pi$
$828$		0			0
$829$	−24.2462			−0.842106			−0.421053	−	0.907036i	$-0.638339\pi$
$829$	−24.2462			−0.842106			−0.421053	+	0.907036i	$0.638339\pi$
$830$		0			0
$831$	29.8617			1.03589
$832$		0			0
$833$	−5.50758			−0.190826
$834$		0			0
$835$	−22.2462			−0.769862
$836$		0			0
$837$			3.12311i			0.107950i
$838$		0			0
$839$			46.7386i			1.61360i	0.590827	+	0.806798i	$0.298802\pi$
$839$			46.7386i			1.61360i	−0.590827	+	0.806798i	$0.701198\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			3.12311i			0.107565i
$844$		0			0
$845$	4.00000	+	12.3693i	0.137604	+	0.425517i
$846$		0			0
$847$			47.6155i			1.63609i
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$		−	26.2462i		−	0.899709i
$852$		0			0
$853$			35.8617i			1.22788i	0.789352	+	0.613941i	$0.210417\pi$
$853$			35.8617i			1.22788i	−0.789352	+	0.613941i	$0.789583\pi$
$854$		0			0
$855$	−6.00000			−0.205196
$856$		0			0
$857$	−30.4924			−1.04160			−0.520801	−	0.853678i	$-0.674367\pi$
$857$	−30.4924			−1.04160			−0.520801	+	0.853678i	$0.674367\pi$
$858$		0			0
$859$	12.0000			0.409435			0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000			0.409435			0.204717	+	0.978821i	$0.434372\pi$
$860$		0			0
$861$	−2.73863			−0.0933324
$862$		0			0
$863$			6.24621i			0.212624i	0.994333	+	0.106312i	$0.0339042\pi$
$863$			6.24621i			0.212624i	−0.994333	+	0.106312i	$0.966096\pi$
$864$		0			0
$865$			23.1231i			0.786209i
$866$		0			0
$867$	13.0000			0.441503
$868$		0			0
$869$			40.9848i			1.39032i
$870$		0			0
$871$	−2.73863	−	17.3693i	−0.0927951	−	0.588537i
$872$		0			0
$873$		−	13.1231i		−	0.444150i
$874$		0			0
$875$	−3.12311			−0.105580
$876$		0			0
$877$		−	18.8769i		−	0.637427i	−0.947851	−	0.318714i	$-0.896749\pi$
$877$		−	18.8769i		−	0.637427i	0.947851	−	0.318714i	$-0.103251\pi$
$878$		0			0
$879$		−	28.7386i		−	0.969330i
$880$		0			0
$881$	28.2462			0.951639			0.475820	−	0.879543i	$-0.342152\pi$
$881$	28.2462			0.951639			0.475820	+	0.879543i	$0.342152\pi$
$882$		0			0
$883$	9.75379			0.328241			0.164121	−	0.986440i	$-0.447521\pi$
$883$	9.75379			0.328241			0.164121	+	0.986440i	$0.447521\pi$
$884$		0			0
$885$	1.12311			0.0377528
$886$		0			0
$887$	18.8769			0.633824			0.316912	−	0.948455i	$-0.397354\pi$
$887$	18.8769			0.633824			0.316912	+	0.948455i	$0.397354\pi$
$888$		0			0
$889$		−	40.9848i		−	1.37459i
$890$		0			0
$891$		−	5.12311i		−	0.171630i
$892$		0			0
$893$	−37.4773			−1.25413
$894$		0			0
$895$		−	16.4924i		−	0.551281i
$896$		0			0
$897$	−18.2462	+	2.87689i	−0.609223	+	0.0960567i
$898$		0			0
$899$		−	6.24621i		−	0.208323i
$900$		0			0
$901$	−26.7386			−0.890793
$902$		0			0
$903$		−	19.5076i		−	0.649172i
$904$		0			0
$905$		−	20.2462i		−	0.673007i
$906$		0			0
$907$	−25.7538			−0.855141			−0.427570	−	0.903982i	$-0.640630\pi$
$907$	−25.7538			−0.855141			−0.427570	+	0.903982i	$0.640630\pi$
$908$		0			0
$909$	−12.2462			−0.406181
$910$		0			0
$911$	−4.00000			−0.132526			−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000			−0.132526			−0.0662630	+	0.997802i	$0.521108\pi$
$912$		0			0
$913$	32.0000			1.05905
$914$		0			0
$915$		−	10.0000i		−	0.330590i
$916$		0			0
$917$			12.4924i			0.412536i
$918$		0			0
$919$	21.7538			0.717591			0.358796	−	0.933416i	$-0.383187\pi$
$919$	21.7538			0.717591			0.358796	+	0.933416i	$0.383187\pi$
$920$		0			0
$921$			31.1231i			1.02554i
$922$		0			0
$923$	−5.75379	−	36.4924i	−0.189388	−	1.20116i
$924$		0			0
$925$			5.12311i			0.168447i
$926$		0			0
$927$	13.1231			0.431019
$928$		0			0
$929$			28.1080i			0.922192i	0.887350	+	0.461096i	$0.152544\pi$
$929$			28.1080i			0.922192i	−0.887350	+	0.461096i	$0.847456\pi$
$930$		0			0
$931$		−	16.5227i		−	0.541511i
$932$		0			0
$933$	8.49242			0.278029
$934$		0			0
$935$	10.2462			0.335087
$936$		0			0
$937$	−20.2462			−0.661415			−0.330707	−	0.943733i	$-0.607287\pi$
$937$	−20.2462			−0.661415			−0.330707	+	0.943733i	$0.607287\pi$
$938$		0			0
$939$	−16.2462			−0.530175
$940$		0			0
$941$		−	14.0000i		−	0.456387i	−0.973616	−	0.228193i	$-0.926718\pi$
$941$		−	14.0000i		−	0.456387i	0.973616	−	0.228193i	$-0.0732819\pi$
$942$		0			0
$943$			4.49242i			0.146293i
$944$		0			0
$945$	−3.12311			−0.101595
$946$		0			0
$947$		−	8.49242i		−	0.275967i	−0.990435	−	0.137983i	$-0.955938\pi$
$947$		−	8.49242i		−	0.275967i	0.990435	−	0.137983i	$-0.0440620\pi$
$948$		0			0
$949$	−7.36932	−	46.7386i	−0.239218	−	1.51720i
$950$		0			0
$951$		−	6.00000i		−	0.194563i
$952$		0			0
$953$	22.9848			0.744552			0.372276	−	0.928122i	$-0.378577\pi$
$953$	22.9848			0.744552			0.372276	+	0.928122i	$0.378577\pi$
$954$		0			0
$955$		−	16.4924i		−	0.533682i
$956$		0			0
$957$			10.2462i			0.331213i
$958$		0			0
$959$	−32.7689			−1.05816
$960$		0			0
$961$	21.2462			0.685362
$962$		0			0
$963$	8.00000			0.257796
$964$		0			0
$965$	25.1231			0.808741
$966$		0			0
$967$			47.6155i			1.53121i	0.643310	+	0.765606i	$0.277561\pi$
$967$			47.6155i			1.53121i	−0.643310	+	0.765606i	$0.722439\pi$
$968$		0			0
$969$		−	12.0000i		−	0.385496i
$970$		0			0
$971$	−37.7538			−1.21158			−0.605788	−	0.795626i	$-0.707142\pi$
$971$	−37.7538			−1.21158			−0.605788	+	0.795626i	$0.707142\pi$
$972$		0			0
$973$		−	51.5076i		−	1.65126i
$974$		0			0
$975$	3.56155	−	0.561553i	0.114061	−	0.0179841i
$976$		0			0
$977$		−	4.73863i		−	0.151602i	−0.997123	−	0.0758012i	$-0.975849\pi$
$977$		−	4.73863i		−	0.151602i	0.997123	−	0.0758012i	$-0.0241514\pi$
$978$		0			0
$979$	16.0000			0.511362
$980$		0			0
$981$			2.87689i			0.0918522i
$982$		0			0
$983$			60.4924i			1.92941i	0.263335	+	0.964704i	$0.415177\pi$
$983$			60.4924i			1.92941i	−0.263335	+	0.964704i	$0.584823\pi$
$984$		0			0
$985$	−16.2462			−0.517647
$986$		0			0
$987$	−19.5076			−0.620933
$988$		0			0
$989$	−32.0000			−1.01754
$990$		0			0
$991$	30.7386			0.976445			0.488222	−	0.872719i	$-0.337645\pi$
$991$	30.7386			0.976445			0.488222	+	0.872719i	$0.337645\pi$
$992$		0			0
$993$		−	7.75379i		−	0.246059i
$994$		0			0
$995$		−	18.2462i		−	0.578444i
$996$		0			0
$997$	38.3542			1.21469			0.607344	−	0.794439i	$-0.292235\pi$
$997$	38.3542			1.21469			0.607344	+	0.794439i	$0.292235\pi$
$998$		0			0
$999$			5.12311i			0.162088i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.g.o.961.3		4
4.3	odd	2		390.2.b.d.181.2	✓	4
12.11	even	2		1170.2.b.f.181.4		4
13.12	even	2	inner	3120.2.g.o.961.2		4
20.3	even	4		1950.2.f.l.649.4		4
20.7	even	4		1950.2.f.o.649.1		4
20.19	odd	2		1950.2.b.h.1351.3		4
52.31	even	4		5070.2.a.bd.1.1		2
52.47	even	4		5070.2.a.bh.1.2		2
52.51	odd	2		390.2.b.d.181.3	yes	4
156.155	even	2		1170.2.b.f.181.1		4
260.103	even	4		1950.2.f.o.649.3		4
260.207	even	4		1950.2.f.l.649.2		4
260.259	odd	2		1950.2.b.h.1351.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.d.181.2	✓	4	4.3	odd	2
390.2.b.d.181.3	yes	4	52.51	odd	2
1170.2.b.f.181.1		4	156.155	even	2
1170.2.b.f.181.4		4	12.11	even	2
1950.2.b.h.1351.2		4	260.259	odd	2
1950.2.b.h.1351.3		4	20.19	odd	2
1950.2.f.l.649.2		4	260.207	even	4
1950.2.f.l.649.4		4	20.3	even	4
1950.2.f.o.649.1		4	20.7	even	4
1950.2.f.o.649.3		4	260.103	even	4
3120.2.g.o.961.2		4	13.12	even	2	inner
3120.2.g.o.961.3		4	1.1	even	1	trivial
5070.2.a.bd.1.1		2	52.31	even	4
5070.2.a.bh.1.2		2	52.47	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000i q^{5} -3.12311i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000i q^{5} -3.12311i q^{7} +1.00000 q^{9} -5.12311i q^{11} +(3.56155 - 0.561553i) q^{13} -1.00000i q^{15} +2.00000 q^{17} +6.00000i q^{19} +3.12311i q^{21} +5.12311 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -3.12311i q^{31} +5.12311i q^{33} +3.12311 q^{35} -5.12311i q^{37} +(-3.56155 + 0.561553i) q^{39} +0.876894i q^{41} -6.24621 q^{43} +1.00000i q^{45} +6.24621i q^{47} -2.75379 q^{49} -2.00000 q^{51} -13.3693 q^{53} +5.12311 q^{55} -6.00000i q^{57} +1.12311i q^{59} +10.0000 q^{61} -3.12311i q^{63} +(0.561553 + 3.56155i) q^{65} -4.87689i q^{67} -5.12311 q^{69} -10.2462i q^{71} -13.1231i q^{73} +1.00000 q^{75} -16.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.24621i q^{83} +2.00000i q^{85} -2.00000 q^{87} +3.12311i q^{89} +(-1.75379 - 11.1231i) q^{91} +3.12311i q^{93} -6.00000 q^{95} -13.1231i q^{97} -5.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{9} + 6 q^{13} + 8 q^{17} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} - 4 q^{35} - 6 q^{39} + 8 q^{43} - 44 q^{49} - 8 q^{51} - 4 q^{53} + 4 q^{55} + 40 q^{61} - 6 q^{65} - 4 q^{69} + 4 q^{75} - 64 q^{77} - 32 q^{79} + 4 q^{81} - 8 q^{87} - 40 q^{91} - 24 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 + 6 * q^13 + 8 * q^17 + 4 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 - 4 * q^35 - 6 * q^39 + 8 * q^43 - 44 * q^49 - 8 * q^51 - 4 * q^53 + 4 * q^55 + 40 * q^61 - 6 * q^65 - 4 * q^69 + 4 * q^75 - 64 * q^77 - 32 * q^79 + 4 * q^81 - 8 * q^87 - 40 * q^91 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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