LMFDB - Embedded newform 608.3.e.b.417.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,3,Mod(417,608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("608.417"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	608.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,-52,0,0,0,0,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$16.5668000731$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 196 x^{18} + 1676 x^{17} + 16346 x^{16} - 161824 x^{15} - 667200 x^{14} + \cdots + 1135285065792 $ x^20 - 8x^19 - 196x^18 + 1676x^17 + 16346x^16 - 161824x^15 - 667200x^14 + 8850712x^13 + 9918758x^12 - 288078272x^11 + 224256138x^10 + 5383493580x^9 - 11116604703x^8 - 53755955160x^7 + 166723956672x^6 + 262105513072x^5 - 1053812084224x^4 - 855143026944x^3 + 3318150008320x^2 + 3262428344448*x + 1135285065792
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.4
Root			$5.68070 + 2.14790i$ of defining polynomial
Character	$\chi$	$=$	608.417
Dual form			608.3.e.b.417.18

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.29580i q^{3} +2.32882 q^{5} +2.95740 q^{7} -9.45390 q^{9} -19.4859 q^{11} +14.0235i q^{13} -10.0041i q^{15} -22.2731 q^{17} +(10.5756 - 15.7847i) q^{19} -12.7044i q^{21} -41.7289 q^{23} -19.5766 q^{25} +1.94987i q^{27} +22.9343i q^{29} -30.9806i q^{31} +83.7075i q^{33} +6.88725 q^{35} -62.1601i q^{37} +60.2423 q^{39} +20.2283i q^{41} +53.2356 q^{43} -22.0164 q^{45} -21.3514 q^{47} -40.2538 q^{49} +95.6807i q^{51} -12.4978i q^{53} -45.3791 q^{55} +(-67.8080 - 45.4305i) q^{57} +19.2741i q^{59} +4.96429 q^{61} -27.9590 q^{63} +32.6583i q^{65} +58.9495i q^{67} +179.259i q^{69} +61.6822i q^{71} +14.5807 q^{73} +84.0972i q^{75} -57.6276 q^{77} -113.346i q^{79} -76.7089 q^{81} +48.9861 q^{83} -51.8699 q^{85} +98.5213 q^{87} -27.8155i q^{89} +41.4733i q^{91} -133.087 q^{93} +(24.6285 - 36.7597i) q^{95} +144.982i q^{97} +184.218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) $ 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		−	4.29580i		−	1.43193i	−0.698134	−	0.715967i	$-0.745986\pi$
$3$		−	4.29580i		−	1.43193i	0.698134	−	0.715967i	$-0.254014\pi$
$4$		0			0
$5$	2.32882			0.465763			0.232882	−	0.972505i	$-0.425184\pi$
$5$	2.32882			0.465763			0.232882	+	0.972505i	$0.425184\pi$
$6$		0			0
$7$	2.95740			0.422486			0.211243	−	0.977434i	$-0.432249\pi$
$7$	2.95740			0.422486			0.211243	+	0.977434i	$0.432249\pi$
$8$		0			0
$9$	−9.45390			−1.05043
$10$		0			0
$11$	−19.4859			−1.77144			−0.885722	−	0.464216i	$-0.846336\pi$
$11$	−19.4859			−1.77144			−0.885722	+	0.464216i	$0.846336\pi$
$12$		0			0
$13$			14.0235i			1.07873i	0.842071	+	0.539367i	$0.181336\pi$
$13$			14.0235i			1.07873i	−0.842071	+	0.539367i	$0.818664\pi$
$14$		0			0
$15$		−	10.0041i		−	0.666942i
$16$		0			0
$17$	−22.2731			−1.31018			−0.655090	−	0.755551i	$-0.727369\pi$
$17$	−22.2731			−1.31018			−0.655090	+	0.755551i	$0.727369\pi$
$18$		0			0
$19$	10.5756	−	15.7847i	0.556608	−	0.830775i
$19$	10.5756	−	15.7847i	0.556608	−	0.830775i
$20$		0			0
$21$		−	12.7044i		−	0.604972i
$22$		0			0
$23$	−41.7289			−1.81430			−0.907150	−	0.420808i	$-0.861747\pi$
$23$	−41.7289			−1.81430			−0.907150	+	0.420808i	$0.861747\pi$
$24$		0			0
$25$	−19.5766			−0.783064
$26$		0			0
$27$			1.94987i			0.0722174i
$28$		0			0
$29$			22.9343i			0.790839i	0.918501	+	0.395420i	$0.129401\pi$
$29$			22.9343i			0.790839i	−0.918501	+	0.395420i	$0.870599\pi$
$30$		0			0
$31$		−	30.9806i		−	0.999375i	−0.866206	−	0.499688i	$-0.833448\pi$
$31$		−	30.9806i		−	0.999375i	0.866206	−	0.499688i	$-0.166552\pi$
$32$		0			0
$33$			83.7075i			2.53659i
$34$		0			0
$35$	6.88725			0.196779
$36$		0			0
$37$		−	62.1601i		−	1.68000i	−0.542584	−	0.840001i	$-0.682554\pi$
$37$		−	62.1601i		−	1.68000i	0.542584	−	0.840001i	$-0.317446\pi$
$38$		0			0
$39$	60.2423			1.54468
$40$		0			0
$41$			20.2283i			0.493373i	0.969095	+	0.246686i	$0.0793417\pi$
$41$			20.2283i			0.493373i	−0.969095	+	0.246686i	$0.920658\pi$
$42$		0			0
$43$	53.2356			1.23804			0.619019	−	0.785376i	$-0.287530\pi$
$43$	53.2356			1.23804			0.619019	+	0.785376i	$0.287530\pi$
$44$		0			0
$45$	−22.0164			−0.489254
$46$		0			0
$47$	−21.3514			−0.454285			−0.227143	−	0.973861i	$-0.572938\pi$
$47$	−21.3514			−0.454285			−0.227143	+	0.973861i	$0.572938\pi$
$48$		0			0
$49$	−40.2538			−0.821505
$50$		0			0
$51$			95.6807i			1.87609i
$52$		0			0
$53$		−	12.4978i		−	0.235808i	−0.993025	−	0.117904i	$-0.962383\pi$
$53$		−	12.4978i		−	0.235808i	0.993025	−	0.117904i	$-0.0376174\pi$
$54$		0			0
$55$	−45.3791			−0.825074
$56$		0			0
$57$	−67.8080	−	45.4305i	−1.18961	−	0.797026i
$58$		0			0
$59$			19.2741i			0.326680i	0.986570	+	0.163340i	$0.0522267\pi$
$59$			19.2741i			0.326680i	−0.986570	+	0.163340i	$0.947773\pi$
$60$		0			0
$61$	4.96429			0.0813817			0.0406909	−	0.999172i	$-0.487044\pi$
$61$	4.96429			0.0813817			0.0406909	+	0.999172i	$0.487044\pi$
$62$		0			0
$63$	−27.9590			−0.443794
$64$		0			0
$65$			32.6583i			0.502435i
$66$		0			0
$67$			58.9495i			0.879844i	0.898036	+	0.439922i	$0.144994\pi$
$67$			58.9495i			0.879844i	−0.898036	+	0.439922i	$0.855006\pi$
$68$		0			0
$69$			179.259i			2.59796i
$70$		0			0
$71$			61.6822i			0.868763i	0.900729	+	0.434382i	$0.143033\pi$
$71$			61.6822i			0.868763i	−0.900729	+	0.434382i	$0.856967\pi$
$72$		0			0
$73$	14.5807			0.199736			0.0998680	−	0.995001i	$-0.468158\pi$
$73$	14.5807			0.199736			0.0998680	+	0.995001i	$0.468158\pi$
$74$		0			0
$75$			84.0972i			1.12130i
$76$		0			0
$77$	−57.6276			−0.748411
$78$		0			0
$79$		−	113.346i		−	1.43475i	−0.696685	−	0.717377i	$-0.745342\pi$
$79$		−	113.346i		−	1.43475i	0.696685	−	0.717377i	$-0.254658\pi$
$80$		0			0
$81$	−76.7089			−0.947023
$82$		0			0
$83$	48.9861			0.590194			0.295097	−	0.955467i	$-0.404648\pi$
$83$	48.9861			0.590194			0.295097	+	0.955467i	$0.404648\pi$
$84$		0			0
$85$	−51.8699			−0.610234
$86$		0			0
$87$	98.5213			1.13243
$88$		0			0
$89$		−	27.8155i		−	0.312534i	−0.987715	−	0.156267i	$-0.950054\pi$
$89$		−	27.8155i		−	0.312534i	0.987715	−	0.156267i	$-0.0499460\pi$
$90$		0			0
$91$			41.4733i			0.455750i
$92$		0			0
$93$	−133.087			−1.43104
$94$		0			0
$95$	24.6285	−	36.7597i	0.259248	−	0.386945i
$96$		0			0
$97$			144.982i			1.49466i	0.664452	+	0.747331i	$0.268665\pi$
$97$			144.982i			1.49466i	−0.664452	+	0.747331i	$0.731335\pi$
$98$		0			0
$99$	184.218			1.86078
$100$		0			0
$101$	150.334			1.48845			0.744226	−	0.667927i	$-0.232818\pi$
$101$	150.334			1.48845			0.744226	+	0.667927i	$0.232818\pi$
$102$		0			0
$103$		−	74.2695i		−	0.721063i	−0.932747	−	0.360532i	$-0.882595\pi$
$103$		−	74.2695i		−	0.721063i	0.932747	−	0.360532i	$-0.117405\pi$
$104$		0			0
$105$		−	29.5863i		−	0.281774i
$106$		0			0
$107$		−	139.704i		−	1.30564i	−0.757511	−	0.652822i	$-0.773585\pi$
$107$		−	139.704i		−	1.30564i	0.757511	−	0.652822i	$-0.226415\pi$
$108$		0			0
$109$			42.5178i			0.390072i	0.980796	+	0.195036i	$0.0624823\pi$
$109$			42.5178i			0.390072i	−0.980796	+	0.195036i	$0.937518\pi$
$110$		0			0
$111$	−267.027			−2.40565
$112$		0			0
$113$		−	109.368i		−	0.967862i	−0.875106	−	0.483931i	$-0.839209\pi$
$113$		−	109.368i		−	0.967862i	0.875106	−	0.483931i	$-0.160791\pi$
$114$		0			0
$115$	−97.1790			−0.845034
$116$		0			0
$117$		−	132.577i		−	1.13314i
$118$		0			0
$119$	−65.8705			−0.553533
$120$		0			0
$121$	258.700			2.13802
$122$		0			0
$123$	86.8966			0.706477
$124$		0			0
$125$	−103.811			−0.830486
$126$		0			0
$127$			40.2362i			0.316821i	0.987373	+	0.158410i	$0.0506369\pi$
$127$			40.2362i			0.316821i	−0.987373	+	0.158410i	$0.949363\pi$
$128$		0			0
$129$		−	228.690i		−	1.77279i
$130$		0			0
$131$	−65.6183			−0.500903			−0.250452	−	0.968129i	$-0.580579\pi$
$131$	−65.6183			−0.500903			−0.250452	+	0.968129i	$0.580579\pi$
$132$		0			0
$133$	31.2762	−	46.6818i	0.235159	−	0.350991i
$134$		0			0
$135$			4.54089i			0.0336362i
$136$		0			0
$137$	168.934			1.23310			0.616549	−	0.787316i	$-0.288530\pi$
$137$	168.934			1.23310			0.616549	+	0.787316i	$0.288530\pi$
$138$		0			0
$139$	−137.539			−0.989487			−0.494743	−	0.869039i	$-0.664738\pi$
$139$	−137.539			−0.989487			−0.494743	+	0.869039i	$0.664738\pi$
$140$		0			0
$141$			91.7214i			0.650506i
$142$		0			0
$143$		−	273.261i		−	1.91092i
$144$		0			0
$145$			53.4099i			0.368344i
$146$		0			0
$147$			172.922i			1.17634i
$148$		0			0
$149$	−50.0197			−0.335703			−0.167851	−	0.985812i	$-0.553683\pi$
$149$	−50.0197			−0.335703			−0.167851	+	0.985812i	$0.553683\pi$
$150$		0			0
$151$		−	204.762i		−	1.35604i	−0.735045	−	0.678019i	$-0.762839\pi$
$151$		−	204.762i		−	1.35604i	0.735045	−	0.678019i	$-0.237161\pi$
$152$		0			0
$153$	210.567			1.37626
$154$		0			0
$155$		−	72.1482i		−	0.465472i
$156$		0			0
$157$	57.6193			0.367002			0.183501	−	0.983020i	$-0.441257\pi$
$157$	57.6193			0.367002			0.183501	+	0.983020i	$0.441257\pi$
$158$		0			0
$159$	−53.6881			−0.337661
$160$		0			0
$161$	−123.409			−0.766517
$162$		0			0
$163$	−97.9480			−0.600908			−0.300454	−	0.953796i	$-0.597138\pi$
$163$	−97.9480			−0.600908			−0.300454	+	0.953796i	$0.597138\pi$
$164$		0			0
$165$			194.939i			1.18145i
$166$		0			0
$167$			84.7060i			0.507222i	0.967306	+	0.253611i	$0.0816182\pi$
$167$			84.7060i			0.507222i	−0.967306	+	0.253611i	$0.918382\pi$
$168$		0			0
$169$	−27.6597			−0.163667
$170$		0			0
$171$	−99.9803	+	149.227i	−0.584680	+	0.872674i
$172$		0			0
$173$			22.3618i			0.129259i	0.997909	+	0.0646294i	$0.0205865\pi$
$173$			22.3618i			0.129259i	−0.997909	+	0.0646294i	$0.979413\pi$
$174$		0			0
$175$	−57.8959			−0.330834
$176$		0			0
$177$	82.7977			0.467784
$178$		0			0
$179$		−	72.7922i		−	0.406661i	−0.979110	−	0.203330i	$-0.934823\pi$
$179$		−	72.7922i		−	0.406661i	0.979110	−	0.203330i	$-0.0651765\pi$
$180$		0			0
$181$		−	350.286i		−	1.93528i	−0.252326	−	0.967642i	$-0.581196\pi$
$181$		−	350.286i		−	1.93528i	0.252326	−	0.967642i	$-0.418804\pi$
$182$		0			0
$183$		−	21.3256i		−	0.116533i
$184$		0			0
$185$		−	144.760i		−	0.782484i
$186$		0			0
$187$	434.011			2.32091
$188$		0			0
$189$			5.76656i			0.0305109i
$190$		0			0
$191$	107.952			0.565194			0.282597	−	0.959239i	$-0.408804\pi$
$191$	107.952			0.565194			0.282597	+	0.959239i	$0.408804\pi$
$192$		0			0
$193$		−	267.636i		−	1.38672i	−0.720593	−	0.693359i	$-0.756130\pi$
$193$		−	267.636i		−	1.38672i	0.720593	−	0.693359i	$-0.243870\pi$
$194$		0			0
$195$	140.293			0.719453
$196$		0			0
$197$	290.715			1.47571			0.737854	−	0.674960i	$-0.235839\pi$
$197$	290.715			1.47571			0.737854	+	0.674960i	$0.235839\pi$
$198$		0			0
$199$	−302.296			−1.51907			−0.759537	−	0.650464i	$-0.774574\pi$
$199$	−302.296			−1.51907			−0.759537	+	0.650464i	$0.774574\pi$
$200$		0			0
$201$	253.235			1.25988
$202$		0			0
$203$			67.8261i			0.334119i
$204$		0			0
$205$			47.1080i			0.229795i
$206$		0			0
$207$	394.501			1.90580
$208$		0			0
$209$	−206.074	+	307.579i	−0.986001	+	1.47167i
$210$		0			0
$211$		−	73.5756i		−	0.348700i	−0.984684	−	0.174350i	$-0.944218\pi$
$211$		−	73.5756i		−	0.348700i	0.984684	−	0.174350i	$-0.0557824\pi$
$212$		0			0
$213$	264.974			1.24401
$214$		0			0
$215$	123.976			0.576633
$216$		0			0
$217$		−	91.6222i		−	0.422222i
$218$		0			0
$219$		−	62.6359i		−	0.286009i
$220$		0			0
$221$		−	312.347i		−	1.41334i
$222$		0			0
$223$			225.342i			1.01050i	0.862973	+	0.505251i	$0.168600\pi$
$223$			225.342i			1.01050i	−0.862973	+	0.505251i	$0.831400\pi$
$224$		0			0
$225$	185.075			0.822557
$226$		0			0
$227$			19.4056i			0.0854872i	0.999086	+	0.0427436i	$0.0136099\pi$
$227$			19.4056i			0.0854872i	−0.999086	+	0.0427436i	$0.986390\pi$
$228$		0			0
$229$	285.505			1.24675			0.623374	−	0.781924i	$-0.285762\pi$
$229$	285.505			1.24675			0.623374	+	0.781924i	$0.285762\pi$
$230$		0			0
$231$			247.557i			1.07167i
$232$		0			0
$233$	−359.060			−1.54103			−0.770515	−	0.637422i	$-0.780001\pi$
$233$	−359.060			−1.54103			−0.770515	+	0.637422i	$0.780001\pi$
$234$		0			0
$235$	−49.7235			−0.211590
$236$		0			0
$237$	−486.910			−2.05447
$238$		0			0
$239$	−182.395			−0.763160			−0.381580	−	0.924336i	$-0.624620\pi$
$239$	−182.395			−0.763160			−0.381580	+	0.924336i	$0.624620\pi$
$240$		0			0
$241$		−	119.880i		−	0.497429i	−0.968577	−	0.248714i	$-0.919992\pi$
$241$		−	119.880i		−	0.497429i	0.968577	−	0.248714i	$-0.0800081\pi$
$242$		0			0
$243$			347.075i			1.42829i
$244$		0			0
$245$	−93.7437			−0.382627
$246$		0			0
$247$	221.358	+	148.307i	0.896185	+	0.600432i
$248$		0			0
$249$		−	210.434i		−	0.845118i
$250$		0			0
$251$	−449.306			−1.79007			−0.895033	−	0.446000i	$-0.852848\pi$
$251$	−449.306			−1.79007			−0.895033	+	0.446000i	$0.852848\pi$
$252$		0			0
$253$	813.125			3.21393
$254$		0			0
$255$			222.823i			0.873815i
$256$		0			0
$257$			469.173i			1.82558i	0.408432	+	0.912789i	$0.366076\pi$
$257$			469.173i			1.82558i	−0.408432	+	0.912789i	$0.633924\pi$
$258$		0			0
$259$		−	183.833i		−	0.709778i
$260$		0			0
$261$		−	216.819i		−	0.830724i
$262$		0			0
$263$	265.109			1.00802			0.504010	−	0.863698i	$-0.331858\pi$
$263$	265.109			1.00802			0.504010	+	0.863698i	$0.331858\pi$
$264$		0			0
$265$		−	29.1051i		−	0.109831i
$266$		0			0
$267$	−119.490			−0.447528
$268$		0			0
$269$			303.491i			1.12822i	0.825700	+	0.564109i	$0.190780\pi$
$269$			303.491i			1.12822i	−0.825700	+	0.564109i	$0.809220\pi$
$270$		0			0
$271$	−200.301			−0.739117			−0.369559	−	0.929207i	$-0.620491\pi$
$271$	−200.301			−0.739117			−0.369559	+	0.929207i	$0.620491\pi$
$272$		0			0
$273$	178.161			0.652604
$274$		0			0
$275$	381.468			1.38716
$276$		0			0
$277$	−160.105			−0.577998			−0.288999	−	0.957329i	$-0.593322\pi$
$277$	−160.105			−0.577998			−0.288999	+	0.957329i	$0.593322\pi$
$278$		0			0
$279$			292.888i			1.04978i
$280$		0			0
$281$			383.009i			1.36302i	0.731808	+	0.681511i	$0.238677\pi$
$281$			383.009i			1.36302i	−0.731808	+	0.681511i	$0.761323\pi$
$282$		0			0
$283$	−226.723			−0.801140			−0.400570	−	0.916266i	$-0.631188\pi$
$283$	−226.723			−0.801140			−0.400570	+	0.916266i	$0.631188\pi$
$284$		0			0
$285$	−157.913	−	105.799i	−0.554079	−	0.371226i
$286$		0			0
$287$			59.8232i			0.208443i
$288$		0			0
$289$	207.090			0.716573
$290$		0			0
$291$	622.815			2.14026
$292$		0			0
$293$		−	429.962i		−	1.46745i	−0.679447	−	0.733724i	$-0.737780\pi$
$293$		−	429.962i		−	1.46745i	0.679447	−	0.733724i	$-0.262220\pi$
$294$		0			0
$295$			44.8859i			0.152155i
$296$		0			0
$297$		−	37.9950i		−	0.127929i
$298$		0			0
$299$		−	585.187i		−	1.95715i
$300$		0			0
$301$	157.439			0.523054
$302$		0			0
$303$		−	645.804i		−	2.13137i
$304$		0			0
$305$	11.5609			0.0379046
$306$		0			0
$307$			208.273i			0.678414i	0.940712	+	0.339207i	$0.110159\pi$
$307$			208.273i			0.678414i	−0.940712	+	0.339207i	$0.889841\pi$
$308$		0			0
$309$	−319.047			−1.03251
$310$		0			0
$311$	135.179			0.434659			0.217329	−	0.976098i	$-0.430265\pi$
$311$	135.179			0.434659			0.217329	+	0.976098i	$0.430265\pi$
$312$		0			0
$313$	−412.048			−1.31645			−0.658224	−	0.752823i	$-0.728692\pi$
$313$	−412.048			−1.31645			−0.658224	+	0.752823i	$0.728692\pi$
$314$		0			0
$315$	−65.1114			−0.206703
$316$		0			0
$317$		−	386.723i		−	1.21995i	−0.792422	−	0.609973i	$-0.791180\pi$
$317$		−	386.723i		−	1.21995i	0.792422	−	0.609973i	$-0.208820\pi$
$318$		0			0
$319$		−	446.896i		−	1.40093i
$320$		0			0
$321$	−600.140			−1.86960
$322$		0			0
$323$	−235.550	+	351.574i	−0.729257	+	1.08847i
$324$		0			0
$325$		−	274.533i		−	0.844718i
$326$		0			0
$327$	182.648			0.558557
$328$		0			0
$329$	−63.1447			−0.191929
$330$		0			0
$331$		−	143.670i		−	0.434047i	−0.976166	−	0.217024i	$-0.930365\pi$
$331$		−	143.670i		−	0.434047i	0.976166	−	0.217024i	$-0.0696349\pi$
$332$		0			0
$333$			587.655i			1.76473i
$334$		0			0
$335$			137.283i			0.409799i
$336$		0			0
$337$		−	162.296i		−	0.481591i	−0.970576	−	0.240795i	$-0.922592\pi$
$337$		−	162.296i		−	0.481591i	0.970576	−	0.240795i	$-0.0774082\pi$
$338$		0			0
$339$	−469.825			−1.38591
$340$		0			0
$341$			603.685i			1.77034i
$342$		0			0
$343$	−263.959			−0.769561
$344$		0			0
$345$			417.461i			1.21003i
$346$		0			0
$347$	−429.273			−1.23710			−0.618549	−	0.785746i	$-0.712279\pi$
$347$	−429.273			−1.23710			−0.618549	+	0.785746i	$0.712279\pi$
$348$		0			0
$349$	463.869			1.32914			0.664568	−	0.747227i	$-0.268615\pi$
$349$	463.869			1.32914			0.664568	+	0.747227i	$0.268615\pi$
$350$		0			0
$351$	−27.3441			−0.0779034
$352$		0			0
$353$	78.3436			0.221937			0.110968	−	0.993824i	$-0.464605\pi$
$353$	78.3436			0.221937			0.110968	+	0.993824i	$0.464605\pi$
$354$		0			0
$355$			143.647i			0.404638i
$356$		0			0
$357$			282.966i			0.792623i
$358$		0			0
$359$	−243.233			−0.677528			−0.338764	−	0.940871i	$-0.610009\pi$
$359$	−243.233			−0.677528			−0.338764	+	0.940871i	$0.610009\pi$
$360$		0			0
$361$	−137.315	−	333.865i	−0.380374	−	0.924833i
$362$		0			0
$363$		−	1111.32i		−	3.06150i
$364$		0			0
$365$	33.9558			0.0930297
$366$		0			0
$367$	−323.083			−0.880335			−0.440167	−	0.897916i	$-0.645081\pi$
$367$	−323.083			−0.880335			−0.440167	+	0.897916i	$0.645081\pi$
$368$		0			0
$369$		−	191.236i		−	0.518255i
$370$		0			0
$371$		−	36.9610i		−	0.0996255i
$372$		0			0
$373$			86.8197i			0.232761i	0.993205	+	0.116380i	$0.0371291\pi$
$373$			86.8197i			0.232761i	−0.993205	+	0.116380i	$0.962871\pi$
$374$		0			0
$375$			445.950i			1.18920i
$376$		0			0
$377$	−321.621			−0.853105
$378$		0			0
$379$		−	20.8856i		−	0.0551071i	−0.999620	−	0.0275535i	$-0.991228\pi$
$379$		−	20.8856i		−	0.0551071i	0.999620	−	0.0275535i	$-0.00877168\pi$
$380$		0			0
$381$	172.847			0.453666
$382$		0			0
$383$		−	310.036i		−	0.809492i	−0.914429	−	0.404746i	$-0.867360\pi$
$383$		−	310.036i		−	0.809492i	0.914429	−	0.404746i	$-0.132640\pi$
$384$		0			0
$385$	−134.204			−0.348582
$386$		0			0
$387$	−503.285			−1.30048
$388$		0			0
$389$	−483.760			−1.24360			−0.621799	−	0.783177i	$-0.713598\pi$
$389$	−483.760			−1.24360			−0.621799	+	0.783177i	$0.713598\pi$
$390$		0			0
$391$	929.430			2.37706
$392$		0			0
$393$			281.883i			0.717260i
$394$		0			0
$395$		−	263.961i		−	0.668256i
$396$		0			0
$397$	−114.551			−0.288543			−0.144271	−	0.989538i	$-0.546084\pi$
$397$	−114.551			−0.288543			−0.144271	+	0.989538i	$0.546084\pi$
$398$		0			0
$399$	−200.536	−	134.356i	−0.502596	−	0.336733i
$400$		0			0
$401$			20.2091i			0.0503968i	0.999682	+	0.0251984i	$0.00802175\pi$
$401$			20.2091i			0.0503968i	−0.999682	+	0.0251984i	$0.991978\pi$
$402$		0			0
$403$	434.458			1.07806
$404$		0			0
$405$	−178.641			−0.441089
$406$		0			0
$407$			1211.24i			2.97603i
$408$		0			0
$409$		−	137.213i		−	0.335485i	−0.985831	−	0.167742i	$-0.946352\pi$
$409$		−	137.213i		−	0.335485i	0.985831	−	0.167742i	$-0.0536477\pi$
$410$		0			0
$411$		−	725.709i		−	1.76571i
$412$		0			0
$413$			57.0013i			0.138018i
$414$		0			0
$415$	114.080			0.274891
$416$		0			0
$417$			590.839i			1.41688i
$418$		0			0
$419$	−69.2314			−0.165230			−0.0826151	−	0.996582i	$-0.526327\pi$
$419$	−69.2314			−0.165230			−0.0826151	+	0.996582i	$0.526327\pi$
$420$		0			0
$421$			96.9103i			0.230191i	0.993354	+	0.115095i	$0.0367174\pi$
$421$			96.9103i			0.230191i	−0.993354	+	0.115095i	$0.963283\pi$
$422$		0			0
$423$	201.854			0.477197
$424$		0			0
$425$	436.031			1.02596
$426$		0			0
$427$	14.6814			0.0343827
$428$		0			0
$429$	−1173.88			−2.73631
$430$		0			0
$431$			473.629i			1.09891i	0.835524	+	0.549453i	$0.185164\pi$
$431$			473.629i			1.09891i	−0.835524	+	0.549453i	$0.814836\pi$
$432$		0			0
$433$			371.336i			0.857588i	0.903402	+	0.428794i	$0.141061\pi$
$433$			371.336i			0.857588i	−0.903402	+	0.428794i	$0.858939\pi$
$434$		0			0
$435$	229.438			0.527444
$436$		0			0
$437$	−441.306	+	658.679i	−1.00985	+	1.50727i
$438$		0			0
$439$		−	576.675i		−	1.31361i	−0.754060	−	0.656806i	$-0.771907\pi$
$439$		−	576.675i		−	1.31361i	0.754060	−	0.656806i	$-0.228093\pi$
$440$		0			0
$441$	380.555			0.862937
$442$		0			0
$443$	−436.152			−0.984542			−0.492271	−	0.870442i	$-0.663833\pi$
$443$	−436.152			−0.984542			−0.492271	+	0.870442i	$0.663833\pi$
$444$		0			0
$445$		−	64.7773i		−	0.145567i
$446$		0			0
$447$			214.875i			0.480704i
$448$		0			0
$449$		−	875.653i		−	1.95023i	−0.221699	−	0.975115i	$-0.571160\pi$
$449$		−	875.653i		−	1.95023i	0.221699	−	0.975115i	$-0.428840\pi$
$450$		0			0
$451$		−	394.166i		−	0.873982i
$452$		0			0
$453$	−879.615			−1.94176
$454$		0			0
$455$			96.5837i			0.212272i
$456$		0			0
$457$	−135.889			−0.297350			−0.148675	−	0.988886i	$-0.547501\pi$
$457$	−135.889			−0.297350			−0.148675	+	0.988886i	$0.547501\pi$
$458$		0			0
$459$		−	43.4296i		−	0.0946179i
$460$		0			0
$461$	−258.134			−0.559943			−0.279972	−	0.960008i	$-0.590325\pi$
$461$	−258.134			−0.559943			−0.279972	+	0.960008i	$0.590325\pi$
$462$		0			0
$463$	83.7030			0.180784			0.0903920	−	0.995906i	$-0.471188\pi$
$463$	83.7030			0.180784			0.0903920	+	0.995906i	$0.471188\pi$
$464$		0			0
$465$	−309.934			−0.666526
$466$		0			0
$467$	210.142			0.449984			0.224992	−	0.974361i	$-0.427764\pi$
$467$	210.142			0.449984			0.224992	+	0.974361i	$0.427764\pi$
$468$		0			0
$469$			174.338i			0.371722i
$470$		0			0
$471$		−	247.521i		−	0.525522i
$472$		0			0
$473$	−1037.34			−2.19312
$474$		0			0
$475$	−207.034	+	309.011i	−0.435860	+	0.650550i
$476$		0			0
$477$			118.153i			0.247700i
$478$		0			0
$479$	226.280			0.472400			0.236200	−	0.971704i	$-0.424098\pi$
$479$	226.280			0.472400			0.236200	+	0.971704i	$0.424098\pi$
$480$		0			0
$481$	871.705			1.81228
$482$		0			0
$483$			530.141i			1.09760i
$484$		0			0
$485$			337.637i			0.696159i
$486$		0			0
$487$			899.572i			1.84717i	0.383394	+	0.923585i	$0.374755\pi$
$487$			899.572i			1.84717i	−0.383394	+	0.923585i	$0.625245\pi$
$488$		0			0
$489$			420.765i			0.860460i
$490$		0			0
$491$	−405.337			−0.825533			−0.412767	−	0.910837i	$-0.635438\pi$
$491$	−405.337			−0.825533			−0.412767	+	0.910837i	$0.635438\pi$
$492$		0			0
$493$		−	510.818i		−	1.03614i
$494$		0			0
$495$	429.009			0.866686
$496$		0			0
$497$			182.419i			0.367041i
$498$		0			0
$499$	194.024			0.388825			0.194413	−	0.980920i	$-0.437720\pi$
$499$	194.024			0.388825			0.194413	+	0.980920i	$0.437720\pi$
$500$		0			0
$501$	363.880			0.726308
$502$		0			0
$503$	623.065			1.23870			0.619348	−	0.785116i	$-0.287397\pi$
$503$	623.065			1.23870			0.619348	+	0.785116i	$0.287397\pi$
$504$		0			0
$505$	350.100			0.693267
$506$		0			0
$507$			118.821i			0.234360i
$508$		0			0
$509$		−	321.176i		−	0.630995i	−0.948926	−	0.315497i	$-0.897829\pi$
$509$		−	321.176i		−	0.630995i	0.948926	−	0.315497i	$-0.102171\pi$
$510$		0			0
$511$	43.1211			0.0843857
$512$		0			0
$513$	30.7782	+	20.6210i	0.0599964	+	0.0401968i
$514$		0			0
$515$		−	172.960i		−	0.335845i
$516$		0			0
$517$	416.051			0.804741
$518$		0			0
$519$	96.0618			0.185090
$520$		0			0
$521$			535.331i			1.02751i	0.857938	+	0.513753i	$0.171745\pi$
$521$			535.331i			1.02751i	−0.857938	+	0.513753i	$0.828255\pi$
$522$		0			0
$523$			873.472i			1.67012i	0.550160	+	0.835059i	$0.314567\pi$
$523$			873.472i			1.67012i	−0.550160	+	0.835059i	$0.685433\pi$
$524$		0			0
$525$			248.709i			0.473732i
$526$		0			0
$527$			690.034i			1.30936i
$528$		0			0
$529$	1212.30			2.29168
$530$		0			0
$531$		−	182.215i		−	0.343155i
$532$		0			0
$533$	−283.672			−0.532218
$534$		0			0
$535$		−	325.345i		−	0.608122i
$536$		0			0
$537$	−312.701			−0.582311
$538$		0			0
$539$	784.380			1.45525
$540$		0			0
$541$	302.002			0.558229			0.279114	−	0.960258i	$-0.409959\pi$
$541$	302.002			0.558229			0.279114	+	0.960258i	$0.409959\pi$
$542$		0			0
$543$	−1504.76			−2.77120
$544$		0			0
$545$			99.0162i			0.181681i
$546$		0			0
$547$			1004.10i			1.83564i	0.396994	+	0.917821i	$0.370054\pi$
$547$			1004.10i			1.83564i	−0.396994	+	0.917821i	$0.629946\pi$
$548$		0			0
$549$	−46.9319			−0.0854861
$550$		0			0
$551$	362.012	+	242.543i	0.657009	+	0.440188i
$552$		0			0
$553$		−	335.209i		−	0.606164i
$554$		0			0
$555$	−621.858			−1.12046
$556$		0			0
$557$	449.650			0.807271			0.403635	−	0.914920i	$-0.367746\pi$
$557$	449.650			0.807271			0.403635	+	0.914920i	$0.367746\pi$
$558$		0			0
$559$			746.552i			1.33551i
$560$		0			0
$561$		−	1864.42i		−	3.32339i
$562$		0			0
$563$		−	829.613i		−	1.47356i	−0.676133	−	0.736779i	$-0.736346\pi$
$563$		−	829.613i		−	1.47356i	0.676133	−	0.736779i	$-0.263654\pi$
$564$		0			0
$565$		−	254.699i		−	0.450795i
$566$		0			0
$567$	−226.859			−0.400104
$568$		0			0
$569$		−	46.4574i		−	0.0816475i	−0.999166	−	0.0408238i	$-0.987002\pi$
$569$		−	46.4574i		−	0.0816475i	0.999166	−	0.0408238i	$-0.0129982\pi$
$570$		0			0
$571$	788.963			1.38172			0.690860	−	0.722988i	$-0.257232\pi$
$571$	788.963			1.38172			0.690860	+	0.722988i	$0.257232\pi$
$572$		0			0
$573$		−	463.740i		−	0.809320i
$574$		0			0
$575$	816.910			1.42071
$576$		0			0
$577$	−25.1604			−0.0436055			−0.0218028	−	0.999762i	$-0.506941\pi$
$577$	−25.1604			−0.0436055			−0.0218028	+	0.999762i	$0.506941\pi$
$578$		0			0
$579$	−1149.71			−1.98569
$580$		0			0
$581$	144.872			0.249349
$582$		0			0
$583$			243.531i			0.417720i
$584$		0			0
$585$		−	308.748i		−	0.527774i
$586$		0			0
$587$	779.591			1.32809			0.664047	−	0.747691i	$-0.268838\pi$
$587$	779.591			1.32809			0.664047	+	0.747691i	$0.268838\pi$
$588$		0			0
$589$	−489.021	−	327.638i	−0.830256	−	0.556261i
$590$		0			0
$591$		−	1248.85i		−	2.11312i
$592$		0			0
$593$	−657.878			−1.10941			−0.554703	−	0.832048i	$-0.687168\pi$
$593$	−657.878			−1.10941			−0.554703	+	0.832048i	$0.687168\pi$
$594$		0			0
$595$	−153.400			−0.257816
$596$		0			0
$597$			1298.60i			2.17521i
$598$		0			0
$599$		−	761.121i		−	1.27065i	−0.772244	−	0.635326i	$-0.780866\pi$
$599$		−	761.121i		−	1.27065i	0.772244	−	0.635326i	$-0.219134\pi$
$600$		0			0
$601$			798.347i			1.32837i	0.747570	+	0.664183i	$0.231220\pi$
$601$			798.347i			1.32837i	−0.747570	+	0.664183i	$0.768780\pi$
$602$		0			0
$603$		−	557.303i		−	0.924218i
$604$		0			0
$605$	602.465			0.995810
$606$		0			0
$607$		−	752.935i		−	1.24042i	−0.784436	−	0.620210i	$-0.787047\pi$
$607$		−	752.935i		−	1.24042i	0.784436	−	0.620210i	$-0.212953\pi$
$608$		0			0
$609$	291.367			0.478436
$610$		0			0
$611$		−	299.422i		−	0.490053i
$612$		0			0
$613$	536.480			0.875171			0.437586	−	0.899177i	$-0.355834\pi$
$613$	536.480			0.875171			0.437586	+	0.899177i	$0.355834\pi$
$614$		0			0
$615$	202.366			0.329051
$616$		0			0
$617$	−317.064			−0.513879			−0.256940	−	0.966427i	$-0.582714\pi$
$617$	−317.064			−0.513879			−0.256940	+	0.966427i	$0.582714\pi$
$618$		0			0
$619$	−351.867			−0.568445			−0.284222	−	0.958758i	$-0.591735\pi$
$619$	−351.867			−0.568445			−0.284222	+	0.958758i	$0.591735\pi$
$620$		0			0
$621$		−	81.3660i		−	0.131024i
$622$		0			0
$623$		−	82.2617i		−	0.132041i
$624$		0			0
$625$	247.659			0.396254
$626$		0			0
$627$	1321.30	+	885.254i	2.10734	+	1.41189i
$628$		0			0
$629$			1384.50i			2.20111i
$630$		0			0
$631$	−371.411			−0.588606			−0.294303	−	0.955712i	$-0.595088\pi$
$631$	−371.411			−0.588606			−0.294303	+	0.955712i	$0.595088\pi$
$632$		0			0
$633$	−316.066			−0.499315
$634$		0			0
$635$			93.7029i			0.147564i
$636$		0			0
$637$		−	564.500i		−	0.886186i
$638$		0			0
$639$		−	583.137i		−	0.912578i
$640$		0			0
$641$		−	256.605i		−	0.400319i	−0.979763	−	0.200160i	$-0.935854\pi$
$641$		−	256.605i		−	0.400319i	0.979763	−	0.200160i	$-0.0641461\pi$
$642$		0			0
$643$	759.280			1.18084			0.590420	−	0.807096i	$-0.298962\pi$
$643$	759.280			1.18084			0.590420	+	0.807096i	$0.298962\pi$
$644$		0			0
$645$		−	532.577i		−	0.825700i
$646$		0			0
$647$	−398.410			−0.615780			−0.307890	−	0.951422i	$-0.599623\pi$
$647$	−398.410			−0.615780			−0.307890	+	0.951422i	$0.599623\pi$
$648$		0			0
$649$		−	375.573i		−	0.578695i
$650$		0			0
$651$	−393.591			−0.604594
$652$		0			0
$653$	−453.692			−0.694781			−0.347391	−	0.937721i	$-0.612932\pi$
$653$	−453.692			−0.694781			−0.347391	+	0.937721i	$0.612932\pi$
$654$		0			0
$655$	−152.813			−0.233302
$656$		0			0
$657$	−137.845			−0.209809
$658$		0			0
$659$			193.340i			0.293383i	0.989182	+	0.146692i	$0.0468625\pi$
$659$			193.340i			0.293383i	−0.989182	+	0.146692i	$0.953138\pi$
$660$		0			0
$661$		−	902.276i		−	1.36502i	−0.730878	−	0.682508i	$-0.760889\pi$
$661$		−	902.276i		−	1.36502i	0.730878	−	0.682508i	$-0.239111\pi$
$662$		0			0
$663$	−1341.78			−2.02380
$664$		0			0
$665$	72.8366	−	108.713i	0.109529	−	0.163479i
$666$		0			0
$667$		−	957.024i		−	1.43482i
$668$		0			0
$669$	968.023			1.44697
$670$		0			0
$671$	−96.7335			−0.144163
$672$		0			0
$673$			486.217i			0.722463i	0.932476	+	0.361231i	$0.117644\pi$
$673$			486.217i			0.722463i	−0.932476	+	0.361231i	$0.882356\pi$
$674$		0			0
$675$		−	38.1719i		−	0.0565509i
$676$		0			0
$677$		−	1034.98i		−	1.52877i	−0.644759	−	0.764386i	$-0.723042\pi$
$677$		−	1034.98i		−	1.52877i	0.644759	−	0.764386i	$-0.276958\pi$
$678$		0			0
$679$			428.771i			0.631474i
$680$		0			0
$681$	83.3626			0.122412
$682$		0			0
$683$		−	100.984i		−	0.147853i	−0.997264	−	0.0739267i	$-0.976447\pi$
$683$		−	100.984i		−	0.147853i	0.997264	−	0.0739267i	$-0.0235531\pi$
$684$		0			0
$685$	393.417			0.574332
$686$		0			0
$687$		−	1226.47i		−	1.78526i
$688$		0			0
$689$	175.263			0.254374
$690$		0			0
$691$	−774.181			−1.12038			−0.560189	−	0.828365i	$-0.689272\pi$
$691$	−774.181			−1.12038			−0.560189	+	0.828365i	$0.689272\pi$
$692$		0			0
$693$	544.806			0.786156
$694$		0			0
$695$	−320.302			−0.460867
$696$		0			0
$697$		−	450.546i		−	0.646407i
$698$		0			0
$699$			1542.45i			2.20665i
$700$		0			0
$701$	1010.71			1.44181			0.720906	−	0.693033i	$-0.243726\pi$
$701$	1010.71			1.44181			0.720906	+	0.693033i	$0.243726\pi$
$702$		0			0
$703$	−981.180	−	657.378i	−1.39570	−	0.935104i
$704$		0			0
$705$			213.602i			0.302982i
$706$		0			0
$707$	444.598			0.628851
$708$		0			0
$709$	−542.239			−0.764794			−0.382397	−	0.923998i	$-0.624901\pi$
$709$	−542.239			−0.764794			−0.382397	+	0.923998i	$0.624901\pi$
$710$		0			0
$711$			1071.56i			1.50711i
$712$		0			0
$713$			1292.79i			1.81317i
$714$		0			0
$715$		−	636.375i		−	0.890035i
$716$		0			0
$717$			783.534i			1.09280i
$718$		0			0
$719$	105.476			0.146699			0.0733493	−	0.997306i	$-0.476631\pi$
$719$	105.476			0.146699			0.0733493	+	0.997306i	$0.476631\pi$
$720$		0			0
$721$		−	219.645i		−	0.304639i
$722$		0			0
$723$	−514.982			−0.712285
$724$		0			0
$725$		−	448.977i		−	0.619278i
$726$		0			0
$727$	859.614			1.18241			0.591206	−	0.806520i	$-0.298652\pi$
$727$	859.614			1.18241			0.591206	+	0.806520i	$0.298652\pi$
$728$		0			0
$729$	800.584			1.09820
$730$		0			0
$731$	−1185.72			−1.62205
$732$		0			0
$733$	1170.01			1.59619			0.798097	−	0.602528i	$-0.205840\pi$
$733$	1170.01			1.59619			0.798097	+	0.602528i	$0.205840\pi$
$734$		0			0
$735$			402.704i			0.547897i
$736$		0			0
$737$		−	1148.68i		−	1.55859i
$738$		0			0
$739$	−742.350			−1.00453			−0.502267	−	0.864713i	$-0.667500\pi$
$739$	−742.350			−1.00453			−0.502267	+	0.864713i	$0.667500\pi$
$740$		0			0
$741$	637.096	−	950.909i	0.859779	−	1.28328i
$742$		0			0
$743$			1306.08i			1.75784i	0.476967	+	0.878921i	$0.341736\pi$
$743$			1306.08i			1.75784i	−0.476967	+	0.878921i	$0.658264\pi$
$744$		0			0
$745$	−116.487			−0.156358
$746$		0			0
$747$	−463.109			−0.619959
$748$		0			0
$749$		−	413.161i		−	0.551617i
$750$		0			0
$751$		−	324.212i		−	0.431707i	−0.976426	−	0.215853i	$-0.930747\pi$
$751$		−	324.212i		−	0.431707i	0.976426	−	0.215853i	$-0.0692533\pi$
$752$		0			0
$753$			1930.13i			2.56325i
$754$		0			0
$755$		−	476.853i		−	0.631593i
$756$		0			0
$757$	−1118.35			−1.47734			−0.738672	−	0.674065i	$-0.764547\pi$
$757$	−1118.35			−1.47734			−0.738672	+	0.674065i	$0.764547\pi$
$758$		0			0
$759$		−	3493.02i		−	4.60214i
$760$		0			0
$761$	696.142			0.914772			0.457386	−	0.889268i	$-0.348786\pi$
$761$	696.142			0.914772			0.457386	+	0.889268i	$0.348786\pi$
$762$		0			0
$763$			125.742i			0.164800i
$764$		0			0
$765$	490.373			0.641010
$766$		0			0
$767$	−270.291			−0.352400
$768$		0			0
$769$	55.8346			0.0726067			0.0363034	−	0.999341i	$-0.488442\pi$
$769$	55.8346			0.0726067			0.0363034	+	0.999341i	$0.488442\pi$
$770$		0			0
$771$	2015.48			2.61411
$772$		0			0
$773$		−	1113.47i		−	1.44045i	−0.693740	−	0.720226i	$-0.744038\pi$
$773$		−	1113.47i		−	1.44045i	0.693740	−	0.720226i	$-0.255962\pi$
$774$		0			0
$775$			606.496i			0.782575i
$776$		0			0
$777$	−789.708			−1.01635
$778$		0			0
$779$	319.298	+	213.925i	0.409882	+	0.274615i
$780$		0			0
$781$		−	1201.93i		−	1.53897i
$782$		0			0
$783$	−44.7190			−0.0571124
$784$		0			0
$785$	134.185			0.170936
$786$		0			0
$787$			104.121i			0.132301i	0.997810	+	0.0661504i	$0.0210717\pi$
$787$			104.121i			0.132301i	−0.997810	+	0.0661504i	$0.978928\pi$
$788$		0			0
$789$		−	1138.86i		−	1.44342i
$790$		0			0
$791$		−	323.447i		−	0.408908i
$792$		0			0
$793$			69.6169i			0.0877892i
$794$		0			0
$795$	−125.030			−0.157270
$796$		0			0
$797$		−	292.129i		−	0.366536i	−0.983063	−	0.183268i	$-0.941332\pi$
$797$		−	292.129i		−	0.366536i	0.983063	−	0.183268i	$-0.0586677\pi$
$798$		0			0
$799$	475.561			0.595196
$800$		0			0
$801$			262.965i			0.328296i
$802$		0			0
$803$	−284.118			−0.353821
$804$		0			0
$805$	−287.397			−0.357015
$806$		0			0
$807$	1303.74			1.61553
$808$		0			0
$809$	−304.463			−0.376344			−0.188172	−	0.982136i	$-0.560256\pi$
$809$	−304.463			−0.376344			−0.188172	+	0.982136i	$0.560256\pi$
$810$		0			0
$811$			816.016i			1.00618i	0.864233	+	0.503092i	$0.167804\pi$
$811$			816.016i			1.00618i	−0.864233	+	0.503092i	$0.832196\pi$
$812$		0			0
$813$			860.452i			1.05837i
$814$		0			0
$815$	−228.103			−0.279881
$816$		0			0
$817$	562.997	−	840.310i	0.689102	−	1.02853i
$818$		0			0
$819$		−	392.084i		−	0.478735i
$820$		0			0
$821$	721.591			0.878917			0.439458	−	0.898263i	$-0.355170\pi$
$821$	721.591			0.878917			0.439458	+	0.898263i	$0.355170\pi$
$822$		0			0
$823$	414.065			0.503117			0.251558	−	0.967842i	$-0.419057\pi$
$823$	414.065			0.503117			0.251558	+	0.967842i	$0.419057\pi$
$824$		0			0
$825$		−	1638.71i		−	1.98631i
$826$		0			0
$827$		−	100.878i		−	0.121981i	−0.998138	−	0.0609903i	$-0.980574\pi$
$827$		−	100.878i		−	0.121981i	0.998138	−	0.0609903i	$-0.0194259\pi$
$828$		0			0
$829$			621.755i			0.750006i	0.927024	+	0.375003i	$0.122358\pi$
$829$			621.755i			0.750006i	−0.927024	+	0.375003i	$0.877642\pi$
$830$		0			0
$831$			687.781i			0.827655i
$832$		0			0
$833$	896.575			1.07632
$834$		0			0
$835$			197.265i			0.236245i
$836$		0			0
$837$	60.4082			0.0721723
$838$		0			0
$839$			115.138i			0.137232i	0.997643	+	0.0686160i	$0.0218583\pi$
$839$			115.138i			0.137232i	−0.997643	+	0.0686160i	$0.978142\pi$
$840$		0			0
$841$	315.016			0.374573
$842$		0			0
$843$	1645.33			1.95176
$844$		0			0
$845$	−64.4144			−0.0762300
$846$		0			0
$847$	765.080			0.903282
$848$		0			0
$849$			973.956i			1.14718i
$850$		0			0
$851$			2593.87i			3.04803i
$852$		0			0
$853$	762.387			0.893772			0.446886	−	0.894591i	$-0.352533\pi$
$853$	762.387			0.893772			0.446886	+	0.894591i	$0.352533\pi$
$854$		0			0
$855$	−232.836	+	347.523i	−0.272323	+	0.406460i
$856$		0			0
$857$			537.227i			0.626869i	0.949610	+	0.313435i	$0.101480\pi$
$857$			537.227i			0.626869i	−0.949610	+	0.313435i	$0.898520\pi$
$858$		0			0
$859$	645.182			0.751084			0.375542	−	0.926805i	$-0.377456\pi$
$859$	645.182			0.751084			0.375542	+	0.926805i	$0.377456\pi$
$860$		0			0
$861$	256.988			0.298477
$862$		0			0
$863$			892.080i			1.03370i	0.856077	+	0.516848i	$0.172895\pi$
$863$			892.080i			1.03370i	−0.856077	+	0.516848i	$0.827105\pi$
$864$		0			0
$865$			52.0765i			0.0602041i
$866$		0			0
$867$		−	889.615i		−	1.02608i
$868$		0			0
$869$			2208.64i			2.54159i
$870$		0			0
$871$	−826.681			−0.949118
$872$		0			0
$873$		−	1370.65i		−	1.57004i
$874$		0			0
$875$	−307.010			−0.350869
$876$		0			0
$877$			873.903i			0.996469i	0.867042	+	0.498235i	$0.166018\pi$
$877$			873.903i			0.996469i	−0.867042	+	0.498235i	$0.833982\pi$
$878$		0			0
$879$	−1847.03			−2.10129
$880$		0			0
$881$	−687.622			−0.780502			−0.390251	−	0.920709i	$-0.627612\pi$
$881$	−687.622			−0.780502			−0.390251	+	0.920709i	$0.627612\pi$
$882$		0			0
$883$	1305.56			1.47855			0.739276	−	0.673403i	$-0.235168\pi$
$883$	1305.56			1.47855			0.739276	+	0.673403i	$0.235168\pi$
$884$		0			0
$885$	192.821			0.217876
$886$		0			0
$887$			57.7531i			0.0651106i	0.999470	+	0.0325553i	$0.0103645\pi$
$887$			57.7531i			0.0651106i	−0.999470	+	0.0325553i	$0.989635\pi$
$888$		0			0
$889$			118.995i			0.133852i
$890$		0			0
$891$	1494.74			1.67760
$892$		0			0
$893$	−225.803	+	337.026i	−0.252859	+	0.377409i
$894$		0			0
$895$		−	169.520i		−	0.189408i
$896$		0			0
$897$	−2513.85			−2.80250
$898$		0			0
$899$	710.520			0.790345
$900$		0			0
$901$			278.364i			0.308951i
$902$		0			0
$903$		−	676.328i		−	0.748979i
$904$		0			0
$905$		−	815.753i		−	0.901385i
$906$		0			0
$907$			837.489i			0.923362i	0.887046	+	0.461681i	$0.152753\pi$
$907$			837.489i			0.923362i	−0.887046	+	0.461681i	$0.847247\pi$
$908$		0			0
$909$	−1421.24			−1.56352
$910$		0			0
$911$		−	662.892i		−	0.727653i	−0.931467	−	0.363826i	$-0.881470\pi$
$911$		−	662.892i		−	0.727653i	0.931467	−	0.363826i	$-0.118530\pi$
$912$		0			0
$913$	−954.537			−1.04550
$914$		0			0
$915$		−	49.6634i		−	0.0542769i
$916$		0			0
$917$	−194.060			−0.211625
$918$		0			0
$919$	−1000.18			−1.08834			−0.544169	−	0.838976i	$-0.683155\pi$
$919$	−1000.18			−1.08834			−0.544169	+	0.838976i	$0.683155\pi$
$920$		0			0
$921$	894.699			0.971443
$922$		0			0
$923$	−865.003			−0.937164
$924$		0			0
$925$			1216.88i			1.31555i
$926$		0			0
$927$			702.137i			0.757429i
$928$		0			0
$929$	1504.37			1.61934			0.809671	−	0.586885i	$-0.199646\pi$
$929$	1504.37			1.61934			0.809671	+	0.586885i	$0.199646\pi$
$930$		0			0
$931$	−425.706	+	635.395i	−0.457257	+	0.682486i
$932$		0			0
$933$		−	580.701i		−	0.622402i
$934$		0			0
$935$	1010.73			1.08100
$936$		0			0
$937$	−961.503			−1.02615			−0.513075	−	0.858344i	$-0.671494\pi$
$937$	−961.503			−1.02615			−0.513075	+	0.858344i	$0.671494\pi$
$938$		0			0
$939$			1770.08i			1.88506i
$940$		0			0
$941$			681.880i			0.724633i	0.932055	+	0.362317i	$0.118014\pi$
$941$			681.880i			0.724633i	−0.932055	+	0.362317i	$0.881986\pi$
$942$		0			0
$943$		−	844.103i		−	0.895126i
$944$		0			0
$945$			13.4293i			0.0142109i
$946$		0			0
$947$	−1441.83			−1.52253			−0.761263	−	0.648443i	$-0.775420\pi$
$947$	−1441.83			−1.52253			−0.761263	+	0.648443i	$0.775420\pi$
$948$		0			0
$949$			204.473i			0.215462i
$950$		0			0
$951$	−1661.29			−1.74688
$952$		0			0
$953$		−	495.317i		−	0.519745i	−0.965643	−	0.259872i	$-0.916320\pi$
$953$		−	495.317i		−	0.519745i	0.965643	−	0.259872i	$-0.0836805\pi$
$954$		0			0
$955$	251.400			0.263247
$956$		0			0
$957$	−1919.78			−2.00604
$958$		0			0
$959$	499.607			0.520967
$960$		0			0
$961$	1.20041			0.00124912
$962$		0			0
$963$			1320.75i			1.37149i
$964$		0			0
$965$		−	623.276i		−	0.645882i
$966$		0			0
$967$	1020.51			1.05534			0.527670	−	0.849449i	$-0.323066\pi$
$967$	1020.51			1.05534			0.527670	+	0.849449i	$0.323066\pi$
$968$		0			0
$969$	1510.29	+	1011.88i	1.55861	+	1.04425i
$970$		0			0
$971$		−	563.750i		−	0.580587i	−0.956938	−	0.290293i	$-0.906247\pi$
$971$		−	563.750i		−	0.580587i	0.956938	−	0.290293i	$-0.0937529\pi$
$972$		0			0
$973$	−406.757			−0.418045
$974$		0			0
$975$	−1179.34			−1.20958
$976$		0			0
$977$			786.557i			0.805073i	0.915404	+	0.402537i	$0.131871\pi$
$977$			786.557i			0.805073i	−0.915404	+	0.402537i	$0.868129\pi$
$978$		0			0
$979$			542.010i			0.553636i
$980$		0			0
$981$		−	401.959i		−	0.409744i
$982$		0			0
$983$			337.555i			0.343393i	0.985150	+	0.171696i	$0.0549248\pi$
$983$			337.555i			0.343393i	−0.985150	+	0.171696i	$0.945075\pi$
$984$		0			0
$985$	677.021			0.687331
$986$		0			0
$987$			271.257i			0.274830i
$988$		0			0
$989$	−2221.46			−2.24617
$990$		0			0
$991$		−	1048.22i		−	1.05774i	−0.848703	−	0.528870i	$-0.822616\pi$
$991$		−	1048.22i		−	1.05774i	0.848703	−	0.528870i	$-0.177384\pi$
$992$		0			0
$993$	−617.176			−0.621527
$994$		0			0
$995$	−703.991			−0.707529
$996$		0			0
$997$	1041.54			1.04467			0.522335	−	0.852741i	$-0.325061\pi$
$997$	1041.54			1.04467			0.522335	+	0.852741i	$0.325061\pi$
$998$		0			0
$999$	121.204			0.121325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.3.e.b.417.4	yes	20
4.3	odd	2	inner	608.3.e.b.417.17	yes	20
8.3	odd	2		1216.3.e.p.1025.3		20
8.5	even	2		1216.3.e.p.1025.18		20
19.18	odd	2	inner	608.3.e.b.417.18	yes	20
76.75	even	2	inner	608.3.e.b.417.3	✓	20
152.37	odd	2		1216.3.e.p.1025.4		20
152.75	even	2		1216.3.e.p.1025.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.e.b.417.3	✓	20	76.75	even	2	inner
608.3.e.b.417.4	yes	20	1.1	even	1	trivial
608.3.e.b.417.17	yes	20	4.3	odd	2	inner
608.3.e.b.417.18	yes	20	19.18	odd	2	inner
1216.3.e.p.1025.3		20	8.3	odd	2
1216.3.e.p.1025.4		20	152.37	odd	2
1216.3.e.p.1025.17		20	152.75	even	2
1216.3.e.p.1025.18		20	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	608.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.29580i q^{3} +2.32882 q^{5} +2.95740 q^{7} -9.45390 q^{9} -19.4859 q^{11} +14.0235i q^{13} -10.0041i q^{15} -22.2731 q^{17} +(10.5756 - 15.7847i) q^{19} -12.7044i q^{21} -41.7289 q^{23} -19.5766 q^{25} +1.94987i q^{27} +22.9343i q^{29} -30.9806i q^{31} +83.7075i q^{33} +6.88725 q^{35} -62.1601i q^{37} +60.2423 q^{39} +20.2283i q^{41} +53.2356 q^{43} -22.0164 q^{45} -21.3514 q^{47} -40.2538 q^{49} +95.6807i q^{51} -12.4978i q^{53} -45.3791 q^{55} +(-67.8080 - 45.4305i) q^{57} +19.2741i q^{59} +4.96429 q^{61} -27.9590 q^{63} +32.6583i q^{65} +58.9495i q^{67} +179.259i q^{69} +61.6822i q^{71} +14.5807 q^{73} +84.0972i q^{75} -57.6276 q^{77} -113.346i q^{79} -76.7089 q^{81} +48.9861 q^{83} -51.8699 q^{85} +98.5213 q^{87} -27.8155i q^{89} +41.4733i q^{91} -133.087 q^{93} +(24.6285 - 36.7597i) q^{95} +144.982i q^{97} +184.218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) \) 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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