LMFDB - Embedded newform 8304.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8304,2,Mod(1,8304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8304 = 2^{4} \cdot 3 \cdot 173 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8304.a (trivial)

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:	yes
Analytic conductor:	$66.3077738385$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 13x^{5} - 8x^{4} + 24x^{3} + 6x^{2} - 13x + 2 $ x^7 - 13x^5 - 8x^4 + 24x^3 + 6x^2 - 13*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2076)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70700$ of defining polynomial
Character	$\chi$	$=$	8304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.13640 q^{5} -1.30202 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.13640 q^{5} -1.30202 q^{7} +1.00000 q^{9} -2.48719 q^{11} +3.75739 q^{13} +1.13640 q^{15} -0.436809 q^{17} -0.587426 q^{19} +1.30202 q^{21} -3.21981 q^{23} -3.70860 q^{25} -1.00000 q^{27} +4.25423 q^{29} +7.37967 q^{31} +2.48719 q^{33} +1.47961 q^{35} -6.01879 q^{37} -3.75739 q^{39} +10.9702 q^{41} +0.537443 q^{43} -1.13640 q^{45} -4.83383 q^{47} -5.30475 q^{49} +0.436809 q^{51} +7.95340 q^{53} +2.82644 q^{55} +0.587426 q^{57} +4.53901 q^{59} -1.40323 q^{61} -1.30202 q^{63} -4.26988 q^{65} -8.68364 q^{67} +3.21981 q^{69} -8.38142 q^{71} +15.0903 q^{73} +3.70860 q^{75} +3.23837 q^{77} -9.66779 q^{79} +1.00000 q^{81} +9.17169 q^{83} +0.496388 q^{85} -4.25423 q^{87} -1.50391 q^{89} -4.89218 q^{91} -7.37967 q^{93} +0.667549 q^{95} +8.68548 q^{97} -2.48719 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} + 6 q^{5} - q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 + 6 * q^5 - q^7 + 7 * q^9	$ 7 q - 7 q^{3} + 6 q^{5} - q^{7} + 7 q^{9} - 4 q^{11} + 4 q^{13} - 6 q^{15} + 7 q^{17} - 9 q^{19} + q^{21} - 17 q^{23} + 5 q^{25} - 7 q^{27} + 6 q^{29} - 10 q^{31} + 4 q^{33} - 9 q^{35} + 6 q^{37} - 4 q^{39} + 7 q^{41} - 11 q^{43} + 6 q^{45} - 22 q^{47} - 8 q^{49} - 7 q^{51} + 20 q^{53} - 21 q^{55} + 9 q^{57} - 18 q^{59} - q^{63} - 2 q^{65} - 8 q^{67} + 17 q^{69} - 14 q^{71} - 14 q^{73} - 5 q^{75} + 13 q^{77} - q^{79} + 7 q^{81} - 7 q^{83} - 16 q^{85} - 6 q^{87} - 6 q^{89} - 10 q^{91} + 10 q^{93} - 31 q^{95} - 17 q^{97} - 4 q^{99}+O(q^{100}) $ 7 * q - 7 * q^3 + 6 * q^5 - q^7 + 7 * q^9 - 4 * q^11 + 4 * q^13 - 6 * q^15 + 7 * q^17 - 9 * q^19 + q^21 - 17 * q^23 + 5 * q^25 - 7 * q^27 + 6 * q^29 - 10 * q^31 + 4 * q^33 - 9 * q^35 + 6 * q^37 - 4 * q^39 + 7 * q^41 - 11 * q^43 + 6 * q^45 - 22 * q^47 - 8 * q^49 - 7 * q^51 + 20 * q^53 - 21 * q^55 + 9 * q^57 - 18 * q^59 - q^63 - 2 * q^65 - 8 * q^67 + 17 * q^69 - 14 * q^71 - 14 * q^73 - 5 * q^75 + 13 * q^77 - q^79 + 7 * q^81 - 7 * q^83 - 16 * q^85 - 6 * q^87 - 6 * q^89 - 10 * q^91 + 10 * q^93 - 31 * q^95 - 17 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.13640	−0.508212	−0.254106	−	0.967176i	$-0.581781\pi$
$5$	−1.13640	−0.508212	−0.254106	+	0.967176i	$0.581781\pi$
$6$	0	0
$7$	−1.30202	−0.492116	−0.246058	−	0.969255i	$-0.579135\pi$
$7$	−1.30202	−0.492116	−0.246058	+	0.969255i	$0.579135\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.48719	−0.749918	−0.374959	−	0.927041i	$-0.622343\pi$
$11$	−2.48719	−0.749918	−0.374959	+	0.927041i	$0.622343\pi$
$12$	0	0
$13$	3.75739	1.04211	0.521056	−	0.853523i	$-0.325538\pi$
$13$	3.75739	1.04211	0.521056	+	0.853523i	$0.325538\pi$
$14$	0	0
$15$	1.13640	0.293416
$16$	0	0
$17$	−0.436809	−0.105942	−0.0529708	−	0.998596i	$-0.516869\pi$
$17$	−0.436809	−0.105942	−0.0529708	+	0.998596i	$0.516869\pi$
$18$	0	0
$19$	−0.587426	−0.134765	−0.0673824	−	0.997727i	$-0.521465\pi$
$19$	−0.587426	−0.134765	−0.0673824	+	0.997727i	$0.521465\pi$
$20$	0	0
$21$	1.30202	0.284123
$22$	0	0
$23$	−3.21981	−0.671376	−0.335688	−	0.941973i	$-0.608969\pi$
$23$	−3.21981	−0.671376	−0.335688	+	0.941973i	$0.608969\pi$
$24$	0	0
$25$	−3.70860	−0.741721
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.25423	0.789991	0.394995	−	0.918683i	$-0.370746\pi$
$29$	4.25423	0.789991	0.394995	+	0.918683i	$0.370746\pi$
$30$	0	0
$31$	7.37967	1.32543	0.662714	−	0.748873i	$-0.269405\pi$
$31$	7.37967	1.32543	0.662714	+	0.748873i	$0.269405\pi$
$32$	0	0
$33$	2.48719	0.432965
$34$	0	0
$35$	1.47961	0.250099
$36$	0	0
$37$	−6.01879	−0.989483	−0.494742	−	0.869040i	$-0.664737\pi$
$37$	−6.01879	−0.989483	−0.494742	+	0.869040i	$0.664737\pi$
$38$	0	0
$39$	−3.75739	−0.601664
$40$	0	0
$41$	10.9702	1.71326	0.856632	−	0.515927i	$-0.172553\pi$
$41$	10.9702	1.71326	0.856632	+	0.515927i	$0.172553\pi$
$42$	0	0
$43$	0.537443	0.0819593	0.0409797	−	0.999160i	$-0.486952\pi$
$43$	0.537443	0.0819593	0.0409797	+	0.999160i	$0.486952\pi$
$44$	0	0
$45$	−1.13640	−0.169404
$46$	0	0
$47$	−4.83383	−0.705086	−0.352543	−	0.935796i	$-0.614683\pi$
$47$	−4.83383	−0.705086	−0.352543	+	0.935796i	$0.614683\pi$
$48$	0	0
$49$	−5.30475	−0.757822
$50$	0	0
$51$	0.436809	0.0611654
$52$	0	0
$53$	7.95340	1.09248	0.546242	−	0.837628i	$-0.316058\pi$
$53$	7.95340	1.09248	0.546242	+	0.837628i	$0.316058\pi$
$54$	0	0
$55$	2.82644	0.381117
$56$	0	0
$57$	0.587426	0.0778065
$58$	0	0
$59$	4.53901	0.590929	0.295464	−	0.955354i	$-0.404526\pi$
$59$	4.53901	0.590929	0.295464	+	0.955354i	$0.404526\pi$
$60$	0	0
$61$	−1.40323	−0.179666	−0.0898328	−	0.995957i	$-0.528633\pi$
$61$	−1.40323	−0.179666	−0.0898328	+	0.995957i	$0.528633\pi$
$62$	0	0
$63$	−1.30202	−0.164039
$64$	0	0
$65$	−4.26988	−0.529613
$66$	0	0
$67$	−8.68364	−1.06088	−0.530438	−	0.847724i	$-0.677972\pi$
$67$	−8.68364	−1.06088	−0.530438	+	0.847724i	$0.677972\pi$
$68$	0	0
$69$	3.21981	0.387619
$70$	0	0
$71$	−8.38142	−0.994692	−0.497346	−	0.867552i	$-0.665692\pi$
$71$	−8.38142	−0.994692	−0.497346	+	0.867552i	$0.665692\pi$
$72$	0	0
$73$	15.0903	1.76618	0.883090	−	0.469203i	$-0.155459\pi$
$73$	15.0903	1.76618	0.883090	+	0.469203i	$0.155459\pi$
$74$	0	0
$75$	3.70860	0.428233
$76$	0	0
$77$	3.23837	0.369046
$78$	0	0
$79$	−9.66779	−1.08771	−0.543856	−	0.839179i	$-0.683036\pi$
$79$	−9.66779	−1.08771	−0.543856	+	0.839179i	$0.683036\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.17169	1.00672	0.503362	−	0.864076i	$-0.332096\pi$
$83$	9.17169	1.00672	0.503362	+	0.864076i	$0.332096\pi$
$84$	0	0
$85$	0.496388	0.0538408
$86$	0	0
$87$	−4.25423	−0.456101
$88$	0	0
$89$	−1.50391	−0.159414	−0.0797069	−	0.996818i	$-0.525398\pi$
$89$	−1.50391	−0.159414	−0.0797069	+	0.996818i	$0.525398\pi$
$90$	0	0
$91$	−4.89218	−0.512840
$92$	0	0
$93$	−7.37967	−0.765236
$94$	0	0
$95$	0.667549	0.0684891
$96$	0	0
$97$	8.68548	0.881877	0.440939	−	0.897537i	$-0.354646\pi$
$97$	8.68548	0.881877	0.440939	+	0.897537i	$0.354646\pi$
$98$	0	0
$99$	−2.48719	−0.249973
$100$	0	0
$101$	17.8750	1.77863	0.889315	−	0.457296i	$-0.151182\pi$
$101$	17.8750	1.77863	0.889315	+	0.457296i	$0.151182\pi$
$102$	0	0
$103$	1.82384	0.179708	0.0898541	−	0.995955i	$-0.471360\pi$
$103$	1.82384	0.179708	0.0898541	+	0.995955i	$0.471360\pi$
$104$	0	0
$105$	−1.47961	−0.144395
$106$	0	0
$107$	−1.36281	−0.131747	−0.0658737	−	0.997828i	$-0.520983\pi$
$107$	−1.36281	−0.131747	−0.0658737	+	0.997828i	$0.520983\pi$
$108$	0	0
$109$	−18.6708	−1.78834	−0.894171	−	0.447725i	$-0.852234\pi$
$109$	−18.6708	−1.78834	−0.894171	+	0.447725i	$0.852234\pi$
$110$	0	0
$111$	6.01879	0.571278
$112$	0	0
$113$	5.63997	0.530564	0.265282	−	0.964171i	$-0.414535\pi$
$113$	5.63997	0.530564	0.265282	+	0.964171i	$0.414535\pi$
$114$	0	0
$115$	3.65897	0.341201
$116$	0	0
$117$	3.75739	0.347371
$118$	0	0
$119$	0.568732	0.0521356
$120$	0	0
$121$	−4.81386	−0.437624
$122$	0	0
$123$	−10.9702	−0.989154
$124$	0	0
$125$	9.89642	0.885163
$126$	0	0
$127$	−9.82875	−0.872161	−0.436080	−	0.899908i	$-0.643634\pi$
$127$	−9.82875	−0.872161	−0.436080	+	0.899908i	$0.643634\pi$
$128$	0	0
$129$	−0.537443	−0.0473193
$130$	0	0
$131$	−9.25251	−0.808395	−0.404198	−	0.914672i	$-0.632449\pi$
$131$	−9.25251	−0.808395	−0.404198	+	0.914672i	$0.632449\pi$
$132$	0	0
$133$	0.764839	0.0663199
$134$	0	0
$135$	1.13640	0.0978054
$136$	0	0
$137$	−10.6870	−0.913052	−0.456526	−	0.889710i	$-0.650906\pi$
$137$	−10.6870	−0.913052	−0.456526	+	0.889710i	$0.650906\pi$
$138$	0	0
$139$	6.64013	0.563209	0.281604	−	0.959531i	$-0.409133\pi$
$139$	6.64013	0.563209	0.281604	+	0.959531i	$0.409133\pi$
$140$	0	0
$141$	4.83383	0.407082
$142$	0	0
$143$	−9.34535	−0.781498
$144$	0	0
$145$	−4.83449	−0.401482
$146$	0	0
$147$	5.30475	0.437529
$148$	0	0
$149$	2.06925	0.169520	0.0847600	−	0.996401i	$-0.472988\pi$
$149$	2.06925	0.169520	0.0847600	+	0.996401i	$0.472988\pi$
$150$	0	0
$151$	−20.8561	−1.69725	−0.848624	−	0.528997i	$-0.822568\pi$
$151$	−20.8561	−1.69725	−0.848624	+	0.528997i	$0.822568\pi$
$152$	0	0
$153$	−0.436809	−0.0353139
$154$	0	0
$155$	−8.38623	−0.673598
$156$	0	0
$157$	−5.28304	−0.421632	−0.210816	−	0.977526i	$-0.567612\pi$
$157$	−5.28304	−0.421632	−0.210816	+	0.977526i	$0.567612\pi$
$158$	0	0
$159$	−7.95340	−0.630746
$160$	0	0
$161$	4.19224	0.330395
$162$	0	0
$163$	21.2538	1.66473	0.832364	−	0.554229i	$-0.186987\pi$
$163$	21.2538	1.66473	0.832364	+	0.554229i	$0.186987\pi$
$164$	0	0
$165$	−2.82644	−0.220038
$166$	0	0
$167$	23.7951	1.84132	0.920658	−	0.390369i	$-0.127653\pi$
$167$	23.7951	1.84132	0.920658	+	0.390369i	$0.127653\pi$
$168$	0	0
$169$	1.11796	0.0859969
$170$	0	0
$171$	−0.587426	−0.0449216
$172$	0	0
$173$	1.00000	0.0760286
$173$	1.00000	0.0760286
$174$	0	0
$175$	4.82866	0.365013
$176$	0	0
$177$	−4.53901	−0.341173
$178$	0	0
$179$	8.67626	0.648494	0.324247	−	0.945972i	$-0.394889\pi$
$179$	8.67626	0.648494	0.324247	+	0.945972i	$0.394889\pi$
$180$	0	0
$181$	−9.58964	−0.712792	−0.356396	−	0.934335i	$-0.615995\pi$
$181$	−9.58964	−0.712792	−0.356396	+	0.934335i	$0.615995\pi$
$182$	0	0
$183$	1.40323	0.103730
$184$	0	0
$185$	6.83973	0.502867
$186$	0	0
$187$	1.08643	0.0794475
$188$	0	0
$189$	1.30202	0.0947078
$190$	0	0
$191$	13.3296	0.964497	0.482249	−	0.876034i	$-0.339820\pi$
$191$	13.3296	0.964497	0.482249	+	0.876034i	$0.339820\pi$
$192$	0	0
$193$	−6.49992	−0.467875	−0.233937	−	0.972252i	$-0.575161\pi$
$193$	−6.49992	−0.467875	−0.233937	+	0.972252i	$0.575161\pi$
$194$	0	0
$195$	4.26988	0.305772
$196$	0	0
$197$	−15.4707	−1.10224	−0.551122	−	0.834425i	$-0.685800\pi$
$197$	−15.4707	−1.10224	−0.551122	+	0.834425i	$0.685800\pi$
$198$	0	0
$199$	−8.11088	−0.574965	−0.287483	−	0.957786i	$-0.592818\pi$
$199$	−8.11088	−0.574965	−0.287483	+	0.957786i	$0.592818\pi$
$200$	0	0
$201$	8.68364	0.612497
$202$	0	0
$203$	−5.53908	−0.388767
$204$	0	0
$205$	−12.4665	−0.870701
$206$	0	0
$207$	−3.21981	−0.223792
$208$	0	0
$209$	1.46104	0.101062
$210$	0	0
$211$	4.24970	0.292562	0.146281	−	0.989243i	$-0.453270\pi$
$211$	4.24970	0.292562	0.146281	+	0.989243i	$0.453270\pi$
$212$	0	0
$213$	8.38142	0.574286
$214$	0	0
$215$	−0.610748	−0.0416527
$216$	0	0
$217$	−9.60845	−0.652264
$218$	0	0
$219$	−15.0903	−1.01970
$220$	0	0
$221$	−1.64126	−0.110403
$222$	0	0
$223$	5.83056	0.390443	0.195222	−	0.980759i	$-0.437457\pi$
$223$	5.83056	0.390443	0.195222	+	0.980759i	$0.437457\pi$
$224$	0	0
$225$	−3.70860	−0.247240
$226$	0	0
$227$	−20.5776	−1.36578	−0.682892	−	0.730519i	$-0.739278\pi$
$227$	−20.5776	−1.36578	−0.682892	+	0.730519i	$0.739278\pi$
$228$	0	0
$229$	4.51093	0.298091	0.149045	−	0.988830i	$-0.452380\pi$
$229$	4.51093	0.298091	0.149045	+	0.988830i	$0.452380\pi$
$230$	0	0
$231$	−3.23837	−0.213069
$232$	0	0
$233$	4.61494	0.302335	0.151167	−	0.988508i	$-0.451697\pi$
$233$	4.61494	0.302335	0.151167	+	0.988508i	$0.451697\pi$
$234$	0	0
$235$	5.49314	0.358333
$236$	0	0
$237$	9.66779	0.627990
$238$	0	0
$239$	−11.2965	−0.730710	−0.365355	−	0.930868i	$-0.619052\pi$
$239$	−11.2965	−0.730710	−0.365355	+	0.930868i	$0.619052\pi$
$240$	0	0
$241$	−29.0353	−1.87033	−0.935165	−	0.354212i	$-0.884749\pi$
$241$	−29.0353	−1.87033	−0.935165	+	0.354212i	$0.884749\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.02830	0.385134
$246$	0	0
$247$	−2.20719	−0.140440
$248$	0	0
$249$	−9.17169	−0.581233
$250$	0	0
$251$	−1.43576	−0.0906242	−0.0453121	−	0.998973i	$-0.514428\pi$
$251$	−1.43576	−0.0906242	−0.0453121	+	0.998973i	$0.514428\pi$
$252$	0	0
$253$	8.00829	0.503477
$254$	0	0
$255$	−0.496388	−0.0310850
$256$	0	0
$257$	−18.1069	−1.12948	−0.564738	−	0.825270i	$-0.691023\pi$
$257$	−18.1069	−1.12948	−0.564738	+	0.825270i	$0.691023\pi$
$258$	0	0
$259$	7.83656	0.486940
$260$	0	0
$261$	4.25423	0.263330
$262$	0	0
$263$	1.08862	0.0671271	0.0335636	−	0.999437i	$-0.489314\pi$
$263$	1.08862	0.0671271	0.0335636	+	0.999437i	$0.489314\pi$
$264$	0	0
$265$	−9.03821	−0.555213
$266$	0	0
$267$	1.50391	0.0920376
$268$	0	0
$269$	−10.3876	−0.633340	−0.316670	−	0.948536i	$-0.602565\pi$
$269$	−10.3876	−0.633340	−0.316670	+	0.948536i	$0.602565\pi$
$270$	0	0
$271$	5.04142	0.306245	0.153122	−	0.988207i	$-0.451067\pi$
$271$	5.04142	0.306245	0.153122	+	0.988207i	$0.451067\pi$
$272$	0	0
$273$	4.89218	0.296088
$274$	0	0
$275$	9.22402	0.556229
$276$	0	0
$277$	22.8667	1.37393	0.686965	−	0.726691i	$-0.258943\pi$
$277$	22.8667	1.37393	0.686965	+	0.726691i	$0.258943\pi$
$278$	0	0
$279$	7.37967	0.441809
$280$	0	0
$281$	−8.68406	−0.518047	−0.259024	−	0.965871i	$-0.583401\pi$
$281$	−8.68406	−0.518047	−0.259024	+	0.965871i	$0.583401\pi$
$282$	0	0
$283$	−19.0137	−1.13025	−0.565124	−	0.825006i	$-0.691172\pi$
$283$	−19.0137	−1.13025	−0.565124	+	0.825006i	$0.691172\pi$
$284$	0	0
$285$	−0.667549	−0.0395422
$286$	0	0
$287$	−14.2834	−0.843125
$288$	0	0
$289$	−16.8092	−0.988776
$290$	0	0
$291$	−8.68548	−0.509152
$292$	0	0
$293$	17.2415	1.00726	0.503629	−	0.863920i	$-0.331998\pi$
$293$	17.2415	1.00726	0.503629	+	0.863920i	$0.331998\pi$
$294$	0	0
$295$	−5.15811	−0.300317
$296$	0	0
$297$	2.48719	0.144322
$298$	0	0
$299$	−12.0981	−0.699649
$300$	0	0
$301$	−0.699760	−0.0403335
$302$	0	0
$303$	−17.8750	−1.02689
$304$	0	0
$305$	1.59463	0.0913081
$306$	0	0
$307$	5.85444	0.334131	0.167065	−	0.985946i	$-0.446571\pi$
$307$	5.85444	0.334131	0.167065	+	0.985946i	$0.446571\pi$
$308$	0	0
$309$	−1.82384	−0.103755
$310$	0	0
$311$	20.6193	1.16921	0.584607	−	0.811316i	$-0.301249\pi$
$311$	20.6193	1.16921	0.584607	+	0.811316i	$0.301249\pi$
$312$	0	0
$313$	−3.52006	−0.198965	−0.0994827	−	0.995039i	$-0.531719\pi$
$313$	−3.52006	−0.198965	−0.0994827	+	0.995039i	$0.531719\pi$
$314$	0	0
$315$	1.47961	0.0833663
$316$	0	0
$317$	−15.3183	−0.860359	−0.430180	−	0.902743i	$-0.641550\pi$
$317$	−15.3183	−0.860359	−0.430180	+	0.902743i	$0.641550\pi$
$318$	0	0
$319$	−10.5811	−0.592428
$320$	0	0
$321$	1.36281	0.0760644
$322$	0	0
$323$	0.256593	0.0142772
$324$	0	0
$325$	−13.9347	−0.772956
$326$	0	0
$327$	18.6708	1.03250
$328$	0	0
$329$	6.29372	0.346984
$330$	0	0
$331$	−1.92001	−0.105533	−0.0527667	−	0.998607i	$-0.516804\pi$
$331$	−1.92001	−0.105533	−0.0527667	+	0.998607i	$0.516804\pi$
$332$	0	0
$333$	−6.01879	−0.329828
$334$	0	0
$335$	9.86805	0.539149
$336$	0	0
$337$	−21.9985	−1.19833	−0.599167	−	0.800624i	$-0.704502\pi$
$337$	−21.9985	−1.19833	−0.599167	+	0.800624i	$0.704502\pi$
$338$	0	0
$339$	−5.63997	−0.306321
$340$	0	0
$341$	−18.3547	−0.993961
$342$	0	0
$343$	16.0210	0.865052
$344$	0	0
$345$	−3.65897	−0.196993
$346$	0	0
$347$	−27.5429	−1.47858	−0.739291	−	0.673386i	$-0.764839\pi$
$347$	−27.5429	−1.47858	−0.739291	+	0.673386i	$0.764839\pi$
$348$	0	0
$349$	2.22349	0.119021	0.0595104	−	0.998228i	$-0.481046\pi$
$349$	2.22349	0.119021	0.0595104	+	0.998228i	$0.481046\pi$
$350$	0	0
$351$	−3.75739	−0.200555
$352$	0	0
$353$	−29.5379	−1.57215	−0.786073	−	0.618134i	$-0.787889\pi$
$353$	−29.5379	−1.57215	−0.786073	+	0.618134i	$0.787889\pi$
$354$	0	0
$355$	9.52461	0.505514
$356$	0	0
$357$	−0.568732	−0.0301005
$358$	0	0
$359$	−27.2754	−1.43954	−0.719771	−	0.694212i	$-0.755753\pi$
$359$	−27.2754	−1.43954	−0.719771	+	0.694212i	$0.755753\pi$
$360$	0	0
$361$	−18.6549	−0.981838
$362$	0	0
$363$	4.81386	0.252662
$364$	0	0
$365$	−17.1485	−0.897594
$366$	0	0
$367$	−20.7995	−1.08572	−0.542861	−	0.839822i	$-0.682659\pi$
$367$	−20.7995	−1.08572	−0.542861	+	0.839822i	$0.682659\pi$
$368$	0	0
$369$	10.9702	0.571088
$370$	0	0
$371$	−10.3555	−0.537629
$372$	0	0
$373$	25.4752	1.31905	0.659527	−	0.751681i	$-0.270757\pi$
$373$	25.4752	1.31905	0.659527	+	0.751681i	$0.270757\pi$
$374$	0	0
$375$	−9.89642	−0.511049
$376$	0	0
$377$	15.9848	0.823258
$378$	0	0
$379$	−19.4119	−0.997121	−0.498560	−	0.866855i	$-0.666138\pi$
$379$	−19.4119	−0.997121	−0.498560	+	0.866855i	$0.666138\pi$
$380$	0	0
$381$	9.82875	0.503542
$382$	0	0
$383$	−8.06608	−0.412158	−0.206079	−	0.978535i	$-0.566070\pi$
$383$	−8.06608	−0.412158	−0.206079	+	0.978535i	$0.566070\pi$
$384$	0	0
$385$	−3.68007	−0.187554
$386$	0	0
$387$	0.537443	0.0273198
$388$	0	0
$389$	−36.2449	−1.83769	−0.918844	−	0.394621i	$-0.870876\pi$
$389$	−36.2449	−1.83769	−0.918844	+	0.394621i	$0.870876\pi$
$390$	0	0
$391$	1.40644	0.0711267
$392$	0	0
$393$	9.25251	0.466727
$394$	0	0
$395$	10.9864	0.552787
$396$	0	0
$397$	−10.8783	−0.545968	−0.272984	−	0.962019i	$-0.588011\pi$
$397$	−10.8783	−0.545968	−0.272984	+	0.962019i	$0.588011\pi$
$398$	0	0
$399$	−0.764839	−0.0382898
$400$	0	0
$401$	−3.54288	−0.176923	−0.0884615	−	0.996080i	$-0.528195\pi$
$401$	−3.54288	−0.176923	−0.0884615	+	0.996080i	$0.528195\pi$
$402$	0	0
$403$	27.7283	1.38124
$404$	0	0
$405$	−1.13640	−0.0564680
$406$	0	0
$407$	14.9699	0.742031
$408$	0	0
$409$	−1.96590	−0.0972076	−0.0486038	−	0.998818i	$-0.515477\pi$
$409$	−1.96590	−0.0972076	−0.0486038	+	0.998818i	$0.515477\pi$
$410$	0	0
$411$	10.6870	0.527151
$412$	0	0
$413$	−5.90986	−0.290805
$414$	0	0
$415$	−10.4227	−0.511629
$416$	0	0
$417$	−6.64013	−0.325169
$418$	0	0
$419$	−14.6616	−0.716265	−0.358132	−	0.933671i	$-0.616586\pi$
$419$	−14.6616	−0.716265	−0.358132	+	0.933671i	$0.616586\pi$
$420$	0	0
$421$	−1.22400	−0.0596539	−0.0298270	−	0.999555i	$-0.509496\pi$
$421$	−1.22400	−0.0596539	−0.0298270	+	0.999555i	$0.509496\pi$
$422$	0	0
$423$	−4.83383	−0.235029
$424$	0	0
$425$	1.61995	0.0785791
$426$	0	0
$427$	1.82703	0.0884163
$428$	0	0
$429$	9.34535	0.451198
$430$	0	0
$431$	−20.0464	−0.965603	−0.482802	−	0.875730i	$-0.660381\pi$
$431$	−20.0464	−0.965603	−0.482802	+	0.875730i	$0.660381\pi$
$432$	0	0
$433$	−31.4225	−1.51007	−0.755034	−	0.655686i	$-0.772379\pi$
$433$	−31.4225	−1.51007	−0.755034	+	0.655686i	$0.772379\pi$
$434$	0	0
$435$	4.83449	0.231796
$436$	0	0
$437$	1.89140	0.0904779
$438$	0	0
$439$	27.1803	1.29724	0.648621	−	0.761111i	$-0.275346\pi$
$439$	27.1803	1.29724	0.648621	+	0.761111i	$0.275346\pi$
$440$	0	0
$441$	−5.30475	−0.252607
$442$	0	0
$443$	13.4027	0.636783	0.318392	−	0.947959i	$-0.396857\pi$
$443$	13.4027	0.636783	0.318392	+	0.947959i	$0.396857\pi$
$444$	0	0
$445$	1.70903	0.0810160
$446$	0	0
$447$	−2.06925	−0.0978724
$448$	0	0
$449$	−28.5130	−1.34561	−0.672805	−	0.739820i	$-0.734911\pi$
$449$	−28.5130	−1.34561	−0.672805	+	0.739820i	$0.734911\pi$
$450$	0	0
$451$	−27.2851	−1.28481
$452$	0	0
$453$	20.8561	0.979906
$454$	0	0
$455$	5.55945	0.260631
$456$	0	0
$457$	−36.3480	−1.70029	−0.850144	−	0.526550i	$-0.823485\pi$
$457$	−36.3480	−1.70029	−0.850144	+	0.526550i	$0.823485\pi$
$458$	0	0
$459$	0.436809	0.0203885
$460$	0	0
$461$	39.4865	1.83907	0.919536	−	0.393006i	$-0.128565\pi$
$461$	39.4865	1.83907	0.919536	+	0.393006i	$0.128565\pi$
$462$	0	0
$463$	−24.2395	−1.12650	−0.563252	−	0.826285i	$-0.690450\pi$
$463$	−24.2395	−1.12650	−0.563252	+	0.826285i	$0.690450\pi$
$464$	0	0
$465$	8.38623	0.388902
$466$	0	0
$467$	18.1652	0.840585	0.420292	−	0.907389i	$-0.361928\pi$
$467$	18.1652	0.840585	0.420292	+	0.907389i	$0.361928\pi$
$468$	0	0
$469$	11.3062	0.522074
$470$	0	0
$471$	5.28304	0.243430
$472$	0	0
$473$	−1.33673	−0.0614627
$474$	0	0
$475$	2.17853	0.0999579
$476$	0	0
$477$	7.95340	0.364161
$478$	0	0
$479$	−23.2728	−1.06336	−0.531681	−	0.846945i	$-0.678439\pi$
$479$	−23.2728	−1.06336	−0.531681	+	0.846945i	$0.678439\pi$
$480$	0	0
$481$	−22.6149	−1.03115
$482$	0	0
$483$	−4.19224	−0.190754
$484$	0	0
$485$	−9.87015	−0.448180
$486$	0	0
$487$	33.8813	1.53531	0.767654	−	0.640865i	$-0.221424\pi$
$487$	33.8813	1.53531	0.767654	+	0.640865i	$0.221424\pi$
$488$	0	0
$489$	−21.2538	−0.961131
$490$	0	0
$491$	24.2132	1.09273	0.546363	−	0.837549i	$-0.316012\pi$
$491$	24.2132	1.09273	0.546363	+	0.837549i	$0.316012\pi$
$492$	0	0
$493$	−1.85828	−0.0836929
$494$	0	0
$495$	2.82644	0.127039
$496$	0	0
$497$	10.9127	0.489504
$498$	0	0
$499$	−8.23598	−0.368693	−0.184347	−	0.982861i	$-0.559017\pi$
$499$	−8.23598	−0.368693	−0.184347	+	0.982861i	$0.559017\pi$
$500$	0	0
$501$	−23.7951	−1.06308
$502$	0	0
$503$	−19.1615	−0.854369	−0.427184	−	0.904165i	$-0.640494\pi$
$503$	−19.1615	−0.854369	−0.427184	+	0.904165i	$0.640494\pi$
$504$	0	0
$505$	−20.3131	−0.903920
$506$	0	0
$507$	−1.11796	−0.0496503
$508$	0	0
$509$	37.3591	1.65591	0.827957	−	0.560792i	$-0.189503\pi$
$509$	37.3591	1.65591	0.827957	+	0.560792i	$0.189503\pi$
$510$	0	0
$511$	−19.6478	−0.869166
$512$	0	0
$513$	0.587426	0.0259355
$514$	0	0
$515$	−2.07260	−0.0913298
$516$	0	0
$517$	12.0227	0.528756
$518$	0	0
$519$	−1.00000	−0.0438951
$520$	0	0
$521$	10.8352	0.474699	0.237350	−	0.971424i	$-0.423721\pi$
$521$	10.8352	0.474699	0.237350	+	0.971424i	$0.423721\pi$
$522$	0	0
$523$	−6.34765	−0.277563	−0.138782	−	0.990323i	$-0.544319\pi$
$523$	−6.34765	−0.277563	−0.138782	+	0.990323i	$0.544319\pi$
$524$	0	0
$525$	−4.82866	−0.210740
$526$	0	0
$527$	−3.22350	−0.140418
$528$	0	0
$529$	−12.6328	−0.549254
$530$	0	0
$531$	4.53901	0.196976
$532$	0	0
$533$	41.2195	1.78541
$534$	0	0
$535$	1.54869	0.0669556
$536$	0	0
$537$	−8.67626	−0.374408
$538$	0	0
$539$	13.1940	0.568304
$540$	0	0
$541$	−9.49447	−0.408199	−0.204100	−	0.978950i	$-0.565427\pi$
$541$	−9.49447	−0.408199	−0.204100	+	0.978950i	$0.565427\pi$
$542$	0	0
$543$	9.58964	0.411531
$544$	0	0
$545$	21.2175	0.908857
$546$	0	0
$547$	19.8769	0.849874	0.424937	−	0.905223i	$-0.360296\pi$
$547$	19.8769	0.849874	0.424937	+	0.905223i	$0.360296\pi$
$548$	0	0
$549$	−1.40323	−0.0598885
$550$	0	0
$551$	−2.49905	−0.106463
$552$	0	0
$553$	12.5876	0.535280
$554$	0	0
$555$	−6.83973	−0.290330
$556$	0	0
$557$	4.79645	0.203232	0.101616	−	0.994824i	$-0.467599\pi$
$557$	4.79645	0.203232	0.101616	+	0.994824i	$0.467599\pi$
$558$	0	0
$559$	2.01938	0.0854108
$560$	0	0
$561$	−1.08643	−0.0458690
$562$	0	0
$563$	6.13462	0.258543	0.129272	−	0.991609i	$-0.458736\pi$
$563$	6.13462	0.258543	0.129272	+	0.991609i	$0.458736\pi$
$564$	0	0
$565$	−6.40924	−0.269639
$566$	0	0
$567$	−1.30202	−0.0546795
$568$	0	0
$569$	−24.2460	−1.01644	−0.508222	−	0.861226i	$-0.669697\pi$
$569$	−24.2460	−1.01644	−0.508222	+	0.861226i	$0.669697\pi$
$570$	0	0
$571$	0.0657417	0.00275120	0.00137560	−	0.999999i	$-0.499562\pi$
$571$	0.0657417	0.00275120	0.00137560	+	0.999999i	$0.499562\pi$
$572$	0	0
$573$	−13.3296	−0.556853
$574$	0	0
$575$	11.9410	0.497974
$576$	0	0
$577$	−22.3901	−0.932110	−0.466055	−	0.884756i	$-0.654325\pi$
$577$	−22.3901	−0.932110	−0.466055	+	0.884756i	$0.654325\pi$
$578$	0	0
$579$	6.49992	0.270128
$580$	0	0
$581$	−11.9417	−0.495425
$582$	0	0
$583$	−19.7817	−0.819273
$584$	0	0
$585$	−4.26988	−0.176538
$586$	0	0
$587$	19.1824	0.791743	0.395871	−	0.918306i	$-0.370443\pi$
$587$	19.1824	0.791743	0.395871	+	0.918306i	$0.370443\pi$
$588$	0	0
$589$	−4.33501	−0.178621
$590$	0	0
$591$	15.4707	0.636380
$592$	0	0
$593$	−42.8578	−1.75996	−0.879980	−	0.475011i	$-0.842444\pi$
$593$	−42.8578	−1.75996	−0.879980	+	0.475011i	$0.842444\pi$
$594$	0	0
$595$	−0.646305	−0.0264959
$596$	0	0
$597$	8.11088	0.331956
$598$	0	0
$599$	31.9383	1.30497	0.652483	−	0.757804i	$-0.273728\pi$
$599$	31.9383	1.30497	0.652483	+	0.757804i	$0.273728\pi$
$600$	0	0
$601$	3.46063	0.141162	0.0705809	−	0.997506i	$-0.477515\pi$
$601$	3.46063	0.141162	0.0705809	+	0.997506i	$0.477515\pi$
$602$	0	0
$603$	−8.68364	−0.353625
$604$	0	0
$605$	5.47045	0.222405
$606$	0	0
$607$	45.5663	1.84948	0.924739	−	0.380602i	$-0.124283\pi$
$607$	45.5663	1.84948	0.924739	+	0.380602i	$0.124283\pi$
$608$	0	0
$609$	5.53908	0.224455
$610$	0	0
$611$	−18.1626	−0.734778
$612$	0	0
$613$	3.35612	0.135552	0.0677762	−	0.997701i	$-0.478410\pi$
$613$	3.35612	0.135552	0.0677762	+	0.997701i	$0.478410\pi$
$614$	0	0
$615$	12.4665	0.502700
$616$	0	0
$617$	16.6322	0.669587	0.334793	−	0.942292i	$-0.391333\pi$
$617$	16.6322	0.669587	0.334793	+	0.942292i	$0.391333\pi$
$618$	0	0
$619$	34.2772	1.37772	0.688859	−	0.724895i	$-0.258112\pi$
$619$	34.2772	1.37772	0.688859	+	0.724895i	$0.258112\pi$
$620$	0	0
$621$	3.21981	0.129206
$622$	0	0
$623$	1.95811	0.0784501
$624$	0	0
$625$	7.29677	0.291871
$626$	0	0
$627$	−1.46104	−0.0583485
$628$	0	0
$629$	2.62906	0.104827
$630$	0	0
$631$	30.0355	1.19570	0.597848	−	0.801610i	$-0.296023\pi$
$631$	30.0355	1.19570	0.597848	+	0.801610i	$0.296023\pi$
$632$	0	0
$633$	−4.24970	−0.168911
$634$	0	0
$635$	11.1694	0.443242
$636$	0	0
$637$	−19.9320	−0.789735
$638$	0	0
$639$	−8.38142	−0.331564
$640$	0	0
$641$	−39.2103	−1.54871	−0.774357	−	0.632749i	$-0.781927\pi$
$641$	−39.2103	−1.54871	−0.774357	+	0.632749i	$0.781927\pi$
$642$	0	0
$643$	−43.2922	−1.70728	−0.853639	−	0.520865i	$-0.825610\pi$
$643$	−43.2922	−1.70728	−0.853639	+	0.520865i	$0.825610\pi$
$644$	0	0
$645$	0.610748	0.0240482
$646$	0	0
$647$	25.1630	0.989258	0.494629	−	0.869104i	$-0.335304\pi$
$647$	25.1630	0.989258	0.494629	+	0.869104i	$0.335304\pi$
$648$	0	0
$649$	−11.2894	−0.443148
$650$	0	0
$651$	9.60845	0.376585
$652$	0	0
$653$	20.8732	0.816832	0.408416	−	0.912796i	$-0.366081\pi$
$653$	20.8732	0.816832	0.408416	+	0.912796i	$0.366081\pi$
$654$	0	0
$655$	10.5145	0.410836
$656$	0	0
$657$	15.0903	0.588727
$658$	0	0
$659$	−27.2993	−1.06343	−0.531716	−	0.846923i	$-0.678452\pi$
$659$	−27.2993	−1.06343	−0.531716	+	0.846923i	$0.678452\pi$
$660$	0	0
$661$	−35.7050	−1.38876	−0.694382	−	0.719607i	$-0.744322\pi$
$661$	−35.7050	−1.38876	−0.694382	+	0.719607i	$0.744322\pi$
$662$	0	0
$663$	1.64126	0.0637412
$664$	0	0
$665$	−0.869159	−0.0337046
$666$	0	0
$667$	−13.6978	−0.530381
$668$	0	0
$669$	−5.83056	−0.225423
$670$	0	0
$671$	3.49011	0.134734
$672$	0	0
$673$	11.2827	0.434917	0.217459	−	0.976070i	$-0.430223\pi$
$673$	11.2827	0.434917	0.217459	+	0.976070i	$0.430223\pi$
$674$	0	0
$675$	3.70860	0.142744
$676$	0	0
$677$	−27.8837	−1.07166	−0.535829	−	0.844327i	$-0.680001\pi$
$677$	−27.8837	−1.07166	−0.535829	+	0.844327i	$0.680001\pi$
$678$	0	0
$679$	−11.3086	−0.433986
$680$	0	0
$681$	20.5776	0.788536
$682$	0	0
$683$	9.60857	0.367662	0.183831	−	0.982958i	$-0.441150\pi$
$683$	9.60857	0.367662	0.183831	+	0.982958i	$0.441150\pi$
$684$	0	0
$685$	12.1447	0.464024
$686$	0	0
$687$	−4.51093	−0.172103
$688$	0	0
$689$	29.8840	1.13849
$690$	0	0
$691$	−15.9394	−0.606363	−0.303182	−	0.952933i	$-0.598049\pi$
$691$	−15.9394	−0.606363	−0.303182	+	0.952933i	$0.598049\pi$
$692$	0	0
$693$	3.23837	0.123015
$694$	0	0
$695$	−7.54582	−0.286229
$696$	0	0
$697$	−4.79190	−0.181506
$698$	0	0
$699$	−4.61494	−0.174553
$700$	0	0
$701$	−22.9952	−0.868517	−0.434258	−	0.900788i	$-0.642990\pi$
$701$	−22.9952	−0.868517	−0.434258	+	0.900788i	$0.642990\pi$
$702$	0	0
$703$	3.53560	0.133348
$704$	0	0
$705$	−5.49314	−0.206884
$706$	0	0
$707$	−23.2735	−0.875292
$708$	0	0
$709$	1.00931	0.0379053	0.0189526	−	0.999820i	$-0.493967\pi$
$709$	1.00931	0.0379053	0.0189526	+	0.999820i	$0.493967\pi$
$710$	0	0
$711$	−9.66779	−0.362570
$712$	0	0
$713$	−23.7611	−0.889860
$714$	0	0
$715$	10.6200	0.397166
$716$	0	0
$717$	11.2965	0.421876
$718$	0	0
$719$	−37.1497	−1.38545	−0.692725	−	0.721202i	$-0.743590\pi$
$719$	−37.1497	−1.38545	−0.692725	+	0.721202i	$0.743590\pi$
$720$	0	0
$721$	−2.37467	−0.0884373
$722$	0	0
$723$	29.0353	1.07984
$724$	0	0
$725$	−15.7773	−0.585953
$726$	0	0
$727$	−41.3170	−1.53236	−0.766182	−	0.642624i	$-0.777846\pi$
$727$	−41.3170	−1.53236	−0.766182	+	0.642624i	$0.777846\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.234760	−0.00868291
$732$	0	0
$733$	−49.6773	−1.83487	−0.917437	−	0.397882i	$-0.869745\pi$
$733$	−49.6773	−1.83487	−0.917437	+	0.397882i	$0.869745\pi$
$734$	0	0
$735$	−6.02830	−0.222357
$736$	0	0
$737$	21.5979	0.795569
$738$	0	0
$739$	6.45298	0.237377	0.118688	−	0.992932i	$-0.462131\pi$
$739$	6.45298	0.237377	0.118688	+	0.992932i	$0.462131\pi$
$740$	0	0
$741$	2.20719	0.0810831
$742$	0	0
$743$	9.61992	0.352921	0.176460	−	0.984308i	$-0.443535\pi$
$743$	9.61992	0.352921	0.176460	+	0.984308i	$0.443535\pi$
$744$	0	0
$745$	−2.35149	−0.0861520
$746$	0	0
$747$	9.17169	0.335575
$748$	0	0
$749$	1.77440	0.0648350
$750$	0	0
$751$	−10.1565	−0.370617	−0.185309	−	0.982680i	$-0.559328\pi$
$751$	−10.1565	−0.370617	−0.185309	+	0.982680i	$0.559328\pi$
$752$	0	0
$753$	1.43576	0.0523219
$754$	0	0
$755$	23.7008	0.862561
$756$	0	0
$757$	33.6725	1.22385	0.611924	−	0.790917i	$-0.290396\pi$
$757$	33.6725	1.22385	0.611924	+	0.790917i	$0.290396\pi$
$758$	0	0
$759$	−8.00829	−0.290682
$760$	0	0
$761$	46.3479	1.68011	0.840056	−	0.542500i	$-0.182522\pi$
$761$	46.3479	1.68011	0.840056	+	0.542500i	$0.182522\pi$
$762$	0	0
$763$	24.3097	0.880072
$764$	0	0
$765$	0.496388	0.0179469
$766$	0	0
$767$	17.0548	0.615814
$768$	0	0
$769$	−0.994208	−0.0358520	−0.0179260	−	0.999839i	$-0.505706\pi$
$769$	−0.994208	−0.0358520	−0.0179260	+	0.999839i	$0.505706\pi$
$770$	0	0
$771$	18.1069	0.652103
$772$	0	0
$773$	−17.9872	−0.646955	−0.323478	−	0.946236i	$-0.604852\pi$
$773$	−17.9872	−0.646955	−0.323478	+	0.946236i	$0.604852\pi$
$774$	0	0
$775$	−27.3683	−0.983097
$776$	0	0
$777$	−7.83656	−0.281135
$778$	0	0
$779$	−6.44421	−0.230888
$780$	0	0
$781$	20.8462	0.745937
$782$	0	0
$783$	−4.25423	−0.152034
$784$	0	0
$785$	6.00362	0.214279
$786$	0	0
$787$	−42.3865	−1.51091	−0.755457	−	0.655198i	$-0.772585\pi$
$787$	−42.3865	−1.51091	−0.755457	+	0.655198i	$0.772585\pi$
$788$	0	0
$789$	−1.08862	−0.0387559
$790$	0	0
$791$	−7.34333	−0.261099
$792$	0	0
$793$	−5.27249	−0.187232
$794$	0	0
$795$	9.03821	0.320552
$796$	0	0
$797$	−19.9294	−0.705934	−0.352967	−	0.935636i	$-0.614827\pi$
$797$	−19.9294	−0.705934	−0.352967	+	0.935636i	$0.614827\pi$
$798$	0	0
$799$	2.11146	0.0746980
$800$	0	0
$801$	−1.50391	−0.0531379
$802$	0	0
$803$	−37.5324	−1.32449
$804$	0	0
$805$	−4.76404	−0.167910
$806$	0	0
$807$	10.3876	0.365659
$808$	0	0
$809$	−24.7811	−0.871257	−0.435629	−	0.900126i	$-0.643474\pi$
$809$	−24.7811	−0.871257	−0.435629	+	0.900126i	$0.643474\pi$
$810$	0	0
$811$	−53.5412	−1.88008	−0.940042	−	0.341058i	$-0.889215\pi$
$811$	−53.5412	−1.88008	−0.940042	+	0.341058i	$0.889215\pi$
$812$	0	0
$813$	−5.04142	−0.176810
$814$	0	0
$815$	−24.1528	−0.846034
$816$	0	0
$817$	−0.315708	−0.0110452
$818$	0	0
$819$	−4.89218	−0.170947
$820$	0	0
$821$	−3.83753	−0.133931	−0.0669654	−	0.997755i	$-0.521332\pi$
$821$	−3.83753	−0.133931	−0.0669654	+	0.997755i	$0.521332\pi$
$822$	0	0
$823$	−6.27847	−0.218854	−0.109427	−	0.993995i	$-0.534902\pi$
$823$	−6.27847	−0.218854	−0.109427	+	0.993995i	$0.534902\pi$
$824$	0	0
$825$	−9.22402	−0.321139
$826$	0	0
$827$	12.8357	0.446339	0.223170	−	0.974780i	$-0.428360\pi$
$827$	12.8357	0.446339	0.223170	+	0.974780i	$0.428360\pi$
$828$	0	0
$829$	−3.79387	−0.131767	−0.0658833	−	0.997827i	$-0.520987\pi$
$829$	−3.79387	−0.131767	−0.0658833	+	0.997827i	$0.520987\pi$
$830$	0	0
$831$	−22.8667	−0.793238
$832$	0	0
$833$	2.31716	0.0802849
$834$	0	0
$835$	−27.0406	−0.935779
$836$	0	0
$837$	−7.37967	−0.255079
$838$	0	0
$839$	−20.8085	−0.718390	−0.359195	−	0.933263i	$-0.616949\pi$
$839$	−20.8085	−0.718390	−0.359195	+	0.933263i	$0.616949\pi$
$840$	0	0
$841$	−10.9015	−0.375915
$842$	0	0
$843$	8.68406	0.299095
$844$	0	0
$845$	−1.27044	−0.0437046
$846$	0	0
$847$	6.26773	0.215362
$848$	0	0
$849$	19.0137	0.652549
$850$	0	0
$851$	19.3793	0.664315
$852$	0	0
$853$	49.8695	1.70750	0.853750	−	0.520684i	$-0.174323\pi$
$853$	49.8695	1.70750	0.853750	+	0.520684i	$0.174323\pi$
$854$	0	0
$855$	0.667549	0.0228297
$856$	0	0
$857$	−22.4004	−0.765182	−0.382591	−	0.923918i	$-0.624968\pi$
$857$	−22.4004	−0.765182	−0.382591	+	0.923918i	$0.624968\pi$
$858$	0	0
$859$	14.3082	0.488191	0.244095	−	0.969751i	$-0.421509\pi$
$859$	14.3082	0.488191	0.244095	+	0.969751i	$0.421509\pi$
$860$	0	0
$861$	14.2834	0.486778
$862$	0	0
$863$	30.9827	1.05466	0.527332	−	0.849659i	$-0.323192\pi$
$863$	30.9827	1.05466	0.527332	+	0.849659i	$0.323192\pi$
$864$	0	0
$865$	−1.13640	−0.0386386
$866$	0	0
$867$	16.8092	0.570870
$868$	0	0
$869$	24.0457	0.815694
$870$	0	0
$871$	−32.6278	−1.10555
$872$	0	0
$873$	8.68548	0.293959
$874$	0	0
$875$	−12.8853	−0.435603
$876$	0	0
$877$	−18.3848	−0.620809	−0.310404	−	0.950605i	$-0.600464\pi$
$877$	−18.3848	−0.620809	−0.310404	+	0.950605i	$0.600464\pi$
$878$	0	0
$879$	−17.2415	−0.581541
$880$	0	0
$881$	19.3633	0.652367	0.326184	−	0.945306i	$-0.394237\pi$
$881$	19.3633	0.652367	0.326184	+	0.945306i	$0.394237\pi$
$882$	0	0
$883$	−18.8775	−0.635278	−0.317639	−	0.948212i	$-0.602890\pi$
$883$	−18.8775	−0.635278	−0.317639	+	0.948212i	$0.602890\pi$
$884$	0	0
$885$	5.15811	0.173388
$886$	0	0
$887$	−5.40161	−0.181368	−0.0906842	−	0.995880i	$-0.528905\pi$
$887$	−5.40161	−0.181368	−0.0906842	+	0.995880i	$0.528905\pi$
$888$	0	0
$889$	12.7972	0.429204
$890$	0	0
$891$	−2.48719	−0.0833242
$892$	0	0
$893$	2.83952	0.0950208
$894$	0	0
$895$	−9.85967	−0.329572
$896$	0	0
$897$	12.0981	0.403942
$898$	0	0
$899$	31.3948	1.04708
$900$	0	0
$901$	−3.47411	−0.115740
$902$	0	0
$903$	0.699760	0.0232866
$904$	0	0
$905$	10.8976	0.362249
$906$	0	0
$907$	10.2266	0.339567	0.169784	−	0.985481i	$-0.445693\pi$
$907$	10.2266	0.339567	0.169784	+	0.985481i	$0.445693\pi$
$908$	0	0
$909$	17.8750	0.592876
$910$	0	0
$911$	−9.53171	−0.315800	−0.157900	−	0.987455i	$-0.550472\pi$
$911$	−9.53171	−0.315800	−0.157900	+	0.987455i	$0.550472\pi$
$912$	0	0
$913$	−22.8118	−0.754960
$914$	0	0
$915$	−1.59463	−0.0527168
$916$	0	0
$917$	12.0469	0.397824
$918$	0	0
$919$	−18.7966	−0.620041	−0.310021	−	0.950730i	$-0.600336\pi$
$919$	−18.7966	−0.620041	−0.310021	+	0.950730i	$0.600336\pi$
$920$	0	0
$921$	−5.85444	−0.192910
$922$	0	0
$923$	−31.4923	−1.03658
$924$	0	0
$925$	22.3213	0.733920
$926$	0	0
$927$	1.82384	0.0599027
$928$	0	0
$929$	29.2330	0.959102	0.479551	−	0.877514i	$-0.340799\pi$
$929$	29.2330	0.959102	0.479551	+	0.877514i	$0.340799\pi$
$930$	0	0
$931$	3.11615	0.102128
$932$	0	0
$933$	−20.6193	−0.675047
$934$	0	0
$935$	−1.23461	−0.0403761
$936$	0	0
$937$	8.70255	0.284300	0.142150	−	0.989845i	$-0.454598\pi$
$937$	8.70255	0.284300	0.142150	+	0.989845i	$0.454598\pi$
$938$	0	0
$939$	3.52006	0.114873
$940$	0	0
$941$	4.68746	0.152807	0.0764034	−	0.997077i	$-0.475656\pi$
$941$	4.68746	0.152807	0.0764034	+	0.997077i	$0.475656\pi$
$942$	0	0
$943$	−35.3221	−1.15024
$944$	0	0
$945$	−1.47961	−0.0481316
$946$	0	0
$947$	53.9053	1.75169	0.875844	−	0.482595i	$-0.160306\pi$
$947$	53.9053	1.75169	0.875844	+	0.482595i	$0.160306\pi$
$948$	0	0
$949$	56.6999	1.84056
$950$	0	0
$951$	15.3183	0.496729
$952$	0	0
$953$	22.8631	0.740610	0.370305	−	0.928910i	$-0.379253\pi$
$953$	22.8631	0.740610	0.370305	+	0.928910i	$0.379253\pi$
$954$	0	0
$955$	−15.1477	−0.490169
$956$	0	0
$957$	10.5811	0.342038
$958$	0	0
$959$	13.9146	0.449327
$960$	0	0
$961$	23.4595	0.756759
$962$	0	0
$963$	−1.36281	−0.0439158
$964$	0	0
$965$	7.38648	0.237779
$966$	0	0
$967$	−22.9524	−0.738100	−0.369050	−	0.929410i	$-0.620317\pi$
$967$	−22.9524	−0.738100	−0.369050	+	0.929410i	$0.620317\pi$
$968$	0	0
$969$	−0.256593	−0.00824295
$970$	0	0
$971$	2.22326	0.0713480	0.0356740	−	0.999363i	$-0.488642\pi$
$971$	2.22326	0.0713480	0.0356740	+	0.999363i	$0.488642\pi$
$972$	0	0
$973$	−8.64556	−0.277164
$974$	0	0
$975$	13.9347	0.446266
$976$	0	0
$977$	37.6589	1.20481	0.602407	−	0.798189i	$-0.294209\pi$
$977$	37.6589	1.20481	0.602407	+	0.798189i	$0.294209\pi$
$978$	0	0
$979$	3.74051	0.119547
$980$	0	0
$981$	−18.6708	−0.596114
$982$	0	0
$983$	47.6347	1.51931	0.759656	−	0.650325i	$-0.225367\pi$
$983$	47.6347	1.51931	0.759656	+	0.650325i	$0.225367\pi$
$984$	0	0
$985$	17.5809	0.560173
$986$	0	0
$987$	−6.29372	−0.200331
$988$	0	0
$989$	−1.73046	−0.0550255
$990$	0	0
$991$	25.0538	0.795861	0.397931	−	0.917416i	$-0.369729\pi$
$991$	25.0538	0.795861	0.397931	+	0.917416i	$0.369729\pi$
$992$	0	0
$993$	1.92001	0.0609297
$994$	0	0
$995$	9.21717	0.292204
$996$	0	0
$997$	−19.5027	−0.617657	−0.308828	−	0.951118i	$-0.599937\pi$
$997$	−19.5027	−0.617657	−0.308828	+	0.951118i	$0.599937\pi$
$998$	0	0
$999$	6.01879	0.190426

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8304.2.a.bf.1.2		7
4.3	odd	2		2076.2.a.g.1.2	✓	7
12.11	even	2		6228.2.a.m.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2076.2.a.g.1.2	✓	7	4.3	odd	2
6228.2.a.m.1.6		7	12.11	even	2
8304.2.a.bf.1.2		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8304 = 2^{4} \cdot 3 \cdot 173 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8304.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.13640 q^{5} -1.30202 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.13640 q^{5} -1.30202 q^{7} +1.00000 q^{9} -2.48719 q^{11} +3.75739 q^{13} +1.13640 q^{15} -0.436809 q^{17} -0.587426 q^{19} +1.30202 q^{21} -3.21981 q^{23} -3.70860 q^{25} -1.00000 q^{27} +4.25423 q^{29} +7.37967 q^{31} +2.48719 q^{33} +1.47961 q^{35} -6.01879 q^{37} -3.75739 q^{39} +10.9702 q^{41} +0.537443 q^{43} -1.13640 q^{45} -4.83383 q^{47} -5.30475 q^{49} +0.436809 q^{51} +7.95340 q^{53} +2.82644 q^{55} +0.587426 q^{57} +4.53901 q^{59} -1.40323 q^{61} -1.30202 q^{63} -4.26988 q^{65} -8.68364 q^{67} +3.21981 q^{69} -8.38142 q^{71} +15.0903 q^{73} +3.70860 q^{75} +3.23837 q^{77} -9.66779 q^{79} +1.00000 q^{81} +9.17169 q^{83} +0.496388 q^{85} -4.25423 q^{87} -1.50391 q^{89} -4.89218 q^{91} -7.37967 q^{93} +0.667549 q^{95} +8.68548 q^{97} -2.48719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} + 6 q^{5} - q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 + 6 * q^5 - q^7 + 7 * q^9	\( 7 q - 7 q^{3} + 6 q^{5} - q^{7} + 7 q^{9} - 4 q^{11} + 4 q^{13} - 6 q^{15} + 7 q^{17} - 9 q^{19} + q^{21} - 17 q^{23} + 5 q^{25} - 7 q^{27} + 6 q^{29} - 10 q^{31} + 4 q^{33} - 9 q^{35} + 6 q^{37} - 4 q^{39} + 7 q^{41} - 11 q^{43} + 6 q^{45} - 22 q^{47} - 8 q^{49} - 7 q^{51} + 20 q^{53} - 21 q^{55} + 9 q^{57} - 18 q^{59} - q^{63} - 2 q^{65} - 8 q^{67} + 17 q^{69} - 14 q^{71} - 14 q^{73} - 5 q^{75} + 13 q^{77} - q^{79} + 7 q^{81} - 7 q^{83} - 16 q^{85} - 6 q^{87} - 6 q^{89} - 10 q^{91} + 10 q^{93} - 31 q^{95} - 17 q^{97} - 4 q^{99}+O(q^{100}) \) 7 * q - 7 * q^3 + 6 * q^5 - q^7 + 7 * q^9 - 4 * q^11 + 4 * q^13 - 6 * q^15 + 7 * q^17 - 9 * q^19 + q^21 - 17 * q^23 + 5 * q^25 - 7 * q^27 + 6 * q^29 - 10 * q^31 + 4 * q^33 - 9 * q^35 + 6 * q^37 - 4 * q^39 + 7 * q^41 - 11 * q^43 + 6 * q^45 - 22 * q^47 - 8 * q^49 - 7 * q^51 + 20 * q^53 - 21 * q^55 + 9 * q^57 - 18 * q^59 - q^63 - 2 * q^65 - 8 * q^67 + 17 * q^69 - 14 * q^71 - 14 * q^73 - 5 * q^75 + 13 * q^77 - q^79 + 7 * q^81 - 7 * q^83 - 16 * q^85 - 6 * q^87 - 6 * q^89 - 10 * q^91 + 10 * q^93 - 31 * q^95 - 17 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more