LMFDB - Embedded newform 768.7.g.g.511.6

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 53x^{6} - 2x^{5} + 2532x^{4} - 772x^{3} - 31349x^{2} - 33880x + 366025 $ x^8 - 2x^7 - 53x^6 - 2x^5 + 2532x^4 - 772x^3 - 31349x^2 - 33880*x + 366025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.6
Root			$3.53301 - 1.46244i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.7.g.g.511.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.5885i q^{3} -85.3891 q^{5} -511.824i q^{7} -243.000 q^{9} +O(q^{10})$	$q+15.5885i q^{3} -85.3891 q^{5} -511.824i q^{7} -243.000 q^{9} -1212.89i q^{11} -2744.91 q^{13} -1331.08i q^{15} -8524.35 q^{17} -10429.2i q^{19} +7978.55 q^{21} +4300.69i q^{23} -8333.71 q^{25} -3788.00i q^{27} -42442.0 q^{29} -57843.0i q^{31} +18907.1 q^{33} +43704.2i q^{35} -83505.3 q^{37} -42788.9i q^{39} +70152.4 q^{41} +28701.0i q^{43} +20749.5 q^{45} +58746.6i q^{47} -144315. q^{49} -132882. i q^{51} -111109. q^{53} +103567. i q^{55} +162575. q^{57} +118257. i q^{59} +53174.3 q^{61} +124373. i q^{63} +234385. q^{65} +42166.1i q^{67} -67041.1 q^{69} -351599. i q^{71} +91202.9 q^{73} -129910. i q^{75} -620786. q^{77} -775208. i q^{79} +59049.0 q^{81} -669966. i q^{83} +727887. q^{85} -661605. i q^{87} +504715. q^{89} +1.40491e6i q^{91} +901683. q^{93} +890541. i q^{95} +1.03116e6 q^{97} +294732. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 1944 q^{9}+O(q^{10}) $ 8 * q - 1944 * q^9	$ 8 q - 1944 q^{9} - 6544 q^{17} + 56632 q^{25} - 33696 q^{33} + 499568 q^{41} - 414712 q^{49} + 375840 q^{57} - 36096 q^{65} + 1962640 q^{73} + 472392 q^{81} + 1694992 q^{89} + 7632752 q^{97}+O(q^{100}) $ 8 * q - 1944 * q^9 - 6544 * q^17 + 56632 * q^25 - 33696 * q^33 + 499568 * q^41 - 414712 * q^49 + 375840 * q^57 - 36096 * q^65 + 1962640 * q^73 + 472392 * q^81 + 1694992 * q^89 + 7632752 * 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
$5$	−85.3891	−0.683112	−0.341556	−	0.939861i	$-0.610954\pi$
$5$	−85.3891	−0.683112	−0.341556	+	0.939861i	$0.610954\pi$
$6$	0	0
$7$	− 511.824i	− 1.49220i	−0.665834	−	0.746100i	$-0.731924\pi$
$7$	− 511.824i	− 1.49220i	0.665834	−	0.746100i	$-0.268076\pi$
$8$	0	0
$9$	−243.000	−0.333333
$10$	0	0
$11$	− 1212.89i	− 0.911261i	−0.890169	−	0.455631i	$-0.849414\pi$
$11$	− 1212.89i	− 0.911261i	0.890169	−	0.455631i	$-0.150586\pi$
$12$	0	0
$13$	−2744.91	−1.24939	−0.624694	−	0.780869i	$-0.714776\pi$
$13$	−2744.91	−1.24939	−0.624694	+	0.780869i	$0.714776\pi$
$14$	0	0
$15$	− 1331.08i	− 0.394395i
$16$	0	0
$17$	−8524.35	−1.73506	−0.867531	−	0.497384i	$-0.834294\pi$
$17$	−8524.35	−1.73506	−0.867531	+	0.497384i	$0.834294\pi$
$18$	0	0
$19$	− 10429.2i	− 1.52051i	−0.649622	−	0.760257i	$-0.725073\pi$
$19$	− 10429.2i	− 1.52051i	0.649622	−	0.760257i	$-0.274927\pi$
$20$	0	0
$21$	7978.55	0.861522
$22$	0	0
$23$	4300.69i	0.353471i	0.984258	+	0.176736i	$0.0565538\pi$
$23$	4300.69i	0.353471i	−0.984258	+	0.176736i	$0.943446\pi$
$24$	0	0
$25$	−8333.71	−0.533357
$26$	0	0
$27$	− 3788.00i	− 0.192450i
$28$	0	0
$29$	−42442.0	−1.74021	−0.870105	−	0.492866i	$-0.835949\pi$
$29$	−42442.0	−1.74021	−0.870105	+	0.492866i	$0.835949\pi$
$30$	0	0
$31$	− 57843.0i	− 1.94163i	−0.239839	−	0.970813i	$-0.577095\pi$
$31$	− 57843.0i	− 1.94163i	0.239839	−	0.970813i	$-0.422905\pi$
$32$	0	0
$33$	18907.1	0.526117
$34$	0	0
$35$	43704.2i	1.01934i
$36$	0	0
$37$	−83505.3	−1.64858	−0.824288	−	0.566171i	$-0.808424\pi$
$37$	−83505.3	−1.64858	−0.824288	+	0.566171i	$0.808424\pi$
$38$	0	0
$39$	− 42788.9i	− 0.721335i
$40$	0	0
$41$	70152.4	1.01787	0.508933	−	0.860806i	$-0.330040\pi$
$41$	70152.4	1.01787	0.508933	+	0.860806i	$0.330040\pi$
$42$	0	0
$43$	28701.0i	0.360987i	0.983576	+	0.180494i	$0.0577695\pi$
$43$	28701.0i	0.360987i	−0.983576	+	0.180494i	$0.942230\pi$
$44$	0	0
$45$	20749.5	0.227704
$46$	0	0
$47$	58746.6i	0.565834i	0.959144	+	0.282917i	$0.0913021\pi$
$47$	58746.6i	0.565834i	−0.959144	+	0.282917i	$0.908698\pi$
$48$	0	0
$49$	−144315.	−1.22666
$50$	0	0
$51$	− 132882.i	− 1.00174i
$52$	0	0
$53$	−111109.	−0.746316	−0.373158	−	0.927768i	$-0.621725\pi$
$53$	−111109.	−0.746316	−0.373158	+	0.927768i	$0.621725\pi$
$54$	0	0
$55$	103567.i	0.622494i
$56$	0	0
$57$	162575.	0.877870
$58$	0	0
$59$	118257.i	0.575798i	0.957661	+	0.287899i	$0.0929568\pi$
$59$	118257.i	0.575798i	−0.957661	+	0.287899i	$0.907043\pi$
$60$	0	0
$61$	53174.3	0.234268	0.117134	−	0.993116i	$-0.462629\pi$
$61$	53174.3	0.234268	0.117134	+	0.993116i	$0.462629\pi$
$62$	0	0
$63$	124373.i	0.497400i
$64$	0	0
$65$	234385.	0.853473
$66$	0	0
$67$	42166.1i	0.140197i	0.997540	+	0.0700986i	$0.0223314\pi$
$67$	42166.1i	0.140197i	−0.997540	+	0.0700986i	$0.977669\pi$
$68$	0	0
$69$	−67041.1	−0.204077
$70$	0	0
$71$	− 351599.i	− 0.982363i	−0.871057	−	0.491182i	$-0.836565\pi$
$71$	− 351599.i	− 0.982363i	0.871057	−	0.491182i	$-0.163435\pi$
$72$	0	0
$73$	91202.9	0.234445	0.117222	−	0.993106i	$-0.462601\pi$
$73$	91202.9	0.234445	0.117222	+	0.993106i	$0.462601\pi$
$74$	0	0
$75$	− 129910.i	− 0.307934i
$76$	0	0
$77$	−620786.	−1.35978
$78$	0	0
$79$	− 775208.i	− 1.57231i	−0.618032	−	0.786153i	$-0.712070\pi$
$79$	− 775208.i	− 1.57231i	0.618032	−	0.786153i	$-0.287930\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	− 669966.i	− 1.17170i	−0.810418	−	0.585852i	$-0.800760\pi$
$83$	− 669966.i	− 1.17170i	0.810418	−	0.585852i	$-0.199240\pi$
$84$	0	0
$85$	727887.	1.18524
$86$	0	0
$87$	− 661605.i	− 1.00471i
$88$	0	0
$89$	504715.	0.715940	0.357970	−	0.933733i	$-0.383469\pi$
$89$	504715.	0.715940	0.357970	+	0.933733i	$0.383469\pi$
$90$	0	0
$91$	1.40491e6i	1.86434i
$92$	0	0
$93$	901683.	1.12100
$94$	0	0
$95$	890541.i	1.03868i
$96$	0	0
$97$	1.03116e6	1.12982	0.564911	−	0.825152i	$-0.308911\pi$
$97$	1.03116e6	1.12982	0.564911	+	0.825152i	$0.308911\pi$
$98$	0	0
$99$	294732.i	0.303754i
$100$	0	0
$101$	927592.	0.900312	0.450156	−	0.892950i	$-0.351368\pi$
$101$	927592.	0.900312	0.450156	+	0.892950i	$0.351368\pi$
$102$	0	0
$103$	− 357299.i	− 0.326979i	−0.986545	−	0.163489i	$-0.947725\pi$
$103$	− 357299.i	− 0.326979i	0.986545	−	0.163489i	$-0.0522750\pi$
$104$	0	0
$105$	−681281.	−0.588516
$106$	0	0
$107$	1.16196e6i	0.948508i	0.880388	+	0.474254i	$0.157282\pi$
$107$	1.16196e6i	0.948508i	−0.880388	+	0.474254i	$0.842718\pi$
$108$	0	0
$109$	741394.	0.572492	0.286246	−	0.958156i	$-0.407593\pi$
$109$	741394.	0.572492	0.286246	+	0.958156i	$0.407593\pi$
$110$	0	0
$111$	− 1.30172e6i	− 0.951805i
$112$	0	0
$113$	−190737.	−0.132190	−0.0660952	−	0.997813i	$-0.521054\pi$
$113$	−190737.	−0.132190	−0.0660952	+	0.997813i	$0.521054\pi$
$114$	0	0
$115$	− 367231.i	− 0.241461i
$116$	0	0
$117$	667012.	0.416463
$118$	0	0
$119$	4.36297e6i	2.58906i
$120$	0	0
$121$	300462.	0.169603
$122$	0	0
$123$	1.09357e6i	0.587665i
$124$	0	0
$125$	2.04581e6	1.04746
$126$	0	0
$127$	819411.i	0.400028i	0.979793	+	0.200014i	$0.0640988\pi$
$127$	819411.i	0.400028i	−0.979793	+	0.200014i	$0.935901\pi$
$128$	0	0
$129$	−447405.	−0.208416
$130$	0	0
$131$	− 3.76018e6i	− 1.67261i	−0.548263	−	0.836306i	$-0.684711\pi$
$131$	− 3.76018e6i	− 1.67261i	0.548263	−	0.836306i	$-0.315289\pi$
$132$	0	0
$133$	−5.33793e6	−2.26891
$134$	0	0
$135$	323453.i	0.131465i
$136$	0	0
$137$	−1.86568e6	−0.725563	−0.362782	−	0.931874i	$-0.618173\pi$
$137$	−1.86568e6	−0.725563	−0.362782	+	0.931874i	$0.618173\pi$
$138$	0	0
$139$	3.37311e6i	1.25599i	0.778218	+	0.627995i	$0.216124\pi$
$139$	3.37311e6i	1.25599i	−0.778218	+	0.627995i	$0.783876\pi$
$140$	0	0
$141$	−915768.	−0.326684
$142$	0	0
$143$	3.32927e6i	1.13852i
$144$	0	0
$145$	3.62408e6	1.18876
$146$	0	0
$147$	− 2.24965e6i	− 0.708212i
$148$	0	0
$149$	3.35713e6	1.01487	0.507434	−	0.861691i	$-0.330594\pi$
$149$	3.35713e6	1.01487	0.507434	+	0.861691i	$0.330594\pi$
$150$	0	0
$151$	− 3.78813e6i	− 1.10026i	−0.835081	−	0.550128i	$-0.814579\pi$
$151$	− 3.78813e6i	− 1.10026i	0.835081	−	0.550128i	$-0.185421\pi$
$152$	0	0
$153$	2.07142e6	0.578354
$154$	0	0
$155$	4.93916e6i	1.32635i
$156$	0	0
$157$	−5.81264e6	−1.50202	−0.751008	−	0.660293i	$-0.770432\pi$
$157$	−5.81264e6	−1.50202	−0.751008	+	0.660293i	$0.770432\pi$
$158$	0	0
$159$	− 1.73202e6i	− 0.430886i
$160$	0	0
$161$	2.20120e6	0.527450
$162$	0	0
$163$	− 5.12374e6i	− 1.18311i	−0.806266	−	0.591554i	$-0.798515\pi$
$163$	− 5.12374e6i	− 1.18311i	0.806266	−	0.591554i	$-0.201485\pi$
$164$	0	0
$165$	−1.61446e6	−0.359397
$166$	0	0
$167$	1.27729e6i	0.274247i	0.990554	+	0.137123i	$0.0437857\pi$
$167$	1.27729e6i	0.274247i	−0.990554	+	0.137123i	$0.956214\pi$
$168$	0	0
$169$	2.70770e6	0.560972
$170$	0	0
$171$	2.53430e6i	0.506838i
$172$	0	0
$173$	−2.87446e6	−0.555160	−0.277580	−	0.960703i	$-0.589532\pi$
$173$	−2.87446e6	−0.555160	−0.277580	+	0.960703i	$0.589532\pi$
$174$	0	0
$175$	4.26540e6i	0.795876i
$176$	0	0
$177$	−1.84344e6	−0.332437
$178$	0	0
$179$	8.99970e6i	1.56917i	0.620024	+	0.784583i	$0.287123\pi$
$179$	8.99970e6i	1.56917i	−0.620024	+	0.784583i	$0.712877\pi$
$180$	0	0
$181$	3.33395e6	0.562242	0.281121	−	0.959672i	$-0.409294\pi$
$181$	3.33395e6	0.562242	0.281121	+	0.959672i	$0.409294\pi$
$182$	0	0
$183$	828905.i	0.135254i
$184$	0	0
$185$	7.13044e6	1.12616
$186$	0	0
$187$	1.03391e7i	1.58109i
$188$	0	0
$189$	−1.93879e6	−0.287174
$190$	0	0
$191$	− 3.66531e6i	− 0.526030i	−0.964792	−	0.263015i	$-0.915283\pi$
$191$	− 3.66531e6i	− 0.526030i	0.964792	−	0.263015i	$-0.0847168\pi$
$192$	0	0
$193$	−8.58154e6	−1.19369	−0.596847	−	0.802355i	$-0.703580\pi$
$193$	−8.58154e6	−1.19369	−0.596847	+	0.802355i	$0.703580\pi$
$194$	0	0
$195$	3.65370e6i	0.492753i
$196$	0	0
$197$	−1.04264e7	−1.36375	−0.681874	−	0.731470i	$-0.738835\pi$
$197$	−1.04264e7	−1.36375	−0.681874	+	0.731470i	$0.738835\pi$
$198$	0	0
$199$	− 4.34498e6i	− 0.551352i	−0.961251	−	0.275676i	$-0.911098\pi$
$199$	− 4.34498e6i	− 0.551352i	0.961251	−	0.275676i	$-0.0889017\pi$
$200$	0	0
$201$	−657305.	−0.0809429
$202$	0	0
$203$	2.17228e7i	2.59674i
$204$	0	0
$205$	−5.99024e6	−0.695317
$206$	0	0
$207$	− 1.04507e6i	− 0.117824i
$208$	0	0
$209$	−1.26495e7	−1.38559
$210$	0	0
$211$	− 5.90463e6i	− 0.628558i	−0.949331	−	0.314279i	$-0.898237\pi$
$211$	− 5.90463e6i	− 0.628558i	0.949331	−	0.314279i	$-0.101763\pi$
$212$	0	0
$213$	5.48088e6	0.567168
$214$	0	0
$215$	− 2.45075e6i	− 0.246595i
$216$	0	0
$217$	−2.96054e7	−2.89729
$218$	0	0
$219$	1.42171e6i	0.135357i
$220$	0	0
$221$	2.33986e7	2.16777
$222$	0	0
$223$	− 1.48155e7i	− 1.33598i	−0.744169	−	0.667992i	$-0.767154\pi$
$223$	− 1.48155e7i	− 1.33598i	0.744169	−	0.667992i	$-0.232846\pi$
$224$	0	0
$225$	2.02509e6	0.177786
$226$	0	0
$227$	− 3.99262e6i	− 0.341334i	−0.985329	−	0.170667i	$-0.945408\pi$
$227$	− 3.99262e6i	− 0.341334i	0.985329	−	0.170667i	$-0.0545923\pi$
$228$	0	0
$229$	−3.71868e6	−0.309658	−0.154829	−	0.987941i	$-0.549483\pi$
$229$	−3.71868e6	−0.309658	−0.154829	+	0.987941i	$0.549483\pi$
$230$	0	0
$231$	− 9.67710e6i	− 0.785071i
$232$	0	0
$233$	5.25478e6	0.415419	0.207710	−	0.978191i	$-0.433399\pi$
$233$	5.25478e6	0.415419	0.207710	+	0.978191i	$0.433399\pi$
$234$	0	0
$235$	− 5.01631e6i	− 0.386528i
$236$	0	0
$237$	1.20843e7	0.907771
$238$	0	0
$239$	7.96267e6i	0.583264i	0.956531	+	0.291632i	$0.0941982\pi$
$239$	7.96267e6i	0.583264i	−0.956531	+	0.291632i	$0.905802\pi$
$240$	0	0
$241$	1.44459e6	0.103203	0.0516017	−	0.998668i	$-0.483567\pi$
$241$	1.44459e6	0.103203	0.0516017	+	0.998668i	$0.483567\pi$
$242$	0	0
$243$	920483.i	0.0641500i
$244$	0	0
$245$	1.23229e7	0.837946
$246$	0	0
$247$	2.86272e7i	1.89971i
$248$	0	0
$249$	1.04437e7	0.676484
$250$	0	0
$251$	920323.i	0.0581995i	0.999577	+	0.0290998i	$0.00926405\pi$
$251$	920323.i	0.0581995i	−0.999577	+	0.0290998i	$0.990736\pi$
$252$	0	0
$253$	5.21625e6	0.322105
$254$	0	0
$255$	1.13466e7i	0.684300i
$256$	0	0
$257$	5.73409e6	0.337804	0.168902	−	0.985633i	$-0.445978\pi$
$257$	5.73409e6	0.337804	0.168902	+	0.985633i	$0.445978\pi$
$258$	0	0
$259$	4.27400e7i	2.46000i
$260$	0	0
$261$	1.03134e7	0.580070
$262$	0	0
$263$	− 1.36973e7i	− 0.752952i	−0.926426	−	0.376476i	$-0.877136\pi$
$263$	− 1.36973e7i	− 0.752952i	0.926426	−	0.376476i	$-0.122864\pi$
$264$	0	0
$265$	9.48752e6	0.509818
$266$	0	0
$267$	7.86774e6i	0.413348i
$268$	0	0
$269$	−1.95267e7	−1.00317	−0.501583	−	0.865110i	$-0.667249\pi$
$269$	−1.95267e7	−1.00317	−0.501583	+	0.865110i	$0.667249\pi$
$270$	0	0
$271$	− 8.79492e6i	− 0.441900i	−0.975285	−	0.220950i	$-0.929084\pi$
$271$	− 8.79492e6i	− 0.441900i	0.975285	−	0.220950i	$-0.0709158\pi$
$272$	0	0
$273$	−2.19004e7	−1.07638
$274$	0	0
$275$	1.01079e7i	0.486028i
$276$	0	0
$277$	−1.32017e6	−0.0621142	−0.0310571	−	0.999518i	$-0.509887\pi$
$277$	−1.32017e6	−0.0621142	−0.0310571	+	0.999518i	$0.509887\pi$
$278$	0	0
$279$	1.40558e7i	0.647208i
$280$	0	0
$281$	−1.77259e7	−0.798893	−0.399446	−	0.916757i	$-0.630798\pi$
$281$	−1.77259e7	−0.798893	−0.399446	+	0.916757i	$0.630798\pi$
$282$	0	0
$283$	4.14391e7i	1.82832i	0.405358	+	0.914158i	$0.367147\pi$
$283$	4.14391e7i	1.82832i	−0.405358	+	0.914158i	$0.632853\pi$
$284$	0	0
$285$	−1.38822e7	−0.599684
$286$	0	0
$287$	− 3.59057e7i	− 1.51886i
$288$	0	0
$289$	4.85271e7	2.01044
$290$	0	0
$291$	1.60742e7i	0.652303i
$292$	0	0
$293$	4.93196e6	0.196072	0.0980362	−	0.995183i	$-0.468744\pi$
$293$	4.93196e6	0.196072	0.0980362	+	0.995183i	$0.468744\pi$
$294$	0	0
$295$	− 1.00978e7i	− 0.393335i
$296$	0	0
$297$	−4.59442e6	−0.175372
$298$	0	0
$299$	− 1.18050e7i	− 0.441623i
$300$	0	0
$301$	1.46899e7	0.538665
$302$	0	0
$303$	1.44597e7i	0.519795i
$304$	0	0
$305$	−4.54050e6	−0.160031
$306$	0	0
$307$	4.35700e7i	1.50582i	0.658124	+	0.752910i	$0.271350\pi$
$307$	4.35700e7i	1.50582i	−0.658124	+	0.752910i	$0.728650\pi$
$308$	0	0
$309$	5.56974e6	0.188781
$310$	0	0
$311$	− 5.31388e7i	− 1.76657i	−0.468837	−	0.883285i	$-0.655327\pi$
$311$	− 5.31388e7i	− 1.76657i	0.468837	−	0.883285i	$-0.344673\pi$
$312$	0	0
$313$	−3.18652e7	−1.03916	−0.519582	−	0.854421i	$-0.673912\pi$
$313$	−3.18652e7	−1.03916	−0.519582	+	0.854421i	$0.673912\pi$
$314$	0	0
$315$	− 1.06201e7i	− 0.339780i
$316$	0	0
$317$	−3.33162e7	−1.04587	−0.522936	−	0.852372i	$-0.675163\pi$
$317$	−3.33162e7	−1.04587	−0.522936	+	0.852372i	$0.675163\pi$
$318$	0	0
$319$	5.14774e7i	1.58579i
$320$	0	0
$321$	−1.81132e7	−0.547621
$322$	0	0
$323$	8.89023e7i	2.63819i
$324$	0	0
$325$	2.28753e7	0.666371
$326$	0	0
$327$	1.15572e7i	0.330528i
$328$	0	0
$329$	3.00679e7	0.844337
$330$	0	0
$331$	2.35633e6i	0.0649758i	0.999472	+	0.0324879i	$0.0103430\pi$
$331$	2.35633e6i	0.0649758i	−0.999472	+	0.0324879i	$0.989657\pi$
$332$	0	0
$333$	2.02918e7	0.549525
$334$	0	0
$335$	− 3.60053e6i	− 0.0957705i
$336$	0	0
$337$	3.61254e7	0.943894	0.471947	−	0.881627i	$-0.343551\pi$
$337$	3.61254e7	0.943894	0.471947	+	0.881627i	$0.343551\pi$
$338$	0	0
$339$	− 2.97330e6i	− 0.0763202i
$340$	0	0
$341$	−7.01571e7	−1.76933
$342$	0	0
$343$	1.36484e7i	0.338221i
$344$	0	0
$345$	5.72457e6	0.139407
$346$	0	0
$347$	− 1.76051e7i	− 0.421357i	−0.977555	−	0.210679i	$-0.932433\pi$
$347$	− 1.76051e7i	− 0.421357i	0.977555	−	0.210679i	$-0.0675674\pi$
$348$	0	0
$349$	−3.02782e6	−0.0712285	−0.0356143	−	0.999366i	$-0.511339\pi$
$349$	−3.02782e6	−0.0712285	−0.0356143	+	0.999366i	$0.511339\pi$
$350$	0	0
$351$	1.03977e7i	0.240445i
$352$	0	0
$353$	−3.74661e7	−0.851755	−0.425877	−	0.904781i	$-0.640034\pi$
$353$	−3.74661e7	−0.851755	−0.425877	+	0.904781i	$0.640034\pi$
$354$	0	0
$355$	3.00227e7i	0.671064i
$356$	0	0
$357$	−6.80120e7	−1.49479
$358$	0	0
$359$	5.14739e7i	1.11251i	0.831012	+	0.556254i	$0.187762\pi$
$359$	5.14739e7i	1.11251i	−0.831012	+	0.556254i	$0.812238\pi$
$360$	0	0
$361$	−6.17226e7	−1.31197
$362$	0	0
$363$	4.68374e6i	0.0979203i
$364$	0	0
$365$	−7.78773e6	−0.160152
$366$	0	0
$367$	1.52811e7i	0.309142i	0.987982	+	0.154571i	$0.0493995\pi$
$367$	1.52811e7i	0.309142i	−0.987982	+	0.154571i	$0.950601\pi$
$368$	0	0
$369$	−1.70470e7	−0.339289
$370$	0	0
$371$	5.68684e7i	1.11365i
$372$	0	0
$373$	−6.19317e7	−1.19340	−0.596701	−	0.802464i	$-0.703522\pi$
$373$	−6.19317e7	−1.19340	−0.596701	+	0.802464i	$0.703522\pi$
$374$	0	0
$375$	3.18910e7i	0.604749i
$376$	0	0
$377$	1.16499e8	2.17420
$378$	0	0
$379$	− 3.43446e7i	− 0.630870i	−0.948947	−	0.315435i	$-0.897849\pi$
$379$	− 3.43446e7i	− 0.630870i	0.948947	−	0.315435i	$-0.102151\pi$
$380$	0	0
$381$	−1.27733e7	−0.230956
$382$	0	0
$383$	− 1.34734e7i	− 0.239817i	−0.992785	−	0.119908i	$-0.961740\pi$
$383$	− 1.34734e7i	− 0.239817i	0.992785	−	0.119908i	$-0.0382601\pi$
$384$	0	0
$385$	5.30083e7	0.928885
$386$	0	0
$387$	− 6.97435e6i	− 0.120329i
$388$	0	0
$389$	1.13679e8	1.93122	0.965610	−	0.259993i	$-0.0837203\pi$
$389$	1.13679e8	1.93122	0.965610	+	0.259993i	$0.0837203\pi$
$390$	0	0
$391$	− 3.66606e7i	− 0.613294i
$392$	0	0
$393$	5.86155e7	0.965683
$394$	0	0
$395$	6.61943e7i	1.07406i
$396$	0	0
$397$	−3.96900e6	−0.0634322	−0.0317161	−	0.999497i	$-0.510097\pi$
$397$	−3.96900e6	−0.0634322	−0.0317161	+	0.999497i	$0.510097\pi$
$398$	0	0
$399$	− 8.32100e7i	− 1.30996i
$400$	0	0
$401$	4.54308e6	0.0704558	0.0352279	−	0.999379i	$-0.488784\pi$
$401$	4.54308e6	0.0704558	0.0352279	+	0.999379i	$0.488784\pi$
$402$	0	0
$403$	1.58774e8i	2.42584i
$404$	0	0
$405$	−5.04214e6	−0.0759014
$406$	0	0
$407$	1.01283e8i	1.50228i
$408$	0	0
$409$	−4.23651e7	−0.619211	−0.309606	−	0.950865i	$-0.600197\pi$
$409$	−4.23651e7	−0.619211	−0.309606	+	0.950865i	$0.600197\pi$
$410$	0	0
$411$	− 2.90831e7i	− 0.418904i
$412$	0	0
$413$	6.05267e7	0.859206
$414$	0	0
$415$	5.72077e7i	0.800406i
$416$	0	0
$417$	−5.25816e7	−0.725146
$418$	0	0
$419$	8.16747e7i	1.11031i	0.831746	+	0.555157i	$0.187342\pi$
$419$	8.16747e7i	1.11031i	−0.831746	+	0.555157i	$0.812658\pi$
$420$	0	0
$421$	7.99635e7	1.07163	0.535816	−	0.844335i	$-0.320004\pi$
$421$	7.99635e7	1.07163	0.535816	+	0.844335i	$0.320004\pi$
$422$	0	0
$423$	− 1.42754e7i	− 0.188611i
$424$	0	0
$425$	7.10395e7	0.925408
$426$	0	0
$427$	− 2.72159e7i	− 0.349574i
$428$	0	0
$429$	−5.18981e7	−0.657325
$430$	0	0
$431$	7.45512e7i	0.931157i	0.885007	+	0.465578i	$0.154154\pi$
$431$	7.45512e7i	0.931157i	−0.885007	+	0.465578i	$0.845846\pi$
$432$	0	0
$433$	−5.35595e7	−0.659740	−0.329870	−	0.944026i	$-0.607005\pi$
$433$	−5.35595e7	−0.659740	−0.329870	+	0.944026i	$0.607005\pi$
$434$	0	0
$435$	5.64938e7i	0.686330i
$436$	0	0
$437$	4.48528e7	0.537458
$438$	0	0
$439$	− 3.84937e7i	− 0.454984i	−0.973780	−	0.227492i	$-0.926947\pi$
$439$	− 3.84937e7i	− 0.454984i	0.973780	−	0.227492i	$-0.0730525\pi$
$440$	0	0
$441$	3.50686e7	0.408886
$442$	0	0
$443$	4.20100e7i	0.483216i	0.970374	+	0.241608i	$0.0776749\pi$
$443$	4.20100e7i	0.483216i	−0.970374	+	0.241608i	$0.922325\pi$
$444$	0	0
$445$	−4.30972e7	−0.489067
$446$	0	0
$447$	5.23325e7i	0.585934i
$448$	0	0
$449$	1.46094e8	1.61396	0.806981	−	0.590577i	$-0.201100\pi$
$449$	1.46094e8	1.61396	0.806981	+	0.590577i	$0.201100\pi$
$450$	0	0
$451$	− 8.50870e7i	− 0.927542i
$452$	0	0
$453$	5.90510e7	0.635233
$454$	0	0
$455$	− 1.19964e8i	− 1.27355i
$456$	0	0
$457$	−1.64317e8	−1.72160	−0.860802	−	0.508941i	$-0.830037\pi$
$457$	−1.64317e8	−1.72160	−0.860802	+	0.508941i	$0.830037\pi$
$458$	0	0
$459$	3.22902e7i	0.333913i
$460$	0	0
$461$	1.01392e7	0.103491	0.0517455	−	0.998660i	$-0.483522\pi$
$461$	1.01392e7	0.103491	0.0517455	+	0.998660i	$0.483522\pi$
$462$	0	0
$463$	− 1.91081e8i	− 1.92519i	−0.270937	−	0.962597i	$-0.587333\pi$
$463$	− 1.91081e8i	− 1.92519i	0.270937	−	0.962597i	$-0.412667\pi$
$464$	0	0
$465$	−7.69938e7	−0.765768
$466$	0	0
$467$	− 1.30526e8i	− 1.28159i	−0.767713	−	0.640793i	$-0.778606\pi$
$467$	− 1.30526e8i	− 1.28159i	0.767713	−	0.640793i	$-0.221394\pi$
$468$	0	0
$469$	2.15817e7	0.209202
$470$	0	0
$471$	− 9.06101e7i	− 0.867189i
$472$	0	0
$473$	3.48111e7	0.328954
$474$	0	0
$475$	8.69140e7i	0.810978i
$476$	0	0
$477$	2.69996e7	0.248772
$478$	0	0
$479$	− 1.71231e8i	− 1.55803i	−0.627003	−	0.779017i	$-0.715719\pi$
$479$	− 1.71231e8i	− 1.55803i	0.627003	−	0.779017i	$-0.284281\pi$
$480$	0	0
$481$	2.29214e8	2.05971
$482$	0	0
$483$	3.43132e7i	0.304523i
$484$	0	0
$485$	−8.80496e7	−0.771795
$486$	0	0
$487$	6.38316e7i	0.552648i	0.961065	+	0.276324i	$0.0891163\pi$
$487$	6.38316e7i	0.552648i	−0.961065	+	0.276324i	$0.910884\pi$
$488$	0	0
$489$	7.98712e7	0.683067
$490$	0	0
$491$	− 1.36774e8i	− 1.15547i	−0.816224	−	0.577735i	$-0.803937\pi$
$491$	− 1.36774e8i	− 1.15547i	0.816224	−	0.577735i	$-0.196063\pi$
$492$	0	0
$493$	3.61791e8	3.01937
$494$	0	0
$495$	− 2.51669e7i	− 0.207498i
$496$	0	0
$497$	−1.79957e8	−1.46588
$498$	0	0
$499$	− 1.94124e8i	− 1.56235i	−0.624312	−	0.781175i	$-0.714621\pi$
$499$	− 1.94124e8i	− 1.56235i	0.624312	−	0.781175i	$-0.285379\pi$
$500$	0	0
$501$	−1.99110e7	−0.158336
$502$	0	0
$503$	− 9.34518e7i	− 0.734317i	−0.930158	−	0.367159i	$-0.880331\pi$
$503$	− 9.34518e7i	− 0.734317i	0.930158	−	0.367159i	$-0.119669\pi$
$504$	0	0
$505$	−7.92062e7	−0.615014
$506$	0	0
$507$	4.22089e7i	0.323877i
$508$	0	0
$509$	−6.32104e7	−0.479331	−0.239665	−	0.970856i	$-0.577038\pi$
$509$	−6.32104e7	−0.479331	−0.239665	+	0.970856i	$0.577038\pi$
$510$	0	0
$511$	− 4.66799e7i	− 0.349838i
$512$	0	0
$513$	−3.95058e7	−0.292623
$514$	0	0
$515$	3.05094e7i	0.223363i
$516$	0	0
$517$	7.12530e7	0.515622
$518$	0	0
$519$	− 4.48084e7i	− 0.320522i
$520$	0	0
$521$	2.37718e8	1.68092	0.840462	−	0.541871i	$-0.182284\pi$
$521$	2.37718e8	1.68092	0.840462	+	0.541871i	$0.182284\pi$
$522$	0	0
$523$	− 5.13720e7i	− 0.359105i	−0.983748	−	0.179553i	$-0.942535\pi$
$523$	− 5.13720e7i	− 0.359105i	0.983748	−	0.179553i	$-0.0574650\pi$
$524$	0	0
$525$	−6.64909e7	−0.459499
$526$	0	0
$527$	4.93074e8i	3.36884i
$528$	0	0
$529$	1.29540e8	0.875058
$530$	0	0
$531$	− 2.87364e7i	− 0.191933i
$532$	0	0
$533$	−1.92562e8	−1.27171
$534$	0	0
$535$	− 9.92189e7i	− 0.647938i
$536$	0	0
$537$	−1.40291e8	−0.905958
$538$	0	0
$539$	1.75038e8i	1.11781i
$540$	0	0
$541$	−1.82128e8	−1.15023	−0.575116	−	0.818072i	$-0.695043\pi$
$541$	−1.82128e8	−1.15023	−0.575116	+	0.818072i	$0.695043\pi$
$542$	0	0
$543$	5.19711e7i	0.324610i
$544$	0	0
$545$	−6.33069e7	−0.391076
$546$	0	0
$547$	6.68154e7i	0.408239i	0.978946	+	0.204120i	$0.0654332\pi$
$547$	6.68154e7i	0.408239i	−0.978946	+	0.204120i	$0.934567\pi$
$548$	0	0
$549$	−1.29213e7	−0.0780892
$550$	0	0
$551$	4.42636e8i	2.64602i
$552$	0	0
$553$	−3.96770e8	−2.34619
$554$	0	0
$555$	1.11153e8i	0.650190i
$556$	0	0
$557$	606923.	0.00351211	0.00175606	−	0.999998i	$-0.499441\pi$
$557$	606923.	0.00351211	0.00175606	+	0.999998i	$0.499441\pi$
$558$	0	0
$559$	− 7.87816e7i	− 0.451013i
$560$	0	0
$561$	−1.61171e8	−0.912845
$562$	0	0
$563$	− 2.47309e8i	− 1.38584i	−0.721012	−	0.692922i	$-0.756323\pi$
$563$	− 2.47309e8i	− 1.38584i	0.721012	−	0.692922i	$-0.243677\pi$
$564$	0	0
$565$	1.62869e7	0.0903009
$566$	0	0
$567$	− 3.02227e7i	− 0.165800i
$568$	0	0
$569$	−1.83163e8	−0.994262	−0.497131	−	0.867675i	$-0.665613\pi$
$569$	−1.83163e8	−0.994262	−0.497131	+	0.867675i	$0.665613\pi$
$570$	0	0
$571$	1.30136e8i	0.699021i	0.936933	+	0.349510i	$0.113652\pi$
$571$	1.30136e8i	0.699021i	−0.936933	+	0.349510i	$0.886348\pi$
$572$	0	0
$573$	5.71365e7	0.303703
$574$	0	0
$575$	− 3.58407e7i	− 0.188527i
$576$	0	0
$577$	2.46187e7	0.128156	0.0640778	−	0.997945i	$-0.479589\pi$
$577$	2.46187e7	0.128156	0.0640778	+	0.997945i	$0.479589\pi$
$578$	0	0
$579$	− 1.33773e8i	− 0.689180i
$580$	0	0
$581$	−3.42905e8	−1.74842
$582$	0	0
$583$	1.34763e8i	0.680089i
$584$	0	0
$585$	−5.69556e7	−0.284491
$586$	0	0
$587$	4.56349e7i	0.225623i	0.993616	+	0.112811i	$0.0359856\pi$
$587$	4.56349e7i	0.225623i	−0.993616	+	0.112811i	$0.964014\pi$
$588$	0	0
$589$	−6.03257e8	−2.95227
$590$	0	0
$591$	− 1.62531e8i	− 0.787360i
$592$	0	0
$593$	1.58370e8	0.759466	0.379733	−	0.925096i	$-0.376016\pi$
$593$	1.58370e8	0.759466	0.379733	+	0.925096i	$0.376016\pi$
$594$	0	0
$595$	− 3.72550e8i	− 1.76862i
$596$	0	0
$597$	6.77316e7	0.318323
$598$	0	0
$599$	− 2.34911e8i	− 1.09301i	−0.837457	−	0.546504i	$-0.815958\pi$
$599$	− 2.34911e8i	− 1.09301i	0.837457	−	0.546504i	$-0.184042\pi$
$600$	0	0
$601$	−6.40513e7	−0.295056	−0.147528	−	0.989058i	$-0.547132\pi$
$601$	−6.40513e7	−0.295056	−0.147528	+	0.989058i	$0.547132\pi$
$602$	0	0
$603$	− 1.02464e7i	− 0.0467324i
$604$	0	0
$605$	−2.56562e7	−0.115858
$606$	0	0
$607$	− 6.98586e7i	− 0.312359i	−0.987729	−	0.156179i	$-0.950082\pi$
$607$	− 6.98586e7i	− 0.312359i	0.987729	−	0.156179i	$-0.0499178\pi$
$608$	0	0
$609$	−3.38626e8	−1.49923
$610$	0	0
$611$	− 1.61254e8i	− 0.706946i
$612$	0	0
$613$	2.62634e7	0.114017	0.0570085	−	0.998374i	$-0.481844\pi$
$613$	2.62634e7	0.114017	0.0570085	+	0.998374i	$0.481844\pi$
$614$	0	0
$615$	− 9.33786e7i	− 0.401441i
$616$	0	0
$617$	2.98936e8	1.27269	0.636345	−	0.771404i	$-0.280445\pi$
$617$	2.98936e8	1.27269	0.636345	+	0.771404i	$0.280445\pi$
$618$	0	0
$619$	1.32500e8i	0.558657i	0.960196	+	0.279328i	$0.0901118\pi$
$619$	1.32500e8i	0.558657i	−0.960196	+	0.279328i	$0.909888\pi$
$620$	0	0
$621$	1.62910e7	0.0680256
$622$	0	0
$623$	− 2.58326e8i	− 1.06833i
$624$	0	0
$625$	−4.44757e7	−0.182172
$626$	0	0
$627$	− 1.97186e8i	− 0.799969i
$628$	0	0
$629$	7.11829e8	2.86038
$630$	0	0
$631$	1.19185e8i	0.474390i	0.971462	+	0.237195i	$0.0762280\pi$
$631$	1.19185e8i	0.474390i	−0.971462	+	0.237195i	$0.923772\pi$
$632$	0	0
$633$	9.20440e7	0.362898
$634$	0	0
$635$	− 6.99687e7i	− 0.273264i
$636$	0	0
$637$	3.96132e8	1.53257
$638$	0	0
$639$	8.54385e7i	0.327454i
$640$	0	0
$641$	7.84005e7	0.297677	0.148838	−	0.988862i	$-0.452447\pi$
$641$	7.84005e7	0.297677	0.148838	+	0.988862i	$0.452447\pi$
$642$	0	0
$643$	− 1.05651e8i	− 0.397410i	−0.980059	−	0.198705i	$-0.936326\pi$
$643$	− 1.05651e8i	− 0.397410i	0.980059	−	0.198705i	$-0.0636736\pi$
$644$	0	0
$645$	3.82035e7	0.142372
$646$	0	0
$647$	2.37697e8i	0.877628i	0.898578	+	0.438814i	$0.144601\pi$
$647$	2.37697e8i	0.877628i	−0.898578	+	0.438814i	$0.855399\pi$
$648$	0	0
$649$	1.43432e8	0.524703
$650$	0	0
$651$	− 4.61503e8i	− 1.67275i
$652$	0	0
$653$	−1.71315e7	−0.0615257	−0.0307628	−	0.999527i	$-0.509794\pi$
$653$	−1.71315e7	−0.0615257	−0.0307628	+	0.999527i	$0.509794\pi$
$654$	0	0
$655$	3.21079e8i	1.14258i
$656$	0	0
$657$	−2.21623e7	−0.0781482
$658$	0	0
$659$	− 3.62898e8i	− 1.26803i	−0.773323	−	0.634013i	$-0.781407\pi$
$659$	− 3.62898e8i	− 1.26803i	0.773323	−	0.634013i	$-0.218593\pi$
$660$	0	0
$661$	−4.90659e8	−1.69893	−0.849465	−	0.527645i	$-0.823075\pi$
$661$	−4.90659e8	−1.69893	−0.849465	+	0.527645i	$0.823075\pi$
$662$	0	0
$663$	3.64747e8i	1.25156i
$664$	0	0
$665$	4.55800e8	1.54992
$666$	0	0
$667$	− 1.82530e8i	− 0.615114i
$668$	0	0
$669$	2.30950e8	0.771330
$670$	0	0
$671$	− 6.44945e7i	− 0.213479i
$672$	0	0
$673$	−1.66234e8	−0.545350	−0.272675	−	0.962106i	$-0.587908\pi$
$673$	−1.66234e8	−0.545350	−0.272675	+	0.962106i	$0.587908\pi$
$674$	0	0
$675$	3.15681e7i	0.102645i
$676$	0	0
$677$	−1.98782e8	−0.640635	−0.320318	−	0.947310i	$-0.603790\pi$
$677$	−1.98782e8	−0.640635	−0.320318	+	0.947310i	$0.603790\pi$
$678$	0	0
$679$	− 5.27772e8i	− 1.68592i
$680$	0	0
$681$	6.22387e7	0.197070
$682$	0	0
$683$	− 795875.i	− 0.00249794i	−0.999999	−	0.00124897i	$-0.999602\pi$
$683$	− 795875.i	− 0.00249794i	0.999999	−	0.00124897i	$-0.000397560\pi$
$684$	0	0
$685$	1.59309e8	0.495641
$686$	0	0
$687$	− 5.79684e7i	− 0.178781i
$688$	0	0
$689$	3.04985e8	0.932439
$690$	0	0
$691$	− 6.26115e8i	− 1.89767i	−0.315778	−	0.948833i	$-0.602266\pi$
$691$	− 6.26115e8i	− 1.89767i	0.315778	−	0.948833i	$-0.397734\pi$
$692$	0	0
$693$	1.50851e8	0.453261
$694$	0	0
$695$	− 2.88027e8i	− 0.857982i
$696$	0	0
$697$	−5.98004e8	−1.76606
$698$	0	0
$699$	8.19139e7i	0.239843i
$700$	0	0
$701$	7.92815e7	0.230154	0.115077	−	0.993357i	$-0.463289\pi$
$701$	7.92815e7	0.230154	0.115077	+	0.993357i	$0.463289\pi$
$702$	0	0
$703$	8.70894e8i	2.50668i
$704$	0	0
$705$	7.81966e7	0.223162
$706$	0	0
$707$	− 4.74764e8i	− 1.34344i
$708$	0	0
$709$	−6.27826e7	−0.176157	−0.0880786	−	0.996114i	$-0.528073\pi$
$709$	−6.27826e7	−0.176157	−0.0880786	+	0.996114i	$0.528073\pi$
$710$	0	0
$711$	1.88376e8i	0.524102i
$712$	0	0
$713$	2.48764e8	0.686309
$714$	0	0
$715$	− 2.84283e8i	− 0.777737i
$716$	0	0
$717$	−1.24126e8	−0.336747
$718$	0	0
$719$	− 3.24833e8i	− 0.873924i	−0.899480	−	0.436962i	$-0.856054\pi$
$719$	− 3.24833e8i	− 0.873924i	0.899480	−	0.436962i	$-0.143946\pi$
$720$	0	0
$721$	−1.82874e8	−0.487918
$722$	0	0
$723$	2.25190e7i	0.0595845i
$724$	0	0
$725$	3.53699e8	0.928154
$726$	0	0
$727$	1.75113e8i	0.455737i	0.973692	+	0.227869i	$0.0731757\pi$
$727$	1.75113e8i	0.455737i	−0.973692	+	0.227869i	$0.926824\pi$
$728$	0	0
$729$	−1.43489e7	−0.0370370
$730$	0	0
$731$	− 2.44658e8i	− 0.626335i
$732$	0	0
$733$	2.79408e8	0.709458	0.354729	−	0.934969i	$-0.384573\pi$
$733$	2.79408e8	0.709458	0.354729	+	0.934969i	$0.384573\pi$
$734$	0	0
$735$	1.92096e8i	0.483789i
$736$	0	0
$737$	5.11428e7	0.127756
$738$	0	0
$739$	5.00369e8i	1.23981i	0.784675	+	0.619907i	$0.212830\pi$
$739$	5.00369e8i	1.23981i	−0.784675	+	0.619907i	$0.787170\pi$
$740$	0	0
$741$	−4.46254e8	−1.09680
$742$	0	0
$743$	− 5.12068e8i	− 1.24842i	−0.781256	−	0.624211i	$-0.785421\pi$
$743$	− 5.12068e8i	− 1.24842i	0.781256	−	0.624211i	$-0.214579\pi$
$744$	0	0
$745$	−2.86662e8	−0.693269
$746$	0	0
$747$	1.62802e8i	0.390568i
$748$	0	0
$749$	5.94721e8	1.41536
$750$	0	0
$751$	− 4.54568e8i	− 1.07319i	−0.843838	−	0.536597i	$-0.819709\pi$
$751$	− 4.54568e8i	− 1.07319i	0.843838	−	0.536597i	$-0.180291\pi$
$752$	0	0
$753$	−1.43464e7	−0.0336015
$754$	0	0
$755$	3.23464e8i	0.751598i
$756$	0	0
$757$	3.55132e8	0.818657	0.409329	−	0.912387i	$-0.365763\pi$
$757$	3.55132e8	0.818657	0.409329	+	0.912387i	$0.365763\pi$
$758$	0	0
$759$	8.13133e7i	0.185967i
$760$	0	0
$761$	7.07962e8	1.60641	0.803204	−	0.595704i	$-0.203127\pi$
$761$	7.07962e8	1.60641	0.803204	+	0.595704i	$0.203127\pi$
$762$	0	0
$763$	− 3.79463e8i	− 0.854272i
$764$	0	0
$765$	−1.76876e8	−0.395081
$766$	0	0
$767$	− 3.24604e8i	− 0.719396i
$768$	0	0
$769$	−5.29332e8	−1.16399	−0.581995	−	0.813193i	$-0.697728\pi$
$769$	−5.29332e8	−1.16399	−0.581995	+	0.813193i	$0.697728\pi$
$770$	0	0
$771$	8.93856e7i	0.195031i
$772$	0	0
$773$	5.71149e8	1.23655	0.618274	−	0.785963i	$-0.287832\pi$
$773$	5.71149e8	1.23655	0.618274	+	0.785963i	$0.287832\pi$
$774$	0	0
$775$	4.82046e8i	1.03558i
$776$	0	0
$777$	−6.66251e8	−1.42028
$778$	0	0
$779$	− 7.31634e8i	− 1.54768i
$780$	0	0
$781$	−4.26450e8	−0.895189
$782$	0	0
$783$	1.60770e8i	0.334904i
$784$	0	0
$785$	4.96336e8	1.02605
$786$	0	0
$787$	1.56385e8i	0.320828i	0.987050	+	0.160414i	$0.0512829\pi$
$787$	1.56385e8i	0.320828i	−0.987050	+	0.160414i	$0.948717\pi$
$788$	0	0
$789$	2.13519e8	0.434717
$790$	0	0
$791$	9.76239e7i	0.197254i
$792$	0	0
$793$	−1.45958e8	−0.292691
$794$	0	0
$795$	1.47896e8i	0.294343i
$796$	0	0
$797$	−6.02539e8	−1.19017	−0.595087	−	0.803661i	$-0.702882\pi$
$797$	−6.02539e8	−1.19017	−0.595087	+	0.803661i	$0.702882\pi$
$798$	0	0
$799$	− 5.00777e8i	− 0.981756i
$800$	0	0
$801$	−1.22646e8	−0.238647
$802$	0	0
$803$	− 1.10619e8i	− 0.213640i
$804$	0	0
$805$	−1.87958e8	−0.360307
$806$	0	0
$807$	− 3.04392e8i	− 0.579178i
$808$	0	0
$809$	−2.30938e8	−0.436164	−0.218082	−	0.975930i	$-0.569980\pi$
$809$	−2.30938e8	−0.436164	−0.218082	+	0.975930i	$0.569980\pi$
$810$	0	0
$811$	− 1.15316e8i	− 0.216185i	−0.994141	−	0.108093i	$-0.965526\pi$
$811$	− 1.15316e8i	− 0.216185i	0.994141	−	0.108093i	$-0.0344743\pi$
$812$	0	0
$813$	1.37099e8	0.255131
$814$	0	0
$815$	4.37511e8i	0.808195i
$816$	0	0
$817$	2.99329e8	0.548887
$818$	0	0
$819$	− 3.41393e8i	− 0.621446i
$820$	0	0
$821$	8.70428e7	0.157291	0.0786454	−	0.996903i	$-0.474941\pi$
$821$	8.70428e7	0.157291	0.0786454	+	0.996903i	$0.474941\pi$
$822$	0	0
$823$	− 1.58502e8i	− 0.284337i	−0.989842	−	0.142169i	$-0.954592\pi$
$823$	− 1.58502e8i	− 0.284337i	0.989842	−	0.142169i	$-0.0454076\pi$
$824$	0	0
$825$	−1.57566e8	−0.280608
$826$	0	0
$827$	− 6.20614e8i	− 1.09725i	−0.836069	−	0.548624i	$-0.815152\pi$
$827$	− 6.20614e8i	− 1.09725i	0.836069	−	0.548624i	$-0.184848\pi$
$828$	0	0
$829$	−1.05341e9	−1.84898	−0.924490	−	0.381206i	$-0.875509\pi$
$829$	−1.05341e9	−1.84898	−0.924490	+	0.381206i	$0.875509\pi$
$830$	0	0
$831$	− 2.05794e7i	− 0.0358617i
$832$	0	0
$833$	1.23019e9	2.12833
$834$	0	0
$835$	− 1.09067e8i	− 0.187341i
$836$	0	0
$837$	−2.19109e8	−0.373666
$838$	0	0
$839$	− 1.80870e8i	− 0.306254i	−0.988207	−	0.153127i	$-0.951066\pi$
$839$	− 1.80870e8i	− 0.306254i	0.988207	−	0.153127i	$-0.0489343\pi$
$840$	0	0
$841$	1.20650e9	2.02833
$842$	0	0
$843$	− 2.76319e8i	− 0.461241i
$844$	0	0
$845$	−2.31208e8	−0.383207
$846$	0	0
$847$	− 1.53784e8i	− 0.253081i
$848$	0	0
$849$	−6.45972e8	−1.05558
$850$	0	0
$851$	− 3.59130e8i	− 0.582724i
$852$	0	0
$853$	−8.06034e8	−1.29869	−0.649346	−	0.760493i	$-0.724957\pi$
$853$	−8.06034e8	−1.29869	−0.649346	+	0.760493i	$0.724957\pi$
$854$	0	0
$855$	− 2.16401e8i	− 0.346228i
$856$	0	0
$857$	4.09751e8	0.650995	0.325497	−	0.945543i	$-0.394468\pi$
$857$	4.09751e8	0.650995	0.325497	+	0.945543i	$0.394468\pi$
$858$	0	0
$859$	− 8.81567e8i	− 1.39084i	−0.718606	−	0.695418i	$-0.755219\pi$
$859$	− 8.81567e8i	− 1.39084i	0.718606	−	0.695418i	$-0.244781\pi$
$860$	0	0
$861$	5.59714e8	0.876914
$862$	0	0
$863$	− 5.12988e8i	− 0.798132i	−0.916922	−	0.399066i	$-0.869334\pi$
$863$	− 5.12988e8i	− 0.798132i	0.916922	−	0.399066i	$-0.130666\pi$
$864$	0	0
$865$	2.45448e8	0.379237
$866$	0	0
$867$	7.56462e8i	1.16073i
$868$	0	0
$869$	−9.40241e8	−1.43278
$870$	0	0
$871$	− 1.15742e8i	− 0.175161i
$872$	0	0
$873$	−2.50571e8	−0.376607
$874$	0	0
$875$	− 1.04710e9i	− 1.56301i
$876$	0	0
$877$	1.12822e8	0.167262	0.0836308	−	0.996497i	$-0.473348\pi$
$877$	1.12822e8	0.167262	0.0836308	+	0.996497i	$0.473348\pi$
$878$	0	0
$879$	7.68816e7i	0.113202i
$880$	0	0
$881$	1.06050e9	1.55090	0.775448	−	0.631412i	$-0.217524\pi$
$881$	1.06050e9	1.55090	0.775448	+	0.631412i	$0.217524\pi$
$882$	0	0
$883$	− 8.09446e8i	− 1.17573i	−0.808961	−	0.587863i	$-0.799970\pi$
$883$	− 8.09446e8i	− 1.17573i	0.808961	−	0.587863i	$-0.200030\pi$
$884$	0	0
$885$	1.57410e8	0.227092
$886$	0	0
$887$	3.59419e6i	0.00515027i	0.999997	+	0.00257514i	$0.000819692\pi$
$887$	3.59419e6i	0.00515027i	−0.999997	+	0.00257514i	$0.999180\pi$
$888$	0	0
$889$	4.19394e8	0.596922
$890$	0	0
$891$	− 7.16199e7i	− 0.101251i
$892$	0	0
$893$	6.12680e8	0.860359
$894$	0	0
$895$	− 7.68476e8i	− 1.07192i
$896$	0	0
$897$	1.84021e8	0.254971
$898$	0	0
$899$	2.45497e9i	3.37884i
$900$	0	0
$901$	9.47135e8	1.29490
$902$	0	0
$903$	2.28993e8i	0.310998i
$904$	0	0
$905$	−2.84683e8	−0.384074
$906$	0	0
$907$	− 7.63921e8i	− 1.02383i	−0.859037	−	0.511913i	$-0.828937\pi$
$907$	− 7.63921e8i	− 1.02383i	0.859037	−	0.511913i	$-0.171063\pi$
$908$	0	0
$909$	−2.25405e8	−0.300104
$910$	0	0
$911$	9.16841e7i	0.121266i	0.998160	+	0.0606330i	$0.0193119\pi$
$911$	9.16841e7i	0.121266i	−0.998160	+	0.0606330i	$0.980688\pi$
$912$	0	0
$913$	−8.12594e8	−1.06773
$914$	0	0
$915$	− 7.07794e7i	− 0.0923940i
$916$	0	0
$917$	−1.92455e9	−2.49587
$918$	0	0
$919$	1.18709e9i	1.52945i	0.644356	+	0.764726i	$0.277126\pi$
$919$	1.18709e9i	1.52945i	−0.644356	+	0.764726i	$0.722874\pi$
$920$	0	0
$921$	−6.79190e8	−0.869385
$922$	0	0
$923$	9.65105e8i	1.22735i
$924$	0	0
$925$	6.95909e8	0.879280
$926$	0	0
$927$	8.68236e7i	0.108993i
$928$	0	0
$929$	3.88302e8	0.484309	0.242155	−	0.970238i	$-0.422146\pi$
$929$	3.88302e8	0.484309	0.242155	+	0.970238i	$0.422146\pi$
$930$	0	0
$931$	1.50509e9i	1.86515i
$932$	0	0
$933$	8.28352e8	1.01993
$934$	0	0
$935$	− 8.82845e8i	− 1.08006i
$936$	0	0
$937$	−7.69038e8	−0.934822	−0.467411	−	0.884040i	$-0.654813\pi$
$937$	−7.69038e8	−0.934822	−0.467411	+	0.884040i	$0.654813\pi$
$938$	0	0
$939$	− 4.96730e8i	− 0.599961i
$940$	0	0
$941$	−6.80625e8	−0.816843	−0.408422	−	0.912793i	$-0.633921\pi$
$941$	−6.80625e8	−0.816843	−0.408422	+	0.912793i	$0.633921\pi$
$942$	0	0
$943$	3.01703e8i	0.359787i
$944$	0	0
$945$	1.65551e8	0.196172
$946$	0	0
$947$	− 4.54633e8i	− 0.535317i	−0.963514	−	0.267659i	$-0.913750\pi$
$947$	− 4.54633e8i	− 0.535317i	0.963514	−	0.267659i	$-0.0862499\pi$
$948$	0	0
$949$	−2.50343e8	−0.292912
$950$	0	0
$951$	− 5.19349e8i	− 0.603834i
$952$	0	0
$953$	6.97601e8	0.805987	0.402994	−	0.915203i	$-0.367970\pi$
$953$	6.97601e8	0.805987	0.402994	+	0.915203i	$0.367970\pi$
$954$	0	0
$955$	3.12977e8i	0.359337i
$956$	0	0
$957$	−8.02453e8	−0.915554
$958$	0	0
$959$	9.54900e8i	1.08268i
$960$	0	0
$961$	−2.45830e9	−2.76991
$962$	0	0
$963$	− 2.82357e8i	− 0.316169i
$964$	0	0
$965$	7.32769e8	0.815428
$966$	0	0
$967$	3.55787e7i	0.0393469i	0.999806	+	0.0196735i	$0.00626266\pi$
$967$	3.55787e7i	0.0393469i	−0.999806	+	0.0196735i	$0.993737\pi$
$968$	0	0
$969$	−1.38585e9	−1.52316
$970$	0	0
$971$	8.23470e8i	0.899477i	0.893160	+	0.449738i	$0.148483\pi$
$971$	8.23470e8i	0.899477i	−0.893160	+	0.449738i	$0.851517\pi$
$972$	0	0
$973$	1.72644e9	1.87419
$974$	0	0
$975$	3.56590e8i	0.384729i
$976$	0	0
$977$	4.25779e7	0.0456563	0.0228281	−	0.999739i	$-0.492733\pi$
$977$	4.25779e7	0.0456563	0.0228281	+	0.999739i	$0.492733\pi$
$978$	0	0
$979$	− 6.12164e8i	− 0.652408i
$980$	0	0
$981$	−1.80159e8	−0.190831
$982$	0	0
$983$	1.45686e9i	1.53376i	0.641793	+	0.766878i	$0.278191\pi$
$983$	1.45686e9i	1.53376i	−0.641793	+	0.766878i	$0.721809\pi$
$984$	0	0
$985$	8.90297e8	0.931593
$986$	0	0
$987$	4.68713e8i	0.487478i
$988$	0	0
$989$	−1.23434e8	−0.127599
$990$	0	0
$991$	1.26328e9i	1.29801i	0.760784	+	0.649005i	$0.224814\pi$
$991$	1.26328e9i	1.29801i	−0.760784	+	0.649005i	$0.775186\pi$
$992$	0	0
$993$	−3.67315e7	−0.0375138
$994$	0	0
$995$	3.71014e8i	0.376635i
$996$	0	0
$997$	8.99928e8	0.908076	0.454038	−	0.890982i	$-0.349983\pi$
$997$	8.99928e8	0.908076	0.454038	+	0.890982i	$0.349983\pi$
$998$	0	0
$999$	3.16318e8i	0.317268i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.7.g.g.511.6		8
4.3	odd	2	inner	768.7.g.g.511.2		8
8.3	odd	2	inner	768.7.g.g.511.7		8
8.5	even	2	inner	768.7.g.g.511.3		8
16.3	odd	4		384.7.b.c.319.7	yes	8
16.5	even	4		384.7.b.c.319.6	yes	8
16.11	odd	4		384.7.b.c.319.2	✓	8
16.13	even	4		384.7.b.c.319.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.7.b.c.319.2	✓	8	16.11	odd	4
384.7.b.c.319.3	yes	8	16.13	even	4
384.7.b.c.319.6	yes	8	16.5	even	4
384.7.b.c.319.7	yes	8	16.3	odd	4
768.7.g.g.511.2		8	4.3	odd	2	inner
768.7.g.g.511.3		8	8.5	even	2	inner
768.7.g.g.511.6		8	1.1	even	1	trivial
768.7.g.g.511.7		8	8.3	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+15.5885i q^{3} -85.3891 q^{5} -511.824i q^{7} -243.000 q^{9} +O(q^{10})\)	\(q+15.5885i q^{3} -85.3891 q^{5} -511.824i q^{7} -243.000 q^{9} -1212.89i q^{11} -2744.91 q^{13} -1331.08i q^{15} -8524.35 q^{17} -10429.2i q^{19} +7978.55 q^{21} +4300.69i q^{23} -8333.71 q^{25} -3788.00i q^{27} -42442.0 q^{29} -57843.0i q^{31} +18907.1 q^{33} +43704.2i q^{35} -83505.3 q^{37} -42788.9i q^{39} +70152.4 q^{41} +28701.0i q^{43} +20749.5 q^{45} +58746.6i q^{47} -144315. q^{49} -132882. i q^{51} -111109. q^{53} +103567. i q^{55} +162575. q^{57} +118257. i q^{59} +53174.3 q^{61} +124373. i q^{63} +234385. q^{65} +42166.1i q^{67} -67041.1 q^{69} -351599. i q^{71} +91202.9 q^{73} -129910. i q^{75} -620786. q^{77} -775208. i q^{79} +59049.0 q^{81} -669966. i q^{83} +727887. q^{85} -661605. i q^{87} +504715. q^{89} +1.40491e6i q^{91} +901683. q^{93} +890541. i q^{95} +1.03116e6 q^{97} +294732. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 1944 q^{9}+O(q^{10}) \) 8 * q - 1944 * q^9	\( 8 q - 1944 q^{9} - 6544 q^{17} + 56632 q^{25} - 33696 q^{33} + 499568 q^{41} - 414712 q^{49} + 375840 q^{57} - 36096 q^{65} + 1962640 q^{73} + 472392 q^{81} + 1694992 q^{89} + 7632752 q^{97}+O(q^{100}) \) 8 * q - 1944 * q^9 - 6544 * q^17 + 56632 * q^25 - 33696 * q^33 + 499568 * q^41 - 414712 * q^49 + 375840 * q^57 - 36096 * q^65 + 1962640 * q^73 + 472392 * q^81 + 1694992 * q^89 + 7632752 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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