LMFDB - Embedded newform 3420.3.h.e.2089.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,3,Mod(2089,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3420.2089");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3420.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$93.1882504112$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241 $ x^12 + 20x^10 + 44x^8 - 270x^6 + 36676x^4 - 71664*x^2 + 687241
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19}\cdot 3 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2089.10
Root			$-0.975196 - 4.19028i$ of defining polynomial
Character	$\chi$	$=$	3420.2089
Dual form			3420.3.h.e.2089.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.75473 - 4.17271i) q^{5} -2.18749i q^{7} +O(q^{10})$	$q+(2.75473 - 4.17271i) q^{5} -2.18749i q^{7} +9.50946 q^{11} +20.4976 q^{13} +6.15792i q^{17} +(9.68651 - 16.3454i) q^{19} +10.9375i q^{23} +(-9.82294 - 22.9893i) q^{25} -28.4243i q^{29} -24.8098i q^{31} +(-9.12776 - 6.02594i) q^{35} +37.4397 q^{37} +81.6583i q^{41} -70.8743i q^{43} -46.1193i q^{47} +44.2149 q^{49} +32.1999 q^{53} +(26.1960 - 39.6802i) q^{55} +94.3206i q^{59} -102.566 q^{61} +(56.4652 - 85.5303i) q^{65} +69.0282 q^{67} +40.5718i q^{71} +102.209i q^{73} -20.8018i q^{77} +73.9145i q^{79} +91.9574i q^{83} +(25.6952 + 16.9634i) q^{85} -57.3634i q^{89} -44.8382i q^{91} +(-41.5207 - 85.4461i) q^{95} -84.3229 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{5}+O(q^{10}) $ 12 * q - 12 * q^5	$ 12 q - 12 q^{5} + 24 q^{11} + 68 q^{19} - 76 q^{25} + 32 q^{35} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 408 q^{85} - 124 q^{95}+O(q^{100}) $ 12 * q - 12 * q^5 + 24 * q^11 + 68 * q^19 - 76 * q^25 + 32 * q^35 + 212 * q^49 + 176 * q^55 - 600 * q^61 + 408 * q^85 - 124 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	2.75473	−	4.17271i	0.550946	−	0.834541i
$5$	2.75473	−	4.17271i	0.550946	−	0.834541i
$6$		0			0
$7$		−	2.18749i		−	0.312499i	−0.987718	−	0.156249i	$-0.950060\pi$
$7$		−	2.18749i		−	0.312499i	0.987718	−	0.156249i	$-0.0499403\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	9.50946			0.864496			0.432248	−	0.901755i	$-0.357720\pi$
$11$	9.50946			0.864496			0.432248	+	0.901755i	$0.357720\pi$
$12$		0			0
$13$	20.4976			1.57674			0.788368	−	0.615204i	$-0.210926\pi$
$13$	20.4976			1.57674			0.788368	+	0.615204i	$0.210926\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.15792i			0.362231i	0.983462	+	0.181115i	$0.0579707\pi$
$17$			6.15792i			0.362231i	−0.983462	+	0.181115i	$0.942029\pi$
$18$		0			0
$19$	9.68651	−	16.3454i	0.509816	−	0.860283i
$19$	9.68651	−	16.3454i	0.509816	−	0.860283i
$20$		0			0
$21$		0			0
$22$		0			0
$23$			10.9375i			0.475541i	0.971321	+	0.237771i	$0.0764167\pi$
$23$			10.9375i			0.475541i	−0.971321	+	0.237771i	$0.923583\pi$
$24$		0			0
$25$	−9.82294	−	22.9893i	−0.392918	−	0.919574i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	28.4243i		−	0.980148i	−0.871681	−	0.490074i	$-0.836970\pi$
$29$		−	28.4243i		−	0.980148i	0.871681	−	0.490074i	$-0.163030\pi$
$30$		0			0
$31$		−	24.8098i		−	0.800315i	−0.916446	−	0.400157i	$-0.868955\pi$
$31$		−	24.8098i		−	0.800315i	0.916446	−	0.400157i	$-0.131045\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.12776	−	6.02594i	−0.260793	−	0.172170i
$36$		0			0
$37$	37.4397			1.01188			0.505941	−	0.862568i	$-0.331145\pi$
$37$	37.4397			1.01188			0.505941	+	0.862568i	$0.331145\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			81.6583i			1.99167i	0.0911903	+	0.995833i	$0.470933\pi$
$41$			81.6583i			1.99167i	−0.0911903	+	0.995833i	$0.529067\pi$
$42$		0			0
$43$		−	70.8743i		−	1.64824i	−0.566415	−	0.824120i	$-0.691670\pi$
$43$		−	70.8743i		−	1.64824i	0.566415	−	0.824120i	$-0.308330\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	46.1193i		−	0.981261i	−0.871368	−	0.490631i	$-0.836766\pi$
$47$		−	46.1193i		−	0.981261i	0.871368	−	0.490631i	$-0.163234\pi$
$48$		0			0
$49$	44.2149			0.902345
$50$		0			0
$51$		0			0
$52$		0			0
$53$	32.1999			0.607546			0.303773	−	0.952745i	$-0.401754\pi$
$53$	32.1999			0.607546			0.303773	+	0.952745i	$0.401754\pi$
$54$		0			0
$55$	26.1960	−	39.6802i	0.476290	−	0.721458i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			94.3206i			1.59865i	0.600896	+	0.799327i	$0.294811\pi$
$59$			94.3206i			1.59865i	−0.600896	+	0.799327i	$0.705189\pi$
$60$		0			0
$61$	−102.566			−1.68141			−0.840707	−	0.541491i	$-0.817860\pi$
$61$	−102.566			−1.68141			−0.840707	+	0.541491i	$0.817860\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	56.4652	−	85.5303i	0.868696	−	1.31585i
$66$		0			0
$67$	69.0282			1.03027			0.515136	−	0.857109i	$-0.327742\pi$
$67$	69.0282			1.03027			0.515136	+	0.857109i	$0.327742\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			40.5718i			0.571433i	0.958314	+	0.285717i	$0.0922316\pi$
$71$			40.5718i			0.571433i	−0.958314	+	0.285717i	$0.907768\pi$
$72$		0			0
$73$			102.209i			1.40012i	0.714082	+	0.700062i	$0.246844\pi$
$73$			102.209i			1.40012i	−0.714082	+	0.700062i	$0.753156\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	20.8018i		−	0.270154i
$78$		0			0
$79$			73.9145i			0.935626i	0.883827	+	0.467813i	$0.154958\pi$
$79$			73.9145i			0.935626i	−0.883827	+	0.467813i	$0.845042\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			91.9574i			1.10792i	0.832543	+	0.553960i	$0.186884\pi$
$83$			91.9574i			1.10792i	−0.832543	+	0.553960i	$0.813116\pi$
$84$		0			0
$85$	25.6952	+	16.9634i	0.302296	+	0.199569i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	57.3634i		−	0.644533i	−0.946649	−	0.322266i	$-0.895555\pi$
$89$		−	57.3634i		−	0.644533i	0.946649	−	0.322266i	$-0.104445\pi$
$90$		0			0
$91$		−	44.8382i		−	0.492728i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−41.5207	−	85.4461i	−0.437060	−	0.899432i
$96$		0			0
$97$	−84.3229			−0.869308			−0.434654	−	0.900598i	$-0.643129\pi$
$97$	−84.3229			−0.869308			−0.434654	+	0.900598i	$0.643129\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−69.2688			−0.685830			−0.342915	−	0.939366i	$-0.611414\pi$
$101$	−69.2688			−0.685830			−0.342915	+	0.939366i	$0.611414\pi$
$102$		0			0
$103$	100.192			0.972739			0.486369	−	0.873753i	$-0.338321\pi$
$103$	100.192			0.972739			0.486369	+	0.873753i	$0.338321\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	52.7344			0.492845			0.246422	−	0.969163i	$-0.420745\pi$
$107$	52.7344			0.492845			0.246422	+	0.969163i	$0.420745\pi$
$108$		0			0
$109$			76.7400i			0.704036i	0.935993	+	0.352018i	$0.114504\pi$
$109$			76.7400i			0.704036i	−0.935993	+	0.352018i	$0.885496\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	210.252			1.86064			0.930320	−	0.366749i	$-0.119529\pi$
$113$	210.252			1.86064			0.930320	+	0.366749i	$0.119529\pi$
$114$		0			0
$115$	45.6388	+	30.1297i	0.396859	+	0.261998i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	13.4704			0.113197
$120$		0			0
$121$	−30.5702			−0.252647
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−122.987	−	22.3411i	−0.983898	−	0.178729i
$126$		0			0
$127$	19.9230			0.156874			0.0784371	−	0.996919i	$-0.475007\pi$
$127$	19.9230			0.156874			0.0784371	+	0.996919i	$0.475007\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.29737			0.0251708			0.0125854	−	0.999921i	$-0.495994\pi$
$131$	3.29737			0.0251708			0.0125854	+	0.999921i	$0.495994\pi$
$132$		0			0
$133$	−35.7554	−	21.1892i	−0.268837	−	0.159317i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	148.188i		−	1.08166i	−0.841131	−	0.540831i	$-0.818110\pi$
$137$		−	148.188i		−	1.08166i	0.841131	−	0.540831i	$-0.181890\pi$
$138$		0			0
$139$	−54.2729			−0.390452			−0.195226	−	0.980758i	$-0.562544\pi$
$139$	−54.2729			−0.390452			−0.195226	+	0.980758i	$0.562544\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	194.921			1.36308
$144$		0			0
$145$	−118.606	−	78.3012i	−0.817974	−	0.540008i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−160.700			−1.07852			−0.539262	−	0.842138i	$-0.681297\pi$
$149$	−160.700			−1.07852			−0.539262	+	0.842138i	$0.681297\pi$
$150$		0			0
$151$		−	285.273i		−	1.88922i	−0.328192	−	0.944611i	$-0.606439\pi$
$151$		−	285.273i		−	1.88922i	0.328192	−	0.944611i	$-0.393561\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−103.524	−	68.3441i	−0.667896	−	0.440930i
$156$		0			0
$157$		−	189.486i		−	1.20692i	−0.797394	−	0.603459i	$-0.793789\pi$
$157$		−	189.486i		−	1.20692i	0.797394	−	0.603459i	$-0.206211\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	23.9256			0.148606
$162$		0			0
$163$		−	152.943i		−	0.938302i	−0.883118	−	0.469151i	$-0.844560\pi$
$163$		−	152.943i		−	0.938302i	0.883118	−	0.469151i	$-0.155440\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−89.7126			−0.537201			−0.268601	−	0.963252i	$-0.586561\pi$
$167$	−89.7126			−0.537201			−0.268601	+	0.963252i	$0.586561\pi$
$168$		0			0
$169$	251.150			1.48610
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−154.463			−0.892851			−0.446425	−	0.894821i	$-0.647303\pi$
$173$	−154.463			−0.892851			−0.446425	+	0.894821i	$0.647303\pi$
$174$		0			0
$175$	−50.2890	+	21.4876i	−0.287366	+	0.122786i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	138.885i		−	0.775892i	−0.921682	−	0.387946i	$-0.873185\pi$
$179$		−	138.885i		−	0.775892i	0.921682	−	0.387946i	$-0.126815\pi$
$180$		0			0
$181$		−	106.605i		−	0.588979i	−0.955655	−	0.294490i	$-0.904850\pi$
$181$		−	106.605i		−	0.588979i	0.955655	−	0.294490i	$-0.0951496\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	103.136	−	156.225i	0.557493	−	0.844458i
$186$		0			0
$187$			58.5585i			0.313147i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	262.954			1.37672			0.688362	−	0.725367i	$-0.258330\pi$
$191$	262.954			1.37672			0.688362	+	0.725367i	$0.258330\pi$
$192$		0			0
$193$	−34.0711			−0.176534			−0.0882670	−	0.996097i	$-0.528133\pi$
$193$	−34.0711			−0.176534			−0.0882670	+	0.996097i	$0.528133\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			293.544i			1.49007i	0.667026	+	0.745035i	$0.267567\pi$
$197$			293.544i			1.49007i	−0.667026	+	0.745035i	$0.732433\pi$
$198$		0			0
$199$	−144.876			−0.728018			−0.364009	−	0.931395i	$-0.618592\pi$
$199$	−144.876			−0.728018			−0.364009	+	0.931395i	$0.618592\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−62.1779			−0.306295
$204$		0			0
$205$	340.736	+	224.947i	1.66213	+	1.09730i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	92.1135	−	155.436i	0.440734	−	0.743711i
$210$		0			0
$211$		−	355.298i		−	1.68388i	−0.539573	−	0.841939i	$-0.681414\pi$
$211$		−	355.298i		−	1.68388i	0.539573	−	0.841939i	$-0.318586\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−295.738	−	195.239i	−1.37552	−	0.908091i
$216$		0			0
$217$	−54.2711			−0.250097
$218$		0			0
$219$		0			0
$220$		0			0
$221$			126.222i			0.571142i
$222$		0			0
$223$	330.367			1.48147			0.740734	−	0.671798i	$-0.234478\pi$
$223$	330.367			1.48147			0.740734	+	0.671798i	$0.234478\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	23.4416			0.103267			0.0516335	−	0.998666i	$-0.483557\pi$
$227$	23.4416			0.103267			0.0516335	+	0.998666i	$0.483557\pi$
$228$		0			0
$229$	−183.710			−0.802229			−0.401114	−	0.916028i	$-0.631377\pi$
$229$	−183.710			−0.802229			−0.401114	+	0.916028i	$0.631377\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			143.648i			0.616515i	0.951303	+	0.308258i	$0.0997459\pi$
$233$			143.648i			0.616515i	−0.951303	+	0.308258i	$0.900254\pi$
$234$		0			0
$235$	−192.442	−	127.046i	−0.818903	−	0.540622i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	79.5051			0.332657			0.166329	−	0.986070i	$-0.446809\pi$
$239$	79.5051			0.332657			0.166329	+	0.986070i	$0.446809\pi$
$240$		0			0
$241$		−	122.504i		−	0.508317i	−0.967163	−	0.254158i	$-0.918202\pi$
$241$		−	122.504i		−	0.508317i	0.967163	−	0.254158i	$-0.0817984\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	121.800	−	184.496i	0.497143	−	0.753044i
$246$		0			0
$247$	198.550	−	335.040i	0.803846	−	1.35644i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−19.2335			−0.0766274			−0.0383137	−	0.999266i	$-0.512199\pi$
$251$	−19.2335			−0.0766274			−0.0383137	+	0.999266i	$0.512199\pi$
$252$		0			0
$253$			104.009i			0.411104i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−60.4198			−0.235097			−0.117548	−	0.993067i	$-0.537503\pi$
$257$	−60.4198			−0.235097			−0.117548	+	0.993067i	$0.537503\pi$
$258$		0			0
$259$		−	81.8989i		−	0.316212i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			23.1886i			0.0881694i	0.999028	+	0.0440847i	$0.0140371\pi$
$263$			23.1886i			0.0881694i	−0.999028	+	0.0440847i	$0.985963\pi$
$264$		0			0
$265$	88.7020	−	134.361i	0.334725	−	0.507022i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			211.221i			0.785208i	0.919708	+	0.392604i	$0.128426\pi$
$269$			211.221i			0.785208i	−0.919708	+	0.392604i	$0.871574\pi$
$270$		0			0
$271$	−371.216			−1.36980			−0.684901	−	0.728637i	$-0.740154\pi$
$271$	−371.216			−1.36980			−0.684901	+	0.728637i	$0.740154\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−93.4109	−	218.616i	−0.339676	−	0.794968i
$276$		0			0
$277$		−	6.90841i		−	0.0249401i	−0.999922	−	0.0124701i	$-0.996031\pi$
$277$		−	6.90841i		−	0.0249401i	0.999922	−	0.0124701i	$-0.00396944\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	7.46967i		−	0.0265825i	−0.999912	−	0.0132912i	$-0.995769\pi$
$281$		−	7.46967i		−	0.0265825i	0.999912	−	0.0132912i	$-0.00423086\pi$
$282$		0			0
$283$			64.0548i			0.226342i	0.993576	+	0.113171i	$0.0361008\pi$
$283$			64.0548i			0.226342i	−0.993576	+	0.113171i	$0.963899\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	178.627			0.622393
$288$		0			0
$289$	251.080			0.868789
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−527.510			−1.80038			−0.900188	−	0.435502i	$-0.856571\pi$
$293$	−527.510			−1.80038			−0.900188	+	0.435502i	$0.856571\pi$
$294$		0			0
$295$	393.572	+	259.828i	1.33414	+	0.880772i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			224.191i			0.749803i
$300$		0			0
$301$	−155.037			−0.515073
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−282.542	+	427.979i	−0.926367	+	1.40321i
$306$		0			0
$307$	385.063			1.25428			0.627139	−	0.778908i	$-0.284226\pi$
$307$	385.063			1.25428			0.627139	+	0.778908i	$0.284226\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−107.903			−0.346956			−0.173478	−	0.984838i	$-0.555501\pi$
$311$	−107.903			−0.346956			−0.173478	+	0.984838i	$0.555501\pi$
$312$		0			0
$313$		−	311.071i		−	0.993838i	−0.867797	−	0.496919i	$-0.834465\pi$
$313$		−	311.071i		−	0.993838i	0.867797	−	0.496919i	$-0.165535\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−213.771			−0.674356			−0.337178	−	0.941441i	$-0.609472\pi$
$317$	−213.771			−0.674356			−0.337178	+	0.941441i	$0.609472\pi$
$318$		0			0
$319$		−	270.300i		−	0.847334i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	100.654	+	59.6488i	0.311621	+	0.184671i
$324$		0			0
$325$	−201.346	−	471.225i	−0.619527	−	1.44992i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−100.886			−0.306643
$330$		0			0
$331$			248.075i			0.749471i	0.927132	+	0.374735i	$0.122266\pi$
$331$			248.075i			0.749471i	−0.927132	+	0.374735i	$0.877734\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	190.154	−	288.034i	0.567623	−	0.859804i
$336$		0			0
$337$	160.776			0.477080			0.238540	−	0.971133i	$-0.423331\pi$
$337$	160.776			0.477080			0.238540	+	0.971133i	$0.423331\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	235.927i		−	0.691869i
$342$		0			0
$343$		−	203.907i		−	0.594480i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			245.342i			0.707038i	0.935427	+	0.353519i	$0.115015\pi$
$347$			245.342i			0.707038i	−0.935427	+	0.353519i	$0.884985\pi$
$348$		0			0
$349$	−220.492			−0.631782			−0.315891	−	0.948796i	$-0.602303\pi$
$349$	−220.492			−0.631782			−0.315891	+	0.948796i	$0.602303\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	414.946i		−	1.17548i	−0.809048	−	0.587742i	$-0.800017\pi$
$353$		−	414.946i		−	1.17548i	0.809048	−	0.587742i	$-0.199983\pi$
$354$		0			0
$355$	169.294	+	111.764i	0.476885	+	0.314829i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	130.150			0.362533			0.181267	−	0.983434i	$-0.441980\pi$
$359$	130.150			0.362533			0.181267	+	0.983434i	$0.441980\pi$
$360$		0			0
$361$	−173.343	−	316.659i	−0.480174	−	0.877173i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	426.488	+	281.558i	1.16846	+	0.771392i
$366$		0			0
$367$			625.628i			1.70471i	0.522965	+	0.852354i	$0.324826\pi$
$367$			625.628i			1.70471i	−0.522965	+	0.852354i	$0.675174\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	70.4370i		−	0.189857i
$372$		0			0
$373$	282.836			0.758273			0.379137	−	0.925341i	$-0.376221\pi$
$373$	282.836			0.758273			0.379137	+	0.925341i	$0.376221\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	582.629i		−	1.54543i
$378$		0			0
$379$			60.1889i			0.158810i	0.996842	+	0.0794049i	$0.0253020\pi$
$379$			60.1889i			0.158810i	−0.996842	+	0.0794049i	$0.974698\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	747.259			1.95107			0.975534	−	0.219848i	$-0.0705563\pi$
$383$	747.259			1.95107			0.975534	+	0.219848i	$0.0705563\pi$
$384$		0			0
$385$	−86.8000	−	57.3034i	−0.225455	−	0.148840i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−160.723			−0.413170			−0.206585	−	0.978429i	$-0.566235\pi$
$389$	−160.723			−0.413170			−0.206585	+	0.978429i	$0.566235\pi$
$390$		0			0
$391$	−67.3520			−0.172256
$392$		0			0
$393$		0			0
$394$		0			0
$395$	308.423	+	203.614i	0.780818	+	0.515479i
$396$		0			0
$397$			394.796i			0.994449i	0.867622	+	0.497224i	$0.165647\pi$
$397$			394.796i			0.994449i	−0.867622	+	0.497224i	$0.834353\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	206.496i		−	0.514953i	−0.966285	−	0.257477i	$-0.917109\pi$
$401$		−	206.496i		−	0.514953i	0.966285	−	0.257477i	$-0.0828910\pi$
$402$		0			0
$403$		−	508.540i		−	1.26188i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	356.031			0.874769
$408$		0			0
$409$			588.126i			1.43796i	0.695031	+	0.718980i	$0.255391\pi$
$409$			588.126i			1.43796i	−0.695031	+	0.718980i	$0.744609\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	206.326			0.499578
$414$		0			0
$415$	383.711	+	253.318i	0.924605	+	0.610404i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	573.699			1.36921			0.684605	−	0.728914i	$-0.259975\pi$
$419$	573.699			1.36921			0.684605	+	0.728914i	$0.259975\pi$
$420$		0			0
$421$			17.8321i			0.0423566i	0.999776	+	0.0211783i	$0.00674176\pi$
$421$			17.8321i			0.0423566i	−0.999776	+	0.0211783i	$0.993258\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	141.567	−	60.4889i	0.333098	−	0.142327i
$426$		0			0
$427$			224.363i			0.525439i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	424.809i		−	0.985636i	−0.870132	−	0.492818i	$-0.835967\pi$
$431$		−	424.809i		−	0.985636i	0.870132	−	0.492818i	$-0.164033\pi$
$432$		0			0
$433$	−309.312			−0.714346			−0.357173	−	0.934038i	$-0.616259\pi$
$433$	−309.312			−0.714346			−0.357173	+	0.934038i	$0.616259\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	178.777	+	105.946i	0.409100	+	0.242439i
$438$		0			0
$439$		−	126.062i		−	0.287158i	−0.989639	−	0.143579i	$-0.954139\pi$
$439$		−	126.062i		−	0.287158i	0.989639	−	0.143579i	$-0.0458612\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	602.717i		−	1.36054i	−0.732964	−	0.680268i	$-0.761863\pi$
$443$		−	602.717i		−	1.36054i	0.732964	−	0.680268i	$-0.238137\pi$
$444$		0			0
$445$	−239.361	−	158.021i	−0.537889	−	0.355102i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			174.732i			0.389157i	0.980887	+	0.194579i	$0.0623339\pi$
$449$			174.732i			0.389157i	−0.980887	+	0.194579i	$0.937666\pi$
$450$		0			0
$451$			776.527i			1.72179i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−187.097	−	123.517i	−0.411202	−	0.271466i
$456$		0			0
$457$		−	766.792i		−	1.67788i	−0.544223	−	0.838941i	$-0.683175\pi$
$457$		−	766.792i		−	1.67788i	0.544223	−	0.838941i	$-0.316825\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	192.820			0.418264			0.209132	−	0.977887i	$-0.432936\pi$
$461$	192.820			0.418264			0.209132	+	0.977887i	$0.432936\pi$
$462$		0			0
$463$		−	440.773i		−	0.951994i	−0.879447	−	0.475997i	$-0.842087\pi$
$463$		−	440.773i		−	0.951994i	0.879447	−	0.475997i	$-0.157913\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	450.985i		−	0.965707i	−0.875701	−	0.482853i	$-0.839600\pi$
$467$		−	450.985i		−	0.965707i	0.875701	−	0.482853i	$-0.160400\pi$
$468$		0			0
$469$		−	150.998i		−	0.321958i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	673.976i		−	1.42490i
$474$		0			0
$475$	−470.920	−	62.1268i	−0.991410	−	0.130793i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−493.345			−1.02995			−0.514974	−	0.857206i	$-0.672198\pi$
$479$	−493.345			−1.02995			−0.514974	+	0.857206i	$0.672198\pi$
$480$		0			0
$481$	767.422			1.59547
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−232.287	+	351.854i	−0.478941	+	0.725473i
$486$		0			0
$487$	111.096			0.228124			0.114062	−	0.993474i	$-0.463614\pi$
$487$	111.096			0.228124			0.114062	+	0.993474i	$0.463614\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−671.679			−1.36798			−0.683990	−	0.729491i	$-0.739757\pi$
$491$	−671.679			−1.36798			−0.683990	+	0.729491i	$0.739757\pi$
$492$		0			0
$493$	175.035			0.355040
$494$		0			0
$495$		0			0
$496$		0			0
$497$	88.7504			0.178572
$498$		0			0
$499$	−813.880			−1.63102			−0.815511	−	0.578741i	$-0.803544\pi$
$499$	−813.880			−1.63102			−0.815511	+	0.578741i	$0.803544\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	901.318i		−	1.79189i	−0.444170	−	0.895943i	$-0.646501\pi$
$503$		−	901.318i		−	1.79189i	0.444170	−	0.895943i	$-0.353499\pi$
$504$		0			0
$505$	−190.817	+	289.038i	−0.377855	+	0.572353i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	92.1471i		−	0.181035i	−0.995895	−	0.0905177i	$-0.971148\pi$
$509$		−	92.1471i		−	0.181035i	0.995895	−	0.0905177i	$-0.0288522\pi$
$510$		0			0
$511$	223.581			0.437537
$512$		0			0
$513$		0			0
$514$		0			0
$515$	276.002	−	418.072i	0.535926	−	0.811790i
$516$		0			0
$517$		−	438.569i		−	0.848297i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	235.653i		−	0.452309i	−0.974091	−	0.226155i	$-0.927385\pi$
$521$		−	235.653i		−	0.452309i	0.974091	−	0.226155i	$-0.0726154\pi$
$522$		0			0
$523$	−290.076			−0.554639			−0.277320	−	0.960778i	$-0.589446\pi$
$523$	−290.076			−0.554639			−0.277320	+	0.960778i	$0.589446\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	152.776			0.289898
$528$		0			0
$529$	409.372			0.773860
$530$		0			0
$531$		0			0
$532$		0			0
$533$			1673.80i			3.14033i
$534$		0			0
$535$	145.269	−	220.045i	0.271531	−	0.411299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	420.460			0.780073
$540$		0			0
$541$	352.993			0.652483			0.326242	−	0.945286i	$-0.394218\pi$
$541$	352.993			0.652483			0.326242	+	0.945286i	$0.394218\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	320.213	+	211.398i	0.587547	+	0.387886i
$546$		0			0
$547$	571.004			1.04388			0.521942	−	0.852981i	$-0.325208\pi$
$547$	571.004			1.04388			0.521942	+	0.852981i	$0.325208\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−464.606	−	275.332i	−0.843205	−	0.499696i
$552$		0			0
$553$	161.687			0.292382
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	245.730i		−	0.441166i	−0.975368	−	0.220583i	$-0.929204\pi$
$557$		−	245.730i		−	0.441166i	0.975368	−	0.220583i	$-0.0707960\pi$
$558$		0			0
$559$		−	1452.75i		−	2.59884i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	623.923			1.10821			0.554106	−	0.832446i	$-0.313060\pi$
$563$	623.923			1.10821			0.554106	+	0.832446i	$0.313060\pi$
$564$		0			0
$565$	579.188	−	877.321i	1.02511	−	1.55278i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			286.095i			0.502804i	0.967883	+	0.251402i	$0.0808915\pi$
$569$			286.095i			0.502804i	−0.967883	+	0.251402i	$0.919108\pi$
$570$		0			0
$571$	213.683			0.374226			0.187113	−	0.982338i	$-0.440087\pi$
$571$	213.683			0.374226			0.187113	+	0.982338i	$0.440087\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	251.445	−	107.438i	0.437295	−	0.186849i
$576$		0			0
$577$			521.531i			0.903866i	0.892052	+	0.451933i	$0.149265\pi$
$577$			521.531i			0.903866i	−0.892052	+	0.451933i	$0.850735\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	201.156			0.346224
$582$		0			0
$583$	306.204			0.525221
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	195.026i		−	0.332243i	−0.986105	−	0.166121i	$-0.946876\pi$
$587$		−	195.026i		−	0.332243i	0.986105	−	0.166121i	$-0.0531244\pi$
$588$		0			0
$589$	−405.525	−	240.320i	−0.688497	−	0.408014i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			333.768i			0.562847i	0.959584	+	0.281423i	$0.0908065\pi$
$593$			333.768i			0.562847i	−0.959584	+	0.281423i	$0.909193\pi$
$594$		0			0
$595$	37.1073	−	56.2080i	0.0623652	−	0.0944672i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			556.121i			0.928415i	0.885726	+	0.464208i	$0.153661\pi$
$599$			556.121i			0.928415i	−0.885726	+	0.464208i	$0.846339\pi$
$600$		0			0
$601$			1015.58i			1.68981i	0.534916	+	0.844905i	$0.320343\pi$
$601$			1015.58i			1.68981i	−0.534916	+	0.844905i	$0.679657\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−84.2127	+	127.561i	−0.139195	+	0.210844i
$606$		0			0
$607$	−916.408			−1.50973			−0.754867	−	0.655878i	$-0.772298\pi$
$607$	−916.408			−1.50973			−0.754867	+	0.655878i	$0.772298\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	945.333i		−	1.54719i
$612$		0			0
$613$		−	265.921i		−	0.433802i	−0.976194	−	0.216901i	$-0.930405\pi$
$613$		−	265.921i		−	0.433802i	0.976194	−	0.216901i	$-0.0695949\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			24.9941i			0.0405091i	0.999795	+	0.0202545i	$0.00644766\pi$
$617$			24.9941i			0.0405091i	−0.999795	+	0.0202545i	$0.993552\pi$
$618$		0			0
$619$	52.7690			0.0852488			0.0426244	−	0.999091i	$-0.486428\pi$
$619$	52.7690			0.0852488			0.0426244	+	0.999091i	$0.486428\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−125.482			−0.201416
$624$		0			0
$625$	−432.020	+	451.646i	−0.691231	+	0.722634i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			230.551i			0.366535i
$630$		0			0
$631$	−439.188			−0.696019			−0.348010	−	0.937491i	$-0.613142\pi$
$631$	−439.188			−0.696019			−0.348010	+	0.937491i	$0.613142\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	54.8825	−	83.1329i	0.0864292	−	0.130918i
$636$		0			0
$637$	906.297			1.42276
$638$		0			0
$639$		0			0
$640$		0			0
$641$			795.892i			1.24164i	0.783953	+	0.620821i	$0.213201\pi$
$641$			795.892i			1.24164i	−0.783953	+	0.620821i	$0.786799\pi$
$642$		0			0
$643$			927.013i			1.44170i	0.693092	+	0.720850i	$0.256248\pi$
$643$			927.013i			1.44170i	−0.693092	+	0.720850i	$0.743752\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			128.287i			0.198280i	0.995073	+	0.0991401i	$0.0316092\pi$
$647$			128.287i			0.198280i	−0.995073	+	0.0991401i	$0.968391\pi$
$648$		0			0
$649$			896.938i			1.38203i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			745.488i			1.14164i	0.821077	+	0.570818i	$0.193374\pi$
$653$			745.488i			1.14164i	−0.821077	+	0.570818i	$0.806626\pi$
$654$		0			0
$655$	9.08337	−	13.7590i	0.0138677	−	0.0210061i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			563.842i			0.855602i	0.903873	+	0.427801i	$0.140712\pi$
$659$			563.842i			0.855602i	−0.903873	+	0.427801i	$0.859288\pi$
$660$		0			0
$661$		−	219.673i		−	0.332335i	−0.986098	−	0.166167i	$-0.946861\pi$
$661$		−	219.673i		−	0.332335i	0.986098	−	0.166167i	$-0.0531392\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−186.912	+	90.8263i	−0.281071	+	0.136581i
$666$		0			0
$667$	310.889			0.466101
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−975.349			−1.45357
$672$		0			0
$673$	−145.649			−0.216418			−0.108209	−	0.994128i	$-0.534512\pi$
$673$	−145.649			−0.216418			−0.108209	+	0.994128i	$0.534512\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1031.23			−1.52323			−0.761616	−	0.648029i	$-0.775594\pi$
$677$	−1031.23			−1.52323			−0.761616	+	0.648029i	$0.775594\pi$
$678$		0			0
$679$			184.455i			0.271658i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1266.00			−1.85358			−0.926791	−	0.375577i	$-0.877445\pi$
$683$	−1266.00			−1.85358			−0.926791	+	0.375577i	$0.877445\pi$
$684$		0			0
$685$	−618.344	−	408.217i	−0.902692	−	0.595937i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	660.020			0.957939
$690$		0			0
$691$	−458.754			−0.663899			−0.331949	−	0.943297i	$-0.607706\pi$
$691$	−458.754			−0.663899			−0.331949	+	0.943297i	$0.607706\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−149.507	+	226.465i	−0.215118	+	0.325848i
$696$		0			0
$697$	−502.846			−0.721443
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−790.423			−1.12756			−0.563782	−	0.825923i	$-0.690654\pi$
$701$	−790.423			−1.12756			−0.563782	+	0.825923i	$0.690654\pi$
$702$		0			0
$703$	362.660	−	611.966i	0.515875	−	0.870506i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			151.525i			0.214321i
$708$		0			0
$709$	−131.077			−0.184875			−0.0924377	−	0.995718i	$-0.529466\pi$
$709$	−131.077			−0.184875			−0.0924377	+	0.995718i	$0.529466\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	271.356			0.380583
$714$		0			0
$715$	536.954	−	813.347i	0.750984	−	1.13755i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	820.977			1.14183			0.570916	−	0.821008i	$-0.306588\pi$
$719$	820.977			1.14183			0.570916	+	0.821008i	$0.306588\pi$
$720$		0			0
$721$		−	219.169i		−	0.303980i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−653.456	+	279.210i	−0.901318	+	0.385118i
$726$		0			0
$727$		−	1121.14i		−	1.54214i	−0.636749	−	0.771071i	$-0.719721\pi$
$727$		−	1121.14i		−	1.54214i	0.636749	−	0.771071i	$-0.280279\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	436.438			0.597043
$732$		0			0
$733$		−	97.7494i		−	0.133355i	−0.997775	−	0.0666777i	$-0.978760\pi$
$733$		−	97.7494i		−	0.133355i	0.997775	−	0.0666777i	$-0.0212399\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	656.420			0.890665
$738$		0			0
$739$	520.032			0.703697			0.351849	−	0.936057i	$-0.385553\pi$
$739$	520.032			0.703697			0.351849	+	0.936057i	$0.385553\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1357.00			−1.82638			−0.913189	−	0.407535i	$-0.866388\pi$
$743$	−1357.00			−1.82638			−0.913189	+	0.407535i	$0.866388\pi$
$744$		0			0
$745$	−442.685	+	670.554i	−0.594208	+	0.900073i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	115.356i		−	0.154013i
$750$		0			0
$751$			431.283i			0.574278i	0.957889	+	0.287139i	$0.0927042\pi$
$751$			431.283i			0.574278i	−0.957889	+	0.287139i	$0.907296\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1190.36	−	785.848i	−1.57663	−	1.04086i
$756$		0			0
$757$			665.943i			0.879713i	0.898068	+	0.439857i	$0.144971\pi$
$757$			665.943i			0.879713i	−0.898068	+	0.439857i	$0.855029\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	792.938			1.04197			0.520984	−	0.853566i	$-0.325565\pi$
$761$	792.938			1.04197			0.520984	+	0.853566i	$0.325565\pi$
$762$		0			0
$763$	167.868			0.220010
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1933.34i			2.52066i
$768$		0			0
$769$	1263.74			1.64336			0.821678	−	0.569953i	$-0.193038\pi$
$769$	1263.74			1.64336			0.821678	+	0.569953i	$0.193038\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	441.297			0.570889			0.285445	−	0.958395i	$-0.407859\pi$
$773$	441.297			0.570889			0.285445	+	0.958395i	$0.407859\pi$
$774$		0			0
$775$	−570.360	+	243.705i	−0.735948	+	0.314458i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1334.74	+	790.985i	1.71340	+	1.01538i
$780$		0			0
$781$			385.815i			0.494002i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−790.670	−	521.983i	−1.00722	−	0.664946i
$786$		0			0
$787$	−367.069			−0.466415			−0.233207	−	0.972427i	$-0.574922\pi$
$787$	−367.069			−0.466415			−0.233207	+	0.972427i	$0.574922\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	459.925i		−	0.581447i
$792$		0			0
$793$	−2102.36			−2.65114
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−923.535			−1.15876			−0.579382	−	0.815056i	$-0.696706\pi$
$797$	−923.535			−1.15876			−0.579382	+	0.815056i	$0.696706\pi$
$798$		0			0
$799$	283.999			0.355443
$800$		0			0
$801$		0			0
$802$		0			0
$803$			971.953i			1.21040i
$804$		0			0
$805$	65.9085	−	99.8344i	0.0818739	−	0.124018i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−823.083			−1.01741			−0.508704	−	0.860942i	$-0.669875\pi$
$809$	−823.083			−1.01741			−0.508704	+	0.860942i	$0.669875\pi$
$810$		0			0
$811$		−	125.683i		−	0.154973i	−0.996993	−	0.0774866i	$-0.975311\pi$
$811$		−	125.683i		−	0.154973i	0.996993	−	0.0774866i	$-0.0246895\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−638.187	−	421.317i	−0.783051	−	0.516953i
$816$		0			0
$817$	−1158.47	−	686.525i	−1.41795	−	0.840300i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−713.601			−0.869185			−0.434592	−	0.900627i	$-0.643108\pi$
$821$	−713.601			−0.869185			−0.434592	+	0.900627i	$0.643108\pi$
$822$		0			0
$823$			823.797i			1.00097i	0.865746	+	0.500484i	$0.166845\pi$
$823$			823.797i			1.00097i	−0.865746	+	0.500484i	$0.833155\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	654.261			0.791125			0.395563	−	0.918439i	$-0.370550\pi$
$827$	654.261			0.791125			0.395563	+	0.918439i	$0.370550\pi$
$828$		0			0
$829$			530.752i			0.640231i	0.947379	+	0.320116i	$0.103722\pi$
$829$			530.752i			0.640231i	−0.947379	+	0.320116i	$0.896278\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			272.272i			0.326857i
$834$		0			0
$835$	−247.134	+	374.344i	−0.295969	+	0.448316i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			917.949i			1.09410i	0.837100	+	0.547049i	$0.184249\pi$
$839$			917.949i			1.09410i	−0.837100	+	0.547049i	$0.815751\pi$
$840$		0			0
$841$	33.0594			0.0393097
$842$		0			0
$843$		0			0
$844$		0			0
$845$	691.850	−	1047.98i	0.818758	−	1.24021i
$846$		0			0
$847$			66.8721i			0.0789517i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			409.495i			0.481192i
$852$		0			0
$853$			600.529i			0.704020i	0.935996	+	0.352010i	$0.114502\pi$
$853$			600.529i			0.704020i	−0.935996	+	0.352010i	$0.885498\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1059.28			1.23603			0.618015	−	0.786167i	$-0.287937\pi$
$857$	1059.28			1.23603			0.618015	+	0.786167i	$0.287937\pi$
$858$		0			0
$859$	734.054			0.854545			0.427272	−	0.904123i	$-0.359475\pi$
$859$	734.054			0.854545			0.427272	+	0.904123i	$0.359475\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	714.413			0.827825			0.413912	−	0.910317i	$-0.364162\pi$
$863$	714.413			0.827825			0.413912	+	0.910317i	$0.364162\pi$
$864$		0			0
$865$	−425.504	+	644.529i	−0.491912	+	0.745121i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			702.886i			0.808845i
$870$		0			0
$871$	1414.91			1.62447
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−48.8710	+	269.034i	−0.0558526	+	0.307467i
$876$		0			0
$877$	−1129.51			−1.28792			−0.643961	−	0.765059i	$-0.722710\pi$
$877$	−1129.51			−1.28792			−0.643961	+	0.765059i	$0.722710\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−161.933			−0.183806			−0.0919029	−	0.995768i	$-0.529295\pi$
$881$	−161.933			−0.183806			−0.0919029	+	0.995768i	$0.529295\pi$
$882$		0			0
$883$		−	199.206i		−	0.225602i	−0.993618	−	0.112801i	$-0.964018\pi$
$883$		−	199.206i		−	0.225602i	0.993618	−	0.112801i	$-0.0359822\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−727.670			−0.820373			−0.410186	−	0.912002i	$-0.634536\pi$
$887$	−727.670			−0.820373			−0.410186	+	0.912002i	$0.634536\pi$
$888$		0			0
$889$		−	43.5814i		−	0.0490230i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−753.837	−	446.735i	−0.844163	−	0.500263i
$894$		0			0
$895$	−579.525	−	382.589i	−0.647514	−	0.427474i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−705.200			−0.784427
$900$		0			0
$901$			198.285i			0.220072i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−444.832	−	293.668i	−0.491527	−	0.324495i
$906$		0			0
$907$	1249.33			1.37743			0.688716	−	0.725031i	$-0.258174\pi$
$907$	1249.33			1.37743			0.688716	+	0.725031i	$0.258174\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	813.909i		−	0.893423i	−0.894678	−	0.446712i	$-0.852595\pi$
$911$		−	813.909i		−	0.893423i	0.894678	−	0.446712i	$-0.147405\pi$
$912$		0			0
$913$			874.465i			0.957793i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	7.21297i		−	0.00786584i
$918$		0			0
$919$	−442.339			−0.481327			−0.240663	−	0.970609i	$-0.577365\pi$
$919$	−442.339			−0.481327			−0.240663	+	0.970609i	$0.577365\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			831.622i			0.900999i
$924$		0			0
$925$	−367.768	−	860.713i	−0.397587	−	0.930501i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	514.468			0.553786			0.276893	−	0.960901i	$-0.410695\pi$
$929$	514.468			0.553786			0.276893	+	0.960901i	$0.410695\pi$
$930$		0			0
$931$	428.288	−	722.709i	0.460030	−	0.776272i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	244.347	+	161.313i	0.261334	+	0.172527i
$936$		0			0
$937$			1147.17i			1.22430i	0.790743	+	0.612149i	$0.209695\pi$
$937$			1147.17i			1.22430i	−0.790743	+	0.612149i	$0.790305\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1355.11i		−	1.44008i	−0.693934	−	0.720039i	$-0.744124\pi$
$941$		−	1355.11i		−	1.44008i	0.693934	−	0.720039i	$-0.255876\pi$
$942$		0			0
$943$	−893.134			−0.947120
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1560.15i		−	1.64746i	−0.566980	−	0.823732i	$-0.691888\pi$
$947$		−	1560.15i		−	1.64746i	0.566980	−	0.823732i	$-0.308112\pi$
$948$		0			0
$949$			2095.04i			2.20763i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	643.715			0.675461			0.337731	−	0.941243i	$-0.390341\pi$
$953$	643.715			0.675461			0.337731	+	0.941243i	$0.390341\pi$
$954$		0			0
$955$	724.368	−	1097.23i	0.758501	−	1.14893i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−324.159			−0.338018
$960$		0			0
$961$	345.476			0.359496
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−93.8565	+	142.169i	−0.0972607	+	0.147325i
$966$		0			0
$967$		−	570.639i		−	0.590113i	−0.955480	−	0.295056i	$-0.904662\pi$
$967$		−	570.639i		−	0.590113i	0.955480	−	0.295056i	$-0.0953384\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	516.280i		−	0.531699i	−0.964015	−	0.265850i	$-0.914348\pi$
$971$		−	516.280i		−	0.531699i	0.964015	−	0.265850i	$-0.0856525\pi$
$972$		0			0
$973$			118.721i			0.122016i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−432.740			−0.442927			−0.221464	−	0.975169i	$-0.571083\pi$
$977$	−432.740			−0.442927			−0.221464	+	0.975169i	$0.571083\pi$
$978$		0			0
$979$		−	545.495i		−	0.557196i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1116.57			1.13588			0.567940	−	0.823070i	$-0.307740\pi$
$983$	1116.57			1.13588			0.567940	+	0.823070i	$0.307740\pi$
$984$		0			0
$985$	1224.87	+	808.633i	1.24352	+	0.820947i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	775.185			0.783806
$990$		0			0
$991$		−	1158.19i		−	1.16871i	−0.811499	−	0.584354i	$-0.801348\pi$
$991$		−	1158.19i		−	1.16871i	0.811499	−	0.584354i	$-0.198652\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−399.093	+	604.523i	−0.401099	+	0.607561i
$996$		0			0
$997$		−	76.0503i		−	0.0762792i	−0.999272	−	0.0381396i	$-0.987857\pi$
$997$		−	76.0503i		−	0.0762792i	0.999272	−	0.0381396i	$-0.0121432\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.3.h.e.2089.10		12
3.2	odd	2		380.3.g.c.189.6	yes	12
5.4	even	2	inner	3420.3.h.e.2089.11		12
15.2	even	4		1900.3.e.e.1101.8		12
15.8	even	4		1900.3.e.e.1101.5		12
15.14	odd	2		380.3.g.c.189.7	yes	12
19.18	odd	2	inner	3420.3.h.e.2089.9		12
57.56	even	2		380.3.g.c.189.8	yes	12
95.94	odd	2	inner	3420.3.h.e.2089.12		12
285.113	odd	4		1900.3.e.e.1101.7		12
285.227	odd	4		1900.3.e.e.1101.6		12
285.284	even	2		380.3.g.c.189.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.g.c.189.5	✓	12	285.284	even	2
380.3.g.c.189.6	yes	12	3.2	odd	2
380.3.g.c.189.7	yes	12	15.14	odd	2
380.3.g.c.189.8	yes	12	57.56	even	2
1900.3.e.e.1101.5		12	15.8	even	4
1900.3.e.e.1101.6		12	285.227	odd	4
1900.3.e.e.1101.7		12	285.113	odd	4
1900.3.e.e.1101.8		12	15.2	even	4
3420.3.h.e.2089.9		12	19.18	odd	2	inner
3420.3.h.e.2089.10		12	1.1	even	1	trivial
3420.3.h.e.2089.11		12	5.4	even	2	inner
3420.3.h.e.2089.12		12	95.94	odd	2	inner

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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3420.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.75473 - 4.17271i) q^{5} -2.18749i q^{7} +O(q^{10})\)	\(q+(2.75473 - 4.17271i) q^{5} -2.18749i q^{7} +9.50946 q^{11} +20.4976 q^{13} +6.15792i q^{17} +(9.68651 - 16.3454i) q^{19} +10.9375i q^{23} +(-9.82294 - 22.9893i) q^{25} -28.4243i q^{29} -24.8098i q^{31} +(-9.12776 - 6.02594i) q^{35} +37.4397 q^{37} +81.6583i q^{41} -70.8743i q^{43} -46.1193i q^{47} +44.2149 q^{49} +32.1999 q^{53} +(26.1960 - 39.6802i) q^{55} +94.3206i q^{59} -102.566 q^{61} +(56.4652 - 85.5303i) q^{65} +69.0282 q^{67} +40.5718i q^{71} +102.209i q^{73} -20.8018i q^{77} +73.9145i q^{79} +91.9574i q^{83} +(25.6952 + 16.9634i) q^{85} -57.3634i q^{89} -44.8382i q^{91} +(-41.5207 - 85.4461i) q^{95} -84.3229 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5}+O(q^{10}) \) 12 * q - 12 * q^5	\( 12 q - 12 q^{5} + 24 q^{11} + 68 q^{19} - 76 q^{25} + 32 q^{35} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 408 q^{85} - 124 q^{95}+O(q^{100}) \) 12 * q - 12 * q^5 + 24 * q^11 + 68 * q^19 - 76 * q^25 + 32 * q^35 + 212 * q^49 + 176 * q^55 - 600 * q^61 + 408 * q^85 - 124 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more