LMFDB - Embedded newform 320.5.b.d.191.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,5,Mod(191,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("320.191");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	320.b (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:	$33.0783881868$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.246034965625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 7x^{6} - 21x^{5} + 49x^{4} - 84x^{3} + 112x^{2} - 192x + 256 $ x^8 - 3x^7 + 7x^6 - 21x^5 + 49x^4 - 84x^3 + 112x^2 - 192*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{28}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.2
Root			$1.95003 - 0.444269i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.5.b.d.191.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.9912i q^{3} +11.1803 q^{5} -78.0345i q^{7} -87.7712 q^{9} +O(q^{10})$	$q-12.9912i q^{3} +11.1803 q^{5} -78.0345i q^{7} -87.7712 q^{9} -95.6447i q^{11} +159.654 q^{13} -145.246i q^{15} +22.5103 q^{17} -324.934i q^{19} -1013.76 q^{21} -204.951i q^{23} +125.000 q^{25} +87.9664i q^{27} -295.747 q^{29} +407.262i q^{31} -1242.54 q^{33} -872.453i q^{35} +2156.28 q^{37} -2074.10i q^{39} +1368.98 q^{41} +1234.22i q^{43} -981.312 q^{45} +1983.43i q^{47} -3688.39 q^{49} -292.436i q^{51} -4595.72 q^{53} -1069.34i q^{55} -4221.28 q^{57} -1389.15i q^{59} -2651.94 q^{61} +6849.19i q^{63} +1784.99 q^{65} +8936.51i q^{67} -2662.56 q^{69} -3375.13i q^{71} -1467.63 q^{73} -1623.90i q^{75} -7463.59 q^{77} +6306.01i q^{79} -5966.68 q^{81} +6104.16i q^{83} +251.673 q^{85} +3842.11i q^{87} +1708.55 q^{89} -12458.5i q^{91} +5290.82 q^{93} -3632.87i q^{95} -1988.16 q^{97} +8394.85i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 328 q^{9}+O(q^{10}) $ 8 * q - 328 * q^9	$ 8 q - 328 q^{9} - 352 q^{13} - 48 q^{17} - 16 q^{21} + 1000 q^{25} - 1200 q^{29} - 1120 q^{33} + 5728 q^{37} + 4896 q^{41} + 400 q^{45} - 5768 q^{49} - 2592 q^{53} + 3840 q^{57} - 7936 q^{61} - 1200 q^{65} + 2256 q^{69} - 14448 q^{73} - 2400 q^{77} - 936 q^{81} - 11200 q^{85} + 23760 q^{89} - 11360 q^{93} - 4368 q^{97}+O(q^{100}) $ 8 * q - 328 * q^9 - 352 * q^13 - 48 * q^17 - 16 * q^21 + 1000 * q^25 - 1200 * q^29 - 1120 * q^33 + 5728 * q^37 + 4896 * q^41 + 400 * q^45 - 5768 * q^49 - 2592 * q^53 + 3840 * q^57 - 7936 * q^61 - 1200 * q^65 + 2256 * q^69 - 14448 * q^73 - 2400 * q^77 - 936 * q^81 - 11200 * q^85 + 23760 * q^89 - 11360 * q^93 - 4368 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 12.9912i	− 1.44347i	−0.692171	−	0.721733i	$-0.743346\pi$
$3$	− 12.9912i	− 1.44347i	0.692171	−	0.721733i	$-0.256654\pi$
$4$	0	0
$5$	11.1803	0.447214
$5$	11.1803	0.447214
$6$	0	0
$7$	− 78.0345i	− 1.59254i	−0.604940	−	0.796271i	$-0.706803\pi$
$7$	− 78.0345i	− 1.59254i	0.604940	−	0.796271i	$-0.293197\pi$
$8$	0	0
$9$	−87.7712	−1.08360
$10$	0	0
$11$	− 95.6447i	− 0.790452i	−0.918584	−	0.395226i	$-0.870666\pi$
$11$	− 95.6447i	− 0.790452i	0.918584	−	0.395226i	$-0.129334\pi$
$12$	0	0
$13$	159.654	0.944698	0.472349	−	0.881412i	$-0.343406\pi$
$13$	159.654	0.944698	0.472349	+	0.881412i	$0.343406\pi$
$14$	0	0
$15$	− 145.246i	− 0.645538i
$16$	0	0
$17$	22.5103	0.0778903	0.0389451	−	0.999241i	$-0.487600\pi$
$17$	22.5103	0.0778903	0.0389451	+	0.999241i	$0.487600\pi$
$18$	0	0
$19$	− 324.934i	− 0.900094i	−0.893005	−	0.450047i	$-0.851407\pi$
$19$	− 324.934i	− 0.900094i	0.893005	−	0.450047i	$-0.148593\pi$
$20$	0	0
$21$	−1013.76	−2.29878
$22$	0	0
$23$	− 204.951i	− 0.387431i	−0.981058	−	0.193715i	$-0.937946\pi$
$23$	− 204.951i	− 0.387431i	0.981058	−	0.193715i	$-0.0620538\pi$
$24$	0	0
$25$	125.000	0.200000
$26$	0	0
$27$	87.9664i	0.120667i
$28$	0	0
$29$	−295.747	−0.351662	−0.175831	−	0.984420i	$-0.556261\pi$
$29$	−295.747	−0.351662	−0.175831	+	0.984420i	$0.556261\pi$
$30$	0	0
$31$	407.262i	0.423789i	0.977293	+	0.211895i	$0.0679634\pi$
$31$	407.262i	0.423789i	−0.977293	+	0.211895i	$0.932037\pi$
$32$	0	0
$33$	−1242.54	−1.14099
$34$	0	0
$35$	− 872.453i	− 0.712206i
$36$	0	0
$37$	2156.28	1.57508	0.787540	−	0.616263i	$-0.211354\pi$
$37$	2156.28	1.57508	0.787540	+	0.616263i	$0.211354\pi$
$38$	0	0
$39$	− 2074.10i	− 1.36364i
$40$	0	0
$41$	1368.98	0.814383	0.407191	−	0.913343i	$-0.366508\pi$
$41$	1368.98	0.814383	0.407191	+	0.913343i	$0.366508\pi$
$42$	0	0
$43$	1234.22i	0.667507i	0.942660	+	0.333754i	$0.108315\pi$
$43$	1234.22i	0.667507i	−0.942660	+	0.333754i	$0.891685\pi$
$44$	0	0
$45$	−981.312	−0.484599
$46$	0	0
$47$	1983.43i	0.897887i	0.893560	+	0.448943i	$0.148200\pi$
$47$	1983.43i	0.897887i	−0.893560	+	0.448943i	$0.851800\pi$
$48$	0	0
$49$	−3688.39	−1.53619
$50$	0	0
$51$	− 292.436i	− 0.112432i
$52$	0	0
$53$	−4595.72	−1.63607	−0.818035	−	0.575168i	$-0.804937\pi$
$53$	−4595.72	−1.63607	−0.818035	+	0.575168i	$0.804937\pi$
$54$	0	0
$55$	− 1069.34i	− 0.353501i
$56$	0	0
$57$	−4221.28	−1.29926
$58$	0	0
$59$	− 1389.15i	− 0.399068i	−0.979891	−	0.199534i	$-0.936057\pi$
$59$	− 1389.15i	− 0.399068i	0.979891	−	0.199534i	$-0.0639428\pi$
$60$	0	0
$61$	−2651.94	−0.712694	−0.356347	−	0.934354i	$-0.615978\pi$
$61$	−2651.94	−0.712694	−0.356347	+	0.934354i	$0.615978\pi$
$62$	0	0
$63$	6849.19i	1.72567i
$64$	0	0
$65$	1784.99	0.422482
$66$	0	0
$67$	8936.51i	1.99076i	0.0960241	+	0.995379i	$0.469387\pi$
$67$	8936.51i	1.99076i	−0.0960241	+	0.995379i	$0.530613\pi$
$68$	0	0
$69$	−2662.56	−0.559243
$70$	0	0
$71$	− 3375.13i	− 0.669535i	−0.942301	−	0.334767i	$-0.891342\pi$
$71$	− 3375.13i	− 0.669535i	0.942301	−	0.334767i	$-0.108658\pi$
$72$	0	0
$73$	−1467.63	−0.275404	−0.137702	−	0.990474i	$-0.543972\pi$
$73$	−1467.63	−0.275404	−0.137702	+	0.990474i	$0.543972\pi$
$74$	0	0
$75$	− 1623.90i	− 0.288693i
$76$	0	0
$77$	−7463.59	−1.25883
$78$	0	0
$79$	6306.01i	1.01042i	0.862997	+	0.505208i	$0.168584\pi$
$79$	6306.01i	1.01042i	−0.862997	+	0.505208i	$0.831416\pi$
$80$	0	0
$81$	−5966.68	−0.909416
$82$	0	0
$83$	6104.16i	0.886073i	0.896504	+	0.443037i	$0.146099\pi$
$83$	6104.16i	0.886073i	−0.896504	+	0.443037i	$0.853901\pi$
$84$	0	0
$85$	251.673	0.0348336
$86$	0	0
$87$	3842.11i	0.507612i
$88$	0	0
$89$	1708.55	0.215698	0.107849	−	0.994167i	$-0.465604\pi$
$89$	1708.55	0.215698	0.107849	+	0.994167i	$0.465604\pi$
$90$	0	0
$91$	− 12458.5i	− 1.50447i
$92$	0	0
$93$	5290.82	0.611726
$94$	0	0
$95$	− 3632.87i	− 0.402534i
$96$	0	0
$97$	−1988.16	−0.211304	−0.105652	−	0.994403i	$-0.533693\pi$
$97$	−1988.16	−0.211304	−0.105652	+	0.994403i	$0.533693\pi$
$98$	0	0
$99$	8394.85i	0.856530i
$100$	0	0
$101$	6012.56	0.589408	0.294704	−	0.955589i	$-0.404779\pi$
$101$	6012.56	0.589408	0.294704	+	0.955589i	$0.404779\pi$
$102$	0	0
$103$	− 10161.1i	− 0.957780i	−0.877875	−	0.478890i	$-0.841039\pi$
$103$	− 10161.1i	− 0.957780i	0.877875	−	0.478890i	$-0.158961\pi$
$104$	0	0
$105$	−11334.2	−1.02805
$106$	0	0
$107$	− 12961.6i	− 1.13212i	−0.824364	−	0.566060i	$-0.808467\pi$
$107$	− 12961.6i	− 1.13212i	0.824364	−	0.566060i	$-0.191533\pi$
$108$	0	0
$109$	−1002.18	−0.0843517	−0.0421758	−	0.999110i	$-0.513429\pi$
$109$	−1002.18	−0.0843517	−0.0421758	+	0.999110i	$0.513429\pi$
$110$	0	0
$111$	− 28012.7i	− 2.27358i
$112$	0	0
$113$	−9956.56	−0.779745	−0.389872	−	0.920869i	$-0.627481\pi$
$113$	−9956.56	−0.779745	−0.389872	+	0.920869i	$0.627481\pi$
$114$	0	0
$115$	− 2291.42i	− 0.173264i
$116$	0	0
$117$	−14013.0	−1.02367
$118$	0	0
$119$	− 1756.58i	− 0.124044i
$120$	0	0
$121$	5493.10	0.375186
$122$	0	0
$123$	− 17784.7i	− 1.17553i
$124$	0	0
$125$	1397.54	0.0894427
$126$	0	0
$127$	21746.1i	1.34826i	0.738613	+	0.674130i	$0.235481\pi$
$127$	21746.1i	1.34826i	−0.738613	+	0.674130i	$0.764519\pi$
$128$	0	0
$129$	16034.0	0.963524
$130$	0	0
$131$	− 26640.6i	− 1.55239i	−0.630491	−	0.776197i	$-0.717146\pi$
$131$	− 26640.6i	− 1.55239i	0.630491	−	0.776197i	$-0.282854\pi$
$132$	0	0
$133$	−25356.1	−1.43344
$134$	0	0
$135$	983.494i	0.0539640i
$136$	0	0
$137$	34924.8	1.86077	0.930385	−	0.366584i	$-0.119473\pi$
$137$	34924.8	1.86077	0.930385	+	0.366584i	$0.119473\pi$
$138$	0	0
$139$	16591.6i	0.858736i	0.903130	+	0.429368i	$0.141264\pi$
$139$	16591.6i	0.858736i	−0.903130	+	0.429368i	$0.858736\pi$
$140$	0	0
$141$	25767.2	1.29607
$142$	0	0
$143$	− 15270.1i	− 0.746738i
$144$	0	0
$145$	−3306.56	−0.157268
$146$	0	0
$147$	47916.6i	2.21744i
$148$	0	0
$149$	10061.9	0.453217	0.226609	−	0.973986i	$-0.427236\pi$
$149$	10061.9	0.453217	0.226609	+	0.973986i	$0.427236\pi$
$150$	0	0
$151$	− 2823.63i	− 0.123838i	−0.998081	−	0.0619190i	$-0.980278\pi$
$151$	− 2823.63i	− 0.123838i	0.998081	−	0.0619190i	$-0.0197221\pi$
$152$	0	0
$153$	−1975.76	−0.0844016
$154$	0	0
$155$	4553.32i	0.189524i
$156$	0	0
$157$	39136.1	1.58774	0.793869	−	0.608089i	$-0.208064\pi$
$157$	39136.1	1.58774	0.793869	+	0.608089i	$0.208064\pi$
$158$	0	0
$159$	59703.9i	2.36161i
$160$	0	0
$161$	−15993.2	−0.616999
$162$	0	0
$163$	− 9169.48i	− 0.345119i	−0.984999	−	0.172560i	$-0.944796\pi$
$163$	− 9169.48i	− 0.345119i	0.984999	−	0.172560i	$-0.0552038\pi$
$164$	0	0
$165$	−13892.0	−0.510267
$166$	0	0
$167$	− 5446.76i	− 0.195301i	−0.995221	−	0.0976507i	$-0.968867\pi$
$167$	− 5446.76i	− 0.195301i	0.995221	−	0.0976507i	$-0.0311328\pi$
$168$	0	0
$169$	−3071.61	−0.107546
$170$	0	0
$171$	28519.9i	0.975338i
$172$	0	0
$173$	16187.9	0.540876	0.270438	−	0.962737i	$-0.412832\pi$
$173$	16187.9	0.540876	0.270438	+	0.962737i	$0.412832\pi$
$174$	0	0
$175$	− 9754.32i	− 0.318508i
$176$	0	0
$177$	−18046.8	−0.576041
$178$	0	0
$179$	3484.14i	0.108740i	0.998521	+	0.0543700i	$0.0173151\pi$
$179$	3484.14i	0.108740i	−0.998521	+	0.0543700i	$0.982685\pi$
$180$	0	0
$181$	12654.0	0.386251	0.193125	−	0.981174i	$-0.438138\pi$
$181$	12654.0	0.386251	0.193125	+	0.981174i	$0.438138\pi$
$182$	0	0
$183$	34451.8i	1.02875i
$184$	0	0
$185$	24108.0	0.704397
$186$	0	0
$187$	− 2152.99i	− 0.0615685i
$188$	0	0
$189$	6864.42	0.192168
$190$	0	0
$191$	− 12436.5i	− 0.340903i	−0.985366	−	0.170452i	$-0.945477\pi$
$191$	− 12436.5i	− 0.340903i	0.985366	−	0.170452i	$-0.0545227\pi$
$192$	0	0
$193$	21537.6	0.578206	0.289103	−	0.957298i	$-0.406643\pi$
$193$	21537.6	0.578206	0.289103	+	0.957298i	$0.406643\pi$
$194$	0	0
$195$	− 23189.1i	− 0.609838i
$196$	0	0
$197$	30840.7	0.794679	0.397339	−	0.917672i	$-0.369934\pi$
$197$	30840.7	0.794679	0.397339	+	0.917672i	$0.369934\pi$
$198$	0	0
$199$	− 57562.7i	− 1.45357i	−0.686866	−	0.726784i	$-0.741014\pi$
$199$	− 57562.7i	− 1.45357i	0.686866	−	0.726784i	$-0.258986\pi$
$200$	0	0
$201$	116096.	2.87359
$202$	0	0
$203$	23078.5i	0.560036i
$204$	0	0
$205$	15305.6	0.364203
$206$	0	0
$207$	17988.8i	0.419818i
$208$	0	0
$209$	−31078.2	−0.711481
$210$	0	0
$211$	− 42485.0i	− 0.954270i	−0.878830	−	0.477135i	$-0.841675\pi$
$211$	− 42485.0i	− 0.954270i	0.878830	−	0.477135i	$-0.158325\pi$
$212$	0	0
$213$	−43846.9	−0.966451
$214$	0	0
$215$	13799.0i	0.298518i
$216$	0	0
$217$	31780.5	0.674902
$218$	0	0
$219$	19066.2i	0.397536i
$220$	0	0
$221$	3593.86	0.0735828
$222$	0	0
$223$	− 14687.7i	− 0.295354i	−0.989036	−	0.147677i	$-0.952820\pi$
$223$	− 14687.7i	− 0.295354i	0.989036	−	0.147677i	$-0.0471796\pi$
$224$	0	0
$225$	−10971.4	−0.216719
$226$	0	0
$227$	− 54734.7i	− 1.06221i	−0.847305	−	0.531106i	$-0.821777\pi$
$227$	− 54734.7i	− 1.06221i	0.847305	−	0.531106i	$-0.178223\pi$
$228$	0	0
$229$	14583.3	0.278090	0.139045	−	0.990286i	$-0.455597\pi$
$229$	14583.3	0.278090	0.139045	+	0.990286i	$0.455597\pi$
$230$	0	0
$231$	96961.0i	1.81708i
$232$	0	0
$233$	−99275.8	−1.82865	−0.914327	−	0.404976i	$-0.867280\pi$
$233$	−99275.8	−1.82865	−0.914327	+	0.404976i	$0.867280\pi$
$234$	0	0
$235$	22175.4i	0.401547i
$236$	0	0
$237$	81922.6	1.45850
$238$	0	0
$239$	− 61775.6i	− 1.08149i	−0.841188	−	0.540743i	$-0.818143\pi$
$239$	− 61775.6i	− 1.08149i	0.841188	−	0.540743i	$-0.181857\pi$
$240$	0	0
$241$	97702.4	1.68218	0.841088	−	0.540899i	$-0.181916\pi$
$241$	97702.4	1.68218	0.841088	+	0.540899i	$0.181916\pi$
$242$	0	0
$243$	84639.6i	1.43338i
$244$	0	0
$245$	−41237.5	−0.687005
$246$	0	0
$247$	− 51877.0i	− 0.850317i
$248$	0	0
$249$	79300.3	1.27902
$250$	0	0
$251$	72156.2i	1.14532i	0.819794	+	0.572659i	$0.194088\pi$
$251$	72156.2i	1.14532i	−0.819794	+	0.572659i	$0.805912\pi$
$252$	0	0
$253$	−19602.5	−0.306245
$254$	0	0
$255$	− 3269.53i	− 0.0502811i
$256$	0	0
$257$	−1740.16	−0.0263465	−0.0131732	−	0.999913i	$-0.504193\pi$
$257$	−1740.16	−0.0263465	−0.0131732	+	0.999913i	$0.504193\pi$
$258$	0	0
$259$	− 168265.i	− 2.50838i
$260$	0	0
$261$	25958.1	0.381059
$262$	0	0
$263$	55537.8i	0.802929i	0.915875	+	0.401464i	$0.131499\pi$
$263$	55537.8i	0.802929i	−0.915875	+	0.401464i	$0.868501\pi$
$264$	0	0
$265$	−51381.7	−0.731673
$266$	0	0
$267$	− 22196.1i	− 0.311353i
$268$	0	0
$269$	60126.3	0.830922	0.415461	−	0.909611i	$-0.363620\pi$
$269$	60126.3	0.830922	0.415461	+	0.909611i	$0.363620\pi$
$270$	0	0
$271$	− 65827.7i	− 0.896334i	−0.893950	−	0.448167i	$-0.852077\pi$
$271$	− 65827.7i	− 0.896334i	0.893950	−	0.448167i	$-0.147923\pi$
$272$	0	0
$273$	−161851.	−2.17165
$274$	0	0
$275$	− 11955.6i	− 0.158090i
$276$	0	0
$277$	−61361.9	−0.799723	−0.399861	−	0.916576i	$-0.630942\pi$
$277$	−61361.9	−0.799723	−0.399861	+	0.916576i	$0.630942\pi$
$278$	0	0
$279$	− 35745.8i	− 0.459216i
$280$	0	0
$281$	−26231.4	−0.332207	−0.166104	−	0.986108i	$-0.553119\pi$
$281$	−26231.4	−0.332207	−0.166104	+	0.986108i	$0.553119\pi$
$282$	0	0
$283$	− 91765.5i	− 1.14579i	−0.819627	−	0.572897i	$-0.805820\pi$
$283$	− 91765.5i	− 1.14579i	0.819627	−	0.572897i	$-0.194180\pi$
$284$	0	0
$285$	−47195.4	−0.581045
$286$	0	0
$287$	− 106828.i	− 1.29694i
$288$	0	0
$289$	−83014.3	−0.993933
$290$	0	0
$291$	25828.6i	0.305010i
$292$	0	0
$293$	−82014.2	−0.955331	−0.477665	−	0.878542i	$-0.658517\pi$
$293$	−82014.2	−0.955331	−0.477665	+	0.878542i	$0.658517\pi$
$294$	0	0
$295$	− 15531.2i	− 0.178469i
$296$	0	0
$297$	8413.52	0.0953816
$298$	0	0
$299$	− 32721.2i	− 0.366005i
$300$	0	0
$301$	96311.9	1.06303
$302$	0	0
$303$	− 78110.3i	− 0.850791i
$304$	0	0
$305$	−29649.5	−0.318727
$306$	0	0
$307$	14373.1i	0.152501i	0.997089	+	0.0762507i	$0.0242949\pi$
$307$	14373.1i	0.152501i	−0.997089	+	0.0762507i	$0.975705\pi$
$308$	0	0
$309$	−132005.	−1.38252
$310$	0	0
$311$	178570.i	1.84623i	0.384520	+	0.923117i	$0.374367\pi$
$311$	178570.i	1.84623i	−0.384520	+	0.923117i	$0.625633\pi$
$312$	0	0
$313$	188132.	1.92032	0.960162	−	0.279444i	$-0.0901501\pi$
$313$	188132.	1.92032	0.960162	+	0.279444i	$0.0901501\pi$
$314$	0	0
$315$	76576.3i	0.771744i
$316$	0	0
$317$	−104987.	−1.04477	−0.522383	−	0.852711i	$-0.674957\pi$
$317$	−104987.	−1.04477	−0.522383	+	0.852711i	$0.674957\pi$
$318$	0	0
$319$	28286.7i	0.277971i
$320$	0	0
$321$	−168387.	−1.63418
$322$	0	0
$323$	− 7314.36i	− 0.0701086i
$324$	0	0
$325$	19956.7	0.188940
$326$	0	0
$327$	13019.6i	0.121759i
$328$	0	0
$329$	154776.	1.42992
$330$	0	0
$331$	107333.i	0.979660i	0.871818	+	0.489830i	$0.162941\pi$
$331$	107333.i	0.979660i	−0.871818	+	0.489830i	$0.837059\pi$
$332$	0	0
$333$	−189260.	−1.70675
$334$	0	0
$335$	99913.2i	0.890294i
$336$	0	0
$337$	48193.1	0.424351	0.212176	−	0.977232i	$-0.431945\pi$
$337$	48193.1	0.424351	0.212176	+	0.977232i	$0.431945\pi$
$338$	0	0
$339$	129348.i	1.12554i
$340$	0	0
$341$	38952.4	0.334985
$342$	0	0
$343$	100461.i	0.853904i
$344$	0	0
$345$	−29768.3	−0.250101
$346$	0	0
$347$	− 158887.i	− 1.31956i	−0.751460	−	0.659778i	$-0.770650\pi$
$347$	− 158887.i	− 1.31956i	0.751460	−	0.659778i	$-0.229350\pi$
$348$	0	0
$349$	155827.	1.27935	0.639677	−	0.768643i	$-0.279068\pi$
$349$	155827.	1.27935	0.639677	+	0.768643i	$0.279068\pi$
$350$	0	0
$351$	14044.2i	0.113994i
$352$	0	0
$353$	−197430.	−1.58439	−0.792197	−	0.610265i	$-0.791063\pi$
$353$	−197430.	−1.58439	−0.792197	+	0.610265i	$0.791063\pi$
$354$	0	0
$355$	− 37735.0i	− 0.299425i
$356$	0	0
$357$	−22820.1	−0.179053
$358$	0	0
$359$	− 12434.1i	− 0.0964774i	−0.998836	−	0.0482387i	$-0.984639\pi$
$359$	− 12434.1i	− 0.0964774i	0.998836	−	0.0482387i	$-0.0153608\pi$
$360$	0	0
$361$	24738.9	0.189830
$362$	0	0
$363$	− 71361.9i	− 0.541568i
$364$	0	0
$365$	−16408.6	−0.123164
$366$	0	0
$367$	112753.i	0.837136i	0.908185	+	0.418568i	$0.137468\pi$
$367$	112753.i	0.837136i	−0.908185	+	0.418568i	$0.862532\pi$
$368$	0	0
$369$	−120157.	−0.882461
$370$	0	0
$371$	358625.i	2.60551i
$372$	0	0
$373$	−129467.	−0.930552	−0.465276	−	0.885166i	$-0.654045\pi$
$373$	−129467.	−0.930552	−0.465276	+	0.885166i	$0.654045\pi$
$374$	0	0
$375$	− 18155.8i	− 0.129108i
$376$	0	0
$377$	−47217.2	−0.332214
$378$	0	0
$379$	259093.i	1.80376i	0.431990	+	0.901878i	$0.357812\pi$
$379$	259093.i	1.80376i	−0.431990	+	0.901878i	$0.642188\pi$
$380$	0	0
$381$	282508.	1.94617
$382$	0	0
$383$	− 191511.i	− 1.30556i	−0.757548	−	0.652780i	$-0.773603\pi$
$383$	− 191511.i	− 1.30556i	0.757548	−	0.652780i	$-0.226397\pi$
$384$	0	0
$385$	−83445.5	−0.562965
$386$	0	0
$387$	− 108329.i	− 0.723308i
$388$	0	0
$389$	−255416.	−1.68791	−0.843954	−	0.536415i	$-0.819778\pi$
$389$	−255416.	−1.68791	−0.843954	+	0.536415i	$0.819778\pi$
$390$	0	0
$391$	− 4613.50i	− 0.0301771i
$392$	0	0
$393$	−346094.	−2.24083
$394$	0	0
$395$	70503.3i	0.451872i
$396$	0	0
$397$	142760.	0.905789	0.452894	−	0.891564i	$-0.350391\pi$
$397$	142760.	0.905789	0.452894	+	0.891564i	$0.350391\pi$
$398$	0	0
$399$	329406.i	2.06912i
$400$	0	0
$401$	194681.	1.21070	0.605348	−	0.795961i	$-0.293034\pi$
$401$	194681.	1.21070	0.605348	+	0.795961i	$0.293034\pi$
$402$	0	0
$403$	65020.9i	0.400353i
$404$	0	0
$405$	−66709.5	−0.406703
$406$	0	0
$407$	− 206237.i	− 1.24503i
$408$	0	0
$409$	−89149.0	−0.532930	−0.266465	−	0.963845i	$-0.585856\pi$
$409$	−89149.0	−0.532930	−0.266465	+	0.963845i	$0.585856\pi$
$410$	0	0
$411$	− 453715.i	− 2.68596i
$412$	0	0
$413$	−108402.	−0.635532
$414$	0	0
$415$	68246.6i	0.396264i
$416$	0	0
$417$	215545.	1.23956
$418$	0	0
$419$	− 316683.i	− 1.80384i	−0.431906	−	0.901918i	$-0.642159\pi$
$419$	− 316683.i	− 1.80384i	0.431906	−	0.901918i	$-0.357841\pi$
$420$	0	0
$421$	−107801.	−0.608217	−0.304109	−	0.952637i	$-0.598359\pi$
$421$	−107801.	−0.608217	−0.304109	+	0.952637i	$0.598359\pi$
$422$	0	0
$423$	− 174088.i	− 0.972946i
$424$	0	0
$425$	2813.79	0.0155781
$426$	0	0
$427$	206943.i	1.13500i
$428$	0	0
$429$	−198376.	−1.07789
$430$	0	0
$431$	122818.i	0.661160i	0.943778	+	0.330580i	$0.107244\pi$
$431$	122818.i	0.661160i	−0.943778	+	0.330580i	$0.892756\pi$
$432$	0	0
$433$	−182503.	−0.973406	−0.486703	−	0.873567i	$-0.661801\pi$
$433$	−182503.	−0.973406	−0.486703	+	0.873567i	$0.661801\pi$
$434$	0	0
$435$	42956.1i	0.227011i
$436$	0	0
$437$	−66595.5	−0.348724
$438$	0	0
$439$	− 284219.i	− 1.47477i	−0.675473	−	0.737385i	$-0.736060\pi$
$439$	− 284219.i	− 1.47477i	0.675473	−	0.737385i	$-0.263940\pi$
$440$	0	0
$441$	323735.	1.66461
$442$	0	0
$443$	− 290002.i	− 1.47772i	−0.673858	−	0.738861i	$-0.735364\pi$
$443$	− 290002.i	− 1.47772i	0.673858	−	0.738861i	$-0.264636\pi$
$444$	0	0
$445$	19102.1	0.0964632
$446$	0	0
$447$	− 130716.i	− 0.654204i
$448$	0	0
$449$	53644.3	0.266091	0.133046	−	0.991110i	$-0.457524\pi$
$449$	53644.3	0.266091	0.133046	+	0.991110i	$0.457524\pi$
$450$	0	0
$451$	− 130935.i	− 0.643730i
$452$	0	0
$453$	−36682.4	−0.178756
$454$	0	0
$455$	− 139291.i	− 0.672820i
$456$	0	0
$457$	51320.3	0.245729	0.122865	−	0.992423i	$-0.460792\pi$
$457$	51320.3	0.245729	0.122865	+	0.992423i	$0.460792\pi$
$458$	0	0
$459$	1980.15i	0.00939881i
$460$	0	0
$461$	−71282.3	−0.335413	−0.167706	−	0.985837i	$-0.553636\pi$
$461$	−71282.3	−0.335413	−0.167706	+	0.985837i	$0.553636\pi$
$462$	0	0
$463$	107296.i	0.500519i	0.968179	+	0.250260i	$0.0805160\pi$
$463$	107296.i	0.500519i	−0.968179	+	0.250260i	$0.919484\pi$
$464$	0	0
$465$	59153.1	0.273572
$466$	0	0
$467$	32462.5i	0.148850i	0.997227	+	0.0744248i	$0.0237121\pi$
$467$	32462.5i	0.148850i	−0.997227	+	0.0744248i	$0.976288\pi$
$468$	0	0
$469$	697357.	3.17037
$470$	0	0
$471$	− 508425.i	− 2.29185i
$472$	0	0
$473$	118047.	0.527632
$474$	0	0
$475$	− 40616.8i	− 0.180019i
$476$	0	0
$477$	403372.	1.77284
$478$	0	0
$479$	33919.3i	0.147835i	0.997264	+	0.0739173i	$0.0235501\pi$
$479$	33919.3i	0.147835i	−0.997264	+	0.0739173i	$0.976450\pi$
$480$	0	0
$481$	344259.	1.48798
$482$	0	0
$483$	207771.i	0.890618i
$484$	0	0
$485$	−22228.3	−0.0944980
$486$	0	0
$487$	324548.i	1.36843i	0.729282	+	0.684213i	$0.239854\pi$
$487$	324548.i	1.36843i	−0.729282	+	0.684213i	$0.760146\pi$
$488$	0	0
$489$	−119122.	−0.498168
$490$	0	0
$491$	− 119081.i	− 0.493946i	−0.969022	−	0.246973i	$-0.920564\pi$
$491$	− 119081.i	− 0.493946i	0.969022	−	0.246973i	$-0.0794358\pi$
$492$	0	0
$493$	−6657.36	−0.0273910
$494$	0	0
$495$	93857.3i	0.383052i
$496$	0	0
$497$	−263376.	−1.06626
$498$	0	0
$499$	− 8360.30i	− 0.0335754i	−0.999859	−	0.0167877i	$-0.994656\pi$
$499$	− 8360.30i	− 0.0335754i	0.999859	−	0.0167877i	$-0.00534394\pi$
$500$	0	0
$501$	−70759.9	−0.281911
$502$	0	0
$503$	400265.i	1.58202i	0.611805	+	0.791009i	$0.290444\pi$
$503$	400265.i	1.58202i	−0.611805	+	0.791009i	$0.709556\pi$
$504$	0	0
$505$	67222.4	0.263591
$506$	0	0
$507$	39903.9i	0.155238i
$508$	0	0
$509$	−271797.	−1.04908	−0.524541	−	0.851385i	$-0.675763\pi$
$509$	−271797.	−1.04908	−0.524541	+	0.851385i	$0.675763\pi$
$510$	0	0
$511$	114526.i	0.438592i
$512$	0	0
$513$	28583.3	0.108612
$514$	0	0
$515$	− 113604.i	− 0.428332i
$516$	0	0
$517$	189705.	0.709736
$518$	0	0
$519$	− 210300.i	− 0.780736i
$520$	0	0
$521$	−347896.	−1.28166	−0.640832	−	0.767681i	$-0.721410\pi$
$521$	−347896.	−1.28166	−0.640832	+	0.767681i	$0.721410\pi$
$522$	0	0
$523$	275580.i	1.00750i	0.863850	+	0.503750i	$0.168047\pi$
$523$	275580.i	1.00750i	−0.863850	+	0.503750i	$0.831953\pi$
$524$	0	0
$525$	−126720.	−0.459756
$526$	0	0
$527$	9167.58i	0.0330091i
$528$	0	0
$529$	237836.	0.849898
$530$	0	0
$531$	121928.i	0.432428i
$532$	0	0
$533$	218563.	0.769346
$534$	0	0
$535$	− 144916.i	− 0.506299i
$536$	0	0
$537$	45263.2	0.156963
$538$	0	0
$539$	352775.i	1.21428i
$540$	0	0
$541$	96023.1	0.328081	0.164041	−	0.986454i	$-0.447547\pi$
$541$	96023.1	0.328081	0.164041	+	0.986454i	$0.447547\pi$
$542$	0	0
$543$	− 164390.i	− 0.557540i
$544$	0	0
$545$	−11204.7	−0.0377232
$546$	0	0
$547$	102032.i	0.341007i	0.985357	+	0.170503i	$0.0545394\pi$
$547$	102032.i	0.341007i	−0.985357	+	0.170503i	$0.945461\pi$
$548$	0	0
$549$	232764.	0.772272
$550$	0	0
$551$	96098.4i	0.316529i
$552$	0	0
$553$	492087.	1.60913
$554$	0	0
$555$	− 313192.i	− 1.01677i
$556$	0	0
$557$	423559.	1.36522	0.682612	−	0.730781i	$-0.260844\pi$
$557$	423559.	1.36522	0.682612	+	0.730781i	$0.260844\pi$
$558$	0	0
$559$	197048.i	0.630593i
$560$	0	0
$561$	−27969.9	−0.0888721
$562$	0	0
$563$	498724.i	1.57342i	0.617325	+	0.786708i	$0.288216\pi$
$563$	498724.i	1.57342i	−0.617325	+	0.786708i	$0.711784\pi$
$564$	0	0
$565$	−111318.	−0.348712
$566$	0	0
$567$	465607.i	1.44828i
$568$	0	0
$569$	237219.	0.732697	0.366349	−	0.930478i	$-0.380608\pi$
$569$	237219.	0.732697	0.366349	+	0.930478i	$0.380608\pi$
$570$	0	0
$571$	− 281100.i	− 0.862161i	−0.902314	−	0.431080i	$-0.858133\pi$
$571$	− 281100.i	− 0.862161i	0.902314	−	0.431080i	$-0.141867\pi$
$572$	0	0
$573$	−161565.	−0.492083
$574$	0	0
$575$	− 25618.8i	− 0.0774861i
$576$	0	0
$577$	404501.	1.21498	0.607489	−	0.794328i	$-0.292177\pi$
$577$	404501.	1.21498	0.607489	+	0.794328i	$0.292177\pi$
$578$	0	0
$579$	− 279799.i	− 0.834622i
$580$	0	0
$581$	476335.	1.41111
$582$	0	0
$583$	439556.i	1.29323i
$584$	0	0
$585$	−156670.	−0.457799
$586$	0	0
$587$	335843.i	0.974675i	0.873214	+	0.487337i	$0.162032\pi$
$587$	335843.i	0.974675i	−0.873214	+	0.487337i	$0.837968\pi$
$588$	0	0
$589$	132333.	0.381450
$590$	0	0
$591$	− 400658.i	− 1.14709i
$592$	0	0
$593$	−139571.	−0.396905	−0.198452	−	0.980111i	$-0.563592\pi$
$593$	−139571.	−0.396905	−0.198452	+	0.980111i	$0.563592\pi$
$594$	0	0
$595$	− 19639.2i	− 0.0554740i
$596$	0	0
$597$	−747809.	−2.09818
$598$	0	0
$599$	− 71633.1i	− 0.199646i	−0.995005	−	0.0998228i	$-0.968172\pi$
$599$	− 71633.1i	− 0.199646i	0.995005	−	0.0998228i	$-0.0318276\pi$
$600$	0	0
$601$	524251.	1.45141	0.725706	−	0.688005i	$-0.241514\pi$
$601$	524251.	1.45141	0.725706	+	0.688005i	$0.241514\pi$
$602$	0	0
$603$	− 784369.i	− 2.15718i
$604$	0	0
$605$	61414.7	0.167788
$606$	0	0
$607$	− 89856.9i	− 0.243879i	−0.992538	−	0.121939i	$-0.961089\pi$
$607$	− 89856.9i	− 0.243879i	0.992538	−	0.121939i	$-0.0389114\pi$
$608$	0	0
$609$	299817.	0.808393
$610$	0	0
$611$	316663.i	0.848232i
$612$	0	0
$613$	−58041.7	−0.154461	−0.0772305	−	0.997013i	$-0.524608\pi$
$613$	−58041.7	−0.154461	−0.0772305	+	0.997013i	$0.524608\pi$
$614$	0	0
$615$	− 198839.i	− 0.525715i
$616$	0	0
$617$	−251038.	−0.659431	−0.329716	−	0.944080i	$-0.606953\pi$
$617$	−251038.	−0.659431	−0.329716	+	0.944080i	$0.606953\pi$
$618$	0	0
$619$	37372.0i	0.0975360i	0.998810	+	0.0487680i	$0.0155295\pi$
$619$	37372.0i	0.0975360i	−0.998810	+	0.0487680i	$0.984470\pi$
$620$	0	0
$621$	18028.8	0.0467502
$622$	0	0
$623$	− 133326.i	− 0.343509i
$624$	0	0
$625$	15625.0	0.0400000
$626$	0	0
$627$	403743.i	1.02700i
$628$	0	0
$629$	48538.6	0.122683
$630$	0	0
$631$	− 81639.6i	− 0.205042i	−0.994731	−	0.102521i	$-0.967309\pi$
$631$	− 81639.6i	− 0.205042i	0.994731	−	0.102521i	$-0.0326908\pi$
$632$	0	0
$633$	−551932.	−1.37746
$634$	0	0
$635$	243129.i	0.602960i
$636$	0	0
$637$	−588866.	−1.45124
$638$	0	0
$639$	296239.i	0.725505i
$640$	0	0
$641$	−72207.4	−0.175738	−0.0878690	−	0.996132i	$-0.528006\pi$
$641$	−72207.4	−0.175738	−0.0878690	+	0.996132i	$0.528006\pi$
$642$	0	0
$643$	221605.i	0.535990i	0.963420	+	0.267995i	$0.0863611\pi$
$643$	221605.i	0.535990i	−0.963420	+	0.267995i	$0.913639\pi$
$644$	0	0
$645$	179266.	0.430901
$646$	0	0
$647$	388806.i	0.928805i	0.885624	+	0.464403i	$0.153731\pi$
$647$	388806.i	0.928805i	−0.885624	+	0.464403i	$0.846269\pi$
$648$	0	0
$649$	−132865.	−0.315444
$650$	0	0
$651$	− 412866.i	− 0.974199i
$652$	0	0
$653$	−32328.2	−0.0758150	−0.0379075	−	0.999281i	$-0.512069\pi$
$653$	−32328.2	−0.0758150	−0.0379075	+	0.999281i	$0.512069\pi$
$654$	0	0
$655$	− 297851.i	− 0.694252i
$656$	0	0
$657$	128816.	0.298427
$658$	0	0
$659$	− 593355.i	− 1.36629i	−0.730282	−	0.683146i	$-0.760611\pi$
$659$	− 593355.i	− 1.36629i	0.730282	−	0.683146i	$-0.239389\pi$
$660$	0	0
$661$	574565.	1.31503	0.657516	−	0.753440i	$-0.271607\pi$
$661$	574565.	1.31503	0.657516	+	0.753440i	$0.271607\pi$
$662$	0	0
$663$	− 46688.5i	− 0.106214i
$664$	0	0
$665$	−283490.	−0.641053
$666$	0	0
$667$	60613.6i	0.136244i
$668$	0	0
$669$	−190810.	−0.426334
$670$	0	0
$671$	253644.i	0.563351i
$672$	0	0
$673$	296688.	0.655043	0.327522	−	0.944844i	$-0.393787\pi$
$673$	296688.	0.655043	0.327522	+	0.944844i	$0.393787\pi$
$674$	0	0
$675$	10995.8i	0.0241334i
$676$	0	0
$677$	348459.	0.760282	0.380141	−	0.924929i	$-0.375875\pi$
$677$	348459.	0.760282	0.380141	+	0.924929i	$0.375875\pi$
$678$	0	0
$679$	155145.i	0.336510i
$680$	0	0
$681$	−711070.	−1.53327
$682$	0	0
$683$	555433.i	1.19067i	0.803479	+	0.595333i	$0.202980\pi$
$683$	555433.i	1.19067i	−0.803479	+	0.595333i	$0.797020\pi$
$684$	0	0
$685$	390471.	0.832162
$686$	0	0
$687$	− 189455.i	− 0.401414i
$688$	0	0
$689$	−733725.	−1.54559
$690$	0	0
$691$	− 417213.i	− 0.873780i	−0.899515	−	0.436890i	$-0.856080\pi$
$691$	− 417213.i	− 0.873780i	0.899515	−	0.436890i	$-0.143920\pi$
$692$	0	0
$693$	655088.	1.36406
$694$	0	0
$695$	185500.i	0.384039i
$696$	0	0
$697$	30816.1	0.0634325
$698$	0	0
$699$	1.28971e6i	2.63960i
$700$	0	0
$701$	−752354.	−1.53104	−0.765519	−	0.643413i	$-0.777518\pi$
$701$	−752354.	−1.53104	−0.765519	+	0.643413i	$0.777518\pi$
$702$	0	0
$703$	− 700650.i	− 1.41772i
$704$	0	0
$705$	288086.	0.579620
$706$	0	0
$707$	− 469187.i	− 0.938658i
$708$	0	0
$709$	−480953.	−0.956775	−0.478388	−	0.878149i	$-0.658779\pi$
$709$	−480953.	−0.956775	−0.478388	+	0.878149i	$0.658779\pi$
$710$	0	0
$711$	− 553486.i	− 1.09488i
$712$	0	0
$713$	83468.6	0.164189
$714$	0	0
$715$	− 170724.i	− 0.333952i
$716$	0	0
$717$	−802539.	−1.56109
$718$	0	0
$719$	− 298189.i	− 0.576811i	−0.957508	−	0.288406i	$-0.906875\pi$
$719$	− 298189.i	− 0.576811i	0.957508	−	0.288406i	$-0.0931251\pi$
$720$	0	0
$721$	−792916.	−1.52531
$722$	0	0
$723$	− 1.26927e6i	− 2.42816i
$724$	0	0
$725$	−36968.4	−0.0703323
$726$	0	0
$727$	104436.i	0.197597i	0.995107	+	0.0987984i	$0.0314999\pi$
$727$	104436.i	0.197597i	−0.995107	+	0.0987984i	$0.968500\pi$
$728$	0	0
$729$	616269.	1.15962
$730$	0	0
$731$	27782.7i	0.0519923i
$732$	0	0
$733$	442720.	0.823988	0.411994	−	0.911187i	$-0.364833\pi$
$733$	442720.	0.823988	0.411994	+	0.911187i	$0.364833\pi$
$734$	0	0
$735$	535724.i	0.991668i
$736$	0	0
$737$	854730.	1.57360
$738$	0	0
$739$	− 316611.i	− 0.579744i	−0.957065	−	0.289872i	$-0.906387\pi$
$739$	− 316611.i	− 0.579744i	0.957065	−	0.289872i	$-0.0936128\pi$
$740$	0	0
$741$	−673945.	−1.22740
$742$	0	0
$743$	− 249871.i	− 0.452626i	−0.974055	−	0.226313i	$-0.927333\pi$
$743$	− 249871.i	− 0.452626i	0.974055	−	0.226313i	$-0.0726671\pi$
$744$	0	0
$745$	112495.	0.202685
$746$	0	0
$747$	− 535769.i	− 0.960145i
$748$	0	0
$749$	−1.01146e6	−1.80295
$750$	0	0
$751$	− 745034.i	− 1.32098i	−0.750835	−	0.660490i	$-0.770349\pi$
$751$	− 745034.i	− 1.32098i	0.750835	−	0.660490i	$-0.229651\pi$
$752$	0	0
$753$	937395.	1.65323
$754$	0	0
$755$	− 31569.2i	− 0.0553821i
$756$	0	0
$757$	602045.	1.05060	0.525300	−	0.850917i	$-0.323953\pi$
$757$	602045.	1.05060	0.525300	+	0.850917i	$0.323953\pi$
$758$	0	0
$759$	254659.i	0.442055i
$760$	0	0
$761$	−46391.1	−0.0801061	−0.0400530	−	0.999198i	$-0.512753\pi$
$761$	−46391.1	−0.0801061	−0.0400530	+	0.999198i	$0.512753\pi$
$762$	0	0
$763$	78204.9i	0.134334i
$764$	0	0
$765$	−22089.6	−0.0377455
$766$	0	0
$767$	− 221784.i	− 0.376999i
$768$	0	0
$769$	−169035.	−0.285841	−0.142920	−	0.989734i	$-0.545649\pi$
$769$	−169035.	−0.285841	−0.142920	+	0.989734i	$0.545649\pi$
$770$	0	0
$771$	22606.7i	0.0380302i
$772$	0	0
$773$	−581948.	−0.973924	−0.486962	−	0.873423i	$-0.661895\pi$
$773$	−581948.	−0.973924	−0.486962	+	0.873423i	$0.661895\pi$
$774$	0	0
$775$	50907.7i	0.0847579i
$776$	0	0
$777$	−2.18596e6	−3.62076
$778$	0	0
$779$	− 444827.i	− 0.733021i
$780$	0	0
$781$	−322813.	−0.529235
$782$	0	0
$783$	− 26015.8i	− 0.0424340i
$784$	0	0
$785$	437555.	0.710058
$786$	0	0
$787$	− 739795.i	− 1.19443i	−0.802080	−	0.597217i	$-0.796273\pi$
$787$	− 739795.i	− 1.19443i	0.802080	−	0.597217i	$-0.203727\pi$
$788$	0	0
$789$	721502.	1.15900
$790$	0	0
$791$	776956.i	1.24178i
$792$	0	0
$793$	−423392.	−0.673281
$794$	0	0
$795$	667510.i	1.05615i
$796$	0	0
$797$	59513.8	0.0936917	0.0468459	−	0.998902i	$-0.485083\pi$
$797$	59513.8	0.0936917	0.0468459	+	0.998902i	$0.485083\pi$
$798$	0	0
$799$	44647.6i	0.0699367i
$800$	0	0
$801$	−149961.	−0.233730
$802$	0	0
$803$	140371.i	0.217694i
$804$	0	0
$805$	−178810.	−0.275931
$806$	0	0
$807$	− 781113.i	− 1.19941i
$808$	0	0
$809$	−1.07133e6	−1.63691	−0.818457	−	0.574568i	$-0.805170\pi$
$809$	−1.07133e6	−1.63691	−0.818457	+	0.574568i	$0.805170\pi$
$810$	0	0
$811$	1.12103e6i	1.70442i	0.523199	+	0.852211i	$0.324738\pi$
$811$	1.12103e6i	1.70442i	−0.523199	+	0.852211i	$0.675262\pi$
$812$	0	0
$813$	−855181.	−1.29383
$814$	0	0
$815$	− 102518.i	− 0.154342i
$816$	0	0
$817$	401040.	0.600820
$818$	0	0
$819$	1.09350e6i	1.63024i
$820$	0	0
$821$	−210197.	−0.311846	−0.155923	−	0.987769i	$-0.549835\pi$
$821$	−210197.	−0.311846	−0.155923	+	0.987769i	$0.549835\pi$
$822$	0	0
$823$	501270.i	0.740068i	0.929018	+	0.370034i	$0.120654\pi$
$823$	501270.i	0.740068i	−0.929018	+	0.370034i	$0.879346\pi$
$824$	0	0
$825$	−155317.	−0.228198
$826$	0	0
$827$	599224.i	0.876149i	0.898939	+	0.438075i	$0.144339\pi$
$827$	599224.i	0.876149i	−0.898939	+	0.438075i	$0.855661\pi$
$828$	0	0
$829$	92189.6	0.134145	0.0670723	−	0.997748i	$-0.478634\pi$
$829$	92189.6	0.134145	0.0670723	+	0.997748i	$0.478634\pi$
$830$	0	0
$831$	797165.i	1.15437i
$832$	0	0
$833$	−83026.8	−0.119654
$834$	0	0
$835$	− 60896.6i	− 0.0873414i
$836$	0	0
$837$	−35825.3	−0.0511375
$838$	0	0
$839$	16114.0i	0.0228918i	0.999934	+	0.0114459i	$0.00364342\pi$
$839$	16114.0i	0.0228918i	−0.999934	+	0.0114459i	$0.996357\pi$
$840$	0	0
$841$	−619815.	−0.876334
$842$	0	0
$843$	340777.i	0.479530i
$844$	0	0
$845$	−34341.6	−0.0480958
$846$	0	0
$847$	− 428651.i	− 0.597499i
$848$	0	0
$849$	−1.19214e6	−1.65391
$850$	0	0
$851$	− 441932.i	− 0.610234i
$852$	0	0
$853$	596015.	0.819142	0.409571	−	0.912278i	$-0.365678\pi$
$853$	596015.	0.819142	0.409571	+	0.912278i	$0.365678\pi$
$854$	0	0
$855$	318862.i	0.436184i
$856$	0	0
$857$	−14006.7	−0.0190710	−0.00953550	−	0.999955i	$-0.503035\pi$
$857$	−14006.7	−0.0190710	−0.00953550	+	0.999955i	$0.503035\pi$
$858$	0	0
$859$	− 221377.i	− 0.300017i	−0.988685	−	0.150008i	$-0.952070\pi$
$859$	− 221377.i	− 0.300017i	0.988685	−	0.150008i	$-0.0479301\pi$
$860$	0	0
$861$	−1.38782e6	−1.87209
$862$	0	0
$863$	1.44444e6i	1.93945i	0.244208	+	0.969723i	$0.421472\pi$
$863$	1.44444e6i	1.93945i	−0.244208	+	0.969723i	$0.578528\pi$
$864$	0	0
$865$	180986.	0.241887
$866$	0	0
$867$	1.07846e6i	1.43471i
$868$	0	0
$869$	603136.	0.798686
$870$	0	0
$871$	1.42675e6i	1.88067i
$872$	0	0
$873$	174503.	0.228968
$874$	0	0
$875$	− 109057.i	− 0.142441i
$876$	0	0
$877$	1.25884e6	1.63671	0.818353	−	0.574715i	$-0.194887\pi$
$877$	1.25884e6	1.63671	0.818353	+	0.574715i	$0.194887\pi$
$878$	0	0
$879$	1.06546e6i	1.37899i
$880$	0	0
$881$	491927.	0.633795	0.316897	−	0.948460i	$-0.397359\pi$
$881$	491927.	0.633795	0.316897	+	0.948460i	$0.397359\pi$
$882$	0	0
$883$	97635.7i	0.125224i	0.998038	+	0.0626120i	$0.0199431\pi$
$883$	97635.7i	0.125224i	−0.998038	+	0.0626120i	$0.980057\pi$
$884$	0	0
$885$	−201769.	−0.257613
$886$	0	0
$887$	− 668572.i	− 0.849769i	−0.905248	−	0.424884i	$-0.860315\pi$
$887$	− 668572.i	− 0.849769i	0.905248	−	0.424884i	$-0.139685\pi$
$888$	0	0
$889$	1.69695e6	2.14716
$890$	0	0
$891$	570681.i	0.718850i
$892$	0	0
$893$	644485.	0.808183
$894$	0	0
$895$	38953.9i	0.0486300i
$896$	0	0
$897$	−425088.	−0.528316
$898$	0	0
$899$	− 120447.i	− 0.149030i
$900$	0	0
$901$	−103451.	−0.127434
$902$	0	0
$903$	− 1.25121e6i	− 1.53445i
$904$	0	0
$905$	141476.	0.172737
$906$	0	0
$907$	− 659118.i	− 0.801214i	−0.916250	−	0.400607i	$-0.868799\pi$
$907$	− 659118.i	− 0.801214i	0.916250	−	0.400607i	$-0.131201\pi$
$908$	0	0
$909$	−527729.	−0.638680
$910$	0	0
$911$	− 852700.i	− 1.02745i	−0.857956	−	0.513724i	$-0.828266\pi$
$911$	− 852700.i	− 1.02745i	0.857956	−	0.513724i	$-0.171734\pi$
$912$	0	0
$913$	583830.	0.700398
$914$	0	0
$915$	385183.i	0.460071i
$916$	0	0
$917$	−2.07889e6	−2.47225
$918$	0	0
$919$	− 1.11737e6i	− 1.32302i	−0.749936	−	0.661511i	$-0.769916\pi$
$919$	− 1.11737e6i	− 1.32302i	0.749936	−	0.661511i	$-0.230084\pi$
$920$	0	0
$921$	186724.	0.220131
$922$	0	0
$923$	− 538852.i	− 0.632508i
$924$	0	0
$925$	269536.	0.315016
$926$	0	0
$927$	891852.i	1.03785i
$928$	0	0
$929$	−59813.4	−0.0693054	−0.0346527	−	0.999399i	$-0.511033\pi$
$929$	−59813.4	−0.0693054	−0.0346527	+	0.999399i	$0.511033\pi$
$930$	0	0
$931$	1.19848e6i	1.38272i
$932$	0	0
$933$	2.31983e6	2.66498
$934$	0	0
$935$	− 24071.2i	− 0.0275343i
$936$	0	0
$937$	−981955.	−1.11844	−0.559220	−	0.829019i	$-0.688899\pi$
$937$	−981955.	−1.11844	−0.559220	+	0.829019i	$0.688899\pi$
$938$	0	0
$939$	− 2.44406e6i	− 2.77192i
$940$	0	0
$941$	−1.07255e6	−1.21126	−0.605629	−	0.795747i	$-0.707079\pi$
$941$	−1.07255e6	−1.21126	−0.605629	+	0.795747i	$0.707079\pi$
$942$	0	0
$943$	− 280573.i	− 0.315517i
$944$	0	0
$945$	76746.5	0.0859400
$946$	0	0
$947$	− 877745.i	− 0.978742i	−0.872076	−	0.489371i	$-0.837226\pi$
$947$	− 877745.i	− 0.978742i	0.872076	−	0.489371i	$-0.162774\pi$
$948$	0	0
$949$	−234313.	−0.260174
$950$	0	0
$951$	1.36391e6i	1.50808i
$952$	0	0
$953$	1.25729e6	1.38436	0.692181	−	0.721724i	$-0.256650\pi$
$953$	1.25729e6	1.38436	0.692181	+	0.721724i	$0.256650\pi$
$954$	0	0
$955$	− 139044.i	− 0.152457i
$956$	0	0
$957$	367478.	0.401243
$958$	0	0
$959$	− 2.72534e6i	− 2.96335i
$960$	0	0
$961$	757659.	0.820403
$962$	0	0
$963$	1.13766e6i	1.22676i
$964$	0	0
$965$	240798.	0.258582
$966$	0	0
$967$	− 506655.i	− 0.541825i	−0.962604	−	0.270913i	$-0.912675\pi$
$967$	− 506655.i	− 0.541825i	0.962604	−	0.270913i	$-0.0873254\pi$
$968$	0	0
$969$	−95022.3	−0.101199
$970$	0	0
$971$	424048.i	0.449755i	0.974387	+	0.224878i	$0.0721982\pi$
$971$	424048.i	0.449755i	−0.974387	+	0.224878i	$0.927802\pi$
$972$	0	0
$973$	1.29472e6	1.36757
$974$	0	0
$975$	− 259262.i	− 0.272728i
$976$	0	0
$977$	−269406.	−0.282240	−0.141120	−	0.989993i	$-0.545070\pi$
$977$	−269406.	−0.282240	−0.141120	+	0.989993i	$0.545070\pi$
$978$	0	0
$979$	− 163413.i	− 0.170499i
$980$	0	0
$981$	87962.8	0.0914031
$982$	0	0
$983$	− 758850.i	− 0.785325i	−0.919683	−	0.392662i	$-0.871554\pi$
$983$	− 758850.i	− 0.785325i	0.919683	−	0.392662i	$-0.128446\pi$
$984$	0	0
$985$	344809.	0.355391
$986$	0	0
$987$	− 2.01073e6i	− 2.06404i
$988$	0	0
$989$	252955.	0.258613
$990$	0	0
$991$	1.22397e6i	1.24630i	0.782102	+	0.623150i	$0.214147\pi$
$991$	1.22397e6i	1.24630i	−0.782102	+	0.623150i	$0.785853\pi$
$992$	0	0
$993$	1.39438e6	1.41411
$994$	0	0
$995$	− 643571.i	− 0.650055i
$996$	0	0
$997$	−1.31108e6	−1.31898	−0.659489	−	0.751714i	$-0.729227\pi$
$997$	−1.31108e6	−1.31898	−0.659489	+	0.751714i	$0.729227\pi$
$998$	0	0
$999$	189681.i	0.190061i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.5.b.d.191.2		8
4.3	odd	2	inner	320.5.b.d.191.7		8
8.3	odd	2		20.5.b.a.11.8	yes	8
8.5	even	2		20.5.b.a.11.7	✓	8
24.5	odd	2		180.5.c.a.91.2		8
24.11	even	2		180.5.c.a.91.1		8
40.3	even	4		100.5.d.c.99.9		16
40.13	odd	4		100.5.d.c.99.7		16
40.19	odd	2		100.5.b.c.51.1		8
40.27	even	4		100.5.d.c.99.8		16
40.29	even	2		100.5.b.c.51.2		8
40.37	odd	4		100.5.d.c.99.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.5.b.a.11.7	✓	8	8.5	even	2
20.5.b.a.11.8	yes	8	8.3	odd	2
100.5.b.c.51.1		8	40.19	odd	2
100.5.b.c.51.2		8	40.29	even	2
100.5.d.c.99.7		16	40.13	odd	4
100.5.d.c.99.8		16	40.27	even	4
100.5.d.c.99.9		16	40.3	even	4
100.5.d.c.99.10		16	40.37	odd	4
180.5.c.a.91.1		8	24.11	even	2
180.5.c.a.91.2		8	24.5	odd	2
320.5.b.d.191.2		8	1.1	even	1	trivial
320.5.b.d.191.7		8	4.3	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.9912i q^{3} +11.1803 q^{5} -78.0345i q^{7} -87.7712 q^{9} +O(q^{10})\)	\(q-12.9912i q^{3} +11.1803 q^{5} -78.0345i q^{7} -87.7712 q^{9} -95.6447i q^{11} +159.654 q^{13} -145.246i q^{15} +22.5103 q^{17} -324.934i q^{19} -1013.76 q^{21} -204.951i q^{23} +125.000 q^{25} +87.9664i q^{27} -295.747 q^{29} +407.262i q^{31} -1242.54 q^{33} -872.453i q^{35} +2156.28 q^{37} -2074.10i q^{39} +1368.98 q^{41} +1234.22i q^{43} -981.312 q^{45} +1983.43i q^{47} -3688.39 q^{49} -292.436i q^{51} -4595.72 q^{53} -1069.34i q^{55} -4221.28 q^{57} -1389.15i q^{59} -2651.94 q^{61} +6849.19i q^{63} +1784.99 q^{65} +8936.51i q^{67} -2662.56 q^{69} -3375.13i q^{71} -1467.63 q^{73} -1623.90i q^{75} -7463.59 q^{77} +6306.01i q^{79} -5966.68 q^{81} +6104.16i q^{83} +251.673 q^{85} +3842.11i q^{87} +1708.55 q^{89} -12458.5i q^{91} +5290.82 q^{93} -3632.87i q^{95} -1988.16 q^{97} +8394.85i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 328 q^{9}+O(q^{10}) \) 8 * q - 328 * q^9	\( 8 q - 328 q^{9} - 352 q^{13} - 48 q^{17} - 16 q^{21} + 1000 q^{25} - 1200 q^{29} - 1120 q^{33} + 5728 q^{37} + 4896 q^{41} + 400 q^{45} - 5768 q^{49} - 2592 q^{53} + 3840 q^{57} - 7936 q^{61} - 1200 q^{65} + 2256 q^{69} - 14448 q^{73} - 2400 q^{77} - 936 q^{81} - 11200 q^{85} + 23760 q^{89} - 11360 q^{93} - 4368 q^{97}+O(q^{100}) \) 8 * q - 328 * q^9 - 352 * q^13 - 48 * q^17 - 16 * q^21 + 1000 * q^25 - 1200 * q^29 - 1120 * q^33 + 5728 * q^37 + 4896 * q^41 + 400 * q^45 - 5768 * q^49 - 2592 * q^53 + 3840 * q^57 - 7936 * q^61 - 1200 * q^65 + 2256 * q^69 - 14448 * q^73 - 2400 * q^77 - 936 * q^81 - 11200 * q^85 + 23760 * q^89 - 11360 * q^93 - 4368 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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