LMFDB - Embedded newform 3420.2.f.c.1369.2

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:	$6$
Coefficient field:	6.0.14077504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 14x^{2} + 4 $ x^6 + 9x^4 + 14x^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			1369.2
Root			$-0.608430i$ of defining polynomial
Character	$\chi$	$=$	3420.1369
Dual form			3420.2.f.c.1369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.58777 + 1.57448i) q^{5} -1.93210i q^{7} +O(q^{10})$	$q+(-1.58777 + 1.57448i) q^{5} -1.93210i q^{7} -5.62981 q^{11} +2.07029i q^{13} -3.42535i q^{17} -1.00000 q^{19} -5.35744i q^{23} +(0.0420411 - 4.99982i) q^{25} +1.09146 q^{29} +3.09146 q^{31} +(3.04204 + 3.06773i) q^{35} +3.28715i q^{37} +11.6107 q^{41} +0.501623i q^{43} +12.6470i q^{47} +3.26701 q^{49} +3.01076i q^{53} +(8.93886 - 8.86401i) q^{55} -2.35109 q^{59} +7.62981 q^{61} +(-3.25963 - 3.28715i) q^{65} +12.2324i q^{67} -14.3511 q^{71} +2.87256i q^{73} +10.8773i q^{77} -4.00000 q^{79} +5.35744i q^{83} +(5.39313 + 5.43867i) q^{85} +8.35109 q^{89} +4.00000 q^{91} +(1.58777 - 1.57448i) q^{95} +3.01076i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5}+O(q^{10}) $ 6 * q + q^5	$ 6 q + q^{5} - 18 q^{11} - 6 q^{19} - 5 q^{25} - 4 q^{29} + 8 q^{31} + 13 q^{35} - 4 q^{41} - 12 q^{49} + q^{55} + 28 q^{59} + 30 q^{61} + 12 q^{65} - 44 q^{71} - 24 q^{79} - 15 q^{85} + 8 q^{89} + 24 q^{91} - q^{95}+O(q^{100}) $ 6 * q + q^5 - 18 * q^11 - 6 * q^19 - 5 * q^25 - 4 * q^29 + 8 * q^31 + 13 * q^35 - 4 * q^41 - 12 * q^49 + q^55 + 28 * q^59 + 30 * q^61 + 12 * q^65 - 44 * q^71 - 24 * q^79 - 15 * q^85 + 8 * q^89 + 24 * q^91 - 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
$5$	−1.58777	+	1.57448i	−0.710073	+	0.704128i
$5$	−1.58777	+	1.57448i	−0.710073	+	0.704128i
$6$		0			0
$7$		−	1.93210i		−	0.730263i	−0.930956	−	0.365132i	$-0.881024\pi$
$7$		−	1.93210i		−	0.730263i	0.930956	−	0.365132i	$-0.118976\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.62981			−1.69745			−0.848726	−	0.528832i	$-0.822630\pi$
$11$	−5.62981			−1.69745			−0.848726	+	0.528832i	$0.822630\pi$
$12$		0			0
$13$			2.07029i			0.574195i	0.957901	+	0.287098i	$0.0926904\pi$
$13$			2.07029i			0.574195i	−0.957901	+	0.287098i	$0.907310\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.42535i		−	0.830768i	−0.909646	−	0.415384i	$-0.863647\pi$
$17$		−	3.42535i		−	0.830768i	0.909646	−	0.415384i	$-0.136353\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.35744i		−	1.11710i	−0.829470	−	0.558552i	$-0.811357\pi$
$23$		−	5.35744i		−	1.11710i	0.829470	−	0.558552i	$-0.188643\pi$
$24$		0			0
$25$	0.0420411	−	4.99982i	0.00840822	−	0.999965i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.09146			0.202679			0.101340	−	0.994852i	$-0.467687\pi$
$29$	1.09146			0.202679			0.101340	+	0.994852i	$0.467687\pi$
$30$		0			0
$31$	3.09146			0.555243			0.277622	−	0.960691i	$-0.410454\pi$
$31$	3.09146			0.555243			0.277622	+	0.960691i	$0.410454\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.04204	+	3.06773i	0.514199	+	0.518541i
$36$		0			0
$37$			3.28715i			0.540404i	0.962804	+	0.270202i	$0.0870905\pi$
$37$			3.28715i			0.540404i	−0.962804	+	0.270202i	$0.912909\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	11.6107			1.81329			0.906645	−	0.421895i	$-0.138635\pi$
$41$	11.6107			1.81329			0.906645	+	0.421895i	$0.138635\pi$
$42$		0			0
$43$			0.501623i			0.0764968i	0.999268	+	0.0382484i	$0.0121778\pi$
$43$			0.501623i			0.0764968i	−0.999268	+	0.0382484i	$0.987822\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.6470i			1.84475i	0.386294	+	0.922376i	$0.373755\pi$
$47$			12.6470i			1.84475i	−0.386294	+	0.922376i	$0.626245\pi$
$48$		0			0
$49$	3.26701			0.466715
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.01076i			0.413560i	0.978388	+	0.206780i	$0.0662984\pi$
$53$			3.01076i			0.413560i	−0.978388	+	0.206780i	$0.933702\pi$
$54$		0			0
$55$	8.93886	−	8.86401i	1.20532	−	1.19522i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.35109			−0.306086			−0.153043	−	0.988220i	$-0.548907\pi$
$59$	−2.35109			−0.306086			−0.153043	+	0.988220i	$0.548907\pi$
$60$		0			0
$61$	7.62981			0.976897			0.488449	−	0.872593i	$-0.337563\pi$
$61$	7.62981			0.976897			0.488449	+	0.872593i	$0.337563\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.25963	−	3.28715i	−0.404307	−	0.407721i
$66$		0			0
$67$			12.2324i			1.49442i	0.664585	+	0.747212i	$0.268608\pi$
$67$			12.2324i			1.49442i	−0.664585	+	0.747212i	$0.731392\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−14.3511			−1.70316			−0.851580	−	0.524224i	$-0.824355\pi$
$71$	−14.3511			−1.70316			−0.851580	+	0.524224i	$0.824355\pi$
$72$		0			0
$73$			2.87256i			0.336208i	0.985769	+	0.168104i	$0.0537645\pi$
$73$			2.87256i			0.336208i	−0.985769	+	0.168104i	$0.946236\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			10.8773i			1.23959i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.35744i			0.588056i	0.955797	+	0.294028i	$0.0949958\pi$
$83$			5.35744i			0.588056i	−0.955797	+	0.294028i	$0.905004\pi$
$84$		0			0
$85$	5.39313	+	5.43867i	0.584967	+	0.589906i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.35109			0.885214			0.442607	−	0.896716i	$-0.354054\pi$
$89$	8.35109			0.885214			0.442607	+	0.896716i	$0.354054\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.58777	−	1.57448i	0.162902	−	0.161538i
$96$		0			0
$97$			3.01076i			0.305696i	0.988250	+	0.152848i	$0.0488445\pi$
$97$			3.01076i			0.305696i	−0.988250	+	0.152848i	$0.951155\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.80536			0.478151			0.239075	−	0.971001i	$-0.423156\pi$
$101$	4.80536			0.478151			0.239075	+	0.971001i	$0.423156\pi$
$102$		0			0
$103$			4.22762i			0.416560i	0.978069	+	0.208280i	$0.0667865\pi$
$103$			4.22762i			0.416560i	−0.978069	+	0.208280i	$0.933214\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.92098i			0.862424i	0.902251	+	0.431212i	$0.141914\pi$
$107$			8.92098i			0.862424i	−0.902251	+	0.431212i	$0.858086\pi$
$108$		0			0
$109$	−5.25963			−0.503781			−0.251890	−	0.967756i	$-0.581052\pi$
$109$	−5.25963			−0.503781			−0.251890	+	0.967756i	$0.581052\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.5088i		−	1.17673i	−0.808596	−	0.588364i	$-0.799772\pi$
$113$		−	12.5088i		−	1.17673i	0.808596	−	0.588364i	$-0.200228\pi$
$114$		0			0
$115$	8.43517	+	8.50640i	0.786584	+	0.793226i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.61810			−0.606680
$120$		0			0
$121$	20.6948			1.88135
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.80536	+	8.00477i	0.698132	+	0.715969i
$126$		0			0
$127$		−	12.2952i		−	1.09102i	−0.838104	−	0.545510i	$-0.816336\pi$
$127$		−	12.2952i		−	1.09102i	0.838104	−	0.545510i	$-0.183664\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.7863			0.942400			0.471200	−	0.882026i	$-0.343821\pi$
$131$	10.7863			0.942400			0.471200	+	0.882026i	$0.343821\pi$
$132$		0			0
$133$			1.93210i			0.167534i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	4.85582i		−	0.414861i	−0.978250	−	0.207430i	$-0.933490\pi$
$137$		−	4.85582i		−	0.414861i	0.978250	−	0.207430i	$-0.0665100\pi$
$138$		0			0
$139$	11.2405			0.953409			0.476705	−	0.879064i	$-0.341831\pi$
$139$	11.2405			0.953409			0.476705	+	0.879064i	$0.341831\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	11.6554i		−	0.974669i
$144$		0			0
$145$	−1.73299	+	1.71848i	−0.143917	+	0.142712i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.62981			0.297366			0.148683	−	0.988885i	$-0.452497\pi$
$149$	3.62981			0.297366			0.148683	+	0.988885i	$0.452497\pi$
$150$		0			0
$151$	−10.3511			−0.842360			−0.421180	−	0.906977i	$-0.638384\pi$
$151$	−10.3511			−0.842360			−0.421180	+	0.906977i	$0.638384\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.90854	+	4.86744i	−0.394263	+	0.390962i
$156$		0			0
$157$		−	7.12708i		−	0.568803i	−0.958705	−	0.284402i	$-0.908205\pi$
$157$		−	7.12708i		−	0.568803i	0.958705	−	0.284402i	$-0.0917949\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.3511			−0.815780
$162$		0			0
$163$			1.21686i			0.0953118i	0.998864	+	0.0476559i	$0.0151751\pi$
$163$			1.21686i			0.0953118i	−0.998864	+	0.0476559i	$0.984825\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.8830i			1.22906i	0.788893	+	0.614531i	$0.210655\pi$
$167$			15.8830i			1.22906i	−0.788893	+	0.614531i	$0.789345\pi$
$168$		0			0
$169$	8.71390			0.670300
$170$		0			0
$171$		0			0
$172$		0			0
$173$			13.7884i			1.04831i	0.851622	+	0.524157i	$0.175620\pi$
$173$			13.7884i			1.04831i	−0.851622	+	0.524157i	$0.824380\pi$
$174$		0			0
$175$	−9.66014	−	0.0812274i	−0.730238	−	0.00614021i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.3511			0.773677			0.386838	−	0.922148i	$-0.373567\pi$
$179$	10.3511			0.773677			0.386838	+	0.922148i	$0.373567\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.17554	−	5.21925i	−0.380514	−	0.383727i
$186$		0			0
$187$			19.2841i			1.41019i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.43517			0.610348			0.305174	−	0.952297i	$-0.401285\pi$
$191$	8.43517			0.610348			0.305174	+	0.952297i	$0.401285\pi$
$192$		0			0
$193$			23.2864i			1.67619i	0.545521	+	0.838097i	$0.316332\pi$
$193$			23.2864i			1.67619i	−0.545521	+	0.838097i	$0.683668\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	8.28116i		−	0.590008i	−0.955496	−	0.295004i	$-0.904679\pi$
$197$		−	8.28116i		−	0.590008i	0.955496	−	0.295004i	$-0.0953211\pi$
$198$		0			0
$199$	18.7863			1.33172			0.665861	−	0.746076i	$-0.268064\pi$
$199$	18.7863			1.33172			0.665861	+	0.746076i	$0.268064\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	2.10881i		−	0.148009i
$204$		0			0
$205$	−18.4352	+	18.2808i	−1.28757	+	1.27679i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.62981			0.389422
$210$		0			0
$211$	−21.7789			−1.49932			−0.749660	−	0.661823i	$-0.769783\pi$
$211$	−21.7789			−1.49932			−0.749660	+	0.661823i	$0.769783\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.789795	−	0.796464i	−0.0538635	−	0.0543184i
$216$		0			0
$217$		−	5.97300i		−	0.405474i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.09146			0.477023
$222$		0			0
$223$			14.5548i			0.974662i	0.873217	+	0.487331i	$0.162030\pi$
$223$			14.5548i			0.974662i	−0.873217	+	0.487331i	$0.837970\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			2.07029i			0.137410i	0.997637	+	0.0687050i	$0.0218867\pi$
$227$			2.07029i			0.137410i	−0.997637	+	0.0687050i	$0.978113\pi$
$228$		0			0
$229$	15.4616			1.02173			0.510867	−	0.859660i	$-0.329324\pi$
$229$	15.4616			1.02173			0.510867	+	0.859660i	$0.329324\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	28.8934i		−	1.89287i	−0.322897	−	0.946434i	$-0.604657\pi$
$233$		−	28.8934i		−	1.89287i	0.322897	−	0.946434i	$-0.395343\pi$
$234$		0			0
$235$	−19.9124	−	20.0805i	−1.29894	−	1.30991i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	26.6181			1.72178			0.860891	−	0.508790i	$-0.169907\pi$
$239$	26.6181			1.72178			0.860891	+	0.508790i	$0.169907\pi$
$240$		0			0
$241$	−19.6107			−1.26324			−0.631619	−	0.775279i	$-0.717609\pi$
$241$	−19.6107			−1.26324			−0.631619	+	0.775279i	$0.717609\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.18726	+	5.14383i	−0.331402	+	0.328627i
$246$		0			0
$247$		−	2.07029i		−	0.131729i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.2522			−0.899594			−0.449797	−	0.893131i	$-0.648504\pi$
$251$	−14.2522			−0.899594			−0.449797	+	0.893131i	$0.648504\pi$
$252$		0			0
$253$			30.1614i			1.89623i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			11.2919i			0.704371i	0.935930	+	0.352185i	$0.114561\pi$
$257$			11.2919i			0.704371i	−0.935930	+	0.352185i	$0.885439\pi$
$258$		0			0
$259$	6.35109			0.394637
$260$		0			0
$261$		0			0
$262$		0			0
$263$			15.3571i			0.946959i	0.880805	+	0.473479i	$0.157002\pi$
$263$			15.3571i			0.946959i	−0.880805	+	0.473479i	$0.842998\pi$
$264$		0			0
$265$	−4.74037	−	4.78040i	−0.291199	−	0.293658i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.4426			0.697665			0.348832	−	0.937185i	$-0.386578\pi$
$269$	11.4426			0.697665			0.348832	+	0.937185i	$0.386578\pi$
$270$		0			0
$271$	12.4161			0.754223			0.377111	−	0.926168i	$-0.376917\pi$
$271$	12.4161			0.754223			0.377111	+	0.926168i	$0.376917\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.236683	+	28.1481i	−0.0142725	+	1.69739i
$276$		0			0
$277$			16.0212i			0.962619i	0.876551	+	0.481309i	$0.159839\pi$
$277$			16.0212i			0.962619i	−0.876551	+	0.481309i	$0.840161\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	24.8851			1.48452			0.742260	−	0.670112i	$-0.233754\pi$
$281$	24.8851			1.48452			0.742260	+	0.670112i	$0.233754\pi$
$282$		0			0
$283$			16.2348i			0.965057i	0.875880	+	0.482529i	$0.160282\pi$
$283$			16.2348i			0.965057i	−0.875880	+	0.482529i	$0.839718\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	22.4330i		−	1.32418i
$288$		0			0
$289$	5.26701			0.309824
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	23.2864i		−	1.36041i	−0.733023	−	0.680204i	$-0.761891\pi$
$293$		−	23.2864i		−	1.36041i	0.733023	−	0.680204i	$-0.238109\pi$
$294$		0			0
$295$	3.73299	−	3.70174i	0.217343	−	0.215523i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.0915			0.641436
$300$		0			0
$301$	0.969184			0.0558628
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.1144	+	12.0130i	−0.693669	+	0.687861i
$306$		0			0
$307$		−	0.964728i		−	0.0550599i	−0.999621	−	0.0275300i	$-0.991236\pi$
$307$		−	0.964728i		−	0.0550599i	0.999621	−	0.0275300i	$-0.00876417\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−18.1638			−1.02998			−0.514988	−	0.857197i	$-0.672204\pi$
$311$	−18.1638			−1.02998			−0.514988	+	0.857197i	$0.672204\pi$
$312$		0			0
$313$			7.30116i			0.412686i	0.978480	+	0.206343i	$0.0661562\pi$
$313$			7.30116i			0.412686i	−0.978480	+	0.206343i	$0.933844\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	7.64135i		−	0.429181i	−0.976704	−	0.214590i	$-0.931158\pi$
$317$		−	7.64135i		−	0.429181i	0.976704	−	0.214590i	$-0.0688416\pi$
$318$		0			0
$319$	−6.14473			−0.344039
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.42535i			0.190591i
$324$		0			0
$325$	10.3511	+	0.0870373i	0.574175	+	0.00482796i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	24.4352			1.34715
$330$		0			0
$331$	22.3129			1.22643			0.613214	−	0.789917i	$-0.289876\pi$
$331$	22.3129			1.22643			0.613214	+	0.789917i	$0.289876\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−19.2596	−	19.4223i	−1.05227	−	1.06115i
$336$		0			0
$337$		−	0.790654i		−	0.0430696i	−0.999768	−	0.0215348i	$-0.993145\pi$
$337$		−	0.790654i		−	0.0430696i	0.999768	−	0.0215348i	$-0.00685528\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−17.4044			−0.942499
$342$		0			0
$343$		−	19.8368i		−	1.07109i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	7.95361i		−	0.426972i	−0.976946	−	0.213486i	$-0.931518\pi$
$347$		−	7.95361i		−	0.426972i	0.976946	−	0.213486i	$-0.0684818\pi$
$348$		0			0
$349$	−15.8777			−0.849915			−0.424957	−	0.905213i	$-0.639711\pi$
$349$	−15.8777			−0.849915			−0.424957	+	0.905213i	$0.639711\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	8.45524i		−	0.450027i	−0.974356	−	0.225013i	$-0.927757\pi$
$353$		−	8.45524i		−	0.450027i	0.974356	−	0.225013i	$-0.0722426\pi$
$354$		0			0
$355$	22.7863	−	22.5955i	1.20937	−	1.19924i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.2787			0.806380			0.403190	−	0.915116i	$-0.367901\pi$
$359$	15.2787			0.806380			0.403190	+	0.915116i	$0.367901\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.52279	−	4.56098i	−0.236734	−	0.238732i
$366$		0			0
$367$		−	25.6331i		−	1.33804i	−0.743245	−	0.669019i	$-0.766714\pi$
$367$		−	25.6331i		−	1.33804i	0.743245	−	0.669019i	$-0.233286\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.81708			0.302008
$372$		0			0
$373$			1.30390i			0.0675132i	0.999430	+	0.0337566i	$0.0107471\pi$
$373$			1.30390i			0.0675132i	−0.999430	+	0.0337566i	$0.989253\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2.25964i			0.116378i
$378$		0			0
$379$	−4.87034			−0.250173			−0.125086	−	0.992146i	$-0.539921\pi$
$379$	−4.87034			−0.250173			−0.125086	+	0.992146i	$0.539921\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			20.6876i			1.05709i	0.848906	+	0.528544i	$0.177262\pi$
$383$			20.6876i			1.05709i	−0.848906	+	0.528544i	$0.822738\pi$
$384$		0			0
$385$	−17.1261	−	17.2707i	−0.872828	−	0.880198i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.252246			0.0127894			0.00639470	−	0.999980i	$-0.497964\pi$
$389$	0.252246			0.0127894			0.00639470	+	0.999980i	$0.497964\pi$
$390$		0			0
$391$	−18.3511			−0.928054
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.35109	−	6.29791i	0.319558	−	0.316882i
$396$		0			0
$397$			28.1665i			1.41364i	0.707395	+	0.706819i	$0.249870\pi$
$397$			28.1665i			1.41364i	−0.707395	+	0.706819i	$0.750130\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.25963			−0.262653			−0.131327	−	0.991339i	$-0.541924\pi$
$401$	−5.25963			−0.262653			−0.131327	+	0.991339i	$0.541924\pi$
$402$		0			0
$403$			6.40023i			0.318818i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	18.5060i		−	0.917310i
$408$		0			0
$409$	25.0533			1.23880			0.619402	−	0.785074i	$-0.287375\pi$
$409$	25.0533			1.23880			0.619402	+	0.785074i	$0.287375\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			4.54253i			0.223523i
$414$		0			0
$415$	−8.43517	−	8.50640i	−0.414066	−	0.417563i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.7022			0.620542			0.310271	−	0.950648i	$-0.399580\pi$
$419$	12.7022			0.620542			0.310271	+	0.950648i	$0.399580\pi$
$420$		0			0
$421$	13.9618			0.680457			0.340228	−	0.940343i	$-0.389496\pi$
$421$	13.9618			0.680457			0.340228	+	0.940343i	$0.389496\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−17.1261	−	0.144005i	−0.830739	−	0.00698528i
$426$		0			0
$427$		−	14.7415i		−	0.713393i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−36.1300			−1.74032			−0.870160	−	0.492770i	$-0.835984\pi$
$431$	−36.1300			−1.74032			−0.870160	+	0.492770i	$0.835984\pi$
$432$		0			0
$433$		−	5.50726i		−	0.264662i	−0.991206	−	0.132331i	$-0.957754\pi$
$433$		−	5.50726i		−	0.264662i	0.991206	−	0.132331i	$-0.0422462\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			5.35744i			0.256281i
$438$		0			0
$439$	34.4811			1.64569			0.822846	−	0.568265i	$-0.192385\pi$
$439$	34.4811			1.64569			0.822846	+	0.568265i	$0.192385\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	28.9562i		−	1.37575i	−0.725830	−	0.687874i	$-0.758544\pi$
$443$		−	28.9562i		−	1.37575i	0.725830	−	0.687874i	$-0.241456\pi$
$444$		0			0
$445$	−13.2596	+	13.1486i	−0.628567	+	0.623303i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−15.5725			−0.734913			−0.367456	−	0.930041i	$-0.619771\pi$
$449$	−15.5725			−0.734913			−0.367456	+	0.930041i	$0.619771\pi$
$450$		0			0
$451$	−65.3662			−3.07797
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.35109	+	6.29791i	−0.297744	+	0.295251i
$456$		0			0
$457$			0.715236i			0.0334573i	0.999860	+	0.0167287i	$0.00532515\pi$
$457$			0.715236i			0.0334573i	−0.999860	+	0.0167287i	$0.994675\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	28.7863			1.34071			0.670355	−	0.742041i	$-0.266142\pi$
$461$	28.7863			1.34071			0.670355	+	0.742041i	$0.266142\pi$
$462$		0			0
$463$		−	22.1055i		−	1.02733i	−0.857991	−	0.513664i	$-0.828288\pi$
$463$		−	22.1055i		−	1.02733i	0.857991	−	0.513664i	$-0.171712\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	37.6645i		−	1.74291i	−0.490478	−	0.871454i	$-0.663178\pi$
$467$		−	37.6645i		−	1.74291i	0.490478	−	0.871454i	$-0.336822\pi$
$468$		0			0
$469$	23.6342			1.09132
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	2.82405i		−	0.129850i
$474$		0			0
$475$	−0.0420411	+	4.99982i	−0.00192898	+	0.229408i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.32461			−0.243288			−0.121644	−	0.992574i	$-0.538817\pi$
$479$	−5.32461			−0.243288			−0.121644	+	0.992574i	$0.538817\pi$
$480$		0			0
$481$	−6.80536			−0.310298
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.74037	−	4.78040i	−0.215249	−	0.217067i
$486$		0			0
$487$		−	14.9425i		−	0.677109i	−0.940947	−	0.338555i	$-0.890062\pi$
$487$		−	14.9425i		−	0.677109i	0.940947	−	0.338555i	$-0.109938\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.81708			−0.262521			−0.131260	−	0.991348i	$-0.541902\pi$
$491$	−5.81708			−0.262521			−0.131260	+	0.991348i	$0.541902\pi$
$492$		0			0
$493$		−	3.73864i		−	0.168380i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			27.7277i			1.24376i
$498$		0			0
$499$	−17.4235			−0.779981			−0.389990	−	0.920819i	$-0.627522\pi$
$499$	−17.4235			−0.779981			−0.389990	+	0.920819i	$0.627522\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			0.490004i			0.0218482i	0.999940	+	0.0109241i	$0.00347731\pi$
$503$			0.490004i			0.0218482i	−0.999940	+	0.0109241i	$0.996523\pi$
$504$		0			0
$505$	−7.62981	+	7.56593i	−0.339522	+	0.336679i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.4811			0.730510			0.365255	−	0.930908i	$-0.380982\pi$
$509$	16.4811			0.730510			0.365255	+	0.930908i	$0.380982\pi$
$510$		0			0
$511$	5.55007			0.245521
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.65629	−	6.71250i	−0.293311	−	0.295788i
$516$		0			0
$517$		−	71.2001i		−	3.13138i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.0533			−0.747117			−0.373559	−	0.927607i	$-0.621863\pi$
$521$	−17.0533			−0.747117			−0.373559	+	0.927607i	$0.621863\pi$
$522$		0			0
$523$		−	34.2777i		−	1.49886i	−0.662084	−	0.749430i	$-0.730328\pi$
$523$		−	34.2777i		−	1.49886i	0.662084	−	0.749430i	$-0.269672\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	10.5893i		−	0.461278i
$528$		0			0
$529$	−5.70218			−0.247921
$530$		0			0
$531$		0			0
$532$		0			0
$533$			24.0376i			1.04118i
$534$		0			0
$535$	−14.0459	−	14.1645i	−0.607257	−	0.612384i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.3926			−0.792227
$540$		0			0
$541$	29.9427			1.28734			0.643669	−	0.765304i	$-0.277411\pi$
$541$	29.9427			1.28734			0.643669	+	0.765304i	$0.277411\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.35109	−	8.28116i	0.357721	−	0.354726i
$546$		0			0
$547$			29.1339i			1.24568i	0.782351	+	0.622838i	$0.214020\pi$
$547$			29.1339i			1.24568i	−0.782351	+	0.622838i	$0.785980\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.09146			−0.0464979
$552$		0			0
$553$			7.72838i			0.328644i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			38.4542i			1.62936i	0.579914	+	0.814678i	$0.303086\pi$
$557$			38.4542i			1.62936i	−0.579914	+	0.814678i	$0.696914\pi$
$558$		0			0
$559$	−1.03851			−0.0439241
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	8.58181i		−	0.361680i	−0.983512	−	0.180840i	$-0.942118\pi$
$563$		−	8.58181i		−	0.361680i	0.983512	−	0.180840i	$-0.0578817\pi$
$564$		0			0
$565$	19.6948	+	19.8611i	0.828566	+	0.835563i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.221120			−0.00926985			−0.00463492	−	0.999989i	$-0.501475\pi$
$569$	−0.221120			−0.00926985			−0.00463492	+	0.999989i	$0.501475\pi$
$570$		0			0
$571$	2.23315			0.0934544			0.0467272	−	0.998908i	$-0.485121\pi$
$571$	2.23315			0.0934544			0.0467272	+	0.998908i	$0.485121\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−26.7863	−	0.225233i	−1.11706	−	0.00939285i
$576$		0			0
$577$		−	16.6225i		−	0.692002i	−0.938234	−	0.346001i	$-0.887539\pi$
$577$		−	16.6225i		−	0.692002i	0.938234	−	0.346001i	$-0.112461\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.3511			0.429436
$582$		0			0
$583$		−	16.9500i		−	0.701998i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	19.2213i		−	0.793347i	−0.917960	−	0.396674i	$-0.870165\pi$
$587$		−	19.2213i		−	0.793347i	0.917960	−	0.396674i	$-0.129835\pi$
$588$		0			0
$589$	−3.09146			−0.127381
$590$		0			0
$591$		0			0
$592$		0			0
$593$			13.0230i			0.534792i	0.963587	+	0.267396i	$0.0861632\pi$
$593$			13.0230i			0.534792i	−0.963587	+	0.267396i	$0.913837\pi$
$594$		0			0
$595$	10.5080	−	10.4200i	0.430787	−	0.427180i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	27.0915			1.10693			0.553464	−	0.832873i	$-0.313306\pi$
$599$	27.0915			1.10693			0.553464	+	0.832873i	$0.313306\pi$
$600$		0			0
$601$	−5.42779			−0.221404			−0.110702	−	0.993854i	$-0.535310\pi$
$601$	−5.42779			−0.221404			−0.110702	+	0.993854i	$0.535310\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−32.8586	+	32.5835i	−1.33589	+	1.32471i
$606$		0			0
$607$			21.0663i			0.855056i	0.904002	+	0.427528i	$0.140616\pi$
$607$			21.0663i			0.855056i	−0.904002	+	0.427528i	$0.859384\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−26.1829			−1.05925
$612$		0			0
$613$			30.2753i			1.22281i	0.791318	+	0.611405i	$0.209395\pi$
$613$			30.2753i			1.22281i	−0.791318	+	0.611405i	$0.790605\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			45.3049i			1.82391i	0.410295	+	0.911953i	$0.365426\pi$
$617$			45.3049i			1.82391i	−0.410295	+	0.911953i	$0.634574\pi$
$618$		0			0
$619$	−18.2331			−0.732852			−0.366426	−	0.930447i	$-0.619419\pi$
$619$	−18.2331			−0.732852			−0.366426	+	0.930447i	$0.619419\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	16.1351i		−	0.646439i
$624$		0			0
$625$	−24.9965	−	0.420396i	−0.999859	−	0.0168158i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	11.2596			0.448951
$630$		0			0
$631$	−11.4087			−0.454173			−0.227086	−	0.973875i	$-0.572920\pi$
$631$	−11.4087			−0.454173			−0.227086	+	0.973875i	$0.572920\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19.3585	+	19.5219i	0.768217	+	0.774704i
$636$		0			0
$637$			6.76365i			0.267986i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.92330			−0.194459			−0.0972293	−	0.995262i	$-0.530998\pi$
$641$	−4.92330			−0.194459			−0.0972293	+	0.995262i	$0.530998\pi$
$642$		0			0
$643$		−	24.4136i		−	0.962779i	−0.876507	−	0.481390i	$-0.840132\pi$
$643$		−	24.4136i		−	0.962779i	0.876507	−	0.481390i	$-0.159868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			11.0424i			0.434123i	0.976158	+	0.217061i	$0.0696472\pi$
$647$			11.0424i			0.434123i	−0.976158	+	0.217061i	$0.930353\pi$
$648$		0			0
$649$	13.2362			0.519566
$650$		0			0
$651$		0			0
$652$		0			0
$653$			28.1665i			1.10224i	0.834426	+	0.551121i	$0.185800\pi$
$653$			28.1665i			1.10224i	−0.834426	+	0.551121i	$0.814200\pi$
$654$		0			0
$655$	−17.1261	+	16.9827i	−0.669173	+	0.663570i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.74037			0.184659			0.0923294	−	0.995729i	$-0.470569\pi$
$659$	4.74037			0.184659			0.0923294	+	0.995729i	$0.470569\pi$
$660$		0			0
$661$	−19.4426			−0.756228			−0.378114	−	0.925759i	$-0.623427\pi$
$661$	−19.4426			−0.756228			−0.378114	+	0.925759i	$0.623427\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.04204	−	3.06773i	−0.117965	−	0.118961i
$666$		0			0
$667$		−	5.84744i		−	0.226414i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−42.9544			−1.65824
$672$		0			0
$673$		−	12.5088i		−	0.482178i	−0.970503	−	0.241089i	$-0.922495\pi$
$673$		−	12.5088i		−	0.482178i	0.970503	−	0.241089i	$-0.0775046\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			40.7265i			1.56525i	0.622497	+	0.782623i	$0.286118\pi$
$677$			40.7265i			1.56525i	−0.622497	+	0.782623i	$0.713882\pi$
$678$		0			0
$679$	5.81708			0.223239
$680$		0			0
$681$		0			0
$682$		0			0
$683$			14.8312i			0.567500i	0.958898	+	0.283750i	$0.0915786\pi$
$683$			14.8312i			0.567500i	−0.958898	+	0.283750i	$0.908421\pi$
$684$		0			0
$685$	7.64538	+	7.70993i	0.292115	+	0.294581i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.23315			−0.237464
$690$		0			0
$691$	13.0958			0.498188			0.249094	−	0.968479i	$-0.419867\pi$
$691$	13.0958			0.498188			0.249094	+	0.968479i	$0.419867\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−17.8474	+	17.6980i	−0.676990	+	0.671322i
$696$		0			0
$697$		−	39.7707i		−	1.50642i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14.0797			0.531785			0.265892	−	0.964003i	$-0.414333\pi$
$701$	14.0797			0.531785			0.265892	+	0.964003i	$0.414333\pi$
$702$		0			0
$703$		−	3.28715i		−	0.123977i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	9.28441i		−	0.349176i
$708$		0			0
$709$	−8.38928			−0.315066			−0.157533	−	0.987514i	$-0.550354\pi$
$709$	−8.38928			−0.315066			−0.157533	+	0.987514i	$0.550354\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	16.5623i		−	0.620264i
$714$		0			0
$715$	18.3511	+	18.5060i	0.686292	+	0.692087i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.70652			0.250111			0.125055	−	0.992150i	$-0.460089\pi$
$719$	6.70652			0.250111			0.125055	+	0.992150i	$0.460089\pi$
$720$		0			0
$721$	8.16816			0.304198
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.0458862	−	5.45712i	0.00170417	−	0.202672i
$726$		0			0
$727$		−	20.5009i		−	0.760337i	−0.924917	−	0.380168i	$-0.875866\pi$
$727$		−	20.5009i		−	0.760337i	0.924917	−	0.380168i	$-0.124134\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.71823			0.0635512
$732$		0			0
$733$		−	42.4323i		−	1.56727i	−0.621220	−	0.783636i	$-0.713363\pi$
$733$		−	42.4323i		−	1.56727i	0.621220	−	0.783636i	$-0.286637\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	68.8661i		−	2.53671i
$738$		0			0
$739$	−44.4352			−1.63457			−0.817287	−	0.576231i	$-0.804523\pi$
$739$	−44.4352			−1.63457			−0.817287	+	0.576231i	$0.804523\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			36.4350i			1.33667i	0.743859	+	0.668336i	$0.232993\pi$
$743$			36.4350i			1.33667i	−0.743859	+	0.668336i	$0.767007\pi$
$744$		0			0
$745$	−5.76332	+	5.71506i	−0.211152	+	0.209384i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	17.2362			0.629797
$750$		0			0
$751$	−29.6107			−1.08051			−0.540255	−	0.841501i	$-0.681672\pi$
$751$	−29.6107			−1.08051			−0.540255	+	0.841501i	$0.681672\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.4352	−	16.2976i	0.598137	−	0.593129i
$756$		0			0
$757$			44.0252i			1.60012i	0.599917	+	0.800062i	$0.295200\pi$
$757$			44.0252i			1.60012i	−0.599917	+	0.800062i	$0.704800\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.9809			−0.941807			−0.470903	−	0.882185i	$-0.656072\pi$
$761$	−25.9809			−0.941807			−0.470903	+	0.882185i	$0.656072\pi$
$762$		0			0
$763$			10.1621i			0.367893i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	4.86744i		−	0.175753i
$768$		0			0
$769$	−2.68308			−0.0967543			−0.0483772	−	0.998829i	$-0.515405\pi$
$769$	−2.68308			−0.0967543			−0.0483772	+	0.998829i	$0.515405\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	39.5328i		−	1.42190i	−0.703244	−	0.710949i	$-0.748266\pi$
$773$		−	39.5328i		−	1.42190i	0.703244	−	0.710949i	$-0.251734\pi$
$774$		0			0
$775$	0.129968	−	15.4568i	0.00466860	−	0.555223i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.6107			−0.415997
$780$		0			0
$781$	80.7940			2.89103
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.2214	+	11.3162i	0.400510	+	0.403892i
$786$		0			0
$787$			9.15783i			0.326442i	0.986590	+	0.163221i	$0.0521883\pi$
$787$			9.15783i			0.326442i	−0.986590	+	0.163221i	$0.947812\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−24.1682			−0.859321
$792$		0			0
$793$			15.7959i			0.560930i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			15.5581i			0.551095i	0.961287	+	0.275547i	$0.0888591\pi$
$797$			15.5581i			0.551095i	−0.961287	+	0.275547i	$0.911141\pi$
$798$		0			0
$799$	43.3203			1.53256
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	16.1720i		−	0.570698i
$804$		0			0
$805$	16.4352	−	16.2976i	0.579264	−	0.574413i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−41.6184			−1.46323			−0.731613	−	0.681721i	$-0.761232\pi$
$809$	−41.6184			−1.46323			−0.731613	+	0.681721i	$0.761232\pi$
$810$		0			0
$811$	−49.1685			−1.72654			−0.863269	−	0.504744i	$-0.831587\pi$
$811$	−49.1685			−1.72654			−0.863269	+	0.504744i	$0.831587\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.91592	−	1.93210i	−0.0671117	−	0.0676784i
$816$		0			0
$817$		−	0.501623i		−	0.0175496i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.9012			0.485154			0.242577	−	0.970132i	$-0.422007\pi$
$821$	13.9012			0.485154			0.242577	+	0.970132i	$0.422007\pi$
$822$		0			0
$823$			19.0704i			0.664754i	0.943147	+	0.332377i	$0.107851\pi$
$823$			19.0704i			0.664754i	−0.943147	+	0.332377i	$0.892149\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	46.7712i		−	1.62639i	−0.581988	−	0.813197i	$-0.697725\pi$
$827$		−	46.7712i		−	1.62639i	0.581988	−	0.813197i	$-0.302275\pi$
$828$		0			0
$829$	−2.37452			−0.0824706			−0.0412353	−	0.999149i	$-0.513129\pi$
$829$	−2.37452			−0.0824706			−0.0412353	+	0.999149i	$0.513129\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	11.1906i		−	0.387732i
$834$		0			0
$835$	−25.0074	−	25.2185i	−0.865416	−	0.872724i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.57252			−0.0542894			−0.0271447	−	0.999632i	$-0.508641\pi$
$839$	−1.57252			−0.0542894			−0.0271447	+	0.999632i	$0.508641\pi$
$840$		0			0
$841$	−27.8087			−0.958921
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.8357	+	13.7198i	−0.475962	+	0.471977i
$846$		0			0
$847$		−	39.9843i		−	1.37388i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	17.6107			0.603688
$852$		0			0
$853$			1.88094i			0.0644021i	0.999481	+	0.0322010i	$0.0102517\pi$
$853$			1.88094i			0.0644021i	−0.999481	+	0.0322010i	$0.989748\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			17.0371i			0.581975i	0.956727	+	0.290987i	$0.0939837\pi$
$857$			17.0371i			0.581975i	−0.956727	+	0.290987i	$0.906016\pi$
$858$		0			0
$859$	−49.8395			−1.70050			−0.850251	−	0.526377i	$-0.823550\pi$
$859$	−49.8395			−1.70050			−0.850251	+	0.526377i	$0.823550\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.1696i		−	0.414258i	−0.978314	−	0.207129i	$-0.933588\pi$
$863$		−	12.1696i		−	0.414258i	0.978314	−	0.207129i	$-0.0664121\pi$
$864$		0			0
$865$	−21.7096	−	21.8929i	−0.738147	−	0.744380i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	22.5193			0.763913
$870$		0			0
$871$	−25.3246			−0.858092
$872$		0			0
$873$		0			0
$874$		0			0
$875$	15.4660	−	15.0807i	0.522846	−	0.509821i
$876$		0			0
$877$			42.2287i			1.42596i	0.701184	+	0.712981i	$0.252655\pi$
$877$			42.2287i			1.42596i	−0.701184	+	0.712981i	$0.747345\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.6683			0.426807			0.213403	−	0.976964i	$-0.431545\pi$
$881$	12.6683			0.426807			0.213403	+	0.976964i	$0.431545\pi$
$882$		0			0
$883$		−	11.5414i		−	0.388400i	−0.980962	−	0.194200i	$-0.937789\pi$
$883$		−	11.5414i		−	0.388400i	0.980962	−	0.194200i	$-0.0622110\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			6.36171i			0.213605i	0.994280	+	0.106803i	$0.0340613\pi$
$887$			6.36171i			0.213605i	−0.994280	+	0.106803i	$0.965939\pi$
$888$		0			0
$889$	−23.7554			−0.796732
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	12.6470i		−	0.423215i
$894$		0			0
$895$	−16.4352	+	16.2976i	−0.549367	+	0.544767i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.37421			0.112536
$900$		0			0
$901$	10.3129			0.343572
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.17554	−	3.14896i	0.105559	−	0.104675i
$906$		0			0
$907$		−	24.9538i		−	0.828576i	−0.910146	−	0.414288i	$-0.864031\pi$
$907$		−	24.9538i		−	0.828576i	0.910146	−	0.414288i	$-0.135969\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	28.6640			0.949680			0.474840	−	0.880072i	$-0.342506\pi$
$911$	28.6640			0.949680			0.474840	+	0.880072i	$0.342506\pi$
$912$		0			0
$913$		−	30.1614i		−	0.998196i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	20.8401i		−	0.688200i
$918$		0			0
$919$	25.0385			0.825944			0.412972	−	0.910744i	$-0.364491\pi$
$919$	25.0385			0.825944			0.412972	+	0.910744i	$0.364491\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	29.7109i		−	0.977947i
$924$		0			0
$925$	16.4352	+	0.138195i	0.540385	+	0.00454383i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27.2744			0.894844			0.447422	−	0.894323i	$-0.352342\pi$
$929$	27.2744			0.894844			0.447422	+	0.894323i	$0.352342\pi$
$930$		0			0
$931$	−3.26701			−0.107072
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−30.3623	−	30.6187i	−0.992954	−	1.00134i
$936$		0			0
$937$			28.7193i			0.938219i	0.883140	+	0.469109i	$0.155425\pi$
$937$			28.7193i			0.938219i	−0.883140	+	0.469109i	$0.844575\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.81708			−0.254829			−0.127415	−	0.991850i	$-0.540668\pi$
$941$	−7.81708			−0.254829			−0.127415	+	0.991850i	$0.540668\pi$
$942$		0			0
$943$		−	62.2037i		−	2.02563i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	40.4886i		−	1.31570i	−0.753148	−	0.657851i	$-0.771466\pi$
$947$		−	40.4886i		−	1.31570i	0.753148	−	0.657851i	$-0.228534\pi$
$948$		0			0
$949$	−5.94704			−0.193049
$950$		0			0
$951$		0			0
$952$		0			0
$953$			7.57857i			0.245494i	0.992438	+	0.122747i	$0.0391703\pi$
$953$			7.57857i			0.245494i	−0.992438	+	0.122747i	$0.960830\pi$
$954$		0			0
$955$	−13.3931	+	13.2810i	−0.433392	+	0.429763i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9.38190			−0.302958
$960$		0			0
$961$	−21.4429			−0.691705
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−36.6640	−	36.9736i	−1.18026	−	1.19022i
$966$		0			0
$967$			15.5195i			0.499075i	0.968365	+	0.249537i	$0.0802786\pi$
$967$			15.5195i			0.499075i	−0.968365	+	0.249537i	$0.919721\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.81708			−0.0583127			−0.0291564	−	0.999575i	$-0.509282\pi$
$971$	−1.81708			−0.0583127			−0.0291564	+	0.999575i	$0.509282\pi$
$972$		0			0
$973$		−	21.7178i		−	0.696240i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	21.0178i		−	0.672420i	−0.941787	−	0.336210i	$-0.890855\pi$
$977$		−	21.0178i		−	0.672420i	0.941787	−	0.336210i	$-0.109145\pi$
$978$		0			0
$979$	−47.0151			−1.50261
$980$		0			0
$981$		0			0
$982$		0			0
$983$			19.2339i			0.613467i	0.951795	+	0.306733i	$0.0992360\pi$
$983$			19.2339i			0.613467i	−0.951795	+	0.306733i	$0.900764\pi$
$984$		0			0
$985$	13.0385	+	13.1486i	0.415441	+	0.418949i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.68742			0.0854549
$990$		0			0
$991$	36.1682			1.14892			0.574460	−	0.818533i	$-0.305212\pi$
$991$	36.1682			1.14892			0.574460	+	0.818533i	$0.305212\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29.8283	+	29.5785i	−0.945621	+	0.937703i
$996$		0			0
$997$		−	35.6955i		−	1.13049i	−0.824923	−	0.565245i	$-0.808782\pi$
$997$		−	35.6955i		−	1.13049i	0.824923	−	0.565245i	$-0.191218\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.f.c.1369.2		6
3.2	odd	2		380.2.c.b.229.1	✓	6
5.4	even	2	inner	3420.2.f.c.1369.1		6
12.11	even	2		1520.2.d.i.609.6		6
15.2	even	4		1900.2.a.k.1.1		6
15.8	even	4		1900.2.a.k.1.6		6
15.14	odd	2		380.2.c.b.229.6	yes	6
60.23	odd	4		7600.2.a.cj.1.1		6
60.47	odd	4		7600.2.a.cj.1.6		6
60.59	even	2		1520.2.d.i.609.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.c.b.229.1	✓	6	3.2	odd	2
380.2.c.b.229.6	yes	6	15.14	odd	2
1520.2.d.i.609.1		6	60.59	even	2
1520.2.d.i.609.6		6	12.11	even	2
1900.2.a.k.1.1		6	15.2	even	4
1900.2.a.k.1.6		6	15.8	even	4
3420.2.f.c.1369.1		6	5.4	even	2	inner
3420.2.f.c.1369.2		6	1.1	even	1	trivial
7600.2.a.cj.1.1		6	60.23	odd	4
7600.2.a.cj.1.6		6	60.47	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)

\(f(q)\)	\(=\)	\(q+(-1.58777 + 1.57448i) q^{5} -1.93210i q^{7} +O(q^{10})\)	\(q+(-1.58777 + 1.57448i) q^{5} -1.93210i q^{7} -5.62981 q^{11} +2.07029i q^{13} -3.42535i q^{17} -1.00000 q^{19} -5.35744i q^{23} +(0.0420411 - 4.99982i) q^{25} +1.09146 q^{29} +3.09146 q^{31} +(3.04204 + 3.06773i) q^{35} +3.28715i q^{37} +11.6107 q^{41} +0.501623i q^{43} +12.6470i q^{47} +3.26701 q^{49} +3.01076i q^{53} +(8.93886 - 8.86401i) q^{55} -2.35109 q^{59} +7.62981 q^{61} +(-3.25963 - 3.28715i) q^{65} +12.2324i q^{67} -14.3511 q^{71} +2.87256i q^{73} +10.8773i q^{77} -4.00000 q^{79} +5.35744i q^{83} +(5.39313 + 5.43867i) q^{85} +8.35109 q^{89} +4.00000 q^{91} +(1.58777 - 1.57448i) q^{95} +3.01076i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5}+O(q^{10}) \) 6 * q + q^5	\( 6 q + q^{5} - 18 q^{11} - 6 q^{19} - 5 q^{25} - 4 q^{29} + 8 q^{31} + 13 q^{35} - 4 q^{41} - 12 q^{49} + q^{55} + 28 q^{59} + 30 q^{61} + 12 q^{65} - 44 q^{71} - 24 q^{79} - 15 q^{85} + 8 q^{89} + 24 q^{91} - q^{95}+O(q^{100}) \) 6 * q + q^5 - 18 * q^11 - 6 * q^19 - 5 * q^25 - 4 * q^29 + 8 * q^31 + 13 * q^35 - 4 * q^41 - 12 * q^49 + q^55 + 28 * q^59 + 30 * q^61 + 12 * q^65 - 44 * q^71 - 24 * q^79 - 15 * q^85 + 8 * q^89 + 24 * q^91 - q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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