LMFDB - Embedded newform 448.6.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, 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("448.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.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:	$71.8519512762$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 198x^{2} + 43x + 5999 $ x^4 - x^3 - 198x^2 + 43x + 5999
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.08658$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.17317 q^{3} +20.6340 q^{5} +49.0000 q^{7} -176.199 q^{9} +O(q^{10})$	$q+8.17317 q^{3} +20.6340 q^{5} +49.0000 q^{7} -176.199 q^{9} +506.585 q^{11} -635.256 q^{13} +168.645 q^{15} -716.795 q^{17} -1380.65 q^{19} +400.485 q^{21} -354.448 q^{23} -2699.24 q^{25} -3426.19 q^{27} +6242.07 q^{29} +1377.52 q^{31} +4140.41 q^{33} +1011.06 q^{35} -11674.8 q^{37} -5192.05 q^{39} -4460.60 q^{41} -21996.6 q^{43} -3635.69 q^{45} +25906.3 q^{47} +2401.00 q^{49} -5858.48 q^{51} -5048.64 q^{53} +10452.9 q^{55} -11284.3 q^{57} +2119.87 q^{59} -8844.91 q^{61} -8633.77 q^{63} -13107.8 q^{65} -50339.9 q^{67} -2896.96 q^{69} -13753.2 q^{71} +39915.1 q^{73} -22061.3 q^{75} +24822.7 q^{77} +66614.9 q^{79} +14813.6 q^{81} -80813.7 q^{83} -14790.3 q^{85} +51017.5 q^{87} +17914.3 q^{89} -31127.5 q^{91} +11258.7 q^{93} -28488.3 q^{95} -154822. q^{97} -89260.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 18 q^{3} + 30 q^{5} + 196 q^{7} + 696 q^{9}+O(q^{10}) $ 4 * q - 18 * q^3 + 30 * q^5 + 196 * q^7 + 696 * q^9	$ 4 q - 18 q^{3} + 30 q^{5} + 196 q^{7} + 696 q^{9} - 484 q^{11} - 686 q^{13} - 2184 q^{15} - 1700 q^{17} - 654 q^{19} - 882 q^{21} + 136 q^{23} + 3304 q^{25} - 14388 q^{27} - 3812 q^{29} + 12748 q^{31} - 1536 q^{33} + 1470 q^{35} - 820 q^{37} + 34704 q^{39} + 22340 q^{41} - 32924 q^{43} + 59070 q^{45} - 2620 q^{47} + 9604 q^{49} - 55788 q^{51} - 22984 q^{53} + 24360 q^{55} + 15132 q^{57} - 108158 q^{59} - 4258 q^{61} + 34104 q^{63} - 95028 q^{65} - 109496 q^{67} + 82392 q^{69} + 54600 q^{71} - 12384 q^{73} - 315198 q^{75} - 23716 q^{77} + 78184 q^{79} + 114384 q^{81} - 115582 q^{83} - 101652 q^{85} - 17772 q^{87} - 31560 q^{89} - 33614 q^{91} - 218040 q^{93} - 67032 q^{95} + 41068 q^{97} - 301284 q^{99}+O(q^{100}) $ 4 * q - 18 * q^3 + 30 * q^5 + 196 * q^7 + 696 * q^9 - 484 * q^11 - 686 * q^13 - 2184 * q^15 - 1700 * q^17 - 654 * q^19 - 882 * q^21 + 136 * q^23 + 3304 * q^25 - 14388 * q^27 - 3812 * q^29 + 12748 * q^31 - 1536 * q^33 + 1470 * q^35 - 820 * q^37 + 34704 * q^39 + 22340 * q^41 - 32924 * q^43 + 59070 * q^45 - 2620 * q^47 + 9604 * q^49 - 55788 * q^51 - 22984 * q^53 + 24360 * q^55 + 15132 * q^57 - 108158 * q^59 - 4258 * q^61 + 34104 * q^63 - 95028 * q^65 - 109496 * q^67 + 82392 * q^69 + 54600 * q^71 - 12384 * q^73 - 315198 * q^75 - 23716 * q^77 + 78184 * q^79 + 114384 * q^81 - 115582 * q^83 - 101652 * q^85 - 17772 * q^87 - 31560 * q^89 - 33614 * q^91 - 218040 * q^93 - 67032 * q^95 + 41068 * q^97 - 301284 * 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
$3$	8.17317	0.524309	0.262154	−	0.965026i	$-0.415567\pi$
$3$	8.17317	0.524309	0.262154	+	0.965026i	$0.415567\pi$
$4$	0	0
$5$	20.6340	0.369111	0.184556	−	0.982822i	$-0.440915\pi$
$5$	20.6340	0.369111	0.184556	+	0.982822i	$0.440915\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−176.199	−0.725100
$10$	0	0
$11$	506.585	1.26232	0.631162	−	0.775651i	$-0.282578\pi$
$11$	506.585	1.26232	0.631162	+	0.775651i	$0.282578\pi$
$12$	0	0
$13$	−635.256	−1.04253	−0.521267	−	0.853394i	$-0.674540\pi$
$13$	−635.256	−1.04253	−0.521267	+	0.853394i	$0.674540\pi$
$14$	0	0
$15$	168.645	0.193528
$16$	0	0
$17$	−716.795	−0.601551	−0.300775	−	0.953695i	$-0.597245\pi$
$17$	−716.795	−0.601551	−0.300775	+	0.953695i	$0.597245\pi$
$18$	0	0
$19$	−1380.65	−0.877405	−0.438702	−	0.898632i	$-0.644562\pi$
$19$	−1380.65	−0.877405	−0.438702	+	0.898632i	$0.644562\pi$
$20$	0	0
$21$	400.485	0.198170
$22$	0	0
$23$	−354.448	−0.139712	−0.0698558	−	0.997557i	$-0.522254\pi$
$23$	−354.448	−0.139712	−0.0698558	+	0.997557i	$0.522254\pi$
$24$	0	0
$25$	−2699.24	−0.863757
$26$	0	0
$27$	−3426.19	−0.904485
$28$	0	0
$29$	6242.07	1.37827	0.689134	−	0.724634i	$-0.257991\pi$
$29$	6242.07	1.37827	0.689134	+	0.724634i	$0.257991\pi$
$30$	0	0
$31$	1377.52	0.257450	0.128725	−	0.991680i	$-0.458912\pi$
$31$	1377.52	0.257450	0.128725	+	0.991680i	$0.458912\pi$
$32$	0	0
$33$	4140.41	0.661848
$34$	0	0
$35$	1011.06	0.139511
$36$	0	0
$37$	−11674.8	−1.40199	−0.700993	−	0.713168i	$-0.747260\pi$
$37$	−11674.8	−1.40199	−0.700993	+	0.713168i	$0.747260\pi$
$38$	0	0
$39$	−5192.05	−0.546610
$40$	0	0
$41$	−4460.60	−0.414413	−0.207206	−	0.978297i	$-0.566437\pi$
$41$	−4460.60	−0.414413	−0.207206	+	0.978297i	$0.566437\pi$
$42$	0	0
$43$	−21996.6	−1.81419	−0.907097	−	0.420922i	$-0.861707\pi$
$43$	−21996.6	−1.81419	−0.907097	+	0.420922i	$0.861707\pi$
$44$	0	0
$45$	−3635.69	−0.267643
$46$	0	0
$47$	25906.3	1.71065	0.855325	−	0.518092i	$-0.173357\pi$
$47$	25906.3	1.71065	0.855325	+	0.518092i	$0.173357\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−5858.48	−0.315399
$52$	0	0
$53$	−5048.64	−0.246879	−0.123440	−	0.992352i	$-0.539393\pi$
$53$	−5048.64	−0.246879	−0.123440	+	0.992352i	$0.539393\pi$
$54$	0	0
$55$	10452.9	0.465938
$56$	0	0
$57$	−11284.3	−0.460031
$58$	0	0
$59$	2119.87	0.0792828	0.0396414	−	0.999214i	$-0.487378\pi$
$59$	2119.87	0.0792828	0.0396414	+	0.999214i	$0.487378\pi$
$60$	0	0
$61$	−8844.91	−0.304347	−0.152173	−	0.988354i	$-0.548627\pi$
$61$	−8844.91	−0.304347	−0.152173	+	0.988354i	$0.548627\pi$
$62$	0	0
$63$	−8633.77	−0.274062
$64$	0	0
$65$	−13107.8	−0.384811
$66$	0	0
$67$	−50339.9	−1.37002	−0.685008	−	0.728535i	$-0.740201\pi$
$67$	−50339.9	−1.37002	−0.685008	+	0.728535i	$0.740201\pi$
$68$	0	0
$69$	−2896.96	−0.0732520
$70$	0	0
$71$	−13753.2	−0.323786	−0.161893	−	0.986808i	$-0.551760\pi$
$71$	−13753.2	−0.323786	−0.161893	+	0.986808i	$0.551760\pi$
$72$	0	0
$73$	39915.1	0.876658	0.438329	−	0.898815i	$-0.355570\pi$
$73$	39915.1	0.876658	0.438329	+	0.898815i	$0.355570\pi$
$74$	0	0
$75$	−22061.3	−0.452875
$76$	0	0
$77$	24822.7	0.477114
$78$	0	0
$79$	66614.9	1.20089	0.600445	−	0.799666i	$-0.294990\pi$
$79$	66614.9	1.20089	0.600445	+	0.799666i	$0.294990\pi$
$80$	0	0
$81$	14813.6	0.250870
$82$	0	0
$83$	−80813.7	−1.28763	−0.643813	−	0.765183i	$-0.722648\pi$
$83$	−80813.7	−1.28763	−0.643813	+	0.765183i	$0.722648\pi$
$84$	0	0
$85$	−14790.3	−0.222039
$86$	0	0
$87$	51017.5	0.722638
$88$	0	0
$89$	17914.3	0.239731	0.119866	−	0.992790i	$-0.461754\pi$
$89$	17914.3	0.239731	0.119866	+	0.992790i	$0.461754\pi$
$90$	0	0
$91$	−31127.5	−0.394041
$92$	0	0
$93$	11258.7	0.134983
$94$	0	0
$95$	−28488.3	−0.323860
$96$	0	0
$97$	−154822.	−1.67072	−0.835360	−	0.549704i	$-0.814741\pi$
$97$	−154822.	−1.67072	−0.835360	+	0.549704i	$0.814741\pi$
$98$	0	0
$99$	−89260.0	−0.915311
$100$	0	0
$101$	98993.4	0.965612	0.482806	−	0.875727i	$-0.339618\pi$
$101$	98993.4	0.965612	0.482806	+	0.875727i	$0.339618\pi$
$102$	0	0
$103$	−167746.	−1.55797	−0.778983	−	0.627045i	$-0.784264\pi$
$103$	−167746.	−1.55797	−0.778983	+	0.627045i	$0.784264\pi$
$104$	0	0
$105$	8263.59	0.0731468
$106$	0	0
$107$	−131142.	−1.10734	−0.553671	−	0.832735i	$-0.686774\pi$
$107$	−131142.	−1.10734	−0.553671	+	0.832735i	$0.686774\pi$
$108$	0	0
$109$	−49878.7	−0.402114	−0.201057	−	0.979580i	$-0.564438\pi$
$109$	−49878.7	−0.402114	−0.201057	+	0.979580i	$0.564438\pi$
$110$	0	0
$111$	−95419.7	−0.735074
$112$	0	0
$113$	155443.	1.14519	0.572593	−	0.819839i	$-0.305937\pi$
$113$	155443.	1.14519	0.572593	+	0.819839i	$0.305937\pi$
$114$	0	0
$115$	−7313.66	−0.0515691
$116$	0	0
$117$	111932.	0.755941
$118$	0	0
$119$	−35122.9	−0.227365
$120$	0	0
$121$	95577.5	0.593461
$122$	0	0
$123$	−36457.2	−0.217280
$124$	0	0
$125$	−120177.	−0.687934
$126$	0	0
$127$	−141092.	−0.776234	−0.388117	−	0.921610i	$-0.626874\pi$
$127$	−141092.	−0.776234	−0.388117	+	0.921610i	$0.626874\pi$
$128$	0	0
$129$	−179782.	−0.951198
$130$	0	0
$131$	133498.	0.679669	0.339834	−	0.940485i	$-0.389629\pi$
$131$	133498.	0.679669	0.339834	+	0.940485i	$0.389629\pi$
$132$	0	0
$133$	−67651.9	−0.331628
$134$	0	0
$135$	−70695.8	−0.333856
$136$	0	0
$137$	66087.7	0.300829	0.150414	−	0.988623i	$-0.451939\pi$
$137$	66087.7	0.300829	0.150414	+	0.988623i	$0.451939\pi$
$138$	0	0
$139$	−131816.	−0.578670	−0.289335	−	0.957228i	$-0.593434\pi$
$139$	−131816.	−0.578670	−0.289335	+	0.957228i	$0.593434\pi$
$140$	0	0
$141$	211737.	0.896909
$142$	0	0
$143$	−321811.	−1.31602
$144$	0	0
$145$	128799.	0.508734
$146$	0	0
$147$	19623.8	0.0749013
$148$	0	0
$149$	−25250.6	−0.0931764	−0.0465882	−	0.998914i	$-0.514835\pi$
$149$	−25250.6	−0.0931764	−0.0465882	+	0.998914i	$0.514835\pi$
$150$	0	0
$151$	−207999.	−0.742369	−0.371184	−	0.928559i	$-0.621048\pi$
$151$	−207999.	−0.742369	−0.371184	+	0.928559i	$0.621048\pi$
$152$	0	0
$153$	126299.	0.436185
$154$	0	0
$155$	28423.6	0.0950276
$156$	0	0
$157$	−129188.	−0.418287	−0.209143	−	0.977885i	$-0.567068\pi$
$157$	−129188.	−0.418287	−0.209143	+	0.977885i	$0.567068\pi$
$158$	0	0
$159$	−41263.3	−0.129441
$160$	0	0
$161$	−17367.9	−0.0528060
$162$	0	0
$163$	−190635.	−0.561996	−0.280998	−	0.959708i	$-0.590665\pi$
$163$	−190635.	−0.561996	−0.280998	+	0.959708i	$0.590665\pi$
$164$	0	0
$165$	85432.9	0.244295
$166$	0	0
$167$	−314255.	−0.871949	−0.435975	−	0.899959i	$-0.643596\pi$
$167$	−314255.	−0.871949	−0.435975	+	0.899959i	$0.643596\pi$
$168$	0	0
$169$	32256.6	0.0868765
$170$	0	0
$171$	243270.	0.636206
$172$	0	0
$173$	−533441.	−1.35510	−0.677550	−	0.735477i	$-0.736958\pi$
$173$	−533441.	−1.35510	−0.677550	+	0.735477i	$0.736958\pi$
$174$	0	0
$175$	−132263.	−0.326469
$176$	0	0
$177$	17326.0	0.0415687
$178$	0	0
$179$	532404.	1.24196	0.620982	−	0.783825i	$-0.286734\pi$
$179$	532404.	1.24196	0.620982	+	0.783825i	$0.286734\pi$
$180$	0	0
$181$	439799.	0.997834	0.498917	−	0.866650i	$-0.333731\pi$
$181$	439799.	0.997834	0.498917	+	0.866650i	$0.333731\pi$
$182$	0	0
$183$	−72290.9	−0.159572
$184$	0	0
$185$	−240896.	−0.517489
$186$	0	0
$187$	−363118.	−0.759352
$188$	0	0
$189$	−167883.	−0.341863
$190$	0	0
$191$	−253509.	−0.502817	−0.251408	−	0.967881i	$-0.580894\pi$
$191$	−253509.	−0.502817	−0.251408	+	0.967881i	$0.580894\pi$
$192$	0	0
$193$	−456826.	−0.882791	−0.441396	−	0.897313i	$-0.645516\pi$
$193$	−456826.	−0.882791	−0.441396	+	0.897313i	$0.645516\pi$
$194$	0	0
$195$	−107132.	−0.201760
$196$	0	0
$197$	−281017.	−0.515902	−0.257951	−	0.966158i	$-0.583047\pi$
$197$	−281017.	−0.515902	−0.257951	+	0.966158i	$0.583047\pi$
$198$	0	0
$199$	−889110.	−1.59156	−0.795780	−	0.605586i	$-0.792939\pi$
$199$	−889110.	−1.59156	−0.795780	+	0.605586i	$0.792939\pi$
$200$	0	0
$201$	−411437.	−0.718312
$202$	0	0
$203$	305861.	0.520936
$204$	0	0
$205$	−92039.7	−0.152964
$206$	0	0
$207$	62453.4	0.101305
$208$	0	0
$209$	−699417.	−1.10757
$210$	0	0
$211$	−265521.	−0.410575	−0.205288	−	0.978702i	$-0.565813\pi$
$211$	−265521.	−0.410575	−0.205288	+	0.978702i	$0.565813\pi$
$212$	0	0
$213$	−112407.	−0.169764
$214$	0	0
$215$	−453876.	−0.669639
$216$	0	0
$217$	67498.3	0.0973068
$218$	0	0
$219$	326233.	0.459640
$220$	0	0
$221$	455348.	0.627137
$222$	0	0
$223$	737166.	0.992666	0.496333	−	0.868132i	$-0.334680\pi$
$223$	737166.	0.992666	0.496333	+	0.868132i	$0.334680\pi$
$224$	0	0
$225$	475604.	0.626310
$226$	0	0
$227$	−1.36368e6	−1.75650	−0.878250	−	0.478202i	$-0.841289\pi$
$227$	−1.36368e6	−1.75650	−0.878250	+	0.478202i	$0.841289\pi$
$228$	0	0
$229$	204640.	0.257871	0.128936	−	0.991653i	$-0.458844\pi$
$229$	204640.	0.257871	0.128936	+	0.991653i	$0.458844\pi$
$230$	0	0
$231$	202880.	0.250155
$232$	0	0
$233$	617695.	0.745391	0.372696	−	0.927954i	$-0.378434\pi$
$233$	617695.	0.745391	0.372696	+	0.927954i	$0.378434\pi$
$234$	0	0
$235$	534550.	0.631420
$236$	0	0
$237$	544455.	0.629638
$238$	0	0
$239$	288474.	0.326672	0.163336	−	0.986571i	$-0.447775\pi$
$239$	288474.	0.326672	0.163336	+	0.986571i	$0.447775\pi$
$240$	0	0
$241$	1.52454e6	1.69082	0.845408	−	0.534121i	$-0.179358\pi$
$241$	1.52454e6	1.69082	0.845408	+	0.534121i	$0.179358\pi$
$242$	0	0
$243$	953638.	1.03602
$244$	0	0
$245$	49542.1	0.0527302
$246$	0	0
$247$	877066.	0.914724
$248$	0	0
$249$	−660504.	−0.675113
$250$	0	0
$251$	263840.	0.264336	0.132168	−	0.991227i	$-0.457806\pi$
$251$	263840.	0.264336	0.132168	+	0.991227i	$0.457806\pi$
$252$	0	0
$253$	−179558.	−0.176361
$254$	0	0
$255$	−120884.	−0.116417
$256$	0	0
$257$	−38044.1	−0.0359298	−0.0179649	−	0.999839i	$-0.505719\pi$
$257$	−38044.1	−0.0359298	−0.0179649	+	0.999839i	$0.505719\pi$
$258$	0	0
$259$	−572063.	−0.529901
$260$	0	0
$261$	−1.09985e6	−0.999382
$262$	0	0
$263$	1.17804e6	1.05019	0.525097	−	0.851042i	$-0.324029\pi$
$263$	1.17804e6	1.05019	0.525097	+	0.851042i	$0.324029\pi$
$264$	0	0
$265$	−104173.	−0.0911259
$266$	0	0
$267$	146416.	0.125693
$268$	0	0
$269$	530015.	0.446588	0.223294	−	0.974751i	$-0.428319\pi$
$269$	530015.	0.446588	0.223294	+	0.974751i	$0.428319\pi$
$270$	0	0
$271$	1.69277e6	1.40015	0.700075	−	0.714069i	$-0.253150\pi$
$271$	1.69277e6	1.40015	0.700075	+	0.714069i	$0.253150\pi$
$272$	0	0
$273$	−254410.	−0.206599
$274$	0	0
$275$	−1.36740e6	−1.09034
$276$	0	0
$277$	267761.	0.209676	0.104838	−	0.994489i	$-0.466568\pi$
$277$	267761.	0.209676	0.104838	+	0.994489i	$0.466568\pi$
$278$	0	0
$279$	−242717.	−0.186677
$280$	0	0
$281$	1.21948e6	0.921319	0.460659	−	0.887577i	$-0.347613\pi$
$281$	1.21948e6	0.921319	0.460659	+	0.887577i	$0.347613\pi$
$282$	0	0
$283$	234277.	0.173886	0.0869429	−	0.996213i	$-0.472290\pi$
$283$	234277.	0.173886	0.0869429	+	0.996213i	$0.472290\pi$
$284$	0	0
$285$	−232839.	−0.169803
$286$	0	0
$287$	−218569.	−0.156633
$288$	0	0
$289$	−906063.	−0.638136
$290$	0	0
$291$	−1.26539e6	−0.875973
$292$	0	0
$293$	1.75507e6	1.19434	0.597168	−	0.802116i	$-0.296293\pi$
$293$	1.75507e6	1.19434	0.597168	+	0.802116i	$0.296293\pi$
$294$	0	0
$295$	43741.3	0.0292642
$296$	0	0
$297$	−1.73566e6	−1.14175
$298$	0	0
$299$	225165.	0.145654
$300$	0	0
$301$	−1.07783e6	−0.685701
$302$	0	0
$303$	809089.	0.506279
$304$	0	0
$305$	−182505.	−0.112338
$306$	0	0
$307$	−1.11032e6	−0.672360	−0.336180	−	0.941798i	$-0.609135\pi$
$307$	−1.11032e6	−0.672360	−0.336180	+	0.941798i	$0.609135\pi$
$308$	0	0
$309$	−1.37101e6	−0.816856
$310$	0	0
$311$	2.69745e6	1.58144	0.790719	−	0.612179i	$-0.209707\pi$
$311$	2.69745e6	1.58144	0.790719	+	0.612179i	$0.209707\pi$
$312$	0	0
$313$	3.33517e6	1.92423	0.962114	−	0.272646i	$-0.0878988\pi$
$313$	3.33517e6	1.92423	0.962114	+	0.272646i	$0.0878988\pi$
$314$	0	0
$315$	−178149.	−0.101159
$316$	0	0
$317$	−3.01830e6	−1.68700	−0.843498	−	0.537133i	$-0.819507\pi$
$317$	−3.01830e6	−1.68700	−0.843498	+	0.537133i	$0.819507\pi$
$318$	0	0
$319$	3.16214e6	1.73982
$320$	0	0
$321$	−1.07184e6	−0.580590
$322$	0	0
$323$	989643.	0.527804
$324$	0	0
$325$	1.71471e6	0.900496
$326$	0	0
$327$	−407667.	−0.210832
$328$	0	0
$329$	1.26941e6	0.646565
$330$	0	0
$331$	2.26736e6	1.13750	0.568748	−	0.822512i	$-0.307428\pi$
$331$	2.26736e6	1.13750	0.568748	+	0.822512i	$0.307428\pi$
$332$	0	0
$333$	2.05708e6	1.01658
$334$	0	0
$335$	−1.03871e6	−0.505689
$336$	0	0
$337$	−28130.5	−0.0134928	−0.00674641	−	0.999977i	$-0.502147\pi$
$337$	−28130.5	−0.0134928	−0.00674641	+	0.999977i	$0.502147\pi$
$338$	0	0
$339$	1.27047e6	0.600432
$340$	0	0
$341$	697829.	0.324985
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−59775.7	−0.0270382
$346$	0	0
$347$	3.55790e6	1.58624	0.793122	−	0.609063i	$-0.208454\pi$
$347$	3.55790e6	1.58624	0.793122	+	0.609063i	$0.208454\pi$
$348$	0	0
$349$	2.81954e6	1.23913	0.619563	−	0.784947i	$-0.287310\pi$
$349$	2.81954e6	1.23913	0.619563	+	0.784947i	$0.287310\pi$
$350$	0	0
$351$	2.17650e6	0.942956
$352$	0	0
$353$	−1.19876e6	−0.512030	−0.256015	−	0.966673i	$-0.582410\pi$
$353$	−1.19876e6	−0.512030	−0.256015	+	0.966673i	$0.582410\pi$
$354$	0	0
$355$	−283783.	−0.119513
$356$	0	0
$357$	−287066.	−0.119209
$358$	0	0
$359$	−224779.	−0.0920492	−0.0460246	−	0.998940i	$-0.514655\pi$
$359$	−224779.	−0.0920492	−0.0460246	+	0.998940i	$0.514655\pi$
$360$	0	0
$361$	−569902.	−0.230161
$362$	0	0
$363$	781171.	0.311157
$364$	0	0
$365$	823607.	0.323584
$366$	0	0
$367$	2.39774e6	0.929261	0.464631	−	0.885505i	$-0.346187\pi$
$367$	2.39774e6	0.929261	0.464631	+	0.885505i	$0.346187\pi$
$368$	0	0
$369$	785954.	0.300491
$370$	0	0
$371$	−247383.	−0.0933115
$372$	0	0
$373$	295184.	0.109855	0.0549277	−	0.998490i	$-0.482507\pi$
$373$	295184.	0.109855	0.0549277	+	0.998490i	$0.482507\pi$
$374$	0	0
$375$	−982227.	−0.360690
$376$	0	0
$377$	−3.96531e6	−1.43689
$378$	0	0
$379$	−2.15015e6	−0.768900	−0.384450	−	0.923146i	$-0.625609\pi$
$379$	−2.15015e6	−0.768900	−0.384450	+	0.923146i	$0.625609\pi$
$380$	0	0
$381$	−1.15317e6	−0.406987
$382$	0	0
$383$	5.11282e6	1.78100	0.890499	−	0.454986i	$-0.150356\pi$
$383$	5.11282e6	1.78100	0.890499	+	0.454986i	$0.150356\pi$
$384$	0	0
$385$	512190.	0.176108
$386$	0	0
$387$	3.87578e6	1.31547
$388$	0	0
$389$	−1.99434e6	−0.668230	−0.334115	−	0.942532i	$-0.608437\pi$
$389$	−1.99434e6	−0.668230	−0.334115	+	0.942532i	$0.608437\pi$
$390$	0	0
$391$	254066.	0.0840436
$392$	0	0
$393$	1.09110e6	0.356356
$394$	0	0
$395$	1.37453e6	0.443262
$396$	0	0
$397$	−191644.	−0.0610267	−0.0305133	−	0.999534i	$-0.509714\pi$
$397$	−191644.	−0.0610267	−0.0305133	+	0.999534i	$0.509714\pi$
$398$	0	0
$399$	−552930.	−0.173875
$400$	0	0
$401$	4.69084e6	1.45677	0.728383	−	0.685170i	$-0.240272\pi$
$401$	4.69084e6	1.45677	0.728383	+	0.685170i	$0.240272\pi$
$402$	0	0
$403$	−875074.	−0.268400
$404$	0	0
$405$	305664.	0.0925991
$406$	0	0
$407$	−5.91426e6	−1.76976
$408$	0	0
$409$	−3.85501e6	−1.13951	−0.569754	−	0.821815i	$-0.692962\pi$
$409$	−3.85501e6	−1.13951	−0.569754	+	0.821815i	$0.692962\pi$
$410$	0	0
$411$	540146.	0.157727
$412$	0	0
$413$	103874.	0.0299661
$414$	0	0
$415$	−1.66751e6	−0.475277
$416$	0	0
$417$	−1.07735e6	−0.303402
$418$	0	0
$419$	−107560.	−0.0299307	−0.0149654	−	0.999888i	$-0.504764\pi$
$419$	−107560.	−0.0299307	−0.0149654	+	0.999888i	$0.504764\pi$
$420$	0	0
$421$	−5.61644e6	−1.54439	−0.772193	−	0.635388i	$-0.780840\pi$
$421$	−5.61644e6	−1.54439	−0.772193	+	0.635388i	$0.780840\pi$
$422$	0	0
$423$	−4.56468e6	−1.24039
$424$	0	0
$425$	1.93480e6	0.519594
$426$	0	0
$427$	−433401.	−0.115032
$428$	0	0
$429$	−2.63022e6	−0.689998
$430$	0	0
$431$	−4.57448e6	−1.18617	−0.593087	−	0.805138i	$-0.702091\pi$
$431$	−4.57448e6	−1.18617	−0.593087	+	0.805138i	$0.702091\pi$
$432$	0	0
$433$	−7.44508e6	−1.90831	−0.954157	−	0.299307i	$-0.903244\pi$
$433$	−7.44508e6	−1.90831	−0.954157	+	0.299307i	$0.903244\pi$
$434$	0	0
$435$	1.05269e6	0.266734
$436$	0	0
$437$	489368.	0.122584
$438$	0	0
$439$	−2.16275e6	−0.535605	−0.267803	−	0.963474i	$-0.586298\pi$
$439$	−2.16275e6	−0.535605	−0.267803	+	0.963474i	$0.586298\pi$
$440$	0	0
$441$	−423055.	−0.103586
$442$	0	0
$443$	5.75516e6	1.39331	0.696655	−	0.717406i	$-0.254671\pi$
$443$	5.75516e6	1.39331	0.696655	+	0.717406i	$0.254671\pi$
$444$	0	0
$445$	369642.	0.0884875
$446$	0	0
$447$	−206377.	−0.0488532
$448$	0	0
$449$	7.25653e6	1.69869	0.849343	−	0.527841i	$-0.176998\pi$
$449$	7.25653e6	1.69869	0.849343	+	0.527841i	$0.176998\pi$
$450$	0	0
$451$	−2.25967e6	−0.523123
$452$	0	0
$453$	−1.70001e6	−0.389231
$454$	0	0
$455$	−642284.	−0.145445
$456$	0	0
$457$	3.83702e6	0.859417	0.429709	−	0.902968i	$-0.358616\pi$
$457$	3.83702e6	0.859417	0.429709	+	0.902968i	$0.358616\pi$
$458$	0	0
$459$	2.45587e6	0.544094
$460$	0	0
$461$	8.45089e6	1.85204	0.926020	−	0.377475i	$-0.123208\pi$
$461$	8.45089e6	1.85204	0.926020	+	0.377475i	$0.123208\pi$
$462$	0	0
$463$	1.84019e6	0.398943	0.199471	−	0.979904i	$-0.436077\pi$
$463$	1.84019e6	0.398943	0.199471	+	0.979904i	$0.436077\pi$
$464$	0	0
$465$	232311.	0.0498238
$466$	0	0
$467$	1.94986e6	0.413725	0.206862	−	0.978370i	$-0.433675\pi$
$467$	1.94986e6	0.413725	0.206862	+	0.978370i	$0.433675\pi$
$468$	0	0
$469$	−2.46666e6	−0.517818
$470$	0	0
$471$	−1.05588e6	−0.219311
$472$	0	0
$473$	−1.11431e7	−2.29010
$474$	0	0
$475$	3.72671e6	0.757864
$476$	0	0
$477$	889566.	0.179012
$478$	0	0
$479$	399165.	0.0794901	0.0397451	−	0.999210i	$-0.487345\pi$
$479$	399165.	0.0794901	0.0397451	+	0.999210i	$0.487345\pi$
$480$	0	0
$481$	7.41646e6	1.46162
$482$	0	0
$483$	−141951.	−0.0276867
$484$	0	0
$485$	−3.19459e6	−0.616681
$486$	0	0
$487$	6.33763e6	1.21089	0.605444	−	0.795888i	$-0.292995\pi$
$487$	6.33763e6	1.21089	0.605444	+	0.795888i	$0.292995\pi$
$488$	0	0
$489$	−1.55809e6	−0.294659
$490$	0	0
$491$	3.67112e6	0.687218	0.343609	−	0.939113i	$-0.388350\pi$
$491$	3.67112e6	0.687218	0.343609	+	0.939113i	$0.388350\pi$
$492$	0	0
$493$	−4.47428e6	−0.829098
$494$	0	0
$495$	−1.84179e6	−0.337852
$496$	0	0
$497$	−673906.	−0.122379
$498$	0	0
$499$	−9.03184e6	−1.62377	−0.811886	−	0.583816i	$-0.801559\pi$
$499$	−9.03184e6	−1.62377	−0.811886	+	0.583816i	$0.801559\pi$
$500$	0	0
$501$	−2.56846e6	−0.457171
$502$	0	0
$503$	−1.08465e7	−1.91148	−0.955739	−	0.294217i	$-0.904941\pi$
$503$	−1.08465e7	−1.91148	−0.955739	+	0.294217i	$0.904941\pi$
$504$	0	0
$505$	2.04262e6	0.356418
$506$	0	0
$507$	263639.	0.0455501
$508$	0	0
$509$	5.09958e6	0.872449	0.436224	−	0.899838i	$-0.356315\pi$
$509$	5.09958e6	0.872449	0.436224	+	0.899838i	$0.356315\pi$
$510$	0	0
$511$	1.95584e6	0.331346
$512$	0	0
$513$	4.73037e6	0.793600
$514$	0	0
$515$	−3.46125e6	−0.575063
$516$	0	0
$517$	1.31238e7	2.15939
$518$	0	0
$519$	−4.35991e6	−0.710491
$520$	0	0
$521$	282838.	0.0456503	0.0228252	−	0.999739i	$-0.492734\pi$
$521$	282838.	0.0456503	0.0228252	+	0.999739i	$0.492734\pi$
$522$	0	0
$523$	−814044.	−0.130135	−0.0650674	−	0.997881i	$-0.520726\pi$
$523$	−814044.	−0.130135	−0.0650674	+	0.997881i	$0.520726\pi$
$524$	0	0
$525$	−1.08101e6	−0.171171
$526$	0	0
$527$	−987396.	−0.154869
$528$	0	0
$529$	−6.31071e6	−0.980481
$530$	0	0
$531$	−373519.	−0.0574880
$532$	0	0
$533$	2.83362e6	0.432039
$534$	0	0
$535$	−2.70598e6	−0.408733
$536$	0	0
$537$	4.35143e6	0.651172
$538$	0	0
$539$	1.21631e6	0.180332
$540$	0	0
$541$	−4.20661e6	−0.617930	−0.308965	−	0.951073i	$-0.599983\pi$
$541$	−4.20661e6	−0.617930	−0.308965	+	0.951073i	$0.599983\pi$
$542$	0	0
$543$	3.59455e6	0.523173
$544$	0	0
$545$	−1.02919e6	−0.148425
$546$	0	0
$547$	−8.24593e6	−1.17834	−0.589171	−	0.808009i	$-0.700545\pi$
$547$	−8.24593e6	−1.17834	−0.589171	+	0.808009i	$0.700545\pi$
$548$	0	0
$549$	1.55847e6	0.220682
$550$	0	0
$551$	−8.61812e6	−1.20930
$552$	0	0
$553$	3.26413e6	0.453894
$554$	0	0
$555$	−1.96889e6	−0.271324
$556$	0	0
$557$	969510.	0.132408	0.0662040	−	0.997806i	$-0.478911\pi$
$557$	969510.	0.132408	0.0662040	+	0.997806i	$0.478911\pi$
$558$	0	0
$559$	1.39734e7	1.89136
$560$	0	0
$561$	−2.96782e6	−0.398135
$562$	0	0
$563$	−9.58996e6	−1.27510	−0.637552	−	0.770407i	$-0.720053\pi$
$563$	−9.58996e6	−1.27510	−0.637552	+	0.770407i	$0.720053\pi$
$564$	0	0
$565$	3.20741e6	0.422701
$566$	0	0
$567$	725869.	0.0948201
$568$	0	0
$569$	−6.56904e6	−0.850592	−0.425296	−	0.905054i	$-0.639830\pi$
$569$	−6.56904e6	−0.850592	−0.425296	+	0.905054i	$0.639830\pi$
$570$	0	0
$571$	−51314.1	−0.00658638	−0.00329319	−	0.999995i	$-0.501048\pi$
$571$	−51314.1	−0.00658638	−0.00329319	+	0.999995i	$0.501048\pi$
$572$	0	0
$573$	−2.07197e6	−0.263631
$574$	0	0
$575$	956739.	0.120677
$576$	0	0
$577$	−1.11070e7	−1.38886	−0.694430	−	0.719560i	$-0.744343\pi$
$577$	−1.11070e7	−1.38886	−0.694430	+	0.719560i	$0.744343\pi$
$578$	0	0
$579$	−3.73372e6	−0.462855
$580$	0	0
$581$	−3.95987e6	−0.486677
$582$	0	0
$583$	−2.55756e6	−0.311641
$584$	0	0
$585$	2.30959e6	0.279027
$586$	0	0
$587$	−9.36125e6	−1.12134	−0.560672	−	0.828038i	$-0.689457\pi$
$587$	−9.36125e6	−1.12134	−0.560672	+	0.828038i	$0.689457\pi$
$588$	0	0
$589$	−1.90187e6	−0.225887
$590$	0	0
$591$	−2.29680e6	−0.270492
$592$	0	0
$593$	1.35675e7	1.58440	0.792198	−	0.610264i	$-0.208937\pi$
$593$	1.35675e7	1.58440	0.792198	+	0.610264i	$0.208937\pi$
$594$	0	0
$595$	−724725.	−0.0839230
$596$	0	0
$597$	−7.26684e6	−0.834469
$598$	0	0
$599$	−1.24270e7	−1.41514	−0.707569	−	0.706644i	$-0.750208\pi$
$599$	−1.24270e7	−1.41514	−0.707569	+	0.706644i	$0.750208\pi$
$600$	0	0
$601$	−2.31586e6	−0.261532	−0.130766	−	0.991413i	$-0.541744\pi$
$601$	−2.31586e6	−0.261532	−0.130766	+	0.991413i	$0.541744\pi$
$602$	0	0
$603$	8.86987e6	0.993399
$604$	0	0
$605$	1.97214e6	0.219053
$606$	0	0
$607$	3.12911e6	0.344706	0.172353	−	0.985035i	$-0.444863\pi$
$607$	3.12911e6	0.344706	0.172353	+	0.985035i	$0.444863\pi$
$608$	0	0
$609$	2.49986e6	0.273131
$610$	0	0
$611$	−1.64571e7	−1.78341
$612$	0	0
$613$	6.56901e6	0.706071	0.353036	−	0.935610i	$-0.385149\pi$
$613$	6.56901e6	0.706071	0.353036	+	0.935610i	$0.385149\pi$
$614$	0	0
$615$	−752256.	−0.0802006
$616$	0	0
$617$	1.77884e7	1.88115	0.940576	−	0.339583i	$-0.110286\pi$
$617$	1.77884e7	1.88115	0.940576	+	0.339583i	$0.110286\pi$
$618$	0	0
$619$	−6.09519e6	−0.639382	−0.319691	−	0.947522i	$-0.603579\pi$
$619$	−6.09519e6	−0.639382	−0.319691	+	0.947522i	$0.603579\pi$
$620$	0	0
$621$	1.21440e6	0.126367
$622$	0	0
$623$	877800.	0.0906098
$624$	0	0
$625$	5.95540e6	0.609833
$626$	0	0
$627$	−5.71645e6	−0.580708
$628$	0	0
$629$	8.36840e6	0.843366
$630$	0	0
$631$	1.65385e7	1.65357	0.826783	−	0.562521i	$-0.190168\pi$
$631$	1.65385e7	1.65357	0.826783	+	0.562521i	$0.190168\pi$
$632$	0	0
$633$	−2.17015e6	−0.215268
$634$	0	0
$635$	−2.91128e6	−0.286517
$636$	0	0
$637$	−1.52525e6	−0.148933
$638$	0	0
$639$	2.42330e6	0.234777
$640$	0	0
$641$	−1.32064e7	−1.26952	−0.634761	−	0.772708i	$-0.718902\pi$
$641$	−1.32064e7	−1.26952	−0.634761	+	0.772708i	$0.718902\pi$
$642$	0	0
$643$	−5.94568e6	−0.567119	−0.283560	−	0.958955i	$-0.591515\pi$
$643$	−5.94568e6	−0.567119	−0.283560	+	0.958955i	$0.591515\pi$
$644$	0	0
$645$	−3.70960e6	−0.351098
$646$	0	0
$647$	−9.93647e6	−0.933193	−0.466597	−	0.884470i	$-0.654520\pi$
$647$	−9.93647e6	−0.933193	−0.466597	+	0.884470i	$0.654520\pi$
$648$	0	0
$649$	1.07389e6	0.100081
$650$	0	0
$651$	551675.	0.0510188
$652$	0	0
$653$	−9.45044e6	−0.867300	−0.433650	−	0.901082i	$-0.642774\pi$
$653$	−9.45044e6	−0.867300	−0.433650	+	0.901082i	$0.642774\pi$
$654$	0	0
$655$	2.75460e6	0.250873
$656$	0	0
$657$	−7.03302e6	−0.635665
$658$	0	0
$659$	−8.78565e6	−0.788062	−0.394031	−	0.919097i	$-0.628920\pi$
$659$	−8.78565e6	−0.788062	−0.394031	+	0.919097i	$0.628920\pi$
$660$	0	0
$661$	313228.	0.0278841	0.0139421	−	0.999903i	$-0.495562\pi$
$661$	313228.	0.0278841	0.0139421	+	0.999903i	$0.495562\pi$
$662$	0	0
$663$	3.72163e6	0.328814
$664$	0	0
$665$	−1.39593e6	−0.122408
$666$	0	0
$667$	−2.21249e6	−0.192560
$668$	0	0
$669$	6.02498e6	0.520464
$670$	0	0
$671$	−4.48070e6	−0.384184
$672$	0	0
$673$	2.95224e6	0.251254	0.125627	−	0.992078i	$-0.459906\pi$
$673$	2.95224e6	0.251254	0.125627	+	0.992078i	$0.459906\pi$
$674$	0	0
$675$	9.24810e6	0.781255
$676$	0	0
$677$	4.36433e6	0.365970	0.182985	−	0.983116i	$-0.441424\pi$
$677$	4.36433e6	0.365970	0.182985	+	0.983116i	$0.441424\pi$
$678$	0	0
$679$	−7.58628e6	−0.631472
$680$	0	0
$681$	−1.11456e7	−0.920949
$682$	0	0
$683$	−5.14494e6	−0.422015	−0.211008	−	0.977484i	$-0.567674\pi$
$683$	−5.14494e6	−0.422015	−0.211008	+	0.977484i	$0.567674\pi$
$684$	0	0
$685$	1.36365e6	0.111039
$686$	0	0
$687$	1.67256e6	0.135204
$688$	0	0
$689$	3.20717e6	0.257380
$690$	0	0
$691$	−2.02588e7	−1.61405	−0.807027	−	0.590514i	$-0.798925\pi$
$691$	−2.02588e7	−1.61405	−0.807027	+	0.590514i	$0.798925\pi$
$692$	0	0
$693$	−4.37374e6	−0.345955
$694$	0	0
$695$	−2.71988e6	−0.213593
$696$	0	0
$697$	3.19733e6	0.249290
$698$	0	0
$699$	5.04852e6	0.390815
$700$	0	0
$701$	−1.99355e7	−1.53226	−0.766128	−	0.642688i	$-0.777819\pi$
$701$	−1.99355e7	−1.53226	−0.766128	+	0.642688i	$0.777819\pi$
$702$	0	0
$703$	1.61188e7	1.23011
$704$	0	0
$705$	4.36897e6	0.331059
$706$	0	0
$707$	4.85068e6	0.364967
$708$	0	0
$709$	−2.08928e7	−1.56092	−0.780461	−	0.625204i	$-0.785016\pi$
$709$	−2.08928e7	−1.56092	−0.780461	+	0.625204i	$0.785016\pi$
$710$	0	0
$711$	−1.17375e7	−0.870766
$712$	0	0
$713$	−488257.	−0.0359687
$714$	0	0
$715$	−6.64023e6	−0.485756
$716$	0	0
$717$	2.35775e6	0.171277
$718$	0	0
$719$	−6.70208e6	−0.483490	−0.241745	−	0.970340i	$-0.577720\pi$
$719$	−6.70208e6	−0.483490	−0.241745	+	0.970340i	$0.577720\pi$
$720$	0	0
$721$	−8.21953e6	−0.588856
$722$	0	0
$723$	1.24603e7	0.886510
$724$	0	0
$725$	−1.68488e7	−1.19049
$726$	0	0
$727$	2.01375e7	1.41309	0.706546	−	0.707668i	$-0.250253\pi$
$727$	2.01375e7	1.41309	0.706546	+	0.707668i	$0.250253\pi$
$728$	0	0
$729$	4.19452e6	0.292324
$730$	0	0
$731$	1.57670e7	1.09133
$732$	0	0
$733$	2.39391e7	1.64569	0.822845	−	0.568266i	$-0.192385\pi$
$733$	2.39391e7	1.64569	0.822845	+	0.568266i	$0.192385\pi$
$734$	0	0
$735$	404916.	0.0276469
$736$	0	0
$737$	−2.55015e7	−1.72940
$738$	0	0
$739$	1.87678e6	0.126416	0.0632081	−	0.998000i	$-0.479867\pi$
$739$	1.87678e6	0.126416	0.0632081	+	0.998000i	$0.479867\pi$
$740$	0	0
$741$	7.16841e6	0.479598
$742$	0	0
$743$	−1.28689e7	−0.855202	−0.427601	−	0.903968i	$-0.640641\pi$
$743$	−1.28689e7	−0.855202	−0.427601	+	0.903968i	$0.640641\pi$
$744$	0	0
$745$	−521019.	−0.0343925
$746$	0	0
$747$	1.42393e7	0.933657
$748$	0	0
$749$	−6.42595e6	−0.418536
$750$	0	0
$751$	2.28520e7	1.47851	0.739255	−	0.673425i	$-0.235178\pi$
$751$	2.28520e7	1.47851	0.739255	+	0.673425i	$0.235178\pi$
$752$	0	0
$753$	2.15641e6	0.138594
$754$	0	0
$755$	−4.29185e6	−0.274017
$756$	0	0
$757$	2.57978e7	1.63622	0.818112	−	0.575059i	$-0.195021\pi$
$757$	2.57978e7	1.63622	0.818112	+	0.575059i	$0.195021\pi$
$758$	0	0
$759$	−1.46756e6	−0.0924678
$760$	0	0
$761$	1.00550e7	0.629391	0.314696	−	0.949193i	$-0.398098\pi$
$761$	1.00550e7	0.629391	0.314696	+	0.949193i	$0.398098\pi$
$762$	0	0
$763$	−2.44406e6	−0.151985
$764$	0	0
$765$	2.60604e6	0.161001
$766$	0	0
$767$	−1.34666e6	−0.0826550
$768$	0	0
$769$	1.06616e7	0.650139	0.325069	−	0.945690i	$-0.394612\pi$
$769$	1.06616e7	0.650139	0.325069	+	0.945690i	$0.394612\pi$
$770$	0	0
$771$	−310941.	−0.0188383
$772$	0	0
$773$	−1.29622e6	−0.0780245	−0.0390123	−	0.999239i	$-0.512421\pi$
$773$	−1.29622e6	−0.0780245	−0.0390123	+	0.999239i	$0.512421\pi$
$774$	0	0
$775$	−3.71825e6	−0.222374
$776$	0	0
$777$	−4.67557e6	−0.277832
$778$	0	0
$779$	6.15853e6	0.363608
$780$	0	0
$781$	−6.96716e6	−0.408722
$782$	0	0
$783$	−2.13865e7	−1.24662
$784$	0	0
$785$	−2.66567e6	−0.154394
$786$	0	0
$787$	1.29627e7	0.746036	0.373018	−	0.927824i	$-0.378323\pi$
$787$	1.29627e7	0.746036	0.373018	+	0.927824i	$0.378323\pi$
$788$	0	0
$789$	9.62829e6	0.550626
$790$	0	0
$791$	7.61673e6	0.432840
$792$	0	0
$793$	5.61878e6	0.317292
$794$	0	0
$795$	−851426.	−0.0477781
$796$	0	0
$797$	2.11336e7	1.17849	0.589247	−	0.807953i	$-0.299425\pi$
$797$	2.11336e7	1.17849	0.589247	+	0.807953i	$0.299425\pi$
$798$	0	0
$799$	−1.85695e7	−1.02904
$800$	0	0
$801$	−3.15648e6	−0.173829
$802$	0	0
$803$	2.02204e7	1.10663
$804$	0	0
$805$	−358369.	−0.0194913
$806$	0	0
$807$	4.33190e6	0.234150
$808$	0	0
$809$	−2.74328e7	−1.47367	−0.736834	−	0.676074i	$-0.763680\pi$
$809$	−2.74328e7	−1.47367	−0.736834	+	0.676074i	$0.763680\pi$
$810$	0	0
$811$	3.45319e7	1.84360	0.921802	−	0.387660i	$-0.126717\pi$
$811$	3.45319e7	1.84360	0.921802	+	0.387660i	$0.126717\pi$
$812$	0	0
$813$	1.38353e7	0.734112
$814$	0	0
$815$	−3.93355e6	−0.207439
$816$	0	0
$817$	3.03696e7	1.59178
$818$	0	0
$819$	5.48465e6	0.285719
$820$	0	0
$821$	−7.22242e6	−0.373960	−0.186980	−	0.982364i	$-0.559870\pi$
$821$	−7.22242e6	−0.373960	−0.186980	+	0.982364i	$0.559870\pi$
$822$	0	0
$823$	−1.80242e7	−0.927589	−0.463795	−	0.885943i	$-0.653512\pi$
$823$	−1.80242e7	−0.927589	−0.463795	+	0.885943i	$0.653512\pi$
$824$	0	0
$825$	−1.11759e7	−0.571675
$826$	0	0
$827$	1.17107e7	0.595412	0.297706	−	0.954658i	$-0.403779\pi$
$827$	1.17107e7	0.595412	0.297706	+	0.954658i	$0.403779\pi$
$828$	0	0
$829$	−2.82504e7	−1.42771	−0.713853	−	0.700295i	$-0.753052\pi$
$829$	−2.82504e7	−1.42771	−0.713853	+	0.700295i	$0.753052\pi$
$830$	0	0
$831$	2.18846e6	0.109935
$832$	0	0
$833$	−1.72102e6	−0.0859358
$834$	0	0
$835$	−6.48432e6	−0.321846
$836$	0	0
$837$	−4.71962e6	−0.232859
$838$	0	0
$839$	1.56383e7	0.766979	0.383489	−	0.923545i	$-0.374722\pi$
$839$	1.56383e7	0.766979	0.383489	+	0.923545i	$0.374722\pi$
$840$	0	0
$841$	1.84523e7	0.899622
$842$	0	0
$843$	9.96704e6	0.483056
$844$	0	0
$845$	665582.	0.0320671
$846$	0	0
$847$	4.68330e6	0.224307
$848$	0	0
$849$	1.91479e6	0.0911699
$850$	0	0
$851$	4.13809e6	0.195874
$852$	0	0
$853$	−1.63205e7	−0.767997	−0.383999	−	0.923334i	$-0.625453\pi$
$853$	−1.63205e7	−0.767997	−0.383999	+	0.923334i	$0.625453\pi$
$854$	0	0
$855$	5.01962e6	0.234831
$856$	0	0
$857$	3.75860e6	0.174813	0.0874065	−	0.996173i	$-0.472142\pi$
$857$	3.75860e6	0.174813	0.0874065	+	0.996173i	$0.472142\pi$
$858$	0	0
$859$	3.09131e7	1.42942	0.714709	−	0.699422i	$-0.246559\pi$
$859$	3.09131e7	1.42942	0.714709	+	0.699422i	$0.246559\pi$
$860$	0	0
$861$	−1.78640e6	−0.0821243
$862$	0	0
$863$	−2.71516e7	−1.24099	−0.620494	−	0.784211i	$-0.713068\pi$
$863$	−2.71516e7	−1.24099	−0.620494	+	0.784211i	$0.713068\pi$
$864$	0	0
$865$	−1.10070e7	−0.500183
$866$	0	0
$867$	−7.40540e6	−0.334581
$868$	0	0
$869$	3.37461e7	1.51591
$870$	0	0
$871$	3.19787e7	1.42829
$872$	0	0
$873$	2.72795e7	1.21144
$874$	0	0
$875$	−5.88868e6	−0.260015
$876$	0	0
$877$	−6.17865e6	−0.271265	−0.135633	−	0.990759i	$-0.543307\pi$
$877$	−6.17865e6	−0.271265	−0.135633	+	0.990759i	$0.543307\pi$
$878$	0	0
$879$	1.43445e7	0.626201
$880$	0	0
$881$	2.26917e7	0.984979	0.492490	−	0.870318i	$-0.336087\pi$
$881$	2.26917e7	0.984979	0.492490	+	0.870318i	$0.336087\pi$
$882$	0	0
$883$	1.63213e7	0.704456	0.352228	−	0.935914i	$-0.385424\pi$
$883$	1.63213e7	0.704456	0.352228	+	0.935914i	$0.385424\pi$
$884$	0	0
$885$	357505.	0.0153435
$886$	0	0
$887$	−1.07946e7	−0.460677	−0.230339	−	0.973111i	$-0.573983\pi$
$887$	−1.07946e7	−0.460677	−0.230339	+	0.973111i	$0.573983\pi$
$888$	0	0
$889$	−6.91351e6	−0.293389
$890$	0	0
$891$	7.50438e6	0.316680
$892$	0	0
$893$	−3.57676e7	−1.50093
$894$	0	0
$895$	1.09856e7	0.458423
$896$	0	0
$897$	1.84031e6	0.0763677
$898$	0	0
$899$	8.59855e6	0.354834
$900$	0	0
$901$	3.61883e6	0.148510
$902$	0	0
$903$	−8.80930e6	−0.359519
$904$	0	0
$905$	9.07480e6	0.368312
$906$	0	0
$907$	−1.68130e7	−0.678622	−0.339311	−	0.940674i	$-0.610194\pi$
$907$	−1.68130e7	−0.678622	−0.339311	+	0.940674i	$0.610194\pi$
$908$	0	0
$909$	−1.74426e7	−0.700165
$910$	0	0
$911$	2.84236e7	1.13471	0.567353	−	0.823475i	$-0.307968\pi$
$911$	2.84236e7	1.13471	0.567353	+	0.823475i	$0.307968\pi$
$912$	0	0
$913$	−4.09390e7	−1.62540
$914$	0	0
$915$	−1.49165e6	−0.0588998
$916$	0	0
$917$	6.54141e6	0.256891
$918$	0	0
$919$	1.80746e6	0.0705960	0.0352980	−	0.999377i	$-0.488762\pi$
$919$	1.80746e6	0.0705960	0.0352980	+	0.999377i	$0.488762\pi$
$920$	0	0
$921$	−9.07482e6	−0.352524
$922$	0	0
$923$	8.73679e6	0.337557
$924$	0	0
$925$	3.15130e7	1.21097
$926$	0	0
$927$	2.95567e7	1.12968
$928$	0	0
$929$	−2.32726e7	−0.884719	−0.442359	−	0.896838i	$-0.645858\pi$
$929$	−2.32726e7	−0.884719	−0.442359	+	0.896838i	$0.645858\pi$
$930$	0	0
$931$	−3.31494e6	−0.125344
$932$	0	0
$933$	2.20467e7	0.829162
$934$	0	0
$935$	−7.49255e6	−0.280285
$936$	0	0
$937$	2.81529e7	1.04755	0.523775	−	0.851857i	$-0.324523\pi$
$937$	2.81529e7	1.04755	0.523775	+	0.851857i	$0.324523\pi$
$938$	0	0
$939$	2.72589e7	1.00889
$940$	0	0
$941$	−5.52770e6	−0.203503	−0.101751	−	0.994810i	$-0.532445\pi$
$941$	−5.52770e6	−0.203503	−0.101751	+	0.994810i	$0.532445\pi$
$942$	0	0
$943$	1.58105e6	0.0578983
$944$	0	0
$945$	−3.46409e6	−0.126186
$946$	0	0
$947$	2.47822e7	0.897977	0.448988	−	0.893538i	$-0.351784\pi$
$947$	2.47822e7	0.897977	0.448988	+	0.893538i	$0.351784\pi$
$948$	0	0
$949$	−2.53563e7	−0.913946
$950$	0	0
$951$	−2.46691e7	−0.884507
$952$	0	0
$953$	3.86371e6	0.137807	0.0689036	−	0.997623i	$-0.478050\pi$
$953$	3.86371e6	0.137807	0.0689036	+	0.997623i	$0.478050\pi$
$954$	0	0
$955$	−5.23089e6	−0.185595
$956$	0	0
$957$	2.58447e7	0.912203
$958$	0	0
$959$	3.23830e6	0.113703
$960$	0	0
$961$	−2.67316e7	−0.933720
$962$	0	0
$963$	2.31071e7	0.802935
$964$	0	0
$965$	−9.42613e6	−0.325848
$966$	0	0
$967$	1.82182e6	0.0626527	0.0313263	−	0.999509i	$-0.490027\pi$
$967$	1.82182e6	0.0626527	0.0313263	+	0.999509i	$0.490027\pi$
$968$	0	0
$969$	8.08852e6	0.276732
$970$	0	0
$971$	−5.00199e7	−1.70253	−0.851265	−	0.524737i	$-0.824164\pi$
$971$	−5.00199e7	−1.70253	−0.851265	+	0.524737i	$0.824164\pi$
$972$	0	0
$973$	−6.45898e6	−0.218717
$974$	0	0
$975$	1.40146e7	0.472138
$976$	0	0
$977$	5.28871e7	1.77261	0.886306	−	0.463101i	$-0.153263\pi$
$977$	5.28871e7	1.77261	0.886306	+	0.463101i	$0.153263\pi$
$978$	0	0
$979$	9.07511e6	0.302618
$980$	0	0
$981$	8.78859e6	0.291573
$982$	0	0
$983$	−4.94239e7	−1.63137	−0.815687	−	0.578494i	$-0.803641\pi$
$983$	−4.94239e7	−1.63137	−0.815687	+	0.578494i	$0.803641\pi$
$984$	0	0
$985$	−5.79849e6	−0.190425
$986$	0	0
$987$	1.03751e7	0.339000
$988$	0	0
$989$	7.79663e6	0.253464
$990$	0	0
$991$	3.07021e7	0.993078	0.496539	−	0.868014i	$-0.334604\pi$
$991$	3.07021e7	0.993078	0.496539	+	0.868014i	$0.334604\pi$
$992$	0	0
$993$	1.85315e7	0.596399
$994$	0	0
$995$	−1.83458e7	−0.587462
$996$	0	0
$997$	−4.47688e6	−0.142639	−0.0713194	−	0.997454i	$-0.522721\pi$
$997$	−4.47688e6	−0.142639	−0.0713194	+	0.997454i	$0.522721\pi$
$998$	0	0
$999$	3.99999e7	1.26808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.bc.1.3		4
4.3	odd	2		448.6.a.bd.1.2		4
8.3	odd	2		224.6.a.g.1.3	✓	4
8.5	even	2		224.6.a.h.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.g.1.3	✓	4	8.3	odd	2
224.6.a.h.1.2	yes	4	8.5	even	2
448.6.a.bc.1.3		4	1.1	even	1	trivial
448.6.a.bd.1.2		4	4.3	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q+8.17317 q^{3} +20.6340 q^{5} +49.0000 q^{7} -176.199 q^{9} +O(q^{10})\)	\(q+8.17317 q^{3} +20.6340 q^{5} +49.0000 q^{7} -176.199 q^{9} +506.585 q^{11} -635.256 q^{13} +168.645 q^{15} -716.795 q^{17} -1380.65 q^{19} +400.485 q^{21} -354.448 q^{23} -2699.24 q^{25} -3426.19 q^{27} +6242.07 q^{29} +1377.52 q^{31} +4140.41 q^{33} +1011.06 q^{35} -11674.8 q^{37} -5192.05 q^{39} -4460.60 q^{41} -21996.6 q^{43} -3635.69 q^{45} +25906.3 q^{47} +2401.00 q^{49} -5858.48 q^{51} -5048.64 q^{53} +10452.9 q^{55} -11284.3 q^{57} +2119.87 q^{59} -8844.91 q^{61} -8633.77 q^{63} -13107.8 q^{65} -50339.9 q^{67} -2896.96 q^{69} -13753.2 q^{71} +39915.1 q^{73} -22061.3 q^{75} +24822.7 q^{77} +66614.9 q^{79} +14813.6 q^{81} -80813.7 q^{83} -14790.3 q^{85} +51017.5 q^{87} +17914.3 q^{89} -31127.5 q^{91} +11258.7 q^{93} -28488.3 q^{95} -154822. q^{97} -89260.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{3} + 30 q^{5} + 196 q^{7} + 696 q^{9}+O(q^{10}) \) 4 * q - 18 * q^3 + 30 * q^5 + 196 * q^7 + 696 * q^9	\( 4 q - 18 q^{3} + 30 q^{5} + 196 q^{7} + 696 q^{9} - 484 q^{11} - 686 q^{13} - 2184 q^{15} - 1700 q^{17} - 654 q^{19} - 882 q^{21} + 136 q^{23} + 3304 q^{25} - 14388 q^{27} - 3812 q^{29} + 12748 q^{31} - 1536 q^{33} + 1470 q^{35} - 820 q^{37} + 34704 q^{39} + 22340 q^{41} - 32924 q^{43} + 59070 q^{45} - 2620 q^{47} + 9604 q^{49} - 55788 q^{51} - 22984 q^{53} + 24360 q^{55} + 15132 q^{57} - 108158 q^{59} - 4258 q^{61} + 34104 q^{63} - 95028 q^{65} - 109496 q^{67} + 82392 q^{69} + 54600 q^{71} - 12384 q^{73} - 315198 q^{75} - 23716 q^{77} + 78184 q^{79} + 114384 q^{81} - 115582 q^{83} - 101652 q^{85} - 17772 q^{87} - 31560 q^{89} - 33614 q^{91} - 218040 q^{93} - 67032 q^{95} + 41068 q^{97} - 301284 q^{99}+O(q^{100}) \) 4 * q - 18 * q^3 + 30 * q^5 + 196 * q^7 + 696 * q^9 - 484 * q^11 - 686 * q^13 - 2184 * q^15 - 1700 * q^17 - 654 * q^19 - 882 * q^21 + 136 * q^23 + 3304 * q^25 - 14388 * q^27 - 3812 * q^29 + 12748 * q^31 - 1536 * q^33 + 1470 * q^35 - 820 * q^37 + 34704 * q^39 + 22340 * q^41 - 32924 * q^43 + 59070 * q^45 - 2620 * q^47 + 9604 * q^49 - 55788 * q^51 - 22984 * q^53 + 24360 * q^55 + 15132 * q^57 - 108158 * q^59 - 4258 * q^61 + 34104 * q^63 - 95028 * q^65 - 109496 * q^67 + 82392 * q^69 + 54600 * q^71 - 12384 * q^73 - 315198 * q^75 - 23716 * q^77 + 78184 * q^79 + 114384 * q^81 - 115582 * q^83 - 101652 * q^85 - 17772 * q^87 - 31560 * q^89 - 33614 * q^91 - 218040 * q^93 - 67032 * q^95 + 41068 * q^97 - 301284 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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