LMFDB - Embedded newform 3700.2.a.n.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,2,Mod(1,3700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3700.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3700 = 2^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3700.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:	$29.5446487479$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.17268201.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 7x^{4} + 4x^{3} + 13x^{2} - 3x - 6 $ x^6 - x^5 - 7x^4 + 4x^3 + 13x^2 - 3x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.45978$ of defining polynomial
Character	$\chi$	$=$	3700.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.85250 q^{3} +2.47048 q^{7} +5.13675 q^{9} +1.45305 q^{11} -0.691767 q^{13} -1.07378 q^{17} +2.15136 q^{19} +7.04705 q^{21} +5.00668 q^{23} +6.09508 q^{27} +4.85250 q^{29} -3.84313 q^{31} +4.14481 q^{33} +1.00000 q^{37} -1.97326 q^{39} -8.90340 q^{41} +1.25195 q^{43} -0.737297 q^{47} -0.896719 q^{49} -3.06297 q^{51} +0.399453 q^{53} +6.13675 q^{57} +5.08846 q^{59} +1.84313 q^{61} +12.6902 q^{63} +10.5297 q^{67} +14.2816 q^{69} -8.84326 q^{71} -7.90223 q^{73} +3.58972 q^{77} +1.16218 q^{79} +1.97595 q^{81} +4.08426 q^{83} +13.8417 q^{87} +15.4744 q^{89} -1.70900 q^{91} -10.9625 q^{93} -10.4542 q^{97} +7.46394 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} - 2 q^{7} + 9 q^{9} - 8 q^{11} + 5 q^{13} + 2 q^{17} + 2 q^{19} + 10 q^{21} - 8 q^{23} + q^{27} + 13 q^{29} - 3 q^{31} - q^{33} + 6 q^{37} + 12 q^{39} + 22 q^{41} - 8 q^{43} + 18 q^{47} + 32 q^{49}+ \cdots + 13 q^{99}+O(q^{100}) $ 6 * q + q^3 - 2 * q^7 + 9 * q^9 - 8 * q^11 + 5 * q^13 + 2 * q^17 + 2 * q^19 + 10 * q^21 - 8 * q^23 + q^27 + 13 * q^29 - 3 * q^31 - q^33 + 6 * q^37 + 12 * q^39 + 22 * q^41 - 8 * q^43 + 18 * q^47 + 32 * q^49 - 5 * q^51 + 3 * q^53 + 15 * q^57 + 13 * q^59 - 9 * q^61 - 44 * q^63 + 27 * q^67 + 38 * q^69 + 20 * q^71 - 41 * q^73 + 3 * q^77 + 14 * q^79 + 30 * q^81 - 16 * q^83 + 29 * q^87 + 16 * q^89 - 28 * q^91 - 4 * q^93 + 23 * q^97 + 13 * q^99

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
$3$	2.85250	1.64689	0.823446	−	0.567395i	$-0.192049\pi$
$3$	2.85250	1.64689	0.823446	+	0.567395i	$0.192049\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.47048	0.933754	0.466877	−	0.884322i	$-0.345379\pi$
$7$	2.47048	0.933754	0.466877	+	0.884322i	$0.345379\pi$
$8$	0	0
$9$	5.13675	1.71225
$10$	0	0
$11$	1.45305	0.438110	0.219055	−	0.975713i	$-0.429703\pi$
$11$	1.45305	0.438110	0.219055	+	0.975713i	$0.429703\pi$
$12$	0	0
$13$	−0.691767	−0.191862	−0.0959308	−	0.995388i	$-0.530583\pi$
$13$	−0.691767	−0.191862	−0.0959308	+	0.995388i	$0.530583\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.07378	−0.260431	−0.130215	−	0.991486i	$-0.541567\pi$
$17$	−1.07378	−0.260431	−0.130215	+	0.991486i	$0.541567\pi$
$18$	0	0
$19$	2.15136	0.493556	0.246778	−	0.969072i	$-0.420628\pi$
$19$	2.15136	0.493556	0.246778	+	0.969072i	$0.420628\pi$
$20$	0	0
$21$	7.04705	1.53779
$22$	0	0
$23$	5.00668	1.04397	0.521983	−	0.852956i	$-0.325192\pi$
$23$	5.00668	1.04397	0.521983	+	0.852956i	$0.325192\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	6.09508	1.17300
$28$	0	0
$29$	4.85250	0.901086	0.450543	−	0.892755i	$-0.351230\pi$
$29$	4.85250	0.901086	0.450543	+	0.892755i	$0.351230\pi$
$30$	0	0
$31$	−3.84313	−0.690246	−0.345123	−	0.938558i	$-0.612163\pi$
$31$	−3.84313	−0.690246	−0.345123	+	0.938558i	$0.612163\pi$
$32$	0	0
$33$	4.14481	0.721519
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−1.97326	−0.315975
$40$	0	0
$41$	−8.90340	−1.39048	−0.695239	−	0.718779i	$-0.744701\pi$
$41$	−8.90340	−1.39048	−0.695239	+	0.718779i	$0.744701\pi$
$42$	0	0
$43$	1.25195	0.190921	0.0954604	−	0.995433i	$-0.469568\pi$
$43$	1.25195	0.190921	0.0954604	+	0.995433i	$0.469568\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.737297	−0.107546	−0.0537729	−	0.998553i	$-0.517125\pi$
$47$	−0.737297	−0.107546	−0.0537729	+	0.998553i	$0.517125\pi$
$48$	0	0
$49$	−0.896719	−0.128103
$50$	0	0
$51$	−3.06297	−0.428901
$52$	0	0
$53$	0.399453	0.0548691	0.0274345	−	0.999624i	$-0.491266\pi$
$53$	0.399453	0.0548691	0.0274345	+	0.999624i	$0.491266\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.13675	0.812832
$58$	0	0
$59$	5.08846	0.662462	0.331231	−	0.943550i	$-0.392536\pi$
$59$	5.08846	0.662462	0.331231	+	0.943550i	$0.392536\pi$
$60$	0	0
$61$	1.84313	0.235988	0.117994	−	0.993014i	$-0.462354\pi$
$61$	1.84313	0.235988	0.117994	+	0.993014i	$0.462354\pi$
$62$	0	0
$63$	12.6902	1.59882
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.5297	1.28640	0.643201	−	0.765697i	$-0.277606\pi$
$67$	10.5297	1.28640	0.643201	+	0.765697i	$0.277606\pi$
$68$	0	0
$69$	14.2816	1.71930
$70$	0	0
$71$	−8.84326	−1.04950	−0.524751	−	0.851256i	$-0.675842\pi$
$71$	−8.84326	−1.04950	−0.524751	+	0.851256i	$0.675842\pi$
$72$	0	0
$73$	−7.90223	−0.924887	−0.462443	−	0.886649i	$-0.653027\pi$
$73$	−7.90223	−0.924887	−0.462443	+	0.886649i	$0.653027\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.58972	0.409087
$78$	0	0
$79$	1.16218	0.130756	0.0653779	−	0.997861i	$-0.479175\pi$
$79$	1.16218	0.130756	0.0653779	+	0.997861i	$0.479175\pi$
$80$	0	0
$81$	1.97595	0.219550
$82$	0	0
$83$	4.08426	0.448306	0.224153	−	0.974554i	$-0.428038\pi$
$83$	4.08426	0.448306	0.224153	+	0.974554i	$0.428038\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	13.8417	1.48399
$88$	0	0
$89$	15.4744	1.64028	0.820142	−	0.572160i	$-0.193894\pi$
$89$	15.4744	1.64028	0.820142	+	0.572160i	$0.193894\pi$
$90$	0	0
$91$	−1.70900	−0.179152
$92$	0	0
$93$	−10.9625	−1.13676
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.4542	−1.06146	−0.530732	−	0.847539i	$-0.678083\pi$
$97$	−10.4542	−1.06146	−0.530732	+	0.847539i	$0.678083\pi$
$98$	0	0
$99$	7.46394	0.750154
$100$	0	0
$101$	−8.90623	−0.886203	−0.443102	−	0.896471i	$-0.646122\pi$
$101$	−8.90623	−0.886203	−0.443102	+	0.896471i	$0.646122\pi$
$102$	0	0
$103$	1.26689	0.124830	0.0624151	−	0.998050i	$-0.480120\pi$
$103$	1.26689	0.124830	0.0624151	+	0.998050i	$0.480120\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.8794	1.24509	0.622547	−	0.782582i	$-0.286098\pi$
$107$	12.8794	1.24509	0.622547	+	0.782582i	$0.286098\pi$
$108$	0	0
$109$	13.2464	1.26878	0.634389	−	0.773014i	$-0.281252\pi$
$109$	13.2464	1.26878	0.634389	+	0.773014i	$0.281252\pi$
$110$	0	0
$111$	2.85250	0.270747
$112$	0	0
$113$	−11.9640	−1.12547	−0.562737	−	0.826636i	$-0.690252\pi$
$113$	−11.9640	−1.12547	−0.562737	+	0.826636i	$0.690252\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.55343	−0.328515
$118$	0	0
$119$	−2.65276	−0.243178
$120$	0	0
$121$	−8.88866	−0.808060
$122$	0	0
$123$	−25.3970	−2.28997
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.73744	−0.686586	−0.343293	−	0.939228i	$-0.611542\pi$
$127$	−7.73744	−0.686586	−0.343293	+	0.939228i	$0.611542\pi$
$128$	0	0
$129$	3.57119	0.314426
$130$	0	0
$131$	−2.17162	−0.189735	−0.0948677	−	0.995490i	$-0.530243\pi$
$131$	−2.17162	−0.189735	−0.0948677	+	0.995490i	$0.530243\pi$
$132$	0	0
$133$	5.31489	0.460860
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.82578	−0.497730	−0.248865	−	0.968538i	$-0.580058\pi$
$137$	−5.82578	−0.497730	−0.248865	+	0.968538i	$0.580058\pi$
$138$	0	0
$139$	−5.00813	−0.424784	−0.212392	−	0.977185i	$-0.568125\pi$
$139$	−5.00813	−0.424784	−0.212392	+	0.977185i	$0.568125\pi$
$140$	0	0
$141$	−2.10314	−0.177116
$142$	0	0
$143$	−1.00517	−0.0840564
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.55789	−0.210971
$148$	0	0
$149$	5.82399	0.477120	0.238560	−	0.971128i	$-0.423325\pi$
$149$	5.82399	0.477120	0.238560	+	0.971128i	$0.423325\pi$
$150$	0	0
$151$	12.4489	1.01308	0.506539	−	0.862217i	$-0.330925\pi$
$151$	12.4489	1.01308	0.506539	+	0.862217i	$0.330925\pi$
$152$	0	0
$153$	−5.51576	−0.445923
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.86082	0.228318	0.114159	−	0.993462i	$-0.463583\pi$
$157$	2.86082	0.228318	0.114159	+	0.993462i	$0.463583\pi$
$158$	0	0
$159$	1.13944	0.0903633
$160$	0	0
$161$	12.3689	0.974808
$162$	0	0
$163$	−8.32188	−0.651820	−0.325910	−	0.945401i	$-0.605671\pi$
$163$	−8.32188	−0.651820	−0.325910	+	0.945401i	$0.605671\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.87274	−0.686593	−0.343297	−	0.939227i	$-0.611544\pi$
$167$	−8.87274	−0.686593	−0.343297	+	0.939227i	$0.611544\pi$
$168$	0	0
$169$	−12.5215	−0.963189
$170$	0	0
$171$	11.0510	0.845091
$172$	0	0
$173$	0.0633788	0.00481860	0.00240930	−	0.999997i	$-0.499233\pi$
$173$	0.0633788	0.00481860	0.00240930	+	0.999997i	$0.499233\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.5148	1.09100
$178$	0	0
$179$	0.304169	0.0227346	0.0113673	−	0.999935i	$-0.496382\pi$
$179$	0.304169	0.0227346	0.0113673	+	0.999935i	$0.496382\pi$
$180$	0	0
$181$	0.546542	0.0406241	0.0203121	−	0.999794i	$-0.493534\pi$
$181$	0.546542	0.0406241	0.0203121	+	0.999794i	$0.493534\pi$
$182$	0	0
$183$	5.25751	0.388647
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.56026	−0.114097
$188$	0	0
$189$	15.0578	1.09529
$190$	0	0
$191$	19.3098	1.39721	0.698604	−	0.715509i	$-0.253805\pi$
$191$	19.3098	1.39721	0.698604	+	0.715509i	$0.253805\pi$
$192$	0	0
$193$	2.41197	0.173618	0.0868089	−	0.996225i	$-0.472333\pi$
$193$	2.41197	0.173618	0.0868089	+	0.996225i	$0.472333\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.8862	−0.918107	−0.459054	−	0.888409i	$-0.651811\pi$
$197$	−12.8862	−0.918107	−0.459054	+	0.888409i	$0.651811\pi$
$198$	0	0
$199$	21.8555	1.54930	0.774648	−	0.632393i	$-0.217927\pi$
$199$	21.8555	1.54930	0.774648	+	0.632393i	$0.217927\pi$
$200$	0	0
$201$	30.0358	2.11856
$202$	0	0
$203$	11.9880	0.841393
$204$	0	0
$205$	0	0
$206$	0	0
$207$	25.7181	1.78753
$208$	0	0
$209$	3.12602	0.216232
$210$	0	0
$211$	−0.678051	−0.0466790	−0.0233395	−	0.999728i	$-0.507430\pi$
$211$	−0.678051	−0.0466790	−0.0233395	+	0.999728i	$0.507430\pi$
$212$	0	0
$213$	−25.2254	−1.72842
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.49437	−0.644520
$218$	0	0
$219$	−22.5411	−1.52319
$220$	0	0
$221$	0.742808	0.0499666
$222$	0	0
$223$	−19.4839	−1.30474	−0.652370	−	0.757901i	$-0.726225\pi$
$223$	−19.4839	−1.30474	−0.652370	+	0.757901i	$0.726225\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.68509	0.443705	0.221852	−	0.975080i	$-0.428790\pi$
$227$	6.68509	0.443705	0.221852	+	0.975080i	$0.428790\pi$
$228$	0	0
$229$	−12.8099	−0.846503	−0.423251	−	0.906012i	$-0.639111\pi$
$229$	−12.8099	−0.846503	−0.423251	+	0.906012i	$0.639111\pi$
$230$	0	0
$231$	10.2397	0.673722
$232$	0	0
$233$	−10.7733	−0.705780	−0.352890	−	0.935665i	$-0.614801\pi$
$233$	−10.7733	−0.705780	−0.352890	+	0.935665i	$0.614801\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.31512	0.215340
$238$	0	0
$239$	4.29404	0.277758	0.138879	−	0.990309i	$-0.455650\pi$
$239$	4.29404	0.277758	0.138879	+	0.990309i	$0.455650\pi$
$240$	0	0
$241$	12.9639	0.835078	0.417539	−	0.908659i	$-0.362893\pi$
$241$	12.9639	0.835078	0.417539	+	0.908659i	$0.362893\pi$
$242$	0	0
$243$	−12.6488	−0.811423
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.48824	−0.0946943
$248$	0	0
$249$	11.6503	0.738311
$250$	0	0
$251$	7.22620	0.456114	0.228057	−	0.973648i	$-0.426763\pi$
$251$	7.22620	0.456114	0.228057	+	0.973648i	$0.426763\pi$
$252$	0	0
$253$	7.27494	0.457372
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.5222	−0.843492	−0.421746	−	0.906714i	$-0.638583\pi$
$257$	−13.5222	−0.843492	−0.421746	+	0.906714i	$0.638583\pi$
$258$	0	0
$259$	2.47048	0.153508
$260$	0	0
$261$	24.9261	1.54289
$262$	0	0
$263$	6.00269	0.370141	0.185071	−	0.982725i	$-0.440749\pi$
$263$	6.00269	0.370141	0.185071	+	0.982725i	$0.440749\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	44.1407	2.70137
$268$	0	0
$269$	25.0161	1.52526	0.762629	−	0.646836i	$-0.223908\pi$
$269$	25.0161	1.52526	0.762629	+	0.646836i	$0.223908\pi$
$270$	0	0
$271$	7.31221	0.444185	0.222093	−	0.975026i	$-0.428711\pi$
$271$	7.31221	0.444185	0.222093	+	0.975026i	$0.428711\pi$
$272$	0	0
$273$	−4.87491	−0.295043
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−28.2585	−1.69789	−0.848944	−	0.528483i	$-0.822761\pi$
$277$	−28.2585	−1.69789	−0.848944	+	0.528483i	$0.822761\pi$
$278$	0	0
$279$	−19.7412	−1.18187
$280$	0	0
$281$	−12.9144	−0.770406	−0.385203	−	0.922832i	$-0.625869\pi$
$281$	−12.9144	−0.770406	−0.385203	+	0.922832i	$0.625869\pi$
$282$	0	0
$283$	−14.4542	−0.859214	−0.429607	−	0.903016i	$-0.641348\pi$
$283$	−14.4542	−0.859214	−0.429607	+	0.903016i	$0.641348\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−21.9957	−1.29837
$288$	0	0
$289$	−15.8470	−0.932176
$290$	0	0
$291$	−29.8206	−1.74812
$292$	0	0
$293$	18.7935	1.09793	0.548964	−	0.835846i	$-0.315022\pi$
$293$	18.7935	1.09793	0.548964	+	0.835846i	$0.315022\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.85643	0.513902
$298$	0	0
$299$	−3.46346	−0.200297
$300$	0	0
$301$	3.09292	0.178273
$302$	0	0
$303$	−25.4050	−1.45948
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.8018	1.13015	0.565076	−	0.825039i	$-0.308847\pi$
$307$	19.8018	1.13015	0.565076	+	0.825039i	$0.308847\pi$
$308$	0	0
$309$	3.61380	0.205582
$310$	0	0
$311$	14.5105	0.822816	0.411408	−	0.911451i	$-0.365037\pi$
$311$	14.5105	0.822816	0.411408	+	0.911451i	$0.365037\pi$
$312$	0	0
$313$	−27.6930	−1.56530	−0.782651	−	0.622461i	$-0.786133\pi$
$313$	−27.6930	−1.56530	−0.782651	+	0.622461i	$0.786133\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.80676	−0.494637	−0.247318	−	0.968934i	$-0.579549\pi$
$317$	−8.80676	−0.494637	−0.247318	+	0.968934i	$0.579549\pi$
$318$	0	0
$319$	7.05091	0.394775
$320$	0	0
$321$	36.7383	2.05053
$322$	0	0
$323$	−2.31009	−0.128537
$324$	0	0
$325$	0	0
$326$	0	0
$327$	37.7854	2.08954
$328$	0	0
$329$	−1.82148	−0.100421
$330$	0	0
$331$	−19.5580	−1.07500	−0.537501	−	0.843263i	$-0.680632\pi$
$331$	−19.5580	−1.07500	−0.537501	+	0.843263i	$0.680632\pi$
$332$	0	0
$333$	5.13675	0.281492
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−33.9345	−1.84853	−0.924266	−	0.381750i	$-0.875322\pi$
$337$	−33.9345	−1.84853	−0.924266	+	0.381750i	$0.875322\pi$
$338$	0	0
$339$	−34.1272	−1.85353
$340$	0	0
$341$	−5.58424	−0.302404
$342$	0	0
$343$	−19.5087	−1.05337
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.21881	−0.226478	−0.113239	−	0.993568i	$-0.536123\pi$
$347$	−4.21881	−0.226478	−0.113239	+	0.993568i	$0.536123\pi$
$348$	0	0
$349$	17.6555	0.945076	0.472538	−	0.881310i	$-0.343338\pi$
$349$	17.6555	0.945076	0.472538	+	0.881310i	$0.343338\pi$
$350$	0	0
$351$	−4.21637	−0.225053
$352$	0	0
$353$	21.7023	1.15510	0.577548	−	0.816357i	$-0.304010\pi$
$353$	21.7023	1.15510	0.577548	+	0.816357i	$0.304010\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.56700	−0.400488
$358$	0	0
$359$	2.87247	0.151603	0.0758015	−	0.997123i	$-0.475848\pi$
$359$	2.87247	0.151603	0.0758015	+	0.997123i	$0.475848\pi$
$360$	0	0
$361$	−14.3717	−0.756403
$362$	0	0
$363$	−25.3549	−1.33079
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.6572	0.660701	0.330351	−	0.943858i	$-0.392833\pi$
$367$	12.6572	0.660701	0.330351	+	0.943858i	$0.392833\pi$
$368$	0	0
$369$	−45.7346	−2.38085
$370$	0	0
$371$	0.986841	0.0512342
$372$	0	0
$373$	5.09508	0.263813	0.131907	−	0.991262i	$-0.457890\pi$
$373$	5.09508	0.263813	0.131907	+	0.991262i	$0.457890\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.35680	−0.172884
$378$	0	0
$379$	−24.5053	−1.25875	−0.629377	−	0.777100i	$-0.716690\pi$
$379$	−24.5053	−1.25875	−0.629377	+	0.777100i	$0.716690\pi$
$380$	0	0
$381$	−22.0710	−1.13073
$382$	0	0
$383$	−2.87538	−0.146925	−0.0734626	−	0.997298i	$-0.523405\pi$
$383$	−2.87538	−0.146925	−0.0734626	+	0.997298i	$0.523405\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.43096	0.326904
$388$	0	0
$389$	−10.8073	−0.547954	−0.273977	−	0.961736i	$-0.588339\pi$
$389$	−10.8073	−0.547954	−0.273977	+	0.961736i	$0.588339\pi$
$390$	0	0
$391$	−5.37610	−0.271881
$392$	0	0
$393$	−6.19455	−0.312474
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.5482	−1.43279	−0.716396	−	0.697694i	$-0.754209\pi$
$397$	−28.5482	−1.43279	−0.716396	+	0.697694i	$0.754209\pi$
$398$	0	0
$399$	15.1607	0.758986
$400$	0	0
$401$	28.3691	1.41669	0.708343	−	0.705868i	$-0.249443\pi$
$401$	28.3691	1.41669	0.708343	+	0.705868i	$0.249443\pi$
$402$	0	0
$403$	2.65855	0.132432
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.45305	0.0720248
$408$	0	0
$409$	7.46524	0.369132	0.184566	−	0.982820i	$-0.440912\pi$
$409$	7.46524	0.369132	0.184566	+	0.982820i	$0.440912\pi$
$410$	0	0
$411$	−16.6180	−0.819707
$412$	0	0
$413$	12.5710	0.618577
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−14.2857	−0.699573
$418$	0	0
$419$	−11.1288	−0.543679	−0.271839	−	0.962343i	$-0.587632\pi$
$419$	−11.1288	−0.543679	−0.271839	+	0.962343i	$0.587632\pi$
$420$	0	0
$421$	−38.1030	−1.85703	−0.928513	−	0.371300i	$-0.878912\pi$
$421$	−38.1030	−1.85703	−0.928513	+	0.371300i	$0.878912\pi$
$422$	0	0
$423$	−3.78731	−0.184145
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.55341	0.220355
$428$	0	0
$429$	−2.86724	−0.138432
$430$	0	0
$431$	27.2146	1.31088	0.655441	−	0.755247i	$-0.272483\pi$
$431$	27.2146	1.31088	0.655441	+	0.755247i	$0.272483\pi$
$432$	0	0
$433$	11.9302	0.573329	0.286665	−	0.958031i	$-0.407453\pi$
$433$	11.9302	0.573329	0.286665	+	0.958031i	$0.407453\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.7712	0.515255
$438$	0	0
$439$	13.8417	0.660626	0.330313	−	0.943871i	$-0.392846\pi$
$439$	13.8417	0.660626	0.330313	+	0.943871i	$0.392846\pi$
$440$	0	0
$441$	−4.60622	−0.219344
$442$	0	0
$443$	−29.7010	−1.41114	−0.705568	−	0.708642i	$-0.749308\pi$
$443$	−29.7010	−1.41114	−0.705568	+	0.708642i	$0.749308\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.6129	0.785765
$448$	0	0
$449$	−6.61844	−0.312343	−0.156172	−	0.987730i	$-0.549915\pi$
$449$	−6.61844	−0.312343	−0.156172	+	0.987730i	$0.549915\pi$
$450$	0	0
$451$	−12.9371	−0.609182
$452$	0	0
$453$	35.5105	1.66843
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.3188	1.27792	0.638960	−	0.769240i	$-0.279365\pi$
$457$	27.3188	1.27792	0.638960	+	0.769240i	$0.279365\pi$
$458$	0	0
$459$	−6.54479	−0.305485
$460$	0	0
$461$	−21.5771	−1.00494	−0.502472	−	0.864593i	$-0.667576\pi$
$461$	−21.5771	−1.00494	−0.502472	+	0.864593i	$0.667576\pi$
$462$	0	0
$463$	−1.20958	−0.0562137	−0.0281069	−	0.999605i	$-0.508948\pi$
$463$	−1.20958	−0.0562137	−0.0281069	+	0.999605i	$0.508948\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.8812	0.642345	0.321172	−	0.947021i	$-0.395923\pi$
$467$	13.8812	0.642345	0.321172	+	0.947021i	$0.395923\pi$
$468$	0	0
$469$	26.0133	1.20118
$470$	0	0
$471$	8.16048	0.376015
$472$	0	0
$473$	1.81914	0.0836443
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.05189	0.0939495
$478$	0	0
$479$	27.1852	1.24212	0.621061	−	0.783762i	$-0.286702\pi$
$479$	27.1852	1.24212	0.621061	+	0.783762i	$0.286702\pi$
$480$	0	0
$481$	−0.691767	−0.0315418
$482$	0	0
$483$	35.2823	1.60540
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10.2171	0.462982	0.231491	−	0.972837i	$-0.425640\pi$
$487$	10.2171	0.462982	0.231491	+	0.972837i	$0.425640\pi$
$488$	0	0
$489$	−23.7381	−1.07348
$490$	0	0
$491$	−41.5453	−1.87491	−0.937455	−	0.348105i	$-0.886825\pi$
$491$	−41.5453	−1.87491	−0.937455	+	0.348105i	$0.886825\pi$
$492$	0	0
$493$	−5.21053	−0.234671
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−21.8471	−0.979977
$498$	0	0
$499$	14.2455	0.637714	0.318857	−	0.947803i	$-0.396701\pi$
$499$	14.2455	0.637714	0.318857	+	0.947803i	$0.396701\pi$
$500$	0	0
$501$	−25.3095	−1.13074
$502$	0	0
$503$	−22.2356	−0.991437	−0.495719	−	0.868483i	$-0.665095\pi$
$503$	−22.2356	−0.991437	−0.495719	+	0.868483i	$0.665095\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−35.7174	−1.58627
$508$	0	0
$509$	43.5439	1.93005	0.965024	−	0.262163i	$-0.0844357\pi$
$509$	43.5439	1.93005	0.965024	+	0.262163i	$0.0844357\pi$
$510$	0	0
$511$	−19.5223	−0.863617
$512$	0	0
$513$	13.1127	0.578940
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.07133	−0.0471169
$518$	0	0
$519$	0.180788	0.00793571
$520$	0	0
$521$	28.0979	1.23099	0.615495	−	0.788141i	$-0.288956\pi$
$521$	28.0979	1.23099	0.615495	+	0.788141i	$0.288956\pi$
$522$	0	0
$523$	−6.53967	−0.285960	−0.142980	−	0.989726i	$-0.545668\pi$
$523$	−6.53967	−0.285960	−0.142980	+	0.989726i	$0.545668\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.12669	0.179761
$528$	0	0
$529$	2.06689	0.0898649
$530$	0	0
$531$	26.1382	1.13430
$532$	0	0
$533$	6.15908	0.266779
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.867640	0.0374414
$538$	0	0
$539$	−1.30297	−0.0561231
$540$	0	0
$541$	−6.13904	−0.263938	−0.131969	−	0.991254i	$-0.542130\pi$
$541$	−6.13904	−0.263938	−0.131969	+	0.991254i	$0.542130\pi$
$542$	0	0
$543$	1.55901	0.0669035
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−35.6966	−1.52628	−0.763138	−	0.646236i	$-0.776342\pi$
$547$	−35.6966	−1.52628	−0.763138	+	0.646236i	$0.776342\pi$
$548$	0	0
$549$	9.46768	0.404071
$550$	0	0
$551$	10.4395	0.444736
$552$	0	0
$553$	2.87115	0.122094
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−44.1070	−1.86887	−0.934437	−	0.356129i	$-0.884096\pi$
$557$	−44.1070	−1.86887	−0.934437	+	0.356129i	$0.884096\pi$
$558$	0	0
$559$	−0.866058	−0.0366304
$560$	0	0
$561$	−4.45063	−0.187906
$562$	0	0
$563$	25.6867	1.08257	0.541283	−	0.840840i	$-0.317939\pi$
$563$	25.6867	1.08257	0.541283	+	0.840840i	$0.317939\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.88155	0.205006
$568$	0	0
$569$	−0.874988	−0.0366814	−0.0183407	−	0.999832i	$-0.505838\pi$
$569$	−0.874988	−0.0366814	−0.0183407	+	0.999832i	$0.505838\pi$
$570$	0	0
$571$	−7.50834	−0.314214	−0.157107	−	0.987582i	$-0.550217\pi$
$571$	−7.50834	−0.314214	−0.157107	+	0.987582i	$0.550217\pi$
$572$	0	0
$573$	55.0812	2.30105
$574$	0	0
$575$	0	0
$576$	0	0
$577$	33.9087	1.41164	0.705819	−	0.708392i	$-0.250579\pi$
$577$	33.9087	1.41164	0.705819	+	0.708392i	$0.250579\pi$
$578$	0	0
$579$	6.88015	0.285929
$580$	0	0
$581$	10.0901	0.418608
$582$	0	0
$583$	0.580423	0.0240387
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−46.4075	−1.91544	−0.957722	−	0.287695i	$-0.907111\pi$
$587$	−46.4075	−1.91544	−0.957722	+	0.287695i	$0.907111\pi$
$588$	0	0
$589$	−8.26794	−0.340675
$590$	0	0
$591$	−36.7580	−1.51202
$592$	0	0
$593$	4.79573	0.196937	0.0984685	−	0.995140i	$-0.468606\pi$
$593$	4.79573	0.196937	0.0984685	+	0.995140i	$0.468606\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	62.3428	2.55152
$598$	0	0
$599$	−1.54282	−0.0630377	−0.0315189	−	0.999503i	$-0.510034\pi$
$599$	−1.54282	−0.0630377	−0.0315189	+	0.999503i	$0.510034\pi$
$600$	0	0
$601$	28.7100	1.17110	0.585552	−	0.810635i	$-0.300878\pi$
$601$	28.7100	1.17110	0.585552	+	0.810635i	$0.300878\pi$
$602$	0	0
$603$	54.0882	2.20264
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−27.3875	−1.11162	−0.555811	−	0.831308i	$-0.687592\pi$
$607$	−27.3875	−1.11162	−0.555811	+	0.831308i	$0.687592\pi$
$608$	0	0
$609$	34.1958	1.38568
$610$	0	0
$611$	0.510038	0.0206339
$612$	0	0
$613$	30.1125	1.21623	0.608117	−	0.793848i	$-0.291925\pi$
$613$	30.1125	1.21623	0.608117	+	0.793848i	$0.291925\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.51196	−0.382937	−0.191469	−	0.981499i	$-0.561325\pi$
$617$	−9.51196	−0.382937	−0.191469	+	0.981499i	$0.561325\pi$
$618$	0	0
$619$	30.7584	1.23628	0.618142	−	0.786067i	$-0.287886\pi$
$619$	30.7584	1.23628	0.618142	+	0.786067i	$0.287886\pi$
$620$	0	0
$621$	30.5161	1.22457
$622$	0	0
$623$	38.2292	1.53162
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.91698	0.356110
$628$	0	0
$629$	−1.07378	−0.0428146
$630$	0	0
$631$	22.9168	0.912302	0.456151	−	0.889902i	$-0.349228\pi$
$631$	22.9168	0.912302	0.456151	+	0.889902i	$0.349228\pi$
$632$	0	0
$633$	−1.93414	−0.0768751
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.620320	0.0245780
$638$	0	0
$639$	−45.4256	−1.79701
$640$	0	0
$641$	−42.5989	−1.68256	−0.841278	−	0.540603i	$-0.818196\pi$
$641$	−42.5989	−1.68256	−0.841278	+	0.540603i	$0.818196\pi$
$642$	0	0
$643$	−8.44016	−0.332847	−0.166424	−	0.986054i	$-0.553222\pi$
$643$	−8.44016	−0.332847	−0.166424	+	0.986054i	$0.553222\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.2212	−0.755664	−0.377832	−	0.925874i	$-0.623330\pi$
$647$	−19.2212	−0.755664	−0.377832	+	0.925874i	$0.623330\pi$
$648$	0	0
$649$	7.39377	0.290231
$650$	0	0
$651$	−27.0827	−1.06145
$652$	0	0
$653$	48.5978	1.90178	0.950890	−	0.309530i	$-0.100172\pi$
$653$	48.5978	1.90178	0.950890	+	0.309530i	$0.100172\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−40.5918	−1.58364
$658$	0	0
$659$	−26.8285	−1.04509	−0.522545	−	0.852612i	$-0.675017\pi$
$659$	−26.8285	−1.04509	−0.522545	+	0.852612i	$0.675017\pi$
$660$	0	0
$661$	−25.9141	−1.00794	−0.503972	−	0.863720i	$-0.668128\pi$
$661$	−25.9141	−1.00794	−0.503972	+	0.863720i	$0.668128\pi$
$662$	0	0
$663$	2.11886	0.0822896
$664$	0	0
$665$	0	0
$666$	0	0
$667$	24.2949	0.940704
$668$	0	0
$669$	−55.5779	−2.14876
$670$	0	0
$671$	2.67815	0.103389
$672$	0	0
$673$	3.87956	0.149546	0.0747730	−	0.997201i	$-0.476177\pi$
$673$	3.87956	0.149546	0.0747730	+	0.997201i	$0.476177\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.68947	−0.333964	−0.166982	−	0.985960i	$-0.553402\pi$
$677$	−8.68947	−0.333964	−0.166982	+	0.985960i	$0.553402\pi$
$678$	0	0
$679$	−25.8270	−0.991147
$680$	0	0
$681$	19.0692	0.730733
$682$	0	0
$683$	−4.76231	−0.182225	−0.0911125	−	0.995841i	$-0.529042\pi$
$683$	−4.76231	−0.182225	−0.0911125	+	0.995841i	$0.529042\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−36.5403	−1.39410
$688$	0	0
$689$	−0.276328	−0.0105273
$690$	0	0
$691$	−34.4542	−1.31070	−0.655350	−	0.755325i	$-0.727479\pi$
$691$	−34.4542	−1.31070	−0.655350	+	0.755325i	$0.727479\pi$
$692$	0	0
$693$	18.4395	0.700459
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.56033	0.362123
$698$	0	0
$699$	−30.7307	−1.16234
$700$	0	0
$701$	37.8935	1.43122	0.715610	−	0.698500i	$-0.246149\pi$
$701$	37.8935	1.43122	0.715610	+	0.698500i	$0.246149\pi$
$702$	0	0
$703$	2.15136	0.0811401
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−22.0027	−0.827496
$708$	0	0
$709$	−2.07659	−0.0779880	−0.0389940	−	0.999239i	$-0.512415\pi$
$709$	−2.07659	−0.0779880	−0.0389940	+	0.999239i	$0.512415\pi$
$710$	0	0
$711$	5.96984	0.223886
$712$	0	0
$713$	−19.2413	−0.720593
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.2487	0.457437
$718$	0	0
$719$	18.1139	0.675533	0.337767	−	0.941230i	$-0.390329\pi$
$719$	18.1139	0.675533	0.337767	+	0.941230i	$0.390329\pi$
$720$	0	0
$721$	3.12982	0.116561
$722$	0	0
$723$	36.9795	1.37528
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.5608	0.688382	0.344191	−	0.938900i	$-0.388153\pi$
$727$	18.5608	0.688382	0.344191	+	0.938900i	$0.388153\pi$
$728$	0	0
$729$	−42.0086	−1.55588
$730$	0	0
$731$	−1.34433	−0.0497217
$732$	0	0
$733$	−11.3513	−0.419270	−0.209635	−	0.977780i	$-0.567228\pi$
$733$	−11.3513	−0.419270	−0.209635	+	0.977780i	$0.567228\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.3001	0.563586
$738$	0	0
$739$	−33.9071	−1.24729	−0.623646	−	0.781707i	$-0.714349\pi$
$739$	−33.9071	−1.24729	−0.623646	+	0.781707i	$0.714349\pi$
$740$	0	0
$741$	−4.24520	−0.155951
$742$	0	0
$743$	30.6460	1.12429	0.562146	−	0.827038i	$-0.309976\pi$
$743$	30.6460	1.12429	0.562146	+	0.827038i	$0.309976\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20.9798	0.767612
$748$	0	0
$749$	31.8182	1.16261
$750$	0	0
$751$	−35.0742	−1.27988	−0.639938	−	0.768427i	$-0.721040\pi$
$751$	−35.0742	−1.27988	−0.639938	+	0.768427i	$0.721040\pi$
$752$	0	0
$753$	20.6127	0.751169
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13.8724	0.504200	0.252100	−	0.967701i	$-0.418879\pi$
$757$	13.8724	0.504200	0.252100	+	0.967701i	$0.418879\pi$
$758$	0	0
$759$	20.7518	0.753242
$760$	0	0
$761$	−1.38470	−0.0501954	−0.0250977	−	0.999685i	$-0.507990\pi$
$761$	−1.38470	−0.0501954	−0.0250977	+	0.999685i	$0.507990\pi$
$762$	0	0
$763$	32.7251	1.18473
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.52003	−0.127101
$768$	0	0
$769$	−36.3472	−1.31071	−0.655357	−	0.755319i	$-0.727482\pi$
$769$	−36.3472	−1.31071	−0.655357	+	0.755319i	$0.727482\pi$
$770$	0	0
$771$	−38.5721	−1.38914
$772$	0	0
$773$	−17.9396	−0.645244	−0.322622	−	0.946528i	$-0.604564\pi$
$773$	−17.9396	−0.645244	−0.322622	+	0.946528i	$0.604564\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.04705	0.252811
$778$	0	0
$779$	−19.1544	−0.686278
$780$	0	0
$781$	−12.8497	−0.459797
$782$	0	0
$783$	29.5764	1.05697
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−22.3205	−0.795641	−0.397821	−	0.917463i	$-0.630233\pi$
$787$	−22.3205	−0.795641	−0.397821	+	0.917463i	$0.630233\pi$
$788$	0	0
$789$	17.1227	0.609583
$790$	0	0
$791$	−29.5567	−1.05092
$792$	0	0
$793$	−1.27501	−0.0452770
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.2845	0.860201	0.430101	−	0.902781i	$-0.358478\pi$
$797$	24.2845	0.860201	0.430101	+	0.902781i	$0.358478\pi$
$798$	0	0
$799$	0.791698	0.0280083
$800$	0	0
$801$	79.4882	2.80858
$802$	0	0
$803$	−11.4823	−0.405202
$804$	0	0
$805$	0	0
$806$	0	0
$807$	71.3584	2.51193
$808$	0	0
$809$	22.4785	0.790303	0.395152	−	0.918616i	$-0.370692\pi$
$809$	22.4785	0.790303	0.395152	+	0.918616i	$0.370692\pi$
$810$	0	0
$811$	−11.1407	−0.391202	−0.195601	−	0.980684i	$-0.562666\pi$
$811$	−11.1407	−0.391202	−0.195601	+	0.980684i	$0.562666\pi$
$812$	0	0
$813$	20.8581	0.731525
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.69340	0.0942301
$818$	0	0
$819$	−8.77869	−0.306752
$820$	0	0
$821$	−36.5507	−1.27563	−0.637814	−	0.770190i	$-0.720161\pi$
$821$	−36.5507	−1.27563	−0.637814	+	0.770190i	$0.720161\pi$
$822$	0	0
$823$	−16.4494	−0.573391	−0.286696	−	0.958022i	$-0.592557\pi$
$823$	−16.4494	−0.573391	−0.286696	+	0.958022i	$0.592557\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0790	0.420028	0.210014	−	0.977698i	$-0.432649\pi$
$827$	12.0790	0.420028	0.210014	+	0.977698i	$0.432649\pi$
$828$	0	0
$829$	30.3250	1.05323	0.526616	−	0.850104i	$-0.323461\pi$
$829$	30.3250	1.05323	0.526616	+	0.850104i	$0.323461\pi$
$830$	0	0
$831$	−80.6073	−2.79624
$832$	0	0
$833$	0.962882	0.0333619
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−23.4241	−0.809657
$838$	0	0
$839$	−5.85098	−0.201998	−0.100999	−	0.994887i	$-0.532204\pi$
$839$	−5.85098	−0.201998	−0.100999	+	0.994887i	$0.532204\pi$
$840$	0	0
$841$	−5.45325	−0.188043
$842$	0	0
$843$	−36.8382	−1.26877
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−21.9593	−0.754529
$848$	0	0
$849$	−41.2307	−1.41503
$850$	0	0
$851$	5.00668	0.171627
$852$	0	0
$853$	44.8679	1.53625	0.768124	−	0.640302i	$-0.221191\pi$
$853$	44.8679	1.53625	0.768124	+	0.640302i	$0.221191\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.1121	0.447901	0.223951	−	0.974600i	$-0.428105\pi$
$857$	13.1121	0.447901	0.223951	+	0.974600i	$0.428105\pi$
$858$	0	0
$859$	−12.6423	−0.431349	−0.215675	−	0.976465i	$-0.569195\pi$
$859$	−12.6423	−0.431349	−0.215675	+	0.976465i	$0.569195\pi$
$860$	0	0
$861$	−62.7427	−2.13827
$862$	0	0
$863$	−37.0117	−1.25989	−0.629947	−	0.776638i	$-0.716923\pi$
$863$	−37.0117	−1.25989	−0.629947	+	0.776638i	$0.716923\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−45.2035	−1.53519
$868$	0	0
$869$	1.68870	0.0572854
$870$	0	0
$871$	−7.28406	−0.246811
$872$	0	0
$873$	−53.7007	−1.81749
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39.6811	1.33994	0.669969	−	0.742389i	$-0.266308\pi$
$877$	39.6811	1.33994	0.669969	+	0.742389i	$0.266308\pi$
$878$	0	0
$879$	53.6085	1.80817
$880$	0	0
$881$	41.8698	1.41063	0.705316	−	0.708894i	$-0.250805\pi$
$881$	41.8698	1.41063	0.705316	+	0.708894i	$0.250805\pi$
$882$	0	0
$883$	−26.0807	−0.877685	−0.438843	−	0.898564i	$-0.644611\pi$
$883$	−26.0807	−0.877685	−0.438843	+	0.898564i	$0.644611\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.2959	−1.05081	−0.525407	−	0.850851i	$-0.676087\pi$
$887$	−31.2959	−1.05081	−0.525407	+	0.850851i	$0.676087\pi$
$888$	0	0
$889$	−19.1152	−0.641103
$890$	0	0
$891$	2.87115	0.0961871
$892$	0	0
$893$	−1.58619	−0.0530799
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−9.87951	−0.329867
$898$	0	0
$899$	−18.6488	−0.621971
$900$	0	0
$901$	−0.428926	−0.0142896
$902$	0	0
$903$	8.82256	0.293596
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−44.5322	−1.47867	−0.739333	−	0.673340i	$-0.764859\pi$
$907$	−44.5322	−1.47867	−0.739333	+	0.673340i	$0.764859\pi$
$908$	0	0
$909$	−45.7491	−1.51740
$910$	0	0
$911$	40.0369	1.32648	0.663240	−	0.748407i	$-0.269181\pi$
$911$	40.0369	1.32648	0.663240	+	0.748407i	$0.269181\pi$
$912$	0	0
$913$	5.93462	0.196407
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.36495	−0.177166
$918$	0	0
$919$	−34.1438	−1.12630	−0.563150	−	0.826355i	$-0.690411\pi$
$919$	−34.1438	−1.12630	−0.563150	+	0.826355i	$0.690411\pi$
$920$	0	0
$921$	56.4847	1.86124
$922$	0	0
$923$	6.11747	0.201359
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.50769	0.213740
$928$	0	0
$929$	−5.63909	−0.185013	−0.0925063	−	0.995712i	$-0.529488\pi$
$929$	−5.63909	−0.185013	−0.0925063	+	0.995712i	$0.529488\pi$
$930$	0	0
$931$	−1.92917	−0.0632258
$932$	0	0
$933$	41.3912	1.35509
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.0293	1.40571	0.702853	−	0.711335i	$-0.251909\pi$
$937$	43.0293	1.40571	0.702853	+	0.711335i	$0.251909\pi$
$938$	0	0
$939$	−78.9943	−2.57788
$940$	0	0
$941$	45.7393	1.49106	0.745529	−	0.666473i	$-0.232197\pi$
$941$	45.7393	1.49106	0.745529	+	0.666473i	$0.232197\pi$
$942$	0	0
$943$	−44.5765	−1.45161
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.16489	0.265323	0.132662	−	0.991161i	$-0.457648\pi$
$947$	8.16489	0.265323	0.132662	+	0.991161i	$0.457648\pi$
$948$	0	0
$949$	5.46650	0.177450
$950$	0	0
$951$	−25.1213	−0.814613
$952$	0	0
$953$	−9.73686	−0.315408	−0.157704	−	0.987486i	$-0.550409\pi$
$953$	−9.73686	−0.315408	−0.157704	+	0.987486i	$0.550409\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	20.1127	0.650151
$958$	0	0
$959$	−14.3925	−0.464758
$960$	0	0
$961$	−16.2304	−0.523561
$962$	0	0
$963$	66.1580	2.13191
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.9448	0.673538	0.336769	−	0.941587i	$-0.390666\pi$
$967$	20.9448	0.673538	0.336769	+	0.941587i	$0.390666\pi$
$968$	0	0
$969$	−6.58954	−0.211687
$970$	0	0
$971$	27.8694	0.894371	0.447186	−	0.894441i	$-0.352426\pi$
$971$	27.8694	0.894371	0.447186	+	0.894441i	$0.352426\pi$
$972$	0	0
$973$	−12.3725	−0.396644
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−56.1379	−1.79601	−0.898005	−	0.439985i	$-0.854984\pi$
$977$	−56.1379	−1.79601	−0.898005	+	0.439985i	$0.854984\pi$
$978$	0	0
$979$	22.4850	0.718625
$980$	0	0
$981$	68.0436	2.17247
$982$	0	0
$983$	−9.70948	−0.309684	−0.154842	−	0.987939i	$-0.549487\pi$
$983$	−9.70948	−0.309684	−0.154842	+	0.987939i	$0.549487\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.19577	−0.165383
$988$	0	0
$989$	6.26813	0.199315
$990$	0	0
$991$	−44.7952	−1.42297	−0.711483	−	0.702703i	$-0.751976\pi$
$991$	−44.7952	−1.42297	−0.711483	+	0.702703i	$0.751976\pi$
$992$	0	0
$993$	−55.7891	−1.77041
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−50.2907	−1.59272	−0.796361	−	0.604822i	$-0.793244\pi$
$997$	−50.2907	−1.59272	−0.796361	+	0.604822i	$0.793244\pi$
$998$	0	0
$999$	6.09508	0.192840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3700.2.a.n.1.6	yes	6
5.2	odd	4		3700.2.d.k.149.2		12
5.3	odd	4		3700.2.d.k.149.11		12
5.4	even	2		3700.2.a.m.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3700.2.a.m.1.1	✓	6	5.4	even	2
3700.2.a.n.1.6	yes	6	1.1	even	1	trivial
3700.2.d.k.149.2		12	5.2	odd	4
3700.2.d.k.149.11		12	5.3	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:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3700.a (trivial)

\(f(q)\)	\(=\)	\(q+2.85250 q^{3} +2.47048 q^{7} +5.13675 q^{9} +1.45305 q^{11} -0.691767 q^{13} -1.07378 q^{17} +2.15136 q^{19} +7.04705 q^{21} +5.00668 q^{23} +6.09508 q^{27} +4.85250 q^{29} -3.84313 q^{31} +4.14481 q^{33} +1.00000 q^{37} -1.97326 q^{39} -8.90340 q^{41} +1.25195 q^{43} -0.737297 q^{47} -0.896719 q^{49} -3.06297 q^{51} +0.399453 q^{53} +6.13675 q^{57} +5.08846 q^{59} +1.84313 q^{61} +12.6902 q^{63} +10.5297 q^{67} +14.2816 q^{69} -8.84326 q^{71} -7.90223 q^{73} +3.58972 q^{77} +1.16218 q^{79} +1.97595 q^{81} +4.08426 q^{83} +13.8417 q^{87} +15.4744 q^{89} -1.70900 q^{91} -10.9625 q^{93} -10.4542 q^{97} +7.46394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} - 2 q^{7} + 9 q^{9} - 8 q^{11} + 5 q^{13} + 2 q^{17} + 2 q^{19} + 10 q^{21} - 8 q^{23} + q^{27} + 13 q^{29} - 3 q^{31} - q^{33} + 6 q^{37} + 12 q^{39} + 22 q^{41} - 8 q^{43} + 18 q^{47} + 32 q^{49}+ \cdots + 13 q^{99}+O(q^{100}) \) 6 * q + q^3 - 2 * q^7 + 9 * q^9 - 8 * q^11 + 5 * q^13 + 2 * q^17 + 2 * q^19 + 10 * q^21 - 8 * q^23 + q^27 + 13 * q^29 - 3 * q^31 - q^33 + 6 * q^37 + 12 * q^39 + 22 * q^41 - 8 * q^43 + 18 * q^47 + 32 * q^49 - 5 * q^51 + 3 * q^53 + 15 * q^57 + 13 * q^59 - 9 * q^61 - 44 * q^63 + 27 * q^67 + 38 * q^69 + 20 * q^71 - 41 * q^73 + 3 * q^77 + 14 * q^79 + 30 * q^81 - 16 * q^83 + 29 * q^87 + 16 * q^89 - 28 * q^91 - 4 * q^93 + 23 * q^97 + 13 * q^99

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