LMFDB - Embedded newform 3120.2.l.o.1249.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1249,3120)]

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, 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("3120.1249");

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.l (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:	$8$
Coefficient field:	8.0.1698758656.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 $ x^8 + 18x^6 + 97x^4 + 176*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.4
Root			$2.16053i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.o.1249.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.23483 + 0.0743018i) q^{5} +2.82843i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(2.23483 + 0.0743018i) q^{5} +2.82843i q^{7} -1.00000 q^{9} +6.46967 q^{11} -1.00000i q^{13} +(0.0743018 - 2.23483i) q^{15} -3.49264i q^{17} -2.21016 q^{19} +2.82843 q^{21} +7.14949i q^{23} +(4.98896 + 0.332104i) q^{25} +1.00000i q^{27} -4.82843 q^{29} +9.65685 q^{31} -6.46967i q^{33} +(-0.210157 + 6.32106i) q^{35} -6.93933i q^{37} -1.00000 q^{39} -0.148604 q^{41} -3.03858i q^{43} +(-2.23483 - 0.0743018i) q^{45} +6.79073i q^{47} -1.00000 q^{49} -3.49264 q^{51} +1.70279i q^{53} +(14.4586 + 0.480708i) q^{55} +2.21016i q^{57} -6.46967 q^{59} +2.66421 q^{61} -2.82843i q^{63} +(0.0743018 - 2.23483i) q^{65} -7.70279i q^{67} +7.14949 q^{69} +11.0879 q^{71} +5.76776i q^{73} +(0.332104 - 4.98896i) q^{75} +18.2990i q^{77} -10.0239 q^{79} +1.00000 q^{81} +2.27171i q^{83} +(0.259509 - 7.80546i) q^{85} +4.82843i q^{87} +9.21104 q^{89} +2.82843 q^{91} -9.65685i q^{93} +(-4.93933 - 0.164219i) q^{95} +8.11091i q^{97} -6.46967 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 16 q^{11} + 4 q^{15} - 24 q^{19} - 4 q^{25} - 16 q^{29} + 32 q^{31} - 8 q^{35} - 8 q^{39} - 8 q^{41} - 8 q^{49} - 8 q^{51} + 36 q^{55} - 16 q^{59} + 24 q^{61} + 4 q^{65} - 8 q^{69} + 24 q^{71} + 4 q^{75} - 24 q^{79} + 8 q^{81} - 40 q^{85} + 8 q^{89} + 32 q^{95} - 16 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 + 16 * q^11 + 4 * q^15 - 24 * q^19 - 4 * q^25 - 16 * q^29 + 32 * q^31 - 8 * q^35 - 8 * q^39 - 8 * q^41 - 8 * q^49 - 8 * q^51 + 36 * q^55 - 16 * q^59 + 24 * q^61 + 4 * q^65 - 8 * q^69 + 24 * q^71 + 4 * q^75 - 24 * q^79 + 8 * q^81 - 40 * q^85 + 8 * q^89 + 32 * q^95 - 16 * q^99

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	2.23483	+	0.0743018i	0.999448	+	0.0332288i
$5$	2.23483	+	0.0743018i	0.999448	+	0.0332288i
$6$		0			0
$7$			2.82843i			1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$			2.82843i			1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	6.46967			1.95068			0.975339	−	0.220713i	$-0.0708383\pi$
$11$	6.46967			1.95068			0.975339	+	0.220713i	$0.0708383\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	0.0743018	−	2.23483i	0.0191846	−	0.577031i
$16$		0			0
$17$		−	3.49264i		−	0.847089i	−0.905875	−	0.423544i	$-0.860786\pi$
$17$		−	3.49264i		−	0.847089i	0.905875	−	0.423544i	$-0.139214\pi$
$18$		0			0
$19$	−2.21016			−0.507045			−0.253522	−	0.967330i	$-0.581589\pi$
$19$	−2.21016			−0.507045			−0.253522	+	0.967330i	$0.581589\pi$
$20$		0			0
$21$	2.82843			0.617213
$22$		0			0
$23$			7.14949i			1.49077i	0.666633	+	0.745386i	$0.267735\pi$
$23$			7.14949i			1.49077i	−0.666633	+	0.745386i	$0.732265\pi$
$24$		0			0
$25$	4.98896	+	0.332104i	0.997792	+	0.0664208i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−4.82843			−0.896616			−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843			−0.896616			−0.448308	+	0.893879i	$0.647973\pi$
$30$		0			0
$31$	9.65685			1.73442			0.867211	−	0.497941i	$-0.165910\pi$
$31$	9.65685			1.73442			0.867211	+	0.497941i	$0.165910\pi$
$32$		0			0
$33$		−	6.46967i		−	1.12622i
$34$		0			0
$35$	−0.210157	+	6.32106i	−0.0355231	+	1.06845i
$36$		0			0
$37$		−	6.93933i		−	1.14082i	−0.821360	−	0.570410i	$-0.806784\pi$
$37$		−	6.93933i		−	1.14082i	0.821360	−	0.570410i	$-0.193216\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	−0.148604			−0.0232080			−0.0116040	−	0.999933i	$-0.503694\pi$
$41$	−0.148604			−0.0232080			−0.0116040	+	0.999933i	$0.503694\pi$
$42$		0			0
$43$		−	3.03858i		−	0.463380i	−0.972790	−	0.231690i	$-0.925575\pi$
$43$		−	3.03858i		−	0.463380i	0.972790	−	0.231690i	$-0.0744255\pi$
$44$		0			0
$45$	−2.23483	−	0.0743018i	−0.333149	−	0.0110763i
$46$		0			0
$47$			6.79073i			0.990530i	0.868742	+	0.495265i	$0.164929\pi$
$47$			6.79073i			0.990530i	−0.868742	+	0.495265i	$0.835071\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−3.49264			−0.489067
$52$		0			0
$53$			1.70279i			0.233897i	0.993138	+	0.116948i	$0.0373112\pi$
$53$			1.70279i			0.233897i	−0.993138	+	0.116948i	$0.962689\pi$
$54$		0			0
$55$	14.4586	+	0.480708i	1.94960	+	0.0648186i
$56$		0			0
$57$			2.21016i			0.292742i
$58$		0			0
$59$	−6.46967			−0.842279			−0.421139	−	0.906996i	$-0.638370\pi$
$59$	−6.46967			−0.842279			−0.421139	+	0.906996i	$0.638370\pi$
$60$		0			0
$61$	2.66421			0.341117			0.170558	−	0.985348i	$-0.445443\pi$
$61$	2.66421			0.341117			0.170558	+	0.985348i	$0.445443\pi$
$62$		0			0
$63$		−	2.82843i		−	0.356348i
$64$		0			0
$65$	0.0743018	−	2.23483i	0.00921600	−	0.277197i
$66$		0			0
$67$		−	7.70279i		−	0.941046i	−0.882388	−	0.470523i	$-0.844065\pi$
$67$		−	7.70279i		−	0.941046i	0.882388	−	0.470523i	$-0.155935\pi$
$68$		0			0
$69$	7.14949			0.860697
$70$		0			0
$71$	11.0879			1.31590			0.657948	−	0.753063i	$-0.271425\pi$
$71$	11.0879			1.31590			0.657948	+	0.753063i	$0.271425\pi$
$72$		0			0
$73$			5.76776i			0.675065i	0.941314	+	0.337533i	$0.109592\pi$
$73$			5.76776i			0.675065i	−0.941314	+	0.337533i	$0.890408\pi$
$74$		0			0
$75$	0.332104	−	4.98896i	0.0383481	−	0.576075i
$76$		0			0
$77$			18.2990i			2.08536i
$78$		0			0
$79$	−10.0239			−1.12777			−0.563886	−	0.825853i	$-0.690694\pi$
$79$	−10.0239			−1.12777			−0.563886	+	0.825853i	$0.690694\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			2.27171i			0.249353i	0.992197	+	0.124676i	$0.0397892\pi$
$83$			2.27171i			0.249353i	−0.992197	+	0.124676i	$0.960211\pi$
$84$		0			0
$85$	0.259509	−	7.80546i	0.0281477	−	0.846621i
$86$		0			0
$87$			4.82843i			0.517662i
$88$		0			0
$89$	9.21104			0.976369			0.488184	−	0.872741i	$-0.337659\pi$
$89$	9.21104			0.976369			0.488184	+	0.872741i	$0.337659\pi$
$90$		0			0
$91$	2.82843			0.296500
$92$		0			0
$93$		−	9.65685i		−	1.00137i
$94$		0			0
$95$	−4.93933	−	0.164219i	−0.506765	−	0.0168485i
$96$		0			0
$97$			8.11091i			0.823538i	0.911288	+	0.411769i	$0.135089\pi$
$97$			8.11091i			0.823538i	−0.911288	+	0.411769i	$0.864911\pi$
$98$		0			0
$99$	−6.46967			−0.650226
$100$		0			0
$101$	−16.1421			−1.60620			−0.803101	−	0.595843i	$-0.796818\pi$
$101$	−16.1421			−1.60620			−0.803101	+	0.595843i	$0.796818\pi$
$102$		0			0
$103$			19.5815i			1.92942i	0.263318	+	0.964709i	$0.415183\pi$
$103$			19.5815i			1.92942i	−0.263318	+	0.964709i	$0.584817\pi$
$104$		0			0
$105$	6.32106	+	0.210157i	0.616873	+	0.0205092i
$106$		0			0
$107$			1.35965i			0.131442i	0.997838	+	0.0657210i	$0.0209347\pi$
$107$			1.35965i			0.131442i	−0.997838	+	0.0657210i	$0.979065\pi$
$108$		0			0
$109$	−3.49264			−0.334534			−0.167267	−	0.985912i	$-0.553494\pi$
$109$	−3.49264			−0.334534			−0.167267	+	0.985912i	$0.553494\pi$
$110$		0			0
$111$	−6.93933			−0.658652
$112$		0			0
$113$		−	12.4320i		−	1.16950i	−0.811213	−	0.584751i	$-0.801192\pi$
$113$		−	12.4320i		−	1.16950i	0.811213	−	0.584751i	$-0.198808\pi$
$114$		0			0
$115$	−0.531220	+	15.9779i	−0.0495365	+	1.48995i
$116$		0			0
$117$			1.00000i			0.0924500i
$118$		0			0
$119$	9.87867			0.905576
$120$		0			0
$121$	30.8566			2.80514
$122$		0			0
$123$			0.148604i			0.0133991i
$124$		0			0
$125$	11.1248	+	1.11289i	0.995034	+	0.0995396i
$126$		0			0
$127$			18.6200i			1.65226i	0.563478	+	0.826131i	$0.309463\pi$
$127$			18.6200i			1.65226i	−0.563478	+	0.826131i	$0.690537\pi$
$128$		0			0
$129$	−3.03858			−0.267532
$130$		0			0
$131$	0.751258			0.0656378			0.0328189	−	0.999461i	$-0.489552\pi$
$131$	0.751258			0.0656378			0.0328189	+	0.999461i	$0.489552\pi$
$132$		0			0
$133$		−	6.25127i		−	0.542054i
$134$		0			0
$135$	−0.0743018	+	2.23483i	−0.00639488	+	0.192344i
$136$		0			0
$137$		−	1.08794i		−	0.0929487i	−0.998919	−	0.0464743i	$-0.985201\pi$
$137$		−	1.08794i		−	0.0929487i	0.998919	−	0.0464743i	$-0.0147986\pi$
$138$		0			0
$139$	0.420314			0.0356506			0.0178253	−	0.999841i	$-0.494326\pi$
$139$	0.420314			0.0356506			0.0178253	+	0.999841i	$0.494326\pi$
$140$		0			0
$141$	6.79073			0.571883
$142$		0			0
$143$		−	6.46967i		−	0.541021i
$144$		0			0
$145$	−10.7907	−	0.358761i	−0.896121	−	0.0297935i
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	14.1265			1.15729			0.578645	−	0.815580i	$-0.303582\pi$
$149$	14.1265			1.15729			0.578645	+	0.815580i	$0.303582\pi$
$150$		0			0
$151$	−6.07717			−0.494553			−0.247276	−	0.968945i	$-0.579536\pi$
$151$	−6.07717			−0.494553			−0.247276	+	0.968945i	$0.579536\pi$
$152$		0			0
$153$			3.49264i			0.282363i
$154$		0			0
$155$	21.5815	+	0.717522i	1.73346	+	0.0576327i
$156$		0			0
$157$			11.7266i			0.935888i	0.883758	+	0.467944i	$0.155005\pi$
$157$			11.7266i			0.935888i	−0.883758	+	0.467944i	$0.844995\pi$
$158$		0			0
$159$	1.70279			0.135040
$160$		0			0
$161$	−20.2218			−1.59370
$162$		0			0
$163$		−	7.31371i		−	0.572854i	−0.958102	−	0.286427i	$-0.907532\pi$
$163$		−	7.31371i		−	0.572854i	0.958102	−	0.286427i	$-0.0924676\pi$
$164$		0			0
$165$	0.480708	−	14.4586i	0.0374231	−	1.12560i
$166$		0			0
$167$		−	21.0420i		−	1.62828i	−0.580670	−	0.814139i	$-0.697209\pi$
$167$		−	21.0420i		−	1.62828i	0.580670	−	0.814139i	$-0.302791\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	2.21016			0.169015
$172$		0			0
$173$		−	18.9706i		−	1.44231i	−0.692776	−	0.721153i	$-0.743613\pi$
$173$		−	18.9706i		−	1.44231i	0.692776	−	0.721153i	$-0.256387\pi$
$174$		0			0
$175$	−0.939333	+	14.1109i	−0.0710069	+	1.06668i
$176$		0			0
$177$			6.46967i			0.486290i
$178$		0			0
$179$	20.4099			1.52551			0.762753	−	0.646690i	$-0.223847\pi$
$179$	20.4099			1.52551			0.762753	+	0.646690i	$0.223847\pi$
$180$		0			0
$181$	24.8934			1.85031			0.925156	−	0.379588i	$-0.123934\pi$
$181$	24.8934			1.85031			0.925156	+	0.379588i	$0.123934\pi$
$182$		0			0
$183$		−	2.66421i		−	0.196944i
$184$		0			0
$185$	0.515605	−	15.5083i	0.0379080	−	1.14019i
$186$		0			0
$187$		−	22.5962i		−	1.65240i
$188$		0			0
$189$	−2.82843			−0.205738
$190$		0			0
$191$	14.9853			1.08430			0.542148	−	0.840283i	$-0.317611\pi$
$191$	14.9853			1.08430			0.542148	+	0.840283i	$0.317611\pi$
$192$		0			0
$193$		−	18.7990i		−	1.35318i	−0.736360	−	0.676590i	$-0.763457\pi$
$193$		−	18.7990i		−	1.35318i	0.736360	−	0.676590i	$-0.236543\pi$
$194$		0			0
$195$	−2.23483	−	0.0743018i	−0.160040	−	0.00532086i
$196$		0			0
$197$		−	1.92857i		−	0.137405i	−0.997637	−	0.0687023i	$-0.978114\pi$
$197$		−	1.92857i		−	0.137405i	0.997637	−	0.0687023i	$-0.0218859\pi$
$198$		0			0
$199$	−10.0239			−0.710572			−0.355286	−	0.934758i	$-0.615617\pi$
$199$	−10.0239			−0.710572			−0.355286	+	0.934758i	$0.615617\pi$
$200$		0			0
$201$	−7.70279			−0.543313
$202$		0			0
$203$		−	13.6569i		−	0.958523i
$204$		0			0
$205$	−0.332104	−	0.0110415i	−0.0231952	−	0.000771173i
$206$		0			0
$207$		−	7.14949i		−	0.496924i
$208$		0			0
$209$	−14.2990			−0.989081
$210$		0			0
$211$	−20.2751			−1.39580			−0.697899	−	0.716197i	$-0.745881\pi$
$211$	−20.2751			−1.39580			−0.697899	+	0.716197i	$0.745881\pi$
$212$		0			0
$213$		−	11.0879i		−	0.759733i
$214$		0			0
$215$	0.225772	−	6.79073i	0.0153975	−	0.463124i
$216$		0			0
$217$			27.3137i			1.85418i
$218$		0			0
$219$	5.76776			0.389749
$220$		0			0
$221$	−3.49264			−0.234940
$222$		0			0
$223$		−	16.1881i		−	1.08403i	−0.840368	−	0.542017i	$-0.817661\pi$
$223$		−	16.1881i		−	1.08403i	0.840368	−	0.542017i	$-0.182339\pi$
$224$		0			0
$225$	−4.98896	−	0.332104i	−0.332597	−	0.0221403i
$226$		0			0
$227$		−	23.0879i		−	1.53240i	−0.642602	−	0.766200i	$-0.722145\pi$
$227$		−	23.0879i		−	1.53240i	0.642602	−	0.766200i	$-0.277855\pi$
$228$		0			0
$229$	4.30886			0.284738			0.142369	−	0.989814i	$-0.454528\pi$
$229$	4.30886			0.284738			0.142369	+	0.989814i	$0.454528\pi$
$230$		0			0
$231$	18.2990			1.20398
$232$		0			0
$233$		−	1.53857i		−	0.100795i	−0.998729	−	0.0503977i	$-0.983951\pi$
$233$		−	1.53857i		−	0.100795i	0.998729	−	0.0503977i	$-0.0160489\pi$
$234$		0			0
$235$	−0.504563	+	15.1761i	−0.0329141	+	0.989983i
$236$		0			0
$237$			10.0239i			0.651119i
$238$		0			0
$239$	−17.8514			−1.15471			−0.577355	−	0.816493i	$-0.695915\pi$
$239$	−17.8514			−1.15471			−0.577355	+	0.816493i	$0.695915\pi$
$240$		0			0
$241$	3.95406			0.254703			0.127352	−	0.991858i	$-0.459352\pi$
$241$	3.95406			0.254703			0.127352	+	0.991858i	$0.459352\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−2.23483	−	0.0743018i	−0.142778	−	0.00474697i
$246$		0			0
$247$			2.21016i			0.140629i
$248$		0			0
$249$	2.27171			0.143964
$250$		0			0
$251$	−3.42284			−0.216048			−0.108024	−	0.994148i	$-0.534452\pi$
$251$	−3.42284			−0.216048			−0.108024	+	0.994148i	$0.534452\pi$
$252$		0			0
$253$			46.2548i			2.90802i
$254$		0			0
$255$	−7.80546	−	0.259509i	−0.488797	−	0.0162511i
$256$		0			0
$257$			3.14772i			0.196349i	0.995169	+	0.0981746i	$0.0313004\pi$
$257$			3.14772i			0.196349i	−0.995169	+	0.0981746i	$0.968700\pi$
$258$		0			0
$259$	19.6274			1.21959
$260$		0			0
$261$	4.82843			0.298872
$262$		0			0
$263$		−	13.7144i		−	0.845669i	−0.906207	−	0.422835i	$-0.861035\pi$
$263$		−	13.7144i		−	0.845669i	0.906207	−	0.422835i	$-0.138965\pi$
$264$		0			0
$265$	−0.126521	+	3.80546i	−0.00777210	+	0.233767i
$266$		0			0
$267$		−	9.21104i		−	0.563707i
$268$		0			0
$269$	−9.84316			−0.600148			−0.300074	−	0.953916i	$-0.597011\pi$
$269$	−9.84316			−0.600148			−0.300074	+	0.953916i	$0.597011\pi$
$270$		0			0
$271$	2.86216			0.173864			0.0869320	−	0.996214i	$-0.472294\pi$
$271$	2.86216			0.173864			0.0869320	+	0.996214i	$0.472294\pi$
$272$		0			0
$273$		−	2.82843i		−	0.171184i
$274$		0			0
$275$	32.2769	+	2.14860i	1.94637	+	0.129566i
$276$		0			0
$277$		−	6.24389i		−	0.375159i	−0.982249	−	0.187580i	$-0.939936\pi$
$277$		−	6.24389i		−	0.375159i	0.982249	−	0.187580i	$-0.0600643\pi$
$278$		0			0
$279$	−9.65685			−0.578141
$280$		0			0
$281$	−31.3869			−1.87239			−0.936193	−	0.351486i	$-0.885677\pi$
$281$	−31.3869			−1.87239			−0.936193	+	0.351486i	$0.885677\pi$
$282$		0			0
$283$		−	2.56496i		−	0.152471i	−0.997090	−	0.0762354i	$-0.975710\pi$
$283$		−	2.56496i		−	0.152471i	0.997090	−	0.0762354i	$-0.0242901\pi$
$284$		0			0
$285$	−0.164219	+	4.93933i	−0.00972747	+	0.292581i
$286$		0			0
$287$		−	0.420314i		−	0.0248104i
$288$		0			0
$289$	4.80150			0.282441
$290$		0			0
$291$	8.11091			0.475470
$292$		0			0
$293$		−	0.834895i		−	0.0487751i	−0.999703	−	0.0243875i	$-0.992236\pi$
$293$		−	0.834895i		−	0.0487751i	0.999703	−	0.0243875i	$-0.00776357\pi$
$294$		0			0
$295$	−14.4586	−	0.480708i	−0.841814	−	0.0279879i
$296$		0			0
$297$			6.46967i			0.375408i
$298$		0			0
$299$	7.14949			0.413466
$300$		0			0
$301$	8.59441			0.495374
$302$		0			0
$303$			16.1421i			0.927341i
$304$		0			0
$305$	5.95406	+	0.197955i	0.340929	+	0.0113349i
$306$		0			0
$307$		−	17.7556i		−	1.01336i	−0.862133	−	0.506682i	$-0.830872\pi$
$307$		−	17.7556i		−	1.01336i	0.862133	−	0.506682i	$-0.169128\pi$
$308$		0			0
$309$	19.5815			1.11395
$310$		0			0
$311$	1.35965			0.0770985			0.0385493	−	0.999257i	$-0.487726\pi$
$311$	1.35965			0.0770985			0.0385493	+	0.999257i	$0.487726\pi$
$312$		0			0
$313$		−	7.75611i		−	0.438401i	−0.975680	−	0.219201i	$-0.929655\pi$
$313$		−	7.75611i		−	0.438401i	0.975680	−	0.219201i	$-0.0703449\pi$
$314$		0			0
$315$	0.210157	−	6.32106i	0.0118410	−	0.356152i
$316$		0			0
$317$			9.38514i			0.527122i	0.964643	+	0.263561i	$0.0848971\pi$
$317$			9.38514i			0.527122i	−0.964643	+	0.263561i	$0.915103\pi$
$318$		0			0
$319$	−31.2383			−1.74901
$320$		0			0
$321$	1.35965			0.0758881
$322$		0			0
$323$			7.71927i			0.429512i
$324$		0			0
$325$	0.332104	−	4.98896i	0.0184218	−	0.276738i
$326$		0			0
$327$			3.49264i			0.193143i
$328$		0			0
$329$	−19.2071			−1.05892
$330$		0			0
$331$	−16.0576			−0.882605			−0.441303	−	0.897358i	$-0.645483\pi$
$331$	−16.0576			−0.882605			−0.441303	+	0.897358i	$0.645483\pi$
$332$		0			0
$333$			6.93933i			0.380273i
$334$		0			0
$335$	0.572331	−	17.2145i	0.0312698	−	0.940526i
$336$		0			0
$337$		−	11.0698i		−	0.603011i	−0.953464	−	0.301506i	$-0.902511\pi$
$337$		−	11.0698i		−	0.603011i	0.953464	−	0.301506i	$-0.0974892\pi$
$338$		0			0
$339$	−12.4320			−0.675212
$340$		0			0
$341$	62.4766			3.38330
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$	15.9779	+	0.531220i	0.860222	+	0.0285999i
$346$		0			0
$347$			29.6127i			1.58969i	0.606811	+	0.794846i	$0.292449\pi$
$347$			29.6127i			1.58969i	−0.606811	+	0.794846i	$0.707551\pi$
$348$		0			0
$349$	−17.1183			−0.916319			−0.458160	−	0.888870i	$-0.651491\pi$
$349$	−17.1183			−0.916319			−0.458160	+	0.888870i	$0.651491\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$			18.3245i			0.975313i	0.873035	+	0.487657i	$0.162148\pi$
$353$			18.3245i			0.975313i	−0.873035	+	0.487657i	$0.837852\pi$
$354$		0			0
$355$	24.7797	+	0.823854i	1.31517	+	0.0437256i
$356$		0			0
$357$		−	9.87867i		−	0.522834i
$358$		0			0
$359$	27.1356			1.43216			0.716082	−	0.698016i	$-0.245933\pi$
$359$	27.1356			1.43216			0.716082	+	0.698016i	$0.245933\pi$
$360$		0			0
$361$	−14.1152			−0.742906
$362$		0			0
$363$		−	30.8566i		−	1.61955i
$364$		0			0
$365$	−0.428555	+	12.8900i	−0.0224316	+	0.674692i
$366$		0			0
$367$		−	15.1855i		−	0.792679i	−0.918104	−	0.396340i	$-0.870280\pi$
$367$		−	15.1855i		−	0.792679i	0.918104	−	0.396340i	$-0.129720\pi$
$368$		0			0
$369$	0.148604			0.00773599
$370$		0			0
$371$	−4.81623			−0.250046
$372$		0			0
$373$			22.1776i			1.14831i	0.818745	+	0.574157i	$0.194670\pi$
$373$			22.1776i			1.14831i	−0.818745	+	0.574157i	$0.805330\pi$
$374$		0			0
$375$	1.11289	−	11.1248i	0.0574692	−	0.574483i
$376$		0			0
$377$			4.82843i			0.248677i
$378$		0			0
$379$	14.7604			0.758191			0.379096	−	0.925358i	$-0.376235\pi$
$379$	14.7604			0.758191			0.379096	+	0.925358i	$0.376235\pi$
$380$		0			0
$381$	18.6200			0.953934
$382$		0			0
$383$			9.33238i			0.476862i	0.971159	+	0.238431i	$0.0766331\pi$
$383$			9.33238i			0.476862i	−0.971159	+	0.238431i	$0.923367\pi$
$384$		0			0
$385$	−1.35965	+	40.8952i	−0.0692940	+	2.08421i
$386$		0			0
$387$			3.03858i			0.154460i
$388$		0			0
$389$	3.09618			0.156982			0.0784912	−	0.996915i	$-0.474990\pi$
$389$	3.09618			0.156982			0.0784912	+	0.996915i	$0.474990\pi$
$390$		0			0
$391$	24.9706			1.26282
$392$		0			0
$393$		−	0.751258i		−	0.0378960i
$394$		0			0
$395$	−22.4016	−	0.744790i	−1.12715	−	0.0374745i
$396$		0			0
$397$		−	7.26777i		−	0.364759i	−0.983228	−	0.182379i	$-0.941620\pi$
$397$		−	7.26777i		−	0.364759i	0.983228	−	0.182379i	$-0.0583799\pi$
$398$		0			0
$399$	−6.25127			−0.312955
$400$		0			0
$401$	−8.40164			−0.419558			−0.209779	−	0.977749i	$-0.567274\pi$
$401$	−8.40164			−0.419558			−0.209779	+	0.977749i	$0.567274\pi$
$402$		0			0
$403$		−	9.65685i		−	0.481042i
$404$		0			0
$405$	2.23483	+	0.0743018i	0.111050	+	0.00369209i
$406$		0			0
$407$		−	44.8952i		−	2.22537i
$408$		0			0
$409$	−11.5815			−0.572666			−0.286333	−	0.958130i	$-0.592436\pi$
$409$	−11.5815			−0.572666			−0.286333	+	0.958130i	$0.592436\pi$
$410$		0			0
$411$	−1.08794			−0.0536639
$412$		0			0
$413$		−	18.2990i		−	0.900434i
$414$		0			0
$415$	−0.168792	+	5.07689i	−0.00828568	+	0.249215i
$416$		0			0
$417$		−	0.420314i		−	0.0205829i
$418$		0			0
$419$	−24.7071			−1.20702			−0.603510	−	0.797355i	$-0.706232\pi$
$419$	−24.7071			−1.20702			−0.603510	+	0.797355i	$0.706232\pi$
$420$		0			0
$421$	−20.7769			−1.01260			−0.506302	−	0.862356i	$-0.668988\pi$
$421$	−20.7769			−1.01260			−0.506302	+	0.862356i	$0.668988\pi$
$422$		0			0
$423$		−	6.79073i		−	0.330177i
$424$		0			0
$425$	1.15992	−	17.4246i	0.0562643	−	0.845218i
$426$		0			0
$427$			7.53552i			0.364669i
$428$		0			0
$429$	−6.46967			−0.312358
$430$		0			0
$431$	−16.6694			−0.802936			−0.401468	−	0.915873i	$-0.631500\pi$
$431$	−16.6694			−0.802936			−0.401468	+	0.915873i	$0.631500\pi$
$432$		0			0
$433$			12.8860i			0.619263i	0.950857	+	0.309631i	$0.100206\pi$
$433$			12.8860i			0.619263i	−0.950857	+	0.309631i	$0.899794\pi$
$434$		0			0
$435$	−0.358761	+	10.7907i	−0.0172013	+	0.517376i
$436$		0			0
$437$		−	15.8015i		−	0.755888i
$438$		0			0
$439$	37.1924			1.77510			0.887548	−	0.460716i	$-0.152407\pi$
$439$	37.1924			1.77510			0.887548	+	0.460716i	$0.152407\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			33.0693i			1.57117i	0.618755	+	0.785584i	$0.287637\pi$
$443$			33.0693i			1.57117i	−0.618755	+	0.785584i	$0.712363\pi$
$444$		0			0
$445$	20.5851	+	0.684397i	0.975829	+	0.0324435i
$446$		0			0
$447$		−	14.1265i		−	0.668161i
$448$		0			0
$449$	10.7448			0.507078			0.253539	−	0.967325i	$-0.418405\pi$
$449$	10.7448			0.507078			0.253539	+	0.967325i	$0.418405\pi$
$450$		0			0
$451$	−0.961416			−0.0452713
$452$		0			0
$453$			6.07717i			0.285530i
$454$		0			0
$455$	6.32106	+	0.210157i	0.296336	+	0.00985232i
$456$		0			0
$457$			21.9419i			1.02640i	0.858270	+	0.513198i	$0.171540\pi$
$457$			21.9419i			1.02640i	−0.858270	+	0.513198i	$0.828460\pi$
$458$		0			0
$459$	3.49264			0.163022
$460$		0			0
$461$	22.0034			1.02480			0.512401	−	0.858747i	$-0.328756\pi$
$461$	22.0034			1.02480			0.512401	+	0.858747i	$0.328756\pi$
$462$		0			0
$463$			32.0208i			1.48813i	0.668106	+	0.744066i	$0.267105\pi$
$463$			32.0208i			1.48813i	−0.668106	+	0.744066i	$0.732895\pi$
$464$		0			0
$465$	0.717522	−	21.5815i	0.0332743	−	1.00082i
$466$		0			0
$467$		−	25.3596i		−	1.17350i	−0.809767	−	0.586752i	$-0.800406\pi$
$467$		−	25.3596i		−	1.17350i	0.809767	−	0.586752i	$-0.199594\pi$
$468$		0			0
$469$	21.7868			1.00602
$470$		0			0
$471$	11.7266			0.540335
$472$		0			0
$473$		−	19.6586i		−	0.903905i
$474$		0			0
$475$	−11.0264	−	0.734003i	−0.505925	−	0.0336783i
$476$		0			0
$477$		−	1.70279i		−	0.0779655i
$478$		0			0
$479$	9.46231			0.432344			0.216172	−	0.976355i	$-0.430643\pi$
$479$	9.46231			0.432344			0.216172	+	0.976355i	$0.430643\pi$
$480$		0			0
$481$	−6.93933			−0.316406
$482$		0			0
$483$			20.2218i			0.920124i
$484$		0			0
$485$	−0.602655	+	18.1265i	−0.0273651	+	0.823083i
$486$		0			0
$487$			22.3327i			1.01199i	0.862536	+	0.505996i	$0.168875\pi$
$487$			22.3327i			1.01199i	−0.862536	+	0.505996i	$0.831125\pi$
$488$		0			0
$489$	−7.31371			−0.330737
$490$		0			0
$491$	−29.8762			−1.34829			−0.674146	−	0.738598i	$-0.735488\pi$
$491$	−29.8762			−1.34829			−0.674146	+	0.738598i	$0.735488\pi$
$492$		0			0
$493$			16.8639i			0.759513i
$494$		0			0
$495$	−14.4586	−	0.480708i	−0.649867	−	0.0216062i
$496$		0			0
$497$			31.3614i			1.40675i
$498$		0			0
$499$	−24.7145			−1.10637			−0.553186	−	0.833058i	$-0.686588\pi$
$499$	−24.7145			−1.10637			−0.553186	+	0.833058i	$0.686588\pi$
$500$		0			0
$501$	−21.0420			−0.940087
$502$		0			0
$503$		−	17.2267i		−	0.768099i	−0.923313	−	0.384049i	$-0.874529\pi$
$503$		−	17.2267i		−	0.768099i	0.923313	−	0.384049i	$-0.125471\pi$
$504$		0			0
$505$	−36.0750	−	1.19939i	−1.60532	−	0.0533721i
$506$		0			0
$507$			1.00000i			0.0444116i
$508$		0			0
$509$	−26.1430			−1.15877			−0.579384	−	0.815055i	$-0.696707\pi$
$509$	−26.1430			−1.15877			−0.579384	+	0.815055i	$0.696707\pi$
$510$		0			0
$511$	−16.3137			−0.721675
$512$		0			0
$513$		−	2.21016i		−	0.0975808i
$514$		0			0
$515$	−1.45494	+	43.7613i	−0.0641122	+	1.92835i
$516$		0			0
$517$			43.9338i			1.93220i
$518$		0			0
$519$	−18.9706			−0.832715
$520$		0			0
$521$	−33.0165			−1.44648			−0.723240	−	0.690597i	$-0.757348\pi$
$521$	−33.0165			−1.44648			−0.723240	+	0.690597i	$0.757348\pi$
$522$		0			0
$523$			32.2751i			1.41129i	0.708564	+	0.705646i	$0.249343\pi$
$523$			32.2751i			1.41129i	−0.708564	+	0.705646i	$0.750657\pi$
$524$		0			0
$525$	14.1109	+	0.939333i	0.615850	+	0.0409958i
$526$		0			0
$527$		−	33.7279i		−	1.46921i
$528$		0			0
$529$	−28.1152			−1.22240
$530$		0			0
$531$	6.46967			0.280760
$532$		0			0
$533$			0.148604i			0.00643674i
$534$		0			0
$535$	−0.101024	+	3.03858i	−0.00436766	+	0.131369i
$536$		0			0
$537$		−	20.4099i		−	0.880752i
$538$		0			0
$539$	−6.46967			−0.278668
$540$		0			0
$541$	−2.21193			−0.0950983			−0.0475491	−	0.998869i	$-0.515141\pi$
$541$	−2.21193			−0.0950983			−0.0475491	+	0.998869i	$0.515141\pi$
$542$		0			0
$543$		−	24.8934i		−	1.06828i
$544$		0			0
$545$	−7.80546	−	0.259509i	−0.334349	−	0.0111161i
$546$		0			0
$547$			31.4822i			1.34608i	0.739605	+	0.673041i	$0.235012\pi$
$547$			31.4822i			1.34608i	−0.739605	+	0.673041i	$0.764988\pi$
$548$		0			0
$549$	−2.66421			−0.113706
$550$		0			0
$551$	10.6716			0.454625
$552$		0			0
$553$		−	28.3517i		−	1.20564i
$554$		0			0
$555$	−15.5083	−	0.515605i	−0.658289	−	0.0218862i
$556$		0			0
$557$		−	35.1209i		−	1.48812i	−0.668112	−	0.744061i	$-0.732897\pi$
$557$		−	35.1209i		−	1.48812i	0.668112	−	0.744061i	$-0.267103\pi$
$558$		0			0
$559$	−3.03858			−0.128518
$560$		0			0
$561$	−22.5962			−0.954012
$562$		0			0
$563$		−	30.3467i		−	1.27896i	−0.768808	−	0.639480i	$-0.779150\pi$
$563$		−	30.3467i		−	1.27896i	0.768808	−	0.639480i	$-0.220850\pi$
$564$		0			0
$565$	0.923718	−	27.7834i	0.0388611	−	1.16886i
$566$		0			0
$567$			2.82843i			0.118783i
$568$		0			0
$569$	−30.8327			−1.29257			−0.646287	−	0.763094i	$-0.723679\pi$
$569$	−30.8327			−1.29257			−0.646287	+	0.763094i	$0.723679\pi$
$570$		0			0
$571$	−21.1832			−0.886490			−0.443245	−	0.896400i	$-0.646173\pi$
$571$	−21.1832			−0.886490			−0.443245	+	0.896400i	$0.646173\pi$
$572$		0			0
$573$		−	14.9853i		−	0.626019i
$574$		0			0
$575$	−2.37438	+	35.6685i	−0.0990183	+	1.48748i
$576$		0			0
$577$			22.6594i			0.943322i	0.881780	+	0.471661i	$0.156345\pi$
$577$			22.6594i			0.943322i	−0.881780	+	0.471661i	$0.843655\pi$
$578$		0			0
$579$	−18.7990			−0.781259
$580$		0			0
$581$	−6.42537			−0.266569
$582$		0			0
$583$			11.0165i			0.456257i
$584$		0			0
$585$	−0.0743018	+	2.23483i	−0.00307200	+	0.0923990i
$586$		0			0
$587$			13.6529i			0.563515i	0.959486	+	0.281758i	$0.0909174\pi$
$587$			13.6529i			0.563515i	−0.959486	+	0.281758i	$0.909083\pi$
$588$		0			0
$589$	−21.3432			−0.879430
$590$		0			0
$591$	−1.92857			−0.0793306
$592$		0			0
$593$			18.7448i			0.769756i	0.922967	+	0.384878i	$0.125757\pi$
$593$			18.7448i			0.769756i	−0.922967	+	0.384878i	$0.874243\pi$
$594$		0			0
$595$	22.0772	+	0.734003i	0.905076	+	0.0300912i
$596$		0			0
$597$			10.0239i			0.410249i
$598$		0			0
$599$	−48.7885			−1.99345			−0.996723	−	0.0808922i	$-0.974223\pi$
$599$	−48.7885			−1.99345			−0.996723	+	0.0808922i	$0.974223\pi$
$600$		0			0
$601$	16.5870			0.676599			0.338300	−	0.941038i	$-0.390148\pi$
$601$	16.5870			0.676599			0.338300	+	0.941038i	$0.390148\pi$
$602$		0			0
$603$			7.70279i			0.313682i
$604$		0			0
$605$	68.9593	+	2.29270i	2.80359	+	0.0932115i
$606$		0			0
$607$		−	29.0642i		−	1.17968i	−0.807520	−	0.589840i	$-0.799191\pi$
$607$		−	29.0642i		−	1.17968i	0.807520	−	0.589840i	$-0.200809\pi$
$608$		0			0
$609$	−13.6569			−0.553404
$610$		0			0
$611$	6.79073			0.274723
$612$		0			0
$613$		−	22.3926i		−	0.904430i	−0.891909	−	0.452215i	$-0.850634\pi$
$613$		−	22.3926i		−	0.904430i	0.891909	−	0.452215i	$-0.149366\pi$
$614$		0			0
$615$	−0.0110415	+	0.332104i	−0.000445237	+	0.0133917i
$616$		0			0
$617$		−	1.63136i		−	0.0656760i	−0.999461	−	0.0328380i	$-0.989545\pi$
$617$		−	1.63136i		−	0.0656760i	0.999461	−	0.0328380i	$-0.0104545\pi$
$618$		0			0
$619$	−37.4485			−1.50518			−0.752591	−	0.658489i	$-0.771196\pi$
$619$	−37.4485			−1.50518			−0.752591	+	0.658489i	$0.771196\pi$
$620$		0			0
$621$	−7.14949			−0.286899
$622$		0			0
$623$			26.0528i			1.04378i
$624$		0			0
$625$	24.7794	+	3.31371i	0.991177	+	0.132548i
$626$		0			0
$627$			14.2990i			0.571046i
$628$		0			0
$629$	−24.2366			−0.966375
$630$		0			0
$631$	−10.7193			−0.426728			−0.213364	−	0.976973i	$-0.568442\pi$
$631$	−10.7193			−0.426728			−0.213364	+	0.976973i	$0.568442\pi$
$632$		0			0
$633$			20.2751i			0.805864i
$634$		0			0
$635$	−1.38350	+	41.6127i	−0.0549026	+	1.65135i
$636$		0			0
$637$			1.00000i			0.0396214i
$638$		0			0
$639$	−11.0879			−0.438632
$640$		0			0
$641$	−31.0183			−1.22515			−0.612574	−	0.790413i	$-0.709866\pi$
$641$	−31.0183			−1.22515			−0.612574	+	0.790413i	$0.709866\pi$
$642$		0			0
$643$		−	12.4515i		−	0.491041i	−0.969391	−	0.245520i	$-0.921041\pi$
$643$		−	12.4515i		−	0.491041i	0.969391	−	0.245520i	$-0.0789589\pi$
$644$		0			0
$645$	−6.79073	−	0.225772i	−0.267385	−	0.00888977i
$646$		0			0
$647$		−	41.6538i		−	1.63758i	−0.574093	−	0.818790i	$-0.694645\pi$
$647$		−	41.6538i		−	1.63758i	0.574093	−	0.818790i	$-0.305355\pi$
$648$		0			0
$649$	−41.8566			−1.64301
$650$		0			0
$651$	27.3137			1.07051
$652$		0			0
$653$		−	3.83095i		−	0.149917i	−0.997187	−	0.0749584i	$-0.976118\pi$
$653$		−	3.83095i		−	0.149917i	0.997187	−	0.0749584i	$-0.0238824\pi$
$654$		0			0
$655$	1.67894	+	0.0558199i	0.0656015	+	0.00218106i
$656$		0			0
$657$		−	5.76776i		−	0.225022i
$658$		0			0
$659$	−41.8926			−1.63191			−0.815953	−	0.578119i	$-0.803787\pi$
$659$	−41.8926			−1.63191			−0.815953	+	0.578119i	$0.803787\pi$
$660$		0			0
$661$	40.8523			1.58897			0.794485	−	0.607284i	$-0.207741\pi$
$661$	40.8523			1.58897			0.794485	+	0.607284i	$0.207741\pi$
$662$		0			0
$663$			3.49264i			0.135643i
$664$		0			0
$665$	0.464480	−	13.9705i	0.0180118	−	0.541754i
$666$		0			0
$667$		−	34.5208i		−	1.33665i
$668$		0			0
$669$	−16.1881			−0.625867
$670$		0			0
$671$	17.2365			0.665409
$672$		0			0
$673$		−	45.2401i		−	1.74388i	−0.489615	−	0.871939i	$-0.662863\pi$
$673$		−	45.2401i		−	1.74388i	0.489615	−	0.871939i	$-0.337137\pi$
$674$		0			0
$675$	−0.332104	+	4.98896i	−0.0127827	+	0.192025i
$676$		0			0
$677$			3.58146i			0.137647i	0.997629	+	0.0688233i	$0.0219245\pi$
$677$			3.58146i			0.137647i	−0.997629	+	0.0688233i	$0.978076\pi$
$678$		0			0
$679$	−22.9411			−0.880399
$680$		0			0
$681$	−23.0879			−0.884732
$682$		0			0
$683$		−	44.8402i		−	1.71576i	−0.513848	−	0.857882i	$-0.671780\pi$
$683$		−	44.8402i		−	1.71576i	0.513848	−	0.857882i	$-0.328220\pi$
$684$		0			0
$685$	0.0808356	−	2.43136i	0.00308857	−	0.0928973i
$686$		0			0
$687$		−	4.30886i		−	0.164393i
$688$		0			0
$689$	1.70279			0.0648712
$690$		0			0
$691$	−18.2891			−0.695750			−0.347875	−	0.937541i	$-0.613097\pi$
$691$	−18.2891			−0.695750			−0.347875	+	0.937541i	$0.613097\pi$
$692$		0			0
$693$		−	18.2990i		−	0.695121i
$694$		0			0
$695$	0.939333	+	0.0312301i	0.0356309	+	0.00118463i
$696$		0			0
$697$			0.519018i			0.0196592i
$698$		0			0
$699$	−1.53857			−0.0581942
$700$		0			0
$701$	36.0667			1.36222			0.681111	−	0.732180i	$-0.261497\pi$
$701$	36.0667			1.36222			0.681111	+	0.732180i	$0.261497\pi$
$702$		0			0
$703$			15.3370i			0.578447i
$704$		0			0
$705$	15.1761	+	0.504563i	0.571567	+	0.0190030i
$706$		0			0
$707$		−	45.6569i		−	1.71710i
$708$		0			0
$709$	11.7769			0.442291			0.221146	−	0.975241i	$-0.429020\pi$
$709$	11.7769			0.442291			0.221146	+	0.975241i	$0.429020\pi$
$710$		0			0
$711$	10.0239			0.375924
$712$		0			0
$713$			69.0416i			2.58563i
$714$		0			0
$715$	0.480708	−	14.4586i	0.0179775	−	0.540722i
$716$		0			0
$717$			17.8514i			0.666673i
$718$		0			0
$719$	−46.7261			−1.74259			−0.871295	−	0.490759i	$-0.836719\pi$
$719$	−46.7261			−1.74259			−0.871295	+	0.490759i	$0.836719\pi$
$720$		0			0
$721$	−55.3847			−2.06264
$722$		0			0
$723$		−	3.95406i		−	0.147053i
$724$		0			0
$725$	−24.0888	−	1.60354i	−0.894636	−	0.0595540i
$726$		0			0
$727$			37.8038i			1.40207i	0.713129	+	0.701033i	$0.247277\pi$
$727$			37.8038i			1.40207i	−0.713129	+	0.701033i	$0.752723\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−10.6127			−0.392524
$732$		0			0
$733$		−	20.9411i		−	0.773477i	−0.922189	−	0.386739i	$-0.873602\pi$
$733$		−	20.9411i		−	0.773477i	0.922189	−	0.386739i	$-0.126398\pi$
$734$		0			0
$735$	−0.0743018	+	2.23483i	−0.00274066	+	0.0824331i
$736$		0			0
$737$		−	49.8345i		−	1.83568i
$738$		0			0
$739$	31.5012			1.15879			0.579396	−	0.815046i	$-0.303289\pi$
$739$	31.5012			1.15879			0.579396	+	0.815046i	$0.303289\pi$
$740$		0			0
$741$	2.21016			0.0811922
$742$		0			0
$743$		−	2.96483i		−	0.108769i	−0.998520	−	0.0543845i	$-0.982680\pi$
$743$		−	2.96483i		−	0.108769i	0.998520	−	0.0543845i	$-0.0173197\pi$
$744$		0			0
$745$	31.5704	+	1.04963i	1.15665	+	0.0384553i
$746$		0			0
$747$		−	2.27171i		−	0.0831176i
$748$		0			0
$749$	−3.84566			−0.140517
$750$		0			0
$751$	−10.0295			−0.365980			−0.182990	−	0.983115i	$-0.558578\pi$
$751$	−10.0295			−0.365980			−0.182990	+	0.983115i	$0.558578\pi$
$752$		0			0
$753$			3.42284i			0.124735i
$754$		0			0
$755$	−13.5815	−	0.451545i	−0.494280	−	0.0164334i
$756$		0			0
$757$			36.4731i			1.32564i	0.748780	+	0.662818i	$0.230640\pi$
$757$			36.4731i			1.32564i	−0.748780	+	0.662818i	$0.769360\pi$
$758$		0			0
$759$	46.2548			1.67894
$760$		0			0
$761$	−9.88085			−0.358181			−0.179090	−	0.983833i	$-0.557315\pi$
$761$	−9.88085			−0.358181			−0.179090	+	0.983833i	$0.557315\pi$
$762$		0			0
$763$		−	9.87867i		−	0.357632i
$764$		0			0
$765$	−0.259509	+	7.80546i	−0.00938257	+	0.282207i
$766$		0			0
$767$			6.46967i			0.233606i
$768$		0			0
$769$	−32.6274			−1.17657			−0.588287	−	0.808652i	$-0.700198\pi$
$769$	−32.6274			−1.17657			−0.588287	+	0.808652i	$0.700198\pi$
$770$		0			0
$771$	3.14772			0.113362
$772$		0			0
$773$		−	44.2950i		−	1.59318i	−0.604519	−	0.796591i	$-0.706635\pi$
$773$		−	44.2950i		−	1.59318i	0.604519	−	0.796591i	$-0.293365\pi$
$774$		0			0
$775$	48.1776	+	3.20708i	1.73059	+	0.115202i
$776$		0			0
$777$		−	19.6274i		−	0.704129i
$778$		0			0
$779$	0.328437			0.0117675
$780$		0			0
$781$	71.7352			2.56689
$782$		0			0
$783$		−	4.82843i		−	0.172554i
$784$		0			0
$785$	−0.871311	+	26.2071i	−0.0310984	+	0.935372i
$786$		0			0
$787$			24.0068i			0.855751i	0.903838	+	0.427875i	$0.140738\pi$
$787$			24.0068i			0.855751i	−0.903838	+	0.427875i	$0.859262\pi$
$788$		0			0
$789$	−13.7144			−0.488247
$790$		0			0
$791$	35.1629			1.25025
$792$		0			0
$793$		−	2.66421i		−	0.0946088i
$794$		0			0
$795$	3.80546	+	0.126521i	0.134966	+	0.00448722i
$796$		0			0
$797$		−	42.4271i		−	1.50285i	−0.659821	−	0.751423i	$-0.729368\pi$
$797$		−	42.4271i		−	1.50285i	0.659821	−	0.751423i	$-0.270632\pi$
$798$		0			0
$799$	23.7175			0.839066
$800$		0			0
$801$	−9.21104			−0.325456
$802$		0			0
$803$			37.3155i			1.31683i
$804$		0			0
$805$	−45.1924	−	1.50252i	−1.59282	−	0.0529568i
$806$		0			0
$807$			9.84316i			0.346495i
$808$		0			0
$809$	12.0495			0.423637			0.211819	−	0.977309i	$-0.432061\pi$
$809$	12.0495			0.423637			0.211819	+	0.977309i	$0.432061\pi$
$810$		0			0
$811$	34.5619			1.21363			0.606816	−	0.794842i	$-0.292446\pi$
$811$	34.5619			1.21363			0.606816	+	0.794842i	$0.292446\pi$
$812$		0			0
$813$		−	2.86216i		−	0.100380i
$814$		0			0
$815$	0.543422	−	16.3449i	0.0190352	−	0.572538i
$816$		0			0
$817$			6.71575i			0.234954i
$818$		0			0
$819$	−2.82843			−0.0988332
$820$		0			0
$821$	−1.92119			−0.0670500			−0.0335250	−	0.999438i	$-0.510673\pi$
$821$	−1.92119			−0.0670500			−0.0335250	+	0.999438i	$0.510673\pi$
$822$		0			0
$823$			36.9246i			1.28711i	0.765399	+	0.643556i	$0.222542\pi$
$823$			36.9246i			1.28711i	−0.765399	+	0.643556i	$0.777458\pi$
$824$		0			0
$825$	2.14860	−	32.2769i	0.0748048	−	1.12374i
$826$		0			0
$827$			20.9501i			0.728507i	0.931300	+	0.364254i	$0.118676\pi$
$827$			20.9501i			0.728507i	−0.931300	+	0.364254i	$0.881324\pi$
$828$		0			0
$829$	3.55255			0.123385			0.0616926	−	0.998095i	$-0.480350\pi$
$829$	3.55255			0.123385			0.0616926	+	0.998095i	$0.480350\pi$
$830$		0			0
$831$	−6.24389			−0.216598
$832$		0			0
$833$			3.49264i			0.121013i
$834$		0			0
$835$	1.56346	−	47.0254i	0.0541057	−	1.62738i
$836$		0			0
$837$			9.65685i			0.333790i
$838$		0			0
$839$	−12.2768			−0.423841			−0.211920	−	0.977287i	$-0.567972\pi$
$839$	−12.2768			−0.423841			−0.211920	+	0.977287i	$0.567972\pi$
$840$		0			0
$841$	−5.68629			−0.196079
$842$		0			0
$843$			31.3869i			1.08102i
$844$		0			0
$845$	−2.23483	−	0.0743018i	−0.0768806	−	0.00255606i
$846$		0			0
$847$			87.2756i			2.99882i
$848$		0			0
$849$	−2.56496			−0.0880291
$850$		0			0
$851$	49.6127			1.70070
$852$		0			0
$853$		−	10.6734i		−	0.365449i	−0.983164	−	0.182724i	$-0.941508\pi$
$853$		−	10.6734i		−	0.365449i	0.983164	−	0.182724i	$-0.0584916\pi$
$854$		0			0
$855$	4.93933	+	0.164219i	0.168922	+	0.00561616i
$856$		0			0
$857$		−	54.5324i		−	1.86279i	−0.364006	−	0.931396i	$-0.618591\pi$
$857$		−	54.5324i		−	1.86279i	0.364006	−	0.931396i	$-0.381409\pi$
$858$		0			0
$859$	−45.6218			−1.55660			−0.778298	−	0.627894i	$-0.783917\pi$
$859$	−45.6218			−1.55660			−0.778298	+	0.627894i	$0.783917\pi$
$860$		0			0
$861$	−0.420314			−0.0143243
$862$		0			0
$863$			18.1798i			0.618849i	0.950924	+	0.309424i	$0.100136\pi$
$863$			18.1798i			0.618849i	−0.950924	+	0.309424i	$0.899864\pi$
$864$		0			0
$865$	1.40955	−	42.3960i	0.0479260	−	1.44151i
$866$		0			0
$867$		−	4.80150i		−	0.163067i
$868$		0			0
$869$	−64.8510			−2.19992
$870$		0			0
$871$	−7.70279			−0.260999
$872$		0			0
$873$		−	8.11091i		−	0.274513i
$874$		0			0
$875$	−3.14772	+	31.4657i	−0.106412	+	1.06374i
$876$		0			0
$877$			23.1611i			0.782096i	0.920370	+	0.391048i	$0.127887\pi$
$877$			23.1611i			0.782096i	−0.920370	+	0.391048i	$0.872113\pi$
$878$		0			0
$879$	−0.834895			−0.0281603
$880$		0			0
$881$	16.7358			0.563843			0.281921	−	0.959438i	$-0.409028\pi$
$881$	16.7358			0.563843			0.281921	+	0.959438i	$0.409028\pi$
$882$		0			0
$883$		−	20.3959i		−	0.686377i	−0.939267	−	0.343189i	$-0.888493\pi$
$883$		−	20.3959i		−	0.686377i	0.939267	−	0.343189i	$-0.111507\pi$
$884$		0			0
$885$	−0.480708	+	14.4586i	−0.0161588	+	0.486021i
$886$		0			0
$887$			58.7395i			1.97228i	0.165912	+	0.986140i	$0.446943\pi$
$887$			58.7395i			1.97228i	−0.165912	+	0.986140i	$0.553057\pi$
$888$		0			0
$889$	−52.6654			−1.76634
$890$		0			0
$891$	6.46967			0.216742
$892$		0			0
$893$		−	15.0086i		−	0.502243i
$894$		0			0
$895$	45.6127	+	1.51649i	1.52466	+	0.0506907i
$896$		0			0
$897$		−	7.14949i		−	0.238715i
$898$		0			0
$899$	−46.6274			−1.55511
$900$		0			0
$901$	5.94723			0.198131
$902$		0			0
$903$		−	8.59441i		−	0.286004i
$904$		0			0
$905$	55.6326	+	1.84962i	1.84929	+	0.0614836i
$906$		0			0
$907$		−	17.9853i		−	0.597192i	−0.954380	−	0.298596i	$-0.903482\pi$
$907$		−	17.9853i		−	0.597192i	0.954380	−	0.298596i	$-0.0965183\pi$
$908$		0			0
$909$	16.1421			0.535401
$910$		0			0
$911$	−55.0269			−1.82312			−0.911561	−	0.411165i	$-0.865122\pi$
$911$	−55.0269			−1.82312			−0.911561	+	0.411165i	$0.865122\pi$
$912$		0			0
$913$			14.6972i			0.486407i
$914$		0			0
$915$	0.197955	−	5.95406i	0.00654421	−	0.196835i
$916$		0			0
$917$			2.12488i			0.0701697i
$918$		0			0
$919$	1.00913			0.0332880			0.0166440	−	0.999861i	$-0.494702\pi$
$919$	1.00913			0.0332880			0.0166440	+	0.999861i	$0.494702\pi$
$920$		0			0
$921$	−17.7556			−0.585066
$922$		0			0
$923$		−	11.0879i		−	0.364964i
$924$		0			0
$925$	2.30458	−	34.6200i	0.0757742	−	1.13830i
$926$		0			0
$927$		−	19.5815i		−	0.643139i
$928$		0			0
$929$	23.1407			0.759222			0.379611	−	0.925146i	$-0.376058\pi$
$929$	23.1407			0.759222			0.379611	+	0.925146i	$0.376058\pi$
$930$		0			0
$931$	2.21016			0.0724350
$932$		0			0
$933$		−	1.35965i		−	0.0445128i
$934$		0			0
$935$	1.67894	−	50.4987i	0.0549071	−	1.65148i
$936$		0			0
$937$			8.72667i			0.285088i	0.989788	+	0.142544i	$0.0455282\pi$
$937$			8.72667i			0.285088i	−0.989788	+	0.142544i	$0.954472\pi$
$938$		0			0
$939$	−7.75611			−0.253111
$940$		0			0
$941$	5.24965			0.171134			0.0855668	−	0.996332i	$-0.472730\pi$
$941$	5.24965			0.171134			0.0855668	+	0.996332i	$0.472730\pi$
$942$		0			0
$943$		−	1.06244i		−	0.0345978i
$944$		0			0
$945$	−6.32106	−	0.210157i	−0.205624	−	0.00683641i
$946$		0			0
$947$		−	17.0420i		−	0.553791i	−0.960900	−	0.276895i	$-0.910694\pi$
$947$		−	17.0420i		−	0.553791i	0.960900	−	0.276895i	$-0.0893055\pi$
$948$		0			0
$949$	5.76776			0.187229
$950$		0			0
$951$	9.38514			0.304334
$952$		0			0
$953$			5.16243i			0.167227i	0.996498	+	0.0836137i	$0.0266462\pi$
$953$			5.16243i			0.167227i	−0.996498	+	0.0836137i	$0.973354\pi$
$954$		0			0
$955$	33.4896	+	1.11343i	1.08370	+	0.0360298i
$956$		0			0
$957$			31.2383i			1.00979i
$958$		0			0
$959$	3.07715			0.0993663
$960$		0			0
$961$	62.2548			2.00822
$962$		0			0
$963$		−	1.35965i		−	0.0438140i
$964$		0			0
$965$	1.39680	−	42.0126i	0.0449645	−	1.35243i
$966$		0			0
$967$			40.3657i			1.29807i	0.760757	+	0.649037i	$0.224828\pi$
$967$			40.3657i			1.29807i	−0.760757	+	0.649037i	$0.775172\pi$
$968$		0			0
$969$	7.71927			0.247979
$970$		0			0
$971$	20.2322			0.649283			0.324642	−	0.945837i	$-0.394756\pi$
$971$	20.2322			0.649283			0.324642	+	0.945837i	$0.394756\pi$
$972$		0			0
$973$			1.18883i			0.0381121i
$974$		0			0
$975$	−4.98896	−	0.332104i	−0.159775	−	0.0106358i
$976$		0			0
$977$			30.4034i			0.972691i	0.873766	+	0.486346i	$0.161670\pi$
$977$			30.4034i			0.972691i	−0.873766	+	0.486346i	$0.838330\pi$
$978$		0			0
$979$	59.5924			1.90458
$980$		0			0
$981$	3.49264			0.111511
$982$		0			0
$983$			50.4250i			1.60831i	0.594422	+	0.804153i	$0.297381\pi$
$983$			50.4250i			1.60831i	−0.594422	+	0.804153i	$0.702619\pi$
$984$		0			0
$985$	0.143296	−	4.31002i	0.00456579	−	0.137329i
$986$		0			0
$987$			19.2071i			0.611368i
$988$		0			0
$989$	21.7243			0.690793
$990$		0			0
$991$	−54.1005			−1.71856			−0.859279	−	0.511507i	$-0.829087\pi$
$991$	−54.1005			−1.71856			−0.859279	+	0.511507i	$0.829087\pi$
$992$		0			0
$993$			16.0576i			0.509572i
$994$		0			0
$995$	−22.4016	−	0.744790i	−0.710180	−	0.0236114i
$996$		0			0
$997$		−	39.3921i		−	1.24756i	−0.781600	−	0.623780i	$-0.785596\pi$
$997$		−	39.3921i		−	1.24756i	0.781600	−	0.623780i	$-0.214404\pi$
$998$		0			0
$999$	6.93933			0.219551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.o.1249.4		8
4.3	odd	2		1560.2.l.e.1249.8	yes	8
5.4	even	2	inner	3120.2.l.o.1249.8		8
12.11	even	2		4680.2.l.f.2809.1		8
20.3	even	4		7800.2.a.bv.1.1		4
20.7	even	4		7800.2.a.bw.1.3		4
20.19	odd	2		1560.2.l.e.1249.4	✓	8
60.59	even	2		4680.2.l.f.2809.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.e.1249.4	✓	8	20.19	odd	2
1560.2.l.e.1249.8	yes	8	4.3	odd	2
3120.2.l.o.1249.4		8	1.1	even	1	trivial
3120.2.l.o.1249.8		8	5.4	even	2	inner
4680.2.l.f.2809.1		8	12.11	even	2
4680.2.l.f.2809.2		8	60.59	even	2
7800.2.a.bv.1.1		4	20.3	even	4
7800.2.a.bw.1.3		4	20.7	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.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.23483 + 0.0743018i) q^{5} +2.82843i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(2.23483 + 0.0743018i) q^{5} +2.82843i q^{7} -1.00000 q^{9} +6.46967 q^{11} -1.00000i q^{13} +(0.0743018 - 2.23483i) q^{15} -3.49264i q^{17} -2.21016 q^{19} +2.82843 q^{21} +7.14949i q^{23} +(4.98896 + 0.332104i) q^{25} +1.00000i q^{27} -4.82843 q^{29} +9.65685 q^{31} -6.46967i q^{33} +(-0.210157 + 6.32106i) q^{35} -6.93933i q^{37} -1.00000 q^{39} -0.148604 q^{41} -3.03858i q^{43} +(-2.23483 - 0.0743018i) q^{45} +6.79073i q^{47} -1.00000 q^{49} -3.49264 q^{51} +1.70279i q^{53} +(14.4586 + 0.480708i) q^{55} +2.21016i q^{57} -6.46967 q^{59} +2.66421 q^{61} -2.82843i q^{63} +(0.0743018 - 2.23483i) q^{65} -7.70279i q^{67} +7.14949 q^{69} +11.0879 q^{71} +5.76776i q^{73} +(0.332104 - 4.98896i) q^{75} +18.2990i q^{77} -10.0239 q^{79} +1.00000 q^{81} +2.27171i q^{83} +(0.259509 - 7.80546i) q^{85} +4.82843i q^{87} +9.21104 q^{89} +2.82843 q^{91} -9.65685i q^{93} +(-4.93933 - 0.164219i) q^{95} +8.11091i q^{97} -6.46967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 16 q^{11} + 4 q^{15} - 24 q^{19} - 4 q^{25} - 16 q^{29} + 32 q^{31} - 8 q^{35} - 8 q^{39} - 8 q^{41} - 8 q^{49} - 8 q^{51} + 36 q^{55} - 16 q^{59} + 24 q^{61} + 4 q^{65} - 8 q^{69} + 24 q^{71} + 4 q^{75} - 24 q^{79} + 8 q^{81} - 40 q^{85} + 8 q^{89} + 32 q^{95} - 16 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 + 16 * q^11 + 4 * q^15 - 24 * q^19 - 4 * q^25 - 16 * q^29 + 32 * q^31 - 8 * q^35 - 8 * q^39 - 8 * q^41 - 8 * q^49 - 8 * q^51 + 36 * q^55 - 16 * q^59 + 24 * q^61 + 4 * q^65 - 8 * q^69 + 24 * q^71 + 4 * q^75 - 24 * q^79 + 8 * q^81 - 40 * q^85 + 8 * q^89 + 32 * q^95 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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