LMFDB - Embedded newform 384.7.b.c.319.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,7,Mod(319,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	384.b (of order $2$, degree $1$, 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:	$88.3407681100$
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} - 4x^{7} + 106x^{6} - 304x^{5} + 4359x^{4} - 8216x^{3} + 73366x^{2} - 69308x + 604693 $ x^8 - 4x^7 + 106x^6 - 304x^5 + 4359x^4 - 8216x^3 + 73366x^2 - 69308*x + 604693
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.2
Root			$-1.23205 - 3.79090i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.7.b.c.319.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.5885 q^{3} -85.3891i q^{5} -511.824i q^{7} +243.000 q^{9} +O(q^{10})$	$q-15.5885 q^{3} -85.3891i q^{5} -511.824i q^{7} +243.000 q^{9} -1212.89 q^{11} +2744.91i q^{13} +1331.08i q^{15} -8524.35 q^{17} +10429.2 q^{19} +7978.55i q^{21} +4300.69i q^{23} +8333.71 q^{25} -3788.00 q^{27} +42442.0i q^{29} +57843.0i q^{31} +18907.1 q^{33} -43704.2 q^{35} -83505.3i q^{37} -42788.9i q^{39} -70152.4 q^{41} +28701.0 q^{43} -20749.5i q^{45} -58746.6i q^{47} -144315. q^{49} +132882. q^{51} -111109. i q^{53} +103567. i q^{55} -162575. q^{57} +118257. q^{59} -53174.3i q^{61} -124373. i q^{63} +234385. q^{65} -42166.1 q^{67} -67041.1i q^{69} -351599. i q^{71} -91202.9 q^{73} -129910. q^{75} +620786. i q^{77} +775208. i q^{79} +59049.0 q^{81} +669966. q^{83} +727887. i q^{85} -661605. i q^{87} -504715. q^{89} +1.40491e6 q^{91} -901683. i q^{93} -890541. i q^{95} +1.03116e6 q^{97} -294732. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−15.5885	−0.577350
$3$	−15.5885	−0.577350
$4$	0	0
$5$	− 85.3891i	− 0.683112i	−0.939861	−	0.341556i	$-0.889046\pi$
$5$	− 85.3891i	− 0.683112i	0.939861	−	0.341556i	$-0.110954\pi$
$6$	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
$11$	−1212.89	−0.911261	−0.455631	−	0.890169i	$-0.650586\pi$
$11$	−1212.89	−0.911261	−0.455631	+	0.890169i	$0.650586\pi$
$12$	0	0
$13$	2744.91i	1.24939i	0.780869	+	0.624694i	$0.214776\pi$
$13$	2744.91i	1.24939i	−0.780869	+	0.624694i	$0.785224\pi$
$14$	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
$19$	10429.2	1.52051	0.760257	−	0.649622i	$-0.225073\pi$
$19$	10429.2	1.52051	0.760257	+	0.649622i	$0.225073\pi$
$20$	0	0
$21$	7978.55i	0.861522i
$22$	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
$27$	−3788.00	−0.192450
$28$	0	0
$29$	42442.0i	1.74021i	0.492866	+	0.870105i	$0.335949\pi$
$29$	42442.0i	1.74021i	−0.492866	+	0.870105i	$0.664051\pi$
$30$	0	0
$31$	57843.0i	1.94163i	0.239839	+	0.970813i	$0.422905\pi$
$31$	57843.0i	1.94163i	−0.239839	+	0.970813i	$0.577095\pi$
$32$	0	0
$33$	18907.1	0.526117
$34$	0	0
$35$	−43704.2	−1.01934
$36$	0	0
$37$	− 83505.3i	− 1.64858i	−0.566171	−	0.824288i	$-0.691576\pi$
$37$	− 83505.3i	− 1.64858i	0.566171	−	0.824288i	$-0.308424\pi$
$38$	0	0
$39$	− 42788.9i	− 0.721335i
$40$	0	0
$41$	−70152.4	−1.01787	−0.508933	−	0.860806i	$-0.669960\pi$
$41$	−70152.4	−1.01787	−0.508933	+	0.860806i	$0.669960\pi$
$42$	0	0
$43$	28701.0	0.360987	0.180494	−	0.983576i	$-0.442230\pi$
$43$	28701.0	0.360987	0.180494	+	0.983576i	$0.442230\pi$
$44$	0	0
$45$	− 20749.5i	− 0.227704i
$46$	0	0
$47$	− 58746.6i	− 0.565834i	−0.959144	−	0.282917i	$-0.908698\pi$
$47$	− 58746.6i	− 0.565834i	0.959144	−	0.282917i	$-0.0913021\pi$
$48$	0	0
$49$	−144315.	−1.22666
$50$	0	0
$51$	132882.	1.00174
$52$	0	0
$53$	− 111109.i	− 0.746316i	−0.927768	−	0.373158i	$-0.878275\pi$
$53$	− 111109.i	− 0.746316i	0.927768	−	0.373158i	$-0.121725\pi$
$54$	0	0
$55$	103567.i	0.622494i
$56$	0	0
$57$	−162575.	−0.877870
$58$	0	0
$59$	118257.	0.575798	0.287899	−	0.957661i	$-0.407043\pi$
$59$	118257.	0.575798	0.287899	+	0.957661i	$0.407043\pi$
$60$	0	0
$61$	− 53174.3i	− 0.234268i	−0.993116	−	0.117134i	$-0.962629\pi$
$61$	− 53174.3i	− 0.234268i	0.993116	−	0.117134i	$-0.0373706\pi$
$62$	0	0
$63$	− 124373.i	− 0.497400i
$64$	0	0
$65$	234385.	0.853473
$66$	0	0
$67$	−42166.1	−0.140197	−0.0700986	−	0.997540i	$-0.522331\pi$
$67$	−42166.1	−0.140197	−0.0700986	+	0.997540i	$0.522331\pi$
$68$	0	0
$69$	− 67041.1i	− 0.204077i
$70$	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	$-0.537399\pi$
$73$	−91202.9	−0.234445	−0.117222	+	0.993106i	$0.537399\pi$
$74$	0	0
$75$	−129910.	−0.307934
$76$	0	0
$77$	620786.i	1.35978i
$78$	0	0
$79$	775208.i	1.57231i	0.618032	+	0.786153i	$0.287930\pi$
$79$	775208.i	1.57231i	−0.618032	+	0.786153i	$0.712070\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	669966.	1.17170	0.585852	−	0.810418i	$-0.300760\pi$
$83$	669966.	1.17170	0.585852	+	0.810418i	$0.300760\pi$
$84$	0	0
$85$	727887.i	1.18524i
$86$	0	0
$87$	− 661605.i	− 1.00471i
$88$	0	0
$89$	−504715.	−0.715940	−0.357970	−	0.933733i	$-0.616531\pi$
$89$	−504715.	−0.715940	−0.357970	+	0.933733i	$0.616531\pi$
$90$	0	0
$91$	1.40491e6	1.86434
$92$	0	0
$93$	− 901683.i	− 1.12100i
$94$	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.	−0.303754
$100$	0	0
$101$	927592.i	0.900312i	0.892950	+	0.450156i	$0.148632\pi$
$101$	927592.i	0.900312i	−0.892950	+	0.450156i	$0.851368\pi$
$102$	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
$107$	1.16196e6	0.948508	0.474254	−	0.880388i	$-0.342718\pi$
$107$	1.16196e6	0.948508	0.474254	+	0.880388i	$0.342718\pi$
$108$	0	0
$109$	− 741394.i	− 0.572492i	−0.958156	−	0.286246i	$-0.907593\pi$
$109$	− 741394.i	− 0.572492i	0.958156	−	0.286246i	$-0.0924075\pi$
$110$	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.	0.241461
$116$	0	0
$117$	667012.i	0.416463i
$118$	0	0
$119$	4.36297e6i	2.58906i
$120$	0	0
$121$	−300462.	−0.169603
$122$	0	0
$123$	1.09357e6	0.587665
$124$	0	0
$125$	− 2.04581e6i	− 1.04746i
$126$	0	0
$127$	− 819411.i	− 0.400028i	−0.979793	−	0.200014i	$-0.935901\pi$
$127$	− 819411.i	− 0.400028i	0.979793	−	0.200014i	$-0.0640988\pi$
$128$	0	0
$129$	−447405.	−0.208416
$130$	0	0
$131$	3.76018e6	1.67261	0.836306	−	0.548263i	$-0.184711\pi$
$131$	3.76018e6	1.67261	0.836306	+	0.548263i	$0.184711\pi$
$132$	0	0
$133$	− 5.33793e6i	− 2.26891i
$134$	0	0
$135$	323453.i	0.131465i
$136$	0	0
$137$	1.86568e6	0.725563	0.362782	−	0.931874i	$-0.381827\pi$
$137$	1.86568e6	0.725563	0.362782	+	0.931874i	$0.381827\pi$
$138$	0	0
$139$	3.37311e6	1.25599	0.627995	−	0.778218i	$-0.283876\pi$
$139$	3.37311e6	1.25599	0.627995	+	0.778218i	$0.283876\pi$
$140$	0	0
$141$	915768.i	0.326684i
$142$	0	0
$143$	− 3.32927e6i	− 1.13852i
$144$	0	0
$145$	3.62408e6	1.18876
$146$	0	0
$147$	2.24965e6	0.708212
$148$	0	0
$149$	3.35713e6i	1.01487i	0.861691	+	0.507434i	$0.169406\pi$
$149$	3.35713e6i	1.01487i	−0.861691	+	0.507434i	$0.830594\pi$
$150$	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
$155$	4.93916e6	1.32635
$156$	0	0
$157$	5.81264e6i	1.50202i	0.660293	+	0.751008i	$0.270432\pi$
$157$	5.81264e6i	1.50202i	−0.660293	+	0.751008i	$0.729568\pi$
$158$	0	0
$159$	1.73202e6i	0.430886i
$160$	0	0
$161$	2.20120e6	0.527450
$162$	0	0
$163$	5.12374e6	1.18311	0.591554	−	0.806266i	$-0.298515\pi$
$163$	5.12374e6	1.18311	0.591554	+	0.806266i	$0.298515\pi$
$164$	0	0
$165$	− 1.61446e6i	− 0.359397i
$166$	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
$171$	2.53430e6	0.506838
$172$	0	0
$173$	2.87446e6i	0.555160i	0.960703	+	0.277580i	$0.0895324\pi$
$173$	2.87446e6i	0.555160i	−0.960703	+	0.277580i	$0.910468\pi$
$174$	0	0
$175$	− 4.26540e6i	− 0.795876i
$176$	0	0
$177$	−1.84344e6	−0.332437
$178$	0	0
$179$	−8.99970e6	−1.56917	−0.784583	−	0.620024i	$-0.787123\pi$
$179$	−8.99970e6	−1.56917	−0.784583	+	0.620024i	$0.787123\pi$
$180$	0	0
$181$	3.33395e6i	0.562242i	0.959672	+	0.281121i	$0.0907062\pi$
$181$	3.33395e6i	0.562242i	−0.959672	+	0.281121i	$0.909294\pi$
$182$	0	0
$183$	828905.i	0.135254i
$184$	0	0
$185$	−7.13044e6	−1.12616
$186$	0	0
$187$	1.03391e7	1.58109
$188$	0	0
$189$	1.93879e6i	0.287174i
$190$	0	0
$191$	3.66531e6i	0.526030i	0.964792	+	0.263015i	$0.0847168\pi$
$191$	3.66531e6i	0.526030i	−0.964792	+	0.263015i	$0.915283\pi$
$192$	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
$195$	−3.65370e6	−0.492753
$196$	0	0
$197$	− 1.04264e7i	− 1.36375i	−0.731470	−	0.681874i	$-0.761165\pi$
$197$	− 1.04264e7i	− 1.36375i	0.731470	−	0.681874i	$-0.238835\pi$
$198$	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
$203$	2.17228e7	2.59674
$204$	0	0
$205$	5.99024e6i	0.695317i
$206$	0	0
$207$	1.04507e6i	0.117824i
$208$	0	0
$209$	−1.26495e7	−1.38559
$210$	0	0
$211$	5.90463e6	0.628558	0.314279	−	0.949331i	$-0.398237\pi$
$211$	5.90463e6	0.628558	0.314279	+	0.949331i	$0.398237\pi$
$212$	0	0
$213$	5.48088e6i	0.567168i
$214$	0	0
$215$	− 2.45075e6i	− 0.246595i
$216$	0	0
$217$	2.96054e7	2.89729
$218$	0	0
$219$	1.42171e6	0.135357
$220$	0	0
$221$	− 2.33986e7i	− 2.16777i
$222$	0	0
$223$	1.48155e7i	1.33598i	0.744169	+	0.667992i	$0.232846\pi$
$223$	1.48155e7i	1.33598i	−0.744169	+	0.667992i	$0.767154\pi$
$224$	0	0
$225$	2.02509e6	0.177786
$226$	0	0
$227$	3.99262e6	0.341334	0.170667	−	0.985329i	$-0.445408\pi$
$227$	3.99262e6	0.341334	0.170667	+	0.985329i	$0.445408\pi$
$228$	0	0
$229$	− 3.71868e6i	− 0.309658i	−0.987941	−	0.154829i	$-0.950517\pi$
$229$	− 3.71868e6i	− 0.309658i	0.987941	−	0.154829i	$-0.0494826\pi$
$230$	0	0
$231$	− 9.67710e6i	− 0.785071i
$232$	0	0
$233$	−5.25478e6	−0.415419	−0.207710	−	0.978191i	$-0.566601\pi$
$233$	−5.25478e6	−0.415419	−0.207710	+	0.978191i	$0.566601\pi$
$234$	0	0
$235$	−5.01631e6	−0.386528
$236$	0	0
$237$	− 1.20843e7i	− 0.907771i
$238$	0	0
$239$	− 7.96267e6i	− 0.583264i	−0.956531	−	0.291632i	$-0.905802\pi$
$239$	− 7.96267e6i	− 0.583264i	0.956531	−	0.291632i	$-0.0941982\pi$
$240$	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.	−0.0641500
$244$	0	0
$245$	1.23229e7i	0.837946i
$246$	0	0
$247$	2.86272e7i	1.89971i
$248$	0	0
$249$	−1.04437e7	−0.676484
$250$	0	0
$251$	920323.	0.0581995	0.0290998	−	0.999577i	$-0.490736\pi$
$251$	920323.	0.0581995	0.0290998	+	0.999577i	$0.490736\pi$
$252$	0	0
$253$	− 5.21625e6i	− 0.322105i
$254$	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
$259$	−4.27400e7	−2.46000
$260$	0	0
$261$	1.03134e7i	0.580070i
$262$	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
$267$	7.86774e6	0.413348
$268$	0	0
$269$	1.95267e7i	1.00317i	0.865110	+	0.501583i	$0.167249\pi$
$269$	1.95267e7i	1.00317i	−0.865110	+	0.501583i	$0.832751\pi$
$270$	0	0
$271$	8.79492e6i	0.441900i	0.975285	+	0.220950i	$0.0709158\pi$
$271$	8.79492e6i	0.441900i	−0.975285	+	0.220950i	$0.929084\pi$
$272$	0	0
$273$	−2.19004e7	−1.07638
$274$	0	0
$275$	−1.01079e7	−0.486028
$276$	0	0
$277$	− 1.32017e6i	− 0.0621142i	−0.999518	−	0.0310571i	$-0.990113\pi$
$277$	− 1.32017e6i	− 0.0621142i	0.999518	−	0.0310571i	$-0.00988737\pi$
$278$	0	0
$279$	1.40558e7i	0.647208i
$280$	0	0
$281$	1.77259e7	0.798893	0.399446	−	0.916757i	$-0.369202\pi$
$281$	1.77259e7	0.798893	0.399446	+	0.916757i	$0.369202\pi$
$282$	0	0
$283$	4.14391e7	1.82832	0.914158	−	0.405358i	$-0.132853\pi$
$283$	4.14391e7	1.82832	0.914158	+	0.405358i	$0.132853\pi$
$284$	0	0
$285$	1.38822e7i	0.599684i
$286$	0	0
$287$	3.59057e7i	1.51886i
$288$	0	0
$289$	4.85271e7	2.01044
$290$	0	0
$291$	−1.60742e7	−0.652303
$292$	0	0
$293$	4.93196e6i	0.196072i	0.995183	+	0.0980362i	$0.0312561\pi$
$293$	4.93196e6i	0.196072i	−0.995183	+	0.0980362i	$0.968744\pi$
$294$	0	0
$295$	− 1.00978e7i	− 0.393335i
$296$	0	0
$297$	4.59442e6	0.175372
$298$	0	0
$299$	−1.18050e7	−0.441623
$300$	0	0
$301$	− 1.46899e7i	− 0.538665i
$302$	0	0
$303$	− 1.44597e7i	− 0.519795i
$304$	0	0
$305$	−4.54050e6	−0.160031
$306$	0	0
$307$	−4.35700e7	−1.50582	−0.752910	−	0.658124i	$-0.771350\pi$
$307$	−4.35700e7	−1.50582	−0.752910	+	0.658124i	$0.771350\pi$
$308$	0	0
$309$	5.56974e6i	0.188781i
$310$	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	$-0.326088\pi$
$313$	3.18652e7	1.03916	0.519582	+	0.854421i	$0.326088\pi$
$314$	0	0
$315$	−1.06201e7	−0.339780
$316$	0	0
$317$	3.33162e7i	1.04587i	0.852372	+	0.522936i	$0.175163\pi$
$317$	3.33162e7i	1.04587i	−0.852372	+	0.522936i	$0.824837\pi$
$318$	0	0
$319$	− 5.14774e7i	− 1.58579i
$320$	0	0
$321$	−1.81132e7	−0.547621
$322$	0	0
$323$	−8.89023e7	−2.63819
$324$	0	0
$325$	2.28753e7i	0.666371i
$326$	0	0
$327$	1.15572e7i	0.330528i
$328$	0	0
$329$	−3.00679e7	−0.844337
$330$	0	0
$331$	2.35633e6	0.0649758	0.0324879	−	0.999472i	$-0.489657\pi$
$331$	2.35633e6	0.0649758	0.0324879	+	0.999472i	$0.489657\pi$
$332$	0	0
$333$	− 2.02918e7i	− 0.549525i
$334$	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
$339$	2.97330e6	0.0763202
$340$	0	0
$341$	− 7.01571e7i	− 1.76933i
$342$	0	0
$343$	1.36484e7i	0.338221i
$344$	0	0
$345$	−5.72457e6	−0.139407
$346$	0	0
$347$	−1.76051e7	−0.421357	−0.210679	−	0.977555i	$-0.567567\pi$
$347$	−1.76051e7	−0.421357	−0.210679	+	0.977555i	$0.567567\pi$
$348$	0	0
$349$	3.02782e6i	0.0712285i	0.999366	+	0.0356143i	$0.0113388\pi$
$349$	3.02782e6i	0.0712285i	−0.999366	+	0.0356143i	$0.988661\pi$
$350$	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
$355$	−3.00227e7	−0.671064
$356$	0	0
$357$	− 6.80120e7i	− 1.49479i
$358$	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
$363$	4.68374e6	0.0979203
$364$	0	0
$365$	7.78773e6i	0.160152i
$366$	0	0
$367$	− 1.52811e7i	− 0.309142i	−0.987982	−	0.154571i	$-0.950601\pi$
$367$	− 1.52811e7i	− 0.309142i	0.987982	−	0.154571i	$-0.0493995\pi$
$368$	0	0
$369$	−1.70470e7	−0.339289
$370$	0	0
$371$	−5.68684e7	−1.11365
$372$	0	0
$373$	− 6.19317e7i	− 1.19340i	−0.802464	−	0.596701i	$-0.796478\pi$
$373$	− 6.19317e7i	− 1.19340i	0.802464	−	0.596701i	$-0.203522\pi$
$374$	0	0
$375$	3.18910e7i	0.604749i
$376$	0	0
$377$	−1.16499e8	−2.17420
$378$	0	0
$379$	−3.43446e7	−0.630870	−0.315435	−	0.948947i	$-0.602151\pi$
$379$	−3.43446e7	−0.630870	−0.315435	+	0.948947i	$0.602151\pi$
$380$	0	0
$381$	1.27733e7i	0.230956i
$382$	0	0
$383$	1.34734e7i	0.239817i	0.992785	+	0.119908i	$0.0382601\pi$
$383$	1.34734e7i	0.239817i	−0.992785	+	0.119908i	$0.961740\pi$
$384$	0	0
$385$	5.30083e7	0.928885
$386$	0	0
$387$	6.97435e6	0.120329
$388$	0	0
$389$	1.13679e8i	1.93122i	0.259993	+	0.965610i	$0.416280\pi$
$389$	1.13679e8i	1.93122i	−0.259993	+	0.965610i	$0.583720\pi$
$390$	0	0
$391$	− 3.66606e7i	− 0.613294i
$392$	0	0
$393$	−5.86155e7	−0.965683
$394$	0	0
$395$	6.61943e7	1.07406
$396$	0	0
$397$	3.96900e6i	0.0634322i	0.999497	+	0.0317161i	$0.0100972\pi$
$397$	3.96900e6i	0.0634322i	−0.999497	+	0.0317161i	$0.989903\pi$
$398$	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
$403$	−1.58774e8	−2.42584
$404$	0	0
$405$	− 5.04214e6i	− 0.0759014i
$406$	0	0
$407$	1.01283e8i	1.50228i
$408$	0	0
$409$	4.23651e7	0.619211	0.309606	−	0.950865i	$-0.399803\pi$
$409$	4.23651e7	0.619211	0.309606	+	0.950865i	$0.399803\pi$
$410$	0	0
$411$	−2.90831e7	−0.418904
$412$	0	0
$413$	− 6.05267e7i	− 0.859206i
$414$	0	0
$415$	− 5.72077e7i	− 0.800406i
$416$	0	0
$417$	−5.25816e7	−0.725146
$418$	0	0
$419$	−8.16747e7	−1.11031	−0.555157	−	0.831746i	$-0.687342\pi$
$419$	−8.16747e7	−1.11031	−0.555157	+	0.831746i	$0.687342\pi$
$420$	0	0
$421$	7.99635e7i	1.07163i	0.844335	+	0.535816i	$0.179996\pi$
$421$	7.99635e7i	1.07163i	−0.844335	+	0.535816i	$0.820004\pi$
$422$	0	0
$423$	− 1.42754e7i	− 0.188611i
$424$	0	0
$425$	−7.10395e7	−0.925408
$426$	0	0
$427$	−2.72159e7	−0.349574
$428$	0	0
$429$	5.18981e7i	0.657325i
$430$	0	0
$431$	− 7.45512e7i	− 0.931157i	−0.885007	−	0.465578i	$-0.845846\pi$
$431$	− 7.45512e7i	− 0.931157i	0.885007	−	0.465578i	$-0.154154\pi$
$432$	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
$435$	−5.64938e7	−0.686330
$436$	0	0
$437$	4.48528e7i	0.537458i
$438$	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
$443$	4.20100e7	0.483216	0.241608	−	0.970374i	$-0.422325\pi$
$443$	4.20100e7	0.483216	0.241608	+	0.970374i	$0.422325\pi$
$444$	0	0
$445$	4.30972e7i	0.489067i
$446$	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
$451$	8.50870e7	0.927542
$452$	0	0
$453$	5.90510e7i	0.635233i
$454$	0	0
$455$	− 1.19964e8i	− 1.27355i
$456$	0	0
$457$	1.64317e8	1.72160	0.860802	−	0.508941i	$-0.169963\pi$
$457$	1.64317e8	1.72160	0.860802	+	0.508941i	$0.169963\pi$
$458$	0	0
$459$	3.22902e7	0.333913
$460$	0	0
$461$	− 1.01392e7i	− 0.103491i	−0.998660	−	0.0517455i	$-0.983522\pi$
$461$	− 1.01392e7i	− 0.103491i	0.998660	−	0.0517455i	$-0.0164785\pi$
$462$	0	0
$463$	1.91081e8i	1.92519i	0.270937	+	0.962597i	$0.412667\pi$
$463$	1.91081e8i	1.92519i	−0.270937	+	0.962597i	$0.587333\pi$
$464$	0	0
$465$	−7.69938e7	−0.765768
$466$	0	0
$467$	1.30526e8	1.28159	0.640793	−	0.767713i	$-0.278606\pi$
$467$	1.30526e8	1.28159	0.640793	+	0.767713i	$0.278606\pi$
$468$	0	0
$469$	2.15817e7i	0.209202i
$470$	0	0
$471$	− 9.06101e7i	− 0.867189i
$472$	0	0
$473$	−3.48111e7	−0.328954
$474$	0	0
$475$	8.69140e7	0.810978
$476$	0	0
$477$	− 2.69996e7i	− 0.248772i
$478$	0	0
$479$	1.71231e8i	1.55803i	0.627003	+	0.779017i	$0.284281\pi$
$479$	1.71231e8i	1.55803i	−0.627003	+	0.779017i	$0.715719\pi$
$480$	0	0
$481$	2.29214e8	2.05971
$482$	0	0
$483$	−3.43132e7	−0.304523
$484$	0	0
$485$	− 8.80496e7i	− 0.771795i
$486$	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
$491$	−1.36774e8	−1.15547	−0.577735	−	0.816224i	$-0.696063\pi$
$491$	−1.36774e8	−1.15547	−0.577735	+	0.816224i	$0.696063\pi$
$492$	0	0
$493$	− 3.61791e8i	− 3.01937i
$494$	0	0
$495$	2.51669e7i	0.207498i
$496$	0	0
$497$	−1.79957e8	−1.46588
$498$	0	0
$499$	1.94124e8	1.56235	0.781175	−	0.624312i	$-0.214621\pi$
$499$	1.94124e8	1.56235	0.781175	+	0.624312i	$0.214621\pi$
$500$	0	0
$501$	− 1.99110e7i	− 0.158336i
$502$	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
$507$	4.22089e7	0.323877
$508$	0	0
$509$	6.32104e7i	0.479331i	0.970856	+	0.239665i	$0.0770377\pi$
$509$	6.32104e7i	0.479331i	−0.970856	+	0.239665i	$0.922962\pi$
$510$	0	0
$511$	4.66799e7i	0.349838i
$512$	0	0
$513$	−3.95058e7	−0.292623
$514$	0	0
$515$	−3.05094e7	−0.223363
$516$	0	0
$517$	7.12530e7i	0.515622i
$518$	0	0
$519$	− 4.48084e7i	− 0.320522i
$520$	0	0
$521$	−2.37718e8	−1.68092	−0.840462	−	0.541871i	$-0.817716\pi$
$521$	−2.37718e8	−1.68092	−0.840462	+	0.541871i	$0.817716\pi$
$522$	0	0
$523$	−5.13720e7	−0.359105	−0.179553	−	0.983748i	$-0.557465\pi$
$523$	−5.13720e7	−0.359105	−0.179553	+	0.983748i	$0.557465\pi$
$524$	0	0
$525$	6.64909e7i	0.459499i
$526$	0	0
$527$	− 4.93074e8i	− 3.36884i
$528$	0	0
$529$	1.29540e8	0.875058
$530$	0	0
$531$	2.87364e7	0.191933
$532$	0	0
$533$	− 1.92562e8i	− 1.27171i
$534$	0	0
$535$	− 9.92189e7i	− 0.647938i
$536$	0	0
$537$	1.40291e8	0.905958
$538$	0	0
$539$	1.75038e8	1.11781
$540$	0	0
$541$	1.82128e8i	1.15023i	0.818072	+	0.575116i	$0.195043\pi$
$541$	1.82128e8i	1.15023i	−0.818072	+	0.575116i	$0.804957\pi$
$542$	0	0
$543$	− 5.19711e7i	− 0.324610i
$544$	0	0
$545$	−6.33069e7	−0.391076
$546$	0	0
$547$	−6.68154e7	−0.408239	−0.204120	−	0.978946i	$-0.565433\pi$
$547$	−6.68154e7	−0.408239	−0.204120	+	0.978946i	$0.565433\pi$
$548$	0	0
$549$	− 1.29213e7i	− 0.0780892i
$550$	0	0
$551$	4.42636e8i	2.64602i
$552$	0	0
$553$	3.96770e8	2.34619
$554$	0	0
$555$	1.11153e8	0.650190
$556$	0	0
$557$	− 606923.i	− 0.00351211i	−0.999998	−	0.00175606i	$-0.999441\pi$
$557$	− 606923.i	− 0.00351211i	0.999998	−	0.00175606i	$-0.000558970\pi$
$558$	0	0
$559$	7.87816e7i	0.451013i
$560$	0	0
$561$	−1.61171e8	−0.912845
$562$	0	0
$563$	2.47309e8	1.38584	0.692922	−	0.721012i	$-0.256323\pi$
$563$	2.47309e8	1.38584	0.692922	+	0.721012i	$0.256323\pi$
$564$	0	0
$565$	1.62869e7i	0.0903009i
$566$	0	0
$567$	− 3.02227e7i	− 0.165800i
$568$	0	0
$569$	1.83163e8	0.994262	0.497131	−	0.867675i	$-0.334387\pi$
$569$	1.83163e8	0.994262	0.497131	+	0.867675i	$0.334387\pi$
$570$	0	0
$571$	1.30136e8	0.699021	0.349510	−	0.936933i	$-0.386348\pi$
$571$	1.30136e8	0.699021	0.349510	+	0.936933i	$0.386348\pi$
$572$	0	0
$573$	− 5.71365e7i	− 0.303703i
$574$	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
$579$	1.33773e8	0.689180
$580$	0	0
$581$	− 3.42905e8i	− 1.74842i
$582$	0	0
$583$	1.34763e8i	0.680089i
$584$	0	0
$585$	5.69556e7	0.284491
$586$	0	0
$587$	4.56349e7	0.225623	0.112811	−	0.993616i	$-0.464014\pi$
$587$	4.56349e7	0.225623	0.112811	+	0.993616i	$0.464014\pi$
$588$	0	0
$589$	6.03257e8i	2.95227i
$590$	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
$595$	3.72550e8	1.76862
$596$	0	0
$597$	6.77316e7i	0.318323i
$598$	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	$-0.452868\pi$
$601$	6.40513e7	0.295056	0.147528	+	0.989058i	$0.452868\pi$
$602$	0	0
$603$	−1.02464e7	−0.0467324
$604$	0	0
$605$	2.56562e7i	0.115858i
$606$	0	0
$607$	6.98586e7i	0.312359i	0.987729	+	0.156179i	$0.0499178\pi$
$607$	6.98586e7i	0.312359i	−0.987729	+	0.156179i	$0.950082\pi$
$608$	0	0
$609$	−3.38626e8	−1.49923
$610$	0	0
$611$	1.61254e8	0.706946
$612$	0	0
$613$	2.62634e7i	0.114017i	0.998374	+	0.0570085i	$0.0181562\pi$
$613$	2.62634e7i	0.114017i	−0.998374	+	0.0570085i	$0.981844\pi$
$614$	0	0
$615$	− 9.33786e7i	− 0.401441i
$616$	0	0
$617$	−2.98936e8	−1.27269	−0.636345	−	0.771404i	$-0.719555\pi$
$617$	−2.98936e8	−1.27269	−0.636345	+	0.771404i	$0.719555\pi$
$618$	0	0
$619$	1.32500e8	0.558657	0.279328	−	0.960196i	$-0.409888\pi$
$619$	1.32500e8	0.558657	0.279328	+	0.960196i	$0.409888\pi$
$620$	0	0
$621$	− 1.62910e7i	− 0.0680256i
$622$	0	0
$623$	2.58326e8i	1.06833i
$624$	0	0
$625$	−4.44757e7	−0.182172
$626$	0	0
$627$	1.97186e8	0.799969
$628$	0	0
$629$	7.11829e8i	2.86038i
$630$	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
$635$	−6.99687e7	−0.273264
$636$	0	0
$637$	− 3.96132e8i	− 1.53257i
$638$	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
$643$	1.05651e8	0.397410	0.198705	−	0.980059i	$-0.436326\pi$
$643$	1.05651e8	0.397410	0.198705	+	0.980059i	$0.436326\pi$
$644$	0	0
$645$	3.82035e7i	0.142372i
$646$	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
$651$	−4.61503e8	−1.67275
$652$	0	0
$653$	1.71315e7i	0.0615257i	0.999527	+	0.0307628i	$0.00979366\pi$
$653$	1.71315e7i	0.0615257i	−0.999527	+	0.0307628i	$0.990206\pi$
$654$	0	0
$655$	− 3.21079e8i	− 1.14258i
$656$	0	0
$657$	−2.21623e7	−0.0781482
$658$	0	0
$659$	3.62898e8	1.26803	0.634013	−	0.773323i	$-0.281407\pi$
$659$	3.62898e8	1.26803	0.634013	+	0.773323i	$0.281407\pi$
$660$	0	0
$661$	− 4.90659e8i	− 1.69893i	−0.527645	−	0.849465i	$-0.676925\pi$
$661$	− 4.90659e8i	− 1.69893i	0.527645	−	0.849465i	$-0.323075\pi$
$662$	0	0
$663$	3.64747e8i	1.25156i
$664$	0	0
$665$	−4.55800e8	−1.54992
$666$	0	0
$667$	−1.82530e8	−0.615114
$668$	0	0
$669$	− 2.30950e8i	− 0.771330i
$670$	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
$675$	−3.15681e7	−0.102645
$676$	0	0
$677$	− 1.98782e8i	− 0.640635i	−0.947310	−	0.320318i	$-0.896210\pi$
$677$	− 1.98782e8i	− 0.640635i	0.947310	−	0.320318i	$-0.103790\pi$
$678$	0	0
$679$	− 5.27772e8i	− 1.68592i
$680$	0	0
$681$	−6.22387e7	−0.197070
$682$	0	0
$683$	−795875.	−0.00249794	−0.00124897	−	0.999999i	$-0.500398\pi$
$683$	−795875.	−0.00249794	−0.00124897	+	0.999999i	$0.500398\pi$
$684$	0	0
$685$	− 1.59309e8i	− 0.495641i
$686$	0	0
$687$	5.79684e7i	0.178781i
$688$	0	0
$689$	3.04985e8	0.932439
$690$	0	0
$691$	6.26115e8	1.89767	0.948833	−	0.315778i	$-0.102266\pi$
$691$	6.26115e8	1.89767	0.948833	+	0.315778i	$0.102266\pi$
$692$	0	0
$693$	1.50851e8i	0.453261i
$694$	0	0
$695$	− 2.88027e8i	− 0.857982i
$696$	0	0
$697$	5.98004e8	1.76606
$698$	0	0
$699$	8.19139e7	0.239843
$700$	0	0
$701$	− 7.92815e7i	− 0.230154i	−0.993357	−	0.115077i	$-0.963289\pi$
$701$	− 7.92815e7i	− 0.230154i	0.993357	−	0.115077i	$-0.0367114\pi$
$702$	0	0
$703$	− 8.70894e8i	− 2.50668i
$704$	0	0
$705$	7.81966e7	0.223162
$706$	0	0
$707$	4.74764e8	1.34344
$708$	0	0
$709$	− 6.27826e7i	− 0.176157i	−0.996114	−	0.0880786i	$-0.971927\pi$
$709$	− 6.27826e7i	− 0.176157i	0.996114	−	0.0880786i	$-0.0280727\pi$
$710$	0	0
$711$	1.88376e8i	0.524102i
$712$	0	0
$713$	−2.48764e8	−0.686309
$714$	0	0
$715$	−2.84283e8	−0.777737
$716$	0	0
$717$	1.24126e8i	0.336747i
$718$	0	0
$719$	3.24833e8i	0.873924i	0.899480	+	0.436962i	$0.143946\pi$
$719$	3.24833e8i	0.873924i	−0.899480	+	0.436962i	$0.856054\pi$
$720$	0	0
$721$	−1.82874e8	−0.487918
$722$	0	0
$723$	−2.25190e7	−0.0595845
$724$	0	0
$725$	3.53699e8i	0.928154i
$726$	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
$731$	−2.44658e8	−0.626335
$732$	0	0
$733$	− 2.79408e8i	− 0.709458i	−0.934969	−	0.354729i	$-0.884573\pi$
$733$	− 2.79408e8i	− 0.709458i	0.934969	−	0.354729i	$-0.115427\pi$
$734$	0	0
$735$	− 1.92096e8i	− 0.483789i
$736$	0	0
$737$	5.11428e7	0.127756
$738$	0	0
$739$	−5.00369e8	−1.23981	−0.619907	−	0.784675i	$-0.712830\pi$
$739$	−5.00369e8	−1.23981	−0.619907	+	0.784675i	$0.712830\pi$
$740$	0	0
$741$	− 4.46254e8i	− 1.09680i
$742$	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
$747$	1.62802e8	0.390568
$748$	0	0
$749$	− 5.94721e8i	− 1.41536i
$750$	0	0
$751$	4.54568e8i	1.07319i	0.843838	+	0.536597i	$0.180291\pi$
$751$	4.54568e8i	1.07319i	−0.843838	+	0.536597i	$0.819709\pi$
$752$	0	0
$753$	−1.43464e7	−0.0336015
$754$	0	0
$755$	−3.23464e8	−0.751598
$756$	0	0
$757$	3.55132e8i	0.818657i	0.912387	+	0.409329i	$0.134237\pi$
$757$	3.55132e8i	0.818657i	−0.912387	+	0.409329i	$0.865763\pi$
$758$	0	0
$759$	8.13133e7i	0.185967i
$760$	0	0
$761$	−7.07962e8	−1.60641	−0.803204	−	0.595704i	$-0.796873\pi$
$761$	−7.07962e8	−1.60641	−0.803204	+	0.595704i	$0.796873\pi$
$762$	0	0
$763$	−3.79463e8	−0.854272
$764$	0	0
$765$	1.76876e8i	0.395081i
$766$	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
$771$	−8.93856e7	−0.195031
$772$	0	0
$773$	5.71149e8i	1.23655i	0.785963	+	0.618274i	$0.212168\pi$
$773$	5.71149e8i	1.23655i	−0.785963	+	0.618274i	$0.787832\pi$
$774$	0	0
$775$	4.82046e8i	1.03558i
$776$	0	0
$777$	6.66251e8	1.42028
$778$	0	0
$779$	−7.31634e8	−1.54768
$780$	0	0
$781$	4.26450e8i	0.895189i
$782$	0	0
$783$	− 1.60770e8i	− 0.334904i
$784$	0	0
$785$	4.96336e8	1.02605
$786$	0	0
$787$	−1.56385e8	−0.320828	−0.160414	−	0.987050i	$-0.551283\pi$
$787$	−1.56385e8	−0.320828	−0.160414	+	0.987050i	$0.551283\pi$
$788$	0	0
$789$	2.13519e8i	0.434717i
$790$	0	0
$791$	9.76239e7i	0.197254i
$792$	0	0
$793$	1.45958e8	0.292691
$794$	0	0
$795$	1.47896e8	0.294343
$796$	0	0
$797$	6.02539e8i	1.19017i	0.803661	+	0.595087i	$0.202882\pi$
$797$	6.02539e8i	1.19017i	−0.803661	+	0.595087i	$0.797118\pi$
$798$	0	0
$799$	5.00777e8i	0.981756i
$800$	0	0
$801$	−1.22646e8	−0.238647
$802$	0	0
$803$	1.10619e8	0.213640
$804$	0	0
$805$	− 1.87958e8i	− 0.360307i
$806$	0	0
$807$	− 3.04392e8i	− 0.579178i
$808$	0	0
$809$	2.30938e8	0.436164	0.218082	−	0.975930i	$-0.430020\pi$
$809$	2.30938e8	0.436164	0.218082	+	0.975930i	$0.430020\pi$
$810$	0	0
$811$	−1.15316e8	−0.216185	−0.108093	−	0.994141i	$-0.534474\pi$
$811$	−1.15316e8	−0.216185	−0.108093	+	0.994141i	$0.534474\pi$
$812$	0	0
$813$	− 1.37099e8i	− 0.255131i
$814$	0	0
$815$	− 4.37511e8i	− 0.808195i
$816$	0	0
$817$	2.99329e8	0.548887
$818$	0	0
$819$	3.41393e8	0.621446
$820$	0	0
$821$	8.70428e7i	0.157291i	0.996903	+	0.0786454i	$0.0250595\pi$
$821$	8.70428e7i	0.157291i	−0.996903	+	0.0786454i	$0.974941\pi$
$822$	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
$827$	−6.20614e8	−1.09725	−0.548624	−	0.836069i	$-0.684848\pi$
$827$	−6.20614e8	−1.09725	−0.548624	+	0.836069i	$0.684848\pi$
$828$	0	0
$829$	1.05341e9i	1.84898i	0.381206	+	0.924490i	$0.375509\pi$
$829$	1.05341e9i	1.84898i	−0.381206	+	0.924490i	$0.624491\pi$
$830$	0	0
$831$	2.05794e7i	0.0358617i
$832$	0	0
$833$	1.23019e9	2.12833
$834$	0	0
$835$	1.09067e8	0.187341
$836$	0	0
$837$	− 2.19109e8i	− 0.373666i
$838$	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
$843$	−2.76319e8	−0.461241
$844$	0	0
$845$	2.31208e8i	0.383207i
$846$	0	0
$847$	1.53784e8i	0.253081i
$848$	0	0
$849$	−6.45972e8	−1.05558
$850$	0	0
$851$	3.59130e8	0.582724
$852$	0	0
$853$	− 8.06034e8i	− 1.29869i	−0.760493	−	0.649346i	$-0.775043\pi$
$853$	− 8.06034e8i	− 1.29869i	0.760493	−	0.649346i	$-0.224957\pi$
$854$	0	0
$855$	− 2.16401e8i	− 0.346228i
$856$	0	0
$857$	−4.09751e8	−0.650995	−0.325497	−	0.945543i	$-0.605532\pi$
$857$	−4.09751e8	−0.650995	−0.325497	+	0.945543i	$0.605532\pi$
$858$	0	0
$859$	−8.81567e8	−1.39084	−0.695418	−	0.718606i	$-0.744781\pi$
$859$	−8.81567e8	−1.39084	−0.695418	+	0.718606i	$0.744781\pi$
$860$	0	0
$861$	− 5.59714e8i	− 0.876914i
$862$	0	0
$863$	5.12988e8i	0.798132i	0.916922	+	0.399066i	$0.130666\pi$
$863$	5.12988e8i	0.798132i	−0.916922	+	0.399066i	$0.869334\pi$
$864$	0	0
$865$	2.45448e8	0.379237
$866$	0	0
$867$	−7.56462e8	−1.16073
$868$	0	0
$869$	− 9.40241e8i	− 1.43278i
$870$	0	0
$871$	− 1.15742e8i	− 0.175161i
$872$	0	0
$873$	2.50571e8	0.376607
$874$	0	0
$875$	−1.04710e9	−1.56301
$876$	0	0
$877$	− 1.12822e8i	− 0.167262i	−0.996497	−	0.0836308i	$-0.973348\pi$
$877$	− 1.12822e8i	− 0.167262i	0.996497	−	0.0836308i	$-0.0266516\pi$
$878$	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
$883$	8.09446e8	1.17573	0.587863	−	0.808961i	$-0.299970\pi$
$883$	8.09446e8	1.17573	0.587863	+	0.808961i	$0.299970\pi$
$884$	0	0
$885$	1.57410e8i	0.227092i
$886$	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
$891$	−7.16199e7	−0.101251
$892$	0	0
$893$	− 6.12680e8i	− 0.860359i
$894$	0	0
$895$	7.68476e8i	1.07192i
$896$	0	0
$897$	1.84021e8	0.254971
$898$	0	0
$899$	−2.45497e9	−3.37884
$900$	0	0
$901$	9.47135e8i	1.29490i
$902$	0	0
$903$	2.28993e8i	0.310998i
$904$	0	0
$905$	2.84683e8	0.384074
$906$	0	0
$907$	−7.63921e8	−1.02383	−0.511913	−	0.859037i	$-0.671063\pi$
$907$	−7.63921e8	−1.02383	−0.511913	+	0.859037i	$0.671063\pi$
$908$	0	0
$909$	2.25405e8i	0.300104i
$910$	0	0
$911$	− 9.16841e7i	− 0.121266i	−0.998160	−	0.0606330i	$-0.980688\pi$
$911$	− 9.16841e7i	− 0.121266i	0.998160	−	0.0606330i	$-0.0193119\pi$
$912$	0	0
$913$	−8.12594e8	−1.06773
$914$	0	0
$915$	7.07794e7	0.0923940
$916$	0	0
$917$	− 1.92455e9i	− 2.49587i
$918$	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
$923$	9.65105e8	1.22735
$924$	0	0
$925$	− 6.95909e8i	− 0.879280i
$926$	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
$931$	−1.50509e9	−1.86515
$932$	0	0
$933$	8.28352e8i	1.01993i
$934$	0	0
$935$	− 8.82845e8i	− 1.08006i
$936$	0	0
$937$	7.69038e8	0.934822	0.467411	−	0.884040i	$-0.345187\pi$
$937$	7.69038e8	0.934822	0.467411	+	0.884040i	$0.345187\pi$
$938$	0	0
$939$	−4.96730e8	−0.599961
$940$	0	0
$941$	6.80625e8i	0.816843i	0.912793	+	0.408422i	$0.133921\pi$
$941$	6.80625e8i	0.816843i	−0.912793	+	0.408422i	$0.866079\pi$
$942$	0	0
$943$	− 3.01703e8i	− 0.359787i
$944$	0	0
$945$	1.65551e8	0.196172
$946$	0	0
$947$	4.54633e8	0.535317	0.267659	−	0.963514i	$-0.413750\pi$
$947$	4.54633e8	0.535317	0.267659	+	0.963514i	$0.413750\pi$
$948$	0	0
$949$	− 2.50343e8i	− 0.292912i
$950$	0	0
$951$	− 5.19349e8i	− 0.603834i
$952$	0	0
$953$	−6.97601e8	−0.805987	−0.402994	−	0.915203i	$-0.632030\pi$
$953$	−6.97601e8	−0.805987	−0.402994	+	0.915203i	$0.632030\pi$
$954$	0	0
$955$	3.12977e8	0.359337
$956$	0	0
$957$	8.02453e8i	0.915554i
$958$	0	0
$959$	− 9.54900e8i	− 1.08268i
$960$	0	0
$961$	−2.45830e9	−2.76991
$962$	0	0
$963$	2.82357e8	0.316169
$964$	0	0
$965$	7.32769e8i	0.815428i
$966$	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
$971$	8.23470e8	0.899477	0.449738	−	0.893160i	$-0.351517\pi$
$971$	8.23470e8	0.899477	0.449738	+	0.893160i	$0.351517\pi$
$972$	0	0
$973$	− 1.72644e9i	− 1.87419i
$974$	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
$979$	6.12164e8	0.652408
$980$	0	0
$981$	− 1.80159e8i	− 0.190831i
$982$	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
$987$	4.68713e8	0.487478
$988$	0	0
$989$	1.23434e8i	0.127599i
$990$	0	0
$991$	− 1.26328e9i	− 1.29801i	−0.760784	−	0.649005i	$-0.775186\pi$
$991$	− 1.26328e9i	− 1.29801i	0.760784	−	0.649005i	$-0.224814\pi$
$992$	0	0
$993$	−3.67315e7	−0.0375138
$994$	0	0
$995$	−3.71014e8	−0.376635
$996$	0	0
$997$	8.99928e8i	0.908076i	0.890982	+	0.454038i	$0.150017\pi$
$997$	8.99928e8i	0.908076i	−0.890982	+	0.454038i	$0.849983\pi$
$998$	0	0
$999$	3.16318e8i	0.317268i

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

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