LMFDB - Embedded newform 3420.2.f.a.1369.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1369,3420)]

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, 0]))

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

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

chi := DirichletCharacter("3420.1369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.f (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:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 4 $ x^4 + 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1369.1
Root			$0.874032i$ of defining polynomial
Character	$\chi$	$=$	3420.1369
Dual form			3420.2.f.a.1369.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{5} -2.82843i q^{7} +O(q^{10})$	$q-2.23607 q^{5} -2.82843i q^{7} +0.763932 q^{11} -5.45052i q^{13} -7.40492i q^{17} +1.00000 q^{19} +1.08036i q^{23} +5.00000 q^{25} -4.47214 q^{29} -4.00000 q^{31} +6.32456i q^{35} +2.62210i q^{37} +6.00000 q^{41} +8.48528i q^{43} +8.48528i q^{47} -1.00000 q^{49} -2.62210i q^{53} -1.70820 q^{55} -1.52786 q^{59} -11.7082 q^{61} +12.1877i q^{65} -11.1074i q^{67} +10.4721 q^{71} +5.24419i q^{73} -2.16073i q^{77} -15.4164 q^{79} -13.7295i q^{83} +16.5579i q^{85} +2.94427 q^{89} -15.4164 q^{91} -2.23607 q^{95} -13.9358i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{11} + 4 q^{19} + 20 q^{25} - 16 q^{31} + 24 q^{41} - 4 q^{49} + 20 q^{55} - 24 q^{59} - 20 q^{61} + 24 q^{71} - 8 q^{79} - 24 q^{89} - 8 q^{91}+O(q^{100}) $ 4 * q + 12 * q^11 + 4 * q^19 + 20 * q^25 - 16 * q^31 + 24 * q^41 - 4 * q^49 + 20 * q^55 - 24 * q^59 - 20 * q^61 + 24 * q^71 - 8 * q^79 - 24 * q^89 - 8 * q^91

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
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	− 5.45052i	− 1.51170i	−0.654743	−	0.755852i	$-0.727223\pi$
$13$	− 5.45052i	− 1.51170i	0.654743	−	0.755852i	$-0.272777\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.40492i	− 1.79596i	−0.440040	−	0.897978i	$-0.645036\pi$
$17$	− 7.40492i	− 1.79596i	0.440040	−	0.897978i	$-0.354964\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.08036i	0.225271i	0.993636	+	0.112636i	$0.0359293\pi$
$23$	1.08036i	0.225271i	−0.993636	+	0.112636i	$0.964071\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.32456i	1.06904i
$36$	0	0
$37$	2.62210i	0.431070i	0.976496	+	0.215535i	$0.0691495\pi$
$37$	2.62210i	0.431070i	−0.976496	+	0.215535i	$0.930850\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	8.48528i	1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$	8.48528i	1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.48528i	1.23771i	0.785507	+	0.618853i	$0.212402\pi$
$47$	8.48528i	1.23771i	−0.785507	+	0.618853i	$0.787598\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.62210i	− 0.360173i	−0.983651	−	0.180086i	$-0.942362\pi$
$53$	− 2.62210i	− 0.360173i	0.983651	−	0.180086i	$-0.0576377\pi$
$54$	0	0
$55$	−1.70820	−0.230334
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.52786	−0.198911	−0.0994555	−	0.995042i	$-0.531710\pi$
$59$	−1.52786	−0.198911	−0.0994555	+	0.995042i	$0.531710\pi$
$60$	0	0
$61$	−11.7082	−1.49908	−0.749541	−	0.661958i	$-0.769726\pi$
$61$	−11.7082	−1.49908	−0.749541	+	0.661958i	$0.769726\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12.1877i	1.51170i
$66$	0	0
$67$	− 11.1074i	− 1.35698i	−0.734609	−	0.678491i	$-0.762634\pi$
$67$	− 11.1074i	− 1.35698i	0.734609	−	0.678491i	$-0.237366\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.4721	1.24281	0.621407	−	0.783488i	$-0.286561\pi$
$71$	10.4721	1.24281	0.621407	+	0.783488i	$0.286561\pi$
$72$	0	0
$73$	5.24419i	0.613786i	0.951744	+	0.306893i	$0.0992894\pi$
$73$	5.24419i	0.613786i	−0.951744	+	0.306893i	$0.900711\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.16073i	− 0.246238i
$78$	0	0
$79$	−15.4164	−1.73448	−0.867241	−	0.497889i	$-0.834109\pi$
$79$	−15.4164	−1.73448	−0.867241	+	0.497889i	$0.834109\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 13.7295i	− 1.50701i	−0.657445	−	0.753503i	$-0.728363\pi$
$83$	− 13.7295i	− 1.50701i	0.657445	−	0.753503i	$-0.271637\pi$
$84$	0	0
$85$	16.5579i	1.79596i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.94427	0.312092	0.156046	−	0.987750i	$-0.450125\pi$
$89$	2.94427	0.312092	0.156046	+	0.987750i	$0.450125\pi$
$90$	0	0
$91$	−15.4164	−1.61608
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.23607	−0.229416
$96$	0	0
$97$	− 13.9358i	− 1.41497i	−0.706730	−	0.707483i	$-0.749830\pi$
$97$	− 13.9358i	− 1.41497i	0.706730	−	0.707483i	$-0.250170\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.23607	−0.521008	−0.260504	−	0.965473i	$-0.583889\pi$
$101$	−5.23607	−0.521008	−0.260504	+	0.965473i	$0.583889\pi$
$102$	0	0
$103$	− 5.86319i	− 0.577717i	−0.957372	−	0.288858i	$-0.906724\pi$
$103$	− 5.86319i	− 0.577717i	0.957372	−	0.288858i	$-0.0932757\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.2681i	− 1.28268i	−0.767258	−	0.641338i	$-0.778380\pi$
$107$	− 13.2681i	− 1.28268i	0.767258	−	0.641338i	$-0.221620\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.3516i	1.53823i	0.639113	+	0.769113i	$0.279302\pi$
$113$	16.3516i	1.53823i	−0.639113	+	0.769113i	$0.720698\pi$
$114$	0	0
$115$	− 2.41577i	− 0.225271i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−20.9443	−1.91996
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	19.5927i	1.73857i	0.494314	+	0.869284i	$0.335419\pi$
$127$	19.5927i	1.73857i	−0.494314	+	0.869284i	$0.664581\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 2.82843i	− 0.245256i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 7.40492i	− 0.632645i	−0.948652	−	0.316322i	$-0.897552\pi$
$137$	− 7.40492i	− 0.632645i	0.948652	−	0.316322i	$-0.102448\pi$
$138$	0	0
$139$	−2.29180	−0.194388	−0.0971938	−	0.995265i	$-0.530987\pi$
$139$	−2.29180	−0.194388	−0.0971938	+	0.995265i	$0.530987\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4.16383i	− 0.348197i
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.6525	−1.03653	−0.518266	−	0.855220i	$-0.673422\pi$
$149$	−12.6525	−1.03653	−0.518266	+	0.855220i	$0.673422\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.94427	0.718421
$156$	0	0
$157$	0.412662i	0.0329340i	0.999864	+	0.0164670i	$0.00524185\pi$
$157$	0.412662i	0.0329340i	−0.999864	+	0.0164670i	$0.994758\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.05573	0.240825
$162$	0	0
$163$	− 8.48528i	− 0.664619i	−0.943170	−	0.332309i	$-0.892172\pi$
$163$	− 8.48528i	− 0.664619i	0.943170	−	0.332309i	$-0.107828\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.4319i	1.34892i	0.738310	+	0.674462i	$0.235624\pi$
$167$	17.4319i	1.34892i	−0.738310	+	0.674462i	$0.764376\pi$
$168$	0	0
$169$	−16.7082	−1.28525
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 8.94665i	− 0.680201i	−0.940389	−	0.340101i	$-0.889539\pi$
$173$	− 8.94665i	− 0.680201i	0.940389	−	0.340101i	$-0.110461\pi$
$174$	0	0
$175$	− 14.1421i	− 1.06904i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.4721	−1.67965	−0.839823	−	0.542860i	$-0.817341\pi$
$179$	−22.4721	−1.67965	−0.839823	+	0.542860i	$0.817341\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5.86319i	− 0.431070i
$186$	0	0
$187$	− 5.65685i	− 0.413670i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.05573	0.221105	0.110552	−	0.993870i	$-0.464738\pi$
$191$	3.05573	0.221105	0.110552	+	0.993870i	$0.464738\pi$
$192$	0	0
$193$	13.9358i	1.00312i	0.865123	+	0.501561i	$0.167241\pi$
$193$	13.9358i	1.00312i	−0.865123	+	0.501561i	$0.832759\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.6491i	0.901212i	0.892723	+	0.450606i	$0.148792\pi$
$197$	12.6491i	0.901212i	−0.892723	+	0.450606i	$0.851208\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	12.6491i	0.887794i
$204$	0	0
$205$	−13.4164	−0.937043
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.763932	0.0528423
$210$	0	0
$211$	15.4164	1.06131	0.530655	−	0.847588i	$-0.321946\pi$
$211$	15.4164	1.06131	0.530655	+	0.847588i	$0.321946\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 18.9737i	− 1.29399i
$216$	0	0
$217$	11.3137i	0.768025i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−40.3607	−2.71495
$222$	0	0
$223$	18.7673i	1.25675i	0.777909	+	0.628377i	$0.216280\pi$
$223$	18.7673i	1.25675i	−0.777909	+	0.628377i	$0.783720\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.86319i	0.389153i	0.980887	+	0.194577i	$0.0623333\pi$
$227$	5.86319i	0.389153i	−0.980887	+	0.194577i	$0.937667\pi$
$228$	0	0
$229$	−7.70820	−0.509372	−0.254686	−	0.967024i	$-0.581972\pi$
$229$	−7.70820	−0.509372	−0.254686	+	0.967024i	$0.581972\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.6491i	0.828671i	0.910124	+	0.414335i	$0.135986\pi$
$233$	12.6491i	0.828671i	−0.910124	+	0.414335i	$0.864014\pi$
$234$	0	0
$235$	− 18.9737i	− 1.23771i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−29.8885	−1.93333	−0.966665	−	0.256046i	$-0.917580\pi$
$239$	−29.8885	−1.93333	−0.966665	+	0.256046i	$0.917580\pi$
$240$	0	0
$241$	−28.8328	−1.85728	−0.928642	−	0.370976i	$-0.879023\pi$
$241$	−28.8328	−1.85728	−0.928642	+	0.370976i	$0.879023\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.23607	0.142857
$246$	0	0
$247$	− 5.45052i	− 0.346808i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.88854	0.371682	0.185841	−	0.982580i	$-0.440499\pi$
$251$	5.88854	0.371682	0.185841	+	0.982580i	$0.440499\pi$
$252$	0	0
$253$	0.825324i	0.0518877i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 17.4319i	− 1.08737i	−0.839288	−	0.543687i	$-0.817028\pi$
$257$	− 17.4319i	− 1.08737i	0.839288	−	0.543687i	$-0.182972\pi$
$258$	0	0
$259$	7.41641	0.460833
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 15.8902i	− 0.979832i	−0.871770	−	0.489916i	$-0.837027\pi$
$263$	− 15.8902i	− 0.979832i	0.871770	−	0.489916i	$-0.162973\pi$
$264$	0	0
$265$	5.86319i	0.360173i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.9443	0.911168	0.455584	−	0.890193i	$-0.349430\pi$
$269$	14.9443	0.911168	0.455584	+	0.890193i	$0.349430\pi$
$270$	0	0
$271$	21.1246	1.28323	0.641614	−	0.767027i	$-0.278265\pi$
$271$	21.1246	1.28323	0.641614	+	0.767027i	$0.278265\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.81966	0.230334
$276$	0	0
$277$	22.2148i	1.33476i	0.744719	+	0.667378i	$0.232583\pi$
$277$	22.2148i	1.33476i	−0.744719	+	0.667378i	$0.767417\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	25.4164	1.51622	0.758108	−	0.652129i	$-0.226124\pi$
$281$	25.4164	1.51622	0.758108	+	0.652129i	$0.226124\pi$
$282$	0	0
$283$	− 2.41577i	− 0.143602i	−0.997419	−	0.0718012i	$-0.977125\pi$
$283$	− 2.41577i	− 0.143602i	0.997419	−	0.0718012i	$-0.0228747\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 16.9706i	− 1.00174i
$288$	0	0
$289$	−37.8328	−2.22546
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.62210i	− 0.153184i	−0.997062	−	0.0765922i	$-0.975596\pi$
$293$	− 2.62210i	− 0.153184i	0.997062	−	0.0765922i	$-0.0244040\pi$
$294$	0	0
$295$	3.41641	0.198911
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.88854	0.340543
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	0	0
$304$	0	0
$305$	26.1803	1.49908
$306$	0	0
$307$	5.45052i	0.311078i	0.987830	+	0.155539i	$0.0497114\pi$
$307$	5.45052i	0.311078i	−0.987830	+	0.155539i	$0.950289\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.70820	0.550502	0.275251	−	0.961372i	$-0.411239\pi$
$311$	9.70820	0.550502	0.275251	+	0.961372i	$0.411239\pi$
$312$	0	0
$313$	− 10.4884i	− 0.592839i	−0.955058	−	0.296419i	$-0.904207\pi$
$313$	− 10.4884i	− 0.592839i	0.955058	−	0.296419i	$-0.0957926\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.94665i	− 0.502494i	−0.967923	−	0.251247i	$-0.919159\pi$
$317$	− 8.94665i	− 0.502494i	0.967923	−	0.251247i	$-0.0808406\pi$
$318$	0	0
$319$	−3.41641	−0.191282
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 7.40492i	− 0.412021i
$324$	0	0
$325$	− 27.2526i	− 1.51170i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	7.41641	0.407643	0.203821	−	0.979008i	$-0.434664\pi$
$331$	7.41641	0.407643	0.203821	+	0.979008i	$0.434664\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	24.8369i	1.35698i
$336$	0	0
$337$	− 16.7642i	− 0.913206i	−0.889671	−	0.456603i	$-0.849066\pi$
$337$	− 16.7642i	− 0.913206i	0.889671	−	0.456603i	$-0.150934\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.05573	−0.165477
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.40802i	− 0.505049i	−0.967590	−	0.252525i	$-0.918739\pi$
$347$	− 9.40802i	− 0.505049i	0.967590	−	0.252525i	$-0.0812608\pi$
$348$	0	0
$349$	−25.4164	−1.36051	−0.680255	−	0.732976i	$-0.738131\pi$
$349$	−25.4164	−1.36051	−0.680255	+	0.732976i	$0.738131\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9.56564i	− 0.509128i	−0.967056	−	0.254564i	$-0.918068\pi$
$353$	− 9.56564i	− 0.509128i	0.967056	−	0.254564i	$-0.0819319\pi$
$354$	0	0
$355$	−23.4164	−1.24281
$356$	0	0
$357$	0	0
$358$	0	0
$359$	29.1246	1.53714	0.768569	−	0.639767i	$-0.220969\pi$
$359$	29.1246	1.53714	0.768569	+	0.639767i	$0.220969\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 11.7264i	− 0.613786i
$366$	0	0
$367$	− 8.48528i	− 0.442928i	−0.975169	−	0.221464i	$-0.928916\pi$
$367$	− 8.48528i	− 0.442928i	0.975169	−	0.221464i	$-0.0710835\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.41641	−0.385041
$372$	0	0
$373$	13.1105i	0.678835i	0.940636	+	0.339417i	$0.110230\pi$
$373$	13.1105i	0.678835i	−0.940636	+	0.339417i	$0.889770\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.3755i	1.25540i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	36.4056i	1.86024i	0.367257	+	0.930120i	$0.380297\pi$
$383$	36.4056i	1.86024i	−0.367257	+	0.930120i	$0.619703\pi$
$384$	0	0
$385$	4.83153i	0.246238i
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.9443	−0.757705	−0.378852	−	0.925457i	$-0.623681\pi$
$389$	−14.9443	−0.757705	−0.378852	+	0.925457i	$0.623681\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	34.4721	1.73448
$396$	0	0
$397$	− 11.7264i	− 0.588530i	−0.955724	−	0.294265i	$-0.904925\pi$
$397$	− 11.7264i	− 0.588530i	0.955724	−	0.294265i	$-0.0950748\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.47214	0.223328	0.111664	−	0.993746i	$-0.464382\pi$
$401$	4.47214	0.223328	0.111664	+	0.993746i	$0.464382\pi$
$402$	0	0
$403$	21.8021i	1.08604i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.00310i	0.0992901i
$408$	0	0
$409$	−17.4164	−0.861186	−0.430593	−	0.902546i	$-0.641696\pi$
$409$	−17.4164	−0.861186	−0.430593	+	0.902546i	$0.641696\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.32145i	0.212645i
$414$	0	0
$415$	30.7000i	1.50701i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.9443	−1.02319	−0.511597	−	0.859225i	$-0.670946\pi$
$419$	−20.9443	−1.02319	−0.511597	+	0.859225i	$0.670946\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 37.0246i	− 1.79596i
$426$	0	0
$427$	33.1158i	1.60259i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.52786	−0.0735946	−0.0367973	−	0.999323i	$-0.511716\pi$
$431$	−1.52786	−0.0735946	−0.0367973	+	0.999323i	$0.511716\pi$
$432$	0	0
$433$	− 28.4906i	− 1.36917i	−0.728933	−	0.684585i	$-0.759983\pi$
$433$	− 28.4906i	− 1.36917i	0.728933	−	0.684585i	$-0.240017\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.08036i	0.0516808i
$438$	0	0
$439$	18.8328	0.898841	0.449421	−	0.893320i	$-0.351630\pi$
$439$	18.8328	0.898841	0.449421	+	0.893320i	$0.351630\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.7295i	0.652307i	0.945317	+	0.326153i	$0.105753\pi$
$443$	13.7295i	0.652307i	−0.945317	+	0.326153i	$0.894247\pi$
$444$	0	0
$445$	−6.58359	−0.312092
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.4721	−0.777368	−0.388684	−	0.921371i	$-0.627070\pi$
$449$	−16.4721	−0.777368	−0.388684	+	0.921371i	$0.627070\pi$
$450$	0	0
$451$	4.58359	0.215833
$452$	0	0
$453$	0	0
$454$	0	0
$455$	34.4721	1.61608
$456$	0	0
$457$	− 37.9473i	− 1.77510i	−0.460710	−	0.887551i	$-0.652405\pi$
$457$	− 37.9473i	− 1.77510i	0.460710	−	0.887551i	$-0.347595\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.8885	−1.11260	−0.556300	−	0.830981i	$-0.687780\pi$
$461$	−23.8885	−1.11260	−0.556300	+	0.830981i	$0.687780\pi$
$462$	0	0
$463$	14.5548i	0.676419i	0.941071	+	0.338209i	$0.109821\pi$
$463$	14.5548i	0.676419i	−0.941071	+	0.338209i	$0.890179\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.7000i	1.42063i	0.703885	+	0.710314i	$0.251447\pi$
$467$	30.7000i	1.42063i	−0.703885	+	0.710314i	$0.748553\pi$
$468$	0	0
$469$	−31.4164	−1.45067
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.48218i	0.298051i
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11.2361	0.513389	0.256695	−	0.966493i	$-0.417367\pi$
$479$	11.2361	0.513389	0.256695	+	0.966493i	$0.417367\pi$
$480$	0	0
$481$	14.2918	0.651650
$482$	0	0
$483$	0	0
$484$	0	0
$485$	31.1614i	1.41497i
$486$	0	0
$487$	10.6947i	0.484624i	0.970198	+	0.242312i	$0.0779057\pi$
$487$	10.6947i	0.484624i	−0.970198	+	0.242312i	$0.922094\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.88854	−0.265746	−0.132873	−	0.991133i	$-0.542420\pi$
$491$	−5.88854	−0.265746	−0.132873	+	0.991133i	$0.542420\pi$
$492$	0	0
$493$	33.1158i	1.49146i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 29.6197i	− 1.32862i
$498$	0	0
$499$	17.1246	0.766603	0.383301	−	0.923623i	$-0.374787\pi$
$499$	17.1246	0.766603	0.383301	+	0.923623i	$0.374787\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 20.2117i	− 0.901193i	−0.892728	−	0.450597i	$-0.851211\pi$
$503$	− 20.2117i	− 0.901193i	0.892728	−	0.450597i	$-0.148789\pi$
$504$	0	0
$505$	11.7082	0.521008
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10.5836	−0.469109	−0.234555	−	0.972103i	$-0.575363\pi$
$509$	−10.5836	−0.469109	−0.234555	+	0.972103i	$0.575363\pi$
$510$	0	0
$511$	14.8328	0.656165
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.1105i	0.577717i
$516$	0	0
$517$	6.48218i	0.285086i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−0.111456	−0.00488298	−0.00244149	−	0.999997i	$-0.500777\pi$
$521$	−0.111456	−0.00488298	−0.00244149	+	0.999997i	$0.500777\pi$
$522$	0	0
$523$	− 24.8369i	− 1.08604i	−0.839720	−	0.543020i	$-0.817281\pi$
$523$	− 24.8369i	− 1.08604i	0.839720	−	0.543020i	$-0.182719\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	29.6197i	1.29025i
$528$	0	0
$529$	21.8328	0.949253
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 32.7031i	− 1.41653i
$534$	0	0
$535$	29.6684i	1.28268i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.763932	−0.0329049
$540$	0	0
$541$	11.7082	0.503375	0.251688	−	0.967809i	$-0.419014\pi$
$541$	11.7082	0.503375	0.251688	+	0.967809i	$0.419014\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.47214	0.191565
$546$	0	0
$547$	8.27895i	0.353982i	0.984212	+	0.176991i	$0.0566364\pi$
$547$	8.27895i	0.353982i	−0.984212	+	0.176991i	$0.943364\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.47214	−0.190519
$552$	0	0
$553$	43.6042i	1.85424i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.0540i	0.849716i	0.905260	+	0.424858i	$0.139676\pi$
$557$	20.0540i	0.849716i	−0.905260	+	0.424858i	$0.860324\pi$
$558$	0	0
$559$	46.2492	1.95613
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 30.0810i	− 1.26776i	−0.773429	−	0.633882i	$-0.781460\pi$
$563$	− 30.0810i	− 1.26776i	0.773429	−	0.633882i	$-0.218540\pi$
$564$	0	0
$565$	− 36.5632i	− 1.53823i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−38.9443	−1.63263	−0.816314	−	0.577608i	$-0.803986\pi$
$569$	−38.9443	−1.63263	−0.816314	+	0.577608i	$0.803986\pi$
$570$	0	0
$571$	25.1246	1.05143	0.525716	−	0.850660i	$-0.323797\pi$
$571$	25.1246	1.05143	0.525716	+	0.850660i	$0.323797\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.40182i	0.225271i
$576$	0	0
$577$	5.65685i	0.235498i	0.993043	+	0.117749i	$0.0375678\pi$
$577$	5.65685i	0.235498i	−0.993043	+	0.117749i	$0.962432\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−38.8328	−1.61106
$582$	0	0
$583$	− 2.00310i	− 0.0829601i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.40182i	0.222957i	0.993767	+	0.111478i	$0.0355586\pi$
$587$	5.40182i	0.222957i	−0.993767	+	0.111478i	$0.964441\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.16073i	0.0887304i	0.999015	+	0.0443652i	$0.0141265\pi$
$593$	2.16073i	0.0887304i	−0.999015	+	0.0443652i	$0.985873\pi$
$594$	0	0
$595$	46.8328	1.91996
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	16.8328	0.686625	0.343312	−	0.939221i	$-0.388451\pi$
$601$	16.8328	0.686625	0.343312	+	0.939221i	$0.388451\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	23.2918	0.946946
$606$	0	0
$607$	− 22.4211i	− 0.910044i	−0.890480	−	0.455022i	$-0.849631\pi$
$607$	− 22.4211i	− 0.910044i	0.890480	−	0.455022i	$-0.150369\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	46.2492	1.87104
$612$	0	0
$613$	− 39.1853i	− 1.58268i	−0.611376	−	0.791340i	$-0.709384\pi$
$613$	− 39.1853i	− 1.58268i	0.611376	−	0.791340i	$-0.290616\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.08347i	0.124136i	0.998072	+	0.0620678i	$0.0197695\pi$
$617$	3.08347i	0.124136i	−0.998072	+	0.0620678i	$0.980230\pi$
$618$	0	0
$619$	−41.1246	−1.65294	−0.826469	−	0.562982i	$-0.809654\pi$
$619$	−41.1246	−1.65294	−0.826469	+	0.562982i	$0.809654\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 8.32766i	− 0.333641i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	19.4164	0.774183
$630$	0	0
$631$	−1.70820	−0.0680025	−0.0340013	−	0.999422i	$-0.510825\pi$
$631$	−1.70820	−0.0680025	−0.0340013	+	0.999422i	$0.510825\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 43.8105i	− 1.73857i
$636$	0	0
$637$	5.45052i	0.215958i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.1115	0.478374	0.239187	−	0.970974i	$-0.423119\pi$
$641$	12.1115	0.478374	0.239187	+	0.970974i	$0.423119\pi$
$642$	0	0
$643$	− 30.7000i	− 1.21069i	−0.795963	−	0.605346i	$-0.793035\pi$
$643$	− 30.7000i	− 1.21069i	0.795963	−	0.605346i	$-0.206965\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.40802i	− 0.369867i	−0.982751	−	0.184934i	$-0.940793\pi$
$647$	− 9.40802i	− 0.369867i	0.982751	−	0.184934i	$-0.0592071\pi$
$648$	0	0
$649$	−1.16718	−0.0458160
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.2148i	0.869331i	0.900592	+	0.434665i	$0.143133\pi$
$653$	22.2148i	0.869331i	−0.900592	+	0.434665i	$0.856867\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−44.9443	−1.75078	−0.875390	−	0.483417i	$-0.839395\pi$
$659$	−44.9443	−1.75078	−0.875390	+	0.483417i	$0.839395\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.32456i	0.245256i
$666$	0	0
$667$	− 4.83153i	− 0.187078i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.94427	−0.345290
$672$	0	0
$673$	− 4.62520i	− 0.178288i	−0.996019	−	0.0891442i	$-0.971587\pi$
$673$	− 4.62520i	− 0.178288i	0.996019	−	0.0891442i	$-0.0284132\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.7302i	1.64225i	0.570746	+	0.821127i	$0.306654\pi$
$677$	42.7302i	1.64225i	−0.570746	+	0.821127i	$0.693346\pi$
$678$	0	0
$679$	−39.4164	−1.51266
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 19.5927i	− 0.749692i	−0.927087	−	0.374846i	$-0.877696\pi$
$683$	− 19.5927i	− 0.749692i	0.927087	−	0.374846i	$-0.122304\pi$
$684$	0	0
$685$	16.5579i	0.632645i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.2918	−0.544474
$690$	0	0
$691$	25.1246	0.955785	0.477893	−	0.878418i	$-0.341401\pi$
$691$	25.1246	0.955785	0.477893	+	0.878418i	$0.341401\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.12461	0.194388
$696$	0	0
$697$	− 44.4295i	− 1.68289i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.875388	−0.0330630	−0.0165315	−	0.999863i	$-0.505262\pi$
$701$	−0.875388	−0.0330630	−0.0165315	+	0.999863i	$0.505262\pi$
$702$	0	0
$703$	2.62210i	0.0988942i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.8098i	0.556981i
$708$	0	0
$709$	−33.4164	−1.25498	−0.627490	−	0.778625i	$-0.715918\pi$
$709$	−33.4164	−1.25498	−0.627490	+	0.778625i	$0.715918\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 4.32145i	− 0.161840i
$714$	0	0
$715$	9.31061i	0.348197i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	27.5967	1.02919	0.514593	−	0.857435i	$-0.327943\pi$
$719$	27.5967	1.02919	0.514593	+	0.857435i	$0.327943\pi$
$720$	0	0
$721$	−16.5836	−0.617605
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−22.3607	−0.830455
$726$	0	0
$727$	− 8.48528i	− 0.314702i	−0.987543	−	0.157351i	$-0.949705\pi$
$727$	− 8.48528i	− 0.314702i	0.987543	−	0.157351i	$-0.0502953\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	62.8328	2.32396
$732$	0	0
$733$	− 33.9411i	− 1.25364i	−0.779162	−	0.626822i	$-0.784355\pi$
$733$	− 33.9411i	− 1.25364i	0.779162	−	0.626822i	$-0.215645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.48528i	− 0.312559i
$738$	0	0
$739$	18.8328	0.692776	0.346388	−	0.938091i	$-0.387408\pi$
$739$	18.8328	0.692776	0.346388	+	0.938091i	$0.387408\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.62210i	0.0961954i	0.998843	+	0.0480977i	$0.0153159\pi$
$743$	2.62210i	0.0961954i	−0.998843	+	0.0480977i	$0.984684\pi$
$744$	0	0
$745$	28.2918	1.03653
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−37.5279	−1.37124
$750$	0	0
$751$	20.5836	0.751106	0.375553	−	0.926801i	$-0.377453\pi$
$751$	20.5836	0.751106	0.375553	+	0.926801i	$0.377453\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.8885	0.651031
$756$	0	0
$757$	38.7727i	1.40922i	0.709597	+	0.704608i	$0.248877\pi$
$757$	38.7727i	1.40922i	−0.709597	+	0.704608i	$0.751123\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.5410	1.54211	0.771055	−	0.636768i	$-0.219729\pi$
$761$	42.5410	1.54211	0.771055	+	0.636768i	$0.219729\pi$
$762$	0	0
$763$	5.65685i	0.204792i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.32766i	0.300694i
$768$	0	0
$769$	38.5410	1.38982	0.694912	−	0.719094i	$-0.255443\pi$
$769$	38.5410	1.38982	0.694912	+	0.719094i	$0.255443\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 5.86319i	− 0.210884i	−0.994425	−	0.105442i	$-0.966374\pi$
$773$	− 5.86319i	− 0.210884i	0.994425	−	0.105442i	$-0.0336258\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 0.922740i	− 0.0329340i
$786$	0	0
$787$	− 24.8369i	− 0.885338i	−0.896685	−	0.442669i	$-0.854032\pi$
$787$	− 24.8369i	− 0.885338i	0.896685	−	0.442669i	$-0.145968\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	46.2492	1.64443
$792$	0	0
$793$	63.8158i	2.26617i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 52.2958i	− 1.85241i	−0.377018	−	0.926206i	$-0.623050\pi$
$797$	− 52.2958i	− 1.85241i	0.377018	−	0.926206i	$-0.376950\pi$
$798$	0	0
$799$	62.8328	2.22287
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.00621i	0.141376i
$804$	0	0
$805$	−6.83282	−0.240825
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.52786	−0.264666	−0.132333	−	0.991205i	$-0.542247\pi$
$809$	−7.52786	−0.264666	−0.132333	+	0.991205i	$0.542247\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.9737i	0.664619i
$816$	0	0
$817$	8.48528i	0.296862i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.9443	−1.35916	−0.679582	−	0.733599i	$-0.737839\pi$
$821$	−38.9443	−1.35916	−0.679582	+	0.733599i	$0.737839\pi$
$822$	0	0
$823$	− 35.9442i	− 1.25294i	−0.779447	−	0.626469i	$-0.784500\pi$
$823$	− 35.9442i	− 1.25294i	0.779447	−	0.626469i	$-0.215500\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.86629i	− 0.273538i	−0.990603	−	0.136769i	$-0.956328\pi$
$827$	− 7.86629i	− 0.273538i	0.990603	−	0.136769i	$-0.0436717\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.40492i	0.256565i
$834$	0	0
$835$	− 38.9790i	− 1.34892i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	49.3050	1.70220	0.851098	−	0.525007i	$-0.175937\pi$
$839$	49.3050	1.70220	0.851098	+	0.525007i	$0.175937\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	0	0
$844$	0	0
$845$	37.3607	1.28525
$846$	0	0
$847$	29.4621i	1.01233i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.83282	−0.0971077
$852$	0	0
$853$	− 27.4589i	− 0.940176i	−0.882619	−	0.470088i	$-0.844222\pi$
$853$	− 27.4589i	− 0.940176i	0.882619	−	0.470088i	$-0.155778\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.3190i	1.06984i	0.844903	+	0.534919i	$0.179658\pi$
$857$	31.3190i	1.06984i	−0.844903	+	0.534919i	$0.820342\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 7.86629i	− 0.267772i	−0.990997	−	0.133886i	$-0.957254\pi$
$863$	− 7.86629i	− 0.267772i	0.990997	−	0.133886i	$-0.0427455\pi$
$864$	0	0
$865$	20.0053i	0.680201i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11.7771	−0.399510
$870$	0	0
$871$	−60.5410	−2.05135
$872$	0	0
$873$	0	0
$874$	0	0
$875$	31.6228i	1.06904i
$876$	0	0
$877$	− 13.9358i	− 0.470579i	−0.971925	−	0.235289i	$-0.924396\pi$
$877$	− 13.9358i	− 0.470579i	0.971925	−	0.235289i	$-0.0756038\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−0.652476	−0.0219825	−0.0109912	−	0.999940i	$-0.503499\pi$
$881$	−0.652476	−0.0219825	−0.0109912	+	0.999940i	$0.503499\pi$
$882$	0	0
$883$	− 52.9148i	− 1.78072i	−0.455253	−	0.890362i	$-0.650451\pi$
$883$	− 52.9148i	− 1.78072i	0.455253	−	0.890362i	$-0.349549\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 49.0547i	− 1.64710i	−0.567247	−	0.823548i	$-0.691991\pi$
$887$	− 49.0547i	− 1.64710i	0.567247	−	0.823548i	$-0.308009\pi$
$888$	0	0
$889$	55.4164	1.85861
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.48528i	0.283949i
$894$	0	0
$895$	50.2492	1.67965
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.8885	0.596616
$900$	0	0
$901$	−19.4164	−0.646854
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.4164	−0.445976
$906$	0	0
$907$	38.5663i	1.28057i	0.768136	+	0.640287i	$0.221185\pi$
$907$	38.5663i	1.28057i	−0.768136	+	0.640287i	$0.778815\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46.4721	1.53969	0.769845	−	0.638231i	$-0.220333\pi$
$911$	46.4721	1.53969	0.769845	+	0.638231i	$0.220333\pi$
$912$	0	0
$913$	− 10.4884i	− 0.347115i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−26.8328	−0.885133	−0.442566	−	0.896736i	$-0.645932\pi$
$919$	−26.8328	−0.885133	−0.442566	+	0.896736i	$0.645932\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 57.0786i	− 1.87877i
$924$	0	0
$925$	13.1105i	0.431070i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.52786	0.246981	0.123491	−	0.992346i	$-0.460591\pi$
$929$	7.52786	0.246981	0.123491	+	0.992346i	$0.460591\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.6491i	0.413670i
$936$	0	0
$937$	43.1915i	1.41101i	0.708707	+	0.705503i	$0.249279\pi$
$937$	43.1915i	1.41101i	−0.708707	+	0.705503i	$0.750721\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	38.9443	1.26955	0.634773	−	0.772698i	$-0.281093\pi$
$941$	38.9443	1.26955	0.634773	+	0.772698i	$0.281093\pi$
$942$	0	0
$943$	6.48218i	0.211089i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 47.6706i	− 1.54909i	−0.632521	−	0.774543i	$-0.717980\pi$
$947$	− 47.6706i	− 1.54909i	0.632521	−	0.774543i	$-0.282020\pi$
$948$	0	0
$949$	28.5836	0.927863
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.5632i	1.18440i	0.805791	+	0.592199i	$0.201740\pi$
$953$	36.5632i	1.18440i	−0.805791	+	0.592199i	$0.798260\pi$
$954$	0	0
$955$	−6.83282	−0.221105
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.9443	−0.676326
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 31.1614i	− 1.00312i
$966$	0	0
$967$	2.41577i	0.0776858i	0.999245	+	0.0388429i	$0.0123672\pi$
$967$	2.41577i	0.0776858i	−0.999245	+	0.0388429i	$0.987633\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40.3607	1.29524	0.647618	−	0.761965i	$-0.275765\pi$
$971$	40.3607	1.29524	0.647618	+	0.761965i	$0.275765\pi$
$972$	0	0
$973$	6.48218i	0.207809i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.9945i	0.799644i	0.916593	+	0.399822i	$0.130928\pi$
$977$	24.9945i	0.799644i	−0.916593	+	0.399822i	$0.869072\pi$
$978$	0	0
$979$	2.24922	0.0718855
$980$	0	0
$981$	0	0
$982$	0	0
$983$	30.2387i	0.964464i	0.876044	+	0.482232i	$0.160174\pi$
$983$	30.2387i	0.964464i	−0.876044	+	0.482232i	$0.839826\pi$
$984$	0	0
$985$	− 28.2843i	− 0.901212i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.16718	−0.291500
$990$	0	0
$991$	38.8328	1.23357	0.616783	−	0.787134i	$-0.288436\pi$
$991$	38.8328	1.23357	0.616783	+	0.787134i	$0.288436\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−35.7771	−1.13421
$996$	0	0
$997$	11.7264i	0.371378i	0.982609	+	0.185689i	$0.0594517\pi$
$997$	11.7264i	0.371378i	−0.982609	+	0.185689i	$0.940548\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.f.a.1369.1		4
3.2	odd	2		380.2.c.a.229.3	yes	4
5.4	even	2	inner	3420.2.f.a.1369.2		4
12.11	even	2		1520.2.d.f.609.2		4
15.2	even	4		1900.2.a.j.1.3		4
15.8	even	4		1900.2.a.j.1.2		4
15.14	odd	2		380.2.c.a.229.2	✓	4
60.23	odd	4		7600.2.a.ce.1.3		4
60.47	odd	4		7600.2.a.ce.1.2		4
60.59	even	2		1520.2.d.f.609.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.c.a.229.2	✓	4	15.14	odd	2
380.2.c.a.229.3	yes	4	3.2	odd	2
1520.2.d.f.609.2		4	12.11	even	2
1520.2.d.f.609.3		4	60.59	even	2
1900.2.a.j.1.2		4	15.8	even	4
1900.2.a.j.1.3		4	15.2	even	4
3420.2.f.a.1369.1		4	1.1	even	1	trivial
3420.2.f.a.1369.2		4	5.4	even	2	inner
7600.2.a.ce.1.2		4	60.47	odd	4
7600.2.a.ce.1.3		4	60.23	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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607 q^{5} -2.82843i q^{7} +O(q^{10})\)	\(q-2.23607 q^{5} -2.82843i q^{7} +0.763932 q^{11} -5.45052i q^{13} -7.40492i q^{17} +1.00000 q^{19} +1.08036i q^{23} +5.00000 q^{25} -4.47214 q^{29} -4.00000 q^{31} +6.32456i q^{35} +2.62210i q^{37} +6.00000 q^{41} +8.48528i q^{43} +8.48528i q^{47} -1.00000 q^{49} -2.62210i q^{53} -1.70820 q^{55} -1.52786 q^{59} -11.7082 q^{61} +12.1877i q^{65} -11.1074i q^{67} +10.4721 q^{71} +5.24419i q^{73} -2.16073i q^{77} -15.4164 q^{79} -13.7295i q^{83} +16.5579i q^{85} +2.94427 q^{89} -15.4164 q^{91} -2.23607 q^{95} -13.9358i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{11} + 4 q^{19} + 20 q^{25} - 16 q^{31} + 24 q^{41} - 4 q^{49} + 20 q^{55} - 24 q^{59} - 20 q^{61} + 24 q^{71} - 8 q^{79} - 24 q^{89} - 8 q^{91}+O(q^{100}) \) 4 * q + 12 * q^11 + 4 * q^19 + 20 * q^25 - 16 * q^31 + 24 * q^41 - 4 * q^49 + 20 * q^55 - 24 * q^59 - 20 * q^61 + 24 * q^71 - 8 * q^79 - 24 * q^89 - 8 * q^91

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