LMFDB - Embedded newform 768.7.g.e.511.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,7,Mod(511,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("768.511");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	768.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$176.681536220$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 5x^{2} + 4x + 16 $ x^4 - x^3 + 5x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.4
Root			$1.28078 + 2.21837i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.7.g.e.511.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.5885i q^{3} +20.0000 q^{5} +529.850i q^{7} -243.000 q^{9} +O(q^{10})$	$q+15.5885i q^{3} +20.0000 q^{5} +529.850i q^{7} -243.000 q^{9} -435.847i q^{11} +341.182 q^{13} +311.769i q^{15} +7682.73 q^{17} -4300.52i q^{19} -8259.54 q^{21} -3175.32i q^{23} -15225.0 q^{25} -3788.00i q^{27} -19409.3 q^{29} -15297.3i q^{31} +6794.18 q^{33} +10597.0i q^{35} -61969.9 q^{37} +5318.50i q^{39} +33740.2 q^{41} +99312.6i q^{43} -4860.00 q^{45} +17726.4i q^{47} -163092. q^{49} +119762. i q^{51} -224406. q^{53} -8716.94i q^{55} +67038.5 q^{57} -199557. i q^{59} -45671.7 q^{61} -128754. i q^{63} +6823.63 q^{65} -496811. i q^{67} +49498.3 q^{69} +452032. i q^{71} -394349. q^{73} -237334. i q^{75} +230933. q^{77} -571742. i q^{79} +59049.0 q^{81} +324465. i q^{83} +153655. q^{85} -302561. i q^{87} -758607. q^{89} +180775. i q^{91} +238462. q^{93} -86010.5i q^{95} +25015.4 q^{97} +105911. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 80 q^{5} - 972 q^{9}+O(q^{10}) $ 4 * q + 80 * q^5 - 972 * q^9	$ 4 q + 80 q^{5} - 972 q^{9} - 5760 q^{13} + 2232 q^{17} - 11664 q^{21} - 60900 q^{25} + 50608 q^{29} - 58320 q^{33} - 212256 q^{37} + 49464 q^{41} - 19440 q^{45} - 139388 q^{49} - 256400 q^{53} + 11664 q^{57} + 80928 q^{61} - 115200 q^{65} - 443232 q^{69} - 38456 q^{73} + 1180224 q^{77} + 236196 q^{81} + 44640 q^{85} - 3319416 q^{89} - 478224 q^{93} - 2464840 q^{97}+O(q^{100}) $ 4 * q + 80 * q^5 - 972 * q^9 - 5760 * q^13 + 2232 * q^17 - 11664 * q^21 - 60900 * q^25 + 50608 * q^29 - 58320 * q^33 - 212256 * q^37 + 49464 * q^41 - 19440 * q^45 - 139388 * q^49 - 256400 * q^53 + 11664 * q^57 + 80928 * q^61 - 115200 * q^65 - 443232 * q^69 - 38456 * q^73 + 1180224 * q^77 + 236196 * q^81 + 44640 * q^85 - 3319416 * q^89 - 478224 * q^93 - 2464840 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	15.5885i	0.577350i
$3$	15.5885i	0.577350i
$4$	0	0
$5$	20.0000	0.160000	0.0800000	−	0.996795i	$-0.474508\pi$
$5$	20.0000	0.160000	0.0800000	+	0.996795i	$0.474508\pi$
$6$	0	0
$7$	529.850i	1.54475i	0.635165	+	0.772376i	$0.280932\pi$
$7$	529.850i	1.54475i	−0.635165	+	0.772376i	$0.719068\pi$
$8$	0	0
$9$	−243.000	−0.333333
$10$	0	0
$11$	− 435.847i	− 0.327458i	−0.986505	−	0.163729i	$-0.947648\pi$
$11$	− 435.847i	− 0.327458i	0.986505	−	0.163729i	$-0.0523523\pi$
$12$	0	0
$13$	341.182	0.155294	0.0776472	−	0.996981i	$-0.475259\pi$
$13$	341.182	0.155294	0.0776472	+	0.996981i	$0.475259\pi$
$14$	0	0
$15$	311.769i	0.0923760i
$16$	0	0
$17$	7682.73	1.56375	0.781877	−	0.623432i	$-0.214262\pi$
$17$	7682.73	1.56375	0.781877	+	0.623432i	$0.214262\pi$
$18$	0	0
$19$	− 4300.52i	− 0.626990i	−0.949590	−	0.313495i	$-0.898500\pi$
$19$	− 4300.52i	− 0.626990i	0.949590	−	0.313495i	$-0.101500\pi$
$20$	0	0
$21$	−8259.54	−0.891863
$22$	0	0
$23$	− 3175.32i	− 0.260978i	−0.991450	−	0.130489i	$-0.958345\pi$
$23$	− 3175.32i	− 0.260978i	0.991450	−	0.130489i	$-0.0416547\pi$
$24$	0	0
$25$	−15225.0	−0.974400
$26$	0	0
$27$	− 3788.00i	− 0.192450i
$28$	0	0
$29$	−19409.3	−0.795821	−0.397910	−	0.917424i	$-0.630264\pi$
$29$	−19409.3	−0.795821	−0.397910	+	0.917424i	$0.630264\pi$
$30$	0	0
$31$	− 15297.3i	− 0.513488i	−0.966479	−	0.256744i	$-0.917350\pi$
$31$	− 15297.3i	− 0.513488i	0.966479	−	0.256744i	$-0.0826497\pi$
$32$	0	0
$33$	6794.18	0.189058
$34$	0	0
$35$	10597.0i	0.247160i
$36$	0	0
$37$	−61969.9	−1.22342	−0.611710	−	0.791082i	$-0.709518\pi$
$37$	−61969.9	−1.22342	−0.611710	+	0.791082i	$0.709518\pi$
$38$	0	0
$39$	5318.50i	0.0896592i
$40$	0	0
$41$	33740.2	0.489549	0.244774	−	0.969580i	$-0.421286\pi$
$41$	33740.2	0.489549	0.244774	+	0.969580i	$0.421286\pi$
$42$	0	0
$43$	99312.6i	1.24911i	0.780983	+	0.624553i	$0.214719\pi$
$43$	99312.6i	1.24911i	−0.780983	+	0.624553i	$0.785281\pi$
$44$	0	0
$45$	−4860.00	−0.0533333
$46$	0	0
$47$	17726.4i	0.170737i	0.996349	+	0.0853685i	$0.0272068\pi$
$47$	17726.4i	0.170737i	−0.996349	+	0.0853685i	$0.972793\pi$
$48$	0	0
$49$	−163092.	−1.38626
$50$	0	0
$51$	119762.i	0.902834i
$52$	0	0
$53$	−224406.	−1.50733	−0.753664	−	0.657260i	$-0.771715\pi$
$53$	−224406.	−1.50733	−0.753664	+	0.657260i	$0.771715\pi$
$54$	0	0
$55$	− 8716.94i	− 0.0523933i
$56$	0	0
$57$	67038.5	0.361993
$58$	0	0
$59$	− 199557.i	− 0.971651i	−0.874056	−	0.485826i	$-0.838519\pi$
$59$	− 199557.i	− 0.971651i	0.874056	−	0.485826i	$-0.161481\pi$
$60$	0	0
$61$	−45671.7	−0.201214	−0.100607	−	0.994926i	$-0.532078\pi$
$61$	−45671.7	−0.201214	−0.100607	+	0.994926i	$0.532078\pi$
$62$	0	0
$63$	− 128754.i	− 0.514917i
$64$	0	0
$65$	6823.63	0.0248471
$66$	0	0
$67$	− 496811.i	− 1.65183i	−0.563791	−	0.825917i	$-0.690658\pi$
$67$	− 496811.i	− 1.65183i	0.563791	−	0.825917i	$-0.309342\pi$
$68$	0	0
$69$	49498.3	0.150676
$70$	0	0
$71$	452032.i	1.26297i	0.775387	+	0.631487i	$0.217555\pi$
$71$	452032.i	1.26297i	−0.775387	+	0.631487i	$0.782445\pi$
$72$	0	0
$73$	−394349.	−1.01371	−0.506853	−	0.862032i	$-0.669191\pi$
$73$	−394349.	−1.01371	−0.506853	+	0.862032i	$0.669191\pi$
$74$	0	0
$75$	− 237334.i	− 0.562570i
$76$	0	0
$77$	230933.	0.505842
$78$	0	0
$79$	− 571742.i	− 1.15963i	−0.814749	−	0.579814i	$-0.803125\pi$
$79$	− 571742.i	− 1.15963i	0.814749	−	0.579814i	$-0.196875\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	324465.i	0.567458i	0.958904	+	0.283729i	$0.0915717\pi$
$83$	324465.i	0.567458i	−0.958904	+	0.283729i	$0.908428\pi$
$84$	0	0
$85$	153655.	0.250201
$86$	0	0
$87$	− 302561.i	− 0.459467i
$88$	0	0
$89$	−758607.	−1.07609	−0.538043	−	0.842918i	$-0.680836\pi$
$89$	−758607.	−1.07609	−0.538043	+	0.842918i	$0.680836\pi$
$90$	0	0
$91$	180775.i	0.239891i
$92$	0	0
$93$	238462.	0.296462
$94$	0	0
$95$	− 86010.5i	− 0.100318i
$96$	0	0
$97$	25015.4	0.0274089	0.0137045	−	0.999906i	$-0.495638\pi$
$97$	25015.4	0.0274089	0.0137045	+	0.999906i	$0.495638\pi$
$98$	0	0
$99$	105911.i	0.109153i
$100$	0	0
$101$	1.60937e6	1.56204	0.781018	−	0.624508i	$-0.214700\pi$
$101$	1.60937e6	1.56204	0.781018	+	0.624508i	$0.214700\pi$
$102$	0	0
$103$	647116.i	0.592203i	0.955156	+	0.296101i	$0.0956867\pi$
$103$	647116.i	0.592203i	−0.955156	+	0.296101i	$0.904313\pi$
$104$	0	0
$105$	−165191.	−0.142698
$106$	0	0
$107$	233041.i	0.190231i	0.995466	+	0.0951156i	$0.0303221\pi$
$107$	233041.i	0.190231i	−0.995466	+	0.0951156i	$0.969678\pi$
$108$	0	0
$109$	525849.	0.406052	0.203026	−	0.979173i	$-0.434922\pi$
$109$	525849.	0.406052	0.203026	+	0.979173i	$0.434922\pi$
$110$	0	0
$111$	− 966015.i	− 0.706342i
$112$	0	0
$113$	−1.21523e6	−0.842216	−0.421108	−	0.907011i	$-0.638359\pi$
$113$	−1.21523e6	−0.842216	−0.421108	+	0.907011i	$0.638359\pi$
$114$	0	0
$115$	− 63506.4i	− 0.0417565i
$116$	0	0
$117$	−82907.1	−0.0517648
$118$	0	0
$119$	4.07069e6i	2.41561i
$120$	0	0
$121$	1.58160e6	0.892771
$122$	0	0
$123$	525957.i	0.282641i
$124$	0	0
$125$	−617000.	−0.315904
$126$	0	0
$127$	− 3.23094e6i	− 1.57731i	−0.614836	−	0.788655i	$-0.710778\pi$
$127$	− 3.23094e6i	− 1.57731i	0.614836	−	0.788655i	$-0.289222\pi$
$128$	0	0
$129$	−1.54813e6	−0.721171
$130$	0	0
$131$	− 1.80119e6i	− 0.801209i	−0.916251	−	0.400604i	$-0.868800\pi$
$131$	− 1.80119e6i	− 0.801209i	0.916251	−	0.400604i	$-0.131200\pi$
$132$	0	0
$133$	2.27863e6	0.968544
$134$	0	0
$135$	− 75759.9i	− 0.0307920i
$136$	0	0
$137$	−50362.3	−0.0195859	−0.00979295	−	0.999952i	$-0.503117\pi$
$137$	−50362.3	−0.0195859	−0.00979295	+	0.999952i	$0.503117\pi$
$138$	0	0
$139$	− 4.23758e6i	− 1.57788i	−0.614471	−	0.788940i	$-0.710630\pi$
$139$	− 4.23758e6i	− 1.57788i	0.614471	−	0.788940i	$-0.289370\pi$
$140$	0	0
$141$	−276328.	−0.0985751
$142$	0	0
$143$	− 148703.i	− 0.0508524i
$144$	0	0
$145$	−388185.	−0.127331
$146$	0	0
$147$	− 2.54235e6i	− 0.800357i
$148$	0	0
$149$	−4.40006e6	−1.33015	−0.665073	−	0.746778i	$-0.731600\pi$
$149$	−4.40006e6	−1.33015	−0.665073	+	0.746778i	$0.731600\pi$
$150$	0	0
$151$	2.17763e6i	0.632488i	0.948678	+	0.316244i	$0.102422\pi$
$151$	2.17763e6i	0.632488i	−0.948678	+	0.316244i	$0.897578\pi$
$152$	0	0
$153$	−1.86690e6	−0.521252
$154$	0	0
$155$	− 305946.i	− 0.0821580i
$156$	0	0
$157$	−5.73506e6	−1.48197	−0.740985	−	0.671522i	$-0.765641\pi$
$157$	−5.73506e6	−1.48197	−0.740985	+	0.671522i	$0.765641\pi$
$158$	0	0
$159$	− 3.49815e6i	− 0.870256i
$160$	0	0
$161$	1.68244e6	0.403147
$162$	0	0
$163$	2.99695e6i	0.692016i	0.938232	+	0.346008i	$0.112463\pi$
$163$	2.99695e6i	0.692016i	−0.938232	+	0.346008i	$0.887537\pi$
$164$	0	0
$165$	135884.	0.0302493
$166$	0	0
$167$	8.03225e6i	1.72460i	0.506400	+	0.862298i	$0.330976\pi$
$167$	8.03225e6i	1.72460i	−0.506400	+	0.862298i	$0.669024\pi$
$168$	0	0
$169$	−4.71040e6	−0.975884
$170$	0	0
$171$	1.04503e6i	0.208997i
$172$	0	0
$173$	2.42976e6	0.469272	0.234636	−	0.972083i	$-0.424610\pi$
$173$	2.42976e6	0.469272	0.234636	+	0.972083i	$0.424610\pi$
$174$	0	0
$175$	− 8.06697e6i	− 1.50521i
$176$	0	0
$177$	3.11078e6	0.560983
$178$	0	0
$179$	− 5.34862e6i	− 0.932573i	−0.884634	−	0.466286i	$-0.845592\pi$
$179$	− 5.34862e6i	− 0.932573i	0.884634	−	0.466286i	$-0.154408\pi$
$180$	0	0
$181$	−9.11072e6	−1.53645	−0.768223	−	0.640183i	$-0.778859\pi$
$181$	−9.11072e6	−1.53645	−0.768223	+	0.640183i	$0.778859\pi$
$182$	0	0
$183$	− 711952.i	− 0.116171i
$184$	0	0
$185$	−1.23940e6	−0.195747
$186$	0	0
$187$	− 3.34849e6i	− 0.512064i
$188$	0	0
$189$	2.00707e6	0.297288
$190$	0	0
$191$	− 1.19063e6i	− 0.170875i	−0.996344	−	0.0854373i	$-0.972771\pi$
$191$	− 1.19063e6i	− 0.170875i	0.996344	−	0.0854373i	$-0.0272287\pi$
$192$	0	0
$193$	2.07789e6	0.289034	0.144517	−	0.989502i	$-0.453837\pi$
$193$	2.07789e6	0.289034	0.144517	+	0.989502i	$0.453837\pi$
$194$	0	0
$195$	106370.i	0.0143455i
$196$	0	0
$197$	2.19700e6	0.287364	0.143682	−	0.989624i	$-0.454106\pi$
$197$	2.19700e6	0.287364	0.143682	+	0.989624i	$0.454106\pi$
$198$	0	0
$199$	6.24588e6i	0.792564i	0.918129	+	0.396282i	$0.129700\pi$
$199$	6.24588e6i	0.792564i	−0.918129	+	0.396282i	$0.870300\pi$
$200$	0	0
$201$	7.74451e6	0.953687
$202$	0	0
$203$	− 1.02840e7i	− 1.22935i
$204$	0	0
$205$	674804.	0.0783278
$206$	0	0
$207$	771603.i	0.0869927i
$208$	0	0
$209$	−1.87437e6	−0.205313
$210$	0	0
$211$	− 1.11658e7i	− 1.18862i	−0.804235	−	0.594311i	$-0.797425\pi$
$211$	− 1.11658e7i	− 1.18862i	0.804235	−	0.594311i	$-0.202575\pi$
$212$	0	0
$213$	−7.04648e6	−0.729178
$214$	0	0
$215$	1.98625e6i	0.199857i
$216$	0	0
$217$	8.10528e6	0.793211
$218$	0	0
$219$	− 6.14730e6i	− 0.585264i
$220$	0	0
$221$	2.62121e6	0.242842
$222$	0	0
$223$	− 658854.i	− 0.0594120i	−0.999559	−	0.0297060i	$-0.990543\pi$
$223$	− 658854.i	− 0.0594120i	0.999559	−	0.0297060i	$-0.00945711\pi$
$224$	0	0
$225$	3.69967e6	0.324800
$226$	0	0
$227$	− 1.04595e7i	− 0.894198i	−0.894484	−	0.447099i	$-0.852457\pi$
$227$	− 1.04595e7i	− 0.894198i	0.894484	−	0.447099i	$-0.147543\pi$
$228$	0	0
$229$	1.94802e7	1.62214	0.811069	−	0.584951i	$-0.198886\pi$
$229$	1.94802e7	1.62214	0.811069	+	0.584951i	$0.198886\pi$
$230$	0	0
$231$	3.59990e6i	0.292048i
$232$	0	0
$233$	1.62947e7	1.28819	0.644095	−	0.764946i	$-0.277234\pi$
$233$	1.62947e7	1.28819	0.644095	+	0.764946i	$0.277234\pi$
$234$	0	0
$235$	354529.i	0.0273179i
$236$	0	0
$237$	8.91257e6	0.669511
$238$	0	0
$239$	− 2.00798e7i	− 1.47084i	−0.677609	−	0.735422i	$-0.736984\pi$
$239$	− 2.00798e7i	− 1.47084i	0.677609	−	0.735422i	$-0.263016\pi$
$240$	0	0
$241$	−5.97568e6	−0.426910	−0.213455	−	0.976953i	$-0.568472\pi$
$241$	−5.97568e6	−0.426910	−0.213455	+	0.976953i	$0.568472\pi$
$242$	0	0
$243$	920483.i	0.0641500i
$244$	0	0
$245$	−3.26184e6	−0.221802
$246$	0	0
$247$	− 1.46726e6i	− 0.0973680i
$248$	0	0
$249$	−5.05791e6	−0.327622
$250$	0	0
$251$	− 1.92078e7i	− 1.21466i	−0.794449	−	0.607331i	$-0.792240\pi$
$251$	− 1.92078e7i	− 1.21466i	0.794449	−	0.607331i	$-0.207760\pi$
$252$	0	0
$253$	−1.38395e6	−0.0854594
$254$	0	0
$255$	2.39524e6i	0.144453i
$256$	0	0
$257$	5.96168e6	0.351212	0.175606	−	0.984461i	$-0.443811\pi$
$257$	5.96168e6	0.351212	0.175606	+	0.984461i	$0.443811\pi$
$258$	0	0
$259$	− 3.28348e7i	− 1.88988i
$260$	0	0
$261$	4.71645e6	0.265274
$262$	0	0
$263$	− 1.92686e7i	− 1.05921i	−0.848243	−	0.529607i	$-0.822339\pi$
$263$	− 1.92686e7i	− 1.05921i	0.848243	−	0.529607i	$-0.177661\pi$
$264$	0	0
$265$	−4.48813e6	−0.241172
$266$	0	0
$267$	− 1.18255e7i	− 0.621278i
$268$	0	0
$269$	1.74390e7	0.895909	0.447955	−	0.894056i	$-0.352153\pi$
$269$	1.74390e7	0.895909	0.447955	+	0.894056i	$0.352153\pi$
$270$	0	0
$271$	2.41844e7i	1.21514i	0.794265	+	0.607571i	$0.207856\pi$
$271$	2.41844e7i	1.21514i	−0.794265	+	0.607571i	$0.792144\pi$
$272$	0	0
$273$	−2.81800e6	−0.138501
$274$	0	0
$275$	6.63577e6i	0.319075i
$276$	0	0
$277$	3.19698e7	1.50418	0.752091	−	0.659059i	$-0.229045\pi$
$277$	3.19698e7	1.50418	0.752091	+	0.659059i	$0.229045\pi$
$278$	0	0
$279$	3.71725e6i	0.171163i
$280$	0	0
$281$	−2.34865e7	−1.05852	−0.529259	−	0.848460i	$-0.677530\pi$
$281$	−2.34865e7	−1.05852	−0.529259	+	0.848460i	$0.677530\pi$
$282$	0	0
$283$	− 7.74978e6i	− 0.341924i	−0.985278	−	0.170962i	$-0.945312\pi$
$283$	− 7.74978e6i	− 0.341924i	0.985278	−	0.170962i	$-0.0546876\pi$
$284$	0	0
$285$	1.34077e6	0.0579189
$286$	0	0
$287$	1.78772e7i	0.756231i
$288$	0	0
$289$	3.48867e7	1.44533
$290$	0	0
$291$	389951.i	0.0158246i
$292$	0	0
$293$	−1.14848e7	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$293$	−1.14848e7	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$294$	0	0
$295$	− 3.99114e6i	− 0.155464i
$296$	0	0
$297$	−1.65099e6	−0.0630194
$298$	0	0
$299$	− 1.08336e6i	− 0.0405284i
$300$	0	0
$301$	−5.26208e7	−1.92956
$302$	0	0
$303$	2.50876e7i	0.901842i
$304$	0	0
$305$	−913434.	−0.0321942
$306$	0	0
$307$	− 2.30562e7i	− 0.796841i	−0.917203	−	0.398421i	$-0.869558\pi$
$307$	− 2.30562e7i	− 0.796841i	0.917203	−	0.398421i	$-0.130442\pi$
$308$	0	0
$309$	−1.00875e7	−0.341909
$310$	0	0
$311$	− 4.24296e7i	− 1.41055i	−0.708935	−	0.705274i	$-0.750824\pi$
$311$	− 4.24296e7i	− 1.41055i	0.708935	−	0.705274i	$-0.249176\pi$
$312$	0	0
$313$	2.01928e7	0.658513	0.329257	−	0.944240i	$-0.393202\pi$
$313$	2.01928e7	0.658513	0.329257	+	0.944240i	$0.393202\pi$
$314$	0	0
$315$	− 2.57507e6i	− 0.0823868i
$316$	0	0
$317$	−5.83594e6	−0.183203	−0.0916016	−	0.995796i	$-0.529199\pi$
$317$	−5.83594e6	−0.183203	−0.0916016	+	0.995796i	$0.529199\pi$
$318$	0	0
$319$	8.45947e6i	0.260598i
$320$	0	0
$321$	−3.63276e6	−0.109830
$322$	0	0
$323$	− 3.30398e7i	− 0.980458i
$324$	0	0
$325$	−5.19449e6	−0.151319
$326$	0	0
$327$	8.19718e6i	0.234434i
$328$	0	0
$329$	−9.39235e6	−0.263746
$330$	0	0
$331$	− 6.94601e7i	− 1.91536i	−0.287828	−	0.957682i	$-0.592933\pi$
$331$	− 6.94601e7i	− 1.91536i	0.287828	−	0.957682i	$-0.407067\pi$
$332$	0	0
$333$	1.50587e7	0.407807
$334$	0	0
$335$	− 9.93621e6i	− 0.264294i
$336$	0	0
$337$	−2.65194e7	−0.692906	−0.346453	−	0.938067i	$-0.612614\pi$
$337$	−2.65194e7	−0.692906	−0.346453	+	0.938067i	$0.612614\pi$
$338$	0	0
$339$	− 1.89436e7i	− 0.486254i
$340$	0	0
$341$	−6.66729e6	−0.168146
$342$	0	0
$343$	− 2.40780e7i	− 0.596676i
$344$	0	0
$345$	989967.	0.0241081
$346$	0	0
$347$	− 5.53620e7i	− 1.32502i	−0.749052	−	0.662511i	$-0.769491\pi$
$347$	− 5.53620e7i	− 1.32502i	0.749052	−	0.662511i	$-0.230509\pi$
$348$	0	0
$349$	−4.23047e7	−0.995206	−0.497603	−	0.867405i	$-0.665786\pi$
$349$	−4.23047e7	−0.995206	−0.497603	+	0.867405i	$0.665786\pi$
$350$	0	0
$351$	− 1.29239e6i	− 0.0298864i
$352$	0	0
$353$	−4.67810e7	−1.06352	−0.531760	−	0.846895i	$-0.678469\pi$
$353$	−4.67810e7	−1.06352	−0.531760	+	0.846895i	$0.678469\pi$
$354$	0	0
$355$	9.04064e6i	0.202076i
$356$	0	0
$357$	−6.34558e7	−1.39466
$358$	0	0
$359$	− 6.79965e7i	− 1.46961i	−0.678276	−	0.734807i	$-0.737273\pi$
$359$	− 6.79965e7i	− 1.46961i	0.678276	−	0.734807i	$-0.262727\pi$
$360$	0	0
$361$	2.85514e7	0.606884
$362$	0	0
$363$	2.46547e7i	0.515442i
$364$	0	0
$365$	−7.88698e6	−0.162193
$366$	0	0
$367$	8.52346e7i	1.72432i	0.506637	+	0.862160i	$0.330889\pi$
$367$	8.52346e7i	1.72432i	−0.506637	+	0.862160i	$0.669111\pi$
$368$	0	0
$369$	−8.19886e6	−0.163183
$370$	0	0
$371$	− 1.18902e8i	− 2.32845i
$372$	0	0
$373$	5.70643e7	1.09961	0.549804	−	0.835293i	$-0.314702\pi$
$373$	5.70643e7	1.09961	0.549804	+	0.835293i	$0.314702\pi$
$374$	0	0
$375$	− 9.61808e6i	− 0.182387i
$376$	0	0
$377$	−6.62209e6	−0.123586
$378$	0	0
$379$	− 5.18779e7i	− 0.952939i	−0.879191	−	0.476469i	$-0.841916\pi$
$379$	− 5.18779e7i	− 0.952939i	0.879191	−	0.476469i	$-0.158084\pi$
$380$	0	0
$381$	5.03653e7	0.910661
$382$	0	0
$383$	7.83098e7i	1.39386i	0.717138	+	0.696931i	$0.245452\pi$
$383$	7.83098e7i	1.39386i	−0.717138	+	0.696931i	$0.754548\pi$
$384$	0	0
$385$	4.61867e6	0.0809347
$386$	0	0
$387$	− 2.41330e7i	− 0.416369i
$388$	0	0
$389$	−9.23410e7	−1.56872	−0.784361	−	0.620305i	$-0.787009\pi$
$389$	−9.23410e7	−1.56872	−0.784361	+	0.620305i	$0.787009\pi$
$390$	0	0
$391$	− 2.43951e7i	− 0.408106i
$392$	0	0
$393$	2.80778e7	0.462578
$394$	0	0
$395$	− 1.14348e7i	− 0.185540i
$396$	0	0
$397$	−3.75593e7	−0.600270	−0.300135	−	0.953897i	$-0.597032\pi$
$397$	−3.75593e7	−0.600270	−0.300135	+	0.953897i	$0.597032\pi$
$398$	0	0
$399$	3.55204e7i	0.559189i
$400$	0	0
$401$	−3.85071e7	−0.597183	−0.298592	−	0.954381i	$-0.596517\pi$
$401$	−3.85071e7	−0.597183	−0.298592	+	0.954381i	$0.596517\pi$
$402$	0	0
$403$	− 5.21916e6i	− 0.0797417i
$404$	0	0
$405$	1.18098e6	0.0177778
$406$	0	0
$407$	2.70094e7i	0.400619i
$408$	0	0
$409$	−1.07783e8	−1.57536	−0.787681	−	0.616083i	$-0.788719\pi$
$409$	−1.07783e8	−1.57536	−0.787681	+	0.616083i	$0.788719\pi$
$410$	0	0
$411$	− 785070.i	− 0.0113079i
$412$	0	0
$413$	1.05735e8	1.50096
$414$	0	0
$415$	6.48931e6i	0.0907933i
$416$	0	0
$417$	6.60574e7	0.910989
$418$	0	0
$419$	7.92976e7i	1.07800i	0.842306	+	0.538999i	$0.181197\pi$
$419$	7.92976e7i	1.07800i	−0.842306	+	0.538999i	$0.818803\pi$
$420$	0	0
$421$	6.67669e7	0.894777	0.447389	−	0.894340i	$-0.352354\pi$
$421$	6.67669e7	0.894777	0.447389	+	0.894340i	$0.352354\pi$
$422$	0	0
$423$	− 4.30752e6i	− 0.0569123i
$424$	0	0
$425$	−1.16970e8	−1.52372
$426$	0	0
$427$	− 2.41992e7i	− 0.310826i
$428$	0	0
$429$	2.31805e6	0.0293596
$430$	0	0
$431$	− 5.07275e7i	− 0.633595i	−0.948493	−	0.316798i	$-0.897392\pi$
$431$	− 5.07275e7i	− 0.633595i	0.948493	−	0.316798i	$-0.102608\pi$
$432$	0	0
$433$	−3.19683e7	−0.393782	−0.196891	−	0.980425i	$-0.563084\pi$
$433$	−3.19683e7	−0.393782	−0.196891	+	0.980425i	$0.563084\pi$
$434$	0	0
$435$	− 6.05121e6i	− 0.0735148i
$436$	0	0
$437$	−1.36555e7	−0.163631
$438$	0	0
$439$	− 7.14885e7i	− 0.844972i	−0.906369	−	0.422486i	$-0.861158\pi$
$439$	− 7.14885e7i	− 0.844972i	0.906369	−	0.422486i	$-0.138842\pi$
$440$	0	0
$441$	3.96314e7	0.462087
$442$	0	0
$443$	7.60820e7i	0.875127i	0.899188	+	0.437563i	$0.144158\pi$
$443$	7.60820e7i	0.875127i	−0.899188	+	0.437563i	$0.855842\pi$
$444$	0	0
$445$	−1.51721e7	−0.172174
$446$	0	0
$447$	− 6.85901e7i	− 0.767960i
$448$	0	0
$449$	9.00650e7	0.994986	0.497493	−	0.867468i	$-0.334254\pi$
$449$	9.00650e7	0.994986	0.497493	+	0.867468i	$0.334254\pi$
$450$	0	0
$451$	− 1.47056e7i	− 0.160307i
$452$	0	0
$453$	−3.39458e7	−0.365167
$454$	0	0
$455$	3.61550e6i	0.0383826i
$456$	0	0
$457$	3.59392e7	0.376548	0.188274	−	0.982117i	$-0.439711\pi$
$457$	3.59392e7	0.376548	0.188274	+	0.982117i	$0.439711\pi$
$458$	0	0
$459$	− 2.91021e7i	− 0.300945i
$460$	0	0
$461$	1.09802e8	1.12075	0.560373	−	0.828240i	$-0.310658\pi$
$461$	1.09802e8	1.12075	0.560373	+	0.828240i	$0.310658\pi$
$462$	0	0
$463$	− 5.53763e7i	− 0.557932i	−0.960301	−	0.278966i	$-0.910008\pi$
$463$	− 5.53763e7i	− 0.557932i	0.960301	−	0.278966i	$-0.0899917\pi$
$464$	0	0
$465$	4.76923e6	0.0474340
$466$	0	0
$467$	− 5.38248e7i	− 0.528484i	−0.964456	−	0.264242i	$-0.914878\pi$
$467$	− 5.38248e7i	− 0.528484i	0.964456	−	0.264242i	$-0.0851217\pi$
$468$	0	0
$469$	2.63235e8	2.55168
$470$	0	0
$471$	− 8.94008e7i	− 0.855615i
$472$	0	0
$473$	4.32851e7	0.409030
$474$	0	0
$475$	6.54755e7i	0.610939i
$476$	0	0
$477$	5.45307e7	0.502442
$478$	0	0
$479$	1.71807e8i	1.56328i	0.623733	+	0.781638i	$0.285615\pi$
$479$	1.71807e8i	1.56328i	−0.623733	+	0.781638i	$0.714385\pi$
$480$	0	0
$481$	−2.11430e7	−0.189990
$482$	0	0
$483$	2.62267e7i	0.232757i
$484$	0	0
$485$	500308.	0.00438543
$486$	0	0
$487$	5.72463e7i	0.495634i	0.968807	+	0.247817i	$0.0797131\pi$
$487$	5.72463e7i	0.495634i	−0.968807	+	0.247817i	$0.920287\pi$
$488$	0	0
$489$	−4.67178e7	−0.399536
$490$	0	0
$491$	− 2.13055e8i	− 1.79989i	−0.436001	−	0.899946i	$-0.643605\pi$
$491$	− 2.13055e8i	− 1.79989i	0.436001	−	0.899946i	$-0.356395\pi$
$492$	0	0
$493$	−1.49116e8	−1.24447
$494$	0	0
$495$	2.11822e6i	0.0174644i
$496$	0	0
$497$	−2.39509e8	−1.95098
$498$	0	0
$499$	− 9.04068e7i	− 0.727611i	−0.931475	−	0.363806i	$-0.881477\pi$
$499$	− 9.04068e7i	− 0.727611i	0.931475	−	0.363806i	$-0.118523\pi$
$500$	0	0
$501$	−1.25210e8	−0.995697
$502$	0	0
$503$	2.29271e8i	1.80155i	0.434290	+	0.900773i	$0.356999\pi$
$503$	2.29271e8i	1.80155i	−0.434290	+	0.900773i	$0.643001\pi$
$504$	0	0
$505$	3.21874e7	0.249926
$506$	0	0
$507$	− 7.34279e7i	− 0.563427i
$508$	0	0
$509$	1.29225e7	0.0979930	0.0489965	−	0.998799i	$-0.484398\pi$
$509$	1.29225e7	0.0979930	0.0489965	+	0.998799i	$0.484398\pi$
$510$	0	0
$511$	− 2.08946e8i	− 1.56593i
$512$	0	0
$513$	−1.62904e7	−0.120664
$514$	0	0
$515$	1.29423e7i	0.0947525i
$516$	0	0
$517$	7.72601e6	0.0559092
$518$	0	0
$519$	3.78762e7i	0.270934i
$520$	0	0
$521$	7.99955e7	0.565656	0.282828	−	0.959171i	$-0.408727\pi$
$521$	7.99955e7	0.565656	0.282828	+	0.959171i	$0.408727\pi$
$522$	0	0
$523$	1.28458e8i	0.897957i	0.893543	+	0.448979i	$0.148212\pi$
$523$	1.28458e8i	0.897957i	−0.893543	+	0.448979i	$0.851788\pi$
$524$	0	0
$525$	1.25752e8	0.869031
$526$	0	0
$527$	− 1.17525e8i	− 0.802969i
$528$	0	0
$529$	1.37953e8	0.931890
$530$	0	0
$531$	4.84923e7i	0.323884i
$532$	0	0
$533$	1.15115e7	0.0760241
$534$	0	0
$535$	4.66083e6i	0.0304370i
$536$	0	0
$537$	8.33768e7	0.538421
$538$	0	0
$539$	7.10832e7i	0.453942i
$540$	0	0
$541$	−2.90048e8	−1.83180	−0.915899	−	0.401410i	$-0.868520\pi$
$541$	−2.90048e8	−1.83180	−0.915899	+	0.401410i	$0.868520\pi$
$542$	0	0
$543$	− 1.42022e8i	− 0.887067i
$544$	0	0
$545$	1.05170e7	0.0649684
$546$	0	0
$547$	− 2.91198e7i	− 0.177921i	−0.996035	−	0.0889604i	$-0.971646\pi$
$547$	− 2.91198e7i	− 0.177921i	0.996035	−	0.0889604i	$-0.0283545\pi$
$548$	0	0
$549$	1.10982e7	0.0670713
$550$	0	0
$551$	8.34700e7i	0.498972i
$552$	0	0
$553$	3.02937e8	1.79134
$554$	0	0
$555$	− 1.93203e7i	− 0.113015i
$556$	0	0
$557$	−1.95014e8	−1.12849	−0.564247	−	0.825606i	$-0.690834\pi$
$557$	−1.95014e8	−1.12849	−0.564247	+	0.825606i	$0.690834\pi$
$558$	0	0
$559$	3.38836e7i	0.193979i
$560$	0	0
$561$	5.21978e7	0.295640
$562$	0	0
$563$	3.06065e8i	1.71510i	0.514404	+	0.857548i	$0.328013\pi$
$563$	3.06065e8i	1.71510i	−0.514404	+	0.857548i	$0.671987\pi$
$564$	0	0
$565$	−2.43046e7	−0.134755
$566$	0	0
$567$	3.12871e7i	0.171639i
$568$	0	0
$569$	−3.27388e8	−1.77716	−0.888578	−	0.458725i	$-0.848306\pi$
$569$	−3.27388e8	−1.77716	−0.888578	+	0.458725i	$0.848306\pi$
$570$	0	0
$571$	1.37293e8i	0.737461i	0.929536	+	0.368731i	$0.120208\pi$
$571$	1.37293e8i	0.737461i	−0.929536	+	0.368731i	$0.879792\pi$
$572$	0	0
$573$	1.85601e7	0.0986544
$574$	0	0
$575$	4.83443e7i	0.254297i
$576$	0	0
$577$	2.84639e8	1.48172	0.740861	−	0.671658i	$-0.234417\pi$
$577$	2.84639e8	1.48172	0.740861	+	0.671658i	$0.234417\pi$
$578$	0	0
$579$	3.23910e7i	0.166874i
$580$	0	0
$581$	−1.71918e8	−0.876583
$582$	0	0
$583$	9.78068e7i	0.493587i
$584$	0	0
$585$	−1.65814e6	−0.00828236
$586$	0	0
$587$	3.11468e8i	1.53992i	0.638090	+	0.769962i	$0.279725\pi$
$587$	3.11468e8i	1.53992i	−0.638090	+	0.769962i	$0.720275\pi$
$588$	0	0
$589$	−6.57865e7	−0.321952
$590$	0	0
$591$	3.42479e7i	0.165909i
$592$	0	0
$593$	−7.54519e7	−0.361831	−0.180916	−	0.983499i	$-0.557906\pi$
$593$	−7.54519e7	−0.361831	−0.180916	+	0.983499i	$0.557906\pi$
$594$	0	0
$595$	8.14139e7i	0.386498i
$596$	0	0
$597$	−9.73636e7	−0.457587
$598$	0	0
$599$	2.08861e8i	0.971800i	0.874014	+	0.485900i	$0.161508\pi$
$599$	2.08861e8i	0.971800i	−0.874014	+	0.485900i	$0.838492\pi$
$600$	0	0
$601$	2.88006e7	0.132672	0.0663359	−	0.997797i	$-0.478869\pi$
$601$	2.88006e7	0.132672	0.0663359	+	0.997797i	$0.478869\pi$
$602$	0	0
$603$	1.20725e8i	0.550611i
$604$	0	0
$605$	3.16320e7	0.142843
$606$	0	0
$607$	− 2.89154e8i	− 1.29290i	−0.762958	−	0.646448i	$-0.776254\pi$
$607$	− 2.89154e8i	− 1.29290i	0.762958	−	0.646448i	$-0.223746\pi$
$608$	0	0
$609$	1.60312e8	0.709763
$610$	0	0
$611$	6.04793e6i	0.0265145i
$612$	0	0
$613$	3.53775e8	1.53584	0.767919	−	0.640548i	$-0.221293\pi$
$613$	3.53775e8	1.53584	0.767919	+	0.640548i	$0.221293\pi$
$614$	0	0
$615$	1.05191e7i	0.0452226i
$616$	0	0
$617$	−1.88929e8	−0.804348	−0.402174	−	0.915563i	$-0.631745\pi$
$617$	−1.88929e8	−0.804348	−0.402174	+	0.915563i	$0.631745\pi$
$618$	0	0
$619$	− 8.45244e7i	− 0.356377i	−0.983996	−	0.178189i	$-0.942976\pi$
$619$	− 8.45244e7i	− 0.356377i	0.983996	−	0.178189i	$-0.0570237\pi$
$620$	0	0
$621$	−1.20281e7	−0.0502253
$622$	0	0
$623$	− 4.01948e8i	− 1.66229i
$624$	0	0
$625$	2.25551e8	0.923855
$626$	0	0
$627$	− 2.92185e7i	− 0.118538i
$628$	0	0
$629$	−4.76098e8	−1.91313
$630$	0	0
$631$	3.49408e8i	1.39074i	0.718653	+	0.695369i	$0.244759\pi$
$631$	3.49408e8i	1.39074i	−0.718653	+	0.695369i	$0.755241\pi$
$632$	0	0
$633$	1.74058e8	0.686251
$634$	0	0
$635$	− 6.46187e7i	− 0.252370i
$636$	0	0
$637$	−5.56440e7	−0.215278
$638$	0	0
$639$	− 1.09844e8i	− 0.420991i
$640$	0	0
$641$	−2.70070e8	−1.02542	−0.512710	−	0.858562i	$-0.671359\pi$
$641$	−2.70070e8	−1.02542	−0.512710	+	0.858562i	$0.671359\pi$
$642$	0	0
$643$	4.94695e8i	1.86082i	0.366518	+	0.930411i	$0.380550\pi$
$643$	4.94695e8i	1.86082i	−0.366518	+	0.930411i	$0.619450\pi$
$644$	0	0
$645$	−3.09626e7	−0.115387
$646$	0	0
$647$	1.19819e7i	0.0442396i	0.999755	+	0.0221198i	$0.00704153\pi$
$647$	1.19819e7i	0.0442396i	−0.999755	+	0.0221198i	$0.992958\pi$
$648$	0	0
$649$	−8.69762e7	−0.318175
$650$	0	0
$651$	1.26349e8i	0.457961i
$652$	0	0
$653$	5.21595e8	1.87324	0.936621	−	0.350345i	$-0.113936\pi$
$653$	5.21595e8	1.87324	0.936621	+	0.350345i	$0.113936\pi$
$654$	0	0
$655$	− 3.60238e7i	− 0.128193i
$656$	0	0
$657$	9.58269e7	0.337902
$658$	0	0
$659$	− 4.10833e8i	− 1.43552i	−0.696292	−	0.717759i	$-0.745168\pi$
$659$	− 4.10833e8i	− 1.43552i	0.696292	−	0.717759i	$-0.254832\pi$
$660$	0	0
$661$	−8.48526e7	−0.293806	−0.146903	−	0.989151i	$-0.546931\pi$
$661$	−8.48526e7	−0.293806	−0.146903	+	0.989151i	$0.546931\pi$
$662$	0	0
$663$	4.08605e7i	0.140205i
$664$	0	0
$665$	4.55727e7	0.154967
$666$	0	0
$667$	6.16306e7i	0.207692i
$668$	0	0
$669$	1.02705e7	0.0343016
$670$	0	0
$671$	1.99059e7i	0.0658891i
$672$	0	0
$673$	−3.39619e8	−1.11416	−0.557079	−	0.830460i	$-0.688078\pi$
$673$	−3.39619e8	−1.11416	−0.557079	+	0.830460i	$0.688078\pi$
$674$	0	0
$675$	5.76722e7i	0.187523i
$676$	0	0
$677$	4.04369e8	1.30320	0.651602	−	0.758561i	$-0.274097\pi$
$677$	4.04369e8	1.30320	0.651602	+	0.758561i	$0.274097\pi$
$678$	0	0
$679$	1.32544e7i	0.0423400i
$680$	0	0
$681$	1.63048e8	0.516266
$682$	0	0
$683$	2.05590e8i	0.645268i	0.946524	+	0.322634i	$0.104568\pi$
$683$	2.05590e8i	0.645268i	−0.946524	+	0.322634i	$0.895432\pi$
$684$	0	0
$685$	−1.00725e6	−0.00313374
$686$	0	0
$687$	3.03667e8i	0.936542i
$688$	0	0
$689$	−7.65633e7	−0.234079
$690$	0	0
$691$	1.25719e8i	0.381037i	0.981684	+	0.190518i	$0.0610169\pi$
$691$	1.25719e8i	0.381037i	−0.981684	+	0.190518i	$0.938983\pi$
$692$	0	0
$693$	−5.61168e7	−0.168614
$694$	0	0
$695$	− 8.47517e7i	− 0.252461i
$696$	0	0
$697$	2.59217e8	0.765534
$698$	0	0
$699$	2.54010e8i	0.743737i
$700$	0	0
$701$	1.36521e8	0.396320	0.198160	−	0.980170i	$-0.436503\pi$
$701$	1.36521e8	0.396320	0.198160	+	0.980170i	$0.436503\pi$
$702$	0	0
$703$	2.66503e8i	0.767072i
$704$	0	0
$705$	−5.52655e6	−0.0157720
$706$	0	0
$707$	8.52724e8i	2.41296i
$708$	0	0
$709$	1.19437e8	0.335119	0.167559	−	0.985862i	$-0.446411\pi$
$709$	1.19437e8	0.335119	0.167559	+	0.985862i	$0.446411\pi$
$710$	0	0
$711$	1.38933e8i	0.386543i
$712$	0	0
$713$	−4.85739e7	−0.134009
$714$	0	0
$715$	− 2.97406e6i	− 0.00813638i
$716$	0	0
$717$	3.13014e8	0.849192
$718$	0	0
$719$	− 2.90874e8i	− 0.782561i	−0.920272	−	0.391280i	$-0.872032\pi$
$719$	− 2.90874e8i	− 0.782561i	0.920272	−	0.391280i	$-0.127968\pi$
$720$	0	0
$721$	−3.42875e8	−0.914807
$722$	0	0
$723$	− 9.31516e7i	− 0.246477i
$724$	0	0
$725$	2.95506e8	0.775448
$726$	0	0
$727$	− 8.94298e7i	− 0.232744i	−0.993206	−	0.116372i	$-0.962873\pi$
$727$	− 8.94298e7i	− 0.232744i	0.993206	−	0.116372i	$-0.0371265\pi$
$728$	0	0
$729$	−1.43489e7	−0.0370370
$730$	0	0
$731$	7.62992e8i	1.95329i
$732$	0	0
$733$	1.28445e8	0.326141	0.163071	−	0.986614i	$-0.447860\pi$
$733$	1.28445e8	0.326141	0.163071	+	0.986614i	$0.447860\pi$
$734$	0	0
$735$	− 5.08471e7i	− 0.128057i
$736$	0	0
$737$	−2.16533e8	−0.540907
$738$	0	0
$739$	7.15451e8i	1.77275i	0.462972	+	0.886373i	$0.346783\pi$
$739$	7.15451e8i	1.77275i	−0.462972	+	0.886373i	$0.653217\pi$
$740$	0	0
$741$	2.28723e7	0.0562154
$742$	0	0
$743$	− 3.45246e8i	− 0.841708i	−0.907128	−	0.420854i	$-0.861730\pi$
$743$	− 3.45246e8i	− 0.841708i	0.907128	−	0.420854i	$-0.138270\pi$
$744$	0	0
$745$	−8.80011e7	−0.212823
$746$	0	0
$747$	− 7.88451e7i	− 0.189153i
$748$	0	0
$749$	−1.23477e8	−0.293860
$750$	0	0
$751$	1.75422e8i	0.414157i	0.978324	+	0.207079i	$0.0663956\pi$
$751$	1.75422e8i	0.414157i	−0.978324	+	0.207079i	$0.933604\pi$
$752$	0	0
$753$	2.99419e8	0.701286
$754$	0	0
$755$	4.35525e7i	0.101198i
$756$	0	0
$757$	−4.03668e8	−0.930544	−0.465272	−	0.885168i	$-0.654044\pi$
$757$	−4.03668e8	−0.930544	−0.465272	+	0.885168i	$0.654044\pi$
$758$	0	0
$759$	− 2.15737e7i	− 0.0493400i
$760$	0	0
$761$	−5.80482e8	−1.31715	−0.658574	−	0.752516i	$-0.728840\pi$
$761$	−5.80482e8	−1.31715	−0.658574	+	0.752516i	$0.728840\pi$
$762$	0	0
$763$	2.78621e8i	0.627250i
$764$	0	0
$765$	−3.73381e7	−0.0834002
$766$	0	0
$767$	− 6.80851e7i	− 0.150892i
$768$	0	0
$769$	5.90678e8	1.29889	0.649444	−	0.760409i	$-0.275002\pi$
$769$	5.90678e8	1.29889	0.649444	+	0.760409i	$0.275002\pi$
$770$	0	0
$771$	9.29335e7i	0.202772i
$772$	0	0
$773$	3.10093e8	0.671357	0.335679	−	0.941977i	$-0.391034\pi$
$773$	3.10093e8	0.671357	0.335679	+	0.941977i	$0.391034\pi$
$774$	0	0
$775$	2.32902e8i	0.500342i
$776$	0	0
$777$	5.11843e8	1.09112
$778$	0	0
$779$	− 1.45100e8i	− 0.306942i
$780$	0	0
$781$	1.97017e8	0.413571
$782$	0	0
$783$	7.35222e7i	0.153156i
$784$	0	0
$785$	−1.14701e8	−0.237115
$786$	0	0
$787$	− 5.73439e8i	− 1.17642i	−0.808708	−	0.588210i	$-0.799833\pi$
$787$	− 5.73439e8i	− 1.17642i	0.808708	−	0.588210i	$-0.200167\pi$
$788$	0	0
$789$	3.00368e8	0.611538
$790$	0	0
$791$	− 6.43890e8i	− 1.30101i
$792$	0	0
$793$	−1.55824e7	−0.0312474
$794$	0	0
$795$	− 6.99630e7i	− 0.139241i
$796$	0	0
$797$	−6.24736e8	−1.23402	−0.617009	−	0.786956i	$-0.711656\pi$
$797$	−6.24736e8	−1.23402	−0.617009	+	0.786956i	$0.711656\pi$
$798$	0	0
$799$	1.36187e8i	0.266991i
$800$	0	0
$801$	1.84341e8	0.358695
$802$	0	0
$803$	1.71876e8i	0.331947i
$804$	0	0
$805$	3.36489e7	0.0645034
$806$	0	0
$807$	2.71847e8i	0.517253i
$808$	0	0
$809$	−9.88696e8	−1.86731	−0.933657	−	0.358169i	$-0.883401\pi$
$809$	−9.88696e8	−1.86731	−0.933657	+	0.358169i	$0.883401\pi$
$810$	0	0
$811$	4.02063e8i	0.753757i	0.926263	+	0.376879i	$0.123003\pi$
$811$	4.02063e8i	0.753757i	−0.926263	+	0.376879i	$0.876997\pi$
$812$	0	0
$813$	−3.76997e8	−0.701563
$814$	0	0
$815$	5.99389e7i	0.110723i
$816$	0	0
$817$	4.27096e8	0.783177
$818$	0	0
$819$	− 4.39284e7i	− 0.0799638i
$820$	0	0
$821$	−6.26365e8	−1.13187	−0.565937	−	0.824448i	$-0.691486\pi$
$821$	−6.26365e8	−1.13187	−0.565937	+	0.824448i	$0.691486\pi$
$822$	0	0
$823$	− 8.23953e8i	− 1.47810i	−0.673652	−	0.739049i	$-0.735275\pi$
$823$	− 8.23953e8i	− 1.47810i	0.673652	−	0.739049i	$-0.264725\pi$
$824$	0	0
$825$	−1.03441e8	−0.184218
$826$	0	0
$827$	− 3.02093e7i	− 0.0534102i	−0.999643	−	0.0267051i	$-0.991498\pi$
$827$	− 3.02093e7i	− 0.0534102i	0.999643	−	0.0267051i	$-0.00850150\pi$
$828$	0	0
$829$	−2.17842e7	−0.0382366	−0.0191183	−	0.999817i	$-0.506086\pi$
$829$	−2.17842e7	−0.0382366	−0.0191183	+	0.999817i	$0.506086\pi$
$830$	0	0
$831$	4.98360e8i	0.868440i
$832$	0	0
$833$	−1.25299e9	−2.16777
$834$	0	0
$835$	1.60645e8i	0.275936i
$836$	0	0
$837$	−5.79461e7	−0.0988208
$838$	0	0
$839$	− 5.51909e8i	− 0.934505i	−0.884124	−	0.467252i	$-0.845244\pi$
$839$	− 5.51909e8i	− 0.934505i	0.884124	−	0.467252i	$-0.154756\pi$
$840$	0	0
$841$	−2.18104e8	−0.366670
$842$	0	0
$843$	− 3.66118e8i	− 0.611136i
$844$	0	0
$845$	−9.42081e7	−0.156141
$846$	0	0
$847$	8.38010e8i	1.37911i
$848$	0	0
$849$	1.20807e8	0.197410
$850$	0	0
$851$	1.96774e8i	0.319286i
$852$	0	0
$853$	4.15243e8	0.669045	0.334523	−	0.942388i	$-0.391425\pi$
$853$	4.15243e8	0.669045	0.334523	+	0.942388i	$0.391425\pi$
$854$	0	0
$855$	2.09005e7i	0.0334395i
$856$	0	0
$857$	−4.44449e8	−0.706121	−0.353061	−	0.935600i	$-0.614859\pi$
$857$	−4.44449e8	−0.706121	−0.353061	+	0.935600i	$0.614859\pi$
$858$	0	0
$859$	4.67604e8i	0.737733i	0.929482	+	0.368866i	$0.120254\pi$
$859$	4.67604e8i	0.737733i	−0.929482	+	0.368866i	$0.879746\pi$
$860$	0	0
$861$	−2.78679e8	−0.436610
$862$	0	0
$863$	5.95476e8i	0.926472i	0.886235	+	0.463236i	$0.153312\pi$
$863$	5.95476e8i	0.926472i	−0.886235	+	0.463236i	$0.846688\pi$
$864$	0	0
$865$	4.85952e7	0.0750835
$866$	0	0
$867$	5.43830e8i	0.834461i
$868$	0	0
$869$	−2.49192e8	−0.379730
$870$	0	0
$871$	− 1.69503e8i	− 0.256521i
$872$	0	0
$873$	−6.07874e6	−0.00913631
$874$	0	0
$875$	− 3.26917e8i	− 0.487993i
$876$	0	0
$877$	−7.21425e8	−1.06953	−0.534765	−	0.845001i	$-0.679600\pi$
$877$	−7.21425e8	−1.06953	−0.534765	+	0.845001i	$0.679600\pi$
$878$	0	0
$879$	− 1.79030e8i	− 0.263609i
$880$	0	0
$881$	−6.77300e8	−0.990497	−0.495248	−	0.868751i	$-0.664923\pi$
$881$	−6.77300e8	−0.990497	−0.495248	+	0.868751i	$0.664923\pi$
$882$	0	0
$883$	− 8.41700e8i	− 1.22257i	−0.791409	−	0.611287i	$-0.790652\pi$
$883$	− 8.41700e8i	− 1.22257i	0.791409	−	0.611287i	$-0.209348\pi$
$884$	0	0
$885$	6.22157e7	0.0897573
$886$	0	0
$887$	− 2.23932e8i	− 0.320882i	−0.987045	−	0.160441i	$-0.948708\pi$
$887$	− 2.23932e8i	− 0.320882i	0.987045	−	0.160441i	$-0.0512917\pi$
$888$	0	0
$889$	1.71191e9	2.43655
$890$	0	0
$891$	− 2.57363e7i	− 0.0363842i
$892$	0	0
$893$	7.62329e7	0.107050
$894$	0	0
$895$	− 1.06972e8i	− 0.149212i
$896$	0	0
$897$	1.68879e7	0.0233991
$898$	0	0
$899$	2.96910e8i	0.408644i
$900$	0	0
$901$	−1.72405e9	−2.35709
$902$	0	0
$903$	− 8.20277e8i	− 1.11403i
$904$	0	0
$905$	−1.82214e8	−0.245831
$906$	0	0
$907$	6.79727e8i	0.910988i	0.890239	+	0.455494i	$0.150537\pi$
$907$	6.79727e8i	0.910988i	−0.890239	+	0.455494i	$0.849463\pi$
$908$	0	0
$909$	−3.91076e8	−0.520679
$910$	0	0
$911$	9.24099e8i	1.22226i	0.791530	+	0.611130i	$0.209285\pi$
$911$	9.24099e8i	1.22226i	−0.791530	+	0.611130i	$0.790715\pi$
$912$	0	0
$913$	1.41417e8	0.185819
$914$	0	0
$915$	− 1.42390e7i	− 0.0185873i
$916$	0	0
$917$	9.54361e8	1.23767
$918$	0	0
$919$	3.27998e8i	0.422595i	0.977422	+	0.211297i	$0.0677689\pi$
$919$	3.27998e8i	0.422595i	−0.977422	+	0.211297i	$0.932231\pi$
$920$	0	0
$921$	3.59410e8	0.460056
$922$	0	0
$923$	1.54225e8i	0.196133i
$924$	0	0
$925$	9.43492e8	1.19210
$926$	0	0
$927$	− 1.57249e8i	− 0.197401i
$928$	0	0
$929$	−8.75671e8	−1.09218	−0.546090	−	0.837727i	$-0.683884\pi$
$929$	−8.75671e8	−1.09218	−0.546090	+	0.837727i	$0.683884\pi$
$930$	0	0
$931$	7.01381e8i	0.869171i
$932$	0	0
$933$	6.61412e8	0.814381
$934$	0	0
$935$	− 6.69698e7i	− 0.0819303i
$936$	0	0
$937$	1.46253e8	0.177782	0.0888908	−	0.996041i	$-0.471668\pi$
$937$	1.46253e8	0.177782	0.0888908	+	0.996041i	$0.471668\pi$
$938$	0	0
$939$	3.14775e8i	0.380193i
$940$	0	0
$941$	−9.23237e8	−1.10801	−0.554006	−	0.832513i	$-0.686901\pi$
$941$	−9.23237e8	−1.10801	−0.554006	+	0.832513i	$0.686901\pi$
$942$	0	0
$943$	− 1.07136e8i	− 0.127761i
$944$	0	0
$945$	4.01414e7	0.0475660
$946$	0	0
$947$	− 4.29334e8i	− 0.505529i	−0.967528	−	0.252764i	$-0.918660\pi$
$947$	− 4.29334e8i	− 0.505529i	0.967528	−	0.252764i	$-0.0813397\pi$
$948$	0	0
$949$	−1.34545e8	−0.157423
$950$	0	0
$951$	− 9.09733e7i	− 0.105772i
$952$	0	0
$953$	−9.16653e8	−1.05907	−0.529537	−	0.848287i	$-0.677634\pi$
$953$	−9.16653e8	−1.05907	−0.529537	+	0.848287i	$0.677634\pi$
$954$	0	0
$955$	− 2.38126e7i	− 0.0273399i
$956$	0	0
$957$	−1.31870e8	−0.150456
$958$	0	0
$959$	− 2.66845e7i	− 0.0302554i
$960$	0	0
$961$	6.53496e8	0.736330
$962$	0	0
$963$	− 5.66291e7i	− 0.0634104i
$964$	0	0
$965$	4.15577e7	0.0462455
$966$	0	0
$967$	− 1.74117e9i	− 1.92558i	−0.270247	−	0.962791i	$-0.587105\pi$
$967$	− 1.74117e9i	− 1.92558i	0.270247	−	0.962791i	$-0.412895\pi$
$968$	0	0
$969$	5.15039e8	0.566068
$970$	0	0
$971$	− 5.75119e8i	− 0.628203i	−0.949389	−	0.314102i	$-0.898297\pi$
$971$	− 5.75119e8i	− 0.628203i	0.949389	−	0.314102i	$-0.101703\pi$
$972$	0	0
$973$	2.24528e9	2.43743
$974$	0	0
$975$	− 8.09741e7i	− 0.0873639i
$976$	0	0
$977$	1.49329e8	0.160125	0.0800625	−	0.996790i	$-0.474488\pi$
$977$	1.49329e8	0.160125	0.0800625	+	0.996790i	$0.474488\pi$
$978$	0	0
$979$	3.30636e8i	0.352373i
$980$	0	0
$981$	−1.27781e8	−0.135351
$982$	0	0
$983$	6.57661e8i	0.692376i	0.938165	+	0.346188i	$0.112524\pi$
$983$	6.57661e8i	0.692376i	−0.938165	+	0.346188i	$0.887476\pi$
$984$	0	0
$985$	4.39400e7	0.0459782
$986$	0	0
$987$	− 1.46412e8i	− 0.152274i
$988$	0	0
$989$	3.15349e8	0.325989
$990$	0	0
$991$	1.37005e9i	1.40771i	0.710341	+	0.703857i	$0.248541\pi$
$991$	1.37005e9i	1.40771i	−0.710341	+	0.703857i	$0.751459\pi$
$992$	0	0
$993$	1.08278e9	1.10584
$994$	0	0
$995$	1.24918e8i	0.126810i
$996$	0	0
$997$	−1.05005e9	−1.05956	−0.529780	−	0.848135i	$-0.677726\pi$
$997$	−1.05005e9	−1.05956	−0.529780	+	0.848135i	$0.677726\pi$
$998$	0	0
$999$	2.34742e8i	0.235447i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.7.g.e.511.4		4
4.3	odd	2	inner	768.7.g.e.511.1		4
8.3	odd	2		768.7.g.c.511.3		4
8.5	even	2		768.7.g.c.511.2		4
16.3	odd	4		384.7.b.d.319.6	yes	8
16.5	even	4		384.7.b.d.319.7	yes	8
16.11	odd	4		384.7.b.d.319.4	yes	8
16.13	even	4		384.7.b.d.319.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.7.b.d.319.1	✓	8	16.13	even	4
384.7.b.d.319.4	yes	8	16.11	odd	4
384.7.b.d.319.6	yes	8	16.3	odd	4
384.7.b.d.319.7	yes	8	16.5	even	4
768.7.g.c.511.2		4	8.5	even	2
768.7.g.c.511.3		4	8.3	odd	2
768.7.g.e.511.1		4	4.3	odd	2	inner
768.7.g.e.511.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15.5885i q^{3} +20.0000 q^{5} +529.850i q^{7} -243.000 q^{9} +O(q^{10})\)	\(q+15.5885i q^{3} +20.0000 q^{5} +529.850i q^{7} -243.000 q^{9} -435.847i q^{11} +341.182 q^{13} +311.769i q^{15} +7682.73 q^{17} -4300.52i q^{19} -8259.54 q^{21} -3175.32i q^{23} -15225.0 q^{25} -3788.00i q^{27} -19409.3 q^{29} -15297.3i q^{31} +6794.18 q^{33} +10597.0i q^{35} -61969.9 q^{37} +5318.50i q^{39} +33740.2 q^{41} +99312.6i q^{43} -4860.00 q^{45} +17726.4i q^{47} -163092. q^{49} +119762. i q^{51} -224406. q^{53} -8716.94i q^{55} +67038.5 q^{57} -199557. i q^{59} -45671.7 q^{61} -128754. i q^{63} +6823.63 q^{65} -496811. i q^{67} +49498.3 q^{69} +452032. i q^{71} -394349. q^{73} -237334. i q^{75} +230933. q^{77} -571742. i q^{79} +59049.0 q^{81} +324465. i q^{83} +153655. q^{85} -302561. i q^{87} -758607. q^{89} +180775. i q^{91} +238462. q^{93} -86010.5i q^{95} +25015.4 q^{97} +105911. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 80 q^{5} - 972 q^{9}+O(q^{10}) \) 4 * q + 80 * q^5 - 972 * q^9	\( 4 q + 80 q^{5} - 972 q^{9} - 5760 q^{13} + 2232 q^{17} - 11664 q^{21} - 60900 q^{25} + 50608 q^{29} - 58320 q^{33} - 212256 q^{37} + 49464 q^{41} - 19440 q^{45} - 139388 q^{49} - 256400 q^{53} + 11664 q^{57} + 80928 q^{61} - 115200 q^{65} - 443232 q^{69} - 38456 q^{73} + 1180224 q^{77} + 236196 q^{81} + 44640 q^{85} - 3319416 q^{89} - 478224 q^{93} - 2464840 q^{97}+O(q^{100}) \) 4 * q + 80 * q^5 - 972 * q^9 - 5760 * q^13 + 2232 * q^17 - 11664 * q^21 - 60900 * q^25 + 50608 * q^29 - 58320 * q^33 - 212256 * q^37 + 49464 * q^41 - 19440 * q^45 - 139388 * q^49 - 256400 * q^53 + 11664 * q^57 + 80928 * q^61 - 115200 * q^65 - 443232 * q^69 - 38456 * q^73 + 1180224 * q^77 + 236196 * q^81 + 44640 * q^85 - 3319416 * q^89 - 478224 * q^93 - 2464840 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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