LMFDB - Embedded newform 112.6.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,6,Mod(1,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("112.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.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:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.6495 q^{3} -46.7492 q^{5} -49.0000 q^{7} +143.103 q^{9} +O(q^{10})$	$q-19.6495 q^{3} -46.7492 q^{5} -49.0000 q^{7} +143.103 q^{9} -666.090 q^{11} -650.640 q^{13} +918.598 q^{15} +1186.89 q^{17} +1565.05 q^{19} +962.826 q^{21} +1100.15 q^{23} -939.515 q^{25} +1962.93 q^{27} +2396.72 q^{29} +2048.46 q^{31} +13088.3 q^{33} +2290.71 q^{35} +1077.54 q^{37} +12784.7 q^{39} +1098.21 q^{41} -16564.3 q^{43} -6689.95 q^{45} +8298.39 q^{47} +2401.00 q^{49} -23321.9 q^{51} +5519.18 q^{53} +31139.1 q^{55} -30752.5 q^{57} +14230.4 q^{59} -14234.7 q^{61} -7012.05 q^{63} +30416.9 q^{65} -19730.4 q^{67} -21617.5 q^{69} -64562.7 q^{71} +28567.0 q^{73} +18461.0 q^{75} +32638.4 q^{77} +30633.4 q^{79} -73344.6 q^{81} +675.946 q^{83} -55486.3 q^{85} -47094.4 q^{87} +125971. q^{89} +31881.3 q^{91} -40251.1 q^{93} -73164.9 q^{95} -22906.8 q^{97} -95319.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 18 q^{5} - 98 q^{7} + 558 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 18 * q^5 - 98 * q^7 + 558 * q^9	$ 2 q + 6 q^{3} - 18 q^{5} - 98 q^{7} + 558 q^{9} - 396 q^{11} - 350 q^{13} + 1656 q^{15} + 1800 q^{17} + 3266 q^{19} - 294 q^{21} - 2088 q^{23} - 3238 q^{25} + 6372 q^{27} + 6696 q^{29} + 20 q^{31} + 20016 q^{33} + 882 q^{35} + 6232 q^{37} + 20496 q^{39} - 6048 q^{41} + 3020 q^{43} + 5238 q^{45} - 11700 q^{47} + 4802 q^{49} - 7596 q^{51} + 9468 q^{53} + 38904 q^{55} + 12876 q^{57} + 43938 q^{59} - 64754 q^{61} - 27342 q^{63} + 39060 q^{65} - 24784 q^{67} - 103392 q^{69} - 97416 q^{71} + 17452 q^{73} - 40494 q^{75} + 19404 q^{77} - 51256 q^{79} - 61074 q^{81} - 117558 q^{83} - 37860 q^{85} + 63180 q^{87} + 84276 q^{89} + 17150 q^{91} - 92280 q^{93} - 24264 q^{95} + 20776 q^{97} + 16740 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 18 * q^5 - 98 * q^7 + 558 * q^9 - 396 * q^11 - 350 * q^13 + 1656 * q^15 + 1800 * q^17 + 3266 * q^19 - 294 * q^21 - 2088 * q^23 - 3238 * q^25 + 6372 * q^27 + 6696 * q^29 + 20 * q^31 + 20016 * q^33 + 882 * q^35 + 6232 * q^37 + 20496 * q^39 - 6048 * q^41 + 3020 * q^43 + 5238 * q^45 - 11700 * q^47 + 4802 * q^49 - 7596 * q^51 + 9468 * q^53 + 38904 * q^55 + 12876 * q^57 + 43938 * q^59 - 64754 * q^61 - 27342 * q^63 + 39060 * q^65 - 24784 * q^67 - 103392 * q^69 - 97416 * q^71 + 17452 * q^73 - 40494 * q^75 + 19404 * q^77 - 51256 * q^79 - 61074 * q^81 - 117558 * q^83 - 37860 * q^85 + 63180 * q^87 + 84276 * q^89 + 17150 * q^91 - 92280 * q^93 - 24264 * q^95 + 20776 * q^97 + 16740 * 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
$3$	−19.6495	−1.26052	−0.630258	−	0.776386i	$-0.717051\pi$
$3$	−19.6495	−1.26052	−0.630258	+	0.776386i	$0.717051\pi$
$4$	0	0
$5$	−46.7492	−0.836275	−0.418137	−	0.908384i	$-0.637317\pi$
$5$	−46.7492	−0.836275	−0.418137	+	0.908384i	$0.637317\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	143.103	0.588901
$10$	0	0
$11$	−666.090	−1.65978	−0.829891	−	0.557926i	$-0.811597\pi$
$11$	−666.090	−1.65978	−0.829891	+	0.557926i	$0.811597\pi$
$12$	0	0
$13$	−650.640	−1.06778	−0.533890	−	0.845554i	$-0.679271\pi$
$13$	−650.640	−1.06778	−0.533890	+	0.845554i	$0.679271\pi$
$14$	0	0
$15$	918.598	1.05414
$16$	0	0
$17$	1186.89	0.996069	0.498035	−	0.867157i	$-0.334055\pi$
$17$	1186.89	0.996069	0.498035	+	0.867157i	$0.334055\pi$
$18$	0	0
$19$	1565.05	0.994591	0.497296	−	0.867581i	$-0.334326\pi$
$19$	1565.05	0.994591	0.497296	+	0.867581i	$0.334326\pi$
$20$	0	0
$21$	962.826	0.476430
$22$	0	0
$23$	1100.15	0.433644	0.216822	−	0.976211i	$-0.430431\pi$
$23$	1100.15	0.433644	0.216822	+	0.976211i	$0.430431\pi$
$24$	0	0
$25$	−939.515	−0.300645
$26$	0	0
$27$	1962.93	0.518197
$28$	0	0
$29$	2396.72	0.529203	0.264602	−	0.964358i	$-0.414760\pi$
$29$	2396.72	0.529203	0.264602	+	0.964358i	$0.414760\pi$
$30$	0	0
$31$	2048.46	0.382844	0.191422	−	0.981508i	$-0.438690\pi$
$31$	2048.46	0.382844	0.191422	+	0.981508i	$0.438690\pi$
$32$	0	0
$33$	13088.3	2.09218
$34$	0	0
$35$	2290.71	0.316082
$36$	0	0
$37$	1077.54	0.129399	0.0646995	−	0.997905i	$-0.479391\pi$
$37$	1077.54	0.129399	0.0646995	+	0.997905i	$0.479391\pi$
$38$	0	0
$39$	12784.7	1.34595
$40$	0	0
$41$	1098.21	0.102029	0.0510147	−	0.998698i	$-0.483754\pi$
$41$	1098.21	0.102029	0.0510147	+	0.998698i	$0.483754\pi$
$42$	0	0
$43$	−16564.3	−1.36616	−0.683081	−	0.730343i	$-0.739360\pi$
$43$	−16564.3	−1.36616	−0.683081	+	0.730343i	$0.739360\pi$
$44$	0	0
$45$	−6689.95	−0.492483
$46$	0	0
$47$	8298.39	0.547960	0.273980	−	0.961735i	$-0.411660\pi$
$47$	8298.39	0.547960	0.273980	+	0.961735i	$0.411660\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−23321.9	−1.25556
$52$	0	0
$53$	5519.18	0.269889	0.134944	−	0.990853i	$-0.456914\pi$
$53$	5519.18	0.269889	0.134944	+	0.990853i	$0.456914\pi$
$54$	0	0
$55$	31139.1	1.38803
$56$	0	0
$57$	−30752.5	−1.25370
$58$	0	0
$59$	14230.4	0.532216	0.266108	−	0.963943i	$-0.414262\pi$
$59$	14230.4	0.532216	0.266108	+	0.963943i	$0.414262\pi$
$60$	0	0
$61$	−14234.7	−0.489807	−0.244904	−	0.969547i	$-0.578756\pi$
$61$	−14234.7	−0.489807	−0.244904	+	0.969547i	$0.578756\pi$
$62$	0	0
$63$	−7012.05	−0.222584
$64$	0	0
$65$	30416.9	0.892958
$66$	0	0
$67$	−19730.4	−0.536970	−0.268485	−	0.963284i	$-0.586523\pi$
$67$	−19730.4	−0.536970	−0.268485	+	0.963284i	$0.586523\pi$
$68$	0	0
$69$	−21617.5	−0.546615
$70$	0	0
$71$	−64562.7	−1.51997	−0.759986	−	0.649940i	$-0.774794\pi$
$71$	−64562.7	−1.51997	−0.759986	+	0.649940i	$0.774794\pi$
$72$	0	0
$73$	28567.0	0.627418	0.313709	−	0.949519i	$-0.398428\pi$
$73$	28567.0	0.627418	0.313709	+	0.949519i	$0.398428\pi$
$74$	0	0
$75$	18461.0	0.378968
$76$	0	0
$77$	32638.4	0.627339
$78$	0	0
$79$	30633.4	0.552239	0.276119	−	0.961123i	$-0.410951\pi$
$79$	30633.4	0.552239	0.276119	+	0.961123i	$0.410951\pi$
$80$	0	0
$81$	−73344.6	−1.24210
$82$	0	0
$83$	675.946	0.0107700	0.00538501	−	0.999986i	$-0.498286\pi$
$83$	675.946	0.0107700	0.00538501	+	0.999986i	$0.498286\pi$
$84$	0	0
$85$	−55486.3	−0.832987
$86$	0	0
$87$	−47094.4	−0.667069
$88$	0	0
$89$	125971.	1.68576	0.842882	−	0.538098i	$-0.180857\pi$
$89$	125971.	1.68576	0.842882	+	0.538098i	$0.180857\pi$
$90$	0	0
$91$	31881.3	0.403583
$92$	0	0
$93$	−40251.1	−0.482582
$94$	0	0
$95$	−73164.9	−0.831751
$96$	0	0
$97$	−22906.8	−0.247192	−0.123596	−	0.992333i	$-0.539443\pi$
$97$	−22906.8	−0.247192	−0.123596	+	0.992333i	$0.539443\pi$
$98$	0	0
$99$	−95319.4	−0.977447
$100$	0	0
$101$	−181474.	−1.77015	−0.885077	−	0.465444i	$-0.845895\pi$
$101$	−181474.	−1.77015	−0.885077	+	0.465444i	$0.845895\pi$
$102$	0	0
$103$	−64772.0	−0.601581	−0.300791	−	0.953690i	$-0.597251\pi$
$103$	−64772.0	−0.601581	−0.300791	+	0.953690i	$0.597251\pi$
$104$	0	0
$105$	−45011.3	−0.398427
$106$	0	0
$107$	148170.	1.25112	0.625562	−	0.780175i	$-0.284870\pi$
$107$	148170.	1.25112	0.625562	+	0.780175i	$0.284870\pi$
$108$	0	0
$109$	−111294.	−0.897237	−0.448618	−	0.893723i	$-0.648084\pi$
$109$	−111294.	−0.897237	−0.448618	+	0.893723i	$0.648084\pi$
$110$	0	0
$111$	−21173.2	−0.163110
$112$	0	0
$113$	−43175.5	−0.318084	−0.159042	−	0.987272i	$-0.550840\pi$
$113$	−43175.5	−0.318084	−0.159042	+	0.987272i	$0.550840\pi$
$114$	0	0
$115$	−51431.2	−0.362646
$116$	0	0
$117$	−93108.5	−0.628817
$118$	0	0
$119$	−58157.8	−0.376479
$120$	0	0
$121$	282625.	1.75488
$122$	0	0
$123$	−21579.3	−0.128610
$124$	0	0
$125$	190013.	1.08770
$126$	0	0
$127$	131449.	0.723182	0.361591	−	0.932337i	$-0.382234\pi$
$127$	131449.	0.723182	0.361591	+	0.932337i	$0.382234\pi$
$128$	0	0
$129$	325480.	1.72207
$130$	0	0
$131$	349458.	1.77916	0.889582	−	0.456775i	$-0.150995\pi$
$131$	349458.	1.77916	0.889582	+	0.456775i	$0.150995\pi$
$132$	0	0
$133$	−76687.5	−0.375920
$134$	0	0
$135$	−91765.2	−0.433355
$136$	0	0
$137$	386434.	1.75903	0.879516	−	0.475869i	$-0.157866\pi$
$137$	386434.	1.75903	0.879516	+	0.475869i	$0.157866\pi$
$138$	0	0
$139$	−17289.3	−0.0758997	−0.0379498	−	0.999280i	$-0.512083\pi$
$139$	−17289.3	−0.0758997	−0.0379498	+	0.999280i	$0.512083\pi$
$140$	0	0
$141$	−163059.	−0.690713
$142$	0	0
$143$	433384.	1.77228
$144$	0	0
$145$	−112045.	−0.442559
$146$	0	0
$147$	−47178.5	−0.180074
$148$	0	0
$149$	−112171.	−0.413917	−0.206959	−	0.978350i	$-0.566357\pi$
$149$	−112171.	−0.413917	−0.206959	+	0.978350i	$0.566357\pi$
$150$	0	0
$151$	−30495.4	−0.108841	−0.0544205	−	0.998518i	$-0.517331\pi$
$151$	−30495.4	−0.108841	−0.0544205	+	0.998518i	$0.517331\pi$
$152$	0	0
$153$	169848.	0.586586
$154$	0	0
$155$	−95763.6	−0.320163
$156$	0	0
$157$	523509.	1.69502	0.847510	−	0.530780i	$-0.178101\pi$
$157$	523509.	1.69502	0.847510	+	0.530780i	$0.178101\pi$
$158$	0	0
$159$	−108449.	−0.340199
$160$	0	0
$161$	−53907.5	−0.163902
$162$	0	0
$163$	439646.	1.29609	0.648043	−	0.761604i	$-0.275588\pi$
$163$	439646.	1.29609	0.648043	+	0.761604i	$0.275588\pi$
$164$	0	0
$165$	−611869.	−1.74964
$166$	0	0
$167$	−279353.	−0.775107	−0.387554	−	0.921847i	$-0.626680\pi$
$167$	−279353.	−0.775107	−0.387554	+	0.921847i	$0.626680\pi$
$168$	0	0
$169$	52038.8	0.140156
$170$	0	0
$171$	223964.	0.585716
$172$	0	0
$173$	99699.4	0.253266	0.126633	−	0.991950i	$-0.459583\pi$
$173$	99699.4	0.253266	0.126633	+	0.991950i	$0.459583\pi$
$174$	0	0
$175$	46036.2	0.113633
$176$	0	0
$177$	−279621.	−0.670867
$178$	0	0
$179$	−329980.	−0.769760	−0.384880	−	0.922967i	$-0.625757\pi$
$179$	−329980.	−0.769760	−0.384880	+	0.922967i	$0.625757\pi$
$180$	0	0
$181$	−505810.	−1.14760	−0.573800	−	0.818995i	$-0.694531\pi$
$181$	−505810.	−1.14760	−0.573800	+	0.818995i	$0.694531\pi$
$182$	0	0
$183$	279706.	0.617410
$184$	0	0
$185$	−50374.3	−0.108213
$186$	0	0
$187$	−790578.	−1.65326
$188$	0	0
$189$	−96183.4	−0.195860
$190$	0	0
$191$	63835.6	0.126613	0.0633067	−	0.997994i	$-0.479835\pi$
$191$	63835.6	0.126613	0.0633067	+	0.997994i	$0.479835\pi$
$192$	0	0
$193$	469355.	0.907001	0.453501	−	0.891256i	$-0.350175\pi$
$193$	469355.	0.907001	0.453501	+	0.891256i	$0.350175\pi$
$194$	0	0
$195$	−597676.	−1.12559
$196$	0	0
$197$	268021.	0.492043	0.246021	−	0.969264i	$-0.420877\pi$
$197$	268021.	0.492043	0.246021	+	0.969264i	$0.420877\pi$
$198$	0	0
$199$	605167.	1.08328	0.541642	−	0.840609i	$-0.317803\pi$
$199$	605167.	1.08328	0.541642	+	0.840609i	$0.317803\pi$
$200$	0	0
$201$	387693.	0.676859
$202$	0	0
$203$	−117439.	−0.200020
$204$	0	0
$205$	−51340.4	−0.0853246
$206$	0	0
$207$	157435.	0.255374
$208$	0	0
$209$	−1.04246e6	−1.65080
$210$	0	0
$211$	−335389.	−0.518612	−0.259306	−	0.965795i	$-0.583494\pi$
$211$	−335389.	−0.518612	−0.259306	+	0.965795i	$0.583494\pi$
$212$	0	0
$213$	1.26862e6	1.91595
$214$	0	0
$215$	774367.	1.14249
$216$	0	0
$217$	−100374.	−0.144702
$218$	0	0
$219$	−561327.	−0.790871
$220$	0	0
$221$	−772240.	−1.06358
$222$	0	0
$223$	−1.02526e6	−1.38061	−0.690305	−	0.723518i	$-0.742524\pi$
$223$	−1.02526e6	−1.38061	−0.690305	+	0.723518i	$0.742524\pi$
$224$	0	0
$225$	−134447.	−0.177050
$226$	0	0
$227$	504226.	0.649473	0.324736	−	0.945805i	$-0.394724\pi$
$227$	504226.	0.649473	0.324736	+	0.945805i	$0.394724\pi$
$228$	0	0
$229$	−1.11939e6	−1.41057	−0.705283	−	0.708925i	$-0.749180\pi$
$229$	−1.11939e6	−1.41057	−0.705283	+	0.708925i	$0.749180\pi$
$230$	0	0
$231$	−641328.	−0.790770
$232$	0	0
$233$	770703.	0.930031	0.465015	−	0.885303i	$-0.346049\pi$
$233$	770703.	0.930031	0.465015	+	0.885303i	$0.346049\pi$
$234$	0	0
$235$	−387943.	−0.458245
$236$	0	0
$237$	−601930.	−0.696106
$238$	0	0
$239$	171646.	0.194374	0.0971871	−	0.995266i	$-0.469015\pi$
$239$	171646.	0.194374	0.0971871	+	0.995266i	$0.469015\pi$
$240$	0	0
$241$	−383779.	−0.425637	−0.212818	−	0.977092i	$-0.568264\pi$
$241$	−383779.	−0.425637	−0.212818	+	0.977092i	$0.568264\pi$
$242$	0	0
$243$	964193.	1.04749
$244$	0	0
$245$	−112245.	−0.119468
$246$	0	0
$247$	−1.01828e6	−1.06201
$248$	0	0
$249$	−13282.0	−0.0135758
$250$	0	0
$251$	1.57046e6	1.57342	0.786708	−	0.617325i	$-0.211784\pi$
$251$	1.57046e6	1.57342	0.786708	+	0.617325i	$0.211784\pi$
$252$	0	0
$253$	−732801.	−0.719755
$254$	0	0
$255$	1.09028e6	1.04999
$256$	0	0
$257$	790656.	0.746715	0.373357	−	0.927688i	$-0.378207\pi$
$257$	790656.	0.746715	0.373357	+	0.927688i	$0.378207\pi$
$258$	0	0
$259$	−52799.7	−0.0489082
$260$	0	0
$261$	342978.	0.311648
$262$	0	0
$263$	−464416.	−0.414017	−0.207008	−	0.978339i	$-0.566373\pi$
$263$	−464416.	−0.414017	−0.207008	+	0.978339i	$0.566373\pi$
$264$	0	0
$265$	−258017.	−0.225701
$266$	0	0
$267$	−2.47527e6	−2.12493
$268$	0	0
$269$	1.99959e6	1.68484	0.842422	−	0.538818i	$-0.181129\pi$
$269$	1.99959e6	1.68484	0.842422	+	0.538818i	$0.181129\pi$
$270$	0	0
$271$	−1.61296e6	−1.33414	−0.667070	−	0.744995i	$-0.732452\pi$
$271$	−1.61296e6	−1.33414	−0.667070	+	0.744995i	$0.732452\pi$
$272$	0	0
$273$	−626452.	−0.508723
$274$	0	0
$275$	625801.	0.499005
$276$	0	0
$277$	−2.08119e6	−1.62972	−0.814860	−	0.579658i	$-0.803186\pi$
$277$	−2.08119e6	−1.62972	−0.814860	+	0.579658i	$0.803186\pi$
$278$	0	0
$279$	293140.	0.225457
$280$	0	0
$281$	−982035.	−0.741927	−0.370964	−	0.928647i	$-0.620973\pi$
$281$	−982035.	−0.741927	−0.370964	+	0.928647i	$0.620973\pi$
$282$	0	0
$283$	1.39622e6	1.03630	0.518152	−	0.855289i	$-0.326620\pi$
$283$	1.39622e6	1.03630	0.518152	+	0.855289i	$0.326620\pi$
$284$	0	0
$285$	1.43765e6	1.04844
$286$	0	0
$287$	−53812.3	−0.0385635
$288$	0	0
$289$	−11140.3	−0.00784609
$290$	0	0
$291$	450107.	0.311590
$292$	0	0
$293$	2.56205e6	1.74348	0.871742	−	0.489965i	$-0.162990\pi$
$293$	2.56205e6	1.74348	0.871742	+	0.489965i	$0.162990\pi$
$294$	0	0
$295$	−665260.	−0.445078
$296$	0	0
$297$	−1.30749e6	−0.860094
$298$	0	0
$299$	−715803.	−0.463037
$300$	0	0
$301$	811651.	0.516361
$302$	0	0
$303$	3.56588e6	2.23131
$304$	0	0
$305$	665463.	0.409613
$306$	0	0
$307$	884855.	0.535829	0.267915	−	0.963443i	$-0.413666\pi$
$307$	884855.	0.535829	0.267915	+	0.963443i	$0.413666\pi$
$308$	0	0
$309$	1.27274e6	0.758303
$310$	0	0
$311$	−1.80120e6	−1.05599	−0.527997	−	0.849246i	$-0.677057\pi$
$311$	−1.80120e6	−1.05599	−0.527997	+	0.849246i	$0.677057\pi$
$312$	0	0
$313$	950366.	0.548315	0.274158	−	0.961685i	$-0.411601\pi$
$313$	950366.	0.548315	0.274158	+	0.961685i	$0.411601\pi$
$314$	0	0
$315$	327807.	0.186141
$316$	0	0
$317$	3.04277e6	1.70068	0.850338	−	0.526237i	$-0.176398\pi$
$317$	3.04277e6	1.70068	0.850338	+	0.526237i	$0.176398\pi$
$318$	0	0
$319$	−1.59643e6	−0.878362
$320$	0	0
$321$	−2.91146e6	−1.57706
$322$	0	0
$323$	1.85755e6	0.990682
$324$	0	0
$325$	611286.	0.321023
$326$	0	0
$327$	2.18688e6	1.13098
$328$	0	0
$329$	−406621.	−0.207110
$330$	0	0
$331$	2.19616e6	1.10178	0.550889	−	0.834579i	$-0.314289\pi$
$331$	2.19616e6	1.10178	0.550889	+	0.834579i	$0.314289\pi$
$332$	0	0
$333$	154200.	0.0762032
$334$	0	0
$335$	922382.	0.449054
$336$	0	0
$337$	−2.41491e6	−1.15832	−0.579158	−	0.815216i	$-0.696618\pi$
$337$	−2.41491e6	−1.15832	−0.579158	+	0.815216i	$0.696618\pi$
$338$	0	0
$339$	848377.	0.400950
$340$	0	0
$341$	−1.36446e6	−0.635438
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	1.01060e6	0.457121
$346$	0	0
$347$	1.08833e6	0.485219	0.242609	−	0.970124i	$-0.421997\pi$
$347$	1.08833e6	0.485219	0.242609	+	0.970124i	$0.421997\pi$
$348$	0	0
$349$	2.79267e6	1.22731	0.613657	−	0.789573i	$-0.289698\pi$
$349$	2.79267e6	1.22731	0.613657	+	0.789573i	$0.289698\pi$
$350$	0	0
$351$	−1.27716e6	−0.553321
$352$	0	0
$353$	2.53134e6	1.08122	0.540610	−	0.841273i	$-0.318193\pi$
$353$	2.53134e6	1.08122	0.540610	+	0.841273i	$0.318193\pi$
$354$	0	0
$355$	3.01825e6	1.27111
$356$	0	0
$357$	1.14277e6	0.474558
$358$	0	0
$359$	−1.09028e6	−0.446480	−0.223240	−	0.974763i	$-0.571663\pi$
$359$	−1.09028e6	−0.446480	−0.223240	+	0.974763i	$0.571663\pi$
$360$	0	0
$361$	−26712.8	−0.0107883
$362$	0	0
$363$	−5.55343e6	−2.21205
$364$	0	0
$365$	−1.33548e6	−0.524694
$366$	0	0
$367$	−188070.	−0.0728879	−0.0364439	−	0.999336i	$-0.511603\pi$
$367$	−188070.	−0.0728879	−0.0364439	+	0.999336i	$0.511603\pi$
$368$	0	0
$369$	157157.	0.0600853
$370$	0	0
$371$	−270440.	−0.102008
$372$	0	0
$373$	−1.79371e6	−0.667545	−0.333772	−	0.942654i	$-0.608322\pi$
$373$	−1.79371e6	−0.667545	−0.333772	+	0.942654i	$0.608322\pi$
$374$	0	0
$375$	−3.73366e6	−1.37106
$376$	0	0
$377$	−1.55940e6	−0.565073
$378$	0	0
$379$	−3.58806e6	−1.28310	−0.641551	−	0.767080i	$-0.721709\pi$
$379$	−3.58806e6	−1.28310	−0.641551	+	0.767080i	$0.721709\pi$
$380$	0	0
$381$	−2.58291e6	−0.911583
$382$	0	0
$383$	3.42457e6	1.19291	0.596457	−	0.802645i	$-0.296575\pi$
$383$	3.42457e6	1.19291	0.596457	+	0.802645i	$0.296575\pi$
$384$	0	0
$385$	−1.52582e6	−0.524627
$386$	0	0
$387$	−2.37040e6	−0.804534
$388$	0	0
$389$	8625.27	0.00289001	0.00144500	−	0.999999i	$-0.499540\pi$
$389$	8625.27	0.00289001	0.00144500	+	0.999999i	$0.499540\pi$
$390$	0	0
$391$	1.30576e6	0.431940
$392$	0	0
$393$	−6.86667e6	−2.24267
$394$	0	0
$395$	−1.43208e6	−0.461823
$396$	0	0
$397$	1.25709e6	0.400306	0.200153	−	0.979765i	$-0.435856\pi$
$397$	1.25709e6	0.400306	0.200153	+	0.979765i	$0.435856\pi$
$398$	0	0
$399$	1.50687e6	0.473853
$400$	0	0
$401$	1.42670e6	0.443070	0.221535	−	0.975152i	$-0.428893\pi$
$401$	1.42670e6	0.443070	0.221535	+	0.975152i	$0.428893\pi$
$402$	0	0
$403$	−1.33281e6	−0.408794
$404$	0	0
$405$	3.42880e6	1.03873
$406$	0	0
$407$	−717741.	−0.214774
$408$	0	0
$409$	−3.06529e6	−0.906073	−0.453036	−	0.891492i	$-0.649659\pi$
$409$	−3.06529e6	−0.906073	−0.453036	+	0.891492i	$0.649659\pi$
$410$	0	0
$411$	−7.59324e6	−2.21729
$412$	0	0
$413$	−697291.	−0.201159
$414$	0	0
$415$	−31599.9	−0.00900670
$416$	0	0
$417$	339726.	0.0956728
$418$	0	0
$419$	248240.	0.0690776	0.0345388	−	0.999403i	$-0.489004\pi$
$419$	248240.	0.0690776	0.0345388	+	0.999403i	$0.489004\pi$
$420$	0	0
$421$	5.96280e6	1.63963	0.819814	−	0.572630i	$-0.194077\pi$
$421$	5.96280e6	1.63963	0.819814	+	0.572630i	$0.194077\pi$
$422$	0	0
$423$	1.18752e6	0.322695
$424$	0	0
$425$	−1.11510e6	−0.299463
$426$	0	0
$427$	697503.	0.185130
$428$	0	0
$429$	−8.51579e6	−2.23399
$430$	0	0
$431$	4.93538e6	1.27976	0.639879	−	0.768476i	$-0.278985\pi$
$431$	4.93538e6	1.27976	0.639879	+	0.768476i	$0.278985\pi$
$432$	0	0
$433$	4.15513e6	1.06504	0.532519	−	0.846418i	$-0.321245\pi$
$433$	4.15513e6	1.06504	0.532519	+	0.846418i	$0.321245\pi$
$434$	0	0
$435$	2.20162e6	0.557853
$436$	0	0
$437$	1.72180e6	0.431299
$438$	0	0
$439$	227955.	0.0564531	0.0282265	−	0.999602i	$-0.491014\pi$
$439$	227955.	0.0564531	0.0282265	+	0.999602i	$0.491014\pi$
$440$	0	0
$441$	343590.	0.0841287
$442$	0	0
$443$	1.98462e6	0.480472	0.240236	−	0.970715i	$-0.422775\pi$
$443$	1.98462e6	0.480472	0.240236	+	0.970715i	$0.422775\pi$
$444$	0	0
$445$	−5.88906e6	−1.40976
$446$	0	0
$447$	2.20410e6	0.521749
$448$	0	0
$449$	2.61077e6	0.611157	0.305579	−	0.952167i	$-0.401150\pi$
$449$	2.61077e6	0.611157	0.305579	+	0.952167i	$0.401150\pi$
$450$	0	0
$451$	−731506.	−0.169347
$452$	0	0
$453$	599220.	0.137196
$454$	0	0
$455$	−1.49043e6	−0.337506
$456$	0	0
$457$	−4.09917e6	−0.918132	−0.459066	−	0.888402i	$-0.651816\pi$
$457$	−4.09917e6	−0.918132	−0.459066	+	0.888402i	$0.651816\pi$
$458$	0	0
$459$	2.32979e6	0.516160
$460$	0	0
$461$	−2.62378e6	−0.575009	−0.287505	−	0.957779i	$-0.592826\pi$
$461$	−2.62378e6	−0.575009	−0.287505	+	0.957779i	$0.592826\pi$
$462$	0	0
$463$	4.28563e6	0.929100	0.464550	−	0.885547i	$-0.346216\pi$
$463$	4.28563e6	0.929100	0.464550	+	0.885547i	$0.346216\pi$
$464$	0	0
$465$	1.88171e6	0.403571
$466$	0	0
$467$	−990118.	−0.210085	−0.105042	−	0.994468i	$-0.533498\pi$
$467$	−990118.	−0.210085	−0.105042	+	0.994468i	$0.533498\pi$
$468$	0	0
$469$	966792.	0.202955
$470$	0	0
$471$	−1.02867e7	−2.13660
$472$	0	0
$473$	1.10333e7	2.26753
$474$	0	0
$475$	−1.47039e6	−0.299019
$476$	0	0
$477$	789812.	0.158938
$478$	0	0
$479$	−2.57976e6	−0.513737	−0.256868	−	0.966446i	$-0.582691\pi$
$479$	−2.57976e6	−0.513737	−0.256868	+	0.966446i	$0.582691\pi$
$480$	0	0
$481$	−701093.	−0.138170
$482$	0	0
$483$	1.05926e6	0.206601
$484$	0	0
$485$	1.07087e6	0.206720
$486$	0	0
$487$	−3.21474e6	−0.614219	−0.307109	−	0.951674i	$-0.599362\pi$
$487$	−3.21474e6	−0.614219	−0.307109	+	0.951674i	$0.599362\pi$
$488$	0	0
$489$	−8.63882e6	−1.63374
$490$	0	0
$491$	7.86108e6	1.47156	0.735781	−	0.677220i	$-0.236815\pi$
$491$	7.86108e6	1.47156	0.735781	+	0.677220i	$0.236815\pi$
$492$	0	0
$493$	2.84465e6	0.527123
$494$	0	0
$495$	4.45610e6	0.817415
$496$	0	0
$497$	3.16357e6	0.574495
$498$	0	0
$499$	1.35382e6	0.243395	0.121697	−	0.992567i	$-0.461166\pi$
$499$	1.35382e6	0.243395	0.121697	+	0.992567i	$0.461166\pi$
$500$	0	0
$501$	5.48914e6	0.977035
$502$	0	0
$503$	−3.85775e6	−0.679851	−0.339926	−	0.940452i	$-0.610402\pi$
$503$	−3.85775e6	−0.679851	−0.339926	+	0.940452i	$0.610402\pi$
$504$	0	0
$505$	8.48376e6	1.48034
$506$	0	0
$507$	−1.02254e6	−0.176669
$508$	0	0
$509$	−1.06060e7	−1.81451	−0.907253	−	0.420585i	$-0.861825\pi$
$509$	−1.06060e7	−1.81451	−0.907253	+	0.420585i	$0.861825\pi$
$510$	0	0
$511$	−1.39978e6	−0.237142
$512$	0	0
$513$	3.07208e6	0.515394
$514$	0	0
$515$	3.02804e6	0.503087
$516$	0	0
$517$	−5.52747e6	−0.909495
$518$	0	0
$519$	−1.95904e6	−0.319246
$520$	0	0
$521$	−9.17989e6	−1.48164	−0.740821	−	0.671703i	$-0.765563\pi$
$521$	−9.17989e6	−1.48164	−0.740821	+	0.671703i	$0.765563\pi$
$522$	0	0
$523$	−9.05585e6	−1.44769	−0.723844	−	0.689964i	$-0.757627\pi$
$523$	−9.05585e6	−1.44769	−0.723844	+	0.689964i	$0.757627\pi$
$524$	0	0
$525$	−904589.	−0.143236
$526$	0	0
$527$	2.43130e6	0.381339
$528$	0	0
$529$	−5.22601e6	−0.811953
$530$	0	0
$531$	2.03642e6	0.313422
$532$	0	0
$533$	−714539.	−0.108945
$534$	0	0
$535$	−6.92681e6	−1.04628
$536$	0	0
$537$	6.48394e6	0.970295
$538$	0	0
$539$	−1.59928e6	−0.237112
$540$	0	0
$541$	1.21783e7	1.78894	0.894468	−	0.447132i	$-0.147555\pi$
$541$	1.21783e7	1.78894	0.894468	+	0.447132i	$0.147555\pi$
$542$	0	0
$543$	9.93891e6	1.44657
$544$	0	0
$545$	5.20292e6	0.750336
$546$	0	0
$547$	9.00451e6	1.28674	0.643372	−	0.765554i	$-0.277535\pi$
$547$	9.00451e6	1.28674	0.643372	+	0.765554i	$0.277535\pi$
$548$	0	0
$549$	−2.03703e6	−0.288448
$550$	0	0
$551$	3.75099e6	0.526341
$552$	0	0
$553$	−1.50103e6	−0.208727
$554$	0	0
$555$	989830.	0.136404
$556$	0	0
$557$	1.54461e6	0.210950	0.105475	−	0.994422i	$-0.466364\pi$
$557$	1.54461e6	0.210950	0.105475	+	0.994422i	$0.466364\pi$
$558$	0	0
$559$	1.07774e7	1.45876
$560$	0	0
$561$	1.55345e7	2.08396
$562$	0	0
$563$	1.18748e7	1.57890	0.789449	−	0.613816i	$-0.210366\pi$
$563$	1.18748e7	1.57890	0.789449	+	0.613816i	$0.210366\pi$
$564$	0	0
$565$	2.01842e6	0.266005
$566$	0	0
$567$	3.59388e6	0.469468
$568$	0	0
$569$	−1.54308e6	−0.199806	−0.0999031	−	0.994997i	$-0.531853\pi$
$569$	−1.54308e6	−0.199806	−0.0999031	+	0.994997i	$0.531853\pi$
$570$	0	0
$571$	7.01812e6	0.900804	0.450402	−	0.892826i	$-0.351281\pi$
$571$	7.01812e6	0.900804	0.450402	+	0.892826i	$0.351281\pi$
$572$	0	0
$573$	−1.25434e6	−0.159598
$574$	0	0
$575$	−1.03361e6	−0.130373
$576$	0	0
$577$	−5.37543e6	−0.672162	−0.336081	−	0.941833i	$-0.609102\pi$
$577$	−5.37543e6	−0.672162	−0.336081	+	0.941833i	$0.609102\pi$
$578$	0	0
$579$	−9.22259e6	−1.14329
$580$	0	0
$581$	−33121.4	−0.00407069
$582$	0	0
$583$	−3.67627e6	−0.447957
$584$	0	0
$585$	4.35274e6	0.525864
$586$	0	0
$587$	2.06682e6	0.247575	0.123788	−	0.992309i	$-0.460496\pi$
$587$	2.06682e6	0.247575	0.123788	+	0.992309i	$0.460496\pi$
$588$	0	0
$589$	3.20594e6	0.380774
$590$	0	0
$591$	−5.26648e6	−0.620228
$592$	0	0
$593$	−5.46947e6	−0.638717	−0.319358	−	0.947634i	$-0.603467\pi$
$593$	−5.46947e6	−0.638717	−0.319358	+	0.947634i	$0.603467\pi$
$594$	0	0
$595$	2.71883e6	0.314840
$596$	0	0
$597$	−1.18912e7	−1.36550
$598$	0	0
$599$	−1.09943e7	−1.25199	−0.625996	−	0.779826i	$-0.715307\pi$
$599$	−1.09943e7	−1.25199	−0.625996	+	0.779826i	$0.715307\pi$
$600$	0	0
$601$	1.58788e7	1.79322	0.896608	−	0.442826i	$-0.146024\pi$
$601$	1.58788e7	1.79322	0.896608	+	0.442826i	$0.146024\pi$
$602$	0	0
$603$	−2.82348e6	−0.316222
$604$	0	0
$605$	−1.32125e7	−1.46756
$606$	0	0
$607$	−5.33262e6	−0.587447	−0.293724	−	0.955890i	$-0.594895\pi$
$607$	−5.33262e6	−0.587447	−0.293724	+	0.955890i	$0.594895\pi$
$608$	0	0
$609$	2.30762e6	0.252128
$610$	0	0
$611$	−5.39926e6	−0.585102
$612$	0	0
$613$	−8.91838e6	−0.958594	−0.479297	−	0.877653i	$-0.659108\pi$
$613$	−8.91838e6	−0.958594	−0.479297	+	0.877653i	$0.659108\pi$
$614$	0	0
$615$	1.00881e6	0.107553
$616$	0	0
$617$	9.63586e6	1.01901	0.509504	−	0.860468i	$-0.329829\pi$
$617$	9.63586e6	1.01901	0.509504	+	0.860468i	$0.329829\pi$
$618$	0	0
$619$	−1.21747e7	−1.27712	−0.638560	−	0.769572i	$-0.720470\pi$
$619$	−1.21747e7	−1.27712	−0.638560	+	0.769572i	$0.720470\pi$
$620$	0	0
$621$	2.15952e6	0.224713
$622$	0	0
$623$	−6.17260e6	−0.637159
$624$	0	0
$625$	−5.94695e6	−0.608968
$626$	0	0
$627$	2.04839e7	2.08087
$628$	0	0
$629$	1.27893e6	0.128890
$630$	0	0
$631$	1.30854e7	1.30832	0.654161	−	0.756356i	$-0.273022\pi$
$631$	1.30854e7	1.30832	0.654161	+	0.756356i	$0.273022\pi$
$632$	0	0
$633$	6.59023e6	0.653719
$634$	0	0
$635$	−6.14513e6	−0.604779
$636$	0	0
$637$	−1.56219e6	−0.152540
$638$	0	0
$639$	−9.23911e6	−0.895113
$640$	0	0
$641$	−441107.	−0.0424032	−0.0212016	−	0.999775i	$-0.506749\pi$
$641$	−441107.	−0.0424032	−0.0212016	+	0.999775i	$0.506749\pi$
$642$	0	0
$643$	4.18888e6	0.399550	0.199775	−	0.979842i	$-0.435979\pi$
$643$	4.18888e6	0.399550	0.199775	+	0.979842i	$0.435979\pi$
$644$	0	0
$645$	−1.52159e7	−1.44012
$646$	0	0
$647$	−1.87822e7	−1.76395	−0.881973	−	0.471300i	$-0.843785\pi$
$647$	−1.87822e7	−1.76395	−0.881973	+	0.471300i	$0.843785\pi$
$648$	0	0
$649$	−9.47874e6	−0.883362
$650$	0	0
$651$	1.97231e6	0.182399
$652$	0	0
$653$	−1.51733e6	−0.139251	−0.0696254	−	0.997573i	$-0.522180\pi$
$653$	−1.51733e6	−0.139251	−0.0696254	+	0.997573i	$0.522180\pi$
$654$	0	0
$655$	−1.63368e7	−1.48787
$656$	0	0
$657$	4.08802e6	0.369487
$658$	0	0
$659$	−1.84809e7	−1.65772	−0.828859	−	0.559458i	$-0.811009\pi$
$659$	−1.84809e7	−1.65772	−0.828859	+	0.559458i	$0.811009\pi$
$660$	0	0
$661$	−1.03952e7	−0.925403	−0.462702	−	0.886514i	$-0.653120\pi$
$661$	−1.03952e7	−0.925403	−0.462702	+	0.886514i	$0.653120\pi$
$662$	0	0
$663$	1.51741e7	1.34066
$664$	0	0
$665$	3.58508e6	0.314372
$666$	0	0
$667$	2.63676e6	0.229486
$668$	0	0
$669$	2.01458e7	1.74028
$670$	0	0
$671$	9.48162e6	0.812973
$672$	0	0
$673$	−1.10398e7	−0.939556	−0.469778	−	0.882785i	$-0.655666\pi$
$673$	−1.10398e7	−0.939556	−0.469778	+	0.882785i	$0.655666\pi$
$674$	0	0
$675$	−1.84420e6	−0.155793
$676$	0	0
$677$	−8.23485e6	−0.690532	−0.345266	−	0.938505i	$-0.612211\pi$
$677$	−8.23485e6	−0.690532	−0.345266	+	0.938505i	$0.612211\pi$
$678$	0	0
$679$	1.12243e6	0.0934298
$680$	0	0
$681$	−9.90780e6	−0.818671
$682$	0	0
$683$	−1.98437e7	−1.62768	−0.813842	−	0.581086i	$-0.802628\pi$
$683$	−1.98437e7	−1.62768	−0.813842	+	0.581086i	$0.802628\pi$
$684$	0	0
$685$	−1.80655e7	−1.47103
$686$	0	0
$687$	2.19955e7	1.77804
$688$	0	0
$689$	−3.59100e6	−0.288182
$690$	0	0
$691$	2.37019e6	0.188837	0.0944185	−	0.995533i	$-0.469901\pi$
$691$	2.37019e6	0.188837	0.0944185	+	0.995533i	$0.469901\pi$
$692$	0	0
$693$	4.67065e6	0.369440
$694$	0	0
$695$	808259.	0.0634730
$696$	0	0
$697$	1.30346e6	0.101628
$698$	0	0
$699$	−1.51439e7	−1.17232
$700$	0	0
$701$	−1.56833e7	−1.20543	−0.602714	−	0.797957i	$-0.705914\pi$
$701$	−1.56833e7	−1.20543	−0.602714	+	0.797957i	$0.705914\pi$
$702$	0	0
$703$	1.68641e6	0.128699
$704$	0	0
$705$	7.62288e6	0.577626
$706$	0	0
$707$	8.89223e6	0.669056
$708$	0	0
$709$	3.68544e6	0.275343	0.137671	−	0.990478i	$-0.456038\pi$
$709$	3.68544e6	0.275343	0.137671	+	0.990478i	$0.456038\pi$
$710$	0	0
$711$	4.38373e6	0.325214
$712$	0	0
$713$	2.25361e6	0.166018
$714$	0	0
$715$	−2.02604e7	−1.48212
$716$	0	0
$717$	−3.37276e6	−0.245012
$718$	0	0
$719$	1.56427e7	1.12847	0.564233	−	0.825615i	$-0.309172\pi$
$719$	1.56427e7	1.12847	0.564233	+	0.825615i	$0.309172\pi$
$720$	0	0
$721$	3.17383e6	0.227376
$722$	0	0
$723$	7.54107e6	0.536522
$724$	0	0
$725$	−2.25175e6	−0.159102
$726$	0	0
$727$	−1.85908e7	−1.30456	−0.652279	−	0.757979i	$-0.726187\pi$
$727$	−1.85908e7	−1.30456	−0.652279	+	0.757979i	$0.726187\pi$
$728$	0	0
$729$	−1.12318e6	−0.0782767
$730$	0	0
$731$	−1.96601e7	−1.36079
$732$	0	0
$733$	2.49466e7	1.71495	0.857476	−	0.514524i	$-0.172031\pi$
$733$	2.49466e7	1.71495	0.857476	+	0.514524i	$0.172031\pi$
$734$	0	0
$735$	2.20555e6	0.150591
$736$	0	0
$737$	1.31422e7	0.891253
$738$	0	0
$739$	2.42944e7	1.63642	0.818211	−	0.574918i	$-0.194966\pi$
$739$	2.42944e7	1.63642	0.818211	+	0.574918i	$0.194966\pi$
$740$	0	0
$741$	2.00088e7	1.33868
$742$	0	0
$743$	−4.16541e6	−0.276812	−0.138406	−	0.990376i	$-0.544198\pi$
$743$	−4.16541e6	−0.276812	−0.138406	+	0.990376i	$0.544198\pi$
$744$	0	0
$745$	5.24388e6	0.346148
$746$	0	0
$747$	96729.9	0.00634248
$748$	0	0
$749$	−7.26032e6	−0.472880
$750$	0	0
$751$	2.36434e7	1.52972	0.764858	−	0.644199i	$-0.222809\pi$
$751$	2.36434e7	1.52972	0.764858	+	0.644199i	$0.222809\pi$
$752$	0	0
$753$	−3.08588e7	−1.98332
$754$	0	0
$755$	1.42564e6	0.0910209
$756$	0	0
$757$	1.50108e7	0.952062	0.476031	−	0.879429i	$-0.342075\pi$
$757$	1.50108e7	0.952062	0.476031	+	0.879429i	$0.342075\pi$
$758$	0	0
$759$	1.43992e7	0.907262
$760$	0	0
$761$	2.92191e6	0.182897	0.0914483	−	0.995810i	$-0.470850\pi$
$761$	2.92191e6	0.182897	0.0914483	+	0.995810i	$0.470850\pi$
$762$	0	0
$763$	5.45343e6	0.339124
$764$	0	0
$765$	−7.94025e6	−0.490547
$766$	0	0
$767$	−9.25887e6	−0.568290
$768$	0	0
$769$	1.42847e7	0.871073	0.435536	−	0.900171i	$-0.356559\pi$
$769$	1.42847e7	0.871073	0.435536	+	0.900171i	$0.356559\pi$
$770$	0	0
$771$	−1.55360e7	−0.941246
$772$	0	0
$773$	−1.09012e7	−0.656186	−0.328093	−	0.944645i	$-0.606406\pi$
$773$	−1.09012e7	−0.656186	−0.328093	+	0.944645i	$0.606406\pi$
$774$	0	0
$775$	−1.92455e6	−0.115100
$776$	0	0
$777$	1.03749e6	0.0616496
$778$	0	0
$779$	1.71875e6	0.101478
$780$	0	0
$781$	4.30045e7	2.52282
$782$	0	0
$783$	4.70459e6	0.274231
$784$	0	0
$785$	−2.44736e7	−1.41750
$786$	0	0
$787$	2.56449e7	1.47593	0.737963	−	0.674841i	$-0.235788\pi$
$787$	2.56449e7	1.47593	0.737963	+	0.674841i	$0.235788\pi$
$788$	0	0
$789$	9.12555e6	0.521875
$790$	0	0
$791$	2.11560e6	0.120224
$792$	0	0
$793$	9.26169e6	0.523007
$794$	0	0
$795$	5.06991e6	0.284500
$796$	0	0
$797$	7.73086e6	0.431104	0.215552	−	0.976492i	$-0.430845\pi$
$797$	7.73086e6	0.431104	0.215552	+	0.976492i	$0.430845\pi$
$798$	0	0
$799$	9.84931e6	0.545807
$800$	0	0
$801$	1.80269e7	0.992748
$802$	0	0
$803$	−1.90282e7	−1.04138
$804$	0	0
$805$	2.52013e6	0.137067
$806$	0	0
$807$	−3.92909e7	−2.12377
$808$	0	0
$809$	−1.87811e7	−1.00890	−0.504452	−	0.863440i	$-0.668305\pi$
$809$	−1.87811e7	−1.00890	−0.504452	+	0.863440i	$0.668305\pi$
$810$	0	0
$811$	9.00729e6	0.480886	0.240443	−	0.970663i	$-0.422707\pi$
$811$	9.00729e6	0.480886	0.240443	+	0.970663i	$0.422707\pi$
$812$	0	0
$813$	3.16939e7	1.68170
$814$	0	0
$815$	−2.05531e7	−1.08388
$816$	0	0
$817$	−2.59240e7	−1.35877
$818$	0	0
$819$	4.56231e6	0.237671
$820$	0	0
$821$	−9.27965e6	−0.480478	−0.240239	−	0.970714i	$-0.577226\pi$
$821$	−9.27965e6	−0.480478	−0.240239	+	0.970714i	$0.577226\pi$
$822$	0	0
$823$	1.08308e7	0.557393	0.278697	−	0.960379i	$-0.410098\pi$
$823$	1.08308e7	0.557393	0.278697	+	0.960379i	$0.410098\pi$
$824$	0	0
$825$	−1.22967e7	−0.629004
$826$	0	0
$827$	2.05230e7	1.04346	0.521731	−	0.853110i	$-0.325286\pi$
$827$	2.05230e7	1.04346	0.521731	+	0.853110i	$0.325286\pi$
$828$	0	0
$829$	−1.42216e7	−0.718724	−0.359362	−	0.933198i	$-0.617006\pi$
$829$	−1.42216e7	−0.718724	−0.359362	+	0.933198i	$0.617006\pi$
$830$	0	0
$831$	4.08944e7	2.05429
$832$	0	0
$833$	2.84973e6	0.142296
$834$	0	0
$835$	1.30595e7	0.648202
$836$	0	0
$837$	4.02097e6	0.198389
$838$	0	0
$839$	9.29934e6	0.456087	0.228043	−	0.973651i	$-0.426767\pi$
$839$	9.29934e6	0.456087	0.228043	+	0.973651i	$0.426767\pi$
$840$	0	0
$841$	−1.47669e7	−0.719944
$842$	0	0
$843$	1.92965e7	0.935211
$844$	0	0
$845$	−2.43277e6	−0.117209
$846$	0	0
$847$	−1.38486e7	−0.663281
$848$	0	0
$849$	−2.74350e7	−1.30628
$850$	0	0
$851$	1.18546e6	0.0561131
$852$	0	0
$853$	−3.07436e6	−0.144671	−0.0723357	−	0.997380i	$-0.523045\pi$
$853$	−3.07436e6	−0.144671	−0.0723357	+	0.997380i	$0.523045\pi$
$854$	0	0
$855$	−1.04701e7	−0.489819
$856$	0	0
$857$	3.45835e7	1.60848	0.804242	−	0.594302i	$-0.202572\pi$
$857$	3.45835e7	1.60848	0.804242	+	0.594302i	$0.202572\pi$
$858$	0	0
$859$	−1.63022e7	−0.753814	−0.376907	−	0.926251i	$-0.623012\pi$
$859$	−1.63022e7	−0.753814	−0.376907	+	0.926251i	$0.623012\pi$
$860$	0	0
$861$	1.05738e6	0.0486099
$862$	0	0
$863$	−2.56962e7	−1.17447	−0.587235	−	0.809416i	$-0.699784\pi$
$863$	−2.56962e7	−1.17447	−0.587235	+	0.809416i	$0.699784\pi$
$864$	0	0
$865$	−4.66087e6	−0.211800
$866$	0	0
$867$	218902.	0.00989012
$868$	0	0
$869$	−2.04046e7	−0.916596
$870$	0	0
$871$	1.28374e7	0.573366
$872$	0	0
$873$	−3.27803e6	−0.145572
$874$	0	0
$875$	−9.31062e6	−0.411111
$876$	0	0
$877$	3.30060e7	1.44908	0.724542	−	0.689230i	$-0.242051\pi$
$877$	3.30060e7	1.44908	0.724542	+	0.689230i	$0.242051\pi$
$878$	0	0
$879$	−5.03429e7	−2.19769
$880$	0	0
$881$	2.26705e6	0.0984060	0.0492030	−	0.998789i	$-0.484332\pi$
$881$	2.26705e6	0.0984060	0.0492030	+	0.998789i	$0.484332\pi$
$882$	0	0
$883$	−1.97779e7	−0.853649	−0.426825	−	0.904334i	$-0.640368\pi$
$883$	−1.97779e7	−0.853649	−0.426825	+	0.904334i	$0.640368\pi$
$884$	0	0
$885$	1.30720e7	0.561029
$886$	0	0
$887$	36468.3	0.00155635	0.000778173	−	1.00000i	$-0.499752\pi$
$887$	36468.3	0.00155635	0.000778173		1.00000i	$0.499752\pi$
$888$	0	0
$889$	−6.44100e6	−0.273337
$890$	0	0
$891$	4.88541e7	2.06161
$892$	0	0
$893$	1.29874e7	0.544997
$894$	0	0
$895$	1.54263e7	0.643730
$896$	0	0
$897$	1.40652e7	0.583665
$898$	0	0
$899$	4.90958e6	0.202602
$900$	0	0
$901$	6.55068e6	0.268828
$902$	0	0
$903$	−1.59485e7	−0.650881
$904$	0	0
$905$	2.36462e7	0.959709
$906$	0	0
$907$	−1.62298e7	−0.655079	−0.327540	−	0.944837i	$-0.606219\pi$
$907$	−1.62298e7	−0.655079	−0.327540	+	0.944837i	$0.606219\pi$
$908$	0	0
$909$	−2.59695e7	−1.04245
$910$	0	0
$911$	2.61699e7	1.04474	0.522368	−	0.852720i	$-0.325049\pi$
$911$	2.61699e7	1.04474	0.522368	+	0.852720i	$0.325049\pi$
$912$	0	0
$913$	−450241.	−0.0178759
$914$	0	0
$915$	−1.30760e7	−0.516324
$916$	0	0
$917$	−1.71234e7	−0.672461
$918$	0	0
$919$	−4.05973e6	−0.158565	−0.0792826	−	0.996852i	$-0.525263\pi$
$919$	−4.05973e6	−0.158565	−0.0792826	+	0.996852i	$0.525263\pi$
$920$	0	0
$921$	−1.73870e7	−0.675421
$922$	0	0
$923$	4.20070e7	1.62300
$924$	0	0
$925$	−1.01237e6	−0.0389031
$926$	0	0
$927$	−9.26907e6	−0.354272
$928$	0	0
$929$	3.69518e6	0.140474	0.0702370	−	0.997530i	$-0.477624\pi$
$929$	3.69518e6	0.140474	0.0702370	+	0.997530i	$0.477624\pi$
$930$	0	0
$931$	3.75769e6	0.142084
$932$	0	0
$933$	3.53927e7	1.33110
$934$	0	0
$935$	3.69589e7	1.38258
$936$	0	0
$937$	−1.91384e7	−0.712126	−0.356063	−	0.934462i	$-0.615881\pi$
$937$	−1.91384e7	−0.712126	−0.356063	+	0.934462i	$0.615881\pi$
$938$	0	0
$939$	−1.86742e7	−0.691160
$940$	0	0
$941$	1.20954e7	0.445294	0.222647	−	0.974899i	$-0.428530\pi$
$941$	1.20954e7	0.445294	0.222647	+	0.974899i	$0.428530\pi$
$942$	0	0
$943$	1.20820e6	0.0442445
$944$	0	0
$945$	4.49649e6	0.163793
$946$	0	0
$947$	1.95969e7	0.710087	0.355044	−	0.934850i	$-0.384466\pi$
$947$	1.95969e7	0.710087	0.355044	+	0.934850i	$0.384466\pi$
$948$	0	0
$949$	−1.85868e7	−0.669945
$950$	0	0
$951$	−5.97890e7	−2.14373
$952$	0	0
$953$	4.23265e7	1.50966	0.754832	−	0.655918i	$-0.227718\pi$
$953$	4.23265e7	1.50966	0.754832	+	0.655918i	$0.227718\pi$
$954$	0	0
$955$	−2.98426e6	−0.105884
$956$	0	0
$957$	3.13691e7	1.10719
$958$	0	0
$959$	−1.89353e7	−0.664852
$960$	0	0
$961$	−2.44330e7	−0.853430
$962$	0	0
$963$	2.12035e7	0.736788
$964$	0	0
$965$	−2.19419e7	−0.758502
$966$	0	0
$967$	−1.39211e6	−0.0478749	−0.0239375	−	0.999713i	$-0.507620\pi$
$967$	−1.39211e6	−0.0478749	−0.0239375	+	0.999713i	$0.507620\pi$
$968$	0	0
$969$	−3.64999e7	−1.24877
$970$	0	0
$971$	−4.41877e7	−1.50402	−0.752009	−	0.659153i	$-0.770915\pi$
$971$	−4.41877e7	−1.50402	−0.752009	+	0.659153i	$0.770915\pi$
$972$	0	0
$973$	847175.	0.0286874
$974$	0	0
$975$	−1.20115e7	−0.404654
$976$	0	0
$977$	−5.60457e7	−1.87848	−0.939239	−	0.343264i	$-0.888467\pi$
$977$	−5.60457e7	−1.87848	−0.939239	+	0.343264i	$0.888467\pi$
$978$	0	0
$979$	−8.39082e7	−2.79800
$980$	0	0
$981$	−1.59266e7	−0.528384
$982$	0	0
$983$	5.00560e7	1.65224	0.826118	−	0.563497i	$-0.190544\pi$
$983$	5.00560e7	1.65224	0.826118	+	0.563497i	$0.190544\pi$
$984$	0	0
$985$	−1.25298e7	−0.411483
$986$	0	0
$987$	7.98990e6	0.261065
$988$	0	0
$989$	−1.82233e7	−0.592428
$990$	0	0
$991$	−1.59116e7	−0.514670	−0.257335	−	0.966322i	$-0.582844\pi$
$991$	−1.59116e7	−0.514670	−0.257335	+	0.966322i	$0.582844\pi$
$992$	0	0
$993$	−4.31534e7	−1.38881
$994$	0	0
$995$	−2.82911e7	−0.905924
$996$	0	0
$997$	4.25995e7	1.35727	0.678635	−	0.734476i	$-0.262572\pi$
$997$	4.25995e7	1.35727	0.678635	+	0.734476i	$0.262572\pi$
$998$	0	0
$999$	2.11514e6	0.0670542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.a.h.1.1		2
3.2	odd	2		1008.6.a.bq.1.2		2
4.3	odd	2		7.6.a.b.1.1	✓	2
7.6	odd	2		784.6.a.v.1.2		2
8.3	odd	2		448.6.a.w.1.1		2
8.5	even	2		448.6.a.u.1.2		2
12.11	even	2		63.6.a.f.1.2		2
20.3	even	4		175.6.b.c.99.2		4
20.7	even	4		175.6.b.c.99.3		4
20.19	odd	2		175.6.a.c.1.2		2
28.3	even	6		49.6.c.d.30.2		4
28.11	odd	6		49.6.c.e.30.2		4
28.19	even	6		49.6.c.d.18.2		4
28.23	odd	6		49.6.c.e.18.2		4
28.27	even	2		49.6.a.f.1.1		2
44.43	even	2		847.6.a.c.1.2		2
84.83	odd	2		441.6.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.b.1.1	✓	2	4.3	odd	2
49.6.a.f.1.1		2	28.27	even	2
49.6.c.d.18.2		4	28.19	even	6
49.6.c.d.30.2		4	28.3	even	6
49.6.c.e.18.2		4	28.23	odd	6
49.6.c.e.30.2		4	28.11	odd	6
63.6.a.f.1.2		2	12.11	even	2
112.6.a.h.1.1		2	1.1	even	1	trivial
175.6.a.c.1.2		2	20.19	odd	2
175.6.b.c.99.2		4	20.3	even	4
175.6.b.c.99.3		4	20.7	even	4
441.6.a.l.1.2		2	84.83	odd	2
448.6.a.u.1.2		2	8.5	even	2
448.6.a.w.1.1		2	8.3	odd	2
784.6.a.v.1.2		2	7.6	odd	2
847.6.a.c.1.2		2	44.43	even	2
1008.6.a.bq.1.2		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q-19.6495 q^{3} -46.7492 q^{5} -49.0000 q^{7} +143.103 q^{9} +O(q^{10})\)	\(q-19.6495 q^{3} -46.7492 q^{5} -49.0000 q^{7} +143.103 q^{9} -666.090 q^{11} -650.640 q^{13} +918.598 q^{15} +1186.89 q^{17} +1565.05 q^{19} +962.826 q^{21} +1100.15 q^{23} -939.515 q^{25} +1962.93 q^{27} +2396.72 q^{29} +2048.46 q^{31} +13088.3 q^{33} +2290.71 q^{35} +1077.54 q^{37} +12784.7 q^{39} +1098.21 q^{41} -16564.3 q^{43} -6689.95 q^{45} +8298.39 q^{47} +2401.00 q^{49} -23321.9 q^{51} +5519.18 q^{53} +31139.1 q^{55} -30752.5 q^{57} +14230.4 q^{59} -14234.7 q^{61} -7012.05 q^{63} +30416.9 q^{65} -19730.4 q^{67} -21617.5 q^{69} -64562.7 q^{71} +28567.0 q^{73} +18461.0 q^{75} +32638.4 q^{77} +30633.4 q^{79} -73344.6 q^{81} +675.946 q^{83} -55486.3 q^{85} -47094.4 q^{87} +125971. q^{89} +31881.3 q^{91} -40251.1 q^{93} -73164.9 q^{95} -22906.8 q^{97} -95319.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 18 q^{5} - 98 q^{7} + 558 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 18 * q^5 - 98 * q^7 + 558 * q^9	\( 2 q + 6 q^{3} - 18 q^{5} - 98 q^{7} + 558 q^{9} - 396 q^{11} - 350 q^{13} + 1656 q^{15} + 1800 q^{17} + 3266 q^{19} - 294 q^{21} - 2088 q^{23} - 3238 q^{25} + 6372 q^{27} + 6696 q^{29} + 20 q^{31} + 20016 q^{33} + 882 q^{35} + 6232 q^{37} + 20496 q^{39} - 6048 q^{41} + 3020 q^{43} + 5238 q^{45} - 11700 q^{47} + 4802 q^{49} - 7596 q^{51} + 9468 q^{53} + 38904 q^{55} + 12876 q^{57} + 43938 q^{59} - 64754 q^{61} - 27342 q^{63} + 39060 q^{65} - 24784 q^{67} - 103392 q^{69} - 97416 q^{71} + 17452 q^{73} - 40494 q^{75} + 19404 q^{77} - 51256 q^{79} - 61074 q^{81} - 117558 q^{83} - 37860 q^{85} + 63180 q^{87} + 84276 q^{89} + 17150 q^{91} - 92280 q^{93} - 24264 q^{95} + 20776 q^{97} + 16740 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 18 * q^5 - 98 * q^7 + 558 * q^9 - 396 * q^11 - 350 * q^13 + 1656 * q^15 + 1800 * q^17 + 3266 * q^19 - 294 * q^21 - 2088 * q^23 - 3238 * q^25 + 6372 * q^27 + 6696 * q^29 + 20 * q^31 + 20016 * q^33 + 882 * q^35 + 6232 * q^37 + 20496 * q^39 - 6048 * q^41 + 3020 * q^43 + 5238 * q^45 - 11700 * q^47 + 4802 * q^49 - 7596 * q^51 + 9468 * q^53 + 38904 * q^55 + 12876 * q^57 + 43938 * q^59 - 64754 * q^61 - 27342 * q^63 + 39060 * q^65 - 24784 * q^67 - 103392 * q^69 - 97416 * q^71 + 17452 * q^73 - 40494 * q^75 + 19404 * q^77 - 51256 * q^79 - 61074 * q^81 - 117558 * q^83 - 37860 * q^85 + 63180 * q^87 + 84276 * q^89 + 17150 * q^91 - 92280 * q^93 - 24264 * q^95 + 20776 * q^97 + 16740 * q^99

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