LMFDB - Embedded newform 1840.2.e.g.369.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} + \cdots + 8 $ x^14 - 2x^11 + 39x^10 - 10x^9 + 2x^8 - 26x^7 + 297x^6 - 116x^5 + 24x^4 - 20x^3 + 64x^2 - 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.8
Root			$1.23362 - 1.23362i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.g.369.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.189375i q^{3} +(-1.60280 + 1.55918i) q^{5} +1.65661i q^{7} +2.96414 q^{9} +O(q^{10})$	$q+0.189375i q^{3} +(-1.60280 + 1.55918i) q^{5} +1.65661i q^{7} +2.96414 q^{9} -0.0759321 q^{11} +1.24146i q^{13} +(-0.295270 - 0.303531i) q^{15} +5.17593i q^{17} +0.792262 q^{19} -0.313721 q^{21} -1.00000i q^{23} +(0.137925 - 4.99810i) q^{25} +1.12946i q^{27} +2.85401 q^{29} -8.90112 q^{31} -0.0143797i q^{33} +(-2.58295 - 2.65521i) q^{35} +6.92065i q^{37} -0.235102 q^{39} +4.64279 q^{41} -1.75255i q^{43} +(-4.75091 + 4.62162i) q^{45} -10.0033i q^{47} +4.25565 q^{49} -0.980193 q^{51} +11.1284i q^{53} +(0.121704 - 0.118392i) q^{55} +0.150035i q^{57} -12.4581 q^{59} -12.0098 q^{61} +4.91041i q^{63} +(-1.93566 - 1.98981i) q^{65} +6.96067i q^{67} +0.189375 q^{69} +7.71650 q^{71} +7.49476i q^{73} +(0.946517 + 0.0261196i) q^{75} -0.125790i q^{77} -15.6615 q^{79} +8.67852 q^{81} +6.68518i q^{83} +(-8.07019 - 8.29597i) q^{85} +0.540478i q^{87} +3.03333 q^{89} -2.05661 q^{91} -1.68565i q^{93} +(-1.26984 + 1.23528i) q^{95} -3.21747i q^{97} -0.225073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) $ 14 * q + 2 * q^5 - 4 * q^9	$ 14 q + 2 q^{5} - 4 q^{9} + 14 q^{11} + 6 q^{15} - 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} + 20 q^{31} + 2 q^{35} - 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} + 14 q^{51} + 38 q^{55} - 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} + 28 q^{71} + 24 q^{75} - 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} + 14 q^{91} + 30 q^{95} - 122 q^{99}+O(q^{100}) $ 14 * q + 2 * q^5 - 4 * q^9 + 14 * q^11 + 6 * q^15 - 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 + 20 * q^31 + 2 * q^35 - 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 + 14 * q^51 + 38 * q^55 - 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 + 28 * q^71 + 24 * q^75 - 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 + 14 * q^91 + 30 * q^95 - 122 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			0.189375i			0.109336i	0.998505	+	0.0546680i	$0.0174100\pi$
$3$			0.189375i			0.109336i	−0.998505	+	0.0546680i	$0.982590\pi$
$4$		0			0
$5$	−1.60280	+	1.55918i	−0.716793	+	0.697286i
$5$	−1.60280	+	1.55918i	−0.716793	+	0.697286i
$6$		0			0
$7$			1.65661i			0.626139i	0.949730	+	0.313070i	$0.101357\pi$
$7$			1.65661i			0.626139i	−0.949730	+	0.313070i	$0.898643\pi$
$8$		0			0
$9$	2.96414			0.988046
$10$		0			0
$11$	−0.0759321			−0.0228944			−0.0114472	−	0.999934i	$-0.503644\pi$
$11$	−0.0759321			−0.0228944			−0.0114472	+	0.999934i	$0.503644\pi$
$12$		0			0
$13$			1.24146i			0.344319i	0.985069	+	0.172159i	$0.0550744\pi$
$13$			1.24146i			0.344319i	−0.985069	+	0.172159i	$0.944926\pi$
$14$		0			0
$15$	−0.295270	−	0.303531i	−0.0762384	−	0.0783713i
$16$		0			0
$17$			5.17593i			1.25535i	0.778477	+	0.627673i	$0.215993\pi$
$17$			5.17593i			1.25535i	−0.778477	+	0.627673i	$0.784007\pi$
$18$		0			0
$19$	0.792262			0.181757			0.0908787	−	0.995862i	$-0.471032\pi$
$19$	0.792262			0.181757			0.0908787	+	0.995862i	$0.471032\pi$
$20$		0			0
$21$	−0.313721			−0.0684595
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	0.137925	−	4.99810i	0.0275850	−	0.999619i
$26$		0			0
$27$			1.12946i			0.217365i
$28$		0			0
$29$	2.85401			0.529976			0.264988	−	0.964252i	$-0.414632\pi$
$29$	2.85401			0.529976			0.264988	+	0.964252i	$0.414632\pi$
$30$		0			0
$31$	−8.90112			−1.59869			−0.799345	−	0.600873i	$-0.794820\pi$
$31$	−8.90112			−1.59869			−0.799345	+	0.600873i	$0.794820\pi$
$32$		0			0
$33$		−	0.0143797i		−	0.00250318i
$34$		0			0
$35$	−2.58295	−	2.65521i	−0.436598	−	0.448812i
$36$		0			0
$37$			6.92065i			1.13775i	0.822425	+	0.568874i	$0.192621\pi$
$37$			6.92065i			1.13775i	−0.822425	+	0.568874i	$0.807379\pi$
$38$		0			0
$39$	−0.235102			−0.0376464
$40$		0			0
$41$	4.64279			0.725082			0.362541	−	0.931968i	$-0.381909\pi$
$41$	4.64279			0.725082			0.362541	+	0.931968i	$0.381909\pi$
$42$		0			0
$43$		−	1.75255i		−	0.267262i	−0.991031	−	0.133631i	$-0.957336\pi$
$43$		−	1.75255i		−	0.267262i	0.991031	−	0.133631i	$-0.0426636\pi$
$44$		0			0
$45$	−4.75091	+	4.62162i	−0.708224	+	0.688950i
$46$		0			0
$47$		−	10.0033i		−	1.45913i	−0.683914	−	0.729563i	$-0.739724\pi$
$47$		−	10.0033i		−	1.45913i	0.683914	−	0.729563i	$-0.260276\pi$
$48$		0			0
$49$	4.25565			0.607950
$50$		0			0
$51$	−0.980193			−0.137254
$52$		0			0
$53$			11.1284i			1.52860i	0.644859	+	0.764301i	$0.276916\pi$
$53$			11.1284i			1.52860i	−0.644859	+	0.764301i	$0.723084\pi$
$54$		0			0
$55$	0.121704	−	0.118392i	0.0164105	−	0.0159639i
$56$		0			0
$57$			0.150035i			0.0198726i
$58$		0			0
$59$	−12.4581			−1.62191			−0.810955	−	0.585108i	$-0.801052\pi$
$59$	−12.4581			−1.62191			−0.810955	+	0.585108i	$0.801052\pi$
$60$		0			0
$61$	−12.0098			−1.53770			−0.768849	−	0.639430i	$-0.779170\pi$
$61$	−12.0098			−1.53770			−0.768849	+	0.639430i	$0.779170\pi$
$62$		0			0
$63$			4.91041i			0.618654i
$64$		0			0
$65$	−1.93566	−	1.98981i	−0.240089	−	0.246806i
$66$		0			0
$67$			6.96067i			0.850381i	0.905104	+	0.425191i	$0.139793\pi$
$67$			6.96067i			0.850381i	−0.905104	+	0.425191i	$0.860207\pi$
$68$		0			0
$69$	0.189375			0.0227981
$70$		0			0
$71$	7.71650			0.915779			0.457890	−	0.889009i	$-0.348605\pi$
$71$	7.71650			0.915779			0.457890	+	0.889009i	$0.348605\pi$
$72$		0			0
$73$			7.49476i			0.877196i	0.898683	+	0.438598i	$0.144525\pi$
$73$			7.49476i			0.877196i	−0.898683	+	0.438598i	$0.855475\pi$
$74$		0			0
$75$	0.946517	+	0.0261196i	0.109294	+	0.00301604i
$76$		0			0
$77$		−	0.125790i		−	0.0143351i
$78$		0			0
$79$	−15.6615			−1.76206			−0.881029	−	0.473063i	$-0.843148\pi$
$79$	−15.6615			−1.76206			−0.881029	+	0.473063i	$0.843148\pi$
$80$		0			0
$81$	8.67852			0.964280
$82$		0			0
$83$			6.68518i			0.733793i	0.930262	+	0.366897i	$0.119580\pi$
$83$			6.68518i			0.733793i	−0.930262	+	0.366897i	$0.880420\pi$
$84$		0			0
$85$	−8.07019	−	8.29597i	−0.875335	−	0.899824i
$86$		0			0
$87$			0.540478i			0.0579454i
$88$		0			0
$89$	3.03333			0.321532			0.160766	−	0.986993i	$-0.448604\pi$
$89$	3.03333			0.321532			0.160766	+	0.986993i	$0.448604\pi$
$90$		0			0
$91$	−2.05661			−0.215592
$92$		0			0
$93$		−	1.68565i		−	0.174794i
$94$		0			0
$95$	−1.26984	+	1.23528i	−0.130282	+	0.126737i
$96$		0			0
$97$		−	3.21747i		−	0.326684i	−0.986569	−	0.163342i	$-0.947773\pi$
$97$		−	3.21747i		−	0.326684i	0.986569	−	0.163342i	$-0.0522275\pi$
$98$		0			0
$99$	−0.225073			−0.0226207
$100$		0			0
$101$	−16.6059			−1.65235			−0.826173	−	0.563417i	$-0.809486\pi$
$101$	−16.6059			−1.65235			−0.826173	+	0.563417i	$0.809486\pi$
$102$		0			0
$103$			7.30241i			0.719528i	0.933043	+	0.359764i	$0.117143\pi$
$103$			7.30241i			0.719528i	−0.933043	+	0.359764i	$0.882857\pi$
$104$		0			0
$105$	0.502831	−	0.489147i	0.0490713	−	0.0477358i
$106$		0			0
$107$		−	0.356936i		−	0.0345063i	−0.999851	−	0.0172532i	$-0.994508\pi$
$107$		−	0.356936i		−	0.0345063i	0.999851	−	0.0172532i	$-0.00549213\pi$
$108$		0			0
$109$	−14.0359			−1.34440			−0.672198	−	0.740371i	$-0.734650\pi$
$109$	−14.0359			−1.34440			−0.672198	+	0.740371i	$0.734650\pi$
$110$		0			0
$111$	−1.31060			−0.124397
$112$		0			0
$113$		−	1.17924i		−	0.110933i	−0.998461	−	0.0554667i	$-0.982335\pi$
$113$		−	1.17924i		−	0.110933i	0.998461	−	0.0554667i	$-0.0176647\pi$
$114$		0			0
$115$	1.55918	+	1.60280i	0.145394	+	0.149462i
$116$		0			0
$117$			3.67986i			0.340203i
$118$		0			0
$119$	−8.57448			−0.786022
$120$		0			0
$121$	−10.9942			−0.999476
$122$		0			0
$123$			0.879230i			0.0792775i
$124$		0			0
$125$	7.57186	+	8.22599i	0.677248	+	0.735755i
$126$		0			0
$127$			19.0097i			1.68684i	0.537255	+	0.843420i	$0.319461\pi$
$127$			19.0097i			1.68684i	−0.537255	+	0.843420i	$0.680539\pi$
$128$		0			0
$129$	0.331890			0.0292213
$130$		0			0
$131$	−4.29043			−0.374857			−0.187428	−	0.982278i	$-0.560015\pi$
$131$	−4.29043			−0.374857			−0.187428	+	0.982278i	$0.560015\pi$
$132$		0			0
$133$			1.31247i			0.113805i
$134$		0			0
$135$	−1.76103	−	1.81030i	−0.151565	−	0.155806i
$136$		0			0
$137$		−	12.2499i		−	1.04658i	−0.852155	−	0.523290i	$-0.824704\pi$
$137$		−	12.2499i		−	1.04658i	0.852155	−	0.523290i	$-0.175296\pi$
$138$		0			0
$139$	2.61437			0.221748			0.110874	−	0.993834i	$-0.464635\pi$
$139$	2.61437			0.221748			0.110874	+	0.993834i	$0.464635\pi$
$140$		0			0
$141$	1.89437			0.159535
$142$		0			0
$143$		−	0.0942666i		−	0.00788297i
$144$		0			0
$145$	−4.57440	+	4.44990i	−0.379883	+	0.369544i
$146$		0			0
$147$			0.805915i			0.0664707i
$148$		0			0
$149$	19.0404			1.55985			0.779927	−	0.625870i	$-0.215256\pi$
$149$	19.0404			1.55985			0.779927	+	0.625870i	$0.215256\pi$
$150$		0			0
$151$	18.3252			1.49128			0.745640	−	0.666349i	$-0.232144\pi$
$151$	18.3252			1.49128			0.745640	+	0.666349i	$0.232144\pi$
$152$		0			0
$153$			15.3422i			1.24034i
$154$		0			0
$155$	14.2667	−	13.8784i	1.14593	−	1.11474i
$156$		0			0
$157$		−	4.37467i		−	0.349137i	−0.984645	−	0.174569i	$-0.944147\pi$
$157$		−	4.37467i		−	0.349137i	0.984645	−	0.174569i	$-0.0558530\pi$
$158$		0			0
$159$	−2.10744			−0.167131
$160$		0			0
$161$	1.65661			0.130559
$162$		0			0
$163$			11.8050i			0.924639i	0.886714	+	0.462319i	$0.152983\pi$
$163$			11.8050i			0.924639i	−0.886714	+	0.462319i	$0.847017\pi$
$164$		0			0
$165$	0.0224205	+	0.0230477i	0.00174543	+	0.00179426i
$166$		0			0
$167$		−	5.19760i		−	0.402202i	−0.979570	−	0.201101i	$-0.935548\pi$
$167$		−	5.19760i		−	0.402202i	0.979570	−	0.201101i	$-0.0644520\pi$
$168$		0			0
$169$	11.4588			0.881444
$170$		0			0
$171$	2.34837			0.179585
$172$		0			0
$173$			2.61834i			0.199069i	0.995034	+	0.0995345i	$0.0317354\pi$
$173$			2.61834i			0.199069i	−0.995034	+	0.0995345i	$0.968265\pi$
$174$		0			0
$175$	8.27989	+	0.228488i	0.625901	+	0.0172721i
$176$		0			0
$177$		−	2.35926i		−	0.177333i
$178$		0			0
$179$	0.627563			0.0469062			0.0234531	−	0.999725i	$-0.492534\pi$
$179$	0.627563			0.0469062			0.0234531	+	0.999725i	$0.492534\pi$
$180$		0			0
$181$	−6.15848			−0.457756			−0.228878	−	0.973455i	$-0.573506\pi$
$181$	−6.15848			−0.457756			−0.228878	+	0.973455i	$0.573506\pi$
$182$		0			0
$183$		−	2.27436i		−	0.168126i
$184$		0			0
$185$	−10.7905	−	11.0924i	−0.793335	−	0.815530i
$186$		0			0
$187$		−	0.393019i		−	0.0287404i
$188$		0			0
$189$	−1.87107			−0.136101
$190$		0			0
$191$	19.1343			1.38451			0.692254	−	0.721654i	$-0.256618\pi$
$191$	19.1343			1.38451			0.692254	+	0.721654i	$0.256618\pi$
$192$		0			0
$193$			0.810900i			0.0583698i	0.999574	+	0.0291849i	$0.00929117\pi$
$193$			0.810900i			0.0583698i	−0.999574	+	0.0291849i	$0.990709\pi$
$194$		0			0
$195$	0.376821	−	0.366566i	0.0269847	−	0.0262503i
$196$		0			0
$197$		−	6.15110i		−	0.438248i	−0.975697	−	0.219124i	$-0.929680\pi$
$197$		−	6.15110i		−	0.438248i	0.975697	−	0.219124i	$-0.0703199\pi$
$198$		0			0
$199$	7.03061			0.498387			0.249193	−	0.968454i	$-0.419834\pi$
$199$	7.03061			0.498387			0.249193	+	0.968454i	$0.419834\pi$
$200$		0			0
$201$	−1.31818			−0.0929772
$202$		0			0
$203$			4.72797i			0.331838i
$204$		0			0
$205$	−7.44146	+	7.23894i	−0.519734	+	0.505589i
$206$		0			0
$207$		−	2.96414i		−	0.206022i
$208$		0			0
$209$	−0.0601581			−0.00416122
$210$		0			0
$211$	−8.12120			−0.559086			−0.279543	−	0.960133i	$-0.590183\pi$
$211$	−8.12120			−0.559086			−0.279543	+	0.960133i	$0.590183\pi$
$212$		0			0
$213$			1.46131i			0.100128i
$214$		0			0
$215$	2.73254	+	2.80899i	0.186358	+	0.191571i
$216$		0			0
$217$		−	14.7457i		−	1.00100i
$218$		0			0
$219$	−1.41932			−0.0959090
$220$		0			0
$221$	−6.42570			−0.432240
$222$		0			0
$223$			2.86268i			0.191699i	0.995396	+	0.0958495i	$0.0305568\pi$
$223$			2.86268i			0.191699i	−0.995396	+	0.0958495i	$0.969443\pi$
$224$		0			0
$225$	0.408829	−	14.8150i	0.0272553	−	0.987670i
$226$		0			0
$227$		−	27.4500i		−	1.82192i	−0.412496	−	0.910960i	$-0.635343\pi$
$227$		−	27.4500i		−	1.82192i	0.412496	−	0.910960i	$-0.364657\pi$
$228$		0			0
$229$	−1.11319			−0.0735613			−0.0367807	−	0.999323i	$-0.511710\pi$
$229$	−1.11319			−0.0735613			−0.0367807	+	0.999323i	$0.511710\pi$
$230$		0			0
$231$	0.0238215			0.00156734
$232$		0			0
$233$		−	21.2957i		−	1.39513i	−0.716523	−	0.697564i	$-0.754267\pi$
$233$		−	21.2957i		−	1.39513i	0.716523	−	0.697564i	$-0.245733\pi$
$234$		0			0
$235$	15.5969	+	16.0332i	1.01743	+	1.04589i
$236$		0			0
$237$		−	2.96590i		−	0.192656i
$238$		0			0
$239$	7.68036			0.496801			0.248401	−	0.968657i	$-0.420095\pi$
$239$	7.68036			0.496801			0.248401	+	0.968657i	$0.420095\pi$
$240$		0			0
$241$	8.42476			0.542687			0.271343	−	0.962483i	$-0.412532\pi$
$241$	8.42476			0.542687			0.271343	+	0.962483i	$0.412532\pi$
$242$		0			0
$243$			5.03188i			0.322795i
$244$		0			0
$245$	−6.82095	+	6.63531i	−0.435774	+	0.423915i
$246$		0			0
$247$			0.983561i			0.0625825i
$248$		0			0
$249$	−1.26601			−0.0802300
$250$		0			0
$251$	2.56210			0.161718			0.0808590	−	0.996726i	$-0.474234\pi$
$251$	2.56210			0.161718			0.0808590	+	0.996726i	$0.474234\pi$
$252$		0			0
$253$			0.0759321i			0.00477381i
$254$		0			0
$255$	1.57105	−	1.52830i	0.0983831	−	0.0957056i
$256$		0			0
$257$			4.39999i			0.274464i	0.990539	+	0.137232i	$0.0438206\pi$
$257$			4.39999i			0.274464i	−0.990539	+	0.137232i	$0.956179\pi$
$258$		0			0
$259$	−11.4648			−0.712388
$260$		0			0
$261$	8.45966			0.523640
$262$		0			0
$263$			24.8510i			1.53238i	0.642613	+	0.766191i	$0.277850\pi$
$263$			24.8510i			1.53238i	−0.642613	+	0.766191i	$0.722150\pi$
$264$		0			0
$265$	−17.3512	−	17.8366i	−1.06587	−	1.09569i
$266$		0			0
$267$			0.574437i			0.0351550i
$268$		0			0
$269$	10.8742			0.663012			0.331506	−	0.943453i	$-0.392443\pi$
$269$	10.8742			0.663012			0.331506	+	0.943453i	$0.392443\pi$
$270$		0			0
$271$	2.60290			0.158115			0.0790575	−	0.996870i	$-0.474809\pi$
$271$	2.60290			0.158115			0.0790575	+	0.996870i	$0.474809\pi$
$272$		0			0
$273$		−	0.389472i		−	0.0235719i
$274$		0			0
$275$	−0.0104729	+	0.379516i	−0.000631542	+	0.0228857i
$276$		0			0
$277$		−	12.1918i		−	0.732537i	−0.930509	−	0.366269i	$-0.880635\pi$
$277$		−	12.1918i		−	0.732537i	0.930509	−	0.366269i	$-0.119365\pi$
$278$		0			0
$279$	−26.3842			−1.57958
$280$		0			0
$281$	5.85068			0.349022			0.174511	−	0.984655i	$-0.444165\pi$
$281$	5.85068			0.349022			0.174511	+	0.984655i	$0.444165\pi$
$282$		0			0
$283$			10.7223i			0.637375i	0.947860	+	0.318688i	$0.103242\pi$
$283$			10.7223i			0.637375i	−0.947860	+	0.318688i	$0.896758\pi$
$284$		0			0
$285$	−0.233931	−	0.240476i	−0.0138569	−	0.0142446i
$286$		0			0
$287$			7.69129i			0.454002i
$288$		0			0
$289$	−9.79021			−0.575895
$290$		0			0
$291$	0.609309			0.0357183
$292$		0			0
$293$			2.72596i			0.159252i	0.996825	+	0.0796261i	$0.0253726\pi$
$293$			2.72596i			0.159252i	−0.996825	+	0.0796261i	$0.974627\pi$
$294$		0			0
$295$	19.9679	−	19.4245i	1.16257	−	1.13094i
$296$		0			0
$297$		−	0.0857623i		−	0.00497643i
$298$		0			0
$299$	1.24146			0.0717955
$300$		0			0
$301$	2.90329			0.167343
$302$		0			0
$303$		−	3.14474i		−	0.180661i
$304$		0			0
$305$	19.2493	−	18.7254i	1.10221	−	1.07222i
$306$		0			0
$307$		−	22.4979i		−	1.28402i	−0.766695	−	0.642012i	$-0.778100\pi$
$307$		−	22.4979i		−	1.28402i	0.766695	−	0.642012i	$-0.221900\pi$
$308$		0			0
$309$	−1.38290			−0.0786703
$310$		0			0
$311$	−17.9597			−1.01840			−0.509200	−	0.860648i	$-0.670058\pi$
$311$	−17.9597			−1.01840			−0.509200	+	0.860648i	$0.670058\pi$
$312$		0			0
$313$			14.4728i			0.818052i	0.912523	+	0.409026i	$0.134131\pi$
$313$			14.4728i			0.818052i	−0.912523	+	0.409026i	$0.865869\pi$
$314$		0			0
$315$	−7.65621	−	7.87040i	−0.431379	−	0.443447i
$316$		0			0
$317$			7.47689i			0.419944i	0.977707	+	0.209972i	$0.0673373\pi$
$317$			7.47689i			0.419944i	−0.977707	+	0.209972i	$0.932663\pi$
$318$		0			0
$319$	−0.216711			−0.0121335
$320$		0			0
$321$	0.0675949			0.00377278
$322$		0			0
$323$			4.10069i			0.228168i
$324$		0			0
$325$	6.20494	+	0.171229i	0.344188	+	0.00949805i
$326$		0			0
$327$		−	2.65806i		−	0.146991i
$328$		0			0
$329$	16.5715			0.913616
$330$		0			0
$331$	27.3565			1.50365			0.751826	−	0.659362i	$-0.229173\pi$
$331$	27.3565			1.50365			0.751826	+	0.659362i	$0.229173\pi$
$332$		0			0
$333$			20.5137i			1.12415i
$334$		0			0
$335$	−10.8529	−	11.1565i	−0.592959	−	0.609547i
$336$		0			0
$337$		−	5.65271i		−	0.307923i	−0.988077	−	0.153961i	$-0.950797\pi$
$337$		−	5.65271i		−	0.307923i	0.988077	−	0.153961i	$-0.0492032\pi$
$338$		0			0
$339$	0.223319			0.0121290
$340$		0			0
$341$	0.675881			0.0366010
$342$		0			0
$343$			18.6462i			1.00680i
$344$		0			0
$345$	−0.303531	+	0.295270i	−0.0163415	+	0.0158968i
$346$		0			0
$347$			15.5293i			0.833658i	0.908985	+	0.416829i	$0.136859\pi$
$347$			15.5293i			0.833658i	−0.908985	+	0.416829i	$0.863141\pi$
$348$		0			0
$349$	−11.0559			−0.591810			−0.295905	−	0.955217i	$-0.595621\pi$
$349$	−11.0559			−0.591810			−0.295905	+	0.955217i	$0.595621\pi$
$350$		0			0
$351$	−1.40218			−0.0748428
$352$		0			0
$353$			6.85381i			0.364792i	0.983225	+	0.182396i	$0.0583852\pi$
$353$			6.85381i			0.364792i	−0.983225	+	0.182396i	$0.941615\pi$
$354$		0			0
$355$	−12.3680	+	12.0314i	−0.656425	+	0.638560i
$356$		0			0
$357$		−	1.62380i		−	0.0859404i
$358$		0			0
$359$	16.3104			0.860832			0.430416	−	0.902631i	$-0.358367\pi$
$359$	16.3104			0.860832			0.430416	+	0.902631i	$0.358367\pi$
$360$		0			0
$361$	−18.3723			−0.966964
$362$		0			0
$363$		−	2.08204i		−	0.109279i
$364$		0			0
$365$	−11.6857	−	12.0126i	−0.611656	−	0.628768i
$366$		0			0
$367$		−	31.6200i		−	1.65055i	−0.564731	−	0.825275i	$-0.691020\pi$
$367$		−	31.6200i		−	1.65055i	0.564731	−	0.825275i	$-0.308980\pi$
$368$		0			0
$369$	13.7619			0.716414
$370$		0			0
$371$	−18.4354			−0.957118
$372$		0			0
$373$			28.2458i			1.46251i	0.682104	+	0.731256i	$0.261065\pi$
$373$			28.2458i			1.46251i	−0.682104	+	0.731256i	$0.738935\pi$
$374$		0			0
$375$	−1.55780	+	1.43392i	−0.0804445	+	0.0740475i
$376$		0			0
$377$			3.54313i			0.182481i
$378$		0			0
$379$	2.98772			0.153469			0.0767345	−	0.997052i	$-0.475551\pi$
$379$	2.98772			0.153469			0.0767345	+	0.997052i	$0.475551\pi$
$380$		0			0
$381$	−3.59997			−0.184432
$382$		0			0
$383$		−	12.5176i		−	0.639621i	−0.947482	−	0.319810i	$-0.896381\pi$
$383$		−	12.5176i		−	0.639621i	0.947482	−	0.319810i	$-0.103619\pi$
$384$		0			0
$385$	0.196129	+	0.201616i	0.00999564	+	0.0102753i
$386$		0			0
$387$		−	5.19480i		−	0.264067i
$388$		0			0
$389$	19.4861			0.987982			0.493991	−	0.869467i	$-0.335538\pi$
$389$	19.4861			0.987982			0.493991	+	0.869467i	$0.335538\pi$
$390$		0			0
$391$	5.17593			0.261758
$392$		0			0
$393$		−	0.812503i		−	0.0409853i
$394$		0			0
$395$	25.1022	−	24.4191i	1.26303	−	1.22866i
$396$		0			0
$397$		−	17.3203i		−	0.869281i	−0.900604	−	0.434640i	$-0.856876\pi$
$397$		−	17.3203i		−	0.869281i	0.900604	−	0.434640i	$-0.143124\pi$
$398$		0			0
$399$	−0.248549			−0.0124430
$400$		0			0
$401$	17.3727			0.867553			0.433777	−	0.901020i	$-0.357181\pi$
$401$	17.3727			0.867553			0.433777	+	0.901020i	$0.357181\pi$
$402$		0			0
$403$		−	11.0504i		−	0.550459i
$404$		0			0
$405$	−13.9099	+	13.5314i	−0.691189	+	0.672379i
$406$		0			0
$407$		−	0.525499i		−	0.0260480i
$408$		0			0
$409$	12.6874			0.627350			0.313675	−	0.949530i	$-0.398440\pi$
$409$	12.6874			0.627350			0.313675	+	0.949530i	$0.398440\pi$
$410$		0			0
$411$	2.31983			0.114429
$412$		0			0
$413$		−	20.6383i		−	1.01554i
$414$		0			0
$415$	−10.4234	−	10.7150i	−0.511664	−	0.525978i
$416$		0			0
$417$			0.495098i			0.0242450i
$418$		0			0
$419$	−4.23468			−0.206878			−0.103439	−	0.994636i	$-0.532985\pi$
$419$	−4.23468			−0.206878			−0.103439	+	0.994636i	$0.532985\pi$
$420$		0			0
$421$	4.17837			0.203641			0.101821	−	0.994803i	$-0.467533\pi$
$421$	4.17837			0.203641			0.101821	+	0.994803i	$0.467533\pi$
$422$		0			0
$423$		−	29.6510i		−	1.44168i
$424$		0			0
$425$	25.8698	+	0.713891i	1.25487	+	0.0346288i
$426$		0			0
$427$		−	19.8956i		−	0.962813i
$428$		0			0
$429$	0.0178518			0.000861892
$430$		0			0
$431$	21.9197			1.05583			0.527917	−	0.849296i	$-0.322973\pi$
$431$	21.9197			1.05583			0.527917	+	0.849296i	$0.322973\pi$
$432$		0			0
$433$		−	20.2195i		−	0.971688i	−0.874046	−	0.485844i	$-0.838512\pi$
$433$		−	20.2195i		−	0.971688i	0.874046	−	0.485844i	$-0.161488\pi$
$434$		0			0
$435$	−0.842702	−	0.866278i	−0.0404045	−	0.0415348i
$436$		0			0
$437$		−	0.792262i		−	0.0378990i
$438$		0			0
$439$	−16.3149			−0.778666			−0.389333	−	0.921097i	$-0.627294\pi$
$439$	−16.3149			−0.778666			−0.389333	+	0.921097i	$0.627294\pi$
$440$		0			0
$441$	12.6143			0.600682
$442$		0			0
$443$		−	28.0868i		−	1.33444i	−0.744859	−	0.667222i	$-0.767483\pi$
$443$		−	28.0868i		−	1.33444i	0.744859	−	0.667222i	$-0.232517\pi$
$444$		0			0
$445$	−4.86181	+	4.72950i	−0.230472	+	0.224200i
$446$		0			0
$447$			3.60579i			0.170548i
$448$		0			0
$449$	24.5071			1.15656			0.578282	−	0.815837i	$-0.303723\pi$
$449$	24.5071			1.15656			0.578282	+	0.815837i	$0.303723\pi$
$450$		0			0
$451$	−0.352537			−0.0166003
$452$		0			0
$453$			3.47033i			0.163050i
$454$		0			0
$455$	3.29634	−	3.20663i	0.154535	−	0.150329i
$456$		0			0
$457$			2.87128i			0.134313i	0.997742	+	0.0671564i	$0.0213927\pi$
$457$			2.87128i			0.134313i	−0.997742	+	0.0671564i	$0.978607\pi$
$458$		0			0
$459$	−5.84601			−0.272868
$460$		0			0
$461$	15.2264			0.709162			0.354581	−	0.935025i	$-0.384623\pi$
$461$	15.2264			0.709162			0.354581	+	0.935025i	$0.384623\pi$
$462$		0			0
$463$			36.6480i			1.70318i	0.524212	+	0.851588i	$0.324360\pi$
$463$			36.6480i			1.70318i	−0.524212	+	0.851588i	$0.675640\pi$
$464$		0			0
$465$	2.62824	+	2.70176i	0.121881	+	0.125291i
$466$		0			0
$467$			14.9765i			0.693031i	0.938044	+	0.346516i	$0.112635\pi$
$467$			14.9765i			0.693031i	−0.938044	+	0.346516i	$0.887365\pi$
$468$		0			0
$469$	−11.5311			−0.532457
$470$		0			0
$471$	0.828456			0.0381732
$472$		0			0
$473$			0.133075i			0.00611879i
$474$		0			0
$475$	0.109273	−	3.95980i	0.00501378	−	0.181688i
$476$		0			0
$477$			32.9861i			1.51033i
$478$		0			0
$479$	4.03398			0.184317			0.0921587	−	0.995744i	$-0.470623\pi$
$479$	4.03398			0.184317			0.0921587	+	0.995744i	$0.470623\pi$
$480$		0			0
$481$	−8.59171			−0.391748
$482$		0			0
$483$			0.313721i			0.0142748i
$484$		0			0
$485$	5.01661	+	5.15695i	0.227792	+	0.234165i
$486$		0			0
$487$			4.28846i			0.194329i	0.995268	+	0.0971643i	$0.0309772\pi$
$487$			4.28846i			0.194329i	−0.995268	+	0.0971643i	$0.969023\pi$
$488$		0			0
$489$	−2.23558			−0.101096
$490$		0			0
$491$	36.8754			1.66416			0.832081	−	0.554654i	$-0.187149\pi$
$491$	36.8754			1.66416			0.832081	+	0.554654i	$0.187149\pi$
$492$		0			0
$493$			14.7721i			0.665303i
$494$		0			0
$495$	0.360747	−	0.350929i	0.0162144	−	0.0157731i
$496$		0			0
$497$			12.7832i			0.573405i
$498$		0			0
$499$	−5.74707			−0.257274			−0.128637	−	0.991692i	$-0.541060\pi$
$499$	−5.74707			−0.257274			−0.128637	+	0.991692i	$0.541060\pi$
$500$		0			0
$501$	0.984298			0.0439752
$502$		0			0
$503$		−	18.7850i		−	0.837583i	−0.908082	−	0.418792i	$-0.862454\pi$
$503$		−	18.7850i		−	0.837583i	0.908082	−	0.418792i	$-0.137546\pi$
$504$		0			0
$505$	26.6158	−	25.8915i	1.18439	−	1.15216i
$506$		0			0
$507$			2.17001i			0.0963735i
$508$		0			0
$509$	20.2669			0.898313			0.449157	−	0.893453i	$-0.351725\pi$
$509$	20.2669			0.898313			0.449157	+	0.893453i	$0.351725\pi$
$510$		0			0
$511$	−12.4159			−0.549247
$512$		0			0
$513$			0.894829i			0.0395077i
$514$		0			0
$515$	−11.3858	−	11.7043i	−0.501717	−	0.515753i
$516$		0			0
$517$			0.759568i			0.0334058i
$518$		0			0
$519$	−0.495850			−0.0217654
$520$		0			0
$521$	6.47593			0.283716			0.141858	−	0.989887i	$-0.454692\pi$
$521$	6.47593			0.283716			0.141858	+	0.989887i	$0.454692\pi$
$522$		0			0
$523$		−	0.414671i		−	0.0181323i	−0.999959	−	0.00906615i	$-0.997114\pi$
$523$		−	0.414671i		−	0.0181323i	0.999959	−	0.00906615i	$-0.00288588\pi$
$524$		0			0
$525$	−0.0432700	+	1.56801i	−0.00188846	+	0.0684335i
$526$		0			0
$527$		−	46.0716i		−	2.00691i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−36.9276			−1.60252
$532$		0			0
$533$			5.76384i			0.249659i
$534$		0			0
$535$	0.556527	+	0.572097i	0.0240608	+	0.0247339i
$536$		0			0
$537$			0.118845i			0.00512854i
$538$		0			0
$539$	−0.323140			−0.0139186
$540$		0			0
$541$	28.5708			1.22835			0.614177	−	0.789169i	$-0.289488\pi$
$541$	28.5708			1.22835			0.614177	+	0.789169i	$0.289488\pi$
$542$		0			0
$543$		−	1.16626i		−	0.0500492i
$544$		0			0
$545$	22.4967	−	21.8845i	0.963654	−	0.937428i
$546$		0			0
$547$			39.0974i			1.67168i	0.548971	+	0.835841i	$0.315020\pi$
$547$			39.0974i			1.67168i	−0.548971	+	0.835841i	$0.684980\pi$
$548$		0			0
$549$	−35.5987			−1.51932
$550$		0			0
$551$	2.26112			0.0963270
$552$		0			0
$553$		−	25.9450i		−	1.10329i
$554$		0			0
$555$	2.10063	−	2.04346i	0.0891667	−	0.0867400i
$556$		0			0
$557$		−	33.9136i		−	1.43696i	−0.695546	−	0.718482i	$-0.744837\pi$
$557$		−	33.9136i		−	1.43696i	0.695546	−	0.718482i	$-0.255163\pi$
$558$		0			0
$559$	2.17572			0.0920232
$560$		0			0
$561$	0.0744281			0.00314236
$562$		0			0
$563$			25.6390i			1.08056i	0.841486	+	0.540278i	$0.181681\pi$
$563$			25.6390i			1.08056i	−0.841486	+	0.540278i	$0.818319\pi$
$564$		0			0
$565$	1.83864	+	1.89008i	0.0773523	+	0.0795163i
$566$		0			0
$567$			14.3769i			0.603773i
$568$		0			0
$569$	13.6257			0.571218			0.285609	−	0.958346i	$-0.407804\pi$
$569$	13.6257			0.571218			0.285609	+	0.958346i	$0.407804\pi$
$570$		0			0
$571$	−0.354294			−0.0148267			−0.00741337	−	0.999973i	$-0.502360\pi$
$571$	−0.354294			−0.0148267			−0.00741337	+	0.999973i	$0.502360\pi$
$572$		0			0
$573$			3.62356i			0.151376i
$574$		0			0
$575$	−4.99810	−	0.137925i	−0.208435	−	0.00575188i
$576$		0			0
$577$		−	18.5870i		−	0.773788i	−0.922124	−	0.386894i	$-0.873548\pi$
$577$		−	18.5870i		−	0.773788i	0.922124	−	0.386894i	$-0.126452\pi$
$578$		0			0
$579$	−0.153564			−0.00638192
$580$		0			0
$581$	−11.0747			−0.459457
$582$		0			0
$583$		−	0.845002i		−	0.0349964i
$584$		0			0
$585$	−5.73755	−	5.89807i	−0.237219	−	0.243855i
$586$		0			0
$587$		−	7.75193i		−	0.319956i	−0.987121	−	0.159978i	$-0.948858\pi$
$587$		−	7.75193i		−	0.319956i	0.987121	−	0.159978i	$-0.0511424\pi$
$588$		0			0
$589$	−7.05202			−0.290573
$590$		0			0
$591$	1.16487			0.0479162
$592$		0			0
$593$			23.3828i			0.960215i	0.877210	+	0.480107i	$0.159402\pi$
$593$			23.3828i			0.960215i	−0.877210	+	0.480107i	$0.840598\pi$
$594$		0			0
$595$	13.7432	−	13.3692i	0.563415	−	0.548082i
$596$		0			0
$597$			1.33142i			0.0544916i
$598$		0			0
$599$	−25.6075			−1.04629			−0.523146	−	0.852243i	$-0.675242\pi$
$599$	−25.6075			−1.04629			−0.523146	+	0.852243i	$0.675242\pi$
$600$		0			0
$601$	4.03750			0.164693			0.0823466	−	0.996604i	$-0.473759\pi$
$601$	4.03750			0.164693			0.0823466	+	0.996604i	$0.473759\pi$
$602$		0			0
$603$			20.6324i			0.840215i
$604$		0			0
$605$	17.6215	−	17.1420i	0.716418	−	0.696920i
$606$		0			0
$607$		−	45.8176i		−	1.85968i	−0.367967	−	0.929839i	$-0.619946\pi$
$607$		−	45.8176i		−	1.85968i	0.367967	−	0.929839i	$-0.380054\pi$
$608$		0			0
$609$	−0.895361			−0.0362819
$610$		0			0
$611$	12.4186			0.502405
$612$		0			0
$613$			29.1714i			1.17822i	0.808052	+	0.589112i	$0.200522\pi$
$613$			29.1714i			1.17822i	−0.808052	+	0.589112i	$0.799478\pi$
$614$		0			0
$615$	−1.37088	−	1.40923i	−0.0552791	−	0.0568256i
$616$		0			0
$617$			17.0562i			0.686659i	0.939215	+	0.343329i	$0.111555\pi$
$617$			17.0562i			0.686659i	−0.939215	+	0.343329i	$0.888445\pi$
$618$		0			0
$619$	36.0620			1.44946			0.724728	−	0.689035i	$-0.241966\pi$
$619$	36.0620			1.44946			0.724728	+	0.689035i	$0.241966\pi$
$620$		0			0
$621$	1.12946			0.0453237
$622$		0			0
$623$			5.02503i			0.201324i
$624$		0			0
$625$	−24.9620	−	1.37873i	−0.998478	−	0.0551491i
$626$		0			0
$627$		−	0.0113925i		−	0.000454971i
$628$		0			0
$629$	−35.8208			−1.42827
$630$		0			0
$631$	−17.4958			−0.696496			−0.348248	−	0.937402i	$-0.613223\pi$
$631$	−17.4958			−0.696496			−0.348248	+	0.937402i	$0.613223\pi$
$632$		0			0
$633$		−	1.53795i		−	0.0611282i
$634$		0			0
$635$	−29.6395	−	30.4687i	−1.17621	−	1.20911i
$636$		0			0
$637$			5.28322i			0.209329i
$638$		0			0
$639$	22.8727			0.904832
$640$		0			0
$641$	−39.9200			−1.57675			−0.788373	−	0.615197i	$-0.789076\pi$
$641$	−39.9200			−1.57675			−0.788373	+	0.615197i	$0.789076\pi$
$642$		0			0
$643$			41.8187i			1.64917i	0.565740	+	0.824584i	$0.308591\pi$
$643$			41.8187i			1.64917i	−0.565740	+	0.824584i	$0.691409\pi$
$644$		0			0
$645$	−0.531953	+	0.517476i	−0.0209456	+	0.0203756i
$646$		0			0
$647$			38.8026i			1.52549i	0.646702	+	0.762743i	$0.276148\pi$
$647$			38.8026i			1.52549i	−0.646702	+	0.762743i	$0.723852\pi$
$648$		0			0
$649$	0.945972			0.0371327
$650$		0			0
$651$	2.79247			0.109445
$652$		0			0
$653$			39.8995i			1.56139i	0.624914	+	0.780694i	$0.285134\pi$
$653$			39.8995i			1.56139i	−0.624914	+	0.780694i	$0.714866\pi$
$654$		0			0
$655$	6.87670	−	6.68955i	0.268695	−	0.261382i
$656$		0			0
$657$			22.2155i			0.866709i
$658$		0			0
$659$	22.9449			0.893808			0.446904	−	0.894582i	$-0.352527\pi$
$659$	22.9449			0.893808			0.446904	+	0.894582i	$0.352527\pi$
$660$		0			0
$661$	19.2759			0.749745			0.374873	−	0.927076i	$-0.377686\pi$
$661$	19.2759			0.749745			0.374873	+	0.927076i	$0.377686\pi$
$662$		0			0
$663$		−	1.21687i		−	0.0472593i
$664$		0			0
$665$	−2.04637	−	2.10362i	−0.0793549	−	0.0815749i
$666$		0			0
$667$		−	2.85401i		−	0.110508i
$668$		0			0
$669$	−0.542121			−0.0209596
$670$		0			0
$671$	0.911930			0.0352047
$672$		0			0
$673$		−	38.0041i		−	1.46495i	−0.680793	−	0.732476i	$-0.738365\pi$
$673$		−	38.0041i		−	1.46495i	0.680793	−	0.732476i	$-0.261635\pi$
$674$		0			0
$675$	5.64515	+	0.155781i	0.217282	+	0.00599602i
$676$		0			0
$677$		−	41.0387i		−	1.57725i	−0.614878	−	0.788623i	$-0.710795\pi$
$677$		−	41.0387i		−	1.57725i	0.614878	−	0.788623i	$-0.289205\pi$
$678$		0			0
$679$	5.33008			0.204550
$680$		0			0
$681$	5.19835			0.199201
$682$		0			0
$683$		−	27.6705i		−	1.05878i	−0.848378	−	0.529390i	$-0.822421\pi$
$683$		−	27.6705i		−	1.05878i	0.848378	−	0.529390i	$-0.177579\pi$
$684$		0			0
$685$	19.0998	+	19.6341i	0.729765	+	0.750181i
$686$		0			0
$687$		−	0.210810i		−	0.00804290i
$688$		0			0
$689$	−13.8155			−0.526327
$690$		0			0
$691$	−33.8078			−1.28611			−0.643056	−	0.765820i	$-0.722334\pi$
$691$	−33.8078			−1.28611			−0.643056	+	0.765820i	$0.722334\pi$
$692$		0			0
$693$		−	0.372858i		−	0.0141637i
$694$		0			0
$695$	−4.19031	+	4.07627i	−0.158948	+	0.154622i
$696$		0			0
$697$			24.0307i			0.910229i
$698$		0			0
$699$	4.03288			0.152538
$700$		0			0
$701$	−46.4878			−1.75582			−0.877909	−	0.478827i	$-0.841062\pi$
$701$	−46.4878			−1.75582			−0.877909	+	0.478827i	$0.841062\pi$
$702$		0			0
$703$			5.48297i			0.206794i
$704$		0			0
$705$	−3.03630	+	2.95366i	−0.114353	+	0.111241i
$706$		0			0
$707$		−	27.5094i		−	1.03460i
$708$		0			0
$709$	13.5298			0.508122			0.254061	−	0.967188i	$-0.418234\pi$
$709$	13.5298			0.508122			0.254061	+	0.967188i	$0.418234\pi$
$710$		0			0
$711$	−46.4229			−1.74099
$712$		0			0
$713$			8.90112i			0.333350i
$714$		0			0
$715$	0.146979	+	0.151090i	0.00549668	+	0.00565046i
$716$		0			0
$717$			1.45447i			0.0543182i
$718$		0			0
$719$	−2.64955			−0.0988115			−0.0494057	−	0.998779i	$-0.515733\pi$
$719$	−2.64955			−0.0988115			−0.0494057	+	0.998779i	$0.515733\pi$
$720$		0			0
$721$	−12.0972			−0.450525
$722$		0			0
$723$			1.59544i			0.0593351i
$724$		0			0
$725$	0.393639	−	14.2646i	0.0146194	−	0.529774i
$726$		0			0
$727$			46.8796i			1.73867i	0.494226	+	0.869334i	$0.335452\pi$
$727$			46.8796i			1.73867i	−0.494226	+	0.869334i	$0.664548\pi$
$728$		0			0
$729$	25.0826			0.928987
$730$		0			0
$731$	9.07108			0.335506
$732$		0			0
$733$			36.1389i			1.33482i	0.744691	+	0.667410i	$0.232597\pi$
$733$			36.1389i			1.33482i	−0.744691	+	0.667410i	$0.767403\pi$
$734$		0			0
$735$	−1.25657	−	1.29172i	−0.0463491	−	0.0476458i
$736$		0			0
$737$		−	0.528538i		−	0.0194690i
$738$		0			0
$739$	22.6473			0.833095			0.416547	−	0.909114i	$-0.363240\pi$
$739$	22.6473			0.833095			0.416547	+	0.909114i	$0.363240\pi$
$740$		0			0
$741$	−0.186262			−0.00684252
$742$		0			0
$743$		−	7.04015i		−	0.258278i	−0.991627	−	0.129139i	$-0.958779\pi$
$743$		−	7.04015i		−	0.258278i	0.991627	−	0.129139i	$-0.0412213\pi$
$744$		0			0
$745$	−30.5180	+	29.6875i	−1.11809	+	1.08766i
$746$		0			0
$747$			19.8158i			0.725021i
$748$		0			0
$749$	0.591304			0.0216058
$750$		0			0
$751$	5.94539			0.216950			0.108475	−	0.994099i	$-0.465403\pi$
$751$	5.94539			0.216950			0.108475	+	0.994099i	$0.465403\pi$
$752$		0			0
$753$			0.485198i			0.0176816i
$754$		0			0
$755$	−29.3715	+	28.5722i	−1.06894	+	1.03985i
$756$		0			0
$757$		−	43.9519i		−	1.59746i	−0.601691	−	0.798729i	$-0.705506\pi$
$757$		−	43.9519i		−	1.59746i	0.601691	−	0.798729i	$-0.294494\pi$
$758$		0			0
$759$	−0.0143797			−0.000521949
$760$		0			0
$761$	22.0290			0.798549			0.399274	−	0.916831i	$-0.369262\pi$
$761$	22.0290			0.798549			0.399274	+	0.916831i	$0.369262\pi$
$762$		0			0
$763$		−	23.2520i		−	0.841779i
$764$		0			0
$765$	−23.9212	−	24.5904i	−0.864871	−	0.889067i
$766$		0			0
$767$		−	15.4663i		−	0.558455i
$768$		0			0
$769$	17.6958			0.638128			0.319064	−	0.947733i	$-0.396632\pi$
$769$	17.6958			0.638128			0.319064	+	0.947733i	$0.396632\pi$
$770$		0			0
$771$	−0.833250			−0.0300088
$772$		0			0
$773$			1.63758i			0.0588997i	0.999566	+	0.0294498i	$0.00937553\pi$
$773$			1.63758i			0.0588997i	−0.999566	+	0.0294498i	$0.990624\pi$
$774$		0			0
$775$	−1.22769	+	44.4887i	−0.0440999	+	1.59808i
$776$		0			0
$777$		−	2.17115i		−	0.0778896i
$778$		0			0
$779$	3.67831			0.131789
$780$		0			0
$781$	−0.585930			−0.0209662
$782$		0			0
$783$			3.22349i			0.115198i
$784$		0			0
$785$	6.82090	+	7.01172i	0.243448	+	0.250259i
$786$		0			0
$787$			27.6760i			0.986543i	0.869875	+	0.493272i	$0.164199\pi$
$787$			27.6760i			0.986543i	−0.869875	+	0.493272i	$0.835801\pi$
$788$		0			0
$789$	−4.70618			−0.167544
$790$		0			0
$791$	1.95354			0.0694598
$792$		0			0
$793$		−	14.9097i		−	0.529459i
$794$		0			0
$795$	3.37781	−	3.28588i	0.119799	−	0.116538i
$796$		0			0
$797$			9.29831i			0.329363i	0.986347	+	0.164681i	$0.0526596\pi$
$797$			9.29831i			0.329363i	−0.986347	+	0.164681i	$0.947340\pi$
$798$		0			0
$799$	51.7761			1.83171
$800$		0			0
$801$	8.99119			0.317688
$802$		0			0
$803$		−	0.569093i		−	0.0200829i
$804$		0			0
$805$	−2.65521	+	2.58295i	−0.0935838	+	0.0910370i
$806$		0			0
$807$			2.05931i			0.0724911i
$808$		0			0
$809$	−48.8800			−1.71853			−0.859264	−	0.511532i	$-0.829078\pi$
$809$	−48.8800			−1.71853			−0.859264	+	0.511532i	$0.829078\pi$
$810$		0			0
$811$	−44.0615			−1.54721			−0.773604	−	0.633669i	$-0.781548\pi$
$811$	−44.0615			−1.54721			−0.773604	+	0.633669i	$0.781548\pi$
$812$		0			0
$813$			0.492926i			0.0172877i
$814$		0			0
$815$	−18.4061	−	18.9210i	−0.644737	−	0.662775i
$816$		0			0
$817$		−	1.38848i		−	0.0485768i
$818$		0			0
$819$	−6.09608			−0.213014
$820$		0			0
$821$	30.1832			1.05340			0.526700	−	0.850051i	$-0.323429\pi$
$821$	30.1832			1.05340			0.526700	+	0.850051i	$0.323429\pi$
$822$		0			0
$823$			32.0171i			1.11605i	0.829826	+	0.558023i	$0.188440\pi$
$823$			32.0171i			1.11605i	−0.829826	+	0.558023i	$0.811560\pi$
$824$		0			0
$825$	−0.0718710	−	0.00198332i	−0.00250223	−	6.90503e-5i
$826$		0			0
$827$		−	20.5192i		−	0.713523i	−0.934196	−	0.356761i	$-0.883881\pi$
$827$		−	20.5192i		−	0.713523i	0.934196	−	0.356761i	$-0.116119\pi$
$828$		0			0
$829$	−38.0551			−1.32171			−0.660854	−	0.750515i	$-0.729806\pi$
$829$	−38.0551			−1.32171			−0.660854	+	0.750515i	$0.729806\pi$
$830$		0			0
$831$	2.30884			0.0800926
$832$		0			0
$833$			22.0269i			0.763188i
$834$		0			0
$835$	8.10399	+	8.33071i	0.280450	+	0.288296i
$836$		0			0
$837$		−	10.0535i		−	0.347499i
$838$		0			0
$839$	24.6946			0.852553			0.426277	−	0.904593i	$-0.359825\pi$
$839$	24.6946			0.852553			0.426277	+	0.904593i	$0.359825\pi$
$840$		0			0
$841$	−20.8547			−0.719126
$842$		0			0
$843$			1.10797i			0.0381607i
$844$		0			0
$845$	−18.3661	+	17.8663i	−0.631813	+	0.614619i
$846$		0			0
$847$		−	18.2131i		−	0.625811i
$848$		0			0
$849$	−2.03054			−0.0696880
$850$		0			0
$851$	6.92065			0.237237
$852$		0			0
$853$		−	23.4770i		−	0.803837i	−0.915675	−	0.401919i	$-0.868343\pi$
$853$		−	23.4770i		−	0.803837i	0.915675	−	0.401919i	$-0.131657\pi$
$854$		0			0
$855$	−3.76397	+	3.66153i	−0.128725	+	0.125222i
$856$		0			0
$857$			20.9632i			0.716090i	0.933704	+	0.358045i	$0.116557\pi$
$857$			20.9632i			0.716090i	−0.933704	+	0.358045i	$0.883443\pi$
$858$		0			0
$859$	35.7273			1.21900			0.609500	−	0.792786i	$-0.291370\pi$
$859$	35.7273			1.21900			0.609500	+	0.792786i	$0.291370\pi$
$860$		0			0
$861$	−1.45654			−0.0496388
$862$		0			0
$863$			19.5880i			0.666784i	0.942788	+	0.333392i	$0.108193\pi$
$863$			19.5880i			0.666784i	−0.942788	+	0.333392i	$0.891807\pi$
$864$		0			0
$865$	−4.08246	−	4.19668i	−0.138808	−	0.142691i
$866$		0			0
$867$		−	1.85403i		−	0.0629660i
$868$		0			0
$869$	1.18921			0.0403412
$870$		0			0
$871$	−8.64139			−0.292802
$872$		0			0
$873$		−	9.53702i		−	0.322779i
$874$		0			0
$875$	−13.6272	+	12.5436i	−0.460685	+	0.424051i
$876$		0			0
$877$		−	49.5791i		−	1.67417i	−0.547074	−	0.837084i	$-0.684259\pi$
$877$		−	49.5791i		−	1.67417i	0.547074	−	0.837084i	$-0.315741\pi$
$878$		0			0
$879$	−0.516230			−0.0174120
$880$		0			0
$881$	21.4406			0.722353			0.361177	−	0.932497i	$-0.382375\pi$
$881$	21.4406			0.722353			0.361177	+	0.932497i	$0.382375\pi$
$882$		0			0
$883$			52.7646i			1.77567i	0.460161	+	0.887836i	$0.347792\pi$
$883$			52.7646i			1.77567i	−0.460161	+	0.887836i	$0.652208\pi$
$884$		0			0
$885$	3.67851	+	3.78142i	0.123652	+	0.127111i
$886$		0			0
$887$			30.1462i			1.01221i	0.862472	+	0.506105i	$0.168915\pi$
$887$			30.1462i			1.01221i	−0.862472	+	0.506105i	$0.831085\pi$
$888$		0			0
$889$	−31.4917			−1.05620
$890$		0			0
$891$	−0.658978			−0.0220766
$892$		0			0
$893$		−	7.92520i		−	0.265207i
$894$		0			0
$895$	−1.00586	+	0.978482i	−0.0336221	+	0.0327071i
$896$		0			0
$897$			0.235102i			0.00784982i
$898$		0			0
$899$	−25.4039			−0.847266
$900$		0			0
$901$	−57.5998			−1.91893
$902$		0			0
$903$			0.549812i			0.0182966i
$904$		0			0
$905$	9.87080	−	9.60217i	0.328117	−	0.319187i
$906$		0			0
$907$			20.3987i			0.677329i	0.940907	+	0.338665i	$0.109975\pi$
$907$			20.3987i			0.677329i	−0.940907	+	0.338665i	$0.890025\pi$
$908$		0			0
$909$	−49.2221			−1.63259
$910$		0			0
$911$	57.7921			1.91474			0.957369	−	0.288869i	$-0.0932792\pi$
$911$	57.7921			1.91474			0.957369	+	0.288869i	$0.0932792\pi$
$912$		0			0
$913$		−	0.507619i		−	0.0167998i
$914$		0			0
$915$	3.54614	+	3.64534i	0.117232	+	0.120511i
$916$		0			0
$917$		−	7.10757i		−	0.234713i
$918$		0			0
$919$	−26.0255			−0.858502			−0.429251	−	0.903185i	$-0.641222\pi$
$919$	−26.0255			−0.858502			−0.429251	+	0.903185i	$0.641222\pi$
$920$		0			0
$921$	4.26055			0.140390
$922$		0			0
$923$			9.57972i			0.315320i
$924$		0			0
$925$	34.5901	+	0.954531i	1.13731	+	0.0313848i
$926$		0			0
$927$			21.6454i			0.710927i
$928$		0			0
$929$	−26.9839			−0.885313			−0.442656	−	0.896691i	$-0.645964\pi$
$929$	−26.9839			−0.885313			−0.442656	+	0.896691i	$0.645964\pi$
$930$		0			0
$931$	3.37159			0.110499
$932$		0			0
$933$		−	3.40112i		−	0.111348i
$934$		0			0
$935$	0.612787	+	0.629930i	0.0200403	+	0.0206009i
$936$		0			0
$937$			35.8201i			1.17019i	0.810964	+	0.585096i	$0.198943\pi$
$937$			35.8201i			1.17019i	−0.810964	+	0.585096i	$0.801057\pi$
$938$		0			0
$939$	−2.74079			−0.0894424
$940$		0			0
$941$	−32.0416			−1.04453			−0.522263	−	0.852785i	$-0.674912\pi$
$941$	−32.0416			−1.04453			−0.522263	+	0.852785i	$0.674912\pi$
$942$		0			0
$943$		−	4.64279i		−	0.151190i
$944$		0			0
$945$	2.99895	−	2.91734i	0.0975560	−	0.0949010i
$946$		0			0
$947$			8.53256i			0.277271i	0.990343	+	0.138635i	$0.0442716\pi$
$947$			8.53256i			0.277271i	−0.990343	+	0.138635i	$0.955728\pi$
$948$		0			0
$949$	−9.30445			−0.302035
$950$		0			0
$951$	−1.41594			−0.0459150
$952$		0			0
$953$		−	42.4054i		−	1.37365i	−0.726825	−	0.686823i	$-0.759005\pi$
$953$		−	42.4054i		−	1.37365i	0.726825	−	0.686823i	$-0.240995\pi$
$954$		0			0
$955$	−30.6684	+	29.8338i	−0.992406	+	0.965398i
$956$		0			0
$957$		−	0.0410397i		−	0.00132662i
$958$		0			0
$959$	20.2933			0.655305
$960$		0			0
$961$	48.2300			1.55581
$962$		0			0
$963$		−	1.05801i		−	0.0340938i
$964$		0			0
$965$	−1.26434	−	1.29971i	−0.0407005	−	0.0418391i
$966$		0			0
$967$		−	10.0988i		−	0.324755i	−0.986729	−	0.162377i	$-0.948084\pi$
$967$		−	10.0988i		−	0.324755i	0.986729	−	0.162377i	$-0.0519162\pi$
$968$		0			0
$969$	−0.776570			−0.0249470
$970$		0			0
$971$	15.7943			0.506863			0.253431	−	0.967353i	$-0.418441\pi$
$971$	15.7943			0.506863			0.253431	+	0.967353i	$0.418441\pi$
$972$		0			0
$973$			4.33099i			0.138845i
$974$		0			0
$975$	−0.0324265	+	1.17506i	−0.00103848	+	0.0376321i
$976$		0			0
$977$			27.0055i			0.863984i	0.901877	+	0.431992i	$0.142189\pi$
$977$			27.0055i			0.863984i	−0.901877	+	0.431992i	$0.857811\pi$
$978$		0			0
$979$	−0.230327			−0.00736128
$980$		0			0
$981$	−41.6043			−1.32832
$982$		0			0
$983$			16.0647i			0.512384i	0.966626	+	0.256192i	$0.0824679\pi$
$983$			16.0647i			0.512384i	−0.966626	+	0.256192i	$0.917532\pi$
$984$		0			0
$985$	9.59066	+	9.85897i	0.305584	+	0.314133i
$986$		0			0
$987$			3.13823i			0.0998910i
$988$		0			0
$989$	−1.75255			−0.0557279
$990$		0			0
$991$	−51.4433			−1.63415			−0.817075	−	0.576532i	$-0.804406\pi$
$991$	−51.4433			−1.63415			−0.817075	+	0.576532i	$0.804406\pi$
$992$		0			0
$993$			5.18066i			0.164403i
$994$		0			0
$995$	−11.2687	+	10.9620i	−0.357240	+	0.347518i
$996$		0			0
$997$			39.2639i			1.24350i	0.783215	+	0.621751i	$0.213578\pi$
$997$			39.2639i			1.24350i	−0.783215	+	0.621751i	$0.786422\pi$
$998$		0			0
$999$	−7.81660			−0.247306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.g.369.8		14
4.3	odd	2		920.2.e.b.369.7	✓	14
5.2	odd	4		9200.2.a.dc.1.3		7
5.3	odd	4		9200.2.a.cz.1.5		7
5.4	even	2	inner	1840.2.e.g.369.7		14
20.3	even	4		4600.2.a.bi.1.3		7
20.7	even	4		4600.2.a.bh.1.5		7
20.19	odd	2		920.2.e.b.369.8	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.7	✓	14	4.3	odd	2
920.2.e.b.369.8	yes	14	20.19	odd	2
1840.2.e.g.369.7		14	5.4	even	2	inner
1840.2.e.g.369.8		14	1.1	even	1	trivial
4600.2.a.bh.1.5		7	20.7	even	4
4600.2.a.bi.1.3		7	20.3	even	4
9200.2.a.cz.1.5		7	5.3	odd	4
9200.2.a.dc.1.3		7	5.2	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.189375i q^{3} +(-1.60280 + 1.55918i) q^{5} +1.65661i q^{7} +2.96414 q^{9} +O(q^{10})\)	\(q+0.189375i q^{3} +(-1.60280 + 1.55918i) q^{5} +1.65661i q^{7} +2.96414 q^{9} -0.0759321 q^{11} +1.24146i q^{13} +(-0.295270 - 0.303531i) q^{15} +5.17593i q^{17} +0.792262 q^{19} -0.313721 q^{21} -1.00000i q^{23} +(0.137925 - 4.99810i) q^{25} +1.12946i q^{27} +2.85401 q^{29} -8.90112 q^{31} -0.0143797i q^{33} +(-2.58295 - 2.65521i) q^{35} +6.92065i q^{37} -0.235102 q^{39} +4.64279 q^{41} -1.75255i q^{43} +(-4.75091 + 4.62162i) q^{45} -10.0033i q^{47} +4.25565 q^{49} -0.980193 q^{51} +11.1284i q^{53} +(0.121704 - 0.118392i) q^{55} +0.150035i q^{57} -12.4581 q^{59} -12.0098 q^{61} +4.91041i q^{63} +(-1.93566 - 1.98981i) q^{65} +6.96067i q^{67} +0.189375 q^{69} +7.71650 q^{71} +7.49476i q^{73} +(0.946517 + 0.0261196i) q^{75} -0.125790i q^{77} -15.6615 q^{79} +8.67852 q^{81} +6.68518i q^{83} +(-8.07019 - 8.29597i) q^{85} +0.540478i q^{87} +3.03333 q^{89} -2.05661 q^{91} -1.68565i q^{93} +(-1.26984 + 1.23528i) q^{95} -3.21747i q^{97} -0.225073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) \) 14 * q + 2 * q^5 - 4 * q^9	\( 14 q + 2 q^{5} - 4 q^{9} + 14 q^{11} + 6 q^{15} - 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} + 20 q^{31} + 2 q^{35} - 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} + 14 q^{51} + 38 q^{55} - 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} + 28 q^{71} + 24 q^{75} - 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} + 14 q^{91} + 30 q^{95} - 122 q^{99}+O(q^{100}) \) 14 * q + 2 * q^5 - 4 * q^9 + 14 * q^11 + 6 * q^15 - 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 + 20 * q^31 + 2 * q^35 - 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 + 14 * q^51 + 38 * q^55 - 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 + 28 * q^71 + 24 * q^75 - 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 + 14 * q^91 + 30 * q^95 - 122 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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